EXCITON-PHONON SYSTEM IN GaAs-Ga1-xAlxAs QUANTUM-WELL WIRES

The binding energies o f l i gh t and heavyhole exciton-phonon systems i n GaAs-Ga A1 As quantum wires are calculated as a function o f the s i z e s o f the I-x x wire for Several values of the heights o f the barrier potential. I t is found that the corrections due t o exciton-phonon coupling are quite s igni f icant . In the last few years some progress has been done in the study of electronic properties of quasi-one-dimensional .semiconductor structures 11,4 ( . In these systems the electron motion is quantized in two directions (y and z ) perpendicular to the wire and is free along the length of the wire (x) . Sakaki 11 I investigated the electron transport of ultra-thin GaAs-Gal-xAlxAs quantum wires and showed that a highmobility effect would be expected. After Petroff, Gossard, Logan and Wiegmann 12 1 fabricated and observed cathodoluminescence which was attributed to transitions involving exciton states. More recently several authors have reported calculations of the mobility of electrons scattered by ionized donors and also by optical and acoustic phonons. The binding energies of hydrogenic impurity placed in a quantum-well wire of GaAs have been calculated and the results are larger than those incomparable two-dimensional quantum wells 13 1 . One of the important features of those one-dimensional semiconductor structure is the presence of excitons which play a fundamental role in the cathodoluminescence spectra of these systems. Because of the energy-band discontinuity at the interface between the two semiconductors the degeneracy of the valence band of GaAs is removed enough that these may be treated as isolated bands, leading as a consequence to two-exciton systems, namely, a heavy-hole exciton and a light-hole exciton. This paper reports the calculation of exciton binding energies in a quantum-well wire of GaAs surrounded by Ga AlxAs with square cross 1-x section and finite height barrier for the confining potential. We have also calculated the effects of electronand holeoptical phonon interaction on the exciton binding energies and showed that the corrections are quite significant. We find that both the magnitude and the behavior of the polaronic contribution are completely different from those recently calculated using infinite barrier potential 141. Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987546 C5-224 JOURNAL DE PHYSIQUE In the framework of the effective-mass approximation the Hamiltonian of this system can be written as: where E is the GaAs band gap, me, myh(mzh) are the band masses of the g + electron and.the hole in the y(z) direction respectively, R . = (yi,zi) -+ and pi = (pyi, pzi), i = e,h are the in-plane projection of the electron and hoie coordinates and momenta; X, Px are the center-of-mass coordinate and momentum, x, px are the electron-hole relative position and momentum, MT = m + m is the total mass along the x direction, xh e u = mxhme/~* is the reduced mass for the x motion, Be = me/MT and Bh mxh/MT. r+ is the Fourier coefficient of the electronand holea phonon interaction 141 and ~(y,z) is the confined potential well for the electron and hole, ~ ( y , z ) = 0 for I I < L/2 and lzl<~/2, V(y,z) = vov for I y l > ~ / 2 andV(y,z) =Voz for yrl > ~ / 2 . a$ is the creation -+ operator for the optical phonons of wave vector q = (q ,6) and frequency wo. X The binding energy of the exciton will be obtained by choosing a product ansatz for the trial wave function. where t(y) and C(z) are the exact ground-state wave functions for finite square-well potentials, X is a variational parameter, U is a unitary transformation which displaces the phonon coordinates and 10) represents the vacuum state. The expectation value of the Hamiltonian E = ( $ 1 ~ 1 ~ ) ~ has then the following variational form: + E = Eg + Ekin + Ecoul pol (3) where E . is the kinetic energy, Ecoul is the coulombic energy and kln E is the polaronic contribution which can be easyly obtained in a PO 1 standard way 151. We then obtain for the coulombic energy, and for the polaronic energy, where the form factor associate with the quasi-one-dimensional confinement of electrons and hole is given by We have numerically minimized the energy expression given by Eq. 3 with respect to the variational parameter A , with and without the presence of the electronand holephonon interactions as a function of the size of the quantum wire for several values of the height of the potential barrier. In the present calculations we have used the following physical parameters: E= 10.9, E~ = 12.5, kuo = 36.77 meV, me = 0.0665 mo 1 1 (mo is the free-electron mass), myhh = (yl + y2) mop my,, = (yl-y2) mot 1 mzhh = (yl 2y2) mop mzRh = (yl + 2y )-lmo, where yl = 6.85 and y2 = 2 2.1 for GaAs; yl = 6.85 3.4 x and y2 = 2.1 1.42 x for Gal-xAlxAs. The masses in the xdirection are the same as in the ydirection. The values of the potential barrier for electrons and holes are taken to be 60 % and 40 % of the energy-band-gap discontinuity AE AE = 1.04 x + g' g 0.47 x2. The results we have obtained for the exciton binding energies with and without the presence of phonons are shown in the figures as a function of the size of the wire and for several values of the potential barrier height. There are several interesting features to be noted from the results of our calculations. Firstly one notices that the values of the exciton binding energies are about twice larger than those in comparable two-dimensional quantum wells 161. These results are consistent with the observed cathodoluminescence in quantum wires by Petroff et a1 , whose observed the value of the exciton binding energy at 8-10 meV higher than that in two-dimensional quantum wells. We may note from the figures that for a given concentration x the values of the exciton binding energies have the same qualitative behavior as those previously obtained for the two-dimensional quantum wells, i.e., the exciton binding energies increase with increasing the size of the wire, reache a maximum value and finally decrease monotonically for larger wires. We also can observe that for a given x the light-hole exciton binding energy is systematically higher value than the heavy-hole exciton energy. As it can be seen, by interchanging the values of the potential barrier heights in the two directions perpendicular to the wire (which are represented in the figures by a pair of values of A1 concentration) we obtain different values for the exciton binding energies: the masses of the holes are not the same in the two dire~.tions perpendicular to the wire. Finally the figures also show that the electronand holeoptical phonons interaction effects are extremaly important, mainly in the limit of thin wire of GaAs. C5-226 JOURNAL DE PHYSIQUE I n conclusion,we have presen ted t h e r e s u l t s f o r t h e e x c i t o n b ind ing e n e r g i e s with and wi thou t t h e presence o f phonons a s a f u n c t i o n o f t h e s i z e of t h e GaAs quantum w i r e and f o r s e v e r a l v a l u e s of the h e i g h t s of t h e p o t e n t i a l b a r r i e r s . We have shown t h a t t h e p o l a r o n i c c o n t r i b u t i o n i s extremaly impor tan t and can n o t be n e g l e c t e d .