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Effect of spin–orbit impurity scattering in d-wave superconductors
Journal of Magnetism and Magnetic Materials 226}230 (2001) 335}336
E!ect of spin}orbit impurity scattering in d-wave superconductors Kazuhiro Kuboki*, Toshifumi Yagi Department of Physics, Kobe University, 1-1 Rokkodai, Nada-ku, Kobe 657-8501, Japan
Abstract The e!ect of a magnetic impurity in the d -wave superconducting (SC) state of the t}J model is studied. The V \W Bogoliubov-de Gennes equation, derived within a slave-boson mean-"eld approximation, is solved numerically at ¹"0. The spin}spin scattering induces spin-triplet p-wave SC order parameters near the impurity, while a local (d #id )-wave SC state with spontaneous currents occurs in the presence of the spin}orbit interaction. When both V \W VW interactions coexist, it turns out that a state which carries "nite spin currents appears. 2001 Elsevier Science B.V. All rights reserved. Keywords: High-¹ superconductivity; t}J model; d-wave symmetry; spin}orbit interaction
The symmetry of the superconducting (SC) state in high-¹ oxides has been studied intensively, and now it is established that the state has a predominantly d V \W character [1]. The e!ects of randomness in such unconventional superconductors can be quite di!erent from those in conventional ones, so that their study is of particular interest. In this paper, we examine the e!ect of a magnetic impurity (such as Ni) in a d -wave SC V \W state of the two-dimensional t}J model which describes the low-energy electronic properties of the High-¹ ox ides. (We consider the case where the concentration of impurities is low, so that the states around the impurities can be treated independently.) We treat the t}J model with the Hamiltonian H"! t (c R c #hc)# J S ) S GH G N H N GH G H 6G H7N 6G H7 #< c R c #J S ) S #H , N N >O N O where t and J are the tranfer integrals and the antiferGH GH romagnetic superexchange interactions, respectively, and they extend over the nearest (t, J) and next-nearest-
* Corresponding author. Fax:#81-78-803-5630. E-mail address: [email protected] (K. Kuboki).
neighbor (t, J) pairs. c is an electron operator within GN the Hilbert space excluding double occupancy, and is the nearest neighbors of the impurity (denoted as S ; S "1) located at a site 0. We take the potential scat tering term < to be <
0304-8853/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 0 8 1 3 - 1
336
K. Kuboki, T. Yagi / Journal of Magnetism and Magnetic Materials 226}230 (2001) 335}336
Fig. 1. Spin currents around the impurity for J "0.6, g"1.2, < "100, "0.189, t"!0.5 and J"0.2. The arrows indi cate only the directions of the currents but not the magnitudes. (An impurity is located at the center of the system.)
ferromagnet bilayer (or multilayer) systems, and these phenomena can be qualitatively explained by the Ginzburg}Landau theory [5]. It was pointed out that the spin}orbit impurity in the d -wave SC state can induce a local (d #id )V \W V \W VW state with a spontaneous (charge) current around the impurity [6]. In the present case of the t}J model, id VW component is actually induced by the spin}orbit interaction when small but "nite t and J are present. This state breaks time-reversal symmetry locally near the impurity. (Note J of the order of J is necessary to have "nite d component in uniform system.) VW Now, we show the results for the case where the spin}spin and the spin}orbit scattering coexist. In this case we "nd spontaneous spin currents (Fig. 1) as well as charge currents. (The spin current is de"ned as the di!erence of the currents of spin-up and spin-down electrons.) Spontaneous currents are possible because the SCOPs have nontrivial phases (other than 0 or ) due to the spin}orbit interaction g, as in the case of gO0 and J "0. In the presence of the spin}spin interaction J , m (and also the p-wave OP) are "nite (Fig. 2) and then the contributions to currents from electrons of opposite spins are not equal. Thus, the spin current can arise when both J and g are "nite. This work was supported in part by a Grant-in-Aid for Scienti"c Research from the Ministry of Education, Sports and Culture of Japan.
Fig. 2. The induced staggered magnetization m. (Note that m is non-dimensional.) Parameters are the same as in Fig. 1.
References
impurity, and the staggered magnetization (m) as well as SCOP's with symmetries di!erent from d is induced V \W [3,4]. Here we note that the presence of m in the d-wave SC state induces the spin-triplet p-wave SCOPs. This is due to the imbalance of the densities of spin-up and spin-down electrons. A similar occurence of p-wave OPs in d-wave superconductors is also seen in superconductor/
[1] D. Scalaplino, Phys. Rep. 250 (1995) 329. [2] P.G. de Gennes, Superconductivity of Metals and Alloys, Addison-Wesley, New York, 1989. [3] M.I. Salkola, A.V. Balatsky, J.R. Scrie!er, Phys. Rev. B 55 (1997) 12648. [4] Tsuchiura, Y. Tanaka, M. Ogata, S. Kashiwaya, Phys. Rev. Lett. 84 (2000) 3165. [5] K. Kuboki, J. Phys. Soc. Jpn. 68 (1999) 3150. [6] A.V. Balatsky, Phys. Rev. Lett. 80 (1998) 1972.
E!ect of spin}orbit impurity scattering in d-wave superconductors Kazuhiro Kuboki*, Toshifumi Yagi Department of Physics, Kobe University, 1-1 Rokkodai, Nada-ku, Kobe 657-8501, Japan
Abstract The e!ect of a magnetic impurity in the d -wave superconducting (SC) state of the t}J model is studied. The V \W Bogoliubov-de Gennes equation, derived within a slave-boson mean-"eld approximation, is solved numerically at ¹"0. The spin}spin scattering induces spin-triplet p-wave SC order parameters near the impurity, while a local (d #id )-wave SC state with spontaneous currents occurs in the presence of the spin}orbit interaction. When both V \W VW interactions coexist, it turns out that a state which carries "nite spin currents appears. 2001 Elsevier Science B.V. All rights reserved. Keywords: High-¹ superconductivity; t}J model; d-wave symmetry; spin}orbit interaction
The symmetry of the superconducting (SC) state in high-¹ oxides has been studied intensively, and now it is established that the state has a predominantly d V \W character [1]. The e!ects of randomness in such unconventional superconductors can be quite di!erent from those in conventional ones, so that their study is of particular interest. In this paper, we examine the e!ect of a magnetic impurity (such as Ni) in a d -wave SC V \W state of the two-dimensional t}J model which describes the low-energy electronic properties of the High-¹ ox ides. (We consider the case where the concentration of impurities is low, so that the states around the impurities can be treated independently.) We treat the t}J model with the Hamiltonian H"! t (c R c #hc)# J S ) S GH G N H N GH G H 6G H7N 6G H7 #< c R c #J S ) S #H , N N >O N O where t and J are the tranfer integrals and the antiferGH GH romagnetic superexchange interactions, respectively, and they extend over the nearest (t, J) and next-nearest-
* Corresponding author. Fax:#81-78-803-5630. E-mail address: [email protected] (K. Kuboki).
neighbor (t, J) pairs. c is an electron operator within GN the Hilbert space excluding double occupancy, and is the nearest neighbors of the impurity (denoted as S ; S "1) located at a site 0. We take the potential scat tering term < to be <
0304-8853/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 0 8 1 3 - 1
336
K. Kuboki, T. Yagi / Journal of Magnetism and Magnetic Materials 226}230 (2001) 335}336
Fig. 1. Spin currents around the impurity for J "0.6, g"1.2, < "100, "0.189, t"!0.5 and J"0.2. The arrows indi cate only the directions of the currents but not the magnitudes. (An impurity is located at the center of the system.)
ferromagnet bilayer (or multilayer) systems, and these phenomena can be qualitatively explained by the Ginzburg}Landau theory [5]. It was pointed out that the spin}orbit impurity in the d -wave SC state can induce a local (d #id )V \W V \W VW state with a spontaneous (charge) current around the impurity [6]. In the present case of the t}J model, id VW component is actually induced by the spin}orbit interaction when small but "nite t and J are present. This state breaks time-reversal symmetry locally near the impurity. (Note J of the order of J is necessary to have "nite d component in uniform system.) VW Now, we show the results for the case where the spin}spin and the spin}orbit scattering coexist. In this case we "nd spontaneous spin currents (Fig. 1) as well as charge currents. (The spin current is de"ned as the di!erence of the currents of spin-up and spin-down electrons.) Spontaneous currents are possible because the SCOPs have nontrivial phases (other than 0 or ) due to the spin}orbit interaction g, as in the case of gO0 and J "0. In the presence of the spin}spin interaction J , m (and also the p-wave OP) are "nite (Fig. 2) and then the contributions to currents from electrons of opposite spins are not equal. Thus, the spin current can arise when both J and g are "nite. This work was supported in part by a Grant-in-Aid for Scienti"c Research from the Ministry of Education, Sports and Culture of Japan.
Fig. 2. The induced staggered magnetization m. (Note that m is non-dimensional.) Parameters are the same as in Fig. 1.
References
impurity, and the staggered magnetization (m) as well as SCOP's with symmetries di!erent from d is induced V \W [3,4]. Here we note that the presence of m in the d-wave SC state induces the spin-triplet p-wave SCOPs. This is due to the imbalance of the densities of spin-up and spin-down electrons. A similar occurence of p-wave OPs in d-wave superconductors is also seen in superconductor/
[1] D. Scalaplino, Phys. Rep. 250 (1995) 329. [2] P.G. de Gennes, Superconductivity of Metals and Alloys, Addison-Wesley, New York, 1989. [3] M.I. Salkola, A.V. Balatsky, J.R. Scrie!er, Phys. Rev. B 55 (1997) 12648. [4] Tsuchiura, Y. Tanaka, M. Ogata, S. Kashiwaya, Phys. Rev. Lett. 84 (2000) 3165. [5] K. Kuboki, J. Phys. Soc. Jpn. 68 (1999) 3150. [6] A.V. Balatsky, Phys. Rev. Lett. 80 (1998) 1972.