Semiconductors and Semimetals. Volume 25: Diluted Magnetic Semiconductors (Semiconductors and Semimetals)

Semiconductors and Semimetals A Treatise

Edited by R . K. Willardson WILLARDSON CONSULTING SPOKANE, WASHINGTON

Albert C. Beer BATTELLE COLUMBUS LABORATORIES COLUMBUS, OHIO

SEMICONDUCTORS AND SEMIMETALS VOLUME 25 Diluted Magnetic Semiconductors Volume Editors JACEK K. FURDYNA DEPARTMENT OF PHYSICS UNIVERSITY OF NOTRE DAME NOTRE DAME, INDIANA

JACEK KOSSUT INSTITUTE O€ PHYSICS POLISH ACADEMY OF SCIENCES WARSAW, POLAND

ACADEMIC PRESS, INC. Harcourt Brace Jovanovich, Publishers

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COPYRIGHT @ 1988 BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL., INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

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9 8 7 6 5 4 3 2 1

To the memory of Professor Jerzy Mycielski (1930-1986)

Contributors Numbers in parentheses indicate the pages on which the authors’ contributions begin.

W. M. BECKER, Department of Physics, Purdue University, West Lafayette, Indiana 47907 (35)

J . K. FURDYNA,Department of Physics, University of Notre Dame, Notre Dame, Indiana 46556 (1) J. A. GAJ, Institute of Experimental Physics, University of Warsaw, 69 Hoza, 00-681 Warsaw, Poland (276) T. GIEBULTOWICZ, Department of Physics, University of Notre Dame, Notre Dame, Indiana 46556, and National Bureau of Standards, Reactor Division, Bld. 235, Gaithersburg, MD 20899 (125) W. GIRIAT,Centro de Fisica, Instituto Venezolano de Investigaciones Cientifcas (IVIC), Caracas IOIOA, Venezuela (1) T. M. HOLDEN,Atomic Energy of Canada Limited, Chalk River Nuclear Laboratories, Chalk River, Ontario, Canada (125) PIETERH. KEESOM,Department of Physics, Purdue University, West Lafayette, Indiana 47907 (73) J. KOSSUT,Institute of Physics, Polish Academy of Sciences, Al. Lotnikow 32/46, 02-668 Warsaw, Poland (1 83) J. MYciELsKI, Institute of Theoretical Physics,. Warsaw University, 00-681 Warsaw, Poland (31 1) SAULOSEROFF, Department of Physics, San Diego State University, San Diego, California 10775 (73) A. K. RAMDAS,Department of Physics, Purdue University, West Lafayette, Indiana 47907 (345) C . RIGAUX,Groupe de Physique des Solides de I’Ecole Normale Supgrieure, 24 rue Lhomond, 75231 Paris Cedex 05, France (229) S . RODRIGUEZ, Department of Physics, Purdue University, West Lafayette, Indiana 47907 (345) P. A. WOLFF,Francis Bitter National Magnet Laboratory, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 (413) xi

Foreword

Diluted magnetic semiconductors have in recent years attracted considerable attention of the scientific community. The present book consists of 10 chapters, each describing a different facet of the physics of these materials. In defining what constitutes the family of diluted magnetic semiconductors (DMS) one might say, to be on the safe side, that any known semiconductor with a fraction of its constituent ions replaced by some species of magnetic ions (i.e., ions bearing a net magnetic moment) is a legitimate member of this group. More realistically, the majority of DMS studied in an extensive way so far involved Mn+' ions embedded in various A"BV' hosts. There is a rationale for this particular choice: (i) Mn+' can be incorporated in sizable amounts in the A"BV' host without affecting substantially the crystallographic quality of the resulting material (e.g., up to nearly 80% of Mn can be accommodated in CdTe); (ii) Mn" possesses a relatively large magnetic moment (S = 5/2), characteristic of a half-filled d shell (from this point of view Euf2, with S = 7/2, might seem to be even more attractive); (iii) Mn+2 is electrically neutral in A"BV' hosts, i.e., it constitutes neither accepting nor donating centers. There are three aspects of DMS properties that make these alloys interesting subjects for scientific investigation (and also give them the potential for future device applications). The first of these concerns semiconducting properties per se: the parameters determining the band structure of the material (e.g., the energy gap) can be varied by changing appropriately the Mn+' mole fraction in the crystal. Similarly, the lattice parameters can be "tuned" by varying the composition of the DMS, this aspect being particularly important in the age of monolithic semiconductor heterostructures, where lattice matching is of obvious importance. In other words, Mn+' affects various properties of the host semiconductor in a manner analogous to, say, Cd in Hgl-,Cd,Te. Secondly, purely magnetic properties of DMS encompass a very broad spectrum of behavior, including paramagnetic, spin-glass and antiferromagnetic properties. It is thus possible to study within one system-say, Cdl-xMnxTe-the development of these magnetic features as a function of the concentration of Mn+'. Spin-spin correlations, for example, and their xiii

xiv

FOREWORD

dependence on the Mn mole fraction, were studied in detail by neutron scattering methods and are described in Chapter 4. Alternatively, DMS offer a possibility of a comparative study of the magnetic behavior of Mn+2spins in different hosts. One is then able to draw conclusions concerning, say, the role played by the anion in the process of mediating the superexchange coupling between two magnetic moments (see, e.g., Spalek et al., 1986, and Samarth, 1986). Thirdly, the interaction which exists between the localized magnetic moments of Mn+2and the conduction and/or valence band electrons (which we shall refer to as the sp-d interaction) results in a series of features which are unique to DMS. The best known (and probably the most spectacular) of these are the huge Faraday rotation of the visible and near-infrared light in wide-gap DMS (see Chapter 7 for details) and the giant negative magnetoresistance in the vicinity of the semiconductor-semimetal transition in Hgl-xMnxTe (dealt with in Chapter 8). As mentioned above, these spectacular features have their common origin in the sp-d exchange interaction. As a result, the band structure of DMS is much more sensitive to the strength of an external magnetic field than in ordinary semiconductors. For this reason DMS are sometimes referred to as “spin amplifiers”. Thus, by means of magnets readily available in the laboratory, one is able to observe phenomena that normally would require the application of megagauss magnetic fields. Since the sp-d interaction causes the semiconducting properties to be strongly sensitive to the details of the magnetic behavior of the Mn+* subsystem, it is thus possible to study the latter indirectly by observing, for example, the optical properties of DMS. The general organization of the present volume reflects these three aspects of DMS physics. The first two chapters describe the crystal structures, methods of preparation, and semiconducting properties (mostly optical) of DMS in the absence of the magnetic field. This is followed by two chapters devoted to the magnetic properties displayed by these materials. The remaining chapters deal with various aspects of the physical properties of DMS that are due to the sp-d interaction of the band electrons with localized magnetic moments. So much for the scope. What about the purpose of the volume? The number of papers published in the literature dealing with DMS, starting with the pioneering works by Delves (1963) and Delves and Lewis (1963), must now exceed 1,000. It is then difficult to make oneself familiar with the field without going to many different sources, scattered throughout theliterature. Thus, the first objective of the present volume is to provide a reasonably full account of the achievements in this field. Secondly, it was thought that, at least in several areas, the stage has been reached warranting formulation of conclusions of a character more general than is appropriate for a typical journal article.

FORE WORD

xv

The work on the book spanned a considerable period of time (as is perhaps unavoidable in the case of any publication involving a large number of contributing authors). The last several years have witnessed a particularly rapid growth of the number of publications concerning DMS. In particular, several important new contributions emerged when the book was already in the middle of its preparation process. For this reason, although they are of unquestionable importance, it was not possible to properly represent them here. To do them partial justice without delaying the appearance of the book, we shall call attention to some of the new developments at least in the Foreword in the form of the list below. (1) Considerable progress is to be noted in the experimental and theoretical studies concerning the relative position of Mn-derived states and various density-of-states features in the valence and conduction bands. The experimental part has made extensive use of the synchrotron radiation photoemission technique (Franciosi etal., 1985a,b, and Wallet al., 1986a,b). The theoretical effort (consisting of a series of papers by Ehrenreich, Hass, Larson, and collaborators) was recently reviewed by Ehrenreich et al. (1987). Based on these studies it was, for example, possible to determine that the dominant mechanism of the Mn-Mn exchange coupling is that of superexchange, and to calculate theoretically the values of the relevant exchange constants. (2) Successful growth by molecular beam epitaxy of superlattices and quantum well systems involving DMS opened a vast field of entirely new opportunities (for a recent review of the work on DMS superlattices, see Furdyna et al., 1987). (3) MaterialsinvolvingMn+2inA1VBV1andA11BV hosts (e.g., Pbl-,Mn,Te and (Cdl-,Mn,)2As3, respectively) were prepared in an attempt to enlarge the family of DMS. Some of the references to these works are given in Chapter 1 in the section devoted to new materials. (4) DMS materials containing substitutional Fe ions instead of the traditional Mn+2(for reviews, see Mycielski, 1986a, and Reifenberger and Kossut, 1986) proved to possess features absent in Mn-based DMS. A considerable enhancement of the low-temperature mobility of electrons in Hgl-,Fe,Se (0.0003 5 x 5 0.01) is the most spectacular example of these new properties. ( 5 ) Studies of the conductivity in the weakly localized regime and in the vicinity of the nonmetal-metal transition have shown that the electronic spin plays an important role in the related processes (see Stankiewicz el al., 1986a,b, Shapira et al., 1985, Wojtowicz et al., 1986, and Sawicki et al., 1986). (6) Investigations of the magnetization induced in DMS by circularly

xvi

FOREWORD

polarized radiation incident on the sample (Krenn et al., 1985) and the “magnetic spectroscopy” (Awschalom and Warnock, 1987) open new possibilities, particularly in the context of time-resolved studies of magnetic systems. Expecting that the book would be read at a rate of one chapter at a time, it was the editors’ decision to give the contributing authors a free hand in their choice of units, notation, etc., provided that they were consistent within their own chapter. For the same reason, a certain degree of overlap between the material contained in some chapters was not frowned upon inshe hope that the contributions would, in this way, be more self-contained. Finally, let us comment on the terminology used in connection with DMS. Quite often these materials also are referred to as semimagnetic semiconductors. Being aware that this is predominantly a matter of taste, we nevertheless prefer the term “diluted magnetic semiconductor”, since this term conveys the physical nature underlying the properties of the material. It refers to a magnetic semiconductor (a name already established, as for EuSe) that has been diluted by the intervening non-magnetic constituent, giving DMS all their physical characteristics. The authors of the chapters contained in this volume were, as a rule, actively involved in the research in the areas which they respectively describe. One of the authors in particular, Professor Jerzy Mycielski, had a very special impact on the development of the physics of DMS. Over and above the contents of his own chapter in this book, his contributions consisted of formulation of many original and important ideas throughout the entire period of DMS studies. The papers of his authorship were among the first that put DMS “on the map”, while his hypothesis concerning a spatial ordering of charges within the donor system in Hg, -xFexSe (Mycielski, 1986b) will remain his last. It was with profound sorrow that we learned of his death in February, 1986, before he could see his contribution in print. To his memory, we would like to dedicate this volume. References Awschalom, D. D., and Warnock, J. (1987). DilutedMagnetic (Semimagnetic)Semiconductors, edited by R. L. Aggarwal, J. K. Furdyna, and S. von Molnar (Materials Research Society Symposia Proceedings, Pittsburgh, PA) Vol. 89, p. 71. Delves, R. T. (1963). J. Phys. Chem. Solids 24, 885. Delves, R. T., and Lewis, B. (1963). J. Phys. Chem. Solids 24, 549. Ehrenreich, H . , Hass, K. C., Larson, B. E., and Johnson, N. F. (1987). Diluted Magnetic (Semimagnetic) Semiconductors, edited by R. L. Aggarwal, J . K . Furdyna, and S. von Molnar (Materials Research Society Symposia Proceedings, Pittsburgh, PA) Vol. 89, p. 187. Franciosi, A . , Chang, S., Caprile, C., Reifenberger, R., and Debska, U. (1985a). J. Vac. Sci. Techno/. A3. 926.

FOREWORD

xvii

Franciosi, A., Reifenberger, R., and Furdyna, J . (1985b). J. Vac. Sci. Technol. A3, 124. Furdyna, J. K., Kossut, J., and Ramdas, A. K. (1987). OpticalPropertiesofNarrow-GapLowDimensionalStructures, edited by C. M. Sotomayor Torres, J. C. Portal, J. C. Maan, and R. A. Stradling (Plenum Press, New York) p. 135. Krenn, H., Zawadzki, W., and Bauer, G. (1985). Phys. Rev. Lett. 55, 1510. Mycielski, A. (1986a). Diluted Magnetic (Semimagnetic) Semiconductors, edited by R. L. Aggarwal, J. K. Furdyna, and S. von Molnar (Materials Research Society Symposia Proceedings, Pittsburgh, PA) Vol. 89, p. 159. Mycielski, J. (1986b). Solid State Commun. 60, 165. Reifenberger, R., and Kossut, J. (1987). J. Vac. Sci. Technol. AS, 2995. Samarth, N. (1986). Ph.D. Thesis, Purdue University, unpublished. Sawicki, M., Dietl, T., Kossut, J., Igalson, J., Wojtowicz, T., and Plesiewicz, W. (1986). Phys. Rev. Lett. 56, 508. Shapka, Y., Ridgley, D. H., Dwight, K., Wold, A., Martin, K. P., Brook, J. S., and Lee, P. A. (1985). Solid State Cornmun. 54, 593. Spalek, J., Lewicki, A., Tarnawski, Z., Furdyna, J. K., Galazka, R. R., and Obuszko, Z. (1986). Phys. Rev. B33, 3407. Stankiewicz, J., von Molnar, S., and Giriat, W. (1986a). Phys. Rev. B33, 3573. Stankiewicz, J., von Molnar, S., and Holtzberg, F. (1986b). J. Magn. Magn. Muter. 54-57, 1217. Wall, A., Caprile, C., Franciosi, A,, Reifenberger, R., and Debska, U. (1986a). J. Vac. Sci. Technol. A4, 818. Wall, A., Caprile, C., Franciosi, A., Vaziri, M., Reifenberger, R., and Furdyna, J. K. (1986b). J. Vac. Sci. Technol. A4, 2010. Wojtowicz, T., Dietl, T., Sawicki, M., Plesiewicz, W., and Jaroszynski, J. (1986). Phys. Rev. Letters 56, 2419.

SEMICONDUCTORS AND SEMIMETALS. VOL. 25

CHAPTER 1

Crystal Structure, Composition, and Materials Preparation of Diluted Magnetic Semiconductors W . Giriat CENTRO DE FISICA INSTITUTO VENEZOLANO D E INVESTIGACIONES CIENTIFICAS (IVIC) CARACAS, VENEZUELA

and J. K. Furdyna DEPARTMENT OF PHYSICS UNIVERSITY OF NOTRE DAME NOTRE DAME, INDIANA

I. INTRODUCTION. . . . . . . . . . . . . . . . . . . . 11. CRYSTAL STRUCTURE AND COMPOSITION OF DMS . . . . . . 1. Overview. . . . . . . . . . . . . . . . . . . . 2 . The A"BV' "Host "Lattice. . . . . . . . . . . . . . 3 . Crystal Structure and Lattice Parameters of DMS . . . . 4. Microscopic Structure of the DMS Lattice . . . . . . . 111. PREPARATION OF DMS CRYSTALS. . . . . . . . . . . . . 5. Preparation of Starting Materials. . . . . . . . . . . 6 . Crystal Growth of DMS Alloys . . . . . . . . . . . IV. THE"NEW"DMS. . . . . . . . . . . . . . . . . . . 1. DMS Films and Superlattices. . . . . . . . . . . . . 8 . A1',FexBV' Alloys . . . . . . . . . . . . . . . . 9. Manganese- and Rare-Earth-Based AIVBV'Alloys. . . . . 10. MisceIIaneous New DMS Systems. . . . . . . . . . . REFERENCES.. . . . . . . . . . . . . . . . . . . .

1

3 3 4 7 11 14

14 17 21

28 28 30 31

32

I. Introduction During the past decade a great deal of attention has been given to semiconductor compounds whose lattice is made up in part of substitutional magnetic ions (Galazka, 1979; Gaj, 1980; Furdyna, 1982; Brandt and Moshchalkov, 1984). Most of these materials are based on A"BV' compounds, with a fraction of the group-I1 constituent replaced by manganese. Cdl-,Mn,Se and Hgl-,Mn,Te are examples of such systems. These ternary

1 Copyright 0 1988 by Academic Press. Inc. All rights of reproduction in any form reserved. ISBN 0-12-752125-9

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W . GIRIAT AND J . K . FURDYNA

alloys (or "mixed crystals") are of interest for several reasons. Firstly, their semiconducting and structural properties, such as the energy gap, the lattice parameter, etc., can be varied in a controlled fashion by varying the composition (much as in non-magnetic ternary semiconductors, e.g., Hgl-,Cd,Te or Gal-,Al,As). Secondly, as dilute magnetic alloys, these materials are of interest for their magnetic properties, e.g., the spin-glass transition (Galazka, 1982; Oseroff, 1982), antiferromagnetic cluster formation (Dolling, 1982), magnon excitations (Ramdas, 1982), and other magnetic effects of current interest. Thirdly, the presence of magnetic ions in the lattice leads to spin-spin exchange interaction between the localized magnetic moments and the band electrons (Gaj, 1980). This interaction affects the energy band and impurity level parameters of these materials (e.g., by enhancing the electronic g-factors), resulting in new physical effects. Such novel and potentially important phenomena as the magnetic-fieldinduced overlap between valence and conduction bands occurring in Hgl -,MnxTe and Hgl-,Mn,Se (Galazka and Kossut, 1980), extremely large Faraday rotation in Cdl-,Mn,Te (Gaj et al., 1978), giant negative magnetoresistance associated with hopping conduction in Hgl -,Mn,Te (Mycielski and Mycielski, 1980), and the magnetic polaron observed in Cdr-,Mn,Se (Nawrocki et al., 1981; Diet1 and Spalek, 1983) are all consequences of the spin-spin exchange interaction. The latter two features-magnetic properties as such, and the exchange interaction with band electrons-distinguish these materials from other semiconductors. For this reason, this group of alloys has been collectively labeled as "semimagnetic semiconductors" or "diluted magnetic semiconductors" (DMS). So far, practically all research on DMS involved A"BV' semiconductor compounds alloyed with manganese, although recently there has been a gradual increase of research activity on A1'BV'-Fe alloys (e.g., Hgl-,Fe,Te, Cdl-,Fe,Te) and AIVBV'-Mn alloys (e.g., Pbl-,Mn,Te). The investigations of these latter materials are still at a relatively early stage. In the present chapter we shall therefore concentrate on the A?,Mn,BV1 alloys. The material is presented as follows. We first review the crystal structure and composition of the various A:'_,Mn,BV' alloys. We show that this can be done in a unified way for both the zinc blende and wurtzite members of this group of materials. We then present methods of preparation of bulk DMS crystals, including materials purification, synthesis of starting binary compounds, and crystal growth. Finally, we discuss-in a much briefer fashion-the more recent developments in the field of DMS, such as A:'_,Mn,BV' epitaxial films and superlattices, A:'-,Fe,BV' alloys, and AIVBV'-based DMS containing substitutional manganese or rare earth ions.

1.

CRYSTAL STRUCTURE AND MATERIALS PREPARATION

3

11. Crystal Structure and Composition of DMS

1. OVERVIEW The entire family of ternary A:’_,MnxBv’ alloys, along with their crystal structures, is presented in schematic form in Fig. 1. The heavy lines in the figure show the composition ranges in which ternary alloys can be formed, with “cub” and “hex” indicating the crystal structure (zinc blende and wurtzite, respectively) of the stable phases. For example, Cdl-,MnxTe forms a ternary alloy of zinc blende structure with x up to 0.77, while Z ~ I - , M ~ , S ~ exhibits zinc blende structure for x < 0.30 and wurtzite structure for 0.30 c x c 0.55. At values of x beyond the heavy lines mixed phases occur. Given the fact that the stable crystal structures of MnTe, MnSe, and MnS differ from the crystal structure of the A”BV’ compounds (Pajaczkowska, 1978), it is truly remarkable that the range of the A1/-,MnxBV1 solid solutions

CdTe

ZiS FIG.1. A diagrammatic overview of the A:’,Mn,BV’ alloys and their crystal structures. The bold lines indicate ranges of the molar fraction x for which homogeneous crystal phases form. “Hex” and “Cub” indicates wurtzite and zinc blende, respectively. Cd,-,Mn,S and Cdl-,Mn,Se also form homogeneous crystalline phases near x = 1.0 in the NaCl structure, but these are not relevant to the present article.

4

W . GIRIAT AND J. K . FIJRDYNA

is so wide. For example, MnTe itself crystallizes in the NiAs structure, yet in the case of Cdl-,Mn,Te the structure of the "host" CdTe (zinc blende) survives for x as high as 0.77. Both crystal structures of the DMS-zinc blende and wurtzite-are very closely related in spite of the differences in symmetry, in that they are both formed with tetrahedral (s-p3) bonding, involving the two valence s-electrons of the group-I1 element and the six valence p-electrons of the group-VI element. Manganese is a transition metal with valence electrons corresponding to the 4s' orbital. Although manganese differs from the group-I1 elements by the fact that its 3d shell is only half-filled, it can contribute its 4s2 electrons to the s-p3 bonding, and can therefore substitutionally replace the group-I1 elements in the A"Bv' wurtzite or zinc blende structures. These properties of manganese are apparently a necessary but not a sufficient condition for forming stable ternary phases over a wide range of compositions, since other transition metal elements-even those which readily form divalent ions-are not nearly as miscible with A"BV' compounds. For example iron (a transition metal with ionic radius comparable to that of manganese) is about an order of magnitude less miscible, and the other transition metal elements even less than that. It would appear, therefore, that the ease with which Mn substitutes for the group-I1 elements in the zinc blende and wurtzite structures (and the well-known predisposition of this element for the divalent state) results from the fact that the 3d orbitals of Mn++ are exactly half-filled. By Hund's rule, all five spins are parallel in this orbital, and it would require considerable energy to add an electron with opposite spin to the atom. In this sense, the 3d5 orbit is "complete," and thus the Mn atom resembles a group-I1 element.

2. THE A"BV' "HOST" LATTICE The physical properties of the A"BV'-based DMS resemble the properties of the "parent" A"BV' binary compounds. It is therefore instructive to review briefly the structural properties of the A1'Bv' lattice before considering their ternary DMS derivatives. Crystallographic properties of the A"Bv' semiconductors are remarkably systematic. With the one exception of HgS, whose stable phase is cinnabar, they all form in either the zinc blende or the wurtzite phase. As has been pointed out, both structures form as a result of tetrahedral s-p3 bonding. Zinc blende is a cubic structure comprised of two interpenetrating fcc sublattices-one consisting of the znions, the other of the cations-shifted with respect t o each other by one-fourth of the body diagonal of the fcc cubic unit cell. The wurtzite structure consists of two interpenetrating hcp sublattices shifted along the c-axis.

1.

CRYSTAL STRUCTURE AND MATERIALS PREPARATION

5

The similarity of the two structures is based on the following: (a) In either structure, each member of one sublattice is tetrahedrally bonded to four members of the other sublattice. Thus Cd is surrounded by four Te atoms in the zinc blende CdTe, and by four Se atoms in the wurtzite CdSe, and within this local configuration it 'is impossible to distinguish between the two crystal structures. (b) The other similarity arises from the fact that wurtzite and zinc blende crystals are based on the hcp and fcc structures, respectively, both of which derive from the two alternate ways to obtain equivalent close packing of spheres. Thus, in both structures a given atom is surrounded by 12 equidistant like atoms. From the above it is obvious that the distance between like atoms d (e.g., the nearest cation-cation distance) and the bond length b (i.e., the nearest cation-anion distance) are identically related in both the zinc blende and the wurtzite structures. This feature will allow us to discuss both structures in a unified way (Yoder-Short et al., 1985). From purely geometrical considerations, we have d=

Gb.

(1)

For the zinc blende case, then, the lattice parameter a is

The wurtzite structure is based on two interpenetrating hcp structures, and is characterized by two lattice parameters, a and c. It is important to realize that for the ideal hcp situation, based on the standard packing of spheres, a and c are not independent: a is just the separation between the centres of touching sphere in the hexagonal plane (i.e., the separation between like atoms of the sublattice in that plane), and c is the separation between repeating planes (which is obviously also a function of the radius of the spheres). In terms of d , it is easily shown that for Wurtzite structure a = d,

(3)

c = &d.

(4)

Now it is truly remarkable how close is the behavior of the wurtzite Ax*Bvx compounds to the ideal case. For example, in the wurtzite crystal CdSe, a = 4.2985 A and c = 7.0150 A (Pajaczkowska, 1978), giving C

- = 1.632, a

6

W. GIRIAT AND J. K . FURDYNA

a

which is very close to the ratio = 1.633 predicted by the ideal close packing of spheres. Furthermore, in those materials which can exist in both cubic and hexagonal crystal phases, the bond length is the same. For example, for cubic ZnS (Pajaczkowska, 1978) a = 514093

and for hexagonal ZnS

A

= f i x 3.825

A,

a = 3.820 A,

c = 6.260 A

=

fi x 3.833 A .

It is useful to express the length of the tetrahedral bond-which underlies the above lattice properties of the tetrahedrally-bonded crystals-in terms of the covalent radii of the participating group-I1 and group-VI elements (Pauling, 1960; van Vechten and Phillips, 1970). The bond length b is then given as the sum of the two radii,

b = rII

+ rvI.

Such a radius is associated with a specific atom independent of the environment; for example, the same covalent radius of Se predicts rather accurately the bond lengths (and thus all other lattice parameters) in the zinc blende HgSe and ZnSe, and in the wurtzite CdSe. We list in Table I the covalent radii for the group-I1 and group-VI elements. It is remarkable that the bond lengths-and thus other lattice parameters-for all zinc blende and wurtzite A1*Bvlcompounds can be immediately generated, to a close approximation, from Table I. In Table I1 we compare the bond lengths calculated from Table I with those measured experimentally. TABLE I TETRAHEDRAL COVALENT R m n (IN S Se Te

Zn

Cd

Hg

Mn

1.127 1.225 1.405 1.225 1.405 1.403 1.326

A)

van Vechten and Phillips, van Vechten and Phillips, van Vechten and Phillips, van Vechten and Phillips, van Vechten and Phillips, van Vechten and Phillips, Yoder-Short el al., 1985

1970 1970 1970 1970 1970 1970

We shall show next that the covalent radius remains meaningful in the context of the Ai'-,Mn,BV1 ternary alloys, providing a very useful guideline to the variation of the lattice parameter with composition in all DMS compounds.

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CRYSTAL STRUCTURE AND MATERIALS PREPARATION

7

TABLE I1

BOND LENGTHSb IN METALCHALCOGENIDES Bond length in

A

Compound

Calculated

Observed*

ZnS ZnSe ZnTe CdS CdSe CdTe HgS HgSe HgTe

2.352 2.450 2.630 2.532 2.630 2.810 2.530 2.628 2.808

2.342 2.454 2.631 2.532 2.633 2.806 2.534 2.634 2.797

*R. W. G . Wyckoff, Crystal Structures, 2nd Edition(Interscience, New York,1963),Vol. 1 , p. 112.

3. CRYSTAL STRUCTURE AND LATTICE PARAMETER OF A:’_,Mn,BV’ DMS Crystal structure of ternary alloys formed by substituting Mn for the group-I1 element in the A1’BV’ lattice is indicated in Fig. 1, where “cub” refers to the zinc blende, and “hex” to the wurtzite structure. As a rule, these DMS retain the crystal structure of the “parent” A1’Bv’ compound. Exceptions to this rule are Znl-,Mn,S and Znl-,Mn,Se: these exhibit the structure of the cubic A1’BV’ host for low Mn content, but above a certain value of x , they form in the wurtzite structure. A natural upper limit on the Mn mole fraction x in DMS is imposed by the fact that the MnBV’binary compounds do not crystallize in the zinc blende or wurtzite structures. It is interesting to note that, although the stable phase of HgS itself is the cinnabar structure (HgS is the only A”BV’ compound to crystallize in this form), the presence of Mn in Hgl-,MnxS appears to stabilize the sphalerite structure of this system (Pajaczkowska, 1978). At this time, relatively little is known about this alloy. The remarkably systematic behavior of lattice parameters observed in the A1’Bv’ compounds, as described in the preceding section, carries over to their A:’-,Mn,BV1 derivatives. Two aspects are particularly worthy of note. First, in all DMS crystallizing in the wurtzite structure, the a :c relationship predicted by the ideal hexagonal close packing of spheres is obeyed. Thus, for Zno.dsMno.ssSe, the observed value is c/a = 1.635, as compared to the = 1.633 (Yoder-Short et al., 1985). Since in the ideal case both ideal

a

W . GIRlAT AND J . K. FURDYNA

a and c are determined by the nearest neighbor distance d of the anion (or cation) sublattice, one concludes that d is the one key parameter which determines the structural dimensions of the ternary DMS lattice. Second, the lattice parameters for all DMS obey Vegard’s law very closely. This is illustrated by Fig. 2, showing the dependence of the three cubic tellurides on composition (Furdyna ef a/., 1983). It is gratifying that an extrapolation to x = I-which indicates the lattice parameter for the hypothetical zinc blende MnTe-is the same for the three alloys. Thus, in spite of the fact that zinc blende MnTe does not exist in nature, one may regard Cdl-,Mn,Te as a true pseudobinary alloy of CdTe and MnTe, and one can write its lattice parameter in the form a = (1 - x)aIr-vI

+ XaMn-VI.

(5)

The behavior illustrated in Fig. 2 for the cubic lattice parameter a is actually true for d independent of structure, so that the Vegard law behavior can be presented in a unified picture for the entire DMS family, wurtzite and zinc blende alike. We show this in Fig. 3. The usefulness of Fig. 3 is made clear by considering, for example, the linear behavior of d observed for the selenides. Note that Hgl-,MnxSe is cubic, Cdl- ,Mn,Se is hexagonal, and ZnI-,Mn,Se changes its crystal structure “in stride” as x increases. Thus, once having established the parameter d for MnSe by extrapolating, e.g., the lattice constant data for the hexagonal Cdl-,Mn,Se to x = 1, we can predict the lattice constant for the cubic Hgl-,Mn,Se, or for either of the two phases 6

.

5

0

5

0.6 0.8 I .o Mole fraction X FIG.2. Lattice parameter as a function of Mn mole fraction x for telluride DMS. Note the Vegard law behavior, and the convergence of the extrapolated lines at a single point (6.334 A ) , which determines the lattice parameter for the zinc blende phase of MnTe. [After Furdyna et a/.(1983), with Hgl-,MnxTe data taken from the paper by Delves and Lewis (1963).]

0

0.2

0.4

1. CRYSTAL STRUCTURE AND MATERIALS PREPARATION

9

MnTe

Mn Se

Mn S

MOLE FRACTION X FIG.3. Mean cation-cation distances d as a function of Mn mole fraction x for A!~,MnxBvl alloys. Analytic expressions correspondingto the figures are given in Table 111. [After YoderShort et at. (1985).]

of Znl-,MnxSe. Two additional points are worthy of note in Fig. 3. First, it is interesting that the mutual relationship of d for HgSe and CdSe in this figure closely resembles that for HgTe and CdTe, despite the difference in structure. This again focuses on the fact that, ultimately, the tetrahedral radii are the meaningful parameters, independent of the lattice environment. Second, we observe that the value of d for Znl-,Mn,Se increases smoothly through the cubic-to-hexagonal transition, suggesting that it is the increase in the behavior of the lattice spacing that forces the onset of the hexagonal structure, rather than the other way around. Analytical expressions for the cation-cation distances d, from which respective lattice parameters can be

10

W. GIRIAT A N D J. K . FURDYNA

TABLE 111

MEANCATION-CATION DISTANCES d IN A:'-,Mn,BV1 ALLOYS

Material

Upper limit of x

Znl-,Mn,S Cdl-,Mn,S Hgl-,MnxS Znl-xMn,Se Cdl-,Mn,Se Hgl-,Mn,Se Znl-,Mn,Te Cdl-,Mn,Te Hgl-xMnxTe

0.60 0.50 0.37 0.57 0.50 0.38 0.86 0.77 0.75

Mean cation-cation distaace d (A) 3.830 4.123 4.139 4.009 4.296 4.301 4.315 4.587 4.568

+ 0.139~

- 0.151~ -

0.167~

+ 0.164~

- 0.123~ - 0.123~

+ 0.168~ - 0.105~ -

0.080~

obtained via Eqs. (2)-(4), are listed in Table I11 for all A:~,Mn,B'" alloys (after Yoder-Short et af., 1985). In analogy with Table 11, we list in Table IV the extrapolated bond lengths b for zinc blende and/or wurtzite MnTe, MnSe, and MnS, along with the covalent radius of manganese obtained from these values and.from the data listed in Table I. The covalent tetrahedral radius for Mn could in principle be obtained from measurements on zinc blende and/or wurtzite phases of MnSe and MnS. These tetrahedrally bonded phases, however, are difficult to prepare, and only limited data are available on the binary compounds. We believe therefore that the value of d for the three manganese chalcogenides obtained by extrapolation of the DMS data to x = 1, as illustrated in Figs. 2 and 3, are more reliable than the literature values obtained from measurements made on the binary compounds MnS and MnSe themselves (Pajaczkowska, 1978). The internal consistency of the extrapolated DMS values further corroborates the reliability of the values listed in Table IV, which ultimately provides the covalent tetrahedral radius of Mn, rMn = (1.326 f 0.018)A (Yoder-Short et al., 1985). It should be noted that, apart from its fundamental importance, the precise knowledge of the lattice parameter is of considerable practical interest in that it provides a convenient determination of crystal composition in ternary alloys. It is a fortunate circumstance in this respect that in the case of all A:'_,Mn,Bv' alloys the linear variation of the lattice parameter with x is very large (contrast this with, e.g., Hgl-,Cd,Te), making this approach to the determination of composition quite reliable. Typical x-ray techniques yield the lattice parameter with precision of 1 part in lo4 or better, thus giving the composition within 1% for most DMS.

1.

CRYSTAL STRUCTURE AND MATERIALS PREPARATION

11

TABLE IV BONDLENGTHS b AND Mn COVALENT RADIUSrMn IN Mn CHALCOGENIDES’ Compound

b (A)

rMn(A)

MnS MnSe MnTe

2.432 2.557 2.746

1.305 1.335 1.341

Average covalent radius of Mn:

1.326

“After Yoder-Short et al., 1985.

4. MICROSCOPIC (LOCAL)

STRUCTURE OF THE

DMS LATTICE

In view of the very consistent adherence to Vegard’s law obtained from x-ray diffraction data described above, one might automatically assume that the crystal structure of a ternary “zinc blende” alloy is strictly zinc blende, i.e., the anions and the cations are distributed over two interpenetrating, geometrically identical fcc sublattices, each with a lattice constant varying linearly with x. Under this assumption, the picture would be identical to the microscopic structure of the parent AxlBvlbinary compound, with the scale adjusted by the Vegard law behavior, and with the cation lattice sites occupied randomly by two kinds of ions. An analogous picture suggests itself for the wurtzite structures. This picture constitutes the foundation of the virtual crystal approximation, frequently used in interpreting the physical properties of ternary alloys. In reality, the microscopic situation is considerably more complicated. Recent extended x-ray absorption fine structure (EXAFS) studies of DMS by Balzarotti et al. (1984) shed important light on the detailed microscopic structure of a disordered DMS lattice (and, by extension, of other tetrahedrally bonded ternary alloys), and we shall examine them in some detail. The EXAFS technique provides detailed information on the immediate atomic environment of a given lattice site. In the case at hand, it is sensitive to the local spatial distribution of the four cations tetrahedrally bound to a given anion. In reporting the results and conclusions which follow from the EXAFS data, we shall, for specificity, refer to Cdl-xMn,Te, on which the measurements were actually performed. The conclusions, however, must apply equally well to other zinc blende ternary alloys (not just DMS) and, indeed, to ternary wurtzite systems as well. Briefly, the EXAFS results carried out on Cdl-xMnxTe for 0 < x < 0.7 indicate that the Cd-Te and the Mn-Te bond lengths in this material remain practically constant throughout the entire range of compositions studied.

12

W. GIRIAT A N D J. K . FURDYNA

The Cd-Te bond length is approximately unchanged from its value in pure CdTe, and the Mn-Te bond length is approximately that predicted for the hypothetical zinc blende MnTe (for which the lattice parameter is predicted in Fig. 2). The actual bond length values are shown as a function of x in Fig. 4. These results are not in contradiction with x-ray diffraction data, which show Vegard’s law behavior for the ternary lattice. X-ray diffraction is not sensitive to the topological details surrounding an individual lattice point, but determines the lattice parameter averaged over all anion and cation sites. It is precisely this averaged a(x) which varies linearly with x. The EXAFS results, however, cannot be reconciled with the interpretation that Vegard’s law is obeyed on the microscopic scale by each of the two interpenetrating fcc lattices (the one occupied by Te atoms, the other bya random distribution of Cd and Mn). Rather, the two sublattices must adjust locally to accommodate the requirement of keeping the respective Cd-Te and Mn-Te bond lengths as nearly constant as possible. While in reality both the anion and the cation lattices are thus distorted from fcc, it is realistic to represent the microscopic behavior in terms of a model proposed by Balzarotti et al. (1984) as follows. The cation sublattice, occupied randomly by Cd and Mn, is to afirst approximation not distorted. It may be regarded, to first order, as an fcc sublattice, obeying Vegard’s law in the spirit of the virtual crystal approximation. The atoms on the anion sublattice, on the other hand, adjust in order to accommodate the requirement that the Cd-Te and the Mn-Te bond lengths remain as constant as possible. The anion sublattice, then, is distorted, and is no longer fcc.

2.80t

-

Cd-Te

T

-

2.75 -

t

n

Q Mn-Te

0.1

0.3

Q

0.5 Mole Fraction X

0 0

0.7

FIG.4. Actual Mn-Te and Cd-Te bond lengths as a function of composition, determined by EXAFS. [After Balzarotti ef al. (1984).]

1. CRYSTAL STRUCTURE AND MATERIALS PREPARATION

13

The justification for the above picture (i.e., for the fact that the two sublattices behave differently) is as follows. In Cdl-,Mn,Te, a given Te site can be surrounded by 0, 1, 2, 3, or 4 Mn nearest neighbors, the remaining ions of the tetrahedral quartet being Cd, respectively. When a given anion is surrounded by four like cations, it will remain in its equilibrium position (assuming nearest neighbor interactions to be dominant). In the three cases of a mixed Cd and Mn environment, however, the anion will move from its equilibrium position so as to adjust its distance as closely as possible to the Cd-Te and Mn-Te bond lengths. Since the Mn-Te bond length is shorter than the Cd-Te bond, the Te ion will be displaced from its equilibrium position towards Mn and away from Cd. The assumption of the model that, unlike the anions, the cation sublattice is not distorted is plausible in that each cation is always surrounded by a tetrad of like atoms (Te in Cdl-,Mn,Te). If a cation were to move to a closer position with respect to one anion, it would automatically move further away from other identical anions. In reality, some distortion of the cation sublattice must also occur, but on the basis of the above argument it is likely to be considerably smaller than the distortion of the anion sublattice and may be neglected at this stage of our understanding. Theoretical calculations for the proposed model carried out by Balzarotti et a/. (1984) are in excellent agreement with observation. The authors show that if all possible configurations of all possible tetrahedra are considered, the average anion sublattice obtained in this way turns out to be exactly the fcc sublattice, with a(x) given by Vegard’s law. X-ray diffraction, by averaging over all possible sites existing in the crystal, measures precisely such an average, Vegard-law-obedient behavior. The ideas advanced by Balzarotti et al. are expected to apply to other random ternary zinc blende alloys of the A:’-,Mn,BV1 type, as well as to analogous wurtzite systems. The constant value of the anion-cation distance that emerges from this picture, gives added strength to the concept of the tetrahedral radius (see earlier discussion) as a physically meaningful concept. While important from a directly crystallographic point of view, these results are also expected to bear on a number of other physical phenomena. For example, in light of these results, the microscopic environment of atoms in the zinc blende and wurtzite ternaries are no longer strictly cubic and hexagonal, respectively, and many optical transitions-particularly those involving local mode excitations-that are forbidden by the ideal symmetry (as manifested in the binary “parent” compound) may be relaxed in these random alloys. Furthermore, magnetic properties of DMS are expected to be dominated by super-exchange interaction, i.e., by Mnf+-Mn++ interactions that are mediated through the intervening anion (Larson et al., 1985; Spalek et al., 1986). The anion-Mn distance, as well as the angle subtended

14

W. GIRIAT AND J . K. FURDYNA

by the Mn++ pair and the anion, are expected to bear directly on this exchange process, and thus the new insights into the bond length are likely to play an important role in understanding the magnetic properties of DMS. 111. Preparation of Bulk DMS Crystals

The literature on the synthesis and crystal growth of A:'_,Mn,BV' ternary alloys is quite extensive. The systems Znl-xMnxS and Cdl-,Mn,S were obtained by Schnaase as early as 1933 by precipitation from dilute aqueous solutions of the binary constituents. Since that time (although mostly after 1960), a large number of studies on synthesis and crystal growth of the A:'_,MnxBv' solid solutions were carried out. We shall not attempt to present a historic survey of the various methods of preparation of these materials. This has been done with admirable completeness in the excellent review by Pajaczkowska (1978). Rather, we shall focus on those techniques which appear to be most successful and practical. In the subject of crystal growth, it is of course difficult, if not impossible, to be entirely objective. It must therefore be stated at the outset that the choice of methods of preparation that we describe are largely flavored by the authors' own experience and their particular assessment of the practical aspects of the procedure, taking into account accessibility, cost, and convenience. 5 . PREPARATION OF STARTING MATERIALS

a. Purification of starting elements

To prepare any semiconducting material, it is of course absolutely necessary to start with high purity elements. Commercially available spectrally pure materials are, unfortunately, very often oxidized to a degree which makes them unsuitable for this purpose. For this reason, we shall first briefly describe a typical method of purification for each of the elements relevant to the A:'_,Mn,Bv' DMS, before discussing the preparation of the compounds themselves. Normally, purity of 6N to 7N is sufficient for preparation of good quality crystals (i.e., adequate for most optical and electrical studies). Mercury can be readily purified to a level of 6 or 7N by double or triple vacuum distillation in quartz tubes. Tellurium and selenium are usually vacuum distilled (twice) and zone refined to attain desired purity. In the case of sulphur, purity better than 6N can be obtained by triple distillation in vacuum. Commercially available cadmium and zinc, even of high nominal purity, are usually highly oxidized as they are delivered, and one must also distill these elements two to three times to make them usable. For storage, materials should be sealed in glass tubes in vacuum.

1. CRYSTAL STRUCTURE AND MATERIALS PREPARATION

15

FIG.5. Arrangement for vacuum sublimitationof Mn: (1) furnace; (2) external ceramic tube; (3) inner ceramic tube; (4) manganese before sublimation; (5) manganese after sublimation; (6) exit to vacuum system.

Manganese constitutes a special case. The commercially available material is typically in the y-Mn form, prepared electrolytically. This form, even with relatively high nominal purity, cannot be used as it stands because it is invariably very heavily oxidized. Even though the oxide layer can be removed by etching (the easiest method is to etch in HNO3), the material oxidizes so rapidly that it is generally difficult to transfer the manganese from the etch to the crucible without prohibitive oxidation. Removal of oxidation by reduction in hydrogen is not practical for the reason that-although higher order Mn oxides (MnO2, Mn203 , and Mn304) can be reduced to MnO at readily accessible temperatures and pressures-the reduction of MnO in H2 is practically impossible, requiring temperatures of about 2000°C and pressure of 150 atmospheres. The most successful method of Mn purification is by sublimation. This not only leaves the troublesome MnO behind, but converts the easily oxidized y-Mn phase to a-Mn, which is much more resistant to oxidation (Kaniewski et al., 1978). A simple arrangement used for vacuum sublimation of Mn is shown in Fig. 5 . Two concentric ceramic tubes are positioned in the furnace, with pieces of manganese placed inside the inner tube, at the hottest zone (marked 4 in the figure). Sublimation is carried out under high dynamic vacuum at a temperature of about 1OOO"C. After sublimation, pure manganese collects on the walls (marked 5 in the figure), and only the MnO powder remains in the hot zone (marked 4) in the inner tube. Such sublimation is generally carried out twice. Immediately after sublimation, the purified manganese is sealed in evacuated glass tubes for storage.

b. Preparation of starting binary compounds In preparing DMS alloys, it is more convenient to start with binary A"BV' and MnB"'compounds than with theelemental constituents. We will therefore briefly describe the preparation of the binary systems relevant to DMS. The preparation of the A"BV1 is relatively easy, largely because the constituent elements have low melting points (see Table V) and high vapor

16

W . GIRIAT A N D J . K . FURDYNA

pressures at relatively low temperatures. Because of these comparatively low temperatures, quartz tubing can be used for the preparation of most of the A"BV' compounds. Stoichiometric amounts of high purity elemental materials are placed in a quartz ampoule, which is then pumped to a vacuum of the order of Torr. The tube with the material is subsequently sealed and heated in such a way as to ensure that there always exists a position within the ampoule where the temperature is less than the lower of the boiling points of the two constituents. After reaction occurs, it is possible to raise the temperature to the melting point of the compound. Melting points of the binary compounds, along with their crystal structure and lattice parameters, are listed in Tabel V. TABLE V MELTING POINTS,LATTICE CONSTANTS, AND CRYSTAL STRUCTURE COMMONMn(Te, Se, S), Hg(Te, Se, S), Cd(Te, Se, S), AND Zn(Te, Se, S) PHASES

OF

m.p. ("C)

4A)

Structure

MnTe

1165

a = 4.1475 c = 6.710

NiAs

MnSe MnS HgTe HgSe HgS CdTe CdSe

1510 1530 670 779 825 1098 1260

NaCl NaCl zinc blende zinc blende cinnabarhinc blende zinc blende wurtzite

CdS

1405

ZnTe ZnSe ZnS

1293 1526 1722

5.462 5.223 6.460 6.084 5.851 6.481 u = 4.2985 c = 7.0150 u = 4.1368 c = 6.7163 6.1037 5.6687 5.4093

wurtzite zinc blende zinc blende zinc blende

Thus, HgTe can be prepared by heating an evacuated sealed ampoule containing Hg and Te so that the highest temperature in the ampoule is ca. 700°C and the lowest is ca. 420°C. This situation is maintained for about 24 hours, which is sufficient for the materials to react. The temperature of the whole ampoule is then raised to 700°C in order to melt the HgTe. Preparation of HgSe proceeds in an analogous manner, except that the melting point of HgSe is 779°C. Reaction of HgS requires higher temperatures, and becomes dangerous because of the high vapor pressure of the elements. Very thick quartz, with external counter pressure, is then recommended. CdTe is

1. CRYSTAL STRUCTURE AND MATERIALS PREPARATION

17

prepared by raising the temperature of the stoichiometric amounts of the constituent elements in an evacuated sealed ampoule from room temperature to 1100°C during 2 to 3 hours. CdSe is prepared by heating the ampoule containing the elements so that the lowest temperature in the ampoule is at 700°C, the reaction occurring below the melting point of CdSe. In the case of CdS, the minimum temperature should be maintained below 440"C, with the high temperature in the ampoule at about 1000°C. Zinc telluride, selenide, and sulfide are prepared like CdS, where now the minimum temperature is 440°C for ZnS, 700°C for ZnSe, and 900°C for ZnTe, and the maximum temperature in the ampoule is held near 1000°C. The A"BV' compounds have the highly desirable property that the vapor pressure of the elements is much higher than that of the compounds. Any unreacted elements can thus be driven off simply by heating the vacuumreacted binary. During this process, unreacted elements evaporate and Can be removed by pumping, so that only the pure stoichiometric compounds remain in the ampoule. This convenient behavior does not occur in other semiconductors, such as the A1"BV, A'"BV1, or Mn compounds. In these materials some of the elements have very low vapor pressures, so that it is not possible to remove them simply by evaporation. To obtain MnS or MnSe, purified Mn is first ground in mortar and pestle to a fine powder and then placed in a quartz ampoule with stoichiometric amounts of Se or S. The ampoule is then sealed off in vacuo, and is placed in a furnace in such a way that the cooler end is at 700°C for MnSe and 450°C for MnS, and the hot end at about 1000°C for both cases. After several days the reaction is complete. MnTe can be prepared much more quickly. Stoichiometric amounts of Mn and Te are placed in a carbonized quartz ampoule in vacuo. The ampoule is inserted into a hot furnace (ca. 1000°C) for about 5 minutes, and is then quickly removed, yielding reacted polycrystalline MnTe.

6 . CRYSTAL GROWTH OF DMS ALLOYS Although similarities exist between the methods of preparation of specific DMS alloys, there are also significant differences because of the wide range of melting points and vapor pressures spanned by this family of materials. For this reason, it is more convenient-at the risk of some repetition-to present the preparation procedure for each alloy system separately. (a) Cdl-,Mn,Te Cdl-,MnxTe crystallizes in the zinc blende structure in the range 0 < x I0.77 (Triboulet and Didier, 1981). For xexceeding this limit, mixed phases occur. The phase diagram for this system is shown in Fig. 6 . The

18

W. GIRIAT AND J. K. FURDYNA

I 150-

Il5Ot Liquid

1100Liquid+solid

I

? w 1050-

I

[r

Solid

3

L

904

Oll

012 013 014 015 d.6 017 018 019

1 3

near-coincidence of the solidus and liquidus curves indicates that this system is ideally suited for growth by the Bridgman method. The melting point, which for pure CdTe occurs just below llOO°C, decreases slightly as the Mn content increases. For example, for x = 0.77 the melting point of Cdl-xMnxTeis about 30°C below that of pure CdTe. Crystal growth by the Bridgman method is typically carried out with a starting temperature of about 1120°C,using a temperature gradient of about 50°C/cm in a vertical furnace. The charge is lowered at a speed of 4 mm/h. High quality monocrystals, with typical dimensions of 15 mm in diameter and 10 cm in length, can be obtained in this manner. The macroscopic quality and homogeneity of the Cdl-*Mn,Te crystals is, at this stage, the best of the DMS alloy family, probably because of the unique coincidence of the solidus and liquidus curves in the phase diagram (Fig. 6). Also, because of this latter feature, the composition of the Bridgman crystals remains close to the starting (i.e., “nominal”) value. In their mechanical properties, single crystals

1. CRYSTAL STRUCTURE AND MATERIALS PREPARATION

19

of Cdl-,Mn,Te in the whole range of compositions strongly resemble those of CdTe, with a tendency to cleave along the (110) faces. Like CdTe, crystals of Cdl-,Mn,Te frequently exhibit twinning (Wu and Sladek, 1982). In terms of their electronic properties, as-grown crystals tend to be p-type, with a high resistivity. One can obtain n-type single crystals, for example, by doping with Ga or In, and by subsequent annealing in the presence of Cd. (b) Cdl-,Mn,Se The phase equilibria of Cdl-,Mn,Se were first investigated by Cook (1968). The phase diagram of this system was determined by lattice constant measurements and differential thermal analysis (DTA). Results of those investigationsare presented in Fig. 7. CdI-,Mn,Se forms a single-phase solid solution for x up to about 0.50, the crystal structure being wurtzite throughout this range of x . The melting point of the solid solution is nearly constant, starting with the value of 1260°C for pure CdSe, and decreasing very slightly as x is increased. The vertical Bridgman technique can be used successfully to grow Cdl-,Mn,Se crystals. The 1260°C melting point is still sufficiently low to permit the use of quartz ampoules, but thick-walled ampoules are recommended. Pre-reacted pure CdSe and MnSe binary compounds are placed in the silica ampoule in desired proportions, and the tube is sealed in good vacuum. The tube is then placed in a furnace at a temperature of 1270°C. The ampoule is lowered at a speed of, typically, 4 mm/h. Single crystals of several cubic centimeters in size can be obtained. All compositions in the range 0 < x < 0.5 can be grown routinely in this manner. Because the solidus L

- I500

Liquid

-

rn

1

I

-

0

- ,I

m

-

-

---

-1200

e

L

-1100

Wurtzite ond rock solt

Wurtzite

Ya-

-1000

I F g

-

- 900 E

-

-

g

1.0

Cd Se

I

I

0.8

I

I

0.6

0 I

0.4

I

I

0.2

I

000

I - 700

0.0

MnSe Mole Fraction X FIG.7. Phase diagram for CdSe-MnSe. [After Cook (1968).]

20

W . GIRIAT AND J. K . FURDYNA

and liquidus lines in the phase diagram are very close, the effect of segregation is not very serious (although it is somewhat more pronounced than in Cdl-,Mn,Te), and the crystals display only a small composition gradient along the direction of growth. (c) Cdl-,Mn,S Single crystal growth of this important system is at a much more preliminary stage than that of the preceding two alloys. The phase diagram of CdS-MnS solid solutions was determined by Cook (1968) and Wiedemeier and Khan (1968). Finely powdered MnS and CdS samples were thoroughly mixed in desired molar ratios and compressed into pellets at a pressure of approximately 13.5 K-bar. The pellets were sealed under high vacuum in quartz tubes, and annealed for approximately 100 hours at each of the following temperatures: 1000", 800°, 700°, and 600°C. The tubes were then taken out of the furnace, and the samples quenched in air. This procedure proved to be sufficient to maintain the equilibrium established at the high temperature, resulting in stable sintered polycrystalline specimens. To ensure that equilibrium was established, selected samples were subjected to additional repeated annealing cycles at 800, 700, and 600"C, for a total of several hundred hours at each temperature. Lattice constants of samples subjected to repeated annealing cycles were identical, within the limits of error, with values for alloys obtained by the original annealing sequence. The temperature-composition phase diagram of the CdS-MnS system is presented in Fig. 8. As can be seen in the figure, the liquidus and solidus are practically indistinguishible on the CdS-rich side of the diagram. In the case at hand, the melting points of the constituent binaries are very high: roughly 1400°C for CdS and about 1600°C for MnS. Because of this, silica can no longer be used in the crystal growth of Cdl-,Mn,S from the melt. Single crystal growth of Cdl-,Mn,S by the Bridgman method was reported by several authors using carbon crucibles, under high inert gas pressure (Ikeda et al., 1968; Komura and Kando, 1975). Compressed pellets were used as starting charges in these investigations. The pellets were first repeatedly annealed at 1000°C in nitrogen/HZS atmosphere. The carbon crucible containing the annealed powder was then placed in a furnace filled with dry Ar gas. The pressure of Ar was between 40 and 100 kg/cm2. After holding the temperature slightly above the melting point of the alloy for approximately one hour, the crucible was lowered out of the furnace at a rate of 30 mm/h. The fusing temperature of the mixed powder was set at 1450°C for the Cd-rich samples (0 5 x I 0.30), and at 1620°C for the Mn-rich composition (0.7 Ix 5 1). This procedure demonstrated that Cdl-,Mn,S forms in the wurtzite structure for Cd-rich compositions, and that the rocksalt phase of this material may form at very high values of x (see Fig. 1).

1.

CRYSTAL STRUCTURE AND MATERIALS PREPARATION

Liquid

II

-

-

,*

..

-

,.'

.,,*-'__,'- -

-

/

/

/

~

0

0

21

;-"I600 - 1500

/

Wurtzite

-

Wurtzite a n d

I

I

l

l

1

I

\

I

I

I

:

,

700

FIG. 8. Phase diagram for CdS-MnS. [After Cook (1968).]

A mixture of both these structures occurs for the intermediate region of x. The size of the crystals obtained in this manner were of the order of several millimeters on the side. It is significant that all the specimens contained more Mn than the starting material, especially the specimens with low Mn content, possibly due to the fact that the volatility of CdS is much higher than that of MnS. The Bridgman crystals were quite homogeneous along the growth directions, as might be expected from the nature of the phase diagram, particularly for the lower values of x . Single crystals of Cdl-,Mn,S in the range of 0 Ix 5 0.50 have also been obtained by chemical transport methods. Pure binary components were mixed in the desired ratio as the starting step. Iodine was used as the transport agent in the amount of about 10 mg/cm3. After one week in the furnace at about 95OoC, small crystals of good quality were obtained, typically in the form of platelets of 5 x 10 x 1 mm in size. The composition of the platelets was usually very close to that of the starting material. Finally, Cdl -,Mn,S crystals have also been grown by the vertical Bridgman method using induction heating and self-sealing graphite crucibles (Debska el al., 1984). By this method crystals of 1 cm3 in size were obtained. While the runs carried out so far were preliminary in nature, it appears that the induction-heated self-sealing approach is highly promising, and warrants further exploration. (For a detailed description of the method, see the discussion of Znl-,Mn,Se, below.)

22

W. GIRIAT A N D J. K . FURDYNA

(d) Znl -,Mn,Te Znl-,MnxTe is reported to form single phase solid solutions over a wider range of Mn concentration than any other member of the DMS family (Pajaczkowska, 1978). Since MnSe and MnS themselves are known to form (albeit with difficulty) stable zinc blende and/or wurtzite phases, while MnTe does not, this wide range of miscibility of Mn in the zinc blende Znl-,Mn,Te is somewhat surprising. The phase diagram of the ZnTe-MnTe system is not exactly known. The melting point of ZnTe is 1295"C, and the melting points of the ternary alloys are expected to be close to that value. As in the earlier examples of DMS growth, the Bridgman method can be used for crystallization of Znl-,Mn,Te, with binary ZnTe and MnTe in appropriate quantities as starting materials. Silica tubes may still be used as ampoules, but-as in the case of Cdl-,Mn,Se-the tubes must be thick-walled because of the high temperatures required. Good quality large crystals have been obtained for low values of x (x < 0. lo), but usually the crystal quality deteriorates as the Mn content increases. For high values of x the ingots consist of relatively small crystals, which are frequently twinned. Typical single crystal size is then about 0.5 cm3, with occasionally larger grains (Furdyna et al., 1983). (e) Znl-,MnxSe

As can be seen from Table 111, MnSe is soluble in ZnSe for x I0.57. In the composition range 0 < x < 0.30 this solid solution crystallizesin the zinc blende structure, and for 0.35 5 x I 0.57 in the wurtzite form. For x > 0.57 mixed phases occur, including various selenides of Mn. Considerable twinning is observed in the zinc blende phase as x approaches the transition value of 0.30. As in most other DMS, pre-reacted binary ZnSe and MnSe are used as starting materials. Taken in desired proportions, they are ground to a powder, mixed, pressed into pellets, and reacted for several days in the temperature range of 1OOO-11OO"C. The resulting sintered compounds are then used as starting material for crystallization. The phase diagram of ZnSe-MnSe is not known, but it is clear that the melting point is slightly over 1500°C throughout the entire range 0 < x < 0.57. Chemical transport is the simplest method to grow single crystals of Znl-,Mn,Se. For this purpose, a tube of inner diameter of about 15 mm and 12to 15 cm long can be used, in which the sintered material is placed, together with iodine (10 mg/cm2) as the transport agent. The tube is then sealed in vacuum and put in a furnace, where the temperature is at 1000°C and 980"C, respectively, at the two ends of the tube. Single crystals can be harvested after one week, size of the crystals being several millimeters in each direction.

1.

CRYSTAL STRUCTURE AND MATERIALS PREPARATION

23

4 6

7 1 C C C

C C

8-

1

5

3 2

FIG.9. Self-sealing graphite crucible used in the growth of Zn, -xMnxSe:(1) graphite crucible; (2) threaded plug; (3) thread; (4) outside silica tube; (5) induction heating coils; (6) powdered material; (7) molten zone: (8) crystallized material; (9) exit to dynamic vacuum. [After Debska et a/. (1984).]

The composition of the crystals is typically quite close to the composition of the original starting sintered materials. The Bridgman method (Twardowski et af., 1983) is made difficult by the high melting point of this system (ca. lSOO"C), where quartz is already too soft to be useful. This can be circumvented by the use of external counterpressures. Normally, argon at about 100 atmospheres is used as the pressurizing gas. Using this method, crystals of Znl-,Mn,Se were grown in the composition range up to x = 0.10. Recently a new and highly promising method of crystal growth of Znl-,Mn,Se was developed (Debska et al., 1984), involving the use of RF induction heating, and utilizing a self-sealing graphite crucible. This method eliminates entirely the need of fused silica ampoules. The crucible assembly consists of a closed graphite tube, into which is fitted a long threaded plug, as shown in Fig. 9. After loading the powdered charge, the graphite crucible

24

W . GIRIAT A N D J. K . FURDYNA

is inverted so that the plug is at the bottom. The crucible is placed at the center of an induction coil, with the bottom half of the plug extending beyond the coil. After the system is evacuated to about Torr, the temperature is slowly raised beyond the melting point of the charge, the graphite acting as the susceptor in the heating process. Some of the material then flows down, and ultimately solidifies in the space between the crucible wall and the plug, thereby sealing the crucible. The center of the crucible is RF-heated to about 100°C above the melting point of the material, and the whole assembly is then lowered out of the stationary induction coil at the rate of about 10 mm/h. The whole process is conducted under dynamic vacuum of 10-4-10-6 Torr. The induction heater in the specific system described by Debska et al. utilized a 20 KW generator operated at about 300 kHz, and produced a hot zone with a sharp thermal gradient. The narrowness of the heater zone is important in view of the fact that graphite has a high thermal conductivity, which can act to “smear out” the temperature gradient. Since the alloy itself has a lower vapor pressure than the constituent elements, it is advisable to pre-synthesize the material at the desired composition prior to the actual crystal growth. A number of good quality Znl-,Mn,Se single crystals have been produced over the entire solid solution range, 0 5 x 5 0.57. Ingots are typically 9 mm in diameter by about 70 to 90 mm, composed of crystals of about 5 to 10 mm in length, with occasional monocrystals as long as 5 to 8 cm. (f) Znl-,Mn,S Znl-xMn,S alloys form single phase zinc blende crystals for x up to = 0.1, and wurtzite crystals for 0.1 < x 5 0.60, as indicated in Table Ill. The phase diagram for this alloy is shown in Fig. 10 (Sombuthawee et al., 1978).

-- -.:--z:--..----

2000@50

2 1600u-

LIQUID

--e--

\

1610

d , ’A /

FIG. 10. Phase diagram for ZnS-MnS. [After Sombuthawee et al. (1978).]

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CRYSTAL STRUCTURE AND MATERIALS PREPARATION

25

ZnS has a melting point of 1722"C, the highest of the A"BV' compounds. The melting point for MnS (rocksalt phase) is 1530"C, also the highest among the Mn-chalcogenides . Because of the high melting points involved, chemical transport is commonly preferred as the crystal growth technique of Znl-,Mn,S. To grow single crystals of this alloy, sintered material (prepared from prereacted binaries ZnS and MnS, in a manner similar to that described for Znl-,Mn,Se) is placed in quartz tubes with inner diameter of about 15 mm, 12 to 15 cm long. Iodine(l0 mg/cm3)is used as the transport agent. The tubes are sealed in vacuum and located in the furnace such that Tm,, = 1000°C and Tmin= 980"C, for a period of one week. For samples with a small Mn content (x < 0.15), monocrystals of the size of 0.5 cm3 have been obtained in this manner. For higher values of x the crystals tend to be smaller (several millimeters in each direction). The composition of the crystals is typically quite close to that of the starting sintered material. (g) HgI-,Mn,Te The preparation of this system over a wide range of compositions has been first investigated in the classical work of Delves and Lewis (1963). As with the other tellurides, the Hg, -,Mn,Te system shows that a surprisingly high fraction of Hg atoms (about 80%) can be replaced by Mn, with the zinc blende structure retained. The phase diagram obtained by Delves and Lewis for the HgTe-MnTe system is shown in Fig. 11. From this figure it is clear that the liquidus-solidus separation is considerable, and standard crystallization is therefore not expected to yield highly homogeneous crystals. In spite of the high segregation coefficient, the majority of Hgl-,Mn,Te monocrystals prepared to date have been grown by the vertical Bridgman method, because of its simplicity. Bridgman crystals are typically prepared as follows. Pre-reacted HgTe and MnTe (or HgTe and elemental Mn and Te in proper amounts) are ground and placed into a thick-walled quartz ampoule with an inner diameter of 10 to 15 mm (Kaniewski et al., 1982). For crystals containing more than 5% Mn, the inner wall of the ampoule is coated with carbon. The ampoules, evacuated to approximately Torr and sealed, are placed in the furnace at ca. 450°C. After 24 hours, the temperature is increased to 820°C and is held at this value over a period of 48 hours to achieve complete solution of manganese. The Bridgman growth is then started. The recommended growth speed is slow, typically about 1 mm/h. Higher growth rates result in a high degree of polycrystallinity. As can be expected from the phase diagram, Hgl-,Mn,Te crystals grown by the Bridgman method have a significant composition gradient along the growth direction due to a large segregation coefficient. The segregation

26

c

W. GIRIAT AND J. K . FURDYNA

600

0.0

HgTe

735%

(

Mole Fraction X

Mn ‘e

FIG.1 1 . Phase diagram for HgTe-MnTe. [After Delves and Lewis (1963).]

effects in the growth of Hg,-,Mn,Te are qualitatively similar to those occurring in the well-known Hg, -,Cd,Te ternary alloy. It should therefore be possible to reduce the composition gradient in the former by employing some of the crystal growth techniques used for preparation of high homogeneity Hg,-,Cd,Te, for example, the traveling zone method (Dziuba, 1979), or the rapid quench-solid state recrystallization approach. The success of these latter techniques in preparation of homogeneous Hgl -,Mn,Te still awaits a systematic study. As-grown Hgl-,Mn,Te (like its sister compound Hgl-,Cd,Te) tends to crystallize with a slight Hg-deficiency. The Hg-vacancies then act as acceptors, rendering the as-grown crystals heavily p-type. Since control of electronic properties is particularly important in the Hg-compounds, this “native” nonstoichiometry is an undesirable property. Stoichiometry can be restored by post-growth annealing of Hgl -,Mn,Te in saturated Hg-vapor. Prescriptions for annealing parameters vary from experimentalist to experimentalist, but typically involve annealing between 18O-22O0C for very long times (usually at least 200 hours for every millimeter of sample thickness). (h) Hgl-,Mn,Se The phase diagram and lattice constants of Hgl -,Mn,Se were investigated by Pajaczkowska and Rabenau (1977a). The solubility limit of the solid

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CRYSTAL STRUCTURE A N D MATERIALS PREPARATION

I

950-

I-

Heating

/

Mn,Hg,-,Se

21

I

Mn,Hg,-,Se+MnSe

I I

HgSe

Oll

0 012 013 6!4 d.5 016 017 d.8 019 MnSe Mole Fraction X

FIG.12. Phase diagram for HgSe-MnSe. [After Pajaczkowska and Rabenau (1977a).]

phase, determined from lattice parameter measurements, is x 2: 0.38 at 700°C. The phase diagram for the system is shown in Fig. 12. In general, crystallization of Hgl-,Mn,Se is similar to that of the Hgl-,Mn,Te system. Again, it is convenient to use the vertical Bridgman method. Since the working temperature does not exceed 9OO”C, thick-walled silica tubes may be used. After crystallization, several large monocrystalline grains are usually obtained, much as for Hgl-,MnxTe. (i) Hgl-,MnxS We know very little about this system. HgS itself occurs in two phases: the cinnabar (a-HgS) and zinc blende (P-HgS) structures. The cinnabar phase is stable below 300°C. Above this temperature a-HgS transforms to the zinc blende P-HgS phase. Pajaczkowska and Rabenau (1977b) found that at 600°C Hgl-,Mn,S forms single-phase solid solutions in the zinc blende structure for x s 0.37 and, using the hydrothermal process, they succeeded in obtaining single zinc blende crystals of this alloy as large as 2 m m in diameter. This system is particularly interesting, since it is the only one among the DMS for which the “parent” A”BV’ compound has a stable room temperature structure other than zinc blende or wurtzite. Thus the presence of Mn appears to stabilize the tetrahedral coordination in this alloy. IV. The “New” DMS

This chapter has been devoted primarily to the now well-established A:?,Mn,BV’ alloys in bulk crystal form. It is important to remember, however, that other DMS systems are steadily gaining the interest of the scientific community. While these new DMS materials have been studied

28

W. GIRIAT A N D J. K . FURDYNA

much less thoroughly than the bulk A:'-,MnxBv' alloys, progress is rapidly being made, particularly in the layered A:!.,Mn,BV' (e.g., MBE films and superlattices), in bulk A:t,FexBV' alloys, and in the A:!!,MnxBV' systems. In concluding this chapter, we therefore wish to mention these new DMS systems briefly, so as to identify the evolving materials trends and to provide a brief literature guide for these emerging DMS activities. 7. DMS FILMSAND SUPERLATTICES

One of the most important-and most rapidly evolving-recent developments in the area of new DMS systems is the successful preparation of DMS films, heterostructures, and superlattices. The first reported DMS film structures involved the quaternary epitaxial layers of Hgl -x-yCdxMnyTe prepared by the close-spaced isothermal vapor transport growth technique (Debska et af., 1981). The field has taken a major step forward when it was demonstrated that Cdl -,Mn,Te films and Cdl -,Mn,Te/Cdl-,Mn,Te superlattices could be readily grown by molecular beam epitaxy (MBE) on a variety of substrates (Kolodziejski et a/., 1984b; Bicknell et a/., 1984; Kolodziejski et a/., 1984a). Since 1984, the field has made truly remarkable progress, including successful preparation by MBE of films and superlattices of Znl-,Mn,Se (Kolodziejski etal., 1984) and of Hgl-,Mn,Te (Harris et al., 1987; Faurie, 1987). In parallel with these developments, atomic layer epitaxy (ALE)-an equilibrium growth method, in contrast to MBE, which involves nonequilibrium growth-has also been shown as a viable technique for the growth of DMS films (Herman et a/., 1984: Tammenmaa et a/., 1985). It is clear that the ability to grow high quality DMS films, superlattices and heterostructures, coupled with the novel properties of DMS materials, opens a rich spectrum of possibilities in basic and applied science. A survey of these opportunities is beyond the scope of this chapter. It should be mentioned, however, that apart from exciting electronic, magnetic, and optical applications (Datta et a/., 1984), epitaxy also opens the way to the growth of entirely new systems that do not form in the bulk. The recent preparation by MBE of zinc blende Znl-,Mn,Se for x > 0.30 and, indeed, of zinc blende binary MnSe layers in thin film form (Kolodziejski et al., 1986)are examples of such new systems. This aspect alone-i.e., the MBE growth of metastable alloys and compounds-is extremely important in its potential to advance the understanding of the crystal growth of magnetic semiconductor alloys, and of semiconductor compounds in general. 8. A:'_,Fe,Bv'

ALLOYS

While bulk growth and the intrinsic properties of the A:'-,Mn,Bv' DMS are now reasonably well understood, the situation is quite different in the

1.

CRYSTAL STRUCTURE AND MATERIALS PREPARATION

29

case of their Fe-based counterparts, such as Hgl-,Fe,Se or Cdl-,Fe,Se. Literature on these materials has only recently begun to emerge and is already reporting rather unique and striking effects. There are two principal features which make these materials especially interesting, both arising from the presence of substitutional Fe in the A"BV' lattice. First, the magnetism of the Fe++ sub-system is quite different from that of Mn++,owing largely to the fact that the permanent magnetic moment of the Fe++ion vanishes at low temperatures. Specifically, it has been shown that in the dilute limited (x < 0.01) A:'_,FexBv' alloys exhibit Van Vleck paramagnetism (Guldner et ai., 1980; Serre et al., 1982; Lewicki et al., 1986). The second extremely important feature is the influence of the Fe atoms on the electrical properties of Hgl-xFexSe and related alloys. It has been a striking feature of the measurements on Hgl-,Fe,Se that the electron concentration n in this material is surprisingly independent of annealing, of x , and of other factors typically affecting n, and remains fixed at n = 5 x 10l8cm-3 in all samples investigated (Serre el al., 1982; Vaziri and Reifenberger, 1958). In particular, electron concentration in this material is remarkably stable in time (Vaziri et al., 1985), quite unlike the case of HgSe or Hgl-,Mn,Se, which can be strongly affected by shelf-life "passive" annealing and/or thermal cycling. This has recently been explained as follows (Mycielski et al., 1986). The Fe2+level lies at about 230 meV above the bottom of the conduction band of HgSe, with Fe2+ acting as a resonant donor. The position of the Fe2+ thus establishes the Fermi level. In this situation, the electron concentration is determined by the band structure and its corresponding density of states (like in a true seminetal, e.g., Bi) rather than by doping or nonstoichiometric defects. This remains true as long as x (the Fe concentration) exceeds the number of electronic states below the Fe2+ level (5 x 10l8cm-3 corresponds to x 2: 0.0003), and accounts for the stability of n. When the Fe concentration x is below this value, n will increase with x. An interesting approach to probing the role of Fe2+ as a resonant donor is to perform studies on the quaternary alloy Hg,-,-,Fe,Cd,Se, as has been recently reported by Mycielski et al. (1986). While the detailed discussion of the electronic structure is beyond the intended scope of this chapter, we stress that in this context the role of iron is primarily to provide electrons to the conduction band (Furdyna, 1986). Hgl -,FexSe also displays remarkably high mobilities at low temperatures, much higher than would occur in HgSe for comparable levels of electron concentrations, indicating a considerable reduction of ionized impurity scattering. This is, further, accompanied by comparatively low values of the Dingle temperature Td (Vaziri and Reifenberger, 1985). It has been suggested by Mycielski (1986) that the ionization of resonant Fe2+ donors occurs in such a way as to maximize the separation in space between the ionized sites,

30

W . GIRIAT AND J. K. FURDYNA

which in turn leads to (at least partial) space ordering. Such an ordered, periodic array of ionized impurities can then be shown to lead to reduced scattering rates as compared to a random distribution of ionized impurities (Pool et al., 1986). . The preparation of A!'_,Fe,Bv' materials is generally similar to their sister Mn alloys (Mizera et al., 1980). The range of x in which Fe can be incorporated into the A"BV' lattice is, however, considerably less than in the case of Mn alloys. For example, the highest values of x reported so far for various recently investigated alloys are -0.06 for Hgl-,Fe,Te (Guldner et al., 1980), 0.12 for Hgl-,Fe,Se (Vaziri et al., 1985), and 0.15 for Cdl-,Fe,Se (Lewicki et al., 1986). The pattern which thus appears to em&+ is that, generally, the higher the melting point of the A"BV' host, the higher is the amount of Fe which can enter substitutionally at the group-I1 sites. The important problem of establishing the limits of miscibility of Fe in specific A1'BV' lattices, and the determination of the corresponding phase diagrams, still awaits a systematic investigation. 9. MANGANESEAND RARE-EARTH-BASED A1"Bv' ALLOYS

Activity has also been recently increasing in the area of A:?,M,Bv' alloys, where M indicates a substitutional magnetic element (either Mn or a rareearth). Lead salts with a fraction of the Pb-sublattice replaced by Mn are the most thoroughly studied members of this group (Pbl-,Mn,S: Karczewski et al., 1982, 1985; Pbl-,Mn,Se: Kowalczyk and Szczerbakow, 1984; Pbl -,Mn,Te: Niewodniczanska-Zawadzka and Szczerbakow, 1980; Pascher et al., 1983; Anderson and Gorska, 1984). Crystal growth methods employed in preparing these alloys were, for the most part, Bridgman growth for bulk crystals and hot wall epitaxy for single crystal films, with growth parameters similar to those which proved successful in the preparation of the parent lead salt binaries. The miscibility of Mn in these systems appears to be smaller than in the A:'_,Mn,BV' alloys, the highest values being those reported for Pbl-,Mn,Te (x = 0.2). No comprehensive study of this aspect has as yet been described for this class of alloys. In addition to the development of the A!!,Mn,BV' alloys, there is increasing activity in lead salt alloys involving rare earths, for example, Pbl -,Eu,Te and Pbl -,Gd,Te. The methods of successful preparation include Bridgman growth (Golacki and Gorska, 1985), hot wall epitaxy of thin film specimens (Krost etal., 1985), and MBE (Partin, 1983,1984). While most of the non-MBE growth has been restricted to x < 0.10, the MBE films showed very high rare-earth miscibility in the lead salt lattice (e.g., x = 0.40 for Pbl-,Yb,Te (Partin, 1983)). Much of the interest in these materials is stimulated by their photovoltaic properties and their potential for diode laser

1. CRYSTAL STRUCTURE AND MATERIALS PREPARATION

31

applications. By comparison, at this stage we know considerably less about their magnetic properties and, in particular, about the nature of exchange interaction between the localized magnetic ions and band electrons in these materials. 10. MISCELLANEOUS DMS ALLOYS

There are two additional cases of DMS alloys that are potentially highly interesting. (Cdl -,MnX)3As2 resembles, in the absence of an external magnetic field, its parent Cd3As2: it is understood to be a zero-gap semiconductor, and can exhibit relatively high electron mobilities. The introduction of Mn brings into play spin-spin exchange interaction between the band electrons and localized magnetic moments (Neve et al., 1981), and in this respect the system should resemble qualitatively the behavior of doped Hgl -,MnxSe and Hgl-,Mn,Te. Comparatively little research has been done on this and related systems (e.g., Zn3Asz-based DMS), but the feasibility of such alloys has been demonstrated and the initial research (Neve et al., 1981) indicates a highly interesting line of investigation. Finally, we wish to point out the very promising DMS opportunities offered by the chalcopyrite structure. Chalcopyrites are tetrahedrally bonded semiconductor compounds which, in their structure and many physical properties, resemble A"BV1 or A"'BV compounds, as discussed in the excellent monograph by Shay and Wernick (1975). CuInSez and CdGeAsz are examples of such systems. The chalcopyrite crystal structure is shown in Fig. 13, side by side with the zinc blende structure, showing the close relationship of the two crystallographic forms. Considering the predisposition of Mn to act as a group-I1 atom, and the close relationship between the chalcopyrite structure and the simple A"BV' tetrahedrally bonded semiconductors, one is immediately tempted to think of incorporating Mn (and other magnetic atoms) into the chalcopyrite lattice. In particular, structures of the form A:'_,Mn,BIVCx (where Mn occupies substitutionally the group-I1 sites of the original A"B"C; lattice); or A:-,B:'f,Mn2,Cx1 (where Mn can occupy either the group-I or the groupI l l sites of the parent cation lattice) immediately suggest themselves. It should also be borne in mind that Fe is readily incorporated into the chalcopyrite structure. Indeed, the original term "chalcopyrite" refers, specifically, to the compound CuFeS2 , the mineral prototype from which the structure class gets its name. Thus Fe, which can be incorporated to only a limited degree into the A:!-,Fe,Bv' alloys, could play a more important role in the chalcopyrite DMS that may be prepared in the future. There are thus vast new possibilities in this family of materials which-like the "established" A:!-,Mn,BV' alloys-is based on the tetrahedral bond. Initial

32

W . GIRIAT A N D J . K . FURDYNA

Zinc Blende

C hatcopyrite

FIG. 13. Comparison of the chalcopyrite structure (one unit cell, shown on right) with zinc blende structure (two unit cells, left).

successes-and surprises-have already emerged. It is, for example, possible to form glassy DMS systems of Cd~-~Mn,GeAsz(Greene ef af., 1986). Given the similarities-as well as the differences-between the A"BV1 and the chalcopyrite lattice, and given the "natural" way of incorporating Fe into the latter, one can expect truly novel and exciting developments involving DMS chalcopyrite materials in the coming decade.

References Anderson, J. R., and Gorska, M. (1984). Solid State Commun. 51, 115. Balzarotti, A., Czyzyk, M., Kisiel, A . , Motta, N. Podgorny, M., and Zimnal-Starnawska, M. (1984). Phys. Rev. B30, 2295. Bicknell, R. N., Yanka, R. W., Giles-Taylor, N. C., Blanks, D. K., Buckland, E. L . , and Schetzina, J . F. (1984). Appl. Phys. Lett. 45, 92. Brandt, N. B., and Moshchalkov, V. V. (1984). Advances in Physics 33, 193. Cook, W.R. (1968). J. American Ceramic Society 51, 5 1 8 . Datta, S., Furdyna, J. K., and Gunshor, R. L. (1985). SuperlatticesandMicrostructures 1,327. Debska, U., Dietl, M., Grabecki, G., Janik, E., Kierzek-Pecold, E., and Klimkiewicz, M. (1981). Phys. Staf. Solidi (a) 64, 707. Debska, U., Giriat, W., Harrison, H . R., and Yoder-Short, D. R. (1984). J . Cryst. Growth 7 0 , 399.

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Delves, R. T., and Lewis, B. (1963). J. Phys. Chem. Solids 24, 549. Dietl, T., and Spalek, J. (1983). Phys. Rev. B28, 1548. Dolling, T.,Holden, T. M., Sears, V. F., Furdyna, J. K., and Giriat, W. (1982). J. Appl. Phys. 53, 7644. Dziuba, 2. E. (1969). J. Electrochemical Society 116, 104. Faurie, J. P. (1987). Proc. NATO Advanced Research Workshop on Optical Properties of Narrow Gap Low Dimensional Structures, St. Andrews (U.K.), July 29-August 1, 1986, edited by C. M. Sotomayor Torres et al. (Plenum, New York), p. 25. Furdyna, J. K. (1982). J. Appl. Phys. 53, 7637. Furdyna, J. K., Giriat, W., Mitchell, D. F., and Sproule, G. (1983). J. Solid State Chem. 46,349. Gaj, J. A. (1980). J. Phys. SOC.Japan 49, Suppl. A, 797. Gaj, J. A., Galazka, R. R., and Nawrocki, M. (1978). Solid State Commun. 25, 193. Galazka, R. R. (1979). Znst. Phys. Conf. Ser. 43, 133. Galazka, R. R. (1982). Proc. Znt. Conf. on Narrow Gap Semiconductors, Linz, Sept. I981, Lecture Notes in Physics 152 (Springer-Verlag. Berlin), p. 294, and references therein. Galazka, R. R., and Kossut, J. (1980). Lecture Notes in Physics 132 (Springer-Verlag, Berlin), p. 245. Golacki, Z., and Gorska, M. (1985). Acta Phys. Polon. A67, 379. Greene, L. H., Orenstein, J., Wernick, J. H., Hull, G. W., and Berry, E. (1986). Bull. Am. Phys. SOC.31, 383, Abstract EN 15. Guldner, Y., Rigaux, C., Menant, M., Mullin, D. P., and Furdyna, J. K. (1980). Solid State Commun. 33, 133. Harris, K. A., Hwang, S., Burns, R. P., Cook, J. W., Jr., and Schetzina, J. F. (1987). Diluted Magnetic (Semimagnetic) Semiconductors, edited by R. L. Aggarwal, J. K. Furdyna, and S. von Molnar (Vol. 89, Materials Research Society Symposia Proceedings, Pittsburgh, PA), p. 255. Herman, A., Jylha, 0. J., and M. Pessa, M. (1984). J. Crystal Growth 66, 480. Ikeda, M., Itoh, H., and Sato, H. (1968). J. Phys. SOC. Japan 25, 455. Kaniewski, J., Witkowska, B., and Giriat, W. (1982). J. Crystal Growth 60, 179. Karczewski, G., Klimkiewicz, M., Glas, I. Szczerbakow, A., Behrendt, R. (1982). Appl. Phys. A29, 49. Karczewski, G., von Ortenberg, M., Wilamowski, Z., Dobrowolski, W., andNiewodniczanskaZawadzka, J. (1985). Solid-State Commun. 55, 249. Kolodziejski, L. A., Sakamoto, T., Gunshor, R. L., and Datta, S. (1984). Appl. Phys. Lett. 33, 799. Kolodziejski, L. A., Bonsett, T. C., Gunshor, R. L., Datta, S., Bylsma, R. B., Becker, W. M., and Otsuka, N. (1984). Appl. Phys. Lett. 45, 440. Kolodziejski, L. A., Gunshor, R . L., Bonsett, T. C., Venkatasubramanian, R., Datta, S., Bylsma, R. B., Becker, W. M., and Otsuka, N. (1985). Appl. Phys. Lett. 47, 169. Kolodziejski, L. A., Gunshor, R. L., Otsuka, N., Gu, B. P., Hefetz, Y., and Nurmikko, A. V. (1986). Appl. Phys. Lett. 48, 1482. Komura, H., and Kando, Y. (1975). J. Appl. Phys. 46, 5294. Kowalczyk, L., and Szczerbakow, A. (1985). Acta Phys. Polon. A67, 189. Krost, A., Harbeck, B., Faymonville, R., Schlegel, H., Fantner, E. J., Ambrosch, K. E., and Bauer, G. (1985). J. Phys. C. 18, 2119. Larson, B. E., Hass, K. C., Ehrenreich, H., and Carlsson, A. E. (1985). Solid State Commun. 56, 347. Lewicki, A., Mycielski, A., and Spalek, J. (1986). Acta Phys. Polon. A69, 1043. Mizera, E., Klimkiewicz, M., Pajaczkowska, A., and Godwod, K. (1980). Phys. Stat. Solidi(a) 58. 361.

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Mycielski, J. (1986). Solid State Commun. 60, 165. Mycielski, A., and Mycielski, J. (1980). J. Phys. SOC.Japan 49, 809. Mycielski, A., and Dzwonkowski, P., Kowalski, B., Orlowski, B., Dobrowolska, M., Arciszewska, M., Dobrowolski, W., and Baranowski, J. M. (1986). J. Phys. C. 19,3605. Nawrocki, M., Planel, R., Fishman, G . , and Galazka, R. R. (1981). Phys. Rev. Lett. 46,735. Neve, J . J., Bouwens, C . J. R., and Blom, F. A. P. (1981). Solid State Commun. 38, 27. Niewodniczanska-Zawadzka, J., and Szczerbakow, A. (1980). Solid State Commun.34, 887. Oseroff, S. B. (1982). Phys. Rev. 25, 6584. Pajaczkowska, A. (1978). Prog. Crystal Growth Charact. 289. Pajaczkowska, A., and Rabenau, A. (1977a). Mat. Research Bull. 12, 2. Pajaczkowska, A., and Rabenau, A. (1977b). Solid State Chemistry 21, 1. Partin, D. L. (1983). J. Vac. Sci. Technol. B1, 174. Partin, D. L. (1984). J. Electron Mater. 13, 493. Pascher, H., Fantner, E. J., Bauer, G., Zawadzki, W., and von Ortenberg, M. (1983). Solid State Commun. 48, 461. Pool, F., Kossut, J., Debska, U., and Reifenberger, R. (1987). Phys. Rev. B35, 3900. Ramdas, A. K. (1982). J. Appl. Phys. 53, 7649 and references therein. Schnaase, H. (1933). Z.Phys. Chem. B20, 89. Serre, H.,Bastard, G., Rigaux, C . , Mycielski, J., and Furdyna, J. K. (1982). Proc. 4th Int. Conf. on the Physics of Narrow Gap Semiconductors, Linz 1981, Lecture Notes in Physics 152 (Springer, Berlin), p. 321. Shay, J. L., and Wernick, J. H. (1975). Ternary Chalcopyrite Semiconductors: Growth, Electronic Properties, and Applications (Pergamon, Oxford). Sombuthawee, C., Bonsall, S. B., and Hummel, F. A. (1978). J. Solidstate Chemistry25,391. Spalek, J., Lewicki, A., Tarnawski, Z., Furdyna, J. K., Galazka, R. R., and Obuszko, Z. (1986). Phys. Rev. B33, 3407. Tammenmaa, M., Koskinen, T., Hiltunen, L . , Niinisto, L., and Leskela, M. (1985). Thin Solid Films 124, 125. Triboulet, R., and Didier, G. (1981). J. Crystal Growth 52, 614. Twardowski, A., Dietl, T., and Demianiuk, M. (1983). Solid State Comm. 48, 845. Vaziri, M., and Reifenberger, R. (1985). Phys. Rev. B32, 3921. Vaziri, M., Debska, V., and Reifenberger, R. (1985). Appl. Phys. Lett. 47, 407. Wiedemeier, H.,and Khan, A. (1968). Trans. Metall. See. A.I.M.E. 242, 1969. Wu, A. Y., and Sladek, R. J. (1982). J. Appl. Phys. 53, 8589. Yoder-Short, D. R., Debska, U., and Furdyna, J. K . (1985). J. Appl. Phys. 58, 4056.

SEMICONDUCTORS AND SEMIMETALS, VOL. 25

CHAPTER 2

Band Structure and Optical Properties of Wide-Gap A:!-,MnxBv' Alloys at Zero Magnetic Field W. M . Becker PHYSICS DEPARTMENT, PURDUE UNNERSITY WEST LAFAYETTE, INDIANA, USA

I . INTRODUCTION . . . . . . . . . . . . . . . . . . . . 1 . General Discussion. . . . . . . . . . . . . . . . . 11. VALENCEAND CONDUCTION BANDSIN WIDE-GAPDMS ALLOYS. 2. Band Structure at Zone Center. . . . . . . . . . . . (a) Basic Features . . . . . . . . . . . . . . . . . (b) Variation of EBwith Mn Concentration. . . . . . . (c) Variation of E8 with Temperature and Pressure. . . . 3 . Band Structure A way From Zone Center . . . . . . . . 111. OPTICAL PROPERTIES OF d-ELECTRONS. . . . . . . . . . . 4. General Considerations . . . . . . . . . . . . . . . 5 . Mn Transitions in Specific DMS Alloys. . . . . . . . . IV. CONCLUDING REMARKS. . . . . . . . . . . . . . . . . 6 . Summary and Discussion . . . . . . . . . . . . . . REFERENCES.. . . . . . . . . . . . . . . . . . . .

35 35 36 36 36 38 50 56 51

51 59

68 68

I0

I. Introduction 1. GENERAL DISCUSSION

In the A~'_,Mn,B"' alloys, manganese substitutes for the cation in A"BV1 semiconducting compounds, and is in the Mn2'3d' configuration. This conclusion stems from studies on both zero-gap compounds such as HgTe, and on wide-gap materials such as ZnS (McClure, 1963; Langer and Ibuki, 1965) and ZnSe (Langer and Richter, 1966). At very low values of x (typically well below 1 atomic percent), the influence of Mn on the band structure can be disregarded, and Mn interactions with carriers can be neglected. When larger mole fractions of Mn (of the order of a few atomic percent or more) are alloyed with the A"BV' host material, however, the situation is drastically altered. For example, the presence of such Mn concentrations leads to new and interesting electrical and optical behavior due to spin-spin exchange interactions between band electrons and localized moments of the magnetic

35 Copyright B 1988 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-752125-9

36

W . M . BECKER

ions, a subject reviewed in detail elsewhere in this volume. Even in the absence of magnetic field, striking alterations in the properties of DMS occur with alloying as a result of large changes in the band structure, or as a consequence of the introduction of Mn-related transitions in the semiconductor at less than band gap energies, For example, the introduction of Mn at large mole fractions may (1) induce a change from semimetal to semiconductor behavior in initially “zero-gap” or “negative gap” materials or (2) move the intrinsic edge through the visible region in wide-gap DMS. In addition to the changes mentioned above, which are traceable to zone center shifts of band edges, variation of properties due to band structure effects throughout the Brillouin zone may be expected. The importance of Mn in the wide-gap materials is particularly striking, since the localized transitions of the Mn ion tend to dominate the optical properties at high Mn concentrations, a complication which must be considered in the use of DMS alloys for device applications. The systematics of effects in narrow-gap DMS are treated elsewhere in this volume. Here we review the properties of the wide-gap A:’-,Mn,BV’ alloys (A = Cd, Zn; B = Te, Se, S). These latter materials are found to exhibit extremely high resistances at low temperatures, thus inhibiting the study of their properties by galvanomagnetic techniques. In fact, nearly all the present experimental knowledge of the band structure of wide-gap DMS alloys is gleaned from investigation of optical effects, e.g., absorption, reflectivity, photoluminescence, and excitation spectroscopy. Here we review a variety of such experimental results that reveal the behavior of these alloys. In Part 11 of this chapter, the dependence of various band structure features on Mn concentration, on temperature, and on pressure are discussed, and tables are provided summarizing the available data. Part I11 presents results which relate the strong optical features seen at less than band gap energies with various Mn intra-ion transitions. Our concluding remarks are given in Part IV. There we pose a number of problems related to the presence of Mn in DMS that are suggested by the investigations summarized in this review and in related work. 11. Valence and Conduction Bands in Wide-Gap DMS Alloys

2. BANDSTRUCTURE AT ZONE CENTER

a. Basic Features For wide-gap DMS, such as Cdl-xMn,Te, Cdl-,Mn,Se, etc., crystallographic studies indicate that single phase regions-either zinc blende or wurtzite-exist over wide ranges of x [see Giriat and Furdyna, this volume]. All results to date suggest that in these regions of solid solution, the band

2.

31

BAND STRUCTURE AND OPTICAL PROPERTIES

gap is always direct. Therefore, the zone center band structure of these materials should be characteristic either of zinc blende or wurtzite crystals, and experimental data, i.e., optical absorption, excitonic emission, etc., have been interpreted accordingly. To understand these data, we first give a brief review of the two types of band structure to be encountered at k = 0. These can be conveniently introduced in terms of the bonding of the A"BV1 host binaries. (For an extended treatment, see Phillips, 1973.) In the A"BV' compounds, the two valence electrons of the group I1 element and the six electrons of the group VI elements are distributed according to the so-called s-p3 orbital bonding configuration. In this picture, the highest-lying valence band is triply degenerate (in the absence of spin-orbit splitting), and the states at the center of the Brillouin zone are bonding combinations of functions which are plike about the nuclei. The lowest conduction band is s-like. Figure 1 gives the results of a band structure calculation using the pseudopotential method; the structure, here obtained for ZnTe (Cohen and Bergstresser, 1966), may be taken as representative of the cubic (i.e., zinc blende) Cd and Zn chalcogenides. The introduction of the spin-orbit interaction alters the bands, principally by splitting certain degeneracies. At k = 0 in the zinc blende structure, the six-fold degenerate (three-fold not counting the two spin states) T1sv state splits into a four-fold r8(J = %)state and a lower, two-fold r7(J = 3)state, with the splitting indicated in Fig. 2 by A,, , the spin-orbit splitting energy. In the "quasi-cubic" model of the valence and conduction band in wurtzite

a4

- 0 W

-I -2

-3

L

r

k

X

K

r

FIG. 1 . Energy band structure of cubic ZnTe as calculated by the empirical pseudopotential method (Cohen and Bergstresser, 1966).

38

W. M. BECKER

2 INCBLENDE

ZINCBLENDE

Aso = Acr = 0

Ac,= 0

(a)

FIG.2. Zinc blende band structure for (a) Aso = A,,

(b)

=

0 , and (b) Aro # 0 , Acr

1

0.

crystals, the valence band states are considered to be derived from the Tlszinc blende valence band state through the action of two perturbations: crystal field splitting and spin-orbit splitting. The quasi-cubic model may be viewed as a reasonable zeroth-order approximation, since the atoms in the zinc blende and wurtzite structures are in nearly similar positions in the unit cell. The model predicts the splitting of the TSstate by the hexagonal crystal field into a rg and r7 state, each state being two-fold degenerate. Figure 3 gives the details of the resulting band structure around k = 0 and also shows the selection rules for optical transitions near the band edge. b. Variation of Eg with Mn Concentration

A review of the literature reveals distinct trends in the behavior of the fundamental gap in DMS. These are, (1) a general increase in gap energy with Mn mole fraction x at fixed temperature T, and (2) approximate agreement of gap energies when results are extrapolated to the same hypothetical x = 1 zinc blende or wurtzite MnB"' compound, dependent only on the particular chalcogenide in the alloy. This behavior is illustrated in Fig. 4 for the two ternaries involving Cd and Zn and the common anion Te, and, as expected, for intermediate quaternary alloys in the CdxZnyMn,Te (x + y + z = 1) system (Brun Del Re et al., 1983). The linear behavior exhibited in Fig. 4 is, of course, only a first order guideline to what may be expected in the various DMS alloys. Based

2.

BAND STRUCTURE AND OPTICAL PROPERTIES

39

WURTZ ITE

FIG. 3 . Band structure and selection rules at k = 0 for wurtzite structure.

on results for many semiconductor ternary alloy systems, a nonlinear dependence of the band gap, namely a downward "bowing" in E, vs x (Thompson and Woolley, 1967), would be anticipated with refined measurements on crystals of good quality. Such bowing is thought to arise from the nonlinear dependence of the band gap on the crystal potential, and on short range fluctuations of the potential in a random substitutional alloy (Van Vechten, 1970). Surprisingly, such nonlinearities, a characteristic of the complete alloy range, are noticeably absent in the wide-gap DMS. Instead, striking downward bowing near x = 0 is seen in several DMS alloys, an unexpected consequence of magnetic fluctuations (Bylsma et al., 1986). Recent results also indicate a further complication in establishing the functional relationship of Eg vs x close to the boundary region between zinc blende and wurtzite phases due to the presence of polytypism. In such cases, Eg may depend on both x and the particular polytype (Brafman and Steinberger, 1966). These refinements, while sometimes clearly observable, are nevertheless small. Experimental determinations of the energy gap rely strongly (1) on the observation of sharp features at the band edge, i.e., excitonic absorption or emission, or (2) on the analysis of the energy dependence of the absorption edge in terms of specific models. For (l), the identity of the transition is obscured at high x due to alloy broadening. In (2), because of extremely strong Mn-related absorption bands, absorption edge data must be obtained on very thin samples and are not often reliable for highly quantitative analysis. For these reasons, the specific transition marking the band gap, e.g., free exciton, bound exciton, etc., although known in the A"BV' host

40

W. M. BECKER

3.0

I

I

I

I

I

I

I

I

I

FIG.4. Variation of the energy gap in Cd,Zn,Mn,Te (x + y + z = 1) with manganese concentration z for fixed x : y ratios. Experimental values: 0 x = 0,. x = 3y, 0 3x = y , M y = 0, A x = y . Line fits ___ are discussed by the authors (Brun Del Re ef al., 1983).

compound, may often not be identified in the DMS. In the wurtzite systems the transition mechanisms may also be obscure, but the dichroism associated with the hexagonal symmetry of these crystals does provide a useful caliper of valence band splittings in the alloys. Although qualitative similarities in behavior are seen in all DMS alloys, it is convenient to present the results alloy-by-alloy, a strategy which will be carried forward through the remainder of this chapter. Table I and Table I1 summarize some representative data for the Cdl-,Mn,BV' and Znl-xMnxBV' alloys (B = Te, Se, or S), respectively. Comments are given below on these and related results in the literature. It should be noted that the column in Tables I and I1 designated as Eg represents the fundamental gap based on strong experimental features, but the specific electronic transition associated with the features may be obscure for x # 0. In the wurtzite region, the (r9, - r7c)and (r7"- r7c) transition energies (see Fig. 3) are designated in the tables by the superscripts I and 11, respectively, where they have been specifically reported in the literature. (a) Cdl-,Mn,Te We begin with Cdl-,Mn,Te, the most extensively studied and probably best understood of the wide-gap DMS. Absorption edge measurements by

TABLE I OF BANDGAPENERGY VERSUS MOLEFRACTION x IN Cdl-xMnxBv' ALLOYS SUMMARY

Eg in eV

Alloy

1.585 + 1.595 + 1.50 + 1.586 + 1.53 + 1.528 + 1.586 + 1.595 +

Cdl-,Mn,l'x

Cdi -,Mn,Se

1.50~

1.592~

+ 1.08~ + 1.23~ Ell 1.725 + 1 . 2 3 ~ E l 1.800 + 1.33x Ell 1.829 + 1 . 3 3 ~ 1.807 +'1.318x 1.821 + 1 . 5 4 ~ EL 1.74 + 1 . 1 6 ~ Ell 1.756 + 1 . 1 3 ~ E L 1.807 + 1 . 3 4 ~ Ell 1.833 + 1 . 3 1 ~ E L 1.818 + 1 . 4 2 ~ 1.70

E L 1.705

Ell

Cdl -,Mn,S

1.51~ 1.587~ 1.44~ 1.393~ 1.26~ 1.316~

1.842

+ 1.40~

Range of x

77 LHe 300 76 300 300 80 10

0 < x < 0.5 0 < x < 0.2 0 < x 5 0.6 0 < x .c 0.7

R. E.M. E.R. P. E.L. P.R.

Gaj ef al. (1978) Twardowski et a/. (1979) Bottka et al. (1981) Vecchi et a/. (1981) Lautenschlager et a/. (1985) Lee and Ramdas (1984) Lee and Ramdas (1984) Lee and Ramdas (1984)

300 297 297 86 86 LN2 2 300 300 80 80 10 10

0

< x c 0.5 0 < x < 0.3 0 < x < 0.3

D.R. & O.T. O.A.

P. R. & O.A. P.R.

Wiedemeier and Sigai (1970) Antoszewski and Kierzek-Pecold (1980) Antoszewski and Kierzek-Pecold (1980) Antoszewski and Kierzek-Pecold (1980) Antoszewski and Kierzek-Pecold (1980) Giriat and Stankiewicz (1980) Wisniewski and Nawrocki (1983) Lee et al. (1986a) Lee ef al. (1986a) Lee et al. (1986a) Lee et al. (1986a) Lee etal. (1986a) Lee et a/. (1986a)

O.A.

Ikeda etal. (1968)

2.45 298 Minimum at x 0.03 dE,/dX 0.9 x 2 0.1

-

Measurement Methods'

T(K)

0 < x < 0.7 0 < x 5 0.7 0 < x I 0.7 0 < x I 0.7

0 < x < 0.3 0 < x < 0.3 0 < x < 0.5 0 I x I0.277 0 < x < 0.5 0 < x < 0.5 0 < x < 0.5 0 < x < 0.5 0 < x < 0.5 0 < x < 0.5

x=o

Reference

-

t Measurement Methods: R. = Reflectivity; E.M. = Excitonic Magnetoabsorption; E.R. = Electroreflectance; P. = Photoluminescence; E.L. = Ellipsometry; P.R. = Piezomodulated Reflectivity; D.R. = Diffuse Reflection; O.T. = Optical Transmission; O.A. = Optical Absorption; M. = Magnetoreflection; M.R. = Modulated Reflectivity.

TABLE I1 SUMMARY OF BANDGAP ENERGYVERSUS MOLEFRACTION x in Znl-xMnxBV' ALLOY.

Eg in eV

Alloy Znl -xMnxTe

2.377 2.381

+ 3.656~ + 0.68~

Minimum for x < 0.01

2.271 2.365 2.376

Zn,-,Mn,Se

+ 0.518~

+ 0.721~

+ 0.820~

2.80

Minimum at x 2.80

- 0.01

M.inimum for 0 < x < 0.2 dE/dx = 0.58

Znl -,Mn,S

3.8 - 3.9

TW) 1.6 2.2 300 80 10

Range o f x 0.01 -c x

< 0.15

x > 0.02

0 5 x I 0.6 0 5 x < 0.6 0 5 x < 0.6

Measurement Methods'

Reference

M. E.M.

Twardowski (1982) Twardowski e? al. (1984)

P.R.

Lee et al. (1986a) Lee et al. (1986a) Lee e f al. (1986a)

2.2

x=o

M.

Twardowski et al. (1983b)

6.5

x=o

R. & P.

Bylsma et al. (1986)

x > 0.3 x=o

M.R.

2.

43

BAND STRUCTURE AND OPTICAL PROPERTIES

Nguyen The Khoi and Gaj (1977) provided one of the first indications that the energy gap increases with x in this material. These authors also showed that the pinning of the edge occurring near 2 eV at high values of x was directly attributable to the onset of Mn2+ intra-ion transitions. This same pinning effect was also seen by Sundersheshu (1980) using photoconductivity were published by Gaj et techniques. The first quantitative results on Eg(x) al. (1978) using reflectivity data. Linear increases in Egus x were recorded by Twardowski et al. (1979) using magnetoabsorption measurements, but in this case, the data was confined to low x values, 0 < x < 0.2. Modulated electroreflectance results by Bottka et af. (1981) provided a survey of Eg(x, T) over nearly the complete single phase region of Cdl-,Mn,Te, and confirmed the linear increase of Eg with x seen by previous investigators. Photoluminescence results by Vecchi et af. (1981) demonstrated that crystal quality was sufficient to give band edge emissions for all x. Finally, Lee and Ramdas (1984), using piezo-modulated reflectivity, were able to separate band edge features from spectra due to the presence of Mn2+ intra-ion transitions (to be discussed in Part 111). Figure 5 gives their results for various Mn concentrations and temperatures. In the figure, the peak labeled A is identified as the free exciton. No indications of band gap bowing are seen in the entire range of solid solution. In particular, there is an absence of the band gap bowing noted in several other DMS at low x (see below). A summary of measured variations of Egwith x is given in Table I. Based on the many experimental results available on Cdl-,Mn,Te, reasonable estimates (to within k0.02 eV) can be given for Egat x = 1 (Le., for zinc blende MnTe), and these are listed in Table 111 for various temperatures. TABLE I11

ESTIMATES OF ENERGY GAP OF HYPOTHETICALX = 1 MnB”‘ ZINC BLENDE (2.b.) OR WURTZITE(w.) SEMICONDUCTORS

E* (x = 1)

Alloy

Cdl-,Mn,Te

Temperature

in eV

2.86 3.05 3.18

(2.b.)

Cd1-,MnrSe ( w . )

I

2.92 3.15 3.24

Znl-,Mn,Te (2.b.) Znl-,MnxSe (2.b.)

II

2.87 3.13 3.30 2.79 3.09 3.19 -3.5 (2.b)

R.T. LNz LHe R.T. LN2 LHe R.T. LN2 LHe

44

W . M. BECKER

G-4

28

C.-.-

t-

1.4 i

0.0

1

Cd ,+MnxTe

300K 80 K

I 0.2

I

I

I

0.4

I

0.6

MANGANESE MOLE FRACTION ( X ) FIG.5 . Variation of the energy gap and Mn transition with Mn concentration in Cdl-,Mn,Te for 10 K, 80 K, and 300 K (Lee and Ramdas, 1984). In the figure, the peak labeled A is identified as the free exciton. The concentration-independent feature Mn2+ is associated with the leading edge of the Mn 6A1(6S)+ 4T,(4G) absorption band.

(b) Cdl-,Mn,Se

The results of powder and single crystal measurements (diffuse reflectance and optical transmission, respectively) by Wiedemeyer and Sigai (1970) gave the first indications of a linear increase in Eg with x in Cdl-,Mn,Se. Optical absorption work by Antoszewski and Kierzek-Pecold (1980) on single crystal samples over a limited range of Mn concentration showed the dichroic effects expected for the wurtzite structure. Stankiewicz (1983) employed modulated

2.

BAND STRUCTURE AND OPTICAL PROPERTIES

45

2

2.6-

C d , - x MnxSe 10 K

2

Y

m

2.4-

0

z a a

u)

w U

c'a

2.2-

w

LL LL

0

>

(3

Q

5w

2.0-

FIG.6. Variation of A and B exciton energies with Mn concentration in Cdl-,Mn,Se at 10 K, 80K,and300K(Leeetal., 1986a). Notshownisanadditionalpointat2.34eVseeninan.r 3 0.5 sample at 10 K, that is due to a MnZ+transition.

electroreflectance to survey Eg(x,T ) for the complete alloy range of Cdl-,Mn,Se. Giriat and Stankiewicz (1980) demonstrated a linear dependence of Eg on x from photoluminescence results in this material. Lee el a/. (1986a) used piezomodulated absorption (see Fig. 6) to separate band edge and Mn-related transitions. Their technique allowed measurement of the (Fv-),?I energy separation, and also gave Eg(x,2") results. Table IV separately lists value of (r7v - Tsv) and As,, including earlier data from measurements confined to the host A"BV' compound alone.

46

W. M. BECKER

Antoszewski and Kierzek-Pecold’s (1980) optical absorption results indicated that the ( r 7 v - r9,) gap is independent of x. In contrast, work by Stankiewicz (1983) and by Morales ei al. (1985) suggested that this gap decreases with x. Finally, Stankiewicz (1983) found that Aso also decreases withx. Values of Eg extrapolated to x = 1 are listed in Table I11 for both polarization directions of the incident light relative to the c-axis. These estimates are probably correct to not more than k0.05 eV, since in the case of Cdl-,Mn,Se, only the range 0 Ix 5 0.5 is accessible for measurement. TABLE IV SUMMARY OF TRANSITION ENERGIES ASSOCIATED WITH THE VALENCEBANDIN VARIOUSDMS ALLOYS WITH WURTZITE STRUCTURE ~

~~~~~~

~

r7v- rgv Alloy

AS0

in eV

in eV

2 6 lo-’ ~ 20 x 10-3

0.436

Cd, -,Mn,Se

(29 t 5 ) x 10-3 (27 k 3) x lo-’ 14 x

x=O O<x<0.3

17 x 10 x 10-3

x=o x =0 x = 0.42 X = 0.20 0.30 0.46

- 2 O x lo-’ 10 x

0.35 0.50

0.440 0.312

Znl-xMnxSe

Range of x

-

~~~~

Measurement Method’ O.A.

O.A.

E.R. E.R.

O.A.

O.A.

Reference Cardona ei al. (1967) Antoszewski and Kierzek-Pecold (1980) Cardona et at. (1967) Stankiewicz (1983) Stankiewicz (1983) Stankiewicz (1983) Morales et at. (1985) Morales et at. (1985) Morales (1985) Morales (1985)

Measurement Methods: O.A. = Optical Absorption; E.R. = Electroreflectance.

(c) Cdl-,Mn,S Cdl-,Mn,S is the least investigated of the Cd-based family of DMS. In this case, pronounced bowing was seen by Ikeda et al. (1968) for low values of x (see Fig. 7), as well as a monotonic increase in Eg vs x for x > 0.1. According to these authors, alloying effects are ruled out as the mechanism for bowing near x = 0. Recent work by Bylsma et al. (1986) on Znl-,Mn,Se at low values of x (see Fig. 8), and by Diouri et al. (1985) on Cdl-,Mn,Te suggest that the bowing in Cdl-xMn,S may be related to s-d and p-d exchange interactions with the Mn magnetic moments. Crystal field and spin-orbit splitting energies are known for CdS, but still remain to be investigated in Cdl-,Mn,S ternary alloys.

2.

41

BAND STRUCTURE A N D OPTICAL PROPERTIES I

I

-

I

I

I

-

2.9

-

-

T = 114’K

-

-_

2 -

-

rn

W

2.3

I

I

I

I

1

K and 298 K

FIG. (Ikeda et al., 1968).

T-6.5K

- PL energy 3,10-

->

4

-Other PL

3.001

0

f”

Y

z W

.* ’ ,

* *

2.90

2.60/

*

I

2.50 0.00 0.10

I

0.20

I

I

I

I

0.30

040

0.50

0.60

O

CONCENTRATION, X FIG. 8. Band gap energy versus Mn mole fraction x in Znl-xMnxSe, taken from photoluminescence and reflectivity maxima at 6.5 K (Bylsma et al., 1986).

48

W . M. BECKER

2.8 Zn I

-

-

0

Mnx Te

300K BOK

+

I

I

I

0.2

I

"

0.4

I

0.6

MANGANSE MOLE FRACTION ( X I

FIG.9. Variation of the energy gap and Mn transition with Mn concentration in Znl-,Mn,Te for 10 K, 80 K, and 300 K (Lee et a/., 1986a). In the figure, the peak labeled A is identified as the free exciton. The concentration-independent feature, labeled Mn2+,is associated with the leading edge of the Mn 6A~(6S) 4T~(4G)absorption band. -t

(d) Znl -,Mn,Te Deviations from linear behavior and possible bowing effects have been seen only in the region x < 0.1 in Znl-,Mn,Te (Twardowski et al., 1984). Linear behavior at higher x is well established in this material from a variety of experimental results (see Table 11), an example of which is given in Fig. 9. (e) Zn, -,Mn,Se Twardowski et al. (1984) found bowing effects in Znl-,Mn,Se near x = 0 from reflectivity data. Bylsma et al., (1986) used photoluminescence and reflectivity measurements in samples grown by the Bridgman technique to establish a minimum in Eg vs x for 0 c x c 0.2, and a linear increase with

2.

49

BAND STRUCTURE AND OPTICAL PROPERTIES

x for x 2 0.3 (see Fig. 8). Apparent scatter in the data for 0.2 Ix I0.3 was traced to the presence of various polytypes found in the vicinity of the zinc blende-wurtzite structural phase transition (x = 0.3). Recently, metastable zinc blende Znl-,Mn,Se epilayers with (100) orientation, free of polytypes, have been grown on GaAs substrates over the complete alloy range by molecular beam epitaxy (Kolodziejski et al., 1986). In these samples (see Fig. lo), photoluminescence results give both a shallow minimum in Eg near x = 0, and a smooth transition to a linear dependence on x at higher Mn concentrations. Extrapolation of Eg to x = 1 gives the same value for both the bulk crystals and the epilayers.

2.701 , 0

,

.

,

,

0.1 0.2 0.3 0.4 0.5 0.8

X

.

,

1

0 . 7 0.8 0.9 1.0

M n PracLion

FIG. 10. Energy of dominant near-band edge features versus Mn mole fraction for MBEgrown zinc blende Znl-,Mn,Se epilayers (Kolodziejski e t a / . , 1986a). (Cross-hatched points are data obtained on bulk crystals by Twardowski el a/., 1983b).

50

W. M. BECKER

(f) Znl-,Mn,S Crystal growth studies show that Znl-xMnxS is zinc blende for 0 5 x 5 0.1 and wurtzite for 0.1 Ix 5 0.43 (Pajaczkowska, 1978). ZnS itself exhibits both the zinc blende and wurtzite modifications, as well as “rotation twinning” and polytypism (Steinberger, 1983), so that sorting out small band structure changes associated with these crystallographic anomalies as a function of x may be quite complicated. As yet, such changes remain unexplored. c. Variation of Eg with Temperature and Pressure

The study of Egin DMS as a function of temperature and pressure has been stimulated by (1) the need to distinguish between band-to-band transitions and Mn2+ intra-ion (d d) or Mn2+-to-band transitions, and (2) the possibility of observing anomalous thermal shifts which may be related to the onset of magnetic phase transitions (Sundersheshu and Kendelewicz, 1982). For (I), the problem arises because of the strength of the Mn-related absorption features near the energy of the fundamental gap. Pronounced differences in the temperature coefficients of the respective transitions then help to identify them separately. With regard to (2), in the usual semiconductor, the temperature variations of the band gaps are well described by an equation given by Varshni (1967), +

aT2 Eg(T) = Eg(0) - _ _ T + 6’ where Eg(T)and Eg(0)are the band gap energies at T and 0 K, respectively, and a and b are constants of the material. This equation predicts (1) that dEg/dT 0 as T 0 K, and (2) that JdE,/dTI approaches a constant value at high temperatures. Anomalous behavior would then consist of departures from this functional dependence on T near the magnetic ordering temperature. Listed in Table V is a summary of results obtained on several DMS alloys. In each case, Egdecreases approximately linearly with Tabove liquid nitrogen temperature, and results are presented in Table V in terms of an average value of the coefficient dE,/dT. In a number of recent observations, however, departures from the Varshni-like behavior (namely, an extra shift to higher energies) below the characteristic spin freezing temperatures have been reported in DMS. The same effect had been seen earlier in magnetic semiconductors such as MnS (Chou and Fan, 1974) and MnTe (Kendelewicz, 1980a). Temperature anomalies of this type are also cited in Table V, where seen in various DMS alloys. +

+

2.

51

BAND STRUCTURE AND OPTICAL PROPERTIES

TABLE V AVERAGED TEMPERATURE COEFFICIENTS dEg/dTv s x ~ TEMPERATURE o~ RANGE77 - dEg/dT

Alloy

in

Cdl-rMn,Te

3.7 3.7

Cd,-,Mn,Se

eV/K

-

8.9 6.6

Range of x 0 0

-

Measurement Methods’ M.E. O.A.

0.6 0.4

4.6 - 7.5 6.8 9.1 3.5 8.5 3.3 - 10

0 - 0.4 0.4* 0.73* 0.1 0.7* 0 - 0.7

4.5

0

-

x

< 0.1

O.A. O.A. O.A. P.R. P.R. O.A. & R.

Reference Bottka et al. (1981) Sundersheshu and Kendelewicz (1982) Abreu et al. (1981) Diouri et a/. (1985) Diouri et al. (1985) Lee and Ramdas (1984) Lee and Ramdas (1984) Bucker et al. (1985)

3.9 3.9 - 7.3 7.3 5.3 - 8.6 6.8 - 7.9

0.28 x > 0.28 0 - 0.35 0.3 - 0.45’

M.E. M.E. M.E. O.A. O.A.

Antoszewski and Kierzek-Pecold (1980) Stankiewicz (1983) Stankiewicz (1983) Stankiewicz (1983) Abreu et al. (1983) Morales et al. (1985)

Cd,-,Mn,S

4.4 - 9.4

0 - 0.4

O.A.

Ikeda e t a / . (1968)

Znl-xMn,Te

4.14 - 4.61 4.3 - 9.8 6 - 9.4

0 - 0.071 0 - 0.6 0.1 - 0.6*

O.A. P.R. O.A.

Twardowski et al. (1984) Lee et al. (1986a) Morales et a/. (1984)

Znl-,MnxSe

4.7 8.5 9.7

0.066 0.36* 0.55*

R. &. P. R. & P.

-

5.5

0.1

R.

< T < 300 K

0.3

-

R. &. P

Bylsma et al. (1986) Bylsma et al. (1986) Bylsma et al. (1986)

~

*Anomalous extra shift to higher energies (“blue shift”) observed below 77 K. +Measurement Methods: M.E. = Modulated Electroreflectance; O.A. = Optical Absorption; P.R. = Piezomodulated Reflectivity; R. = Reflectivity; P. = Photoluminescence.

Pressure-induced changes in the optical properties of DMS materials have also been pursued, (1) in order to establish various features of the band structure, and (2) again, in order to separate band structure effects from those due to localized Mn-related transitions. Results related to the variation of Eg with pressure are listed in Table VI. For measurements at low pressure, linear pressure coefficients suffice to describe the results. Fitting of the data for much higher pressures, that may involve phase transitions (rendering the samples opaque), would require inclusion of higher order terms in P. In such

52

W. M. BECKER

cases, an expression usually used is

E,(P) = Eg(0)+ yP

+ 6Pz,

(2)

where Eg(0)is the band gap at P = 0. Values of the coefficients y and 6 are given in Table VI for a number of DMS. We now examine the temperature and pressure dependence of Eg for specific DMS alloys. (g) Cdl -,Mn,Te The temperature dependence of the band gap of Cdl-,Mn,Te was first investigated systematically by Bottka et al. (1981). These researchers found that for T > 100 K, (dEg/dTIis constant for x < 0.15,but increases linearly with x thereafter. The authors suggested that the sharp change in slope at x = 0.15 is related to lattice mode softening at the percolation threshold of the Mn spin system. Sundersheshu and Kendelewicz (1982) carefully examined Eg vs T for anomalous optical behavior expected to accompany a magnetically induced shift in the band gap. While such effects were not detected, they found instead a sharp drop in the values of b [see Eq. (I)] above x = 0.3, and took it as indicative of a magnetically induced widening of the gap. Later, Bucker et al. (1985)used reflectivity to survey dEg/dTfor all x. In contrast to the electroreflectance results of Bottka et al. (1981),they found only a uniform increase in the magnitude of the temperature coefficient with x. Lee and Ramdas (1984)noted that (dEg/dT(for 10 4 T I80 K is greater than IdEg/dT)for 80 IT 5 300 K for an x = 0.7 sample, a first indication of the sought-for anomalous effect presumably associated with the magnetic phase transition in this material. Stronger evidence for this behavior was obtained by Diouri et al. (1985)from optical absorption measurements (see Fig. 11). Working with very thin layers ground and polished from the bulk, these investigators found shifts of 45 meV and 17 meV to higher energies in x = 0.7 and x = 0.4 samples, respectively, as compared to Eg(0) values predicted from higher temperature results using Eq. (1). Turning now to pressure effects, Ambrazevicius et al. (1984)were able to observe the pressure variation of Eg by the use of samples thin enough to allow measurements at the level of absorption coefficients greater than lo3cm-'. With these precautions, positive values of the pressure coefficient dEg/dPwere recorded for all x . At lower levels of the absorption coefficient (smaller than lo2cm-I), the sign of the coefficient becomes negative between x = 0.4 a n d x = 0.5.This latter result is interpreted in terms of intra-ion d-d transitions of Mn2+. Similar sign changes were observed by Wei Shan et al. (1985)when comparing behaviors of x = 0.1 and x = 0.5 samples at low levels of absorption.

TABLE VI OF EBAT ROOMTEMPERATURE FROM ABSORPTION EDGEMEASUREMENTS PRESSURE DEPENDENCE

Alloy Cdt-,Mn,Te

dE,/dP eV/MPa 8 1.9 1.3 5.9 4.6

Ea(0)' in eV

y' in

eV/MPa

1.483 1.618 2.161 2.115

8.3 7.1 - 4.9* -5.1'

Znl-,Mn,Te

2.21 2.265 2.28 2.37

10.4 10 10.6 8.2

Znl-,Mn,Se

2.688 2.648 2.615 2.756

7.12 6.1 6.30 6.34

dt in eV/GPaZ

- 4.0

- 3.9

- 28 - 26 - 54 - 80 - 15 - 11 - 18 - 27

Fitting parameter in Eq. (2). *Negative values of y are indicative of absorption effects involving Mn2' 3d5 levels.

X

Reference

0

0.4 0.52 0.6 0.7 0 0.1 0.5 0.615

Babonas ef al. (1971) Ambrazevicius et al. (1984) Arnbrazevicius et al. (1984) Ambrazevicius et al. (1984) Arnbrazevicius et al. (1984) Wei Shan et a/.(1985) Wei Shan et a/. (1985) Wei Shan el al. (1985) Wei Shan et al. (1985)

0

Ves et al. (1986a) Ves el a/.(1986a) Ves e f al. (1986a) Ves et al. (1986a)

0

Ves et al. Ves et al. Ves et al. Ves ef at.

0.01 0.1 0.3 0.1 0.15 0.25

(1985) (1986b) (1986b) (1986b)

54

W.

1

I

I

M. BECKER

1

I

I

I

FIG. 11. Absorption edge as a function of temperature for (a) Cdo.Z,Mno.73Te, and (b) Cdo.sMno.4Te (Diouri et af., 1985). The dotted curves show the extrapolation of Eg obtained from high temperatures according to Eq. (1).

2.

BAND STRUCTURE AND OPTICAL PROPERTIES

55

(h) Cdl -,MnxSe As in Cdl-,Mn,Te,

the temperature coefficient- of the energy gap,

- dEg/dT, of Cdl-,MnxSe increases with increasing x . In Stankiewicz’s

(1983) survey of this phenomenon over the concentration range 0 5 x S 0.4, the coefficient changes rapidly between x = 0.1 and x = 0.3, but is otherwise constant at the ends of the interval. An interesting bowing of Eg(T)occurs in an x = 0.42 sample at low temperatures and is possibly related to an interference effect between band-to-band transitions and excitations of Mn” states. Changes in - dEg/dTwith x were seen by Abreu et al. (1983) using isoabsorption measurements at low values of absorption coefficient. An abrupt drop in the magnitude of the coefficient athigh x reported by these authors is traceable to the onset of Mn2+ d-d transitions, now known (Morales et al., 1985) to have a weaker temperature dependence than bandto-band transitions. Morales et ul. (1985) noted that IdEg/dTI for for 77 < T < 300 K for an x = 0.5 10 < T < 77 K is greater than 1-1 sample, an indication of an anomaly similar to that observed in Cdl -,Mn,Te, as discussed earlier. (i) Cdl-,MnxS Optical absorption measurements carried out on Cdl-,Mn,S by Ikeda et al. (1968) showed that absorption coefficients as large as lo3cm-’ must be achieved in order to measure intrinsic behavior. Results were obtained only at 114 K and 298 K; linear behavior is presumed between these temperatures in listing the results given in Table V for this material. (j) Znl -,MnxTe

Twardowski et al. (1984) found little variation in -dE,/dT for x < 0.1. The piezomodulated reflectivity results of Lee et al. (1986a), obtained from data taken at 80 K and 300 K, show that IdEg/dTI increases by a factor of two between x = 0 and x 5 0.5. Similar to the observations on Cd,-,Mn,Te and Cdl-,MnxSe, Morales Tor0 et ul. (1984) showed that in Zn1-,MnxTe 1-1 for 10 < T < 77 K is greater than 1-1 for 77 < T < 300K at high x, with reverse behavior seen at low x . Ves et al. (1986a) carried out optical absorption measurements on Znl-,Mn,Te under hydrostatic pressure in a diamond anvil cell. As expected, Eg increases with increasing pressure, but strong nonlinearities were observed. Results were limited to low-x material (x 5 0.3) because of the deterioration of the absorption feature associated with Eg at high x and P i n Znl-,Mn,Te, and also because in this material the pressure at which the phase transition occurs decreases with increasing x.

56

W. M. BECKER

(k) Znl-,Mn,Se Bylsma et al. (1986) found a monotonic increase in - dEg/dTwith x for Znl-,Mn,Se samples in the range 0 < x < 0.55. A deviation from Varshnilike behavior, namely, an extra shift of E,(T) to higher energies at low temperatures and large Mn concentrations, was also seen to occur in this material. Ves et al. (1986b) carried out optical absorption measurements at high pressure on samples with x values lying in the zinc blende region (0 < x < 0.3). As in Znl-,Mn,Te, Eg was found to increase with pressure. Pronounced nonlinearities were observed, but values of 6, the quadratic pressure coefficient [see Eq. (2)], were smaller than in Znl-,Mn,Te. (1) Znl-,Mn,S At the time of this writing, no published information on the temperature and pressure dependence of Eg in Znl-,Mn,S is available.

3. BANDSTRUCTURE AWAYFROM ZONECENTER Reflectivity measurements for energies above the fundamental edge have proved indispensable in probing the electronic structures of semiconductors deep into the bands or away from k = 0. In DMS materials, the technique has been applied to several alloys over the available solid solution regions. In this work, there was the expectation of being able to detect sharp spectral features at high x. This was based on earlier findings that these features persist over large alloy ranges in other A”BV’ mixed crystals, such as Hgl -,CdxTe and Znl -,Cd,Te. First indications of reflectivity features in DMS associated with transitions away from k = 0 were seen by Kendelewicz and Kierzek-Pecold (1978) in their work on the narrow-gap alloy Hgl-,Mn,Te. In contrast to the behavior in non-DMS A”BV’ ternary alloys, however, a blurring of the reflectivity spectral structures occurred with growing Mn concentration. This effect was also noted in studies on the wide-gap DMS Cdl-,Mn,Te (Kendelewicz, 1980b; Kendelewicz, 1981; Zimnal-Starnawska et al., 1984; Bucker et al., 1984) and on Znl-,Mn,S and Znl-,Mn,Se (Zimnal-Starnawska et al., 1984). Except for the persistence of the fundamental edge reflectivity feature, other details, such as the El and El + A1 transitions, disappear for x z 0.3. The rate of deterioration of the reflectivity spectra with increasing Mn content is highest for Znl-,Mn,S and lowest for Cdl-,Mn,Te, according to ZimnalStarnawaka et al. (1984). These authors explain the blurring of the spectra in terms of localized Mn2+3ds “spin down” states lying at higher energies (spin splitting of = 6 eV; see also Podgorny and Oleszkiewicz, 1983).

2.

BAND STRUCTURE AND OPTICAL PROPERTIES

57

111. Optical Properties of d-Electrons

4. GENERAL CONSIDERATIONS

The introduction of Mn not only changes the band structure of the host A"BV' semiconductor, but it also introduces new optical transitions. These include prominent absorption and emission features, which can be correlated with the expected splittings of Mn multiplets in a crystalline field. Below is given a brief summary of the origin of the multiplets and their splittings, leading to the hierarchy of states usually observed in wide-gap A:t,Mn,BV' alloys. The reader is referred to standard references (e.g., Tanabe and Sugano, 1970; Abragam and Bleaney, 1970; Griffith, 1971; Ballhausen, 1961; McClure, 1959) for more complete details. Significant understanding of this aspect of DMS can be gained by considering an isolated Mn ion in an A"BV' lattice. A freeMn atom has the 3d54s2 ground state configuration for the outermost shells. When the atom is put into a DMS, it substitutes for the cation. The two 4s electrons form the bonds with the surrounding ions, and the five 3d electrons interact strongly with the electrical potential of the nearest neighbor anions. The total Hamiltonian is H = Ho + V + H,,, where Ho is the many-electron Hamiltonian of a free ion without spin-orbit interaction, Vis the contribution of the crystal field, and H,, is the spin-orbit interaction. Since the magnitude of V satisfies the inequalities H,, < V < H,, V can be treated as a perturbation. We now review very briefly the spectroscopic notation used in describing the ground and excited states of both the free ion, and the ion when in a ligand field of an appropriate symmetry. For the free ion, the state is given by ""L, where S is the total spin, and L is the total orbital angular momentum. In the ground state of a free Mn2+ ion, the electron spins are aligned due to Hund's rule, giving S = f; this state is designated as 6S ( L = 0). When the Mn2+ion is put into a crystal field that is not strong enough to break Hund's rule, the 'S state remains as the ground state. Since it is spherically symmetric, it transforms as A1 (in group theoretical notation), and is therefore unsplit by the crystal field. In the free ion state, spin reversal of a single 3d electron gives rise to four quartet states corresponding to S = 5; 4G ( L = 4), 4P ( L = l), 4D ( L = 2), and 4F ( L = 3). The 4Gstate has nine-fold orbital degeneracy (2L + 1 = 9). This degeneracy, when lifted by a crystal field of cubic symmetry, leads to four states labeled (in group theoretical notation) 4T1,4T2, 4A1, and 4E. The degeneracies of the states are as follows: 4Tl-three fold; 4Tz-three fold; 4A~-one fold; 4 E - t ~ o fold. Using the same method, it is found that the 4D state splits into 4E and 4T2, and the 4P state does not split, but transforms as 4T1. State 4Fsplits into 4T1, 4T2,and 4A2; 4A2 is non-degenerate, and the other states have the degeneracies indicated earlier.

58

W. M. BECKER

Using these eigenstates as basis, the total Hamiltonian matrix for the Mn2+ ion in an A"BV1host (without spin-orbit interaction) can be written and , 4A1, and 4Estates. diagonalized, yielding all the energy levels of the 4 T ~4T2, The energy associated with each state is usually given as a function of the crystal field parameter, Dq, and the Racah parameters B and C (Tanabe and Sugano, 1954). The parameter Dq is related to the orbital splitting of a single d electron in a cubic crystal potential, and B and Care combinations of radial integrals over the interelectronic distance. Tanabe and Sugano (1954) carried out this calculation for a cubic field, and all the energy matrices were collected by McClure (1959). The solution of the secular equation of these matrices gives the appropriate energies of the states. Shown in Fig. 12 is the energy level diagram appropriate for splitting of states of the d5 configuration in a cubic field (Tanabe and Sugano, 1970). For the usual values of Dq/B, the 4Gstates are the lowest excited states, and are therefore the first to become optically accessible in DMS alloys as the band gap widens. Transitions between the states are governed by strict selection rules. For example, in the free ion case, transitions between the 6Sground state and the 4Gexcited states are forbidden both by the AS = 0 selection rule and by the

E/B

,

2::

'A2 'TI

4 T ~ 6A,(dc3 d'yz) 4T1( d r 4 d y )

10 -

-

'Tp(dr5)

6A I

6S

6s

0

obs (Mn IU)

1

2

3

4 Dq/B

FIG. 12. Splitting of states of the dS configuration by a crystal field of oh symmetry (Tanabe and Sugano, 1970). Similar splitting is expected for a crystal field of Td symmetry.

2.

BAND STRUCTURE AND OPTICAL PROPERTIES

59

parity selection rule. The A S = 0 rule is, however, broken by spin-orbit interaction, and the parity rule fails in a crystal field which lacks inversion symmetry, such as Td ,leading to observation of relevant features in both zinc blende and wurtzite DMS alloys. Electric dipole transitions will therefore occur through this and other perturbations of the environment of the ion. Details of Mn2+optical transitions in A"BV' semiconductors have been the subject of intense investigation for many years, but results were predominantly confined to samples with Mn in extreme dilution, In this regime, details of vibrational interactions, Jahn-Teller effects, and other phenomena requiring high resolution could be studied because of the rich structure of the spectra. Such structure disappears rapidly with alloying for x z 0.01, and only broad excitation and emission features remain as the Mn concentration is further increased. Another limitation for study of these transitions in some DMS is that the spectroscopic "window" for observation of Mn-related transitions-the fundamental gap-may open only for high x values or for high pressure. As a result of both these limitations, identification of observed absorption and excitation features associated with Mn has been somewhat controversial in DMS. 5 . Mn TRANSITIONS IN SPECIFIC DMS ALLOYS

In Table VII are listed Mn-related absorption/excitation bands seen in various DMS, the intra-ion transitions from which they derive and, in several selected cases, also crystal field parameters employed in the analysis of the data. In all A:'_,Mn,B"' DMS where Eg exceeds 2.2eV, a broad (FWHM = 0.1 eV) photoluminescence band is seen in the yellow region of the visible part of the spectrum. This transition is generally accepted as the 4T1(4G) 6A~(6S)transition, and is the emission associated with the excitation bands listed in Table VII. Below, we briefly comment on the results listed in the table. Several high pressure measurements that confirm the identity of various d-d transitions seen at zero pressure, and in some cases reveal higher energy transitions not previously accessible by optical techniques, are also discussed. +

(m) Cdl-,Mn,Te The first optical effects seen in Cdl-,Mn,Te related to Mn transitions were those of Nguyen The Khoi and Gaj (1977), who associated an absorption edge at =2.0eV with the onset of the 6A1(6S)+ 4T1(4G)transition. From photoluminescence results on thick samples (=1 mm), Tao et af. (1982) and Moriwaki et af. (1982) suggested that both the 6A~(6S) 4T~(4G)and 6A~(6S) 4T2(4G)excitation features were present at high x . However, it was not until samples thinned to the micron range were employed that the -+

+

TABLE VII MnZ+ABSORPTION/EXCITATION BANDMAXIMA(IN eV)

IN

DMS ALLOYS

_______~

Alloy Cdl-xMn,Te

6Ai

+

4T1

- 2.2

- 2.2

Cdl-,Mn,S Znl -,MnxTe

Znl-,Mn,Se

Znl-,Mn,S

-2.5

- 2.43* - 2.4*

6Ai

+

4Al, 4E

Other

- 2.63*

-2.63*

- 0.48 - 0.4

-2.3 (Threshold) 2.43

2.58

- 2.3

-2.3 -2.33

2.34 2.38 2.4 2.42

- 2.36 -2.38 2.40

-

B(cm-l) 720 713 713

~(cm-') 750 749 75 1

2.22(z.p.) 2.35

2.48 2.57 2.5 2.54

2.44(z.p) 2.53

X

0.4 5 x I0.7 0.4 Ix 5 0.7 0.7 0.7 0.7 0.4 5 x 5 0.7

- 2.6*

- 2.43*

-2.2

Cdl-,Mn,Se

6A1 + 4T2

2.72 -2.576 - 2.605 -2.745 C(cm-') Dq(cm-l) 2664 743(Td), 330(0h) 1638 750(Td), 333(0h) 2637 913(G), 406(0h) 2.655 2.68 2.70 2.9 2.70 2.93 C(cm-') Dq(cm-') 2730 836(%), 372(0h) 2772 830(G), 369(0h) 2782 795(G), 353(0h) 2.635 (z.P.) 2.69

T a o et al. (1982) Moriwaki et al. (1982) Rebmann et a/. (1983) Lascaray et at. (1983) Diouri et al. (1985) Lee and Ramdas (1984) Morales et al. (1985) Ikeda et al. (1968)

-0.1 -0.5 0.6

-

Morales Toro et al. (1984) Morales Toro et al. (1984) Morales Toro et al. (1984)

-0.1 0.5 -0.6

-

Morales Toro et al. (1984) Morales Tor0 et al. (1984) Morales Toro et al. (1984)

0.001 0.23 0.35 0.5

Langer and Richter (1965) Morales (1985) Morales (1985) Morales (1985)

0.23 0.35 0.50

Morales (1985) Morales (1985) Morales (1985)

2.815 (z.P.), 3.13 (z.P.) <1.0 mole% x = 0.216 2.96, 3.13

* Assignments ambiguous because of broadening and overlapping of bands. (z.P.) = zero phonon line.

Reference

Langer and Ibuki (1965)

2.

61

BAND STRUCTURE AND OPTICAL PROPERTIES

absorption band features (see Fig. 13) due to Mn were clearly revealed (Rebmann ef af., 1983, and Lascaray el a[., 1983). Some uncertainty in separately identifying the bands is encountered at high x because of broadening and overlap effects, as indicated in Table VII. As mentioned earlier, the Mn intra-ion transitions in DMS alloys can be conveniently discussed in terms of the Tanabe-Sugano diagram for the d5 configuration. Based on this diagram, the energy of the 6A1(6S) 4T1(4G) transition for Mn2+decreases with the increase of the crystal field parameter Dq. In a cubic field, Dq is proportional to R-' in the approximation of a point charge ligand, where R is the distance between the transition metal ion and the ligand ion. Therefore, the application of pressure should give a decrease in the energy of the 6A1(6S) -+ 4T1(4G) transition, and in the associated 4T1(2S)-+6A1(4G)emission (that is shifted to longer wavelengths due to lattice relaxation). Investigation of the pressure shifts of (1) the 2.2 eV Mn absorption edge (Muller et af., 1983, Ambrazevicius et al., 1984) and (2) the 2.0 eV photoluminescence band (Miiller er al., 1983)gave negative pressure coefficients, consistent with the above picture of the d-d transitions. Also, the magnitudes of the coefficients, 3-8 x eV/GPa, were explained using known compressibility results for the host A1'Bv' semiconductors and reasonable estimates of the pressure dependence of Dq and the Racah parameters B and C (Miiller et af., 1983). Piezomodulated reflectivity studies (Lee and Ramdas, 1984) also yield phase shift data (see Fig. 14) near the Mn absorption edge, that are consistent with the hydrostatic pressure results. -+

(n) Cdl -,Mn,Se

-

Moriwaki ef af. (1983) observed a photoluminescence band at about 2.13 eV in an x = 0.45 sample of Cdl-*MnxSe, with an accompanying excitation threshold at ~ 2 . eV. 3 Both results were ascribed to the 6A1(6S) 4T1(4G) transition. Morales er al. (1985) carried out optical absorption measurements near the upper limit of the solubility region of this alloy (x = 0.5). By using thin samples, it was possible to see-as shown in Fig. 15-both the Mn2+ threshold (up to =200cm-') and the region of intrinsic absorption (the latter identified by the onset of dichroism associated with the wurtzite structure). Thus, optical features directly traceable to Mn2+intra-ion transitions have been demonstrated in the two wide-gap A1'Bv' host crystals with Eg < 2.0 eV by alloying with Mn within the range of solid solution. (0)Cdl -,Mn,S

Successful growth of single crystal Cdl-,Mn,S up to x = 0.4 was reported by Ikeda et a/. (1968). For x = 0.4, and using T = 114 K, these researchers

62

W. M. BECKER

CMT 10

-I 6000 5000 -

4000

-

3000 -

2000

-

l8ooo

19000 20000 21000 22000 E (ern-')

FIG.13. Absorption spectra for (a) Cdo.,Mno.,Te(CMT10) and (b) Cdo,4Mno,6Te(CMT20) at 5 K (Lascaray el al., 1983). Only A, a band due to overlapping 6A~(6S) -+ 4T1(4G) and 6A,(6S)-* 4T2(4G)transitions is seen in the x = 0.6 sample. In the x = 0.7 sample, band B, due to 6A1(6S) 4Al,4E(4G)transitions is seen because of the widening of the forbidden band with increasing x. -+

2.

BAND STRUCTURE AND OPTICAL PROPERTIES

63

PHOTON ENERGY ( eV 1 FIG.14. Piezomodulation spectrum of Cd0.35Mn0.35Teat T = 80 K (Lee and Ramdas, 1984). The opposite signs of the derivative signals of the A and Mn2+ features are consistent with the direction of energy shifts observed using hydrostatic pressure.

were able to detect three Mn2+absorption bands below the intrinsic edge (see Table VII), that, for this composition, corresponds to Eg = 2.85 eV. (P) Znl-,Mn,Te Zn-based DMS alloys are special in that all of them have gaps Eg > 2.0 eV, allowing the study of Mn transitions in even the low-xlimit. In Znl-,Mn,Te, results were first obtained by Skowronski et al. (1978), who observed absorption bands associated with Mn on a sample with x = 0.1. Morales Tor0 et al. (1984) extended the alloy range of investigation up to x = 0.6 (see Fig. 16) and found a total of four Mn absorption bands in samples with the highest Mn concentrations at liquid helium temperatures. Of these, the three lowest bands are ascribed to the 4G transitions, while the origin of the highest-lying band remains in doubt. The values of Dq (listed for both Td and Oh symmetry) and the Racah parameters given in Table VII show changes with x that are consistent with the known composition-related parameter changes. Ves et al. (1986a) carried out room temperature hydrostatic pressure measurements on samples with x I0.6. Four absorption features, identified as Mn intra-ion transitions, were shown to have negative pressure

64

W. M. BECKER

I

Cd, X

g a

I

I

-,Mn, Se

-

I

RT

I

I

LN,

rl

LHe

0.45

10'

t Ioo

1.90 2.00

2.10

2.20

2.30

2.40

2.50

2.60

PHOTON ENERGY ( eV 1 FIG.15. Optical absorption coefficient versus photon energy in Cdl-,Mn,Se for an x = 0.45 sample at RT, LNz, and LHe temperatures (Morales et a / . , 1985). The different polarizations of the incident light are shown in the figure. Band gap dichroism is seen over the complete range of a at room temperature, but only appears for a's greater than 2 5 x 102cm-' at low temperatures.

coefficients, as in Cdl-,Mn,Te. The authors accounted for the magnitude and sign of the coefficient associated with the lowest transition, 6A1-+ 4T1, by using crystal field theory. This involved estimates of changes in the Racah parameters based on the pressure shift of the two highest, nearly degenerate absorption features, identified as 6Al 4A1,4E. Piezomodulated reflectivity studies for 0 Ix I0.5 (Lee et al., 1986a) give phase shift data for the -2.2 eV Mn absorption edge, that are consistent in sign with the hydrostatic pressure results for Znl-,Mn,Te. +

2.

BAND STRUCTURE AND OPTICAL PROPERTIES

I

2.2

I

2.3

I

I

I

2.4

2.5

2.6

65

I

2.7

2.8 2.9

hv(eV) FIG. 16. Optical absorption coefficient versus photon energy in Znl-,MnxTe for an x = 0.6 sample at RT, LN2, and LHe temperatures (Morales Toro et a/., 1984). (See Table VII for assignment of transitions.)

(4)Znl -,MnXSe Four Mn absorption bands and a shoulder have been seen (Morales Toro, 1985) at liquid helium temperature in Znl-,Mn,Se for x = 0.5 (Figs. 17 and 18). The absorption band at ~ 2 . 9 eV 3 and the shoulder at ~ 3 . 1 eV 0 involve multiplets of, as yet, unknown origin. Dichroism associated with band-toband transitions is clearly indicated at all temperatures for absorption coefficient values greater than lo3cm-', as seen in Fig. 18. The dichroic effects in the Mn absorption region and, in several cases, distinct energy shifts with polarization of the Mn bands are unexpected for Td symmetry. An explanation for this result appears to be outside the limits of ligand theory. Ves et al. (1986b) carried out room temperature absorption measurements as a function of hydrostatic pressure on Znl-,MnxSe for 0 Ix 5 0.25.

66

W. M. BECKER

I Znl-xMnxSe

I

3 P 0°3

X - 0.5

LHe

*PI; 0

z

11;

0

8

0

ZI 8

lo-''

I.&

I.QO

2.k

2.10

2.10

2.k

PHOTON ENERGY ( e V

2.AO

1

3.00 32!0

FIG. 17. Optical absorption coefficient versus photon energy in Znl-,MnxSe for an x = 0.5 sample at LHe temperature (Morales Toro, 1985). (See Table VII for assignment of transitions.) The different polarizations of the incident light are shown in the figure.

Transitions to the lowest excited states 6A~(6S) 4T1(4G) and 6A~(6S) 4T~(4G) were not seen due to their weak absorption relative to background. Figures 19 and 20 show results for x = 0.15 and x = 0.25 samples, respectively. In these, the band identified as the 6 A ~ 4A1,4E(4G)[Epeak(P = 0) = 2.7eVI first appears at moderate pressure as the band gap opens. As in Znl-xMn,Te (Ves et al., 1986a), this band is found to have a negative pressure coefficient. Two additional absorption bands are revealed at 2.7 eV and at 2.95 eV. The lower of these shows a small negative pressure coefficient, and is attributed to the transition 6A~(6S) [4T1(4D) + 4Tz(4P)]. A tentative identification of a 6A1(6S) 4E(4D)transition is made for the 2.95 eV band. -+

-+

+

-+

-+

2.

67

BAND STRUCTURE AND OPTICAL PROPERTIES

lo4 -

-

I

RT

I

I

1

LN2

LHe

-

-

-

-

'02

2.k 2.k

3hO

3.k

3.20

PHOTON ENERGY ( eV1

FIG.18. Optical absorption coefficient versus photon energy in Znl-,Mn,Se for an x = 0.5 sample at RT, LN2, and LHe temperatures (Morales Toro, 1985). The different polarizations of the incident light are shown in the figure. Band gap dichroism is seen for absorption coefficients greater than lo3cm-', but dichroism is also present in the region of Mn absorption.

(r) Znl-,Mn,S Early work on ZnS doped with Mn (McClure, 1963) showed that five absorption bands with their associated zero-phonon lines could be seen. The later work of Neumann (1971) on Znl-,MnxS (reported by Gumlich, 1981) for 0 c x I 0.2 revealed the characteristic broadening-and ultimate disappearance-of spectroscopic structure, as discussed in the beginning of this section. As yet, the highest two bands observed in this alloy remain unidentified.

68

W . M. BECKER

L--

22-

26 2.8 ENERGY (eV)

24

3.0

32

FIG.19. Absorption coefficient versus photon energy at various hydrostatic pressures for an

x = 0.15 sample of Znl-,Mn,Se (Ves ef al., 1986b). The curves for p 2 4.02GPa have an

arbitrary shift of the ordinate scale to allow for a clear display of the results.

IV. Concluding Remarks 6. SUMMARY AND DISCUSSION In zeroth approximation, the energy gap in the various DMS discussed above increases linearly with increasing Mn content, following a relationship of the form E(x) = E(0) + [E(x = 1) - E(O)]x.The extrapolated results at the hypotherical x = 1 zinc blende or wurtzite limit are seen to depend only on the particular chalcogen in the MnBV'compound. Normal bowing effects in the energy gap usually seen in ternary alloy systems are unexpectedly absent in the wide-gap DMS. The bowing effects seen at low x in some DMS and explained in terms of s-d and p-d interactions (ByIsma et a/., 1986) seem to be unique to these systems. The dependence of the temperature coefficient of the band gap has been extensively studied in several DMS, but there appears to be considerable divergence of results among different investigators. A stimulus for further investigation in this area would be to have good theoretical guidelines, which are, so far, unavailable.

2.

A-l 2.5 27 2.9 2.3

O-O2.l

69

BAND STRUCTURE AND OPTICAL PROPERTIES

3.0

ENERGY (eV1

FIG.20. Absorption coefficient versus photon energy at various hydrostatic pressures for an > 0.001 GPa have an arbitrary shift of the ordinate scale.

x = 0.25 sample of Znl-xMnJSe (Ves ef a / . , 1986b). All curves for p

Departures from Varshni-like behavior of the energy gaps below the characteristic spin-freezing temperature seem to be present in several DMS. Unfortunately, some-of the clearest results related to this problem (Diouri et al., 1985) depend on isoabsorption measurements on very thin samples, rather than on observation of the thermal shift of sharp spectral features. Further experimental work on this effect by less ambiguous techniques is needed to clarify the status of this result. Both in the work on the energy gap, and on states away from k = 0, the blurring of experimental features remains a strongly limiting factor in the investigation of the band structure of these materials. The blurring effect needs to be better understood, and new results should be sought on samples prepared by crystal growth methods other than the Bridgman technique. The presence of Mn-related absorption/excitation bands and an associated emission feature at =2eV appears to be well established in all wide-gap DMS. Further, the identification of the three lowest lying absorption/ excitation bands and the 2eV emission in terms of ligand theory seems satisfactory, in spite of the disappearance of spectral detail with increasing

70

W. M. BECKER

Mn content. Much less certain is the origin of the transitions seen at higher energy. This problem, that arose early in the study of Mn at very low values of x (McClure, 1963), remains an active area of theoretical research (Richardson and Janssen, 1986). More experimental work is certainly needed, but present techniques, such as optical absorption at high pressure, seem unlikely to impinge on this problem. Strong evidence exists for the hybridizing of the ground state of the Mn ion with valence band states at energies well below the band edge. Such hybridization does not appear to affect the existence and observation of the lowest energy localized d-d transitions of the ion. Further experimental and theoretical work is needed to understand the coexistence of these two effects. Finally, charge transfer states, such as Mn'+3d6 levels, have been predicted recently for DMS materials (Ehrenreich et al., 1986), but experimental evidence for such states is slight. Recent results by Lee et al. (1986b) using reflectivity at high magnetic fields rule out this mechanism as being responsible for the 2 eV absorption/excitation feature in CdI-,Mn,Te. The possibility exists that short-lived charge transfer states are located in the gap; these will probably require investigation by time-resolved spectroscopy to verify their presence. References Abragam, A., and Bleaney, B. (1970). Electron Paramagnetic Resonance of Transition Ions (Clarendon Press, Oxford). Abreu, R. A., Stankiewicz, J., and Giriat, W. (1983). Phys. Stat. Sol. (a) 75, K153. Ambrazevicius, G., Babonas, G., Marcinkevicius, S., Prochukhan, V. D., and Rud, Yu V. (1984). Solid State Commun. 49, 65 1. Antoszewski, J., and Kierzek-Pecold, J. (1980). Solid Sate Commun. 34, 733. Babonas, G. A., Bendoryus, R. A., and Shileika, A. Yu.(1971). Sov. Phys.-Semicond. 5,392. Ballhausen, C. J. (1962). Introduction to Ligand Field Theory (McGraw-Hill Book Company, Inc., NY.) Bottka, N., Stankiewicz, J., and Giriat, W. (1981). J. Appl. Phys. 52, 4189. Brafman, O.,and Steinberger, I. T. (1966). Phys. Rev. 143, 501. Brun Del Re, R., Donofrio, T., Avon, J., Majid, J., and Woolley, J. C. (1983). I1 Nuovo Cimento 2D, 1411. Bucker, R., Gumlich, H. E., and Krause, M. (1985). J. Phys. C: Solid State Phys. 18, 661. Bylsma, R. B., Becker, W. M., Kossut, J., Debska, U., and Yoder-Short, D.(1986). Phys. Rev. B 33, 8207. Cardona, M., Shaklee, K. L., and Pollak, F. H. (1967). Phys. Rev. 154, 696. Chou, H . H., and Fan, H. Y. (1974). Phys. Rev. B 10, 901. Cohen, M. L., and Bergstresser, T. K. (1966). Phys. Rev. 141, 789. Diouri, J., Lascaray, J. P., and El Amrani, M. (1985). Phys. Rev. B. 31, 7995. Ehrenreich, H . , Hass, K. C., Johnson, H. F., Larson, B. E., and Lempert, R. J. (1986). Proceedings of the 18th International Conference on the Physics of Semiconductors, Stockholm, Sweden. Gai, J. A., Galazka, R. R., and Nawrocki, M. (1978). Solid State Commun. 25, 193. Giriat, W.,and Stankiewicz, J . (1980). Phys. Stat. Sol. (a) 59, K79.

2.

BAND STRUCTURE AND OPTICAL PROPERTIES

71

Griffith, J. S. (1971). The Theory of Transition-Metal Ions (Cambridge University Press, London). Gumlich, H.-E. (1981). Journal of Luminescence 23, 73. Ikeda, M.,Itoh, K., and Sato, H. (1968). J. Phys. SOC.Japan 25, 455. Kendelewicz, T., and Kierzek-Pecold, E. (1978). Solid State Commun.25, 579. Kendelewicz, T. (1980a). Proceedings of the Xth Polish Seminar on Semiconductor Compounds, Jaszowiec, Poland. Kendelewicz, T. (1980b). Solid State Commun. 36, 127. Kendelewicz, T. (1981). J. Phys. C: Solid State Phys. 14, L407. Kolodziejski, L. A., Gunshor, R. L., Venkatasubramanian, R., Bonsett, T. C., Frohne, R., Datta, S., Otsuka, N., Bylsma, R. B., Becker, W. M., andNurmikko, A. V. (1986). J. Vac. Sci. Technol. B4, 583. Langer, D., and Ibuki, S. (1965). Phys. Rev. 138,A809. Langer, D. W., and Richter, H. J. (1966). Phys. Rev. 146, 554. Lascaray, J. P., Diouri, J., El Amrani, M., and Coquillat, D. (1983). Solid State Commun. 47,709. Lautenshlager, P., Logothetidis, S., Vina, L., and Cardona, M. (1985). Phys. Rev. B32,3811. Lee, Y. R., and Ramdas, A. K. (1984). Solid State Commun. 51, 861. Lee, Y. R., Ramdas, A. K., and Aggarwal, R. L. (1986a). Proceedings of the 18th International Conference on the Physics of Semiconductors, Stockholm, Sweden. Lee, Y. R., Ramdas, A. K., and Aggarwal, R. L. (1986b). Phys. Rev. B 33, 7383. McClure, D.S. (1959). “Electronic Spectra of Molecules and Ions in Crystals” in Solid State Physics, Vol. 9 (Academic Press, NY). McClure, D. S. (1963). J. Chem. Phys. 29, 2850. Morales, J. E., Becker, W. M., and Debska, U. (1985). Phys. Rev. B 32, 5202. Morales Toro, J. E., Becker, W. M., Wang, B. I., Debska, U., and Richardson, J. W. (1984). Solid State Commun. 52, 41. Morales Toro, J. E. (1985). Ph.D. Thesis, Purdue University, unpublished. Moriwaki, M. M., Becker, W. M., Gebhardt, W., and Galazka, R. R. (1982). Phys. Rev. B 26, 3165. Moriwaki, M. M., Tao, R. Y.,Galazka, R. R., Becker, W. M., and Richardson, J. W. (1983). Physica 117B & 118B, 467. Muller, E., Gebhardt, W., and Rehwald, W. (1983). J. Phys. C: Solid StatePhys. 16,L1141. Neumann, E. (1971). Thesis, Technische Universitat Berlin, D83, unpublished. Nguyen The Khoi and Gaj, J. A. (1977). Phys. Stat. Sol. (bJ83, K133. Pajaczkowska, A. (1978). Prog. Cryst. Growth Charact. 1, 289. Parsons, R. B., Wardzynski, W., and Yoffe, A. D. (1961). Proc. Roy. SOC.A 262, 120. Phillips, J. C. (1973). Bonds and Bands in Semiconductors (Academic Press). Podgorny, M., and Oleszkiewicz, J. (1983). J. Phys. C: Solid State Phys. 16, 2547. Richardson, J. W., and Janssen, G. J. M. (1986). Proceedingsof theMaterials Research Society Fall Meeting, Boston, Massachusetts. Rebmann, G., Rigaux, C., Bastard, G., Menant, M., Triboulet, R., and Giriat, W. (1983). Physica 117B & 118B,452. Skowronski, M., Baranowski, J. M., and Ludwicki, L. J. (1978). Polska Akademie Nauk (PAN) Nr. 75, 127. Stankiewicz, J. (1983). Phys. Rev. B 27, 3631. Steinberger, I. T. (1983). “Polytypism in Zinc Sulphide,” in Crystal Growth and Characterization of Polytype Structures (Pergamon Press). Sundersheshu, B. S . (1980). Phys. Stat. Sol. (a) 61,K155. Sundersheshu, B. S., and Kendelewicz, T. (1982). Phys. Stat. Sol. (aJ 69, 467.

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Tanabe, Y . , and Sugano, S . (1954). J. Phys. Soc. Japan 9, 753. Tanabe, Y . , and Sugano, S. (1970). Multiplets of Transition Metal Ions in Crystals (Academic Press, New York). Tao, R . Y . , Moriwaki, M. M., Becker, W. M., and Galazka, R. R. (1982). J. Appl. Phys. 53, 3772. Theis, D. (1977). Phys. Stat. Sol. (b) 79, 125. Thompson, A. G., and Woolley, J. C. (1967). Can. J. Phys. 45, 255. Twardowski, A., Nawrocki, M., and Ginter, J. (1979). Phys. Stat. Sol. (b) 96, 497. Twardowski, A. (1983a). Phys. Lett. 94A, 103. Twardowski, A.,Dietl, T., and Demianiuk, M. (1983b). Solid State Commun.48,845. Twardowski, A., Swiderski, P., von Ortenberg, M., and Pauthenet, R. (1984). Solid State Commun. 50, 509. Van Vechten, J . A. (1970). Proceedings of the Tenth International Conference on the Physics of Semiconductors, Cambridge, Massachusetts, 602. Varshni, Y. P. (1967). Physics 34, 149. Vecchi, M. P., Giriat, W., and Videla, L. (1981). Appl. Phys. Lett. 38, 99. Ves, S.,Strossner, K., Christensen, N. E., Chul Koo Kim, and Cardona, M. (1985). SolidState Commun. 56, 479. Ves, S . , Strossner, K . , Gebhardt, W., and Cardona, M. (1986a). Phys. Rev. B 33, 4077. Ves, S.,Strossner, K., Gebhardt, W., and Cardona, M. (1986b). SolidState Commun. 57,335. Wei Shan, Shen, S. C., and Zhu, H. R. (1985). Solid State Commun.55, 475. Wiedemeier, H.,and Sigai, A. G. (1970). J. Electrochem. SOC. 117, 551. Wisniewski, P.,and Nawrocki, M. (1983). Phys. Stat. Sol. (b) 117, K43. Zimnal-Starnawska, M., Podgorny, M., Kisiel, A., Giriat, W., Demianiuk, M., and Zmija, J . (1984). J. Phys. C: Solid State Phys. 17, 615.

SEMICONDUCTORS AND SEMIMETALS. VOL. 25

CHAPTER 3

Magnetic Properties: Macroscopic Studies Saul Oseroff DEPARTMENT OF PHYSICS, SAN DIEGO STATE UNIVERSITY SAN DIEGO, CALIFORNIA, USA

and Pieter H. Keesom DEPARTMENT OF PHYSICS, PURDUE UNIVERSITY WEST LAFAYETTE, INDIANA, USA

INTRODUCTION. . . . . . . . . . . BACKGROUND. . . . . . . . . . . RESULTSAND ANALYSIS. . . . . . . 1. Electron Paramagnetic Resonance. 2. Magnetic Susceptibility . . . . . 3 . Specsfic Heat . . . . . . . . . 4. Remanent Magnetization . . . . IV. CONCLUDING REMARKS.. . . . . . REFERENCES.. . . . . . . . . . . I.

11. 111.

. . . .

. . . .

. . . .

. . . .

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 14 80 80 91 107 112 117 119

I. Introduction The purpose of this work is to review the magnetic properties of diluted magnetic semiconductors (DMS). These are ternary compounds of the form AI-,Mn,C, where A = Zn, Cd, Hg are group-I1 elements of the periodic table, C = S , Se, Te are group-IV elements and xis the mole fraction of the magnetic ion MnZ+.The properties of interest will be electron spin resonance, magnetization, and the contribution to the specific heat due to the magnetic Mn ions incorporated substitutionally in the lattice (neutron scattering data, that are also of importance as far as magnetic properties of DMS are concerned, are discussed elsewhere in this volume). These properties can be calculated when the energy level scheme of the Mn ions is known. Vice versa, when the above properties are known from the measurements, they can be used to deduce the energy level scheme. The divalent manganese ions have a 3d5 configuration and 6S5/2 ground state. A spherically symmetric S state cannot be affected by a crystalline electric field. However, it is known that a crystal field splitting of the ground state of Mn2+ does occur. This is because it is not a pure 6Sstate, but contains admixtures of higher lying levels.

73 Copyright 0 1988 by Academic Press, Inc. All rights ot reproduction in any form reserved. ISBN 0- 1 2 - 7 9 125-9

74

SAUL OSEROFF AND PIETER H. KEESOM

The mechanisms of such mixing involve higher order processes that take into account crystal fields, spin-orbit coupling, spin-spin interaction, relativistic effects, etc. (Watanabe, 1957; Gabriel et al., 1961). However, the admixtures in question are relatively small and often can be neglected. We shall return to this problem later in this chapter. 11. Background

In this section, we shall try to give a list of general problems that are encountered when studying magnetic properties of DMS with increasing number of magnetic ions incorporated in the host semiconductor matrix. Thus, we start with an examination of the properties of an isolated Mn2+ ion in the 11-VI lattice. We then proceed to describe the interation of the ion with other Mn2+ ions. The latter interaction will be the dominant effect determining the magnetic properties of concentrated systems. We shall also give expressions appropriate to describe various quantities of interest to us in this chapter for each regime of Mn2+ concentration. For concentrations of Mn below 0.1 atomic percent, x < 0.001, the interaction between the Mn ions can probably be neglected. Thus, in any analysis, only isolated paramagnetic Mn ions with a ‘ S m ground state need to be considered. The presence of a crystal electric field will remove part of the sixfold degeneracy of the ground state. A crystal field cannot remove all the degeneracy of a 6 S ~ / 2state; at most it will split it into three Kramers doublets. Because of the complexity of the mechanisms involved in the splitting, it is difficult to perform direct calculations. Therefore, spin Hamiltonian method is usually used, Abragam and Bleaney (1970). The crystal field Hamiltonian for Mn2+ can be written as an even polynomial of fourth degree in the projection of the spin operators &, and $. For example, for an orthorhombic lattice, the crystal field Hamiltonian for Mn2+ is:

sy,

where D , E , and a are crystal field parameters to be determined from the experiment. The first term accounts for the tetragonal field with thezdirection taken as the axis of symmetry; the second term accounts for deviations from the axial symmetry, and the last term gives the contribution of the cubic field. Most of the DMS discussed here have a zinc blende structure, two fcc lattices displaced from each other by one quarter of a body diagonal, or a wurtzite structure composed by two hcp lattices. For a zinc blende structure only the last term on Eq. (1) is different from zero. In such a case, the 6S5/2 ground state splits into a doublet and a quadruplet separated by 3a. The order of magnitude of the splitting in a typical DMS is smaller than 0.1 K.

3.

MAGNETIC PROPERTIES : MACROSCOPIC STUDIES

75

When an external magnetic field H is applied, the degeneracy is completely removed. The EPR spectra for the Mn*+,then, consist of five resonance lines which correspond to the fine structure transitions AM = + 1. Their energy level scheme depends on the intensity of the crystal field and external magnetic field H, as well as on the angle between the magnetic field and the crystal axis. Because the Mn nucleus has a magnetic moment Z = 5/2, the interaction of the electron magnetic moment and nuclear magnetic moment will split each of the fine structure energy levels (labeled by M ) into six additional levels. A simplified spin Hamiltonian for Mn2+ in a cubic host, that includes the hyperfine interaction is:

where the first term accounts for the electronic Zeeman interaction, the second for the cubic crystal field, and the third for the hyperfine interaction. In Eq. (2), we have ignored smaller contributions, such as the superhyperfine interaction and the nuclear Zeeman interaction. For a more complete discussion on the subject, see Abragam and Bleaney (1970). When the energies of the Mn levels are known, the magnetization M , susceptibility x and specific heat C follow directly. For low concentrations, the uncoupled Mn ion levels in zero magnetic field will only show up at temperatures such that the value of k T is smaller or comparable than their crystal field and or hyperfine splitting, less than 0.1 K. If a sufficiently strong magnetic field (compared with the crystal field splitting) is applied, H 2 lOkOe, then the Mn ions give a Schottky contribution to M , x, and C characteristic of a system with essentially six evenly spaced levels. The susceptibility for the temperatures above 1 K and low concentration follows a Curie law:

where N Mis~the number of Mn ions/unit volume, g is the Lande factor, PB is the Bohr magneton, S is the total spin, and kg is the Boltzmann constant. When the Mn concentration increases above x = 0.005, the probability that Mn ions will form interacting clusters increases rapidly. The probability for the occurrence of clusters of various sizes can be calculated as a function of concentration, assuming a random distribution of Mn atoms for a given structure of the lattice. Probability tables for the clusters, and corresponding spin Hamiltonians and eigenvalues for cases of nearest, next-nearest, and third nearest-neighbor interactions can be found in the literature (Kreitman and Barnett, 1965; Okada, 1980, Nagata et al., 1980). We show these probabilities for singles, pairs, and triples for the fcc and hcp lattices in

76

SAUL OSEROFF AND PIETER H . KEESOM

TABLE I PROBABILITY THAT

FOREIGN ATOMIS I N A SINGLE, DOUBLE, AND TRIPLE CLUSTER ASSUMING ONLY THE NEAREST-NEIGHBOR INTERACTIONS‘

A

fcc

Cluster Type

hCP

(1 - x)l2 12x(l - X ) l 8 18?(1 - ~)’~[5(1- x) 24?(1 - x)”

S D‘ Tb

z1

(1 12X(1 - X ) ’ 8 i8X2(1 - Xy3[2 + 5(1 - X)I 3x2(1 - ~ ) ~ ‘ + [ 16(1 - X) + (1 - x ) ~ ]

+ 21

~

“After Kreitman and Barnett. 1965.

TABLE I1 PROBABILITY THAT A FOREIGN ATOMIS IN A SINGLE, DOUBLE, OR TRIPLE CLUSTER ASSUMING THE NEAREST-NEIGHBOR AND NEXT-NEAREST-NEIGHBOR INTERACTIONS‘ fcc

Cluster Type

(1 - x)’8 12x(1 - X)26 6x(1 - x ) ~ ’ 902(1 - x ) ’ ~ 92(1 - x)41[4 72u2(1 - x)37[1 24x2(1 - x ) ~ ’

+ (1 - x)]

+ (1 - x ) ~

0

36x2(1 - x)34 0

(1 - x)l8 12x(l - x)26 6x(1 - x ) ~ ’ 9x2(1 - x)”[l 9(1 - x)] 9 2 ( 1 - x)39[2(1 - x)2 + 2(1 - X ) 3 + 11 18x2(1 - ~ ) ~ ’ [+3 5(1 - x)] 3x2(1 - xI2’[l + 6(1 - x)’ + (1 - x ) ~ ]

+

0 36x2(1 - x ) ~ ~

0

‘After Kreitman and Barnett, 1965.

Tables, I, 11, and 111. The corresponding Hamiltonians and eigenvalues for singles, pairs, and triples are given in Table 1V. The susceptibility and specific heat C for a system consisting of clusters of various sizes can be written as:

x

where xi and Ci are the contributions from each cluster type including isolated ions, and Ni is the number (per unit volume) of occurrence of clusters of the i-th type. The susceptibility and the specific heat contributions of each

3.

77

MAGNETIC PROPERTIES: MACROSCOPIC STUDIES

TABLE I11 PROBABILITY THAT A FOREIGN ATOMIS IN A SINGLE, DOUBLE, OR TRIPLE CLUSTER ASSUMING INTERACTIONS UP TO 3RD-NEAREST-NEtGHBORS FOR fcc LATTICE' Cluster Type

Probability

S

(1 - x)42 12x(l - x)5' 6x(1 - x)62 24x(1 182(1 - x)74 9x2(1 - x ) ~ + * 36x2(1 - x)" 36x2(1 - x)'~+ 144x2(1- x ) ~ ' + 252x2(1 - x)93+ 180x2(1- x)94 72x2(1 - x ) ~ ' 144x2(1 - x)" + 144x2(1- x)" + 216x2(1- x)'4 72x2(1 - x ) ' ~+ 144x2(1- x)*' 24x2(1 - x)~' 24x2(1 - x)'~ 36x2(1 - x)~' 72x2(1 - x)" 72x2(1 - x)" 72x2(1 - x)71 72x2(1 - x)'3

Dl L? D3

z' T,'

z3

After Okada, 1980.

of the clusters are given by X , ( T )= 1

s%B

c

3k~T

s

w + 1)(2S + l)exp(-Es/ksT) cs(2s + l)exp(-&/kBT)

d ES exp(- E s / k T ~) Cj(T) = a T exp( - &/kB T ) '

I

(6) (7)

where Es is the energy for the case of zero magnetic field. In the presence of a magnetic field H, Es in Eqs. (6) and (7) should be replaced by E, = Es - g,uBmH where m is the magnetic quantum number. In that case, an additional summation over rn (- S 5 m IS) must be performed. As the concentration increases the mean cluster becomes more extended until, at a certain critical concentration x = x, , its size becomes comparable with the size of the sample. The value of xc , known as the critical concentration for the site-percolation problem, depends on the lattice structure and the number of neighboring magnetic ions for which the exchange interactions are significant. For example, if interactions up to the lst, 2nd, or 3rd neighbors are taken into account in an the fcc lattice, we have xc = 0.195,

78

SAUL OSEROFF AND PETER H. KEESOM

TABLE IV HAMILTONIANS AND EIGENVALUES FOR SINGLE, DOUBLE, AND TRIPLE CLUSTERS'?

Type

Hamiltonian

Eigenvalue

After Okada, 1980.

0.136, and 0.061 respectively (see e.g., Domb and Dalton, 1966; Frisch et al., 1961).

DMS systems usually form a simple crystallographic phase with a zinc blende or wurtzite structure for a wide range of concentrations. In Table V, we give the crystal structures and approximate ranges of composition in which good quality single-phase samples can be obtained. The DMS can sometimes be obtained for higher values of x than that given in Table V, but the crystals tend to be of poorer quality (see e.g., Pajaczkowska, 1978; Furdyna, 1982). In all the systems given in Table V, the maximum concentration of Mn is well above the nearest-neighbor percolation critical value (xc = 0.2). Thus in principle, a long magnetic ordering can be expected below certain temperatures, In these materials it was found that the nearest-neighbor interaction between Mn ions is antiferromagnetic (AF) (Brumage et al., 1964; Kreitman et al., 1966; Davydov, 1980; Nagata et al., 1980; Galazka et al., 1980;

3.

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79

TABLE V CRYSTAL STRUCTURE AND RANGEOF COMPOSITION FOR THE DMS STUDIED Material Znl-,Mn,S

Znl-,Mn,Te Cdl-,Mn,S Cdl-,Mn,Se Cdl -,MnxTe Hgl-,MnxS Hgl-,Mn,Se Hgl-,MnxTe

Crystal Structure

Range of Composition

Zinc blende Wurtzite Zinc blende Wurtzite Zinc blende Wurtzite Wurtzite Zinc blende Zinc blende Zinc blende Zinc blende

x < 0.10 0.10 s x s 0.43

x s 0.30 0.30 < x < 0.55 x s 0.75 x s 0.45 x s 0.50 x s 0.70 x ~0.37 x s 0.30 x 5 0.50

Shapira et al., 1984). A spin-glass phase with only AF interactions was predicted for an fcc lattice owing to the frustration of AF ordering (DeSeze, 1977; Aharony, 1978; Villain, 1979). The mechanism of the frustration can be visualized as follows: consider three spins forming an equilateral triangle and assume that a pair of spins is aligned antiparallel. This orientation results in a lowering of the energy of that pair. What about the third spin? If it is aligned antiparallel-and hence favorably coupled-to one of the spins in the original pair, it is of necessity aligned parallel to the other one. This impossibility of satisfying strict antiferromagnetic configuration amongst all nearest neighbors is referred to as frustrated antiferromagnetism. It is currently believed that some form of frustration is necessary for a system to form the spin-glass state. Thus, two main ingredients for a system to show a spin-glass (SG)behavior are the frustration of long magnetic ordering and a random distribution of impurities. By similar arguments, frustration is also expected for an hcp lattice. Most of the systems exhibiting the SG behavior studied so far are diluted solutions of transition ions in metallic matrices (Cannella and Mydosh, 1972; Fischer, 1976; Anderson, 1979; Mydosh, 1981). However, in a metallic SG, it is not always easy to separate the contribution of the conduction electrons from that of the localized moments, e.g., in the case of specific heat (Kondo, 1965). Therefore, for a better understanding of SG it is desirable to study simpler systems. One possibility is to use frustrated systems with only AF interactions. As was mentioned above, DMS are appropriate candidates for this purpose. In the case of DMS with large electronic energy gaps, no free electrons are available at low temperatures to mediate the

80

SAUL OSEROFF AND PIETER H. KEESOM

Ruderman-Kittel-Kasuya-Yosida interaction between the magnetic ions. Then if the SG phase is observed in these systems it must be explained in terms of a mechanism with predominantly short ranged interactions. It is the current opinion that these interactions in DMS are mainly due to the super exchange mechanism (Larson et al., 1985; Lewicki et al., 1985; Spalek et al., 1986). For zero-gap or narrow-gap DMS, another mechanism was suggested leading to long range indirect AF interactions between Mn2+ions. This is the so-called Bloembergen-Rowland mechanism, where the interaction is mediated by carriers from the valence band virtually excited to the conduction band (Bloembergen and Rowland, 1955; Bastard and Lewiner, 1979a,b; Lewiner et al., 1980; Liu, 1982). Summing up the introductory remarks, magnetic properties of DMS may show many facets: starting from that due to the isolated ion, through the isolated cluster, the spin-glass phase and, ultimately-for very concentrated DMS-long range ordered system (see also Giebultowicz and Holden, this volume). Sometimes, as in the case of Cdl-xMnxTe, it is possible to observe all these phases in one DMS system. This situation offered by DMS is rather unique: one is then able to follow the development of the magnetic behavior with increasing concentration of the magnetic constituent in a matrix that is relatively simple from the viewpoint of the electronic band structure. In the following sections, we shall give a review of existing experimental data on spin resonance, magnetic susceptibility, magnetization and specific heat obtained for various DMS, along with their detailed analysis aimed at conclusions of a more general nature. 111. Results and Analysis

This section is divided into several parts, each devoted to a single physically measureable phenomenon or quantity. We start with electron paramagnetic resonance, to be followed by magnetic susceptibility, specific heat, and magnetization. In each case, we first review the available experimental data in the given field, and then give a detailed theoretical analysis.

PARAMAGNETIC RESONANCE 1. ELECTRON a. Experimental Results

The EPR spectra of Mn2+in various II-VI compounds for x < 0.001 have been studied in detail by several authors (Matarrese and Kikuchi, 1956; Dorain, 1958; Matamura, 1959; Lambe and Kikuchi, 1960; Hall er al., 1961; Kikuchi and Azarbayejani, 1962; Schneider et al., 1963; Title, 1963a,b, 1964; Estle and Holton, 1966; Falkowski, 1967; Deigen et al., 1967; Leibler et al., 1970a,b, 1971, 1973). The main features of the spectra are: the gyromagnetic

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MAGNETIC PROPERTIES : MACROSCOPIC STUDIES

81

factor g is nearly equal to that of the free electron, 1.99 < g c 2.02; a well resolved isotropic hyperfine structure is observed consisting of six lines, possibly due to configurational interaction between the 3s23p63d54sconfiguration with the 3s23p63d5configuration (Abragam, 1950); and a fine structure that results from an anisotropic contribution. The values of the gyromagnetic factor g, the hyperfine structure constant A , and the crystal field parameters are obtained by fitting the data to the energy spectrum corresponding to a spin Hamiltonian analogous to that given by Eq. (2). As the concentration of Mn increases, the spectrum changes its shape. For x > 0.005, the hyperfine lines initially broaden, eventually becoming a structureless single line for x 1 0.02 (Ishikawa, 1966; Deigen et al., 1967; Leibler et al., 1977; Oseroff et al., 1979, 1980a,b; Andrianov et al., 1980; Oseroff, 1982; Manoogian et al., 1982; Webb et al., 1983; Sayad and Bhagat, 1985). The EPR linewidth AH,, first narrows as x increases and passes through a minimum at x = 0.03. Van Wieringen (1955), after calculating the second and fourth moments of the spectrum, suggested that the variation of the shape of the spectrum may result from the narrowing effect of the hyperfine splitting due to the exchange interaction with other Mn2+ions. The effect of exchange interaction on the linewidth has also been discussed by Bleaney (1970). After reaching the minimum at x = 0.03, the line broadens monotonically with concentration (Leibler et al., 1976, 1977; Oseroff, 1982; Manoogian et ai.,1982; Webb et al., 1983; Sayad and Bhagat, 1984). Different mechanisms which contribute to the broadening of the line with concentration have been discussed by Kittel and Abrahams (1953); Swarup (1959); Deigen et al. (1967); Abragam and Bleaney (1970); and Sayad and Bhagat (1985). Values of the gyromagnetic factor g and of the linewidth AH,, at room temperature, defined as the peak-to-peak in the first derivative of the absorption line, are given in Fig. 1 for Cdl-,MnxTe as a function of concentration and frequency. Similar behavior was found for other DMS studied. At high temperatures, the linewidth and the gyromagnetic factor g were found to be independent of microwave frequency used (Oseroff, 1982; Webb et al., 1983; Sayad and Bhagat, 1985). For concentration above x = 0.03, a significant increase in the linewidth with decreasing temperature was observed for all the DMS studied so far by EPR (Leibler et al., 1976, 1977; Grochulski et al., 1979; Oseroff et al., 1979, 1980a,b; Mullin et al., 1981; Oseroff, 1982; Manoogian et al., 1982; Webb et af., 1983; Sayad and Bhagat, 1985; Kremer and Furdyna, 1985a,b). This can be understood as an increase (as the temperature is lowered) of the internal field due to the presence of finite clusters. A typical dependence of the linewidth AH,, on the temperature can be seen in Figs. 2 and 3 for Cd,-,Mn,Te and Cdl-,Mn,Se. When the linewidth AH,, is of the order of

82

SAUL OSEROFF AND PIETER H. KEESOM

I"""' I

"/ 1 40.05

P

P

i

I

6'

-0.05

X

FIG. 1. Room temperature peak-to-peak linewidth AH,, and g-factor as a function of concentration. Linewidth (0)35 GHz and (A) 11 GHz, from Webb et al. (1983); (0)9.2 GHz, from Oseroff (1982); g-factor: (0)35 GHz and (A)11 GHz, from Webb et a[. (1983). [After

Webb er af. (1983).]

the resonance field, H R,special care must be taken to obtain meaningful data for the width and the position of the line. Because the radio frequency (r.f.) field is linearly polarized in a typical microwave experiment, the experimental signal contains, in addition to the resonance signal induced by the resonanceactive circularly polarized component of the r.f. field, also a contribution due to the resonance-inactive-circular component. Thus, to obtain the true resonance line shape from the raw data, the unwanted component has to be subtracted. Such a correction was made when preparing Figs. 2 and 3. It has been also found that the resonance field H R of the EPR line in DMS shifted to lower magnetic fields with decreasing temperature (Oseroff, 1982; Manoogian et al., 1982; Kremer and Furdyna, 1983, 1985a,b). As the shift is found to be almost independent of the microwave frequency, it is attributed to be due to an internal field rather than to a change in the gyromagnetic factor itself (Oseroff, 1982). The observed change in the position of the resonance line, Hi = H , - HR(whereN, is the magnetic field corresponding to g = 2) is shown in Fig. 4 for Cdl-,Mn,Te and Cdl-,Mn,Se. The difficulty to perform EPR measurements when o n < 1, where o is the microwave frequency and T is the relevant relaxation time, was successfully overcome by using the microwave Faraday rotation technique based on

3.

MAGNETIC PROPERTIES : MACROSCOPIC STUDIES

83

9000-

7000 n

Q)

0

Y

P P

I

a

FIG. 2. Temperature dependence of the peak-to-peak EPR linewidth for Cdl-,MnxTe (0.15 5 x s 0.60). The solid lines are least-squares fits to the data using Eq. (10). [After Oseroff (1 982).]

the dispersion associated with EPR, as discussed by Kremer and Furdyna (1983, 1985b). With this method, Kremer and Furdyna (1983, 1985a,b) performed measurements of the dynamic magnetic susceptibility at 35 GHz down to W T < 0.2. They measured AH,, and H R for concentrations well above the percolation limit, at temperatures below the spin-glass temperature (as revealed by the cusp in the zero field cooled susceptibility, c.f., further in this Chapter). It is to be noted that no anomaly was observed either in AH,, or HR at the spin-glass temperature Tg.

b. Analysis (a) Low Mn Concentrations (x s 0.03) For extremely small concentrations of Mn, x < 0.001, the interaction between the Mn ions is negligibly small. For the wide gap DMS, the gyromagnetic factor and the crystal field contribution were found to be larger for Mn2+ in the zinc than in the cadmium compounds with the same anion, whereas the hyperfine structure parameter A was found to be smaller in the zinc compound (see, for example, Kikuchi and Azarbayejani, 1962, and Title, 1963b). Kikuchi and Azarbayejani (1962) noted that the ionic radius of the Mn2+ ion was intermediate in size between the ionic radii of Zn2+and Cd2+.Then

84

SAUL OSEROFF AND PIETER H. KEESOM

j 2/

3.

MAGNETIC PROPERTIES : MACROSCOPIC STUDIES

85

T( K) FIG. 4. Temperature dependence of the internal magnetic field Hi for Cdl-,Mn,Se, and Cdl-,Mn,Te measured at 9 and 35 GHz. Solid lines are least-squares fits to the data using Eq. (12). H i is defined as Hi = Ho - H E , where Ho is the resonance field corresponding to g = 2, and H R is the measured resonance field. [After Oseroff (1982).]

anions than occurs in cadmium compounds. This results in a larger value of the crystal field parameter, a, and a smaller value of hyperfine constant, A , for the zinc compounds. This means that the bonds in the Mn2+-doped zinc compounds are more covalent than in cadmium compounds. A change in the resonance parameters was also observed, when going from Te to Se to S (Woodbury and Ludwig, 1961; Kikuchi and Azarbayejani, 1962; Title, 1963a,b). The gyromagnetic factor and the crystalline field parameters decreased, and the hyperfine structure constant increased when going from Te to Se to S (Title, 1963b). These tendencies appear to be related to changes in the covalent character of the bonds for different anions (Van Wieringen, 1955; Matamura, 1959; Fidone and Stevens, 1959; Hall et al., 1961; Kikuchi and Azarbayejani, 1962; Title, 1963a,b). A decrease of the g-factor and an increase of the hyperfine structure constant was also observed when the lattice spacings were reduced in the narrow gap DMS (Leibler et al., 1971, 1973). The g-factor for the narrow gap DMS was also found to be smaller than the g-factor for free electrons (Leibler et al., 1970a,b, 1971, 1973). As the concentration of Mn2+ increases, the EPR spectra change their shape. Deigen ef al. (1967) studied the concentration dependence of Cdl-,Mn,Te for lo-' 5 x 5 0.03, and found that the profile of the EPR line changed from almost Gaussian to Lorentzian when going from x = 0.005

86

SAUL OSEROFF AND PIETER H . KEESOM

to x 3 0.03. They found that within experimental error, the values of the hyperfine structure constant, A , and the crystal field parameter, a, were independent of concentration, at least forx < 0.008. Van Wieringen (1959, who studied EPR of ZnS:Mn, calculated second and fourth moments of the spectrum, and suggested that the variation observed in its shape was due to the narrowing of the hyperfine splitting by the exchange interaction between Mn2+ions. Ishiwaka (1966) analyzed the EPR spectra of exchange-coupled Mn2+ ions in ZnS and CdS using a spin-Hamiltonian of the form:

X = gpBusHSz

+

A

N

+ C JijSi i,j

*

Sj,

(8)

zzl

where S, and Z, are the z-components of S = S; and I = xEl I ; , respectively, N is the number of Mn2+ ions coupled together by exchange interaction and J i j is the exchange integral. The first, second, and third terms in Eq. (8) represent the Zeeman, hyperfine and isotropic exchange interaction, respectively. Ishikawa (1966) concluded that the EPR spectra for an exchange-coupled pair of Mn2+ions are described satisfactorily by the spin Hamiltonian given by Eq. (8) and that anisotropic terms due to the crystal field can be neglected. Oseroff (1984) analyzed the spectrum for Cdl -,Mn,Te with 0.002 s x 5 0.01, in the context of the overlap of isolated Mn2+ ions and coupled pairs of Mn2+. It turned out that the EPR results are not sufficiently sensitive to separate unambiguously the contributions due to singles and pairs of MnZ+. To study the effect caused by the distortion on the crystal lattice due to replacing Cd (ionic radius = 0.97 A)by Mn (ionic radius 3 0.80 A),Koh et al. (1984) simulated this system by studying the EPR of alloy Cdl-,Hg,Te:Mn. As the ionic radii of Cd and Hg (ionic radius 1.10 A) differ almost as much as that of Cd and Mn, these authors argued that the lattice distortion in the case of Cdl -,Mn,Te can be expected to be of the same order as in Cdl-,Hg,Te:Mn, with the advantage that in the case of latter-because of the low Mn concentration-the effect of the distortion can be investigated in the highly resolved EPR spectrum of nearly isolated Mn2+. Koh et al. (1984) found similar values for the crystal field splitting for Cdl-,H&Te:Mn to those found previously for very diluted Mn systems in the 11-VI compounds, and on this basis concluded that relatively large concentrations of Mn do not produce sufficient lattice distortions to cause a significant change in the crystal field splitting. (b) High Mn Concentrations (x 2 0.03) For x above =0.03, the room temperature EPR linewidth broadens with increasing concentration, as shown in Fig. 1. Webb et al. (1983) and Sayad

3.

MAGNETIC PROPERTIES: MACROSCOPIC STUDIES

87

and Bhagat (1985) suggested that the broadening of the resonance lines arises from a distribution of internal fields, or rather their projections along the applied field. The observed broadening of the linewidth with decreasing temperature was analyzed by several authors. Leibler et al. (1977), Grochulski et al. (1979), Oseroff et al. (1979) and Manoogian et al. (1982) analyzed the temperature dependence of AH,, in terms of a theory first used for antiferromagnets above the Nee1 temperature (see Huber, 1972 and Seehra, 1972). The broadening of the linewidth was described using Huber's expression: AH,, = A T( - 2T , 7

+ B,

(9)

where a denotes a critical exponent, B the temperature-independent linewidth, and T, the temperature of the order-disorder transition. Equation (9) gives a good fit to the data for temperatures well above TC . However, the predicted divergence at T = T , was not observed when AH,, was measured by using the Faraday rotation technique (Kremer and Furdyna, 1983,1985b). The EPR data analyzed with Eq. (9) could be separated into three regions of concentration. For x < 0.15, the values of T , were found to be smaller than 1 K. For x > 0.25, T, was found to be substantially larger and increased with the concentration. In the transition region, 0.15 S x 5 0.25, there occurred a large and rapid change in T , for all DMS samples studied so far. As can be noticed, the percolation critical point xc calculated for nn falls into this region of concentration. The values of T , determined from Eq. (9) were found to be several times larger than the temperature G where the cusp in the ZFC (zero field cooled) susceptibility was observed. For Cdl-,Mn,Te, Oseroff et al. (1979) found that the critical exponent CY increased with the concentration until x = 0.2, and then it became almost independent of the concentration. They also observed that B, the high temperature linewidth, increased with concentration. Instead, for Cd,Zn,Mn,Te (x + y + z = 1)) Manoogian et al. (1982) obtained the best fit with B = 0 and a = 0.33 f 0.08 for the whole range of concentrations studied. Oseroff et al. (1980a,b) and Oseroff (1982) analyzed their linewidth data using an equation similar to Huber's expression (Dormann and Jaccarino, 1974): AHp, = A [

'AH

IaAH

T - GAH

+ B[; + 11,

where GAHis the order-disorder temperature, CYAH is the critical exponent, 0denoting the Curie-Weiss temperature, and B ( W T + 1) is the high temperature linewidth that becomes dominant for T %- GAH(see Figs. 2 and 3).

88

SAUL OSEROFF AND PIETER H . KEESOM

Using Eq. (lo), the authors obtained values of GAHthat were jn good agreement with those that follow from the dc susceptibility measurements at the same field, = 3.3 kOe. The values of T ~ A Hbecame substantially larger for x > 0.29 than for the low concentration region, and increased (within the large experimental error, linearly) with concentration. The values of T f a ~ were calculated by using only the low temperature data. Oseroff et al. (1980a,b) and Oseroff (1982) justified their analysis by arguing that the high temperature data bear no relation with the critical phenomena, the high temperature behavior being accounted for by the second term of Eq. (10). These authors found that the critical exponent increased with concentration for x < 0.30, and for larger concentration, it became less dependent on it. It is not clear at this point if the behavior observed for C X A His just an artifact of the model or not. The solid lines in Figs. 2 and 3 are a least-squares fit of the experimental results with Eq. (10). Values of &H and CXAHfor a wide range of concentration are given in Tables VI and VII for Cdl-,Mn,Te and Cdl -,Mn,Se, respectively. Webb et af.(1983) and Sayad and Bhagat (1985) used adifferent expression to adjust the data for AHpp(T).Theirs is an empirical expression used first by Bhagat et al. (1981) to fit the temperature dependence of AH in metallic spin-glasses and in re-entrant ferromagnets. They expressed AHpp(T ) as

A H ~ ~ (=T ro ) + rl exp(-T/&), where To is the high temperature linewidth, and

r 1

(1 1)

and To are empirical

TABLE VI OF THE CURIE TEMPERATURE 8; THE EXTRAPOLATED TEMPERATURE FOR THE MAXIMUM OF x, f i ( H 0) AFTER ZERO-FIELD COOLING; THE TEMPERATURES fimAND E H ~AND , CRITICAL EXPONENTS CYANAND C X HOBTAINED ~ FROM THE TEMPERATURE DEPENDENCE OF THE EPR LINEWIDTH’

VALUESAS A FUNCTION OF CONCENTRATION FOR Cdl-,Mn,Te

+

1

2 5 10 15 20 30 40 53 60

3 8 22 35 I0 100 170 230 310 350

“ A f t e r O s e r o f f , 1982.

e l

el

2 + 0.5 8+1 12 + 2 18 ? 2 23 3

*

0.01 0.35 0.85 4+2 9 2 2 13 k 3 20 f 3 25 + 3

0.05 0.2

0.3 0.4 0.5 0.6 1.5 1.5 1.6 1.5

1.7 2.2

3.

89

MAGNETIC PROPERTIES : MACROSCOPIC STUDIES

TABLE VII VALUESAS A FUNCTION OF CONCENTRATION FOR Cd,-,Mn,Se OF THE CURIE TEMPERATURE 0, THE EXTRAPOLATED TEMPERATURE FOR THE MAXIMUM OF x, E(H 0) AFTER ZERO-FIELD COOLING; THE TEMPERATURES f i AND ~ &,yi, ~ A N D CRITICAL EXPONENTS CYAHA N D CYH~OBTAINED FROM THE TEMPERATURE DEPENDENCE OF THE EPR LINEWIDTH' +

(at. To)

-e(+io) (K)

1

3

2 5 11 15 20 23 25 30 35 41 45

10

E{H+O) (K)

26


80

4

I20 135

1.8 3.0

0.5

+ 0.5

220

9.3

*1

330

15

+2

EAH

EHi

(K)

(K)

41 e l 41 -0.3 -1 -1 3.5 + 1 6+1 9+1 11.5 t 1

- 0.2 - 0.3 - 0.4 -1

ffAH(k0.2)

0.2 0.3 0.4 0.5 0.6 I .o 1.3 1.8 1.9 2.0

CXH,(+~.~)

2.2 2.9 32 5.4

"After O s e r o f f , 1982.

parameters associated with the freezing of the spins. Figure 5 shows the agreement between Eq. (11) and the results for Cdo.-loMn0.30Te.Figure 6 shows the agreement between Eq. (11) and the data obtained by other authors. Webb et al. (1983) and Sayad and Bhagat (1985) interpreted the increase of AH,, with decreasing temperature as a result of the spins slowing down when the freezing temperature was approached from above. Oseroff et al. (1980a,b), Oseroff (1982), and Manoogian et al. (1982) measured a shift of the resonance field HRwith temperature. Oseroff (1982) attributed the observed shift to an internal field Hi,because it was found to be independent, within the experimental error, of the microwave frequency used. The increase of H i with decreasing temperature was analyzed with an expression similar to Eq. (10) in which the second term accounting for Hi at high temperature was neglected. The expression used for Hi was:

where HO denotes the field corresponding to g = 2 , H R the measured resonance field, C a suitable constant to be obtained from the experiment, Z H the ~ transition temperature obtained from the change of HR, and CYH~the critical exponent. Equation (12) fits the data only for temperatures above

90

SAUL OSEROFF AND PIETER H. KEESOM

1 04,

I

I

I

1

FIG.5 . “Excess linewidth” for Cdo.7oMno.3oTeas a function of temperature, fitted with Eq. ( 1 1 ) with To = 650Oe. r1 = 4500Oe, and To = 5OK. [After Webb et al. (1983).)

=0.30,p0.55

I 10’

0

x= 0.20,y=o

40

80

120 160 200 240 280

T( K) FIG.6 . “Excess linewidth” as a function of T fitted with Eq. ( 1 1) for Cdo.s,,Mno.z,,Te (0). from Oseroff (1982); Cdo.s~sZno.17~Mn0.30Te (O), C ~ O . I S Z ~ O , ~ S M ~ (a), O . J and OT~ Zn0.70Mn0.3oTe(0)from Manoogian et al. (1982). [After Webb et al. (1983).]

E H ~As . for the linewidth AHpp,no divergence was observed for Hi at

T = TfHi (Kremer and Furdyna, 1983, 1985b). The solid lines shown in Fig. 4 are least-squares fit to the data using Eq. (12). Values for G H and ~ CYH~are given in Tables VI and VII for CdI-,Mn,Te and Cdl-,Mn,Se, respectively, As mentioned, Kremer and Furdyna (1983, 1985b) found that for Cdl-,Mn,Te, there was no anomaly in either AHpp or in the shift of HR at the temperature where the ZFC dc susceptibility shows a cusp, contrary to the case of metallic spin glasses (Schultz ef a[., 1980 and Oseroff et al., 1983).

3.

MAGNETIC PROPERTIES: MACROSCOPIC STUDIES

91

2. MAGNETIC SUSCEPTIBILITY

a. Experimental Results At high temperatures the dc susceptibility for the DMS can be approximated by a Curie-Weiss law,

K

=

C / ( T - O ) + Xd,

(13)

where 0 is the Curie-Weiss temperature, C the Curie constant, and x d the diamagnetic susceptibility of the host. In Figs. 7 to 9, the inverse suscepti, S ~Hgl-xMnxTe , is given as a bility x-’ for Cdl-,MnxTe, C ~ I - ~ M ~ and function of the temperature and of the Mn concentration. As the temperature decreases below 50 K, the inverse susceptibility x - l shows a characteristic departure from a Curie-Weiss law (Brumage et al., 1964; Kreitman et al., 1966; Deigen et al., 1967; Savage et al., 1973; Sondermann, 1976, 1979, 1980; Andrianov et al., 1976; Sondermann and Vogt, 1977a,b; Pajaczkowska and Pauthenet, 1979; Davydov et al., 1980; Nagata et al., 1980; Galazka et al., 1980; Oseroff et al., 1980a,b; Oseroff

..

120 1 4 o m

....

T( K1 FIG.7. The temperature dependence of the inverse susceptibility x - l of Cdl-,MnxTe. The low-temperature data (open circles) were taken after ZFC for increasing temperature in a 30 Oe field. The high-temperature data (full circles) were obtained in an 8.5 kOe field. [After Oseroff (1982).]

92

SAUL OSEROFF AND PIETER H. KEESOM

CdMnSe

100

I

I

200

100

0

3

T( K) FIG. 8. Inverse susceptibility x-' of Cdl-,Mn,Se as a function of temperature. The low temperature data (open circles) were taken after ZFC for increasing temperature in a 30 Oe field. The higher temperature data (fun circles) were obtained in an 8.5 kOe field. [After Oseroff (1982).]

3.0r

2.5h

E

a,

I

H g,-,Mn,Te

2.0-

1

I

I

X

a:0.01; b:0.02: c:0.03 e0.1 1 f0.14 g:O.16 h:0.22 i:0.25

i0.35

0)

FIG. 9. Inverse susceptibility x-' vs. T for Hgl-,Mn,Te. [After Nagata et al. (1980).]

3.

MAGNETIC PROPERTIES: MACROSCOPIC STUDIES

93

and Acker, 1981; Heidrich, 1981; Oseroff, 1982; Escorne and Mauger, 1982; McAlister et al., 1984; Oseroff and Gandra, 1985; Spalek et al., 1986) which can be discussed in the framework of a cluster model using Eq. (4), as in, e.g., Kreitman et al. (1966); Simpson (1970); Savage et al. (1973); Nagata et al. (1980; Galazka et al. (1980); Okada (1980); Shapira et al. (1984). For still lower temperatures, T z 1.5 K, and at concentrations of Mn x k 0.17, it was found that the susceptibility depends on whether the sample was cooled in the presence or the absence of an external magnetic field H (Nagata et al., 1980; 1981; Galazka et al., 1980; Oseroff et al., 1980a,b; Oseroff and Acker, 1981; Heidrich, 1981; Escorne et al., 1981; Khattak et al., 1981; Oseroff, 1982; Escorne and Mauger, 1982; Yang et al., 1983; Amarasekara et al., 1983; McAlister et al., 1984; Oseroff and Gandra, 1985). When the sample is cooled in zero field (ZFC), and then a magnetic field is applied at the lowest temperature, the susceptibility measured in an increasing temperature presents a maximum at a certain temperature r f ,that depends both on x and the magnetic field H. The susceptibility for Znl-,MnxTe, Cdl-,Mn,Te, and Cd,-,Mn,Se, showing these irreversible effects, is given in Figs. 10-12, respectively, as a function of x and the temperature near r f . Most of the susceptibility data were taken for T 2 1.5 K. For this region of temperatures, a cusp was observed only for x 2 0.17. This was the fact that seemed to support the original idea that a spin glass (SG) transition in DMS occurred only above the critical concentration derived for the first nearest-neighbor percolation, x, 0.2 (Nagata et al., 1980; Galazka et al., 1980; Oseroff, 1982; Yang et al., 1983; Amarasekara et al., 1983; McAlister

-

1.1,

,

I

T( K) FIG. 10. Comparison of field-cooled (0)and zero-field-cooled (El) magnetizations for Znl-,Mn,Te with x = 0.37, measured in 200 Oe. The values of magnetization are normalized to their values at Tr. [After McAlister et 01. (1984).]

94

SAUL OSEROFF AND PIETER H. KEESOM

0.141

\

O.Ol0L----L---I----I------L--J 4 8 12 16

20

T( K) FIG. 11. ZFC and FC susceptibility data of Cdl-xMnxSe and Cdl-,Mn,Te as a function of T near Tr for H = 30 Oe. The arrow (-+) indicates ZFC, (+-) indicates FC. [After Oseroff (1982).] 111

0.

Cdl-xMnxTe

X

30.225 :I 0.30 30.40

:10.60

*.

I

3;

5

10

15

20

25

T( K) FIG.12. ZFC and FC susceptibility data of Cdl-,MnxTe as a function of Tnear f i observed in a 15 Oe field. [After Oseroff and Gandra (1985).]

3.

MAGNETIC PROPERTIES : MACROSCOPIC STUDIES

95

et al., 1984). However, more recently, by performing the measurements at very low temperature, T s 0.1 K, a cusp was also observed in the susceptibility for x c 0.17 (Brandt et al., 1981. 1982, 1983; Novak et al., 1981, 1984, 1985). In Fig. 13, the susceptibility in Hgl-,MnxTe for various values of concentration is given as a function of temperature and the magnetic field. The field cooled susceptibility, normalized to the concentration of Mn, is given in Fig. 14 for Cdl-,Mn,Se and Cdl-,MnxTe. Also the cusp in the ZFC susceptibility was observed for low values of x in (Cdl-,Mn,)3Asz, (De Jonge et al., 1983). The fact that SG behavior is found well below the critical concentration calculated for the nearest-neighbor interactions indicates that the interactions with more distant Mn ions should be included, particularly at very low temperatures. When the external magnetic field is increased, the ZFC susceptibility maximum becomes weaker, broader, and shifts to lower temperature (Oseroff, 1982; Brandt et al., 1983). This effect can be seen in Figs. 13 and 15. Values of T f obtained in ZFC samples as a function of magnetic field are given in Fig. 16 for CdI-,Mn,Te and Cdl-,Mn,Se. Low field a.c. susceptibility was measured for Hgl-,Mn,Te by Otto et al. (1980) and Mycielski et al. (1984), and for Cdl-,Mn,Te by Oseroff (1984)

T(K)

FIG. 13. Temperature dependence of the susceptibility x for Hg,-,Mn,Te in different magnetic fields. Right scale x for x = 0.034: curves (1) H = 0.5 Oe, (2) 50 Oe, and (3) 100 Oe. Left scale x for x = 0.02; curve (4) 0.5 Oe, and curve ( 5 ) 50 Oe. [After Brandt et a/. (1983).]

96

SAUL OSEROFF AND PIETER H. KEESOM

X

b

l-

0’

I

0.5

I

1.o

I

1.5

T( K) FIG.14. Field-cooled susceptibility normalized to the concentration of Mn, for Cdl-,Mn,Te (open symbols) and Cdl-,Mn,Se (closed symbols). [After Novak et al. (1985).]

FIG. 15. The behavior of the susceptibility near Tr in Cdo.ssMnod e for three different magnetic fields. Arrows pointing to increasing Tindicate ZFC; arrows in decreasing-Tdirection correspond to FC susceptibility data. The temperatures of the cusps are indicated by vertical arrows. [After Oseroff (1984).]

3.

MAGNETIC PROPERTIES : MACROSCOPIC STUDIES

97

1 .o

0.8

-5 0.6 h

0

z

k-

0.4

0.2 o

I

2

0.53

,

0.45

4

6

8

HWOd FIG. 16. The ratio F(H)/Tr(O)as a function of H. F ( H ) is the temperature at which the ZFC susceptibility exhibits a maximum at a given H , with F(0) being the freezing temperature obtained by extrapolating to zero magnetic field. [After Oseroff (1982).]

and Ayadi et al. (1986). They found a small shift of the maximum to higher temperatures when the frequency of measurement was increased. A similar behavior was also observed in metallic and other insulating SG systems (see, e.g., Mulder et al., 1981a,b and Goldfarb and Patton, 1981).

b. Analysis (c) Low Mn Concentrations The magnetic susceptibility for x < 0.001 and temperatures above 1 K follow the Curie-Weiss law given by Eq. (13), with a Curie-Weiss temperature @ 6 1 K (Deigen et al., 1967; Andrianov et al., 1976; Oseroff, 1984). The diagmagnetic susceptibility contribution of the lattice, Xd , is significant for this range of concentration. Values for the diamagnetic susceptibility for 11-VI compounds are collected in Table VIII. Magnetization measurements performed down to 10 mK for CdTe with 0.05% Mn by Novak et al. (1981) showed a departure from the Curie-Weiss law below 0.1 K. At these low values of x, it seems unlikely that this departure results from the presence of finite clusters, like pairs, triples, etc. That would require significant long range interactions, i.e., cluster formation involving Mn2+ ions that are separated by more than twenty coordination spheres. The deviation is possibly due to the presence of hyperfine interaction and the existence of a small but non-zero crystal field splitting.

98

SAUL OSEROFF AND PIETER H. KEESOM

TABLE VIII DIAMAGNETIC SUSCEPTIBILITY FOR ~

THE ~

11-VI HOSTSOF THE DMS _ _ _ _ _ _ _ _ _ _

Compound

- ~ ~ ( 1 0emu/g) -~

Reference

CdTe

3.5 3.1 3.3 3.3 3.8 3.5 3.6 3.2 3.3 3.6 2.3 2.1 2.3

Ivanov-Omskii et al. (1972a) Candea el al. (1978) Aresti et al. (1979) Singh and Singh (1980) Aresti et al. (1979) Singh and Singh (1980) Singh and Singh (1980) Singh and Singh (1980) Singh and Singh (1980) Ivanov-Omskii et al. (1972b) Singh and Singh (1980) Singh and Singh (1980) Singh and Singh (1980)

CdSe CdS ZnTe ZnSe ZnS HgTe HgSe HgS

The high temperature susceptibility for x 2 0.01 has been investigated by several authors. For Hgl-,MnxTe, it was studied by Savage et al. (1973); Andrianov et al. (1976); Davydov et al. (1980); and Nagata et al. (1980). From the high temperature data, T > IOOK, they concluded that 0 is negative for all values of concentration, i.e., that the exchange interaction between the Mn2+ions is antiferromagnetic. Similar results were reported by Pajaczkowska and Pauthenet (1970) for Hgl-,Mn,Se and Hgt-,Mn,S, and by Heidrich (1981) and McAlister et al. (1984) for Znl-,MnxTe. Oseroff (1982) studied Cdl-,Mn,Te and Cdl-,Mn,Se and reported positive values of 0 for x < 0.04, and negative values for larger concentrations. However, more recently, Oseroff (1984) measured Cdl-,Mn,Te for x = 0.005, 0.01, and 0.05 and deduced negative values for 0:- 1 K, - 3 K, and -22 K respectively and 0 = -26K for Cdo.wMno.osSe. The disagreement was caused by the use of smaller and incorrect values of Xd in the earlier analysis (Oseroff, 1982). The later results (Oseroff, 1984) lead to the general conclusion that the Curie-Weiss temperature obtained from the high temperature data is negative (i.e., indicative of antiferromagnetic interactions) for all values of concentration in the DMS. (We remark parenthetically that positive values of 0 were reported for Hgl-,Mn,Te with x < 0.07 by Sondermann and Vogt (1977a,b) and Sondermann (1979); for Hgl-,MnxSe with x < 0.03 by Sondermann (1980); and for Cdl-,Mn,Te with x < 0.04 by Sondermann (1976). Those authors concluded that the exchange coupling for high temperature is ferromagnetic for small concentrations of Mn2+ in DMS. However, all recent high temperature data on DMS consistently indicate that the Curie-Weiss temperature is indeed

3.

MAGNETIC PROPERTIES : MACROSCOPIC STUDIES

99

negative. It is likely that the use of an erroneous value of Xd may have been responsible for the early interpretation of Sondermann and Vogt.) From the Curie-Weiss temperature, it is possible to obtain an estimate for the exchange interaction J. For antiferromagnets and ferromagnets, @ =

2S(S + l)x 3kB

1K ZKJK,

where S is the spin of Mn2+and ZK the number of K-th nearest neighbors of a given atom. For the case of an fcc or hcp lattice with only nearest-neighbor interactions z1 = 12. With S = 5, Eq. (14) is then further reduced to 0 = 70XJdk~.Values for the exchange interaction between nearest-neighbors JI obtained from different sources, aregiven in Table IX. The exchangeconstant JI was recently determined in a very direct experiment, where steps on the magnetization vs. external magnetic field curve have been observed (Aggarwal etaf., 1984, Shapiraetaf., 1984, Aggarwal etaf., 1985). Parenthetically, these experiments confirmed also that the distribution of Mn2+ ions in the host lattice is statistical. We shall return to this problem later in this section. As seen in Figs. 7 to 9, for temperatures below 100 K and above the spinglass temperature, the curves x - l vs. Tshow a continuous down-turn toward the origin. The temperature at which these deviations take place increases with the concentration. For low concentration, x rs 0.05, this magnetic behavior can be analyzed in terms of a spin-cluster model (see, for example, Kreitman and Barnett, 1965; Simpson, 1970; Nagata et af., 1980; Galazka et af.,1980; Okada, 1980; Shapira et af., 1984). For higher values of concentration, the contribution of clusters larger than triples becomes important and a quantitative analysis on the basis of the model is complicated because of mathematical difficulties. The principal assumptions made in the cluster model are, first, that the Mn ions are randomly distributed in the fcc or hcp sublattice, which allows one to calculate the probabilities for the occurrence of clusters of various sizes and types for each crystal lattice (see, for example, Kreitman and Barnett, 1965; Yamaguchi and Sakamoto, 1969; Okada, 1980). We tabulate these probabilities for different clusters in Tables I, 11, and 111. Second, the cluster model assumes that the exchange interaction between the Mn ions is given by a Heisenberg Hamiltonian, X = - 2JSiSj. Analytic expressions for the Hamiltonians of the clusters and the eigenvalues for the exchange energy are given in Table IV and discussed in detail by Brumageetaf. (1964), Kreitmanetaf. (1966), Nagataetaf. (1980), andOkada (1980). Finally, the cluster model neglects interactions between the clusters. Thus one can assume that the total susceptibility of the system is the sum of the susceptibilities of the individual clusters. The total susceptibility is then given by Eq. (4).

-

100

SAUL OSEROFF AND PIETER H . KEESOM

TABLE IX NEAREST-NEIGHBOR EXCHANGE CONSTANT FOR THE DMS: (*) DETERMINED BY USINGTHE VALUEOF THE FIRSTSTEPOBSERVED IN THE MAGNETIZATION DATA IN THE REFERENCE QUOTED,(**) DETERMINED BY USINGTHE CURIE-WEISS e REPORTEDIN THE REFERENCE QUOTED. TEMPERATURES Compound CdMnTe CdMnTe CdMnTe CdMnTe CdMnTe CdMnSe CdMnSe CdMnS CdMnS ZnMnTe ZnMnTe ZnMnTe ZnMnSe ZnMnS HgMnTe HgMnTe HgMnTe HgMnTe HgMnSe HgMnS

- Ji/ke (K)

Reference

-7

Gaj et al. (1979) Ching and Huber (1984) Giebultowicz et a/. (1985) Oseroff (1984) Aggarwal et al. (1985) Shapira et af. (1984) Oseroff (1984) Kreitrnan e t a / . (1966) Nawrocki er al. (1984) McAlister el a/. (1984) Twardowski et nf. (1984a) Shapira (1985) Shapira et al. (1984) Brumage et af. (1964) Davydov et al. (1980) Savage er of. (1973) Nagata et ul. (1980) Jaczynski et af. (1978) Pajaczkowska and Pauthenet (1979) Pajaczkowska and Pauthenet (1979)

-6

-7.5 -1 7.7 5 0.3* 8.3 5 0.7* -8 -4 -8 15** -9 - 10* 13* 13 16 15 -7** -8 8** 13**

-

-

-

Brumage et al. (1964), Savage et al. (1973), Nagata et af. (1980), Galazka et al. (1980), Oseroff (1982), and Shapira et af. (1984) analyzed their susceptibility data for low concentration and low temperature using a cluster model with only nearest-neighbor (nn) interactions. This assumption was made first by Brumage et a/. (1964) because the next nearest-neighbor (nnn) interaction between Mn ions was believed to be much smaller than the nn interaction. This assumption is supported by Kreitman et al. (1966), Bastard and Lewiner (1979b), Davydov (1980), Escorne and Mauger (1982), and Keesom (1983, who concluded that the interaction between nearest neighbors in DMS is about one order of magnitude larger than the interactions between more distant Mn ions. While it is difficult to calculate the interaction for large (realistic) clusters, valuable, qualitative insights into the mechanisms governing inter-ion interactions, and thus the overall behavior of the magnetic susceptibility of DMS, can be obtained from calculations involving pairs, triples, etc., such as those described above.

3.

MAGNETIC PROPERTIES: MACROSCOPIC STUDIES

101

(d) Distribution of Mn in the Host Lattice The nature of the distribution of Mn2+ ions throughout the lattice is important for quantitative understanding of magnetic properties of DMS. Nagata et al. (1980), Galazka et a f . (1980), and Oseroff (1982) fit their susceptibilitydata, using for the exchange integral, J I / ~ Bthe, value obtained from specific heat, -0.7K for Hgl-,Mn,Te (Nagata el al., 1980) and -0.55K for Cdl-,Mn,Te (Galazka et al., 1980). They found that the calculated susceptibility differed substantially from the data when a random distribution with only nn interactions was assumed. To obtain a better fit, they proposed that the Mn ions were not randomly distributed, resulting in an increase of finite clusters at the expense of the isolated Mn ions. However, recently, Shapira et al. (1984) and Aggarwal et a f . (1984, 1985) showed that the Mn ions are randomly distributed in the host, and the reason for the disagreement with the previous authors is that the magnitude of J l / k ~ for the DMS is about - 10 K, instead of less than - 1 K used by Galazka et al. (1980). It is known from the energy level scheme that for J c 0 and for zero magnetic field the ground state for a pair is non-magnetic, i.e., its total spin STis zero. For closed and open triplets the ground state, under similar conditions, have a total spin ST = 5 and f respectively. The magnetization, for T < 4 K and H < 100 kOe, obeys the following phenomenological equation (Gaj et al., 1979; Heiman et al., 1983, 1984) where B5/2 is the Brillouin function for S = 5, and M S and % are phenomenological parameters. The saturation magnetization MS, is lower than the saturation value MOfor a concentration x of free Mn ions with S = f and g = 2. Shapira et al. (1984) defined an effective concentration R by the relation X/x = Ms/Mo, where R is the concentration that would have given rise to a magnetization MS if the Mn spins were fully aligned. Knowing the ground states for the pairs and triples, Shapira et al. (1984) found that for H < 100 kOe and T 2 K, the experimental ratio .Wx for x < 0.05 is accounted well by a random distribution of Mn ions with only nn interactions, and can be written as:

where 9 ,P3, and P4 are the probabilities for singles, open triplets and closed triplets. When the concentration of Mn increases above = 0.05, larger clusters than triplets should be included. Kreitman et al. (1966) suggested that an upper limit for the contribution of these larger clusters can be estimated by assuming that clusters of four or more spins will be replaced by nn

102

SAUL OSEROFF AND PIETER H. KEESOM

o.8ro\... y.... \I Y '. ...

--..+

I2

..._.

..._.. '.....&

0 CdMnTe

0 CdMnSe

0 CdMnS

A ZnMnSe

FIG. 17. Comparison between experimental results and theoretical predictions for R/x in various materials; (0)Gaj et al. (1979); (0)Heiman et al. (1984); (0)Heiman et al. (1983), Nawrocki et al. (1984); (+) Twardowski et a/. (1984a); (A) Heiman et al. (to be published); Twardowski et al. (1984b); (V) Shapira and Foner (unpublished). The solid line represents Eq. (16), which ignores clusters larger than triples. The dashed line represents Eq. (17), which includes an estimate for these larger clusters. [After Shapira et a/. (1984).]

quintuplets with a ground state ST = 5. Then Eq. (16) transforms into:

x

- = PI + 4 1 3 X

+ P4/15 + (1 - PI - P2 - P3 - P4)/5,

(17)

where PZis the probability for nn pairs, etc. In Fig. 17, the ratio .Vx vs. x and its fit to Eqs. (16) and (17) are given for several DMS. The statistical nature of Mn distribution was also confirmed by Barilero et al. (1986) in Zn -,MnnTe. Even though Shapira et al. (1984) and Aggarwal et al. (1984, 1985) were able to fit the magnetization data by taking into account only interactions between nearest-neighbors, there is experimental evidence that for temperatures below 2 K the interactions between farther Mn neighbors become noticeable (see, e.g., Larson et al., 1986). When long range interaction is assumed, the problem becomes difficult to handle because of the mathematical complications involved. In that case, only the number of isolated Mn ions can be easily estimated from Ns = Nx(1 - x)",

(18)

where Ns is the number of isolated ions, N i s the total number of cations,

3.

MAGNETIC PROPERTIES: MACROSCOPIC STUDIES

103

x is the concentration, and n is the number of cation sites with which the

interaction takes place. Several authors reported data that support the existence of interactions between remote Mn ions. Assuming a random distribution, Kreitman et af. (1966) found that the best fit for the susceptibility data of Mn2+in CdS was obtained when one included interactions up to fourth-neighbor. Davydov et al. (1980), using Eq. (18), concluded that the number of isolated Mn ions necessary to fit the low temperature susceptibility data for HgTe with x = 0.01 1 was such that the interaction between the Mn ions needed to be extended to at least seven spheres of coordination. Bastard and Lewiner (1980) reproduced qualitatively the susceptibility data for Hgl -,MnxTe obtained by Andrianov et af.(1976) for low values of concentration and low and intermediate temperatures, by using a two spin-cluster model in which the interactions between first and second neighboring pairs of spins were treated exactly, while the interactions between these clusters and the remaining spins were accounted for by using the molecular field approximation. Novaketaf. (1981,1984,1985) measured Cd~-,Mn,TeandCd~-,Mn,Se with 0.005 5 x s 0.15 down to 10 mK. For x z 0.01, they found that, if a random distribution is assumed, the number of cation sites n,estimated using Eq. (18), is about 60. The presence of a spin-glass behavior for concentrations well below the nn percolation limit supports the existence of interaction between Mn ions beyond the nn interactions. (e) High Concentrations of Mn All zero field cooled (ZFC) susceptibility data for T L 1.3 K and x z 0.20 showed a maximum in the susceptibility at a temperature Tf that depended on the concentration and the intensity of the magnetic field. The sharpness of the cusp for small magnetic fields, H s 50 Oe, was not the same for all the DMS. This fact may be related to the homogeneity of the samples, as small fluctuations in the composition can result in a broadening of the cusp. In Fig. 18 magnetic phase diagrams for several DMS are given. If a linear extrapolation of Tf is made from the high concentration region, it intersects the abscissa at x = 0.17, a value close to the percolation critical value calculated for nn. The agreement between this lower limit and the critical concentration calculated for nearest neighbors is not surprising, since it was obtained by extrapolating from temperatures of the order of the exchange interaction between nn, JJnn/kBI = 10K, which considerably exceed the values expected for the exchange interaction between farther neighbors, IJfn/kBI s 1 K, c.f., Kreitman et af. (1966), Davydov et a f . (1980), Escorne and Mauger (1982), Akbarzadeh and Keesom (1985). Thus, for T > 2 K, DMS behave approximately according to the original suggestion of DeSeze (1 977) regarding the existence of insulating spin-glasses with only antiferromagnetic interactions.

104

2l ;$$

SAUL OSEROFF AND PIETER H. KEESOM I

Cd,-,Mn,Te

10

0

Zn

-Y,

20-

t=

10-

P

0

/

/d

40

M n SeB’

-

/

/

20

f

/

9’

2ol

/

20 40 x(at.96)

rZn,-,Mn,S

-

10

SG

//

SG 0

20 40 x( at .%)

1 0 ° L x(at.%)

FIG. 18. Paramagnetic (P)-spin glass (SG) phase diagram, Tr vs. x, for several DMS. from Galazka et al. (1980), (x) from Oseroff (1982) and (0)from Escorne Cdl-,Mn,Te (0) and Mauger (1982); Cdl-,Mn,Se (+) from Oseroff (1982) and (0)from Amarasekara et al. (1983); Cdl-,Mn,S (0)from Yang et al. (1983); Hgl-,Mn,Se (0)from Khattak et al. (1981); Znl-,Mn,Se (0)from Akbarzadeh et al. (1983); Znl-,Mn,Te (0)from McAlister et al. (1984); Znl-xMn,S (0)Yang et al. (1983).

As mentioned in the previous subsection, Novak et al. (1981, 1984, 1985) measured the field cooled susceptibility for Cdl-,Mn,Te and Cdl-,Mn,Se for 0.005 5 x 5 0.15, and found that the field cooled susceptibility shows a distinct kink, or knee for these low concentrated samples, as illustrated in Fig. 14. Brandt et al. (1981, 1982, 1983) measured the susceptibility for Hgl-,Mn,Te with 0.02 s x s 0.075 down to T = 0.04 K and observed a definite cusp in the susceptibility, as shown in Fig. 13. These results appear to indicate that for temperatures such that IJfn/k~Iz T the interaction between farther Mn ions becomes increasingly important. If the data reported by these last authors are indeed associated with a spin-glass transition, the phase diagrams for Cdl-,Mn,Te, Cdl-,Mn,Se, and Hgl-,Mn,Te can be extended below x = 0.17 for low temperatures. Such diagrams are shown in Figs. 19-21. For x z 0.17, the mechanism suggested t o account for the SG transition is a short range mechanism (probably superexchange: see Spalek et al., 1985 and Larson et al., 1985), where the main contribution comes from the

3.

MAGNETIC PROPERTIES: MACROSCOPIC STUDIES

105

25 Te

Cd,-,Mn,

20 -

15-

5 c-10-

5-

0

0.40

0.20

0.60

X

FIG. 19. Paramagnetic (P)-spin glass (SG) phase diagram, f i vs. x, for Cdl-,Mn,Te with 0.05 < x < 0.60. The solid line is a guide for the eyes. The values of f i were obtained from data reported by Novak et al. (1984), Oseroff (1982), Galazka et al. (1980), and Oseroff and Gandra (1985).

antiferromagnetic interaction between nearest-neighbors. If interactions with farther neighbors are present, the concentration at which the mean cluster becomes infinite is highly reduced. When interactions with 2nd or 3rd neighbors are included in an fcc lattice, the values calculated for x, are reduced from 0.195 to 0.136 and 0.061, respectively (Domb and Dalton, 1966). Under these conditions, it appears reasonable to expect a spin-glass behavior for concentrations below = 0.17. Another exchange mechanism besides the short-range mechanism is also possible in the open-gap semiconductors (Bloembergen and Rowland, 1955). In this mechanism, the localized spins interact via virtual excitation of an electron from the filled valence band to the empty conduction band. Bloembergen and Rowland have shown that the effectiveness of this interaction decreases exponentially with the inter-spin distance and with the square root of the energy of the forbidden gap. Bastard and Lewiner (1979a,b), Lewiner et a!. (1980), and Ginter et al. (1979) studied the BloembergenRowland mechanism for symmetry-induced zero-gap semiconductors,

106

SAUL OSEROFF AND PIETER H. KEESOM

X

FIG. 20. Paramagnetic (P)-spin glass (SG) phase diagram, for Cdl-,Mn,Se with 0.05 < x < 0.50. The solid line is a guide for the eyes. The values of Tr were obtained by Novak e f ai. (1985), Oseroff (1982), and Amarasekara e f a/. (1983).

251

I

I

I

X

FIG.21. Paramagnetic (P)-spin glass (SG)phase diagram, for Hgl -,Mn,Te. (A) from Brandt e t a / . (1983), (x) from Mycielski et a/. (1984) and Otto e t a / . (1980), a.c. magnetic susceptibility data; (0)from Mycielski et a/. (1984), Faraday rotation data. [After Mycielski ef al. (1984).]

3.

MAGNETIC PROPERTIES: MACROSCOPIC STUDIES

107

especially the case of Hgl-,MnxTe at low Mn concentration, and found that it gave rise to an indirect long-ranged exchange interaction of antiferromagnetic sign. Because of the exponential damping, this long-range indirect interaction can be expected to be small in wide-gap semiconductors. For x < 0.20, the temperatures of the cusp for narrow-gap Hgl-,Mn,Te are systematically larger than for Cdl-,Mn,Te or Cdl-,Mn,Se. This may be understood as follows: for Hgl-,Mn,Te, both mechanisms, the Bloembergen-Rowland and the short-range cluster mechanism, contribute. Instead, for the wide gap DMS, only short-range mechanism will result in a significant contribution. 3. SPECIFIC HEAT a. Experimental Results

The specific heat, C , of DMS has been measured in the temperature range between 0.3 K and 50 K and in magnetic fields up to 30 kOe (Nagata et al., 1980, 1981; Galazka et al., 1980). In the case of the pure 11-VI host (i.e., x = 0), only the lattice vibrations contribute to the specific heat, and an extrapolation of the low temperature specific heat to T = 0 determines the Debye temperature. Values of the Debye temperature for different 11-VI systems are collected in Table X. TABLE X VALUESOF THE DEBYE TEMPERATURE FOR

Zn Cd

Hg

DIFFERENT DMS (IN KELVINS) S

Se

Te

260

223

202

174

158 140

1i a 144

144

The low temperature specific heat in DMS is substantially greater than that in non-DMS. The data for Cdl-,Mn,Te at zero magnetic field are given in Fig. 22 as an example. The excess specific heat (Cex= C - CII-”I)has its origin mainly in the magnetic properties of the Mn ions. Small changes in the lattice vibrations spectrum can be also expected, but with the lattice contribution being very small below 2 K (as compared to C,), these changes can be safely neglected. As shown in Fig. 23, for Mn concentrations greater than x = 0.2, C e x behaves very similarly to the magnetic contribution found in metallic spinglasses (Wenger and Keesom, 1975, 1976); that is, C e x is linear in T i n the

108

SAUL OSEROFF AND PIETER H . KEESOM

0.1 0.5 1

3 5 10 30 T( K)

FIG. 22. Specific heat vs. Tfor Cdl-,Mn,Te at H

0.25

/.

-0

a20kG 028kG

a',

=

0. [After Galazka el a/. (1980).]

d

A'

FIG.23. Specific heat for Cdl-,Mn,Te, x = 0.2, 0.3, and 0.5, and pure CdTe. Dashed lines show the linear behavior of C vs. T for x = 0.2 and 0.3. [After Galazka et a/. (1980).]

3.

MAGNETIC PROPERTIES: MACROSCOPIC STUDIES

109

region below and slightly above the spin-glass freezing temperature T f . No anomalies are detected in the vicinity of Tf . In addition, only a weak magnetic field dependence is observed for these samples. Figure 24 shows an additional feature observed in Cdo.sMn0.2Se:Cex is history dependent (Amarasekara et al., 1983). The value of Cexmeasured in zero magnetic field depends on whether the sample was cooled in the absence of the magnetic field, or cooled in a magnetic field and then demagnetized at the lowest temperature. The lower inset of Fig. 24 shows two such sets of results. For very high Mn concentration, Cexshows a broad maximum. For example, for x = 0.70, such a maximum occurs around 36 K. The appearance of such a maximum is interpreted as a paramagnet-antiferromagnetic phase transition. Theantiferromagnetism which makes itself evident through the Cex maximum may be associated with antiferromagnetically ordered clusters. Finally, at the lowest temperatures, a T-' term can be distinguished in the samples with x > 0.3. It is magnetic field independent and corresponds to a nuclear specific heat. This arises when the 3d electrons polarize the 1s core electrons, that in turn produce a field of = 400 kOe at the Mn nuclei. I

Cd,-,MnxSe

0.3-

++

X=O.2O 9 H=O aH=O(demagnetized) f ' + H=POkG +

+

.

-

. .

+A

+-

f

T( K) FIG.24. Excess specific heat C,, vs. T for Cdo.8oMno.zoSe in fields of 0 and 20 kOe. Also shown are measurements in zero field after demagnetization from 20 kOe. The insert shows the zero-field data at low temperature. [After Arnarasekara et af. (1983).]

110

SAUL OSEROFF AND PIETER H. KEESOM

FIG.25. Excess specific heat C,, vs. Tfor Zno.99Mn0.01Se in H = 0, 10 kOe, and 20 kOe. The solid lines indicate the calculated values for Cex.[After Keeson (1986).]

As an example of the excess specific heat for low concentration samples (x < 0. lo), the results for Zn0.99Mn0.01Seareshown in Fig. 25 (Keesom, 1986). In zero magnetic field, C,, decreases monotonically with T ( T 2 0.3 K). In magnetic fields H = 10 kOe and 20 kOe, the excess specific heat Cexshows a pronounced maximum at about 1 . 1 K and 2.2 K respectively.

b. Analysis ( f ) Low Mn Concentrations

As mentioned above, the spin-glass paramagnetic phase boundary originally indicated the concentration of x = 0.17 as a lower limit for the existence of the spin-glass phase (Nagata et al., 1980; Galazka et af., 1980; Oseroff, 1982; McAlister et al., 1984). This is close to the critical concentration for the fcc (= 0.195) and hcp (= 0.204) lattices when calculated for the nearest-neighbors interactions. Therefore, in various attempts to understand Cex, only nn interactions were assumed. In one approach, it was assumed that the presence of large concentrations of Mn would result in small distortions of the lattice which would greatly increase the crystal field splitting (Amarasekara et al., 1983). However, recent EPR measurements by Oseroff (1984) and Koh et al. (1984) showed that this approach is incorrect and that crystal splitting is within the experimental error, independent of

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MAGNETIC PROPERTIES: MACROSCOPIC STUDIES

111

the concentration of Mn. Also, as mentioned before, in an attempt to fit the < 1 K, a non-random distribution of Mn ions data with a value of (J,,n/k~( was assumed (Nagata et al., 1980; Galazka et al., 1980). It was shown, however, by Shapira et al. (1984) that the exchange integral for the DMS is Jnn/kB = -lOK, and that the data can be fitted satisfactorily with a completely random distribution. It may be concluded, therefore, that the ground state splitting of cluster systems involving only nn is so large that they cannot contribute to the excess specific heat Cexbelow 3 K for fields H < 100 kOe. When the interacting ions are farther apart than the nearest neighbor distance, a spectrum of different values of the interaction energy has to be used. This, together with the different types of clusters, pairs, triples, etc., makes it nearly impossible to do a meaningful quantitative analysis of the excess specific heat, and a qualitative approach becomes necessary. The value of C,, decreases monotonically above 0.3 K. The value at 0.3 K allows us then to deduce the lower limit of the number of ions forming of a cluster. Moreoever, the maximum of C,, at a given field allows us to estimate the number of single ions present in these samples. In Fig. 25 are shown the experimental data for Zn0.99Mno.olSe, along with the values of C e x calculated using these estimates. The agreement is quite good. The value of the interaction energy is about 0.2 K, much smaller than that expected for the nearest-neighbor interaction, and in agreement with the values estimated by Kreitman et al. (1966). The probability of an ion to be completely isolated (a “single”) is given by Eq. (18). For the Zn and Cd-based DMS an estimate of n gives about 50, while for the Hg-systems n is about 100. While the two former compounds are wide-gap semiconductors, the Hg compound in these low concentration ranges is a zero-gap semiconductor. This fact may be responsible for the difference in the values of n. The values of n estimated from the low temperature susceptibility (Kreitman et al., 1966; Davydov et al., 1980; Novak et al., 1981, 1984, 1985) and specific heat agree very well. (g) High Mn Concentrations For x > 0.2, Cexshows a linear dependence on Tin the region T 5 r f ,and shows no anomaly around Tf (Galazka et al., 1980; Nagata et al., 1980; Amarasekara et al., 1983). A similar behavior was found in metallic spinglasses (Wenger and Keesom, 1975, 1976; Nagata et al., 1979). It was found for many DMS systems that the field dependence of C,, decreases as the concentration x increases. This can be understood by assuming the presence of loosely bound spins, that contribute to C e x when they break away from Mn clusters and whose number decreases as the concentration is increased. For Cdo.~1Mno.20Se, Amarasekara et al. (1983) observed that C e x measured at low temperature and in zero external field depended on how the sample

112

SAUL OSEROFF AND PIETER H. KEESOM

was cooled. That is, it depended on whether the cooling took place in zero field or whether it was done in the presence of a magnetic field and then, after the field was turned off, the actual measurement was performed. The differences between the two cases are shown in the inset of Fig. 24. The authors interpreted this effect as resulting from loose (uncoupled) spins that were bound to large clusters by the magnetic field and then remained frozen after the field was turned off. This suggestion is supported by the fact that a smaller effect was observed as x increased, until for x = 0.40 the effect was not observed at all, agreeing with an almost complete absence of loosely bound spins. 4. REMANENT MAGNETIZATION

One of the characteristic features of SG systems is the presence of a timedependent remanent magnetization ( M R)below a certain temperature TR (Tholence and Tournier, 1974; Guy, 1978; Gray, 1980; Oseroff et al., 1982). In metallic SG, TRwas found to be within a few percent of the temperature Tf ,where a cusp in the low field susceptibility is observed (Chamberlin et al., 1981; Oseroff et al., 1982, 1983). The value of MR depends on the history of the sample. If the sample is cooled from T > TR to Tz < TR in the presence of an external magnetic field, and the remanence is measured after the magnetic field is turned off, the M R is known as the thermoremanent magnetization (TRM). On the other hand, if the sample is cooled from 5 > TRto Tz < TRin zero field, and a field is turned on and after a certain time turned off again, then the MR obtained is known as the isothermal remanent magnetization (IRM) (see Tholence and Tournier, 1974). The remanent magnetization M Rin DMS was measured as a function of time t , magnetic field, temperature and concentration of Mn ions by two different techniques. One set of data was obtained by the EPR method, where the magnetization M and the EPR spectrum were measured simultaneously by observing the EPR spectra of two thin p-doped silicon spin EPR markers glued to the sample (see Schultz et al., 1980 and Oseroff, 1982). In Fig. 26 the increase and decay of TRM and IRM with time, obtained by the EPR method, is given for Cdo.6oMno.~oSe.The other set of data for MRin DMS was obtained by using conventional magnetometers. While there remains no doubt about the existence of remanence phenomena in DMS the results of M R reported by different authors are in considerable, quantitative disagreement. The M R data reported by Oseroff (1982) and Oseroff and Gandra (1985) show behavior similar to that observed in other metallic or non-metallic spin-glasses. However, the MR data reported by Escorne et al. (1981) and Escorne and Mauger (1982) suggest that the freezing process of the Mn spins in DMS can be analyzed in terms of

3.

113

MAGNETIC PROPERTIES: MACROSCOPIC STUDIES

h

rr, (u

Y

0.3

0 L N

I

I

d

d

0.2 I

I

?

0 L

r

I

0.1 d

I 2

1

C I 0;

10

/Y

L _ _ L

25 50 100

---1 I

500 1000

5000

t(s)

FIG. 26. EPR measurements of the increase (i) and the decreae (d) of TRM and IRM as a function of time for Cdo.aoMno.40Seat 4.2 K, normalized to the difference in linewidth between H F C obtained , after FC in the resonance field and HZFC(25), measured after ZFC with the field on for 25 seconds. [After Oseroff (1982).]

clustering, supporting the idea that the DMS behave like mictomagnetic compounds rather than spin glasses. Figure 27 shows the time dependence of TRM for Cdo.60Mn~.4~Te. The evolution with time of the ZFC susceptibility for Cdo.60Mno.40Teis given in Fig, 28. The temperature dependence of TRM in Cd, -,MnxTe is given in Fig. 29. This figure shows that MRvanishes at TRwhich is close to r f . The results shown in Figs. 27 and 29 are in contradiction with the temperature dependence of M R reported for Cdl -xMnxTe by Escorne and Mauger (1982). They observed the remanence well above T f and found that MR did not depend on time on the scale of one-half hour. Also, Galazka et al. (1980) did not detect, within spans of 30 min, any time dependences for the zero-field-cooled magnetization. Possible explanations for this disagreement will be discussed later in this section. As mentioned above, the remanent magnetization depends also on the strength of the magnetic field applied. An example of such dependence is presented in Fig. 30 for Cd0.6Mn0.4Te. The study of the remanent magnetization MRmay provide further insights into whether the ZFC cusp observed in DMS should be associated with a SG transition or whether their behavior is rather more typical of mictomagnets, where the cusp results from cluster freezing. As mentioned above, only few measurements of M R in DMS were reported so far, and the data disagreed. Oseroff (1982) and Oseroff and Gandra (1985) fitted the time dependence of MR for Cdl-,Mn,Te and Cdl-,Mn,Se with a log(t) law, MR(t) = M R ( t o ) k s l o g t / t o ,

(19)

114

SAUL OSEROFF AND PIETER H. KEESOM

0.7 6 -

I

cdO.60Mn0.40Te 0)

3 0.74-

-

9

z

$. 0.72-

z oc !-

&

0.70-

7

0.68

FIG.28. Time dependence of the ZFC susceptibility for Cdo.60Mno.40Te,measured in 18.5 Oe at 5.5 K . [After Oseroff and Gandra (1985).]

where to was the initial time and S the slope of the logarithmic decay. Figure 31 gives S vs. H in Cdo.soMn0.40Tefor the thermo-remanent magnetization (TRM) and the isothermal remanent magnetization (IRM). A similar behavior was observed in metallic and insulating SG (see Guy, 1978, Ferre et al., 1980, and Oseroff et al., 1982). Kinzel(l978) and Dasgupta et al. (1979) performed Monte Carlo calculations of a SG system consisting of an Ising model with random interactions

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MAGNETIC PROPERTIES: MACROSCOPIC STUDIES

115

T( K) FIG.29. TRM vs. Tfor Cdl-xMnxTe, obtained by FC to T = 1.8 K in 100 Oe and measured after the field was turned off for 3 minutes. The arrows indicate the maximum of ZFC susceptibility for the three concentrations measured. [After Oseroff and Gandra (1985).]

and approximated the decay of MR by a power law of the following form: MR(f) = M R ( ~ O ) ~ - ~ ,

(20)

where CY is an exponent to be determined from experimental data. Oseroff and Gandra (1985) attempted to describe their data using Eq. (20), and Fig. 32 gives CY vs. Nobtained in this manner for Cdo.60Mno.40Te. The observed dependence of (Y on magnetic field for TRM and IRM agreed well with the calculations of Kinzel (1978) and Dasgupta et al. (1979), and also with the results obtained for metallic spin glasses (Tovar et af., 1985). However, Escorne et af. (1981) and Escorne and Mauger (1982) measured MR for CdI-,MnxTe and reported that it did not depend on time on the scale of onehalf hour. Another conflicting point is that Oseroff (1982) and Oseroff and Gandra (1985) measured MRvs. Tand found that MRvanished at a temperature close to r f . Corresponding TRM vs. Tfor Cdl-,Mn,Te is given in Fig. 29, similar to the dependence found in canonical SG. Instead, Escorne et af. (1981) and Escorne and Mauger (1982) reported that M Rfor Cdo.,oMno.3oTe was almost

116

SAUL OSEROFF AND PIETER H. KEESOM I

I

I

,

I

I

Cd0.60Mn0,40Te

-

TRM + IRM

I

I

h

t

T=6K Cd0.60 Mn0.40Te

0.3

"0

10

20

30

H (kOe)

40

50

FIG. 31. Variation of the slope S for the log(t) decay of TRM (0)and IRM (0)for Cdo.soMno.aoTeas a function of magnetic field. The slope was obtained from the change of M R between 1 0 0 s and 1ooOs. [After Oseroff and Gandra (1985).)

3.

MAGNETIC PROPERTIES: MACROSCOPIC STUDIES 1

I

I

I

I

117

I

CdO 60M "0 40Te

T = 6K

.I

*

u

' 0

10

20

30

40

50

H (kOe) FIG. 32. Field dependence of the exponent a of the power law M ( t ) = M(t,)t-" for Cdo.mMno.4oTemeasured at T = 6 K, (0)IRM data and (0)TRM data. [After Oseroff and Gandra (1985).]

temperature independent below T f and did not disappear at T f , but a measurable remanence persisted up to 30 K, well above T f . So, on the one hand, the MR data reported by Oseroff (1982) and Oseroff and Gandra (1985) supports the idea that DMS behave like spin glasses. But, on the other hand, Escorne et al. (1981) and Escorne and Mauger (1982) data suggest that the whole freezing process of the Mn spins can be analyzed in terms of clustering, and that the DMS behave as mictomagnetic compounds and not as SG. Neglecting experimental errors in taking or analyzing the data, possible reasons for the differences can be: the use of inhomogeneous samples by Escorne et al. (1981) and Escorne and Mauger (1982), or that the data was taken under different physical conditions, like cooling in different magnetic fields or measuring after different times once the magnetic field was turned off. This point, however, requires further clarification. IV. Concluding Remarks The bulk of the data concerning magnetic susceptibility, specific heat and EPR accumulated so far provides a fairly thorough understanding of magnetic properties of DMS. It is now quite firmly established that the distribution of Mn ions throughout the host semiconductor lattice obeys random statistics. Also, the values of nearest neighbor exchange constants J n n are known rather accurately. It is quite likely (although this fact requires further confirmation) that superexchange is the dominant mechanism

118

SAUL OSEROFF AND PIETER H. KEESOM

leading to the coupling of Mn magnetic moments. Correlations between values of Jnnand lattice parameters noted in various DMS (Spalek et al., 1986) are consistent with the above notion. The fact that DMS are available in rather wide ranges of Mn2+concentration enabled one to trace the development of specific features in various magnetic properties with increasing numbers of magnetic constituent incorporated in the host lattice. In particular, for dilute samples (x c 0.001), the data are quite well understood. In this range of concentrations, the changes of the values of the gyromagnetic factor, hyperfine coupling constant and crystal field parameters can be explained within one of the models proposed: distortions caused by the different ionic radii of Zn2+, Mn2+,Cd2+,and Hg2+;changes in the covalent character of the bonds for Te, Se, and S, and variations of the band structure. The high temperature susceptibility for these diluted samples showed a Curie-Weiss behavior with B < 1 K. For temperatures below 0.1 K, a departure from a Curie-Weiss law was found for CdI-,Mn,Te and Cdl-,Mn,Se for x as small 0.0005. This may be a consequence of the presence of hyperfine and crystal field splittings. As x increases, the interaction between the Mn becomes evident, a change in the shape of the individual resonances lines in the EPR spectra beginning to manifest itself for x > 0.002. The spectrum becomes a structureless single line with a narrower width for all the DMS with x = 0.02. Contradictory values of 0 for x < 0.05 reported on the basis of the high temperature susceptibility data seem to result from the use of inhomogenous samples or incorrect values for the host diamagnetic susceptibility. More recent studies indicate rather consistently that the exchange interaction between the Mn2+ ions is antiferromagnetic for all values of concentration. This conclusion is confirmed by a very direct determination of Jnn by Shapira et al. (1984) and Aggarwal et al. (1984, 1985). The low concentration data (x 5 0.1) can be analyzed in terms of a spincluster model. It is now clear that a good fit to the specific heat and susceptibility data can be obtained for x s 0.10 and T z 2 K by taking into account only the interactions between Mn nearest-neighbors, with JJn,/kb)= 1 0 K for all the DMS, and assuming a random distribution of Mn2+ ions in the lattice. For lower temperatures, interactions with farther neighbors (fn) should be included. Even accepting that the value of ( J f , , k ~5l 1 K, a systematic analysis of the data including farther-neighbors is missing in the literature. (Recently, an attempt to do such an analysis in the context of magnetization steps was done by Larson et af., 1985.) The broadening and shift observed for the EPR line with x > 0.03 and decreasing temperature can be understood qualitatively by an increase of an internal field due to the presence of finite clusters. However, a quantitative theory of these phenomena is still lacking.

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Above x k 0.17 with T 2 1.3 K the EPR, susceptibility and specific heat data show several features that can be associated with spin glass transition. All DMS show a cusp for the ZFC susceptibility measured at low field. The temperature dependence of the maximum with magnetic field is similar to other systems exhibiting a spin glass behavior. The specific heat shows a linear temperature dependence for T < Tf and no anomaly around 3.For x < 0.17, the low temperature magnetization results appear to indicate a similar behavior to those observed in the high concentration region. However, at this point there is not enough data to assure that DMS show a spin-glass behavior in the whole range of concentrations in which these materials exist. Conspicuous in this field is our lack of understanding of the phenomena related to remanent magnetization in the spin glass phase. Also, the question of the nature of the antiferromagnetic ordering in very concentrated samples (x 5: 0.60) requires still further investigation (for discussion of this problem, see the contribution by Giebultowicz and Holden in this volume). It is to be hoped that from the current intense activity a deeper and more thorough understanding of magnetic properties of DMS and of mechanisms underlying these will shortly emerge.

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Bhagat, S. M., Spano, M. L., and Lloyd, J. N. (1981). Solid State Commun.38, 261. Bloembergen, N., and Rowland, T. J. (1955). Phys. Rev. 97, 1679. Brandt, N. B., Moshchalkov, V. V., Skrbek, L., Taldenkov, A. N., and Chudinov, M. (1981). JETP Letr. 35, 401. (Pis’ma v Zh. Eksper. Teor. Fiz. 35, 326). Brandt, N. B., Tsidil’kovskii, I. M., Moshchalkov, V. V., Ponikarov, B. B., and Skrbek, 1. (1982). Sov. J. Low. Temp. Phys. 6. (Fiz. Nizkikh Temper 6). Brandt, N. B., Moshchalkov, V. V., Orlov, A. O., Skrbek, L., Tsidil’kovskii, I. M., and Chudinov, S. M. (1983). Sov. Phys. JETP57, 614. (Zh. Eksper. Teor. Fiz. 84, 1050). Brumage, W. H., Yarger, C. R., and Lin, C. C. (1964). Phys. Rev. 133, A765. Candea, R. M., Hudgens, S. J., and Kastner, M. (1978). Phys. Rev. Bl8, 2733. Cannella, V., and Mydosh, J. A. (1972). Phys. Rev. B6, 4220. Chamberlin, R. V., Hardiman, M., and Orbach, R. (1981). J. A p p l Phys. 52, 1771. Ching, W. Y., and Huber, D. L. (1984). Phys. Rev. B30, 179. Dasgupta, C., Ma, S., and Hu, C. (1979). Phys. Rev. 820, 3837. Davydov, A. B., Noskova, L. M., Ponikarov, B. B., and Ugodnikova, L. A. (1980). Sov. Phys. Semicond. 14, 869. (Fiz. Tekh. Poiuprov. 14, 1461). Deigen, M. F., Zevin, V. Ya., Maevskii, V. M., Potykevich, I. V., and Shanina, B. D. (1967). Sov. Phys. Solid State 9, 773. (Fiz. Tvevdogo Tela 9, 983). DeJonge, W. J. M., Otto, M., Denissen, C. J . M., Blom, F. A . P., van der Steen, C., and Kopinga, K. (1983). J. Magn. Magn. Mater. 31-34, 1373. De Seze, L. (1977). J. Phys. C10, L353. Domb, C., and Dalton, N. W. (1966). Proc. Phys. Soc (London) 89, 859. Dorain, P. B. (1958). Phys. Rev. 112, 1058. Dormann, E., and Jaccarino, V. (1974). Phys. Lett. A48, 81. Escorne, M., and Mauger, A. (1982). Phys. Rev. B25, 4674. Escorne, M., Mauger, A , , Triboulet, R., and Tholence, J . L. (1981). Physica 107B, 309. Estle, T. L., and Holton, W. C. (1966). Phys. Rev. 150, 159. Falkowski, K. (1967). Acta Phys. Polon. 32, 831. Ferre, J . , Pommier, J., Renard, J. P., and Knorr, K. (1980). J. Phys. C 13, 3697. Fidone, I . , and Stevens, K. W. H. (1959). Proc. Phys. SOC. (London) A73, 116. Fischer, K. H. (1976). Physica (Utrecht) 86-88, 813. Frisch, H. L., Sonenblick, E., Vyssotsky, V. A,, and Hammersley, J. M. (1961). Phys. Rev. 124, 1021. Furdyna, J. K. (1982). J. Appl. Phys. 53, 7637. Gabriel, J. R., Johnston, D. F., and Powell, M. D. J. (1961). Proc. Roy. SOC. (London) A264, 503. Gaj, J . A., Planel, R., and Fishman, G. (1979). Solid Stare Commun.29, 435. Galazka, R. R., Nagata, S., and Keesom, P. H. (1980). Phys. Rev. B22, 3344. Giebultowicz, T., Minor, W., Kepa, H., Ginter, J., and Galazka, R. R. (1982). Magn. Magn. 30, 215. Giebultowicz, T. M., Rhyne, J. J., Ching, W. Y., and Huber, D. L. (1985). To be published. Ginter, J., Kossut, and Swierkowski, L. (1979). Phys. Stat. Sol. (b) 96, 735. Goldfarb, R. B., and Patton, C. E. (1981). Phys. Rev. B24, 1360. Gray, E. M. (1980), J. Magn. Magn. Mater 15-18, 177. Grochulski, T., Leibler, K., Sienkiewicz, A., and Galazka, R. R. (1979). Phys. Stat. Sol. (b) 91, K73. Guy, C. N. (1978). J . Phys. F8, 1309. Hall, T. P. P., Hayes, W., Williams, F. I. B. (1961). Proc. Phys. Soc (London) A78, 883. Heidrich, H . (1981). Phys. St. Sol. (a) 67, 163. Heiman, D., Shapira, Y., and Foner, S. (1983). Solid Stare Commun. 45, 899.

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Heiman, D., Shapira, Y., Foner, S., Khazai, B., Kershaw, R., Dwight, K., and Wold, A. (1984). Phys. Rev. B29, 5634. Huber, D. L. (1972). Phys. Rev. B6, 3180. Ishikawa, Y. (1966). J. Phys. SOC. Japan 21, 1473. Ivanov-Omskii, V. I., Kolomiets, B. T., Ogorodnikov, V. K., Rud, Yu. V., and Tsmots, V. M. (1972a). Phys. Stat. Sol. (a) 13, 61. Ivanov-Omskii, V. I . , Kolomiets, B. T., Ogorodnikov, V. K., Smekalova, K. P., and Tsmots, V. M. (1972b). Phys. Stat. Sol. (a) 14, 51. Jaczynski, M., Kossut, J., and Galazka, R. R. (1978). Phys. Stat. Sol. (b) 88, 73. Keesom, P. H. (1985). Private communication. Keesom, P. H., (1986). Phys. Rev. B33, 6512. Khattak, G. K., Amarasekara, C. D., Nagata, S., Galazka, R. R., and Keesom, P. H. (1981). Phys. Rev. B23, 3553. Kikuchi, C., and Azarbayejani, G. H. (1962). J. Phys. SOC. Japan 17, Suppl. B-I, 453. Kinzel, W. (1979). Phys. Rev. B19, 4595. Kittel, C., and Abraham, E. (1953). Phys. Rev. 90, 238. Koh, A. K., Miller, D. J., and Grainger, C. T. (1984). Phys. Rev. B29, 4904. Kondo, J . (1965). Prog. Theor. Phys. 33, 575. Kreitman, M. M., and Barnett, D. L. (1965). J. Chem. Phys. 43, 364. Kreitman, M. M., Milford, F. J., Kenan, R. P., and Daunt, J. G. (1966). Phys. Rev. 144,367. Kremer, R. E., and Furdyna, J. K. (1983). J. Magn. Magn. 40, 185. Kremer, R. E., and Furdyna, J. K. (1985a). Phys. Rev. B31, 1 . Kremer, R. E., and Furdyna, J . K. (1985b). Phys. Rev. B32, 5591. Lambe, J., and Kikuchi, C. (1960). Phys. Rev. 119. Larson, B. E., Hass, K. C., and Ehrenreich, H. (1985). Solid State Commun. 56, 347. Larson, B. E., Hass, K. C., and Aggarwal, R. L. (1986), Phys. Rev. 833, 1789. Leibler, K., Giriat, W., Checinski, K., and Wilamowski, 2. (1970a). Proc. XVICong. Ampere, Bucuresti 1970, p. 993. Leibler, K., Giriat, W., Checinski, K., and Wilamowski, 2. (1970b). Proc. XVICong. Ampere, Bucuresti 1970, p. 987. Leibler, K., Giriat, W., Wilamowski, Z., and Iwanowski, R. (1971). Phys. Stat. Sol (b)47,405. Leibler, K., Giriat, W., Checinski, K., Wilamowski, 2.. and Iwanowski, R. (1973). Phys. Stat. Sol. (b) 55, 447. Leibler, K., Biernacki, S., Sienkiewicz, A., and Gaiazka, R. R. (1976). Proc. XIX Congresse Ampere, Heidelberg 1976. Leibler, K., Sienkiewicz, A., Checinski, K., Galazka, R., and Pajaczkowska, A. (1977). Proc. of the International Conf. on Phys. of Narrow Gap Semiconductors (Rauluszkiewicz, J., Kaczmarek, E., and Gorska, M., eds.) P W N Publishers, Warsaw 1978, p. 199. Lewicki, A., Spalek, J., Furdyna, J. K., and Galazka, R. R. (1985). ProceedingsInternat. Conf. Magnet., San Francisco. ( J . Magn. Magn. Muter. 54-57, 1221 (1986).) Lewiner, C., Gaj, J. A., and Bastard, G . (1980). J. Physique C5, 289. Liu, L. (1982). Phys. Rev. B26, 975 and 6300. Manoogian, A., Chan, B. W., Brun del Re, R., Donofrio, T., and Woolley, J. C. (1982). J. Appl. Phys. 53, 8934. Matamura, 0. (1959). J. Phys. SOC.Japan 14, 108. Matarrese, L. M., and Kikuchi, C. (1956). J. Phys. Chem. Solids 1, 117. McAlister, S. P., Furdyna, J. K., and Giriat, W. (1984). Phys. Rev. B29, 1310. Mulder, C. A. M., van Duyneveldt, A. J., van der Linden, H. W. M., Verbeek, B. H., van Dongen, J. C. M., Nieuwenhuys, G. J., and Mydosh, J. A. (1981a) Phys. Lett. 83A, 74. Mulder, C. A. M., van Duyneveldt, A. J., and Mydosh, J. A. (1981b). Phys. Rev. B23, 1384.

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Mullin, D. P., Galazka, R. R., and Furdyna, J. K. (1981). Phys. Rev. 824, 355. Mycielski, A., Rigaux, C., Menant, M., Dietl, T., and Otto, M. (1984). Solid State Commun. 50, 257. Mydosh, J. A. (1981). Lecture Notes in Physics 149, Springer-Verlag. Nagata, S.,Keesom, P. H., and Harrison, H. R. (1979). Phys. Rev. B19, 1633. Nagata, S.,Galazka, R. R., Mullin, D. P., Akbarzadeh, H., Khattak, G . D., Furdyna, J. K., and Keesom, P. H. (1980). Phys. Rev. B22, 3331. Nagata, S., Galazka, R. R., Khattak, G. D., Amarasekara, C. D., Furdyna, J. K., and Keesom, P. H.(1981). Physica 107B, 311. Nawrocki, M., Planel, R., Mollot, F., and Kozielski, M. J. (1984). Phys. Stat. Sol. (b) 123,99. Novak, M. A., Oseroff, S., and Symko, 0. G. (1981). Physica 107B, 313. Novak, M. A., Symko, 0. G., Zheng, D. J., and Oseroff, S.; (1984). Physica 126B,469. Novak, M. A., Symko, 0. G., Zheng, D. J., and Oseroff, S. (1985). J. Appl. Phys. 57, 3418. Okada, 0.(1980). J. Phys. Soc. Japan 48, 391. Oseroff, S. B. (1982). Phys. Rev. 25, 6584. Oseroff, S. B. (1984). Unpublished results. Oseroff, S. B., and Acker, F. (1981). Solid State Commun. 37, 19. Oseroff, S. B., and Gandra, F. (1985). J. Appl. Phys. 57, 3421. Oseroff, S. B., Calvo, R., and Giriat, W. (1979). J. Appl. Phys. 50, 7738. Oseroff, S. B., Calvo, R., Giriat, W., and Fisk, Z. (198Oa). Solid Stafe Commun. 35, 539. Oseroff, S. B., Calvo, R., Fisk, Z., and Acker, F. (1980b). Phys. Lett. 80A, 311. Oseroff, S.,Mesa, M., Tovar, M., and Arce, R. (1982). J. Appl. Phys. 53, 2208. Oseroff, S., Mesa, M., Tovar, M., and Arce, R. (1983). Phys. Rev. B27, 566. Otto, M., D i d , T., Mycielski, A., Dobrowolska, M., and Dobrowolski, W. (1980). Proc. of the X Conf. On Physics of Semicond. Compounds, Jaszowiec, Poland 1980, p. 225. Pajaczkowska, A. (1978). Prog. Cryst. Growth and Charact. 1, 289. Pajaczkowska, A., and Pauthenet, R. (1979). J. Magn. Magn. Muter. 10, 84. Price, M. H. L., and Stevens, K. W. H. (1950). Proc. Phys. SOC.A63, 36. Savage, H., Rhyne, J. J., Holm, R., Cullen, J. R., Carroll, C . E., and Wohlfarth, E. P. (1973). Phys. St. Sol (b) 58, 685. Sayad, H. A., Bhagat, S. M. (1985). Phys. Rev. B31, 591. Schneider, J., Sircar, S. R., and Rauber, A. (1963). Z . Naturf. 18a, 980. Schultz, S.,Gullikson, E. M., Fredkin, D. R., andTovar, M. (1980). Phys. Rev. Left.45, 1508. Shapira, Y. (1985). Private communication. Shapira, Y . , and Foner, S. (1985). Unpublished. Shapira, Y . , Foner, S., Ridgley, D. H., Dwight, K., and Wold, A. (1984). Phys. Rev. B30,4021. Seehra, M. (1972). Phys. Rev. 86, 3186. Simpson, A. W. (1970). Phys. Stat. Sol. (b) 40, 207. Singh. S., Singh, P. (1980). J. Phys. Chem. Solids. 41, 135. Sondermann, U. (1976). J. Magn. Magn. Muter. 2, 216. Sondermann, U. (1979). J. Mugn. Magn. Mafer. 13, 113. Sondermann, U. (1980). J. Magn. Magn. Muter. 21, 228. Sondermann, U.,and Vogt, E. (1977a). J. Mugn. Magn. Muter. 6,223. Sondermann, U., and Vogt, E. (1977b). Physicu 86-88B,419. Spalek, J., Lewicki, A., Tarnawski, Z., Furdyna, J. K., Galazka, R. R., and Obuszko, 2. (1986). Phys. Rev. B33, 3407. Swarup, P. (1959). Can. J. Phys. 37, 848. Tholence, J. L., and Tournier, R. (1984). J. Physique 35, C4-229. Title, R. S. (1963a). Phys. Rev. 130, 17. Title, R. S. (1963b). Phys. Rev. 131, 2503.

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SEMICONDUCTORS AND SEMIMETALS, VOL. 25

CHAPTER 4

Neutron Scattering Studies of the Magnetic Structure and Dynamics of Diluted Magnetic Semiconductors T. M . Giebultowicz DEPARTMENT OF PHYSICS, UNIVERSITY OF NOTRE DAME NOTRE DAME, INDIANA, USA AND REACTOR DIVISION, NATIONAL BUREAU OF STANDARDS GAITHERSBURG, MARYLAND, USA

and T. M. Holden ATOMIC ENERGY OF CANADA LIMITED, CHALK R N E R NUCLEAR LABORATORIES, CHALK RN ER, ONTARIO, CANADA

INTRODUCTION.. . . . . . . . . . . . . . . . . . . 1 . Motivafion and Introductory Remarks . . . . . . . . . 2 . Theory of Neutron Scattering in Systems with Short-Range Magnetic Order . . . . . . . . . . . . . . . . . . 3 . Theoretical Models of Spin Correlation Phenomena and Spin Dynamics in DMS Systems. . . . . . . . . . . . 11. POWDER DIFFRACTION EXPERIMENTS ON Znl-,Mn,Te. . . . . 4. Elernentsof the Experimental Technique. . . . . . . . . 5 . Results of Experiments . . . . . . . . . . . . . . . 111. NEUTRONSCATTERING STUDIES OF Cdl -,Mn,Te. . . . . . . 6 . Samples and Elements of Experimental Technique. . . . . 7 . Dvfraction Studies of Cdl -,Mn,Te Single Crystals. . . . . 8 . Inelastic Neutron Scattering in Cd0. 35M no.ssT e . . . . . . IV. CONCLUDING REMARKS. . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . I.

125 125 128 135 140 140 141 155 155 157 173 177 179

I. Introduction 1 . MOTIVATION AND INTRODUCTORYREMARKS

Thermal neutron scattering provides an extremely powerful method for investigating a number of important properties of solids. Since the de Broglie wavelength of thermal neutrons ( A 10-8cm) is of the same order of

-

125 Copyright 0 1988 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-752125-9

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magnitude as the interatomic spacings in condensed matter, pronounced interference effects take place in the scattering of neutrons from solids, and such effects can yield information about the structure of the scattering specimen. Moreover, the energy of thermal neutrons (10-100 meV) is comparable to typical energies of excitations in solids. The relative energy changes undergone by neutrons during inelastic scattering events involving creation and/or annihilation of these excitations are therefore large, and thus readily measurable. This makes neutron scattering an excellent tool also for studies of dynamic phenomena in condensed matter. In any material medium, the neutrons are scattered by nuclear forces, giving information about the structure and dynamics of the crystal lattice. In addition, however, the neutron also possesses a magnetic moment y = 1.913 ,UN (nuclear magnetons) which can interact with atomic magnetic moments, if such moments are present in the material. Elastic scattering arising due to this interaction depends on the static correlations between the magnetic moments of the atoms. Measurements of this effect can therefore be used as a method for studying the magnetic order (such as spin structures in magnetic crystals, or short-range ordering phenomena in spin-glasses). Inelastic magnetic scattering, on the other hand, depends in general on the spin-spin correlations both in space and in time. This mode of the scattering can yield information concerning collective magnetic excitations, such as the energies and the dispersion relations for the magnon modes, or the various spin relaxation processes occurring in disordered magnetic systems. Often neutron scattering is the only direct means of probing the magnetic structure in the case of antiferromagnets, or of determining the dispersion relations for the spin waves over a broad range of wave vectors. Neutron scattering studies in diluted magnetic semiconductors (DMS) are important for two reasons. First, as discussed in other chapters in this volume, many novel physical effectsseen in these materials arise because the conduction and/or the valence band electrons are strongly coupled via the spin-spin exchange interactions to the localized moments of the magnetic atoms. A detailed knowledge of the properties of the magnetic ion subsystem, and especially the knowledge of the distribution of the local exchange fields and their spatial and temporal fluctuations, are therefore essential for the understanding of the microscopic mechanisms underlying these effects. Here, neutron scattering methods are indeed quite helpful and, as mentioned above, are sometimes the only way of acquiring the necessary information in this context. Secondly, DMS are in their own right interesting magnetic materials, often referred to as “frustrated” antiferromagnets. This term describes a system where the energies of all pairs of antiferromagnetically bonded spins cannot be minimized simultaneously. A prototypical example of a frustrated

4. NEUTRON DIFFRACTION STUDIES

127

antiferromagnet is the face centered cubic (FCC) array of spins, each spin interacting antiferromagnetically only with its nearest neighbors. The A:'_,Mn,BV' DMS are very close realizations of such a system. The consequences of the frustration of the spin lattice are far reaching in the context of magnetic properties of these materials. For example, as a consequence of frustration, the ground state of the system is highly degenerate, there is no long range order in three dimensions, and the mechanism of spin excitations becomes very complex. Frustration may also result in a spin-glass behavior of the system. Frustrated FCC systems were subjects of many recent theoretical analyses and modelling studies (see, e.g., Villain, 1978; Grest and Gabl, 1979; de Seze, 1979; Alexander and Pincus, 1980; Ching and Huber, 1981, 1982a,b, 1984; Fernandez et al., 1983), where various aspects of this general problem are discussed. Since the cubic members of the DMS family, Cdl-,Mn,Te and Znl-,Mn,Te, resemble closely the prototype FCC frustrated antiferromagnets considered in these theoretical works, the results obtained in both elastic and inelastic neutron scattering experiments on these crystals can therefore be directly compared with the predictions of the models, providing valuable feedback for further development of the theories. In this chapter, we give a review of neutron scattering studies performed on Cdl-,Mn,Te and Znl-,Mn,Te. Since some readers of this book may be unfamiliar with neutron scattering methods, certain basic elements of neutron scattering theory are given in Sec. 2, together with a brief explanation of the data analysis procedures, in order to establish a common language. We then follow, in Sec. 3, with a short description of the models of diluted antiferromagnets used in the interpretation of the experimental results. Cdl-,Mn,Te can be obtained in single crystal form for Mn concentrations 0 s x 4 0.7. On the other hand, Znl-,Mn,Te samples used in the experimental studies described in this chapter were available only as polycrystals. Thus, different experimental techniques and data processing procedures had to be applied in these two cases. Although the materials are quite similar as far as their magnetic properties are concerned, we chose to discuss them in separate sections, hoping that this will contribute to the clarity of the presentation. Part I1 of this chapter is devoted to ZnI-,Mn,Te, where the experimental findings obtained by neutron scattering on powder samples are presented. The analysis of the short-range antiferromagnetic order observed in this material yielded direct information about the spin-spin correlation coefficients between magnetic neighbors as a function of their separation, concentration of Mn ions, and the temperature. Then, in Part 111, we discuss neutron diffraction experiments on single crystals of Cdl-,Mn,Te. In addition to the information similar to that presented in Part 11, these experiments led to the observation of the anisotropy of the magnetic

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T. M . GIEBULTOWICZ AND T. M. HOLDEN

correlation length and permitted the study of the dynamics of the spin system by inelastic scattering methods. 2. THEORYOF NEUTRONSCATTERING IN SYSTEMS WITH SHORTRANGEMAGNETIC ORDER

a. Basic Definitions The number of neutrons counted in a detector in an experiment to measure both the energy and the angular distributions of neutrons scattered by a sample is proportional to the double differential scattering cross section, dza/dQdc'. The incident (scattered) neutrons are characterized by wave vector k(k') and energy The solid angle into which the neutrons are scattered is designated by Q. The momentum transfer, hQ, and energy transfer, h o , to the sample are given by &(&I).

hQ = hk - hk',

and h o = E - E'.

The number of neutrons counted by a detector in a diffraction experiment, to measure only the angular distribution of scattering by a sample, is proportional to the differential scattering cross section, do/dQ, which is an integral over all scattered neutron energies,

The measured double differential scattering cross section is related to the physical properties of the sample through the scattering function S(Q, o), d2a k' -- - S(Qt a ) . gad&' k

(4)

The scattering function is the time Fourier transform of the correlation function ( F ~ F Q ( ~that ) > is , determined explicitly by the Hamiltonian for the sample, as follows (Marshall and Lovesey, 1971; Sears, 1978):

' j+m

s ( Q , 0)= 277

(F&(O)FQ(t))=

v'a'ua

-m

dt e-'"''(F&O)&(t))

pyPa((v', ( Y ' I F Q I Y , O!)12E?iwa'mf.

(5)

(6)

The states of the sample are specified by a and the energy transfer to the sample is the difference of the initial and final states, h a , , , = Em<- Em. The correlation function is averaged over the initial states of the sample

4.

NEUTRON DIFFRACTION STUDIES

129

characterized by probability of occupation, P,, and the initial spin state, v, of the neutron occupied with probability P, . The scattering amplitude, including both nuclear and magnetic scattering, is given by

FQ(O)=

c (bt

-

2yref(Q)s ' Si)e'Q'RI

In Eq. (7,b, is the nuclear scattering length of atom I , S?'is the projection of the electronic spin perpendicular to the scattering vector Q, s is the neutron spin and f ( Q ) is the magnetic form factor. Rt denotes the position of site I in the lattice. The neutron gyromagnetic ratio and classical electron radius are denoted by y and r e , respectively. In the case of a pure magnetic material, such as a type-I11 antiferromagnet which we will discuss later, the scattering function at a particular wave vector Q has the form of a delta function at the characteristic frequency of the elementary excitation (a spin wave in this case), W Q , where

S(Q,W )

-

-

OQ).

(9)

The strength of the scattering depends on the distribution of spin deviation in the two sublattices, on the spin-wave frequency and on the temperature. For magnetic material diluted with non-magnetic impurities, the characteristic frequencies are decreased and the excitations have finite lifetimes and are no longer described by a delta function. The static structure factor S(Q) with which we will be concerned in the interpretation of diffraction experiments is defined by +m

S(Q)

=

-m

S(Q, 0)d o

=

Uii(W~(0)).

(10)

The differential scattering cross section,

is equal to the static structure factor in the limit of very high incident energies m. For this reason, it is important in diffraction measurements of S(Q) to employ high incident neutron energies. For an alloy of the type Al-,M,B, where there is a random substitution of magnetic atoms, M, in place of the non-magnetic atoms, A, the differential scattering cross section is given by E

-+

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T . M . GIEBULTOWICZ AND T. M. HOLDEN

For a structure with one formula unit per unit cell, FQ =

1I [VI(FM.I - FA,/) + FA,/ + FB,I].

(13)

The independent quantity r,q is unity if an M atom occupies the lth lattice site and zero otherwise. The individual scattering amplitudes are:

F M ,=~ [- bM FA,I= -bAe

+ 2yref(Q)eiQ'R's sf], *

iQ.Ri

,

(14)

(15)

FB,I= -bBe i Q . U b + r )

(16)

The average in Eq. (12) has to be taken over the statistical distribution of magnetic atoms, the electron spin direction, the neutron spin direction (assuming the beam is unpolarized) and, for a powder sample, over all orientations of the crystal lattice with respect to the incident beam. The nuclear scattering length b~ is taken to be identical on each M atom, and it is assumed for the present that the nuclear incoherent scattering is absent, but it is an easy matter to include it at a later stage.

b. Single Crystal Diffraction

On carrying out the first two averages, with the assumption that V I and Si] are statistically independent, the resulting expression for the cross section in a single-crystal is

with

G(Q) = ~ S ( + S 1)

C ( ~ -dS-j)eiQ'R[.

1#0

(18)

Here h = (h, k,I) labels the reciprocal lattice sites, and Kh are corresponding reciprocal lattice vectors. is the volume of the primitive unit cell, x is the manganese concentration and Fh is the unit cell structure factor, Fh = x b -t ~ (1 - X)bA + bgeiKh.'.

(19)

The scattering is made up of three parts: the nuclear Bragg scattering from the site-averaged scattering length, the diffuse nuclear component coming from the site-to-site fluctuation in occupancy on the A-sites, and the magnetic diffuse scattering corresponding to the correlations between spins

4. NEUTRON DIFFRACTION STUDIES

131

on neighboring sites. By subtracting the nuclear diffuse component, the magnetic diffuse part can be isolated and analyzed to give the spatial (including direction) distribution of spin correlations in the crystal. It is sometimes convenient (particularly when considering long-range magnetic ordered systems) to express the magnetic scattering cross section in terms of the unit cell structure factor. In the case of a simple antiferromagnet this form factor can be written as Fh,M

a

c PeXP[i& r

'

r],

(20)

where p = f1 for the two antiparallel spin directions, and the sum extends over all spin sites in the magnetic unit cell. Consider now the unit cell of the type-111 antiferromagnetic structure, depicted in Fig. lb. One can readily check that in this case, the magnetic superstructure reflections occur at h' being odd integer, and for k' and I' being even or odd integers. Here the symbols h', k',I' refer to the tetragonal unit cell (2a x a x a). Usually, both magnetic and nuclear reflections are described in terms of a common notation involving the cubic indices. In the latter notation, the magnetic reflection indices are: h is an odd half-integer, while k and I are a combination of odd and even integers.

c. Powder Diffraction On carrying out the powder average, the differential cross section is found to be

where 26' is the scattering angle, Oh is the Bragg angle and Mh is the multiplicity of the h-th reflection. In reality, the Debye-Scherrer peaks have a finite width due to experimental resolution, so for comparison with experiment, the delta-function may be replaced by an appropriate Gaussian. The function of interest, that measures the correlations between spins, is 1#0

with At =

1

S(S

+ 1 ) ( S o - SI),

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T. M. GIEBULTOWICZ AND T. M. HOLDEN

and

The spherical Bessel functions j o , j 2 arise from averaging over the spin directions and from the axial symmetry of the spin tensor with respect to the vectors RI.The z-axis for each value of I is the vector joining the positions of the zeroth and I sites. If the absolute cross section is measured, it is possible to determine both S and the spin correlation functions. For the range of Q values studied in powder diffraction experiments, jz(QZ?l) is approximately equal to -jo(QR1), so that in effect only A I - Br can be determined by fitting Eq. (21) to experimental results of the accuracy normally achieved. Similar expressions for the magnetic correlations in amorphous materials have been used by Nagele et al. (1978). It is convenient to reinterpret the parameters ( A - BI)in Eq. (22) as the sum over all sites with the same R I , i.e., as the correlation within a given coordination shell. The quantity A I - BI is usually treated as a fitting parameter for each shell of neighbors and measures the size and range of the correlations. It is instructive and straightforward to derive A / for each shell of neighbors on the basis of what appears to be the ideal spin structure observed in DMS, the type-I11 antiferromagnet. The parameter BI depends on the spin direction in the ideal structure. In the absence of information about the actual spin direction we often cannot calculate B I , but the ideal spin-spin correlation coefficients A I, including multiplicity, can be derived (see, e.g., Table I1 discussed later in this chapter). The additional contribution to the total cross section from the incoherent nuclear scattering from each component is given by d&nc

--

dC2

- "OF 4n

+ xu$ + (1

- x)ajy].

The contribution to the diffuse intensity from thermal diffuse scattering (i.e., the sum of the phonon scattering) may be written in the incoherent approximation as

where e - 2 Wis an average Debye-Waller factor. From the magnetic diffuse scattering (MDS), we can obtain the wavevector-dependent susceptibility via the relation, valid at high temperatures

4.

NEUTRON DIFFRACTION STUDIES

133

in the quasi-elastic approximation (Marshall and Lovesey, 197 l),

The static susceptibility per formula unit, allowing for spin correlations, is therefore given by

d. PhenomenologicaI Approach

In this section we give an outline for a simplified scheme of data analysis which has been employed in the case of the single crystal studies described in Part I11 of this chapter. As pointed out in the preceding section, a fit of Eq. (21) to experimental points can be used as a direct method for determining the short-range spin correlation parameters. This method has been successfully employed in the analysis of powder diffraction results for Znl-,Mn,Te (see Sec. 5 ) . The neutron diffraction studies of Cd,-,Mn,Te, on the other hand, have been carried out only on single crystals. A similar analysis, making use of Eq. (17), could in principle serve as a straightforward method for extracting the short-range spin correlation coefficients in the single crystal case. However, to obtain meaningful results for single crystals, the method requires accurate scanning over a large three-dimensional volume of Q-space. Such measurements, being extremely time consuming, have not yet been undertaken in the case of Cdl-,Mn,Te. For this reason we will discuss the results obtained for Cdl-,Mn,Te in terms of a simpler model of magnetic short-range order, that makes use of only one parameter: the correlation length K. The underlying assumption of the model is that an approximate description of spin-spin correlations in a system with short-range order can be obtained by convoluting the correlations for a perfect magnetic long-range order by means of a phenomenological “correlation damping function”. Thus, we adopt here a method often used in the theory of critical neutron scattering in the vicinity of a second order phase transition. Although the critical spin fluctuations are a dynamic phenomenon, such a quasi-static picture is applicable when the process to be described is relatively slow in comparison with the time of the neutron interaction with the spin system (which is of the order 10-’2-10-13s). Because of this formal analogy, the calculation of the neutron scattering cross section (outlined briefly below) is practically the same as in the theory of the critical scattering (see, e.g., AlsNielsen, 1976; Lovesey, 1984; Marshall and Lovesey, 1971; Izyumov and Ozerov, 1970).

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T. M. GIEBULTOWICZ AND T. M. HOLDEN

On the basis of the above assumption, the pair correlation function in a system with short-range order can be written as

(SO * S r ) S R = D(RI)(SO * s I ) P ,

(29)

where subscripts SR and P stand for short-range and perfect order, respectively, and D(R/)is an “envelope” function that describes the decrease of the correlations with the distance RI = IRo- RII between the spins forming the pair. For the sake of simplicity, let us also assume that no correlations exist between the crystallographic directions and spin orientations, i.e., that the distribution of spin directions SOin various clusters comprising the system is perfectly random. The average value (Sd S i ) in Eq. (18) thus becomes $<SO S r ) for any direction in Q-space, which in turn yields the cross section for elastic magnetic diffuse scattering in the following form:

-

If we consider only the reciprocal space points Q = QO + q in the vicinity of the magnetic superstructure points QO characterizing the “initial” perfect order, then Eq. (30) simplifies to:

Equation (31) shows that the diffuse scattering intensity is described by the Fourier transform of the correlation damping function D(R). This fact makes it possible to calculate the shapes of the diffuse scattering maxima for various types of correlation damping by carrying out relatively easy Fourier inversions. For instance, in the case of D(R) in the form R-‘ exp(- KR)

(324

which are often employed in the theory of critical scattering, one obtains the following lineshapes, respectively: 1 44) = q2

1 + 7 K

(Lorentzian lineshape)

(334

4.

NEUTRON DIFFRACTION STUDIES

135

or

Additionally, it can be concluded on the basis of Eq. (31) that any symmetric damping function D(R) = D(-R) leads to a distribution of the scattered intensity that is symmetric with respect to a given reciprocal lattice point. So far, we have assumed that D(R)is isotropic. In general, however, the rate of correlation damping may very well be different for various crystallographic directions. This effect is clearly observed in Cdl -,Mn,Te. Shortrange spin correlations that depend on direction can be simply modeled by replacing the isotropic exponent K Rin Eq. (32) by an anisotropic form, e.g., [ d x 2 + ~,?y' + K$?]'", where K , , K ~ and , K* are correlation lengths for the three principal axes. By applying an uncomplicated transformation of variables, one can then easily obtain the following lineshapes:

or I

instead of the isotropic expressions (33a) and (33b), respectively. We can thus conclude by saying that (i) the short-range magnetic order described by simple types of correlation damping leads to a broadening of the neutron diffraction peaks, but not to their shift from the ideal reciprocal lattice positions; and (ii) the width and the shape of the diffuse maxima in Q-space provide information on range and the anisotropy of spin-spin correlations. 3. THEORETICAL MODELS OF SPINCORRELATIONS AND SPINDYNAMICS IN DILUTED MAGNETIC SEMICONDUCTORS

a. Magnetic Structure and Excitation Spectra of FCC Antiferrornagnets Spin structures for Mn2+ions in DMS can be discussed with the aid of the Hamiltonian

136

T. M. GIEBULTOWICZ AND T. M. HOLDEN

where the first term describes isotropic exchange between Mn ions, labeled byj, on sublattice k (1 or 2). The second term introduces an anisotropy field that gives a gap at q = 0 in the excitation spectrum. For non-cubic magnetic structures, this can arise from dipole-dipole interactions or from a Dzialoshinski-Moriya interaction that is allowed (Keffer, 1962) in the type111 structure. The positive value of the exchange interaction constant in Eq. (35) signifies an antiferromagnetic coupling. The stability of magnetic structures of the face-centered lattices of spins was discussed by Smart (1966) in terms of molecular field theory. For collinear spin arrangements, there are four domains of stability: ferromagnetism, and three types of antiferromagnetism, labelled I, 11, and 111, that are determined by the sign and relative magnitude of the first and second neighbor exchange parameters J I , J 2 . The phase diagram for face-centered cubic lattices and the type-111 magnetic structure are shown in Fig. 1. Antiferromagnetism of type-I (AF-I) is stable for J I antiferromagnetic and J2 ferromagnetic, whereas AF-111 is stable for both J I and J2 antiferromagnetic. Since the second neighbors exchange appears to be small in the compounds with which we are concerned, the AF-I1 phase is not encountered. As we shall see, the observed correlations in DMS strongly resemble those of AF-111, indicating that J2 is antiferromagnetic. In fact, the analysis of single crystal experiments, discussed later, provides a quantitative measurement of 5 2 . Cubic P-MnS is one of the few materials that exhibits AF-I11 in the pure state. Hastings et al. (1981) have shown experimentally that the magnetic phase transition is of first order. Theoretical arguments given by Hastings et al. suggest that the transition will always be first order in this structure. The stability of various magnetic structures was also investigated by Ter Haar and Lines (1962), who showed that the spin-wave frequency along the tetragonal axis in reciprocal space would be zero in the absence of J 2 , implying that the ground state is unstable with respect to spin fluctuations. In spin-wave theory for a type-I11 antiferromagnet, an antiferromagnetic J2 raises the spin-wave frequency along the tetragonal axis, whereas a ferromagnetic J2 gives imaginary frequencies. Spin-wave theory thus gives the same stability criteria as molecular field theory. The spin-wave dispersion relation for FCC AF structures takes the form (ha)'

=

A+A

-

B'B,

(36)

with A

=

2SJ( 1lq) - 2SJ( 110) + 2SJ( 120) + HA,

(37)

B

=

2SJ(12q).

(38)

4.

NEUTRON DIFFRACTION STUDIES

137

PHASE DIAGRAM FOR MAGNETIC ORDER IN THE (J, ,J,) PLANE FOR AN F.C.C. LATTICE

ANTIFERROMAGNETIC ORDER OF THE THIRD KIND ON AN F.C.C. LATTICE (b)

FIG.1. (a) Phase diagram for magnetic order in the ( J I , J z )plane for a face-centered cubic lattice. JI and JZ are first and second neighbor exchange constants, and a positive sign denotes an antiferromagnetic exchange interaction. (b) Unit cell for antiferromagnetic order of type-I11 on an fcc lattice. Open and closed circles denote the opposing spin sublattices.

The functions J(kk’q)are the Fourier transforms of the real space exchange interactions defined by

and have to be worked out for the appropriate spin structure. For the AF-111 structure, with the y-axis, as the tetragonal axis we have

138

T. M. CIEBULTOWICZ AND T. M. HOLDEN

J(l lq) = 2J1(eiTrycos ncx + e-iTcy cos n[J

+ 2Jzfcos 2ncx + cos 2nc& J(12q) = 2J1(e-jTfycos xcx + eiTrycos nl,

(40)

with There are three antiferromagnetic domains corresponding to the tetragonal axis being along x,y , or z directions in a macroscopic specimen. Thus the observed neutron scattering from spin-wave excitations at (1, ),0) which corresponds to q = 0 for one domain would have contributions from the other two domains. With this background in the structure and dynamics of FCC antiferromagnets, we proceed to discuss the properties of the disordered FCC lattice, as described by Ching and Huber (1982a,b).

6 . Ching-Huber Modeis By means of numerical simulation methods, Ching and Huber (1982a,b) have investigated the dynamics of FCC lattices of spins, interacting only via nearest neighbors isotropic Heisenberg exchange, for both the fully occupied lattice and the lattice diluted with non-magnetic ions. For the fully occupied lattice, (001) sheets of spins order antifferomagnetically under the influence of JI. However, the spin direction varies randomly from layer to layer, so that there is no three dimensional magnetic order. Each spin does experience the same local molecular field, -4SJ1, from its neighbors in the same plane. Presumably, because of the two-dimensional order, the excitations within this structure are well defined in energy and wave vector, and follow a dispersion relation that is-practically identical with that of a type-I11 AF. The numerical calculation also gives a finite density of magnon modes at zero energy. The presence of these modes indicates that the ground state of the disordered AF lattice is unstable with respect to spin fluctuations. In the diluted FCC array, there is a distribution of local molecular fields about the mean value, and the latter shifts to lower energies according to the approximate form for x > 0.5,

HM= -4SJ

+ 2SJ(1 - x).

(43)

There is no evidence in the distribution of local fields that the Heisenberg system, unlike the Ising system (Grest and Gabl, 1979), freezes into a spinglass phase.

4.

NEUTRON DIFFRACTION STUDIES

139

The calculations of the dynamics show that the excitations fall in energy as the dilution increases, with an accompanying increase in lifetime. The average spin-wave energy is nearly equal in magnitude to the frequency of the type-I11 AF structure after taking the cubic average and scaling linearly with concentration. Ching and Huber (1984) have also calculated S(Q) numerically for diluted FCC antiferromagnets with J2 = 0.1 J1 . They find that S(Q) is a maximum at wave vectors like (1, i,0), which is an ordering wave vector of the type-I11 A F structure, and rapidly decreases with wave vector offset from (1, &, 0). Close to the characteristic wave vector, t, they found that S(Q) could be closely represented by the following analytical form

rather than by the Ornstein-Zernicke form alone

The first term in Eq. (44) corresponds to a spatial exponential decrease of the form e-x'', and the second to a spatial form (l/r)e-"I'at large values of r.

c. Numerical Calculations of the Correlations Our understanding of the spin structure and dynamics of disordered materials has improved with the recognition of the multicritical character of the approach to long range order. For ferromagnetism or antiferromagnetism on lattices where the nearest neighbors of a given site are not nearest neighbors of each other, long range order may be approached along the concentration axis, at the percolation concentration, or along the temperature axis. Experiments on CrFe alloys (Burke et al., 1983) and on Znl-,Mn,F2 crystals (Cowley et al., 1980) have verified the power laws in concentration and temperature that emerge from this theory. The picture becomes more complex for an FCC lattice where nearest neighbors of a given site are nearest neighbors of each other, and where there are competing second neighbors interactions. In the percolation case, clusters are isolated by nonmagnetic sites, whereas for the FCC lattice, clusters are isolated by frustrated links at the boundaries. That is, there are situations where the spin at a given site would like to point up in the molecular field of another, and the net molecular field at the site is zero. Numerical calculations showed that the ferromagnetic transition was pushed away from the percolation concentration by the frustrated links (Binder et al., 1979), Eu,Srl-,S appearing to be a physical realization of this situation (Maletta ef al., 1982). A spin-glass

140

T. M. GIEBULTOWICZ AND T. M. HOLDEN

state may, however, be established above the percolation concentration, i.e., once an infinite cluster comes into existence. The concentration at which long range order occurs may be pushed to even higher concentrations when both the first and second neighbors interactions are antiferromagnetic. The physical realizations of this scheme of exchange interactions are probably Cdl-,MnxTe and Znl-xMnxTe. Again a spin-glass state appears to be established above the percolation concentration (Escorne and Mauger, 1982; Galazka ef al., 1980). At the present time, there is no definitive evidence for true long range order in these two DMS alloys, at least for x I0.75. 11. Powder Diffraction Experiments on Znl-,Mn,Te

4. ELEMENTS OF

THE

EXPERIMENTAL TECHNIQUE

The powder diffraction measurements on Znl-,Mn,Te were made with the L3 triple-axis spectrometer at the NRU reactor, Chalk River, in the diffractometer mode, with no analyzing crystal. The (113) plane of a squeezed germanium crystal served as monochromator at a fixed take-off angle of 48.69", corresponding to a neutron wavelength A = 1.406 A or an incident energy E = 41.4 meV (10 THz). The mosaic spread of the monochromator was 0.2"and the angular collimation before and after the sample was 0.32". It was desirable to have good resolution in these experiments in order to be able to extract information from the widths of the nuclear Bragg peaks about sample homogeneity via the spread in d-spacings. It was also important to minimize the resolution correction to the characteristic widths of the sharp peaks, that develop as a function of concentration and temperature near the ordering wave vectors for the type-I11 AF structure. In these experiments, it is vital to keep EO high and to achieve good wave vector resolution by collimation. If the high wave vector resolution is achieved by decreasing the neutron energy, the diffractometer integrates less adequately over the whole energy spectrum; what is then measured is the-scattering within a certain energy window which is only part (and an unknown part) of S ( Q ) . Typical low angle nuclear Bragg peaks had widths (full width at half maximum, FWHM) of 0.4".Measurements were made over a range of scattering angles (20) from 5 to 100" to establish the coherent scattering. The powder samples were packed in thin-walled A1 cans (inner diameter 0.7 cm, length 5.0 cm) and weighed to find the effective density. A typical packing fraction was about 0.6. Each sample was placed in a variable temperature cryostat, permitting measurements to be made between 2 and 300 K. An empty-can run in the cryostat was made in order to measure the background. The Bragg scattering from the A1 can give a useful measure of the instrumental width of the powder peaks and permited extraction of

4.

NEUTRON DIFFRACTION STUDIES

141

information about sample homogeneity from the widths of the Bragg peaks of Znl-xMn,Te. The observed counts must be corrected for multiple scattering and self-shielding. Following Sears (1975a, b), the observed counts C are related to the true cross section by

where Kis the instrumental normalization factor, H(k0, k') is the absorption factor (= 10%) which is independent of angle for our experimental arrangement, and m is the multiple-scattering correction which is =1.6% for Znl-xMnxTe samples. With improved accuracy of the coherent and incoherent scattering cross sections (Koester, 1977) due to the individual components of the alloy, the analysis of the Bragg peak intensities permits the calculation of the concentration of Mn atoms in the samples as well as an effective Debye-Waller factor and the normalization factor to put the observed diffuse scattering on an absolute scale, The relative intensities of the three families of peaks in the zinc-blende structure (h + k + 1 = 4n, 4n k 2, and 4n -t 1, where n is an integer) are very sensitive to concentration, since the scattering lengths of zinc and manganese have opposite signs, The accuracy in measurement of concentration is in fact about +- 1 at.9'0. The effective Debye-Waller factor permits an estimate to be made of the thermal diffuse scattering via Eq. (26). The agreement between intensity calibrations at different temperatures was k 5 % . The normalization factor was also measured with a cylindrical vanadium sample of known diameter and irradiated length, correcting the observed counts for self-shielding and multiple scattering (8.7%). Agreement between the internal Bragg scattering normalization and with vanadium was within 5%. 5. RESULTSOF EXPERIMENTS The diffraction pattern for Zno.435Mno.565Te taken with neutrons of wavelength 1.4064 A at 4.2 K (Holden et al., 1982) for scattering angles between 5 and 75" is shown in Fig. 2. We see the expected pattern of Bragg peaks for the zinc-blende structure together with A1 powder peaks from the sample can, superposed on a diffuse background. The principal feature of the diffuse scattering is a broad peak centered on 14.5", which may be indexed as (1, i,0) in the zinc-blende structure. This peak is clearly much wider than the experimental resolution in reciprocal space given by the width of the Bragg peaks. Two other diffuse peaks are identifiable. The peak near 23.3", which is partly overlaid by the adjacent (111) nuclear peak of the zinc-blende structure, may be indexed as (1, *, 0) and the peak near

142

T. M. CIEBULTOWICZ AND T. M . HOLDEN

50001 i I

v, I-

z

3

4000

3000

z

0 IT

'r w

z

'

2000

:

I

5

1

I

.

x = 0.565 T=42K

(Ill)'

A = I3863 6

(220)

II

1

.1

I

3

I

10

I

15

I

20

SCATTERING

I

25

I

30

I

35

40

ANGLE 2 8

FIG.2. Neutron diffraction pattern for Zn,~.43~Mno.s6sTe at 4.2 K, showing Bragg scattering from the zinc-blende structure, the aluminum sample can, and strong magnetic diffuse scattering [After Dolling et al. (1982).] which peaks at 14.5' (Q = 1.1 k').

30.5" may be indexed as (1, *, 2). Similar diffraction patterns were recorded for Zno.6~4Mno.376Te, Zn0.406Mno.s94Te (Dolling el d., 1982), and Zno.32Mno.68Te. For the x = 0.68 sample, there was evidence in the diffraction pattern for small amounts of other phases, such as MnTez and hexagonal MnTe. The manganese concentration, Debye-Waller factor, lattice spacing and intensity scaling factors, obtained by fitting to the integrated powder diffraction intensities and angles are collected in Table I. The table also contains estimates, derived from the (220) and (442) nuclear reflections of Znl-,Mn,Te, of the homogeneity of the samples based on the widths of these Bragg peaks. Excess width, above the experimental resolution, was assumed to originate from a spread of lattice parameters in the sample. The homogeneity of the three samples with lowest c is excellent by this criterion. However, there appear to be concentration fluctuations in the x = 0.68 sample, which may be connected with the appearance of impurity phases. The temperature dependence of the diffuse scattering between 5 and 33" for the concentrations x = 0.376,0.594, and 0.68 is shown in Figs. 3, 4, and 5 . In each case the dashed-dot lines give the nuclear diffuse scattering,

TABLE I MANGANESE CONCENTRATION, TEMPERATURE, LATTICEPARAMETERS, DEBYE-WALLER FACTOR,AND BRAGGPEAKDATAINDICATING HOMOGENEITY OF THE Zn,-,Mn,Te SAMPLES STUDIED.

37.6 k 0.3 59.4 f 0.2 59.3 k 0.3 68.0 & 0.7 56.7 f 0.05 56.3 k 0.6 56.5 k 0.6

77.6 240 4 300 4 300 17

6.176 6.236 6.230 6.263 6.218 6.234 6.218

0.0051 t O.OOO4 0.0131 t 0.0006 0.0047 f 0.0008 0.0156 t 0.0021 0.0035 f 0.0017 0.0162 t 0.0026 0.0051 0.0021

*

"The instrumental width for this reflection is 0.435 f 0.02". bThe instrumental width for this reflection is 0.903 k 0.04".

0.47 f 0.02

0.90

0.04

0.06 rt 0.04

0.0 t 0.15

0.45 k 0.02 0.53 k 0.02

0.92 f 0.04 0.95 f 0.04

0.04 0.05 0.097 f 0.20

0.03 t 0.06 0.048 f 0.050

_+

...

I600 1200 -

X =0.376

4.2 K M=3

. -. . '--0.5

000 400 -

-

- 1.0

/ -

*

-

-

--- - ----

- - - - - - - - - - - - - - - - - - - - - - - - - - - -_ -

-

c

& 0 u

-

c

C

3 I

0

ln

1200

1

I

I

I

*... ...

000-

0 V

400 -

0

1200 -

800 400-

0

Q

77 K

-

I-

z 2

I

1

. . .. - . , - ... _ :. ---- I 5

-----

-

I 10

1

_ . . . a

I

1

1

I

I

- .. - .

I-

r0.5 u w

1

e.

--

-.

ul ul ul

0

I

- 1.0

300 K

....

.*

- . - - -... . *

0.5

- - - - - - - - - - - - - - - - - --- - - - - - -_ _ I

I

15

20

I 25

I 30

-2 z 0

- - - - - - - - - - - - - - - - - - - - - - - - - -___

-- --------1

* -

-1.0

7;

0

50

cl

?

..800 -

.?

+-\

X *0.!594 M=I

400

- 2.0

4.2 K

t

-

Y

-- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - I

0 In

I

1

I

I

s

.-

C

(200)

I I

I

I

a

-2.0 7.

L

78 K

-

I-

z

(111)

u

c

.*

400-

*

m

4

.*

1.0

-- - - _- - - - - - - - _ _ - _ - - -- - - _ - -- - - - - - - - - - - - - - - - - - - 0.

I

I

I I

I

I

I

I

400 -

I

I

.. - - .

I

~*

-.. .

-- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 0-

I

5

I 10

-2.0

240 K ------v

I 15

I 20

1 25

z

0

~

I 30

-1.0

V W

In In

g a u

: 2000 z 3

g

I

1000

1

1000

78.0K

.... - _T1.0

. . - * - -

-

. - .'-<. .. . .\ - . ..

- 2.0

.-.. .

_ _ _ _ _ _ _ _ - - ------------ -------_

(2005 -

.-. .. j I . 0

- - - - --

2

0

I-

0

w

v)

4.

NEUTRON DIFFRACTION STUDIES

147

including both incoherent and thermal diffuse scattering, that accounts for the results at high angles, where the magnetic scattering is negligible. As the temperature is raised the diffuse peak near 14.5" weakens, becomes broader and shifts to lower angles. The peak indicates the existence of magnetic correlations between manganese spins that are strongest at low temperatures. The correlations persist to high temperatures even for the x = 0.376 sample, although they are not very long-ranged, since the 300 K data are ill-described by a MnZ+form factor alone. The development of the spectrum as a function of concentration and temperature can be quantitatively described by the growth of correlations of a site with many shells of neighbors. There does not appear to be a tendency for either clustering or superlattice formation on the (Mn, Zn) sublattice. There are no sharp features persisting to room temperature in this system, and in each case the spectrum evolves smoothly to a high temperature state with short-ranged correlations. Little chemical short-range ordering is expected, since MnZ+and Znz+ ions have similar ionic sizes and the transition metal lattice is interpenetrated by the Te lattice. Thermally driven short-range order would involve interchanging transition metal and tellurium ions and this process is strongly inhibited by the chemical bonding. The question of long-range magnetic order, that is, a phase transition to an ordered antiferromagnetic state, can be definitely settled for x = 0.594. The width of the main magnetic diffuse peak certainly does not approach that of a Bragg peak at the lowest temperatures. For x = 0.68, this conclusion is not so clear cut. It may be possible, though unlikely, that a small fraction of the sample shows long-range order, since the top 30% of the main peak has the FWHM of a Bragg peak. This portion could be associated with the concentration spread noted earlier for this sample. It is clear, however, that the bulk of the sample does not show long-range order. The analysis of the data supports this view. The maximum cross section observed for all the samples is shown as a function of temperature in Fig. 6. In every case there is a gradual increase in the cross section as the temperature is decreased, with saturation below 20 K. By contrast, the static susceptibility of these alloys shows the cusp behavior of spin-glasses, and the position of the cusp, Tg (McAlister et al., 1984), is also shown in Fig. 6. There is no anomaly in the neutron scattering at T g . It is well-known that the cusp in the susceptibility, that is a characteristic of spin-glass behavior, becomes rounded in an applied magnetic field. The neutron cross section was therefore measured in a magnetic field above or below Tg . The x = 0.594 sample was mounted in a superconducting magnet capable of providing fields up to 4T in a direction normal to the horizontal scattering plane. No magnetic field effects were observed either on field

-

148

T. M. GIEBULTOWICZ A N D T . M. HOLDEN

-

I

V

c .-

c ¶

Y

n Y

I

I

0.68 0.594 o x - 0.565 x x = 0.376 0

X=

o x=

TEMPERATURE

-

(K)

FIG.6. Maximum magnetic cross section obtained from the magnetic counts observed in the vicinity of the ( 1 , i,0) diffuse peak for the four samples studied, as a function of temperature. In each case the intensity increases smoothly with decreasing temperatures, and saturates below 10 K . Arrows indicate spin-glass temperatures for these samples. [After Holden et al. (1982).]

cooling in 2 T from 7 0 K through Tg = 31 & 1 K, or on applying this field below Tg . Thus the correlation length is not a function of magnetic field, at least up to 2T. The observed angular variation of the magnetic diffuse scattering was fitted to the theory described in Sec. 3 to obtain the magnetic correlations between a spin and successive shells of neighbors. The solid curves in Figs. 3, 4, and 5 represent least squares fits of the observed scattering to the term in Eq. (21) with the definition of g(Q) given in Eq. (22). The non-magnetic terms were fixed by the high angle scattering data, where magnetic scattering is negligible, and the variable parameters were the Mn2+ spin and the shell correlations. The parameters derived from this procedure are collected in Table 11. For x = 0.376, this description proved successful over the whole temperature range. For x = 0.565,0.594, and 0.68,the quality of the fit was

41

?

0

Y

I

T

tl +I

?N

0 0

I

3

1

4

909

2 m

0909???df 0 0 0 0 0 0 0

I

I

I

I

I

I

I

tl ti +I +I +I +I +I ?99???09

e * m r 4 r 4 - 0

I

I

l 1 + 1

I

r 4 - - - - m - - - -

+I $1

tI +I +I +I t i +I +I +I

09dN +I +I +I

0 0 0

?W? W N O

l

l

l

tl l

l

/

1

tl tl tl

I

l

l

-!?Z

l

+I +I

I

tl

l

09b.N"

+I +I +I +I

I

I

I

I

?P-1'4

9 m W

T

--=

9 9 9 9 9 9 9 9 9 9 0 0 0 0 0 0 0 0 0 0

7%" 0 0 0

+I +I +I

099Y m m o

I

???

- 0 0

+I +I

?'4? 0 0 0

1?09

r4-0

- m m

-00

I

l

+I +I +I t I

0 0 0 0

232

W ? ? N 909N1 r4"O

I

+I +I +I tl

0 0 0 0

t ? N N

0 0 0 0

- 0 - 0

+I +I +I +I +I +I +I

0 0 0 0 0 0 0

F w N 0 0 0 0

??f????

?dP-d:Yb.W 0 0 0 0 0 0 0

2

v,

150

T. M. GIEBULTOWICZ AND T. M. HOLDEN

unsatisfactory below about 100K, as can be seen from the corresponding values of the goodness of fit parameter, x, and from inspection of Figs. 4 and 5 . This occurs because a limited number of shell terms cannot reproduce the sharp peaks found experimentally. For more than seven shell terms in the fitting process, the errors on the shell correlations were larger than the magnitudes. In spite of the poor quality of fit at low temperatures the fitted parameters are, we believe, physically significant. The average value of the manganese spin from these experiments is 2.4 k 0.2 which is consistent with the half filled shell spin of $, expected for Mn2+.The shell correlations reproduce the sign sequence of correlations calculated for the type-I11 antiferromagnet, A:". The magnitudes, AI - BI, are a large fraction of the ideal correlation for each shell, A!", and the fraction diminishes as the distance from the chosen origin increases. However, it is not possible to establish the functional form of the decay of the correlations uniquely from the data. The spatial Fourier transform of an Ornstein-Zernicke correlation function is a Lorentzian of FWHM 2 x 1 . The polycrystalline average of a Lorentzian only gives a weak logarithmic divergence at aQ/271 = (i,1, 0), and Figs. 2, 3, 4 display diffuse peaks that are much sharper than logarithmic. An inspection of the data rules out the Ornstein-Zernicke form, and forces consideration of longer ranged diminution of the correlation with distance from the origin, or other explanations. Other methods of analysis were carried out to model the sharp peaks in the diffuse scattering. Guided by the observation that the sign sequence of the correlations was the same as for an ideal type-I11 antiferromagnet, the values of A1 were assumed to be the ideal values, A:", but modified by a damping function of the form e-xlR'. With 336 shells of neighbors, extending to about 85 A from the origin, the calculated peak widths were as sharp as those observed. However, the calculated main peak always occurred at a Q / 2 n = whereas the observed peak shifted to lower wave vectors as the temperature increased, or as the concentration diminished. In addition, the cross section at wave vectors below the main peak was always overestimated. This failing suggested that a pure exponential description of the fall-off for the nearest shells of neighbors was also inaccurate. The most satisfactory analysis was one in which the correlations with the first three shells of neighbors were allowed to vary independently and the correlations with the 4th to the 336th shell were decribed by an e-'lR' envelope superposed on the ideal type-I11 correlations. This approximation corresponds to a small q approximation to S(q), given by l/(d + K ~ ) and ~ , makes the connection with the calculation of Ching and Huber (1983). The parameters derived from the fitting procedure are given in Table 111. The goodness of fit parameter, x, is smaller than in previous analysis, Table 11, but the first

m,

4. NEUTRON

151

DIFFRACTION STUDIES

three shell parameters are in good agreement with those previously obtained. The dashed curves in Figs. 3, 4, and 5 represent the best fit to this model. The analysis of single crystal measurements of Cdl-xMnxTe, discussed later in this chapter, suggests that the best description of the decrease of spin correlations with distance is provided by an anisotropic Lorentzian function. It may be shown that the functional form of the wave vector dependence of the intensity in a powder experiment is altered if an anisotropic Lorentzian function is used instead of an isotropic Lorentzian. This may well be the reason why the isotropic Lorentzian gave an unsatisfactory description of the powder data. TABLE I11 MANGANESE CONCENTRATION, TEMPERATURE, SPINCORRELATIONS AND INVERSE CORRELATION FROM FITTING Znl-,Mn,TE NEUTRON DIFFRACTION DATATO EQ. (22). x LENGTHS OBTAINED INDICATESTHE QUALITY OF THE FITS. X

(at. '70)

37.6 56.5

59.4

T

(K) 4.2 4.2 15 25 35 45 55 77 300 4.2 25 50

68.0

78 240 4.2 40 78 170 300

Ai - B I

A2 - B2

-5.OfO.l -3.3 *O.l -3.4*0.1 -3.4 f 0.1 - 3.2 0.1 -3.2 f 0.1 - 2.8 i0.1 -2.5 f 0.1 - 1.8 i 0.1 - 2.6 f 0.1 -2.8 f 0.1 -2.6 f 0.1 - 1.6 f 0.1 -0.30 i 0.1 - 2.8 f 0.2

*

-2.6 & 0.1 -2.6 i 0.1 -0.9 f 0.1

-0.4 f 0.1

A3 - B3

KlaNN

X

2.0 0.2 1.7 +- 0.1 1.8 i 0.1 1.6 f 0.1 1.5 0.1 1.3 0.1 1.0 f 0.1 2.1 f 0.2 -0.3 C 0.1 1.6 i 0.1 1.5 f 0.1 1.1 i 0.1 0.7 f 0.1 0.2 f 0.1 1.4 f 0.3

3.4 f 0.1 4.6 f 0.2 4.5 i 0.2 4.1 f 0.2 3.6 f 0.2 2.9 f 0.2 2.6 f 0.2 1.05 f 0.04 0.3 f 0.2 3.6 i 0.2 3.3 0.2 2.0 f 0.1 1.8 f 0.2 0.6 f 0.1 5.2 0.2

0.62 f 0.02 0.48 f 0.01 0.49 f 0.01 0.54 i 0.01 0.64 f 0.02 0.74 f 0.02 0.86 rt 0.03 0.9 1.7 f 0.2 0.53 i 0.01 0.56 f 0.01 0.86 i 0.02 1.23 f 0.04 7f9 0.38 f 0.01

1.2 1.3 1.1 1.2

0.1 f 0.1 0.1 f 0.1

1.9 f 0.2 1.4 f 0.1 0.7 f 0.1

* *

1.5 5 0.1 0.9 i 0.1

*

*

3.6 f 0.2

0.68 i 0.02 1.15 f 0.04 1.8 f 0.2

0.3

*2

1.1

1.o

1.1

0.7 1.O 1.5

1.3 0.8 0.7 3.4 2.5

1.9 1.4 0.9

At high temperatures, above about 80 K, KI would not be expected to be well determined, since the shell analysis alone gives a good description of the cross section in this limit. Nevertheless, fits up to 300 K are included in Table 111 for completeness. The results are summarized in Fig. 7. Analysis of the development of correlations in ferromagnets (Burke et al., 1983) and antiferromagnets (Cowley et al., 1980) near the percolation

152

0.4 -o

0.2-

0.o 0

10

20

30

40

50

60

TEMPERATURE ( K )

70

00

90

FIG. 7. The inverse correlation range ( K I U N N ) , deduced by fitting the data to a model with exponential fall-off of the ideal correlations with the 4th-336th shell. The solid line is a guide to the eye. The numbers labelling various symbols denote the Mn mole fractions.

concentration shows that the inverse correlation length can be broken into a part determined by geometry (and, in our case, exchange interactions) and a part governed by temperature alone, as follows: Kill"

= (K/UNN)G

+ (K/UNN)T,

where UNN is the nearest neighbor separation of manganese sites. The geometric parts of K/U" are shown in Fig. 8, where they are compared with

4.

153

NEUTRON DIFFRACTION STUDIES

estimates of Ching and Huber (1983). In the previous analysis (Holden et al., 1982) the inverse correlation length was obtained from the FWHM of the (1, 0) peak under the assumption that the peak was Lorentzian in shape. The present analysis is probably superior because it includes fitting all the data, not just the peak. The variation of K I is in excellent agreement with the numerical estimates of Ching and Huber (1983), bearing in mind that a powder experiment averages over two correlation lengths, corresponding to intra-planar and inter-planar effects. The variation of K I suggests that any transition to an ordered state will occur at concentrations well in excess of x = 0.7. The temperature dependent parts (K/a”)TOf the inverse correlation length for the three highest manganese concentrations fall on a common curve that flattens at low temperatures (Fig. 9). This low temperature flattening could be a cross-over from Heisenberg to Ising interactions along the links between already large clusters (Cowley, 1980).

t,

CALCULATED AND EXPERIMENTAL INVERSE CORRELATION LENGTH O

’

*

3

0 X

o

X

x

0

ox

0

0

I 0.1

I

0.2

I

0.3

I

I 0.6 CONCENTRATION X I 0.4

0.5

I 0.7

I 0.8

I 0.9

J

1.0

FIG.8. Comparison of the zero temperature inverse correlation lengths squared, calculated for Znl-,Mn,Te with JZ = 0.1 J I (Ching and Huber, 1983), with the experimental result for powder sample. There are expected to be two inverse correlation lengths, corresponding to coupling between planes and within planes, and the experimental result on a powder is an average over both.

154

T. M. GIEBULTOWICZ AND T. M. HOLDEN

TEMPERATURE DEPENDENCE ON INVERSE CORRELATlON LENGTH

:

0.9 -

0.8 -

0.70.6

0 0.68 0.594 0 0.565

-HEISENBERG LINEAR CHAIN J, = I I

K

-

0.5(K,Q")f

0.4-

0.3-

0.20.1

TEMPERATURE ( K

FIG. 9. The temperature dependent portion (Kl(INN)T of the inverse correlation length for x = 0.68, 0.594, 0.565 which lies on a common curve. The solid curve is the result for a Heisenberg linear chain with J I = 11 K, and is not in close agreement with experiment.

Similar results have been obtained for ferromagnetic CrFe (Burke et al., crystals (Cowley et al., 1980). However, in both these cases the experiments were carried out near the percolation concentration, where the clusters are ramified, i.e., major clusters are joined by linear pathways of adjacent sites occupied by manganese ions. The temperature dependence of the inverse correlation length for Znl -,Mn,F2 was successfully modelled as a Heisenberg linear chain. Here we are far from the percolation threshold, and the one-dimensional linear chain model is not very plausible. In any case, the slope of the variation of KlaNN with temperature is steeper (Fig. 9) than the variation of KIaNN calculated for a Heisenberg linear chain with classical spins (Fisher, 1964), making use of J I = 11 K deduced from the high temperature static susceptibility (McAlister et al., 1984). 1983) and antiferromagnetic Zn1-,Mn,F2

4.

NEUTRON DIFFRACTION STUDIES

155

It is of interest to compare the behavior of the inverse correlation length in Znl-xMn,Te and EuxSrl-,S as they pass through the spin-glass temperature characterized by the cusp in &. In the case of Eu0.szSro.d (Maletta et al., 1982), in which no long range ferromagnetic order occurs, the correlation length drops to a minimum, corresponding to a range of correlations of about 400 A , just above &. Below & the inverse correlation length increases, corresponding to a decrease in the size of the correlated regions. In the present case the inverse correlation length does not exhibit any change at Tg.

111. Neutron Scattering Studies of Cdl-xMnxTe

6. SAMPLES AND ELEMENTS OF EXPERIMENTAL. TECHNIQUE Because of the extremely high neutron absorption cross section of cadmium (ou = 2650 barns for I = 1.08 A, see Bacon, 1975), in practice only single crystal diffraction studies are possible in the case of Cdl -*Mn,Te, the powder signals being too weak. Fortunately, this material can be obtained in single crystal form in the whole zinc-blende composition range ( x s 0.70). The earliest diffraction data for this alloy were obtained using very thin crystal plates (-0.1 cm) for x = 0.60-0.70. However, the increase of absorption with increased Cd content did not permit extending these studies to lower x values. Fortunately, samples prepared from the low-absorbing Il4Cd isotope created new opportunities for neutron scattering studies, and three such samples (x = 0.65, 0.44, and 0.35) have been investigated in various diffraction experiments. The largest of these specimens, a Cd0.3&ln0.6sTe crystal with a volume of 1.5 cm3, also made it possible to study inelastic magnetic scattering in a DMS system for the first time. In the above experiments, the crystal axis [OOl] was set vertical, because this orientation permitted observation of the largest number of type-I11 antiferromagnetic superstructure reflections (i.e., the type of ordering seen in Cdl-xMnxTe). The inelastic studies and most of the diffraction measurements were carried out with crystal spectrometers (see Brockhouse, 1961) in either the triple-axis mode of operation, or the diffractometer mode, after the analyzing crystal was removed. With a diffractometer, a given momentum transfer vector Q may be selected by setting both the angle that the crystal makes with the incident beam and the angle of scattering. The magnetic component of the observed intensity provides in this case a direct measure of the magnetic structure factor S(Q) (see Eqs. (10) and (1 1)). Using the triple axis configuration, one can observe inelastic scattering for a chosen momentum transfer QO and energy transfer 0 0 . The intensity of magnetic

156

T. M. GIEBULTOWICZ AND T. M. HOLDEN

scattering in this case is given by:

where R(Q - Qo, o - 0 0 ) is the instrumental resolution function, and other symbols are consistent with Eqs. (4) and ( 5 ) . A commonly used technique, employed also in studies of Cdo.35Mno.ssTe, is the constant-Q method, in which Q is kept fixed, and the energy is scanned over the required range by varying the energy of the incident beam (mi) while the analyzer energy is kept fixed. If the scattering is normalized with a counter with a “I/v” characteristic (i.e., with the yield inversely proportional to the velocity of incident neutrons) in the incident beam, then the scattering function S ( Q , w ) can be obtained from raw data after relatively uncomplicated resolution corrections (see Brockhouse et al., 1963). In neutron diffraction experiments, special problems may be caused by the presence of twinning in the samples. Most of the Cdl-,Mn,Te crystals grown by the Bridgman technique exhibit this type of imperfection. One of the lowabsorbing crystals (with x = 0.44) that were studied revealed only a small twinned grain, that could be easily identified and separated. Unfortunately, the Cdo.3&ln0.65Te sample used for diffraction and inelastic scattering studies was found to contain a large number of twinning defects. In order to show how this affects neutron scattering results-and how such undesirable effects may be avoided-let us first explain the mechanism of the twinning. For simplicity, we consider the FCC structure, although the same arguments will apply to the zinc-blende structure. As is well known, the FCC lattice can be viewed as a system of hexagonal close-packed atomic layers, stacked along the [I 1I] axis with the sequence ABCABCABC. Twinning occurs when one of these planes is wrongly stacked, and starting from this point the sequence is reversed: ABCABACBA. The “twin” crystal has the same [ 11I] axis, but is rotated about this axis by 180”. In Q-space such a situation corresponds to two “interpenetrating” reciprocal lattices. In each of these coordinate systems, the indices of the lattice points belonging to the other system can be found by a simple transformation:

For example, (h’,k’, Z’) = (2,2,0) and (4,2,0) correspond to (h, k , I ) = (2,2,0), and (0,2,4), respectively. This shows that some of the Bragg reflections from both components coincide. At the same time, (2,2,0) and $), meaning that new reflections appear (4,z, 0) lead to ($, $, f)and (-$,

y,

4.

NEUTRON DIFFRACTION STUDIES

157

at “parasitic” positions. In diffraction experiments, the interest is focused on the scattering in the vicinity of the reciprocal lattice points. The above example shows therefore that, if such points are appropriately selected, one can avoid the interference of scattering components from the twinned grains. This procedure is not applicable to measurements of inelastic scattering which is distributed throughout the (Q, o)space. One can, however, take advantage of the fact that some Q vectors are equivalent for both components, as can be readily checked using Eq. (47). In the (001) plane such points lie along the [ilO], [120], and [210] axes. Inelastic spectra measured in these cases are not “contaminated” with scattering components corresponding to other Q-vectors. 7 . DIFFRACTION STUDIES OF Cdl -,Mn,Te SINGLE CRYSTALS

a. Room Temperature Experiments Magnetic interactions in Cdl -,Mn,Te were the focal point of neutron diffraction studies performed on this compound. For that reason, most of the experiments have been carried out at low temperatures. Room temperature data do not provide much information about the properties of the spin system in Cdl-,Mn,Te. The measurements show in this case only the characteristic pattern of nuclear Bragg reflections for the zinc-blende structure, and a search in other areas of Q-space does not reveal detectable maxima indicative of a superstructure of, e.g., magnetic nature. An example of room temperature data is shown in Fig. 1Oc. These experiments are consistent with the findings of other studies on Cdl-,Mn,Te (e.g., magnetic susceptibility measurements, or EPR studies; see Oseroff and Keesom, this volume), which indicated the existence of a paramagnetic phase in this temperature region for all compositions studied (0 c x 5 0.70). However, room temperature results also provide us with an important additional conclusion: there is no observable chemical clustering, or shortrange atomic ordering in Cdl-,Mn,Te. Effects of this type are known to occur quite often in alloys and other mixed compounds, and may also produce additional maxima in X-ray and neutron diffraction patterns (atomic ordering phenomena and methods for studying these features are reviewed, e.g., in the book by Warren, 1969; some aspects of neutron diffraction in this context are also discussed by Bacon, 1975). In general, the intensity of diffuse scattering due to short-range atomic order is proportional to ( b -~ b~)’,where b A and b g denote the scattering amplitudes of atoms A and B comprising the mixture. Hence, effects in diffraction are strongest when the difference @A - b ~is)relatively large. This is, indeed, the case in Cdl-,Mn,Te. In fact, for all Mn-based DMS compounds the difference

158

T. M. GIEBULTOWICZ AND T. M . HOLDEN

FIG.10. The contour map of the scattered neutron intensity for a sector of the (001) plane of the reciprocal space for Cdo.56Mno.wTeat (a) 4.2 K, (b) 78 K, and (c) 295 K. The Bragg nuclear reflections are shown by black dots, and the crossed circles show the ideal positions of the typeI11 superstructure reflections. The plots were obtained by multiple scanning parallel to the [OlOl axis within the trapezoidal area marked as ABCD in (a); the upper part of the plot is the “mirror” image with respect to the line BC. Plot (b) shows that magnetic maxima begin to form at 78 K. As can be seen in (c), no maxima occur at 295 K. The shape of the equal intensity lines, however, suggest that very weak residual magnetic correlations persist even at this temperature. [After Giebultowicz et al. (1986b).]

4. NEUTRON DIFFRACTION STUDIES

159

( b A - bB) is large because manganese has, unlike most other elements, a negative neutron scattering amplitude (see Bacon, 1975). The absence of observable maxima in Cdl-,Mn,Te thus proves clearly that no significant deviations from a random distribution of Cd and Mn atoms are found in this compound. Neutron scattering observations remain here in agreement with the results of X-ray studies and recent EXAFS experiments (Balzarotti et al., 1985) on Cdl-,Mn,Te, that also do not detect local ordering effects. This conclusion is of considerable value for magnetic studies, assuring us that the random distribution of atoms (which has been commonly assumed) is indeed a good description of the systems studied.

b. Magnetic Ordering at Low Temperatures

The pattern of zinc-blende Bragg reflections in Cdl-xMnxTe does not change significantly with temperature. However, the measurements at low temperatures reveal additional diffuse maxima in the scattered intensity (Giebultowicz et al., 1981; Giebultowicz et al., 1984; Steigenberger et al., 1986; Steigenberger and Schaerpf, 1986). First indications of such peaks can be seen at temperatures as high as 100 K. The scattering intensity gradually increases with further decrease of temperature and tends to saturate in the lowest temperature region (see Fig. 10 and Sec. 7g). For all crystals in the Mn concentration range 0.44 < x < 0.70 studied by neutron diffraction the peak intensity levels off before reaching 4.2 K. Even in the saturated region, the maximaare relativelyweak: in the most concentrated samples (x = 0.65-0.70) the intensities of the magnetic peaks are only 0.3-0.5'70of the intensity of the strongest nuclear Bragg reflections and as low as 0.1'70for x = 0.44 (see Figs. 11 and 12). Since Cdl -,MnxTe contains Mn++ions with non-zero magnetic moments, it was natural to assume that the new maxima arise due to magnetic ordering effects in the crystals. This interpretation is supported by the following additional findings. Other effects indicating the onset of spin correlations, such as deviations from the Curie-Weiss law in the magnetic susceptibility and broadening of EPR lines, emerge in the same temperature region (see Oseroff and Keesom, this volume). Atomic ordering may be ruled out, since this process in alloys typically occurs at 100-200" below the melting temperature (Warren, 1969), and melting temperature for Cdl -,Mn,Te is 1400 K. It should be added that recently the magnetic nature of the diffuse peaks in Cdl-,MnxTe has been confirmed directly in a spectacular way by neutron diffraction experiments with polarization analysis (Steigenberger and Schaerpf, 1986), that allows a direct separation of the magnetic and nuclear components in neutron scattering at a given wavevector Q.

160

T . M. GIEBULTOWICZ AND T. M. HOLDEN

” 2.8

2.9

3.0

3.1

3.2

WAVEVECTOR COMPONENT Q y FIG.11. Diffuse magnetic scattering in 114Cdo.,sMno.~sTe single crystal at 6 K. The plots show the results of radial (or “0 - 28”) scans through points in the reciprocal space with coordinates (4,1, O), and ($, 3 , O ) . The shaded contour is a nuclear Bragg reflection shown for comparison. [After Giebultowicz et al. (1982).]

c. The Type of Order The most basic information that can be obtained by studying magnetic diffraction patterns is the determination of the type of magnetic structure. As in the case of ZnI-,Mn,Te, all results obtained in experiments on various Cdl -,Mn,Te specimens indicate that the low-temperature spin configuration is closely related to the type-I11 antiferromagnetic ordering of the FCC lattice, shown in Fig. 1. Characteristic superstructure reflections of a type-I11 antiferromagnet occur at reciprocal space points Q = (2n/a)(h,k , I), h , k , I being combinations of one even integer, one odd integer, and one odd halfinteger. Diffraction measurements performed on Cdl-rMn,Te samples with 0.35 < x < 0.70 always reveal diffuse maxima at points satisfying this rule.

I

o (3/2,I,<)scan

I

(A 1

( 312, I

,Oo0t

+ <,O)scan A (3/2+<,I,O)scan

I

2000 -

I

o ( 3/2,l,( )scan

I

I

(8)

(312, I + <,O)scan

.

f

Q - VECTOR

COMPONENT,

FIG. 12. Scans through reflection points (i,1,O) along three cubic axes for Cdl-,MnxTe single crystals, with (a) x = 0.65 and (b) 0.44 at 4.2 K. The ellipses in the figures show the resolution function width (FWHM) within the experimental plane (with the scale on the x and y axes the same as on the horizontal axes in the main plots). The vertical resolution width (i.e., along the z axis) was in each case slightly larger than the longer ellipse axis. The shaded contours show the (200) nuclear reflections for comparison. Note that the horizontal scales are different in both parts of the figure. [After Giebultowicz ef al. (1986a).]

162

T. M. GIEBULTOWICZ AND T. M. HOLDEN

Single crystal techniques offer considerable advantages in the structure determination over the powder method used in studies of Znl-,Mn,Te. In the latter method, diffracted intensity is measured as a function of the Q-vector length, but is averaged over all directions in Q-space. Thus, one cannot distinguish between Bragg reflections and magnetic peaks whenever they happen to overlap. As can be seen from the data displayed in Sec. 2, in practice such overlap limits the observable Q-range to the first three or four magnetic maxima. This problem does not exist in single crystal measurements. Since zinc-blende Bragg reflections are centered at Q-points with all h , k,1indices being even-only or odd-only integers, the magnetic maxima are always well separated from the Bragg peaks, thus making magnetic peaks distinguishable even for high indices (see Fig. 11). Up to twenty inequivalent peaks have been observed in Cdo.3sMno.asTe(Giebultowicz el al., 1982a), with no detectable shift of the peak centers from the points characteristic for the type-111 superstructure. Recent studies of a “sister” wurtzite-structured material Znl-,Mn,Se (Giebultowicz el al., 1987a),however, revealed a small deviation of magnetic peaks from the ideal superstructure positions. This fact suggests that the description in terms of a symmetric damping function D(R) (see Sec. 2d) may not always be appropriate in DMS compounds. The type of spin ordering seen in these experiments is consistent with Anderson’s theory of antiferromagnetic structures in FCC lattices (Anderson, 1953; see also Smart, 1965). The phase diagram illustrating findings of this theory is shown in Fig. la. Type-I11 ordering occurs if the exchange constants for nearest ( J 1 )and next-nearest neighbors (Jz)are both antiferromagnetic (i.e., positive in the convention used by us), and fulfill the condition 0 < JZ < )JI. According to our present knowledge of the Mn-Mn exchange in cubic DMS, all these interactions are, indeed, antiferromagnetic (see, e.g., Oseroff and Keesom, this volume). The ratio of J z / J I is still not very well known, and the existing estimates vary from 0.1 to 0.25 (see, e.g., Galazka et al., 1980; Escorne and Mauger, 1982; Larson et al., 1985; Larson et al., 1986; we discuss this question in Sec. 7f).; All these values satisfy the requirement for type-111 ordering. On the other hand, the lowest estimate of J z / J ~places Cdl-,Mn,Te quite close to the boundary between type-I11 and type-I orderings in the phase diagram, and this fact may lead to the question whether a tendency toward formation of type-I structure does not occur in this material. This possibility cannot be entirely excluded a priori, since Anderson’s theory may not be assumed to apply rigorously to diluted lattices. An answer to this question is also of interest from the point of view of computer simulation studies related to DMS compounds. Experiments on single crystals can easily resolve this problem, because type-I11 and type-I superstructure maxima (the latter occurring at Q = (27r/a)(i, 0, 0) or (2n/a)(), i, 0)) should be positioned far away from each other in Q-space.

4.

NEUTRON DIFFRACTION STUDIES

163

Scans carried out over such points show convincingly that there is no detectable type-I component in Cdl-,Mn,Te.

d. Short-Range Character of the Magnetic Ordering The magnetic maxima seen at the type-I11 superstructure points in Cdl-xMn,Te are characterized by Lorentzian lineshapes and are decidedly broader than the corresponding Gaussian instrumental resolution function. The intrinsic width of these lines decreases with growing x and with lowering temperature, but in all cases stabilizes at a finite value for the lowest temperatures. Such behavior indicates that the type-I11 antiferromagnetic correlations between the Mn ions remain relatively short-ranged even for x as high as 0.70. In the case of x = 0.70, the low temperature linewidth corresponds to a correlation length of only a few lattice constants. The above result for Cdl-,MnXTe shows a close analogy with the findings in Znl-,Mn,Te. As pointed out in Section 5 , the sharpness of the magnetic peaks observed in Znl-,MnXTe with x = 0.68 may suggest that long-range ordered domains begin to form in this concentration region. Since the maxima seen in Cdl-,Mn,Te with x = 0.65-0.70 are equally sharp, such possibility should be also taken into consideration in the case of this material. Detection of a long-range ordered component would have an important implication for the still open problem of the magnetic phase diagram of Cdl-,Mn,Te (see Secs. 7f and 7g). The issue can probably be clarified considerablyby the following simple analysis of the peak intensities observed in Cdl-,Mn,Te. Taking into account the nuclear scattering amplitudes of Cd, Mn, and Te, and the scattering amplitude corresponding to the magnetic moment of Mn++ions (approximately $ 4 3 ,see Spalek et al., 1986)-al1 this information can be found, e.g., in the book by Bacon, 1975-one concludes that in the case of the long-range order in Cdl-,Mn,Te with x = 0.60-0.70, the intensity of Bragg nuclear (I,,)and Bragg magnetic (Z,) scattering should be of similar magnitudes (as is observed, e.g., for a type-I11antiferromagnet p-MnS, see Corliss et al., 1956).On the other hand, if the long-range ordered domain comprise only a part of the total crystal volume, the ratio of intensities ZJZn gives us a rough estimate of the upper limit for the fraction of long-range ordered inclusions. The data obtained for a crystal with x = 0.65 presented in Fig. 11 and 12a show that Zm/I" is very small, of the order of 0.01. All measurements for x c 0.70 yield Zm/Zn values of the same order of magnitude, which practically eliminatesthe possibility of long-range effects in CdI-,Mn,Te with x I0.70. Similar estimates cannot be made in the case of the results obtained for powdered Zno.32Mn0.68Te. In single crystal experiments, one observes the scatteringcorresponding to a single point of the Q-space(or, rather, to a small finite volume surrounding this point), whereas in a powder measurement

164

T. M . GIEBULTOWICZ AND T. M. HOLDEN

the observed effect is additionally averaged over a whole sphere in Q-space. Since Bragg maxima possess a negligibly small width in Q-space, and diffuse maxima are broadened in all three dimensions, the ratio Zm/Zn in the powder method is enhanced compared to that obtained in a measurement on a single crystal, and may be relatively large even in the absence of long-range ordered phase. Thus, the relatively strong diffuse maximum (1, *, 0) seen in 2no.32Mno.68Tecannot be taken as a firm indication of long-range ordering effects.

e. Detailed Analysis of Magnetic Peak Profiles The experiments presented in Secs. 7b-7d have established that: (i) the magnetic order in Cdl-,Mn,Te is short-ranged, and (ii) positions of the diffuse magnetic peaks agree with those characteristic of the type-I11 antiferromagnetic superstructure. These features are fairly well accounted for by a simple picture of “damped” magnetic correlations in a type-I11 antiferromagnet. As mentioned earlier (Sec. 2d), detailed analysis of the profiles of the diffuse maxima can shed additional light on the nature of the shortrange magnetic correlations of spins in DMS materials. The most intriguing property of the magnetic peaks seen in Cdl -xMnwTe is the dependence of their widths on the direction in Q-space. A good illustration of this feature is given in Fig. IOa, showing a contour map of the distribution of the scattered neutron intensity in a section of the (001) plane of Q-space for Cd0.S6Mn0.44Teat 4.2 K. One can see that the equal intensity lines are quite elongated, indicating clearly that the correlation length in Cdl-,Mn,Te is not isotropic. Studies of this phenomenon have been reported independently by two teams of workers (Giebultowicz et al., 1985; Giebultowicz et al., 1986a,b; and Steigenberger et al., 1986-hereafter, referred to as GRCHG, and SLG, respectively). Although the experimental results obtained in these works are quite similar, two entirely different explanations of the origin of the lineshape anistropy have been proposed. We present below both explanations, as well as a short comparative discussion. The model of anisotropic magnetic correlations proposed by GRCHG is a result of combined computer simulation studies and of a phenomenological analysis of experimental data. Since the computer simulations are described in Secs. 3 and 7f, we present here only the latter aspect of the model. The basic observation is that the peaks are wider along the directions in Q-space corresponding to their half-integer reflection indices (e.g., the (1, i,0) peak is broader along [OIO]axis, (3,1,O) along [loo], etc.). Such behavior can be clearly seen in Fig. 10a and is characteristic of a large number of reflections that have been observed in samples with x = 0.44 and x = 0.65. This feature reflects the symmetry inherent in the type-I11 antiferromagnetic structure.

4. NEUTRON DIFFRACTION STUDIES

165

The configurational symmetry in this type of ordering is tetragonal (Fig. lb). That is, the structure consists of (100)-type planes of atoms with antiferromagnetically ordered spins, with the planes stacked along the tetragonal axis (x-axis in Fig. la) with an ABABABAB sequence (where the bars denote antiparallel spin orientation). A closer look at the arrangement of neighboring spins shows that there is a considerable difference between the “inplane” and “interplane” spin coupling: within the plane each spin interacts antiferromagnetically with four nearest-neighbors. On the other hand, the interactions with nearest neighbors in adjacent planes sum up to zero, the coupling between the A-A, B-B planes being maintained only through the second-neighbors interaction, which is relatively weak (see the equilibrium conditions for type-111 order, Fig. la). This shows that a perfect type-111 configuration can be considered as a system of weakly coupled ordered planes. In a short-range ordered system the correlation length should therefore be larger within the planes than for the “interplanar” direction. Consequently, one may expect stronger broadening of the diffraction peaks along the direction in the Q-space corresponding to the “interplanar” axis in real space. Let us notice now that the half-integer reflection index also refers to such an axis (see Eq. (20) and the discussion following it). The fact that the direction of the maximum broadening is not the same for all the peaks can be understood if we realize that the underlying tetragonal symmetry allows three inequivalent configurations of short-range ordered “domains” (with the interplanar directions parallel to [loo], [OlO], and [OOl]). Each of these gives rise to a separate group of peaks. The above arguments can be additionally tested by analyzing the peak profiles along all three cubic axes (see Fig. 12). It turns out that the profiles for both “in-phase” directions [OlO] and [OOl] are practically the same and are clearly narrower than those for the “interplane” axis [loo], in agreement with the expected behavior. Contrary to the above interpretation, Steigenberger et al. (1986) attribute the observed anisotropy of magnetic correlation length to imperfections in the crystallographic structure of Cdl-,Mn,Te. The samples used in their study exhibited strong twinning effects. Closer examination by X-ray and neutron diffraction led the authors to the conclusion that the structure of the samples-although closely related to the zince-blende structure-is trigonally deformed, with one of the [l 111cubic axes preferred, thus imposing on the system a hexagonal symmetry. Further analysis of the data suggests that Mn atoms tend to avoid the tetrahedral coordination by moving toward the (1 11) plane perpendicular to the twinning axis. As a result of such displacement, there are two types of nearest neighbors-“in-plane” and “out-ofplane”-the distances between which are different. This leads, further, to an anisotropy of the exchange interactions. It was finally concluded that the

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anisotropy of exchange interactions induced in this way gives rise to the different peak broadening for various Q-space directions. Both interpretations outlined above agree in ascribing the lineshape anisotropy to the anisotropy of exchange interactions in the system of Mn ions. In one case, however, the anisotropy is of “configurational” type (GRCHG), whereas in the other case it is associated with an actual change of the Mn-Mn exchange constant (SLG). Two facts seem to favor the first mechanism. First, the lineshape anisotropy was also found to occur in a sample that does not exhibit twinning effects (x = 0.44, see Figs. 10a and 12b, and Sec. 6 for details) and the peak profiles obtained for a twinned sample with x = 0.65 show practically an identical character (Fig. 13). Secondly, the explantion given in the paper by Steigenberger et al. (1986)

J

z

P

tJ w K

0

TEMPERATURE ( K 1 FIG. 13. Temperature variation of the correlation length if! = KC’ in Cdo.&In0.65Te, deduced from the half-width (FWHM) of the magnetic diffuse (1, +,0) maximum, and of the integrated intensity of this maximum. [After Steigenberger el a/. (1986).]

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167

implies that any observable effect caused by the exchange interaction anisotropy of the type dicussed should be symmetric with respect to the twinning axis, i.e., to a [llll-type axis; however, all experiments indicate a symmetry with respect to [100]-type axes. Although the result obtained for the twinning-free sample seems here to be particularly convincing, it still does not rule out the possibility that SLGtype anisotropy may also occur in cases of very high density of defects. The problem undoubtedly requires further study. The two different interpretations of the lineshape anisotropy stress the importance of careful checking of the quality of the samples prior to diffraction measurements. The studies of diffuse peak lineshapes provide also information about the decay of the spin correlations with distance. Analysis of the data for x = 0.44 and x = 0.65 samples has shown that a satisfactory description of the observed profiles can be obtained in terms of an anisotropic Lorentzian + (q; + q$)/~:], where the physical meaning function Z(q) oc 1/[1 + of the adjustable parameters K I Iand K~ is that of “interplanar” as well as the “in-plane” inverse correlation ranges, respectively. In view of the results for Znl-,Mn,Te and computer simulation results (see Sections 3b and 7f), attempts have been made to describe the lineshape in terms of a sum of Lorentzian and “squared-Lorentzian” components (see Eq. (34)). In some cases this form led, indeed, to a slightly better fit than a single Lorentzian line, although the contribution of the squared-Lorentzian component did not exceed 20%. This fact suggests that the decrease of spin-spin correlations with distance in Cdl-,Mn,Te resembles the Ornstein-Zernicke damping, in some contrast to the findings of experiments on Znl-,Mn,Te (Part 11), and the results of computer simulations by Ching and Huber (1983), that indicated quite sizable or even predominant “squared-Lorentzian” components of the lineshapes. Another piece of information of interest for the studies of magnetism in Cdl -,Mn,Te is the dependence of the correlation length on the Mn concentration in the material. Unfortunately, our knowledge of this dependence is still incomplete. Separate determination of K I I and K~ has been possible only for two low-absorbing specimens with x = 0.44 and x = 0.65 ( K I I= 0.110 A-’, K~ = 0.060 A-’, and K I I = 0.031 A-’, K~ = 0.017 A-’, respectively). In experiments involving strongly absorbing samples, K = 0.030 A-’has been obtained for x = 0.70, and K = 0.042 A-’for x = 0.60 (Giebultowicz et al., 1982a; Giebultowicz et al., 1982b). The latter measurements have been carried out using ‘‘6 - 20” scans (i.e., scans along the reciprocal lattice vector), so that the values obtained correspond neither to K ( nor to K~ but rather to a “mean” inverse correlation range ( K < ~ K c ql),analogous, to a certain extent, to the values of K inferred from the powder technique (see Part 11).

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The values of K in Cdl-,Mn,Te are somewhat lower than the results obtained in the experiments on Znl-,Mn,Te and also in the model studies by Ching and Huber. We discuss this question in Sec. 7f. It should be noted that the low-temperature diffraction data obtained for Cdl -,Mn,Te indicate an identical qualitative character of the short-range spin ordering for 0.44 Ix I0.70. the only concentration-dependent effect seen in this x-range is the change of the magnitude of the correlation range. In particular, the anisotropy K ~ / K Idoes I not vary with x within the experimental error (see Fig. 12). Such behavior remains in strong contrast with the interpretation of the magnetic specific heat and magnetic susceptibility data for Cdl-,Mn,Te in the paper by Galazka et al. (1980), who suggested the existence of a boundary between two distinct magnetic phases in this concentration region.

J Comparison of Lo w-Temperature Diffraction Data with Model Results Many experimental facts presented in Secs. 7b-e can be explained on the basis of the results of computer simulation studies. Ground state configurations simulated using the method of Ching and Huber (techniques of numerical modelling have been presented in Sec. 3b) for a system with 5 2 = 0.151 represent the model of the low-temperature magnetic phase in Cdl -,Mn,Te. Results of the simulations in this case were found to reproduce the short-range type-111 spin ordering seen in this material. Analysis of the static structure factor S(Q) for the simulated system along various directions in the Q-space in terms of Eqs. (37)-(38) led Ching and Huber (1983) to predict the anisotropy of the correlation range. Thus the experimental evidence presented in Sec. 7e, especially the data obtained on twinning-free samples, confirm the validity of the Ching-Huber model. The model not only helps us to understand the mechanism of the anisotropy, but also offers the possibility to estimate the next-nearest neighbors exhange constant 5 2 in Cdl-,Mn,Te. The knowledge of this parameter is essential for the verfication of the theories of Mn-Mn interactions in DMS systems (Larson et al., 1985 and Spalek et al., 1986). The ratio of the inverse correlation ranges K I I / K should ~ be sensitively dependent on the ratio of the exchange constants 5 2 / J 1 . Simulations carried out on a system with x = 0.65 for various values of J 2 / J 1 show, indeed, quite strong correlation between these quantities: for 5 2 / 5 1 = 0, K I I / K is ~ close to 3, and decreases gradually to K I I / K * = 1 for 5 2 / 5 1 = 0.3 (Giebultowicz et al., 1986a). Comparison of the calculated data with the experimental value of K I I / K ~obtained from the (1, 0) maximum in Cd0.35Mno.65Te yields 5 2 / 5 1 = 0.12.

+,

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169

As noticed in Sec. 7e, the absolute values of K and the details of the lineshapes obtained by the method of Ching and Huber show some discrepancies from the observed data. These differences may be caused by “finite size effects’’ in the simulations: because of the finite dimensions of the “supercell” used (i.e., 12a x 12a x 12a), the model data reflect primarily the character of the correlations between relatively close spin pairs. On the other hand, in the case of the measured results, the contribution of more distant neighbors appears to be significant. For example, as can be estimated from Eq. (32a), for Ornstein-Zernicke damping the pairs with RI IK - ~give rise to only -26% of the peak intensity. For Cdo.3sMno.asTe, however, K ; ~= 9a. Secondly, a description in terms of K independent of the distance between the neighbors is certainly an approximation. The fact that the results obtained for Znl-,Mn,Te (see Sec. 5) are in a better agreement with the model simulations seems to support these arguments. A comparison with the theory has been made in that case on the basis of correlation coefficients for the few first shells of neighbors (determined directly from powder measurements), that better corresponds to the situation described by the model. In view of these shortcomings of the model, the accuracy of the determination of JZ may be questioned. However, test calculations carried out for systems of various sizes suggest that the ratio K ~ / K I isI decidedly less sensitive to finite size effects than the absolute value of K (Ching and Huber, 1985). In simulations carried out by Ching and Huber (1983) for the model systems with 0.40 Ix I0.80, the short-range character of the spin ordering persisted up to x = 0.80. The variation of K in this range of x does not show any anomaly, suggesting the existence of two different phases, in agreement with the conclusions drawn from diffraction data for Cdl-,Mn,Te. Similarly, Monte Carlo simulations do not indicate a phase boundary as proposed by Galazka et al. (1980). Studies done by the latter method for Cdl -,MnxTe (using a “supercell” of 4000spins, with JZ = 0.1J1) in the range 0.70 I x I 1 showed only a short-range type-I11 ordering for concentrations lower than x = 0.85 (Giebultowicz, 1986). Above this concentration, the systems exhibited a long-range (non-collinear) order at low temperatures. Since in this case the scale of long-range order is necessarily the scale of the supercell, what the model suggests is that a phase boundary may occur even at x higher than 0.85. g . Variation of Magnetic Scattering with Temperature

So far we have considered in detail the magnetic correlations only in the region of low temperatures. We now turn to the discussion of their temperature behavior. This temperature dependence can provide valuable insights into the mechanisms of the spin-glass formation seen in Cdl -,Mn,Te.

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Our discussion of this problem will, of necessity, be limited to crystals with relatively high Mn concentrations ( x = 0.6-0.7), since the relevant measurements have only been carried out for such samples. A typical temperature behavior reported by Steigenberger et al. (1986) for Cd0.35Mn0.65Teis shown in Fig. 13. Two quantities are displayed in this figure as a function of temperature: (a) the correlation length (11 = KC’ obtained by fitting a Lorentzian shape to the experimental (1, $, 0) diffuse magnetic maximum, and (b) the integrated intensity P ( T ) , i.e., the area under this particular maximum. Let us note that the correlation length remains constant in the lowest temperature range, and that it decreases rapidly above T = 30 K. The integrated intensity, on the other hand, does not show any pronounced anomaly at T = 30 K, but it decreases monotonically at temperatures above 10 K. At low temperatures (below 10 K), the intensity also displays a saturated character. A similar temperature behavior of the correlation length and of the integrated intensity was found for other Cdl-,Mn,Te samples with x values between 0.6 and 0.7 (Giebultowicz et al., 1981 and Giebultowicz et al., 1984). Specifically, it was found that the low-temperature P ( T )data can, to a good approximation, be described in terms of a squared Brillouin function for S = 3 (see Fig. 14). By fitting this function to the experimental data, a temperature TOB = (45 f 0.5) K was obtained in the case of the sample with x = 0.65. It has also been established that the temperature dependence of the inverse correlation length can be described by a power law K 0: ( T - TO”)’. For example, for x = 0.65 it was found the To” = (32.8 f 0.6) K and v = 0.68 f 0.05. Galazka et al. (1980) in their well-known paper suggested, on the basis of the magnetic susceptibility and specific heat data on Cdl -,Mn,Te, that above x = 0.6 in this material there occurs an “antiferromagnetic” phase transition (as opposed to the transition to the spin-glass state taking place for 0.17 Ix I0.60). This point.of view was shared by many later works. Let us reexamine this notion in the context of the presently available results of neutron scattering measurements. There are indeed certain similarities between P(T ) and K ( T )dependences observed in Cdl -,Mn,Te and those found in simple antiferromagnets exhibiting a second order phase transition, e.g., MnF2, RbMnF3, and KMnF3. It is, for example, observed that the intensity of the magnetic reflections seen in the latter materials below the phase transition temperature To varies typically as the square of the Brillouin function. This behavior can be easily understood: the intensity of the reflection is proportional to the square of the magnetization of the antiferromagnetic sublattice. This quantity, in turn, follows the Brillouin curve (see Bacon, 1975; Smart, 1966). Although the long-range order disappears above TO,weak magnetic maxima

4.

171

NEUTRON DIFFRACTION STUDIES

I

I I 1 Cd1-x MnXTe (312 10 )reflection 8 0 X 50.70

1

0 0 X = 0.65

o X=0.63

T (K)

FIG. 14. The variation of the integrated intensity of the magnetic diffuse (1, +,0) maximum (related to the intensity of the (200) nuclear Bragg reflection) for Cdl-,MnxTe single crystals with x = 0.60, 0.63, 0.65, and 0.70. [After Giebultowicz ef al. (1981).]

persist to higher temperatures due to fluctuations in the spin system. The inverse correlation length in true antiferromagnets obeys a power law K oc ( T - To)” for T 1 To. The values of the exponent v found in the Heisenberg antiferromagnets are close to 8 (Als-Nielsen, 1975; Marshall and Lovesey, 1971), similar to the value obtained for Cdl-,Mn,Te. On the other hand, thefinite value of K at low temperatures definitely rules out the possibility of any truly long-range order. One is then tempted to visualize the magnetic ordering phenomena in Cdl-,Mn,Te with x 1 0.6 as similar to those occurring in very fine antiferromagnetic powders, where the correlation length cannot exceed the size of the grains, while the behavior of other characteristics still resembles that of the bulk crystal. Thus one thinks of Cdl -,Mn,Te as consisting of magnetic “clusters” or short-range ordered domains, into which the spin system breaks up due to dilution. In view of the above discussion, the existence of the antiferromagnetic phase in Cdl-,Mn,Te above x = 0.6 appears doubtful. This conclusion implies that the spin-glass state persists at least up to x = 0.7. It is difficult to give firm evidence for this claim on the basis of the neutron diffraction data alone. This is because there is at present no theory of the temperature

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T. M. GIEBULTOWICZ A N D T. M. HOLDEN

behavior of neutron scattering in spin-glasses in the vicinity of the transition to this state. In fact, even the question of whether the paramagnetic-spinglass transition is sharp or continuous is still a subject of debate. Nevertheless, there is an agreement as to the point that such a transition involves gradual slowing down of the spin relaxation processes and occurs within a finite temperature range. (For a review of recent studies of spin-glasses, including neutron scattering experiments, see e.g., Huang, 1985). The neutron scattering data in Cdl-,Mn,Te can indeed be interpreted in a way consistent with this picture. For instance, the constant value of K observed in Cdo.35Mno.asTe below 30 K indicates the existence of a “frozen” phase, whereas the “high temperature” phase sets in above 40K, with the most important freezing processes taking place between 30 K and 40 K. The above interpretation of the ordered state seen in Cdl-,Mn,Te at low temperatures is further supported by the results of Monte Carlo simulations. Studies of the FCC diluted Ising systems (Grest and Gabl, 1979), and of the Heisenberg systems (Fernandez et af., 1983) with only nearest-neighbor interactions, revealed the presence of spin-glass effects. The above systems can serve as the simplest models of magnetic interactions occurring in cubic DMS. It must be said here that the models with only nearest-neighbor interactions do not always reproduce the magnetic properties of DMS. For example, in the case of a fully occupied magnetic lattice it was found, both for the Ising Hamiltonian (Phani et a/., 1980) and for the Heisenberg Hamiltonian (Giebultowicz, unpublished), that the antiferromagnetically ordered phase of type-I sets in. It was only after invoking the interactions with the more distant neighbors that the type-I11 antiferromagnetic ordering was obtained (see Sec. 7f). Detailed Monte Carlo studies of such systems (with JZ = J1/10) have been performed for two concentrations of Mn: x = 0.70 and x = 1.0. For x = 1.O (which may be viewed as the model of /3MnS), the simulations resulted in a long-range type-I11 structure at low temperatures, with a first-order transition to the paramagnetic phase Giebultowicz and Furdyna, 1985). These results are in good agreement with observations on /3-MnS (Hastings et al., 1981). For x = 0.70, the model correctly describes the spin ordering as well as the temperature variation of the magnetic specific heat and magnetic susceptibility seen in Cdl -,Mn,Te (Giebultowicz et a/., 1984 and Giebultowicz, 1986). In order to determine the type of transition in the case of x = 0.70, the inverse relaxation time T-’ was also calculated from the self-correlation function (S(0) * S(t)>of the spins, and was studied as a function of temperature. Its behavior was found to be ~ different from that characteristic of the first order transition (where T - has a discontinuity at To), as well as of the second order transition (where T-’ = 0 for T < To, and obeys a power law above To). the computer simulation results suggest rather that 5 - l depends logarithmically on temperature,

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NEUTRON DIFFRACTION STUDIES

173

similarly to the result of Monte Carlo modelling of spin-glasses by Binder and Schroder (1976). Monte Carlo simulations carried out for x between 0.70 and 1.O indicate that there is in fact an antiferromagnetic (i.e., long-range) order in the high 4 concentration region. This antiferromagnetic phase exhibits additionally a “reentrant” behavior (i.e., with a decrease of the order with decreasing temperature). This is in agreement with the original conclusions of de Seze (1977). The Monte Carlo studies show, however, that the long-ordered phase appears at Mn mole fraction considerably higher than those suggested by Galazka et al. (1980) in their pioneering study, namely above x = 0.85. The differences observed by Galazka et al. (1980) in the magnetic behavior of samples with x = 0.60 and x = 0.70 do indicate, however, that the processes occurring with the spin system are still not fully understood, and require further studies.

8. INELASTICNEUTRON SCATTERING STUDIES

OF

Cd0.3sMn0.65Te

Because of difficulties in the preparation of large, non-absorbing Cdl-,MnxTe samples, studies of inelastic neutron scattering in this material have so far been performed on only one single crystal, Cd0.3sMn0.6sTe. An example of the data obtained for this specimen is displayed in Fig. 15 (Giebultowicz et al., 1984). The scans were done at T = 1.8 K in the constantQ mode, for Q = (2n/u)(l, 0.5,O) and (2n/a)(l, 0.4,O). The data reveal broad inelastic maxima on the neutron energy-loss side of a strong line centered at zero energy. The pronounced elastic component arises in this case from the maximum of diffuse scattering at Q = (2n/a)(l, 4,O). The inelastic peaks cannot be attributed to phonon scattering, because, in this region of reciprocal space, the phonon energies in Cdo.&ln0.6sTe are far above the energy range covered by the measurements (Gebicki and Nazarewicz, 1978). Thus, the data clearly indicate the existence of magnon-like excitations. Earlier Raman scattering studies in Cdl -,Mn,Te (Ramdas, 1982; Grynberg and Picquart, 1981; see also Ramdas and Rodriguez, this volume) clearly exhibited the existence of magnon modes at low temperatures. Results of measurements of the inelastic neutron spectra in Cd0.3sMn0.6sTe at Q = (2n/u)(l, 0.5,O) and several other points along the [lo01 and [OlO] directions in Q-space, up to Q = (2n/a)(2,0.5,0) and (2n/a)(l, 1,0), respectively (corresponding to the boundary of the Brillouin zone for the type-I11 antiferromagnetic structure), are displayed in Fig. 16b, together with the magnon energy determined in the Raman experiments. All scans displayed in the figure were done at T = 4.2 K, using the variable incident energy spectrometer mode. The plot shows the positions of the maxima of the observed inelastic spectra, and their linewidths. As can be seen in the

174

T. M. GIEBULTOWICZ AND T. M . HOLDEN

800 In

0 0 0 (D

\

m

I-

z

600-

3

8

400 -

I. 0

I 1

I 2

I 3

1

4

I 5

I 6

ENERGY TRANSFER (meV 1

FIG. 15. Inelastic neutron scattering spectra for Cdo.~jMn0.6jTeat T = 1.8 K for two different wavevectors. The lines are guides for the eye. [After Giebultowicz el at. (1984).]

figure, the excitation energy increases away from the superstructure point Q = (2n/a)(l, *, 0) in both directions, but the growth is faster for the “inplane” direction [ 1001. Such behavior, reflecting the features of spin-wave dispersion relations in type-Ill structure (see Sec. 3a, Eqs. (36)-(42)), confirms that the excitations in the spin system in Cdo.3sMno.ssTe have the character of propagating magnon modes. Unusually large linewidths of the inelastic spectra observed in Cdo.35Mno.ssTe indicate at the same time that the magnon excitations are strongly damped. The experimental data cannot therefore be interpreted quantitatively in terms of Eqs. (36)-(42), which apply to the case of undamped spin-waves. For the purposes of quantitative analysis one can, however, utilize the model of spin dynamics in diluted FCC antiferromagnets developed by Chiang and Huber (see Sec. 3b), which enables one to calculate the dynamic structure factor S(Q, a), Eqs. (5)-(6). As we note in Sec. 6, the theoretical dynamic structure factor can be directly compared with the inelastic neutron scattering data obtained by the variable incident energy method. Problems such as excitation damping and contributions of different domain types (see the comment in Sec. 3a) are inherently solved by this modelling technique.

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NEUTRON DIFFRACTION STUDIES

175

112.0 1 ( I,I Wavevector

FIG. 16. Results of model calculation (a) and measurements (b) of the inelastic magnetic scattering in Cd0.35Mno.65Te for various momentum transfers. The circles and the squares indicate, respectively, the calculated and measured positions of the spectral maximum, and the bars in both plots show the spectrum width at half maximum (not the standard deviation!). The energy of the modes observed in Raman experiments (see text) is shown in (b) by the arrow. [After Giebultowicz et al. (1986a).]

The initial step in the modelling method of Ching and Huber is a numerical simulation of the ground state spin configuration, which is not known apriori. It must be checked subsequently that the results agree with the actual spin arrangement in the material. In the case of Cdl-,Mn,Te, a comparison with diffraction data indeed confirms that the numerical procedure reproduces satisfactorily the short-range antiferromagnetic order seen in this compound (see Sec. 7f). Results of the calculation of S(Q, o)for a model of Cd0.3sMn0.65Te(i.e., for x = 0.65 and JZ = 0.1J1)displayed in Fig. 16a show that the theory describes the measured dynamic characteristics quite well. The fit of the model values to the measured data indicates that JI (the only free parameter in the calculations) in Cdl-,Mn,Te is (14 f 1) K. The results presented in this section have not been corrected for twinning effects in the sample (see Sec. 6 ) . In general, the scattering vector Q for a twinned grain is different from the remainder of the crystal, so that the scattering contributions from such grains may lead to some distortion of the measured spectra. It is primarily for this reason that the simplified

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T . M. GIEBULTOWICZ AND T . M. HOLDEN

method of data analysis (concerning only the widths and positions of the maxima, but not the exact profiles) has been applied. In order to carry out a detailed analysis of the inelastic lineshapes, in a more recent experiment (Giebultowicz et al., 1986c), the scans and calculations were done for the Qpoints lying along the [120] and [ l i O ] directions in the (001) plane. As explained in Sec. 6, the scattering vectors for the two twinned components are equivalent in these two special cases, making corrections for the twinning unnecessary. Examples of fits of the calculated S(Q, w ) curves to the measured data (including corrections for the instrumental effects and for the background) are displayed in Fig. 17. As clearly illustrated by these plots, only for Q-vectors in the vicinity of the superstructure point ( 2 n / a ) ( l , 0) is there a certain discrepancy between the measured and the calculated

4,

ENERGY TRANSFER (meV )

FIG. 17. Comparison of the experimental and theoretical inelastic magnetic neutron scattering lineshapes for Cdo.&n0,65Te for several Q-vectors. The circles show the intensities measured using a variable incident energy spectrometer operation mode vs. a fixed monitor count rate. The solid lines show the fits of the theoretical lineshapes (convoluted with the instrumental resolution) to these data. The values of the nearest-neighbor exchange constant JI (the only adjustable parameter) obtained from each fit are displayed on the plots. [Giebultowicz e l al. (1986c).]

4.

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177

lineshapes. At all other investigated points the model of Ching and Huber provides an excellent description of the experimental data. The values of the exchange constant J1obtained from these fits are very close in each case. The average result for all runs, JI = (13.8 f 0.2) K, is in very good agreement with the value J1 = (13.8 k 0.3) K determined by Lewicki et al. (1986) (see also Spalek et al., 1986), from the high-temperature magnetic susceptibility, and close to the value J1 = (12.6 & 0.6) K reported by Larson et al. (1986) for very dilute systems ( x < 0.05). The lack of quantitative agreement for Q = (27c/a)(l, 3,O)is probably due to some shortcomings of the model, such as the size of the model “supercell” being too small. It can be concluded that the excitation mode observed in neutron diffraction is probably the same as that seen in Raman measurements. The measured energies are in fair agreement in both cases. The point in Q-space, Q = (2n/a)(l, *,O), at which the mode is observed in the neutron case is consistent with the interpretation of the Raman data in terms of a onemagnon process at the zone center proposed by Ramdas (1982). Thus the neutron scattering data do not confirm the interpretation in terms of a twomagnon Raman process (Grynberg and Picquart, 1981), primarily by showing that there are no excitations in Cd0.35Mn0.6sTe with energies sufficiently low to be consistent with such a description.

IV. Concluding Remarks As we have shown in this chapter, the information obtained from neutron scattering experiments on single crystals and on powdered samples is complementary, and both techniques are equally valuable. For example, the powder data in Znl-,Mn,Te yielded the value of the magnetic moment of the Mn2+ ions. This quantity, in principle, can also be extracted from the single crystal data, provided that the absolute value of the scattering cross section is determined. This is, however, not easy in practice because of problems associated with extinction effects (see Bacon, 1975). Moreover, such determination would be further complicated due to the twinning effects that commonly occur in Cdl-,Mn,Te and Zn, -,Mn,Te monocrystals. Similarly, such important parameters as the correlation coefficients (A - Br) were much more easy to deduce from the powder data. The analysis of the single crystal data is, in principle, also capable of yielding this information (even with additional details not available in the powder method-see Eqs. (16)-(17) in Sec. 2b), but to collect the necessary data one would in this case have to perform scans in three different directions in Qspace-a rather laborious and time consuming task. On the other hand, the single crystal method is the only one enabling us to study anisotropic effects, such as the anisotropy of the correlation length K - observed ~ in Cdl-,Mn,Te.

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The results of the low temperature neutron scattering studies in Znl-,Mn,Te and Cdl-,Mn,Te clearly show that both substances are quite similar as far as their macroscopic magnetic properties and, in particular, the nature of the spin ordering, are concerned. Apart from minor differences, the behavior in both materials can be described quite well in terms of the “zero-temperature” model of Ching and Huber (described in Sec. 3), which takes into account both first- and second-nearest neighbor exchange interactions. The model is particularly sensitive to the ratio of these two exchange constants, J 2 / J 1 . Neutron diffraction data in Cdl-,Mn,Te yield J 2 / J 1 = 0.12, in agreement with an earlier estimate of Escorne and Mauger (1982) who deduced from their magnetic susceptibility measurements the value J2/J1 = 6. The powder data in Znl-,MnxTe do not provide such an accurate determination of J d J 1 , but they do suggest that its value is about 0.1, in accord with that obtained for Cdl-,Mn,Te. In a recent determination, Denissen (1986) found that his magnetic susceptibility data in Znl-,MnxTe were consistent with an exchange constant which depends on the Mn-Mn distance as J(R) R-6.8. This relationship corresponds to the ratio JdJ1 = 0.095, in good agreement with the estimates based on the ChingHuber model and on the neutron scattering data. The conclusion emerging from the neutron scattering studies that the magnetic ordering in both Znl-,MnXTe and Cdl-,Mn,Te are of the same type is fully consistent with our present views concerning the nature of the exchange interactions in these materials. It was shown rather convincingly (Larson et al., 1985; Spalek et al., 1986)that the dominant mechanism of this interaction is that of the superexchange, involving the mediation of the chalcogen ion. Since this is a common factor in both materials of interest to us here, the similarities displayed by the magnetic properties of Znl -,MnxTe and Cdl-,Mn,Te are not surprising. The observed behavior of the inelastic neutron scattering in Cd0.3sMn0.6sTedemonstrates that the Ching-Huber model is also successful in describing the dynamic properties of the spin system in DMS. In particular, the value of J I , determined from inelastic scattering data by the Ching-Huber analysis, is in very good agreement with independent determinations of this quantity from the high temperature susceptibility data and from the positions of the magnetization steps. We add parenthetically that recent inelastic neutron scattering experiments on very dilute (x = 0.01 -0.05) samples of Znl-,Mn,Te, Znl-,Mn,Se, and Znl-,Mn,S by Corliss et al. (1986) and Giebultowicz et al. (1987b), involving transitions between various energy levels of a Mn-Mn pair, rendered precise values of the exchange constant JI, also in very good agreement with previous determinations. Furthermore, the inelastic neutron scattering experiments involving magnons helped to decide-by favoring the one-magnon process-which of

-

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the two alternative explanations of the origin of the magnon mode observed in Raman scattering studies in Cdl-,Mn,Te is more likely. All these successes of the Ching-Huber model tempted its authors to apply their approach to the case of the heat capacity calculations (Ching and Huber, 1984), again with good results. To summarize the results of the neutron scattering investigations in Cdl-,Mn,Te and Znl-,Mn,Te, let us repeat that both materials exhibit the same short-range antiferromagnetic ordering of type-111, at least for 0.35 C x < 0.7. The temperature behavior of the magnetic scattering observed in all investigated samples shows features typical of spin-glasses, thus providing evidence against the spin-glass antiferromagnet phase boundary in the vicinity of x = 0.6. Unfortunately, the existing neutron scattering data do not give us much insight into the mechanism of the spinglass transition itself. This is because of the fact that the methods employed so far were sensitive to the spin-spin correlations in space, while it is the temporal behavior of the correlations that is of importance in the context of spin-glasses. Methods involving high energy resolution, such as the time-offlight method and spin-echo spectrometry (see Mezei, 1982), are very promising in this context. Experiments of this kind should be undertaken on DMS in order to extend our understanding of the nature of the various clustering and spin-glass effects that take place in these materials.

References Aggarwal, R. L., Jasperson, S. N., Becla, P., andGalazka, R. R. (1985). Phys. Rev. 832,5132. Alexander, S . , and Pincus, P . (1980). J. Phys. A 13, 263. Als-Nielsen, J. (1976). In “Phase Transitions and Critical Phenomena,” 5A (C. Domb and S. M. Green, eds.), Academic Press, London. Anderson, P. W. (1053). Phys. Rev. 79, 705. Bacon, G. E. (1975). “Neutron Diffraction.” Clarendon Press, Oxford. Balzarotti, A., Motta, T., Kisiel, A., Zimnal-Starnawska, M., Czyiyk, M. T., and Podgorny, M . (1975). Phys. Rev. B 31, 7526. Binder, K., Kinzel, W., and Stauffer, D. (1979). Z . Phys. B 36, 161. Binder, K., and Schroder, K. (1976). Phys. Rev. B 14, 2142. Brockhouse, B. N. (1961). In “Inelastic Scattering of Neutrons from Solids and Liquids,” Vol. 1 (IAEA, Vienna, 1961). p. 147. Burke, S. K., Cywinski, R., Davis, J. R., and Rainford, B. D. (1983). J. Phys. F: Met. Phys. 13, 451.

Brockhouse, B. N., Becka, L. N., Rao, K. R., Woods, A. D.(1963). In “Second Symposium on Inelastic Scattering of Neutrons from Solids and Liquids,” Vol. 11 (IAEA, Vienna, 1963), p. 23. Ching, W. Y., and Huber, D. L. (1981). J. Appl. Phys. 52, 1715. Ching, W. Y., and Huber, D. L. (1982a). Phys. Rev. B 25, 5761. Ching, W. Y., and Huber, D. L. (1982b). Phys. Rev. B 26, 6164. Ching, W. Y., and Huber, D. L. (1983). Unpublished. Ching, W. Y., and Huber, D. L. (1984). Phys. Rev. B 30, 179.

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Ching, W. Y., and Huber, D. L. (1985). Unpublished. Corliss, L., Elliott, N., and Hastings, J. (1956). Phys. Rev. 104, 924. Corliss, L. M., Hastings, J. M., Shapiro, S. M., Shapka, Y., and Becla, P. (1986). Phys. Rev. B 33, 608. Cowley, R. A. (1980). Phil. Trans. Roy. SOC.Lond. B290, 583. Cowley, R. A,, Shirane, G., Birgeneau, R. J., Svensson, E. C., and Guggenheim, H . J. (1980). Phys. Rev. B 22, 4412. Denissen, C. M. J. (1986). “Analysis of the Magnetic Properties of Semimagnetic Semiconductors”, Ph.D Thesis, Technical University, Eindhoven. The Netherlands, unpublished. de Seze, L. (1977), J. Phys. C 10, L 353. Dolling, G., Holden, T. M., Sears, V. F., Furdyna, J. K., and Giriat, W. (1982). J . Appl. Phys. 53, 7644. Escorne, M., and Mauger, A . (1982). Phys. Rev. B 25, 4674. Fernandez, J. A . , Farah, H. A . , Poole, C. P., and Puma, M. (1983). Phys. Rev. B 27, 4274. Fisher, M. E. (1964). Am. J. Phys. 32, 343. Galazka, R . R., Nagata, S., and Keesom, P. H. (1980). Phys. Rev. B 22, 3344. Giebultowicz, T., Kepa, H., Buras, B., Clausen, K., and Galazka, R. R. (1981). Solid State Commun. 40, 499. Giebultowicz, T. M., Lebech, B., and Buras, B. (1982a). Unpublished. Giebultowicz, T., Minor, W . , Buras, B., Lebech, B., and Galazka, R. R. (1982b). Phys. Scr. 25, 731. Giebultowicz, T., Minor, W., Kepa, H., Ginter, J., andGalazka, R. R. (1982~).J. Magn. Magn. Muter. 30, 215. Giebultowicz, T., Lebech, B., Buras, B., Minor, W., Kepa, H., and Galazka, R. R. (1984). J. Appl. Phys. 55(6), 2305. Giebultowicz, T. M., Rhyne, J. J., Ching, W. Y., and Huber, D. L. (1985). J. Appl. Phys. 57(1), 3415. Giebultowicz, T. M., and Furdyna, J. K. (1985). J. Appl. Phys. 57(1), 3312. Giebultowicz, T. M. (1986). J. Magn. Magn. Mat. 54-57, 1287. Giebultowicz, T. M., Rhyne, J. J., Ching, W. Y., Huber, D. L., and Galazka, R. R. (1986a). In preparation. Giebultowicz, T. M., Rhyne, J . J., Ching, W. Y., Huber, D. L., and Galazka, R. R. (1986b). J. Magn. Magn. Mat. 54-57, 1149. Giebultowicz, T. M., Rhyne, J. J., Ching, W . Y . , Huber, D. L . , Lebech, B., and Galazka, R. R. (1986~).In preparation. Giebultowicz, T. M., Rhyne, J. J., Furdyna, J . K., and Debska, U. (1987a). Proc. 31st Conf. Magn. Magn. Muter, Baltimore, 1986, to appear in J. Appl. Phys. Giebultowicz, T. M., Rhyne, J. J., and Furdyna, J. K. (1987b). Proc. 31sr Conf. Magn. Magn. Marer, Baltimore, 1986, to appear in J. Appl. Phys. Gebicki, W . , and Nazarewicz, W. (1978). Phys. Stat. Sol. (b) 86, K135. Grest, C. S., and Gabl, E. F. (1979). Phys. Rev. Lett. 43, 1182. Grynberg, M., and Picquart, M. (1981). J. Phys. C 14, 14677. Hastings, J. M., Corliss, L. M., Kunnmann, W . , and Mukamel, D. (1981). Phys. Rev. B. 24, 1388. Holden, T. M., Dolling, G., Sears, V. F., Furdyna, J. K., and Giriat, W. (1982). Phys. Rev. B 26, 5074. Huang, C. Y. (1985). J. Magn. Magn. Muter. 51, 1. Izyumov, Ya., and Ozerov, R. P. (1970). “Magnetic Neutron Diffraction,” Plenum Press, New York.

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Keffer, F. (1962). Phys. Rev. 126, 896. Koester, L. (1977). In “Neutron Physics (Springer Tracts in Modern Physics)” (G. Hohler, ed.), Vol. 80, p. 1. Springer, Berlin. Larson, B. E., Hass, K. C., Ehrenreich, H., and Carlsson, A. E. (1985). Solid State Commun. 56, 347. Larson, B. E., H a s , K. C., and Aggarwal, R. L. (1986). Phys. Rev. B 33, 1789. Lewicki, A., Spalek, J., Furdyna, J. K., and Galazka, R. R. (1986). J. Mugn. Mugn. Muter. 54-57, 1221. Lovesey, S. W. (1984). “Theory of Neutron Scattering from Condensed Matter.” Clarendon Press, Oxford. Maletta, H., Aeppli, G., and Shapiro, S. M. (1982). Phys. Rev. Lett. 48, 1490. Marshall, W., and Lovesey, S. W. (1971). “Theory of Thermal Neutron Scattering.” Clarendon Press, Oxford. McAlister, S. P., Furdyna, J. K., and Giriat, W. (1984). Phys. Rev. B. 29, 1310. Mezei, F. (1981). J. Appl. Phys. 53, 7654. Nagele, W., Knorr, K., Prandl, W., Convert, P,, and Buevoz, J. L. (1978). J. Phys. C. 11, 3295. Phani, M. K., Leibowitz, J. L., and Kalos, H. M. (1980). Phys. Rev. B 21, 4027. Ramdas, A. K. (1982). J. Appl. Phys. 53, 7649. Sears, V. F. (1975a). Adv. Phys. 24, 1. Sears, V. F. (1975b). fiTucl.Instrum. Methods 123, 521. Sears, V. F. (1978). “Theory of Thermal Neutron Scattering.” Atomic Energy of Canada Ltd., Publ. #6326. Smart, J. S. (1966). “Effective Field Theories in Magnetism.” Saunders, Philadelphia (1966). Spalek, J., Lewicki, A., Tarnawski, Z., Furdyna, J. K., and Galazka, R. R. (1986). Phys. Rev. E 33, 3407. Steigenberger, U., Lebech, B., and Galazka, R. R. (1986). J. Mugn.Mugn. Mu?. 54-57, 1285. Steigenberger, U., and Schaerpf, 0. (1986). In “Neutron Scattering,” Proceedings of the International Conference on Neutron Scattering, Sante Fe, August 1985, North Holland, Amsterdam, p. 87. Ter Haar, D., and Lines, M. E. (1962). Proc. Roy. SOC. 255, A1049. Villain, J. (1979). Z. Phys. B 33, 31. Warren, €3. E. (1969). X-ruy Diffruction, Addison-Wesley, Reading.

SEMICONDUCTORS AND SEMIMETALS, VOL. 25

CHAPTER 5

Band Structure and Quantum Transport Phenomena in Narrow-Gap Diluted Magnetic Semiconductors J. Kossut INSTITUTE OF PHYSICS, POLISH ACADEMY OF SCIENCES WARSAW, POLAND

I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . INTERACTION BETWEEN MOBILES OR p CARRIERS AND 11. EXCHANGE LOCALIZED 3d OR 4f ELECTRONS IN SEMICONDUCTORS. .... 1. Interaction Hamiltonian . . . . . . . . . . . . . . 2 . Exchange Constants . . . . . . . . . . . . . . . . 111. BANDSTRUCTURE OF NARROW-GAPDILUTED MAGNETICSEMICONDUCTORS IN QUANTIZING MAGNETIC FIELDS . . . . . . . IV. TRANSPORT MEASUREMENTS IN THE QUANTUM REGIME: BAND STRUCTURE MODEL. . . 3. Hgl-,MnxTe and Hgl -, M n, Se . . . . . . . . . 4. (Cdl-,Mn,),As2 . . . . . . . . . . . . . . . 5. Pbl-xMn,Te and Pbl -, M n, S. . . . . . . . . . V. TWO-DIMENSIONAL ELECTRON GASIN Hgl-,Mn,Te AND Hgl-,,Cd,MnYTe . . . . . . . . . . . . . . . 6. Experiments . . . . . . . . . . . . . . . . 7. Energy Levels of 2d Electron Gas. . . . . . . . 8. Future Possibilities . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . CONFIRMATION OF THE

. . .

. . . . . . . . . . . . . .

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183 185 185 186 190 204

204 214 216 211 217 219 224 225

I. Introduction In this chapter, we deal with narrow-gap diluted magnetic semiconductors

(DMS),known also as semimagnetic semiconductors. We focus our attention

on their band structure in the presence of a strong, quantizing magnetic field. The small values of the effective masses, characteristic of narrow-gap semiconductors, make them particularly suitable for investigations in the 183 Copyrighl 0 1988 by Academic Press. Inc. All rights of reproduction in any lorm reserved. ISBN 0-12-752125-9

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quantum regime. In fact, the diluted magnetic semiconductors with a narrow forbidden gap were probably the first to be studied experimentally (Delves, 1963 and 1966; Morrissy, 1973). Their puzzling properties remained largely unexplained until the model of their band structure has been proposed (Jaczynski et al., 1978; Bastard et al., 1978; see also the review paper by Galazka and Kossut, 1980). The model, based on the k * p approximation, assumes that two electronic subsystems can be distinguished. The first contains mobile delocalized electrons from the conduction and/or uppermost valence band. These electrons, which can be described in terms of the virtual crystal approximation, are mostly responsible for electrical and optical properties. The second subsystem consists of electrons from the 3d shells of, typically, Mn ions. It is assumed that these produce localized magnetic moments. The degree of the localization of the magnetic ions is still disputed. Only recently the calculation based on the coherent potential approximation started to give insight into this problem (Hass and Ehrenreich, 1983). The latter electronic subsystem is responsible for the magnetic properties of DMS. The interaction between these two subsystems produces a spectrum of anomalies in the electrical and optical properties of diluted magnetic semiconductor. The reciprocal influence of mobile electrons is more difficult to detect, because it would require much greater electron concentrations than usually met in semiconductors. Such an influence has been recently detected in a quaternary mixed crystal Pbl -x,Sn,MnyTe (Story et al., 1985). We shall assume throughout this chapter that no spontaneous magnetization exists in the materials considered, i.e., the ordering of the localized moments is always induced by an external magnetic field. The plan of this chapter is the following: Part I1 discusses in detail the interaction between two electronic subsystems. Next, Part I11 presents models of the band structure in a magnetic field for narrow gap DMS with the zinc blende lattice structure (e.g., Hgl-,Mn,Te). Part IV deals with experimental results for various DMS (including those with NaCl and tetragonal crystal structures) obtained in the course of investigations of quantum transport phenomena, mostly the Shubnikov de Haas effect. In this chapter we shall put stress on those features that are unique for the diluted magnetic semiconductors and that support the proposed model of the band structure. The final Part V is devoted to a discussion of a preliminary investigation of quasi two-dimensional electronic systems in diluted magnetic semiconductors. We do not attempt to quote here the values of all important band structure parameters determined for the narrow-gap diluted magnetic semiconductors. We refer the readers to the tables of these quantities compiled by Galazka and Kossut (1983).

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11. Exchange Interaction Between Mobile s or p Carriers

and Localized 3d or 4f Electrons in Semiconductors

1. INTERACTION HAMILTONIAN The exchange interaction between mobile, s-like band electrons and localized electrons from partially filled 3d (or 4f) shells was first considered in transition (or rare-earth) metals. Early ideas of Zener (1951a-c) and Vonsovskii and Turov (1953) led to the development of the so-called s-d (or s-f) model. The case of rare-earth metals was particularly stimulating in this development, since magnetic properties of these materials could hardly be explained in terms of direct exchange interaction involving the overlap of 4f wave functions localized on neighboring rare-earth atoms. The localization of these wave functions is so strong (thus their overlap so small) that the estimated values of the direct exchange constants are far too small to account, e.g., for magnetic transition temperatures observed. Therefore indirect coupling between the localized magnetic moments of rare-earth ions via conduction electrons was invoked in order to explain the observed magnetic behavior. Several authors dealt with the problem of the form of this coupling (see, e.g., Vonsovskii and Turov, 1953; Kasuya, 1955; Abrahams, 1955; Yosida, 1957; Mitchell, 1957). It was found that the exchange interaction between the conduction electrons and 3d or 4f electrons (i.e., those responsible for the formation of localized magnetic moments) can be represented by a simple Heisenberg-like formula

where s is the spin operator of an s-electron at position r, S, is the total spin operator of a 3d (or 4f) shell at R, and J(r - R,) is an appropriate exchange constant (a constant term has been omitted in Eq. (1)). It was argued that the function J(r - R,) is strongly peaked around Rn and quickly vanishes away from this point, reflecting the localized nature of the 3d (or 4f) electrons. The form given by Eq. (1) is not the most general one, since several assumptions have been made during its derivation. First, a constant number of electrons in the 3d (4f) shell associated with a given ion was assumed. In other words the magnitude of the localized spin was taken to be constant. It is possible to generalize the Hamiltonian and avoid this limitation (e.g., Vonsovskii and Svirskii, 1964; Irkhin, 1966). However we shall restrict our attention to the simpler case, assuming a constant value of the localized spin, as experimental data obtained on diluted magnetic semiconductors seem to suggest.

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The second assumption made when deriving Eq. (1) concerns the validity of the description of the localized moments in terms of the spin eigenvalues. Such description often requires the use of a total angular momentum J, instead of the spin only. It was shown by Liu (1961) that even in such a case a simple form involving the scalar product can be retained if S, in Eq. (1) is replaced by J, and the coupling coefficient J(r - R,) is slightly redefined:

Conversely, when mobile band electrons are subject to a strong spin-orbit interaction (as is commonly the case in semiconductors, e.g., in the valence band p-like states in zinc blende III-V and II-VI compounds), it is more proper to label their states using eigenvalues of total angular momentum j. Then, the coupling Hamiltonian (that in this case corresponds to p-d coupling) can be conveniently expressed as:

All expressions quoted so far are isotropic. It can be shown that this is only an approximation, valid if spatial extensions of 3d (4f) wave functions are small compared to the de Brolie wavelength of mobile s- or p-electrons. Only then the mobile carriers do not “feel” an anisotropic character of 3d (40 functions. This condition is sometimes not easy to meet in metals. Therefore, a generalization of Eq. (1) was found by Kaplan and Lyons (1963) that is explicitly anisotropic. In semiconductors, however, the mobile carriers which usually play a role, are characterized by much smaller values of momenta k than in metals; that is, they have longer de Brolie wavelengths. This fact justifies the use of the isotropic expression (1) in our further considerations of diluted magnetic semiconductors.

2. EXCHANGE CONSTANTS We shall now turn our attention to the nonlocal “potential” J(r - Rn) appearing in Eq. (1). All papers quoted in the preceding section defined rather the Fourier transform of J(r - Rn) than this function itself. This transform can be written as J(k, k’)

= -2

.r

q&(r)+;(r’)V(r

- rr)~,lk’(r’)+d(r)d3rrd3r

(4)

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where W n k denotes the spin-independent part of the wave function of a mobile band electron in the n-th band with a wave vector k, 6 d is an appropriate wave function of 3d (or 4f) electrons (possibly integrated over all spatial variables but one) and V(r - r’) is the electrostatic interaction between the electrons. One may interpret Eq. (4) as a direct exchange interaction between the particles in question. A problem arises concerning the actual dependence of J(k, k’) on the mobile electron momenta k and k‘. This dependence is difficult to extract from Eq. (4) in a general case. In metals, one often argues that, since only the electrons from the Fermi surface are important in the majority of physical situations, it is possible to replace k and k’ in Eq. (4) by their values at this surface; that is, for a spherical Fermi surface they can be replaced by a single parameter k~ .Thus, one is left with only one exchange constant. Its value has been estimated theoretically on the basis of Eq. (4) by several authors (e.g., Izyumov and Noskova, 1962; Kasuya, 1966; Kasuya and Lyons, 1966; Watson and Freeman, 1966 and 1969). These estimations give values in the range 10-3-10-’ eV. Slightly different reasoning has to be invoked in the case of semiconductors, which is of interest to us here. A most widely used description of band electrons in semiconductors is in terms of the k * p perturbation approach (Luttinger and Kohn, 1955). Usually, one is interested only in electrons that are in the vicinity of band extrema. Therefore, the wave function W n k is sought as a combination of the wave functions at various band edges un (n is the band index; for the sake of simplicity let us assume now that the band extrema occur at k = 0),

where an(k) are coefficients of the combination. The sum in Eq. (5) can often be restricted to only a few terms. This is particularly true in the case of narrow-gap semiconductors. Then the coefficients an(k) can be found exactly together with their explicit dependence on the wave vector k. Inserting wave functions ( 5 ) into (4), we obtain

(6) As we can see a part of the k, k’ dependence is already taken out from the integral. The remaining k-dependence is of lesser importance. As we

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mentioned above, the wave vectors of band electrons usually encountered in semiconductors are rather small in magnitude. Because of the strongly localized character of the f$d functions, the main contribution to the integral in Eq. (6) comes from the region of small r and r’. In this region the exponential factors in the integrand can be approximated by 1 . So, the whole dependence of J(k, k’) on k, k’ is now given by the factors an*(k)and a,,(k’). One may now define a set of basic exchange constants, all of them independent of k , k’ (Kossut, 1976)

Jnn,= - 2 uE(r)f$2(r’)V(r - rf)un,(r‘)f$d(r)d3rd3r‘.

(7)

Moreover, when symmetry arguments are invoked, only the “diagonal” elements J,, can be shown to be nonvanishing. We shall denote them by J, . Two of these basic exchange constants are very commonly encountered in diluted magnetic semiconductors. They involve s-like electrons from what usually is a conduction band of rs symmetry in semiconductors with the zinc blende lattice, and p-like electrons from the Tg symmetry band. Traditionally they are denoted, respectively, by a and p in the literature dealing with diluted magnetic semiconductors:

where ur6 = S&’/’S(r) and tirs = Qo’/’X(r)while SZO stands for the volume of a unit cell. The elements as ( S I J I S ) , etc., commonly used in the literature, represent symbolically the integration indicated by Eq. (7) with appropriate

wave functions un of the electron at the band edge. In the course of various investigations of DMS it was established (see other chapters in this volume) that the constants a and p differed in sign: j? turns out to be positive (“antiferromagnetic”) while a is negative (“ferromagnetic”). Usually the absolute value of p is greater than a. On the basis of Eq. (7), on the other hand, one could expect a single sign (“ferromagnetic”) for both constants, Recently, two attempts to solve this puzzling fact have been published (Semenov and Shanina, 1981 and Bhattacharjee et al., 1983). The first of these two papers notes that, apart from the term given by Eq. (4), there is also a contribution arising from the overlap integral of the 3d wavefunction and wavefunctions of the band electron (this contribution also leads to a

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189

general form of the interaction Hamiltonian (3)). This contribution is opposite in sign to that given by the direct exchange mechanism represented by Eq. (4). This is of particular importance for p-like electrons, which usually form the valence band in semiconductors, while for s-like conduction band electrons it can be practically neglected. Therefore, it can explain the difference in sign of a and p. The importance of the overlap term in the case of p-like electrons follows from the fact that these electrons originate mainly from the anions (e.g., Te ions in Cdl-xMnxTe). Therefore, the corresponding wave functions are peaked at the anion lattice sites (say, in a tight binding scheme), which are the nearest neighbors of a given Mn ion. Thus, the overlap of these functions and 3d wave functions of Mn may be appreciable. On the other hand, the s-like electrons originate mainly from cation sublattice atoms, i.e., either from Mn atoms themselves or from more distant Cd or Mn ions. In the former case their wave functions are orthogonal to the 3d functions (same ion). In the latter situation the overlap produced must be smaller because of a greater distance between the ions. It is then concluded that in the case of the constant a the direct (ferromagnetic) exchange mechanism dominates, while the overlap contribution determines the value of p. Indeed, estimates by Semenov and Shanina (1981) give an order of magnitude for IayI p = 0.1 + 1 eV and a < 0. Note that our definition (1) differs in sign from the definition used by Semenov and Shanina. A different approach to the problem of sign of a and p is taken in the paper by Bhattacharjee et al. (1983). It stresses the importance of hybridization of the localized d and the valence-band p electrons. On the basis of the virtual bound state approach (Anderson, 1961; Friedel, 1958), it is shown that the Anderson Hamiltonian (which by the Schrieffer and Wolff (1966) transformation is equivalent to the s-d model) for reasonable model parameters leads to a positive (“antiferromagnetic”) exchange constant p for the p-like valence electrons. The hybridization is not effective in the case of a. This is partly because the virtual bound state due to d electrons falls into the valence band and is energetically quite distant from the conduction band. The energy denominators appearing in the theory reduce then quite strongly the contribution to a due to the hybridization. Also, the relevant hybridization matrix elements are smaller for s-like than for p-like electrons. Thus the normal direct exchange mechanism is mainly responsible for the value and sign of a , while an interplay between direct exchange and hybridization determines /3. It is difficult to decide at the present stage, which of the two approaches describes better the physical reality. None of the experiments (e.g., experiments on samples under hydrostatic pressure, which would affect the overlap integrals discussed by Semenov and Shanina, 1981) could distinguish

=

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between the models. One is, then, still in a situation in which the relevant exchange constants have to be treated as phenomenological quantities, with values treated as adjustable parameters of the given model of a diluted magnetic semiconductor.

111. Band Structure of Narrow-Gap Diluted Magnetic Semiconductors in Quantizing Magnetic Fields

The majority of papers dealing with narrow-gap diluted magnetic semiconductors is devoted to alloys based on 11-VI mercury compounds, e.g., Hgl-,Mn,Te or Hgl-,Mn,Se. They were also the first studied experimentally in an extensive manner. Therefore, we describe here the band structure of these materials. We shall discuss briefly the band structure of tetragonal (Cdl-,Mn,)3Asz and DMS based on lead chalcogenides (or rather only the modifications of the general model associated with their different crystallographic symmetries), when we describe in the following sections the experimental results obtained for these materials. Early investigations (Morrissy, 1973; Delves, 1963and 1966)indicated that semiconducting properties of Hgl -,Mn,Te cannot be entirely understood in terms of existing theoretical models that could be successfully applied to such semiconducting mixed crystals as Hgl -,Cd,Te. The unexplained features of Hgl -,Mn,Te were a non-monotonic temperature behavior of the Shubnikovde Haas oscillation amplitude (Morrissy, 1973) and the value of the spinsplitting of electrons and its temperature dependence observed in magnetooptical study of Bastard et al. (1978) (see also Rigaux, this volume). On the other hand, when no strong magnetic field was applied, the band structure of Hgl-,MnxTe appeared to be qualitatively the same as in Hgl-,Cd,Te, as indicated in the study of Stankiewicz et al. (1975). As in other narrow-gap materials with the zinc blende lattice symmetry, the minimum of the conduction band and the maximum of the valence band occur at the central point r of the Brillouin zone. There are three bands with similar energies (see Fig. 1): a fourfold degenerate band of TSsymmetry, and doubly degenerate r6 and r7 symmetry bands. The sequence of these bands varies with the crystal composition. For low values of Mn mole fractionx, we deal with the so-called inverted band ordering (or symmetry-induced zero-gap configuration) depicted in Fig. la. For increasing values of x, r6 valence band shifts upwards, and at x = 0.065 (at 4.2 K), it coincides with the position of the TS band (Fig. 1b). Above this semimetal-semiconductor transition point, r6 band starts to form the conduction band (Fig. lc) while the TSband becomes the light and heavy hole band. The energy gap Eg defined as the energy difference between TSand r6 levels is then positive, and we are dealing with

5.

QUANTUM TRANSPORT IN DMS

191

x A

FIG.1 . The energy level scheme in the vicinity of the conduction and valence band edges in narrow-gap semiconducting mixed crystals with zinc blende structure: (a) inverted (zero-gap) band structure; (b) semimetal-semiconductor transition point; (c) open-gap structure.

the standard semiconducting band ordering. The variation of the energy gap with the crystal composition in Hgl-,Mn,Te is shown in Fig. 2. The r7 symmetry band (or spin-orbit split-off band) remains a deeper valence band. It is well known from the work of Kane (1957) that the proper description of a narrow-gap semiconductor requires that at least the three bands r6, r7 and l-8 , because of their energetic proximity, are considered at the same level of accuracy in a k p treatment of the band structure. The remaining more remote bands, often referred to as “higher bands”, may be treated in an approximative way. Their influence on the shape of the conduction and valence bands is usually evaluated with accuracy to terms proportional to k2, where k is the electronic wave vector. Only in the case of HgSe, and of mixed crystals similar in composition to HgSe, does one have to include in the set of closely spaced (quasi-degenerate) bands also the nearest higher band of r15 (no spin notation) symmetry, as found by Mycielski et al. (1982). This fact is connected with the relation EE = 2/3A, where A is the spin-orbit splitting (or the distance between TS and r7 bands), that is fulfilled in HgSe. However, a very steep slope of the Eg vs. x relationship in Hgl-xMn,Se (Dobrowolska et al., 1981) limits this “anomalous” region to x s 0.0005. Also, the modifications due to such enlargement of the quasi-degenerate set of bands concern mainly the r6 band (light holes) and therefore are possibly important in an analysis of the interband magnetoabsorption

-

192

J. KOSSUT

500

x

400 300

-> 2 0 0

M

* X

E

I

0 1 00

W

I

0

0.05

I

0.10

I

0.15

1

0.20

0 un

-100

-200

A

X

A

ml 54

? I

-3 00 X

FIG.2. Variation of the energy gap with composition of Hg,-,Mn,Te. Kossut (1983).]

[After Galazka and

experiments. In this chapter, which is focused mainly on the magnetotransport phenomena in n-type DMS, we shall neglect this complication in further considerations. There exist in the literature on narrow-gap semiconductors the simplified models of their band structure in a presence of a strong magnetic field. In particular, the so-called quasi-Ge model of Luttinger (1 956) includes into the quasi-degenerate set of states only the Tg band. The three-band model (e.g., Kacman and Zawadzki, 1971) although considering in detail all closely positioned r6, l-7 and TSbands, neglects totally the influence of the higher bands. The advantage of these models lies in the fact that they can be solved analytically. However, their applicability is only of limited range.

5 . QUANTUM TRANSPORT IN DMS

193

In particular, the spin-splitting of electronic states, a quantity of primary importance in the case of diluted magnetic semiconductors, is only approximately rendered by these models. Moreover, when the exchange interaction between electrons from the r6, G , T8 bands and 3d electrons forming magnetic moments localized on Mn ions is taken into account, even these simpler models cannot, in general, be solved in an analytic way without further simplifications. Therefore, we shall base our further calculations on the model of Pidgeon and Brown (1966) that, although only numerically soluble for standard narrow-gap semiconductors, takes fully into account all the important bands. The Pidgeon-Brown model treats within the k p approximation the bands r6, r7, Ts as quasi-degenerate and accounts for the higher bands in a perturbative way by means of several parameters. As is usually the case when starting a k * p calculation, one has to define the set of basis functions at the band extrema (often called the Luttinger-Kohn amplitudes). We choose this set in a standard form for the crystals of zinc blende symmetry.

where the Luttinger-Kohn amplitude S transforms under the operations from the rd group as an atomic s-like function and the amplitudes X , Y , and Z as atomic p x , p y , p,-like functions, respectively. The symbols t and 1 denote the Pauli spinors. The matrix elements of the k p Hamiltonian in the

-

194

J . KOSSUT

presence of the external magnetic field form an 8 x 8 matrix. Rearranging the sequence of the basis functions to u1, u3, u5, u 7 , u2, u 4 , US, Us we can write this matrix (Leung and Liu, 1973) as

Db,and Dcare defined by Eqs. (14), (15), and (16) where the matrices Da, with the following notation

c = -h2, mo

s=-

eH hc '

with mo being the free electron mass.

I

Eg + 2CsF(N + 9)

+ Cs(N + 1) + (F + +)ck:

iP&a

+

5.

195

QUANTUM TRANSPORT IN DMS

D, = i

- -Pk,

43

- C a y 3 k,a

- 3 N y 3 kza+

0

I

The operators a and a+ in Eqs. (14)-(16) are the harmonic oscillator lowering and raising operators, respectively, defined as

196

J . KOSSUT

and N = a’a.

P is the interband momentum matrix element

and the parameters y 1 , y 2 , y 3 , K , and F describe the coupling to the higher bands. The eigenvectors of the matrix (11) can be sought in the form

Fn

=

(LyLz)-”2exp[i(kyy

+ k,z)]

where 4 n ( c ) is the harmonic oscillator eigenfunction of the variable [ = x - ky/S. Acting with the matrix (1 1) on the eigenvector given by Eq. (21) we obtain the matrix which can be diagonalized numerically in order to find the eigenvalues for a given magnetic field and harmonic oscillator quantum number n and k,. Let us note further that if we put k, = 0 (i.e., we are interested in the energy positions of the Landau subband minima), the whole matrix D can be partially decomposed into two noninteracting matrices Da and Db. The matrix D,,Eq. (16), is then equal to zero. We are then left with two independent eigenproblems, each 4 x 4 dimensional, the solution of which gives us in general two sets (a- and b-set) of eigenenergies. Note that, in the case of the Landau quantum number n = - 1 one has to put all coefficients aj in Eq. (21) equal to zero except for a4. The single energy level obtained in such case, which is very easy to find, corresponds to the uppermost heavy hole valence band Landau level. Similarly, when n = 0 one has to put a1 = a3 = a6 = as = 0 and for n = 1: a3 = 0. Having described in detail the model of the band structure of a narrow-gap semiconductor, let us return to the problem of a diluted magnetic semiconductor. As mentioned before one could not hope to describe the band structure in this case simply by an appropriate choice of a set of parameters

5 . QUANTUM TRANSPORT IN DMS

197

defined previously in this section. This fact is probably best illustrated by a simultaneous analysis of the effective mass and spin splitting of the conduction electron. The model described above predicts that the effective mass at the bottom of the conduction band is determined mainly by the ratio Eg/P2, the dependence on other parameters being less significant. Similarly, the electronic g-factor, that describes the spin splitting, is roughly determined by mo/m* (where we assumed A = 00). Thus, for a given crystal composition, i.e., for a given Egand P (Pbeing only slightly sensitive to the actual value of Mn mole fraction x), both the effective mass and the g-factor should be interrelated. The measured values of the effective masses in HgI-,Mn,Te were indeed in accordance with these predictions (Bastard et al., 1978; Jaczynski et al., 1978). However, the g-factors, as determined experimentally, turned out to be much greater (easily by a factor of 2) than these expectations and, moreover, they depend rather strongly on the temperature. Clearly, the band structure model was inadequate to account for these observations. The fact that only the spin properties seemed to be anomalous immediately suggested that the interaction with localized moments of Mn ions involving conduction electron spin is the source of the problem. As mentioned in Part I1 of this chapter, it is possible to represent this interaction by the Heisenberglike Hamiltonian H s d = C j J(r - Rj)S S j , with S j and s standing for spin operators of thej-th Mn ion and the band electron, respectively. Calculating the matrix elements of H s d with the basis functions, Eq. (lo), one obtains an additional matrix which must be added to Eq. (1 1) before the eigenvalues of our problem are found:

-

where

DL =

198

J . KOSSUT

0

DL =

where ST = SJ?f iSj’ and, as in Part 11, the only nonvanishing and independent exchange constants are defined as

Here QOis the unit cell volume, and NO = V / Q , is the number of cation sublattice sites. The matrix (22) is of rather complex form. The problem is further complicated by the fact that after addition of (22) to (1 l), its solutions can no longer be sought as a simple vector-column of harmonic oscillator eigenfunctions. The matrix (22) can be, however, considerably simplified when an approximation analogous to the mean field approximation from the physics of magnetism is invoked. Namely, the wave functions of conduction and valence band electrons are spread over the whole volume of the crystal. Therefore, one may say that these electrons interact with all magnetic moments present in a sample, thus “feeling” in effect the average value of the localized moments. One may thus say that the band electrons are subject to a certain average or resultant field of the localized moments. It is therefore permissible to carry out an averaging procedure over all possible states of Mn moments already at the level of the Hamiltonian matrix. For a paramagnetic system only the S, component of the localized spin does not vanish after the averaging procedure. This very major approximation was confirmed a posteriori by Gaj et af. (1979) who determined experimentally a direct proportionality between spin splittings and magnetization that is proportional to (Sz>x(note that C,(S~>/NO = x(S,>). After this averaging procedure the entire off-diagonal blocks DL in Eq. (22) vanish and one is left with two nearly diagonal 4 x 4 blocks DL and Di,. The matrix resulting from the addition of Eq. (1 1) and Eq. (22), with DL blocks

5.

QUANTUM TRANSPORT IN DMS

199

omitted, may then be solved in terms of the harmonic oscillator wavefunctions. The new parameters a and p are treated as adjustable parameters (their fitted values are discussed further in this chapter). The values of ( S z ) which depend on the crystal composition and also on the temperature and magnetic field strength are either calculated assuming some simple model (see, e.g., Galazka and Kossut, 1980) or taken from an independent experiment measuring the magnetization of the samples. An example of the diagonalization is shown in Fig. 3, where a series of the conduction and heavy hole Landau levels is shown for zero-gap Hgl-,Mn,Te at two temperatures. The strong temperature dependence of the band structure occurs via the terms involving ( S z ) . While the band structure at T = 30 K (Fig. 3b) resembles that of nonmagnetic Hgl-,Cd,Te with corresponding energy gap, the situation at low temperatures T = 1.4 K in Fig. 3a is qualitatively different. We shall enumerate the most important of these differences below.

Ordering of the Landau Levels The first thing to be noted in Fig. 3 is an enhancement of the low temperature spin splitting (marked by thick solid arrows), which was achieved by putting a positive value of p. As we pointed out earlier, such an enhancement was indeed observed experimentally. Note also that the higher of the spin-split Landau levels (b-set levels) are shifted so strongly upwards in low temperature that they cross the Landau levels from the m e t of solutions with a greater harmonic oscillator quantum number n. This feature constitutes a unique attribute of narrow-gap DMS: the spin splitting, being ruled mainly by the exchange interaction of the band carriers with the localized magnetic moments, may easily exceed the cyclotron splitting that is due to the effect of the external magnetic field on the orbital degrees of freedom of an electron or a hole. This is particularly true in the case of the heavy holes for which the cyclotron splitting is rather small because of the large value of their effective mass. Therefore, DMS may be viewed as systems where “amplification of the spin properties’’ takes place. The degree of this amplification is, furthermore, tunable by the temperature.

Conduction and Valence Band Overlap A second striking feature to be noted in Fig. 3 is the existence of the band overlap at low temperature. The presence of strong magnetic field lifts the degeneracy of the conduction and heavy hole bands in zero-gap semiconductors. In diluted magnetic semiconductors, the uppermost heavy hole subband (indicated by an arrow in Fig. 3) is shifted upwards, contrary to the case of, e.g., Hgl-,Cd,Te where this shift is directed downwards so that a

200

J. KOSSUT

-3

I

-3

-2

I

-2

-I

-I

0

0

I

2

I

3

I

2

FIG. 3. The Landau subband energies in Hg,-,Mn,Te, x = 0.025 calculated at a constant magnetic field and at (a) T = 1.4 K and (b) T = 30 K as a function of the wave vector k~ along the magnetic field direction. The field is applied along the ( 100) crystallographic axis. Thick vertical arrows indicate spin splitting of the lowest conduction-band Landau level. The valence band Landau level bu(- 1) is indicated by thin arrow. [After Dobrowolska et al. (1979); reprinted with permission of Pergamon Press, Ltd.]

5 . QUANTUM TRANSPORT IN DMS

201

real energy gap opens. Therefore, it may happen that the band gap in DMS opens only at elevated temperatures. The magnitude of the band overlap is a result of an interplay between this exchange interaction-induced shift and the movement of the lowest conduction band that occurs also in the upward direction on the energetic scale. For very small values of Eg(close to the semimetal-semiconductor transition), the conduction band levels (due to the smallness of their effective mass) move up faster than the top of the valence band, and the overlap may not occur even at very low temperature. Also, for sufficiently high magnetic fields the shift of the top of the valence band reverses its sign and the level beings to move downwards, as it does in standard narrow-gap semiconductors. Then the energy gap induced by the field starts to open. This strange behavior of the top of the valence band may be qualitatively understood when one bears in mind how ( S , ) depends on the temperature and magnetic field. Let us assume for a moment that we are dealing with a perfect paramagnet where the magnetic moments do not interact with each other. Then ( S , ) varies as 1/T, so that at higher temperatures the additional terms proportional to (S,) may become negligible compared to “normal” terms brought by Eq. (11). The magnetic field tends to align the magnetic moments along its direction. When all the moments are already aligned a saturation is reached and ( S , ) = - 512. The “normal” terms in Eq. ( l l ) , on the other hand, are directly proportional to the field. A value of the magnetic field may thus be reached when the contribution of these normal terms wins over the exchange induced contribution. For the top of the valence band such a return to “normalcy” occurs if the following condition is met:

The above general remarks remain true even if the interaction between the localized moments becomes appreciable and ( S , ) cannot be described by a simple Brillouin function applicable to non-interacting spin systems. Shape of the Heavy Hole Landau Levels

Another feature in which diluted magnetic semiconductors differ from their nonmagnetic counterparts, is existence of very pronounced “camel backs” in the heavy hole Landau levels with large quantum number n. Such a “camel back” shape of the heavy hole Landau subbands is characteristic also for ordinary narrow-gap semiconductors. However, in diluted magnetic semiconductors it is particularly large at low temperatures. Since this shape is enhanced by the ( S , ) terms, the magnitude of the “camel backs” decreases with the temperature, as can be seen in Fig. 3.

202

J. KOSSUT

Reduction of the g-Factor above the Semimetal-to-Semiconductor Transition Finally, let us turn our attention to the region “open-gap” of diluted magnetic semiconductors, i.e., to compositions above the semimetalsemiconductor transition (that occurs, e.g., at x = 0.065 in Hgl-,Mn,Te at 4.2 K). The conduction band is then of r6 symmetry and the electronic wave functions are mainly of s-like character. Therefore, one may expect that the modifications of the band structure due to the s-d interaction will be mainly described by the constant a. The sign of a (determined experimentally) is in our notation negative, i.e., opposite to the sign of p. Therefore, the exchange contribution to the electronic g-factor in the open-gap DMS is also opposite in sign to that contribution in zero-gap DMS. Since the exchange induced contributions in both zero-gap and open-gap narrow gap DMS constitute a sizable part of the total spin splitting of the conduction electrons, one can, consequently, expect a reduction of the electronic spin splitting at very low temperature in the latter. The absolute value of a is found to be smaller than that of /3, thus this reduction is expected to be less pronounced than the enhancement observed in the case of zero-gap materials. Figure 4

40

->

-E“

30

20

10

T=5 K

0

2

I

3

4

!

B(T)

(a) FIG.4. The energies of the minima of the conduction subbands ( k =~0) in Hg,-,Mn,Te (x = 0.1). calculated at (a) T = 5 K, (b) T = 15 K and (c) without the contribution of the

exchange interaction (i.e., for

01

=

p

= 0) as a function of an external magnetic field.

5.

203

QUANTUM TRANSPORT IN DMS

40

3c

2c

10 T = 15 K

0

I

2

I

2 B(T)

(c) FIG.4 (continued)

3

4

3

4

204

J . KOSSUT

presents results of numerical diagonalization of matrices (1 1) and (22) for Hgl-,Mn,Te with x = 0.1. Only the minima of the conduction band Landau levels (k, = 0) are shown. For the sake of comparison, the Landau levels in the “nonmagnetic” counterpart (the same band parameters, but for a = /?= 0) are given in Fig. 4c. One may observe that the spin splitting is indeed greatly reduced. For T = 5 K, even the level ordering is opposite, which corresponds to a very small but positive value of the g-factor. So, there is again a qualitative difference with non-magnetic narrow-gap semiconductors where the g-factors determined mainly by the spin-orbit interaction are generally negative. The positive g-factors-and their sign reversal with increasing temperature- have been, in fact, observed experimentally in the magnetooptical experiments (see Rigaux, this volume). IV. Transport Measurements in the Quantum Regime: Confirmation of the Band Structure Model The study of transport phenomena in strong quantizing magnetic fields (in particular, the Shubnikov-de Haas effect) is a useful tool of investigating the band structure. Since the Landau quantization is strong in semiconductors with small effective masses, the method is particularly well suited for narrowgap semiconductors. In the present section, we shall describe the experimental data on the quantum transport phenomena, stressing these aspects that reflect the unique band structure character of diluted magnetic semiconductors. The sequence of the sections will be the following: first we shall focus our attention on the zinc blende materials Hgl-,Mn,Te and Hgl-,MnxSe, for which the greatest amount of data is available. Next we proceed to tetragonal DMS and lead salts containing manganese (e.g., (Cdl-,MnX)3As2 and Pbl-,Mn,Te), which are only at the initial stages of investigation. Our attention throughout this section will be limited to n-type samples. We will thus, not discuss in this chapter the experiments on the quantum transport phenomena in p-type samples of zero-gap Hgl -,Mn,Te (see, e.g., Sawicki et al., 1982 and Ponikarov et af., 1981). The peculiarities observed in these samples are mainly associated with the pinning of the Fermi level by the acceptor states degenerate with the conduction band. 3. HgI-,MnxTe

AND

Hgl-,MnxSe

The quantum transport study of Hgl-,Mn,Te by Morrissy (1973) was the first t o reveal anomalies, characteristic for what later become known as semimagnetic or diluted magnetic semiconductors. However, the anomalies remained largely unexplained until the band structure model as described in Section I11 was developed by Bastard et al. (1978) and Jaczynski et al. (1978).

5.

QUANTUM TRANSPORT IN DMS

205

Further in this section we shall try t o describe the most striking differences between the quantum transport phenomena observed in diluted magnetic and nonmagnetic semiconductors, and we shall discuss how the anomalies occurring in DMS can be understood. a. Temperature Dependence on the Shubnikov-de Haas Peak Positions

The magnetic field positions of the maxima (or minima) observed in the Shubnikov-de Haas (SdH) effect are mainly determined by the crossing of the Fermi level with consecutive Landau levels. Their temperature dependenCe is, in usual materials, only very slight. It was, therefore, quite surprising to observe a very strong temperature shift of the maxima of the SdH effect in Hg,-,Mn,Te (Jaczynski et al., 1978). The shift was particularly pronounced in the case of the spin split peaks. The small temperature variation of the effective mass of conduction electrons could not possibly account for the observed shifts. These shifts are quite naturally explained in terms of the band structure model incorporating the s-d exchange terms presented in Section I11 of this chapter. They are due to a strong temperature dependence of the terms proportional to ( S , ) , that govern the spin splittings of the Landau levels, and also, but to a much lesser degree modify the positions of the unsplit Landau levels. Similar strong temperature dependence was also observed in later Shubnikov-de Haas effect studies both in Hgl-,Mn,Te by Byszewski et af. (1979) and Hgl-,Mn,Se by Takeyama and Galazka (1979), Byszewski et af. (1980) and Lyapilin et al. (1983). It should be mentioned that in the case of the very last oscillation, i.e., that occurring at the highest magnetic field (often labeled 0 - ) , the Shubnikov-de Haas peak exhibits a relatively strong dependence on the temperature also in nonmagnetic semiconductors. This has its source in the partial lifting of a strong degeneracy of the electron gas. However, in diluted magnetic semiconductors, as noted in Hgl-,MnxSe by Lyapilin et al. (1983), the 0peak observed in intense magnetic fields may shift towards smaller fields when the temperature is increased. This is exactly the opposite direction of the temperature shift of the 0- peak to that observed in nonmagnetic semiconductors. Hence, the usual explanation may again not be applied to the case of diluted magnetic semiconductors. On the other hand, both the direction of the 0- shift and its magnitude can be readily explained by the variation of the Landau level (n = 0, set b) with temperature due to the exchange terms proportional to (&). Probably the most striking feature in the SdH behavior of DMS was the evolution the temperature of a well resolved doublet structure of Shubnikovde Haas peaks that remained unsplit at lower temperatures. This can be clearly seen in Fig. 5 in the case of the oscillation maxima associated with

206

J . KOSSUT

160

-2 1 2 0 E w

80

40

0

3

6

9

12

15 B(T1

10

21

24

27

18

21

24

27

(a)

160

z120

E

W

80

40

0

3

6

9

12 15 B(T)

(b) FIG.5 . The Landau levels at k~ = 0 in Hg,-,Mn,Te (x = 0.02) calculated at (a) T = 4 K and (b) T = 36 K using the value of /3 = 0.6 eV and a = - 0.4 eV. Open circles show the positions of the maxima of the Shubnikov-de Haas oscillations observed by Byszewski et al. (1979). Arrows indicate the peak which splits at elevated temperatures. Note that at the lower temperature the energy gap opens only above 8T.

5.

QUANTUM TRANSPORT IN DMS

207

a, (2) and b, (1) Landau levels-sometimes labeled as 2' and 1- levels-in Hgl-,Mn,Te (x = 0.02). This is contrary to the case of nonmagnetic semiconductors where the increase of the temperature leads usually to a smearing out of the observed maxima. Again, this anomalous feature is naturally accounted for by the exchange-induced modifications of the band structure. The spin-splitting at the lower temperature is so big that the b, (1) spin-split Landau level nearly coincides with the a, (2) level. Thus, a single Shubnikovde Haas maximum is observed. When the temperature is raised, the value of ( S , ) and, thus the spin splitting is reduced. Consequently, the be (1) sublevel is sufficiently separated from the a, (2) sublevel for thz doublet structure to appear. Similar behavior was also observed in the case of Hgl-,Mn,Se crystals by Lyapilin et al. (1983). The temperature dependence of the Shubnikov-de Haas maxima provides an abundance of information that, in principle, enables a precise determination of the band structure parameters. The analysis aimed at such determination requires, however, a detailed knowledge of the (S,),particularly of its dependence on the temperature, composition x and magnetic field. This can be obtained independently from the measurements of the magnetization. Lack of reliable measurements of this quantity made the interpretation of SdH data rather difficult, and led to inaccuracies of the values of the band structure parameters determined at the initial stages of research on Hgl-,MnxTe and Hgl-,Mn,Se. We shall return to this problem later in this section. b. T h e m0- Oscillations The strong temperature dependence of the energies at which the Landau levels occur suggests that at a constant magnetic field (that provides the Landau quantization), the temperature sweep will cause the crossing of various Landau levels with the Fermi energy, if the carrier concentration in a sample is properly chosen. These crossings will then lead to successive peaks in the magneto-resistivity of the sample observed as a function of temperature, similar to those observed by sweeping the magnetic field at a constant temperature in the usual Shubnikov-de Haas experiment. Such peaks in the resistivity were observed in Hgl-,Mn,Te by Dobrowolska et al. (1979) as shown in Fig. 6 . The phenomenon was named the thermo-oscillation of magnetoresistance. An experiment similar in nature but involving the oscillations of the absorption of sound was suggested theoretically by Lyapilin and Karyagin (1980). Although the thermo-oscillations are difficult to interpret quantitatively, they serve to illustrate rather dramatically the unique temperature dependence of the Landau quantization of DMS owing to the contributions associated with the magnetization in these materials.

208

J. KOSSUT

FIG. 6. Resistivity in Hgl-,Mn,Te (x = 0.009) versus temperature showing thermooscillation maxima. The arrows indicate the theoretically expected positions of the maxima. [After Dobrowolska et al. (1979).]

c. The Amplitude of the Shubnikov-de Haas Oscillations

The temperature dependence of the amplitude of the magnetoresistance oscillations is an important feature of the Shubnikov-de Haas effect, since it provides a source of information concerning the electron effective mass at the Fermi level. Usually, in nonmagnetic semiconductors, the amplitude of a given magnetoresistance maximum decreases monotonically when the temperature is increased. It was therefore quite surprising to observe (Jaczynski et al., 1978) that in Hgl-,MnxTe, the amplitude behaved in a drastically non-monotonic manner (see Fig. 7). It dropped quickly with increasing temperature, passed a region of very small values where the oscillations were hardly detectable, and became appreciable once again at even higher temperatures. Then, after passing through a maximum, it started to decrease again. Only the last portion of this behavior, in the highest temperature range, could be consistently understood in terms of the theory usually applicable to nonmagnetic semiconductors. A very similar temperature dependence of the Shubnikov-de Haas amplitude was observed in Hgl-xMnxSe (Takeyama and Galazka, 1979; Lyapilin et al., 1983). The theory of the Shubnikov-de Haas effect (see, e.g., Roth and Argyres, 1966) gives the following expression, conveniently expressed in the form of

5.

209

QUANTUM TRANSPORT IN DMS

15

-s

-d 1 0

0 Y

-

+ 3

.-n E d

5

0

2

6 TIK)

4

8

10

FIG.7. The amplitude of the Shubnikov-de Haas oscillation in Hgl-,Mn,Te (x = 0.02) as a function of temperature.The solid line is calculated using the temperature-dependent g-factor. [After Jaczynski et al. (1978).]

‘

a harmonic series, for the amplitude behavior of the SdH oscillations:

-hPOp-

r=l

f i X r exp(- arTD/H) cos(mr) cos[ 2 n r e - y ) f GsinhX,

:]

(29)

where a = 2n2m*kdeA, TD is the Dingle temperature describing phenomenologically the broadening of the Landau levels, X , = arT/H, v is the spin splitting factor defined as v

=

1 m* -g*2 mo’

with m* and mo standing for, respectively, the effective mass and the free electron mass. The oscillatory character of the magnetoresistance is given by the last cosine factor in Eq. (29) with the fundamental period PSH.For spherical Ferrni surfaces, this can be related to the electron concentration N by

Often only the first few terms of the expansion (29) are of importance, because of the exponential damping factor involving TD. The temperature

210

J. KOSSUT

behavior of the amplitude is given by the factor XJsinhX, in Eq. (29), provided that there is no implicit temperature dependence of the remaining coefficients in Eq. (29). Now, in a diluted magnetic semiconductor, as we have pointed out in the previous sections, the temperature dependence of the g-factor can be quite remarkable, and thus the cos(nvr) term in Eq. (29) can influence the temperature behavior of the amplitude quite significantly. When the argument of the cosine is equal to i d 2 , where i is an odd integer, the amplitude may vanish. For the first harmonic in Eq. (29), i.e., for r = 1 , the condition for the zeros can be written explicitly as m0

g* = - i ; m*

i

=

* l , *3.

When theg-factors given by the band structure calculation for Hgl -,Mn,Te were calculated, it turned out that, indeed, it is possible to meet condition (30) in the temperature range where the Shubnikov-de Haas effect was studied. A superposition of the factors Xl/sinh XI and cos(nv) gave a good description of the amplitude vs. temperature curve (see the solid line in Fig. 7). The condition (30) suggests, further, that more than one zero of the amplitude can be expected as a function of the temperature. In fact, two zeros were detected in Hgl-,Mn,Se by Lyapilin et al. (1983). The temperatures at which both amplitude zeros occurred could be well accounted for by the simple explanation presented above. Let us note that condition (30) corresponds to the situation where the Landau spin-split sublevels are equally spaced in energy. This is an easy situation for the collision broadening of the levels to smear out the quantum oscillations, i.e., to produce a zero of the amplitude. So, one expects an occurrence of a clear zero in relatively impure samples with rather high values of the Dingle temperature TD. Such values enable one to neglect all higher terms in the harmonic series, Eq. (29). For purer samples, when more terms in the harmonic expansion are important, only a minimum of the amplitude vs. temperature relationship can be expected (Kossut, 1978). The analysis of the positions of the zeros of the amplitude as a function of temperature can serve as a rather precise method for determining the gfactor, as well as the exchange constants a and fl. This method was actually employed in the case of (Cdl-,Mn,)sAst mixed crystals (see further in this part). Recently, Reifenberger and Schwarzkopf (1983) have suggested a different explanation for the beating of the amplitude of the Shubnikov-de Haas effect observed by them in Hgl-,MnxSe with small x and also in HgSe at weak magnetic fields. They suggest that the beating effect, which can be interpreted as a zero of the amplitude at a certain magnetic field, is due to

5.28

5.

QUANTUM TRANSPORT IN DMS

211

the magnetic breakdown between the closed-orbits in the spin-split conduction band. The splitting results from the lack of inversion symmetry characteristic for the zinc blende lattice structure. Actually, the fact that the beating effect is also found by Reifenberger and Schwarzkopf (1983) in HgSe leads them to reject the explanation in terms of the cos(nv) factor described above. In order to account for the temperature shift of the beat fields observed in Hgl-,Mn,Se samples, Reifenberger and Schwarzkopf (1983) conclude that substitution of Mn for Hg atoms makes the linear k terms in the k * p Hamiltonian also sensitive to the temperature in this material.

d . Conduction and Valence Band Overlap The degeneracy of the conduction and heavy hole valence band edges in zero-gap HgTe and Hgl-,Cd,Te is lifted when the magnetic field is applied and a real energy gap, induced by the field, opens. As we have seen in Part I11 of this chapter, the situation in semimagnetic Hgl-,Mn,Te and Hgl-,Mn,Se is quite different. The degeneracy of the Ts level is, of course, also lifted, but in these DMS a strong upward shift of the uppermost Landau level of the heavy hole band may result in an overlap of the bands (see Fig. 5a). The extent of the overlap can be calculated analytically and is given by, AE = Eb,(-1) - EnC(o)

4eH

1'2

2 where we assumed that A = OD. The overlap corresponds to a positive AE, while for an open gap the value of AE is negative. The value of the band overlap is a resultant of several factors. First there is an upward movement of the lowest conduction band level a,(O) with the magnetic field, determined mainly by the effective mass (or, more precisely, by the cyclotron frequency) characterizing the conduction band. The variation of this level with the magnetic field is modified by exchange interaction to a lesser degree by the exchange interaction than that of the uppermost valence band Landau level bv(- 1). The latter level, whose energy in nonmagnetic semiconductor decreases with increasing magnetic field, may be shifted in DMS to considerably higher energies if the

212

J. KOSSUT

following condition is fulfilled:

Since ( S , ) has a tendency to saturate in the magnetic field while the right hand side of Eq. (32) is linear in the field, the above condition can only be met in relatively small field region, if at all. Of course, the lower the temperature, the easier it is to satisfy the inequality (32). Because of antiferromagnetic interactions between the localized magnetic moments in Hgl- .Mn,Te and Hgl- ,MnxSe, the value of x(S,) at a given field and temperature is a decreasing function of x for samples with the Mn mole fraction greater than x = 0.05. At the same time, the rate at which the a,(O) level moves up in energy with the field increases with x in zero-gap materials, because the effective mass at the bottom of the conduction band becomes smaller. Therefore, the existence of the overlap is limited to rather low Mn contents region. It is estimated that in both HgI- .Mn,Se and Hgl- .MnxTe, the highest value of x for which the overlap may be observable is about 0.04-0.05. The detailed knowledge of the magnetization (i.e., of x ( S , ) ) is of course, quite crucial for this estimate. The use of values of x(S,) taken from more or less substantiated theoretical models of magnetization can lead to an overestimation of the extent of the overlap. The existence of the band overlap may lead to a substantial redistribution of carriers between the two bands in question. This, in turn, is reflected as an anomaly of various transport coefficients in very pure samples. In fact, the analysis of the conductivity tensor components by Sandauer and Byszewski (1982) gave the first direct evidence for the band overlap. It was found that the warping asymmetry of the heavy hole band, proportional to y 2 - ~ 3 , is also observable as the anisotropy of the band overlap in Hgl- .Mn,Te (Sandauer et al., 1983).

e. Spin Dependent Scattering The presence of localized magnetic moments not only modifies the band structure of diluted magnetic semiconductors, but also gives rise to an additional scattering mechanism of the carriers. The contribution of this mechanism is not easy to observe experimentally, because it is quite difficult to separate it from the contributions of other, often more important, scattering mechanisms. Recently, existence of spin dependent scattering has been evidenced in Hgl- .Mn,Te under quantum-limit conditions by Wittlin et al. (1980a). In their experiment, the sample was placed in a strong magnetic field, such that only the lowest conduction band level was occupied. The field

5.

QUANTUM TRANSPORT IN DMS

213

also caused a total alignment of the localized Mn spins, i.e., the region of saturation ( S t ) = - $ was reached. The sample was then illuminated by a microwave radiation with energy chosen in such a way that it correponded to the condition of electron paramagnetic resonance of the Mn-subsystem. The resonant absorption by the Mn ions and the change of the polarization of the localized moments associated with it were visible as an increase of the resistivity of the sample. The possibility of other mechanisms (e.g., bolometric effect, influence of the optical excitation of the carriers between the Landau levels, etc.) contributing to the observed increase was ruled out by the authors. On the other hand, when the resistivity change connected with the sudden variation of the spin dependent scattering at the resonance condition is calculated (Wittlin et al., 1980b), the correct magnitude of the observed effect is obtained.

f, Values of Exchange Constants Obtained from Quantum Transport Measurements The results of the early studies of the Shubnikov-de Haas effect in Hgl-.MnxTe by Jaczynski et al. (1978) led to rather high values of the exchange constants: /.? = 1.4 eV and a = - 0.7 eV. These values were in agreement with the original results of magnetooptical measurements of Bastard et al. (1978). However, it must be stressed that in both studies, the magnetization of the investigated samples was, in fact, unknown. Moreover, the interpretation of the Shubnikov-de Haas oscillation pattern (Jaczynski et al., 1978) assumed that the Fermi level remained constant in the entire range of magnetic fields investigated. This may be a good approximation when many Landau levels are below the Fermi level E F , but becomes doubtful in higher fields, when only few Landau levels remain below E F . Unfortunately, this is exactly the region of fields where the spin splitting of the SdH maxima was resolved, that is, where the subsequent analysis was sensitive to the choice of a and p. Substantial error could therefore be present in the determination of a and p by Jaczynski et al. (1978). Later investigations of the Shubnikov-de Haas effect (Sandauer and Byszewski, 1982), free of the two above mentioned approximations, put the value of /3 = 0.8-0.9 eV and (Y = -0.3 eV. These values are considerably smaller than those originally proposed and are much closer to p = 0.6 eV, (Y = - 0.4 as determined from intraband (Pastor et al., 1979) and interband (Dobrowolska and Dobrowolski, 1981) magneto-absorption. A similar situation is also true for Hgl- .Mn.Se. The investigation of the material by Takeyama and Galazka (1979) yielded values of p = 1.4 eV and a = - 0.9 eV for x = 0.01 and x = 0.016, while for the x = 0.066 sample, = 0.9 eV and a = 0.3 eV. Later, when the magnetization of the samples was studied as well, it was found that the observed spin splitting is described

214

J. KOSSUT

b y p = 0.9 eV and a = - 0.35 eV (Byszewski et al. , 1980). Again, these later values are in better agreement with the magnetooptical determination of Dobrowolska et al. (1981) where p = 0.7 eV and a = - 0.4 eV. Recently, Lyapilin et al. (1983) found, by an analysis of the spin splitting of the n = 1 Landau level observed in the Shubnikov-de Haas experiments in Hg,- *Mn,Se, that the value of p is still smaller: p = 0.28 eV. However, in their analysis Lyapilin et al. (1983) again did not use experimental values of magnetization, and, moreover, they employed a simplified band structure model. Although there seems to be a trend toward reconciliation, the values of the exchange constants a and p in Hgl- .Mn,Te and Hgl- .MnxSe, as determined by various authors, do show a considerable disagreement. This is a situation distinctly different from that of wide gap DMS, where a and p are known with a greater accuracy. This unsatisfactory state of our knowledge is, at least partially, due to the complicated nature of the band structure of narrow gap DMS, where both spin and orbital quantization have to be simultaneously considered. Undoubtedly, this situation will be improved in a foreseeable future. 4. (Cdl- .Mnx)3As2

The very complex crystal structure of semiconducting Cd3As2 (symmetry (2): can be approximated fairly well by a cubic structure with an additional tetragonal distortion (see Kildal, 1974; Blom, 1980). Bodnar (1978) found that treating this distortion as a perturbation in the k p scheme provides a reasonably good description of the band structure of the compounds in question. This model is a generalization of the three band approach to zinc blende semiconductors (Kacman and Zawadzki, 1971) in the sense that it takes similar basis functions into the quasi-degenerate set of states and neglects the effect of the higher bands entirely. The k p perturbation Hamiltonian contains, as mentioned above, an additional parameter 6 describing the axial distortion, that lifts the degeneracy of the conduction and heavy hole bands characteristic for the inverted band structure of zinc blende compounds. There appears a real-although small-energy gap. Of course, the presence of the axial crystal field leads to a strong anisotropy of the band structure noted by Wallace (1979). The calculation of the Landau level in the presence of exchange interaction proceeds quite analogously to the method described in detail for zinc blende compounds in Section I11 of this chapter (Neve et al., 1982). The main difference between the present case and previous calculations is that the tetragonal distortion introduces, instead of a single constant p describing the exchange interaction of p-like electrons with magnetic moments of Mn ions,

-

5.

215

QUANTUM TRANSPORT IN DMS

two such constants PI and fill which, in general, may have different values. This fact may contribute additionally to the anisotropy of the g-factor of (Cdl- xMnx)3Asz. Much less experimental information is available on this diluted magnetic semiconductor system than in the case of Hgl- *Mn,Te. The Shubnikov-de Haas effect was investigated in this material by Neve et af. (1981). Because of relatively high electron concentration in these crystals (of the order of 5 x 10" ~ m - ~it) was , impossible to determine the variation with x of such parameters as the energy gap and the momentum matrix elements P I ,PII (again, because of the tetragonal distortion, the model contains two independent momentum matrix elements). However, it was noted that the pattern of the oscillations was quite distinct from that in pure Cd3Asz crystals. In particular, the oscillations exhibited a strong beating effect, with the position of the nodes sensitive to temperature. When the amplitude at a fixed field was plotted versus the temperature, it was noted that two zeros were observed (see Fig. 8) that, when interpreted as being due to the cos(nvr) term (see subsection IV.3c of this chapter), rendered the values of the exchange constants PI = = 4.9eV and a = -3.4eV. These values are remarkably greater than those in diluted magnetic semiconductors with the zinc blende lattice structure.

x=O.Ol

A

0

2

I x.0.02

8-1.53'1

4

6

B=1.90T

8

10

T (K)

FIG.8. Temperature dependence of the Shubnikov-de Haas amplitudes in (Cdl- .Mnx)3Asz (x = 0.01). The points represent experimental data, and the solid lines are fitted. In the insert the SdH amplitude vs. Tcurves are shown for two samples, both with x = 0.02 but differing in electron concentration:(a) n = 3.9 10" cm-3 and (b) n = 5.45 10'' cm-'. [After Neve et al. (1981). Reprinted with permission of Pergamon Press Ltd.]

216

J. KOSSUT

The anisotropy of the conduction band was also studied in (Cdl- .Mnx)3As2 by Blom et al. (1983) who studied the Shubnikov-de Haas effect in monocrystalline samples. The angular dependence of the effective mass was found to be qualitatively similar to that in Cd3As2. The anistropy of the g-factor was found to be smaller than predicted by a simple theory of Neve et al. (1982), and even less pronounced than in Cd3Asz. This fact led Blom et al. (1983) to conclude that there must be a strong dependence of the energy gap Eg and the crystal field parameter 6 on the crystal composition given by the Mn mole fraction x. 5. Pbl- .Mn,Te

AND

Pbl- .Mn,S

In this section, we are dealing with diluted magnetic semiconductors based on lead chalcogenides, e.g., Pbl- .Mn,Te. These semiconductors crystalize in the NaCl cubic structure. Again the calculation of the Landau levels in the presence of the exchange interaction can be done (within k * p approach) along the general lines given in Section I11 for zinc blende compounds (see, e.g., NiewodniczanskaZawadzka, 1983). The calculation is however more complicated because of the anisotropic nature of the band structure of the host material, e.g., PbTe (Adler el af., 1973). It is only for the magnetic field oriented along some special directions with respect to the crystallographic axes (e.g., HI(( 1 1 1 )) that the problem is relatively easy to solve. Preliminary investigation of the Shubnikov-de Haas effect in Pbl- .Mn,Te by Niewodniczanska-Zawadzka et al. (1982) appeared to indicate that the presence of Mn ions modifies the band structure of this material in a manner predicted by the model outlined in Section 111. A similar conclusion, although much more cautiously worded, was reached in the intraband magnetoabsorption study of Niewodniczanska-Zawadzka et al. (1 983). The position of the magneto-absorption lines and of the Shubnikov-de Haas maxima in the recent study of Elsinger (1983) show practically no temperature dependence, contrary to strong temperature variation of the magnetization. In fact, these data can be interpreted in terms of the k p theory disregarding the contribution of the exchange interaction. However, in order to explain the observed spectra, a large zero-field spin splitting of the valence band has to be assumed (Pascher et al., 1983). It was found that the value of this splitting depends on the temperature. A similar spin splitting at H = 0 was also discovered in the study of Pbl- xMn,S p-n junction lasers by Karczewski and Kowalczyk (1983). It is suspected that the off-diagonal terms involving the neglected components S, and S, of the localized moment are responsible for the zero-field splitting. It is still not clear why the influence of the terms proportional only to (S,),

-

5.

QUANTUM TRANSPORT IN DMS

217

so pronounced in DMS with the zinc blende structure, is not observable in Pbl-*Mn,Te and Pbl-.Mn,S. This may be related to the fact that the exchange constants for Pbl- .Mn,Te are very small, as inferred by Toth and Leloup (1970), from the shape of the spin paramagnetic resonance line. The value found in this study (0.07 eV) is by an order of magnitude smaller than that characteristic of, e.g., Hgl- .Mn,Te. The question arises whether the lack of the splittings proportional to ( S , ) is not related to the fact that the band edges in Pbl-,Mn,Te and Pbl-,Mn,S occur at the L point of the Brillouin zone. In Cdl-,Mn,Te it was found (see Gaj, this volume) that there is a great reduction of such splittings at the L point, while at the zone center they are large and easy to observe. The solution to this problem requires further study. V. Two-Dimensional Electron Gas in Hgl-,Mn,Te and Hgl-x-yCd,MnyTe The space charge layers in semiconductors have been intensively studied in recent years (see, e.g., review by Ando et al., 1982). It was therefore natural to extend the studies of diluted magnetic semiconductors in this direction. So far, the inversion layers in p-Hgl- *MnxTe and a quaternary system Hgl - ,,Cd,Mn,Te were investigated. These preliminary studies were performed by the quantum transport method and concerned the quasi twodimensional electrons either on the surface of the semiconductor (Grabecki et al., 1984a), or confined in a thin layer surrounding a grain boundary within the semiconductor (Grabecki et al., 1984b). Here we shall discuss-after a very brief description of the experiments mentioned above-a simple model of the two-dimensional energy subbands that allows a semiquantitative interpretation of the data.

6. EXPERIMENTS In the first series of experiments by Grabecki et al. (1984a), the MIS structures were prepared by placing a thin sheet of Mylar as an insulator on the Hgl- .Mn,Te surface. The p-type samples were used with the Mn mole fraction x = 0.1, that corresponded to an open energy gap of the order of about 100 meV (c.f., Fig. 2). The relatively high value of x was chosen to limit the effects of tunneling of electrons from the surface inversion layer to the valence band in the bulk of the crystal. It was found that the conductivity at low temperatures is dominated by a flow of electrons within a thin quasi two-dimensional inversion layer produced by an attracting surface potential. This potential restricts the electronic motion in the direction perpendicular to the surface. The motion

218

J. KOSSUT

in this direction becomes quantized: a series of surface or “electric” subbands is then produced. The electronic motion parallel to the surface remains unaffected in a first approximation. The concentration of the electrons in the inversion layer could be changed by means of an applied gate voltage. When a sample is placed in a strong magnetic field and energy levels of electrons in the layer become further quantized (Landau quantization), an oscillatory structure in the conductivity vs. the gate voltage V, curves can be recorded. Two periods of oscillation were detected, indicating that at least two surface subbands were populated by electrons. This observation reflects a small density of states of the conduction band in narrow-gap semiconductors related to the smallness of the effective mass. By studying the p vs. V, dependences at various temperatures, a remarkable shift of the resistivity maxima was noted. Two maxima that were clearly resolved at T = 5 K approached each other when the temperature was raised. At about 10K only single maximum was observed, but at still higher temperature, about 14 K, a doublet structure became visible again. This effect was interpreted as being due a temperature shift of the Landau levels so characteristic for diluted magnetic semiconductors (see Section IV.3a of this chapter). However, it was possible to draw only rather qualitative conclusions from these preliminary experiments on the MIS structures. This was because of the rather low electrons mobilities in the inversion layers, reflecting severe problems with the proper preparation of the surfaces. A more quantitative interpretation became possible quite recently when research began on a new quasi two-dimensional system of electrons. It was noted that the grain boundaries, often found in Hgl- .Mn.Te crystals grown by the Bridgman method, can be a source of electronic confinement (Grabecki et al., 1984b). Similar systems of two-dimensional electrons were already studied in Ge by Vul and Zavaritskaya (1979) and Uchida et at. (1983) (see also a review by Seager, 1982). The physical reason for the electron confinement is the presence of charge traps at the grain boundary. These may be associated with dislocations existing at the boundary, as is the case in Ge (Uchida et at., 1983). The samples containing a single boundary plane (often referred to in the literature as bicrystals) were carefully cut out of a slice of Hg,- .Mn.Te material with x = 0.1 where special etching techniques made the boundaries between monocrystalline grains visible. It was found by Grabecki et at. (1984b) that, although the single grain samples prepared from the same ingot exhibited typical p-type behavior, the low temperature conductivity of samples containing the grain boundary was due to electrons. Further, it was also observed that these electrons possessed features of the quasi two-dimensional gas: the Landau quantization of their states depended only on the component of the magnetic field normal to the

5.

QUANTUM TRANSPORT IN DMS

219

grain boundary. The mobility of electrons in the layer at the boundary was quite high (of the order of 104cm2/Vs), enabling observation of the Shubnikov-de Haas oscillations in magnetic fields as low as 0.2T. Two or three periods of the oscillations were observed, indicating again that several electric subbands were occupied by 2d electrons. The electron concentrations and effective masses at the Fermi level were determined for each of the subbands and are listed in Table I. They are compatible with results of a calculation based on the model to be described in the following section. The results of this calculation are also listed in Table I . No spin splittings of the Landau levels were observed in the Shubnikov-de Haas oscillations studied in the samples of Hgl- .Mn,Te ( x = 0.1). This fact is due to the smallness of electronic g-factor, which at low temperatures, is greatly reduced by the exchange interaction in this open-gap material. This arises because the large and negative contribution to the g-factor due to the spin-orbit interaction is nearly compensated by the exchange induced terms proportional to the magnetization. Parenthetically, this fact enabled a precise determination of the effective masses from the temperature study of the oscillation amplitude. To avoid the compensation of the g-factor, Grabecki et al. (1985) prepared (x = 0.23, y = 0.02) also samples from quaternary Hgl- ,-,Cd,Mn,Te containing the grain boundaries. The energy gap in these samples (=200 meV) is open mainly by the presence of the Cd atoms. The effects of the exchange interaction due to the presence of the small number of Mn atoms are here more pronounced than in Hgl- .Mn,Te (x = 0. I), because the reduction of the magnetization in the latter material is greater. This results from a stronger antiferromagnetic coupling between localized Mn moments in the less diluted case. In fact, the Shubnikov-de Haas oscillations of the 2d electrons in Hgl- ,-,Cd,Mn,Te samples show well-resolved spin splittings in the available range of magnetic fields. This enabled the first determination of the g-factor of the 2d electrons in a diluted magnetic semiconductor. Not surprisingly, the g-factor observed showed a strong temperature dependence (see Fig. 9). It is also well accounted for by the model of the subband structure (see next section). The effective mass at the Fermi level and the population of the lowest electric subband (only one period of oscillation was observed) in Hgl- x-yCdxMnyTeare given in Table I. 7. ENERGY LEVELS OF 2d ELECTRON GAS

The model described below is based on semiclassical WKB approximation. It has been shown by Zawadzki (1983) that this simple approach can give a correct insight into the complex scheme of energy levels of a two-dimensional electron gas in narrow-gap semiconductors. Here we present a simplified

TABLE I EXPERIMENTAL AND THEORETICAL VALUES OF 2d ELECTRON CONCENTRATION, n: AND THE EFFECTIVE MASSESAT THE FERMILEVEL( E F ) mf IN THE r-TH ELECTRIC SUBBAND IN SAMPLES CONTAINING GRAINBOUNDARY. E , IS THE CALCULATED VALUEOF THE BOTTOM(n: = 0) OF THE r-m SUBBAND. THE FITTEDVALUE OF THE POTENTIAL SLOPE uo IS GIVEN. THESPATIAL EXTENSION OF THE r-TH STATE, W,, WAS ESTIMATED FROM Wr = E r / U o

Hgo.s9Mn0.11Te

110

0 1 2

H g 0 . 9 0 M n o .l o T e

85

0

5.3

1 0

Hg,-.-,Cd,Mn,Te (X =

0.23, y = 0.02)

205

15.4 f 2 6.0 + 0.5 1.8 0.2

5.0 f 1 3.3 + 0.5 2.2 i 0.2

12.18 5.98 1.87

3.60 3.01 2.40

57.2 111.8 151.3

163 319 432

2.6

0.2

5.29

2.49

52.7

192

1.3 f 0.1

2.0 f 0.1

1.34

1.85

102.1

371

3.2

2.6 f 0.4

3.2

2.4

44.2

161

+ 0.3 f 0.1

f

172

7.0 lo4

122

5.5 lo4

80.9

5.5 lo4

5.

QUANTUM TRANSPORT IN DMS

221

T(K) FIG.9. Theg-factor of 2d electrons at agrain boundary in Hgo Kdo.z,Mno.o2Teas a function of the temperature. The symbols with error bars represent experimental data, based on the splitting of the magnetoresistance maximum at approximately H = 4.5T.Solid lines show the results of the calculation based on the model described in the text. [After Grabecki ef a/.(1985).]

version of the model, neglecting the presence of the spin-orbit split-off band r7,aiming at the analytic form of solutions. Actual calculations, the results of which are given in Table I, were done without making this approximation. However, the influence of the higher bands was neglected in both versions. We start with the Hamiltonian in the matrix form given by Eqs. (13)-(16) with y1 = y2 = y3 = F = K = 0. The presence of the confining potential at the surface or at the grain boundary requires an addition of an appropriate potential U to the diagonal terms of these matrices. When the z-axis is chosen perpendicular to the surface (or to the grain boundary), the confining potential depends only on z. In the present model, very simple forms of this potential were assumed. In the case of the surface layer, the potential was taken to be a triangular well with infinite height on the insulator side; in the case of the grain boundary, it was assumed in the form of a symmetric triangular potential well. The slope of the potential is the only fitting parameter of the model. The other band structure parameters can be taken from electronic studies of bulk crystals. This simple form of the confining potential is probably the most severe approximation of the model. We assume also that the magnetic field is parallel to the z-axis. Finally, the exchange terms involving the constants a! and must be added to the Hamiltonian. The matrices obtained in this way represent two sets of differential equations for the envelope functions. These

222

3. KOSSUT

functions can be sought in a factorized form @(x,y)$(z),with the dependence on x , y variables separated from the z variable. Moreover, @(x,y ) can be chosen as a harmonic oscillator wave function, as in the case of Landau quantization. Thus we are left with a set of differential equations involving only d/dz, i.e., the z-component of the momentum p z . In the spirit of the semiclassical approximation, we may now treat p , as a classical quantity. It can be then expressed in the following form (neglecting small terms proportional to [U, p,], x8(Sz>Pp,/E~A and x/3<Sz)Eg):

where 2 P2 A=-3 A2'

at

= [E f

a!'

- &f

-

(34) - P'

[

U(z)] Eg +

&+

7+ E f

-

w].

(35)

In Eqs. (34) and ( 3 9 , Pdenotes the interband momentum matrix element and a' = + X a ! ( S Z ) ,

(36)

8' = +x/3<S,>,

(37)

while E denotes the eigenvalue of energy that we are looking for. Plus and minus indices in Eqs. (33) and (35) refer, respectively, to solutions for predominately spin-up and spin-down states in the conduction band. The parameter cf is defined by

with

where n = 0, 1, ... denotes the Landau quantum number, and s = (hc)/(eH). The WKB quantization condition can be written in our case as p Z d z = hn(r

+ $),

r = 0, 1 , 2 , ...

(40)

where za = 0 and q5 = $ for the case of the surface layer potential, while zo = - zo and $ = for the layer at the grain boundary. Here zo denotes the classical turn point of the motion in a confining potential well, i.e., it is

5.

QUANTUM TRANSPORT IN DMS

223

defined by PZ(Z0) =

0

The smallest root of Eq. (41) corresponds to solutions for the conduction band 20’ =

f ayl - Ef

E

uo

where we have explicitly used the form of the potential assumed

for the surface layer, and

U(z) = Uolzl

(44)

for the grain boundary potential. The integration indicated by Eq. (40) can be carried out analytically, giving the following equations

where

a*

=E 7

a’ - ~ f ,

b* = Eg + and

G=

E

+ ~f F B’ -, 3

[

1 for the surface layer .2 for the grain boundary layer

Equation (45) has now to be solved numerically for the energies of the n-th Landau level with spin (+) or (-) in the r-th electric subband. A similar calculation can be carried out when no external magnetic field is applied. The result is formally very similar to that obtained for InSb, with the quantity ~f redefined (cf., Zawadzki 1983):

where m3 = h2(4P2/3Eg)-’ and kll is the momentum within the 2d layer. Solving Eq. (45) with the definition (49) of e f , we may calculate the position

224

J. KOSSUT

of the Fermi level ~ f related . to the electron concentration ns in the r-th subband. 1 nd = -kl;. 2n By calculating the derivative of ~ fwith . respect to kll, we may calculate the effective mass at the Fermi level. Theoretical values of the effective mass listed in Table I were calculated in this manner. By choosing a proper value of the potential slope U o ,one can now fit the electron concentrations in each of the observed subbands and the effective masses to the experimental data, assuming a single Fermi level. A fair agreement can be reached indicating a proper choice of the shape of the potential (and its slope) at the grain boundary. By solving Eq. (45) with given by Eq. (38) for a fixed magnetic field, one can finally determine the value of the g-factor of the nth Landau level as

The g-factors calculated in this way give a fairly good description of the values observed in Hgl- x-yCdxMnyTe.At lower temperatures, they turn out to be positive due to the large contribution of the exchange interaction terms. As T increases (and ( S , ) decreases), the value of the g-factor drops, becoming nearly zero at the highest temperatures at which the oscillations were studied. Let us mention also that the values of the g-factor for the 2d gas of electrons at the grain boundary are slightly different from those expected for bulk electrons. The above results indicate that the influence of the localized magnetic moments also manifest itself in the properties of 2d electrons in diluted magnetic semiconductors. 8 . FUTURE POSSIBILITIES

Other 2d electronic systems in diluted magnetic semiconductors, e.g., those in quantum wells produced by heterostructures and/or superlattices, seem to offer quite interesting possibilities. One of those, noted by von Ortenberg (1982), makes use of the temperature and magnetic field variation of the band edges. Thus, if a superlattice is grown with alternating layers of nonmagnetic (e.g., Hgl- .Cd,Se) and diluted magnetic (e.g., Hgl- xMnxSe) semiconductors, then it should in principle be possible to change the relative position of the band edges in adjacent layers by means of varying the field and/or the temperature. It is argued that the resulting minigap of a superlattice could be tuned by these external factors. However, such possibilities are still be be explored.

5.

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225

References Abrahams, E. (1955). Phys. Rev. 98, 387. Adler, M. S., Hewes, C. R., and Senturia, S. D. (1973). Phys. Rev. B7, 5186. Anderson, P. W. (1961). Phys. Rev. 124, 41. Ando, T., Fowler, A. B., and Stern, F. (1982). Rev. Mod. Phys. 54, 437. Bastard, G., Rigaux, C., Guldner, Y . , Mycielski, J., and Mycielski, A. (1978). J. dePhysique 39, 87. Bhattacharjee, A. K., Fishman, G., and Coqblin, B. (1983). I n “Physics of Semiconductors” (Proc. 16th Internat. Conf. on the Physics of Semiconductors, Montpellier, 1982, M. Averous, ed.) Part I, p. 449, North Holland, Amsterdam. Blom, F. (1980). I n “Narrow Gap Semiconductors Physics and Applications” (W. Zawadzki, ed., Proceedings International School, Nimes, 1978, Lecture Notes in Physics, vol. 133, p. 191, Springer-Verlag). Blom, F. A. P., Neve, J. J., and Nouwens, P. A. M. (1983). In “Physics of Semiconductors” (M. Averous, ed., Proc. 16th Internat. Conf. Montpellier, 1982) Part I, p. 470, North Holland, Physica 117-118 B + C). Bodnar, J. (1978). In “Physics of Narrow Gap Semiconductors” (J. Rauluszkiewicz, M. Gorska, E. Kaczmarek, eds., Proceedings of International Conference, Warsaw, 1977) p. 311, P.W.N. Publishers. Byszewski, P., Szlenk, K., Kossut, J., and Galazka, R. R. (1979). Phys. Stat. Sol. (b) 95, 359. Byszewski, P., Cieplak, M. Z., and Mongird-Gorska, A. (1980). J. Phys. C13, 5383. Delves, R. T. (1963). J. Phys. Chem. Solids 24, 885. Delves, R. T. (1966). Proc. Phys. SOC.87, 809. Dobrowolska, M., Dobrowolski, W., Galazka, R. R., and Kossut, J. (1979). Solid State Cornmun. 30, 25. Dobrowolska, M., and Dobrowolski, W. (1981). J. Phys. C14, 5689. Dobrowolska, M . , Dobrowolski, W., Galazka, R. R., and Mycielski, A. (1981). Phys. Star. Sol. (b) 105, 477. Elsinger, G. (1983). Thesis, Montanuniversitat Leoben, unpublished. Friedel, J. (1958). Nuovo Cimento 52, 287. Gaj, J. A., Planel, R., and Fishman, G. (1979). Solid State Commun. 29, 435. Galazka, R. R., and Kossut, J. (1980). I n “Narrow Gap Semiconductors Physics and Applications” (W. Zawadzki, ed., Proc. Internat. School, Nimes, 1979) Lecture Notes in Physics, vol. 133, p. 245, Springer-Verlag. Galazka, R. R., and Kossut, J. (1983). In “Landolt-Borstein New Series,” Group 111 (0. Madelung, M. Schulz, and H. Weiss, eds.) vol. 17b, p. 302, Springer-Verlag. Grabecki, G., Dietl, T., Kossut, J., and Zawadzki, W. (1984a). SurJ. Sci. 142, 588. (Proc. 5th Internat. Conf. on Electron Properties of Two-Dimensional Systems, Oxford, 1983.) Grabecki, G., Dietl, T., Sobkowicz, P., Kossut, J., and Zawadzki, W. (1984b). Appl. Phys. Lett. 45, 1214. Grabecki, G., Dietl, T., Sobkowicz, P., Kossut, J., and Zawadzki, W. (1985). Acta Phys. Polon. A67, 297. Hass, K. C., and Ehrenreich, H. (1983). J . Vac. Sci. Techno/. 1, 1678. Irkhin, Yu. P. (1966). Zh. Eksp. Teor. Fiz. 50, 379. (Sov. Phys. JETP23, 253.) Izyumov, Yu. A., and Noskova, L. M. (1962). Fiz. Tverdogo Tela 4, 217. (Sov. Phys. Solid State 4, 153.) Jaczynski, M., Kossut, J., and Galazka, R. R. (1978). Phys. Stat. Sol. (b) 88, 73. Kacman, P., and Zawadzki, W. (1971). Phys. Stat. Sol. (b) 47, 629. Kane, E. 0. (1957). J. Phys. Chem. Solids 1, 249.

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Kaplan, T. A., and Lyons, D. H. (1963). Phys. Rev. 129, 2072. Karczewski, G., and Kowalczyk, L. (1983). Solid State Commun. 48, 653. Kasuya, T. (1955). Progr. Theor. Phys. 16, 45, 58. Kasuya, T. (1966). In “Magnetism” (G. T. Rado, and H. Suhl, eds.) vol. 2B, p. 215. Kasuya, T., and Lyons, D. H. (1966). J. Phys. SOC. Japan 21, 287. Kildal, H. (1974). Phys Rev. B10, 5082. Kossut, J. (1976). Phys. Stat. Sol. (b) 78, 537. Kossut, J . (1978). Solid State Commun. 27, 1237. Leung, W., and Liu, L. (1973). Phys. Rev. BS, 3811. Liu, S. H. (1961). Phys. Rev. 121, 451. Luttinger, J. M. (1956). Phys. Rev. 102, 1030. Luttinger, J. M., and Kohn, W. (1955). Phys. Rev. 97, 869. Lyapilin, I. I., Ponornarev, A. I., Kharus, G. I., Gavaleshko, N. P., and Maryanchuk, P. D. (1983). Zh. Eksp. Teor. Fiz. 85, 1638. (Sov. Phys. JETp58, 953.) Lyapilin, I. I., and Karyagin, V. V. (1980). Fiz. Tverdogo Tela 22,206. (Sov. Phys. SolidState 22, 118.) Mitchell, A. H. (1957). Phys. Rev. 105, 1439. Morrissy, C. (1973). Ph.D. Thesis, Oxford, unpublished. Mycielski, A., Kossut, J., Dobrowolska, M., and Dobrowolski, W. (1982). J. Phys. C15,3293. Neve, J. J., Bouwens, C. J. R., and Blom, F. A. P . (1981). Solid State Commun. 38, 27. Neve, J . J., Kossut, J., van Es, C. M., and Blom, F. A. P. (1982). J. Phys. C15, 4895. Niewodniczanska-Zawadzka, J . , Kossut, J . , Sandauer, A., and Dobrowolski, W. (1982). In “Physics of Narrow Gap Semiconductors” (E. Gornik, H. Heinrich, and L. Palmetshofer, eds., Proc. Internat. Conf. Linz, 1981)LectureNotesinPhysics,vol. 152, p. 326, SpringerVerlag. Niewodniczanska-Zawadzka, J., Elsinger, G., Palmetshofer, L . , Lopez-Otero, A., Fantner, E. J., Bauer, G., and Zawadzki, W. (1983). In “Physics of Semiconductors” (M. Averous, ed., Proc. 16th Internat. Conf. Montpellier, 1982) part I, p. 458, North Holland (Physica 117-118 B + C). von Ortenberg, M. (1982). Phys. Rev. Lett. 49, 1041. Pascher, H., Fantner, E. J., Bauer, G., Zawadzki, W., and von Ortenberg, M. (1983). Solid State Commun. 48, 461. Pastor, K., Grynberg, M., and Galazka, R. R. (1979). Solid State Commun. 29, 739. Pidgeon, C. R . , and Brown, R. (1966). Phys. Rev. 146, 575. Ponikarov, B. B., Tsidilkovskii, I. M . , and Shelushinina, N. G . (1981). Fiz. Tekh. Poluprov. 15, 296. (Sov. Phys. Semicond. 15, 170.) Reifenberger, R., and Schwarzkopf, D. A. (1983). Phys. Rev. Lett. 50, 907. Roth, L., and Argyres, P. (1966). In “Semiconductors and Semimetals” (R. K. Beer, and A. C. Willardson, eds.) vol. 1, p. 159, Academic Press. Sandauer, A. M., and Byszewski, P. (1982). Phys. Stat. Sol. (b) 109, 167. Sandauer, A. M., Dobrowolski, W., Kossut, J., and Galazka, R. R. (1983). In “Physics of Semiconductors” (M. Averous, ed., Proc. 16th Internat. Conf. Monpellier, 1982) part I, p. 455, North Holland (Physica 117-118 B + C). Sawicki, M., Dietl, T., Plesiewicz, W., Sekowski, P., Sniadower, L., Baj, M., and Dmowski, L. (1982). In “Application of High Magnetic Fields in Physics of Semiconductors” (G. Landwehr, ed., Proc. Internat. Conf., Grenoble, 1982). Springer-Verlag. Schrieffer, J. R.,and Wolff, P. A. (1966). Phys. Rev. 149, 491. Seager, C. H. (1982). In “Grain Boundaries in Semiconductors” (H. J. Leamy, G . E. Pike, C. H. Seager, eds.) North Holland. Semenov, Yu. G . , and Shanina, B. D. (1981). Phys. Stat. Sol. (b) 104, 631.

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Stankiewicz, J., Giriat, W., and Bien, M. V. (1975). Phys. Stat. Sol. (b) 68, 485. Story, T., Lewicki, A., and Szczerbakow, A. (1985). Acta Phys. Polon. A67, 317. Takeyama, S.,and Galazka, R. R. (1979). Phys. Stat. Sol. (b) 96, 413. Toth, G., and Leloup, Y. Y. (1970). Phys. Rev. B1, 4573. Uchida, S., Landwehr, G., and Bangert, E. (1983). Solid State Commun. 45, 9869. Vonsovskii, S. V., and Svirskii, M. S. (1964). Zh. Eksp. Teor. Fiz. 47, 1354. (Sov. Phys. JETP 20, 914.) Vonsovskii, S. V., and Turov, E. A. (1953). Zh. Eksp. Teor. Fiz. 24, 419. Vul, B. M., and Zavaritskaya, E. T. (1979). Zh. Eksp. Teor. Fiz. 76, 1089. (Sov. Phys. JETP 49, 551.) Wallace, P. R. (1979). Phys. Stat. Sol. (b) 92, 49. Watson, R. E., and Freeman, A. J. (1966). Phys. Rev. 152, 566. Watson, R. E., and Freeman, A. J. (1969). Phys. Rev. 178, 725. Wittlin, A., Knap, W., Wilamowski, Z., and Grynberg, M. (1980a). Solid State Commun. 36, 233. Wittlin, A., Grynberg, M., Knap, W., Kossut, J., and Wilamowski, Z. (1980b). J. Phys. SOC. Japan 49 (suppl. A), 635. (Proc. 15th Internat. Conf. Phys. Semicond. Kyoto, 1980.) Yosida, K. (1957). Phys. Rev, 106, 893. Zawadzki, W. (1983). J. Phys. C16, 229. Zener, C. (1951a). Phys. Rev. 81, 440. Zener, C. (1951b). Phys. Rev. 82, 403. Zener, C. (1951~).Phys. Rev. 83, 299.

SEMICONDUCTORS A N D SEMIMETALS, VOL. 25

CHAPTER 6

Magnetooptics in Narrow Gap Diluted Magnetic Semiconductors C. Rigaux GROUPE DE PHYSIQUE DES SOLIDES DE

L’ECOLE NORMALE SUPERIEURE PARIS, FRANCE

INTRODUCTION. . . . . . . . . . . . . . . . . . . . THEORETICAL DESCRIPTION OF ELECTRONIC STATESIN NGDMS 1 . Band Structure . . . . . . . . . . . . . . . . . . 2. Landau Levels . . . . . . . . . . . . . . . . . . 3. Magnetooptical Transitions . . . . . . . . . . . . . 111. INTERBANDMAGNETOOPTICS. . . . . . . . . . . . . . . 4. Zero Gap Hg,-,Mn,Te. . . . . . . . . . . . . . . 5. Narrow Gap H g l - , M n x T e . . . . . . . . . . . . . . 6 . Other Semimagnetic Compounds. . . . . . . . . . . IV. INTRAEIANDMAGNETOOPTICS . . . . . . . . . . . . . . 7. Far IR Magnetospectroscopy in Zero Gap DMS. . . . . 8 . Harmonics of the Cyclotron and Combined Resonances in Narrow Gap Semiconductors . . . . . . . . . . . . V. RESULTS OF MAGNETOOPTICAL STUDIES.. . . . . . . . . . 9. Band Parameters . . . . . . . . . . . . . . . . . 10. Exchange Parameters . . . . . . . . . . . . . . . VI. CONCLUDING REMARKS. . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . I. 11.

229 23 1 232 233 237 240 24 1 248 252 257 257 264 267 267 269 27 1 272

I. Introduction Diluted Magnetic Semiconductors (DMS) are characterized by the presence of magnetic ions of a transition element, that enter substitutionally into the semiconductor lattice. The interest in this class of magnetic semiconductors arises from the possibility of controlling the dilution of magnetic ions and the well known band structure of the host semiconductor (typically a 11-VI compound). Thus DMS offer an unique opportunity of gradually incorporating the exchange phenomena, characteristic of magnetic semiconductors, into the electronic properties of classical semiconductors. Narrow-gap DMS are ternary random alloys obtained by substituting magnetic ions (Mn”, Fez+) for Hgz+ in mercury chalcogenides (HgTe, HgSe). Among them, the system Hg1-xMnxTe has been studied particularly 229 Copynght 8 1988 by Academic Press. Inc. All rights o f reproduction in any form reserved. ISBN 0-12-752125-9

230

C. RIGAUX

thoroughly. Both groups of compounds exist in zinc blende structure, forming a single crystallographic phase up to x = 0.5 for Hgl-,Mn,Te and x = 0.3 for Hgl-,Mn,Se. The Te (or Se) atoms occupy the sites of one of the fcc sublattices, whereas mercury and manganese atoms are more or less randomly distributed over the second fcc lattice sites. The delocalized electronic states responsible for the semiconducting properties of these compounds originate from s-(cations) and p-(anions) type orbitals. In the absence of magnetic field, the band structure of Hgl-,Mn,Te or Hgl-,MnxSe is very similar to that of non-magnetic ternary alloys such as Hgl-,Cd,Te or Hgl-,Cd,Se, displaying either a zero gap or a narrow gap semiconductor band structure, with the interaction gap depending on composition. In Hgl -,Mn,Te, the interaction gap separating s(I-6)- and p(rg)-like bands varies from -0.3 t o 1 . 1 eV in the range of composition 0 Ix I0.5. The semiconducting properties of these alloys may be controlled by varying the composition. The essential peculiarities of DMS originate from the unfilled d shells of transition elements. The d electrons tightly bound to the Mn2+ions give rise to localized magnetic moments that are responsible for the magnetic properties of these compounds: the magnetic susceptibility of Mn-based DMS follows a Curie-Weiss law which evidences antiferromagnetic interactions (Spalek et al., 1986) between magnetic ions, even in diluted alloys (Galazka, 1982). The existence of a spin-glass phase, at low temperature, is established from several experiments in HgMnTe (Nagata et al., 1980; Brandt et al., 1983; Mycielski et a/., 1984) and HgMnSe (Khattak et a/., 1981). In the presence of an external magnetic field, the electronic properties of DMS are strongly influenced by the exchange coupling between d electrons and mobile carriers. In NGDMS, the direct effect of magnetic field on the delocalized carriers leads to the Landau quantization, as these compounds are characterized by small effective masses. Moreover, the magnetic field orients the localized magnetic moments, creating a non-vanishing average component of the Mn spins along its direction. The exchange interactions between a delocalized electron and the Mn spins drastically affect the spin properties of mobile carriers in quantizing magnetic field. The effect of magnetic field on electronic states in NGDMS has been analyzed theoretically (Bastard et al., 1978; Jaczynski et a/., 1978; Galazka and Kossut, 1980) in terms of a model where, using the virtual crystal and molecular field approximations, the exchange interactions are incorporated into the framework of the Pidgeon-Brown formalism (Pidgeon and Brown, 1966). Energies of Landau spin sublevels are then modified by exchange contributions involving the d electron magnetization and become temperature dependent.

6.

MAGNETOOPTICS IN NARROW GAP DMS

231

The role of exchange interactions on electronic properties of DMS has been discussed in several review articles (Galazka, 1978; Bastard et al., 1980; Galazka and Kossut, 1980; Gaj, 1980; Mycielski, 1981; Dietl, 1981; Furdyna, 1982; Brandt and Moshchkalkov, 1984). Magnetooptics provide a particularly useful technique for investigating the exchange phenomena and the localized spin ordering in wide gap (Gaj et al., 1978) and narrow gap DMS (Bastard et al., 1980). Near and far infrared (IR) spectroscopy involving interband and intraband transitions between Landau levels have revealed striking magnetooptical anomalies in zero and narrow gap semimagnetic semiconductors. These are related to the effect of exchange on delocalized electronic states: the occurrence of anomalously large and temperature-dependent gyromagnetic factors of conduction electrons (positivein the case of narrow gap semiconductors) and the existence of a magnetically induced overlap between the valence and conduction bands in zero gap HgMnTe are among the most spectacular manifestations of these exchange phenomena. The theoretical analysis of the magnetooptical data not only yields the fundamental band structure parameters (energy gap, effective masses)-similarly as in non-magnetic semiconductors-but also provides access to the magnetization of the subsystem of localized spins. In the latter context, for example, the Faraday rotation, which is directly related to the magnetization, has been used to observe the spin-glass phase transition in Hgl-,Mn,Te (Mycielski et al., 1984). In this chapter the most important magnetooptical experiments in zinc blende NGDMS are reviewed, with emphasis on the striking manifestations of exchange phenomena. The theoretical description of electronic states in the presence of a magnetic field and the determination of magnetooptical transitions are presented in Part 11. Interband magnetooptics in zero and narrow gap DMS are reviewed in Part 111, with emphasis on the differences between magnetic and non-magnetic ternary alloys of similar band structure. FIR magnetospectroscopy, leading to the observation of intraband and interband r8 r8 transitions in zero gap DMS, are described in Part IV. The results of magnetooptical investigations (band and exchange parameters) are summarized in Part V and are compared with those obtained from quantum transport (Kossut, this volume). +

11. Theoretical Description of Electronic States In semimagnetic semiconductors, such as HgMnTe, one may distinguish two sets of electronic states: -Localized d states originating from unfilled d shells of the transition element are characterized by spatially localized wave functions centered around the magnetic ions. The d electrons give rise to localized magnetic

6 .I

232

C . RIGAUX

moments. The 3d5 electrons of Mn form a ‘SW multiplet, whose degeneracy is not lifted by the cubic ligand field. This configuration, corresponding to L = 0, S = 5, J = 8, and the Land6 factor gMn = 2, were experimentally confirmed by electron paramagnetic resonance (Leibler et al., 1978), helicon enhanced spin resonance (Holm and Furdyna, 1977; Mullin et al., 1981) and static magnetic susceptibility measurements (Savage et al., 1973; Andrianov et al., 1976; Spalek et al., 1986). -Extended states giving rise to wide bands originate from outer shells, i.e., 6s2(Hg) or 4sZ(Mn) and 5p4(Te). In random substitutional alloys, the electronic states are described within the virtual crystal approximation (VCA), which replaces the crystalline potential of Hg (6s2) and Mn (4s’) by the weighted average of these atomic constituents. The translational periodicity of the lattice is then restored, and the electrons are described in terms of Bloch states. In practice, the VCA consists of describing the electronic band structure of ternary random alloys in terms of k p models developed for pure crystal components, with the band parameters depending on composition.

-

1. BAND STRUCTURE In II-VI semiconductors, the band edges of relevance are located at the Brillouin zone center (r point), where the states transform according to the representation of the Td crystal symmetry group. The six p-like states are decoupled by the spin-orbit interaction into a quadruplet Ts (J = 4) and a doublet r7 (J = the latter forming a lower-lying valence band, with the energy A = Er, - Er, determined by the spin-orbit splitting. In HgTe, the s-like band, transforming as r6 (J = *), lies below the Ts quadruplet, leading to a symmetry-induced zero-gap semiconducting configuration; the Ts levels then form the degenerate valence and conduction band edges, separated from the r6 valence band by the interaction gap EO = Er6 - Ers = -0.3eV at 4.2 K (Fig. 1). CdTe and (hypothetical) cubic MnTe are wide gap semiconductors. The Tsquadruplet forms two degenerate valence bands and the conduction band edge belongs to r6 representation. The fundamental energy gap (1.6 eV in CdTe) is estimated to be 3.2 eV in cubic MnTe (Gaj et al., 1979). The band structure of ternary alloys Hg, -,Cd,Te and Hg, -,Mn,Te evolves continuously from the zero gap configuration to an open-gap configuration, as shown in Fig. 1. The interaction gap EO , defined as a difference between energies of the r.5and rs states, increases with the alloy composition. These systems undergo a zero gap to semiconductor transition that occurs, at 4.2 K, for x = 0.075 in Hgl-,Mn,Te and x = 0.165 in Hgl-,Cd,Te. In the absence

i),

6.

233

MAGNETOOPTICS IN NARROW GAP DMS

-

F

T

\

/

\

“positive gop”

$i

v

\‘“ I

A

9

‘0

4-

I

I

I

I

I

1

0.02 0.04 006 0.08 0.10 0.12 Mn content ( X I

I

*

0.14

FIG. 1 . Band structure of Hgl-,Mn,Te at the center of the Brillouin zone.

of magnetic field, the electronic states in NGDMS may be described in terms of the Kane band structure model (Kane, 1957).

2. LANDAULEVELS In the presence of an applied magnetic field, the electronic states in DMS are obtained by including the exchange interaction as an additive term X,, in the proper effective mass Hamiltonian XO. For zinc blende narrow gap semiconductors, the most accurate effective mass Hamiltonian XO is provided by the eight-band Pidgeon-Brown model (1966). In this model the k p interactions between closely spaced r6, r7, Ts bands are treated exactly and the interactions between these states, and the more remote higher bands are included up t o order k2. The magnetic Hamiltonian XO, in the absence of exchange, consists of an 8 x 8 matrix D,that reduces to two 4 x 4 matrices Da and Db ,at the r point when inversion asymmetry and warping in Tg bands are neglected. The matrix D has the form:

-

in the basis of the band edge Bloch functions ur(J, MJ) associated with

r 6

234

(f

= 1,2),

C . RIGAUX

Ts (f

= 3,

..., 6 ) , r7(I

i.;(;,-;)

rU2(;-;,)

= 7,8) levels:

= -&(Xi

iY)t

+ Zl)

= ijS1)

Here S , X , K 2 are periodic functions that transform like atomic functions s, p x , pr , p E ,respectively, under the operations of the Td group at the

r

point. XOinvolves the following band parameters: the interaction gap EO = Er, - Er,, the spin-orbit splitting energy A = Er, - Er,, the energy of the s-p interaction Ep = (2/mo)I ( S ( p , l X )12, and modified Luttinger parameters yl, 7, K describing the interaction between Ts and distant bands. The exchange interaction between 3d electrons and delocalized electrons may be described in the vicinity of the r point (Kossut, 1976) by an Heisenbergtype Hamiltonian

where Si is a localized spin at the site Ri (Si = 5);0 is the spin of the mobile electron, and J(r - Ri) is an exchange integral rapidly varying over a unit cell. Since the mobile electron has a spatially extended wave function, it interacts simultaneously with a larger number of Mn ions. This allows one to replace the Mn spin operators by their thermal average ( S ) . In the absence of magnetic field ( S ) vanishes. When the magnetic field H is applied along the z direction, ( S ) has only a non-vanishing component (S,) .

6.

MAGNETOOPTICS IN NARROW GAP DMS

235

Thus, by using the VCA and the molecular field approximation, the exchange perturbation becomes:

where R denotes the sites of the cation sublattice. ( S , ) is directly related to the magnetization M of Mn ions: M

=

-NOxgMnpB(Sz),

(4)

where No is the number of unit cells per unit volume, x is the Mn molar fraction and p~ is the Bohr magneton. For 3d5 electrons of Mn, ( S , ) = -Sfnorm;where S = 5 and fnom denotes the normalized magnetization. For non-interacting magnetic moments,

where B5/2(X)is the Brillouin function for a spin S = 5: B,(x) =

2S+1

2

~

coth

2S+1 ~

2

x

-

1 x -coth-. 2 2

In the basis (l), the exchange Hamiltonian, Eq. ( 3 ) , consists of two 4 x 4 matrices :

where /3Aa/P

0

0 -3AdP

Mb=[

0 0 0

0

-2Afi

(7)

0 0 A 0 0 -3A 2Afi 0

The off diagonal r7 - TSterms in Eq. (7) may be neglected in the case of large spin-orbit splittings. Here M, and Mb involve the s-d and p-d exchange integrals (Y = (SIJ(r)lS),B = (XIJ(r)IX) = (YIJ(r)lY)= ( Z I J ( r ) l Z ) ,that are the only non-vanishing matrix elements of J(r) involving S , X , X Z

236

C . RIGAUX

functions. The exchange parameter appearing in Eq. (7), A = &NoPx(S,) = - &No/3xfnorm, is directly proportional to the magnetization M , Eq. (4). The Landau levels are obtained by solving the two decoupled matrix equations:

(DO- Eaf))lva = 0 ,

(Db

-

Ebf))Wb =

0,

(8)

where Da = D, + Ma and D b = Db + Mb include both band structure and exchange contributions. The wave functions v / o , V b are given in terms of envelope functions f n ( r ) and band edge Bloch functions, Eq. (1):

wa(n) = a i nf n u l vb(n)

=

+ a 3 n f n - i ~ + a 5 n f n + i ~ 5+ a 7 n f n + i ~ 7 , + h n f n - 1 ~ 4 + b6nfn+lU6 + b8nfn-lu8.

(9)

b2nfn~2

For n > 0 , f n ( r ) is the Landau function fn(r) =

exp(i(k,y + kzz))+n(x + A2k,>,

where +n is the n-th harmonic oscillator eigenfunction of characteristic length A = (hc/eH)"2. If n < 0, fn 0. For each quantum level n L 1, there exist four solutions in each a and b ladder that describe r6, TS (light and heavy), and r 7 Landau levels. The eigenvalues and eigenvectors are obtained by solving numerically Eq. (8) for a given set of band and exchange parameters, i.e., at fixed temperature and magnetic field. For the levels n = -1 the solutions are analytical:

where hoo is the free electron cyclotron energy. The exchange contribution induces an energy shift of the levels which is three times larger for b( - 1) than for a( - 1). Approximate expressions of the Landau level energies obtained in the parabolic limit are useful for understanding the influence of exchange phenomena. In the r 6 band, the energies of a and b ladders corresponding to nt (or a) and n l (or b) levels, respectively, are

where m?6 and g?6 are the band-edge effective mass and gyromagnetic factor

6.

MAGNETOOPTICS IN NARROW GAP DMS

237

resulting from the k * p interactions:

g *r , -2

(

z.,:*>*

I------

The last term in Eq. (11) represents the s-d exchange contribution. The exchange effect leads to a modification of the effective g-factor, that can be expressed in the form:

For the l-8 bands, the energies of a and b ladders are given in the parabolic limit by the analytical expressions of the Luttinger model (Luttinger, 1959), in which the parameter K is replaced by R = K - 2A/hw0. In the classical limit (i.e., large n), the l-8 band-edge effective masses are

and the light l-8-band gyromagnetic factor is

As indicated by the above expressions (12)-( 16), the exchange perturbation does not change the effective masses, but contributes an additional term to the effective gyromagnetic factors that depend on temperature and magnetic field through fnorm. The effect of exchange on the magnetic sublevels thus results essentially in a temperature and field dependent shift, the orbital motion in quantizing magnetic field remaining unaffected. 3. MAGNETOOPTICAL TRANSITIONS The interaction between the electromagnetic radiation of frequency o and the band electron is described by the perturbation Hamiltonian (Roth el al., 1959):

238

C. RIGAUX

Here EOis the amplitude and E the unit vector in direction of the electric field of the radiation, and ll is the operator - ih V + e A / c , where A is the vector potential due to the external magnetic field (H 11 d ) . The probability for a direct transition of an electron from an initial state IZ) to a final state IF) is proportional to the square of the modulus of the matrix element nt = (FIE* IIlZ). The absorption coefficient is given by:

where I/ is the crystal volume, q is the refractive index, and g(E1) and (EF) denote the occupation factors for initial and final states, respectively. We consider direct transitions between Landau levels described within the modified Pidgeon-Brown model by wave functions, Eq. (9), of the form:

#I(m) =

C Ajffrn;, ujr ;

j'

yF(n) =

Cj p j f n j u j .

As the envelope function f(r) and the vector potential A(r) are slowly varying over the unit cell, after taking into account the orthogonality of the bandedge Bloch functions the matrix element connecting initial and final states is:

The dominant contribution to the transition results from the first term. The absorption coefficient is finally obtained by summing Eq. (17) over initial ( m , ky ,k,) and final ( m , ky , k,) states:

with the condition

ha

=

EF(n, k,) - EI(m, kz).

(19)

Magnetoabsorption peaks occur for the k,-values corresponding to the minima of the energy distance EF@,k,) - EI(m, k,), usually at k, = 0 for interband transitions. By expressing the transition operator E ll in the form &-TI+ + &+lT-+ &,lT,, where E* = ( E ~f k y ) / f i , lT* = (n, f i r I y ) / f i , and using the Landau level wave functions (9), the selection rules for E + , E - , cZ radiation polarizations are derived. These are listed in Table I .

-

6.

239

MAGNETOOPTICS IN NARROW GAP DMS

TABLE I SELECTION RULESFOR INTERBAND MAGNETOOPTICAL [m. n DENOTELANDAUQUANTUM TRANSITIONS a , b CORRESPOND TO A GIVENSET OF NUMBERS; SOLUTIONS, EQ.(9)]

Polarization

Selection rules

E-

E+

o(m) b(m)

+

+

b(n) n a(n) n

= =

m m

+1 -

1

All allowed interband magnetooptical transitions are included in Table I: l-8 and Ts" l-; transitions in zero gap semiconductors (the latter, i.e., to transitions resulting entirely from the s-p coupling, which induces an s-type component in the TS electronic wave function), and l-8,ighl + r6 and rghcavy-+ r6 in open gap semiconductors. At a given field, the relative transition probabilities are determined by IEjnmnlZ, with the dominant contribution, resulting from the first term in Eq. (18), proportional to E p . The relative strengths of the magnetooptical transitions depend on the Landau level eigenvectors, and can be estimated (for k, = 0) by solving numerically Eq. (8) for a given set of band and exchange parameters. For r6 TS transitions in zero gap semiconductors (and TS r6 in open gap configuration), there exist only one series of dominant transitions for each polarization of the incident radiation. The intraband transitions whose matrix elements do not vanish at k, = 0 are also contained in Table I. The cyclotron resonances, allowed for E polarization, correspond to intra-set transitions (a a or b b) with An = + 1. Even for cyclotron resonance, the intraband part of the matrix element, Eq. (18), is always negligible as compared to the first term resulting from the s-p interband coupling. The combined resonances, allowed for E /I H, correspond to inter-set transitions (a b) with An = + 1. These "spin flip" transitions arise from both spin-orbit effects (mixing spin up and down states in the TS band-edge Bloch functions) and s-p interband coupling in wo and vb wave functions. The mechanism also accounts for the occurrence of the electric dipole spin resonance (EDSR corresponding to a + b with An = 0) in the E + polarization. As this transition takes place at k, # 0, the full eight-band Hamiltonian r6

-+

-+

+

--t

-+

+

240

C . RIGAUX

(Weiler, 1981) or the simplified three band model (Kacman and Zawadzki, 1971) are required to describe the Landau levels away from the r point and the electric dipole spin resonance absorption (McCombe, 1969). The inversion asymmetry of the zinc blende lattice leads to additional that have been neglected in the above terms in the magnetic Hamiltonian KO, formulation as these terms are usually very small. The inversion asymmetry effects lead to induce the electric dipole spin resonance and the combined resonance for any polarization. Of the two possible processes for the excitation of intraband spin-flip resonances, the s-p interband coupling was, until recently, thought to be the dominant mechanism in HgCdTe compounds (McCombe et al., 1970,1971). However, more recently, it was clearly demonstrated in the case of InSb by Chen et al. (1985) that the lack of inversion symmetry in the zinc blende lattice is indeed the dominant mechanism which allows the transitions involving spin-flip resonances by electric dipole excitation.

111. Interband Magnetooptics

Interband Magnetooptical studies in NGDMS were carried out by low temperature IR magnetoabsorption experiments (typically between 200 and 370 meV) performed on very thin unoriented samples, for both Faraday and Voigt geometries using 0 and l 7 polarizations, respectively. Oscillatory magnetotransmission spectra observed at photon energies larger than the interaction gap are associated with interband r6 T8 (zero gap) and l-8 r 6 (open gap) magnetooptical transitions. Several semimagnetic compounds were investigated. The most extensive interband experiments were performed on the Hgl -,MnxTe system, which has been studied in both zero and narrow gap configurations, over a wide range of compositions (0.001 Ix I0.17), at helium temperatures (Bastard et al., 1977,1978) and between 10 and 65 K (Dobrowolska and Dobrowolski, 1981). The r6 r S magnetoabsorption data obtained at 4.2 and 2 I( for dilute HgMnTe alloys ( x I0.015) were the first experimental results to demonstrate striking magnetooptical anomalies characteristic of NGDMS. These were correlated with exchange phenomena by comparison with the spectra of non-magnetic Hgl -,CdxTe alloys of similar band structure (Bastard et al., 1978). Far IR magnetooptics carried out in the same + T pC composition range have enabled the observation of interband transitions in the low energy spectrum (Pastor et al., 1979). Interband magnetoabsorption data have been also reported for zero and narrow gap Hgl-,MnxSe compounds (Dobrowolska et al., 1981), as well as for very dilute semimetallic Hgl-,Fe,Te mixed crystals (Serre et al., 1981, 1982). +

+

-+

6.

MAGNETOOPTICS IN NARROW GAP DMS

241

4. ZEROGAPHgl-,MnxTe

a. Experimental Manifestations of Exchange Interactions Interband r6 Ts magnetoabsorption spectra in zero gap HgMnTe alloys are illustrated in Figs. 2-4 by plots of transition energies vs. magnetic field. The experimental lines are identified with the dominant transitions according to the selection rules: +

&-(or 0 - ) : ar6(n) ar,(n +

+ 1);

&+(oro+):br6(n + 1) 4 brs(n); E

(1 H: br6(n +

1)

+

(20)

ar8(n).

The line identification is indicated in the figures. Only for very diluted alloys (x = 0.0014), the relative ordering of the transitions is identical to that observed in the spectrum of HgTe (Guldner et al., 1973) or Hgl-,Cd,Te alloys (Guldner et al., 1977; Rigaux, 1980). For x L 0.004, a change of regular transition sequence appears, resulting in an inversion of the relative positions of the circular polarization transitions ar6(n) ar,(n + 1)(0-) and br,(n) 4 brs(n - l)(a+)(Figs. 2-4). Comparison between Hgl-,MnxTe and Hgl -,Cd,Te alloys of identical interaction gaps shows that this modification originates from anomalous spin effects in the semimagnetic compounds, whereas the energy separation between consecutive transitions for a given polarization (i.e., the period of oscillations of a given spectrum) remains +

H( kG 1 FIG. 2. Energies of r6 Ts transitions vs. magnetic field for Hgl-,Mn,Te of Mn content x = 0.004 at 4.2 K . Symbols are experimental. Solid lines represent theoretical fits using the modified Pidgeon-Brown model. The quantum number n identifying the transitions according to Eq. (20) is indicated for each line. [After Bastard et 01. (1978).] +

242

C . RIGAUX

0

10

20

30

50

40

H(kG)

FIG. 3. Energies of rs

--f

l-8

transitions vs. magnetic field for Hg,-,Mn,Te of Mn content

x = 0.004 at 2 K. Symbols are experimental. Solid lines represent theoretical fits using the

modified Pidgeon-Brown model. The quantum number n identifying the transitions according to Eq. (20) is indicated for each line. [After Bastard et al. (1978).]

360

I

I

I,.

I

I

'

3 20

2 v

W

280

HW) FIG. 4. Energies of r,5 + Tg transitions vs. magnetic field for Hgl-,Mn,Te of Mn content x = 0.008 at 2 K . Symbols are experimental. Solid lines represent theoretical fits using the modified Pidgeon-Brown model. The quantum number n identifying the transitions according to Eq. (20) is indicated for each line. [After Bastard ef 01. (1978).]

identical for both compounds. These features are illustrated in Fig. 5 by plots of the period ha and the spin splitting = Eb(l) - Ea(l)of the n = 1 conduction Landau level, directly measured form the energy difference at a given field, between the r7+ and E 11 H transitions, c.f., Eq. (20). The quantity ha, representing in the parabolic limit the sum of the cyclotron energies in r' and Ta conduction bands, has comparable values for HgMnTe and HgCdTe, demonstrating the similarity of the band parameters in both families. The strong enhancement of the spin splitting in the semimagnetic

6. I

243

MAGNETOOPTICS IN NARROW GAP DMS

H=20kG

I

I

20

2 E

Y

UJ

10

0

I

I

I

-300

-250

-200

I

e, ( meV 1

- 150

I

-100

FIG.5. Spin and cyclotron splittings, Sc(l) and An; respectively, as a function of EO in HgCdTe (open symbols) and HgMnTe (black symbols) at 4.2 and 2 K, for H = 20 kG. [After Bastard et al. (1978).]

compounds provides direct evidence of the modification of electron spin sublevels induced by the exchange interaction. Another manifestation of exchange interaction in zero gap HgMnTe is the 0 spectrum observation of a new and intense magnetoabsorption line in the ' of alloys for x 2 0.004 (Fig. 6). This line, quantitatively interpreted as being 0 polarization, provides direct due to br6(0) br,(-l) allowed for ' evidence for the emergence of the br8(- 1) heavy hole level above the Fermi energy. One of the most remarkable properties of r6 * Ts magnetooptical spectra is their strong temperature dependence in striking contrast to HgCdTe alloys. On decreasing the temperature, an energy shift is observed for the circular ' , negative for 0-), but E 1) H transitions are polarization lines (positive for o unaffected. The temperature behavior is illustrated in Fig. 6 by the shift of the resonance field of the inter-valence transition br6(0) br,(- 1). The observed temperature dependence of l-6 Ts transitions results in an enhancement of the conduction electron spin splitting between 4.2 and 2 K (Fig. 5). This effect originates entirely from the exchange contribution, which induces an (&)-dependent term in the gyromagnetic factor (11.2, Eq. (16)). A remarkable peculiarity of the spectra is the opposite curvature exhibited by the u+ and o- transitions in the energy diagrams E ( H ) (Figs. 3, 4). This bowing effect, particularly pronounced at 2K, gives rise to an unusual crossing of the lines associated with br,(n) * br,(n - I )(o') and ar,(n) ar,(n + l)(a-)transitions. All these manifestations of the exchange interaction are particularly pronounced at very low temperature (4.2 and 2 K). In the temperature range -+

+

-+

+

244

C. RIGAUX

I

0 10

I

I

20

30

1 LO

I

50

H (kG)

FIG. 6 . Magnetotransmission spectrum at hw = 243.3 meV for HgMnTe of Mn content 0' polarization, observed at T = 4.2 and 2 K . [After

x = 0.015, in the Faraday geometry and Bastard et a / . (1978).]

10-65 K, the results of Dobrowolska and Dobrowolski (1981) show that the magnetooptical transitions conserve the same sequence as in the spectrum of HgTe, and the inter-valence transition is not observed. The main effect of the exchange interaction consists in the temperature dependence of the spin splitting Sc(n), which is seen to decrease with increasing temperature.

b. Theoretical Interpretation Interband magnetooptical spectra were successfully interpreted within the framework of the Pidgeon-Brown model generalized to include the s-d and p-d exchange contributions (Part 11, section 2 ) . A numerical fitting procedure of the r6 + Ts transition energies to the experimental lines was achieved at a fixed magnetic field. The variables of the minimization procedure were restricted to Ep(or P = ( - ih /m )(Sl p x l X ))and the exchange parameters. The spin-orbit splitting and the Luttinger parameters were fixed to the values obtained for HgCdTe alloys of low composition: A = 1 eV; y1 = 3; p = 0; K = -1.65 (Guldner et al., 1977). The interaction gap EO was determined for each composition from the extrapolated zero field energy of the r6 r8 transitions and then adjusted to obtain the best theoretical fits. For dilute HgMnTe compounds ( x < 0.015), EPwas found between 18.3 and 18.8 eV ( T = 4.2 and 2 K), in excellent agreement with the results +

6.

MAGNETOOPTICS IN NARROW GAP DMS

245

obtained for HgCdTe, which demonstrates the similarity of the band structure in both alloy systems. By fitting low temperature magnetoabsorption data, Bastard et al. (1978) determined the ratio r = C Y /of~ exchange integrals and the relative magnetization A ( H, T ) . The parameter r was found to be independent of composition and equal to -0.5 for k 5 0.015. The magnetic field dependence of A ( H ) obtained from the theoretical fits for 4.2 K is shown in Fig. 7. For extremely dilute alloys only (cO = -293 meV, x = 0.0014), the variation A ( H ) follows in relative magnitude the Brillouin function BS/Z(gMnpUgH/kB T ) , which enabled the evaluation of exchange integrals NoP = 1.5 eV and NOCY = -0.75 eV for this particular composition. Using these values, the normalized magnetization fnorm(H, T ) for diluted compounds was deduced from A ( H , T ) . The results, presented in Fig. 16 (see Part 111, section 6b), agree quantitatively with direct magnetization measurements at 4.2K for x = 0.01 and with low field d c magnetic susceptibility data for x = 0.012 (Nagata et al., 1980). The comparison between theory and experiments, at 4.2 and 2 K , is illustrated in Figs. 2, 3, and 4. Excellent agreement is found at both temperatures for a common set of parameter (except A , which is a function of temperature and magnetic field) (Fig. 7). In particular, the curvatures of 'rc and cr- lines originating from the effect of magnetization on the Landau level energies are well accounted for by the model, and the experimental position of the inter-valence transition is quantitatively interpreted (Figs. 2-4). Band-edge effective masses and gyromagnetic factors of conduction electrons for very dilute HgMnTe and HgCdTe alloys (in the parabolic approximation) are shown in Fig. 8. The results show the similarity of

OL 0

'

I

I WG)

I

I

]

50

FIG.7. Relative magnetizationA as a function of the magnetic field, observed at T = 4.2 K for Hg,-,MnxTe of different compositions. [After Bastard et a/. (1978, 1981).]

246

C . RIGAUX

0.035

1 -I

0.0251.

i-"

0 -300

, &@",j -250

-200

FIG.8. Band edge effective masses and gyromagnetic factors for the conduction band in zerogap HgCdTe (crosses) and HgMnTe alloys. Open circles are for T = 4.2 K, and black circles for T = 2 K. [After Bastard (1978).]

effective masses and the strong enhancement of the gyromagnetic factor induced by exchange interactions in the semimagnetic system.

c. MagneticaIIy Induced Semimetal-to-Semiconductor Transition The T8 Landau level spectrum ( k =~0) calculated from the band and exchange parameters is reproduced in Fig. 9 for several Mn compositions. The magnetic field dependence of the energy levels exhibits at 4.2 and 2 K quite interesting peculiarities originating from exchange phenomena: the spin-spin interaction induces a magnetic-field-dependent shift of the l-8 levels (positive for b(n),negative for a(n)),leading to an enhancement of the spin splitting Sc(n) = b(n) - a(n), which may exceed the cyclotron energy. For x > 0.004, the levels b(n) overlap a(n + l), and the usual sequence of the Landau levels becomes modified. The strong upwards shift AE = -3A of the heavy hole level brs(-l) predicted by the model (11.2, Eq. (10)) is responsible for the occurrence of the transition br,(O) br8(-l). The overlap of the lowest electronic level ar,(O) by the uppermost valence state -+

6.

MAGNETOOPTICS IN NARROW GAP DMS

241

brs(- 1) leads to a semimetallic configuration induced by the magnetic field L 0.004). In the high field region, the exchange contributions to ars(0) and brs(- 1) energies reach a saturation, and the field behavior of these levels is entirely governed by the band structure contribution, i.e., ar,(O) increases 1) decreases with field. For some critical field a crossing appears and hS(between the uppermost valence and lowest conduction levels, and a magnetic-field-dependent gap is opened between the Ta bands. Thus a magnetically induced semimetal-to-semiconductor transition occurs in zero gap Hgl-xMnxTe alloys. For x = 0.004 and 2 K, this transition takes place at about 37 kG (Fig. 9). (x

20

2 E

10

Y

w

0

20

10

0 0

10

20

30 40

0

H (tG)

10

20

30 40

FIG. 9. Magnetic field dependence of Ts Landau level energies at kz = 0, for HgMnTe of several Mn compositions, for 4.2 and 2 K. [After Bastard et al. (1978).]

248

C. RIGAUX

5 . NARROW GAP Hgl-,Mn,Te

a. TS

+

r 6

Magnetoabsorption

Semiconducting Hgl-xMnxTe alloys of narrow but positive band gap (100 IEg I300meV) were investigated by low temperature interband magnetoabsorption (Bastard et al., 1981; Dobrowolska and Dobrowolski, 1981). The TS-, rb magnetooptical spectra again show novel features, that contrast sharply with the results obtained in non-magnetic zinc blende semiconductors (Rigaux, 1980). A reduction of the energy gap appears in applied magnetic field, since the lowest energy transition-observed for 0-polarization-decreases in energy with increasing magnetic field (Fig. lob). Semiconducting HgMnTe alloys are characterized by the near-coincidence of 0-and E 1) H transitions, except for the two lowest-lying transitions (labelled - 1 , O in Fig. 10). This feature is illustrated in the magnetotransmission spectra shown in Fig. 11 for an alloy of energy gap E, = 219 meV for 4.2 K. The TS + r6 magnetooptical transitions depend weakly on temperature, in contrast to the behavior observed in zero gap HgMnTe: the lowest-lying transitions are only slightly shifted to lower energies with decreasing temperature whereas the other lines (nearly coincident in 0- and E I( H spectra) are practically unaffected. These characteristic features of narrow gap semimagnetic semiconductors are explained by the effect of exchange on TSand r6 Landau levels. Quantitative analysis of the transition probabilities (that depend on the exchange contribution) indicates that the transition from heavy hole Landau levels are expected to be much weaker for the o+ polarization than for 0- and E I( H polarizations. For E 1) H, transitions from heavy hole ladders are the most intense, the dominant ones corresponding to: ar,(n)

+

bs(n

+

l),

TI 2

-1.

(21)

For the 0- polarization, transitions from heavy and light valence ladders are expected to be strong, the dominant series corresponding to:

br,(n) --* br6(n + l),

n

L

-1.

(22)

Experimental lines observed for the E ( J Hand 0- polarizations are identified according to Eqs. (21) and (22), respectively (Fig. 10). The nearcoincidence (for n > 1) of the 0 - and E (( H transitions demonstrates the fact that the final states in both polarizations are identical, whereas the initial states belong to heavy hole ladders ahh(n) and bhh(n), which lie close to each other. The lowest-lying transitions originating from bra(- 1) and ar,(- 1) are most sensitive to the exchange mechanism, as expected from the model (11.2,

6.

MAGNETOOPTICS IN NARROW GAP DMS

249

.i‘ T=2K

27

li Y

230

210 10

0

20

30

40

50

H(kG)

(4

2w

0

10

20

30

40

50

H(kG)

(b) FIG. 10. Energies of Ts + Ts transitions vs. magnetic field for Hgl-,MnxTe of x = 0.128 (EO = 219 meV), at 2 K, (a) E I( H polarization, and (b) 0 - polarization. Points are experimental. Solid lines represent theoretical fits obtained within the modified Pidgeon-Brown model. The quantum number n identifying the transitions according to Eqs. (21) and (22) is indicated for each line. The line marked “Imp” corresponds to the acceptor -+ br,(O) transition. [After Bastard er a/. (1981).]

250

C. RIGAUX

I

I

I

I

H(kG)

FIG. 1 1 . Magnetotransmission spectra for 0 - and E )I H polarizations, observed at the photon energy hw = 265.3 meV at 4.2 K for Hgl-xMn,Te with x = 0.128. [After Bastard et a/. (1 98 I).]

Eq. (10)). This explains the observed temperature and magnetic field dependence of bra(- 1) -+ br,(O)(a-)and ars(- 1) br,(O)(E (1 H) transitions (Fig. 10). Comparison between theoretical and experimental transition energies vs. magnetic field is shown in Fig. 10 for x = 0.128 (EO = 219 meV). The band parameters eo ,E p , and the exchange parameters r, A(H, 7')were determined at 4.2 and 2 K in the range of composition 0.12 5 x 5 0.17. The results are presented in Part V. The magnetization parameter A ( H ) varies almost linearly with the magnetic field (Fig. 7) and is weakly temperature dependent in this range of composition. Hgl-,Mn,Te semiconductors are usually p-type. The relatively high density of acceptors (Nu = 5 x 10'6cm-3) present in the investigated compounds makes possible the observation of acceptor-to-conduction Landau level transitions. An intense absorption line satellite of the interband br,(O) br,(l) transition of weaker intensity (labelled Imp in Fig. lob), extrapolates at zero field to about 5 meV below the energy gap. This line may result from the transition between an acceptor state (bound to br,(O))and the conduction band Landau level br,( 1). +

-+

6.

MAGNETOOPTICS IN NARROW GAP DMS

b. Effect of Exchange on

r 6

251

Landau Levels

The exchange interaction affects the Landau level spectrum of the narrow (positive) gap HgMnTe in a striking fashion. The magnetic field dependence of the conduction Landau level edges, for a semiconductor of EO = 219 MeV shown in Fig. 12, illustrates the new important features originating from the exchange phenomena. In the conduction band, the relative positions of the levels ar,(n) and br,(n), corresponding, respectively, to nt and nl electron spin states, are inverted with respect to the usual ordering in zinc blende semiconductors, so that in the present case E&) > Eb(n). This effect is a direct consequence of the s-d exchange interactions, which shift the spin sublevels upward for a(n) and downward for b(n). Thus, in semimagnetic alloys with positive energy gap, the gyromagnetic factor of conduction electron is positive. This is in complete contrast to zero gap HgMnTe and to non-magnetic narrow gap semiconductors, such as InSb or HgCdTe, where the electron gyromagnetic factors are always negative. The origin of the sign reversal of the g-factor occurring in these materials can be understood from the analytical expression (14) for the effective gyromagnetic factor in the parabolic limit. The first term gc* resulting from the k * p contribution is negative, whereas the second term, representing the s-d exchange contribution, is positive (a < 0). Thus, with increasing magnitude of the exchange interactions, the effective gyromagnetic factor

301

s

2 w

I

0

€

20[1

0

.--+a

b

FIG. 12. Magnetic field dependence of Landau levels of the conduction band at k, = 0 for HgMnTe with Mn content x = 0.128 calculated at T = 2 K . The labels a, b correspond to two sets of solutions of Eq. (8); the Landau quantum numbers are given at each curve. [After Bastard et al. (1981).]

252

C . RIGAUX

g& will go through zero and subsequently reverse sign when the exchange

contribution becomes dominant. Large andpositive conduction g factors are observed in open gap semimagnetic semiconductors, e.g., for EO = 219 meV, g?6 = 100 at 2 K. This is a unique situation for narrow gap semiconductors, associated exclusively with exchange interactions. As one can see in Fig. 12, the electron spin splitting becomes comparable to the cyclotron energy. In the valence band, the most spectacular effect caused by the exchange interactions is the strong positive energy shift exhibited by the uppermost heavy hole level br&- 1). This feature, accompanied by a downward shift of the first conduction electron level br,(O), leads to the reduction of the energy gap in an applied magnetic field. 6. RESULTSFOR OTHERSEMIMAGNETIC COMPOUNDS a. HgMnSe

Interband magnetooptics in HgMnSe was carried out by Dobrowolska et al. (1981) for several Mn compositions (0.011; 0.075; 0.115). Magnetotransmission measurements were made at temperatures between 10 and 40 K , in the spectral region 230-370meV, on n-type samples with electron concentration ne = 10” ~ m - Interband ~ . r6 Tg transitions observed for (T and ll polarizations in zero gap Hgo.9g9Mno.ollSeare identified according to the selection rules (20), by referring to the spectrum of HgSe (Dobrowolska et al., 1980; Mycielski ef al., 1982). Due to the high carrier concentrations in HgMnSe alloys, the first conduction subband edges are fully populated, precluding the observation of the lowest-lying transitions br,(O) bra(- I), ar,(O) ar8(l), that are the most sensitive to the exchange effect (Fig. 13). The experimental spectra obtained at different temperatures were quantitatively interpreted on the basis of the modified Pidgeon-Brown model: for x = 0.01 1, the magnetization was calculated in terms of a model of isolated and nearest-neighbor interacting Mn pairs. The values of the exchange integrals for r6 and TSbands, the exchange coupling constant JO between pairs, and the band parameters (EO ,P ) were evaluated by fitting r6 Tg transition energies as a function of magnetic field. Higher band parameters and the spin-orbit splitting energy A = 387 meV were taken to be equal to the values obtained for HgSe (Dobrowolska et al., 1980a). Comparison between the calculated and the experimental transition energies vs. magnetic field is illustrated in Fig. 13, at 10 and 40 K, for the values of the exchange integrals NOD = 0.7 eV; NO(Y= -0.4 eV; JO = 2.3 K. The observed temperature dependence of the transitions shows a decrease of the conduction spin splitting with increasing temperature. Figure 14 shows the gyromagnetic factor g? of the n = 1 conduction Landau level vs. magnetic field at several temperatures for HgSe and for Hgo.9g9Mno.ollSe. -+

-+

-+

6.

z

z

o

~

l

20I

'

MAGNETOOPTICS IN NARROW GAP DMS

I 40

'

I 60

253

'

HlkGI -

FIG. 13. Magnetic field dependence of Ts Ts transitions for Hgl-,MnxSe of Mn content x = 0.01 1. Symbols are experimental, full circles, open circles, and crosses denoting the 0-,oi, and E ( 1 H polarizations, respectively. (a) T = 10 K. (b) T = 40 K. Solid lines are theoretical fits obtained using the modified Pidgeon-Brown model. Transitions are identified according to the notation of the three-band model (n' = a(n); n- = b(n)). [After Dobrolowska el at. (1981).] -+

H/kGl--

FIG.14. Magnetic field dependence of the electron g* factor calculated for the n = 1 Landau level for HgSe (dashed lines) and Hg0.9~Mno.ollSe(solid lines). [After Dobrolowska et al. (1981).]

254

C . RIGAUX

The temperature dependence of gf in HgSe results solely from the energy gap variation, whereas the decrease of gf observed between 10 and 4 0 K in the semimagnetic compound is the consequence of the exchange interaction. In the semiconducting region, the magnetoabsorption spectra of Hgl-*MnxSe (x = 0.075 and 0.115) are weakly temperature dependent. The TS+ r6 magnetooptical spectra are interpreted by assuming that the exchange integrals a,p are independent of composition and equal to the values obtained for x = 0.011. The values of ( S z ) deduced by fitting l-8 rs magnetoabsorption spectra considerably differ from those obtained from susceptibility measurements on samples of similar composition (Pajaczkowska et al., 1979; Dobrowolska et al., 1980b). This disagreement indicates that further and more extensive study of Hgl-,Mn,Se crystals is necessary in order to determine reliable values of a and p in this material. -+

b. HgFeTe New DMS containing substitutional Fe2+ions were recently synthetized (for x I 0.02) at Purdue University. The first magnetooptical study carried out by low temperature interband magnetoabsorption in zero gap Hg, -,Fe,Te alloys have emphasized the semimagnetic character of these compounds (Serre et al., 1981, 1982). The r6 l-8 magnetooptical spectra obtained for the rn and ll polarizations show (similarly as in HgMnTe) exchange-enhanced spin splittings, as well as the occurrence of the inter-valence transition br,(O) -+ h 8 ( - 1 ) in the rn+ spectrum of alloys with ~ E O I > 260meV. In complete c6ntrast to HgMnTe, however, the magnetooptical spectra are independent of temperature between 4.2 and 2 K. The r6 r8 magnetooptical spectra of very dilute Hgl -*FexTe alloys were quantitatively interpreted in the framework of the modified Pidgeon-Brown model and the exchange parameters r = -0.3 and A ( H ) were determined. The negative values of r and A imply that a < 0 and fl > 0, similarly as in the case of HgMnTe, leading to exchange-enhanced gyromagnetic factors in both the r6 and the Ts conduction bands. The exchange effects in Hgl-,Fe,Te are, however, much weaker than in Hgl-xMn,Te alloys of comparable interaction gap, as indicated by the values of A ( H ) obtained for both alloy systems: e.g., for EO = -230 meV, at H = 2T and 2 K, A is about twice as large in HgMnTe as it is in HgFeTe. The most striking peculiarities of HgFeTe compounds are the linear field dependence of A ( H ) and its invariance with temperature. The results obtained for several values of the Fe content also show a linear dependence of the magnetization on composition, as illustrated in Fig. 15 by plotting - A / H vs. the relative interaction gap energies. This implies that in dilute HgFeTe alloys, ( S , ) is independent of composition, in sharp contrast with -+

+

6.

--

255

MAGNETOOPTICS IN NARROW GAP DMS I

I

I

I

1

1-

-

-

I

%r E c 0

u-

0.3

0 1 0

I

10

1

20

I

30

H(kG)

I

40

I

50

I

60

FIG. 16. Normalized magnetization &,, vs. magnetic field determined at 2K for Hg,-,Mn,Te alloys of several Mn concentrations. (A: x=0.0014; B: x=0.004; C: x = 0.008; D x=O.Ol; E x = 0.015. Dots are experimental data obtained form interband magnetooptics. [After Bastard ef a/. (1978).] Continuous lines represent theoretical fits obtained by using the generalized Brillouin function given in Eq. (6) with Treplaced by T + r0. The value of the fitting parameter %(x) for each composition is indicated in the figure. [After Bastard and Lewiner (1980).]

the pronounced decrease of ( S , ) observed in HgMnTe with increasing values of x-see Fig. 16 (cf. also, Oseroff and Keesom, this volume). The most important property of Hgl-,Fe,Te alloys is the linear variation of ( S , ) with magnetic field, and ips independence of temperature between 2 and 4.2 K. These observations are corroborated by dc magnetic susceptibility (Mullin, 1980) which is found independent of temperature in Hg, -,Fe,Te alloys of comparable Fe content. This important feature

256

C. RIGAUX

drastically contrasts with the results obtained in dilute Hg, -*Mn,Te alloys, where magnetization manifests a strong temperature dependence. The behavior observed on Hgl-,FexTe alloys originates from paramagnetism of Fe2+ions in the zinc blende lattice (Serre et al., 1982). The d shell of the Fe2+ ions has 6 electrons, and in vacuum the ground state corresponds to S = 2 and L = 2. The cubic crystal field lifts the resulting 25-fold degeneracy into a 10-fold-degenerate level and a 15-fold-degenerate excited state separated by A = 0.3 eV from the ground state (Mahoney et al., 1970). The spin-orbit coupling AL .S (with A = 12 meV) further lifts the degeneracy of the ground state into 5 equidistant levels with an energy separation equal to 6A2/A, the degeneracy of the levels being 1, 3, 2, 3, and 1, respectively. Finally, the Zeeman termpB(L, + gFeSz)Hisintroduced. The resulting spectrum of the d level of an Fe2+ ion is shown in Fig. 17. The ground state is a singlet, that is, there is no permanent moment at zero magnetic field. Consequently, there is no linear Zeeman shift. The quadratic Zeeman shift

Eo(H) = EO - 4-PB2 A H 2 3 A physically corresponds to a field-induced magnetic moment, that varies linearly with magnetic field and arises from the virtual transitions between the ground state and the excited states. The magnetic susceptibility x = 83 p2~ ( A / A 2 is) therefore independent of magnetic field and of temperature, as long as the contribution of the first excited triplet can be neglected 1

b

1,

\

/ /

2

IL

1

L

'I

-

1

H

c

6.

257

MAGNETOOPTICS IN NARROW GAP DMS

(i.e., kT Q 6A2/A). Thus at low temperatures, in contrast with the CurieWeiss paramagnetism of Mn2+ions, the Fe2+magnetism is of the Van Vleck type and arises from the lack of a permanent moment at zero field. IV. Intraband Magnetooptics 7. FARIR MAGNETOSPECTROSCOPY IN ZEROGAPDMS In non-magnetic zero gap semiconductors, FIR spectroscopy has been successfully applied in carrying out detailed studies of the Fg valence and conduction bands. Magnetotransmission experiments performed in the submillimeter and FIR region on unoriented n-type samples of HgTe (Tuchendler et al., 1973) and Hgl-,CdxTe alloys (Kim and Narita, 1976; Guldner et al., 1977) with low electron densities ( h a > EF) led to the observation of intraband electronic and interband Ti' Ti transitions. Measurements performed on oriented p-type samples with magnetic field applied along the principal crystallographic directions were especially useful for determining the Fg band anisotropy in HgTe (Pastor et al., 1978). HgSe is characterized by large electron densities (n, = 10'' ~ m - leading ~ ) to an opacity of the crystal at long wavelengths, due to plasma cut-off. FIR measurements carried out on HgSe at frequencies larger than the plasma frequency have provided observation of intraband electronic transitions (plasma-shifted cyclotron resonance for E I H, combined resonance for E I( H, and electric dipole spin resonance for both E 11 H and E I H). The interband Ti' -+ Fitransitions requiring photon energies larger than the Fermi energy (typically 14 meV) could also be excited in the investigated spectral region (7.61-12.85 meV) (Pastor et al., 1980, 1981). In zero gap DMS, FIR magnetooptics provides one of the best experimental methods for studying the behavior of Tg Landau levels in the presence of magnetic exchange. Pastor et al. (1979) observed at 4.2K a strong influence of the exchange effects on the positions of the magnetotransmission minima associated with intraband and interband Ti --* r; transitions in dilute Hgl -,Mn,Te compounds with low electron densities. Similar features were reported by Kim et al. (1982) in quaternary Hgl-,-,Cd,Mn,Te compounds for several Mn compositions. In zero gap Hgl-xMn,Se, FIR magnetooptical experiments were particularly difficult due to the high electron concentration (typically 10'' cmP3), leading to a plasma edge cutoff at wavelengths about 100pm. Using samples of exceptionally low electron , by appropriate annealing, concentration (5 x 1OI6~ m - ~ ) prepared Witowski et al. (1982) performed a detailed study of intraband spin-flip resonances in zero-gap Hgo.97Mn0.03Se.The strong influence of exchange on the conduction spin sublevels is experimentally manifested by a dramatic +

258

C . RIGAUX

temperature dependence of the spin-flip transitions between 1.9 and 6.5 K. Theoretical analysis of these results yields conduction band and exchange parameters for dilute HgMnSe alloys.

a. Influence of the Exchange Interactions on the TS Magnetoabsorption Spectrum

+

TS

(a) Zero Gap Hgl-,MnxTe Alloys Using molecular gas lasers (HCN, 3.68meV; DCN, 6.36meV; HzO, 10.45meV) as light sources, Pastor et al. (1979) have performed FIR magnetotransmission measurements at 4.2 K in the Voigt geometry, using linearly polarized radiation E I(H and E IH, on oriented Hgl-,Mn,Te crystals of low Mn content (x I 0.018). The evolution of the magnetotransmission spectra with composition, shown in Fig. 18 at the photon energy ho = 3.68 meV, shows a shift of the transmission minima to lower magnetic field as x increases. It was possible to identify the resonances by comparison with similar data obtained by HgTe (Pastor et al., 1978). The electric dipole spin resonance a(0) -,b(0)and the combined resonance (CB) a(0) -+ b(1) are identified in Fig. 18 in the E 11 H spectrum. Figure 19 shows the observation of cyclotron resonance for E I H. Several interband Ti + Titransitions were also observed for E 11 H and E I H. The assignment of the lines made according to the selection rules (Table I), is indicated in Figs. 18 and 19 using the notation of the Luttinger model (Luttinger, 1956). Since the resonance

0

rb

m

30

io

50

MAGNETIC FIELD (kG1

FIG. 18. Magnetotransmission spectra at frw = 3.68 meV for Hgl-,Mn,Te alloys of low x observed in the E 1) H polarization. Experimental lines are identified as follows according to the notation of the Luttinger model: CB: combined resonance ~ ( 1 + ) &2(2);SR: spin resonance &](I).+ &$(I); E: d ( 2 ) EZ(~); 8:$(2) &](I);A : d ( 2 ) E I ( ~ ) .[After Pastor ef a/. (1979).1 -+

+

-+

6 . MAGNETOOPTICS IN

0

1.o

0.5

259

NARROW GAP DMS

15

2.0

x (”/) FIG.19. Comparison of transmission minima observed at I = 118.6pm in several oriented Hgl-,MnxTe samples (H 11 (1 11)) with the predictions of the theory. (a:transitions observed for E 11 H; 0: transitions observed for E I H; 0 : transitions observed for both E (1 H and E 1 H.) [After Pastor et d.(1979).1

field varies linearly with the photon energy for the three different laser wavelengths, the non-parabolicity effects in r s bands are expected to be negligible. The experimental data were thus analyzed on the basis of the Luttinger model, with the effect of anisotropy of the Ts bands ( y 2 # y 3 ) included. In this model, the warping contribution introduces a mixing of the Ts Landau levels at kH = 0, and the k p Hamiltonian D thus becomes an infinite matrix. The exchange interaction between mobile carriers and Mn2+ ions was introduced into the model by using an additional Heisenberg-type Hamiltonian Xex,(Eq. (2)). In the representation of the matrix D,the

-

260

C. RICAUX

exchange interaction X,, has the form of an infinite matrix that depends only on the exchange parameter A = - & Z V ~ f l x f ~The ~~~ effective . Hamiltonian D + X,, was solved numerically, and the calculated transition energies were fitted to the experimental data, treating NODas the only fitting parameter. In the analysis, the normalized magnetization was taken from interband magnetooptical data, and the Luttinger parameters were estimated from those obtained for HgTe (Pastor et al., 1978), taking into account the composition dependence of the energy gap. Comparison of the experimental resonances for different compositions with the prediction of the Luttinger model is shown in Fig. 19, obtained from theoretical fits of the data that yielded the value of Nofl as 650 meV. (b) Hgl-,-,Cd,Mn,Te The quaternary system Hgl -,,Cd,Mn,Te offers the interesting and important possibility of independently varying the energy gap and the magnetic ion concentration in DMS. The first magnetooptical study of quaternary alloys of low Mn content ( y < 0.01) was reported by Kim et al. (1982). FIR magnetotransmission measurements were carried out at 4.2 and 1.6 K, in both Faraday and Voigt geometries, on n-type samples of low electron concentration (8 x 1014~ m - ~Energies ). of the resonance lines vs. magnetic field are shown in Fig. 20 for alloys with interaction gaps EO = - 26 and - 56 meV, respectively. The FIR spectra consist of several Ts" TsC transitions, as well as the spin, combined, and cyclotron resonances. In the low field region, the crossing of the cyclotron and spin resonances lines directly demonstrates crossing of a(1) by b(0) conduction levels. The band edge effective masses and gyromagnetic factors of conduction electrons are estimated from these experiments for several compositions. The g factors are found to be twice as large in the quaternary semimagnetic alloys as those in Hgl -,Cd,Te alloys ofidentical interaction gap, whereas the effective masses remain comparable in both alloys. -+

b. Spin-Flip Resonances in Zero Gap Hgl-,Mn,Se Intraband spin-flip transitions have been studied in zero gap Hg, -,Mn,Se alloys (x = 0.032) by FIR magnetospectroscopy carried out with the use of an optically pumped FIR laser (Witowski et al., 1982). Transmission measurements were performed in magnetic fields up to 2.5T in both Voigt and Faraday geometries, at wavelengths between 96.5 and 170pm, in the temperature range between 1.9 and 6.5 K. In the Faraday configuration, two clearly resolved absorption lines, shown in Fig. 21 for A = 119 pm, are identified as the electric dipole spin resonance (EDSR) transitions a(n) b(n). A strong absorption line observed in the Voigt E 1) H spectrum (Fig. 22) is attributed to the combined resonance allowed for this polarization. -+

FIG. 20. Energies of transmission minima vs. magnetic field for quaternary Hg,-,-,Cd,Mn,Te alloys of interaction gaps = -26meV (a) and EO = -56meV (b). CR, CO, SR, and 11.2.3 denote cyclotron resonance, combined resonance, spin resonance, and interband transitions, respectively. [After Kim et al. (1982).]

EO

262

C . RIGAUX

FARADAY I

L

1

I

I.o

0.5

I.5

MAGNETIC FIELD (7) FIG. 21. FIR magnetotransmission spectrum obtained for Hg,-,Mn,Se (x = 0.032) in the Faraday configuration. ( T = 4.2K; 1 = 118.8pm.) [After Witowski et al. (1982).] I c:

50-

1

I

I

119pm

I

-

?

MAGNETIC FIELD (TI FIG. 22. FIR magnetotransmission spectrum obtained at 118.8pm on Hgl-,Mn,Se ( x = 0.032) in the Voigt E 11 H configuration, at several temperatures. [After Witowski et al. (1982).]

As pointed out previously, the electric dipole spin-flip transitions in zinc blende narrow gap semiconductors are allowed, in the presence of spin-orbit interaction, as a consequence of either s-p interband coupling or of inversion asymmetry. One of the most spectacular features of these experiments is the strong temperature dependence exhibited by the spin-flip transitions. The temperature behavior is illustrated in Fig. 22 by the shift of the combined resonance between 2 and 6.5 K. For EDSR, the effects are even stronger, the resonance

6.

MAGNETOOPTICS IN NARROW GAP DMS

263

field shifting by about 50% in the same temperature range. This feature is evident in Fig. 24, which shows the temperature variation of the spin-flip resonance at 119pm. The strong temperature shift of the spin-flip transitions represents a direct influence of the (&)-dependent term on the energy scheme of the T8 spin sublevels due to the p-d exchange interaction. The experimental data were quantitatively interpreted in terms of the modified Pidgeon-Brown model. A generalized Brillouin function for an effective temperature T(see, Gaj et al., 1979) was used to describe the thermal average of the Mn spins. The parameter To, as well as the fundamental band structure parameters EO and P , were treated as adjustable parameters in the numerical fitting procedure of the transition energies vs. magnetic field. The values of the remaining band parameters were taken from those determined earlier for HgSe (Pastor et al., 1981; Dobrowolska et al., 1980a) and the values of the exchange integrals a,p were taken from the quantum transport results on dilute HgMnSe alloys (Byszewski et al., 1980). The best theoretical fit of the transition energies vs. magnetic field to the experimental line positions at 4.2K is shown in Fig. 23. Two groups of transitions are distinguished: at higher energies are combined resonances, corresponding to a@’) b(n’ + l), and at lower energies occur the EDSR transition a(n) b(n). The quantum numbers n, n’, ascribed to each transition, are indicated in the figure. Since the Fermi level lies at about 16 meV above the r point, several spin-flip transitions are possible in the investigated spectral region. From the fitting procedure, Witowski et a/. determined(at4.2K)~o= -83.1 meV,P= 5.58 10-8eVcm,and%= 7.23K for x = 0.032. The temperature dependence of the resonance positions for several wavelengths was subsequently quantitatively interpreted by fitting %(T) as the only fitting parameter. Comparison between the calculated and experimental line positions for I = 119pm is shown in Fig. 24 indicating excellent agreement. From theoretical fits performed at different wavelengths, the temperature dependence of the parameter %( T ) was estimated. The results show that the parameter To( T ) increases from 6 K to about 11 K between 2 and 6.5 K. Conduction band Landau levels at the r-point, obtained from the analysis of spin-flip transitions in Hgo.97Mno.o~Se at T = 1.6 and 6 K, are depicted in Fig. 25. The b(n)levels are the most strongly affected by the temperature, their energies increasing when the temperature is lowered, similarly as in zero gap HgMnTe alloys. The behavior of the T8 Landau levels explains the important shift of the EDSR at I = 119pm (a(0) b(0);Fig. 22), as well as the transformation of the transition 4 2 ) b(2) into a(1) b(1) observed at L = 163pm when the temperature is raised from 1.6 to 6 K. As one can see in Fig. 23, the spin splitting b(n) - a(n) exceeds the cyclotron splitting a(n + 1) - a(n) in the investigated temperature range.

+

+

+

+

-+

+

264

C. RIGAUX

I

O.8.5

I.o

1.5

MAGNETIC FELD

20

2.5

3bO

(TI

FIG.23. Transition energies vs. magnetic field for EDSR and combined resonance at 4.2 K observed in Hgl-,MnxSe (x = 0.032). Symbols are experimental. Solid lines represent the best fits to the experimental data. Unprimed and primed integers refer to EDSR and combined resonance transitions, respectively. [After Witowski et at. (1982).]

The temperature dependence of the parameter To obtained from the analysis of spin-flip transitions was used to calculate the magnetic susceptibility. A rather satisfactory agreement is found between zealcand direct dc susceptibility measurements (Khattak er al., 1981). 8. CYCLOTRON AND COMBINED RESONANCE HARMONICS IN n-TYPE HgMnTe SEMICONDUCTORS

Intraband magnetooptical studies in n-type narrow band gap Hgl -,Mn,Te (x = 0.11, eg = 165 meV) were reported by Grynberg et al., 1982). Using C02 laser lines (1 15-135 meV), magnetotransmission and magnetophotoconductivity experiments have been carried out at 1.8 Kin fields up to 18Tesla,

6.

265

MAGNETOOPTICS IN NARROW GAP DMS 1

2.2

-

2.0

-

I

1

I

1

1

2 0

-

-I-

1.8-

-

Q

1.6-

-

1.4-

-

z 1.2-

-

J

wLL

9 I-

w

0

1 I .o

I

0.8: I

I

I

1

L

I

1

2

3

4

5

6

7

in the Voigt configuration with unpolarized radiation. The measurements were performed on In-doped samples of electron concentration ne = 8 x 1015cm-3 obtained after appropriate annealing. Pronounced oscillatory magnetotransmission minima, which coincide with magnetophotoconductivity minima, are associated with intraband magnetooptical transitions within the conduction band Landau levels. Using the modified PidgeonBrown model, with the band parameters obtained from interband magnetooptics on very similar HgMnTe crystals (except for EO and A ( H ) ) , the observed resonances were identified as the harmonics of the cyclotron and combined resonances. For magnetic field lower than 8 Tesla, transitions take place from the lowest-lying Landau level b(0) to high quantum levels b(5) and b(6) (cyclotron resonance harmonics) and to a(4), 43,and 4 6 ) (combined resonance harmonics). At very high fields (H L 8 T ) , the magnetic freezing of the conduction electrons on donor states has been experimentally manifested by the observation of transitions from the donor ground state bound to the lowest Landau level to high quantum levels a@),

266

C . RIGAUX

r

I

I

I

I

I

MAGNETIC FIELD ( T I

FIG. 25. Magnetic field dependence of Tg conduction Landau level energies at k, = 0, calculated for 1.9 K (solid lines) and 6 K (dashed lines). The broken arrows show how the magnetic field of the EDSR transition shifts between 1.9 and 6 K for I = 118.8 pm.The solid arrows show the transformation of the EDSR from 4 2 ) -,b(2) at 1.9 K to a ( ] ).+ b(1)at 6 K for I = 163pm. [After Witowski e t a / . (1982).]

b(n) (or to excited donor states bound to these levels). The sensitivity of the combined resonance harmonic transitions b(0) a(n) to the exchange parameter A has made it possible to determine the magnetic field variation A ( H ) up to 18 Tesla. Satisfactory agreement is also found with the interband magnetoabsorption data obtained at lower fields for HgMnTe of similar composition. Low temperature magnetoabsorption and stimulated spin-flip Raman scattering experiments in n-type Hgo.g9Mno.l1Tewere also reported by Geyer and Fan (1980). Weak magnetotransmission structures observed at photon energies smaller than the energy gap were attributed to cyclotron resonance harmonic transitions coupled with LO phonon emission. A line observed in scattering measurements under applied magnetic field was found to be due to spin-flip scattering, and the magnitude of the Land6 factor of conduction electron was estimated as [gel = 45. The observed spin-flip scattering was determined to be stimulated on the basis of its intensity dependence on the +

6.

MAGNETOOPTICS IN NARROW GAP DMS

261

laser excitation intensity and the calculated photon occupation number of the scattered radiation. V. Results of Magnetooptics 9. BANDPARAMETERS

The variation of the interaction gap EO with composition obtained from magnetooptics and other experiments is shown in Figs. 26 and 27 for HgMnTe and HgMnSe, respectively. A non-linear composition dependence is found for Hgl-,MnxTe, with EO(X) varying twice as fast as it does in Hgl-,Cd,Te: thus, to obtain a positive energy gap of 100 eV at 4.2 K, one would require x = 0.22 in Hgl-,CdxTe while in the case of Hgl-,Mn,Te the same energy gap occurs for x = 0.11. The zero gap-to-semiconductor transition for HgMnTe and HgMnSe takes place at x 5: 0.075 and x 3 0.06, respectively, at cryogenic temperatures. Finally, the temperature coefficient d d d T for HgMnTe is positive, twice as large as in HgCdTe, and nearly linear in x up to x = 0.15 (Dobrowolska and Dobrowolski, 1981).

*O 0 X

*

2100 oOl x

*

u

0

5

x

Hg,,

Mn, TI .

10

15

(%I

FIG.26. Interaction gap vs. composition in Hg,-,MnlTe alloys for T = 4.2 K. Black circles are from Bastard et a1 (1978, 1981), open circles from Dobrowolska et at. (1981), crosses from Jaczynski et at. (1978), and triangles from McKnight et at. (1978).

268

C. RIGAUX

-200 -070

005

--"'OO

1

5

X

FIG. 27. Energy gap vs. composition in Hgl-,Mn,Se for T = 10 K . Open circles are from interband magnetooptical data (Dobrowolska et al., 1981 ) ; full circles are from quantum transport (Takeyama and Galazka, 1979).

- I

u

-I -300 -200 -100

o

100

200

300C&meV

FIG.28. The momentum matrix element P v s . the interaction gap for Hgl-,Mn,Te alloys for T = 4.2 K. 0 are from Bastard et al. (1978, 1981); 0 are from Dobrowolska el al. (1981); 3 are from Jaczynski er al. (1978). and C

The Kane matrix element P = - i h / m ( S IpzlZ> was determined from theoretical fits of interband and intraband experiments for several compositions. The results are presented in Figs. 28 and 29 together with those obtained from Shubnikov-de Hass measurements. In both Mn-based systems, the interband momentum matrix element P shows a systematic decrease with x , unlike the behavior observed for Hgl-,Cd,Te (Guldner et al., 1978).

6.

269

MAGNETOOPTICS IN NARROW GAP DMS

0 h

5 2

80

0

-

0

90

0

c

U

a

470,

t

.

..

I

.

I

8

.

I

I

I

0

++4I

0

-

10. EXCHANGE PARAMETERS

The energies of magnetic levels depend on the exchange parameters a,p, and ( S , ) through thequantities a(&) andp(S,) (Section 11.2, Eq. (7)). Thus a typical theoretical analysis of magnetooptical data does not usually permit a separate determination of exchange integrals and of the average component of the localized spins. From interband magnetooptics, one obtains the ratio r = a / p of the exchange integrals and A ( H ) ,the relative magnitude of (S,). There are indications that in the HgMnTe system exchange integrals depend on composition. The values of r obtained in the composition ranges x I0.015 and 0.12 Ix 5 0.17 are significantly different (r = -0.5 & 0.05 and -0.8 +. 0.1, respectively; cf. Bastard et al., 1978 and 1981). Only in the very dilute region (x = 0.0014), where the experimental field variation A ( H ) follows the Brillouin function, was it possible to extract the values of Noa and NO^ from the values of A and r. For larger Mn content, A ( H )departs from the Brillouin law, and the exchange parameters and (S,) cannot be estimated without additional information. Jaczynski et a/. (1978) and Dobrowolska et al. (1981) assumed that ( S z ) may be described by a model of isolated spins and nearest-neighbor-interacting spin pairs, with a statistically random spatial distribution over the fcc lattice, and estimated the exchange coupling constant JObetween the pairs by fitting the magnetization data. Theoretical fits of quantum transport and interband magnetooptics

TABLE I1 EXCHANGE CONSTANTS IN NARROWGAP DMS. (IBMO: INTERBAND MACNETOOPTICS; FIRMO: FARINFRARED MAGNETOOPTICS.) THE CONSTANTS MODEL(PB) OR THE LUTTINGER MODELOF THE BANDSTRUCTURE WEREDERIVEDUSINGTHE PIDCEON-BROWN

T (K)

r

NOD(ev)

NOCU (eV)

0.0014 0,004-0.02 0.12-0.17 0.02; 0.03; 0.04 0.007; 0.01 1 x < 0.02

2-4.2 2-4.2 2-4.2 4.2 10-65 4.2

-0.5 -0.5 -0.8

1.5

- 0.75

1.4 0.6 0.65

- 0.7

Hg,-,Mn,Se

0.011; 0.075; 0.115 0.01 0.016 0.066 0.026; 0.047 0.011; 0.013

10-40 1.6-30 1.6-30 1.6-30 4.2 4.2

0.7 1.54 1.35 0.92 0.9 0.9

- 0.4 - 0.9 - 0.96 -0.32 - 0.35 -0.35

Hgl-,Fe,Te

5 0.016

2-4.2

Compound Hgl-,Mn,Te

Composition x

- 0.3

- 0.4

Experiments IBMO (PB) Bastard et al. (1978) IBMO (PB) Bastard et a / . (1978) IBMO (PB) Bastard et al. (1981) Quantum transport (PB) Jaczynski et a / . (1978) lBMO (PB) Dobrowolska & Dobrowolski (1981) FIRMO (Luttinger) Pastor et al. (1981) IBMO (PB) Dobrowolska et a / . (1981) Quantum transport (PB) Takeyama & Galazka (1979) Quantum transport (PB) Takeyama & Galazka (1979) Quantum transport (PB) Takeyama & Galazka (1979) Quantum transport (PB) Byszewski et al. (1980) Quantum transport (PB) Byszewski et al. (1980) IBMO (PB) Serre et al. (1982)

6.

MAGNETOOPTICS IN NARROW GAP DMS

271

performed with this method lead to quite different values of exchange integrals for dilute HgMnTe. The values of r, Nos, and NoP deduced from different experiments for several semimagnetic compounds of various compositions are collected in Table 11. The origin of the large discrepancies was not yet elucidated. This between the different determinations of (Y and /I points out the necessity of determining ( Sz)from magnetization measurements on samples investigated by magnetooptical experiments, to obtain accurate values of exchange integrals in NGDMS. VI. Concluding Remarks

Magnetooptical experiments in NGDMS demonstrate the str6ng influence of paramagnetic ions on electronic properties of these materials when an external magnetic field is present. Most of the experimental data obtained from magnetooptics and quantum transport (Kossut, this volume) agree very well with the predictions of the theoretical model, justifying the applicability of VCA and the molecular field approach used in treating carrier-ion interactions in random substitutional diluted magnetic alloys. Another important aspect of NGDMS, considered in this volume by Mycielski, is the magnetic behavior of impurity states evidenced by IR and submillimeter magnetospectroscopy . A particularly interesting property of DMS is the proportionality of the exchange-induced spin splitting to the magnetization, allowing an optical access to the magnetic properties of the localized spins. As is well known, the antiferromagnetic interactions between Mn2+ions lead to the formation of a spin-glass phase due to the frustration mechanism of the fcc lattice (de Seze, 1970; see also Oseroff and Keesom, this volume). The large interband Faraday rotation, directly related to the magnetization in DMS, permits the observation at weak fields of the formation of the spin-glass phase in Mnbased compounds (Kierzek-Pecold et al., 1984; Mycielski et al., 1984). In these experiments, the temperature dependence of the Faraday rotation, measured at fixed photon energies below the fundamental absorption edge exhibits characteristic kinks, may be related to the paramagnetic-to spinglass transition in Hgl -xMnxTe and quaternary Hgl -x-y MnxCdyTealloys. The transition temperatures Z(x) observed by means of the low field Faraday rotation and susceptibility measurements are in good mutual agreement. The phase diagram Z(x) obtained from both types of experiments is shown in Fig. 30. Faraday rotation experiments demonstrate the fact that the spin-glass transition occurs at lower temperature in quaternary alloys of larger energy gap than in HgMnTe compounds of identical Mn composition. This may indicate that the antiferromagnetic interaction between Mn2+ions due to the indirect exchange (Bloembergen-Rowland process) could be significant in

272

C. RIGAUX

2o

t

I

"0

010 0.20 0.30 0.40 0.50 X

FIG. 30. Magnetic phase diagram of Hgl-xMnxTe, P and S denoting the paramagnetic and spin-glass regions, respectively. The origin of the data points is as follows: crosses: ac susceptibility (Mycielski et al., 1984); open circles: Faraday rotation experiments (Mycielski et al., 1984); open triangles: dc susceptibility (Nagata et al., 1980); full triangles: Brandt et al. (1983); stars: Faraday rotation data for Hgo.s~Mn0.~5Cdo.zoTe and Hgo.~oMno.&do.~oTe(Mycielski et al., 1984).

the mechanism responsible for the spin-glass formation. This unique opportunity of having access to magnetic properties by studying the magnetooptical properties of the materials, as exemplified by the above remarks, is particularly attractive, and undoubtedly will be pursued further in the future.

References Andrianov, D. G., Gimelfarb, F. A., Kushnir, P. I., Lopatinskii, I . E., Pashkovskii, M. V., Savelev, A. S . , and Fistul, V. I . (1976). Sov. Phys. Semicond. 10,66. (Fit. Tekh. Pafuprov. 10, 111.) Bastard, G., Rigaux, C., and Mycielski, A. (1977). Phys. Stat. Sol. (b) 79, 585. Bastard, G . , Rigaux, C., Guldner, Y., Mycielski, J., and Mycielski, A. (1978). J. Physique 39, 87. Bastard, G. (1978a). These de Doctorat d'Etat, Paris (unpublished). Bastard, G. (1978b). (14th Int. Conf. Phys. Semicond.) Institute ofphysics Con$ Series 43 (Wilson, London), 231. Bastard, G . , and Lewiner, C. (1979). Phys. Rev. B 20, 4256. Bastard, G., Gaj, J. A,, Planel, R., and Rigaux, C. (1980). J. Physique 41, C5, 247. Bastard, G., Rigaux, C., Guldner, Y . , Mycielski, A., Furdyna, J. K., and Mullin, D. P. (1981). Phys. Rev. B 24, 1961. Brandt, N. B., and Moshchalkov, V. V. (1984). Adv. Phys. 33, 193. Brandt, N . B., Moshchalkov, V. V . , Orlov, A. O . , Skrbek, L., Tsidil'kovskii, I. M., and Chudinov, S. M. (1983). Sov. Phys. JETP57, 614. (Zh. Eksp. Teor. Fiz. 84, 1050.)

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Byszewski, P., Cieplak, M. Z., and Mongird-Gorska, A. (1980). J. Phys. C 13, 5383. Chen, Y .-F., Dobrowolska, M., Furdyna, J . K., and Rodriguez, S. (1985).Phys. Rev. B32,890. Davydov, A. B., Noskova, L. M., Ponikarov, B. B., andugodnikova, L. A. (1980). Sov. Phys. Semicond. 14, 869. (Fiz. Tekh. Poluprov. 14, 1461.) De Seze, L. (1970). J. Physique 10, L353. Dietl, T. (1981). Solid State Science 24, 344. Dobrowolska, M., Dobrowolski, W., and Mycielski, A. (1980a). SolidState Commun. 34,441. Dobrowolska, M., Dobrowolski, W., Otto, M., Dietl, T., and Galazka, R. R. (1980b). (15th Int. Conf. Phys. Semicond.) J. Phys. SOC.Japan 49, Suppl. A 477. Dobrowolska, M., and Dobrowolski, W. (1981). J. Phys. C 14, 5689. Dobrowolska, M., Dobrowolski, W., Galazka, R. R., and Mycielski, A. (1981). Phys. Stat. Sol. (b) 105, 477. Furdyna, J. K. (1982). J. Vac. Sci. Technol. 21, 220. Gaj, J. A., Byszewski, P., Cieplak, M. Z., Fishman, G., Galazka, R. R., Ginter, J., Nawrocki, M., Nguyen The Koi, Planel, R., Ranvaud, R., Twardowski, A. (1978). (14th Int. Conf. Phys. Semicond.) Institute of Physics Conf. Series 43 (Wilson, London), p. 1113. Gaj, J . A., Planel, R., and Fishman, G. (1979). Solid State Commun. 29, 435. Gaj, J. A. (1980). (15th Int. Conf. Phys. Semicond.) J. Phys. SOC. Japan 49, Supple. A, 747. Series 43 (Wilson, Galazka, R. R. (1978). (14th Int. Conf. Semicond.) Instituteo~PhysicsCon~. London), 133. Galazka, R. R., and Kossut, J. (1980). In “Narrow Gap Semiconductors: Physics and Applications,” Lecture Notes in Physics 133, p. 245. Springer, Berlin. Galazka, R. R., Nagata, S., and Keesom, P. H. (1980). Phys. Rev. B 22, 3344. Galazka, R. R. (1982). In “Physics of Narrow Gap Semiconductors” (4th Int. Conf.), Lecture Notes in Physics 152, p. 294. Springer, Berlin. Geyer, F. F., and Fan, H. Y. (1980). IEEE J. of Quantum Electron. QE16, 1365. Grynberg, M., Martinez, G., and Brunel, L. C. (1982). Solid State Cornmun. 43, 153. Guldner, Y., Rigaux, C., Grynberg, M., and Mycielski, A. (1973). Phys. Rev. B 8, 3875. Guldner, Y., Rigaux, C., Mycielski, A., and Couder, Y. (1977). Phys. Stut. Sol. (b) 81,615 and Phys. Stut. Sol. (b) 82, 149. Holm, R. T., and Furdyna, J. K. (1977). Phys. Rev. B 15, 844. Kacman, P., and Zawadzki, W. (1971). Phys. Stat. Sol. (b) 47, 629. Jaczynski, M., Kossut, J., and Galazka, R. R. (1978). Phys. Stat. Sol. (b) 88, 73. Kane, E. 0. (1957). J. Phys. Chem. Sol. 1, 249. Khattak, G. D., Amarasekara, C. D., Nagata, S., Galazka, R. R., and Keesom, P. H. (1981). Phys. Rev. B 23, 3553. Kim, R. S., and Narita, S. (1976). Phys. Stat. Sol. (b) 73, 741. Kim, R. S., Mita, Y., Takeyama, S., and Narita, S. (1982). I n “Physics of Narrow Gap Semiconductors” (4th Int. Conf.), Lecture Notes in Physics 152, p. 316. Springer, Berlin. Kierzek-Pecold, E., Szymanska, W., and Galazka, R. R. (1984). Solid State Commun. 50,658. Kossut, J. (1976). Phys. Stat. Sot (b) 78, 537. Leibler, K., Sienkiewicz, A., Checinski, K., Galazka, R. R., and Pajaczkowska, A. (1978). I n “Physics of Narrow Gap Semiconductors” (3rd Int. Conf.), p. 199. Polish Scientific Publishers, Warsaw. Luttinger, J. M. (1953). Phys. Rev. 102, 1030. Mahoney, J., Lin, C., Brumage, W., and Dorman, F. (1970). J. Chem. Phys. 53, 4286. McCombe, D. B. (1969). Phys. Rev. 181, 1206. McCombe, D. B., Wagner, R. J., and Prinz, G. A. (1970). Phys. Rev. Lett. 25, 87. McCombe, D. B., and Wagner, R. J. (1971). Phys. Rev. B 4 , 1285. McKnight, S. W., Amirtharaj, P. M., and Perkowitz, S. (1978). Solid State Commun.25,357.

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Mullin, D. P. (1980). Ph.D. Thesis, Purdue University (unpublished). Mullin, D. P., Galazka, R. R., and Furdyna, J. K. (1981). Phys. Rev. B 24, 355. Mycielski, A., Kossut, J., Dobrowolska, M., and Dobrowolski, W. (1982). J. Phys. CIS, 3293. Mycielski, A., Rigaux, C., Menant, M., Dietl, T., and Otto, M. (1984). Solid State Commun. 50, 257. Mycielski, J. (1981). In “Recent Developments in Condensed Matter Physics,” p. 725. Plenum Press, New York. Nagata, S., Galazka, R. R., Mullin, D. P., Akbarzadeh, H., Khattak, G. D., Furdyna, J. K., and Keesom, P . H. (1980). Phys. Rev. I3 22, 3331. Pajaczkowska, A., and Pauthenet, R. (1979). J. Magn. Mat. 10, 84. Pastor, K., Grynberg, M., andCouder, Y. (1978). In “Physicsof Narrow GapSemiconductors,” p. 97. Polish Scientific Publishers, Warsaw. Pastor, K., Grynberg, M., and Galazka, R. R. (1979). Solid State Commun. 29, 739. Pastor, K., Jaczynski, M., and Furdyna, J. K. (1980). J. Phys. SOC.Japan 49, Suppl. A, 779. Pastor, K., Jaczynski, M., and Furdyna, J. K. (1981). Phys. Rev. B 24, 7313. Pidgeon, C. R., and Brown, R. N. (1966). Phys. Rev. 146, 575. Rigaux, C.(1980). In “Narrow Gap Semiconductors: Physics and Applications.” Lecture Notes in Physics 133,p. 110. Springer, Berlin. Roth, L. R., Lax, B., and Zwerdling, S. (1959). Phys. Rev. 114, 90. Savage, H., Rhyne, J. J., Holm, R. T., Cullen, J . R., Carroll, C. E., and Wohlfarth, E. P . (1973). Phys. Stat. Sol. (b) 58, 685. Serre, H. (1981). These de 3eme cycle, Paris (unpublished). Serre, H., Bastard, G., Rigaux, C., Mycielski, J., and Furdyna, J. K. (1982). In “Physics of Narrow Gap Semiconductors” (4th Int. Conf.), p. 321, Lecture Notes in Physics 152. Springer, Berlin. Sondermann, U., and Vogt, E. (1977). J. Magn. Magn. Mat. 6, 223. (Physica 86-88B,419.) Sondermann, U. (1979). J. Magn. Magn. Mat. 13, 113. Spalek, J., Lewicki, A., Tarnawski, Z., Furdyna, J. K., Galazka, R. R., and Obuszko, Z. (1986). Phys. Rev. B. 33, 3407. Takeyama, S., and Galazka, R. R. (1979). Phys. Stat. Sol. (b) 96, 413. Tuchendler, J., Grynberg, M., and Couder, Y. (1973). Phys. Rev. B 8, 3884. Weiler, M. (1981). In “Semiconductors and Semimetals” (R. K. Willardson and A. C. Beer, eds.), vol. 16,p. 119. Academic Press, New York. Witowski, A., Pastor, K., and Furdyna, J. K. (1982). Phys. Rev. B 26, 931.

SEMICONDUCTORS AND SEMIMETALS. VOL . 25

CHAPTER 7

Magnetooptical Properties of Large-Gap Diluted Magnetic Semiconductors J . A . Guj INSTITUTE OF EXPERIMENTAL PHYSICS. WARSAW UNIVERSITY WARSAW. POLAND

I . INTRODUCTION:POSSIBILITIES OPENEDBY STUDIES OF LARGE GAPDMS . . . . . . . . . . . . . . . . . . . . . 1 . Optical Access to Magnetic Properties. . . . . . . . . 2 . Studies of Excitons in Magnetic Field . . . . . . . . . 3 . Precise Investigation of Selection Rules . . . . . . . . TECHNIQUES . . . . . . . . . . . . . . I1 . EXPERIMENTAL 4 . Faraday Rotation . . . . . . . . . . . . . . . . . 5 . Reflectivity . . . . . . . . . . . . . . . . . . . 6 . Transmission Measurements . . . . . . . . . . . . I . Photoluminescence . . . . . . . . . . . . . . . . 8 . Other Techniques. . . . . . . . . . . . . . . . . 111. EXCHANGE SPLITTING OF THE FREEEXCITON GROUND STATE . 9 . Splitting Valuesand Selection Rules . . . . . . . . . . . 10. Relation of Splittings to Magnetization of the Ion System 1 1 . Transitions from the Spin-Orbit Split-Off Component of the Valence Band . . . . . . . . . . . . . . . . . IV . THEORETICALDESCRIPTIONOF BASIC MAGNETOOPTICAL PROPERTIES OF DMS . . . . . . . . . . . . . . . . . 12. Hamiltonian of the Problem and Typical Approximations 13 . Band EIectron States Split by Exchange . . . . . . . . 14. Zeeman Splitting of the Free Exciton Ground State . . 15 . Faraday Rotation . . . . . . . . . . . . . . . . . 16. Analysis of the Mean Field Approximation in Terms of the Magnetic Polaron . . . . . . . . . . . . . . . . . V . DETAILED MAGNETOABSORPTION STUDIES OF EXCITON STATES . . 17. Experiment Results . . . . . . . . . . . . . . . . 18 . TheoreticalAnalysis . . . . . . . . . . . . . . . . VI . EXCITON LUMINESCENCE . . . . . . . . . . . . . . . . VII . TRANSITIONS AWAYFROM THE BRILLOUIN ZONECENTER. . . 19. Experimental Data . . . . . . . . . . . . . . . . 20 . Attempts of Theoretical Description . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . .

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293 296 291 300 300 303 303 305 305 306 308

Copyright 0 1988 bv Academic Press. Inc . All rights of reproduction in any form rexrved . ISBN 0-12-752125-9

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I. Introduction: Possibilities Opened by Studies of Large Gap DMS Diluted magnetic semiconductors known also as semimagnetic semiconductors, offer a wide range of new possibilities in studies of magnetism and semiconductor physics. From among those possibilities we shall mention some proper to large gap semimagnetic semiconductors. 1.

OPTICAL

ACCESSTO MAGNETIC PROPERTIES

As we shall see below, magnetooptical effects such as Zeeman splitting or Faraday rotation in large gap DMS are often proportional to the magnetization of the crystal. This property enables one to study magnetic properties of DMS by means of optical measurements. A beautiful example has been supplied by Komarov et al. (1977), who measured Faraday rotation in Mndoped CdTe under microwave excitation, as a function of magnetic field. As the magnetic field reached the resonance value for Mn++ ions, the microwaves heated efficiently the Mn++spin system, causing a strong decrease of its magnetization and-as a consequence-a sharp drop in Faraday rotation (Fig. 1). That experiment is in fact a form of optical detection of magnetic resonance (ODMR). Another important example of such possibilities is the application of Faraday rotation to spin-glass phase studies. It has been used by Kett et al. (1981) and by Kierzek-Pecold et al. (1984) in Cdl-,Mn,Te.

FIG. 1 Faraday rotation of Cdl-xMn,Te versus magnetic field with microwave excitation for two wavelengths (1 and 2). Note a drop in rotation at 3.5 kOe (resonance field). [After Komarov et al. (1977).]

7.

MAGNETOOPTICAL PROPERTIES

277

2. STUDIES OF EXCITONS IN MAGNETIC FIELD

Exciton studies in DMS may contribute to the solution of a very general problem: that of a hydrogen atom in an arbitrary magnetic field. The problem in its general form has received little attention among atomic physicists, since in order to obtain a magnetic characteristic energy ( A o , ) comparable to the Coulomb characteristic energy (Rydberg), a magnetic field unavailable on earth would be needed. A perturbation treatment is thus sufficient (except in the case of recently studied highly excited Rydberg states). In an exciton in a semiconductor, due to small values of effective masses and to high values of the dielectric constant, these two energies can be made comparable in a field available in the laboratory. However, the problem is usually more difficult than in the hydrogen atom because of the complicated structure of the energy bands involved. Semimagnetic semiconductors, on the other hand, due to large exchange splittings, offer an advantageous possibility to create excitons of holes and electrons from simple nondegenerate bands. We shall consider this in more detail in Part V. 3. PRECISE INVESTIGATION OF SELECTION RULES

Due to high values of the Zeeman splittings in semimagnetic semiconductors, absorption lines observed in different polarizations are sufficiently separated to resolve and study weak lines due to forbidden transitions (they are, typically, lo2times weaker than the allowed transitions). This possibility has been used by Nguyen The Khoi et al. (1983), who explained the observed asymmetrical splitting of the absorption edge of Cdl-,Mn,Te in a magnetic field by a small contribution from forbidden transitions. This contribution has been measured as a function of magnetic field. The origin of the deviation from strict selection rules remains still an open question. 11. Experimental Techniques 4. FARADAY ROTATION

Standard Faraday rotation measurements in DMS are greatly facilitated by the fact that, due to the strong exchange field of the magnetic ions acting on charge carriers of the host semiconductor, Zeeman splittings of optical transitions are very large under typical experimental conditions, giving rise to a strong influence of the magnetic field on the real part of the dielectric function. Thus in a typical experiment, polarization direction of the incident light may rotate many times (Gaj, 1981) before leaving the sample. In such a case, it is sufficient to use a simple transmission setup (Fig. 2) and, without

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bothering to turn the polarizers, to plot the intensity of the transmitted light as a function of magnetic field. Subsequent minima of the resulting curve correspond to multiples of of the rotation angle. Furthermore, varying the wavelength at a fixed magnetic field one obtains also an oscillating curve, permitting a quick determination of the whole Faraday rotation spectrum from a single recorder trace (Fig. 3). In some experiments, however, it is desirable to measure small rotation angles, of the order of minutes of arc. This is the case in the spin-glass phase magnetooptical studies, where use of a small magnetic field is of a great importance. Small rotations occur also in reflection of polarized light from a crystal placed in a magnetic field-the so-called magneto-optical Kerr effect. In small angle measurements various modulation techniques prove useful. An example is shown in Fig. 4. An oscillating plate consisting of two pieces of polarizing foil with mutually perpendicular polarization directions is placed in the optical system in such a way that the outgoing light passes alternatively through two pieces of the foil. If the incident light is polarized along a direction at n / 4 to polarization directions of both pieces of foil and for zero Faraday rotation, no modulation will arise. For small deviations of the outgoing light polarization from the equilibrium direction, a modulation Md

sc

P

FIG. 2. Experimental setup for Faraday rotation measurements in semimagnetic semiconductors. The elements of the setup are designated as follows: Sc-source, P-polarizers, C-superconducting coil, S-sample, Md-modulator, M-monochromator, D-detector, E-electronics.

0.77

078 0.79 2 (pm)

0

FIG. 3. Intensity versus wavelength in a Faraday rotation measurement in Cd,-,Mn,Te, at a fixed magnetic field of about 1 T.

7.

MAGNETOOPTICAL PROPERTIES

219

-

FIG.4. Oscillating plate of a polarization modulator. Polarization directions in two parts of

the plate marked by arrows. The incident light polarization is shown by the bottom arrow.

signal will be proportional to the deviation angle. In such a manner rotations up to a few degrees can be measured with an accuracy depending on the mechanical stability of the system. Accuracy of 1’ is achieved without any special precautions. With Verdet constants of the order of a few deg/cm.Gs and in samples a couple mm thick, it is possible in this technique to use fields of one Gs or less, values quite adequate in spin-glass studies. 5 . REFLECTIVITY

Reflectivity measurements are commonly used in DMS to determine Zeeman splittings of the free exciton ground state. Since only the positions of the reflectivity structures are usually of interest, absolute values of the reflection coefficient are not determined. Relative reflectivity measurement does not present any difficulties. The only precaution to be taken is to avoid photoluminescence excitation. This may be easily done with suitable filters (usually it is convenient to place the monochromator after the sample). A typical experimental setup is shown in Fig. 5 . Figure 6 shows an example of magnetoreflectivity results in Cdl -,Mn,Te. 6. TRANSMISSION MEASUREMENTS

The transmission configuration usually allows the experimenter to obtain more detailed and precise information on optical transitions than the reflectivity. In a transmission spectrum of Cdl-,Mn,Te shown in Fig. 7, besides the ground exciton state, an excited (2s) state is clearly visible in the absence of a magnetic field. Application of a field of 5T (Twardowski, 1981b) produces a great number of absorption peaks, invisible in a straight-forward reflectivity spectrum. In spite of its informative value, the transmission technique is used less frequently, because it requires much more effort. Since

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FIG.5 . Experimental setup for magnetoreflectivity measurements (Sc-source, L-lenses, F-filter, S-sample, C-superconducting coil, Q-quarter wave plate, P-linear polarizer, M-monochromator, D-detector, and E-electronics).

I64

I63

I62

I61

PHOTON ENERGY ( e V )

FIG.6. Reflectivity spectra of Cdl-,MnxTe at 1 . 1 Tin the Faraday configuration (a' and upolarizations) and in the Voigt configuration (n polarization). [After Gaj el of. (1979).]

the absorption coefficient in the excitonic region of adirect gap semiconductor is of the order of 105-106cm-', samples with a thickness below 1pm are necessary. Such samples have been obtained by local etching by Twardowski et al. (1979), using a method known in electron microscopy, as follows. A 200pm slice of the crystal is placed in an etching solution, with both faces protected by plastic masks that cover the faces of the slice except for two small circular areas, available for etching. By carefully controlling the process of etching, a spot of about 20-50pm in diameter and a thickness sufficiently small to allow light to pass can be obtained. Figure 8 shows the etching arrangement and the resulting sample form.

7.

MAGNETOOPTICAL PROPERTIES

281

c+ B=5T

1.61

1.62

1.63 1.64 1.65 1.66 1.67 ENERGY [eV]

FIG.7. Transmission spectra of Cdl-,Mn,Te at zero magnetic field and at 5T.Temperature 1.8 K. [After Twardowski (1981b).] Besides the principal magnetically split absorption peaks A, B, C, D, a rich structure is visible.

FIG.8. Sample etching arrangement. The optical system is used to interrupt etching when the sample becomes sufficiently thin to transmit light (Sc-source, L-lens, E-etching solution, B-bar of a magnetic stirrer, S-sample, P-plastic screens, M-microscope of low magnification).

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M1

sc

L

-

FIG. 9. Experimental setup for magnetotransmission measurements (Sc-source, L-lens, S-sample, C-superconducting coil, M1, M2--monochromators, Q-quarter wave plate, P-polarizer, D-detector, E-electronics).

Inhomogeneous thickness of samples obtained in this manner does not permit quantitative absorption measurements, but it allows to determine positions of the absorption peaks. Results of Fig. 7 have been obtained on such a sample. In such experiments, due to the small size of the transmitting region of the sample, special care must be taken to avoid photoluminescence excitation. A convenient way to do this is to use two monochromators. The first one, with its slits open wide, transmits from the light emitted by the source a band corresponding to the width of the spectrum to be taken. The second one, placed after the sample, is used to scan the spectral range. Figure 9 shows schematically such an experimental setup. 7. PHOTOLUMINESCENCE

In photoluminescence experiments on mixed crystals, it is of particular importance to know exactly the energy gap of the excited region of the sample. Therefore in photoluminescence measurements on semimagnetic semiconductors, it is desirable to do a complementary measurement of absorption or reflectivity on the same sample, under the same experimental conditions. As for photoluminescence itself, a number of methods have been used with pulsed or CW-excitation, using dye, Argon ion, or Helium-Neon lasers. 8. OTHERTECHNIQUES

Apart from the principal experimental methods discussed above, and Raman scattering described in detail in Chapter 9 in this volume, a number of techniques were used in magnetooptical studies of wide gap DMS. Among them must be mentioned optical detection of magnetic resonance (ODMR) in photoluminescence (Komarov et al., 1980; Malyavkin, 1983) and Faraday rotation (Komarov et al., 1977), modulated reflectivity (Ginter et at., 1983) and photoconductivity (Lindstrom e f al., 1983).

7.

MAGNETOOPTICAL PROPERTIES

283

111. Exchange Splitting of the Free Exciton Ground State

9. SPLITTING VALUESAND SELECTION RULES Zeeman effect on the free exciton ground (1s) state has been studied in a number of semimagnetic semiconductors. Results vary depending on crystal structure of the semiconductor.

a. Cubic Structure Magnetoreflectivity and magnetotransmission have been studied in a number of cubic (zinc blende) large gap DMS, such as Cdl-,Mn,Te (Komarov et al., 1977; Twardowski et al., 1979; Gaj et al., 1978, 1979; Rebman et al., 1983), Znl-,Mn,Te (Komarov et al., 1978; Twardowski etaf., 1984) or Znl-,MnxSe (Komarov et af., 1980b; Twardowski et al., 1983). The exciton ground state in those alloys splits in six components: four (denoted A through D) visible in the Q polarization (electric field of light perpendicular to the external magnetic field) and two (E and F) visible in the 72 polarization (electric field of light parallel to the external magnetic field). In the Faraday configuration (light propagation along the external magnetic field), the four a-components exhibit a' (A and B) or a- (C and D) circular polarizations (see also Fig. 16). The whole splitting pattern is approximately symmetric relative to the zero field position. Components B and C are significantly weaker than A and D. The lowest-lying component A is much sharper (especially at high fields) than the remaining ones. The features described above are visible in Figs. 6 and 7 for reflectivity and transmission, respectively, of Cdl -,Mn,Te. For a given semiconductor in a given magnetic field and temperature, the Zeeman pattern can be characterized by splitting values of two pairs of acomponents: strong (A and D) and weak ones (B and C). The splitting of the two n components, E and F, is not independent. It equals one half of the difference between splittings A-D and B-C. The ratio of the two latter splittings has been found to be independent of magnetic field or temperature for a given semiconductor. The polarization selection rules are quite strong. However, absorption measurements allow one to observe very small deviations from perfect selection rules. Nguyen The Khoi el al. (1983) have observed a small (of the order of 1%) admixture of Q- polarization in some a+-polarized Zeeman components in Cdl-,Mn,Te. Figure 10shows the positions of the Zeeman components of the free exciton ground state in a sample of Cdl-,Mn,Te as a function of magnetic field. Splittings increase linearly at low fields and tend to saturate at higher field values. That behavior can be examined in detail in Fig. 11, where the splitting

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of the strong D components A and D is plotted vesus magnetic field for various compositions. In crystals of low Mn mole fraction x , the splitting increases with x and the saturation is clearly visible, while at higher x values the curves show only a slight deviation from linearity. At higher temperatures the values of the splittings becomes smaller and the tendency to saturate is weaker.

FIG. 10. Energy of the Zeernan components of the exciton versus magnetic field for CdI-,Mn,Te in the Faraday configuration. Continuous lines are drawn to guide the eye. [After Twardowski (1981b).]

FIG. 11. Splitting of the two strong u components of the exciton Zeernan structure of Cdl-,Mn,Te versus magnetic field at 1.4 K. Manganese concentration values are given in percentage. [After Gaj e t a / . (1979a).]

7.

285

MAGNETOOPTICAL PROPERTIES

b. Hexagonal Structure Magnetoreflectivity and magnetoabsorption measurements have also been reported on hexagonal (wurtzite) semimagnetic semiconductors, such as Cdl-,Mn,Se (Arciszewska and Nawrocki, 1982; Aggarwal et al., 1983; Komarov et al., 1980c) or Cdl-,Mn,S (Gubarev, 1981). The results show a behavior generally more complicated than in the cubic case. A relatively simple situation can be obtained by applying the magnetic field along the hexagonal axis of the crystal. The resulting splitting pattern and selection rules are then analogous to those in cubic crystals', except for a displacement of the two strong o components relative to the remaining ones due to the crystal axial anisotropy (see Fig. 12). The variation of the effect with composition and temperature are also like in cubic semimagnetic semiconductors. For an arbitrary direction of the magnetic field, the results are more complicated.

-2

2.31

I

I

I

MAGNETIC FIELD f T ) FIG. 12. Energy of the Zeeman components of the exciton versus magnetic field for Cdl-,Mn,Se in the Faraday configuration. Magnetic field is parallel to the crystal hexagonal axis. [After Arciszewska and Nawrocki (1984).] CdS seems to be an exception. The strong u+ component, being narrower (as in other alloys) is placed above its u- counterpart (Gubarev, 1981).

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10. RELATIONSHIP OF THE SPLITTINGS TO THE MAGNETIZATION OF THE IONSYSTEM

The behavior of the Zeeman splittings as a function of the magnetic field, temperature and composition resembles (particularly at low compositions) the magnetization of a paramagnet. Therefore it is natural to look for a relationship between the magnetization of the ion system present in the crystal and the Zeeman splittings. Experiments performed on a number of wide gap DMS show a simple proportionality of the splittings to the magnetization. Figure 13 shows that relationship for Cdl-,Mn,Te. The plot of Fig. 13 has been obtained from data measured on many samples, with compositions varying from x = 0.005 up to x = 0.30. Expressing the magnetization of the ion system by a mean value of the spin component along the magnetic field per cation x ( S , ) , we can write for the splitting: AE

=

AX( S z ) .

The coefficient L determined for Cdl-,Mn,Te from the results presented in Fig. 13 is 1.10eV. In other wide gap semimagnetic semiconductors similar values of that coefficient have been obtained. It must be pointed out that the proportion described above is only approximate. A more detailed discussion will follow in Part IV I

I

1

3

I

,

x 0.5 -0

0 +

I

2 5 0 10 A 20

30

I -10'~

-10-

-10-

SPIN PER UNIT CELL

FIG. 13. Splitting of the two strong u components of the exciton Zeeman structure of Cdl-,Mn,Te versus magnetization, expressed in terms of the mean value of the Mn spin (per cation). The straight line corresponds to a proportionality with a slope of 1.10 eV. Manganese concentration is indicated in percentage. [After Gaj et a/. (1979b).]

7.

MAGNETOOPTICAL PROPERTIES

11. TRANSITIONS FROM THE SPIN-ORBIT SPLIT-OFF OF THE VALENCEBAND COMPONENT

An exciton state can also be created with a hole from the lower component of the valence band, split off by the spin-orbit interaction. Such a state is more or less difficult to observe depending on the value of the spin-orbit splitting A. It is quite easy to see in CdS (A = 0.065 eV), fairly easy in CdSe (A = 0.45 eV) whereas in CdTe (A = 0.95 eV) it takes some effort to observe it. In an alloy, disorder introduces an additional broadening, making its observation more difficult. In CdTe an exciton absorption line associated with transitions from the r7 valence band has been observed by Twardowski et al. (1980). Substituting 2% of Cd by Mn broadens the structure enough to make the absorption peak disappear (only a threshold remains, as seen in Fig. 14). Any magnetooptical measurements on such an ill-defined structure must be inaccurate. Within

I

1

I

I

2.550

2.600

I

[eV]

ENERGY FIG. 14. Transmission versus photon energy for CdTe ( B = 0) and Cdo.9~Mn~.o~Te (B = 0 and 4 T ) . Straight lines were drawn to determine transition energies (marked by arrows). [After Twardowski ef al. (1980).]

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GAJ

that poor accuracy, Faraday configuration measurements show a splitting of the two components occurring in the g+ and 0- polarizations, respectively, with the value of the splitting proportional to the magnetization in Cdl-,Mn,Te (Twardowski et al., 1980) and in CdMnSe for magnetic field parallel to the hexagonal axis (Arciszewska and Nawrocki, 1984). IV. Theoretical Description of Basic Magnetooptical Properties of DMS

12. HAMILTONIAN OF THE PROBLEM AND TYPICAL APPROXIMATIONS We shall look for solutions of the eigenproblem of a band electron in a given external magnetic field and a given state of the magnetic ion system. Such an approach can be justified if the band electron wave function is sufficiently delocalized to make an influence of the electron on each of the magnetic ions negligible. The Hamiltonian for a band electron will have the form,

H

=

Ho

+ HB + HC + Hex + Heh

(1)

where the five terms in the Hamiltonian correspond to various interactions influencing the band electron state. Relative contributions of each of the terms may vary depending on the crystal in question and on experimental conditions. H, is here a (one electron) Hamiltonian of the electron in a perfect crystal:

H,

P2

=-

2m

+ V(r)

The term

e HB = - - p ' A mc

+ g/fBb'B

(3)

describes (in a linear approximation) the direct influence of the magnetic field on the carrier state. This term is responsible for Landau quantization and spin splitting of Landau levels (this gives, of course, only a small fraction of the total spin splitting in DMS). The Coulomb term HC represents energy of the carrier interacting with an impurity or with another carrier (e.g., electron-hole interaction producing exciton states). The term Hex represents the fundamental interaction in a semimagnetic semiconductor: the carrier-ion exchange. Following Kossut (1976) and Bastard el al. (1978) we shall express it in a Heisenberg form:

Hex =

-c J(r n

- R,)s~

-G

(4)

7.

MAGNETOOPTICAL PROPERTIES

289

where the index n runs over all the magnetic ions. The term Heh represents the electron-hole exchange interaction and may be of importance in exciton problems. The difference between the present case and a non-magnetic semiconductor lies in the presence of the ion-carrier exchange Hamiltonian Hexin Eq. (1). Unfortunately Hex does not possess the translational symmetry of a perfect crystal. To avoid that difficulty, one commonly involves the molecular field approximation. Strictly speaking, this is combined with a virtual crystal approximation: to every cation site we attribute a fraction of the magnetic ion spin equal to the fraction of cation sites really occupied by magnetic ions. Those spins, instead of being treated properly as operators, are represented by numbers, i.e., by thermal average values. Thus in Eq. (4),all the manganese spin operators S n are replaced by their (common) average value ( S n ) and the sum over the magnetic ions is replaced by the sum over all the cation sites multiplied by the magnetic ion mole fraction x. Considering a magnetically isotropic crystal placed in a magnetic field (taken along the z-axis), we have ( S ” ) = ( S y ) = 0, and Eq. (4),then becomes

He, =

-Ci

J(r - R i ) x a Z ( S Z ) .

(5)

It must be pointed out that, within the framework of the effective mass theory, the above Hamiltonian has the form identical to the spin part of HB (Zeeman energy of the electron spin). Therefore the introduction of the exchange term He, does not in principle create any new formal difficulties (within the approximation outlined above) compared to the non-magnetic case. Let us make now another essential approximation, neglecting the term HB. This is justified in wide gap DMS when the temperature is not excessively high, i.e., when the magnetic field effects due to carrier-ion exchange are strong compared to “nonmagnetic” effects described by H B . Now the Landau quantization is absent from the problem, and we may speak of the electron wave vector k in the presence of a magnetic field. In what follows we present a k * p calculation of the energy bands, using the following further approximations: (1) The k * p term is included as a second order perturbation (Luttinger model). In the absence of exchange, this leads to parabolic bands; (2) Warping of the valance band is neglected, as well as effects due to lack of inversion symmetry of the crystal; (3) Spin-orbit splitting of the valence band is assumed to be large compared to the ion-carrier exchange energy.

290

J. A . GAJ

13. BANDELECTRON STATES

SPLIT BY

EXCHANGE

Let us consider a cubic zinc blende crystal. Under the above assumptions, we can treat separately the conduction band and each component of the valence band. Following Bir and Pikus (1972), we shall use as basis vectors for the conduction (r6)band:

I),

+)C

= st

In the chosen bases, matrices of Hexare diagonal:

(Wv3/2IHexlWu3/2)

where

=

3 O 0 0

B

O

O B O 0 - B 0

0

O O 0

-3B

7.

MAGNETOOPTICAL PROPERTIES

291

Here No denotes the number of unit cells per unit volume, and a = (slJls) and p = ( X l J l X ) are exchange integrals for the conduction and valence bands, respectively. Within the assumed approximations the total Hamiltonian will now consist of on!y two terms, H = Ho + H e x . (13) In the k * p calculation the term H , will be represented by

-

Let us point out that at k = 0, where the k p term vanishes, the presence of exchange produces two-fold or four-fold and equal splittings of bands, with the splitting values proportional to x( S z ) and therefore (if diamagnetic effects are neglected) to the magnetization of the crystal. The solutions of the problem follow immediately for the conduction, as well as the lower valence bands. In those cases the entire Hamiltonian matrices (Hexand k * p terms) are diagonal, where the energies Ec = E,"

h2k2 ++- 3A 2mc

and

correspond to the wavefunctions y/,+(k) = sfeik' y;(k) = sleikr y&(k)

=

1

- [(X + iY)l

d3

-i vui/2(k) = - [(X - i Y ) t

\r5

+ Zt]eIb - Zl]eih

for the two bands, respectively. In the case of the TSvalence band, the k p Hamiltonian matrix is

-

(18)

292

J. A. GAJ

where in the spherical approximation

F = Mk2 + *N(k? + k,?) G

=

(A4+ f N ) k 2 - SN(k:

+ k,?)

N

I=JTZ (kx - iky)2, M and N being combinations of interband momentum matrix elements (Bir and Pikus, 1972). Although in a general case the diagonalization of the total Hamiltonian is rather complicated, some particular situations are simple to solve, e.g., the band structure in the vicinity of k = 0 for a given non-zero magnetization. The k p term may then be treated as a perturbation, producing four equally split anisotropic bands:

-

Ei,4 = F I+- 3B, E2,3 = G f B ,

or, in terms of light and heavy hole masses,

(--

h2k2 + 3 h2 - Gh2) ( k : E2,3 = -2mlh 4 2mrh

+ k,?) f B.

\--I

We thus obtain four bands whose constant energy surfaces are ellipsoids: cucumber-(E1,4) or disc-like (E2,3). The diagonalization of the k p Hamiltonian alone produces the well known spherical and parabolic light and heavy hole bands:

-

h2k2 Ehh(k) = E," + Mk2 = E," - ---, 2mhh

The wavefunctions corresponding to Eq. (23) and (24) can be easily obtained by a suitable rotation of the basis functions. Note that for k along the z-axis the Hamiltonian given by Eq. (19) is diagonal, considerably simplifying the calculations (see, e.g., Kane, 1957).

7.

MAGNETOOPTICAL PROPERTIES

293

t 0

- 10 -20

3 +lC;

2

1

0

1

2

3

Z("= 100)-

FIG. 1s. Energy band structure of Cdl-,Mn,Te in dimensionless energy ( E ) and wave vector

(K)coordinates at two values of the angle between the wave vector and the magnetization: 90" (KJ and 10". IAfter Gaj e t a / . (1978).]

In a case of a small magnetic field [and hence a small exchange term B, Eq. (12)] He, can be treated as a perturbation. A straightforward first order calculation produces highly anisotropic splittings of light and heavy hole bands: Elh(k,

B) = Eih(k) f B d 4

Ehh(k,

-

3 COS2 8

B) = E l h ( k ) f 3 8 C O S 6

(25)

(26)

where B is the angle between the wave vector k and the magnetization direction. A numerical diagonalization of the Hamiltonian matrix in a general case produces a highly nonparabolic and anisotropic band structure. This is shown in Fig. 15 for two directions of the wave vector: at angles 90" and 10" to the magnetization direction. 14. ZEEMANSPLITTING OF FREEEXCITON GROUNDSTATE

The problem of exciton states in the presence of an exchange field has been treated, e.g., by Cieplak (1980), who followed an approach developed by Altarelli and Lipari (1973). We shall not enter here into the details of the complete exciton calculation, but will simply note that in the effective mass approximation the Coulomb potential of the electron-hole interaction HC does not mix the basis functions, enabling us to think of an exciton with a hole from one of the exchange-split valence subbands and an electron from one of the conduction subbands, if the electron-hole exchange is to be neglected. This allows us to describe excitonic magnetooptical splittings in terms of interband transitions. Remembering that direct exciton creation

294

J . A. GAJ

occurs practically at k = 0 we can, according to Eqs. (4)and (21), work out a Zeeman scheme presented in Fig. 16 and in Table I. The relative intensities are, of course, proportional to the corresponding interband matrix elements. The data of Table I are in good agreement with the experimental results obtained in Cdl-,Mn,Te if the exchange parameters are: N,a = 0.22 eV and N,P = -0.88eV. For excitons involving holes from the r7 band, the experimental results are not very precise, but within experimental error they agree with the above model (Twardowski et al., 1980). In other cubic semimagnetic semiconductors similar agreement has been found. Table I1 gives exchange parameter values for various large gap DMS, obtained from a comparison of magnetooptical splittings with magnetization. In hexagonal semiconductors the situation is more complicated. In the first rough approximation (the so-called quasi-cubic model), one can use the same wavefunctions as for a cubic crystal, adding a diagonal term corresponding to a crystal (axial) field. If the magnetic field is parallel to the crystal c-axis and the spin-orbit splitting is large compared to the crystal field and to the exchange splittings, the model outlined above can be applied, the only difference consisting in a displacement of the ,;I +*> states relative to the I$, k $>ones due to the crystal field splitting. However, experimental results of Arciszewska and Nawrocki (1982) show clearly that the quasi-cubic approximation is inadequate in Cdl-xMnxSe. In particular, the relation between splittings A-D, B-C, and E-F mentioned in Section 9 is not satisfied (see Fig. 12). To improve the model, an anisotropy of the spin-orbit interaction may be introduced. That additional parameter allows for a reasonably accurate description of the experimental data in Cdl-,Mn,Se, using a single

V

cond. G band

valence bands

G r,

FIG.16. A schematic representation of band splittings in a cubic (zinc blende) semimagnetic semiconductor. Transitions allowed in the Faraday configuration are marked by arrows.

7.

295

MAGNETOOPTICAL. PROPERTIES

TABLE I CALCULATED ENERGIES AND RELATIVE INTENSITIES OF THE ZEEMAN COMPONENTS OF THE FREEEXCITON IN A ZINCBLENDE SEMIMAGNETIC SEMICONDUCTOR States Valence

Conduction

Energy (Relative to zero field position)

3B B B -B -B -3B -B -B B B

Polarization

- 3A

U+

+ 3A -

U+

3A

R

+ 3A

R

- 3.4 + 3A + 3A - 3A + 3A - 3A

U U

-

U+

R

n 0

Relative intensity

Symbol

3 1 2 2 1

3 2 1 I 2

A B

E F

C

D

-'

A = --x NO(Y(S') B = sxNoP(S'> Attention: (S') is normally negative!

TABLE I1 EXCHANGE CONSTANTS IN VARIOUS LARGEGAPDMS DETERMINED FROM MAGNETOREFLECTNITY OR MAGNETOABSORPTION COMBINED WITH MAGNETIZATION MEASUREMENTS

~~

~

Crystal structure

Naa

NaP

Compound

[evl

[evl

Cdl-,Mn,Te

zinc blende

0.22

-0.88

Gaj el al. (1979)

Znl-xMnxTe

zinc blende

0.19

- 1.09

Twardowski et a/. (1984)

Znl-xMnxSe

zinc blende

0.29

- 1.4

Twardowski et al. (1983)

Cdl-,MgSe

wurtzite

0.23

-

Cdl-,Mn,S

wurtzite

Reference

1.27

Arciszewska and Nawrocki (1982)

5.6"

Gubarev (1981)

'A positive value was obtained as a consequence of defining the Heisenberg Hamiltonian without a minus sign [see Eq. (4)]. Both the sign and the unexpectedly high absolute value must be regarded with caution (see also the footnote in Section 9).

296

J . A. GAJ

exchange constant 3/ for all the valence band functions (see Table 11). In principle, two different values px= ( X I J I X ) and pz = ( Z l J l Z ) should be allowed, since the X and Z functions are no longer equivalent. For an arbitrary direction of the magnetic field, the exchange splittings are no longer proportional to the magnetization, since the two perturbations, crystal field and exchange field, are now represented by non-commuting Hamiltonians of comparable magnitude. The situation at k = 0 resembles the eigenproblem in a cubic semiconductor k # 0, which has rather complicated non-linear solutions, as mentioned above. In general, magnetooptical studies of hexagonal DMS also permit one to find (by extrapolation) parameters of the band structure of the host nonmagnetic semiconductor. The assumption that px = pz can be discarded if transitions from the spin-orbit split-off component of the valence band are also studied. 15. FARADAY ROTATION Using the results of the previous sections, one can now calculate Faraday rotation spectra. This has been done using the excitonic absorption model developed by Elliott (1957). The model predicts a series of discrete hydrogenic lines corresponding to creation of consecutive exciton bound states of energies En = Eg - R*/nZ (R*-exciton binding energy) and of oscillator strength varying as l / n 3 . Above the energy gap, absorption is a continuous function of the photon energy, with zero slope at Eg,and slowly rising above that value to approach a square-root curve at high energies ( A o - Eg S= R*). The value of the absorption coefficient at ho = Eg is equal to the value obtained by an infinitesimal broadening of the discrete lines at A o E g . In reality a broadening must always be expected, so that only a limited number of states of the hydrogenic series can be distinguished (two states are observable in Fig. 7). There is no particular difficulty in calculating the Faraday rotation spectra using the full Elliott model. However, a very simplified approach, consisting in the calculation of two first nonvanishing moments of the circular dichroism, gives quite an adequate description of the experimental rotation spectra near the band gap (Gaj, 1981). In the calculation, performed in a linear (low field) approximation, the line broadening, as well as any dichroism above the energy gap, has been neglected. The resulting solution for the rotation angle 8 is:

7.

MAGNETOOPTICAL PROPERTIES

297

0

c 0

c 0

a

035

0.20 log ( E l l eV1

FIG. 17. Faraday rotation spectrum of Cdl-,Mn,Te fitted to Eq. (4): x = 0.05,T = 77 K, = 0.12, Eg = 1.66eV.

KO

where

KO

d(i 4)

is the extinction coefficient at tzo = E g ,

Eo = Eg - 2R*[

f

n=l

4 n

-

2

n = n ~

-

i 41 n

n=1

= Eg - 1.14R* (28)

I is the sample thickness and c is the velocity of light. The resulting solution should, of course, not be used too close to the exciton structure. In Fig. 17 an experimental rotation spectrum has been fitted with Eq. (27) by adjusting two parameters, KO and Eg,producing good agreement and reasonable values of these parameters. (Reflection data give the same value of Eg). The values of A and B were calculated from Eq. (12) using magnetic susceptibility data of Oseroff (1982). 16. ANALYSIS OF THE MEAN FIELDAPPROXIMATION IN TERMSOF THE MAGNETIC POLARON

In a detailed analysis of magnetooptical and magnetoreflectivity data, Zeeman splittings show small deviations from strict proportionality to magnetization. In order to explain those deviations we must go beyond the molecular field approximation and admit a mutual interaction between the ions and the band electrons. The simplest way to do this is by looking for eigenfunctions (in a magnetic field) of an electron interacting equally strongly with N magnetic ions. We take a limited (several tens) number of ions and we assume that the

298

J . A. GAJ

= I

N.32

gpB/ kT=0

n

A

..........

-----

ll 0-

(b) FIG. 18. Exciton absorption spectra calculated by Gaj and Golnik (1985): (a) zero field, no broadening; (b) broadening included.

7.

MAGNETOOPTICAL PROPERTIES

299

electron wave-function is localized. Formally the problem has much in common with the model of the bound magnetic polaron used extensively in physics of magnetic semiconductors (see Chapter 10 in this volume). Attempts to apply that model to magnetoabsorption of semimagnetic semiconductors have been done by von Ortenberg (1984) and by Gaj and Golnik (1985). We shall follow here the approach of the latter work. The physical meaning of the model is represented by the splitting of an electron (or exciton) energy by the exchange field of spin fluctuations of magnetic ions. Even in the absence of a net macroscopic magnetization, those fluctuations will lead to a non-zero splitting of band electron states. Figure 18a shows a diagram of the density of states (that corresponds to optically allowed transitions) for an exciton in a CdTe-type semiconductor interacting with N = 32 ions, each with a spin s = $ (the ion-ion interaction is neglected). Each configuration of the ion system (given by quantum numbers of the total spin Sand its projection M o n a quantization axis) generally gives rise to six eigenstates of the coupled system. It is reasonable to assume that the state of magnetic ions does not change during the optical transition (a “magnetic Frank-Condon principle”). Without an external magnetic field the optical absorption spectrum will be identical with the density of states curve in Fig. 18a. The presence of a magnetic field, on the other hand, will favor states with a large total spin S of the ion system. It may be interesting to note that within this model the position of the center of gravity of the absorption spectrum for each polarization follows exactly the molecular field result. The above model gives a qualitative explanation of the experimentally observed deviation from the mean field behavior in wide gap DMS, when supplemented by a phenomenological broadening large enough to remove the zero-field splitting that is not observed in those compounds. Results (shown in Figs. 18b and 19b) are consistent with experimental data (Fig. 19a).

FIG. 19. Splittingof the two strong acomponents of the excitonic Zeeman structure measured in Cd,-,Mn,Te (a), compared with a calculation for N = 32 Mn ions interacting with the exciton (b). [After Gaj and Golnik (1985).]

300

J. A. G k l

The model discussed above includes a localization of the exciton. The nature of this localization remains an open question. One of the possible explanations is to view it as due to alloy composition fluctuations. V. Detailed Magnetoabsorption Studies of Exciton States 17. EXPERIMENTAL RESULTS

Transmission studies on thin samples reveal many more details than the energy of the exciton ground state and its Zeeman components. We shall present here a survey of experimental results obtained mainly on Cdl-,MnxTe by Twardowski and coworkers (1979, 1981, 1982). In the magnetoabsorption spectra shown in Fig. 7, above the principal components of the exchange split exciton ground states A1, . .., D1, a number of weaker lines are visible that can be denoted A2, ..., As, Bz,..., B4, etc. We have thus four sets of absorption lines Ai through Di. Evolution of all those lines with magnetic field is shown in Fig. 20. As discussed above, the energy of the ground state of each set (A,, ...,DI) is determined by the magnetization of the sample and by the exchange parameters.

1,700

-2 Y

>

0

a W

z W

1,650

1

2

3

4

5

MAGNETIC FIELD (TESLAS)

FIG.20. Positions of the features of the magnetoabsorption of the exciton in Cdl-xMnxTe versus magnetic field. [After Twardowski (1981a).]

7.

MAGNETOOPTICAL PROPERTIES

301

Let us limit our discussion to the case of strong exchange effects (x 2 0.02 at liquid He temperatures). It turns out that, within each set, the positions of the lines related to the principal (i = 1) line depend only on the magnetic field, and not on magnetization. These two types of behavior, governed by magnetization (ground states of sets) or by magnetic field (relative position within sets) can be seen in Figs. 21 and 22. In Fig. 21, where magnetization is varied by changing the temperature at a given magnetic field, constituent lines of the sets A and B follow the variation of the corresponding ground states A1 and BI. The magnetization dependent behavior is eliminated in Fig. 22, where the relative positions within sets A and B are plotted as a function of magnetic field. The data from various compositions (and thus various magnetization values) coincide well in the plot. Such a distinction is not possible in CdTe or in very diluted crystals ( x I0.01), where the exchange energy is comparable to, or smaller than, the cyclotron energy.

FIG. 21. Positions of magnetoabsorption peaks in Cdt-,Mn,Te at a fixed magnetic field B = 5T,versus the splitting of the strong u components A I and D1. Magnetization is varied by changing the temperature and the composition. [After Twardowski (1981a).]

302

J . A. GAJ I

I

1

I

I

B

B

5'10 *

2

1Oo/o

6

3

4

5

B(TJ

(b) FIG.22. Line positions within sets Ai (a) and Bi (b) versus magnetic field. Continuous curves correspond to a calculation described in Section 18. [After Twardowski (1981a).]

7.

MAGNETOOPTICAL PROPERTIES

303

18. THEORETICAL ANALYSIS

The magnetotransmission data reviewed above support the idea that the exchange and Coulomb terms may be considered independently, as brought up in Section 14. In order to include excited excitonic states, we can no longer continue neglecting the orbital part of H B , that must be now added to the Coulomb terms HC . We are thus left with the problem of a simple exciton created from a pair of non-degenerate bands. It should be pointed out that due to a difference in form between the valence subbands of mj = +* and those of mj = +*, two types of solutions may be expected in this case, consistently with the experiment. A calculation along these lines has been performed by Twardowski and Ginter (1982), who used an improved version of the adiabatic method. Comparison of their theoretical results with experimental data is shown in Fig. 22, where continuous lines represent results of the calculation. The agreement is quite good, except for a certain difference in slope for line A1 . Line A2, interpreted as originating from a 3d-2 exciton state, is not included in the calculation. VI. Exciton Luminescence

Most of the luminescence measurements on semimagnetic semiconductors have been performed in order to study bound magnetic polaron (BMP) effects. Those studies are considered in this volume in Chapter 10. However, there exists a phenomenon particularly suited to be considered in this chapter, i.e., the destabilization of exciton complexes in a magnetic field. This effect has been reported by Planel et al. (1980), Ryabchenko et al. (1981), and Golnik et al. (1983). In Fig. 23 photoluminescencespectrain the excitonicregion of Cdl -,Mn,Te are presented, as measured by Planel et al. (1980). The principal line has been identified as originating from the recombination of an exciton bound to a neutral acceptor-an AoX complex. A free exciton line X is also visible. Application of a magnetic field causes a shift of the A'Xline towards lower energies-much slower than that of the lowest free exciton component observed in emission or reflectivity (Fig. 24). At a certain field the two lines meet. When the magnetic field continues to increase, the A'Xline follows the X line as it shifts. This behavior has been interpreted as a destabilization of the bound exciton state by exchange effects. At zero field, the A'Xenergy is lower then the energy of a dissociated system A' + Xby a binding energy that consists of a Coulomb part and an exchange (magnetic polaron) part. Application of a magnetic field, lowers the energy of the ground hole of both the bound and the dissociated systems by affecting the exchange contributions.

304

J. A. GAJ

-.-

H = I IT H = 5.6T

I .oo

L

0 00 ENERGY ( e V )

FIG. 23. Luminescence spectra of Cdo,9~Mno.~Te at 1 . 4 K observed at three values of the magnetic field. The low energy shoulder on the 5.6Tline appears at high fields ( B > 4T) and is not explained. [After Planel et al. (1980).] I 6700,

k., \,'"\-.. ."-a,\ I

I

I

I

XMn=

5%

0

X%

*@.

-2 1650 > c3

LL

a '.

I605

-

0

V

O

W Z W

I630

x-x-x-+x_x

0 . 0 ' 0 ' 0-0-0

I

I

I

I

I

-

7.

MAGNETOOPTICAL PROPERTIES

305

Now, the A' + X system is affected more strongly by that additional energy, since both its holes are free to adjust the direction of their spins to the magnetic field direction, the orbital parts of their wavefunctions being different. The two holes in A'X have a symmetrical orbital wavefunction, and the Pauli principle makes such a free adjustment impossible. In addition, the presence of an external magnetic field reduces magnetic polaron effects. Therefore, if the exchange effects are strong enough, at a certain field value the net binding energy will vanish and the bound state will disappear. Detailed studies of this phenomenon can supply information on excitonic wave functions, on the values of the exchange constant for bound states and, of course, on the magnetic polaron effects. At this moment no precise theoretical interpretation is as yet available. VII. Transitions Away from the Brillouin Zone Center 19. EXPERIMENTAL DATA

The only magnetooptical results off the Brillouin zone center in a large gap semimagnetic semiconductor are those obtained at the L point in Cdl-xMnxTe. Those transitions give rise to reflectivity structures at energies denoted commonly by E I = 3.5 eV and El + A1 = 4 eV. Figure 25 shows a relevant fragment of the reflectivity spectrum of Cdl -xMnxTe after ZimnalStarnawska et af. (1984). The first magnetoreflectivity measurements reported by Dudziak et af. (1982) have already shown that the Zeeman splitting of the El structure is much smaller than the splittings at the r point. In fact it was not possible to detect any Zeeman structure at the L point; there was only a small difference in the structure position when measured with the (T' or the (T- polarizations indicating that the splitting (if any) was much smaller than the structure width. To resolve it, a modulation technique has been applied. Figure 26 shows an experimental setup with an elasto-optical modulator used in a later work (Ginter et af., 1983, see also Coquillat et af., 1986). Results obtained on Cdl-,Mn,Te samples can be summarized as follows: (1) At low (pumped liquid He) temperatures, the structures El and

+ A1 exhibit small splittings of opposite sign. (2) The splitting of the El structure is of the same sign and is reduced in magnitude in comparison with the strongest splitting (between the two strong (T components) observed at the point by a factor of = 15 common for all compositions and temperatures. (3) The splitting of El + A1 (opposite in sign to that of El) is also reduced in magnitude (probably even more) as compared to the splitting at the r point. This result is only qualitative.

El

306

J. A. GAJ

00

0 05

0 12 0 2

03 04

......- ............. I

I

2.0

I

05

......................... I I I

4.0

-,.-

06

"'1

0.7

b.

I

6.0

8.0

Energy ( e V )

FIG 25. Reflectivity curves of Cdl-,MnxTe at room temperature. El and EI + At peaks are marked by arrows. [After Zimnal-Starnawska er af. (1984).]

There is clearly a need for systematic and precise magnetooptical measurements of these transitions in other materials and for a better theoretical model.

20. ATTEMPTS OF THEORETICAL DESCRIPTION It must be said that up to the present moment there exist no satisfactory explanation of the observed strong reduction of the splitting at the L point. Certain features of the observed effects have been explained by Ginter et al. (1983) who used a simple tight binding model to find a relationship between the exchange splittings and the selection rules at the L point, and those at the point. The energy bands in the model are developed from the atomic 5s

7.

L

MAGNETOOPTICAL. PROPERTIES

O

307

PM

FIG. 26. Experimental setup for modulated reflectivity measurements (Sc-source, L-lenses, Q-fused quartz plate used to eliminate the linear polarization of the light emitted by the source, PR-reflecting prism, S-sample, C-superconducting coil, PM-photoelastic modulator, P-polarizer, M-monochromator). [After Ginter ef al. (1983).]

states of Cd (or 4s states of Mn) and 5p states of Te. Besides a periodic potential of the (virtual) crystal, spin-orbit interaction and the ion-carrier exchange are taken into account. The calculation predicts a rather complicated structure due to the existence of four, generally non-equivalent, band extrema. In the experiment this structure cannot be resolved because of the breadth of the reflectivity peaks. An averaged splitting found in the calculation is reduced only four times compared to the splitting at the r point, in contrast to the experimental value of 16. The signs of the splitting obtained in the calculation agree with the experiment. It is clear that the presented model lacks some essential physical features to account for the experimental observation. However, the paper by Ginter eta/.(1983) points out a number of reasons that lead to the splitting reduction at the L point. These reasons, which should remain valid even in a new improved theory, are as follows: (1) A mixing of the s-like and the p-like states occurring in the conduction band produces an effective exchange integral equal to an average (possibly weighted) of a and p and, due to their opposite signs, leads to their partial cancellation. (2) The component of the magnetic field perpendicular to the k vector produces no splitting of the heavy hole states. This effect has already been found in a k p calculation (Gaj et al., 1978),valid near the rpoint. Combined with the necessity of averaging over the four equivalent band extrema, it leads to a reduction of the exchange splittings of the valence states involved.

-

308

J. A. GAI

(3) r he selection rules are much less strict at the L-point than they are at the center of the Brillouin zone.

Acknowledgments It is the author’s pleasure to thank Dr. J. Ginter for many valuable discussions, as well as Ms. M. Arciszewska, Drs. M. Nawrocki, and A. Twardowski for permission to reproduce their unpublished results.

References Aggarwal, R. L., Jasperson, S. N., Stankiewicz, J., Shapka, Y., Foner, S., Khazai, B., and Wold, A. (1983). Phys. Rev. B28, 6907. Altarelli, M., and Lipari, N. 0. (1973). Phys. Rev. B7, 3798. Arciszewska, M., and Nawrocki, M. (1982). In “Physics of Semiconducting Compounds” (1 1th Conf.), p. 225,. Pol. Acad. Sci., Warsaw, Poland. Arciszewska, M., and Nawrocki, M. (1984). Unpublished. Bastard, G., Rigaux, C., Guldner, Y., Mycielski, J., and Mycielski, A. (1978). J. Physique 39, 87. Bir, G. L., and Pikus, G. E. (1972). Simmetria i deformatsionnye effekty vpoluprovodnikakh, Nauka, Moscow, Sections 23 anll 24 (Symmetry and Strain-induced Effects in Semiconductors, Wiley, 1974). Cieplak, M. Z. (1980). Phys. Stat. Sol. (b) 97. Coquillat, D., Lascaray, J. P., Dejardins-Deruelle, M. C., Gaj, J. A., andTriboulet, R. (1986). Solid State Commun. 59, 25. Dudziak, E., Brzezinski, J., and Jedral, L. (1982). In “Physics of SemiconductingCompounds” (11 Conf.), p. 166. Pol. Acad. Sci., Warsaw, Poland. Elliott, R. I. (1957). Phys. Rev. 108, 1384. Gaj, J. A., Ginter, J., and Galazka, R. R. (1978). Phys. Stat. Sol. (b) 89, 655. Gaj, J. A., Byszewski, P., Cieplak, M. Z., Fishman, G., Galazka, R. R., Ginter, J., Nawrocki, M., NguyenTheKhoi, Planel, R., Ranvaud, R., and Twardowski, A. (1979a). In “Physics of Semiconductors” (14th Int. Conf.), p. 1113, Inst. Phys. Conf. No. 43. Gaj, J. A., Planel, R., and Fishman, G. (1979b). Solid State Commun. 29, 435. Gaj, J . A. (1981). In “Exchange Interaction of Paramagnetic Ions With Band Electrons in Cdl-,MnxTe,” Editions of University of Warsaw (in Polish). Gaj, J . A., and Golnik, A. (1985). Acta Phys. Polon. A67, 307. Ginter, J . , Gaj, J. A., and Le Si Dang (1983). Solid State Commun. 48, 849. Golnik, A., Ginter, J., and Gaj, J. A. (1983). J. Phys. C16, 6073. Gubarev, S. I. (1981). Zh. Eksp. Teor. Fiz. 80, 1174. (Sov. Phys. JETP53, 601.) Kane,. E. 0. (1957). J . Phys. Chem. Sol. 1 , 249. Kett, H., Gebhardt, W., Krey, U., and Furdyna, J. K. (1981). J. Magn. Magn. Muter. 25,215. Kierzek-Pecold, E., Szymanska, W., and Galazka, R. R. (1984). Solid State Commun. 50,658. Komarov, A. V., Ryabchenko, S. M., Terletskii, 0. V., Zheru, I. I., and Ivanchuk, R. D. (1977). Zh. Eksp. Teor. Fit. 73, 608. (Sov. Phys. JETP46, 318.) Komarov, A. V., Ryabchenko, S. M., and Vitrikhovskii, N. I. (1978), Pis’mz v Zh. Eksp. Teor. Fiz. 27. (JETP Letter 27, 413.) Komarov, A. V., Ryabchenko, S. M., Terletskii, 0. V., Ivanchuk, R. D., and Savitskii, A. V. (1980a). Fiz. Tekh. Popuprovod. 14, 17. (Sov. Phys. Semicond. 14, 9.)

7.

MAGNETOOPTICAL PROPERTIES

309

Komarov, A. V., Ryabchenko, S. M., and Terletskii, 0. V. (1980b). Phys. Stat. Sol. (b) 102, 603. Komarov, A. V., Ryabchenko, S. M., Semenov, Y. G., Shanina, B. D., and Vitrikhovskii, N. 1. (1980~).Zh. Eksp. Teor. Fiz. 79, 1554. (Sov. Phys. JETP52, 783.) Kossut, J. (1976). Ph.D. Thesis, Institute of Physics, Polish Academy of Sciences, Warsaw. Lindstrorn, M., Kuivalainen, P., Heleskivi, J., and Galazka, R. R. (1983). Physica 117B and 118B, 479. Malyavkin, A. V. (1983). Phys. Stat. Sol. (b) 115, 353. Nguyen The Khoi, Ginter, J., and Twardowski, A. (1983). Phys. State Sol. (b) 117, 67. Ortenberg, M. von (1984). Solid State Commun. 52, 111. Oseroff, S. B. (1982). Phys. Rev. B25, 6584. Planel, R., Gaj, J., and Benoit a la Guillaume, C. (1980). J. Physique 41, C5-39. Rebman, G., Rigaux, C., Bastard, G., Menant, M., Triboulet, R., and Giriat, W. (1983). Physica 117B and 118B, 452. Ryabchenko, S. M., Terletskii, 0. V., Mizetskaya, I. B., and Oleinik, G. S. (1981). Fiz. Tekh. Poluprovodn. 15, 2314. (Sov. Phys. Semicond. 15, 1345.) Twardowski, A. (1981a). Ph.D. Thesis, University of Warsaw. Twardowski, A. (1981b). Unpublished. Twardowski, A,, Nawrocki, M., and Ginter, J. (1979). Phys. Stat. Sol. (b) 96, 497. Twardowski, A., Rokka, E., and Gaj, J. A. (1980). Solid State Commun. 36, 927. Twardowski, A., Dietl, T., and Demianiuk, M. (1983). Solid State Commun. 48, 845. Twardowski, A., Swiderski, P., Ortenberg, M. von, and Pauthenet, R. (1984). Solid State Commun. 50, 509. Twardowski, A., and Ginter, J. (1982). Phys. Stat. Sol. (b) 110, 47. Zimnal-Starnawska, M., Podgorny, M., K i d , A., Giriat, W., Demianiuk, M., and Zmija, J. (1984). J. Phys. C17, 615.

SEMICONDUCTORS AND SEMIMETALS. VOL. 25

CHAPTER 8

Shallow Acceptors in DMS: Splitting, Boil-Off, Giant Negative Magnetoresistance J. Mycielskif INSTITUTE OF THEORETICAL PHYSICS, WARSAW UNIVERSITY WARSAW, POLAND

. .

.

I. INTRODUCTION . . . . . . , . . . . . . . . . . 1. Shallow and Resonant Acceptors in DMS in the Absence of Magnetic Field . . . . . . . . . . . . . . . , . 2. Effect of Exchange Interactionon Band Structure. . . . 11. THEORY OF SHALLOW ACCEPTORS IN THE ABSENCE OF MAGNETIC FIELD . . . .. . . . . ... .. . .. . . ... 3. Acceptor Hamiltonian . . . . . . . , . . . . . . . 4. Variational Procedure . . . . . . . . . . . . . 111. ACCEPTORS IN WIDE-GAP DMS IN A MAGNETIC FIELD.. . . . 5. Exchange Splitting. . . . . . . . . , . . . . 6. ExperimentalResults. . . . . . . . . . . IV. ACCEPTORS IN NARROWGAP DMS IN A MAGNETIC FIELD. . . . 7. Acceptor Hamiltonian in Ultra-Quantum Limit. . . . 8. Acceptor States in Ultra-QuantumLimit; Boil-Off Effect. . 9. Arbitrary Magnetic Fields. . . . . . . . . . . . . . 10. Experimental Binding Energies. .,.... . ., . IN p-TYPENARROW-GAP DMS V. IMPURITYMAGNETOCONDUCTIVITY 1 1 . Hopping Magnetoconductivityand Giant NegativeMagnetoResistance in Hgl-,Mn,Te . . . . . , . . . 12. Nonmetal-to-Metal Transition . . . . . . . . . . , . VI. RESONANT ACCEPTORS IN ZERO-GAP DMS . . . . . . . . 13. MagnetoopticalData. . . . . . . . . . . . . . . . 14. MagnetotransportData. . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. .

. . . . . .

.

.

. .

. . . .

311 312 313 3 16 316 318 322 323 324 325 325 321 329 330 333 333 331 338 339 341 342

I. Introduction

Probably the best known diluted magnetic semiconductors (DMS) that can be made p-type are Cdl-xMnxTe and Hgl-,MnxTe (see the review articles: Galazka, 1979; Galazka and Kossut, 1979; Gaj, 1980; Bastard et al., 1980; Mycielski, 1981; Dietl, 1981; Galazka, 1982; Furdyna, 1982a,b; Grynberg, i ( 1930- 1986)

311 Copyright 0 1988 by Academic Press. Inc. All rights of reproduction in any form IeSeNed. ISBN 0-12-752125-9

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J . MYCIELSKI

1983; Mycielski, 1983 and 1984). Both are A:'_,Mn,BV' systems of the zincblende cubic structure, and both have wide conduction and valence bands. However, Cd, -,Mn,Te is a wide-gap semiconductor, while Hgl -,MnxTe is a narrow-gap semiconductor (for x > 0.075), or a zero-gap semiconductor (for x c 0.075). For specificity, in this chapter we will only consider acceptors in the above materials. We limit our considerations to shallow .acceptors, and ignore more complex systems like excitons bound to acceptors (for excitons see: Gaj, this volume). Taking into account the effect of the exchange interaction of a hole with magnetic ions on the acceptor states, we will neglect the feedback: ordering of spins of magnetic ions by their interaction with the hole, i.e. the bound magnetic polaron effect (for magnetic polarons see: Wolff, this volume). 1. SHALLOW AND RESONANTACCEPTORS IN DMS IN THE

ABSENCE OF

A

MAGNETIC FIELD

In the absence of magnetic field the spins of magnetic ions are not ordered. Thus there is no net exchangeinteraction of these ions with a hole. The acceptor states in DMS are then similar to those in non-magnetic semiconductors. The theoretical ionization energy EA of a shallow acceptor in CdTe is 87.4meV (Baldereschi and Lipari, 1973), and should be the same in Cdl-,Mn,Te, as the parameters of the valence band (Twardowski and Ginter, 1982) and the static dielectric constant (Gebicki and Nazarewicz, 1980) were found to be independent of composition in this material. In a photoluminescence experiment values of 57.8, 58.8, 108, and 147 meV were found for EA in CdTe and were interpreted as corresponding to the substitutional impurities Li, Na, Ag, and Cu, respectively (Molva et al., 1982). In Cdl-,MnxTe the absorption due to acceptor photoionization was measured and analyzed by Wojtal et a/. (1979). The photoionization spectrum was fitted using a formula derived by the quantum defect method (Bebb, 1969). This gave a rather high quantum defect for the ground state, and ionization energies of 165 and 170meV for x of 0.05 and 0.10, respectively. Thus, the acceptor is not very shallow. In the narrow-gap Hgl -,Mn,Te, the theoretical ionization energy of a shallow acceptor increases withx, and is about 9 meV for x = 0.15 (Mycielski and Mycielski, 1980). From transport measurements the value 9.4 meV was obtained for x = 0.15 (Wojtowicz and Mycielski, 1984), while the photoconductivity experiment revealed-for the same value of x-a deeper acceptor, with the ionization energy about 30 meV (Wrobel et al., 1984). A second acceptor level-with an ionization energy about 25 meV-was also reported in Hgl-,Mn,Te with a wider gap (Anderson et al., 1983). Acceptors are believed to be due to mercury vacancies or foreign impurities.

8.

SHALLOW ACCEPTORS IN DILUTED MAGNETIC SEMICONDUCTORS

313

In the zero-gap Hgl-,Mn,Te resonant acceptor states do exist in the conduction band. Two levels were found in magnetooptical experiments, with the zero-field energies 0.9 meV-observed at very low x-and 2.2 meV-observed at x r 0.008 (Bastard et al., 1979). They were identified as Ao (impurity) and A1 (mercury vacancy) levels, respectively, found also in non-magnetic zero-gap Hgl-,CdxTe (Guldner et af., 1977). A level with the energy 1.5meV was also reported by Dobrowolska et al. (1980) and Dobrowolska and Dobrowolski (1981). From magnetotransport measurements at low fields and for x = 0.06 (i.e., close to the zero- to open-gap transition), the resonant level was observed at 4 meV or 7 meV (Sawicki et al., 1983 and Sawicki and Dietl, 1983). 2. EFFECT OF EXCHANGE INTERACTIONON BANDSTRUCTURE

In the absence of a magnetic field (i.e., also of a net exchange interaction), the DMS we consider have the band structure of non-magnetic semiconducting alloys such as Hgl-,Cd,Te. They have a direct band gap (increasing with x) at the r-point of the Brillouin zone. In the open-gap case, the symmetry of the conduction band and upper (light and heavy holes) and lower valence bands is given by the double Td point group representation r6, I‘s , and r7,respectively. In the zero-gap case (Hgl-,Mn,Te for low x), the positions of r6 and r8 levels are reversed, and the light-hole TS band of the open-gap structure becomes the conduction band. In any case, the r7 level is substantially split off from the r8 level due to a strong spin-orbit coupling. The basis for the rs level (in which we are primarily interested in this chapter) for the total angular momentum component ranging from to -9, are a1 = I+,$) = 2 - 1 / 2 ( ( x + i Y ) T ) , (1) @3

I$, $) = i6-”21(X + iY)1 - 2Zt), = It, -+) = 6-’”1(X - iY)T + 2ZJ.),

a 4

=

=

I+,

-+> = i 2 - 9 x - i Y ) l ) ,

(2) (3) (4)

where the position-dependent functions X, Y, Z transform under the operations of 7A point group as p-type functions, and have average square modulus equal to 1. We will now take into account the exchange interaction of the band electrons with the 3ds-electrons of magnetic Mn2+ ions. This interaction is of Heisenberg form (Kossut, 1976)

314

J. MYCIELSKI

where s and S; are the spin operators (in units of h ) of the band electron and of the i-th magnetic ion, respectively, and the summation is over all lattice sites occupied by magnetic ions. The exchange constant J is a short-range function of distance. Using the mean-field and virtual-crystal approximations we can write Eq. (5) in the form (Bastard et a/., 1978 and Jaczynski et al., 1978)

which has the periodicity of the crystal lattice, as the summation is now over all metal lattice sites. Here x is, as before, the molar fraction of the magnetic component, and ( S ) a v is the spin of the magnetic ion, averaged thermally and over all magnetic ions. Thus, ( S ) a v determines the macroscopic magnetization of DMS. In the absence of an external magnetic field (S), = 0. In the presence of a magnetic field ( S ) a v increases with decreasing temperature. In the following, we take the external magnetic field parallel to the z-direction. Thus, (S)avhas a non-vanishing z-component (Sz)av that tends to -5 for strong fields and low temperatures. The exchange Hamiltonian, Eq. (6), written in the basis given by Eqs. (1)-(4) has the form 3Bexch 0 0 Hexch

with

=

k

0

Bexch

0

-Bexch

0

Bexch

P

0

= ;XNoP(Sz)av

0 -3Bexch

,

j

(7)

(8)

(9) where NOis the number of unit cells in unit volume. It is worth noting that &xch may be modified to include the Luttinger parameter K of the valence band (Luttinger, 1956). It follows from Eq. (7) that the effect of exchange interaction results in splitting the I-" level into equally spaced singlets. This splitting dependsthrough (S,)av-On the external magnetic field and the temperature, and can approach values as large as 100 meV. = (ZIJIZ),

a. Wide-Gap DMS In a wide-gap DMS like Cdl-xMnxTe, because of the high effective masses of holes, the exchange splitting is much higher than the Landau and Zeeman splittings. It is, therefore, justified to perform the k p calculation using Eq.(7) while neglecting the direct effect of the external magnetic field on the

8. SHALLOW ACCEPTORS

IN DILUTED MAGNETIC SEMICONDUCTORS

315

spin splitting and the orbital motion of the valence electrons (Gaj eta/., 1978 and Gaj, this volume). One then obtains a dramatic dependence of the splitting of the valence bands on both the direction and the absolute value of the electron wave-vector k. Of particular interest for the following are dispersion relations close to the r point, i.e., for

where mh is the light-hole mass. Neglecting warping, the four non-degenerate valence bands are:

where mhh is the heavy-hole mass and k, is the wave-vector component perpendicular to the magnetic field. Thus, the constant energy surfaces are rotational ellipsoids, cucumber-like for the highest and the lowest bands (f3)and disc-like for the intermediate bands (fts).For mhh S- mm,the mass anisotropy is particularly strong for the bands -t+, being equal to 3mhh/4mm. Since > 0 for both DMS under consideration, i.e., &xch c 0, E-3/2(k) corresponds to the highest-lying valence band.

b. Narrow- and Zero-Gap DMS In a narrow- or zero-gap DMS, like Hgl-,Mn,Te, one cannot neglect the direct effect of the external magnetic field on the spin splitting and the orbital motion of the valence electron, as the low effective masses yield high cyclotron frequencies and high g-factors. The exchange splitting then has to be treated together with Landau and Zeeman splittings. This is done by adding the exchange interaction, Eq. (7)-written in a larger basis, including also r6 and r7functions-to the Pidgeon-Brown k p Hamiltonian (Pidgeon and Brown, 1966) of a zinc-blende type semiconductor in a magnetic field. Inversion asymmetry and warping are neglected in this model. The numerical solution of the modified Pidgeon-Brown Hamiltonian (Bastard et al., 1978; Jaczynski et al., 1978; Bastard et al., 1981) fits very well the observed positions of the Landau levels in the narrow- and zero-gap DMS. In what follows, one of these Landau levels- referred to as bv(- 1)-will be of particular interest. This level originates from Ts and its energy has an analytic form, &,(-I) = -3hUJ~3/2 + 3 f i W o h . - 3Bexch, (13)

-

316

J. MYCIELSKI

where coo is the free-electron cyclotron frequency and ~ 1 3 / 2is the cyclotron frequency corresponding to the transverse effective mass of the bands E,3/z(k) given by Eq. (1 1). The wave function of bu(-1) is a ground-state Landau envelope multiplied by @4. For the usual values of parameters (i.e., p > 0), bu(-l) is the highest valence band level in both the open- and zerogap DMS and is well-separated from other valence levels. It is the uppermost heavy-hole Landau subband. From Eq. (13) and from the form of the wave function, one can see that bu(-l) is the first Landau level originating from E-3/z(k) valence band. In spite of the fact that it corresponds to a heavy-hole Landau subband (in the z-direction), it is well-separated from other valence levels due to a low transverse effective mass of the E-3/2(k) valence band, and to the high value of &ch. It should be stressed, however, that other Landau levels cannot be calculated in a narrow- or zero-gap DMS from the simple expresssion, Eq. (1 1). Because of the -33Bexch term in Eq. (13), &,(-I) increases with the magnetic field (even for x = lop3) up to the saturation of (at a few T ) and only then starts to decrease slowly. In a zero-gap DMS at moderate fields, it may overlap in energy with the lowest conduction-band levels. 11. Theory of Shallow Acceptors in the Absence of a

Magnetic Field We will present here the theory of shallow acceptors in a diamond or zincblende-type open-gap semiconductor in the absence of an external magnetic field, following the variational method of Schechter (1962). Being less accurate than other methods (see Baldereschi and Lipari, 1973), it is, however, more analytic. We present here Schechter’s theory in a simplified and more explicit (while equivalent) version due to Mycielski and Rigaux (1983). 3, ACCEPTOR HAMILTONIAN

In the absence of magnetic field, a shallow acceptor has a four-fold degenerate ground state of Ts symmetry, with the wave functions of the form 4

v/; =

C mj4j;, j= 1

i = 1, ..., 4,

where 4j;(r)are slowly-varying envelope functions. The functions orthonormal:

(14) v/i

are

8.

SHALLOW ACCEPTORS IN DILUTED MAGNETIC SEMICONDUCTORS

317

We are using the valence band Hamiltonian as derived by Luttinger and Kohn (1955), adding to it the Coulomb term. The expectation value of the acceptor Hamiltonian (i.e., an eigenvalue if wi is an eigenstate) for the wave function of the form given by Eq. (14) (say, wl) is

where

E

is the dielectric constant of the crystal and Dii

Di2

Di3

D = ( Dt2

D22

0

D?3

0

0

DT3

-Diz Dii

D22

-D:2

in which the Di, are 3 x 3 matrices given below: - A - )B Dii

=

-A D22

0 -A - iB 0

+ )B

0 0

=

-ti(3B2

~

1

=3

i

+B

0

-A

=

+ )B 0

D12

-A

+ C2)'/'

- (31/2/2)B -443~' + 0

-A -B

-ti(3B2 + C2)'/2 -$(3B2 + C2)'/2 - i ( 3 B 2 + C2)'/2 0 0 0

- )i(3B2 + C2)"2 (31/2/2)B

0

")

.

(21)

0

Here A , B, and Care material constants (see Dresselhaus et al., 1955). Signs of the elements of D12 and D13 involving C 2 correspond to the case of the negative value of the constant N defined by Dresselhaus et al. (1955). If N > 0, the signs should be opposite. The physical meaning of A , B, and C can be seen from the dispersion law of electrons in the valence band: E(k)

=

Ak2 -+ [B2k4+ C*(k;k;

+ k,k: + k,"kx')]"2.

(22)

A and B are real but C may be imaginary (i.e., C 2may be negative). In order

318

J. MYCIELSKI

to have E(k) real and negative for all k f 0, the following conditions must be satisfied: C 2 L -3B2,

A < - [ B +~ f max(0, c ~ ) ] " ~ .

(23) (24)

If C = 0, there is no warping of the valence band. From Eq. (22) we then obtain

+ IBI),

(25)

= h2/2(IAI - 1B1).

(26)

m/i, = A2/2(IAI mhh

4. VARIATIONAL PROCEDURE

We assume now that the wave functions of the acceptor ground state are of the form (see Schechter, 1962 and Kaczmarek, 1966):

where CI, c2, c3, rl , and r2 are real parameters, and rl and r2 are positive. The functions defined by Eq. (27)-(30) fulfill the orthonormalization condition, Eq. (15), if 2 3 mlr 1 + 9 n c X + (:)ncfr? = 1. (31)

8. SHALLOW

ACCEPTORS IN DILUTED MAGNETIC SEMICONDUCTORS

319

As one can see from Eqs. (27)-(30), the envelope functions are composed from an s-type part (that proportional to CI) and two d-type parts (those proportional to cz and c3). The latter parts do mix different @ j in a given v / i . In order to determine the parameters CI, CZ, c3, r l , and r2, and the ionization energy EA, we have to maximize expression (16). From Eqs. (16)-(21)and Eq. (27) we get-after some algebra-E[yl] as a quadratic form of dimensionless quantities a1 , at, and a3, with coefficients depending on dimensionless PI and pz , where

ez

pz = -

EJAI

” *

The quantities d , a$,and a: are the norms of the s-type and the two d-type parts of the envelope, respectively (c.f., condition (31)). Dividing E [ ~ / by I ] d + a: + a: and maximizing the ratio with respect to a2/a1 and 0 3 / 0 1 , we obtain

where the dimensionless quantities f,F,and

r;l

are given by

320

J. MYCIELSKI

The sign of adal is that of B , and a3/al is positive. If the constant Ndefined by Dresselhaus et al. (1955) is positive, nothing is changed in our results except for adal being negative. It follows from Eq. (23) and Eq. (24) that (43)

O
In the absence of warping (i.e., for C = 0) we have from Eqs. (25), (26), and (42)

where =

(45)

mhh/mlh

From Eqs. (38) and (39) we then obtain (a3/a2)2 = 9 .

(46)

The inequality (24) imposes some limitations on the value of ( C/ B) 2 ,at a given q, namely, max(-3,2q - 5 ) < (c/B12 < (max[O,# q

- I)<$ - q)11-'.

(47)

In order to maximize E [ y / l ] given by Eq. (37), we have to maximize a given q. This can be done numerically for all values of q in the range defined by Eq. (43). The optimal values of PI and p2 and the maximal value o f f (all being now functions of q ) are shown in Fig. 1.

f(p1, p2; q) for

8.

FIG.1.

SHALLOW ACCEPTORS IN DILUTED MAGNETIC SEMICONDUCTORS

321

T

The maximum value of the functionfdefined by Eq. (40)and the corresponding values of the variables p1 and p2, for different values of the parameters q. [After Mycielski and Rigaux (1983).]

17 FIG.2. The function F(p1, pz; q)-giving the acceptor radii [Eqs. (38), (39), and (401-as a function of the material parameter q defined by Eq. (42). The values of p1 and p2 were chosen so as to maximize the acceptor energy, Eq. (37). [After Mycielski and Rigaux (1983).]

Inserting the optimal values of p1 and pz in Eq. (41), we obtain the function F(q) shown in Fig. 2. Substituting f = f(q) and F = F(q) in Eqs. (37)-(39), we obtain the acceptor binding energy EA and the norms of the two d-type parts of the envelope. To calculate EA for CdTe-and therefore also for Cdl-,Mn,Te with moderatex(see Gebicki and Nazarewicz, 1980; Twardowski and Ginter,

322

J . MYCIELSKI

1982)-we take the parameters values used by Baldereschi and Lipari (1973): & = 9.7,

] A1 = 5.29h2/2mo,

IBI = 3.78h2/2mo,

ICI

= 5.45h2/2rno,

where mo is the free-electron mass. From Eq. (42), Fig. 1, and Eq. (37), we then obtain the acceptor ionization energy, EA = 79 meV.

Baldereschi and Lipari (1973) obtained EA = 87.4meV by their more accurate method. For Hgl-,Mn,Te with x = 0.085, 0.11, 0.15, we have, respectively, the energy gap of about 40, 140, 300 meV, and (c.f. McKnight et al., 1978 and Bastard et af., 1981) E

= 15,

( A (= (136.3,41.1, 20.8)h2/2mo,

IB( = (133.4,38.2, 17.9)h2/2mo,

c = 0. This yields

EA = (3.8,6.9, 8.9) meV for the three compositions considered. It should be noted that the acceptor theory presented above is valid only for EA much lower than the energy gap T6-T8. However, this condition is satisfied even for Hgl-,MnxTe with x as low as 0.085. 111. Acceptors in Wide-Gap DMS in a Magnetic Field In the wide-gap DMS (like Cdl-,Mn,Te), because of the high effective masses of holes, the acceptor binding energy (in the absence of the exchange interaction) is much higher than the exchange splitting 61BexfhIof the l-8 level. This exchange splitting is, in turn, much higher than the Landau splitting. It is, therefore, justified to treat the exchange interaction as a perturbation in the acceptor problem, neglecting the direct effect of the magnetic field on the orbital motion (i.e., on the envelope functions). This was done by Mycielski and Rigaux (1983). In the following, we present the results of this approach.

8. SHALLOW

ACCEPTORS IN DILUTED MAGNETIC SEMICONDUCTORS

323

5. EXCHANGE SPLITTING We will write now the exchange Hamiltonian (7) in the basis of the acceptor ground state functions, Eq. (14) and Eqs. (27)-(30). Using also Eq. (33) and Eq. (34), we obtain after some algebra,

k

01Bexch

Hexch

where

=

0 0 -02Bexch

0 0 2 Bexch

0 0

0

0

’

(48)

- 01Bexch

0 1

= 3 - 2at - +a$,

(49)

02

= 1 - 2d.

(50)

It follows from Eq. (48) that the functions (14) and (27)-(30) are eigenfunctions also in the presence of exchange. However, the fourfold degenerate ground state of the acceptor is split by exchange. Comparing Eq. (7) with Eqs. (48)-(50), one can see that the acceptor splitting is smaller than the splitting of the free-carrier Tg state, and that the split levels are not equidistant. This quenching of the acceptor splitting results from the fact that the acceptor wave function is composed of different functions of the Tg level (with d-type envelopes). Substituting Eqs. (38) and (39) into Eqs. (49) and (50), we obtain 01 and 02 as the functions of the material parameters and (C/B)2: 01=30 2 = 1 -

12

+ (8/3)(C/B)2 F(V) 5 + (C/B)’ 9

5

+

4 F(r;l). (C/B)’

These functions are shown in the form of a map in Fig. 3. The region of points representing all possible materials is limited by Eq. (47). Comparing again Eq. (7) with Eq. (48), we can see that the acceptor ionization energy in the presence of exchange is reduced, and equals EAexch =

EA - (3

- 01)IBexchI.

(53)

Using the material parameters given in Section 4, we get from Fig. 3 for Cdl-*Mn,Te (for moderate x ) 01

= 2.25,

02

= 0.83,

324

J . MYCIELSKI

I

0

I

I .o

0.5

I

I

I5

2.0

1

2.5

7 FIG.3. Acceptor splitting as a function of the material parameters. Bold lines: the boundaries defined by Eq. (47) of the region of points representing all possible materials, C = 0 line corresponding to the absence of warping. Thin continuous lines correspond to the various values of 6 1 and broken lines to the various values of 02 as marked. Open circle is for Cdl-,Mn,Te, and the cross is for Hgo.sfMno.l5Te. [After Mycielski and Rigaux (1983).]

and for Hgo.~~Mno.~sTe CT~

= 2.26,

02

=

0.75.

It should be noted that the latter material is a narrow-gap DMS and, therefore, the rigorous validity of the present acceptor splitting theory is doubtful in this case even for low magnetic fields. The matrix elements of the optical transitions from the split acceptor ground state to the r6 conduction-band Landau levels may be of interest in the context of interband magnetoabsorption. They were calculated by Mycielski and Rigaux (1983) and are independent of exchange, since Hexch does not mix the acceptor ground state wave functions, Eq. (14), Eqs. (27)-(30). 6 . EXPERIMENTAL RESULTS For Cdl-xMn,Te, the quenching (by about 20%) of the acceptor ground state exchange splitting given by the theory (see Section 5) was confirmed only indirectly by Jaroszynski et al. (1983). They observed an increase of the activation energy of extrinsic conductivity with increasing x (in the absence of magnetic field) in p-type conductivity measurements at different temperatures. As we have already mentioned (see Section l ) , there should be no such

8. SHALLOW

ACCEPTORS IN DILUTED MAGNETIC SEMICONDUCTORS

325

dependence for the acceptor binding energy in Cdl-,Mn,Te in the absence of exchange. Thus, the effect was ascribed to the formation of bound magnetic polaron and magnetization fluctuations at higher values of x. Assuming 20% quenching of the acceptor ground state splitting, calculations taking account of the polaron and fluctuation effects were found to be in a reasonable agreement with the experimental data (Jaroszynski et al., 1983).

IV. Acceptors in Narrow-Gap DMS in a Magnetic Field In narrow-gap DMS (like open-gap Hgl-,Mn,Te with low x), because of the low effective masses of holes the exchange splitting is comparable with Landau splitting. Thus, the theory presented in Part 111 can hardly be applied in this case. This very involved acceptor problem can often be simplified, however, due to the fact that at already moderate magnetic fields (of a few T ) ,the Landau and exchange splittings are higher than the acceptor binding energy EAat zero field, as the latter is low for small effective masses of holes. We consider this situation below. 7. ACCEPTOR HAMILTONIAN IN ULTRA-QUANTUM LIMIT

In Section 2, we have discussed the highest valence subband in a narrowgap DMS in the presence of an external magnetic field: that is, a heavy-hole (in z-direction) band originating from the level b u ( - l ) ,of energy given by Eq. (13). It is separated from lower subbands roughly by h 0 1 3 / 2 or 21B,,,1,l. If 2IBexch1, h013/2 B EA,

(54)

the acceptor wave functions may be constructed from the wave functions of this single subband (“ultra-quantum limit”), at least for the acceptor states of energies above the bv(-l) level (Mycielski and Mycielski, 1980). In the symmetric gauge A = )Bz(- y , x , 0 ) (where Bz is the magnetic induction), an orthonormal electron wave function of the subband bu(-1) has the form (see, e.g., Gawron and Mycielski, 1984): ymkZ =

I

’’’ e x p ( i k Z z ) ~ , ( r ~ 4 ,

LT

Xm(r) = [ ( 2 1 ~ ) ’ / ~ L ]I m - ’ ( !)-’I2 exp(imr$)(p/21/2L)1m1 exp( -p2/4L2),

(55)

(56)

where p and r$ are cylindrical coordinates; 02m>

-00;

(57)

= (tic/eB,)’/’ is the magnetic length (e > 0); L, is the dimension of the periodicity box in the z-direction; kz = 2nKz/Lz;and Kzis an integer. The

L

326

J. MYCIELSKI

electron energy in the state given by Eq. ( 5 5 ) is

The problem of constructing an impurity state from the states defined by Eqs. (55)-(58) of a single subband was considered by many authors. It seems that the simplest method consists of deriving some effective one-dimensional wave equations describing the impurity states (see, e.g., Elliott and Loudon, 1960; von Ortenberg, 1973; Mycielski and Mycielski, 1980; Gupta et al., 1982; and Gawron and Mycielski, 1984). One starts with the acceptor (i.e., a valence electron interacting wih a negative ion) eigen-equation H+

eL

Y = EY,

(59)

where H is the valence electron Hamiltonian in the presence of a magnetic field and of exchange, i.e., the Hamiltonian with the eigenstates (55)-(58). The impurity wave function is taken in the form

Inserting this into Eq. (59), multiplying by P $ ' k i , integrating over r, multiplying by exp(ikiz), and summing over k:, we show that a given acceptor is composed of y m k , with a given m. The normalized acceptor wave function has the form yms = Tms(~)~m(r)@4,

(61)

where the normalized envelope function Tms(z) and the acceptor state energy fulfill the following effective one-dimensional wave equation :

Ems

<

Here = 2/2'/'~, 6 = A / a h h , and a h h = h 2 & / e Z m h h (the effective Bohr radius for heavy holes), Ems = 2[Eb,,(-1) - E m s ] / h O h h , O h h = e&/Cmhh (the cyclotron frequency for heavy holes), and

where M ( z , b, x ) is the confluent hypergeometric series (for special functions,

8.

SHALLOW ACCEPTORS IN DILUTED MAGNETIC SEMICONDUCTORS

327

we use the notation of Abramowitz and Stegun, 1965). It is worth noting that 6 is the ratio of the magnetic length to the heavy-hole Bohr radius. For m = 0, one obtains

h(It0 = ( 2 W 2 exp(t2)[1 - erf(ltl)l,

(64)

where erf(x) is the error function. It follows from Eq. (63) that Vm(l 0, that V(l(l)decreases with increasing and that, for JtI+ 1 , V m ( J ( 1 ) 2 2”’Itl. As V(ltl) is symmetric, the bound eigenstate of Eq. (62) is either symmetric or antisymmetric. Equation (62) with Vm given by Eq. (63) may be rather easily solved numerically. However, it is useful to obtain an analytic expansion of Eq. (63) for small I
h(ltl)= (27W2[1 - (2/771/z)ltll,

(65)

Thus, Vm(1tI) decreases with increasing ImJand tends to 0 for ImJ 00. The selection rules for optical transitions between the functions defined by Eq. (61) were discussed by Gawron and Mycielski (1984). -+

8. ACCEPTOR STATESIN

THE

THE BOIL-OFFEFFECT

ULTRA-QUANTUM LIMIT;

For an isotropic effective mass in a non-magnetic semiconductor the quantum limit regime-i.e., Landau level spacing larger than the impurity binding energy-implies 6 < 1. If, however, the effective mass of the carrier in the direction perpendicular to the magnetic field is much smaller than that in the parallel direction, it is possible-for not too strong a magnetic fieldto be in the quantum limit, and yet to have 6 1. This follows from the fact that the Landau levels spacing is determined by the transverse effective mass, not by the longitudinal one involved in 6. The quantum-limit regime with 6 %- 1 was already discussed for nonmagnetic materials with strong effective mass anisotropy, e.g., Bil-,Sb, (Akimoto and Hasegawa, 1967; Beneslavskii et a/., 1974; Beneslavskii and Entralgo, 1975). Acceptors in DMS that we consider here provide another example of this situation (Mycielski and Mycielski, 1980). In Section 2, we have discussed the very strong exchange-induced anistropy of the valence band [see Eq. (1 l)] and, consequently, a strong separation of the Eb,(- level from other Landau levels. We will show in the following that the decrease of the transverse mass leads to a decrease of the acceptor binding energy in

+

328

J . MYCIELSKI

the presence of an external magnetic field, i.e., the so-called “boil-off effect”. It follows from Eqs. (62), (65), and (66) that if

(67)

d B 1,

then the envelope function Tms vanishes, except when ( 1 4 1, at least for the ground state of Eq. (62) and not too high a value of 1m 1. One can then replace Vmin Eq. (62) by its expansions: Eq. (65) (form = 0) or Eq. (66) (for m < 0). In the following we consider both these cases.

a. States with m

=

0.

Substituting Eq. (65) into Eq. (62), we obtain an equation of the Airy type. Its eigenfunctions (vanishing for both 03 and ( - 00, and continuous with the first derivative at = 0) are of the form

<-+ +

<

Tos(()

= (sgn

+

Ai(25/661/31(l+ xs)Cs,

(68)

and the eigenvalues are EOs

= - ~ ~ ) 1 / 2 6 (+ 1

21/6n-1/2d-1/3 Xs) .

(69)

Here Ai(x) is the Airy function, C, is a normalization factor, and xs are zeros of the Airy function (odd s) or of its derivative (even s): xs 3 - 1.02, - 2.34, -3.25, -4.09, -4.82, -5.52, ... for s = 0, 1,2, 3,4, 5, .... The envelope function To, is symmetric for even s and antisymmetric for odd s. It follows from Eq. (69) that the ground state is the symmetric state s = 0. Equation (69) can also be written in the form’

whereEAh = e4mhh/2h2&2 is the shallow acceptor binding energy (for BZ = 0) calculated for the heavy-hole (i.e., longitudinal) effective mass. Because of Eq. (67), the acceptor binding energy in the ultra-quantum limit is much lower than EAh (which is not much higher than acceptor ionization energy EA when the energy gap is not too low), EA(&)

=

Eoo - &,,(-I)

Q EAh.

(7 1)

We have thus obtained the boil-off effect, the opposite of freeze-out observed in non-magnetic semiconductors. The transverse radius of the acceptor, which is now 2A, is larger (see Eq. (67)) than the acceptor radius in the absence of the magnetic field (= a h ) , and the wave function is strongly In the corresponding equation for s = 0 in the paper by Mycielski and Mycielski (1980) the = -0.65 was mistakenly replaced by -2.73. coefficient 21/6n-1/2xo

8.

SHALLOW ACCEPTORS IN DILUTED MAGNETIC SEMICONDUCTORS

329

anisotropic. It should be noted, however, that the decrease of ionization energy and the increase of the transverse radius occur before reaching the ultra-quantum limit as defined by Eq. (54). In the ultra-quantum limit, the ionization energy increases and the transverse radius decreases with increasing magnetic field, as can be seen from Eq. (70). It follows from Eqs. (54) and (70) that En@,) is higher than the acceptor binding energy calculated for the transverse effective mass in Eq. (11). It is also worth noting that EA(Bz)depends only weakly on mhh, since EAh/d is mass-independent

.

6. States with m < 0 Substituting Eq. (66) into Eq. (62), we obtain an equation of the harmonicoscillator type. Its eigenfunctions and eigenvalues are

where s = 0, 1,2, ..., Hs(x) is the Hermite polynomial, and C m , is a normalization factor. Tms is symmetric for even s and antisymmetric for odd s. It follows from Eqs. (69) and (73) that Em0 - Eb,(-1) is at least about two times smaller than the binding energy Em - Eb,(-l). 9. ARBITRARY MAGNETIC FIELDS

The problem of calculating the acceptor states in a narrow- or medium-gap DMS in moderate magnetic fields ( 5 1 T ) is very involved, because the condition (54) is not fulfilled and 21Bexchl, A u 1 3 / 2 , and EA may all be comparable. This problem was considered by Gawron and Trylski (1982) and Gawron (1986a). They have included the exchange interaction into the effective mass Hamiltonian for the valence band in magnetic field, and used the spherical tensor operators and reduced matrix elements technique (see Baldereschi and Lipari, 1973 and Lipari and Altarelli, 1980). In the variational procedure the envelope functions were assumed to be sums of products of spherical harmonics and exponentials of distance. Their

330

J. MYCIELSKI

numerical results show the development of four levels originating from the four-fold degenerate acceptor ground state. 10. EXPERIMENTAL BINDINGENERGIES One can expect that the strange behavior of the acceptor in narrow-gap DMS in a magnetic field should affect substantially the low-temperature magnetoresistance of p-type samples. In fact, a strong drop of the resistivity of Hgl -,Mn,Te in magnetic field was observed at low temperatures by Delves (1966) and Morrissy (1971). In the extrinsic range a very strong negative dc magnetoresistance was observed in several p-type open-gap Hgl -,Mn,Te samples with a net acceptor concentration of the order of 10’6cm-3 by Mycielski and Mycielski (1980) for 0.12 c x c 0.17 and Wojtowicz and Mycielski (1983) for 0.081 Ix I0.11. Anderson et at. (1983) observed a strong negative ac magnetoresistance and a decrease of the ac Hall coefficient of p-Hgl-,Mn,Te with 0.12 < x c 0.20 in the extrinsic range. In the extrinsic p-type conductivity range the negative magnetoresistance and the decrease of the Hall coefficient are due to a decrease of the acceptor ionization energy with increasing magnetic field (the boil-off effect). Such a decrease was demonstrated by the dc resistivity p vs. inverse temperature and the ac Hall-coefficient vs. inverse temperature plots (Anderson et al., 1983). It should be stressed, however, that the slope of Inp vs. l / T a t a given magnetic field does not determine quantitatively the value of the acceptor ionization energy at this field, since the ionization energy itself depends on (at non-zero magnetic field). This was taken into the temperature via (S,), account recently by Wojtowicz and Mycielski (1984). They have measured the transverse magnetoresistance and the Hall effect of p-type Hgl -,Mn,Te in the composition range 0.085 Ix 5 0.15 and for the acceptor concentrations from 6 x 1015cm-3 to 3 x 10” ~ m - in~ the , magnetic field B, up to 7Tand in a broad range of temperatures. In analyzing the data they assumed that the acceptor ionization energy EA depends on Bz and T only through (S,), , thus neglecting the direct effect of the magnetic field on EA. Let us denote by B,(Mz, T ) the magnetic induction at which (S,), (i.e., also the magnetization M,(B,, T)of the sample) has a given value at temperature T: Mz[Bz(Mz

3

n,TI = Mz

*

(74)

One can then determine EA(M,) from the formula

if one knows from experiment the conductivity tensor component axx(Bz,T) and the magnetization Mz(Bz,T ) as functions of B, and T. Using the

8. SHALLOW

ACCEPTORS IN DILUTED MAGNETIC SEMICONDUCTORS

331

experimental data for Mz(Bz,T) in Hgl-,Mn,Te (Dobrowolski et al., 1982), Wojtowicz and Mycielski (1984) calculated EA(M,) from the measured aXx(BZ, T ) using Eqs. (74) and (75). Their results are presented in Fig. 4 for three samples with different compositions x and different acceptor concentrations. For each sample the Bz(Mz,T)-scale is also given (for T = 15 K). Lower slope of the experimental curves at low MZmay be due to surface effects. The values of EA for Mz = 0 are in reasonable agreement with the values calculated at the end of Section 4. However, EAfor samples 3 and 5 does not tend to EA(B,)of sample 1, as it roughly should in the ultra-quantum limit according to Eq. (70). The sharp decrease of EA for these two cases is probably due to the acceptoracceptor interaction (becoming stronger if the wave functions expand), since the acceptor concentrations are relatively high in these samples. II

'

x.0.15

10.

No I

I

2

0

0.5

I .o

I .5

M, [emu41 FIG.4. Acceptor ionization energy in Hgl-xMnxTe as a function of magnetization. For each composition x, the magnetic induction corresponding to a given value of magnetization at 15 K is also given. Full circles are the experimental results of Wojtowicz and Mycielski (1984). The acceptor concentration of the samples 1, 3, and 5 are 6 x lof5,4 x and 9 x 1016cm-3, respectively. Open circles correspond to Eq. (37), broken line to Eq. (53), and dotted line to Eq. (70).

332

J . MYCIELSKI

For sample 1 , the experimental data in Fig. 4 are compared with the dependence of EA on M , given by Eq. (53), with GI calculated at the end of Section 5 and Bexchobtained from Eq. (8) with NO@= 0.65 eV (see Kossut, this volume, and Rigaux, this volume). From Eqs. (70) and (26), and the material parameters given at the end of Section 4, we have calculated EA in the ultra-quantum limit. These values of EAare shown in Fig. 4 as a function of magnetic induction B, for sample 1 . The agreement of both low- and highfield theory with experiment is reasonable, in spite of the fact that the applicability of Eq. (53) to Hgl-,Mn,Te is questionable (see Section 5 ) and the first part of the ultra-quantum limit condition (54) is not fulfilled: i.e., at a few Tesla the quantity 2/&xchl is slightly lower than EA at zero field. The second part of Eq. (54) and the condition (67) for the validity of Eq. (70) are fulfilled: for several Tesla Aco13/2 is about twice as high as EA in zero field, and 6 is about 5 . It should also be noted that the method of determining EA based on Eq. (75) is not well-justified for strong magnetic fields, which influence EA not only through Mz but also directly (i.e., through orbital magnetic quantization). It is interesting that the decrease of EAwith increasing magnetization (i.e., with decreasing temperature at a given magnetic field) may be so strong for samples with low x and high acceptor concentration that the concentration of free holes will increase in the extrinsic range with decreasing temperature. This effect was observed by Wojtowicz and Mycielski (1984) in sample 5 , both in the conductivity and the Hall constant measurements. Some information on the acceptor states in narrow-gap Hgl -,Mn,Te were obtained from the magnetotransmission measurements of p-type samples at liquid helium temperatures (Rigaux et a/., 1980 and Bastard et a/., 1981). A line identified as the transition from an acceptor level to the b,(l) Landau state in the conduction band was observed. The acceptor level was found to decrease with respect to the top of the valence band at B, = 0 as the magnetic field increased (see Fig. 5 ) . The transmission measurement in the far infrared (at B, = 0) has shown an absorption line at about 5 meV, which may correspond to the transition from the valence band to this acceptor level (Rigaux eC al., 1980). At 4T, this level is about 12meV below the bu(-l) Landau level. Thus, it is not the shallow acceptor ground state, but probably one of the excited states. The measurements of impurity photoconductivity of p-type Hgl -,Mn,Te with x = 0.15 at the temperature 4.2K and in magnetic field up to 8.3T revealed a deeper acceptor, with a zero-field ionization energy of about 30 meV (Wrobel et al., 1984). The ionization energy decreases strongly in magnetic field, and at 4Tis only about 17 meV. At higher fields the ionization energy starts to increase with field. The strong decrease of the ionization energy in magnetic field, much stronger than for shallow acceptors, may be

8.

SHALLOW ACCEPTORS IN DILUTED MAGNETIC SEMICONDUCTORS

0

333

20

40 B,(kG) FIG. 5 . Magnetic field dependence of the acceptor level observed in Hgl-,MnxTe with x = 0.128 (corresponding to an energy gap of 219 rneV). Zero energy is taken to be at the top of the valence band at B, = 0. [After Bastard et al. (1981).]

interpreted as resulting from a much lower exchange splitting of the acceptor ground state, i.e., much lower 01 (see Eq. (53)). It is natural to expect a lower value of 01 for deeper acceptors, since in that case the mixing of different functions of the r8level in the acceptor wave function should be stronger (see discussion in Section 5 ) .

V. Impurity Magnetoconductivity in p-Type Narrow-Gap DMS The strange behavior of the acceptor ground state in a narrow-gap DMS in the presence of a magnetic field affects the conductivity not only in the extrinsic, but also in the hopping or impurity-band range of temperatures. A strong drop of the resistivity of Hgl-,MnxTe in magnetic field at very low temperatures was observed already by Delves (1966) and Morrissy (1971).

11. HOPPINGMAGNETOCONDUCTIVITY AND THE GIANT NEGATIVE MAGNETORESISTANCE IN Hg, -,Mn,Te

It can be expected that the increase of the transverse radius of the acceptor wave function in magnetic field should lead to a strong increase of the p-type hopping conductivity in narrow-gap DMS and, furthermore, that the conductivity should become anisotropic due to the magnetic-field-induced anisotropy of the overlap integral of neighboring acceptors. This was observed and interpreted for narrow-gap Hgl-,Mn,Te (0.12 < x < 0.17) by Mycielski and Mycielski (1980). The samples were p-type and partially compensated, with net acceptor concentration of the order of 10l6~ m - The ~ . liquid-helium-temperature dc resistivity at 7 T was typically two orders of magnitude lower than for zero magnetic field, and the transverse resistivity

334

J. MYCIELSKI

in magnetic field was lower than the longitudinal. It was also observed that the activation energy for dc hopping conductivity decreases strongly with magnetic field. Anderson et al. (1983) observed a more than one order of magnitude decrease of the ac hopping resistivity of p-Hgl -,Mn,Te (x = 0.19) at liquid helium temperatures in a magnetic field of 2.6T, and a drop of the ac hopping activation energy. The giant negative dc hopping magnetoresitance of Hgl-,MnXTe was measured and analyzed in more detail by Wojtowicz and Mycielski (1983). The samples used were narrow-gap (0.081 Ix I0.11) partially compensated p-type, with acceptor concentration of the order 10’6cm-3. The transverse and longitudinal magnetoresistance was measured at temperatures down to 1.4 K and magnetic fields up to 7T. The transverse resistivity vs. reciprocal temperature for one of the samples is shown in Fig. 6 for different magnetic fields. The hopping conductivity dominates below about 7 K. At 1.4 K, as B, increases to 7T, the resistivity drops by nearly seven orders of magnitude! Both transverse and longitudinal resistivities of another sample are shown as functions of magnetic field in Fig. 7, for different temperatures. One can observe the anisotropy of the resistivity in magnetic field (the transverse resistivity being smaller) and, ultimately an increase of resistivity in very strong magnetic fields. The latter effect is expected when the true ultra-quantum limit is finally reached, as the acceptor radius decreases slowly in this limit with increasing magnetic field (see discussion at the end of Section 8a). In the hopping regime, the resistivity may be expressed in the form (Miller and Abrahams, 1960) P = ~3 e x p ( - ~ d k ~ T ) ,

(76)

where p3 depends on the overlaps of the wave functions of neighboring acceptors, and the activation energy ~3 is due to the potential differences between different acceptor sites. Wojtowicz and Mycielski (1983) assumed that both p3 and e3 depend only on magnetization Mz (although this is not fully justified for a high field, where the field can also influence the wave function directly), and derived them from the observed p(B,, T )and the data on M,(B,, T)of Dobrowolski et al. (1982), using the procedure described (in connection with EA)in Section 10. We present their results in Table I for the sample in Fig. 6. The magnetic induction corresponding to each value of magnetization at 4.2 K is also given. The large decrease of p3 with increasing magnetization is due to the increasing transverse radius of the acceptor, i.e., strongly increasing overlaps. It is worth noting that for x = 0.11, the zero-field radius rl of the (dominating) s-type part of the acceptor envelope, and the radius r2 of the d-type part are, respectively, 54 A and 45 A (see Eqs. (27)-(30), (3 9 , (36),

8.

SHALLOW ACCEPTORS IN DILUTED MAGNETIC SEMICONDUCTORS

335

10’

loc

10’

lo4

-t6

10:

c

Y

Q

lo2 10‘

Ioo

lo-;

7

FIG.6 . Transverse resistivity as a function of temperature and magnetic field for a sample of p-type Hgl-,MnxTe withx = 0.11. The hopping resistivity region corresponds approximately to T - l > 0.1 (above the “knee” of the data).

(42), Fig. 1 , and the material parameters at the end of Section 4). On the other hand, the transverse radius of the acceptor wave function in the ultraquantum limit, i.e., 2A, is 98 A at 6.8T The decrease of e3 in the magnetic field may be caused by the approach of the nonmetal-metal transition (see Section 12) or may correspond to the destroying action of the magnetic field on the bound magnetic polaron and on fluctuations of the magnetization (see Wolff, this volume). The latter

336

J . MYCIELSKI

field and 'wicz and

interpretation of the decrease of ~3 in the magnetic field was proposed by Diet1 e t a / . (1983) for an n-type DMS (wide-gap Cdl-,Mn,Se). The polaron effect is additionally reduced due to the increase of the acceptor radius. The decrease of ~3 with increasing acceptor radius may also be due to the increasing role of the variable-range hopping (see Mott, 1968) and/or the correlated (multi-hole) hopping (see Pollak, 1980). In an alloy, a part of ~3 is given by fluctuations of the composition x in space (see Gelmont et a/., 1974 and Shlimak eta/., 1977, for non-magnetic semiconductors, and Mycielski eta/.,

8.

SHALLOW ACCEPTORS IN DILUTED MAGNETIC SEMICONDUCTORS

337

TABLE I TRANSVERSE HOPPINGRESISTIVITY PARAMETERS FROM THE SAMPLE OF p-Hg,-,MnxTe IN FIG. 6 (x = 0.1 1) AS FUNCTIONS OF MAGNETIZATION.“ M Z

Iemukl 0 0.5 1 .o 1.5 2.2

BAM,, 4.2 K) IT1 0 1.1

2.4 4.0 6.8

P3

E3

[Qcm]

[meVl

175 112 32 7.8 1.8

1.45 1.1 0.7 0.4 0.2

After Wojtowicz and Mycielski (1983).

1984, for DMS). If the radius of the wave function increases, 63 will decrease as the fluctuations are averaged over a bigger volume. Solving Eq. (62) [with Eq. (64)] numerically, computing the overlap integral, and using percolation theory, Gawron (1986b) has calculated recently the (positive) magnetoresistance in the ultra-quantum limit, as well as the anistropy of the resistivity. The anisotropy is of the order of 2 at 4T, and drops to less than unity at about 10T. 12. NONMETAL-METAL TRANSITION The increase of the overlap of the neighboring acceptor wave functions with increasing magnetic field may induce the Mott nonmetal-to-metal transition. This transition corresponds, in principle, to the vanishing of the energy gap between the band of localized acceptor ground states and the empty band of mostly delocalized positive acceptor states (i.e., acceptors with two holes; see Mott, 1974). Before the transition occurs, two conductivity mechanisms are possible: excitation of holes from the localized ground states to the positive acceptor band (more precisely, to the states above the mobility edge of this band), and the hopping between the localized ground states themselves. The conductivity activation energy of the former mechanism is denoted by c t , and of the latter by 6 3 . As 8 3 < 8 2 , the hopping conductivity dominates at lower temperatures. For different values of 6 2 the conductivity extrapolated to T = 00 is roughly the same and is given by amin =

0.026e2N,“3/tr,

(77)

i.e., the “minimum metallic conductivity,” NJ’3 being the critical acceptor concentration at which the Mott transition occurs, i.e., at which the conductivity activation energy vanishes.

338

J. MYCIELSKI

The Mott nonmetal-to-metal transition occurs at a given ratio of the mean distance between the randomly distributed acceptors to the radius of their wave function: (78) where aA is an average acceptor radius. In non-magnetic semiconductors one approaches this transition by increasing the acceptor concentration. In narrow-gap p-type DMS, one can also bring about this transition by applying a magnetic field, thus increasing the acceptor radius. The magnetic-field-induced nonmetal-to-metal transition was observed by Wojtowicz and Mycielski (1983) in a partially compensated p-Hgl-xMn,Te sample with a net acceptor concentration 8 x 10'6cm-3. The transverse and longitudinal resistivities vs. reciprocal temperature for this sample are shown in Fig. 8 for different magnetic fields. The conductivity activation energy vanishes at about 2T. The conductivities for lower fields, extrapolated to T = 03, are roughly the same. Thus, the activation energy is interpreted as ~ 2 Measurements . at lower temperatures and about 2T yield o l m i n = 0.9 SZ-' cm-' while from Eq. (77) one gets omin = 2.7 0-' cm-', in order-of-magnitude agreement. From Eq. (78), we obtain ffA = 50A,which is also reasonable, although somewhat low. It is worth observing that at the nonmetal-to-metal transition the activation energy in the extrinsic range (i.e., EA > 0) still exists. Only at higher fields does it disappear, as the impurity band merges into the valence band. In the ultra-quantum limit, the acceptor radius will eventually begin to decrease with increasing magnetic field. Thus, one can expect a reverse metalto-nonmetal transition (similar to that known for non-magnetic semiconductors) to occur at very high magnetic fields. N C l 3 a ~ 0.2,

VI. Resonant Acceptors in Zero-Gap DMS

In a zero-gap semiconductor, donor and acceptor states are degenerate with the valence and conduction bands, respectively. Thus, they may exist only as resonant states, not bound states. The width of a resonant state depends on the density of states of the band with which it overlaps, and thus may be very large when the effective mass is large. This happens to resonant donors in A"BV' zero-gap semiconductors-they do not exist as quasidiscrete states. However, the density of states in the conduction band is low, and thus resonant acceptor states are observed. For non-magnetic semiconductors a great deal of experimental and theoretical work was doneon the resonant acceptors (see, e.g., Trzeciakowski, 1982, and the review paper by Bastard, 1979). The effect of magnetic field on resonant states was also discussed (Gortel et ai., 1979). However, the

8. SHALLOW

ACCEPTORS IN DILUTED MAGNETIC SEMICONDUCTORS

339

10'

loi

E

10'

Y

Q

loa

10-

I

0.1

I

0.2

I

0.3 0.4 T-' [ K-' ]

I

I

0.5

0.6

J

FIG. 8. Transverse ( I )and longitudinal (11) resistivities as a function of temperature for several values of magnetic field for a sample of p-type Hgl-,Mn,Te with x = 0.081, having an acceptor concentration of 8 X and a compensation donor concentration of 7 X 101scm-3. [After Wojtowicz and Mycielski (1983).]

effect of exchange interaction on resonant acceptors in DMS was not yet studied theoretically, in spite of the fact that a considerable amount of experimental data on these states were already obtained from both optical and transport measurements.

DATA 13. MAGNETOOPTICAL Several magnetoabsorption measurements were performed on zero-gap Hgl-,Mn,Te (Bastard el al., 1978, 1979; Dobrowolska et al., 1980;

340

J. MYCIELSKI

Dobrowolska and Dobrowolski, 1981). Transition from b(0) Landau level in r.5 valence band to a resonant acceptor state was observed. The zerofield energy of this state was estimated to be about 1 meV (Bastard el al., 1978, 1979) or 1.5meV (Dobrowolska et al., 1980; Dobrowolska and Dobrowolski, 1981). Bastard el al. (1979) found a strong quenching of the acceptor g-factor: the acceptor state does not split or shift in the magnetic field. However, such a shift was observed by Bastard et al. (1978), Dobrowolska ef al. (1980), and Dobrowolska and Dobrowolski (1981). Electronic transitions from the ra-band Landau levels to resonant acceptor states cannot be observed in magnetic field at liquid helium temperatures for x 2 0.008, since the bu(-1) Landau level of the r8 valence band (see section 2) is situated above the acceptor level. Thus, the latter state is occupied (Bastard ef al., 1978, 1979). For higher temperatures and/or lower x , however, the exchange interaction is weaker (-3Bexch in Eq. (13) is smaller) and the bu(-1) level is located below the acceptor state, which may thus be unoccupied. Bastard et al. (1 979) peformed submillimeter resonance experiments in Hgl-,Mn,Te as a function of magnetic field at liquid helium temperatures for different values of x. For very low values of x, a transition from a r8valence Landau state to the (unoccupied) acceptor level was observed, together with r8 + TSinterband transitions (Fig. 9). The (extrapolated) zerofield energy of this level was about 0.9 meV. Thus, the level was identified as the A0 (impurity) level, observed before in non-magnetic zero-gap Hg,-,Cd,Te (Guldner el al., 1977). For x L 0.008, the transition from the

0

10

20

30

40

Bz (kG)-+ FIG. 9. Submillimeter absorption lines vs. magnetic field for zero-gap Hg,-,Mn,Te with x = 0.0014, at T = 1.7 K in the Faraday configuration. The data represent two transitions

between the r8-valence and the rs-COndUCtiOnLandau levels, and the transition from a revalence Landau level to an (impurity) acceptor level. [After Bastard ef af. (1979).]

8.

SHALLOW ACCEPTORS IN DILUTED MAGNETIC SEMICONDUCTORS

341

2 I

t W

-I

-2

0

10

20

B, (kG)+ FIG. 10. Submillimeter absorption line vs. magnetic field for zero-gap Hgl-,Mn,Te with x = 0.008, observed at T = 1.7 K in the Faraday configuration. The absorption corresponds to a transition from an acceptor level to the bu(-l) valence Landau level. [After Bastard et al. ( 1979).]

occupied acceptor state to the bv(- 1) valence Landau level was observed (Fig. 10). From an extrapolation of the transition energy to zero field (i.e., to negative energies), the zero-field acceptor energy level was estimated to be 2.2 meV. Therefore, the level was identified as the A1 (mercury vacancy) level, observed also in non-magnetic zero-gap Hgl-xCdxTe (Guldner et al., 1977). The existence of a transition from this level to the b4-I) level indicates that the acceptor wave function involves a contribution from the function (P4 = 15, -$) of the Tg basis. The energy of A1 was found to be nearly independent of magnetic field. 14. MAGNETOTRANSPORT DATA In a partially compensated p-type zero-gap semiconductor with a low conduction band density of states (i.e., a low conduction band effective mass), the Fermi level is usually pinned to the resonant acceptor level (in the absence of magnetic field). Measuring the low field Hall coefficient and the period of the Shubnikov-de Haas oscillations, and knowing the conduction effective mass, one can determine the position of the Fermi level, i.e., the resonant acceptor energy (Sawicki et al., 1983; Sawicki and Dietl, 1983). Energies of about 4 meV and 7 meV were reported for x 0.06 (i.e., for the

=

342

J. MYCIELSKI

r 6 band only slightly below the TSbands), and it was observed that the energy increases with increasing magnetic field. Conductivity in the impurity band of resonant acceptors was observed by Davydov et af. (1980), Ponikarov et af. (1981), and Davydov et af. (1981). The conductivity increases strongly in the field range from 1 to 2T, in particular in the transverse direction. This effect was interpreted as the decrease of spin-disorder scattering in the impurity band. However, it may also be interpreted by the increase of the mean radius of the acceptor wave function, as discussed earlier in this chapter. Transformation of resonant acceptor states in zero-gapp-type Hgl -,Mn,Te into non-resonant states was investigated by magnetotransport measurements in high magnetic fields and/or under high pressures. Temperatures down to 30 mK and hydrostatic pressures up to 1.5 GPa were used (Sawicki et al., 1983). For x 4 0.06, the r6 band is situated only slightly below the TS bands. A high hydrostatic pressure can then reverse the relative band positions thus opening the gap of the material and transforming the resonant states into the non-resonant acceptors. The magnetic field may open a gap between the TSconduction and the TSvalence band (possibly after initially overlapping them by an upward shift of the bu(-1) level; see Section 2), and also make the acceptor states non-resonant. The latter effect was observed by Byszewski et af. (1979), Ponikarov et af. (1981), Davydov et af. (1981), Sawicki et af. (1983), and Sawicki and Dietl (1983). The field intensity necessary to open the gap decreases from about 10T for low (but non-zero) x to about 1T for x = 0.065. After opening the gap, hopping conductivity involving non-resonant acceptors was observed (Ponikarov et af., 1981; Davydov et al., 1981). It decreases sharply with increasing (strong) magnetic field, due to shrinking of the acceptor wave function. The transverse conductivity is then lower than the longitudinal conductivity. In a sample with x = 0.065 (where the r6band is only slightly below the TSbands) and with a net acceptor concentration of 2 x 10’6cm-3m, the acceptors were found to be in a metallic phase after opening the gap by the magnetic field of about 1T (Sawicki and Dietl, 1983). For magnetic fields higher than 3T, they show a positive magnetoresistance.

References Abramowitz, M., and Stegun, I. A . (1965). Eds. of “Handbook of Mathematical Functions”. Dover, New York. Akimoto, O., and Hasegawa, H. (1967). J. Phys. SOC. Japan 22, 181. Anderson, J. R., Johnson, W. B., and Stone, D. R. (1983). J. Vac. Sci. Technol. A l , 1761. Baidereschi, A., and Lipari, N. 0. (1973). Phys. Rev. B8, 2697. Bastard, G. (1979). In “Physics of Semiconductors, 1978” (B. L. H. Wilson, ed.), Conf. Ser. 43, p. 231. The Institute of Physics, Bristol.

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343

Bastard, G., Rigaux, C., Guldner, Y., Mycielski, J., and Mycielski, A. (1978). J. Physique 39, 87. Bastard, G . , Rigaux, C., Couder, Y., Thome, H., and Mycielski, A. (1979). Phys. Stat. Sol. (b) 94, 205. Bastard, B., Gaj, J. A., Planel, R., and Rigaux, C. (1980). J. Physique C5, 247. Bastard, G., Rigaux, C., Guldner, Y., Mycielski, A., Furdyna, J. K., and Mullin, D. P. (1981). Phys. Rev. B24, 1961. Bebb, H. B. (1969). Phys. Rev. 185, 1 1 16. Beneslavskii, S. D., and Entralgo, E. (1975). Zh. Eksp. Teor, Fiz. 68,2271. (Sov. Phys.: JETP 41, 1135.) Beneslavskii, S. D., Brandt, N. B., Golyamina, E. M., Chudinov, S. M., and Yakovlev, G. D. (1974). Zh. Eksp. Teor. Fiz. Pis’ma 19, 256. (Sov. Phys. JETP Letters 19, 154.) Byszewski, P., Szlenk, K., Kossut, J., and Galazka, R. R. (1979). Phys. Star. Sol. (b) 95, 359. Davydov, A. B., Ponikarov, B. B., andTsidilkovskii, I. M. (1980).Phys. Stat. Sol. (b) 101,127. Davydov, A. B., Ponikarov, B. B., and Tsidilkovskii. I. M. (1981). Fiz. Tekh. Poluprov. 15, 881. (Sov. Phys. Semicond. 15, 504.) Delves, R. T. (1966). Proc. Phys. Soc. 87, 809. Dietl, T. (1981).In “Physicsin High Magnetic Fields” (N. Miura, ed.), p. 344. Springer, Berlin. Dietl, T., Antoszewski, J., and Swierkowski, L. (1983). Physica B117 and 118, 491. Dobrowolska, M., and Dobrowolski, W. (1981). J. Phys. C: Solid St. Phys. 41, 5689. Dobrowolska, M., Dobrowolski, W., Otto, M., Dietl, T., and Galazka, R. R. (1980). J. Phys. Soc. Japan 49, Suppl. A, p. 815. Dobrowolski, W., Ortenberg, M. von, Sandauer, A. M., Galazka, R. R., Mycielski, A., and Pauthenet, R. (1982). In “Physics of Narrow Gap Semiconductors” (E. Gornik, H. Heinrich, and L. Palmetshofer, eds.), p. 302. Springer, Berlin. Dresselhaus, G., Kip, A. F., and Kittel, C. (1955). Phys. Rev. 98, 368. Elliot, R. J., and Loudon, R. (1960). J. Phys. Chem. Solids 15, 196. Furdyna, J. K. (1982a). J. Vac. Sci. Technol. 21, 220. Furdyna, J. K. (1982b). J. Appl. Phys. 53, 7637. Furdyna, J. K. (1985). Solid State Commun. 53, 1097. Gaj, J. A. (1980). J. Phys. Soc. Japan 49, Suppl. A, p. 797. Gaj, J. A., Ginter, J., and Galazka, R. R. (1978). Phys. Stat. Sol (b) 89, 655. Galazka, R. R. (1979). In “Physics of Semiconductors, 1978” (B. L. H. Wilson, ed.), Conf. Ser. 43, p. 133. The Institute of Physics, Bristol. Galazka, R. R. (1982). In “Physics of Narrow Gap Semiconductors” (E. Gornik, H. Heinrich, and L. Palmetshofer, eds.), p. 294. Springer, Berlin. Galazka, R. R., and Kossut, J. (1979). In “Proceedings of the International School of NarrowGap Semiconductors”, Nimes, Lecture Notes in Physics, 133, p. 245. Springer-Verlag. Gawron, T. R. (1986a). J. Phys. C19, 21. Gawron, T. R. (1986b). J. Phys. C19, 29. Gawron, T. R., and Mycielski, J. (1984). Phys. Stat. Sol. (b) 125, 341. Gawron, T. R., and Trylski, J. (1982). In “Physics of Narrow Gap Semiconductors” (E. Gornik, H. Heinrich, and L. Palmetshofer, eds.), p. 312. Springer, Berlin. Gelmont, B. L., Gadzhiev, A. R., Shklovskii, B. J., Shlimak, J. S., and Efros, A. L. (1974). Fiz. Tekh. Poluprov. 8, 2377. (Sov. Phys. Semicond. 8, 1549.) Gebicki, W., and Nazarewicz, W. (1980). Phys. Stat. Sol. (a) 61, K135. Gortel, 2. W., Szymanski, J., and Swierkowski, L. (1979). In “Physics of Semiconductors, 1978” (B. L. H. Wilson, ed.), Conf. Ser. 43, p. 261. The Institute of Physics, Bristol. Grynberg, M. (1983). Physica 117B and 118B, 461. Guldner, Y., Rigaux, C., Mycielski, A., and Couder, Y. (1977). Phys. Stat. Sol. (b) 81, 615.

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Gupta, 0. P., Joos, B., and Wallace, P. R. (1982). Phys. Rev. B25, 1101. Jaczynski, M., Kossut, J., and Galazka, R. R. (1978). Phys. Stat. Sol. (b) 88, 73. Jaroszynski, J., Dietl, T., Sawicki, M., and Janik, E. (1983). Physica 117B and 118B,473. Kaczmarek, E. (1966). Acta Phys. Polonica 30, 267. Kossut, J . (1976). Phys. Stat. Sol. (b) 78, 537. Lipari, N. O . , and Altarelli, M. (1980). Solid State Commun. 33, 47. Luttinger, J . M. (1956). Phys. Rev. 102, 1030. Luttinger, J. M., and Kohn, W. (1955). Phys. Rev. 97, 869. McKnight, S. W., Amirtharaj, P. M., and Perkowitz, S. (1978). Solid State Commun. 25, 357. Miller, A., and Abrahams, E. (1960). Phys. Rev. 120,745. Molva, E., Chamonal, J . P., and Pautrat, J. L. (1982). Phys. Stat. Sol. (b) 109, 635. Morrissy, J. H. (1971). Ph.D. Thesis, Oxford University. Mott, N. F. (1968). Phil. Mag. 17, 1259. Mott, N. F. (1974). “Metal-Insulator Transitions”. Taylor and Francis, London. Mycielski, A., and Mycielski, J. (1980). J. Phys. SOC.Japan 49, Suppl. A, p. 807. Mycielski, J . (1981). In “Recent Developments in Condensed Matter Physics” (J. T. Devreese, ed.), Vol. 1, p. 725. Plenum Press, New York. Mycielski, J. (1983). In “Application of High Magnetic Fields in Semiconductor Physics” (G. Landwehr, ed.), p. 431. Springer, Berlin. Mycielski, J. (1984). Progr. Cryst. Growth Character 10, 101. Mycielski, J., and Rigaux, C. (1983). J. Physique 44, 1041. Mycielski, J., Witowski, A., Wittlin, A., and Grynberg, M. (1984). In “Proceedings 17th Intern. Conf. on Physics of Semiconductors, San Francisco” (D. J. Chadi and W. A. Harrison, eds.). Springer, Berlin, 1415. Ortenberg, M. von (1973). J. Phys. Chem. Solids 34, 397. Pidgeon, C. R., and Brown, R. N. (1966). Phys. Rev. 146, 575. Pollak, M. (1980). Phil. Mag. B42, 781. Ponikarov, B. B., Tsidilkovskii, I. M., and Shelushinina, N. G. (1981). Fiz. Tekh. Poluprov. 15, 296. (Sov. Phys. Semicond. 15, 170.) Rigaux, C., Bastard, G., Guldner, Y., Rebmann, G., Mycielski, A., Furdyna, J. K., and Mullin, D. P . (1980). J. Phys. SOC. Japan 49, Suppl. A, p. 81 1. Sawicki, M., and Dietl, T. (1983). In “Physics of Semiconducting Compounds” (R. R. Galazka and Rauluszkiewicz, eds.), Polish Academy of Sciences, Institute of Physics, Proc. Conf. in Physics, Vol. 6, p. 400. Ossolineum, Wroclaw. Sawicki, M., Dietl, T., Plesiewicz, W., Sekowski, P., Sniadower, L., Baj, M., and Dmowski, L. (1983). I n “Application of High Magnetic Fields in Semiconductor Physics” (G. Landwehr, ed.), p. 382. Springer, Berlin. Schechter, D. (1962). J. Phys. Chem. Solids 23, 237. Shlimak, I. S., Efros, A. L., and Yantchev, I. Ya. (1977). Fiz. Tekh. Poluprov. 11,257. (Sov. Phys. Semicond. 11, 149.) Trzeciakowski, W. (1982). J. Phys. C.: Solid St. Phys. 15, 1199. Twardowski, A., and Ginter, J. (1982). Phys. Stat. Sol. (b) 110, 47. Wojtal, R.,Golnik, A., and Gaj, J. A. (1979). Phys. Stat. Sol. (b) 92, 241. Wojtowicz, T., and Mycielski, A. (1983). Physica 117B and 118B. 476. Wojtowicz, T., and Mycielski, A. (1984). Acta Phys. Polonica A67, 363. Wrobel, J., Kuchar, F., and Meisels, R. (1984). Acta Phys. Polonica A67, 369.

SEMICONDUCTORS AND SEMIMETALS, VOL. 25

CHAPTER 9

Raman Scattering in Diluted Magnetic Semiconductors A . K . Ramdas and S . Rodriguez DEPARTMENT OF PHYSICS, PURDUE UNIVERSITY WEST LAFAYETTE, INDIANA

I. 11.

111. IV.

V.

VI.

INTRODUCTION . . . . . . . . . . . . . . . . . . . . INELASTIC LIGHTSCATTERING PRINCIPLES. . . . . . . . . . EXPERIMENTAL TECHNIQUES. . . . . . . . . . . . . . . RAMAN SCATTERING BY VIBRATIONAL EXCITATIONS. . . . . . 1 . Phonons in Perfect Crystals . . . . . . . . . . . . . 2 . Phonons in Mixed Crystals. . . . . . . . . . . . . . 3. Experimental Results and Discussion. . . . . . . . . . MAGNETIC EXCITATIONS . . . . . . . . . . . . . . . . 4. General Considerations . . . . . . . . . . . . . . . 5 . Paramagnetic Phase . . . . . . . . . . . . . . . . 6 . Magnetically Ordered Phase . . . . . . . . . . . . . 7. Spin-Flip Raman Scattering . . . . . . . . . . . . . VIBRATIONAL,ELECTRONIC AND MAGNETICEXCITATIONS IN SUPERLATTICES . . . . . . . . . . . . . . . . . . . . REFERENCES.. . . . . . . . . . . . . . . . . . . .

345 346 350 353 353 351 365 313 313 316 389 396 406 410

I. Introduction In this chapter the focus is on Raman scattering studies on the localized and collective excitations in Mn-based 11-VI diluted magnetic semiconductors (DMS), the excitations being vibrational, electronic or magnetic in character. As has been extensively discussed elsewhere in this volume, the DMS alloys possess a unique combination of semiconducting and magnetic properties. In common with their non-magnetic counterparts (e.g. Hgl-,Cd,Te), they exhibit the well-known composition-dependent vibrational spectra (Barker and Severs, 1975), electronic band structure (Zanio, 1978) and transport properties (e.g. carrier mobilities). In addition, in a Mn-based 11-VI alloy like Cdt-,Mn,Te, new properties arise which can be traced to the substitutional Mn2+ions randomly replacing the group I1 element (Cd, Zn, or Hg). In these tetrahedrally co-ordinated semiconducting alloys with the zinc blende or the wurtzite structure, the ground state of Mn2+ is (ignoring small crystal 345 Copyright 0 I988 by Academic Press. Inc. All rights 01 reproduction in any form reserved.

ISBN 0-12-752125.9

346

A. K . RAMDAS AND S . RODRIGUEZ

field effects); these ions thus have an effective magnetic moment of 5.92 Bohr magnetons @B). The exchange interaction between the Mn2+ ions and the band electrons produces large spin splittings at the conduction band minimum and valence band maximum, in turn resulting in the large Zeeman splitting of the free exciton (Gaj et al., 1978a) and the giant Faraday rotation (Gaj et al., 1978b). When the Mn2+ concentration (x) exceeds a certain critical value, long range magnetic ordering sets in below a certain temperature (Galazka et al., 1980)-a consequence of the anti-ferromagnetic interaction between neighboring Mn2+ions. Raman scattering has proved to be very fruitful in discovering and delineating vibrational modes, localized and collective; electronic transitions within the Zeeman multiplet of the ground state of Mn2+ split by an external magnetic field; the spin-flip of electrons bound to donors in large, effective mass orbits; and the collective magnetic excitations in the low temperature, magnetically ordered phases. In this chapter, the theoretical and experimental background on Raman scattering are first presented in general terms. Each specific topic with its relevant theory is then developed along with the presentation of experimental results and their interpretation. 11. Inelastic Light Scattering: Principles

In its interaction with material media, the reflected and transmitted electromagnetic radiation of wavelength much larger than the interatomic distances are accompanied by weak scattered radiation in all directions. The incident radiation at an angular frequency OL induces oscillating electric dipole moments in the medium and the scattered radiation field is an appropriate superposition of the radiation from the individual dipoles. The macroscopic polarization P of the system is to first order in EL,the incident electric field, x being the electric susceptibility tensor. From this picture one expects scattered radiation at the same frequency as that of the incident wave, its intensity being proportional to the fourth power of the frequency; the scattering is thus elastic. However, the medium has internal degrees of freedom of frequencies, say 0 1 , either quiescent or already in a state of excitation. The electric susceptibility is a function of the internal co-ordinates (lattice vibrations, electronic excitations, magnetic excitations, ...) Qi = Qjo cos 0 i t . Expanding x in power series in Qi one obtains an induced polarization vibrating at frequencies OL f mi to first order, W L k o;k o j to second order and so on to higher orders. The scattered radiation is thus modulated and, when analyzed into

9.

347

RAMAN SCATTERING

its spectral components, exhibits side bands at WL i w ; , WL i w; i w j , ... besides the “central” or “Rayleigh” line at OL . The side bands are referred to as Raman lines except when the internal excitation is an acoustic wave in which case they are the “Brillouin” components. The components at OL - O;(OL + a;) correspond to the creation (annihilation) of a single quantum of the internal excitation of energy h a ; ; the nomenclature Stokes (anti-Stokes) is used to describe them. Raman lines can also occur associated with muItipIe excitations of the medium and are labeled as second, third, ... order depending on the number of such quanta simultaneously excited or absorbed. Inelastic light scattering, especially since the advent of lasers as exciting monochromatic light sources, is a powerful tool in the study of matter in all states of aggregation. For general background and additional references, we cite Hayes and Loudon (1978), Rodriguez and Ramdas (1985), and the series edited by Cardona (1975) and by Cardona and Guntherodt (1982a, 198213, and 1984). The quantum theory of light scattering can be developed in close analogy to the corresponding classical picture described above. Within the framework of the electric-dipole approximation, the scattering cross section per unit solid angle 0 resulting in the scattering of a photon of frequency OL , wave vector kL and polarization &L into a state characterized by the corresponding quantities os ,ks ,and &S , while the scattering system experiences a transition from Jvo) to Iv), is

Here

5 (Eyv (v

=

1d 1 v -

) ( v 1 d 1 yo) E., - AWL

+

1 1

1 I v’

( v d yo) ( v d E,,l

- E,

+h

))’

o ~

(3)

the states Iv) are the energy eigenvectors with eigenvalues E, and d, the electric dipole moment operator of the system. The quantities n~ and ns are the photon populations in the incident and scattered beams, respectively. The summation in Eq. (3) extends over all states Iv‘) of the scatterer. Often, it is convenient to describe the motion of the scattering system as consisting of two or more types of excitations coupled by very small interactions. Typical examples are molecules with weakly coupled electronic, vibrational and rotational motions, and crystals where one can, in a first approximation, disregard the electron-phonon interaction. In such cases, the states Iv) described above are the stationary states of the system taking due account of the coupling between all modes of motion. To fix the ideas, we think of crystals in which the electron-phonon interaction Hepis sufficiently

348

A . K . RAMDAS A N D S. RODRIGUEZ

small to allow a description of the system by states 4, written as products of electronic and vibrational states. Then, the energy eigenstates appearing in Eq. (3) are

where the EoYare the energies associated with the 4,. Clearly each Eo, is the sum of electronic and phonon energies. The sum omits the v' = v. An equivalent procedure is to take He, and the electron-photon interaction as perturbations. Then, the scattering cross section is expressed as a sum reflecting the use of third order perturbation theory as in Loudon (1963, 1964). For example, a Raman process associated with emission of an optical phonon can be viewed as the virtual creation of an electron-hole pair (step l), emission by the electron or by the hole of an optical phonon (step 2, Hep) followed by electron-hole recombination with emission of a scattered photon (step 3). This process is illustrated in Fig. 1. The steps involved in this description are implicit in the nature of the states Iv) in Eq. (4) that explicitly take the electron-phonon interaction into account. Equation (3) for CY,~,, contains matrix elements between the exact states I VO), I v), and I v' ) as well as the energy eigenvalues E, . These are related to 6,0, 4v, 4,~and EoYby standard perturbation theory. It is sometimes useful to view C Y as~ being ~ ~modulated ~ by the internal motions. The amplitude of the polarization giving rise to a Raman line at OL - wi is of the form

Here we consider the matrix elements of d to be slowly varying functions of Qi. We note that when h w is ~ close to an electronic excitation energy there is an enhancement in the amplitude of the scattering cross section. This occurs also when hws is close to the excitation energy E,! - E,. The dipole approximation is valid for molecules whose dimensions are much less than the wavelengths of the incident and scattered photons. It is also valid for crystals whose primitive cells have diameters small compared to these wavelengths. The proof for this result makes use of the translational symmetry of the crystal. The stationary states of the crystal are classified by their wave vectors q confined within the fundamental Brillouin zone. Upon translation by a lattice vector n, a stationary state tyg becomes eiq"'ty9.Now,

9.

349

RAMAN SCATTERING

\

\

4

k'

'

\

FIG. 1 . Schematic diagram representing inelastic light scattering in crystals with the creation of a phonon. The dashed lines represent photons, the solid lines with arrows directed towards the right (left) correspond to electrons (holes) and the wavy line indicates a phonon.

the interaction between the radiation field and the scattering system changes by factors eikr'" and e-jks'" upon translation. If q and qo are the wave vectors of the states I v ) and I VO), respectively, the amplitude of the resulting transition probability is a lattice sum, each term being proportional to ei(-q

+ kr- ks +qo).

II

Thus, we find, as a first selection rule, the law of conservation of crystal momentum, namely kL+qo=ks+q+G

where G is a vector of the reciprocal lattice. For visible light, /kLl and (ksl are negligible compared to [GIexcept for G = 0; therefore, this conservation law reduces to kL

=

ks

+ q - 90.

(5)

The transition probability is, therefore, proportional to the square amplitude of a matrix element containing an integration over the primitive cell alone. If the diameter of this cell is small compared to 1kLI-l and Iksl-', the dipole approximation is clearly valid. In many circumstances, the energies of the excitations involved are small compared to the energies AclkLl and Aclksl of the incident and scattered photons. Then JkLJ Jksl and, setting qo = 0,

-

e

19) = 21k~Isin-, 2

350

A. K. RAMDAS AND S. RODRIGUEZ

where 8 is the scattering angle. Thus in first order Raman scattering, only states whose wave vectors are near the center of the fundamental Brillouin zone can be excited (or de-excited). A large number of investigations accessible to Raman scattering involve collective or localized excitations with frequencies in the infrared. For this reason an excitation observed with the absorption or emission of a photon is labeled “infrared active” whereas that seen in a Raman spectrum, “Raman active”. Raman and absorption spectroscopy are complementary tools in the study of the internal motions of centro-symmetric molecules and crystals. (See, for example, Bhagavantam and Venkatarayudu, 1969.) In addition to wave vector conservation, Raman transitions are governed by selection rules appropriate to the second rank polarizability tensor a. In this context symmetry arguments can be used to deduce selection rules including details of the polarization features for different scattering geometries. A transition from a state vo belonging to the irreducible representation ry0 of the space group of the crystal to a state v belonging to r, occurs only if rY* x ,?I x rvocontains the totally symmetric representation. Here r, is the representation generated by the components of a. We also note that, for Raman shifts associated with visible exciting radiation, since the wave vector of the excitation responsible is near the center of the Brillouin zone, one is justified in treating these “zone-center” excitations using the point-group of the crystal. 111. Experimental Techniques

In Raman and Brillouin spectroscopy, a sample under study-maintained in an environment with specific characteristics like temperature, magnetic field, pressure, . ..-is illuminated with monochromatic light of frequency W L . The scattered radiation is analyzed with a spectrometer or an interferometer equipped with a detection system. Since Raman features occur with frequencies os = WL k mi,it is often desirable to measure the Raman shifts for several values of OL . In most Raman phenomena OL - os is independent of oL; if the frequency shift does depend on OL as in Brillouin scattering and, in very special circumstances, in Raman lines, such experiments will reveal them. Luminescence features, which occur at fixed frequencies, are also discriminated from Raman features by doing such experiments. The measurement of resonances in the intensity of a given Raman line provides valuable insights into the electronic transitions which are selectively favored in transitions to or from intermediate levels. A tunable monochromatic source is thus valuable in light scattering studies. Scattering geometry and the intensity of the Raman line for different polarization features of the incident and scattered radiation allow selection rules to be established.

9.

351

RAMAN SCATTERING

In Table I we list the various ion lasers currently available and the wavelengths of the visible radiations they provide. Monochromatic lines in the range 1-12 pm are generated with Nd:YAG, CO, and C02 lasers. Frequency doubling with non-linear crystals is a procedure often exploited to obtain exciting lines in a desired range. Tunable dye lasers with suitable dyes, pumped with a Ar+ laser, a Kr’ laser, a frequency doubled 1.06 pm line of the Nd:YAG laser, or a nitrogen laser allow a complete coverage from the blue to the red. Though CW operation is by far the standard in Raman spectroscopy, pulsed lasers are exploited in time resolved as well as in stimulated Raman studies. TABLE I LASERAND VISIBLE OUTPUT WAVELENGTHS USED BRILLOUIN SPECTROSCOPY Laser Krypton ion He-Ne Dye laser

Argon ion

Output wavelength

IN

RAMANAND

(A)

5208, 5309, 5682, 6471, 6764, 7525, 7993 6328 4060-4650 (Stilbenes 3) 5800-6500 (Rhodamine 590) 6100-7500 (DCM) 7800-9000 (Styryl 9) 4579, 4658, 4727, 4765, 4880, 4965, 5017, 5145

Since Raman signals are typically orders of magnitude weaker than the “parasitically” scattered, unshifted laser radiation, spectrometers have to be able to reject it efficiently, thus allowing Raman lines to stand out. Double and triple grating monochromators are especially designed to address this problem; with currently available spectrometers it is possible to observe Raman lines with shifts as small as a few wave numbers, and even Brillouin components under favorable experimental conditions. Holographic gratings are particularly effective in rejecting efficiently the parasitic radiation. Brillouin components and sharp Raman lines can be experimentally studied by isolating a small spectral range with an optical filter and performing further spectral analysis with a scanning Fabry-Perot interferometer. Multipassing of a Fabry-Perot has proved a significant measure to enhance its contrast, and tandem operation allows an advantageous increase of the free spectral range. Scattered radiation after its spectral analysis is typically detected with thermoelectrically cooled photomultipliers followed by photon counting electronics. The state-of-the-art data acquisition and data processing systems incorporate microcomputers. These can be organized to multi-scan a given

352

A. K . RAMDAS AND S. RODRIGUEZ

spectral range and the data can be displayed on a multichannel analyzer, X-Y or chart recorder or a monitoring screen and plotted on a plotter after the data are suitably processed. Figure 2 shows the schematic diagram of the experimental set up used in the Raman scattering experiments in the authors’ laboratory. Variable temperature cryostats operating from room temperature down to 1.8 K using liquid helium as a coolant; uniaxial stress attachments and hydrostatic stress apparatus (e.g. the diamond anvil cell); variable temperature cryostats incorporating a superconducting magnet-such equipment currently are in an advanced and sophisticated state of development and are commercially available or readily fabricated. We refer the reader to Chapter 2 of Hayes and Loudon (1978) for a very readable account of Raman and Brillouin spectrometers. For a comprehensive description of Raman spectrometers, we recommend the article by Hathaway (1971). A complete description of a piezo-electrically scanned tandem Fabry-Perot interferometer appears in Sandercock (1982).

-

-

RATEMETER

r

MICROCOMPUTER SYSTEM

PLOTTER

FIG.2. Schematic diagram of the experimental setup used in Raman scattering experiments. PR stands for polarization rotator, A for analyzer, DP for Dove prism, and HWP for half-wave plate. L1, L2, and L3 are lenses. [After Peterson (1984).]

9.

RAMAN SCATTERING

353

IV. Raman Scattering by Vibrational Excitations It is important to establish the nature of the vibrational spectrum of a solid since it influences almost all physical properties, e.g., linewidths of optical transitions, heat capacity, transport phenomena, . ... In a perfect crystal with full translational symmetry, the atomic vibrations can be described in terms of collective excitations characterized by wave-vectors q, confined to the fundamental Brillouin Zone (BZ). Associated with imperfections, chemical or structural in nature, localized vibrations occur. The microscopic theory of structural phase transitions draws upon the occurrence of normal modes whose frequencies tend to zero as a function of an external macroscopic parameter (temperature, pressure, ...) causing a displacive phase transition. In this section we review the theoretical background on lattice dynamics of perfect and mixed crystals (alloys). This is followed by a discussion of the vibrational Raman scattering in DMS. We note here that even though inelastic neutron scattering provides, in principle, information about the vibrational spectrum throughout the BZ, Raman scattering gives more precise frequencies when a given mode appears in the Raman spectrum. It should be emphasized that for the Cd-based DMS crystals observation of neutron scattering is difficult due to the large neutron absorption cross section of the naturally occurring Cd. The physical considerations underlying Raman scattering are additional motivations for its study. 1. PHONONS IN PERFECT CRYSTALS Consider the motion of the atoms in a crystal with fatoms per primitive cell. Let the displacements from their equilibrium positions be characterized by the coordinates U n n where n designates a particular cell of the crystal and a (a, = 1,2,3, ..., 3f) corresponds to any one of the 3f Cartesian components of the displacements of the f atoms within the cell at n. To first order in the displacements, the equations of motion are

We take a crystal with lattice translations a l , az, 8 3 , having dimensions Noaj(i = 1,2,3), and use periodic boundary conditions with periods Noaj. The quantities Cnal(n- d)in Eq. (7) are real coefficients, symmetric in na, and n‘ar and depend only on n - n’ in order to satisfy translational symmetry. Assuming solutions of these equations in the form Unol

=

-

(NMu)-1’2eaexp[i(q n - at)],

(8)

354

A. K. RAMDAS AND S. RODRIGUEZ

where N = N i is the number of primitive cells, we obtain

C [Caar(q)- o26,,!1e,,

=

a’

o

where Caa,(q)= C(MolMol~)-”2Ca,~(n - n’)exp[-iq n’

(9)

- (n - nf)l.

(10)

Here ea is a component of a 3f-dimensional vector determined by the eigenvalue problem represented by Eq. (9); it is usually convenient to select an orthonormal set of eigenvectors. We note that C,,I(q), which can be regarded as a Fourier transform of the force constants C,,f(n - n’), is independent of n. Further, the 3f x 3fmatrix with components given by Eq. (10) is Hermitian by virtue of the symmetry of the force constants; it is customary to refer to this matrix as the dynamical matrix. The set of linear equations (9) has a non-trivial solution only if o satisfies the secular equation IC,,(q)

-

02601a‘I

= 0.

(1 1)

For each wave vector q, Eq. (1 1) gives 3f solutions for the frequency o(q) yielding 3f branches of the dispersion curves of the vibrational spectrum of the crystal, The values of q are restricted to the reduced Brillouin zone to avoid redundancy. For crystals containing f atoms per primitive cell, there are 3 acoustic branches and 3(f - 1) optical branches. For acoustic modes, o is linear in q near q = 0 and vanishes at the center of the Brillouin zone. However, optical branches have a non-vanishing frequency at the zone center. Except for degenerate polar optical phonons in crystals free from improper symmetry operations (Pine and Dresselhaus, 1969, 1971; Grimsditch et a/., 1977; Imaino el al., 1980), the frequency w(q) is of the form

for small values of q. The phonon excitations of crystals, like other stationary states, can be classified according to the irreducible representations of the group of the wave vector q. In the first order Raman effect, only phonons with wave vectors near q = 0 need be studied. The eigenvectorsof the dynamical matrix of each optical phonon corresponding to q = 0 generate one of the irreducible representations of the point group of the crystal. The group theoretical classification of the optical phonons is carried out as follows (Bhagavantam and Venkatarayudu, 1969, p. 140). The 3fdisplacement of the atoms in a primitive cell generate a 3f-dimensional representation of the point

9.

355

RAMAN SCATTERING

group of the crystal whose character is easily determined. The trace of a matrix associated with a particular symmetry operation is the sum of the diagonal elements associated with atoms that remain in fixed positions or move to sites related to the initial position by a lattice translation; each of these atoms contributes (f1 + 2 cos d), 4 being the angle of rotation and the upper (lower) sign corresponding to a proper (improper) rotation by 6.The atoms, which are mapped into a non-equivalent position upon a symmetry operation, give vanishing contributions since the non-zero matrix elements associated with their coordinates are off-diagonal. This representation contains those generated by the optical phonons and by the acoustic phonons at q = 0. The latter correspond to the three uniform translations of the crystal as a whole and thus generate a vector representation of the group. Thus, we can subtract from the character of the 3f-dimensional representation the character of the polar vector to yield the representation generated by the optical modes. In molecules, the procedure is the same but we must also subtract the characters associated with a rigid rotation (pseudo-vector representation). In crystals this is not legitimate since a rigid rotation violates the periodic boundary conditions. For convenience we give in Tables I1 and I11 the character tables for the irreducible representations of the groups Td and CSv,the point groups of crystals having the zinc-blende and wurtzite structures, respectively. The last column of these tables gives functions generating the different irreducible representations, selected in such a manner that these representations are automatically unitary. The procedure outlined above shows that in zinc-blende crystals at q = 0, there are three-fold degenerate F2 optical and F2 acoustic phonons. Because of the partially ionic character'of zinc-blende crystals, the F2 phonons are split (for q small but nonvanishing) into a longitudinal (LO) and two transverse (TO) branches. The FZoptical modes are both Raman and infrared active. TABLE I1

CHARACTER TABLEFOR

&

E

8C3

3Cz

AI

1

1

1

A2 E FI F2

1

1 -1

1

2 3 3

0 0

2 -1

-1

60d

6s4

1

1 -1

-1

0

0

-1 1

1

-1

Td Basis functions

x 2 +Y 2 + Z 2 222 - x2 - Y2, A ( X 2 - Y2) Rx,Ry,Rz x, Y , Z ; Y Z , Z X , X Y

X , Y, Z are the components of a polar vector field along the cubic axis x, y, z . R,, Ry , and R , are the components of a pseudo-vector along the cubic axes.

356

A. K. RAMDAS AND S. RODRIGUEZ

TABLE 111 CHARACTER TABLEFOR

A1

1 1 1 1

Az B1

Bz El EZ

2 2

1 1 -1 -1

-2 2

1 1 1 1 -1 -1

1 -1 -1 1

1

1 -1

-1

1

z; xz

R,

-1

1 -1

0 0

1

-1

~

c 6 v

0 0

+ Y2; z2

Y(Y2 - 3 x 7 X ( X 2 - 3Y2) X , K Z X , ZV; R , , Ry xz-Y Z , - 2 X Y

~~

~~

X , Y, and Z are the components of a polar vector field with respect to a coordinate system x , y , z in which 1is along c6.. and the xz plane is a Ud R,, R,, and R, are the components of a pseudo-vector along the coordinate axes.

The wurtzite structure, having four atoms per primitive cell, has optical phonons with symmetries A, ,El , B1, Ez ,there being two of each of the last two irreducible representations. Of these A1 and E1 are both Raman and infrared active, E2 is only Raman active, whereas B1 is active in neither. In order to explain the polarization features of the Raman spectra of phonons we determine the form of the Raman scattering tensor a, assumed to be symmetric away from resonance, by reducing the representation it generates into its irreducible components. This reduction is r, = A1 E FZ for Td and r, = 2A1 + El + EZ for C6". Next we use a theorem in group theory which we quote: the inner product of two vectors (or quantum mechanical states if we are dealing with a dynamical system) is zero unless the vectors belong to the same row of identical, unitary irreducible representations. As an example, we deduce the form of a for the E modes in the zinc-blende structure. Clearly, for the E mode belonging to the row 22' - X' - Y z ,

+ +

+ aYy+ aZZ= 0, axx- aYy= 0, and a,,

aXx

= aZx= av = 0.

The quantity 2aZz- axx- or,, does not vanish. If we denote it by 66 we find ayy = - b, azz = 2b. We find the components of a for the second row by noting

aXx=

f i ( ~ ~ x-x ~ l y y )=

while ffxx

+ f f y y + ffzz = 2ffzz - ffxx

-

ffyy

6b = ay, = ffzx =

ffp

= 0.

These equations yield cyZz = 0 and aXx= -ayy = b f i . The forms of a for the transitions from the totally symmetric ground state A1 to the

9.

357

RAMAN SCATTERING

TABLE IV RAMANTENSORS FOR

AI a 0 0 10 a 0 1 O O a

1-i fib

E 0 - b0

Td F2

0 0 0 2b : l , 2 Z 2 - X 2 - Y 2 1 0O 0d O d / , X

0

0

I:

FI 0

:I,R.

0

--c0

O O d

0

0

-c

O d O

o

c

o

-fib

d O O

Note that the tensors associated with A l , E, and FZare symmetric whereas those with FIare antisymmetric. TABLE V RAMAN TENSORS

FOR c6"

Symmetric AI

EI

Antisymmetric E2

A2

EI

0 0

0 0 10 b

0 - b , Y,Ry 01

allowed final states in first order Raman scattering are displayed in Tables IV and V for Td and c 6 u , respectively. The symmetry classification of the lines in an experimental spectrum can be arrived at on the basis of polarization studies for a variety of scattering geometries.

2. PHONONS IN MIXEDCRYSTALS A number of physically interesting aspects of crystal physics have their origin in the presence of imperfections, e.g., foreign atoms, lattice defects, .... Alloys of two or more compounds forming a homogeneous crystalline

358

A . K . RAMDAS AND S. RODRIGUEZ

phase also exhibit novel physical phenomena. These systems lack strict translational symmetry. The lattice dynamics of such crystals is the subject of this sub-section. We illustrate our discussion of mixed crystals with the results on alloys of two compound semiconductors like CdTe and MnTe in atomic proportions of ( 1 - x)/x. The resulting compound is designated by Cdl-,Mn,Te. It has been found that for x 5 0.7, it has the zinc-blende structure with Mn atoms replacing at random sites Cd atoms (See Chapter 2). We investigate first the small vibrations of a crystal containing imperfections located at the atomic sites of the perfect crystal. We follow procedures and nomenclature similar to those in Sec. IV. 1. To second order in the small displacements from equilibrium, una,the potential energy is of the form

where the coefficients Ciit(n, n’) = C$&, the crystal were perfect

n), are the force constants. If

Cik)(n,n‘) = Caal(n- n’),

(14)

because of the translational symmetry. The equations of motion of the lattice are

MAA’iina

=-

C

n’a’

Cik$(n,n’)unsa,,

(15)

where Mi;) is the mass associated with the a-th degree of freedom in the cell n. Periodic solutions of Eq. (15) of the form (0) - i d

una(t) = unae exist if n‘a’

Equation (17) constitutes a set of 3Nf linear homogeneous equations in the 3Nf variables u!i’?- A non-trivial solution exists only for those values of o making the determinant of the coefficients of the u!i? equal to zero, i.e.,

IMAL)U2Saa1Snnl- Ci3(n,n’)l = 0.

(18)

We can regard CiAj(n, n‘) and Mn(A)Snn*Saa* as the elements of the matrices C(l) and M(’) and ui:) as the components of the column vector do)in 3Nf dimensions. The eigenvalue problem in Eq. (17) can be expressed as (M(’)02 - C(o))u(o)= 0.

(19)

There are, in general, 3Nf eigenvalues o2and an equal number of orthogonal eigenvectors (M(1))1’2u(o)satisfying Eq. (19).

9.

RAMAN SCATTERING

359

As seen in Sec. IV. 1, in a perfect crystal, the size of the eigenvalue problem can be reduced, invoking the translational symmetry, to one involving a Hermitian matrix in 3f dimensions. In fact, in such a case Mi:) = M, and Ci;j(n, n') = CPPl(n- n'). For each of the N wave vectors q in a quasi-continuous distribution within the fundamental Brillouin zone of the crystal, we must solve the eigenvalue problem in Eq. (9) in 3f dimensions. The eigenvalues are designated by wl(q) and the eigenvectors by ePs(q), (s = 1,2, ..., 3f) which can be chosen to satisfy the orthogonality condition

The general solution for the displacement of the atoms in the lattice is of the form

with QJq, t ) =

- iws(q)Qs(q, t ) .

(22)

allow The orthogonality of the vectors e5(q) and of the functions N-1/2eiq*" us to solve for the coordinates QS(q,t ) in terms of u,,(t). We obtain

For imperfect crystals, we follow the procedure given in Lifshitz and Kosevich (1966) which is most appropriate when the deviations from the ideal crystal are small as is the case when we have a single foreign atom in an otherwise perfect lattice. We consider the 3Nf-dimensional matrix appearing in Eq. (19) and write it in the form M(1)02 - C(1) = Mw2 - C + M1/2AM1/2 (24) where C and M are the values of C(l) and M(') for the perfect crystal. We have already shown how to factor the eigenvalue problem for the perfect crystal into Nsimilar problems in 3f dimensions. The matrix A is defined by Eq. (24). A convenient algorithm for the case of a localized imperfection consists in noting that A possesses elements most of which vanish except for those in the vicinity of the lattice site of the imperfection. For that purpose we remark that the matrix o2- M-1/2CM-'/2 is non-singular for those values of w z which differ from wf(q) for all s and q, i.e., for frequencies that do not coincide with those of the vibrational spectrum of the ideal crystal. Let G(w) be the inverse of w 2 - M-1/2CM-1/2.The matrix G is called the

360

A . K . RAMDAS A N D S. RODRIGUEZ

resolvent operator or the Green function of the problem. Multiplying Eq. (19) by M-’12 on the left, and then by G and using Eq. (24) we obtain (1

+ G(co)A)M~/~u(O) = 0.

(25)

The frequency spectrum of the imperfect solid is obtained by solving the determinantal equation 11

+ G(o)Al

=

0.

(26)

In the representation generated by the “normal coordinates” Qs(q, t ) , the matrix G(o)is diagonal with elements given by

The matrix elements of G in the representation generated by MJ’2ui:) are

The matrix A can be rewritten in the form

A=

@ ~ - 1

-

1)02

- ~ - 1 / 2 ( @ ) - c)M-’/~.

(29)

We consider now the case of a foreign atom of mass M ’ , less than M , the mass of the atom in a crystal that it replaces. If we assume further that the force constants remain the same, A is a matrix having zeros everywhere except at three positions along the diagonal in which the value of the matrix element is M‘ - M o2c 0. A = M The eigenfrequencies of the imperfect crystal are obtained equating a 3 x 3 determinant to zero, namely

where the indices CY and a’are those corresponding to the atom replaced by the foreign atom of mass M ‘ . On the basis of the above theory the presence of a Mn atom replacing Cd in CdTe will result in a high-frequency local mode. On the other hand, Cd in the hypothetical MnTe of zinc-blende structure can have only a gap mode. The variation of the local mode of Mn in CdTe as a function of x can be calculated, at least for small x , by considering the mutual interactions of the local modes centered around the different Mn ions. This problem has been

9.

361

RAMAN SCATTERING

discussed by Maradudin and Oitmaa (1969). We give first a simple approach similar to that used by Frohlich (1958) in the context of the long-wavelength polar modes in an ionic crystal. A more complete theory is presented later. When the lattice is excited at frequencies near 0 0 , the angular frequency of the local mode, the optical modes of the host are quiescent. If u is the displacement of the Mn atom in a long-wavelength excitation, we have

d2u MI= -M'w;u dt2

+ zeEp,

where Ep is the local electric field at the position of a Mn ion and ze is the effective charge of the Mn ion upon displacement. We have assumed here that the local mode has an inertia equal to that of the free Mn atom ; in any case, (z/M') can be viewed as a phenomenological constant. The local electric field Ep with the system having uniform polarization P is

Ep = E

4n + -P, 3

(33)

where E is the macroscopic electric field inside the material. Each Mn ion gives rise to a dipole moment zeu. Since the probability that a cell be occupied by a Mn atom is x , the contribution to the polarization is (xzeu/uo), where uo is the volume of the primitive cell. However, the solid experiences an electronic polarization, which for frequencies below the optical absorption edge gives rise to a screening of the dipole moments associated with the local modes. This results in a polarization of the medium given by

where n is the index of refraction and = n2 is the so-called high-frequency dielectric constant. Neglecting the effect of retardation, we note that the fields E, P, and D = E + 4nP must obey the equations of electrostatics, i.e., V D = 0 and V x E = 0. In a wave with wave vector q these quantities vary as exp(iq - r - i d ) . Thus q D = 0 and, for a longitudinal wave, this yields D = 0, and hence

-

-

E = -4nP.

-

(35)

For a transverse wave, on the other hand, V * E = - 4niq P = 0 which, combined with V x E = 0, results in E = 0. Hence, for transverse waves, 4n Ep = -P. 3

362

A . K . RAMDAS A N D S. RODRIGUEZ

and for longitudinal waves

8n Ep=--P 3

.

(37)

Thus, from Eqs. (32)’ (34)’ (36), and (37), we find that for transverse (TO) waves

while for longitudinal (LO) waves

For

Eqs. (38) and (39) yield

respectively. The theory outlined above has the advantage of simplicity and provides a satisfactory account of the two-mode behavior of Cdl-,Mn,Te. It neglects the polarization due to ionic displacements of the Cd and Te atoms and all electronic polarizabilities except for the use of the index of refraction n. We outline below the theory proposed by Genzel et al. (1974), that also considers the displacements (for q = 0) of all three types of atoms and the corresponding restoring forces. The model is based on the following equations of motion for an A B 1 - G mixed crystal of zinc-blende symmetry, taking into account nearest-neighbor and second-neighbor interactions:

mAUA = -(I - x)fb(uA -

UB)

- Xfc(UA - UC)

+ [(l - x)eb + eclEr, mBuB = -fb(uB mciic

=

UA)

- xfs(uB - UC) - ebEt,

-fC(uc - UA)- (1 - x)Js(uc - UB)- ecEe,

(42) (43) (44)

9.

363

RAMAN SCATTERING

and

ffA

+ UC vo

Ej.

(45)

In these equations, m ,u, and a are the masses, displacements, and electronic polarizabilities of the ions A, B, and C, respectively. The volume of the primitive cell, vo ,will vary with composition and is given by vo = a3/4, where a is the lattice parameter of the zinc-blende crystals. The subscripts b and c on the force constants f and the Szigeti-effective charges, e, refer to the compounds AB and AC, respectively. Equations (42)-(45) are those of Genzel et al. (1974), with additional terms involving the second-neighbor force constant between the B and C ions, fs. Based on the experimental evidence that the extrapolation of the frequencies of the phonon modes in Cdl-,Mn,Te and Znl-xMnxTe to x = 1 yields the same values, the force constants between the Mn and Te ions, f c , must be the same in both alloy systems as x + 1. Hence, it is necessary to let the force constants exhibit a dependence on the lattice parameter. This dependence is approximated by a linear function

where ac is the lattice parameter of the AC compound (cubic MnTe in this case), a(x) is the lattice constant of the alloy with Mn concentration x and F and 0 are constants. The alloy systems, Cdl-,MnxTe and Zn1-,MnxTe, have similar chemical bonding, and, hence, 0 is assumed to be the same for both. Since the local electric field is given by Eq. (33), the relationship between Ep and the electric polarization P is Eo

=

P/(

(47)

where [ = 3/4n and -3/8;rr for transverse and longitudinal modes, respectively. As demonstrated by Genzel et al. (1974), the microscopic parameters in Eqs. (42-45) are related to macroscopic parameters according to the usual Born-Huang procedure (1968).

364

A . K . RAMDAS A N D S . RODRIGUEZ

Here fib and p c denote the reduced masses of ions A and B, and of ions A and C, respectively; v b and vc are the corresponding volumes of the primitive cells. The static and high frequency dielectric constants of the crystals AB and AC are given by &Ob ,&mb ,E ~ and ~ E, ~ and ~ the , corresponding TO phonon In addition to the force frequencies at zero wave vector are o m and constants, the polarizabilities and effective charges should also exhibit an xdependence, since the nearest-neighbor distances change with x. To a first approximation, these effects will be neglected and the force constants are assumed to have a linear dependence on the lattice parameter. The eigenfrequencies of Eqs. (42)-(44) have the form where

and = VOt

In the limits of x

=

-

CYA

- (1

-

x)aB

-

XaC.

(58)

0 and x = 1, the frequencies of the impurity modes will

9.

365

RAMAN SCATTERING

be given by the following expressions: x = 0:

Fe = [-&-(I

arc

+ Fs

a, - ab

+4 3 7 4

1/2

’

(59)

The microscopic parameters of Eqs. (54)-(58) are determined from the BornHuang relationships of Eqs. (48)-(53) and the boundary conditions of Eqs. (59) and (60).

3. EXPERIMENTAL RESULTS AND DISCUSSION Figure 3 shows the room temperature Raman spectrum of Cdo.6MnoaTe (Venugopalan et al., 1982). The scattering configuration is y’(z’z’)x’ where x‘,y ‘ , and z’ are along [loo], [Oll], and [Oil], respectively; it allows the observation of the Raman active phonon features having A I , E, and FZ symmetries. The first order Raman spectrum of pure CdTe consists of a pair of LO-TO lines of F2 symmetry occurring at 140 and 171 cm-’, respectively (Mooradian and Wright, 1968; Selders et al., 1973). We note that the alloy exhibits a rather intense and quasicontinuous spectrum below 130 cm-’, in addition to the two pairs of relatively sharp lines, one at 143 and 158 cm-’ and the other at 189 and 203 cm-’. This clearly shows that the mixed crystal has a richer and more complex spectrum as compared to that of CdTe. 30

0

4

,

1

I

I

I

I

I

I

50

I00 150 200 RAMAN SHIFT (cm-I) FIG. 3. Raman spectrum of Cdo.6Mno.aTeat 295 K, excited with 6764 A Kr’ laser line. x’, y’, and z’ are along [loo], [Oll], and [Oil], respectively. The polarization geometry and the allowed phonon symmetries are also indicated. [After Venugopalan et at. (1982).]

366

A. K. RAMDAS A N D S. RODRIGUEZ

a. Zone Center Opticai Phonons

Polarization features as well as concentration dependence of the spectra provide further insight into their origin. To the extent that the mixed crystal as a whole can be considered to have Td point group symmetry, the spectrum in Fig. 3 can be analyzed in terms of the different Raman active species viz., A 1 , E, and F2. Such studies show that the 143-158cm-’ and the 189203cm-’ pairs are LO-TO split FZ modes. In other words, Cdl-,Mn,Te exhibits a “two-mode” behavior (Barker and Sievers, 1975). Based on the zone center LO-TO F2 modes of pure CdTe, one can identify the lower frequency pair as the respective TO and LO modes characteristic of the CdTe component of the mixed crystal; they are labeled “CdTe-like” lines. Similarly the higher frequency pair can be regarded as the “MnTe-like” TO and LO modes characteristic of the MnTe component of the alloy. The variation of these LO and TO modes as a function of x is shown in Fig. 4. It is seen that as x -+ 0, the MnTe-like modes merge toward one that can be regarded as the triply degenerate local mode of Mn in pure CdTe. Similarly, the CdTe-like modes are expected to become degenerate for x = 1, at which point they must correspond to the gap mode of Cd in the “hypothetical” MnTe crystal with point group Td. The extrapolations shown in

00

02

04

06

08

10

08

06

04

02

00

X FIG.4. The frequencies of the Cdl-,Mn,Te and Znl-,MnxTe zone-center optical phonons

at T = 80 K. The curves were generated using the MREI model described in the text. [After Peterson er a/. (1986).]

9.

RAMAN SCATTERING

367

Fig. 4 beyond x = 0.7 reflect these assumptions. We note here that Cdl -xMn,Te occurs in a homogeneous, zinc-blende phase over 0 Ix 5 0.75; pure MnTe crystallizes in the hexagonal nickel arsenide structure with C& symmetry and in this context the zinc-blende structure for MnTe as the end member of Cdl-,Mn,Te is hypothetical. Experimental results for the vibrational Raman spectra for Znl -*Mn,Te, another zinc-blende DMS, are presented in Figs. 5 and 6 (Peterson et al., 1986). We first consider the Raman lines which can be traced to the zonecenter LO and TO phonons of ZnTe as x 0 and those that evolve from the impurity mode of Mn in ZnTe as x increases. As can be seen in Fig. 5 , an inflection labeled I appears on the low frequency side of the LO mode in Znl-,Mn,Te, x = 0.05; we identify it with the band mode of Mn in ZnTe, -+

RAMAN SHIFT (crn-’1 FIG. 5. Room-temperature Raman spectrum of Znl-,Mn,Te for (a) x = 0.003 and (b) x = 0.05; AL = 6471 A. The scattering geometriesare (a) z’(y’y’)x’ and (b) x’(y’y‘)z‘,where x ’ , y’,and z’ are along [100], [Oll], and [Oli],respectively. The labeling of the phonon features is discussed in the text. [After Peterson el a/. (1986).]

368

A . K . RAMDAS AND S. RODRIGUEZ

30

90

I50

210

270

RAMAN SHIFT (cm-’)

FIG.6 . Room-temperature Raman spectrum of Znl-,Mn,Te, x = 0.37; l r = 6471 A. The scattering geometries are (a) x’(y’y’)z’ and (b) x’(y‘x‘)z’, where x’, y ’ , and z’ are along [loo], 10111, and [Oli]. respectively. [After Peterson ei al. (1986).]

since it lies between the TO and LO frequencies. As the Mn concentration is increased, the impurity mode splits into a transverse (TO1)and a longitudinal component (LOZ), as illustrated in Fig. 6 for x = 0.37. The frequencies of the TO and LO modes as a function of x are shown in Fig. 4. In contrast to that observed for Cdl-xMn,Te, the Znl -,Mn,Te phonons exhibit a mixed mode behavior intermediate to the one- and two-mode situations. For x 0, one observes the TO and LO modes of ZnTe and the band mode ZnTe :Mn. As x is increased, the band mode splits into the longitudinal mode LO2 and the transverse mode T01. The vibrational modes that evolve from the LO and TO phonons of ZnTe are designated as LO1 and TO2. As x 1, the LO1 mode, which is the LO phonon of ZnTe, evolves into the LO mode of MnTe. The TO1 phonon, one of the components evolving from the impurity mode +

-+

9.

RAMAN SCATTERING

369

ZnTe :Mn, becomes the TO vibrational mode of MnTe. The two remaining phonons, TO2 and L02, merge to become the gap mode of Zn in the zincblende MnTe (MnTe: Zn). As can be seen in Fig. 4, the extrapolation of the Cdl-xMn,Te and Znl-,Mn,Te phonons to x = 1 results in the same values for the frequencies of the TO and LO modes for the hypothetical zincblende MnTe. This effect has also been observed for other physical characteristics of these alloys. The lattice parameters (Furdyna et al., 1983) and the energy band gaps (Lee, Ramdas, and Aggarwal, 1987; Brun del Re et al., 1983) of Cdl-xMn,Te and Znl-xMnxTe extrapolate to the same values for x = 1. The curves in Fig. 4 were determined from the modified random element isodisplacement (MREI) model. The fundamental assumptions of the random element isodisplacement model are that in the long wavelength limit (q 0), the anion and cation of like species vibrate with the same phase and amplitude, and that the force each ion experiences is provided by a statistical average of the interaction with its neighbors. This MREI model is a modification of that developed by Genzel et al. (1974) which emphasizes the use of the local field and is completely defined by the macroscopic parameters of the pure end members. The modified theory is described in the theoretical discussion given above and embodied in Eqs. (42)-(60). The additional constraint that the frequencies of the LO and TO modes for Cdl-,Mn,Te be equal to the corresponding modes of Znl-,MnxTe when x = 1 allows one to incorporate second-neighbor force constants and a linear dependence of the force constants on the lattice parameter into the model without resorting to microscopic fitting parameters. The only fitting parameters necessary for this model are the frequencies of the TO and LO modes of MnTe and of the gap modes MnTe :Cd and MnTe :Zn. The frequencies of the vibrational modes in Cdl-,Mn,Te and Znl-,Mn,Te as a function of x (Fig. 4) were determined from the macroscopic parameters of Table VI. The resulting force constants are also given in the table. The magnitude of the second-neighbor force constants is about one third that of the nearest-neighbor constants. As illustrated in Fig. 4, the curves generated from this MREI model follow the experimental results quite well, except for the TO1 modes for which there is a significant discrepancy in the curvature of the theory and the experimental results. However, considering the simplifying assumptions of this model, it is gratifying to note that the theory provides an adequate description of the phonon frequencies in CdI-xMn,Te and ZnI-,MnxTe over the entire composition range. The theoretical model used by Venugopalan et al. (1982) for interpreting the composition dependence of the optical phonon frequencies in Cdl-xMnxTe is a simplification of the model presented here and, though strictly valid only for small x, gave an adequate description of the distinct two mode behavior in Cdl-,Mn,Te.

-

310

A. K . RAMDAS AND S. RODRIGUEZ

TABLE VI PARAMETERS IN THE MREI MODEL Cdl -,Mn,Te

(experimental)"

wTo(CdTe) = 147 Cm-' wLo(CdTe) = 173 cm-' w,(CdTe :Mn) = 195 cm-' &o(CdTe)= 9.6 o = 6.486 - 0 . 1 4 5 ~ Ab

Znl-,Mn,Te

(experimental)"

wTo(ZnTe) = 181 Cm-' wLo(ZnTe) = 210cm-' wI(ZnTe : Mn) = 108 cm-' &o(ZnTe)= 10.1 o = 6.103 + 0 . 2 3 8 ~Ab

Resultant force constants [lo6a.m.u.(cm-')z]: = 1.72 Fc~-T = ~1.85 FZ"-T~= 1.57 FE.~"-T~

"fitting parameters" wTo(MnTe) = 185 cm-' wLo(MnTe) = 216 cm-' wI(MnTe :Cd) = 147 cm-' or(MnTe: Zn)= 171 cm-'

Fcd-Mn = 0.58

Fz"-M"= 0.34

T = 80K.

bFrom Furdyna et al. (1983).

b. Low Frequency Phonon Features

The room temperature Raman spectrum of Znl-,Mn,Te, x = 0.003, shown in Fig. 5(a) is essentially that of ZnTe, which has been studied in detail (Weinstein, 1976). The prominent features of the low frequency two phonon density of states spectrum have been attributed to two transverse acoustic phonons at point L of the Brillouin zone [2TA(L)], two TA modes at X [2TA(X)J, the difference mode of the transverse optical and acoustic phonons at X [TO-TA(X)J, and two acoustic phonons of the second type at approximately point K [2A(K, 2)]. In addition to the transverse optical (TO) and the longitudinal optical (LO) phonons, there are two higher frequency features that have been identified as the combination of the LO phonon with transverse acoustic phonons at points L and X of the Brillouin zone, LO + TA(L) and LO + TA(X), respectively. The Raman shifts of the low frequency features are given in Table VII. TABLE VII Low FREQUENCY PHONON FEATURES IN Znl -,Mn,Te

0.003 0.05

0.13 0.19 0.37 0.60 0.70

53

54

73 72

50

69

84 82 82

109 109 108 108 107 101

99

(IN

cm- ')

119 119 119 120 120 120 120

150

150 148 146 143 140 138

9.

371

RAMAN SCATTERING

The room temperature vibrational Raman spectrum for Zn, -xMnxTe,

x = 0.05, is shown in Fig. 5(b). The low frequency Raman spectrum for x = 0.05 is similar to that for x = 0.003, except that the 2TA(L) mode has

decreased in frequency and there is an increase in the relative Raman intensity in the region of the difference mode TO-TA(X). The label B reflects the emergence of a new feature very close to the TO-TA(X) mode and the evidence for this is apparent in the spectra of the higher composition samples. In addition, weak disorder-induced one-phonon features are observable having Raman shifts < 75 cm-'. The trends continue as the Mn concentration is increased to x = 0.13 and x = 0.19 with the TA(L), TA(X), and A(K, 2) phonons decreasing in frequency. The intensity of the vibrational mode B continues to increase. The low frequency disorder-induced features are attributed to the one-phonon density of states with the peaks being identified as TA(X) and A(K, 2); these features also increase in intensity as x increases. When the heavier Cd atom is replaced by Mn in Cdl-,Mn,Te, one expects that below the TO mode frequency of CdTe, there should appear new features corresponding to band modes. This is borne out by the pronounced low frequency modes seen in Figs. 3 and 7 in the 20-130 cm-' range. These features persist over the composition rangex = 0.4-0.7, but are considerably weaker for x 5 0.3. Also, for any given x , the higher the sample temperature the greater is their intensity. The presence of Mn at random sites within the lattice effectively destroys the translational periodicity in the mixed crystals. Thus, although only zone-center (q 0) optical phonons are allowed in the first-order Raman spectrum of a perfect crystal, the q 0 selection rule is relaxed in the alloys. Consequently, they may exhibit one-phonon Raman scattering due to optic and acoustic phonons with all possible q vectors spanning the Brillouin zone. As a substitutional impurity, Mn retains the site symmetry Td in the mixed crystal. Hence, the new impurity-induced features of the spectrum should again belong to the Raman-active representations of &, viz., A I , E, and Fz. Figure 7 displays the striking polarization behavior of these low frequency modes. The asymmetric peak centered at 42cm-' in Fig. 7(a) exhibits a predominant polarization characteristic of transverse F2 modes. While investigating the side-band absorption associated with the local mode of Be in CdTe, Sennett et al. (1969) performed a shell model calculation of the weighted density of one-phonon states, S(o)/02, for CdTe. We note that in the 20-130 cm-' range the main features of Fig. 7(a) bear a striking similarity to their calculated function S(o)/oz,shown as an inset in the same figure. The one-phonon density of states in CdTe has also been determined experimentally by Rowe et al. (1974)using neutron inelastic scattering. Their results reveal a prominent peak at -40cm-' due to transverse acoustic (TA)

-

-

372

A. K . RAMDAS AND S . RODRIGUEZ

151

I

I

50

I

I

too

I

I

I50

I

I

I

200

RAMAN SHIFT (cm-’) FIG.7. Polarized Raman spectra of Cdo.sMn0.4Te.(a) y’(x’z’)x’;(b) y(xz)y. The significance of the inset in (a) is discussed in the text. The crystallographic axes are the same as in Fig. 3 for (a). For (b), x, y , z are the cubic axes. [After Venugopalan et al. (1982).]

-

phonons and a less intense peak at 110 cm-’ due to longitudinal acoustic (LA) phonons. Furthermore, their comparison of the frequency distributions of CdTe and InSb shows that as compared to InSb, the optic and LA modes of CdTe show a clear “softening”, but the TA modes lie closely within the same range of frequencies for both compounds. We therefore conclude that the peak in Fig. 7(a) at 42 cm-’ originates due to disorder-activated, firstorder Raman scattering by TA phonons and that it reflects the corresponding TA phonon density of states in the mixed crystal.

9.

RAMAN SCATTERING

373

The Raman spectra of Figs. 5 and 6 demonstrate that the low frequency disorder-induced features become more intense in Znl -,Mn,Te as the disorder in the lattice increases. The lack of strict translational symmetry in the alloy allows the observation of Raman features associated with nonzero wave vector excitations. However, since the masses of Zn and Mn are more closely matched, the deviation from strict translational symmetry is not as severe as in Cdl-xMnxTe, for which there is a large difference in the masses of Cd and Mn. This conclusion is based on the observation that the onephonon density of states features dominate the low frequency Raman spectra in Cdl-,Mn,Te, in contrast to the situation that prevails for Znl-,Mn,Te.

V. Magnetic Excitations 4. GENERAL CONSIDERATIONS

The incomplete d-shell of the magnetic atoms in a DMS gives rise to a variety of properties in which their localized magnetic moments play important roles, either individually or collectively through their mutual interactions. For sufficiently low concentrations, x , of the magnetic ions, or at temperatures above a critical temperature TN, the material is in a paramagnetic phase. The temperature TN is, of course, a function of x . A transition to an ordered phase occurs below TN.In these materials, the interaction between neighboring magnetic ions is antiferromagnetic, giving rise to a spin-glass or to an antiferromagnetic phase. Galazka et al. (1980), using specific-heat and magnetic-susceptibility measurements, found that Cdl-,Mn,Te crystals are paramagnetic at all temperatures for x < 0.17. For compositions in the range 0.17 < x < 0.75, the crystals are paramagnetic at high temperatures, and, as the temperature is lowered, a spin-glass is obtained for 0.17 < x < 0.60, while, for 0.60 <x < 0.75, a paramagnetic-to-antiferromagnetic phase transition occurs. The theory of Raman scattering by magnetic excitations is similar to that by phonons studied in Sec. IV. However, because of the axial nature of the magnetic field and of the magnetization M, the selection rules for Raman scattering differ from those associated with symmetric polarizability tensors. In a magnetic system, the electric polarizability is a functional of the magnetization as well as of the other variables describing internal modes of motion. Thus we write

P

=

x(M) * EL.

(61)

Several microscopic mechanisms for the dependence of x on M can be envisioned. Magnetostriction can conceivably be one but it is likely to yield

374

A. K . RAMDAS A N D S. RODRIGUEZ

extremely small scattering cross sections. Exchange interactions with itinerant or localized electrons, having energies comparable to electrostatic interactions, are expected to be important. This suggests that the Raman features associated with magnetic excitations should exhibit strong resonance enhancement when the energy of the quantum ~ O Lof the exciting radiation is near the energy of an electronic transition, e.g., the direct energy gap. The modulation of P resulting from magnetic excitations is obtained from a Taylor series expansion of the functional x(M). We write p

= p(0) + p(l) + p(2) +

...

(62)

where the successive terms are independent of M, linear in M, second order in M, ... . The restriction imposed by the Onsager reciprocity relations, the lack of absorption in the frequency region of interest and the cubic symmetry require that (see, for example, Landau and Lifshitz, 1960)

P(')= iGM x EL,

(63)

where G is a constant. The scattering cross-section for a Raman process is proportional to J E S * P12,

where, as before, ES is the direction of polarization of the scattered radiation. Thus the scattering cross-section involving a magnetic excitation in the first order is of the form o = CI(&sx EL) MI2,

(64)

where C is an appropriate function of OL and os. Thus Raman scattering does not occur when the polarizations of the incident and scattered radiation are parallel. In the presence of a magnetic field H,M varies according to the Bloch equation dM - - yM X H, dt y being the gyromagnetic ratio, i.e., the ratio of the magnetic-moment and the angular-momentum densities. For electrons y = -ge/2mc is negative. Considering only inelastic scattering, we need only keep the time-dependent components of M. The solutions of Eq. (65), taking H parallel to the z-axis, are such that Mx f iMy

vary as exp[riyHl], respectively. Since Mz and M 2 are constants of the motion, we can write M

=

(Msin 8 cos at, Msin 8 sin at, Mcos e),

(66)

9.

375

RAMAN SCATTERING

where

S2

=

-7H

(67)

is the Larmor frequency and 8 the angle between M and H. We consider incident radiation propagating parallel to the z-axis selected along H. For circularly polarized radiation 6+ and 6- we write

EL = (2i ij)Eo exp[- i w ~ t ] .

(68)

Using Eq. (63), we obtain

P(l) = r&GMEosin0 exp[-

~ ( W L=F

S2)tI.

(69)

This shows that in this geometry there is a Stokes line with polarization (6+, 2) and an anti-Stokes line with (6-,2). In a similar way, if the incident wave propagates at right angles to H but is polarized along H,

EL = E02 exp[- i ~ ~ f ]

(70)

and

P(') = ~ G M Esin o 8[(f - ij)exp[-

- (f + ij)exp[-

~(OL

~(WL -

Q)t]

+ S2)t]].

(71)

Thus Stokes and anti-Stokes lines occur in the geometries (2,6-) and (2, 6+), respectively. The two cases described above are, of course, related to each other by time reversal symmetry. In order to describe the magnetic excitations observed in Raman scattering in terms of microscopic models, it is useful to consider the Hamiltonian of the MnZ+ions interacting with one another and with either band electrons or electrons bound to donors. We designate the spin of a Mn2+ ion at the site R ; by S; and the spin of the electron by s. In the presence of a magnetic field H, the Zeeman energies of the Mn2+ ion and of the electron are g,uB H Si and g*pB H s where pug is the Bohr magneton and g and g* are the Land6 g-factors of Mn2+and the electron, respectively. In addition there are exchange interactions between Mn2+ions at Ri and R j o f the form -2JijSi S j and between an electron and the MnZ+ions: The Hamiltonian of an electron in mutual interaction with Mn2+ ions is

-

H =

a

--(Y

C Si*Sly/(Ri)12+ g*j.lBH'S + gj.lBH*C S; - C 2JijSj.Sj. i

i

i<j

(72) Here yl(Ri) is the electronic wavefunction normalized over the primitive cell and evaluated at R ; ; aNo is the s-d exchange integral, NObeing the number of primitive cells per unit volume.

376

A. K . RAMDAS AND S . RODRIGUEZ

5. PARAMAGNETIC PHASE We now consider Raman transitions between Zeeman sublevels of the individual Mn2+ ions in an external magnetic field, the sample being in its paramagnetic phase. In this phase the exchange interaction between Mn2+ ions is smaller than the thermal energy kg T and the ions can be considered as being independent of one another. The 6S5/2 ground state of the Mn2+ion has a total spin S = 5 / 2 , orbital angular momentum L = 0 and total angular momentum J = 5 / 2 . In this subsection we will discuss Cdl-,Mn,Te as an illustrative example. The cubic crystalline field (site symmetry 5)splits the six-fold degenerate ground state into a Tg quadruplet state at + a, and a r7 doublet at -2a, where 3a is the crystal field splitting. From electron paramagnetic resonance (EPR) experiments, Lambe and Kikuchi (1960) obtained 3a = 0.0084 cm-' for Mn2+in CdTe. This crystal field splitting is too small to be observed with the resolution of a standard Raman spectrometer and we treat the ground state of MnZ+in Cdl-,Mn,Te as an atomic 'S5/2 level. The application of an external magnetic field, H, results in the removal of the six-fold degeneracy of the ground state, the energy levels being E(ms) = gpBHms. Here ms, the projection of S along H, has the values - 512, - 312, ..., + 5 / 2 . These energy levels form the Zeeman multiplet of the ground state of Mn2+. In the paramagnetic phase, Raman scattering associated with spin-flip transitions between adjacent sublevels of this multiplet has been observed by Petrou et al. (1983). The results in Cdl-,Mn,Te are shown in Fig. 8 for x = 0.40. As can be seen, a strong Stokedanti-Stokes pair is observed with a Raman shift of UPM = 5.62 f 0.02cm-' at room temperature and H = 60 kG. Taking H and incident light parallel to 2 and denoting light polarization with positive and negative helicity b+ and b- , respectively, the Stokes line is observed in the (I?,, 2) configuration, whereas the anti-Stokes line is seen in (6-, 2). When the incident light propagates at right angles to H(2), the Stokes component appears in the polarization (i?,b-),while the anti-Stokes is observed in (2,&+). Within experimental error the frequency shift is linear in H . With the energy separation between adjacent sublevels of the Zeeman multiplet given by AE = gPBH = ~ U P M ,it is found that g = 2.01 f 0.02. The Raman line at UPM in Cdl-,Mn,Te has been observed for a variety of compositions ranging from x = 0.01 to x = 0.70. Following the arguments given by Fleury and Loudon (1968) one can consider, as a possible mechanism for the UPM Raman line, a two step process having as the intermediate state one of the excited states of the Mn2+ ion (L = 1, S = 5/2). Figure 9 shows such mechanisms for the Stokes and antistokes components of the UPM line. For the Stokes component an incident photon of energy tzu~ and polarization 6, induces a virtual electric

9. 40 r

377

RAMAN SCATTERING

- F,z, A

__

h

u- , z

T.300 K H.60 kG

10

5 0 5 RAMAN SHIFT ( c m ' )

10

FIG.8. Stokes (S) and anti-Stokes (AS) Raman lines at WPM resulting from Ams = 1 spinflip transitions within the Zeeman multiplet of MnZ+in Cdl-,Mn,Te, x = 0.40: -(a+, 8; (a_,2). The wavelength of the exciting laser line A L = 6764 A; the applied magnetic field H = 60 kG; the temperature T = 300K; x , y , and z are along [OOI], [ITI], and [IIO], respectively. [After Petrou et al. (1983).]

_._.

dipole transition between an initial and an intermediate state which differ by AmJ = + 1; it is followed by a second electric dipole transition between the intermediate and final states with AmJ = 0. This is accompanied by the emission of a scattered photon of polarization Zand energy AUS = AUL - AUPM . At the end of this process the Mn2+ ion is in an excited state within the Zeeman multiplet differing from the initial state by Ams = + 1. The appearance of the Stokes component in the (2, &) configuration is also illustrated in Fig. 9. Similar processes can be visualized for the anti-Stokes component having the (a_,2) or the ( Z , 6 + ) polarization. This mechanism correctly predicts the experimentally observed polarization characteristics of the Stokes and anti-Stokes components of the WPM Raman line. We note that

378

A. K . RAMDAS A N D S . RODRIGUEZ

L=0 s= 512 mJ=mS J = 512

mJ+I

I '

FIG.9. Raman mechanism for the WPM line involving the internal transitions of the Mn2+ ion. The arrows indicate virtual electric-dipole transitions. The energy level scheme is not to scale and the energy difference between the excited and ground states EOis much greater than ~ W P M . [After Petrou et al. (1983).]

the selection rules are immediate consequences of conservation of angular momentum of the system comprised of the Mn2+ions and the photon field. It should also be pointed out that the Stokes scattering process in the ( 8 + ,2) configuration is the time reversed conjugate of the (?,a+)anti-Stokes process 2) anti(Loudon, 1978); in the same manner, the (2,8-)Stokes and the (6-, Stokes processes are related by time reversal. Exploiting the variation of the band gap with manganese concentration and/or temperature, it is possible to match the band gap of several samples with the energy of one of the discrete lines of a Kr+ laser. In addition, a dye laser can also be used to achieve resonant conditions. It is found that the intensity of the OPM line increases by several orders of magnitude as the laser photon energy approaches that of the band gap. The observation of this resonant enhancement in the intensity of the OPM Raman line prompts the consideration of a mechanism involving interband transitions. It involves the Mn2+-bandelectron exchange interaction described by the first term in the Hamiltonian in Eq. (72) (Gaj et al., 1978a). The term Si s can be written as

-

Si* s

= S?)$(Z)

+

+Si(+)$(-) + +&W$(+).

(73)

here S f * ) and s(*) are the spin raising and lowering operators for a Mn2+ion and band electron, respectively, and S?) and s(') are the corresponding projections of spin along 2. The second term of Eq. (73) raises the spin of a Mn2+ion while simultaneously lowering the spin of a band electron, i.e., (mS)Mn2+lmJ)e

+

ImS

+

l)MnZ+ImJ- 1 ) e .

(74)

379

9. RAMAN SCATTERING

In a similar fashion, the third term lowers the spin of an ion while raising the spin of a band electron, i.e., ImS)Mn2+(mJ)e4ImS.- 1)Mn2+(mJ-b 1)e.

(75)

Hence, these terms can induce simultaneous spin-flips of the band electrons on the one hand and the Mn2+ ions on the other, corresponding to Arns(Mn2') = f 1 and AmJ(e)= =F 1. In Fig. 10 we show the above mechanism for both the Stokes and the antiStokes component and the two right-angle geometries considered above. In the presence of a magnetic field the Ts valence band splits into four subbands with m~ = - 3/2, - 1/2, + 1/2, and + 3/2, and the l-6 conduction band splits into ~ T Z J= + 1/2 and - 1/2 subbands. The possible processes for the Stokes 2) configuration are shown in Fig. 10a. In component appearing in the (6+,

Stokes

anti -Stokes

FIG.10. Raman mechanism for the WPM line involving the band electrons: CB and VB refer to the conduction and valence bands, respectively, which are labeled by the electronic quantum number r n .~The single arrows indicate virtual electric-dipole transitions, while the double arrows refer to transitions induced by the electron-Mn*+ exchange interaction. The allowed (c), while those for the antipolarizations for the Stokes component are (8,.2): (a) and (i,&-): Stokes are (&-, .?): (b) and (2, &+): (d). [After Petrou et al. (1983).]

380

A. K . RAMDAS AND S . RODRIGUEZ

the first process an incident photon of polarization 8+ is absorbed, raising an electron to the conduction band with AmJ = + 1 and creating a hole in the valence band. In the second step the excited electron interacts with a Mn2+ ion via the second term of Eq. (73), resulting in Arns(Mn2+) = + 1 and ArnJ(e) = - 1. Finally, the electron and hole recombine emitting a photon of energy h a s = A ~ -L hap^ of polarization 1;the band electrons have thus returned to their ground state, but leaving the MnZ+ion excited to the next sublevel of the Zeeman multiplet. In the other two processes shown in Fig. 10a, the hole, rather than the excited electron, interacts with the Mn2+ ion, resulting, however, in identical polarization selection rules. In the same manner the Stokes component in the (2,8-) configuration follows from Fig. 1Oc. The anti-Stokes processes for the (8-,0)and the (1,8+)geometries are shown in Figs. 10b and 10d, respectively. All the observations discussed above and the predictions of the microscopic models considered are in accord with the general phenomenological selection rules embodied in Eqs. (68)-(71). We also note that this entire phenomenon is electron paramagnetic resonance observed as Raman shifts, i.e., it is Raman-EPR. In the spirit of the above discussion one can also visualize a mechanism in which the Mn2+-electron and Mn2+-holeexchange interactions result in a similar Raman line but with a shift of h p M . The processes resulting in a Raman shift of 2WpM are shown in Fig. 11. For example, in Fig. l l a , a photon of polarization 8, is absorbed, virtually exciting an electron from the mJ = - 1/2 valence subband to the mJ = + 1/2 conduction subband. The excited electron then interacts with a Mn2+ion resulting in Ams(Mn2+) = + 1 and ArnJ(e) = - 1, while the hole in the valence band interacts in the same manner yielding Ams(Mn2+) = + 1 and AmJ(e) = - 1. Figures 1l b and 1l c show how a Stokes shift of 20pM might arise exclusively through a Mn2+valence electron exchange interaction; from Eq. (73) it is clear that such an interaction will involve two successive Mn2+ spin flips. Due to the extended wavefunctions of both the conduction electron and the hole it should be more probable that this process involves two Mn2+ ions rather than just one. Similarly, the anti-Stokes components of the mechanism are those presented in Figs. l l d - l l f . Energy conservation requires that the energy of the scattered photon be h o s = AWL T 2wPM. The polarization selection rules predicted by this mechanism are (8+, 8-)for the Stokes and (8-,&+) for the anti-Stokes components of the Raman line at h P M . This mechanism is expected to be significant only close to resonance. The Raman line at 20PM has been observed under resonant conditions for a variety of compositions. The results for Cdl-,Mn,Te with x = 0.10 are shown in Fig. 12; in this case, resonant conditions were achieved by maintaining a sample temperature of 120 K and using the 7525 A line of the Kr+

9.

RAMAN SCATTERING

Stokes

381

(&+,&-I

+ 1/2

-3/2 -112

+ 1/2 +3/2

(a 1

(C)

anti - Stokes (&-,&+) + 1/2

- 3/2

- 1/2 + 1/2

+3/2

(d) FIG. 1 1 . Raman mechanism for the Ams = i=2 spin-flip transition, i.e., the 2mPM line, involving the band electrons. The Stokes line is allowed in (6+, 6-)polarization, whereas the anti-Stokes appears in (6-, 6+).[After Petrou et nl. (1983).]

laser as the exciting radiation. The forward scattering geometry was used in order to obtain the (a,, 6-) and (6-, 6+)configurations. As can be seen in Fig. 12, the Stokes and the anti-Stokes lines appear with the expected polarizations. Under the resonant conditions of this experiment the OPM line is extremely intense (- lo6 counts/sec); although this line is forbidden for these polarizations, it is not surprising that the leakage observed is quite strong. It is known that electrons and holes in polar crystals interact strongly with zone center longitudinal optical (LO) phonons through the Frohlich interaction (see Hayes and Loudon, 1978). An LO phonon can be created or annihilated as a result of such an interaction. Referring to the mechanism responsible for the OPM line, shown in Fig. 10, one can visualize a fourth step

382

A . K . RAMDAS A N D S. RODRIGUEZ

RAMAN SHlFTkm-') FIG.12. Stokes and anti-Stokes components of 2WPM in Cdl-,Mn,Te, x = 0.10, recorded for the (a+,&) and (a_,8+)polarizations. The sample temperature T = 120 K, H = 60 kG, and A L = 7525 A; x , y , and z are along [I lo], [TlO], and (0011, respectively. The WPM line appearing in the spectra is due to leakage. [After Petrou ef al. (1983).]

in which the excited electron or hole interacts with the lattice and creates or annihilates an LO phonon. Such a mechanism would result in a scattered photon with a Raman shift of WLO f W P M . The net result for the Stokes process with a shift of WLO + WPM is that an LO phonon is created and a Mn2+ion is excited to the next sublevel of the Zeeman multiplet. A Stokes shift of WLO - WPM corresponds to the creation of an LO phonon and the de-excitation of a Mn2+ion to the next lower sublevel of the multiplet. The W L O + OPM Stokes Raman line is expected to appear in the (&+ ,2) or the ( f , b - ) configurations, whereas the WLO - WPM Stokes line is allowed for (8- , 2) or (2,8+). In addition to the above, the creation or annihilation of two LO phonons in a similar process is also possible. An electron or a hole can create or annihilate two LO phonons in a single step. Another possibility is that an electron as well as a hole each separately create an LO phonon. In either case, the scattered photon can have the following Raman shifts: 2 ~ ~ f0WPM , , WLO, + WLO, f WPM or 2 ~ ~ f0W,P M , where LO1 refers to the CdTe-like and LO2 to the MnTe-like zone center longitudinal optical phonons (Venugopalan et al., 1982).As before the Stokes lines involving the excitation of a Mn2+ion are expected in the (a+,2) or (2, 8-) polarization geometries, while those involving a de-excitation of a Mn2+ ion appear in (8-,2) or (2, 8+).

9.

RAMAN SCATTERING

383

The new lines described above should occur only under conditions of bandgap resonance. Under such conditions we have indeed observed the new Raman lines with shifts of WLO f WPM in Cdl-,MnxTe for a variety of compositions. The Raman spectra in the region of the longitudinal and transverse optical (TO) vibrational modes are shown in Fig. 13 for Cdl-,MnxTe with x = 0.10. The 7525 A Kr+ laser line was used to excite the spectra. The zero magnetic field LO and TO phonon spectrum is shown in Fig. 13b. Here the CdTe-like TO and LO and the MnTe-like LO modes are quite distinct, while the MnTe-like TO appears as a shoulder to the LO. The corresponding Raman spectra, recorded in the presence of a magnetic field of 60 kG and in the (2,6+) and (2,8-) configurations, are presented in Fig. 13a and 13c, respectively. The additional Raman lines with Stokes shifts WLO f up^ are clearly present with the proper polarization characteristics. The sample temperature was 120K; at this temperature, the sublevel occupation probability ratio of adjacent levels in the Zeeman multiplet of Mn2+is exp(- h ~ p ~ / Tk )g= 0.94. Thus the intensities of the WLO + WPM and the oL0- WPM lines are expected to be approximately equal. The WLO f WPM lines can be observed only when the exciting photon energy is

I I

24

z

H=GOkG

61d

L A ~~

"130

170

210

Ix)

170

~

21

RAMAN SHIFT (ern-') FIG. 13. Raman spectra of Cdl-,MnxTe, x = 0.10, showing the combination lines WLO, f WPM and W L O ~f W P M , where LO1 and LO2 are the "CdTe-like" and "MnTe-like" LO phonons. The sample temperature T = 120K,H = 60 kG, and IL = 7525 A; x, y, and z are along [IlO], [Ool], and [ITO]. respectively. [After Petrou e t a / . (1983).]

384

A . K . RAMDAS A N D S. RODRIGUEZ

strongly resonant with the band gap. There is no evidence of corresponding Raman lines associated with the TO phonons, which would have shifts of WTO f O P M . This supports the assumption that the Frohlich interaction is responsible for the appearance of the new features. The scattering amplitude for the Raman mechanism discussed above is proportional to the magnitude of the exchange coupling between the Mn2+ ions and the band electrons which is especially strong in these alloys. Adapting Loudon’s theory for optical phonons (Loudon, 1963, 1964), the scattering cross-section for such a three-step process can be written as being proportional to (ns

-k

1 , O(HeRlns,b > ( m s -I- 1 , b(ffexlms,a)(nL - 1, alffeRlnL, 0) (Ob

+ WPM

- WL)(Wo - U L )

+ 5 additional terms

l2

. (76)

Here, H e R is the electron-photon interaction Hamiltonian, He, is the exchange Hamiltonian describing the exchange interaction between 3dlocalized states of Mn2+and conduction (valence) electrons, n~ , ns denote occupation numbers of the incident and scattered photon modes, respectively, and ha,, hub are intermediate excitation energies of the virtual electron-hole pair. The other five terms in Eq. (76) are the remaining permutations of the three steps of the Raman process. It is clear that the first term will dominate when W o WL or O b (WL - WPM) = W S , producing a double peak in the frequency dependence of the cross section. The condition of “in resonance” results from the matching of the incident photon energy with that of an electronic excitation whereas “out resonance” occurs when the scattered photon energy equals the energy of such a transition (Barker and Loudon, 1972). In Eq. (76) h a , and hub are the energies of excitation of two states of the exciton with angular momenta differing by one unit. For example, following Twardowski et al. (1979), for a (2, 3+)Stokes line

-

-

+ B, hub = Ex - 3A + 3B hwa

=

Ex - 3A

(77)

where Ex is the exciton energy and A = +xNocY(SP”>,

B

=

+xNo/3(S,Mn>.

(78)

In Eqs. (77), it is assumed that the electron-hole exchange and correlation energy is insensitive to H. In Eqs. (78) a(P) is the exchange integral of

9.

385

RAMAN SCATTERING

,

Cd,-,Mn Te

X=0.05

PHOTON ENERGY ( e V FIG.14. The photoluminescence spectra of Cdc.ssMno.o5Teat T = 5 K with H = 0 for (a)and (b), H = 60 kG for (c). I L = 7525 A for (a) and (c), A L = 6764 A for (b), and laser power PL = 25 mW for all cases. Raman features are denoted by 'R'. In (c) the features between 1.55 and 1.6eV are also displayed on a scale reduced by 10. [After Peterson et al. (1985a).]

the 3d states of Mn2+ and conduction (valence) electrons and <S,"n)is the average value of the component of Mn2+ spin along H given by (5/2)&/2(g,uBH/kB T ) in the paramagnetic phase. Thus the energies h&,b are characterized by an effective g factor that is a non-linear function of (H/T). In Fig. 14(a) and (b), we show the photoluminescence spectrum of Cdl-xMnxTe,x = 0.05, at 5 Kin the absence of a magnetic field. The feature labeled X is attributed to free exciton recombination (Plane1 et al., 1980) and has an energy of Ex = 1.665 eV, an increase of 70 meV from the value in CdTe (Zanio, 1978, p. 100). The energy of this exciton varies linearly with Mn2+ concentration (Lee and Ramdas, 1984) in Cdl-,Mn,Te and the value

386

A . K . RAMDAS AND S. RODRIGUEZ

measured here is in good agreement with that expected. The feature labeled AoX is attributed to the exciton bound to a neutral acceptor (Plane1 et al., 1980) with a binding energy of 9meV. The feature at 1.608eV, when corrected for the change in the energy gap, corresponds to the 1.54 eV feature in CdTe which may result from free electron to acceptor transitions (Zanio, 1978, p. 151); the features at 1 3 3 7 and 1.566 eV are the LO phonon replicas of this transition. Similarly, the series of luminescence peaks beginning at 1 . 5 eV are the LO phonon replicas of the transitions associated with a vacancy-donor complex (V&D’) (Zanio, 1978, p. 139), where the donors in this case are presumed to be interstitial copper atoms (Zanio, 1978, p. 144) introduced during the crystal growth. As the magnetic field is increased the free exciton peak increases in intensity and shifts to lower energy sweeping across the AoX feature (Plane1 ef al., 1980). At sufficiently high magnetic fields a splitting in the free exciton feature is observed (Ryabchenko et al., 1981). Two of these are present in Fig. 14(c); the low energy component labeled X + appears in the 6+polarization, while the other, X , , is polarized along 2. The feature associated with the free electron to acceptor transition and its LO phonon replicas shift by 14 meV in a magnetic field of 60 kG. On the basis of the luminescence spectra, appropriate choices of laser wavelength, sample temperature and magnetic field can be made to achieve conditions of “in resonance” or “out resonance” which can selectively enhance specific features in the Raman spectrum. In the experiments reported here, the OLO - WPM line, i.e., the Raman shift with the creation of an LO phonon and the de-excitation of the MnZ+by Ams = - 1 involves a virtual transition at an energy tiw, for the incident radiation polarized along i and another at hob for scattered radiation having 6+polarization. With the 7525 A ( ~ O L = 1.648 eV) laser line, the in resonance condition is nearly fulfilled; with A and B in Eq. (78) for x = 0.05, T = 5 I<, and H = 60 kG, we estimate the energies of X , and X , , the components which move to lower energies with increasing H , to be 1,.636eV and 1.603 eV, respectively (here we use Noa = 220 meV and NoP = - 880 meV given by Gaj et al. (1979)). Thus, it is clear that the in resonance condition is satisfied even more closely whereas the out resonance for OLO - OPM can now be realized. The results in Fig. 15 show the resonantly enhanced lines at O P M , 2 0 P M , 3 W P M , and 4UpM. We discuss some of the underlying physical considerations later. Here we emphasize the dramatic enhancement in the intensity of the OLO, - OPM line as illustrated in Figs. 16 and 17. The spectra were recorded with the incident polarization along 2, unanalyzed scattered radiation and ks I/ H. The relatively broad feature at 220 cm-’ is the X + luminescence feature attributed to the exciton component at Ex - 3A + 3B. In addition to the LO1, LO1 f PM, LOz, and LO2 f PM lines, that have been reported before (Petrou et al., 1983), two additional features with Raman shifts of

-

-

9.

387

RAMAN SCATTERING I

,

Cd - M n xTe

OO:

10

20

RAMAN SHIFT ( cm-' FIG. 15. The Stokes Raman lines at nWPM ( n = 1 , ...,4) resulting from Ams = + n spinflip transitions in the Zeeman multiplet of the ground state of the 3d shell of Mn2+ in Cdo.g~Mn~.,,~Te. Exciting wavelength AL = 7525 A, A = 90mW; applied magnetic field H = 60 kG and temperature T = 5 K. The spectrum for shifts greater than 15 cm-' is the average of ten scans. [After Peterson et al. (1985a).] OLO, f 2WpM are observed. A direct consequence of the out resonance conditions is the pronounced enhancement in the intensity of the LO1 - PM line in Fig. 16 with respect to that of LO1 + PM. In the scattering geometry and the polarization conditions used, the preferential enhancement results from the fact that the polarization of the scattered light for LO1 + PM is 6 - , while that for LO1 - PM is 6 + ,matching that of the X+transition. Under non-resonant conditions, for T = 5 K and H = 60 kG, the intensity of LO1 + PM would be five times greater than that of LO1 - PM, as calculated from the Boltzmann factor. This enhancement of LO1 - PM becomes even more pronounced under exact out resonance achieved by decreasing the magnetic field to 35 kG and moving the X + luminescence feature under the LO1 and the LO1 - PM Raman lines; this is illustrated in Fig. 17 where the LO1 - PM Raman line is more intense than the LO1 line. The LO2 - PM, L02, and LO2 + PM Raman lines show similar effects as can be seen from a comparison of Figs. 16 and 17. In Fig. 18, the resonance conditions are controlled by keeping H = 60 kG but varying the temperature over the range 4.5 K - 2 5 K . As can be clearly seen, the resonance enhancement at T = 11 K and H = 60 kG is almost identical to that at T = 5 K and H = 35 kG. We note that the Raman shift OLO f OPM is insensitive to temperature variation while the position of the Zeeman component of the exciton and, hence, the resonance conditions are strongly temperature dependent for the reasons already emphasized.

388

A. K . RAMDAS A N D S . RODRIGUEZ

Cd - Mn Te X =

0.05 5K

60 k G

L O , t PM [rLOl

0' 140

180

t 2PM

220

RAMAN SHIFT ( cm-l) FIG. 16. The Raman Spectrum of Cd0.~5Mno.o5Tein the region of the LO phonons with = 30 mW. Incident light polarized along L 11 H ( 1 6 , T = 5 K , H = 60 kG, A L = 7525 A, and and scattered light unanalyzed. [After Peterson et al. (1985a).]

We now discuss the origin of the Raman lines in Fig. 15 with shifts of 3 0 p M and 4 0 p M , corresponding to Ams = 3 and 4, respectively. Mutiple spin-flip Raman scattering from electrons bound to donors (Oka and Cardona, 1981a, b) has been reported in the literature and explained invoking either an exchange coupling among donor spins (Economou et al., 1972) or multiple scattering (Wolff quoted by Geschwind and Romestain, 1984). The multiple spin-flip features in DMS can be accounted for in terms of excitations within neighboring pairs of Mn2+ions coupled antiferromagnetically and assuming an anisotropic exchange interaction between the ground state multiplet of one and an excited state of the other. In fact, a pair of neighboring Mn2+, say 1 and 2, in their 'S5/2 states in a magnetic field fill 2 can be described by the Hamiltonian H = gpBH& -

J(s2 - y ) ,

(79)

9.

389

RAMAN SCATTERING

LO1 - PM

Cd,-,Mn, Te X = 0.05

5K

35 k G

J LO,-2PM

40

180

2 20

RAMAN SHIFT ( cm-' 1 FIG. 17. The Raman spectrum of Cdo.9~Mn0.05Tein the region of the LO phonons with H = 35 kG and other conditions as in Fig. 16. [After Peterson et a/. (1985a).]

where J is a MnZ+-Mn2+exchange interaction and S = S(') + S(2). In a Raman transition, one of the MnZ+ions can experience a virtual excitation to a L = 1 state. The exchange interaction between a Mn2+ion in its ground state and another in an excited state will, in general, lack rotational invariance. Thus, virtual transitions to intermediate states need not conserve the total (pair + the photon) angular momentum and S, can change by more than one unit. Such transitions followed by a final transition to one having a value of S equal to that of the initial state but having S, differing by AS, yields a Raman shift of gpB H ASz, i.e., a multiple of OPM ,independent of J .

6 . MAGNETICALLY ORDERED PHASE As mentioned earlier in this Section, Cdl -*Mn,Te exhibits a magnetically ordered low temperature phase for x > 0.17. The transition from the paramagnetic to the magnetically ordered phase is accompanied by the

390

A. K . RAMDAS AND S . RODRIGUEZ

RAMAN SHIFT ( cm-'1 FIG. 18. Temperature variation of the resonance Raman scattering in Cd0.95Mn0.05Tein the region of the LO phonons with H = 60 kG, and other conditions as in Fig. 16. [After Peterson et al. (1 985a).J

appearance of a new Raman feature at low temperatures as shown in Fig. 19 (Venugopalan et al., 1982). Since this excitation is associated with magnetic order, it is attributed to a magnon. A distinct magnon feature was observed in Cdl-xMn,rTe for the composition range 0.40 5 x I0.70. The magnon feature is absent when the incident and the scattered polarizations are parallel and appears when they are crossed in agreement with Eq. (64) and shown in Fig. 20. This was found to be the case for several crystallographic orientations as well as for polycrystalline samples. Such a behavior, irrespective of the crystallographic orientation, is exhibited only by an excitation whose Raman tensor is antisymmetric. As the temperature is increased, the Raman shift of the magnon, CUM, decreases, and above a characteristic Nee1 temperature TN(x) the feature is no longer observable. For x = 0.70, the temperature dependence of CUM follows a Brillouin function.

9.

01

I

391

RAMAN SCATTERING

I

I

I

I

50

I

1

I

I

100

I

I

I

I

I50

I

I

200

I

RAMAN SHIFT (cm-1) FIG. 19. Rarnan spectra of Cdo.3MnO.TTe in the paramagnetic and anti-ferromagnetic phases; A L = 6764A. The phase transition occurs at Ti 40K. The scattering geometry corresponds to VH + VV. Here VH and VV denote incident light vertically polarized (V) and scattered light analyzed horizontally (H) and vertically (V), respectively, the scattering plane being horizontal. (a) T = 295 K and (b) T = 5 K . Mdenotes the peak due to rnagnon scattering. [After Venugopalan er al. (1982).]

-

The coordinates describing the magnons, the elementary excitations of a system of interacting magnetic dipole moments can be regarded as the Fourier components of the magnetization M(r, t ) , i.e., the coefficients M, in the expansion

M(r, t ) =

c M4 exp(iq 4

*

r - iw,t).

(80)

In first order Raman scattering, only the long wavelength magnons can be excited. In an antiferromagnetic system these excitations can be described by classifying the spins into those which, in the state of equilibrium, point in one direction and those pointing in an anti-parallel direction. This classification gives rise to magnetizations MI and M2 where MI is the magnetic moment per unit volume of the spins of the first class and Mz that of the second. In equilibrium MI + M2 = 0. Now the agents responsible for the preferential orientation of a spin of type 1 are those of type 2 and conversely, the former are in a molecular field of the form - AM2 + HY’ where the first term, called the exchange field, is isotropic and the second points in a preferred crystallographic direction and i s , thus, called the anisotropy field.

392

3 3.6 -

-

3.2 -

-

- 2.8-

-

0 v)

al

\

$ =

2.4-

40.0 K-

mg 2.0-

34.0 K -

0

Y

>-

5z

1.6-

32.0 K-

1.2-

-

30.0 K

W

k

z

20.0 K

-

0.8 0.4

\

-

01

5.0 K \L-44

I

I

-p-----,--~5.0 I

I

K I

I

The equations of motion of M I and M2 are dMi - ~ M xI (H - AM2 dt

+ Ha")

dMz - - - yMz x (H - AM1 dt

+ Ha")

--

and

where H is an externally applied magnetic field. Supposing, for simplicity, that H and Hf) = - Ha) are parallel to a direction that we take as the z-axis, taking the exchange fields - AMz, 1 as approximately + AMs = +HE where

9.

393

RAMAN SCATTERING

Ms is the saturation value of M I we find

-M ;

d dt

=

d -M? dt

=

iy(H -t HA-k H E ) M 7 ~

M$

(83)

r i y ( H - HA - HE)M+ f iyHEMt

(84)

where M t and M? are MI, f iM1, and MzX f iMzy, respectively. Equations (83) and (84) have non-trivial solutions of frequency u& given by

a&

=

fyH + I y l ( H i

+ 2HAHE)’”.

(85)

Thus when H = 0, a long wavelength magnon of frequency Iyl(H2 + 2 H A H ~ ) ’ ’ occurs, * manifests itself in Raman scattering and is the M line in Fig. 19. The polarization features of this line are those predicted by Eq. (64) and illustrated in Fig. 20. We have here the Raman-antiferromagnetic resonance (Raman-AFMR). We assume that HAlike HEis proportional to the saturation values of M I or MZ given by

M S = (hgpBs/a3)BS(r),

(86)

where pug is the Bohr magneton, a the lattice constant, and Bs(y)the Brillouin function with

The Nee1 temperature, TN,is then given by kBTN = (2x/3a3)g2p$(S

+1)~.

(88)

A numerical solution of Eq. (86) yields M S as a function of temperature, T, and in turn the variation of COM with T. The best fit for the data shown in Fig. 21 for x = 0.7 is given by TN= 4 0 K . The magnon feature in the presence of an external magnetic field of 60 k G is shown in Figs. 22a and 22c for Cdl-,Mn,Te, x = 0.70. The spectrum shown in Fig, 22a was recorded at T = 5 K with 21. = 5682 A in the (8-,2) polarization while the spectrum in Fig. 22c was observed in the (8+,2) configuration. In the following, we discuss the Stokes components of these Raman features. The Raman shifts of the peaks of the features in Fig. 22 are OM- = 8.5 cm-’, COM = 12 cm-’, and OM+ = 15.5 cm-’. It can be shown that a one magnon Raman line in an antiferromagnet should split into two components of equal intensity separated by 2gPB H at T = 0 K, if H is along the anisotropy field, HA. The observed spacing between M+ and M - of 7 cm-* for 60 k G is significantly smaller than 2 g p ~ H= 11 cm-‘. The

394

A . K . RAMDAS A N D S. RODRIGUEZ I

-

I

1

I

I

Cd ,+ MnxTe A x.0.7

TEMPERATURE (K)

FIG.21. Temperature dependence of the magnon peak frequency in the magnetically ordered phases of Cdl-,Mn,Te. Triangles: x = 0.7, antiferromagnetic phase; circles: x = 0.4, spinglass phase. The solid curve passing through the points for x = 0.7 is calculated for TN = 40 K solving Eq. (87) for MS as a function of T 5 TN and assuming WM is proportional to MS and equal to 12.5 cm-’ at T = 0 K. [After Venugopalan el a/. (1982).]

polarization characteristics of OM+ and O M - are those expected. These results are independent of the crystal orientations with respect to the applied field suggesting that HA is small compared to the applied field. In Venugopalan et al. (1982), HEfor x = 0.7 was calculated from the observed transition temperature TN = 40 K to be 208 kG,while HAwas determined to be 36 kG as deduced from h c o ~= gpB(H2 + HAH HE)'", strictly applicable to an antiferromagnet exhibiting long range order. While the value for HE is reasonable, that for HA must be viewed as too large in the context of the experimental results (Petrou et al., 1983). We have described the Raman features associated with magnetic excitations appearing in the paramagnetic state as well as in the magnetically ordered phase. The exchange interaction between MnZ+ions is negligible compared to k s T at high temperatures becoming more important as the temperature is lowered. It is of interest to investigate the effects of temperature on the OPM line, particularly the effect of lowering the temperature below TN,the transition temperature characterizing the magnetically ordered phase. The temperature evolution of the OPM line is shown for Cdl-*Mn,Te, x = 0.70, in Fig. 23. The spectra were recorded for a magnetic field of 60 kG using the (&+ ,2) configuration. As the temperature is lowered, the OPM line

-

9. 1

AS

(a)

(&.a H=GOkG

(C)

($.

395

RAMAN SCATTERING I

S

M-

AS

2)

H=GOkG

M-

24

I

I

12 0 12 24 RAMAN SHIFT (cm-')

FIG. 22. Effect of theommagnetic field on the magnon feature of Cd,-,Mn,Te, x = 0.70, at T = 5 K with A L = 5682 A. x , y , andz are along [110], [TlO], and [Ool], respectively: (a) (6-, t), H = 60 kG; (b) (6+, 2). H = 0; (c) (a+,2). H = 60 kG. Owing to imperfect polarizationresults, 2) configuration. leakage of the fairly strong featureM+ appears as a small shoulder in the (6-, [After Petrou et a/. (1983).]

initially broadens and then moves towards higher Raman shifts as a consequence of the increased importance of the exchange interaction. An increase in Raman shift is observed at temperatures well above TN 40 K. As the temperature is lowered through and below TN, the line becomes the magnon component observed in the (a+,2) configuration.

-

396

A. K . RAMDAS A N D S. RODRIGUEZ

I I

OOK

300K I

xi6

6K

I

I

xi6

56c4 ~ 6 c i

10

0

10

10

20

1 -

10

20

RAMAN SHIFT (crn-')

FIG.23. Evolution of the Raman line at WPM of Cdl-xMn,Te, x = 0.70, into the magnon feature as the temperature is lowered from room tempefature to below the Ntel temperature. The spectra were recorded with H = 60 kG, I L = 6764 A in the (a,, 2) polarization; x , y , and z are along [IlO],[TlO], and [@I], respectively. [After Petrou et a/. (1983).]

Finally, the conclusion that the magnetic feature observed in the magnetically ordered phases is a one magnon excitation was initially deduced from its polarization characteristics and temperature behavior (Venugopalan et al., 1982); this is supported by the results of Ching and Huber (1982a, 1982b). The fact that this feature shows a splitting in the presence of a magnetic field and that the OPM line of the paramagnetic phase, clearly associated with a single ion excitation, evolves smoothly into the higher energy component of the magnon provides a strong confirmation of this interpretation. In the same spirit, one might expect a two magnon feature associated with the 2 0 p M line; however, given the intensity of the 2 0 p line ~ compared to that of the OPM line, the intensity of such a feature would preclude its observation. A two magnon feature similar to that seen in MnF2 by Fleury and Loudon (1968) would have symmetric polarization characteristics; such a feature has also not been observed in Cdl -,Mn,Te. In the light of these experimental results, the two magnon interpretation advanced by Grynberg and Picquart (1981) to explain the feature at OM is clearly excluded. 7. SPIN-FLIPRAMANSCATTERING Since the first observations of electron spin-flip Raman scattering in the narrow band gap semiconductor InSb by Slusher et al. (1967) followed by

9.

RAMAN SCATTERING

397

that in the wide band gap semiconductor CdS by Thomas and Hopfield (1968), there has been a continuing interest in this magneto-optical phenomenon. An important contribution to this field is the first demonstration of the spin-flip Raman laser by Patel and Shaw (1970). The success of the InSb spin-flip Raman laser as a practical source of magnetic field tuned, coherent radiation in the 5 to 15 pm spectral region (Patel and Shaw, 1970; Mooradian et al., 1970) can be traced to the large effective g-factor of the electron, 1g*I = 50. In contrast, the wider band gap II-VI semiconductors CdS, CdSe, and CdTe have rather small spin splittings (Thomas and Hopfield, 1968; Walker et al., 1972) characterized by 1g*I 5 2, since their conduction and valence band spin-orbit interactions are small in comparison to their energy band gaps. When detectable, spin-flip Raman scattering provides a practical means of probing the electronic structure of semiconductors, as dramatically illustrated in DMS. The large Raman shifts associated with spin-flip scattering from electrons in DMS were first observed in the narrow gap Hgl-,Mn,Te by Geyer and Fan (1980). The first evidence of a finite spinsplitting of the electronic level in the absence of a magnetic field was reported by Nawrocki et al. (1980, 1981) in the wide gap diluted magnetic semiconductor Cdl-,Mn,Se. The effects of Mn concentration and the antiferromagnetic coupling among the Mn*+ ions on the spin-flip Raman shifts were first observed (Peterson et al., 1982) in a study of Cdl-,Mn,Te. These studies established the nature of Raman scattering associated with the spinflip transitions of electrons bound to donors in DMS. The large Raman shifts depend not only on the applied magnetic field, but also on temperature and manganese concentrations. The far-infrared absorption spectra (Dobrowolska et al., 1982) of Cdl-,Mn,Se also provided evidence for these spin-flip transitions. Following these initial reports, there have been several investigations (Alov et al., 1981, 1983; Heiman et al., 1983a,b, 1984a,b; Douglas et al., 1984; Peterson et al., 1985b) of spin-flip Raman scattering in DMS. The extensive results currently available allow a detailed comparison with the theory of Diet1 and Spalek (1982, 1983). The Raman spectra of Cdl-,Mn,Te(Ga), x = 0.03, are shown in Fig. 24 for the (6+,2) and (6-,2) polarization configurations with T = 40 K and H = 60 kG. The two Stokes features labeled PM and SF are present only in (a+,i),while the corresponding anti-Stokes features appear only in the (6-, 2) configuration. The observed width of the PM line is instrument limited, while that of the SF feature is 23 cm-'. The PM feature is associated with the spin-flip transitions. The SF feature of Fig. 24 is attributed to spin-flip Raman scattering from electrons bound to gallium donors. It has the same polarization characteristics as those of the PM line appearing (Peterson et al., 1982)in the (a+,2)

398

”0

20

I0

0

-10

RAMAN SHIFT ( cm-’

-20

-3

I

FIG. 24. Raman spectra of Cdl-,Mn,Te(Ga), x = 0.03, showing the Ams = f 1 transitions within the Zeeman multiplet of MnZf(PM) and the spin-flip of electrons bound to Ga donors (SF). kcps = 10’ countslsec. [After Peterson et a/. (1985b).]

or (&a_)polarizations for Stokes scattering and in (8-, 2) or (2, a+)for antiStokes. As illustrated in Fig. 25, the peak Raman shift of the spin-flip feature, 6,exhibits a strong .dependence on both temperature and magnetic field. The primary source of the spin splitting of the electronic level is the exchange coupling with the Mn2+ ions (first term of Eq. (72)) with the Zeeman effect (second term of Eq. (72)) making a relatively small contribution. Hence, the Raman shift should be approximately proportional to the magnetization of the Mn2+ ion system, which amplifies the effect of the magnetic field on the electron. As can be seen in Fig. 25, a finite Raman shift is observed for zero magnetic field. This effect is attributed by Diet1 and Spalek (1983) to the “bound magnetic polaron” (BMP). The electron localized on a donor in a diluted magnetic crystal polarizes the magnetic ions within its orbit, creating a spin cloud that exhibits a net magnetic moment.

9.

I- Cd,-,Mn,Te

(Ga 1

X = 0.03

-

A

'E

0

Y

90 -

+

z rn

+ + +

+

0

0

0

-

+

I-

LL

399

RAMAN SCATTERING

0 0

+

60-

A

0

A

+

1

A

0 A

0

A

LL +

o A

0

0

x

X

x

20

;

1

x

:40Ki

x

40

60

MAGNETIC FIELD ( k G ) FIG.25. Magnetic field and temperature dependence of the Raman shift associated with the spin-flip of electrons bound to donors in Cdl-,Mn,Te(Ga), x = 0.03. [After Peterson el uf. (1985b).]

An additional effect on the binding energy of the electron bound to the donor originates from thermodynamic fluctuations of the magnetization and the resulting spin alignment of the magnetic ions around the donor. According to Eq. (72), the spin splitting of the donor energy levels in DMS arises from the combined effect of the magnetization of the Mn2+ions and the external field H. Due to the strong s-d coupling, the effect due to the magnetization dominates. The Raman shift associated with spin-flip scattering from the donor states has the form

where MOis the macroscopic magnetization. The magnetization is proportional to the thermal average of the Mn2+spin projection along H multiplied

400

A . K . RAMDAS A N D S. RODRIGUEZ

by the density of Mn2+ ions contributing to the magnetization, yielding Ao

=

X C Y N O ( S+~ g*pBH. ~)

(90)

Here R is the concentration of Mn2+ ions that contribute to the magnetization. For small x, the crystal is paramagnetic and the thermal average of the Mn" spins is

where B5/2 is the Brillouin function BJ for J = 5. The compositional dependence of spin-flip Raman scattering has two sources. Within a DMS system, such as Cdl-,Mn,Te, the properties of the spin splitting should show a strong dependence on the Mn concentration. For a givenx, these properties should also vary from one DMS system to another. The spin-flip Raman shifts for Cdl-,Mn,Te at T = 1.8 K are shown in Fig. 26 as a function of magnetic field and composition. The results for x = 0.01 show the saturation behavior characteristic of the paramagnetic phase. As the Mn concentration is increased to x = 0.03 and x = 0.05, the Raman shifts increase and the effects of saturation are still clearly evident, I

160 C d

I

,-, MnT,e

I

I

A

'E 120 0

LL

I

+

+s

80

z a

o

A

O

A

+

I

Q

I

I

0 O

Ox:o.05

A

A

A

+

+

+

+

--

+

+x=o

A A

x=o.3

A A

0

A

A A

+O +O

I

+

oX=0.20 A

__

I

+

+

A

X = 0.03

A

I

+

+

__ h

I

+X=O.Q- Cd,-,Mn,Te T=1.8K

n +n

z

I

+

+

I-

V,

+

+

v

I

+ +

T = 1.8K

-

I

A

40

0

i

i

20

i

l

40

l

l

6

0

I

0

l

l

I

20

l

l

40

I

l

60

l

MAGNETIC FIELD ( k G ) FIG.26. Magnetic field and composition dependence of the peak spin-flip Raman shift in the Cdl-,Mn,Te samples at T = 1.8 K . [After Peterson et a/. (1985b).]

9.

RAMAN SCATTERING

401

but less pronounced. For x = 0.10, the deviation from the paramagnetic behavior is quite evident. For H = 60 kG, the Raman shift for x = 0.10 is only four times that for x = 0.01. As x exceeds 0.10, the Raman shifts for a given field actually decrease; note that the shifts for the x = 0.20 sample lie below those for the x = 0.10 sample. And the Raman shifts for the Cdl-,Mn,Te(Ga), x = 0.30, sample are significantly smaller than those for the x = 0.10 and x = 0.20 samples. These trends have their origin in the decrease of the mean magnetic field due to the increasing antiferromagnetic pairing of MnZ+neighbors. In order to discuss the experimental results on the bound magnetic polaron, it is useful to review the theory for it as developed by Dietl and Spalek (1982, 1983). The electron is assumed to be bound to a shallow donor interacting through an s-d coupling with a paramagnetic subsystem of localized magnetic moments. Only the large polaron case is considered, allowing the continuous-medium, effective-mass, and molecular-field approximations. This model assumes that the donor electron interacts with a large cloud of spins behaving classically. Neglecting field induced anisotropy, Dietl and Spalek derive an effective BMP Hamiltonian given by

where ED(^) is the binding energy of the electron arising from the Coulomb potential, A is the magnitude of the spin-splitting with A, parallel to the effective local magnetic field, defining the direction of the spin quantization of the electron,

is the magnetic field induced component of A and

is the characteristic BMP energy for an s-type wave function with an effective Bohr radius a. Here Mo is the magnetic field induced magnetization and x is the magnetic susceptibility. At a finite temperature, a range of A beyond the minimum of H ( A )is accessible to the system. The probability of a thermal fluctuation of the magnetization giving rise to a specific A is

I : [: :

P(A) = Cexp --

(95)

402

A. K . RAMDAS A N D S. RODRIGUEZ

where C is a normalization constant. The probability distribution for A = 1 A ( ,P(A), is given by integrating (95) over all angles, yielding

In the absence of a magnetic field, P(A) reduces to

where

The theory of Dietl and Spalek results in an intensity distribution characterizing the spin-flip Raman line, Z(A), to be P(A), given by Eqs. (96) or (97), multiplied by the probability that the donor electron has its spin aligned parallel (antiparallel) to the effective field A(H, T ) .

where Cis a constant related to the scattering cross section, and f refers to the Stokes and anti-Stokes components of the line. The peak position of the Raman line, 6, satisfies 'F

("4"'

- dAocoth - & p k ~ T

) - 4&pk~T=

0,

(100)

which shows an asymmetry in 6 between the Stokes and anti-Stokes components. The final expression for the Raman intensity as calculated by Dietl and Spalek (1983) and Heiman et al. (1983a) does not include the effects of the magnetic field and temperature on the matrix elements of the Raman tensor. Since Raman scattering proceeds through intermediate states, the Raman matrix elements exhibit an angular dependence such that the cross section for scattering from a donor electron with a given A depends not just on \A1 as implicit in Eq. (99), but also on the angle between A and A0 11 H. This angular dependence has a small effect on the peak position 6. However, in the light of the approximations that made the problem tractable and led to the result of Eq. (92), this small correction can be neglected and Eq. (100) serves as a good approximation for 6 .

9.

RAMAN SCATTERING

403

In order to calculate 6 , the magnetic field induced spin-splitting, A0 and the characteristic energy of the bound magnetic polaron, c P , must first be determined. For a weakly antiferromagnetic system,

where we use the notation of Heiman et al. (1983a) 35 R(cYNo)~ w:=---.-

96 na3No

As demonstrated by Diet1 and Spalek (1983), the dependence of the effective Bohr radius (a) on X, TAF,H , and T can be determined using a variational technique. The values of the effective mass m* and the static dielectric constant K are needed in order to evaluate the radius a using this procedure. Since the effects of composition on these parameters are not known, only an approximate value for a can be deduced. Therefore, in generating the theoretical fits discussed here, it is convenient to treat WO,and hence a, as a third adjustable parameter along with R and TAF.The resulting values of a may then be compared with the known Bohr radius for x = 0. The low field data and the associated theoretical curves are plotted in Fig. 27 for Cdl-,Mn,Te(Ga), x = 0.03, 0.05, and 0.10. As can be seen in Fig. 27(a), a zero-field shift of 3.5 cm-' was observed in Cdl-,Mn,Te(Ga), x = 0.03, at T = 1.8 K. The data for the other two samples, x = 0.05 and x = 0.10, give evidence of zero-field shifts of - 4 and -6cm-'. The magnetic field and temperature dependence of the data, particularly for x = 0.10, are well described by these curves. The zero-field spin-flip Raman spectra of the Cdl-,Mn,Se, x = 0.10, sample are shown in Fig. 28 for T = 1.8, 5 , 10, and 20 K. As can be seen in the figure, the Raman shift of the spin-flip feature decreases for higher temperatures. Except for the temperature, the experimental conditions were identical for the four scans. The four theoretical curves are best fits generated from Eqs. (97), (98), and (99) for TAF= 1.28 K and WO= 0.63 meV, using the same value for Cin Eq. (99). The predicted variation of the peak intensity with temperature agrees quite well with the experimental results.

a 0 a

? ?;I

*wg u &

*

l3

?

0

0

3 15

0

5

10

15

0

5

10

15

MAGNETIC FIELD ( k G ) FIG. 27. Magnetic field and temperature dependence of the peak spin-flip Raman shift in the Cdl-,Mn,Te(Ga), x = 0.05, 0.10, and 0.20, samples at low field. [After Peterson et al. (1985b).]

a N

9. RAMAN SCATTERING

405

406

A. K. RAMDAS A N D S . RODRIGUEZ

VI. Vibrational, Electronic, and Magnetic Excitations in Superlattices During the past decade or so the fabrication of heterostructure of semiconductors by techniques such as molecular beam epitaxy (MBE), or metaloorganic chemical vapor deposition (MOCVD) has been a major technological breakthrough. Single quantum well, multiple quantum wells, superlattices and such heterostructures possess properties and exhibit phenomena not encountered in the bulk. Such “synthetic,” “modulated” or “engineered” structures are fascinating in the context of fundamental physics just as much as for their technological importance. Electronic and optical properties of multilayer structures have brought out several phenomena unique to superlattices, e.g., Brillouin zone folding effects, plasma dispersion in layered electron gases and multiple quantum well effects. Much of the focus till recently has been on 111-V semiconductors and their ternary alloys (Ploog and Dohler, 1983; Dingle, 1975; Klein, 1986). In the past few years heterostructures of DMS-superlattices-have been successfully fabricated (Kolodziejski et al., 1984; Bicknell et al., 1985; Datta et al., 1985). DMS superlattices offer the exciting prospect of tuning the electronic potential within the individual layers, after fabrication, using external parameters such as temperature and magnetic field. It has also been pointed out that (Ortenberg, 1982) by “spin doping” a superlattice, a tunable electronic energy gap can be created at the zone boundary. Here we present illustrative results that demonstrate the effectiveness of the Raman scattering technique for investigating various structural aspects of DMS superlattices (Venugopalan et al., 1984). Figure 29 shows the low frequency spectrum of a Cdl-,Mn,Te/ Cd, ,Mn,Te superlattice recorded at 295 K . The experimental results discussed here are on films grown on a GaAs (001) substrate. A buffer layer of either CdTe or Cdl-,Mn,Te, -2pm in thickness, was first deposited on the substrate; this was followed by the epitaxial growth of the superlattice layers of Cdl -,MnxTe, with. alternating Mn concentrations in adjoining layers. The experiments were performed with samples containing 270 alternating layers of Cd0.5Mn0.5Teand Cdo.g9Mno.] lTe; transmission electron microscopy indicates a layer thickness of 59A. In this spectral region, bulk crystals corresponding to the composition of either layer show no discrete Raman lines due to phonons (Venugopalan et al., 1982). However, in the superlattice we observe very distinct Stokes and anti-Stokes components at 9.2, 1 1 , 19, and 20.7 cm-’. These new signatures, characteristic of the superlattice, can be attributed to the Brillouin zone folding effect resulting from the new period (0) imposed along the growth direction of the sample. Further, the point group symmetry of the superlattice with its growth axis along 11111 (denoted hereafter as 2) is reduced to C3”, whereas bulk

9.

501

-2b

407

RAMAN SCATTERING

-10

10

0

20

RAMAN SHIFT (cm-')

FIG. 29. Stokes (S) and anti-Stokes (AS) components of the folded longitudinal acoustic x = 0.5 and y = 0.11. [After branch of the superlattice Cdl-,Mn,Te/Cdl,Mn,Te, Venugopalan et al. (1984).]

crystals of Cdl-xMnxTe possess the higher symmetry Td, for 0 Ix 5 0.75. Hence phonons propagating along L with wave vectors 191 = 2nN/D, (N = 1,2,3, ...) within the extended zone are now mapped onto the center of the folded Brillouin zone. In addition, the lower symmetry of the superlattice causes the splittings observed at q 0 for these phonons. Similar Raman scattering results on GaAs-A1As and GaAs-Gal -,ALAS superlattices grown along [lo01 have been reported by Colvard et a/. (1980, 1985) and Sapriel et al. (1983). We consider the superlattices as an elastic continuum containing two alternating layers with densities p1 and p2, their bulk longitudinal acoustic (LA) velocities along L being 01 and 02, respectively. Then, the angular frequency ( 0 )of LA phonons traveling along L and the magnitude of their wave vector (4)are related (Rytov, 1956) by

-

where dl and d2 are the respective thicknesses of the adjacent layers, = dl + d2, and a = (PIu1/p2u2). In the backscattering geometry

D

408

A . K . RAMDAS A N D S. RODRIGUEZ

A.

employed in this experiment, q = 5.437 x lo5 cm-'; dl = d2 = 59 Using the interpolated values of the densities and the elastic constants for crystals with x = 0.11 and 0.5, we find LY = 1.064, u1 = 3.445 x lo5cm/sec and u2 = 3.466 x lo5 cm/sec. The solutions of Eq. (105) then predict that the first pair ( N = 1) of lines due to the folded LA phonon branch should be seen at 8.8 and 10.8 cm-'. The excellent agreement with the experimental values, viz., 9.2 and 1 1 .O cm-', demonstrates the sensitivity of this technique to the growth parameters of the superlattice. The higher frequency pair of lines in Fig. 29 at 19 and 20.7 cm-' are assigned to the second fold ( N = 2) of the LA phonon branch. For 59 A layer thickness, from Eq. (105), these are expected at 18.5 and 20.5cm-', respectively. Since AL here is very close to the energy gap of the superlattice, the presence of the N = 2 nodes indicates a resonant enhancement as noted by Colvard et al. (1985). Striking resonance effects observed for longitudinal optic (LO) phonons are illustrated in Fig. 30 where a magnetic field of 60 kG shifted the onset of the photoluminescence peak to the region beyond 600cm-'. The two prominent peaks in the 160-210 cm-' range arise from the fundamental LO modes characteristic of the "two-mode" behavior of Cdl-,Mn,Te alloys

1

-

I

I

I

I

SL

1500

I

I

I

325 - 4

H=60KG

v)

CL

0

v

>

c

1000

v)

z W

k

z

500

0

200

400

600

RAMAN SHIFT ( cm-' FIG. 30. Resonant scattering by longitudinal optic phonons, their overtones and combinations in the superlattice of Cdl-,MnxTe/Cdl,Mn,Te, x = 0.5 and y = 0.11. The line labels denote the following assignments: (1) L o ] , (2) L o z , (3) 2 L o 1 , (4) Lo1 + L o z , ( 5 ) 2L02, (6) 3LO,, (7) 2LO1 + L02, (8) LO1 + 2L02, (9) 3LOz, (10) 4J-01, (11) 3 L 0 1 + LOz, (12) 2L01 + 2LOz. LO1 and LOz are, respectively, at 167 and 199 cm-'. [After Venugopalan et al. (1984).]

9.

409

RAMAN SCATTERING

(Venugopalan et al., 1982). Their respective positions are 167 (LOI) and 199

(L02) cm-I. No zone-folding effects were seen for the LO phonons,

presumably due to the flatness of the dispersion curves of these modes. The additional groups of peaks seen here, centered at 370, 530,and 700 cm-', originate from strong resonant scattering by overtones and combinations of the two fundamental LO phonons. The clear observation of second and third overtones in a very thin (1.6,um) sample, where the effective scattering volume is necessarily small, attests to the resonant enhancement. It is probable that the frequencies of all these features are influenced by any strains present within the superlattice. We further note that although transverse optic modes are also allowed in the experimental geometry, their intensity was too weak to be detected. With 7525A excitation-that lies below the energy gap of the superlattice-none of the features, including LO modes, could be seen. Based on these facts and that the excitation employed for the spectrum shown in Fig. 30 is above the energy gap, the selective enhancement seen for the LO modes and their overtones suggests an underlying Frohlich mechanism. The sloping background beyond 640 cm-' is due to the onset of the photoluminescence peak. Under an applied magnetic field, the position of the luminescence peak shifts towards lower photon energy with an effective g-factor of 100 at 5 K; these data, demonstrate that the very large electronic g-factors first reported by Gaj et al. (1978a) for Cdl-,Mn,Te alloys are also realized in DMS superlattices. We finally discuss the question of magnetic excitations within a DMS superlattice. As noted earlier, Galazka et af., (1980) have shown that Cdl -,Mn,Te alloys possess a spin-glass phase at low temperatures for the composition range 0.17 Ix 5 0.60. In single crystals with x = 0.5 and 0.4, Venugopalan et af. (1982) detected, in the absence of an external magnetic field, a distinct magnon Raman line characteristic of the spin-glass phase. In the superlattice, there appears to be no evidence for such a zero-field excitation. Instead, with an applied field of 60 kilogauss, one observes an intense peak originating from the Raman-paramagnetic resonance (OPM) between the Zeeman sublevels of the ground state of MnZ+ions (Petrou et af.,1983). In addition, the first and second overtones of OPM are also easily detectable. Venugopalan et af. (1984) attribute this to the possibility that even at 5 K the 59 A thick layers (with x = 0.5) of the superlattice may lack the spin-glass ordering, although it does occur for the same composition of the bulk alloy. If so, this would also signify that the dimensional constraints created in a superlattice can exert a decisive influence on magnetic phase transitions. In summary, this study illustrates that Raman scattering can serve as a sensitive, diagnostic probe of diverse structural aspects of DMS superlattices.

-

-

410

A. K . RAMDAS AND S. RODRIGUEZ

Acknowledgments The support from the National Science Foundation during the preparation of the chapter is gratefully acknowledged (Grant No. DMR-84-03325).

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412

A . K . RAMDAS A N D S . RODRIGUEZ

Petrou, A., Peterson, D. L., Venugopalan, S., Galazka, R. R., Ramdas, A. K., and Rodriguez, S. (1983). Phys. Rev. B27, 3471. Pine, A. S., and Dresselhaus, G. (1969). Phys. Rev. 188, 1489. Pine, A. S., and Dresselhaus, G. (1971). Phys. Rev. B4, 356. Planel, R., Gaj, J., and Benoit a la Guillaume (1980). J. Phys. (Paris) Colloq. 41, C5-39. Ploog, K., and Dohler, G. H. (1983). Advan. Phys. 32, 285. Rodriguez, S, and Rarndas, A. K. (1985). “Inelastic Light Scattering in Crystals,” Highlights of Condensed Matter Theory, Proceedings of the International School of Physics “Enrico Ferrni” (Course LXXXIX) (F. Bassani, F. Furni, and M. P. Tosi, eds.), pp. 369-420. North Holland, Amsterdam. Rowe, J . M., Nicklow, R. M., Price, D. L., and Zanio, K . (1974). Phys. Rev. B10, 671. Ryabchenko, S. M., Terletskii, 0. V., Nizetskaya, I. B., and Oleinik, G. S. (1981). Fiz. Tekh. Poluprovodn. 15, 2314; Sov. Phys. Semicond. 15, 1345. Rytov, S. M. (1956). Akust. Zh. 2, 71; Sov. Phys. Acoust. 2 , 68. Sandercock, J. R. (1982). “Light Scattering in Solids” (M. Cardona and G. Giintherodt, eds.), Vol. 3, p. 173. Sapriel, J., Michel, J. C., Toledano, J. C., Vacher, R., Kervarec, J., and Regreny, A. (1983). Phys. Rev. B28, 2007. Selders, M., Chen, E. Y., and Chang, R. K. (1973). Solid State Commun. 12, 1057. Sennett, C. T., Bosomworth, D. R., Hayes, W., and Spray, A. R. L. (1969). J. Phys. C2,1137. Slusher, R. E., Patel, C. K . N., and Fleury, P. A. (1967). Phys. Rev. Lett. 18, 77. Thomas, D. G., and Hopfield, J. J. (1968). Phys. Rev. 175, 1021. Twardowski, A., Nawrocki, M., and Ginter, J. (1979). Phys. Stat. Solidi. (b) 96, 497. Venugopalan, S., Petrou, A., Galazka, R. R., Ramdas, A. K., and Rodriguez, S. (1982). Phys. Rev. B25, 2681. Venugopalan, S., Kolodziejski, L. A., Gunshor, R. L., and Ramdas, A. K. (1984). Appl. Phys. Lett. 45, 974. Walker, T. W., Litton, C. W., Reynolds, D. C., Collins, T. C., Wallace, W. A., Gorrell, J . H., and Jungling, K. C. (1972). “Proceedings of the XI International Conference on the Physics of Semiconductors, Warsaw, 1972,” p. 376. Elsevier, New York. Weinstein, B. A. (1976). “Proceedings of the XI11 International Conference on the Physics of Semiconductors, Rome, 1976” (F. A. Fumi, ed.), p. 326. Tipografia Marves, Rome. Zanio, K. (1978). “Semiconductors and Semimetals” (R. K. Willardson and A. C. Beer, eds.), Vol. 13. Academic Press, New York.

SEMICONDUCTORS AND SEMIMETALS, VOL. 25

CHAPTER 10

Theory of Bound Magnetic Polarons in Semimagnetic Semiconductors P. A . Worff FRANCIS BITTER NATIONAL MAGNET LABORATORY MASSACHUSETTS INSTITUTE OF TECHNOLOGY CAMBRIDGE. MASSACHUSETTS

I. INTRODUCTION . . . . . . . . . . . . . POLARONS . . 11. FREEVERSUSBOUNDMAGNETIC 111. THEBMP MODEL. . . . . . . . . . . .

IV.

V.

VI . VII. VIII.

IX. X.

. . . . . . . . . . . .

. . . . . .

THE BMP PARTITION FUNCTION . . . . . . . . . . . . 1. Zero Magnetic Field Case . . . . . . . . . . . . . 2 . Finite Field Case. . . . . . . . . . . . . . . . . 3. Soluble Polaron Model . . . . . . . . . . . . . . THERMODYNAMIC PROPERTIES OF BMP. . . . . . . . . . 4. Internal Energy . . . . . . . . . . . . . . . . . 5 . Magnetization . . . . . . . . . . . . . . . . . 6 . Spin Correlation. . . . . . . . . . . . . . . . . THEDIETL-SPALEK FORMALISM. . . . . . . . . . . . . TIMEDEPENDENT SPIN-SPINCORRELATION FUNCTION.. . . COMPARISON OF THEORY WITH EXPERIMENT. . . . . . . . I . Brief Review of Magnetic Semiconductor Work . . . . 8. Optical Evidence f o r BMP in Semimagnetics. . . . . . 9. Analysis of Spin-Flip Raman Scattering Experiments . . 10. Analysis of Acceptor-BMP Experiments. . . . . . . FOR FREEMAGNETIC POLARONS . . . . . . . . EVIDENCE . . . . . . . . . . . . . . . . . . . . CONCLUSION 11. Current Status of BMP Theory. . . . . . . . . . . 12. Directions f o r Future Work . . . . . . . . . . . . REFERENCES.. . . . . . . . . . . . . . . . . . .

413 415 417 420 420 423 424 425 42.5 428 430 430 433 436 436 438 440 44s 449 45 1 45 1 45 1

452

1. Introduction In semiconductors containing magnetic ions, there is generally a sizable exchange interaction between carrier spins and those of the ions. Exchange causes novel spin-dependent phenomena in such materials including giant 413 CopYnEht 0 1988 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-752125-9

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P. A. WOLFF

spin-splittings of the bands, large Faraday rotations, and magnetic polarons. Free magnetic polarons-magnetization clouds associated with unbound carriers-have not yet been conclusively demonstrated. On the other hand, there is abundant evidence that carriers localized at impurities can induce sizable magnetizations in their vicinity. These ferromagnetic complexes, termed bound magnetic polarons (BMP), often have net moments as large as 25 Bohr magnetons. BMP control many properties of magnetic and semimagnetic semiconductors; they are responsible, for example, for prominent features in the luminescence spectra of semimagnetics and for the spectacular metalinsulator transition observed in EuO. They were first studied in the europium chalcogenides, mainly via transport and magnetization measurements; there is now an extensive literature on that subject (see Kasuya and Yanase, 1968; Yanase, 1972; Kuivalainen et al., 1979; Nagaev, 1983 for bibliographies). None of these experiments, however, unambiguously determines microscopic parameters-such as binding energy, moment, or size-of the BMP. Direct observation of BMP in Eu-chalcogenides is impeded by their complicated band structure, relatively poor electronic and optical properties, and by competing magnetic interactions. The situation is more favorable in the semimagnetic semiconductors with large, direct bandgaps and simple band structures; examples are Cdl -,Mn,S, Cdl-,Mn,Se, and Cdl-,Mn,Te. These crystals, in contrast to the EuX compounds, have highly structured luminescence spectra whose features can be associated with well-known centers; a study of the temperature, composition, and magnetic field variation of the features gives microscopic information concerning BMP interactions in such complexes as the acceptor, the acceptor-bound exciton (A'X), and the donor-bound exciton (D'X). Semimagnetic alloys also favor light scattering experiments since, in direct gap crystals, carrier scattering cross sections are enhanced by large factors as the laser frequency approaches the gap. This resonance facilitates spin-flip Raman scattering (SFRS) studies, which provide key evidence concerning magnetic fluctuations in donor-BMP. Two optical experiments (Golnik et al., 1980; Nawrocki et a/., 1980) were the stimulus for recent advances in the theory of BMP. The theoretical work, by Diet1 and Spalek (DS; 1982, 1983) and by Heiman, Wolff, and Warnock (HWW; 1983b), has given new insight into the behavior of BMP. The crucial point was first enunciated by DS. They showed that, at high temperature, magnetic fluctuations determine the behavior of BMP. As temperature is lowered, the BMP continuously evolve from a fluctuation-dominated regime to a collective regime in which the carrier and magnetic ion spins are strongly correlated with one another; the system then behaves like a large classical magnetic dipole whose moment is many Bohr magnetons. The transition

10. THEORY

OF BOUND MAGNETIC POLARONS

415

cannot be described by mean field theory, which predicts a phase transition in the finite BMP system, as well as other unrealistic behavior. This article discusses the theory of BMP in semimagnetics, with emphasis on the interpretation of DS. The formalism used will be that of HWW which is rather different from that of DS. DS approach the BMP problem microscopically but use the continuous medium approximation in the calculation of the partition function; the central ingredient in their analysis is a classical, Landau free energy functional. HWW, on the other hand, base their calculations on a microscopic Hamiltonian, with suitable approximations to evaluate the partition function and correlation functions. Despite their dissimilarities, the two theories give identical results in the continuum limit where they can legitimately be compared. The HWW approach is chosen here because it is somewhat simpler than that of DS, and because it naturally handles the problems of spin saturation and centers containing a small number of spins. Detailed comparisons of theory with experiment will be made wherever possible. For the simpler centers, the agreement is excellent; it fully substantiates the DS picture of a gradual transition from the fluctuationdominated to the collective regime. Other, untested, predictions of the theory are also strengthened by its overall agreement with experiment. Yet, despite these successes, the theory is not complete. There is another class of BMP problems-those of spin relaxation, spin diffusion, and spin kinetics generally-that are just beginning to be addressed. These problems cannot be fully solved in the semiclassical approximation common to the DS and HWW theories; instead they require a quantum mechanical theory. The development of such a theory, and of experiments to test it, is an important next step in the BMP problem. 11. Free Versus Bound Magnetic Polarons

The simplest possible magnetic polaron-the free magnetic polaronconsists of a carrier trapped, via the exchange interaction, in a magnetic potential well created by locally aligning the spins of magnetic ions. The stability of such complexes was investigated by Kasuya et al. (1970). Their analysis shows that the polaron energy is determined by a balance of two terms of opposite sign: a positive kinetic energy due to carrier localization, and a negative energy due to the exchange well. Free polarons are stable in certain regions of the polaron parameter space, unbound in others. The existence of free magnetic polarons in a given material thus depends, in detail, on the parameters that characterize it. Calculations suggest that they are generally not stable in dilute (x < 0.05) semimagnetic alloys. The energetics of BMP are quite different from those of free magnetic

416

P . A. WOLFF

polarons. As described in the introduction, BMP involve a carrier (or carriers) trapped at an impurity by electrostatic forces. The kinetic energy of localization is balanced by the Coulomb attraction of the impurity. Spin alignment of nearby magnetic ions is favored by the exchange interactionbut this effect is not the primary mechanism of localization, as it is in the free polaron. Nevertheless, significant spin alignment can occur in BMP, even when the strength of the exchange interaction is well below that required to stabilize the free magnetic polaron. The preceding argument suggests that carriers localized at impurities in magnetic or semimagnetic semiconductors will generally show BMP effects, whereas free magnetic polarons should be rarer. Experiments on semimagnetics confirm this view; BMP effects have been observed in several ways in a number of semimagnetic crystals. It is only recently, however, that evidence for free polarons in semimagnetics has been presented (Golnik et al., 1983). The evidence is suggestive but not yet conclusive; we will return to it in Part IX. The analysis of Kasuya et al. (1970) implies another important difference between free and bound magnetic polarons. The former, if stable, must be spin saturated; BMP, on the other hand, need not be. The requirement of spin saturation in free polarons is implied by the expression for the polaron free energy. Kasuya et al. (1970) assume an exchange interaction of the form

where (r,s) are the coordinate and spin of the trapped carrier, and ( R j , Sj) those of a magnetic ion at site R j . The free energy is calculated in the molecular field approximation, assuming a variational form for the carrier wave function. For the semimagnetic case, the free energy becomes:

where a is the radius of the carrier wave function, x the susceptibility of the magnetic ion system, and NO = a,' is the unit cell density of the crystal. This expression is valid until the spin correlation function, ( s - S j ) , begins to saturate, i.e., approaches its maximum value, of $ for the Mn2+ ion case. Equation (2) becomes negative for sufficiently small values of a. If saturation has not occurred prior to this crossover point, the polaron is bound and will continue to increase its binding energy by decreasing its radius until saturation occurs. Though approximate-since it relies on mean field theory-this argument is qualitatively correct.

10.

THEORY OF BOUND MAGNETIC POLARONS

417

111. The Bound Magnetic Polaron Model

The simplest bound magnetic polaron is the donor-BMP in a semimagnetic semiconductor of moderate (say x c 0.1) magnetic ion concentration. The single electron in such a complex moves in a nondegenerate conduction band, and has an extended orbit that is well-described by the effective mass approximation. Moreover, for small x, the magnetic ion susceptibility in such crystals as Cdl-,MnxS and Cdl-,Mn,Se is nearly Curie-like in the temperature range of interest. Both of these facts simplify the analysis. Thus, the donor case will be used to develop the theory; these ideas will later be extended, in a less rigorous way, to acceptor-BMP. Donor-BMP have been extensively studied via SFRS in CdMnS (Alov et al., 198 1 , 1983; Douglas et al., 1984; Nawrocki et al., 1984; Heiman et al., 1983a), CdMnSe (Nawrocki et al., 1980; Heiman et al., 1983b; Plane1et al., 1984), and ZnMnSe (Douglas et al., 1984; Heiman et al., 1984), which are n-type as grown; there has also been one report (Peterson et al. , 1982) of SFRS by donor-bound electrons in normally p-type CdMnTe. These experiments were an important stimulus to the theory, by providing the first clear evidence for spin fluctuation effects in BMP. The donor-BMP in semimagnetics is not, however, sufficiently strongly coupled to exhibit the full range of BMP behavior; even at 2 K there is only moderate spin correlation in these complexes. Complete correlation, i.e., spin saturation, is illustrated by the more complicated, but strongly interacting, acceptor-BMP case. The model Hamiltonian of the donor-BMP is:

where m* is the effective mass, EO the dielectric constant, and a the exchange constant (Gaj, 1980) for the conduction band. In applying Eq. (3) to wurtzite crystals, such as CdMnS or CdMnSe, we will ignore the small ( = 5 % ) anisotropy of m* and E O . A more serious approximation is the neglect of direct Mn2+-Mn2+ spin interaction in such materials. Experimentally, Cdl -,Mn,X crystals with small x-values have nearly Curie-like magnetic susceptibilities (Kreitman et al., 1966; Nagata et al., 1980; Galazka et al., 1980) in the temperature range 4-40 K, but with magnitudes corresponding to effective Mn2+concentrations (2)somewhat smaller than x . For example, for x = 0.05 one finds R = 0.03. This behavior, which has caused some confusion in the literature, has recently been shown (Shapira et al., 1984; Aggarwal et al., 1984) to be the result of antiferromagnetically-coupled Mn2+ spin clusters (singlets, doublets, triplets, etc.). Isolated Mn2+spins, with no nearest neighbor Mn2+ ions, give a full Curie contribution to the susceptibility; MnZ+pairs (doublets) are antiferromagnetically coupled and make no

418

P . A. WOLFF

contribution at low temperatures; and there is a small contribution from the two types (open and closed) of spin triplets. This model quantitatively explains the observed variation of R with x , if Mn2+ ions are assumed randomly distributed on the cation sublattice. From the BMP viewpoint, it implies that Mn2+doublets and triplets are essentially “locked out” of the problem by strong, nearest neighbor, Mn2+-Mn2+interactions. The BMP sees only the isolated Mn2+ ions, whose concentration is R. This argument is valid for small x (x ?. 0. l), where most Mn2+ions are in clusters of small size. In that range, however, it enables one to treat the BMP problem-via the R trick-as if there were no direct Mn2+-Mn2+interactions. The resulting model Hamiltonian [Eq. (3)] contains the essential physics of the BMP in its purest form and, at the same time, quantitatively explains the SFRS experiments in CdMnS, CdMnSe, and ZnMnSe. It should always be borne in mind, however, that Eq. (3) involves important approximations; in particular, one cannot expect it to be valid for x > 0.1. The Schrodinger equation, X F = EF, determines the modulating function in the effective mass approximation; the complete wave function is v/ = FUO, where uo is the band edge Bloch function. The exchange potential in the effective mass Schrodinger equation mixes orbital and spin degrees of freedom; for example, exchange can scatter an electron from one orbital state to another while flipping its spin. In practice, such scattering occurs infrequently, as will be demonstrated below. It is then reasonable to seek a solution of the Hartree form:

In this approximation the orbital wave function, q5(r),is determined by the combined actions of the Coulomb potential and the coherent average of the exchange potential; fluctuations of the exchange potential about its average value, which cause scattering, are ignored. To test this Hartree-like approximation, it is useful to make estimates of the spin flip scattering rate. A Born approximation calculation for a free electron in a semimagnetic crystal gives the following expression for the spin flip mean free path:

_1 - S(S + 1) R((oIN0)2(rn*)2

A-

6n

A4No

(5)

With R = 0.1 and parameters appropriate to CdMnSe NO = 0.26 eV and m* = 0.12), Eq. (5) predicts A = 2 x loM4 cm. This value is two orders of magnitude larger than the circumference of the Bohr orbit in CdSe (or CdMnSe). Thus, in a semimagnetic crystal an electron in a donor-BMP circles the

10. THEORY

OF BOUND MAGNETIC POLARONS

419

impurity 100 times-establishing a well-defined orbit-before experiencing an exchange scattering. This estimate is the main justification for Eq. (4). Comparison of theory with experiment also suggests that the Hartree approximation is accurate. With the aid of Eq. (4)one can derive separate, but coupled, Schrodinger equations for q5 and x by integrating Eq. (3) against x* or +*. They are:

and

where and

Eqs. (6) and (7) could, in principle, be solved self-consistently to determine the individual spin states of the BMP. The number of such states is enormous, however, of order 6NwhereN 100 is the number of Mn2+spins within a donor orbit. Many of these states are accessible to the system, even at low temperatures; the properties of the BMP are thus determined by an ensemble average over low lying states. Self consistent equations for the ensemble averaged, orbital wave function are derived by minimizing the free energy with respect to $(r). One requires

-

subject to the normalization condition jJ+(r)IZd3r = 1, where G - kTln ( Z ) and

=

where = l/kB T . The resulting Schrodinger equation for +(r)has the same form as Eq. (6). However, (s * S j ) is now defined as the ensemble average:

420

P. A. WOLFF

over the spin Hamiltonian:

Eqs. (6) and (12) determine 4(r) self consistently. Their solution is usually calculated-or estimated-variationally by assumin a simple form for W ) , such as the hydrogenic wave function +(r) = e-r'u/ na ,calculating G from Eq. (1 l), and minimizing it with respect to a.

$-J

IV. The BMP Partition Function

1. ZEROMAGNETIC FIELDCASE

The spin partition function [Eq. (1 l ) ] determines the thermodynamic properties of BMP. Evaluation of this formula is a primary task of the theory. Throughout the calculation the wave function 4(r) will be assumed known. In donor-BMP, where the exchange interaction is relatively weak, 4(r) is nearly the same (+1-2%) as the donor wave function of the binary host compound. On the other hand, for acceptors, a variational calculation is required to determine the optimum size of the BMP. Subsequent analysis is simplified by rewriting the spin Hamiltonian in the form: %spin

= -a

C [ ( ~ * S j ) l 4 ( R j )=I ~- C] j

j

[Kj(s.Sj)]E

-(S*r),

(14)

The expression for the coupling constant, Kj, involves the exchange interaction and the square of the BMP wave function. Both factors are smaller in donor-BMP than in acceptor-BMP; when combined they imply that polaron interactions in donors are an order of magnitude weaker than in acceptors. The quantity l- is proportional to the effective field experienced by the electron in the BMP. It determines the Zeeman splitting of the complex. r is a vector operator whose components do not commute with one another; that fact prevents an exact evaluation of the trace in Eq. (1 1). On the other hand, if r were a classical variable, the trace over spin variables could be calculated via the standard identity:

10.

THEORY OF BOUND MAGNETIC POLARONS

421

This identity is not correct when r is an operator. In the Mn2+case, however, the individual spins are fairly large (S = 8) and combine in the BMP to give a net moment of many Bohr magnetons. Thus, it should be a good approximation to treat r as a classical variable. An exactly soluble model of the BMP, discussed in Part IV, Section 3, suggests that this approximation is correct to order N-l”, where Nis the number of Mn2+spins within the BMP orbit. The classical approximation gives the following expression for the partition function:

where 55/2(x)

= i [ e5/2x

+

e 3 / 2 ~ + e l / 2 ~+ e-1/2X

+

e-3/2~

+ e-5’2x].

(19)

5 5 / 2 ( ~ )is an even, analytic function. This fact can be used to calculate 2. After performing the angular integrations, the expression is rewritten in the form:

and evaluated by displacing the A-contour upwards in the complex plane, above the line 1 = i/3/2, to make the y-integration convergent. Integration over y gives the relation:

Finally, the A-integral is calculated by moving the contour back to the real

422

P. A. WOLFF

axis. Only the pole at A = iB/2 contributes. The result is:

It is interesting to note that the first factor in this expression is the partition function for an Ising model of the BMP-with Hamiltonian XIsing

=

C[~j(szsjz)l. j

Equation (22) is a fairly complicated function of temperature because the alignment of each MnZ+ion varies nonlinearly with (PKj/2), and saturates when the exchange field exceeds thermal energies (BKj S 1). Saturation occurs at low temperatures in acceptor-BMP, but not in donor-BMP. In discussing the latter, it is often convenient to simplify the expression for 2. For small y , the distribution of internal fields appearing in Eq. (18) is Gaussian:

where

wd'=

(;:)

- C(K,?). j

-

Equation (23) is valid well beyond the fluctuation range ( y WO),but fails before y reaches its maximum value, $X(CYNO). Substitution of Eq. (23) into Eq. (18) yields the following formula for the partition function in the Gaussian approximation:

(25)

Equation (25) is similar to that of DS, and will later be used to make a connection between the DS and HWW theories.

10.

423

THEORY OF BOUND MAGNETIC POLARONS

2 . FINITEFIELDCASE Eqs. (18)-(25) can easily be extended to the finite magnetic field case. When fields are present, the Hamiltonian takes the form: Xspin

=-

c [Kj(s Sj)] - /.fBg*(s Bo) - PBgMn c (sj Bo). *

*

*

j

(26)

j

The partition function becomes:

(27) where b*

pBg*Bo

and

bMn

PBgMnBo.

(28)

The angular integration in Eq. (27) is complicated, but simplifies in the Gaussian limit. One then finds:

where

The effective field (beff)experienced by the carrier is the sum of the external field (b*)and the internal, exchange field defined by:

424

P. A. WOLFF

At low temperatures, the exchange field is often much larger than the applied field. In CdMnSe, for example, aNo = 0.26 eV; if x = 0.05 and T = 4 K, the dimensionless factor in brackets in Eq. (31) then has the value 100. Field enhancement factors of that size are not unusual in semimagnetics. Such materials act as amplifiers of external magnetic fields, at least insofar as their effect on carrier spin degrees of freedom is concerned. It should be clearly understood, however, that the exchange interaction does not affect carrier orbital motion. 3. SOLUBLE POLARON MODEL Several authors (Planel, 1982; Golnik et al., 1983; Ryabchenko and Semenov, 1983) have discussed an exactly soluble model of BMP. Though unrealistic, it provides a valuable test of the classical approximation. The model assumes that a donor-bound electron interacts with N Mn2+ ions through an exchange interaction of constant strength: N

x = -K C

j= 1

=

(s.Sj)

K 2

- - [(S * s)2 - S 2 - s2],

(32)

where S = Cj”,,(S,). This Hamiltonian would follow from Eq. (13) if NMn2+ ions were distributed on a spherical shell surrounding the impurity. The operators J = ( S + s), J z , and S 2 commute with the model Hamiltonian. Thus, it has eigenvalues

E(J, J z , S ) = - - J(J K2 [

+ 1) - S(S + 1) - -431 ,

(33)

that are independent of J z . For a given S , the allowed values of the total angular momentum are J = ( S f i).The corresponding energies are: KS E ( + , J , , S ) = --, 2

In these two groups of levels the electron spin is aligned parallel and antiparallel to the internal field, respectively. SFRS causes transitions between them, with S conserved. The partition function for the model is:

z

=2

C(D(S)[(S+ 1)eflKS/’ + Se-”K/2(S+1) 1 1 9

(35)

10.

THEORY OF BOUND MAGNETIC POLARONS

425

where D ( S ) is the number of Mn2+spin states, of fixed S,, with total spin S . Yanase and Kasuya (1968) describe a technique for calculatingD (S);when 1 Q S Q 5/2N, it can be shown that: 3/2

D(S) = -!(L)(2s + 1)e-6S2/35N. \/?T 35N The partition function then becomes:

+

~ ( 2 s l)e-PKS/2(S+l) 6S*/3SN 1e1. +

(37)

This summation over S is similar in form to the integral over y in Eq. (25). We now compare the classical partition function for the model to the exact result [Eq. (35)]. If S is assumed to be a classical variable, the trace over the carrier spin in the partition function can be evaluated with Eq. (17); one finds Z = Tr[e°K(S'S)] I :

[

b(S)(ZS + 1)cosh b K Y ] j .

The function of S under the summation sign in this formula differs from that appearing in Eq. (35) by terms of order S - ' . Since the main contribution to the sums comes from terms with S N"' , or larger, we conclude that the classical approximation to the partition function is accurate to order N-"2.

-

V. Thermodynamic Properties of BMP 4. INTERNALENERGY

Much of BMP behavior is determined by statistical mechanics, and can be calculated from the partition function [Eqs. (22), (25) and (29)l. Thermodynamic functions demonstrate, in particular, the gradual transition from the fluctuation-dominated regime to the collective regime mentioned above. Consider first the internal energy, U,of a BMP. It is an especially important thermodynamic variable, directly measured by luminescence studies of acceptor-BMP. U can be calculated from the usual statistical mechanical relation:

Differentiation of Eq. (22) produces a complicated, not-too-meaningful formula. The result simplifies, however, in the unsaturated limit. One then

426

P. A. WOLFF

finds [from Eq. (25)l:

u = - pWo2 -[ 4

1.

12 + (pwo)2 4 + (pwo)2

This expression can be seen to be identical to that of DS [their Eq. (4.11)] by making the replacement p W,’ + 4&,,and assuming a Curie susceptibility. Equation (40) has different temperature variations in the high (pWOe 1) and low ( ~ WBO 1) temperature regimes. To understand the meaning of these two regimes, we note that there are two ways in which the BMP can lower its internal energy to increase binding. At high temperature, though (r) = 0, there are sizable fluctuations of r about its mean value-of size because the BMP contains a finite number of Mn2+ spins. As temperature is reduced the carrier spin, which relaxes rapidly compared to those of the Mn2+ions, gradually aligns with the instantaneous fluctuations of r, causing a decrease in U. Throughout this spin alignment process, as will be shown below, the magnitude of r remains essentially unchanged, determined solely by the statistics of Mn2+ spin orientation. Hence the term “fluctuation regime”. When alignment of the carrier spin is nearly complete-for B WO= 2-the system can further reduce its energy by forcing the Mn2+ spins to adopt statistically unfavorable, but energetically favorable, configurations with gradually increases throughout this BMPlarger values of formation regime, where fl WO> 1. Ultimately, at sufficiently low temperatures, the Mn2+ spins may saturate, i.e., ( s - s,) 5/4. This interpretation of the temperature variation of U can be tested by rewriting the expression for the internal energy as a product of two factors, one describing the carrier spin alignment process, the other the gradual lengthening of r:

a-

m.m

--.)

(r2)is calculated from Eq. (25) with the aid of the classical approximation r. One finds

for

10.

427

THEORY OF BOUND MAGNETIC POLARONS

This expression implies = when pW0 = 0 (high temperature remains essentially constant until regime). AS ~ W Oincreases, ~ W=O1; thereafter it increases approximately as pW0/2. Conversely, the spin orientation factor, 2(s T ) / m i n Eq. (41), grows linearly with (pW0) throughout the fluctuation regime (0 c ~ W 1) range. Thus, though the transition is a gradual one, it has two distinct regimes-a higher temperature one dominated by magnetic ion spin fluctuations, whose internal fields align the carrier spin; and a lower temperature one controlled by true BMP interactions that gradually increase the net moment of the complex. Not surprisingly, in the high temperature (pWo 6 1) regime, there are large (percentagewise) fluctuations of the BMP energy about its mean value. In the Gaussian limit,

-

4 =“:[1+

(

(4+/32WgZ) - 4+pZwgZ 4pW0

>’I

’

(43)

and

On the other hand, in the BMP regime the energy is quite sharply defined since

In donor-BMP, the exchange interaction is barely strong enough to force such complexes into the collective regime at T = 4 K. A Cd0.9oMn0.1oSe sample (Heiman et al., 1983b), for instance, has Wo = 0.6 meV, implying PWo = 1 at T = 8 K. It is not surprising, therefore, that SFRS experiments on CdMnSe (discussed in Part VIII, Section 3) show only faint indications of BMP formation in the temperature range T 2 2 K. If the Mn2+susceptibility of such systems were strictly Curie-like one could, in principle, reach the BMP regime by going to lower temperatures. In practice, this idea does not work because next nearest neighbor Mn2+-Mn2+exchange interactions cause the Mn2+susceptibility to saturate. Phenomenologically, this effect is described by making the replacement T + ( T + TAF)in the Curie law susceptibility, where TAF= 2.3 K in Cdo.goMno.loSe. Since magnetization fluctuations of a thermodynamic systems are proportional to its susceptibility (Landau and Lifshitz, 1977), this change also requires a modification

428

P . A. WOLFF

of the parameter, W,” -+ W 2 = W,”(T/T + TAF),that controls them in the Gaussian limit [Eq. (25)l. With that replacement, there is little further increase in donor-BMP bindng energy or spin correlation below T = 2 K . The situation is quite different in acceptor-BMP, where Wo = 5 meV and BMP formation commences when T = 30-50 K. In many cases, the Mn2+ spins near the center of the hole orbit are fully saturated at low temperatures. Such complexes behave like a single, giant magnetic dipole whose moment is 2 0 - 5 0 ~ Their ~ . properties will be discussed, in connection with experiments on acceptor-BMP, in Part VIII, Section 10. 5 . MAGNETIZATION Magnetization is another important thermodynamic property of BMP; in the Gaussian limit it is determined by the formula

X

where [from Eq. (30)]

The second factor in Eq. (44) is the BMP contribution to the magnetization. At low temperatures, the BMP “moment” is much larger than that of the carrier (geffP g*). The extra moment is provided by Mn2+ spins in the vicinity of the BMP, whose alignment with the magnetic field is enhanced by the exchange interaction. In the low field limit, the BMP has susceptibility:

This expression can be shown to be identical to Eq. (4.8) of DS, via the replacements discussed after Eq. (40). Note that XBMP is a smooth function

10.

THEORY OF BOUND MAGNETIC POLARONS

429

of temperature, without singularities at finite T. We will see below that, in contrast, a mean field theory of BMP predicts a spurious, finite-temperature divergence of X B M P . That prediction is incorrect, since a finite system cannot undergo a phase transition. Experimentally, there is no evidence of singularities in X B M P . As BO increases, Eq. (44) describes the gradual saturation of BMP magnetization. The general formula is complicated, but has simple and interesting limits when /3 WO6 1 and j?WO% 1. For p WO6 1. One finds:

This result follows directly from mean field theory (MFT), which assumes that each type of spin aligns in the external field plus the average exchange field created by the other type. The MFT equations for the BMP are:

and <Sjz>

=

SP[~M~ + Kj(sz)I,

when Mn2+ spins are not saturated. Elimination of formula for ( s z ) :

(49) (Sj,) gives

the MFT

When ~ W6O1, the second term in the argument of the hyperbolic tangent is small compared to the first. If it is neglected, Eqs. (48) and (49) are equivalent to Eq. (47). The approximations s (Sj) = (s Sj) = (s) Sj ,used in deriving Eqs. (48) and (49), eliminate the spin-spin correlations responsible for BMP formation. Thus, Eq. (50) cannot give correct results for ~ W O F )l . In fact, it predicts a divergence in K B M f~ o r b WO= 2 and finite (sz) in zero applied field if ~ W >O2. Neither effect is physical. To correctly describe the BMP susceptibility in the BMP regime ~ Ws Ol), we must return to Eq. (44). When ~ W sO1, Eq. (44) implies:

-

-

where S(q) =

[

I:

ctnh(q) - -

430

P. A, WOLFF

is the Langevin function that determines the spin alignment of a classical moment in a field. In our case, the classical moment is that of the BMP, whose spins are aligned with one another via the exchange interaction.

6. SPINCORRELATION Finally, we consider the spin correlation function, (s * Sj), for the BMP. Note that (s * Sj) = - aX/aK,. For such an operator-the derivative of the Hamiltonian with respect to a c-number parameter-one can use the cyclic property of the trace to prove that:

Here B5/z(x)is the Brillouin function defined by 5/2&/2($X)

5bz(X)/55/2(X).

(54)

This formula is based on Eq. (22) which includes the effects of spin saturation. When PKj a 1, the second term, in Eq. (53) is small and (s S,) + $ 8 5 / 2 -, $ as anticipated. In the fluctation regime, where PWO Q 1, one finds a small correlation: ( s - Sj) = 105/48(pKj) 4 1. This statement seems, at first glance, to contradict our earlier assertion that there can be considerable alignment of the carrier spin (with the fluctuating internal field) in the PWo e 1 regime. Actually, both statements are true. To reconcile them, remember that the fluctuating moment of the BMP involves of the total number of Mn2+ spins within only a small fraction, 0 (1/N1”), it. Thus, even full alignment of the carrier spin does not represent, on average, a high degree of spin correlation. For pWo Q 1, one expects ( s - Sj) = l/N’/’. VI. The Dietl/Spalek Formalism

Equations (40) and (46) of the preceding section are identical-assuming a Curie-law MnZ+ suceptibility-to the corresponding formulas in the Dietl-Spalek paper [their Eqs. (4.11) and (4.8)]. This fact suggests that, in the Gaussian limit, the two theories are equivalent. T o confirm that

10. THEORY

OF BOUND MAGNETIC POLARONS

431

relationship, which is obscured by their different formalisms, we recast our expression for 2 into a form that makes apparent its connection to the DS formulas. The DS theory uses a continuous medium approximation (Spalek, 1980) to describe MnZ+ magnetization. To make contact with their work, we partition the space surrounding the BMP into cells, each small compared to the volume of its orbit. We will also assume that each of the cells contains a sizable number of MnZ+ions. The two conditions can only simultaneously be met if there are many ions within the BMP orbit, i.e., if (4na3N0/3)XP 1. This inequality is fairly well satisfied in donor-BMP systems. For example, in n-Cdl-,Mn,Se with x = 0.05 (X = 0.03),one finds (4na3N0/3)X= 180. The continuum approximation should work well under such conditions. However, it is a poor approximation for acceptor-BMP, since their orbits usually contain about ten Mn2+ions. Equation (18) for Zcan be rewritten as a product of traces over the various cells:

Here yc(c = 1,2, ..., N ) is the contribution of Mn2+spins in cell “c” to the internal field. In the last line of the equation we have made the approximation Kj = Kc = constant for all j E c. To evaluate the cell spin traces in Eq. ( 5 9 , we assume (i) that Mn2+spins are not saturated and (ii) that each cell contains sufficiently many spins so that its spin statistics are Gaussian. One then finds:

432

P. A. WOLFF

=

=

! !

d3A

-( [ 5 5 / 2 ( - iAKc)]“ceiX’Yc)

(27CI3

Here W? = 35/12ncK:, and nc is the average number of Mn2+ spins in cell “c”. yc is related to the magnetization density in cell “c” via the expression

With this change of variables, Eq. (55) takes the form Z =

where

P(y)= (const.)!

5

d3M(rl)

!

P(y)d3y.

(57)

d3M(rN)[cosh(PTY)

uc is the volume of cell “c”, and x = 35/12[(pBgMn)/kT]xNO is the Curie susceptibility of the unpaired Mn2+ spins. In the continuum limit, Eq. (58) becomes a functional integral over suitably-weighted “paths” of M(r). With the change of variables y A (to match DS’s notation) one finds: +

p[M(r)l4A - A[M(r)llBM(r), where

and

10.

THEORY OF BOUND MAGNETIC POLARONS

433

These formulas are identical to Eqs. ( 3 . 2 ) , (3.3) and ( 3 . 5 ) of the DS paper for the case of a Curie-law, MnZ+susceptibility. The DS theory and the preceding analysis rely on the Gaussian approximation. Neither is correct for systems such as acceptor-BMP, in which spin saturation occurs. However, for donor-BMP, which do not saturate, they provide an accurate description of the behavior of the complex. DS also point out an interesting relationship between the mean field theory of the BMP and the Gaussian approximation to its partition function [Eq. (25)l:

The integrand in this expression has a maximum at the value of y determined by the formula:

The maximum is relatively sharp at low temperatures (PWO>> I), where BMP formation occurs. In that limit, @ym) % 1 and the factor 2 on the right hand side of Eq. (63) become unimportant; the formula is then equivalent to the mean field theory (MFT) result [Eq. (SO)] in zero magnetic field. At higher temperatures, on the other hand, the full equation is required to avoid the spurious phase transition predicted by MFT. In this regime, the distribution of y-values is broad and dominated by fluctuations. To summarize, MFT predicts the low temperature spin alignment of BMP, and the high temperature response to weak fields. It is totally inadequate in describing fluctuations or the gradual transition from the fluctuationdominated to the collective regime. VII. The Spin-Spin Correlation Function

Thermodynamic variables, such as the internal energy or magnetization discussed in Part V, can be used to interpret certain experiments concerning BMP. Others, however, require a knowledge of time-dependent properties. For example, the spin flip Raman scattering (SFRS) spectrum, S ( o ) , is the Fourier transform of the time-dependent, spin-spin correlation function (Wolff et al., 1977):

434

P. A. WOLFF

Here (with A = 1)

s(t) = ei"rs(0)e-ixt,

(66)

and 2 is given by Eq. (18) or Eq. (27). The unit vector (Y is determined by the geometry of the experiment and by the laser frequency. For our purposes, a can be viewed as a constant-involving neither the electron nor the Mn2+ spin variables. In the semiclassical approximation d3y6(y -

r - b*)eflz(Sj'bMd

The time dependent operator in this equation can be evaluated by rewriting (s * a) in terms of stepping-up and stepping-down operators relative to the quantization direction y. One finds:

This identity can now be used to evaluate the electron spin trace in Eq. (67). A tedious, but straightforward, calculation yields the result:

10. THEORY OF

BOUND MAGNETIC POLARONS

435

Finally, the trace over (Sj) is evaluated after substituting Eq. (69) into Eq. (67). In the Gaussian approximation, one finds (Heiman et al., 1983)

where

[1:

]

C = Z-'(6) exp -iVN(pb~~)' .

The integral in Eq. (70) is similar to that determining the partition function in the Gaussian approximation [Eq. (29)]. This fact substantitates the DS hypothesis that the integrand of the partition function determines the spin flip spectrum, with the two terms in the cosh(By/2) factor corresponding to Stokes and anti-Stokes scattering, respectively. This statement will become clearer in Part VIII, where Eqs. (69) and (70) are compared with SFRS experiments. We will also see there that the factor T [ t ; y , p l in Eq. (69) predicts striking polarization effects in SFRS. Their observation provides a stringent test of the theory. The Fourier transform of Eq. (70) determines the spin flip spectrum:

We are primarily interested in the second and third terms of this formula, that describe Stokes and anti-Stokes SFRS. The first, zero-frequency term can be shown (Romestain et al., 1975) to cause Faraday rotation. Equation (71) contains no damping. The electron spin precesses freely, without spin relaxation, in an effective field (beff)that is the sum of the external field and an internal, exchange field of the MnZ+ions [Eqs. (30) and (31)]. Damping is eliminated by the classical approximation which ignores the (s'S7) terms in the exchange interaction. In practice, Eq. (71) reproduces the measured SFRS spectra almost perfectly. The theory succeeds because the Mn2+spins relax slowly at low temperatures; thus, the electron spin has ample time to align with the slowly fluctuating effective field.

436

P. A. WOLFF

VIII. Comparison of Experiment With Theory 7. BRIEFREVIEWOF MAGNETIC SEMICONDUCTOR WORK Research on magnetic semiconductors was stimulated nearly twenty years ago by the discovery of novel electronic, optical, and magnetic phenomena not encountered in conventional semiconductors. These effects are caused by the carrier-local moment exchange interaction. They include: (i) Giant red-shifts of the band gap in ferromagnetic semiconductors such as EuO (Busch et al., 1964; Busch and Wachter, 1966a; Schoenes and Wachter, 1974a) and CdCrzSe4 (Harbeke and Pinch, 1966; Busch et al., 1966b). (ii) Giant Faraday rotations in ferromagnetic semiconductors (Ahn and Shafer, 1970; Tu et al., 1972; Schoenes et al., 1974b). (iii) A metal-insulator transition in Eu-rich EuO (Oliver et al., 1970; Oliver et al. , 1972; Torrance et ai., 1972). (iv) Large resistivity maxima in the vicinity of magnetic phase transitions, accompanied by negative magnetoresistance (Heikes and Chen, 1964; von Molnar and Methfessel, 1967). (v) Substantial variation of the paramagnetic Curie temperature with x in Eul-xGdxSe and Eul-,Gd,Se (Holtzberg et al., 1964, 1965). The first two phenomena listed result from the rigid, exchange-induced shift of the energy bands. No polaron effects need by invoked to explain them. On the other hand, the behavior of the conductivity and susceptibility suggest carrier localization by magnetic polaron formation. Susceptibilities and magnetization fluctuation amplitudes are known to diverge at second order magnetic phase transitions. The large susceptibilities favor polaron creation and carrier trapping. Spatial fluctuations of the exchange potential can scatter or localize carriers even when polaron formation is not possible. The divergences are suppressed by magnetic fields, that usually produce a negative magnetoresistance. It is important to realize that these conductivity effects are huge. For example, at the metal-insulator transition in EuO the resistivity changes by ten orders of magnitude in a small temperature interval. The mobility of Eul-,Gd,Se decreases by about a factor 1000 near its antiferromagnetic transition temperature. Magnetic fields in the 10T range increase the conductivity by comparable factors. Theorists have developed a variety of magnetic polaron models to explain these remarkable results. Kasuya and Yanase (1968; see also Yanase and Kasuya, 1968) attribute the unique magnetic and transport properties of dilute Eul-,Gd,Se and Eul-xLa,Se alloys to BMP formation at the Gd3+or

10.

THEORY OF BOUND MAGNETIC POLARONS

431

La3+ donors. Experimentally, the binding energy of the polaron is about 0.5 eV; this fact implies that its wave function is localized, mainly confined to the central Gd3+(or La3+)ion and its twelve nearest neighbor Eu2+ions. Kasuya and Yanase call this complex a “giant spin molecule”. It is rather different from the more extended BMP found in semimagnetics. In the latter, most of the BMP binding energy is provided by the Coulomb interaction with corrections due to exchange, whereas in the former exchange is dominant. Yanase and Kasuya (1968) calculate the magnetic energy, statistical weights, and magnetization of such BMP in detail, assuming tight binding wave functions. The spin problem can be solved exactly because there are only two exchange constants (Kj in our language)-that of the carrier with the central Gd3+ ion, and the carrier-nearest neighbor Eu2+ coupling. In other respects, the analysis is similar to that of Part IV, Section 1 . It gives a good fit to the susceptibility data (Holtzberg et al., 1964, 1965) with reasonable choices of the exchange constants, and also explains the resistivity maximum (von Molnar and Methfessel, 1967) as a transition from band conduction to hopping conduction between impurities. Though fluctuation effects are implicit in the theory, they are not readily apparent from the data, nor are they emphasized by Yanase and Kasuya. Unfortunately, there is no measurement that gives microscopic information concerning fluctuations, comparable to that provided by SFRS in n-type semimagnetics, for the BMP in EuI-,Gd,Se or Eul-xLa,Se. Development of such experiments could elucidate a fascinating, tightly-coupled BMP complex. The BMP in Eu-rich EuO are less well understood. In that case, it is believed (Oliver et al., 1970, 1972) that oxygen vacancies which bind two electrons are responsible for doping of the crystal. For impurity concentrations in the 1-5 x lOI9 cm-3 range, there is an abrupt transition from band conduction to an activated conductivity regime. The transition occurs at T = 50K, below the Curie point (69 K) of the undoped material. The resistivity for T < K can exceed that for T > Tt by factors larger than 10”. Transport experiments show that the resistivity change is caused by a large decrease in free carrier concentration. Theories and models of the metal-insulator transition (Torrance et al., 1972; Leroux-Hugon, 1972; Nagaev and Grigin, 1974; Kubler and Vigren, 1975; Leroux-Hugon, 1976; Kuivalainen et al., 1979; Mauger, 1983) agree in ascribing it to electron localization via BMP formation at oxygen vacancies. However, the detailed electronic structure of these centers is still in doubt. It is not yet known, for example, whether the ground state is a singlet or a triplet. Analysis of the structure of the BMP is hindered by the exceedingly complicated, many-body nature of the metal-insulator transition.

438

P . A. WOLFF

8. OPTICAL EVIDENCE FOR BMP

IN

SEMIMAGNETICS

Optical experiments have given the most convincing evidence for the existence of BMP in semimagnetic semiconductors. Two techniques are used: spin flip Raman scattering and luminescence. When observable, SFRS gives detailed information concerning the BMP whose interpretation is relatively straightforward since the center under study involves a single carrier plus associated Mn2+ spin cloud. Luminescence lines that exhibit BMP effects are, in contrast, the result of recombination by complexes, such as excitons bound to neutral impurities (a three body problem) or donor-acceptor pairs (whose energy of emission depends on the separation of donor and acceptor). Neither spectrum is easy to interpret. Unfortunately, to date SFRS has only been seen in donor-BMP where polaron effects are weak. The observation of SFRS by more strongly coupled acceptor-BMP may be precluded, in cubic crystals, by valence band degeneracy. Here, nature conspires against us since the only wide gap semimagnetic that naturally occurs p-type is cubic p-CdMnTe; (CdMn)S and (CdMn)Se, with simpler, wurtzite valence band structures, are invariably n-type. The luminescence spectra of lightly-alloyed Cdl-,Mn,S, Cdl-,Mn,Se, and Cdl-,Mn,Te crystals with x I0.05 contain features that are similar, though less sharp, to those observed in the parent binary compoundsincluding free exciton lines, bound exciton lines, and donor-acceptor pair (DAP) lines. The bound exciton lines are generally those of excitons bound to neutral acceptors (AOX) or neutral donors (D'X). These identifications are made by comparing alloy luminescence spectra to that of the binary. In Cdl-,Mn,Te, for example, the spectra contain a prominent line that continuously evolves (Plane1 et al., 1980) into the known bound exciton (A'X) line of pure CdTe as x -,0. Thus, in the dilute alloys, at least, one can be sure that this feature is caused by A'Xrecombination. Its position determines the binding energy of the exciton to the neutral acceptor. Measurements of this energy were performed by Golnik et al. (1980). They found, in contrast to the CdTe case, that in the semimagnetic alloys, the A'X binding energy is a rapidly varying function of temperature; at low temperatures, it exceeds that in CdTe by a sizable factor. The data are shown in Fig. 1. Large temperature variations of impurity energies are not ordinarily observed in semiconductors. Golnik et al. attribute them to BMP formation around the bound exciton complex, and give a qualitative interpretation of the results. This important experiment was one of the first to give convincing evidence for BMP in semimagnetics. Unfortunately, the complexity of the three body problem has so far prevented the development of a quantitative theory of the AOX-BMP.

There have also been studies (Huber et al., 1983) of D'X luminescence in

10.

THEORY OF BOUND MAGNETIC POLARONS

,

C31-, Mn,Te

o

x

0 I

0

1

439

l

=0.05

I

z0.2 zo.3

i

aa

a 4

30

10

I

I

I0

I

I

30 T(K)

I

50

FIG. 1 . Variation of AoX binding energy with temperature in (Cd,Mn)Te alloys. Solid line: theory of GGNPB; dashed line gives AoX binding in pure CdTe. [After Golnik et al. (1980).]

(CdMn)Se. Here, again, one faces a three body problem in interpreting the data. Experimentally, (Cdl-,Mn,)Se samples with x = 0.05 have a DoX binding energy larger than that of CdSe, and nearly temperature independent. The increased binding is attributed to carrier spin alignment in the exchange field caused by Mn2+ spin fluctuations. On the other hand, in crystals with x = 0.10 the DoX spectrum abruptly shifts and broadens as temperature is lowered from T = 10 K to 2 K, an unexpected result ascribed to polaron formation. The complexity of the AoX and DoX centers has forced investigators to seek luminescence techniques for studying isolated donor- or acceptorBMP. That goal can be achieved via DAP luminescence, though at a considerable experimental price. In DAP emission the energy of radiation from a given donor-acceptor pair varies, because of the residual Coulomb interaction between their charges, with the distance R, between them. The complete DAP line is the superposition of radiation from all pairs. Usually it is too broad to give information concerning BMP formation. This convolution can be unfolded, experimentally, by studying DAP spectra as a function of delay time after pulsed excitation. The effective distance at

440

P. A. WOLFF

which pairs recombine is related to the delay time (t) by the approximate expression:

e2 R ( t ) = -ln(t/to), EO ED

(72)

where ED is the donor binding energy and to = 10-”-lO-’o sec. Nhung and Plane1 (1983) have performed a remarkable series of measurements of this type in CdMnTe, inferred acceptor-BMP binding energies from them, and interpreted the results via the DS theory. We will discuss their work in some detail in Part VIII, Section 4. SFRS is the other important optical method for studying BMP. In the (CdMn) chalcogenides, it was first studied by Nawrocki et al. (1980). Their observation of a finite, zero-field spin splitting was unambiguous evidence for BMP effects. It motivated recent theoretical work on BMP and stimulated further experiments. We will see below that the SFRS data clearly demonstrate the role of fluctuations in BMP. 9.2 ANALYSIS OF SPINFLIP RAMAN SCATTERING EXPERIMENTS

SFRS in wide gap semimagnetics has been observed in CdMnSe, CdMnS, CdMnTe, and ZnMnSe. In most cases, the scattering is caused by electrons bound to donors. The spin flip line is identified by its variation with magnetic field; typical data showing that variation in (CdMn)Se are illustrated in Fig. 2. The large initial slope and large saturation value of the spin splitting imply that electrons in CdMnSe have g-values at low temperatures in excess of 100, as compared to a g* below two in CdSe. Figure 3, which shows the low field data on an expanded scale, clearly indicates a finite spin flip frequency in zero field, whose value is nearly independent of temperature. This effect is unique to semimagnetics. In conventional semiconductors the spin flip Raman frequency shift varies linearly with field near BO = 0. The existence of this splitting implies that electrons in donor-BMP feel an internal field, due to Mn2+spin fluctuations, even when BO = 0. Though, on average, the Mn2+ magnetization within the donor orbit vanishes, it does so by fluctuating in time about zero. The electron spin, whose relaxation is rapid, aligns with the fluctuating Mn2+ magnetization to produce a net spin-spin correlation responsible for the zero field splitting. To compare SFRS data with theory, it is necessary to perform the integrals in Eq. (71). The result depends on the geometry of the experiment (through the vector a)and is complicated. However, it has simple limits for large and small Bo.They are

10.

THEORY OF BOUND MAGNETIC POLARONS

441

These formulas interpolate smoothly into one another, and the latter can be shown to give a good approximation to the spectrum over the whole field range below saturation (0 < b~~ < k T ) . Equation (74) is essentially the formula used by DS [their Eq. (3.25)] in analyzing.spin flip spectra.

FIG.2. Variation of spin-flip energy with magnetic field in (Cd,Mn)Se. [After Heiman et al.

(1 983 b).]

442

P. A. WOLFF I

>

0

T=l 9 K

0

3 4 69 I28

x A

0

0

I

2

I

I 4

6

+ o 8

-

18 0 283 I0

B (kG) FIG. 3. Low field spin-flip energies in (Cd,Mn)Se. [After Heiman et ul. (1983b).]

The theoretical curves in Figs. 2 and 3 were calculated from Eq. (74). They illustrate the magnetic field variation of the Stokes peak of that spectrum. Figure 4 compares the complete spectrum at zero field [Eq. (73)] with experiment. The agreement is excellent in both cases. DS achieve comparable fits to the data of Nawrocki et al. (1980). The fits determine two parameters:

and the temperature TAFused to correct the Mn2+ susceptibility for next nearest neighbor interactions (Part V, Section 2). For Cdo.~oMno.loSe,the parameters are WO= 0.66 f 0.07 meV and T A =~ 2.3 K. The value of WO Calculated from Eq. (75) is WO= 0.56 f 0.06meV and is in reasonable agreement with experiment. At the lowest temperature ( T = 2 K) achieved in these measurements, ~ W =O4. The system is then just beginning to enter the polaron regime. At higher temperatures fluctuations control its behavior. In particular, the zero field splitting is temperature independent as implied by the discussion after Eq. (42). Temperature independent energy shifts are characteristic of the fluctuation regime.

10. THEORY OF

443

BOUND MAGNETIC POLARONS

C d , _ , Mnx S e

'

O

-

r

-

0

-

x = o 10

-

2

1

3

4

STOKES SHIFT ( m e V ) FIG. 4. Zero field SFRS spectrum of (Cd,Mn)Se. Points are experimental; solid line the theory. [After Heiman el al. (1983b).]

A further test of the theory is provided by studies of polarization ratios in SFRS. Two groups (Alov et al., 1983; Plane1 et al., 1984) have pointed out that, in a particular geometry, polarization of SFRS measures the fieldinduced spin alignment of BMP in wurtzite crystals. The appropriate configuration is one in which BolltllZn, the incident laser beam propagates normal to t with Eo 11 t,and one studies the circular polarization of scattered light propagating parallel to P. Since light polarized parallel to 2 does not couple to the A-valence band, the main channel for light scattering in this geometry is via virtual transitions to the B-valence band. This fact simplifies the analysis and provides optimal polarization signals. To estimate the polarization, we make use of Eq. (71). It can be shown that the vector a = (2i ij) for right- and left-circularly polarized scattered light. The angular integrals in the expression for the Stokes spectrum become:

"

x4

)at12

11

ly*a*(2

i y - ( a * x a**)

Y

Y2

--

+

where 6' is the angle between the vector y and the &axis. The polarization

444

P . A. WOLFF

ratio is:

Here the average is over the angular weighting function in Eq. (76). For example: ex@pdp=

sinh(x)

2( 4 X

9

where x = abed W?, and C(x) is the Langevin function. Similarly 1 2

- (1

+ cos2#) =

ex@(l+ p 2 )d p = X

Hence

Alov et al. (1983) have derived a formula of precisely this form, but with argument x = /3MeffBoin a more intuitive way, and shown that it fits the field dependence of their polarization data. The BMP moments (Meff) they infer vary, with SFRS frequency shift, from 2 5 , to ~ ~5 6 ~ This ~ . frequency variation agrees with that predicted by Eqs. (78)-(80), and thus fully accounts for the polarization data. The experiment also implies that the SFRS line is inhomogeneously broadened with different frequency shifts corresponding to BMP with different effective moments and internal fields. It is illuminating to briefly review the arguments of Alov et al. (1983) and Plane1 et al. (1984) leading to Eq. (80). Both groups emphasize that the donor-bound electron experiences an effective field that, because of Mn2+ spin fluctuations, need not be parallel to the applied field. As a consequence, the electron spin wave functions take the form: XT = cos(8/2))t) - sin(B/2)Jl), XJ = sin(8/2)lt)

+ cos(8/2)11),

(81)

where the spinors It), 1.1) are referred to the c-axis. These states are then used, in conjunction with B-valence band edge wave functions

10.

THEORY OF BOUND MAGNETIC POLARONS

445

to calculate spin flip matrix elements. One can easily show that the US,+1/2 channels generate negatively or positively polarized SFRS, with amplitudes cos2(O/2) or sin2(O/2). Equation (80) follows if it is assumed that the angular probability distribution has the classical form, e-’’BMeffBo. These polarization experiments clearly indicate that BMP have large ( 2 5 - 5 0 ~ ~moments ) associated with them, whose alignment by a field follows the classical law-but with a different effective moment at each point in the SFRS spectrum. Thus, in the fluctuation regime, where the spectrum is relatively broad [Eq. (43)], it is not correct to speak of the BMP moment as if it were a rigid, time invariant entity. Whenp WO4 1, each BMP can have a wide range of moments and gradually fluctuates, in time, over many values. The situation is different in the collective regime. When ~ W%-O1, the distribution of effective fields is sharply peaked about the value The BMP then has a well-defined moment, whose average alignment is truly described by the classical Langevin formula [Eq. (51)].

a.

10. ANALYSIS OF ACCEPTOR-BMP EXPERIMENTS

Acceptor-BMP are potentially more interesting than donor-BMP because they can exhibit the full range of possible BMP behavior, from the fluctuation-dominated regime at high temperatures to the fully saturated collective regime at low temperatures. In a typical wide gap semimagnetic crystal, the coupling constant for acceptor-BMP is WO= 6meV, as compared to 0.7 meV in the corresponding donor-BMP. Thus, in the acceptor case, the transition from the fluctuation-dominated regime to the collective regime occurs at T = 3 0 K , where the MnZ+ susceptibility is Curie-like and the theoretical ideas of Parts I11 and IV should apply. Unfortunately, it is not easy to study the properties of the simple acceptorBMP. SFRS of holes has not yet been seen in semimagnetics, and most luminescence features are produced by three-body complexes whose wave functions are exceedingly difficult to calculate. The donor-acceptor pair (DAP) line is an exception; it involves the recombination of electrons and holes bound to fairly well separated donors and acceptors. Moreover, since the coupling constant of acceptor-BMP is much larger than that of donorBMP, any polaron effects observed in DAP luminescence can be attributed to the acceptor.

446

P. A. WOLFF

The time resolved DAP luminescence technique, developed by Nhung and Planel (NP, 1983) has made possible direct optical measurements of the acceptor-BMP binding energy in CdMnTe. Their results for a series of alloys are shown in Fig. 5 . These data have two important features: (1) acceptorBMP binding energies are substantially larger, at high temperatures, than that of the acceptor in CdTe and (2) there is a gradual increase in binding energy with decreasing temperature. NP ascribe the high temperature energy difference to Mn2+ spin fluctuations and suggest that BMP formation is responsible for the increase in binding energy at lower temperatures. In their original publication, NP tested this picture with a calculation that made several assumptions concerning the acceptor-BMP, namely: (1) Mn2+ magnetization is a continuous function of position, M(r). (2) The hole wave function is hydrogenic. (3) Mn2+ magnetization is linear in effective field to saturation, and constant thereafter. (4) The magnetic entropy can be calculated in the classical, Gaussian approximation.

Since then two groups (Wolff and Warnock, 1984 and Nhung et al., 1984) have developed formalisms which do not require assumptions (l), (3), and

I0 010 A

8

0

400

0

I

10

I

20

T(K)

-

I

30

5 '10

2 010 0 '10

I

40

FIG.5. Acceptor-BMP energies vs. temperature in (Cd,Mn)Te alloys. Dashed line: theory of Nhung and Planel; solid line: theory of Warnock and Wolff. [After Nhung and Planel (1983).]

10. THEORY OF

441

BOUND MAGNETIC POLARONS

(4). The analysis parallels that of Part IV. For the valence band one postulates, in the single band hydrogenic approximation, a spin Hamiltonian of the form:

x = - J Cj [(s

*

Ci [ K j ( s

~ j ) ~ $ ( ~ j z= ) l~ ]

*

~ j ) ] .

(83)

where the hole spin s = and $(r) = e - ' / " / G is the acceptor wave function. The radius a is determined variationally. The spin partition function, in the classical approximation, is then: 2 = Tr {s.sjl

5

-1 - Tr

S[y -

S[y -

C (KjSj)]eacs")d3y

C (KjSj)][&20Y

+ e1/20Y + e-'/2@Y+ e - 3 / 2 8 Y ] d 3 y

{SJI

= ( 6 ) N j d '(2n)3 dd'yeiX.ynIss/z(-UKj)][e3/20y

+ e1/20Y + e-1/2@r + e-3/2i37

11.

(84)

This expression is the analogue, for a spin$ carrier, of Eq. (18). The integrals can again be evaluated as outlined in Part IV, Section 1. The result is:

a formula quite similar to Eq. (22). Wolff and Warnock (1984) calculated BMP energies from this expression by making the continuum approximation, though in principle the discrete sum could have been evaluated. Their results are indicated in Fig. 5 along with earlier calculations of NP. The only adjustable parameter (a = 13 A ) in this fit was determined from the acceptor binding energy measured by N P in CdTe. Other quantities @NO,X) are known from independent measurements. Both of the calculations illustrated in Fig. 5 imply that Mn2+ spins within the acceptor orbit are fully aligned, with the hole spin, at low temperatures. The net moment of the BMP is then . radius of the acceptor wave function could, in principle, about 20 p ~ The vary with temperature via the changing exchange interaction. However, this effect is found to be small. The very recent calculations of Nhung et al. (1985) are compared with their experiments in Fig. 6. These results were obtained by numerically integrating an expression for the internal energy derived from Eq. (85):

448

P. A. WOLFF

u=--a(ln z) aP = -

S ~ i ~ y y ~ ( y ) [ + e +~ /+e’/2’r ~ ’ y - )e-1/28r

- )e-3/28r]

jd3y~(y)[e3/2P +~e 1 / 2 6 ~+ e-l/Wr + e - 3 / 2 8 ~ ~

(86)

A finite number of Mn2+ions was randomly distributed throughout the BMP to give the correct x-value, and the sum over j was cut off at R, = 1.5a = 15 A. Physically, R , is the distance beyond which the hole exchange interaction is no longer strong enough to break up antiferromagnetic, Mn2+-Mn2+ spin alignments. Though the calculations outlined above differ in detail, they contain similar physics. It is remarkable, and gratifying, that first principles calculations can predict theEvs. Tcurve for a center as complicated as the acceptorBMP. The detailed shape of this curve is sensitive to the cutoff R,;in this regard, the Nhung et al. (1985) calculation is the more realistic of the two. Both calculations use hydrogenic wave functions for the acceptor. Errors in the binding energy caused by this approximation are minimized by choosing the acceptor radius to give the correct energy in CdTe. However, other properties of the acceptor-BMP may be poorly described by a wave function that does not incorporate the degeneracy of the valence band.

I

I

0

I

I

I

I

10

20

30

40

Temperature ( K ) FIG. 6. Comparison of theory and experiment for acceptor-BMP energies. The solid curve is the exact, numerical calculation; the triangles are experimental points for Cd0.9sMno.osTe.The dashed curve corresponds to a truncated Gaussian approximation. [After Nhung et al. (1985).]

10.

THEORY OF BOUND MAGNETIC POLARONS

449

It would be interesting to repeat the Nhung-Planel experiments in CdMnS, where the valence band exchange parameter, PNo, is reported to be 5 eV (Gubarev, 1981). This coupling is anomalously large compared to other semimagnetics, where PNo = 1 eV. The large interaction should produce striking polaron effects in CdMnS.

IX. Evidence for Free Magnetic Polarons in Semimagnetics Exciton luminescence in semimagnetic crystals with x < 0.05 is known to be caused by excitons bound to neutral impurities (DoXor AOX). Recently, Golnik et al. (1983) have done an extensive series of exciton luminescence experiments in Cdl-,Mn,Te samples with x ranging to larger values. They show that the luminescence spectrum changes dramatically between x = 0.05 and 0.10. The AoX emission, that dominates the spectrum for x < 0.05 becomes weak and broad for x > 0.10. A new line nearer the band edge, labelled L2, emerges at x = 0.05 and becomes the strongest feature in the spectrum for x 2 0.10. Golnik et al. suggest that the L2 line is caused by magnetically localized excitons. Localization could result from Mnz+ spin fluctuations, polaron formation, or a combination of the two effects. Several trends in the data suggest magnetic localization of excitons: (1) As temperature increases, the L2 line evolves into the free exciton line. This result is implied by Eq. (2), which shows that the binding energy of a free polaron decreases with increasing temperature. (2) With increasing magnetic field, which smooths out magnetic fluctuations, the L2 line evolves into the free exciton line. (3) Curves of L2 energy vs. temperature have considerable structure at the spin-glass transition in crystals with x = 0.2, 0.3, and 0.4. In addition, the magnetically localized exciton hypothesis can explain the different variations with temperature of the intensities of the L1 (AOX) and L2 line. These arguments are not conclusive, but make a persuasive case for free polarons. It is not easy to think of other mechanisms for the anomalies Golnik et al. observe. Other experiments also give hints of free polaron formation. Warnock et al. (1984) have seen large optical pumping signals by exciting CdMnSe and CdMnTe crystals with a tunable laser whose energy was a few meV below the free exciton, but little polarization with higher energy excitation. Their measurements suggest that the spins of free excitons relax rapidly, by exchange scatterings, whereas those of excitons created in magnetic fluctuations are prevented, energetically, from relaxing by the sizable local exchange field. They infer, from the energy of the subsequent Stokes-shifted radiation, that the magnetic spin “trap” is most probably a free polaron.

450

P. A. WOLFF

This interpretation is supported by a magnetic field experiment, which narrows the exciton luminescence line by a factor of six, and shifts it towards the free exciton. The stability of free polarons can be tested with the analysis of Kasuya et al. (1970).As they indicate, the polaron becomes stable when its free energy vanishes: AF=

[

2mh:a2

]

35 R(JNol2 = o. 48 (8na3No)kT

Here we have used Eq. (40),in the collective limit, to evaluate the exchange energy where the wave function has been assumed hydrogenic. Equation (87) for AF remains valid until Mn2+ spin saturation occurs. That condition is not precisely defined since Mn2+spins at different positions in the BMP orbit saturate at different temperatures. Roughly speaking, however, we may consider saturation to occur when PKj > 1 for a Mn2+ ion at the center of the BMP. To avoid saturation we require:

If the radius a is eliminated from these equations, one obtains the inequality:

Note that this criterion, for the stability of free polarons, is quite sensitive to temperature and x-value. Electrons have small masses and small exchange constants (JNo= 0.25 eV). For such carriers, Eq. (89) cannot be satisfied at any temperature above T = 1 K. Thus, free electrons do not form polarons in the known semimagnetics. Holes (or excitons) have larger exchange constants and larger masses; both factors favor polaron formation. With (JNo) = 1 eV, m* = 1, and T = 10 K, Eq. (89) is satisfied for R > 0.05. This critical value of R agrees surprisingly well, considering the crudeness of the approximations, made in deriving Eq. (89), with the R determined by Golnik et al. from the L1 -,L2 crossover of the luminescence spectra. We conclude, therefore, that there is considerable circumstantial evidence for free magnetic polarons, and that the evidence is not inconsistent with the criterion of Kasuya et al. for their stability.

10.

THEORY OF BOUND MAGNETIC POLARONS

451

X. Conclusion

11. CURRENT STATUS OF BMP THEORY The theory outlined in Parts IV and V predicts the equilibrium properties

of BMP in detail. Excellent agreement with experiment confirms the Dietl-

Spalek picture of their behavior, whose key feature is the gradual transition from a high temperature, fluctuation-dominated regime to a low temperature, collective one. The fluctuation regime is exemplified by donor-BMP and has been thoroughly studied via SFRS in CdMnS and CdMnSe. On the other hand, only BMP involving holes have exchange interactions strong enough to produce the complete range of possible BMP effects, culminating in spin saturation at low temperatures. Unfortunately, the centers containing holes that are easy to study experimentally (AOX,D o X ) involve three carriers-two holes and one electron or vice versa. To date, the complexity of the resulting theoretical problem has prevented quantitative comparisons of theory with experiment for such complexes. A theoretical advance in this direction could provide several interesting, new tests of the BMP model. An exception to these statements is the elegant, but complicated, time resolved DAP luminescence technique of Nhung and Plane1 (1983). Their measurements determine binding energy versus temperature for the simple acceptor-BMP in Cdl -xMnxTe,and clearly demonstrate BMP formation at low temperatures. Though theory, including the effects of spin saturation, gives energies in surprisingly good agreement with their measurements, there are still some uncertainties in the analysis. The DS and HWW theories of BMP use a classical approximation to calculate Mn2+spin statistics. Carrier spins are assumed to follow the slowly varying internal field of the Mn2+ions. This approach accurately predicts the equilibrium properties of BMP, but is inadequate for treatment of Mn2+or carrier spin relaxation processes. 12. DIRECTIONS FOR FUTURE WORK

Recent advances in our understanding of polaron structure and energetics have begun to stimulate interest in other properties of polarons in semimagnetic semiconductors, including: (i) Spin dynamics. (ii) Free polarons. (iii) Polarons in the spin-glass regime. Harris and Nurmikko (1983) have used picosecond optical techniques to explore the polaron formation process in CdMnSe. They observe a sizable frequency shift of the DoX luminescence feature about 400 psec after

452

P. A. WOLFF

excitation, and propose that this delay is the BMP formation time. Optical pumping experiments (Warnock et af., 1984) show that free carrier spin relaxation rates are much faster than those of polarons in CdMnSe. These preliminary, but tantalizing, results suggest a thorough study of BMP spin dynamics; the problem seems naturally suited to recently-developed picosecond optical spectroscopy techniques. On the theoretical side, a fully quantum mechanical theory of BMP is required. The work of Yanase and Kasuya (1968) suggests that such a theory may not be inordinately difficult, since they have rigorously solved a quantum mechanical BMP problem with spatially varying exchange constants. Hopefully, a complete quantum mechanical theory will clarify the roles of spin relaxation and spin diffusion in the BMP formation process. The luminescence studies of Golnik et af. (1983) demonstrate a dramatic change in the spectrum of Cdl-,MnxTe as x increases beyond x = 0.07. The properties of the new L2 feature they observed are consistent with those expected from recombination of a magnetically self-trapped exciton. That interpretation is supported by optical pumping experiments (Warnock et al., 1984). However, further measurements to confirm this hypothesis , and better theoretical estimates of the free polaron energy, are needed. In particular, for crystals with larger x-values, competition between the strong carrier-Mn2+ exchange interaction, which tends to align Mn2+spins parallel to that of the carrier, and the direct Mn2+-Mn2+interaction which favors antiferromagnetic or spin-glass alignment, will play an important role in determining the polaron binding energy. Cdl-xMnxTecrystals with x 5; 0.2 have a spin-glass phase at low temperatures. Golnik et al. (1983) see substantial structure, in their measurements of the energy of the L2 feature versus temperature, at temperatures near the spin-glass transition in samples withx = 0.2,0.3, and 0.4.These remarkable observations imply that a localized exciton, whose center-of-mass wave function may be no larger than 1OA in radius, can sense the Mn2+spin-glass phase. It would be interesting to exploit this opportunity, by using the strong (exchange fields approaching 100T), localized magnetic probe that nature here provides us, as a tool to study the spin-glass phase in semimagnetics. References Aggarwal, R . L., Jasperson, S., Shapira, Y . , Foner, S., Sakikabara, S., Goto, T., Miura, N., Dwight, K., and Wold, A. (1985). Proc. XVII Intl. Conf. Phys. of Semiconductors, Sun Francisco, I984 (J. D. Chadi and W. A. Harrison, eds.), p. 1419. Springer. Ahn, K., and Shafer, M. (1970). J. Appl. Phys. 41, 1260. Alov, D., Gubarev, S., Timofeev, V., and Shepel, B. (1981). JETP Letfers 34, 71. (Pis’ma v Zh. Eksp. Teor. Fiz. 34, 76.)

10.

THEORY OF BOUND MAGNETIC POLARONS

453

Alov, D., Gubarev, S., and Timofeev, V. (1983). Sov. Phys. JETP57, 1052. (Zh. Eksp. Teor. Fiz. 84, 1806.) Busch, G., Junod, P., and Wachter, P. (1964). Phys. Letters 12, 11. Busch, G., and Wachter, P. (1966a). Phys. Condensed Matter 5, 232. Busch, G., Magyar, B.. and Wachter, P. (1966b). Phys. Letters 23, 438. Dietl, T., and Spalek, J. (1982). Phys. Rev. Letters 48, 355. Dietl, T., and Spalek, J. (1983). Phys. Rev. B28, 1548. Douglas, K., Nakashirna, S., and Scott, J. (1984). Phys. Rev. B29, 5602. Gaj, J. (1980). Proc. X V Intl. Conf. Phys. of Semiconductors, Kyoto; J. Phys. SOC. Japan, Suppl. A49, 797. Galazka, R., Nagata, S., and Keesom, P. (1980). Phys. Rev. B22, 3344. Golnik, A., Gaj, J., Nawrocki, M., Planel, R., and Benoit a la Guillaume, C. (1980). Proc. X V Intl. Conf. Phys. of Semiconductors, Kyoto; J. Phys. SOC. Japan, Suppl. A49, 819. Golnik, A,, Ginter, J., and Gaj, J. (1983). J. Phys. C16, 6073. Gubarev, S. (1981). Sov. Phys. JETP53, 601. (Zh. Eksp. Teor. Fiz. 80, 1174.) Harbeke, G., and Pinch, H. (1966). Phys. Rev. Letters 17, 1090. Harris, J., and Nurrnikko, A. (1983) Phys. Rev. Letters 51, 1472. Heikes, R., and Chen, C. (1964). Physics 1, 159. Heiman, D., Shapira, Y., and Foner, S. (1983a). Solid State Commun. 45, 899. Heiman, D., Wolff, P., and Warnock, J. (1983b) Phys. Rev. B27, 4848. Heiman, D., Shapira, Y., and Foner, S. (1984). Solid State Commun. 51, 603. Holtzberg, F., McGuire, T., Methfessel, S., and Suits, J. (1964). Phys. Rev. Letters 13, 18. Holtzberg, F., McGuire, T., Methfessel, S., and Suits, J. (1965). Proc. Intl. ConJ Magnetism, Nottingham (The Institute of Physics and the Physical Society, London, 1965). Huber, C., Nurmikko, A., Gal, M., and Wold, A. (1983). Solid State Commun. 46, 41. Kasuya, T., and Yanase, A. (1968). Rev. Mod. Phys. 40, 684. Kasuya, T., Yanase, A., and Takeda, T. (1970). Solid State Commun. 8, 1543. Kreitman, M., Milford, F., Kenan, R., and Daunt. J. (1966) Phys. Rev. 144, 367. Kiibler, J., and Vigren, D. (1975). Phys. Rev. B11, 4440. Kuivalainen, R., Sinkkonen, J., Kaski, K., and Stubb, T . (1979). Phys. Stat. Solidi(b)94, 181. Landau, L., and Lifshitz, E. (1977). Statistical Physics (Pergamon Press, New York), Chap. 12. Leroux-Hugon, P. (1972). Phys. Rev. Letters 29, 939. Leroux-Hugon, P . (1976). J. Magn. Magn. Muter. 3, 165. Mauger, A. (1983). Phys. Rev. B27, 2308. Nagaev, E., and Origin, A. (1974). Phys. Stot. Solidi (b)65, 457. Nagaev. E . (1983). Physics of Magnetic Semiconductors (MIR Publishers, Moscow). Nagata, S., Galazka, R., Mullin, D., Akbarzadeh, H., Khattak, G., Furdyna, J., and Keesom, P. (1980). Phys. Rev. B22, 3331. Nawrocki, M., Planel, R., Fishman, G., and Galazka, R. (1980). Proc. X V I n t l . Conf. Phys. of Semiconductors, Kyoto; J. Phys. SOC.Japan Suppl. A49, 823. Nawrocki, M., Planel, R., Molot, F., and Kozielski, M. (1984). Phys. Stat. Solidi (b)123, 99. Nhung, T., and Planel, R. (1983). Proc. XVI Intl. Conf. of Semiconductors, Montpellier; Physica 117B-l18B, 488. Nhung, T., Planel, R., Benoit A la Guillaume, C., and Battacharjee, A. (1985). Phys. Rev. B31, 2388. Oliver, M., Kafalas, J., Dirnmock, J., and Reed, T. (1970). Phys. Rev. Letters 24, 1064. Oliver, M., Dimmock, J., McWhorter, A., and Reed, T. (1972). Phys. Rev. B5, 1078. Peterson, D., Petrou, A., Datta, M., Ramdas, A., and Rodriguez, S. (1983). Solid State Commun. 43, 667.

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Planel, R., Gaj, J., and Benoit a la Guillaume, C. (1980). Proc. ist Intl. Meeting on Magnetic Semiconductors, Montpellier; J. Phys. CON.41, C5-39. Planel, R. (1982). Proc. XVI Intl. Conf. on Appl. of High Magnetic Fields in Semiconductor Physics, Grenoble; Lecture Notes in Physics 111, p. 309. Springer, Berlin. Planel, R., Nhung, T., Fishman, G., and Nawrocki, M. (1984). J. de Physique 45, 1071. Romestain, R., Geschwind, S., and Devlin, G. (1975). Phys. Rev. Letters 35, 803. Ryabchenko, M., and Semenov, Yu. (1983). Sov.Phys. JETP57, 825. (Zh. Eksp. Teor. Fiz. 84, 1419.) Schoenes, J., and Wachter, P . (1974a). Phys. Rev. B9, 3097. Schoenes, J., Wachter, P., and Rhys. F. (1947b). Solid State Commun. 15, 1891. Shapira, Y., Foner, S., Ridgley, D., Dwight, K., and Wold, A. (1984) Phys. Rev. B30,4021. Spalek, J. (1980). J. Mag. Magn. Muter. 15-18, 1289. Torrance, J., Shafer, M., and McGuire, T . (1972). Phys. Rev. Letters 29, 1168. Tu, K., Ahn, K., and Suits, J. (1972). IEEE Trans. on Magnetic 8 , 651. von Molnar, S., and Methfessel, S. (1967). J. Appl. Phys. 38, 959. Warnock, J., Heiman, D., Wolff, P., Kershaw, R., Ridgley, D., Dwight, K., Wold, A., and Galazka, R. (1985). Proc. XVII Intl. Conf. Phys. of Semiconductors, Sun Francisco ( J . D. Chadi and W. A. Harrison, eds.), p. 1407. Springer. Wolff, P., Ramos, J., and Yuen, S. (1977). Theory of Light Scattering in Condensed Matter, edited by Bernard Bendow, Joseph L. Birman, and Vladimir M. Agranovich (Plenum, New York). Wolff, P., and Warnock, J. (1984). J. Appl. Phys. 55, 2300. Yanase, A. (1972). Internat. J. Magnet. 2, 99. Yanase, A., and Kasuya, T. (1968). J. Phys. SOC.Japan 25, 1025.

Index A

B

Acceptors, see also Valence band, in DMS alloys acceptor-bound magnetic polaron, in Cd,-,MnxTe, 445-449 luminescence, from acceptor-donor pairs, 439-440 resonant acceptor states, 312 shallow acceptors, 316-322 Hamiltonian, 3 16-318 variational treatment, 3 18-322 Cdl -,Mn,Te, acceptor parameters, 321-322 Hgl-,Mn,Te, acceptor parameters, 322 Acceptors, in magnetic field, see also Valence band, in DMS alloys narrow-gap DMS alloys, 325-333 arbitrary magnetic fields, 329-330 binding energy, Hgl-,Mn,Te, 330-333 hopping conduction, p-type Hgl-,Mn,Te, 333-339 magnetic boil off, 327-328 resonant states, zero-gap DMS alloys, 338-342 ultra-quantum limit, 325-329 wide-gap DMS alloys Cdl-,Mn,Te, 325 spin splitting, exchange-induced, 322-325 Antiferromagnetic order, see also Correlation length, magnetic; Exchange interaction, d-d; Neutron scattering, magnetic; Short-range order, magnetic in FCC lattices, 136 long range, absence of in Cdl-,Mn,Te, 163 in Znl-xMnxTe, 147 short range in Cdl-,MnxTe, 159-173 in Znl-,MnxTe, 141-155

Band structure, see also Energy gap; Exchange interaction, sp-d; Overlap of bands, exchange-induced; Spin splitting; Valence band, in DMS alloys anisotropy of valence band, exchangeinduced, 292-293, 315, 318-320, 327 band parameters effective mass, in narrow-gap DMS alloys, 237, 246 g-factor, effective, 202, 221, 224, 237, 246, 253, 428 for Hgl-,MnxSe, 253, 263, 267-270 for Hgl-,MnxTe, 246, 267-270 momentum matrix element, 196, 268-269, 292 for 2-dimensional electron gas, 220-224 narrow-gap DMS alloys, in magnetic field Landau levels, 199-204, 206-207, 247, 251, 266 Pidgeon-Brown model, 193-199, 233-237 quasi-Ge model, 192 three band model, 192 narrow-gap DMS alloys, in zero magnetic field, near r-point, 190-191, 232-233 wide-gap DMS alloys, in magnetic field, 288-295, 313-315 wide-gap DMS alloys, in zero magnetic field, 36-39 near r-point, 38-39 away from zone center, 37, 56, 305 Bonding, tetrahedral, 4, 6 Bond length A"Bvl compounds, 7 ternary DMS alloys, 11-14 tetrahedrally bonded MnBV' compounds,

455

11

456

INDEX

Bound magnetic polaron acceptor-bound polaron, 445-449 in Cdl-,MnxSe, 405, 440-445 in Cdl-,Mn,Te, 399, 404, 446-449 Dietl-Spalek theory, 401-403, 430-432 donor-bound polaron, 398-405, 440-445 effect on exciton spin splitting, 297-299 in Eul-,Gd,Se, Eu~-,ta,Se, 436 in EuO, 437 Hartree approximation, 418-419 internal energy, 425-428 magnetization, 428-430 partition function, 420-424 Raman scattering, spin-flip, 398-405, 440-445

soluble model of, 424, 425 spin fluctuation effects, 299, 401, 425, 439,444-445

spin-spin correlation in, 430, 433-435 C

Cation-cation distance, see Lattice parameters (Cd1-,MnJ3Asz, 3 1 exchange integral, sp-d interaction, 215 Shubnikov-de Haas effect, 215 Cdl-,Mn,S crystal growth, 20-21 crystal structure, 3, 7-10, 79 energy gap, 46-47, 55 exchange integral, d-d interaction, 100 exchange integral, sp-d interaction, 295 exciton, spin splitting, 285 lattice parameter, 8-10 magnetic phase diagram, 104 Mn d-electron transitions, 60-62 spin-flip Raman scattering, 417 Cdl -,Mn,Se bound magnetic polaron, 405, 440-445 crystal growth, 19-20 crystal structure, 3, 7-10, 79 Curie-Weiss temperature, 89 electron paramagnetic resonance, 80, 84 energy gap, 44-45, 55 exchange integral, d-d interaction, 100 exchange integral, sp-d interaction, 295 exciton, spin splitting, 285 free magnetic polaron, 449-450 lattice parameter, 8-10

magnetic phase diagram, 104, 106 magnetic susceptibility, 92 Mn d-electron transitions, 60-62 specific heat, 109, 111 spin-flip Raman scattering, 403, 405, 414, 417, 440-445

spin-glass transition, 89, 94, 97, 104 Cdl-,Mn,Te acceptors, in magnetic field, 325 antiferromagnetic short range order, 159-173

correlation length, 164-170 band edge, spin splitting, 277, 280-281, 293-294

bound magnetic polaron, 399, 404, 446-449

crystal growth, 17-19 crystal structure, 3, 7-10, 79 Curie-Weiss temperature, 88 electron paramagnetic resonance, 80, 82-86

energy gap, 40, 44, 52 EXAFS, 11-13 exchange integral, d-d interaction, 100, 168, 177-178

exchange integral, sp-d interaction, 294-295

exciton, spin splitting, 281, 283-284, 294, 300

Faraday rotation, 276, 297 free magnetic polaron, 449-450 lattice parameter, 8-10 magnetic phase diagram, 104-105 magnetic susceptibillity, 91 magnons, 173-177, 390-396 Mn d-electron transitions, 59-61 Mn++ pairs, 388-389 neutron scattering, 155-172 inelastic, 155, 173-177 phonons, 365-373 specific heat, 107-108 spin dynamics, 173-177 spin-flip Raman scattering, 397-401, 404

spin-glass transition, 88, 94, 97, 105 superlattice, 28, 406-409 twinning, 156, 165, 175 Chalcopyrite DMS alloys, 31 Correlation length, magnetic, 133 anisotropy, 135, 164-169

457

INDEX

dependence on composition Cdl-,Mn,Te, 167 Zn,-,Mn,Te, 150-155 temperature dependence Cdl-,MnrTe, 170 Znl -xMnr.Te, 150- 155 Covalent radii of elements, 6, 11 Crystal structure, A:L,MnxBV1 alloys, 3-4, 7-14, 79

Curie-Weiss temperature, 88-89, 99-100 Cyclotron resonance, see Magnetooptical transitions, intra-band

D Debye temperature, 107 Diamagnetic susceptibility, A"BV' compounds, 91, 98 Dingle temperature, 209, see ufso Shubnikov-de Haas effect Donor-acceptor pair luminescence, 439-440

E Electron paramagnetic resonance, in DMS alloys, 80-90, see also Raman scattering, magnetic excitations in Cdl-,Mn,Se, 84-85, 89 in Cdl-,Mn,Te, 82-86, 88-90 g-factor, of Mn++ ions, 82 internal field, 82, 89 linewidth, effect of d-d exchange interaction, 82-90 in quaternary DMS alloys, 90 Energy gap bowing effects, 38-39 Cdl-,Mn,S, 47 sp-d exchange contribution, 39, 47-49 Znl-xMn,Se, 47, 49 Znl-,MnxTe, 48 concentration dependence, 41, 42 Cdl-,Mn,S, 46, 47 Cdl-xMnxSe, 44-45 Cdl-,Mn,Te, 40, 44 Hgl-,Mn,Se, 268 Hgl-,Mn,Te, 192, 267 Znl-xMnxSe, 47-49 Znl-,Mn,Te, 48

MnB"' compounds, 43 pressure dependence, 51-53 Cdl-,Mn,Te, 52 Znt-,MnxSe, 56, 68-69 Znl-xMn,Te, 55 temperature dependence, 50-51 Cdl-,Mn,S, 55 Cdt-,Mn,Se, 55 Cdl-xMn,Te, 52 Znl -xMnxSe, 56 Znl-,Mn,Te, 55 Epitaxy, A:'_,Mn,Bv' alloys, 28 EXAFS, in ternary DMS alloys, 11-13 Exchange integral, nearest-neighbor d-d interaction Cdl-,Mn,S, 100 Cdl-,MnxSe, 100 Cdl-,Mn,Te, 100, 168, 177-178 Hgl-,Mn,S, 100 Hgl-,MnxSe, 100 Hgl-,Mn,Te, 100 Znl.,Mn,S, 100 Znl-,Mn,Se, 100 Znl_,Mn,Te, 100, 154 Exchange integral, sp-d interaction, 188-198, 296, 375 (Cdl-,Mnx)3As2, 215 Cdl-,Mn,S, 295 Cdl-,Mn,Se, 295 Cdl-,Mn,Te, 294-295 definition, 188, 198, 235, 291, 296, 314 Hgl-,Mn,Se, 213-214, 270 Hgl-,Mn,Te, 213, 270 Pb1-,MnxBV' alloys, 217 physical interpretation, 188-189 Znl-,MnxSe, 295 Znl-,Mn,Te, 295 Exchange interaction, d-d, see also Antiferromagnetic order; Exchange integral, nearest-neighbor d-d interaction; Raman scattering, magnetic excitations; Spin-glass transition Bloembergen-Rowland mechanism, 80, 105, 107, 271

Curie-Weiss temperature, 99-100 effective Mn concentration, 101-103, 417-41 8

electron paramagnetic resonance, effect on, 81-85, 88-90

458

INDEX

Hamiltonian, 86 for specific cluster forms, 78 interaction beyond nearest neighbor, 77, 103, 118 superexchange, 13, 80, 84 Exchange interaction, ion-carrier, see Exchange interaction, sp-d Exchange interaction, sp-d, see also Spin splitting; Bound magnetic polaron Hamiltonian, 185-186, 197, 234-235, 288-289, 313-314, 375, 378, 416-417, 420 basis vectors, 193, 196, 234, 290, 313 diagonalization, 194-198, 235-236, 290-291, 314 eigenvalues, 197-198, 235-236, 290-293, 314-315 Pidgeon-Brown model, modification of, 197-199, 233-237 valence band, effect on, 292-293, 313-318 Exchange splitting, see Exchange interaction, sp-d; Spin splitting Excitons, in magnetic field in Cdl-,MnxS, 285 in Cdl-,Mn,Se, 285 in Cdl-,Mn,Te, 281, 283-284, 294, 300 complexes, in Cdl-,Mn,Te, 303-305 excited states, 300-303 luminescence, in Cdl-,Mn,Te, 303-305 magnetic polaron, effect of, 297-299 selection rules, magnetooptical transitions, 283, 294-295 from spin-orbit split-off valence band, 287 spin splitting, ground state, 277, 279, 283, 286, 293-296 in Znl-,Mn,Se, 283 in Znl-xMnxTe, 283 Extended x-ray absorption fine structure, 11-13

F Faraday rotation Cdl-,MnxTe, 276, 297 experimental technique, 277-279 Hgl-,Mn,Te, 271

magnetic properties, measurement of, 271-272, 276 theoretical model, 296-297 Fe-based DMS, 28-30, 254-257 Free magnetic polaron, 415-416, 449-450

G g-factor, effective, 202, 237, 428 Hgl-,Mn,Se, 253 Hgl-,MnxTe, 246 2-dimensional electron gas, 221, 224

H Hgl-,-,Cd,Fe,Se, 29-30 Hg~-,-,Cd,Mn,Te magnetooptical studies, 260-261 spin-glass transition, 27 1 thin film, 28 2-dimensional electron gas, 217, 219, 221 Hgl-,FexSe, 29-30 Hgl-,Fe,Te energy gap, 254 magnetooptics, 254-257, 270 Van Vleck paramagnetism, 256-257 Hgl-,Mn,S crystal growth, 27 crystal structure, 3, 7-10, 79 exchange integral, d-d interaction, 100 lattice parameter, 8-10 Hgl-,Mn,Se band parameters, 253, 263, 267-269 combined resonance, 264 conduction-valence band overlap, exchange-induced, 21 1-212 crystal growth, 26-27 crystal structure, 3, 7-10, 79 energy gap, 268 exchange integral, d-d interaction, 100 exchange integral, sp-d interaction, 213-214, 270 g-factor, 253 lattice parameter, 8-10 magnetic breakdown, 210-21 1 magnetic phase diagram, 104 momentum matrix element, 269 Shubnikov-de Haas effect, 204-21 1 spin splitting of Landau levels, 205

459

INDEX

Hgl-xMnxTe acceptors, in magnetic field, 330-333 band parameters, 246, 267-269 band structure, near r-point, 191-192, 233, 267 combined resonance, 258-264 conduction-valence band overlap, exchange-induced, 199-200, 21 1-212 crystal growth, 25-26 crystal structure, 3, 7-10, 79 cyclotron resonance, 264 effective mass, 246 energy gap, 192, 267 exchange integral, d-d interaction, 100 exchange integral, sp-d interaction, 213, 270 Faraday rotation, 271 g-factor, 246 hopping conduction, 333-338 lattice parameter, 8-10 magnetic phase diagram, 104, 106, 272 magnetic susceptibility, 92 momentum matrix element, 268 negative magnetoresistance, 333-338 nonmetal-metal transition, 338 resonant acceptor states, 339-342 shallow acceptors, 322 Shubikov-de Haas effect, 204-209 spin-glass transition, 95, 106, 272 spin splitting of Landau levels, 199-206 stimulated spin-flip Raman scattering, 266 2-dimensional electron gas, 217-219 Hopping conduction, in magnetic field in Hgl-,Mn,Te, p-type, 333-339 magnetoresistance, giant negative, 334-339 anisotropy of, 336-337

I Impurity band conduction, see Hopping conduction

K Kerr effect, magnetooptical, 278

L Landau levels, see Band structure Lattice parameters A"BV' compounds, 4-6 A'l',Mn,BV' alloys, 7-10 Lattice vibrations. see Phonons

M Magnetic breakdown, 210-21 1 Magnetic excitations, see also Spin dynamics magnons, in Cdl-,Mn,Te, 173-177, 390-396 neutron scattering, inelastic, 173-177 Raman scattering, 373-375, 390-396 Magnetic phase diagram, see specific DMS alloys Magnetic polaron, see Bound magnetic polaron; Free magnetic polaron Magnetic susceptibility, see also Spin-glass transition Cdl-,Mn,Se, 92 Cdl-,Mn,Te, 91 Curie-Weiss form, 91, 97 Hgl-,Mn,Te, 92 irreversible effects, 93-94, 96 of specific cluster configurations, 76-77 Magnetization, 198, 235, 314, 400-401, 428 Brillouin function, 235 effective Mn concentration, 101-102, 403, 418 modified, 101 equations of motion, for magnetic excitations, 373-375 in Hgl-,Fe,Te, 254-255 magnetic polaron correction, 297-299 mean field approximation, 198, 289, 314, 400 remanent, 112-117 spin splitting, relation to, 198, 235, 286 Magnetooptical transitions, see a h Excitons, in magnetic field narrow-gap DMS, 237-240 inter-band, 241-250 intra-band, 257-266 selection rules, 239 wide-gap DMS away from zone center, 305-308 selection rules, 283, 294-295

460

INDEX

Magnetoresistance, negative, in Hgl-,Mn,Te, 333-337, see also Hopping conduction, in magnetic field Magnons, see also Magnetic excitations; Raman scattering; Spin dynamics neutron scattering, CdI-,Mn,Te, 173-177 Raman scattering, Cdl-,Mn,Te, 390-396 Manganese covalent radius, 6, 11 distribution in A:L,Mn*BV1 lattice, 76-77, 101-102, 111, 147, 157 frustrated antiferromagnetic lattice, 79 percolation, nearest-neighbor, 77-78, 93, 103, 110 probability, specific cluster forms, 76-77 effective concentration, for magnetization, 101-102, 400, 418 similarity to group-I1 elements, 4 Manganese d-states charge transfer states, 70 hybridization, p-d, 70 intra-ion transitions, 58-68 pressure shifts Cdl-xMnxTe, 61 Znl-,Mn,Se, 65, 68, 69 Znl-*Mn,Te, 63 Racah parameters, 58-60 Znl-,Mn,Se, 60 Znl-*Mn,Te, 60 splitting and degeneracy, 57 Tanabe-Sugano diagram, 58 Mean field approximation, 198, 289, 314 magnetic polaron correction, 297-299 Mn++-Mn++interaction, see Exchange interaction, d-d MnS bond length, 11 lattice parameter, tetrahedrally bonded phases, 8-10 synthesis, 17 MnSe bond length, 11 lattice parameter, tetrahedrally bonded phases, 8-10 synthesis, 17 thin films, zinc blende, 28 MnTe bond length, 11

lattice parameter, hypothetical zinc blende phase, 8-10 synthesis, 17 Mott transition, see Nonmetal-metal transition, exchange-induced

N Neutron scattering, magnetic Cdl-,Mn,Te diffuse scattering, 155-172 inelastic scattering, 155, 173-177 in short-range-ordered systems, 128-135 theory, for magnetic short range order powder, 131-135 single crystal, 130, 135-140 Znl-xMn,Te, diffuse scattering, 140-155 Neutron scattering, nuclear Bragg scattering, 130 Cdl-,MnxTe, 157 Znl-xMnxTe, 141-143 diffuse, 130, 157 Nonmetal-metal transition, exchangeinduced, 337-338, see also Hopping conduction, in magnetic field

0 Optical absorption, in zero magnetic field Varshni equation, 50 Mn d-electron transitions, 59-69 Overlap of bands, exchange-induced, 199-201, 2 11, 246-247

P Pbl-,Mn,BV1 alloys, 30 exchange integrals, sp-d interactions, 217 Shubnikov-de Haas effect, 216 Percolation, see Manganese, distribution in A{~,Mn,B"' lattice Phonons, see also Raman scattering in Cd,-,Mn,Te, 365-373 in Znl-,MnxTe, 366-373 Preparation of ATTBV' compounds, 15-17 of Mn chalcogenides, 17 Purification of elements, 14-15

INDEX

Q Quantum oscillations, see Shubnikov-de Haas effect

R Raman scattering cross section, 347 experimental technique, 350-352 magnetic excitations magnons, in Cdl-,Mn,Te, 390-396 Raman antiferromagnetic resonance, 393 Raman electron paramagnetic resonance, 376-389 theory, 373-376 phonons mixed crystals, 357-365 perfect crystals, 353-357 selection rules, 348-350 spin-flip, 396-405, 414, 417, 440-445 Cdl-,Mn,S, 417 Cdl-,Mn,Se, 403, 405, 414, 417, 440-445 Cdl-,Mn,Te, 397-401, 404, 414 Hgl-,Mn,Te, 266 Znl-,Mn,Se, 417 tensors zinc-blende structure, 356-357 wurtzite structure, 356-357 theory, 346-350 Rare-earth-based DMS alloys, 30 Remanent magnetization, 93-94, 112-1 17 Resonant states, see Acceptors

S

Semiconductor-semimetal transition, see Overlap of bands, exchange-induced; Zero-gap Hg, -xMnxBV'alloys Short-range order, magnetic anisotropy, 135, 164-169 correlation length, 133 in Cdl-,Mn,Te, 157-173 in Znl-,Mn,Te, 141-155 models, for DMS alloys, 139, 168 neutron scattering from, 128, 135 phenomenological description, 133

461

Shubnikov-de Haas effect, 204-21 1 in (Cdl-,Mnx)3As2, 215 in Hgl-,Mn,Se, 204-211 in Hgl-,Mn,Te, 204-209 in Pb,-,Mn,Bv' alloys, 216 temperature dependence, of peak positions, 205-208 thermo-oscillations, in Hg, -,Mn,Te, 207 in 2-dimensional electron gas, 218 Specific heat, in DMS alloys, 107-112 Spin-dependent scattering, 212 Spin dynamics in Cdl-,Mn,Te, 173-177 models, for DMS, 138, 174 Spin-flip Raman scattering, see Raman scattering, spin-flip Spin-flip transitions, see Magnetooptical transitions Spin fluctuations, 299, 401, 427, 439, 444-445, see also Bound magnetic polaron Spin-glass transition below percolation concentration, 95-96, 103-106 Cdl-,Mn,Se, 92-96, 106 Cdl-,MnxTe, 94-96, 105 determination by Faraday rotation, 271-272, 276 effect of magnetic field, 97 frustrated antiferromagnetic lattice, 79 Hgl-,Mn,Te, 95, 106, 271-272 irreversible effects, 93-94, 96, 112-1 17 magnetic phase diagram for specific DMS alloys, 104-106, 272 percolation, nearest neighbor, 77-78, 103, 110 remanent magnetization, 112-1 17 Znl-,Mn,Te, 93 Spin-orbit splitting, valence band CdS, 287 CdSe, 287 CdTe, 287 Spin splitting, see also Bound magnetic polaron; Exchange interaction, sp-d of band edge, in DMS alloys, 292-296 in Cdl-,Mn,Se, 285 in Cdl-,Mn,Te, 277, 280-281, 283-284 relationship to magnetization, 286 of excitons, in DMS alloys, 277, 279, 283. 296

462

INDEX

excited state, 300-303 ground state, 293-296, 384 of Landau levels, 199-206 Hgl-,Mn,Se, 205 Hg -,Mn,Te, 199-206 2-dimensional electron gas, in DMS alloys, 219 of valence band, in DMS alloys, 314-315, 323 away from zone center, in Cdl-,Mn,Te, 305-308 Spin waves, in type-Ill antiferromagnets, 136 Stimulated spin-flip Raman scattering, 266 Superexchange, in A:'-,Mn,BV' alloys, 13, 80, 104 Superlattices, A!!,Mn,BV'-based, 28 Cdl -,Mn,Te/Cdl -,,Mn,Te, 406-409 Raman scattering, 406-409 Rytov formula, 407

T Tetrahedral bond, 4, 6 Tetrahedral radius, 6, 11 Twinning Cdl-,MnxTe, 156, 165, 175 zinc-blende crystals, 156 Two-dimensional electron gas, in DMS alloys, 217-224

V Valence band, in DMS alloys, see also Acceptors effect of sp-d exchange narrow-gap DMS, 315-316 wide-gap DMS, 314-315 anisotropy, exchange-induced, 292-293, 315, 318-320, 327-329 Van Vleck paramagnetism, in A!'-,FexBV' alloys, 257

2

Zeeman splitting, see Spin splitting Zero-gap Hg,-,Mn,BV' alloys conduction-valence band overlap, exchange-induced, 199-201, 21 1, 246-247 magnetooptical studies, 241-247, 260-265 resonant acceptor states, 312, 338-342 transition to open-gap, 190-192, 232-233, 267 Znl -,Mn,S crystal growth, 24-25 crystal structure, 3, 7-10, 79 exchange integral, d-d interaction, 100

exciton, spin splitting, 283 lattice parameter, 8-10 magnetic phase diagram, 104 Mn d-electron transitions, 60, 67 Znl-,Mn,Se crystal growth, 22-24 crystal structure, 3, 7-10, 79 energy gap, 47-49, 56, 68-69 exchange integral, d-d interaction, 100 exchange integral, sp-d interaction, 295 exciton, spin splitting, 283 lattice parameter, 8-10 magnetic phase diagram, 104 Mn d-electron transitions, 60, 65, 67-69 specific heat, 110, 111 superlattice, 28 thin film, 28 Zn 1-xMn,Te crystal growth, 22 crystal structure, 3, 7-10, 79 energy gap, 48, 55 exchange integral, d-d interaction, 100, 154 exchange integral, sp-d interaction, 295 lattice parameter, 8-10, 143 magnetic correlation length, 150-155 magnetic phase diagram, 104 Mn d-electron transitions, 60, 63 neutron scattering, 140-155 phonons, 366-373 short-range antiferromagnetic order, 141-155