Elastic scattering of light in polaron gas

- - hL this work we study the elastic scattering of light with the sblglc-partiele excitation by a system of electrons interacting with phonons--a polaron gas or liquid.. Solid line: el

I L NUOV() CIMENTO VOL 52B, N 2 11 Agosto 1979

Elastic Scattering of Light in Polaron Gas

~'~GUYE.'~" ~BA ~:.N, ~GUYEN VAN HIEU,

~TGUYEN TOAN THA~'G a n d ~GUYEN AI VIET

Institute o/ Physics, ~ghia Do Tu Liem - Ilao~>i, VietTtam,

(ric(;vuto il 10 Ft.bbl'aio 1979)

S u m m a r y - - hL this work we study the elastic scattering of light with the sblglc-partiele excitation by a system of electrons interacting with phonons a polaron gas or liquid We observe a significant enhance- ment of the ,~cattering due to the strong electron-phonon coupling The order ~,f this (,nhancemcnt effect depend,~ ou the polarization properties

of the inci(h,Ht and scattering lights We also prove that due to a Ward- tyI)(' id~.ntity the contribution of the A"-term in the IIamiltonian to the ,~('~tWring amplitude is not affe(.ted by the eh, etron-phonorL interaction

T h e electronic Rt~mazl s c a t t e r i n g b y free c h a r g e carriers in solids a n d p l a s m ~ was s t u d i e d in m a n y w o r k s (~ o~) I t is well k n o w n t h a t t h e r e are t w o kinds of

s c a t t e r i n g processes: t h e s c a t t e r i n g on collective e x c i t a t i o n s (plasmons) a n d

t h e s c a t t e r i n g w i t h single-particle e x c i t a t i o n ( S P E ) ( ~ , , : o : 1 ) I n t h e lowest

o r d e r ()f t h e e l e c t r o m a g n e t i c i n t e r a c t i o n t h e c o n t r i b u t i o n s t o t h e s c a t t e r i n g

a m p l i t u d e c o m e f r o m t h e F e y n m a n d i a g r a m s in fig la) a n d b) I n a n e l e c t r o n

(1) l) F 1)urn)IS and V GILINSKI: Phys lr 133, A 1308, A 1317 (1964)

(~) l' M PLATZ.~tAN and N TZOAR: Phys Rec., 136, A 11 (1964)

(3) P ~I I)LATZ3IAN: Phys Rev., 139, A379 (1965)

(4) I ) A WoI, FI.': Phys Rec Lett., 16, 225 (1966); Phys Rev., 171, 436 (1968) (s) A Moomki)IA~- and G B WRIGIIT: Phys Rev J, ett., 16, 999 (1966); 18, 608 (1967) (0) E BI'R,~TEIN, A PINCZUK and S IWASA: Phys Rev., 157, 611 (1967)

(7) A 3IooRAI)IAN and A L ),IcWHoRTER: Phys Rec Lett., 19, 849 (1967)

(s) D C IIA.~tILTO.~ ~ and A L ~[cWItoRTER: ill Zight Scattering Spectra el ~%lids,

edited by G B ~VRIGItT (New York, N Y , 1969), p 309

(9) Z F ,~COTT, T C DAM:EN, ,]' RUVALDS and A ZAWAD()Vr Phys Rev B, 3,

267

2 6 8 NGUY~N BA AN, NGUYEN VAN I[IEU, NGUYEN TOAN TIIANG and NGUY~N AI VIET

gas or liquid, u n d e r t h e isotropic p a r a b o l i c dispersion law f o r t h e e l e c t r o n

p 2

t h e s c a t t e r i n g m e c h a n i s m r e p r e s e n t e d b y t h e d i a g r a m in fig l a ) is s c r e e n e d

b y t h e C o u l o m b i n t e r a c t i o n for t h e s c a t t e r i n g w i t h t h e S P E , a n d t h e m a t r i x

e l e m e n t s of t w o d i a g r a m s in fig lb) cancel in t h e limit v > 0, w h e r e v is t h e

e l e c t r o n v e l o c i t y in t h e u n i t s y s t e m with ]g ~ c ~ 1, w h i c h will be used in this work

Fig 1 - Fcynman diagrams in the absence of clcctron-phonon interaction Solid line: electron line; wavy line: photon line

I n t h e solid-state p l a s m a t h e s i t u a t i o n m a y be different: t o g e t h e r with

t h e s c a t t e r i n g on t h e charge d e n s i t y f l u c t u a t i o n (CDF) as in t h e classical l)lasma

1295 (1971)

(z0) p ~[ PLATZMAN and N TZOAR: Phys l~ev., 182, 510 (1969)

(11) S S JFrA: Phys Rev., 182, 815 (1969); Nuovo Cimento, 63 B, 331 (1969) (12) p C KWOK, J W F Woo and S S JHA: Phys Rev., 182, 671 (1969) (la) A PINCZUK, L BRILLSON, E BtYRSTEIN and E ANASTASSAKIS: Phys Rev Lett.,

27, 317 (1971)

(14) p M PLATZMAN, P EISENB/.;ItGER and N TZOAR : in Light Scattering in Solids, edited

by M BALKANSKI (Paris, 1971), p 80

(~5) p j COLW~LL and M V KLEIN: in fight Scattering in Solids, edited by M I~,~.L- KA.~SKI (Paris, 1971), p 102; Phys Rec B, 6, 1198 (1972)

(ts) A R VASCONC):LLO and R LvzzI: Nuovo Cimento, 23 B, 335 (1974)

(17) j DOEHLER, P J COL~VELL and S A SOLIN: Phys Rev B, 9, 636 (1974) (18) 5 DOEIILER: Phys Rev B, 12, 2917 (1975)

(z~) K P JAIN and M BALKA~'SKT: in Light Scattering in Solids, edited by M BAL- KANSKI, R C C LEITE and S P S PORTO (Paris, 1975), p 106

and S P S PORTO (Paris, 1975), p 119

('I) M 5OUA-~'E, R B~.S~.RMA~', K P J x I s and M BALKANSKI: in Light Scattering

in Solids, edited by M BALKANSKI, R C C LEITE and S P S PORTO (Paris, 1975),

p 125

E L A S T I C SGATTEIr O F L I G [ I T IN I ' O L A R O N G A S 269 there exists also the scattering on the energy density fluctuation (EDF) and

t h e spin density fluctuation (SI)F) due to the nonparabolicity of the dispersion law and the spin-orbit coupling, and two latter scattering mechanisms m a y not be screened b y the Coulomb interaction (4,u,12) Moreover, due to the pre- sence of the virtual i n t e r b a n d transitions, the m a t r i x elements of the two diagrams in fig lb) do not cancel, b u t can exhibit some e n h a n c e m e n t at the

p h o t o n energy near the value of the b a n d gap (4,an,a3)

Together with the scattering ou t h e electrons there exists also the scat- tering of light with the emission of a phonon As the result of t h e inter- ference of the latter process with the scattering on the collective excitations,

we have the creation of new e l e m e n t a r y e x c i t a t i o n s - - t h e plasmon-phonon coupling modes (5,~o,~das), while the interference of t h e scattering with the

S P E and t h a t with the p h o n o n emission yields the so-called resonant and antiresonant frequencies (~) which were ~lso observed e x p e r i m e n t a l l y (~,2~)

In this work we s t u d y the ICama.n scattering of light in an electron liquid with electron-phonon interaction in a different context We shall investigate the influence of the electron-phonon interaction (m the electronic :R~man scattering with the S P E , b u t we shall not consider the processes with the emis- sion of real phonons The initial a n d final states of the scattering process are the systems of electrons interacting with p h o n o n s ~ t h e polarons, b u t without real phonons I n other words, we s t u d y the elastic scattering of light in a polaron liquid We suppose t h a t there is only one kind of optical phonons with m o m e n t u m - i n d e p e n d e n t energy ~Q, and assume the Frohlich interaction

H a m i l t o n i a n (2~) with coupling constant v/~ The energy of the bare electrons

is assumed to be determined b y eq (1)

Denote by k, k' a n d p, p' the m o m e n t a of the initial and final photons and electrons, respectively b y (o, ~o' a n d E , E ' their energies and b y ~e and ~' the

p h o t o n polarization unit vectors:

(~k) = ( F k ' ) = O

In tile limit k = k', p = p' the m a t r i x elements of the diagrams in fig ]a), b) equal

( 2 ) Jr,<, _ ( ~ , ~ ) ,

m

( 3 ) MI0 ~ - - ( ~ ' n ) ( ~ )

p ' - / " - m <,> - ( p - k ) 2 / 2 m - - i 5

(22) tI FROHLICI[, It PELZER and S ZIENAN: Phil Mag., 41, 221 (1950)

2 7 0 /~-GUYEN B A A N , N G U Y E N V A N ] I I E U , N G U Y E N T O A N T H A N G a n d N G U Y ] E N A I VI:ET

I n the second order of the electron-phonon interaction we have the diagrams

in fig 2a)-c) The m a t r i x elements of the diagrams in fig 2a) are the cor- rections to t h a t of the diagram in fig l a ) in the given order: two first d i a ~ a m s

a~

/4" / ]

c}

Fig 2 - Feynman diagrams in the second order of electron-phonon interaction Solid line: electron line; ~vavy line : photon line; dashed line: phonon line

in fig 2a) yield the renormalization of the wave functions of the external elec- trons (with the change of their dispersion law), while the third one yields the corresponding vertex ~ o t e t h a t as the result of t h e wave function renor- realization the one-particle initial and final states become those of the polarons

ELASTIC SCATT:ERING OF LIGIIT IN POLARON GAS 271

T h e explicit calculation shows t h a t in t h e limit k = k', p = p ' the two first

d i a g r a m s in fig 2a) h a v e t h e m a t r i x element

~ U

while the m a t r i x element of the third is

- ( ~ ' ~ ) : ; _ ,

m - V i - - v " / u "

where

u-" == 2rnD

These m a t r i x elements cancel e x a c t l y and, therefore, do not c o n t r i b u t e to the

s c a t t e r i n g a m p l i t u d e I t is worth noticing t h a t this cancellation is the conse- quence of a W a r d - t y p e i d e n t i t y a n d takes place in a n y order of electron-

p h o n o n interaction

4-

Fig 3 - Two-photon vertex with renormalized external electron line Single solid line : bare-electron line; double solid line: dressed-electron line; wavy line: photon line; dashed line: phonon line

I n d e e d the m a t r i x e l e m e n t of the d i a g r a m of the f o r m in fig l a ) with the ronormalized external electron lines a n d t h e m o s t general v e r t e x (fig 3) equals

(4)

I n this f o r m u l a

(5)

e 2 ~ : = - ( ~ ' ~) Z , ( p ) z , ( p )

m

z,(p) :: [1 - i ~z-(C' P!]-'

is the wave function renormalization constant,

(6) Z,(p) = 1 + ~[d~(p), p ; #'(p), p]

is the expression of the v e r t e x represented b y the diagrams in fig 4 with equal initial and final electron momenta~ p = p', and energies d'(p) = #(p')~ 27(E, p)

is the compact self-energy p a r t of the electron represented b y the diagrams

+ ~

Fig 4 - Two-photon vertex with bare external electron lines Solid line: electron line; wavy line: photon line; dashed line: phonon line

\

Fig 5 - Compact self-energy part of the electron Solid line: electron line; dashed line: phonon line

in fig 5, and d'(p) is the energy of the dressed electron (polaron) which satisfies the equation

p~

I n the second order we have

(8) # ( P ) ~ m + i ~ ~m,P ~Tm - a ~ - P

7~

7~

-~ , p > u

I t is easy to check t h a t in a n y order of the electron-phonon interaction we have

t h e W a r d - t y p e i d e n t i t y

~ 2 ( E , p)

F r o m eq (9) it follows the exact cancellation of the high-order diagrams in fig 3 in the limit k = k', p = p', and the m a t r i x element of all diagrams in

ELASTIC SCATT~ERING O'F L I G H T IN POLARON GAS

fig 3 equals

e2

273

I t is worth noticing t h a t the contribution from each diagram of the t y p e in

fig 2a) is v e r y large at the m o m e n t u m satisfying the condition

p 2 ~_~ ~ 2

Similarly the diagrams in fig 2b) yield the renormalization of the external- electron wave functions and of the electron Green's function in the lowest- order m a t r i x element of the diagrams in fig lb)~ a n d also the appearance of

the single photon-electron interaction v e r t e x I'~[E,p; E ' , p ' ] I n the limi$

k ~ k', p p' the contribution of the diagrams in fig lb) a n d fig 2b) to the scattering a m p l i t u d e equals

(11)

(~2

Mb = - - ~'~o, Zl(p){ff ~[~(p) ~- co, p -]- k; #(p), p]

m

9

G[#(p) + ~o,p -[- k]F~[~(p),p; #(p) -4- o~,p + k] +

where G(E, p) is the electron Green's function in the presence of the electron-

p h o n o n interaction The v e r t e x can be written in the following general f o r m : (12) l ~ ( E , p ; E ' , p ') ~ 89 ~ - p ' ) ~ Z 3 ( E , p ; E ' , p ' ) -~ 8 9

where Z3(E, p; E', p') is symmetric u n d e r the substitution p (-+ p', E ~- E '

I n the one-particle a p p r o x i m a t i o n we h a v e

- # ( p ) - i o "

Inserting expressions (12) and (13) into eq (11), we obtain

(14) M~ e~ p~ ~z3[#(p), p; ~(p) § ~, p § k] gl(p § k)

_~ Z3[F(p), p; #(p) 09, p k]Zl(p k!~

#(p) ~o # ( p k) i0 J

274 I ~ ' G U Y E N B A A N , N G U Y E N V A N H I E U , N G U Y E N T O A N T I [ A N G and N G U Y E N A I V I E T

I n the second order in the electron-phonon interaction we have

z~[#(p), p, # ( p ) =t= co, p + k] : : 1 :~- 2 p v ~

P

I t is easy to verify t h a t even in the limit

P

k < < p , - - = v < < l

m

t h e r e is no cancellation of the two terms on the r.h.s, of eq (14), and this m a t r i x element is of the order

This contribution is mainly due to the presence of the single photon-clectron interaction vertex On the other hand, the corresponding scattering mechanism

is n o t screened b y the Coulomb interaction Therefore, the introduction of the vertex F,(E, p; E', p') yields the e n h a n c e m e n t of the scattering with the S P E ,

if the constant a is not small

The same conclusion is true ~lso for the diagrams with the two-particle intermediate states in fig 2c) I n the limit k ~ k', p ~ p' their m a t r i x element can be written in the general form

(17) Mc : m 4 ((~'~) A(p, ~)) § (~' n)(~n) B(p, ~o)} , e 2 (x

of the variables p~ a n d v 2 we h a v e different expressions for these functions

F o r example, in the domain u 2 > v ~ § p2 we have

E L A S T I C S C A T T E R I N G O F L I G H T I N P O L A R O N GAS 275

p ~ v ~ J V u ~ + v ~

p 2 u 2

§ § 2 p ~ - u:'~ V u ~ - - v" - - p~

3 ~1 - - 2 p ~ - ~ u ~ u ~ § v 2 - p~

I n particular, at p = 0 and v 2 < u 2

(:~o) A ( 0 , (,~) = ~ V (2u - - V U - 8 U 8 v~ - V u ~' + v 0 ,

:Note t h a t the corresponding scattering mechanisms is also not screened by the Coulomb interaction Therefor% we conclude t h a t the strong electron-phonon interaction leads to a significant enhancement of the elastic scattering of light with the SPE in a polar gas or liquid: the scattering amplitude is of the same order as the Thompson scattering amplitude on the free electron without the screening by the Coulomb interaction, multiplied by ~:

M c " ~ ~ M a

We have discussed two possible mechanisms of enhancement of the elastic scattering of light in a polaron gas or liquid with the SPE due to electron- phonon interaction This effect is very large in the case of the strong coup- ling between electrons and phonons ( a ~ l ) I n the case of the scattering of unpolarized light the cross-section in the presence of strong electron-phonon coupling would be of the same order as t h a t of the scattering in a free-electron

phonons m a y lead to increase the cross-section about e~ times, where s is the dielectric constant of the medium In case of the scattering of the linearly polarized light with scattered light linearly polarized perpendicularly to the polarization of the incident light (~'~ = 0)7 we have another enhancement factor: from eqs (2), (10), (14), (19) it follows t h a t the corresponding amplitude

order ( p / m ) M ~ when there is no electron-phonon interaction

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