a. virosztek and k. maki- phason dynamics in charge and spin density waves

8/3/2019 A. Virosztek and K. Maki- Phason dynamics in charge and spin density waves

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JOURNAL DE PHYSIQUEIVColloque(2, upplkment au Journal de PhysiqueI, Volume 3,juillet 1993

Phason dynamics in charge and spin density wavesA. VIROSZTEKand K. MAKI*Research Institute for Solid State Physics, R0. ox 49,1525 Budapest, Hungary*Department of Physics andAstronomy, University of Southern California, Los Angeles, C A 90089, U .A.

Abstract, --- Phason propagaton in charge and spin density waves in presence of long range Coulombinteraction are studiedwithin mean field theory. In charge &nsity wave the longitudinal phason splitsintoanacoustic mode and an optical mode in the presence of Coulomb interaction. Except at lowtemperatures(T 0.2TJ he acoustic mode exhausts most of the optical weight. In spin density wavethere is no longitudinal acoustic mode; the phason becomes the plasmon. On the other hand in thetransverse limit, the plasmon decouples completely from the phason both in charge and spin densitywaves.

1. Introduction.It is well known that the long range Coulomb interaction will m w he longitudinal phason mode strongly[I]. However, most works published on this subject [21are incomplete due to unnecessary approximation.We shall report here our recent study of the phason dynamics 131 of both charge and spin density waves(CDWandSDW). As a model we take a quasi-one dimensional system vrahlich Hamiltonian[4] for CDWandYamaji model [5] for SDW) supplemented by the long range Coulomb interaction

wherenq s the electron densitywith momentumQ . As we shall see the condensate densityf, which dependsboth ono he frequency and Q the momentum plays the crucial role in the following analysis. In thelongitudinal limit (i-e. 4 parallel to the most conducting direction) we obtain an acoustic modewith thephason velocity decreasing rapidlywith increasing temperature in CDW while there will be no phason inSDW. In particular this phason mode is obse~vedecently by neutron scattering for a single crystal ofMo% [a. n the transverse limit, on the other hand, the Coulomb effect disappears completely from thephason propagator in contrary to an early analysis of the electric conductivity in SDW [7].

2. Phason Propagator in CDW.We shall first consider the phason propagator in CDW. In the presence of the long range Coulombinteraction the phason propator is given by

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jp4:1993241
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JOURNAL DE PHYSIQUE IV-1 -1a2 1-a= ( A ) A . 2 - 2 - ( ) )

where A2 is the imaginary part of the order parameter and the effect of the Coulomb interaction isincorporated within mean field theory [8]. Here we limit ourselves to the longitudinal case and in the laststep we took%=w, where a+, = (47~2n /m)'I2 is the plasma frequency. The functionf is the generalizedcondensate density given by

[ - k c + t h (if3ACh+) sh2(+ -I$~)- (1- 2) (c/2A121' for a l

wherea =a /c=d v q anda for a s 1

th+o = (a-l for a > l

We recover familiar expressions in the adiabatic limit ( a , c u 2A (T) )

and0 2lim [ d + ~ e ~ hth($~ch+)fd = a + m

Alsof = 1 fora = 1 ndependent of temperature. The temperature dependence off, andfd are shown in Fig.1. The phason massm* is given by [91

It is important to note that the phason mass m* has different temperature dependence depending on whichlimit you are n. As already pointed out byTakadaand his collaborators [2], the longitudinal phason consistsof 2 modes, which is determined from

where substitutedEq (7) in the pole of Eq (2)


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3. Acoustic Mode.One of the solutionsis givenby

where m*lm defined inEq (7) has to be used. Then for not too low temperatures (say T>0 3 T) we haveo < and Eq (9) simplifiesas a=vo q with

where suffixsmeans the static limit(a < 1). The temperature dependence of v4 is shown in Fig. 2. Asseenfrom Fig. 2, v,+ increases rapidly with indecreasing ternpaatweand ultimately it mergeswith another

Fig. 1 Fig. 2Fig. 1.- The condensate densityf, (the static limit) andfd (the dynamic limit) are shown as function of thereduced temperature t.Fig. 2. - he ongitudinaland the transverse phason velocitiesv+ andvg inCDWare shown as functionof reduced temperature.mode anddisappears at low temperatures (TCL0.2 TC). Itisof interest to consider thespectralweight. Totalspectral weight of phasonis given from Eq (2) as

independent of c, Tand%. The acoustic pole gives, onthe otherhad,


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JOURNAL DE PHYSIQUE IV

which almost exhausts the spectral weight as long as 1 -f,>>mlm: (i.e. T 0.3T,).For an arbitrary cj the acoustic mode is given by

where c, = vq, and cl = vlql and v l is the Fermi velocity in the transverse direction.In the transverse limit (i.e. c,,=0)where the Coulomb interaction is completely decoupled, the phason

velocity v+ldepends only weakly on T hroughm,* (7')as shown in Fig. 2.4. Optical Mode.At T=OK another solution of Eq (8) is [2]

The optical frequency is almost independent of TforT c 0.2T,and then start to increasewith increasingtemperature. At the same time the optical weight of this mode decreases very rapidly. At T=OK,he opticalmode almost exhausts the optical weight.

5. Phason Propagator in SDW.Sincem*lm = 1 practically for SDW, the phason propagator is now given by

where ( )means the angular average andc = c r / + J 5 1 ; l ~ ~ ~ ~

In the longitudinal limit (cl =0)Eq (15)educes toD,$ (q, a) = (2112f1 a; 1 fi + c2- CI?)(c2 x (a;+ c2-m 2 ~ l (17)

Since l e 1, 4 ( Aa) has only the plasmon pole in the longitudinal limit. More generdy in the limito c Eq (15) an be rewritten as


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where cos6 = q, / q. The pole is then given by

which always has two solutions. One at high frequencies (i.e.o >> 2A(T))

and another below the quasi-particle energy gap;

where fd is the condensate density in the dynamical limit and

At T=OK wherefd = 1, the second mode is almost gapless when c,


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JOURNAL DE PHYSIQUE IVAcknowledgements.One of us (KM) thanks the Research Institute for Solid State Physics at Budapest for thekind hospitalitywhere apart of this work was done. The presentwork is suppartedby National Science Foundation underGrant No DMR 92-18317, the Hungarian National Research Fund No. OTKA 2944 and T4473. Also thiswork is sponsoredby the US - Hungary Science and Technology Joint Fund under Project No 264.

References[I] LEE P.A. and FUKWAMA H., Phys. Rev. B 17 (1987) 542.121 NAKANE Y. and TAKADA S., J. Phys. Soc. Jpn. 54 (1985) 977; WONG K.Y. andTAKADAS.,Phys. Rev. B 36 (1987) 5476.[3] VIROSZTEKA. andMAKI K., Phys. Rev. Len. 69 (1992) 3265; Synrh. Meta ls 57 (1993) 4678 andPhys. Rev. B 48 (1993).[4] LEE P.A., RICE T.M. and ANDERSONP.W., Solid Srate Commun 14 (1974)703.[5] YAMAJI K., J. Phys. Soc. Jpn. 51 (1982) 2787 and 52 (1983) 1361.[q HENNIONB., POUGET J.P. and SAT0 M. Phys. Rev. Lett. 68 (1992) 2374.[7] MAKI K. and GR- G., Phys. Rev.Len. 66 (1991) 782.[8] W A N O F FL.P. and FALKO 1.1.. Phys. Rev. 136 (1964) 1170.191 MAKIK, and VIROSZTEK A., Phys. Rev. B 36 (1987) 511.[lo] VIROSZTEKA. andMAKI K., Phys. Rev.B 37 (1988) 2028;MAKIK. and VIROSZTEKA., Synrh.Metals 55 (1993) 2774.

a. virosztek and k. maki- phason dynamics in charge and spin density waves

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