, LCvai, xi .... 5.. . ‘ 5 $3 «firms». . .... .3 awfiwum.fip$:ftfi _ ‘ fir ! .flr ”mama... Man? i. ‘ u tun: A .. ammuuhuwuf. 4. :3 x. 4a.. H‘— 1. .41.! :59 3.1.50... 1 at}! ..vxfiafilgnfi: ' u . .4 17‘s....3 1135.: .u... ‘ fin : ... 1 l L...) figuvstau‘l. 35:3} ...; . i, it: €93.13 .....annmvuo. u ......tvar‘rti q. .11 .. 5.9302! .. o - llstv . i .....5 Hang: .1. .. .. 1.5.5 . ‘ ”5.» the... 4.: :37 .l 0.45:. 9.... ! .71. V 2...! 3 u i flux. .3. ; Ll: i213 IS A ()4 XL) _. C) C) This is to certify that the thesis entitled The Usadel Formalism And Nazarov's Circuit Theory presented by MARKUS A. HOINKIS has been accepted towards fulfillment of the requirements for M. S . degree in PhYSiCS ZYWQW Major professor N Birge Date 08/07/02 0-7639 MS U is an Affirmative Action/Equal Opportunity Institution LIBRARY Michigan State University PLACE IN RETURN BOX to remove this checkout from your record. To AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 6/01 cJCIRC/DateDuepssop. 1 5 THE USADEL FORMALISM AND NAZAROV’S CIRCUIT THEORY By Markus A. Hoinkis A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Physics and Astronomy 2002 ABSTRACT THE USADEL FORMALISM AND NAZAROV’S CIRCUIT THEORY By Markus A. Hoinkis The conductance of small normal metal structures adjacent to a superconductor is determined by coherent Andreev reflection [5]. In this thesis I give a brief summary of the quasiclassical theory that describes phase—coherent transport. Two different ways to calculate the dc electrical conduc- tance of hybrid nanostructures are presented. The first method, which I will refer to as the Usadel formalism, is based on solving the Usadel equation in an angular pa- rameterization. The second method, the circuit theory developed by Nazarov, makes use of the quasiclassical theory’s formal similarity to the circuit theory for normal metal structures. It enables us to treat hybrid nanostructures as a circuit with two types of resistive elements: tunnel junctions and diffusive conductors. Several examples are treated in detail with both methods and it is shown how the circuit theory can be implemented numerically. Finally, a comparison between the two methods is made. Contents LIST OF FIGURES 1 Introduction 1.1 1.2 1.3 1.4 2 The 2.1 2.2 2.3 2.4 3 The 3.1 3.2 3.3 The Proximity Effect ........................... Andreev Reflection ............................ The Diffusive Limit ............................ The Diffusion of Andreev Pairs ..................... Theory of Nonequilibrium Superconductivity Definitions ................................. Boundary Conditions ........................... The Expression for the Current and the Filling Function ....... The Kinetic Equations .......................... Usadel Formalism The 9-Parameterization .......................... Physical quantities in terms of the proximity angle 6 ......... 3.2.1 The Density of States ....................... 3.2.2 The Conductance of a Diffusive Wire .............. 3.2.3 The Conductance of a Tunnel Junction ............. 3.2.4 Summary ............................. Examples ................................. 3.3.1 Approximations .......................... 3.3.2 Semi-Infinite Diffusive Wire ................... 3.3.3 Finite Wire ............................ 3.3.4 Finite Wire with Finite Spin-F lip Scattering Time ....... 3.3.5 Finite Wire with Finite Superconducting Gap ......... 3.3.6 Finite Wire with a Tunnel Barrier ................ iii thump-H (DOOOUQ 12 15 15 17 18 18 20 20 21 21 29 29 31 3.3.7 Double Tunnel Junction 4 N azarov’s Circuit Theory 4.1 Zero Energy Circuit Theory . . 4.1.1 Derivation ........ 4.1.2 Summary of the Rules . 4. 1.3 Examples ........ 4.2 Novel Circuit Theory ...... 4. 2.1 Derivation ........ 4.2.2 Analytical Example. . . 4.2.3 Numerical Implementation .................... 5 Conclusion A Mathematica Source Code BIBLIOGRAPHY iv 38 38 39 46 48 50 50 55 57 61 64 67 List of Figures 1.1 1.2 2.1 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 4.1 Andreev reflection: Two electrons of the normal metal form a Cooper pair in the superconductor 4:) electron gets reflected as a hole . . . . 2 Retro-reflection .............................. Normal distribution function f, odd and even distribution functions f L and f7, and fL + fT for a reservoir at potential V. Note that fL + fT = 1 — 2f .................................... 11 Graphical representation of the proximity angle at zero energy . . . . 17 Semi-infinite wire ............................. 22 Energy (space) dependence of 6 in the complex plane ......... 23 Energy (space) dependence of the density of states ........... 24 Finite wire ................................. 24 Energy dependence of 6 in the complex plane at :1: = L / 2 ....... 26 Energy dependence of the density of states 11(6) at a: = L/ 2 ...... 26 Differential conductance obtained using the non-linear Usadel equation (full line) or using the linear approximation (dashed line) (taken from [2]) 27 Differential conductance obtained using the linear approximation for various temperatures ........................... 28 Differential conductance versus temperature obtained using the linear approximation ............................... 28 Differential conductance with spin-flip scattering (taken from [2]) . . 30 Differential conductance with finite superconducting gap (taken from [2]) 3O Diffusive wire with a tunnel barrier ................... 31 Differential conductance of a tunnel barrier in series with a diffusive wire in units of the normal-state conductance GN. The linear approx- imation was used. The ratio of the resistances is RT/RD = 1 ...... 35 Diffusive wire with two tunnel barriers. G1 and 02 are the conductiv- ities of the tunnel barriers ......................... 36 Spectral vectors on both ends of a diffusive wire ............ 46 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 NS circuit ................................. Spectral vectors on the unit sphere. The tunnel junction and the dif- fusive wire are shown in accordance with the Remark in Section 4.1.2. NS circuit ................................. Spectral vectors .............................. Discretization of a diffusive wire. Connections to the ground represent the leakage currents ............................ Double tunnel junction with corresponding circuit diagram. The resis- tance of the wire is much smaller than the tunnel resistances. Connec- tion to the ground represents the leakage current. ........... Differential conductance of the double tunnel junction in units of the normal state resistance G N = GT / 2 ................... Tunnel junction in series with a diffusive wire and corresponding circuit diagram. Connections to the ground represent the leakage currents Differential conductance for RT/RD = 50 calculated with the circuit theory (full line) and with the Usadel formalism (dashed line) in units of the normal state resistance GN. (Lines almost coincide.) ...... Differential conductance for RT/RD = 1 calculated with the circuit theory (full line) and with the Usadel formalism (dashed line) in units of the normal state resistance G N ..................... vi 52 55 58 58 60 6O Chapter 1 Introduction 1.1 The Proximity Effect The proximity effect is the generic name for the phenomena appearing at the interface of a normal metal (N) with a superconductor(S). Although this term also covers the modifications induced in the superconductor, I want to concentrate on describing the effects induced in the normal metal. To this end I will take a closer look at the mechanism of charge transfer between a normal metal and a superconductor. In the next Section I will explain that normal electrons with energies below the gap cannot drain into the superconductor. This implies that Andreev reflection is the only mechanism of charge transfer for low energies. It is important to understand that concentrating only on the resistance of a NS interface itself is not the whole story. Andreev reflection is a coherent phenomenon and interference between the incoming electron and the outgoing hole persist in the normal metal at mesoscopically large distances from the interface [5]. 1 .2 Andreev Reflection Andreev reflection is the reflection of an electron as a hole at a normal-supercon- ducting boundary. n(e) > 71(6) Figure 1.1: Andreev reflection: Two electrons of the normal metal form a Cooper pair in the superconductor 4:) electron gets reflected as a hole In Figure 1.1 the density of states for a normal metal is shown on the left side, while the density of states for a superconductor according to the semiconductor model is shown on the right side. They are plotted versus the energy 6 measured with respect to the Fermi level. Since nearly all states of the superconductor for e < —A are occupied, only negligibly many electrons can go from the normal metal into this energy region. Similarly few electrons can go from the normal metal into the superconductor for e > A, because nearly all of these states are empty in the normal metal. So the only possibility to get a relevant number of electrons from the normal metal into the superconductor for temperatures and voltages small compared to the energy gap is to form Cooper pairs at the Fermi level. An electron above the Fermi level can therefore go into the superconductor by grabbing a partner from below the Fermi level. These two electrons then form a Cooper pair at the Fermi level. When dealing with situations involving superconductivity, one rather likes to think in terms of excitations. The situation before the Andreev reflection can be described by only one excitation, namely the electron above the Fermi level in the normal metal. The situation after the process can be characterized by one hole-like excitation below the Fermi level. Therefore one can equivalently say that the electron-like excitation gets reflected as a hole-like excitation. During a normal reflection only the component of the velocity perpendicular to the interface gets reversed. In contrast to this, also the parallel component reverses when an electron gets Andreev reflected. Therefore it is also called retro—reflection. The reversal of the velocity is perfect at the Fermi level (6 = 0). But if we consider an electron with a slightly different energy 6, the reversal of the direction is not perfect. To understand this it is helpful to look at the wave-vector dispersion of a free electron as shown in Figure 1.2. The electron’s wave-vector kc = kp + q is larger than the Fermi wave-vector kp. The reflected hole has the same energy as the electron, but its wave-vector is in first order k. 2 RF — q. Therefore, the wave-vector mismatch is 2q = kFE/EF. 1.3 The Diffusive Limit In this paper I focus on the case of a mesoscopic normal wire in the diffusive limit. In this regime electrons undergo a lot of elastic scattering, e.g. scattering at impurities, boundaries or lattice defects. These scattering processes have in common that the Figure 1.2: Retro-reflection electrons do not lose their phase information. Coherence of the electrons is thus not destroyed. This is in contrast to inelastic scattering, where the electrons lose their phase information. Examples for inelastic scattering are the electron-phonon and electron—electron interaction. Since there is a lot of elastic scattering, the k-vector is not a good quantum number anymore and it is better to think of energy eigenstates than of momentum eigenstates. The important length scales can be ordered in the following way [2]: 1. << L << La, (1.1) where [6 is the elastic mean free path, L is the sample length and L,p is the phase- breaking length due to inelastic scattering. 1.4 The Diffusion of Andreev Pairs The wave—vector mismatch is responsible for a phase difference of the electron and the hole. This phase difference will be of the order of 7r after an energy dependent distance L6 = (/hD/e, (1.2) where D is the diffusion constant. Furthermore, after travelling this distance the two trajectories will be separated by a distance of the order of the Fermi wavelength. It is clear that L6 plays the role of an energy dependent coherence length. At zero energy the Andreev pair is coherent up to the phase-breaking length, ch' At non-zero energy the pair is only coherent until it reaches the smaller of these two length scales. The Thouless energy 6C is the energy that corresponds to a certain length L: _hD 5c One can interpret the Thouless energy as the energy that is related to the time an electron needs to diffuse from one end of the sample to the other, treating the electrons’ paths as a. random walk. The following example might be helpful to get an idea how big this energy is: The Thouless energy for a typical metallic thin film with a diffusion constant of 100 cm2/s of 1 pm length is equivalent to ca. 75mK. Chapter 2 The Theory of Nonequilibrium Superconductivity 2. 1 Definitions The proximity effect can be treated with the semiclassical theory of nonequilibrium superconductivity. This theory describes a system of interacting electrons in terms of its correlation functions G) and G<, defined as the matrices [7]: G>(x,t;x',t’) ___ —z<( cT(x,t)c x’ t,’) cT(x,t)cl(x’,t’) )>, (2.1) ( [(x’,t’) —c[(x,t)c[(x’,t’) .< , , . 0*(x, t)cT(x,t) cT(x’,t’)c[(x,t) )> x 'x = —z I . G ( ’t’ ’t) <( —c[(x’,t’)c] x,t) —c[(x’,t’)c[(x,t) ’ (2 2) where C]? are the usual creation-annihilation operators, and () stands for an averaging over the dynamic state of the system. The “hat” denotes a 2 x 2 matrix. With these correlation functions we are able to define the retarded, advanced and Keldysh Green functions: R(x,e) = %(.7:[t9(t1—t2)(G>(x1,t1;x2,t2)—G<(x1,t1;x2,t2))]> (2.3) A(x,€) = —$(}"[0(t2—t1)(G>(x1,t1;x2,t2)—G<(x1,t1;x2,t2))]> (2.4) in“) = %(f[(G>(x1,t1;x2,t2)+G<(x1,t1;x2,t2))]) (2.5) Here () is the disorder-averaging operation, i.e. a spherical average over the Fermi surface is taken. The 6 function selects t1 — t2 > O for R and t1 — t2 < 0 for A. .7 denotes the Fourier transform in time and space and 1/ is the density of states at the Fermi level. The diagonal elements of R and A describe normal correlations between two elec- trons; a normal correlation corresponds to the excitation of an electron, or in other words, to creating one particle-like and one hole-like excitation. The off-diagonal elements describe superconducting-type correlations, that is subtracting or adding a pair of particles. The Green functions can be grouped together in a 4 x 4 matrix, called the global Green function G, which obeys the normalization condition G2 = I: G(x,e) = ( I: if ) (2.6) The “check” denotes a 4 x 4 matrix. With this definition we can formulate the basic equation governing the electron transport, called the Usadel equation [7]: 71 7—Sf hDV (CV6?) +2; [H,G] — [rzérmé] = 0. (2.7) Here D is the diffusion constant and Tsf is the spin-flip scattering time. Square brackets denote a commutator of two matrices. Further symbols used in this equation: - f- 0 . . . T,- = < 0' . > , where T,- are the Pauli matrices. Ti . H O . - . . . . 6 z'A‘ H—(O 1:1),Whel‘eH—ETZ+LTIR6[A(1L‘)]+2TyIm[A($)] — (2A 6 ), e is the energy measured with respect to the Fermi- level and A is the superconducting pair potential. Inelastic scattering can be included in this theory by adding an imaginary part to the energy: e ——> e :l: zit/Tin. Here the + and — are to be used in the retarded and advanced part of H, respectively. To incorporate the effect of a magnetic field, the derivative operator V must be replaced by its covariant form (V + ZifiAfZ). The advanced and retarded Green function are redundant because of the electron- hole symmetry. They are related through [7] A = 4212173.. (2.8) At zero energy this can be written as A = 2.1%.. (2.9) 2.2 Boundary Conditions The retarded Green function in a normal reservoir can be found using Equation (2.7). Since in a reservoir the Green function G is constant, the first term equals zero. Neglecting spin-flip scattering leaves us with [EL C] = 0. (2.10) This implies the following condition for R. [H,R] =0 (2.11) In the normal metal, the pair potential A is zero, so the above condition reads _ E 0 R11 R12 _ 0 “R12 o—llo )(R. gal—2e . >- Therefore R has to be diagonal. It follows from the normalization condition G2 = 1 that also R2 = 1. Together with the fact that R is traceless we obtain the following result for the retarded Green function in a normal reservoir: . 1 0 R: < 0 _1) (2.13) In a superconducting reservoir we again keep only the second term of Equation (2.7) for the same reasons as before. But this time, the pair potential A is nonzero, and we consider energies much smaller than A. Then we get the condition __ A A _ 0 2A)“ R11 R12 we [(2, m... )1. (2.14) Since we can write A as |A|e""*° [7], we get ( €wR21 —‘ 64901212 6i“°(R22 - R11) ) = 0 (2 15) 6—w(Rii — 322) 6450312 — €wR21 That implies that R11 2 R22 2 0, because R is traceless. From the normalization of R we get Rf, + R12R21 = 1. Thus, we can write R12 = 61" and R21 = 6‘“, where X is real. It follows from Equation (2.15) that O = eW‘X) — e"i(‘p‘X) = 22' sin(90 — x). Therefore, we can finally write the retarded Green function in a superconducting reservoir as 1%:(6; 6: ). (2.16) For the general case, that is without the restriction 6 << A, the expression for R in a superconducting reservoir will be derived in Section 3.1. 2.3 The Expression for the Current and the Filling Function The advanced and retarded Green functions A and R contain the equilibrium proper- ties of a structure, while the Keldysh Green function R determines the propagation of the nonequilibrium carriers in the structure. Therefore, one must determine If in order to get the electric current through a structure. The basic expression for the electric current through a diffusive wire is [7] 1 = fl / deTr 1+. (var + awn] (2.17) 88 ’ where 0 is the normal-state conductivity and S is the cross—section area of the wire. A similar expression describes the electric current across a tunnel junction [7]: US . A A A A A A A A 1 = g; [deTr [7. (121K2 + K1212 — 122K. — 19.41)] , (2.18) where the subscripts 1 and 2 refer to the different sides of the tunnel junction. The normalization condition for the global Green function G leads to RR + If A = O. This allows us to express the nonequilibrium Green function If in terms of the equilibrium Green functions A and R [7]: fr = Rf — M, (2.19) where f is called the filling matrix. This matrix can be taken to be diagonal [4], and is related to the distribution function for electrons, f (6), and the distribution function for holes, 1 — f (-€), in the following way: f: < 1— ENE) 2f(—2) _ 1 ) (220) For practical calculations, it is convenient to decompose the filling matrix in an even and odd part with respect to the energy, called the transverse and longitudinal part, respectively [3]: f= M + fri'z, (2.21) fL(€) = ‘fo—flv fT(€) = fT(—€) (222) The odd part has the symmetry that corresponds to a change in temperature; the even part is a measure for the deviation of the electric potential from equilibrium. 10 1 . 1 ‘40» l —év 0 el/ l ...: l A: o p j . -1 . . . . —eV 0 eV 6 ‘ 1‘ 1 fr . Q ’7‘ 1 L -1 . . —eV 0 eV 6 1. J: + 0 ..j -1 . 1 . —eV 0 eV 6 Figure 2.1: Normal distribution function f, odd and even distribution functions fL and f1, and fL + fT for a reservoir at potential V. Note that fL + f7 = 1 — 2f. The boundary conditions for fL and fT in both a normal and superconducting reservoir at potential V are: 1 e 6—eV ——- t 1 t h . fL 2(ani 2kBT + an 2kBT) (2 23) 1 e+eV e—eV =— t.h———t l ) 2.24 h 2(an ZkBT am 21:31“ ( ) The distribution functions are plotted in Figure 2.1. 11 2.4 The Kinetic Equations The kinetic equations can be derived from the Keldysh part of the Usadel equation [1], which reads hDV (RVR + RVA) + [H, K] = 0. (2.25) Insertng [1’ = Rf — fA leads to A A +v2f — R(V2f)A — R(vf)VA— (Vf)AVA— f(v hD[(VR)2f + (VR)va — (v R)(Vf)A + R(V2R) f + 2R(VR)V vf AV] (2.26) +RRf— RfA— RfR + fAH = To rewrite this we use the retarded part and the advanced part of the Usadel equation: hD((VR)2+RV2R) + [RR] = 0 (2.27) A hD((VA)2+AV2A) + [H,A] = 0 (2.28) We can then write D[V2f + R(VR)vf — (vf)AvA — V(R(vf)A)] +R [H,f] — [H,f] A = 0. (2.29) In the following I will only consider the situation of a normal metal where A = 0. Thus, R = 61 and [Rf] = 0. Now we use the decomposition of f into an even and odd part to extract the two kinetic equations from the above equation. For the first one, we multiply the equation by fz and take the trace: 11~[+(v‘2(fL + rzfr) + R(VR)V(fL + tzfr) —V(fL + rsz)AVA — V(RV(fL + rsz)A))] = 0. (2.30) 12 With cyclic permutations and using the fact that RA oc I, we can bring this into the form 2V2fT + "n [71. (RVR — AvA)] VfL —Tr [V (R%.A+.)] VfT — Tr [RRAR] szT = v (2VfT — Tr [Ra—.Aa] Vfr) + Tr [a (RVR — Av )] VfL = 0. (2.31) For the second kinetic equation we only take the trace of Equation (2.29) without multiplying by f2. The kinetic equations can be written in terms of spectral diffusion coefficients DL/T: V (DTVfT) + 2IijVfL = 0 (2.32) V (DLVfL) + 2IijVfT = 0. (2.33) with the energy dependent spectral quantities 1 . . . . . 21ij = E11 +z(RVR—AVA)], (2.34) 1 A . . DT = Z11 l—Ri-zAi-z], (2.35) . 1 r . . 19L = Z11 _1—RA]. (2.36) j E is a spectral supercurrent density, and is conserved. To see this we subtract the advanced component of the Usadel equation (2.27) from the retarded one (2.27): 110 ((vR)2 + RVQR — (VA)2 — AV2A) + [H, R — A] = 0 (2.37) In a normal metal R = e I, and the last commutator vanishes. Thus v (RVR — AVA) = 0, (2.38) and With this, we can rewrite the kinetic equations as conservation laws: V [DTVfT + 2IijVfL] = 0 (2.40) V[DLVfL +2IIIIjEVfT] = 0 (2.41) The first conserved quantity is the charge current density j(€), the second is the heat current density jQ(e). In this thesis I treat only situations with a single superconducting reservoir, hence the superconducting phase can be taken to be constant and there is no supercurrent: jE=0 14 Chapter 3 The Usadel Formalism 3.1 The 6-Parameterization For most purposes it is very convenient to parameterize the retarded and advanced Green function in terms of a complex proximity angle 6. In the literature many different parameterizations are used. I have based this Chapter on S. Gue’ron’s Ph.D. thesis [3] and will stick to the formulation used therein: cos6 e’i‘p sin6 ) R = cos 672 + 62126 (cos (on. + 81“ 9071/) 2 ( 6W9 sin 6 — cos 6 — cos 6* e‘i‘p sin 6* ei‘p sin 6* cos 6* ) (3'1) A = — cos 6%. + 32716“ (cos go‘fz + sin nyl = ( If we replace the retarded Green function R in the Usadel equation (2.7) by this notation, then we can extract the two conditions out of the retarded part; in one dimension, they read: 2 . ,. 2 hDd 6 + [if _ <1 + £261.16 + %AI> )cos6] sin6+A(:I?)C089 = 0 (32) 78}? 7'31 2 (1.17 h and d ([69 2e , 2 E [(11:13 + EA1)s1n 6] — 0 (3.3) I will refer to Equation (3.2) as the Usadel equation, as well. The symbols used are: 15 6: Complex proximity angle 6: Energy measured with respect to the Fermi level 7'51: spin-flip scattering time o to: Superconducting phase Ax: x-component of the vector potential 0 A(:r): Pair potential The boundary conditions for the retarded and advanced Green functions derived in Section 2.2 translate into the following conditions for 6: In a normal reservoir 6 = 0, whereas in a superconducting reservoir 6 = 7r/ 2 for 6 = 0. To find the value of 6 in a superconducting reservoir for nonzero energy, one can assume that the order parameter A is constant, and the Usadel equation reduces to tan6 = 2%. (3.4) Thus, A _{ g+iarctani forltl < A _ iarctani— for|e| > A ' (3.5) Tunnel junctions can be included by applying the following boundary conditions, valid for constant phase (,0: d d C . 01E61 = 023:;62 2 TS: 8111(62 — 01) (3.6) Here the subscripts 1 and 2 refer to the two sides of the tunnel junction, and GT denotes the conductance of the tunnel junction. To get a feeling for the complex proximity angle 6, it is helpful to represent it graphically. At zero energy, 6 is real and can be interpreted as the latitude on a 16 unit sphere (see Figure 3.1). The representative point of a superconducting reservoir lies then on the equator, with a longitude equal to the superconducting phase (p. A normal reservoir has a proximity angle of zero at all energies and is thus represented by the north pole. >2 v l 0 0 ],1.. 8 Figure 3.1: Graphical representation of the proximity angle at zero energy From this interpretation one can see that 6 has the following physical meaning: It is a measure of the “superconducting-like behavior” of the metal at a given point. 3.2 Physical quantities in terms of the proximity angle 6 In this Chapter I will consider the physical setup of a one-dimensional system, consist- ing of diffusive wires and tunnel junctions that connect a superconducting reservoir at 2: = 0 with a normal one at a: = L. The superconducting reservoir is at zero potential, while the normal one is kept at a voltage V. Since there is only one superconducting reservoir, the phase (,9 is constant. 17 3.2.1 The Density of States The local density of states can be expressed in terms of 6 as 71(6) = nNRe(cos 6), (3.7) where "N is the normal density of states at the Fermi level [3]. 3.2.2 The Conductance of a Diffusive Wire An expression for the electric current in a diffusive wire in terms of the proximity angle 6 can be obtained starting from Equation (2.17). The next step is to express If through the filling function f and the Green functions R and A according to Equation (2.19). Writing R and A in terms of the proximity angle 6 finally leads to the following expression for the spectral current1 of a diffusive wire with constant phase: [(6) = :—S cosh21m(6(e,x))%fr(e,x) (3.8) 8 This is similar to the result in the absence of the proximity effect with a renormalized, energy dependent diffusion constant [3] DT(e, r) = D cosh2 Im(6(e, 1)). (3.9) We see that the diffusion constant and therefore also the conductivity is always larger than in the normal state. Since 6 is strongly energy dependent, so is the conductivity enhancement. The total current is obtained by integrating over energy: I = /d€1(€) (3.10) 1With spectral current I denote currents per energy slice. In this Chapter I will not always emphasize that we are dealing with electric currents. However, do not confuse the electric current with the matrix current, which will be introduced in Chapter 4 18 We can do that as long as the contributions to the current at different energies are conserved. However, this is only true in the absence of inelastic scattering, which would mix correlations at different energies. I will now derive how to express the current through the values of the filling function in the reservoirs [4]. To this end we use the fact that the spectral current does not depend on the spatial coordinate :6. Hence we can write L 1 2 L fT(€’ L) — fT(€’0) 2 /0 dz ffng(€’$)_ — a; 1(6)/odxcoshzlni(6(e 36)) (3.11) The electric current through a diffusive wire now reads Q—ZS/df fT(€L)_f1T(€O) (3.12) /0L:6d mcoshz Im(6(€, :L‘)) Taking the derivative of I with respect to the voltage V, we get the differential conductance. Since fT(e, L) is the only term in Equation (3.12) that depends on V, we can write d_1_§:d€/ / d 1 ah]; L). cosh2 Irn1 (6(6, 56)) (3.13) At zero temperature, we can avoid the integration over energy, as I will explain in the following. In the normal reservoir the transverse filling function fT has the shape as plotted in Figure (2.1). Taking the derivative with respect to V at zero temperature gives the following result: dfT(Ev L) dV 2 6 (6(6 + eV) + 6(6 — eV)) (3.14) Thus we get for the differential conductance of a diffusive wire at zero temperature: fi_ 50 (3.15) dV /dr 1 0 I cosh2 Im(6(eV,:6)) 19 3.2.3 The Conductance of a Tunnel Junction The spectral electric current through a tunnel junction in terms of the proximity angle 6 can be found in a very similar way as the current in a diffusive wire, only that we have to start from Equation (2.18). The result can be decomposed into a quasiparticle, Andreev and Josephson contribution [7]: [(6) = qu(6) + IA(€) + IJ(€) (3.16) The spectral quasiparticle current at a certain energy is proportional to the density of states on both sides of the tunnel junction: 1,, (6) = (7:;- (fm — fm) Re[cos 61]Re[cos 62] (3.17) The spectral Andreev current reads GT =$ 1,4(6) (fm — fT’2)R€[SlI161]R€[SII162]. (3.18) If the phase is constant, there will be no Josephson current. For zero temperature we can simplify these results and integrate over energy using the 6-functions of Equation (3.14): 54‘]; = GT [Re[cos 61(6V)]Re[cos 62(eV)] + Re[sin 61(eV)]Re[sin 62(6V)]], (3.19) where V is the voltage across the junction. 3.2.4 Summary It is clear now what we have to do in order to calculate the electric current through a system. First we have to solve the Usadel equation, so that we get 6 at every point :6 and at every energy 6. According to Equation (3.9), we then also have the renormalized diffusion constant. Since we know the transverse filling function in 20 the reservoirs, we can finally use Equation (3.12) to calculate the current through a diffusive wire. The current through a tunnel junction is given by Equations (3.16) to (3.18). At zero temperature we can avoid the integration over energy using Equations (3.15) and (3.19). 3.3 Examples 3.3.1 Approximations The Usadel equation can be simplified using the following assumptions: For the physical situation that there is only one superconducting reservoir involved, the phase (,0 is constant and the corresponding term can therefore be dropped. Furthermore, in the normal metal the pair potential A(:6) is zero, so we can drop this term, too. Assuming that there is no magnetic field present, we can set the vector potential A to zero. In addition, I will make the approximation that there is no inelastic scattering. This guaranties that the energy is real and does not have an imaginary part. I also assume that the spin-flip scattering time is infinite and we can drop the corresponding term. We are then left with the following equation: 2 d 6 . . hDd—IL‘Z 'l‘ 226 81n6 = 0 (3.20) 3.3.2 Semi-Infinite Diffusive Wire In this section, I will solve Equation (3.20) for the case of a semi-infinite normal wire connected to a superconducting reservoir (see Figure 3.2). In order to integrate the second order differential equation (3.20), I multiply it 21 Figure 3.2: Semi-infinite wire with 3%: (126 d6 . . d6 hDEE—zfi + 226 811105 = 0 Now we can integrate once: E E 2 + 276(~ cos 6) — const(6) 2 d3: — Inserting the boundary condition 6],woo = 65 const(6) = —2’i€ EP— 512 2--2i6(cos6— 1) 2 dz _ Using a trigonometric identity yields hD d6 2 , , 2 0 —2— (a) — —426(Sin 2)- We can now solve for £2: d6 _ _ —876 sin 6 dm " RD 2 This differential equation can be easily solved by separating the variables: / d6 __ ]—876/d$ sing _ hD Integrating this, we obtain 6 —27'6 ln(tan 4) —— —]] 752: + const(6). 22 (3.21) (3.22) = 0 determines the constant: (3.23) (3.24) (3.25) (3.26) (3.27) (3.28) The constant can be determined by inserting the boundary condition 6|x=0 = 5- (where I assume that 6 << A): 6 ——2 '_ ln(tan 4) = — T113633 + ln(tan 78:) (3.29) Finally, this can be solved for 6: 6(1), 6) = 4arctan ](\/2 — 1) exp ((2 — 1) i ] , (3.30) 6c where Cc is the Thouless energy CC = hD/TQ. The energy dependence of 6 is plotted in Figure 3.3 in the complex plane. The reduced energy 6 / EC increases from zero to infinity. Since this parameter also includes the coordinate :6, this plot can also be interpreted as showing the spatial dependence of 6 at a given energy. lme 1 u%=0 1 n/2 Re 6 ( Figure 3.3: Energy (space) dependence of 6 in the complex plane The density of states, obtained from Equation (3.7), is plotted versus 6 / 6C in Figure 3.4. Again, this parameter includes both energy and space dependence. Notice that there are no states at zero energy, but there is no finite gap. Since we have considered energies much smaller than the superconducting gap energy, this gap does not come into play here. 23 n(e)/nN. 1 2 A 3 21 5 eV/eC Figure 3.4: Energy (space) dependence of the density of states 3.3.3 Finite Wire In this Section I will treat the case of a diffusive wire between a superconducting and a normal reservoir (see Figure 3.5). The finite wire can be treated in the same way as the semi—infinite wire, however, there is one complication. The boundary condition at Figure 3.5: Finite wire :2: = L does not include information about the derivative of 6, and as a consequence of that, the differential equation cannot be solved analytically. In order to get an analytical result one can linearize the Usadel equation: 2 72,1)fl + 22'69 = 0 (3.31) due? I want to emphasize that the necessary assumption 6 << 1 is not well justified in this example. However, it is helpful to go through this example, to become familiar with the technique. This approximation is better suited for the example in the next 24 section, where a tunnel barrier is present. It is easy to see that the general solution of Equation (3.31) reads _2ieat - _2’i€$ 0=AeV 5D +Be ED. (3.32) When we introduce L6 = (MD/e as the length corresponding to the energy 6, we can write 9 = Ae(1_ilm/L‘ + Be“(1”i)I/L€. (3.33) The boundary conditions 6(33 = 0) = 7r/2 and 6(27 = L) = 0 lead to the following conditions for A and B: 7r A B = — + 2 AeU—ilL/L‘ + Be’u’ilL/L‘ = 0 (3.34) If we solve that for A and B we get (l-ill/Lc -(1-i)1‘/Le 9 = 3 e + e (3.35) 2 1 _ €2(1—i)L/Lc 1 _ e—-2(1—z')L/L6 ' The proximity angle 6 and the density of states Tl(€) are plotted in Figures 3.6 and 3.7. The results look similar to the ones for the semi-infinite wire except for the fact that we have to plot the results at a certain distance :15. Now that we have calculated 6, we can get the differential conductance at zero temperature using Equation (3.15): d1 50 (3.36) a? = [14613: 1 o cosh21m(6(e,:r)) This conductance is plotted in Figure 3.8 versus the applied voltage as a dashed line. The full line was obtained solving the non-linear Usadel equation. 25 ImO K\a% y%=0 1 11/2 Re 9 ( Figure 3.6: Energy dependence of 6 in the complex plane at :1: = L/ 2 ii L2 3 4‘ L 5 eV/ec Figure 3.7: Energy dependence of the density of states 72(6) at a: = L/ 2 It is interesting to follow the path of the differential conductance coming from high voltages. It is intuitively understandable that the differential conductance equals the normal-state conductance for high voltages, because there the system is driven far out of equilibrium and superconductivity is suppressed. Lowering the voltage therefore increases the conductivity. It is not so obvious that the differential conductance comes down again to the normal-state conductance for no applied voltage. The following argument shows why the conductance equals the normal state conductance at this 26 1.15 ~ 1-1 r 2 o ~T> .- ------------- 1.05— -' """ 1 l l l l 0 5 10 15 20 eV/eC Figure 3.8: Differential conductance obtained using the non-linear Usadel equation (full line) or using the linear approximation (dashed line) (taken from [2]) point: At zero energy, the proximity angle is real in both reservoirs, namely 0 and 7r/2. The Usadel equation at zero energy reads (126/(1232 = 0, therefore 6 is just a linear interpolation between the two reservoirs and is real everywhere. Thus, the conductance, which is proportional to cosh2 Im6, is unchanged. The maximum of the curve corresponds roughly to a 15% conductance enhance- ment at a voltage of ca. Sec / e. This implies that the temperature must not be higher than roughly EC/kB in order not to have this effect washed out by thermal fluctuations. To get the differential conductance at nonzero temperatures, we can use Equation (3.13). Writing out dfT/dV leads to 1 1 1 « , + . . 3.37 :8A13T0d6/6/0Ld1, (coshz(‘.+e" ) cosh2("2‘ 6" )) ( ) ZkBT kBT cosh2 Iml(6(€,:1:)) In Figure (3.9) the differential conductance is plotted versus the bias voltage for var- ious temperatures. The effect of finite temperature is to smear out the curve. The spectral conductance is averaged over an energy window of approximately 4kBT. Fig- 27 1.15 1.1 " .di dv 1.05) T/eC = O, 1, 2, 5, 10 5 10 15 20 25 eV/eC Figure 3.9: Differential conductance obtained using the linear approximation for var- ious temperatures 1.15 1.1“ ,/GN gr (IV 1.05» 5 10 15 20 25 kBT/éc Figure 3.10: Differential conductance versus temperature obtained using the linear approximation 28 ure (3.10) shows a plot of the differential conductance versus the temperature without a bias voltage.2 The fact that the conductance comes down again, when lowering the temperature below ca. 566, is called the reentrance effect. Similar to before, this phenomenon is counter-intuitive, because normally one expects superconductivity to become stronger as the temperature is lowered. 3.3.4 Finite Wire with Finite Spin-Flip Scattering Time I will now briefly show how the previous results change if we take spin-flip scattering into account. Then the appropriate Usadel equation reads , 2 f h—Dd—g + ie sin6 — —l— cos6 sin6 = 0. (3-38) 2 dIL'2 7st" The resulting differential conductance is plotted in Figure 3.11 for various values of Lsf / L. Here, Lsf is the length that corresponds to the spin—flip scattering time: Lsf = (/DTSf (3.39) The results are relatively unaffected as long as Lsf > L. If this is not the case, Lsf acts as an effective sample length. The maxima of the differential conductances are shifted, because they are no longer located with respect to CC = fiD/L2 but with respect to the energy hD/Lgf. The reason for this is that the Andreev pairs lose their coherence after travelling this distance. 3.3.5 Finite Wire with Finite Superconducting Gap If we want to include a finite superconducting gap, which frees us from the restriction of small temperature and small bias, we cannot use the boundary condition in a superconductor, 6 = 7r / 2, any more. We must rather work with the full expression as 2The corresponding lVlathematica file can be found at www.pa.msu.edu/people/hoinkis 29 1.15 _ IOandS Lsf/L : 1 \ z 1.] - g 0 5 st; ' 1.05 - 0.2 ,/ 0.1 1 1 1 1 i O 5 10 15 20 25 eV/ec Figure 3.11: Differential conductance with spin-flip scattering (taken from [2]) 1 2 2.5 - Alec: Z 2 - Q3 5 2T; M 1 5 _ 10 ' 50, ’ 20 100 , ,,,-_ flandoo , )M “R“ ~~ L=~ 0 5 10 15 20 25 eV/ec Figure 3.12: Differential conductance with finite superconducting gap (taken from [2]) derived in Section 3.1: A iarctan i for|e| > A (3'40) _{ §+iarctan£ forlel < A Figure 3.12 shows the differential conductances calculated using this boundary condi- tion. The palpable result is that we get peaks at energies equal to the superconducting gap. These peaks are due to the singularity in the density of states at A in a super- conductor. 30 3.3.6 Finite Wire with a Tunnel Barrier Zero Energy The conductance of a diffusive wire in series with a tunnel barrier, as illustrated in the upper part of Figure 3.13, can be easily solved for small temperature and bias voltage. The first term in the Usadel Equation (3.20) is of the order of the Thouless 111/2 9m 0: .: 0 Lx Figure 3.13: Diffusive wire with a tunnel barrier energy 6,, = D/Lz, whereas the second term is of the order of e. We can drop this term when we only consider energies much smaller than the Thouless energy. That includes both small voltages and temperatures. The Usadel equation then reads 329 rind—$2 = 0. (3.41) This implies that the first derivative of 6 with respect to :c is constant. The boundary condition at the tunnel barrier is given by Equation (3.6). The tunnel barrier is adjacent to the superconducting reservoir, where 6 = 7r / 2, so we can write GT d ? cos 6m — —UE0' (3.42) 31 Here 6m denotes the value of the proximity angle at the tunnel barrier on the side of the diffusive wire. Since 6 is real in both reservoirs, 6 must be real everywhere. In the lower part of Figure 3.13 the spatial behavior of 6 is plotted schematically. In the normal reservoir 6 = O; the problem of finding 6 at every point 10 therefore reduces to the transcendental equation GT 9m —S— cos 6m 2 a—L—. (3.43) To get the spectral current through the system we can use Equation (3.12) for the diffusive wire and Equations (3.18) and (3.17) for the tunnel junction: 50 L 1 —1 10(6) : 2_€ (an _ me) (A d$COSll2 Im(6(€, :13») (3.44) IA(€) = (5—: (me — fTs) Re[sin 6m]Re[sin 63] (3.45) qu(e) = (72:1 (me — fTs) Re[cos 6m]Re[cos 68] (3.46) Here the subscripts s and n refer to the reservoirs and m refers to the point between the tunnel junction and the diffusive wire. These equations can be simplified using the following facts: 6 is real, sin 63 = 1 and cos 68 = 0. Then we have: 10(6) = :32; (an _ me) (fodecosh2 Iml(6(€ $))) (3.47) 1A(6) = 92—61(me — fTs) Re[sin am] (3.48) LIME) = 0 (3.49) Since the spectral current I = ID = I A is conserved , we can eliminate me: La' 1 [(6) f0 'xcosh21m(6(e,:r))+ 1 SO GTRe[sin 6m] = Elgffifn - fTs) (350) 32 To get the total electric current, we have to integrate over the energy. The differential conductance for V ——> 0 reads 1 1 (1an _ d . 1 v_.() —/ E L 1 2—8 (1V v_.() (3 5 ) d /0 xcosh21m(6(e,:r)) + 1 So GTRe[sin 6m] d—V At zero temperature the derivative (1an / (1V gives a 6-function. Therefore we can use the results for E = 0. Then, 6 is real and we can write a ... 1 (1V V_.0 — RT R D + sin 6m (3.52) In other words, the resistance of the diffusive wire and the renormalized resistance of the tunnel junction add in series. The renormalization factor 1/ sin 6m remains to be determined from Equation (3.43), which can be rewritten as cos 6m __ 6m _ —, 3. RT RD ( 53) This result has two simple limits: If RT << RD, then 6m z 7r/2 and d1 1 — = —. 3.54 —.0 RD + RT ( ) In the other case where RT >> RD, we can approximate 6m x RD/RT and 31716", x 6",. With this we get d—1 = 52. (3.55) 32, This result is very interesting, because the conductance is proportional to resistance of the diffusive wire. The physical explanation for this is that the electrons are confined by the disorder in the wire and get repeatedly backseattered at the interface. The different trajectories can add up coherently and so the probability for tunnelling of pairs is increased. 33 Finite Energy If we want to calculate the differential conductance at finite temperature or with a finite bias voltage, we cannot drop the energy term in the Usadel equation. But if we have a strong tunnel barrier, we can at least linearize the Usadel equation, because 6 will then be small everywhere. We already now from Section 3.3.3 how the solution to the linearized Usadel equation looks like (Equation (3.33)): 6— — Ael ””6 +Be1 ”/14 (3.56) Only the boundary conditions are different. For :1: = 0, where we denote the proximity angle by 6m, we can use Equation (3.42) to write 2'— 1 07 L. (A — B)— :5— cos (A + B). (3.57) The boundary condition at the normal reservoir (:1: = L) reads AeU—W 1.. + Be‘Al‘ilf/L‘ = 0. (3.58) These are two coupled transcendental equations for A and B. We have to solve them numerically for every energy if we want to proceed. After that we can use Equation (3.51) to get the differential conductance: 5g _ [.1 1 idfrn dV _ 70‘1“"526 dV cosh2 Inf( ((66.36)) + 1 So GTRe[sin 6m] 1 h21(6+___6_l ) + 1 2(6— 61 ) = /d6 cos 2,”) 7. com 213T (3.59) 816.37“ /0L:rd cosh2 Im( ((66,117)) + 1 So GTRe[sin(A + B)] In Figure 3.14 the result of integrating the above equation at zero temperature is plotted. 34 d_I (1V eV/6C Figure 3.14: Differential conductance of a tunnel barrier in series with a diffusive wire in units of the normal-state conductance GN. The linear approximation was used. The ratio of the resistances is RT/RD = 1. 3.3.7 Double 'Ihnnel Junction Zero Energy As a last example in this chapter, I want to treat the case with tunnel barriers on both ends of the diffusive wire (see Figure 3.15). The tunnel resistances should be much larger then the resistance of the wire. The approach will be very similar to the example before. First we will calculate 6 at every point in the structure. After that, we can get the differential conductance by setting equal the spectral electric currents through the two tunnel barriers. For this example, I only want to consider the regime where the temperature and the applied voltage are much smaller then the Thouless energy. This means that we can drop the energy term in the Usadel equation, as before, and write d26 150— = 0. (3.60) (11:2 The boundary conditions at the tunnel barriers can be combined in the following 35 6 ‘l to|=| A Y 0 L 15 Figure 3.15: Diffusive wire with two tunnel barriers. 01 and 02 are the conductivities of the tunnel barriers. equation: 95,1 sin (6,. — 6(x = 0)) = —a—x6 = G— sin (6(a: = L) — 6”) (3.61) Since 05/ L >> GI, G2, we can assign the constant value 6m to the proximity angle in the wire. Hence, the proximity angle comes down from its value of 7r/ 2 in the superconductor to O in the normal reservoir only by two jumps at the tunnel barriers. The height of these jumps can be determined by inserting the values for 6 in the reservoirs into Equation (3.61): 01 cos 6m = Gz sin 6", (3.62) We can transform this, using sin2 6m + cos2 6m 2 1, into 02 JORGE. This equation determines 6",. The lower part of Figure 3.15 illustrates the spatial cos 6m = (3.63) behavior of 6. 36 The spectral current through a tunnel barrier is given by Equations (3.17) and (3.18). When we insert the values of 6 in the two reservoirs, we see that there is no quasiparticle current through the first tunnel barrier and no Andreev current through the second. 11(6) = $81— (me — fTs) Re[sin 6m] (3.64) [2(6) = (2:: (an — me)Re[cos 6m] (3.65) Here 11 and [2 denote the currents through the two tunnel barriers. Since the spectral current is conserved, these two currents are the same, and we can eliminate firm. We get 1 1 1(6) (GlRe[sin 6m] + GgR€[COS 6m] ) = i(an — fTS) - (3°66) As in the previous example, we can now integrate over energy to get the total current. The differential conductance for V —+ 0 reads dV 1 1 d n — / d6 1 2— ;; (3.67) VTO 01 Re [lsin6 m] +,,,GgRe[cos6 ] e V"0 Again, we use the fact that ifll gives us a 6- function. Then we can insert our 6 obtained for zero energy: fl dV 1 z 0102 COS am (368) = 1 1 cos 6, V—+0 G —3m + G Glsin6m + G2 cos6m 28m m 1 We can now use Equation (3.62) to write: 611 Gng cos 6m __ = 3.69 Finally, we insert cos 6m from Equation (3.63): .11 _ 0303 (3 70) W M _ (01+ G3)“2 ' 37 Chapter 4 Nazarov’s Circuit Theory The Usadel formalism has the disadvantage that the differential equation can only be solved analytically for very simple examples. Nazarov’s circuit theory is a very elegant way to calculate the resistance of a structure without the need to solve a differential equation. However, there are also disadvantages. The circuit theory published by Nazarov in 1994 [5] is only valid in the regime where the energy of the electrons is smaller than the Thouless energy. This limits the temperature range and the magnitude of the applied voltage. I will refer to this theory as the zero energy circuit theory. The novel circuit theory published in 1999 [6] is freed from this limitation, but it is mostly suited for numerical calculations and gives analytic results only in few cases. 4.1 Zero Energy Circuit Theory In this chapter, I will first show how Nazarov’s zero energy circuit theory can be derived from the quasiclassical theory of nonequilibrium superconductivity. Then I will explain the rules of the theory and use them to go through two simple examples of how to apply the circuit theory. 38 4. 1. 1 Derivation The physical situations that can be treated with this circuit theory consist of nor- mal and superconducting reservoirs connected through either diffusive wires or tunnel junctions. The superconducting reservoirs are all biased at the same voltage in or- der to avoid ac Josephson effects. Let us set this voltage to zero. However, the superconducting reservoirs are allowed to have a phase difference. In order to derive the rules for the circuit theory, we start with the Usadel equation: 50v (ave) +1 [11,6] _ h [565,6] = 0 (4.1) 2Tsf Using the Einstein relation 0 = 6201/, where a is the conductivity and 1/ is the density of states at the Fermi level, we can write the above equation as a conservation law . ' - 1 v . Vj + 821/ 1H — —sz‘fz, G = 0, (4.2) L1 Tsf where the matrix current densityj is defined as j = 0(x)GVG’. (4.3) Note that the matrix current density is a vector in real space, and is printed boldface. Its three components are each a 4 x 4 matrix. The important approximation in this circuit theory is to neglect the second and third term in Equation (4.1). In the following I will show when this is justified. The first term in Equation (4.1) is of the order of the T houless energy 6C = D/L2, L being the system size. The second term in Equation (4.1) is of the order of 6 in the normal metal, where A = 0. This implies that we can neglect the second term in the normal metal as long as 6 << EC. This is true if T << 6C and V << Go. The third term can be neglected if the Thouless energy 6C is large compared to 71/ Tsf. In this 39 approximation we can rewrite Equation (4.2) as Vj = 0. (4.4) If there are tunnel interfaces in the structure, the Green function G is in general different on both sides of the interface. According to [5], the boundary conditions read : g X ' - N1: 12—? [G1(X).Gz(x)]. (4.5) where N is a vector normal to the interface at point x, g(x) is the conductance of the interface per unit area at the same point, and the subscripts 1 and 2 refer to the two sides of the interface. The Equations (4.3) and (4.4) resemble the equations describing the conductivity of a normal metal: Vj = O, (4.6) j = —0(x)Vu. (4.7) Here j is the electric current density, and u is the electrostatic potential. Equation (4.5) is similar to the equation describing the voltage drop at an interface in the normal state situation: N5 = g(x) [151(X) — ”200] (4-8) The circuit theory for normal metals can be derived from Equations (4.6) to (4.8). Our goal is now to get a circuit theory including superconducting reservoirs from the similar Equations (4.3) to (4.5). It is possible to separate the equations for the advanced and retarded Green functions, determining the equilibrium properties, from the equations for the Keldysh Green function, determining the nonequilibrium properties [5]. 40 The two Definitions (4.3) and (4.5) for the matrix current in a diffusive conductor and a tunnel junction allow to combine the corresponding Equations (2.17) and (2.18) for the electric current into the following expression: j..,(x) = {31; / den [6,j“’(e, 5)] , (4.9) where j", (6, :r) is the Keldysh component of the matrix current density. In this chapter, I will explicitly add the subscript “e1” if a current is an electric current, so that it will not be confused with a matrix current. Starting from Equation (4.3), we can now derive an expression for the electric current in a diffusive wire. According to Equation (4.9), we have to calculate the Keldysh part of the matrix current density: 3" = a (RVK + KVA) (4.10) From Equation (2.19) we obtain: 3" = a (R(V1?)f + RRVf — R(Vf)xi — 1?va + 1?va — f/iVA) (4.11) Using R2 = 1, which follows from the normalization condition for G, and replacing A by —fszfz according to Equation (2.9), we can write 3" = 0 (RH/1‘2) f + vf + R(Vf)7‘zR%Z — fszVsz). (4.12) Here I made use of the fact that fzfz = 1. Now we can take the trace to get the electric current: jel(x) = i / d6Tr [6, (MW?) f + vf + 11(Vf)%zf2+z — fu‘zvfzr—J] (4.13) Since the trace is a invariant under cyclic transformations, we can write j,.(x) = é/dsn [6.1%(vfz)f+ avf + 621%,.szth — aRvRaaf] . (4.14) 41 Using again 627‘} = 1 and R2 = 1, we are left with jel(x) = Egg/(16% [2657f] . (4.15) After inserting Equation (2.21), this reads 1.4x) = g, / Mr [272qu + 64.)] . (4.15) The term with f1, drops out, because 7'; is traceless: 1.1x): 207, / 46% (4.17) To measure a deviation of the quasiparticle distribution from the equilibrium, it is helpful to define 1 ((x) = —2— / d6 fT. (4.18) e In a normal system, ((x) is the local electrostatic potential. The final expression for the electric current density in a diffusive wire in terms of this C reads jel(x) = —0(x)V((x). (4.19) Now I will similarly derive an expression for the electric current through an inter- face, starting from Equation (4.5). The Keldysh component of this equation reads N3“ = 9—(2’9 (13,162+ R142 _ KK _ K24), (4.20) where the subscripts 1 and 2 refer to the two sides of the interface. Expressing K in terms of f, we obtain 2 x - . - . ~ ~ ~ . .. . . - NJK = %)(R132f2 — R1f2A2 + R1f1A2 - f1A1A2 — R2R1f1+ R2f161— R2f261 + 132621211)- (421) 42 The advanced Green function A will now be replaced by —fZsz again: 4. _ 46:)”. NJ — —2‘(RiR2f2 + le2TzR2Tz — R1f1T2R2Tz — fiTzR1R27'z _ R2R1f1 _ R2f1FzR1f-z + R2f2f-2R1f-z + f2+zR2R1Tz) (422) To get the electric current, we use Equation (4.9): . 1 . 2 NJ.,,(x) = g[c1611 [TzNJK] (4.23) By applying cyclic permutations, we obtain Nje1(X) = gé:) /d6T1‘[’f'z (R1122 + R2121) (f2 — f1)] . (4.24) In the following I will parameterize B = s f. Underline denotes a vector in the Pauli matrix space: s I: = 5161. + syfy + 5272.. This is possible because If is traceless. Considering the normalization condition G2 = I, it is clear that s is also normalized: £2 = 1. In this notation the following rule holds: A (61:1) (.8562) = 61 '62 + ii (21 X .85) (4-25) Applying this to Equation (4.24) gives us . 9(X) - ~ ~ N1..(x) = 5; [411525.445 — 11)] . (4.26) Expressing the filling matrices through their longitudinal and transverse parts leads to . X A . - .. . N1e1(x) = %—l f 4611 [7.24.5.2 (11,21 + fm. — 11.11 — fag] e x = age—)fdééléflfre — fT,1l- (427) Using the Definition (4.18), we finally obtain Nje1(X) = 9(X)§1§21C1(X) — (2001- (428) 43 Summarizing, we can see that the equations describing the electric current in a system with superconducting reservoirs (4.19) and (4.28) are very similar to the equations in a normal system (4.7) and (4.8). Also the conservation of current is valid, as one can easily see from Equation (4.4): Vie1(X) = 0 (4-29) The only difference is that we have to replace the electrostatic potential u(x) by ((x), and that the electric current through an interface is renormalized by the factor §1§2- Before we can discuss the rules of the circuit theory, we have to find a way to calculate this factor. Therefore, we take a look at the retarded part of the matrix current densityj in a diffusive wire, which is according to its Definition (4.3): A 33 = 0 (RVR — (yam) (4.30) Using the above mentioned notation, we can write that as 3R = 0164) (2174) — (6V4) (+811. (4.31) Applying rule (4.25) gives us jR = 01' (s x V§)i. (4.32) We now define a matrix current density in the Pauli matrix space j through 3’? = ij - i. (4.33) With that we can write down the final expression 1 = 0(X)§ >< V5. (4.34) To get a similar expression for the matrix current density at an interface, we start with the retarded part of Equation (4.5): Njfi : €- [131,132] 44 The final expression for the matrix current density in the Pauli matrix space is there- fore The matrix current density _j_ is also conserved, since it is proportional to the retarded part of the matrix current densityj, which is conserved according to Equation (4.4): Vj = 0 (4.37) As a result of the conservation of matrix current j, the Equations (4.34) and (4.36) can be discretized into a circuit theory. Let us consider a diffusive wire of length l and cross section S. Then the matrix current _I_ that corresponds to the matrix current density 1' can be written as d 1 = _ , . _ 505 X dx§ (4 38) Taking into account that the matrix current is conserved, it is evident from Figure 4.1 that the spectral vector 6 changes at the same rate from one end of the wire to the other. Thus, the matrix current is S 1 = 121$ a, (4.39) l [$5.1 X £2 We can rewrite that as £1 X £2 R01 2 arccos(§1§2), (4.40) — 61-64162)? where RD is the resistance of the wire. 45 Figure 4.1: Spectral vectors on both ends of a diffusive wire For a tunnel junction, we simply have to multiply Equation (4.36) with the cross section area S to obtain a similar relation: 4.1.2 Summary of the Rules The Equations (4.19) and (4.28) together with the conservation of physical current allow to formulate rules very similar to the circuit theory of normal systems. The magnitude of electric current is proportional to a drop of C over an element. This C takes the part that the electrostatic potential has in normal systems, although it is not the same, since the mechanisms of transport are not identical. we have two different kinds of resistive elements: Whereas transport in the dif- fusive wire does not depend on spectral vectors, the current in a tunnel junction is renormalized by a factor s13). This factor can be obtained with help of the matrix current _I_. The boundary conditions for the spectral vectors in a normal or super- conducting reservoir can be found from Equations (2.13) and (2.16), respectively. In a normal reservoir, R = $2 and, thus, s = (0,0,1). In a superconducting reservoir, A R = cos 99%,, + sin (of?) and, thus, s = (cos (,0, sin 99, 0). 46 In the following I will summarize the rules of the circuit theory [5]: (I). The conductance is the same as for a normal circuit with renormalized tunnel conductivities. The renormalization factor is given by the scalar product of spectral vectors belonging to two shores of the tunnel junction. The following rules determine the spectral vectors: (II). The spectral vector in a normal reservoir is (0,0,1) (north pole of the hemi- sphere). The spectral vector in a superconductor is (cos (,0, sin (0, 0) (equator of the hemisphere). (III). The matrix current 1 is perpendicular to both spectral vectors at the ends of an element. Its magnitude is given by either I = G'Da for a diffusive conductor, or I 2 GT sina for a tunnel junction, 61 being the angle between the spectral vectors at the ends. (IV). The matrix current 1 is conserved in nodal points. Remark An equivalent way of formulating these rules can be found in [3]. Instead of a matrix current, the author uses forces acting on representative points on the unit sphere to determine the spectral vectors. Springs pulling towards the equator or the north pole represent the tunnel junctions and the diffusive wires. The stiffness of the springs is given by the conductances of the wires and the tunnel junctions. The springs representing diffusive wires lie on the surface of the unit sphere, whereas tunnel junctions are represented by springs stretched along the cord joining the end points on the sphere. 47 4. 1.3 Examples Tunnel Junction and Diffusive Wire in Series A simple example is a tunnel junction in series with a diffusive wire between a super- conducting and a normal reservoir as pictured in Figure 4.2. RT 3 RD N 24 —® Figure 4.2: NS circuit The resistance of this circuit can be found applying rule (I): R = + RD (4.42) Here RT and RD are the resistances of the tunnel junction and the diffusive wire, respectively, and §m and s3 are the spectral vectors as shown in Figure 4.3. 5N _ RD §S Figure 4.3: Spectral vectors on the unit sphere. The tunnel junction and the diffusive wire are shown in accordance with the Remark in Section 4.1.2. To determine the factor §m§s = cos a, we make use of the conservation of the matrix current in the nodal point. According to rule (III), the magnitude of the 48 matrix current in the tunnel junction is given by sina I = . 4.43 RT < ) For the diffusive wire we get 7r / 2 — a I = —. 4.44 RD ( > Therefore, we get the following relation to implicitly determine a: sina : 7r/2—a (445) RT RD Note that this is the same result as the one we obtained in Section 3.3.6. Equations (3.52) and (3.53) are identical to the above result, we only need to replace 6m by 7r / 2 — a. Double Tunnel Junction In the next example, two tunnel junctions are in series between a superconducting and a normal reservoir (see Figure (4.4)). If the resistance of the metal between the two junctions is much smaller than the junction resistances, we can regard it as a node. N R‘ R2 8 @— E Figure 4.4: NS circuit From rule (I) we get the resistance of the structure: R: 121+ R2 = R1+ 112 (4.46) stm §m§s sina cosa The spectral vectors and a are shown in Figure (4.5). 49 R2 Figure 4.5: Spectral vectors Rule (III) gives us the magnitude of the matrix current: sin 04 cos a I = = 4.47 R2 R1 ( ) Replacing cos a by \/1 — sin2 a, we can eliminate sina out of Equations (4.46) and (4.47). The final result is then (121+ RE)”2 R = 12le (4.48) Note again that this is the same result as the one we obtained in Section 3.3.7 (Equation (3.70)), only in terms of resistances instead of conductances. 4.2 Novel Circuit Theory 4.2. 1 Derivation The Novel circuit theory is not restricted to energies smaller than the Thouless energy. It can be derived from the Usadel equation, which was written as a conservation law in Section 4.1.1: Vj + 821/ 3H — i'szI'fz, G = 0 (4.49) h 27'Sf 50 Here the matrix current density is j = U(X)GVG. (4.50) But this time, we do not drop the second term. This term is called leakage current for the following reasons: It consists of a part proportional to the energy and a part proportional to the superconducting pair potential. The first part describes the decoherence between electrons and holes, while the latter part describes the conversion between quasiparticles and Cooper pairs. In other words, this terms describes the leakage of coherence and the leakage of quasiparticles [6]. The last term, describing spin flip scattering, could in principle be included in the theory, but I will also drop that term for simplicity. The underlying idea of the novel circuit theory is to discretize diffusive conductors, so that we get a circuit of connected nodes instead of a continuous diffusive wire. Then we can calculate the Green functions in every node numerically and can therefore obtain the electric current through the system. To discretize a one-dimensional wire I will take a set of nodes at the coordinates 11:,- such that the Green functions G,- E G (513,) in neighboring nodes are close to each other. Between two nodes there is a connection of length a and a resistor R,- is associated with each connection (see Figure 4.6). In this geometry the conservation law can be written as - 2 [1.1+1+ 11,1—1 + Enid [£1.01] = 0, (4-51) where In,“ is the matrix current between two adjacent nodes. In order to get this matrix current, we interpolate between two neighboring points :6,- and 113,-“: 61(5) 2 6:. + Mm — as.) (4.52) 0. 51 leakage Figure 4.6: Discretization of a diffusive wire. Connections to the ground represent the leakage currents Then, the matrix current between the nodes 2 and 2 + 1 reads V V V i.,.-+1 : a (a, + 9559C, _ 3%)) 9:59 (453) a v Ignoring the term of second order in (CH1 — 0,) we can write v Ii,i+1 = | s: G. (6:.-+1 — 6'1.) , (4.54) :17 4‘ where R,- is the resistance of the connection between the two nodes. To write this expression in a more symmetric form, we make use of the normalization of G: V " ' G2 — Ci V 01 — Ci 1: 02 = (01 + —+—la—($ — 5133)) (01+ +(f — 13)) (4.55) In first order, this gives 11s 1 G40: = ‘2' (Gram + Gi-HGz'I- (4-56) With this we can rewrite Equation (4.54) as in.“ = 5—1— Gi,Ci+1 . (4.57) R, 52 This looks like the expression for the matrix current through a tunnel junction of resistance R,- (cf. Equation (4.5)). It is now clear, what the central idea of the novel circuit theory is: A diffusive wire can be approximated by a series of resistive elements similar to tunnel junctions but with a leakage current. Therefore, it is also easy to treat systems that include tunnel barriers: We simply replace some connections of the resistive network with the resistances of the tunnel barriers. However, there is one important difference: There is no leakage current for the current through a tunnel junction. The physical reason for this is that an electron crosses a tunnel junction without any time delay and therefore does not experience decoherence. To obtain the Green function in the nodes, we make use of the fact that we can now write the conservation law (4.51) in the commutator form 0 3 [Ch 9,] , (4.58) where Q,- is defined as v 01+} G151 2621/0 .. I. z 4. 9 2R.- + 2 114-1 h ( 59) The components of C,- are denoted by calligraphic letters: ' R2 [Ci 9: _ ( 0 A: ) (4.60) Equation (4.58) allows us to express the Green function G,- in terms of 9,. Using the fact that the retarded and advanced Green functions are traceless we find that A,- is proportional to A,- and R,- is proportional to 72,-. Since the advanced and retarded Green functions are normalized, we get A1 : “ii/a) azaw, am) where a 2 VA? and b = V7331 . 1/2 1In the algorithm described below it is advantageous to use a = (Tr [21,2] /2) to make sure to 53 To find the Keldysh Green function we explicitly write the Keldysh component of Equation (4.58): 0 = RI€+fwi—7%f(-IC/i = RIC + aRA — 2‘le — 16.51 (4.62) Now, we use the relation RK + If A and get A A AA A A (2‘ + 5)er = RIC — ICA. (4.63) Multiplying both sides with R leads to the final result Rle—RICA. 4.64 2‘ + a ( ) Equations (4.58), (4.61) and (4.64) allow us to determine the Green function in a node from the Green functions in the adjacent nodes. This is the basis for the analytic example and the algorithm presented at the end of this chapter. In the following paragraph 1 will describe how one can get the differential conductance once the Green functions are calculated. The electric current can be obtained starting from 1..l = i d6Tr 2.1““ , (4.65) 86 where IK is the Keldysh component of the matrix current (cf. Equation (4.9)). One can get the expression for the electric current very similar to the way presented in the derivation of the zero energy circuit theory: Inserting the matrix current found above and using the relationship (2.8) between A and R we find the electric current as 1 1:2... = 2; / deG.,.-+1(6) (fm — fT,i+1), (4.66) divide by a scalar 54 where G,,,+1(6) is an energy dependent conductance: 1 Gi,2+1(€) = 8—R:Tr [(A4 + Ai) (1‘12“ + Ai+1)] (4-67) Again, at zero temperature this leads to a simple result for the differential conduc- tance: dIi,-i+1 (iv = Gi,i+1(€V) (4.68) 4.2.2 Analytical Example The outlined theory does usually not lead to analytic results. The reason for this is that we need many nodes to approximate a diffusive wire and therefore we get a system of too many equations to solve them analytically. However, if we consider the situation shown in Figure 4.7 where the resistance of the wire is much smaller than the resistances of the tunnel junctions, we can reduce it to a circuit with only three nodes and thus solve it easily. leakage Figure 4.7: Double tunnel junction with corresponding circuit diagram. The resis- tance of the wire is much smaller than the tunnel resistances. Connection to the ground represents the leakage current. The basic idea is that the Green function is constant in the intermediate normal metal. Therefore, we can neglect the matrix current proportional to the commutator of the Green functions at both ends and we only have to consider the leakage current through the intermediate wire. To simplify the formulas, I will consider only the case 55 where the energy 6 is much smaller than the pair potential A. Then, A 6 0 ....(0 _,). (4...) The advanced Green functions in the reservoirs read A 0 1 (1 0) (.70. and A —1 0 A3—( 0 1). (4.71) Thus, we can write A—101+1—10 2621/6160 2‘21:le0 2RT01 5 0—6 _ 1 —1 + 2iGT/h 1 — 277. < 1 1— 2iET/h >’ (4.72) where 7 = e2uaRT is a typical escape time from the node. The advanced Green function in the node can now be simply found by normalizing: (4.73) .42 = ((1— 2’ifT/h)2 +1)‘1/2( 71+ 2mm 1 ) 1 1— 2267/h Now we can use Equation (4.67) to calculate the conductance of the first tunnel junction: G1 = 8—11,;11‘[(/i1 + 21]) (.42 + [13)] (4.74) Using (A1 + A]) = 221,, we get: _ 1 . 2 -1/2 . —1+22'6T/f2 1 01(6) _. firm] ((1 — 2267/71) +1) Tr ( 1 1— 2267/12 ) . 2 —1/2. —1—2i€7’/h 1 + ((1+ 2267/17.) +1) Tx( 1 1+ 2267/12 _ 1 ‘ _ 2 -1/2 1 1— 2iGT/h _ ETI [ ((1 — 2267/6) +1) ( —1 + 2iET/h 1 > 56 + ((1+ 2mm)? +1)” ( 1 ”QM/71)] —1—22'6T/h 1 = 5127““ — 2767/72)? +1)”2 + ((1+ 2767/77)? +1)"”2] _—. RLTR9[((1+2iET/n)2+1)"/2] (4.75) In the same way we get the conductance of the second tunnel junction as __1_ e I+2iET/h 02m— RTR [((1+2ier/h)2+1)1/2]' (4'76) The differential conductance of the whole system at zero temperature is given as d1 = 1 . (4.77) _ 1 1 (1V 01(ev') + Gz(eV) Figure 4.8 presents plots of G 1, G2 and d1 / dV. Note that we obtain the same result for zero bias as in the zero energy circuit theory, but with this new method we also get the differential conductance for nonzero energies. 4.2.3 Numerical Implementation In this Section I will show how the described theory can be implemented numerically. I choose the example of a diffusive wire in series with a tunnel junction, which was already treated in Section 3.3.6 in the linear approximation. Again I will only consider energies much smaller than the Thouless energy 6C. Figureh4.9 shows the physical situation and the corresponding circuit diagram. The diffusive wire is approximated by a series of 72—1 tunnel junctions with tunnel resistances R’T = RD/(n — 1), where RD is the normal state resistance of the diffu- sive wire. These elements are in series with the tunnel junction with resistance RT. There is a leakage current in the nodes 2 through n, but no leakage current in node 1, because the real tunnel junction is adjacent to this node. In the nodes 1 and n+1 57 Gg/GN /GN, Gi/GN, s|~ 2L dV .05. ‘i H‘ilsflfléu‘2.5‘”3 eVT/h Figure 4.8: Differential conductance of the double tunnel junction in units of the normal state resistance GN = GT / 2 leakage Figure 4.9: Tunnel junction in series with a diffusive wire and corresponding circuit diagram. Connections to the ground represent the leakage currents 58 the Green functions have the reservoir values. The algorithm calculates the Green functions in the nodes 2 through 71 at a given energy in the following way: Initially, an approximation for G is stored in memory. Then the program calculates g” in every node using Equation (4.59). Note that for node 2 no leakage term will be added. Af- ter that the next approximation for G is obtained from Equations (4.61) and (4.64). This is repeated until the Green functions stop to change significantly. After that the conductance for two adjacent nodes is calculated from Equation (4.67). The differen- tial conductance for the whole structure is obtained by adding the conductances up in series. This procedure is repeated for a certain number of energy values, where the final value of G at a certain energy is used as an initial value for the next energy. The algorithm is implemented in Mathematica, because it is very well suited to produce and interpolate functions and it is very convenient to display the results. The source code can be found in the Appendix or at www.pa.msu.edu/people/hoinkis. Figure 4.10 compares this algorithm’s result to the results obtained with the Us- adel formalism in linear approximation for RT/RD = 50. Then the linear approxima- tion is well justified. The two curves can hardly be distinguished. The difference in the results is below 3% in the shown energy range; it would be smaller for an even greater ratio RT/RD. The number of nodes for this example was 90. The precision of the circuit theory result can be increased by using more nodes, but the calculating time is proportional to the number of nodes in the third power [6]. Figure 4.11 shows the same plot for RT/RD = 1. Here the linear approximation gives worse results and a offset is clearly visible. 59 l/GN 41. av eV/eC Figure 4.10: Differential conductance for RT/RD = 50 calculated with the circuit theory (full line) and with the Usadel formalism (dashed line) in units of the normal state resistance GN. (Lines almost coincide.) eV/6C Figure 4.11: Differential conductance for RT/RD = 1 calculated with the circuit theory (full line) and with the Usadel formalism (dashed line) in units of the normal state resistance GN. 60 Chapter 5 Conclusion The Usadel formalism and the Nazarov circuit theory have different advantages and disadvantages. The Usadel formalism gives analytic results in simple cases and is therefore well suited to provide understanding of these examples. The results are not only valid at zero energy and can therefore demonstrate how the current depends on the bias voltage and the temperature. At zero energy the circuit theory is an elegant means to calculate the current through a structure. The same results can be obtain with the Usadel formalism, however, the circuit theory leads much quicker to the goal since all the important steps are already laid out, and one must simply follow the given steps. Admittedly, zero energy results are usually not sufficient, because of the finite temperature in real experiments. Furthermore, it is desirable to obtain results with an applied voltage. Therefore, the zero energy circuit theory is an alternative to the Usadel formalism only in a very restricted field. Unfortunately, the Usadel formalism does often not give analytic results when the physical situation is more complex. Then, the problem has to be tackled numerically, and in those cases, the novel circuit theory is a powerful alternative to the Usadel formalism. The big advantage of the circuit theory is that it is very flexible. Once the 61 algorithm has been implemented, changes of the physical situation can be included with very little effort. Different geometries can quickly be simulated, and the cal- culating time does not increase extraordinary when additional physical features are included. 62 APPENDIX 63 Appendix A Mathematica Source Code emin = 10“-4; emax = 10; nsteps = 50; estep = emax/(nsteps - 1); (* algorithm runs from emin to emax, doing nsteps *) p = -7; (* 10‘p determines when Green function is constant *) n = 4; (* n : number of nodes *) g Table[n - 1, {n}]; g[[1]] = 1; (* g : normal state conductances of circuit elements *) Gn = 1/(Sum[1/g[[j]]. {j. 1. n}]); (* Gn : normal state resistance of whole structure *) Gd = 1/(Sum[1/g[[j]], {3. 2. n}]); (* Gd : normal state resistance of diffusive wire *) G = Table[, {j, 1, n}, {i, 1, nsteps}, {w, 1, 2}]; (* G : conductances according to circuit thy. *) Ginterp = Table[, {j, 1, n}]; (* Ginterp : interpolation of G in the energy interval *) An = DiagonalMatrix[{-1, 1}]; Rn = -An; Kn = DiagonalMatrixff2, -2}]; (* An, Rn, Kn : Definition of normal reservoir values *) Table[An, {n + 1}]; Table[Rn, {n + 1}]; Table[Kn, {n + 1}]; >A9<£flt> as," II II * , R, K : Green functions in the nodes 1 through n + 1 *) = Table[, {n + 1}]; (* Abak : backup of A *) Ar = Table[, {n + 1}]; Br = Table[, {n + 1}]; Kr = Table[, {n + 1}]; (* Ar, Rr, Kr : caligraphic Green functions *) a = Table[, {n + 1}]; r = Table[, {n + 1}]; (* a, r : normalization factors of Ar, Rr *) 64 (*green fctns in nodes 1 and n+1 are set to reservoir values:*) A[[n + 111 = An;, R[[n + 1]] = Rn;. K[[n + 11] = Kn; A[[1]] = {{O, 1}, {1, 0}};. R[[1]] = Afflll; K[[1]] = DiagonalMatrix[{O, 0}]; For[i = 1; e = emin, i < nsteps + 1, e += estep; i ++, (* 1 counts the energy runs *) H = DiagonalMatrix[{e, -e}]; difsum = 1; (* difsum : indicator if Green function is constant *) WhileEdifsum > 10“p, (* j is used to denote the nodes . j = 2 is treated differently from j = 3...n, because there i=2: Ar[[jl] erfjll Krffjll .5(g[[j-1]])A[[j-1]]+ .5(g[[j-1]l)R[[j-1]]+ .5(g[[j-1]l)K[[j-1]]+ is no leakage current at j = 2 *) .5(g[[j]])A[[j+1]]; .5(g[[j]l)R[[j+1]]; .5(g[[j]])K[[j+1]]; a[[j]] Sqrt[.5 Tr[Ar[[j]] . Ar[[j]]]l; r[[j]] Sqrt[.5 TrEerfjll . erfjllll; Abak[[j11 = A[[j1]; A[[j]] NEAr[[j]]/a[[j]]]; R[[j]] NERrEIjll/rffjlll; Kfljl] N[(Kr[[j]] - R[[j]].Kr[[jl].A[[j11)/(a[[jll + r[[j]])]; For[j = 3, j < n + 1, j ++, Ar[[j]] .5(g[[j-1]])A[[j-1]]+ .5(g[[j]])A[[j+1]]- I Gd H/(n—Q); RrIEjll .5(g[[j-1]])R[[j-1]l+ .5(g[[j]])R[[j+1]]- I Gd H/(n-Q); Kr[[j]] .5(g[[j-1]])K[[j-1]]+ .5(g[[j]])K[[j+1]J; a[[j]] Sqrt[.5 Tr[Ar[[j]].Ar[[j]]]]; r[[j]] Sqrt[.5 Tr[Rr[[j]].Rr[[jll]]; Abak[[j]] = A[[j]]; A[[j]1 = NEArEEjll/aEEjlll; a[[j]] = NERrEEj]]/r[[j]]]; K[[j1] = N[(Kr[[j]] - R[[j]].Kr[[j]].A[[jll)/(a[[j]] + r[[j1])]; ]; difsum = Sum[Abs[A[[j,1,1]] - Abak[[j,1,1]]] + Abs[A[[j,1,2]1 - AbakEEj,1,2]]]. {j.2,n}]; 1; (*Green fctns are now constant -> conductances can be calculated:*) Forfj = 1, j < n + 1. j ++. G[[j,i]] = {9, .125g[[j]]Tr[(A[[j]]+Conjugate[TransposeEAffjllll). (A[[j+1]]+Conjugate[TransposefAf[j+1]]]])1}; 1; ] For[j = 1, j < n + 1, j ++, Ginterp[[j]] = Interpolation[G[[j]]];] 65 BIBLIOGRAPHY 66 Bibliography [1] W. Belzig, F.K. Wilhelm, C. Bruder, and G. Schon. Quasiclassical green’s func- tion approach to mesoscopic superconductivity. Superlattices and Microstructures, 25(5/6):1251—1289, 1999. [2] H. Courtois, P. Charlat, Ph. Gandit, D. Mailly, and B. Pannetier. The spectral conductance of a proximity superconductor and the re-entrance effect. J. Low Temp. Phys., 116(187), 1999. [3] S. Guéron. Quasiparticles in a diflusive conductor: Interaction and pairing. Ph.D. thesis, Université Paris 6, 1997. [4] C. J. Lambert and R. Raimondi. Phase—coherent transport in hybrid supercon- ducting nanostructures. J. Phys. Condens. Matter, 10:901—941, 1998. [5] Yu. V. Nazarov. Circuit theory of andreev conductance. Phys. Rev. Lett, 73(10):1420—1423, 1994. [6] Yu. V. Nazarov. Novel circuit theory of andreev reflection. 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