Advanced Semiconductor Heterostructures Novel Devices Potential Device Applications and Basic Prope

( 1 ^

U,;

v|/( =iP(a+b(a-J))k

.

(3.13)

Substituting this representation into the Schrodinger equation, HxFl = EWl, and taking into account that AhxFl = 0 , we easily find the coefficients of the linear combination:

U

E+\A

_

__i—; fc =— 1 ; E = E + Y3k2. (3.14) E(E + A) E(E + A) Vector column \|/( takes an especially simple form in the basis of coupled momenta (3.3) due to the diagonal representation of the spin-orbit coupling operator a-J (3.4) fl =

i

0

0

E V,=iP 0 ^J— o

,

Ok;

E +A

1

o

4

.

((2kz+qt)/S k= (kz-qz)/fi .

(3.15)

(^,)/V2 v

'

J

E. Normalizing spinor amplitudes Ui, and U\ also include the spatial dependence of the envelopes Uhl °= exp(ikr). We can use the projector (3.7) to obtain the quasiparticle dispersions. For heavy states Eh(k) = ^Sp(AhH)

= -(y2+yi)k2,

(3.16)

while for the light bands the dispersion relations can be obtained from the equation del[(l-Ah)(H-E)]

= (E,,+y0k2-E) s

+ P2k2-^l—-0. E(E + A)

(3.17)

This equation is of third order in k2 and results in three different types of light eigenstates, though only two solutions, f/j and *F/2. are physically relevant. They describe, respectively, the electron/light-hole and the spin-orbit split-off bands. The spurious solution f^ will be discussed in section 3.2 below. In a heterostructure, spherical symmetry of the bulk energy spectrum is reduced to the axial symmetry group Cv. We can significantly simplify the eigenvalue problem using the reflection symmetry ft^z with respect to the plane formed by the heterostructure growth axis z and the electron in-plane momentum K, chosen here along the jc-axis, so that k = (K, 0, kz). This transformation is equivalent to the product of the inversion, /, and the

14

Electron-Phonon

Interactions

in Intersubband Laser Heterostructures

953

rotation by angle n about the y-axis, cOyK. In basis (3.3) corresponding matrix represen­ tation is la

co,

= (e0 - e3) exp(/;r£ ) = (e0 + e3) ia

(3.18)

We can choose the normalizing spinors U\ and Uh as eigenstates of the operator ay, ayULh=TjULh,

77 = ± 1 ,

(3.19)

thus introducing in our model the eigenstate polarization rj. From (3.18) it follows that this quantum number correlates with the eigenstate parity with respect to the xzreflection. In basis (3.3), the matrix representations of the xz-reflection a^z and the timereversal operation Tdiffer only by the complex conjugation operator 5R: (3.20)

»r,5R

From (3.19) it thus follows that yiUn= U.n, so that the time-reversal operation changes the sign of the eigenstate polarization r/. The inversion operation / does not change TJ, therefore, this quantum number can be used both in Kramers degeneracy condition E{K,Tj) =

E{-K-rj),

(3.21)

E(K-r1),

(3.22)

and in the two-fold degeneracy condition E(K,rj) =

characteristic of the eigenstates in heterostructures with inversion symmetry. In asymmetrical heterostructures, such as InAs/GaSb DQW, this degeneracy is lifted and (3.22) fails. So defined, the state polarization significantly simplifies the matrix representation for
** =

Uu

** =

vf A ;

/

«/ =

o

(3.23)

-2k2-irjK (2kz-iTjK)/y[6E

< 1^ Hi

*ti=iP

Vii)

(kz+irjK)/S(E K/JIE

+ A)

(3.24)

3.2. Phenomenological boundary conditions At a heterointerface A-B, the phenomenological description of a mesoscopic state by an effective Hamiltonian (3.2) should be accomplished with appropriate boundary conditions (BC) for the A- and B-parts of the multicomponent wave function !P(3.1). The most common approach to the matching procedure requires the component-bycomponent continuity of the envelopes y/„, thus assuming uniform microscopic structure of the contacting materials. At the A-B interface (z = 0) this approach entails the BC in the form22 (3.25) )zA^A ~ JzB^B ■ VA=VB,

15

954

M. V. Kisin, M. Dutta & M. A. Stroscio

The second condition provides for the current continuity across the interface. The flux operator matrix, j , can be defined so that the current density }=W+jW continuity equation flu/ +w divj = = iW + HW~i(HW)+W .

satisfies the

(3.26)

BC (3.25) reveal significant drawback when degenerate spectrum of the constituent semiconductors is involved. In this situation, the transverse components of the momentum operator, pxy, directly enter the current operator. This makes BC (3.25) dependent on the mesoscopic structure of the electron state, in this case - on the in-plane electron wave-vector K. Microscopic uniformity also implies the identity of the A-B basis states, which is inconsistent with the actual diversity of the effective Hamiltonian parameters in constituent semiconductors. Finally, the most important disadvantage of the conventional BC (3.25) is the lack of the adjustable parameters required in any kind of the phenomenological models to allow fitting in experimental data. Several authors have already suggested more general phenomenological BC,23"26 which reject the unjustified assumption of the component-by-component continuity of the envelopes and make no specific assumptions about the basis functions on each side of the heterointerface. Here, we shall restrict our analysis with the simplest diagonal form of the boundary conditions ¥nA = Fn¥nB '> V'nA = Gn¥nB

■

(3-27)

Each envelope y/n is a spinor in basis (3.3), so that BC parameters F„ and G„ should be considered as 2x2 matrices. The internal structure of these blocks, however, is severely restricted by axial symmetry assumed for the heterostructure energy spectrum. Invariance under rotations about z-axis leads to the diagonal structure of each 2x2 block, since the matrix of the rotation operator a>zv is diagonal, see (3.6) and (3.9). The symmetry under jtz-reflection a>xz (3.18) makes diagonal elements equal, while the time-reversal symmetry T (3.20) imposes the real values for these matrix elements. So, finaly, each 2x2 block in BC is represented by a real number, F„ or G„, multiplied by 2x2 unity matrix of the Kramer's subspace, 8m>. It is important that BC with such a matrix structure do not mix electron states with different polarizations n. This significantly simplifies the eigenvalue problem: 4-component analytical solutions (3.23-3.24) can be used together with phenomenological BC (3.27) in reduced 4x4 matrix representation. Phenomenological matching parameters, F„ and G„, are independent on the mesoscopic characteristics of the electron state, though they are interrelated due to the current continuity condition at the A-B interface JlA=JzB-

(3-28)

In model (3.2) the continuity equation (3.26) is satisfied by the expression

1 V+ 272

^(p.^^M^L^,

(3.29)

where h.a. stands for Hermitian adjoint. The z-component of the flux j is represented in basis (3.3) as

16

Electron-Phonon

-iP ro

Interactions

c c

^

in Intersubband Laser Heterostructures

'>o (vivo - cc) + >/3 (ViVi + W r

--

+ KYi +Y3W3V1 ~c-c-) -^YiWliPx

955

c c

- )

-i(rxPy)W\+c.c\.

(3.30)

Here, cc. stands for complex conjugate. An arbitrary mesoscopic state expansion includes 4 independent partial solutions, one heavy and three light, W=Wh+Wn

(3.31)

+Wn+y,

therefore, each wave-function component in (3.30) can be varied independently. The current continuity (3.28) then imply the following relationships between BC parameters and parameters of the constituent Hamiltonians

P

A

hGo =

YOB

F G

\ \

YlA

= F2Gi

=

YOA

YlB

F,G3

YiA

(Y2+Y3)B

(3.32)

(Y2+Y3)A

One matching parameter, for instance F0, can be conveniently treated as a phenomenological parameter of the BC. It should be noted that BC in general form (3.27) can be used only if all partial solutions have been included in the expansion (3.31). However, the third k2 root of the dispersion equation (3.17), k?3 x-P /y0y3 »k}x,k}1, gives spurious solution with a matrix structure following from (3.24):

*»«:

1 Yo

\l2 1

Un(z).

(3.33)

0 This solution appears due to the influence of remote bands treated perturbatively and is beyond the applicability of the Kane model. Therefore, when considering the mesoscopic electron states, we have to truncate both the expansion (3.31) and the BC set (3.27) providing them only for smooth envelopes. Assuming t//3 « Un, U/2, we can neglect spurious amplitudes in expansion (3.31) and omit them in the matching condition for envelopes (3.27). First three matching conditions for envelope derivatives with n = 0,1,2 contain large terms k^Un. These terms are cancelled out from the linear combination of the boundary conditions with n = 1,2 since, according to (3.32), we have F\ = F2, and consequently G] = G2. Remaining boundary condition with n = 0 determines unphysical amplitude Ua, which we do not consider explicitly, and thus may be skipped. As a result, we arrive at the following truncated BC set for smooth envelopes only (3.34) WnA = FnWnB \

{VI-J2V'2)A=GM-J2V'2)B-> VIA = GIV'IB ■

17

(3.35) (3.36) (3.37)

956

M. V. Kisin, M. Dutta & M. A.

Stroscio

Similar truncation procedure should be used any time we encounter strongly evanescent solutions, which is the case, for example, for spin-split or heavy-hole bands in type-II heterostructures, which will be considered in the next section. 3.3. Electron energy spectrum of basic heterostructures Analytical representation of the Kane model eigenstates completed with phenomenological BC provides the tool for a semianalytical treatment of the electron confinement in the most basic intersubband laser heterostructures. Narrow-gap semiconductor materials used in the mid-infrared lasers are characterized by large spinorbit splitting of the valence band A. In the limit A»E, small envelops with n = 2 can be omitted in (3.24). Split-off light solution ¥i2 becomes spurious, therefore, an arbitrary mesoscopic state should be represented by a superposition of smooth wave packets of only two partial solutions - one light and one heavy, !F« %\ + %. Correspondingly, the matching condition (3.36), which determines the spurious amplitudes, should be skipped. Large values of the valence band offsets simplify the boundary conditions even more. Let us assume that the mesoscopic state 'Fis mostly localized in layer B and the energy of the state is far off the valence band top in the A-material. The heavy solution can then be treated as a spurious one in the layer A. Proceeding with our truncation scheme, small amplitude UM can be neglected and BC (3.37) skipped, so that the A- and B-parts of the mesoscopic state Ifare matched at the interface by BC (3.35) only. As a first example, we consider the energy spectrum of an asymmetric narrow-gap type-I single-QW heterostructure with layer sequence A1-B-A2; see Figure 9a, inset. We do not consider strain-induced and space-charge effects here. In the flat-band

(a)

(b)

Fig. 9. Band diagram of an asymmetric type-II InAs/GaSb DQW heterostructure. Asymmetric type-I QW heterostructure is shown schematically in the inset, (a) "Leaky" DQW heterostructure modeling a single stage of an intersubband cascade laser. Direct interband tunneling depopulation of the lower lasing subband through the heterostructure leaky window S is shown by a bold arrow (I"). (b) Isolated type-II DQW heterostructure with first electron-like and first light-hole-like subbands anticrossed. Interband LO-phonon assisted depopulation process (rpi,) is shown by an arrow.

18

Electron-Phonon

Interactions

in Intersubband Laser Heterostructures

957

approximation, the wave function of a confined state is represented in the QW layer B as a superposition of light and heavy solutions with opposite k

( 3 - 38 )

^B = 2>//,(**/,/>)> l,h,±

while in the barrier regions Al and A2 we consider only smooth evanescent light solutions with kAl—iAAi and kA2=iAA2- Matching ¥AI, ^B, and yA2 by BC (3.35) with n = 0,1,3, we easily obtain the dispersion equation in the form D(E) = \ ^

+ (Xj +!,)£, cot/?, - k } ~ ~ > 1 - cos Si cos Bu HZi^+tM^otfa-^kt-^+^k, , Pl. Ph = 0 . (3.39) sin Pi sin Pf, Here

, Pi(h) = ki{h)aB . ^ = £* > A±=X±7]K,

77 = ^ ( 1 - ^

EB , 3K'-1 ( ^ = 3 27 ^ ; c F £ . ' ' 4/c2 +T2" V

+ 0 , and

1? ..

\

EAYIB

Indexes 1 and 2 in Eq. (3.39) refer, respectively, to interfaces Al-B and B-A2. The energy E in each layer is referenced to the top of the valence band, and, according to the dispersion equations (3.16) and (3.17), with a good approximation we have k

hB +K2 ~ -2mhBEB ; kfA(B) + K2 * 2mlA{B)EA(B)(EAiB)

- E^^/Eg^

. (3.40)

In an asymmetrical QW, linear terms ~rjK determine subband "spin" splitting, which is more pronounced in valence band, where £, < 0. In the case of a symmetrical QW, the terms with TJK are cancelled out so that the double degeneracy of the energy spectrum is restored. For conduction-band states, the electron confinement is described by a simplified dispersion equation obtained in the limit mh » mt. This is exactly the first line of Eq. (3.39). It is important that the dispersion equation (3.39) holds the same general form even for more complex heterostructures and, therefore, can be conveniently used in a great variety of situations. In this work, as an example, we shall study asymmetric InAs/GaSb DQW heterostructure shown in Figure 9a. This structure models a single stage of a cascaded active region of a type-II intersubband laser.27,28 In the next section we consider in more details the quasibound eigenstates situated in the heterostructure "leaky window", which is the overlap of the InAs conduction and GaSb valence bands. Right half-space layer of InAs, A2, models the injector region of a cascade laser heterostructure. For quasibound states, we assume only the outgoing wave with an amplitude if to be present there. The utmost left barrier region, adjacent to the InAs QW Al, models an AlSb barrier layer with £gAisb » £ginAs • Here, even the light solution (3.24) becomes strongly evanescent, with kz» K and y/\ » ^o» V3, so that, according to (3.35), zero BC should be imposed on y/^M and ^.AI a t t n e AlSb/InAs interface. Note, that zero boundary conditions cannot be used for all envelopes in InAs QW even in the limiting case £gAisb-»°°- At least one matching condition (in this case - for y/{) should be truncated to exclude from explicit consideration the amplitude of strongly evanescent solution in the barrier region. "Leaky" DQW heterostructure of Figure 9a looks 19

958

M. V. Kisin, M. Dutta & M. A.

Stroscio

0.1 0.2 0.3 0.4 in-plane wave vector, nrrf1

0.1 0.2 0.3 0.4 in-plane wave vector, nm"1

0.5

Fig. 10. Subband structure in the leaky window 8 of an InAs/GaSb isolated DQW (QA=12 nm, aB = 10 nm, 5 = 150 meV). Solid and dashed lines represent subbands with opposite polarizations fj. (a) Anticrossing between light-particle subbands in the limit m* —> oo. (b) Complete picture of subbands including four heavyparticle subbands HI -H4; mA = 0.5mo.

substantially different from type-I single-QW heterostructure shown schematically in the inset, nevertheless the matching of the wave function components at the interfaces still leads to the dispersion equation in the same general form (3.39), with only changes for X\ and A\: D(E) = 0;

^ = kM cotkMaA]

; X2=-iklA2.

(3.41)

For InAs/GaSb/InAs triple QW heterostructure with an additional InAs QW layer, which models the so-called W active region of type-II lasers,' the dispersion equation can be obtained in the same fashion D(E) = 0 ; A, = kM

cotkMaA\

COt ^M2fl42' > ^2 - ^ M 2

(3.42)

In the limit f—>0 this equation describes also asymmetric type-I DQW heterostructure. Finally, in the limit A2 -»°° we obtain simple dispersion equation for an asymmetric two-layer Table 2 InAs/GaSb DQW heterostructure isolated on both InAs Parameters GaSb sides by AlSb barrier layers (Figure 9b): 29.6 28.8 ticoio, meV «

k

lA COt PlA+

—

+ ka COt Pm

C khBcotfr hB

2

— 2 0.

(3.43)

ficoro, meV

26.7

27.7

e» eo Es, meV

12.25 14.77 360 0.027 0.5

14.44 15.62 680 0.051 0.5

mi

mh

Using the above dispersion equations makes the analysis of the subband structure very illustrative. Since at K = 0 the light- and heavy-particle subbands are decoupled, it is convenient to consider them first separately. This separation can be accomplished formally in the limit mh —» 00, which eliminates the terms with f from the dispersion equation. Figure 10 shows exemplary calculation of the subband structure in the leaky window of an isolated

20

Electron-Phonon Interactions in Intersubband Laser Heterostructures 959

InAs/GaSb DQW heterostructure schematically represented in Figure 9b. Parameters used in calculations are represented in Table 2.10 Solid and dashed lines represent subbands with opposite polarizations rj, which are split due to the lack of inversion symmetry in this heterostructure. Note, that this splitting is much more pronounced for Btype (light-hole) states. Strong anticrossing between the light subbands is readily seen in Figure 10a and can be easily traced in the complete picture of the electron energy spectrum shown in Figure 10b. In this heterostructure, the top of the first light-hole subband LB1 is above the bottom of the first In As-related electron-like subband LAI, so that the subbands anticrossing occurs at a final value of the in-plane wave vector K. In intersubband laser design, the lover lasing states (subband LAI) should be located in the upper part of the leaky window. Figure 11 shows on the enlarged scale the subband alignment for a smaller value of the InAs QW width. With aA decreasing, the electron­ like subband moves toward the upper edge of the leaky window so that the anticrossing finally occurs at K « 0. Here, we cannot assign any definite type (electron- or hole-like) to the states of the subband extremities Ll±, and linear terms dominate the subband dispersion. Note, that at the anticrossing this linear subband splitting is equally distributed between the Ll± subbands. It is important that the anticrossing gap is smaller than LO-phonon energy in constituent semiconductors, hco^o « 30 meV, so that the tunneling depopulation of the lower lasing state can be favorably complemented with phonon-assisted transitions, which will be considered in the next section. 4. Subband Depopulation in Type-II Laser Heterostructures In this section, we use the results of the previous parts to analyze the lower lasing state depopulation in type-II cascade laser heterostructures. InAs/GaSb/AlSb material system

Fig. 11. Subband structure in the upper part of the leaky window for an InAs/GaSb isolated DQW heterostructure; aA= 8.5 nm, aB = 10 nm.

is very promising for implementation of high-temperature mid-infrared intersubband lasers covering the 3-5 urn atmospheric window. Comparing with type-I QCL,12 higher

21

960

M. V. Kisin, M. Dutta & M. A.

Stroscio

conduction band offset at InAs/AlSb interface allows extension of the laser operation at shorter wavelength, simultaneously reducing the leakage current. Cross-gap alignment between InAs and GaSb allows also better blocking of the injected electrons in the upper lasing states, while the lower lasing state depopulation can be favorably accomplished by two efficient processes: direct interband tunneling through the InAs/GaSb "leaky window" and LO-phonon assisted interband electron transition. In type-II lasers,27"29 the direct interband tunneling has always been considered as a basic depopulation mechanism whereas the interband LO-phonon assisted tunneling is habitually treated as an inefficient one due to a symmetry difference between the initial and final electron states involved in the transition.30 Here, we make a comparative study of these two processes and show that symmetry constraint for LO-phonon emission can be essentially removed in coupled InAs/GaSb quantum wells by significant nonparabolicity and bandmixing effects inside the heterostructure leaky window. 4.1. Interband tunneling in InAs/GaSb "leaky" heterostructure One of the attractive features of type-II "broken gap" heterostructures for the intersubband cascade laser design is the opportunity of direct interband tunneling depopulation of the lower lasing states located in the heterostructure leaky window.27,28 The rate of this process can be easily evaluated using the dispersion equation (3.41). Imaginary term A2 corresponding to the outgoing electron wave in the collector region makes the energy eigenvalues in the leaky window complex with imaginary part -ihT/2. The quantity T represents the interband tunneling depopulation rate, which is proportional to the inverse lifetime of the quasibound electron state. Figure 12 shows two upper light-type energy levels (K = 0) in the leaky window S of the InAs/GaSb DQW heterostructure depicted in Figure 9a. The level positions e are counted from the InAs conduction band minimum. Levels LAI and LB1 demonstrate typical anticrossing behavior, changing the type of the state localization (A- or B-layer) beyond the anticrossing point; see also Figure 13. At the anticrossing point, the electron density spreads equally over both coupled quantum wells, so that the interband tunneling rates (or the energy level widths) are equal for the upper (L-) and lower (L+) anticrossing levels;

8

9

10

11

12

13

InAs quantum well width aA, nm

8

9

10

11

12

13

InAs quantum well width aA, nm

Fig. 12. Anticrossing between two upper light-type energy levels in the leaky window. Position of the energy levels s in the leaky window (a) and the energy level width hT (b) as a function of the InAs QW width aA at two values of GaSb QW width, aB = 16 nm (solid lines) and aB = 10 nm (dashed lines).

22

Electron-Phonon Interactions in Intersubband Laser Heterostructures 961

see Figure 12b. Before the anticrossing, the quasibound A-type states are substantially narrower than the B-type states, since the tunneling from the A-levels, localized mostly in InAs layer, requires two interfaces to penetrate. All these features of the quasibound states in the leaky window S can be easily inferred and analyzed from the phase of the electron reflection coefficient.31,32 Restoring the amplitude of the incoming wave, IT, in the right half-space InAs-layer, the reflection coefficient can be expressed through the function D(e) from Eq. (3.41)

Die) D(e)

R(£)

(4.1)

Since there is no propagating wave in the left barrier region, the latter relation readily follows also from the time-reversal symmetry of the reflection process. In the vicinity of a resonant state, the phase of the reflection coefficient undergoes n phase shift and the peak of the phase derivative with respect to energy has Lorentzian shape. The value of the full peak width at half maximum represents the inverse lifetime r of the quasibound state. Figure 14 illustrates the evolution of the light-type quasibound levels (K = 0) in the heterostructure leaky window, 0 < s < S, as the GaSb quantum well width aB increases. The InAs quantum well width is constant, aA=12 nm, thus keeping the bottom of the subband LAI in the middle of the leaky window. In subplot (a) the narrow peak LAI corresponds to the lowest electron-like level localized mostly in InAs quantum well A. Much wider peak LB1 corresponds to the highest light-hole subband in the GaSb upper state

Fig. 13. Redistribution of the electron density at the level anticrossing in InAs/GaSb DQW. InAs QW width aA = 10 nm; GaSb QW width aB is changed from 10 nm to 20 nm in 2 nm increments. Corresponding redistribution is depicted by arrows.

23

962

M. V. Kisin, M. Dutta & M. A.

Stroscio

quantum well B. When the A- and B-type levels anticross (b), the decay rates of the light quasibound states become comparable. This situation, when occurs in the middle of the leaky window, reveals the highest rate of the interband tunneling process. The broadened peaks L1+ and L I - represent symmetrical and antisymmetrical combinations of A- and B-type levels under the anticrossing condition. After the anticrossing, the LAI lewel width shrinks, see subplot (c). The anticrossing broadening can be readily traced also in the case of anticrossing between LAI and the next light-hole-like level, LB2; shown in subplot (d), peaks L2±. This subplot demonstrates also significant narrowing of the Btype levels when they are verging the leaky window edges. The decrease in InAs QW width aA pushes the lower lasing level LAI upwards, simultaneously decreasing the level width associated with the interband tunneling. Since the uppermost position of the LAI subband reduces the thermal backfilling of the lower lasing states, it would be beneficial to keep LAI as high as possible in the leaky window. However, it is readily seen from Figure 12 that in the most important range, 8nm
0

50 100 electron energy e, meV

150

Fig. 14. Light quasibound states in the leaky window S = 150 meV. All curves are normalized to the highest peak value and labeled with the full peak width at half maximum. Each subplot is labeled with the GaSb QW width aB. The width of the InAs QW is aA = 12nm.

24

Electron-Phonon

Interactions

in Intersubband

Laser Heterostructures

963

subbands can be adjusted by proper design of the GaSb QW width aB, therefore, for the depopulation of the lower lasing states in subband LAI we may employ a competitive depopulating transitions LA1-»LB1 assisted by LO-phonon emission.32'33 4.2. Phonon enhancement of the depopulation process To demonstrate the possibility of the electron-LO-phonon resonance in interband transitions, we consider an isolated InAs/GaSb DQW heterostructure shown schematically in Figure 9b. This model structure explicitly reveals some important features of the depopulation process determined by the heterostructure asymmetry. Strain and space-charge related effects are not included in these exemplary calculations. To optimize the depopulation of the lowest electron-like subband, LAI, we need to provide for anticrossing with the highest light-hole-like subband, LB1, which also enhances the electron-phonon wave function overlap in the phonon emission process.15 This subband anticrossing can be arranged in the upper part of the leaky window if InAs QW width is about aA ~ 9 nm. We shall calculate and compare the LO-phonon emission rate for two different values of the InAs QW width, aA=9 nm and aA=8.5 nm, the later allowing for the higher position of the subband LAI; see Figure 12. u

•

8

(a)

6

(b)

•

1 R1

I

R2

R3

4 2 -

^as^*^ I

3a 1a -

u

I

0.5-

■

3bi

1b

A 6

2a

7

\ 2bi V

8

9

10

GaSb quantum well width d B , nm Fig. 15. Electron-phonon resonance in InAs/GaSb DQW heterostructure. Upper subplot shows LO-phonon emission rate r ph calculated for two values of InAs QW width aA = 9 nm (curve set a) and aA = 8.5 nm (curve set b). Smaller width of the InAs QW provides for the higher position of the lower lasing states in the leaky window. Three regions of the phonon emission rate, R1-R3, correspond to three different type of resonant transition in Brillouin zone. Lower subplot shows the total effective density of the final electron states D(E) for the phonon emission transitions. Level broadening r is taken 0.1 meV (thick solid lines), 1 meV , and 5 meV (thin lines in each curve set).

25

964

M. V. Kisin, M. Dutta & M. A.

Stroscio

in-plane wave vector K, nnf1

energy in leaky window e, meV

Fig. 16. Electron energy spectrum of an InAs/GaSb DQW with layer widths a\ = 8.5 nm and as- 9 nm. (a) subband diagram in the upper part of the leaky window illustrating three basic interband transitions assisted by the LO-phonon emission. The most important resonant transition, Rl, is indirect in Brillouin zone. (b) Effective density of states for electron subbands participating in the phonon-emission transitions. Level broadening parameter is r = 0.1 meV (solid lines), 1 meV (dashed lines), and 5 meV (dashed-dotted lines).

The upper limit for the phonon-assisted depopulation rate, r ph , can be obtained assuming the final states for electron transitions to be unoccupied. Since the LO-phonon energies are very close in both constituent materials (see Table 2), we neglect polar mode confinement and calculate the phonon emission rate using model bulk-like phonon spectrum and Eq. (2.13). The coupling constant can be simply averaged with respect to the layer widths: fieff=(fiA<*A+PB<'B)/(<'A+aB)-

(4-2)

Index / in Eq. (2.13) counts nondegenerate spin-split subbands. Since the spin is entangled in the multiband electron state envelopes % and If/; the electron-phonon interaction couples the initial electron state with the final states in either spin-split subbands. The integration over the q - directions in Eq. (2.13) returns factor % for each subband, which would recover usual factor 2n in case of non-split subbands. The width of the quasibound electron states in leaky window is determined by interband tunneling and in the upper part of the window is about 1 meV; see Figure 12. This small level broadening can be included in the r ph rate calculations by using Lorentzian lineshape function with half-width T instead of the ^-function. With electron-phonon coupling constant j3 = 1, the last integral in (2.13) characterizes the effective density of the final states Dj available in the subband Ej for electron transitions assisted by LO-phonon emission. Figure 15 shows the depopulation rate r ph calculated as a function of the GaSb Q W width aB for two different values of the InAs QW width. For a narrower InAs QW, the lower lasing subband LAI has moved higher in the leaky window, and, as a result, all the resonances occur at larger values of the GaSb QW width, aB. The increase of aB in the range from 5 nm to 10 nm, while keeping the energy position of the initial electron-like subband LAI practically unchanged, makes it possible to scan the final states in the hole­ like subbands Ej, here - the light subband LB1 and the heavy subband H2, which thus move toward the upper part of the heterostructure leaky window. Figure 15 clearly

26

Electron-Phonon Interactions in Intersubband Laser Heterostructures 965

demonstrates three distinctive regions in the LO-phonon emission rate, R1-R3, which are related to three consecutive resonances. The first, most remarkable resonance, Rl, corresponds to the onset of the LA1-»LB1 phonon-assisted transition, as illustrated in Figure 16a. For brevity, the initial electron state in the LAI subband is taken with zero inplane momentum, so that the electron wave vector in the final state is equal to the emitted phonon wave vector, K/=q. The anticrossing between LAI and LB1 subbands results in resonant penetration of the LAI subband states into the adjacent GaSb QW B and ensures sufficient electron-phonon overlap for this indirect interwell A->B transition. Rl-related transitions are indirect also in the /C-space, since the top of the upper light-hole-like 0.12

0.2

0.3

phonon wave vector q, nrrr' Fig. 17. Electron-phonon overlap integral I(q) for LA1-»LB1 (solid line) and LA1->H2 (dashed line) transitions in an InAs/GaSb DQW with layer widths aA = 9 nm and aB = 8 nm.

subband LB 1 is displaced to the final value of K due to the subband spin splitting inherent to asymmetric heterostructures.34"35 This splitting is especially strong in heterostructures based on the narrow-gap semiconductor materials.36 In Figure 16a, for convenience, we show the split subbands with only one sign of the spin polarization. The Kramers degeneracy condition (3.21), imposed on any system by time-reversal symmetry, should be used here to restore the complete subband structure. Note, that the final momentum transfer is important for the high phonon emission rate in Rl resonance, because the optimum value of the electron-phonon overlap I(q) can be engineered close to the peak of the effective density of the final electron states at the top of LB 1 subband; see Figure 16b and corresponding peaks la and lb in the lower subplot of Figure 15. With aB increasing, phonon-assisted LA1-»LB1 transitions become less efficient, firstly, because of corresponding decrease of the effective final density of states away of the top of the LB 1 subband, and secondly, due to the suppression of the electron-phonon overlap I(q) both at small and at the large momentum transfers. This can be readily seen from Figure 17, showing the electron-phonon overlap integrals for interband transitions

27

966

M. V. Kisin, M. Dutta & M. A.

Stroscio

as a function of the transferred (phonon) wave vector q = Kf. For a "vertical" transition with zero momentum transfer (q = 0), the electron-phonon overlap integral l{q) is reduced simply to the overlap between orthogonal initial and final electron states, and thus vanishes. This explains the pronounced minimum in the phonon emission rate between the peaks Rl and R2. The final electron states for the latter resonance belong to the ring of the LB1 subband saddle points, which are again characterized by a high efficient density of states; see Figure 16b and peaks 2a and 2b in the lower subplot of Figure 15. For aK = 9 nm, the peak of the effective density of states is more pronounced and is also favorably located at small negative values of Kf, where the electron-phonon overlap for L A 1 ( 0 ) H > L B 1 ( / 0 electron transition is again significant; see Figure 17. For aA = 8.5 nm, both the density peak 2b and electron-phonon overlap are less pronounced and, correspondingly, the peak rate value R2 is relatively small. Finally, the increase of the phonon emission rate in the region R3 corresponds to the onset of electron transitions into the H2 subband. The overall rate of the phonon emission is significantly depressed here because the most important transitions to the subband top, characterized by the highest effective density of states (peaks P3a and P3b in the lower subplot of Figure 15) correspond to inefficient vertical transitions with nearly zero electron/heavy-hole overlap integral l{q). Most of the R3 rate is actually determined by the transitions to the H2subband states with Kj< -0.1 nm"1, where the LA1-H2 overlap significantly increases due to the light-heavy state mixing effect in the H2 subband, see Figure 17. Finally, we compare the LO-phonon emission rates calculated for two different values of the InAs QW width; see the upper plot in Figure 15. For a narrower InAs QW, represented by curve set (b), the lower lasing subband LAI has been relocated to the extreme upper part of the leaky window. Still, the overall magnitude of the main peak Rl remains practically unchanged. This means that phonon-assisted depopulation can be conveniently employed even when the lower lasing level is designed near the upper edge of the heterostructure leaky window, where direct interband tunneling depopulation becomes inefficient. This design is beneficial for the laser performance providing for the highest value of the matrix element for intrawell optical lasing transition and simultaneously preventing thermal backfilling of the lower lasing states. 5. Conclusions LO-phonon assisted scattering determines intersubband electron relaxation in novel quantum well semiconductor lasers. We describe a simple phenomenological model, which allows comprehensive analysis of the LO-phonon mediated processes in intersubband laser heterostructures. As an application of our approach, we consider the process of the lower lasing level depopulation in type-II InAs/GaSb intersubband laser active region. We show that LO-phonon assisted transitions can significantly enhance the depopulation process comparing with direct interband tunneling through the heterostructure leaky window, traditionally considered as the main depopulation mechanism in type-II lasers. Inclusion of the phonon-assisted depopulation into the laser design scheme allows relocation of the lower lasing level into the upper part of the leaky window thus ensuring higher oscillator strength for the lasing transition and simultaneously preventing thermal backfilling of the lower lasing states.

28

Electron-Phonon Interactions in Intersubband Laser Heterostructures 967

Acknowledgments The authors gratefully acknowledge the collaboration with Prof. G. Belenky and Prof. S. Luryi. This work was supported by US Army Research Office. References 1.

2.

3.

4. 5.

6. 7. 8.

9. 10.

11. 12. 13.

14. 15. 16. 17. 18.

J.P. Sun, G.I. Haddad, M. Dutta, and M.A. Stroscio, "Quantum well intersubband lasers", in Advances in Semiconductor Lasers and Applications to Optoelectronics, Selected Topics in Electronics and Systems 16, World Scientific, Singapore, 2000, 21-53. S. Das Sarma, "Quantum many-body aspects of hot-carrier relaxation in semiconductor microstructures", in Hot Carriers in Semiconductor Nanostructures, Academic, New York, 1996,53-85. M. Dutta and M.A. Stroscio, "Advanced semiconductor lasers: phonon engineering and phonon interactions", in Advances in Semiconductor Lasers and Applications to Optoelectronics, Selected Topics in Electronics and Systems 16, World Scientific, Singapore, 2000, 419-431; for a general discussion of phonon processes in nanostructures see M.A. Stroscio and M. Dutta, Phonons in Nanostructures, Cambridge University Press, Cambridge, 2001. L.D. Landau and E.M. Lifshitz, "Electrodynamics of Continuous Media", Pergamon, Oxford, 1984. J.K. Jain and S. Das Sarma, "Role of discrete slab phonons in carrier relaxation in semiconductor quantum wells", Phys. Rev. Lett. 62 (1989) 2305-2308; S. Das Sarma, V.B. Campos, M.A. Stroscio and K.W. Kim, "Confined phonon modes and hot-electron energy relaxation in semiconductor microstructures", Semicond. Sci. Technol. 7 (1992) B60-B66. M. Born and K. Huang, "Dynamical Theory of Crystal Lattices", Oxford University Press, Oxford, 1954. K.J. Nash, "Electron-phonon interactions and lattice dynamics of optic phonons in semiconductor heterostructures", Phys. Rev. B 46 (1992) 7723-7744. SeGi Yu, K.W. Kim, M.A. Stroscio, G.J. Iafrate, J.P. Sun, and G.I. Haddad, "Transfer matrix method for interface optical-phonon modes in multiple-interface heterostructure systems", J. Appl. Phys. 82 (1997) 3363-3368. K.W. Kim and M.A. Stroscio, "Electron - optical-phonon interaction in binary/ternary heterostructures" J. Appl. Phys. 68 (1990) 6289-6292. E.O. Madelung. Semiconductors: Group IV Elements and III-V Compounds, Springer, New York, 1991; S. Adachi. Physical Properties of III-V Semiconductor Compounds, Wiley, NY, 1992. M.V. Kisin, M.A. Stroscio, G. Belenky, V.B. Gorfinkel, and S. Luryi, "Effects of interface phonon scattering in three-interface heterostructures", J. Appl. Phys. 83 (1998) 4816-4822. J. Faist, F. Capasso, D.L. Sivco, C. Sirtori, A.L. Hutchinson, and A.Y. Cho, "Quantum cascade laser", Science 264 (1994) 553-556. J.-P. Leburton, F.H. Julien, and Yu. Lyanda-Geller, "Advanced concepts in intersubband unipolar lasers", in Advances in Semiconductor Lasers and Applications to Optoelectronics, Selected Topics in Electronics and Systems 16, World Scientific, Singapore, 2000, 317-342. L.F. Register, "Microscopic basis for a sum rule for polar-optical-phonon scattering of carriers in heterostructures", Phys. Rev. B 45 (1992) 8756-8759. M.A. Stroscio, M.V. Kisin, G. Belenky, and S. Luryi, "Phonon enhanced inverse population in asymmetric double quantum wells", Appl. Phys. Lett. 75, 3258 (1999). N.C. Constantinou, "On the interaction of electrons with electromagnetically active excitations in bulk polar semiconductors", J. Phys.: Condens. Matter 3 (1991) 6859-6864. M. Zaluzny, "Intersubband absorption line broadening in semiconductor quantum wells", Phys. Rev. B 43 (19912) 4511-4514. M.V. Kisin, M.A. Stroscio, G. Belenky, and S. Luryi, "Electron-plasmon relaxation in quantum wells with inverted subband occupation", Appl. Phys. Lett. 73 (1998) 2075-2077.

29

968

M. V. Kisin, M. Dutta & M. A. Stroscio

19. G.L. Bir and G.E. Pikus, "Symmetry and Strain-Induced Effects in Semiconductors", Wiley, New York, 1976. 20. E.O. Kane, "The kP method", in: Semiconductor and Semimetals, eds. R.K. Willardson, A.C. Beer, 1, Academic, New York, 1996. 21. M.V. Kisin, B.L. Gelmont, and S. Luryi, "Boundary condition problem in the Kane model", Phys. Rev. B 58 (1998) 4605-4616. 22. G. Bastard, "Wave Mechanics Applied to Semiconductor Heterostructures", Les Editions de Physique, Les Ulis, 1988. 23. A.A. Grinberg and S. Luryi, "Electron transmission across an interface of different onedimensional crystals", Phys. Rev. B 39 (1989) 7466-7475. 24. T. Ando, S. Wakahara, and H. Akera, "Connection of envelope functions at semiconductor heterointerfaces", Phys. Rev. B 40 (1989) 11609-11618. 25. B. Laikhtman, "Boundary conditions for envelope functions in heterostructures", Phys. Rev. B 46(1992)4769-4774. 26. M.V. Kisin, "Phenomenological analysis of the boundary conditions for the wave function used in the Kane model", Sov. Phys. Semiconductors, 27 (1993) 274-278. 27. R. Q. Yang, and J.M. Xu, "Population inversion through resonant interband tunneling", Appl. Phys. Lett. 59(1991)181-183. 28. H. Ohno, L. Esaki, and E.E. Mendez, "Optoelectronic devices based on type-II polytype tunnel heterostructures",-4/?p/. Phys. Lett. 60 (1992) 3153-3155. 29. J.R. Meyer, C.A. Hoffman, and F.J. Bartoli, "Type-II quantum-well lasers for the midwavelength infrared", Appl. Phys. Lett. 67 (1995) 757-759. 30. Yu. B. Lyanda-Geller and J.-P. Leburton, "Phonon-assisted transmission in resonant interband tunneling devices", Appl. Phys. Lett. 67(1995) 1423-1425. 31. E. Anemogiannis, E.N. Glytsis, and T.K. Gaylord, "Quantum state engineering based on electromagnetic analogies and numerical methods for semiconductor intersubband lasers", in Advances in Semiconductor Lasers and Applications to Optoelectronics, Selected Topics in Electronics andSystems 16, World Scientific, Singapore, 2000, 389-418. 32. M.V. Kisin, M.A. Stroscio, S. Luryi, and G. Belenky, "Interband tunneling depopulation in type-II InAs/GaSb cascade laser heterostructure", Physica E 10 (2001) 576-586. 33. M.V. Kisin, M.A. Stroscio, G. Belenky, and S. Luryi, "Interband phonon-assisted tunneling in InAs/GaSb heterostructures", Physica B, (2002, in print); M.V. Kisin, M.A. Stroscio, G. Belenky, S. Luryi. "Electron-Phonon Resonance in InAs/GaSb type-II Laser Heterostructures". Appl. Phys. Lett. 80, (March 2002, in print). 34. G. Goldoni, and A. Fasolino, "Spin splitting in asymmetric double quantum wells", Phys. Rev. Lett. 69(1992)2567-2570. 35. M.V. Kisin, "Spin structure of the surface states and circular photogalvanic effect in heterojunctions." Sov. Phys. Semicond. 26 (1992), 441-445; M.V. Kisin. "The spin polarization of conduction electrons in boundary states", JETP Lett. 53 (1991) 306-309. 36. M.V.Kisin, "Interface states in the conduction band of an abrupt heterojunction", Sov. Phys. Semicond. 23 (1989) 180-183; M.V. Kisin. "Electron-type interface states in inverted heterojunction", Sov. Phys. Semicond. 24(1990) 1233-1235.

30

International Journal of High Speed Electronics and Systems, Vol. 12, No. 4 (2002) 969-994 © World Scientific Publishing Company

QUANTUM DOT INFRARED DETECTORS AND SOURCES P. BHATTACHARYA and A. D. STIFF-ROBERTS Solid State Electronics Laboratory, Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109-2122, USA SANJAY KRISHNA Center for High Technology Materials, Department of Electrical Engineering and Computer Engineering, University of New Mexico, Albuquerque, NM 87106, USA S. KENNERLY Sensors and Electron Devices Directorate, US Army Research Laboratory, Adelphi, Maryland 20783, USA

InAs/GaAs quantum dot devices have the potential to be the leading technology for infrared detection and emission, which are necessary for many military and domestic applications. Quantum dot infrared photodetectors yield higher operating temperatures, lower dark currents, and more wavelength tunability. They also permit the detection of normal-incidence light. Quantum dot infrared sources are also expected to yield higher operating temperatures, in addition to lower threshold currents and higher modulation bandwidths. After a brief review of the history of infrared detection and emission, the optical and electrical characteristics of self-organized In(Ga)As/GaAs quantum dots grown by molecular beam epitaxy are discussed, followed by results for the quantum dot detectors and emitters that have been developed at the University of Michigan, Ann Arbor. Keywords: Self-organized quantum dots; infrared detectors; infrared sources; intersubband devices.

1. Introduction Two hundred years have passed since 1800, when Sir William Herschel first dis­ covered infrared light. Since that time, infrared detection and emission have become cornerstones of the technological age. Infrared light is so appealing because it allows one to see when visible light does not. This so-called "night vision" is very familiar due to the frequent depiction of infrared imaging in popular culture, especially science fiction movies. However, infrared detectors and emitters are also crucial for an abundance of other applications. Infrared detectors are necessary for military targeting and tracking, law enforcement, medical diagnoses, space science, and even art. On the emitter side, there is an increasing demand for infrared sources for

31

970

P. Bhattacharya

et at.

optical IR spectroscopy, monitoring chemical species and pollutants, point-to-point atmospheric communication, remote controls, fiber optic telecommunication, and optical radars. Infrared light is important for these various applications because of two main reasons. First of all, the blackbody radiant emittance for objects with a temperature less than or equal to 1000 K peaks in the infrared wavelength range. Another reason infrared light is so important is that due to its longer wavelength, it does not have the same scattering/absorption characteristics as visible light. The earth's atmos­ phere has transparent windows where infrared light is not significantly absorbed by the carbon dioxide and water that are present. These wavelength ranges are the midwavelength infrared (MWIR) from 3-5 fim, the long-wavelength infrared (LWIR) from 8-14 ^m, and the far infrared (FIR) from 14-25 fim. These atmospheric windows make many of the infrared applications that are in use today possible. Quantum dot devices, comprised of self-organized In(Ga)As/Ga(Al)As quantum dots, have recently demonstrated very promising results as infrared detectors 1 - 1 8 and emitters. 1 9 - 2 3 These devices offer several favorable attributes. In terms of infrared detection, quantum dot infrared photodetectors (QDIPs) are inherently sensitive to normally incident infrared light, they can potentially achieve hightemperature operation (> 100 K), and they have a low dark current. 24 ' 25 As for quantum dot infrared emitters, the advantages of high-temperature operation, low threshold current, and high modulation bandwidth associated with typical quantum dot interband lasers are still expected in these devices. In addition, the electron relaxation time from the higher energy states of the quantum dot to the ground state is such that intersubband lasing is possible by inducing a population inversion through interband spontaneous emission. The details of these advantages, as well as device fabrication and performance will be discussed later. This chapter will review: (i) a brief history of infrared detection and emission, (ii) the optical and electrical characteristics of self-organized In(Ga) As/GaAs quan­ tum dots grown by molecular beam epitaxy (MBE), (iii) results for the quantum dot detectors and emitters that have been developed at the University of Michigan, Ann Arbor, and finally, (iv) a projection for the future of these devices.

2. Historical Background In 1800, Sir William Herschel discovered infrared light while conducting the following classic experiment. Sunlight was directed through a prism in order to obtain the visible spectrum of light. A thermometer was placed in the path of the different colors, and the temperature was measured as a function of energy (or light frequency). Much to his surprise, Herschel discovered that just outside of the spectrum, at a frequency below that of red, the thermometer measured the highest temperature, even though he could see no light. This newly discovered invisible light was called ultrared, and later infrared (or below-red) in 1870. During the 1800s, the development of the first thermal detectors of infrared radiation occurred. In

32

Quantum Dot Infrared Detectors and Sources

971

1830, a thermocouple was first used to measure radiant heat, and in 1880, the first bolometer was used to measure absorption in the earth's atmosphere. One hundred years after the discovery of infrared light, in 1900, Max Planck correctly described the radiant behavior of a blackbody. This significant accomplishment not only introduced the idea of quanta of energy, which eventually lead to the modern topic of quantum mechanics, but it also ushered in the era of microscopic science as opposed to macroscopic science. As a result, the 20th century is full of developments in light sources and semiconductor/solid state microelectronic devices. In addition to those mentioned above, a list of some significant scientific achievements related to infrared detection and emission can be found in Ref. 26. There are two main classes of infrared detectors: thermal and photon detectors. Thermal detectors are made of materials whose physical properties change in the presence of radiant heat. The most common thermal detectors are: (i) thermo­ couples, which experience a change in voltage at the junction of two different solid state materials; (ii) bolometers, which experience a change in the resistance of bulk metal; and (iii) pyroelectric detectors, which experience a change in the surface charge of a material. Thermal detectors, which are readily available commercially, are less expensive than photon detectors, and they offer uncooled operation, which means that they are more portable for field applications. However, thermal detectors generally have a slower response to changing input, and they cannot offer two-color detection in a single device, a requirement for achieving high-resolution imaging. Thus photon detectors are often preferred for more sophisticated applications. Photon detectors, which directly detect incident infrared radiation, can be either photoconductive (generate a corresponding photocurrent) or photovoltaic (generate a corresponding photovoltage). Photon detectors are usually made of semiconductor materials, be it three-dimensional bulk material (like HgCdTe), semiconductor heterostructures (like Type II InSb superlattices), or low-dimensional heterostructures (like III-V quantum well- and quantum dot-infrared photodetectors). All photon detection technologies are limited in that the detecting element must be cooled, which means that a dewar, cryostat, or thermoelectric cooler of some type is required. Photon detectors must be cooled in this way because the energy corresponding to infrared light is so small that the dark current generated at higher temperatures easily drowns any real signal created by the detector. The advan­ tages and disadvantages of three types of photon detectors; mercury cadmium telluride detectors, 2 7 - 2 9 quantum well infrared photodetectors, 3 0 - 3 3 and quantum dot infrared photodetectors, 1 - 1 8 are discussed below. Mercury cadmium telluride (MCT) detectors are intrinsic, bulk material detectors, which means they detect IR light corresponding to the energy gap of the semiconductor. MCT detectors are the industry standard, used in most state-of-theart infrared imaging cameras in the MWIR and LWIR ranges. However, there are some drawbacks to this technology. First, there are difficulties in growing MCT, such as the requirement for effusion cell temperature feedback/control during growth for consistent material composition. Moreover, MCT experiences nonuniform dopant

33

972

P. Bhattacharya

et al.

incorporation, which leads to variations in device responsivity, adversely affecting the pixel operability of large area focal plane arrays. Another disadvantage, due to Auger recombination processes that severely reduce photoexcited carrier lifetimes, is that the MCT detector requires an operating temperature less than or equal to 80 K.28 The GaAs/AlGaAs quantum well infrared photodetector (QWIP) is an alterna­ tive technology that detects infrared (IR) light through intersubband transitions in the conduction band. The required operating temperature for QWIPs (< 60 K) is lower than for MCT detectors because thermionic emission in MCT, for equivalent device parameters, is approximately five orders of magnitude less than in a QWIP. 2 8 ' 3 3 The QWIP benefits from mature III-V growth and processing techniques. QWIPs are also extremely uniform across a large area, which increases the pixel operability in a focal plane array. However, QWIPs require the fabrication of random reflectors at the top of detector pixels in order to allow lateral incidence since they cannot detect normally incident light due to polarization selection rules. 30 In(Ga)As/Ga(Al)As quantum dot infrared photodetectors (QDIPs), which also detect light through intersubband transitions in the conduction band, as shown in Fig. 1, benefit from the same advantages as the QWIP in terms of III-V growth and processing. In addition, there are three main advantages of the QDIP: normal-incidence detection, high-temperature operation, and low dark current. The polarization selection rules corresponding to three-dimensional electron con­ finement in quantum dots allow QDIPs to detect normally incident light. 5 ' 9 ' 13 High-temperature operation in QDIPs results from a large electron relaxation time from the higher energy states of the quantum dot to the ground state. As a result, photoexcited carriers that escape from the dot have a higher probability of contributing to the photocurrent before relaxing back into the ground state. 11 ' 24 Thus, these increased relaxation times can lead to improved responsivity of the QDIP, which in turn allows higher operating temperatures. QDIPs are also

ph

E

c i

>m^^ 9 LAE

I GaAs

«

0.14eV

•

:

'■

3aAs

(

InAs Fig. 1. Intersubband transition within the conduction band of an InAs/GaAs quantum dot for IR detection.

34

Quantum Dot Infrared Detectors and Sources 973

theoretically predicted to have lower dark currents than MCT detectors and QWIPs. 25 This is predicted because the three-dimensional quantum confinement of the electron wave function leads to the equality of the photoionization energy and the activation energy of thermionic emission in quantum dots. The main disadvantage of the QDIP is the random variation of dot size, shape, and com­ position due to the Stranski-Krastanow growth mode. As a result, the uniformity of quantum dots across a large area is reduced. Another growth-related issue is that nonuniform dopant incorporation adversely affects the responsivity of the QDIP, as in the MCT detector. As far as infrared emission is concerned, the most abundant source of infrared light is the sun. However, sunlight is not a convenient source for laboratory exper­ iments and practical applications. Instead, blackbody sources such as the Nernst glower (typically made from zirconia, yttria, or thoria) and the globar (made from silicon carbide) are often used. These sources usually emit light from the visible range to the far infrared (~ 30 /xm). Two other important sources of infrared light are solid-state light emitting diodes (LEDs) and lasers. While there are many types of LEDs and lasers that emit light in the near-infrared (from 0.9 to 2 /xm), there are very few that emit light in the MWIR, LWIR, and FIR ranges. Due to the small energy spacing that corresponds to infrared light in the desired range, bulk material is usually not suitable for such devices. Instead, there is a heavy dependence on quantum well and quantum dot devices, which are relatively new developments, especially when compared to the long history of infrared detection. Early reports on light emitting diodes 3 4 - 3 6 were followed by an account of the quantum cascade laser, 3 7 - 3 9 a novel unipolar semiconductor laser based on intersubband transitions in quantum wells. MIR emission has also been reported in quasi one-dimensional wires grown by molecular beam epitaxy (MBE) and in optically pumped quantum fountain lasers. 40,41 Recently, room temperature photoluminescence has been reported in the 3-4 /xm range using PbSe/PbSrSe multiple quantum well structures. 42 Self-organized quantum dots are expected to display MIR emission and absorption characteristics as the energy spacing of the bound states in these dots lies in the MIR regime. Vorob'ev et al. have reported the observation of weak MIR emissions from interband InGaAs/GaAs quantum well and InGaAs/AlGaAs quantum dot lasers. 19 More recently, Krishna et al.20~22 and Grundmann et al.,23 have reported mid-infrared emission (12 and 16 /xm) from near-infrared quantum dot lasers.

3. Self-Organized Quantum Dots for Devices The realization of a high density of small ( « 100-200 A) and uniform quantum dots (QDs) has been elusive. The most direct approach, that is, epitaxial growth of a quantum well followed by controlled etching, provides the requisite size and uniformity. However, surface defects produced by the etching process reduce the radiative efficiency to levels that are not suitable for lasers, or other types of quan-

35

974 P. Bhattacharya et al.

turn dot devices.43 Recently, self-organized quantum dots have proven to be the structures which best approach desired device properties. 4 4 - 4 6 The use of strain to produce self-assembled quantum dot structures is now a well-accepted approach and is widely used in III-V semiconductors and other material systems. Much progress has been made in the area of growth, 4 7 - 5 0 where the focus has been size control, as well as optical characterization, 51-59 where the focus has been application to quantum dot detectors, sources, and other types of optoelectronic devices. The use of defect-free strain-induced self-organized quantum dots provides several advantages. Due to the pyramidal shape of these dots and the complicated strain tensor with a strong hydrostatic component within them, large modulation of the interband photon energy can be produced. For example, InAs/GaAs quantum dot lasers emit at « 1 /xm, a wavelength much smaller than that corresponding to the bandgap. Thus, another means of tunability is introduced. It has been shown that highly lattice-mismatched In(Ga)As epitaxially grows on GaAs in the so-called Stranski-Krastanow growth mode, where self-organized islands are formed after a few monolayers of layer-by-layer growth. 60 From RHEED measurements during molecular beam epitaxy (MBE) of InGaAs on GaAs and from energy minimization considerations in a unit cell of the growing layer, it has been determined that for a misfit / > 1.8%, the island mode of growth is preferred. 61 For typical growth parameters used in MBE or metal-organic vapor phase epitaxy (MOVPE), an array of pyramidal islands of widths from 10-40 nm and heights from 5-8 nm are formed. Elastic relaxation on the facet edges, renormalization of the surface energy of the facets, and interaction between neighboring islands via the substrate are the driving forces for self-organized growth. As will be described later, there are considerable strain fields within the pyramidal dots, in the substrate un­ derneath, and in the overlayer, if the latter is grown. In situ atomic force microscopy (AFM) studies during growth of InAs on GaAs have given valuable insights into the evolution of the size distribution between dots, as growth proceeds, and a tendency for eventual size equalization. 49 Careful studies of growth in the InAs-GaAs system have also shown that there exists a relatively narrow range of deposition parameters where the islands are small (« 10 nm), have very similar size and shape, and form dense arrays. 45 Interaction of the islands via the substrate also makes their lateral ordering favorable.62 By virtue of their size and shape, the self-organized islands best approach the desired properties of zero-dimensional quantum dots. An AFM image of an array of Ino.4Gao.6As/GaAs dots grown by MBE at 540°C and a rate of 0.25 monolayers/sec is shown in Fig. 2(a). From this image, the dot density is estimated to be 5 x 1010 c m - 2 . The pyramids have a base diagonal of 20 nm and a height of 7 nm. The cross-sectional transmission electron microscope (TEM) image of a single InAs dot grown by MBE at 500°C is shown in Fig. 2(b). If a layer of InGaAs dots is covered with a thin layer of GaAs and another InGaAs growth cycle is initiated, the dots in the second sheet are formed exactly on top of the dots in the first layer and this trend continues, resulting in a 3D array of

36

Quantum Dot Infrared Detectors and Sources

(a)

975

(b)

Fig. 2. (a) AFM image of a single layer of exposed self-organized In(Ga)As quantum dots. Dot density estimated from this image is 5 x 10 1 0 c m - 2 ; and (b) cross-sectional T E M image of a single InAs quantum dot.

vertically aligned and electronically coupled dots. 63 ' 64 Such multiple layer quantum dots (MLQDs) are very useful for device applications. The optimum growth condi­ tions for multilayer dots have been described in detail previously. 58,59 Some salient features, relevant to detector and emitter operation, are reiterated here. Usually, a smaller thickness of InGaAs needs to be deposited for subsequent quantum dot layers. This is because the wetting layer thickness progressively decreases. For example, in the case of Ino.4Gao.6As/GaAs quantum dots, seven monolayers (MLs) of InGaAs need to be deposited for the first dot layer, and subsequent dot layers are formed with 3-5 MLs of InGaAs, depending on the GaAs barrier layer thickness.

4. Electronic Spectra and Carrier Dynamics in Self-Organized Quantum Dots Bandstructure calculations of individual Ino.4Gao.6As/GaAs quantum dots based on an eight-band, k • p formalism, including the strain distribution in the dots, predicts the bandstructure shown in Fig. 3. 6 5 There are two electron levels and several hole levels confined in the dots. In real quantum dot ensembles, these discrete levels are inhomogeneously broadened due to the size variation of the dots. In addition, level splittings occur due to interdot coupling, causing the formation of bands of electronic levels around the central excited- and ground-state levels. The excited level in each dot has a two-fold degeneracy due to the symmetry of the dot geometry. In the four vertically coupled dot configuration, the excited levels form a band of eight levels each of which has a spin degeneracy of two. The ground state band consists of four levels, each with a spin degeneracy of two. Photoluminescence data depicting the ground state and excited state transitions in Ino.4Gao.6As/GaAs dots are shown in Fig. 4. There are higher order electron states in dots with different compositions. Hence, the electron interband, as well as intersubband, energies can greatly vary with dot size, dot composition, and the heterostructure band offsets.

37

976

P. Bhattacharya

et al.

Eight-band model

1.5

^ >

1 1.2499eV

1.1003eV

\ 0.5

0 Fig. 3.

Theoretical eight-band, k • p bandstructure of a single Ino.4Gao.6As/GaAs quantum dot.

Energy (eV) Fig. 4. Photoluminescence measurements of Ino.4Gao.6As/GaAs quantum dots. Excited state transitions are visible in addition to ground state transitions with increasing excitation intensity. The curves of largest, median, and smallest areas correspond to 2 K W / c m 2 , 100 W / c m 2 and 0.1 W / c m 2 , respectively.

Femtosecond pump-probe differential transmission spectroscopy (DTS) mea­ surements have been performed on four-layer Ino.4Gao.6As/GaAs quantum dot heterostructures at temperatures > 10 K for a range of excitation levels. 66 ' 67 Electron-hole pairs are generated in the barrier region of the dots using a 100 fs, 800 nm pump beam. The DTS signal at the ground and excited state transition energies are then measured as a function of the delay between the pump and probe pulses. Since the DTS signal is proportional to the occupation number of each level, the relaxation times are obtained directly using this technique. In terms of QDIP operation, the results from the DTS measurements suggest that at temperatures of 77 K and higher, there is a significant electron lifetime ( « hundreds of picoseconds) in the higher-lying states. With an applied transverse bias, the lifetime can become even larger. When IR photons are absorbed by the QDIP, electrons are excited to the higher-lying states directly, or they are raised to the dot excited states, from where they are emitted to the higher-lying states. The probability of these electrons (which contribute to the photocurrent) relaxing back into the ground state is small, particularly at high temperatures. It is important

38

Quantum Dot Infrared Detectors and Sources 977

to realize that the increased lifetime of electrons in the higher-lying states does not reduce the dark current in the same way increased carrier lifetime does in small bandgap junction IR detectors; however, photocurrent and responsivity are favorably impacted.

5. Quantum Dot Infrared Detectors and Focal Plane Arrays The lateral QDIP and the vertical QDIP are two general device structures that have been studied. The lateral QDIP, which operates much like a field-effecttransistor, conducts photocurrent through lateral transport of carriers across a high-mobility channel. AlGaAs barriers, which provide this high-mobility channel, are also necessary to modulation-dope the quantum dots. Since the major con­ tributions to the dark current in lateral QDIPs are due to interdot tunneling and hopping conduction, these devices have demonstrated lower dark currents and higher operating temperatures than vertical QDIPs. 12 The vertical QDIP conducts photocurrent through vertical transport of carriers. In this case, the quantum dots are directly doped to provide free carriers during photoexcitation, and an AlGaAs barrier can be included in the vertical device to block dark current created by thermionic emission.16 Both types of devices can be grown by solid source molecu­ lar beam epitaxy (MBE) or metal-organic vapor phase epitaxy (MOVPE). Typical MBE growth conditions for both devices are related below. For the lateral QDIP, a 1 /xm GaAs buffer layer is grown at 620° C on a semiinsulating (100) GaAs substrate, followed by a 300 A Alo.1sGao.82As barrier, which forms a high-mobility channel with the subsequent 500 A GaAs spacer layer. The 18% AlGaAs layer is silicon-doped (n = 1 x 10 17 c m - 3 ) in order to provide free carriers to the quantum dots by modulation-doping. Next, the substrate tempera­ ture is decreased to 500°C, and 2.2 ML of InAs are deposited to form the quantum dots. A 500 A GaAs cap layer is then grown over the dots, and this sequence is repeated nine times for a total of ten InAs/GaAs quantum dot layers bordered by an 18% AlGaAs layer on either side. After the final 18% AlGaAs barrier deposition, a silicon-doped (n = 2 x 10 18 c m - 3 ) GaAs contact layer is grown for the two top metal contacts of the device. The device heterostructure is shown in Fig. 5(a). For the vertical QDIP, a 0.5 pm silicon-doped (n = 2x 10 18 c m - 3 ) GaAs contact layer is deposited on a semi-insulating (100) GaAs substrate at 620°C, followed by a 250 A intrinsic GaAs buffer layer. The substrate temperature is decreased to 500°C, and 2.2 ML of InAs are deposited to form the directly-doped quantum dots (n = l x 10 18 c m - 3 ) . A 250 A intrinsic GaAs cap layer is grown on top of the InAs in order to complete the quantum dot barrier. This sequence of growth is then repeated nine times for a ten-layer InAs/GaAs quantum dot active region. After the final GaAs layer is grown, the substrate temperature is increased to 620° C, and 400 A of intrinsic Alo.3Gao.7As are deposited in order to form a current-blocking barrier at the top of the device. Finally, a 0.1 /zm silicon-doped (n = 2 x 10 18 c m - 3 ) GaAs top contact layer is grown, as shown in the device heterostructure in Fig. 5(b).

39

978

P. Bhattacharya

0.05 urn

et al.

GaAs Contact

(n=1e17crrr3)

300 A 18% AIGaAs Barrier (n=1 e17ctrr 3 )

500 A

GaAs Spacer (i)

2.2 ML

InAs Quantum Dots

500 A

GaAs Spacer (i)

0.1 pm GaAs n=2x10l,cm-3 400 A Al o 3Ga07As spacer I

x10 3

250 A

GaAs spacer

WML

InAsQD

n=1x10"cm^

300 A 18% AIGaAs Barrier (n=1e17cnr )

250 A

GaAs

I

1 jcm

0.5 pm

GaAs

n=2x10"cm- J

GaAs Buffer (i)

}

X10

S.I. GaAs Substrate

S. I. GaAs Substrate

(a)

(b)

Fig. 5. Molecular beam epitaxy heterostructures for (a) a modulation-doped lateral QDIP and (b) a directly-doped vertical QDIP.

(a)

(b)

Fig. 6. SEM micrograph of fabricated device for (a) a lateral QDIP with optical area 8.4 x 10 3 /im 2 and (b) a vertical QDIP with optical area 2.83 x 10 5 /jm 2 .

The lateral and vertical QDIPs are then fabricated using standard photolithog­ raphy and wet-etching techniques. The lateral QDIP requires a two-step process. First, the two top Ni/Ge/Au/Ti/Au metal contacts with interdigitated fingers are evaporated, followed by a recess etch in order to prevent shorting of the device. Second, a mesa etch is performed in order to define the active region of the device. The vertical QDIP requires a three-step process. The first step comprises metal evaporation for the top ring contact. Second, a mesa etch ( « 1 /zm) around the top contact defines the active region. Third, the metal evaporation is repeated for the bottom ring contact around the device mesa. Ohmic contacts are achieved in both devices by annealing at 400°C for approximately one minute. The fabricated lateral and vertical devices are shown in Figs. 6(a) and 6(b), respectively. While there are performance advantages in using lateral QDIPs, it will be very difficult to fabricate these devices as focal plane arrays since each pixel requires three contacts (or three bump bonds), two for the top lateral contacts and one for a common ground. Therefore, it is necessary to improve the performance of the vertical QDIP since it is much more compatible with commercially available read-out circuits. In the remainder of this section, the performance of a vertical InAs/GaAs QDIP with a current-blocking Alo.3Gao.7As barrier is examined.

40

Quantum Dot Infrared Detectors and Sources

979

Dark current, spectral response, and blackbody response measurements are con­ ducted in order to characterize the vertical QDIP. Typical dark current-voltage characteristics are shown in Fig. 7 for a range of temperatures from 78 K to 295 K. The dark current of this QDIP (J dar k = 1.7 pA, Vhias = 0.1 V, T = 100 K) is much lower than that measured in a similar Ino.15Gao.s5As/GaAs QWIP device (-Jdark = 10 /iA, Vbias = 0.1 V, T = 60 K). 3 1 This reduction in dark current is due to the AlGaAs barrier at the top of the device heterostructure. The asymmetry in the I-V curves is also due to the AlGaAs barrier in that it only blocks current near the top contact. The activation energy (Ea), determined by considering the linear sections of the Arrhenius plots of the dark current, is shown as a function of bias in Fig. 8. The asymmetry of the activation energy is also a direct result of

-2.0

-1.0

0.0

1.0

2.0

Bias Voltage (V) Fig. 7.

Dark current-voltage characteristics for temperature range from 78 K to 295 K.

380

-♦-activation energy (meV) •

cutoff energy (meV)

Arrhenius Equation: lnl(tart< = l n A - ( E . / k B ) ( 1 / T ) 20 -4.0

-2.0 0.0 Bias Voltage (V)

2.0

Fig. 8. Activation energy, E& (♦), as a function of bias voltage, as calculated from Arrhenius plots. The calculated activation energies show reasonably good agreement with the measured cutoff energies (•) of the spectral response for several bias voltages.

41

980

P. Bhattacharya

et al.

the asymmetry in the heterostructure. As shown in Fig. 8, these activation energies agree with the cutoff energies measured in the spectral response of the detector, which is discussed next. A Fourier Transform Infrared (FTIR) spectrometer with a broadband (1 to 20 /xm), high intensity source is used to determine the spectral response of the QDIP at normal-incidence. The spectral response is obtained for a detector temperature of 78 K and a bias range from —1.0 V to 0.25 V. Figure 9(a) depicts the spectral response at a bias of 0.1 V. The peak wavelength, Apeak! is 3.72 /xm, and the linewidth, AA/A, is 0.3, most likely a bound-to-continuum intersubband transition. 5 The spectral characteristics (Apeak and AA/A) of the vertical QDIP change with bias, as shown in Fig. 9(b). As the bias becomes less negative, Apeak blue-shifts to shorter wavelengths and AA/A decreases. The blue-shift of the peak wavelength is due to the decrease in band bending of the conduction band as the bias voltage nears zero volts. The significant decrease in AA/A is not a function of device operation, but rather results from the strong atmospheric absorption that occurs below 3 /xm. The blackbody response of the vertical QDIP is measured as a function of detec­ tor temperature and bias voltage. A calibrated, 800 K blackbody source is used to determine the absolute responsivity of the QDIP to normally incident IR radiation, and a germanium block is used to filter out near-IR radiation (< 1.8 ^m) emitted by the blackbody. The QDIP photocurrent signal and noise are measured with a Fast Fourier Transform (FFT) Analyzer. A flat-band noise spectrum is desired be­ cause it indicates that the dominant noise mechanism is generation-recombination (GR) noise, as assumed in most theoretical calculations. The QDIP is character­ ized at 78 K, 100 K, 125 K, and 150 K. For temperatures greater than 150 K, the signal-to-noise ratio measured by the FFT analyzer is less than one. The best 1.00

0.80

0.60

0.40 -

0.20

0.00 2.25

3.75

5.25

-1.1

6.75

Wavelength (urn)

-0.8

-0.5

-0.2

0.1

Bias Voltage (V)

(a)

(b)

Fig. 9. Relative spectral response of vertical QDIP at a bias of 0.1 V and temperature of 78 K and (b) bias voltage dependence of the peak wavelength, Apeak, and the FWHM linewidth, AA/A, at 78 K.

42

Quantum Dot Infrared Detectors and Sources

3x109

981

Detector Temp = 100 K Blackbody Temp = 800 K

I

jg 2x1O9 £ o 1x109 a. -0.9

-0.3

0.3

100

Bias Voltage (V)

125

Temperature (K)

(a)

(b)

Fig. 10. (a) Peak responsivity, Rpeak, and peak specific detectivity, D*, as a function of bias volt­ age calibrated by an 800 K blackbody at a temperature of 100 K; and (b) the detector temperature dependence of D".

device performance is measured at 100 K, and the bias-dependent # pe ak and D* values for the device at this temperature are shown in Fig. 10(a). The responsiv­ ity, and therefore, the detectivity are relatively low at negative biases because the AlGaAs barrier prevents carriers from being collected at the contact under reverse bias. Even the maximum -Rpeak value, 2 mA/W for a bias of 0.3 V at 100 K, is low because of the AlGaAs barrier, which blocks photocurrent as well as dark current. The responsivity quickly increases for positive biases, however, and at a low forward bias, a large D* can be obtained before the dark current increases and drives down the signal-to-noise ratio. A maximum D* of 2.94 x 109 cmHz 1//2 /W at a bias of 0.2 V is obtained at 100 K, and this is a significant milestone in the performance of normal-incidence, vertical QDIPs. The temperature dependence of the maximum D* values is shown in Fig. 10(b). Since the blackbody response measurement does not consider the wavelength of IR light that is detected, the peak values .Rpeak and D* shown in Fig. 10 are calculated using a blackbody-to-peak conversion factor. This conversion factor, T, which is inversely proportional to the relative response (per watt) measured by the spectral response, multiplies both the responsivity and detectivity calculated from the signal and noise values obtained during the blackbody response measurement: ^Ipeak — I X

D*

rx

^pho

(i)

-* incident ■*photo

v-^-detector^/noiseJ

(2)

incident

where 7photo is the measured photocurrent, incident is the photon power incident on the detector determined from the blackbody calibration, ^detector is the approx­ imate optical area of the device, A / n o j s e is the bandwidth over which the noise

43

982

P. Bhattacharya

et al.

voltage is measured, and 7 no j se is the measured noise current. Another important quantity that must be calculated is the photoconductive gain in the quantum dot detector. The gain mechanism in these vertical QDIPs is due to the increased carrier relaxation times in the excited states of quantum dots, which decrease the capture probability of free carriers in quantum dots. 11 ' 24 The photoconductive gain for a QWIP can be expressed in terms of the capture probability by 11,68,69 : 9 =

(3)

Np(l + p) '

where p is the capture probability (p
0

-0.5

Bias Voltage (V)

Fig. 11.

Photoconductive gain as a function of bias voltage for a detector temperature of 100 K.

44

Quantum Dot Infrared Detectors and Sources

983

0 1 Bias Voltage (V)

(a)

(b)

Fig. 12. (a) Room temperature (300 K) dark current I-V curves for each pixel in an address­ able 4 x 4 QDIP array; and (b) SEM micrograph of a (9 x 9) interconnected, nonaddressable InAs/GaAs QDIP array with 40 (am mesa size and 120 fim pitch.

While raster-scan imaging should work with a single detector, a small array is actually used because it is easier to collect infrared light over a larger area. Also, the average photocurrent from an array should be larger, and therefore, much easier to distinguish from background noise. Standard photolithography and wet-etch processing are used to fabricate (4 x 4), (9 x 9), and (13 x 13) individually address­ able and nonaddressable (interconnected) arrays of vertical, mesa-shaped QDIPs. The pixel diameter is 40 um, and the array pitch is 120 /xm. The photomicrograph of an interconnected, (9 x 9) QDIP array is shown in Fig. 12(b). The entire array effectively behaves as a single detector with a very large optical area and a single photocurrent signal. Examples of images obtained by the vertical QDIP array through this rasterscan technique are shown in Fig. 13. Figure 13(a) shows the image of a 20 W

(a)

(b)

(c)

Fig. 13. Raster-scanned images obtained from the (13 x 13) QDIP array at 80 K: (a) 20 W broadband infrared globar source through a circular pinhole, (b) heated graphite furnace igniter through a circular hole and linear slit, and (c) heating element from a hot plate, shown schemat­ ically in the inset, partially showing two metal strips.

45

984

P. Bhattacharya et al.

broadband infrared globar source through a circular pinhole. Figure 13(b) is the image of a 700° C furnace igniter shielded by an aluminum block with a circular pinhole and a linear slit. Figure 13(c) depicts the partial image, in this case limited by the field-of-view of the scanning mirrors, of the heating element of a hot plate at 500°C. Portions of the heated strips (schematically shown in the inset) are seen as the bright regions. These results indicate that, in spite of the low responsivity of the QDIPs, they can be used for imaging. With progressive improvement of device performance, they should be applicable to focal plane arrays.

6. Quantum D o t Infrared Sources The electron energy level spacing between (i) the ground state and the first excited state and (ii) the ground state and the GaAs conduction band edge in In0.4Ga0.eAs/GaAs self-organized quantum dots are about 60-80 meV and 230-250 meV, respectively, as determined from theory and experiments. 59,65 From analysis of the small-signal modulation of quantum dot interband lasers, electron relaxation times from higher-energy states to the ground state at room temper­ ature have been estimated to be as long as 30-50 ps. 24 This is supported by DTS measurements, 66 ' 67 as described in Sec. 4. These favorable relaxation times invoke the possibility of intersubband lasing in quantum dots. This was first suggested by Singh, who proposed the use of an external interband laser to rapidly depopulate the ground state electrons by stimulated emission, thus creating a favorable nonequilibrium carrier distribution between the ground and excited states for MIR emission. 70 Recently, Kastalsky et al., have theoretically analyzed a simi­ lar dual-color laser using a three-level carrier rate equation. 71 Also, Krishna et al. recently demonstrated intersubband stimulated emission in interband quantum dot lasers. 22 For population inversion to occur, the energy relaxation time between the upper level excited states and the ground state, T„I, should be as long as possible, and the lifetime of electrons in the ground states, TI, should be very short. Providing a high density of coherent photons in the cavity, which can greatly reduce the interband electron-hole recombination time, T st j m , can decrease the lifetime of the electron in the ground state. A high density of coherent photons can be made available in the intersubband laser cavity by simultaneous interband lasing due to current injection. In order to examine population inversion between the ground state and the excited dot states, the following rate equations are solved self-consistently: dnu _ J nugu(l — fi) n i g i ( l — /„) —— _ -qln I at e T„I T\u nPhunu9u(l - fu- fh) _ „

46

TU

,.■.

Quantum Dot Infrared Detectors and Sources 985

dnx

,

.J

nugu{l-f-i)

= (1 - r)m)- H at e +

ni#i(l - fu) T\u

TUI

rcPhirciffl(l-/l ~fh)

=

nigxhfh T\

0

,5x

n where 77jn is the injection efficiency; n p hi and nphu &re the photon occupation number of the ground state and the upper level excited states that are involved in the lasing process, respectively; n\ and nu are the density of ground states and excited states, respectively; and f\ and fu are the occupational probabilities of the ground state and excited states, respectively. The degeneracy of the ground state (<7i = 2) and the excited states (gu = 4) 65 are taken into account in the rate equa­ tion calculations. Interband recombination times for the ground state, T\ (= 700 ps) and the excited state, TU (= 250 ps) have been derived from time-resolved photoluminescence measurements, 59 and an intersubband relaxation time of r„i = 60 ps is assumed. A thermal distribution of holes is used (fh = 0.45). When the ground state photon occupation number is zero, f\ > fu for all values of the injection current, and no population inversion is possible. However, when the ground state photon occupation number is increased to 50, / i is pinned at a value of ~ 0.5, whereas /„ increases linearly with the current. Again, when the number of photons in the excited state (nph«) is increased to 50, / i > /„ for all values of injection. This is consistent with the fact that if nph« = 50, interband lasing occurs from the excited state in the dot and no MIR emission results. The overlap integral and the intersubband gain are calculated for various injection levels in accordance with the equation below 72,73 :

,(M = *f%*f

i

Neexp

(Jto-Eif\ mK) _ m))].

(6)

' mlcnre0hu y/lA&ra \ 1.44cr2 / The results of the calculation are shown in Fig. 14. It is evident that gains up to 170 c m - 1 can be achieved even if an inhomogeneous broadening of 20 meV is assumed for the interband transition. Nishi et al. have reported photoluminescence linewidths of 21 meV at room temperature. 74 The intersubband population inver­ sion and lasing processes, together with the bipolar recombination, are illustrated in Fig. 15. The results described here are performed on multi-dot layer, single-mode, ridge waveguide, interband lasers (A ~ 1 /zm) grown by solid-source molecular beam epitaxy. The waveguide is designed for near-infrared emission. The gain region consists of a four dot-layer stack of Ino.4Gao.6As dots, separated by 15 A GaAs barriers in the middle of a GaAs waveguide and surrounded by 1 ^m Alo.3Gao.7As outer cladding layers and appropriate GaAs contact layers. Lasers are fabricated using standard photolithography, lift-off techniques, and a combination of dry and wet etching. The width of the waveguide is 3 /im, and the length of the laser varied from 400-600 ^m. The output light is directed through a band pass filter, after which the light is coupled to a liquid nitrogen cooled MCT detector, which can detect radiation from 5-26 /xm. Bandpass filters are used to

47

986

P. Bhattacharya

et al.

180 J=3.0 J=4.0

.

T21

/

J=5.0

80

NP„= 50 = 60 ps

/"

<j= 20 meV AE = 50 meV

/

J=6.0

V.

/

91=2

\g2=4

O

-20 0"

2CiV\ 40

60

.//&

-O

-120

-220

J expressed in k A / c m 2

\^

Intersubband energy separation (meV)

Fig. 14. Calculated intersubband gain in Ino.4Gao.6As/GaAs quantum dots using a two-photon rate equation model. The curves with the larger positive gain, smaller positive gain, smaller negative gain, and larger negative gain correspond to J = 6.0, J = 5.0, J = 4.0, and J = 3.0, respectively.

©

.1-2 p s

A/v

t21~100ps^

FIR

0 t

A.

slim= t , « 6 p s /\.fc.~lum

W

6 ps

^ Fig. 15. Schematic illustration of electronic bound states, approximate carrier relaxation times and t h e intersubband population inversion process.

48

Quantum Dot Infrared Detectors and Sources

l/lth

987

Wavelength (um)

(a)

(b)

Fig. 16. (a) Spontaneous emission as a function of injected current. The best fit to the MIR emission (Jx) is with x = 2.5; and (b) radiation from a blackbody source at (i) T — 420 K and (ii) T = 200 K, corresponding t o heat sink temperatures of 300 K and 80 K, respectively. The dotted line represents the room temperature F I R emission from the interband quantum dot laser at / = 1.30/ t h .

select the MIR output. Figure 16(a) depicts the MIR signal amplitude as a function of injection current at T = 17 K and T = 300 K. Spectral measurements of the MIR output are performed on the quantum dot lasers at T = 80 K and T = 300 K. The lasers are wire bonded and mounted in a cold finger cryostat with a ZnSe window and are biased with a low frequency positive pulse ( / = 10 Hz, duty cycle = 25%). A silicon filter is used to block the interband signal at 1 /an, and MIR emission is measured as a function of injection bias using an FTIR spectrometer. The recorded data is corrected for the ambient blackbody background response of the system. At T = 300 K, no emission is observed when the laser is biased below threshold. However as the threshold bias is reached, a broad peak attributed to intersubband transitions and centered around 12 /j,m is observed. This peak increases in amplitude until / = 1.2/th, and then it remains almost constant in magnitude. To confirm that the observed peaks are not due to thermal heating of the device, the data are analyzed by considering emission from a blackbody source. Local temperatures at the laser mirrors have been measured to be about 120°C higher than the temperature of the heat sink in GaAs-based quantum well lasers. 75 Using a value of A T = 120°C, the temperature of the laser mirrors is estimated to be about 420 K and 200 K when the heat sink is at 300 K and 80 K, respec­ tively. The blackbody curves corresponding to these two temperatures are shown in Fig. 16(b), in addition to the observed room temperature spontaneous emission from the interband QD laser.

49

988

P. Bhattacharya

et al.

For the observation of stimulated intersubband emission, plasmon-enhanced waveguides are designed and grown by MBE. The confinement factor, T, for the intersubband mode is 3.7 x 1 0 - 4 . The waveguide loss, aw, is calculated to be 6.84 c m - 1 . The width of the waveguide varies from 20-60 (im, and the length varies from 800-1200 fim. Therefore the devices are multimode laterally. The experiments reported here are performed on 60 fim wide and 1.2 mm long devices. Interband lasing occurs from the ground state in the dot (A = 1.07 fim) with a threshold density of 380 A/cm 2 . The measured light (power)-current characteristics are shown in Fig. 17(a). The threshold in the MIR output occurs at 1.6 times the inter­ band laser threshold. The additional carriers injected after the interband laser reaches threshold recombine to provide the high coherent photon density required for intersubband gain. 21 The intersubband threshold current density is 1.1 kA/cm 2 . In essence, the device converts the more readily available near-IR photons to the more difficult to obtain mid-IR photons. While device heating prevents measure­ ments at higher injection currents, these devices seem to demonstrate intersubband gain and dominant stimulated emission with a distinct threshold. In an ideal spherical quantum dot, one would not expect any polarization dependence due to the symmetry of the dot shape. However since self-organized dots are very asymmetric, with the base almost three times larger than the height, a polarization dependence of the output is expected. The polarization dependence of the MIR emission is measured using a mid-infrared polarizer. The intersub­ band emission is found to be strongly TE polarized, as shown in Fig. 17(b). In TE polarization, the electric field vector lies in the plane of the quantum dots, whereas

•nterbanc\'. ;

^.1.5

> -3IS o

ar6.84cm~* Jth= 1.1 kA/cm2

TE - TM

T=285K 1=770 mA L=0.8mm Ridge width= 50 urn

ZJ CO

TE

-

k_

s.

3

o

L~

d)

5 £0-5

n

•

^

J

*

—-""

^

intersubband

0.5 1 Current (A)

1.5

(a)

8.5

12.5

16.5

(b)

Fig. 17. (a) Light-current characteristics of the device showing a distinct threshold in the MIR emission. The interband emission is also shown for reference; and (b) the MIR output reveals a dominant T E polarization mode, which is consistent with the predictions of the eight-band k • p model.

50

Quantum Dot Infrared Detectors and Sources 989

in TM polarization, the electric field lies along the growth direction. The TE polar­ ization of the emission is in agreement with the polarization dependence obtained using the eight-band k • p model. 65 Although very fine and narrow features can be resolved in the emission spectrum beyond threshold, the general shape of the spectrum still remains broad. This could be due to several reasons. The emission has always been observed to be very broad in a Fabry-Perot cavity, even in quantum cascade lasers. Design of a distributed feedback cavity could enhance the spectral purity of the output. Moreover since the ridge is very broad (60 pm), the emission is highly multimode in nature. The output power is low, and this can be enhanced with improved device design and appropriate coatings of the laser facets. Also multiple periods of QD layers could be incorporated in the active region to enhance the gain and confinement factor. The area fill factor of self-organized quantum dots is 0.25-0.3, and this low value contributes to the low confinement factor. With changes in growth techniques, the dot density can be increased by an order of magnitude.

7. Future Prospects Phillips has made a recent comparison of the performance characteristics of MCT detectors, QWIPs, and QDIPs. 76 It is apparent from this analysis that QDIPs have the potential to outperform MCT devices. The reduction of the dark current in the vertical QDIPs, as described earlier, is very promising, but with the present heterostructure design, the photocurrent and responsivity are also reduced. In fact, increasing the responsivity of the devices is the biggest challenge. To achieve this, it is necessary to grow many more dot layers (w 50) without generating dislocations. Another option is to consider resonant cavity devices. Nonetheless, the demonstra­ tion of D* approaching 10 10 c m H z ^ / W at T = 100 K for a bias voltage of 0.2 V is very encouraging. The lateral QDIPs appear promising, as well, and further work is needed to exploit their full potential. As outlined in this chapter, QDIP focal plane arrays are yet to be realized. Single-pixel imaging is demonstrated, and this can be followed by the characterization of linear arrays. The device properties, and in particular, the responsivity, must improve before QDIPs can be incorporated in large focal plane arrays. Another important consideration for array applications is the spatial dot uniformity. The self-organization process by which the dots are formed inherently introduces a size nonuniformity, and it is yet to be seen how this affects array performance. While great strides have been made with interband quantum dot lasers in terms of threshold current, temperature dependence, tunability of output wave­ length, output power, and modulation bandwidth, the development of intersubband devices, as described in this article, is still in a nascent stage. The intersubband quantum dot light emitter described here is a bipolar device that converts interband photons to intersubband photons with a weak efficiency. Nonetheless, stimulated emission is observed, which indicates that with proper heterostructure and device

51

990

P. Bhattacharya et al.

design, a significant amount of coherent light can be obtained at I R frequencies. Ultimately, a unipolar device, like t h e q u a n t u m cascade laser, 3 7 is desirable. How­ ever, the design of multiple periods with strained q u a n t u m dots is going to be a challenge. Nonetheless, the hurdles are not insurmountable. Acknowledgments T h i s work is being supported by t h e Army Research Office under G r a n t s DAAD1901-1-0462 and DAAD 19-00-1-0394 (DARPA program), as well as t h e National Science F o u n d a t i o n under Grant E C S 9820129. References 1. K. W. Berryman, S. A. Lyon, and M. Segev, "Mid-infrared photoconductivity in InAs quantum dots", Appl. Phys. Lett. 70 (1997) 1861. 2. J. Phillips, K. Kamath, and P. Bhattacharya, "Far-infrared photoconductivity in selforganized InAs quantum dots", Appl. Phys. Lett. 72 (1998) 2020. 3. S. Kim, H. Mohseni, M. Erdtmann, E. Michel, C. Jelen, and M. Razeghi, "Growth and characterization of InGaAs/InGaP quantum dots for mid-infrared photoconductive detector", Appl. Phys. Lett. 73 (1998) 963. 4. S. Maimon, E. Finkman, and G. Bahir, "Intersublevel transitions in InAs/GaAs quan­ tum dots infrared photodetectors", Appl. Phys. Lett. 73 (1998) 2003. 5. D. Pan, E. Towe, and S. Kennedy, "Normal-incidence intersubband (In, Ga)As/GaAs quantum dot infrared photodetectors", Appl. Phys. Lett. 73 (1998) 1937. 6. S. Sauvage, P. Boucaud, J. M. Gerard, and V. Thierry-Mieg, "In-plane polarized intraband absorption in InAs/GaAs self-assembled quantum dots", Phys. Rev. B 58 (1998) 10562. 7. S. J. Xu, S. J. Chua, T. Mei, X. C. Wang, X. H. Zhang, G. Karunasiri, W. J. Fan, C. H. Wang, J. Jiang, S. Wang, and X. G. Xie, "Characteristics of InGaAs quantum dot infrared photodetectors", Appl. Phys. Lett. 73 (1998) 3153. 8. Q. D. Zhuang, J. M. Li, H. X. Li, Y. P. Zeng, L. Pan, Y. H. Chen, M. Y. Kong, and L. Y. Lin, "Intraband absorption in the 8-12 /mi band from Si-doped vertically aligned InGaAs/GaAs quantum-dot superlattice", Appl. Phys. Lett. 73 (1998) 3706. 9. A. Weber, O. Gauthier-Lafaye, F. H. Julien, J. Brault, M. Gendry, Y. Desieres, and T. Benyattou, "Strong normal-incidence infrared absorption in self-organized InAs/InAlAs quantum dots grown on InP(OOl)", Appl. Phys. Lett. 74 (1999) 413. 10. N. Horiguchi, T. Futatsugi, Y. Nakata, N. Yokoyama, T. Mankad, and P. M. Petroff, "Quantum dot infrared photodetector using modulation doped InAs self-assembled quantum dots", Jpn. J. Appl. Phys. 38 (1999) 2559. 11. J. Phillips, P. Bhattacharya, S. W. Kennedy, D. W. Beekman, and M. Dutta, "Self-assembled InAs-GaAs quantum-dot intersubband detectors", IEEE J. Quan­ tum Electron. 35 (1999) 936. 12. S. W. Lee, K. Hirakawa, and Y. Shimada, "Bound-to-continuum intersubband photoconductivity of self-assembled InAs quantum dots in modulation-doped heterostructures", Appl. Phys. Lett. 75 (1999) 1428. 13. L. Chu, A. Zrenner, G. Bohm, and G. Abstreiter, "Normal-incident intersubband photocurrent spectroscopy on InAs/GaAs quantum dots", Appl. Phys. Lett. 75 (1999) 3599. 14. D. Pan, E. Towe, and S. Kennerly, "Photovoltaic quantum-dot infrared detectors", Appl. Phys. Lett. 76 (2000) 3301.

52

Quantum Dot Infrared Detectors and Sources 991

15. H. C. Liu, M. Gao, J. McCafferey, Z. R. Wasilewski, and S. Fafard, "Quantum dot infrared photodetectors", Appl. Phys. Lett. 78 (2001) 79. 16. S. Y. Wang, S. D. Lin, H. W. Wu, and C. P. Lee, "Low dark current quantum-dot infrared photodetectors with an AlGaAs current blocking layer", Appl. Phys. Lett. 78 (2001) 1023. 17. A. D. Stiff, S. Krishna, P. Bhattacharya, and S. Kennerly, "High-detectivity, normalincidence, mid-infrared (A « 4 fim) InAs/GaAs quantum-dot detector operating at 150 K", Appl. Phys. Lett. 79 (2001) 421. 18. A. D. Stiff, S. Krishna, P. Bhattacharya, and S. Kennerly, "Normal-incidence, hightemperature, mid-infrared InAs-GaAs vertical quantum-dot infrared photodetector", IEEE J. Quantum Electron. 37 (2001) 1412. 19. L. E. Voro'bev, D. A. Firsov, V. A. Shalygin, V. N. Tulupenko, Yu. M. Shernyakov, N. N. Ledentsov, V. M. Ustinov, and Zh. I. Alferov, "Spontaneous far-IR emission accompanying transitions of charge carriers between levels of quantum dots", J. Ex­ perimental and Theor. Phys. Lett. 67 (1998) 275. 20. S. Krishna, O. Qasaimeh, P. Bhattacharya, P. J. McCann, and K. Namjou, "Roomtemperature far infrared emission from self-organized InGaAs/GaAs quantum dot laser", Appl. Phys. Lett. 76 (2000) 3355. 21. S. Krishna, P. Bhattacharya, P. J. McCann, and K. Namjou, "Room-temperature long-wavelength (A = 13.3 fxm) unipolar quantum dot intersubband laser", Electron. Lett. 36 (2000) 1550. 22. S. Krishna, P. Bhattacharya, J. Singh, T. Norris, J. Urayama, P. J. McCann, and K. Namjou, "Intersubband gain and stimulated emission in long wavelength (I = 13 mm) intersubband quantum dot emitters", IEEE J. Quantum Electron. 37 (2001) 1066. 23. M. Grundmann, A. Weber, K. Goede, V. M. Ustinov, A. E. Zhukov, N. N. Ledenstov, P. S. Kop'ev, and Zh. I. Alferov, "Mid-infrared emission from near-infrared quantum dot lasers", Appl. Phys. Lett. 77 (2000) 4. 24. D. Klotzkin, K. Kamath, and P. Bhattacharya, "Quantum capture times at room temperature in high-speed Ino.4Gao.6As-GaAs self-organized quantum-dot lasers", IEEE Photonics Technol. Lett. 9 (1997) 1301. 25. V. Ryzhii, "The theory of quantum-dot infrared phototransistors", Semiconductor Set. Technol. 11 (1996) 759. 26. J. Caniou, Passive Infrared Detection: Theory and Applications, Kluwer Academic Publishers, Boston, 1999. 27. A. Rogalski, "Assessment of HgCdTe photodiodes and quantum well infrared photoconductors for long wavelength focal plane arrays", Infrared Phys. Technol. 40 (1999) 279. 28. A. Rogalski, Infrared Detectors, Gordon and Breach Science Publishers, Australia, 2000, pp. 155-650. 29. J. Piotrowski and W. Gawron, "Ultimate performance of infrared photodetectors and figure of merit of detector material", Infrared Phys. Technol. 38 (1997) 63. 30. B. F. Levine, "Quantum-well infrared photodetectors", J. Appl. Phys. 74 (1993) R l . 31. S. D. Gunapala and K. M. S. V. Bandara, Homojunction and Quantum-Well Infrared Detectors, eds. M. H. Francombe and J. L. Vossen, Academic Press, San Diego, 1995, pp. 113-237. 32. J. L. Pan and C. G. Fonstad, Jr., "Theory, fabrication, and characterization of quan­ tum well infrared photodetectors", Mater. Sci. Eng. R, Reports: review j . 28 (2000) 65. 33. M. Z. Tidrow, "Device physics and state-of-the-art of quantum well infrared photo­ detectors and arrays", Mater. Sci. Eng. B 74 (2000) 45.

53

992 P. Bhattacharya et al.

34. M. Helm, P. England, E. Colas, F. DeRosa, and S. J. Allen, Jr., "Intersubband emis­ sion from semiconductor superlattices excited by sequential resonant tunneling", Phys. Rev. Lett. 6 3 (1989) 74. 35. S. Sauvage, Z. Moussa, P. Boucaud, and F. H. Julien, "Room temperature infrared intersubband photoluminescence in GaAs quantum wells", Appl. Phys. Lett. 70 (1997) 1345. 36. C. Sirtori, F . Capasso, J. Faist, D. L. Sivco, A. L. Hutchinson, and A. Y. Cho, "Quan­ tum cascade unipolar intersubband light emitting diodes in the 8-13 /um wavelength region", Appl. Phys. Lett. 66 (1995) 4. 37. J. Faist, F. Capasso, D. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, "Quantum cascade laser", Science 264 (1994) 553. 38. J. Faist, F. Capasso, D. L. Sivco, A. L. Hutchinson, C. Sirtori, and A. Y. Cho, "Quan­ tum cascade laser: A new optical source in the mid-infrared", Infrared Phys. Technol. 36 (1995) 99. 39. J. Faist, F. Capasso, C. Sirtori, D. Sivco, A. L. Hutchinson, S.-N. G. Chu, and A. Y. Cho, "Mid-infrared field-tunable intersubband electroluminescence at room temper­ ature by photon-assisted tunneling in coupled-quantum wells", Appl. Phys. Lett. 64 (1994) 1144. 40. M. Grayson, D. C. Tsui, M. Shayegan, K. Hirakawa, R. A. Ghanbari, and H. I. Smith, "Far-infrared emission from hot quasi-one-dimensional quantum wires in GaAs", Appl. Phys. Lett. 67 (1995) 1564. 41. O. Gautheir-Lafaye, P. Boucaud, F. H. Julien, S. Sauvage, S. Cabaret, and J. M. Lourtioz, "Long-wavelength (~ 15.5 //m) unipolar semiconductor laser in GaAs quan­ tum wells", Appl Phys. Lett. 71 (1997) 3619. 42. P. J. McCann, K. Namjou, and X. M. Fang, "Above-room-temperature continuouswave mid-infrared photoluminescence from PbSe/PbSrSe quantum wells", Appl. Phys. Lett. 75 (1999) 3608. 43. H. Benisty, C. M. Sotomayor-Torres, and C. Weisbuch, "Intrinsic mechanism for the poor luminescence properties of quantum-box systems", Phys. Rev. B 44 (1991) 10945. 44. L. Goldstein, F. Glas, J. Y. Marzin, M. N. Charasse, and G. Leroux, "Growth by molecular beam epitaxy and characterization of InAs/GaAs strained-layer superlattices", Appl. Phys. Lett. 47 (1985) 1099. 45. D. Leonard, M. Krishnamurthy, C. M. Reaves, S. P. Denbaars, and P. M. Petroff, "Direct formation of quantum-sized dots from uniform coherent islands of InGaAs on GaAs surfaces", Appl. Phys. Lett. 63 (1993) 3202. 46. J. Pamulapati, P. K. Bhattacharya, J. Singh, P. R. Berger, C. W. Snyder, B. G. Orr, and R. L. Tober, "Realization of in-situ sub two-dimensional quantum structures by strained layer growth phenomena in the I n x G a i _ x A s / G a A s system", J. Electron. Mater. 25 (1996) 479. 47. S. Guha, A. Madhukar, and K. C. Rajkumar, "Onset of incoherency and defect introduction in the initial stages of molecular beam epitaxy growth of highly strained I n x G a i _ x A s on GaAs (100)", Appl. Phys. Lett. 57 (1990) 2110. 48. J. M. Moison, L. Leprince, F. Barthe, F. Houzay, N. Lebouche, J. M. Gerard, and J. Y. Marzin, "Self-organized growth of InAs/GaAs quantum boxes", Appl. Surface Sci. 9 2 (1996) 526. 49. N. P. Kobayashi, T. R. Ramachandran, P. Chen, and A. Madhukar, "In-situ, atomic force microscope studies of the evolution of InAs three-dimensional islands on GaAs (001)", Appl. Phys. Lett. 68 (1996) 3299.

54

Quantum Dot Infrared Detectors and Sources 993

50. Q. Xie, P. Chen, A. Kalburge, T. R. Ramachandran, A. Nayfonov, A. Konkar, and A. Madhukar, "Realization of optically active strained InAs island quantum boxes on GaAs (100) via molecular beam epitaxy and the role of island induced strain fields", J. Crystal Growth 150 (1995) 357. 51. J. Y. Marzin, J. M. Gerard, A. Izrael, and D. Barrier, "Photoluminescence of single InAs quantum dots obtained by self-organized growth on GaAs", Phys. Rev. Lett. 73 (1994) 716. 52. S. Fafard, D. Leonard, J. L. Merz, and P. M. Petroff, "Selective excitation of photoluminescence and the energy levels of ultrasmall InGaAs/GaAs quantum dots", Appl. Phys. Lett. 65 (1994) 1388. 53. G. Wang, S. Fafard, D. Leonard, J. L. Merz, and P. M. Petroff, "Time-resolved optical characterization of InGaAs/GaAs quantum dots", Appl. Phys. Lett. 64 (1994) 2815. 54. M. Grundmann, "InAs/GaAs quantum dots radiative recombination from zerodimensional states", Phys. Status Solidi 188 (1995) 249. 55. D. I. Lubyshev, P. P. Gonzales-Borrero, E. Marega, E. Petitprez, N. Lascala, and P. Basmaji, "Exciton localization and temperature stability in self-organized InAs quantum dots", Appl. Phys. Lett. 68 (1996) 205. 56. R. Heitz, M. Grundmann, N. N. Ledenstov, L. Eckey, M. Veit, D. Bimberg, V. M. Ustinov, A. Egorov, A. E. Zhukov, P. S. Kop'ev, and Z. I. Alferov, "Multiphonon relaxation processes in self-organized InAs/GaAs quantum dots", Appl. Phys. Lett. 68 (1996) 361. 57. M. Grundmann, N. N. Ledentsov, O. Stier, D. Bimberg, V. M. Ustinov, P. S. Kop'ev, and Z. I. Alferov, "Excited states in self-organized InAs/GaAs quantum dots: Theory and experiment", Appl. Phys. Lett. 68 (1996) 979. 58. K. Kamath, P. Bhattacharya, and J. Phillips, "Room temperature luminescence from self-organized I n x G a i _ x A s / G a A s (0.35 < x < 0.45) quantum dots with high size uniformity", J. Crystal Growth 1 7 5 - 1 7 6 (1997) 720. 59. K. Kamath, N. Chervela, K. K. Linder, T. Sosnowski, H. T. Jiang, T. Norris, J. Singh, and P. Bhattacharya, "Photoluminescence and time-resolved photolumines­ cence characteristics of I n x G a i _ x A s / G a A s self-organized single- and multiple-layer quantum dot laser structures", Appl. Phys. Lett. 71 (1997) 927. 60. I. N. Stranski and L. V. Krastanow, Akad. Wiss. Lett. Mainz Math. Natur. 146 (1939) 797. 61. P. R. Berger, K. Chang, P. Bhattacharya, J. Singh, and K. K. Bajaj, "Role of strain and growth conditions on the growth front profile of I n ^ G a i - x A s on GaAs during the pseudomorphic growth regime", Appl. Phys. Lett. 53 (1988) 684. 62. V. A. Shchukin, N. N. Ledentsov, P. S. Kop'ev, and D. Bimberg, "Spontaneous ordering of arrays of coherent strained islands", Phys. Rev. Lett. 75 (1995) 2968. 63. Q. Xie, A. Madhukar, P. Chen, and N. P. Kobayashi, "Vertically self-organized InAs quantum box islands on GaAs (100)", Phys. Rev. Lett. 75 (1995) 2542. 64. G. S. Solomon, J. A. Trezza, A. F . Marshall, and J. S. Harris, Jr., "Vertically aligned and electronically coupled growth induced InAs islands in GaAs", Phys. Rev. Lett. 76 (1996) 952. 65. H. Jiang and J. Singh, "Strain distribution and electronic spectra of InAs/GaAs self-assembled dots: An eight-band study", Phys. Rev. B 56 (1996) 4696. 66. T. Sosnowski, T. Norris, H. Jiang, J. Singh, K. Kamath, and P. Bhattacharya, "Rapid carrier relaxation in Ino.4Gao.6As/GaAs quantum dots characterized by differential transmission spectroscopy", Phys. Rev. B 57 (1998) R9423.

55

994

P. Bhattacharya el al.

67. J. Urayama, T. B. Norris, J. Singh, and P. Bhattacharya, "Observation of phonon bottleneck in quantum dot electronic relaxation", Phys. Rev. Lett. 86 (2001) 4930. 68. H. C. Liu, "Noise gain and operating temperature of quantum well infrared photodetectors", Appl. Phys. Lett. 61 (1992) 2703. 69. W. A. Beck, "Photoconductive gain and generation-recombination noise in multiplequantum-well-infrared detectors", Appl. Phys. Lett. 63 (1993) 3589. 70. J. Singh, "Possibility of room temperature intra-band lasing in quantum dot structures placed in high-photon density cavities", IEEE Photonics Technol. Lett. 8 (1996) 488. 71. A. Kastalsky, L. E. Vorobjev, D. A. Firsov, V. L. Zerova, and E. Towe, "A dual-color injection laser based on intra- and inter-band carrier transitions in semiconductor quantum wells or quantum dots", IEEE J. Quantum Electron. 37 (2001) 1356. 72. H. T. Jiang and J. Singh, "Conduction band spectra in self-assembled InAs/GaAs dots: Comparison of effective mass and an eight-band approach", Appl. Phys. Lett. 71 (1997) 3239. 73. J. Singh, Physics of Semiconductors and Their Heterostructures, McGraw-Hill Inc., New York, 1993. 74. K. Nishi, H. Saito, S. Sugou, and J.-S. Lee, "A narrow photoluminescence linewidth of 21 meV at 1.35 ^m from strain-reduced InAs quantum dots covered by In0.2Gao.sAs grown on GaAs substrate", Appl. Phys. Lett. 74 (1999) 1111. 75. G. Abstreiter, "Micro-Raman spectroscopy for characterization of semiconductor devices", Appl. Surface Set. 50 (1991) 73. 76. J. Phillips, "Evaluation of the fundamental properties of quantum dot infrared detectors", to be published in J. Appl. Phys. 91 (2002).

56

International Journal of High Speed Electronics and Systems, Vol. 12, No. 4 (2002) 995-1024 © World Scientific Publishing Company

Generation of Terahertz Emission Based on Intersubband Transitions Qing Hu Department of Electrical Engineering and Computer Science and Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

In this chapter, we present our work on the development of coherent THz sources based on intersubband transition in quantum-well structures. The main focus is on electrically pumped or quantum-cascade structures, which have been quite successful in generating coherent radiation at mid-infrared frequencies. Relevant issues, such as various depopulation intersubband scattering rates, the role of complex phonon spectra, and coherent vs. incoherent tunneling are discussed in details. Optically pumped sources, including optical parametric amplifiers, and both intersubband and interband pumped THz emitters, are also investigated for their feasibility in generating coherent THz radiations. Key words: THz (Terahertz), intersubband, quantum wells, quantum-cascade lasers, LO-phonon, electron-electron scattering 1.

Introduction Terahertz (1-10 THz, or 4-40 meV, or 30-300 um) frequencies are among the most underdeveloped electromagnetic spectra, even though their potential applications are promising for spectroscopy in chemistry and biology, astrophysics, plasma diagnostics, remote atmospheric sensing and imaging, noninvasive inspection of semiconductor wafers, and communications. This underdevelopment is primarily due to the lack of coherent solid-state THz sources that can provide high radiation intensities (greater than a milliwatt). The THz frequency falls between two other frequency ranges in which conventional semiconductor devices have been well developed. One is the microwave and millimeter-wave frequency range, and the other is the near-infrared and optical frequency range. Semiconductor electronic devices that utilize the transport of free charge carriers (such as transistors, Gunn oscillators, Schottky-diode frequency multipliers, and photomixers) are limited by the transit time and parasitic RC time constants. Consequently, the power level of these classical devices decreases as \lf, or even faster, as the frequency / increases above 1 THz. Semiconductor photonic devices based on quantum-mechanical interband transitions, however, are limited to frequencies higher than those corresponding to the semiconductor energy gap, which is higher than 10 THz even for narrow-gap lead-salt materials. Thus, the frequency range of 1-10 THz is inaccessible for conventional semiconductor devices.

57

996

Q. Hu

Semiconductor quantum wells are human-made quantum-mechanical systems in which the energy levels can be designed and engineered to be of any value. Consequently, unipolar lasers based on intersubband transitions (electrons that make lasing transitions between subband levels) were proposed for long-wavelength sources as early as the 1970s.' This device concept has been realized in the successful development of quantum-cascade lasers (QCL) at mid-infrared wavelengths.2 Recent development has extended the operating wavelengths of the QCLs to as long as 24 urn 3 These achievements provide a great inspiration for the development of intersubband lasers in the THz frequencies below the Reststrahl band. Free-carrier absorption increases at long wavelengths as A.2, which could cause a significant cavity loss at the THz frequencies. By using thick (>10 urn) active regions or metallic waveguides for mode confinement, in combination of a low doping concentration (<10' /cm3), the cavity loss can be reduced to below 50 cm"', as verified recently in experiments.4 Thus, the key to achieve lasing is to obtain a sufficient level of gain to overcome this moderate level of cavity loss. The intersubband emitters are known to have a large joint density of states, because the two subbands, for example £ 3 and E2, track each other in the &-space. Thus electrons emit photons at the same energy regardless of their initial momentum if nonparabolicity is ignored. Therefore, the peak gain is related to the inverted population density An = n3n2 in a simple linear fashion, that is, g = (An/0(2e2oVher1/2eoc) zj/Af.

(1)

In Eq. (1), t is the thickness of the mode confinement region, and thus Anlt is the threedimensional inverted population density within t. zy is the radiative dipole moment between the two subbands, and Af is the FWHM linewidth of spontaneous emission. From Eq. (1), it is straightforward that a large peak gain can be achieved by a large dipole moment z(/, a narrow emission linewidth A/, and an appreciable level of population inversion An. The optimizations of the first two parameters are usually closely related, as a large dipole moment (accomplished by a strong spatial overlap of the two subband wavefunctions) tends to yield a narrow emission linewidth. Measured emission linewidth as narrow as 0.7 meV has been achieved with the calculated dipole moment greater than 50 A.5'6 Even with a moderate level of population inversion of An ~ 109/cm2, the estimated peak gain will be greater than 100 cm"1, which should exceed the cavity loss by a comfortable margin. Therefore the main challenge in the development of THz intersubband lasers is. to achieve population inversion between two narrowly separated subband levels. In this chapter, we review our recent work on the development of intersubband THz emitters. In Section 2, we will discuss our efforts in developing electrically pumped THz emitters, which include two types of structures. One utilizes electron-LO-phonon scattering, and the other utilizes resonant tunneling to depopulate the lower radiative level in order to achieve population inversion. Related issues, such as the role of interface and confined phonon modes, coupling between subbands near anticrossing (or resonance), and rate-equation analysis will also be discussed. In Section 3, we will review our investigations of optically pumped intersubband THz sources, including both intersubband-pumped and interband-pumped structures.

58

Generation of Terahertz Emission Based on Intersubband Transitions 997

2.

Electrically pumped intersubband THz emitters

Figure 1 illustrates schematics of emitters based on intersubband transition. Fig. 1(a) illustrates a scheme where the intersubband radiative transition takes place within one quantum well. This scheme is known as "intrawell" transitions. Fig. 1(b) illustrates a scheme where the intersubband transition takes place between two wells, which is known as "interwell" transitions. The intrawell scheme is conceptually straightforward and it has the advantages of a larger radiative dipole moment and a narrower emission linewidth. The interwell scheme, on the other hand, has the advantage that it allows the use of electron-LO-phonon scattering to depopulate the lower radiative state. We have made detailed investigations on both schemes.

(a)

lb)

Figure 1. (a) Intrawell scheme of intersubband-transition lasers, (b) Interwell scheme of intersubband lasers.

2.1 THz emitters using electron-LO-phonon scattering for depopulation Following the design principle of the original QCLs, we have designed intersubband THz emitters based on a three-level system. The top two subband levels, E3 and E2, form the radiative pair, while the ground state £y is at > hcoLO below E2. Since it is energetically allowed, the fast electron-LO-phonon E2-^Et scattering will help to keep the population in E2 low, and therefore maintain a population inversion between E3 and E2. However, because E3-E2 < E2-Et > ha)L0 for THz emitters, it is difficult to implement this three-level system based on an intrawell transition scheme, in which both E3 and E2 are primarily located in a single well. The bottom of this well would have to be raised relative to the rest of the structure so that E2-Et > hcoL0. Raising the bottom of the well would require adding aluminum to the well material GaAs, which would cause a significant alloying scattering and result in a broad emission linewidth. Our design of the three-level systems is based on a scheme that the radiative transition takes place in a coupled double-well structure. A third well, which is much wider than the two wells, contains the ground-state level Eh In our first design, the wavefunctions of E3 and E2 are primarily localized in separate wells, thus the E3—>E2 transition is spatially diagonal. This design offers the advantage of a high selectivity in

59

998

Q. Hu

injection into E3 and removal from E2, because of the spatial separation of the two wavefunctions. However, the diagonal nature of die radiative transition is quite sensitive to scattering due to interface roughness and alloy in the barrier, and thus the emission showed rather broad spectra (A/~ 3-5 THz).7 It is well known that in a coupled double-well structure, the wavefunctions of the two lowest levels are spatially extended with a strong overlap at the anticrossing. Because of this spatial overlap, both levels are subject to the same interface and alloy scattering, and thus the emission linewidth of radiative transition between the two levels is reduced. In an improved structure, whose band structure and wavefunctions are shown in Fig. 2, we have taken advantage of this feature to enhance the strength of the radiative E3->E2 transition.8

Figure 2. Schematic of a three-level system based on a triple quantum-well structure shown inside the dashed box. On the right is the dispersion relation between the energy and the transverse momentum.

In the structure shown in Fig. 2, the core is a three-well module of GaAs/Al0.3Gao.7As heterostructures (inside the dashed box), with three barriers Bi (4.5 nm), B 2 (2.8 nm), B 3 (5.6 nm) and three wells W, (8.8 nm), W2 (5.9 nm), and W3 (6.8 nm). The collector barrier B] is center 8-doped at a level of 6xl010/cm2 in order to provide dynamic charges. Under the designed bias of 51 mV per module, the ground state E,' (not shown) of a previous module is aligned with E3. Thus, the upper level E3 can be selectively populated via resonant tunneling. At this bias, the energy separation E32 ~ 11 meV (corresponding to 2.67 THz), and the dipole moment z32 ~ 30 A. The energy separation E21 ~ 40 meV > hcoLO , enabling electron-LO-phonon E2—>Ei scattering for depopulation. DC transport measurement confirmed our design. Fig. 3 shows the I-V curve and the differential conductance of a MQW structure with 30 modules. The device is cooled to ~5 K. The current increases monotonically with the bias below 2 V. The conductance

60

Generation

of Terahertz Emission Based on Intersubband

Transitions

999

i

Figure 3. Measured dc current density and differential conductance versus the bias voltage of a THz emission device with 30 triple-well modules. The conductance peaks at the designed bias of -1.7 V with a current density of 254 A/cm2. Also plotted are the measured THz emission power versus bias and current (inset).

Ml

Bias supply -

Reference

Cryostat Lock-in Si bolometer LN2

_n_.

LHe

Parabolic mirror

Parabolic mirror


LHe KO~

Device

Signal

B.S. Mirror y r Mi r Fixed mirror Mirror positior (step-scan)

Moving mirror I

PC FTIR

Figure 4. THz emission measurement set-up that uses an external Fourier transform spectrometer to spectrally resolve the emitted THz signals.

61

1000

Q. Hu

peaks at 1.7 V, which is close to the designed bias (51x30 mV) with the extra voltage likely due to the contact resistance. The conductance reaches its maximum at this bias due to the resonant tunneling of the E^—>E3 alignment. In order to measure the intersubband THz emission and resolve its spectra, we constructed a set-up that included a Fourier transform infrared spectrometer (FTIR). The system's schematic is shown in Fig. 4. We have improved this system and perfected our measurement techniques so that THz emission measurements can be routinely performed on our emitters with output power levels of only several pW. Our emission spectra reveal a clear peak due to the ES-^E2 intersubband emission. A representative spectrum taken at 5-K device temperature is shown in Fig. 5(a), which was taken at the designed bias of 1.6 V (-30x51 mV). The measured peak frequency of 2.57 THz (10.6 meV) is close to the designed value of 11.3 meV. The FWHM linewidth is as narrow as 0.47 THz (1.9 meV). In order to verify the intersubband origin of the measured emission spectra, we have measured emission spectrum at a high bias of 4.0 V at which the energy levels are severely misaligned. The spectrum is shown in the inset of Fig. 5(a), and it bears little resemblance to the main figure. Spectra were also taken with the cold stage cooled with liquid nitrogen to 80 K. One taken at a bias of 1.6 V is shown in Fig. 5(b). The main peak is essentially the same as the one measured at 5 K, with a slightly broader linewidth of 0.52 THz (2.14 meV). The linewidth measured at 80 K is expected to be similar to that at 5 K, since nonparabolicity is negligible for THz intersubband emitters. Nevertheless, our experimental verification is encouraging for the development of intersubband THz sources at elevated temperatures.

80 K

6

8

10

Frequency (THz) Figure 5. Spectrally resolved THz intersubband spontaneous emission taken at (a) 5-K and (b) 80-K bath temperature under 1.6-V bias. The inset shows the spectrum for 4.0-V bias, clear evidence that emission results from intersubband transitions.

62

Generation of Terahertz Emission Based on Intersubband Transitions

1001

2.2. Role of interface and confined phonon modes Just like the electronic wavefunctions forming discrete subbands in MQW structures, phonon spectra also become discrete, forming spatially localized interface and confined modes. Despite this parallel analogy, bulk LO-phonon mode has been used to calculate the scattering times in QCLs. This practice may be justified for mid-infrared QC lasers, in which E3-E2 » ha>w> so that the sum-rule yields the same result as obtained from using the bulk mode. However, as Dutta and Stroscio recently pointed out,9 such a practice may be questionable for THz intersubband lasers, because E3-E2< hcow and E2-E1~h(Ow. We have investigated the role of the complex phonon spectra on the intersubband scattering rates.10 In the model used in our calculations, the phonon modes are described by the potential <))(r) resulting from the polarization field created by atomic displacements in a polar semiconductor. Each material layer of index i is described by a dielectric function Ei(co) as given by the Lyddane-Sachs-Teller relations, which vanishes at the LOphonon frequencies. For lattice vibrations, since there are no free charges, the phonon potential must satisfy e(co)V2(r) = 0. Two types of solutions exist: interface modes for which V2(j)(r) = 0, and confined modes for which e(co) = 0. For the confined modes, e(co) = 0 and therefore co = toLo, where coLO is the bulk LOphonon frequency in the layer of interest. Since coLo changes at the heterointerfaces, the potential must vanish there, and <|)(r) can be described in terms of sine wave modes in z. A representative potential of several confined phonon modes is shown in Fig. 6. For the interface modes, e(co)*0, and the modes have frequencies CO^COLO, and the phonon frequencies depend on the transverse momentum q. The potential solution is a linear combination of exponential terms peaked at the interfaces, hence the name of "interface mode". For our GaAs/AlGaAs quantum-well structures, there are usually two "GaAslike" and one "AlAs-like" modes associated with each GaAs/AlGaAs interface. Thus, for the six interfaces in our triple-well structures shown in Fig. 2, there are a total 18 interface modes. A representative dispersion relation and potential profile of the interface phonon modes is shown in Fig. 7. The total scattering rate is the sum of the contributions from these 18 interface modes and all the confined modes (30 lowest confined modes were used in our calculations with the contributions from the higher modes negligible). We used the transfer matrix approach11 to account for the electromagnetic boundary conditions and obtain the mode potentials and dispersion relations for the interface modes. As it turns out, the 12 "GaAs-like" modes are clustered around 33-36 meV, close to the bulk GaAs LO-phonon energy. The 6 "AlAs-like" modes are clustered around 4547 meV, close to the bulk AlAs LO-phonon energy, as illustrated in Fig. 7(a). Special care was taken to ensure a proper normalization of each mode, which was verified by limiting cases.10 In order to address the key issue raised in Ref. [9], namely the optimum subband separation E2h we have calculated the maximum total scattering rate as a function of E2i, as shown in Fig. 8. The rate shows two peaks, one at -35 meV due to the "GaAs-like" modes and the other at -47 meV due to the "AlAs-like" modes. As a comparison, we also include the scattering rate calculated using the bulk GaAs LO-phonon mode. It is clear that the scattering rate is -30% lower than the bulk mode if E21 ~ 36 meV, because of the exclusion of the "AlAs-like" modes. However, increasing E2l to -47 meV does not increase the total scattering rate appreciably. This is because that at E2i - 47 meV, even

63

1002

Q. Hu

though all the phonon modes participate in the scattering, the rapid decrease in the strength of the "GaAs-like" modes away from their resonance (at -36 meV) diminishes most of the benefit gained from including more active phonon modes. „xid1 3 cd

■B 4-

j io 2

■

i\ /

i

i\ ;

: <

:

: I

I

I' !■

/

: ,

;

:

■

'

o

10

15

20 z (nm)

25

30

35

40

Figure 6. Phonon potentials for the confined "GaAs-like" modes for m = 1 (solid lines) and m ■■ 2 (dashed lines) at q = 0.3 nm'1. The vertical lines represent heterointerfaces. 1 0.8 0.6 0.4 0.2' 0 : 0.2 0.4 0.6 0.8 1

0.2 0.3 q (nnr1)

200

GaAs-like

(b)

A

A

/'• W^=-- y \

'

' ^-J&y4^J5^e»-»-—-to-**-

\M/\ \.. '1

VJ 'i"-s : /

;W3; AA/2

100

y

Wi

100 200 300 Z (Angstroms)

400

500

Figure 7. (a) Dispersion relation of all the 18 interface modes; (b) phonon potential profiles of the 12 "GaAslike" interface modes associated with our triple quantum-well device.

In Fig. 9, we plot the key figure of merit t 3 (l -121/132), which is proportional to the population inversion kn32, as a function of the scattering time T32. As E2i increases from 36 to 47 meV, the increase in AnJ2 is marginal at a given t32. Even this marginal increase should be taken with a grain of salt. As E2i increases from 36 to 47 meV, there will be an additional 11 meV energy dissipation per electron. This extra energy dissipation will further raise the electronic temperature. As a result, T32 decreases because of LO-phonon scattering of hot electrons. Fig. 9 shows that a reduction in T32 could undo any advantage

64

Generation

of Terahertz Emission

Based on Intersubband

Transitions

1003

gained by increasing £ 2 ; to 47 meV. The conclusion from our analysis is that in electrically pumped THz intersubband emitters, because the barriers are thin and Al concentrations are low (x < 0.3), the contribution from the higher -energy "AlAs-like" modes only barely make up for the loss in the strength of "GaAs-like" modes. Thus, increasing E2i from 36 to 47 meV will only yield at the best a marginal (if any at all) improvement in population inversion A%2- For optically pumped intersubband THz lasers, however, because of thicker barriers with high Al concentrations (x > 0.4) and relatively narrow wells, the influence of the AlAs-like modes will be stronger; therefore the benefit of setting E2l ~ 47 meV may be more significant, as suggested in Ref. [9]. A recent development of an optically pumped intersubband Raman laser may have provided evidence that suggests important contributions of the AlAs-like interface modes to the intersubband scattering process.12

IF+confined GaAs bulk

°30

35

40

45 (meV) 21

50

55

60

Figure 8. Maximum scattering rate versus subband separation E2i for the three-level structure shown in Fig. 1. Maximum scattering rate calculated with GaAs bulk modes is present for comparison.

CO -« Q.

20 10 0 E21=36 meV E21=47 meV

10 20 10

30

40 x32(ps)

50

60

70

80

Figure 9. Plot of the quantity Tj( 1-T21/T32), which is proportional to the population inversion An32, versus the lifetime T32 for the structure shown in Fig. 1.

65

1004

Q. Hu

2.3. Intrawell THz emitters using resonant tunneling for depopulation Fig. 10(a) shows the calculated band diagram of another type of THz intersubband emission structure that we investigated recently.6 In this structure, the radiative transition takes place between two subband levels primarily located in one quantum well. We consider this scheme as "intrawell" as opposed to the interwell scheme discussed earlier. It is well known that the intrawell scheme yields a larger dipole moment and a narrower emission linewidth, because the subband separation is less sensitive to impurity and interface roughness scattering.5

1

n

><65

6.9

Bias per module (mV) Figure 10. (a) Computed conduction band profile and squared magnitude wavefunctions for the intrawell device. Plot of the (b) subband energies with respect to the injection level n = 1', and (c) dipole matrix elements vs. applied bias. The dashed line in (b) and (c) indicates the designed operating bias.

66

Generation of Terahertz Emission Based on Intersubband Transitions

1005

The core of the structure shown in Fig. 10(a) is a coupled double-well module shown inside the dashed box. Sixty-five nominally identical modules are cascade connected. Under the designed bias of -20 mV/module, the lower level E2 in the wide well is aligned with the lowest level Et in the narrow well, which in turn is aligned with the upper level in the wide well of the following module. The radiative transition takes place between E3 and E2, which have a strong spatial overlap as can be seen in Fig. 10(a). The depopulation of E2 is facilitated through "resonant tunneling" from E2 to Eh which deserves special attention and will be discussed in the following. The calculated dipole moment z32 is as large as -6.0 nm at the designed bias (shown in Fig. 10(c)), as a result of the strong radiative coupling in this intrawell scheme. Fig. 11 shows measured emission spectra taken at a bias of 1.5 V and 2.0 V, which are somewhat greater than the designed bias (65x20 mV = 1 . 3 V). Both spectra show a clear peak at 21 meV (-5.04 THz) due to the 3-»2 radiative transition, with a FWHM linewidth as narrow as 0.7 meV (0.18 THz). As expected, due to the intrawell nature of the radiative transition, the peak displays no appreciable Stark shift for the bias range of 0.8-2.25 V. The measured emission frequency is close to the calculated subband separation E32 ~ 18.5 meV, and the narrow linewidth indicates the high quality of the MBE growth.

.0 300

1

Frequency (THz) 3 4

2

5

6

7

(a) 1.5 V, 146 mA, (39 A/crrf)

0.7 meV (0.17 THz)

/UAA

A

W

0.75 meV (0.18 THz) 0.2 0.4 Current (A)

-

^ v \

10 15 20 Photon energy (meV)

/"v ^ ~ 25

30

Figure 11. Electroluminescence spectra at 5 K for applied biases (a) V = 1.5 V and (b) V = 2.0 V. The inset displays the emitted optical power vs. current.

67

1006

Q. Hu

The term "resonant tunneling" usually refers to electron transport between two subband levels close in energies. There has been a long debate on whether this process should be analyzed as intersubband scattering between two spatially extended wavefunctions (the scenario of coherent resonant tunneling), or as tunneling between two spatially localized states (the scenario of incoherent sequential tunneling). This question is not merely academic, but is crucial for the successful development of intersubband THz lasers, as the two transport mechanisms yield very different depopulation rates. For coherent resonant tunneling, the electron transport is facilitated by fast (<1 ps) electronelectron intersubband scattering. For incoherent sequential tunneling, the depopulation rate is determined by the barrier transparency, which can be much slower than the first scenario. 0

10

Bias per module (mV) 20 30

1.5 Bias (V)

40

2

Figure 12. Current (a) and conductance (b) for several values of applied magnetic fields at 4.2 K. (c) Position of A/ = 2 peaks in G-B plots and associated energy difference near the anticrossing of subband 2 and 1. The solid lines are the corresponding calculated energy difference E32 and E31.

68

Generation

of Terahertz Emission

Based on Intersubband

Transitions

1007

In the coherent picture, the energy difference between any pair of two subbands has a finite minimum value, known as the anticrossing gap A, which characterizes how strongly the two subbands are coupled. In contrast, in the incoherent scenario, the two subband wavefunctions are spatially localized in different wells. Their energy levels can be degenerate and thus may be arbitrarily close at resonance. Based on the coherent model, we have calculated A2] = min LE2-E/I ~ 2.5 meV. Even though current -field (I-B) magneto-tunneling spectroscopy has been successfully used to resolved subband separations of -20 meV,13'14 the resolution of the anticrossing gaps of only a few meV requires the measurement of conductance G - dl/dV to enhance the energy resolution and sensitivity. For magnetotransport measurements, devices were processed into 150x150 urn2 mesas, and mounted in liquid helium inside a magnet dewar. Magnetic fields up to 8 T were applied along the growth direction (parallel to current flow). Current-voltage and conductance-voltage characteristics are shown in Fig. 12.

3 ■•—»

c 0

k_

O

0

2 4 6 Magnetic field (T)

2 4 6 Magnetic field (T)

Figure 13. Current (a) and conductance (b) vs. magnetic field (BIIJ) for several applied biases. The curves are offset for visibility. The arrows in (b) indicate the resonance peaks that correspond to 2-1 anticrossing.

At B = 0, the conductance curve flattens at 1.8-2.1 V, as n = 3 becomes aligned first with n = 2' and then with the injector state n = 1'. At 2.25 V, n = 1' becomes severely misaligned with n = 3, and a negative differential resistance (NDR) occurs. While this bias voltage is somewhat greater than the calculated design bias of 1.3 V, the difference

69

1008

Q. Hu

is partially due to parasitic series resistance. As B is increased from 0 to 7.8 T, the current decreases due to the reduction in phase space for intersubband scattering. Additionally, the NDR shifts to lower voltages with increasing B. The formation of the Landau levels sharpens the density of states, and thus less misalignment of the n - V and 3 is tolerated before the device enters a NDR region. Fig. 13 shows both I-B and G-B curves measured in a voltage range near the E2-E1 anticrossing. For example, at V - 1.90 V, the I-B maxima corresponding to the change of Landau level index A/ = 2,3,4 are seen at 5.47, 3.64, and 2.73 T, which correspond to an energy separation of 19 meV. This value of energy separation is close to the calculated intersubband spacing. While the peaks in the G-B curves generally mirror those in the I-B curves, however, at V>1.8 V the G-B data reveal the more complicated structure of an anticrossing between two A/ = 2 peaks (indicated by the arrows in Fig. 13(b)). As the bias is increased, the magnitude of the lower-energy peak decreases while that of the higherenergy peak increases. The limited resolution at B~A T largely obscures much of the anticrossing effect for the A/ =3 peaks. The dependence of these A/ = 2 conductance peaks on bias is shown in Fig. 12(c), and it displays a typical anticrossing behavior between a Stark shifted peak and another fixed at -6.0 T (-21 meV). This 21-meV peak corresponds to the 3—>2 transition, which agrees well with the emission data shown in Fig. 11. The Stark shifted peak corresponds to the 3->l transition, as the two wavefunctions are spatially more separated than those of E3 and E2. At 1.9 V, the peaks in G-B are minimally separated by 0.5 T, which corresponds to an anticrossing gap of 1.7 meV. This value agrees reasonably well with the calculated value of 2.5 meV, which was calculated assuming a totally coherent scenario. In the presence of scattering, which is unavoidable in any real devices, the minimum energy separation of two subbands should be somewhat smaller than the calculated value based on a coherent model. In addition, the overall dependence of the conductance peaks on the bias agree reasonably well with the calculated energy differences E31 and £j2> a s shown in Fig. 13(c). Our work shows that in the particular structure shown in Fig. 10(a), the E2-^E, depopulation should be modeled as intersubband scattering between two spatially extended states.

2.4 Transport issues of electrically pumped THz intersubband emitters The results presented in Sections 2.1 and 2.3 showed unambiguous evidence of intersubband emission at THz frequencies with good characteristics (narrow linewidths and fidelity at elevated temperatures). The key issue in achieving lasing now is in a careful design and implementation of a transport scheme, involving injection into the upper radiative subband, intersubband scattering, and removal from the lower radiative subband level, in order to achieve an appreciable degree of population inversion between the two radiative levels. In principle, all the above-mentioned transport processes should be correctly modeled as intersubband scattering, in order to obtain a quantitative measure of the relative rates of various processes. Fig. 14 illustrates the scattering processes of several mechanisms. The radiative transition and electron-acoustic-phonon scattering are slow processes with the time scale longer than ~ns. Thus, they can be ignored in rate-equation analysis, although the

70

Generation

of Terahertz Emission

Based on Intersubband

Transitions

1009

radiative transition rate is crucial in determining the spontaneous emission efficiency and gain of the active medium, and the electron-acoustic-phonon scattering is important in dissipating excess electronic energies. The processes due to electron-LO-phonon scattering and electron-electron scattering have some features that are unique for THz intersubband emitters. In contrast to mid-infrared QCLs, where the energy separation between the two radiative subband levels is always greater than the LO-phonon energy h®w, thus electron-LO-phonon scattering is always energetically allowed. At THz frequencies, even though the subband energy separation is less than h(0LO, however, hot electrons with excess kinetic energy will subject to electron-LO-phonon scattering. Clearly, this scattering process depends on the excitation level of the devices. Because of the long-range Coulomb interaction, the electron-electron scattering is inversely proportional to the square of the momentum (or the energy) involved. Thus, even though this process has been more or less ignored in the analysis of mid-infrared QCLs, it plays a crucial role in the intersubband scattering in THz intersubband emitters. As can be seen from the insets in both Fig. 3 and Fig. 11 (b), the power-current relation is sublinear. In fact, the power-current (P-I) relation can be well described by a P °c I1/2 relation for the PI curve shown in Fig. 11(b) at 5 K. This P °c \m relation is a direct consequence of a lifetime reduction due to the electron-electron scattering at high injection levels. A s P « n 3 ,1 °c n3/x3, and T3 °= l/n3, and hence the P <* \m dependence.

XAC ~ 1 ns for acoustic phonon scattering

Trad ~ 1 US for

spontaneous radiation

te-e x A/n ~ 1 -100 ps forn~10io-10H/cm2 important for THz

T L o~1-10psif A £ (OLO, and °° if A<

OOLO

Figure 14. Schematics of various intersubband scattering processes, namely: radiative transition, electronacoustic-phonon, electron-LO-phonon, and electron-electron scattering.

71

1010

Q. Hu

Clearly, the rate of electron-electron scattering depends on the populations of the involved subband (roughly proportional to the subband population), thus this scattering process also depends on the excitation levels. Both electron-LO-phonon scattering and electron-electron scattering rates increase with the excitation level. At high excitation levels (corresponding to high current densities and bias fields), the electronic temperature could be significantly higher than the lattice temperature, resulting in an increase of hot electron scattering by emitting LO-phonons. Also, at high injection levels, the increased electron population at the upper level increases the electron-electron scattering rate, resulting in a decrease of the injection efficiency. Both processes are detrimental for the establishment of a population inversion. Thus, a successful design of THz intersubband lasers cannot simply increase the injection level by increasing the doping concentrations and using thinner barriers. Rather, a delicate trade-off must be balanced between a sufficient injection level and not being overwhelmed by the two detrimental processes at high injection levels. The mechanisms of depopulation of the lower radiative level is different for the two structures shown in Fig. 2 and Fig. 10(a). The first one (in Fig. 2) relies on electron-LOphonon scattering, while the latter relies on electron-electron scattering. As a result, the rate analysis is quite different for the two structures. In the structure shown in Fig. 2, because of the large energy separation of E2-E1 = 40 meV, back filling process of 1 —»2 is negligible at cryogenic temperatures. In the structure shown in Fig. 10(a), however, E2-E! ~ 2 meV, back filling is an important process. We analyze the two structures separately in the following. THz emitters using electron-LO-phonon scattering for depopulation In this structure, if we ignore the 1 —>2 back scattering, then the population inversion is given by: T An = n3 - n2 = " (T 3 2 - T 3 1 ) > 0 , if T32 >T 31 e T32 +T 31

(2)

where ty is the scattering time from ith to jth subband, and J is the injection current density. It is clear from Eq. (2) that a population inversion is achieved if x32 > x3], regardless of other parameters. However, in order to achieve a large degree of population inversion, the ratio x3i/(x31+x32) should be optimized by making x31 » x32. The scattering time x2i is mainly determined by electron-LO-phonon scattering, as the energy separation AE2] is designed to be close to the LO-phonon energy ha>L0 . In the particular structure shown in Fig. 2, because of the relatively thick barrier Bi, the lifetime x2i is as long as -27 ps (calculated using the bulk GaAs LO-phonon mode). x32, however, is mainly due to electron-electron scattering and LO-phonon scattering of hot electrons. Both processes depend on the injection levels and it is difficult to accurately calculate this scattering time. From the measured peak current density and designed doping concentration, we can estimate that x32 to be -20 ps, with a large uncertainty due to the uncertainty in doping concentration and nonspecular tunneling from 1' to 2. Given our later investigation on the electron-electron scattering time (-10 ps), we believe that it is unlikely that population inversion is achieved in the structure shown in Fig. 2, because of the long lifetime of x2i. In principle, this problem can be solved straightforwardly by using a thinner barrier B],

72

Generation of Terahertz Emission Based on Intersubband Transitions

1011

which will yield a shorter lifetime T2\- In practice, however, because of the strong spatial overlap of the wavefunctions of E3 and E2 that is needed to achieve a narrow emission linewidth and a large dipole moment, a short T2I will result in a proportionally short T31, which will be detrimental in two ways. First, it will reduce the prefactor in Eq. (2), which will quantitatively reduce the degree of population inversion. Second and perhaps the worst, it will result in a significant parallel current channel due to the 3—A scattering process. This parallel current channel will not contribute anything to the desired injection channel, but will generate additional heat that raises the electronic temperature. As a consequence, the LO-phonon scattering of hot electrons in E3 will significantly shorten the scattering time x32, which could qualitatively destroy population inversion, as can be seen fromEq. (2). The dilemma in the design of a suitable structure similar to that shown in Fig. 2 illustrates the competing requirements of a strong coupling between the two radiative subband levels and a selective coupling of one of them to a different level. For E3, it is desirable that it is strongly coupled to the injector level Ei'. For E2, it is desirable that it is strongly coupled to the ground-state level E,. Meanwhile, it is desirable that E3 has a minimum coupling with E b and E2 has a minimum coupling with E]'. The latter requires a high-quality interface, so that tunneling is mostly specular, and the simultaneous conservation of energy and transverse momentum reduces the E]'—>E2 process at the designed bias. The former can be achieved by adding an additional buffer well between E2 and E). This buffer well will largely block the E3—>E] process while still allowing a fast depopulation of E2. We are actively pursuing this approach in our current investigation. THz emitters using electron-electron scattering for depopulation Because of the small energy separation AE2) = 2 meV and thus electron -LO-phonon scattering is not energetically allowed, the depopulation of E2 in structure shown in Fig. 10(a) relies on resonant tunneling. As discussed in Section 2.3, this process should be correctly modeled as electron-electron scattering between two spatially extended states E2 and E|. Because of the small energy separation of AE21, the rate of electron-electron scattering is very fast (~1 ps, depending on the subband populations). Also, because of the small energy separation AE2i, the back filling process 1—>2 is almost as likely as the forward scattering process of 2—>1 at elevated temperatures. To the first degree of approximation, we model the two subbands E2/E! as a coherent "doublet" that can be characterized by a single chemical potential u.. This model assumes that the intra-doublet scattering is much faster than the inter-doublet scattering, so that electrons in the doublet reach a quasi-equilibrium. The total electron density n is splited in three ways among the three subband levels E3, E2, and Ej, with the population on E2 and Ej determined by U. In this simplistic model, from the measured current density (assuming the injection tunneling is specular), it is possible to estimate the relative subband populations if the lifetime on the upper subband level t 3 is known. Fig. 15 shows the result of a measurement that is intended to extract the value of lifetime x3. Since the emission power P °= n3, and the current density J <* «3/x3, the ratio of the two is given by:

73

1012

Q. Hu

7

- T , -

+ T^exp

(Ei2-/zcoL0^ v

k T

»

(3) j

where xee is the scattering time due to electron-electron scattering, and xL0 is the scattering time due to electron-LO-phonon scattering when the electrons have sufficient kinetic energy on E3 for LO-phonon scattering. XLO is calculated to be -0.4 ps using the bulk GaAs LO-phonon mode. In Fig. 15, we plot measured PIJ vs. 1/T. As can be seen from the figure, PIJ decreases at high temperatures due to the lifetime reduction from LO-phonon scattering of hot electrons. PIJ is flat at low temperatures with the level set by the electron-electron scattering, which is approximately temperature independent. From Fig. 15, we can infer the electron-electron scattering time to be approximately 17 ps. The current density and the upper-level population is related through J - en3/x3, or n3 = (J/e)x3. From Fig. 12(a), the peak current density J ~ 100 A/cm2, which yields a upperlevel population n3 =1.1 xl010/cm2. The total two-dimensional electron density is measured from a C-V measurement to be ND = n1+n2+«3 ~ 2 xl010/cm2. Thus, a population inversion rc3 > n2 can be inferred from these measurements. However, devices with structures similar to that shown in Fig. 10(a) did not achieve lasing, even when they were integrated with low-loss THz cavities formed by metal waveguides. We suspect that the extracted value of x3 is much longer than its actual value, and consequently no population inversion is established in the structure. The reason for this discrepancy is unclear to us at the moment. A recent transport analysis based on a three-dimensional Monte Carlo simulation yielded a much shorter electron-electron scattering time (<10 ps) for a structure similar to the one shown in Fig. 10(a).15 The same Monte Carlo simulation analysis correctly predicted the measured current density, illustrating its accuracy in analyzing various scattering processes. In collaboration with Prof. S. Goodnick at Arizona State University, we are currently developing our own Monte Carlo simulation package to further investigate the transport processes in various intersubband THz emitters. Our investigations on the emission and transport properties of the two intersubband structures shown in Fig. 2 and Fig. 10(a) indicate that they show good radiative characteristics. However, further refinement in the design of their transport properties is needed to achieve a population inversion, while still preserving their good radiative characteristics. For the structure shown in Fig. 2, the key challenge is to achieve a short lower-level lifetime x2, while keep the upper-level lifetime x3 relatively long. This can be achieved by using additional buffer wells to reduce the coupling between the upper level E3 and the ground state E]. For the structure shown in Fig. 10(a), an obvious direction for improvement is to increase the injection current density J, as n3 = (7/e)x3. An increase in current density will increase n3 and decrease n2-N0- (n3+ni), both are desirable changes to help establishing a population inversion n 3 > n2.

74

Generation of Terahertz Emission Based on Intersubband Transitions

100

50

25

T(K)

16.6

12.5

1013

10 24 22 20 18

o£$J*--~.
0)

14 LO-phonon activation

1

12~~ ■10

0.02

0.04

0.06 1/T (K1 )

0.08

Figure 15. Measured light power/current (171) ratio, which is proportional to the lifetime of the upper level, as a function of the device temperature. The9 roll-off of L/I at high temperatures is due to thermally activated LO-phonon scattering. L/I is approximately constant at low temperatures, which gives a measure of the electron-electron scattering time scale of -17 ps.

3.

Optically pumped intersubband THz emitters

Although electrical pumping is preferred because of its simplicity in operations, optical pumping offers advantages in selectivity of pumping and simplicity in designs. Many types of lasers were first invented using optical pumping and then achieved electrical pumping only after the involved technology became mature. In parallel with our main effort in the development of electrically pumped intersubband THz lasers, we have also investigated the possibility of using optical pumping to achieve THz lasing. We summarize key results in this section. 3.1 Intersubband pumped THz optical parametric oscillators (OPOs) We started our investigation of optically pumped THz sources with the development of optical parametric oscillators (OPOs). OPOs are commonly used near-infrared sources at wavelengths longer than ~1 u.m where the frequency is below the bandgap energy of typical semiconductor diode lasers. They have the advantage of frequency tunability (by temperature or incident angle). Their operation requires a large nonlinearity in the dielectric constants (x<2)) and phase-matching condition. Because of the large dipole moments associated with intersubband transitions, it has long been recognized that quantum-well structures possess large %<2> values at frequencies near the intersubband

75

1014

Q. Hu

resonances. Experimentally, it has been demonstrated that % is as large as 10 m/V, which is orders of magnitude larger than the value for bulk GaAs materials.1617 It was this large nonlinearity associated with the intersubband transitions that motivated us to pursue the development of intersubband-pumped THz OPOs, whose schematic is illustrated in Fig. 16. The active core of the device is a coupled double-well structure with three subband energy levels. The energy separation of these subband levels correspond to the frequencies of the pump C0p, THz signal cos, and the idler (0|. As a result, all these three beams are at resonance with intersubband transitions, yielding a large value of x<2> for the intended frequency down conversion process. Quasi-phase matching in this device can be achieved by applying different voltages on periodic placed electrodes, the resulting Stark shift could significantly reduce the value of %<2) in the region that the reverse OPO process takes place, thus extending the length of the active gain medium.

co„

i

t t i 1

to„

nonlinear device

P

GO,

^s

♦

p

s E 2

CO.

C t 1

(a)

(b)

Figure 16. (a) Schematic of an OPO. involving three subband energy levels.

(b) Its implementation using a coupled double-well structure

In a theoretical analysis, we have discovered that in contrast to near-infrared OPOs in which all. the three beams suffer very little attenuation losses, the losses for the intersubband OPOs are heavy for all the three beams due to a large-value %(1) process that is associated with the large dipole moments in intersubband transitions. As a result of this heavy attenuation, plane waves are no longer good approximations to describe the three propagating beams. In other words, the momentum of each beam is no longer a good quantum number. Consequently, phase matching condition, which is the mathematical statement of momentum conservation, is no longer strictly required in a gain medium that the pump and idler beams suffer heavy attenuation.18 This result could help to simplify the design and implementation of OPOs based on intersubband transitions. Using reasonable device parameters, we estimate that the threshold gain is -50 cm"1, which requires a pump power level of-100 W. Such a pump power should be easily obtainable from a Q-switched C0 2 laser.

76

Generation

of Terahertz Emission Based on Intersubband

Transitions

1015

3.2 Intersubband optically pumped THz lasers Both optically pumped lasers and OPOs are frequency down converters, which requires the use of nonlinear elements. Using the analogy and language of circuit elements, one utilizes a nonlinear reactance (OPOs) and the other one utilizes a nonlinear resistance (optically pumped lasers). For OPOs, the reactive nonlinear down conversion conserves both energy (through the Manley-Rowe relation) and momentum (through the phase matching condition). For optically pumped lasers, neither photon energy nor photon momentum is conserved, reflecting their dissipative nature in the down conversion process. In practice, since no phase matching condition is required, optically pumped lasers are much easier to operate than OPOs, as the pumped beam can be aligned arbitrarily with respect to the output signal beam. At near-infrared wavelengths (A, > 1 |im), there are no real energy levels with long lifetimes available in commonly used semiconductor materials. Thus, OPOs that utilize the reactive nonlinearity (associated with virtual states) are commonly used. In the case when real energy levels are available, optically pumped lasers have always been the choice of approach for their easy operations. For the intersubband structure depicted in Fig. 16(b), clearly there are real energy levels with sufficiently long lifetimes, thus optically pumped lasers should be easier to develop than OPOs. It was this analysis that led us to the path of developing optically pumped THz lasers by intersubband pumping. Unlike OPOs, in which all the three beams at C0p,ft>s,and c^ play important roles in the outcome of the devices, in optically pumped lasers only the radiative pair involves in the optical transition and the rest of the states provide nonradiative scattering channels for achieving population inversion. Thus, OPOs will only need three energy levels while optically pumped lasers will need a minimum three levels but often more levels for better performance. We have investigated intersubband structures with three-, four-, and fivelevels, with the main focus on the four-level structures.19'20 Fig. 17(a) shows a intersubband structure with three-levels, and Fig. 17(b) shows a structure with fourlevels.

Position (nm)

Position (nm)

Figure 17. (a) Schematic of a three-level intersubband system using a coupled double-well structure, (b) Schematic of a four-level intersubband system using a coupled triple-well structure.

77

1016 Q. Hu

The three-level system is conceptually simpler and the analysis is straightforward. The radiation efficiency 7] is given by: ft) r"" r, =a „^2LlL'hL

°3l"l

rod

(4)

K

'

where 031 is the pump cross section for the 1—»3 excitation, n\ is the electron density on Ei, co's are intersubband transition frequencies, and T'S are intersubband transition times. With reasonable device parameters, we estimate that the (spontaneous) radiation efficiency for the three-level system is ~4xl0"10, which is quite low, largely because of the short lifetime x3 as a result of many fast nonradiative relaxation channels for electrons to come down from E3. It is difficult to significantly improve r\1L by modifying the design of the three-level system. Increasing the doping concentration to increase n\ will cause a greater level of free-carrier absorption. Increasing the cross section 031 will also not work. The resulting stronger overlap between wave functions in levels Ei and E3 leads to a faster LO-phonon scattering rate I/T31. It is also difficult to increase the Manley-Rowe factor 0)32/0)3! because the only practical pump source (a C0 2 laser) operates in a narrow frequency range -110-135 meV. Here is where the power of quantum well as an "artificial atom" comes to play. With quantum wells we have the ability to design wave functions and phonon scattering rates. However, the simple three-level scheme does not give us enough flexibility to take advantage of this power. By using more levels, for example a four-level system, we could tailor the wave functions to achieve a longer upper-level lifetime while still preserving a large pump cross section. Fig. 17(b) illustrates the structure of a four-level system. In this structure, electrons are optically pumped from Ei to E4. E4-E3 is designed to be close to hco LO so that electrons quickly decay from E4 to E3. The radiative transition takes place between E 3 and E2 with a photon energy hco ~ 30 meV. Electrons then quickly depopulate from E2 to E! because E2-E] > hcoL0. The spontaneous emission efficiency of this four-level system is given by:

w

41

T

43

T

32

At first sight it looks like this result is worse than the three-level case because of the additional factor T4,ot/T43. Actually it is this extra ratio that allows us to bypass a major deficiency of the three,-level design. We can now design the scattering rates such that I/T43 is the dominant contribution to lAc4tot so that T4tot/X43 - 1. Thus the absorption of the pump O41M1 is now effectively decoupled from the LO-phonon scattering l/x3tot. We can increase the dipole moment 241 to increase o 41 without affecting l/t3tot. This gives us the extra flexibility to increase the emission efficiency over the simpler three-level design. Fig. 18(a) shows a calculation of the emission efficiency as a function of electron density ti\ for both the three-level and the four-level structures. Clearly, the four-level scheme gives an order of magnitude improvement at the electron density of 2.5xl0 u /cm 2 . The maximum efficiency for the four-level structure occurs when E4-E3 = hco 10, at which the rate I/T43 i s t n e highest. The same physics that allows us to improve the spontaneous emission efficiency also leads to a larger population inversion and therefore gain. Fig. 18(b) shows the calculated

78

Generation

of Terahertz Emission

Based on Intersubband

Transitions

1017

gain for 1 W of pump power and a 6-meV linewidth. The active region is composed of 100 modules of the three-well structures separated by 200-A barriers. It is apparent that the gain of the four-level design scheme is an order of magnitude greater than the threelevel design at moderate electron densities. Extending this design analysis one step further, we have also analyzed a five-level structure, with the lower radiative level separated from the level below by ~/ZG) LO . Because of the enhanced depopulation rate at this resonance, the population on the lower radiative level can be kept low even with a fast relaxation from the upper radiative level. This feature is attractive as it is important for high-temperature operations. At elevated temperatures, the LO-phonon scattering of hot electrons significantly increases the relaxation scattering between the two radiative subband levels.21

64-E3<36meV

E4-E3 - 36™V

E<-EJ>3flmeV

35 .

E 4- E g - 36moV

E 4 - E 3 < 36rr»V

I

30 4-Level Design

>v

25

•p

I

E 4 -E 3 > 36nwV

4-Levef Design

\^^^

20

1

—

3-Level Design

3-Level Design

j ^

0

^ * * * ^

^^^^^

10

5

^ 1

^ I.S

2

2.5

3

3.5

4

4.S

S

Electron Density (x10 11 cm~ 2 )

Figure 18. (a) Comparison of emission efficiency for the three-level and four-level designs, Comparison of gain for the three-level and four-level design.

80

90 100 110 120 130 140

(b)

150

Frequency (meV) Figure 19. Mid-infrared transmission measurement of the four-level quantum-well structure. The measured E4-E, peak position is at 119.8 meV with a FWHM of 13.4 meV.

79

1018

Q. Hu

We have grown a four-level structure whose band diagram is shown in Fig. 17(b). Fig. 19 shows the absorption spectrum of the device. The absorbance is obtained by taking the ratio of the transmitted beam with its electrical field in the plane of the quantum wells and perpendicular to the plane. The main peak at 120 meV in the absorption spectrum is due to the l-»4 intersubband transition, which occurs at the designed photon energy of 119.5 meV (corresponding to X ~ 10 um). The THz emission measurements were carried out using the set-up shown in Fig. 20. In this set-up, the input C0 2 laser beam and the output THz beam are at orthogonal angle so that the output is not contaminated by the strong C0 2 laser beam. The measurements are more difficult than those of electrically pumped THz emitters, because both the input and output need to be well aligned and the output power levels are weaker than those from the electrically pumped emitters. Nonetheless, we have obtained emission spectra that showed strong evidence of THz emission due to intersubband transitions.

Figure 20.

Emission measurement set-up for the optically pumped intersubband THz emitters.

Fig. 21 shows representative emission spectra. The solid curve corresponds to the pump being tuned to resonance, while the dashed line is the spectrum when the pump is tuned off resonance. At off resonance, only bulk absorption can take place and the

80

Generation

of Terahertz Emission

Based on Intersubband

Transitions

1019

emission mainly results from blackbody radiation. Thus the dashed curve in Fig. 21 provides an estimate on the background of thermal radiation. In the solid curve, there is a prominent peak centered at -31 meV, which is close to the designed value of 3—»2 transition. A Lorentzian curve fit gives a FWHM of ~7 meV, which is a relatively narrow linewidth compared to the room-temperature linewidth of the absorption spectrum. As can be seen from Fig. 21, the 3—»2 emission peak lies on top of a broad blackbody spectrum. The strong blackbody radiation is a result of low emission efficiency of the (intersubband) optically pumped THz emitters.

E v»

O (D Q. CO C

o w w

'£

Hi

10

20

30

40

50

Frequency (meV) Figure 21. THz emission spectra. 0.2-W of cw pump power was coupled into the sample. The solid curve corresponds to when the pump frequency is tuned to the 1—>4 transition resonance, while the dashed curve corresponds to the case of off resonance. The main emission peak of the solid curve is at 30.8 meV with a FWHMof7meV.

3.3 Interband optically pumped THz emitters Our investigation discussed in section 3.2 showed some interesting results, namely that we can explore subband engineering to improve THz emission efficiency and gain. However, it also showed the drawbacks of (intersubband) optically pumped THz emitters. The much stronger blackbody radiation shown in Fig. 21 compared to the electrically pumped THz emitters shown in Figs. 5 and 11 indicates that the emission efficiency of the optically pumped emitters is much lower than their electrically pumped counterparts. The main reason of this low emission efficiency is the inseparable coupling between the pumping and the relaxation processes. In order to have a large pump cross section, the ground-state level and the top level must have a strong spatial overlap, which will result in a very fast relaxation due to electron-LO-phonon scattering. It was to overcome this challenge that motivated us to explore the feasibility of interband optically pumped THz lasers.22 The core of these structures is a three-well

81

1020

Q. Hu

module, as shown in Fig. 22. In these structures, the pump process (which is due to interband excitations) and the relaxation process (which is mostly due to intersubband scatterings) are largely decoupled. Consequently, a relatively long lifetime of the upper subband level can be engineered along with a high pump efficiency. The less favorable Manley-Rowe relation of the interband pumping compared to that of the intersubband pumping will not result in additional heating, as a large portion of the pump photon energy will be removed from the devices via interband lasing at a slightly lower photon energy. The absence of doping will likely yield narrower emission linewidths and lower cavity losses. Furthermore, the absence of a selection rule in the interband pumping scheme allows a convenient surface pumping without using diffractive gratings.

Figure 22. Computed conduction and valence band profiles and squared magnitude wave functions for the interband optically pumped THz emitters. Layer thicknesses in nm are displayed above the layers. In the valence band, heavy-hole subbands are indicated with solid lines, light-hole subbands with dashed lines.

As far as the intersubband transitions in the conduction band are concerned, the structure is essentially a four-level system crc4, with the top two conduction subbands

82

Generation of Terahertz Emission Based on Intersubband Transitions 1021

forming the radiative pair for THz emission. Electrons are optically pumped to c4, which acts as the upper level for a radiative transition into c3. The lower level c3 is strongly coupled to the first excited level c2 in the wide well. These two levels are at anticrossing and form a doublet with a strong spatial overlap. The calculated energy separation E43 is 18.6 meV (corresponding to -4.6 THz) with a dipole moment Z43 = 36 A, and E32 ~ 3 meV. In order to obtain a fast depopulation of the c-ilc2 doublet, the energy separation E2i is designed to be slightly greater than hcoLO- Scattering down from the higher-energy subbands, electrons will pile up in Ci until the interband lasing threshold is reached for the transition between C\ and the highest valence subband. We used a two-band k«p model to calculate the valence subband levels and wave functions,23,24 and from which we can compute the interband radiative transition strengths. The intersubband relaxation processes in the conduction band were calculated self-consistently including both electron-LO-phonon scattering and electron-electron scattering. The rates of electron-electron scattering were calculated using a numerical Hartree code.25 The main results are presented in Fig. 23. In Fig. 23(a), the electron-LOphonon scattering times of several processes are plotted as functions of electron temperature. As can be seen from the figure, the 2—>l and 3—»1 always have the shortest times because the wave functions of the involved subbands have strong spatial overlaps. Fig. 23(b) shows the populations of the four conduction subbands vs. the electron temperature. It appears that a population inversion (n4 > n3) exists at temperatures as high as 70 K. Fig. (c) and (d) show the calculated emission efficiency and gain vs. the electron temperature. Note that even with an unfavorable Manley-Rowe relation, the calculated emission efficiency is approximately two orders of magnitude higher than that for an intersubband pumped emitter (see Fig. 18(a)). This is largely because the decoupling between the pumping and relaxation processes in the interband pumped scheme. The upper-level lifetime T4 can be made much longer than that in intersubband pumped schemes. Finally, the estimated value of gain is -50 cm"' at 50 K at a pumping level of 30 kW/cm2, which should be achievable, at least in a pulsed mode. In summary, we have investigated both electrically and optically pumped intersubband THz emitters with the goal to develop coherent THz laser sources. By far, the electrically pumped structures have shown much greater promise in terms of their radiation characteristics and our understanding of the transport processes in the devices. We strongly believe that with careful fine tuning as discuss in Section 2, THz lasing should be achieved in the near future. Needless to say, such a development will have a qualitative impact on the science and technology in the THz electromagnetic spectrum, which is currently severely underutilized mainly because of lack of coherent sources.

83

1022

Q. Hu

Carrier Temperature (K) Figure 23. (a) Electron-LO-phonon scattering times and (b) subband populations versus temperature for a pump intensity of 30 kW/cm2. In (c) the temperature dependence of the THz emission efficiency is displayed for several pump intensities, (d) Temperature dependence of the intersubband gain for a pump intensity of 30 kW/cm2 and spontaneous emission linewidth of 2 meV.

Acknowledgments The author would like to thank several of his current and former students, B. Xu, I. Lyubomirsky, B. S. Williams, H. Callebaut, and S. Kumar for their contributions during various stages of the project. He would also like to thank M. R. Melloch and J. L. Reno for providing high-quality MBE wafers for the project. This work has been supported by AFOSR, NASA, NSF, and ARO.

84

Generation

of Terahertz Emission

Based on Intersubband

Transitions

1023

References 1. R. F. Kazarinov and R. A. Suris, "Possibility of amplification of electromagnetic waves in a semiconductor with a superlattice," Sov. Phys. Semicond. 5, (1971) 707— 709. 2. J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, "Quantum cascade laser," Science 264, (1994) p. 477. 3. R. Colombelli et al, "Far-infrared surface-plasmon quantum-cascade lasers at 21.5 and 24 urn wavelengths," Appl. Phys. Lett. 78, (2001) 2620-2622. 4. M. Rochat, M. Beck, J. Faist, and U. Oesterle, "Measurement of far-infrared waveguide loss using a multisection single-pass technique," Appl. Phys. Lett. 78, (2001)1967-1969. 5. M. Rochat, J. Faist, M. Beck, U. Oesterle, and M. Ilegems, "Far-infrared (k = 88 urn) electroluminescence in a quantum cascade structure," Appl. Phys. Lett. 73, (1998) 3724-3726. 6. B. S. Williams, H. Callebaut, Q. Hu, and J. Reno, "Magnetotunneling spectroscopy of resonant anticrossing in terahertz intersubband emitters," Appl. Phys. Lett. 79, (2001)4444-4446. 7. B. Xu, Q. Hu, and M. R. Melloch, "Electrically pumped tunable THz emitters based on intersubband transition," Appl. Phys. Lett. 71, (1997) 440-442. 8. B. S. Williams, B. Xu, Q. Hu, and M. R. Melloch, "Narrow-linewidth terahertz intersubband emission from three-level systems," Appl. Phys. Lett. 75, (1999) 29272929. 9. M. Dutta and M. A. Stroscio, "Comment on 'Energy level schemes for far-infrared quantum well lasers'," Appl. Phys. Lett. 74, (1999) p. 2555; Q. Hu and I. Lyubomirsky, Appl. Phys. Lett. 74, (1999) p. 3065. 10. B. S. Williams and Q. Hu, "Optimized energy separation for phonon scattering in three-level terahertz intersubband lasers," J. Appl. Phys. 90, (2001) 5504-5511. 11. S. G. Yu, K. W. Kim, M. A. Stroscio, G. J. Iafrate, J. -P. Sun, and G. I. Haddad, "Transfer matrix method for interface optical-phonon modes in multiple-interface heterostructure systems," J. Appl. Phys. 82, (1997) 3363-3367. 12. H. C. Liu etal. Appl. Phys. Lett. 78, (2001) 3580-3582. 13. J. H. Smet, C. G. Fonstad, and Q. Hu, "Magneto-tunneling spectroscopy in Wide InGaAs/InAlAs Double Quantum Wells," Appl. Phys. Lett. 63, (1993) 2225-2227. 14. J. Ulrich, R. Zobl, W. Schrenk, G. Strasser, K. Unterrainer, and E. Gornik, "Terahertz quantum cascade structures: Intra- versus interwell transition," Appl. Phys. Lett. 11, (2000) 1928-1930. 15. R. Kohler, R. C. Iotti, A. Tredicucci, and F. Rossi, "Design and simulation of terahertz quantum cascade lasers," Appl. Phys. Lett. 79, (2001) 3920-3922. 16. C. Sirtori, F. Capasso, J. Faist, L. N. Pfeiffer, and K. W. West, "Far-infrared generation by doubly resonant difference frequency mixing in a coupled quantum well two-dimensional electron gas system," Appl. Phys. Lett. 65, (1994) 445—447. 17. E. Rosencher and Ph. Bois, "Model system for optical nonlinearity: asymmetric quantum wells," Phys. Rev. B44, (1991) p. 11315. 18. I. Lyubomirsky and Q. Hu, "Optical parametric oscillators without phase matching," J. Opt. Soc. Am. B14, (1997) 984-988.

85

1024

Q. Hu

19. I. Lyubomirsky and Q. Hu, "Energy level schemes for far-infrared quantum well lasers," Appl. Phys. Lett. 73, (1998) 300-302. 20. I. Lyubomirsky, Q. Hu, and M. R. Melloch, "Measurement of far-infrared intersubband spontaneous emission from optically pumped quantum wells," Appl. Phys. Lett. 73, (1998) 3043-3045. 21. I. Lyubomirsky, Ph. D. thesis, "Toward Far-infrared Quantum Well Lasers," MIT (1999) (unpublished). 22. H. Callebaut, Master's thesis, "GaAs/AlGaAs Far-infrared Quantum Well Lasers," MIT (2001) (unpublished). 23. J. M. Luttinger and W. Kohn, "Motion of electrons and holes in perturbed periodic fields," Phys. Rev. 97, (1955) 869-883. 24. D. A. Broido and L. J. Sham, "Effective masses of holes at GaAs/AlGaAs heterojunctions," Phys. Rev. B31, (1985) 888-892. 25. The code used in our calculations of electron-electron scattering is from P. Harrison, "Quantum wells, wires, and dots," John Wiley and Sons, Chichester, UK, (1999).

86

International Journal of High Speed Electronics and Systems, Vol. 12, No. 4 (2002) 1025-1038 © World Scientific Publishing Company

MID-INFRARED GaSb-BASED LASERS WITH TYPE-I HETEROINTERFACES D. V. DONETSKY, R. U. MARTINELLI*, G. L. BELENKY Slate University of NY at Stony Brook, Stony Brook NY 11794 *Sarnoff Corporation, Princeton, NJ 08543

The design of room-temperature, InGaAsSb/AlGaAsSb diode lasers has evolved from the first double-heterojunction lasers described in 1980 that operated in the pulsed-current mode to presentday continuous-wave (CW), high-power, quantum-well diode lasers. We discuss in detail recent results from type-I-heterostructure, GaSb-based CW room-temperature diode lasers. The devices operate within the wavelength range of 1.8 to 2.7 urn, providing output powers up to several Watts. We analyze the factors limiting device performance.

1. Introduction High-power mid-infrared diode lasers operating at room-temperature have numerous applications, including countermeasures, LIDAR, free-space communications, pumping solid-state lasers, biomedical applications, compact trace-gas sensors and analyzers. The first mid-infrared diode lasers developed in mid-60s were based on lead salt alloys. These lasers were the only commercial product available for a number of years with major application in environmental monitoring and laser absorption spectroscopy. Today, lead-salt lasers still demonstrate continuous wave (CW) operation only at temperatures less than 225 K with output power less than one mW1. III-V compounds provide another option for mid-infrared diode laser design. They have an established epitaxial growth technique, relatively low thermal resistance and low series resistance. The choice for high-power and single-frequency applications in the wavelength range up to 2 \\m is InP-based lasers. Multi-watt CW operation was demonstrated with strained InGaAsP/InP quantum-well (QW) lasers2. Extension to longer wavelengths within InGaAsP/InP system meets the principal limitation of increasing lattice mismatch. Heavily strained InAs QWs and a strain-compensated technique were implemented to extend the wavelength range. As a result InP-based lasers with 2.2 um room-temperature operation3 and lasers with 2.52 um operation at 190 K4 were demonstrated. Narrow-gap III-V antimonides can be utilized to overcome the long-wavelength limitation of InP-based systems. Similar lattice constants of InAs, GaSb and AlSb allow lattice-matched epitaxial grows of a variety of heterostructures based on ternary and quaternary solid alloys. The main advantage of quaternary alloys is that both the average

87

1026

D. V. Donetsky, R. U. Martinelli

& G. L.

Belenky

lattice constants and bandgaps can be varied independently, so that different compositions can be fabricated on a suitable lattice matched substrate. For example, quaternary alloy InGaAsSb grown on a GaSb substrate might have a band gap within the range from 0.72 to 0.25 eV, which corresponds to wavelengths of 1.7 to 5 urn. The conduction-band-offset ratio AEc/AEg is close to unity for InGaAsSb/GaSb interface. For some InGaAsSb compositions this interface forms staggered (type-II) band alignment, as shown in Figure la. In this case the transition energy is smaller than bandgap and a longer lasing wavelength can be achieved for composition chosen. However, the spatial separation of the electrons and the holes in this type-II structure lowers the radiative recombination rate, and hence, the optical gain.

InGaAsSb

GaSb

InGaAsSb

©0©

AlGaAsSb

©e©

£v

©©©

©©©L. (b)

(a)

Figure 1. Optical transitions (arrows) in heterostructures with type-II (a) and type-I (b) band alignments.

QW lasers based on type-I band alignment, as shown in Figure lb, offer higher optical gain per unit inversion fraction, owing to the greater overlap of electron and hole wave functions. Such an alignment occurs when an AlGaAsSb alloy with a high Al content is grown on an InGaAsSb layer. In this paper only InGaAsSb/AlGaAsSb diode lasers grown on GaSb substrates with a type-I band alignment are considered. The aim of this work is to outline the basic designs of type-I high-power, room-temperature diode lasers operating above 2 urn.

2. Progress in the Design of Type-I GaSb-based Diode Lasers Although III-V ternary and quaternary semiconductor alloys are characterized by broadly tunable spectral responses, the miscibility gaps and the lattice-constraint limit the useful range of bandgaps in most of these systems. This is particularly true of InGaAsSb, which exhibits a broad miscibility gap for compositions lattice-matched to GaSb. Very stable alloys can, however, be obtained in the composition range close to GaSb and InAs. Figure 2 shows the phase diagram, along with the miscibility gap, for InGaAsSb. Also shown are solid lines of constant bandgap energy and dashed lines of constant lattice constant. Lattice matching requires maintaining approximately constant ratio of III-V components. For InxGai.xAsySbi_y on GaSb x/y should be about 0.9. Lattice-matched InGaAsSb on GaSb becomes unstable with the increase of x and y when x approaches 0.25, which is known as a miscibility gap problem.6

88

Mid-Infrared

GaSb-Based Lasers with Type-I Heterointerfaces

1027

The first InGaAsSb/AlGaAsSb lasers were grown in 19807 using liquid phase epitaxy (LPE). Broad-area double heterostructure (DH) lasers operated at room temperature in the pulsed mode. Realization of CW operation was possible using a narrow-stripe design. The first CW GaSb-based laser was grown by LPE and operated at 2.34 um with a roomtemperature threshold current of 80 mA.8 Figure 3 reflects the progress in the reduction of the threshold current density over the years. The reduction of the threshold current density in DH lasers was achieved mainly through better optical confinement with an increase of the Al content in the cladding layers. Lattice-matched AlGaAsSb alloys become more difficult to grow with increasing Al content because incorporation of a sufficient percentage of As for lattice matching requires the growth temperature to be significantly increased.

InSb

t

InAs

6Qv<>-3eV

//

x

GaSb

y "►

GaAs

Figure 2. Illustration of a miscibility gap (shaded) in InxGai.xAsySbi.y system. Solid lines show the compositions with constant bandgaps. Dashed lines denote the lattice-matched compositions.

Alloys with a composition inside the miscibility gap cannot be grown from the liquid phase, which practically limits the maximum operating wavelength of lattice-matched lasers to about 2.4 um.9 The first DH lasers obtained by molecular beam epitaxy (MBE) were reported in 1986.10 The growth in vacuum improved the structure uniformity, and increased the laser efficiency. Metastable InGaAsSb alloys with extended range of composition can be grown by MBE at significantly lower temperatures. Thus, 3-um DH lasers with intended composition of x = 0.54 and y = 0.48 in active region operated in pulse up to 255 K were demonstrated." The InGaAsSb/AlGaAsSb system encountered problems of compositional control and reproducibihty of the designed structures. The incorporation of group-V fluxes is much smaller than unity and depends on a number of technological parameters. The best 2.2 2.3 um DH lasers with x = 0.16 and y = 0.16 in the active region and 75 % Al in the claddings were fabricated after multiple growth iterations. These lasers demonstrated a

89

1028

D. V. Donetsky,

R. U. Martinelli

& G. L.

Belenky

room-temperature threshold current density of 940 A/cm2, 0.9 W/facet pulsed12 and 10.5mW/facet CW output power.13 The typical characteristic temperature T0 for DH lasers was 50 K near room temperatures due to the domination of Auger recombination.

1Tio4

2.78um 2.2um •(10) ^ ■ 2 ^ m 2.3um

o

'e

<9>

Q103 ■e

1

o

510 2

I

t-

. LPE y

">2'.2^m 2.7um. <13)\ r (29)-0.6um 2.1um i 1.9urrU2-3*i (14) 2.26um

, . , . -2.05 1 MBEj> M W r > ( 2 2 ) u m : ■ ■ ■ ■ ■

I ■ ■ ■ ■ ■ ■ ■ ■

■

1980 1985 1990 1995 2000 Years Figure 3. Record room-temperature threshold current densities in GaSb-based diode lasers over the years. The sharp decrease in threshold current density in 1992 was due to transition from DH to QW structures. Numbers in parentheses indicate references.

Employing compressively strained QWs increased the differential gain, reduced the threshold current and the Auger recombination rate. The first QW lasers14 reported in 1992 consisted of an active region with five 0.8 % compressively strained QWs of the same composition as in Ref. 12, separated by lattice-matched barriers with 20 % Al and surrounded by cladding layers with 90 % Al. In most later laser designs, 90 % Al in AlGaAsSb cladding layers were utilized, providing the maximum index difference between the barrier and cladding layers for better optical confinement. The separation of optical and carrier confinements using QW structures; effected a two- to three-fold reduction of both internal loss and threshold current density. As a result, a broad-stripe 2.1-um laser demonstrated 190 mW/facet CW at room temperature. Pulsed operation was possible up to 150 °C owing to high characteristic temperature T0 of 115 K. This is another advantage of InGaAsSb/AlGaAsSb system for high-temperature laser applications. For comparison, in both DH and QW InGaAs/InGaAsP devices, T0 values are typically about 55 K. The CW power of diffraction-limited ridge-waveguide lasers was limited to 40 mW, while the beam quality of broad-area lasers suffered from filamentation. In order to increase the amount of useful, single-spatial-mode power, the first tapered laser emitting at 2-um was designed in 1993.15 As high as 200 mW CW power in the near-diffractionlimited central lobe of the far field were achieved. A better understanding of the growth of antimonide-based epitaxial materials, careful measurement and control of lattice matching and intentional strain significantly improved the mid-infrared laser performance.16 These improvements resulted in the demonstration of 1.9-um 5-QW lasers with room-temperature pulsed thresholds as low as 143 A/cm2 and a 1.3 W CW single-ended output power.17

90

Mid-Infrared GaSb-Based Lasers with Type-I Heterointerfaces 1029

The longest emission wavelength of 2.78-um for room-temperature III-V type-I lasers was demonstrated in 1995.18 Four-QW InGaAsSb/AlGaAsSb structures with 10.5 nm QW width, composition of x = 0.24 and y = 0.16, and 25 % Al in barriers were designed with compressive strain up to 0.3 %. The lasers operated in pulsed mode up to 60 °C with a characteristic temperature of 58 K. Devices of similar design operated CW up to -39°C at 2.7 urn.19 In the QW lasers described above the total QW width was comparable with the waveguide width, corresponding to the maximum QW confinement. Optical loss from free-carrier absorption in the cladding layers becomes greater for the long-wavelength lasers. Decreasing the fraction of the optical mode that propagates in the cladding layers, especially in the p-cladding layer, is essential to the reduction of the internal optical loss. Broadening the separate confinement heterojunction layers (SCH) accomplishes this, and results in an exponential reduction of the internal loss, while the QW confinement factor decreases insignificantly. A nearly three-fold increase of the external efficiency was demonstrated in 1996 in 2-um 5-QW lasers when the waveguide width was increased from 0.12 urn to 0.88 um.20 An ultra-low internal loss of 2 cm"1 made it possible to increase the cavity length up to 2 mm and to reduce the thermal resistance without considerable degradation of the external efficiency. A record for 2-um GaSb-based lasers CW output power of 1.9 W was demonstrated using a single-QW structure.21 The laser structure consisted of a 1 % compressively strained 10-nm thick, InGaAsSb QW with x = 0.19 and y = 0.02 surrounded by lattice-matched AlGaAsSb layers with 25 % Al in SCH. By adopting a broadened waveguide (BW) design, single-QW diode lasers emitting at 2.05 (im22 exhibited room-temperature threshold current densities as low as 50 A/cm2, one of the lowest values reported for any room-temperature diode laser. A 10-nm QW with x = 0.22 and y = 0.01 had an estimated compressive strain of 1.4 %. Compositionally graded regions were introduced between the layers to reduce the series resistance. The internal quantum efficiency and internal loss coefficient were 95 % and 7 cm"1, respectively. Based on this structure a tapered laser produced a CW diffractionlimited power of 0.6 W23 and an array of nine tapered lasers output a peak power of 3W. 24 Providing ultra-low internal loss, the BW design increases the role of recombination in the barriers and can decrease the internal efficiency. A small valence-band offset at the InGaAsSb/AlGaAsSb interface increases the role of thermionic hole emission into the barrier, especially for QW compositions with high As contents. A compressive strain decreases heavy-hole valence-band edge and contributes to the valence-band discontinuity. Essentially, the hole confinement can be improved by increasing the Al concentration in barriers. Thus, 2-um devices were designed similar to devices in Ref. 20 with only the difference being Al content in the barrier of 40 %, compared with 25 % in the original structure. Two-QW devices demonstrated an enhanced internal efficiency of 80 % and a saturated gain of 30 cm"'/QW, compared with 60 % and 20 crn'/QW observed in Ref. 20. At temperatures below 50 °C, similar 2-um 4-QW lasers demonstrated a record value of T0 = 140 K for long-wavelength lasers.25 Authors26 suggested increasing the As concentration in the QW in order to improve the electron localization, while the Coulomb interaction compensates the negative effect of the shallower QW in the valence band. The QW laser structure was grown with high contents of both In (x = 0.35) and As (y = 0.16). The 0.8-um active region consisted of three strained 10-nm QWs and four 30-nm AlGaAsSb barriers with a 35 % Al content, surrounded by SCH layers of the same composition. Based on the decrease in the external

91

1030

D. V. Donetsky, R. U. Martinelli & G. L. Belenky

efficiency with increasing cavity length, the estimated internal efficiency was greater than 75 %, and the internal loss was less than 7 %. The maximum output power reached 45 mW/facet for a 5-mn-wide ridge-waveguide laser operated in a single spatial mode. At room temperature the characteristic temperature was 120 K. The lasers demonstrated CW operation up to 130 °C with an increase of the emission wavelength from 2.26 um at room temperature to 2.43 um at 124 °C. Among the lowest threshold current densities of 120 A/cm2 for single-QW lasers were demonstrated by 2.26-um lasers.27 The compressively strained (2.5%) single-QW and triple-QW structures with 10-nm-wide QWs and QW composition of x = 0.30 and y = 0.06 were separated by 20-nm AlGaAsSb barriers with 28 % Al content. Degradation of the internal efficiency with increasing QW number was not observed; the internal efficiencies were 65 % and 69 % for single-QW and triple-QW structures, respectively. A single-ended power of 240 mW CW was obtained from a triple-QW laser. Reproduction of the designs described above is very difficult in the InGaAsSb material system due to the multiple growth problems listed above. In a number of works only one of the design parameters was intentionally varied, while the others were unchanged. Among these are the strain adjustment28 with variation of only the As content in either the QW or the barrier, variation of the Al contents in the barriers25, and variation of the QW number.25'27 The variation of In composition in the quasi-ternary InGa(As)Sb system to be described in the section below was the most successful with respect to increasing the wavelength and the maximum output power.

3. High-power Room-temperature CW Diode Lasers Operating in the Wavelength Range of 2.3 - 2.6-pjn A new approach in the design of InGaAsSb/AlGaAsSb QW lasers that has lead to CW room-temperature lasing up to 2.7 um was suggested in Ref. 29. In the InGaAsSb QW, only the In composition was increased while holding the As composition constant at about 2 %. This value of As was used previously for 2-um lasers. The authors call these QWs quasi-ternary since the As level was maintained at a very low value and not intentionally varied during MBE growth. The compressive strain in the QWs increased from 1.5 to 2.3 % with increasing In content. This approach eliminated degradation of the QW material observed in earlier attempts to achieve longer wavelengths when 1 % strain was maintained by increasing the compositional values for both In and As simultaneously. The diode structures consisted of two QWs located in the central part of the undoped 0.8-(am-thick AlGaAsSb waveguide layer having 30 % Al content. A double-QW structure was chosen to avoid the fast gain saturation previously observed for 2-um lasers with a single-QW active region. For most of the structures the distance between the QWs was approximately ten times greater than the QW thickness. The AlGaAsSb cladding layers with 90 % Al were 2-um thick. The n-cladding layer was doped to n = 2xl0 1 7 cm" 3 using Te. The p-cladding layer had a Be step-doping profile with p = 5 x 1017 cm"3 in 0.2-um-thick layers adjacent to the waveguide and 5 x 1018 cm"3 in the remaining 1.8 um. The GaSb cap layer was doped p = 4 x 1019 cm."3 The In composition was intentionally varied from x = 0.25 to 0.40, while the QW thicknesses were in the range of 10 nm to 20 nm. A photoluminescence (PL) study was

92

Mid-Infrared

GaSb-Based Lasers with Type-I Heterointerfaces

1031

conducted on the identical structures. The PL intensity decreased by factor of two over the spectral range of 2.3 - 2.6 urn, and a more significant decrease was observed for structures with wavelength above 2.6 um. Gain-guided Fabry-Perot lasers with 100-umwide stripe contacts and 1-mm or 2-mm cavity lengths were characterized. Figure 4 shows differential efficiency and threshold current density for 2-mm-long-cavity diodes prepared from five wafers. For lasers with In-composition in the QWs exceeding 0.35, a sharp increase of threshold current density and rapid decrease of differential efficiency were observed. This is shown in Figure 4 for 2.7-um devices where the In-composition was 0.38. For the case of x = 0.40, the measured room-temperature threshold current density was 12 kA/cm2. From these results and the results of the PL studies, it was concluded that the strain relaxation and the generation of dislocations near the QWs starts at In compositions exceeding 0.35.

1200 500|—•— 2.3 um 2.5 um

L=2mm

400l~*^~

Pulsed

2 6

^

m

I 40

CW .

20

17°C 0

2.3 2.4 2.5 2.6 2.7 Lasing Wavelength (um)

1000 2000 3000 4000 5000 Current (mA)

Figure 4. Threshold current density (pulsed and CW) and differential efficiency (pulsed) for the lasers of different wavelengths.25

Figure 5. CW output power for three diode lasers of different wavelengths.29

At T < 20°C for lasers with wavelengths shorter than 2.7-um, the laser parameters were weakly dependent on QW composition. The threshold current density in the pulsed regime increased slowly from 230 to 300 A/cm2, while the wavelength increased from 2.3 to 2.6 um. Output power characteristics measured in the pulsed regime were linear up to ten times the threshold current. Corresponding values of differential efficiency were independent of wavelength in the wavelength range of 2.3 - 2.6 um and close to 30 % for all devices with 2-mm-long cavities. Figure 5 shows CW output powers of 500, 250, and 160 mW that were obtained for 100-um-stripe-width lasers emitting at wavelengths of 2.3, 2.5 and 2.6-um, respectively. In contrast to pulsed-current measurements, the CW output powers tend to saturate with increasing current. The maximum output powers were limited by the power saturation due to overheating of the active region. To clarify the factors limiting the maximum external efficiency to 30 %, evaluation of both internal optical loss and internal efficiency were necessary. The method of extrapolating the inverse external efficiency to zero cavity length becomes uncertain in

93

1032

D. V. Donetsky,

R. U. Martinelli

& G. L.

Belenky

low-loss structures and underestimates both parameters. The value of the total loss was determined from the modal gain spectra shown in Figure 6. These spectra were obtained from the contrast of Fabry-Perot fringes in the amplified spontaneous emission spectrum using both Hakki-Paoli and Cassidy approaches. To select a single lateral mode a Michelson interferometer was utilized together with spatial filtering of the far field emission.30 The values of total loss were determined through the TE-TM-gain-crossing method previously applied to telecommunication lasers.31 Similar total losses were obtained from the long-wavelength modal gain saturation values32 (see Figure 6). The internal loss, a^t, and the device internal efficiency, n^,,, were calculated using measured slope efficiency and known device mirror loss am. For a 2-mm-cavity-length device with antireflection/high reflection coating ctm was estimated to be 8 cm"1, resulting in calculated values of aint = 9 cm"' and r)in, = 51 %. Thus, improvement of the internal efficiency is a very important task in the device optimization. Direct measurements show that the leakage of holes from the SCH layer into the n-cladding layer is insignificant and, hence, does not affect the internal efficiency.30 On the other hand, spontaneous emission measured in the pulsed-current from the side of the laser, normal to the optical-mode axis, continues to increase above the threshold, indicating an increase in the carrier concentration. In this case, enhancement of recombination processes with concentration necessarily leads to the efficiency degradation.

Figure 6. Current dependencies of the modal gain spectra in 2.3-um InGaAsSb QW lasers. Total optical loss value was determined from intersection point of TE and TM modal gain spectra at threshold and confirmed by the gain saturation value at longer wavelengths.

Figure 7. Temperature dependencies of the modal gain spectra in 2.3-um InGaAsSb QW lasers. Rate of the gain broadening with temperature (FWHM) and temperature dependence of internal loss are shown in the inset.

Figure 7 shows the temperature dependencies of the modal gain spectra in 2.3-um lasers. No significant increase of the internal loss was observed. A noticeable broadening of the gain spectra with increasing temperature and current causes gain suppression and

94

Mid-Infrared

GaSb-Based Lasers with T)jpe-I Heterointerfaces

1033

degrades the laser high-temperature performance. This broadening is characterized by the full-width at half-maximum of the gain spectra (FWHM) (see inset in Figure 7). The threshold current temperature dependence for these lasers was quite different above room temperature, and the characteristic temperature T0 was at least twice as large for 2.3-um lasers as it was for 2.6-um lasers as shown in Figure 8. The high value of characteristic temperature (T0 = 110 K) observed at T < 65 °C for 2.3-um lasers was identified with dominating monomolecular non-radiative recombination: the dependences of the spontaneous emission on current S (I) from the side of the chip for 0.98-um lasers were at least five times steeper than that for longwavelength devices of the same size. This dependence was linear, S xl, with a transition to S oc I' at higher temperatures and longer wavelengths, indicating that Auger recombination becomes dominant. Low T0 values of 40 to 50 K for 2.3-um lasers at T > 65 °C and for 2.6-um lasers in the whole temperature range studied were identified with the predominance of the Auger recombination process. 3 The detailed calculation of Auger recombination has shown that the main Auger process in these structures is the process with hole excitation from the QW into the continuum.34 The internal efficiency for 2.3-um lasers (Figure 9) demonstrated weak temperature dependence: rii, decreased by 6 % in the range of T = 15 to 90 °C.

% 20

40

60

80 100 120 Temperature ( C)

Temperature (°C)

Figure 8. Temperature dependence of the threshold current for 2.3 and 2.6 urn lasers with cavity length of 2 mm measured in pulsed regime.

Figure 9. Temperature dependence of the internal efficiency for 2.3-um lasers with cavity length of 2 mm. The external efficiency was measured in pulsed regime.

Index-guided ridge-waveguide lasers with 5-um-wide ridges were fabricated from double-QW structures with an In content x = 0.25 - 0.35 in the QWs.35 A 5-um-wide ridge was chosen, which, according to calculations,35 provides a single-spatial-mode operation. The ridge was chemically etched down to the waveguide, since this configuration reduces lateral losses for the zero-order mode, while maintaining high lateral losses for the higher-order modes. The comparison of measured and calculated lateral far-field patterns confirmed the single-spatial-mode operation of the devices. CW

95

1034

D. V. Donetsky,

R. U. Martinelli

& G. L. Be.le.nky

powers in the range of 3 - 15 mW were obtained for wavelengths of 2.3 to 2.6 um at room temperature (Figure 10). In short-cavity lasers a twofold jump in the differential efficiency can be observed. This increase takes place when the lasing switches from le lhh to 2e - 2hh transitions. The switching to shorter wavelength confirms that the carrier concentration continues to rise when the current exceeds the threshold for le - lhh transitions. During pulsed testing, the current at which lasing levels changed was independent of pulse duration, indicating a non-equilibrium carrier distribution. Ridge-waveguide devices with 0.25-mm cavity lengths demonstrated singlelongitudinal-mode, i. e., single-frequency, operation with a side-mode suppression ratio of 22 - 25 dB. Within regions of single-mode operation that are typically 0.5 to 2 nm wide, the wavelength can be tuned continuously by changing the temperature or current. Figure 11 shows gain spectra of 2.35-u.m ridge-waveguide laser with a cavity length of 0.5 mm.36 The shaded area indicates the device total loss (transparency level). The inset in Figure 11 shows the modal gain maximum as a function of the energy of quasiFermi level separation, AsF. Assuming that the total losses are the same for all current injection levels, the AsF value (transparency energy) can be estimated from intersection of the high-energy tail of the gain spectra and the transparency level. Since the change in AeF is approximately proportional to the change in carrier concentration, data in the inset present the behavior of the differential gain within a wide range of current injection levels. These data show that the modal gain peak for the higher-energy transition rises more rapidly than does the peak for the lower-energy transition.

20 Ridge-waveguide CW T=15°C

I

15

• /

n d =17% ». = 2.27nm ,

1

0.5 mm J

10

; ri d =10.5%

y*

' \ = 2.46 |im X• ■ "

SM

■ -v. 100

1 mm

. , . 200

300

)A6

400

0.50 0.54 0.58 Photon Energy (eV)

0.62

Current (mA)

Figure 11. Gain spectra for a 0.5 mm cavity length, 2.35-u.m, ridge-waveguide laser. The gain peaks resulting from the le-lhh and 2e-2hh transitions have the same magnitude. The shaded region indicates the device total loss. The inset shows the gain maxima of the corresponding peaks versus quasi-Fermi level separation.

Figure 10. CW output power for ridge-waveguide lasers with different cavity length of 0.5 mm and 1 mm. Two-fold dependence arises from wavelength switching from le-lhh to 2e-2hh transitions.35

Adjusting temperature, it is possible to maintain the comparable magnitude of gain for both transitions. Due to the small difference between transition energies (40 meV) the gain spectrum at threshold, 73 mA, covers essentially the whole interval between peaks.

96

Mid-Infrared

GaSb-Based Lasers with Type-I Heterointerfaces

1035

A wide optical gain bandwidth (FWHM > 350 nm) makes these lasers promising candidates as active elements in external-cavity tunable laser systems.

4. Conclusion The chart below summarizes current CW performance of mid-IR lasers with electrical pumping and illustrates the position of type-I diode lasers with respect to diode lasers utilizing type-II band alignment. Only lasers with CW output power more than 10 mW are shown. One can see that type-I lasers are superior to type-II lasers in the wavelength range up to 2.6-um, operating at room temperatures. On the other hand type-II lasers offer comparable output powers in the range of 3.25 - 3.4 -um with liquid nitrogen

Wavelength (|J,m) Figure 12. CW performance of GaSb-based lasers with electrical injection operating in the range of 1.9 to 3.4 p.m.

cooling. CW operation requires a cryogenic cooling; and the maximum CW operating temperature is still well below room temperature. Extension to longer wavelengths within the InGaAsSb system is challenging owing to lattice-mismatching and miscibility-gap limitations. Utilization of a strain-compensation technique might extend the operating wavelength range up to 3 um and fill out the gap in the spectrum between type-I and type-II lasers shown above.

97

1036 D. V. Donetsky, R. U. Martinelli & G. L. Belenky

Acknowledgement The authors appreciate to United States Air Force Office of Scientific Research for support, grant No F-49620-01-10108.

References 1

Z. Feit, M. McDonald, R. J. Woods, V. Archambault and P. Mak, "Low threshold PbEuSeTe/PbTe separate confinement buried heterostructure diode lasers", Appl. Phys. Lett. (1996), 68,738. 2 J. Major, J. S. Osinski, and D. F. Welsh, "8.5 W CW 2.0 urn InGaAsP laser diodes", Electron. Lett. (1993) 29, 2112. 3 J. S Wang, H. H. Lin, L. W. Sung, "Room-temperature 2.2 pm InAs-InGaAs-InP highly strained multiquantum-well lasers grown by gas-source molecular beam epitaxy", IEEE J Quant. Electron. (1998)34(10), 1959-1962. 4 R. U. Martinelli, and T. J. Zamerowski, "InGaAs/InAsPSb diode lasers with output wave;length at 2.52 \im",Appl. Phys. Lett. (1990) 56, 125. 5 Y. Cuminal, A. N. Baranov, D. Bee, P. Grech, M. Garcia, G. Boissier, A. Joullie, G. Glastre, and R. Blondeau, "Room-temperature 2.63 pm GalnAsSb/GaSb strained quantum-well laser diode", Semicond. Sci. Technol. (1999) 14, 283-288. 6 F. Karouta, A. Marbeuf, A. Joullie, and J. H. Fan, "Low temperature phase diagram of GalnAsSb", J. of Crystal Growth (1986) 79, 445. 7 N. Kobayahi, Y. Horikoshi, and C. Uemura, "Room temperature operation of InGaAsSb/AlGaAsSb laser at 1.8 pm wavelength", Japan. J. Appl. Phys. (1980) 19, L30-L32. 8 A. N. Baranov, T. N. Danilova, B. E. Dzhurtanov, A. N. Ilmenkov, S. G. Konnikov, A. M. Litvak, V. E. Usmanskii, and Yu. P. Yakovlev, "CW lasing in GalnAsSb/GaSb buried channel laser (T = 20 °C, X = 2.0 pm)", Sov. Tech. Phys. Lett. (1988) 14, 727-729. 9 A. E. Bochkarev, L. M. Dolginov, A. E. Drakin, P. G. Eliseev, and B. N. Sverdlov, "Continuouswave lasing at room temperature in InGaSbAs/GaAlAsSb injection heterostructutre emitting in the spectral range of 2.2-2.4 pm", Sov. J. Quant. Electron. (1988) 18, 1362-1363. 10 T. H. Chiu, W. T. Tsang, J. A. Ditzenberger, and J. P. van der Ziel, "Room temperature operation of InGaASSb/AlGaSb double heterostructure lasers near 2.2 urn prepared by molecular beam epitaxy", Appl. Phys. Lett. (1986)49, 1051-1052. " H. K. Choi, S. J. Eglash, and G. W. Turner, "Double-heterostructure diode lasers emitting at 3 pm with a metastable GalnAsSb active layer and AlGaAsSb cladding lasers", Appl. Phys. Lett. (1994)64(19), 2474-2476. 12 H. K. Choi, and S. J. Eglash, "High-efficiency high-power GalnAsSb-AlGaAsSb doubleheterostructure lasers emitting at 2.3 pm", IEEE J. Quant. Electron. (1991) 27, 1555-1565. 13 H. K. Choi, and S. J. Eglash, "Room-temperature cw operation at 2.2 pm of GalnAsSb/AlGaAsSb diode lasers grown by molecular beam epitaxy", Appl. Phys. Lett. (1991) 59(10), 1165-1167. 14 H. K. Choi and S. J. Eglash, "High-power multi-quantum-well GalnAsSb/AlGaAsSb diode lasers emitting at 2.1 pm with low threshold current density", Appl. Phys. Lett. (1992) 61(10), 1154-1156. 15 H. K. Choi, J. N. Walpole, G. W. Turner, S. J. Eglash, L. J. Missaggia, and M. K. Connors, "GalnAsSb-AlGaAsSb taper lasers emitting at 2 pm", IEEE Photon. Technol. Lett. (1993) 5(10), 1117-1119. 16 G. W. Turner, H. K. Choi, D. R. Calawa, J. V. Pantano, and J. W. Chludzinski, "Molecular-beam epitaxy growth of high-performance midinfrared diode lasers", J. Vac, Sci., Technol. B (1994) 12(2), 1266-1268.

98

Mid-Infrared GaSb-Based Lasers with Type-I Heterointerfaces

17

1037

H. K. Choi, G. W. Turner, and S. J. Eglash, "High-power GalnAsSb-AlGaAsSb multiplequantum-well diode lasers emitting at 1.9 urn", IEEE Photon. Technol. Lett. (1994) 6(1), 7-9. 18 H. Lee, P. K. York, R. J. Menna, R. U. Martinelli, D. Z. Garbuzov, S. Y. Narayan, and J. C. Connolly, "Room-temperature 2.78 urn AlGaAsSb/InGaAsSb quantum-well lasers", Appl. Phys. Lett. (1995)66(15), 1942-1944. 19 D. Z. Garbuzov, R. U. Martinelli, R. J. Menna, P. K. York, H. Lee, S. Y. Narayan, and J. C. Connolly, "2.7 um InGaAsSb/AlGaAsSb laser diodes with continuous-wave operation up to 39°C", Appl. Phys. Lett. (1995) 67(10), 1346-1348. 20 D. Z. Garbuzov, R. U. Martinelli, H. Lee, P. K. York, R. J. Menna, J. C. Connolly, and S. Y. Narayan, "Ultralow-loss broadened waveguide high-power 2 um AlGaAsSb/InGaAsSb/GaSb separate-confinement quantum-well lasers", Appl. Phys. Lett. (1996) 69(14), 2006-2008. 21 D. Z. Garbuzov, R. U. Martinelli, H. Lee, R. J. Menna, P. K. York, L. A. DiMarco, M. G. Harvey, R. J. Matarese, S. Y. Narayan, and J. C. Connolly, "4W quasi-continuous-wave output power from 2 um AlGaAsSb/InGaAsSb single-quantum-well broadened waveguide laser diodes", Appl. Phys. Lett. (1997) 70 (22), 2931-2933. 22 G. W. Turner, H. K. Choi, M. J. Manfra, "Ultralow-threshold (50A/cm2) strained singlequantum-well GalnAsSb/AlGaAsSb lasers emitting at 2.05 um", Appl. Phys. Lett. (1998) 72 (8), 876-878. 23 H. K. Choi, J. N. Walpole, G. W. Turner, M. K. Connors, L. J. Missaggia, and M. J. Manfra, "GalnAsSb-AlGaAsSb tapered lasers emitting at 2.05 urn with 0.6-W diffraction-limited power", IEEE Photon. Technol. Lett. (1998) 10(7), 938-940. 24 J. N. Walpole, H. K. Choi, L. J. Missaggia, Z. L. Liau, M. K. Connors, G. W. Turner, M. J. Manfra, and C. C. Cook, "High-power high-brightness GalnAsSb-AlGaAsSb tapered laser arrays with anamorphic collimating lenses emitting at 2.05 um", IEEE Photon. Technol. Lett. (1999) 11(10), 1223-1225. 25 T. Newell, X. Wu, A. L. Gray, S. Dorato, H. Lee, and L. F. Lester, "The effect of increased valence band offset on the operation of 2 um GalnAsSb-AlGaAsSb lasers", IEEE Photon. Technol. Lett. (1999) 11(1), 30-32. 26 D. A. Yarekha, G. Glastre, Aperona, Y. Rouillard, F. Genty, E. M. Skouri, G. Boissier, P. Grech, A. Joullie, C. Alibert, and A. N. Baranov, "High temperature GalnSbAs/GaAlSbAs quantum-well single mode continuous wave lasers emitting near 2.3 um", Electron. Lett. (2000) 36(6), 537-538. 27 C. Mermelstein, S. Simanowski, M. Mayer, R. Kiefer, J. Schmitz, M. Walter, and J. Wagner, "Room-temperature low-threshold low-loss continuous-wave operation of 2.26 um GalnAsSb/AlGaAsSb quantum-well laser diodes", Appl. Phys. Lett. (2000) 77(11), 1581-1583. 28 S. Simanovski, N. Herres, C. Mermelstein, R. Kiefer, J. Schmitz, M. Walter, J. Wagner, G. Weimann, "Strain adjustment in (GaIn)(AsSb)/(AlGa)(AsSb) QWs for 2.3 - 2.7um laser structures", J. of Crystal Growth (2000) 209, 15-20. 29 D. Z. Garbuzov, H. Lee, V. Khalfin, R. Martinelli, J. C. Connolly and G. L. Belenky, "2.3 2.7um room temperature CW operation of InGaAsSb-AlGaAsSb broad waveguide SCH-QW diode lasers", IEEE Photon. Technol. Lett. (1999) 11(7), 794-796. 30 D. V. Donetsky, G. L. Belenky, D. Z. Garbuzov, H. Lee, R. U. Martinelli, G. Taylor, S. Luryi, and J. C. Connolly, "Direct measurements of heterobarrier leakage current and modal gain in 2.3 um double QW p-substrate InGaAsSb/AlGaAsSb broad area lasers", Electron. Lett. (1999) 35, 298-299. 31 L. J. P. Ketelsen, "Simple technique for measuring cavity loss in semiconductor lasers", Electron.Lett. (1994) 30 (17), 1422-1424; P. A. Andrekson, N. A. Olson, T. Tanbun-Ek, R. A. Logan, D. L. Coblentz, and H. Temkin, "Novel technique for determining internal loss of individual semiconductor laser", Electron. Lett. (1992) 28 (2), 171-172. 32 E. A. Avrutin, I. E. Chebunina, I. A. Eliashevttch, S. A. Gurevich, and G. E. Shtengel, "TE and TM optical gain in AlGaAs/GaAs single-quantum-well lasers", Semicond. Sci. Technol. (1993) 8 (1), 80-87.

99

1038

D. V. Donetsky, R. U. Martinelli & G. L. Belenky

33

D. Garbuzov, M. Maiorov, H. Lee, V. Khalfin, R. Martinelli, and J. Connolly, "Temperature dependence of continuous wave threshold current for 2.3 - 2.6 urn InGaAsSb/AlGaAsSb separate confinement heterostructure quantum well semiconductor diode lasers", Appl. Phys. Lett. (1999) 74(20), 2990-2992. 34 A. D. Andreev, D. V. Donetsky, "Analysis of temperature dependence of the threshold current in 2.3-2.6-um InGaAsSb/AlGaAsSb quantum-well lasers", Appl. Phys. Lett. (1999) 74(19), 27432745. 35 D. Garbuzov, R. Menna, M. Maiorov, H. Lee, V. Khalfin, L. DiMarco, D. Capewell, R. Martinelli, G. Belenky, and J. Connolly, "2.3 - 2.7 u,m room temperature CW operation of InGaAsSb/AlGaAsSb broad-contact and single-mode ridge-waveguide SCH MQW diode lasers", SPIEPhotonics West {\999) v. 3628, 124-129. 36 D. V. Donetsky, D. Westerfeld, G. L. Belenky, R. U. Martinelli, D. Z. Garbuzov, and J. C. Connolly, "Extraordinarily wide optical gain spectrum in 2.2 - 2.5 urn In(Al)GaAsSb/GaSb quantum-well ridge-waveguide lasers", J. of Appl. Phys. (2001) 90 (8), 4281-4283.

100

International Journal of High Speed Electronics and Systems, Vol. 12, No. 4 (2002) 1039-1056 © World Scientific Publishing Company

ADVANCES IN QUANTUM-DOT RESEARCH A N D TECHNOLOGY: T H E PATH TO A P P L I C A T I O N I N BIOLOGY

M. A. STROSCIO*'t and M. DUTTAt 'Department of Bioengineering, 'Department of Electrical and Computer Engineering, University of Illinois at Chicago, 851 S. Morgan Street, Chicago, llinois 60607, USA

This account presents a survey of recent advances in quantum-dot research and technol­ ogy and highlights trends of key significance as they relate to the emerging applications of quantum-dot technology in biology. Keywords: Quantum dots; advanced heterostructures; biotags; integration of nanostructures with biological structures; photoluminescence.

1. Introduction There is currently a major international awareness of the potential applications of semiconductor quantum-dot technology to biology.1 Indeed, semiconductor quantum-dot nanocrystals have been prepared and studied experimentally for their potential as fluorescent biological labels having narrow, tunable and symmetric emission spectra with properties that are potentially superior to those of conven­ tional dyes.2 Moreover, fluorescent semiconductor quantum dots have been covalently coupled to biomolecules as a step in their use in ultrasensitive biological detection applications where there is the potential for exploiting the clear advantage of the multiplexed optical coding of biomolecules.3 In parallel efforts, there have been recent advances in the programmed assembly of DNA functionalized semi­ conductor quantum dots 4 as well as in techniques for functionalizing the surfaces of gold nanoparticles 5 and for devising a method for designing particles with the physical properties of a chosen nanoparticle composition but with the extensivelystudied surface chemistry of gold.6 In parallel activities, peptides are being identified that have semiconductor binding specificity for directed nanocrystal assembly.7 All of these recent advances portend major advances in the uses of semiconductor quantum dots in biology. This paper surveys recent efforts to understand the electrical, optical, and mechanical properties of these semiconductor quantum dots with special emphasis on understanding the physical properties of quantum dots underlying these biological applications. A number of reviews on the growth, 8 characterization, 9 electron-phonon interactions, 10 and size- and shape-dependent properties 11 of these

101

1040

M. A. Stroscio & M. Dutta

quantum dots have been published in recent years and there are also several texts now available on the theory and experimental characterization of semiconductor quantum dots. 1 2 - 1 4 Moreover, there is an extensive literature covering many topics underlying the basic properties of quantum dots such as the optical lattice vibra­ tions of polar semiconductor quantum dots, 15 the effective-phonon approximation of polarons in ternary mixed crystals, 16 techniques for one-phonon Raman scatter­ ing from arrays of semiconductor nanocrystals, 17 and the theoretical models of the optical linewidths of individual quantum dots. 18 The electrical, optical, and mech­ anical properties of GaAs-based quantum dots have been studied extensively 19-29 and these dots have been applied in forefront applications such as the singlephoton detection in the far-infrared portion of the electromagnetic spectrum. 30 In addition, the electrical, optical, and mechanical properties of InAs-based quan­ tum dots have been studied extensively 31-40 and these dots have been applied in forefront applications such as the mid-infrared second-harmonic generation in p-type InAs/GaAs self-assembled quantum dots, 41 mid-infrared absorption and photocurrent spectroscopy of InAs/GaAs self-assembled quantum dots, 42 InAs quantum dots field effect transistors, 43 self-assembled InAs/GaAs quan­ tum dots intersubband detectors, 44 InAs-based infrared photodetectors, 45,46 and InGaAs/GaAs quantum dot lasers. 47 These studies are paralleled by other works on a variety of quantum-dot materials including GaSb/GaAs, 48 GaN, 49 P b S , 5 0 - 5 3 CdTe, 5 4 - 5 6 I n P , 5 7 - 5 9 and PbSe. 60 The synthesis and characterization of the struc­ tural properties of CdSe nanocrystals have received considerable attention 6 1 - 6 9 and these studies have been the basis for research on the electrical, optical and mech­ anical properties of CdSe-based quantum dots. 7 0 - 9 1 Moreover, the magnetic, 92 and infrared 93,94 properties of these CdSe quantum dots have been investigated and they have been applied to realize CdSe-based single-electron transistors. 95 Many of the applications of quantum dots in electronics and optoelectronics depend on a detailed understanding of the properties of semiconductor quantum dots. 9 6 - 9 8 The recent realization of quantum-dot lasers 97 ' 98 provides an excellent example of quantum dots in complex optoelectronic devices. In this paper, the focus is on highlighting recent developments in quantum-dot research and technology that are of special relevance to potential applications in biology. 2. Recent Developments in the Application of Quantum Dots to Biology Potential applications of semiconductor quantum-dot technology to biology1 are currently receiving the attention of the international research community as a result of several relatively recent discoveries and initiatives. Specifically, semicon­ ductor quantum-dot nanocrystals have been prepared and studied experimentally in light of their potential as fluorescent biological labels having narrow, tunable and symmetric emission spectra with properties that are potentially superior to

102

Advances in Quantum-Dot Research and Technology 1041

those of conventional dyes. 2 Furthermore, fluorescent semiconductor quantum dots have been covalently coupled to biomolecules as a step in their use in ultrasensi­ tive biological detection applications where there is the potential for exploiting the clear advantage of the multiplexed optical coding of biomolecules.3 As noted by Mattoussi et al.,69 organic fluorophores generally have narrow excitation spectra, and frequently exhibit broad emission bands with red tailing; as a result of spectral overlap, it is difficult to make simultaneous quantitative evaluation of the relative amounts of different probes present in the same sample. Moreover, the variation of the emission and absorption spectra of dye-tagged bioconjugates requires the use of chemically distinct molecular labels that must be synthesized and conjugated. In promising parallel efforts, there have been recent advances in the programmed as­ sembly of DNA functionalized semiconductor quantum dots 4 as well as in techniques for functionalizing the surfaces of gold nanoparticles 5 and for devising a method for designing particles with the physical properties of a chosen nanoparticle com­ position but with the extensively-studied surface chemistry of gold.6 In still other promising activities, peptides are being identified that have semiconductor binding specificity for directed nanocrystal assembly.7 These recent advances and discoveries portend major advances in the uses of semiconductor quantum dots in biology. In a particular noteworthy development in the application of quantum dots in the study of biology, Bruches et al.2 have prepared CdSe-based quantum-dot semiconductor nanocrystals as fluorescent biological labels. As will be discussed in Sec. 6, an important discovery motivates the use of CdSe quantum dots in biological applications; this is, it was discovered62 that a thin ZnS capping on a 2.7-to-3.0-nm diameter CdSe quantum dot passivates the core CdSe quantum dot removing surface traps and resulting in a very high quantum yield of 50% at room temperature. Within this general scheme, nanometer-sized quantum dots are detected through laser-stimulated photoluminescence, and biomolecules attached to the quantum dots are used for purposes such as to recognize specific analytes including viruses, DNA, or proteins. As compared with conventional fluorophores, these flourescent nanocrystals were found to have a narrow, tunable, symmetric emission spectrum as well as to be photochemically stable. Bruches et al.2 used core—shell particles of CdSe-CdS enclosed in silica shells in order to make them soluble in water. The utility of these nanocrystals for biological staining was demon­ strated by using them to fluorescently label mouse fibroblast cells with silica-coated CdSe—CdS nanocrystals of two different diameters so that they would fluoresce at two wavelengths: the smaller 2-nm-core nanocrystals emitted green radiation with a spectral maximum at 550 nm and a quantum yield of 15%, and the larger 4-nmcore nanocrystals emitted red radiation with a spectral maximum at 630 nm and a quantum yield of 6%. The surfaces of these nanocrystals were modified to interact selectively with the biological sample by (a) electrostatic and hydrogen-bonding interactions or (b) specific ligand-receptor interactions. In case (a), nanocrystals coated with trimethoxysilylpropyl urea and acetate groups were observed to bind

103

1042

M. A. Stroscio & M. Dutta

with high affinity in the cell nucleus. In case (b), the avidin-biotin interaction was used by Bruches et al.2 to specifically label the F-actin filaments with the 4-nmcore red-emitting nanocrystals. In this latter case, biotin was bound covalently to the nanocrystals and these nanocrystals were then use to label fibroblasts that had been incubated in streptavidin and phalloidin-biotin. To image these samples, both laser-scanning-confocal-fluorescence and conventional wide-field microscopes were used. Bruches et al.2 further noted that: (a) these nanocrystals had long fluores­ cence lifetimes of hundreds of nanoseconds; (b) additional future developments such as in situ hybridization, direct immunolabeling, and incorporation of nanocrystals into microspheres will be of utility in applications such as immunocytobiology and cytometry; and (c) InP-based and InAs-based nanocrystals could be used as far-red or infrared emitting nanoprobes. In another noteworthy development in the application of quantum dots in the study of biology, Chan and Nie 3 used highly luminescent ZnS-capped CdSe quantum dots covalently coupled to biomolecules for ultrasensitive biological detection. Within this general scheme, nanometer-sized quantum dots are detected through laser-stimulated photoluminescence, and biomolecules attached to the quantum dots are used for purposes such as the recognition of specific analytes including viruses, DNA, or proteins. In particular, these investigators linked quantum dots to antibodies and used them to label specific target proteins in human cancer cells. In this research, Chan and Nie3 used mercaptoacetic acid for solubilization and cova­ lent protein attachment. Chan and Nie 3 found that (a) the mercapto group binds to a Zn atom when it reacts with the ZnS-capped CdSe quantum dots in chloroform, (b) the polar carboxylic acid group renders the quantum dots water soluble, and (c) the free carboxyl group is available for covalent coupling to biomolecules — such as nucleic acids, peptides, and proteins — by cross-linking to reactive amine groups. Using such covalently attached proteins, Chan and Nie 3 demonstrated that the ZnScapped CdSe quantum dots were biocompatible in vitro and living cells. Moreover, they obtained fluorescent images from cultured HeLa cells that had been incu­ bated with mercapto-quantum-dots — as control samples — and with tranferrinquantum-dot conjugates; they found that in the absence of transferrin, no quantum dots were observed inside the cell. Chan and Nie 3 further investigated the use of quantum dot labels for, sensitive immunoassay. In this latter work, fluorescent im­ ages were obtained of quantum-dot-immunoglobulin G (IgG) conjugates that were incubated with bovine serum albumin (BSA) (0.5 mg/ml) and with a specific polyclonal antibody (0.5 /zg/ml); it was found that the polyclonal antibody recognized the Fab fragments of the immunoglobulin and led to extensive aggregation of the quantum dots, and — in contrast — well-dispersed, primarily single-quantum-dots were detected in the presence of BSA. In a major advance underlying the application of highly-fluorescent CdSe-based quantum dots, Mitchell, Mirkin, and Letsinger 4 have developed techniques for producing DNA-functionalized quantum dots. This development portends greatly expanded applications of quantum dots since the attributes of DNA include

104

Advances in Quantum-Dot Research and Technology 1043

exceptional binding specificity, ease of synthesis, and extensive programmability — as a result of nucleotide sequence — that opens the way to the directed assembly of quantum dots in two- and three-dimensional geometries. In their work, Mitchell, Mirkin, and Letsinger4 have used 3-mercaptopropionoc acid to passivate the quan­ tum dot surface and to act as a pH trigger for controlling water solubility and subsequent oligonucleotide surface immobilization. In their approach, 4 an excess of 3-mercaptopropionoc acid, specifically in the amount of 0.10 mL, is then used to react with a suspension of CdSe/ZnS quantum dots — coated with a mix­ ture of trioctylphosphine oxide (TOPO)/trioctylphosphine (TOP) — in 1.0 mL of N,N-dimethylformamide to form propionic acid functionalized quantum dots. The coating mixture of T O P O / T O P on the CdSe quantum dots results from the common methods 4 ' 62 ' 69 ' 93 of preparing highly luminescent quantum dots. Unfor­ tunately, TOPO/TOP-coated quantum dots are soluble only in nonpolar solvents. These quantum dots exhibit the characteristic vco band indicating the presence of surface bound propionic acid. Mitchell, Mirkin, and Letsinger 4 found that these quantum dots were still essentially insoluble in water but that their solubility was enhanced by deprotonating the surface bound mercaptopropionic acid with 4-(dimethylamino)-pyridine. This deprotonating procedure resulted in quantum dots that were readily soluable in water and were stable for up to a week at room temperature. The work of Mitchell, Mirkin, and Letsinger 4 is especially noteworthy because they report the first successful modification of semiconductor nanocrystals with single-stranded DNA, the generation of DNA-linked quantum dot assemblies, and a preliminary study of the optical properties of these structures. In yet another potentially revolutionary development, Mattoussi et al.69 have demonstrated the self-assembly of CdSe-ZnS quantum dot bioconjugates using engineered recombinant proteins. These protein-molecule-conjugated luminescent CdSe/ZnS core-shell nanocrystals have potential applications as bioactive fluo­ rescent probes in imaging and sensing as well as in other diagnostic applications such as immunoassay. In their approach, 65 Mattoussi et al. use a chimeric fu­ sion protein that binds electrostatically to the oppositely charged surface of the capped quantum dots. Mattoussi et al.69 developed a conjugation method based on self-assembly utilizing electrostatic attractions between negatively-charged lipoic acid capped CdSe-ZnS quantum dots and engineered bifunctional recombinant proteins consisting of positively charged entities — containing a leucine zipper — genetically fused with desired biologically relevant molecules. The sequence of steps employed by Mattoussi et al.69 to prepare water-soluble CdSe-ZnS coreshell nanoparticles follows: TOPO/TOP capping groups were exchanged with dihydrolipoic acid groups by suspending 100-300 mg of TOPO/TOP-capped dots after size selection precipitation in 150-500 /xL of dihydrolipoic acid; after di­ lution with about 1.5 mL of dimethylformamide, deprotonation of the terminal lipoic acid -COOH groups was carried out by adding potassium tert-butoxide; the resulting precipitate of nanoparticles and released T O P O / T O P reagents was sedimented by centrifugation; and the sediment was then dispersed in water and

105

1044

M. A. Stroscio & M. Dutta

centrifugation/filtration was used to remove T O P O / T O P resulting in a clear dispersion of alkyl-COOH capped nanocrystals. The resulting stable aqueous quantum-dot dispersions were found to have the emission characteristics of the initial nanoparticles; that is, a photoluminescent yield of about 10-20%. Mattoussi et o/.69 then conjugated these lipoic-acid-capped core-shell nanopar­ ticles with maltose-binding-protein-basic leucine zipper (MBP-zb) protein in 5 mM sodium borate at pH 9. Table 1 summarizes representative functionalization schemes as well as the intended applications.

Table 1. Compound semiconductor

Quantum-dot functionalization and application. Functionalization

Application

CdSe-based quantum-dot semiconductor nanocrystals: fluorescent biological labels

Trimethoxysilylpropyl urea & acetate groups 2

Binds with high affinity to cell nucleus due to electrostatic and hydrogen-bonding interactions

CdSe-based quantum-dot semiconductor nanocrystals: fluorescent biological labels

Avidin for avidin-biotin interaction 2

Specifically label the F-actin filaments

CdSe-based quantum-dot semiconductor nanocrystals: fluorescent biological labels

Mercaptoacetic acid (mercapto group binds to Zn) (polar carboxylic acid renders QD water soluable) (free carboxyl group is available for covalent coupling to biomolecules) 3

Coupling to biomolecules (by cross-linking to reactive amine groups) —

CdSe-based quantum-dot semiconductor nanocrystals: coated with trioctylphoshhine oxide (TOPO)/trioctylphosphine (TOP)

Excess of mercaptopropionic acid in solution of dimethylformamide with QDs in suspension to form propionic functionalized QDs 4

Binding to singlestranded DNA

CdSe-based quantum-dot semiconductor nanocrystals: coated with trioctylphoshhine oxide (TOPO)/trioctylphosphine (TOP)

Chimeric fusion protein binds electrostatically to oppositely charged surface of QD

Binding DNA

Here the conjugation method is based on the electrostatic self-assembly of: negatively-charged lipoic acid capped CdSe-ZnS QDs and engineered bifunctional 69 recombinant proteins consisting of positively charged entities — containing a leucine zipper — genetically fused with desired biologically relevant molecules

106

- nucleic acids - peptides - proteins

Generation of DNAlinked QD assemblies

Advances in Quantum-Dot

Research and Technology

1045

3. Recent Developments in GaAs-Based Quantum Dots As stated previously, the electrical, optical, and mechanical properties of GaAsbased quantum dots have been studied extensively 19-29 and these dots have been applied in forefront applications such as the single-photon detection in the farinfrared portion of the electromagnetic spectrum. 30 Bulk GaAs crystals are one of the most-studied of the direct-bandgap semiconductors. Moreover, bulk GaAs has bandgap of approximately 840 nm. While the growth and optical properties GaAs quantum dots have been studied extensively 19-29 they do not exhibit quantum yields as high as ZnS-passivated CdSe nanocrystals to be discussed in Sec. 6. The energy bandgap of bulk GaAs is in the near infrared portion of the electromagnetic spectrum. As is well-known in the semiconductor community, the energy bandgaps of direct-gap semiconductors determine the minimum energies of photon-emitting or photon-absorbing optical transitions between states in the valence and conduction bands. In Table 2, the energy bandgap of GaAs is compared with those of other III-V compound semiconductors.

Table 2. Energy bandgap of GaAs is compared with those of other III-V compound semiconductors. III-V Compound semiconductor

Bandgap (eV) (direct bandgaps unless otherwise specified)

Temperature: Room temperature is denoted by RT

BN cubic

6.45 (indirect) 14.5 (direct)

RT RT

BN hexagonal

5.2

RT

InSb

0.25

72 Kelvin

BP

2.4

RT

BAs

6.2

RT

A1P

3.6 (direct) 2.5 (indirect)

RT 4 Kelvin

AlAs

2.15 (indirect) 3.03 (direct)

RT RT

AlSb

1.6 (indirect) 2.30 (direct)

RT RT

GaN

3.44 (direct)

RT

GaP

2.27 (indirect) 2.78 (direct)

RT RT

GaAs

1.424 (direct)

RT

GaSb

0.75

RT

GaN

2.5

RT

InP

1.344

RT

InAs

0.354

RT

107

1046

M. A. Stroscio & M. Dutta

4. Recent Developments in InAs-Based Quantum Dots The electrical, optical, and mechanical properties of InAs-based quantum dots have been studied extensively 31-40 and these dots have been applied in fore­ front applications such as the mid-infrared second-harmonic generation in p-type InAs/GaAs self-assembled quantum dots, 41 mid-infrared absorption and photocurrent spectroscopy of InAs/GaAs self-assembled quantum dots, 42 InAs quantum dot field effect transistors, 43 self-assembled InAs/GaAs quantum dot intersubband detectors, 44 InAs-based infrared photodetectors, 45 ' 46 and InGaAs/GaAs quantum dot lasers. 47 Selected applications of quantum dots in electronics and optoelectronics depend on a detailed understanding of the properties of semi­ conductor quantum dots. 9 6 - 9 8 This situation is illustrated clearly by the case of InAs-based quantum dots where the recent realization of quantum-dot lasers 97 ' 98 provides an excellent example of quantum dots in complex optoelectronic devices. Boucaud et al.39 have observed far-infrared absorption in systems of two InAs quantum dots separated by a GaAs barrier; in these "InAs/GaAs quantum dot molecules", the electronic coupling between the dot states results in intraband absorption in the THz frequency range. Quantum dots that are fluorescent in response to irradiation in the infrared portion of the electromagnetic portion of the spectrum may find specialized applications in the study. 5. Recent Developments in GaSb-, GaN-, PbS-, CdTe-, I n P - , and PbSe-Based Quantum Dots While investigations of GaAs-based and InAs-based semiconductor quantum dots have been extensive, many other semiconductor quantum dot systems have been studied in light of their unique properties. As example, selected works have focused on understanding the properties quantum-dot materials including GaSb/GaAs, 4 8 GaN, 49 P b S , 5 0 - 5 3 CdTe, 54 " 56 InP, 5 7 " 5 9 and PbSe. 60 The Raman scattering measurements of de Paula et al.5i are of special significance since they reveal phonon confinement effects in CdTe nanocrystals that are a function of the quantum dot radius. Moreover, de Paula et al.54 observed longitudinal-optical (LO) phonon modes and surface phonons in CdTe quantum dots as well as some overtones com­ binations of both types of modes. Damping of coherent acoustic phonon modes in PbS has been observed experimentally 51 and the general features of these observations have been modeled 50 via the elastic continuum model. 6. Recent Developments in CdSe-Based Quantum Dots As discussed previously, the synthesis and characterization of the structural pro­ perties of CdSe nanocrystals have received considerable attention 6 1 - 6 9 and these studies have been the basis for research on the electrical, optical and mechanical properties of CdSe-based quantum d o t s . 7 0 - 9 1 In addition, the magnetic, 92 and infrared, 93 ' 94 properties of these CdSe quantum dots have been investigated and they have been applied to realize CdSe-based single-electron transistors. 95

108

Advances in Quantum-Dot Research and Technology 1047

Table 3. Energy bandgap of CdSe is compared with those of other II-VI compound semiconductors. II-VI Compound semiconductor

Bandgap, (eV) (direct bandgaps unless otherwise specified)

Temperature: Room temperature is denoted by RT

CdS hexagonal

2.4 Eg (A) 2.5 Eg (B) 2.55 Eg (C)

RT RT RT

CdS cubic

2.5

RT

CdSe hexagonal

1.75 Eg (A) 1.771 Eg (B) 2.17 Eg (C)

RT RT RT

CdSe cubic

1.9

RT

CdTe

1.49

RT

HgO

2.19

RT

HgS

2.10

RT

HgSe

-0.061

RT

Much of the recent activity directed at the use of quantum dots for biological applications is motivated by: - the excellent size control that can be achieved for CdSe nanocrystals; - the discovery62 that a thin (0.6 ± 0.3 nm) ZnS capping on a 2.7-to-3.0-nm diam­ eter CdSe quantum dot passivates the core CdSe quantum dot removing surface traps and resulting in a 50% quantum yield at room temperature; - the development of novel methods for conjugating protein molecules to lumines­ cent CdSe-ZnS quantum-dots for use as bioactive fluorescent probes. 66 The existence of this high quantum yield is a primary motivation for the exten­ sive attention give to CdSe-based quantum dots in Sec. 2 as well as by the interna­ tional research community working to realize the broad application of quantum-dot diagnostics in biology. In Table 3, the energy bandgap of CdSe is compared with those of other II-VI compound semiconductors. Klein et al.89 have derived expressions for the longitudinal-optical (LO) and surface-optical phonon modes in CdSe nanospheres and have modeled the size dependence of electron-phonon coupling in these semiconductor nanospheres. In the same paper, 89 these authors have compared their results with experi­ ments. These experiments are reported to confirm the size independence of the predicted 89 electron-phonon coupling constant and also manifest the presence of the SO modes. Such SO modes have been studied by a number of authors for CdSe quantum dots as well as for a variety of other semiconductor quantum dots. 99

109

1048 M. A. Stroscio & M. Dutta

7. Implementing Quantum Dot Technology in Biological Applications The enormous potential for using quantum dots in biological applications is not limited t o their uses as biological tags. In particular, the authors have conceived a number of ideas for using the photoelectric properties of quantum dots as active electronic interfaces with biological systems. As an example, in classic experi­ ments performed many years ago, 100 Hodgkin and Huxley showed that the voltagedependent switching — opening and closing — of K and Na ion channels are associated with the action potential 101 and that a 4 mV change 100 in the potential causes an e-fold change in the ratio of the open probability to the closed probability. This corresponds to a free-energy difference, VAq, with Aq = 6e and it is clear very few electrons are needed to switch such an ion channel. Accordingly, a quantum dot or an ensemble of quantum dots functionalized to bind to the membrane protein of such an ion channel could supply the electrons needed to switch such an ion channel upon photoexcitation. Thus the basic photoelectric effects in bulk semi­ conductors may have potential applications where the semiconductor materials are quantum dots that can be tailored through dimensional confinement to realized tunable electronic and optical properties. 98,99 ' 102 To realize such applications of quantum dots in biological systems it is neces­ sary to consider a number of properties of quantum dots that are influenced by dimensional confinement or that are influenced by their interactions in aqueous solutions. These properties include: (a) the photoluminescent spectrum; (b) the binding energies of carriers; (c) the dielectric properties of the quantum dot and of the surrounding materials; (d) the energy band structure of quantum dots; and (e) the linewidth-influencing acoustic properties of quantum dots. As is well known, the photoluminescent spectra and binding energies of quantum dots are influenced by dimension confinement and by their incorporation in polar and nonpolar semi­ conductor, polymer, and aqueous environments. 45 ' 103-109 The optical linewidths of quantum dots are also influenced by phonon-assisted processes that depend on the spherical nature of the vibrational modes in quantum dots. 110 Moreover, a number of coatings have been developed to ensure that specific functionalized quantum dots will be mobile in aqueous environments and will be able to bind to analytes. Silica2 and polymer coatings 51 have proven highly successful in this regard. Such coatings are especially critical in the case of polar quantum dots, such as CdSe. Clearly, all of these physical phenomena need to be addressed in more detail to pursue quantumdot applications in biology. Moreover, these phenomena provide an indication of the rich variety of research investigations underlying the application of quantum dots in biology. The enormous potential payoffs from the use of quantum dot technology in biology, are illustrated further by recent work on the interfacing of quantum dots and nerve cells 111 as well as by the observation of electrically-stimulated neurite growth. 112 Indeed, it has been demonstrated that now. commonly-known

110

Advances in Quantum-Dot Research and Technology 1049

quantum-dot growth 113 technology may serve as the basis for fabricating structures which may be subsequently coated with peptides that lead to the attachment 111 of CdS quantum-dot complexes to cells. To ensure that the quantum dots bind in close proximity to the cell, recognition-molecule-directed interfacing between semiconductor quantum dots and nerve cells is facilitated through the use of short peptide sequences 111 that result in an estimated maximum separation between the quantum dots and cells of about 3 nm. Clearly, these techniques portend the potential widespread use of quantum-dot technology for realizing intimate con­ tact between semiconductors and cells. Moreover, these techniques offer means of potentially achieving reproducible and reliable neuron-semiconductor interfacing even in situations where neuronal growth is expected. In conclusion, potential applications of semiconductor quantum dots in biology include the use of quantum dots as biotags as well as their use as electrical interfaces with neurons. To realize the potentially revolutionary applications of quantum dots in biology, it is essential that the optical, electrical, and acoustical properties of quantum dots be understood in biological environments. A first step to providing this broad understanding, is defined by efforts to model and measure the properties of quantum dots in aqueous environments. Even for this rather restricted set of environmental conditions, a rich and diverse group of physical phenomena must be taken into account in order to render quantum-dot technology useful in a broad range of biological applications. Acknowledgments The authors are appreciative of the encouragement and contributions of Prof. Michael Shur; his insights have had a major impact on this paper. The authors thank Prof. Philippe Guyot-Sionnest of the University of Chicago for identifying several key trends in quantum-dot research. In addition, the authors have benefitted from discussions with Prof. Chad Mirkin of Northwestern University on the general topic of receptor-analyte interactions as well as with Prof. Raphael Tsu of the University of North Carolina of Charlotte on the topic of the Penn model as well as other related dielectric effects in quantum dots. Moreover, the authors are indebted to Dr. Dan Johnstone and Dr. Todd Steiner of the Air Force Office of Sci­ entific Research and Dr. John Carrano of the Defense Advanced Research Projects Agency for their encouragement regarding our research on nanostructures. This research was supported, in part, under AFOSR Grant F49620-02-1-0224. References 1. E. Klarreich, "Biologists join the dots", Nature 413 (2001) 450-452. 2. M. Bruchez, Jr., M. Moronne, P. Gin, S. Weiss, and P. A. Alivisatos, "Semiconductor nanocrystals as fluorescent biological labels", Science 281 (1998) 2013-2016. 3. W. C. W. Chan and S. Nie, "Quantum dot bioconjugated for ultrasensitive nonisotropic detection", Science 281 (1998) 2016-2018; M. Han, X. Gao, J. Z. Su,

111

105O

4. 5. 6. 7.

8. 9. 10. 11. 12.

13. 14. 15. 16. 17.

18. 19. 20. 21. 22. 23. 24. 25.

M. A. Stroscio & M. Dutta

and S. Nie, "Quantum-dot-tagged microbeads for multiplexed optical coding of biomolecules", Nature Biotechnology 19 (2001) 631-635. G. P. Mitchell, C. A. Mirkin, and R. L. Letsinger, "Programmed assembly of DNA functionalized quantum dots", J. Am. Chem. Soc. 121 (1999) 8122-8123. K. J. Watson, J. Zhu, S. T. Nguyen, and C. A. Mirkin, "Hybrid nanoparticles with block copolymer shell structures", J. Am. Chem. Soc. 121 (1999) 462-463. Y. W. Cao, R. Jin, and C. A. Mirkin, "DNA-modified core shell Ag/Au nanoparti­ cles", J. Am. Phys. Soc. 123 (2001) 7961-7962. S. R. Whaley, D. S. English, E. L. Hu, P. F. Barbara, and A. M. Belcher, "Selection of peptides with semiconductor binding specificity for directed nanocrystal assembley", Nature 405 (2000) 665-668. A. Ekimov, "Growth and optical properties of semiconductor nanocrystals", J. Lu­ minescence 70 (1996) 1-20. O . I . Micic and A. J. Nozik, "Synthesis and characterization of binary and ternary III-V quantum dots", J. Luminescence 70 (1996) 95-107. T . Takagahara, "Electron-phonon in semiconductor nanocrystals", J. Luminescence 70 (1996) 129-143. Y. Ohfuti and K. Cho, " Size- and shape-dependent optical properties of assembled semiconductor spheres", J. Luminescence 70 (1996) 203-211. L. Banyai and S. W. Koch (eds), Semiconductor Quantum Dots, Series of Atomic, Molecular, and Optical Physics, Vol. 2, World Scientific, Singapore, New Jersey, London, Hong Kong, 1993. D. Binberg, M. Grundmann, and N. N. Ledentson, Quantum Dot Heterostructures, John Wiley & Sons, Chichester, 1999. U. Woggon, "Optical properties of semiconductor quantum dots", Springer Tracts in Modern Physics, Vol. 136, Springer, Berlin, 1996. R. Englman and R. Ruppin, "Optical lattice vibrations in finite ionic crystals: I", J. Phys. C (Proc. Phys. Soc.) 1 (1968) 614-629. X. X. Liang and J. Yang, "Effective-phonon approximation of polarons in ternary mixed crystals", Solid State Commun. 100 (1996) 629-634. V. Vasilevskiy, A. G. Rolo, and M. J. M. Gomes, "One-phonon raman scatter­ ing from arrays of semiconductor nanocrystals", Solid State Commun. 104 (1997) 381-386. X.-Q. Li and Y. Arakawa, "Optical linewidths in an individual quantum dots", Phys. Rev. B 60 (1999) 1915-1920. X.-Q. Li, H. Nakayama, and Y. Arakawa, "Phonon bottleneck in quantum dots: Role of lifetime of the confined optical phonons", Phys. Rev. B 59 (1999) 5069-5073. T. Takagahara, "Theory of exciton dephasing in semiconductor quantum dots", Phys. Rev. B 60 (1999) 2638-2652. H. Gotoh, H. Ando, and T. Takagahara, "Radiative recombination lifetime of excitons in thin quantum boxes", J. Appl. Phys. 81 (1997) 1785-1789. X.-Q. Li and Y. Arakawa, "Ultrafast energy relaxation in quantum dots through defect states: A lattice-relaxation approach", Phys. Rev. B 56 (1997) 10423-10427. M. El-Said, "Study of the energy level-crossings in GaAs/Ala;Gai_ x As quantum dots", Solid State Commun. 97 (1996) 971-971. T. Inoshita and H. Sakaki, "Electron-phonon interaction, and the so-called phonon bottleneck effect in semiconductor quantum dots", Phys. B 227 (1996) 373-377. M. Grundmann, J. Christen, N. N. Ledentsov, J. Bohrer, D. Bimberg, S. S. Ruvimov, P. Werner, U. Richter, U. Gosele, J. Heydenreich, V. M. Ustinov, A. E. Zhukov, P. S. Kop'ev, and Zh. I. Alferov, "Ultranarrow luminescence lines from single quan­ tum dots", Phys. Rev. Lett. 74 (1995) 4043-4046.

112

Advances in Quantum-Dot Research and Technology 1051

26. I. Vurgaftman, Y. Lam, and J. Singh, "Carrier thermalization in sub-threedimensional electronic systems: Fundamental limits on modulation bandwidth in semiconductor lasers", Phys. Rev. B 50 (1994) 14309-14326. 27. L. P. Kouwenhoven, S. Jauhar, J. Orenstein, P. L. McEuen, Y. Nagamune, J. Motohisa, and H. Sakaki, "Observation of phonon-assisted tunneling through a quantum dot", Phys. Rev. Lett. 73 (1994) 3443-3446. 28. I. Vurgaftman and J. Singh, "Effect of spectral broadening and electron-hole scattering on carrier relaxation in GaAs quantum dots", Appl. Phys. Lett. 64 (1994) 232-234. 29. L. Banyai, P. Gilloit, Y. Z. Hu, and S. W. Koch, "Surface-polarization instabilities of electron-hole pairs in semiconductor quantum dots", Phys. Rev. B 45 (1992) 14136-14142. 30. S. Komiyama, O. Astafiev, V. Antonov, T. Kutsuwa, and H. Hirai, "A single-photon detector in the far-infrared range", Nature 403 (2000) 405-407. 31. B. Legrand, J. P. Nys, S. Grandidier, A. Lemaitre, J. M. Gerard, and V. ThierryMieg, "Quantum box size effect on vertical self-alignment studied using crosssectional scanning tunneling microscopy", Appl. Phys. Lett. 74 (1999) 2608-2610. 32. S. Sauvage, P. Boucaud, F. Glotin, R. Prazeres, J.-M. Ortega, A. Lemaitre, J.-M. Gerard, and V. Thierry-Mieg, "Saturation of intraband and electron relaxation time in n-Doped InAs/GaAs self-assembled quantum dots", Appl. Phys. Lett. 73 (1998) 3818-3820. 33. B. Legrand, S. Grandidier, J. P. Nyes, D. Stievenard, J. M. Gerard, and V. ThierryMieg, "Scanning tunneling microscopy and scanning tunneling spectroscopy of self-assembled InAs quantum dots", Appl. Phys. Lett. 73 (1998) 96-98. 34. H. Zhou, A. V. Nurmikko, C.-C. Chu, J. Han, R. L. Gunshor, and T. Takagahara, "Observation of antibound states for exciton pairs by four-wave-mixing experiments in a single ZnSe quantum well", Phys. Rev. B 58 (1998) R10131-R10134. 35. S. Sauvage, P. Boucaud, F. H. Julien, J.-M. Gerard, and V. Thierry-Mieg, "Intra­ band absorption in n-doped InAs/GaAs quantum dots", Appl. Phys. Lett. 71 (1997) 2785-2787. 36. S. V. Nair and T. Takagahara, "Weakly correlated exciton pair states in large quan­ tum dots", Phys. Rev. B 53 (1996) R10516-R10519. 37. M. Grundmann, O. Stier, and D. Bimberg, "InAs/GaAs pyramidal quantum dots: Strain distribution, optical phonons, and electronic structure", Phys. Rev. B 52 (1995) 11969-11981. 38. J.-Y. Marzin, J.-M. Gerard, A. Izael, D. Barrier, and G. Bastard, "Photoluminescence of single InAs quantum dots obtained by self-organized growth on GaAs", Phys. Rev. Lett. 73 (1994) 716-719. 39. P. Boucaud, K. S. Gill, J. B. Williams, M. S. Sherwin, W. V. Schoenfeld, and P. M. Petroff, "Saturation of THz-frequency intraband absorption in InAs/GaAs quantum dot molecules", Appl. Phys. Lett. 77 (2000) 510-512; P. Boucaud, J. B. Williams, K. S. Gill, M. S. Sherwin, W. V. Schoenfeld, and P. M. Petroff, "Terahertz-frequency electronic coupling in vertically coupled quantum dots", Appl. Phys. Lett. 77 (2000) 4356-4358. 40. S. Ruvimov, P. Werner, K. Scheerschmidt, U. Gosele, J. Heydenreich, U. Richter, N. N. Ledentsov, M. Grundmann, D. Bimberg, V. M. Ustinov, A. Yu Ergorov, P. S. Kop'ev, and Zh. I. Alferov, "Structural characterization of (In,Ga)As quantum dots in a GaAs matrix", Phys. Rev. B 51 (1995) 14766-14769. 41. T. Brunhes, P. Boucaud, S. Sauvage, F. Glotin, R. Prazeres, J.-M. Ortega, A. Lemaitre, and J.-M. Gerard, "Midinfrared second-harmonic generation in p-type InAs/GaAs self-assembled quantum dots", Appl. Phys. Lett. 75 (1999) 835-837.

113

1052

M. A. Stroacio & M. Dutta

42. S. Sauvage, P. Boucaud, T. Brunhes, V. Immer, E. Finkman, and J.-M. Ger­ ard, "Midinfrared absorption and photocurrent spectroscopy of InAs/GaAs selfassembled quantum dots", Appl. Phys. Lett. 78 (2001) 2327-2329. 43. G. Yusa and H. Sakaki, "InAs quantum dot field effect transistors", Superlattices and Microstructures 25 (1999) 247-250. 44. J. Phillips, P. Bhattacharya, S. W. Kennedy, D. W. Beekman, and M. Dutta, "Self-assembled InAs-GaAs quantum-dot intersubband detectors", IEEE J. Quan­ tum Electron. 35 (1999) 936-943. 45. V. V. Mitin, V. I. Pipa, A. V. Sergeev, M. Dutta, and M. Stroscio, "Highgain quantum-dot infrared photodetector", Infrared Phys. Technol. 42 (2001) 467-472. 46. V. Ryzhii, I. Khmyrova, V. Mitin, M. Stroscio, and M. Willander, "On the detectivity of quantum-dot infrared photodetectors", Appl. Phys. Lett. 78 (2001) 3523-3525. 47. R. L. Sellin, Ch. Ribbat, M. Grundmann, N. N. Ledentsov, and D. Bimberg, "Closeto-ideal device characteristics of high-power InGaAs/GaAs quantum dot lasers", Appl. Phys. Lett. 78 (2001) 1207-1209. 48. Ph. Lelong, K. Suzuki, G. Bastard, H. Sakaki, and Y. Arakawa, "Enhancement of the Coulomb correlations in Type-II quantum dots", Phys. E 7 (2000) 393-397. 49. K. Tachibana, T. Someya, Y. Arakawa, R. Werner, and A. Forchel, "Roomtemperature lasing oscillation in an InGaN self-assembled quantum dot laser", Appl. Phys. Lett. 75 (1999) 2605-2607; O. I. Micic, S. P. Ahrenkiel, D. Bertram, and A. J. Nozik, "Synthesis, structure, and optical properties of colloidal GaN quan­ tum dots", Appl. Phys. Lett. 75 (1999) 478-480. 50. M. A. Stroscio and M. Dutta, "Damping of nonequilibrium acoustic phonon modes in a semiconductor quantum dot", Phys. Rev. B 60 (1999) 7722; for the normalization of these modes see, M. A. Stroscio, K. W. Kim, S. Yu, and A. Ballato, "Quantized acoustic phonons in quantum wires, and quantum dots", J. Appl. Phys. 76 (1994) 4670-4975. 51. T. D. Krauss and F. W. Wise, "Coherent acoustic phonons in a semiconductor quantum dot", Phys. Rev. Lett. 79 (1997) 5102-5105. 52. T. D. Krauss, F. W. Wise, and D. B. Tanner, "Observation of coupled vibrational modes of a semiconductor nanocrystal", Phys. Rev. Lett. 76 (1996) 1376-1379. 53. P. Hoyer and R. Konenkamp, "Photoconduction in porous Ti02 sensitized PdS quantum dots", Appl. Phys. Lett. 66 (1995) 349-351. 54. A. M. de Paula, L. C. Barbosa, C. H. B. Cruz, O. L. Alves, J. A. Sanjurjo, and C. L. Cesar, "Quantum confinement effects on the optical phonons of CdTe quantum dots", Superlattices and Microstructures 23 (1998) 1104-1106. 55. C. R. M. de Oliveira, A. M. de Paula, F . O. Filho, N. Plentz, J. A. Medeiros, L. C. Brabosa, O: L. Alves, E. A. Menezes, J. M. M. Rios, H. L. Fragnito, C. H. B. Cruz, and C. L. Cesar, "Probing of the quantum dot size distribution in CdTe-doped glasses by photoluminescence excitation spectroscopy", Appl. Phys. Lett. 66 (1995) 439-441. 56. V. Esch, B. Fluegel, G. Khitrova, H. M. Gibbs, J. Xu, K. Kang, S. W. Koch, L. C. Liu, S. H. Risbud, and N. Peyghambarian, "State filling, Coulomb, and trapping effects in the optical nonlinearity of CdTe quantum dots in glass", Phys. Rev. B 42 (1990) 7450-7453. 57. O. I. Micic, S. P. Ahrenkiel, and A. J. Nozik, "Synthesis of extremely small InP quantum dots and electronic coupling in their disordered solid films", Appl. Phys. Lett. 7 8 (2001) 4022-4024; O. I. Micic, J. Sprague, Z. Lu, and A. J. Nozik, "Highly efficient band-edge emission from InP quantum dots", Appl. Phys. Lett. 68 (1996) 3150-3152.

114

Advances in Quantum-Dot Research and Technology 1053

58. U. Banin, G. Cerullo, A. A. Guzelian, C. J. Bardeen, A. P. Alivisatos, and C. V. Shank, "Quantum confinement and ultrafast dephasing in InP nanocrystals", Phys. Rev. Lett. B 55 (1997) 7059-7067. 59. H. Giessen, B. Fluegel, G. Mohs, N. Peyghambarian, J. R. Sprague, O. I. Mimic, and A. J. Nozik, "Observation of the quantum confined ground state in InP quantum dots", Appl. Phys. Lett. 68 (1996) 304-306. 60. G. Springholz, V. Holy, M. Pinczolits, and G. Bauer, "Self-organized growth of three-dimensional quantum-dot crystals with FCC-like stacking and tunable lattice constant", Science 282 (1998) 734-737. 61. M. Danek, K. F. Jensen, C. B. Murray, and M. G. Bawendi, "Electrospray organometallic chemical vapor deposition — A novel technique for preparation of II-VI quantum dot composites", Appl. Phys. Lett. 65 (1994) 2795-2797. 62. M. Hines and P. Guyot-Sionnest, "Synthesis and characterization of strongly lumi­ nescing ZnS-capped CdSe nanocrystals", J. Phys. Chem. 100 (1996) 468-471. 63. M. Danek, K. F. Jensen, C. B. Murray, and M. G. Bawendi, "Synthesis of luminescent thin-film CdSe/ZnSe quantum dot composites using CdSe quantum dots passivated with an overlayer of ZnSe", Che. Mater. 8 (1996) 173-180. 64. D. E. Fogg, L. H. Radziloski, B. O. Dabbousi, R. R. Schrock, E. L. Thomas, and M. G. Bawendi, "Fabrication of quantum dot-polymer composites: Semiconduc­ tor nanoclusters in dual-function polymer matrices with electron-transporting and cluster-passivating properties", Macromolecules 30 (1997) 8433-8439. 65. E. Empedocles, D. J. Norris, and M. G. Bawendi, "Photoluminescence spectroscopy of single CdSe nanocrystalhte quantum dots", Phys. Rev. Lett. 77 (1996) 3873-3876. 66. C.-C. Chen, A. B. Herhold, C. S. Johnson, and A. P. Alivisatos, "Size depen­ dence of structural metastability in semiconductor nanocrystals", Science 276 (1997) 398-401. 67. J. R. Heine, J. Rodriguez-Viejo, M. G. Bawendi, and K. F. Jensen, "Synthesis of CdSe quantum dot-ZnS matrix thin films via electrospray organometallic chemical vapor deposition", J. Crystal Growth 195 (1998) 564-568. 68. X. Peng, L. Manna, W. Yang, J. Wickham, E. Scher, E. Kadavanich, and A. P. Alivisatos, "Shape control of CdSe nanocrystals", Nature 404 (2000) 59-61. 69. H. Mattoussi, J. Matthew Mauro, E. R. Goldman, G. P. Anderson, V. C. Sundar, F. V. Mikulec, and M. G. Bawendi, "Self-assembly of CdSe-ZnS quantum dot bioconjugates using an engineered recombinant protein", J. Am. Chem. Soc. 122 (2000) 12142-12150. 70. C. Wang, M. Shim, and P. Guyot-Sionnest, "Electrochromic nanocrystal quantum dots", Science 291 (2001) 2390-2392. 71. P. Palinginis and H. Wang, "High-resolution spectral hole burning in CdSe/ZnS core/shell nanocrystals", Appl. Phys. Lett. 78 (2001) 1541-1543. 72. M. Shim and P. Guyot-Sionnest, "Intraband hole burning of colloidal quantum dots", Phys. Rev. B 64 (2001) 245342-1-5. 73. C. E. Finlayson, D. S. Ginger, and N. C. Greenham, "Optical microcavities using highly luminescent films of semiconductor nanocrystals", Appl. Phys. Lett. 77 (2000) 2500-2502. 74. V. I. Klimov, Ch. J. Schwarz, D. W. McBranch, C. A. Leatherdale, and M. G. Bawendi, "Ultrafast dynamics of inter- and intraband transitions in semiconduc­ tor nanocrystals: Implications for quantum-dot lasers", Phys. Rev. B 60 (1999) R2177-R2180. 75. E. Poles, D. C. Selmarten, O. I. Micic, and A. J. Nozik, "Anti-stokes photolumi­ nescence in colloidal semiconductor quantum dots", Appl. Phys. Lett. 75 (1999) 971-973.

115

1054

M. A. Stroscio & M. Dutta

76. B . Alperson, I. Rubinstein, and G. Hodges, "Energy level tunneling spectroscopy and single electron charging in individual CdSe quantum dots", Appl. Phys. Lett. 75 (1999) 1751-1753. 77. P. Guyot-Sionnest and M. A. Hines, "Intraband transition in semiconductor nanocrystals", Appl. Phys. Lett. 72 (1998) 686-688. 78. J. Rodriguez-Viejo, K. F. Jensen, H. Mattoussi, J. Michel, B. O. Dabbousi, and M. G. Bawendi, "Cathodoluminescence and photoluminescence of highly luminescent CdSe/ZnS quantum dot composites", Appl. Phys. Lett. 70 (1997) 2132-2134. 79. M. E. Schmidt, S. A. Blanton, M. A. Hines, and P. Guyot-Sionnest, "Polar CdSe nanocrystals: Implications for electronic structure", J. Chem. Phys. 106 (1997) 5254-5259. 80. S. A. Blanton, M. A. Hines, and P. Guyot-Sionnest, "Photoluminescence wandering in single CdSe nanocrystals", Appl. Phys. Lett. 69 (1996) 3905-3907. 81. B. O. Dabbousi, M. G. Bawendi, O. Onitsuka, and M. F. Rubner, "Electrolumines­ cence from CdSe quantum-dot/polymer composites", Appl. Phys. Lett. 66 (1995) 1316-1318. 82. D. M. Mittleman, R. W. Schoenlein, J. J. Shiang, V. L. Colvin, A. P. Alivisatos, and C. V. Shank, "Quantum'size dependence of femtosecond electronic dephasing and vibrational dynamics in CdSe nanocrystals", Phys. Rev. B 49 (1994) 14435-14447. 83. D. J. Norris, A. Sacra, C. B. Murray, and M. G. Bawendi, "Measurement of the size dependent hole spectrum in CdSe quantum dots", Phys. Rev. Lett. 72 (1994) 2612-2615. 84. R. W. Schoenlein, D. M. Mittleman, J. J. Shiang, A. P. Alivisatos, and C. V. Shank, "Investigation of femtosecond electronic dephasing in CdSe nanocrystals using quantum-beat-suppressed photon echoes", Phys. Rev. Lett. 70 (1993) 10141017. 85. U. Woggon, S. Gaponenko, W. Langbein, A. Uhrig, and C. Klingshirn, "Homoge­ neous linewidth of confined electron-hole-pair states in II-VI quantum dots", Phys. Rev. B 4 7 (1993) 3684-3689. 86. T. Takagahara, "Electron-phonon interactions, and exciton dephasing in semi­ conductor nanocrystals", Phys. Rev. Lett. 71 (1993) 3577-3580. 87. S. Nomura and T. Kabayashi, "Exciton-LO-phonon couplings in spherical microcrystallites", Phys. Rev. B 45 (1992) 1305-1316. 88. G. Scamarcio, M. Lugara, and D. Manno, "Size-dependent lattice constant in CdSj-xSex nanocrystals embedded in glass observed by Raman scattering", Phys. Rev. B 4 5 (1992) 13792-13795. 89. M. C. Klein, F. Hache, D. Ricard, and C. Flytzanis, "Size dependence of electronphonon coupling in semiconductor nanospheres: The case of CdSe", Phys. Rev. B 4 2 (1990) 11123-11132. 90. M. G. Bawendi, W. L. Wilson, L. Rothberg, P. J. Carroll, T. M. Jedju, M. L. Steigerwald, and L. E. Brus, "Electronic structure and photoexcited-carrier dynamics in nanometer-size CdSe clusters", Phys. Rev. Lett. 65 (1990) 1623-1626. 91. P. Roussignol, D. Ricard, C. Flytzanis, and N. Neurothy, "Phonon broadening and spectral hole burning in very small semiconductor particles", Phys. Rev. Lett. 62 (1989) 312-315. 92. S. A. Blanton, R. L. Leheny, M. A. Hines, and P. Guyot-Sionnest, "Dielectric disper­ sion measurements of CdSe nanocrystal colloids: Observation of a permanent dipole moment", Phys. Rev. Lett. 79 (1997) 865-868. 93. M. Shim and P. Guyot-Sionnest, "n-type colloidal semiconductor nanocrystals", Nature 4 0 7 (2000) 981-983.

116

Advances in Quantum-Dot Research and Technology 1055

94. D. S. Ginger, A. S. Dhoot, C. E. Finlayson, and N. C. Greenham, "Long-lived quantum-confined infrared transitions in CdSe nanocrystals", Appl. Phys. Lett. 77 (2000) 2816-2818. 95. D. L. Klein, R. Roth, A. K. L. Lim, A. P. Alivisatos, and P. L. McEuen, "A singleelectron transistor made from a cadmium selenide nanocrystal", Nature 389 (1997) 699-701. 96. P. Bhattacharya, "Quantum dot semiconductor lasers", in Advances in Semicon­ ductor Lasers and Applications to Optoelectronics, Selected Topics in Electronics and Systems, Vol. 16, eds. M. Dutta and M. A. Stroscio, World Scientific, Singapore, New Jersey, London, Hong Kong, 2000, pp. 235-261. 97. A. E. Zhukov, V. M. Ustinov, and Z. I. Alferov, "Device characteristics of lowthreshold quantum-dot lasers", in Advances in Semiconductor Lasers and Applica­ tions to Optoelectronics, Selected Topics in Electronics and Systems, Vol. 16, eds. M. Dutta and M. A. Stroscio, World Scientific, Singapore, New Jersey, London, Hong Kong, 2000, pp. 419-431. 98. M. Dutta, and M. A. Stroscio, "Advanced semiconductor lasers: Phonon engineering, and phonon interactions", in Advances in Semiconductor Lasers and Applications to Optoelectronics, Selected Topics in Electronics and Systems, Vol. 16, eds. M. Dutta and M. A. Stroscio, World Scientific, Singapore, New Jersey, London, Hong Kong, 2000, pp. 419-431; see also M. A. Stroscio and M. Dutta, Phonons in Nanostructures, Cambridge University Press, Cambridge, 2001. 99. R. M. de la Cruz, S. W. Teitsworth, and M. A. Stroscio, "Phonon bottleneck effects for confined longitudinal optical phonons in quantum boxes", Superlattices and Microstructures 13 (1993) 481-486; R. M. de la Cruz, S. W. Teitsworth, and M. A. Stroscio, "Interface phonons in spherical GaAs/Al(x)Ga(l — x)As quantum dots", Phys. Rev. B 52 (1995) 1489-1492; M. P. Chamberlain, C. Trallero-Giner, and M. Cardona, "Theory of one-phonon Raman scattering in semiconductor microcrystallites", Phys. Rev. B 51 (1995) 1680-1683; J. C. Marini, B. Stebe, and E. Kartheuser, "Exciton-phonon interaction in CdSe and CuCl polar semiconductor nanospheres", Phys. Rev. B 50 (1994) 14302-14311; E. Roca, C. Trallero-Giner, and M. Cardona, "Polar optical vibrational modes in quantum dots", Phys. Rev. B 49 (1994) 13704-13711; P. A. Knipp and T. L. Reinecke, "Classical interface modes in quantum dots", Phys. Rev. B 46 (1992) 10310-10319; D. Romanov, V. Mitin, and M. Stroscio, "Polar surface vibration strips on GaN/AIN quantum dots and their in­ teraction with confined electrons", Phys. E12 (2002) 491-494; A. M. Alcalde and G. Weber, "Scattering rates due to electron-phonon interaction in CdSi_ x Sea; quantum dots", Semiconductor Sci. Technol. 15 (2000) 1082-1086. 100. A. L. Hodgkin, The Conduction of the Nervous Impulse, Liverpool University Press, Liverpool, 1964. 101. J. Howard, Mechanics of Motor Proteins and the Cytoskeleton, Sinauer Associates, Inc. — Publishers, Sunderland, Massachusetts, 2001. 102. V. Mitin, V. Kochelap, and M. Stroscio, Quantum Heterostructures for Microelec­ tronic and Optoelectronics, Cambridge University Press, Cambridge, 1999. 103. C. A. Leatherdale and M. G. Bawendi, "Observation of solvatochromism in CdSe colloidal quantum dots", Phys. Rev. B 63 (2001) 165315-1-6. 104. R. Tsu and D. Babic, "Doping of a quantum dot", Appl. Phys. Lett. 64 (1994) 1806-1808; R. Tsu, D. Babic, and L. Ioriatti, "Simple model for the dielectric constant of nanoscale silicon particle", J. Appl. Phys. 82 (1997) 1327-1329. 105. D. Babic, R. Tsu, and R. F. Green, "Ground-state energies of one- and two-electron silicon dots in an amorphous silicon dioxide matrix", Phys. Rev. B 45 (1992) 14150-14155.

117

1056

M. A. Stroscio & M. Dutta

106. D. R. Penn, "Wave-number-dependent dielectric function of semiconductors", Phys. Rev. 128 (1992) 2093-2097. 107. L.-W. Wang and A. Zunger, "Pseudopotential calculations of nanoscale CdSe quan­ tum dots", Phys. Rev. B 5 3 (1996) 9579-9582. 108. D. B. I r a n Thoai, Y. Z. Hu, and S. W. Koch, "Influence of the confinement potential on the electron-hole-pair states in semiconductor microcrystallites", Phys. Rev. B 42 (1990) 11261-11266. 109. C. A. Leatherdale, C. R. Kagan, N. Y. Morgan, S. A. Empedocles, M. A. Kastner, and M. G. Bawendi, "Photoconductivity in CdSe quantum dot solids", Phys. Rev. B 62 (2000) 2669-2679. 110. A. Tamura, K. Higeta, and T. Ichinokawa, "Lattice vibrations and specific heat of a small particle", J. Phys. C: Solid State Phys. 15 (1982) 4975-4991. 111. J. O. Winter, T. Y. Liu, B. A. Korgel, and C. E. Schmidt, "Recognition molecule directed interfacing between semiconductor quantum dots and nerve cells", Adv. Mater. 13 (2001) 1673-1677. 112. C. E. Schmidt, V. R. Shastri, J. P. Vacanti, and R. Langer, "Stimultion of neurite outgrowth using an electrically conducting polymer", Proc. Natl. Acad. Sci. USA 94 (1997) 8948-8953. 113. H. M. Chen, X. F. Huang, L. Xu, J. Xu, K. J. Chen, and D. Feng, "Self-assembly, and photoluminescence of CdS-Mercaptoacetic clusters with internal structures", Superlattices and Microstructures 27 (2000) 1-5.

118

International Journal of High Speed Electronics and Systems, Vol. 12, No. 4 (2002) 1057-1081 © World Scientific Publishing Company

HIGH-FIELD ELECTRON TRANSPORT CONTROLLED BY OPTICAL PHONON EMISSION IN NITRIDES S. M. K O M I R E N K O AND K. W. KIM Department of Electrical and Computer Engineering, North Carolina State University, Raleigh, North Carolina 27695-7911 V. A. KOCHELAP Institute of Semiconductor Physics, National Academy of Sciences of Ukraine, Kiev-28, 252650, Ukraine M. A. STROSCIO Department of Electrical and Computer Engineering, University of Illinois at Chicago, Chicago, Illinois 60707 We have investigated the problem of electron runaway at strong elec­ tric fields in polar semiconductors focusing on the nanoscale nitride-based heterostructures. A transport model which takes into account the main features of electrons injected in short devices under high electric fields is developed. The electron distribution as a function of the electron mo­ menta and coordinate is analyzed. We have determined the critical field for the runaway regime and investigated this regime in detail. The elec­ tron velocity distribution over the device is studied at different fields. We have applied the model to the group-Ill nitrides: InN, GaN and AIN. For these materials, the basic parameters and characteristics of the high-field electron transport are obtained. We have found that the transport in the nitrides is always dissipative. However, in the runaway regime, en­ ergies and velocities of electrons increase with distance which results in average velocities higher than the peak velocity in bulk-like samples. We demonstrated that the runaway electrons are characterized by the extreme distribution function with the population inversion. A three-terminal heterostructure where the runaway effect can be detected and measured is proposed. We also have considered briefly different nitride-based smallfeature-size devices where this effect can have an impact on the device performance. 1. I n t r o d u c t i o n Group Ill-nitride semiconductors and heterostructures have unique fundamental material properties which make t h e m very attractive for a number of optoelectronic and high-power, high-frequency applications. In these materials t h e energy bandgaps range from 1.9 eV (InN) to 6.2 eV (AIN) and t h e effective masses are relatively small: 1 0.11 m 0 (InN), 0 . 2 m 0 (GaN), 0 . 4 8 m 0 (AIN) for t h e electrons and about (0.3 — 0.5) m 0 for the holes ( m 0 is the free electron mass). T h e electron-polar-op-

119

1058 S. M. Komirenko et al.

tical-phonon coupling in nitrides is much stronger in comparison with that of the other III-V compounds. The energies of optical phonons in these materials are large (about 90 — 100 meV) and the Frohlich phonon-coupling constants are estimated to be a = 0.22, 0.41 and 0.74 for InN, GaN and A1N, respectively (for GaAs a = 0.075 ). The group Ill-nitrides are characterized by large peak velocities in the steadystate regime. The velocities are estimated to be 1 4.3x 107 cm/s (InN), 3.1 x 107 cm/s (GaN), 1.7 x 10 7 cm/s (A1N). The materials are expected to manifest negative differential resistance with threshold electric field of about 70 — 500 kV/cm. The breakdown fields for these materials are in the MV/cm-range. Other remarkable properties include high intrinsic pyro- and piezo- fields which provide a new type of doping, high thermal conductivity, etc. The steady-state high-field transport and, particularly, the velocity - electric field characteristics of group Ill-nitrides were studied in a number of works. 2 - 8 One can sort out these works into two main approaches. The first approach 2 is based on analytical calculations assuming a drifted Maxwellian function for the carriers with two parameters, the electron temperature and the drift velocity. These parameters are determined from the-energy and momentum balance equations. The method can be justified for large electron concentrations and dominant electron-electron scattering. The second approach 3 - 6 is the numerical Monte-Carlo method, which can be applied also for low carrier concentrations when the distribution function cannot be reduced to the drifted Maxwellian. The low-carrier concentration case was also considered analytically 7 ' 8 for the modest fields. All these studies have been performed for the steady-state regime in uniform and infinite materials and structures. The transient time- or space-dependent transport in the nitrides and the velocity overshoot effect9,10 were evaluated using the Monte-Carlo technique 11,12 and a nonstationary balance-like-equation procedure. 13,14 It was found that the onset of the velocity overshoot takes place in the subpicosecond-time-scale and ten-nanometerspatial scale regimes at fields which correspond to the peak velocities. It is expected that maximum velocities of about (6 — 8) x 107 cm/s may be achieved for InN and GaN. 1,11 In general, the cited papers have illustrated the great potential of group IIInitrides for high-power and high-frequency electronics. However, some interesting and practically-important high-field effects have not yet been investigated in detail. In the present paper, we report the results of detailed investigation of the electron runaway effect in small-size group Ill-nitride devices. The runaway effect, is a well-known hot-electron phenomenon, 15 ' 16 which arises in polar crystals with predominant carrier scattering on polar optical phonons. Be­ cause of the Coulomb nature, the rate of such a scattering is inversely proportional to the square of the momentum change during a scattering event, and,, thus, de­ creases for electrons with high momenta and energies. The runaway effect occurs at very high electric fields, when a sufficient number of carriers stream into the high energy states. Above a critical field, the momentum and energy gained by the elec­ trons from the field can not be relaxed to the lattice. The carriers then runaway to higher energies. The subsequent scenarios are different for two cases. In a bulk-like sample the electron runaway has to be stabilized by one of the following effects: a breakdown due to impact ionization of impurities or across the gap, a nonparabolicity or transfer to upper valleys, 17 an additional scattering present at high energies, etc. In a short sample the previously-mentioned mechanisms can be avoided and

120

High-Field Electron Transport Controlled by Optical Phonon Emission

1059

the electron transport can occur in the runaway regime. In the following discussion, we analyze the latter case. To illustrate three different regimes possible in a short sample, in Fig. 1 we present the energy-coordinate diagram. The energy w and the coordinate ( are dimensionless: w — W/fiuj, (, = eZE/fuv, where Z is the coordinate, W is the kinetic energy of electrons corresponding to motion along the Z-axis, w is the op­ tical phonon frequency, E is the applied field, and e is the elementary charge. The electrons are injected with an energy Wi at the point ( = 0. The dimen­ sionless potential energy is — £. Scattering with emission of the optical phonons is assumed to be the dominant relaxation process. In this {w, ^-representation, the transport regimes at different fields can be illustrated on the same diagram.

Fig. 1. Illustration of different electron transport regimes in a sample under an electric bias: 1 - low-field transport regime, 2 - runaway regime, 3 - ballistic regime. The first regime is indicated by 'trajectory'-l and corresponds to the case when in a steady-state the averaged parameters of the carriers are independent on the £ co­ ordinate. In this regime the electrons, on average, regularly lose the energy gained from the field by emitting the optical phonons. The second regime is the runaway regime at a field above the threshold. 'Trajectory'-2 illustrates this case, when the electrons, despite the optical phonon scattering, gain steadily the energy from the field. Both, the first and second cases, are multi-scattering regimes. Note, some­ times the runaway regime is called as the quasi-ballistic regime. For comparison we present also the ballistic motion, 'trajectory'-3, when no scattering (emission) occurs. Besides very distinct coordinate dependences of the average energy and ve­ locity, these three regimes differ considerably in terms of their electron distribution functions, as illustrated in Fig. 1 for a given cross-section of the sample. Distri­ bution 2 corresponds to the electrons running away from the local bottom of the conduction band. Note that the runaway effect is different from another short-distance-scale phe­ nomenon, the overshoot effect. In the steady-state, the overshoot is manifested if the

121

1060

S. M. Komirenko

et al.

momentum relaxation length is much smaller than that of the energy relaxation. 9 ' 10 Particularly, it can be observed even in nonpolar materials, for example, in silicon. The runaway effect does not require an inequality between the momentum and energy relaxation lengths (times) and can occur only for polar materials. 18 In general, the runaway effect is more pronounced for crystals with stronger electron-optical phonon coupling. It was found and treated by the quantum mechan­ ical approach under arbitrary strong coupling, 19 as well as in the classical framework of Boltzmann transport theory. 16 The group Ill-nitrides represent a class of materi­ als with relatively strong electron-phonon coupling, they are able to operate at high electric fields and, thus, are expected to manifest a considerable runaway effect. The previously-mentioned peculiarities of runaway electron transport - high av­ erage velocities which are characteristic for quasiballistic transport, and extremely nonequilibrium distribution functions - are important for the operation of devices based on the group-Ill nitrides. The former feature is favorable for high-speed (high-frequency) devices. The latter attribute is important for vertical transport devices where hetero-barriers control the electric current. One of the examples is the induced base transistor (see Ref. 20 and references therein). In this paper, we develop a kinetic approach to the problem of the runaway effect. This approach allows us to calculate exactly the distribution function of the electrons and all characteristics of their transport in short samples at arbitrary electric fields. Then we apply the results to the group Ill-nitrides. 21 Our approach is based on the following assumptions: the steady-state regime is achieved; electron motion is semiclassical and the scattering is due to the optical phonons; and phonon absorption is negligible, i.e., the lattice temperature T is small comparing to the optical phonon energy. The rest of the paper is organized as follows. In Section 2, we introduce the main definitions and formulate the basic equations. In Section 3, we find the exact solutions to the kinetic equations. In Section 4, we present the numerical results on spatial dependences of the distribution function and the main electron characteris­ tics, including those for the runaway regime. In Section 5, we apply these results to the group Ill-nitrides materials and discuss new phenomena brought by the run­ away electrons. In Section 6, we analyze briefly the impact of the runaway effect on nitride-based nanoscale devices and suggest a special heterostructure design to measure and study the effect. Finally, the conclusions are made in Section 7. 2. The Basic E q u a t i o n s 2.1. The Boltzmann equation We start with the Boltzmann transport equation for the distribution function,

J(P,R,t): dt

dP\

dRj

8Rm

122

\dt)coll

High-Field Electron Transport Controlled by Optical Phonon Emission

1061

Here P , R are the electron momentum and coordinate, U(f) = —eEZ is the poten­ tial energy (e > 0), E is the electric field, m is the effective mass, and (-Jr) is V

/ coll

the collision integral. P2 Let us define the total energy £ = ^ + U(Z) and the energy associated with pi electron motion along the Z-coordinate W = ^ + U(Z). The characteristic energy is the phonon energy hu> and the characteristic momentum is Po = y/^mfko. Now, we introduce dimensionless momentum, potential and total energies: ^_

P

_ U_

Po'

_W

_ £_

TUJJ '

huj '

hu)

Let lo be a characteristic length which we will specified later. Then, we can define dimensionless coordinate z, field e, and potential energy u(z): Z

eElo

. ,

Now, we rewrite the Boltzmann equation as: aJ ( _ 9 ^ 4.o<>Z„ _ mis. (§£')

~

e2

Pnfa (_k

* / ( ^ F { [e" w / f c B T 7(^) ~ 7(p)] %

2

L)

,i

" P' 2 + 1)

+ [7(P0 - cft-/fc«r7(p)] *(P2 -P'2 -1)},

(i)

where the collision integral is written down for scattering of the nondegenerated electrons by equilibrium polar optical phonons. T is the temperature; /to and «oo are the low frequency and high frequency permittivities of the crystal. Then we set e2P$m ( 1 ft2Pn

, e2m ( 1

1\

\ K oo

7i2 V«oo

«o/

1 K0

that is, ;

0

=a

B

^22_

(2)

with a s being the effective Bohr radius. The function f(P, z) should be normalized:

where n(z) is the local electron concentration. It is convenient to introduce a new function P3 / = 2

(2^t)3

/

3

with the normalization / d pf = n(z). For low temperatures, one can simplify Eq. (1) neglecting the phonon absorption: df

(

du\

df

,,^1

123

3 2 12 f
1062 S. M. Komirenko et al.

1 fd3p'f(p')6(p2-p,2 + (p-i?)2

+ l)

If

-f(p)a(p) + r(p).

(3)

i

(4}

The coefficients, a(p) and r(p) are: @{p2 - 1)

a(p)

In

r(p±,Pz) = J

p + yV - 1 p- y V - 1 > P= VP±+Pi,

V?2+i

D(p±,pz,p'z)

(Pi)2 + l,P'J

dp

(5)

D{p±,pz,p'z)

= y/1 + 4P2(P2 + (p'z)2) - 4pzp'z(2p2 + 1) + 4p2 .

Q(a;) is the Heaviside function. In Eq. (3), the variables are p and z, or, because of the symmetry, pj_,pz and z. Note, Eq. (3) is an integro-differential equation with partial derivatives with respect to two variables: z, pz. 2.2. New variables It is convenient to use the following transformations in Eq. (3): P± -> P± , Pz -» w = p2z + u(z), f(p±,Pz,z) /(P.L,P*, 0

z -> C = ez,

r±

(6)

-> J (p±,iw,0>

= e(pz) ^+(pj.,p2 - c, o + e(- P a ) .F-(P±,P2 - C O .

where ± corresponds to pz > 0 and p 0 < 0, respectively. Then, in new variables we will designate: a(p±, y/w + < ) = «4(p±, u>, C) > r ( p i , V ^ + 0 = ft*(p±,w,0Finally, we obtain the Boltzmann equation in the form: ±

—

3C

= --

. -T7 (P±,w,C) + —

2£ v/iF+C

2e

/., ,

A —•

V^+T

(7)

Now, instead of Eq. (3) we come to a pair of equations for the distribution func­ tions T+ and T~ describing the electrons moving to the right and to the left, re­ spectively. Below we shall find relationships between these function in the turning points. Remarkably, the dimensionless Eqs. (7) contain the derivative with respect to a single variable (. Two other variables, w and p±, enter just as parameters. Then, in these equations there is the only 'controlling' parameter - the dimensionless field e. 2.3. Boundary conditions For Eqs. (7), one needs to impose the boundary conditions. We will analyze a sample of a finite dimension in the z-direction (z > 0) and suppose that the electrons are

124

High-Field Electron Transport Controlled by Optical Phonon Emission

1063

injected at the point z — C, = 0. Then, we suppose that there is no injection from the collector electrode and reflection of the electrons from this electrode is absent. First, let us assume that the distribution function of the injected electrons is: f(inj)(p±,Pz)

= 0) = -n05(pl)8(pz -Pi); n i.e., the electrons are injected with the initial momentum pz = p^. We restrict our analysis to the case pi < 1. This injection distribution corresponds to the injected current J = ^PiTio, where no is the concentration of the injected electrons. We can rewrite the function in the equivalent form through the total electron energy e: f{inj) = f n 0 5{p\ +p2z- p2) S(pz - pi) = In0 S(e - p2) 6(pz - p^ . In terms of the new variables, we obtain the boundary condition for the coupled Eqs. (7): T+(p±,w,C

= f(px,pz,z

= 0) = -no5(pl

+ w-p2i)6(V^-pi),

f-(P±,w,(

= 0)=0.

(8)

An arbitrary distribution of injected electrons has the form: fiinj)(p±,Pz\p±,o,Pi) The function f(p±,pz\p±,o,Pi) cause Eq. (3) is linear.

= -n0S(e-(plfi+p2))S(pz-Pi).

(9)

is, in fact, the Green's function of the problem be­

3. Solutions of the Kinetic Equation 3.1.

General structure of solutions

For the case of the boundary conditions (8), we can expect that emission of dispersionless optical phonons should lead to electron energies descending down the ladder of the total energy e: e0 = p\ , ei = p2 - 1,

,es=p2 - s ,

where s numerates the energy stairs. This implies the following structures for the distribution function:

/(px,pz,o = J]/,(pj-,p„c) = X] 5 (pi + ^-c-p J 2 + s)^(p-L,pZ!c). (io) ^(p±,w,C) = ^ J " s ( p x , t i ; , C ) = $ 3 ^ p i + w - p f + s ) ^ ( p X ) w , C ) .

(11)

Below, we will show that the functions in the form of Eqs. (10) and (11) are the solution to Eqs. (3) and (7). The J-functions in Eqs. (10) and (11) imply that <j>s and $ 5 are functions of two arguments. For what follows, it is convenient to introduce for the s t,l -stair the functions Rf: ^±(p±,w,0=S(pl+w-p2

+ s)Rf(p±,w,C).

125

(12)

1064

S. M. Komirenko et at.

The J-functions in Eqs. (10) and (11) imply that for the sth-energy stair, the func­ tions $ s and <j>s are non-zero functions at £ > s + p\ — p2. Now, the distribution function shall be analyzed for each stair s. 3.2. Relationships between f+, f~ and Jr+,

T~

Eqs (10) and (11) allow one to find relationships between distribution functions <j>3 and $ s with positive and negative momenta pz. Since pi > 0, in the {p±,pz,Qvariables, we obtain that for a given p± the turning point, where pz —> 0, is £ = Q = p\+s—p2. Near the turning points one can neglect any scattering and consider that the electron fluxes toward and backward the turning points, Q, should be equal. Thus, in the {p±,pz, C}-variables we obtain: Pzf~(puPz,C)

=Pzf+(p±,Pz,0,

<->Ct,Pz-»0.

(13)

From the latter equation, one can not conclude that / ~ = / + at the turning points because pz —> 0 and both functions, / - and / + , diverge. However, we can use Eq. (13) for the functions in the form of Eq. (10). Indeed, at a given p±, a coordinate ( can be the turning point only for a single energy stair s. Thus, near a turning point we shall keep only one of the contributions in Eq. (10): f±(p±,pz,() ~ 2 2 S(pl+P z-( + s-P i)f(p±,Pz,0Then, S(P1+P2Z-C

+ S-P2)=

y

J 2

.

2y/C,-S+pj-p]_ Now, from Eq. (13) we can find that t(P±,Pz,0=7(P±,Pz,0 In the {p±,wX}"-variables, form: $f(p±,w,0

at

C^Ct

(p2->0).

(14)

the turning point is (t = —w and Eq. (14) takes the = K(P±>w,0

at

£-+
(15)

Formula (15) can be considered, in fact, as the boundary condition to Eq. (7) for the function '.F+ in the form (11) at s > 0. For the function ,F~ we can use another boundary condition, JT=0,

(16)

at the collector electrode. The latter condition implies the absence of electrons moving from this electrode. Thus, Eqs. (8), (15) and (16) compose the full set of the boundary conditions for the distribution functions Tf. With these boundary conditions, we complete the formulation of the problem. Now, we shall show that the problem can be solved exactly. Consider the structure of Eqs. (7). These are still integro-differential linear equations, which can be thought

126

High-Field Electron Transport Controlled by Optical Phonon Emission

1065

as a set of coupled equations for a set of the energy stairs. At a given w and p± corresponding, say, to the energy stair s, the last integral term, proportional to H, depends on the distribution function of the electrons with energy w + p\ + 1, i.e., on the distribution on the previous energy stair (s — 1); see Eqs (3), (5). For the stair s = 0, according to the boundary condition of Eq. (8), we obtain 7^± = 0 and, thus, the equation for FQ reduces to linear homogeneous differential equation, which can be solved easily. With a known TQ, one can calculate the functions Tl1 , which enter the equations for the electrons on the next energy stair s = 1. For s = 1, Eqs. (7) become linear nonhomogeneous differential equations and Tx can be found analytically. This allows one to calculate the functions IZf, obtain simple equations for Tf, etc. In the following discussion, we will follow the described recurrent procedure of finding the solutions of Eqs. (3). 3.3. The zero energy stair s = 0 For this step of calculations, we get: dF+(P±,W,()

_

1

A{p_L,W,()

• ^ ( p ± , w , C ) , ^o = ° -

(17)

A general solution of this equation for TQ is
Fo(p±, w, C) = C{pL,w) exp

dC

(18)

where C can be found from the boundary condition of Eq. (8): C(p±,w) =

rf(p±,w,(

= 0) = 6(pl + w - p ? ) 5 ( v ^ - f t ) •

(Here we omit inessential multiplier 2no/7r coming from Eq. (8)). The function A becomes: At

,\

y/p'2± + W + C + y/pj + W + C ~ 1 y/p\ +W + C- yJV\ + W + C - 1

Q Pi + w + C ~ 1 , yjp]_ +W + C

Using relationship 6(y/w — Pi) = 2pi$(w — pf) we can rewrite Eq. (18) as: MP±, W, 0 = S(e - pf) $£{p±,w,

$o(p±,w,()

= 2pi6(w - pf)exp

„ .. ,

e(c+pf-i) ,

with

v ^ T C x / C + P?

127

C),

(19)

(20)

1066 S. M. Komirenko et al. In the {pj_,Pz>C}- var i a bl es we obtain: /o(p±,Pz,C) = s(Px +PZ

4>o(P-L,Pz,() = 2piS{p2z ~

(21)

-C-Pi)n(P±,Pz,0,

C-P2i)exp

Bo{CX

(22)

3.4- The first stair s = 1 Next we have to calculate ri(p±,pz,()

using Eq. (5)

Pi S(P2 - C + 1 - P2)

n(p±,pz,0

2^/pJTCD

Px,Pz,VPi+(

■5 £,*«-*•

exp

Here we take into account that \Jp2 + 1 = \Jp\ + £. This equation can be applied for both signs of pz. The next step is the calculation of TZ^ according to Eq. (12):

nt(puw,0

-Ri

= 6(pi+w +

e{w + c)

Pi

(p±,w,£)

l-p*)Rt(px,w,(),

exp

2^/pJTCD p±,±VwT£,y/pi+t\

- 1 /

(23)

C

B0(C)dC

Finally, for s — 1 Eq. (7) takes the form: ±

^(P,,W,C)=_^^X^0^(^!W;C) d(

IE

+

y/vT+T,

^^(PX^0 2e

;

(24)

sJw + C,

where the inhomogeneous terms are already found. Now, both functions Tf not zero. The solution to the equation for T^ is

are

J T ( p x , ^ 0 = *(pi+u' + l-Pi)*r(PJ-»w>0.

x ST'<%"R"X+cP

ex

P [- Te f

dC'"Bi«'")} ■

(25)

with

Bi(0

e(( +

2 P

-2)

V^TCy/c+p'i-i

In

x/C + ^ - i - V C + p f - 2 .

A general solution of Eq. (24) for the function T^ is Jr^r(p±,w,C)

= S{p± + w + 1 - p 2 ) x

\c{px,w)exp

128

4/>*«'> +

High-Field Electron Transport Controlled by Optical Phonon Emission

2e exp

2£ Jw

J-w

y/w + C"

1067

2e J_w

where C is an arbitrary function, which we can determine from the conditions at the turning point (15). The result is C(p_\_,w) =
$i"(pj.,«>,C) = 2e exp

x

d( .„RT(p±,w,C")

s/>H (/-

dC"Bi(C")

Vw + C"

,X(PX,^C")

exp

1/VB I ( O]}.

(26)

To complete this step one needs to compute f±(p±,pz,() through ^(pxjivX)After calculation of R2(p±,Pz,C) with the help of Eqs. (5) and (12) one can do the next step. 3.5. The arbitrary stair s The general procedure for the s'^-energy stair can be formulated as follows. Suppose we have found the distribution for the (s — l ^ - s t e p :

?ti(p±,w,0

= 5{pl+w-p1 + (S- l))$±_i(Px^.C).

Then we have to compute

fs-i{p±,Pz,C):

fs-i(p±,Pz,0 = 5(p±+pl-(-P2i+(s-i)){e(Pz)$ti(p±,pl-(,0 + e(-p 2 )*,-_ 1 (p J .,p2-c,c)} and Rf(p±,pz,()

(see Eqs. (5), (12)):

Rf(p„Pz,C) = I

dPz

D(p±,pz,p,}

, K-iWp2-(pz)2

J-Jp2

dp\ +1

R7(P. -,Pz,0 = /

h(p'z)2-co

+

D(pi_,pz,p'z)

^+ld

, $UWP2 Pz

-dPz P2 + l

+ Pz > 0;

- iP'z)2 +1,{P' Z ?

-C,0

D(p±,pz,p'z)

Jo

L

(27)

?T7Z „/N D(p±,Pz,p' z)

129

' Pz < °-

1068

S. M. Komirenko

et al.

where we can substitute ^/p2 + 1 —> \/C + p f Bs as

_

(s — 1). Then we define the function

B;(,.0-^-''V»h. V ^ + CvC + Pi - «

The following integrals for $^(p±,w,^) complete the calculations of .7\s(pi,w,C) for the energy stair s: = Y£eXV

®s (P±,w,0

i

x /

r<"

dC

Vw + C"

$+(RL,W,0

{/:

ny^

exp

= YeeXV

iJ_JC"Bs(C")

(28)

iVBX)

lL

^RTfo.y.n...

i£«-B.(0

Vvi + ("

(29)

dC"'B8(C"')

Vw + C"

Then one can find fs in accordance with Eq. (27) and compute the results for the next energy stair and so on. The final formulae contain two dimensionless parameters: the initial momen­ tum pz = Pi, and the electric field e. However, having the distribution function f(p.L>Pz,(\Pii£) calculated at a given e and a given pi from the assumed range 0 < Pi < 1, one can easily obtain the results for any initial momentum. Indeed, let us compare motion of two groups of electrons with the initial momenta p, and pi, respectively. From the energy diagram presented in Fig. 1, it follows that the electrons of the second group reach the momentum equal to p, (without scattering) in the point C = p2 —f\- Using £ as the reference point and redefining the potential energy of the second group as — (C — C) o n e can obtain equivalent dynamics for both groups at ( > max{0, Q. Thus, if the distribution function of the first group of electrons, f(p±,pz,(\Pi,£), is found, one can calculate the distribution for the second group: 'D'

f(p±,PzX\Pi,e)

—

—

= — f(P±,Pz,C-(\Pi,e),

C>max{0,C}.

Pi

For the {p_i_,w,(}-variables, the corresponding transformation is 59'

F(p±,w,(\pi,e)

—

= —F(px,w Pi 130

—

+ C,C ~ t\Pi,£) ■

High-Field Electron Transport Controlled by Optical Phonon Emission

1069

Thus, we can conclude that the only essential 'controlling parameter' of the prob­ lem is the dimensionless electric field e. 4. Numerical Calculations We have calculated the distribution functions in the form of Eqs. (10), (11) for different e. The results for the factors $ s are depicted in Fig. 2.

Fig. 2. Factors 5>3 as functions of pz and ( at different electric fields: a: £ = 0.2; b: s = 0.8; c: e = 3.

131

1070 S. M. Komirenko et al.

Note, that the factor $o is proportional to S(p2z - ( - pf), and, consequently, it cannot be plotted explicitly. Nevertheless, one can integrate $o over pz and obtain the electron concentration on the 0-stair, no(C)-

Fig. 3. Populations of the energy stairs versus the distance for different fields. The panels a, b and c correspond to those of Fig. 2.

132

High-Field Electron Transport Controlled by Optical Phonon Emission 1071

Since the (5-function implies that at a given £ all electrons have the same momentum Pz = yPi+C, instead of $o((,Pz), we plot in Fig. 2 lines no((,pz). Consider Fig. 2 a, where the calculations are presented for the small electric field e = 0.2. Eight energy stairs are shown. One can see that at a given ( and for each energy stair, the factors $ s are strongly asymmetric functions of pz with sharp peaks near the maximum of pz: pz,max = \/pf + ( — s. The number of electrons moving against the field is very small. As expected for small fields, after one or two steps the functions <3?s almost reproduce themselves. For ( in the interval [s — p\, s + 1 — pf], the electrons occupy approximately two energy stairs, s and ; - f l . This is illustrated by Fig. 3 a, where the populations of different energy stairs ns(() are shown. All this is in the agreement with the qualitative conclusions made in the Introduction: for small fields, distance-independent steady dissipative transport occurs (schematically presented by 'trajectory'-1 in Fig. 1). When the electric field increases, a strong redistribution of the electrons over the momenta pz and between the energy stairs occurs. This is illustrated by Fig. 2 b presented for e = 0.8 at s = 0,1,..,7. The functions $ s become much smoother, though they remain well peaked at pz,ma.x- The shapes of these functions are considerably different for different s. At a fixed £, the population by the electrons of the stairs with high energies (small s) becomes larger as shown in Fig. 3 b. The self-similarity of the dependences n s (£) is destroyed. The distribution depends strongly on the coordinate £. This behavior can be interpreted as the development of the runaway effect. 1

1

1

1

1

1

Fig. 4. The average velocity-distance dependences for different fields. The low-field steady-state regime is shown by horizontal dotted line.

133

1072 S. M. Komirenko et al.

Further increase of the field magnitude by a factor of 4 (e = 3) changes the distribu­ tion dramatically. In Fig. 2 c and Fig. 3 c one can see that the 0-stair remains the most populated. Electrons are distributed over two or three stairs (s = 1..3), while the lower energy stairs are almost empty. The electron kinetic energy grows with the distance. A well-developed runaway effect and quasiballistic electron transport are obviously reached under this field. The evolution of the runaway effect can be studied by analyzing the average electron velocity as function of £ for different electric fields. The results obtained for electrons injected into the sample with Maxwellian distribution at T=300 K are shown in Fig. 4 in terms of the dimensionless average velocity

vz

[2mJ

V0

V m

All dependences v(() are oscillating as a result of sequential single-optical phonon emission. For the small field, e = 0.2, the velocity-distance dependence has a regular behavior, oscillating around the value 0.6. No runaway is observed.

Fig. 5. Formation of the population inversion with increase in the field. The results are obtained for £ = 7. Note, this value is close to the average velocity of the one-dimensional electron motion in an electric field under optical phonon emission v = 0.5. 10 This is due to the strong anisotropy of the distribution function even at small fields. At the field e = 0.5, the velocity oscillations are smaller, the velocity magnitudes are larger. The

134

High-Field Electron Transport Controlled by Optical Phonon Emission

1073

tendency of velocity increase with the distance is observed. This result indicates the onset of runaway. Thus, the threshold field for the effect is about e = 0.5. At £ = 0.6, 0.8, and 1.2 the increase in v with distance becomes more pronounced. When the field increases well above the threshold, e — 3, the velocity oscillations are almost suppressed and the velocity magnitudes grow considerably with the distance. For comparison, in this figure we present the dependence v = y fy + (, which is characteristic for purely ballistic transport. The line v = 0.5 indicates the smallfield dissipative transport. Although under the well developed runaway effect the velocity magnitude is below that of the ballistic case, the transport can reasonably be interpreted as quasiballistic. Yet another remarkable property of the well-developed runaway effect is the ex­ tremely nonequilibrium distribution of the electrons over the momentum and energy. In Fig. 5 we show the results of the calculations of populations of the energy stairs at different electric fields for the distance C, — 7. These results demonstrate the formation of population inversion with increasing field. At small fields, the carrier distribution is a decreasing function of the kinetic energy. When the field increases up to the threshold value e = 0.5, the distribution becomes wider and all energy stairs become populated. At e = 0.8, population inversion between the two lowest energy stairs s = 6, 7 is formed. Above this value, the population inversion becomes more pronounced. At e = 1.2, for instance, only the populations of energy stairs s = 0,1,2 are not inverted with respect to each other. At even higher fields, the populations of all energy stairs becomes inverted as illustrated in the Figure for £ = 3. 5. The Runaway Effect in the Group Ill-Nitrides To apply these results to particular materials we start with calculations of the char­ acteristic parameters lo, Eo, and Vo- All variables were scaled to these parameters. For the group Ill-nitrides, we use material parameters collected in Ref. 12. The re­ sults are given in Table 1. Table 1. The characteristic parameters for group Ill-nitrides and GaAs. Material InN GaN A1N GaAs

h, 'A. Eo, kV/cm Vo, 107 cm/s Too, meV A, eV 90 35 12 525

99 257 828 6.9

5.3 4 2.7 4.4

89 91 99 36

2.2 1.9 0.7 0.3

Let us estimate the conditions under which our model can be applied to the nitrides. As discussed in the Introduction, generally, the electrons are coupled stronger to the optical phonons. We have omitted the optical phonon absorption processes. The condition for this to be valid is the small number of equilibrium optical phonons, i.e.,

135

1074

S. M. Komirenko et al.

flu

e

fc T

e

< 1.

(30)

If inequality (30) holds, the scattering with emission of optical phonons dominates for the electrons with the kinetic energy greater than hu (the so-called active en­ ergy region) and vanishes for energies less than Tiw (the passive energy region). 22 Note, at each energy stair studied there are electrons passing the passive energy region. We need to find the criterion indicating that one can neglect the scattering processes in the passive region. Let these scattering processes be characterized by the momentum scattering time TP. In an external field E, the time required to gain the kinetic energy equal to hu is reg = V^mhw/eE. If this energy gain time is much less than r p , the scattering in the passive region can be neglected. Thus, the model discussed in this paper can be applied for the fields: „

£ » -

y/2mhuj

erp

.

.„,.

(31)

Using the data of Table 1, one can see that for the nitrides, inequality (30) is met for temperatures at and below 300 K. To perform numerical estimates of the criterion (31) we can use the relationship between r p and the mobility [i: fi = eTp/m, where omitted mechanisms, scattering by acoustic phonons and impurities, contribute to the mobility. In terms of /J, the criterion of Eq. (31) is fi 3> ^ = --§- . From the data of Table 1, for the ratio -j^2- we obtain 540 cm2 /Vs, 160 cm2 /Vs and 35cm2/Vs for InN, GaN and A1N, respectively. Estimations of contributions to the electron mobilities from the mechanisms other than the optical phonon scattering give values considerably higher than these numbers. For example, in GaN the mobility restricted by acoustic phonon scattering is higher than 6000 cm2/Vs at temperatures below 300 K.1 Next, our model is developed for the simple parabolic electron energy band. Nonparabolicity of the conduction band as well as the presence of the upper valleys are neglected. Thus, the model is restricted by consideration of moderate kinetic energies of the electrons. For the electron runaway, we should restrict the analysis to the case of not very large electric bias and (or) device length. It is well known that for the bulk materials the increase of the effective mass with the energy due to the band nonparabolicity stabilizes the energy gain-loss balance and prevents the development of the runaway effect.16 In a short sam­ ple, the nonparabolicity also would tend to stabilize the velocity increase with the distance. In the Kane model of a spherical T-valley, the nonparabolicity is accounted by the formula ^ = £& (1 + 0£k), where £& is the total kinetic en­ ergy and /3 is the nonparabolicity coefficient. The electron velocity is V{£k) = ^/2m£k(l + /3£fc)/(l + 2(3£k). For the well-developed runaway effect, the electron energy is of the order of the applied bias eUapi, i.e., £k ~ eUapi- Thus, to avoid the influence of nonparabolicity we have to restrict the value of the applied voltage. Let us require the accuracy for the calculation of the velocity to be better than 20%, then we find the boundaries for the biases: UCr>np = 0.43 V, Ucr>np = 0.9 V and Ucr,nP = 4 V for InN, GaN and A1N, respectively.

136

High-Field Electron Transport Controlled by Optical Phonon Emission

1075

Alternatively, in order to be valid, our model has to be restricted by consideration of electrons with energies below the energies of the upper valleys. Introducing the intervalley spacing A, we obtain the corresponding condition :Uapi < Ucr^v = A/e. Using the parameters A presented in Table 1 and comparing the characteristic biases we can conclude that for InN and GaN the main restriction to our model comes from the nonparabolicity effect, while in A1N it comes from the intervalley transfer effect. In terms of the optical phonon energies, we are restricted to 5 energy steps for InN, 10 steps for GaN and 7 steps for A1N. It is important that the restrictions in the applied voltage, U < Ucr = min{Ucrinp,Ucr^v}, implies a restriction to the length of the device L. Indeed, the runaway effect occurs at E > 0.5Eo, where Eo = hu/elo- Since E = Uapi/L, we obtain L < 2Ucr/Eo- This results in the following restrictions: L should be less that LRAE = 860 A, 700 A, ami 170 A for InN, GaN, and A1N, respectively. A well-developed runaway effect can be measured in shorter devices. For comparison, in the case of GaAs the inequality (30) requires the temperature to be below 80 K. At E = Eo, the criterion of Eq. (31) can be easily satisfied for fj. > 6400 cm/Vs. Then, we obtain UCT,np « Uiv = 0.27 F and LRAE « 4000 A, which corresponds to about 7 energy steps for GaAs. Under these restrictions our model can be applied to GaAs. 23 Now we shall return to the discussion of the runaway effect in the nitrides. The characteristic electric fields given in Table 1 are far below the breakdown fields. The other discussed conditions can be met for nanoscale devices. Prior to considering runaway-effect devices, we need to make several remarks about the parameters presented in Table 1. First, one can see that the characteris­ tic lengths lo defined by Eq. (2) are very small. This length is a mean path of the electrons between two sequential events of optical phonon emission at the electric fields E ss Eo (s « 1). The small values of lo and large characteristic fields mean that in the nitrides a purely ballistic transport regime cannot occur in practice. Ac­ tually, the transport is always dissipative. This is in great contrast to GaAs, where the length lo is considerably larger and the characteristic field is much smaller. In high fields (E » Eo) the mean path can be of order of IQ X E/EO, i.e., it can reach the micrometer scale. At these conditions, nearly ballistic, non-dissipative transport regimes can be realized in real heterostructures at least for low temperatures. 24 ' 25 Then, from the data presented in Figs. 2 - 5 and Table 1, it follows that in a short nitride-based device the electron velocities up to 5 x 107 cm/s — 108 cm/s can be reached in the runaway regime. These results are in the agreement with data obtained by the Monte-Carlo method in Ref. 12 with all scattering mechanisms included. There, the transient high-field transport was calculated in the time do­ main and then, using the time dependence of the average velocity, the relationship between the velocity and the coordinate has been recalculated. The comparison of these results with results obtained in the framework of our model is possible for small distances, since we restricted our analysis to short samples. 26 Besides the velocity-distance dependences, we also obtained strong anisotropy of

137

1076

S. M. Komirenko et al.

electron quasiballistic motion toward the collector and formation of the inversion population of the electron states. The existence of the latter effects is not limited by the parabolic electron energy dispersion, they should persist at higher electric biases when the nonparabolicity is considerable. One can show that in such a case the population inversion becomes even more pronounced. 6. T h e Runaway Effect in Nanoscale Devices 6.1. The runaway-effect diode The simplest vertical device operating in the runaway effect regime can be thought as a n+ — i — n+ heterostructure diode with doped narrow-gap n + -contact regions and the wide-gap i-base. For example, it can be n + -InGaN-i-GaN-n + -InGaNheterostructure with the GaN-base. If the base length is about 590 A and the applied voltage Uapi = 0.9 V (e = 0.6), we obtain 4 x 10 7 cm/s for the electron velocity averaged over thebase and 0.17ps for the transit time (the corresponding cut-off frequency v = 940 GHz). If the length of the base is decreased by a factor of, L = 295 A (e = 1.2), the average velocity and the transit time are: 7.2 x 10 7 cm/s and 0.06ps {v = 2.7THz), respectively. For a device with a 140 A-length A1N base, at an applied voltage 0.7V (e = 0.6), we estimate the average velocity and the transit time to be 2.5 x 10 7 cm/s and O.OAps (y = ATHz), respectively. 6.2. The runaway-effect electron spectrometer In thin III-V-compound layers, ballistic electrons can be observed and studied by using the hot-electron spectroscopy method at low temperatures, when the collisionless regime occurs. 24 ' 25 The method exploits a three-terminal device where the emitter and the base are separated by a thin barrier and a relatively wider bar­ rier is placed between the base and collector electrodes. The emitter barrier serves for the hot electron injection controlled by the base voltage. The collector barrier is designed to control the energy of electrons traversing the device by the basecollector voltage. Typically, the active part of the device is about, or smaller than, 1000 A for AlGaAs-GaAs and GaAs-AlGaAs-InGaAs structures. Using the princi­ ple of the injection of ballistic electrons, a variety of monopolar hot-electron devices (transistors) has been proposed for ultra-fast operation. 10,27 Similar three-terminal devices can be exploited to observe and measure the run­ away effect in the nitrides. For this case, the emitter barrier has to be wide enough to satisfy the conditions of the effect discussed previously. The base and the collec­ tor barrier have to be as thin as possible to avoid scattering and perturbation of the electron distribution, and to facilitate control of the emitter-base and collector-base currents. For example, to measure the runaway effect in GaN, an InGaN-GaNInGaN-GaN-InGaN heterostructure can be used. In such a heterostructure, three doped narrow-gap n + -InGaN layers are contacts. An «-GaN emitter barrier has width of about 700 A. The collector barrier is an i-GaN layer of the width below 100 A and designed to isolate electrically the base and collector (in general, it can

138

High-Field Electron Transport Controlled by Optical Phonon Emission

1077

be an InAlGaN narrow layer, etc.). In such a device, runaway electron transport will be realized in the biased emitter barrier. The collector barrier will control the energy of collected electrons. Using this spectrometer, one can observe the runaway effect by measuring the distribution of collected electrons at different emitter-base biases. When the conditions of the effect are fulfilled, devices with identical layers, but wider emitter barriers, should demonstrate higher energies of collected electrons at the same emitter-barrier field. 6.3. The runaway effect in other heterostructure devices Since group Ill-nitrides have a potential for high power devices which have to operate under high biases, it is expected that the runaway effect may occur in the active regions of these devices. Examples of such devices are induced base transistor, 20 the heterostructure bipolar transistor, 28 hot electron transistor, 29 etc. Consider, for example, the induced base transistor proposed in Ref. 20. In this device, a heterostructure with a AlGaN emitter barrier with a width of about 100 A, operating under electric fields up to 1 MV/cm is assumed. From the considerations given above, it follows that the runaway effect should occur under these conditions and, in particular, this effect should determine the emitter-base transit time. Then, the collector InAlGaN barrier is designed in such a way that only high energy electrons can be collected. The energy distribution of the electrons traversing the emitter barrier is critically important for the current gain of the device. Thus, the runaway effect will impact the overall device performance. In the heterostructure hot-electron transistors proposed in Ref. 29, the InGaNcollector barriers are of about 1000 — 1500 A wide and should accommodate 2 — 5 V. This means that the runaway effect should certainly occur in this part of the device and affect the emitter-collector transit time. The same is valid for other types of high-speed transistors with relatively thick collector barriers including heterostruc­ ture bipolar transistors (see Ref. 28 and references therein), real-space transfer devices, 10 ' 27 ' 30 etc. The operation speed and overall performance of these devices can be optimized when the behavior of runaway electrons is taken into account. 6.4- The runaway effect for high-frequency generation The devices operating in the runaway-effect regimes can be exploited for traditional high-frequency circuit applications (see, for example Refs. 10,27). Besides, the stud­ ied peculiarities of the high-field electron transport can be used for direct generation of electric oscillations. Indeed, we found that at electric fields of the order of the critical field e « 0.5 the electron distribution function consists of contributions cor­ responding to a few groups of the nonequilibrium electrons with different energies and distinctly different velocities. In other words, the high-field regimes produce a few electron groups drifting with different velocities. From plasma physics 31 ' 32 it is well known that a two-stream distribution (or a multiple-stream distribution) of charged particles is unstable with respect to electrostatic oscillations. Thus, one

139

1078

S. M. Komirenko et al.

can suggest that electric oscillations can be generated under the analyzed high-field regimes. Another possibility to utilize the runaway-effect for ultra-high-frequency oscilla­ tions is to generate coherent optical lattice vibrations via the stimulated Cerenkov effect when the electron drift velocity exceeds the optical phonon phase velocity: v > hu)/q.33~35 To generate coherent optical optical phonons via the Cerenkov effect it is necessary to reach electron velocities exceeding at least 107 cm/s.36 Analysis shows that in bulk materials it is practically impossible to meet this requirement and compete with the large rate of phonon losses. Nanoscale structures with ultra-high speed electron transport in the runaway-effect regime represent a class of devices, where the Cerenkov criterion can be satisfied and the electric current can induce an optical phonon instability. Indeed, using the data of Table 1 for fields about 0.5 — 0.8 we can easy estimate that the Cerenkov criterion is met for the phonon wavevectors of the range of (3 — 5) x 1 0 - 6 cm~1. For this range of phonon wavevectors, the nonequilibrium (hot) electrons efficiently interact with the optical phonons. These conditions provide for generation of coherent optical phonons. It is a general result that a spatially nonhomogeneous distribution of the elec­ trons in nanoscale devices leads to a coupling between longitudinal electrostatic oscillations and the total alternative current (the particle current plus displacement one) through the device. Thus, the discussed instabilities of plasmons and optical phonons will produce an current instability in the external circuit at the plasmon and optical phonon frequencies. 7. Conclusions In this paper, we have revisited the long-standing problem of electron runaway under strong electric fields in polar, semiconductors and have focused on the short-length group-Ill nitrides heterostructures. We have proposed a model which accounts the main features of the motion of electrons injected in a short device under high electric fields. The Boltzmann equation is solved analytically. The analysis has resulted in the determination of the electron distribution as a function of the electron momenta and the distance from the plane of injection. The model has a single 'controlling' parameter - the dimensionless electric field. We have determined the critical field for the runaway regime and studied this regime in detail. The electron velocity distribution over the device as a function of the field is calculated. We have analyzed the criteria and limitations, which allow one to apply the model to the particular nitrides - InN, GaN and A1N. For these materials, we have obtained the basic parameters and characteristics of high-field electron transport. We have estimated the mean path of the electrons between two sequential optical phonon scattering events to be of the order of tens of angstroms. Thus, generally, the transport in the nitrides is always dissipative. However, for the runaway-effect regime we have found that the electrons progressively gain energy and velocity. As a result, the average velocity can reach higher values than the peak velocity in bulk­ like samples. These results are in agreement with published numerical results on

140

High-Field Electron Transport Controlled by Optical Phonon Emission

1079

the transient electron transport in the nitrides calculated by using of Monte Carlo technique. Then, we have shown that the runaway electrons are characterized by the extreme distribution function with population inversion. We have suggested a three-terminal heterostructure to observe and measure the runaway effect and considered briefly different nitride-based small-feature-size de­ vices where this effect may have an impact on the device performance. Finally, we have established that the runaway regime is the main high-field transport regime in the high-speed group-Ill nitride nanoscale devices. Acknowledgment This work was supported by the ONR under MURI program, Grant No. N0001498-1-0654 and by the ARO. V.A.K. would like to acknowledge the support from ERO of US Army (Contract N68171-01-M-5166). References 1. S. J. Pearton, J. C. Zolper, R. J. Shul, and F. Ren, "GaN: Processing, defects, and devices" J. Appl. Phys., 86 (1999) 1-78. 2. D. K. Ferry, "High-field transport in wide-band-gap semiconductors", Phys. Rev. B, 12 (1975) 2361-2369. 3. M. A. Littlejohn, J. R. Hauser, and T. H. Glisson, "Monte-Carlo calculation of velocity-field relationship for gallium nitride", Appl. Phys. Lett., 26 (1975) 625627. 4. B. Gelmont, K. Kim, and M. Shur, "Monte-Carlo simulation of electron-transport in gallium nitride", J. Appl. Phys., 74 (1993) 1818-1821. 5. N. S. Mansour, K. W. Kim, and M. A. Littlejohn, "Theoretical-study of electrontransport in gallium nitride", J. Appl. Phys.,77 (1995) 2834-2836. 6. U. V. Bhapkar and M. S. Shur, "Monte Carlo calculation of velocity-field char­ acteristics of wurtzite GaN", J. Appl. Phys., 82 (1997) 1649-1655. 7. B. L. Gelmont, M. S. Shur, and M.A. Stroscio, in Proceedings of ISDRS-1997, Univ. of Virginia, (1998) 389-392. 8. N. A. Zakhleniuk, C. R. Bennett, B. K. Ridley, and M. Babiker, "Multisubband hot-electron transport in GaN-based quantum wells", Appl. Phys. Lett, 75 (1998) 2485-2487. 9. D. K. Ferry, Semiconductors (Macmillan, New-York, 1991). 10. V. V. Mitin, V. A. Kochelap, and M. Stroscio, Quantum Heterostructures for Microelectronics and Optoelectronics (Cambridge University Press, Ney-York, 1999). 11. B. E. Foutz, L. E. Eastman, U. V. Bhapkar, and M. S. Shur, "Comparison of high field electron transport in GaN and GaAs", Appl. Phys. Lett, 70 (1997) 2849-2851.

141

1080

S. M. Komirenko et al.

12. B. E. Foutz, S. K. O'Leary, M. S. Shur, and L. E. Eastman, "Transient electron transport in wurtzite GaN, InN, and A1N", J. Appl. Phys., 85 (1999) 7727-7734. 13. E. W. S. Caetano, R. N. Costa Filho, V. N. Freire, and J. A. P. da Costa, "Velocity overshoot in zincblende and wurtzite GaN", Solid State Commun., 110 (1999) 469-472. 14. C. G. Rodrigues, V. N. Freire, A. R. Vasconcellos, and R. Luzzi, "Velocity over­ shoot onset in nitride semiconductors", Appl. Phys. Lett., 76 (2000) 1893-1895. 15. R. Stratton, "The influence of interelectronic collisions on conduction and break­ down in covalent semi-conductors", Proc. Roy. Soc. (London), A242 (1957) 355-373. 16. E. M. Conwell, High Field Transport in Semiconductors (Academic Press, New York, 1967). 17. The former case is achieved in narrow-gap materials like InSb, where the small effective mass facilitates the acceleration and the band-gap is less than the intervalley distances. In GaAs the critical runaway field is about 3kV/cm, i.e., it al­ most coincides with the threshold field of electron transfer from the central to sa­ tellite valleys. 18. In the low-dimensional structures other scattering mechanisms can lead to the runaway effect, see, for example, B. K. Ridley and N. A. Zakhleniuk, "Hot elec­ trons under quantization conditions. 1. Kinematics", J. Phys.:Condensed Mat­ ter, 8 (1996) 8525-8537; A. P. Dmitriev, V. Yu. Kachorovskii, M. S. Shur, and M. Stroscio, "Electron runaway and negative differential mobility in two-dimen­ sional electron gas in elementary semiconductors", Solid State Commun. 113 (2000) 565-568. 19. K. K. Thornber and R. P. Feynman, "Velocity acquired by an electron in a finite electric field in a polar crystal", Phys. Rev. B 1 (1970) 4099. 20. M. S. Shur, A. D. Bykhovski, R. Gaska, M. A. Khan and J. W. Yang, "AlGaNaN-AlInGaN induced base transistor", Appl. Phys. Lett, 76 (2000) 3298-3300. 21. Applicability of an approach used in this paper for nitrides has been analyzed recently in S. M. Komirenko, K. W. Kim, M. A. Stroscio, and M. Dutta, "Applica­ bility of the Fermi golden rule and the possibility of low-field runaway transport in nitrides", J. Phys.: Condensed Matter 13 (2001) 6233-6246, where a possibili­ ty of realization of low-field runaway in GaN and A1N has also been suggested. 22. I. I. Vosilius, I. B. Levinson, "Generation of optical phonons and stronglyanisotropic galvano-magnetic effects under high electric fields", Zh. Eksp. Teor. Fiz. 50, (1966) 1660-1665 [Soviet Phys. JETP 25 (1967) 672]. 23. Note that in 1 //-length GaAs diode the electron transport with sequential singlephonon emission was observed at the electric fields below the runaway threshold: T. W. Hickmott, P. M. Solomon, F. F. Fang, and F. Stern, "Sequential SinglePhonon Emission in GaAs-AlxGal-xAs Tunnel Junctions",Phys. Rev. Lett., 52 (1984) 2053-2056; see also A. E. Belyaev, S. A. Vitusevich, R. V. Konakova, T. Figielski, A. Makosa, T. Wosinski and L. N. Kravchenko, "Tunnel current oscil-

142

High-Field Electron Transport Controlled by Optical Phonon Emission

1081

lations in a GaAs/AlAs double-barrier heterostructure" JETP Lett., 60 (1994) 416-419. 24. A. F. J. Levi, J. R. Hayes, P. M. Platzman, and W. Weigmann, "Injected-HotElectron Transport in GaAs", Phys. Rev. Lett, 55 (1985) 2071-2073. 25. M. Heiblum, M. L. Nathan, D. C. Thomas, and C. M. Knoedier, "Direct Ob­ servation of Ballistic Transport in GaAs", Phys. Rev. Lett, 55 (1985) 2200-2203. 26. It is necessary to note that velocity oscillations with the distance, as presented in Fig. 4, could not be reproduced in the approach undertaken in Ref 12. Mainly, it is because of the averaging procedure of the Monte Carlo method in the time domain and further recalculation of the velocity as a function of the distance. General discussion of free electron nights interrupted by sudden optical phonon emission is dated back to papers by W. Shockley, Bell. Syst. Techn. J., 30 (1951) 990 and P. J. Price, IBM J. Res. Develop., 3 (1959) 191. 27. S. M. Sze, ed. High-Speed Semiconductor Devices (Wiley, New York, 1990). 28. L. S. McCarthy, I. P. Smorchkova, H. Xing, P. Kozodoy, P. Fini, J. Limb, D. L. Pulfey, J. S. Speck, M. W. Rodwell, S. P. DenBaars and U. K. Mishra, "GaN HBT: toward an RF device", IEEE Electron. Devices, 48 (2001) 543-551. 29. P. Asbeck, private communication. 30. Z. S. Gribnikov, K. Hess, and G. A. Kosinovsky, "Nonlocal and nonlinear trans­ port in semiconductors: Real-space transfer effects", J. Appl. Phys., 77 (1995) 1337-1373. 31. J. R. Pierce, J. Appl. Phys., 19 (1948) 231. 32. M. C. Steel and B. Vural, Wave Interactions in Solid State Plasmas (McGill Hill, New-York, 1969). 33. V. L. Gurevich, Soviet Phys., Solid State, 4 (1962) 1380. 34. I. A. Chaban, A. A. Chaban, Soviet Physics, Solid State, 6 (1964) 2411. 35. J. B. Gunn, "Quantum Theory of Lattice-Wave Amplification in Semiconduc­ tors", Phys. Rev., 138 (1965) A1721-A1726. 36. S. M. Komirenko , K. W. Kim, V. A. Kochelap, I. Fedorov, M. A. Stroscio, "Coherent optical phonon generation by the electric current in quantum wells", Appl. Phys. Lett, 77 (2000) 4178-4180; "Generation of coherent confined LO phonons under the drift of two-dimensional electrons", Phys. Rev. B, 63 (2001) 165308.

143

This page is intentionally left blank

International Journal of High Speed Electronics and Systems, Vol. 12, No. 4 (2002) 1083-1100 © World Scientific Publishing Company

COOLING B Y I N V E R S E N O T T I N G H A M EFFECT WITH RESONANT TUNNELING

Y. YU, R. F. GREENE, and R. TSU University

Electrical and Computer Engineering, of North Carolina, Charlotte, North Carolina,

USA

The inverse Nottingham Effect (INE) cooling involves emission of electrons above the Fermi level into the vacuum. Our scheme involves t h e use of a Double Barrier Reso­ nant Tunneling (DBRT) section positioned between the surface and the vacuum for a much increased emission, and to provide energy selectivity for assuring cooling, without surface structuring such as tips and ridges leading to current crowding and additional heating. Unlike resonant tunneling from contact-to-contact, where barrier heights and thicknesses are controlled by the choice of heterojunctions, the work function at the surface dictates the barrier height for tunneling into the vacuum. The calculated field emission via resonant tunneling gives at least two orders of magnitude greater than without resonance, however, without work function lowering, the large gain happens at fairly high field. The use of resonance to enhance cooling by INE results in an important byproduct, an efficient cold-cathode field emitter for vacuum electronics. Keywords: Inverse Nottingham Effect; resonant tunneling; cold cathode emitter; field emission.

1. Introduction The original Nottingham effect dealt with the additional thermal effects beyond the nonohmic behavior field emission from metal tips. 1 The Inverse Nottingham Effect (INE) cooling involves emission of electrons above the Fermi level into the vacuum. 2 - 4 Our scheme involves the use of a Double Barrier Resonant Tunneling (DBRT) section positioned between the surface and the vacuum for a much in­ creased emission, and to provide energy selectivity for assuring cooling, without surface structuring such as tips and ridges leading to current crowding and ad­ ditional heating. Two approaches appeared: (1) The G-T, Greene-Tsu, scheme consists of inserting a double barrier resonant tunneling section, 5-7 between the surface of the semiconductor to be cooled and the vacuum. (2) The K-L, KorotkovLikharev, scheme consists of a step inserted between the semiconductor and the vacuum. Under the application of a high electric field, the step forms a triangular quantum well for resonance tunneling. 4 The G-T scheme offers better flexibility in design possibility for optimizing cooling while keeping the electric field at the surface to a "safe limit". The K-L scheme is simple, however, because the step is a barrier material, usually formed by alloying resulting in lower mobility and hence

145

1084

Y. Yu, R. F. Greene & R. Tsu

lower efficiency. There is a basic principle of symmetry when resonant tunneling is involved. The maximum transmission at resonance requires a symmetrical struc­ ture. It is easy to understand the need for a symmetrical structure. Imagine that a resonant cavity for photons, the Fabry-Perot interferometer having one surface reflectivity much larger than the other, then it is not possible to build up the field from coherent interference without a symmetrical structure. In DBRT, if one de­ signs a symmetrical structure without a bias voltage applied, the applied voltage will destroy the symmetry. Therefore it is important to design the barrier structures as symmetrical as possible at the operating resonant condition. Thus it is necessary to design a symmetrical potential profile at the operating voltage. 7 Unlike resonant tunneling from contact-to-contact, where barrier heights and thicknesses are con­ trolled by the choice of heterojunctions, for tunneling into the vacuum, the work function at the surface dictates the barrier height. And this is the difficulty in optimizing tunneling into the vacuum via DBRT, because typically the first barrier height is ~ few tenth of eV above the quantum well, but the 2nd barrier is deter­ mined by the work function, usually several eVs above the quantum well. One may assume that a very thick 1st barrier can match the high work function of the 2nd barrier. In principle it is possible, however, the quantum state with thick barriers, instead of a resonant state, is practically an eigenstate, having almost no energy width to support a large tunneling current. Therefore, we have concluded that the best way is to search for a 2nd barrier with very low work-function.7 For this reason, under the HERETIC DARPA/ARO program, we have been pushing the structure with Ill-nitrides. In view of the conflicting claims of negative electron affinity for AlGaN with [Al]/[Ga] over 60%, and also it is doubtful that such material may have good mobility, we calculated cases where we do not need the low work function. Basically, we know that resonant tunneling can enhance field emission. We want to see how much gain over the usual Fowler-Nordheim tunneling. It is indeed encour­ aging that the calculated field emission via resonant tunneling is several orders of magnitude above the F-N tunneling at a surface field of several 107 V/cm. And if we consider a surface field of up to ~ 2 x 107 V/cm, it is 2-4 orders of magni­ tude higher. For cooling, it is important to extract the emission current with the Pierce-electrode, 3-8 which is difficult to implement, so that some of the power at the anode can bejecovered as in traveling-wave tube designs. 8 Removing the heat at the anode is identical to removing the heat from the heat exchanger in any air conditioner. Even if all these practical problems cannot be overcome, there is a hi'ge byproduct. Field emission with resonance results in an efficient cold-cathode field emitter for vacuum electronics, which is particularly important in high power TWT!

2. Background Originally G-T scheme targeted the use of a p-type semiconductor to produce surface inversion as in the MOSFET to pull electrons above the Fermi level into the vacuum, 2 while the K-L scheme targeted the use of n-type. 4 We exclude the

146

Cooling by Inverse Nottingham

Effect with Resonant

Tunneling

1085

Fig. 1. G - T (Greene-Tsu) scheme: Band profile of p-doped inversion layer under high electric field with tunneling into the vacuum via a DBRT structure. The barrier width may be much reduced with NEA as shown at left for higher tunneling current into the vacuum. Also shown is trapping level at the interface, Nt, to be incorporated for increased replenishment of electrons pulled out into the vacuum.

usual approach of surface structuring such as tips and ridges, 9 because tips result in current crowding which leads to heating and the problem of robustness. This is the main reason why we decided on the use of DBRT to enhance field emission into the vacuum. For the convenience of the reader, the original G-T scheme for ]>doped semiconductors method in Fig. 1 of Ref. 2 is shown below as Fig. 1. It was shown that the average energy per electron above the Fermi level at the surface representing cooling is (E-EF)

= A~1[1.5kBT+(Eco-EF)],

(1)

in which A^1 term represents a reduction factor due to the presence of tun­ neling upsetting the balance between generation and recombination at thermal equilibrium.2 Also for the convenience of the reader, the K-L scheme taken from Ref. 4 is reproduced in Fig. 2. Figure 2 of Ref. 7 that was missing,7 is reproduced as Fig. 3, showing without NEA (negative electron affinity), the 2nd barrier, the one next to the vacuum is so much thicker except under extremely high field, so that tunneling current is quite small as in Fig. 3(c). Whenever one barrier is much thicker than the other, the transmission coefficient is much reduced. As mentioned that the restoration of symmetry is key to the success of high gain at resonance in a DBRT structure, the transmission T and the maximum T, TM presented in Ref. 7 are repeated here especially, due to printer's error, the figures were missing in Ref. 7.

147

1086

Y. Yu, R. F. Greene & R.

Tsu

"7TT"

(7-1 eV b~4eV

-U

£~3kBT

...MC....

d- 4 nm

fc

>

dzns = 3 n m mzns = 0.42 m 0

(7=1.0eV Xzns = 2.7eV m si = 0.2 m0

composite emitter

emitter

(a)

(b)

Fig. 2. Band profile of the proposed emitter (a) no field, (b) with field. Horizontal lines in (b) show quantized sub-bands. Arrows show RT via the sub-bands. The lowest sub-band is placed at few ksT above the Fermi level so that cooling occurs with removal of hot electrons of the emitter. 4

Vacuum Field

Al,Ga,.xN (a)

(b)

(c)

Fig. 3. Three cases: (a) no applied voltage with NEA (negative electron affinity), (b) with applied voltage and NEA, and (c) same as (b) but without NEA, resulting in a large barrier marked as B; depending on the value of the applied voltage, the tunneling current into the vacuum is generally substantially reduced.

The reflectivity and phase shift of a single barrier are, 7 R =

(A;2 + a2)2 \{k + a2)2 + 4k2a2F2) 2

cj) = tan- l

(2)

'

2kaF (a 2 - k2)

(3)

in which a2 aig - k2, and F (1 + e-2ab)/{l - e~2ab) and k2 = 2m*E/h2 al = 2m*V0/h2, where V0 and b being the barrier height and width respectively. The transmission, T for well-width w is T = TiT 2 /[(l - i?!)(l - i? 2 )/(l + RXR2 - 2{RlR2)1'2

148

cos$]

(4)

Cooling by Inverse Nottingham

Effect with Resonant

Tunneling

1087

Fig. 4. TM versus the ratio R2/R1 for various R\. Hi and R2 are the reflectivity at the 1st and 2nd-barrier. Note that for Ri > 0.95, TM is only significant for R2/R1 > 0.9.

in which $ = <j>\ + 2 + 2kw. At resonance $ = 2n-K, so that the maximum T is T M = TiT2/[(l

- Ri)(l - R2)/(l

+ (R1R2)112)] ■

(5)

The calculated TM is shown in Fig. 4 for various R\ and R2 ■ Note that near unity TM only occurs for Ri ~ #2- This is because unity TM is obtained when the two reflectivities are equal for maximum interference. As long as the structure is symmetrical, the transmission at resonance is always unity. Whenever one of the Ri and R2 is high, the other R must also be high to give a large TM- For relatively low value of the reflectivity R, Q (Q is the quality factor of a resonating system denned by number of cycles a photon is confined before it decays to 1/e of the initial value) is low, and it does not take many traversals to produce a relatively high transmission. This is why we need thin barriers to produce high throughput. Since thin barrier needs low work function or NEA, Ill-nitrides seems to be the best choice at this point. With the Q for each R obtained from Ref. 10, the corresponding energy window A E is shown in Fig. 5 with E ~ 0.5 eV for the resonant energy. Significant values of AE, can only be obtained with the barrier width of 4-9 A (the lower value of 4 A is dictated by the size of the unit cell). That means we need to fabricate the barriers of Al x Gai_ x N, for example, by not more than a couple of unit cells. Unity transmission at resonance with thicker barriers with very narrow energy linewidth AE does not lead to large currents. Thus we have only two ways to reduce the 2nd barrier: using low work function materials for the 2nd barrier or by applying a huge electric field to reduce the vacuum barrier. Figures 4 and 5 are really very important for optimizing the design of RTDs, but for RTFE, resonant tunneling field emission, the vacuum level is fixed by the work function, so that there is not much one can do about it. Optimization amounts to picking the right materials for the construction of the DBRT structure.

149

1088

Y. Yu, R. F. Greene & R. Tsu

0 . 3 -,

Q

: Q 2

0.1

R. Fig. 5.

t-O

AE and Q versus reflectivity R, lower scale; and barrier-width 6(A), upper scale.

The best parameters for large resonant tunneling into the vacuum can be summarized: (1) The reflectivity for the two barriers should be almost equal at the operating voltage. The effective barrier must not be too thick otherwise AE is too small. (2) The barrier material next to the vacuum must be almost NEA so that the effective thickness of this barrier is relatively low. Without NEA, it would be necessary to apply a large electric field to effectively lower the 2nd barrier to match the first. Direct computations for the tunneling current into the vacuum at resonance have been performed in this work. We found that the estimate for the thermal current for cooling of ~ 300 — 2000 W/cm 2 in Ref. 2, is too optimistic because in the previous estimate, the difference in the effective mass in the semiconductor and the free electron mass in the vacuum was not properly accounted for. 3. Computation of the RTFE We have undertaken the direct computation of RTFE, resonant tunneling field emission, for n-type semiconductors with the barrier structures. Although we have stated several times that Ill-nitrides appear to be the most promising materials, we selected silicon for the calculation because as we shall see that we are far from obtaining practical design optimization partly because resonant tunneling into the vacuum is far more complex than resonant tunneling from contact-to-contact. 5 ' 6 We want to establish some rules governing high emission at a given electric field with the available material parameters. We start with the Tsu-Esaki approach 5 by integrating the transmission T given by the matrix relating the input at the left contact to the output at the right contact with the appropriate distribution and

150

Cooling by Inverse Nottingham

10'

1

|

■ ■ i

0

i

|

i

i

1

i

■ |

■ ■ ■ ■ |

2

Effect with Resonant

i—■

3

i

■ |

4

■ ■ ■ i

Tunneling

1089

|

■ ■ ■ ■ |

5

6

E(eV) Fig. 6. Transmission versus energy in eV with and without applied voltage Va = 6 V correspond­ ing to an electric field in the vacuum of 7.5 x 10 6 V/cm. 1st Barrier height = 0.5 eV and 2nd Barrier height (vacuum side) = 4 eV.

density of states functions for the total tunneling current density, the sum of jw, from left to right; and jri, from right to left, or

• 3T

emk T

B

= -^W

r ^ v ,

J0

| T | ln

l + e( g /- g '>/ fc *T

(

[1

+

elB/-Bl-.v.)/l.BT)

^ lEi+eVajlP

V —ETdEl ■

fR.

(6)

Equation (6) contains an extra term, under the square root, a correction term added in by Coon and Lieu. 10 This correction term is not so important in tunneling from contact-to-contact, but significant for tunneling from contact-to-vacuum where Va may be quite large. Before we present cases of interest, let us define some terms useful for the remaining of this article. For emission into the vacuum useful as a cold cathode, we need the total current jr defined by an integration for E = 0 to oo. For cooling, because we only remove the hot electrons, the current jjj is defined by an integration for E = Ep to oo. Figure 6 gives the comparison of the transmission coefficients of a structure, 5-5-7 nm (the 1st barrier width, b, of 5 nm, and quantum well width, w, of 5 nm, and a vacuum barrier, the 2nd barrier of w = 7 nm), and one without resonant tunneling having only the 2nd barrier of 7 nm, the F-N tunneling. For better visualization, portion of Fig. 6 is plotted in linear scale shown in Fig. 7. It is important that we discuss these two figures because the transmission is rather different from the conventional tunneling from contact-to-contact. Take, for

151

1090

Y. Yu, R. F. Greene & R. Tsu

1.0-1

I-

* r-

E(eV) Fig. 7.

Transmission versus energy with linear plot in the range of interest where T*T is large.

example, the 5-5-7 case, the three peaks near E ~ 0 represent the resonance via the quantum well states. The peak values are so small because the structure is very asymmetrical due to the high 2nd barrier of 4 eV. In fact we thought that the computation was incorrect because we failed to find the resonant tunneling peaks first. After realizing that the value of these peaks may be so small that we missed them, we proceeded to look for them, and we found them as shown in Fig. 6. For E > 0.5 eV, there are small structures due to interference just above the 0.5 eV of the 1st barrier height. There are no substantial resonant states until the energy is above the vacuum level, at E > 4 eV. The large oscillation of T*T between 0 and 1 comes from resonant states in the vacuum due to the vacuum barrier. This point is not too familiar to most although several years ago similar phenomena were treated regarding the physics of "Quantum Step". 11 In fact these resonant states in the vacuum can.be calculated very simply from kn = nir/B, with B being the distance between the surface of the semiconductor and the anode placement in the vacuum. With an applied +V with respect to the cathode, T*T shifts to the left. This shift is same as resonant tunneling from contact-to-contact. For a symmetrical structure, the spectrum is shifted by an amount in energy very close to V/2. Why is the magnitude of the oscillations so huge? What is happening is the fact that at higher energy, the 1st barrier and the well region play almost no role. The bulk of the oscillation is due to the vacuum barrier. The discontinuity of the barrier is further augmented by the difference of the effective mass in the semiconductor, and free electron mass in the- vacuum. These oscillations are nearly the same for the case shown in dotted having only the vacuum barrier. Figure 7 shows a linear

152

Cooling by Inverse Nottingham

Effect with Resonant

Tunneling

1091

Ev (V/cm) Fig. 8. The calculated total tunneling current density versus the applied electric field for three cases: dotted for the vacuum barrier of 7 nm only, dashed for 1-5-7 nm, and solid for 10-5-7 nm.

plot to give a better feeling what is happening. Note that as the applied voltage increases, these large oscillations of T*T move toward the origin as discussed. When they pass the Fermi level of the left contact, substantial tunneling current appears. Again we emphasize that it only occurs at high electric field, > 107 V/cm. This type of oscillation of resonant tunneling via the quantized vacuum state cannot be obtained by the use of WKB perturbation usually used for computing tunneling. In fact Mimura et al. showed the comparison of tunneling emission from metal-oxide cathode computed from the exact numerical results and the WKB approximation clear demonstrated the absence of oscillation with the WKB method. 12 Figure 8 shows the calculated tunneling current from Eq. (6). Our results are only valid provided the quantum well region is shorter than the coherence length which is ~ 10 nm, because no scattering is accounted as in the case of including dissipation factors. 13 With a large applied voltage, most part of the barrier moves below the Fermi energy of the contact so that the length we deal with for these resonant states is the sum of the width of the quantum well and a significant part of the barrier width. Earlier we discussed the K-L scheme with a semiconductor step forming a triangular quantum well under a large applied voltage in a mate­ rial most likely formed by alloying. Alloys are usually poorer in mobility and thus having shorter mean-free-path. Now we have the same problem because our quan­ tum well under a large voltage consists in addition to the quantum well part, a part of the barrier. For this reason, it is far better to have as large a barrier as possible, which cannot be transformed into a quantum well under a large electric field. A wider barrier width but lower barrier height presents extra effective barrier at low field. However, at high field, much of this extra width has a band-edge energy

153

1092

Y. Yu, R. F. Greene & R. Tsu

Ibl

W

IB

4^—EVAC$.

tWPZ'Ss'

- E ,

EK<

( E c -+- E , )

+Va

Fig. 9. A schematic potential profile for the DBRT structure under bias of +Va. Note that the work function <j>o is reduced by A 0 ~ in the figure.

moving below E = 0 so that electrons tunneling through the vacuum barrier see a much lowered vacuum level. This is the mechanism for the lowering of the effective work function regardless whether having resonance or not. Resonance results in additional gain in the tunneling process. To make this extremely important point clearer, we show in Fig. 9 a schematic used for explaining the origin of the large tunneling current at high field — effectively lowering the electron affinity! A bias of +Va is applied in Fig. 9, producing a vacuum electric field of FQ which effectively lowers by A4> = Fo(b + w)/e from <j>o. Obviously making b + w large is what lowers the effective . However if w + b > mean free path of the electrons, the electrons would be near Ec, and no reduction of <j>o is possible. Therefore, the additional gain in tunneling results from electrons resonantly tunnel via higher energy levels, the vacuum resonant states above (f>o, lowered to the position of the Fermi level in the contact. In this scheme, the 1st barrier merely serves to keep the high dop­ ing in the contact from going into the undoped regions which can be lowered by an applied electric field. In this respect, the scheme explored by Mumford and Cahay 14 using hot electrons injected from a metal into a wide bandgap semiconductor, reso­ nantly tunnel through a thin semimetal layer to effectively lower the work function bears similar principles. 14 Only that our present results show that a much simpler scheme for the lowering of the work function is possible: injecting electrons into the quantum well maintaining its energy by avoiding scattering to resonantly tunneling through the quantized vacuum states! The computed currents, the total current density jr, and the hot current density JH, for 7 nm case, the lower trace; and those for 10-5-7 nm, the upper trace, using Si as an example, are shown in Fig. 10. Note that, at a field of ~ 1.5 x 10 7 V/cm, the 10-5-7 nm case reaches ~ 10 3 A/cm 2 , which is more than 10 12 times greater than the case of 7 nm, without the lowering of the effective work function. Therefore

154

Cooling by Inverse Nottingham

1.0X107

1.2X10 7

1.4X107

Effect with Resonant

1.6X10 7

1.8x10 7

Tunneling

1093

2.0x10 7

Ev (V/cm) Fig. 10. Computed currents, total current density jx, and hot current density ju, for 7 nm case, lower trace; and for 10-5-7 nm, the upper trace, for Si. The two arrows indicate the positions we have calculated the temperature for cooling.

our results not only can serve the INE cooling scheme, but also can serve many applications requiring cold cathodes. 15 ' 16 For better visualization, part of the computed results is shown in Fig. 11 with linear scale for the total current densities. The oscillations are much better shown in a linear plot. It is possible to utilize the oscillation for modulation of field emission, for example, turning on and off of a TWT. In fact, these oscillations may be also used to digitize the output of a TWT. It may also be designed with a two-step scheme, one at much lower field for digitization, and amplified by another high power TWT. A discussion on the placement of the anode is in order. Since barrier-width is reduced by creating a triangular barrier with an applied voltage, as long as the effective barrier width at energy for tunneling is same, it does not matter how far away the anode should be placed. Farther placement of the anode results in more kinetic energy gain after tunneling. Since we assumed that this extra gain in the kinetic energy may be recovered by the Pierce-electrode, we simply place the electrode sufficiently close to reduce computational complexity. In measurements however, it is difficult to design the anode extractor only 7 nm from the front surface. The barrier height used in our calculation is 0.5 eV for silicon may be achieved by epitaxially grown (Ba x Sri_ x )203, 1 7 as well as recent success in a superlattice

155

1094

Y. Yu, R. F. Greene & R. Tsu

5x10"

4x10

m* = 0.26m e> n = 1018cm"3 V1 = 0.5 eV, electron affinity = 4 eV

3x10

2x10 -

10jT7nm j T 10-5-7 nm

1x10 -

0-i

-i—i—|—i

1x10

2x10

7

3x10

1

1

1

r-

4x10

5x10

Ev (V/cm) Fig. 11.

Oscillations in the resonant tunneling current at high field for two cases.

structure involving monolayers of oxygen, the Si/O superlattice barrier. 18 There­ fore, the calculation using silicon as an example is not just to prove the principle, rather, the structure may be realized. Actually, the Ill-nitride system seems even more promising because GaN, particularly GaAlN is robust and possibly having low work function. To summarize, at a tolerably high electric field, the use of resonant tunneling can indeed produce a huge gain in field emission over the case without resonant tunneling even without surface treatment to lower the work function! For practical considerations, we focus our search for the design of getting a relatively high current peak at a field in the low 107 V/cm regimes. According to Heinz Busta, formerly of Sarnoff, and Professor Binh of France, a field of 2 x 10 7 V/cm may be a practical limit. Our calculations do show that proper choice of design parameter allows us to keep the electric field at the surface to this limit.

4 . Calculation of Cooling For the calculation of the temperature drop from the INE scheme, we start with

V-JQ

+

dQ dt

156

5>

(7)

Cooling by Inverse Nottingham

Effect with Resonant

Tunneling

1095

where the right side represents the sum of dissipation — output via emission, explicitly, / _, Q = {I R)\oss — (J r #V')emission •

(8)

At steady state and in one dimension, -g* = 0. We assume, for a first order approx­ imation, ^ w 0, leading to ^ - « 0, or £) Q = 0. Without these assumptions, we need to solve the differential equations. In terms of the loss per electron per unit area for the first term in Eq. (2), and the emission term per electron per unit area for the second term in Eq. (2), there results

in which J? and J # are the total current (integration form E = 0 to oo); and the hot electron current (integration from E = Ep to oo) respectively; V is the applied voltage (anode voltage applied in the vacuum near the surface), p is the resistivity of the semiconductor, I is the thickness of the semiconductor, n is the electron density of the semiconductor (The barrier and quantum well are undoped), and N(j is the effective density of states. Then

W = (*-*,)/A.{(^)(£)}.

do)

To account for the Wiedemann-Pranz law relating the thermal and electrical conductivities, i.e., K/a = {-nkB)2 T/3e, the term J^V should be reduced to JQ s JHV -

0.lkBT

JT

where JQ, is the power flow of hot electrons into the vacuum taking into account the Wiedemann-Franz law. Including a thermal load, Eq. (10) becomes

kBT - (EC - B,)/* { ({4pl

J + T"Z^^)

(V) } '

<">

At the DARPA HERETIC Principal Investigators' Meeting, Atlanta, GA, on May 23-25, 2001, we have presented the temperature cooling for three materials, Si, GaAs and GaN, in a presentation 19 : "Inverse Nottingham effect cooling of semi­ conductors with resonant tunneling". At that time, we have used a good estimate of the effect of resonance on J # , from known results of resonant tunneling for contactto-contact, rather than direct calculation involving putting an anode in the vacuum. Table 1 presented at the HERETIC Meeting is listed below for comparison. The parameters in the above chart such A - 1 and CM is defined in Ref. 2. We compare several cases shown in Table 2 with direct computation. The notations, 1st*, and 2nd*, refer to the points of operation marked by arrows in Fig. 10. For F-N field emission with a single barrier formed by the vacuum level under an applied field at the same operating points. Top chart is for the case without the thermal load. The bottom chart refers to thermal load of 300 W/cm 2 . Without

157

1096

Y. Yu, R. F. Greene & R. Tsu

Table 1. Computed cooling power JQ for p-type with inversion, and n-type: for Si, GaAs and GaN using an estimation scheme from resonant tunneling from contact-to-contact. 1 8 p-type with inversion : N^ = 10 1 8 c m - 3 , C M (Brown) = 0.013, C M (Chang-Esaki-Tsu) = 1 0 " 4 , A " 1 (Si) = 0.29, A " 1 (GaAs and GaN) = 0.44 ns

(cm"3)

JQ ( W / c m 2 )

18

Si GaAs GaN

1.8 x 10 4.6 X 10 1 7 3.8 x 10 1 7

360 1280 1030

JQ (C) 324 1042 962

Nc/ns 22.6 1.02 6.8

T (K) 296 150 187

n-type : Same values for A - 1 and C M as for p-type with inversion ns(cm-3)

JQ ( W / c m 2 )

18

Si GaAs GaN

1.01 X 10 2.02 x 10 1 7 0.99 x 10 1 7

Table 2.

325 880 1900

JQ (C) 303 843 1723

Nc/ns 40.3 2.33 2.6

T (K) 299 138 107

Results of direct computation for tunneling into the vacuum.

Without thermal load

Resonance Without Resonance

1st* 2nd* 1st* 2nd*

PL ( W / c m 2 )

PH ( W / c m 2 )

T(K)

A T (K)

40 22563 6.29 x 1 0 ~ 2 4 9.80 x 1 0 ~ 1 9

2442 67450 6.62 x 1 0 - 1 1 8.38 x 10~ 8

143 233 33 39

157 67 267 261

PL ( W / c m 2 )

PH ( W / c m 2 )

T(K)

A T (K)

340 22863 300 300

2442 67450 6.62 x 1 0 _ n 8.38 x 1 0 - 8

197 233 NA NA

103 67 NA NA

W i t h 300 W / c m 2 of thermal load

Resonance Without Resonance

1st* 2nd* 1st* 2nd*

thermal load, there is always cooling, even for the F-N case. What it means is that, without thermal load, as long as one takes out the hot electrons, cooling should result, regardless of how low is the current level! This is because we have not included any losses in the calculations. In these charts, NA refers to situation that our theory does not apply or no cooling is possible. The situation is very different when thermal loss is included. With the inclusion of 300 W/cm 2 of thermal load, F-N emission cannot cool because the remover of power via field emission is less than the thermal load. What is interesting is the fact that there is an optimum cooling. Going from the 1st to 2nd, while PH is much increased, but cooling is reduced because the loss due to current flowing through the substrate is even higher. We shall also compare the results for Pu obtained previously with the present direct calculation: Previously, PJJ = 325 W/cm 2 while the direct calculation gives 2442 W/cm 2 . What is most encouraging is the fact that a cooling of 103 K is still

158

Cooling by Inverse Nottingham

Effect with Resonant

Tunneling

1097

possible with 300 W/cm 2 of thermal load, employing resonant tunneling operating at an electric field ~ 1.5 x 107 V/cm, which is manageable in practice. The term PL refers to the contribution of loss from J\pt, which is quite negligible compared to the thermal load used at the operating point 1st, however, the loss is huge at the 2nd operating point. The term PH refers to the rate of energy removed via field emission of the hot electrons above the Fermi level. 5. Discussion Our calculated results show that INE can be realized. What is exciting is the fact that resonant tunneling into the vacuum gives ~ 1 0 3 - 4 times higher emission current over the case without resonance at an electric field quite manageable. Even if INE cooling scheme is ultimately proved to be impractical due to the difficulty with the Pierce-electrode, the huge gain in the emission current should have a strong impact in cold cathode vacuum electronic devices. Since the DBRT structure is planar, it should be much more robust than geometrical structuring. As pointed out before, the low field emission reported by Binh and Adessi, 20 require further discussion. First of all, inserting TiO-2 between the metal and the vacuum, structurally their scheme is similar to the K-L scheme shown in Fig. 2. The calculation in the K-L scheme is based on resonant tunneling, which takes into account the quantized sub-bands in the well created by the applied high electric field. However, if losses and dampings are high, quantized sub-bands cannot exist so that the two models are structurally identical. The question remains why Binh and Adessi measured low emission current at such low electric field. Heinz Busta thought that the low current field emission comes from defects inadvertently intro­ duced. It is likely that the high dielectric TiO"2 results in "smoothing" the localized defects rendering the appearance of more uniform emission. We have calculated the K-L scheme from Eq. (6) using the same parameters as in Ref. 3. The total current density is compared with our model as well as the F-N case without resonant tun­ neling in Fig. 12. Note that the high emission occurs at a field of over 3 x 107 V/cm. As we have pointed out before that usually the mobility of a step using alloying is much reduced so that losses should be included. Nevertheless, it is impressive that such a simple structure with a step is capable of producing a current less than a factor of ten below the DBRT case! The large gain in the tunneling current compared to the case without shown in dotted is again due to the lowering of the effective work-function. The additional gain between 2.75 — 3.3 x 107 V/cm is due to resonance gain. We have computed se­ veral cases for GaN-AlGaN-GaN system seems to be the best possible choice for high current emission discussed previously. Figure 13 gives the calculated total current density versus electric field in the vacuum for GaN-Alo.sGao.sN-GaN with b / w / B cases as shown. A single vacuu*n barrier of 2 nm is also shown for comparison. The effective mass for the barrie/ and quantum well are taken to be 0.22mo and the barrier height is taken to be 0.8 eV. The internal build-in field, ~ 2-3.6 x 106 V/cm,

159

1098

Y. Yu, R. F. Greene & R. Tsu

10 1 10'I

,/' n.

\rJ

\/../'

■~/

\

/""

iy

'^'

10*1

r

10 7 i

E

10

'l

10 5 1

10* "j

10s] 102-

L 7 nm

r1 J . =•

2.5x10

-

. !

J

-^

jT

\{J-D-I

nm

j _ d.-o-1 nm rv-L !

1

T

i

■

i

3.5x10

3.0x10

4.0x10

■

i

4.5x10

Ev (V/cm) Fig. 12. Comparison of the total tunneling current density between the K-L scheme, our DBRT case, with t h e F-N case without resonant tunneling.

10' ~\ 10' 10 8 10 7 10 6 10 5 10 4 10 3 10 2 10 1 10°

io-1 102

2-2-2 nm 2-4-2 nm 4-4-2 nm 2nm

10"3 10* 10* 10^

Iff 7 io"8

io*

— i —

2x107

1x10

3x10

Ev (V/cm) Fig. 13. Calculated total current density versus electric field in the vacuum for GaN-Alo.5Gao.5NGaN with b / w / B cases as shown. A single vacuum barrier of 2 nm is also shown for comparison. The effective masses for the barrier and quantum well are taken to be 0.22 and the barrier height is taken to be 0.8 eV. The internal build-in field, ~ 2 — 3.6 x 10 6 V/cm, has not be-m included.

160

Cooling by Inverse Nottingham

Effect with Resonant

Tunneling

1099

has not been included. The calculated current density near a field of 1.5 x 107 V/cm is 104 A/cm 2 . Considering the robustness of the Ill-nitride system, this is the best choice thus far. In fact we predict that the ideal emitter may be achieved by adding a second barrier of few monolayers of A1N at the surface of GaN both as a protection against oxidation as well as for further lowering the effective work function. Such a system cooperating at a reduced temperature (the field emitter will be self-cooled by the INE cooling process) will be an ideal cold cathode for TWT and other applications requiring high current protected from oxidation of the emitter. 6. Conclusion Let us discuss what is universally known in the field emission community. Most researchers agree that for a field > 2 x 107 V/cm, the usual Fowler-Nordheim emission would result in substantial emission current. At a significantly lower field, work-function lowering such as Ce on Si so that useful emission can take place much below 1 x 10 7 V/cm, for avoiding surface degradation. Our samples measured by Heinz Busta at Sarnoff, as well as the samples from K-L method designed by Kostantin K. Likharev and fabricated by Wiley P. Kirk (Texas A & M University) using ZnS on Si and measured by Vu Thien Binh (Laboratoire d'Emission Electronique, Departement de Physique des Materiaux, UMR-CNRS, Universite Claude Bernard Lyon 1) showed structure in emission current at field of ~ 1 x 106 V/cm or lower, which may be caused by some inadvertently introduced surface contaminants. What we know now is the fact that the main factor in obtaining significant emission current at relatively low applied electric field still needs low work-function at the surface. Since work function lowering has been an intensive research for more than 30 years, we want to see whether it is possible to design field emission via resonant tunneling without work function lowering, capable of producing substantial emis­ sion at an electric field below 1 x 107 V/cm. We are reassured that calculations show that it is indeed possible! Resonant tunneling not only can improve transmission by a mechanism similar to photon resonance cavity, the appearance of higher energy states due to quantum confinement effectively lowers the work function at the semi­ conductor vacuum interface. Even if INE cooling would be too difficult practically, the huge increase in the tunneling current into the vacuum would open the door for vacuum electronics with efficient cold cathode. Recently, we have fabricated two samples, one with Si and a second one with GaN for measurements in V. T. Binh's laboratory in France. Preliminary results indicated that the huge enhancement in the resonant tunneling current into the vacuum is close to predicted value. 21

Acknowledgments The support of this work by the DARPA-HERETIC program, DARPA/ARODAAD 19-99-0154, is gratefully acknowledged.

161

1100

Y. Yu, R. F. Greene & R. Tsu

References 1. W . B. Nottingham, Phys. Rev. 59 (1941) 907-908. 2. R. Tsu and R. F. Greene, "Inverse Nottingham effect cooling in semiconductors", Electrochem. Solid-State Lett. 2 (1999) 645-647. 3. R. F. Greene and R. Tsu, Workshop on Microelectronic Thermal Management, Sys­ tem Planning Corporation, DARPA ETO, E. Towe and E. Brown, Arlington, VA, December 11-12, 1997. 4. A. Korotkov and K. Likharev, Appl. Phys. Lett. 75 (1999) 2491-2493. 5. R. Tsu and L. Esaki, Appl. Phys. Lett. 22 (1973) 562. 6. L. L. Chang, L. Esaki, and R. Tsu, Appl. Phys. Lett. 24 (1974) 593-595. 7. R. Tsu, ECS Proc. 98, 19 (1999) 3-16. 8. A. S. Gilmore, Jr., Microwave Tubes, Artech House Inc., 1986. 9. Y. Yamaoka, S. Kanemaru, and J. Itoh, J. Appl. Phys. 35 (1996) 6626-6628. 10. D . D. Coon and H. C. Liu, Appl. Phys. Lett. 47 (1985) 172-174. 11. H. Shen, F. H. Pollak, and R. Tsu, "Optical properties of quantum steps", Appl. Phys. Lett. 57 (1990) 13-15. 12. H. Mimura, Y. Abe, J. Ikeda, K. Tahara, Y. Neo, H. Shimawaki, and K. Yokoo, J. Vac. Sci. Technol. B 16, 2 (1998) 803-846. 13. R. Tsu, J. Non-Crystalline Solids 114 (1989) 708-710. 14. P . D. Mumford and M. Cahay, J. Appl. Phys. 79 (1996) 2176-2179. 15. I. Brodie and C. A. Spindt, Adv. Electro. Electron Phys. 83 (1992) 1-106. 16. S. Iannazzo, Solid-state Electron 36 (1993) 30-320. 17. Private communication from William Shelton of Oakridge National Lab. 18. R. Tsu, A. Filios, C. Lofgren, K. Dovidenko, and C. G. Wang, Electrochem and Solid State Lett. 1 (1998) 80-82. 19. R. Tsu and R. F. Greene, DARPA HERETIC Principal Investigators' Meeting, Atlanta, GA, May 23-25, 2001. 20. V. T. Binh and C. Adessi, Phys. Rev. Lett. 85 (2000) 864-867. 21. To be published.

162

International Journal of High Speed Electronics and Systems, Vol. 12, No. 4 (2002) 1101-1133 © World Scientific Publishing Company

T H E P H Y S I C S OF SINGLE ELECTRON T R A N S I S T O R S

M. A. KASTNER Department of Physics, Massachusetts Institute of Technology Cambridge, MA 02139, USA

The single electron transistor (SET) is a nanometer-size device that turns on and off again every time one electron is added to it. In this article, the physics of the SET is reviewed. The consequences of confining electrons to a small region of space are that both the charge and energy are quantized. We review how the charge states and energy states of the confined electrons, sometimes called an artificial atom, are measured, and how the precision of these measurements depends on temperature. We also discuss the coupling of electrons inside the artificial atom to those in the leads of the SET, which results in the Kondo effect. We review measurements of the Kondo effect, which demonstrate that the Anderson Hamiltonian provides a quantitative description of the SET. Keywords: Single Electron Transistor, Nanoelectronics, Artificial Atom, Mesoscopic Physics.

1

Introduction

The interplay of technology and science is the engine that has powered economic growth and improved living conditions for half a millennium. New scientific dis­ coveries have led to new technology, and new technology makes possible new discoveries. The discovery of the transistor, for example, has led to the in­ formation revolution. In turn, the technology developed to make very small transistors has given birth to nano-science. The topic of this chapter is one example of the new science we have discovered by using the techniques created by the electronics industry to confine electrons to nanometer dimensions. At the end of the 1970's many physicists began asking whether electrons would behave in unusual ways if they were constrained to move in one or two dimen­ sions. The field effect transistor accomplished two-dimensional confinement in a natural way. I h e electrons in the inversion layer at the surface of Si have a single wave-function for motion perpendicular to the surface at low temper­ ature, making their motion strictly two-dimensional. Studies of the resulting two-dimensional electron gas (2DEG) in strong magnetic fields led to the dis-

163

1102

M. A. Kastner

covery of the quantum Hall effect. The invention of modulation doping, in an effort to make higher speed electronics, made possible very high mobility 2DEGs and led to the discovery of the fractional quantum Hall effect. Constraining electrons to move in one instead of two dimensions revealed a host of new phenomena. Making very narrow wires revealed the behavior called weak localization, and showed that the interactions between electrons also changes. Closely related is the non-statistical behavior called universal conductance fluc­ tuations. [1] Given that restricting electrons' motion in one or two dimensions had such dramatic consequences, it is not surprising that confining them in all three di­ mensions also led to new physics. Physicists showed that the confinement leads to the quantization of charge. As a result, a transistor made by confining elec­ trons to a small region of space, isolated from metallic leads by tunnel junctions behaved in a very unusual way. Whereas a conventional field effect transistor turned on only once when electrons are added to it, the new kind of transistor turned on and off again with the addition of only one electron. This is why it was called a single electron transistor (SET). However, the confinement not only quantizes the charge but also the energy of the electrons. Thus, the confined electron system behaves like an atom, with quantized excitation energies. In recent years we have extended the analogy to atoms even further. We have found that the interaction between the confined electrons in the SET and those in its leads is quantitatively described by a theory designed for magnetic impurities in metals. Thousands of papers have been written about SETs and related structures called quantum dots. I do not propose to review this literature. Rather I present the background that someone new to the field would need to begin reading the literature. Several reveiws contain much of the material presented here [2, 3, 4, 5, 7, 6] At the end of this chapter, I provide a brief history of the SET.

2

Quantization of Charge and Energy

The simplest SET to understand is the kind made first, by Fulton and Dolan. [8] It consists of a small aluminum particle, which is oxidized. Two aluminum leads are deposited on opposite sides of the particle, so electrons must tunnel, quantum mechanically, to move from one lead, onto the metal particle, and onto the other lead. The entire structure is created on an insulator, below which there is a metal gate. Fulton and Dolan used heavily doped Si as the gate metal and a thermally grown oxide as the insulator. To understand the behavior of the metallic SET, consider first a single tunnel junction, shown in Fig. la. With a small bias voltage Vds applied, the Fermi

164

The Physics of Single Electron Transistors 1103

energy in the left lead is higher than that in the right lead by eVas. We call this bias the drain-source voltage to be consistent with transistor nomenclature. The current through the tunnel junction is given by I = j T(E)[f(E)

- f(E - eVds)}dE

(1)

where T(E) is the transmission coefficient for the tunnel barrier at energy E and f(E) = [exp(E - n)/kT + 1 ] _ 1 is the Fermi-Dirac distribution function. The term involving —f{E — eVds) results from electrons tunneling from right to left; it offsets part of the tunneling from left to right. For metallic leads, T(E) is a slowly varying function of E, so one obtains a conductance that varies slowly with Vds. If there are two tunnel junctions in series, with a large piece of metal

Figure 1: a) A single tunnel barrier between two metal leads. The current flows when a small drain-source voltage is applied, b) Double tunnel barriers. If the central metal particle is ve:y large, this behaves like two single barriers in series, but when the particle is small, tunneling is prohibited because of the Coulomb gap.

165

1104

M. A.

Kastner

between them as in Fig. lb, little changes. However, if one has a small metal particle between two tunnel barriers, also shown in Fig. lb, the conductance is radically different. The transfer of an electron from the left lead onto the metal particle results in charging of the capacitances of the two tunnel junctions, and this costs energy. Classically, the energy cost is U/2 = e2/2C where C is the total capacitance of the two junctions. If the metal particle is small, C is also small, and U can become larger than kT. One can readily make particles with sizes of order 100 nm. Using the expression for the capacitance of a sphere of this size, it is clear that the capacitances can be of order 1 0 - 1 6 farad, giving charging energies of order meV. This charging energy gives rise to a gap in the transimission spectrum T(E). Since it costs energy U/2 to add an electron, it also costs energy U/2 to remove one, or to add a hole. Thus, there is a gap in the spectrum of size U, as illustrated in Fig. lb. At temperatures for which kT -C U and at voltages for which eV
(2)

where Vs is the voltage on the gate electrode, Cg is the capacitance bewteen the the gate and the metal particle and C is the total capacitance between the par­ ticle and all electrodes nearby. Equation 2 describes a parabola with minimum at Qo, the charge that would reside on the particle, were the charge a continuous variable. However the charge is quantized because of the confinement, so that only discrete values of the charge, equal to multiples of e, are allowed. These are the points on the parabolas in Figure 2. When Qo = Ne the minimum energy conincides with one of the allowed values of the charge, as illustrated in the left­ most digagram in Fig. 2. As Vs is increased, the minimum shifts, but the lowest energy allowed state does not change at first; it still correponds to N electrons. As shown in the second diagram from the left, the energy to add an electron is

166

The Physics of Single Electron Transistors

1105

now less than the energy to remove one. However, when Qo = (AT + l/2)e the state with N electrons has the same energy as the state with N + 1 electrons. At this value of Vs the charge can fluctuate with no cost in energy, so current can flow at zero temperature. We have discussed this so far by ignoring the quantization of the energy resulting from confinement of electrons in a small region of space. Actually, the same condition that leads to the quantization of charge also leads to quantization of energy, as discussed below. In Fig. 2 we have sketched energy levels on the isolated particle. The charge can only fluctuate when one of the energy levels of the particle coincides with the Fermi energies in the leads. More precisely, the iVth peak in the conductance occurs when the state of the particle containing N electrons is degenerate with the state containing N + 1

Q0=-We Q0= - ( N + l / 4 ) e Q„=-(N+l/2)e Q0=-(N+3/4)e

(N+l)

N

Charge

Increasing Gate Voltage Figure 2: Top: total energy as a function of charge for four values of the gate voltage. Were the charge on the artificial atom continuous, the charge would have the value Qo- Instead, the allowed values of the charge are shown as points on the parabolic energy curves. Bottom: energy level spectrum for the corresponding gate voltages. There is a gap for tunneling at the Fermi energy (dashed line) for every case except Qo = ~(N + l/2)e, for which current can flow at very small bias even at zero temperature.

167

1106

M. A.

Kastner

Figure 3: Schematic drawing of a field-effect transistor and an SET. Wires are connected to source and drain contacts to pass current through the 2DEG at the GaAs/AlGaAs interface. Wires are also connected to the confining electrodes t o bias them negatively and to the gate electrode that controls the electrostatic energy of the confined electrons.

electrons. Were the gate the only electrode contributing to the electrostatic energy of the droplet, the gate voltage at which the Nth peak occured multiplied by the charge of the electron e would be the energy difference between the two states. Since there are several electrodes near the droplet, the energy change caused by Vg is aeVg where a = Cg/C. Therefore, a conductance peak occurs when aeVg{N) = E(N + l)-E{N), (3) apart from a constant, where E(N) is the total energy of the droplet with N electrons. A schematic of one kind of SET is shown in Figure 3. Part of the confinement comes from the use of a modulation-doped field-effect transistor shown in the upper part of the figure. The active layer of GaAs is separated from the metal gate electrode by an insulator, in this case AlGaAs. The AlGaAs is doped with Si, which donates electrons. These fall into the GaAs, because their energy is lower in the latter material. The resulting positive charge on the Si atoms creates a potential that holds the electrons at the GaAs/AlGaAs interface, creating a two dimensional electron gas (2DEG). The source and drain contacts allow one to drive electrons from an external circuit through the 2DEG. To make an SET one needs to confine the electrons in all three dimensions. The 2DEG is confined perpendicular to the GaAs/AlGaAs interface by the strong electric field of the donors; as shown in the second panel of Figure 3, the confinement

168

The Physics of Single Electron Transistors

1107

Figure 4: Electron micrograph of the top surface of the SET used in the exper­ iments of Goldhaber-Gordon et al. [9, 10].

in the other two directions is accomplished with electric fields imposed by very small confinement electrodes. Figure 4 shows an electron micrograph of such confinement and gate electrodes for one of the smallest SETs made in this way so far. [9, 10] The region surrounded by electrodes appears to be a few hundred nanometers in diameter. However, the droplet of electrons confined in it is considerably smaller. We estimate that these SETs have about 50 electrons confined to a droplet about 100 nm in diameter. The GaAs/AlGaAs structure is grown with molecular beam epitaxy. The electrodes are fabricated using electron beam lithography. A negative voltage on the electrodes creates a potential similar to the one sketched in Figure 5; the negative voltage repels electrons from underneath the confinement electrodes and creates saddle point potential barriers under the constrictions. For the remainder of our discussion, we assume that the voltage on these constriction electrodes is fixed, resulting in a fixed confinement potential. However, the voltage on an additional electrode, the gate, is varied to adjust the potential of the electrons confined in the potential well. When the voltage on the gate electrode is increased, the potential minimum, in which the electrons are trapped, becomes deeper. This causes the number of trapped electrons to increase. However, unlike a conventional transistor, in which the charge increases continuously, the charge in the trap increases in discrete steps, and this is reflected in the conductance between source and drain. Figure 6 shows the conductance as a function of gate voltage Vg for a SET made by U. Meirav [11], larger than the one in Figure 4. The conductance

169

1108

M. A.

Kastner

Figure 5: Sketch of the electrostatic potential energy experienced by an electron moving at the interface between GaAs and AlGaAs in Fig. 1.

is measured by applying a very small voltage Vds between drain and source, small enough that the current is proportional to Vds. As seen in the figure, the conductance increases and decreases by several orders of magnitude almost periodically in Vg. A calculation of the capacitance between the gate electrode and the droplet of confined electrons shows that the voltage between two peaks or two valleys is just that necessary to add one electron to the droplet. From our previous discussion, it should be clear that the nearly periodic conductance peaks result from the quantization of charge. Indeed, at zero temperature the charge on the isolated region between the tunnel barriers increases by e every time the gate voltage increases enough to pass through one of the peaks. This is illustrated in Figure 7, which shows how the charge would increase with Vg at zero tempearture. But under what conditions is charge quantized? This may seem like a silly question. Since the discovery of the electron we have known that its charge is quantized. However, because the wave functions of electrons in conductors are extended over macroscopic distances, the charge in any small volume is not quantized. It is the localization of electrons to a small region of space that quantizes their charge. However, the degree of localization depends on the transmission of the tunnel barriers. A very elegant argument tells us how resistive the tunnel barriers must be for charge to be quantized. One simply demands that the RC time constant for an electron to tunnel off the droplet into the leads be great enough that the energy uncertainty is less than the charging energy. If the tunneling resistance is R, this condition is RC > h/U or approximately R > h/e2, the fundamental unit of resistance that enters,

170

The Physics of Single Electron Transistors 1109

0.03

130

Figure 6: Conductance of a SET as a function of the gate voltage. The spacing between the peaks is the voltage necessary to add one electron to the artificial atom. These results are for a device of the kind studied by Meirav et al. [11]

3e 2e

Figure 7: Sketch of the charge on the artificial atom as a function of gate voltage. The charge increases by one electron every time a conductance peak (Fig. 6) is passed.

171

1110 M. A. Kastner

for example, in the quantum Hall effect. Thus, while the calculation of the charging energy is entirely classical, Planck's constant determines whether the charging energy is present or not. This condition is valid at T = 0, independent of C and therefore of the size of the artificial atom. Of course, thermal charge fluctuations can overcome this localization, so charge quantization is observable only at temperatures kT < U, which means that it is easier to see the effects in smaller artificial atoms, which have larger U. As already mentioned, in addition to charge quantization, energy quantization is important when electrons are confined to small volumes. Interestingly, the criteria for charge and energy quantization at T = 0 are exactly the same. Whereas U is the energy to add an extra electron to the artificial atom, there is a typical level spacing Ae necessary to excite the artificial atom with fixed number of electrons. Furthermore, the levels of the artificial atom are not perfectly sharp, but rather have typical width T. The level width is caused by lifetime broadening, because an electron in a level on the artificial atom can tunnel into the leads and, therefore, does not remain on the artificial atom forever. Alternatively, one can say that the eigenstates of the system are mixtures of localized states on the artificial atom and extended states in the leads. Clearly, energy quantization means that Ae > T. Following Thouless [12] one argues that the current through the SET for a single quantum level is the charge of the electron divided by the time t for an electron in a single quantum state t o traverse the artificial atom while in that level. If ^ is the density of states in the artificial atom, then ^ e V d s is the number of current-carrying channels between the Fermi energy in the source and that in the drain. Thus, the current is given by

The width gives the traversal time, t = h/T and ^ - = 1/Ae, so the condition for energy quantization is R = Vds/I > h/e2, the same as for charge quantization. However, while the conditions for charge and energy quantization at zero tem­ perature are the same, charge quantization often survives to higher tempera­ tures. The charge „quantization can be observed when kT < U, but energy quantization requires kT < Ae. Since U > Ae for most SETs made to date, en­ ergy quantization is more difficult to observe than charge quantization. Energy quantization can be observed by measuring the variations between peak posi­ tions for data like those in Figure 6. Alternatively, the energy level spectrum can be measured directly by observing the tunneling current at fixed Vg as a function of Vas- Suppose we adjust Vg so that Qo = —Ne (Fig. 8a) and then be­ gin to increase Vds(Fig. 8b). The Fermi level in the source rises in proportion to Vds relative to the drain, so it also rises relative to the energy levels of the artifi­ cial atom. Current begins to flow when the Fermi energy of the source is raised just above the first quantized energy level of the atom. As the Fermi energy is

172

The Physics of Single Electron Transistors 1111

a

u $ Ae

Figure 8: Sketch of the effect of a finite drain-source voltage on energies of an SET. (a) Kb = 0, (b) finite Vds

173

1112

M. A.

Kastner

raised further, higher energy levels in the atom fall below the Fermi energy, and more current flows because there are additional channels for the electron to use for tunneling onto the artificial atom. One measures the energies by measuring the voltage at which the current increases, or, equivalently, the voltage at which there is a peak in the derivative of the current, the differential conductance, dI/dVds. One of the most beautiful experiments of this kind is shown in Figure 9. [13] This is for an SET made in a very different way from that in Fig. 4, such that the confining potential is almost perfectly circular. The diamonds with very low differential conductance are regions within which only one charge state is stable. Data like those in Figure 6 would be obtained by moving along the vertical axis at Vas = 0. The boundary of the diamonds corresponds to the threshold for changing the charge of the artificial atom. One can overcome the charging energy by changing the source-drain voltage as well as by changing Vs. The diagonal lines outside the diamonds correspond to excited energy levels of the artificial atom. In this case, the artifical atom is so small that Ae ~ U so the peak spacings at Vds = 0 are far from constant and reflect the shell structure of the artifical atoms. One can even see the effects of exchange, that is, the filling follows Hund's rule, making certain values of TV more stable than others. [13] So far we have identified three different energy scales that are important in understanding SETs: U, Ae and V. These are the typical energies, respectively, t o add an electron to the artificial atom, to excite the artificial atom with a fixed number of electrons and the broadening of the artifical atom's energy levels by quantum mechanical tunneling to the leads. Although the Coulomb blockade model is often adequate to estimate U, we know that it, like Ae and Y has a quantum mechanical origin. These three energies result in the density of states sketched in Figure 10. On the left is the energy level structure assuming V = 0 for a case in which Ae > U. To our knowledge, no SET has been made in this limit yet, but it is, in principle, possible, and it makes the physics simpler to explain. On the right is the density of states with the levels broadened by the interaction between the artificial atom and the leads. In most cases the transmission coefficient T in Equation 1 is proportional to the density of states, so the broadening of the levels can be observed directly. Figure 11 shows data of Foxman et al. for a range of gate voltages over which the coupling to the leads changes considerably. The reason is that as the gate voltage is increased, its capacitive coupling to the constrictions (see Figs. 3) causes the tunnel barriers to be reduced. In the limit that Vds "C kT, [f(E) — f(E — eVds)] equals —eV
£_|dCT(£)£2fW^)

2

where the inverse cosh comes from df/dE.

174

(5)

Thus, the conductance is the con-

The Physics of Single Electron Transistors

1113

Figure 9: Differential conductance on a gray scale as a function of both gate and drain-source voltage. The dark diamonds correspond to regions in which there is a gap to current flow. For a SET made of metal all diamonds would have identical size and there would be no variations of conductance outside the diamonds. Semiconductor SETs have diamonds of different sizes and peaks in differential conductance outside the diamonds, corresponding to excited states. These results are from Kouwenhoven et al. [13] who have made SETs so per­ fect that shell structure gives more stability (larger diamonds) for the electron numbers indicated.

175

1114

M. A.

Kastner

1

I

u

Ae n

u g(E)

Figure 10: Schematic energy level diagram for isolated artifical atom (left) and resulting density of states when tunneling to the leads is allowed (right). Filled states are shaded and empty states are not. In the case illustrated the level spacing is greater than the charging energy. The various energies are described in the text.

volution of the zero-temperature line shape T{E) with the derivative of the Fermi-Dirac distribution function. When the intrinsic width of the resonance is much less than kT, T{E) may be replaced by a delta function so that the line shape is simply the derivative of the Fermi distribution. If the line shape is Lorentzian a better description is [14] T(E) =

7rr/2 (r/2)2 + [(eaVg - Eres) - Ef

176

(6)

The Physics of Single Electron Transistors

1115

Figure 11 shows conductance as a function of Vg on a logarithmic plot. As the gate voltage is increased, it not only induces the addition of charge to the artificial atom, but it also reduces the size of the potential barriers that limit the tunneling rate. As a result, T increases with Vg. Figure 11 also shows two peaks on an expanded horizontal scale. The peak at lower Vg is so narrow that the derivative of the Fermi-Dirac distribution provides an excellent fit to the data over the entire peak. However, for the peak at higher Vg the FermiDirac distribution is adequate only near the center of the peak. In the tails, the Lorentzian dominates. The fit shown, using Eq. 6 in Eq. 5, provides an excellent fit over the entire temperature range. The peak height of the derivative of the Fermi-Dirac distribution decreases as T _ 1 , and the width increases as T. Thus the peak conductance of a narrow peak in the single-level regime decreases with increasing temperature and the width increases. Calculating the full width at half maximum (FWHM) from Equations 6 and 5 one finds that the FWHM is 3.5fcT+r so the natural line width may be determined by measuring the FWHM as a function of temperature. An example is shown in Figure 12. [10] Equation 5 is appropriate when kT < Ae. At higher T, the current is carried by multiple levels of the artificial atom rather than a single one. It turns out that the line shape in this multi-level transport regime is indistinguishable from df/dE, but the temperature dependence is quite different. As mentioned already the contribution of a single level decreases as T _ 1 . However, the number of levels contributing to the current in the multi-level regime increases as kT, so the peaks for multi-level conductance have a height that is independent of T. While the width in both regimes increases linearly with T, there is a subtle change in slope at the crossover from single to multilevel transport. [16] SETs made with metals are usually in the multilevel transport limit because the density of states at the Fermi energy is very high compared with semiconductor SETs. Thus, there are four energy scales that are important in determining whether the charge and energy are quantized, T, e, U, and, of course, kT. Both ernergy and charge quantization require that the level width of the artificial atom be less than the level spacing, i.e. T < Ae. This is the same as the condition that the conductance be less than e2/h. However, this condition is the dominant one only at zero temperature. As T is increased from zero, the conductance is dominated by a single level and the effects of energy quantization can be observed, in differental conductance, for example. However, when kT exceeds Ae one crosses over to the multilevel regime, in which the energy fluctuations are

177

1116 M. A. Kastner

280

285

290

295

300

10-l

10 JS

%> 10~3 o 10

10

282.5

282.9 291.0 Vg (mV)

292.2

Figure 11: (a)Zero-bias conductance of an SET as a function of gate voltage on a log scale, from Foxman et al. [14]. (b) At low Vg the zero-temperature width of the resonances are so small that the line shape is well-described by the Fermi-Dirac distribution function (solid line), (c) At high Vg the tunnel barriers are lower, T is larger and the Lorentzian tails of the peaks can be seen. The solid curve in (c) is a fit using Equations 5 and 6. These data were taken in a magnetic field of 2.53T. The peak-height alternation results from spin splitting, as discussed by Klein et al. [15]

178

The Physics of Single Electron Transistors 1117

Figure 12: Full width at half maximum of a conductance peak as a function of temperature. These data are from Reference [10].

large, but the charge fluctuations can still be small. Thus, energy quantization is lost at kT ~ Ae, but charge quantization persists until kT ~ U.

3

Kondo Effect

As we have seen, thinking of the confined droplet of electrons in a SET as an artificial atom is very helpful. However, the coupling of the states in the artificial atom to those in the leads, would, at first, appear to destroy the analogy. In fact, the Anderson model, designed to explain the coupling of natural atoms to metals, provides a quantitative description of the coupled electronic system that makes up the entire SET, including the leads [17, 18, 19, 20, 21]. The Anderson model was developed to address one of the most challenging problems of 20th century physics: the behavior of a metal containing a magnetic impurity. At high temperatures the spin of the impurity is independent of the spins of the electrons in the metal, but at low temperatures many-body correlations lead to a singlet state, in which the spin of the impurity is screened by those of the conduction electrons. The incipient formation of this singlet as the temperature is lowered results in strong scattering of the conduction electrons near the Fermi energy and a consequent increase in the resistance.

179

1118

M. A.

Kastner

How the singlet state evolves with temperature and the consequences of this evolution for magnetization and conductivity is called the Kondo problem. Because the coupling between the localized electron and those in the metal is strong for electronic states with a range of energies throughout the conduction band, the Kondo problem could not be solved with perturbation theory but re­ quired the invention of renormalization group theory. Indeed, Wilson points out the importance of the Kondo problem in his Nobel address.[22] However, when theory became adequate to account for the experiments on magnetic impurities in metals, it also made predictions that could not be tested with those systems. In particular, there were several parameters in the Anderson Hamiltonian to which the theory was very sensitive. For magnetic impurities these were deter­ mined by the properties of the elements involved and were therefore fixed for a particular chemical composition. It appeared, therefore, that some aspects of the theory could never be verified. Beginning in the late 1980's theorists proposed that the Kondo effect should also arise in nanometer-sized structures that allow tunneling between localized states and metal leads. [17, 18] They argued that were a localized state to contain an unpaired electron, it would be analogous to a magnetic impurity, and the electrodes would be analogous to the surrounding metal. It was predicted that because scattering increased rather than reduced transport for tunneling, the Kondo effect would increase the conductance instead of the resistance at low temperature. With this modification, the powerful theoretical machinery that was invented to explain magnetic impurities in metals could be used to predict the conductance in resonant tunneling through localized states. Following the invention of SETs, detailed predictions were made about the mod­ ification of their characteristics by the Kondo effect. [20, 21] However, many years passed without experimental realization of the phenomenon. The first clear demonstration was the work of Goldhaber-Gordon et al. [9], who made the effect dramatic by making smaller SET's than ever before. Soon thereafter several other groups observed Kondo phenomena in larger SETs.[23, 24, 25, 26] It now appears that Kondo physics occurs very often in nanostructures. It has been seen in SETs that are relatively large as well as small; it has been seen for the standard case, in which the degeneracy causing the effect results from spin 1/2, as well as for cases in which the degeneracy is threefold [27]; and it has been seen in carbon nanotubes[28] as well as semiconductor nanostructures. The Kondo effect is more easily observable when the coupling between the artifi­ cial atom and the leads is strong, that is, when F is large. We vary F by adjusting the voltages on the gates that form the constrictions. For large enough T and at low enough temperature, we find that the conductance peaks occur in pairs as seen in Fig. 13. For an SET in which the artificial atom is sufficiently small and has low sym-

180

The Physics of Single Electron Transistors

1119

Figure 13: Conductance as a function of gate voltage when the tunnel barriers are adjusted to be more open than in Fig. 6. The pairing of the peaks is the consequence of the Kondo effect.

metry, all spatial degeneracies are lifted. For this case one would expect that increasing V& would cause electrons to enter successively higher-lying energy levels, two electrons (of opposite spin) for each level. Although each level can accommodate two electrons the Coulomb interaction results in an energy cost U to add a second electron when a level is singly occupied. Thus, one expects the typical energy to add an electron to alternate between U and U + Ae. As we show below, the pairing of peaks in Fig. 13 is primarily the result of the Kondo effect. Nonetheless, the two closely spaced peaks in a pair result from the same localized spatial state, and we next focus on one such pair of peaks. The theory of the SET, including the Kondo effect, is based on the Anderson model, in which the SET is approximated as a single localized state coupled by tunneling to two electron reservoirs.[20] The state can contain zero, one, or two electrons. Adding the first electron takes an energy eo referenced to the Fermi energy in the leads, but the second electron requires eo + U. Tunneling of electrons from the localized state into the leads gives rise to broadening of the localized-state energies, with full width T. Making the voltage 7 g on a nearby electrode (the

181

1120

M. A.

Kastner

-f»/r

10

0.5

0.4--

0.3 -

- lOOmK — 200mK - 300mK - 500mK -*- 800mK - lOOOmK - 1500mK - 2200mK - 3000mK - 3800mK

0.2

A

k .L f,\f \1 1 ♦

-

M Al

-

A

Ail M

ikll\

i

m

-

b-

W\\ ~v *_

■ ,

140

-130

-120

-110 -100 V, (mV)

^

-90

Figure 14: Conductance versus plunger gate voltage Vg (or eo/T) at various temperatures. The vertical dashed lines mark gate voltages at which two charge states are degenerate. Between the dashed lines the number of electrons confined in the SET is odd, and the Kondo effect enhances the conductance. Figure 12 shows the width versus temperature for one of the peaks at high temperatures; the extrapolation to T = 0 gives T. (From Reference [10])

plunger gate in Fig. 4) more negative increases eo. Thus, with a change in Vg one can change from the empty orbital regime, eo > 0, to the mixed valence regime, eo ~ 0, to the Kondo regime, eo < 0. Each of these regimes can be seen in conventional systems only by finding the right impurity and the right host metal, but in the SET we can explore all of them in a single experiment. As mentioned above, current flow is possible at T — 0 only when two charge states of the droplet are degenerate, i.e. eo = 0 or eo + U = 0. Two of these charge degeneracy points occur at the values of Vg marked by vertical dashed lines in Fig. 14; the left degeneracy point corresponds to a change from even to odd occupancy of the level, and the right one to a change from odd to even. At high temperatures all conductance peaks broaden with increasing T, as seen in Fig. 14. The data of Figure 12 are obtained from those in Fig. 14. Using Figure 12 we determine the conversion between gate voltage and energy, allowing us to determine eo and U from the peak positions. Furthermore, as discussed above, the plot of width versus T gives a straight line that can be extrapolated to T = 0 to give T.

182

The Physics of Single Electron Transistors

1121

Figure 15: Schematic energy level d i a g r a m for an SET. T h e states below the Fermi energy Ep in the drain and source are filled. A singly occupied level at eo gives rise to the sharp Kondo resonance in the density of states a t Ep. T h e doubly occupied level is higher t h a n eo by U. T h e tunneling between the localized states a n d the leads gives the density of states resulting from each level a width r .

T h u s , as T decreases from high t e m p e r a t u r e , as seen in Fig. 14, the reduction in thermal broadening results in a decrease in conductance between the peaks. However, theory predicts t h a t a t sufficiently low T the conductance increases again because of the Kondo effect. T h e coupling between the localized spin and the spins of the electrons in the leads gives rise, a t T = 0, to a sharp peak in the density of states a t the Fermi energy. This peak is illustrated in Figure 15. There is one such peak for each lead, resulting in a set of states at the Fermi level strongly coupled from one lead, t h r o u g h the localized state, to the other lead. At T = 0 this coupling is predicted t o give rise to perfect transmission (for symmetric coupling t o the two leads) a n d , consequently, a conductance equal to 2e 2 //i.[18] As T is increased from zero the conductance is predicted to decrease, approximately logarithmically, when T ~ TK, where the Kondo t e m p e r a t u r e Tx is a measure of t h e binding energy of the singlet. This qualitative behavior can be seen in Figure 14. In fact, the theory predicts t h a t the t e m p e r a t u r e dependence of conductance for any Vg between, b u t not too close to, the charge degeneracy points is uni­ versal at low T.[29] Indeed, we find t h a t all d a t a in this region collapse onto the curve predicted by numerical renormalization group theory, as shown in Figure 16. T h e d a t a have been scaled by the Kondo t e m p e r a t u r e TK and the zero-temperature extrapolation of the conductance Go- To carry out the fits

183

1122 M. A. Kastner

0.8

,=0.6 D

f :„/r

= -0.74, T = 280 ueV e 0 /r=-0.91 e„/r=-1.08

0.4

e' 0 /r=-0.98, r = 215ueV 0.2

e 0 /r=-1.00, NRG results

10

10

T/TK

10"

10

Figure 16: Conductance versus temperature for various values of eo on the right side of the left-hand peak in Fig. 14. The solid curve represents the results of numerical renormalization-group calculations. By fitting the latter we extract TK and Go for each set of data. (From Reference [10])

we have constructed an empirical form [10] that provides a good fit to the nu­ merical renormalization group calculations of Costi and Hewson.[29] As shown in Fig. 17, TK and Go determined in this way agree strikingly well with the predictions of theory. For eo -C —T scaling theory [30] predicts TK

,TV£o(eo+U)/rU

(7)

Since eo, U, and T are determined independently, TK is, in principle, predicted with no adjustable parameters. In fact, the exponent in Eq.7 is in excellent agreement with the prediction, but the prefactor differs by a factor ~ 3, which we consider good agreement, given the sensitivity to the exponent. We have adjusted the prefactor for Figure 17. The zero temperature extrapolation of the conductance Go agrees with the form predicted from the Friedel sum rule. [21] Theory predicts that Go approaches the unitarity limit Go = 2e2/h at T = 0 for symmetric tunneling to the two leads. We have normalized the data to allow for lower values of the zero-temperature conductance caused by asymmetry of the tunnel barriers. Figures 16 and 17 are remarkable. The theory was developed to describe the

184

The Physics of Single Electron Transistors

1123

Figure 17: (a) Kondo temperature determined from the scaling of Fig. 16. The solid curve is the prediction of scaling theory, Eq. 7. Inset: Expanded view of the left side of the figure showing the quality of the fit. (b) Values of Go extracted from the data of Fig. 16. The solid line is the prediction of Wingreen and Meir[21]. The values have been normalized to account for asymmetries of the tunnel barriers: Gmax = 0A9e2/h for the left peak and 0.37e 2 //i for the right peak. (From Reference [10])

185

1124 M. A. Kastner

behavior of magnetic impurities in metals, but one could never test the predicted dependence on the parameters £o a n d T because for magnetic impurities these parameters are fixed by the composition of the material. On the other hand, in the SET these parameters are controlled by electrode voltages and can be varied continuously. Thus, with the SET we have verified predictions of the renormalization group and scaling theories that have never been tested before. The data in Fig. 14 are for experiments that are very close to equilibrium: the source-drain voltage Vds is kept less than ksT/e. For a magnetic impurity in a metal, it is impossible to disequilibrate the system, but for the SET it is straightforward. Wingreen and Meir[21] have shown that this makes it possible to explore a new kind of Kondo physics. When the Fermi levels of the two leads are substantially different, there are two peaks in the density of states resulting from the coupling of the spin of the artificial atom, separately, to each of the two leads (see Fig. 15). With the application of a drain-source voltage the Fermi levels are separated, and the excess conductance resulting from the Kondo coupling decreases. This is revealed by measuring the derivative of the current with respect to Vds as a function of Vds- Figure 18 shows the results of such an experiment for the same type of device as studied under equilibrium conditions. The Kondo effect gives rise to a sharp peak at zero bias. Thus, an electrochemical potential difference between the leads destroys the Kondo coupling. Surprisingly, a magnetic field restores it. In a magnetic field, the peak in the density of states splits into two, one above and one symmetrically below the Fermi energy, one for spin up and one for spin down. When one of these peaks for the left lead lines up in energy with the Fermi energy in the right lead, or vice versa, enhanced conductance is observed. Figure 18 shows the resulting splitting of the differential conductance peak with magnetic field. Goldhaber-Gordon et al.[9] and Cronenwett et al.[23] find that the magnitude of the splitting in magnetic field is in approximate agreement with theory. The differential conductance can be displayed in a different way. This is shown in Figure 19 where dl/dV^s is plotted on a gray scale as a function of both Vds and V^. The broad diagonal bands correspond to Coulomb charging steps in the conductance. Whenever the combination of gate and drain-source voltages are sufficient to increase the number of electrons on the droplet of the SET such a step occurs and gives rise to a peak in dl/dV. However, the sharp feature near Vds = 0 for odd electron number results from Kondo physics. Unlike the Coulomb charging features, the Kondo peak remains at Vds = 0 when the gate voltage is changed because the peak in the density of states is tied to the Fermi energy in the leads. The peak splits with magnetic field as in Figure 18. We note that the Kondo peak in differential conductance has been observed in other structures, in which the conductance is limited by tunneling through localized states.[49, 32, 33, 34] However, the SET is unique in allowing us to vary the crticial parameters in a single structure.

186

The Physics of Single Electron Transistors 1125

1.0

Zero-bias peak in differential conductance

0.8

90 mK 0T

90 mK 4T

300 mK 0T

90 mK 6T

600 mK 0T

90 mK 7.5 T

0.6 0.4 0.2 1.0 0.8 0.6 T3

0.4 0.2 1.0 0.8 0.6 0.4 0.2

-°0 4 -0.2 0

0.2 0.4 -0.2 Vds (mV)

0

0.2

0.4

Figure 18: Differential conductance as a function of drain-source voltage. The left three panels show that the sharp peak at zero bias decreases rapidly with in­ creasing T. The other panels show that the peak splits with increasing magnetic field.

187

1126 M. A. Kastner

Figure 19: Differential conductance as a function of drain-source voltage and gate voltage on a gray scale. The broad diagonal bands correspond to the peaks for addition of extra electrons and the diamonds centered on Vds = 0 correspond to low conductance regions where the number is fixed (compare Figure 9). The left panel shows the pronounced Kondo peak at zero drain-source voltage when the number of electrons is odd (center diamond) but not when it is even (top and bottom diamonds). The right panel illustrates the splitting of the Kondo peak in a magnetic field.

188

The Physics of Single Electron Transistors

1127

The experimental work has stimulated many theoretical papers containing new predictions that should be tested. The review by Kastner and GoldhaberGordon [6] contains many references to recent work. In particular Glazman et aZ. [36] have predicted that the Kondo effect persists even when the dot is quite open to the leads. They point out that an important role is played by spin-charge separation in the Kondo effect, and that the spin may be quantized even if the charge is not. They predict a non-monotonic temperature depen­ dence of the conductance in this regime. The evolution of the differential conductance with magnetic field has been pre­ dicted by Costi.[38] and by Moore and Wen[37]. Costi, using Wilson's numerical renormalization group method, predicts that there is a critical magnetic field, below which there is no splitting in the nonlinear conductance. This critical field increases with temperature and results in a peak in the conductance as a function of T for high field but not for low field. Moore and Wen predict more complex behavior. They expect that the splitting of the nonlinear con­ ductance peak in a magnetic field is not exactly linear in magnetic field. As a result, the overall width of the double peak, they say, should increase more rapidly than linear with field. They also predict unusual asymmetric peaks once g/iH » kBTK. More generally, the nonlinear conductance peak shape should be measured pre­ cisely at various temperatures. The variety of theoretical papers on this subject shows the widespread interest in this issue. This has not been done before, in part because the base temperature for most measurements is not much lower than TK when eo is well into the Kondo regime. In particular, the measure­ ments of Goldhaber-Gordon et al. had a lowest electron temperature of about 100 mK, although the base temperature of the refrigerator was about 25 mK. Another topic that has attracted great theoretical effort is the effects of AC modulation on the Kondo effect.(See Reference [6] for references.) Measure­ ments have been made at Delft with relatively large SETs.[39]. These show the suppression of the Kondo effect but not the photon sidebands predicted by theory. Clearly this is a fruitful area for study with small SETs. In summary, we now have solid evidence that the Anderson Hamiltonian pro­ vides a quantitative description of single electron transistors, including the sub­ tle Kondo effect. Indeed, it is remarkable that the SET provides a way of tuning the parameters in this many-body Hamiltonian that is not available for natural systems. We expect that studies of non-equilibrium and high frequency Kondo phenomena will provide new and deep insights in the next few years.

189

1128

4

M. A. Kastner

History of the SET

The effects of charge quantization were first observed in tunnel junctions con­ taining metal particles as early as 1968 [40]. Later, the idea that the Coulomb blockade can be overcome with a gate electrode was proposed by a number of authors [41, 42, 43, 44], and Kulik and Shekhter [45] developed the theory of Coulomb-blockade oscillations, the periodic variation of conductance as a func­ tion of gate voltage. Their theory was classical, including charge quantization but not energy quantization. However, it was not until 1987 that Fulton and Dolan [8] made the first SET and observed the predicted oscillations. They made a metal particle connected to two metal leads by tunnel junctions, all on top of an insulator with a gate electrode underneath. Since then, the capaci­ tances of such metal SETs have been reduced to produce very precise charge quantization. The first semiconductor SET was fabricated accidentally in 1989 by ScottThomas et al. [46] in narrow Si field effect transistors. In this case the tunnel barriers were produced by interface charges. Shortly thereafter Meirav et al. [11] made controlled devices similar to those of Goldhaber-Gordon et al. citedavid, albeit with an unusual heterostructure with AlGaAs on the bottom instead of the top. In these and similar devices the effects of energy quantization were easily observed. [47, 14, 48] Only in the past few years have metal SETs been made small enough to observe energy quantization. [49] Foxman et al. [14] also measured the level width T and showed how the energy and charge quantization are lost as the resistance decreases toward h/e2. In most cases the potential confining the electrons in an SET is of sufficiently low symmetry that one is in the regime of quantum chaos: the only quantity that is quantized is the energy. In this case there is a very sophisticated approach, based in part on random matrix theory, for predicting the distributions of peak spacings and peak heights for data like those in Fig. 4. [50, 51, 52] There are challenging problems in this arena that are still unsolved. In particular, there is great interest in how the interplay of exchange and level spacing determines the spin of-a small metal SET. [53] As already mentioned, the data of Figure 9 are for an SET of sufficiently high symmetry that angular momentum in the plane of the 2DEG is conserved, so shell structure is apparent. Another way to eliminate the scattering that destroys angular momentum conservation is to apply a magnetic field perpen­ dicular to the 2DEG. At sufficiently high fields elegant patterns are seen in the single-electron-peak positions as a function of field. [54] The evolution of Coulomb charging peaks with magnetic field have been inter­ preted with various degrees of sophistication, imitating the development of the theory of atoms. First one tries the "constant interaction model"., in which elec-

190

The Physics of Single Electron Transistors 1129

trons are treated as independent except for a constant Coulomb charging energy. This gives only a qualitative picture of the physics. In order to be quantitative, one needs to at least treat the electron-electron interactions self-consistently (analogous to the Thomas-Fermi model) [55], and for some cases one needs to include exchange and correlations. In particular, it is found that electrons in an SET undergo a series of phase transitions at high magnetic field. [15]. One of these is well described by Hartree-Fock theory, but others appear to require additional correlations. The future of research on SETs looks very bright. There are strong efforts around the world to make the artificial atoms in SETs smaller, in order to raise the temperature at which charge quantization can be observed. These involve self-assembly techniques [56] and novel lithographic and oxidation methods [57] whereby artificial atoms can be made nearly as small as natural ones. This is, of course, driven by an interest in using SETs for practical applications. However, as SETs get smaller, all of their energy scales can be larger, so it is very likely that new phenomena will emerge.

Acknowledgements The work at MIT is the result of the heroic efforts and brilliant insights of a wonderful group of students, postdocs and faculty who have collaborated with M. A. Kastner. It has been a great privilege to work with them and to be part of the mesoscopic physics community throughout the world. This work was supported by the US Army Research Office under contract DAAG 55-98-1-0138 and by the National Science Foundation under grant number DMR-9732579.

References [1] P. A. Lee and T. V. Ramakrishnan, "Disordered Electronic Systems", Rev. Mod. Phys. 57 (1985) 287-337. [2] M.A. Kastner, "Artificial Atoms", Physics Today, 46 (1993) 24-31. [3] M. A. Kastner, "The single-electron transistor", Rev. Mod. Phys. 64 (1992) 849-858. [4] U. Meirav and E.B. Foxman, Semiconductor Science and Technology 11 (1995) 255-284. [5] R.C. Ashoori, ""Electrons in Artificial Atoms", Nature 379 (1996) 413-419.

191

1130

M. A.

Kastner

[6] M. A. Kastner and D. Goldhaber-Gordon, "Kondo physics with single elec­ tron transistors", Solid State Communications 119 (2001) 245-252. [7] C. W. J. Beenakker, H. van Houten and A. A. M. Staring, in Single Charge Tunneling edited by H. Grabert and M. H. Devoret, NATO ASI Series B, Plenum, New York,1991 [8] T.A. Fulton and G.J. Dolan, Phys. Rev. Lett. 59 (1987) 109-112. [9] D. Goldhaber-Gordon, H. Shtrikman, D. Mahalu, D. Abusch-Magder, U. Meirav, and M. A. Kastner, "Kondo Effect in a Single Electron Transistor", Nature 391 (1998) 156-159. [10] D. Goldhaber-Gordon, J. Gores, M. A. Kastner, H. Shtrikman, D. Mahalu, and U. Meirav, "From the Kondo Regime to the Mixed-Valence Regime in a Single-Electron Transistor," Phys. Rev. Lett. 81 (1998) 5225-5228. [11] U. Meirav, M.A. Kastner, and S.J. Wind, "Single-Electron Charging and Periodic Conductance Resonances in GaAs Nanostructures", Phys. Rev. Lett. 65 (1990) 771-774. [12] D. J. Thouless, Phys. Rev. Lett. 39 (1977) 1167-1169. [13] L.P. Kouwenhoven, T. H. Oosterkamp, M. W. S. Danoesastro, M. Eto, D. G. Austing, T. Honda, and S. Tarucha, Science 278 (1997) 1788-1792. [14] E. B. Foxman, P. L. McEuen, U. Meirav, N. S. Wingreen, Y. Meir, P. A. Belk, N. R. Belk, M. A. Kastner, and S. J. Wind, "Effects of Quantum Levels on Transport through Coulomb Island", Phys. Rev. B 47 (1993) 10020-10023. [15] 0 . Klein, C. de C. Chamon, D. Tang, D.M. Abusch-Magder, S.-G. Wen, M.A. Kastner and S.J. Wind, "Exchange Effects in an Artificial Atom at High Magnetic Fields", Phys. Rev. Lett. 74 (1994) 785-788. [16] E.B. Foxman, U. Meirav, P.L. McEuen, M.A. Kastner, O. Klein, P.A. Belk, D.M. Abusch and S.J. Wind, "Crossover From Single- Level to Multilevel Transport in Artificial Atoms", Phys. Rev. B 50 (1994) 14193-14199. [17] T. K. Ng and P."A. Lee, "On-site Coulomb repulsion and resonant tunnel­ ing," Phys. Rev. Lett. 61 (1988) 1768-1771. [18] L.I. Glazman and M.E. Raikh, "Resonant Kondo Transparency of a barrier with quasilocal impurity states," JEPT Lett. 47 (1988) 452-455. [19] S. Hershfield, J. H. Davies, and J. W. Wilkins, Phys. Rev. Lett. 67 (1991) 3720-3723. [20] Y. Meir, N.S. Wingreen, and P.A. Lee, "Transport through a Strongly In­ teracting Electron System: Theory of Periodic Conductance Oscillations," Phys. Rev. Lett. 70 (1993) 2601-2604.

192

The Physics of Single Electron Transistors 1131

[21] N.S. Wingreen and Y. Meir, Phys. Rev. B49 (1994) 11040-11052. [22] K. G. Wilson, Rev. Mod. Phys. 55 (1983) 583-600; K. G. Wilson, Rev. Mod. Phys. 47 (1975) 773-840. [23] S. M. Cronenwett, T. H. Oosterkamp, and L. P. Kouwenhoven, "A Tunable Kondo Effect in Quantum Dots", Science 281 (1998) 540-544. [24] J. Schmid, J. Weis, K. Eberl, and K. v. Klitzing, "A quantum dot in the limit of strong coupling to reservoirs" Physica B 256-258 (1998) 182-185. [25] F. Simmel, R. H. Blick, J. P. Kotthaus, W. Wegscheider, and M. Bichler, "Anomalous Kondo Effect in a Quantum Dot at Nonzero Bias", Phys. Rev. Lett. 83 (1999) 804-807. [26] L .P. Rokhinson, L. J. Guo, S. Y. Chou, and D. C. Tsui, "Kondo-Like ZeroBias Anomaly in Electronic Transport through an Ultrasmall Si Quantum Dot", Phys. Rev. B 60 (1999) R16319-R16321. [27] S. Sasaki, S. De Franceschi, J. M. Elzerman, W. G. van der Wiel, M. Eto, S. Tarucha, and L. P. Kouwenhoven, "Kondo effect in an integer-spin quantum dot", Nature 405, (2000) 764-767. [28] J. Nygard, D. H. Cobden, and P. E. Lindelof, "Kondo physics in carbon nanotubes", Nature 408, (2000) 342-346. [29] T. A. Costi and A. C. Hewson, J. Phys. Condens. Matter 6 (1994) 25192558. [30] F. D. M. Haldane, "Scaling Theory of the Asymmetric Anderson Model", Phys. Rev. Lett. 40 (1978) 416-419. [31] D.C. Ralph and R.A. Buhrman, Phys. Rev. Lett. 72, (1994) 3401-3404. [32] J. Applebaum, Phys. Rev. 154 (1967) 633-643. [33] L. Y. L. Shen and J. M. Rowell, Phys. Rev. 165 (1968) 566-577. [34] P. Nielsen, Phys. Rev. B 2 (1970) 3819-3833. [35] H. Schoeller and J. Konig, "Real-time Renormalization Group and Charge Fluctuations in Quantum Dots" Phys. Rev. Lett. 84 (2000) 3686-3689. [36] L. I. Glazman, F. W. J. Hekking and A. I. Larkin, , "Spin-Charge Separa­ tion and the Kondo Effect in an open Quantum Dot", Phys. Rev. Lett. 83 (1999) 1830-1833. [37] J. E. Moore and X. G. Wen, "Anomalous Magnetic Splitting of the Kondo Resonance", Phys. Rev. Lett. 85 (2000) 1722-1725. [38] T. A. Costi, "The Kondo Effect in a Magnetic Field and the Magnetoresistivity of Kondo Alloys", Phys. Rev. Lett. 85 (2000) 1504-1507.

193

1132

M. A.

Kastner

[39] J. M. Elzerman, S. DeFranceschi, D. Goldhaber-Gordon. W. van der Wiel, and L. Kouwenhoven, "Suppression of the Kondo Effect in a Quantum Dot by Microwave Radiation", J. Low Temperature Physics 118 (2000) 375-389. [40] H. R. Zeller, and I. Giaver, Phys. Rev. 181 (1969) 789-799. [41] K. K. Likharev, IBM J. Res. Dev. 32 (1988) 144 [42] D. V. Averin and K. K. Likharev, in Mesoscopic Phenomena in Solids, edited by B.L. Altshuler, P. A. Lee and R. A. Webb, Elsevier, Amsterdam 1991 [43] M. Amman, K. Mullen and E. Ben-Jacob, J. Appl. Phys. 65 (1989) 339-346. [44] L. I. Glazman and R. I. Shekhter, J. Phys. Condens. Matter 1 (1989) 58115815. [45] I. O. Kulik and R. I. Shekhter, Zh. Eksp. Teor. Fiz.68 (1975) 623-640. [Sov. Phys. JEPT 41 (1975) 308-325.] [46] J. H. F. Scott-Thomas, S. B. Field, M. A. Kastner, H. I. Smith, and D. A. Antoniadis, "Conductance Oscillations Periodic in the Density of a OneDimensional Electron Gas," Phys. Rev. Lett. 62 (1989) 583-586. [47] A.T. Johnson, L.P. Kouwenhoven, W. de Jong, N.C. van der Vaart, C.J.P.M. Harmans, and C.T. Foxon, Phys. Rev. Lett. 69 (1992) 1592-1595. [48] J. Weis, R.J. Haug, K. v. Klitzing, and K. Ploog, Phys. Rev. Lett. 71 (1993) 4019-4022. [49] D. C. Ralph, C. T. Black, and M. Tinkham, Phys. Rev. Lett. 78 (1997) 4087-4090. [50] Y. Alhassid, P. Jacquod, A. Wobst, Phys. Rev. B. 61 (2000) R13357R13360 and references therein [51] A. M. Chang, H. U. Baranger, L. N. Pfeiffer, K. W. West, and T. Y. Chang, Phys. Rev. Lett. 76 (1996) 1695-1698. [52] J. A. Folk, S. R. Patel, S. F. Godijn, A. G. Huibers, S. M. Cronenwett, and C. M.. Marcus, Phys. Rev. Lett. 76 (1996) 1699-1702. [53] P. W. Brouwer^ X. Waintal, and B. I. Halperin, Phys. Rev. Lett. 85 (2000) 369-372. [54] P L . McEuen, E.B. Foxman, U. Meirav, M.A. Kastner, Y. Meir, N.S. Wingreen, and S.J. Wind, "Single-Electron Charging and Periodic Con­ ductance Resonances in GaAs Nanostructures", Phys. Rev. Lett. 66 (1991) 1926-1929; P.L. McEuen, E.B. Foxman, J. Kinaret, U. Meirav, and M.A. Kastner, N.S. Wingreen, and S.J. Wind, "Self-consistent addition spectrum of a Coulomb island in the quantum Hall regime", Phys. Rev. B 45 (1992) 11419-11422; P.L. McEuen, N.S.. Wingreen, E.B. Foxman, J. Kinaret, U. Meirav, M.A. Kastner, Y. Meir, and S.J. Wind, Physica B 189 (1993) 70-79.

194

The Physics of Single Electron Transistors 1133

[55] D. B. Chklovskii, BI. Shklovskii, and L. I. Glazman, Phys. Rev. B 46 (1992) 4026-4034. [56] C. B. Murray, D. J. Norris, M. G. Bawendi, J. Am. Chem. Soc. 115 (1993) 8706-8715. [57] Y. Takahashi, M. Nagase, H. Namatsu, K. Kurihara, K. Iwdate, Y. Nakajima, S. Horiguchi, K. Murase, and M. Tabe, Electronics Letters 31 (1995) 136-137.

195

This page is intentionally left blank

International Journal of High Speed Electronics and Systems, Vol. 12, No. 4 (2002) 1135-1145 © World Scientific Publishing Company

CARRIER CAPTURE AND TRANSPORT WITHIN TUNNEL INJECTION LASERS: A QUANTUM TRANSPORT ANALYSIS LEONARD F. REGISTER,1 WANQIANG CHEN,1 XIN ZHENG1 and MICHAEL STROSCIO2 'Microelectronics Research Center Department of Electrical and Computer Engineering, The University of Texas at Austin, Austin, TX 78758 2

Departments ofBioengineering and Electrical and Computer Engineering University of Illinois at Chicago, 851S. Morgan St., Chicago, IL 60091

Hot electron distributions within the active region of quantum well lasers lead to gain suppression, reduced quantum efficiency, and increased diffusion capacitance, greater low-frequency roll-off and high-frequency chirp. Recently, "tunnel injection lasers" have been developed to minimize electron heating within the active quantum well region by direct injection of cool electrons from the separate confinement region into the lasing subband(s) through a tunneling barrier. Tunnel injection lasers, however, also present a rich physics of transport and scattering, and a correspondingly rich set of challenges to simulation and device optimization. In this work, some of the fundamental physics of carrier capture and transport that should be addressed for optimization of such lasers is elucidated using Schrodinger Equation Monte Carlo (SEMC) based quantum transport simulation. In the process, qualitative limitations of the Golden-Rule of scattering in this application are pointed out by comparison. Specifically, a Golden-Rule-based analysis of the carrier injection into the active region of the ideal tunnel injection laser would suggest approximately uniform injection of electrons among the nominally degenerate quantum well states from the separate confinement region states. However, such an analysis ignores (via a random-phase approximation among the final states) the basic real-space transport requirement that injected carriers still must pass through the wells sequentially, coherently or otherwise, with an associated attenuation of the injected current into each subsequent well due to electron-hole recombination in the prior well. Transport among the wells then can be either thermionic, or, of theoretically increasing importance for low temperature carriers, via tunneling. Coherent resonant tunneling between wells, however, is sensitive to the potential drops between wells that split the energies of the lasing subbands and (further) localizes the electron states to individual wells. In this work such transport issues are elucidated using Schrodinger Equation Monte Carlo (SEMC) based quantum transport simulation.

1. Introduction In conventional quantum well lasers, carriers are in injected from the separate confinement layer into the upper energy ranges of quantum wells. They then dissipate energy and fall to the bottom of these wells via phonon emission. Only then are they in a position to contribute to lasing. However, the time required to reach the bottom of the well can be comparable to or larger than the lifetime of the carriers at the bottom of the well prior to recombination. Thus, the carriers may not have enough time to come to equilibrium with lattice, and a hot electron distribution within the quantum wells results [1]. Hot carriers distributions, in turn, lead to gain suppression, reduced quantum efficiency, and increased diffusion capacitance, greater low-frequency roll-off and highfrequency chirp [1,2,3]. Recently, "tunnel injection lasers" have been developed to minimize electron heating within the active quantum well region by direct injection of cool electrons from the separate confinement region into low-energy states of the quantum well active region through a tunneling barrier, where, in particular, phonon

197

1136

L. F. Register et al.

assisted tunnel injection minimizes back injection from the active region to the separate confinement region [1]. Although the physics of carrier transport in QW lasers is already quite complex [2,3], tunnel injection lasers pose still additional challenges to device simulation and optimization. Even in bulk semiconductors, the dominant polar-optical phonon scattering mechanism is anisotropic. In tunnel injection lasers the capture process depends critically on the overlap of localized carrier wave-functions of nominally separate regions and the polar-optical modes of the crystal, themselves localized [4]. Still, while more difficult to evaluate in this heterostructure geometry, one might expect to be able to address these challenges using the Golden Rule of scattering. In this work, however, an alternative, quantum transport approach to issue is taken, and in the process it is demonstrated that a Golden-Rule-based analysis would lead to not only quantitative errors but also qualitative errors pertinent to the basic operating principles of tunnel injection lasers.

2. The Breakdown of Golden-Rule Analysis The quantum well region is designed such that the subbands of the quantum wells are nominally degenerate and, thus, delocalized among the wells. A Golden-Rule-based analysis of the phonon assisted carrier injection into the quantum well active region, therefore, would suggest a corresponding delocalized injection of electrons into the typically multiple quantum wells. For example, such an analysis would predict that the last well in the active region would be filled as quickly and efficiently as the first whether the active region were composed of two wells or—in deliberate exaggeration—a hundred wells. It should be intuitively clear that the qualitative accuracy of this Golden-Rule analysis must break down at some point, and, indeed, that point is when the second well is introduced as will be shown below. That it would be difficult to maintain such degeneracy among multiple wells, is both true and irrelevant to the above point, although is certainly an important additional consideration as also will be demonstrated. Thus, as powerful a tool as the Golden Rule is, it cannot be relied upon in this application ... at least not without great care. In fact, the Golden rule analysis overlooks—in part, via a random-phase approxima­ tion among the final states—the basic real-space transport requirement that injected carriers still must pass through the wells sequentially, coherently or otherwise, with an associated attenuation of the injected current into each subsequent well due to electronhole recombination in the prior well. The required transport among the wells then can be either thermionic, or, of theoretically increasing importance for low temperature carriers, phonon-assisted or coherent tunneling. However, as alluded to above, coherent tunneling between wells is sensitive to even small potential drops that split the energies of the lasing subbands and localize the electron states to individual wells. While bad for transport, such localization again improves the accuracy of the Golden Rule.

3. Schrodinger Equation Monte Carlo In this work these quantum transport issues are addressed using Schrodinger Equation Monte Carlo (SEMC) [5]. SEMC provides a qualitatively and quantitatively accurate, non-perturbative, current conserving treatment of coherent transport and incoherent/ phonon-mediated transport due to real scattering processes, including the dominant process of long-range polar-optical phonon scattering. SEMC already has been used to

198

Carrier Capture and Transport within Tunnel Injection Lasers

1137

study the effects of phase-coherence and phase-breaking on carrier capture by quantum wells [6,7]. The SEMC method is described in detail in Ref. 5; a brief summary is provided here. Phase breaking and energy dissipation within this Schrodinger Equation-based method are modeled via the exchange of probability among oscillator degrees of freedom within a many-body electron-phonon system just as in the true carrier-phonon scattering. For phonon scattering, a set of Schrodinger Equations is defined for the charge carrier corresponding to an "initial" state and many (e.g., 100s or 1000s of) "final" states separated from the initial state by the emission or absorption of one phonon. Coupling potentials between the initial and final states are provided by Monte Carlo sampling of the (spatial correlation functions of the) true carrier-phonon interactions. A probability source to the initial state is provided by an open boundary in the carrier coordinates or coupling to a prior phonon state; probability sinks are provided by open boundaries in the carrier coordinates of both the initial and final states and/or, as required for bound final states in this work, complex "self-energy" potentials in the final states representing still subsequent scattering. The resulting set of equations is solved self-consistently to find the many-body carrier-phonon wave-function from which any physical observable (transmission, reflection and capture probabilities, self-energies/scattering rates, currents in real-space or "phonon-space," etc.) can be obtained. Probability, energy and even phase information are inherently conserved with respect to full-many body system, but with respect to the carrier alone, the interaction is inelastic and phase breaking. This procedure precisely emulates scattering, both real and virtual, in the true carrierphonon system to first-order, and to higher orders within the accuracy of the estimated final-state self-energies. Scattering is neither local in position nor time. Indeed, the calculations of this work are time-independent (propagating) energy eigenstate calculations in the coupled carrier-phonon system. "Initial" and "final" only indicates the direction of probability current flow. This basic procedure can be repeated sequentially to trace carriers through an unlimited number of scattering "events." The old initial state becomes the source, a new intermediate state is selected by Monte Carlo sampling from among the old final states according to the probability flow to/through the final states, and a new set of final states is generated each with its own complex self-energy potentials. Note that SEMC can accurately be referred to as a non-equilibrium Green's function method. The "Schrodinger Equation Monte Carlo" moniker was chosen as an indicator of the unique method of treating scattering within a carrier-phonon wave-function/ Green's function based approach, as opposed to a density-matrix ("two-particle") based Green's function analysis.

4. Model Tunnel Injection Laser Structures The design objective of the tunnel injection laser is to deliver, in particular, "cool" electrons as uniformly as possible among multiple quantum wells for recombination with holes. The goal of this work is to study the essential physics of this process within quantum transport formalism, and compare the results with the predictions of a Golden Rule analysis. To this end, the quantum well/barrier structure to be simulated has been simplified by reducing the number of wells to two, one "leading well" and one "trailing well," between which disparities in the carrier densities and capture rates and the reasons for those disparities may be more readily identified. Two structures are considered, modeled after those discussed in Ref. 1, as illustrate in Fig. 1(a) and (b) and

199

1138

L. F. Register et al.

(a)

n

(b)

39meW

39meV^

Fig. 1. Schematic of a high barrier system. Note that the bound state is approximately one phonon energy below the well.

Table. 1. Well/barrier width w (nm) conduction band edge Ec .effective mass m* (units of m,,) and nonparabahcity y (eV 1 ) for model, tunnel injection laser systems with (a) high and (b) low interwell potential barriers. (a)

sep. conf.

w Ec m*

r

NA 0 0.67 0.61

(b)

sep. conf.

w £= m*

NA 0 .067 0.61

r

injection barrier 3 1000 0.14 0.25

leading well

injection barrier 3 1000 0.14 0.25

leading well

7 -117 .047 1.02

7 -101 .047 1.02

interwell barrier 7 100 .071 0.57

trailing well

sep. conf.

7 -100 .047 1.02

NA 100 .071 0.57

interwell barrier 7 0 .067 0.61

trailing well

sep. conf.

7 -87 .047 1.02

NA 100 .071 0.57

0.2

-j

0.15 0.1

< CD

:-

i

0.05

c Q)

O Q.

^» -j

0 ;

5" 3 in

-0.05

I

-0.1

; -0.15

C 3 O

:

5

10

15 20 25 30 35 40 position (nm)

Fig. 2. Delocalized ground-state subbands of the double quantum well structure of Fig 1(a) and Table 1(a)

200

Carrier Capture and Transport within Tunnel Injection Lasers

1139

parameterized in Tables 1(a) and 1(b), respectively. Both of these structures are designed to have nominally degenerate ground state energies in the two wells 39meV below the conduction band edge of the electron injection side of the separate confinement region. Figure 2 shows the ground-state wave-functions for the structure of Fig. 1(a) confirming the delocalization of the wave-functions (calculated using precisely the same discretization of Schrodinger's equation as employed in SEMC absent the phonon scattering). The only significant difference between the two structures is the height of the barrier between the two wells; one equal to the height of the band edge on the hole injection side of the separate confinement region, the other equal to the height of the band edge on the electron injection side of the separate confinement region. The well depths were adjusted accordingly to maintain the aforementioned degeneracy. Polar optical phonon scattering due to GaAs bulk modes was considered within the active region in this work. Consideration of the full spectrum of interface and confined phonon modes [4] would be necessary ultimately for quantitative accuracy, but not for the discussion of essential physics addressed in this work.

5. Simulation Results and Discussion As noted above, a Golden Rule-based analysis would suggest uniform injection of electrons into the ground states of the two wells from low-energy incident electrons. However, as shown in Fig. 3, there is a decided segregation of the charge after the initial capture/scattering event toward the leading well, and a significant fraction of the charge

>

uf

position (nm) Fig. 3. Carrier distribution (probability density) after the first scattering event as a function of position and well normal component of energy for a thermal distribution of electrons incident from the left, with the conduction band edge shown for reference. The brighter the region, the greater the probability density.

201

1140

L. F. Register et al.

captured in the second well enters hot into the excited state of that well, from higher energy incident electrons passing over the wells. The dim secondary probability distribution peaks near the bottoms of each well slightly above the primary probability distribution peaks are also supplied through scattering from electrons passing over the well, and also represent hot electrons despite being in the ground-state subbands. The differing effective masses and non-parabolicities of the well and barrier materials result in differing confinement energies as a function of total electron energy for the captured electrons. This effect is also primarily responsible for the apparent broadening of the distributions in energy, although there is a component due to true collision broadening. Furthermore, the those electrons that are found in the ground state of the second well at low energies after one scattering event get there by tunneling from the first well after the scattering event, as shown in Figure 4. In short, in contrast to a Golden Rule analysis, essentially all phonon-assisted injection of cool carriers into the quantum well active region is into the leading well. Only via subsequent tunneling or other transport process, do carriers reach the trailing well. Over time, however, the inter-well tunneling will reduce if not eliminate this segregation, as cool carriers filter into the trailing well, as shown in Fig. 5. However, if a small additional voltage bias is applied, here 26mV (1 kBT at 300K) across the active region for this work, the degeneracy of the subbands is lifted and the resonant tunneling process is attenuated, again resulting in strong segregation of the carriers toward the leading well, as shown in Fig. 5. In this latter case, most electrons entering the trailing well must do so over top of the barrier, that is enter hot—even by a Golden Rule analysis. Reducing the barrier height for the structure of Fig 1(b), eliminates much of this problem, however, by allowing penetration of the incident carrier wave-functions all of the way across the active region. As a result, carriers may be captured from the incident states via phonon scattering directly into the second well leading to a nearly uniform distribution of electrons between the two wells after the initial capture process at flat band, as shown in Fig. 6(a), or under additional bias, as shown in Fig. 6(b). Of course, with the lowered barrier height between the wells, all types of subsequent interwell transport are also be enhanced. The low barrier system performed well in another regard. For the high barrier system here and in past simulations of carrier capture for single conventional wells the fraction of electrons injected past the wells without scattering can be quite high [6,7]— ultimately resulting in diffusion capacitance and dark current. However, as seen in Fig. 7, for the low barrier system the fraction of carriers that leak beyond the well prior to scattering is quite low., This results from a combination of efficient capture for low relatively low energy electrons and the offset between the conduction band-edge in the separate confinement region on the electron injection side and that on the hole injection side. Similarly, the tunnel barrier to electron injection should minimize hole transport beyond the active region. Further, in both systems considered here, the tunneling barrier may serve to allow hot electrons injected from the cladding layer more time to cool within the separate confinement region before entering the active region. 6.

Conclusion

A study of the essential physics of carrier capture and transport in tunnel injection lasers has bee performed using a quantum transport approach, Schrodinger Equation Monte Carlo. These results illustrate how tunnel injection lasers can offer advantages over more

202

Carrier Capture and Transport within Tunnel Injection Lasers

1141

(a)

(b)

(c)

Fig. 4. (a) The real space current flow into (and through) the active region in the initial/incident electron state, (prior to scattering) as a function of position and the interface normal component of the electrons energy E„. (b) The current flow from the initial/incident electron state to the final/captured electrons states, that is in "phononspace," which is localized primarily to the leading well, (b) The subsequent resonant real-space coherent tunneling current from the first well to the second. The dark spot in (c) is backwash from toward the center of the leading well from the slightly off-center flow of probability into the well exhibited in (b).

203

1142

L. F. Register et al.

>

position (nm) Fig. 5. The cumulative proability densities within the wells through 10 scattering events (including the incident state) as a function of position and normal component of energy at "flatband"

>

5

position (nm) Fig. 6. The cumulative proability densities within the wells through 10 scattering events as a function of position and normal component of energy under a small additional bias producing a 26 meV potential drop between wells.

204

Carrier Capture and Transport within Tunnel Injection Lasers

1143

(a)

> ■Si-

(b)

>

position (nm) Fig. 7. Carrier distribution (probability density) after the first scattering event as a function of position and well normal component of energy (a) at flat-band, and (b) under additional bias (26mV across active region).

205

1144

L. F. Register et ai.

> uf

position (nm) Fig. 8. Real-space current flow into the active region as a function position and normal component of energy.

conventional lasers by, as intended, lowering the carrier injection energy and by, in addition, reducing leakage currents. In addition, although designed with electron transport in mind, these systems also offer advantages for hole transport as suggested above. However, it also has been demonstrated that a Golden-Rule analysis of capture can be misleading, failing to address essential transport issues such as those that clearly differentiate the model systems of Figs. 1(a) and (b). Indeed, more generally, anytime there is a real-space current flow of significance produced by Golden Rule scattering, the results should be considered with caution [6,7] as further evidenced by this work. It also has been shown that interwell transport may be quite sensitive to the small voltage drops between wells, particularly in high-barrier systems. Design optimization should address these effects as well—although not necessarily requiring as rigorous a transport approach once the essential physics, has been identified. Finally, for quantitative analysis, the richer energy spectrum of the full set of phonon modes [4], as compared to that of bulk modes used in this qualitative work, should be considered along with the electrostatic self-consistency that drives the inter-well potential drops.

Acknowledgments This work was supported by the U.S. Army Research Office and Battelle.

206

Carrier Capture and Transport within Tunnel Injection Lasers 1145

References [1] P. Bhattacharya, Int. J. High-Speed Electronics and Systems 9 (1998) 847-866. [2] M. Grupen and K. Hess, Appl. Phys. Lett. 70 (1997) 808-810. [3] M. Grupen and K. Hess, IEEE J. Quantum Electronics 34 (1998) 120-140. [4] SeGi Yu, et al. J. Appl. Phys. 82 (1997) 3363-3367, and references therein. [5] L. F. Register, Int. J. High-Speed Electronics and Systems 9 (1998) 251-279. [6] L. F. Register and K. Hess, Appl. Phys. Lett. 71 (1997) 1222-1224. [7] L. F. Register, Int. J. High-Speed Electronics and Systems 16 (2000) 365-388.

207

This page is intentionally left blank

International Journal of High Speed Electronics and Systems, Vol. 12, No. 4 (2002) 1147-1158 © World Scientific Publishing Company

T H E I N F L U E N C E OF E N V I R O N M E N T A L EFFECTS O N T H E ACOUSTIC P H O N O N S P E C T R A I N QUANTUM-DOT HETEROSTRUCTURES

SALVADOR RUFO*, MITRA DUTTA*, and MICHAEL A. STROSCIO*>t * Department

of Electrical & Computer Engineering, and t Department of Bioengineering, University of Illinois at Chicago, IL 60607, USA

We present calculations of the acoustic phonon spectra for a variety of quantum dots and consider the cases where the quantum dots are both free-standing and embedded in a selection of different matrix materials — including semiconductors, plastic, and water. These results go beyond previous calculations for free-standing quantum dots and demonstrate that the matrix material can have a large effect on the acoustic phonon spectrum and consequently on a variety of phonon-assisted transitions in quantum-dot heterostructures. Keywords: Quantum dot; acoustic phonons; environmental effects; quantum-dot matrix.

1. Introduction Quantum-dot (QD) heterostructures are used widely in a number of advanced semiconductor devices including quantum-dot semiconductor lasers. In addition, recent investigations of the uses of quantum dots as biological t a g s 1 - 6 and as active electrical contacts to neurons 7 portend extensive applications of quantum dots in biology, bioengineering and medicine. In all of these applications, it is recog­ nized that the confined phonons in quantum dots play a major role in phononassisted transitions as well as in determining the luminescent spectra of quantum dots. In this paper, we present calculations of the frequency spectra of confined acoustic phonons in quantum dots for a number of quantum dots with different heteromaterials. The quantum-dot heterostructures under consideration have been selected in view of their potential electronic, optoelectronic, and biological appli­ cations. In addition, we have determined how the acoustic phonon frequencies in these quantum dots are affected by the presence of different matrix materials. The case of quantum dots in aqueous environments has been considered since it is of great interest in biological applications. It is important to see how these frequencies are affected by the matrix materials because acoustic phonons have been found to influence properties — such as the width of the photoluminescence (PL) — which are critical to many envisioned applications. Acoustic phonons have been reported

209

1148 S. Rufo, M. Dutta & M. A. Stroscio

to be responsible for the width of the lowest exciton transition in nanocrystals. 8,9 These transitions in nanocrystals often have a linear temperature dependence attributed to the thermal excitation of acoustic phonons. 10 In other works, acousticphonon sidebands found in the emission spectra 11 lead to broadening of the PL and influence energy relaxation in QDs. Also, work with PbS QDs by Krauss and Wise shows that the deformation potential coupling of excitons to acoustic phonons in QDs sets a lower limit for the optical line width. 12 Knowledge of the acoustic phonon spectra is therefore Essential to understanding the electronic and optical properties of quantum-dot heterostructures. These acoustic phonon spectra have been calcu­ lated and are presented in this paper for a number of quantum-dot heterostructures of interest in electronic, optical, and biological applications.

2. Basic E q u a t i o n s Lamb's theory of the vibrations of a homogeneous, free-standing elastic body of spherical shape is the basis for the calculations presented in this account. The equation of motion of a three-dimensional body is given by 13 ' 14 : f)2Tt P - ^ = (A + M)V(V-D) +

MV

3

D,

where D is the displacement and A and \x are Lame's constants. For stress-free boundary conditions at the surface of the sphere there are two different types of modes: spheroidal modes and torsional modes. Using this theory, Tamura et al.13 have developed a model for spherical quantum dots under stress-free boundary conditions, and Alcalde et a/.15 extended the theory to obtain the modes for the case of spherical quantum dots embedded in an infinite matrix.

2 . 1 . Free-standing

spherical

quantum

dots

The case of free-standing quantum dots based in Lamb's theory has been used 11 ' 13 ' 14 to model QDs in which the displacement u in the spherical quantum dot at a distance R equal to the radius of the QD is not restricted, and the nomal stress on the surface of the QD is zero. The spheroidal mode is given by the eigenvalue equation 13 :

2 ^ + (,-i,p + ,)[s^si- ( i + ,)]}£^a-i^ 7

J v? ) t 2 - mil ; , +o 2)] M ^ '' - W + (l- 1)(2Z + 1)VJ2 l .+t j[r, 21(1 -i vl)(l :;\ "

where jl? R Q

, and

210

r\ = '

J?-R Ct

= 0,

Environmental Effects on the Acoustic Phonon Spectra 1149

C; and Ct are the longitudinal and transverse sound speeds and they depend on the material properties, R is the radius of the quantum dot, and ujlg is the frequency of the spheroidal mode for the mode In. The torsional mode is given by:

_d_ (JM) dr)

V

0,

I > 1.

)

This mode does not depend on the material properties, and produces oscillation of the spherical quantum dot without dilatation. For both cases, I is the angular momentum. The eigenvalue for n = 0 corresponds to the surface mode for torsional and spheroidal modes, which have large amplitudes near the surface. The modes with n > 1 are generally referred to as the inner modes.

2.2. Spherical

QDs embedded

in an infinite

matrix

Quantum dots may be fabricated in a glass matrix or they may be fabricated as free-standing structures and be immersed in aqueous environments. The influ­ ence of a glass matrix on the spheroidal and torsional modes has been studied by Alcalde et al.15 They obtained the torsional and spheroidal modes by applying the boundary conditions on the surface between the matrix region and the inner region. The spheroidal modes are given by the eigenvalue equation 12 £>^ = 0, where D^ is the determinant of the 4 x 4 matrix and is given by 15 : dn

du

1(1 - 1) _ , , 2 _ , .' + -Jl+1(r) r

}■

■&{**-■ ±^-tjl(r)

1{ c ?'*< (<) +

"°2 ' °3 '

(I-I)

ill

dia = 2 • C\ ■ C\ ■ 1(1 + 1)

di4 =

-2-Cf

efei =

-xf1

i nr q~\JTq (l-l)

-1(1-1) K,(s) -

2pp Kt(q) -

Jlip)

2Kl+1(s)

-Jl+iip)

Kl+1(q)

[(/-I) J , ( r ) - J I + 1 ( r ) r

r V 2r *22

Z23

d24

o/-i2

r*l

i nr s\2~s

-c?

\2(l2-l) 2 p \ L P2

Ki(3) -

s J

Ji(p) +

Kl+1(s)

-Ji+i(p)},

= °2 " \[Tq{ [^-q^1 ~ X] Kliq) + \Kl+^} ' 211

1150

S. Rufo, M. Dutta & M. A. Stroscio

Mr),

^31

d32 = — C 4

■Ki{8),

d3s = 0, d34 = 0, du — 0, d42 = 0, r<2 /-<2 °2 ' ^3

di3

■Mp) >

■Ki(q),

c/44 —

where ,ln

, ,., .In

P = 7HR> ^t

9=

c m

w

,ln

U#

r=-*rR, C] ^l

-,

s=^-R. CT ^l

Here, i denotes the inner (sphere) material and m denotes the matrix material. Jj is the Bessel function of the first kind, and Ki is the modified Bessel function of the second kind. The parameters C\, C2, C3, and C4 are given by: C i = \ —j >

C,=

C2 =

I Xi + 2p,i

and Ci

!Xm + 2fj, 7)

Mi

where p is the density, and A and // are the Lame's constants. The torsional modes are obtained by the eigenvalue equation D^ = 0, where D^ is the determinant of the 2 x 2 matrix 15 :

(i-i) s

D- =

nr

~^\h^Mv) p y Ip Ci-C7 :2 •

nr

- \h-Ji+i(p) (I-I) y lp Ki{q) -

Kl+1(q)

-MP) '^-Kl(q) 2q

where p, q, C\, and Ci have the same values as for the spheroidal case. 3. R e s u l t s In these calculations, QD materials have been selected in view of their envisioned electronic, optoelectronic and biological applications; specifically, CdSe, CdS, Ge, GaN, GaSb, Si, Ag, Au, and PbS are considered in this work. We calculate the torsional and spheroidal modes for all these QDs in the stress-free case. In addition,

212

Environmental

Effects on the Acoustic Phonon Spectra

1151

the acoustic modes are determined for each of these QDs embedded in water, SiC>2, Ge02, and Lucite, a plastic material. In each case, the torsional mode w^P and the spheroidal modes UJ^°, UI^1, and wjj° are calculated, where u\n

=

In the following tables the value of x Table 1.

.Ct R

x\n

is given for all of these different cases.

CdSe QD under stress-free boundary conditions and embedded in different materials.

INFINITE MATRIX-)-

H20

Stress-free

Si02

Lucite

ZnS

Ge02

MODE4. v

10

5.79

1.41

3.09

2.83

3.31

1.3

x°s°

2.4585

1.7845

2.396

5.255

2.3586

1.8

xf xf

5.9589

5.199

2.7917

8.5011

3.1674

5.205

3.7091

0.54972

2.0261

1.4329

2.0

0.552

Table 2. CdS QD under stress-free boundary conditions and embedded in different materials. INFINITE MATRIX-)-

Stress-free

Si02

H20

ZnS

Ge02

Lucite

MODE;

xi?

5.79

1.32

2.99

2.73

X°s°

2.4585

1.796

5.226

5.2736

2.3766

1.81

v01

5.9589

5.213

8.49

8.5151

3.0

5.2

3.6882

0.525

1.7834

1.2442

1.7834

0.489

Xs

v10

Xs

3.16

1.2

Table 3. Ge QD under stress-free boundary conditions and embedded in different materials. INFINITE MATRIX-)-

Stress-free

H20

Si02

Ge02

Lucite

MODE4. v10

5.79

0.7

2.43

2.1

0.66

xf xf

2.4585

1.78

1.22

0.696

1.78

3.9397

5.205

1.88

1.93

5.22

l

3.2969

0.3

1.0555

0.72297

0.2826

AT

x s°

213

1152

S. Rufo, M. Dutta & M. A.

Stroscio

Table 4. GaN QD under stress-free boundary conditions and embedded in different materials. INFINITE MATRIX-*Stress-free

H20

Si02

Ge02

A1N

Lucite

MODEI

xf

5.79

0.59

1.68

1.41

2.82

0.4

x°s°

2.4654

1.76

1.86

1.89

5.26

1.77

01

4.0585

5.199

5.241

5.283

7.9135

5.22

0.19

0.67804

0.462

1.2802

0.1748

v

•vlO

3.3318

Table 5. GaSb QD under stress-free boundary conditions and embedded in different materials. INFINITE MATRIX-*Stress-free

H20

Si02

Ge02

0.88

2.77

2.45

AlSb

Lucite

MODE4. 5.79

2.95

0.83

x¥ xf xf

2.4724

1.78

5.23

1.27

5.29

1.79

4.4218

5.199

8.31

1.90

8.295

5.205

x}?

3.4227

0.38

1.34

0.92

1.2352

0.3545

Table 6. Si QD under stress-free boundary conditions and embedded in different materials. INFINITE MATRIX-*Stress-free

H20

Si02

Lucite

MODE4. v10 XT

5.79

0.73

2.58

0.71

voo

2.4724

1.85

1.41

1.87

xf

4.1982

5.265

1.94

5.265

xl°

3.3668

0.26

0.8488

0.2377

214

Environmental

Effects on the Acoustic Phonon Spectra

Table 7. Ag Q D under stress-free boundary conditions and embedded in different materials. INFINITE MATRIX-* Stress-free

H2O

Si02

GeC>2

Lucite

MODE4.

xj?

5.79

1.05

2.58

2.27

0.86

X°s°

2.4654

1.7598

5.172

1.59

1.76

v01

5.9589

5.185

8.4592

1.67

5.19

3.6672

0.41741

1.5228

1.0825

0.39045

Xs v

10

Table 8. Au QD under stress-free boundary conditions and embedded in different materials. INFINITE MATRIX-)Stress-free

H2O

SiC>2

GeC>2

Lucite

2.1

0.78

MODE4. v10

XT

5.79

0.96

2.45

voo

Xs

2.4585

1.7385

5.1777

1.197

1.74

xf

5.9589

5.1417

8.439

1.7605

5.2

xl°

3.751

0.39944

1.4779

1.0645

0.3725

Table 9. PbS QD under stress-free boundary conditions and embedded in different materials. INFINITE MATRIX-* Stress-free

H20

Si02

Ge02

ZnS

Lucite

5.79

0.87

2.74

2.4

2.97

0.82

2.4625

1.77

5.2

1.23

1.81

1.77

4.7103

5.19

8.46

1.86

5.25

5.175

3.4815

0.39

1.397

0.9836

0.7859

0.3635

MODE4. v10

XT

xf xf 1 0

x,

215

1153

1154

S. Rufo, M. Dutta & M. A.

Table 10.

MATERIAL CdS15'16

Stroscio

Material properties of the materials used in the above calculations. Ci (10 5 c m / s ) (Longitudinal speed)

Ct (10 5 cm/s) (Transversal speed)

P (g/cc)

4.25

1.86

4.82

15 16

3.59

1.50

5.81

16

5.25

3.25

5.33

4.261

2.467

5.619

17

8.945

5.341

2.331

Ag

16

3.65

1.66

10.0

Au

17

19.488

CdSe Ge

GaSb 1 7 Si

Si02

3.361

1.239

15 16

5.95

3.76

2.2

15 16

3.43

2.14

3.6

Ge02

'

.

18

5.45

2.98

4.075

AlSb 1 7

4.851

2.73

4.36

H2O 1 9

1.483

—

1.0

ZnS

GaN

20

A1N 20 AlAs PbS

21

17

Lucite 2 2

8.0

4.87

6.15

11.12

6.27

3.255

19.34

11.10

3.717

3.754

2.088

7.500

2.68

1.26

1.18

These calculations were made using the values for the longitudinal and transverse acoustic speed, and the density as shown in Table 10. The case of CdSe is of special interest since CdSe QDs have been shown to have high quantum efficiencies and can be made to radiate throughout the visible spec­ trum by changing the radius of the quantum dot. Moreover, Woggon et al.n have observed ujg1 and w^0 phonon sidebands in the PL spectra of CdSe quantum dots. In view of the clear importance of these sidebands in modifying the PL spectra of CdSe QDs, the energy of the spheroidal acoustic phonon mode corresponding to w^1 as a function of quantum dot radius, R, is illustrated in Fig. 1 for stress-free quantum dots as well as for QDs embedded in a number of different matrix materials. These results illustrate clearly the influence of the matrix material on the acoustic phonon frequencies. Figure 2 depicts the energy of the torsional acoustic phonon mode cor­ responding to Wy° as a function of the QD radius. In this case, there are major differences in the subject acoustic phonon frequencies for in the stress-free and aqueous-environment cases. This finding is of significance in the application of QDs as biotags since knowledge of the PL spectrum of the CdSe QD is essential for this application.

216

Environmental

Effects on the Acoustic Phonon Spectra

1155

CdSe Quantum Dot 8FT

1

6 -

i

BBO Stress-Free Plastic — ZnS H X * Water .,..+ ,. S i 0 2 Ge02

"*--..,_

1

1.6

2.2

1.8

2.4

2.8

2.6

R(nra) Fig. 1. Energy in meV of spheroidal acoustic phonon mode corresponding to u j 1 as a function of quantum dot radius R in nanometers for several cases of interest.

1

1

CdSe Quantum Dot 1 1

1

TSk

"-a

1 BSD Stress-Free Plastic ZnS vx-x Water 4...,..,.

^ ^ - S .

1

- : : ■

S i 0 2

Ge02

- . . , . . .

; : :^:- ■. r+r. - • - . : 7 - : — - . .

_...

—~*~.^,.-,. 1

1.6

1.8

l

i

1

i

i

2.2

2.4

2.6

2.8

R(ron) Fig. 2. Energy in meV of torsional acoustic phonon mode corresponding to u>^ as a function of quantum dot radius R in nanometers for several cases of interest.

The results of Tables 1-9 and Figs. 1 and 2 make it immediately obvious how the optical emission spectra of these quantum dots will be modified as a result of the different acoustic phonon spectra presented in Tables 1-9. As in the case of CdSe, phonon sidebands will result in broadened optical emission spectra with the

217

1156

S. Rufo, M. Dutta & M. A.

Stroscio

increased widths being given by the energies of the acoustic phonons responsible for the sidebands (for a general discussion of phonon confinement in quantum dots, the reader is referred to Ref. 23). The importance of the matrix material is also manifestly obvious from the results of Tables 1-9. For example, aqueous environ­ ments produce large changes in the acoustic phonon energies which will, in turn, result in significant changes in the sidebands — and optical bandwidths — when QDs are placed in aqueous environments. 4. Conclusion These results demonstrate that acoustic phonon spectra of quantum-dot heterostructures depend on the matrix materials encasing the quantum dots. We have found that the matrix material can have a major impact on the luminescent line width as a result of the spectral features introduced by phonon sidebands. In addition, it is shown that the acoustic phonon spectra of quantum dots change significantly when free-standing quantum dots are introduced into aqueous envi­ ronments. This last result is especially significant in view of the role that phononassisted transitions play in influencing the electronic and optical properties of quantum dots as well as the pervasive role of aqueous environments in biological applications. For example, in the case of CdSe QDs which are being researched for possible use as biotags, it is important to know that the acoustic phonons, although not affected when these QDs are embedded in SiC>2 or ZnS, change notably when these QDs are embedded in GeC>2 or in the plastic Lucite, or when they are intro­ duced into an aqueous medium. It is of great interest to observe that for all the cases studied (except the spheroidal mode for 1 = 0 and n = 1) the eigenfrequencies exhibit important changes in the case of aqueous medium. In summary, these results demonstrate that the matrix material can have a large effect on the acoustic phonon spectrum and consequently on a variety of phonon-assisted transitions in quantum-dot heterostructures. Acknowledgments The authors are appreciative of the encouragement and contributions of Prof. Michael Shur of the Rensselaer Polytechnic Institute. The authors thank Prof. Philippe Guyot-Sionnest of the University of Chicago for identifying several key trends in quantum-dot research. In addition, the authors have benefited from dis­ cussions with Prof. Chad Mirkin of Northwestern University on the general topic of receptor-analyte interactions as well as with Prof. Raphael Tsu of the University of North Carolina of Charlotte on the topic of the Penn model as well as other related dielectric effects in quantum dots. The authors also gratefully acknowledge extremely useful communications with Dr. Augusto Alcalde of the Departamento de Fisica, Universidade Federal de Sao Carlos, Brasil. Moreover, the authors are in­ debted to Drs. James W. Mink, Rajinder Khosla, and Usha Varshney of the National

218

Environmental Effects on the Acoustic Phonon Spectra 1157

Science Foundation, Drs. Dan Johnstone and Todd Steiner of the Air Force Office of Scientific Research and Dr. John C a r r a n o of t h e Defense Advanced Research Projects Agency for their encouragement regarding our research on nanostructures. This research was supported, in part, under A F O S R grant F49620-02-1-0224. References 1. E. Klarreich, "Biologists join the dots", Nature 413 (2001) 450-452. 2. M. Bruchez, Jr., M. Moronne, P. Gin, S. Weiss, and P. A. Alivisatos, "Semiconductor nanocrystals as fluorescent biological labels", Science 281 (1998) 2013-2016. 3. W. C. W. Chan and S. Nie, "Quantum dot bioconjugated for ultrasensitive nonisotropic detection", Science 281 (1998) 2016-2018; M. Han, X. Gao, J. Z. Su, and S. Nie, "Quantum-dot-tagged microbeads for multiplexed optical coding of biomolecules", Nature Biotechnol. 19 (2001) 631-635. 4. G. P. Mitchell, C. A. Mirkin, and R. L. Letsinger, "Programmed assembly of DNA functionalized quantum dots", J. Am. Chem. Soc. 121 (1999) 8122-8123. 5. K. J. Watson, J. Zhu, S. T. Nguyen, and C. A. Mirkin, "Hybrid nanoparticles with block copolymer shell structures", J. Am. Chem. Soc. 121 (1999) 462-463. 6. Y. W. Cao, R. Jin, and C. A. Mirkin, "DNA-modified core shell Ag/Au nanoparticles", J. Am. Phys. Soc. 123 (2001) 7961-7962. 7. S. R. Whaley, D. S. English, E. L. Hu, P. F. Barbara, and A. M. Belcher, "Selection of peptides with semiconductor binding specificity for directed nanocrystal assembley", Nature 405 (2000) 665-668. 8. M. Shim and P. Guyot-Sionnest, "Intraband hole burning of colloidal quantum dots", Phys. Rev. B 64 (2001) 245342-1-5. 9. T. Takagahara, "Electron-phonon interactions and excitonic dephasing in semicon­ ductor nanocrystals", Phys. Rev. Lett. 7 1 (1993) 3577-3580. 10. A. P. Alivisatos, A. L. Harris, N. J. Levinos, M. L. Steigerwald, and L. E. Brus, "Electronic states of semiconductor clusters: Homogeneous and inhomogeneous broadening of the optical spectrum", J. Chem. Phys. 89 (1988) 4012. 11. U. Woggon, F. Gindele, O. Wind, and C. Klingshirn, "Exchange interaction and phonon confinement in CdSe quantum dots", Phys. Rev. B 54 (1996) 1506-1509. 12. T. D. Krauss and F. W. Wise, "Coherent acoustic phonons in a semiconductor quantum dot", Phys. Rev. Lett. 79 (1997) 5102-5105. 13. A. Tamura, K. Higeta, and T. Ichinokawa, "Lattice vibrations and specific heat of a small particle", J. Phys. C: Solid State Phys. 15 (1982) 4975-4991. 14. A. Tanaka, S. Onari, and T. Aral, "Low-frequency Raman scattering from CdS microcrystals embedded in a germanium dioxide glass matrix", Phys. Rev. B 47 (1993) 1237-1243. 15. A. M. Alcalde, G. E. Marques, G. Weber, and T. L. Reinecke, "Electron-acousticphonon scattering rates in II-VI quantum dots: Contribution of the macroscopic deformation potential", Solid State Comm. 116 (2000) 247-252. 16. N. N. Ovsyuk and V. N. Novikov, "Influence of a glass matrix on acoustic phonons confined in microcrystals", Phys. Rev. B 53 (1996) 3113-3118. 17. O. L. Anderson, "Determination and some uses of isotropic elastic constants of polycrystalline aggregates using single-crystal data", Phys. Acoustics III B (1965) 43-95. 18. CRC Handbook of Chemistry and Physics, (Editor-in-Chief David R. Lide, 81st Ed. 2000-2001, CRC Boca Raton, London, New York, Washington DC).

219

1158 S. Rufo, M. Dutta & M. A. Stroscio

19. Speed of Sound, (8/15/2002) http://230nscl.phy-astr.gsu.edu/hbase/sound/souspe2.html. 20. NSM Archive — Physical properties of semiconductors (7/18/2002) http://www.ioffe.ru/SVA/NSM/Semicond/. 21. I. Vurgaftman, J. R. Meyer, and L. R. Ram-Mohan, "Band parameters for III-V compound semiconductors and their alloys", J. Appl. Phys. 89 (2001) 5815-5875. 22. Ultrasonic properties of various materials (8/9/2002) http://www.cnde.iastate.edU/ncce/UT_CC/Sec.7.l/Plastics.html. 23. M. A. Stroscio and M. Dutta, Phonons in Nanostructures, Cambridge University Press, Cambridge, 2001.

220

International Journal of High Speed Electronics and Systems, Vol. 12, No. 4 (2002) 1159-1171 © World Scientific Publishing Company

Q U A N T U M DEVICES W I T H MULTIPOLE-ELECTRODE — H E T E R O J U N C T I O N S HYBRID STRUCTURES

RAPHAEL TSU University

Electrical and Computer Engineering, of North Carolina at Charlotte, Charlotte NC 28223,

USA

Since the introduction of the man-made superlattices and quantum well structures, the field has taken off and developed into Quantum Slab, QS; Quantum Wire, QW; Quantum Dot, QD; and Nanoelectronics. This rapidly expanding field owes its success to the development of epitaxially grown heterojunctions and heterostructures to confine carriers in injection lasers. Meanwhile, the advancement of lithography allows potentials to be applied in nanoscale dimension leading to the possibility of quantum confinement without heterostructures. Actually, quantum states in the inversion layer of field effect transistors, FETs, formed by the application of a large gate voltage appeared several years before the introduction of the superlattices and quantum wells. The quantum Hall effect was first discovered in the Si inversion layer. This chapter, Multipole-Electrode Heterojunction Hybrid Structure, MEHHS, discusses hybrid structures of heterojunc­ tions and applied potentials via multipole-electrodes for a much wider variety of struc­ tures for future quantum devices. The technology required to fabricate these electrodes, to some degree, is routinely used in the double-gate devices. Few specific examples are detailed here, hopefully, to stimulate a rapid adoption of a hybrid system for the formation of quasi-discrete states for quantum devices. Keywords: Quantum devices; multipole-electrode; heterojunctions; hybrid structures; quasi-discrete states.

1. Introduction Since the introduction of the man-made superlattices and quantum well structures, 1 - 3 the field has taken off and developed into Quantum Slab, QS; Quantum Wire, QW; Quantum Dot, QD; and Nanoelectronics.4 In fact, the superlattices owe their successes to the rapid development of heterojunctions and heterostructures to confine carries in injection lasers.5""7 Meanwhile, the advancement of lithography allows potentials to be applied in nanoscale dimen­ sion leading to the possibility of quantum confinement without heterostructures. 8,9 Actually, quantum states in the inversion layer of field effect transistors, FETs, formed by the application of a large gate voltage appeared several years before the introduction of the superlattices and quantum wells.10 The quantum Hall effect was first discovered in th Si inversion layer. 11 Apart from devices, many fundamental phenomena connected with quantum conductance effects will not be discussed in

221

1160

R. Tsu

Fig. 1.

Schematic cross section of an ideal double-gate F E T , taken from Ref. 13.

this paper, however, we should point out that most experiments in Aharonov-Bohn effect, conductance oscillation in a magnetic field, etc. have already embodied the use of hybride-electrodes. 12 This chapter discusses hybrid structures of heterojunctions and applied potentials via multipole-electrodes for a much wider variety of structures for future quantum devices. The technology required to fabricate these electrodes, to some degree, is routinely used in the double-gate devices targeted for improving efficiency of CMOS devices. 13 Figure 1 shows a typical DGFET taken from Ref. 13. Note that the double-gate structures could very well be the gates needed to confine electrons provided the dimension is below the inelastic meanfree-path of the channel. We shall see later how similar this structure is to the schemes proposed in this work involving multipole-electrodes. Resonant tunneling via man-made double barrier heterostructures, RTD, 2 was introduced ten years before working devices appeared. 14 Quantum cascade lasers, QCL, incorporating the principles of superlattices appeared almost 25 years after the introduction of,the man-made superlattices. 15 The main advantage of using double-gate CMOS, DG-FET, is for control of fields in the active region of extremely short channel devices, and more importantly, doping becomes unnecessary as in RTDs. In spite of the fact that DG-FET originates from the working engineers of silicon technology, the adoption into the mainstream of ULICs may very well be ten years away. On the other hand, it must be recognized that the ever decreasing in the size of IC structures, the goal stressed here in the use of heterojunction - multipoleelectrodes for quantum devices may very well be reached via the silicon technology. In short, research in silicon technology and quantum devices are converging towards each other, but the full implementation may be many years away. Few specific examples are detailed here, hopefully, to stimulate a rapid adoption of a hybrid

222

Quantum Devices with Multipole-Electrode — Heterojunctions Hybrid Structures

1161

system for the formation of quasi-discrete states, the resonant states, for quantum devices. We shall go into some details why multipoles are important. The multipole expansion of an electric field has a very important feature: higher the multipole, faster the field falls of with distance. In fact this is essentially the reason why heterojunctions can be much more abrupt than pn-junctions. Sometimes we con­ sider quantum confinement by geometrical boundaries as a separate means to achieve quantum states, however, geometrical confinement cannot take place with­ out band-edge alignments of heterostructures. For example, a microwave resonator is formed by a section of the waveguide with geometrical constriction. And the constriction is precisely the results of discontinuities such as difference in dielec­ tric constants or difference between an insulator and conductor. Although the origin of confinement may be ultimately traced to potentials, for the purpose of describing and characterizing the fundamental mechanisms for engineering designs, it is very useful to distinguish confinement by heterostructures, and confinement by multipole-electrodes, as well as a hybride scheme using both. 12 2. Examples of Heterojunction — Multipole-Electrode Hybrid Structures Next we shall discuss the hybrid system with mutipole-electrodes applied to a quantum slab formed by heterojunctions for a much wider variety of structures for future quantum devices.8 The rapidly expanding field of man-made quantum structure owes its success to the development of epitaxially grown heterostructures. Meanwhile, the advancement of lithography allows potentials to be applied in nanoscale dimension leading to the possibility of quantum confinement without heterostructures. 16,17 In principle, multipole-electrodes can provide confinement as well as control of symmetry for specific device functions. Such potentials may be designed for (i) arbitrary geometry, (ii) produce softer scattering and (iii) be dy­ namic, e.g., turning the device on and off. An example of a hybrid system with multipole-electrodes applied to a quantum slab formed by heterojunctions is shown in Fig. 2, taken from Ref. 8. The difference between the electrodes in Fig. 2 and the double-gates in Fig. 1 is the multipole nature with (-1—) pair of electrodes. Note that the potentials at the top, t = 0 and the bottom, t = d, are ideal for confining an electron, but at the center, t = d/2, the confinement is much weaker. The thickness of the quantum slab, d, should be less than the inelastic scattering length of the material forming the QS, usually no more than few tens of nm. The maximum separation of the (+ +) pair and ( ) pair should be determined by the breakdown field of the material, generally no more than ~ 107 V/cm. Figure 3 shows a case where the thickness of the slab is much greater than the separation of the (+ +), and (— —) pairs of electrodes. If the bottom electrodes are moved far from the quantum slab, the confining potential near the top at t = 0 is nearly the same but at t = d is naturally reduced. Therefore for d ~ 10 nm, the bottom

223

1162

R. Tsu

-1V

+1V w —•I m>. *—

-1V t=0

T

.",

U

*

J ,

I

QS

■ i

,■ . ■ ' I '

■' i i , I

■.

\

'••■', i - t i i 11 ■ i : ' M ' I J ' i \ \l

, i I ' >.'

i

i v ij '/

i

v V

■ > \ . 11 ; /

t = dl2

d >»

t=d

\ \ 1,

VAm

-0.8-0.4-0.2 0 0.2 0.4 0.8

(a)

(b)

Fig. 2. (a) Potential inside a QS (quantum slab) of thickness d with multipole-electrodes on top and bottom of the slab for effecting quantum confinement, (b) Potential versus distance at three levels: t = 0, t = d/2, and t = d. Note that the confining potentials are reduced at the middle of the slab.

-1V

J

+1V ^•'j'im\\,

i J j

• i ,- i Q S ■>'

I

-1V 7>n

k

L

■ I i \ \ \ i' / > H i i \ V •/ / i U

I

9d

-1V

+1V

t = 10d

-1V

(a)

(b)

Fig. 3. (a) Showing the potential inside a QS with multipole-electrodes placed on top and bottom. (b) Potential versus distance at the bottom of the QS, t = d, and "delta-function like" potential spikes versus distance when the electrodes are separated by lOd. Note that in this case, confinement at t = d almost disappears because these spikes are not really delta functions with peak position fixed at + 1 V and - 1 V.

electrodes should not be placed more than few d's away. The requirements that the top + 1 V and —IV electrodes must be sufficiently separated limited by the break down voltage, together with the minimum distance separating the top and bottom electrodes; present a severe technical challenge though not impossible. Figure 3 shows the case where the bottom electrodes are placed lOd below the top electrodes,

224

Quantum Devices with Multipole-Electrode

— Heterojunctions

Hybrid Structures

1163

the confinement at t — d almost disappears because these potential spikes are not delta functions. The multipole-electrodes in Figs. 2 and 3 are similar to most buried gates currently being investigated for high efficient CMOSFET. The fabrication is complex but doable. Wong details an excellent account for the self-aligned doublegate fabrication process. 18 The major difference between what is presented here and the DGFET is the multipole nature of the electrodes discussed earlier. It is quite possible that this hybrid confinement scheme may be realized in pursuing high-end silicon technology rather than researcher investigating nanoelectronic devices.

3. Some Fundamental Issues: Mainly Difficulties In bulk crystalline solids, symmetry manifests in the periodic part of the Bloch func­ tion, which depends on the wave vector k of the Brillouin zone. The symmetry of the man-made quantum state is determined by the symmetry of the heterostructure and the applied potentials. As in atomic and molecular physics, for two electrons occupying a given state, a symmetric state goes with the antisymmetric spin func­ tion, the singlet; while the antisymmetric state goes with the three symmetric spin states, the triplets. The applied potential can change the symmetry into neither symmetric nor antisymmetric, resulting in complex spin functions. More over, the boundary conditions due to band-edge offset and together with the dielectric dis­ continuity due to the difference of the dielectric functions of the region of interest, the quantum dot, QD or quantum well QW, constitute a formidable task in compu­ tational physics of the problem. For example, the dielectric discontinuity was fully included in the treatment of the doping of a quantum dot 19 ' 20 and the capacitance of a quantum dot. 21 However in these treatment, a rather simple wave function was assumed, that the electron wave function goes to zero at the boundary. We know that is not the case because RTD will never work with this type of simplified boundary conditions. We shall simply point out the involvement. If an electron in­ duces a potential at the dielectric mismatch, there will be inside and outside (for example, outside refers to the matrix) resulting in Coulomb energy. However, if the induced electron wave function originates from another site of the matrix, even the Heisenberg exchange term should be included. All these are quite negligible in QW structures, but magnified by a large factor in QD structures simply due to the small volume of the active region. When size is sufficiently reduced, a Poisson distribution replaces Gaussian statistics so that fluctuations from the mean from inadvertent defects become unavoidable, and switching results. This is a classical problem associated with random processes, which prevents the implementation of redundancy and robustness. The familiar concept of RC time constant is still appli­ cable except that capacitance is not simply given by the geometry. 21 The quantum mechanical definition of capacitance in terms of stored energy includes kinetic as well as electrostatic energy. Therefore, capacitance is much reduced in QD, not only due to the reduction of dielectric constant, but also due to the increased kinetic energy inside the QD. Lastly, we should remember that atomic physics is based

225

1164

R. Tsu

on spherically symmetric potentials. One needs to carefully examine each case for arbitrary symmetry. 9 Nevertheless, as in all cases of theoretical and computational issues, one resorts to simple solvable geometry to obtain guideline for engineering design. Input/output, I/O, is the most difficult problem for these nanoscale devices.22 First of all, when one speaks of a voltage applied at a particular region, terminal, or contact, an equal potential surface is implied. As size decreases to the nanoscale region, one cannot rely on doping. For example, solid solubility limits the maximum doping concentration, usually ~ 0.1%. Silicon, for example, has 5 x 1022 atoms per cm 3 . In a volume space defined by 10 nm linear dimension, there are 50 dopants at 0.1% doping level. However, for a doping density of 10 18 /cm 3 , there is only one dopant in the volume. The doping density fluctuates wildly so that device cannot function. Therefore, it is safe to say that metallic contact or semi-metallic contact must be used to ensure I/O to the active quantum device, if nothing more than the demand of electrostatic. ■■ Many dramatic results have been claimed in the work of single electron transis­ tor for example. 23 First of all these results take advantage of the relatively larger size operating at very low temperatures, or using STM probes for smaller devices operating at higher temperatures. Both cases represent bench top experiments, far from being able to apply to the world of electronic devices. Meanwhile there is a steady stream of results and claims involving nanoscale particles in optical applications. 24 However, they are far from any real devices. First of all, photons, with its large wavelength, invariably contact many particles in parallel. A parallel system is intrinsically unstable such as optical "blinking". Let us use a very sim­ ple argument against system in parallel. A resistor with two contacts is basically a parallel system. Then why is it stable? It is stable because resistor is operating in the linear range, the Ohm's law regime, so that the conductance is proportional to the area of the contact. Almost ten years ago, Tsu and Nicollian took up the investigation of tunneling via nanoscale silicon particles, ~ 3 nm in size, between two large contacts. 2 5 - 2 7 Many strange phenomena occurred. First of all, there is the traditional tunneling via resonant state of the Si QDs resulting in delta-function like conductance structure which is to be expected. The charging and discharging of the QD, the so-called Coulomb blockade is also to be expected. Even the hys­ teresis curve is to be expected from inadvertent trapping of electrons. But, what is most disturbing is the conductance oscillations in time, or more precisely, on/off switching of the conductance. Figure 4 shows an unpublished data of the on/off switching of conductance with very slow switching at a reverse bias of —12.25 V. Details are found in Ref. 28. Note that the on time is constant over the entire trace, but there is a small decrease in the off time going from the top to the bottom, covering more than half an hour, possibly due to heating. Changing the bias to —12.35 V, there is a dramatic change of the switching shown in Fig. 5. Again there is a small reduction of the switching period going from the top to the bottom covering 50 minutes. The change in the

226

Quantum Devices with Multipole-Electrode

Ji

— Heterojunctions

fl h N N ft h_i

16

24

Hybrid Structures

1165

^

1 8

i_J^^_X_ii_i_jL_l__^_A. 2'6

K

^^^^L__, L_i\_^^_^__^L__^^_^_^L_^__;^ 32

'

3*5

""

Time (min) Fig. 4. Conductance measured at 1 MHz showing switching in time of many ~ 3 nm Si-particles, QDs in parallel a t a reverse bias voltage of 12.25 V.

Time (min) Fig. 5. Conductance measured at 1 MHz showing switching in time of many ~ 3 nm Si-particles, QDs, in parallel a t a reverse bias voltage of 12.35 V.

applied bias is less than 1% results in such a dramatic change. Note that the time spectra are quite different from the telegraph noise spectrum, 29 or conductance fluctuation,30 shown in Fig. 6 taken from Ref. 30. The most obvious difference is the regularity of the spectra. In fact, we had the chart recorder going for more than 6 hours! Basically, resonant tunneling is very nonlinear having NDC, which, to no one's surprise, should induce instability. Therefore, one must find a way to avoid that if this phenomenon is universal whenever contacting many nearly identical entities in parallel. As one goes into the realm of nanoscale dimension, it is hard to

227

1166

R. Tsu

(a)

0

„ A

20

40

60

80

100

time (min) Fig. 6. Time spectra of conductance without photons (a), and with photons (b) taken from Ref. 30.

imagine an actual device actually having STM probes as input/output, nor can we imagine giving up multiplicity for redundancy and robustness or reliability issues. Perhaps what one needs to consider is limiting to a handful of nanoscale particles in parallel so that ways can be designed to circumvent the switching instability. Facing with such a range of difficulties, how do we proceed from laboratory to industrial applications? Actually among those theoretical and practical difficulties discussed in this section, the hardest roadblock is the I/O to a nanoscale device. We need to identify an area where I/O is not a problem, or means to overcome these prob­ lems are at hand. Before we proceed to discuss this point in the next section, it is appropriate to point out that I/O is not a problem for resonant tunneling devices in­ volving quantum wells because these devices have planar contacts. Now in the light of what was just presented, we may ask why DBRTDs, double-barrier-resonanttunneling-diodes, generally do not run into the switching problems. In the case of superlattices, theoretical models, 31 and experimental verifications32 in transport in superlattices point out the source of instability, the formation of electric domain. And recently, the chaotic transport was theoretically investigated, tracing the time oscillations to sequential resonant tunneling, even in RTDs with sequential reso­ nant tunneling. 33 Nevertheless, our data shows that time oscillations do not occur in DBRTDs, and prominently show up in resonant tunneling via many quantum dots in parallel. Perhaps the answer lies in 2D behavior of the resonant states in RTDs where the transverse direction is governed by free particle wave function. This 2D behavior may prevent efficient interactions with traps. What is apparent is the experimental fact that time oscillation, particularly the extremely regular switching shown in Figs. 4 and 5, is typical of resonant tunneling via many nanoscale particles in parallel. Therefore, further experimental studies must be conducted in a system where parameters are all under control. We suggest that a single DBRTD should be experimentally compared to a number of DBRTDs with nanoscale dimensions in the

228

Quantum

Devices with Multipole-Electrode

— Heterojunctions

Hybrid Structures

1167

transverse direction so as to break up the momentum conservation in the transverse plane. Experiments should be with III-V at low temperatures so that these QD-like DBRTDs may be lithographically fabricated and connected in parallel. The observa­ tion of time instability and/or oscillations in time should be the goal. Furthermore, trapping sites can be introduced to see if the instability is indeed due to stronger coupling between the resonant states and the trapping states. In fact, the original paper on the observation of slow oscillations26 did point out the role of traps and another possibility, the self-trapping due to Coulomb effects (charging effects) of oc­ cupied states. In other words, suppose resonant tunneling results in the occupation of a given state, the Coulomb energy, charging results in switching. Incidentally, this second self-trap on/off switching is nothing but what is proposed in a theoret­ ical model of chaotic transport in sequential resonant tunneling. 33 In fact, we like to take this opportunity to point out that in the literatures of resonant tunneling, two models are used. The first is the so-called Tsu-Esaki model of direct coherent tunneling, 2 and the second is the sequential tunneling, 34 which seems to account for losses due to scattering. In fact, Tsu had incorporated losses in the coherent formulation by introducing a dissipative, non-Hermitian term into the Hamiltonian operator, and using the Green function instead of the eigen-state formulation with equally successful prediction as to the linewidth contribution in RTD. 35 Although both coherent tunneling and sequential tunneling can take into account of scat­ tering, the fundamental physics is very different. Perhaps in the context of what is discussed here in this section, finally one can distinguish the two processes: coherent resonant tunneling does not lead to instability while sequential resonant tunneling does. 33 4. Q u a n t u m Waveguide w i t h M E H H S Facing the serious I/O problem of quantum devices utilizing the MEHHS, we iden­ tify the Electron Quantum Waveguide, EQW, as a system of application having no serious I/O problem. But before we discuss in more detail of the issues involved, a brief account of EQW is presented. Figure 7 shows the coordinates used for the derivation of the waveguide modes, for a section with a and b, the transverse dimension and z, the propagation direction.

Fig. 7. A section of an Electron Waveguide with coordinates shown. Either the top and bottom, or the sides may include MEHHS, The potential V in the derivation may be the multipole-electrodes, or an extra superposed potential for control.

229

1168

R.

Tsu

Following the usual waveguide theory, the propagating wave-vector kz including the potential energy eV is given by

kl = 2-^(E + eV)-klm,

(1)

in which the transverse momentum vector fctnm is

<2>

* - = (=)'+(T)'The density of states, DOS for a cross-sectional area A is 1

MsM

DOS where fczM is the maximum value for a given set of (m,n), E = EF, for ^(EF

E and eV. At T — 0,

+ eV)
(4)

fc^ is pure imaginary, propagation is not possible. Counting only the propagating modes, the current density is 2e v ^ fkzM 1 dE n,m

leading to the current,

/-££<* + ">-£*.■

(5)

n.m

,„;j.u /ni_ _ 2e Since the conductance G is denned by G = Jdl^ , with Go — h- ,

G = J2G°e(EF-eV-k?™),

(6)

in which 6 is the unit step function, having a series of steps depending on how many modes, (m, n) are included. The factor of 2 in G 0 is for the two spins. Thus for spin polarized case, there should be Go (+) and Go (—) without the factor 2. It is important to recognize that for single mode operation, only one step in G appears, depending on the condition given by Eq. (4). The origin of these extra modes is clearly due to the inclusion of modes (m, n) coming from the incident electrons having transverse energy. In fact, as we have pointed out before that the applied voltage in the region, for example, the top and bottom electrodes may be different; the extra transverse degree of freedom even without the transverse momentum from the excitation can give rise to extra modes. Thus, the EQW is a multi-purpose electronic device. The quantized conductance steps in units of Go = 2e2/h for the transport of electrons in constricted geometry was first pointed out by Landauer, 36 ' 37 and

230

Quantum

Devices with Multipole-Electrode

— Heterojunctions

Hybrid Structures

1169

often mistaken as ballistic behavior or even some mechanism of energy loss in these systems. However, these conductance steps are entirely a wave phenomenon: the dependence of the longitudinal component of the wave vector kz on the potential without dissipation. Remember that a photon propagating in dissipationless free space has a resistance, the wave impedance, of 377 ft. In most photon waveguides and microwave waveguides; only single mode is present by limiting the frequency of the exciting incident photon. In EQW, the above derivation shows that, unlike photons, electrons from a contact involve those on a Fermi surface with electrons in all directions. For example, a cone of electrons in a solid angle Aft emerging from a spherical Fermi surface involves electrons both in a given direction such as the z-direction as well as in the transverse directions. Therefore, in general, multimode operation dominates over a single mode operation. With the application of a potential V, many transverse modes are included resulting in propagation having a sum of modes given by the expression for G. It should be apparent that this sum of modes is totally controllable by the potential V, thus, a very useful electronic device, not only as filters, but also as selectable tunneling. The I/O of conventional microwave waveguides is no child's play. Many great minds were involved in the excitation, coupling, and matching of microwaves using waveguides during World War II, conducted mainly at MIT and Bell Telephone Labs. Remember, the electron waveguide involves even more complex mathematics. Some of the most fundamental means for excitation, coupling, and matching form some of the current preoccupation of this author. Complicated mathematics and experimentations aside, at this stage, we see no fundamental difficulties in realizing some aspect of the EQW for industrial applications. 5. Conclusions and Future Outlook The Multipole-Electrode-Heterojunction Hybrid Structure, MEHHS, is ready for possible applications. The system, which is ready, may very well be the elec­ tron quantum waveguide, where fabrication techniques for heterostructures are sufficiently mature and high-resolution lithography is sufficiently advanced to allow the formation of this hybrid system. Some of the fundamental issues such as excitation and coupling as well as switching phenomenon need intense investigation. Lastly, in recent years, the research community has feverishly pursued the lure of Quantum Computing (QC). Basically, the binary bits are replaced by Q-bits, a sort of quaternary bits, which includes phase of the state into logical gates of a classical computing scheme. 38,39 Mathematically a complete set can represent any function. However, because of the need for input/output, devices are not totally isolated and one is forced to deal with resonance states instead of eigenstates of the system. To­ gether with unavoidable losses and finite bandwidth, serious problems exist to this day in the implementation of the concept. Nevertheless, we mention QC- devices because it may become important someday in the future, and more importantly, the use of EQW with MEHHS may be a vehicle for reaching the dream of QC.

231

1170 R. Tsu

Acknowledgments T h e Army Research Office, spread out in a number of years, has been t h e primary support of the work described in this article. References 1. 2. 3. 4. 5. 6. 7. 8.

9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33.

L. Esaki and R. Tsu, IBM J. Res. Dev. 14 (1970) 61-65. R. Tsu and L. Esaki, Appl. Phys. Lett. 22 (1973) 562-564. L. L. Chang, L. Esaki, and R. Tsu, Appl. Phys. Lett. 24 (1974) 593-595. U. Meirav, M. A. Kastner, and S. J. Wind, Phys. Rev. Lett. 65 (1990) 771-773. H. Kroemer, Proc. IEEE 5 1 (1963) 1782-1783. Z. I. Alferov and R. F. Kazarinov, USSR Patent 181 (1965) 737. I. Hayashi, "Heterostructure lasers", IEEE Trans. Electron Device ED-31 (1984) 1630. R. Tsu, "Fundamental issues for heterojunctions multipole-electrode hybrid confine­ ment", Proc. ECS Int. Symp. Adv. Lum. Material & Quantum Confinement, ed. M. Cahay, Philadelphia, May 13-14, 2002. R. Tsu and T. Datta, Proc. 26th ICPS Edinborgh-2002, to be published. F. Stern and W. E. Howard, Phys. Rev. 163 (1967) 816-835. K. von Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45 (1980) 494-496. S. Washburn and R. A. Webb, Rep. Prog. Phys. 55 (1992) 1311-1383. H.-S. Wong, K. Chan, and Y. Taur, IEDM Tech. Digest (1977) 427-430. T. C. L. G. Sollner, W. D. Goodhue, P. E. Tannenwald, C. D. Parker, and D. D. Peck, Appl. Phys. Lett. 4 3 (1983) 588-590. J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, "Quantum cascade laser", Science 264 (1994) 533-555. A. M. Song et al., Phys. Rev. Lett. 80 (1998) 383-385. Yu. V. Sharvin, Sov. Phys. JETP 21 (1965) 655-656. H.-S. P. Wong, IBM J. Res. Dev. 46 (2002) 133-168. R. Tsu, D. Babic, and L.. Ioriatti, J. Appl. Phys. 82 (1997) 1327-1329. R. Tsu and D. Babic, Appl. Phys. Lett. 64 (1994) 1806-1808. D. Babic, R. Tsu and R. F . Greene, Phys. Rev. B 45 (1992) 14150-14155. R. Tsu, Nanotechnology 12 (2001) 625-628. M. A. Reed, J. N. Randall, R. J. Aggarwel, R. J. Matyi, T. M. Moore, and A. E. Wetsel, Phys. Rev. Lett. 60 (1988) 535-537. See for example in "Spectroscopy of isolated and assembled semiconductors nanocrystals", eds. L. E. Brus, Al. L. B. Efros, and T. Itoh, J. Lumin. 70 (1996) 1. Q. Y. Ye, R. Tsu, and E. H. Nicollian, Phys. Rev. B 44 (1991) 1806-1811. R. Tsu, X. L. Li, and E. H. Nicollian, Appl. Phys. Lett. 65 (1994) 842-844. D. W. Boeringer and R. Tsu, Phys. Rev. B 51 (1995) 13337-13343. See a review in: R. Tsu, Appl. Phys. A 71 (2000) 391-402. K. S. Rails et al., Phys. Rev. Lett. 52 (1984) 228-230. N. J. Long, S. Yin, P. M. Echternach, and G. Bergmann, Phys. Rev. B 50 (1994) 2693-2695. O. M. Bulashenko, M. J. Garcia, and L. L. Bonilla, Phys. Rev. B 53 (1996) 1000810018. Y. Zhang, J. Kastrup, R. Klann, K. H. Ploog, and H. T. Grahn, Phys. Rev. Lett. 77, (1996) 3001-3003. M. Zwolak, D. Ferguson, and M. D. Ventra, "Chaotic transport in low-dimensional superlattices", Phys. Rev., to be published.

232

Quantum Devices with Multipole-Electrode — Heterojunctions Hybrid Structures

1171

34. S. Luryi, Appl. Phys. Lett. 47 (1985) 490-492. 35. See for example, R. Tsu, J. Non-crystalline Solids 114 (1989) 708-710, and references therein. 36. R. Landauer, IBM J. Res. Dev. 1 (1957) 223-231. 37. R. Landauer, Phil. Mag. 21 (1970) 863-867. 38. D. Deutsch, Proc. R. Soc. Lond. A 400 (1985) 97-117. 39. See for example, J. Mullins, in the editorial of IEEE Spectrum (February 2001).