Splitting of donor-acceptor ground state in zink-blende type semiconductors

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Splitting of donor-acceptor ground state in zink-blende type semiconductors

C. Benoit À La Guillaume

To cite this version:

C. Benoit À La Guillaume. Splitting of donor-acceptor ground state in zink-blende type semicon-

ductors. Journal de Physique I, EDP Sciences, 1991, 1 (3), pp.317-321. �10.1051/jp1:1991107�. �jpa-

00246324�

Classificafion

Physics

Abstracis 71.55 72.80

Shom Communication

Splitting of donor-acceptor ground state in zinc.blende type sendconductors

C. Benoit la Guillaume

Groupe

de

Physique

des

Solides(*),

lbur

23,

Universitfs Paris VII et Paris

VI,

2

place Jussieu,

75251 Paris Cedex

05,

France

(Received

30 October199t~

accepted

in

final form

21

December1990)

R4sum4. La

perturbation

coulombienne exerc6e par un donneur neutre sur

l'accepteur

neutre le

plus proche produit

un dfdoublement de l'ftat de base de la

paire

dans )es semiconducteurs de type blende. Un bon accord entre le mod61e et (es donn6es

exp6rimentales

est obtenu pour Cdlb et GaA8.

On discute )es facteurs

possibles

de d6saccord dans Znse _et GaP

Abstract. The Coulomb

perturbation

of a neutral donor on the nearest neutral acceptor

gives

rise to a

splitting

in zinc-blende type semiconductor8. Good agreement ^of

experimental

data vith the model is found in Cdlb and GaA8. Possible _reasons of

discrepancy

found in Zhse and GaP are

discussed.

t. Intluduction.

In _a

previous publication [II,

it was shown that

acceptor ground

state in zinc-blende

(ZB) _type

semiconductors _was

split by

the

quadrupolar

moment of the

perturbing potential weighted by

the

spatial

dhtribution of hole around the

acceptor

^{site. In this}

paper,

we

give

an

application

to the

splitting

of

acceptor ground

state

perturbed by

the nearest neutral donor

[2];

^{data from} selective

pair

luminescence have been

published

for several ZB semiconductors

[3-~;

this allows

meaningfull comparbon

^with the model. In section

2,

the

theory

h

presented

and

application

to several

ZB-type

semiconductors is

given

in section 3.

(*)

Unitf de Recherche assoc16e _au Centre National de la Recherche

Scientifique.

318 JOURNAL DE PHYSIQfJE I N°3

2.

Splitting

of

donor-acceptor Q3A) ground

state.

Using

the notation of reference

[II,

the

splitting

^{SF of} an

acceptor perturbed by

a

potential V(r)

reads:

SF

=

4CiC2 / _V(r)

exp (-2rla) _{[z~ (z~}

+

y~) /2) ^d~r (1)

where

Cl

and

C2

are normalization coefficients of the Kohn-Schechter wavefunction and _{a =}

2/(a/~

₊

_{ap~ ),}

al and _a2

being

the Bohr radii of thin

multicomponent

wavefunction. In _{our case,} the

perturbation

is that of _a neu~al donor [6]. The

corresponding potential

caused

by

the

positive

localized

charge

screened

by

the donor electronic cloud b:

V(r)

_{= 211y}

(1

+

ad/r)

_exP

(-2rlad) (2)

where

fly

is the effective

Rydberg

and _ad the effective Bohr radius of the donor

(noting

that

211yad

=

e21e,

with _e the electronic

charge

and _e the dielectric

constant). Integration

of

(I)

can be

performed using bipolar

coordinates

d3r

-

rir2/R dridr2d~ (see

inset of

Fig. I):

CO R+r2

SF =

8CiC2w/R _dr2 r2V(r2) R-r~( dri no (ri, r2) exp(-2ri la) (3)

where

Q (ri, r2)

_"

3R~ /8

+

r)/4 3r(/4

₊ 3

(r) r()~ / _(8R~)

The

integral

over

dri

can be done

analytically

and the last one over

dr~

has to be done numer-

ically.

£ 2

$

r, r2

(

A°~

Z ~

CdTe

4O 60 _R Ill 80

Fig.

I.

Acceptor splitting

in Cdlb as a function of

donor-acceptor separation.

The _{curves are} the result of

present

^model for an effective _mass acceptor ^{Bohr radius}

(curve I)

and for a Bohr radius reduced

(curve 0.9).

The

experimental points

are from reference [3]. The inset shows the

bipolar

coordinate system ^{used in}

equation (3).

3.

Application

to several

ZB.type

semicondoctors.

Calculation of SF as a function of R have been

performed

^for several

semiconductors, using

the

parameters

^{of table} [7~;

they

are

compared

to available

experimental

data.

Figures

I and 2 show the results for Cdlb and

GaAs,

the

experimental

data

coming

kom selective

pair

luminescence

[3, 4];

^the calculations are done for nominal values of the

parameters,

^{I-e- for} "eflective mass" donor

and

acceptor.

^{In order to allow for}

deeper _acceptor,

the calculations _were

repeted assuming

_a re-

duction of the

acceptor

^{Bohr radius}

by

a factor

0.9;

this causes a reduction of the

splitting by

about 159b _at

large pair separation.

^We can see that the

agreement

^between

theory

and

experiment

^is

quite good, conceming

the R

dependence

as well as the absolute value.

TMngs

become worse at

short dhtance

~fig. 2),

but our

perturbation approach

h

probably

inaccurate in that case.

lhble I. Material

parameters

^{used in this calculadon}

gram Re( / ).

_eo is the static dielectric _con- stant ~i, 72, 73 the valence band

Luthnger parameters,

_p ^{is the}

strength cfthe spherical spin-orbit

in-

teracfion

(p

₌

(673

+

4~2) /5~i

b the cubic

anisotrcpy (b

₌

(~3 ~2) /71)

,

neglected

in the

present

paper ^and m~ the elec~on

effective

mass.

eo 71 72 73 P ^'b me

9.

GaP

~

(

^~

~

~~~~

g + +

=

#

50

Rp~ll)

Fig.

Z The _{same as}

figure I,

but for GaAS.

Experiments

are from reference [4].

Figure

3 shows our results for Znse as

compared

to the

eTpedments

of reference

[5]:

^{in that}

case there is _a

dhcrepancy concerning

the absolute

value,

the

experimental

data

being

lower than

theory by

a factor 2.5. One

possible

_reason for such _a failure could be the

stronger

electron

(hole)

32o JOURNAL DE PHYSIQfJE I N°3

LO

phonon coupling

ⁱⁿ

Znse;

_on the one

hand,

this _can be _{a source} of

inaccuracy

in the deter- mination of the valence band

parameters;

in

addition,

to our

knowledge,

there is _no theoretical

treatment of the

acceptor

states

including

at the same tiJne

polaron coupling

and the

complexity

of ZB valence band.

p ^Znse

Em

# h

= j

5

#

~~ ~~

RpA Ill ^~~

Fig.

3. The _{same as}

figure I,

but for Zhse.

Experiments

are from reference [5].

Finally,

_we consider

again

^the case of close

pain

in GaP

[8, 9]. Morgan

introduced _a model where the

acceptor ground

state was

split by

the

inhomogeneous

stress caused

by

the nearest

substitutional donor. Thb model

produced good

fit to the data with reasonable values of the pa- rameters

(deformation potential); however,

_a

puzzling point

_was that the

experimental splittings

were not

dependent _upon

the size of the

donor, along

the series of substitutional donors

O, S, Se,

16 [9]. We have tried _our model also for

GaP;

the results _are shown in

figure

4 with the

eTperi-

mental data of reference

[8].

^One can see that the

prediction

of _our model h _too

large by

a factor about 2. But _our model is

certainly

inaccurate for several reasons:

first,

the calculation _{assume a}

spin-orbit splitting

at the

top

^{of the valence band much}

larger

than the

acceptor binding

_energy,

a

point poorly

verified in GaP

Second,

_as in

Znse,

the

strength

of electron

(hole)-LO phonon coupling might

be _{a cause} of

discrepancy. Third,

the model _{assumes a}

spherical

donor wavefunc-

tion;

this _was the _case for the three first

examples,

but not in GaP with the

complicated

camel's back structure around the X

point

at the bottom of its conduction band. Notice that a more re-

alistic model of the electron cloud would allow for some

anisotropy

of

SF;

such _an

anisotropy,

observed in the

experiment (see Fig. 4),

was

explained

in

Morgan's

model

by

the

anisotropy

of the deformation

potential.

4. Conclusion.

16

conclude,

among ^the two sources of

acceptor splitting

which have been

considered,

electrostatic

perturbation

_or strain field of the

donor,

our calculation shows that the first one

gives always

a

significant

contribution. In the case of Cdlb and

GaAs,

it is

clearly

the dominant _one;

attempts

^to

fit the data with the second _one in references

[3,

4]

requires

unrealhtic values of the deformation

potential.

On the other

hand,

the _case of GaP remains open: both _sources could

contribute;

let

us add that the

signs

of these contributions

might

be

different, giving

rhe to some

compensation.

For

completeness,

the existence of the electron-hole

exchange splitting

is mentionned at

last,

but its

magnitude

h smaller

[10].

@ Gap

%

' %,

%

I %/

I §

II §

II §

> '

§

~ II 'T §

E (( "'_"' %Wm

~p Ii 11,1 ~

~ '

l I

I j'j

t T

l ~

l" j'

m

I I"

~

l '" '>

~

~ (~

((l

11',j

'

~

j

~

'

lj

j

I

j ,

l

l I

1

~~ ~~ ~~

RO~ Ii) ^~~

Fig.

4.

Experimental

data

((, )

^ofS, C D-A

pair splitting

in GaP from reference [8]. The hatched _curve

gives

order of

magnitude

of the pre8ent model.

References

[I]

BENOrr A_LA GUIUAUME C., ^Solid State Commttn. 4s

(1983)

51.

[2] The

validity

of the _treatment of acceptor

ground

state

splitting

in reference

[I]

was in fact re8tricted to

quadrupolar perturbation

of uniaxial symmetry. This is indeed the _case for the

application

_pre-

sented here.

[3] NEU

G.,

LEGROS R. and &%BERING

Y.,

L Lttminescence 24/25

(1981)

159.

[4] LEYMARIE

J., Thesis, Nice,

France

(1989).

[5j TEws H. and NEU G.,

Phys.

Rev 825

(1982)

1253.

[6~ The _same

perturbative approach

was

previously

used

by

BARU

VG.,

PETRov VA. and SANDOMIRSXiI

VB.,

SOY

Phys.

Semicond 9

(1976)

1344. But

they

^{did consider} the

perturbation by

an ionised

donor.

[7j BALDERESCHI A. and LIPARI

N.o., Phys.

^Rev 88

(1973)

^2697.

[8] MORGAN ^{TN. and} MATER

H., Phys.

Rev Lett. 27

(1971)

1200.

[9] MORGAN TN., Proc. of the XI Int. Conf. _on the

Physics

of

Semicond.,

lAbr8aw, Ed. Baranowski

(1972)

_p. 989.

[10] ^Cox R-T and DAVIES

J-J-, Phys.

Rev 834

(1986)

8591.

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