Theoretical study of superconducting proximity-contact systems by use of the quasi-classical Green's function 準古典的グリーン関数による超伝導近接系の研究

博士論文

We have derived the quasi-classical Green's function in the triple layer system including semi-infinite superconductor by extending the AAHN formulation for double layer system in the clean limit and have studied some superconducting proximity contact systems. Following AAHN, we first constructed a solution of the Green's function for the superconducting finite triple layer system in a form including the spatial evolution operator within the quasi-classical approximation. Taking the limit of both the layer sizes LL, and LR to infinity ( keeping the center layer size L finite ), we have obtained the solution of the quasi-classical Green's function for the semi-infinite triple layer system. The present formulation has a great advantage in computing the self-consistent pair potential. In the conventional quasi-classical Green's function method, one has to solve the Eilenberger equation under the restriction of the normalization condition as well as of the boundary condition[23]. In numerical calculations according to that program, one needs sophisticated techniques to find converging solutions at infinities. In the present formulation, we have obtained an explicit form of the Green's function which already satisfies the boundary condition and is written by quantities converging at infinities. This reduces the numerical efforts very much. One of the applications of the present formulation is a study of the point contact experiment. Taking account of the reflection coefficients at the point contact and at the normal-superconducting interface, we have calculated the Andreev reflection. It is found that the Andreev reflection, which is closely related to the differential conductance, can have double peak structure as a function of the incident energy below the bulk energy gap 06 ΔSbulk. This happens because of a finiteness of the normal layer and finite interfacial reflection coefficients. It was found that the differential conductance also can has double peak structure as a function of the bias voltage. Next, we have obtained the density of states of the N region for the superconducting-normal-superconducting proximity contact system. It is found that the density of states has a structure below the bulk energy gap ΔSbulk, in a similar manner to the Andreev reflection in the N'-N'-S system. This structure comes from time de Gennes- Saint-James bound state which is originated by the Andreev reflection at time N-S interface. Also we have calculated the total density of states. This total density of states can be detected by scanning tunneling spectroscopy. The spatial variation of the density of states, which was detected in a STM experiment, has been realized by taking account, of the pairing interaction of the N region. We have also studied the s-wave superconductor and the d-wave superconductor junction. The self-consistent calculation of the pair potential can be achieved by use of the present quasi-classical formulation. The pair potential near the interface are suppressed due to the proximity effect. When the supercurrent flows through time junction, it has been found from the numerical results, that time spatial variation of the phase φz of the pair potential in the s-wave vs d-wave junction is different from that in the s-wave vs s-wave junction near the interface. A sign of the spatial derivative of the phase near the interface is opposite to that of the bulk region. Also, even if time reflection coefficient is zero, the phase shows a jump at the interface. To interpret these unordinary behavior, we have tried to analyze this junction by use of the G-L expansion devised for the S-S junction. It was, however, found that an obtained boundary condition is not consistent. Time anomalous behavior of the pair potential is remarkable at lower temperature. It indicates that such behaviors may not be reproduced by the G-L expansion. This problem is still to be examined in future study. We have studied some systems in the clean limit and in equilibrium. All actual systems, however, are not in the clean limit and not in equilibrium. It is important to take account of the impurity effect and extend the formulation to the non-equilibrium system.

Contents 1 Introduction / p1 2 The quasi-classical Green's function / p8 　2.1 The Bogoliubov-de Genncs equation and the Andreev equation / p8 　2.2 The quasi-classical Green's function for a finite triple layer system1G / p16 　2.3 The quasi-classical Green's function of a semi-infinite triple layer system. / p22 　2.4 The Green's function and physical quantities / p25 　2.5 The self-consistent pair potential / p29 3 Andreev reflection in the normal-normal-superconducting triple layer system / p35 　3.1 Andreev reflection in the normal-superconducting double infinite system / p37 　3.2 Normal-normal-superconducting proximity contact system / p46 4 The local density of states of the superconducting-normal-superconducting triple layer system / p59 　4.1 The de Gennes-Saint-James bound state / p59 　4.2 The superconducting-normal-superconducting semi-infinite triple layer sys-tem / p63 5 The infinite double layer system with s-wave and d-wave superconductor / p73 6 Summary / p94 7 Acknowledgment / p97