Enhancement of energy relaxation rates in semiconductor superlattices

Solid-State Electronics Vol. 32, No. 12, pp. 1675-1679, 1989

0038-1101/89 $3.00+ 0.00 Copyright © 1989 Pergamon Press plc

Printed in Great Britain. All rights reserved

ENHANCEMENT OF ENERGY RELAXATION RATES IN S E M I C O N D U C T O R SUPERLATTICES M.P~ Chamberlain and M. Babiker Physics Department, University of Essex, Colchester CO4 3SQ, UK

ABSTRACT Intersubband transitions are considered for the case of semiconductor superlattices with transitions proceeding by the emission of LO phonons and (at appropriate carder densities) by the emission of couple.d phonon-plasmons. The variation of the emission rates with the superlattice well width (for fixed barrier width dl = 100A) are calculated assuming different values of electron density. At low densities it is shown how an enhancement of the LO phonon rate can occur at specific well widths for transitions between the lowest two subbands of a GaAs/GaA1As superlattice. The rate exhibits a variation with well width of up to two orders of magnitude and similar results are borne out by calculations involving the infinite quantum well. At higher densities when the coupling between the LO phonons and the plasmons is taken into account the results show an enhancement due to the plasmons together with a screening of the Fr6hlich interaction. Keywo~s: Superlattice, inter-subbandtransition, LOphonons, coupled phonon-plasmons.

Energy relaxation of hot electrons is a subject of considerable importance for all microstructures based upon semiconductor quantum wells and superlattices. In these systems hot carders, created e.g. by photo-excitation or by electrical injection, initially occupy so called delocalised states which have energies that lie above a typical barrier energy of the heterostructure and correspond to motion perpendicular to the layers. Furthermore, if such structures are based on polar materials, as is often the case, the excited carriers relax energy primarily by emission of polar LO phonons. Energy relaxation is currently being investigated by both theory (Babiker and co-workers 1987, 1988, Sawaki 1986, Wendler and Haupt 1987 and Trallero Giner and Comas 1988) and experiment (Seilmeir and co-workers 1987, Abstreiter and coworkers 1988, Shah and co-worders 1985, Ryan 1984). This work concentrates on calculating the rate of LO-phonon mediated transitions in a superlattice initially for the case of low electron densities. We compare the superlattice results with those for the corresponding infinite potential quantum well. At higher electron densities where the plasma oscillations of the electrons are of a similar frequency to the LO phonon frequency there is a strong phonon-plasmon coupling (Richter 1982) which consequently modifies the Fr6hlich interaction (Ridley 1988). Therefore, we have also calculated transition rates in a superlattice at higher electron densities via the emission of coupled phonon-plasmons. As we describe later the results for each case exhibit an interesting variation with increasing well width but a common feature that is worthy of particular emphasis is the existence of well defined threshold well widths at which rates are enhanced and beyond which no emission occurs. These features are determined by both the superlattice electron structure and the relevant characteristics of the Coulomb field quantum mediating the transition. The theory leading to these conclusions is outlined as follows. We describe the electron band structure in a superlattice by the effective mass approximation (Bastard 1981, Altarelli 1987). For a superlattice with barrier width dl, well width d2, and barrier energy Vo the dispersion relation is cosQd = cOsKldlcosK2d2 - ~-(Y + ~)sinKldlSinK2d2 1

(1)

where d = d l + d2, KI -- ([2m~(E-Vo)/h2] - K2) I/2, K2 -- ([2m2*Effi2] - K2) I/2, and Y = Klm~]K2m~' • Q and KII are the perpendicular and parallel components of the electronic wavevector and m~ and rn~ are the effective masses. The labels I and 2 refer respectively to the GaAIAs and GaAs. This dispersion relation is used to determine the variations of the energy levels with the well width. If we consider a GaAs,/Ga0.7AI0.3As superlattice with a fixed barrier of Ga0.7Al0.3As, width dl = 100A, the variations of energy levels with the well width d2 are shown in Fig. I. We have used the effective masses mT = 0 088m~ and m~ = 0 063 me and barrier energy Vo = 0.19 eV. The energies correspond to subband minima when the electron v~avevector parallel to the mterfaces is zero. The determmatton of the corresponding electron eigenfunctions has been outlined in a previous paper (Bablker and Ridley 1986). The longitudinal optical phonons are described by a dispersive Born and Huang continuum model for ionic materials (Born and Huang 1968, Babiker 1986). We consider a bulk phonon approximation which does not include the effect of the heterostructure on the phonons. According to the dispersive continuum model the Hamiltonian for the LO phonons is .

•

.

.

.

1 (47~tx)2 1 {(i~)2 + o 2L E 2 _ v 2 ( V • E ) 2 } H = f d3x ~-

1675

.

.

(2)

1676

M . P . CHAMBERLAINand M. BABIKER

(g O 13 ')%1 1t2

where eL is the characteristic LO mode frequency, v is an acoustic velocity describing dispersion and cx =

I

L eoe.~ j

eo and e,,. are the static and high frequency dielectric constants respectively. Associated with the electric field E is the Coulomb scalar potential • such that E = - V ~. In terms of the phonon annihilation operator {a(k,,q)} and creation operator {a*(k,,q)} the scalar potential • can be written as O=fd2k,,fdq

{Cpei(k'~'x"+qZ) a(kH,q)-H.C. }

(3)

Using the procedure of canonical quantisation the Hamiltonian forces the phonon normalisation factors Cp to take the form C

f

h(eo-~)o~

],/2

P = 14rC2eoe.~ (k2 + q2)j

(4)

The corresponding dispersion relation is m2 = o ~ - v2(kt2 + q2)

(5)

We now consider transitions from an initial state IKipQi,{0}> corresponding to the first excited subband to all possible final states of the form IK~,Qf,{k,,q}>. Such a final state consists of an electron in the ground state subband plus one LO phonon {ktl,q} emitted in the transition. The transition rate is given by the Fermi Golden rule as follows:F(KII,Q) = ~ ;d2kllfdqld2K~ fdQflM(Kii,Qi,K~,Qf, kll,q )1) 2 8(Ei-Ef-hto)

(6)

M(KII,Qi,K~,Qf,klI,q) = ;

(7)

where Ej = (fiKi,)2/2m * + Ej(0,Q)

and the delta function ensures energy conservation, eO represents the coupling between the electron and the scalar potential associated with the electric field of the ionic motion. We concentrate on transitions in which the initial state is defined by the energy level at the bottom of the first excited subband as shown in Fig. 1. This corresponds to Kit = 0 and Qi = n/d and we take into account all possible final states that conform with the requirements of energy and momentum conservation. After some algebra we find that the transition rate, as a function of GaAs well width (d2), can be written as

F(d2) = 4n2m*g2 fi3 -~J" dq { ICiCi.pCpl21Zl+Z212}kFko

(8)

k2 2m* {E2 (0,rc/d) - El(0,~/d-q) - fi~L} o = ---~--

(9)

where

In equation (8) CI and CI.p are superlattice electronic factors for the initial and final states, Cp is the phonon factor given by equation (4), and Z1 and Z2 come from the integration over layers of component 1 and component 2. For comparison we also consider transition rates in an infinite quantum well. The electron wave-functions are sine functions completely contained within the well in the z direction and plane waves propagating parallel to the interfaces. At subband minima when the in-plane electron momentum is zero the variation of electron energy levels with well width are shown in Fig. 1. The phonons are again described by the bulk phonon-model and the electron transition rate in the infinite well is calculated using a similar procedure to be

c~, F(d2) = Fo4.9d2

f

lit + ii212 d(qd2) (k,d222 + q2 d22)

(10)

-oo Here I 1 and 12 come from the integration over z and F o is the emission rate for bulk GaAs; Fo=

(e.o-e.**)e2 (2m*~L~ 1/2 heoe.o L - " ~ ) =7.74x 10t2s -1

The results for the superlattice and the infinite well transitions are shown in Fig. 2.

(11)

Enhancement of energy relaxation rates

1677

The superlattice transition rate decreases to a minimum for d2 in the region of 50A and then increases until it attains a value close to Fo for a limiting well width d2 ~- 180A. It is inappropriate to calculate the intersubband transition rate between the lowest two subbands for well widths greater than 180A because the energy separation is less than fi~)L, the energy required for the emission of an LO phonon. Fig. 2 shows that the ratio of the transition rate at d2 = 180/~ to that at d2 about 50A is approximately 50.

[

/

i

t_o

I

I

:

.~

/

I 0.8

!

~

'

•

/

/

!

i

;

l

/,/

0 t,

02 v

20

GO dz(~l~).,~j

140

180

0

50

100 150. Welt Width I A

200

Figure 2 Figure 1 Variation of the lowest subband minima with well width Transition rates via the emission of LO phonons d2 in GaAs/Ga0.7Al0.3As superlattices, for which the barrier and in units of Fo from the fu'st excited subband width is fixed at dl = 100A, and in infinite potential quantum for semiconductor superlattices, full curve, and wells. All energies are in units of the superlattice well depth infinite potential quantum wells, broken curve. Vo = 0.19 eV. The labels 0 and ~ show values of Q(dl+d2) Fois the rate for phonon emission in bulk GaAs F o = 7.74 x 1012 s "l at the band edges• The dependence on well width of the transition rate is can be largely explained in terms of the variation of the in" plane phonon-wavevector 1%(q) given by equation (9)• 1%(q) itself depends on the energy subband separation. From Fig. 1 we can see that ford2 = 50A the energy subband separation is at a maximum and for d2 = I80A, it is exactly equal to the phonon quantum of energy hOL leading to a zero value of 1%. The transition rate depends on the phonon factor Cp which is proportional to 1/(k~+q2). It then follows that the minimum transition rate occurs for maximum 1%at d2 about 50A and the enhancement of the transition rate is realised when 1%is zero at d2 = 180A. The transition rate for the infinite potential well is also shown in Fig. 2 for comparison. By contrast the rate in this case monotonically increases with increasing well width from small values at small widths to a maximum Fo at the threshold. For small well widths the separation of energy levels in the infinite well is extremely large, by comparison to the superlattice, and so the value of the in-plane phonon wavevector is also large, leading to very small transition rates. As the energy level separation decreases with increasing well width there is a gradual increase of the transition rate to a maximum value Fo. This maximum value of the transition rate again occurs when the energy level separation is equal to the phonon quantum of energy fit0L. This is equivalent to f 3h 2 ]1/2 d2 = ~Sm---~fi~~ = 222/~

(12)

Ridley (1982) calculated the transition rate for an infinite quantum well via LO phonon emission using a momentum conservation approximation in the z-direction. Our results agree remarkably well with those of Ridley although he did not explore variations with well width beyond 120A. Subsequent investigations appear to suggest that the momentum conservation approximation is valid for the case of the infinite potential weU up to the threshold well width of 222A. At higher electron densities the FrOhlich interaction of the electrons with the LO phonons is modified since the LO phonons couple to the plasma oscillations of the electrons. This phonon-plasmon coupling is particularly strong at those electron densities for which the LO phonon frequency and the plasmon frequency are comparable. The electron density when these two frequencies are equal is given by

n=( 2m(~

(13)

k 4~e2 ) For bulk GaAs this occurs when the electron density n ---5 x 1017 cm-3. We have calculated transition rates in the superlattice which occur via the emission of coupled phonon-plasmons that are appropriate at higher electron densities. The treatment of coupled phonon-plasmons in bulk polar semiconductors and semiconductor heterostructures is

1678

M . P . CHAMBERLAIN and M. BABIKER

described elsewhere (Chamberlain and Babiker, 1989a,b). The Hamiltonian for these modes is derived from the energy conservation relation and for a coupled phonon-plasmon of frequency COthis Hamiltonian is given by

H=

d 3 x ~ A2

+ t 0 p2 -

i~2 +

E 2 - U2(V.E) 2

(14)

where A2 and U 2 are given by A2 =

v2

[~2CO2p Eo*

4ng2(CO2 . (o2)2

o)4

(15)

U2 = (1- CO2p/co 2) v 2 + (0)2/0)2- 1) 132

(16)

COpis the plasma frequency COp= (4xne2/m*v.~,.)l/2,and g-2 = 47~(eo.~.)COZTwith ~ = (e~/Eo)O~. [~ is given by 4 3V~/5 with VF the Fermi velocity of the degenerate carrier system. When one takes the limit of low electron densities i.e. COp~ 0 and [3 --* 0 then this Hamiltonian can be seen to reduce to the pure LO phonon Hamiltonian of equation 2. The coupled phonon-plasmons should now be represented by a Coulomb potential ~ such that

(1)(x) = 2.,aJdkllJdq(Cx(kll,q)e i(k" .x,l+ q z ) a k ( k l l , q ) - H . C . )

(17)

),=1,2

a. with ak(k,,q) and a~(k,,q) the phonon-plasmon annihilation and creation operators. The label ~. = 1,2 designates the branch which, in asymptotic regions, can be described as either phonon-like or plasmon-like. Canonical quantisation requires the energy normalisation factors C~.(k,,q) to be 1/2

_( hu2

I

a,2

Cx(k,,q) = [4x2(k~+q2)A 2 CO

(18)

These coupled phonon-plasmon modes are longitudinal in nature with total electric displacement field zero leading to the following dispersion relation:(04. (02(0~ + COp2.(V2. ~2)k2)+ ( ( 0 2 4 " k2v2co~ + k2~2CO2L-v2~2k4) = 0

(19)

1.6

~)~

/ ~ / / / / /

(Jr I 1.2

/

/

/

I-0

0~ Figure3.

2!5 5:0 7g

|0.0 1}5 lgO 175

k(cm-') x10"

Dispersion curves (COversus k) for coupled phonon plasmons in bulk OaAs (full curves), for electron density n = 5 x 1017 cm -3. Also shown are the dispersion curves for single component fields:- LO phonons, dotted curve; plasmons, dashed curve.

Enhancement of energy relaxation rates

1679

The dispersion relation splits into the two branches labelled ~ = 1,2 as shown in Fig. 3 for bulk GaAs where n = 5 x 1017 cm -3 so COp= 0.91 COL;[3 = ~ ~ V F --- 3.5 x 107 cms "1 and v = 4.7 x 105 cms -1 an acoustic velocity. For COp> o~L the upper branch corresponding to ~, = 2 is the plasmon-like branch and ~. = 1 is the phonon-like branch. However, at low densities COp<< O~L,~, = 2 corresponds to the phonon-like branch and since in this region there is a weak coupling, only this branch should be retained. We consider the same superlattice system as before and evaluate transition rates from the bottom of the first excited subband to all possible states in the ground state subband via the emission of coupled phonon-plasmon modes. The evaluation of the transition rate is analogous to before and is again given by equations (9) and (10), but with different integration and phonon-plasmon factors.

a

b

Ir ,r:r

//,/ //

3.0 2.s 2.0

050

1.(~

..#,.. i...,I ,l''P" "°

/ 0 /

i

25

i

50

i

i

i

I

i

75 100 125 150 175 200 0 25 50 75 ~0 125 150 175 200 dz(~) d2 (R) Figure 4. Coupled phonon-plasmon mediated electron transition rates and their variation with well width (t2. Results for two doping densities are shown corresponding to (a) top --- COL,(b) o)p --- 1.5 COL. The contributions from individual phonon-plasmon branches are distinguished as - t - lower branch, and ,~ upper branch. The variations of transition rate FI mediated by branch 1 and 1"2 from branch 2 with the well width d2 are displayed in Fig. 4 for two different values of COp(i.e. different carrier densities). The curves are of a similar form as before with a minimum rate at (t2 ---50A and then a maximum for each branch at a threshold well width where the energy level separation is equal to the phonon-plasmon branch energy. In Fig. 4(a) where the density corresponds to o)p = 0.9 COL we see that the results from the two branches are essentially of the same magnitude and exhibit similar variation with well width except in the threshold regions. The contributions from each branch become progressively dissimilar as the carrier density increases which can be seen in Fig. 4(b) for COp_-,-1.5 COL. For these larger electron densities the g = 1 phonon-like branch exhibits only a small transition rate since the LO-phonon-electron Fr~hlich interaction is effectively screened under these circumstances. In conclusion we have shown in this article that electron transition rates in the infinite potential well model and in semiconductor superlattices, for the emission of LO phonons and LO-phonon-plasmons, vary by a factor of up to 100 for different well widths. We have found that the largest transition rate occurs when the electron energy level separation matches either the energy I~0~Lof the LO phonon or the energy hflz. of the coupled phonon-plasmon branch. In the near future, we anticipate presenting results of transition rate calculations which take into account the effect of the heterostructure on the coupled phonon-plasmon modes.

References Abstreiter G, Egeler T, Beeck S, Seilmeier A, Hiibner H J, Weimann G, Schlapp W (1988) Surf Sci 196 613 Altarelli M (1987) Interfaces, Quantum Wells and Superlattices, eds. R. Taylor and C.R. Leavans, Plenum, p. 43. Babiker M (1986) J. Phys C. Solid State Phys 19 683 Babiker M, Chamberlain M P, Ridley B K (1987) Semi Cond. Sci. Technol. 2 582 Babiker M, Chamberlain M P, Ghosal A, Ridley B K (1988) Surf. Sci. 196 422 Bastard G (1981) Phys. Rev. B 24 693 Born M and Huang K (1968) Dynamical Theory of Crystal Lattices (Oxford: Clardenon) Chamberlain M P and Babiker M (1989a) under preparation Chamberlain M P and Babiker M (1989b) under preparation Lassnig R (1984) Phys. Rev. B 30 7132 Richter W (1982) Proc. Antwerp Advanced Study Institute, Priorij, Corsendonk, Belgium Ridley B K (1982) J. Phys. C: Solid State Phys. 15 5899 Ridley B K (1988) Quantum Processes in Semiconductors 2nd end (Oxford: Clarendon) Ryan J (1984) Physica 127B 343 Sawaki N (1986) J. Phys. C: Solid State Phys. 19 4965 Seilmeier A, Hiibner H J, Abstreiter G, Weinmann G and Schlapp W (1987) Phys. Rev. Lett. 59 1347 Shah J, Pinczuk A, Gossard A C and Wiegmann W (1985) Phys. Rev. Lett. 54 2045 Trallero Giner C and Comas F (1988) Phys. Rev. B37 4583 Wendler L and Haupt R (1987) Phys. Status Solidi (b) 143 487 SSE

32/12"--OO