00381098/88 $3.00 + .00 Pergamon Press plc
Solid State Communications, Vol. 65, No. 10, pp. 11851187, 1988. Printed in Great Britain.
I N T E R A C T I O N O F OPTICAL P H O N O N S W I T H E L E C T R O N S IN GaAs Q U A N T U M WIRES M.H. Degani and O. Hip61ito Departamento de Fisica e Ci~ncia dos Materiais Instituto de Fisica e Quimica de Sho Carlos, Universidade de S~.o Paulo, Caixa Postal 369, 13560  Silo Carlos, SP, Brasil
(Received 11 September 1987 by R.C.C. Leite) The energy and the effective mass of an electron in a quantumwell wire of GaAs surrounded by Ga~_xAlxAs is calculated in the presence of electronLOphonon interaction using a variational approach, The polaron mass is found to be dramatically dependent on the sizes of the wire, and also that its magnitude is greater than that in comparable two and threedimensional semiconductor structures.
R E C E N T L Y , A G R E A T N U M B E R of experiments have reported measurements of polaronic effects in quasitwodimensional semiconductor structure [15]. Although the enhancement of the polaron effect in twodimensional systems with respect to the bulk case is by now well established, cyclotronresonance experiments in GaAs heterojunctions have revealed a polaronic correction much smaller than that in the corresponding threedimensional bulk GaAs. In order to explain this result, Das Sarma [6] worked out a model calculation in which the form of the electron subband wave function as well the screening of the electronopticalphonon interaction were taken into account. A generalization of this model was done by Degani and Hiprlito [7] by including in the theory the interaction of the electrons with the interface phonons and the screening within the randomphaseapproximation formalism. It was shown that these effects reduce the effective interaction between electrons and phonons and the results appeared in very good agreement with the experimental measurements. With the recent advances in semiconductor technology it has been possible to confine electrons in extremely thin semiconducting wires, so called, quantumwell wires [8, 9]. In these quasionedimensional semiconductor structures the electron motion is free along the length of the wire and it is quantized in the two dimensions perpendicular to the wire. Recently much theoretical work has been done to understand the electronic problem of such systems. Several authors have reported calculations of mobility of electrons scattered by ionized donors as well as by the optical and acoustic phonons [1012]. The binding energies for the bound states of hydrogenic impurity and excitons associated with the lowest electron and hole subbands have also been calculated [1315]. Therefore, one of the important features of these new
onedimensional semiconductor structures not addressed by any experimental and theoretical investigations until now is the polaronic correction to the electron energy and the effective mass. The purpose of this is to report a first calculation of the polaron effects is a quantumwell wire which is realized by embedding ultra thin wire of GaAs in a confining barrier material of Ga~_xAlxAs. By using the RayleighSchr6dinger perturbation theory we have calculated the polaronic energy and effective mass correction as a function of the size of the wire. For the phonon system we have used the socalled bulkphonon approximation instead of the confined phonon modes, i.e., the quasionedimensional confined electrons are interacting with the bulk phonons of the relevant semiconductor material via the Fr6hlich Hamiltonian. In the calculation we have neglected any nonparabolicity corrections of the electron band mass m. We will show that an enhancement of the polaronic corrections as compared to the bulk GaAs result is obtained. This result which contrasts the twodimensional one make explicit the fact that the dimensionality effects play a fundamental role in this system. In the effective mass approximation an electron interacting with the optical phonons of GaAs in a quantumwell wire is described by the Fr6hlich Hamiltonian as H 
p2 2m + ~ hC°L°aqaq + ~ q
(Fq eiQ'R e tqxx aq +
q
H.C.) + V(y, z),
(1)
where p is the electron momentum, m is the freeelectron mass, R = (y, z) is the inplane projection of the electron coordinates, a~ is creation operator for the optical phonons of wave vector q  (Q, qx) and fre
1185
1186
I N T E R A C T I O N OF OPTICAL P H O N O N S W I T H E L E C T R O N S
quency (DLO. V ( y, z) is the electron confining potential well which will be taken to be V( y, z) = 0 for [y[ < Ly and Izl < L~ and V(y, z) = + ~ otherwise. Fq is the Fourier coefficient of the electron LOphonon interaction given by
fflO~LO( Fq 
h ~1/4(4~'~1/2 \2megL~/ \   ~  J '
~
Vol. 65, No. 10
Here F(Q) is the form factor for the quasionedimensional system which is given by
F.,.~,..mz(O) = f d~
a2ym), (~)nnzm z (Q, ~) , (~2 + Q2),/2
(8)
where (2)
where c( is the usual Fr6hlich coupling constant. The method of calculation [l 6] consists in subjecting the Hamiltonian H to a canonical transformation S which removes the electron coordinate x,
S = exp(i~q~xa~a,).,
(3)
Next, the expectation value of the transformed Hamiltonian is evaluated by choosing for the ground state of the coupled electronphonon system a wave function I~b) which is a product of an electron wave function and a coherent phonon state,
I~> = 4,.y(y)4,.:(z) e 'qxx UlO>, ny, n~ = 1 , 2 , 3 . . . . .
(4)
G,,.my(~) = f dr] ~y(~])l~m),(r]) ei¢n,
(9)
and
Bhang(Q, ~) = f dr] dr]' 4~ .z(n)4%(r])4~ .~(r] )4%(r] ) x exp( ~
+ CZlr]  r]'l).
(10)
In order to obtain the ground state energy and effective mass correction of the polaron we have numerically evaluated the expressions given by equation (6) and (7) for the case nz = ny = 1. The result we have got for the energy shift as a function of the size Lz of the quantumwell wire for several values of Ly are plotted in Fig. 1. For the sake 0fcomparison we have also shown the results for the bulk polaron energy (E = mh~oLO) and for the purelytwo dimensional system (E =  n~/2hCOLO). As we can see from this figure, the polaronic energy can be either larger or
where q~.~(y) and q~.~(z) are the electron wave functions for the motion along the y and zdirections respectively in an infinite rectangular well potential. 10> is the phonon vacuum state and U is a unitary transformation which displaces the phonon coordinates
2.0 1.8 >
1.6 Z hi
with fq to be determined variationally. We get the following expressions for the polaronic energy shift AE%,,~) and the effective mass correction AM~,~,,~) for the (r/y, nz) electronic subband of the system,
2D
Z 0 r~
1.4 O I00,~
1.2
AE(.~,.;) 
2 fdQ
rt
my,mz
Fnyn~mym~(a) 2 (n/Ly) 2 (my2  ny) + (zC/Lz)2(m~
1.0
3D
n~) + Q2 + 1' (6)
O.8
i
0
and
8o~m
E fde
my,mz
[(n/Ly)2(m 2


Q2F.,.~m/.,(Q) n~) + (n/L~) 2 (m;'  n22) +
02
+ 1]3 • (7)
i
i
i
I00 200 300 4 0 0 500 Lz (A)
Fig. 1. Polaron energy shift in GaAs quantumwell wires in units of afiO~LOas a function of the size of the wires. As we can see the curves approach the purely twodimensional result (2D) in the limit when one of the sizes is zero while the other one tends to infinity. The bulk limit (3D) is also recovered at large values of both sizes of the quantum wire.
INTERACTION OF OPTICAL PHONONS WITH ELECTRONS
Vol. 65, No. 10
1187
and does not screen the longranged Coulomb fields of the phonons. In conclusion, we have calculated the polaron energy shift and the electron effective mass correction in the GaAs quantumwell wires with rectangular cross section as a function of the size of the wires. We find that both the energy and the polaron mass increase with decreasing one size of the wire. The results approach the value expected for twodimensional quantum wells of finite thickness by expanding one side of the wire while keeping the other fixed. In contrast to the twodimensional semiconductor structures, we verified that the electronphonon interaction is not screened in quantum wires. For wires of small dimensions the polaronic contribution is much larger than that in comparable two and threedimensional systems.
1.0 0.9 O.B
O.7 ~, o.6 g o.5 g o.4 o.3 02 o.I
REFERENCES o.o o
I00 200 300 400 500 Lz 1~)
Fig. 2. Polaron mass correction in GaAs quantumwell wires in units of ~ m as a function of the size of the wires. The curves approach the purely twodimensional (2D) or the threedimensional (3D) results when one of the sizes is zero and the other one goes to infinity (2D) or both sizes tend to infinity (3D). smaller than that of the corresponding twodimensional semiconductor structure depending on the sizes Ly and Lz of the quantum wire. We also note that the values for the energy approach asymptotically the purely twodimensional regime in the limit when Ly = 0 and Lz tends to infinity. The threedimensional case is also recovered at large values of the quantumwire sizes. In Fig. 2 we show the polaronic mass correction AM/~tm, as a function of Lz for several values of Ly. Again the bulk (AM = ~m/6) as well as the twodimensional (AM = item~8) results are plotted for comparison. From this figure we may note that the values of the polaronic mass in quantum wires of small dimensions are much larger than those in comparable twodimensional wells. It is interesting to stress that the effect of the electronphonon screening which plays an important role in the understanding of polaron effective mass in GaAs heterostructures, has no influence in quantumwell wires of small sizes. In fact, in the quasione dimensional systems the plasma frequency c%(tit% ~ 5 meV [17]) is much smaller than the phonon frequency 09LO~O~LO = 36.2 meV). Then, the electron gas cannot oscillate as fast as the phonons
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