Raman scattering by the lattice of ionic crystals containing impurities

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Raman scattering by the lattice of ionic crystals containing impurities

L.E. Gurevich, I.P. Ipatova, A.A. Klotchichin

To cite this version:

L.E. Gurevich, I.P. Ipatova, A.A. Klotchichin. Raman scattering by the lattice of ionic crystals contain- ing impurities. Journal de Physique, 1965, 26 (11), pp.742-745. �10.1051/jphys:019650026011074200�.

�jpa-00206345�

RAMAN SCATTERING BY THE LATTICE OF IONIC CRYSTALS CONTAINING IMPURITIES

By

^{L. E.}

GUREVICH,

^{I. P. IPATOVA and A. A.}

KLOTCHICHIN,

A. F. Ioffe

Physical

^Technical Institute, U. S. S. R. Academy ^of Sciences, Leningrad, U. S. S. R.

Résumé. 2014 Théorie de la diffusion Raman induite par la présence d’impuretés dans les cristaux du type ^{NaCl. On} montre que l’effet Raman du premier ordre n’est _pas produit par l’altération de la masse, mais par celle ^de la constante de force.

Abstract. - Theory ^{of the}

impurity-induced

^Raman scattering ⁱⁿ

crystals

with NaCl structure.

It is shown that the first-order Raman effect is not produced by alteration of the mass defect but

by

change in the force constant.

JOURNAL DE PHYSIQUE 26, NOVEMBRE 1965,

The Raman

scattering gives

^the

possibility

^of

studying

the vibration

spectrum

^{of the}

crystal.

Recent

experimental

^work

[1]

^{on the}

impurity

induced Rau1an

scattering

renewed the interest in this field. This

scattering

^can exist when the

perfect

^{lattice does not}

give

any kind of first order

scattering [2].

In the case of neutron

scattering

^the

impurity

effect is

always

^{a small} correction to the host lattice effect

[3].

The difference is due to the different

type

of neutron and

photon

interactions with

crystal,

^{and due to the}

practically

^{zero value}

of the

photon

mornentum : ka - k’a « 1

(a being

the lattice

constant,).

^{As a}

result,

^{the neutron}

cross section is

proportional

^{to the}

squared

^trans-

ferred momentum while it is ^not the case for

light

cross section.

Further,

^{the neutron cross section}

depends

^{on the kind of}

isotope.

^{It leads to the}

incoherent

scattering

^{and to the}

necessity

^to

average ^over the

isotopic composition. Light

cross section does not

depend

^{on the}

isotopic

^com-

position. Finally,

^some additional selection rules

can be established due to the small value of

light

momentum. For

examples,

^{the first} order Raman

scattering

^{in rocksalt}

type, crystals

^vanishes

because of the existance of the center of inver- sion

[2].

Stekhanov and co-workers have observed a first order Raman

spectrum

^{in such}

crystals

^{in the} pre-

sence of

impurities [1].

Our _paper deals with the

theory

^{of the}

impurity

induced Raman

scattering

ⁱⁿ

crystals

^{with rocksalt}

structure. It is shown that the mass defect does not lead to first order Raman

scattering

^{as it does}

not affect the electron

subsystem.

The force constant defect

changes

^the

properties

of the election

suhsystem :

local electron states appear and

impurity scattering

^{of electrons}

becomes

important.

^{In this case} the first order continuous ^Raman

spectrum

exists. If the

crystal

vibration

spectrum

^{contains branches with dis-}

persion ^At exceeding

^the

phonon damping T’at,

^the

frequency dependence

^{of the}

corresponding

^contri-

bution to the first order

spectrum

^{follows the one-}

phonon

distribution function. The cross section

temperature dependence

^{is the usual one in this}

case. But if the

dispersion

^{of some} branch is less

or

comparable

^with

phonon damping,

^the

spectrum

does not

repl"oduce

the distribution function. At

high temperatures

^{some new cross section}

temp-

rature

dependance

can occur in addition to the usual one.

The influence of the force constant defect ^on the Raman

scattering

^{due to the local and}

quasi-local

vibrations is not considered in this paper.

I. Mass defect. ^-

Generally speakirig,

^{we are}

interested in some

temperature

^ellects

produced by phonon damping. Therefore,

it is convenient to describe the

phonon subsystem by

^{the retarded}

Green function

Dt-t,,I(q ; co).

^The electron sub-

system

^can be characterised

phenomenologically hy

the electronic

polarisability

^tensor

[2].

^One

can _get

the

explicit expression

^{for the}

polarisa- bilit,y

^{with the}

help

^of

microscopic theory.

^The

expression

^is

always

very

complicated

^{and we shall}

not use it as far as

possible.

The

one-phonon

differential cross section per

atom,

per unit solid

angle

^{and unit}

frequency

^inter-

val,

^can be written

by analogy

^with

Cowley [4],

who considered the

perfect

^{lattice case, in the}

following

^form

Here

P"P(O)

^{is an electron}

polarisability ;

^{6), w’}

and _P, e’ are

frequences

^and

polarisation

^{vectors of}

incident and scattered

light ; N( ù)

^{is the mean}

phonon occupation

^number

obeing

^the

relationship

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019650026011074200

743

The retarded Green function

DR (q ; co)

^{is an}

analytic

continuation irt o of the

corresponding temperature

^{Green function}

Dtt,(q itùn). ^Taking

into account the

phonon

interaction with mass

defect,

lB1a.radudin has found the

following

expres- sion for

Dlt,(q iCùn)

Here

where _coqt is

phonon frequency

^{of the branch t and}

momentum q.

In linear

approximation

^with

respect

^{to the}

impurity

concentration

Cd, Htt,(q ; imn)

is repre- sented

by

^{the set of}

diagrams

ⁱⁿ

figure

^{1. In the}

diagram

the wavy line characterizes ^a

phonon,

^a

dot

corresponds

^{to the vertex of the}

phonon-imyu- rity

interaction

FIG. 1.

A dasbed line

represents

the average ^over the random distribution of

impurities.

In

(6) bqt, b’

^are

phonon

annihilation and crea-

tion

operators;

&+ ==

(M’- M+) IM’, wbere

^{M’ is}

the mass of

impurity

^atom.

(6)

^contains

only

^the

positive

^ions

polarisation

^vectors

§+i(q)

^{as we}

assume that the defect is substituted in

place

^{of a}

positive

^{ion. Summation over I includes all the}

defect

positions.

As it ^was shown in

[r;], 1-It,(q ; i mn)

^is

where N is the whole

quantity

^of

elementary

^cells.

The

poles

^of

nu,(q ; iCa>.) corresponds

^{to the local}

and

quasi-local frequencies

^of the lattice

spectrum,.

The substitution of the first item from

(3)

^into

(1)

leads the usual

perfect

^lattice

scattering

^{.For cubic}

crystals

^{of NaCl}

type

^{it vanishes. In}

fact, P’0(0)

is a linear function of the ion

displacement.

^The

inversion

changes

^the

sign

^of

displa;einent,

^{but it}

does not

change

^the

sign

^of

Pab(0).

^Therefore

P-0(0) = 0..

^,

The substitution of the second item from

(3)

into

(1)

would describe the contribution of local and

quasi-local

vibrations. But the presence of

IIu,(q ; iúJn)

^does-not

change

^the

properties

^of

P"0(0).

All Born’s

arguments

^remain valid. This result is

a

quite natural,

as a niass defect does not affect the electron

subsystem

^and

polarisability.

2. Electron

scattering by impurities.

^{-- The}

situation

changes essentially

ⁱⁿ the presence of the force constant defect.

When

acting

^{on the}

phonon subsystem

^this

defect

produces

^{local and}

quasi-local phonon states [6].

On

^{the other} hand these defects

change

the pro-

perties

of the electron

subsystem :

the electrons can

be scattered

by impurities

and besides local elec- tron states can appear. In the

present

^{case the}

electron

polarisability depends

^{on the momentum} q transferred to

impurity.

^It

gives

^{rise to the first}

order Raman

scattering

^of the host lattice. This paper deals

just

with this effect.

We

shall consider in detail the

scattering

^of

electrons

by impurities.

^{The result is}

analogous,

in the case of local electron states. For the inves-

tigation

^of

Pt’O(q)

^{one should calculate the matrix}

elements of the

electron-impurity

interaction in the

representation

^{of Bloch functions}

uip(r) :

P’O(q) depends

^{on the}

impurity position ^Re.

One should t,ake this fact into account when avera-

ging

^over the random

irnpurity

distribution.

In linear

approximation

ⁱⁿ

impurity

^concern-

tration

Cd,

^{one can}

get

The

q-dependence

^of

Pjfi3(q)

^{does not}

permit

^Born

and

Huang

considerations.

Thus,

the first order Raman

scattering

exists in the

presence

^of

impu-

rities.

In

analogy

^with

[7]

^{let us take} into account the finite lifetime of

phonons

^{due to} the anharmonic interaction. As a

result,

8-function in

(10)

^is

substituted

by

^the « distributed))

expression

where

rqt is a phonon damping.

One can transform

(10)

^into

Here the first

integral

^{is taken over the constant}

frequency

^surface mqi ^=-

D,

^dS

being

^{an element}

of area on the surface. The second

integral

^is

over the

frequency.

The electron

polarisability

^{tends to zero with}

q --->- 0.

Therefore,

the maximum momenta are of

importance

ⁱⁿ

integral.

^{We assume that even at}

q - qmax all matrix elements in

prf3(q)

^{are slow}

functions of q. This

assumption

^seerns

quite

^rea-

sonable as the distribution function of some

optical branch,

^t.hat

always

^has rather low

dispersion,

^{is a}

sharp

function of

frequency. Then,

^{it is}

possible

to take

Pxi3(q...)

^{out of the}

integral

^{over the}

surface

(13) depends essentially

^{on the}

dispersion At and

on the

phonon damping rqi

^{of some} branch. It is

possible

^to

investigate

^two limits for the case of

optical

^branches.

I.

r qt ^At.

The distribution function

gt{ (0)

has no very

sharp

^maxima

produced by

^{the low}

dispersion optical

branch. The

replacement

^{of the}

distributed

expression (11) by 8-function,

^allowed

in this _case,

yields

Taking

^the

integral

^over

Q,

^{one can}

finally get

Thus,

^if

gi(co)

^is

sufficiently

^{smooth function of}

trequency,

^{the cross} section follows ^its

shape,

^and

it

depends

^{on the}

temperature

^{in the usual way.}

I I.

rqt > At.

^The distributed

expression (11)

can not be taken out of the

integral.

^{As to}

g( Cù),

it can have a very

sharp

^maxima

produced by optical

branches with very ^low

dispersion.

^Let

S2

be a

position

^{of such a maximum. Then. at}

û) - w ⁼

Q

^{the cross section is}

The

order

of

magnitude

estimation

gives

The cross section behaves at the maximum

as rqma=t and

decreases with

growling temperature.

Far from

no,

^the estimation

gives

The

comparison

^of

(17)

^and

(18)

shows that the

cross section at (ù - co’ ^-

Qo

^is

larger

^{than the}

cross sect,ion far from

Qo by

the factor

(1, jS?o)2.

That

is,

^{there is a} maximum which does not repro- duce the distribution function. It has

approxi- mately

^{a Lorentz}

shape.

This maximum has an

additional

temperature dependence

^connected

with

r qt.

The cross section

(17) ^differs

^{from the cross sec-}

tion

(15) by

^{the factor}

(Atf]Pqt)-

The consideration of electron local states leads to the result of the

type (1.3). Plt’O

^{is the}

polarisa- bility

^of

impuritv

^{electrons in this case. In deno-}

minators of

PaB arising

^{from the}

perturbations theory,

the electron energy gap

Eg

^is

replaced by

the ionisation _{energy of}

impurity ^E,...Bt

^{the fre-}

quency ^riear transition oi the

impurity

^{the cross}

section is

larger by

the factor

(E,IFi)2

^{than that}

far from this transition.

The

present experimental

^{data do not allow to}

distinguish

^which

possibility

is realized.

Stekhanov and

Eliashberg

have observed in KBr

a

peak

^{at 126} cm-1 that is not

possible

^to

interpret

as a combination of

frequencies

in the second order Raman

spectrum [1]. They interpret

^this

peak

as the first order Raman

scattering

^induced

by impurities (KBr

^Raman

frequency

^{is 106}

^ent-11).

Its

shape

^{can be}

explained by

the existence of the

745

transversal

optical

^{branch of low}

dispersion [8, 9].

The

temperature dependence

^{of this}

peak

^was

not measured. Its

knowledge

^could

help

^revea-

ling

the real situation.

The neutron

investigation

^{of NaI}

phonon

spec- trum

[8, 9] ^showed

that in this

crystal

^{the trans-}

versal

optical

branch has lower

dispersion

^than

for KBr. It is

quite

reasonable ^to

expect

^{that at}

least in NaI the ^second case takes

place.

Up

^{to now we} considered

optical

^branches

only.

[13]

shows that acoustic branches

produces

^the

continuous first order

spectrum

^{too. This} spec- trum has no

sharp

maxima because of the relati-

vely high dispersion

^{of the} branches. At first

this

spectrum

^was observed and

interpreted by Gross,

Pavinski and Stekhanov

[10].

Note ajout6e a la correction, ^{relative à la} communication

sur : The second-order Raman spectrum ^{of calcium} fluoride,

par R. S. KRISHNAN et N. KRISHNAMURTY, p, 633. - Dr. J. P. Russel has

recently

^{shown that the observed}

second-order spectrum of calcium fluoride is not genuine

and is the fluorescence spectrum due to Er3+ ions. We wish to state that the interpretation ^{of the} observed fea- tures with our critical point phonon branches is still

possible but with different selection rules which do not forbid most of the combinations. (Loudon,

1964)

^and thereby

improving

^the agreement between the observed and calculated

frequency

^shifts.

Dr. J. P. RUSSEL (Private Communication).

Dr. R. LOUDON, ^Proc. Phys. Soc., 19s4, 84, ^379.

REFERENCES

[1] STEKHANOV (A. I.) and ELIASHBERG (M. B.), ^Solid

State Phys., 1960, 2, ^2354.

ELIASHBERG (M. B.), Thesis, Tartu, 1965.

[2] ^BORN (M.) and HUANG (K.), Dynamical theory ^of

Crystal

Lattice, Oxford, Clarendon Press, 1954.

[3] ^KAGAN

(Ju.)

and IOSILEVSKY (Ja.), ^{J. E. T.} P., 1963, 44. 1375.

[4] ^COWLEY (R. A.), ^Proc. Phys. Soc., 1964, 84, 281.

[5] ^MARADUDIN (A. A.), Astrophysics ^and Many-Body Problem, ^New York, 1963.

[6] ^LIFSHITZ (I. M.), ^Nuovo Cim., 3, Suppl., 1956, 4, 716.

MONTROLL (E. W.) ^{and POTTS} (R. B.), Phys. ^Rev., 1955, 100, ^525.

[7] ^GUREVICH (L. E.), ^IPATOVA (I. P.) and KLOTCHICHIN

(A. A.), Proceedings ^{of the} International Confe-

rence on Semiconductor Physics, ^Paris, 1964, p. 1052.

[8] ^WOODS (A. ^D. B.), ^COCHRAN (W.) and BROCKHOUSE

(B. N.), Phys. Rev., 1960, 119, ^980.

[9] ^WOODS (A. ^D. B.), BROCKHOUSE (B. N.), ^COWLEY (B. N.) and COCHRAN

(W.),

Phys. Rev., 1963, 131, 1025, 1031.

[10] ^GROSS (E. F.), ^PAVINSKY (P. P.) and STEKHANOV

(A. I.), Usp. ^Fiz. Nauk., 1951, 43, 536.