Attenuation of ultrasound in p-wave superconductors

Solid State Communications, Vol. 60. No. 6, pp. 535-539, 1986. Printed in Great Britain.

0038-1098/86 $3.00 + .00 Pergamon Journals Ltd.

ATTENUATION OF ULTRASOUND IN p-WAVE SUPERCONDUCTORS

K. Scharnberg, D. Walker, H. Monien and L. Tewordt

Abteilung fiir Theoretische Festk6rperphysik, FB Physik der Universit/it Hamburg, Jungiusstrafie 11, D-2000 Hamburg 36, West Germany

and

R.A. Klemm

Exxon Research and Engineering Co, Annandale, NJ 08801, USA

(Received 16May 1986 by B. Miihlschlegel)

The theory of ultrasound attenuation in impure BCS superconductors is extended to unitary p-wave states. For the ABM state, but not for the polar state, a set of parameters can be found such that the theoretical predictions agree reasonably well with the experimental data on UPt3.

IT SEEMS NOT UNLIKELY that the unusual superconducting properties of the heavy fermion compounds UPt3 and UBe~3 are due to a novel type of superconducting pairing. In the absence of a microscopic theory of superconductivity for these materials, we can only hope to prove the presence of non-BCS type pairing by choosing some model pairing interaction and comparing the predictions derived from it with all the available experimental evidence.

This approach has been used already by several authors [1 -6 ] , assuming essentially the p-wave interaction known from the theory of superfluid 3He. In the absence of an external magnetic field [7], the self- consistency equation for the order parameter derived from the paring interaction has several solutions, so that one can specify various superconducting states with distinctive features. The ABM state with points of nodes and the polar state with lines of nodes have been studied most extensively, even though the polar state appears to be no solution of the self-consistency equation, when spin-orbit interaction and crystal field effects are included in the pairing interaction [8, 9].

Unfortunately, in between modelling the superconducting state and arriving at predictions for, say, transport coefficients, a number of physical assumptions and computational approximations have to be introduced.

Consider, for example, the electronic contribution to the attenuation of ultrasound. Before this can be calculated for any particular superconducting state, one ihas to assume some kind of interaction mechanism between sound wave and conduction electrons.

It has been argued that the interaction mechanism operating in heavy fermion materials is quite different from that in any other metal [10, 11]. The temperature dependence of the sound attenuation in the supercon-

ducting state resulting from this particular interaction mechanism has not yet been studied in any detail [12].

In the present investigation we have made the usual assumption that deformation potential coupling and collision drag are responsible for the damping of longitudinal sound by conduction electrons. At least within the NFE approximation both these coupling mechanisms represent simply scattering of electrons off moving ion cores [13]. In the case of transverse waves, only collision drag is taken into account. The electromagnetic interaction can be ignored because the experiments are done in the hydrodynamic regime ql ~ 1.

A microscopic calculation of the attenuation coefficient from the phonon self-energy [I 3] would have the benefit of elucidating the role of Migdal's theorem, which might not be applicable in heavy fermion metals, because the Fermi velocity is believed to be not much greater than the sound velocity.

However, it is much more economical to treat the displacement field of the sound wave classically and to use linear response theory to obtain the attenuation coefficient. The use of linear response theory precludes, of course, an explanation of the amplitude dependence of the attenuation coefficient observed in UPt3 [14].

The derivation of the well-known formulae for the attenuation coefficient is independent of the particular superconducting state under consideration. One finds, assuming complete screening [15],

~ , im/( [n, h] ) (q, ~ , ) aL = pV~ ~< [n, n] > (q, a~,) < [0rO, n] > (q, ~o,)

h

- < [ O r O , h] > (q, ~o,)~, (1) /

with

535

536 ATTENUATION OF ULTRASOUND IN p-WAVE SUPERCONDUCTORS Vol. 60, No. 6

The term in h involving the current density j is usually neglected because it is smaller than the leading term by a factor vL/vF. This approximation might break down in heavy fermion systems.

(Ye can be further simplified by the observation that the leading term in each correlation function is due to contributions from wave vectors far away from the Fermi surface, which are not affected by the superconducting transition. Linearizing with respect to the corrections to these leading terms gives [ 151 :

This presents a great simplification because now we only need to calculate the imaginary parts of the correlation functions. However, the transition from (1) to (2) also requires the assumption vL < v,.

For transverse waves with only collision drag as interaction mechanism one finds [ 161:

QT = - 2 Im ( [Crd, &q^] ) (q, cd,),

where 6 is the direction of polarization, The calculation of the correlation functions tar

superconducting states with unitary order parameters is straightforward. It proceeds along the same lines as in the case of BCS superconductors.

Electron-impurity scattering is taken into account self-consistently. This leads to a renormalization of the Matsubara frequencies o,, which makes it impossible to perform the sum over w, exactly, as in the pure limit [6]. Instead, we do the energy integral approximately using the residue theorem and assuming a constant density of states near the Fermi surface. This approximation can only be valid, if the width of the peak in the normal density-of-states near EF is much larger than T,. If only s-wave scattering is considered, order parameters representing E> 0 pairing are not renormalized. In addition to self-energies appearing in the single particle Green functions, vertex corrections due to electron- impurity scattering arise in the calculation of the correlation functions.

The resulting expressions for oL and cyT are rather lengthy when the sound frequency w, is retained every- where. If the effects of pair breaking by the sound wave were to be studied, the complete expressions would have

to be used. Since w, is much less than the order parameter amplitude we expect only a very small contribution to (YL,T from pair breaking and we therefore take

the limit o, + 0. With these approximations and limitations we obtain

after analytic continuation:

(YL = cY; (1 - 3(@ ” Q)Z }2I

with

/ =

i

1 + 1~21’ - lAWI 1cz2 - INK%” i

(5)

CX~ and a$ are the longitudinal and transverse normal state attentuation coefficients in the limit ql< 0 and F is the normal state scattering rate.

Except for the vertex corrections, Hirschfeld et al. [4] have used the same expressions taking the limit ql -+ 0 and specializing to propagation and polarization parallel and perpendicular to the symmetry axis of the order parameter.

One might think that the vertex corrections should be negligible in the hydrodynamic limit qZ+ 0. For an isotropic order parameter the integrand I is angle independent so that the vertex corrections do indeed vanish. However, for anisotropic order parameters this is no longer the case, even for ql = 0. In this limit, integration with respect to the azimuthal angle can be performed analytically. One can then see that in the case of longitudinal waves the vertex corrections make their largest contribution for 0, = 0 (propagation along the

symmetry axis of the order parameter). For 0e = 7r/2 the contribution is smaller by a factor l/4 and for cos 0,

Vol. 60, No. 6 ATTENUATION OF ULTRASOUND IN p-WAVE SUPERCONDUCTORS

1 5} O q r = 01nT c

,:,Ai

10

00[_ , 0q=095 O0 T/T c 05 10

Fig. 1. Attenuation of longitudinal sound in the ABM state for three different propagation directions. Electron- impurity scattering has been treated in the Born approximation 5N = 0.

= 1/X/~- (propagation along a (1 1 1) direction) it van- ishes completely. For transverse waves the vertex corrections vanish for propagation and polarization parallel or perpendicular to the symmetry axis of the order parameter. These are the cases considered by Hirschfeld et aL [4]. For other directions, however, vertex corrections are important.

When the electron self-energy and the vertex corrections are calculated within the Born approximation, one finds for the ABM state with ql = 0 the results shown in Fig. 1. For ql > O, at` does vanish for T--> 0, but for small ql the results shown in Fig. 1 are changed only at very low temperatures. Except for the strong anisotropy, this kind of temperature dependence of a L has been anticipated by Kadanoff and Pippard [17]. They expected a gradual transition to the clean limit attenuation because the mean free path of the thermally excited quasiparticles should increase as the temperature is lowered due to the reduction in phase space available for elastic scattering.

Since these theoretical predications are nowhere near the experimental data, one must ref'me the theory or conclude that the unusual superconducting properties of heavy fermion materials have nothing to do with p- wave superconductivity [2, 3].

Fortunately, Pethick and Pines [3] have noticed that the theoretical results will be drastically changed, when the electron self-energy is calculated in the t-matrix approximation with phase shifts 8N close to 7r/2. Argu- ments have been given why the phase shifts should be as large as that in heavy fermion metals [3 -5 ] .

For arbitrary phase shifts and s-wave scattering only, the renormalized frequency is given by

.~+ = 60+ + iF g(~+) c°s2 8N + S in2 t~Ng 2 (~+)" (7)

Here, only that part of the t-matrix has been kept which

537

is odd in w [4, 5]. g(~+) is an abbreviation for

dr2 c~+

The density-of-states resulting from (7) with ~n "~ 7r/2 is finite even at co = 0 for both ABM and polar state [4, 5, 8, 23]. Therefore, the phase space argument of Kadanoff and Pippard [17] does not apply and the quasiparticle mean free path remains finite even at T = 0.

Calculating the vertex corrections in a consistent fashion we find for the quantity C in (4) and (5):

c = ,4 + - ( 8 ) fj~2 , '

1 --A~-4-£I

with

A = cos28N + sin26N Ig(w+) 12 [ COS2 ~N -~- sin2 ~Ng 2 (~+)12 ,

2sin 8N COSSNImg(~ +) B = icos2~ N + sin2~Ng2(~.)l 2 . (9)

The integrand I ' differs from (6) in the sign of IA(p^)l in the numerator of the coherence factor.

If one wanted to obtain the temperature dependence of aL and aT analytically, one would have to use a number of rather doubtful approximations [3, 5]. One reason is that 1/Plmx/~2+ --IA(/0)L 2 is quite a compli- cated function of o~ and/~, even in the unitarity limit 8 s = rr/2, where it is supposed to be very close to its BCS or normal state value of 1.

Since in the self-consistent t-matrix approximation (7) ImP+ > 0 for all c~, one does not run into numerical difficulties even at the lowest temperatures. We have, therefore, evaluated at` and a r numerically with great accuracy. In order to obtain results near T e one has to also solve the self-consistency equation for the order parameter amplitude for each state and each degree of purity considered.

In addition to the Born approximation (SN = 0), for which results are displayed in Fig. 1, we performed calculations only in the unitarity limit. The results for the attenuation coefficients in this limit are shown in Figs. 2 and 3.

It has already been noted by Hirschfeld et al. [4], that one finds rather striking agreement between the experimental results for a T in UPt 3 [18] and the theoretical results for rather clean superconductors in the polar state with its axis parallel to the ~-axis (Fig. 2). Hirschfeld et al. concluded that at, for such a polar state was also in reasonable agreement with experiment.

However, including vertex corrections we fmd that

538 ATTENUATION OF ULTRASOUND IN p-WAVE SUPERCONDUCTORS Vol. 60, No. 6

0:7 a o .

05

0C

A r =3C, I~T c 6N= ~2 / ~I qL = 00

c " ' /~'S/" / / /

/

/

0 05 T/Tc 10

Fig. 2. Attenuation of transverse sound in the polar state (full curves) and the ABM state. Curves marked "]1" and "±" represent results for polarization directions # parallel and perpendicular to the symmetry axis / o f the order parameter, resp.. The propagation direction 4 is perpendicular to l. The dashed curves are obtained when the angle between 4 and [ is zr/4. If 0 is normal to the plane spanned by [ and q, the curve marked. "n" results. If 0 is rotated into the plane spanned by [ and q, curve "c" results.

aL(T ) is roughly linear over a wide temperature range for propagation both parallel and perpendicular to the polar axis (Fig. 3). For0q = 0, aL(T ) even has a negative curvature at intermediate temperatures.

This is in stark contrast to experiments [14, 1 9 - 21].

For 0 a = 7r/4 the calculated aL(T ) looks somewhat like the experimental data. To make this comparison more quantitative we tried to represent the calculated curve by a power law. A simple power law a oc T y fits any of our curves only at very low temperatures. A very good representation over quite a wide temperature range can be achieved using expressions of the form

a/a ° = A ( T / T c - - B ) v + C. (10)

For 0q = Tr/4 we find that A = 0.88, B = 0.2, C = 0.035 and 7 = 1.35 gives a good fit for 0.2 < TIT c < 0.7. This exponent 7 is still rather small compared to the ones observed experimentally but, most of all, it is unlikely that the experiments were done in this geometry.

Assuming a larger scattering rate P brings the theoretical predictions for u z more in line with experiment by increasing the (positive) curvature. For r = 0.1 7rTc there is a substantial electronic attenuation at T = 0. Since in the processing of experimental data this would probably be subtracted off together with the residual attenuation due to any other source, we rescaled our curve for 0q = 0 such that a L ( T = 0 ) = 0. Fitting the resulting curve for 0.2 < TIT c < 0.8 by an expression of the form (10), we find 7 = 1.33. Without the rescaling we would have 7 = 1.28.

We see that even for this rather large scattering rate,

[ = 001~Tc. 6N= n;2 . " c~ q[ = 00 / " ,

' 6 N ~ 0 0 "'/ rrt2 /~

0 / " O0 ,,"

i ./ ~lZ~ ." "

O0 05 T/Tc 10

Fig. 3. Attenuation of longitudinal sound in the polar state (full curves) and the ABM state (dashed curves) for different directions 4 of sound propagation relative to the symmetry axis l of the order parameter.

which causes a noticable depression of Tc(ATclT°e = --0.25) , we cannot reproduce the T 2 or T a behavior observed experimentally [14, 1 9 - 2 1 ] .

In the case of a polar state, where quasiparticles with/7 • 4 ~- 0 contribute most strongly to the attenuation, the introduction of a ffmite ql has very little effect for propagation parallel to the polar axis. In fact, the results for 0 a = 0 shown in Fig. 3 are somewhat increased in the presence of finite ql. For propagation perpendicular to the polar axis aL drops more rapidly just below Te when finite values of ql are taken into account. I f a clean material is considered (P = 0.0DrTc), az is again linear for 0.2 < TIT c < 0.8, but with a smaller slope than for ql = 0. Propagation at intermediate directions is hardly affected at all. In order to find a noticable change in aL(O q = zr/2) (~--15%), one has to choose ql as large as 0.5. Taking ql = 0.035, which is a more realistic value for dean UPta [20], we find essentially no change from the results obtained for ql = O.

From the results obtained in the Born approximation we infer that the theoretical prediction for a z and a T will increase, when phase shifts ~N < rr/2 are assumed.

We must, therefore, conclude that the polar state does not provide a consistent picture for the sound attenuation in UPt 3 and UBe13, no matter how the scattering rate F, the phase shifts 8n , the propagation direction 0q, or ql are varied.

The ABM state yields results for aT shown in Fig. 2, which are in complete disagreement with experiment, unless it is assumed that the symmetry axis of the order parameter is in the basal plane of UPt3. For propagation along the symmetry axis one does not obtain the desired linear behavior, but if 4 and # form an angle of rr/4 with the axis of the order parameter an approximately linear temperature dependence is found. When # is rotated by

Vol. 60, No. 6 ATTENUATION OF ULTRASOUND IN p-WAVE SUPERCONDUCTIONS 539

rr[2, the resulting a T can be fitted in the form (10) with 3' = 1.5. The difference in a T for the two polarizations is largely due to vertex corrections.

Assuming a somewhat larger scattering rate F would increase the exponent 3' and thus improve the agreement with experiment. A larger exponent 3' would probably also result if the inelastic scattering present in the normal state [18] were taken into account properly [22].

For longitudinal waves one also finds reasonable agreement between theory and experiments, which have been done with ~ II 0, if it is assumed that the axis of the ABM state is in the basal plane so that 0q = rr/2. In this case we predict a very large anisotropy, which is reduced if either P or ql is increased (Fig. 3).

ABM - like states with their axes in the basal plane (perpendicular to the propagation direction of longitudinal sound), which belong to the E, representation, are allowed by symmetry [23] and seem to provide a consistent picture of the sound attenuation data in UPt3 and UBela available at present. The attenuation peaks observed just below Tc [14, 21] require for their explanation novel interaction mechanisms [12, 24] not considered in this paper.

Acknowledgements - One of the authors (K.S.) grate- fully acknowledges useful discussions with Drs. V. Miiller, P. Hirschfeld, D. Einzel and Prof. G. Eilenberger.

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Attenuation of ultrasound in p-wave superconductors