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Vortex shadow on surface of chiral p-wave superconductors
Physica C 470 (2010) S888–S889
Contents lists available at ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
Vortex shadow on surface of chiral p-wave superconductors Takehito Yokoyama a,*, Christian Iniotakis b, Yukio Tanaka c, Manfred Sigrist b a
Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland c Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan b
a r t i c l e
i n f o
Article history: Accepted 20 October 2009 Available online 25 October 2009 Keywords: Vortex Chirality Chiral superconductor
a b s t r a c t We study the local density of states at the surface of a chiral p-wave superconductor in the presence of an Abrikosov vortex which is situated not too far from the surface. It is shown that the presence of the vortex leads to a zero-energy peak of the density of states at the surface if its chirality is the same as that of the superconductor, and to a gap structure for the opposite case. Ó 2009 Elsevier B.V. All rights reserved.
Unconventional superconductors are characterized by a sign or general phase change of their gap function as a function of momentum. One important consequence of the sign change of the gap function is the possible existence of Andreev bound states at the surface of the superconductor [1–3]. The formation of Andreev bound states increases the local zero-energy quasiparticle density of states (DOS) at the surface, leading to a pronounced zero-bias conductance peak in the tunneling conductance observable both in singlet d-wave superconductors like the cuprates and in triplet p-wave superconductors such as Sr2RuO4 [4–11]. For the case of d-wave superconductors it is well-known, that an applied magnetic field or an applied electric current result in a split of this zero-bias conductance peak, since the zero-energy spectral weight of the bound states is effectively Doppler shifted towards higher energies [5,12–14]. The same effect also appears for an Abrikosov vortex, which is pinned not too far from the boundary. Here, the zero-energy DOS is suppressed in a shadow-like region ‘behind’ the vortex [15,16]. Regarding the chiral p-wave superconducting phase as it is likely realized in Sr2RuO4, a further aspect appears. The chiral pwave state characterized by the vector dðkÞ ¼ ð0; 0; kx iky Þ breaks time reversal symmetry [17–19]. In this paper, we will show that the influence of Abrikosov vortex on the surface density of states is selective for the chirality. The quasiparticle density of states at the surface increases or decreases depending on the orientation of vorticity with respect to chirality. In the calculations, we use the quasiclassical Eilenberger theory of superconductivity [20,21] in the so-called Riccati-parametriza-
* Corresponding author. E-mail address: [email protected] (T. Yokoyama). 0921-4534/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2009.10.021
tion [22], which allows to achieve numerically stable solutions for the quasiclassical propagators in spatially inhomogeneous systems. We consider a superconducting half space x > 0 exhibiting a gap function of chiral p-wave symmetry. Technical details of the calculation are given in Ref. [23] (see also Ref. [24]). In the following, N, E and d denote normalized density of states, the quasiparticle energy with respect to the Fermi level, and an effective scattering parameter that corresponds to an inverse mean free path, respectively. For all numerical calculations, we fix this value as d ¼ 0:1D0 . Now let us show the results. We study the case of a single Abrikosov vortex line parallel to the z-axis. For such a vortex in the bulk of a chiral p-wave superconductor, the interplay between vorticity and chirality has been intensively studied already [25]. In the present work, however, we focus on important surface effects, which appear at the boundary due to the presence of the vortex. We assume the vortex to be at a distance xV from the boundary. With y denoting the coordinate along the boundary, the vortex shall sit at yV ¼ 0. In the following, we consider two different cases: The chiralities of the p-wave state and the vortex are the same (a) or opposite (b). Our results for the corresponding local zero-energy DOS near the surface of the p-wave superconductor are shown in Fig. 1. The Abrikosov vortex position is fixed at a distance of xV ¼ 3n from the boundary with n ¼ hv F =D0 denoting the coherence length. Apart from zero-energy bound states in the vortex core, there are also bound states at the surface of the p-wave superconductor. Far away from the vortex, these surface bound states have the spectral weight of NðE ¼ 0Þ ¼ 1. However, as can be seen quite clearly in Fig. 1, the local DOS drastically changes in a shadow-like region behind the vortex. If vortex and p-wave state have the same chirality, the bound states are strongly
T. Yokoyama et al. / Physica C 470 (2010) S888–S889
S889
In summary, we have studied the DOS at the surface of a chiral p-wave superconductor in the presence of an Abrikosov vortex. We clarified that the weight of low-energy surface bound states gets either suppressed or increased, depending on the orientation of both chirality and vorticity. These characteristic effects could open an alternative way to chirality sensitive probes, in contrast to phase or spin sensitive setups, which have been intensively used in the scientific community already. Acknowledgement T.Y. acknowledges support by the JSPS. References
Fig. 1. NðE ¼ 0Þ in a chiral p-wave superconductor in the presence of a vortex, which is situated at a distance of xV ¼ 3n from the surface. The vortex and the pwave state have (a) the same chirality and (b) the opposite chirality.
enhanced (a) and for opposite chirality they are suppressed (b). This can be understood as a consequence of the interplay of Doppler shifts by the vortex and the superconductor.
[1] C.R. Hu, Phys. Rev. Lett. 72 (1994) 1526. [2] Y. Tanaka, S. Kashiwaya, Phys. Rev. Lett. 74 (1995) 3451. [3] L.J. Buchholtz, M. Palumbo, D. Rainer, J.A. Sauls, J. Low Temp. Phys. 101 (1995) 1099. [4] J. Lesueur, L.H. Greene, W.L. Feldmann, A. Inam, Physica C 191 (1992) 325. [5] M. Covington, M. Aprili, E. Paraoanu, L.H. Greene, F. Xu, J. Zhu, C.A. Mirkin, Phys. Rev. Lett. 79 (1997) 277. [6] S. Kashiwaya, Y. Tanaka, Rep. Prog. Phys. 63 (2000) 1641. [7] M. Yamashiro, Y. Tanaka, S. Kashiwaya, Phys. Rev. B 56 (1997) 7847. [8] C. Honerkamp, M. Sigrist, J. Low Temp. Phys. 111 (1998) 895. [9] M. Matsumoto, M. Sigrist, J. Phys. Soc. Jpn. 68 (1999) 994. [10] Z.Q. Mao, K.D. Nelson, R. Jin, Y. Liu, Y. Maeno, Phys. Rev. Lett. 87 (2001) 037003. [11] T. Yokoyama, Y. Tanaka, J. Inoue, Phys. Rev. B 72 (2005) 220504(R). [12] M. Fogelström, D. Rainer, J.A. Sauls, Phys. Rev. Lett. 79 (1997) 281. [13] M. Aprili, E. Badica, L.H. Greene, Phys. Rev. Lett. 83 (1999) 4630. [14] Y. Dagan, G. Deutscher, Phys. Rev. Lett. 87 (2001) 177004. [15] S. Graser, C. Iniotakis, T. Dahm, N. Schopohl, Phys. Rev. Lett. 93 (2004) 247001. [16] C. Iniotakis, S. Graser, T. Dahm, N. Schopohl, Phys. Rev. B 71 (2005) 214508. [17] Y. Maeno, H. Hashimoto, K. Yoshida, S. Nishizaki, T. Fujita, J.G. Bednorz, F. Lichtenberg, Nature (London) 372 (1994) 532. [18] K. Ishida, H. Mukuda, Y. Kitaoka, K. Asayama, Z.Q. Mao, Y. Mori, Y. Maeno, Nature (London) 396 (1998) 658. [19] A.P. Mackenzie, Y. Maeno, Rev. Mod. Phys. 75 (2003) 657. [20] G. Eilenberger, Z. Phys. 214 (1968) 195. [21] A.I. Larkin, Yu.N. Ovchinnikov, Zh. Eksp. Teor. Fiz. 55 (1968) 2262 [Sov. Phys. JETP 28 (1969) 1200]. [22] N. Schopohl, K. Maki, Phys. Rev. B 52 (1995) 490. N. Schopohl, condmat:9804064. [23] T. Yokoyama, C. Iniotakis, Y. Tanaka, M. Sigrist, Phys. Rev. Lett. 100 (2008) 177002. [24] M.A. Silaev, T. Yokoyama, J. Linder, Y. Tanaka, A. Sudbø, Phys. Rev. B 79 (2009) 054508. [25] N. Hayashi, Y. Kato, Phys. Rev. B 66 (2002) 132511; N. Hayashi, Y. Kato, J. Low Temp. Phys. 131 (2003) 893.
Contents lists available at ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
Vortex shadow on surface of chiral p-wave superconductors Takehito Yokoyama a,*, Christian Iniotakis b, Yukio Tanaka c, Manfred Sigrist b a
Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland c Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan b
a r t i c l e
i n f o
Article history: Accepted 20 October 2009 Available online 25 October 2009 Keywords: Vortex Chirality Chiral superconductor
a b s t r a c t We study the local density of states at the surface of a chiral p-wave superconductor in the presence of an Abrikosov vortex which is situated not too far from the surface. It is shown that the presence of the vortex leads to a zero-energy peak of the density of states at the surface if its chirality is the same as that of the superconductor, and to a gap structure for the opposite case. Ó 2009 Elsevier B.V. All rights reserved.
Unconventional superconductors are characterized by a sign or general phase change of their gap function as a function of momentum. One important consequence of the sign change of the gap function is the possible existence of Andreev bound states at the surface of the superconductor [1–3]. The formation of Andreev bound states increases the local zero-energy quasiparticle density of states (DOS) at the surface, leading to a pronounced zero-bias conductance peak in the tunneling conductance observable both in singlet d-wave superconductors like the cuprates and in triplet p-wave superconductors such as Sr2RuO4 [4–11]. For the case of d-wave superconductors it is well-known, that an applied magnetic field or an applied electric current result in a split of this zero-bias conductance peak, since the zero-energy spectral weight of the bound states is effectively Doppler shifted towards higher energies [5,12–14]. The same effect also appears for an Abrikosov vortex, which is pinned not too far from the boundary. Here, the zero-energy DOS is suppressed in a shadow-like region ‘behind’ the vortex [15,16]. Regarding the chiral p-wave superconducting phase as it is likely realized in Sr2RuO4, a further aspect appears. The chiral pwave state characterized by the vector dðkÞ ¼ ð0; 0; kx iky Þ breaks time reversal symmetry [17–19]. In this paper, we will show that the influence of Abrikosov vortex on the surface density of states is selective for the chirality. The quasiparticle density of states at the surface increases or decreases depending on the orientation of vorticity with respect to chirality. In the calculations, we use the quasiclassical Eilenberger theory of superconductivity [20,21] in the so-called Riccati-parametriza-
* Corresponding author. E-mail address: [email protected] (T. Yokoyama). 0921-4534/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2009.10.021
tion [22], which allows to achieve numerically stable solutions for the quasiclassical propagators in spatially inhomogeneous systems. We consider a superconducting half space x > 0 exhibiting a gap function of chiral p-wave symmetry. Technical details of the calculation are given in Ref. [23] (see also Ref. [24]). In the following, N, E and d denote normalized density of states, the quasiparticle energy with respect to the Fermi level, and an effective scattering parameter that corresponds to an inverse mean free path, respectively. For all numerical calculations, we fix this value as d ¼ 0:1D0 . Now let us show the results. We study the case of a single Abrikosov vortex line parallel to the z-axis. For such a vortex in the bulk of a chiral p-wave superconductor, the interplay between vorticity and chirality has been intensively studied already [25]. In the present work, however, we focus on important surface effects, which appear at the boundary due to the presence of the vortex. We assume the vortex to be at a distance xV from the boundary. With y denoting the coordinate along the boundary, the vortex shall sit at yV ¼ 0. In the following, we consider two different cases: The chiralities of the p-wave state and the vortex are the same (a) or opposite (b). Our results for the corresponding local zero-energy DOS near the surface of the p-wave superconductor are shown in Fig. 1. The Abrikosov vortex position is fixed at a distance of xV ¼ 3n from the boundary with n ¼ hv F =D0 denoting the coherence length. Apart from zero-energy bound states in the vortex core, there are also bound states at the surface of the p-wave superconductor. Far away from the vortex, these surface bound states have the spectral weight of NðE ¼ 0Þ ¼ 1. However, as can be seen quite clearly in Fig. 1, the local DOS drastically changes in a shadow-like region behind the vortex. If vortex and p-wave state have the same chirality, the bound states are strongly
T. Yokoyama et al. / Physica C 470 (2010) S888–S889
S889
In summary, we have studied the DOS at the surface of a chiral p-wave superconductor in the presence of an Abrikosov vortex. We clarified that the weight of low-energy surface bound states gets either suppressed or increased, depending on the orientation of both chirality and vorticity. These characteristic effects could open an alternative way to chirality sensitive probes, in contrast to phase or spin sensitive setups, which have been intensively used in the scientific community already. Acknowledgement T.Y. acknowledges support by the JSPS. References
Fig. 1. NðE ¼ 0Þ in a chiral p-wave superconductor in the presence of a vortex, which is situated at a distance of xV ¼ 3n from the surface. The vortex and the pwave state have (a) the same chirality and (b) the opposite chirality.
enhanced (a) and for opposite chirality they are suppressed (b). This can be understood as a consequence of the interplay of Doppler shifts by the vortex and the superconductor.
[1] C.R. Hu, Phys. Rev. Lett. 72 (1994) 1526. [2] Y. Tanaka, S. Kashiwaya, Phys. Rev. Lett. 74 (1995) 3451. [3] L.J. Buchholtz, M. Palumbo, D. Rainer, J.A. Sauls, J. Low Temp. Phys. 101 (1995) 1099. [4] J. Lesueur, L.H. Greene, W.L. Feldmann, A. Inam, Physica C 191 (1992) 325. [5] M. Covington, M. Aprili, E. Paraoanu, L.H. Greene, F. Xu, J. Zhu, C.A. Mirkin, Phys. Rev. Lett. 79 (1997) 277. [6] S. Kashiwaya, Y. Tanaka, Rep. Prog. Phys. 63 (2000) 1641. [7] M. Yamashiro, Y. Tanaka, S. Kashiwaya, Phys. Rev. B 56 (1997) 7847. [8] C. Honerkamp, M. Sigrist, J. Low Temp. Phys. 111 (1998) 895. [9] M. Matsumoto, M. Sigrist, J. Phys. Soc. Jpn. 68 (1999) 994. [10] Z.Q. Mao, K.D. Nelson, R. Jin, Y. Liu, Y. Maeno, Phys. Rev. Lett. 87 (2001) 037003. [11] T. Yokoyama, Y. Tanaka, J. Inoue, Phys. Rev. B 72 (2005) 220504(R). [12] M. Fogelström, D. Rainer, J.A. Sauls, Phys. Rev. Lett. 79 (1997) 281. [13] M. Aprili, E. Badica, L.H. Greene, Phys. Rev. Lett. 83 (1999) 4630. [14] Y. Dagan, G. Deutscher, Phys. Rev. Lett. 87 (2001) 177004. [15] S. Graser, C. Iniotakis, T. Dahm, N. Schopohl, Phys. Rev. Lett. 93 (2004) 247001. [16] C. Iniotakis, S. Graser, T. Dahm, N. Schopohl, Phys. Rev. B 71 (2005) 214508. [17] Y. Maeno, H. Hashimoto, K. Yoshida, S. Nishizaki, T. Fujita, J.G. Bednorz, F. Lichtenberg, Nature (London) 372 (1994) 532. [18] K. Ishida, H. Mukuda, Y. Kitaoka, K. Asayama, Z.Q. Mao, Y. Mori, Y. Maeno, Nature (London) 396 (1998) 658. [19] A.P. Mackenzie, Y. Maeno, Rev. Mod. Phys. 75 (2003) 657. [20] G. Eilenberger, Z. Phys. 214 (1968) 195. [21] A.I. Larkin, Yu.N. Ovchinnikov, Zh. Eksp. Teor. Fiz. 55 (1968) 2262 [Sov. Phys. JETP 28 (1969) 1200]. [22] N. Schopohl, K. Maki, Phys. Rev. B 52 (1995) 490. N. Schopohl, condmat:9804064. [23] T. Yokoyama, C. Iniotakis, Y. Tanaka, M. Sigrist, Phys. Rev. Lett. 100 (2008) 177002. [24] M.A. Silaev, T. Yokoyama, J. Linder, Y. Tanaka, A. Sudbø, Phys. Rev. B 79 (2009) 054508. [25] N. Hayashi, Y. Kato, Phys. Rev. B 66 (2002) 132511; N. Hayashi, Y. Kato, J. Low Temp. Phys. 131 (2003) 893.