Binding to a pair of marginal potentials : a test for the stability of N- in GaP

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Binding to a pair of marginal potentials : a test for the

stability of N- in GaP

C. Benoît À La Guillaume

To cite this version:

C. Benoît À La Guillaume.

Binding to a pair of marginal potentials :

a test for the

sta-bility of N- in GaP. Journal de Physique Lettres, Edp sciences, 1983, 44 (1), pp.53-58.

�10.1051/jphyslet:0198300440105300�. �jpa-00232142�

Binding

to

a

_pair

of

_marginal

_{potentials :}

a

test

for the

_stability

of N- in

GaP

C. Benoit à la Guillaume

Groupe de _Physique des Solides de l’E.N.S. _(*), Tour _23, Université de Paris VII, 2 Place Jussieu,

75251 Paris Cedex 05, France

(Re~u le 17

_septembre

1982, revise le 12 novembre, accepte le 16 novembre

₁₉₈₂₎

Résumé. 2014 On montre _{que le comportement} des

_paires

distantes NN- fournit un test de stabilité de N-. On _propose une méthode améliorée de calcul de

l’énergie

de NN-, ainsi que du comportement

asymptotique

à

_grande

_{distance R pour} un

potentiel près

de la limite de liaison. On démontre que

l’énergie

de liaison à une

_paire

de

_potentiels

«

_marginaux

» est une fonction universelle de R, et

indé-pendante

de la nature du _potentiel. On conclut que N- n’est pas stable dans GaP. Abstract 2014 It is shown that _a test for the

stability

of N- can be found from the behaviour of _pairs

NN-i at _{large separation.} An

_improved

method of calculation of the energy of NN-i is

_presented

and its

_asymptotic

behaviour at _large

_{pair separation R}

is obtained for _potential near the limit of

vanishing binding.

It is shown that the

_binding

energy to a _pair of «

_marginal

»

_potentials

is a universal

function of R, independent of the nature of the

_potential.

It is concluded that N- is unstable in GaP. Classification

Physics

Abstracts 71.55F - 78.55D

1. Introduction. - N substituted

to P in GaP is the most

_thoroughly

studied shallow isoelec-tronic centre

_[1].

N

_being

more

electronegative

than P causes an attractive

_short-range

_potential

for the electron

_[2]

but one does not know if N- is stable or not in GaP. In this paper, we _propose

to find a test of

_stability

of N- from the

_{study ofNN,’,}

an electron bound to a

pair

of

nitrogen;

good experimental

data about

_NN~

have been obtained

_by

Cohen and

_{Sturge [3]}

from excitation

spectroscopy

of the luminescence of excitons bound to

_pairs.

_Here,

we _propose an

improved

method,

yet very

simple,

to

_compute

the

_ground

state of

_NN¡-

as a function of the

_pair

sepa-ration and the

_depth

of the localized isoelectronic

_potential.

In

addition,

we found an

_asymptotic

law for the

_binding

_{energy E}

to a

_pair

of

_{« marginal»}

_potentials

at

_{large pair separation R :}

ER 2

= _const.,

independent

of the

nature

of the

potential.

A

simple

test for N -

stability

is derived.

2. Theoretical calculation of

_NNi

_ground

state. - Let

V(r)

the isoelectronic

_potential

spheri-cally symmetric.

It is assumed of constant

_shape

but with a

_prefactor

_{( 1}

+

_{e). V(r)}

is chosen such that

_being

short-range,

the

_binding

_energy of the electron is

_{just vanishing :}

this is our

defi-nition of a «

_{marginal » potential.}

Actual

potential

(1

+

_{E) V(r)}

has a

_ground

state of energy

E(s)

for e > 0.

(*) Laboratoire associe au CNRS.

L-54 JOURNAL DE PHYSIQUE - LETTRES

For a

pair

at

_separation

R,

we use a variational

method,

with a trial wave function of the

_bonding

type : ’

~,

centred on

impurity

i

_(i

=

1,2)

is the solution for _a

potential

(1

₊

yy) ~(r):

where 11

is the variational

_parameter.

This trial wave function is exact in the two

_limiting

cases R = 0 where 1

+ 11 =

2(1

₊

e)

give

the

_solution,

and R -+ oo

_{where 11}

= _a Thus it is

expected

to be

quite good

at _any value

of

R ;

it

gives

much better results then the

_{choice 0 - exp( -}

_rla)

as used in

_{[4]. The}

variational

energy

E

1 is :

where

Numerical calculations as a function of e and R have been

performed

in the case of a _square well

-

77~

potential :

_{V(r) _ -}

for r

b,

V(r)

= 0 for _{r >} b

(b

is the radius of the localized

potential).

4b

The results are shown in Section 4.

We find that the minimum

_Elm

_{of E,}

is obtained for a

_{value r~}

=

1. such that

E, m ~ ~(~m)’

This is not

_surprising

for the

_following

reason : outside the two localized

_potentials,

the

_ground

state wave function of NN- is a linear combination of

plane

waves _{of energy}

_Eo,

where

_Eo

is the

_ground

state _energy. The ansatz

₍₁₎

outside the two localized

_potentials

is also a linear

combi-nation of

_plane

waves of _energy

E(q).

If

(1)

is a

good

_ansatz, then

clearly

one should have

_Eo

=

E( r~m).

3.

_Asymptotic

behaviour. - We consider the

case ~ =

0,

7! -~ _oo. Since the ansatz

(1)

is exact

at ~ -~ oo, we can use the «

_{physical »}

_argument

of the end of Section 2 and

_replace

the

minimi-zation of

_{E1, by El}

=

E( r~)

in

equation (3),

hence

.

Indeed,

we show in the

appendix

that we can arrive at the same result

_by

a direct

(but

more

_tedious)

derivation from

_equation

_(3).

We take ? = _m = 1.

To evaluate

A,

B and

_C,

we

consider fo

=

r~o ;

fo

is solution of

The normalization is chosen such that

_fo

= 1 _{at r} -+ oo. For a small _r~,

_f~

is

_{essentially equal}

to

_fo

~n

as

_usually,

the main contribution comes from the

_region

outside the

_{potential, introducing}

a

factor N 2 -

k .

So _we have

factor N~ =

y2013.

So we have

~n

=

Nfo/r

inside the

_potential,

(6)

4>~

= N

exp( - kr)/r

outside the

potential.

(6)

Then one obtains

where

and

IA 1

has

the

dimension of a

_length.

It can be considered as the « effective » radius of the

_marginal

potential

(for

a _{square well :}

IA ’ - -

8

&/77

~).

IB

can be

computed using

(5)

This condition does not introduce any limitation on V for the

_following

reason : if when r -+ oo,

/-

1 -

M~thenr~ -

r-’and ~ -

~~~~~

_-~ Oif~ >

0,

and so ~should

_decay

faster

fo

-- I - ur -S then

r

dfo dr ,~

r -S a n d y ~,

r S+2~

’ r dfo ~ dr p

if s >

₀

and so V should

_decay

faster

~"

dr dr

than

~’~.

This is a _necessary condition for a

_marginal

_{potential [5].}

7~

has the dimension of a

length (7~

= -

77~

&/12

for _{a square}

well).

The

approximate

_way we

have

_computed

C

_implies

that V should

decay

faster than

r - 3.

For

_large

_R,

₍₆₎

show that A > B ~> C. This comes from the _power

dependence

in

R,

and not

from the

_exponential

_term, because at the end it

_happens

that kR remains finite. For c =

0,

equation

(4)

reduces to

Now

from the differentiation of

_equation

₂

we

_get

~4 =

20132013"hence,

using

(7)

and

E -

2013

2

Now,

from the

_{differentiation}

of

_equation

₍₂₎

we

_get

A

d

hence,

using (7)

and

_~~I)

= -

k ,

since k = 0

_{at ~ = 0.}

Then

₍₈₎

becomes

L-56 JOURNAL DE PHYSIQUE - LETTRES

The numerical result is kR = 0.567 _or

Thus,

the

_{binding of}

a

_particle

to a

_pair

_of

«

_{marginal » potentials}

_is, at

_large

_pair

_separation,

a

universal function of R, independent of

the nature

_of

the potential.

For e

small,

(4)

can be

_approximated

as

This shows that the relevant

_parameter

to measure a distance to the «

_{marginal »}

case is

_EIA

(rather

than

_e).

In

_particular,

if e 0 the distance

_Ro

at which the

_binding

to a

_pair

vanishes is

4.

_Comparison

with N in GaP. -

_Figure

1

_gives

the results of NN -

_ground

state _energy calcu-lation for a _square well

potential,

as a function of R and for five values of e

_{(from -}

10

_;~

to 10

_/~).

The

_asymptotic

law

₍₁₀₎

has also been drawn. On the same

_graph,

we have

_plotted

the

experi-mental

_{points coming}

from reference

_[3].

The

_only

_parameter of interest is the electron effective

mass in GaP. Since we are interested in _very shallow states, there is no

_ambiguity.

We have taken m* =

0.5,

which

gives

associated with K = 11 _an electron

Rydberg

of 55 meV

(the

_energy of the

valley

orbit

_splitted

states of the

_donor).

In reference

_[3],

Cohen and

_Sturge

measure the excitation

spectrum of

the luminescence of an exciton bound to a

_pair

_NN¡.

The

_pair

_separation

is

Ri

=

d i/2,

where d is the lattice _constant

( ~

5.4

A

in

GaP).

The excited _states

correspond

_to nS

orbitals of the hole wave function. The ionization

_limit,

_{corresponding}

to

_{NN~ ,}

is obtained

_by

an

extrapolation procedure,

the

_binding

_energy

_{of NNi}

is

_{EGap - ENNi .}

Fig. 1. -

Log-Log

plot

of the

_{pair separation R}

versus the _{binding energy of NN- for different} values of a

( - 10

_{%, - 5}

_%, 0, 5

_{%, 10 %).}

The

_{experimental points correspond}

to

_NNi

in GaP _(i = 1 _to 7).

In indirect _gap semiconductors an accurate value of

_EGap

is difficult to obtain. The data known

accurately

is _{the energy of the bottom of the free exciton band.}

Here,

we have taken a value of the exciton

_Rydberg

of 20

meV,

the error

_being

estimated to ± 2 meV. This error would affect to

some extend the

_position

of the

_points

i = 5 to 7 and the

_{log-log plot}

of

_figure

1,

but it would not

affect the main conclusion : the

_{experimental points}

at

_{large R}

_(i

= 4 _to

7)

_are

clearly

_on the left

of the

_asymptote.

This shows

_{unambiguously}

that B 0 for N in GaP. Also the

_slope

at

_{large R}

also

_typical

of a

_region

e ~ 2013

_0.08,

and has

_nothing

to do with a

_{particular shape}

of the

isoelec-tronic

_potential.

From

₍₁₂₎

one can evaluate the

separation R

at which the

_binding

of NN-vanishes. One

_{find ’s R -}

24

A

_(i

~

40).

Let _us recall

why

the

experimental

data

stop

at i = 7. This

is because the nS excited states are broadened when

_they

are in resonance with the free exciton

continuum. ,

In

_figure

_1,

the theoretical curves were

_computed

with a radius b = 2.4

A.

Changing

b would

just

move the theoretical curves

_parallel

to

the R - 2

asymptote. This

_corresponds

to the fact that the relevant

_parameter

of the N isoelectronic

_potential

we can extract is

_eIA,

which is about

0.04

A -1.

For

_{ENN- larger}

than 40

_meV,

the model based on a

_simple

_parabolic

band structure is

no

_longer

accurate. The deviation for i =

3,2

is

equivalent

_{to an} increase of m* for

deeper

levels.

The _very

_peculiar

situation

_{of NN}

could be

related,

we

_believe,

to an

_overlap

of the

potentials

of

the two

_{impurities, producing}

some non-linear effect. This is the

only

reason

_why

we choose a

value of b such that 2 b is between the

_{pair separation}

in

_NN1

_{and NN2.}

However, a smaller value

of b ( ~

1.5

_A)

is

_predicted

in first

_principle

theoretical calculation of isoelectronic

_potentials

_(2).

5. Conclusion. - We have shown that for

an isoelectronic

_potential

near the limit of

vanishing

binding,

the

_study

of

_{pairs (NNi) provides}

an

_unambiguous

test for the

_stability

_{of N -. This} proce-dure is not sensitive to the exact

_shape

of the electronic

_potential.

On the other

hand,

one can _try

to deduce the

_stability

of N- from the known

_binding

_energy of the exciton bound to N. An

attempt

to do it

_through

a

_simple

Hartree self-consistent method has

_predicted

a

_positive

value of

8

[7].

This

_{simply points}

out the

_necessity

to include electron-hole correlation in such calculations.

Acknowledgments.

- I thank P. Nozieres for his

encouragement and critics.

Appendix.

- To deduce

formally

(4)

from

_(3),

we have to _compute the

_{overlap integral}

D. We assume

_that,

at

_large

R and small _ri, the main

_part

comes from the

_region

outside the

_potentials.

Then from

₍₆₎

The volume element is dr =

2013~

dr,

dr2

de¡>

and in the

_{quadrant rl r2}

the domain of

integration

is limited

_{by r,}

_{+ r2} > R

_and

_{~i 2013}

_r2

_I

R

Using

r 1 + r2 = x and r 1

-

r2 = y

Retaining

the main

_significant

terms in

₍₃₎

we obtain :

L-58 JOURNAL DE PHYSIQUE - LETTRES

Differentiation with _respect to k

_gives

which is

_equivalent

to

_(9).

References

[1] For a _review, see _{e.g. CZAJA, W.,}

_{Festkörperprobleme}

11 ₍₁₉₇₁₎ 65.

[2]

For a discussion on isoelectronic

_potentials,

see _{BALDERESCHI, A.,} J. Lumin. 7

₍₁₉₇₃₎

79.

[3]

COHEN, E. and STURGE, M. D.,

Phys.

Rev. B 15 ₍₁₉₇₇₎ 1309.

[4]

THUSELT, F., KREHER, K. and WÜNSCHE, H. J., Solid State Commun. 36 ₍₁₉₈₀₎ 563.

[5]

LANDAU, L. and LIFCHITZ, E.,

Mécanique Quantique (ed.

Mir. _{Moscow) 1967, p. 143.} [6] ALLEN, J. W., J. _Phys. C 1 ₍₁₉₆₉₎ 1136.

[7]

BENOÎT À LA _GUILLAUME, C., 16th Int. _Conf. on the

_{Physics of}

Semiconductors,

Montpellier

1982. To be