Behera, S N

Ojha, M S

Roul, G C

IACS Institutional Repository

l«iMy./'ftys.76A(l). 79-81 (2002)U P  Aan international journalTheory of ultrasonic attenuation in heavy fermion compoundsM S Ojha'*, G C Roul*' and S N Behera^"S, C S College (Autonomous), Pun-7S2 (X)l, Orissa. India "Condensed Matter Physics Group, G M, College (Autonomous), Sambalpiir-768 (K)4, Orissa, India'^Institute of Physics. Sachivnlaya Marg, Bhiibaneswar'75l 005. Orissa, India E-mail mojhaC^iopb res.intract . The mechanism  of the ultiusonic attenuation in heavy fermion compounds is not yet studied clearly both experimentally and L'lically A microscopic theoretical model is proposed here to study the attenuation in the compounds like UPt,, CcCu,Si„ CeRu.Si. in their normal :s Wc consider the Periodic Anderson Model and incorporate phonon coupling lo the hybridisation between the conduction electrons and Litnns as well as to the /-clecirons alone The phonon Green's function is calculated by Zubarev technique The temperature dependence of the iNoniL attenuation coefficient ( rjr) is calculated from the imaginary part of the phonon self-energy and the velocity of sound in the dynamic and wavelength limit The dependence of the parameters like the electron-phonon coupling parameters (g, 0 . hybridisation (v), position of /-level (c/), lomb correlation (m), frequency ( ro ), is investigated numerically through plotswords Ultrasonic alleniialion, heavy fermions, electron phonon interactionS Nos. 74 25 Ld, 75 30 Mb, 63 20 KrIntnxlucdon>n aflci ihe discovery of heavy fermion superconductivity, i.isonic data have revealed power law dependence for ;nuahon coefficicnls for 7’ < and a maximum around - 0.5 K for UPI3 is observed 11 -31. The allenualion maximum urs exactly where the elastic constant and the velocity of ind exhibit minima [4]. It is also observed that the pronounced asonic allenualion peak of URu^jSij coincides with the middle llic elastic constant step indicating that the maximum of a  ursal 7; «1.2K.The first attenuation measurements in zero magnetic field jwing Kondo effect were done by Muller et al [5]. A peak in attenuation was seen at 10-12K [6 ]. Similar low temperature Mniiiics were observed in the elastic constants and the ocity of sound for compounds CeCu^ (7^ = 4K), CcCuSij (7* U Ru2Si2 (7^ = 70K). CeBe,, (7^ = 340K) [7].fhalnieicr [8] has considered Griincisen parameter coupling '^ ound waves to the bands to explain the clastic anomalies in  ^ icmperature quasi-particlc resime. Others have considered '"^'^'n-dcpendence of the hybridisation to investigate the V temperature anomaly in sound velocity of heavy fermionsystems by using a mean field approach |9]. Rout and Das 11 0 | have calculated velocity of sound of superconducting heavy fermion systems. In this paper, a microscopic theoretical model is proposed lo investigate the ultrasonic attenuation in nonnal stale heavy fermion systems. The attenuation Miillcr peak [5] can be described in terms of a narrow resonance peak with width of ~ 2 meV in the density of slate lying above the Fermi surface resulting from the hybridisation of /-level and the conduction band.2. FormalismThe system is described by the Hamiltonian7{ = -H + 77^ , , ( 1 )7/r^ (/x.a K^.(7 fK,a^H U / 2) (2)‘’^ responding Author80 MS  Ojha, G C Rout and S N Beherafk.a fK+q,a i^r,ry) ^  ) “ ^k'-q\a' fk \a ' f  ^k\a' ^+ /2 (9 ) ( /< . , .a / r .a ) ]K + * 4 ) .  (3)(4)P ' = f w f k ' . c - -In the long wavelength limit, ^ >0. One hascu = and (Oq = Vq^  where and v are the bare and renormali longitudinal sound velocities. The velocity of sound is gi byV=Vn 1 + ^nx(<o,q)1/2(On^ 0  ’ ^f~p  and are the Hamiltonians representing Periodic Anderson Model, elcctron^phonon interaction and free phonon term. Here arc the creation(annihilation) operators of d-  and /- electrons and phonons respectively,  ^ and V'are the conduction electron energy, position of/-level and hybridisation respectively. U is the waves of frequency co and at a temperature T is obtained fr Coulomb interaction./, (^) and f^(q) are the phonon coupling to the imaginary part of self energy through hybridisation and /-electron respectively.The double time Green’s function of Zubarev type is defined asThe ultrasonic attenuation coefficient a{(0,T) for soi4/ra (o ),T )= ------ ImxUo.q).« where the -ve sign indicates absorption of energjy. To siAqO) ; ) ] / .  (5 ) ultrasonic attenuation, the parameters are made dimensionwith respect to Debye frequency O)  ^. \Applying Dyson's approximation, the phonon Green function can be written as (On^ (O (0 ,(O = , c = ,  ^= ---- ’ ^ =(On (Or (Or (Or 0) rw = ,d^ = d + un, V = -----, ^ = ---- -^----(O fj (O p  (O p/,(0 ) CD, CD,3. Results and discussion(Or (OnD^ ,i(co) =(cD, /;r)[cD^ - cd,^  - I ^ ( cd)] (6)where phonon self energy is given by I^(CD) = 4;rcD„;f,,/cD).^  1 + /2<9) (^ 4  + r , )+/2'(<7)^6]- (7)The velocity of sound is evaluated numerically under half-fil r,(k,q ,co)  s (i  = 3 to 6 ) represent the electron response band situation. The Fermi level is taken at the middle of functions. They are defined by dropping k,q,co  in terms of band with = 0. The dimensionless parameters involve a 's and P 's as numerical calculations are the phonon coupling strengththe ratio (r) between the phonon coupling to hybridisation the/-lcvcl, the position of the bare/-level ( d ), hybridisation Coulomb interaction (u). The temperature variation of ultrasonic attenuation of sound of the heavy fermion systc shows unusually high anomaly at low temperatures for ccr physical parameters. The anomaly is discussed below.Figure 1 shows the variation of ultrasonic attenuation \ temperature. As phonon coupling (^) to the hybridisai (8) increases, the ultrasonic absorption increases and as a roi peak height increases near temperature t = 0.05. With De frequency » 200K, this corresponds to fluctuai temperature T*^ lOK. The phonon coupling increases hybridisation between /-electron and conduction band results in increase of mixed valence behaviour. When effeer ,  = « ( « "  + Cf r 4 = « ( a ‘'+ a ' ’) ; ^ '  » ,  . r ^ = « a '  ;( /3 "+ i3 '’)Tft = « a ‘ ; P' .These operators are written in abreviated form as^   ^ k-q,(T f k . a '  ^  ”  f k - q , o  ^k.a^  ~~ fk-q,c fk,a*  ® ”  ^k-iq,a ^'k,aTheory of ultrasonic attenuation in heavy fermion compoundsovcl lies below the Fermi level, the attenuation decreases with Lfease of phonon coupling (r) to the /-electron alone.ii;iin! I The variation of ultrasonic atlcnualion with tcinperatiiic tor (Ici-ni values of = 0 020, 0 022. 0 024. 002.S, 0.026, 0028 and fixed lilies of r = 0, d = -0 9 4 , v = 0 05, ii = I 0. e = 0(XI5, /) =0.12.6 006-09^eil values of s  = 0 026, r  = 0, v = 0 05, m = I 0. <■ = 0.(X)5, /) = 0 12Figure 2 shows the variation of attenuation peak with temperature. As /-level moves from d = -  0.85 to -  0.95 from above, towards the Fermi level =0, the attenuation increases and attenuation maximum shifts to lower temperature. The shifting of /-level in the conduction band changes the hybridisation between d- and /-electrons. This leads to change in the density of states near Fermi level and changes the fluctuation temperature (P). Hence, the ultrasonic attenuation and softening clastic constant of several heavy fermion compounds can be explained.AcknowledgmentThe authors (M.S.O. and G.C.R.) gracefully acknowledge the financial support of U.G.C., New Delhi vide letter No. F-PSO- 035/99-00 (ERO) dated 08.2.2000.References111 1: Bucher, B BalUigg, IJ J Bishop and C M Vertna J. Appl. Pliw., 57 2060 (1985)12] V Milller, D Maurer. E W Schcidl, Ch Roth. K LUders, E Bucher and II E Bdinmcl Solid Suite Commun. 7 219 (1986)12] B Golding, D J Bishop, H Ballogg, W II Hacminerle, Z Fisk and H R On P h\i Rev Lett 55 2479 (1985)[4 1 C Jill, I) M Lee, S W Lm, H K Shormu and D G Hinks J Low Tenifi P Im  89 557 (1992)|5J V Muller. D Miiurei, K de Grool, E Bucher and H E Bbmmel Phys Rev Leu 56 248 (1986)[6] S W Lin. I Kouroudis, A C M  Jensen, P Wyder, B Lulhi, f) G Hink.s, J B Kciterson. M Levy and B K Sharina Physka B 223/ 224 185 (1996)|7 |  B Lulhi, G Brulls. P Thalmcier, B Wolf, D Finnsterbusch and I Kouroudis J. Low Temp Phv.s 95 257 (1994)18] P T h a lm e ie r i P Im  C20 1119 (1987)|91 R Wo|ciechow.ski, G A Gehring and L E Major J Phyy Condens. Muitei 6 9707 (1994)I ID] G C Roul and S Das Solid Slate Commun. 109 177 (1999)

Theory of Ultrasonic Attenuation in Heavy Fermion Compounds

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Theory of Ultrasonic Attenuation in Heavy Fermion Compounds

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