A tight-binding calculation of the chemical shift in trigonal selenium and tellurium

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A tight-binding calculation of the chemical shift in trigonal selenium and tellurium

M. Bensoussan, M. Lannoo

To cite this version:

M. Bensoussan, M. Lannoo. A tight-binding calculation of the chemical shift in trigonal selenium and tellurium. Journal de Physique, 1977, 38 (8), pp.921-929. �10.1051/jphys:01977003808092100�.

�jpa-00208660�

A TIGHT-BINDING CALCULATION OF THE CHEMICAL SHIFT

IN TRIGONAL SELENIUM AND TELLURIUM

M. BENSOUSSAN

C.N.E.T., 196,

^rue

de Paris,

⁹²²²⁰

Bagneux,

^France

and M. LANNOO

Equipe

^de

Physique

^{des Solides du L.A. au C.N.R.S. N°}

^253, I.S.E.N., 3,

^rue

F.-Bads,

59046 Lille

Cedex,

^France

(Reçu

^le

9 fevrier 1977, accepté

^{le 4 mai}

1977)

Résumé. ²⁰¹⁴ Nous présentons ^un calcul détaillé du tenseur de

déplacement chimique

^{dans le}

sélénium et le tellure trigonaux. ^{Ce calcul est fondé sur un modèle en} liaisons fortes où les interactions,

entre orbitales moléculaires convenablement définies, ^sont prises ^en compte par ^un traitement de

perturbation

^{au second ordre. La} partie

diagonale

^{du tenseur} peut ^être complètement

interprétée

^en utilisant des valeurs raisonnables pour les

paramètres.

^{Les résultats sont en accord avec les} spectres optiques

qui

font ressortir un

important

^caractère liant dans la bande de valence

supérieure.

^{Avec ces} valeurs pour les paramètres ^{on ne} peut pas rendre compte ^du signe ^{du terme non}

diagonal.

^Les

raisons

possibles

^{de ce désaccord sont discutées.}

Abstract. ²⁰¹⁴ A detailed calculation of the chemical shift tensor in trigonal selenium and tellurium is presented. It is based on a

tight-binding

model where interactions between

suitably

defined mole- cular orbitals are taken into account by second order perturbation theory. ^The diagonal part ^{of the}

tensor can completely ^be

interpreted

using reasonable values for the parameters. ^{The results are in} agreement ^with

optical

results which show that there is a

substantial

^{amount of} bonding character in the upper valence band. With these values for the parameters ^{the model cannot} reproduce ^the sign ^of

the non diagonal ^{term. Possible}

reasons

for this discrepancy ^{are discussed.}

Classification

Physics ^Abstracts

8.138 ^- 8.520 ^- 8.662 ^- 8.810

1. Introduction. ^- N.M.R.

spectra

obtained from

trigonal

tellurium exhibit three distinct lines

[ 1 ]

^each

showing

^an

anisotropic

shift with respect ^{to the orien-} tation of the

magnetic

^{field. A} similar result is obtained for

trigonal

^selenium

[2].

From this it follows that there

are three distinct sites in the

primitive

cell of these

crystals.

^{In fact}

they

^are

equivalent

^but are

differently

oriented relative to the

magnetic field. ^However, although

^the structures of the two

crystals

^are

the same,

the

anisotropic

^{behaviour of the N.M.R.}

^lines,

^which

is

interpreted

^{in terms of a} chemical shift tensor a, ^are

quite

different. The N.M.R.

properties

^{of a}

nucleus,

characterized

by

^{a, are related to} its local electronic

configuration.

^The

applied magnetic

^field

partially dequenches

the orbital momentum of the valence shell which in turn creates on the nuclei an

hyperfine

^field

related to the local symmetry

[3].

Therefore the che- mical shift is sensitive to any

perturbation

^{of the che-}

mical bonds around a

given

^{atom and its}

study

pro- vides information on the local electronic distribution.

A

possible

theoretical

description

^of the chemical shift tensor a in

trigonal

selenium and tellurium has

been

proposed recently [4].

^{It was} obtained with a

purely molecular

model based on the

assumption

^of

perfectly decoupled

bonds. The orientation of the atomic

hybrids

^was deduced from the direction of the

principal

^{axes of a,}

leading

^{however to} orbitals which did not

point

towards the nearest

neighbours.

^{Such a}

result

clearly

shows that the situation cannot be

easily

described

by

^a

simple a priori

^model.

Our aim in this work

is

^{then to} undertake such a

calculation

using

^a

tight-binding approximation.

^We

want to derive a

simple

^model

capable

^of

describing

the results in selenium arid tellurium

which

are

quite

different from each other. For this we start from the most natural definition of a molecular model in terms of a pure ^s

band,

^a pure p

bonding band,

^a pure p

lone-pair

band and then the _p

antibonding

^{band. As we}

shall show this does not

predict

^correct results for 6, and it is _necessary to go

further, taking

^{into account}

the interactions of these molecular states

by

^second

order

perturbation theory.

In a first part ^we recall the

experimental

^results

concerning

the chemical shift tensor. We then detail a

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01977003808092100

922

theoretical formulation in ^terms of the _p character of the electron

population.

This is determined from a

model of the band structure which allows the

popula-

tions in the different bands to be calculated.

Finally

the last part compares the theoretical and

experimen-

tal results and a discussion of

possible

refinements is

presented.

2.

Summary

^{of the}

experimental

^{results. - For}

simplicity

^we

split

the chemical shift tensor a into an

isotropic part

Qo and ^an

anisotropic

part

Au,

ao

being

such that the trace of Au is

equal

^{to zero. Let us}

call

A, B,

^C the three sites of the

primitive

^cell

(Fig. 1).

FIG. 1. ^- Chain structure and definition of the axes in trigonal

selenium and tellurium. a) Unit cell : the Z axis is perpendicular

to the plane ^XY containing ^{the three atoms} A, B, ^C. b) Right ^hand variety. c) ^{Left hand} variety.

From

symmetry

considerations the tensor Ac cor-

responding

^{to site A can} be written :

Here the axes

O.,,y,z

^{are defined on}

^figure

^{1 and the}

^plus

or minus

sign correspond respectively

^to the left hand

or

right

^hand

variety.

If we call u the unit vector

along

^the direction of the

magnetic

^field

Ho,

^the

change

^{in Au}

along

this direction is

given by :

From this it is _easy to write down the contributions of the sites

A,

^{B and C to}

Au,

for rotations of

Ho

^around

the

Ox, Oy

^{and Oz axes. A} fit of these

expressions

^with

the

experimental

results allows the parameters ^occur-

ring

in eq.

(1)

^to be determined. However

only H, J,

^K

can be determined

unambiguously from experiment.

The

magnitude

^{of L can also} be determined but its

sign

cannot be obtained without

knowledge

^{of the}

crystal-

line

variety

^{as well as the}

crystalline

orientation of the

axes of the

sample

^under

study.

^{This has}

proved possible

^for

trigonal

^tellurium

[5]

^{from a detailed}

analysis

^{of the}

etchpits (L

^{is found to be}

negative),

^but

not for

trigonal

^selenium

[2]

^where

only ! I L is

^known.

The

experimental

^{values of}

^{H, J, K,}

^{L are}

given

ⁱⁿ

table I.

TABLE I

Experimental

^values

of

the elements

of

_{u xyz}

For the

following

theoretical

study,

it is useful to associate with each atom a local system ^{of axes}

OX, OY,

^OZ defined in

figure

^{1. In this} system ^{the tensor}

Aa(A)

for site A can be written :

Here

again

^{the + or -}

sign

^are associated with the

right

and left hand varieties. One can now relate

H, J, K,

^{L to abcd}

by :

0 is the

angle

between OZ and the c axis. It is

equal

^to

63027’ for selenium and 62017’ for tellurium.

The results are

given

^{in table II.} For selenium there

are two sets of values

corresponding

^{to the two}

possible signs

^{for L. There is a}

striking

difference between selenium and tellurium which cannot be

explained

from

geometrical arguments,

^{the local} environment

being

^{the same.}

TABLE II

Experimental

^values

of

^{the elements}

of

axyz

3. Formulation of the chemical shifl - The che- mical shift results from the contribution of several terms.

Usually

^{the most}

important

^{is due to the}

second order interaction of the electron orbital moment with the nuclear

magnetic

^{moment. It can}

be written in the form

[4] :

4/p

^are the conduction band states of

energy Ep, 03C8q

the valence band states of _energy

Eq.

^The

^origin

^{for r}

is taken at the atomic site.

For a

given pair

^{of narrow}

bands,

one can use an

average gap

approximation,

which allows a to be reduced to the

simpler

^{form :}

the sum

running

^{over the different} valence and conduc- tion bands of _average

energies Ev

^and

Ec;

^{k and k’}

are the wave vectors of the Bloch states within each band. In a

tight-binding

^{treatment one writes :}

where

the I i, a >

^are pure atomic states, ⁱ

being

^the

site

index, a

^{the orbital index.}

In the

following

^{we shall}

only

^{consider intra-}

atomic contributions to

(6).

^This

considerably simpli-

fies the calculation and can be considered as a first step ^{towards a}

complete

treatment.

Thus,

^{for atom} i,

a(i)

reduces to :

with

If one details the matrix elements of

L,

^{one then} obtains the

expressions given

^{in table III where XYZ}

now

represent

^the

corresponding

p atomic orbitals.

To _go further it is now necessary ^to have a model for the band structure which allows all

populations

^{N to}

be calculated.

TABLE III

Exprcssion pf a

ⁱⁿ

term of the populations

avec

4. A band structure model. ^- We choose the most

simple description, including

^{all nearest}

neighbours

interactions as well as some second nearest

neighbours

interactions within a chain. We start from a so-called molecular

approximation

from which the band struc- ture can be most

easily

^discussed.

We start from the

following

^atomic

orbitals s )

and the three _p

orbitals ( pX ), py X ! Pz X

^XYZ

being

defined in

figure

^1. However instead of

working with ) px ) and ! I py >

^we

prefer

^to build their sum and their difference which we

denotes p1 ) and I P2 ) respectively.

These have the

advantage

^of

pointing approximately

towards the nearest

neighbours

^{in the}

chain. The molecular

approximation

is defined

by

including

^the

following

matrix elements :

924

where h is the

hamiltonian, ( - fl)

^{the resonance inte-}

gral

^{between nearest}

neighbour hybrids

^{of the same}

bond; Es

^and

Ep

^{are the free atom s and p}

^energies.

All

overlap integrals

^are

neglected.

^{Within this}

model,

^{one ends} up with four distinct flat bands : a

pure ^s

band,

^a p

bonding band,

^a _{pure pz} ^{band and}

finally

^a p

antibonding

^{band as}

pictured

^on

figure

^2.

This model thus

gives

^a

simplified

^{but correct}

picture

of the electronic structure. The six valence electrons per ^atom fill the three lowest bands. This is in _agree- ment with

photoemission [6]

^and

pseudopotential

results

[7].

^The

bonding

^and

antibonding

^{orbitals are :}

at

energies

In order to

derive

^{a more}

precise

band structure,

one must include additional interactions.

However,

if the molecular model has some

meaning,

^{their main}

effect must be to widen the bands as shown on

figure

²

while the

coupling

between different bands must be weak so that it can be treated

by perturbation theory.

In this work we have included all

nearest-neighbour interactions,

^{within a} two center

approximation [8],

where there are

only

^four

independent

^{terms i.e. ss, 66,}

s6, ⁿⁿ in the usual notation. Some details are

given

ⁱⁿ

appendix

^A

concerning

the interactions in terms of the molecular orbitals. In this

appendix

it is also shown

that fl

^is

practically equal

^{to the au} term, while the ⁿⁿ term

approximates

^the width of the _p bands.

We shall not derive here a detailed band structure.

Our aim is instead to calculate the effect of these additional interactions on the chemical shift.

Optical

transitions will also be discussed in a

qualitative

manner.

For,

within the molecular

model,

^{there is}

practically

^no oscillator

strength

^between

the I Z >

FIG. 2. ^- The formation of bands from the molecular model.

band and the

antibonding

band. This is contrary ^{to the}

experimental finding

that there is an

important peak

in

82(E) [9]

^which may

only

be attribuable to this transition. This can

only

^be

understood

if there exists

a strong

coupling

^between

the Z )

band and the

bonding

^{or s band.}

As we shall see

later,

^{when one}

applies perturbation theory

^to calculate the

populations,

^{there is a first}

order term

only

^{when one includes an} interaction between second nearest

neighbours

^{within a chain. We}

shall

only

retain the most

important

^one,

given by (Fig. 1)

This term,

although small,

^can

give

^a substantial contribution

compared

^to

nearest-neighbour

^terms

which

only

contribute to second order.

5. Second order

perturbation theory.

^{- Atomic}

populations

^{Nv and Nc} of the valence and conduction

bands, respectively,

^are calculated

by

second order

perturbation theory

^{via a Green’s} function for- malism

[10].

^The hamiltonian of the molecular model is h and the total hamiltonian ^H

equals

^{h + V. The}

resolvent

G = (E - H) -1

^{can be related}

to g = (E - h)

and

Vby

^the

following

power series

expansion :

Any

^matrix

element ( ia ) I G ia ’ )

^{can be}

expanded

in a power series of the

following independent

^terms

of g :

Here i is a site index. The

origin

^{of the}

energies

^{is taken}

at the _p atomic

level,

^L1

being equal

^to

Ep - ^ES.

ia I G ia’ ) presents simple

^and

multiple poles

at E ⁼

0, + fl, -

^{L1 which are} the molecular levels defined above. The

multiple poles

describe the

widening

of these levels into bands. One can calculate the

population Nai,ai

^{in a} band derived from the molecular level E =

Ev, by

^{a method}

analogous

^to

that used

by Deccarpigny

and Lannoo

[11].

^One

obtains :

This

gives

^a

fairly

direct method for

evaluating

^the

populations,

with the condition that the _power series

expansion

^is convergent. ^{We shall assume that this is} the case and evaluate _eq.

(14)

^to second order. The

complete

derivation of all

G,.

^is

given

ⁱⁿ

appendix

^B.

In the same

appendix,

^the

populations appearing

ⁱⁿ

table

III are

expressed

^{in terms of n, a,}

3S6,

^{L1 and y}

(here,

^for

simplicity

^we call n and a the TTTC and uu

interactions, Psa being

^{the s-a}

one).

The formal results for the

populations

of interest

are

given

^{in table IV in terms of two} parameters :

x ⁼

(n/u)2,

^{a =}

[0.31(P;a/uL1) - y/2 a]

for the dia-

gonal part

^and

N y

^{= + 0.184}

/x

^{for the}

^non-diago-

nal

part (the -

^{and +}

signs

^{are related to the}

right

and left hand

variety respectively).

^{Some much}

smaller terms have been

neglected.

^The

sign

^{of oc will}

depend

^{on the ratio}

Ba/yA.,

which is in some respects ^a

measure of the relative

importance

^{of the}

coupling

between second

nearest-neighbours

^{within a chain.}

TABLE IV

Populations

^{in the}

different

^{bands. For}

Nyz

^the

upper and lower

signs respectively correspond

^{to the}

right

^{hand and}

left

^{hand varieties.}

6. Discussion of the chemical shift tensor. ^- The chemical shift tensor can be obtained

directly

^from eq.

(8)

and tables III and IV. To first order in x and _a, one obtains for the

right

^hand

variety :

with

Substracting

^the

isotropic

part :

one obtains :

This can be

compared

^{with the}

experimental

^values

of table II. Before

doing

^{this in} detail it is worth

mentioning

^the

following points :

- All terms in x, i.e.

(n/u)2

^{in the}

diagonal

part ^are

essentially

^{due to the} interactions

of I Z >

^orbitals

with B ) and I A >

^orbitals.

- The result for the molecular model

(with

^{x = 0}

and a ⁼ 0 in eq.

( 17))

^cannot

explain

the values of the

diagonal

^{terms and}

gives

^{zero for the non}

diagonal

one. It is thus

completely

unrealistic.

Let us now discuss

separately

^the

diagonal

^{terms of}

tellurium and selenium. The

non-diagonal

^{terms will}

be examined afterwards.

a)

TELLURIUM. ^- From

photoemission

spectra

[5]

we

take fl -

^{3 eV. We take}

to take account of the contraction of the wave

functions in the solid

compared

^{to the free} atom,

as shown for covalent solids

[12].

Fitting

^the

diagonal

^terms thus leads to :

From this we deduce a ratio

1t/u equal

^{to 0.31. This}

926

is

satisfactory owing

^to the fact that

usually 7r/a

^{is of}

order 0.33 to 0.5. This is in

good

agreement ^{with the} value 0.27 found elsewhere

[7].

^{From a one can also}

deduce _y,

taking

reasonable values for the other

parameters,

^{i.e. :}

This set of values leads to y rr 0.22 eV which is

quite

coherent with the values obtained for the nearest-

neighbours’

interactions.

A further confirmation of the coherence of this model is

provided by

^the

isotropic part

Qo of the chemical shift

given

ⁱⁿ

(16)

and which is found to

equal -

5 190 p.p.m. This is in

extremely good agreement

with the result of

Willig

^and

Sapoval [13]

deduced from a theoretical calculation of the Te che- mical shift in CdTe. Moreover this last value has allowed an

interpretation

for all tellurium

compounds

to be

given.

b)

SELENIUM. - As we do not know the

sign

^{of L}

we shall consider two cases. We

take B~

^{4 eV}

[6].

In

analogy

^{with tellurium we} obtain for the two cases :

Both cases thus lead to reasonable values.

The essential difference between Se and Te then

comes from the ratio

7c/r (n/a)Te (7r/a)s.).

^This

indicates that the

bonding

character of the upper valence band is much stronger ^{in Se than in Te. In fact} this turns out to be

exactly equal

^{to x which we find to}

be of the order of 20

%

for Se. This value is in _very

good agreement

with the result 25

%

^{of a recent numerical}

computation [14].

^{For Te we thus}

expect

^{that this}

bonding

character is reduced to about 10

%.

These results are confirmed

by optical spectra.

^In both

materials,

^the

E2(E) spectra

^{exhibit a broad}

peak

at

energies

^{between E = 1.5 and 2.0 eV} for Te and 2.5 and 4.5 eV for Se. The low _energy

part

^{of these}

peaks

may

only

be attributed to transitions from the _upper valence band to the conduction band. However if one

computes

the oscillator

strengths

associated with transitions from

pure I Z >

^{states to the} lowest excited states

[15] they

^{are found to be} very weak. One then reaches the conclusion that this

peak

^{must be attri-}

buted to a non,

negligible bonding

character in the upper valence band.

If we now consider the

non-diagonal

^{term, the}

agreement

^{is not so}

good.

^To obtain the correct

sign

ⁱⁿ

both cases x should

roughly

^be greater ^than

1/3

^which

is not coherent with the values obtained from the

diagonal

^terms. However if this were the _{case, per-} turbation

theory

^{would no}

longer

be valid. We are

then led to conclude that our model is still too

approxi-

mate and that it is _necessary to introduce further refinements. For this we believe that the two most

important

limitations of this work are the

following.

We have assumed that the average gaps ^were all

given by

the molecular model of

figure

^{2 and are thus}

fixed

by

the value of

fl.

^In

practice,

detailed results obtained for the band structure

[6, 7]

show that the distance between the ^two _upper valence bands is smaller than the distance between the _upper valence band and the lower conduction band.

Preliminary

calculations

correcting

for this effect show that while it

improves

the value for the

non-diagonal

^{term, the}

agreement

^{is less}

good

^{for the}

diagonal

^one.

Our central

approximation

^{has been to retain}

only

matrix elements

corresponding

^to the site for which 6

is calculated. While this is

certainly

^{correct for the term}

containing L/r3,

^{there is no clear reason}

why

^this

should also be true for the matrix element of L. Even if one retains

only

intra-atomic matrix elements a more

general expression

is obtained for a :

This is

evidently

^an extension of _eq.

(8).

^{Terms with}

j =

ⁱ

might

^be

important

^{for nearest}

neighbours.

^Their

importance

^is

proportional

^{to intersite}

populations.

They

^{are much more}

complicated

^to evaluate and this needs further work.

7. Conclusion. ^- In this work we have

developed

a method of

calculating

the chemical shift tensor in

trigonal

selenium and

tellurium, using

^a

tight-binding approximation.

^It

clearly

appears that a molecular model cannot

explain

the observed features. For this it is _necessary to take into account the interactions between molecular states. We have done this to second order in

perturbation theory using

^{an intrasite}

approxi-

mation. We have then shown that it is

quite possible

^to

explain

all the results

concerning

^the

diagonal

^{terms of}

the chemical shift tensor for

quite

reasonable values of the interatomic interactions. Furthermore these results

can be related in a coherent manner to the relevant

optical properties.

The main conclusion to be drawn is that there is a non

negligible

^{amount of}

bonding

character in the _upper valence band which is of the order of 20

%

for Se and 10

%

for Te. For

Se,

^{this is in} fair agreement with the result of OPW calculations.

We have also shown that the same values of the

parameters

^cannot

explain

^the

sign

^{of the non-}

diagonal

^terms.

Therefore,

^{it is} necessary ^to include additional terms in the

expression

of the chemical

shift, going beyond

^the

purely

^intrasite

approxima-

tion. This is

presently being

^examined.

We believe that this work demonstrates that the chemical shift is

extremely

^{sensitive to} any pertur-

bation of the chemical bonds around a

given

^atom.

Any

such

perturbation

^can

severely modify

the interband

coupling

^{which we have shown to be}

responsible

^for

the most

important

^observed

properties

of the che- mical shift tensor. For this _reason, it will be

interesting

to extend this

type

of calculation to selenium and tellurium

compounds

^and try ^to understand their local

configurations.

Appendix

^{A :} interatomic interactions between nearest

neighbours.

^{- All terms can} be reduced to the follow-

ing

matrix elements :

the

labelling

of the sites

being

^{detailed on}

figure

^3.

FIG. 3. ^- Local environment along ^a chain. On each atom i there

are two bonding ^orbitals Bi, Bi - 1, ^two antibonding ^orbitals Ai, Ai - 1 and ^{two free atom orbitals} Zi ^and Si.

We can now express all these elements in terms of _{a, 7r,}

Psa

^and

Pss

which denote here the absolute values of the usual ca, nn, ^s6 and ss two center

integrals.

^From

figure

^{4 one can} show that :

928

FIG. 4. ^- Chain structure and detailed definition of local axes.

TABLE V

Nearest-neighbours’

interactions

along

^{a chain}

It is a

simple

^{matter to} express all

independent

^{terms of}

(A. 1)

^{in terms of our} four distinct parameters. ^The

corresponding

^{results are}

given

^{in table V.} All results are

expressed

^as functions of the

angles

^{2 0 and} T

approxi- mately equal

^{to 1040 and}

100°, respectively,

^{in Se as well as in Te.}

The most

important

^term

fl

of the molecular model can be calculated as a function of J and n. One finds :

Appendix

^{B : intrasite}

populations.

^- As mentioned in the text one has to calculate the matrix elements

Gxx, Gyy, Gzz

^and

Gyz at a given ^site,

^to second order

perturbation theory.

^{For this we use :}

They

^{can be}

expressed

^{in terms of}

(13)

^and of the second

nearest-neighbour

interaction _y defined in

(11).

^This

leads to

We can then calculate the

populations using (14)

^and express them as functions

of a,

n,

Bso, Pss

^{and y. The final}

result is

given

^{in table IV after}

neglecting

^{terms which are}

clearly

smaller than the others.

References

[1]

BENSOUSSAN, M., ^J. Phys. & Chem. Solids 28 (1967) ^1533.

[2] KOMA, A., TANAKA, S., Solid State Commun. 10 (1972) ^823.

[3] RAMSEY, ^N. F., Phys. ^{Rev. 78} (1950) ^699.

[4] BENSOUSSAN, M., ^J. Phys. &, Chem. Solids 35 (1974) ^1661.

[5] KOMA, A., TAKIMOTO, E., TANAKA, S., Phys. ^{Stat. Sol. 40} (1970) 239.

[6] SHEVCHIK, ^N. J., CARDONA, M., TEJEDA, J., Phys. ^{Rev. B 8} (1973) ^2833.

[7] KRAMER, B., MASCHKE, K., LAUDE, ^L. D., Phys. ^{Rev. B 8} (1973) ^5781.

SCHLÜTER, M., JOANNOPOULOS, ^J. D., COHEN, M. L., Phys.

Rev. Lett. 33 (1974) ^89.

[8] HULIN, M., ^J. Phys. & Chem. Solids 27 (1966) ^441.

[9] WEISER, G., STUKE, J., Phys. Stat. Sol. 35 (1969) ^747.

[10] LANNOO, M., J. Physique ³⁴ (1973) ^869.

[11] DECARPIGNY, ^J. N., LANNOO, M., ^J. Physique ³⁴ (1973) ^651.

[12] MAUGER, A., LANNOO, M., Phys. ^Rev. (to ^be published).

[13] WILLIG, A., SAPOVAL, B., Proc. of the XIII International Conference on the Physics ^of Semiconductors, ^Rome (1976).

[14] KRUSIUS, P., VON BOEHM, J., STUBB, T., Phys. Stat. Sol. 67

(1975) 551.

[15] DECARPIGNY, J. N., LANNOO, M., BENSOUSSAN, M. (to ^be published).

[16] DECARPIGNY, ^J. N., LANNOO, M., Phys. ^{Rev. B 14} (1976) ^538.