Elastic scattering of light in polaron gas

IL 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 and ~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 enhancement 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.

The electronic Rt~mazl sca t te r ing by free charge carriers in solids and plasm~

was s tudied in m a n y works (~..o~). I t is well k n o w n t h a t there are two kinds of

sca t te r ing processes: the sca t te r ing on collective exci ta t ions (plasmons) and the sca t te r ing wi th single-part icle exc i ta t ion (SPE)(~, , :o :1) . I n the lowest

order ()f the e lec t romagne t ic in te rac t ion the con t r ibu t ions to the sca t te r ing

ampl i tude come f rom the F e y n m a n d iagrams in fig. la) and b). I n an e lectron

(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 the isotropic parabol ic dispersion law for the e lec t ron

p 2

(1) E(p) -- 2m '

t he sca t te r ing m e c h a n i s m represen ted b y the d iag ram in fig. l a ) is screened

b y the Coulomb in te rac t ion for the sca t te r ing wi th the S P E , and the ma t r i x

e lements of two d iagrams in fig. lb) cancel in the l imit v --> 0, where v is t he

e lectron ve loc i ty in the un i t sys t em with ]g ~ c ~--1, which will be used in

this work.

a) b)

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

I n the sol id-state p lasma the s i tua t ion m a y be different : t oge the r with

t he sca t te r ing on the charge dens i ty f luc tuat ion (CDF) as in the 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. (20) ]~. CERDEIRA: in Light Scattering in Solids, edited by M. BALKANSKI, R. C. C. L~;ITE and S. P. S. PORTO (Paris, 1975), p. 119. ('I) M. 5OUA-~'E, R. B~.S~.RMA~', K. P. Jx 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 OF LIG[ IT IN I 'OLARON GAS 269

there exists also the scattering on the energy density f luctuation (EDF) and the spin density fluctuation (SI)F) due to the nonparabolici ty of the dispersion law and the spin-orbit coupling, and two la t ter scattering mechanisms m ay not be screened by the Coulomb interact ion (4,u,12). Moreover, due to the presence of the vir tual in terband transitions, the mat r ix elements of the two diagrams in fig. lb) do not cancel, bu t can exhibit some enhancement a t the photon energy near the value of the band gap (4,an,a3).

Together with the scattering ou the electrons there exists also the scattering of light with the emission of a phonon. As the result of the interference of the lat ter process with the scattering on the collective excitations, we have the creation of new elementary exc i ta t ions - - the plasmon-phonon coupling modes (5,~o,~das), while the interference of the scattering with the SPE and tha t with the phonon emission yields the so-called resonant and ant iresonant frequencies (~) which were ~lso observed exper imental ly (~,2~).

In this work we s tudy the ICama.n scattering of light in an electron liquid with electron-phonon interact ion in a different context . We shall investigate the influence of the electron-phonon interact ion (m the electronic :R~man scattering with the SPE, but we shall not consider the processes with the emission of real phonons. The initial and final states of the scattering process are the systems of electrons interacting with p h o n o n s ~ t h e polarons, bu t without real phonons. In other words, we s tudy the elastic scattering of light in a polaron liquid. We suppose tha t there is only one kind of optical phonons with momentum-independent energy ~Q, and assume the Frohlich interaction Hamil tonian (2~) with coupling constant v/~. The energy of the bare electrons is assumed to be determined by eq. (1).

Denote by k, k' and p, p' the momenta of the initial and final photons and electrons, respectively by (o, ~o' and E , E ' their energies and by ~e and ~' the photon polarization unit vectors:

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

In tile limit k = k', p = p ' the matr ix elements of the diagrams in fig. ]a), b) equal

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

(3 ) MI0 ~ - - ( ~ ' n ) ( ~ ) m /2m o~ - - (p -~- k)2/2m - - iO

-t- , n =p/ !p l . 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 BA AN, N G U Y E N VAN ] I I E U , N G U Y E N TOAN T H A N G a n d NGUY]EN A I VI:ET

I n the second order of the electron-phonon interact ion we have the diagrams in fig. 2a)-c). The matr ix elements of the diagrams in fig. 2a) are the cor- rections to tha t of the diagram in fig. la ) 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- t rons (with the change of their dispersion law), while the third one yields the corresponding vertex. ~ o t e tha t as the result of the wave funct ion renor- realization the one-particle initial and final states become those of the polarons.

ELASTIC SCATT:ERING OF LIGIIT IN POLARON GAS 271

The explicit calculation shows t h a t in the l imit k = k', p = p ' the two first

d iagrams in fig. 2a) have the ma t r ix element

. . . . . . ~ U m '2 x / i - ~ '

while the ma t r ix element of the third is

e" ~ 1 - ( ~ ' ~ ) : ; . . . . _ ,

m - V i - - v " / u "

where

u-" == 2rnD.

These ma t r ix elements cancel exact ly and, therefore, do not contr ibute to the scat ter ing ampli tude. I t is worth noticing t ha t this cancellation is the conse- quence of a W a r d - t y p e ident i ty and takes place in any order of electron- phonon 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.

Indeed the ma t r ix e lement of the d iagram of the fo rm in fig. la ) with the

ronormalized external electron lines and the mos t general ver tex (fig. 3) equals

(4)

In this formula

(5)

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

m

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

2 7 2 _~-GVY~.~ ~ BA AN, N G U Y E N VAN I I I E U , ~ ' G U Y E N TOAN T H A N G a n d X G U ~ . W AI V I ~ T

is the wave function renormalization constant,

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

is the expression of the ver tex represented by 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 par t of the electron represented by the diagrams

+ . . ~

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

/ \ / \ [ t ~ + . . .

\ ~, /

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~ (7) ~(P) = Tram + i ~ [~(p), p ] .

In the second order we have

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

7~ arcsin ~- , p < u ,

7~ -~ , p > u .

I t is easy to check tha t in any order of the electron-phonon interact ion we have

the Ward- type ident i ty

~2(E, p) (9) ).(E,p;E,p) =- - i 8E

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 mat r ix element of all diagrams in

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

fig. 3 equals

e2 (101 Me = - - (~ '~) .

273

I t is worth noticing tha t the contr ibut ion from each diagram of the type in fig. 2a) is very large at the momen tum 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 funct ion in the lowest- order mat r ix element of the diagrams in fig. lb)~ and also the appearance of the single photon-electron interaction ver tex I'~[E,p; E' ,p '] . In the limi$ k ~ k', p -- p' the contr ibution of the diagrams in fig. lb) and fig. 2b) to the scattering ampl i tude 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 funct ion in the presence of the electron- phonon interaction. The ver tex can be wri t ten in the following general form:

(12) l ~ ( E , p ; E ' , p ') ~-- �89 ~-p ' )~Z3(E,p;E' ,p ' ) -~ � 8 9

where Z3(E, p; E', p') is symmetr ic under the substi tut ion p (-+ p ' , E ~-. E ' . In the one-particle approximat ion we have

(13) G(E, p) Z,(p)

- # (p) - i o "

Insert ing 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

18 - I I N u o v o Ciraenlo B .

274 I~ 'GUYEN BA AN, N G U Y E N VAN H I E U , N G U Y E N TOAN TI [ANG and N G U Y E N AI V I E T

In the second order in the electron-phonon interaction we have

(15) ~t a

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

{(~z.. ) ( u2~: V2'p2 ! " V u ~p �9 -- 2 arcsin-P § 2 ! arcsm _ _ _ -1-

~/, : ~ V 2

§ v ' ~ - - p ~ - - V u 3 - - p i} v ~ 2m(o P

I t is easy to verify tha t even in the limit

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

there is no cancellation of the two terms on the r.h.s, of eq. (14), and this matr ix element is of the order

(16) Mb ~ aM~.

This contr ibution is mainly due to the presence of the single photon-clectron interact ion vertex. On the other hand, the corresponding scattering mechanism

is not screened by the Coulomb interaction. Therefore, the introduction of the ver tex F,(E, p; E', p') yields the enhancement of the scattering with the SPE, if the constant a is not small.

The same conclusion is t rue ~lso for the diagrams with the two-particle intermediate states in fig. 2c). In the limit k ~ k', p ~ p' their matr ix element

can be writ ten in the general form

(17) e 2 (x

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

where A(p, w) and B(p, o)) are scalar functions. Depending on the values of the variables p~ and v 2 we have different expressions for these functions.

For example, in the domain u 2 > v ~ § p2 we have

u 3 [ u 4 p (u~--v~)~arcsin / u : (18) A(p, (o) - - - 2 p h ~ arcsin . . . . . p V 2 ~ p 2 V 2 __ V 2

(u 2 -~- v~) ~ p 2p 2 -- u 2 ~/uiL- p~ § arcsin § 2 - -

p2v2 ~ § v 2 v 2 p

ELA S TIC S C A T T E R I N G OF L I G H T IN POLARON GAS 275

(19) B ( p , ,,)) u3/[ 8 p2 § v"-- u ~ (u~-- v~-)2] arcsin ~ § pv'-[[ v'~ + 3 -p~--v~ _1 V u ' ~ - v ~-

p . , V 2 _ _ U ~ § 8 + 3

V 2 (u~-~-v~)2-] arcsin .~P

p ~ v ~ J V u ~ + v ~

p 2 u 2 - - 2 8 v 2 § u4 ]arcsinP- §

p'%~J u

§ § 2p ~ - u:'~ V u ~ - - v" - - p~

v2 ] "10 3 ~1 - - 2 p ~ - ~ u ~ u ~ § v 2 - p~

\ v ~ ] P §

In particular, at p = 0 and v2< u 2

(:~o) 8 U 8

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

(21) B(0 , o~) = 0 .

:Note tha t the corresponding scattering mechanisms is also not screened by the Coulomb interaction. Therefor% we conclude tha 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 coupling between electrons and phonons ( a ~ l ) . In 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 tha t of the scattering in a free-electron gas without screening. Therefore, the strong coupling of electrons with phonons may 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 tha t the corresponding amplitude is of the order Ma in the case of strong electron-phonon coupling and of the order ( p / m ) M ~ when there is no electron-phonon interaction.

2 7 6 NGUYEN BA A~', NGUYEN VAN HIEU, NGUYEN TOAN TI.IASIG a n d NGUYEK AI VIET

�9 R I A S S U N T 0 (*)

I n qucs to lavoro si s t u d i a lo s c a t t e r i n g e las t ico de l ia luce con ecc i taz ione a pa r t i ce l l a s ingo la da p a r t e di u n s i s t em a di e l e t t r on i i n t e r a g e n t i con i fonon i - - urt gas o u n liquid() d i po la ron i . Si osserva u n s igni f ica t ivo a u m e n t o dello s c a t t e r i n g d o v u t o al fo r te accoppia- m c n t o c lc t t rone- fonone . L ' o r d i n e di ques to cffet to di a u m e n t o d ipende dal le p r o p r i e t h di po lar izzaz io lm del la luce i n c i d e n t e e d ispersa . Si p r o v a a n c h e cite pe r un ' i den t i t , r del r i pe di W a r d il c o n t r i b u t e del t e r m i n c A" n c l l ' h a m i l t o n i a n a a l l ' amp iezza di s c a t t c r i n g n o n ~ in f luenza to d a l l ' i n t e r a z i o n e e lc t t rone- fonone .

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Elastic scattering of light in polaron gas

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