Advances in Energy Transfer Processes

Figure 16 reports the decay of the donor luminescence following an exciting pulse, for the case cjc0=2 and n=8 (dipole-quadrupole interaction). Figure 17 reports the donor luminescence yield for n = 6, 8 and 10 obtained by integrating the solution of Eq. 23712. Note that the smaller is x0 (the greater hopping rate), the weaker is the luminescence yield of the donors. Let us cite a theorem that applies to Laplace transforms: "Given two functions fi(0) and /2(d) with Laplace transforms Fj(s) and F^fs), respectively, then 9 F1is)F2is) = L.T.JM9 - & )/ 2 «7 )dff 0

where L.T. = Laplace transform."

(238)

64

N/N0 °0.4

N/N0

N/Nn

Fig. 17. Yield of donor luminescence obtained by integrating the solution of Eq. 237. (Reproduced from [12])

Let us now use this theorem to deal with Eq. 237. If we set x = ax„ 1 = 0, — = a- = ad X

X„

(239)

X

Eq. 237 can be written as follows e 0(0) = p(6)e-ae + aje~a(e-ff)p(6 - 0 )(0 )dff o If, in addition, we call

/1(0) = e-oyp(0), f2(0) = m

(240)

65 then F1(s) = p(s + a),

F2(s) =
and, because of Eq. 238, e

\e-<6-ff)p(e o

- 9 )
Therefore Eq. 240 gives (s) = p(s +a) + ap(s + a)(f)(s)

Pi5 + a)

W=

l-ap(s

(241)

+ a)

<j>(0) is the inverse Laplace transform of <j>(s) in Eq. 241 above. We may want to check the general validity of Eq. 237. We note that «'>—j-^-»p(0

(242)

Let us now assume that no acceptors are present or, that WDA = const, as in the SternVolmer model. In this case we expect that the hopping of excitation should have no effect. Let p(r) = «?-(" r> T

(243)

= ax0

(244)

Then

p ( r ) = e -«/o = e-e/rxi+a) p(t-t)

= e~('~niZ"

= e -( 1 + a X'/r) e -(l+aX''/T)

66 and Eq. 237 becomes m

= e-^Xtlr) +

ae-(l+aXt/T)

ra+aXtlr)^

)d(

(245)

X

which gives 1 + a ,,., . \ + a -(i + a )(i/r) -
=

e

^ » nj_ _£0+£) e -(l+«X«/T)f e a + ««X»'/r)^ / , )
(246)

0

On the other hand Ml dt

=

_ l ± £ e - a + « ) ( * / r ) _ flO + a) e - ( i + a X ( / r ) r ( i + a ) ( r v v w w J v x Q

= _l±£ e -0+«X»/T) _ l ± £ ^ 0 + l + £ e - 0 + f l X f / r ) T r

+

«0(r) r

=

+

a T

_IW r

or m

= e-(t/T)

= m

(24?)

as expected.

In this section we consider the migration of excitation energy among donors as a random walk process. The excitation moves from donor to donor at random time intervals of average value ToAs the energy reaches a donor, the probability of transferring this energy to an acceptor changes. These changes are also random. After an excitation jump, as long as the excitation energy lingers on a particular donor, this donor finds itself in the same situation as a donor in the absence of energy migration; the average excitation probability is a function of the types already examined in the previous chapter. When evaluating the value that the average probability of donor excitation can take at a particular time, starting from an initial time at which the system is excited, we have

67 to sum over the various possibilities: 1 jump, two jumps, three jumps, etc. This sum is actually an integral. Appropriate mathematical manipulations lead to a renewal equation, solvable numerically, in which the average probability of donor excitation at time t depends on its value at time t-1'. 5. 3. Comparison of the Two Models We summarize now the results found for the two models. 1. Diffusion Model

_i_W!Yl T C \2J\T


1/2 1 + 10.87X + 15.5JC 2 3 / 4

1 + 8.743*

0

(248)

where x = DxR-2[-\

=-%rjt213

(n = 6only)

(249)

2. Random Walk (Hopping Model) , t


(250)

T. . *

For any p(t). Watts compared Eq. 248 and Eq. 250 and found agreement, as shown in Figure 18, for the case n = 6, CA = 2cQ and x=0

: = 0.l{i

2/3

2/3

x = 1.33

0

10

68

1 T HOPPING MODEL DIFFUSION MODEL

r

CA-2Ca

X-DC^T^tt/T)'*

Fig. 18. Decay of donor luminescence according to the diffusion model and to the hopping model. (Reproduced from [12])

Summary A parameter of importance is the time To, the average time it takes the excitation to jump. A small value of x0 is related to fast diffusion; a large value to slow diffusion. Examples are given in which the data obtained by the two approaches are presented and compared. 5.4. Regimes of Migration among Donors - Summary There are three regimes of migration among donors. 1. No Diffusion, en is small. The decay of donor excitation is given by p(t). p(t) is the average of exponentials corresponding to the various transfer rates wDA(R). An extreme case is that of a donor which is far away from all acceptors; the donor decay function is p(t) = e •(t/T)

(251)

where r = intrinsic lifetime of donor. The other extreme case is that of a donor which has an acceptor as nearest neighbor; in this case

69 p(t) = exp

(252)

twDA(Rm) T

Where Rm = minimum D - A distance. l/2 3/2 2. Diffusion-Limited Decay. CY> is greater. For t « C D / /i r iID 1/2

for the case of no migration. For t » C^

-,.

, p(t) is the same

3/2

ID

p ( 0 = e~('/T)~*D'

(253)

t* = CQA ID represents a boundary between the transfer without migration region and the transfer with migration region. 3. Fast Diffusion, CD is still greater. An increase in the concentration of donors produces a faster migration of energy, i.e. a larger D, and a larger KD. The characteristic time t* becomes shorter and shorter until it reaches the value WpA(Rm), Rm being the shortest D - A distance. Any additional increase of CD has effect on D, but no effect on KD which takes a saturated value 23 . In these conditions the decay of the donor luminescence is purely exponential, since the fast diffusion averages out all the different donor environments. In this case we set KD=KcA

(254)

where K = const., independent of CD, or, taking into account the possibility that a large fraction of acceptors are excited KD = KnA

(255)

where n& = concentration of ground state acceptors. In this case the evolution of the excited donor population follows the rate equation ±n(r(t) dt

=

-^l-KnA(t)nd,(t) x

(256)

Rate equations have been used for explaining energy transfer in rare-earth doped systems 4. Summary

(1)

Three regimes of migration are of interest: No migration. This regime is present when the concentration of donors is low. The deviation of the donors' decay from exponentiality is due to the different

70

(2)

(3)

environments in which the donors find themselves; some are near acceptors, some are far from acceptors. Migration-limited decay. A higher concentration of donors favors the migration of excitation energy among them. A critical time is considered whose value is the greater the higher is the donor-acceptor transfer rate, and the smaller is the diffusion coefficient. This time separates the first part of the donors' decay, that is unaffected by energy migration, from a part in which migration place a predominant role in feeding the acceptors. Fast migration. At even higher donor concentration the diffusion coefficient becomes much larger and the critical time approaches the minimum transfer time between a donor and an acceptor. The decay is exponential. The different donors' environments are averaged out by the rapidly moving excitation: all the donors decay at the same rate. An additional increase of D has no effect on the decay pattern of the donors' excitation. This decay, however, still depends on the concentration of acceptors, and becomes faster as this concentration increases.

References 1. 2. 3. 4. 5. 6. 7. 8. 9.

10. 11. 12. 13. 14. 15. 16. 17. 18.

H. Eyring, J. Walter and G. F. Kimball, Quantum Chemistry, Wiley, New York (1944), p. 351. H. Margenau, Phys. Rev. 38, 347 (1931). H. Margenau, Rev. Mod. Phys. 11, 1 (1939). B. C. Carlson and G. S. Rushbrooke, Proc. Camb. Phil. Soc. 46, 626 (1950). P. M. Levy, Phys. Rev. 177, 509 (1969). A. J. Freeman and R. E. Watson, Phys. Rev. 127, 2058 (1962). 7. E. Clementi, IBM J. of Res. and Dev. 9, 2 (1965). B. Di Bartolo, Optical Interactions in Solids, Wiley, New York (1968), p. 456. R. C. Powell, B. Di Bartolo, B. Birang and C. S. Naiman, in Optical Properties of Ions in Crystals, H. M. Crosswhite and H. W. Moos, eds., Interscience, New York (1967), p. 207. T. Miyakawa and D. L. Dexter, Phys. Rev. Bl, 2961 (1970). F. Auzel, in Radiationless Processes, B. Di Bartolo, ed., Plenum Press, New York (1980), p. 213. R. K. Watts, in Optical Properties of Ions in Solids, B. Di Bartolo, ed., Plenum Press, New York and London (1975), p. 307 F. Perrin, Compt. Rend. 178,1978 (1928). O. Stern and M. Volmer, Physik Z. 20,183 (1919). Th. Forster, Z. Naturforsch, 4a, 321 (1949). Th. Forster, Discussions Faraday Soc. 27, 7 (1959). M. D. Galanin, Zh. Eksperim, i Teor. Fiz. 28, 485 (1955) [English translation: Soviet Phys. JETP 1, 317 (1955)]. M. Inokuti and F. Hirayama, J. Chem. Phys. 43,1978 (1965).

71 19. 20. 21. 22. 23. 24.

F. Reif, Fundamentals of Statistical and Thermal Physics, McGraw Hill, New York (1965), p. 483 M. Yokota and O. Tanimoto, J. Phys. Soc. of Japan 22, 779 (1967). R. K. Watts, notes from 1974 NATO ASI lectures. A. I. Burshtein, Soviet Physics JETP 35, 882 (1972). R. K. Watts and H. J. Richter, Phys. Rev. 6, 1584 (1972). J. T. Karpick and B. Di Bartolo, J. of Luminescence 4, 309 (1971).

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T H E O R Y OF E N E R G Y T R A N S F E R P R O C E S S E S I N SOLIDS: MODES, COHERENT- A N D NON-MARKOVIAN PROPERTIES

RALPH v. BALTZ Institut fur Theorie der Kondensierten Materie Universitat Karlsruhe, D-76128 Karlsruhe, Germany

ABSTRACT An introduction to the description of energy transfer processes in condensed matter is given with special emphasis on the definition of the energy- and energy-current-density, relevant modes, coherent/incoherent and Markovian/ non-Markovian dynamics.

1.

Introduction

According to our present understanding of nature the notion of energy forms the basis of physics1 as well as other natural sciences, industry, economy, and society. 2 Quoting Freeman J. Dyson:3 "Even within the framework of physical science energy has a transcendent quality. On many occasions when revolutions in thought have demolished old sciences and created new ones, the concept of energy has proved to be more valid and durable than the definitions in which it was embodied". The central role of energy originates - apart from its conservation - that its change is accomplished with the change of at least one further extensive quantity, e.g. linear and angular momentum (p,L), position (r), charge (Q), vector potential (A), polarization (V), magnetization (M), Entropy (S), Volume (V), or particle number N, etc. 4 dE = vdp - Frfr + Q,dL + QdQ+jdA

+ SdV + ridM + TdS - pdV +

fidN..., (1)

73

74 Table. 1. Examples of energy transfer. system vacuum

gases

fluids

solids

modes of energy transport electromagnetic radiation kinetic energy of massive particles internal (or chemical) energy flow, convection density waves type-II collisions between atoms heat flow particle flow flow of chemical energy convection, diffusion, . . . charge transport localized states collective excitations

examples solar energy flux particle reactions excited atoms atmosphere sound waves He-Ne laser pump house heating system hydraulic system oil pipe line electric power lines photosynthesis phonons, excitons . . .

where the intensive variables v (velocity), F (force), etc. have their usual meaning.* With exception of r and V, all other extensive variables have a density and a current and, thus, they play a different role: r labels different spatially separated subsystems of size V which are described by densities of energy, momentum, etc. In a perceptive view p , L, Q, . . . play the part of "energy-carriers". A review on energy transfer processes up to 1983 can be found in the Proceedings of a previous summer institute on the same topic 5 whereas this volume 6 documents the progress obtained. Since 1969 the INSPEC database provides more than 53000 entries on energy transfer processes, yet only about 1100 papers are related to coherence and 130 study (non-) Markovian properties. The latter subjects are of interest in the excitonic energy transport in molecular crystals 7 and, in particular, for the description of ultrafast processes in optically excited semiconductors. 8 The scope of this article is to give an introduction of the description of the conservation and flow of energy (Chapter 2.), construction of energy density and energy current density of the systems of interest, and examples of (normal-) modes (Chapter 3.), description of (ir-)reversible and (in-)coherent dynamics (Chapter 4.), and (non-)Markovian (memory function) properties (Chapter 5.). Various examples will be given or can be found in this volume. 6

•Notation: Vectors and tensors in boldface, the magnitude of a vector v is denoted by the same letter, e.g. v = |v|. Matrices and operators are set with a hat, e.g. a. Electromagneticfieldsare set in calligraphic style, e.g. S, B.

75 2.

Conservation and Flow of Energy Quoting Richard P. Feynman 9 "If a cat were to disappear in Pasadena and at the same time appear in Erice, that would be an example of the global conservation of cats. This is not the way cats are conserved. Cats or charge or baryons are conserved in a much more continuous way. If any of these quantities begin to disappear in a region, they begin to appear in a neighbouring region. Consequently, we can identify a flow of charge out of a region with disappearance of charge inside a region. This identification of divergence of a flux with the time rate of change of a charge density is called a local conservation law. A local conservation law implies that the total charge is conserved globally, but the reverse does not hold. However, relativistically it is clear that nonlocal global conservation laws cannot exist, since to a moving observer the cat will appear in Erice before it disappears in Pasadena. flow of energy

Figure 1: Schematic of theflowof energy through the surface of a volume. If there is a change of energy in a certain portion of space it may either flow across the surface of V or it is exchanged with another system which is not explicitely considered here, see Fig.l. Like cats the rate of change of an extensive variable G t is governed by a relation of the form

f = -J G + £ G ,

(2)

where ^ and E G (=production rate of G) depends on the volume V of the portion of space and Ja (=current) depends on the surface of V. For a conserved quantity, such as electrical charge or energy E G = 0 whereas for entropy T,s > 0. Sometimes, however, merely the energy balance of a subsystem is considered, e.g. the electromagnetic field may gain or loose energy by interaction with the mechanical system of the charges so that E# ^ 0 if T,E refers to the electromagnetic field alone. As the size and shape of the volume V considered in Eq. (2) is arbitrary, a local formulation of a conservation law can be derived by using Gauss' law ^ M

+ divJG(r,i) = a G ( r , t ) .

^G denotes the name as well as the quantity.

(3)

76 pG(r, t),jG(r, t) and ffG(r, t) respectively denote the density, current density, and production rate density of G. In addition, the relations of these quantities to the state variables have to be specified. It is an old question whether an energy current can be imagined as energy moving with a well-defined velocity10 yet a solution has been worked out only recently for some cases. 11 In a kinematic interpretation, one assumes a decomposition of the form j E — PEVE, like it is known to hold for the particle current of a one-component fluid, JAT = PNVN, where, pN, vN are state variables of the fluid which can be measured independently. In particular, vN is the velocity of a reference frame in which j N is locally zero. For several components the current density consists of several terms

PN =Y2pi, i

iN = Yl PiVi '

(4)

i

where z labels different particle numbers TV*. Note, j ^ = PWVJV rnay appear as selfevident, -nevertheless, it does not define a particle transport velocity v # which, in general, can be measured independently and, hence, such a velocity is of no physical relevance. This holds for the energy-current velocity, too. The situation is even more queer for the momentum balance of the electromagnetic field. For a pure electrostatic field the momentum density (=£ x %/c) is zero everywhere, whereas, the momentum-current denstity (= negative Maxwellian stress tensor) has nonzero (diagonal) components, see Landau-Lifshitz 14 (Vols. 2,8). In some special cases, however, v £ = JE/PE is identical with the group velocity of a wave puis. Furthermore, a decomposition of the form of Eq.(4) may sometimes be appropriate in terms of plane-wave modes of the system which play the part of particle numbers degree of freedom.

3.

Density, Current, and Modes of Energy Transfer

In this chapter, we will construct and list of results for the energy density, energy current density and (normal-) modes of basic systems and equations which are frequently used to describe energy transfer processes. Normal modes of a system are defined as particular solutions of the respective (homogeneous, linear) field equations supplemented by appropriate boundary conditions. (In some cases, such modes could even be chosen as complex functions even though the physical fields are real.) Their importance and utility lies in the fact, that arbitrary field configurations can be expanded in terms of these modes which themselves represent a type of stationary states. In Quantum Mechanics these modes are the eigenstates of the Hamiltonian. The following chapter gives some selected examples. For a discussion of nonlinear waves see, e.g. the review article by Bishop et al. 12

77 3.1.

Scalar Waves Propagation of scalar (complex) waves are described by the wave-equation

^£- A )* ( r ' i ) = s ( r ' f ) '

(5)

where s(r, t) describes an external source. The state of a wave-field is fixed by i]/(r,to) = $ ( r ) and 4>(r, t0) = 9 ( r ) , where $ ( r ) , 6 ( r ) are two arbitrary functions. The homogeneous part of Eq. (5) is (form-) invariant under time reversal and, thus, describes reversible processes. By standard manipulations the energy density, currentand production rate densities can be found

d*(r,t)

Pfi(r,£)

2 c2

dt

jE(v,t)

=

oE(T,t)

= pm®(^^s(T,t)^

2

1 + -/3m|grad*(r,t)|2,

-p^R^^lgrad^r,*))

,

,

(6) (7)

(8)

where pm denotes the mass-density (or another appropriate quantity) and 5R means real part, see e.g. the textbook by Barton. 13 Expressions of this form are relevant to all scalar waves, though the physical significance of the individual terms may be different. For example, for an elastic continuum \I> denotes the displacement of a volume element, c = \jEjpm is the velocity of sound, and E is the bulk modulus, see Landau-Lifshitz 14 (Vol.7). For a nonviscous compressible fluid, on the other hand, \I/ is the velocity potential, v = —grad?/> and Eqs.(6-7) can be rewritten as M r > *) = ^PmV2 + pme ,

j B ( r , t) = ( -pmv2

+ pmw J v .

(9)

Here the energy density is just the sum of the kinetic and internal energy e (per unit mass), w = e +p/pm is the enthalpy (per unit mass), and p is the pressure, see Landau-Lifshitz 14 (Vol. 6). J B consists of two terms: the term pEv represents the energy which is "transmitted convectively" whereas the term pv is traditionally described as "the work per sectional area" which is done by the fluid.11 Next we consider the modes of the (homogeneous) wave-equation Eq. (5) (s = 0) by separation of the variables r, t *K(r,i) = e-^**K(r).

(10)

K labels different mode functions which obey the (homogeneous) Helmholtz-equation

NT)'

*«(r) = 0.

(11)

78 An arbitrary solution of the (homogeneous) wave-equation can be decomposed in modes * ( r , t) = ] T [AWc-*"-* + A^e+^t]

*K(r),

(12)

K

where the two independent functions A^ are fixed by the initial conditions. For real fields, these functions are not independent. The simplest modes are plane waves, l'k(r) = exp(ikr), where K = k is the wave vector and wk = ck. Remarkably, the energy current of such a mode is just energy density pE(r,i) = pmk2 times c, j B ( r , t ) = pB(r,t)ck (see end of Chapter 2.). As an example, we consider the propagation of (longitudinal) waves in onedimension (no source term), where the general solution can be explicitely stated in terms of two arbitrary functions f^(x), which are fixed by the initial conditions. •9{x, t) = /<+) (x - ct) + /<-> (x -ct).

(13)

Individually, f^\x) describe wave-packets which are propagating in ±x directions. Tab. 2. summarizes the results for two special cases. Table 2. Solutions of the one-dimensional wave equation (D'Alembert-solution). Left: Displacement-, right: momentum-excitation at t = 0. #(a;,0) = $ ( z ) ,

*(z,0)=0

*(a;,0) = 0, 1

*(*,*)

- [$(x - ct) + $(x + ct)]

PE(x,t)

Pm $'2{x-ct)+$'2{x + ct) 4 Pm ®2(x - ct) - $' 2 (a; + ct) c 4

3E{x,t)

*(x,0)=6(x)

PX+Ct

j- / WW 2 ^[e (x-ct) + e2(x + ct)] ^ [e\x - a) - e2(x + a)} c

The prime denotes differentiation with respect to the argument. These solutions explicitely show that the energy current can be decomposed into two parts of opposite directions, each moving with velocity c,11 see Fig.2. The mean square displacement with respect to pB{x,t) is < x2 > (t) = (ct)2. Finally, we notice a property of wave propagation in two dimensions which is in strange contradiction with our daily experience. In three-dimensions, an initial pulse of finite duration propagating off the source will always create a wave packet with a leading as well as a trailing edge. In two dimensions, however, there is no trailing edge and, hence, an observer will find an infinite afterglow, see Fig. 3. Remarkably, this property holds in all space dimensions of even order d = 2,4, 6 etc. For spherical or cylindrical waves and solutions of the inhomogeneous wave equation in terms of Green-functions, see e.g. Barton's book. 13

79 1.2 1.0

f

^0.8 & 0.6 9"

*

i t = 15

I

I

in

*!

0.6

;t = 0

0.4 0.2

0

10

l\

J

-10

20

0

x 0.40 0.35

0.5

:t = 0

0.30

0.4

0.25

0.3

0.20 0.15

t = 15

0.2

t=15

0.10

0.1

0.05

10

-10

20

Figure 2: Time evolution of an initial wave-packet and its energy-density. Initial conditions: *(x,0) = *(x) (left) and *(x,0) = Q(x) (right). $(x),0(x) are Gaussians centered at x = 0. Note, the dotted curves in right upper graph display the velocity. (Dimensionless quantities.)

d=2

Y

d=3

r = ct

r^r=c,

/ dark \ ct a)

ct

I

r

b)

Figure 3: Pulse propagation in d=2 dimensions shows an infinite afterglow whereas pulses in d=3 dimensions display a leading as well as a trailing edge.

80 3.2.

Diffusion

Consider a conserved quantity with density p with a diffusive flow, j = —Z?gradp (=Fick's law) where D is the diffusion constant. By construction, p obeys the diffusion equation | | = DAp. For notational convenience, we shall write \P (which is assumed to be real) instead of p, where ^ may describe the density of particles, charge, energy, etc., and allow for an external source term s(r, t).

dt

•£>Atf(r,i) = s ( r , i ) .

(15)

The state of this system is fixed by ^ ( r , 0) = $ ( r ) . In contrast to the wave-equation, Eq. (15) is not symmetric with respect to time-reversal and, thus, describes irreversible processes. The modes of the (homogeneous) diffusion equation {s = 0) are defined as *K(r,i)=e^**K(r).

(16)

K labels different mode-functions which are again solutions of the (homogeneous) Helmholtz-Eq. (11) where (LUK/C)2 has to be replaced by \K/D. An arbitrary solution of Eq.(15) (with s(r, t) = 0) can be decomposed as

*(r,t) = X)A (t e- A - t \P (t (r),

(17)

where the real coefficients AK are determined by the initial condition. A very useful solution of Eq.(15) is the "heat pole" which belongs to ^ ( r , 0 ) <5(r), see Fig.4. <J(r,t) = (4wDt)~*e"nii,

t>0,

(18)

where d = 1, 2, 3 is the space dimension. The mean square displacement with respect to Eq.(18) is < x2 > (t) = 2Dt. Remarkably, this result holds in any dimension if a;

Figure 4: Time evolution of the "heat pole".

81 refers to one of the components of the d-dimensional position vector. The solution Eq.(18) is also called the propagator and denoted by K(r,t) as it propagates the initial state to finite times

#(r, t) = UD(t)V{T, 0) = J K(T - r', t)*(r', 0)dV .

3.3.

(19)

Electromagnetic Field

The state of the electromagnetic field (EMF) in a vacuum is described by two vector fields £,B whose time-evolution is determined by the Maxwell-equations, see Landau-Lifshitz 14 (Vol. 8) —-c2curlB, = j ( r , i ) , div £ = —p{r,t), dt e0 e0 (20) ^ + curl £ = 0, div B = 0. dt The two equations containing the divergences represent "auxilary conditions" (as p(v, t) is assumed to be prescribed) rather than dynamic equations, c = 1/^toMo is the speed of light and p, j denote the electrical charge density and current density, respectively. The Maxwell-equations are (form-) invariant under time reversal t —» —t, j —> —j, p —»• p, £ —> £, B —¥ —B and, therefore, describe reversible processes. By standard manipulations of Eqs. (20) a conservation law is derived

9iv^

2+

^ s J + d i v l £ x ^ = -j£i

(21)

which describes the energy-balance of the EMF. Thus, by comparison with Eq.(3) pE(r,t)

= ^£2

+ ^-B2,

jE(r,t)

=£ x - ,

aE(r,t)

=-j£.

(22)

Traditionally, j B ( r , t) is called the Poynting-vector and 0£;(r,i) is interpreted as the work done by the EMF on the charges. A variety of different modes of the EMF is known which are used for the description of fields in free space and for microwave- or laser-resonators. In infinite space the simplest modes are transverse, linear polarized, travelling plane-waves £{v,t) = £ 0 c o s ( k r - w i ) ,

B(r,t) = -k x £(r,t), c

(23)

where k is a unit vector along the wave vector k, k • £0 = 0, and w = ck. The energy density and energy current density of this mode are pE{r, t) = e0£o cos 2 (kr - Lot), j E ( r , t) = pE{r, t) c k .

(24)

82 Note, for this mode the energy current density is particularity simple: it is just energy density times the velocity of light. In many cases the flow of energy occurs along rather unexpected paths so that the Poynting-form of the energy current density has been frequently questioned. In particular, the nonzero energy flow in (crossed) static and magnetic fields appears to be "unphysical" and other forms of j E (which still are in accordance with Eq. (21)) have been proposed. 15 Nevertheless, there are convincing arguments that the Poynting form of the energy current (which is also the momentum density times c) is correct, see Vol. II, 17-5, 27-4, and 27-11 of the Feynman-lectures. 16

3.4-

Electromagnetic Field in Matter

According to the atomic structure of matter the microscopic fields are spatially strongly fluctuating on an atomic scale. Instead of the true microscopic fields, averaged (macroscopic) fields are used, which are likewise denoted by £, B, see LandauLifshitz14 (Vol. 8). ^M-curlH(r,i)

=

-j(r,i),

div2>(r,<)

=

p(v,t),

(25>

dBlv t) " r

' + c u r l £(r,t)

=

0,

divB(r,t)

=

0.

The averaged total charge- and current- densities in matter are separated in three groups: • those who are counted explicitely as />, j , • "displacement charges/currents" pd = —div V, u = V, and • "magnetization currents" j m = curl M, (pm = 0 as div j m = 0). The polarization V and magnitization M fields combine with £, B to two new fields V — e0£ + V and W = B/p,o — M.. For a tutorial on the dielectric descriptions of semiconductors see, e.g., Ref.27 By standard manipulations of Eqs.(25) a conservation law is derived

^ ^

+

f«-) + d i v ( £ ^, = - j £ - , f - w W ^ ,

pa)

which is identified with the conservation of energy. Note, the appearance of H, whereas the averaged microscopic magnetic field is B\ Note also, that the rhs of Eq. (26) may contain dissipative as well as non-dissipative parts. Some parts of the latter can be combined with £ x % to form an energy-current vector of the combined system EMF + matter. 17

83 For nondispersive, nonconducting matter, V = e0(e—1)£, M = (p — l)'H, j = 0, where e, /i are assumed to be constant. In this case, we may define an energy balance of the combined system EMF + Matter (denoted by a tilde) by PE

= T;£V

+ ipH.

]E = £ * U ,

OE

= 0,

(27)

where V = e0e£ and B = HQ/JM. For example, for a plane monochromatic wave, Eq.(23), pE{r,t)

= e0e£ (r,£),

jB(r,t) =

pE(r,t)-k,

(28)

where n = \ft is the refractive index. If dispersion is present, the situation is different. As an example, we consider uncoupled isotropically and harmonically bound charges —e with density n and displacement s. 18 Omitting spatial dispersion, the polarization V = —nes obeys 92

d

,\ „,

ne 2 „.

.

.

(29)

When multiplying Eq.(29) by V we recover Eq. (26) which may be rewritten in the form of Eq.(3) with e

A£2 2

PE ]E

=

oE H — B/n0.

*Ln2+™

+

2

2 V dt J

(30)

SxH,

(31)

-7•ne'\dt)

(32)

For a harmonic field as given by Eq.(23) the polarization is V(t) = 5R [e0x(u)£oe-iut]

where x( w )

2

(33)

,

= e w

( ) ~~ 1 denotes the electrical susceptibility

xM

(34)

WQ — w2 — i'yui

and w2 = (ne2/rrK0) is the square of the plasma frequency, see Fig.5. The time averaged energy- and current density of the combined EMF and matter are PE

=

j£0

l + |£(w)| + ^ L |

x H

|

2

.

3E = -^£ln{u)c.

(35)

84

Figure 5: Dielectric function for harmonically bound charges.

n = K-y/e is the real part of the refractive index. Outside the stop-band where n(uj) ^ 0, and in the limit of zero damping, Eq.(35) can be rewritten as e

PE

o C2 i ^d[n(u)u)] ;£ln(w)- du>

(36)

Note, Eq.(36) is not a general result, in particular, it does not hold for finite damping, or in the region of anomalous dispersion.

3.5.

Energy Transport Velocity and "Superluminal" Pulses

Presently there is a remarkable interest concerning the velocity of energy transport and the interpretation of "superluminal" pulses in dispersive media and wave guides, see e.g. a special issue of the "Annalen der Physik". 20 Although such phenomena are not new a great fuss arose. In particular, it was well known that the propagation of a puis with a Gaussian envelop 23-25 is essentially different compared to a semi-infinite envelop, which was initially studied by Sommerfeld and Brillouin. 21 Pulse propagation experiments usually measure the cross-correlation function between the pulse travelled through matter along distance L and a reference pulse, see Fig.6. From that a delay/advance distance As = cAt a pulse velocity in matter c/(l + As/L) is deduced. It is well known that the group velocity vgr{k) =

dcu(k) dk

d[n(ui)uj] dw

n(u) + um'(u>) '

(37)

describes the propagation in a linear dispersive, nonabsorbing medium outside the stop band, where u(k) = ck/n. (If the refractive index is complex, one extrapolates Eq.(37) replacing n(u) by its real part.) In the region of normal dispersion {n'(ui) > 0) the group velocity is always smaller than c. However, in regions of strong anomalous dispersion (n'(uj) < 0), vgr can exceed c or even become negative. The common belief is that the meaning of Eq.(37) breaks down and the behaviour of the pulse becomes much more complicated. This is indeed true for the semi-infinite sinussoidal puis, 22 for

85

GOPtt SAMPLE (17K)

<S2•

&

%

^O^n

Figure 6: Left: Schematic of a Gaussian pulse propagating in a dispersive medium and of the experimental arrangement to measure the cross-correlation function. Right: a sample of the crosscorrelation data as the laser is tuned through the exciton line of the GaAs : N sample, N = 1.5 x 1017cm~3. According to Chu and Wong.23 a Gaussian shaped pulse, however, the situation is different. As discussed by Garrett and McCumber, 24 and in more detail by Oughstun and Balictsis, 25 the leading edge of the pulse is less attenuated than the trailing edge so that the pulse maximum speedsup at the expense of the pulse height, see Fig.6. For a Fourier transform limited puis whose spectral width is much less than the width of the absorption line, the pulse propagates with the group velocity as given by Eq.(37) and even the shape and width of the pulse can remain almost intact after it emerges from the sample. Moreover, for undamped, harmonically bound electrons with a single resonance frequency the group velocity is identical with JE/PE', compare Eq.(36) with Eq.(37). The physics of puis propagation in wave guides below the cut-off frequency is similar, yet the pulse reshaping and attenuation is due to reflection rather than by absorption. 19 The propagation of an electron wave-packet through a potential barrier was studied by Krenzlin et al., see Ref.20 (p. 732). Note, there is no indication of advance propagation of the leading pulse wing which, trivially, can never be surpassed. In addition, the time separation of two pulses is not affected so that there is no superluminal transmission of information, even if vgr > c. If a physically motivated definition of an energy transport velocity is desired, it ought to be based on a (local) Lorentz-transformation to a moving frame (denoted by a prime) of vanishing energy current

£

• e

B

•Bf

7 (£ + V x B), V x£ 1\B

(38) (39)

86 where 7

= 1 - (V/c)2.

Equating £' x B' = 0, we obtain in the limit V < c ~ 2 ^ ~ e 0 / i 0 £ 2 + £2'

(40)

Remarkably, this velocity is numerically only, but one half of the "self evident" form.

3.6.

Quantum

Mechanics

In quantum physics, expressions for particle probability density and its currentdensity are well known from text books, yet expressions for energy-density and energy-current-density are less familiar. To construct these quantities, we first consider a single particle in one dimension in a time-dependent potential V(x,t). The wave function ip(x,t) obeys the Schrodinger-equation

tt»*M = iMM,

£=4 S

+

^ -

(4i)

Under a time-reversal operation Tip = ijj*, so that all expectation values remain unchanged. Hence, Eq. (41) describes reversible processes, see Landau-Lifshitz 14 (Vol.3, §7). First, we find the energy-density starting from the expectation value of the Hamiltonian H =

J ip*(x, t)Hijj(x, t)dx=

f pE(x, t) dx.

(42)

By partial integration we rewrite the integrand such that it is intrinsically positive, hence

pE{x t)=

2

'L

Q_,./„ ,> dip{x,t) dx

2

+ V(x,t)\i;(x,t)f .

(43)

Next, we study the variation of < H > with respect to time. Elimination of V^by Eq.(41) and performing a partial integration we obtain for the integrand of ^ < H > I =

i— 4m2

<^<>{*-0-4+^>(i',)r

(44)

Now, the task is to rewrite the integral over I to extract the current-density via —§^3E{X) + 0E(x,t). For the first term of Eq.(44) we get dx •> -1

_^LA/^1^_ C C \ 4m 2 dx \ dx dx2

J

(45)

87 By partial integration, we find that the second term compensates the third one. Finally, the desired results for the energy-current and energy-production rate densities are 3E{x,t)

3

f h3 dip* d2tp \ I QITTL dx

dx

I

2

aE(x,t)

=

^ W i , f ) |

2

.

(47)

For time-independent potentials, aE{x,t) = 0, as the energy is conserved. The generalization to three-dimensions is obvious by replacing Jj by grad . As an example, we consider a free particle in a plane wave state tp(x, t) = ,7-exp(ikr), where Vo is the normalization volume. pB(r,t)

= ^P-,

jj5(r,t)=pB(r,t)—,

VQ

(48)

Til

where -E(k) = h2k2/2m is the energy of the particle. The Schrodinger-equation of a free particle bears some analogy to the (homogeneous) diffusion-equation in "imaginary time", to —• its- Eq.(17) can be viewed as the result of the application of the time-evolution operator Uo(t) on the initial state * ( r , i ) = £z>(t)*(r,0),

UD{t) = e-Bt,

(49)

where H = —DA is the "Hamiltonian". However, there is a profound difference between the diffusion equation and the Schrodinger-equation. Eq.(41) describes reversible processes and the solution can also be propagated backwards in time whereas it cannot for the (irreversible) diffusion-equation, because there are arbitrary large positive eigenvalues A^. In mathematical terms, the Schrodinger time evolution operator, Us{t) = exp(—itH/h), generates a group of unitary transformations, whereas Un(t), Eq.(19) is not unitary and generates only a semigroup which has no inverse element.

3.7.

Collective Excitations in Solids

Condensed matter is a strongly interacting many body system which, in chemical terms, forms macromolecules with unsaturated bonds which allows for unlimited aggregation of particles. Due to the strong interaction between the (bare) particles (electrons, nuclei) no exact treatment is possible. However, in many cases a number of relevant features of the ground state as well as for the low-lying excitations have

88 been recognized. For a review on basic notions in solid state physics see Anderson's book. 29 For our purposes, the most important properties are: Ground state properties: • broken symmetry: crystal, ferromagnet, superconductor . . . • rigidity: mechanical, magnetic, gauge . . . Low energy excitations: • behave like a gas of weakly interacting (quasi-) particles, • single particle excitations resemble the bare particles, at least at not too strong interactions (=adiabaticity), • collective excitations are dynamically equivalent to the creation/absorption of bosons. Broken symmetry means that the ground state of the system has a lower symmetry than the Hamiltonian. For example, the Hamilton of a solid is invariant under translations and rotations whereas the crystal state is not. (This is different for fewbody systems, e.g. the hydrogen atom where the ground state has the full symmetry of the Hamiltonian.) Rigidity refers to external perturbations. For example, condensed matter (fluids, solids) is almost incompressible, in addition, solids are rigid with respect to shear deformations. As an example of gauge-rigidity we consider the London-equations of a (type I) superconductor | j ( r , t ) = — £{T,t), Ov

curlj(r,t) = - ^ B ( r 1 t ) ,

Tft

(50)

if li

where n is the density, m the mass, and —e the charge of the electrons. Remarkably, these equations remain unchanged when replacing the electrons by Cooper-pairs with density n/2, charge - 2 e , and mass 2m. We compare Eqs. (50) with the quantum mechanical result for the particle current of an electron in a magnetic field of vector potential A, B = curl A: JN{r,t) = ~ ZiTThZ

(*Vad *-cc)-

— \V(r,t)\2A(r,t).

(51)

Tit

For JV particles, * = \P(ri, r 2 , . . . rjv) summation and integration of Eq.(51)on JV - 1 coordinates has to be included. If the many electron wave function is not affected by the magnetic field Eq.(51) (times —e) reduces to just the second London-equation (in the Coulomb-gauge, div A = 0).

89 The concept of quasi-particles was originally developed by Landau who realized that there is a continuous mapping of the low energy excitation spectrum with the strength of the interparticle interactions, see Landau-Lifshitz, 14 (Vol. 5). Amazingly, this description holds even in relatively strong interacting systems like in metals or in liquid helium 3He, 4He. As an example of single-particle excitations we consider the transformation of free electrons to Bloch electrons in crystals, see Fig.7. In terms of quasi-particles, the excited states are described by occupation numbers na = 0,1,2..., where a labels the one-particle states (or "modes") with energy eQ. Hence, the excitation energy of the total system becomes

E({na}) -E0 = Y^ tana + Eint.

(52)

a

E0 is the ground state energy and Eint({na}) contains nonlinear terms in na which describe interactions between quasi-particles. For a translationally invariant system or a crystal, a is identical with the wave number k, in addition, there may be several branches of wave-like excitations which will be labelled by an additional index (= v). Collective modes are equivalent to a set of (uncoupled) harmonic oscillators with frequencies ua. Quantization of these oscillators directly leads to Eq.(52) which is dynamically equivalent to a system of bosons with energies hu>a, see Fig.8. In contrast to massive bosons, however, (e.g. .ffe-atoms in liquid Helium) these quasi-particles can be easily created and destructed. An appropriate description of such processes is not possible within the "ordinary" wave-function formulation of quantum mechanics, and "second quantization"* is needed.30 In the extended zone scheme quasi-momentum ?ik plays almost the same role as real momentum. The two most important properties of particles and quasi-particles are • The transport velocity of energy and momentum is given by vT = — — . dp

(53)

• The overall (quasi-) momentum and energy of particles and quasi-particles is conserved, see Fig.9. Unfortunately, there is no general rule under which conditions such a scenario exists, and how to find the collective variables. Some examples will be given in the next sections. A survey on the dynamics and spectroscopy of collective excitations in solids, 32 and a collection of common and different properties of particles and quasiparticles 28 can be found in the Proceedings of two previous Erice-Schools. *The name "second quantization" is misleading. Besides h no second quantum constant arises. A better name is "occupation number representation", yet it is used in a different sense as position or momentum representation.

90

-KK-K-KK O M K

'AY.

Figure 7: Left: Energy (quasi-)momentum relation of free electrons in an empty lattice (thin line) and in a weak periodic potential (solid line), where K = 2im/a, n = 0 , ± 1 , ± 2 . . . are reciprocal lattice vectors. Right: Energy bandstructure of GaAs (in the reduced zone scheme). According to Cohen and Chelikowski.31

energy bosons

oscillators

0--

o -

Figure 8: Equivalence of a system of N (noninteracting) bosons with single—particle energies e 0 and occupation numbers na and an infinite (uncoupled) set of harmonic oscillators with frequencies ua = £a/Ti- Note that the zero-point energies of the oscillators are omitted. Dots symbolize particles, crosses excited states, respectively. N = 6. According to Ref.28

impurity

neutron

b)

c)

Figure 9: Examples of interactions between particles and quasi-particles. (a) Excitation of a phonon by neutron scattering, (b) scattering of a phonon by impurities, and (b) decay of a phonon due to anharmonic interactions. Energy and momentum are conserved at each particle/quasi-particle vertex whereas the impurity takes momentum but no energy.

91

D haq n m ') ^•<)-VMMW-0-VtfWV>ftAO Wilt C * w A w O - W W W U O ^

1 2 H a I-

3 I

4 I

N-l I

N I

H-q

Figure 10: (a) Linear monoatomic chain with equal masses m and nearest neighbour springs D and periodic boundary conditions, (b) frequency spectrum for N = 10 "atoms". Note, there is no vibrational q = 0 mode (at u> = 0); this degree of freedom is taken over by the common translational motion. According to Ref.28 A.

Phonons

Phonons are quantized vibrations of a crystal-lattice. For notational simplicity, we consider first a linear chain of equal masses m and nearest neighbour springs, see Fig.10. Starting from the Hamiltonian in terms of canonical momenta pj and displacements Uj of the masses at sites j = 1 , . . . N, {pj, Uf} 5jf N

j=i

1

2m^

i. +

(54)

2D{u^

we first find the collective momenta Pk and coordinates Qk (55) {a, b} denotes the Poisson-bracket symbol, k = |j^/s, K = 0, ± 1 , ± 2 . . . ± 7V/2 is the wave-vector which is restricted to the first Brillouin-zone, and a is the lattice constant. Eq. (55) is canonical, {Pk,Qk'} = &kk' and transforms Eq. (54) to a set of uncoupled oscillators with frequencies u>k H(P,Q)

E ^ + ! '-QkQl

k>

W

* =

\

S

(56)

In d dimensions, a vibrating lattice has d acoustic (uk = cs|fc| for k —¥ 0) and d(s — 1) optical branches (u>k ^ 0, k —• 0), where s denotes the number of (inequivalent) atoms in the (primitive) unit cell, v = 1,1...d • s, see Fig.11. The quantization of uncoupled oscillators, Eq. (56), is almost trivial. The state

92

(1,0,0)

L

\

9?

\

F 0.2 0.4 0.6 0.8

L

f

\L T \

Jd

0

(1,1,0) £.

(i.i.i)

1.0 0.8 0.6 0.4 0.2

ifTi' l//\ 0

0.2 0.4

b)

a)

Figure 11: Structure and Brillouin-zone (a), and phonon dispersion curves (b) for potassium (bcc, one atom per primitive unit cell). Along the horizontal axis we plot q, q/\/2, and q/\/Z for the (1,0,0), (1,1,0), and (1,1,1) directions, respectively. According to Cowley et al.33 of each oscillator (labelled by k) is fixed by quantum numbers nk = 0,1, 2..., hence E({nk}) = Y, *"* (n* + \)

•

(57)

This is already of the form of Eq. (52), and we may identify e(k) = htok with the energy of the quasiparticles which are called phonons. To describe arbitrary phonon states or interaction processes between ghonons it is convenient to use creation and anihilation operators ak,a[ rather than Pk,Qko-k =

mukQk + iPk /n

fe

f^ ^ t \ak,ak,

5kk' •

(58)

These are also called ladder operators as their repeated application on \n > creates the "ladder" of stationary states n = 0 , 1 , . . . , where a+ \ n >= %/n + 1 | n + 1 >, a\n>= s/n\n-l> "climb" up/down the ladder by one "rung".

E H

=

K e'k>a(alk + ak), 2NmuJk

^^ tiu}kakak .

(59) (60)

The zero point energy of the oscillators has been omitted as it is not relevant for the dynamics of the system.

93 Note, there are three different types of particles in the game: • N coupled "bare" particles of mass m described by Pj,Uj, • N uncoupled harmonically bound particles of unit mass described by

Pk,Qk,

• phonons of unlimited number described by 1ik,a[. In addition, we can consider phonon states which are not even eigenstates of the phonon number operator (see example below).

(61)

N = J^^k

For example, we consider two particular phonon states: One-phonon state:

| <j>(k) > = ^ 0 ( f c ) | l f c > = ^(fc)aJ,|0 >. k

k

For notational simplicity, |0 > = \{nk} > with nk = 0 for all k. a£|0 > = |l/t > represents a steady state of the oscillator #fe in excited state nk = 1, all other oscillators being in the ground state. By construction, \(k) > is an eigenstate of the phonon number operator with eigenvalue 1: N\(j>(k) > = l\(j>(k) >. The time-evolution of this state is e~'Ukt\lk >, hence (f>(k) —> 4>{k,t) = <j)(k)e~lh}kt which may be interpreted as wave function of a phonon (-wave packet) in momentum spaced For such a state, the expectation value of the position operator of mass # j is < \v,j\<j> > = 0 for all times so that it cannot be the quantum analog of a classical wave, it is "quantum noise". 00

a-state:

n

\a >= V"* e - ? ' 0 ' —y=\n >. 71—0

Here k is fixed, n = nk, and will be suppressed for notational simplicity, and a = |a|e , , p denotes a complex number. Again, the time evolution of | a > is easy to find just by replacing a by a(t) = a • e~'w*. Remarkably, these states describe almost classical (wave-like) motion of the masses in the chain < a(t)\uj\a(t)

>oc |a| cos(kja — uikt — (p).

(62)

Hence, \a\ and ip fix the amplitude and phase of the wave, a-states are not eigenstates of the phonon number operator and the expectation value and uncertainty are =

\a\2,

AN = \a\.

(63)

§In contrast to massive quantum particles it is not possible, however, to define a wave function in position space. The reason is that for phonons (or photons) u> <x \k\, which does not permit a Fouriertransformation from k to r space, see Landau-Lifshitz14 (Vol. 4).

94 For large amplitudes AN/ < N >-> 0. In addition, AujApj = §, withy timeindependent uncertainties. Thus, in real space, \a > describes a Gaussian wavepacket. Although these states have been already introduced by Schrodinger they are nowadays called "coherent" states, Glauber-states, or just a-states. In quantum optics they play a fundamental role for the description of laser radiation and coherence phenomena. 34

-77777 77777 Figure 12: Interacting two-level systems (a) and their pseudo-spin analogon (b). Left: ground state, right: a localized excited state.

B.

Excitons

Excitons are propagating electronic exitations in matter. Traditionally, one considers two limiting cases: • Frenkel-excitons are almost localized atomic or molecular excitations in molecular crystals with little coupling to neighbouring units. • Wannier-excitons, on the other hand, are loosely bound hydrogen-like electronhole pairs in semiconductors. These pairs are delocalized over many lattice sites so that a continuum description is possible. For simplicity, we consider only Frenkel-excitons and approximate the spectrum of molecular excitations by two-level systems, see Fig.l2a. As the Coulomb interaction between the molecules is isotropic, the coupled molecules are described by a Heisenberg model for (pseudo-) spin |

H = J2 ^e0ajz - Y^ Ji&di > i

( 64 )

v

where d{dj denotes the scalar product of the spin vector operator with Pauli-matrices (ax,dy,az) at site j . The broken symmetry of the ground states is evident from Fig. 12b. Due to the interaction, the excitation of an isolated molecule (as described by a spin flip at site j) may spread off to neighbouring sites. To bring Eq.(64) to the quasiparticle form Eq. (52), we first transform to exciton creation/annihilation operators b^bj by the Holstein-Primakoff transformation 30 (omitting the site index for notational simplicity)

dM=as±iZv,

a^ = (l-W2X

?(-> = (? (+) ) f •

(65)

95 2.00

E(k)

1.75

,.

^

1.50

\

/" '--

1.25

\

1.00

\

0.75

0.25 -7t

-7C/2

0

JC/2

Jt

ka

Figure 13: Energy (quasi-)momentum relation of Frenkel-excitons. (Dimensionless quantities, eo : 1,/i! = 0.25.) Operators at different sites commute. From [oj- , Sj-, '] = 4azj5jj> we obtain [6j, 6|,] 5if, hence "3>"j bj,b], describe bosonic excitations. (66)

H(b\b) = 5 > 6 & + ! > « • ' * & • ' j=i

Higher order terms in bj,bj describe interactions between excitons which can be omitted at low exciton densities. Due to the long-range structure of the Coulombinteraction many contributions to the coupling elements hjji have to be taken into account in a realistic description. 35 In crystals hif = hj^ji discrete Fourier-transformation (i.e. transformation to Bloch states) b* =

E4VN e

ikja

ak,

\ak

>«t] ='

(67)

diagonalizes the exciton Hamiltonian, Eq. (66) H@,a)

= ^2
c(fc) = e0 + J2 V ^ ' " •

k

(68)

J

In particular, for nearest neighbour interactions, j = ± 1 , e(k) = €0 — 2\hi\ cos ka,

(69)

where, as usual, hi < 0 is assumed. Eq.(69) is the standard tight-binding result of the linear chain with nearest neighbour couplings, see Fig.13. For a discussion of Wannier-excitons, see e.g. the contributions by Klingshirn. 36 ' 37

96 4.

Description of Energy Transfer Processes

4-.1.

States, Processes, and Reversible Dynamics

Processes are described as transitions between states (labelled by ip). In a state all physical quantities (= observables) have a definite (expectation) value which will be denoted by < G >^. Note, this definition does not necessarily imply that < G >^ is "sharp", i.e. has vanishing uncertainty AG = 0, where AG = V < G 2 > — < G > 2 . There are two distinct classes of states: • pure (or ideal) states which have zero entropy • mixed (or statistical) states of nonzero entropy. Irreversibility and loss of coherence are intimately connected with increase of entropy. Hence, the transformation of pure states into mixed states plays a fundamental role in the description of energy transfer processes in macroscopic systems. According to the dogma of classical physics there are (infinitely many pure) states in which all observables have a sharp value. Hence, AG ^ 0 is always caused by errors, either in the state preparation or in the measurement of G. In quantum physics the situation is different. There are particular states in which a specified observable G has sharp values, AG = 0. These states are the eigenstates of G: G\ip > = g\tp >. But there are no states in which AG n = 0 for all Gn. Therefore, AG„ ^ 0 is not necessarily caused by errors! In classical mechanics pure states are specified by the values of coordinates and velocities (or momenta) of all particles which defines a point in phase space: i1 = (Pi?) = (Pi>QV,p2,^, is just the functional value of G. The dynamics of these states is governed by the Hamiltonequations of motion PW

=

M

= -

^

,

m

= {H,q} = ^

(70)

which underline the central role of the Hamiltonian (= energy) as the generator of time-evolution. Mixed states, on the other hand, are a statistical mixture of pure states and are described by a positive definite, normalized phase space density p(p,q)>0

< G > = [ [G(p,q)p(p,q)dpdq,

If

The dynamics of mixed states, p(p,q) -4 p(p,q,t) equation

p{p,q)dpdq=l. (71) is governed by the Liouville-

dp{p, q, t) + {H,p} = 0, dt

(72)

97

Figure 14: Time evolution in classical mechanics: (a) pure state, (b) mixed state which represents a bundle of trajectories. which follows from Eq.(70) as each phase space point initiates an individual trajectory, see Fig.14. In particular, for a single particle in one-dimension we have dp(p, x, t) dt

^<9p + Fd£ = dx dp

Q

(73)

where v is the velocity and F = —V'(x) the force acting on the particle. In quantum mechanics pure states are specified by state-vectors | ip > [or wave functions tp(x,t)], observables are described by hermitian operators G, and the (expectation) value of G is defined by < G >,/,=< 1>\G\ip >=

fip*{x,t)Gil){x,t)dx.

(74)

The dynamics of pure states is determined by and the Schrodinger equation ih-\i>(t)

>= HW)

(75)

>•

A mixed state, as in classical physics, is a statistical mixture of pure states, where the phase space density is replaced by a density operator

p[t) = ^2pn\n>
^ p „ = l,

p„>0.

(76)

| n > denote an arbitrary set of state-vectors (which might not even be orthogonal!) with (real) positive weights pn, and the expectation value Eq.(74) is replaced by =tr

(pG) = Y^ < a\pG\a > = ^

pn < n\G\n > .

(77)

where a labels an arbitrary base. The last part of Eq. (77) can be interpreted as the "usual" quantum average of G with respect to the pure states \n > with a classical average with weights p n put on top. Eventually, we mention that the density operator

98 a) E(r 0 ,t 0 )

f^- mirror 2 "E, |

V ir

t screen

E

i

I

— ** mirror 1

J

10

15

20 cm At

Figure 15: Schematic of the Michelson interferometer (a) and visibility curve of the red Cadmium line (b). Al is the difference in path length of mirrors 1,2. At = Al/c is the coherence time. According to Born and Wolf.44 of a pure state \ip > is the projector p = \ip > < tp\. The dynamics of a quantum mixture follows the v. Neumann-equation^ (78) Mixed states have nonzero entropy S = —kBtr (plrip) .

(79)

In the classical case the trace is replaced by phase-space integration. Noticeable, both the Liouville- and the v. Neumann equations describe reversible processes, —gp- = 0! We are able to describe states of nonzero entropy, but not irreversibility!

4-2.

Coherence and Correlation

The notion of coherence originates from optics where it describes space-time correlations of the electromagnetic field. Familiar consequences are the appearance of interference fringes in a double-slit experiment or other oscillatory phenomena. A prominent instrument to study coherence phenomena is the Michelsoninterferrometer, see Fig. 15. Field on screen E:

£ = £i + £2

C [£in(r0, t0 + 7i) + £in(r0, t0 + r 2 )]

Intensity:

I = \£\2

C 2 [G(1,1) + G(2 ) 2) + 25RG(1,2)]

Correlator:

G(r2,i2;ri,ti

<£<->( r 2 ) t 2 )£<+)( r i i t l ))

(80)

'At first sight Eq.(78) looks like a Heisenberg equation of motion of p(i). However, the sign is different and p(t) is in the Schrodinger-picture! In fact, Eq. (78) expresses the conservation of probability &&• = 0.

99 C is a suitable constant and S^ contains the positive/negative frequency components of the electrical field (oc exp^iuikt). In a mode-expansion (e.g. free propagation of the EMF from a star to the earth) we have £{t, t)=Y,

^ k e i ( k r - W k t ) +cc = £{+){r, t) + S(~] (r, t)

(81)

so that the correlator at equal space points, r 2 = ri, can be written as G(t2,ti) = E l ^ | 2 e - i W k ( t 2 - t l ) .

(82)

For example, for a Gaussian line of width AUJ centered at u>0 the normalized correlation function becomes (G(t,t) = 1): A(UJ)

G(h,ti)

— exp =

(w - wo)2

(83)

2ALJ2

e-"" 0 ^-' 1 ' exp

;{Aw(t2

-

h)f

(84)

For such a line the fringe-contrast ("visibility" in the Michelson interferometer (see Fig.15) is defined by V

••

+ In

exp

-[Au,{t2 - h)}2

(85)

where the explicit result refers to the Gaussian line. Eqs.(83,84) explicitely show an example of the famous Wiener-Khinchine theorem: Filtering in k-space (=restriction of frequencies and/or k-directions) increases the coherence-time or coherence-area (perpendicular to k). This is just a property of the Fourier-transformation, At oc 1/ALO,

etc.

Analogous to optics, coherence can be defined for any quantity which is additive and displays a phase or has a vector character, e.g. electrical and acoustic "signals", electromagnetic fields, wave-functions, etc. Coherence is intimately connected with reversibility, yet the opposite is not always true. At first sight, a process might appear as fully incoherent or random, nevertheless it may represent a highly correlated pure state. A beautiful example is the spin- (or photon-) echo 27 ' 41 which is related to a superposition of many sinusoidal field components with fixed (but random) frequencies. At t = 0 these components have zero phase difference and combine constructively to a nonzero total amplitude. Later, however, they develop large random phase differences and add up more or less to zero so that the signal resembles "noise". Nevertheless, there are fixed phase relations between the components at every time. By certain manipulations at time T a time-reversal operation can be realized which induces an echo at time t = 2T, which uncovers the hidden coherent nature of the state.

100

Figure 16: Resistance of a metallic magnesium film as a function of magnetic field. For elastic electron-impurity scattering there are two equivalent time-reversed paths which lead to an increase of the resistance by coherent backscattering ("weak localization"). A magnetic field leads to a phase shift between both paths which uncovers the finite phase-coherence time of the electrons originating from the (temperature dependent) inelastic electron-phonon scattering. According to Bergmann.45 Another nice example in this respect, is the phenomenon of "weak localization" of conduction electrons in disordered materials, see Fig. 16. Echo phenomena are always strong indications of hidden reversibility and coherence! Problem: Consider a sum of many (co-)sinusoidal functions with fixed (but randomly chosen) frequencies w„ /(*)

=

^costont,

(86)

un = (l + r„)27r.

The distribution of rn is assumed to have a zero mean with probability P(r). (a) Study f(t) numerically for a box distribution P(r) = Q(a2—r2)/2a and a Gaussian distribution. (b) Compare with the analytical result /(*) = (cos(l + r)27rf) = Jtje 2 ™' f P(r)e2wirtdr\

.

(87)

[Hint: A Gaussian distributed random variable can be generated numerically by summing many random numbers of any distribution (Central Limit theorem)].

101 4-3.

Irreversible

Dynamics

The phenomena of irreversibility is ubiquitous in all macroscopic systems. Nevertheless, the old question how to describe irrevesibility and decoherence on a microscopic (reversible) basis has no simple answer and, possibly the final answer has still not been found. 38 ' 39 Irreversibility, i.e. the increase of entropy with time intimately connected with the transformation of pure states into mixed states. In addition, it is not possible to include dissipative forces in the Hamiltonian itself. Presently, the dogma of theoretical physics is that irreversibility arises from the coupling of a system to its infinite environment ("heat bath") or by the boundary conditions which, in an infinite system, can also spoil time-reversal symmetry. " A real expert in the field of traditional thermodynamics, however, will consider such a "trick" as mean deception! The dynamics of the total system with Hamiltonian Htot = Hsys + Hbath + Hint

(88)

obeys the reversible Liouville/v. Neumann equation, where Hint describes the interaction between the system and the bath. For electrons in a solid the bath may be realized by the phonon system which is held by a constant temperature T. The task is to eliminate the bath variables from the equations of motion of the total system, and to construct equations for the system of interest. Note, although the basic equations are reversible the dynamics of the system will be not. For finite systems, however, there is always a finite Poincare recurrence time. Nakajima and Zwanzig40 have derived a general procedure to eliminate the bath variables, at least in principle, but it is very difficult to do it in practice. The resulting equations are irreversible, nonMarkovian, and have a memory behaviour. Some simple examples will be given in Chapter 5. In many cases, however, various approximate treatments are used, some of which will be discussed in more detail in the next sections. a. Relaxation time approximation. Phenomenologically, a relaxation term is added to the Liouville/v. Neumann equations which mimics the system-bath interaction: d

M+t-[H,p\

= -±(p(t)-Peq),

(89)

where r is an appropriate relaxation-time and peq describes the equilibrium state. Obviously, this equation is no longer symmetric with respect to timereversal and, thus, describes irreversible processes. Note, Eq. (89) conserves "An example is a lossless coaxial cable with (real) impediance Z and infinite length. At one of its ports, this cable is indistinguishable from an ohmic resistor with resistance Z, although there is no dissipation involved.

102 the normalization trf>(t) = 1. In general, peq refers to a local rather than to global equilibrium in order not to violate particle number conservation. The relaxation time approximation is expected to be good for weak coupling and not too small times. A nice application of this approximation has been given by Mermin 43 who included damping in the Lindhard dielectric function e(q, w). b. Master equation (incoherent limit). This description considers only the diagonal elements of the statistical operator (with respect to the eigenstates of the isolated system) Pn(t) =< n\p\n >; nondiagonal elements are considered to be zero. -pm(i) = ^[rmnpn(i)-rnmpm(i)]

(90)

n

The transition rates Tmn can be obtained from the "Golden Rule" and obey the symmetry relation Tmnexp(—Em/kBT) = Tnmexp(—En/kBT), where T is the temperature of the bath. 57 c. Stochastic Liouville/v. Neumann equations. The interaction of an electronic system (like excitons) and the phonons in a crystal is approximately treated as a heat bath pushing the electrons or excitons in a stochastic manner. This description is justified if the temperature is not too low, otherwise polaron states are formed. However, it is not possible to take the reaction of the electron or exciton on the phonons into account. There is action but not reaction! For example, for Frenkel-excitons the interaction part of the Hamiltonian Eq. (66) is replaced by

Hint = Yifmn(t)blbn,

(91)

mn

where fmn(t) is assumed to descibe a Gaussian stochastic process with zero mean value. The diagonal element fnn(t) describes fluctuation of the exciton energy e at lattice site n, whereas the non-diagonal elements represent the stochastic variations of the (coherent) interaction matrix element hjj> in Eq.(66). For details we refer to the book by Kenkre and Reineker7 (see p. 120 ff). d. Kinetic (Boltzmann) equation We consider a gas of (quasi-) particles with weak short range interactions. Instead of the (classical) phase space distribution function p ( r i , p i . . . rjv,Piv) the one-particle distribution function / ( r , p, t) = / / p(i, p; r 2 , p 2 ; . . . rN, pN; t) dr2 d p 2 . . . rfrjy dpN

(92)

103 is used which obeys the kinetic equation ^

^

+v ( p ) ^ M

+

F ( p , r ) ^ ^ = C;

(93)

where v is the velocity, Eq.(53), and F an (external) force acting on the particles. Boltzmann had the ingenious idea to approximate the interactions between the particles, as summarized by C, by the concept of collisions, i.e. short range, instantaneous gain-loss processes with no memory to previous states (=Markov-process). Collision term:

C = Nin(r, p, t) - Nout(r, p,t).

Nin/aut count the number of particles scattered in/out dp at p . It is almost a miracle how this equation has withstood all criticsms and how it could be adapted to quantum mechanics, [ f(r,p,t) becomes the Wigner-function, in thermal equilibrium, / is the Fermi/Bose function.] In contrast to the Liouville equation, Eq.(93) is no longer symmetric under time-reversal: the Ihs changes sign whereas the rhs does not. Therefore the Boltzmann equation describes irreversible processes. Cohen and Thirring 52 give a historical survey and discussions of the celebrated Boltzmann equation, whereas Landau-Lifshitz 14 (Vol. 10) deal with physical kinetics in the wide sense of the microscopic theory of nonequilibrium processes. For example, for elastic scattering of electrons by impurities in a metal or semiconductor the collision term becomes** C(f) = j M p ' -»• p ) / ( r , p , t) - w(p -> p ' ) / ( r , p , t)] dp'

(94)

with the intrinsic scattering rate (in Born approximation) ™(p -> p') = Nimp^-1<

p'|V i m p |p >\25{ep - e P 0 .

(95)

For electron-phonon interaction JVj„, Nmt each contain two terms describing the absorption and emission of a phonon, 58 see Fig.9a.

"Note that Eq.(94) contains no "blocking" factors (1 — / ) for final scattering states, in accordance with Kohn and Luttinger, 47 but contrary to naive thinking and many textbooks. The (1 — / ) cancel if I f k, k') = Wfk^k) is symmetric but would lead to erroneous results for skew-scattering. 48 In an independent particle description no products / p / p < can occur as these describe particle-particle interactions. The situation is different for (inelastic) electron-phonon interaction where the blocking factors are necessary.

104 0.200 0.175 0.150 ^ 0.125

0.06

0.100 0.04

0.075 0.050

0.02

0.025

Figure 17: Spatio-temporal evolution of the energy per site, hm(t). Local momentum excitation at m = 0. According to Vasquez-Marquez et al. 49

4-4-

Examples

A. Linear chain Energy transport in a classical monoatomic chain, see Fig. 10, has been considered already by Hamilton and Schrodinger46 and more recently by Vasquez-Marquez et al., 49 who derived analytical results for the energy transport. In particular, for momentum excitation at t = 0 and site m = 0 the local energy \hm(t) = pE{ma,tj\ pi 2M

hm(t)

^2m W

+ o 4 - i W + JL+i{r))

shows an oscillating, nonexponential decay in time to neighbouring sites. denotes a Bessel-function,

51

r = Wj,* is the dimensionless time, UJL = yjij^

(96) Jm(T) is the

largest phonon frequency, and M is the mass of the particles. Note, the main peaks in hm(r) resemble those of the string, see Fig.2, yet there is no trailing edge and the energy between the peaks propagates off rather slowly. From the mean square displacement < m2 >-¥ (CT)2 we deduce a (dimensionless) mean propagation velocity c = l / v ^ , where c2 is just the mean square of the group velocity. B.

Two-level-system We study a system with two base states represented by 1>=

1\ 0

,„ { 0 12 >= 1

(97)

and a coupling parameterized by a real, positive constant e. The Hamiltonian of this system and its eigenstates \I >, \II > and energies Ef = -e, En = e are given by H =

0 -t

-e 0

" - ; * ( !

'" > - ; * ( - .

(98)

105 A realization of such a system is a spin 1/2 in a magnetic field in x-direction, H = —HBB -a ( a = (jyx,oy,az) denote the Pauli-matrices) or a particle in a double-well potential. Further examples can be found in the Feynman-Lectures 16 (Vol. III). (a) Schrodinger dynamics The Schrodinger equation for \ip(t) > |V>(i)>=Cl(t)|l>+c2(t)|2>

(99)

yields two coupled differential equations for the amplitudes ihdi(t) = -ec2{t),

ihc2{t) = -ec^t)

(100)

which can be easily solved by insertion. For \ip(0) >= |l > we obtain Ci(*)=cos(^t),

c2(i)=lsin(^t)J

(101)

where LO0 = 2e/h is the transition frequency between levels 1,11. As a result, the probability finding the system in (base) states |1 >, |2 > and expectation values of the spin vector are P1{t)

=

| C l (t)| 2 = c o s 2 ( ^ t ) ,

(102)

P2(t)

=

|c2(t)|2=sin2(^t),

(103)

< ax > = 2$tc*1{t)c2{t) = 0 ,

(104)

< av > = 29ct(t)c 2 (t) = sin(w 0 t),

(105)



=

2

2

| C l ( t ) | - | c 2 ( t ) | = cos(o;ot).

(106)

Note, that < ax > is time independent as ax commutes with H and, thus, is conserved. (b) Relaxation dynamics It is convenient to expand the density operator with respect to the eigenstates |7 >, \II > of H because peg is diagonal in this representation

»=(-.'!)• *.-iCo A ) Z = exp(/3e) + exp(—/3e) is the partition function and /? = l/(kBT) temperature. The general form of the density operator is

Pit) =(B*

IBA)-

(-> is the inverse

(108)

106 A is real, 0 < A < 1, whereas B may be complex. Initially, p(0) = |1 > < 1|, hence 4(0) = 5(0) = 1/2. Writing Eq. (89) in a matrix form, we obtain for the (1,1) and (1,2) components of p

Mi) + yA(t)

=

,/e

§<

(109)

B(t) + ( 7 - iu0) B(t) = 0,

(110)

where 7 = 1/r. As a result, iult,

1

A{i)

1

B(t)

. t + I e , £ [1 — e"-If } .

(Ill)

w

-7 ot

(112)

In order to compare with the previous example we have to transform vectors I/J and matrices M (base |1 >, |2 >) to xj), M (base \I >, \II >): t/, = Ui>,

M = UM&,

£=y!(j

\ )

•

(H3)

For example, ax = az, ay = —oy, az = ax, hence

=

[1 - e"7*] tanh(/3e),

(114)



= =

e" 7 'sin(w 0 <), e~ 7 ' cos(w 0 i),

(115) (116)

A(«)

=

^[l+e-7tcos(w0i)] ,

(117)

P&)

=

^[l-e"7tcos(w0t)] ,

(118)

-e

(119)

< H > =

.

(c) Master equation As states |1 >, |2 > have equal energy the transition rates must be equal, I \ 2 = T 2 i = F and so that the master equations Eqs.(90) become P1(t)=T(P2-P1),

P2(t) = T(P1-P2).

(120)

By conservation of probability, Pi + P2 = 1 these equations can be easily solved, thus Pi(t) = \[l

+ e-™],

P2(t) = \ [ l - e ^ } .

(121)

Results of coherent, fully incoherent, and relaxation dynamics are displayed in Fig. 18.

107 1.4 1.2

1

-° V

£ 0.8 \ a. 0.6 0.4 0.2 0.5

1.0

1.5

2.0

2.5

3.0

T

Figure 18: Dynamics of the two-level-system. Coherent (solid line), relaxing (dashed line), and fully incoherent motion (dashed-dotted line). C. Exciton chain (a) Schrodinger dynamics Instead of expanding the wave function into the stationary states of Eq.(68), we try to solve the (time-dependent) Schrodinger equation for a single exciton directly. In site representation, the amplitudes cm(t) of the state vector

M*)

> =

I] C mW|lm > ,

Cm(0) = <5m,0

(122)

m

obey the coupled set of differential equations ihcm{t) = e 0 c m (t) - hi [c m+ i(i) + cm^i(t)} .

(123)

Using an "innocent looking" Ansatz cm{t)=ime-i^lnCm{T),

T=uot,

wo = M / f t

Eq.(123) reveals a well known recursion relation of Bessel-functions

(124) 51

C'm{r) =

^[Cm_!-Cm+1(r)],

(125)

C m (r)

aJm{T + T0) + pYm(r + r 0 ) ,

(126)

=

a,(3,T0 being constants, but j3 = 0 as Cm(r) must be finite. Imposing c m (0) = 5m>0 as an initial condition, we obtain Cm(t) = ^e-^J^LJot),

Pm(t) = \Cm{t)f

(127)

For large times T > 1 , Pm(t) -* 2cos(wo* — rn.7r/2 — ir/4)/(ircJot) displays a damped, oscillatory behaviour, see Fig. 19. In contrast to the two-level-system there is no Poincare recurrence time because of the infinite system size. Note also, the decay of

108 the oscillations is not due to dissipation. The mean square displacement is wave-like, < (ma)2 > (t) = (ci) 2 , where the square of the propagation velocity c = ato0/\/2 is identical with the mean square of the exciton group velocity. (b) Master equation rvious success su Motivated by the previous we intend to solve the set of Master-equations dPm(t) dt

T [{Pm+1 - i V i ) + (Pm+1 - P m _ x )]

(128)

by an appropriate Ansatz Pm(t) = e-TZm{T),

T

= 2Tt.

(129)

Indeed, this reveals a recursion relation for (modified) Bessel-functions 51 Z'm(r)

=

\[Zm+l(r)

+ Zm^(r)]

,

Zm(j)

=

aIm(T + T0) + pKm(T + T0).

(130) (131)

Again, /3 = 0 as Pm must be finite. Imposing Pm(0) = 5m$ the solution is Pm(t) = e-2VtIm(2Tt).

(132)

For large times r > 1, P m (i) -> l/(-\/47iT t), which resolves the diffusive character of the process, < (ma)2 > = 2Dt, D = Ya2 is the diffusion constant. Problems: 1.) Calculate the entropy, Eq.(79), of the damped two-level system as a function of time. [Hint: Evaluate the trace in eigenbasis of /?(£)]• 2.) Show that the coherent and fully incoherent dynamics of a two-level system can be described in terms of a generalized master equation with a memory kernel Y(t — t') dPi(t) dt

Jr(t-tl)[P2(t')-P1(t')}dt'.

(133)

Which T(t — t1) corresponds to the relaxing system? 3.) An interesting, yet pathological case (of Ca,b), is a cluster of molecules with equal coupling strength between all TV sites. [Hint: use ^ncn =const, ^2nPn = 1]-

109 0.14 0.12

T=10

0.10

\ 0.08 0.06 0.04 0.02

1 , .

0.14 0.12

T = 10

0.10

i °08 0.06 0.04 0.02

III,Figure 19: Spatio-temporal evolution of the energy per site in the excitonic chain Pm(r) (= probability finding an exciton at m). (Top) coherent, (bottom) incoherent motion, according to Eqs.(127,132).

D. Gas of (quasi-) particles (a) Approach to equilibrium We consider a homogeneous, dilute gas of hard disks in two-dimensions. At t = 0 all particles are located on a square grid and have the same magnitude but random directions of momentum (velocity). By numerical integration of the Newton-equations, 53 / dp is obtained by counting the number of particles in dp. As time proceeds, the particle collisions tend to realize a Maxwellian distribution /o(p) = /oexp

2mkBT

(134)

which is accompanied by a monotonous increase of entropy. The time interval for reaching equilibrium roughly corresponds to 200 collisions. However, this state is highly correlated and apparently only describes "molecular chaos": Reversing all particle velocities after 50 or 100 collisions reproduces the initial state, see Fig. 20. Therefore,

-fcs / /

f(p,r,t)£nf{p,T,t)drdp

(135)

obviously does not give the correct (change) of entropy. (Note the difference between

110

r*

-1

- o o

X o o

0 eq.

0

O m <

• •

i

»-

•• -• • •

o

r

o i i i

V

"TWS'dbooooon««rtOOooooo(to

20

40

**••

60

time

Figure 20: Eta-function H = —S/ICB) of a system of 100 hard disks in a square with periodic boundary conditions showing kinetic (open circles) and antikinetic (full circles) evolutions for velocity inversions taking place at 50 and 100 collisions. According to Orban and Bellemans.53 Eq.(79) and Eq.(135).) For longer times, however, numerical errors come into play and destroy reversibility. An analytical solution of the Boltzmann equation is very difficult as the collision integral C(f) is nonlinear in / . In the vicinity of thermal equilibrium, however, a linearized description by a relaxation time approximation is possible (homogeneous conditions, no external forces)

^M

=

-LJl

dt T so that the distribution approaches equilibrium in an exponential manner.

(136) '

(b) Noninteracting particles We consider (quasi-) classical point-like particles without mutual interaction moving in a random array of stationary scatterers ("pinball machine"). Realizations of such a model are • slow neutrons in a heavy medium, • gas mixture with large mass ratio (=Lorentz gas), • electrons in "anti-dot" semiconductor systems, • elastic electron impurity scattering in a metal. We study the case of electrons in a metal or doped semiconductor. In thermal equilibrium, C(/o) = 0, where / 0 is the Fermi-function

/o(p) =

V—T>

(137)

and n is the chemical potential (Fermi-energy). An applied electrical field (parallel to the z-axis) £z(t) = £0Rexp(-iuJt) drives the system out of equilibrium. For isotropic,

111 elastic scattering of electrons by impurities the scattering kernel is independent of the scattering angle £1 so that the kinetic equation becomes

d

J^l + ( _ e) g oC -^ffi) = I J _L [f{SVj t) _ f{n> t)] dtf .

(138)

In linear order with respect to £0, f = /o(p) + / i ( p ) exp(—iui), we obtain for the first order correction. f („\ /l(P) =

eVz

: 7~!u

d/o(p)c , 5 £o + : • op 7 — IUJ

ncHV, (139)

< / I ( P ) > denotes the angular average of / i ( p ) which can be calculated selfconsistently from the above equation. 7 = 1/r. (The current induced by the external field, however, does not depend on < / i ( p ) >). As a result, we have M<j)e-**

=

^X>,/i(p)e-iwt =

ffIJM£oe-i''',

(140)

p

ne 2

/ \ aD(u)

=

1

. (141)

— .

m 7 — IUJ CTD(O;) is the Drude-conductivity. For a detailed study of the Lorentz-model see Hauge's article. 54 (c) Plasma oscillations A one-component plasma consists of one type of mobile particles (electrons) which are imbedded in an homogeneous positive neutralizing background (density p+). As the Coulomb interaction between the particles is long-range it cannot be treated by the concept of collisions. A reasonable approximation, however, can be formulated within the concept of a mean field £mf, which describes the coherent motion of the particles. £mf is treated on equal footing with an applied electrical field (if any), g/(r,p,t) QI +

v

df(v,p,t) ^

+ (-e)£m/(r,t)

div £mf(T, t) = - [p+(r, t) + p_] ,

9/(r,p,t) — = 0,

curl £mf(r, i) « 0 .

(142) (143)

These are the Landau-Vlassov equations. We are looking for a self-sustained plane wave solution of the linearized equations of the form / ( r , P, t) = /o(p) + / i ( p ) e ^ r ~ " " " .

(144)

(Note the difference between wave vector q and electron momentum p.) For small wave vectors q, the dispersion of the plasma wave becomes wp(«) = wp + ^q2

...,

(145)

112 where u^ = ne2/(me0) is the square of the plasma frequency, (3 = 3vp/5, and vF is the Fermi velocity of the electrons. For q -> 0, there is no damping of the plasma wave in the Landau-Vlassov approximation. For wave vectors q « kp, however, plasmons can decay into electron-hole pairs (Landau-damping), although Eq.(142) is invariant under time-reversal! For a survey on plasmons, see, e.g., Ref.55

5. 5.1.

M e m o r y Functions and non-Markovian Behaviour Memory function concept

The dynamics of closed systems as studied in the previous sections is determined by a set of first order differential equations with respect to time. Given the state at time t0 these equations determine the state at all later times, see e.g. the Hamiltonian equations Eq.(70). In addition, in classical mechanics damping may often be phenomenologically included by introducing a friction-force —7oi. For instance, for a particle with mass M in a potential V(x) the Newton-equation reads Mx(t) = -MJox

- ^ ^

+ /(i)

,

(146)

where f(t) is an external force. Actually, the friction-force has to be supplemented by an additional fluctuating force f(£), (with zero mean) which represents e.g. the collisions of molecules impinging on the (heavy) particle. This is the Langevin formulation which not only describes the damped average motion but also the fluctuations around it, see van Kampen. 57 In a more detailed description of open systems the heat bath is often modelled by a system of harmonic oscillators and the bath degrees of motion are eliminated to construct an equation of motion for the system (particle) alone. This will be illustrated by two examples in the following sections. In such a description the reaction of the bath on the system particle is also taken into account which leads to an equation of the form Mx + f

M 7 ( i - t') x{t') dt' + ^ ^

= f(t).

(147)

This equation is nonlocal in time and the memory function y(t — t') determines how far the "history"of x(t') (t' < t according to causality) influences the present x(t). With "/(t - t') = 7 0 5(i - t') we are back at Eq.(146). Note, x(t0) and x(t0) do no longer determine the state of the particle, at least not in the same sense as it has been used in Chapters 3. and 4. Additional assumptions for t —>• — oo are needed. Equation (147) describes a non-Markovian process which will be discussed in the next section. As an example, we consider a harmonic oscillator with frequency UJ0 under the

113 action of a monochromatic force, f(t) = /ocos(wt). As a result, the solution is x(t)

= »[xMfoe-**]

X(UJ)

=

±--2

T{u)

=

/

,

(148)

l

,

(149)

dt = r 1 (w) + ir 2 (a;).

(150)

2

/>oo

7 (t)e

iwt

Jo Note, memory effects do not only lead to a frequency dependent scattering rate (= I \ ) but to a shift in the resonance frequency (= r 2 ) , too. Memory effects have an interesting consequence on the frequency dependence of conduction electrons (mass m, density n, w0 = 0). Expressing the electron current as j(t) = —enx(t), f(t) = —e£(t), we obtain for the conductivity aM = — „ , * . , (151) m 1 (u) — IUI which is a generalization of the Drude result Eq.(141), see Fig.22. Deviations the measured conductivity of conduction electrons from the Drude result are always indications of memory effects. For experimental evidence, see, e.g., measurements by Dressel et al. 56 on some organic conductors. In quantum physics, the inclusion of dissipation requires more care because quantum systems are described by a Hamiltonian which, in the absence of time dependent external potentials, ensure the conservation of energy. On the other hand, dissipative forces cannot be included in the Hamilton itself. A successful and rather general approach is by the concept of an (infinite) reservoir and elimination of the reservoir degrees of freedom. For details and applications we recommend the book by Dittrich et al. 50 which gives an excellent introduction and overview on the relation of quantum transport and dissipation. Master and Boltzmann equations with memory kernels have been respectively studied by Kenkre 5 ' 42 and Hauge. 54 A. Caldeira-Leggett model As an example how to eliminate the bath variables we consider a classical particle of mass M and coordinate q, which is bilinearly coupled to a set of harmonic oscillators ("bath"), see Ingold's article in Ref.50 (p. 213). H H

-

=

Hs + Hbath + Hint,

=

V & 1M + (?)>

(152) (153)

H

»^ = Y,^: + lm^xl

(154)

i

Hmt

=

-qJ^CiXi i

+q

2

^ — ^ ' i

*

(155)

114

with suitable constants for CJ,WJ, rrij. This model has been used by several authors and is nowadays known as the Caldeira-Leggett model. The equations of motion of the coupled system are:

Mq + V'(q)+qJ2~-2=J2C'Xi' i

miU)

i

(156)

i

set + ujfxi = —q(t).

(157)

nil

As the bath represents a system of uncoupled oscillators their equation of motion can be easily solved in terms of the (unknown) driving term ~ q(t). Xi(t) = Xi(0) cos(uiit) + ^ - ^ sin(wji) + / rrii

~^— sin[ui(t - t')]q(t')dt'.

(158)

Jo rriiU){

Inserting this solution into the Newton-equation of the particle, we obtain Mq(t) + [ My{t - t')q(t') dt' + V'(q) = £(t), Jo

(159)

where 7(f — t') is the damping kernel (memory function) and f (t) is a fluctuating force (which depends on the initial conditions of the bath variables and is not stated here) 7 (t)

=

-

r

^ -

7T JO

'(«)

cos(tut)dcj,

(160)

U

= -E^^"^).

(161)

i

For a finite number of bath oscillators the total system will always return to its initial state after a finite (Poincare) recurrence time or may come arbitrarily close to it. For N —> oo, however, the Poincare time becomes infinite simulating dissipative behavior (see also Chapter 4.3.). We therefore first take the limit N —> oo and consider the spectral density of bath modes as a continous function. Frictional damping, y(t) ~ joS(t), is obtained for J(ui) ~ 7oW- A more realistic behavior would be the "Drude" form J( W ) = 7 o w - ^ x h r ,

7(*)=7o7ue- 7 D *,

(162)

which behaves as in the friction case for small frequencies but goes smoothly to zero for uj > 7£), see Fig.21. B. Rubin model A rather nontrivial yet exactly solvable model is obtained by a linear chain (see Chapters 3.7.A and 4.4.A), with one mass replaced by a particle of (arbitrary) mass

115

Figure 21: Spectral density of the bath oscillators (left) and memory functions (right). Solid lines: Caldeira-Leggett model with a Drude form, dashed dotted lines: Rubin model. (Dimensionless quantities, M0/M = 1).

2.0

b

£ 1.0 0.5

Figure 22: Real and imaginary parts of the electrical conductivity for the Caldeira-Leggett and Rubin models. Notation as in Fig.21. Mo, see Fig. 10. The left and right semi-infinite wings of the chain serve as a bath to which the central particle is coupled. As a result the damping kernels are

7(*) =

M0

Ji{ujLt)

(163) 2

I »

=

M

I —u V>i*}\sgn(u))

+ iuj ,

Ul\
Mo M

(164) (165)

where J\ (a;) is a Bessel-function. In contrast to the Drude case the memory function shows oscillations and decays merely algebraically for large times, see Fig. 21. 7(*)

M0

l2u)Lsm[u)Lt — 7r/4]

(166)

The oscillations in j(t) are connected with the upper cut-off in J(UJ) at the maximum phonon frequency uiL. For details, see Fick and Sauermann 40 (p. 255).

116

5.2.

Stochastic Processes and Non-Markovian

Behavior

Many phenomena depend on time in an extremely complicated way which is far beyond the possibility of observation and calculation. A possibility to circumvent this dilemma in theoretical physics is to replace the system by a suitable chosen ensemble of systems all having the same equations but different initial microstates. Therefore, state variables are turned into stochastic variables. By additional drastic assumptions about the nature of the stochastic processes this does simplify matters enormously. In many cases, these processes are modelled by Markovian-processes, which (in appropriate variables), lead to equations of motion which are local in time. Examples are the derivation of the Master and Boltzmann equations. In many cases, however, memory effects show up which indicate non-Markovian behaviour, see e.g. Wegener's article 8 in this volume. To elucidate the physics of these phenomena and their description we follow van Kampen 57 who gives an excellent introduction and overview on stochastic processes in physics and chemistry. The definition of a stochastic variable X consists of specifying • the set of all possible values ("sample space") and • the probability distribution P(X) over this set P(x)>0, The average of f(X)

fp{x)dx

= l.

(167)

is defined as =

f f(x)P(x)dx.

(168)

(For discrete X, the integral is understood as a sum). A prominent example is the Gauss-distribution p(X)

= \U—e-^'2,

<x2>=o.

(169)

V 27T<7

A stochastic process is defined as a function of X by some mapping X —> Y which is a function of time Yx(t)=f(X,t).

(170)

Such a quantity is also called a random function. On inserting for X one of its possible values x, an ordinary function of time results, which is called a sample function or a realization of the process, see Fig. 23. Averages are defined by

= =

JYx(x,t)Px{x)dx, fYx{x,t1)Yx{x1t2)Px(x)dx.

(171) (172)

117

A/*V^^^^'

1

t'

#3

Figure 23: Schematic of sample functions. The latter is [apart from a redefinition of Yx{t) —> Yx(t) — < Yx{t) >] the autocorrelation function of Yx(t). A stochastic process can also be specified by a hierarchy of distribution functions Pn{yu *i; 2/2, h;...; yn, tn), n = 1,2,.... Px{y, t) is the probability density for Yx(t) to take the value y at time t, P2 is the joint probability that Yx(t) has the value y\ at t\ and y2 at t2, etc. Functions Pn are symmetric in all arguments yi,tt but ij 7^ tk is implied. The conditional probability Pi\i(y2, *2|l/i, *i) i s the probability density for >x(£) to take the value y2 at t 2 , given that its value at tx is j / x . More generally one may fix the values at k different previous times t\,t2,.. .tk and ask for the joint probability of Yx at t other later times tt+i, • • • tk+i- This leads to the general definition of the conditional probability P£|fc(fc + 1 ; . . . k + e\l, 2,...k)

=

p (l 2

k)

'

^

where "i" is a shorthand for yi,U. Like Pi, Pn, Pt\k is non-negative and normalized. (Notice the difference in sequence of the arguments in joint and conditional probabilities). A process is called a Gaussian process if all its P„ are (multivariate) Gaussian distributions. For a stationary process all P n depend on time-differences alone and Pi(y,t) is time-independent. A Markov-process is defined as a stochastic process for any set of successive times ti
• • -yn-l,tn-l)

= Pl|l(2/n,*n|j/n-l,*n-l) •

In short: Markov processes don't have a memory. A Markov process is fully determined by Pi(y,t)

(I74)

and Pi|i(t/2,*2|2/i, h) {h > h)

118

Figure 24: Schematic of sample functions contributing to the conditional probability. as the whole hierarchy of joint probabilities Pn can be reconstructed from them, e.g.

Pi{y2,h) = J Plll{y2,t2\yut1)P{yut1)dy1, P2{yi,ti;y2,t2) P3(yi,ti;y2,t2;y3,t3)

= Pi\i{y2,h\yuti)

(175)

Pi(yi,h),

(176)

= P2{yuU;y2,t2)Pi\2(y3,h\yi,U;y2,t2), =

Pi{yiM)Pi\i{y2Myuti)Pi\MMyiM

(177) •

(178)

Continuing this algorithm one finds successively all Pn. This property makes Markov processes so managable and, fortunately, Markov processes are by far the most important stochastic processes in physics and chemistry. One of the best known Markov processes is the Wiener-process which describes the diffusive behavior of the position of a Brownian particle JMvi.O) Pi\i{y2,t2\yi,t1)

=

(179)

6{Vl), 1 ^4nD(t2

- h)

exp

(Z/2 - V\f 4i?(*2 - *l) J

(180)

In particular, we have for the time-dependent (i.e. non-stationary) probability, finding the particle at y at time t Pi(y,t)

=

1 -e

iDt

(181)

An example of a stationary Markov-process is the Ornstein-Uhlenbeck process which describes the velocity of the Brownian particle, see van Kampen. 57 The concept of a Markov process is not restricted to one-component processes but applies to processes with r components as well. However, if one ignores s > 0 components of an r component Markovian process, the resulting r — s component stochastic process will - in general - not be Markovian! The art of the physicist is to find such variables that are needed to make the description (approximately) Markovian. In physics "non-Markovian" is used almost synonymously for "memory behaviour" , yet the reader should beware of several pitfalls. Quoting van Kampen 57

119 When a physicist talks about a process he/she normally refers to a certain phenomenon involving time. Concerning a process in this way it is meaningless to ask whether or not the process is Markovian, unless one specifies the variables to be used for its description ... The criterion Eq. (174) is a condition on all distributions Pn of the hierarchy. It is impossible to aver that a process is Markovian if only information about the first few Pn is available ... Surprisingly, any deterministic process, like the Hamiltonian dynamics in phase space, Eq.(70), or the v.Neumann dynamics, Eqs. (78,89), is Markovian ... Sometimes "non-Markovian" equations of the form

¥^- = JtG{t,1?)P(!?)tf

(182)

are produced, yet one cannot be sure that the earlier values of P(t) are indispensable for knowing its future ... Problem: Consider the non-Markovian Eq.(147) with an exponential memory function j(t) = 7O7D exp(—7£>i)0(£). Show by differentiation that this equation can be transformed to a system of three first order differential equations with no memory for the composite variable £ = (x, x, x).

6.

Final Remarks

In this article the standard equations and their typical solutions and properties for (energy) transport in condensed matter were provided. We found that reversibility and coherence are often - yet not always - accompanied by oscillations in the physical quantities of interest. A large number of important questions and techniques, however, have only been touched and perhaps not even mentioned. For example, for semiconductors the standard theory to describe coherent phenomena are the Semiconductor-Bloch equations which describe the coherent motion of the interband amplitude as well as the electron/hole kinetics, see Haug and Koch. 58 A short survey about such problems has been given previousely27 and is not repeated here. The interplay between transport and coherence is most pronounced in systems where extremly short length scales (~ lnm) or ultra fast time scales (~ lfs) play a crucial role. It turns out to be very difficult to make significant generalizations of the Boltzmann equation to go beyond the limits set by the "old master" himself. To properly describe phenomena of this kind, a quantum theory of nonequilibrium processes is needed which has only recently been developed. For details, we refer to Fick and Sauermann, 40 Dittrich et al., 50 and Haug and Jauho. 59

120 7.

Acknowledgements

Many thanks to colleagues with whom I have discussed problems of energy transfer particularly Profs. P. Wiirfel and F. Herrmann. I thank also Mrs. R. Schrempp for typing the major part of the manuscript, Mrs. M. Bielfeldt for preparing the figures, and Dipl. Phys. A. Mildenberger for his help with computer problems.

REFERENCES 1. M. Planck, Das Prinzip der Erhaltung der Energie, (Teubner, Leipzig, 1887). 2. Scientific American, Sept. 1971 Energy and Power. 3. Freeman J. Dyson, Energy in the Universe, in Ref.2 p. 51. 4. H.B. Callen, Thermodynamics,

(Wiley, 1965).

5. Energy Transfer Processes in Condensed Matter, Nato ASI series B114, edited by B. Di Bartolo, (Plenum, 1985). 6. Advances in Energy Transfer Processes, edited by B. Di Bartolo, World Scientific, this volume. 7. V.M. Kenkre and P. Reineker, Exciton Dynamics in Molecular crystals and Aggregates, Springer Tracts in Modern Physics, Vol. 94, (Springer, 1982). 8. M.Wegener, Non-Markovian

Scattering Processes in Semiconductors in Ref.6

9. From discussions at the International School of Subnuclear Physics, Erice (1964). Near the entrance of the Feynman Lecture Hall, Ettore Majorana Center for Scientific Culture, San Rocco, Erice, Sicily, Italy. 10. (a) H. Hertz, Untersuchungen iiber die Ausbreitung der elektrischen Gesammelte Werke, 2, 2nd edn, (Ambrosius Barth, Leipzig, 1884). (b) G. Mie, Sitzungsber. konigl. Akad. der Wiss. 107 (1898) 1128.

Kraft.

11. F. Herrmann, Eur. J. Phys. 7 (1986) 198. 12. A.R. Bishop, J.A. Krumhansl, and S.E. Trullinger, Solitons in Condensed Matter: A Paradigm, Physica I D (1980) 1. 13. G. Barton, Elements of Green's functions and Propagation, (Oxford, 1991). 14. L.D. Landau and E.M. Lifshitz, Course of Theoretical Physics, Vol. 1-10, (Butterworth and Heinemann, 1998). 15. (a) J. Slepian, J. Appl. Phys. 13 (1942) 512. (b) C. O. Hines, Can. J. Phys. 30 (1952) 123. (c) C.S. Lai, Am. J. Phys. 49 (1981) 841.

121 (d) U. Backhaus, Der Energietransport durch elektrische Strome und elektromagnetische Felder, Naturwissenschaft und Unterricht, Bd. 20, Westarp-Wiss., (Wolf Graf von Westarp, 1993). 16. R.P. Feynman, R.B. Leighton, and M. Sands, The Feynman-Lectures Vol. I-III, (Addison Wesley, 1964).

on Physics,

17. D.F. Nelson, Phys. Rev. Lett. 76 (1996) 4713. 18. R. Loudon, J. Phys. A3 (1970) 233. 19. A. Enders and G. Nimtz, J. Phys. I France 2 (1992) 1693. 20. Ann. Phys. Lpz. 7 (1998) Nr. 7-8. 21. L.P. Brillouin, Wave Propagation and Group Velocity, (Academic, 1960). 22. Th. Ankel, Z. Physik, 144 (1956) 120. 23. (a) S. Chu and S. Wong, Phys. Rev. Lett. 48 (1982) 738. (b) A. Katz and R.R. Alfano, Phys. Rev. Lett, bf 49 (1982) 1292. (c) S. Chu and S. Wong, ibid p. 1293. 24. C.G.B. Garrett and D.E.McCumber, Phys. Rev. A l , 305 (1970). 25. K.E. Oughstun, C M . Balictsis, Phys. Rev. Lett. 77 (1996) 2210. 26. Ultrafast Dynamics of Quantum Systems: Physical Processes and Spectroscopic Techniques, NATO ASI series B Physics: Vol 372, edited by B. Di Bartolo, (Plenum, 1999). 27. Dielectric Description of Semiconductors: Bloch-Equations, in Ref.26

From Maxwell-

to

Semiconductor

28. R.v.Baltz and C.F. Klingshirn, Quasiparticles and Quasimomentum,

in Ref.26

29. P.W. Anderson, Basic Notions in Condensed Matter Physics, Frontiers in Physics, (Benjamin, 1963). 30. C. Kittel, Quantum Theory of Solids, (Wiley, 1963). 31. M.L. Cohen and J. Chelikowsky, Electronic Structure and Optical Properties of Semiconductors, 2nd edn., Solid State Sci., Vol. 75, (Springer, 1989). 32. Spectroscopy and Dynamics of Collective Excitations in Solids, Nato ASI series B: Physics Vol. 356, edited by B. Di Bartolo, (Plenum, 1997). 33. R.A. Cowley, A.D.W. Woods, and G. Dolling, Phys. Rev. B 6 4 (1966) 189. 34. M.O. Scully and M.S. Zubairy, Quantum Optics, (Cambridge, 1997). 35. B. Di Bartolo, Fundamental Interactions Leading to Energy Transfer, in Ref.6 36. C.F. Klingshirn, Light Matter Interaction: Experimental Aspects, in Ref.26 37. C.F. Klingshirn, Energy Transfer in Luminescent Processes in Bulk Semiconductors and Systems of Reduced Dimensionality, in Ref.6

122 38. W. H. Zurek, Decoherence and the Transition from Quantum to Classical, Physics Today, (Oct., 1991). 39. I. Prigogine, From Being to Becoming - Time and Complexity in Physical Sciences, (Freeman, 1980). 40. E. Fick and G.Sauermann, The Quantum Statistics of Dynamic Processes, Solid State Sciences 86, (Springer, 1990). 41. L. Allen and H. Eberly, Optical Resonance and Two-Level Atoms, (Wiley, 1975). 42. V.M.Kenkre, S.Raghavan, A.R.Bishop, and M.I.Salkola, Phy. Rev. B53 (1996) 5407. 43. N.D.Mermin, Phys. Rev. B l (1970) 2362. 44. M.Born and E.Wolf, Principles of Optics, (Pergamon, 1964). 45. G.Bergmann, Physics Reports, 107 (1984) 1. 46. (a) W.R.Hamilton, Proc. R.Irish Acad. 267 (1839) 341. (b) E. Schrodinger, Ann. Phys. Lpz. 44 (1914) 916. 47. W. Kohn and J.M. Luttinger, Phys. Rev. 108 (1957) 590. 48. L.E. Ballentine and M. Huberman, J. Phys. CIO (1977) 4991. 49. J. Vasquez-Marquez, M. Wagner, M. Montagna, O. Pilla, and G. Viliani, Physica B172 (1991) 355. 50. Quantum Transport and Dissipation, T. Dittrich, P. Hanggi, G. Ingold, B. Kramer, G. Schon, and W. Zwerger, (Wiley-VCH, 1998). 51. M. Abramowitz and I. Stegun, Handbook of Mathematical Tables, (Dover, 1970). 52. The Boltzmann equation, Edited by E.G.D.Cohen and W.Thirring, Acta Phys. Austriaca., Suppl. X, (Springer, 1973). 53. -J. Orban and A. Bellemans, Phys. Lett. 24A, 620 (1967). 54. E.H. Hauge, What can we learn from Lorentz models?, in Transport Lecture Notes in Physics, 3 1 , 337, (Springer, 1974).

Phenomena,

55. R.v. Baltz, Plasmons and Surface Plasmons in Bulk Metals, Metallic and Metallic Heterostructures, in Ref P

Clusters,

56. M. Dressel, A. Schwartz, G. Griiner, and L. Degiorgi, Phys. Rev. Lett.77 (1996) 398. 57. N.G. van Kampen, Stochastic Processes in Physics, (North Holland, 1983). 58. H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, (World Scientific, third edition, 1994). 59. H. Haug and A.-P. Jauho, Quantum Kinetics in Transport and Optics of Semiconductors, Solid State Sciences 123, (Springer, 1996).

E N E R G Y T R A N S F E R PROCESSES I N ATOMS A N D MOLECULES W . D E M T R O D E R , M. K E I L , T . P L A T Z , H. W E N Z Universitat

Kaiserslautern,

Erwin-Schrodinger-Str. D-67653 Germany E-mail: [email protected]

Kaiserslautern,

In this paper different mechanisms for intra- and inter-molecular energy transfer in excited atoms or molecules are discussed and experimental techniques for their investigations are presented. These techniques which involve cw excitation as well as time resolved laser spectroscopy differ for vibrationally excited molecules in their electronic groundstate from those involving electronically excited states. The advantages and limitations of the experimental methods and the amount of information, extracted from them, are outlined.

1

Introduction

Energy transfer in atoms or molecules in the gas phase can proceed in many different ways: We distinguish between intra- and inter-molecular transfer and between radiative and radiationless transitions. While intra-molecular transfer involves the redistribution of excitation energy among the different degrees of freedom within the same molecule and can be found in excited free atoms or molecules, the inter-molecular transfer demands interactions between different atoms or molecules which occur in the gas phase when the atoms approach each other closely and where the energy is transferred from one collision partner to the other during collisions. Examples of intra-molecular transfer are the autoionization of doubly excited atoms, which is caused by the interaction between two excited electrons of the same atom, or the exchange of energy between the vibrational kinetic energy of a vibrating or rotating molecule and its electronic energy. Generally speaking: Intra-molecular energy transfer is based on the coupling between different degrees of freedom or between different parts of a system 1 where the total energy and angular momentum of the system are preserved. Investigations of inter-molecular energy transfer give information on the interaction potential between the colliding particles, which might be electrons, nuclei, atoms or molecules. If the colliding particles have internal structure, the interaction potential generally depends on the internal states of the collision partners. For example, reactive collisions can be generally greatly enhanced by vibrational excitation of the reactands 2 .

123

124

The information drawn from the experiments is based on measurements of energy transfer rates between initial and final states, their dependence on density and temperature of the gaseous sample, from time resolved spectroscopy and also from measurements of broadening and shifts of spectral lines 3 . The most detailed information about state-to-state energy transfer can be gained from a combination of molecular beam technology with laser spectroscopy including time resolution. Here the initial state and the final state are both well defined and, in case of collisional transfer, also the kinematics of the process can be inferred from the angular distribution of the collision products. The following sections will illustrate these aspects by several examples. 2

Energy Transfer in Doubly Excited Atoms

There are several ways of exciting atoms to states with a total energy above the ionization energy Fig. 1. One possibility is the excitation of an inner shell electron into an empty state of an outer shell, which can be achieved by electron collisions or by photon absorption (Fig. la). The inner shell vacancy can be filled by an electron from the higher occupied shells. When the energy E\ released by such a transition is transferred to another electron with a binding energy E\, < Ei which is smaller than the released energy E\, this electron will have enough energy to escape the atom, i.e. the atom will be ionized. The autoionization rate depends on the correlation between the two electrons involved. Another way is the successive excitation of two outer electrons ei and e2 into Rydberg states where the sum of the two excitation energies E\ + Ei > IP exceeds the ionization potential IP of a single electron. Then the interaction between the two excited electrons can lead to energy transfer, where one electron is deexcited and the other is ionized (Fig. lb). Measurements of energies and lifetimes of these doubly excited states gives detailed information on the correlation between two electrons in different quantum states of the atom. The correlation describes, how the wavefunction and the energy of one electron is changed by the presence of the other electron. Such 'planetary atoms' have been therefore extensively studied over the past 4 5

years ' . 3

Vibrational Energy Redistribution in Molecules

In their lowest vibrational levels polyatomic molecules can be well described by weakly coupled harmonic oscillators, because the potential is nearly harmonic at these low energies and the total vibrational energy is to a good

125 Autoionization

g I P 2 9dn'd-

IP1 Ba+

••UM--"

6S

6snd-

6s*

Ba

6s*

Figure 1. Autoionization by inner shell excitation (a) or by simultaneous excitation of two valence electrons (b). The atomic orbital model is only schematic.

approximation the sum of the energies of the participating normal vibrational modes. A normal mode qi is a vibration, where all nuclei of the molecule oscillate at the same frequency and pass through the equilibrium configuration in Fig. 2a) at the same time. The situation changes with increasing excitation energy, where the potential becomes more and more anharmonic. The potential V(qi, ...,73^-6) in which the N nuclei of a molecule oscillate can be expanded into a Taylor series of the (3./V-6) normal coordinates of a nonlinear molecule, which gives

126

AE

^c

v-, a)

v2

v3

b)

Figure 2. a) Comparison of harmonic and real potential for a diatomic molecule, b) Level scheme of normal vibrations for a nonlinear triatomic molecule.

"<« ••~> = * + ?(5i) 0 « + 5?(ss;)„« f + dV qiqjqk + ••• -T

(1)

3

where VQ = 0 represents the potential minimum at the equilibrium configuration qi = 0 and therefore the first derivative is (dV/dqi)o = 0. In a harmonic potential the third sum in (1) vanishes and there is no coupling among the normal vibrations. In an anharmonic potential, however, the cubic term is non zero and mixes different normal modes, since the potential energy and the restoring force depends on the products of different normal mode amplitudes. This anharmonicity leads therefore to an increasing coupling between different vibrational modes and the total vibrational energy is no longer equal to the sum of the normal mode energies (Fig. 2b). In fact the model of vibrational normal modes can no longer be applied and the motion of the nuclei may become very complicated, in some cases with strong nonlinear

127 o..08 o..07 0,.06 C 0..05 o .04 'B. 0 .03 o o .02
Trichlorethylen

s°

0 -0.01 6040

6060

6080

6100

6120

v / cm"1 Figure 3. Overtone spectrum (21/5+^3) of trichlorethylene

6140

C2HCI3.

couplings even chaotic 6 ' 7 . If by optical absorption on an overtone transition many vibrational quanta of a specific vibrational mode are excited, the excitation energy may be redistributed among all vibrational modes, due to the strong vibrational coupling. In a rotating molecule furthermore Coriolis interaction can couple different vibrational modes. With increasing energy the coupling probability increases, which means that the time scale for redistribution of the energy becomes shorter because of two reasons: 1. The level density of vibrational levels increases strongly with increasing vibrational energy and therefore the energy separation between the optically excited level and neighboring perturbing levels becomes smaller, enhancing the interaction probability. 2. The coupling strength becomes stronger due to the increasing anharmonicity of the potential. Both effects lead to perturbations between levels in different vibrational states which cause shifts of absorption lines and the appearance of new lines which can be monitored by high resolution overtone spectroscopy. In case of dense spectra the separation between the lines may become smaller than the linewidth. In such cases broad absorption features appear with a spectral width depending on the number of interaction levels and the coupling strength (Fig. 3). We will now discuss experimental techniques which allow studies of such couplings resulting in energy transfer.

128

axis of rotation

grating

opt. diode

Figure 4. Widely tunable single-mode diode laser with external cavity and wavelength control.

4

Overtone Spectroscopy

High lying vibrational levels within the electronic ground state of molecules can be reached by overtone transitions, where more than one vibrational quantum is excited. Since the absorption cross section of overtone transitions is smaller by several orders of magnitude than that of fundamental vibrational transitions, the detection techniques must be sufficiently sensitive. Two examples out of several possible spectroscopic techniques are presented. These are modulated absorption spectroscopy using single mode tunable lasers and optothermal spectroscopy of molecules in a collimated molecular beam with a cold bolometer as detector. 4.1

Frequency Modulated Absorption Spectroscopy

The radiation source is a single-mode continuously tunable GaAssemiconductor laser in an external cavity in Littman configuration consist-

129

m

LD "

m

ro

LD-roRF

RF ™LD

ro

LD

+

ro

RF

without Absorption

m

LD

m

LD +

m

RF

with Absorption

Figure 5. Basic principle of absorption spectroscopy with phase-modulated laser beam.

ing of an optical grating, a tiltable mirror and one endface of the laser diode (Fig. 4). The other endface is antireflection-coated. If the tilting axis of the mirror is correctly chosen (close to the intersection of mirror- and grating planes), mode-hop-free continuous tuning over about 300 GHz can be achieved 8 . The laser output beam (zero-order diffraction from the grating) is sent through a multiple path absorption cell and the detected transmitted intensity is compared with a reference intensity obtained from a beam splitter before the absorption cell. The signal-to-noise ratio can be drastically increased by modulating the incoming laser beam. This modulation is achieved with an electrooptic modulator. The advantage of the modulation is illustrated by Fig. 5. The phase modulation generates two sidebands at frequencies WLD ± WRF with equal amplitudes but opposite phases. Detection of the transmitted signal by a lock-in amplifier tuned to the modulation frequency URF selects the two beat signals between the carrier and the two sidebands and gives therefore two contributions with opposite signs which cancel each other if there is no absorption. Even if the laser intensity fluctuates, these fluctuations cancel, which reduces the background noise by at least two orders of magnitude. If one of the sidebands falls onto an absorption line, the balance is perturbed and the lock-in gives a signal with a magnitude proportional to the absorption I0-a(cu)-L. The best results are obtained, if the modulation frequency QJRF/2-K is comparable to the width Av of the absorption lines which is generally the Dopplerwidth (Av « 1 GHz). By forming the ratio of the lock-in output signal to the reference detector signal the absorption signals can be normalized to constant input intensity I0. The disadvantage of the one-tone technique is the high modulation frequency LORF which cannot be handled by conventional lock-in detectors. An even higher sensitivity can be achieved with two-tone modulation

a p p o p

laser intensity [a. u.]

130

-

2

-

1

0

1

2

-

frequency [GHzJ

a)

\

f: IE 2

-

1

f

V 0

1

2

frequency [GHz]

c)

t k 0.4

A

0.2

switching of he HFmo dulation by the second, low frequeny f I

^ ^ ^ f e

0.0

V"

-0.2 -0.4

-2

i

{:: 0.2

loo -

2

-

1

-

1 0 frequency [GHz]

1

2

lock in signal on the low detection frequency ii

'

2.

~

A

0

1

2

frequency [GHz] laser spectrum without (a) and with (b) HF modulation

0.B

I: 5:: d)

frequency [GHz]

detector intensity by scanning the absorption line without (c) and with (d) HF modulation

Figure 6. Absorption profiles obtained with an unmodulated tunable single-mode laser (upper part), with a frequency modulated laser (lower part), and by amplitude modulation of the frequency modulation.

techniques. The principle of this method is explained in Fig. 6. The upper part shows the absorption profile, if the wavelength of the unmodulated laser is tuned over the absorption lines. The lower part gives the absorption profile obtained with the modulated laser, where the upper sideband, the carrier and the lower sideband are tuned successively over the absorption line and the transmitted intensity is detected. The absorption

131

4000

Absorption 0.05%

2000


0

-

c 2 .Q

JO .53, -2000 c •? - 4 0 0 0

AbswpUonO.05%

absorption signal without modulation

-6000 200

400 600 laser wavelength [arb. units]

800

Figure 7. Overtone spectrum of a rotational line of the (1,2,l)i -(0,0,0) band of H 2 0 . Main part: with modulation; insert: without modulation.

profile in the middle part is the difference between the upper and the lower profiles. It is obtained, when the modulation is periodically switched on and off and a lock-in detector synchronized at the switching frequency monitors the difference (two-tone modulation spectroscopy). For illustration of the increase in sensitivity the absorption signal on the overtone transition (l,2,l)<-(0,0,0) of H2O molcules is shown in Fig. 7 with and without modulation. The gain in achievable signal-to-noise ratio due to the modulation is at least two orders of magnitude 9 . Another technique uses instead of the high frequency phase modulation wavelength modulation by slowly modulating the length of the laser cavity. In this case the laser wavelength is modulated across the absorption line while it is continuously tuned across the spectrum 10 . The sensitivity is comparable to that of two-tone frequency modulation, but the experimental setup is easier.

132 Dewar

Bolometer

reservoir reflection arrangement

Figure 8. Experimental arrangement for optothermal spectroscopy.

4-2

Optothermal Spectroscopy

The spectral resolution of the modulation spectroscopy in a multipath cell is limited by the Dopplerwidth of the absorption lines. Sub-Doppler spectroscopy is possible by optothermal spectroscopy in a collimated molecular beam 11 . The basic principle is depicted in Fig. 8. The molecules are excited into higher rotational-vibrational levels by a tunable single-mode laser which crosses the molecular beam perpendicularly several times in multiple reflection arrangement. The excited molecules with radiative lifetimes longer than their flight time travel to the detector, which is a bolometer, cooled to 1.6 K, stick at its surface and transfer their excitation energy to the bolometer. This increases its temperature by AT and decreases its electrical resistance by dR <0 (2) dT ^ ^ A small current I is sent through the device with resistance R. If the laser is chopped at a frequency / , the bolometer delivers an ac signal to a lock-in amplifier, which is proportional to with

AU(f)

= I • AR(f)

oc AP

(3)

133 =i

M

Fourier spectroscopy

T — • —

6150,75

" " • — i — • " "

6150,80

— i — | — i — i — i — i — | — i — i — i — i — | — i — 1 > -

6150,85

6150,90

v/crri

6150,95

1

optoacoustic spectroscopy 375 MHz - frequency markers

optothermal spectroscopy (SNR>1000)

I

-L.

Figure 9. Comparison of different spectroscopic techniques illustrated by measurements of the Q branch in the Vf,+vg overtone band of ethylene C2H4.

where A P is the laser power absorbed by the molecules and transferred to the bolometer. The minimum detectable power is about 10~ 14 W! The technique is therefore very sensitive. Fig. 9 shows for illustration a comparison between identical sections of the overtone spectrum of ethylene C2H4 around 1.5 /mi, measured with a Fourier spectrograph (2 hours exposure time) with optoacoustic detection and with optothermal detection. The gain in spectral resolution and in S/N ratio is obvious 12 . With increasing excitation energy the density of lines increases strongly. Often lines are broadened and shifted from their expected positions, due to interactions with other nearby levels. From the shifts and the broadening of spectral lines these intramolecular interactions can be therefore inferred. The

134

true wavefunctions of the interacting levels can be written as linear combinations * = J2ici^i °f t n e wavefunctions of the different levels | i) involved. The absorption probability of the transition from the ground state | g) to these interacting levels is proportional to the square

^2=i(*9i^iEc^)i2

(4)

i

of the matrix element M and therefore depends on the mixing coefficients d. In case of interactions between the optically excited level and other levels, which are directly not accessible from the ground state (dark levels) the matrix elements do not vanish for transitions to the dark levels, since their wavefunction contains contributions of the accessible level. In this way the forbidden transitions to the dark levels borrow oscillator strength from the allowed transitions and become partly allowed. This means that extra lines appear in the spectrum which give direct information on these otherwise dark levels. 5

Intramolecular Transitions in Electronically Excited Molecules

It often happens that the excitation energy of levels in excited electronic states is lower than the dissociation energy of the electronic ground state. Examples are the triatomic molecules NO2, SO2, CS2 and many more polyatomic molecules. In such cases interactions may occur between low rovibronic levels of the excited state and high vibrational levels of the electronic ground state. These interactions result in intra-molecular energy transfer between the coupled levels, i.e. the molecule transfers part of its electronic energy into vibrational or rotational energy, which means from the electronic cloud to the kinetic energy of the nuclei 13 . The interaction may be caused by Coriolis coupling (in a rotating molecule), spin-orbit coupling (for instance between singlet- and triplet states) or vibronic coupling, where the vibrational mode of the molecule is changed, or electrostatic coupling where the energy transfer causes a change of the electron distribution. Famous examples are dye molecules used as the active medium in dye lasers. Here vibrational levels in an excited singlet state are populated by optical pumping. These levels are rapidly transferred into the lowest vibronic level of this excited singlet state by collisions with the solvent molecules. Then radiationless transitions into excited triplet states can occur, induced by spin-orbit coupling. This energy transfer depletes the excited sin-

135

a=60°

Qb

Figure 10. Schematic cut ( Q a = 0 , Q s =const.) through the potential energy surfaces of the ground and an excited electronic state of an X3 molecule.

glet levels and competes with the radiative decay into the singlet ground state thus diminishing the gain of the laser transitions 14 . It is therefore essential for efficient dye laser operation to keep the rate of radiationless transitions as low as possible and to remove the long living excited triplet states which can absorb the laser radiation thus introducing extra losses. We will here discuss another example where such intra-rnolecular energy transfer occurs, namely the alkali clusters Na3 15 and Li3 16 , which have been studied in detail in our group. These small clusters can be regarded as floppy molecules which continuously change their geometric structure by vibrations. In the equilateral triangle geometry the electronic ground state and several excited states are doubly degenerate because of symmetry. In such cases the Jahn-Teller theorem states that the potential minimum

136

is not at the configuration of highest symmetry (here: equilateral triangle) but the molecule is stabilized in a configuration of lower symmetry (spontaneous symmetry breaking 1 7 ) . In Fig. 10 the potential energy is plotted as a function of the normal coordinate Qb of the symmetry lowering bending vibration. Two potential curves intersect each other at the point of highest symmetry. If the potential energy is plotted against both Jahn-Teller active normal coordinates (Qb, Qa (asymmetric stretch)) the potential surface shows a conical intersection at the equilateral triangle configuration. The lower sheet resembles a Mexican hat with 3 shallow minima and potential barriers. In Fig. 11 this lower sheet of the electronic ground state X2E' of Li3 is shown as an energy contour diagram. The minima correspond to an obtuse isosceles triangle the barriers to an acute isosceles triangle. The point of conical intersection (equilateral triangle) has an energy of 500 c m - 1 with respect to the minima. The combination of the bending and asymmetric stretch vibration results in a continuous change of the molecular geometry, following the closed path shown in Fig. 11. This movement is called pseudorotation because it does not represent a real rotation of the nuclear frame but the obtuse angle of the triangle moves from one nucleus to the other. It can be regarded as a synchronous rotation of the three nuclei with a phase shift of ^ against each other around points which represent the three corners of the equilateral triangle. This means the axes of the main principle moments of inertia rotate. If the vibrational energy is much lower than the saddle points the system can be regarded as localized in the three equivalent minima. However, it can tunnel through the barriers where the tunnel frequency increases with increasing vibrational energy. Above the barrier the system is no longer localized but represents a nearly free pseudorotor. The tunneling results in a splitting of the rotational-vibrational energy levels, similar to the inversion splitting in the NH3 molecule. Since there are three minima one might expect a splitting into three components. However, it turns out that two of these components are degenerate. The pseudorotation therefore splits the vibrational levels into a nondegenerate component with vibronic A symmetry (in the DZH symmetry group) and a degenerate vibronic E component. The interesting point is now that the electronic wave function experiences a phase change of n (not of 2ir\) when the system completes a full cycle of pseudorotation around the conical intersection. This 'Berry phase' 18 can be experimentally verified in the following way: Since the total wavefunction (which can be written as a product of electronic and nuclear wavefunctions in the Born-Oppenheimer approximation) must be unambiguous (i.e. single valued at each point of the phase space), the nuclear part of the wavefunction also has to show a

137

Figure 11. Cut through the lower adiabatic potential energy surface of the electronic A state of 2 1 Li3 calculated ab initio by W. Meyer. Starting at the centre from the D^h geometry, the coordinates denote asymmetric deformations where all three nuclei are displaced by the same distance, but at angles that differ by 27r/3 from nucleus to nucleus. The triangles show the molecular geometries that result for selected points of the diagram.

phase change of ir. This means that the energetic order of the E and A pseudorotation components is reversed with the 'Berry phase' compared to the situation without the 'Berry phase'. The experimental verification is based on sub-Doppler laser spectroscopy of Li 3 or Na 3 15 in a collimated supersonic beam using resonant two-photon ionization. A tunable dye laser excites the molecules into levels of the A2E" state and a cw argon laser ionizes the excited molecules. In case of Li 3 , in order to separate the spectra of the different isotopomers 7 Li 7 Li 7 Li, 6 Li 7 Li 7 Li, 6 Li 6 Li 7 Li and 6 Li 6 Li 6 Li a quadrupole mass spectrometer is used and the spectra are recorded mass selective. Because of the high spectral resolution the rotational lines of the A2E" 4- X2E' transition could be all resolved. Figure 12 shows for illustration a section of the A2E"(0,1,0) <- X2E'(0,0,0) vibrational band of 2 1 Li 3 . From the rotational analysis of such spectra the geometric structure of the system can be obtained. Since the pseudorotation components with E symmetry exhibit a different rotational pattern they can be clearly distinguished from the A components.

138

LLJIJ^ I— 14687

1 14688

•

II

11

1 14689

14690

wave number [cm'1] Figure 12. RTPI spectrum with rotational resolution of a part of the A2 E" X2E'(0,0,0) band of 2 1 L i 3 .

(0,1,0)

Therefore the energetic order of A and E components can be clearly recognized from the rotationally resolved spectra and the existence of the 'Berry phase' can be verified. The pseudorotation causes a periodic change of the principal axes of inertia. Since the total angular momentum of the rotating molecule is fixed in space the rotational energy of the molecule suffers a periodic change. The total energy of the molecule has, of course, to be constant. The pseudorotation therefore induces a periodic energy transfer between vibration and rotation of the molecule.

6

Lifetime Measurements

Time-resolved measurements of the fluorescence emitted from selectively excited levels yield not only radiative transition probabilities but also cross sections of quenching collisions which deplete the excited level. From the time derivative of the population density

139

n 1/i:[10V]

wV'n Jn 6.0

NaK: D 1 it v' = 7; J' = 23 x-t = 20 ns

5.5

5.0

6-1015n[cm"3]

Figure 13. Level scheme and Stern-Vollmer-plot of the pressure dependent effective lifetime of the excited level (v'k=7, J'k=13) in the D 1 !!,, state of NaK.

dNk = -AkNk - RkNk (5) dt where Ak = £V Aki is the total radiative decay rate of level | k) and Rk = NB0kVr is the total radiationless decay rate induced by collisions with molecules B at a density NB, collision cross section ak and mean relative velocity vr we obtain

Nk(t)=Nk(0)-eT'tt

(6)

with = Ak + NBakvT

T

eff

(7)

Plotting the inverse effective lifetime versus the density NB of collision partners (Stern-Vollmer-plot) yields the radiative transition probability Ak and the collision cross section ak (Fig. 13).

140

parent line

51

I 54 I

52 I

50 I

4B I

(6 I

44 42 40 I f I

46 44 I I

49

47

F 42 I

|

45

C

43

41

39

L _ i

I

3? 35

I

I

33

31 29

1 I I

40 38 36 34 32 30 28 26 24 I I I 54 52 50 4B 46 44 42 40 38 36 I I I I I I I I I I

Figure 14. Collisional satellites in the fluorescence spectrum of the selectively excited level B1IIu(v' = 6, J' = 43) of the Na2 molecule. The parent line is recorded with 20 times reduced sensitivity.

Measurements of Collisional Energy Transfer A m o n g Excited States While lifetime measurements give information about the radiative decay rate and the total cross section for collisional quenching of selectively excited molecular levels, the final state of the collisional transition is not defined. Collisional energy transfer rates between specified initial and final levels in electronically excited states can be obtained in the following way: If the excited molecule undergoes collision induced transitions with AJ' or Av' from the initially excited level (v',J') to other rotational or vibrational levels | n) = | v' + Av', J' + A J') the dispersed fluorescence spectrum shows, besides the 'parent lines' emitted from the primarily excited level, collision-induced satellite lines which correspond to fluorescence lines emitted from these collisionally populated levels (Fig. 14).

141

The intensity ratio of a satellite line on a radiative transition | n) —v\ m) to the 'parent line' on the transition | k) ->| i) is given by

hi

Nk • A^

The population Nn can be obtained from the equilibrium condition dN -L^=Nk.Rkn-Nn-An at

=0

(9)

and the intensity ratio

A-ki ' An

yields the collision-induced transfer rate Rkn if the radiative transition probabilities Anm and A^ are known and the total radiative decay probability An — — has been obtained from lifetime measurements. The individual cross sections
(11) 19

.

Collisional Transfer in Ground States of Molecules

Collisional transfer rates between rotational-vibrational levels in the electronic ground states of molecules cannot use laser-induced fluorescence for detection but can be measured with time-resolved optical double resonance techniques (Fig. 15). A short pulse of the pump laser, tuned to the transition | i) —>| fc) depletes the lower level population Ni, which is filled up again by collisional transfer from neighboring nondepleted levels. The time dependence of the population Ni(t) can be monitored with a weak pulsed probe laser, tuned to a transition | i) —>\ j) which measures the absorption a(t) = Ni{t)-Oij at variable time delay At either directly or via the probe laser-induced fluorescence intensity which is proportional to the population Ni(t). If the probe laser is tuned to a transition starting from a neighboring level | J " 4- A J ) , the time dependent absorption of molecules in this level gives the individual contribution of the collision induced net transfer rate | J!' + A J ) -•! Jf) to the level | i) depleted by the pump.

142 (V k \ Jk')

( V i , Ji')

pump Laser

(Vf, Ji")

i l pump

Llfl.

\

I 1

1 / 1 / T

/

rel.

Figure 15. Measurements of collisional population transfer in electronic ground states by time-resolved optical-optical double resonance techniques.

The most detailed information about collisional transfer in ground state molecules can be gained from crossed beam experiments 20 . Molecules M\ in a collimated beam collide with atoms or molecules Mi in another beam, which crosses the first beam perpendicularly (Fig. 16). The molecules M\ scattered under an angle •& are detected state-selective by laser-induced fluorescence with a probe laser tuned to a transition | _;') —»| m). The intensity of the

143

probe laser

pump laser •&\

molecules

I m> —

—

J

pump

depletion of I i >

detection of molecules in I j >

Figure 16. State selective measurements of differential cross sections in crossed molecular beams.

fluorescence is proportional to the population Nj of scattered molecules Mi in level \ j). If a pump laser depletes a selected level | i) of the molecules Mi before they enter the collision region, this level cannot contribute to the population Nj after the collision. Therefore, the difference ANj of the population Nj with the pump laser on and off yields the individual collisional transfer rate from level | i) to level | j). Since the scattering angle defines the impact parameter, such a 'complete scattering experiment', where initial and final states and the scattering angle are measured, gives the most detailed information on the dependence of an individual collision-induced transition | i) —»| j) (e.g. a

144

rotational transition J ->• J 4- A J, or a vibrational transfer v -> v + Av) on the impact parameter, the collision partners and the relative velocity. With other words it tells us, at which distance between the collision partners the individual transitions have maximum probability. 9

Conclusion

The examples, given in the preceding sections illustrate, that laser spectroscopy is a powerful tool for the detailed investigation of the various energy transfer channels in atoms or molecules. In intra-molecular transitions the total energy of a molecule and its total angular momentum is preserved. The transitions cause a redistribution of energy and angular momentum among the different degrees of freedom, for example between the kinetic energy of the nuclei and the energy of the electron cloud. In case of collisional energy transfer either the kinetic energy of the relative motion of the collision partners is transferred into excitation energy of one or both of the partners or the internal energy of one partner is transferred to the other (inter-molecular energy transfer). These collisional energy transfer processes form the molecular basis of all chemical reactions and together with energy transfer by absorption or emission of photons form the basis of biological life. It is, therefore, worthwhile to study them thoroughly.

References 1. K.K. Lehmann, G. Scoles, Intramolecular Dynamics From EigenstateResolved Infrared Spectra, Ann. Rev. Phys. Chem. 45, 241 (1994) A. Beil, D. Luckhaus, M. Quack, J. Stohner, Ber. Bunsenges. Phys. Chem. 101, 311 (1997) 2. St.R. Leone, State-Resolved Molecular Reaction Dynamics, Ann. Rev. Phys. Chem. 35, 109 (1984) 3. see for instance the Proceedings of the Int. Conf. On Spectral Line Shapes, Vol. I-X, Library of Congress Catalogue 1979-99 4. I.C. Percival, Planetary Atoms, Proc. Roy. Soc. London A353, 289 (1977) 5. J. Boulmer, P. Camus, P. Pillet, J. Opt. Soc. Am. B4, 805 (1987) 6. A. Delon, R. Jost, J. Chem. Phys. 95, 5701 and 5687 (1991) 7. A. Holle, J. Main, G. Wiebusch, H. Rottke, K.H. Welge in: Atomic Spectra and Collisions, ed. By K.T. Taylor, M.H. Nayfeh, C.W. Clark (Plenum Press, New York 1988)

145

8. H. Wenz, R. GroBklofi, W. Demtroder, Laser und Optoelektronik 28, Febr. 1996, p. 58 9. R. Grofiklofi, P. Kersten, W. Demtroder, Appl. Phys. B58, 137 (1994) 10. P.C.D. Hobbs, Appl. Opt. 36, 803 (1997) 11. R.E. Miller, Infrared Laser Spectroscopy in: G. Scoles, ed.: Atomic and Molecular Beam Methods, p. 192 (Oxford Univ. Press, 1992) 12. T. Platz, W. Demtroder, Chem. Phys. Lett. 294, 397 (1998) 13. V. May, 0 . Kuhn, Transfer Phenomena in Molecular Systems (Wiley VCH, Weinheim 1999) 14. F.P. Schafer, ed.: Dye Lasers (Springer, Berlin, Heidelberg 1977) 15. H. von Busch, Vas Dev, H.-A. Eckel, S. Kasahara, J. Wang, W. Demtroder, P. Sebald, and W. Meyer, Phys. Rev. Lett. 8 1 , 4584 (1998) 16. H.-G. Kramer, M. Keil, C.B. Suarez, W. Demtroder, W. Meyer, Chem. Phys. Lett. 299, 212 (1999) 17. LB. Bersuker, The Jahn-Teller Effect, Plenum Press, New York 1984 18. M. Berry, The geometric phase, Scientific Am. 12, 26 (1988) 19. K. Bergmann, W. Demtroder, J. Phys. B5, 1386 and 2098 (1972) 20. K. Bergmann, U. Hefter, J. Witt, J. Chem. Phys. 7 1 , 2726 (1979); 72, 4277 (1980)

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ADVANCES IN THE TECHNIQUES FOR THE STUDY OF ENERGY TRANSFER

Daniele Hulin Laboratoire d'Optique Appliquee ENSTA, Ecole polytechnique, Palaiseau, France

1. Introduction When an external source of energy excites a material, there is a transfer of energy from the source to the sample. This newly excited state is usually not an equilibrium state and the acquired energy is transferred to the surrounding world. This mechanism can be studied either through the relaxation of the initially excited state or through the apparition of newly excited states. Monitoring these phenomena after an excitation gives an insight on the fundamental mechanisms that govern the material. In this chapter, the excitation will be an optical excitation, although many features could be generalized to other sources of excitation. Recent progresses in the techniques of observation have opened new fields of investigation and the purpose of this chapter is to provide a (non-exhaustive) overview of the most-frequently-used recent techniques. How can we monitor an excited state and its evolution after excitation? In a simplified manner, one can say that there is two different ways to perform this study. A change of the properties of the material, mainly but not exclusively the optical properties, can be analyzed or an emitted signal (spontaneous, stimulated emission) can be recorded. The separation between the two methods is not as abrupt as it is presented here. This chapter will first present different recent techniques used to monitor the changes in the optical properties. Then it will discuss techniques allowing to study emitted signals. Finally, phase-related techniques will also be described.

2. Pump-probe spectroscopy : absorption 2.1 Experimental methods The optical excitation is often referred as "the pump". In order to monitor the changes of the optical properties resulting from the excitation, light again has to be used. This light is referred as "the probe". In this paragraph, we will be concerned by the changes in absorption.

147

148 In order to follow the energy transfer, the absorption changes have to be time-resolved. This is usually done in two different ways:

pump

•4

cw probe

H

Time-gated detector

(a)

pump

Timeintegrating detector

(b)

After a pulsed excitation, the temporal evolution of the sample absorption can be recorded through the change of the transmitted probe beam in function of time by a timeresolving detector (part (a) of the figure). This can be realized through a photodiode, a streak camera or an optical gate. In the second case (part (b) of the figure), the detector is time-integrating (long response time), but the temporal resolution is provided by the fact that the probe beam is also a light pulse. If the probe pulse reaches the sample before the excitation (situation 1), it sees an unexcited sample; if it arrives at a delay x after the excitation (situation 3), it records the sample situation that has already evolved after excitation. This sequence can be repeated and the delay x can be varied on purpose. This method is analogous to the stroboscopic technique. The choice between the two methods depends on the need in time resolution. For times longer than few tens of picoseconds, the first solution may be easier to handle. However, if one wants to take advantage of the recent advances in ultrashort-pulse lasers, only the second method can be used. Lasers with ultrashort pulses are mainly based on oscillators using Kerr-lens modelocking and Titanium-doped Sapphire crystal. They deliver currently pulses as short as 15 femtosecondes (15xlO"15 s) with a repetition rate of 15 MHz. Because of the short duration, the pulses have a quite large spectral width (40 nm) and this allows already to do some spectral measurements. However they are centered around 800 nm and this restricts the range of study.

149 In order to overcome this limitation and also to obtain a larger intensity per pulse, amplification chains are used to boost the energy from the picojoule level up to the kilojoule level but with only few shots per day! For most of the studies this extreme performance is not necessary so that the repetition rate after amplification can remain large enough to accumulate the signal over a reasonable time. Following the desired energy per pulse, it varies from 10 Hz to more than 100 kHz (Rulliere, 1998). In the pump-probe experiment, the time resolution depends on the pulse duration and on the precise determination of the delay x. This last requirement is fulfilled by splitting the initial pulse in order to avoid any jitter between pump and probe pulses. The two pulses have different optical paths before reaching the sample. These paths can be varied on a controlled manner. The zero delay (coincidence between pump and probe pulses) is determined through the help of physical phenomena occurring only when the two pulses overlap. The signal is recorded in function of the delay x, providing the evolution of the absorption after excitation.

In the above scheme, pump and probe have the same wavelength since they originate from the same laser pulse through the beam splitter (BS). It may happen quite often that a different wavelength is needed for the pump or for the probe. In order to change the wavelength of the laser pulse, different methods can be used. Three examples are given below: •

Raman effect may be efficient enough to shift the laser toward longer wavelengths:

150 G^laser

•

^

tOiaser " tOphonon

Parametric generation is more and more widely used (Nisoli, 1994; Yakovlev, 1994; Seifert, 1994). Parametric effects are based on the use of non-linear quadratic crystals having large second-order susceptibility (%<2)) such as BBO (b-barium borate), LBO (lithium triborate), KTP (KtiOP4) or some non-centrosymetric organic crystals. In such a medium, two incident fields at frequencies u)i and (1)2 can produce a new radiation at frequency W3 = Wi + (1)2 or (ti'3 = a>i - CO2 • If C0i = 002 as stated above, this leads to the second-harmonic generation. The reverse process is also possible: starting with the frequency (fliaser, one can obtain: COlaser = tt>l + CO2

If the laser frequency is around 800 nm like for most of the Ti:Sapphire lasers, only wavelengths in the infrared can be reached with this method. To obtain visible radiation, second harmonic generation has to be used, either before the parametric generation (2 a w * ) or after (2a>i); the two ways are almost equally utilized, the choice depending on the required result because neither coi or (02 can be close to zero for an efficient process. The most spectacular and easiest way to get broadband tunability is to focus a high-intensity pulse in a transparent medium such as water, glass or undoped sapphire (even liquid nitrogen surrounding the sample in a cryostat !). When the intensity reaches 1013W/cm2, the beam undergoes self-focusing and self-phase modulation (Salin, 1992; Ranka, 1998). A continuous spectrum from 200 nm to 1500 nm can then be produced and the "white pulse" has the same duration as the incident laser pulse. Self-phase modulation can be described by writing the phase <J> of the electromagnetic field: (t, z) = Wot - kz = coo (t - n v z /c) co= 9<5(r,z)/ dt =coo+

d{-nvzlc)ldt

where n is the refractive index, v the frequency and c the speed of light. Usually, the second term is not supposed to be time-dependent and (1)= coo. This is just an approximation since n can also be time dependant: n = n0 + n 2 I(t) The approximation co= coo remains valid when the intensity is not very high because n2 is very small. This is no more true when the intensity is high and has a value rapidly varying in time: co=(Oo - n 2 v z / c

dl(t)/dt

The frequency extends around the central frequency (Oo, both towards smaller values during the rising edge of the pulse and larger values when the intensity goes back to zero. When looking at the above equation, it is easy to understand why this effect requires both short pulse duration and high-intensity peak to be observed and fruitfully used.

151 2.2 Induced absorption The study of intermediate states can be done by monitoring the induced absorption from excited levels towards higher-energy lying states.

;^A# probe •}.

excited states -

pump

. !

l*CN(B2IT>

ft

3895 A

ground state

This induced absorption occurs only when the exited level is populated. The photon energy at which it occurs provide an information on the energy of this level, as long as the upper excited state is well known. A good example is given by the chemical reaction of the above figure (Dantus, 1987 and 1988) concerning the dissociation of a molecule: (I—CN)* -> (I •••• CN)* - • I + CN The pump pulse is used to excite bound ICN molecules to an upper repulsive state. The probe pulse is used to follow the birth of the CN fragments as a function of time. By probing the induced absorption at different wavelengths, one can monitor the passage of the dissociating molecule in a given transient state of separation. This method is quite general as long as a well-known upper excited state exists. This is not useful when the upper state is a continuum of states, like a conduction band in solids or a continuum of ionization for atoms.

152 2.3 Induced transparency Induced transparency occurs: (i)

when all the electrons than could be excited toward an upper level are no more available because they are already excited

(ii)

when the upper level is fully occupied, or both.

In this description stays underlying the fact that an electron is a particle ruled by the Fermi law so that a quantum state cannot be occupied by more than one particle (fermion). This special feature will allow monitoring accurately the electron energy relaxation.

The observation of a "hole burning" in the absorption spectrum reveals occupied excited states. Let consider the case represented in the above figure. The presence of many excited levels gives rise to a broad absorption line resulting from the collection of a large number of narrower lines. When one of the upper levels is occupied by an electron, the magnitude of the corresponding absorption line is reduced. This shows up in part (b) of the figure as a hole in the absorption spectrum. This hole lasts as long as the corresponding excited state is occupied. When an energy relaxation towards lower lying states occurs, the hole in the absorption spectrum move to lower energies. This method is a powerful tool to follow energy transfers. Results in hole-burning type experiments may easily be misinterpreted due to the socalled "coherent artefact". It is worthwhile to discuss these phenomena.

153

(1) creation of a grafting inside the material

Hole-burning experiments require two light beams, an intense one for the pump, a much weaker one for the probe. Very often they are not collinear to allow the discrimination of the weak probe beam with respect to the strong pump beam on the detector. When the two pulses arrive simultaneously on the sample, they interfere and write a grating (refractive or absorption grating). Due to the presence of this grating, auto-diffraction of the beams (1) and (2) will occur. In particular, the beam (2) is diffracted in the directions (3) and (1). If (2) is the pump beam, a very small part of the pump beam is diffracted in the direction of the probe, but since the pump is much more intense than the probe, this corresponds to sizeable number of photons. This results in an increased signal collected by the detector at the pump wavelength and can be considered as a decrease of the sample absorption seen by the probe at the same wavelength, a source of erroneous interpretation. Different solutions have been proposed to characterize and overcome this coherent artefact: (i)

two beams interfere when they have parallel polarizations: if they have orthogonal polarizations, there is no more modulation of the electromagnetic field amplitude on the overlapping spot. However, there is still a spatial modulation of the light field direction. In some materials, this leads to a change from a population grating toward a polarization grating;

(ii)

the two beams interfere when they overlap temporally inside the material, but this effect may exist on a time scale longer than the temporal overlapping. The first beam reaching the sample creates inside the material a polarization with a lifetime usually called the coherence time: P 1 (t) = % E 1 e x p [- t / Uherence]

When the second beam reaches the sample at a delay t after the first one, the total polarization reads:

154

P(t)= P,(t)exp[-T/ ^coherence ] +

JMt)

There is again a writing of a grating with reduced amplitude and autodiffraction can still occurs. This effect, which can lead sometimes to difficult interpretations, can be overcomed when probing from a different ground state, avoiding the spectral coincidence between the hole-burning and the diffracted beam (Camescasse, 1996) 2.4 Stimulated emission Stimulated emission using pump and probe beams is quite similar to the spectroscopic methods described above. Let's again consider the following schematic situation:

Q "

v

m

^ t f i p e r a y transfer

V>

pum

mulated emi ss

§

a.

~u ground state

(2) ^H

1

V

...

t=o

time

The pump beam populates a first excited level (1) from which there is relaxation towards lower-lying states (2, 3,...) through energy transfer. We have previously seen how a change in the absorption coefficient can provides information on the energy path of the relaxing electron. Luminescence from the excited states towards the ground state may be too weak to be used. Stimulated emission can provide an alternative solution, especially when accurate time resolution is needed. A probe beam at a fixed wavelength, for example at fao = E2Eground» will experience gain when the corresponding level (2) is fully occupied. The rise time of the signal reflects the time needed to fill this level and the decay time provides information on its lifetime. The above description corresponds to a stochastic regime. The ultrafast time resolution allows now to monitor coherent evolution and its decay into an incoherent regime. This situation is well examplarified in case of wavepacket excitation. This occurs for example when the equilibrium configuration of a molecule is not the same in the ground state and in the excited state. Just after an electronic excitation, the molecule is not in its stable configuration and will relax to it. This decay may give rise to oscillations around the equilibrium before the molecule reaches its stable position. The occurrence of this phenomenon requires that the excitation is much shorter that the vibration period otherwise

155 everything is blurred. Going back to an electronic description, one can say that a short pump pulse, due to its quite broad spectral width, excites coherently different vibration levels in the excited state. Theses states evolves with their own frequency, giving rise to wavepacket oscillations.

nuclear coordinate

Wave-packet oscillations are very useful in molecular physics, as a way to explore energy surfaces corresponding to nuclear motion in chemical reactions. They are also present in Rydberg atoms or in quantum wells structures. In this latter case, they can generate an emission in the far-infrared or even in the mid-infrared [Bonvalet, 1996].

3. Pump-probe spectroscopy : refractive index The above paragraph has been devoted to the changes in the imaginary part of the dielectric constant. Monitoring the changes in the real part, i.e. the refractive index, can also provide information on the material evolution after excitation and hence on the energy transfers. Here again, the pump-probe method will be used: the intense beam excites the sample and the delayed weak one probes the evolution of the refractive index. There are different methods that are commonly used for this purpose. 3.1 Reflectivity at the Brewster angle At the Brewster angle, for the correct polarization of the light beam, there is a complete extinction of the reflected beam. If the excitation changes the material refractive index, the Brewster angle is modified and a signal appears now in the direction corresponding to the previous Brewster angle (unexcited sample). This method has for advantage to give, in principle, a background-free signal and therefore an increased sensitivity. This is not always true due to possible scattering on a rugged surface.

156 3.2 Thin film : shift of the Fabry-Perot fringes In a thin transparent or almost transparent film, light undergoes many internal reflections. The output beams, either in the forward or the backward directions, result from the interference between the different internal paths, leading to the Fabry-Perot fringes. The interference is constructive when the optical path inside the film for a round trip (2ne) is a multiple of the wavelength. When there is a modification of the refractive index n, the spectral fringes shift. Monitoring this shift lead directly to the change of n. A simultaneous induced absorption may perturb this measurement. In this case, it is important to monitor both the reflected and the transmitted beams, so that one can decide what is due to the absorption (decrease of the sum Itransmitted + Irefiected) and what is due to a change of repartition between transmitted and reflected beams. 3.3 Refractive index gratings Two pump beams reaching simultaneously the sample interfere and write a grating inside the sample. If the pump wavelength is within the transparency spectral range, there is no production of a population having a lifetime usually longer than the laser pulse duration. However, the electromagnetic field still gives rise to an induced polarization inside the material: P=% E In the spectral transparency region, at first order, the material has an instantaneous response time and the polarization follows the temporal profile of the electromagnetic field; % is a tensor but does not have a temporal dependence. The figure below describes a simple situation where two pump beams write an instantaneous refractive grating and a probe beam "reads" the grating. This belongs to the large category of four-wave-mixing experiments (Heichler, 1985). The magnitude of the diffracted signal varies with the amount of the refractive index change resulting from the induced polarization. This is a background-free signal leading to a large sensitivity.

In this experimental scheme, it can happen that there is a possible transfer of energy from the light towards lattice vibrations.

157 3.4 Z-scan This is an other method to measure the change of the refractive index whatever is the origin of this change. In the following we will concentrate on changes induced by a light beam, but the principle can be easily generalyzed to other situations. Here again, this method is based on the Kerr effect, that is a change of n in function of the light intensity: n (z , r) = n0 + n21(z , r) Usually, t\2 is so small that the last term of the above equation is negligible. This no more true when I is large, for example for ultra-short pulses, and especially at the focal point after a converging lens. The laser beam is transversally limited and for simplicity we can assume that it has a gaussian shape. The variations of n will follow this transverse profile leading to an equivalent additional lens. Exactly at the focal point of the glass lens, this additional lens has no influence on the beam propagation, but slightly before or after this point, the beam convergence or divergence is affected. If a pinhole or a ring diaphragm is placed further, the transmission of the beam will be modified. The non-linear refraction can be monitored from the beam profile distortion analysis. This method is called Z-scan because the sample is moved back and forth across the focal point. sam

P,e

diaphragm

detector

*

z •

The method described above is static and cannot be used directly to follow energy transfer. However, if the beam in the above figure is a probe beam (to weak to affect the sample dielectric constant) and if the Kerr effect is now produced by an external pump beam, it is easy to see how the above method can be used to perform time-resolved changes of the refractive index. In this case, the sample is at a fixed position and only the delay between the pump and the probe is varied.

158 3.5 Mach-Zender interferometer

A way to measure the change An of the refractive index is to monitor the change of the optical path Anl in a sample of length 1. This term will affect the phase of the probe travelling electromagnetic wave. This is obtained through the interference with a reference beam in a Mach-Zender type interferometer. pump

V

•*

•

sample

^ A _^

-^ time-integrating **BS2

detector

/

BSf

This is a very sensitive method able to detect changes of n as small as 10"6. The drawback is that the optical paths have to be stable with the same precision.

4. Time-resolved luminescence Luminescence provides direct information on the population in an excited state that relaxes towards a lower-energy state through the emission of a photon. The signals may be weak, but this is usually a back-ground-free method and progresses in the detector performance allow almost reaching the level of one photon. The limitation of this experimental method is that it provides information only on the population without insight on phase properties. In order to learn about the dynamics in the energy transfer, time-resolved luminescence has to be performed.

4.1 Synchronized streak camera The principle is to convert the emitted photons into electrons on a photocathode like in photomultiplier tubes. Electrons are accelerated through a voltage and reach a phosphor screen. During their flight, they are deflected by a perpendicular voltage linearly increasing in time (voltage ramp) so that the last arriving photons experience the largest voltage and then the largest deflection. On the phosphor screen the temporal distribution of the initial photons to be analyzed appears as a spatial distribution of the collected electrons.

159 The major difficulty is to start the deflecting voltage ramp at the right time, especially if repetition and accumulation are necessary due to the weakness of the signal. Jitter problems become important and are often the limitation on the time resolution of streak cameras. For example, a 200 fs time resolution (with poor can be obtained on a one shot basis while it is few picoseconds with the same camera after accumulation. A solution has been

photoconductor

recently proposed to overcome this difficulty. The voltage ramp is synchronized with the exciting laser pulse through the help of a photoconductor that acts as switch when it is illuminated by the laser. This will allow to go routinely below the picosecond resolution. 4.2 Optical-Kerr effect gate This method, as well as the two following ones, relies on the use of non-linear effects (Duguay, 1976). In the present case, it originates from the induced anisotropic change of the refractive index n in a material in presence of a polarized light pulse. For example, n is more affected in the direction of the polarization of the intense laser pulse (Kerr effect).

160 The isotropic Kerr medium is placed between two crossed polarizors, so that in absence of external light beam, no signal can pass trough the system. When a strong beam with adequate polarization illuminates the medium, the induced anisotropy turns the signal polarization and allows a part to pass through the second polarizor towards the detector. This process occurs only during the presence of the strong light beam, leading to a temporal gating. The medium can be glass, liquids or crystals, with a compromise between the maximum efficiency and the time response of the medium. The sensitivity of this method is not important and relies on the efficiency of the cross polarizors to block the signal in absence of excitation. However, this is the preferred technique when the signal to analyze lies in the blue or in the UV (Hulin, 1985). Since there is no spectral selection, having also simultaneously all the signal wavelengths is quite convenient. 4.3 Parametric amplification sampling spectroscopy (PASS)

A weak beam, like a luminescence signal, can be amplified in a non-linear crystal with large second-order susceptibility by a strong beam trough a parametric interaction. This occurs only during the coincidence of the two beams inside the crystal. This fact can be used for a temporal discrimination (Hulin, 1986; Ledoux, 1987). If the signal under analyze is amplified at a delay x by a factor 104 during a time At (pulse duration of the strong beam), the collected signal on a time-integrating detector is: Itomi = J Kt) dt + 104At I(T) When the delay T is varied, the first term of the right part of the above equation remains constant while the second one sketches the time dependence of the signal I . Although this method is not background free, it is useful as long as the first term remains small with respect to the second one because the signal is amplified in the process. The fact that, due to the phase-matching conditions, there is both a spectral selection and an angular selection can be regarded as an inconvenient or as an advantage. This has been used recently to discriminate the first ballistic photons travelling trough a diffusive medium with respect to the scattered photons arriving later and with different angles.

161

4.4 Time-resolved luminescence by up-conversion This is the most widely used method as long as the signal wavelength is not too small and a good time resolution is needed. The up-conversion method is again an optical sampling technique that is based on the %(2) non-linear optical properties of crystals like LJIO3, LBO or BBO for instance (Shah, 1988). When in such a crystal the luminescence signal and a reference pulse are present simultaneously, frequency mixing occurs, which results in the generation of an up-converted signal.

minescence, (Oj reference, C0„ _ t v.,,H*^-w^*«'ri';')i:>

non-linear crystal

upconverted signal ( 0 , = CO, +
The limitations of this method are the moderate sensitivity linked to the efficiency of the upconversion process and the lack of adequate crystals for signals in the UV spectral range. The advantage is that the collected signal at frequency CO2 is background-free and that the time resolution is in principle limited only by the reference pulse duration. In reality, one has also to take into account the group velocity dispersion inside the non-linear crystal. Due to phasematching requirement, only one wavelength of the signal can be monitored for a fixed angle of the crystal, the tunability being obtained by varying the crystal angle.

5. Fourier transform spectroscopy To have a full insight of what is happening inside a material and hence to learn the maximum on the energy transfer, it is useful to monitor the induced polarization inside the sample. P(t) = £ J R(t - f) E(t') dt' °c e 0 £3 X (w) E(oo) exp(-icot) dm

The radiated field is the time-derivative of the polarization : ER(t)«= 3 P / 3 t

162 From the complete knowledge of this field, one can deduce the absorption coefficient, the refractive index and the dephasing time. Moreover, the advantage to know E(t) in amplitude and phase is that E((o) can be easily deduced through a Fourier transform of E(t) and viceversa. Usually, experimental methods allow only to record the amplitude since a detector is sensible to the light intensity I°= | E | 2 where the phase information is lost. Interferometric methods are needed to obtain both amplitude and phase (Lepetit, 1995; Fittinghoff,1996).

sample

L

z

W2L ^ spectrometer

WI It delay stage

detector

&RP2

lime-inlegialing detector

Let us briefly exemplify the principle of Fourier-transform spectroscopy by describing the above experience. The reference pulse Eo(t) and the unknown signal E(t) propagate in the same direction (along the same beam) after the last half-reflecting plate (HRP2). The signal E(t) is the experimental signature of the phenomena under study inside the sample; it could for example be linked to a pump-probe experiment where the pump (not shown on the figure) changes the transmission of the probe beam. A delay x can be introduced between the signal and the reference trough moveable mirrors on the reference path. The time-dependence of the total electromagnetic field I(t,t) is usually too rapid to be time-resolved. On a time-integrating detector this leads to: I ( T ) = I |Eo(t-x) + E ( t ) | 2 d t I ( T ) = J IEo(t) 12dt +J | E ( t ) | 2 d t + 2RejE* 0 (t-T)E(t)dt

163 The last term is a correlation product and contains more information than the two other terms: it depends on the delay x between the reference pulse and the phase information is not lost. Therefore, by performing a Fourier transform of this last term, the direct product E*0(G)) E(co) can be obtained. This method relies on the variation of the delay x. Since the reference pulse is supposed to be well-known, the signal E(co) can then be deduced in amplitude and phase, allowing to go back to the full determination of E(t) (Chemla, 1994; de Boeij, 1995). By replacing the time-integrating detector by a spectrometer and a time-integrating detector, one can read directly on this detector : I(co) = |Eo(co) + E(Q))|2 I(co) = I Eo(co) 12 + I E(co) 1 2 + 2 Re E*0(co) E(co) The last terms appears as fringes (Froehly, 1973) for which the spacing is inversely proportional to the delay x. It provides a measurement of the complex electric field in the frequency domain. A simultaneous acquisition of the whole interferogram is possible for only one delay. The delay x cannot be too large, otherwise the ensemble spectrometer + detector could not have enough spectral resolution to resolve the fringes. On the other side, x can be longer than the pulse duration: even if there is no temporal overlap of the two pulses, a spectral interference is still possible inside the spectrometer. In general, interferometric measurements are powerful methods. They require a wellcharacterized reference pulse, but this is not a major difficulty. They can be used also in 4wave-mixing experiments where they allow to record directly the time-dependence of the diffracted signal. References Bonvalet A., Nagle J., Berger V., Migus A., Martin J.L., Joffre M. : Phys. Rev. Lett. 76, 4392 (1996). F. X. Camescasse, A. Alexandrou and D. Hulin, Phys. Rev. Lett. 77, 5429 (1996) D.S. Chemla, J.-Y. Bigot, M.-A. Mycek, S. Weiss and W. Schafer Phys. Rev. B50, 8439 (1994) M. Dantus, M.J. Rosker and A.H. Zewail, J. Phys. Chem. 89, 6128 (1988) M. Dantus, J. Rosker, and A.H. Zewail, J. Chem. Phys. 87, 2395 (1987) W.P. de Boeij, M.S. Pshenichnikov and D.A. Wiersma, Chem. Phys. Lett. 238, 1 (1995) M.A. Duguay, Prog. Opt 14, 161, 1976

164 D.N. Fittinghoff, J.L. Bowie, J.N. Sweetser, R.T. Jennings, M.A. Krumbugel, K.W. DeLong, R. Trebino and I.A. Walmsley, Opt. Lett. 21, 884 (1996) C. Froehly, A. Lacourt and J.C. Vienot, J.Opt.(Paris) 4,183 (1973) H.J. Heichler, P. GUnter, D.W. Pohl, Laser Induced Dynamic Gratings Springer, Berlin, Heidelberg 1985 D. Hulin, A. Mysyrowicz, A. Migus and A. Antonetti, J. Lum.30, 290 (1985) D. Hulin, A. Migus, A. antonetti, I. Ledoux, J. Badan, J.L. Oudar and J. Zyss, Appl. Phys. Lett. 49, 76 (1986) I. Ledoux , J. Badan, J. Zyss, A. Migus, D. Hulin, J. etchepare, G. Grillon and A. Antonetti, J. Opt. Soc. Am. B 4, 987 (1987) L. Lepetit, G. Cheriaux and M. Joffre, JOSA B 12, 2467 (1995) M. Nisoli, S. DE Silvestri, V. Magni, O. Svelto, R. Danielus, A. Piskarkas, G. Valiulis and A. Varanavicius, Opt. Lett. 19, 1973 (1994) C. Ruliere (ed) Femtosecond laser pulses: principles and experiments, Springer Verlag Berlin Heidelberg New York, (1988) F. Salin, J. Watson, J.F. Cormier, P. Georges and A. Brun, Ultrafast Phenomena VIII, Springer Ser. Chem. Phys., Vol. 55 (Springer, Berlin, Heidelberg 1992) p. 306 J.K. Ranka and A.L. Gaeta, Opt. Lett. 23, 534 (1998) F. Seifert, V. Petrov and F. Noack, Opt. Lett. 19, 837 (1994). J. Shah, IEEE J. Quant. Electr.24, 276 (1988) V. V. Yakovlev, B. Kohler and K. R. Wilson, Opt. Lett.19, 2000 (1994)

L I G H T P R O P A G A T I O N IN M A T T E R : FROM POLARITONS TO PHOTONIC CRYSTALS

C. Klingshim Institut fur Angewandte Physik Universitat Karlsruhe Kaiserstr. 12 D-76128 Karlsruhe, Germany

Abstract In a first part of this contribution we calculate the optical properties of matter using the classical approach of the model Lorentz-oscillators, derive then the dispersion relation of light in matter and introduce the concept of polaritons when going to the quantum mechanical description. In the second chapter the polariton concept is applied to three-dimensional ordered and amorphous matter like crystals, gases or alloys. The resonances by which we replace the model Lorentz oscillator are exctions, optical phonons, plasmons as well as transitions in the atomic electron system of Na vapour or in 57Fe nuclei. The third chapter is devoted to systems of reduced dimensionality like exciton polaritons in semiconductor quantum wells, surface polaritons or cavity polaritons. The dielectric mirrors used to form the cavity allow in the fourth chapter an easy introduction of the concept of photonic (or more precisely polaritonic) crystals and ~ bandstructures. This concept will be elucidated with several examples. The contribution will close with a short conclusion and outlook.

1.

Introduction

In this section we introduce the basic ideas of the polariton concept using a mechanical model.

/./.

Some General Aspects

The aspect of energy transfer treated in this contribution to the school concerns the propagation of light through matter. Actually this specific topic and the general one have been treated already in a preceeding school of this series ' ' 2 . A comparison of ' ' 2 with the present book allows to see, which topics did not change very much since then

165

166 and can be considered as „mature" like the concept of polaritons in bulk semiconductors or insulators and which ones are new like cavity polaritons or photonic crystals. The concept of polaritons has been introduced at the end of the fifties and research on photonic crystals at the end of the eighties. Some pionieering work for both topics are and , respectively. However, generally there are for every new field of research some precursors and we shall see examples for this statement here, too. The reserach on polaritons is documented by a large number of publications, which increased over the last three decades from about 650 per decade to now 1200 per decade. The field of photonic materials resulted in approximately 500 publications so far.

1.2. The Model ofLorentz

Oscillators

The optical properties of matter like the spectra of the complex dielectric function e(co) = Si(co) + i£2(co), of the complex index of refraction fi(co) = n(co) + iK(co) or of the reflectivity of a single surface at normal incidence Ri(co) can be easily derived with the Lorentz- or Lorentz-Drude model and Maxwell's equations. The basic idea is, that coupling of the electromagnetic radiation field to matter occurs in simpliest approximation by electrical dipole interaction, where the two charges forming the dipole are attatched to each other by a harmonic force. Therefore we consider a three-dimensional array of harmonic model oscillators consisting of a mass m bound by a spring with spring constant P to its place. In Fig. 1 we show for simplicity an one dimensional array only, but it does not need much phantasy to extend this model to three dimensions. We have used in Fig. 1 a spatially periodic arrangement of the oscillators. This is however not a necessary prerequisite. It is only necessary to have many (almost) identical oscillators per coherence volume of light. Since X3 is a lower limit of this coherence volume of light, this condition is always fulfilled, if N • X3 »

1

(1)

where X is the wavelength of the light and N the density of oscillators. Under this condition the light does not resolve the individual oscillators but an effective medium theory is appicable . The slight disorder, which arises for a non-crystallin system under the condition of (1) results in weak Rayleigh-scattering. The investigation of this phenomenon itself contains a lot of information on the optical properties. See e.g. 7. If we elongate all oscillators in phase (Fig. lb) or in antiphase (Fig. lc) they will oscillate with the same frequency coo = (P/rn)"2. In the first case the amount of the wavevector k = 2%X'1 is zero, in the second one it is n/a. The dispersion relation m(k) is a horizontal line. See Fig. 2a. Consequently the group velocity vg vanishes which describes the velocity with which the „center of mass" of a wave paket propagates and which is given by

g

5k

(2a)

167

while the phase velocity vPh, given by _0>(k)

(2b)

has a finite value. For this topic see also

Fig. 1. The model of the Lorentz oscillator, unexcited (a) a collective excitation with X => oo (b) or with the minimal physically meaningful wavelength X^a (c); the excitation of a wave paket (d) and an improved version used to describe spatial dispersion (e). According to 3 ' 5 .

168

* E=Uco

* E=1ico

0>n

2 71

7C

a

•W:

Fig. 2: The dispersion relation for the ensemble of uncoupled oscillators of Fig. la - d (a) and for the coupled ones of Fig. le (b). According to 3 ' 5 .

The vanishing of the group velocity is immediately clear for the wavepaket sketched in Fig. Id. If a more realistic model is required, which includes a k-dependence of the eigenfrequency, the so-called spatial dispersion (flo(k), it can be modeled by a coupling between the Lorentz oscillators. See Figs, le and 2b. The coupling between the oscillators results in finite vg and a non-local response of the medium 9. In order to couple the oscillators to the electric field of a light wave, we give every mass a small electric charge e and place for neutrality reasons the opposite charge in the equilibrium position of the oscillator. The elongation of the oscillators x(t) is then connected with a dipole moment p(t) given by P(t) = ex(t)

(3)

In principle we could also introduced a magnetic dipole moment, but since the magnetic coupling is usually much weaker than the electric one, we neglect the magnetic dipole-, electric and magnetic quadropole and higher transitions in this article, if not explicitely stated otherwise. We assume now that an electromagnetic wave is falling on the (threedimensional) ensemble of model oscillators, which propagates in z-direction and has an electric field polarized in x-direction i.e. E = E0 exp [i(kzz - cot)], E0 = ( E0, 0, 0)

(4)

169

Then the equation of motion for each oscillator is given by mx(t) + 2ymx(t) + (3x(t) = eE0 exp (- icot)

(5)

We introduce here a phenomenological damping term y, which is justified for quasistationary conditions, on which we concentrate here. For the regime in which this approximation is valid see e.g. two contributions to this book ' . Eq. (5) is nothing but the well-known equation for a damped, driven, harmonic oscillator. The solution is for the case of weak damping the sum of a transient feature (actually the general solution of the homogeneous differential equation) plus the driven oscillation X(t) = XtransientO) + X P exp(-ifflt)

(6)

Since we are not interested in the transient features here (for this topic see e.g. the contributions 8 " 12 ) we concentrate on the second term on the r.h.s. of (6) and obtain the well know resonance term for xP(co). We define the polarizability a of every oscillator by d =^ E„

(7)

and obtain e2

I m „^ « =— —— (») co0-co -lmy The polarization of the medium P(co) i.e. the dipole moment per unit volume is then given by P = Nexp

(9)

A comparison of the material equation in its general form D = e0E + P

(10a)

where D is the electric displacement (see e.g. 5) and its linear approximation used in the following if not stated otherwise D = s0s(co)E yields with Eqs. (8, 9) for the dielectric function e(co)

(10b)

170

, . , Ne 2 /me„ e(©) = l + — j—^(D0-a) -icoy

„_ (11)

The numerator of the r.h.s. of Eq. (11) is usually called oscillator strength f. If we have several eigenfrequencies Oo; in the system, their contributions simply add in linear optics, resulting in e(co) = l + S —

^—

i C0oi-G)

(12)

-1C0Y,

For very close lying resonances, the sum in Eq. (12) may be replaced by an integral. We note that every term in the sum of Eq. (12) tends to zero for co » co0i and to a constant value f; Ia20{ for co «co 0 1 . If we consider the simple case of a single resonance, spectrally well separated from the others we can rewrite Eq. (12) as e(co)=e b [l+ 2 f / f b , j V. co0-co -icoyy

(13)

where the so-called background dielectric constant 8b includes the „1" of Eq. (12) and the constant contributions of all higher resonances, while the contributions of all lower laying resonances are neglected, following the argument given above. In principle several corrections have to be included like local fields, which alter f and coo, but the overall shape of Eq. (11) to (13) is not changed 5. Optical anisotropy can be included by a directional dependence of the parameters 8b, f, co0 and y which leads to a tensor character of s(co)5. Spatial dispersion results in a dependence of s on two independent variables co and k. We do not go into details of these topics here but refer the reader for details to 5 and references given therein. In Fig. 3a and b we show the real and imaginary parts of e(co) according to Eq. (13) for vanishing and weak damping y. For co => 0 si(co) starts with the static dielectric constant es = eb + {/a>I, goes through the resonance and approaches 8b for high frequencies from below. For the highest resonance in a given system 8b is 1. The imaginary part s2 (co) has a Lorentzian shape and develops towards a 8-function centered at coo for vanishing damping. The resonancefrequencycoo is also called transverse eigenfrequency, while a longitudinal wave may exist at the frequency co = COL for which e(co = COL) vanishes. This becomes clear when inserting a plane wave ansatz E = E0 exp [i(kr - cot)] in the Maxwell equation divD = p = 0

(14)

171 for vanishing free space charge density p. With Eq. (10b) and executing the Nabla operator we obtain 80ikE(co)E = 0

(15)

Eq. (15) is fulfilled for k 1 E. This is the usual way to show that electromagnetic radiation in vacuum is a transverse wave with (16)

E l k l B l E

where B is the magnetic flux density. In vacuum this is the only solution since e = 1. In matter there is for co = coL also the solution S((D) = 0. This means, that at this frequency a longitudinal wave may exist. A closer inspection of Maxwell's equations shows, that this is a wave with longitudinal E and P fields which are antiparallel to each other according to Eq. (10a) and vanishing magnetic fields.

£i

(b)

Y=0 Y = 0.2 A LT

ji

1 1 1 1

Y = 0.2A LT

1 1 I

/

\ \\\\ <»0

<%.

Fig. 3: Real (a) and imaginary part (b) of e(co) for weak and vanishing damping. From

Fig. 4 gives the real and imaginary parts of the refractive index n(ffl) = n(co) + k(co). It is connected with the dielectric function e(co) simply by fl(co) = (BCCO))1'

(17)

172

On the other hand it is related to the real and imaginary parts of the wavevector by Re{k} = —n(co) c

(18a)

Im{k} = — K ( O )

(18b)

and

consequently n(co) describes the oscillatory or wave-like propagation, while K(CO) gives the decrease of the amplitude in the direction of propagation. It is connected with the absorption (or more precisely extinction coefficient) for the intensity of the light a(co) by a(ro)=2 —K(CO)

(19)

since the intensity (or precisely energy flux density) of the light is proportional to its amplitude squared.

n = Re(n)

K K == lm(n)

(a) /

Y=0 T = 0-2A LT

T=0 Y=0.2 ALT

v? ^ (D 0

(DL

0)

Fig. 4: Real (a) and imaginary part (b) of fi(co). From 5 .

It should be noted, that n(co) and consequently Re{k} vanish for vanishing damping between (Oo and coL. Consequently there exists no propagating mode in this frequency interval and an incident light beam is totally reflected already for normal incidence in agreement with Snellius' law of refraction.

173

This is shown in Fig. 5, where we plot the reflctivity for the light intensity of a single surface and normal incidence R±(a>). This quantity is given Fresnel's formulare as (n(g>)-l)2+K2(cp) R±(o>) =

(20)

(n(co) + l) 2 +K 2 (o)

The stop-band or reststrahlbande between co0 and COL is clearly seen. The minimum of Ri(co) occurs where n(co) = 1.

>-A LT

Fig. 5: Reflection spectrum in the vicinity of a resonance for normal incidence. From 5

Increasing damping y tends to wash out all spectral features in e(co), n(a>) and R(co). It should be mentioned that a finite oscillator strength f is for small damping necessarily connected with a finite splitting between coo and coL according to (O2L-(O20=f/Eb

(21)

and the so-called Lyddane-Sachs-Teller relation (22)

174

For a discussion of Eq. (22) see also 6. A frequency dependent, complex electrical conductivity o(co) can be included by extending the model of Lorentz oscillators to the Drude-Lorentz model. Formally this can be done by considering a term like Eq. (5) with a vanishing restoring force Px(t). Such a term then corresponds to the relaxation approximation to desribe electric conductivity. In Eq. (11) to (13) this results in a term with vanishing eo0 and an oscillator strength given by the plasma frequency <

= Ne 2 (e 0 mr'

(23)

eventually modified by a background dielectric constant eb 5 ' 8 ' 9 . N is in this case the density offreecarriers. Inserting eoo = 0 in Eq. (21) shows, that the plasma frequency is the longitudinal eigenfrequency and that the region of high reflectivity in Fig. 5 extends from zero to COPL. There are other arguments to explain that the transversefrequencyof a plasma oscillation is zero while the longitudinal one corresponds to copL. One is, that the shear stiffness of every gas, including a gas of free carriers, is zero and consequently thefrequencya>T, while thefinitecompressibility enters in the longitudinal frequency. Another one, which is illustrated in Fig. 6 goes as follows. In Fig. 6a we show the resulting space charge pattern of a longitudinal plasmon wave. The restoring force eE || k leads to the finite longitudinal eigenfrequency and the antiparallelity of P and E mentioned above is obvious. In Fig. 6b we show the charge and field distributions which one should have for a finite transverse eigenfrequency. Acutally this pattern does not develop for bulk material (i.e. thickness » wavelength) and therefore we crossed it out. The acutal field distribution which develops is given in Fig. 6c i.e. the bulk of the sample is field free, explaining the vanishing transverse bulk co0 = 0 eigenfrequency. Instead surface plasmon modes appear. These modes will be treated in subsection 3.2.. 1.3. The Dispersion ofLight in Matter and the Polariton Concept To obtain the dispersion relation of light in matter there are different ways. One is to deducefromMaxwell's equations the so-called polariton equation 8(co)=^ CO

Together with Eq. (13) this gives an implicite representation of co(k) according to

(24)

175 Other ways follow Eq. (18a, b) together with Fig., 4 or the concept of avoided crossing of two coupled oscillators which is discussed in detail e.g. in 5 ' 13, w . The result is in all cases the same and is shown in Fig. 7, for simplicity for vanishing damping. The case of finite damping is treated e.g. in x 5 ' 16 and the references given therein.

++ + P + . *~~+ +

+ —± + £

=

- " - _--jT ++ + ~~- - ~ - + + * + ,-> •_

-^—r^ - - - _ E + + +

sample

a

Fig. 6: The electric field in a longitudinal plasma wave (a) the situation one should have for a finite transverse plasma frequency but which does not exist (b) and the actual field distribution in this situation (c).

In vacuum the dispersion of light is linear with equal group- and phase velocities vg = vPh = c = (e0uo)"1/2

(26)

The quantization of light in vacuum does not change its dispersion relation. We know, however, e.g. from the external photo-electric effect, that light in vacuum can exchange energy with other systems only by integer multiples of hm as expected from the quantization of a harmonic oscillator. These energy units or -quanta are called photons. Every

176

photon carries a momentum h k. Cosequently we define the photons as the quanta of the electromagnetic radiation in vacuum. The dispersion of light in matter looks more complicated. This is essentially due to the fact, that light in matter is an electromagnetic wave accompanied by a polarization wave, whenever e(co) or n(a>) are different from one. From the internal photoeffect we know, that this mixed state is also quantized into integer multiples of energy to accompanied by a quasi-momentum h k. For a discussion of the similarities and differences of momentum and quasi-, crystal- or pseudomementum of the quanta of elementary excitations in solids see e.g.8'15. We call the quanta of the mixed state of electro-magnetic and polarization wave „polaritons". The dispersion relation of the polariton in Fig. 7 starts on the lower polariton branch (LPB) with photon-like behaviour. The term photon-like is used, because the dispersion relation is linear like for photons but the slope is different and given by ftc(es)"2, where the factor (es ) " 2 is resulting from the accompaning polarization wave.

photon UPB

tico

energy laco

AM

1

long. lacor taco - -

LPB

wave vector k Fig. 7: The dispersion relation of light in vacuum (---) and in matter (—) for vanishing damping. From

177 Then the dispersion relation bends over to the behaviour of the resoncance. Depending on what this resonance is, one speaks e.g. of the phonon-like, exction-like, etc. part of the LPB. There is a finite longitudinal transverse splitting ALT (see Eq. (21)), eventually a longitudinal eigenmode and an upper polariton branch starting at ha L and bending over to a photon-like behaviour again, however with a steeper slope c(£b)"1/2 (see Eq. (22)). It should be mentioned that in the quantum-mechanical description the oscillator strength f contains the transition matrix element squared. The quantum-mechanical description of the polaritons proceeds via the diagonalization of the Hamiltonian containing the number operators for the photon-field B k (co)B k (co), of the Lorenztoscillator field Ck(co)Ck(co) and their interaction according to H = z M k ) B 1 > ) B k ( f f l ) + lE(k)C + k (a>)C 1 >) + k

k

Zgk,kC+k,((o)Bk(co)+c.c.

(27)

kk'

The gkk are the coupling coefficients. More details on this topic can be found e.g. in 16. The „user-friendly" aspect is, that the dispersion relation of light in matter remains the same inedpendent if we use the classical description in terms of an ensemble of Lorentz oscillators or the quantum-mechanical one in terms of polaritons. The difference is „only" the prefactor h to both co and k, more prescisely to Re{k} which describes the propagation. If not stated otherwise, we mean with k in the following the real part Actually one uses often even a mixture i.e. E = ha as a function of k. Sometimes one can hear the statement, that the polariton concept works only for ordered or crystalline matter. Indeed this concept has been introduced e.g. in 3 for a periodic arrangement of oscillators. But, as we will show in the following section 2, it can be easily extended to non-crystalline matter like amorphous solids, alloys, liquids and gases, as long as inequality Eq. (1) is fulfilled. For the opposite regime one faces rather a scattering problem. The only difference is, that in ordered matter the quasi momentum h k is conserved only modulo integer multiples of unit vectors in reciprocal or k space G = hibi + h 2 b 2 + h 3 b 3

(28a)

where the bj are connected with the primitive translation vectors aj, in real space by b,=27r

'— V(ajxak)

andc.p.

(28b)

which is one of the reasons why h k is called quasi-momentum. Eq. (28a) allows to restrict all dispersion relations to the first Brillouin zone. See e.g. every textbook on solid state physics like 17. In disordered matter, k is a good quantum number, as long as Eq. (1)

178

holds, i.e. an effective medium description is adequate. For increasing k the concept of k gets more and more vague when one leaves the effective medium regime. However, this is not a serious problem in our case. In the IR and VIS range X is much longer than the interparticle distances in solids and liquids and also for not too diluted gases. In the X- and y-ray region the refractive tends to the value in vaccum n = 1 indicating the transition from polaritons to photons, except for the spectral vicinity of well defined resonances as shown in section 2. This above statement is nicely illustrated by the following facts. In the IR, VIS and UV the refraction of light is equally well described by Snellius's law, which is based on the conservation of the momentum parallel to the interface h ky according to ENoether's theorem 5, for crystalline quartz SiC>2 and fused, amorphous glass or silica. However, in the X-ray regime where Eq. (1) does not longer hold the crystal gives still sharp reflection beams according to Ewald's construction i.e. kout

=

ki„ + Or,

| kout | =

| kj n |

(29)

in elastic scattering, while the pattern is blurred for the amorphous material, especially for higher diffraction orders, indicating that the concept of the conservation of the quasimomentum h k in matter starts to break down. 2.

Application of the Polariton Concept to Three-Dimensional Matter In this section we apply the concept of model oscillators to three-dimensional or bulk materials, starting with crystals and proceeding to disordered matter. The Lorentz oscillator will be replaced by the quanta of the elementary excitations in solids like phonons, excitons, plasmons or magnons, by (two-level) transitions in atoms or by transitions in the nuclei, which couple to the light field. The justification to replace the harmonic Lorentz oscillators in the above mentioned way follows from the close relation between the excited states of a harmonic oscillator and the occupation numbers of a Boson field as detailed e.g. in 8 ' 9 . There are, however, also some deviations between a two-level system and the harmonic Lorentz oscillator. An ideal harmonic oscillator can be driven into higher and higher quantum states, which still remain equally spaced in energy, while a two level system will be saturated by the compensation of absorption and stimulated emission, when ground and excited states are equally populated. If we restrict ourselves to the low density regime, also an ensemble of two or more level systems will be perfectly modeled. Presently there is a trend to describe the optical properties of matter by the optical - and the semiconductor Bloch equations. See e.g. 8"10,16. This theory is necessary for the understanding or modeling of transient features especially in the regime of (ultra-)short timeresolved spectroscopy as shown e.g. in 8" or 18. This theory is also capable to reproduce the linear optical properties discussed in this contribution and the elementary excitations in solids like the excitons. However, it is a rather complicated theory which allows in many cases numerical solutions only. Since it is in addition also not a „complete" theory, e.g. since it treats generally the electric- and

179

polarization fields on a classical (Maxwellian) basis and the electronic excitations on a quantum mechanical level, we recommend to use the didactically simple and intuitive model developped above, where it is applicable, and this is a wide range of linear and partly even of non-linear optics, and to restrict the use of the much more complex semiconductor Bloch equations to the situations where there are really necessary. 2.1. Polaritons in Crystalline Solids In this section we present polaritons in crystalline solids which result from the coupling and mixing of the electromagnetic radiation with the quanta of the elementary excitations like excitons, phonons, plasmons or magnons. We start with exciton-polaritons and treat them in same detail to introduce various techniques to measure the polariton dispersion, which will be used later on. Excitons are the quanta of the elementary excitation in the electronic system of semiconductors and insulators. In simple approximation they can be considered as a hydrogen- or positronium atom like system of a negatively charged crystal electron in the conduction band and a positively charged hole in the valence band, bound together by Coulomb interaction. The excitonic Ryelberg energy is modified by the reduced mass of electron and hole and by the dielectric function compard to the value of Hydrogen. Instead of 13.6eV these corrections result in values between 5 and 200meV for typical semiconductors. The onset of the ionization continum reached for main quantum numbers nB => oo coincides with the energy of the band gap. We present here data only for direct gap semiconductors with conduction and valence band extrema at k = 0, the so-called Ypoint. As an example we show in Fig. 8 the bandstructure, the reflection and transmission spectra and the polariton dispersion of excitons in CdS. There are three exciton series with holes in the A, B and C valence bands (Fig. 8a). The first two series overlap energetically and are shown in Fig. 8b and c in reflection and transmission. There is some dichroism and birefringence since the A exciton couples strongly to the radiation field only for light polarized with E 1 c where c is the crystallographic axis of the hexagonal CdS crystals. The dispersion of the m = 1 AT5 exciton polariton in Fig. 8d corresponds to the dispersion relation of Fig. 7 with the inclusion of a k-dependence of the eigenfrequency co0(k) which represents just the kinetic energy of the center of mass motion of excitons. The ne = 1 BTs resoncance shows a complication, namely the appearance of an intermediate polariton branch, which results from a mixing of the dipole allowed singlet T5 states and the dipole forbidden Y2 triplet states by the k-linear term shown schematically for the T7 valences bands in Fig. 8a. This intermediate polariton branch is the reason for the small dip in the reflection spectrum of the B exciton indicated in Fig. 8b by an arrow. The dip coincides with the energy at which the refractive index of the intermediate polariton branch has the value one. The experimental points in Fig. 8d result from reflection and hyper-Raman spectroscopy. For details see 22.

180 band structure

(b) —1—'

1 1 ' 1

l r i

i — i — r * ~ r "1

IT

1 1 1 1 1

BnB=l

AnB= 1

CdS B n B = 2;n B = 3 A n B == 2;n B = 3

"it

^A

,

i

,

,

,

flic*

A/

,

1

2.55

2.56

i

Elc

i

J—i-_l

L

2.57

\ i

,

,

,

,

2.58

X

2.59

(ev;

10-'

.p A

10"2

10"J

104 2.54

2.55

156

2.57

2J8

photon energy (eV)

159

2.60

5

10

15

20

k (10 5 cm"1)

Fig. 8: The bandstructure of CdS in the vicinity of the T-point (a) the reflection (b) and transmission spectra (c) in the vicinity of the A- and B-exciton resonances showing various members of the hydrogen like series up to n B = 4 and the dispersion of the n B = 1 A and B exciton polaritons (d). From 1 9 , 2 0

181

Fig. 9 shows the results of a four wave mixing experiment (FWM) in reflection from21. In FWM one excites a polariton by afirstpulse (ki, ha,) and probes its decay by the interference with a second delayed pulse (k2, ha 2 ). Generally one observes the diffracted order in the direction 2k2 - ki in transmission and reflection. For details of this We choose here the reflection geometry since transmission expetechnique see e.g. riments in thicker samples involve propagation and absorption effects, which makes their interpretation rather complicated. See e.g. 21 and examples below. The lOOfs pulses used in the experiment are spectrally so broad that they excite simultaneously the nB = 1 Aand B-exciton resonances. Consequently a beating is observed with a period corresponding to the splitting between the nB = 1 A- and B-excitons of about 15meV. In additon there is a slow modulation with a period of about 4.4ps corresponding to an energy splitting of about lmeV. This is just the splitting between the lower and the intermediate polariton branches of the ne = 1 B-exciton resonance in the region where both dispersion curves are rather flat and almost parallel, i.e. where the (combined) density of states is high. 1 LI i r r r n i rp"' i i i i i i 11 i f i i i I T ri [ i i i"i i i i i i pTT'rri r i i | i n n n i'T| I I I n r r r r

• 0.1

^ c p

0.01

4K In

42 c ^ •e ra t——i

CdS

: . • •

, . 0.001

• I . . I l l }I . I l l 1

0.0001 -

3

-

I

2

1 1 • • I

-

1

1I

0

1

'

t

» • • ' • ' •

2

'

3

'

I »

11

'

4

Timedelay x [ps] Fig. 9: The FWM signal in reflection in the spectral region of nB = 1 A and B excitons in CdS. From 21.

The overall decay of the signal results from damping. The value deduced here (< lmeV) coincides with data deduced from a quantitative fit of the reflection spectra 5'22.

182 A first experimental technique to measure directly the dispersion relation of polaritons is discussed in Fig. 10. If one measures the refraction of a light beam as a function of Aco at a prism with know angle a (see inset) it is possible to deduce the spectrum of the refractive index n(Aco ). Data are shown in Fig. 10 for the first three dipole-allowed exction resonances in CdS. It should be noted, that in the lowest resonance n(ha>) could be followed up to values of 25! The dispersion relation is closely related to n(fao ) via Eq. (18). Especially the lowest resonance reproduces thus the exciton polariton resonance including the k dependence of a>o, if one just interchanges the x and y axes.

400

n 100 25

2.55

2.56

2.57

2.58

irco, (eV) Fig. 10: The spectrum of the refractive index of exciton polariton resonances in CdS deduced from the refraction at a thin prism. From 23.

Now we present three examples of propagation effects in the vicinity of the exciton polariton resonances.

183 In CuCl the ALT of the lowest exction resonance is about 5meV. With a Fourierlimited pulse of about lps duration and lmeV width it is therefore possible to scan the group velocity over the resonance. Results shown in Fig. 11. The right hand side gives the polariton dispersion, the left the group velocity deduced from the time of flight through a sample of a few urn thickness.

3.22 II i i i i i i i i i \/

i i i mn|

1 i i IIIIII

1 i I MINI—I

I I IIIII

3.21 -

3.19 ••

(a)3.18

i

i

i

•

i

i

•

•

i

io io-k (x 106 cm"1)

Fig. 11: The exciton polariton dispersion in CuCl (a) and the measured (•) and calculated (—) group velocity (b). From 24.

The calculated spectrum of the group velocity is just the slope of the polariton dispersion relation Eq. (2a). There is excellent agreement between experiment and theory and it should be noted that vg can be as low as 5 • 10"5c in the resonance region. The semiconductor Cu 2 0 has also a direct band gap at k = 0. Since both bands have the same parity only excitons with P-type envelope function have a small oscillator strength in dipole approximation. The na = 1 singlet- (or ortho-) exciton couples to the radiationfield only in quadropol approximation with an extremely small oscillatorstrength, which is two to three orders of magnitude smaller than e.g. in the examples CdS and CuCl above 25. Consequently one may think, that the strong coupling or polariton-picture is not necessary to describe the interaction of theses resonance with light and one can limit oneself to the perturbative or weak coupling case. For many applications this statemtent is true, but there are also experimental data, which can be described only in the polariton picture 26. We discuss these so-called propagation quantum-beats with Fig. 12. A ps pulse

184

is send on a Cu20 sample with a thickness in the mm range. Its spectral width AE is in contrast to the above example such that it covers the whole energetic region around the resonance i.e. AE » ALT-

I

'

•

'

10 r

'

I

•

'

Cu 2 0 T=2K

i\i

J

, sj \ i \

experiment

"' \ i \

10

theory

Laser

10

I

w a v e

vector

0%

f

\

.$%:

10 _l

I

I

I

500

I

I

L_

1000 time (ps)

1500

2000

2500

•

Fig. 12: The temporal evolution of a ps pulse travelling through a sample of Cu 2 0 in the spectral region of the nB = 1 ortho exciton resonance. From le.

After transmission through the sample, the pulse has a length exceeding a ns and shows a temporal modulation, the period of which increases with time. The explanation in the polariton picture is straight forward. As shown on the l.h.s. of Fig. 12, the pulse excites the upper and lower polariton branches simultaneously. There are, as indicated, always pairs of states on the lower and upper polariton branches which propagate with the same group velocity. During the propagation the interference between these two states oscillates periodically between constructive and destructive, and they arrive with a certain relative phase at the other side of the sample and radiate a photon field into the vacuum. An inspection of the dispersion relation shows immediately, that the pairs with propagate fast have a short oscillation- or beatperiod and the ones which are slow have a

185

long period. This explains quantitatively the experimentally observed behaviour The overall decay of the curve follows from the damping. A theoretical fit to the data gave ALT « 5ueV and Tij « 0.9ueV. Similar propagation quantum beats have been observed also with free and bound exciton resonances in InSe or CdS 27. The third example of propagation phenomena is observed in a platelet type CdSi. xSex crystal involving the lower polariton branch only. Alloy semicondcutors like CdSi. xSe* show exciton localization effects at their band edge due to disorder induced by spatial fluctuation of the composition x. For a recent review see28 and references therein The spectrally resolved FWM signal (see Fig. 13a) shows for increasing delay a periodic modulation of the signal intensity. The period increases very rapidly with increasing photon energy. The spectrum of the laser-pulse covered the tail of the localized exciton states in the alloy, i.e. only states on the lower polariton branch of this strongly inhomogenously broadened resonance have been excited. The plan-parallel surfaces of the platelet-type sample formed a Fabry-Perot resonator, exhibiting a clearly modulated reflection spectrum (Fig. 13b). The interpretation of the data in Fig. 13a in first approximation is as follows. The first pulse creates a polariton wave-paket which propagates through the sample. If the second pulse arrives without time delay there is optimal overlap between the wave-pakets and a strong diffracted signal e.g. in direction 2k2 - ki. If the second pulse comes with some delay, the spatio-temporal overlap between the two pulses decreases and the signal drops. However, with further increasing delay, signal maxima occur for delays which correspond to integer multiples of round trip times of the first paket in the Fabry-Perot resonator, since then optional overlap between the two pulses is recovered. The long dephasing times of localized excitons in CdSi.xSex and the good reflectivity of the sample surfaces favour this effect, which has been addressed shortly in a theoretical work 30. If the above model is correct, one can deduce the group velocity from Fig. 13a See the open circles in Fig. 13c. The group velocity can be deduced independently from the data of Fig. 13b since the Fabry-Perot modes are equally spaced in k by Ak = 7t/d where d is the geometrical thickness of the sample. These data are shown by closed squares in Fig. 13c. The good agreement gives strong support for the above interpretation. The data are nothing but the derivative of the dispersion curve of the lower polariton branch. The upper one is not accessible by this method because it falls in the absorbing regime of the sample. A further argument comes from the fact, that the modulation disappears, when the temporal pulse length in the sample is made longer than the round trip time by spectral filtering 29. Similar propagation effects have been observed in the stimulated emission of CdSi.xSex samples31. However, there must be additional effects in this experiment, which are not yet fully understood and which involve e.g. excitation induced modifications of the optical properties of the samples, since we observe e.g. that the modulation period at one and the same energy changes slightly with the spectral width and the center frequency of the laser pulse29. Now we mention shortly the polaritons formed with other resonances in crystalline solds.

186

2.085

2.090

2.095

2.100 \

energy (eV)

2.105

\ d o m a i n e of extracted time cuts

1.70

1.75

1.80

1.85

1.90

1.95

2.00 2.05 2.10 2.15

energy (eV)

i

•

•I

i

c

T

i

-

O)

2.0x10

, 7

-

I

•

8

-

I i

"

^

o p/VM-signal

• 2.04

Fabry-Perot modes ( E i c) 2.05

IT

2.06

<

2.07

i

2.08

' 2.09

' 1

2.10

energy (eV)

Fig. 13: The spectrally resolved FWM signal close to the exction resonance in CdSi.xSex as a function of the delay x between pulses 1 and 2 (a), the Fabry-Perot modes of the sample recorded in reflection (b) and the group velocitiy as a function of energy deduced from data in (a) and (b). From 29 .

187 Fig. 14 shows the dispersion of the phonon polariton in GaP in the range of the optically active phonon branches. The experimental data on the lower polariton and on the longitudinal optical phonon branch have been deduced from Raman-spectroscopy in close forward direction. The variation of the angle between the incident and the scattered light beams allows to vary the wave vector of the quasiparticle created in the Raman process. The resonance reproduces exactly the one deduced for the ensemble of model oscillators in Fig. 7. A detailed description of this method and more data are found e.g. in 33

y

UPB

D A

I

i A

1

l

long. opt.

•

48

i

i

tlCOL

i

i

•3C

• i

o4.07°

&f—^"

i

,-JlCGr, > | 44

y^ x i.74°

~

• 6.18°

e=

P^0.66°

40

. -

; "

LPB

36 i

i

i

i

'

i

0.5

1.0

1.5

2.0

2.5

3.0

"

wave vector (1 O^cm"1) Fig. 14: The phonon-polariton dispersion in GaP. From3 An alternative possibility to measure the dispersion of phonon polaritons, based on the interference of two short laser pulses in the sample with wave vectors ki and k2 is described e.g. in The observation of propagation quantum beats for TFfz pulses centered around the optical phonon resonance can be expected in the near future.

188

For plasmon polaritons the lower branch coincides with the x-axis in a ha over k plot since the transverse eigenfrequency is zero. The high reflectivity stop-band extends consequently from zero to ha pl as shown for semiconductors with various doping levels e.g. in 36. The coupling between plasmons and optical phonons leads to so-called plasmon-phonon mixed states. For an example in heavily doped ZnO:Ga see e.g.37. In metals the plasmon frequency is generally situated in the UV (ha PL «(10 + 5)eV depending on the material). Consequently the reststrahlbande extends over the whole visible spectrum and explains the high reflectivity of simple metals. The reflection spectrum can be modified, however, by the superposition of a resonance due to band-to-band transitions and the plasmon resonance. This effect causes a drop of reflectivity in the spectral vicinity of the former similarly as for the above mentioned plasmon-phonon mixed state in semiconductors. If this dip in the reflectivity of metals occurs in the visible, they are coloured like Cu or Au. In ferri- or antiferromagnets there exist also „optical" magnon branches. Usually their coupling to the radiation field is very weak, but there are few reports on magnon polaritons and their dispersion relations like38. 2.2. Polaritons in Disordered Matter The purpose of this section is to demonstrate, that exactly analogous phenomena are observed in disordered matter in various spectral regions, and that consequently the polariton concept can be logically extended to these cases, too. Actually we have seen the first example for a disordered solid already in connection with the CdSi-xSex data of Fig. 13, since an alloy is strictly speaking a disordered system which can be considered as ordered only in the virutal crystal approximation. In order to convince the reader that the polariton concept holds also for completely disordered systems with inhomogenously broadened resonances, we consider the case of Na vapour. Na atoms have a strong intraatomic absorption form the 3s to the 3p state. In luminescence this transition gives rise to the well-known yellow emission of Na-lamps. The disperion of light in Na and other metal vapours has been measured already since the end of the last century, using the prism method introduced in Fig. 10, with the only difference that the prism is not formed by keeping the vapour in a prism-shaped container, but by changing the density of the vapour across the light beam. This transverse concentration gradient is created e.g. by heating on one side and cooling on the other. In Fig. 15 we give an example for the dispersion of the refractive index of Navapour in the vicinity of the 3 s -> 3p resonance from39. Of course the concept of photons was hardly known at that time and the idea of polaritons was still some decades ahead. But since we agree, that the energy of light is quantized also in Na-vapour into integer multiples of ha, and that each quantum carries a (quasi-)momentum Tik Fig. 15 represents nothing but the dispersion of polaritons as follows also from a comparison of the data in Fig. 15 with the ones for the exciton-polariton in Fig. 10. So it is not surprising, that propagation quantum beats in analogy to Fig. 12 have been observed when sending a short pulse through a tube containing Na vapour of constant density and temperature as shown in Fig. 16. The interpretation of the phenomenon

189 is the same as in Fig. 12. Above we have stated, that the dispersion relation of light in matter is the same using the classical model of Lorentz oscillators or the quantum mechanical description in the polariton picture. Consequently the data of Fig. 15 and 16 can be described also by classical optics. For Fig. 15 this is given e.g. in 41 for Fig. 16 the procedure is the following. The short incident, Fourier limited pulse is decomposed into its various frequency components. Every component propagates through the sample according to the real- and imaginary parts of the complex index of refraction. See Eq. (18). After the passage through the sample, the various components are added again with their modified amplitudes and phases and are back-transformed from the frequency to the time domain. The result is the dotted line in Fig. 16, which reproduces the data quite well.

D Linien

P red

oranjit

yellow

green

blue

K

CO

o

photon energy wave length Fig. 15: The spectrum of the real part of the refractive index in Na-vapour. From ;

Very recently it has been found 40 that the group velocity of a light pulse in a cold and dense Na vapour can be close to the resonance as low as 17m/s!

190 As a last example we give the results of an experiment where a short y-ray pulse from a synchrotron centered around the Mosbauer transition at 14.4keV(!) of 57Fe is transmitted through a "Feo.ss Cr0.25-Nio.2 alloy sample. See the data in Fig. 17 from 42. Again we see all the properties of polariton propagation quantum beats, which we know already from Figs. 12 and 16. However, the polarization results now from a transition within the nuclei of 57Fe and the photon energy is not in the eV but in the 1 OkeV region.

40

80

120

time (psec) Fig. 16: The propagation quantum beats of a light-pulse travelling through Na-vapour of different optical thickness. From4011. To summarize, we can state that the polariton concept is well established experimentally and theoretically for three-dimensional crystalline and non-crystalline matter as long asEq. (1) holds.

3.

Polaritons in Systems of Reduced Dimensionality

In this chapter we treat systems of reduced dimensionality like quantum-wells and -wires, surface polaritons and cavity polaritons.

191

Fig. 17: Temporal behaviour of a short pulse of synchrotron radiation transmitted through 57 Fe 55Cr 25Ni 20 foils of various thicknesses. From 42.

3. J. Quantum Wells and Wires In simplest approximation a quantum well is obtained, if one grows a thin layer (« lOnm) of a semiconductor with small band gap (e.g. GaAs) between layers of a material with larger bandgap (like Gai.yAlyAs). An arrangement of the conduction- and valenceband edges as shown schematically in Fig. 18a, in which electron and hole are confined in the same material, is called a quantum well of type I. We show in Fig. 18a also the envelope function of electrons and holes in the well. In the plane normal to the growth- or zdirection, electrons and holes can move as free quasiparticles. This is why such a system is also called quasi two-dimensional. There are exctions formed by the Coulomb-

192 interaction between electrons and holes for every transition between conduction- and valence subbands. They show up as peaks in the absorption spectrum of Fig. 18b. The appearance of heavy-hole (hh) and of light hole (lh) excitons results from a peculiarity of the bulk band structure of the parent materials. For details see 5.

CB U

I

li

n —9 — n - 1 "z~ ' i

nt=lhh

Eg I

1

1

Eg II

11,= ID) -

n, = 2hh

'' VB

h--'*(a)

100 x (11 nm InGaAs, 10 nm InP)

i

1.2

.2 o £-* ^-<

0.8

D O O

o

o en

unbound) .

0.4

T=10K (b)

J0.75

_i

1.00

1.25

tico(eV) Fig. 18: The band-alignment in a type I quantum well and the envelope functions of some quantized levels (a) and the absorption spectrum of a Ini.yGayAs/InP multiple quantum well sample (b). From 5 .

193

Quasi one-dimensional quantum wires can be formed e.g. by nanolithography and etching of quantum wells, forming structures, which are typically 50nm wide and several lOOnm long and in which carriers and excitons can move freely only in one dimension. For a recent data-collection and a review of the optical properties of such so-called quantum structures see e.g. . Now we shall shortly outline, that the polariton concept is valid also for such structures of reduced dimensionality. A thick stack of barriers and well with dimensions » X in all directions (where X is the wavelength of the light) and a periodicity of wells and barriers in growth direction which is « X can be considered as a strongly anisotropic bulk material13 in analogy e.g. to mica or layered semiconductors. A single quantum well (or a few of them in a multiple quantum well sample) can be considered as a layer of oscillators. It has the simplifying property that a polarization excited in the well with a light beam impinging normal to the well cannot propagate. This fact makes the interpretation of experiments especially easy if e.g. two delayed incident pulses are used in a four-wave mixing experiment since propagation effect of the polarization excited in a well are excluded in contrast to the bulk samples mentioned above. This is eventually one of the reasons, that many of the recent time resolved experiments have been performed in multiple quantum well structures or thin (< X) layers. But even here, some propagation effects can appear e.g. through the polarization induced coupling between neighbouring wells or for situations, where the distance between the wells becomes comparable to X u. An especially favourable situation for the investigation of polariton modes in quantum wells is the propagation of the light beam in the plane of the well. This situation has been investigated in detail in theory 45'46. We show in Fig. 19 schematically the polariton dispersion of the lowest optically allowed exciton resonance in a GaAs quantum well between Ali.yGayAs barriers. The realistic assumption has been made, that the value of the dielectric function of the barrier equals the background dielectric constant 6b of the well material. See also Eq. (13). The refractive index of the well is larger than that of the barrier for ha coL the refractive index of the well is smaller than that of the barrier. The wave is no longer guided (see Fig. 19c) and is radiated in about 10 to 20ps into the barrier material by diffraction. Therefore these modes are called radiant(or anti-guided) waves. Actually the situation is not so simple as in the calculation of Fig. 19a, where perfectly flat interfaces have been assumed so that there is a strict conservation law for the k-vector in the plane of the well. In reality there are well width fluctuations and other imperfections, which localize excitons and partly relax k-conservation (Fig. 19d). Such localized exciton states can decay, among others radiatively, with a time constant in the range of 100 to 500 ps 5'47. The population of the radiant and of the localized exciton states depends in thermal equilibrium on temperature and explaines the linear increase of the luminescence decay time in the temperature range up to about 100K observed in most but not all quantum well samples47'48. In spite of these imperfections it was possible to observe near the exciton resonance a decrease of the group-velocity in time-of-flight experiments to values around 0.1c 49. Though this value is not so small as in the data of Fig. 11, it is a clear indication,

194 that the polariton concept is valid also in this case. Furthermore there are caluclations of the polariton dispersion in individual quantum wires 50 and experiments for periodic quantum wire arrays 51, so that we can state, that the polariton concept is well established also for these structures of reduced dimensionality. Some further experiments to verify this concept have been proposal recently in 13.

tlCO .radiant J k modes|^-Suidedmodes-^

1

1.612

/ r5hh

V

(eV)

1.610

I

.1

I

I

I

I

(10 c m )

>

k„

a barrier

QW

Jr \

J\

y
\

T

/

Fig. 19: The calculated polariton dispersion for the r 5hll exciton resonance of a single 5nm wide GaAs quantum well between Gao7Al03As barriers. Schematically according to 46a (a) and a guided, an antiguided wave and a localization site (b, c, d).

195 3.2. Surface Polaritons For almost every wave, which propagates in the bulk, there exists a surface-(or interface-)mode, which propagates along the surface and has amplitudes, which decay exponentially on both sides of the surface over a distance comparable to its wavelength. Examples are e.g. surface acoustic waves 52 or surface polaritons. We concentrate here on the latter and limit the discussion to the simplest situation, namely the ideally flat interface between a bulk material and vacuum (air). As can be deduced from the boundary conditions resulting from Maxwells equations, the dispersion relation of a surface polariton is situated in the frequency intervall between the transverse and the longitudinal eigenfrequencies, i.e. in the stop band. This fact prevents a coupling of the surface polariton into the bulk, since there are no propagating modes in thisfrequencyinterval. Furthermore the dispersion relation to (k(|) starts at to0 with k;i = co/c and reaches for larger kg values thefrequencyat which e(co) = -1, i.e. close to the longitudinal eigenfrequency ". The fact that kg > co/c prevents the radiation into vacuum. Since the surface polariton is confined to the two-dimensional surface plane, it is also not possible to observe it simply in reflection, transmission or luminescence. However, there are special techniques of k space spectroscopy for surface polaritons. One possibility is to send a light beam into a prism of e.g. fused silica so that total internal reflection occurs at the prism base. The wave vector ky of the evanescent wave propagating is this situation on the vacuum side of the base can be tuned to some degree by the angle under which the incident beam hits the base. The energy to can be tuned trivially. If both kg and to of the evanescent wave match a point of the surface polariton dispersion of the material under investigation, the evanescent wave can couple to this surface polariton if the investigated material is brought sufficiently close to the prism base. Sufficiently" means in this context a distance < X. When the evancescent wave couples to the surface polariton, the reflectivity at the prism base is no longer total but has values significantly below 100%. This so-called attenuated or frustrated total reflection (ATR) method has been used to measure the dispersion of surface phonon polaritons in GaP and of surface plasmon polaritons in a highly doped InSb semiconductor in Fig. 20a, b. In both cases the dispersion relation agrees with what has been said above. Since the transverse plasma frequency is zero (see above with Fig. 6) the surface plasmon polariton branch extends from frequency zero to just below a>pL. A sketch of a surface plasmon polariton is given in Fig. 6c. Examples for the dispersion of exciton surface polaritons can be found e.g. in 56. An alternative way to couple to surface polaritons is to produce a grating on the surface of the material under investigation with a grating period A, which modifies the conservation law for k to k|| + m27t/Awithm = 0, +1,+2, ...

(30)

and allows for a suitable combination of to , kj m and A a coupling to surface polariton modes.

196 Evidently the concept of polaritons is also well established for the restricted, quasi-twodimensional geometry of surface- (or interface-)polaritons.

- HooL iKQr

licor

1.5

2.0

2.5

wave vector k (k^)

/

.,..—

1

/ / •~0

OU>0

0
00,

/A?

/°

n-InSb

•

.a

a

theory O O experiment

•a

b 1

2

3

normalized wave vector k (c / C0PL)

Fig. 20: The dispersion relation of the surface phonon polariton in GaP (a) and of the surface plasmon polariton in n-doped InSb (b). From54,55.

3.3. Cavity Polaritons Cavity polaritons are the mixed states of Fabry-Perot resonator or -cavity modes and some excitation of matter, generally excitons in quantum wells.

197

. ^

f

\

1 1 1

— R ---T

1

^ ^ m

>V m+1

Fig. 21: Two plan-parallel mirrors forming a Fabry-Perot resonator (a), the spectra of transmission and reflection as a function of the phase shift per round trip 8 for R « 1 (b) the geometry of oblique incidence (c), the dispersion relation of photons in vacuum (d) and in a Fabry-Perot resonator (e).

198

To approach the problem, we introduce first the eigenmodes of a Fabry-Perot resonator or an etalon, assuming for the moment that the resonator contains nothing (i.e. vacuum or air). A Fabry-Perot resonator is in simplest case an arrangement of two plan-parallel, lossless mirrors with a reflectivity R < 1 of each at a distance d. See Fig. 21a. An incident plane, monochromatic wave will be partly reflected and partly transmitted at each of the two mirrors. In most cases the partial waves in the resonator will interfere destructively resulting in a reflectivity of the whole structure R,ot « 1 a transmission T,„t « Rtot -1 » 0 and a very low field amplitude in the resonator. See Fig. 21b. There are, however, special situations namely when an integer number of half-waves fits into the resonator, i.e. if m - = d or kj_ =m— 2 d

m = l,2,3,...

(31)

Then all partial waves in the resonator interfere constructively, the field amplitude in the resonator can exceed considerably the incident one, and the transmission of the whole arrangement is close to unity (Fig. 21b). This situation is shown schematically in Fig. 21a. We can reformulate this fact by saying that the eigenmodes of the Fabry-Perot resonator occur for discrete values of the normal component of the wave vector of the incident light beam k.

=m-

m = l,2,...

(32)

For oblique incidence shown in Fig. 21c the wave-vector of the incident light beam k can be decomposed in components parallel and normal to the plane of the resonator k = k i + k||

(33)

The normal component of k has to fulfil Eq. (32) for the eigenmodes of the resonator while a conservation law holds for ka due to the translational invariance of the problem in the two-dimensional plane of the resonator. This situation is shown in Fig. 21d for light in vacuum, i.e. for photons. The dispersion relation to (k) is a cone with slope h c. The cross section of this cone with a Fabry-Perot mode with a fixed ki is a parabola as shown in Fig. 21e. This figure gives the dispersion of light in an empty Fabry-Perot resonator as a function of kn for a given m. If one tilts a Fabry-Perot resonator in a parallel light beam away from normal incidence i.e. from kp = 0, the photon energy of the transmitted light shifts to higher values, i.e. to shorter wave-length. This effect has been demonstrated during the lecture. If a semiconductor physicist (and the author belongs to this community) sees a parabolic dispersion relation, he thinks automatically about the concept of effective masses, beeing defined as 5 '"

199 -i *-2 5 2 to(k) "U=ft gk2

,,.. ( 34 )

An inspection of this concept gives for this situation the physically reasonable result, that the „effective mass" of photons in a Fabry-Perot resonator is given by the energy equivalent mass of the photons in the minimum of the parabolic dispersion to 0 i.e. by m e f f c 2 =to 0

(35)

In reality the mirrors used in Fabry-Perot resonators are generally not simple surfaces but dielectric- or Bragg-mirrors. The way how they work will be outlined only in chapter 4. In a next step, we bring some matter into the resonator, which has an eigenmode at a certain frequency to M . Since such a resonance is generally connected with absorption, and since absorption deteriorates thefinesseof a Fabry-Perot resonator 5, only a thin layer of matter is recommended. Therefore one places usually one, or a few quantum wells in the resonator at the position of an antinode of the Fabry-Perot mode and with an eigenfrequency of the lowest free exciton resonance close to the one of the Fabry-Perot mode. In Fig. 22a we show schematically the dispersion relations of the Fabry-Perot resonator mode to pp (ky) and of the matter to M (ky). The curvature of the exciton resonance to M is negligible compared to the one of to ^ . If both eigenmodes do not couple, their dispersion relation cross. If there is a finite coupling, we observe again the anti-crossing behaviour, and the dispersion describes mixed states of the Fabry-Perot or cavity mode and of the exction resonances, the so-called cavity polaritons. In Fig. 22b we show an example in which the luminescence has been observed as a function of the angle relative to the normal of the cavity. A variation of this angle is equivalent to a variation of ky. The fact, that both branches of the cavity polariton show up in luminescence under non resonant band-to-band excitation proofs that they are really the quanta of a mixed state. A Fabry-Perot resonator mode alone would not luminesce at to „, if light with a different photon energy is sent on the resonator. The system of Fig. 21b consists of the hh exciton resonance in Ini.yGayAs quantum wells in a 3 A/2 cavity with dielectric Bragg-mirrors on both sides 57b, but there are many other cavity-polariton systems known in literature. A small selection from III-V and II-VI compounds is given in" To conclude, a necessary prerequisite for the appearance of cavity polaritons should be mentioned. The dephasing time of the excitation in matter T2 must be much longer than the round-trip time of light in the resonator. Otherwise the coupled modes cannot develop, because the excitation in matter loses too quickly the phase coherence to the Fabry-Perot mode. The lowest exciton resonances of quantum wells fulfill usually the above condition at low temperature and density with T2 values of a few and up to some ten picoseconds 43. For increasing excitation, T2 becomes shorter, e.g. by excitation induced dephasing by exciton-exciton colloissions or the transition to an electron-hole plasma and the characteristic cavity polariton mode structure dissappears 58.

200 theory without coupling -•

•

•—

-10° 1.35-

experiment and theory with coupl. 0

external angle 10° 30°

L

hco

1.32

-

Fig. 22: Schematic drawing of the dispersion relations of a Fabry-Peropt resonator mode h(£> ^ (k|) and of a resonance in matter h(Q M (kj) for vanishing and finite coupling (a) and experimental luminescence data for the exction cavity-polariton mode from57b (b).

4.

Photonic Crystals

After having established the polariton concept for bulk matter and for structures of reduced dimensionality, we investigate the consequences of a spatially periodic modulation of the optical properties. This will lead us to the concept of photonic (more precisely polaritonic) crystals. In prinicple it is however not so terribly new but only a repetiton of what is known as Ewald-Bloch theorem in X-ray diffraction or in the bandstructure of crystalline matter.

4.1. Introduction of the Basic Concept To introduce the concept of photonic crystals with photonic bandstructures we consider a dielectric- or Bragg-mirror and the Fabry-Perot resonator formed from two dielectric mirrors with a cavity in between. It is long known, that a high reflectivity close to unity can be obtained over a certain frequency interval, if one produces a stack of layers of different refractive indices ni and nn, where each layer has a thickness d; of a quater wavelength, i.e.

201 d

1

AL=_5L

4

i=i,n

(36)

2n,co

The partial waves reflected at every interface interfere under this condition in a way, that high reflectivity occurs over a certain intervall around co. The spectral width of this high reflectivity band increases with increasing difference of the n; and the reflectivity converges to unity with increasing number of pairs of layers, provided, the layers are lossless. If a cavity is placed between two of these dielectric mirrors, with a thickness given by an integer number of XJ2 or X

CTt

r d c a v l t y =m^ =m m^—=

2

(37)

n co

one obtains a Fabry-Perot resonator with a spectrally very narrow dip and peak in the reflection- and transmission-spectra respectively. These things are known since many years and recipies can be found in textbooks like 59. Now we approach the problem from another point of view, assuming that the reader is familiar with the properties of an electron in a periodic potential as outlined e.g. in 5, 17

We write down Schrodinger's equation and the wave equation of light deduced from Maxwell's equations in linear approximation here for the electricfieldstrength E - ^ A < K r ) + V(r)cKr) = E<|)(r) 2m

(38a)

and AE--U(co)u(co)E = 0 c

(38b)

We rewrite these equations, execute the second time devirative in Eq. (38b) for the usual exp(-icot) ansatz, replace e(co) by n2(co) and assume that we are dealing with non-magnetic materials i.e. u(co) = 1. Thus we obtain 2m , A<))+—(E-V)<|):=0 "ft7

(38c)

and AE + ^ - n 2 E = 0 c

(38d)

202

For simplicity we neglect all types of absorption, assuming that V and n are real quantities. Evidently there is a very close analogy between Eq. (38c) and Eq. (38d). The term E - V corresponds to n2 and since the eigenenergy E is for stationary solutions of Eqs. (38a, c) just a constant, we can identify in this analogy essentially the potentail V with the square of the refractive index n. A difference is, that the eigenfunction <|>(r) of Eqs. (38a, c) is scalar quantity while E in Eqs. (38b, d) is a vector field. We consider first the solutions for spatially constant V(r) = V0 and n(co, r) = no. Then the solutions are plane waves with a dispersion relation typical for massive and massless particles see Fig. 23a, b. For the Schrodinger equation we obtain E=2m

forE>V0

(39a)

and for the wave equation E = hm=h — k

(39b)

Eq. (39b) corresponds to the dispersion of photons for no = 1 and to the lower, photon like polariton branch for no * 1. See Fig. 7. Now we consider a spatially periodic variation of V(r) and of n2(r) in one dimension as shown on the l.h.s. of Fig. 23c. This is the wellknown Kronig-Penney potential for electrons. Its consequence is the appearance of a forbidden region or energy gap in the dispersion relation as shown schematically in Fig. 23d. The wave vectors at which theses gaps occur are integer multiples of 7i/a, where a is the period of the potential. The physical reason of the appearance of the gap is the constructive interference of all reflected waves propagating e.g. in negative x-direction if a plane wave propagates over the periodic potential in positive x-direction. The superposition of the propagating and counterpropagating waves results in a standing wave. This standing wave can have for the same kx its antinodes at the maxima or at the minima of the periodic potential resulting in two different eigenenergies for the same kx with an energy gap between. A further consequence of the periodic potential is, that k is conserved only modulo integer multiples of reciprocal lattice vectors G = mra/a, m = ±1, ±2. See also Eqs. (28a, b). Consequently the outer branches of the dispersion reflection can be shifted into the first Brillouin zone -7t/a < k < 7t/a with the help of suitable G or it can be periodically continued as shown by the dashed lines in Fig. 23d and e. Since Eqs. (38c) and (38d). are equivalent, we obtain the same effect for the dispersion of the lower polariton branch if we introduce a spatially periodic modulation of n2 as shown in Fig. 23 e. This is our first example for a bandstructure for light in periodically modulated matter a so-called photonic-bandstructure and the periodic structure in n is consequently called a photonic (here one-dimensional) crystal. Since the light quanta in this structure are accompanied by a polarization wave where ever n(r) * 1, the term „polaritonic" would be more adequate than „photonic" with the general definition of the

203

term polariton introduced here. However, the „photonic" seems already well established and therefore difficult to change and consequently we adopt it here for the following, too.

E~k'

V

E = hco~k

n

Fig. 23: The dispersion relation of the stationary equations (36c, d) for constant V and n2 (a, b), the periodic potential of the Kronig-Penney model with a „defect" (c) the resulting dispersion relations (d, e) and the reflectivity in the vicinity of the band gap (f).

204

Since there are no propagting modes in the sample in the energy interval of the gap in this simple one dimensional model, we expect a reflection R « 1 in this region as shown in Fig. 23f. This is nothing but the high reflectivity region of the dielectric mirror discussed above. We continue now the discussion, by introducing a defect in the periodic potential for electrons as shown schematically on the r.h.s. of Fig. 23c. If this defect has s suitable shape concerning e.g. its width or depth, it can give rise to a localized electron state in the otherwise forbidden gap. If the parameters of the defect potential are not suitably chosen, the energy of the localized state falls in the region of the allowed energy bands, merges with them and is therefore difficult to defect. The same happens for light. The „defect-state" falls in the forbidden gap e.g. if the width of the defect fulfills the condition 31,37, resulting in a spectrally sharp dip in the reflection and a peak in the transmission spectrum which is nothing but a Fabry-Perot resonator mode e.g. in Fig. 21b. Now we use our phantasy or our physical imagination to extend the onedimensional system of Fig. 23 to three dimensions. The result will be a three-dimensional bandstructure for light in matter, which starts with a linear, photon-like branch around k = 0. The additional complication is, that the light field has vector character so that eventually different bandstructures exist for orthogonal polarizations in some analogy to the various transversal phonon branches. 4.2. Examples One of the aims is to create a photonic crystal with a true, three-dimensional band gap i.e. a frequency intervall in which no propagating mode exists for all directions and values of k. For this purpose a large difference of n2 in the periodic structure is necessary. Systems, which obtained considerable interest recently, are face-centered cubic (fee) arrangements of closely packed air-spheres (ni = 1) in a silicon matrix (nn = e"2 = (11 9)1/2). In other words a system in which the „voids" between the spheres arefilledwith Si while the spheres themselves are empty space60'61. Fig. 24a gives the band structure for various directions of high symmetry in the first Brillouin zone. The largest part of the complexity arises from the backfolding of the dispersion of higher zones into this reduced scheme, but the splitting of the branches e.g. at various points of the zone boundary and especially the appearance of a true gap, shown in the resulting density of states in Fig. 24b around coa/rac = 0.8 are a result of the spatially modulated refractive index. The lattice constant is given by a. There is a similar natural system namely opal. It consists of Si02 spheres forming a fee lattice and the voids between the spheres are partly filled with H20. The difference between the refractive indices is not sufficient, to produce a true gap (Fig. 24c) but for every direction of k (i.e. direction of observation) there are some frequency intervals in which propagation of light is not possible. This results in a colour, which depends on the direction of illumination and of observation. Actually this effect is known as opalescence, a property of a natural photonic crystal. There can be many exciting effects forseen. If a luminescent atom is brought into a photonic crystal, which would emit at a photon energy, which is situated in a true gap of

205

the photonic structure, the atom cannot radiate light, because there are no propagating modes. The luminescent atom can emitt a light quantum, but it stays localized in the surrounding of the atom, can be reabsorbed and so on. In this sense spontaneous emission can be supressed allowing exciting new insights in quantum-electro-dynamics.

wave vector

:

0.9

2" 0.8

go,

b

[

b

I

I

I /

E? 0.6 0.5

x>

J

K

0.1 0.1

0.2

*

!l

0.4

0.5

0.6

0-7

cs

b

0.5

%

0.4

00

0.3

c w *

,-

/ ,"

//

s

1

M1 •si 1/

0.3

S

3 £> o.a

i

.

O.S

,-v'

s«

y

/

0.1

0.7

02

0.3

0.4

0.5

06

0.7

C0a/2rrc

COa / 2nc

energy

Fig. 24: The calculated photonic band structure of a close packed fee lattices of air spheres in Si (a), the resulting density of states (b) and the density of states for a close packed fee lattice of Si0 2 spheres in air corresponding to natural opal (c). The quantities a and c stand for the lattice constant and the vacuum speed of light, respectively. From 61.

206

Another, application oriented use would be to influence the bandstructure by external fields e.g. by cladding the air spheres with a liquid crystal, which changes its refractive index under the influence of an external electric field, allowing to manipulate the bandstructure so ' 61 and thus the refelction, transmission or luminescence spectra. Similar phenomena can be predicted with ferro electric photonic structures. In Fig. 25 we present another system of a two-dimenisonal photonic structure 60' 62 63 ' namely a hexagonal array of tubes in silicon. This structure has a true twodimensional bandgap around coa/27tc « 0.4 which is narrower for the electric field polarized parallel to tubes than for orthogonal polarization. A chain of defects, here missing holes, acts for frequencies in the gap as a wave guide. The waveguide structure in Fig. 25a has much narrower curvatures than classical waveguides or fibers can have. A realization of a beam splitter is shown in Fig. 25b and of a resonator in Fig. 25c. The central defect „localizes" the lightfield,which is contacted by the two waveguides.

Fig. 25: A hexagonal periodic array of holes in silicon with a wave-guide (a) a beam splitter (b) and a resonator (c). From 62. In a similar way a ridge waveguide of Si on SiC>2 has been converted in a FabryPerot resonator by a periodic arrangement of holes, which act as Bragg mirrors and a central part as cavity60'62'6i. We present here as a last example the building of a one-dimensional photonic structure from photonic atoms64. The concept of a photonic atoms is explained with Fig. 26. A quantum well is grown in a cavity consisting of two stacks of dielectric Bragg mirrors as explained already in section 3.3. The luminescence is shown in the lowest trace of Fig. 26b and we see the by now well known effect, that the emission maximum is shifted away from the exction resonance of the quantum well without cavity. In a next step a three dimensional resonator is formed by etching a small, square mesa of widht w. See Fig. 26a. The natural reflectivity of the side walls resulting from refractive indices n around three is sufficient to form a resonator with three-dimensional confinement for light as seen from the increasing spacing of the emission peaks with decreasing w in Fig. 26b.

207

I i

i

i

i

i

t

i

i

i

[

PL = 200 W/cm2

T = 2K

i

i

i

i' i

i

i

i

excjton

J

1.412

1.400

1.410

1.420

energy (eV) b

1.400

1

2

3

4

5

lateral size (/im)

Fig. 26: A photonic atom consisting of a square mesa prepared from a structure containing a quantum. well in a cavity with stacks of Bragg-mirrors on both sides (a) the luminescence of the mesa with w => oo and for finite w (b) and the coincidence of the measured and calculated emission peaks as a function of w (c). From6".

208

The spectral position of the emitted quanta is given by

to=^(k2+k*+kJ)1/J

(40a)

where ko is the wavevector denned by the Bragg mirror cavity while kK and ky are given by k i =(m i +l)—, w

mi =0,1,2,...; i = x,y

(40b)

Fig. 26c shows the excellent agreement between the experimental data and the results of Eqs. (40a, b) for various modes64. The next step is to couple two of the photonic atoms to a photonic molecule by a bridge between them shown in Fig. 27a ". The two lowest modes have even and odd parity in close analogy to the binding and antibinding states in a H2 molecule. Increasing coupling realized in Fig. 27c by an increase of the width of the bridge at a constant length of lum leads to a splitting of the otherwise degenerate levels without and with one node line in each dot. The natural extension is of course to couple more and more photonic atoms in a linear array. Actually one observes a splitting into more and more discrete levels as expected for coupled harmonic oscillators66. These levels develop with increasing number of coupled dots N > 10 into a one-dimensional bandstructure with small gaps at the borders of the Brillouin zones 66 while an unmodulateed photonic wire shows a smooth dispersion relation 61. More information on photonic structures and their possible applications can be found e.g. in the reviews60,68 and references given therein. 5.

Conclusion and Outlook

The concept of polaritons as the quanta of light in matter has been developped for three-dimensional ordered and disordered matter, for structures of reduced dimensionality like surfaces, quantum wells or cavities and for the spectral range from the IR through the visible to the soft y-ray regime. A recent „offspring" of thisfieldare structures with an artificially created periodic modulation of the refractive index in one, two or three dimensions, so-called photonic or polaritonic crystals. These structures bring phenomena, which are usually only known from X-ray diffraction 69 at natural crystals, into the cm-, um-wave or IR and visible range of the electromagnetic spectrum and make these phenomena much more pronounced because larger variations of the refractive index can be realized in the latter spectral ranges. Exciting new experimental findings, theoretical insights and applications can be expected from thisfieldof research in the future.

209

0++

0- +

1++

1- +

1- +

1MT

.

1.4M

IMS

' 1 channel length: 1pm 1 " " " ^ a

\M*

• *

*

m -» .

-«~<J~-»

a

• o-

1.441

channel width [ym|

Fig. 27: A photonic „molecule" made by coupling two „atoms" with a bridge (a) calculations of the field distribution of the lowest six coupled states (b) and their observation as a function of the bridge width (c). From 65 .

210 Acknowledgements: Stimulating discussions with my academic teachers, with many colleagues and with my coworkers about the topics treated here are gratefully acknowledged. Without trying to be complete I should like to mention Profs. Drs. K. Hummer, M. Wegener and R. v.Baltz (Karslruhe), Prof. Dr. H. Stolz (Rostock), Prof. Dr. E. Mollwo (f) and Prof. Dr. R. Helbig (Erlangen), Prof. Dr. A. Forchel (Wurzburg) or Prof. Dr. D. Frohlich (Dortmund). I should likte to express my special thanks to Prof. Dr. A. Forchel (Wurzburg), Dr. U. van Biirck (Miinchen) and last but not least Dr. K. Busch (Karlsruhe) for making available to me their excellent and valuable results prior to publication. References 1. 2. 3. 4. 5. 6.

7.

8. 9. 10. 11. 12. 13. 14. 15.

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NON-MARKOVIAN SCATTERING PROCESSES IN SEMICONDUCTORS

MARTIN WEGENER Institut fur Angewandte Physik, Universitat Karlsruhe, Kaiserstrafie 12 D-76128 Karlsruhe, Germany

ABSTRACT Often we think about scattering processes in solids in terms of classical point-like particles that bounce against each other in instantaneous collisions. In particular the only time scale that appears in the corresponding mathematical description, e.g. in the Boltzmann-equation or in Fermi's golden rule, is the mean free time between collisions. This picture is obviously in contradiction with the quantum mechanical, i.e. the wave-like nature of the collective excitations, which obey the time-dependent Schrodinger equation. Quantum mechanics tells us that any interaction between collective excitations is nothing but an interference phenomenon, including obviously the scattering processes that are responsible for decoherence and 'dissipation'. Intuitively, we can think about the dynamics of such many-body systems in terms of a finite duration of scattering processes which can be described by non-Markovian relaxation. Here we give an introduction into this growing field of quantum kinetics in semiconductors. 1.

Introduction

In the first semester of any physics course we learn that the energy transfer from a harmonic oscillator to its surrounding can be well described by a phenomenological Stokes damping parameter 73. Stokes damping means that the corresponding force acting on the oscillator at time t is proportional to its velocity at the same time t the dynamics is local in time. The resulting damped harmonic oscillator equation for the displacement P is one of the paradigmas of physics: P(t) + 7sP(t) + n2P(t)

= 0.

(1)

fi is the eigenfrequency of the undamped oscillator. An example of such a system is shown in Pig. 1. While the phenomenological Stokes damping can properly describe many aspects of the damped oscillator, it completely fails in other situations. Let us analyze the case where the oscillator motion is started by an impulse and then is stopped at the origin, P = 0, for a short but finite time, thus P = 0. For these two initial conditions the second order differential equation, Eq. 1, determines that P = 0 from thereon, which is in sharp contradiction to the experimental observation. It is clear that the water

215

216

Figure 1: Scheme of a simple mechanical system showing non-Markovian relaxation. in the tank, which is still in motion, acts on the oscillator and sets it in motion as well. If we wish to write down an appropriate equation of motion for the oscillator only, a more general form has to be used, j(t - t') P(t') dt1 + Q2P(t) = 0 .

Pit) + f

(2)

J — oo

This equation is non-Markovian, it is nonlocal in time and the memory function •y(t - £') determines how far the history (causality => t' < t) of the system influences its present state. For j(t - t') = 2^sS(t - 1 ' ) , i.e. immediate loss of the memory, Eq. 1 is recovered. For the simple example 7(t - H) oc e- ( '-*' )/T -'

(3)

we have the characteristic memory time constant r m for the decay of the memory. In the limit rm —> 0, the exponential decay of P is recovered. On the other hand, if we had written down Newton's law for the oscillator (without phenomenological damping) coupled to the Navier-Stokes equation for the fluid in the tank, we clearly would have obtained an equation of motion which is local in time. This is the same in quantum mechanics. Consider that we prepare a quantum mechanical system, e.g. a semiconductor, in a known state at time t = 0. It is clear that from thereon its wavefunction follows the time-dependent Schrodinger equation. Thus the evolution of its wavefunction at time t is completely determined by the state of the crystal at the same time t - the dynamics is again local in time. The semiconductor, however, is such a complicated many-body system that its Schrodinger equation is not solvable, and it is necessary to restrict oneself to treat only certain degrees of freedom of the problem, for example the electrons in a conduction and a valence band. With this approach, we have to consider that this subsystem interacts with other degrees of freedom, e.g. vibrations of the nuclei or other electrons, which are modeled as a thermal bath. Then the equation of motion of the subsystem becomes nonlocal in time, i.e. the evolution at time t not only depends on the state of the system at time t but also on its history. This is the regime of quantum kinetics 1 . 2> 3 . While the classical oscillator model suggests that memory effects arise due to nonequilibrium effects within the bath, we will see later that this is not necessary in quantum mechanics.

217

Figure 2: Scheme of an optical excitation process in which a photon with energy hCl takes an electron from the valence to the conduction band. This leads to an occupation / at state k in the conduction band. Before we come back to non-Markovian relaxation in solids, we first have to discuss briefly the Markovian relaxation, i.e. the Stokes-like energy transfer from the system to a bath in a solid. Consider an electron in the state with wavevector k in the conduction band of a two-band model semiconductor with parabolic energy dispersion depicted in Fig. 2. As this electron is clearly not in the lowest energy state, it eventually relaxes down to the bottom of the conduction band parabola. For the dynamics of the occupation function / at state k (the probability that state k is occupied, a number between 0 and 1 for fermions), it is often argued along the lines of the radioactive decay: The change of / , df, is proportional to the occupation itself, proportional to the discussed time intervall dt and proportional to some specific energy relaxation rate 71 and we obtain / + 7i / = 0.

(4)

As a result, the occupation / decays exponentially with rate 71 or equivalently with time constant 7\ = 7-f1. If we actually want to prepare such a nonequilibrium initial condition, we can employ optical excitation with a short laser pulse shown in Fig. 2. The optical transition at frequency fi means that we take an electron from the fully occupied valence band and transfer it to the conduction band. Because of the small photon momentum (large speed of light) these transitions look almost vertical in the Brillouin zone. It is intuitively clear (and will also be shown later) that the corresponding polarization follows a harmonic oscillator equation with eigenfrequency fi. The phenomenological damping rate of this coherent oscillation is called the phase relaxation rate 72 or decoherence rate. T2 = 72-1 is the dephasing time. The free oscillation P + Q2P = 0. (5) can be rewritten to p + iQp = 0 (6) where P is simply the real part of the complex quantity p which we want to call the optical transition amplitude.

218 Back to non-Markovian relaxation. Actually, how valuable is the concept of such decay rates ji and 72 ? Later we will discuss quantum mechanical models for the interaction of crystal electrons with lattice vibrations - a problem which is far from being easily solved. To get a first feeling at this point already, we consider a simple physical approximation: the fluctuation model. 1.1.

The Fluctuation

Model

Lattice vibration means that the local lattice constant fluctuates as a function of time. As a result, the local band structure also fluctuates, in other words: The eigenfrequency fl(£) becomes time-dependent. We can write it in the form

n(i) = n0 + sn(t)

(7)

and choose Q0 such that the average fluctuations SCl are zero, i.e. (S^t(t)} = 0. At this point, the average (...) is a purely classical one and can be interpreted as an ensemble average over many individual experiments. Let us see what the influence of these fluctuations is on the transition amplitude p. Formal integration of Eq. 6 with Eq. 7 immediately leads us to p{t) = p(0) (exp ( - i j * [Oo + Sn(t')} dt'^j) = p(0) e- ifio( fl - i f (5Q(t')) dt' --[*[* L

Jo

»

'

(Sn(t')Sn(t11))

2 Jo JO •

„

dt'dt" + ... 1 . '

(8)

J

= 0 In the second line we have employed a Taylor expansion of the exponential function. In this line we face a correlation function, (5£l(t')6Cl(t")), for the first time. If we consider these frequency fluctuations as a random process, the correlation function is similar to a combined probability in an experiment where we throw two dice. Let us consider three cases i)-iii). i) If the two events of throwing dice are independent, the probability of throwing, let us say first a six and then a one, V(6,1), is simply the product of the probabilities throwing a six, V(6), times the probability of throwing a one, V(l), and thus given by ^(6,1) = •p(6) , P(l) = i • | . Thus for actually different fluctuation events, i.e. t' ^ t", we have (<5ft(t')<5ft(i")} = (5fl(t')} • (Sn(t")) = 0. In the usual jargon we say that the correlation function can be factorized. Altogether we have {SQ(t')5fl(t")} = 272 S(t — t'). Introducing this in Eq. 8, the second integral collapses due to the 5 function, and we can identify these terms as the first terms of a Taylor-expansion of an exponential function, i.e. ... [1 - - 2721 + ...} = ... exp(—72i), or follow a more lengthy and complete calculation 4 , and obtain p(t) = p(0) e~mt

e-

0)

219 which can clearly be translated back into the differential equation of a damped harmonic oscillator p + (in + 7 2 ) p = o. (10) This is nothing but Eq. 6 with damping. We conclude that uncorrelated fluctuation events, called Markov processes, lead t o an exponential decay on the average. ii) The result is different if two fluctuation events are not independent. In the picture of dice we can for example imagine that the two dice are connected by a string. Then the same probability as above is given by V{6,1) / 7?(6) P ( l ) . P(6,1) will depend on the particular interaction of the events. Coming back to our model this means that (SQ(t')SQ(t")) is finite even for t' ^ t". Consequently, we do not obtain an exponential decay but rather a decay which depends on the specific form of the correlation function (5Q(t')5Q(t")). A solvable example is (5fl(i')5fl(i")) = a2 exp(|£' — t"\/tc), with the correlation time tc and the root mean square of the frequency fluctuations a. After some mathematical manipulations this leads to 5 p(t) = p(0) e- 1 ""' e " G « ,

(11)

with G(t) = a21\ [exp(-i/t c ) + t/tc - 1] .

(12) 2

We obviously recover the exponential decay of case i) for £c —» 0 with 72 = a tc. For finite correlation times, however, the decay starts Gaussian, i.e. p(t) oc exp(—a2 t2/2), and becomes exponential for times much longer than tc. iii) Another interesting case is a coherent oscillation of the lattice - a nonequilibrium (and nonthermal) situation for the bath. Assume we kick the phonons by a short excitation pulse centered at t = 0. This means that we have an electronic eigenfrequency O(i) which oscillates in real-time with the frequency of the lattice uio (e.g. the longitudinal optical (LO) phonon frequency) and some amplitude A, i.e. 9.{t) = n 0 + A sin(w L ot).

(13)

Integration of Eq. 6 with Eq. 13 immediately leads us to p{t)

=

p(0) exp [- ^

i (n 0 + Asin(wLOt'))

=

_A p(0) e- in °* exp fi — cos(w LO t)

=

p(0)e- i " o ( '•,

•

l + i

dt'

A

COSIWLO*)

+

(14)

wLo In the last step we have performed a Taylor expansion of the exponential function. As a result, the optical transition amplitude p exhibits oscillations for coherent phonons - in contrast to the exponential decay of p imposed by thermal phonons as discussed in i). The fact that coherent phonons have been observed experimentally 6 tells us

220 that the underlying idea of the fluctuation model, an electronic eigenfrequency which is modulated in real-time by the lattice displacement, is not just an academic concept but reality. In order to avoid confusion, it is important at this point already to point out that the experiments discussed in Sect. 4 cannot be interpreted as such coherent phonon oscillations, but rather have a distinctly different origin. We will come back to this point in more detail in Sect. 4.5. 2.

Theory

For a semiconductor in which Bloch-electrons couple to phonons, to other Blochelectrons, and to an external laser field which is roughly resonant with the semiconductor band edge, the Hamiltonian H is well established 2 ' 7. It is given by H = He + .HLO + i^e-LO + -He-light + # e - e •

(15)

In terms of creation cvk and annihilation cUtk operators for electrons we have the single-particle energy H e = Yl Ev,kc\kcv,k • (16) In the following, we use the effective mass approximation, which is illustrated in Fig. 2. The index v runs over the valence (v) and conduction (c) band with states at wavevector k and energy Ev^. For short times only the longitudinal optical (LO) phonons are important, acoustic phonons are minor corrections. To simplify things, we approximate that the LO-phonons have no dispersion in the relevant part of the Brillouin-zone. In terms of creation aq and annihilation aq operators for the phonons at wavevector q, the LO-phonon energy is

#LO = J2 hujh0aqaq ,

(17)

where the zero point energy of the phonon modes has been subtracted. The electronphonon Frohlich-interaction energy is1- 2 #e-LO = h Y, ( & X c l - « c * + ggaqc\+qck)

,

(18)

q,k

with the Frohlich-matrix element gq oc 1/q. For clarity, we have suppressed the band index v. An example of such scattering processes is visualized in Fig. 3(a). Notice that wavevector conservation is built into this interaction Hamiltonian while energy conservation for the individual scattering event is not given due to 'energy-time uncertainty'. In other words: The electron cannot 'know' the phonon oscillation period for times shorter than this period (see Fig. 4). The effective strength of the electronphonon interaction is usually described by the dimensionless Frohlich-constant a1. It can be expressed by quantities which are altogether accessible experimentally V^WLO Ve(°°)

/

221

Figure 3: (a) Scheme of an electron-phonon interaction process oc c\_qa\ck. The electron at wavevector k and energy E emits a phonon with wavevector q and energy RWLO and is scattered into a state with wavevector k' and energy E'. While wavevector conservation is given, i.e. k' = k - q, for short times 'energy-time uncertainty' leads to violation of energy conservation for the individual scattering event, i.e. E' ^ E - hui^o. (b) Scheme of an electron-electron interaction process. Two electrons with wavevectors k and k1, respectively, exchange a momentum q. This process is proportional to ck_„ck,+qCk'c^. For short times, energy conservation is again violated, i.e. we have E(k - q) + E(k' + q) ^ E(k) + E(k'). ER is the exciton Rydberg energy, e(0) and e(oo) are the static and the high-frequency dielectric constants, respectively, a scales as |
(20)

with the classical electric field E and the dipole matrix element dcv for a transition from the conduction to the valence band. Eq. 20 is equivalent to the energy of an electric dipole in a static electric field. He^e is the electron-electron Coulomb interaction 1 ' 2 which is illustrated in Fig. 3(b), #e-e = ;r Z

Yl

yqc\-qci+qCk'Ck.

(21)

k,k',q^a

For clarity, we have suppressed the band index v. Vq oc l/q2 is the Fourier-transform of the V(r) oc 1/r Coulomb potential. The simplest consequence of the Coulomb interaction Eq. 21 is that, due to the interaction of an optically excited, negatively charged conduction band electron with the positively charged hole in the valence band, a combined two-particle state, the exciton, is formed. In linear optics (and only in linear optics, see Sect. 3.1), the exciton is analogous to the hydrogen atom 2 . 7 . So far it has not been possible to solve this many-body Hamiltonian Eq. 15 exactly for conditions corresponding to the experiments described below. Approximate numerical solutions have, however, been obtained for electron-phonon scattering 1 3 - 2 1 as well as for electron-electron scattering 2 2 - 2 8 . They will be discussed and compared with experiments in Sects. 4 and 5.

222

Figure 4: An electron is optically excited at time t = 0 and 'observes' the phonon oscillation. After half an optical phonon cycle, O/LO* = n (upper row), equivalent to t = 57.5 fs in GaAs, the Fourier-spectrum (RHS) is very broad, the electron cannot 'know' the phonon frequency UJ-LO- For longer times, WLO* = 27r (middle row) and u)i,ot = Sn (lower row), a clear peak at oi = O>LO develops in the spectrum. Table 1: Relevant material parameters of GaAs and ZnSe. Remember that we have ft = 658 fs meV or h = 4134 fs meV. Material

-Ek

^B

ftwLO

2TT/O;LO

6(0)

e(oo)

a

bulk GaAs bulk ZnSe

4.2 meV 19meV

12.5 nm 3.9 nm

36meV 31.8 meV

115 fs 131 fs

12.9 8.8

10.9 5.7

0.06 0.43

2.1.

Material

Parameters

For a better readability of Sects. 4 and 5 it is useful to summarize a few relevant material parameters at this point. Table 1 gives the excitonic Rydberg energy S R , the excitonic Bohr radius r B , the LO-phonon energy ftu>Lo, the phonon oscillation period 27T/WLO, the static dielectric constant e(0), the high-frequency dielectric constant e(oo), and the Frohlich electron-phonon coupling constant a for bulk GaAs and bulk ZnSe. 2.2.

The Phonon-Assisted

Optical

Bloch-Equations

Let us consider a simple model of electron-phonon quantum kinetics, which allows the experimentalist to follow the mathematics rather easily and obtain an intuition for the underlying physics. H = He + Hi,o + -He-LO + #e-light = h£lc[ci + hu/ho^a

+ hg [a)c\ci + acjcij — dE (c\c0 + c\c\j .

(22)

The electronic ground state 0 of the fermionic two-level system is at zero energy, the excited state 1 at energy Ml. Again, the zero point energy of the phonon mode has been subtracted. Only scattering from the upper electronic level back into the upper level via emission or absorption of phonons is included. E.g. cJa^Ci takes

223 away one electron at state 1 (ci), generates a phonon (a*) and creates an electron at state 1 (c{). These processes obviously violate energy-conservation for the individual scattering process. Thus they would not contribute in Fermi's golden rule or the Boltzmann-equation, b o t h of which use the Markov approximation. These processes are, however especially important in the regime of quantum kinetics. We will see in a moment that this violation of energy conservation is very closely related to the non-Markovian character of relaxation and thus to the correlations of scattering events discussed above. From hereon we follow two different approaches. First we discuss analytical solutions within the short excitation pulse limit, which are nonperturbative with respect to the electron-phonon coupling. Secondly, we compare these results with numerical solutions following the so-called hierarchy approach. This offers the possibility i) to translate the results of the simple model to the usual language of the more complex many-body problem (which cannot be solved analytically) in terms of particle correlation functions and ii) to test the validity of truncation schemes. The analytical solution is not at all trivial to obtain. For its derivation we refer the reader to Ref. 29. Exciting the system from thermal equilibrium at zero temperature with a weak (linear optics) and 5-shaped optical pulse, i.e. E(t) = E05(t), we have for the polarization p{t) = (c0Ci) p(t) = l^p. "•

e-*2 Q(t) £ 9—- c-i
(23)

n

-

which shows explicitly the well-known result of phonon sidebands at integer multiples of the phonon frequency above the polaron shifted energy Q = f2 — <JJLO g2 with the normalized electron-phonon coupling strength g = g/ui-^o- This result is schematically depicted in Fig. 5(a). The complete analytical solution for linear as well as for nonlinear optics and arbitrary initial temperature of the system is given in Ref. 29. The nonlinear solution is also given and discussed at the end of Sect. 4.6. The physical meaning of the one-phonon sideband is that an electron does not lose its phase after it has emitted a phonon. Its initial and final state can interfere and as a result we observe a beating with the phonon frequency. Similarly, n-phonon emission processes lead to oscillation components with n times the phonon frequency. We can also see from Eq. 23 that the strength of the n-phonon sideband scales oc (g2)n oc an, and thus it rapidly decreases with increasing n if the electron-phonon coupling is small, i.e. if j 2 C l . The semiconductor GaAs is indeed in the regime of weak electron-phonon coupling and we can safely account for n = 0 and n = 1 processes and ignore others. This is quite different for ZnSe, where the electron-phonon coupling is much stronger. The transition amplitude p versus time according to Eq. 23 is depicted in Fig. 5(b) for the case of extremely strong electron-phonon coupling, 5 = 5. g = 1 and g = 0.5 are shown for comparison. Notice that we can indeed obtain a relaxation behavior of p for short times. For long times, i.e. at wLo * = 27T, however, the system exhibits a reoccurance behavior. Such behavior is much more general than this model suggests. The largest relevant energy separations in any solid determine the time scale of a first

224 (a)

(b) M i l

1.0

A,:

0.8

i

\ \

0.6 0.4 0.2 \

0.0

1

. tlUl

LO

"l 1 1 1 11

-0.5

/ ;

A

""" 1 1 1

• \\ -

L

0.0 0.5 1.0 TIME wLOt / 2n

1.5

Figure 5: (a) Scheme of the energy levels and optical transitions of the phonon-assisted optical Bloch-equations. (b) Solutions of the polarization p(t) for linear response, i.e. Eq. 23, for g = 0.5 (dashed line), g = 1 (dotted line), and g = 5 (solid line). decay of an observable under investigation, the smallest relevant energy spacings give the time after which the system comes very closely back to its initial state. In real macroscopic solids, the smallest energy spacings are so tiny that it takes times comparable to the age of the universe, thus this phenomenon is of little practical relevance. It shows, however, that n o t r u e irreversibility is possible in closed a n d finite q u a n t u m m e c h a n i c a l s y s t e m s . Let us relate this analytical result to the discussion on non-Markovian relaxation in Sect. 1. In other words: Which equation of motion will we obtain if we write down an equation of motion for the polarization p only ? For any observable O we have the Heisenberg equation of motion, — ihO = [H, C]_. From this it is straightforward to compute the equation of motion for the density matrix Pkn

(c\Cj(<J)kan).

(24)

Note that the elements of p are again correlation functions, this time describing correlations between the electron properties (the c' s) and the phonon properties (the a's). For example, the quantity p\\ = (c\c\a)a) is the correlation function (see Sect. 1) between the electronic occupation number c\c\ and the phonon occupation number a)a. The usual transition amplitude (polarization) is nothing but the element V = Poo i *he occupation number of the upper electronic level is / = pJJ. With the anticommutators [ c n , c t j + = 5nm and the commutator [a,a']_ = 1, we obtain the infinite, coupled, and open set of equations pE, + i(fi + ( n - * H o ) P h , 1

+ih- dE(p™-(&)+ig(-

nplU

n01 Pk,n+l

Pk+l,n)

(25)

225 and

+itr1dE

Ptn + H« - fcHoPte = (pZ - pfn) + ig (k p\\n - n p ^ )

(26)

and similar equations for p°° and p£°. The terms

"(pft-piJ.) in Eq. 25 are the only origin of optical nonlinearities in the phonon-assisted optical Bloch-equations. They are often called Pauli-blocking terms because they are an immediate result of the Fermi-statistics of the two-level system. For k = n = 0, they are identical to the inversion factor (cjco) - (cjci), the difference between the occupations of levels 0 and 1. In this simple model, relaxation is nothing but the physics of coupled oscillators. We excite one oscillator, say the electronic transition amplitude PQJ = (c'0Ci), which couples to the one-phonon excitation p®\ = (c'QCi a), which couples to the two-phonon excitation p^\ = (clci a2) and so forth. In the complete Hamiltonian Eq. 15, energy thus spreads into many degrees of freedom, which eventually looks like an irreversible decay. How can we solve this open set of equations? If we truncate at a certain sufficiently large number of phonons involved, i.e. 0 < n, k < n m a x = fcmax, a n d assure that the solution does not depend on the particular choice of this cut-off, we have numerically solved the dynamics. Corresponding results have been obtained and will be presented and compared with experiments in Sect. 4.6. Alternatively, we can employ different levels of Born-approximations. A simple and educative limit is linear optics and weak electron-phonon coupling at T = OK. Here we have p™ = 1 and pJJ = 0. Furthermore we can factorize within the second Born-approximation (e.g. p1\ = (c'0cia^a) x (cjcx) (a^a); if the reservoir remains in thermal equilibrium at T = 0 we also have that the Bose-factor is zero, i.e. (ala) = 0), which means that we account for one-phonon processes only. We obtain the following simple form for the transition amplitude p(t) = p§J and the scattering amplitude s(t) = ipjj} p + gs = ih^dE s + IWLOS = gp

(27) (28)

with E = Eexp(—iQ,t), p = pexp(-iQt), and s = sexp(-if2t), we have switched to the so-called rotating frame 2 . Again, these two equations describe nothing but the physics of two coupled pendula, one of which is the electronic transition, the other one the phonon oscillator. Formal integration of Eq. 28 introduced into Eq. 27 gives

p{t) + g2 f

e-WL°('-(''p(t') dt' = ih~ldE .

(29)

226 Again, the reason for the non-Markovian form of the dynamics ist that we have written an equation of motion for the electrons only. The combined equations for electrons and phonons, i.e. for p and s, are local in time. The solution of Eq. 29 can again easily be written as a main oscillation and a one-phonon sideband. This corresponds approximately to the n = 0 and n = 1 contribution of Eq. 23. This statement becomes exact in the limit of weak electron-phonon coupling, i.e. g —> 0. How good are these simple phonon-assisted optical Bloch equations when applied to semiconductors? While this model accounts for an atom- or molecule-like two-level system only, the semiconductor has bands of interacting electron states (Fig. 2). A scattering process that takes an electron from one state, emits a phonon with energy 7io;Lo, and brings the electron back to the same state has a deficit HA — HCJ^O in the energy balance. From the phonon-assisted Bloch-equations we have learned that this leads to an oscillation with frequency A = w L0 - We can generalize this finding intuitively: A scattering process which brings an electron from state k to state k' and which has a deficit in the energy balance of HA leads to an oscillation with frequency A. The semiconductor has bands of states and thus we have a continuous distribution of frequencies A. For long times, their sum will average to zero, except for the ones which do not oscillate at all, i.e. the ones with frequency A = 0. In other words: Only those scattering processes survive in the long time limit which conserve energy for the individual scattering event. This result is nothing but Fermi's golden rule. These energy conserving processes are often called real scattering processes. They lead to real energy and phase relaxation of Bloch-electrons due to their coupling to the phonons. The energy non-conserving processes become relevant on a short timescale and are called virtual scattering processes. In the phononassisted Bloch-equations, these virtual scattering processes lead to phonon sidebands. While nobody has seen such phonon sidebands in the linear absorption spectrum of GaAs or ZnSe, one expects intuitively that these phonon sidebands are still there, yet somehow hidden. The nonlinear techniques discussed in Sect. 4 will allow us to detect these hidden sidebands experimentally. To summarize the findings of Sect. 2.2 : The physical meaning of the occurence of phonon sidebands is that a scattering process needs a finite time. In the semiconductor, the width of the phonon sideband is given by the duration of the scattering event, or equivalently by the memory time r m . Observation of such oscillations in itself is thus evidence for non-Markovian scattering. Only for rm = 0 the decay becomes exponential. 3.

Experimental Techniques

First experiments on semiconductors using sub-lOfs optical pulses have already been published several years ago 3 0 ' 3 1 . They were performed at room temperature and showed no indication of non-Markovian relaxation. While these experiments used low repetition rate ( « 10 kHz) laser sources, recent advance in the generation of ultrashort pulses directly from laser oscillators provide us with high stability and high repetition rate ( « 100 MHz) sources32 of very short, even of sub-lOfs, pulses. These lasers are

227

(a) 2q2-Qi

(b) q 3 +q 2 - qi l21.

A

(0

\ i ^ 4 l AA ^ ^ q 2

-11'

K

2q P -q4

2" 1

Figure 6: Scheme of the three different types of four-wave-mixing (FWM) experiments discussed in this article, (a) Two-pulse experiment, (b) three-pulse experiment, and (c) coherent control configuration. suitable for experiments on GaAs based systems. For the experiments on ZnSe we simply frequency-double the output of the laser oscillator. Group velocity dispersion is more pronounced in the blue, thus we have to compensate the optics of the experiment and the cryostat window by a prism sequence and a grating (or dispersive mirror) sequence. In this fashion we can obtain blue 10.0fs optical pulses 33 . These are the shortest optical pulses generated in this spectral region by any means to date. An important point to notice is that a short pulse alone does not help for anything because optical detectors with a time resolution even close to 10 fs are not available. Todays best detectors, so-called streak-cameras, have a time resolution of about 500 fs. Thus all the experiments discussed below are correlation techniques (except for the up-conversion discussed in Sect. 3.4). A first optical pulse excites the system and a second, time-delayed pulse probes certain aspects of the system in a stroboscopic fashion. Pump/probe geometries probe the population of the system. We will not say much about them here (an introduction can e.g. be found in Chaps. 4,5 and 12 of Ref. 2). The following wave-mixing techniques selectively probe the quantum mechanical coherence of the system.

228 3.1.

Two-Pulse

Experiments

The geometries of three different four-wave-mixing ( F W M ) experiments are shown in Fig. 6. For the two-pulse experiment (Fig. 6(a)), two pulses with wavevectors q[ and q2 respectively, arrive at the sample at times £i and t2. The diffracted signal in direction 2
a

single homogeneous line and by T x A £ H W H M >

h/2

for an inhomogeneously broadened line. It is tempting, yet wrong, to interpret this result as one form of the 'time-energy uncertainty' relation. We will come back to this aspect in Sect. 5.1. '•'In gas media this notion refers to undesired and extrinsic effects like e.g. Doppler-broadening of an absorption line. In semiconductor physics this effect is an intrinsic property of the band continua. Bad semiconductor samples can show additional extrinsic inhomogeneous broadening due to spatial inhomogeneities. The latter effect has no importance for the experiments discussed in this article.

229 (a)

(b)

Figure 7: Scheme of the result of a two-pulse four-wave-mixing experiment (a) on a single homogeneously broadened two-level system and (b) on an inhomogeneously broadened twolevel transition. This corresponds to dE > Vp in Eq. 30. Both the real-time behavior of the diffracted light intensity for characteristic time delays and the time-integrated intensity (see signal on the left hand side) are depicted.

(a)

(b)

Figure 8: Scheme of the result of a two-pulse four-wave-mixing experiment (a) on a single exciton resonance and (b) on an inhomogeneously broadened exciton transition. This corresponds to dE dE + Vp (30) sometimes referred to as the renormalized Rabi-energy dE. The prefactor V can be interpreted as an exchange interaction energy. Originally, this local field m o d e l was introduced phenomenologically. It can, however, also be derived from the Hamiltonian Eq. 15 for -ffe-LO = 0, in Hartree-Fock (mean-field) approximation for He_e (which leads to the semiconductor Bloch-equations 2 . 7 ), and assuming that the exciton is excited exclusively. We refer the reader to the extensive corresponding textbook discussion in Chap. 6 of Ref. 2. Fig. 24 illustrates the local field model. The analysis

230

gives that the behavior depicted in Fig. 7 is substantially modified as depicted in Fig. 8, showing that t h e coherent radiation of excitons is distinctly different from that of two-level systems. The real-time behavior of the diffracted FWMsignal shows a finite rise time and a shifted maximum. This is because an exciton is still driven by the field of the other excitons - even if the laser pulse is over. Also, FWM-signals at negative time delays occur, which is another immediate indication of such interaction effects. We will find them in many of the data shown in Sect. 4 (see e.g. Fig. 13 (ZnSe) or Fig. 15 (GaAs)). Intuitively, this aspect can be understood as follows. Let us consider the diffraction direction 2q2 — q\ where we take two photons out of beam 2 and one photon out of beam 1. Without interaction (V = 0), only light, i.e. an E-field, can be diffracted from the grating that is formed by the two pulses. For positive time delay (i 2 i > 0), pulse 1 comes before pulse 2. Pulse 1 creates a polarization. What remains from that polarization interferes with pulse 2 and forms the grating. A second photon from the laser pulse 2 can be diffracted from this grating. For negative time delay (<2i < 0) pulse 1 comes after pulse 2. Pulse 2 creates a polarization and what remains from that polarization interferes with pulse 1 and forms the grating. As pulse 2 has, however, already passed the sample, no photon from pulse 2 can be diffracted from this grating in direction 2
3.2.

Coherent

Control

Let us now consider the coherent control geometry shown in Fig. 6(c). In a general sense, coherent control of a quantum mechanical system means that we are able to prepare the system in any desired initial state wavefunction \P. If this wavefunction is represented in a complete orthonormal system of eigenfunctions n in the form

9 = Y,an^Vn4>n,

(31)

n

this is equivalent to the ability to control the real amplitudes an and the phases ipn deliberately. While such complete coherent control is indeed possible in state-of-theart experiments on atomic systems, most experiments on semiconductors have used a much simpler form of coherent control which consists of excitation with a pair of phase-locked pulses. Here two phase-locked pulses 1 and 1' are separated by the time delay t\v = t\ — ty, which needs to be kept constant within ± 40 attoseconds corresponding to a motion of one of the mirrors in the interferometer of ± 6 nm, and

231

Figure 9: The Pancharatnam screw {left), a detail of that part of the coherent control experiment which generates the phase-locked pulse pairs. Their power spectra versus time delay tn< are depicted on the right-hand side. propagate collinearly in the direction gi = qy. An educative way to look at these pulse pairs is to think in frequency, u>, space (Fig. 9). A small temporal separation means that we see interference fringes in the spectrum which are very broad, a larger time delay t\y leads to a rapid oscillation in the spectrum. Small variations in tn> merely shift the spectrum while large variations significantly change the spacing of the nodes. Consequently, we can adjust the time delay £n< so that we have a node in the spectrum at some frequency 0. This means that this optical transition is specifically not excited while other states can be excited. In the time domain the first pulse excites the system and the phase of the second pulse is such that it de-excites the system via destructive interference. Interaction with the third pulse 2 propagating along wavevector q2 with the timedelay t2y = t2 — tv leads to two nonequivalent diffracted FWM-signals with wavevectors 2q2 — Qi and 2gi — q2, respectively. The detailed set-up for generating phase-locked pulse pairs, the Pancharatnam screw41, is quite simple (Fig. 9). A reference laser, e.g. a Helium-Neon laser, is alligned in a plane parallel to the femtosecond pulses in the two arms of a modified Michelson interferometer. In one arm the polarization is rotated to polarization parallel to the plane of incidence by introducing a A/2 retardation plate. Behind the interferometer, the equally vertically polarized beams are transformed into left and right circularly polarized beams, respectively, by a A/4 retardation plate. These oppositely circularly polarized beams, ideally of equal intensity, lead to linearly polarized light. The plane of polarization depends only on their relative phase. After passing an analyzer (A) which includes an angle