Elastic Scattering of Light in Polaron Gas
I L NUOV() CIMENTO VOL 52B, N 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 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 WardtyI)(' 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 plasm~ was studied in m a n y works (~ o~) I t is well k n o w n t h a t t h e r e are two 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 excitations (plasmons) a n d t h e scattering w i t h single-particle e x c i t a t i o n (SPE)(~,,:o:1) I n t h e lowest order ()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 to 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 an electron 133, A 1308, A 1317 (1964) (1) l) F 1)urn)IS and V GILINSKI: Phys lr (~) 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, 267 268 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 parabolic dispersion law for t h e e l e c t r o n p2 (1) E(p) 2m ' 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 screened 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 with t h e S P E , a n d t h e m a t r i x elements of t w o d i a g r a m s in fig lb) cancel in t h e limit v > 0, where v is t h e electron 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, which will be used in this work a) b) Fig - 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 the situation 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 fluctuation (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/.;ItGERand 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~,~.LKA.~SKI (Paris, 1971), p 102; Phys Rec B, 6, 1198 (1972) (ts) A R VASCONC):LLOand 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) DOEIILER: Phys Rev B, 12, 2917 (1975) (z~) K P JAIN and M BALKA~'SKT:in Light Scattering in Solids, edited by M BALKANSKI, 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 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 269 I ' O L A R O N GAS there exists also the scattering on the energy density fluctuation (EDF) and the 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 presence of the virtual interband transitions, the m a t r i x elements of the two diagrams in fig lb) not cancel, b u t can exhibit some enhancement at the photon energy near the value of the b a n d 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 latter process with the scattering on the collective excitations, we have the creation of new elementary 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 the 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 experimentally (~,2~) In this work we study 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 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, b u t without real phonons I n other words, we study 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 Hamiltonian (2~) with coupling constant v/~ The energy of the bare electrons is assumed to be determined b y eq (1) Denote by k, k' and p, p' the m o m e n t a of the initial and final photons and electrons, respectively b y (o, ~o' and 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,