Fogelström, M.

Löfwander, T.

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arXiv:cond-mat/0503391v1  [cond-mat.supr-con]  16 Mar 2005Low-Temperature Thermal Conductivity of Superconductors With Gap NodesTomas Löfwander1, 2 and Mikael Fogelström31Institut für Theoretische Festkörperphysik, Universität Karlsruhe, 76128 Karlsruhe, Germany∗2Department of Physics & Astronomy, Northwestern University, Evanston, Illinois 60208, USA3Applied Quantum Physics Laboratory, MC2, Chalmers University of Technology, S-412 96 Göteborg, Sweden(Dated: March 16, 2005)We report a detailed analytic and numerical study of electronic thermal conductivity in d-wavesuperconductors. We compare theory of the cross over at low temperatures from T -dependence toT 3-dependence for increasing temperature with recent experiments on YBa2Cu3O7 in zero magneticfield for T ∈ [0.04K, 0.4K] by Hill et al., Phys. Rev. Lett. 92, 027001 (2004). Transport theory,including impurity scattering and inelastic scattering within strong coupling superconductivity, canconsistently fit the temperature dependence of the data in the lower half of the temperature regime.We discuss the conditions under which we expect power-law dependences over wide temperatureintervals.Low-temperature measurements of electronic trans-port properties have given a wealth of information aboutnodal quasiparticles deep in the superconducting phaseof unconventional superconductors [1, 2, 3, 4, 5, 6, 7, 8,9, 10, 11, 12, 13]. The ultra-low temperature regime isof great interest because response functions such as thezero-frequency charge conductivity, thermal conductiv-ity, and thermoelectric response function, are believed tobe limited by elastic impurity scattering and obey powerlaws [14, 15, 16, 17]. Thermal conductivity is of particu-lar importance in this context, since theory predicts uni-versality in the sense that the low-temperature asymp-totic does not depend on the properties of the impuritypotential [18]. For higher temperatures, of the order ofthe low-energy impurity band width γ and higher, ther-mal conductivity is not universal and is sensitive to thedetails of impurity scattering [16].The universal character of the T → 0 thermal con-ductance was studied experimentally in great detail forseveral different cuprate materials for a wide range ofdoping levels [2, 4, 5, 6, 9, 10, 11]. Measurements ofhow the universal limit is approached as temperature islowered was only recently reported by Hill et al [12] forYBa2Cu3O7. After subtracting a phonon contribution∝ T 3, they found an electronic contribution of the formκel/T = (κ0/T )(1+kT2), with κ0/T = 0.16 mW/(K2cm)and the coefficient k = 19.2K−2. This form is consistentwith a Sommerfeld-type expansion of κel(T ) ([16] andbelow). However, the large value of k implies an unre-alistically clean sample if impurity scattering is in theunitary limit. On the other hand, for Born limit scat-tering, the concentration required to fit the large slopeis unrealistically large and should lead to a severe reduc-tion of the critical temperature which is not seen. Theirplausible conclusion was therefore that scattering mightbe in between the unitary and Born limits.In this paper we present results for thermal conduc-tivity including effects of elastic impurity scattering andinelastic electron-boson scattering. We focus on howthe universal limit is approached and make a detailedcomparison with the experimental results of Hill et al[12]. Our results can be summarized as follows. (1) Themain effect of inelastic scattering is an effective mass-dependence of the universal (with respect to impurityscattering) T → 0 thermal conductance [19]. The coef-ficient k is however effective mass independent. (2) Wecan fit the data [12] in the lower half of the T−intervalwith a best fit for a phase shift slightly below 90o. (3)The discrepancy at higher T is related to the split ofthe low-energy impurity band, which makes it hard toobserve power laws over wide T−intervals in ultra-cleanmaterials.The thermal conductivity κ(T ) of a superconductorhas an electronic contribution κel(T ), a phononic partκph(T ), and possibly contributions from other existingexcitations. We will only consider κel(T ). To quan-tify the effects of inelastic scattering on the same footingas elastic impurity scattering we make a self-consistentstrong-coupling calculation using quasi-classical Eliash-berg theory [20] including impurity scattering in thet̂−matrix approximation. We consider a two-spectramodel for electron-boson interactions that generate bothd-wave superconductivity and an inelastic scattering selfenergy. One spectrum gives coupling in the s-wave andd-wave channels, but is attractive only in the d-wave.Hence, this part is responsible for pairing but also con-tributes to inelastic scattering. The second spectrumcouples only in the s-wave channel (repulsively) and givesan incoherent background scattering. Even though themodel is general we choose parameters that generates ad-wave superconductor with properties close to that of aweakly coupled superconductor, with a spectroscopic gap∆0 = 2.59 Tc (Tc = 0.13ωmode) [21]. The temperature de-pendence κel(T ) is displayed in Fig. 1 for several impurityconcentrations nimp (Γ0 = nimp/πNf ) and near-resonantimpurity scattering (phase shift δ0 = 89.5o). We showin panels (a) and (b) the electron-boson coupling func-tions, χ2I(ω), and the resulting density of states, NS(ǫ).The imaginary part of the inelastic self energy gives aT−dependent contribution to the scattering rate , Γin(T )20 0.5 1 1.5T/Tc01234Γ0=0.0008 TcΓ0=0.0083 TcΓ0=0.0837 TcΓ0=0.9005 Tc0 5 10ω/ωmode02460 2.5 5 7.5 10ε/Tc00.511.522.5 pairing spectra background spectraκel(T)/κN(Tc;Γ0=0)NS(ε;Γ0)/Nf(a)χ2I(ω)(b)FIG. 1: Temperature dependence of the electronic contribution tothe thermal conductivity for elastic scattering rates as indicated,for a phase shift δ0 = 89.5o. Insert (a): the two electron-bosoncoupling spectra used in the calculations. Insert (b): single-particledensity of states in the superconducting state at T = 0.[22, 23, 24], while the contribution from elastic impurityscattering, Γ = Γ0 sin2 δ0, is T−independent. The rela-tive importance of the two contributions leads to a cross-over temperature T ∗. Impurity scattering dominates inthe low−T region, while inelastic scattering dominates inthe high−T region. The cross-over is reflected in κel(T )as the peak at T ∼ T ∗ in Fig. 1 [19, 24].Although the imaginary part of the inelastic self-energy Σ̂Rin is small for T ≪ T∗, the real part remainsimportant. By using general symmetry relations we canwrite down the renormalization of the energy from bothelastic and inelastic scattering asǫ̃R = ǫ −12Tr[τ̂3Σ̂R(ǫ)]= iγ + (1 + λ)ǫ + O[ǫ2]. (1)Here is γ solely due to impurity scattering. The slope ofthe real part of ǫ̃R(ǫ) has contributions from both impu-rity scattering and inelastic scattering, λ = λimp + λin.Because ℜΣRin(ǫ) is linear in energy up to the mode-energy, ωmode > ∆0 ≫ T ∗, an effective theory can bemade for T ≪ T ∗ by dividing the transport equation by1 + λin evaluated in the superconducting phase. The in-elastic part is then interpreted as an effective mass viam∗/m = 1 + λin. The contribution from impurity scat-tering on the other hand, has a non-trivial energy depen-dence for ǫ . ∆0, and can not be divided out. Withinthe effective theory, we use Eq. (1) and the method ofGraf et al [16] to write down the low−T formκel(T )T=π232Nfv2fπµ∆0[1 +7π215ã2T 2γ2]+O[(Tγ)4]. (2)Here are Nf and vf the effective density of states atthe Fermi level and the effective Fermi velocity, while∆0 is the spectroscopic gap and µ is the opening rateof the gap function at the node. We now find thatthe remaining effect at T ≪ T ∗ of inelastic scatteringwithin strong-coupling superconductivity is a modifica-tion of the T → 0 asymptotic of the thermal conductance.When we write explicitly, Nf → N ∗f = N0f (1 + λin), andvf → v∗f = v0f/(1 + λin), one factor 1 + λin remains inthe denominator. Within the bare theory, this result canbe traced back to that the spectroscopic gap ∆0 in theweak-coupling limit is replaced by the strong couplingoff-diagonal function W (0) [19]. In our model W (ǫ) isa weak function of energy and to a good approximationW (0) ≈ ∆0(1 + λin). This effective mass dependenceis in sharp contrast to the normal state, in which caseκNel (T ) = T (π2/3)Nfv2f/2Γ is independent of m∗, be-cause of canceling factors of (1 + λin) in Nf , v2f and Γ inaccordance with general strong-coupling results for trans-port coefficients [25]. In this sense, transport propertiesof quasiparticles in the low-energy nodal impurity bandis quite different from quasiparticles in the normal state.Effective-mass factors from ã and γ cancel in the coef-ficient k = 7π215 (ã/γ)2 in Eq. (2) and it is only dependenton properties of the impurity potential. In addition tothe electron-hole symmetric part above, also the electron-hole asymmetric part of the impurity self-energy enters[26]. Only the imaginary part12Tr[ℑΣ̂Rimp(ǫ)]= c ǫ + O[ǫ3]. (3)contributes to the conductivity through ã2 = (1+λimp)2+2c2 [17]. The presence and importance of c represents thefact that electron-like and hole-like quasiparticles havedifferent transport lifetimes above and below the Fermilevel, except in the strict unitary and Born limits forwhich it vanishes.We now discuss the experiment [12] in terms of thepresented effective theory and its low−T expansion inEq. (2) with the parameters {γ, λimp, c} computed self-consistently [17]. Starting for simplicity with s-wave im-purity scattering, we present in Fig. 2(d) several com-binations of impurity concentrations and phase shiftsthat give good fits to the data in the lower half of theT−interval. In fact, there is a broad range in parameterspace for which the slope k = 19.2K−2 can be repro-duced [blue dash-dotted lines in Fig. 2(a)-(c)]. Similarly,we can find many combinations yielding other values ofk (black and red lines). However, not all combinationsof parameters produce a consistent fit, in the sense thatthe expansion Eq. (2) represents over a wide tempera-ture range the full numerically computed thermal con-ductivity shown in Fig. 2(d). The deviation from trueT 2-behavior is also seen in the figure: the three curvesstart at low T with the same coefficient, k = 19.2K−2,but deviate at higher T . This is also the reason for thediscrepancy between theory and experiments in the up-per half of the T−interval. The part in parameter space386o87o88o89o90o10-310-210-11Γ0✳ [K]k = 1 K-2k = 5 K-2k = 19.2 K-286o87o88o89o90o01234γ✳  [K]86o87o88o89o90os-wave phase shift δ0-0.500.511.520 0.1 0.2 0.3 0.4T [K]0.10.20.30.40.50.6κel(T)               Tempirical fit ~T2experimental data(a) (b)(c)(d)(I)(II)(III)(I):    Γ0✳=0.0138K, δ0=89.4ο(ΙΙ): Γ0✳=0.0108K, δ0=89.5o(III): Γ0✳=0.0082K, δ0=89.6ολimp✳c✳FIG. 2: (a)-(c) The low-temperature parameters in Eqs. (1) and (3) for three values of the T 2-coefficient in Eq. (2) as function ofscattering phase shift. (d) Three sets of parameters that represents our best fits to the data of Ref. [12]. The blue shaded region indicatesthe temperature range where a good fit can be made. The dashed curves are from a strong-coupling calculation while the underlying solidlines are generated by the effective theory. Curve (II) is the low−T part of the black solid line in Fig. 1. In all cases k = 19.2K−2. Notethat the material appears cleaner in the effective theory as Γ∗0= Γ0/(1+λin) with Γ0 = nimp/πNf . We normalized κel(T )/T to the T → 0extrapolated value 0.16mW/(K2cm). The energy scale is Tc = 90K.for which the Sommerfeld expansion is applicable can bebroadly defined as phase shifts for which λimp is negative.Let us discuss in more detail why Eq (2) workswell only in a restricted low-T interval. Recall thatthe thermal conductivity can be written as κel(T ) =∫∞−∞dǫK(ǫ)(−∂f∂ǫ), where f is the Fermi function. Thederivative of the Fermi function is sharply peaked at theFermi level in an interval of order T , which leads us toexpand all quantities around ǫ = 0, as in Eqs. (1) and(3), and do a Sommerfeld expansion κel(T ) = K(0) +∑∞n=1d2nK(ǫ)dǫ2n∣∣∣ǫ=0anT2n, where the for us relevant coeffi-cients a1 = π2/6 and a2 = 7π4/360. However this schemeworks well only if the function K(ǫ) is slowly varying onthe scale T . In our case we can write K(ǫ) = ǫ2κ̃(ǫ) andstudy the energy dependence of κ̃(ǫ), see Fig. 3(d). Thefine structure at low energies for phase shifts below 90o,can intuitively be understood by studying the phase shiftdependence of the self-energy. In a d-wave superconduc-tor with a single impurity scattering strongly (near theunitary limit) there is a resonance in the t̂−matrix at [27]ǫres ≈ ±π cot δ0/[2 ln(8/π cot δ0)] where the sign corre-sponds to the electron and hole sectors. The width ofthe resonance is smaller than the resonance energy by afactor 1/ ln(8/π cot δ0) ≪ 1. Within transport theory weemploy an impurity averaging technique [28] to obtainthe self-energy for a material with a small concentrationof randomly distributed impurities. As a result of theconfiguration average, the single impurity resonance isbroadened into an impurity band of width γb, centeredaround an energy slightly shifted from the single impu-rity resonance ǫres → ǫb. The impurity bands are shownin Fig. 3(a)-(c). For clean systems with a phase shiftdeviating substantially from 90o, the resulting bands aresplit into two parts symmetrically positioned above andbelow the Fermi level. The critical point is when theslope at zero energy of the real part of the self energychanges sign, i.e. when λimp = −ℜΣR3 (0) turns posi-tive. The structures in the self-energy lead to structuresalso in the response functions at low temperatures on thescale T ∼ ǫb ≪ γb, see Fig. 3(d), and the applicability ofEq. (2) is drastically reduced in temperature by the newconstraint T ≪ ǫb.In a more realistic model, impurity scattering isanisotropic. The main technical complication in this caseis the necessity to take care of vertex corrections. Theyare small in the limit T ≪ γ [18], but are important forhigher temperatures. In particular they contribute to thecoefficient ã in Eq. (2). For definiteness we have includedp-wave and d-wave scattering channels and considered ar-bitrary phase shifts [29]. In general we obtain a pair ofresonances, one electron-like and one hole-like (symmet-rically positioned on each side of the Fermi level), foreach phase shift. This fact was also noted recently inRef. [30]. The main conclusion relevant in the presentdiscussion is that the impurity self-energy aquires morestructures (related to the new resonances) and the rangeof applicability of Eq. (2) is not improved.In summary, let us list our results and make a few re-marks. (1) Inelastic scattering contributes to the T → 0asymptotic (i.e. κ0 ∝ T ), but plays no role for theT 3 coefficient (i.e. k). (2) The low−T half of the ex-perimental data can be consistently fitted with param-eters representing a wide range of impurity concentra-4-6-3036-40 -20 0 20 40ε / Γ0-9-6-30-0.4 -0.2 0 0.2 0.4ε / Tc00.10.2-0.1 0 0.1 0.2ε / Tc NS(ε)               Nfκ(ε)∼90odfdε-Im ΣR3(ε)Γ080o(a)(c) (d)(b)Re ΣR3(ε)Γ0FIG. 3: (a)-(b) The impurity scattering self-energy and (c) thedensity of states for Γ0 = 0.01Tc and phase shifts ranging from 90o(red dashed line) to 80o (thick black line) in steps of two degrees.(d) The kernel κ̃(ǫ) of the thermal conductivity contains structuresat low energies for phase shifts below 90o because of the split im-purity bands seen in (a)-(c). Also shown is the derivative of theFermi function which has a peak at the Fermi level of width ∼ T .Here is T = 0.005Tc = 0.45K for Tc = 90K.tions and phase shifts. The best fit to the large coeffi-cient k = 19.2K−2 is obtained for a rather clean materialwith a scattering phase shift slightly below the unitarylimit (δ0 < 90o). (3) However, we find deviations inthe high−T region. The reason is that the expansionEq. (2) has a limited range of applicability, smaller thanthe temperature range in which the T 3 term was foundexperimentally to dominate. Indeed, in an earlier discus-sion of the data [12] in Ref. [31], model parameters werechosen so that the expansion Eq. (2) was inapplicable.In their work the unit-term Eq. (3) was neglected, whichchanged their results by about a factor two. We con-clude that a clear cross over between T−dependence toT 3-dependence, with the respective power laws holding inwide T−intervals, is most easily observable in dirty to in-termediately clean, but not in ultra-clean, materials. 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Low-Temperature Thermal Conductivity of Superconductors with Gap Nodes

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Low-Temperature Thermal Conductivity of Superconductors with Gap Nodes

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