Line width and exciton-impurity scattering in Cu2O

377

Journal of Molecular Structure, 45 (1978)377-387 0 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

LINE

WIDTH

AND

S. NIKITINE,

S.G.

Laboratoire (sssoci& rue

5,

EXCITON-IMPURITY

de au

de

ELKOMOSS

SCATTERING

and

Spectroscopic

C.N.R.S.

no

C.

et

IN

Cu20

SCHWAB

d'optique

du

Corps

232),

Universite

Louis

67000

Strasbourg

(France)

l'tlniversitg,

Solide

Pasteur,

ABSTRACT The

variation

series nal to

Ag

in +

the

the

of

Cu20

ions

as (ref.

must

1) is

acquire

increases.

This

component

Lorentzian of

the

and

the

line

line

line

0 = 1014 Further

of

excitons

cm2

and

can

R = 3.2

discussion

of

in

with

results

leads

Ag+

N.

of

cm-l

The

a shape

concentrations

N

good

cross-section

to

that

convolution

= 0.57

be calculated. The obtained -5 10 cm for a concentration

these

shown of

surprinsingly

a collision

similar

is

t," N + 0) and

different

is

It

the

Av

variable

calculated

substitutio-

a theory

from

width

for

comparison

yellow

concentration

AWL

lines

path

the

the

N of of

lines

theoretical this

when

of

theory.

width

1)

From

ground

extrapolation

(ref.

the

on

line

by

1 line

concentration

is obtained

a constant

of

n =

broadening

shape

experiment

component

the

the

interpreted

shape of

of

experimental

agreement. free

of

a Voigt

Voigt

from

width

collision

a Gaussian (obtained

line

a function

Lorentzian

line

the

and

mean

values

are 18 cm-3 N = 10 I information.

useful

INTRODUCTION The been

width

of

studied

the

The

choice

forbidden

a)

Ag

of

1S line very

impurity

the

yellow

carefully

concentration

exciton

both (ref.

as

series

has

a function

1) and

of

of

temperature

2).

width

residual that

50 K or

of

this

quadrupole

the

tures, and

1,

experimentally

substitutional (ref.

n =

the

so,

line

transition

of

the

strains

in

width

is particularly

of

line the the

and

is very

depends crystal

line

favourable.

on

the

only.

varies

narrow.

The At

low

concentration

It has

slightly

been with

line

is

a

temperaof

shown

impurities :

temperature

below

378 b)

that

width

in

a systematic

It has been

lattice

the

tions. the

Ag+

analogy

The

line

Toyozawa's being The

of

of width

(ref.

substituted

some

dispersion

in

AwR

figure

AgZO

sites

this

3)

interband

of in

Cu the

line +

not

affect

the

line

Ag

not

width is

than Auls

are not substituted

lattice

the the

with

higher

suggested

J;lO concenrrorion

points. to

zero.

that

I

400 (ppm)

of the 1s absorption concentration.

this

account

exciton

on terms

1 level.

linear tends

This

of

CuzO.

concentration

It

in

in disloca-

substantially

n =

approximately

N tends

of

depend

scattering,

1

200

Fig. 1. Variation of the Ag impurity

do

rather located + in Cu sites on

the

does

energy

sites

been

are

located

experimental

as conclntration 1. It has

Ca

impurities

with

1s line

higher the

of

Si,

but

to be lattice

of

considerably

variation

Cu"

likely

the

Mg,

that these

on

is very

like

way.

suggested

of Cu20

ions

width

impurities

other

of

in to

Ag+

spite

of

a residual

variation

extrapolated

is

shown

width

10

line

width

in CuzO

as

function

379

corresponds As

the

predominantly

impurity

Lorentzian The the

of

1s line

ning atoms. line

In

The

paper

with by

lines

is

concentration

this

way

of

shape

increases

by

our

due

the

scattering we

on

of

The

ambiguous

as no

calculations velocity

is

of

of

of

the

(ref.

line

2)

of

the

develops

an

then

the

line.

rather

a

this

for

between

these

neutral

gas of

the

these

lines c of

of

excitons

somewhat

is known.

velocities.

limit

in

impurities.

velocity

problem

of

broade-

cross-section

is however

two

explained

1) on

velocity the

width

component

the

extreme

be

neutral

(ref.

collision

the

Lorentz

provided

of

two

on

of

on

Lorentzian

experiments

effective

of

atoms

1 excitons

n =

treatment

given

theories

considered

that

of

theoretical

included

the

the

the variation + Ag in Cu20 can

atomic

impurities,

determination

are

of of

previous

isoelectric

is known.

show.that

suggest

scattering

a determination

excitons

to

concentration

theory

to

is

a transposition

interpretation

allows

a Gaussian

shape.

aim

a simple

to

The

The

actual

cases.

THEORY In the one

has

’ a) AVR b)

transposition to make

the is

the

residual

as

of

of

line

a Gaussian

c) charge dered

a function

the

of

substitution

distribution as

a neutral

responsible increases. suggested

for The

has

be

the

of

and

shape

with

AvR

impurity. broadening

scattering

of

the

and

of

Voigt

excitons

,

therefore

a width

from

the

component. line.

independent,

The

the

Let first

second

not change the + has to be consiAg

component

is

alone

line on

deformation

= AuG

shaped

Lorentzian

the

has

resulting

concentration of Ag+ , + + site does in a Cu

the

shape

a Lorentzian

a Ag

So

to

one

this

to excitons,

:

concentration

lattice

due

collisions

Gaussian

a Voigt

in the

the

to be

Gaussian

width

of

of

assumptions

component

be the experimental *%i component is supposed to is only

theory

predominantly

a pure

experimental

convolution

Lorentz

following

line

considered

the

of

Ag+ as concentration N + substitutional atoms Ag

of

the

lattice

of

a spectral

around

the

is Ag+

ion. It

is well

collision

known

gives

that

the

the

1) the

line

is broadened

2)

the

line

becomes

3) the

line

is

values

of N.

broadening

following

effects

(Lorentz

with

increasing

line

due

to

concentration

broadening),

asymmetric,

shifted

with

respect

to

its

position

at very

small

:

380 The An

first

effect

asymmetry

negative result

of

nent the

Lorentz

of

AvV.

line

The

(in cm-')

width

at the

experimental

and

the

top

AU

the of

shape

experimental

AvG

third

effect

measurements.

= AvR

been

considered

= 0.57

line

value

and

line,

of Avis.

variable

for

different

The

Voigt

shape

the

absorption

Avv

have

has

not

been

a Gaussian

cm -1 determined

now

and

AvV

value

from

a Lorentzian line

has

a

coefficient

to be value

concentrations line

a

compo-

measured

A given

or

as being

of

to N + 0,

the

positive

be

The resultant Voigt shape L' value of Kmax, the absorption the

accuracy.

It can only -1 0.25 cm .

than

a convolution

value

of

the

a reliable

it has

be

is smaller

a AWL formula

as

extrapolated

the of

our of

width

width

shape,

the

result

line line

but

Finally, it

a constant

with

it cannot

if it exists, is the

observed

observed,

samples, of

shape

component

been

been

accuracy

experimental

width

has

collisions. the

that

line of

has

different

within

concluded The

also

for

observed

only

compared

with

of Kmax(lS) corresponds

N.

is given

by

the

following

of

the

Gaussian

: eJ 2 dy

a2+(w-y) with

the

K

G max component.

following is the

Avn+AvL

notations.

absorption

coefficient

at the

top

_ v

a=

Iln2

AvG

(2)

2(v-vo) W=

\/Rn2 AVG

is the natural Avn -1 exciton in cm .

THE

COLLISION

If Av, shape the This

lJ=LI

a:d

line

CROSS-SECTION

is determined the

Gffective is given c Au-

_

N

v

width

from

experimental cross by

the

section formula

and

v o is the

eigen

frequency

of

the

(3 the one,

comparison

of

it is possible

(5 for

excitons

on

the to

theoretical calculate

neutral

line the

impurities.

:

(3)

to

381 of the excitons, Here c is the velocity of light, v is the velocity the impurity being supposed to be at rest. AvL is expressed in cm-' 2 can arise from the assumption and u is given in cm . some difficulty to be made i)

as

If one

the

=

the

assumes

exciton

VT

to

gas

value that

then

"T

ex

of

Boltzman (ref.

5,

masses, ii)

the

6) where

me

ion

Mex

106

in thermal

equilibrium

with

and

10

at T = 10 K (4)

6

atT=4K

supposed

is

to be

is expressed are

mh

excitons by

This

are

the

the

very

in gm;

Mex

effective

gives

created

optically

conservation

large.

= me

k is

the

f mh

electron

and

hole

of

with

momentum

a certain

of

the

velocity

photon

in the

:

nh

(5)

n is the photon

refractive

outside

the

index

known

that

10-11

set

thermal

electrons so when

energy

of

as

there

in

a very

of

this The

This

must

small

process

not

observed

thermalized

the

yellow

authors

series

ans

long

life

extent

time

damping

of

in this

remembered is based

on and

the

the

are

length

"cold".

of

It

is

to

the

in about

with

respect

phonons. a much

However slower

are

knowledge,

an

the

process,

available

only

accurate

theory

treated.

probably

lifetime

is very

wave

n = 2.49,

which

been

has

The

forbidden.

because

some

create

certainly

to have

i and

energy

of phonons,

not, seem

the

can

is

recently

the

a certain

are excess

measured

to

with

excitons

only

of

to be

The

been

account

collisions

To

number.

X the

VT.

and

excitons

crystal;

A = 6098

an

an absorption

of

malized It has

have

lattice

is a dipole

knowledge been

the

does

1s state line

they

of cold be

than

the

For

(and excitons)

or

thermalization

of

crystal.

= 2.105 cm set-l. ??Y -_This velocity is smaller

by

are

x"ex

Here

V

and

1.55

= 0.98

impurity

constant

is given

=-

ex

excitons

respectively.

crystal. V

the

However

which

the

=

“T

mass

v.

:

q

the

of

a long

of

this

luminescence weak.

1s state

of

It is the

life

line

this

likely

excitons

time.

has

to our

line

that

has

on

are

ther-

of

lines

case.

that

Lorentz

the

behaviour

phase

change

theory of

of broadening

a classical

induced

by

the

oscillator collisions

382

during

the

life

collisions

may

thennalization sical G,

time also of

of

the

representation

which

should

the

change

exciton the

be

oscillator the

is

in

treatment

can

be

of

vex

for

would our

the

best

we

that

have

WITH

with

tical

curve

every

sample

the

the

it can

be

values

component figure

1 and

tration. calculated the

tical

by

curve

It has a certain

have

to be

been

The are

on

on

in Table

above

clas-

velocity

explanation

Note

that

has

which

taking

treatment no

sens

value

velocities

of

vT

is

and

in the

vex-

same

fit

Av,

the

given

for

for

for

the

same

sample.

to probable some

this

errors of

I in which

and In

of

the

the

samples

the

values

of

values

mean of

have the

of

concencan

be

velocities

8 of

of our

we

and

the

theore-

Table of

I.

Avis

have

and

also

concentration

order

of

a Lorentizian

section

fitting

on

curve

possible

Therefore

accuracy

Gaussian

adjusting

sample

cross

sample

same

less

a

experimental

the

surpri-

extrapolation

samples

experimental the

of

N only.

each

the

is

somewhat

by

both

2 shows

of

of

for

for

acceptable.

effective

above

points that

the

theore-

relative

fitting

is

and

adjusted

convolution

all

with

samples

the

quite

is

The

points

this it

Kmax

Awv

samples

a AvL

obtained

is obtained

of

account

only

cm

and

which

-1

of

sample.

experimental

the

follows.

several

samples

still

Figure

for of

the

as value

every

concentration

best

AvL

for

AuG

from

the the

the

emphasized

lead

mean

a

that

which

The

for

for

= 0.57

experimEEta1

taken.

results

given

AvG

formula

on

so

this

mechanical

is performed AulS

as being

value

parts

the

sure

both

curves.

most

obtained

to be

dispersion

evaluations

samples

v Tandv.

and

in different

some

obtain

the

(5 for

with For

are

Auv

a value

exciton

to

not

a constant

sample. for

this

a certain

time and

In

is obvious

neglected

values

compared

depending

From

of

place.

a quantum

determined

N

considered

to

in

experiment

with

supposed

AuL

It

this

oscillator

consideration.

experimental

a breadth

in order AvV breadth Avis

of

then

of

of

component

the

the

impurity

Even

acquire

simply

values

calculated is

take

is theoretically

experimental of

to

equivalent

exciton are

the

EXPERIMENT

with

from

good.

The

it

corresponding

singly good

THE

comparison

concentrations

to

as

considered

is determined fitted

the

in

of

(3).

statistical

collisions

However

COMPARISON The

ground

velocity

mean

case.

on

will

formula


likely

exciton

taken

changes.During

velocity

values

of

AvL

determinationsand 10 %.

considered

oscillator

strength

a

383 TABLE

1

Calculated AvG

= 0.567

no of sample

values

fv,

of

AvL,

and

(J for

some

samples

N~10-l~

fvxlOg

AvI, in

ax1014

cm-l for

8

with

cm -1 .

v=vT

cm2 for

v=vex

I

1.3

1.91

0.4

0.942

4.99

10 I

1.6

1.77

0.51

0.976

5.17

10 II

1.6

2.00

0.77

1.473

7.8

3R6

3.25

2.26

1.09

1.027

5.44

65t 9'E "32-

Fig.

2. Calculated the

curve

data

of

-

Calculated

000

Experimental

with

AvG

1s absorption curve data

line

using of

cm -1

= 0.567 eq.

ref.

in Cu20 (1) 1.

and

experimental

for

the

sample

no

8.

384 f

is also

carried

given.

out

This

gives

collision

5.85

lo-l4

it has

factor

THE

of

2 or

1) It has of

increasing the

= the

INTERNAL shown

small

in direction very

pure

of

the

width on

been

necessarily

evaluated

the

shape

of

the

Further

on

it has

natural

line

magnitude

This

line

line

comparable exceptional a) same

the

sample

We

the

the

obtained Av

V

theoretical

for

a Lorentz for was

a

in the

Stronger by

stresses

Haken

(ref.

in strength shape

split 2) that

and

of

assumption

possibly

the

line

that

also

width.

yellow

applied

a component This

This

due

correction

correction

is predominent

is

in

crystals. AuR

is much should

larger be

is expected

than

several

for

the

tried

of the

the

orders

of

absorption

all

adjust

N the

other the

experimental

Gaussian

width

samples

of

theoretical

curve

component

crystals, of

a much

comparable

approximately curve

to

to

of this

:

other

of

concentration

as compared

component

samples

Voigt

the

1s line

have

shape

attempt as

1s line

contain

of a given

large

In a first

= AvR

is

is created.

ways

AvG

neutral atoms

Artificially

character

that

width

in two

introducing

resultant

to

=

for

spectrometer.

10 8 of

the

the

our

however

pure

sample

width

those

for

abnormally

curve

effective

with

diameter

Gaussian

of the

noted

Doppler

concentration.

shape

collision

random

the

Gaussian

in very

an exciton

is

a(~,,)

line.

at

of

about

which No

for

suggested

justifies

but

to be

In an'exceptional

the

Voigt

by

width

in which

2) of

to be

Gaussian

smaller.

i and

an

been ppm.

diameter.

of

the

distributed

responsible

line

and

stresses.

been

= AVG. The component AvR *"R to the limitedwidth of the slit has

100

cm*

atomic

atomic

can

not

has

10 to

101*

collision

broaden

It has

crystals.

curves

about

STRESSES

first

be

= 1.11

the

depends

stresses

can

from

= 21.05

the

that

crystals

experimental

;.

results

than

in a triplet.

residual

d(vT)

that

3 higher

been

o(vT)

48.37

noted

stresses

line

for

with

OF

Cu20

with varying

of

of

d(ve,)

to be

INFLUENCE

series

value

diameter

cm',

comparison

atoms

comparison

concentrations

a mean

mean In

The

for

did

the

same

not

fit

and

was

obtain

larger

width

concentration. as the

kept

to

of

the

a good A$

Though

fit

than the

it happens that 1s ' experimental points well. Av

385 b) same

In a second as

width,

The

Gaussian

obtain attempt

the

fit

1) of

broad tal

the

very

Gaussian

comparable

was

of

given

such

AvlS

theoretical

and

curve

was

given

a line

width

AuvW

In this

and

the

concentration.

the

as

to

second

experimental

attempt

line

of an

section

it

component

is

width

of

as the

abnormally to

:

considered

large

is tempted

indicates

~0110~s

sample

width

suggest

abnormally

Av&.

that

high

is due

Refering

this

abnormally

internal

acciden-

stresses.

ii)

This

= AvR

AVG

iii) the

renforces

The

lack

abnormal

THE

MEAN can

on

formula

:

g=l

success width

be

week of

of

the

this

BETWEEN

mean

obtained

impurities

TWO

the

first

residual

attempt

particular in the

path

from

the

between above

the

a)

with

crystals. indicates

crystal

is not

that

due

two

COLLISIONS

exciton-impurity of

free

CJ. For

path

colli-

collisions

is given

by

gives

The

second

mean

9. = 3.5 value

distance

THEORETICAL The charge

10

-5

VALUES

the

v = vT

to be more

impurities

OF THE of

for

excitons

on

II = 6.6

realistic

which

EXCITON

and

is of

as the

10

-6

for

v = vex.

compared

to

order

10 -6 cm .

of

the

IMPURITY

COLLISION

CROSS-SECTION

impurities

obviously

depends

on

the

impurity.

a) Kachlishvili

(ref.

for

interstitial

charged

his

calculation,

the

corrected.

cm

appears

between

scattering of

of

the

(6)

TNCJ

This

to

crystal.

values

mean

line

in purest

EXCITONS-IMPURITY

free

at rest

that tensions

concentration

PATH

of the

exciton

of

line

suggestion

residual

in the

FREE

A value sions

the

indicates

fluctuations

be

component

samples

between

two

large

component

Lorentzian

good.

of these

this

of

however

between

abnormally

a Gaussian

point

the

lines

agreement

to be

conclusion

i) The to

a good

happens

The

the

component

again

points

attempt

for

Applied

7) has

calculated

impurities.

electron-hole to

Cu20

with

collision

However exciton

this

cross-sections author

interaction

interstitial

Agf

neglects

which

ions

this

in

should theory

386

gives as

a cross-section

that

given

substitutional impurity. does

Ag+

seem

b) We have a neutral

tried

perturbation

CU'

ion.

the

ionic

is

the

lattice

rAg

apply

case

but

order

to a non

mentioned

our

above gives

this

only

to

charged 'theory

an order

of

scattering

of

an exciton

on

in this

situation can be due to c. ion is substituted to a an Ag

when

is due

Ag+,

same

not

correspond

a theory

scattering

: for

the

does

comparable.

The

perturbation

radii

difficulty

of such

perturbation

These

however

difficult To

which

is of

theory

to

the

+ = 1.13

considerable difference of + and for Cu , rCu+ = 0.96 :

i

8).

The the

This

which

which

to

to develop

of

cm2 this

correction

applicable

impurity.

the

lo-l4

in Cu20,

of the

to be

of a case

of magnitude

1.3

I. However

ions

So besides

not

(ref.

of

in Table

knowledge

be

considered

of

this

with

problem

a new from

this

other

problem

to find are

parameter

forms

problem

or

is

calculations.

been

solved

It is hoped

that

a partial

in another

of

on models. which

yet

be published

not

reliable

in progress

in the

experiments has

precautions.

will

is

Calculations

potential. introduce

to.evaluate

our

calculations

and

has

to

solution

paper.

CONCLUSIONS Considering tion

of the

of Cu20 sections Cu20

been

have

been

involves

A mean

free

evaluated

Finally Gaussian be

due

width

determined

path from

it has character

The

the

the

to a mean

stresses.

a

of

of

the

been

for

two

thermal

are

line

the

to be

two

pure due

cross-

impurity

of

and

series

realistic.

values

Lorentz

varia-

yellow

effective

VT has

residual

of very

of a broadening

the

absorbing

between

of the

the

a neutral

of

of the

considerations

of

on

extreme

of

surprisingly

value

character

that

treatment

1s line

velocity

exciton

shown

of the value

the

an exciton

changes

these

of

results

classical

phase

a classical

realistic

scattering

of

of AvlS

of N the that

in Cu20.

a consequence

been

shown

of the

exciton

which

simplicity and

as a function

It has

the

the shape

the

in

velocity

of

considered

line

width

as

theory

oscillator.

collisions appears line

has

to be width

crystals to residual

is

also

realistic.

AvR likely

and to

accidental

Schwab, A. Goltzene. a. Meyer and S. Nikitine, Phys. Stat. Sol., @al60 (1973) 651. 3.C. Merle, S. Nfkitine and H- Haken, Phys. Stat. Sol., (b161 (1974) 229Y. Tuyozawa, Progress in Theor. Phys., 20 (1958) 53. A.C.G. Mitchell and M.W. zemansky. Resanance Radiation and Excited Atoms, Cambridge Univ. Press, London, 1973.. A. Goltzene, C. Schwab and H.C. Wolf, Solid State Corn., 3.0 (X976)1565. J-W. Hodby, T.E. Jankins, C!. Schwab, H. Tamura and D. Trivich, 3. Phys. c, 9 (1976) 1429. Z.S. KachlishviLi, Sov, Phys. Sol. State, 3 (1962) 1554. Handbook of Chem. and Phys., 35th ed. Edited by Ch. D. Hodgeman, by Chemical Rubber Publishing Co, Cleveland, Ohio, pubfished

C.

U.S.A.

(J.953).