Threshold anomaly in Raman light scattering from degenerate Fermi systems

Solid State Communications,

Vol.10, pp.1101—1105, 1972. Pergamon Press.

Printed in Great Britain

THRESHOLD ANOMALY IN RAMAN LIGHT SCATTERING FROM DEGENERATE FERMI SYSTEMS C.S. Ting*t and J.L. Birman*f Department of Physics, New York University, Washington Square, New York, N.Y. 10003 and Elihu Abrahams*

—

Physics Department, Rutgers University, New Brunswick, New Jersey, 08903

(Received 29 November 1971 by E. Burstein)

A power-law divergence is calculated in the inelastic scattering cross section for light near the absorption threshold in a metal or degenerate semiconductor. The intermediate state interaction between the virtual hole and conduction electrons is responsible for the anomaly.

3 on anomalies of the soft X-ray RECENT work’ absorption coefficient of metals and degenerate semiconductors4 has shown that final-state interactions between the degenerate sea of conduction electrons and the core state hole left behind by the interband optical excitation produce a powerlaw singularity in the absorption spectrum at threshold. Here we show that the same type of singularity occurs in the resonant Raman scattering of light when the incident or scattered frequency is close to one of the interband excitation energies of the system. 13 with a flat core band We adopt the model (infinitely heavy hole) and a parabolic conduction band. The inelastic scattering processes we shall discuss involves an interband excitation in the intermediate state and a one-plasmon (matrix element M2) or a single-particle (Al3) excitation in the final state. The electron gas readjusts to the transient potential caused by *

Research supported in part by the National Science Foundation,

t Research supported in part by the Army Research Office, Durham. 1101

the temporary appearance of the virtual core state hole; this gives rise to singular behavior, just as in the absorption case. The Raman cross section at zero temperature for a cubic crystal may be written5 2

e

do =

d~Id~

2

2

-

Cu 2

—i

mec 0 ( ~)

—

2

(A1. A2) ~ ~(a

0(

~

—

—

1 X

+

Cu) 1

4 ~

+

2 ~

2

M2~ + M~

(1)

where me is the electron mass, A 1, A2 are the incoming and outgoing photon polarization vectors, ~ = (s2/2m) — ~ is the excitation energy of a conduction electron of momentum ~, effective mass m and chemical potential ~, = — wz and ~ are the energy and momentum transfer, O(~)is the step function, and (~,w) is the dielectric function of conduction electrons and will be evaluated in the random phase approximation. In equation (1), M1 1 is the intraband matrix element and M2, M9 are the interband ones depicted in Figs. 1 and 2, where the solid and dotted lines are conduction

1102

THRESHOLD ANOMALY IN RAMAN LIGHT SCATTERING cu

1

-

Vol.10, No.12

q

~jW2~9÷

~~W2

w-w2

(b)

(a)

,q

W- W2

(d)

(c)

FIG. 1. Diagrams for the one-plasmon matrix element Al2(~,~1,w2).

1-W, $ $

W

f

-q

(uI1

~ &-

w,

s-q

(b) (a’) FIG.2. Diagrams for the single-particle matrix element M 3(~,c~ 1,c~~2). 4 electron deeplines holeare propagators respectively, the singleand wavy incoming and outgoing photons, the double wavy line denotes a conductionband plasmon excited by its interaction with the intermediate state electron or deep hole via the screened coulomb potential 47Te2/q2E(q,w). The shaded triangle is the electromagnetic vertex A which is renormalized by conduction electron— deep hole interactions (see Fig. 3). In the case of weak coupling, all the singular effects are c.ntained in A and it is sufficient3 to use the unrenormalized parquet expansion2 for

its the evaluation. be represented by graphs ofThus, Fig. 3the in vertex which may the black dot denotes V, the strength of the electron—hole interaction.6 Furthermore, we may use unrenormalized deep hole propagators throughout. We note that the A defined in Fig.3 and the matrix elements of Figs. 1, 2 are a mutually self- consistent7 choice of approximation. The integral equation corresponding to Fig. 3 may be solved to logarithmic accuracy.24 The

Vol.10, No.12

THRESHOLD ANOMALY IN RAMAN LIGHT SCATTERING

(a)

U - W

1

WI

pI

1103

-

WI,pI Cu)

U

U1

-

II

+

+ <&~~_‘,_

Ii

-

U

U

U-U1

FIG. 3. Diagrammatic representation of the integral equation for the vertex A(W,W1). 8 is

leading term is

result

<~>~q2

~o

A(w,w1)

if ~

=

> c~

“2

—

a

~

2)

2m

k

{ ~i

—

2

4~~W1_~)l W

—~

ifW1 ~

A(~k,Cu)

2;

+2~~

2~1~D~

A(~k,Wl)

_______________

me —

~

____

=

(2)

I

WDj ~

~-~1

6

p is the density of states at the where surface g = pV, and ~c is a cutoff of order p~. We Fermi have written the dependence of A on frequency near threshold (WD). If both cu and w, are far from threshold, A is essentially unity.

1

{i4(W2~D)1

3

i~k_W2+WD_1~J(4)

where

is the interband momentum matrix element. In the incoming resonance region, where CL)

For the case of one plasmon in the final state we have w = — cu2 ~ w~ where ‘(q,w) has the plasmon pole. Then Al3 may be neglected in equation (1). After integrating over cc we have the result

~

~1~~

q2ci’ — do e 2 S(q,cc 1,cc2) dci = [m~c2j CL)1 2j 2 4~e (3) where ci’ is the optically active volume and

~

2

1 WD, CL)2 ~ WD — W~, we may set ~ 1 and use equation (2) for A(~k,w). The quantity S which appears in equation (3) has the following form:

S with

1

=

C±(w) =

+

2C(W1) f(cc)

C~(cc1)g(cu1),

4~e2 1 ____

±

~~p~L2mecup

~

+ W

(5)

-

3cup 2g

S

The graphs in Fig. 1 are

f(cc)

=

evaluated with the help of equation (2) and the result expanded in powers of the small ratio kFq/mcu~ where kF is the Fermi momentum. The

g(cc1)

=

=

1-I- 47Te2M

2

.

~o/cc~

cos[2~g0(w

—

2/q21 —

WD

-

—

21 (cc

~)

—

1 — 1.

CL)D)]

—

1

1104

THRESHOLD ANOMALY IN RAMAN LIGHT SCATTERING

Similarly, in the outgoing resonance region cc 2 ~ CL)~, we have

indicates that this condition is met. We note also

= 1 + 2C~(w2)f(w2) + C~(w2)g(w2). (6) Substituting equation (5) or (6) into equation (3), we find the most singular term proportional

S

to

I ~/cc,

2

—

WD

I.

We discuss briefly the single particle final state. In this case, in the resonance region, it is easy to show that M3 is the dominant term. It can be evaluated from Fig.3. Substituting the result into equation (1) and carrying out the summation over ~‘, we >find 2/m~ cc a simple form in the region

1 — > cc. 2o — e2 }2 CL) d A2) m2ccci’ 2j cct2 (A 2~2q dIldw — [mec <~>2

2

I m~(cu~

—

~

Vol.10, No.12

that because C+(cc2)/C_(w1) 5 for the reasonable choice w~ 2~the outgoing resonance is likely to be much larger thantwo-band the incoming certain 9 the modelone. is aInfair approximation. semiconductors, Then5

2/2m ~ ~ [(rn~/m) — 1]PE~, where E 9 is the separation between the top of the valence band and the bottom of the conduction band. From this, it follows that C~are of order (102 -‘-j 10~)in equations (6) and (7) so that in this case also, the singular interband term can be large compared to the intraband contribution which is represented by 1 in equation (5). Our calculation shows that very close to threshold in a metal the anomaly is a power law, with exponent twice that in the absorption case. We found both an ‘in’ and an ‘out’ resonance

(7)

4g

cc 1)

WD

—

CL)1

5 have previously found this Platzman Tzoar result but and without the final singular factor which arises from the intermediate state interaction between conduction electrons and the deep hole. =

The above results can be extended to the case of a degenerate semiconductor. In this case, the valence hole has a mass m*. From reference 4, we can expect that the only change is to replace the divergent factors WD — CL) 1 appearing 2fi/.L) in equations (5)—(7) by max( CL)D — CL)1 g where ~ = m/m*. This divergence can also be broadened by the finite lifetime of the valence hole and impurity scattering. A discussion of these effects for metals and semiconductors may be found in references 2 and 4. For example, in typical doped semiconductors g = 0.1—0.2, and one could have g2p.f3 10~eV. The broadening of the resonance is of order /3/i where T is the

,

impurity collision time.

‘0 1,

E

—

~

0

‘o

I

~

I

~

I

I

I

I

_______________________________ I nc~ent

Photon

Energy

w 1

We have obtained a singular enhancement of the resonant Raman scattering due to intermediate state interactions. In order that these effects be observable in the one-plasmon it is that at least 2/2m~cc~ ~ case, 1 in order the necessary that

coefficient (C±) of the singular term be at least of order unity. A very rough estimate for a typical metal having an absorption coefficient 10~cm1

FIG. 4. A plot of the scattering cross section vs. the incident photon energy cc1 for a plasmon. This follows equations (3) and (5) with ~o = 5, cc~= 10, ~.c= 5, and g = 0.4. anomaly for the plasmon and single particle case. From equations (5), (6) and (3) we determined the line shape of the scattered radiation, as well as the relative magnitudes near the ‘in’ and ‘out’

Vol.10, No.12

THRESHOLD ANOMALY IN RAMAN LIGHT SCATTERING

resonances. We find that the out resonance intensity is more enhanced than the in. A sketch of the calculated results is given in Fig. 4. According to our results the interband contribution from the one-plasmon scattering process is much larger than the intraband contribution by a factor

I ç~/(cc12

~

— WD) which can produce a very large enhancement of the scattering cross-section. This will make the observation of the plasmon scattering much easier in the threshold region. If the predicted scattering anomaly is found there 4 would confirmation of the be model for be all another these anomalies. It should used’ noted that not only is the out resonance more enhanced than the in, but it should be relatively easier to observe experimentally, since the

1105

incident radiation is not required to be monochromatic in order to observe an enhancement in the out channel. For degenerate semiconductors the power law anomaly will be broadened owing to both finite mass and lifetime effects. A complete calculation of these effects is not yet available so the exact line shapes cannot be quantitatively given, but only the estimates cited above (based on reference 4). However, it is our opinion that any theory (by of the line shape of resonance scattering plasmons, single particles, Raman phonons, or mixed modes) in degenerate materials must properly take into account a contribution from the intermediate state (many-body) interaction mechanism.

REFERENCES 1.

MAHAN G.D., Phys. Rev. 163, 612 (1967).

2.

ROULET B., GAVORET

3.

NOZIERES P., GAVORET

4.

GAVORET

5.

PLATZMAN P.M. and TZOAR N., Phys. Rev. 182, 510 (1969).

6.

V is the strength of the 6-function electron—hole interaction used to approximate the average screened potential between them. See reference 2 for more details. We assume here that s-wave scattering only is important so that g = pV > 0.

7.

BAYM G., Phys. Rev. 127, 1391 (1962).

8.

We have written only the real part of A. For the determination of imaginary parts, see references 2, 3.

9.

See, for example, KANE E.O., J. Phys. Chen?. Solids 1, 249 (1957).

J.,

J.

and NOZIERES P., Phys. Rev. 178, 1072 (1969).

J.

and ROULET B., Phys. Rev. 178, 1084 (1969).

NOZIERES P., ROULET B. and COMBESCOT M., J. Phys. 30, 987 (1969).

La section efficace pour Ia diffusion Raman au seuil de l’absorption dans ur. metal ou dans un semiconducteur dégénéré est calculee. Ou trouve une singulanité qui suit un loi en puissance. L’anomalie est due ~ l’interaction dans l’état intermédiaire entre le trou virtuel et les electrons de conduction.