Phonon anomalies in layered superconductors with unconventional pairing

PHYSICA ELSEVIER

PhysicaC261 (1996) 137-146

Phonon anomalies in layered superconductors with unconventional pairing A.G. Yashenkin *, D.N. Aristov, S.V. Maleyev Petersburg Nuclear Physics Institute, Gatchina, St. Petersburg 188350, Russia

Received 5 January 1996

Abstract

We calculate the electron contribution to the phonon damping in layered superconductor with nodes in the spectrum. In the long-wavelength limit this damping is gapless and quadratically depends on the momentum. Only the phonons propagating along some particular directions are damped. Being connected with the symmetry of the order parameter, the angular dependence of damping may be used for its identification, which is illustrated by examples. The phonon Kohn anomaly in the superconducting state of Laz_xSrxCuO 4 is also discussed. We attribute its "fine structure" previously observed for the magnetic structure factor by Won and Maki to the presence of low-energy crossover scale.

1. Introduction During recent years theoretical predictions conceming the d-wave nature of the pairing state in high-T~ superconductors [1] was multiply examined in experiments without unambiguous results. A certain success has been achieved most recently when angle-resolved photoemission spectroscopy (ARPES) [2,3] and SQUID-interferometry data [4] have allowed (i) to determine the positions of the "experimental zeros" of the gap function on the Fermi surface (ARPES), and (ii) to prove the altering of the gap sign in the Brillouin zone (Josephson junctions experiments). However, the situation is far from being absolutely clear. First, there exist ambiguities with interpretation of SQUID experiments [4]. Second, even assuming the sign-altering gap we cannot

* Corresponding author. Fax +7 812 71 31963; e-mail [email protected].

uniquely judge about location of the nodal points since some of the ARPES data are mutually contradictory (cf. Refs. [2,3]). Thus, the additional independent methods of the gap identification are needed. On the other hand, there are two oddities of phonons in the high-Tc superconductors. The first one: though many experiments demonstrate the strong electron-phonon coupling in these compounds [5], the typical inelastic neutron scattering (INS) data do not reveal the phonon line broadening for low-lying branches. Second, while the "incommensurate" peaks in the dynamical magnetic structure factor of La2_xSrxCuO 4 (LSCO) compound [6] are usually associated with the Kohn anomaly [7], no experimental evidences for the phonon one are known yet. In the present paper we evaluate the phonon damping in the two-dimensional (2D) superconductor with d-wave-like pairing appearing due to the electron-phonon interaction. This problem has been studied [8] mainly numerically in terms of q-plots of

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138

A.G. Yashenkin et al./Physica C 261 (1996) 137-146

the polarization operator at fixed frequency or as softening/hardening and broadening/narrowing of the phonon spectra relative to the normal state. It is obvious that this approach is convenient for the comparison with experiment. On the contrary, we perform some direct analytical calculations and concentrate mostly on the analysis of origin for peculiarities obtained. We show that the damping of long-wavelength acoustic phonons in layered superconductors with nodes in the spectrum is gapless and reveals k 2 momentum dependence. This damping takes place only for phonons propagating along some specific directions determined by the symmetry of the order parameter; therefore, the angular dependence of damping may be used for its identification. Keeping in mind definite compounds, we present these dependencies for a few of the model spectrum parametrizations. We also argue in favor of the phonon Kohn anomaly in the superconducting state of LSCO and associate its "fine structure" discussed for the magnetic susceptibility by Won and Maki [9] with the presence of low-energy crossover scale. The outline of the paper is as follows. In Section 2 we give the minimal formalism needed to read this paper. In the third section we obtain the expression for damping of long-wavelength phonons and discuss the origin of its peculiarities. In Section 4 we utilize the results of the previous section for the analysis of particular high-T~ compounds from the point of view of the possibility of identifying the symmetry of the order parameter. Section 5 is devoted to the treatment of the phonon Kohn anomaly. The last section contains a discussion of the results obtained in the previous sections and conclusions.

is helpful to rewrite Eq. (1) for frequencies close to the resonant ones, to ~ __ tok, in the form

1 oJk w) - - 2 t o ( k ) - i T ( k )

D(k,

t o ( k ) = tok[1 + ~ 1 Re H ( k , =

'

~tok Im

-T- t o '

_+ wk)],

H(k, + tok),

(2)

(3)

where ~o(k) and T ( k ) stand for the renormalized frequency and the damping of phonons, respectively, and the analiticity of the phonon Green function within the upper semiplane of complex variable to is provided by the oddness of function Im H. Below we shall consider only the contribution to H(k, to) arising from the electron-phonon (EP) interaction. The Hamiltonian of this interaction looks as follows c

*

c.c.}.

(4)

kqoHere C*q~ and b~ are electron and phonon creation operators and gk stands for the matrix element of electron-lattice interaction. Hereafter we are interested mainly in the acoustic phonons for which gk2 _~ r.Ok f 2 , where f is the dimensional EP coupling constant assumed for the simplicity to be momentum-independent. In the superconducting state the Hamiltonian (4) obtains the form [1 1] , ~ ' = E g k [ ( ( U q U q + k-['Uquq+k)

kq

×(<, OLq+k, , [ + Ogq,[ OLq+k,r ) --]-( UqUq+ k -- UqUq+k) × ( 0 l : $ OLq+kT "~ Ol.q+ + k $ OLq,L)}b~ --~- c . c . ] , (5)

2. Theory We start from the Dyson equation for the phonon Green function D(k, to) [10]

D - ' ( k , 00)=Dol(k, a s ) - I I ( k , to).

(1)

Here Do(k, 0~) = to2(to2 _ to2)-! is the bare phonon propagator with ~0k being the bare phonon frequency and H(k, w) is the polarization operator appearing due to the phonon's various interactions. It

where new fermionic operators o%~ are connected with the old ones Cq~ via the Bogolyubov transform ~qT = UqCqT -- UqCt-q,l. ' ¢Yq,l. = gqC-q,l. + UqCtq$ "

(6)

Here 2__ ~ / q - ~-

1 -[-

Eq ] ,

Uq2 : -~ 1 -

(7)

139

A.G. Yashenkin et al. / Physica C 261 (1996) 137-146

<=>



Fig. 1. One-loop diagrams giving the first nonvanishing contribution to the phonon self-energy.The lines with one (two) arrows denote the normal (anomalous) Green functions.

by thermally activated excitations while the second one corresponds to the phonon decay into the couple of excitations. We shall discuss this second term since it survives at zero temperature.

3. A c o u s t i c p h o n o n s

with SCq and Aq being the normal-state electron dispersion and the gap function, respectively, while the excitation spectrum is given by ~'q : [ ~2 + I Aq]211/2. In the one-loop approximation (see Fig. 1) the EP interaction described by Eq. (5) contributes to the polarization operator as follows

f2"n" q~ [ 'q'q+k--AqAq Im II ( k, t o ) = - - - ~ 1. . . .

f2T n(k,

E{Gq(ito,,,)Gq+k(ito.,+.)

ito.) = - - 7

First, we consider the damping of long-wavelength acoustic phonons, i.e. the phonons with dispersion tok = uk, k ~ O. Here u = u(n) is the phonon velocity, n = k / k . At T = 0, to > 0 the imaginary part of the polarization operator (10) is given by

~q~q+k

qnl

--Fq(iton,)V+q+,(iton, +n)}"

(8)

Here i to. is the imaginary Matsubara frequency. Normal G and anomalous F electron Green functions in Eq. (8) are given by Z~q iw n + ~q Gq(ito.) = 2 + 2, Fq(ito.) = - 2_.}_ C.On Eq2" ton .S'q

(9) After the summing over the intermediate frequencies and analytical continuation ito n ~ to + i8 Eq. (8) transforms to II(k,

w) =

-~f2_ ([ ~

x

, q , q + kk_° _° qASq+k qz ~ q +

1 +

2(nq -- %+k) e q + k - eq -

to-

i6

X~(~q+kqLEq--to).

toth(k) = m{q}i n iLs q +k + Sq].

OAq ~F ~

~=0'

/'Yg

~

(13) ~=0'

1 - nq - nq+ k

[ eq+ k + Sq +

(12)

For the isotropic BCS pairing state the threshold is equal to 2 A, so that in the old theory EP interaction produces no damping of long-wavelength acoustic phonons. However, it is not the case for the d-wavelike pairings extensively discussed nowadays. There exist the " n o d a l " points q; in the excitation spectrum of such superconductors stemming from the intersections of Fermi surface with zero-gap lines. Presence of these points strongly affects the lowfrequency and low-temperature behavior in this case. Really, let us expand the normal-state spectrum and the gap function near such a point

~q~q+k -- AqAq+k ] ~q'~q+k

(11)

This function reveals a threshold feature at

+ 1. . . .

X[

+k

1-

nq

to -

-- nq+

i6

k

S q + k "~- ,~q "~- to "~- i6

I

I'

(lO)

where nq denotes the Fermi function. The first term in Eq. (10) describes the inelastic phonon scattering

where ~ = q - qi- From the general reasons one can expect that the " g a p velocity" Vg is of order of the gap value (we put the lattice parameter to be unity) and much smaller than the Fermi one: vg ~ A << v F. We see, that in the vicinities of the nodal points the superconducting excitations are gapless thus providing absence of threshold in Eq. (11) at k --> 0.

140

A.G. Yashenkin et a l . / Physica C 261 (1996) 137-146

Next, the excitations have a linear phonon-like dispersion, eq = v~, with strongly anisotropic velocity whose sharp angular variation is a consequence of difference between vg and VF:

U(tt) = [ ( P F n ) 2 + (Pgn)2] 1/2

Eq. (11) may be written in the vicinity of the ith nodal point as follows

(14)

• Using expansions (13), one can easily calculate Im H at k, ~ o ~ 0 Im H

f2 ----~ 32 .

'

;i-1

I b'F X Vg

¢¢.*)2-- ( ~ / ¢ ) 2 - - (g,'~k)2 ' (15)

where the sum runs over all nodal points. Consequently, for the long-wavelength acoustic phonon's damping we get y ( k ) = a g ( n ) k 2,

f2

(16)

U(n)

/,/2(n)

-- (/2~ n) 2

a~C(n) = ~ ~. ]v~ × v~l ~u-~n;--- ~

(17)

Let us now discuss the results obtained. First, one should mention that the damping is gapless and depends quadratically on the momentum. For nonsuperconducting metals such a behavior is typical within the long-wavelength limit kl << 1, where l is the mean free path of electrons [12]. The quadratic dependence of damping appears in that case due to the viscosity of the electron liquid. Meanwhile, in our case k2-dependence stems from the two-dimensionality and also from the linear phonon-like dispersion of the excitations near the nodal points. Moreover, as we discuss below (see Section 5), we actually consider the opposite case, i.e. kl >> 1. The strong angular dependence is another characteristic feature of damping described by Eqs. (16) and (17). The physical reasons for it are as follows. It is known that the propagating sound wave is able to create some excitations with linear dispersion only if their velocities are lower than the sound one (cf. Ref. [10]). Otherwise, no sound absorption due to this process appears. In our case the kinematic condition stemming from the argument of 6 function in

2

2" 1/2

)

(18)

Since for the high-T~ compounds Vg < u << v F, this condition may be satisfied for some small wavevectors 7/ and k, only if both the excitations eq+k and 8q created by the sound wave are moving almost normally to the vector of Fermi velocity v Fi (fixed by pairing and Fermi surface geometry). In the meantime, two simultaneous requirements, v Fi _L?/+ k and v iF _L~, are fulfilled, when k 'M' q or k $ $ ~. Hence, we conclude, that only the long-wavelength acoustic phonons propagating along some particular directions in the momentum space are damped. These directions are located within the narrow sectors around the normals for the Fermi velocity at the nodal points. The damping takes place because of phonon's decay into the couple of superconducting excitations with momenta lying near the nodal points. This phenomenon has the same nature as the Cherenkov effect in the strongly anisotropic media. One should emphasize here that the mentioned sharp anisotropy of the acoustic phonon line broadening is caused only by the anisotropy of the excitation dispersion near the nodal points while the effects induced by the anisotropy of the EP interaction itself (say, through the coupling constant, f = f ( n ) ) are not discussed here. On the other hand, there are no reasons to expect such a tangible anisotropy as that found above from any other source. We mention also, that the coefficient d ( n ) given by Eq. (17) formally diverges for directions wherein I u(n) l = I vi(n)[. This divergence is the artifact of the linear expansion (13) for the spectra of excitations we use. In particular, it means that the true answer along these directions should include the curvature of the Fermi surface. This cuts the divergence, whereupon the damping for such lines increases with k slower than quadratically [13]. The general picture, however, persists, including the fact that along directions I u(n) l = I vi(n) I the damping will be maximal.

A.G. Yashenkin et al./Physica C 261 (1996) 137-146

4. Phonon damping: comparison for different pairings

the model which accounts for nearest-neighbors and next-nearest-neighbors overlaps ~k=--2t[COS

Below we shall show that the angular dependence of the sound absorption may be regarded as an independent tool for the identification of the pairing state. To clear up this fact, let us consider now some particular model spectra addressing to the concrete compounds. Since in all cases studied the k2-depen dence of damping takes place, we will operate immediately with the coefficient ~ ( n ) determined by Eq. (17). We note also, that the analytical expression for it may be easily found for any model we will utilize; however, we restrict ourselves by figures and brief analysis. At first, we shall address to LSCO compound. For the normal-state spectrum parametrization we choose

141

k x--I-COS ky]

+ 4 f cos k x cos ky + tz.

(19)

The relatively small second hopping term (t' ~ 0.1 0.2t,) results in this case in the Fermi surface of altering sign of curvature [7]. Since there are no grounds to expect a deformation of the square symmetry for LSCO (cf. below), we write the superconducting order parameter in the simplest dx2_y2 form A

+ = 3-[cos kx- cos ky].

(20)

The angular dependence of the coefficient ~¢ for this spectrum parametrization within the first quad-

0.030

0.008

(a) 0.024

(b)

0.006

0.018

A

&

<

0.005

<

0.012

0.008

O.OOE

0.002

0.000 0.00

0

0,000 0.00

1.57

0.030

c~

1.57

0.005

(c)

0,024

(d)

0,004

0.018

0.008

<

< 0,012

0.002

0.006

0.001

0.000 0.00

@

1.57

0.000 0.00

@

1.57

Fig. 2. Angular dependence of the coefficient A given by Eq. (16) in the first quadrant of the Brillouin zone for (a) LSCO-like Fermi surface and dx2_y2 gap function, (b) Y-123 - like Fermi surface and d + s gap function, (c) Y-123 - like Fermi surface and extended-s gap function and (d) X - Y - anisotropic Fermi surface determined by Eq. (21) and dx2_y2 gap function.

142

A.G. Yashenkin et al. / Physica C 261 (1996) 137-146

rant of the Brillouin zone is shown in Fig. 2a (the same picture is reproduced in all other quadrants because of the square symmetry of the spectrum). We see that the ranges of nonzero damping occur along the diagonals of the Brillouin zone. Their widths are determined by the conditions: U2

1 2 sin2k0 7/I

-

[8t2(1- ~r/t2)-½/12] sin2k0 > cos2(q~q +__-n'/4), t

k0 = c o s - ' ~

[1 -- ~1 -- /.ztt///2 ] ,

(21)

where q~k is the polar angle of vector k. Estimates show that for the realistic set of parameters the ~pk-widths of this sectors should be of the order of 0.1 rad which seems to be experimentally resolvable, say, in the INS experiments. Let us discuss now the case of YBa2Cu307_ a (Y-123) compound. The normal-state spectrum for this material is usually described with the use of Eq. (19) with large enough second hopping term ( f ~0.4 - 0.5t) which results in the so-called "45°-angle rotated Fermi surface". As for the gap function, the widely accepted one is again the dx2_y~-wave. It would result in qualitatively the same q~k-dependence of d , as for LSCO. In the meantime, as it has been argued in Refs. [14] and [15] in virtue of analysis of experimental facts, the C u - O chains could produce the breaking of the square symmetry within the CuO 2 planes. In particular, the dx2_y2 pairing state may acquire a minor admixture of the extended s-wave component [15] /1 A~ + = -~- [COS k x ac cos k y ] ,

(22)

so that the resulting gap function will be given by /1d+s = /ix COS k x -- Z~y cos k y .

(23)

with /1, v~/1y. This will lead to a shift of zero-gap lines from the diagonals. The angular dependence of the coefficient d in Eq. (16) for the "45°-angle rotated Fermi surface" and d-I-s gap function is shown in Fig. 2b. We see the displacement of the damped phonon's region from the diagonals of the Brillouin zone caused by rotation of vectors of Fermi velocities at the nodal points [16].

These results should be contrasted with the qOk-dependence of ag for Y-123 Fermi surface and the purely extended s-wave gap function (22). As shown in Fig. 2c, the doubling of sectors of nonzero damping arising from the doubling of the nodal points is clearly seen in this case. All such regions are symmetrically displaced to both sides of the diagonal direction (cf. Ref. [3]). We shall sketch now the experimental situation with other high-Tc compounds. The data on Bi2Sr2CaCu208+ ~ (Bi-2212) are somewhat similar to those on Y-123. Namely, while the Stanford ARPES group has spoken out in favor of the pure dx2_y2 pairing in Bi-2212, [2] the recent experiments of the Argonne group [3] reveal some shift of zerogap lines from (Tr, 7r) directions, so it looks rather as a mixed state or even as the extended-s one. The most probable origin for such distortions of the simple dx2 y2 picture is the chains influence. There exists the compound without chains T12Ba2CuO6+ a (T1-2201), wherein no experimental evidences for a mixed state has been obtained yet, while the general d-wave-like picture of the pairing ( A c t cos 20) has been established on the basis of the high-resolution SQUID-interferometry [17]. Therefore, one can suggest, that the single-layered superconductors without C u - O chains correspond to the pure dx2_y2 pairing, while some mixed state appears for superconductors with chains. However, the issue is still questionable, since (i) the experiments partially contradict each other, and (ii) reliable data are provided only by two kinds of experiment such as ARPES and SQUID-interferometry. At the same time, it worth noting that the reasons leading to the mixed pairing state should produce the X - Y asymmetry of the conducting band, as well. This effect can be modelled by introducing the anisotropy parameter • into the normal-state spectrum [15] ~:k = - 2 t [ c o s k x + ( 1 + • ) c o s + 4t' cos kx cos

ky-t-I.L.

ky] (24)

This also should lead to shifts of the centers of sectors with nonzero phonon damping from the diagonals of the Brillouin zone, even in the case of pure dx2_y2 pairing state (see Fig. 2d).

A.G. Yashenkin et aL /Physica C 261 (1996) 137-146

5. Kohn anomaly The purpose of this section is to draw attention to the studying of the phonon Kohn anomaly in LSCO. At present there are no direct experimental signs of it. Their discovering would be very important from the point of view of independent confirmation for the mentioned below theoretical interpretation of incommensurate peaks in the spin susceptibility of this material. In the previous two sections we discussed the damping of the long-wavelength acoustic phonons. In the meantime, due to the large number of atoms in the unit cell the high-T~ compounds demonstrate the rare richness of the phonon spectra [18]. The lowest branches of LSCO spectrum were reported to be of the order of 15-20 meV (which is somewhat larger than the expected gap value for this compound A 10-12 meV) while the highest ones amount to 80 meV. For concreteness, hereafter we shall discuss only the LSCO system. The normal-state dynamical magnetic structure factor of this compound reveals the so-called "incommensurate" structure at large momenta: four peaks centered symmetrically around the 2D antiferromagnetic vector Q0 [6]. Such behavior has been attributed to the two-dimensional Kohn anomaly, corresponding to the wavevectors connecting the points of Fermi surface with antiparallel

"IT

0,

"IT

/

\

Fig. 3. Schematic drawing of the LSCO-like Fermi surface (solid line) and the nodes of dx2_y2 gap function (dashed lines). The vector of internodal transfer Q~, and the incommensurate vector Q~ are denoted by arrows and the direction of scan along the Brillouin zone edge is shown by the thick arrow.

143

Fermi velocities (see, e.g., Ref. [7]). Though the whole line around Q0 should occur on which this phenomenon takes place (see Fig. 3), the only positions where it appears to be visible in the experiment are the incommensurate ones because of the maximal prefactor for these momenta. Up to some diminishing in the intensity, the above picture persists in the superconducting state, as well [19]. From the theoretical side, d-wave superconductivity induces (i) the so-called "fine structure" of the Kohn anomaly, and (ii) the strong variation of the threshold function (12) in the relatively small momenta region near the antiferromagnetic wavevector [9]. All peaks in theory appear to be narrower and sharper than in the experiment. Because of the similarity of the expressions for magnetic structure factor and phonon damping the same incommensurate behavior should be typical for the latter. Now we shall consider this problem in detail. We start from discussion of variation of function tOth(k) n e a r the antiferromagnetic wavevector. For the model spectrum given by Eq. (19) the incommensurate peaks are located at Q~ = (Tr, 2 cos-l/z/ 2t) and symmetry related points. The value of threshold at these points for A~+ gap function (20) is given by toth(Qs) = d /z = 3.5 meV, 2t

(25)

which is essentially less than the energy of the acoustic phonon with this momentum: ¢o(Q8)-- 15 meV [18]. The threshold function Wth(k) varies along the line of Kohn anomalies from this value to zero at the vector of internodal transfer Q v = (2k0, 2k0) (see Fig. 3). Thus, the phonons with large momenta belonging to the line of anomalies will be damped in the superconducting state. On the other hand, the threshold value for the antiferromagnetic wavevector is larger than 2 za (cf. Ref. [9]); hence, the acoustic phonons are not damped at k = Q0Under these circumstances, it would be very instructive to study experimentally the changes in the acoustic phonon line width as a function of momentum, scanning from Q~ to Q0, i.e. along the zone edge (see Fig. 3). In reality, our estimates show (see also below), that in the superconducting state of

144

A.G. Yashenkin et al./Phy s i c a C 261 (1996) 137-146

LSCO the line of Kohn anomalies should be although presented, and the most pronounced ones should correspond the incommensurate vectors. At the same time, quite close to them (at the antiferromagnetic wavevector) the acoustic phonon line should not be broadened; therefore, it would be possible to investigate the fast changes in the line width. Now we shall discuss the "fine structure" of the Kohn peaks which appears in the superconducting state. For the magnetic dynamical structure factor this issue has been investigated by Won and Maki [9]. For simplicity, let us consider this phenomenon for the wavevector of internodal transfer Qv" In order to obtain the leading term for the Kohn anomaly, one should expand the spectra of both the excitations in Eq. (11), Sq and eq+k, in the vicinities of two nodal points connected by vector Qv, leaving the first and the second terms in expansion of the conducting band 2 2mll

}1/2

2m ± (26)

where symbols _1_ and [[ are introduced to denote the transverse and the longitudinal with respect to the Fermi surface components of vector ~ and the tensor of the inverse effective-mass (in the chosen coordinates the off-diagonal components of tensor t~-l are equal to zero at this point). The quantities involved into Eq. (26) are related to the spectrum parameters of the model given by Eqs. (19) and (20) as follows v F = 2~-t~/1 - t z t ' / t 2 sin k o,

Vg = A / V ~ sin k 0,

mll- ,±l = 2t~l - t*t'/t 2 cos k o + 4t' sin2ko -

.

(27)

It may be easily demonstrated that for all the momenta for which the energy of excitation is less or equal to the phonon energy to(Qv)-~ 15 meV, the term 7 2 / 2 m ± in Eq. (26) can be omitted, since the spectrum dispersion along the normal to the Fermi surface will be determined by the leading linear term of this expansion. As for the direction along the Fermi surface, the situation appears to be somewhat different.

Simple arguments show that for small enough q l < q , , where

q* = V2mlI/VF << 1,

(28)

one can neglect the Fermi surface curvature in Eq. (26). As a result, we return to the spectrum discussed in previous sections, eq = [(VFT±) 2 + (Vg~ll)]1/2. In this case all the spectrum dispersion along the Fermi surface direction stems from the gap one. Calculation of damping at k ~- Qe then yields g~

7(k)-

1

to2 - (VFk)2

64 [VFXI)g[ ~to2 (15F~£)2 (Vg~)2 ' (29)

where 7 c ± < 7 , . The formula (29) describes the peaking of y ( k ) in the closest vicinity of the internodal vector. On the contrary, at 7 , -< 7 ± << 1 the gap term in Eq. (26) becomes inessential, so that the spectrum dispersion along the Fermi surface is determined in this range by the second-order term in the expansion of the conducting band, arising from the finite Fermi surface curvature Sq--- VF71 +

.

(30)

Up to the modulus sign the latter expression has a "quasinormal" character, which immediately gives the conventional Kohn anomaly (for details of the theory of the normal-state 2D Kohn anomaly see Ref. [7]). The energy scale to, corresponding to the crossover from linear spectrum dispersion to the quasinormal one is given by

to, = v2mll ,

(31)

which is much less than the acoustic phonon frequency in this momentum region; therefore, there exists room for both these regimes developing at 7c± < q , and 7¢± > q , , respectively. Thus, the physical picture is as follows. The fine structure of the Kohn anomaly in the superconducting state is determined by two circumstances. First, there exists a crossover from one spectrum behavior to another, which results in different regimes of

A.G. Yashenkin et al./Physica C 261 (1996) 137-146

formation of the anomaly in the closest vicinity of the transfer wavevector and away from it. Second, the energies of both the excitations in Eq. (11) are positive and in order to obtain an anomaly we cannot put to = 0 in the 6 function. This leads to some shift (proportional to to) of the maximum of the "quasinormal" contribution from the transfer vector (cf. Refs. [7] and [9]; this point agrees with experiment, see Ref. [20]), while the " n o d a l " one is centered exactly at vector Qv (see Eq. (29)). The fine structure thus also may result in the asymmetry of experimentally observed peaks in the magnetic INS experiments. We have discussed the developing of the Kohn anomaly near the vector Qv, which is the wavevector of intemodal transfer and a particular example of the Kohn vector. One can easily generalize our treatment to any other Kohn vector. It will result in some minor changes. Say, the effective-mass tensor in Eq. (26) will possess off-diagonal components. Also, the expansion (13) for Aq will contain the constant term

a q = ½toth(Q) + vg~,

(32)

where Q is the Kohn vector and ~ = q - Q / 2 , which will lead to the appearance of the additional small energy crossover scale. At the same time, the whole picture described in this section is preserved for the general case, as well.

6. Concluding remarks We discussed above only the properties of the phonon damping which are proportional to the imaginary part of the polarization operator / / ( k , to). Up to the w-independent term the asymptotic behavior of R e / / can be restored from I m H via the wellknown Kramers-Kronig relations. Say, the value of R e / / defining the long-wavelength acoustic phonon frequency renormalization could be found by altering the sign inside the square-root in Eq. (15) and then putting w = tok. The fast changes of the phonon line width thus will be accompanied by frequency jumps. From the experimental side, it may appear to be more convenient to study the effects by investigating these latter jumps. Let us discuss now the range of applicability for our equations.

145

In this paper we calculated the phonons damping in the lowest (one-loop) approximation completely ignoring influence of impurities. This is correct only in the clean limit tok'/'imp ~. 1, which restricts the momentum range of applicability of our results for long-wavelength phonons from below. In the opposite case tok~'imp ~ 1, multiple phonon scattering by impurities should be taken into consideration; this may essentially change the behavior as it occurred in the ordinary metals [12]. The role of impurities for phonons in d-wave superconductors has been studied numerically in Ref. [8]. At the same time, analytically the problem appears to be rather complex, mainly because of the open question with the leading contribution to the diagrammatic series for the electron-impurity scattering [21], yielding the nontrivial low-frequency asymptotic behavior of density of states. [22] On the other hand, in the previous section we demonstrated that the region of applicability for Eq. (27) is limited by the criterion of the correctness of the linear approximation for the spectrum near the nodal points. On the energy scale this condition looks as to, > t o . . It is valid also for formulas obtained for the long-wavelength acoustic phonons, which restricts from above the momentum range of applicability for results obtained in that case. We would like to emphasize here that the energy scale w , as a scale of transition from the " n o d a l " regime to the quasinormal behavior has quite general origin. It should manifest itself not only as a fine structure of the Kohn anomaly but also in crossovers in temperature and frequency dependencies of various integral characteristics of d-wave superconductors such as density of states, penetration depth, spin-lattice relaxation rate, etc. These problems are beyond the scope of our study. To summarize, we have studied the electron-phonon contribution to the damping of phonons in the case of layered superconductors with nodes in the spectrum. For the long-wavelength acoustic phonons we obtained gapless kZ-dependence of damping and its strong angular variation stemming from the sharp anisotropy of the excitation spectrum near the nodal points. A sizeable damping was found only for the phonons propagating along some specific directions in momentum space. Since these directions are different for different gap functions, we discussed the

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possibility of using the phonon data as an independent method for identification of the symmetry of order parameter complemental for the ARPES and SQUID-interferometry. We also discussed the phonon Kohn anomaly in the superconducting state of LSCO compound. We argued in favor of experimental searching of this anomaly in the INS experiments with the use of scan along the Brillouin zone edge. We demonstrated in details the developing of the anomaly in the superconducting state and attributed their "fine structure" as arising from the presence of energy crossover scale lying well below the phonon frequency.

Acknowledgements We would like to acknowledge useful conversations with M. Lavagna, F. Onufrieva, M. Brauden, V. Gurevich, G. Halliulin, A. Ivanov, K. Kikoin, A. Mishchenko, and A. Rumiantsev. We also thank for support of this work from the International Science Foundation and the Russian Government (Grant No. R3Y300), from the Russian Foundation for Basic Researches (Grant No. 93-02-2224), and also from the Russian Federal Program of Neutron Studies of Condensed Matter.

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