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Ultrasonic attenuation in clean anisotropic superconductors
Physica 135B (1985) 27-29 North-Holland, Amsterdam
ULTRASONIC ATTENUATION IN CLEAN ANISOTROPIC SUPERCONDUCTORS S.N. C O P P E R S M I T H
Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA R.A. KLEMM
Exxon Research and Engineering Co., Annandale, NJ 08801, USA Recently, the ultrasonic attenuation of UPt 3 in the hydrodynamic regime has been found to be proportional to T 2 at low temperatures T. In the opposite (clean) limit, ql>>1, we find that the ultrasonic attenuation for an anisotropic superconductor has a temperature dependence that depends not only on the nodal structure of the gap but also on the relative orientation of the sound wave and the nodes. Both power law and exponential temperature dependences nodes, depending upon the sound wave orientation relative to the nodes. In some directions, finite experimental resolution must be taken into account.
Recent experiments [1] on UPt 3 have indicated that the ultrasonic attenuation a s in the superconducting phase behaves as T 2 at low temperatures T. Above T c, these experiments were in the hydrodynamic regime in which the longitudinal wavelength 2 r r / q of the sound is much larger than the electronic mean free path I. The results were interpreted [1] as evidence for an anisotropic state of the polar type; i.e., with lines of zeroes of the gap on the Fermi surface. However, the same results have also been interpreted as evidence for an axial-like state in which the gap vanishes at points on the Fermi surface [2]. Regardless of the resolution of this controversy, the parity of the pair state remains to be determined definitively. H e r e we examine the temperature dependence of % in the limit that all impurity scattering can be neglected (ql >> 1). This limit can be examined by performing the experiments with cleaner samples or at higher frequencies than have been used. Even for "dirty" samples, at low temperatures an anisotropic gap reduces the quasiparticle density of states and may cause I to rise (via Fermi's golden rule). The advantage in choosing this limit is that theoretical complications arising from electronic impurity scattering do not play a role. The calculations involve straightforward extensions of the well-known BCS results [3]. We find that in this non-hydrodynamic limit, strong anisotropies of a s
are expected for both axial- and polar-like states. A polar-like state can have either an anisotropic linear temperature dependence or can exhibit both linear and exponential behavior, depending upon the sound direction q. An axial-like state has an a~ that can be either quadratic or exponential in temperature, depending upon q. Our starting point is eq. (64) of Balian and Werthamer [4]. This equation describes a s for a lattice anisotropy included. However, we do not expect these complications to affect the asymptotic temperature dependence, which is determined by the density of states. They found that the longitudinal a s for a sound wave of wavevector q and frequency OJq = v~lql obeys 2
e~ 2 ~f
OlsOC(-Oq~ k (~kk) - ~ k ~ ( E k + q - Ek-hC.oq)
(1)
for small q, where e k is the normal state energy relative to the Fermi energy, Ak is the anisotropic gap, E~ = e~ + A~ is the square of the quasiparticle energy in the superconducting state, and f ( E ) is the Fermi function. In this clean limit, conservation of m o m e n t u m (k' = k + q) and conservation of energy ( E k = E k, - h t o q ) place severe restrictions on the allowed values of m o m e n t u m k for the quasiparticles that are scattered by the sound wave. For instance, consider the "light fermion"
0378-4363 / 85 / $03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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S, N. Coppersmith and R. A. Klemm / Ultrasonic attenuation in clean anisotropic superconductors
limit Vs/Vv ~ 1, so that Wq can be set equal to zero in the delta functions. In this case, the delta function in eq. (1) requires q . k = 0. (We assume A o ~ E v, for simplicity*). Thus, the experiment only samples states in the plane with k normal to q. In real "heavy-fermion" systems similar considerations hold, and though k and q are no longer perpendicular, they are forced to be at a fixed relative angle. These arguments (which are valid for both normal and superconducting states) have been used to understand the temperature dependence and the anisotropy of a s of tin [5]. Let us first consider the possible p-wave states, assuming vs/vv~ 1, The isotropic, or BalianWerthamer, (BW) state was shown previously [4] to give an isotropic a s with an exponential asymptotic temperature dependence, identical to that for the BCS state. In the polar state, IAkl 2 = A~(T) cos2(0), where 0 is the angle k makes with the z axis (see fig. 1). Note that [zlk] vanishes everywhere in the x - y plane. Hence, if q[[z, it is possible that in the measurement the gap is sampled only where it vanishes, as shown in fig. 2a, and a s equals its normal state value a n. A sampled "slice" of a generic case with q- z f 0 is shown in fig. 2b. The gap vanishes along two points of the slice, resulting in an a s z T. When v~ is not negligibly small, the sampled region of the
Fig. I. Polar state quasi-particle energy surface. * We a s s u m e A o ( T ) / E v ~ 1 in these calculations. For T~ ~ T v, this is not true and very interesting complications arise. This will be discussed elsewhere.
Fermi surface may have no points of vanishing gap, as shown in fig. 2c. For the case in which v~ can be neglected and qllz, a finite beamwidth results in an average over the finite solid angle with width 00, yielding as~T/O o. Hence, the magnitude of the linear temperature dependence should be highly anisotropic. For an axial state, [Ak[2 = A2o(T) sin2(0), which vanishes on isolated points on the Fermi surface, as shown in fig. 3. For v J r v ~ 1, if the sound is directed normal to the plane in which the nodes exist, an infinitely narrow beam would result in a linear temperature dependence of a s. A finite beam resolution yields a T2/Oo behavior for a~. For any other sound direction, a s would be exponential in temperature, as the gap would be everywhere non-vanishing, as shown in fig. 2c. Hence, we expect a highly anisotropic a s for an axial state, varying between exponential and quadratic temperature dependences, depending upon the sound direction. In the case of a d-wave superconductor, we note that one example of a polar-like state is shown in fig. 4. For this state, there are two lines of nodes of zlk on the Fermi surface. For a "light fermion" material, two possibilities arise, as shown in fig. 5. For 0 less than a critical value, the sampled cross-section of the gap never vanishes, as shown in fig. 5a. For larger 0 values, the gap vanishes at four points on the cross-section, as shown in fig. 5b. These cases result in exponential and linear temperature dependences of as, respectively. In this paper we have ignored the effects of possible domain structures, which would cause the gap to vary spatially. If many domains are present with random orientations, a s would involve the average of all possible slices of the quasiparticle energy surface. This would result in a linear temperature dependence for a polar-like state and a quadratic temperature dependence for an axial-like state, respectively. This temperature dependence of a S is in agreement with that predicted to occur in the hydrodynamic limit by Rodriguez [2]. We emphasize, however, that if only one domain is present, our results predict a highly anisotropic ultrasonic attenuation: for certain polar-like states, a~ varies between exponen-
S.N. Coppersmith and R . A . Klemm / Ultrasonic attenuation in clean anisotropic superconductors
29
0 (a}
(b)
(c)
Fig. 2. Cross-section of the quasi-particle energy in which the gap vanishes (a) everywhere; (b) at two points; and (c) nowhere.
"\.._._.j (a)
(b)
Fig. 5. Cross-sections of a d-wave polar state energy surface in which the gap (a) never vanishes (b) vanishes at four points.
Fig. 3. Axial state quasi-particle energy surface.
Acknowledgements We would like to acknowledge useful discussions with Anil Khurana, Ad Pruisken, Mark Robbins, and Clare Yu. Work at Brookhaven was supported by the Division of Materials Sciences, USDOE under contract no. DE-AC0276CH00016.
References
Fig. 4. Quasi-particle energy surface for a polar-like state with d-wave symmetry.
tial and linear temperature dependences, while for axial states, a~ varies between exponential and quadratic temperature dependences, depending upon the direction of q.
[1] D.J. Bishop, C.M. Varma, B. Batlogg and E. Bucher, Phys. Rev. Lett. 53 (1984) 1009. [2] J. Rodriguez, Phys. Rev. Lett. 55 (1985) 250. [3] See for example G. Rickayzen, in: Superconductivity, R. Parks, ed. (Marcel Dekker, New York, 1969) p. 100. [4] R. Balian and N.R. Werthamer, Phys. Rev. 131 (1963) 1553. [5] R.W. Morse, T. Olsen and J.D. Gavendra, Phys. Rev. Lett. 3 (1959) 15. L.T. Claiborne and N.G. Einspruch, Phys. Rev. Lett. 15 (1965) 862.
ULTRASONIC ATTENUATION IN CLEAN ANISOTROPIC SUPERCONDUCTORS S.N. C O P P E R S M I T H
Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA R.A. KLEMM
Exxon Research and Engineering Co., Annandale, NJ 08801, USA Recently, the ultrasonic attenuation of UPt 3 in the hydrodynamic regime has been found to be proportional to T 2 at low temperatures T. In the opposite (clean) limit, ql>>1, we find that the ultrasonic attenuation for an anisotropic superconductor has a temperature dependence that depends not only on the nodal structure of the gap but also on the relative orientation of the sound wave and the nodes. Both power law and exponential temperature dependences nodes, depending upon the sound wave orientation relative to the nodes. In some directions, finite experimental resolution must be taken into account.
Recent experiments [1] on UPt 3 have indicated that the ultrasonic attenuation a s in the superconducting phase behaves as T 2 at low temperatures T. Above T c, these experiments were in the hydrodynamic regime in which the longitudinal wavelength 2 r r / q of the sound is much larger than the electronic mean free path I. The results were interpreted [1] as evidence for an anisotropic state of the polar type; i.e., with lines of zeroes of the gap on the Fermi surface. However, the same results have also been interpreted as evidence for an axial-like state in which the gap vanishes at points on the Fermi surface [2]. Regardless of the resolution of this controversy, the parity of the pair state remains to be determined definitively. H e r e we examine the temperature dependence of % in the limit that all impurity scattering can be neglected (ql >> 1). This limit can be examined by performing the experiments with cleaner samples or at higher frequencies than have been used. Even for "dirty" samples, at low temperatures an anisotropic gap reduces the quasiparticle density of states and may cause I to rise (via Fermi's golden rule). The advantage in choosing this limit is that theoretical complications arising from electronic impurity scattering do not play a role. The calculations involve straightforward extensions of the well-known BCS results [3]. We find that in this non-hydrodynamic limit, strong anisotropies of a s
are expected for both axial- and polar-like states. A polar-like state can have either an anisotropic linear temperature dependence or can exhibit both linear and exponential behavior, depending upon the sound direction q. An axial-like state has an a~ that can be either quadratic or exponential in temperature, depending upon q. Our starting point is eq. (64) of Balian and Werthamer [4]. This equation describes a s for a lattice anisotropy included. However, we do not expect these complications to affect the asymptotic temperature dependence, which is determined by the density of states. They found that the longitudinal a s for a sound wave of wavevector q and frequency OJq = v~lql obeys 2
e~ 2 ~f
OlsOC(-Oq~ k (~kk) - ~ k ~ ( E k + q - Ek-hC.oq)
(1)
for small q, where e k is the normal state energy relative to the Fermi energy, Ak is the anisotropic gap, E~ = e~ + A~ is the square of the quasiparticle energy in the superconducting state, and f ( E ) is the Fermi function. In this clean limit, conservation of m o m e n t u m (k' = k + q) and conservation of energy ( E k = E k, - h t o q ) place severe restrictions on the allowed values of m o m e n t u m k for the quasiparticles that are scattered by the sound wave. For instance, consider the "light fermion"
0378-4363 / 85 / $03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
28
S, N. Coppersmith and R. A. Klemm / Ultrasonic attenuation in clean anisotropic superconductors
limit Vs/Vv ~ 1, so that Wq can be set equal to zero in the delta functions. In this case, the delta function in eq. (1) requires q . k = 0. (We assume A o ~ E v, for simplicity*). Thus, the experiment only samples states in the plane with k normal to q. In real "heavy-fermion" systems similar considerations hold, and though k and q are no longer perpendicular, they are forced to be at a fixed relative angle. These arguments (which are valid for both normal and superconducting states) have been used to understand the temperature dependence and the anisotropy of a s of tin [5]. Let us first consider the possible p-wave states, assuming vs/vv~ 1, The isotropic, or BalianWerthamer, (BW) state was shown previously [4] to give an isotropic a s with an exponential asymptotic temperature dependence, identical to that for the BCS state. In the polar state, IAkl 2 = A~(T) cos2(0), where 0 is the angle k makes with the z axis (see fig. 1). Note that [zlk] vanishes everywhere in the x - y plane. Hence, if q[[z, it is possible that in the measurement the gap is sampled only where it vanishes, as shown in fig. 2a, and a s equals its normal state value a n. A sampled "slice" of a generic case with q- z f 0 is shown in fig. 2b. The gap vanishes along two points of the slice, resulting in an a s z T. When v~ is not negligibly small, the sampled region of the
Fig. I. Polar state quasi-particle energy surface. * We a s s u m e A o ( T ) / E v ~ 1 in these calculations. For T~ ~ T v, this is not true and very interesting complications arise. This will be discussed elsewhere.
Fermi surface may have no points of vanishing gap, as shown in fig. 2c. For the case in which v~ can be neglected and qllz, a finite beamwidth results in an average over the finite solid angle with width 00, yielding as~T/O o. Hence, the magnitude of the linear temperature dependence should be highly anisotropic. For an axial state, [Ak[2 = A2o(T) sin2(0), which vanishes on isolated points on the Fermi surface, as shown in fig. 3. For v J r v ~ 1, if the sound is directed normal to the plane in which the nodes exist, an infinitely narrow beam would result in a linear temperature dependence of a s. A finite beam resolution yields a T2/Oo behavior for a~. For any other sound direction, a s would be exponential in temperature, as the gap would be everywhere non-vanishing, as shown in fig. 2c. Hence, we expect a highly anisotropic a s for an axial state, varying between exponential and quadratic temperature dependences, depending upon the sound direction. In the case of a d-wave superconductor, we note that one example of a polar-like state is shown in fig. 4. For this state, there are two lines of nodes of zlk on the Fermi surface. For a "light fermion" material, two possibilities arise, as shown in fig. 5. For 0 less than a critical value, the sampled cross-section of the gap never vanishes, as shown in fig. 5a. For larger 0 values, the gap vanishes at four points on the cross-section, as shown in fig. 5b. These cases result in exponential and linear temperature dependences of as, respectively. In this paper we have ignored the effects of possible domain structures, which would cause the gap to vary spatially. If many domains are present with random orientations, a s would involve the average of all possible slices of the quasiparticle energy surface. This would result in a linear temperature dependence for a polar-like state and a quadratic temperature dependence for an axial-like state, respectively. This temperature dependence of a S is in agreement with that predicted to occur in the hydrodynamic limit by Rodriguez [2]. We emphasize, however, that if only one domain is present, our results predict a highly anisotropic ultrasonic attenuation: for certain polar-like states, a~ varies between exponen-
S.N. Coppersmith and R . A . Klemm / Ultrasonic attenuation in clean anisotropic superconductors
29
0 (a}
(b)
(c)
Fig. 2. Cross-section of the quasi-particle energy in which the gap vanishes (a) everywhere; (b) at two points; and (c) nowhere.
"\.._._.j (a)
(b)
Fig. 5. Cross-sections of a d-wave polar state energy surface in which the gap (a) never vanishes (b) vanishes at four points.
Fig. 3. Axial state quasi-particle energy surface.
Acknowledgements We would like to acknowledge useful discussions with Anil Khurana, Ad Pruisken, Mark Robbins, and Clare Yu. Work at Brookhaven was supported by the Division of Materials Sciences, USDOE under contract no. DE-AC0276CH00016.
References
Fig. 4. Quasi-particle energy surface for a polar-like state with d-wave symmetry.
tial and linear temperature dependences, while for axial states, a~ varies between exponential and quadratic temperature dependences, depending upon the direction of q.
[1] D.J. Bishop, C.M. Varma, B. Batlogg and E. Bucher, Phys. Rev. Lett. 53 (1984) 1009. [2] J. Rodriguez, Phys. Rev. Lett. 55 (1985) 250. [3] See for example G. Rickayzen, in: Superconductivity, R. Parks, ed. (Marcel Dekker, New York, 1969) p. 100. [4] R. Balian and N.R. Werthamer, Phys. Rev. 131 (1963) 1553. [5] R.W. Morse, T. Olsen and J.D. Gavendra, Phys. Rev. Lett. 3 (1959) 15. L.T. Claiborne and N.G. Einspruch, Phys. Rev. Lett. 15 (1965) 862.