Ehrenfest Relations for Ultrasound Absorption in Sr 2 RuO 4

Transcription

1 917 Progress of Theoretical Physics, Vol. 107, No. 5, May 2002 Ehrenfest Relations for Ultrasound Absorption in Sr 2 RuO 4 Manfred Sigrist Theoretische Physik, ETH-Hönggerberg, 8093 Zürich, Switzerland (Received January 18, 2002 We examine the Ehrenfest relations for the elastic constants of the spin-triplet superconducting state in Sr 2 RuO 4. It is shown on a phenomenological level that the relations for different sound modes are not independent. This result may be used to analyze experimental data of sound velocity renormalization in the superconducting state. Finally, the relation between couplings of the order parameter and uniaxial strain is discussed from a microscopic point of view for the particular band structure of Sr 2 RuO 4. Much effort has been invested in identifying the unconventional superconducting state of Sr 2 RuO 4. It is widely accepted now that chiral p-wave pairing with the basic gap structure d(k =ẑ(k x ±ik y is realized in this compound. 1, 2 This state breaks time reversal symmetry and possesses a two-dimensional complex order parameter, belonging to the representation E u of the tetragonal crystal point group D 4h,ifwe assume strong spin-orbit coupling. 3 The multi-dimensional order parameter gives rise to a number of low-energy collective modes that are related to those in superfluid 3 He. 4 6 However, in contrast to the superfluid, in Sr 2 RuO 4 the crystal field and spin-orbit coupling lead to gaps in these modes, so that they may lead to observable effects only close to the onset of superconductivity. In this paper, we would like to examine the effect of the order parameter modes of the chiral p-wave phase on the ultrasound absorption and, in particular, the renormalization of the elasticity constants at the onset of superconductivity. Recently, various experimental data have been published showing strong renormalization effects It is the aim of this paper to discuss various Ehrenfest relations that allow us to analyze the data for consistency with the complex order parameter of the chiral p-wave state. For a qualitative description of the order parameter dynamics close to we use a Landau-Khalatnikov-type of approach based on the Ginzburg-Landau theory of the superconducting state and the basic coupling between the order parameter and the lattice strain. Close to the onset of superconductivity, we may ignore the multi-band structure of Sr 2 RuO 4 and describe the superconducting phase by a single set of order parameters d(k =ẑ(η x k Fx + η y k Fy, reflecting the basic symmetry of the condensate. The Ginzburg-Landau free energy density, ignoring gradient terms, has the well-known form f s = a η 2 + β 1 η 4 + β 2 2 (η 2 x η 2 y + η 2 xη 2 y +β 3 η x 2 η y 2, (1 where a = a (T and β i are phenomenological coefficients. The elastic energy density is f el = 1 2 [c 11(ɛ 2 xx + ɛ 2 yy+c 33 ɛ 2 zz +2c 12 ɛ xx ɛ yy +4c 66 ɛ 2 xy

2 918 M. Sigrist +2c 13 (ɛ xx + ɛ yy ɛ zz +4c 44 (ɛ 2 xz + ɛ 2 yz], (2 where the coefficients c ij represent the six independent elastic constants of a tetragonal lattice, and ɛ ij = 1 2 (( u i/ x j +( u j / x i is the strain with u as the local displacement vector. The terms describing the coupling between strain and order parameter have the general form f c ={r 1 (ɛ xx + ɛ yy +r 2 ɛ zz } η 2 + r 3 ɛ xy (ηxη y + η x ηy + r 4 (ɛ xx ɛ yy ( η x 2 η y 2, (3 where the r i are coupling constants which are related to the pressure dependence of. 7 9 For convenience we use from now on the abbreviations ɛ 1 = ɛ xx + ɛ yy, ɛ 2 = ɛ zz, ɛ 4 = ɛ xx ɛ yy, ɛ 3 = ɛ xy (4 for the basic strain components appearing in the coupling terms. The second-order coupling (connected with the second derivative of with respect to pressure also plays an important role and can be incorporated by replacing the elastic constants in Eq. (2 by c ij c ij = c ij (1+λ ij η 2, (5 where λ ij denotes a further set of coupling constants. The total free energy density f is the sum of the three terms Eqs. (1 (3, including these c ij. We discuss microscopic aspects of the coupling constants r j and λ ij below. The dynamics of both the displacement vector u and the order parameter η are described by the equations of motion ρ 2 u t 2 = j ( x j f ( u/ x j and η t = 1 f τ 0 η, (6 where ρ is the density of the mass and τ 0 is a phenomenological relaxation time, which we assume to be a constant close to. Here, we use the standard Landau- Khalatnikov formulation of overdamped dynamics of the order parameter. We ignore the spatial modulation of the order parameter, assuming sound wavelengths much longer than the coherence length. A sound wave is described by the plane wave form u(r,t = ũ exp(ik r iωt. There are three normal modes of the superconducting order parameter that couple to the strain. From the parameterization (η x,η y = η(cos θ, e iγ sin θ, we find fluctuations of the modulus η, the angle θ and the relative phase γ around their equilibrium values: η = η 0 + η, θ = θ 0 + θ, γ = γ 0 + γ (7 with η 2 0 = 2a/(4β 1 β 2 + β 3, θ 0 = π/4 and γ 0 = ±π/2. Obviously, θ is the fluctuation of the relative magnitude of the two order parameter components and the mode γ is equivalent to the clapping mode of 3 He. Linearizing the equation of motion in these fluctuations, we find a set of equations that we can solve for different sound modes.

3 Ehrenfest Relations for Ultrasound Absorption in Sr 2 RuO Now, choosing a sound wave, characterized by its propagation direction k and polarization ũ, we can solve this set of equations and determine the renormalized k -vector of the sound propagation. We consider first the case of a longitudinal sound wave along the [100]-direction, i.e. k =(k, 0, 0 and ũ =(ũ, 0, 0. As a result we obtain for the renormalized wavevector, k = ω r1 [ ṽ 1 c 11 (4β 1 β 2 + β 3 1 iωτ 1 r 2 ] 3 1 +, (8 2c 11 (β 2 β 3 1 iωτ 2 where ṽ 1 = v 1 (1+λ 11 η 2 and v 1 = c 11 /ρ is the sound velocity in the normal state. The effective relaxation times are τ 1 = τ 0 4a and τ 2 = τ 0(4β 1 β 2 + β 3. (9 8a(β 2 β 3 The renormalized sound velocity v and the absorption coefficient α are defined by v = respectively, which leads to ω Re k and α = Im k, (10 δv = λ 11 η 2 r1 2 1 v 1 c 11 (4β 1 β 2 + β 3 1+ω 2 τ1 2 r c 11 (β 2 β 3 1+ω 2 τ2 2, (11 α = 1 [ r1 2 ω 2 τ 1 c 11 v 1 4β 1 β 2 + β 3 1+ω 2 τ1 2 r4 2 ω 2 ] τ 2 + 2c 11 (β 2 β 3 1+ω 2 τ2 2. (12 The sound velocity experiences a renormalization due to the coupling to the order parameter modes, the fluctuations of the modulus and of the angle θ. These corrections lead to a jump-like feature in v immediately below. In addition, there is a change in the temperature derivative v due to the renormalization of the elastic constant. Which feature is the dominant one in the sound velocity renormalization is a matter of microscopic details and cannot be decided within our approach. Moreover, we find that the coupling to the order parameter leads to a non-resonant absorption close to. The temperature dependence of α suggests the presence of an absorption peak. However, we ignore absorption for the moment, and comment on it below. In correspondence with the usual presentation experimental sound absorption data we analyze now the behavior of the elastic constants c ij rather than the velocities. The renormalization of the elastic constants (equivalent to the renormalization

4 920 M. Sigrist of the sound velocities acquires the form of a discontinuity at in the limit ω 0 limit, given by 2r 2 1 c 11 = 2r2 4. (13 4β 1 β 2 + β 3 β 2 β 3 We would like now to relate the right-hand side to measurable quantities. First, the coupling constants r j are connected with the change of by the variation of the strain, = r 1 ɛ 1 a and = r 4 ɛ 4 a, (14 where the first relation corresponds to the volume change induced by in-plane strain, and the second is a volume-conserving uniaxial strain that leads to a splitting of the transition temperature into a higher and lower branch in the two-dimensional order parameter space. 18 This is the basis for the splitting of the phase transition under symmetry-lowering uniaxial strains and applies also to ɛ 3. Note that the uniaxial strain ɛ 2 preserves the symmetry. Furthermore, the specific heat jump at is given by C 2a 2 =. (15 4β 1 β 2 + β 3 These quantities can be used to write the Ehrenfest relation c 11 = C [ (c 2 ( ] 2 c 4β 1 β 2 + β 3 +. (16 ɛ 1 ɛ 4 β 2 β 3 Note that the factor (4β 1 β 2 + β 3 /(β 2 β 3 is of order 1, and exactly equal to 2 in the case of a weak-coupling approach, assuming a single cylindrical Fermi surface. Next, we turn to the transverse sound mode with k =(k, 0, 0. Among the two possible modes, we first consider the in-plane mode T1 with ũ =(0, ũ, 0. Here, the renormalization of the corresponding elastic constant leads to the relation c 66 = C ( c ɛ 3 2 4β 1 β 2 + β 3 8β 2, (17 where (4β 1 β 2 +β 3 /8β 2 is 1/2 in the weak-coupling case. The transverse mode with polarization along the z-axis does not couple to any of the three order parameter modes, and therefore no discontinuity in the elastic constants is expected. Now we consider sound propagation along [1,1,0] and obtain ĉ L = C [ (c 2 ( ] 2 c 4β 1 β 2 + β 3 + ɛ 1 ɛ 3 8β 2 (18 for the longitudinal mode, where ĉ L =(c 11 +c 12 +2c 66 /2. For the in-plane transverse mode, the corresponding elastic constant ĉ T1 =(c 11 c 12 /2 shows the jump ĉ T1 = C [ (c ] 2 4β 1 β 2 + β 3. (19 ɛ 4 β 2 β 3

5 Ehrenfest Relations for Ultrasound Absorption in Sr 2 RuO Table I. Table of first-order couplings of ultrasound to the different modes of the order parameter of the chiral p-wave state. Here η represents the modulus, θ the orientation and γ the relative phase of the two-component order parameter. denotes a finite coupling and the absence of coupling to the corresponding order parameter mode. k polarization η θ γ c ij [100] L c 11 T1 c 66 T2 c 44 [110] L (c 11 + c 12 +2c 66 /2 T1 (c 11 c 12 /2 T2 c 44 [001] L c 33 T c 44 Again, the z-axis polarized mode has no coupling. All discontinuities are connected with specific modes of the order parameter. The coupling of the order parameter modulus to longitudinal sound modes and the resulting discontinuity of the corresponding elastic constants is a standard behavior of any (conventional superconductor. The discontinuity in c 66 and ĉ T1, however, is a unique result of the presence of a multi-component pairing state, as non-trivial collective modes of the order parameter couple to the ultrasound (Table I. It is interesting that the clapping mode, associated with the relative phase γ, couples also to longitudinal modes for the tetragonal system for certain directions. The clapping mode has been analyzed also using a cylindrical symmetric microscopic model by various groups who have come to conflicting conclusions about the coupling to longitudinal sound waves. 4 6 While several unknown parameters enter the different Ehrenfest relations, there is a simple relation between some of them: c 11 ĉ T1 = ĉ L c 66 = C ( 2 c, (20 where the first equality is between the two combinations of jumps in c ij and the second is expressed in terms of quantities that are determined by different measurements. Finally, the longitudinal mode along the z-axis yields c 33 = C ɛ 1 ( 2 c, (21 ɛ 2 which is independent of the other Ehrenfest relations. Now, we consider the second-order effect due to the coupling in Eq. (5, which leads to a discontinuity of the derivative of c ij with respect to the temperature at. Here, the couplings λ ij are related to the second derivative of with respect to the different strains. A straightforward calculation yields the differences in slope of c ij above and below ( c11 = C ( 2 ɛ ɛ 2, (22 4

6 922 M. Sigrist ( c66 ( ĉl ( ĉt1 = C = C 2, (23 ɛ 2 3 ( 2 ɛ ɛ 2, (24 3 = C 2 ɛ 2. (25 4 Obviously, a relation similar to that in Eq. (20 exists between these quantities, ( c11 ( ĉt1 ( ĉl = = C ( c66 2 ɛ 2, (26 1 Both relations, which again can be expressed in terms of measurable quantities. Eqs. (20 and (26, provide a test for the analysis of experimental data. We now consider the T2-modes of in-plane propagating modes with ũ =(0, 0, ũ. In this case, the elastic constants do not exhibit a discontinuity at. The relevant component c 44 behaves as ( c44 = C 2 ɛ 2, (27 5 where ɛ 5 = ɛ zx cos φ+ɛ yz sin φ, with φ the direction angle of the in-plane propagation vector. Note that ɛ 5 is invariant under rotations around the z-axis. A correction to this result is caused by spin-orbit coupling. For this set of strains, a coupling of the chiral p-wave state to other p-wave states is possible. These states have a lower bare than the chiral p-wave state, due to spin-orbit coupling. To be concrete we assume that the state d = η(ˆxk x ŷk y has the next highest. The corresponding coupling term with ɛ 5 is f so = r so [ η (ɛ zx η x ɛ zy η y +c.c.]. (28 The free energy for η is f =ã(t η 2 +. From this coupling, we obtain a correction to Eq. (27 of the form ( c44 = C 4rso 2 ãa. (29 It is rather difficult to give an estimate of the magnitude of this correction. Interestingly, the modes induced by this transverse ultrasound mode correspond to fluctuations in the spin direction of the Cooper pair state. This mode is not soft, even very close to the onset of superconductivity, due to the lower transition temperature of η. We now turn briefly to the thermal expansion coefficient, which exhibits a discontinuity at the superconducting phase transition as well. We define α = ɛ 1 / for the in-plane expansion and α = ɛ 2 / for the z-axis expansion. The corresponding relations for the discontinuities α( are α = C and α = C. (30 c 11 ɛ 1 c 33 ɛ 2

7 Ehrenfest Relations for Ultrasound Absorption in Sr 2 RuO This provides an independent determination of the strain dependence of, including sign (see below. While our phenomenological discussion provides various relations among measurable quantities, it does not allow us to obtain a relation among the different coupling constants, such as r j and λ ij. They are determined by the details of the electronic properties. Within the weak-coupling approach, these constants are determined from the response of the condensation energy to the deformation of the Fermi surface. The influence of weak strain can be incorporated qualitatively into the electronic band structure via ε(k,ɛ ij =ε(k+ i,j Λ ij k Fi k Fj ɛ ij, (31 where the Λ ij are coupling constants that we assume to be all of similar magnitude. In a single-band theory, this may be incorporated as a correction to the second-order term in the Ginzburg-Landau theory. The coupling of the order parameter to the strain of first order involves the Fermi surface average of the form N (k F k Fµ k Fν k Fi k Fj FS ɛ ij (η µη ν +c.c., (32 where N (k F denotes the derivative of the density of states N(k F with respect to the chemical potential at the Fermi surface. Analogously, the second-order coupling has the form N(k F k Fµ k Fν k Fi k Fj k Fi k Fj FS ɛ ij ɛ i j (η µη ν +c.c.. (33 Sr 2 RuO 4 has three bands that contribute to its metallic and superconducting state. There is evidence from experimental and theoretical studies that there is one band that is dominant for superconductivity in Sr 2 RuO 4, the so-called γ-band derived from the 4d-t 2g orbital d xy of Ru The band structure calculations show that the Fermi surface of the γ-band approaches van Hove singularities along the [100]- directions. Hence, the density of states and the particle-hole asymmetry (related to N are large for these directions. Thus, the coupling between the order parameter and strain should be dominated by the properties of this band. Based on Eqs. (32 and (33, we can therefore state the following qualitative relations: ɛ < 3 ɛ 4 and 2 ɛ 2 3 < 2 ɛ 2 4 ; (34 i.e. the renormalization for c T1 is larger than for c 66 at both orders. These inequalities are apparently satisfied in the results of recent experiments by Okuda et al. 13 and can be added to the evidence for the dominance of the γ-band. Furthermore, we can conclude that the observation of a splitting of the superconducting transition under in-plane uniaxial stress is more likely to be observed for the ɛ 4 -strain. 18, A further point to test experimentally is related to the discontinuity of the thermal expansion, α and α which test the coupling of the strain in-plane and along Note that the normal state elastic constants c 66 and (c 11 c 12 /2 have very similar magnitude so that the inequalities also apply to the derivatives of with respect to the stress instead of strain.

8 924 M. Sigrist the z-axis. A simple analysis of the distortion effect of the RuO 6 octahedra under these strains shows, that the filling of the γ-band is lowered (raised for the strain in the plane (along the z-axis. 17 Hence, the first-order coupling lowers (raises the, leading to α < 0 and α > 0. Direct measurement of the strain dependence of indicates that this relation should hold. 13, 17 Finally, we would like to comment on the non-resonant ultrasound absorption. As shown in Eq. (12, we expect, in principle, the appearance of an absorption peak immediately below, originating from the linear coupling to the order parameter modes. Such peaks have been observed in the heavy Fermion superconductors UBe 13 and UPt 3 which have also unconventional Cooper pairing. We may argue that the absorption peak would be difficult to observe in Sr 2 RuO 4 because of the sharp decrease of ultrasound absorption below when the quasiparticle gap opens. However, the same argument is applicable to UBe 13 and to UPt 3. We would like to speculate here that the absorption in the two heavy Fermion superconductors may be enhanced by domain wall fluctuations. 9, 19 Domain walls of a degenerate superconducting phase can couple to ultrasound, if the order parameter breaks crystal field symmetry. 9, 19 This may indeed be the case in UBe 13 and UPt 3. On the other hand, the chiral p-wave state preserves crystal field symmetry and does not possess this type of coupling, so that this form of absorption is absent in Sr 2 RuO 4. In summary, we would like to emphasize that the results of ultrasound data in Sr 2 RuO 4 are subject to a set of Ehrenfest relations that relate the discontinuities of the elastic constants c ij and thermal expansion α associated with various sound modes. We believe that these relations should be helpful in analyzing the experimental data and may help in determining whether there are missing elements in our present understanding of the superconducting phase. Since these relations are based on symmetry, they should be independent of microscopic details and multiband features. Nevertheless, the dominant band (most likely the γ-band may leave significant traces. Finally, we would like to emphasize again the prospects of probing spin rotation modes of the Cooper pairs with ultrasound because of spin-orbit coupling. Acknowledgements I would like to thank D. Agterberg, C. Lupien, B. Lüthi, Y. Maeno, H. Matsui, L. Taillefer, and N. Toyota for many stimulating discussions. This work was supported by the Swiss Nationalfonds and in part a Grant-in-Aid from the Japanese Ministry of Education, Culture, Sports, Science and Technology. References 1 Y. Maeno, Physica B (2000, Y. Maeno, T. M. Rice and M. Sigrist, Physics Today 54, No. 1 (2001, T. M. Rice and M. Sigrist, J. Phys. Cond. Mat. 7 (1995, L D. Fay and L. Tewordt, Phys. Rev. B 62 (2000, 4036, and references therein. 5 S. Higashitani and K. Nagai, Physica B (2000, 539; Phys. Rev. B 62 (2000, H.-Y. Kee, Y.-B. Kim and K. Maki, Phys. Rev. B 62 (2000, 5877.

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