EXCITON BINDING-ENERGY IN A GAAS/ALXGA1-XAS QUANTUM-WELL WITH UNIFORM ELECTRIC-FIELD

Exciton

binding

energy

in

a

GaAs/Al„Ga&

„As

quantum

well with

uniform

electric

field

Der-San Chuu and Yu-Tai Shih

Institute

of

Electrophysics, National Chiao Tung University, Hsinchu, Taiwan, Republic ofChina

(Received 18December 1990;revised manuscript received 7 May 1991)

The efFects ofa uniform electric field on the binding energies ofexcitons and the subband energies in a GaAs/Al Gal Asquantum well are studied bya perturbative variational approach. Our calculation is

based on an e8'ective-width infinite-well model. The eftective-mass mismatch isalso taken into account. Our results show that the electric field causes a large shift ofthe subband energy and exciton peak posi-tion. The calculated results are compared with the data observed from an optical-absorption experi-ment. Satisfactory agreement isobtained.

I.

INTRQDUCTIQN

Recently, growth

of

alternately thin-layered

semicon-ductor heterostructures with controlled thicknesses and relatively sharp interfaces has become possible. More re-cently, much effort has been devoted to the study

of

the

exciton states in these one-dimensional periodic

struc-tures and quantum wells. ' ' A knowledge

of

the exci-ton binding energy is crucial to the interpretation

of

the photoluminescence spectra and photoluminescence exci-tation spectra, which are used to determine the electronic

properties

of

heterostructures. The exciton binding ener-gy is also important because it is now possible to make the layers so thin that some quantum-confinement effects

of

electrons and holes can readily be seen.

For

example, in bulk GaAs, the exciton resonances can be observed only at low temperature. At room temperature, excitonic

resonances are very weak and thus nonresolvable. How-ever, in multiple quantum wells, one remarkable phenomenon happens

—

the exciton resonances can be clearly observed at room temperature. When the electric field was applied in the direction perpendicular tothe lay-ers

of

molecular-beam-epitaxy- (MBE-) grown multiple-quantum wells in a p-i-n doped structure, the field-induced absorption was observed to have a large shift to

lower energies accompanied by some broadening. '

This exceptional quantum-confinement Stark efFect has received a lot

of

attention for devices because it isa very large electroabsorption effect even at room temperature.

This efFect makes possible the fabrication

of

low-energy

optical modulators and switches. Many authors ' have devoted their efforts to studying the mechanism

of

the quantum-confinement. Stark effect. Most

of

the

theoretical works are usually based on the application

of

the conventional perturbation method, the traditional variation principle, Monte Carlo techniques, and exact

numerical approaches. A method based on the Monte Carlo technique and variation principle was developed by Singh ' and Singh and Hong to study the ground-state problem in an arbitrary quantum well. An extensive cal-culation on several variants

of

the single-subband model

of

the infinite and finite wells with interwell Coulomb effects were reported recently by Mo and Sung. Gal-braith and Duggan 'reported a variational calculation

of

the external-field efFect on the exciton binding energy in a double-quantum-well model. Zhu investigated the effects

of

the heavy- and light-hole mixing on the exciton spectra

of

a quantum well in the presence

of

the electric field by taking account the different exciton spinor com-ponents. The agreements with the experimental data

of

these theoretical works are, in general, only qualitative.

It

is, therefore, interesting to employ different theoretical

treatments tostudy the exciton binding energy

of

aGaAs

quantum well with uniform electric field.

In the present work, we propose a simple but much more efficient approximation method to study the bind-ing energy

of

the exciton in atype-I heterostructure with an electric -field.

For

the purpose

of

illustration,

numeri-calcalculation will be performed only forthe GaAs quan-tum well. This is because the experimental data for the

GaAs/Al„Ga& As quantum well isabundant.

II.

THKGRY

We consider a Wannier exciton in a GaAs quantum well sandwiched between two semi-infinite Al Ga& As slabs in the presence

of

a uniform electric field

I'

along the positive z direction (i.

e.

,perpendicular to the material layers). The origins

of

distance and

of

electrostatic po-tential are chosen at the center

of

the well.

To

write down the Schrodinger equation, it is convenient to retain the z-direction-motion Hamiltonian but transform the in-plane motion Hamiltonian into plane-relative coordi-nates:

x

=x,

—

xl, and y

=y,

—

y&. This is because the ex-citon in-plane

_(x,

_y) motion is free except for the Coulomb potential. Then under the effective-mass ap-proximation _{and by dropping} the plane

center-of-mass-coordinates part, the Schrodinger equation can be ex-pressed as

where the exciton Hamiltonian His

EXCITON BINDING ENERGY IN AGaAs/Al„Ga&

„As.

. .

8055 g2 $2 $2

+

2p& Bx By

+

V

B(z„L

—

/4)+F.

,

_„

8 fi 8

₊

eFz, Bz, 2m,

*,

Bz,

+V,

B(z,

I.

—

/4)—

B A B

—

eFz~ 2

The

H,

orHI, are Hamiltonians for an electron or ahole in a one-dimensional confinement potential well acting under a uniform electric field. The

H

_„

isthe Hamiltoni-an for a two-dimensional hydrogen atom. The

Schrodinger equations for

_H„Hh,

and

H

„can

be solved separately.

For

H

_„,

the eigenfunctions canbe solved exactly as

2

k[x +y

+(z,

—

zl, )

]'~

(2) _eime (p,

8)

=

N„

ao

n+

—,' lml

where

+,

,

is the exciton envelope wave function,

E,„,

is the exciton energy and Egp isthe band-gap energy

of

the

material that forms the well, and X exp

pk ao(n

+

—,

')

2lml ~n+Iml m,zmI,& Pj. mel+mh3. 2pk ao(n

+

—,

')

(5a)

H=H,

+Hh+H„+H'(A,

)+E,

=Ho(A,

}+H'(A,

)+E,

„,

where

Ho=H, +H&+H

_{~, and}

H

e

=—

K

h

=—

+eFz,

+

V,

B(z,

L

/4),

—

B

r'

a 2m eFzh

+

Vq

B(—

zh

L

/4),

—

(4a) (4b) Q2 B2 B2 2pi Bx By A, 2 H'(A,)

=

k(

2+

2)1/2 A,e

k(x'+y')'

'

e2

k[x +y

+(z,

—

zz)

]'~

(4c) (4d) is the in-plane reduced mass. _Egp will be assumed to

remain constant under the electric field. The variable

r,

(rh ) is the electron (hole) position vector, m,

*,

(ml*,

,

)

is the electron (hole) effective mass along the z axis,

m,*~ _(m&~) is the electron (hole)

x,

y plane effective mass.

These masses, are, in general, different in the wells and in the barriers, hence they are zdependent. The kinetic en-ergy has been written, for z-dependent masses, in a way which restores the Hermitian character

of

the

Hamiltoni-an. V, (Vh )is the barrier height confining the electron

(hole).

B(x)

is the unit step function:

B(x)

=0

for

x

(0,

B(x)

=1

for

x

)

0,

and

L

isthe well width. The dielectric constant k contained in the electron-hole Coulomb

in-teraction term is assumed

to

be z independent. We shall neglect image effects tosimplify the problem.

Equation (2) cannot be solved exactly; the usual method for solving this problem is the variational

ap-proach. ' _{Here, we shall employ adifferent}

approxi-mation approach, namely, the perturbative variational method (denoted as PVM} to study this problem. This

method has been successfully applied

to

calculate the im-purity level

ofboth

isotropic and anisotropic

semi-conductors. Following the treatment

of

the PVM, a term

of

—

A,e

/k(x +y

)'

isnow added and subtracted from

H.

H

isthen divided into three terms:

with n

=0,

1,2, 3,

. .

.

and m

=0, +1,

+2,

+3,

.

.

.

and

_X„

isthe normalization constant

2 ao(n

+

—,

')

(n

—

fm f)! (2n

+1)[(n

+

fm f)!] '_1/2 (5b)

The eigenenergies for

H„are

pyA, e v _2k2g2(n

₊

1₎₂ 2 A,

R*

(n

+

—,

')

(5c)

where

R

*

=

p~e /2k A' _{is the eff'ective Rydberg}

calculat-ed with transverse effective reduced mass _p~.

For

the Hamiltonians

H,

and H&, one can see that the potential-energy term in Eqs. (4a) and (4b) tends to

+

ao

as z goes to

+

~,

so that the solutions

of

the Schrodinger

equation are restricted by a boundary condition that

re-quires the wave function to vanish at infinity. On the other hand, the potential tends to

—

~

as z goes to

—~,

so that the system does not, strictly speaking, have true bound states.

'

In other words, the carrier initially confined in a well can always lower its potential energy by tunneling out the well when the field is not zero.

However, provided the electric field is not too strong, we still expect to see resonances associated with the so-called

"quasiband" states.

'

Although the actual state

of

an isolated quantum well with an electric field is quasibound,

it may be approximated by a bound state

if

the applied field isnot too strong so that the lifetime ~

of

the carrier

in the well is long enough. Thus one can neglect the

es-cape

of

the carrier outside the quantum well.

'

' _We

will use the bound-state approximation

to

study the sub-band energies in a quantum well.

For

the

GaAs/Ga,

Al As superlattice, the conduction-band-edge offset Vo is about

85% of

the band-gap difference between GaAs and Ga&

Al„As,

thus electrons in the

conduction band

of

GaAs are located, in general, in a

quantum well with finite barrier height Vo.

Therefore, for our problem the Schrodinger equation

for subband energies

of

the electric (or the hole) in a finite-barrier-height quantum well with a uniform electric field can be expressed as

d A d

H,

PF( )

= —

—

P~(z)

—

qFz(bF(z)+ V,P~(z) dz 2m~ dz

dip(z)

QF(z) Z=L/2 where

=&zPF(z»

(6a) _2m,

'

—1/2 (Vo

+

qFL /2

Fz

—

)

M„~z

I&

I,

/2

*'"'=

_m„~z

_&L/2

and the confining potential V, is defined as

0

for~z~)L/2

Vo for~z~

~L/2

.

(6b)

(6c)

where

We shall employ a simple perturbation approach

pro-posed by Lee and Mei ' tosolve

Eq.

(6). The main idea

of

this approach is to treat the finite-height-barrier potential-well problem as the approximation

of

the

corre-sponding infinite-height-barrier potential-well problem with a broadened well width

L

+25,

where 5 is the penetration depth.

To

find out the penetration depth, let us first assume

the electric field is so weak that the bound-state approxi-mation holds. Then the wave function inside the quan-tum well can be approximated as

PF(z)=c

exp

j

K(z)dz

(7a)

0

for

—

L/2

—

5L

~z

~L/2+5~

V

otherwise

.

(9c)

The solution

of

Eqs.(6a) and (9c)can be expressed as

P„(z)

=a„Ai(rl„)+P„Bi(g„)

with the boundary conditions

Nn(z)~z=L/2+5& Nn(z) ~z= L/2—

s&—

(10a)

(10b)

where

Ai(g„)

and

Bi(g„)

are Airy functions;

a„,

I3„are

normalization constants and

If

we now move the well walls at

—

L/2

and

L/2

to

and

L/2+5+,

then outside the new well walls the probability for finding the electrons are rather

small. This is equivalent to saying that for the first-order approximation one can replace the finite-well problem with well walls located at

L/2

—

and

L

/2 by an infinite-well problem with well walls located at

L/2

—

—

5L and

L/2+5~,

as shown in

Fig. 1.

Thus, our problem

corre-sponds to solving Eq.(6a) with Vz defined as

K(z)

=

1/2 2m~ (Vo qFz

Ez

)—

—

_(7b) 2m

(efiF

) 1/3

(E„z+qFz)

. (10c)

The integral contained in

Eq.

(7a) can be evaluated easily. One gets

Equation (10b)yields

Ai(g„+

)Bi(g„)

—

Ai(g„)Bi(il„+

)

=0,

(10d)

2 P_F

(z)=

c exp

3qF

X 3qF

(V

E

)3/2 (Vo qFz

Ez

)—

(7c)

—

where rI„+ _{or r)„} is given by Eq. (10c)with z

=L/2+5~

or

—

L/2

—

6L and the eigenvalues

E„can

be obtained by solving

Eq.

(10d)numerically.

The

Ez

contained in Eqs. (9a) and (9b) is the total

eigenenergy for the finite-well problem [Eqs.(6a) and (6c)]

and is actnally unknown. In the

first-order-The first derivative

of

PF(z) can be expressed as

1/2

dP~(z) 2m,

"

=4

(z) (Vo qFz

Fz

)—

dz

Therefore, the penetration depth at the boundary

z=

L

/21s

dyF(z) F Z= —L/2 —1/2 2m (Vo

—

qFL /2

Fz

)—

(9a) —_L/2-s I /2 L~2+ ~zR

In a similar procedure, the penetration depth at the boundary z

=L

/2 can be obtained as

FIG.

1. A finite-height quantum we11acted on by an electric

EXCITON BINDING ENERGY IN A GaAs/Al„Ga&

„As.

.

.

8057 approximation calculation, the

Ez

is approximated by

the eigenenergy

Ez

'

_of

_{the corresponding}

infinite-barrier-height quantum well. The value

of

q contained in Eqs. (9a) and (9b) is

—

e for the electron, and e for the

hole. One can define the effective well width for the

elec-tron

(v=e)

or hole

(v=h)

as

'11',

_„,

_{'(z„zl„p,}

_B,

A,)

=N„

(p,

6,

A,

)P,

„(z)gl,

„(z),

(12)

where

@„(p,

B,

A.)and P

„(z)

(v=e

for the electron and

v=h

for the hole) are eigenfunctions obtained in Eqs. (5a) and (10a),respectively. The first-order energy correction

is defined as

+effv

~

+

~vL

+

~vR

Now from the above discussion, the solution for total

unperturbed Hamiltonian Ho can be expressed as

correction to the unperturbed energy

E,

"„',(X)

=E.

,

„(X)+E„+E„,

+E„„

where

E„„(A,

)

=-(n

+

—,

')

is the eigenenergy for the two-dimensional hydrogen

atom;

E,

,

(El,

,

)isthe eigenenergy for the electron (hole)

in a finite quantum well with a uniform electric field ap-plied in the direction perpendicular to the well walls; and

E,

is the band-gap energy

of

the well-acting material. Therefore, the total eigenenergy for the total Hamiltoni-an

His

EE,

'„",

(A)=(e'

'(A)IH'(A)le'

'(A))

.

(13)

E

(A)

E

_y

(A)+E

+El,

+Es p+bE

l(A) (14) Since the second-order perturbation correction is in

gen-eral much smaller than the first-order correction, the ex-pression in

Eq.

(13) can be treated as the total-energy

I

For

the ground state, the energy correction b,

E,

'„",(A,)can

be evaluated as

4g

2 _L/2+5 R

aE'.

„".

,

(X)=

'„

f,

_{, ;}

dz,

ly„(z)I'

L/2+6 _ao X

f

dzl, lpga,

F(z)l'

'

₊

lz,

—

zl, I

—

—

lz,

—

zp,I

H

hL 4A,_IZ,

—

_z„

I ao

4xlz,

—

z„

I ao (15) and

m„=

(%,

„,

lm,",

(z,

)

lq,

„,

)

In the practical calculation, the wave function

4,

_„,

con-tained in _m,~or m&~is replaced by the unperturbed wave

function

4',

_„,

'(z,

)

or 4',

„,

'(zl, ), respectively. The

parame-ter A, contained in

Eq.

(14)can be determined _by

requir-ing the fast convergence

of

the perturbation series, and is equivalent tosetting

aE,

'„",(X)

=0

. (16)

After obtaining Xfrom the above equation, one can

sub-stitute this A,into

Eq.

(14)to get the total eigenenergy

of

exciton

_E,

_„,

(A. ) and the binding energy

of

the exciton:

Ell

=E,

_i+El(

_+Es,

p

—

E,„„where

E,

l (E~l) is the

confinement energy

of

the first electron (hole) subband. where

Hi(x)

and

_N,

(x)

are the Struve and Neumann functions

of

order

1.

The effective Bohr radius

ao=kA'

_/pie

and effective Rydberg R

=pie

/(2k

A'2)

are related with the in-plane reduced mass:

pi=m,

iml,

i/(m,

i+mal,

i),

where the equivalent in-plane effective masses m,~and m&~ are defined as follows:

III.

RESULTS AND DISCUSSIONS

+

3 4m_{hh, z} 1 4m_1h,z m1h (17a) mhh,

l

1 4mhh,

.

+

3 4m 1h, Mhh,

i=

1 3

4M.

.

+

4M-.

_,

1 3

4M.

.

.

+

4M.

.

.

(17b)

The Eg p for GaAs and

Al„Ga1

As are assumed to be

1.

424 eV and

1.

424+

1.

247x eV, respectively.

If

one uses

the

65%

—

35/o rule ' _{to share the difference between}

GaAs and

Al„Ga1

As band gaps, then one can obtain

The values

of

the physical parameters used in this work are taken from some previous works. ' ' _The

effective masses for the electron, light hole, and heavy hole inside the well are m,

,

=

0.

0665mo, m1h,

=0.

088mo, and mhh,

=0.

45mo, where mo is the electron rest mass. Those in the barrier are

M,

,

=(0.

0665+0.

0835x)mo, Mll,

,

=(0.

088+0.

049x)mo, and Ml,

„,

=(0.

45+0.

31x)mo, where

x

is the Al

compo-sition which is taken as

x

=0.

32.

The effective mass

of

the holes are anisotropic. In the plane motion, they are

t0'

E I~ ₄ lU

2—

ppA I i ~OQ T20 ELp (

kv/c~)

80'

«E~TRr

C I 20 gQ

f

the heavy kV/cm) dePe ell width&: ndence o ' -geld (in foUI we Electn

-"

(in

me»

o ding en

_f

sition x

=

OO

1.

- xclton bin 250A (Alcompo ]50, 200 and 2500 A.

.

1.

=95,

. rgy de-ell widths: he binding energ l for our

fi Ure that the from th gu field streng

0

ecan note the electrlc-as ng lectric fie ses

for»cre

nt valu re sjgnlfican a constan . Ch mo e oac ease 1sm n bin in hea~y- o corn h calcUlat fol _g d to decl larger e

c

r

exten-ies

we«

fou

the

excito»

energ increases an

t

m we&&. ln l ger quan

sion»

250 o+ C LLl

~

&00 F50 95A CL t00 I ~00 &0 ) FIEL I 20 50 ce o

f

the mean depende

ll;dths:

eld (gnkV/ o )

f

~foUr w Elect -tension (in

=p.

32).

FI .

-e&cqton osition x

=

heavy-hole-e plane e -e L,

=95,

150, 2 ~ tric constant 1S

0

14.eV

~a&s)

The dielec r and 26eV

a"d

_"

k

=1

b roun electro g _. fi ]d stre g ~ sexac t

cal-of

the e ested by p energy

d-iect,rlc a rev'o e-fun

'

n

t

rs as sugge electron en r P _. 16 _{Our re} h electric e hen the hyslcal pa ult shows . fi ld increase '

«o»ca»y

_".

'd

by Ref-h sthat latlon Rs the e

l6

even w ses mono btal hjs show result agr _. Strong as 2 f r a

st«nge.

field»

a ble even e]ectrlc

t.

n ls fe].la th

"

. fi

t

roxjIna lo d the first . Rn jn nl feld

~e

a so l 2 Rnd 3 o l trlc field pe p 0 0 fir e

z)0

regl so that they ld, though

till has alarg . for the ho the eec amp 1 le, an excl ds ate s 0region or t ez te ls stl

r

the electro

the third sta for

energy shift for

t

.

_f

the elec-the d region ~ well. wl ths o ated Rs nger

f

the egect recalcUl he var)at o hole _(L~tr,h

L

95 A )and

t"

d,

trength fo r sl;ghtly tron

(L

eff'e lectric-6e ll idths are a func tion

of

the

11'ective vve 25

(L

—

133 esults show

of

Mille It'ective wld ases slight4' 1,the

electri«e

err, h s increa

oft

e electron 1 decrease t n dePth er-ho es enetra whl e e let- R that

«hea

y

f

h nd-side Pe the decreases

e,

eases more th

~,

- n becau~~

t

e rapidly y the case tron inc tion deP

f

the cas 0 o

f

the elect

.

ical with resp

direction n ersymm

'

ihe negative .

for ihe hole no long

_.

~ Shjft ln

t

d ctjon or osltlon ltlve-z su an have _P ~ ~ the pOS shift 'n

ll ag'ect the carr

electron h._ft

_of

_I,

&Wl

Sitlon»

ron, light hese Po

f

the ~lect r les.

and ene1-g_{les 0}

~ ecrease sUbb h t these energles e calc hole show

t

R Snce the e e and hea y field increases.

f

the electron he electric than those o bband as

t

e islarger lower su a,n energy than the energies h l becomes field 1s large,

'

hat

of

heavy the absorption ole are p shifts o

f

ene gy

t

the excjton P These large an expect t fie]d ln-tive. tra

of

ex ' energy ton, sowe

ca

as the e spectra h ft to lowel

of

the as

a].

arge s

'

d d pendence o reases. the elect _. ~RAS _q

c

shows .

r

y ln a rlC- fle Uantum Figure 2 . blndlng ene gy h le exclton heavy I t&0

EXCITON BINDING ENERGY IN AGaAs/Al Ga&

„As.

.

.

8059 $480— 1470— E C)

~

t4$0 O CL o

t450—

C) t440

GaAs quantum well for several values

of

L

and

I.

The

electric-field-induced increase

of

the exciton size in the layer plane will make the exciton resonances weaken and

broaden. This isconsistent with the observation

of

Wood

et

al.

' Figure 4presents acomparison between our

cal-culated values and the measurements

of

the Stark shift

of

the heavy-hole-exciton and light-hole-exciton peak with the electric field applied in the GaAs-Al GaI As quan-tum well for

x

=0.

32 and

L

=95

A.

The agreement be-tween theory and experiment is better for the

heavy-hole-exciton case. The discrepancy in the

light-hole-exciton case might be due to the overestimation

of

light-hole subband energy. Since the light-hole efFective mass is small, it has a larger ground-state energy; thus, the as-sumption that the barrier height is much larger than the carrier energy might not be very suitable for the case

of

the light-hole exciton.

IV. SUMMARY

't430 l

20 40 60 80 100

ELECVRIC FrELO (&V&cm)

FIG.

4. Variations ofheavy-hole- (dotted line) and light-hole-(solid line) exciton peak positions (in meV) with electric field (in

kV/cm) for temperature T

=300

K, well width L

=95

A, and Al composition x

=0.

32). The experimental data are taken from Ref. 37).

The subband states and the exciton binding energies have been studied by an approximation method. The variations

of

the exciton binding energies with the

electric-field strength and the well width are obtained. The mean in-plane exciton extension is found to increase with the electric-field strength. The excellent agreement

of

our calculated heavy-hole-exciton peak position with the experimental data shows that our bound-state as-sumption is suitable forheavy holes.

The electric-field dependence

of

the exciton binding en-ergy can also be understood by the variation

of

the mean in-plane exciton extension

_(p)

=

(%',

_„,

~p~%',

„,

).

Figure 3

presents the well width and the electric-field dependence

of

the mean in-plane heavy-hole-exciton extension in a

ACKXOWI.EDCMEm'S

This work was supported partially by the National

Sci-ence Council, Taiwan,

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