Ground state energy of an exciton bound to an ionized donor impurity in two dimensional semiconductors

Superlattices

and Microstructures,

451

Vol. 5, No. 3, 1989

GROUND STATEETNEXGYOF AN EXITON ROUND To AN IONIZED DONOR IMPDRITY IN 'IUDDIMENSIONALSEXICO-RS B. St&be Centre Lorrain d'optique et d'Electronique des Solides Universite de Metz et Ecole Superieure d'Electricite 2 rue Edouard Belin, 57078 Mets Cedex 3, France and L. Stauffer Laboratoire de Physique du Solide Faculte des Sciences et Techniques,

Universite

de Haute Alsace

4 rue des FrPres Lumiere, 68093 Mulhouse Cedex, France (Received 8 August 1988)

The ground state energy of the (D+ ,X)-complex corresponding exciton

bound to an ionized donor is calculated

semiconductors mass

ratio.

in

variationally

the whole range of the electron to hole

to an

for

2D

effective

Using a 55-term Hylleraas type wave function,

we

found

that the complex remains stable for mass ratios up to 0.88. This value is about two times larger than that, 0.36, we obtain with the same wave function in the 3D case. for

semiconductors

Bound dicted tons

The

excitons (BE) states have been pre-

weakly bound to donors or and

more favorable conditions

are realized in

strictly

2D

and quantum well structures than in bulk materials.

in 1958 when the analogy between

atomic

As a result,

the detection of the (D+,X)-complex

molecular hydrogenic

exci-

acceptors systems

recent ability to grow semiconductor

layered

and

new opportunities

WC3

devices.

In

sophisticated

structures offers

in semiconductor

realized'. Since then, various types of BE have

due to their quasi two-dimensional

the excitons have binding energies8'9

semiconductors'.

rably

In particular, variational calculations3*' performed tion

show that the neutral-donor

acceptor for

within the effective mass approximaand

effec-

tive mass ratio 0 = mE/mE,

Similar hold

than in

the

(2D) nature, conside-

three-dimensional

bulk ones and give rise to excitonic pho-

toluminescence

neutral

BE remain stable against dissociation

all values of the electron-to-hole

(3D)

greater

and

the quantum well (QW) structures,

been observed and studied theoretically

in bulk

many

physics

in

published

lines up to room temperature. conclusions

the case of BE. results

are

may be In

expected very

fact,

available

to few

concerning

whereas a critical mass ratio UC exists above which the ionized donor BE (D+,X) is unstable5-7. As a consequen-

impurity luminescence in QW structures. Most papers are related to neutral BE. In this case,

ce, the localization energy of the latter is often very small so that its experimental observation in bulk semiconductors may be difficult, even at low temperature.

theoretical studiesr7'r8 confirm that the

0749-6036/89/030451+03$02.00/0

the ding

experimental

observationslO_r6

and

the

binenergies are enhanced compared to the 2D-

case. We

expect also a more extended

0 1989

Academic

range

of

Press Limited

452

Superlattices

the

stability of the (D+,X)-complex

semiconductors.

in layered

But, to our knowledge,

no study

has been devoted up to now to this type of In particular,

the lack of estimated values of

its binding energy makes the experimental tification less easy. We have therefore undertaken the electronic in

structure of the

semiconductor

iden-

calculation

the

model

limit of very small

QW

of its bin-

widths.

This

normalized tion

to unity,

constants.

coordinates s=r

to

test the validity of our wave function by comparison with previous 3D studies516. This limit

s, t, u define

-uStZu,

ob-

tained in the 2D and 3D limits.

We have calculated for

the

ground the 2D

in

and 3D limits. Within the effective mass approximation, tron

the the

u

the ground state energy

several values of the effective mass ratio by using a 55-terms wave function

rized

by

the condition

rapidly

converging

l+m+n 5 6

characteusing

the

iterative method previously

The total energy values have been

decimal point.

We use the same method and the same varia-

hole

impurity.

variation method.

described".

those

and

coefficients clmn as well as factor k have been determined by

obtained with six significant

function to calculate

(2)

linear

scaling

are expected to lie between

wave

u = rh,

where r, and rh are the electron coordinates relative to the ionized The

be

elliptic

O
refore

energy of the (D+,X)-complex

the

t = re - reh,

widths

tional

zero. to

olmn being the normaliza-

also to very large QW widths. Thethe binding energies for finite QW

corresponds

state

or

Xlmn are assumed

:

e + 'eh*

0 < s,

surfaces or inter-

The 3D study has been done in order

faces.

functions

re-

may also be used to understand what hap-

pens in doped semiconductor

Vol. 5, No. 3, 1989

1, m, n are positive integers basis

(D+,X)-complex

: the in two limit situations ding energy strictly 2D and 3D cases. The former corresponds to

The

the study of

We present here the

QW.

of a variational

sults

BE.

where

and Microstructures,

figures after the

The (D+,X)-complex

is only sta-

ble if its energy E is lower than the energy ED of the neutral donor ground state.

Our results obtained in the 2D and the 3D cases are reported in Fig. 1.

we represent the complex by an elec-

(e) and a hole (h) bound to a fixed ioni-

zed donor impurity 2D limit,

(D+).

We assume that in the still

the electron and the hole may

be caractzrized by isotropic effective masses * and mh as it is usual in bulk isotropic me semiconductors. Further, we do not take into account

other effects which may arise from the

detailed band structure. It is more difficult

to determine

the mean

value of the ground state energy in the 2D case because the system has no more spherical

symme-

try. Thus we do not use the best wave functions From

the

latter it appears that the electron-to-hole

and

obtained in the previous 3D studies. electron-to-ionized give of take

donor Coulomb interactions

an essential contribution the complex.

Therefore

the corresponding

to the

binding

it is important

distances into

to

account

in the wave function. We choose a Hylleraas-type centered

on

ionized donor. It reads $(s,t,u)

wave function"

the electron rather than

on

the

:

u=mz/m*h

= @(ks,kt,ku)

Figure

1. Ratio of the ground state energy E of

the (D+,X)-complex

@(s,t,u) = 1 c1mn Xlmn

ED as a function Xlmn = olmn exp(-s/2)

slumt"

(1)

to the neutral

donor energy

of o = mz/rni in the 2D and 3D

cases with ED(2D)/ED(3D)

= 4.

Superlattices

and Microstructures,

Tn the 3D case, the complex remains stable if the mass ratio is lower than 0.365, close to obtained6

0.426,

the value, wave

function

constitutes

previously. therefore

in this case.

approximation

Our

a

good

We can reasonably hold

3. B.

St&be

and

G.

Munschy,

Solid

State

Commun. 35, 557 (1980). 4. G. and

Staszewska, L.

Suffczynski,

M.

Wolniewicz,

J.

W. Ungier

Phys. C 17, 5171

(1984). Rotenberg and J. Stein, Phys. Rev. 182,

in

5. M.

we obtain a critical mass

6. T.

that the same conclusion will

expect

453

Vol. 5, No. 3, 1989

1 (1969).

the 2D case. In the 2D case,

ratio close to 0.88. Thus the (D+,X)-complex by far more stable in the 2D than in the

is 3D

Skettrup,

Gorzkowski, 7. L.

Suffczynski

M.

and

W.

Phys. Rev. B 5, 512 (1971).

Stauffer and G.

Munschy,

Phys.

Stat.

case. The 2D energies are equal to about four times those obtained in the 3D case. In particular, in the limit of a zero mass ratio, the

Sol. (b) 112, 501 (1982). 8. G. Bastard, E.E. Mendez, L.L. Chang and L.

2D and 3D energy values are respectively

9. R.P.

to

2.6314 and 0.5731 in units of twice the

donor energy. As a result, ior

equal

the

realized

more favorables

(D+,X)-complex

in strictly 2D semiconductors

superlattices,

are

and

than in bulk semiconductors.

is of considerable

result

conditions QW This

relevance to the

QW

where the energies are expected

to be comprised between the 2D and 3D limits. In the present study we have restricted ourselves to isotropic spherical and non degenerate electron and hole bands. This approximation

becomes

questionable

for materials with In this case ,j = 3/2 type hole band structure. the best result would probably result from the use of an experimental

"mean" hole mass deduced

from the observed exciton spectra.

0.

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