LONGITUDINAL-OPTICAL-PHONON EFFECTS ON THE EXCITON BINDING-ENERGY IN A SEMICONDUCTOR QUANTUM-WELL

Longitudinal-optical-phonon

efFects

on

the exciton

binding

energy

in

a

semiconductor

quantum

well

Der-San Chuu

Institute ofPhysics, National Chiao Tung University, Hsinchu, Taiwan, Republic of China Win-Long Won

Department ofElectrophysics, National Chiao Tung University, Hsinchu, Taiwan, Republic of China Jui-Hsiang Pei

Department

oj

Science and Mathematics Education, National Hsin Chu Normal College, Hsinchu, Taiwan, Republic of China

(Received 29October 1993)

The longitudinal-optical-phonon effect on the exciton binding energies in a quantum well con-sisting of a single slab of GaAs sandwiched between two semi-infinite slabs ofGa~ Al As are studied. Byusing the Lee-Low-Pine unitary transformation, the Hamiltonian can beseparated into two parts which contain the phonon variables and exciton variables, respectively, providing that the virtual phonon-electron and virtual phonon-hole interactions are neglected. A trial wave function,

which is able to reproduce the correct exciton binding energy in a quantum well, is obtained by using a perturbative variational technique. The trial wave function consists ofa product of the envelope function in the z direction (perpendicular tothe layers) for electron and hole and a purely two-dimensional exciton wave function. The dependence of the ground-state binding energy, which

includes the effect ofelectron-phonon interaction on the well width, is investigated. Itis found that the correction due to polaron effects on the exciton binding energy is quite significant for a well

width ofseveral hundred angstroms and the effects ofeither surface phonons or bulk phonons on the binding energy of the heavy-hole exciton is always larger than that of the light-hole exciton. Our results are compared with some previous results, and satisfactory agreements are obtained.

I.

INTRODUCTION

During the past decade it has become possible to grow systems consisting of alternate layers of two diferent

semiconductors with controllable thickness and relatively sharp interfaces using epitaxial crystal growth techniques such as molecular beam epitaxy and metal-organic chem-ical vapor deposition. This situation allows the exper-imental studies of excitonic states to become possible. One ofthe most interesting features of photoexcited sys-tems isthat the electrons and the holes interact strongly

at low temperature their kinetic energy islow enough for

them to combine and form bound states (free exciton).

This free exciton is analogous

to

a

hydrogen atom or,

more exactly, to positronium, since the mass of

a

hole is typically about the same as the mass ofan electron. In

some semiconductors an exciton can exist for a relatively long period oftime

—

more than 10 psec in silicon before the electron-hole pair annihilates. The recombination in-volves a characteristic emission which can be detected

to obtain information. The exciton in the quantum well

behaves like

a

quasi-two-dimensional hydrogen atom; be-cause that the exciton binding energy is observed

to

in-crease asthe quantum well width decreases. The binding energy for an exciton is typically 100—1000times smaller

than that of ahydrogen atom.

Themost extensively studied heterostructure isthe one consisting of alternate layers

of

GaAs and Gaq Al As with layer thicknesses varying &om

a

few monolayers

to

more than 400A.. The quantum states and band structure

of

a

GaAs-Gaq Al As quantum well has been observed by many authors.

Both

ofthese two compounds have zinc blende lattices. For instance, the direct gap ofGaAs is

at

K

=

0 where the conduction band has

a

nonde-generate minimum and. the valence band has

a

threefold-degenerate maximum if the spin is neglected. The in-clusion of spin and spin-orbit interactions modi6es the bands by splitting the sixfold-degenerate valence states

into _{an upper fourfold (}

1

=

3/2) state and alower twofold

(1=

1/2) state separated by spin-orbit splitting

A.

Sev-eral recent studies have shown that Gaq Al As has a

direct band gap

at

the

I

point for Al concentrations less

than about 40%

(x

=

0.

4).

The conduction and valence

band discontinuities at the interface have been shown to

be about 85'% and 15%, respectively, of the band gap

difFerence between the two semiconductors. Thus elec-trons and holes in the GaAs matrix 6nd themselves in a

potential well whose height depend on Al concentration

in the surrounding Gaq Al As layers. The transport

and optical measurements of this heterostructure have

been investigated. The optical spectra are often

domi-nated by excitonic transitions from hole subbands to

tron subbands. The intersubband optical (or

magneto-optical) transitions in GaAs-Gaq Al As quantum wells

have been observed. Theabsorption,

magnetoabsorp-tion, and luminesence excitation spectra in high quality

samples clearly display exciton features. Most

of

the

at-tention has been focused onthe excitons formed between

the lower-lying electron and hole subbands, the latter be-ing split into heavy-hole and light-hole subbands.

The binding energies

of

the exciton in

a

quantum well have been studied extensively by many authors. '

The binding energies of the ground heavy-hole and

light-hole excitons are still matters ofexperimental and

theo-retical controversy. 2' ' _{The magneto-optical}

observa-tions give binding energies which are consistently larger

than those deduced &om the absorption or luminescence

excitation experiments. ' _{In previous works, the hole}

was usually treated as a particle with either the heavy-hole mass or the light-hole mass. Most previous calcu-lations employed the variational approach. ' _Some

works ' _{included the nonparabolicity}

_of

_the

conduc-tion band and the degeneracy

of

the valence band.

It

is well known that an electron staying in

a

low-lying level of the conduction band of

a

polar crystal will

in-teract strongly with the longitudinal optical mode of

lat-tice vibrations. We may picture the electron as it

moves through the crystal accompanied by a cloud of

phonons. Since the

III-V

materials used in producing typical quantum well structures are polar crystals, an electron weakly bound in this system will interact with

the phonons of the host semiconductor and so increase

the donor binding energy. On the basis of the strong coupling scheme, Ercelebi and Ozdincer calculated the

ground state binding energy of the exciton-phonon sys-tem in GaAs/GaA1As quantum well structures and found

that the corrections due

to

electron-phonon coupling are

rather significant. Degani and Hipolito also found the

polaronic contribution

to

the exciton binding energies is quite significant and increases with decreasing well thick-ness. Riicker et aL calculated the electron-LO-phonon

scattering rate in quasi-two-dimensional systems, based

on afully microscopic description of the phonon spectra

and concluded that interface phonons are ofgreat

impor-tance.

As mentioned above, most previous theoretical studies employed the variation principle

to

construct the trial

wave functions. However, the construction

of

variational

trial wave functions relies heavily on physical intuition. And the errors involved in the construction are usually

dificult

to

estimate. Since

a

good knowledge

of

the

ex-citon binding energy is essential for an accurate

inter-pretation of the experimental observations, therefore, it

should be interesting

to

employ

a

more reliable method

to

study the binding energy of the exciton system more

accurately. In this work we concentrate on the effect

of

electron-phonon coupling on the binding energy

of

Wan-nier excitons in quantum well structures. Wewill employ

the perturbative variation technique

to

construct

a

trial

wave function which is separated into z and the

x

—

y

coordinates. The interaction between the electron and

surface phonon and the electron and bulk-longitudinal phonon will be taken into account. Lee-Low-Pines

trans-formation will be applied

to

separate the electron and phonon variables. The reason that we take

a

separable form wave _{function is partly} because

of

the numerical cal-culation involved using such

a

wave function is simpler. Another reason is because it has been shown that the

separable trial function is able

to

yield reasonably good results for thin layers.

II.

THEORY

The Hamiltonian

of

an exciton associated with

ei-ther the heavy-hole band or the light-hole band in a

GaAs slab sandwiched between two semi-infinite slabs

of

Gaq Al As interacting with the longitudinal optical

phonon can be expressed within the &amework

of

effec-tive mass approximation as

H

=

H,

+

Hh+ H,

h+

Hsp+

Hsp

+Hsp

—h

+

HBp

+

HBp—e

+

HBp —h& where h2

H,

=

—

V',

_+V,

2m.

Hh

=-

Vh+V,

„,

2mh

—

e2 eo

f~.

-ra]'

Hsp

=

)

fuu,

at

aq, q

H,

h —— 2z,

H».

=)

r,

_L

(4)

+H.c.

2z

)

r

em*( a——,₎ e—e*( a+—,)

+H.c.

HBp

=

)

hush„bI„

A: HBp

=

)

Wy

8(z,

_{) cos}

k,

z,

e' "'"'~~bic

+

H.

c.

, k HBp g

=

)

Wgg 0(zg) cosk zg e' """~~ bit„+ H.

c.

,

(io)

where V, _{(z )}and Vh, (zg) are the well potentials seen by

the electron and hole, aq is the annihilation operator for

optical surface (SO) phonons

of

wave vector

_q

=

(q„,q, )

and frequency ur„bg, is the annihilation operator for the

optical bulk

_(BO)

phonons

of

vector

k

=

(k„,

k,

) and

fre-quency u~, Hsp is the surface phonon energy, and

Esp,

sur-face phonon and the electron (hole) in the well. HBp is the bulk phonon energy and HBp ~-(Hnp g) is the

in-teracting hamitonian between the bulk phonon and the

electron (hole) in the well. And the interaction strengths are42 44 fzrhcu, e21

'

I'~

=z

(

s'Aq

₎

m,

@~=e

(E~

Ep) 2h tdb

j

=e,

h

where V is the crystal volume, co and e are the static

and high &equency dielectric constants, and

~,

and ~b are the dispersionless SO and

BO

phonon energies.

Now applying the Grst and second Lee-Low-Pines transformation4 for the two-dimensional system

fop

—

1 s

—

11

(op+1

s

+1)

(12)

Ui

—

exp

i

r,

„.

)

—

qataz

—iz, .

)

k„b&

bie,

(18)

k L —L 0( ) 0)

z)

L(z(L

2

orz(

U2

—

—

exp z

r"

ll

)

qa~ a~ zz_i'i

)

_k„bi,bz, ,

(19)

1 U~ gk k

~'I

(14)

Us

—

—

exp ~

)

(a

fs

—

aq

f*)

U4

—

—

exp

)

(b&tgq

—

bz,g&)

(20)

(21)

h

)

1/2

U~

=

brut 4~zj~ I

(2m~us)

where_tron and

r,

holeandpositions and one obtainsrp, are the in-plane projection of the

elec-K:

U4 U3 U2 U~

K

Uy U2 U3 U4) (22) h2q2 h2q2

H

=H, +Hg+H,

g+)

hzd,

+

+

I(a + f*)(a

+f

₎ 2m

2m')

+)

I'

e—

~l.

l(a

+f)+H.

c.

+)

I'

e ~l

_~l(a

+f)+Hc

h2k2 h2k2 &

+)

.

I h

s+

"

₊

"

I (bi,

+»)(bA:+»)

2m,

2m'

₎

+)

W,i,

8(z,

) cos

k,

z,

(bi,

+

g~)

+

H.

c.

+

)

Wgg 0(zg)

cosk,

zz, (by

+

gA,)

+

H.

c.

In the above derivation, we have neglected the terms involving the virtual phonon-electron and virtual phonon-hole

interaction. The trial wave function for our system is assumed as a product ofthe exciton part and phonon part:

I

~)

=

4(

) I &), (24)

where

P(r)

depends on the exciton coordinates and I

g)

depends on the phonon coordinates. Since the major

contribution to the energies ofthe polaron system comes from the Coulomb interaction between the electron and hole, the realistic energies for the polaron states can be obtained only ifthe excitonic part can be solved more accurately.

To achieve this goal we shall perform a perturbative variational approach

to

obtain more accurate eigenenergies for

the excitonic

part.

Before we perform this, let us first manage the phonon part. The I

g)

can be expressed as

I

g)

=

P

(~(a+)

I

0),

and I 0) is the phonon vacuum

state.

For the low-lying polaron states, I

g)

can be taken

as I

0).

Then (

H')

=

(o I 4

'(

)

H'

0( ) I o)

.

(

hzq2 h2qz

_l

=(&()

IH.

+H +H.

—.

l@())+).

Ih

.

+

+

2m 2m',

₎

.

(

h2kz h2kz

₎

+)

r,

P.

,

f,

+H.

c.

++I'

_Pg,

f,

+H.

c.

+)

I h

~+

"+

"

Iqgq~ 2m, 2m',

₎

+

_Q

Wg

a,

ggz

+

H

c

+)

Wig

ai

i

.

gz.

+

H.

c.

, (25) (26)

where —L/2&z,zp,&L/2 l _q(z,—

-)

—_q(z,

+-)

L

x

I@(r)I

x dx,

dy, dz, dxg dyI,dzg, (27)

H

_y —— H'(A)

=

8'

₎

Ae'

y')

spaz'+

y2'

Q2

-h2

02 2

+

Vh (zh), 2mh ~zh g2

(

g2 2p~

(Bx

,

+

8

At. 2

(39)

(4o) (41) Phq

=

—_L/2&z„zp,&L/2 2z L ~q( ~—_{g) ~}—q(

~+g)

L

L

xl&(

)

l'

x

dz,

dy, dz, de~dy~ dz~, (28)

in the above equations and A is treated as

a

parameter

which can be varied tomake the perturbation term H'(A)

as small as possible. _py is the reduced mass correspond-ing

to

heavy

₍₊₎

or light ₍

—

_{) hole bands} in the plane perpendicular

to

the z direction. The potential wells seen by the electron V,

(z,

) and by the holes Vh (zh) are assumed

to

be square wells with well width

L:

—L/2&z,,zg&L/2

e(z,

)cos

k,

z,

~

P(r)

~'

xdx,

dy, dz,dzh, dye dz~, (29) O,

iz,

i

&L/2

V

iz i&L/2

0, [zh

[&L/2

Vh~(

h)—

(42) (43) —L/2&zan, zp,&L/2 8(zh) cos

k,

zh ~

P(r)

~

xdx,

dy, dz,dx~dyh, dz~.

The parameters fq,

f',

gh, and gh can be obtained by

minimizing the

_(H

_{) with} respect to the parameters _fq,

f

',

gh, and gh, respectively,

Here we have chosen, without loss

of

generality, the origin

of the coordinate system

to

be

at

the center ofthe GaAs

well. The heights of the potential well V, and Vj, are

determined Rom the Al concentration in Gaq Al As, using the following expression for the total band gap discontinuity: 1'q

_&"

+

_{1'q ~hq} $2qQ fPq2 'I 2m, 2m'

~~k

ek

+

~ak

~ak h~k~ h~k~ b _{2m 2m}

fq=(f;)',

gh

=

(gh)*

(31)

(33)

(34)

AE

=

1.

155x

+

0.

37x eV. e2 z2

(Ll

(H')

—

-

R~

—

~ Eppp

(a

j

(44)

The value ofV, and Vh are 85%and

15'

of

EEg,

respec-tively. Now let us roughly estimate the magnitude of the term

H'

by the uncertainty relation for the ground state exciton:

Using these parameters, one obtains finally

(H')

=

(&(r) I

H.

+

Hh+

H.

hI

&(r))-)

-

I F'q

I'

I

&.

q+

&hq

I'

a

s

+

"".'+

_2m,

"'"

_2m', 2

-

~ lVeh o'eh

+

lVhh chh ~ h _k„

+

h k„ k 2m, 2m'

Now let us turn

to

the excitonic part. TheHamiltonian

of

an exciton associated with either the heavy-hole band or

the light-hole band in

a

GaAs slab sandwiched between

the two semi-infinite slabs

of

GaAS-Waq Al As is

mod-A

ified by adding

a

term

"'

to

and then subtracting

~2+@2

the same term f'rom the Hamiltonian and rearranging as follows:

for small well width

L

(&

a

where

L

is the GaAs well width, a

=

ep7i2/p~e is the transverse efFective Bohr

radius, and

R

=

pye4/2sp52 is the three-dimensional ef-fective Rydberg calculated with transverse reduced mass

py.

Hence as

_(L/a)

((

1,the term

H'

in

H

can be taken as a small perturbation. The introduction

of

the varia-tional parameter Aensures that the term

H'

can be made

as small as possible. One can note that the eigenfunc-tions

of

Ho can be solved exactly. The motions of the

electron or hole along the z direction are just those

of

the one-dimensional square well potential. The solution

to

the transverse part

H

_„

is

just

the two-dimensional hydrogen problem. For illustration, we shall consider the

ground state only; it isstraightforward

to

obtain the

ex-cited states. Now the ground state eigenfunction and

eigenvalue for

_{H, (H,}

_„)

and

H

„can

be expressed as

where

H

=

Hp(A)

+

H'(A)

=

H,

.

+

H,

„+

H.

„+

H'(A), (36)

(37)

where

~'=A,

cos

k,

z„

/

z.

) &L/2

B,

exp(

—K,

/

z,

/), /

z,

f

)

L/2 (45)

—

h2 02

H„=,

+

V,

_(z.),

2m, Oz

(38)

2m,

E,

h2 (46)

2M,

(V,

—

E,

) 62

k, L

K

=

k,

tan

2 (48)

and

E,

isthe ground state energy ofelectron in potential

well

of

height V,.

M,

is interpolated effective mass of

electron in the Gaq Al As material and

m,

=

0.

067mp, m+

—

—

0.

45mp,

=

0.

082mp,

M,

=

(0.

067+

0.

083x)mp, M+

=

(0.

45+

0.

2x)mo,

M

=

(0.

082

₊

0.

068x)mp, cp

—

—

12.5,

(»)

(52) (53) (54) (55) (56) (57) l. t'4&1

c".

(s»4)

=

—

I

—

I 2i)'

(a)

Ep

—

—

-4A

B.

(49) where mp isthe free electron mass, Hence the wave

func-tion ofthe unperturbed part for the lowest subband

ex-citon can be written as

It

was known that Ga~ Al As is direct for

x

(

0.

45 the longitudinal effective masses in the GaAs region (de-noted by m) and in the GaAli As region (denoted by

M)

are

@(a,4,

z)

=

f.

(z.

)

fi

(za)@oo(S,

4)

(58)

and the first order perturbation to the ground state en-ergy is

«,

"(")

=

j

&z.

))'*lz.

₎₎

f

&z.

f~)z~))

H'("))O~o)S

₄₎₎

p~p&4

a2 2

dz.

f.

(z,

)I' dz, lfi,(z),)

'

—4A Ae

~dp—

—4A

pe

dp

p'+

Iz.

—

zi,l' (59) ]6$2e2 OO

dz.

I

f.

(z.

)

I'

dz.

I

fs (zs)

I' P —OO —OO a 7t &4&I

z.

—

_«

I )

(4&

lz.

—

_«

I )

-+

lz.

—

«I

——

Iz

—

«I

4

'

2

'

₍

a

₎

g a

)

(60)

where

_{Hi (x)}

and Ni

(x)

are the Struve and Neumann function oforder

1.

Based on the first order perturbation energy, the fast

convergence condition requires KEYED l(Ap)

=

0, which

yields the optimum value Ap for the variational

parame-ter.

Hence the binding energy of the ground state exci-ton without the polaron effect, which is defined as

—

Ep,

is 4Ap2B.

as

L

decreases. This result is similar to that obtained previously. Furthermore, the value ofAp for the

heavy-hole exciton islarger than that for light-hole exciton. The

variation ofbinding energy with well width for x

=

0.

15

is displayed in

Fig.

2. One can note that the binding energy decreases as the well width increases. This is because as L is reduced, the exciton wave function is

090--—

--

——

III.

RESULTS

AND

DISCUSSIONS

0.85—

0.80—

—

—

heavy-hole

light-hole

We have calculated the binding energy of the heavy

hole

_(EIi~)

and the light hole

(E~I,

) exciton of GaAs quantum wells for different Al concentrations

x

0.

15,

0.

2,

0.

25,

0.3,

0.

4 as a function of the well width

L.

The physical parameters are adopted from the previous works. The reduced masses in the x

—

y plane for the heavy-hole

(J

=

3/2) and the light-hole

(J

=

1/2) exci-tons are taken as

0.

04mp and

0.

051mp, respectively. The

reduced mass associated with

1

=

3/2 band is smaller

than that

of

J=

1/2 band.

In

Fig.

1 we display the variation ofAp as afunction of

the well width

L

for both heavy-hole exciton and

light-hole exciton with x

=

0.15.

The range of widths

con-sidered is between 30and 300 A. One can see from

Fig.

1 that for a given value of x, the value

of

Ap increases

E GJ m CL C: O 070 Cg 0.65— 060-- 055-0 50 100 150 200 width(A) 250 300 350

FIG.

1.

Thevariation ofAoasafunction ofthe well width L for both heavy-hole and light-hole excitons with concentration x

=

0.15.

13 12— light-hole

——

heavy-hole 10— I E o) ₉ tD C UJ O) c 8 C tD 50 100 150 200 Width(A) 250 300 350

FIG.

2. The variation ofheavy-hole and light-hole exciton binding energy (meV) as afunction of the well Lfor concen-tration

x=0.

15.

13

compressed in the quantum well, thus leading

to

increase binding. When

L

is larger than a certain value of

L,

the

spilling

of

the wave function becomes more important and this makes the binding energy get closer

to

the bulk value. One can note &om the 6gures that the binding energy

of

heavy-hole exciton issmaller than that

of

light-hole exciton. This is because the conduction-band

non-parabolicity is enhanced by quantum con6nement.

Ac-cording

to

the magneto-optics observation

of

Rogers et al. the heavy-hole effective mass for motion in

a

layer plane iswell thickness dependent and decreases consider-ably for decreasing well widths due

to

the decoupling

of

the light- and heavy-hole subbands, while the light hole exhibits electronlike dispersion relations with

a

large ef-fective mass inthe layer plane. This makes the binding energy ofheavy-hole exciton smaller than that

of

the light

hole exciton. In

Fig.

3we plot the variations ofbinding

energies with respect

to

the well width

L

for heavy- and light-hole excitons with

x

=

0.

15,

0.

2,

0.

25,

0.3.

We 6nd

that the binding energies decrease as

L

increases and the

variation with

L

isalmost independent

of

the concentra-tion

z.

The effects of the surface phonon and bulk phonon

on the exciton binding energy for both heavy-hole and

light-hole exciton in

a

quantum well for Al concentration

z

=

0.

15 and

0.3

as a function of the well width

L

are presented in Figs. 4 and

5.

One can see that the Al

con-centrations cannot signi6cantly influence exciton binding

energies. Table

I

presents our calculated exciton binding energies with and without the phonon effect for several well widths and concentration

x

=

0.15.

We also list the

percentage

of

the phonon effect in the last column

of

Ta-ble

I.

One can see &om Table

I

that the polaron effect on

the exciton binding energy is in general smaller for

nar-12 g4 C v' CJ 0l a 10 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I 30 90 120 150 180 ₀ 240 270 300 WelIWidth(A)

FIG.

3. The variation ofheavy-hole and light-hole binding energies (meV) as a function ofwell width Lfor concentration

13 10— 12 9 lD E 8— Cl C I) C 7 C LQ 6— 11 (D E 10— Q) e 9 O) T3 C Q3 5— Is « iI Ii I III ii I III « II iI iII i ii I iI II 4 0 50 100 150 200 250 300 350 400 450 Width(A) 12 13 50 100 L 150 200 250 Width(A) 300 350 y10 E 9 Q C 8 C C CQ x=0.3 12—

)

E ~10--e 9 CD

8-CO 4I, II 1 II I II III I iI Ii I iI I I Ii I I i I iI i I I I i I I 50 100 150 200 250 300 350 400 450 (b) Width(A)

FIG.4. The variation ofheavy-hole exciton binding energy (meV) as a function ofwell width _(A)for Ep (curve 1), Ep+SO

(curve 2),Ep+BO (curve _3),and Ep+SO+BO (curve 4).

5

0 250 300

I

50 100 150 200 350

WIdth(A)

FIG.5. The variation oflight-hole exciton binding energy (meV) asa function ofwellwidth _(A)for Ep (curve 1), Ep+SO

(curve 2),Ep+BO (curve _{3), Ep+SO+BO} (curve 4).

row well width and becomes more pronounced as the well width becomes wider for either heavy-hole or light-hole

exciton. Table

II

presents

a

comparison ofour calculated

results with recently observed

data.

In order to make

a

clear comparison, our results are calculated for some

spe-cific well widths and concentrations. One may note that the phonon effect on the heavy-hole exciton is larger than that on the light-hole exciton and the total binding en-ergy oflight-hole exciton is larger than that ofheavy-hole

exciton. This may be due

to

the effective mass difference

as we mentioned above. One may also note from Table

II

that, although the observed data have some degree of

uncertainty, our results agree satisfactorily with most re-cent observations, ' _{except the experimental data}

ofMaan e.t

al'4

Figures 6 and 7show the plot

of

the variation of per-centage of effects of surface-phonon, bulk-phonon, and

the total phonon effects on heavy-hole and light-hole

ex-citon binding energies. Some interesting results can be

noted in these two figures. The effect of the surface phonon decreases very fast for both cases of very large and very small well widths. Forvery large well width the reason for the decrease of the surface phonon effect on

the total exciton binding energy is due

to

the large

pos-sibility ofthe exciton existing in the interior of the bulk semiconductor and thus reduce the interaction strength

of

the exciton and the surface phonon. On the contrary,

the decrease ofthe surface phonon effect on the total

ex-citon binding energy for very small well width is caused

by the leakage

of

the exciton wave function out of the quantum well because the well potential height is finite.

For the intermediate well width, the surface phonon

ef-fect on the total binding energy increases with well width for smaller well width, but after a maximum value the

surface phonon effect decreases very fast for larger well width. This might be due

to

the smaller the well width,

the more pronounced SO phonon modes, which in turn increases the importance of the exciton-surface phonon coupling. However, the wave function begins

to

leak out

of

the quantum well for well widths shrunk to a much smaller value so that the surface phonon effect reaches a

maximum value for a well width of about 80 A.

It

may

be worthwhile

to

analyze here the competition between

the surface phonon and the bulk phonon modes. One

can see from the figures that the surface phonon effect grows larger than the bulk phonon effect when the width isreduced

to

a

certain value. Therefore, the influence of

the phonon effect on the exciton binding energy is

dom-inated by the surface phonon as the well width becomes

small while the polaron effect is dominated by the bulk phonon as the well width becomes large. This is because asthe well width decreases, the exciton isclose

to

the

het-erojunction and the confinement on the exciton becomes

important which makes the interface phonon modes

be-comes more pronounced. This in turn makes the

TABLE

I.

The binding energy (meV) ofheavy hole and light hole exciton as afunction of the

well width _(A) with and without the phonon effect.

E'

and Eo are the binding energies with and without the phonon effect.

Width A 30 40 50 60 70 80 90 100 150 200 250 290 300 30 40 50 60 70 80 90 100 150 200 250 290 300 Exciton Eo 10.28

9.

61

9.

09 8.66 8.31 8.01 7.74 7.51 6.67 6.10 5.68 5.42 5.35 12.30

11.

39 10.68 10.12

9.

65

9.

26 8.93 8.64 7.62 6.97 6.51 6.21 6.15

Binding energy (meV)

Exciton with phonon effect

E'

Heavy-hole exciton x

=

0.15 10.66 10.25 9.92 9.63 9.36 9.12 8.89 8.68 7.85 7.26 6.80 6.50 6.43 Light-hole exciton

z

=

0.15 12.42

11.

63

11.

05 10.60 10.23 9.92 9.66

9.

42 8.52 7.89 7.41 7.10 7.03

(&' —

&s)/&' (%%uo)

3.63 6.27 8.44 10.07

11.

28 12.19 12.90 13.45 15.12 15.94 16.41 16.64 16.68 0.95 2.04

3.

30 4.54 5.68 6.82 7.54 8.28 10.56

11.

60 12.17 12.50 12.44

the interaction between exciton and bulk phonon. One

can also note that the inBuence

of

the bulk phonon

ef-fect on the exciton binding energy increases with well

width and thus the

total

phonon effect (including the surface and bulk phonon efFects) increases with increasing well width. Our calculated percentage

of

phonon efFects

reach

a

saturation value of about

17.

5% for heavy-hole

exciton and

13.

5% for light-hole exciton. One can see

from Figs. 6 and 7 that the percentage

of

the efFect of

either surface phonon or bulk phonon on the heavy-hole

exciton are always larger than that on the light-hole

exci-ton. This is because the efFective mass

of

the heavy hole

TABLE

II.

Comparison ofour calculated results with some observed data.

Width (A) 75 75 75 75 92 92 100 100 110 110 112 112 Reference 8. Reference 15. 'Reference 25. Reference 14. 'Reference 50. 0.4 0.4 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.3

0.

3 Hole level Hg Hg Lg Hg Lg Hg Lg Hg Lg Hg Hg

Experimental data (meV)

10.5—

11.

5

11.

3—12.3 10—12,

9.3'

11, 11

9.

5—10.5

11.

2—12.2 13 10 8-9.5, 8.4' 9,b

11'

12'

Our results (meV)

9.

56 10.45

9.

43 10.31

9.

01

9.

80 8.87

9.

63 8.69

9.

42 8.56

9.

31

18_I— 16 14-O g t

.

12'-0 k 10— (g 0 CL P 0 8 G) CD 6--C (U O I h 4 CL t-2 r 0.— I-x=0 15

——

curve 1 curve 2 curve 3 14 ( 12— O C)

~10-= 8I-6— Q) C C e 4— O I P 0—

/

//'

/

x=0. 15

——

curve 1 curve 2 curve 3 I I I I I I . I I I I I I I I I I I I I I I I

~~~

J 0 50 100 150 200 250 300 350 400 450 width(A) 50 100 150 200 250 300 350 ( & ) width(A) 18 p ———

16;

4 o 12t P 0 10'-P 0 8 E Q o 6 CL 4 I-P P" I-0I 0 100 200 Width(A) x=0.3

——

curve 1 curve 2 curve 3 300 400 14

I—

12 I-~10 C-(g 8— 3. 0 Ch rO C O L Q

2-50 x=0.3

—

—

curve 1 curve 2 ———_{curve 3} 100 150 200 250 300 350 ('t,_{) Width(A)}

FIG.

6. The variation ofpercentage of phonon effect on the heavy-hole exciton asafunction ofwell width (A) for SO (curve 1),BO (curve 2),aud SO+BO (curve

3).

FIG.7. The variation of percentage of phonon effect on the light-hole exciton as afunction ofwell width (A) for SO (curve 1),BO (curve 2),and SO+BO (curve

3).

along the zdirection is heavier than that ofthe light hole; therefore the heavy hole is bound more tightly than light hole and yields a smaller interaction range. Therefore,

a larger effect will be over a short distance for the sur-face phonon effect. The same reason can be applied

to

the bulk phonon case, so that the effect of

BO

phonon on the heavy-hole exciton isalso larger than that on the

light-hole exciton. Ercelebi and Ozdincer considered the electron (hole) lattice interaction and introduced a

variational trial wave function in their calculation. They

obtained an enhancement on the binding energy which increases as the well width becomes larger and larger. However, their results yield more than 30%phonon ef-fect as the well width becomes larger than 150A.. Their

results seem to overestimate the phonon effect. Degani and Hipolito studied the phonon effect on the binding energy of exciton and obtained a result of26—

20%

effect

ofphonons as the well width ranges from 10to 150 A..

Comparing with the recent observations as listed in

Ta-ble

II,

our calculated results seem to be more reliable.

The results show that polaronic effects are important and cannot be neglected. Our results predict that the correc-tion due

to

the polaron effect on the exciton binding en-ergy ranges Rom several percent in the small well width

to 17%in the bulk case.

It

has also been found that the energy correction due to the polaron effect isdominated by the SO phonon when the well width is suKciently small while the electron and bulk phonon interaction be-comes important when the well width becomes larger.

The effects of either electon-surface-phonon interaction

or electron-bulk-phonon on the binding energy are larger for the heavy-hole exciton. This may be ascribed

to

the

effective mass difference for the heavy hole and the light hole. We also found that the

total

binding energy of the light-hole exciton is larger than that

of

heavy-hole

exci-ton. This is consistent with recent observations and other

theoretical calculations.

IV.

SUMMARY

ACKNOWLEDGMENT

Ia.this work we studied the longitudinal optical surface and bulk phonon effects on the exciton binding energy.

This work is partially supported by the National

L.

L.

Chang, L.Esaki, and

R.

Tsu, Appl. Phys. Lett. 24,

593(1974).

R.

Dingle, W. Wiegmann, and C.H. Henry, Phys. Rev.

Lett.

33,

827 (1974).

R.

Dingle, in Festkorperprobleme/Advances in Solid State Physics, edited by H.

R.

Queisser (Vieweg, Braunschweig,

1975),Vol. 15, p. 21.

J.

Batey,

S.

L.Wright, and D.

J.

DiMaria,

J.

Appl. Phys.

57,484 (1985).

H.

J.

Lee,L.

Y.

Juravel,

J.

C.Woolley, and A.

J.

S.

Thorpe, Phys. Rev.

B 21,

659 (1980).

R.

C.Miller, D. A. Kleinman, W.

T.

Tsang, and A. C.

Gossard, Phys. Rev.

B

24, 1134

(1981).

C.Weisbuch,

R.

C.Miller,

R.

Dingle, A. C.Gossard, and

W.Wiegmann, Solid State Commun.

37,

219

(1981).

P. Dawson, K.

J.

Moore, G. Duggan, H.

I.

Ralph, and C.

T.

Foxon, Phys. Rev.

B

34,6007(1986).

K.

J.

Moore, P. Dawson, and C.

T.

Foxon, Phys. Rev.

B

34,

6022 (1986).

G. Duggan, H.

I.

Ralph, and K.

J.

Moore, Phys. Rev.

B

32,8395(1985).

E.

S.Koteles and

J.

Y.Chi, Phys. Rev.

B

37,

6332(1988).

J.

Lee,

E.

S.Koteles, and M. O.Vassell, Phys. Rev.

B 33,

5512(1986).

F.

Y.

Juang,

Y.

Nashimoto, and P. K. Bhattacharya,

J.

Appl. Phys. 58, 1986(1985).

J.

C. Maan, G. Belle, A. Fasolino, M. Altrarelli, and K.

Ploog, Phys. Rev.

B

30,

2253 (1984).

D. C. Rogers,

J.

Singleton,

R.

J.

Nicholas, C.

T.

Foxon, and

K.

Woodbridge, Phys. Rev.

B

34,4002 (1986).

J.

A. Brum and G. Bastard,

J.

Phys. C

18,

L789 (1985). G.Bastard,

E. E.

Mendez, L.L.Chang, and L.Esaki, Phys.

Rev.

B

26, 1974(1982).

R.

L.Greene,

K.

K.

Bajaj,

and D.

E.

Phelps, Phys. Rev.

B 29,

1807(1984).

R.

L.Greene and K. K.

Bajaj,

Solid State Commun. 45,

831(1983).

Y.

Shinozuka and M.Matsuura, Phys. Rev.

B

29, 3717(E) (1984).

C.Priester, G. Allen, and M. Lannoo, Phys. Rev.

B 80,

7302 (1984).

T.

F.

Jiang, Solid State Commun. 50, 589(1984).

G. D. Sanders and

Y.

C. Chang, Phys. Rev.

B

31,

6892

(1985).

G. D. Sanders and

Y.

C. Chang, Phys. Rev.

B 32,

5517

(1985).

U. Ekenberg and M. Altarelli, Phys. Rev.

B 35,

7585

(1987).

R.

Dingle, A. C.Gossard, and W.Wiegmann, Phys. Rev.

Lett.

34,

1327(1974).

R.

C. Miller, A. C. Gossard, D. A. Kleinman, and O. Munteanu, Phys. Rev.

B

29,3740 (1984).

R.

C. Miller, D. A. Kleinman, and A. C.Gossard, Phys.

Rev.

B

29,7085 (1984).

M. H. Meynadier, C. Delalande, G. Bastard, M. Voos,

F.

Alexandre, and

J.

L.Lievin, Phys. Rev.

B

31,

5539(1985).

T. F.

Jiang and D.S.Chuu, Physica

B

1B4,287(1990).

X. X.

Liang, S. W. Gu, and D.L.Lin, Phys. Rev.

B

84,

2807(1986).

A. Ercelebi and M. Tomak, Solid State Commun. 54, 883

(1985).

A. Ercelebi and U.Ozdincer, Solid State Commun. 57,441

(1986).

M. H. Degani and O. Hipolito, Phys. Rev.

B 35,

4507

(1988).

S. W.Gu and H. Sun, Phys. Rev.

B 37,

8805 (1988).

C.Guillemot and

F.

Clerot, Superlatt. Microstruct. 8, 263

(1990).

X.

X.

Liang and

X.

Wang, Phys. Rev.

B

43,5155

(1991).

K.

T.

Tsen, K.

R.

Wald,

T.

Ruf, P.

Y.

Yu, and H.Morkoc, Phys. Rev. Lett. 67, 2557

(1991).

H. Rucker,

E.

Molinari, and P. Lugli, Phys. Rev.

B

44,

3463

(1991).

C. Y.Chen, P.W.Jin, and S.Q.Zhang,

J.

Phys. C4,4483

(1992).

Y.

C.Lee, M.N. Mei, and K. C.Liu,

J.

Phys. C

15,

L469

(1982).

J.

Sak, Phys. Rev.

B

B, 3981(1972).

M.Matsuura, Jpn. Phys. Soc.

41,

394 (1976).

R.Zheng and S.Gu, Solid State Commun. 59, 331 (1986).

T.

D. Lee,

F.

E.

Low, and D.Pines, Phys. Rev.

90,

297

(1953).

P.Lawaetz, Phys. Rev.

B

4,3460

(1971).

Handbook of Mathematical Functions, edited by M. Abramowitz and

I.

A. Stegun (Dover, New York, 1965).

K. K.

Bajaj

and C.H. Aldrich, Solid State Commun.

35,

163(1980).

A. Baldereschi and N. O. Lipari, Phys. Rev.

B 3,

439

(1971).

S.Tarucha, H. Okamoto,

Y.

Iwasa, and N. Miura, Solid State Commun. 52,815 (1984).