Medium-range order and cohesive energy in Ge x Se 1-x glasses

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Submitted on 1 Jan 1990

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Medium-range order and cohesive energy in Ge x Se 1-x

glasses

P. Tronc

To cite this version:

Medium-range

order and

cohesive

_{energy in}

_{GexSe1 - x}

_glasses

P. Tronc

Direction de

_{l’Enseignement Supérieur}

des _{Télécommunications,} 46 rue Barrault, 75634 Paris

Cedex 13, France

(Requ

le 3 août 1989,

accepte

le 28 novembre

₁₉₈₉₎

Résumé. 2014 On

présente

l’étude de la structure

_{électronique}

des verres

_GexSe1-x

(x ~

_{0,33 )}

effectuée à l’aide d’une

_{approximation}

de liaisons fortes dans un modèle de Hartree auto-cohérent. Les résultats

_indiquent

_que le terme

_significatif

dans

_l’énergie

de cohésion,

quand

on

compare les différentes structures

_possibles

d’un verre de

_composition

déterminée,

provient

du

transfert de

_{charges électroniques}

des atomes de

_germanium

vers les atomes de sélénium. L’existence de liaisons Ge-Ge étant exclue _par les résultats des mesures de diffusion Raman, le

calcul de

_l’énergie

de cohésion en fonction de l’ordre à moyenne distance confirme la tendance

des atomes de

_germanium

à diffuser dans la matrice de sélénium. Cette tendance

_empêche

la

formation de

_séquences

Ge-Se-Ge aussi

_longtemps

que la concentration en

_germanium

le _permet.

Par

_exemple

les verres

_{GeSe4 (x}

=

0,2)

sont constitués

principalement

de

séquences

Ge-Se-Se-Ge, donc de tétraèdres

_GeSe4.

Les tétraèdres

_{GeSe4/2 qui}

_partagent une arête _augmentent

l’énergie

de cohésion du verre

_{GeSe2 ;}

au contraire les tétraèdres

_{GeSe4 qui}

_partagent une arête

ne maximisent _pas

_l’énergie

de cohésion du verre

_GeSe4.

Abstract. 2014 The

study

of the electronic structure of

_GexSe1-x

_glasses

_{(x ~}

_{0.33 )}

is

_presented

using

the

_{tight-binding approximation}

with a self-consistent Hartree model. The results indicate that the

_leading

term in cohesive energy, when the

_possible

structures of a

_glass

with a

_given

composition

are

_compared,

is due to the transfer of electronic

_charges

from Ge atoms to Se atoms.

_Excluding

Ge-Ge bonds from Raman

_scattering

_{measurements, the} calculation of the

cohesive _energy versus

_medium-range

order confirms the

_tendency

of Ge atoms to diffuse into the

selenium matrix so

_preventing,

as

_long

as the

_germanium

concentration makes this

_possible,

the formation of Ge-Se-Ge sequences. For

example GeSe4 glasses

(x

=

0.2)

_are

mainly

built from

Ge-Se-Se-Ge sequences i.e., from

_GeSe4

tetrahedra.

_{Edge-sharing GeSe4/2}

tetrahedra are shown to enhance cohesive _{energy in}

_{glassy GeSe2 ;}

on the _contrary

_{edge-sharing GeSe4}

tetrahedra do not maximise cohesive _{energy in}

_{glassy GeSe4.}

Classification

Physics

Abstracts

61.40D - 71.25M

1. Introduction.

The structure of

_{GexSel _ x}

_glasses

_{(x = 0.33 )}

_{has,,been extensively}

studied

_by

Raman

spectroscopy

[1, 2].

Even if the

_assignment

of the Raman

_peaks

is still considered

controversial,

it is

_generally

admitted that the coordination numbers of Ge and Se atoms are

equal

to 4 and 2

_respectively

and that Ge-Ge bonds are

_{forbidden ;}

in

_addition,

Ge atoms

exhibit a

_tendency

to diffuse into the selenium matrix. This last result is deduced

_directly

in reference

_[1]

from the

_assignment

of the 215

cm-1

Raman

_peak

to _{Ge-Se-Ge sequences ;} it is also consistent with the measurements

_reported

in reference

_[2]

in which this

peak,

called the

companion peak,

is

_assigned

to

_{edge-sharing GeSe4~2}

tetrahedra for the

_peak

_appears

_only

when x r 0.15 and remains weak when x 0.25.

_GeSe2

_glasses

are therefore built from

GeSe4~2

tetrahedra and

_GeSe4

_{glasses mainly}

from

_GeSe4

tetrahedra. This model of medium-range order is consistent with

reported

measurements of the

_{optical absorption edge}

_[1].

The existence of

_{edge-sharing GeSe4~2}

tetrahedra which has been

_recently

_proposed

for the

_glasses

[2]

stresses the close

_{relationship existing}

between the structures of

_{crystalline GeSe2,}

in which such tetrahedra were

_formerly

found

_[3],

and those of

_glasses

with

_neighboring

_{compositions.}

The purpose of the

present

paper is to demonstrate

_{theoretically}

some of the

_previous

experimental

results

_{by calculating}

_{the cohesive energy of the}

_glasses

versus their different

possible

structures. It is assumed that the coordination numbers of Ge and Se atoms have the values cited above and that Ge-Ge bonds are forbidden. It will be shown that Ge-Se-Ge sequences are excluded

_{statistically}

when x = 0.2 and remain as scarce as the amount of

germanium

makes it

_possible

when 0.2 x ,

0.33,

and that

_{edge-sharing GeSe4~2}

tetrahedra

enhance the cohesive energy of

germanium-rich glasses.

The _{cohesive energy is the energy} difference between isolated atoms and the solid. In a self consistent Hartree model the different contributions to _{the cohesive energy} come from :

i)

the delocalization of valence electrons which form energy bands in the

solid ;

ii)

the

_dependence

of the vibration

_frequencies

versus the local structure of the

_{lattice ;}

iii)

the appearance of electric

charges

on the atoms due to the

_{electronegativity}

difference between

_germanium

and selenium.

The most

_probable

structure _{is that which maximises the cohesive energy} at the

_liquidus

temperature

of the

_compound.

A

_{tight-binding}

_{approximation}

will be used with a Hukel Hamiltonian. This method is not an elaborate one and could not

_provide

detailed features of the electronic bands in the

_solids,

but it should be noticed that to calculate the cohesive energy,

only

the

_averaged

_{energy of the valence electrons is needed.}

_Furthermore,

to determine the structure of a

_glass

with a

_{given composition only}

the

_change

of the cohesive

energy versus

_medium-range

_{order is necessary.}

2. Electronic _{energy bands.}

Let

_Ha

be the

_{single-electron}

Hamiltonian

_{(Hukel Hamiltonian)}

~ i , J)

is the J-th

_sp3

_hybridized

orbital on the i-th atom, 4

_{AG (4 dS )}

_{the energy difference} between the 4s and

_4p

electronic levels of

_Ge(Se)

_{atom, {3 os ({3 ss)}

the resonance

_integral

between two

_{first-neighbors}

_(Ge-Se

and Se-Se

_{respectively)}

and

_{2 So}

_{the energy difference} between the

_sp3

levels of Ge and Se. Coordination numbers of Ge and Se atoms are 4 and 2

respectively.

Se atoms have two lone

_pairs.

The

_density

of electronic states

_{n (E )}

is

_expanded

into moments

_Mq,

the

_origin

_{of energy}

being equidistant

from the

_Sp3

levels of the Ge and Se atoms

Using

trace invariance under a

_unitary

transformation it is _easy to show that :

in which

_{Nq ’ n’ p}

is the mean

_number,

_per atom of the

solid,

of closed

paths

including

q-m-n chemical

bonds,

m

_transfers,

on a definite atom, from one

_hybridized

sp3

orbital to

another,

and n

_stops

on a

_hybridized

sp3

orbital

_(among

_{them p}

take

_place

on Se

atoms) ;

/3

is

_equal

to

_/3

_ss

_{or f3}

_Gs

_depending

on whether the bond is between 2 Se atoms or one Se atom and one Ge

_{atom ; A}

is

_equal

to

_~s

or

_{AG according}

to whether the transfer takes

place

on a Se atom or a Ge atom.

f3GS being

unknown,

it will be assumed

_{[9] that ~ Gs}

= 1

_{(f3GG + f3ss)}

_ - 4.55 eV.

2

Mq

can be

expanded

into

_increasing

orders of A and

_So,

and,

through integration,

the

cohesive _{energy formula is}

_expanded

into

_increasing

orders of

_d/~i

and

_{S~/~3 .}

The

_expansion

will be limited to _{square values of A and}

_So.

Closed

_paths

allow then at most the second nearest

_neighbors

of each atom to be

reached ;

this accuracy is consistent with the conclusions which have been drawn from the Raman

_experiments

_[1].

Moments

_{corresponding}

to a Se atom imbedded in a chain of Se atoms with Ge atoms at both ends cannot

_depend

on the other chains terminated on these two Ge atoms, for Ge atoms can be bonded

_only

to Se atoms.

Conversely,

moments

_{corresponding}

to a Ge atom can differ from one another

_only

because

of the second nearest

_neighbors

of this Ge atom. From these

considerations,

it is

_particularly

interesting

to examine

regular compounds,

i.e.

_compounds

built from identical

chains,

the constitutive chain of every

compound being respectively :

-These

_compounds

have the

_following

chemical formulae :

_{GeSe2, GeSe4,}

_GeSe6,

GeSeg

and

_GeSelo.

Moments

_{corresponding}

to Ge atoms are the same in

_regular

GeSe4

and in

_{regular compounds}

with

_higher

amounts of

_{selenium ;}

moments

_{corresponding}

to the central Se atom in the chains of

_{regular GeSelo}

are the same as those of Se atoms in pure selenium. Table I

displays

the coefficients of the different terms in the cohesive energy

~

(So )

of the

_{regular compounds}

_(per

Ge

_atom)

and pure selenium

(per

atom) .

For

_example,

Table I. -

Coefficients of

the

_differents

terms

_existing

in the cohesive

_{energy 0}

_{(So )}

of

the

regular compounds (per

Ge

atom)

and in pure selenium

(per

atom).

Results show

that,

if the

_present

_{single-electron}

model alone is taken into account, the most stable structure _{for any}

_glass

is the

_{phase segregation}

_{between pure selenium and pure}

GeSe2.

Each time one or more Se atoms are introduced into a Ge-Se-Ge sequence

_(which

is the constitutive element of

_GeSe2),

the cohesive energy is decreased

by

an amount

a 1 :

. q /~ ~ ,,"}

81

has to be

_compared

with

_{kT p, T p being}

the

_liquidus

_temperature

of the

_compound.

The

phase diagram

of the Ge-Se

_system

has been

_{investigated extensively}

_[4-7].

_Figure

1

reproduces

a

_{phase diagram compiled}

from the different authors.

kTj

is between 42 meV

(x = 0.08 )

and 87 meV

_(x

=

0.33 ).

61

is therefore very small when

compared

with

kT~ ;

consequently

where our above result alone is taken into account it would

_support

a

completely

random model to connect

_GeSe4~2

tetrahedra with Se atoms for x ~ 0.33. 3. Vibrational energy and

entropy.

Three

_peaks

_{appear in the Raman}

_spectra

of the

_glasses

at 195

cm-1,

215 cm-1 and 250 cm-1. We have

_assigned

these

_{peaks respectively}

to _{the Se-Ge-Se sequences, the} Ge-Se-Ge sequences and the Se atoms which are not bound to two Ge atoms

_{[1]. Introducing}

one or more Se atoms into a _{Ge-Se-Ge sequence}

_changes

a 3d 215 cm-1 oscillator into a 3d

250 cm-1 oscillator

_(in

the Einstein

model,

which is used

_presently,

atoms are assumed to vibrate

_{independently).}

The mean _{energy of} a harmonic oscillator with characteristic

frequency w

is,

at

_temperature

T

The

_liquidus

_temperature of

_glasses

with x * 0.33 is between 491 K and 1 013 K

_{(Fig. 1).}

Changing

a 3d 215 cm-1 oscillator into a 3d 250 cm-1 oscillator decreases the cohesive energy of the

glass

by ~ 2

which is

always

less than 1 meV. Even if very low energy Raman

peaks

were

_present

in the

_spectra

the induced

_change

in

₈₂

would be

negligible.

It can

Fig. 1. - Phase

_diagram

of _{Ge-Se system.}

_a)

218 °C

_{(4) ; b)}

580 °C

_(5),

578 °C

_(6),

573 °C

_{(7) ; c)}

630 °C

_(5),

603 °C

_{(7) ;}

_d)

627 °C

_{(7) ; e)}

660 °C

_(6),

651 °C

_{(7) ; f)}

666 °C

_(6),

661 °C

_{(7) ; g)}

740 °C

_(5,

6,

7) ;

h)

900 °C

_(6),

890 °C

_{(7) ; x)}

at. % Ge = 8

(4) ;

y)

_at. % Ge = 38

(5), 40-42 (6), 43 (7) ; z)

_at. %

Ge = 88-89

(6),

86

(7) ; (4) :

Dembovsky

_et al. ;

(5)

Vinogradova et

al. ;

(6)

Ross et al. ;

(7) :

Quenez et

al.

introducing

one or more Se atoms in a Ge-Se-Ge sequence decreases the cohesive energy

by

an

_{amount 8 = 81 + 82}

which is at most of the order of 2.5

meV,

therefore very small when

compared

with

_kTp.

To allow for the

_tendency

not to form _{Ge-Se-Ge sequences which has} been shown

_by

Raman

_spectroscopy

a

_negative

value of 8 would be _necessary.

Before

_improving

the model

_{by introducing}

electron-electron interactions and to be more

precise,

it is necessary to take into account the

_entropy

terms

_arising

from the

_topology

of the lattice and the vibrations.

i)

TOPOLOGICAL ENTROPY. - Let P

_{and Q be}

the numbers of Se and Ge atoms in the lattice and N the numbers of chains

_including

two or more Se atoms but terminated

_by

Ge atoms. The

_topological

_entropy

_{ST (N )}

is

_{given by :}

ii)

VIBRATIONAL ENTROPY. - The

_entropy

of a harmonic oscillator at

_temperature

T is :

with

The vibrational

_entropy

_{Sv (lV )}

of the

_glass

is therefore

Sl (T ), S2(T)

and

_{S3 (T )}

are the

_entropies

of the

_{195 cm-1,}

215cm-1

and 250 cm-1

oscillators. _{The energy E of the}

_{glass being}

taken as zero when

_{complete segregation}

into pure selenium and _pure

_GeSe2

is achieved

_(i.e.,

when N =

0),

the

The

_equilibrium

at the

_liquidus

_temperature

is obtained

_by

_{minimising U(N )}

versus N. One

obtains :

if 8 = 0 and the

_entropy

_{term ~ = 3 (S3 - SZ )/k}

is

_{neglected, n}

is

_equal

to _{no =}

2 x (1- 3 x )/1- x.

The

_GeSe4/2

tetrahedra are then

_randomly

connected with the

remaining

Se atoms

_[1].

If the

_intensity

of the 215 cm-1

_peak

is assumed to be

_proportional

to the number of Ge-Se-Ge sequences, then this

intensity

varies as

x2ll - x (Fig.

_2,

curve

_a).

"

Fig.

2. - 215 cm-1 Raman

peak intensity,

a)

GeSe4/~

tetrahedra

_randomly

connected with the

remaining

Se atoms.

_b)

Phase

segregation

into _{pure selenium and pure}

_GeSe2-

_c)

_{Ge-Se-Ge sequences}

as scarce as the

_germanium

amount makes it

_possible.

₍₀₎

_{Experimental point}

_{(1) ; (1) :}

Tronc et al. It can be

_easily

shown

that ~

is small and

_{approximately equal}

to - 0.4. As a

_result,

if 5 is

positive

and

_large

_compared

with

_kTe

then n = 0 and a

_phase

_segregation

takes

_place

into pure selenium and pure

GeSe2

(Fig.

2,

curve

b).

On the other

hand,

if 5 is

negative

and

_large

compared

with

_kTe,

then there are no _{Ge-Se-Ge sequences when x _ 0.2 and there is} a linear

dependence

of the

_intensity

of the 215 cm-1 Raman

_peak

versus x when 0.2 x _

0.33

_(Fig.

_2,

curve

c).

The

_{experimental points}

[1]

are located close to the c curve

(Fig. 2).

A

value of 5 can therefore be deduced. From the data at x = 0.20 it can be seen

_that,

at

_point

pl, the number of Ge-Se-Ge sequences is maximal and therefore also the 215 cm-1 Raman

peak intensity,

which is assumed to be

_proportional

to the number of _{above sequences.} The difference in the

intensity

of the

peak,

when

_compared

to

_point

_{pl, is}

_proportional

to n.

no

corresponding

to

_point

_P2

_(Fig.

_2,

_{curve d),}

one finds :

and

The cohesive energy calculations

performed

in sections 2 and 3 do not therefore

_sufficiently

match the

_experimental

results. It should be noticed that the electrical

_{charges appearing}

on

4. Electron-electron interactions.

Positives

_charges

_{appear, in} the

_glasses,

on Ge atoms and

_negative

_charges

on Se atoms due to

the

_{electronegativity}

difference between both elements. These

_{charges :}

i)

induce a

_Madelung

_energy

_by

_{electric interactions between atoms,} and

ii) modify

the electron-electron interaction energy within each _{atom ;} the

_So

_parameter

introduced in section 2 now

_depends

on the electronic

charges

and is

_quoted

as S.

4.1 CALCULATION OF THE ELECTRIC CHARGES APPEARING ON THE ATOMS.

1)

Within the molecular

_{approximation}

₍₄

=

0)

the solid is

equivalent

to _a set of Ge-Se

bonds,

Se-Se bonds and lone

pairs

(2

on _{every Se}

_atom).

Since A =

0,

_an electron _cannot be

released from a chemical bond or from a lone

_pair.

There is no

_charge

transfer

_along

a Se-Se

bond.

Along

a Ge-Se bond the

single-electron

Hamiltonian

_Hl

is

_part

of the Hfckel Hamiltonian

_Ho

_(see

Sect.

_2).

_HI

_{== J3GsI1/l’Ge) 1/I’sel I}

+

_{J3Gsl1frse) ’P’Gel I}

+

SI1/I’Ge)

~

’pGe

I

- S

11/1’ Se) 1/1’ Se I ; I P Ge)

and

_{11fr Se)}

are the two

_{i , J)}

Sp3

_hybridized

orbitals involved in the Ge-Se bond. The molecular orbital is

_{11fr) == (1}

+

t2)-1~ i ’~’Ge

+

_{1/1’ Se) ,}

the energy of the electron

E ( h ) _ ( ’~Y’ ~ H1 ~ ’~ )

and the

probabilities

of the presence of the electron on the Ge atom and the Se atom are

_respectively

_{I (}

1/1’ Ge

I

1JI’) 12

and

_{I (}

_{1fr Se}

I

_1/1’)

_12.

A = t +

₍₁

+

t 2}1l2 ~

in which t =

S / {3 GS,

is obtained

by minimizing

E(A ).

The

pair

of

opposite

spin

electrons involved in the Ge-Se bond induces a

_{positive charge con}

_{the Ge atom, where}

E = - t 1- t 2

+ 0 (t5) in

electronic

_charge

units,

and the

_opposite

_charge

on the Se atom.

~

2

2)

The calculation of electric

charges

appearing

per bond can be extended to the

_complete

Huckel Hamiltonian

_{Ho by}

use of a

_procedure

described elsewhere

_{[10, 11].}

The Green

operator g

(E )

of the electron in the solid

_{is g}

_{(E ) _ ~}

_{P k/E -}

_E~,

where

_Ek

is the

_eigenvalue

k

of

_Ho

and

_P~

the

_projector

on the

corresponding

_subspace.

The

_probability

of

_finding

an

electron with wave function

_’~k~

in state

_{I 00~}

is therefore the residue of

_{Ek pole}

of

~ (A 0 I 9

(E )

I ~ o ) ~

Ho

can be written

_Ho

=

HI

+ V , V

being

_a

perturbation

Hamiltonian,

taking

into account the 4 terms :

If G

and g

are the Green _operators of

Ho

and

_Hl,

one _gets

¡

It is to be noticed

that,

for

_equal

values

_{of t, s}

is

_larger

in

_regular

_GeSe4

than in

_{GeSe2 by}

an

amount 20133/8(J~//3~s)~

The

_{charges existing}

in both lattices are

_displayed

in

_figure

3.

A

_relationship

has therefore been established between E and S. Another will be established

by

evaluating

the electron-electron interactions between atoms

_{(Madelung energy)}

and within each atom.

_Combining

both relations will

_provide

a self-consistent calculation of - and S.

u~ / - , /

Fig.

3. - Electric

_charges

in

_(a)

_{GeSe2 ;}

_(b)

_regular

GeSe4-4.2 MADELUNG ENERGY. - The

_density

of

_{Ge,,Se 1 glasses}

_{(x * 0.33 )}

_{varies very little} with

x 12.

Moreover Ge and Se atoms _{have very close} mass numbers and the bond

_lengths

in

crystalline germanium

[13],

selenium

_[14]

and

_GeSe2

_[3]

are all

_{approximately equal}

to 0.24 nm. It can therefore be assumed that Ge and Se atoms _{occupy the} same volume in the

glasses

and that all the bond

lengths

are

_equal

to 0.24 nm. It will also be assumed that all bond

angles

are

_equal

to that

_existing

in a

_regular

tetrahedron

_{(approximatively}

109°

28’ ;

this value is very close to that

_existing

in a

_regular

_pentagon,

i.e.,108°) ;

_actually,

in

_crystalline

selenium

[13]

and

_crystalline

_GeSe2

_[3],

the values of bond

_angles

are

_spread

about both sides of the former value. Nevertheless the

_{approximation proposed}

above appear to be consistent with the accuracy of the

tight-binding approximation.

The contributions to

_Madelung

_energy can

be divided into three

_{parts :}

i)

electric interactions within a

_tetrahedron,

which

_provide

the

_major

_part

of the

_Madelung

energy,

ii)

interactions of a tetrahedron with its first nearest

_neighbor

tetrahedra. Some

averaged

distances have to be used to calculate this

_part,

and

iii)

interactions with other tetrahedra. This

_part

is taken as zero for their relative

orientations are random.

For

_regular

_GeSe4

interactions between first nearest

_neighbors

tetrahedra

_depend

_strongly

on the

_spatial

_{configuration}

of the Ge-Se-Se-Ge sequence. The

_Madelung

_{energy has been}

calculated for three

_{configurations}

I,

II and

_III,

_{of the Ge-Se-Se-Ge sequence}

_{(Fig. 4).}

Fig.

4. -

Recently Sugai

proposed

the existence of

_edge-sharing

_GeSe4/Z

in the

_glasses

_[2].

Such tetrahedra do exist in

_{crystalline GeSe2}

_{[3] (the}

ratio of the number of Se atoms on

edge-sharing

double bonds to those on the

_{corner-sharing single}

bonds

being

1/3).

To check the effect of

_edge-sharing

tetrahedra in our model the

_Madelung

_{energy has} been calculated for

glassy GeSe2

without

_edge-sharing

_GeSe4/2

tetrahedra

_(which

will be called

_{corner-sharing}

GeSe2)

and for

_{glassy GeSe2}

in which each

_GeSe4,2

tetrahedron shares one

_edge

with one

other

_GeSe4/2

tetrahedron

_{(edge-sharing}

_GeSe2)

_{(Figs. 5a}

and

_b).

_Using

_equivalent

definitions for

_regular

_{glassy GeSe4,}

it can be noticed that

_{configurations}

_I,

II and III

_belong

to

_{corner-sharing}

_{regular GeSe4}

_{(Fig. 5c).}

We have also calculated the

Madelung

energy for

Fig.

5. - Lattices of

(a)

corner-sharing GeSe2 ;

(b) edge-sharing

GeSe2 ;

(c)

corner-sharing regular

GeSe4 ; (d) edge-sharing regular

GeSe4-edge-sharing regular

GeSe4

(Fig.

5d), by assuming

that the two nearest

_neighbor

tetrahedra which share one corner are in

_{configuration}

I. The results are listed in table II.

_Edge-sharing

tetrahedra increase the

_Madelung

_energy as

_compared

to

_{corner-sharing}

tetrahedra in

GeSe2-In

_{regular GeSe4}

the

_Madelung

_{energy decreases}

_rapidly

when one _goes from

_{configuration}

I

to

_{configurations}

II and III.

_Edge-sharing

_GeSe4

tetrahedra decrease the

_Madelung

energy in

configuration

I but slow the decrease when one _goes to

_{configurations}

II and III.

4.3 ELECTRON-ELECTRON INTERACTION ENERGY WITHIN EACH ATOM. - Let J be the interaction _{energy of} a

_pair

of electrons in the fourth shell. J is assumed to be

independent

of the number of electrons

_existing

in the shell.

_{If q}

is the mean number of electrons in the shell and - e the

_charge

of the

_electron,

the intra-atomic interaction

_potential

is

_Vi

=

-

(q -1 )

J/e. J

is

approximately

the difference between the second and first ionization

energies

or the difference between the first ionization energy and the electronic

_affinity.

The

two values are not

_{exactly equal.}

To obtain the best

_{approximation}

the first value was chosen for Ge atoms

_(which

bear a

_positive

_charge)

and the second for Se atoms :

_JG

=

8.5 eV and

_Js

= 8.8 eV

[8,

15,

16].

From the above

_assumption

a

_relationship

between S and E can be drawn

_{immediately :}

Table II. -

Madelung

energy

(per

Ge atom in the

_GeSe2

and

_GeSe4

_compounds

and _per Se atom in _pure

_{selenium), S, e}

and r values.

with

one

_{gets :}

Combining

these

_{relationships}

with those

_previously

obtained in 4.1

_provides

a

self-consistent determination of Sand £

_(Fig.

_6).

The results are listed in table II. It can be

_readily

seen that E increases with

_EM.

5. Cohesive _energy.

Let us denote the number of valence electrons

_present

within an isolated atom

_by

p o and within the same atom in the

_{glass by}

p. To make the calculation clearer we introduce

fictitious atoms with the same A values as real atoms but with

_sp3

levels,

when these atoms are

isolated,

that are identical to those of the real atoms in the

_glass.

We

_assign

the

_prime

index to

Fig.

6. variations of 8 versus t =

So/~i ~s

deduced from

a)

Green operator formalism

(GeSe2) ; b)

Green _operator formalism

_{(regular GeSe4) ; c)}

electron-electron interactions

(GeSe2) ; d)

electron-electron interactions

_{(regular GeSe4) .}

intraatomic interactions

_{+ E}

interatomic interactions glass

The electronic contribution T to _{the cohesive energy is}

_then,

_per

tetrahedron,

in

GeSe2

in

_{regular GeSe4}

and in

Se,

per atom,

The numerical results are listed in table II.

6. Discussion.

It is now

_possible

to demonstrate some features

concerning

the

medium-range

order in

GexSel _ x glasses.

From Raman

scattering

it has been shown that :

i)

Ge atoms exhibit a

_tendency

to diffuse into the selenium

matrix,

i.e.,

to avoid the formation of Ge-Se-Ge sequences as

_long

as the

_germanium

concentration of the mixture makes it

_possible

_{[1] ;}

and

ii)

Edge-sharing

GeSe4,2

does exist in the

_glasses

_[2].

The ratio of such tetrahedra increases with x and has a maximum at x = 0.33.

From

_i)

it can be

_predicted

that the

_glass

with x = 0.2

presents

a lattice structure close to that of

_{regular GeSe4,}

_{which is built from Ge-Se-Se-Ge sequences,}

_{i.e., GeSe4}

tetrahedra. Within the

_theory

_presented

_{in this paper the}

_glass

with x = 0.2 would have the structure of

regular GeSe4

if Z were

_positive

and would

_present

a

_{phase segregation}

into

_GeSe2

_{and pure} selenium if Z were

_negative.

As shown in section

_3,

terms

_arising

from _{electronic energy}

_bands,

lattice vibrations and

entropy

are _{very small}

_compared

to the 1’ and introduce corrective terms of the order of a few

tens of meV. From section

3,

it is clear that a value of the order of 3.2

_~7~,

i.e.,

225 meV

(with

7~

= 820

K)

is

expected

for Z. If

corner-sharing

GeSe2

only

is taken into account, Z is

positive

for all the Ge-Se-Se-Ge

_{configurations.}

The minimum of

Z,

300

_meV,

_{i.e. very} close

to the 225 meV

_expected

value,

is obtained for

configuration

III

_(Fig.

7,

curve

_a).

The chains should therefore

_present

an extended structure, two consecutive Ge atoms

_being

located as

far as

_possible

from each other

_{(Fig. 4).}

If a certain

_proportion

of

_edge-sharing

_GeSe4,2

tetrahedra in

_GeSe2

_glasses

is considered to be

_possible,

the 225 meV value of Z is obtained for a

_{configuration}

between I and III. The chains then _present a more or less folded structure.

The maximal

_{possible proportion}

of

_{edge-sharing GeSe4,2}

tetrahedra

_corresponds

to the lattice of

_{edge-sharing GeSe2 glass :}

each

_GeSe4,Z

tetrahedron shares one

_edge

with one

Fig.

7. - Variations of Z

with Ge-Se-Se-Ge sequences

configuration

in

_{regular GeSe4 glasses : a)}

edge-sharing GeSe4/2

tetrahedra are not taken _{into account ;}

_b)

the lattice of

_{GeSe2 glasses}

is assumed to be

that of

_{edge-sharing GeSe2 ;}

_c)

the lattice of

_{GeSe2 glasses}

is assumed to be that

_{proposed by Sugai}

_(3).

configuration

I

_{approximately}

_(Fig.

7,

curve

_b).

The chains should

_present

a _very folded

structure

_{(configuration}

I,

Fig.

4),

but without

_edge-sharing

GeSe4

tetrahedra which would decrease the cohesive energy of the

glasses.

In

_crystalline

_GeSe2

there are

_equal

amounts of

_edge-sharing

and

_{corner-sharing}

_GeSe4,2

tetrahedra

_[3].

_{Sugai proposed}

a _{very close}

_proportion

in

GeSe2

glasses

[2].

If we assume this result the 225 meV value is obtained for a

_{configuration}

between

_{configurations}

I and II

(Fig.

7,

curve

_c).

_Taking

into account the

_possible

existence of

_edge-sharing

_GeSe4

tetrahedra would

_{only slightly change}

the

_{configuration}

of the _{Ge-Se-Se-Ge sequence.} From the

_previous

conclusions

_concerning

_glasses

with x = 0.2 various conclusions

concerning

the other

glasses

can be

_reasonably

drawn :

i)

in

_glasses

with x 0.2 there is no

_{appreciable proportion}

of Ge-Se-Ge sequences. As a

matter of

_fact,

it is clear that a structure where a Ge atom has

1,

2 or 3 Ge atoms as second

nearest

_neighbors

is less stable than a structure

_excluding

Ge atoms as second nearest

neighbors

(the Madelung

energy and the electric

charges appearing

on the atoms would

decrease).

ii) Conversely,

for

_glasses

with 0.2 x _

_0.33,

the

_proportion

of Ge-Se-Ge sequences is

minimal as

_compatible

with the amount of

germanium,

the other

_existing

_sequences

_being

Ge-Se-Se-Ge.

The main feature

_{provided by}

our calculations is that Ge-Se-Se-Ge _{sequences may} induce a

larger

cohesive _{energy than Ge-Se-Ge sequences.} The

_physical

_origin

of this result can be

understood

_{by noticing}

that E _{may be}

_larger

in

_{regular GeSe4}

than in

_GeSe2

_{(Tab. II) :}

the

only

(13 6

+

~~ JS

within two Se atoms each

_having

a

_{charge - E (Ge-Se-Se-Ge}

_sequence)

(Fig. 3b).

This is less than

_{(13 e}

+ 2

_sb

_Js

within one Se atom

_having

a

_{charge -}

2 E

(Ge-Se-Ge

_{sequence) (Fig. 3a).}

7. Conclusion.

The calculation of _{the cohesive energy of}

_{GexSel - x glasses}

allows us to

_explain

two

_important

features

_concerning

the

_medium-range

order of these

_{glasses :}

the

_tendency

of Ge atoms to diffuse into selenium matrix and the existence of

_edge-sharing

_GeSe4,2

tetrahedra. We

needed,

to find a suitable

_description

of the electronic

_effects,

to take into account the transfer of electronic

_charges

from Ge atoms to Se atoms. Such a model would

_probably

also

be useful in the

_study

of the structure of other

_{chalcogenide glasses}

and

_especially

of the

GexSl _ x glasses.

These

glasses

have much in common with the

_{GexSel _ x glasses,}

as do the

germanium sulphide crystals

with the

_germanium

selenide

_crystals.

Nevertheless,

a difference

does arise because of the difference in the sizes and the masses of the selenium and sulfur

atoms and

_consequently

of the

_germanium

and sulfur atoms. It is

_probably

these size and mass

difference which increase the

_difficulty

of

_{preparing homogeneous GexS1 _ x glasses}

over a

wide range of x values.

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