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

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

M. Degani, O. Hipólito

To cite this version:

M. Degani, O. Hipólito. EXCITON-PHONON SYSTEM IN GaAs-Ga1-xAlxAs QUANTUM- WELL WIRES. Journal de Physique Colloques, 1987, 48 (C5), pp.C5-223-C5-226.

�10.1051/jphyscol:1987546�. �jpa-00226750�

Colloque suppl6ment au n0ll, Tome novembre

EXCITON-PHONON SYSTEM IN G ~ A S - G ~ , - ~ A ~ ~ A ~ QUANTUM-WELL WIRES

D e p a r t a m e n t o d e ~ l / s i c a e C i S n c i a dos M a t e r i a i s , I n s t i t u t o d e

~ l / s i c a e ~ u l / m i c a d e

C a r l o s , U n i v e r s i d a d e d e

P a u l o , C . P . 369, 13560

C a r l o s ,

P a u l o , B r a s i f

ABSTRACT - The binding energies o f l i g h t - and heavy- hole exciton-phonon systems i n GaAs-Ga I-x A1 As quantum wires are calculated as

a function o f the s i z e s o f the wire

h e i g h t s o f the b a r r i e r p o t e n t i a l . I t

found

t h e corrections due t o exciton-phonon coupling are q u i t e s i g n i f i c a n t .

In the last few years some progress has been done in the study of electronic properties of quasi-one-dimensional .semiconductor struc- tures 11,4 ( . In these systems the electron motion is quantized in two directions (y and

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 high-

-

mobility 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 se- veral 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 in- comparable two-dimensional quantum wells 13 1 . One of the im- portant features of those one-dimensional semiconductor structure is the presence of excitons which play a fundamental role in the cathodo- luminescence spectra of these systems. Because of the energy-band discontinuity at the interface between the two semiconductors the de- generacy 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 electron- and hole- optical phonon in- teraction on the exciton binding energies and showed that the correc- tions 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,

(yi,zi) and pi -+

(pyi, pzi), i

e,h are the in-plane projection of the elec-

tron 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 electron- and hole-

phonon interaction 141 and ~ ( y , z ) is the confined potential well for the electron and hole, ~ ( y , z )

for I I

L/2 and l z l < ~ / 2 , V(y,z)

vov for I y l

~ / 2 andV(y,z) = V o z for

> ~ / 2 . a$ is the creation

-+

-

operator for the optical phonons of wave vector q

(q ,6) and fre-

quency wo.

The binding energy of the exciton will be obtained by choosing a pro- duct ansatz for the trial wave function.

where t(y) and C(z) are the exact ground-state wave functions for finite square-well potentials,

is a variational parameter, U is a unitary transformation which displaces the phonon coordinates and 1 0 ) 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

We then obtain for the coulombic energy,

and for the polaronic energy,

We have numerically minimized the energy expression given by Eq. 3 with respect to the variational parameter

with and without the presence of the electron- and hole- phonon interactions as a function of the size of the quantum wire for several values of the height of the poten- tial barrier. In the present calculations we have used the following physical parameters:

10.9,

kuo

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

2.1 - 1.42 x for Gal-xAlxAs.

The masses in the x- direction are the same as in the y- direction. 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 com- parable 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 perpendicu-

lar to the wire. Finally the figures also show that the electron- and

hole- optical phonons interaction effects are extremaly important,

mainly in the limit of thin wire of GaAs.

C5-226

JOURNAL DE PHYSIQUE

I n c o n c l u s i o n , w e h a v e p r e s e n t e d t h e r e s u l t s f o r t h e e x c i t o n b i n d i n g e n e r g i e s w i t h and w i t h o u t t h e p r e s e n c e 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 t h e h e i g h t s o f t h e p o t e n t i a l b a r r i e r s . We h a v e 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 e x t r e m a l y i m p o r t a n t and c a n n o t b e n e g l e c t e d .

HEAVY

-

Heavy-hole and light-hole exciton binding energies as a function of the size of the quantum wire for different values of the height of the potential barriers. The numbers in the parenthesis indicate the A1 concentration in the directions y and z respectively. The solid and dashed curves correspond to the case with and without the electron- and hole- phonon interactions.

REFERENCES

1. SAKAKI, H . , J p n . J . Appl. Phys.

19

2 . PETROFF, M.; GOSSARD, A . C . ; LOGAN, R . A . a n d WIEGMANN, W . , Appl.

Phys. L e t t .

41

3. BRUM, J . A . , S o l i d S t a t e Commun.

54

4. D E G A N I , M.H. a n d H I P ~ L I T O , O . , Phys. Rev. B

2

5. DE BODAS, E.L. a n d H I P ~ L I T O , O . , Phys. Rev. B

27

6. DEGANI, M . H . a n d H I P ~ L I T O , O . , Phys. Rev. B

35