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update previously added training data with the corrected version
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<front>arXiv:cond-mat/0106014v1 [cond-mat.supr-con] 1 Jun 2001 <lb/>Tunneling conductance of normal metal / d x 2 −y 2 -wave <lb/>superconductor junctions in the presence of broken time reversal <lb/>symmetry states near interfaces <lb/>Y. Tanuma * , Y. Tanaka <lb/>Department of Applied Physics, Nagoya University, Nagoya 464-8063, Japan <lb/>S. Kashiwaya <lb/>Electrotechnical Laboratory, 1-1-4 Umezono, Tsukuba, 305-0045, Japan <lb/>(October 30, 2018) <lb/>Abstract <lb/>In order to clarify the influence of (the presence of) the broken time-reversal <lb/>symmetry state (BTRSS) induced near the interface, tunneling conductance <lb/>spectra in normal metal / d x 2 −y 2 -wave superconductor junctions are calcu-<lb/>lated on the basis of the quasiclassical Green&apos;s function method. The spatial <lb/>dependence of the pair potential in the superconductor side is determined <lb/>self-consistently. We discuss two types of the symmetry on the BTRSS; i) <lb/>d x 2 −y 2 +is-wave state and ii) d x 2 −y 2 +id xy -wave state. It is shown that the am-<lb/>plitude of the subdominant component (is-wave or id xy -wave) is quite sensitive <lb/>to the transmission coefficient of the junction. As the results, the splitting of <lb/>the zero-bias conductance peak due to the BTRSS inducement is detectable <lb/>only at junctions with small transmission coefficients for both cases. When <lb/>the transmission coefficients are relatively large, the explicit peak splitting <lb/>does not occur and the difference in the two cases appears in the height of <lb/></front>
<front>arXiv:cond-mat/0106014v1 [cond-mat.supr-con] 1 Jun 2001 <lb/>Tunneling conductance of normal metal / d x 2 −y 2 -wave <lb/>superconductor junctions in the presence of broken time reversal <lb/>symmetry states near interfaces <lb/>Y. Tanuma * , Y. Tanaka <lb/>Department of Applied Physics, Nagoya University, Nagoya 464-8063, Japan <lb/>S. Kashiwaya <lb/>Electrotechnical Laboratory, 1-1-4 Umezono, Tsukuba, 305-0045, Japan <lb/>(October 30, 2018) <lb/>Abstract <lb/>In order to clarify the influence of (the presence of) the broken time-reversal <lb/>symmetry state (BTRSS) induced near the interface, tunneling conductance <lb/>spectra in normal metal / d x 2 −y 2 -wave superconductor junctions are calcu-<lb/>lated on the basis of the quasiclassical Green&apos;s function method. The spatial <lb/>dependence of the pair potential in the superconductor side is determined <lb/>self-consistently. We discuss two types of the symmetry on the BTRSS; i) <lb/>d x 2 −y 2 +is-wave state and ii) d x 2 −y 2 +id xy -wave state. It is shown that the am-<lb/>plitude of the subdominant component (is-wave or id xy -wave) is quite sensitive <lb/>to the transmission coefficient of the junction. As the results, the splitting of <lb/>the zero-bias conductance peak due to the BTRSS inducement is detectable <lb/>only at junctions with small transmission coefficients for both cases. When <lb/>the transmission coefficients are relatively large, the explicit peak splitting <lb/>does not occur and the difference in the two cases appears in the height of <lb/> 1 <lb/> the zero-bias peaks. <lb/>PACS numbers: 74.50.+r, 74.20.Rp, 74.72.-h <lb/></front>

<page>1 <lb/></page>

<front>the zero-bias peaks. <lb/>PACS numbers: 74.50.+r, 74.20.Rp, 74.72.-h <lb/></front>

<note type="footnote">Typeset using REVT E X <lb/></note>
<note type="footnote">Typeset using REVT E X <lb/></note>

<page>2 <lb/></page>

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<page>10 <lb/></page>

<body>above suppression effect is most significant. In Fig. 1(b), Re[∆ d (x)] and Im[∆ s (x)] are plot-<lb/>ted for various Z with T s /T d = 0.2 and θ = π/4. Even if the BTRSS becomes to be most <lb/>stable at θ = π/4, when the height of barrier is small, the magnitude of the subdominant <lb/>imaginary component of ∆ s (x) is not induced at all. The induced imaginary component of <lb/>∆ s (x) is enhanced with the increase of Z. <lb/>The spatial dependence of the pair potentials near the interface with the intermediate <lb/>angle (θ = π/6) is shown in Fig. 2 for various height of barrier. In such a case, both <lb/>Im[∆ d (x)] and Re[∆ s (x)] becomes nonzero and the spatial dependence is much more complex <lb/>as compared to that for θ = 0 or θ = π/4. The amplitudes of Im[∆ d (x)], Re[∆ s (x)], and <lb/>Im[∆ s (x)] are enhanced for larger magnitude of Z, where the suppression of Re[∆ d (x)] is <lb/>significant. However, the amplitudes of Im[∆ d (x)] and Re[∆ s (x)] are one order smaller than <lb/>that of Im[∆ s (x)]. <lb/>Next, we look at the magnitude of subdominant components of the pair potential at <lb/>the interface, Im[∆ s (0)], Im[∆ d (0)], and Re[∆ s (0)], for various T s , Z, and θ. As shown in <lb/>Fig. 3(a), the magnitude of Im[∆ s (0)] increases monotonically with T s for fixed θ and Z, <lb/>and it is enhanced for larger magnitude of Z. In other words, the amplitude of Im[∆ s (0)] is <lb/>sensitive to the transmission probability of the junctions. In Fig. 3(b), Im[∆ s (0)] is plotted <lb/>as a function of θ for sufficiently larger magnitude of Z(= 5.0). For θ = 0, i.e., junction <lb/>with (100) interface, the magnitude of Im[∆ s (0)] is negligibly small near the interface even <lb/>at the larger magnitude of T s . The magnitude of Im[∆ s (0)] is a monotonically increasing <lb/>function with the increase of θ and has a maximum at θ = π/4. As seen from Figs. 3(c) and <lb/>3(d), both the magnitude of Im[∆ d (0)] and Re[∆ s (0)] is enhanced and has a maximum at a <lb/>certain θ. In the intermediate θ, i.e., θ = 0 or θ = π/4, the magnitude of ∆(φ + , x) and that <lb/>of ∆(φ − , x) does not coincide any more, the interference with the quasiparticle and the pair <lb/>potential becomes complex. Then, not only Im[∆ s (0)] but also Re[∆ s (0)] and Im[∆ d (0)] <lb/>become nonzero. <lb/>Using self-consistently determined pair potentials, let us look at the normalized tunneling <lb/>conductance σ T (eV ). In order to clarify the temperature T dependence of σ T (eV ), we choose <lb/>11 <lb/>T = 0 in the left panels of Fig. 4 and T = 0.05T c in the right panels. Only for θ = 0, line <lb/>shape of σ T (eV ) is similar to that of the bulk density of states of d x 2 −y 2 -wave superconducting <lb/>state. In other cases, σ T (eV ) has a zero bias enhanced line shape. As clarified in previous <lb/>literatures 4 , when θ deviates from zero, since the injected and reflected quasiparticles have <lb/>a chance to feel the sign change of the pair potentials, zero-energy ABS is formed at the <lb/>interface. This zero-energy ABS causes the ZBCP when the magnitude of T s is small. With <lb/>the increase of the magnitude of T s , the zero energy ABS is unstable and s-wave subdominant <lb/>component is induced which breaks time reversal symmetry and it blocks the motion of the <lb/>quasiparticles. Then, the energy levels of bound state shift from zero and the local density <lb/>of states has a zero-energy peak splitting. The resulting σ T (eV ) has a ZBCP splitting as <lb/>shown in Fig. 4(b). However, with the increase of T , the slight splitting of ZBCP fades out <lb/>due to smearing effect by finite temperature and the resulting σ T (eV ) has a rather broad <lb/>ZBCP [see Fig. 4(c) and Fig. 4(d)]. <lb/>Next, we concentrate on how σ T (eV ) is influenced by the transmission probability of the <lb/>junctions, i.e., the magnitude of Z. In Fig. 5, σ T (eV ) with θ = π/4 is plotted for T = 0 (left <lb/>panels) and T = 0.05T c (right panels). For the junctions with high transmissivity, σ T (eV ) <lb/>has a ZBCP, [see Fig. 5(a)] and the magnitude of σ T (0) is firstly enhanced with the increase <lb/>of Z. In this case, the predominant d x 2 −y 2 -wave component only exists near the interface as <lb/>shown in Fig. 1(b). However, with the increase of Z, σ T (eV ) starts to have a ZBCP splitting <lb/>at a certain value of Z, where the magnitude of subdominant component Im[∆ s (x)] at the <lb/>interface becomes the same order as that of the predominant component Re[∆ d (x)]. For <lb/>sufficiently larger magnitude of Z, σ T (eV ) has a ZBCP splitting, [see Fig. 5(b) and 5(c),] <lb/>and the magnitude of σ T (0) decreases with the increase of Z. However, the above obtained <lb/>results are influenced by finite temperature effect. The right panels of Fig. 5 is shown <lb/>for the tunneling conductance in T /T c = 0.05. The slight enhanced structure of σ T (eV ) <lb/>at eV = ±∆ 0 in Fig. 5(a), 5(b), and 5(c) is invisible due to the smearing effect by finite <lb/>temperature [see Fig. 5(c), 5(d), and 5(e)]. With the increase of the magnitude of Z, σ T (eV ) <lb/>has a ZBCP with tiny dip even at Z = 2.5, where the order of the amplitude of Im[∆ s (0)] is <lb/></body>
<body>above suppression effect is most significant. In Fig. 1(b), Re[∆ d (x)] and Im[∆ s (x)] are plot-<lb/>ted for various Z with T s /T d = 0.2 and θ = π/4. Even if the BTRSS becomes to be most <lb/>stable at θ = π/4, when the height of barrier is small, the magnitude of the subdominant <lb/>imaginary component of ∆ s (x) is not induced at all. The induced imaginary component of <lb/>∆ s (x) is enhanced with the increase of Z. <lb/>The spatial dependence of the pair potentials near the interface with the intermediate <lb/>angle (θ = π/6) is shown in Fig. 2 for various height of barrier. In such a case, both <lb/>Im[∆ d (x)] and Re[∆ s (x)] becomes nonzero and the spatial dependence is much more complex <lb/>as compared to that for θ = 0 or θ = π/4. The amplitudes of Im[∆ d (x)], Re[∆ s (x)], and <lb/>Im[∆ s (x)] are enhanced for larger magnitude of Z, where the suppression of Re[∆ d (x)] is <lb/>significant. However, the amplitudes of Im[∆ d (x)] and Re[∆ s (x)] are one order smaller than <lb/>that of Im[∆ s (x)]. <lb/>Next, we look at the magnitude of subdominant components of the pair potential at <lb/>the interface, Im[∆ s (0)], Im[∆ d (0)], and Re[∆ s (0)], for various T s , Z, and θ. As shown in <lb/>Fig. 3(a), the magnitude of Im[∆ s (0)] increases monotonically with T s for fixed θ and Z, <lb/>and it is enhanced for larger magnitude of Z. In other words, the amplitude of Im[∆ s (0)] is <lb/>sensitive to the transmission probability of the junctions. In Fig. 3(b), Im[∆ s (0)] is plotted <lb/>as a function of θ for sufficiently larger magnitude of Z(= 5.0). For θ = 0, i.e., junction <lb/>with (100) interface, the magnitude of Im[∆ s (0)] is negligibly small near the interface even <lb/>at the larger magnitude of T s . The magnitude of Im[∆ s (0)] is a monotonically increasing <lb/>function with the increase of θ and has a maximum at θ = π/4. As seen from Figs. 3(c) and <lb/>3(d), both the magnitude of Im[∆ d (0)] and Re[∆ s (0)] is enhanced and has a maximum at a <lb/>certain θ. In the intermediate θ, i.e., θ = 0 or θ = π/4, the magnitude of ∆(φ + , x) and that <lb/>of ∆(φ − , x) does not coincide any more, the interference with the quasiparticle and the pair <lb/>potential becomes complex. Then, not only Im[∆ s (0)] but also Re[∆ s (0)] and Im[∆ d (0)] <lb/>become nonzero. <lb/>Using self-consistently determined pair potentials, let us look at the normalized tunneling <lb/>conductance σ T (eV ). In order to clarify the temperature T dependence of σ T (eV ), we choose <lb/></body>

<page>11 <lb/></page>

<body>T = 0 in the left panels of Fig. 4 and T = 0.05T c in the right panels. Only for θ = 0, line <lb/>shape of σ T (eV ) is similar to that of the bulk density of states of d x 2 −y 2 -wave superconducting <lb/>state. In other cases, σ T (eV ) has a zero bias enhanced line shape. As clarified in previous <lb/>literatures 4 , when θ deviates from zero, since the injected and reflected quasiparticles have <lb/>a chance to feel the sign change of the pair potentials, zero-energy ABS is formed at the <lb/>interface. This zero-energy ABS causes the ZBCP when the magnitude of T s is small. With <lb/>the increase of the magnitude of T s , the zero energy ABS is unstable and s-wave subdominant <lb/>component is induced which breaks time reversal symmetry and it blocks the motion of the <lb/>quasiparticles. Then, the energy levels of bound state shift from zero and the local density <lb/>of states has a zero-energy peak splitting. The resulting σ T (eV ) has a ZBCP splitting as <lb/>shown in Fig. 4(b). However, with the increase of T , the slight splitting of ZBCP fades out <lb/>due to smearing effect by finite temperature and the resulting σ T (eV ) has a rather broad <lb/>ZBCP [see Fig. 4(c) and Fig. 4(d)]. <lb/>Next, we concentrate on how σ T (eV ) is influenced by the transmission probability of the <lb/>junctions, i.e., the magnitude of Z. In Fig. 5, σ T (eV ) with θ = π/4 is plotted for T = 0 (left <lb/>panels) and T = 0.05T c (right panels). For the junctions with high transmissivity, σ T (eV ) <lb/>has a ZBCP, [see Fig. 5(a)] and the magnitude of σ T (0) is firstly enhanced with the increase <lb/>of Z. In this case, the predominant d x 2 −y 2 -wave component only exists near the interface as <lb/>shown in Fig. 1(b). However, with the increase of Z, σ T (eV ) starts to have a ZBCP splitting <lb/>at a certain value of Z, where the magnitude of subdominant component Im[∆ s (x)] at the <lb/>interface becomes the same order as that of the predominant component Re[∆ d (x)]. For <lb/>sufficiently larger magnitude of Z, σ T (eV ) has a ZBCP splitting, [see Fig. 5(b) and 5(c),] <lb/>and the magnitude of σ T (0) decreases with the increase of Z. However, the above obtained <lb/>results are influenced by finite temperature effect. The right panels of Fig. 5 is shown <lb/>for the tunneling conductance in T /T c = 0.05. The slight enhanced structure of σ T (eV ) <lb/>at eV = ±∆ 0 in Fig. 5(a), 5(b), and 5(c) is invisible due to the smearing effect by finite <lb/>temperature [see Fig. 5(c), 5(d), and 5(e)]. With the increase of the magnitude of Z, σ T (eV ) <lb/>has a ZBCP with tiny dip even at Z = 2.5, where the order of the amplitude of Im[∆ s (0)] is <lb/></body>

<page>12 <lb/></page>

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