Hole generation of prepeaks in diffraction patterns of glasses

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Hole generation of prepeaks in diffraction patterns of glasses

Jean Dixmier

To cite this version:

Jean Dixmier. Hole generation of prepeaks in diffraction patterns of glasses. Journal de Physique I,

EDP Sciences, 1992, 2 (6), pp.1011-1027. �10.1051/jp1:1992188�. �jpa-00246582�

Classification

physics

Abstracts

61.40 61.40D

Hole generation of prepeaks in diffraction patterns of glasses

Jean Dixmier

Laboratohe de

Physique

des Solides de Bellevue, C-N-R-S-,

place

Aristide Briand, 92195 Meudon Cedex, France

(Received 6 January 1992,

accepted

in

final form

10 March 1992)

Abstract. It is shown that most of the covalent

glass

atomic structures _can be derived from the Dense Random

Packing

of Hard

Spheres ID-R-P-H-S-

model

by introducing successively

_a series of inflated subnetworks

consisting

^of holes. From this

point

of view the _so called First

Sharp

Diffraction Peak IF-S-D-P-) of their diffraction pattem appears ^as a hole

generated prepeak

of the lst

neighbor

component ^{of the} D-R-P-H-S- Structure Factor. The various Medium

Range

Order (M.R.O.) features which _can modulate the

prepeak

characteristics _are reviewed local

topology,

network relaxation, chemical

ordering, homogeneity.

A tentative

relationship

is made between

prepeak

behavior and

physical

property variations with _a

special emphasis

on the

optical absorption

spectra ^{of covalent}

glasses.

Introduction.

Not

surprisingly,

the first

ambiguous

weak

peak

which appears

occasionally

_on the low q range of various

Metglass (M.G.)

structure factors

S(q) (q

=

4 _w sin b/A

),

_as well _as the

strong composition dependent

_« First

Sharp

Diffraction Peak _»

(F.S.D.P.)

of _some covalent

amorphous materials,

have attracted much attention since the

beginning

of

glass

studies.

Indeed,

all disordered materials _are

concemed,

from metallic to covalent

glasses,

from

single

^element materials to

multi-component amorphous alloys.

Medium

Range

Atomic Order and Chemical Order _are the usual

key

words which _are associated with the F-S-D-P-

designation.

The fact that the

strength

of _one, and

only

_one diffraction halo

might

_vary

considerably

_{or even}

disappear depending

_on the

sample composition, preparation

conditions

or more

astonishingly,

the type ^{of radiation chosen} for _a

given

diffraction

experiment ii-e-

X- ray ^{or neutron}

source),

has often

given

_a

mysterious

character _to its hidden

physical origin (See Fig, I).

Besides a renewed interest in the

understanding

of

prepeak

structural

origins,

_a few studies have also been devoted

recently

to the

relationship

between

physical property

^{variations and} their influence _on the

prepeak profiles.

At the

present

time new

key

words should be added to the

previous

_ones such _as minimum

minimorum,

for the

configuration

_energy, relaxation

phenomena,

texture and _more

generally speaking

defects in

amorphous

materials.

In this paper we would like to

present

an overview of the

subject

without

attempting

at _an exhaustive

bibliography.

We would like to demonstrate how _a

generalized prepeak concept

4

Qi

3 »~

~~2

~

~*

i

* +

O ₂ 4 6 8 12

o i~~

Fig. 1. Revealing _a

prepeak by

_a

weighting

_contrast between different atomic

scattering lengths

_: X-

ray and neutron diffraction _structure factors of Co80P20. In the _neutron

experiment

the P-P pair

contribution _to the total

S(q)

is enhanced from I.4

IX-Ray)

to 10.2 9b. Hence the

bump

at q = 2

h~

' is

generated by

_{an average} P-P distance of 3.5

h

_{due to chemical}

ordering.

can

help

to model the _structure of

glasses

and _to

classify

the different

possible physical origins

of its

profile

variation.

Finally,

the connection between

prepeaks

and

physical properties

is undertaken with

special emphasis

_on the

optical

characteristics of

amorphous

semiconductors.

1.

Prepeak

and

packing

fraction. The

origins

of the hole model idea.

To _our

knowledge

the first paper that suggests ^{that the F.S.D.P.}

might

arise from the appearance of _{a « zone} of low atom occupancy » is

Veprek

et al. 1981

[1]. Later,

in

1985,

an

overview of

prepeak problems

^is

presented by

Moss and Price

[2].

All materials

exhibiting prepeaks

_are

rewiewed,

from covalent to metallic

glasses (M.G.).

In their

opinion

the clue to the appearance ^{of the F-S-D-P- would arise} from the

packing

of basic structural _or molecular

units associated with Medium

Range

Atomic Order

(M.R.O.).

Before

discussing

the

physical origin

of

prepeaks

let _us first establish _a definition of what shall be considered _a

prepeak

or a F-S-D-P-

In the first studies

conceming glasses, liquid

and

amorphous materials,

before the

calculation of Radial Distribution Functions

(R.D.F.),

the formula

generally applied

to the first diffraction halo

position

_{was :}

X~

= 1.23 _x A/2 _x sin

b~

^{where is the}

X-ray wavelength.

This

procedure

_was

supposed

to

yield

the first

neighbor pair

distance between close _atoms

X~. However,

_as Guinier reports

(3),

this formula must be used with caution.

Indeed,

while the first

peak

of metallic

glass

pattems

yields

the _true first

neighbor

of _a close

packed network,

the _same

procedure applied

to the

amorphous

Si pattem ^leads to a

pair

distance 1.65 times

larger

than

expected. Applying

the _same formula to the second

peak position

leads _{to a} value closer to the atomic diameter but there is still a

discrepancy

of about 20 fb. When various _structure factors of metallic

glasses

became available it

appeared

that

they

all

exhibited the _same

profile

with the well-known «shoulder» on the second halo. The

corresponding

Radial Distribution Function

(R.D.F.)

indicated that this

typical

feature _arose from the hidden 5-fold symmetry ^{of what should be described} as the Dense Random

Packing

of Hard

Spheres.

At this

point,

it is worthwhile to compare

(Fig. 2A)

the M-G-

S(Q)

with those of

amorphous

tetracoordinated elements and of

binary amorphous glasses mixing

2 elements with different valencies. In the latter _case, it _can be _a mixture of elements of the fourth

column of the

periodic

table with either elements of the fifth column

(coordination

number

3)

or of the sixth column

(coordination

number

2), obeying

the 8-N rule. We have chosen

GeSe2

since its

X-ray

diffraction pattem ^is almost

equivalent

to a

single

element

pattem,

^due to the almost

equal

size and diffusion factor values of Ge and Se _atoms. The abscissae have been

normalized in order to eliminate the atomic size differences.

The

important

feature to note here is that the

fight

side of all three _curves,

beyond

the

peaks

marked

Qi,

have almost the _same

profile, particularly

the

typical

shoulder _on the second halo. This

particularity

is well

explained

if _one remembers that the covalent

angle

inside _an

elementary

tetrahedron is 109°28'which is close _to the value of the pentagon

angle, 108°,

encountered in the

pseudo

icosahedral structure of the D.R.P.H.S. model

[4].

The structure factor

S(Q)

of

glasses being govemed by

_very few

pair distances,

the _common

part

^{of the 3} curves is

mainly

made up ^{of the first}

neighbor

distance ri_ ^{and the 1.65}

ri second distance which _{are common to} all materials.

Following

this

observation,

the idea

emerged early

that the

peaks

of covalent

glass

pattems ^situated on the left side of

Qi

should be derived from the

D-R-P-H-S-, S(Q).

Therefore,

in the

following

discussion _we shall call

Qo Prepeaks,

all the

peaks

which _are

located _on the low

Q

side of

Qi referring

to the lst

neighbor peak

contribution to

S(Q).

Both

experimental

results and model calculations have

suggested

^such a relation.

On the

experimental

side

Mangin

et al.

[5]

have studied

amorphous Feji

_{_~~Si~}

alloys,

with _x

varying

from 0.20 to I. The first

peak

of the pure Si

S(Q), emerged progressively

_as a left shoulder of

Qi

_on a.Fesi

S(Q), increasing

with Si content.

The continuous transformation from the D-R-P-H-S- _to the disordered tetracoordinated

network,

the _so called Random Covalent

Packing (R.C.P.)

is

accompanied by

a drastic

reduction in the local atomic

density,

which measures the number of _atoms per

A~.

_{Its value falls from 0.075 to 0.045.}

The

packing

fraction is

equal

to ~ =

l/6(wpo «~),

«

being

the atomic diameter. It is therefore reduced from 0.64 _to 0.30.

Using

the

Debye

formula for the calculation of

S(q)

_:

S(q)

=

I ₊ I/N

£

sin

q.R,/q.R,

,

with

only

two

distances,

and

taking

into account the reduction from 12 to 4 first

neighbors, Mangin

et al. show that the emergence of the

prepeak Qo

is linked _to the reduction of the

packing fraction,

which in _retum favors the 2nd distance

weight

on the

S(Q) Profile.

Without

using

the word

explicitly,

the hole model is

already implicit.

On the

modelling

side _a similar

approach

has been used

by

Chaudhari et al. in 1975

[6].

Using

the coordinates of the models of Steinhardt _et al.

(derived by

relaxation of the Polk

model) [7]

and Connell Temkin

[8]

for the disordered tetracoordinated network _on the _one

hand,

and the Bennett model for the D-R-P-H-S- on the other hand

[9] they

showed that _one

can transform the first model into the second and vice _versa.

In the first

transformation,

the four _atoms of the first shell _are removed in order to reach as new first

neighbors

the second shell of 12 balls, while

increasing

the

sphere

diameter.

Conversely,

the second transformation is based _on the decoration of _some tetrahedra of the D-R-P-H-S-

by placing

_one atom at their center,

consequently reducing

the

sphere

diameter.

The

Qo Peak

becomes

Qi

in the first transformation while

Qi

becomes

Qo

in the second.

S(i) A

~(%) B

a~ a a

3

al

1.5

)ljl)11)1

~_~ ~_

=C;=~;;S

>' ^3~

q~/~~T)~li

2

^{~'~'' '}

l

I=I

AFT.5

ij

,

(=1.65A,f2

al ~l §=2 Apl _~~$

~

~~ b a~ b

~

l

Ql _I A~=2

@

Rpl.65A,=3 o

~

ai

a~ V~~[~+~ #-',

O c ^,

,

'

0 2

Ii /%~

o j 2 3

ij Iii

Fig.

2.

IA)

From

metglasses

to covalent

glasses

the structure filiation _a a.Ni-P b _: a.Si _;

c :

a.GeSe2.

All _curves have been rescaled _to Qi I-e- the first neighbor contribution _to the total S(q). (B) Generation of

Prepeaks:

the hole model transformation of the D-R-P-H-S- into bond

oriented structures. Calculated S(q) with the

Debye

formula:

S(q)=Ai(sinqRj/qRj)+

A~(sin qR~/qR~)

₊

Ai(sin qRi/qR~).

(R values _are the ratios of

experimental

^values to the lst

neighbor

distance one). IA ^values are the relative

strengths

of the

experimental

G jr)). Insets equivalents in 2 dimensions (not related to the indicated R and A

values)

_: a) 2d close

packed

network _; b) the first hole subnetwork transforms _a 6 fold coordinated model into _a 3 fold oriented bond network _; c) the inflated 2nd hole subnetwork transforms the

previous

3 fold

connectivity

into _a mixture of altemate 3 and 2 _ones.

Another

approach using

_a decoration

procedure

has been

performed by

Sadoc _et al, on

various

polytopes (polyhedra

in 4d

space).

For

example,

if _{a vertex} is

put

at the centre of _one among five of the 600 tetrahedral cells of the

[3-5] polytope,

one transforms its 12-fold local coordination into _a four-fold coordinated

polytope (so

called _« 240

»),

which contains

only

even

rings

_as in the C,T, model

[10].

A recent contribution to the renewed

general

interest for

prepeak studies,

_an

explicit

hole model for the tetravalent

glass

structures has been

recently proposed by Bl£try [11]. Bl£try

uses a

computer algorithm,

which creates _a close

packed

mixture of

spherical

holes and _atoms of the _same atomic size and concentration.

Using

the Bathia-Thomton formalism _to calculate the total

S(q)

of

binary alloys

the

repartition

of holes is

optimized

to

yield

_a maximum chemical

partial

order between these _atoms and holes. The calculated

S(q)

shows

clearly

^the

Qo Prepeak.

Nevertheless the shoulder of the

Q2 Peak disappears.

Similar to the hole

chemically

ordered

approach

of

Bldtry

for tetravalent monoatomic

networks,

S. R. Elliott attributes the

S(Q)

F-S-D-P- of covalent

alloy glasses

to

chemically

ordered holes around cation centered structural units

[12].

Much earlier Warren has

already

described

a.SiO~

_{as a}

more or less ordered

packing

of

Si04

tetrahedra

[13]

and he succeeded in

reproducing

theoretically

the first main halo of the

diagram.

Following

the

qualitative approach

of

Mangin

_et al, it is

tempting

to

apply

it _{to a} mixture of 2 elements of different valencies. Since _we have to include those contributions of the M-R-O-

which should be

responsible

for _{a new}

prepeak formation,

let _us compare the

pair

correlation

function

G(r)

derived from the

experimental S(Q)

of

a.Co~ooP~o, a.Si,

and

a.Gese~

(Fig. 3).

The

advantage

of

comparing G(r)

instead of R.D.F. is that

G(r)

is limited to the fluctuations of the atomic order around the _mean atomic

density.

Hence the

comparison

is _not affected

by

the

packing

fraction reduction.

At this

point

we would like _to find the M-R-O-

pair

distances which

might

be related _to the

new hole

model,

which is

imposed by

the mixture of 4-2 coordination of Ge and Se. We have

3

72 t c

~

~ l 3

ri/ri

Fig.

3. Pair correlation Function G jr) for _: a) the metglass N180P20 (continuous line) b) amorphous Si (stars) ; c)

amorphous

Gese~ (after 32)

(interrupted

line). The first hatched _{zone at a} 1.65 times the

atomic diameter

Rj

is generating

Qo

of a.Si while the 2nd hatched

zone at about 1.652 _times Rj is at the

origin

of

Qi

of a.GeSe2.

already

_seen that the decrease of the first shell occupancy from M-G- to a.Si resulted in _an increase of the relative 2nd

peak height (first

hatched _zone of

Fig. 3). Similarly,

_{a new}

enhanced domain characterizes the

G(r)

of

a.GeSe2

_as

compared

with that of a.Si

(second

hatched

zone).

In order to introduce the _remote distances of the second hatched _zone in the

simplified

theoretical

S(Q)

_we have added _{one more term to} the

Debye

formula _as

compared

with what has been done

by Mangin

et al.

Nevertheless,

this

approach

remains

qualitative

since

G(r) actually

consists of _a continuum of

pair

distances rather than discrete

peaks.

This remark has

already

been made in the paper of

Veprek

et al.

II-

So the A

i,

A~

^{and A}_~

pair frequency

values, that we have used in the

calculation,

_are somewhat

arbitrary

_even if

they

_are

supposed

to represent ^{the relative}

strength

of the 3

peaks

on

G(r).

It is shown in

figure

28 that the introduction of _{a new}

component

at 2.6 ri

generates,

as

expected,

_{a new}

prepeak

that _we shall denote

Q(

since it should be considered _{as a}

prepeak

bis of the M-G-

S(Q).

How _{can we}

describe,

based _on the D-R-P-H-S-

model,

the 2nd hole model

ground

rules ?

Recalling

the

previous procedure,

I-e-

reducing

from 12 to 4 the close

neighbors,

and

taking

into account the almost flat covalent

angle

Ge-Se-Ge

(144°

in

average)

we

procede

_as follows

(Fig. 4)

_: after the lst hole

repartition

has been made we repeat ^it on the 12 2nd

neighbors

of the tetracoordinated network which has

just

been created. In

doing

_{so we} create _{a new} hole

subnetwork with _a

larger

cell

parameter

^and a 4,2 mixture of

bonding connectivity.

/

Fig.

4. The schematic hole inflation full circles _: 4 fold coordinated _{atoms ;} stars _: 2 fold coordinated atoms open circles _: holes. The reduction of 12 to 4

neighbors

in the lst shell is

repeated

in the 2nd shell.

A two dimensional

equivalent

model is shown in the insets of

figure

28. From the close

packed (necessarily ordered)

2d network _{one can} form _a subnetwork with _a coordination number of three

by connecting

the atoms left _over from the

regular

first hole distribution. The

procedure

is then

repeated

on the atomic subnetwork

having

the _same parameter as the lst hole subnetwork. One _{can see} that _we generate a new

larger

hole subnetwork which is made up of _an

altemating

3-2 bonded

connectivity.

In summary, one can conclude

that,

at least

qualitatively,

the

general

features of covalent

glass

models _can be derived from the

D-R-P-H-S-, through

_a kind of self similar hole distribution

procedure,

which generates at each step a

corresponding prepeak.

Therefore the

prepeak

formation does _not

require

_a

priori

additional

physical origin hypotheses

such _as 2d entities _{or «}

superstructural

_» units which should

be,

if

anything,

_a

by-product

of the hole model.

This is not the _case for another

intriguing prepeak

case, The

S(Q)

of

amorphous

phosphorous,

arsenic and antimonium _are shown in

figure

5. An

amazing point

to notice is

that,

while _a very

strong prepeak Q(

is visible _on the a-P

S(q)

its relative

intensity

is

drastically

da

P

ao a~

ao

°'

As

Ql ao

Q<

1

-1

Fig.°(~)

c)

(after 16) : _the

vanishing of

Qo _{going a-P} to a.Sb.

reduced _to the a-As _case and that it

disappears permanently

from the a.Sb structure factor. It is

worthwhile, nevertheless,

to add that

beyond Q

i the Sb

S(q)

has _a

profile

which has

nothing

to do with the

« shoulder

»

shape

of the

previous diagrams.

When

going

down

through

the Vth column it is indeed well known that the covalent character of the elements is reduced in

favor of _a metallic

tendency.

The atomic _structure

modelling

of these

single

element

glasses

has been _a

puzzling

task for

a

long

time. The

problem

of the

prepeak generation

is

complicated by

the existence of _a 2d

layered

structure in the

crystalline

state. Puckered

layers

of covalent bonded 3 atom

triangular pyramids

are

packed

with _a

layer spacing depending

on the

intensity

of the Van der Waals bonded _atoms

belonging

_{to two} different

layers.

Many

studies have been devoted to this

subject [14-18].

Smith et al.

[17] pointed

_out the very

strange

fact that

using

_a

micro-crystalline

model of orthorhombic _or rhombohedral

arsenic does _not

yield

the calculated

S(q)

with _a

Qi Prepeak. Nevertheless,

this

ghostlike prepeak might

be created

by slightly increasing

the

interlayer spacing

of the

microcrystalline

model.

Then it becomes clear that the

prepeak origin is,

in this _{case, more a} texture

problem

than _a

topological

_one. The

breaking

of the

puckered layer packing

leaves _some kind of _caverns

separating

two

microlayers

whose lateral

expansion

is at _a nanometric scale. For this _reason the sth column _{case can} be classified in the

packing

fraction

dependent prepeak

class.

2.

Prepeak profile

modifications _: their

relationship

to the medium range order fluctuations.

Now that _we have learned how to

generate

_a

prepeak

_on

S(q) by reducing

^the

packing

fraction of atoms

through

a uniform distribution of holes in _{matter a new}

question

arises _: for _a

given packing

fraction what _are the

parameters

that _can

modify

the

position

of q and

intensity profile

of the

corresponding prepeak

? This

question

is

obviously

related to the

metastability

of

glasses

and to the

expected

existence of _a minimum minimorum for their free

enthalpy.

It is also

likely,

a

priori,

that the

physical properties

of

glasses

^{will be}

quite dependent

_on the

M-R-O- and

consequently

on the

prepeak

behavior. This is the aim of this _new section.

2.I PREPEAK

Qo

AND CHEMICAL ORDER IN METALLIC GLASSES REVEALING THE PREPEAK

BY A WEIGHTING EFFECT OF THE SCATTERING LENGTHS BETWEEN THE ALLOY CONSTITUENTS.

In this _case,

Qo

is _not

generated by

the hole model. M-G- _are

close-packed

structures.

Even if the differences of sizes between the

alloy

constituents

might weakly

contribute _to the

prepeak formation,

its main

origin

arises from the

affinity

between hetero-atoms

yielding

chemical order electronic fluctuations.

Historically

the term _«

prepeak

_» has been used Flrst in the M-G- _case while the F-S-D-P-

appeared

in

amorphous

covalent elements and

alloys.

The

prepeak

of M-G. is

always

a weak shoulder of

Qi

even in the _case of _an

optimized

contrast

experiment

where the constituents have very different

scattering lengths.

The M-G-

prepeak

_was revealed for the lst time in _a.

Co~oP~o, by

Sadoc et al,

[19]- Using coupled X-ray

and neutron diffraction

experiments,

the

partial

R-D-F- _were

derived, showing clearly

_a chemical

affinity

between hetero-atoms-

Many

_«

partials

_» have since been derived for various M-G-

composed

either of metals and metalloids _or of 2 different metallic transition

elements such _as

Cu, Zr,

Ti etc.

Preparation dependent inhomogeneities

have been studied _as

well,

but the scale of the volume involved is

larger

than that

required

for

prepeak modifications,

I-e. 20,50

/k

_{instead of} 5-10

/k- Hence,

the

composition

and

density

fluctuations within materials such _as

Cu~oZr4o prepared by

cathodic

sputtering (as compared

to melt spun

samples)

_are revealed

by

the

presence of _{a new «}

prepeak

_» ⁱⁿ the Small

Angle Scattering

domain

IS-A-S.) [20].

As far _as

magnetic amorphous alloys

_are

concemed,

M-R-O-

compositional

in-

homogeneities

_are

responsible

for both nuclear and

magnetic

order diffraction

rings

at low

angles.

Boucher _et al- have

extensively

studied this

problem

in rare,earth based

alloys, by

neutron S-A-S-

experiments [21]-

In

a-Tb~~Cu7~

for

example,

_a

ring

_appears at q = 0-2

i~

which is

interpreted

_as

coming

from small aggregates ^{of Tb}

separated by

50

/k-

Let _us remark _at this

point

the very

complex,

but

interesting

transition from

point

defects to

large

scale

inhomogeneities

in the direct space which _are detected in the

reciprocal

_space

within the _{« no} man's land domain _» between S-A-S- and

Large Angle Scattering IL-A-S-)-

It is not _a

simple

task to

correctly

describe the passage from

topological

order to the

long-

range ordered

compositional

_waves of _a

spinodal-like decomposition.

^{In the}

reciprocal

_space

the observed _«

prepeaks

_» at small

angles

_are

generated by

constructive interferences between beams issued from individual

particles

instead of atoms.

It is worth

making

_a

comparison

with

equivalent phenomena

in the

crystalline

state. It is well known that well defined

scattering rings

_appear ^{sometimes in the S-A-S- domain} when

long

_range electronic fluctuations establish a 3d

periodic superlattice,

_as in the

spherical

Guinier-Preston

(G-P-)

_zone of

Al-Ag alloys [22].

2.2 PREPEAK _AND TOPOLOGY i THE EVEN FOLD RING

(E.F.R.)

AND THE ODD-EVEN FOLD

RING

IO-E-F-R-

MODELS FOR DISORDERED TETRACOORDiNATED MATERIALS. Since the first

attempts

to construct disordered models for a.Si _or a-Ge

using

sticks and balls it has

become clear that there _were 2

possible

solutions to connect the

elementary

tetrahedra without

introducing

too much strain energy into the models.

The

generated rings

_can include either _{more or} less atoms than the 6 membered _« chair _{» or}

« boat

» like

puckered rings

of the

crystalline

structures. As

already mentioned,

Polk built the first model

involving 5, 6,

7 membered

rings

that _we shall denote here _as the O-E-F-R-

model,

while Connell and Temkin

proposed

_a

homogeneous

six membered

ring

model denoted here

as the E-F-R- model.

It _was believed that while the O-E-F-R- model

might

well

represent

^{the local} structure of a-Si _or

a-Ge,

the E-F-R- _was better

adapted

to III-V

materials,

where the local electrical

neutrality

could

only

be

produced by using

_an altemate _even

ring

solution. If the _atom

coordinates of the 2 models _are introduced in the

Debye

^formula their theoretical

S(q)

_are derived _: what kind of differences should appear on their

respective profiles

?

They

are, in

fact,

concentrated in the relative

positions

of the various diffraction halos. We _{can see} that _an

opposite

shift _moves

Qi

and

Qo

towards each

other,

when

going

from the E-O-F-R- _to the E-F.R, model

S(q) (Fig- 6)-

~O al

220 311

ill

Q (

ill

O ₁ 2 3 4

°(i~~)

Fig.

6. Theoretical

S(q)

for the O-E-F-R- (Polk) model _: continuous line and the E-F-R- (Connell- Temkin) model _: dashed lines. The vertical _arrows indicate the

positions

^{of the}

crystalline

d

spacings

in Si. The horizontal _arrows show the shifts of the

Qo

and

Qi Peaks.

In table I _we have

gathered

the ratios

Qi/Qo

for various

experimental S(q)

_as

compared

with the theoretical values for the E-O-F-R- and E-F-R- models. It is rather

amazing

that

experimental

_cases

reproduce clearly

the 2

simple

ball and stick models

[23].

As

expected,

the E-F-R- model is encountered in

compounds

such _as GaAs-

More

spectacular

is the _case of

a-Sii _~Sn~ alloys

^whose low Sn contents

S(Q)

_are of the O-E-F-R-

type

^{while the E-F-R-}

progressively

dominates when

approaching

the

equiatomic composition, revealing

_very

simply

_an obvious chemical order

[24].

Coming

back to

figure 6,

it is worthwhile _to focus _on the

Qo

location as

compared

with the I11 line of the diamond close

packed interspacing,

The O-E-F-R- model

Qo

is

definitely

on

the low _q side of the 111

line,

while _on the

contrary

the E-F-R- model

Qo

has drifted _on its

higher

_q side. The atomic radius

being

the _same, such _a difference

suggests

that the

interspacing

distance between fictive dense

planes

is

larger

in the first model than in the

second,

_so that the local

density (say

inside the first 3

shells)

should be greater ^{in the} E,F-R-

network. At the _same

time,

due _to the

uniqueness

of the 6 fold

variety

of

rings,

_a

large

dihedral

angle

statistical distribution

yields

_more strain energy into the E-F-R, model.

Table I. The ratio

Q i/Qo for

tetracoordinated models and

experimental amorphous

elements and

alloys.

Qi/Qo Qi/Qo

References

(models) (experimental)

O-E-F-R-

(Polk) [7]

Relaxed 1.82

[6]

a.Si

IS-P-)

^1.736

[27]

a.Si _: H

IS-P-)

^1.797

[27]

a.Si _: H

(G-D-)

1.849

[27]

a-Si~osnio

^1.796

[24]

E-F-R-

(Connell-Temkin) [8]

Relaxed 1.76

[28]

Unrelaxed 1.67

[28]

a.GaAs 1-71

[23]

a-Ga-P 1.70

[23]

a-Gasb 1.70

[23]

a-InP 1.67

[23]

a-

Si5oSn5o

⁶⁶

[24]

These overall features induce model

dependent

constraints _on the medium range order level which _are

expected

to be reflected _on the band _structure characteristics,

Ching

et al.

[25]

have studied this effect

theoretically, using

the well known

technique

of

orthogonal

linear combinations of atomic orbitals

(OLCAO).

The calculation shows _a

large

band gap difference between the 2 models. The greater ^{the local}

distortion,

the smaller the

bandgap (2.19

eV for O-E-F-R- and 1.24 eV for

E-F-R-)- Despite

the

likely over-simplified approximations

of the

calculation,

it is demonstrated

qualitatively

that

large

electronic property differences _can be

generated by

two different

topological

M-R-O-

building rules,

_even

leading

to an atomic

assembly having

at a

macroscopic

scale the _same

packing

fraction. At the

experimental

level it

can be stressed that _a very

simple

criterion I-e- the

Qi/Qo

ratio _can

yield significant

information

regarding

the M-R-O,

2.3. PREPEAK _{AND NETWORK RELAXATiON :} a.Si AND a.Si-Ge ALLOYS. In this section _we

enter an

important

field of interest with

respect

to _our purpose since it deals with _a direct correlation between the

Qo Peak

^{behavior and} some

physical

property ^variations

depending

on the

glass preparation

conditions. We shall

develop

^{in detail the} case of tetracoordinated materials which _are of both theoretical and

practical

interest.

A considerable amount of studies has been devoted

during

the

past

²⁰ years ^to

a-Si,

a.Si _: H thin films _as well _as to their

alloys

with C and Ge elements.

Finding

the free

enthalpy

minimum minimorum of the system ^{is crucial for} some

applications

such _as the determination of the maximum

efficiency

of

photocells.

Indeed the

photocarriers

lifetime is

drastically

reduced

by

the localized _states in the valence and conduction band tails

generated by

the atomic disorder.

A

4

Q

+M 3

~

~

W_~ 2 ~

~

I i

b

~'o.5

_{I-o I.5}

Energy ie.V.)

B

i

u

/

#

#

_~

~ j

~

'fo.,

Energy le.V.I

Fig.

7.

Optical

_spectra obtained

by

Photo-Deflection

Spectroscopy

(P.D.S.). IA)

Amorphous

Si a) _pure Si IS-P-) b) a.Si

: H (G.D.) c) a.Si _: H IS-P-) _{; a} and b

samples

_are the _{same as} those of

figure

9. (B)

Amorphous GeSe2

a)

deposited

film b) annealed film _{; a} and b

samples

are the _same

as those of

figure

9.

Among

the various methods of

deposition,

the

decomposition

of silanes

by

the

glow discharge technique (G-D.),

and the reactive cathodic

sputtering IS-P-)

_are

typical

methods of

preparing

a-Si:H

alloys.

Due to the native

hydrogen

formed in the S-P- method this

technique yields

_a great

variety

of H

bondings

such _as

SiH, SiH~,

^SiH clusters, etc. but

unfortunately

the energy of the extracted atoms from the target ^is such that

they produce

a

permanent bombardment of the

deposit inducing

irradiation defects.

It is therefore useful to make a

comparison

with the

sample

characteristics

prepared by

the G-D- method which is not so

energetic.

It has been

proposed

that besides the reduction of the a-Si electronic gap ^states

through

the

dangling

bond

saturation,

H _atoms also

play

a role in

network

relaxation,

I,e. minimize the band

tailing

effect.

The tail _{states are}

conveniently

characterized

by optical absorption

_spectra. This is the

reason

why

_we have undertaken

coupled X-ray

and Photodeflection

Spectroscopy (P.D.S.)

studies _on

samples prepared

under the _same conditions.

The P-D-S-

technique

_measures the

optical absorption

under the

absorption edge,

due to gap states and transition from extended _states in the _one band _to localized states in the tail of the other band. The

optical

_spectra of both

hydrogenated

^and

nonhydrogenated samples

are

shown in

figure

7A _{: a} drastic reduction of the subband gap

optical absorption

is observed when a.Si is

hydrogenated.

This reduction is nevertheless _more substantial in G-D-

samples

than in

samples

of S.P. The

optical

order

parameter

^is

Eo,

the inverse

slope

value of the linear

part

of the

absorption edge (Urbach tail).

The _more relaxed the

material,

the smaller the value of

Eo,

I-e- the steeper ^the

edge.

On the structural

side,

_{a very}

simple

relevant order parameter ^{is the coherence}

length L,

I-e- the size of the volume in which

X-rays

« see » an ordered network. L is derived from the

Full Width _at Half Maximum

IF-W-H-M-)

of

Qo-A(2b) by

the formula:

L

= 0-9

A/(A(2 b)

_{x cos}

bo)-

We have shown in

(26)

that L

actually

decreases

linearly

with

Eo (Fig- 8)- However,

the

bonding

_type of H

(identified by

infra,red

spectroscopy) plays

a determinant role in the network modification. The atomic fraction of H

being larger

than several percent, ^the saturation of the

dangling

H bonds is achieved in all materials.

w

w .

Y _

@~

o

deterioration of the structural and

optical

order

parameters

: here

SiH2 dihydrides

dominate the I,R.

spectra.

In another

study (27)

is has been stated that the network relaxation also modifies the

prepeak position (Fig- 9a).

As

pointed

out in the

previous

section the

Qo

location relative to the c,Si I I I d

spacing

is _an indirect _measure of the bound distortions. Thus _{one can}

predict

that all

parameters releasing

the strain energy should shift

Qo

towards lower _q values. It _can be _seen that the

prepeak

of the

hydrogenated sample

is in fact shifted from the

right

to the left side of the theoretical 111 line. But

contrary

to the

topological effect,

the

Qi position

is not

modified

by

the relaxation

phenomenon.

Now,

if

alloys

a-SiGe _: H _are

prepared by

these 2

techniques, Qi

follows _a linear

Vegard's law, independent

of the

preparation conditions,

while

Qo

on the

contrary

is

quite dependent

on them

(Fig- 10)-

A numerical simulation has been

performed by

Mosseri

[28]

of the minimization of the strain energy in a-Si-Ge

alloys.

The

alloys

_were made up

by randomly substituting

Ge atoms for Si atoms

ii-e-

with _a

larger

atomic

diameter)

within the O-E-F-R- and E-F-R- models. If the theoretical

predictions

_agree

quite

well with the

experimental

behavior of

Qi (reflecting

the _mean atomic size

variation,

_a function of the

composition),

the

Qo

evolution with _{x on} the

contrary

^exhibits

large preparation dependent

fluctuations _as

compared

with the theoretical

predictions.

The

alloy

network

topology

^remains of the O-E-F-R-

type

^{but the}

slopes

_as well _as the absolute values of the

Qo positions obey

different linear laws.

These results suggest ^{the interest of}

investigating

the M-R-O- differences in the direct space associated with their localized counterparts on

Qo

in the

reciprocal

_space. This has been done

by

anomalous diffusion

scattering experiments

_on the Ge K

edge

which

yield

the differential R.D.F, It is shown that H reduces the dihedral

angle

distribution between

adjacent

tetrahedra in the

alloy network, reducing

the number of

pair

distances between the 2nd and 3rd shells of

atoms. It is also confirmed that the M-R-O- atomic order in G-D-

samples

is _more relaxed than in S-P.

[29],

Finally

the

sharpness

of

Qo

_as well _as the ratio

Qi/Qo

of

amorphous

tetracoordinated semiconductors

S(q)

should _{serve as} very

simple

tests for the calibration of the

degree

of reduction of the

optical

and electronical active defects in these materials.

j~

,

Qi ~

Q

°0

= ~

i I ^d

I I

~ ~

~ ~

~ ~

uJ uJ

~ ~

~ ~

+ &

b

+ +

0 lo 20 30 40

e d_eg_] ⁴ i~ 2D 2e ~

Fig-

9.

Experimental X-Ray

pattems ^of :

IA) amorphous

Si films, a) a-Si _:H _; b) a.Si : the small vertical _arrow marks the Ii ii) c.Si line. F-W-H-M- is the Full Width at Half Maximum which is used to calculate the coherence

length,

L of

figure

8-

(B) Amorphous GeSe2

films. a)

deposited GeSe2i

b) annealed GeSe2.

i

2.05~

-Si,

~

~~

i

~3

o

Fig. 10. The effects of

hydrogen

and the modes of

preparation

_{on Si~i_~~Ge~} films

Qj

and

Qo

variations with _x.

Squares sputtered

_pure Si _; full _squares

sputtered hydrogenated

Si

IS-P-)

full circles _:

hydrogenated

Si

decomposed

from silanes (G.D.) _; dotted lines _: theoretical simulations after

(27)

(P stands for Polk and CT for

Connell-Temkin)-

One _{can see} that

Qi obeys

the same law for all

samples,

which is in agreement ^{with the} theoretical results, besides _a

slight

shift. On the contrary,

Qo

^is

strongly dependent

_on the H content and the

preparation

conditions. Their behaviors _are also

different from the calculated values.

2.4 PREPEAK

Q(

AND GLASS HOMOGENEITY. THE _{PHASE SEPARATION IN} IV VI _GLASSES.

We shall _now focus _{on a more} remote

part

of the M-R-O- _at about three times the first

neighbor

distance. Henceforth we are

dealing

with the second

prepeak Q(

^{of the} D-R-P-H-S- At this near-nanometric scale the M-R-O- _can be

govemed

not

only by topological

features but also

by

local

composition

fluctuations _as well _as

departure

^from chemical

ordering.

This

situation is encountered when

alloying

elements A of the Ivth _or Vth columns with those B of the Vlth column. Let _us examine the IV VI _case. Numerous studies have been devoted to the

study

of the atomic structure of

Si,S, Si-Se, Ge,Se, GeTe,

etc. It has often been observed in

Ai

_~B~

alloys

such _as Gese that _a

Q( prepeak

is

progressively emerging

_on the left side of

Qo

when the

composition approaches

the stoechiometric

AB2

value

[30].

The Gese _case has received considerable attention in the

past.

^{Its first}

sharp

diffraction

peak (Q()

has been considered _{as «} anomalous _»

by Phillips [3 II

and

necessarily

associated

with molecular

clusters,

whose center-to-center

spacing

should be of the order of 5

I

as

derived from the

Q( position. Ruling

out a

microcrystalline

model since the F-S-D-P- subsists above the

melting point,

a

layer-like

structure is nevertheless

proposed

_{as a} reminiscence of the

high

T

crystalline

from of

Gese~-

It is the well known _«

outrigger

_» raft model which

connects

GeSe4/~

tetrahedra but breaks the chemical order at the

edge

of the

raft, making

Se- Se wrong bonds. This model _was

opposed

_{to a}

strictly chemically

ordered _one

connecting

also

GeSe412

^{tetrahedra but linked} to form _a 3d network,

In _a recent

study [32]

Fischer-Colbrie _et al. tried to elucidate the

problem by studying

the structure evolution of

Gese~ glasses

with the

sample

thickness. As _a

starting assumption

the

layer

structure should be favored in very thin films

la

few

I thickness),

while _a

likely isotropic spherical tendency

should be

expected

in bulk

samples.

Despite

the

high precision

of the

experimental

data

(high

flux of the

X-ray synchrotron

radiation _source and

grazing incidence)

the authors did _not detect

significant

differences between the spectra. ^After a detailed

investigation

of the M-R-O- and the failure of

matching

the

S(q) by layered models, they finally

obtain _a

good

fit with _a 3d

spherical microcrystalline

model except ^{for the}

Qi Position

which

requires

_a

slight

increase of the

crystal layer spacing

of the

experimental

value. We have

carefully compared

the

Q(

values for their thin

sample

and bulk material. The

position

is shifted

slightly

from I

i~

_{to 0-94}

i~

_{for the thin}

samples.

So the

required interspacing

increase should

again

^be

larger

^for the thin film

S(q) matching.

More

recently,

_an intermediate _case has been obtained in

relatively

thick

evaporated

films of I micrometer

[33].

It _can be _seen in

figure

98 that the

deposited

film

Q(

_can be

roughly

decomposed

into 2

subpeaks

which tum out to be located at 0-85

i~

_{and I}

i~

~,

respectively.

Remembering

the above remark it is

likely

that the smaller

angle component corresponds

to the

beginning

of the

deposit

when it _was very

thin,

while the

larger angle

component ^should be attributed to _an almost bulk _state. But the

important

fact to underline is that, after

annealing

at T

= 330 °C the lst component is

drastically

reduced and appears as a weak shoulder of the

Q( peak

at I

i~

~-

We _can conclude from this that the

possible decomposition

of

Q(

into 2

subpeaks

which behave

differently

on

annealing clearly

demonstrates that 2 different solutions for the M-R-O-

can exist, Are

they representative

of the 2d and 3d models ? More work is needed _{to answer} this

question.

The

investigation

of

physical properties might bring

about _some

help

_to make _a choice.

Unfortunately, divergent

conclusions _are drawn

through

ultra-sonic results made

by

Bellessa et al.

[34] being interpreted only

in _a 2d

framework,

and X-P-S-

experiments by Theye

et al.

which rule _out the

outrigger

raft model since _no wrong Se-Se bonds _are detected

by

this

technique.

The

optical spectral

behavior _on

annealing

exhibits _an increase in the

optical

_gap of 0.3 eV

accompanied by

_an increase of the

pseudo

_gap defect state contribution _to the low energy

optical absorption (Fig- 78).

In this case, the evolution is very different from what

happened during

the relaxation

phenomena

of a-Si and a-Si-Ge

alloys

which _were

accompanied by

a

decrease in the gap defects without any

change

in their gap width. This is _a

paradoxal behavior,

since

annealing

of _a vapor

quenched

material

usually

releaves strains and local defects.

It is also

unexpected

that the

Qi Profile

exhibits _a

complex

structure in the

as-deposited

state. This is nevertheless consistent with the observed evolution of

annealing

towards _a

single

stoichiometric

phase

which is known to

correspond

to the

largest optical

_gap of the

alloy.

Conclusion.

We have _seen that the atomic structure of covalent

glasses

_can be derived from the disordered

network of _a

compact

^hard

sphere packing,

The hidden 5 fold

symmetry, goveming

the

general

features of the M-G-

pattems

^{is retained}

during

the reduction of the

packing fraction, through

_a

duality

transformation from _a

pseudo-icosahedral

environment towards _a

pseudo-

dodecahedral environment.

Prepeaks

_appear at

decreasing

_q value

positions according

_to the homothetic inflation of the hole

generation procedure.

Remembering

that the low _q domain of diffraction

pattems

^{reflects the} medium and

long

range

order, prepeak profiles

and

positions

are very sensitive to the

topology,

relaxation

phenomena

and _to the

homogeneity

of covalent

glasses.

The various

origins

of local order fluctuations do _react

differently

to

S(q) affecting

^either

the whole

diagram

_or

only

the

prepeak shape and/or position.

At the _same

time, physical properties

exhibit

parallel

fluctuations.

Prepeak

behaviors reflect at least

qualitatively

these variations.

The difficulties

begin

when

quantitative

evaluations _are undertaken in theoretical studies.

Beyond qualitative trends,

the _true gap, the valence and conduction band

profiles

and tail width need _to be

calculated,

_{a more}

precise participation

of the

M-R-O-, involving

several

hundreds _{or even} thousands of _atoms. Are theoretical calculations able to fulfil these

implements

? New

approaches

_are in progress

taking

into account _more

precisely specific bonding

and dihedral

angle

distortions

[35].

It is _now reasonable to foresee that fruitful

developments

_on the band _structure

knowledge

of covalent

glasses

will arise from the _use of M-R-O- details

provided by

fits of the _exact

profiles

of

prepeaks-

Acknowledgements.

I would like to thank R. Mosseri for fruitful discussions and

help

in the

generation

of the

graphics

of the 2D models.

References

[ii VEPREK S., BEYELER H. U., Philos.

Mag.

B 44 (1981) 557.

[2] Moss S- C., PRICE D. L.,

Physics

of disordered materials

(Plenum Publishing Corporation,

1985).

[3] GUINIER A., Thdorie et

Technique

de la

Radiocristallographie

(Dunod, Paris) p. 452.

[4] SADOC J- F., DIXMIER J. and GUINIER A., J. Non

Crysl.

Solids 12

(1973)

46.

[5] MANGIN Ph., MARCHAL G., RODMACQ B-, JANOT Chr., Philos. Mag. 36 (1977) 643.

[6] CHAUDHARI P., GRACzYK J. F., HENDERSON D-, SrEINHARDT P.

(1975)

727.

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