Analysis of field-angle dependent specific heat in unconventional superconductors: a comparison between Doppler-shift method and Kramer-Pesch approximation
aa r X i v : . [ c ond - m a t . s up r- c on ] O c t Analysis of field-angle dependent specific heat in unconventionalsuperconductors: a comparison between Doppler-shift method andKramer-Pesch approximation
Nobuhiko Hayashi a,b , Yuki Nagai c,d , Yoichi Higashi e a Nanoscience and Nanotechnology Research Center (N2RC), Osaka Prefecture University, 1-2 Gakuen-cho, Sakai 599-8570, Japan b CREST(JST), 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan c Department of Physics, University of Tokyo, Tokyo 113-0033, Japan d JST, TRIP, Chiyoda, Tokyo, 102-0075, Japan e Department of Mathematical Sciences, Osaka Prefecture University, 1-1 Gakuen-cho, Sakai 599-8531, Japan
Abstract
We theoretically discuss the magnetic-field-angle dependence of the zero-energy density of states (ZEDOS) in super-conductors. Point-node and line-node superconducting gaps on spherical and cylindrical Fermi surfaces are consid-ered. The Doppler-shift (DS) method and the Kramer-Pesch approximation (KPA) are used to calculate the ZEDOS.Numerical results show that consequences of the DS method are corrected by the KPA.
Key words:
Unconventional superconductor, Field-angle dependent measurement, Specific heat
PACS: C ( T ) / T , is proportional to the ZEDOS in the low tem-perature limit T → N ( E )in the DS method is given as [3, 4, 12] N ( E ) ∝ Re *Z dS F | E − δ E | p ( E − δ E ) − | ∆ | + DS , (1) where δ E = m v F · v s is the DS energy. m , v F , and v s are the electron mass, the Fermi velocity, and thecirculating superfluid velocity around a vortex, respec-tively. The field-angle dependence is taken into ac-count via v s , which is perpendicular to the magneticfield H (i.e., v s ⊥ H ). Around a single vortex, | v s | = ~ / mr ( r is the radial distance from the vortex center). h· · ·i DS = R r a ξ rdr R π d α · · · , is the spatial integrationaround the vortex in the cylindrical coordinates ( r , α, z )with ˆ z k H . Here, ξ is the coherence length and r a isthe cuto ff length with r a /ξ = √ H c / H , ( H c ≡ Φ /πξ , Φ = π r a H , and Φ is the flux quantum). dS F is anarea element on the Fermi surface (FS) [e.g., dS F = k sin θ d φ d θ for a spherical FS in the spherical coordi-nates ( k , φ, θ ), and dS F = k F ab d φ dk c for a cylindrical FSin the cylindrical coordinates ( k ab , φ, k c )]. The pair po-tential is ∆ ≡ ∆ Λ ( k F ), where ∆ is the maximum gapamplitude and Λ ( k F ) represents the gap anisotropy onthe FS. Λ and v F are functions of the position k F on theFS.On the other hand, the DOS in the KPA is givenas [13] N ( E ) = v F0 η π ξ *Z dS F | v F | λ h cosh( x /ξ ) i − λ/π h ( E − E y ) + η + KPA . (2) Preprint submitted to Physica C December 5, 2018 ere, λ = | Λ | , and h· · ·i KPA ≡ R r a rdr R π · · · d α/π r a isthe real-space average around a vortex in the cylindri-cal coordinates ( r , α, z ) with ˆ z k H . x = r cos( α − θ v ), y = r sin( α − θ v ), and E y = ∆ λ y /ξ h . θ v ( k F , α M , θ M ) isthe angle of v F ⊥ in the plane of z =
0, where α and θ v aremeasured from a common axis [12, 15]. v F ⊥ is the vec-tor component of v F ( k F ) projected onto the plane normalto ˆ H = ( α M , θ M ). Here, the azimuthal (polar) angle of H is α M ( θ M ) in a spherical coordinate frame fixed to crys-tal axes. | v F ⊥ ( k F , α M , θ M ) | ≡ v F0 ( α M , θ M ) h ( k F , α M , θ M )and v F0 is the FS average of | v F ⊥ | represented as [12], v F0 = R dS F | v F ⊥ | / R dS F . The coherence length is ex-pressed as ξ = ~ v F0 /π ∆ . The smearing factor is set as η = . ∆ .We numerically calculate the ZEDOS N (0) for thefollowing four combinations: the line-node or thepoint-node gap on the cylindrical or the spherical FS.Here, the line-node gap is the d -wave pairing Λ = cos 2 φ sin θ for the spherical FS and is Λ = cos 2 φ forthe cylindrical FS. The point-node gap is the s + g wave Λ = (1 + sin θ cos 4 φ ) / Λ = (1 + cos ( k c /
2) cos 4 φ ) / T region. Taking into account acontribution of a vortex core [13], the KPA yields morequantitative results with the same computational loadcompared to the DS method.One of us (Y.N.) acknowledges support by Grant-in-Aid for JSPS Fellows (204840). References [1] Y. Matsuda, K. Izawa, I. Vekhter, J. Phys.: Condens. Matter 18(2006) R705. N / N M AX α M [rad](a) Line-node gap on spherical FSKPADS 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1 1.02 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 α M [rad](b) Point-node gap on spherical FSKPADS Figure 1: Azimuthal field-angle α M dependence of the ZEDOS for thespherical FS. The polar angle θ M = π/
2. The cuto ff length r a = ξ . N / N M AX α M [rad](a) Line-node gap on cylindrical FSKPADS 0.75 0.8 0.85 0.9 0.95 1 1.05 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 α M [rad](b) Point-node gap on cylindrical FSKPADS Figure 2: Azimuthal field-angle α M dependence of the ZEDOS for thecylindrical FS. The polar angle θ M = π/
2. The cuto ff length r a = ξ .[2] T. Sakakibara, A. Yamada, J. Custers, K. Yano, T. Tayama, H.Aoki, K. Machida, J. Phys. Soc. Jpn. 76 (2007) 051004.[3] G. E. Volovik, JETP Lett. 58 (1993) 469.[4] I. Vekhter, P. J. Hirschfeld, J. P. Carbotte, E. J. Nicol, Phys. Rev.B 59 (1999) R9023.[5] H. Won, S. Haas, D. Parker, S. Telang, A. V´anyolos, K. Maki,AIP Conf. Proc. 789 (2005) 3.[6] P. Miranovi´c, N. Nakai, M. Ichioka, K. Machida, Phys. Rev. B68 (2003) 052501.[7] H. Kusunose, J. Phys. Soc. Jpn. 73 (2004) 2512; Phys. Rev. B70 (2004) 054509.[8] M. Udagawa, Y. Yanase, M. Ogata, Phys. Rev. B 71 (2005)024511.[9] A. B. Vorontsov, I. Vekhter, Phys. Rev. Lett. 96 (2006) 237001;Phys. Rev. B 75 (2007) 224501; Phys. Rev. B 75 (2007) 224502;I. Vekhter, A. B. Vorontsov, Physica B 403 (2008) 958.[10] T. R. Abu Alrub and S. H. Curnoe, Phys. Rev. B 78 (2008)104521.[11] G. R. Boyd, P. J. Hirschfeld, I. Vekhter, A. B. Vorontsov, Phys.Rev. B 79 (2009) 064525; S. Graser, G. R. Boyd, C. Cao, H.-P.Cheng, P. J. Hirschfeld, D. J. Scalapino, Phys. Rev. B 77 (2008)180514(R).[12] Y. Nagai, Y. Kato, N. Hayashi, K. Yamauchi, H. Harima, Phys.Rev. B 76 (2007) 214514.[13] Y. Nagai, N. Hayashi, Phys. Rev. Lett. 101 (2008) 097001.[14] Y. Nagai, N. Hayashi, Y. Kato, K. Yamauchi, H. Harima, J.Phys.: Conf. Ser. 150 (2009) 052177.[15] Y. Nagai, Y. Ueno, Y. Kato, N. Hayashi, J. Phys. Soc. Jpn. 75(2006) 104701..[2] T. Sakakibara, A. Yamada, J. Custers, K. Yano, T. Tayama, H.Aoki, K. Machida, J. Phys. Soc. Jpn. 76 (2007) 051004.[3] G. E. Volovik, JETP Lett. 58 (1993) 469.[4] I. Vekhter, P. J. Hirschfeld, J. P. Carbotte, E. J. Nicol, Phys. Rev.B 59 (1999) R9023.[5] H. Won, S. Haas, D. Parker, S. Telang, A. V´anyolos, K. Maki,AIP Conf. Proc. 789 (2005) 3.[6] P. Miranovi´c, N. Nakai, M. Ichioka, K. Machida, Phys. Rev. B68 (2003) 052501.[7] H. Kusunose, J. Phys. Soc. Jpn. 73 (2004) 2512; Phys. Rev. B70 (2004) 054509.[8] M. Udagawa, Y. Yanase, M. Ogata, Phys. Rev. B 71 (2005)024511.[9] A. B. Vorontsov, I. Vekhter, Phys. Rev. Lett. 96 (2006) 237001;Phys. Rev. B 75 (2007) 224501; Phys. Rev. B 75 (2007) 224502;I. Vekhter, A. B. Vorontsov, Physica B 403 (2008) 958.[10] T. R. Abu Alrub and S. H. Curnoe, Phys. Rev. B 78 (2008)104521.[11] G. R. Boyd, P. J. Hirschfeld, I. Vekhter, A. B. Vorontsov, Phys.Rev. B 79 (2009) 064525; S. Graser, G. R. Boyd, C. Cao, H.-P.Cheng, P. J. Hirschfeld, D. J. Scalapino, Phys. Rev. B 77 (2008)180514(R).[12] Y. Nagai, Y. Kato, N. Hayashi, K. Yamauchi, H. Harima, Phys.Rev. B 76 (2007) 214514.[13] Y. Nagai, N. Hayashi, Phys. Rev. Lett. 101 (2008) 097001.[14] Y. Nagai, N. Hayashi, Y. Kato, K. Yamauchi, H. Harima, J.Phys.: Conf. Ser. 150 (2009) 052177.[15] Y. Nagai, Y. Ueno, Y. Kato, N. Hayashi, J. Phys. Soc. Jpn. 75(2006) 104701.