Energy spectrum of stationary states for glassy semiconductors

Journal of Non-Crystalline Solids 90 (1987) North-Holland. Amsterdam

ENERGY

M.D.

SPECTRUM

69 - 72 69

OF STATIONARY

STATES

FOR GLASSY

BALMAKOV

Lenigrad

State

University,

199164

Leningrad,

Universitetskaya

The energy spectrum of homogeneous stationary is shown to represent a set of alternating The criterion underlying the band classification ity of the states in question.

There exist

is

a fundamental

besides

their

first

to

the

level

spectrum All

know

a large

densed

states

The form

can

stationary

bands

a band

(1-2,

may the

become

realized,

lie

energy

in Within

for gaps

a band,

occurring

when

the

multiplicity

in

structure

do

it,

structure

of

one

the

has

energy

determined

in the

use to

the

place

way

polymorphic and

liquid

systems (7-8).

within

E

Fig.1).

there

states,

is

The

states

Such

(see

another,

stationary

equilibria.

of con-

modifications

structures enclosed

inhomogeneous

in

the

bodies

homogeneous

following

various

by

macroscopic

conveniently

amorphous

phase

are

studying

etc. which

in

may

particular,

6-7). varies from

insignificantly,

one

band

to

fluctuation

of term

Therefore

m atoms.

a rule,

4-5,

a spatial

electron

In

non-crystalline

in

over

to

addiabatic

to

are

instance, (2-3,

answer

corresponding

may be completely

there

crossing

is

due

band

structure

semiconductors

can

In glassy,

the

glassy

one spectrum

homogeneous

one

USSR

semiconductors

to

a system

E.

represented

5-6).

homogeneous,

glassy

the

of

corresponding

to

and

energy

atoms,

energy

schematically

3-4,

overlap,

Besides

m of the

states

corresponding

bands

of

number

be

properties states-in

There

7/9,

of a macroscopic system and forbidden bands. is the degree of homogene-

In order governing

nab.

system.

stationary

c=E/m.

amorphous

relationships

thermodynamic of

rates

why

state allowed

counterparts.

a macroscopic

equilibrium

containing

question

crystalline general

of

distribution

the

SEMICONDUCTORS

of

physically

U,(fi), fragments

of on

the

For

the the

their

vector same

chemical in

the

composition the

and

typical' $1

specifying

changes

amorphous

is

minima t

position

substantial the

structure

nonequivalent

with

depending

more

another.

of

which the

coordinates differ,

non-crystalline

as semi-

conductor. Structural

transformations

relatively

low-energy

different even

minima radio

0022-3093/87/$03.50 (North-Holland

emission

Physics

photons of

the

potential

can

be

0 Elsevier Publishing

are

accompained

and

represents

observed

U (I). ml .

Science Publishers Division)

by the

emission

essentially Therefore

B.V.

(absorption)

tarnsitions in

the

transformation

of between range

70

In tion

order of

ductor,

is

number

tribution the in

in

a melt

where

are

is

rotational at

glass.

Therefore

values

of

mean

m;(T),

energy

at

of

vibrational

the

reduced

low,

this

the

will

hinder

crystallization

to

occur

as

the

is

In

this

lowered,

case,

(1-q)

rate

as one

decreasing of

the

band

line

structures.

will

freeze

glass-forming

take

capacity

place,

of

the

in

bottom

non-crystalthe

melt

melts

their

the

homogeneous

with

the (9.X

tempera-

approaches

As result, out

the

structural

will

of

atom.

for

nonequilibrium

transformations

one

density

equilibrium Figure).

tempera-

to

Tt

required

(g)

the

is

the the

rearrangements

ture

into

internal exception

realized

structural

the

creasing

to

temperature

N,(m;(TttO))

FIGURE 1 Energy spectrum

which

close

energies,

liquidus

the disIndeed,

primarily

for

the the

only

their

as well'.

are

those

not

but

N,(E),

formasemicon-

know

minima,

Um are

E(T)

the

a glassy to

such

which

T, with

and

of

energy,

potential

ture

understand

essential of

structures

the

If

fully

structure

it

total

e

to

the

equilibria

transforming

increases

with

de-

criterion

Quantum-mechanical U,(E),

and

viable

here.

use

a Gaussian minima,

N,(E)

of the

which

distribution, exp(om), the

are

N,(E)

independent.

line

we obtain

that, We assume

statistically

E, at

methods

after

difficult and

crystalline

(1).

fragments

of

Therefore

for

and where

to

K

a>O.

for

apply

Model suffuciently the

structure

Ecr

calculation

the

there be

changed,

of

should

large

of that can

the

size

approxiamtion

evaluation

Assuming

to

approaches

(w)

of number exists

be more to

N,(E) of

one

non-crystal-

a minimum and

that

be can

energy

U,(6,,)=0,

an expression

= b(E)

+

E yJzmn(1

- @--

h - m,) Yfi

(2)

M. Il. BolrlIukol~ E where

e(E)

tion,

and Eq.

= k

e I -m h are

y and

(2)

can

be

x2 -2-

6(E)

the

used

the

part

Z,(T) Eq.

of

the

(3)

Dirac's

the

ofsroriolrur~~

delta the

states

function,

Gaussuian

following

71

o(E)

is

step

func-

distribution.

expression 2

q)kxp

statistical

leads

m(a

- F + f) (3)

E,-hm - $(-I} Yfi

(1 for

specmo,,

of

obtain

E,-hm --+Y YJiii

= 1 +

formations.

is

parameters

to

{I-@( Z,(T)

dx,

/ hergv

us

sum which

to

the

accounts

following

for

structural

trans-

relation

for

T < u2/h

for

cg 2 0 and

T < T,,

for

cg < 0 and

T > u2/h,

for

cg 2 0 and

T > T,,

0 i(T)

= 2 h-5

L

here

Es+&

9

T, =

E9 ES

If

(4)

0 <

<

E

Therefore

within

energies,

E

9.

tEs)

(5)

=h-yJZFi,

(6)

= 2yvE.

(7)

or

for

E

9(

4a

Ed

nentially

+2/E

cg t cs

mcm and

if

< E then E

<

E

the <

density

Nm(ms)

then

cgtEs

it

(2)

will

will

decrease

increase

expo-

exponentially.

9

homogeneous For the

showing

9

find

the

model, energy

and gap

non-crystalline

the

the

and

cs

vibrational

is

the

and

width

of

the

rotational band

of

of

glass-forming

ability

(l),

one

can

obtain

from

(4)-(7)

relation

that

(I the

to the

neglecting width,

structures.

criterion

= 1 -

Similar to

is

following K(S)

this

the

relation

t

25 t

2fq7n))-2

quantity

way

K depends

it of

(8)

was the

done parameters

only

on

elsewhere a,h,y

one

parameter:

1,2,

expressions

with

such

(4)-(7) experimentally

can

be used measur-

able

quantities

before

(C;)

and

CI = q/T,

as melting

temperature

after

melting:

+ 0.5(C;

h = q + Tm(C; Y' where

(C';)

T,,

the

q,

heat

capacities

- C;),

(10)

- C;),

(11)

(12) quantities of

evaluation

are

Substance

q,

the

C;,

CG are

parameters

of

presented

in

a

h,

reduced

the

the

eV

0.13

0.13

0.11 0.24 0.11 0.24 0.05 0.03

0.07 0.09 0.05 0.06 0.02 0.02

3.1

Te TC Na

For

Si02

band

and

width of

For

non-glass

and

cS and

As2Se3 of

possessing

the

magnitude

homogeneous than

forming cg are

of

the

substance the

same

atom.

Eqs. The

gap

eV

(lo)-(12) results

0.26 0.22 0.43 0.20 0.33 0.06 0.04

permit of

this

good

non-crystalline

structures

(Te,

the TL,

of

criterion

Na)

the

5

0.003 0.003 0.026 0.015 0.081 0.022 0.012

a comparatively

width,

order

eV

E , 9

%’

1.3

2.0 3.4 1.0 1.0

one question.

eV

Y*

0.5

As2Te3

in

Table.

As2Se3

Se

to

model

Si02

orders

heat

T,m,

evaluation

the

melting

glass-forming

K (1) value

0.01 0.01 0.06 0.07 0.25 0.34 0.28

of

magnitude.

REFERENCES 1)

M.D.

Balmakov,

Vestnik

Len.

Univ.,

No.10,

(1983),

104.

2)

M.D.

Balmakov,

Vestnik

Len.

Univ.,

No.18,

(1985), 105.

is not K is

0.36 0.37 0.62 0.66 0.85 0.89 0.87 capacity

larger

by

exceeding close

two 40%.

to

go%,