Andreev Reflection 36

Andreev Reﬂection

Yasuhiro Asano Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan. Email: [email protected]

October, 2019 1

1Update version is available at http://zvine-ap.eng.hokudai.ac.jp/˜asano/pdf1/lecture.pdf Contents

1 Introduction 4 1.1 Overview of superconducting phenomena ...... 4

2 Theoretical tools 10 2.1 Harmonic oscillator ...... 10 2.2 Free electrons in a metal ...... 14 2.3 Coherent state ...... 16

3 Mean-ﬁeld theory of superconductivity 20 3.1 Pairing Hamiltonian ...... 20 3.2 Mean-ﬁeld approximation ...... 22 3.3 Gap equation and thermodynamic properties ...... 24 3.4 Cooper pair ...... 27 3.5 Magnetic properties ...... 29

4 Andreev reﬂection 35 4.1 Conductance in an NS junction ...... 36 4.2 Retroreﬂectivity ...... 42 4.3 Josephson current in an SIS junction ...... 43

5 Unconventional superconductor 49 5.1 Conductance in an NS junction ...... 51 5.2 Surface Andreev bound states ...... 53 5.3 Josephson current in an SIS junction ...... 56 5.4 Andreev bound states in an SIS junction ...... 57 5.5 Topological classiﬁcation ...... 59

6 Various Josephson junctions 64 6.1 Singlet-singlet junction ...... 65 6.2 Triplet-triplet junction ...... 68 6.3 Singlet-triplet junction ...... 70 6.4 Chiral superconductors ...... 73 6.5 Helical superconductors ...... 74 6.6 Noncentrosymmetric superconductor ...... 75

A Pair potential near the transition temperature 76

B Landauer formula for electric conductance 78

2 CONTENTS 3

C Mean-ﬁeld theory in real space 83

D Reﬂection coeﬃcients in one-dimensional junctions 86 Chapter 1

Introduction

Superconductivity is a macroscopic quantum phenomenon discovered by H. K. Onnes in 1911. He and his colleague observed that the resistivity of Hg disappeared suddenly at 4.2 Kelvin (K) [1]. This experimental finding opened a new era of low temperature science. Superconduc- tors exhibit various unusual response to an electromagnetic field such as perfect diamagnetism, persistent current and flux quantization. In 1957, Bardeen-Cooper-Schrieffer (BCS) explained the microscopic mechanism of superconductivity [2]. According to BCS theory, two electrons on the Fermi surface form a pair due to an attractive interaction mediated by phonons at a low temperature. The electron pair is called Cooper pair. It was shown that the coherent condensation of a number of Cooper pairs causes the unusual response to electromagnetic fields. The mean-field theory by Bogoliubov [3] enabled us to describe the thermodynamic properties of a superconductor as well as the linear response to electromagnetic fields. Before publishing the BCS theory, however, a phenomenological theory by London and that by Gintzburg-Landau (GL) explained the electromagnetic properties of superconductors successfully. The assumptions introduced in these phenomenological theories have been justified by the microscopic theory of superconductivity. The purpose of this note is to explain the electric transport properties in superconducting hetero structures such as the differential conductance in a normal-metal/superconductor (NS) junction and the Josephson effect in a superconductor/insulator/superconductor (SIS) junction. To shortcut, we will explain minimum necessary items to understand the Andreev reflection and skip details of the BCS, GL, and London theories. Therefore, readers should study standard textbooks for these contents [4, 5, 6, 7, 8, 9, 10, 11]. We, however, begin with superconducting phenomena observed in experiments to make clear theoretical requirements to explain these phenomena.

1.1 Overview of superconducting phenomena

Let us begin a brief overview of superconducting phenomena with the discovery of superconductivity in 1911. Low temperature physics might be a trend in science at the beginning of the 20th century. The last inactive gas He became liquid at 4.2 K in 1908. A natural question would be what happen on materials near the zero temperature. The resistance R in a metal was an issue in this direction. At that time, it was accepted that a charged particle (electron) moving freely in a metal carries the electric current J under the bias voltage V . The Ohm’s law V = JR holds true in metals for wide temperature range. According to kinetic theory of gas by Boltzmann, every particle stops its motion at the zero temperature. By extension, a theory indicated that

4 CHAPTER 1. INTRODUCTION 5

Figure 1.1: (Left) Historic plot by Onnes[1]. The resistance in Hg vanishes at 4.2 K. (Right) A superconductor excludes magnetic flux. a metal might be an insulator at the zero temperature because all electrons stop their motion. According to the theory, the resistivity would be infinitely large at the zero temperature. On the other hand, on the basis of experimental results at that time, a group of scientists inferred that the resistivity would go to zero at the zero temperature. An electron is scattered by the thermal vibration of ions in a metal, which is a source of the resistance. At zero temperature, therefore, a biased electron would move freely at zero temperature. The experiments by Onnes could settle the debate over. The experimental data on Hg, however, showed the disappearance of the resistivity at a temperature above zero as shown in Fig. 1.1. This was the first quantum phenomenon that we observed in our history. The phenomenon was named superconductivity. The critical temperature dividing the two transport phases is called the transition temperature Tc. A series of experiments reported anomalies in thermodynamic property of a superconductor at Tc. The specific heat shows a jump at Tc, which suggested the rapid decrease of the entropy and the development of an order below Tc. We summarize briefly the unusual response of a superconductor to magnetic field in what follows. The first phenomenon is called the Meissner effect. An external magnetic field H threats a normal metal at a room temperature. Far below the transition temperature T ≪ Tc, the metal excludes a weak external magnetic field from its interior as shown in Fig. 1.1. The magnetic flux density in a superconductor B is zero as represented by B = H + 4πM = 0, (1.1) where M denotes the magnetization of a superconductor. The magnetic susceptibility defined by M = χH results in 1 χ = − . (1.2) 4π A superconductor is a perfect diamagnet far below Tc in a weak magnetic field. The magnetic response of a multiply connected superconductor is more unusual. We apply a magnetic filed to a ring at T > Tc as shown In Fig. 1.2, where we illustrate the cooling process under a magnetic field. When a temperature is decreased down below Tc, a ring-shaped superconductor excludes the magnetic flux from its arm. After switching off a magnetic field at T ≪ Tc, magnetic flux are trapped at a hole of the ring. According to the Maxwell equation 4π ∇ × H = j, (1.3) c CHAPTER 1. INTRODUCTION 6

H H

T < T T > Tc c Φ H = 0

J T < Tc

Figure 1.2: Persistent current at a ring-shaped superconductor. At a higher temperature than Tc, a uniform external magnetic field is applied as shown in the upper left figure. Keeping the magnetic field, the superconductor is cool down to a temperature much less than Tc as shown in the upper right figure. The magnetic field is excluded from the superconductor due to the Meissner effect. Then the external magnetic field is switched off as shown in the lower figure. Quantized magnetic flux Φ is left in the ring. The current persists as far as T ≪ Tc. the magnetic flux left in the hole generates the electric current circulating the ring. The current is called persistent current because it does not dump at all even after passing a long enough time. The results imply that the superconducting current flows in equilibrium while the ring accommodates magnetic flux. We have already known an example of similar effect in an external magnetic field H. The weak orbital diamagnetism M = χH of a mental generates the electric current in equilibrium as shown in Fig.1.3. The susceptibility χ is negative in a metal. Together with Eq. (1.3), we find cχ j = ∇ × M. (1.4) 4π The electric current is represented as the rotation of a vector. The total transport current can

H j

M C

Figure 1.3: Schematic picture of the diamagnetic current in a metal. be calculated as Z Z I cχ cχ J = dS · j = dS · ∇ × M = dl · M. (1.5) 4π S 4π C We find J = 0 because it is possible to take the integration path l outside the metal where M = 0. Namely, the current described as j ∝ ∇ × M does not flow out from the metal. As we will see later on, the supercurrent can be represented as j ∝ A and flows out from a superconductor. In addition to the persistency of the current in a superconducting ring, the CHAPTER 1. INTRODUCTION 7 magnetic flux at a hole is quantized as πℏc Φ = nϕ , ϕ = , (1.6) 0 0 e where n is an integer number. Quantum mechanics tells us that the quantization of a physical value is a result of a boundary condition of wave function. As we will see later on, the macroscopic wave function of Cooper pairs must be single-valued in real space, which explains the flux quantization. The perfect screening of a magnetic filed is realized when the magnetic field is weak enough. In higher magnetic fields, the magnetic response depends on types of superconductors. The type I superconductor excludes magnetic fields up to the thermodynamical magnetic field Hc as shown in Fig. 1.4(a). All the superconducting properties disappears for H > Hc. The magnetic properties of the type II superconductor is characterized by two critical magnetic fields: HC1 and HC2 as shown in Fig. 1.4(b). Such a type II superconductor excludes magnetic field for

H < HC1 and become a normal metal for H > HC2 . In the intermediate magnetic ﬁelds for

HC1 < H < HC2 , the superconductor is superconductive but allows the penetration of magnetic flux as quantized vortices. The magnetic flux is quantized as ϕ0 at each vortex. As a result, the profile of the magnetic flux becomes inhomogeneous in real space. Near HC2 , a number of vortex form a hexagonal structure called Abrikosov lattice. Ginzburg-Landau showed that the boundary of the two types superconductors are described by a parameter κ = λL/ξ0, where ξ0 is called coherence length and λL is the London’s penetration length. The meaning of the two length scales√ will be explained√ in Chap. 3. The type I (type II) superconductors are characterized by κ < 1/ 2 (κ > 1/ 2).

(a) H 0 H c

B vortex Abrikosov Lattice H

(b) H H 0 c1 H c2

Figure 1.4: B − H curve in a superconductor: (a) type I and (b) type II.

Quantum mechanics also tells us that the wave function can be a complex number

ψ(r) = |ψ(r)|eiθ(r). (1.7)

The amplitude |ψ(r)|2 is proportional to the probability that we observe a particle there. A constant phase does not play any role in the Schroedinger picture for a quantum particle. As we will see later on, the superconducting state is described phenomenologically well by the macroscopic wave function, p ψ(r) = n(r) eiθ(r). (1.8) CHAPTER 1. INTRODUCTION 8

23 Here ψ(r) describes a quantum state consisting of NA Cooper pairs with NA ∼ 10 being the Avogadro number. Namely, NA Cooper pairs share the common phase θ and form a phase coherent condensate. This causes the unusual response to the electromagnetic field. The Josephson effect is a highlight of superconducting phenomena [12]. The electric current flows between the two superconductors in equilibrium. Figure. 1.5 shows a typical Josephson junction where an insulating film is sandwiched by two superconductors. Today we know that the current can be represented as

J = Jc sin(φL − φR), (1.9) where Jc is called critical current and φL(R) is called superconducting phase at the left (right) superconductor.

superconductor superconductor

ϕL ϕR

Figure 1.5: Two superconductors are separated by a thin insulating barrier. The electric current ﬂows through the junction in the presence of the phase diﬀerence across the junction.

What is the superconducting phase? Why does the phase diﬀerence carry the electric current? We would understand the essence of superconductivity if we could answer these questions clearly. As a ﬁrst step to reach the goal, let us consider an argument by Feynmann. He knew that the superconducting condensate including NA electrons behaves as if it were a coherent object. This view might be enable us to introduce a wave function of the superconducting condensate √ √ iφL iφR ψL = nLe , ψR = nRe , (1.10) where nL(R) is the electron density in the superconducting condensate at the left (right) superconductor and φL(R) is the phase of the wave function. The two superconducting condensates interact with each other through a thin insulating barrier. Let us introduce the tunneling Hamil- tonian between the two superconductors. The rate equation would be given by ∂ ∂ iℏ ψ = T ψ , iℏ ψ = T ψ (1.11) ∂t L I R ∂t R I L where TI is the tunneling amplitude of the insulating barrier. The electric current through the barrier might be given by

∂ † j = −e n , n = ψ ψ . (1.12) ∂t L L L L The time derivative of density is calculated as

∂ † † i † i † n =∂ ψ ψ + ψ ∂ ψ = T ψ ψ − T ψ ψ , (1.13) L t L L L t L ℏ I R L ℏ I L R ∂t i √ − − − TI √ i(φL φR) − i(φL φR) − − =ℏTI nLnR e e = 2 ℏ nLnR sin(φL φR). (1.14) CHAPTER 1. INTRODUCTION 9

The electric current results in √ 2e − j = ℏ TI nLnR sin(φL φR). (1.15) The obtained expression explains well the Josephson eﬀect. Feynmann got a clear view of superconductivity. However, his assumption suggests that the ground state of many electrons should have a uniform phase. It would be not easy for us to imagine such ground state of electrons in a metal. Chapter 2

Theoretical tools

Today we have already known that superconductivity is a many-body eﬀect of electrons in a metal and that the attractive interactions between two electrons play essential role in stabilizing the superconducting state. To understand physics of superconductivity, we need to use several theoretical tools. The purpose of this section is to explain what tools we need and how to use them. The ﬁrst tool we need is the creation and the annihilation operators of a particle.

2.1 Harmonic oscillator

The Hamiltonian of a harmonic oscillator in one-dimension reads, p2 1 ℏ d H = x + mω2x2, p = , (2.1) 2m 2 x i dx in the Schroedinger picture. The eigen energy is given by 1 E = ℏω n + , n = 0, 1, 2, ··· . (2.2) 2

Let us introduce a length scale x0 which balances the kinetic energy and the potential energy as, ℏ2 1 2 2 2 = mω x0. (2.3) 2mx0 2 p The solutions x0 = ℏ/mω is the characteristic length scale of the Hamiltonian. By applying the scale transformation x = x0q, the Hamiltonian and the commutation relation becomes ℏ2 2 ℏ 2 − d 1 2 2 2 ω − d 2 H = 2 2 + mω x0q = 2 + q , (2.4) 2mx0 dq 2 2 dq d [q, p] =i, p = −i . (2.5) dq When we deﬁne two operators as 1 1 d a =√ (q + ip) = √ q + , (2.6) 2 2 dq 1 1 d a† =√ (q − ip) = √ q − , (2.7) 2 2 dq

10 CHAPTER 2. THEORETICAL TOOLS 11 we ﬁnd the commutation relations as 1 [a, a†] = [(q + ip)(q − ip) − (q − ip)(q + ip)] , (2.8) 2 1 = q2 − iqp + ipq + p2 − q2 − iqp + ipq − p2 = −i[q, p] = 1, (2.9) 2 [a, a] =0. (2.10)

The original operators are represented in terms of a and a† by 1 1 q = √ (a + a†), p = √ (a − a†). (2.11) 2 2i Using the commutation relation, it is possible to derive h i 1 p2 + q2 = −(a − a†)(a − a†) + (a + a†)(a + a†) , (2.12) 2 h i 1 = −a2 + aa† + a†a − (a†)2 + a2 + aa† + a†a + (a†)2 , (2.13) 2 =2a†a + 1. (2.14)

Thus the Hamiltonian is diagonalized as 1 H = ℏω a†a + . (2.15) 2

The expectation value should be ⟨a†a⟩ = n so that the expectation value of H coincides with Eq. (2.2). The operatorn ˆ = a†a is called number operator and satisﬁes the relations

[ˆn, a] =a†aa − aa†a = a†aa − (1 + a†a)a = −a, (2.16) [ˆn, a†] =a†aa† − a†a†a = a†(1 + a†a) − a†a†a = a†. (2.17)

Let us assume that |ν⟩ is the eigenstate ofn ˆ belongs to the eigenvalue ν,

′ nˆ|ν⟩ = ν|ν⟩, ⟨ν |ν⟩ = δν,ν′ . (2.18)

It is easy to conﬁrm the relations

The results suggest that a|ν⟩ is the eigenstate ofn ˆ belonging to ν − 1 and that a†|ν⟩ is the eigenstate ofn ˆ belonging to ν + 1. Therefore, it is possible to describe these states as

a|ν⟩ = β|ν − 1⟩, a†|ν⟩ = γ|ν + 1⟩, (2.21) with β and γ being c-numbers. Together with the Hermite conjugate the relations

⟨ν|a† = β∗⟨ν − 1|, ⟨ν|a = γ∗⟨ν + 1|, (2.22) the norm of these states are calculated as

⟨ν|a†a|ν⟩ = |β|2⟨ν − 1|ν − 1⟩ = |β|2, ⟨ν|aa†|ν⟩ = |γ|2⟨ν + 1|ν + 1⟩ = |γ|2. (2.23) CHAPTER 2. THEORETICAL TOOLS 12

Simultaneously, the left hand side of these equations can be represented by ⟨ν|a†a|ν⟩ = ⟨ν|n|ν⟩ = ν, ⟨ν|aa†|ν⟩ = ⟨ν|n + 1|ν⟩ = ν + 1, (2.24) √ √ because of the definition in Eq. (2.18). Thus we find β = ν, γ = ν + 1, and √ √ a|ν⟩ = ν|ν − 1⟩, a†|ν⟩ = ν + 1|ν + 1⟩. (2.25) The relation ν ≥ 0 must be true because ν is the norm of a|ν⟩ as shown in Eq. (2.24). The recursive relation Eq. (2.25) √ a|ν⟩ = ν|ν − 1⟩, (2.26) p a2|ν⟩ = ν(ν − 1)|ν − 2⟩, (2.27) p aM |ν⟩ = ν(ν − 1) ··· (ν − M + 1)|ν − M⟩, (2.28) suggests that ν − M can be negative if we can chose large enough M. To avoid such unphys- ical situation, we impose the boundary condition of the series of the recursive relation. The eigenvalue ofn ˆ include zero so that a|0⟩ = 0, (2.29) delete the physical state. The relation in Eq. (2.29) defines the vacuum of a particle |0⟩. Finally we obtain √ √ a|n⟩ = n|n − 1⟩, a†|n⟩ = n + 1|n + 1⟩, nˆ|n⟩ = n|n⟩. (2.30)

These desired relations represent the mechanics of harmonic oscillators. The operator a (a†) is called annihilation (creation) operator because it decreases (increases) the number of oscillator by one. The ground state is then given by |0⟩ where no harmonic oscillator exists. The m-th excited state |m⟩ describes the state in which m oscillators are excited on the vacuum as (a†)m |m⟩ = √ |0⟩. (2.31) m! Next we discuss the wave function in the Schroedinger picture which can be obtained by multiplying ⟨q| to the relation a|0⟩ = 0. The representation becomes 1 1 q 1 d 0 = ⟨q|a|0⟩ = √ ⟨q|q + ip|0⟩ = √ ⟨q|q + (d/dq)|0⟩ = √ ⟨q|0⟩ + √ ⟨q|0⟩. (2.32) 2 2 2 2 dq We obtained the diﬀerential equation d ⟨q|0⟩ = −q⟨q|0⟩. (2.33) dq The solution is given by

− 1 q2 ⟨q|0⟩ = C e 2 . (2.34) The left hand side is the deﬁnition of the wave function at the ground state and the right hand side is the Gauss’s function. The wave function at the excited states are calculated as † 1 d − 1 q2 ⟨q|1⟩ = ⟨q|a |0⟩ = √ q − C e 2 , (2.35) dq 2 n † n 1 d − 1 q2 ⟨q|n⟩ = ⟨q|(a ) |0⟩ = √ n q − C e 2 . (2.36) 2 dq CHAPTER 2. THEORETICAL TOOLS 13

This sequence generates the series of Hermite’s polynomial. The observable values are calculated as the average of operators. Here the average has double meanings: the average in quantum mechanics and the average in statistical mechanics. The average of operator Q is deﬁned by h i h i 1 ⟨Q⟩ = Tr e−βHQ , Ξ = Tr e−βH , (2.37) Ξ where H is the Hamiltonian under consideration and β = 1/kBT corresponds to the inverse of temperature with kB being the Boltzmann constant. To proceed the calculation, we deﬁne Tr in the average and explain several identities for the trace of matrices. Let us assume that |p⟩ is the eigen state of H belonging to the eigenvalue of ϵp, X ′ H|p⟩ = ϵp|p⟩, ⟨p |p⟩ = δp,p′ , |p⟩⟨p| = 1. (2.38) p

When we choose |p⟩ as the basis of the trace, the partition function can be calculated as h i X X − H − H − Ξ = Tr e β = ⟨p|e β |p⟩ = e βϵp . (2.39) p p

The ﬁrst theorem says that value of the trace remains unchanged under the unitary transformation, X X X ∗ ∗ | ⟩ | ⟩ ′ q = Aq,p p , Aq,pAq′,p = Ap,q′ Ap,q = δq,q . (2.40) p p p

The second equation is the deﬁnition of the unitary matrix of A, (i.e., A†A = A†A = 1). The trace of Q is transformed as " # X X X X X X ⟨ |Q| ⟩ ⟨ ′| ∗ Q | ⟩ ∗ ⟨ ′|Q| ⟩ ⟨ |Q| ⟩ p p = q Ap,q′ Ap,q q = Ap,q′ Ap,q q q = q q . (2.41) p p q,q′ q,q′ p q

The second theorem says the values of trance remain unchanged under the cyclic permutation of operators

Tr[ABC] = Tr[CAB] = Tr[BCA]. (2.42)

It is easy to conﬁrm the relations X X Tr[AB] = Ap,qBq,p = Bq,pAp,q = Tr[BA], (2.43) p,q p,q Tr[(AB) C] =Tr[C (AB)]. (2.44)

By using these theorems, we try to calculate the average of number operator in the Hamiltonian X † † H ′ ′ = ϵp ap ap, [ap, ap′ ] = δp,p , [ap, ap ] = 0, (2.45) p where p indicates a quantum state. The average of the number operator is calculated by 1 1 1 ⟨a†a ⟩ = Tr[e−βHa†a ] = Tr[a e−βHa†] = Tr[e−βH eβHa e−βH a†]. (2.46) p q Ξ p q Ξ q p Ξ q p CHAPTER 2. THEORETICAL TOOLS 14

The Heisenberg operator is represented as

βH −βH βH −βH I(β) = e aqe , ∂βI = e [H, aq]e . (2.47) From the relation X X X H † † − † H − aq = ϵp ap ap aq = ϵp ap aq ap = ϵp( δp,q + aq ap) ap = aq ϵqaq, (2.48) p p p we ﬁnd [H, aq] = −ϵqaq. By substituting the results into Eq. (2.47), we obtain the diﬀerential equation

∂βI(β) = −ϵqI(β). (2.49) The solution is obtained as

−ϵqβ I(β) = ape . (2.50) The average can be represented as

† 1 − H † − 1 − H † − † − − ⟨a a ⟩ = Tr[e β a a ]e ϵqβ = Tr[e β (δ + a a )]e ϵqβ = ⟨a a ⟩e ϵqβ + δ e ϵqβ. p q Ξ q p Ξ p,q p q p q p,q (2.51) We reach the ﬁnal results of average, ⟨a†a ⟩ =δ n (ϵ ), (2.52) p q p,q B q 1 1 ϵp − nB(ϵq) = ϵ β = coth 1 , (2.53) e q − 1 2 2kBT where nB is the Bose-Einstein distribution function. The results are consistent with a fact that Eq. (2.45) describes the commutation relation of bosons.

2.2 Free electrons in a metal

The Hamiltonian for free electrons in a metal is given by Z X ℏ2∇2 H = dr ψ† (r) − − ϵ ψ (r), (2.54) 0 α 2m F α α=↑,↓ n o † ′ − ′ ′ ψα(r), ψβ(r ) = δ(r r )δα,β, ψα(r), ψβ(r ) = 0, (2.55) {A, B} = AB + BA. (2.56) To diagonalize the Hamiltonian, we apply the Fourier transformation of operator X 1 ik·r ψα(r) = √ c e , (2.57) V k,α vol k where Vvol is the volume of the metal. The basis functions satisfy the normalization condition and completeness, Z 1 i(k−k′)·r dr e = δk,k′ , (2.58) Vvol X Z 1 ik·(r−r′) 1 ik·(r−r′) − ′ e = d dk e = δ(r r ), (2.59) Vvol (2π) k CHAPTER 2. THEORETICAL TOOLS 15

where d represents the spatial dimension of the metal. The anticommutation relation of ck,α can be derived as Z Z { † } 1 ′ { † ′ } −ik·r ip·r′ ck,α, cp,β = dr dr ψα(r), ψβ(r ) e e , (2.60) Vvol Z 1 −i(k−p)·r =δα,β dr e = δα,βδk,p, (2.61) Vvol {ck,α, cp,β} =0. (2.62)

Substituting the Fourier representation, the Hamiltonian becomes, Z X X ′ ℏ2∇2 H 1 −ik ·r † − − ik·r 0 = dr e ck′,α ϵF e ck,α, (2.63) Vvol 2m α=↑,↓ k,k′ X Z X X X 1 i(k−k′)·r † † = dr e ck′,α ξk ck,α = ξk ck,αck,α, (2.64) Vvol α=↑,↓ k,k′ α=↑,↓ k

2 2 where ξk = ℏ k /(2m)−ϵF is the kinetic energy of an electron measured from the Fermi energy. Let us check the average of number operator given by

† 1 † 1 † 1 † ⟨c c ⟩ = Tr[e−βHc c ] = Tr[c e−βHc ] = Tr[e−βHI(β)c ], (2.65) k,α p,λ Ξ k,α p,λ Ξ p,λ k,α Ξ k,α βH −βH I(β) =e cp,λe . (2.66)

−βξp Since [H, cp,λ] = −ξpcp,λ, it is possible to replace I(β) by cp,λe as discussed in Eq. (2.50). The average is then described as

† 1 − H † − ⟨c c ⟩ = Tr[e β c c ]e βξp , (2.67) k,α p,λ Ξ p,λ k,α 1 − H † − = Tr[e β (δ δ − c c )]e βξp , (2.68) Ξ α,λ k,p k,α p,λ − † − βξp − ⟨ ⟩ βξp =δα,λδk,pe ck,αcp,λ e . (2.69) Therefore, we reach the expression

† ⟨c c ⟩ =δ δ n (ξ ), (2.70) k,α p,λ α,λ k,p F k 1 1 − ξk nF (ξk) = βξ = 1 tanh , (2.71) e k + 1 2 2kBT where nF is the Fermi-Dirac distribution function. Eqs. (2.61) and (2.62) are the anticom- ⟨ ⟩ ⟨ † ⟩ mutation relation of fermion operators. It is easy to show ck,αcp,λ = 0 and ck,αcp,λ = [1 − nF (ξk)]δα,λ δk,p. 3 23 In usual metals such as Au, Al, and Pb, a crystal with volume of 1 cm contains NA ∼ 10 −3 metallic ions. Therefore, the density of free electrons is about NA cm . At T = 0, NA electrons occupy all the possible states from the band bottom up to the Fermi energy ϵF . The Fermi energy is typically 104 Kelvin and is much larger than the room temperature of 300 K. 8 10 The Fermi velocity vF is about 10 cm/s is 1/100 of the speed of light c = 3× 10 cm/s. Electrons on the Fermi surface have such high kinetic energy due to their statistics. In addition, the degree of the degeneracy at the Fermi energy is large. The quantum mechanics suggests that the large degree of degeneracy in quantum states are a consequence of high symmetry CHAPTER 2. THEORETICAL TOOLS 16 of Hamiltonian. Generally speaking, highly degenerate quantum states are fragile under the various perturbations breaking the symmetry. In metals, electrons in conduction bands always interact with another bosonic excitations such as phonon and magnon. In addition, the Coulomb repulsive interaction works between two electrons. Band electrons may break symmetry and decreases their energy by means of theses interactions. Indeed a number of simple metals becomes ferromagnetic or superconductive at a low enough temperature. Fig. 2.1 shows the periodic table of elements, where elements colored in red or pink are superconductors and those in blue are ferromagnets. The table implies that superconductivity is a common feature among metals at a low temperature. Time-reversal symmetry is broken due to the short range repulsive interaction in a ferromagnet and translational symmetry is broken due to the coupling with the lattice distortion in a charge density wave compound. The phase transition of the second kind describes well the appearance of such ordered phases at a low temperature. The transition to superconducting phase is also a phase transition of the second kind which is caused by the attractive interaction between two electrons. The superconducting phase breaks gauge symmetry spontaneously. In what follows, we will see the physical consequence of the breaking gauge symmetry.

2.3 Coherent state

Superconductivity is a macroscopic quantum phenomenon. This means that the superconducting state including NA electrons is described well by a macroscopic wave function Ψ. The Feyn- mann’s explanation of the Josephson eﬀect suggests that the wave function can be decomposed into √ Ψ = Neiθ. (2.72)

The phase θ is uniform in a superconductor. In the ﬁeld theory, let us assume that the Ψ is an operator obeys the bosonic commutation relation

[Ψ, Ψ†] = 1, [Ψ, Ψ] = 0. (2.73)

As a result, the number of particle N and the phase θ are also operators which do not commute to each other. The commutation relation results in √ √ √ √ [Ψ, Ψ†] = Neiθe−iθ N − e−iθ N Neiθ = N − e−iθNeiθ. (2.74)

We will show that the relation

[N, θ] = i, (2.75) is a suﬃcient condition for the the right hand side of Eq. (2.74) being unity. Beginning with Eq. (2.75), we ﬁrst prove the relation

[N, θm] = miθm−1. (2.76)

Eq. (2.75) corresponds to the case of m = 1. At m → m + 1,

[N, θm+1] =Nθmθ − θm+1N = (miθm−1 + θmN)θ − θm+1N, (2.77) =miθm + θm(i + θN) − θm+1N = (m + 1)iθm. (2.78) CHAPTER 2. THEORETICAL TOOLS 17

Therefore, a commutator becomes " # ∞ ∞ ∞ X (−1)m X (−1)m d X (−1)m d [N, e−iθ] = N, θm = m i θm−1 = i θm = i e−iθ, (2.79) m! m! dθ m! dθ m=0 m=0 m=0 =e−iθ. (2.80)

Thus Eq. (2.74) results in [Ψ, Ψ†] = 1. Although Eq. (2.74) is only a sufficient condition, it has an important physical meaning. Eq. (2.75) suggests that N and θ are canonical conjugate to each other. Thus it is impossible to fix the two values simultaneously. Namely if we fix the number of electrons, the phase is unfixed. On the other hand, when we fix the phase, the number of electrons is unfixed. The latter corresponds to the superconducting state. It would be easier to understand the difference between the two state when we imagine the classical coherent light so called laser beam. Light is described as a state of number of photons. Let us introduce the bosonic annihilation (creation) operator of a photon a (a†) with the vacuum a|0⟩ = 0. The coherent state |C⟩ is defined as the eigenstate of the annihilation operator,

a|C⟩ = α|C⟩, (2.81) where the eigenvalue α = |α|eiφα is a complex number. The coherent state is decomposed into a series of ∞ X (a†)n |C⟩ = cn|n⟩, |n⟩ = √ |0⟩. (2.82) n=0 n! According to the definition, we find X∞ √ X∞ X∞ a|C⟩ = cn n|n − 1⟩ = α cm|m⟩ = α cn−1|n − 1⟩, (2.83) n=1 m=0 n=1 √ and obtain the relation cn n = cn−1α. By applying the recursive relation repeatedly, the coefficients of the series is represented as α α α αn cn =√ cn−1 = √ √ cn−2 = ··· = √ c0, (2.84) n n n − 1 n! ∞ X αn |C⟩ =c0 √ |n⟩. (2.85) n=0 n! with a constant c0 which is determined by

X∞ X∞ ∗ n m X∞ 2n (α ) α |α| 2 ⟨C|C⟩ = |c |2 √ √ ⟨n|m⟩ = |c |2 = |c |2e|α| = 1. (2.86) 0 0 n! 0 n=0 m=0 n! m! n=0 The coherent state is then described as

X∞ n X∞ n − 1 |α|2 α − 1 |α|2 (αa) |C⟩ =e 2 |n⟩ = e 2 √ |0⟩, (2.87) n! n! n=0 n=0 − 1 |α|2 iφ † 1 iφ † 2 1 iφ † n =e 2 1 + (|α| e α a ) + (|α| e α a ) + ··· + (|α| e α a ) + ··· |0⟩. (2.88) 2 n! CHAPTER 2. THEORETICAL TOOLS 18

The number of photon is not ﬁxed in the coherent state. In addition, all photons are generated on the vacuum with the same phase of φα. Such coherent state is realized in the classical coherent light (laser light). Incoherent photons, on the other hand, should be described as " # X X | ⟩ iϕλ1 2 iϕλ2 ··· | ⟩ Sun = 1 + aλe + aλe + 0 , (2.89) λ λ where λ represents the wavelength. Neither the phase nor the photon number are ﬁxed. The gentle sunshine may be described by such an incoherent state. Ginzbrug-Landau theory of superconductivityp described phenomenologically the equation of motion for the wave function Ψ(r) = n(r) eiθ(r). In the microscopic theory, the normal ground state of free electrons in a metal is called Fermi’s Sea and is described by Y | ⟩ † | ⟩ N = ck,α 0 . (2.90) |k|≤kF ,α=↑,↓

When the number of states for |k| ≤ kF is N, the number of electron is ﬁxed at 2N. According to the BCS theory [2], the superconducting state is described by Y | ⟩ iφ † † | ⟩ S = uk + vke c−k,↓ck,↑ 0 , (2.91) |k|≤kF

2 2 with uk +vk = 1. A pair of two electrons are created with the phase at φ independent of k. The superconducting condensate is described by the coherent superposition of such electron pairs. It is clear that the number of electrons is not ﬁxed in the superconducting state in Eq. (2.91). CHAPTER 2. THEORETICAL TOOLS 19

Superconducting Elements

H He

Li Be BC N O F Ne

Na Mg AlSi P S Cl Ar

K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn GaGe As Se Br Kr

Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe

Cs Ba La Hf Ta W Re Os Ir Pt Au Hg TlPb Bi Po At Rn

Fr Ra

La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb

Ac Th Pa U Np Pu

Al Superconducting

Si Superconducting under high pressure or thin film

Rb Mettalic but not yet found to be superconducting

F Nonmetallic elements

Ni Elements with magnetic order

He Superfluid

Figure 2.1: The periodic table of elements. Elements colored in red or pink are superconductors. Elements show the Bose-Einstein condensation are not indicated. Chapter 3

Mean-ﬁeld theory of superconductivity

The mechanism of superconductivity in metals was explained by Bardeen, Cooper, and Schrief- fer in 1957. They showed that the energy of the normal state in Eq. (2.90) is higher than that in the superconducting state in Eq. (2.91) when the weak attractive interaction works between the two electrons [2]. They applied the variational wave function method in their original pa- per. Today, however, we often use the mean-field theory proposed by Bogoliubov to describe the superconducting condensate instead of the variational method. The mean-field theory is advantageous when we calculate various physical values and generalize the theory to inhomogeneous superconductors or superconducting junctions. According to BCS, a phonon mediates the attractive interaction between two electrons. They showed that the Fermi surface is fragile in the presence of the attractive interactions between two electrons. It is clear that repulsive Coulomb interaction mediated by a photon always works between two electrons. In the Coulomb gauge, the coulomb interaction is instantaneous in time, whereas the attractive interaction is not instantaneous because the velocity of phonon vph ≈ 100 m/sec is much slower than vF . Thus such retardation effect in the attractive interaction enables two electrons to form a pair called a Cooper pair. The classical image of the pairing mechanism may be explained as follows. An electron comes to a certain place (say r0) at a certain time (say t = 0) in a metal and attracts ions surrounding it. The coming electron stays near r0 with in a time scale given by te ∼ ℏ/ϵF . On the other hand, it takes tph ∼ 1/ωD ≫ te for ions to move toward r0, where ωD is called Debye frequency. At t = tph, the electron has already moved away from r0. But remaining ions charge the place positively, which attracts another electron. Two electrons on the Fermi surface form a Cooper pair due to such an attractive interaction nonlocal in time. The classical image of the interaction is illustrated in Fig. 3.1

3.1 Pairing Hamiltonian

The interaction between two electrons is described by the Hamiltonian X Z Z 1 † H = dr dr′ψ† (r)ψ (r′)V (r − r′)ψ (r′)ψ (r). (3.1) I 2 α β β α α,β

20 CHAPTER 3. MEAN-FIELD THEORY OF SUPERCONDUCTIVITY 21

+e +e

+e e +e + +e 1 1 +e +e +e +e

e −e − 2 2 electron (a) (b)

Figure 3.1: The schematic image for electron-phonon interaction. (a) The ﬁrst electron modiﬁes the lattice structure locally and leaves away. (b) The local positive charge attracts the second electron.

The total Hamiltonian we consider is H = H0 +HI , where H0 is given in Eq. (2.54). By applying the Fourier transformation in Eq. (2.57), we obtain H)0 in Eq. (2.64) and X X 1 † † H = V c c c c , (3.2) I q k1+q,α k2−q,β k2,β k1,α 2Vvol α,β k1,k2,q 1 X V (r) = V eiq·r. (3.3) V q vol q

We assume the property of Vq as

• Vq is independent of q.

• V works between two electrons at k2 = −k1 and β =α ¯. • The scattering event is possible in a small energy window near the Fermi level characterized by ℏωD. Under these conditions, the interaction Hamiltonian becomes X 1 ′ † † H ′ ′ ↑ − ↓ I = V (k, k )c−k ,↓ ck ,↑ ck, c k, , (3.4) Vvol k,k′ ′ − −| | ℏ −| ′ | ℏ V (k, k ) = g Θ( ξk + ωD)Θ( ξk + ωD), (3.5) where g > 0 is a constant and the negative sign indicates that the interaction is attractive. The total Hamiltonian X X X 1 † g ′ ′ † † H − ′ ′ ↑ − ↓ = ξk ck,α ck,α c−k ,↓ ck ,↑ ck, c k, , (3.6) Vvol Vvol k,α k k′ X′ X = Θ(−|ξk| + ℏωD), (3.7) k k is called pairing Hamiltonian or BCS Hamiltonian. The interaction at the final expression in real space is given by V (r − r′) = −gδ(r − r′) which s short-range in space and instantaneous in CHAPTER 3. MEAN-FIELD THEORY OF SUPERCONDUCTIVITY 22 time. These properties might be different qualitatively from the effective interaction mediated by a phonon. In this sense, the BCS Hamiltonian is a model Hamiltonian that describes many- electron states in the presence of a specialized attractive interaction between two electrons on the Fermi surface. As we will see below, however, the mean-field theory of the BCS Hamilto- nian explains well the superconducting phenomena observed in experiments. This suggest that the attractive interaction between two electrons is indispensable to superconductivity and its dependence on space-time plays only a minor role.

3.2 Mean-ﬁeld approximation

The interaction term in Eq. (3.6) is nonlinear. It is impossible to have an exact solution of the ground state. Here, instead of solving Eq. (3.6) exactly, we apply the mean-ﬁeld approximation to the interaction term. We introduce the average of operators

X′ iφ g ∆e ≡ ⟨ck,↑ c−k,↓⟩, (3.8) Vvol k which is called pair potential and plays a central role in superconductivity. Generally speaking, the right-hand side is a complex number. Thus, the left-hand side is decomposed into the amplitude and the phase of the pair potential. Before solving the mean-field Hamiltonian, the meaning of the pair potential should be clarified. The product of two annihilation operators decreases the number of electrons by two. If we consider a ground state in which the number of electrons N is fixed, the quantum average results in

⟨N|ck,↑ c−k,↓|N⟩ ∝ ⟨N|N − 2⟩ = 0. (3.9)

Namely, such a number-ﬁxed state cannot be the ground state. A possible candidate of the ground state might be the coherent state of the laser light, where product of electron operators is replaced by a boson operator by ak = ck,↑ c−k,↓. Since

ak|C⟩ ∝ |C⟩, (3.10) the pair potential can have a nonzero value in the coherent state. Thus the pair potential determines the characteristic properties of the ground state automatically. The next task is to linearize the BCS Hamiltonian by using the mean ﬁeld in Eq. (3.8),

c ↑ c− ↓ =⟨c ↑ c− ↓⟩ + [c ↑ c− ↓ − ⟨c ↑ c− ↓⟩] , (3.11) k, k, k, k, h k, k, k, k, i † † ⟨ † † ⟩ † † − ⟨ † † ⟩ c−k,↓ ck,↑ = c−k,↓ ck,↑ + c−k,↓ ck,↑ c−k,↓ ck,↑ . (3.12)

These are the identities. The first term on the right-hand-side is the averaged value. The second term is considered to be the fluctuations from the average. In the mean-field theory, we assume that the fluctuations are much smaller than the average and expand the nonlinear term within CHAPTER 3. MEAN-FIELD THEORY OF SUPERCONDUCTIVITY 23 the first order of the fluctuations. The interaction term becomes X X ′ † † ′ ′ ′ ↑ − ↓ c−k ,↓ ck ,↑ ck, c k, , (3.13) ′ Xk k h i X ′ † † † † † † ′ ⟨ ′ ′ ⟩ ′ ′ − ⟨ ′ ′ ⟩ ⟨ ↑ − ↓⟩ ↑ − ↓ − ⟨ ↑ − ↓⟩ = c−k ,↓ ck ,↑ + c−k ,↓ ck ,↑ c−k ,↓ ck ,↑ ck, c k, + [ck, c k, ck, c k, ] , k′ k (3.14) ( " #) ( " #) −iφ X′ −iφ iφ X′ iφ Vvol∆e † † Vvol∆e Vvol∆e Vvol∆e = + c ′ c ′ − + c ↑ c− ↓ − , g −k ,↓ k ,↑ g g k, k, g k′ k (3.15) 2 2 X′ h i Vvol∆ Vvol −iφ iφ † † ≈ − + ∆e c ↑ c− ↓ + ∆e c ′ c ′ . (3.16) g2 g k, k, −k ,↓ k ,↑ k On the way to the derivation, we have used the complex conjugation of the pair potential

X′ −iφ g ⟨ † † ⟩ ∆e = c−k,↓ ck,↑ . (3.17) Vvol k The mean-ﬁeld Hamiltonian for superconductivity has the form

X X′ h i 2 † −iφ iφ † † Vvol∆ H = ξ c c − ∆e c ↑ c− ↓ + ∆e c c + , (3.18) MF k k,α k,α k, k, −k,↓ k,↑ g k,α k " # X h i iφ 2 † ξk ∆e ck,↑ Vvol∆ = ck,↑, c−k,↓ −iφ ∗ † + . (3.19) ∆e −ξ− c− ↓ g k k k, To diagonalize the mean-ﬁeld Hamiltonian, we solve the eigen equation, iφ ξk ∆e a a −iφ − ∗ = E . (3.20) ∆e ξ−k b b The solutions are given by uk −iφ , (3.21) vke for E = Ek and −v eiφ k , (3.22) uk for E = −Ek with s s q 1 ξ 1 ξ 2 2 k − k Ek = ξk + ∆ , uk = 1 + , vk = 1 . (3.23) 2 Ek 2 Ek

The results are summarized as iφ iφ ξk ∆ uk −vke Ek 0 uk vke ∗ − ∗ = −iφ − − −iφ . (3.24) ∆ ξ−k vke uk 0 Ek vke uk CHAPTER 3. MEAN-FIELD THEORY OF SUPERCONDUCTIVITY 24

By substituting the results into Eq. (3.19), the Hamiltonian is diagonalized as " # X h i γ ↑ 2 H † Ek 0 k, Vvol∆ MF = γk,↑, γ−k,↓ † + , (3.25) 0 −E−k γ− ↓ g k k, X 2 † † Vvol∆ = E (γ γ ↑ + γ γ− ↓ − 1) + , (3.26) k k,↑ k, −k,↓ k, g k with " # " # iφ γk,↑ uk vke ck,↑ † = − −iφ † , (3.27) γ−k,↓ vke uk c−k,↓ " # " # iφ ck,↑ uk −vke γk,↑ † = −iφ † . (3.28) c−k,↓ vke uk γ−k,↓

The last relationships are called Bogoliubov transformation. The operator γk,α is the annihilation operator of a Bogoliubov quasiparticle and obeys the fermionic anticommutation relations { † } { } |˜⟩ γk,α, γp,β = δk,pδα,β, γk,α, γp,β = 0, γk,α 0 = 0. (3.29)

The last equation deﬁnes the ’vacuum’ of the Bogoliubov quasiparticle, where |0˜⟩ represents the superconducting ground state. The creation of a Bogoliubov quasiparticle describes the elementary excitation from the superconducting ground state.

3.3 Gap equation and thermodynamic properties

The remaining task is to determine the pair potential ∆ in a self-consistent way. By the deﬁnition of the pair potential Eq. (3.8) and the Bogoliubov transformation in Eq. (3.28), we obtain

X′ D E iφ g − iφ † iφ † ∆e = (ukγk,↑ vke γ−k,↓)(vke γk,↑ + ukγ−k,↓) , (3.30) Vvol k X′ X′ iφ g iφ g ∆e Ek = ukvke (1 − 2nF (Ek)) = tanh . (3.31) Vvol Vvol 2Ek 2kBT k k On the way to the last line, we have used the relations, ⟨ † ⟩ ⟨ ⟩ γk,αγp,β = nF (Ek)δk,pδα,β, γk,αγp,β = 0. (3.32) The amplitude of ∆ should be determined by solving Eq. (3.31). Thus Eq. (3.31) is called the gap equation. We note that the equation has a solution only for g > 0 (attractive interaction). To proceed the calculation, we introduce the density of states per volume per spin, 1 X N(ξ) = δ(ξ − ξk). (3.33) Vvol k The gap equation becomes, p ! Z ℏ ωD N(ξ) ξ2 + ∆2 1 =g dξ p tanh . (3.34) 0 ξ2 + ∆2 2kBT CHAPTER 3. MEAN-FIELD THEORY OF SUPERCONDUCTIVITY 25

At the zero temperature T = 0, the integral can be carried out as

Z ℏ Z ℏ ωD N(ξ) ωD dξ 1 =g0 dξ p ≃ gN0 p , (3.35) 2 2 2 2 0  ξ + ∆ 0 ξ + ∆ s ℏω ℏω 2 =gN ln  D + D + 1 ≃ gN ln(2ℏω /∆), (3.36) 0 ∆ ∆ 0 D where N0 is the density of states at the Fermi level per spin per unit volume. The amplitude of the pair potential at T = 0 is then described as

− 1 gN ∆0 = 2ℏωDe 0 . (3.37)

As discussed in a number of textbooks, it is impossible to expand the right-hand side with respect to the small parameter gN0 ≪ 1, which implies the instability of the Fermi surface in the presence of a weak attractive interaction. Just below the transition temperature Tc, it is possible to put ∆ → 0 into the gap equation,

Z ℏ ωDdξ ξ 1 =gN tanh , (3.38) 0 ξ 2k T 0 B c Z ℏ ℏ ∞ ωD ωD −2 =gN0 ln tanh − gN0 dξ ln(ξ) cosh (ξ), (3.39) 2kBTc 2kBTc 0 ℏωD 4γ0 2ℏωDγ0 ≃gN0 ln + ln = gN0 ln . (3.40) 2kBTc π πkBTc

On the way to the last line, we considered ℏωD ≫ Tc and used Z ∞ −2 dx ln(x) cosh (x) = ln(4γ0/π), (3.41) 0 ln γ0 ≃0.577 Eular constant. (3.42)

From the last line, we obtain 2ℏω γ γ D 0 −1/g0N(0) 0 Tc = e = ∆0. (3.43) πkB πkB

The relation of 2∆0 = 3.5kBTc has been conﬁrmed in a number of metallic superconductors. Figure 3.2 show the dependence of the pair potential on temperature. As shown in Appendix A, the pair potential just below Tc depends on temperature as s r 8 Tc − T ∆ = πkBTc . (3.44) 7ζ(3) Tc

Therefore,

∂∆ , (3.45) ∂T T =Tc jumps at T = Tc as shown in Fig 3.2. CHAPTER 3. MEAN-FIELD THEORY OF SUPERCONDUCTIVITY 26

0.4

0.3 c T

k 0.2 Δ / 2 π 2 π / Δ

0.1

0.0 0.0 0.5 1.0 T / T c

Figure 3.2: The amplitude of the pair potential is plotted as a function of temperature.

By using the energy of superconducting state in Eq. (3.26), it is possible to calculate the partition function, specific heat of a superconductor,     X X  − − 2  ΞS = exp  Ek(nk,σ 1) + Vvol∆ /g /kBT , (3.46) n =0,1 k,σ k,σ Y Y 2 Ek/kB T −Ek/kB T 2 = exp(−Vvol∆ /gkBT ) e (1 + e ) . (3.47) k k The thermodynamic potential of the superconducting state per volume is represented by h i 1 ∆2 1 X −Ek/kB T 2 ΩS = −kBT log ΞS = − Ek + kBT log(1 + e ) . (3.48) Vvol g Vvol k The specific heat is then calculated as X ∂ΩS 2kB SS = − = − (1 − fk) log(1 − fk) + fk log fk, (3.49) ∂T Vvol k X X 2 2 ∂SS 2 ∂fk 2 ∂fk Ek 1 ∂Ek CS =T = Ek = − − , (3.50) ∂T Vvol ∂T Vvol ∂Ek T 2 ∂T k k 1 Ek fk = 1 − tanh . (3.51) 2 2kBT At the transition temperature, the specific heat is discontinuous, which implies the rapid decrease CHAPTER 3. MEAN-FIELD THEORY OF SUPERCONDUCTIVITY 27

of entropy at T = Tc. The diﬀerence in the speciﬁc heats per volume is described as Z ∂f(ξ) 1 ∂∆2 ∆C = C − C =2 dξN(ξ) − − , (3.52) S N ∂ξ 2 ∂T 8π2 =N k T . (3.53) 0 7ζ(3) B C

By using the specific heat in the normal state at T = Tc 2π2 C = N k T , (3.54) N 3 0 B C we obtain ∆C 12 = ≈ 1.42. (3.55) CN (Tc) 7ζ(3) The left hand-side of the equation is a universal value independent of materials. The free energy in the normal state ΩN is obtained by putting ∆ → 0 in ΩS. The difference between the free energies corresponds to the condensation energy of the superconducting state, " # − 1 ∆2 E 1 + e Ek/kB T Ω = Ω − Ω = tanh k + |ξ | − E − 2k T log , (3.56) SN S N k k B −|ξ |/k T Vvol 2Ek 2kBT 1 + e k B where we have used the gap equation. At T = 0, the condensation energy is calculated to be Z ∞ 2 2 ∆ 1 2 Hc ΩSN = 2N0 dξ + ξ − E = − N0∆ = − . (3.57) 0 2E 2 8π The last equation relates the condensation energy to the thermodynamic critical field.

3.4 Cooper pair

The pair correlation function is a two-point function in real space, X 1 · ′· ⟨ ⟩ ⟨ ′ ⟩ ik r1 ik r2 F (r1, r2) = ψ↑(r1) ψ↓(r2) = ck,↑ ck ,↓ e e , (3.58) Vvol k,k′ X 1 ′ · − ′ · ⟨ ′ ⟩ i(k+k ) r i(k k ) ρ/2 = ck,↑ ck ,↓ e e , (3.59) Vvol k,k′ where we introduce the coordinate r + r r = 1 2 , ρ = r − r . (3.60) 2 1 2 In a uniform superconductor, the pair correlation function is independent of r, which requiers k′ = −k. As a result, correlation function depends only on the relative coordinate as X 1 ik·ρ F (ρ) = ⟨ck,↑ c−k,↓⟩e . (3.61) Vvol k A Cooper pair with two electrons at the same place ρ = 0 is linked to the pair potential in Eq. (3.8). We note that the pair potential is the product of the pair correlation function and CHAPTER 3. MEAN-FIELD THEORY OF SUPERCONDUCTIVITY 28 the attractive interaction. In the BCS Hamiltonian, the interaction is short range in real space V (ρ) = −gδ(ρ). The pair potential satisﬁes the relation,

∆↑,↓ = g⟨ψ↑(r1) ψ↓(r1)⟩ = −g⟨ψ↓(r1) ψ↑(r1)⟩ = −∆↓,↑. (3.62)

Due to the Pauli’s exclusion principal, two electrons must have the opposite spin to each other. Above relation shows that the pair potential is antisymmetric under the commutation of two spins. Thus, a Cooper pair belongs to spin-singlet s-wave symmetry class. Although the pair potential is deﬁned by the correlation function at ρ = 0, the correlation function itself has a ﬁnite correlation length. The correlation function is calculated at T = 0 as a function of ρ = |ρ|,

1 X′ ∆eiφ F (ρ) = eik·ρ. (3.63) Vvol 2Ek k By replacing the summation by integration in three-dimension, Z Z Z X 2π π k0 1 1 2 = 3 dϕ dθ sin θ dkk , (3.64) Vvol (2π) k 0 0 0 the correlation function is expressed as Z iφ k0 ∆e 2 sin(kρ) q 1 F (ρ) = 2 dkk , (3.65) 4π 0 kρ 2 2 ξk + ∆ Z iφ k0 ≈∆e ℏ { ℏ }q 1 2 dk(kF + ξ/ vF )sin (kF + ξ/ vF )ρ . (3.66) 4π ρ 0 2 2 ξk + ∆

Since the integrand is large at ξk = 0, we have expanded as k ∼ kF + ξ/ℏvF in the second line. The running variable is transformed as k → ξk,

Z ℏ ∆eiφ ωD 1 F (ρ) = dξ(kF + ξ/ℏvF )sin{(kF + ξ/ℏvF )ρ}p , (3.67) 2ℏ 2 2 4π vF −ℏωD ξ + ∆ Z ℏ iφ ωD ℏ ≈ ∆e sin(kFpρ) cos(ξρ/ vF ) 2 2 dξkF . (3.68) 4π ℏvF 0 ξ2 + ∆2

The transformation t = ξ/∆ changes the integral as,

Z ℏ m∆eiφ ωD/∆ sin(k ρ) cos(tρ/πξ ) F (ρ) = dt F √ 0 , (3.69) 2π2ℏ2 2 0 t+ 1 iφ mkF ∆e sin(kF ρ) ρ = 2 2 K0 , (3.70) 2π ℏ kF ρ πξ0 ℏvF ξ0 = , (3.71) π∆0

−x where K0 is the modiﬁed Bessel function in the second kind and K0(x) ∼ e for x ≫ 1. The characteristic length ξ0 is called coherence length or Pippard length and describes the spatial −6 size of a Cooper pair. In typical metallic superconductors, ξ0 is about 1 × 10 m. CHAPTER 3. MEAN-FIELD THEORY OF SUPERCONDUCTIVITY 29

3.5 Magnetic properties

The pair correlation function becomes non uniform and depends on the center-of-mass coordinate r in the presence of magnetic ﬁeld. We consider pairing correlation function, p iθ(r) Ψ(r) ≡ ⟨ψ↑(r) ψ↓(r)⟩ = ns(r)e . (3.72)

This relation defines the macroscopic wave function of superconducting condensate described by the density of Cooper pairs ns and the phase of pairs θ. The wave functions should obey a phenomenological Schroedinger equation, ℏ2 e∗ 2 − ∇ − i A Ψ(r, t) = iℏ∂ Ψ(r, t), (3.73) 2m∗ ℏc t where m∗ = 2m and e∗ = 2e are the effective mass and the effective charge of a Cooper pair, respectively. Let us focus on the phase θ of the wave function and assume the density ns is uniform in superconductor for a while. In the absence of the vector potential, the energy is estimated as Z Z ℏ2 ℏ2n E = dr∇Ψ∗ · ∇Ψ = s dr(∇θ)2. (3.74) 2m∗ 2m∗ The results suggest that the phase gradient costs the kinetic energy. Thus θ should be uniform in the ground state, (i. e., ∇θ = 0). The current density of such condensate is described as − ∗ℏ ∗ ∗ ie ∗ ∇ − e − ∇ e ∗ j = ∗ Ψ iℏ A Ψ + iℏ A Ψ Ψ , (3.75) 2m c c e∗ℏ e∗ eℏ 2e = n ∇θ(r) − A = n ∇θ(r) − A . (3.76) m∗ s ℏc m s ℏc

The current is carried by the spatial derivative of the phase, which explains the Josephson eﬀect. The supercurrent density on the left-hand side is an observable, whereas the vector potential on the right-hand side depends on the gauge choice. The superconducting phase is necessary to be transformed as θ → θ + 2e/(ℏc)χ under the gauge transformation A → A + ∇χ. Therefore, Z 2e r θ(r) − dl · A(l), (3.77) ℏc is referred to as a gauge-independent phase.

Meissner effect We consider a superconductor in a weak magnetic field. The magnetic field in the superconductor should be described by the Maxwell equation, 1 4π ∇ × H + ∂ E = j. (3.78) c t c

In a static magnetic ﬁeld, ∂tE = 0 and H = ∇ × A. Together with Eq. (3.76), the equation becomes 4π eℏn 2e ∇ × H = s ∇θ − A . (3.79) c m ℏc CHAPTER 3. MEAN-FIELD THEORY OF SUPERCONDUCTIVITY 30

By multiplying ∇× from the left and use relations

∇ × ∇ × H = −∇2H, ∇ · H = 0, ∇ × (∇θ) = 0, (3.80) we obtain 4π(2n )e2 −∇2H = − s H. (3.81) mc2 We try to solve the equation at a surface of a superconductor as shown in Fig. 3.3(a), where the superconductor occupies x ≥ 0 and x < 0 is vacuum. The magnetic ﬁeld is applied uniformly in the y direction as H = H0yˆ in vacuum. The magnetic ﬁeld in the superconductor H = H(x)yˆ obeys s 2 2 d − 1 ≡ mc 2 H(x) 2 H(x) = 0, λL 2 , (3.82) dx λL 4πnee where we have used a relation 2ns = ne with ne is the electron density. The solution with a boundary condition H(0) = H0

− x c H − x λ 0 λ H(x) =H0e L , j = − e L zˆ, (3.83) 4π λL indicates that the penetration of the magnetic field is limited in the range of λL and that the current flows at the surface of the superconductor to screen the applied magnetic field. The char- −8 acteristic length λL is called London’s length is about 10 m in typical metallic superconductors.

y z d >> λL Vac. S d H0 Φ l

λL x 0 (a) (b)

Figure 3.3: (a) A magnetic ﬁeld at a surface of a superconductor. (b) The integration path l inside a superconducting ring.

At the surface 0 < x < λL, the electric current in the z direction under the magnetic ﬁeld in the y direction feels the Lorentz force in the x direction,

1 H2 − 2x 1 0 λ f = j × H = e L x = − ∂xHHx. (3.84) c 4πλL 4π The force points to the center of the superconductor. The current does not move because the superconducting condensate pushes the current back. To push the current and to move it from the center (x = ∞) to the surface (x = 0) against the Lorentz force, the superconductor works Z Z x=0 1 0 H2 0 H2 0 −2x/λL 0 W = − dr · f = dx∂xHH = e = . (3.85) x=∞ 4π ∞ 8π ∞ 8π CHAPTER 3. MEAN-FIELD THEORY OF SUPERCONDUCTIVITY 31

The superconducting condensate decreases the free energy below Tc by ΩS − ΩN < 0. The condensation energy must be larger than the energy of the magnetic ﬁeld

H2 Ω − Ω > 0 , (3.86) N S 8π to keep superconductivity. Otherwise the current and the magnetic fields penetrate into the metal. As a result, superconductivity disappears. The magnetic field Hc satisfies H2 Ω − Ω = c , (3.87) N S 8π 2 is called thermodynamic critical magnetic field. The left hand side is given by N0∆0/2 at T = 0 as we have discussed in Sec. 3.3.

Flux quantization A superconducting ring can accommodate magnetic flux Φ in its hole at the zero temperature as shown in Fig. 3.3(b). When the thickness of the arm d is much larger than λL, both the electric current and the magnetic field are absent at the center of the arm. Let us integrate the current in Eq. (3.76) along such a closed path along the ring, I I I eℏn 2e dl · j(l) = s dl · ∇θ − dl · A = 0. (3.88) m ℏc The right-hand side is calculated using the Stokes theorem, I Z Z 2e 2e 2e 2e dl · A = dS · (∇ × A) = dS · H = Φ. (3.89) ℏc ℏc ℏc ℏc The left hand side should satisfy I dl · ∇θ(l) = θ(2π) − θ(0) = 2πn, (3.90) because the wave function must be single-valued in quantum mechanics. Thus, we find that the magnetic flux passing through the hole is quantized by πℏc Φ = nϕ , ϕ = = 2 × 10−7gauss · cm2, (3.91) 0 0 e where ϕ0 is called flux quanta.

Vortex as a topological matter Superconductors are classified into two categories in terms of their magnetic property. Phe- nomenologically, λ κ = L (3.92) ξ0 √ characterizes the boundary. The type I superconductor characterized by κ ≤ 1/ 2 excludes the external magnetic field up to H < Hc and keeps superconductivity. Once H goes over Hc, superconductivity disappears. The threshold Hc is called the thermodynamic√ critical field. On the other hand, the type II superconductor characterized by κ > 1/ 2 have two critical fields

Hc1 and Hc2 ,( Hc1 < Hc2 ). The superconductor excludes perfectly the external magnetic ﬁeld CHAPTER 3. MEAN-FIELD THEORY OF SUPERCONDUCTIVITY 32

in a weak magnetic ﬁeld H < Hc1 . The superconductivity disappears in a strong magnetic ﬁeld

H > Hc2 . In the intermediate region Hc1 < H < Hc2 , a magnetic ﬁeld penetrates into the superconductor as quantized magnetic vortices. According to textbooks, these critical ﬁelds are represented as ϕ0 ϕ0 λL Hc2 = 2 ,Hc1 = 2 log . (3.93) 2πξ0 2πλL ξ0

2 When the ﬂux quanta ϕ0 passes through the area of ξ0 and ﬂuctuates the phase of the condensate, the condensate cannot keep the phase coherence any longer.

|ψ|

ξ0

Η λL r (a) (b)

Figure 3.4: The superconducting phase around a vortex core for r ≠ 0 is illustrated in (a). (b) The spatial proﬁle of the macroscopic wave function and the magnetic ﬁeld around a vortex core.

Let us focus on a magnetic flux at the origin of two-dimensional plane and the flux is represented by the vector potential ϕ A = e , (3.94) 2πr θ where we used the polar coordinate in two dimension x = r cos θ and y = r sin θ. Using the relation 1 1 1 ∇f(r, θ) =∂rfer + ∂θfeθ, ∇ · A = ∂r(rAr) + ∂θAθ, (3.95) r r r 1 1 ∇ × A = ∂ (rA ) − ∂ A e , (3.96) r r θ r θ r z the magnetic field is calculated as 1 ϕ H = ∂ r = 0. (3.97) z r r 2πr

Namely, the magnetic ﬁeld is zero at r ≠ 0. On the other hand, the magnetic ﬂux is calculated as I Z 2π ϕ dl · A = dθr = ϕ, (3.98) 0 2πr CHAPTER 3. MEAN-FIELD THEORY OF SUPERCONDUCTIVITY 33

where dl = dθ r eθ indicates the integration path enclosing the ﬂux. The Stokes theorem does not hold true because of the singularity in the vector potential at the origin. At r ≫ λL, the wave function may be described by √ iφ(r) Ψ(r) = nse , (3.99) which carries the current eℏn 2e j(r) = s ∇φ − A , (3.100) m ℏc

Due to the Meissner effect, the current is absent far from the origin r ≫ λL. The line integral on a closed loop gives the quantization of the flux ϕ = ϕ0n as already been mentioned. As a consequence of the quantization, the wave function has the form √ inθ Ψ(r) ≈ nse , (3.101) at Aθ = nϕ0/(2πr). The phase of wave function winds by 2πn times along a loop enclosing nϕ0 flux. The integer number n is a topological invariant called the winding number. In Fig. 3.4(a), we illustrate the superconducting phase far from the origin, where the phase φ(θ) = nθ is indicated by arrows for n = 1. Namely, the superconducting phase is linked to the angle in real space coordinate. At the origin r = 0, however, it is impossible to determine the direction of the arrow. To relax the situation, ns(r) vanishes at r = 0 as shown in Fig. 3.4(b). The coherence length ξ0 characterizes the spatial variation of the wave function, whereas the London length characterizes the variation of magnetic fields. The place for ns(r) = 0 (a node in the wave function) is called vortex core. The energy of the superconducting condensate around a vortex with nϕ0 is calculated as Z ℏ2 ie ie E = dr ∇ + A ψ∗ · ∇ − A ψ, (3.102) n 4m ℏc ℏc Z ℏ2 √ n nϕ √ n nϕ = dr n (−i) − 0 e−inθ n i − 0 einθ , (3.103) 4m s r 2πϕ s r 2πϕ Z Z 0 0 ℏ2n 2π R 1 = s n2 dθ drr , (3.104) 16m r2 0 ξ0 ℏ2πn R = s n2 log . (3.105) 8m ξ0

The point is that the energy is proportional to n2. Here let us consider a superconductor in which the magnetic flux Nϕ0 stays for Hc1 < H < Hc2 . There would be two choices for Nϕ0 flux to accommodate in the superconductor: one giant flux with the winding number n = N or N flux quanta with the winding number n = 1 at each. The energy of the former is given by EN and that of the latter is NE1. In the case of N ≫ 1, the latter case is more stable because of EN ≫ NE1. Therefore, the magnetic flux passes through a superconductor as a number of flux quanta. Finally, we briefly discuss a topological aspect of a vortex. We first consider a simple defect in a crystal lattice as shown in Fig. 3.5 (a), where a filled circle corresponds to an atom in the crystal. An open circle coming from the left is a vacancy. The vacancy comes into the crystal at the 1st time-step, moves to the right at 2nd and the 3rd time-step, and leaves away from the crystal. There is no difference between the crystal structures before the vacancy coming in and CHAPTER 3. MEAN-FIELD THEORY OF SUPERCONDUCTIVITY 34

Before 1 2 3 After (a) Propagation of a lattice defect

Before 1 2 3 After (b) Prapagation of a vortex core

Figure 3.5: The propagation of a defect from left to right through a uniform system. (a) A filled circle corresponds to an atom in a crystal. An open circle coming from the left is a vacancy. (b) An arrow represents a superconducting phase at the place. An open circle from the left indicate a vortex core. those after the vacancy leaving away. The vacancy is a local defect in the crystal and affects only its neighboring atoms. Next we consider a vortex core with n = 1 in a superconductor. Since the vortex core is a topological defect, its propagation changes the global phase configuration as illustrated in Fig. 3.5 (b). Before the core coming, a superconductor is in the ground state indicated by a uniform phase at φ = 0. At the 1st time-step, the superconductor changes the phase configuration drastically to accommodate the vortex. The winding number of the phase configuration becomes n = 1 at the second time-step. At the 3rd time-steps, the arrows tend to point to the left. After the vortex leaving, the superconductor is in the ground state with a uniform phase at φ = π. In the process, the winding number in the superconducting state changes as 0 → 1 → 0. The propagation of a topological defect changes the phase configurations of whole the superconductor. Chapter 4

Andreev reﬂection

The penetration of a Cooper pair from a superconductor to a non-superconducting material modifies the electromagnetic properties of the material. Such effect is called proximity effect. In the mean field theory of superconductivity, the Andreev reflection [13] describes the penetration of a Cooper pair into a metal. In the Bogoliubov-de Gennes picture, the Andreev reflection represents the conversion between an electron and a Cooper pair. Since a Cooper pair is charged particle, the propagation of a Cooper pair through a junction interface affects drastically the low energy charge transport such as the differential conductance in normal-metal/superconductor (NS) junctions and the Josephson current in superconductor/insulator/superconductor (SIS) junctions. In SIS and NS junctions, the pair potential is zero in a normal metal and in an insulator. To analyze the electric current in such a non-superconducting segment, we begin this section with the mean-field Hamiltonian in real space, Z X H † MF = dr ψα(r) ξ(r) ψα(r) Z α † iφ † −iφ + dr ψ↑(r) ∆(r)e ψ↓(r) + ψ↓(r) ∆(r)e ψ↑(r), (4.1)   ψ↑(r) Z   ψ↓(r) 1 † † ˇ   = dr[ψ↑(r), ψ↓(r), ψ↑(r), ψ↓(r)] HBdG(r)  †  , (4.2) 2  ψ↑(r)  † ψ (r) ↓ iφ ˇ ξ(r)ˆσ0 ∆(r)e iσˆ2 HBdG(r) = −iφ , (4.3) ∆(r)e (−i)ˆσ2 −ξ(r)ˆσ0 ℏ2∇2 ξ(r) = − + V (r) − ϵ , (4.4) 2m F whereσ ˆj for j = 1 − 3 is the Pauli’s matrix in spin space andσ ˆ0 is the unit matrix. The derivation is shown in Appendix. C. Eq. (4.3) has 4 × 4 structure because of the spin and electron-hole degrees of freedom. The 2×2 space at the upper-left of Eq. (4.3) is the Hamiltonian of an electron (particle) and that at the lower-right is the Hamiltonian of a hole (anti-particle). The pair potential hybridizes the two spaces. The hole-degree of freedom is introduced in the Bogoliubov-de Gennes picture in such a way as Z X Z X 1 † dr ψ† (r)ξ (r)ψ (r) = dr ψ† (r) ξ (r) ψ (r) − ψ (r) ξ∗ (r) ψ (r). (4.5) α α,β β 2 α α,β β α α,β β α,β α,β

35 CHAPTER 4. ANDREEV REFLECTION 36

The relation holds exactly even when ξ includes spin-dependent potentials and vector potentials. The spin structure of the pair potential is represented by iσˆ2 which means that the pair potential is antisymmetric under the permutation of two spins. The equation uν(r)ˆσ0 uν(r)ˆσ0 HˇBdG(r) = Eν (4.6) vν(r)(−i)ˆσ2 vν(r)(−i)ˆσ2 is called Bogoliubov-de Gennes (BdG) equation. It is possible to derive the similar equation ∗ − ∗ − vν(r)( i)ˆσ2 vν(r)( i)ˆσ2 HˇBdG(r) ∗ = −Eν ∗ , (4.7) uν(r)ˆσ0 uν(r)ˆσ0 because of particle-hole symmetry of the Hamiltonian. 0 σ0 ˇ ∗ 0 σ0 − ˇ HBdG(r) = HBdG(r). (4.8) σ0 0 σ0 0 The wave function satisﬁes the orthnormal property and the completeness. Z ∗ ∗ dr [uν(r)uλ(r) + vν(r)vλ(r)] = δν,λ, (4.9) X ∗ − uν(r)ˆσ0 ∗ ′ ∗ ′ vν(r)( i)ˆσ2 ′ ′ uν(r )ˆσ0, vν(r )iσˆ2 + ∗ vν(r ) iσˆ2, uν(r )ˆσ0 vν(r)(−i)ˆσ2 u (r)ˆσ0 ν ν ′ = 1ˇ4×4δ(r − r ). (4.10)

The Bogoliubov transformation is given by   ψ↑(r) " #   X ∗ †  ψ↓(r)  uν(r)ˆσ0 γ ↑ v (r)(−i)ˆσ2 γ− ↑  †  ν, ν ν,   = − + ∗ † , (4.11) ψ↑(r) vν(r)( i)ˆσ2 γν,↓ uν(r)ˆσ0 γ− ↓ † ν ν, ψ↓(r)     ∗ u 0 0 −v γν,↑ ν   X  ∗  γ ↓  0 uν vν 0   ν,  =  ∗   †  . (4.12) 0 −vν u 0  γ− ↑  ν ν ν, v 0 0 u∗ † ν ν γ−ν,↓

4.1 Conductance in an NS junction

The Andreev reﬂection occurs at an interface between a superconductor and a normal metal. In what follows, we will consider the transport property of an NS junction, where a normal metal (x < 0) is connected to a superconductor (x > 0) at x = 0. The BdG Hamiltonian in Eq. (4.3) is separated in two 2 × 2 Hamiltonian. Here we focus on the Hamiltonian for an electron with spin ↑ and a hole with spin ↓. The BdG Hamiltonian of the junction becomes ξ(r) ∆(r) Hˆ (r) = , (4.13) BdG ∆∗(r) −ξ∗(r) with ℏ2∇2 ξ(r) = − + v δ(x) − ϵ , ∆(r) = ∆eiφΘ(x). (4.14) 2m 0 F CHAPTER 4. ANDREEV REFLECTION 37

In a normal metal, the BdG equation is divided into two equations, ℏ2∇2 ℏ2∇2 − − ϵ u(r) = Eu(r) and + ϵ v(r) = Ev(r). (4.15) 2m F 2m F

The eigenvalues and the wave functions are obtained easily. The solutions are represented as 1 0 eikx f (ρ) and eikx f (ρ), (4.16) 0 p 1 p respectively. Here the ﬁrst (second) wave function is the solution of an electron (a hole) belonging 2 2 2 to E = ξk (E = −ξk) with ξk = ℏ (k + p )/(2m) −√ϵF . The wave function in the transverse ip·ρ direction to the interface is given by fp(ρ) = e / S with S being the cross section of the junction. These dispersions are illustrated in Fig. 4.1(a), where we plot the energy of an electron ξk and that of a hole −ξk for a propagationq channel p as a function of the wave number in the x 2 − 2 direction k. At the Fermi level k = kx = kF p , the two dispersions are degenerate because

ξkF = 0. In a superconductor, two solutions given by iφ/2 iφ/2 uk e ikx −vk e ikx −iφ/2 e fp(ρ) and −iφ/2 e fp(ρ), (4.17) vk e uk e belong to E = Ek and E = −Ek, respectively. The amplitudes of wave function is given by Eq. (3.23). The dispersion of Ek are illustrated in Fig. 4.1(b). The pair potential hybridizes the dispersion of an electron and that of a hole. The effects of the hybridization are most dominant at the Fermi level, (i. e., k = ±kx). The results for |k| > kx are the dispersion of an electronlike quasiparticle because of uk > vk. On the other hand, the results for |k| < kx are the dispersion of an holelike quasiparticle. In this book, an electronlike (a holelike) quasiparticle is refereed to as an electron (a hole) for simplicity. Eqs. (4.16) and (4.17) are the eigenstate vectors a function of the wave number k. Next we consider a situation where an electron at the Fermi level comes into the junction from the normal metal as indicated by ]alpha in Fig. 4.1(a). Due to the potential barrier at x = 0, the incoming wave is reflected into the electron branch as indicated by A. Simultaneously, the incoming wave is transmitted to the superconducting side. Since the pair potential hybridizes the electron and hole branches, two outgoing waves are possible as indicated as C and D. In the normal metal, therefore, the outgoing wave from the hole channel is also possible as indicated by B. The last process is called Andreev reflection where an electron is reflected as a hole by a superconductor. To describe the transmission and reflection processes at the junction interface, we need the wave function for an incoming and those for outgoing waves at an energy E. In a normal metal, the wave number calculated by the relation,

ℏ2k2 ℏ2p2 ±E = ξ = + ϵ − ϵ , ϵ = . (4.18) k 2m p F p 2m

The wave number is calculated as k+ in the electron branch with r 2mE 2 2 2 k± = k2 ± , k = k − p . (4.19) x ℏ2 x F

The wave numbers at four branches in a normal metal are illustrated in Fig. 4.1(a). The wave number at A becomes −k+. In the hole branch, the wave number at B is k− because of the CHAPTER 4. ANDREEV REFLECTION 38

relation E = −ξk. The wave function in a normal metal is represented by X 1 0 − 1 − 0 ϕ (r) = α eik+x + β e ik−x + A e ik+x + B eik−x f (ρ). L 0 1 0 1 p |p|

The wave number is positive at the left-going hole branch, which is explained by the group velocity of a quasiparticle. Since E = ξk in the electron branch, the velocity is deﬁned as 1 ℏ ve(k, p) = [∂ ξ xˆ + ∂ ξ ] = [k xˆ + p] , (4.21) ℏ k k p k m where xˆ is the unit vector in the x direction. In the hole branch, the velocity is calculated as 1 ℏ vh(k, p) = [−∂ ξ xˆ − ∂ ξ ] = − [k xˆ + p] . (4.22) ℏ k k p k m The velocity of a hole is negative in the x direction for k > 0. In a superconductor, the relation E = Ek is transformed to p 2 2 ξk = ±Ω, Ω = E − ∆ . (4.23)

Therefore, we obtain r 2mΩ q± = k2 ± . (4.24) x ℏ2

In the electron branch as indicated by C in Fig. 4.1(b), we put ξk → Ω in Eq. (4.23) and the wave number is q+. In the hole branch as indicated by D in Fig. 4.1(b), however, we put ξk → −Ω in Eq. (4.23) and the wave number is −q−. Thus the wave function on the right-hand side of the junction interface is represented as X u v − u − v ϕ (r) = Φˆ C eiq+x + D e iq−x + γ e iq+x + δ eiq−x f (ρ), R v u v u p |p|

The group velocity in a superconductor is calculated as a function of k,

1 ℏ ξk v(k, p) = [∂kEk xˆ + ∂pEk] = [k xˆ + p] (4.27) ℏ m Ek

In the electron (hole) branch, ξk is replaced by Ω (−Ω). The group velocities are then calculated to be ℏ Ω ve(±q , p) = [±q xˆ + p] , (4.28) + m + E ℏ − h Ω v (±q−, p) = [±q− xˆ + p] , (4.29) m E The group velocity in the x direction is indicated by arrows in Fig. 4.1. For E < ∆, the wave number in Eq. (4.24) become complex, which suggests that a quasiparticle at C and D CHAPTER 4. ANDREEV REFLECTION 39

E k −ξ k ξ k -q - q q q+ k k k + - - - + - - - k+ E E α γ D δ C ABβ Δ k k −k 0 −kx 0 kx x kx (a) normal metal (b) superconductor

N S N S e i ϕ 0 2Δ 0 i ϕ i ϕ e -i ϕ e e (c) Andreev reflection (d) Andreev reflection from an electron to a hole from a hole to an electron

Figure 4.1: The dispersion relation in a normal metal (a) and that in a superconductor (b). The arrow points the direction of the group velocity. The two Andreev reflection processes are schematically illustrated in (c) and (d). in Fig. 4.1(b) do not propagate to x = ∞. In this case, all the transport channels become evanescent mode. The boundary conditions for the wave functions are given by, ϕ (0, y) = ϕ (0, y), (4.30) L R 2 ℏ d d − ϕR(r) − ϕL(r) + v0ϕR(0, y) = 0. (4.31) 2m dx x=0 dx x=0 The first condition represents the single-valuedness of the wave function. The second condition is derived from the BdG equation Z Z δ δ lim dx HBdGϕ(r) = E lim dx ϕ(r) = 0, (4.32) → → δ 0 −δ δ 0 −δ and implies the current conservation law. See also Appendix B for details. The wave number p is conserved in the transmission and reflection processes because of translational symmetry in the ∗ transverse direction. It is easy to confirm the statement by multiplying fp(ρ) to the Eq. (4.30) and (4.31) and carrying out the integration with respect to ρ. The boundary conditions are represented in a matrix form, α A C γ + = Φˆ Uˆ + Φˆ Uˆ , (4.33) β B D δ α A C γ k¯τˆ − k¯τˆ = Φˆ Uˆ k¯τˆ + 2iz − Φˆ Uˆ k¯τˆ − 2iz , (4.34) 3 β 3 B 3 0 D 3 0 δ ˆ u v mv0 U = , z0 = 2 , (4.35) v u ℏ kF CHAPTER 4. ANDREEV REFLECTION 40

whereτ ˆ3 is the third Pauli’s matrix in particle-hole space. We have used the relation k¯ ≈ k±/kF ≈ q±/kF which are valid at a low energy E ≪ ϵF . At first, we calculate the transport coefficients for a quasiparticle incoming from the normal metal. They are defined as A ree reh α = nn nn , (4.36) B rhe rhh β nn nn ee eh C tsn tsn α = eh hh . (4.37) D tsn tsn β By putting γ = δ = 0, the resulting coefficients are given by ∗ | |2 −iφ | |2 iφ ee 2rnΩ hh 2rnΩ he tn ∆e eh tn ∆e rnn = , rnn = , rnn = , rnn = , (4.38) ΞNS √ √ΞNS ΞNS √ √ ΞNS −iφ/2 ∗ iφ/2 ee tn e 2E E + Ω hh tn e 2E E + Ω tsn = , tsn = , (4.39) ΞNS√ √ ΞNS√ √ ∗ −iφ/2 − ∗ iφ/2 − he rntn e 2E E Ω eh rntn e 2E E Ω tsn = , tsn = , (4.40) ΞNS ΞNS ¯ − k iz0 2 tn = , rn = , ΞNS = 2Ω + |tn| (E − Ω) (4.41) k¯ + iz0 k¯ + iz0 where tn (rn) is the transmission (reflection) coefficient in the normal state. By putting α = β = 0, the transport coefficients for a quasiparticle incoming from the superconductor become, C ree reh γ = ss ss , (4.42) D rhe rhh δ ss ss ee eh A tns tns γ = eh hh . (4.43) B tns tns δ with ∗ −| |2 −| |2 ee 2rnΩ hh 2rnΩ he tn ∆ eh tn ∆ rss = , rss = , rss = , rss = , (4.44) ΞNS r ΞNS ΞNS r ΞNS iφ/2 ∗ −iφ/2 ee 2tnΩe E + Ω hh 2tnΩe E + Ω tns = , tns = (4.45) ΞNS r2E ΞNS 2rE ∗ −iφ/2 − ∗ iφ/2 − he 2tnrne Ω E Ω eh 2tnrne Ω E Ω tns = , tns = . (4.46) ΞNS 2E ΞNS 2E The scattering matrix of the junction interface is then represented by       ee eh ˜ee ˜eh A α rnn rnn tns tns      he hh ˜he ˜hh   B  S  β  S  rnn rnn tns tns    =   , =  ˜ee ˜eh ee eh  . (4.47) C γ tsn tsn rss rns ˜he ˜hh he hh D δ tsn tsn rss rns The transmission coefficients are correctedp by the ratio of the velocity in the outgoing channel and that in the incoming channel, (i.e., vout/vin). The results are given by s Ω t˜ij = Re tij , (4.48) sn E sn s E t˜ij = Re tij . (4.49) ns Ω ns CHAPTER 4. ANDREEV REFLECTION 41

For E > ∆, all the channels in a superconductor are propagating with the velocity of |vS| = vkx (Ω/E). In a normal metal, all the channels are always propagating with the velocity of | | vN = vkx . For E < ∆, on the other hand, all the transport channels in a superconductor are ij ij † evanescent, which results in t˜sn = 0 and tñs = 0. The unitarity of the scattering matrix SS = 1 implies the current conservation law. The Andreev reflection is a peculiar reflection process of a quasiparticle by a superconductor. When the energy of an incident electron is smaller than the gap E < ∆, an incident electron cannot go into a superconductor because of a gap at the Fermi level. In the superconducting gap, however, a number of Cooper pairs condense coherently at a phase of φ. A Cooper pair is a composite particle consisting of two electrons and a phase as shown in Fig. 4.1(c). An incident electron has to find a partner of a pair and get the phase of eiφ so that the electron can penetrate into the superconductor as a Cooper pair. As a result of such Cooper pairing, the empty shell of the partner is reflected as a hole into the normal metal. Simultaneously, the conjugate phase e−iφ is copied to the wave function of a hole. Thus an electron is converted to a Cooper pair at the Andreev reflection. The inverse process, the reflection from a hole to an electron, is also called the Andreev reflection as shown in Fig. 4.1(c). A hole cancels an electron of a Cooper pair. As a consequence, the remaining electron is reflected into the normal metal with the phase of eiφ. At E = 0, the wavenumber of an electronlike quasiparticle (electron) is calculated as i ∆ ∆ q+ = kF 1 + = kF + i , (4.50) 2 ϵF ℏvF where we put p = 0 for simplicity. The inverse of the imaginary part characterizes the penetration length of an electron and is equal to the coherence length ξ0 at T = 0. Namely, an incident electron decreases its amplitude and condenses as a Cooper pair in the superconducting gap. Such event happens at the interface of a superconductor and a normal metal. At E < ∆, there is no propagating channel in a superconductor. Thus the conservation law | ee |2 | he |2 rnn + rnn = 1, (4.51)

2 holds true. When the normal transmission probability is much smaller than unity |tn| ≪ 1, the | he | → | ee | → Andreev reflection is suppressed rnn 0 and the normal reflection is perfect rnn 1. On | ee | the contrary at tn = 1, the normal reflection is absent rnn = 0 and the Andreev reflection is | he | → perfect rnn 1. The reflection coefficients √ he −i arctan( ∆2−E2/E) −iφ rnn =e e , (4.52) √ eh −i arctan( ∆2−E2/E) iφ rnn =e e , (4.53) includes only the phase information at tn = 1. The Blonder-Tinkham-Klapwijk formula or Takane-Ebisawa formula enables us to calculate the differential conductance of a NS junction in terms of the reflection coefficients,

dI 2e2 X − | ee |2 | he |2 GNS = = 1 rnn + rnn . (4.54) dV eV h p eV =E We plot the conductance as a function of the bias voltage in Fig. 4.2, where the vertical axis is normalized to the conductance of the junction in the normal state, 2e2 X G = |t |2 = R−1. (4.55) N h n N p CHAPTER 4. ANDREEV REFLECTION 42

At |tn| = 1, the conductance is twice of its normal value at the subgap regime E < ∆ due to the perfect Andreev reﬂection. 2e2 X 4e2 X 4e2 2e2 G = 2 |rhe|2 = = N ,G = N = R−1, (4.56) NS h h h c N h c N p p where Nc is the number of propagating channels of the junction. (See Appendix B for details) An incident electron is converted to a Cooper pair perfectly, which increases the number of the charge passing through the interface by factor 2. On the other hand in the limit of |tn| ≪ 1, the conductance spectra become E GNS → GN √ , (4.57) E2 − ∆2 which is proportional to the density of states in a uniform superconductor. The results shown in Fig. 4.2 at tn → 0 correspond to the spectra measured by scanning tunneling microscopy (STM) or scanning tunneling spectroscopy (STS) experiments.

! '

%0 %1%$ " $ )*+,%-%& '(

%& # %0$%1%# %0 %1%2 $ $ ." .# $ # " %*+%-%/

Figure 4.2: The diﬀerential conductance of a NS junction is plotted as a function of the bias voltage across the junction for several choices of the potential barrier hight z0.

4.2 Retroreﬂectivity

The Andreev reﬂection has quite unusual properties also in real space. When the wave number of an incident electron is (k, p), the wave number at outgoing waves are summarized as

A :(−k, p),B :(k, p),C :(k, p),D :(−k, p). (4.58)

The trajectories of outgoing waves are illustrated in the top panel in Fig. 4.3, where we consider the group velocities in Eqs. (4.21), (4.22), (4.28) and (4.29). The velocity component perpendicular to the interface changes sign in the normal reflection, whereas all the velocity components change signs in the Andreev reflection. As a result, a reflected hole traces back the original trajectory of an incoming electron (retroreflectivity) as shown in the first panel in Fig. 4.3. An electron-hole pair on the Fermi level (E = 0) connected by the Andreev reflection are retroreflective exactly in the presence of time-reversal symmetry. Therefore the two trajectories are CHAPTER 4. ANDREEV REFLECTION 43 retroreflective to each other even in the presence of impurities as shown the second panel in Fig. 4.3. Although the pair potential is zero in the normal metal, a Cooper penetrating from the superconductor stays there. Such effect is called proximity effect. The presence of a Cooper in a normal metal is described by a strongly correlated electron-hole pair in the Bogoliubov-de Gennes picture. This fact is very important to understand the Josephson effect through a dirty metal. In the absence of time-reversal symmetry, the retroreflective is lost. A good example might be the motion of a quasiparticle in the magnetic field perpendicular to the two-dimensional plane as shown in the bottom panel in Fig. 4.3. The Lorentz force acts on a charged particle moving in a magnetic field

F = mr¨ = er˙ × H. (4.59)

The charge of a hole has the opposite sign to that in the electron. Simultaneously, the velocity of a hole is just the opposite to that of an electron. Thus the Lorentz force acts in the same direction in the two branches. In the bottom panel in Fig. 4.3, an electron and a hole depart at a same point on the NS interface and meet again at an another point on the interface. Such correlated motion of an electron and a hole is responsible for Aharonov-Bohm like eﬀect in the magnetoconductance of a ballistic NS junction [14].

4.3 Josephson current in an SIS junction

The Josephson effect is a highlight of superconducting phenomena. The dissipationless electric current flows between two superconductors at the zero bias voltage. In this section, we try to understand the Josephson effect in terms of the Andreev reflection. A typical example of Josephson junction may be a SNS junction as shown in Fig. 4.4, where two superconductors sandwich a normal metal. The Josephson current is calculated based on Furusaki-Tsukada formula [17], e X X ∆ J = kBT [rhe(p, ωn) − reh(p, ωn)] . (4.60) ℏ Ωn p ωn In this formula, the Josephson current is expressed in terms of two Andreev reflection coefficients rhe and reh. The reflection coefficient rhe represents the process where an electron incoming from the left superconductor is reflected as a hole after traveling whole the junction as shown in Fig. 4.4. This process carries a Cooper pair from the left superconductor to the right superconductor. The coefficient reh represents the conjugate process to rhe. The formula suggests that the net current should be described by the subtraction of the two coefficients. Since the direct-current Josephson effect happens in equilibrium, coefficients are a function of a Mat- subara frequency ωn = (2n + 1)πkBT/ℏ, where n is an integer number and T is a temperature. The analytic continuation in energy is summarized as p 2 2 E → iℏωn, Ω = E − ∆ → iΩn, (4.61) p ℏ2 2 2 with Ωn = ωn + ∆ . The conductance is represented by the square of reflection coefficients in Eq. (4.54). The Josephson current in Eq. (4.60) is represented using reflection coefficients themselves. The two superconductors exchange their phase information through the phase degree of freedom of the transport coefficients. The transmission and reflection coefficients appear in the Landauer formula for conductance which is a basic formula in mesoscopic physics. The formula in Eq. (4.60) as well as Eq. (4.54) bridges physics of superconductivity and physics of mesoscopic transport. CHAPTER 4. ANDREEV REFLECTION 44

Normal metal Superconductor

A C

h B e D

e h

h e e 1

Figure 4.3: The trajectory of outgoing waves. The Andreev reflection is retroreflective on the Fermi surface in the presence of time-reversal symmetry. A hole traces back the trajectory of an incoming electron as shown in the two panels from the top. In magnetic fields, the retroreflectivity is lost as shown in the bottom panel.

Our next task is to calculate the Andreev reﬂection coeﬃcients in an SIS junction by solving the BdG equation, ξ(r) ∆(r) u(r) u(r) = E , (4.62) ∆(r)∗ −ξ(r) v(r) v(r) ( ∆eiφL x < 0, ∆(r) = (4.63) ∆eiφR x > 0,

The wave function in the left superconductor is given by u v − u − v ϕ (r) =Φˆ eikxxα + e ikxxβ + e ikxxA + eikxxB f (ρ), (4.64) L L v u v u p s s eiφj /2 0 1 Ω 1 Ω Φˆ = , u = 1 + n , v = 1 − n , (4.65) j −iφj /2 0 e 2 ℏωn 2 ℏωn where j = L or R, A and B are the amplitude of outgoing waves and α and β are the amplitude of incoming waves. In the right superconductor, the wave function is represented with two CHAPTER 4. ANDREEV REFLECTION 45

Superconductor Normal Superconductor Ek Ek Ek

k x k k k

ϕ ϕ L R

rhe

reh

Figure 4.4: The dispersion relation at each segment of a SNS junction. The two Andreev reﬂection processes which carry the Josephson current. amplitudes C and D as u v − u − v ϕ (r) =Φˆ eikpxC + e ikpxD e ikpxγ + eikpxδ f (ρ). (4.66) R R v u v u p

By substitute the wave functions into the boundary conditions in Eqs. (4.30) and (4.31), we obtain the relationship among the amplitudes of wave functions, α A C γ ΦˆUˆ + = Uˆ + Uˆ , (4.67) β B D δ h i h i α A C γ ΦˆUˆ k¯τˆ − k¯τˆ = Uˆk¯τˆ + 2iz Uˆ − Uˆk¯τˆ − 2iz Uˆ , (4.68) 3 β 3 B 3 0 D 3 0 δ − ei(φL φR)/2 0 u v Φˆ = − − , Uˆ = , (4.69) 0 e i(φL φR)/2 v u

By eliminating C and D under γ = δ = 0, we obtain A r r α = ee eh , (4.70) B rhe rhh β 2 |tn| ∆ rhe = [ℏωn(cos φ − 1) + iΩn sin φ] , (4.71) 2iΞSS 2 |tn| ∆ reh = [ℏωn(cos φ − 1) − iΩn sin φ] , (4.72) 2iΞSS h i φ Ξ =(ℏω )2 + |t |2∆2 1 − |t |2 sin2 , φ = φ − φ , (4.73) SS n n n 2 L R where we have used the relations p ∆ (ℏω )2 + ∆2 uv = , u2 − v2 = n . (4.74) 2iℏωn ℏωn CHAPTER 4. ANDREEV REFLECTION 46

The Josephson current is represented as

e X X |t |2∆2 J = sin φ k T n . (4.75) B 2 2 2 2 2 φ ℏ ℏ ω + ∆ 1 − |tn| sin p ωn n 2

The identity X 1 1 y kBT 2 2 = tanh (4.76) (ℏωn) + y 2y 2kBT ωn enables us to carry out the summation over Matsubara frequency. Finally, we reach at  q  2 2 φ X | |2 ∆ 1 − |tn| sin e q tn ∆  2  J = ℏ sin φ tanh . (4.77) − | |2 2 φ 2kBT p 2 1 tn sin 2

First we consider the tunneling limit of |tn| ≪ 1 which represents an SIS junction. Using the normal resistance in Eq. (4.55), the Josephson current becomes π∆ ∆ ∆ J = 0 tanh sin φ. (4.78) 2eRN ∆0 2kBT

Eq. (4.78) was first derived by Ambegaokar-Baratoff [18] and has explained well the Josephson effect observed in experiments. By using the dependence of ∆ on temperature in Fig. 3.2, the amplitude of the Josephson current is plotted as a function of temperature in Fig 4.5. The Josephson current increases with the decrease of temperature and saturates at a low temperature.

0.6

T / T =0.0001

c 1.0

0.4

0.1

0.5

0.8 ]

] 0.2 N N

0.8

0.6 / 2eR 0 / 2eR 0.0 0

[ C

J 0.4 J [

-0.2

0.2 -0.4

-0.6

0.0

-1.0 -0.5 0.0 0.5 1.0

0.0 0.2 0.4 0.6 0.8 1.0

T / T

Figure 4.5: Left: the amplitude of the Josephson current in a SIS junction (|tn| ≪ 1) is plotted as a function of temperature. Right: the current-phase relationship in a S-constriction-S junction (|tn| = 1).

Next we consider a junction through a constriction by choosing tn = 1. Using the Sharvin resistance in Eq. (4.56), the Josephson current becomes ! π∆ ∆ φ ∆ cos φ J = 0 sin tanh 2 . (4.79) eRN ∆0 2 2kBT CHAPTER 4. ANDREEV REFLECTION 47

This formula was ﬁrst derived by Kulik-O’melyanchuck [19]. The results at T = 0 suggest the fractional current-phase relationship as sin(φ/2). The Josephson current is discontinuous at φ = ±π. Just below Tc, the current-phase relationship is sinusoidal as displayed in Fig. 4.5. Generally speaking, the Josephson current is decomposed into a series of X J = Jn sin(nφ), (4.80) n=1 when the junction preserves time-reversal symmetry at φ = 0. The current-phase relationship in Eq. (4.79) is described as ∞ φ X 8(−1)n+1 sin = sin(nφ). (4.81) 2 (2n + 1)(2n − 1) n=1

Namely, not only the lowest order coupling term J1 but also higher order terms Jn for n = 2, 3, ··· contribute to the Josephson current. The Andreev reﬂection processes for the lowest order term are analyzed in what follows. In Fig. 4.6(a), we show the Andreev reﬂection process for the

S XS he hh eh ee t ( L ) t ( L ) t ( ) t ( L ) ns ns ns L ns he he r ( R ) eh reh r ( ) nn rnn ( R ) nn ( R ) nn R t hh ( L ) he ee eh sn ϕ t ( L ) t ( L ) t sn ( L ) ϕ R sn sn L (a1) elecron (a2) (a3) (a4) hole

he he eh J reh r ( R ) r ( L ) r ( R ) 1 nn ( L ) nn − nn nn

(b)

− J2

(c)

Figure 4.6: (a) The process include the Andreev reflection at the right interface once. The material at the central segment is indicated by X. The transmission coefficient of X are tX in ∗ the electron branch and tX in the hole branch. (b) The reflection processes contribute to J1. (c) The schematic illustration of the Andreev reflection processes in J2.

Josephson current, where a quasiparticle suﬀers the Andreev reﬂection at the right interface once. The two processes are possible for rhe in Eq. (4.60) as shown in (a1) and (a2). Figures in (a3) and (a4) represent the processes for reh. The contributions of these four processes to the Josephson current are summarized as A1 as X X h e ∆ hh · ∗ · he · · ee he · · eh · ∗ · he A1 = kBT tsn (L) tX rnn(R) tX tns(L) + tsn(L) tX rnn(R) tX tns(L) ℏ Ωn p ωn i − ee · · eh · ∗ · hh − eh · ∗ · he · · eh tsn(L) tX rnn(R) tX tns (L) tsn(L) tX rnn(R) tX tns(L) , (4.82)

∗ where tX (tX ) is the transmission coefficient of a material X in the electron (hole) branch. We have assumed that the material X preserves time-reversal symmetry. Among the transport CHAPTER 4. ANDREEV REFLECTION 48 coefficients shown in Eqs. (4.38)-(4.46), it is possible to confirm the relations

∆ ∆ tee(L) · · thh(L) − teh(L) · · teh(L) = 2reh (L), (4.83) ns Ω sn ns Ω sn nn ∆ ∆ thh(L) · · tee(L) − the(L) · · tee(L) = 2rhe (L). (4.84) ns Ω sn ns Ω sn nn These relationships hold true even after the analytic continuation. Thus we obtain X X h i 2ie eh · ∗ · he · − he · · eh · ∗ A1 = ℏ kBT rnn(L) tX rnn(R) tX rnn(L) tX rnn(R) tX . (4.85) p ωn

The reflection processes indicated by this equation are illustrated in Fig. 4.6(b). The processes include the Andreev reflection once at the left interface and once at the right interface. The transport coefficients include tn which is the transmission coefficients of the interface between the superconductor and the material X. To proceed the calculation let us choose tn ≪ 1. X X 2 e | 2 |2 ∆ A1 = kBT tX tn sin φ. (4.86) ℏ Ωn p ωn

The results correspond to J1 sin φ. In the n-th order term in Eq. (4.80) are derived from the processes including the Andreev reﬂection n times at the left interface and n times at the right 2n interface. In addition, n th order process is proportional to |tX | because a quasiparticle travel 2 the X segment 2n times. When |tX | ≪ 1 as it is in the insulator, the higher harmonics is negligible. In Eq. (4.79), the higher order terms contribute to the Josephson current because of tn = 1. Fig. 4.6(c) shows the reﬂection process which contribute to J2 sin(2φ).

Note As already mention in the formula of Eq. (4.60) that the Josephson current is described by the Andreev reflection coefficients. Although the current should be gauge invariant, the reflection coefficients depend on a gauge choice. The user of Eq. (4.60) should pay attention to the gauge choice with which the Andreev reflection coefficients are calculated. In this text, Eq. (4.60) is correct when we calculate the transport coefficients in Eq. (4.70) from the wave functions in Eqs. (4.64) and (4.66). Chapter 5

Unconventional superconductor

In previous chapters, we have assumed that a superconductor is a simple metal such as Al, Pb and Nb. A Cooper pair in such a superconductor belongs to spin-singlet s-wave symmetry class. We ﬁrst generalize the pair potential within the mean-ﬁeld theory,

∆α,β(r1, r2) =g(r1 − r2) ⟨ψα(r1)ψβ(r2)⟩ = −g(r1 − r2) ⟨ψβ(r2)ψα(r1)⟩ , (5.1)

= − ∆β,α,(r2, r1), (5.2) where the attractive interaction g(r1 − r2) = g(r2 − r1) is symmetric under the permutation of real space coordinate and we have used the anticommutation relation of fermion operators. The negative sign on the right-hand side should be derived from the permutation of spins or the permutation of spatial coordinates of two electrons. This relation enables us to categorize superconductors into two classes: spin-singlet even-parity and spin-triplet odd-parity. By applying the Fourier transformation X X 1 ik·r 1 ik·r ψα(r) = √ ck,αe , g(r) = gke , (5.3) V Vvol vol k k the pair potential is represented as X 1 iq·ρ i(k+k′)·R i(k−k′)·ρ/2 ∆ (R, ρ) = g e ⟨c c ′ ⟩e e , (5.4) α,β V 2 q k,α k ,β vol q,k,k′ r + r R = 1 2 , ρ = r − r . (5.5) 2 1 2 Here we put k′ = −k because we consider a uniform superconductor. Namely the pair potential is independent of the center-of-mass coordinate of two electrons R. As a result, we have an expression of the pair potential, X X 1 ⟨ ⟩ i(k+q)·ρ 1 ik·ρ ∆α,β(ρ) = 2 gq ck,αc−k,β e = ∆α,β(k)e . (5.6) V Vvol vol k,q k with the Fourier component 1 X ∆ (k) = g(k − p)f (p), f (k) = ⟨c c− ⟩. (5.7) α,β V α,β α,β k,α k,β vol p In the weak coupling theory, two electrons on the Fermi level form a Cooper pair due to the attractive interaction. As a result, the pair potential can be deﬁned only near the Fermi level.

49 CHAPTER 5. UNCONVENTIONAL SUPERCONDUCTOR 50

Therefore, k and p in Eq. (5.7) are limited as the momenta on the Fermi surface. To proceed the argument, let us consider a superconductor in two-dimension and replace the summation by the integration, Z Z 1 X 2π dθ → dξN(ξ) , (5.8) Vvol 2π k 0 where N(ξ) is the density of states per spin per volume. Z Z 2π ′ dθ ′ ′ ∆α,β(θ) = dξN(ξ) g(θ, θ ) fα,β(ξ, θ ), (5.9) 0 2π where θ is an angle on the two-dimensional Fermi surface as shown in Fig. 5.1. The the pairing function is decomposed into the Fourier series as X (c) (s) fα,β(ξ, θ) = fα,β(ξ, 0) + fα,β(ξ, n) cos(nθ) + fα,β(ξ, n) sin(nθ). (5.10) n=1

The transformation of k → −k is described by θ → θ + π. The component fα,β(ξ, 0) is the (c) (s) even-parity s-wave. We ﬁnd that fα,β(ξ, 2m) and fα,β(ξ, 2m) are even-parity functions for m (c) (s) being an integer. In addition, fα,β(ξ, 2m + 1) and fα,β(ξ, 2m + 1) are odd-parity functions. When g(θ, θ′) = g is independent of angles, the pair potential is independent of θ because the integral over θ′ extracts the s-wave component. The resulting relation is the gap equation in the BCS theory Z

∆α,β = dξN(ξ)g fα,β(ξ, 0). (5.11)

(c) At n = 1, fα,β(ξ, 1) cos(θ) belongs to px symmetry because of kx = kF cos θ. Simultaneously, (s) fα,β(ξ, 1) sin(θ) belongs to py symmetry because of ky = kF sin θ. To describe p-wave pair potential, we should choose

g(θ, θ′) = 2 cos(θ − θ′) = 2 cos θ cos θ′ + 2 sin θ sin θ′, (5.12) where the ﬁrst (second) term extracts px-wave (py-wave) component from the pairing function. In the similar manner, the d-wave pair potential is described by

g(θ, θ′) = 2 cos(2θ − 2θ′) = 2 cos(2θ) cos(2θ)′ + 2 sin(2θ) sin(2θ′). (5.13)

The ﬁrst (second) term extracts dx2−y2 -wave (dxy-wave) component from the pairing function. The interaction potential determines the symmetry of the pair potential. The unconventional pair potentials are illustrated in Fig. 5.1 on the Fermi surface. Generally speaking, the pairing function has richer symmetry variety than the pair potential. The BdG equation for a spin-singlet unconventional superconductor in momentum space is given as ξ σˆ ∆ eiφ iσˆ uˆ uˆ k 0 k 2 k = k E , (5.14) −∆∗ e−iφ iσˆ −ξ∗ σˆ vˆ vˆ k −k 2 −k 0 k k  ∆ s 2 − 2 2 ∆k =  ∆(kx ky)/kF dx2−y2 . (5.15) 2 ∆(2kxky)/kF dxy CHAPTER 5. UNCONVENTIONAL SUPERCONDUCTOR 51

The pair potential depends on the direction of the wavenumber on the Fermi surface. We consider a time-reversal spin-triplet superconductor in unitary state. In such case, BdG Hamiltonian becomes ξ σˆ ∆ eiφ id · σˆσˆ uˆ uˆ k 0 k 2 k = k E , (5.16) ∆∗ e−iφ iσˆ d · σˆ −ξ∗ σˆ vˆ vˆ k −k 2 −k 0 k k ∆kx/kF px ∆k = , (5.17) ∆ky/kF py where d is a real unit vector in spin space.

ky + - + - + θ - + kx + + + - - - d 2 2 Fermi surface s px py dxy x - y

Figure 5.1: The pair potentials are illustrated on the the two-dimensional Fermi surface.

5.1 Conductance in an NS junction

We consider an NS junction as shown in Fig.4.1. The junction is uniform in the y direction. The wave function in a normal metal is α 0 − A − 0 ϕ (x) = eikxx + e ikxx + e ikxx + eikxx, (5.18) L 0 β 0 B q 2 − 2 where the wave number in the y direction is ky and kx = kF ky is the wave number in the x direction at the Fermi level. We have omitted the wave function in the y direction fky (y) and have neglected the corrections to the wave number of the order of ∆/ϵF . The amplitude of wave function, X↑ X = , (5.19) X↓ has two spin components for X being α, β, A, and B in Eq. (5.18). This is also true for C and D in Eq. (5.20). Thus ”0” in Eq. (5.18) is 2 × 1 null vector in spin space. The right-going wave in a superconductor is represented as u σˆ + 0 ikxx v−s−σˆS −ikxx ϕR(x) =Φ † e C + e D , (5.20) − v+s+σˆS u σˆ0 s s 1 Ω± 1 Ω± u± = 1 + , v± = 1 − , (5.21) 2 E 2 E q ± ∆ 2 2 ∆± =∆(±kx, ky), s± = , Ω± = E − ∆±, (5.22) |∆±| iσˆ2 singlet σˆS = (5.23) id · σˆσˆ2 triplet CHAPTER 5. UNCONVENTIONAL SUPERCONDUCTOR 52 where C and D represent the amplitudes of out-going waves. Using the boundary conditions in Eqs. (4.30) and (4.31), it is possible to derive a relation α A kτ U −1 − U −1kτ + 2iz U −1 = − kτ U −1 + U −1kτ + 2iz U −1 , (5.24) 3 3 0 β 3 3 0 B u+σˆ0 v−s−σˆS U = Φ † . (5.25) v+s+σˆS u−σˆ0 The reflection coefficients defined by, A rˆ rˆ α = ee eh (5.26) B rˆhe rˆhh β are calculated as − ∗ − rn(1 Γ+Γ−) rn(1 Γ+Γ−) rêe = 2 σˆ0, rˆhh = 2 σˆ0, (5.27) 1 − |rn| Γ+Γ− 1 − |rn| Γ+Γ− 2 −iφ 2 iφ |tn| Γ+e † |tn| Γ−e rˆhe = 2 σˆS, rêh = 2 σˆS, (5.28) 1 − |rn| Γ+Γ− 1 − |rn| Γ+Γ− with v± ∆± Γ± ≡ s± = . (5.29) u± E + Ω±

In an unconventional superconductor junction, the two pair potentials ∆+ and ∆− enter the transport coefficients. In particular, the Andreev reflection at E = 0 shows qualitatively different behavior depending on the relative sign between the two pair potentials. To see this, let us consider two limiting cases: ∆+ = ∆− and ∆+ = −∆−. The pair potentials in s-, py-, and dx2−y2 -wave symmetry satisfy the former relation. The latter relationship ∆+ = −∆− holds true for the pair potentials in px- and dxy-wave symmetry. By considering a relation

lim Γ± = −is±, (5.30) E→0 we find that the reflection probabilities at E = 0 in the tunneling limit become | |2 2 2 tn 2 2 lim |rˆhe| = σˆ0, lim |rêe| = |rn| σˆ0 (5.31) E→0 2 E→0 for ∆+ = ∆−. The zero-bias conductance is zero for ∆+ = ∆− in the tunnel limit |tn| → 0. In the case of ∆+ = −∆−, on the other hand, we find Γ+Γ− = 1 at E = 0, which results 2 2 lim |rˆhe| =σ ˆ0, lim |rêe| = 0. (5.32) E→0 E→0 The Andreev reflection is perfect at E = 0 independent of the potential barrier at the interface. Fig. 5.2 shows the differential conductance of an NS junctions consisting of un unconventional superconductor. The conductance in the tunnel limit z0 = 5 approaches the density of states at the uniform superconducting state in py- and dx2−y2 -wave symmetries. The V-shaped spectra around the zero bias in Fig. 5.2(b) and (d) are a result of nodes of the superconducting gap on the Fermi surface. On the other hand, the conductance in the tunnel limit shows a large peak at the zero bias for px- and dxy-wave symmetries. [15] The sign change of pair potential is a general feature of all unconventional superconductors. Thus the zero-bias anomaly observed in STM/STS experiments in a superconductor suggests that the pairing symmetry of the superconductor would be unconventional. CHAPTER 5. UNCONVENTIONAL SUPERCONDUCTOR 53

$(7+$4 !$ ! $(6+$405 0 -$5 &$ &$ ! ! $1#2# $1#2# ()*+$,$% ()*+$,$% $ " " $ " " &' $ &' $% 3 $% 3 # # -! -" # " ! -! -" # " ! $)*, $)*,

$(.+$/0 (4+$/5 &$ $1 2# &$ ! # ! $1#2# $ ()*+$,$% ()*+$,$% $ " $" $ " " &' &' $% $% 3 $3 # # -! -" # " ! -! -" # " ! $)*, $)*,

Figure 5.2: The conductance spectra are calculated for an NS junction consisting of an unconventional superconductor for several choices of the barrier parameter z0. The results for z0 = 5 correspond to the tunnel spectra measured in STS experiments.

5.2 Surface Andreev bound states

We explain the reasons of the zero-bias anomaly appearing in the conductance spectra by tracing the trajectories of an incident electron into the NS interface. [16] In Fig. 5.3, we decompose the Andreev reflection process into a series of multiple reflection events by the pair potential and by the barrier potential. In the electron branch, the transmission and reflection coefficients of ∗ ∗ the barrier are tn and rn, respectively. In the hole branch, they are given by tn and rn. At the first order process, an electron transmits into the superconductor with the coefficient tn. The penetration of such an electron into a superconductor is limited spatially by the coherence length ξ0 from the interface. Thus an electron is reflected as a hole by the pair potential ∆+. The Andreev reflection coefficients in this case is obtained by putting tn = 1 into Eq. (5.28) as

(0) −iφ † (0) iφ rˆhe =Γ+ e σˆS, rêh = Γ− e σˆS. (5.33) ∗ Finally a reflected hole transmits back into the normal metal with tn. The Andreev reflection coefficient at the first order process is summarized as (1) ∗ · (0) · rhe = tn rˆhe tn. (5.34) The reflection sequence is ordered from the right to the left of the equation. In the second order process, a quasiparticle in the 1st process is reflected by the pair potential twice more. Simultaneously, a quasiparticle reflected by the barrier twice. The resulting reflection coefficient becomes h i (2) ∗ (0) · · (0) · ∗ (0) · rhe = tn rˆhe rn rêh rn rˆhe tn. (5.35) CHAPTER 5. UNCONVENTIONAL SUPERCONDUCTOR 54

By repeating the argument, it is possible to calculate m-th order reflection coefficient as h i m−1 (m) ∗ · (0) · · (0) · ∗ (0) · rhe = tn rˆhe rn rêh rn rˆhe tn. (5.36)

Since

lim Γ± = −is±, (5.37) E→0 the Andreev reﬂection probability at E = 0 is given by

∞ 2 X | |2 | |4 − | |2 m − m rhe = tn (1 tn ) ( s+s−) σˆ0, (5.38) m=0 4 |tn| σˆ0 = 2 2 . (5.39) |1 + (1 − |tn| )s+s−|

When the two pair potentials have the same sign (i. e., s+s− = 1), the Andreev reﬂection probability in Eq. (5.38) is represented by the summation of an alternating series. The zero-bias conductance represented as

2 X | |4 | 2e 2 tn GNS E=0 = 2 2 , (5.40) h (2 − |tn| ) ky vanishes in the tunnel limit |tn| → 0. When the two pair potentials have the opposite sign (i. e., s+s− = −1), on the other hand, the Andreev reﬂection probability is unity independent of the potential barrier. The resulting conductance

2e2 G | = 2N , (5.41) NS E=0 h c is the twice of the Sharvin conductance. The relative sign of the two pair potentials s+s− plays a crucial role in the interference effect of a quasiparticle near a surface of a superconductor. Namely s+s− = 1 causes the destructive interference effect. The constructive interference effect with s+s− = −1 enables the resonant transmission of a quasiparticle (resonant conversion of an electron to a Cooper pair) through the potential barrier. Thus, it is expected that the resonant states at E = 0 would be bound at the surface of an unconventional superconductor. The zero-bias peak in the conductance could be interpreted as the local density of states at the surface. It is possible to confirm this argument by seeking a bound state in terms of the wave function in Eq. (5.20). Here we show more simple expression of Eq. (5.20) which represents the right-going wave in the superconductor at E, σˆ 0 ikxx Γ−σˆS −ikxx −x/ξ0 ϕR(x) = † e C + e D e . (5.42) Γ+σˆS σˆ0

Here we assume that E < |∆±| and φ = 0. The imaginary part of wave number is represented explicitly to describe bound states at the surface. At x = 0, the wave function should satisfy σˆ0 Γ−σˆS C ϕR(0) = † = 0. (5.43) Γ+σˆS σˆ0 D CHAPTER 5. UNCONVENTIONAL SUPERCONDUCTOR 55

N S N S r^ 1st order r^ 3rd order he he t tn n ^ rhe r* tn* n rn ^ rhe rn* ^ rn reh N S ^ tn* r^ 2nd order rhe eh t n ^ rhe rn* rn

tn* ^ reh

Figure 5.3: The Andreev reﬂection process is decomposed into a series of reﬂections by the pair potential and by the barrier potential.

The condition is satisﬁed when

Γ+Γ− = 1. (5.44)

In the case of ∆+ = ∆− = ∆, E = ∆ is a solution of this equation. The wave function of such solution, however, is not localized at the interface. In the case of ∆+ = −∆− = ∆, E = 0 is a solution of this equation. The wave function of such a bound state is described by σˆ −x/ξ0 0 ϕR(x) =C sin(2kxx) e − † , (5.45) iσˆSs+ where C is a constant. Today, the surface Andreev bound states are called topologically protected bound states at a surface of a topologically nontrivial superconductor. We will address this issue in Sec. 5.5. CHAPTER 5. UNCONVENTIONAL SUPERCONDUCTOR 56

5.3 Josephson current in an SIS junction

Let us consider an SIS junction, where two unconventional superconductors are identical to each other. In the left superconductor, the wave function is given by u σˆ + 0 ikxx v−s−σˆS −ikxx ϕL(x) =Φˇ L † e α + e β v s σˆ u−σˆ0 + + S u−σˆ0 −ik x v+s+σˆS ik x + † e x A + e x B , (5.46) v−s−σˆS u+σˆ0 s s 1 Ωn,± 1 Ωn,± u± = 1 + , v± = 1 − , (5.47) 2 ℏωn 2 ℏωn q ∆± ± ℏ2 2 2 ∆± =∆( kx, ky), s± = , Ωn,± = ωn + ∆±. (5.48) |∆±|

Here we also omitted the wave function in the y direction fky (y). In the same way, the wave function on the right is represented by, u σˆ + 0 ikxx v−s−σˆS −ikxx ϕR(x) =Φˇ R † e C + e D . (5.49) − v+s+σˆS u σˆ0 The phase of superconductor is represented iφj /2 ˆ ˇ e σˆ0 0 Φj = −iφ /2 (5.50) 0ˆ e j σˆ0 with j = L or R. The Josephson current is represented by a formula [20, 21] " # X X † e ∆+σˆSrˆhe ∆−σˆSrêh J = kBT Tr − . (5.51) 2ℏ Ωn,+ Ωn,− ωn ky As discussed at the end of Chap. 4, the expression of the formula depends on the gauge choice. The formula is correct when the Andreev reflection coefficients are calculated from the wave functions in Eqs. (5.46) and (5.49). Applying the boundary conditions in Eqs. (4.30) and (4.31), we obtain the Andreev reflection coefficients. Here we consider simple case where the two pair potential is represented as ∆+ = s+∆ and ∆− = s−∆. The resulting reflection coefficients become   † 2 2 ∆ σˆ  |t | {(cos φ − 1) ℏω + i sin φ Ω } + |r | (1 − s s−) ℏω  + S  n n n n n + n o  rˆhe = 2 , (5.52) 2i 2 2 2 2 2 1−s+s− 1+s+s− |tn| {(ℏωn) + ∆ cos (φ/2)} + |rn| ℏωn + Ωn  2 2  2 2 ∆−σˆS  |tn| {(cos φ − 1) ℏωn − i sin φ Ωn} + |rn| (1 − s+s−) ℏωn  rêh =  n o  . (5.53) − 2 2i | |2 { ℏ 2 2 2 } | |2 ℏ 1 s+s− 1+s+s− tn ( ωn) + ∆ cos (φ/2) + rn ωn 2 + Ωn 2

The Josephson current becomes, " p # e X |t |2∆ ∆ 1 − |t |2 sin2(φ/2) J = sin φ p n tanh n . (5.56) ℏ 2 1 − |t |2 sin2(φ/2) 2kBT ky n

The results are identical to the Josephson current in Eq. (4.77). For s+s− = −1, on the other hand, the coeﬃcients are represented as † σˆ ∆ |t |2 {(cos φ − 1) ℏω + i sin φ Ω } + 2|r |2ℏω rˆ = S + n n n n n , (5.57) he 2i ℏ2ω2 + ∆2|t |2 cos2(φ/2) n n 2 2 σˆ ∆− |t | {(cos φ − 1) ℏω − i sin φ Ω } + 2|r | ℏω rˆ = S n n n n n . (5.58) eh ℏ2 2 2| |2 2 2i ωn + ∆ tn cos (φ/2) The Josephson current is calculated to be, e X |t |∆ ∆|t | cos(φ/2) J = sin φ n tanh n . (5.59) ℏ 2 cos(φ/2) 2kBT ky

At T = 0, the current-phase relationship is fractional irrespective of tn because of the resonant transmission of a Cooper pair via a Andreev bound state at the interface. In Fig. 5.4(a), we plot the amplitude of the Josephson current as a function of temperature for s-, px-, and dxy- wave SIS junction. The results for px- and dxy-wave junctions show that the Josephson current increases rapidly with the decrease of temperature for T < 0.5Tc. In Fig. 5.4(b), we show the current-phase relationship in a px SIS junction. The relation is sinusoidal just below Tc, whereas is becomes factional at a low temperature T = 0.001Tc. Such unusual behavior is called low-temperature anomaly of Josephson eﬀect.

%$)$$# #$ %2$%3%4)$ %5:7%1. ! %$)$# %567 * %-.0

$ + " $ %$)#

%(%& %1. ' %& %&%(%& # %-."/0" " %,%(%,'%3%$)4% %1 $ 0 $ $)$ $)" $) $)+ $)* #)$ $)$ $)" $) $)+ $)* #)$ %8%(%9 %,%(%,'

Figure 5.4: The Josephson critical current is plotted as a function of temperature for s-, px- and dxy- symmetry in (a). The current-phase relationship in a px-wave SIS junction is shown for several choices of temperatures in (b). The transmission probability of the barrier is about 0.026 as a results of choosing z0 = 5.

5.4 Andreev bound states in an SIS junction

We calculate the energy of the Andreev bound states which localize the interface of an SIS junction. Here we focus only on a simple case described by ∆+ = ∆s+ and ∆− = ∆s−. We CHAPTER 5. UNCONVENTIONAL SUPERCONDUCTOR 58 consider 2 × 2 the BdG Hamiltonian by block diagonalizing the original 4 × 4 Hamiltonian in terms of spin degree of freedom. In the left superconductor, the wave function localizing at the interface is described by − E + iΩ −ikxx E iΩ ikxx x/ξ0 ϕL(x) =Φˇ L e A + e B e , (5.60) ∆s− ∆s− p iφj /2 2 2 e 0 Ω = ∆ − E , Φj(x) = − . (5.61) 0 e iφj /2 The wave function on the right-hand side becomes, − E + iΩ ikxx E iΩ −ikxx −x/ξ0 ϕR(x) =Φˇ R e C + e D e . (5.62) ∆s+ ∆s− (5.63) They are connected with each other at x = 0 by the boundary conditions in Eqs. (4.30) and (4.31). From the Boundary conditions, we obtain the equation     χk(E + iΩ) −χk(E − iΩ) (k − 2iz0)(E + iΩ) −(k + 2iz0)(E − iΩ) A  ∗ ∗    χ k∆s− −χ k∆s (k − 2iz )∆s −(k + 2iz )∆s− B  + 0 + 0    = 0,  χ(E + iΩ) χ(E − iΩ) −(E + iΩ) −(E − iΩ)   C  ∗ ∗ χ ∆s− χ ∆s+ −∆s+ −∆s− D (5.64) p 2 2 2 iφ/2 z0 = mv0/ℏ kF , k = kx/kF , Ω = ∆ − E , χ = e , φ = φL − φR. (5.65) The determinant of the matrix is proportional to 2 −| |2 2 − 2 2 2 − 2 − | |2 k E + iΩ cos φ + E Ω + z0 E Ω s+s− E + iΩ = 0. (5.66)

Using the normal transmission coeﬃcient tn, the energy of the bound states are calculated to be 1/2 φ 1 + s s− E = ±∆ |t |2 cos2 + + (1 − |t |2) . (5.67) n 2 2 n

In the case of s-wave junction (i.e., s+s− = 1), the energy of the Andreev bound states becomes h i φ 1/2 E = ±∆ 1 − |t |2 sin2 . (5.68) n 2 The negative sign must be chosen for occupied bound states which carry the Josephson current. The Josephson current 2e d J = E(φ), (5.69) ℏ dφ is identical to Eq. (5.56) at T = 0. The energy of the Andreev bound states for a px-wave junction (i.e., s+s− = −1) results in φ E = ±∆|t | cos . (5.70) n 2 The Josephson current calculated from Eq. (5.69) is identical to Eq. (5.59) at T = 0. The energy of the Andreev bound states at a Josephson junction are plotted as a function of the phase diﬀerence in Fig. 5.5. In both Eqs. (5.68) and (5.70), the occupied energy branch takes its minimum at φ = 0. CHAPTER 5. UNCONVENTIONAL SUPERCONDUCTOR 59

!"#

%+,-+%.%#"$ #"$

#"# %&%'%(

#"$

!"# !"# #"$ #"# #"$ !"# %)%'%*

Figure 5.5: The energy of Andreev bound states at a Josephson junction for |tn| = 0.5. The results in Eq. (5.68) plotted with red lines are 2π periodic, whereas those in Eq. (5.70) shown with black lines are 4π periodic.

5.5 Topological classiﬁcation

A topologically nontrivial state of material is a novel concept in condensed matter physics. Quantum states of an electron in solid can be characterized in terms of topological numbers. When we apply the periodic boundary condition to obtain a unique solution of Schroedinger equation, the wave number k indicates place of the Brillouin zone and the wave function ϕl(k) describes properties of the quantum state labeled by l. In other words, ϕl(k) is a map from the Brillouin zone to the Hilbert space. We will calculate a topological number (winding number) to characterize nodal unconventional superconductors preserving time-reversal symmetry at the end of this section. Before turning into details, let me summarize important features of topological numbers.

1. To deﬁne a topological number, the electronic state must have gapped energy spectra the Fermi level stays in the gap.

2. The topological number Z is an integer number which can be deﬁned in terms of the wave functions of all occupied states below the gap.

3. Topological number remains unchanged as far as the gap opens.

As the energy spectra are gapped at the Fermi level, topological materials can be an insulator or a superconductor. On the basis of these features, it is possible to reach an important conclusion. Let us consider a surface of a topologically nontrivial material at x = 0 as shown in Fig. 5.6(a). The topological number is Z ̸= 0 in the topological material x > 0 and it is zero (trivial) in vacuum x < 0. Since Z is an integer number, it jumps at x = 0. To change the topological CHAPTER 5. UNCONVENTIONAL SUPERCONDUCTOR 60 number, the gap must close at the surface. More speciﬁcally, electronic states at the Fermi level are necessary at x = 0. This argument explains the existence of metallic states at a surface of a topological material. In addition, the number of such surface states at the Fermi level is identical to |Z|. This fact is called bulk-boundary correspondence.

vac. SC

x 0 (a) (b)

Figure 5.6: (a) A surface of a superconductor at x = 0. (b) The dispersion relation of a px-wave superconductor.

In Chap. 5, we have discussed physics of the Andreev bond states at a surface of an unconventional superconductor. In this section, we will show that an unconventional superconductor is a topologically nontrivial material and that the surface Andreev bound states mean nothing other than topologically protected bound states. To do so, let us focus on a spin-triplet px-wave superconductor in two-dimension. Z " # ∂x ξ r) ∆0 u (r) u (r) ( ikF ν ν dr ∂x ∗ = Eν , (5.71) ∆0 −ξ (r) vν(r) vν(r) ikF where the superconducting phase is ﬁxed at φ = 0. We consider 2 × 2 BdG equation by focusing on spin ↑ sector with choosingσ ˆS =σ ˆ3 in Eq. (5.23). The Bogoliubov transformation becomes X ∗ ψ(r) uν(r) vν(r) γν † = ∗ † . (5.72) ψ (r) vν(r) u (r) γ ν ν ν Here we omit spin ↑ from the operators and functions. We solve the BdG Hamiltonian on a two-dimensional tight-binding lattice as X ′ ′ ′ ξ r, r ) ∆(r, r ) u(r ) u(r) ( = E , (5.73) −∆∗(r, r′) −ξ∗(r, r′) v(r′) v(r) r′ ′ ξ(r, r ) = − t(δr,r′+xˆ + δr,r′−xˆ + δr,r′+yˆ + δr,r′−yˆ ) + vrδr,r′ − ϵF δr,r′ , (5.74) ′ ∆0 ∆(r, r ) =i (δ ′ − δ ′− ), (5.75) 2 r,r +xˆ r,r xˆ where t is the hopping integral between the nearest neighbor lattice sites, r = jxˆ + myˆ, and the on-site potential vr is set to be zero. At ﬁrst, we apply the periodic boundary condition in both the x and the y directions. Under these periodic boundary conditions, the wave function is represented in the Fourier series as r r X 1 1 u(r) = u eikxj eikym, k = 2πn/L, k = 2πl/M, (5.76) kx,ky L M x y kx,ky CHAPTER 5. UNCONVENTIONAL SUPERCONDUCTOR 61 with −L/2 < n ≤ L/2 and −M/2 < l ≤ M/2, where L and M is the number of lattice sites in the x and y direction, respectively. The wave function representsq the eigen state of the BdG ± 2 2 Hamiltonian. The energy eigenvalue is calculated as E = ξk + ∆k with,

ξk = −2t(cos kx + cos ky − 2) − ϵF , ∆k = ∆0 sin kx. (5.77)

In Fig. 5.6(b), we plot the energy eigenvalues as a function of ky, where we choose ∆0 = t, ϵF = 2t, L = 500 and M = 200. The results are shown with black dots. The pair potential has nodes on the Fermi surface at ky = ±π/2 for the current parameter choice. The results in Fig. 5.6(b) suggest the eigenvalues at E = 0 between the two nodal points. However, the analytical results do not describe such zero-energy states. In the analytic calculation, we apply the periodic boundary conditions in both x and y directions which means that the shape of a superconductor is two dimensional spare. There is no edge or surface in such a superconductor. To have edges or surfaces, we need to apply the hard wall boundary condition in the x direction. When we introduce the hard wall potential at j = 0 and j =, the energy eigenvalues of the BdG Hamiltonian can be calculated only numerically. Roughly speaking, the results become the black dots shown in Fig. 5.6(b) plus the red dots between the nodal points. Namely a surface of a px-wave superconductor in the x direction hosts bound states at the zero energy. It is possible to have surfaces in the y direction by introducing the hard wall potential at m = 0 and m = M. In this case, however, the energy spectra remain unchanged and no extra zero-energy state appears. There is no bound state at surfaces of a px-wave superconductor in the y direction. By repeating the argument in Sec. 5.2, it is possible to conﬁrm the presence of the Andreev bound states at a surface of a px-wave superconductor in Fig. 5.6(a). The wave function of the right-going wave in the superconductor is given by r 2 πly 1 ikxx Γ− −ikxx −x/ξ0 ϕS(x) = e C + e D e sin , (5.78) Γ+ 1 W W ∆± kx ∆± Γ± = q , ∆± = ∆(±kx, ky) = ±∆0 , s± = , (5.79) 2 2 kF |∆±| E + E − ∆± where we consider surface states localized at x = 0 by assuming E < ∆0 and we apply the hard wall boundary condition in the y direction. From the boundary condition ϕS(x = 0) = 0, Γ+Γ− = 1 is necessary to have the surface bound state. We ﬁnd that a bound state stays at E = 0 when

s+s− = −1, (5.80) is satisﬁed. The wave function of the bound state is given by, χ∗ ϕ (r) =A f (r), (5.81) E=0,l χ l r − 2 πly f (r) = sin(k x) e x/ξ0 sin , (5.82) l x W W with χ = e−iπ/4, A being a constant, and l = 1, 2, ··· . The Bogoliubov transformation in Eq. (5.72) connects the operator of the surface state and the operator of an electron at each propagating channel l, ∗ ∗ ψl(r) χ χ γl † = fl(r) † . (5.83) ψl (r) χ χ γl CHAPTER 5. UNCONVENTIONAL SUPERCONDUCTOR 62

Here, we briefly mention an unusual property of the Andreev bound state. In Eq. (5.83), it is possible define an operator of the bound state † γl(r) = fl(r)(γl + γl ). (5.84) Because of the definition, the operator satisfies † γl(r) = γl (r). (5.85)

The last relation is called Majorana relation. An equal-spin-triplet px-wave superconductor hosts Majorana Fermions at its surface. The presence of more than one Majorana Fermion is responsible for the anomalous proximity effect of a spin-triplet superconductor [22, 23, 24]. The remaining task is the calculation of a topological number. We apply the periodic boundary condition in all directions because topological numbers are defined by using the wave functions of all bulk states below the gap. However, it is impossible to apply naive topological classification to a px-wave superconductor because the superconducting gap has nodes on the Fermi surface as shown in Figs. 5.1 and Fig. 5.6(b). To topologically characterize such a nodal superconductor, we apply a theoretical prescription called dimensional reduction [25]. We fix ky at a certain point other than the nodal points ky = ±π/2 and consider the one-dimensional Bril- louin zone of −π < kx ≤ π. In this case, one-dimensional Brillouin zone is identical to a sphere 1 in one-dimension S because both kx = −π and kx = π indicate an identical quantum state. Fig. 5.6(b) suggests that the superconducting gap opens in such a one-dimensional Brillouin zone. We have already known the solutions of the BdG equation s s −vk sgn(∆k) 1 ξk 1 ξk ψk = , uk = 1 + , vk = 1 − , (5.86) uk 2 Ek 2 Ek with k = (kx, ky). The wave function an eigenstate is specified by an angle of −π < αk ≤ π as ξ ∆ sin − αk cos(α ) ≡ k , sin(α ) ≡ k , ψ = 2 . (5.87) k k k − αk Ek Ek cos 2

The function −π < αk ≤ π represents the mapping from the one-dimensional Brillouin zone −π < kx ≤ π to the eigenstates as illustrated in Fig. 5.7. It is possible to deﬁne the winding number in terms of the mapping, Z π W 1 − (ky) = dkx ∂kx ( αk), (5.88) 2π −π where W represents how many time αk encircles the one-dimensional sphere while kx encircles the one-dimensional Brillouin zone once. In Fig. 5.7(b), we show the dispersion in the normal state and the pair potential in Eq. (5.77) in one-dimensional Brillouin zone. At xi1, we choose a ky so that the dispersion is always above the Fermi level and that the channel is an evanescent mode. On the other hand, at ξ2, we choose a ky so that the dispersion comes across the Fermi level and the channel becomes propagating mode. The wave number −π/2 < ky < π/2 in Fig. 5.6(b) correspond to the propagation channels and those π/2 < |ky| < π describe the evanescent channels. The pair potential ∆k is an odd function of kx. In Fig. 5.7(c), we plot the mapping function αk for the two channels. In an evanescent channel, α1 depends on kx only slightly. As a consequence, the winding number is zero. On the other hand in a propagating channel, α2 changes by −2π while kx encircle the Brillouin zone once, which results in W = 1. CHAPTER 5. UNCONVENTIONAL SUPERCONDUCTOR 63

S1 S1

π π Π (S1 ) = Z −π 1 −π

−π < k x < π −π < α k < π (a)

6 ξ1 / t α 1 2 4

ξ2 / t π α1 2 / ∆ k / t k 0 α

(b) -2 -1 (c) -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 / π / π kx kx

Figure 5.7: (a) The winding number Z counts the number how many time αk encircles the one-dimensional sphere while kx encircles the one-dimensional Brillouin zone once. (b) The dispersion of two channels are shown. ξ1 is the dispersion of an evanescent channel and is away from the Fermi level E = 0. ξ2 is the dispersion of a propagating channel and comes across the Fermi level. The pair potential ∆k is an odd function of kx. (c) The mapping function αk is plotted as a function of kx. For an evanescent channel, α1 remains around zero, which results in W = 0. For a propagating channel, α2 changes by −2π while kx moves from −π to π.

To summarize, we ﬁnd

1 − s+s− ∆± W(ky) = s+ , s± = , ∆± = ∆(±kx, ky), (5.89) 2 |∆±| where s± is the sign of the two pair potentials deﬁned in Eq. (5.79). In other words, a nodal superconductor is topologically nontrivial for s+s− = −1 which is identical to the condition for appearing the surface Andreev bound state in Eq. (5.80). Chapter 6

Various Josephson junctions

In Secs. 4.3 and 5.3, we discussed the Josephson effect between the two superconductors which are identical to each other. The Josephson effect, however, depends sensitively on the relative difference in symmetries between the two superconductors. The purpose of this Chapter is to demonstrate such rich properties of the Josephson current. The wave function in the two superconductors should be described by u σˆ ˆ L+ 0 ikxx v˜L−SL −ikxx ϕL(x) =Φˇ L † e α + e β v˜ Sˆ u −σˆ L+ L L 0 ˆ uL−σˆ0 −ik x v˜L+SL ik x + † e x A + e x B , (6.1) v˜ −Sˆ u σˆ L L L+ 0 u σˆ − ˆ − ˇ R+ 0 ikxx v˜R SR ikxx ϕR(x) =ΦR ˆ† e C + e D , (6.2) v˜R+SR uR−σˆ0 with s s 1 Ωn,j± 1 Ωn,j± uj± = 1 + , vj± = 1 − , v˜j± = vj± sj± (6.3) 2 ℏωn 2 ℏωn q ∆ ± ± j ℏ2 2 2 ∆j± =∆j( kx, ky), sj± = , Ωn,j± = ωn + ∆j±, (6.4) |∆j±| for j = L or R. The pair potential is described as iσˆ2 singlet ∆ˆ j± = Sˆj∆j,±, Sˆj = . (6.5) idj · σˆσˆ2 triplet Here we assume that the d vector is independent of wavenumber for simplicity. By substitute the wave functions into the boundary conditions in Eqs. (4.30) and (4.31), C α C Uˇ =Φˇ Uˇ + Uˇ , (6.6) R D L1 β L2 D C α C Uˇ k τˆ =Φˇ Uˇ (k τˆ − 2iz ) + Uˇ (−k τˆ − 2iz ) , (6.7) R 3 D L1 3 0 β L2 3 0 D ! ! uL+σˆ0 v˜L−SˆL uL−σˆ0 v˜L+SˆL UˇL1 = † , UˇL2 = † , (6.8) v˜L+Sˆ uL−σˆ0 v˜L−Sˆ uL+σˆ0 L ! L ˆ ˇ uR+σˆ0 v˜R−SR UR = ˆ† , (6.9) v˜R+SR uR−σˆ0

64 CHAPTER 6. VARIOUS JOSEPHSON JUNCTIONS 65

By eliminating C and D, we ﬁnd A α rˆ rˆ α = − Yˇ −1Yˇ = ee eh (6.10) B AB ab β rˆ rˆ β he hh k k Yˇ = (Tˇ Vˇ − Vˇ Tˇ ) + iz Vˇ , Yˇ = (Tˇ Vˇ + Vˇ Tˇ ) + iz Vˇ , (6.11) ab 2 3 1 1 3 0 1 AB 2 3 2 2 3 0 2 ˆ ˆ ˇ ˇ −1 ˇ ˇ aˆ1 b1 ˇ ˇ −1 ˇ ˇ aˆ2 b2 V1 = UR ΦUL1 = , V2 = UR ΦUL2 . (6.12) cˆ1 dˆ1 cˆ2 dˆ2

The Andreev reflection coefficients can be calculated as h i −1 − −1 ˆ | |2 −1ˆ −1 − | |2 −1 rˆhe = cˆ2 d2 + r2 aˆ2 b2 cˆ2 cˆ1 rn aˆ2 aˆ1 , (6.13) h i h i −1 −ˆ−1 | |2 ˆ−1 ˆ−1ˆ − | |2 ˆ−1 ˆ rêh = b2 aˆ2 + rn d2 cˆ2 b2 b1 rn d2 d1 . (6.14)

The electric Josephson current is given by [21], " # X X ˆ ˆ † e ∆L+rˆhe ∆L−rˆeh J = kB T Tr − . (6.15) 2ℏ ΩL+ ΩL− ωn ky

6.1 Singlet-singlet junction

We first consider Josephson junctions consisting of two spin-singlet superconductors. The results of the Andreev reflection coefficients are summarized as

† (iσˆ2) rˆhe = [(1 − ΓR+ΓR−)(ΓL− − ΓL+) Ξ 2 2 ∗ 2 + |tn| ΓR+χ + ΓR−ΓL+ΓL−(χ ) − ΓR+ΓR−ΓL+ − ΓL− , (6.16) iσˆ2 rêh = [(1 − ΓR+ΓR−)(ΓL+ − ΓL−) Ξ 2 ∗ 2 2 + |tn| ΓR−(χ ) + ΓR+ΓL+ΓL−χ − ΓR+ΓR−ΓL− − ΓL+ , (6.17) Ξ(φ) =(1 − Γ Γ −)(1 − Γ Γ −) R+ R L+ L 2 2 ∗ 2 + |tn| ΓR+ΓR− + ΓL+ΓL− − ΓR+ΓL+χ − ΓR−ΓL−(χ ) , (6.18) −i∆j± Γj± = . (6.19) ℏωn + Ωj± The expression above can be applied to any singlet-singlet junctions. We first consider a junction where an s-wave superconductor stays on the left and a dxy-wave superconductor stays on the right. In such a junction, uj± and vj± are independent of ±. We also assume a relation ∆L = ∆ and ∆R = ∆sgn(kxky) for simplicity. The Andreev reflection coefficients in such case are calculated be

† 2 (iσˆ2) |tn| ∆R(ℏωni sin φ + Ω cos φ) − ∆LΩ rˆhe = 2 , (6.20) i 2ℏωnΩ + |tn| ∆L ∆R i sin φ 2 iσˆ2|tn| ∆R(ℏωni sin φ − Ω cos φ) − ∆LΩ rˆeh = 2 . (6.21) i 2ℏωnΩ + |tn| ∆L ∆R i sin φ CHAPTER 6. VARIOUS JOSEPHSON JUNCTIONS 66

!" !5 $ ! # $,-.$/$'$0 $,9.$ 12 $3 $4$5 $ ! #

!# ! $'$% & $+'$+&$4$ !# $%$'$% $%

6 !5 ! ! !" !* !) !( #! ! !" !* !) !( #! $+$'$+& $7$'$8

Figure 6.1: The Josephson critical current is plotted as a function of temperature for a junction consisting of a s-wave superconductor and a dxy-wave superconductor in (a). The current-phase relationship is shown for several choices of temperatures in (b).

The Josephson current results in X 4 2 e |t | ∆ sin(2φ) 1 D − 1 D J = − n √ tanh s − tanh s+ , (6.22) ℏ − 2 Ds− 2kBT Ds+ 2kBT k 4 1 a √ y √ ± − 1 + a 1 a 2 D ± =∆ , a = |t | | sin φ|. (6.23) s 2 n

Just below Tc, the current is approximately represented as 3 X 1 e∆ ∆ 4 J ≈ − |tn| sin(2φ). (6.24) 12 ℏ 2kBT ky

The lowest harmonic J1 in Eq. (4.80) is missing [26]. Roughly speaking, J1 terms is given by X h i ∝ ˆ ˆ † J1 Tr ∆L∆R , (6.25) ky which is sensitive to the pairing symmetry in the two superconductors. In the present case the s-wave pair potential is independent of ky, whereas dxy-wave pair potential proportional to ky is an odd function of ky. Therefore, J1 = 0 as a result of the summation over propagating channels. We also note that J2 < 0 in most junctions. At T = 0, the current becomes X e∆ 2 1 J ≈ − |tn| p cos(φ)sgn(φ). (6.26) ℏ 2 1 + |tn| | sin φ| ky Figure ??(a) shows the Josephson critical current versus temperature. The results show the low temperature anomaly because of the Andreev bound states originated from at a surface of a dxy-superconductor. The current-phase relationship is shown in Fig. 6.1 (b) for several choices of temperature. The results also show an unusual current-phase relationship at a low temperature. The Josephson eﬀect in unconventional superconductor depends sensitively on relative con- ﬁgurations of the two pair potentials. In Fig. 6.2, the two d-wave potentials are oriented by β in the opposite manner to each other. The pair potentials are represented by

∆L = ∆(cos 2θ cos 2β + sin 2θ sin 2β), ∆R = ∆(cos 2θ cos 2β − sin 2θ sin 2β), (6.27) CHAPTER 6. VARIOUS JOSEPHSON JUNCTIONS 67

β β d-wave

Figure 6.2: The pair potentials in the two superconductors are oriented by β in the opposite manner to each other. The Josephson critical current is plotted as a function of temperature for several choices of β. The current-phase relationship at β = π/10 is shown for several temperatures in (b). The results at T = 0.15Tc is ampliﬁed by ﬁve for better visibility.

where kx = kF cos θ and ky = kF sin θ. The maximum amplitude of the Josephson current is plotted as a function of temperature for several choices of β in Fig. 6.2(a). At β = 0, the Josephson current is saturate at a low temperature. At β = π/4, the Josephson effect shows the low temperature anomaly as we already discussed in Sec. 5.3. The results for intermediate angles indicate the nonmonotonic dependence on temperature as shown in β = π/10 and π/8.[27] The first term of the pair potential in Eq. (6.27) is common in ∆L and ∆R. The two pair potentials at the Andreev reflection process are represented as ∆+ = ∆(θ) and ∆− = ∆(π − θ). Since cos 2θ = cos 2(π − θ), such component gives the conventional Josephson current saturate at low temperature. On the other hand, the second term of the pair potential in Eq. (6.27) changes the sign in ∆L and ∆R. Namely the phase of such component on the right is shifted by π as eiπ = −1. In addition, such component generates the surface Andreev bound states because sin 2θ = − sin 2(π − θ), which results in the low temperature anomaly in the Josephson effect. The Josephson current at β = π/10 is almost vanishes around T = 0.15Tc as shown in Fig. 6.2(a). The current-phase relationship for β = π/10 is plotted for several temperatures in Fig. 6.2(b). At T = 0.5Tc, the relation is sinusoidal as usual. The results at T = 0.1Tc is also sinusoidal but the current changes its sign. Such junction is called π-junction because the minimum of the junction energy stays at φ = ±π. The lowest coupling component J1 is positive at T = 0.5Tc and decreases its amplitude with decreasing temperature. At T = 0.15Tc, J1 changes the sign to negative and increases its amplitude rapidly with decreasing temperature. The current- phase relationship at T = 0.15Tc shows the absence of J1 and J ≈ −J2 sin 2φ with J2 > 0. The nonmonotonic dependence of the Josephson critical current on temperature is a unique character to unconventional superconductor junctions, where two pair potentials are oriented as the mirror image of each other. CHAPTER 6. VARIOUS JOSEPHSON JUNCTIONS 68

6.2 Triplet-triplet junction

The Andreev reflection coefficients in spin-triplet junctions have complicated structure due to spin degree of freedom of a Cooper pair. It is not easy to obtain simple analytical expression of the Andreev reflection coefficients for general cases. When d vectors in the two superconductors † † are identical to each other, the expression in Eq. (6.16)-(6.18) are valid with (iσˆ2) → (id·σˆσˆ2) inr ˆhe (iσˆ2) → (id · σˆσˆ2) inr êh. In this situation, for example, the nonmonotonic dependence of the Josephson current on temperature discussed in Fig. 6.2 can be seen also p-wave SIS junctions. The results are demonstrated in Fig. 6.3. The characteristic features in p-wave mirror junctions are essentially the same as those in d-wave mirror junctions. Thus the misalignment between the two d vectors features phenomena unique to spin-triplet junctions.

β β

p-wave

Figure 6.3: The results of the mirror type of junction for spin-triplet p-wave symmetry. The Josephson critical current versus temperature (a) and the current-phase relationship at β = π/10 in (b). The results at T = 0.09Tc is ampliﬁed by ten for better visibility.

To discuss eﬀects of relative spin conﬁguration of two superconductors on the current, we consider same spin-triplet superconductors on either side of the junction.

∆ˆ L+ =∆kidL · σˆσˆ2, ∆ˆ L− = s∆ˆ L+, (6.28)

∆ˆ R+ =∆kidR · σˆσˆ2, ∆ˆ R− = s∆ˆ R+, (6.29) with s = 1 or −1. The spin vectors dL and dR point diﬀerent directions in spin space and are CHAPTER 6. VARIOUS JOSEPHSON JUNCTIONS 69

normalized as |dL| = |dR| = 1. The reﬂection coeﬃcients for s = 1 are calculated as

| |2 ˆ † tn ∆L+ 2 2 rˆhe = Ω (X cos α − ℏωn) + |tn| K−(X − ℏωn cos α) 2iΞP +ΞP − 2 2 +iΩ(∆ cos φ + ℏωnX − |tn| K−)n · σˆ , (6.30) | |2 tn 2 ∗ 2 ∗ rˆeh = Ω (X cos α − ℏωn) + |tn| K−(X − ℏωn cos α) 2iΞ Ξ − P + P 2 ∗ 2 ∗ −iΩ(∆ cos φ + ℏωnX − |tn| K−)n · σˆ ∆ˆ L−, (6.31) 2 ∆k K± = (cos φ ± cos α),X = ℏω cos φ + iΩ sin φ, (6.32) 2 n n dL · dR = cos α, n = dR × dL, (6.33) s ± 2 2 2 φ α Ξ ± =(ℏω ) + P ,P ± = ∆ 1 − |t |2 sin . (6.34) P n t± t k n 2

The Josephson current in Eq. (6.15) results in X 2 2 e |t | ∆ sin(φ + α) P sin(φ − α) P − J = n k tanh t+ + tanh t . (6.35) ℏ 4 Pt+ 2kBT Pt− 2kBT ky

Just below Tc, the current becomes e X J ≈ |t |2 ∆ sin φ cos α, (6.36) 2ℏ n k ky and is proportional to dL · dR. As discussed in Eq. (6.25), the first order coupling vanishes when the pairing functions in the two superconductors are orthogonal to each other in spin space. Since s = 1, the Josephson current is saturate at a low temperature well below Tc. The Andreev reflection coefficients have a term proportional to n·σˆ, which is a characteristic feature of spin-triplet superconducting junctions. These reflection terms represent the flow of spin supercurrents. The spin current flowing through the junction can be given by [28, 29] " # X X ˆ ˆ ∗ ˆ † ∗ ˆ † 1 ∆L+ rˆhe σˆ +r ˆhe ∆L+ σˆ ∆L− rêh σˆ +r êh ∆L− σˆ J s = − kB T Tr − . (6.37) 8 ΩL+ ΩL− ωn ky

By substituting the Andreev reﬂection coeﬃcients in Eqs. (6.30) and (6.31), the spin current is represented by X − n | |2 2 sin(φ + α) Pt+ − sin(φ α) Pt− J s = tn ∆k tanh tanh (6.38) 8 Pt+ 2kBT Pt− 2kBT ky

At T ≲ Tc, the results become

J s ∝ n cos φ sin α. (6.39)

The spin supercurrent polarized in n = dR ×dL direction in spin space flows across the junction, which is a unique property of spin-triplet superconductor junctions. The spin flow does not need the gradient in the phase but the gradient in the d vector. The current-phase relationship of the charge current in Eq. (6.35) is plotted for several α in Fig. 6.4(a), where the temperature is fixed CHAPTER 6. VARIOUS JOSEPHSON JUNCTIONS 70

!"# &&-&(&,&.&! !"# &!"* !"$ !"$ &!"% &!"/ &!"/ !"% !"% &!") !9

! &!"* !"! !"! &-&(&,&.&!:&) &&(&' &!"# 9 &'&(&' !"% &' !"% &)"0 &!"0 !"$ !"$ &)"; &123&452678 &)"! &1<3&9=>? &)"* !"# !"# )"! !"* !"! !"* )"! )"! !"* !"! !"* )"! &+&(&, &+&(&,

Figure 6.4: The Josephson charge current (a) and spin current (b) in spin triplet SIS junctions, where α is the relative angle between the two d vectors in the two superconductors. The current is calculated at gn = 0.1 and T = 0.1Tc.

at T = 0.1Tc and the transmission probability of the barrier is chosen as 0.1. The amplitude of the current decreases with the increase of α from 0. At α = π/2, the lowest order coupling vanishes as indicated by Eq. (6.36). For α > π/2, the junction becomes π-junction. The flip of the d vector in spin space is compensated by the π-phase shift in gauge space. Fig. 6.4(b) shows the spin current in Eq. (6.38) plotted as a function of φ for several α. The spin current is an even function of φ as the spin flow does not need the breakdown of time-reversal symmetry. Instead of the phase difference, the gradient in the d vector generates the spin current. For s = −1, the Andreev reflection coefficients are calculated as

† |t |2∆ˆ n L+ ℏ 3 | |2ℏ 2 − ℏ 2 − ℏ rˆhe = 2( ωn) + tn ωn ∆k(1 + cos φ cos α) ( ωn) ωnX cos α 2iΞM+ΞM− 4 2 2 − |tn| K+(X + ℏωn cos α) +i|tn| Ω(ℏωnX − |tn| K+)n · σˆ , (6.40) | |2 tn ℏ 3 | |2ℏ 2 − ℏ 2 − ℏ ∗ rˆeh = 2( ωn) + tn ωn ∆k(1 + cos φ cos α) ( ωn) ωnX cos α 2iΞ Ξ − M+ M 4 ∗ 2 ∗ 2 ∗ − |t | K (X + ℏω cos α) −i|t | Ω(ℏω X − |t | K )n · σˆ ∆ˆ −, (6.41) n + n n n n + L ± 2 2 φ α Ξ ± =(ℏω ) + M ,M ± = |t |∆ cos . (6.42) M n t± t n k 2

We only supply the ﬁnal results of Josephson electric current and spin current which are obtained by substituting Pt± in Eqs. (6.35) and (6.38) by Mt±. The resonant transmission through the topological bound states causes the low-temperature anomaly in both the charge current and the spin current [29].

6.3 Singlet-triplet junction

The lowest order coupling between a spin-singlet superconductor and a spin-triplet superconductor is absent when the junction area is free from any spin-dependent potentials. To understand the property, let us consider a junction consisting of spin-singlet s-wave superconductor, a spin- triplet p-wave superconductor, and a material X in between. We begin this section with a CHAPTER 6. VARIOUS JOSEPHSON JUNCTIONS 71 phenomenological expression of the lowest order Josephson current in Eq. (4.85), X X h i ie · ˆh · · ê − · ê · · ˆh J = ℏ kBT Tr rêh(L) tX rˆhe(R) tX rˆhe(L) tX rêh(R) tX . (6.43) p ωn In what follows, we neglect effects of topological bound states on the Josephson current and

^ ^e t h ^ t ^ ^ X ^ r X r ( ) r r eh ( L ) he R − he ( L ) eh ( R ) t^h t^e X X

Figure 6.5: The Andreev reflection process carrying the Josephson current. consider the reflection process in Fig. 6.5. The Andreev reflection coefficient is given in Eq. (5.28) in a NS junction,

2 2 |t| Γ+ |t| Γ− −i∆± rˆ = , rˆ = , Γ± = . (6.44) he − 2 eh − 2 1 Γ+ 1 Γ− ωn + Ω± Here we assume that that the transmission probability of the interface between the superconductor and the material X is much smaller than unity |t| ≪ 1. In Eq. (5.28), 1 − Γ+Γ− describes the resonant transmission due to the surface Andreev bound state. In Eq. (6.44), however, we − 2 − 2 substitute it by 1 Γ+ inr ˆhe and by 1 Γ− inr êh by hand to suppress the resonant tunneling. The resulting reflection coefficients are

∆ † − ∆ rˆ (L) = − i|t|2 (iσˆ ) e iφL , rˆ (L) = −i|t|2 iσˆ eiφL , (6.45) he 2Ω 2 eh 2Ω 2 † (∆ id · σˆσˆ ) ∆−id · σˆ σˆ 2 + 2 −iφR 2 2 iφR rˆhe(R) = − i|t| e , rˆeh(R) = −i|t| e , (6.46) 2Ω+ 2Ω− ∆± =∆(±kx, p). (6.47)

ê ˆh In Eq. (6.43), tX (tX ) is the transmission coefficient of a material X in the electron (hole) branch. These coefficients are related to each other as h e − ∗ tX (k, p) = [tX (k, p)] , (6.48) because of particle-hole symmetry. When both two superconductors belong spin-singlet s-wave symmetry, the formula in Eq. (6.43), the Andreev reflection coefficients in Eq. (6.45), and ê tX = t0σˆ0, recover the Ambegaokar-Baratoff formula in Eq. (4.78) with 2e2 X R−1 = G = |t|4|t |2. (6.49) N N h 0 p In what follows, we consider a spin-singlet s-wave superconductor on the left and a spin-triplet p- wave superconductor on the right. For simplicity, we focus only on px-wave of py-wave symmetry. ê When the transmission coefficients of X is independent of spin tX = t0σˆ0, it is easy to show that

J ∝ Tr [d · σˆ] = 0. (6.50) CHAPTER 6. VARIOUS JOSEPHSON JUNCTIONS 72

The spin of a Cooper pair is unity in a triplet superconductor, whereas it is zero in a singlet superconductor. The lowest coupling vanishes because such pairing states are orthogonal to each other in spin space. Thus the interface must be magnetically active to have the lowest order Josephson coupling. The ﬁrst example of a magnetically active layer is a ferromagnetic barrier described by

HFI = δ(x)[v0σˆ0 + v · σˆ] , (6.51) where m is the magnetic moment. The transmission coeﬃcients of such a barrier are calculated as 1 tˆ =k(k + iz + iz · σˆ)−1 = k [k + iz − iz · σˆ] , (6.52) X 0 Ξ 0 1 rˆ =(k + iz + iz · σˆ)−1(−iz − iz · σˆ) = (k + iz )(−iz ) − |z|2 − k iz · σˆ , (6.53) X 0 0 Ξ 0 0 mV0 mv z0 = 2 , z = 2 , k = kx/kF > 0, (6.54) ℏ kF ℏ kF 2 2 Ξ =(k + iz0) + |z| . (6.55)

By substituting the transmission coeﬃcients, the Josephson current is represented as X X 2 · ie | |4 ∆ k d z ∆+ iφ − ∆− −iφ J = kBT t 2 i e i e . (6.56) ℏ Ω 2|Ξ| Ω+ Ω− p ωn

We find that J = 0 for py-wave junction because ∆± is an odd function of ky. The selection rule in the orbital part also affects the Josephson effect. For px-wave symmetry, by assuming ∆± = ±∆, the current becomes e∆ ∆ X k2 J = − |z| cos α cos φ tanh |t|4 , (6.57) ℏ 2k T 2|Ξ|2 B p where α is the angle between d and m [30]. The magnetic moment m ∥ d can supply spin angular momentum ±1 to a Cooper pair. The current-phase relationship is also unusual as the current is finite even at the zero-phase difference. Generally speaking, the flow of electric current needs the breakdown of time-reversal symmetry. In this junction, the magnetic moment breaks the time-reversal symmetry. The second example of a magnetically active potential is spin-orbit coupling due to the nonmagnetic barrier potential,

HSO = δ(x)[v0σˆ0 + λex · (i∇ × σˆ)] , (6.58)

The transmission coeﬃcient is given by

ˆe 1 − − 2 2 2 2 tX (k, p) = k [k + iz0 iλSO(kyσˆ3 kzσˆ2)] , ΞSO = (k + iz0) + λSO(ky + kz ) (6.59) ΞSO where λSO is the dimensionless coupling constant. The transmission coeﬃcients of a hole is calculated to be

ˆh 1 − − tX (k, p) = ∗ k [k iz0 iλSO(kyσˆ3 + kzσˆ2)] . (6.60) ΞSO CHAPTER 6. VARIOUS JOSEPHSON JUNCTIONS 73

The current is calculated as X X 2 ie | |4 ∆ k z0λSO ∆+ iφ − ∆− −iφ − J = kBT t 2 e e (kyd3 kzd2). (6.61) ℏ Ω |ΞSO| Ω+ Ω− p ωn

For px-wave superconductor, we ﬁnd that J = 0 because the summation over p in Eq. (6.61) is zero. In the case of py-wave superconductor, on the other hand, the current becomes X 2 2 e∆ ∆ k z0λSO k d3 J = − sin φ tanh |t|4 y . (6.62) ℏ 2k T 2k |Ξ |2 B p F SO The lowest order Josephson coupling depends sensitively on magnetic potential at the interface, the direction of d vector and the orbital symmetry of spin-triplet superconductor. [31] Therefore, measurement of the lowest Josephson coupling is a good symmetry tester of spin- triplet superconductors.

6.4 Chiral superconductors

In the following sections, we discuss transport properties of unconventional superconductors which has been discussed in recent studies. The chiral superconductors are characterized by the pair potential of inθ ∆e iσˆ2, n = ±2, ±4,..., ∆k = inθ (6.63) ∆e id · σˆ σˆ2, n = ±1, ±3,..., where kx = kF cos θ and ky = kF sin θ. An integer n is called Chern number or TKNN number in solid state physics. We consider a superconductor in two-dimension because the Chern number is defined in even dimension. When we consider a surface of a chiral superconductor at x = 0, the wave function at the surface of superconductor for E < ∆ can be given by √ √ 2 2 2 2 E + i ∆ − E E − i ∆ − E − − ϕ (x, y) = eikxxC + e ikxxD eikyye x/ξ0 , (6.64) R −inθ † − n inθ † ∆e σˆS ∆( 1) e σˆS inθ in(π−θ) where we have taken into account the relation ∆+ = ∆e and ∆− = ∆e . From the boundary condition at x = 0, the energy of the bound states are obtained for n = 1, 2 and 3, E =∆ sin θ (n = 1), (6.65) E = − ∆ cos(2θ) sgn(sin 2θ)(n = 2), (6.66) E =∆ sin(3θ) sgn(cos 3θ)(n = 3). (6.67) Figure 6.6 shows the dispersion of the bound states. The number of the bound states at E = 0 is |n| according to the bulk-boundary correspondence. These bound states carry the electric current along the surface of a superconductor as schematically shown in Fig. 6.6(d). When the sign of the Chern number is inverted, the dispersion changes its sign and the electric current flows the opposite direction. The tunnel spectra of a chiral superconductor can be calculated from the reflection coefficients of a NS junction, − | |2 † rn(1 Γ+Γ−) tn Γ+σˆS rêe = 2 σˆ0, rˆhe = 2 , (6.68) 1 − |rn| Γ+Γ− 1 − |rn| Γ+Γ− ∗ p ∆ − + ∆ 2 2 Γ+ = , Γ− = , Ω± = E − |∆±| . (6.69) E + Ω+ E + Ω− CHAPTER 6. VARIOUS JOSEPHSON JUNCTIONS 74

chiral y superconductor

x (d)

Figure 6.6: The dispersion of the bound states at a surface of a chiral superconductor. (a) chiral p-wave n = 1, (b)chiral d-wave n = 2, and (c)chiral f-wave n = 3.

iφ The phase factor of a superconductor e is also included in the pair potential ∆±. These expression are valid for any superconductors. The tunnel spectra are displayed in Fig. 6.7. The tunnel spectra for a chiral p-wave superconductor have a dome-shaped broad peak below the gap as shown in the results with z0 = 5 in Fig. 6.7(a). For a chiral d-wave superconductor, the tunnel spectra have a shallow broad dip at the subgap energy region as shown in Fig. 6.7(b). The results for a chiral f-wave case shwon in Fig. 6.7(c) shows almost no subgap structure in the tunnel spectra. The pair potential for a chiral p-wave symmetry in two-dimension is given by

kx + iky ∆ˆ k = ∆ i σˆ3 σˆ2, (6.70) kF ∗ where d vector points the third direction in spin space. By taking into account ∆+ = ∆(kx − iky)/kF and ∆− = −∆(kx − iky)/kF , we ﬁnd that the Josephson current X 2 2 e∆ ∆ k z0λSO k J = cos φ tanh |t|4 y , (6.71) ℏ 2k T 2k |Ξ |2 B p F SO is proportional to cos φ reﬂecting the breaking time-reversal symmetry.

6.5 Helical superconductors surface state spectra, absence of chiral electric current, helical spin curent, tunnel conductance, Josephson eﬀect (parallel and antiparallel) CHAPTER 6. VARIOUS JOSEPHSON JUNCTIONS 75

!" !" !" %-4/%012345%: %-./%012345%6 %-0/%012345%9 %7 %8%# !# !# # !# ' '

' $!" $!" $!" %7#%8%$ )%& )%& )%& '(% '(% '(% $!# $!# $!# %& %& %& %7#%8%" #!" #!" #!"

#!# #!# #!# * *$ # $ * *$ # $ * *$ # $ %+%)%, %+%)%, %+%)%,

Figure 6.7: The conductance spectra of a NS junction consisting of a chiral superconductor. (a) chiral p-wave n = 1, (b)chiral d-wave n = 2, and (c)chiral f-wave n = 3.

6.6 Noncentrosymmetric superconductor surface bound state spectra, s+p, d+pI, d+pII Appendix A

Pair potential near the transition temperature

The dependence of ∆ on temperature near Tc can be calculated analytically. Using an identity, ∞ X 1 1 a k T = tanh , ω = (2n + 1)πk T/ℏ, (A.1) B ℏ2ω2 + a2 2a 2k T n B n=−∞ n B

We ﬁrst derive a relation from the gap equation,

Z ω Z ℏωD XD ∞ −1 1 E 1 (gN0) = dξ tanh = kBT dξ 2 2 2 2 , (A.2) 0 E 2kBT −∞ ℏ ω + ξ + ∆ ωn n p 2 2 with E = ξ + ∆ . We introduce a high-energy cut-oﬀ ℏωD to either the summation or the integration so that the summation and integration converges. At T = Tc, by putting ∆ = 0, the equation results in, Z XωD ∞ XωD XNTc −1 1 π 1 (gN0) =kBTc dξ 2 2 = kBTc = , (A.3) −∞ ω + ξ |(2n + 1)πkBTc| n + 1/2 ωn n ωn n=0

N NT N XT 1 Xc 1 XT 1 = + − , (A.4) n + 1/2 n + 1/2 n + 1/2 n=0 n=0 n=0 N XT 1 T ≈ + log , (A.5) n + 1/2 T n=0 c where we have used a relation XN 1 1 ≈ γ + log N + 2 log 2 + O , (A.6) n + 1/2 N 2 n=0 ℏωD ℏωD NTc = ,NT = , (A.7) 2πkBTc 2πkBT

76 APPENDIX A. PAIR POTENTIAL NEAR THE TRANSITION TEMPERATURE 77

for N ≫ 1 with γ = 0.577215 ··· being the Eular constant. Near T ≲ Tc, ∆ is much smaller than Tc. it is possible to expand the integrand with respect to ∆/ωn,

ω Z XD ∞ 1 (gN )−1 =2k T dξ , (A.8) 0 B ℏ2 2 2 2 −∞ ωn + ξ + ∆ ωn>0 ω Z XD ∞ 1 ∆2 ≈2k T dξ − + ··· , (A.9) B ℏ2 2 2 ℏ2 2 2 2 −∞ ωn + ξ ( ωn + ξ ) ωn>0 XωD 2 XNT 2 X∞ 1 − 1 ∆ 1 − ∆ 1 =πkBT 3 = 3 . (A.10) ℏωn (ℏωn) 2 n + 1/2 πkBT (2n + 1) ωn>0 n=0 n=0 Using the relation,

∞ ∞ X 1 X 1 2m − 1 ζ(n) = , = ζ(m), (A.11) jn (2n + 1)m 2m j=1 n=0 we reach at T ∆ 2 7ζ(3) log = − . (A.12) Tc πkBT 8 Then we obtain s r 8 Tc − T ∆ = πkBTc . (A.13) 7ζ(3) Tc

The results are desired expression of order parameters in the mean-field theory. It is possible to confirm that lim − ∂ ∆ → −∞ reflecting the phase transition of the second kind. T →Tc T Appendix B

Landauer formula for electric conductance

The derivation of the Landauer’s conductance formula is intuitive. To measure the conductance, we consider one-dimensional junction as shown in Fig. B.1, where a sample is connected two perfect read wires which are connected to reservoirs. The perfect lead wire is rather theoretical lead wire which is free from any scattering events. The reservoir is always in equilibrium characterized by the chemical potential and absorbs any incoming waves. In Fig. B.1, the chemical potential on the left (right) reservoir is µL (µR). The sample is shorter than any inelastic scattering lengths. An electron at the left reservoir can go through the sample and reach the right lead lead wire sample wire μL (scatterer) T R μR

Figure B.1: The junction to deﬁne the conductance in the Landauer formula. reservoir. Since all the states with energy below µR are occupied, an electron at the energy window µL − µR = eV ≪ µR can penetrate into the left lead wire and the sample. The electron number in such energy window is estimated as eV N = , (B.1) πℏvF

−1 where (πℏvF ) is the density of states per unit length per spin in one-dimensional conductor. Among them, an electron with positive velocity vF penetrate into the sample and N eV 1 vF T = vF T, (B.2) 2 πℏvF 2

78 APPENDIX B. LANDAUER FORMULA FOR ELECTRIC CONDUCTANCE 79 can reach the right reservoir because the transmission probability of the sample is T . The electric current is calculated as N e2 J = e v T = TV. (B.3) 2 F h The conductance deﬁned by the relation J = GV results in

e2 G = T. (B.4) h The product of the velocity and the density of states gives a constant, which is a characteristic feature in one-dimensional conductor. Thus the formula can be applied to quasi-one-dimensional structures. The formula in higher dimensions are explained in the next paragraph. The transmission probability T and the reflection probability R are the characteristic physical values of waves. The current conservation law implies T + R = 1. The interference effect of an electron is taken into account through these probabilities. In addition, the lead wires are necessary to define the transmission probability. The Landauer’s conductance formula can be generalized to the junctions which have more than one transport channel in the lead wires. The formula for such case is given by,

e2 X G = T , (B.5) h m m where Tm is the transmission probability of an electron which is incoming to the sample from a propagating channel indicated by m as shown in Fig. B.2. This Appendix explains what the transport channels in the formula and how to calculate the transport probabilities instead of deriving the Landauer formula. Let us consider a conductor in quasi one-dimension, where the conductor is confined within a finite region in the transverse direction to the current. The Schrödingerequation is described by, ℏ2∇2 − − ϵ + V (r) ϕ(x, y) = Eϕ(x, y), (B.6) 2m F where V (r) symbolically represents scattering potential in the sample. As shown in Fig. B.2(a), the sample is connected to two lead wires where the transport channels are defined. To define the transport channel clearly, we consider the lead wires are free from any scatterers. We consider the periodic boundary condition in the transverse direction to the current (y direction) for a while. The solution is described as ipy ℏ2 ikx e 2 2 2πm ϕ(x, y) = e fp(y), fp(y) = √ ,Ek,p = k + p , pm = , m = 0, ±1, ±2,.... W 2m W (B.7)

In Fig. B.2(b), we plot the energy Ek,p as a function of the wave number in the x direction. The kinetic energy in the y direction is quantized due to the confinement as shown with m = 0, m = ±1..... Thus the dispersion shows the subband structure, where m specifies a subband. The horizontal line indicates the Fermi level. At the Fermi level, the relation Ek,p = ϵF defines the wave number in the x direction

2 2mϵF − 2 km = ℏ pm. (B.8) APPENDIX B. LANDAUER FORMULA FOR ELECTRIC CONDUCTANCE 80

E lead wiresample lead wire k p m l tl,m rl,m 5 k l (a) 4

2 y W |m|=1 (c) (b) m=0 x x=0

Figure B.2: Two conductors are separated by a potential barrier at x = 0. The width of the junction is W .

The wave function is plane wave in the x direction for km being real. Such a channel can carry the current and is called propagating channel. On the other hand for km being imaginary, the wave function dumps exponentially in the x direction for. Such a channel is called an evanescent channel because it cannot carry the current. In this picture, nine subbands m = 0, ±1, ±2, ±3, ±4 have are propagating channels and the subband with |m| ≥ 5 are the evanescent channels. The transport coefficients are defined for the propagating channels. Next we consider a situation where an electron is incident from a channel of pm at the left lead wire. The incident electron is transmitted to the right lead wire partially and reflected to the left lead wire partially. The wave functions can be described as X L ikmx −iklx ϕm(x, y) =e fm(y) + rl,me fl(y), (B.9) X l R iklx ϕm(x, y) = tl,me fl(y), (B.10) l where tl,m is the transmission coefficient to the l-th channel at the right lead wire and rl,m is the reflection coefficient to the l-th channel at the left lead wire. Generally speaking, the out going channel l is not necessary to identical to the incoming channel m because an incident electron are scattered random potentials in the sample. The transmission and the reflection probabilities are defined by X X ∗ ∗ Tm = tl,mtl,m,Rm = rl,mrl,m. (B.11) l l

The current conservation law implies Tm + Rm = 1. These transport coeﬃcients are calculated from the boundary condition of wave function. We consider a practical situation where the sample is represented by the potential barrier of v0δ(x) as shown in Fig. B.2(c). A boundary condition for wave function L R ϕm(0, y) = ϕm(0, y), (B.12) means that the wave function must be single-valued. By substituting Eqs. (B.9) and (B.10) into ∗ the boundary condition and carrying out the integration along y after multiplying fn(y), we ﬁnd

δn,m + rn,m = tn,m. (B.13) APPENDIX B. LANDAUER FORMULA FOR ELECTRIC CONDUCTANCE 81 due to the orthnormality Z W ∗ dy fn(y)fl(y) = δn,l. (B.14) 0 To derive the second boundary condition, we integrate the Schr¨odingerequation at an interval of −γ < xγ and take the limit of γ → 0. Z Z γ ℏ2 ∂2 ∂2 γ lim dx − + − ϵ + v δ(x) ϕ(x, y) = lim dx Eϕ(x, y). (B.15) → 2 2 F 0 → γ 0 −γ 2m ∂x ∂y γ 0 −γ Since the right-hand side becomes zero, we obtain ℏ2 ∂ R ∂ L R − ϕ (x, y) − ϕ (x, y) + v0ϕ (0, y) = 0. (B.16) 2m ∂x x=0 ∂x x=0 By substituting Eqs. (B.9) and (B.10) into the boundary condition and carrying out the inte- ∗ gration along y after multiplying fm(y), we ﬁnd the relation

k¯m δn,m − k¯n rn,m =(k¯n + 2iz0)tn,m, (B.17) ℏ2 2 mz0 ¯ km kF z0 = 2 , km = , = ϵF . (B.18) ℏ kF kF 2m From (B.13) and (B.17), the transport coeﬃcients are calculated as

k¯m −iz0 tn,m = δn,m, rn,m = δn,m. (B.19) k¯m + iz0 k¯m + iz0 They satisfy the conservation law

2 2 |tm,m| + |rm,m| = 1. (B.20)

Since the junction has translational symmetry in the y direction, the transport coeﬃcients are diagonal with respect to the transport channels. The conductance results in h i 2e2 2e2 X G = Tr tˆtˆ† = |t |2, (B.21) h h m,m m where we multiply 2 to the conductance by taking the spin degeneracy into account. Since pm = 2πm/W , the summation over m can be replaced by the integral for wave number in the y direction, Z X kF W W kF = dp = = Nc, (B.22) 2π − π m kF where [··· ] means the Gauss’s symbol and Nc represents the number of propagating channels.

Normal transport coeﬃcients through a magnetic potential barrier APPENDIX B. LANDAUER FORMULA FOR ELECTRIC CONDUCTANCE 82

Let us consider the transmission of an electron through an insulating barrier between two metals. The Hamiltonian reads, ℏ2∇2 Hˆ (r) = − − ϵ + V δ(x) σˆ + V · σˆδ(x), (B.23) N 2m F 0 0 whereσ ˆj for j = 1 − 3 is the Pauli’s matrix in spin space. The wave function of an electron on either side of the interface can be described as h i ikxx −ikxx ip·ρ ψL(r) = ae + Ae e , (B.24) h i −ikxx ikxx ip·ρ ψR(r) = be + Be e , (B.25) α↑ β↑ A↑ B↑ a = , b = , A = , B = , (B.26) α↓ β↓ A↓ B↓

2 2 2 ℏ2 where r = (x, ρ), kx + p = kF = 2mϵF/ , a and b are spinor representing the amplitudes of incoming wave into the interface. On the other hand, A and B are spinor representing the amplitudes of outgoing wave from the interface. The two wave functions are connected by the the boundary conditions at x = 0,

ψ (0, ρ) = ψ (0, ρ), (B.27) L R ℏ2 ∂ ∂ − ψR(r) − ψL(r) + (V0 + V · σˆ) ψL(0, ρ) = 0. (B.28) 2m ∂x ∂x x=0 The boundary conditions relate the outgoing waves and incoming ones as ˆ′ A rˆN tN a = ˆ ′ , (B.29) B tN rˆN b 1 tˆ =tˆ′ = k(k + iz + iz · σˆ)−1 = k [k + iz − iz · σˆ] , (B.30) N N 0 Ξ 0 1 rˆ =ˆr′ = (k + iz + iz · σˆ)−1(−iz − iz · σˆ) = (k + iz )(−iz ) − |z|2 − k iz · σˆ , N N 0 0 Ξ 0 0 (B.31)

mV0 mV z0 = 2 , z = 2 , k = kx/kF > 0, (B.32) ℏ kF ℏ kF 2 2 Ξ =(k + iz0) + |z| . (B.33)

The current conservation law implies

ˆ ˆ† † tNtN +r ˆNrˆN =σ ˆ0. (B.34) The transport coeﬃcients in the hole branch of BdG picture results in

1 ∗ ∗ tˆN = k [k − iz0 + iz · σˆ ] = tˆ , (B.35) e Ξ∗ N 1 2 ∗ ∗ rˆN = (k − iz0)(iz0) − |z| + k iz · σˆ =r ˆ . (B.36) f Ξ∗ N Appendix C

Mean-ﬁeld theory in real space

The Hamiltonian for an electrons in the presence of interactions is represented by

H = HN + HI, (C.1) Z X H † N = dr ψα(r)ξα,β(r)ψα(r), (C.2) α,β ( ) ℏ2 ie 2 ξˆ (r) = − ∇ − A − ϵ + V (r) σˆ + V (r) · σˆ + iλ(r) × ∇ · σˆ, (C.3) α,β 2m ℏc F 0 0 Z Z X 1 † H = dr dr′ ψ (r′) ψ† (r) −g(r − r′) ψ (r) ψ (r′), (C.4) I 2 β α α β α,β where V0 is the spin-independent potential, V is the Zeeman potential, λ represents the spin- orbit potential, and g(r − r′) = g(r′ − r) > 0 is the attractive interaction between two electrons. The pair potential is deﬁned as

∆ (r, r′) = − g(r − r′) ψ (r) ψ (r′) , (C.5) α,β D α β E D E ∗ ′ − − ′ † ′ † − ′ † † ′ ∆α,β(r, r ) = g(r r ) ψβ(r ) ψα(r) = g(r r ) ψα(r) ψβ(r ) . (C.6) The pair potential satisfies the antisymmetric property ′ ′ ∆α,β(r, r ) = −∆β,α(r , r), (C.7) because of the anticommutation relation among the electron operators. The interaction Hamil- tonian is decoupled by the mean field approximation. The operators are replaced by the order parameters as " # ∗ ′ ∗ ′ † ∆ (r, r ) † ∆ (r, r ) ψ (r′) ψ† (r) = − α,β + ψ (r′) ψ† (r) + α,β , (C.8) β α g(r − r′) β α g(r − r′) ∆ (r, r′) ∆ (r, r′) ψ (r) ψ (r′) = − α,β + ψ (r) ψ (r′) + α,β . (C.9) α β g(r − r′) α β g(r − r′) The first term is the average of the operator and the second term represents the fluctuations. The mean-field expansion is carried out up to the first order of the fluctuations, Z Z X ∗ ′ 2 1 † |∆ (r, r )| H → dr dr′ ψ† (r)∆ (r, r′) ψ (r′) − ψ (r) ∆∗ (r, r′) ψ (r′) − α,β . I 2 α α,β β α α,β β g(r − r′) α,β (C.10)

83 APPENDIX C. MEAN-FIELD THEORY IN REAL SPACE 84

As a result, the mean-ﬁeld Hamiltonian of superconductivity is represented as Z Z X h i 1 ′ † † H = dr dr ψ (r), ψ (r), ψ↑(r), ψ↓(r) MF 2 ↑ ↓ α,β   ′ ψ↑(r )  ′  δ(r − r′) ξˆ(r′) ∆(ˆ r, r′)  ψ↓(r )  × ∗ ′ ′ ∗ ′  † ′  . (C.11) −∆ˆ (r, r ) −δ(r − r ) ξˆ (r )  ψ↑(r )  † ′ ψ↓(r )

The BdG equation reads, Z X − ′ ˆ ′ ˆ ′ ′ ′ δ(r r ) ξ(r ) ∆(r, r ) uˆν(r ) uˆν(r) ˆ dr ∗ ′ ′ ∗ ′ ′ = Eν, (C.12) −∆ˆ (r, r ) −δ(r − r ) ξˆ (r ) vˆν(r ) vˆν(r) α,β   uν,↑,1(r) uν,↑,2(r)   ˆ Eν,1 0 uˆν(r)  uν,↓,1(r) uν,↓,2(r)  Eν = , =   . (C.13) 0 Eν,2 vˆν(r) vν,↑,1(r) vν,↑,2(r) vν,↓,1(r) vν,↓,2(r)

The wave function ∗ vˆν(r) ∗ (C.14) uˆν(r) belongs to the eigenvalue of −Eˆν. The wave function satisﬁes the orthonormality and the completeness, Z h i † † uˆ (r) dr uˆ (r), vˆ (r) ν = δ , (C.15) λ λ vˆ (r) λ,ν Z ν h i ∗ † † vˆ (r) dr uˆ (r), vˆ (r) ν = 0, (C.16) λ λ uˆ∗(r) ν X h i ∗ uˆν(r) † ′ † ′ vˆν(r) T ′ T ′ ˇ − ′ uˆν(r ), vˆν(r ) + ∗ vˆν (r ), uˆν (r ) = 14×4 δ(r r ). (C.17) vˆν(r) uˆ (r) ν ν

Let us assume that V , λ are uniform, and A = V0 = 0. It is possible to derive the mean-ﬁeld Hamiltonian in momentum space by substitute the Fourier transformation X 1 ik·r ψα(r) =√ ψ e , (C.18) V k,α vol k X 1 ′ ∆ (r − r′) = ∆ (q)eik·(r−r ), (C.19) α,β V α,β vol q APPENDIX C. MEAN-FIELD THEORY IN REAL SPACE 85 into Eq. (C.11). Z Z X X h i 1 ′ 1 † † − ′· H ′ ′ ik r MF = dr dr ψk′,↑, ψk′,↓, ψ−k ,↑, ψ−k ,↓ e 2 Vvol ′ α,β k,k   " P # ψk,↑ ′ ˆ ′ 1 ˆ iq·(r−r′)   δ(r − r ) ξ(r ) ∆(q)e · ′  ψk,↓  × P Vvol q ik r  †  1 ∗ −iq·(r−r′) ′ ∗ ′ e , (C.20) − ∆ˆ (q)e −δ(r − r ) ξˆ (r )  ψ−k,↑  Vvol q † ψ−k,↓ Z X X h i 1 1 † † − ′· ′ ′ ik r = dr ψk′,↑, ψk′,↓, ψ−k ,↑, ψ−k ,↓ e 2 Vvol ′ α,β k,k   ψ ↑  k,  ˆ ˆ ψ ↓ ξ(r) ∆(k) ik·r  k,  × ∗ ∗ e  †  , (C.21) −∆ˆ (−k) −ξˆ (r)  ψ−k,↑  † ψ −k,↓   " # ψk,↑ X X h i   1 † ξˆ ∆(ˆ k)  ψk,↓  k  †  = ψk,↑, ψk,↓, ψ−k,↑, ψ−k,↓ ∗ ˆ∗ , (C.22) 2 −∆ˆ (−k) −ξ−  ψ−k,↑  α,β k k † ψ−k,↓ ℏ2k2 ξˆ =ξ σˆ + V · σˆ − λ × k · σˆ, ξ = − ϵ . (C.23) k k 0 k 2m F The examples of the pair potential are shown below,   ∆iσˆ2 : s-wave ˆ ¯2 − ¯2 ∆(k) =  ∆(kx ky)iσˆ2 : dx2−y2 -wave , (C.24) ∆2k¯x k¯yiσˆ2 : dxy-wave ¯ for spin-singlet even-parity order with kj = kj/kF for j = x, y, z and   ∆k¯ id · σˆ σˆ : p -wave  x 2 x ¯ − ¯ ˆ ∆(kyσˆ1 kxσˆ2)iσˆ2 : 2D Helical p-wave ∆(k) =  ¯ ¯ · , (C.25)  ∆(kx + iky)d σˆ σˆ2 : 2D chiral p-wave ∆(k¯xσˆ1 + k¯yσˆ2 + k¯zσˆ3)iσˆ2 : Helium3 B phase for spin-triplet odd-parity order. Appendix D

Reﬂection coeﬃcients in one-dimensional junctions

NS junction The reflection coefficients of an NS junction is calculated from the boundary conditions for the wave function. The boundary conditions are represented in a matrix form, α A C + = Φˆ Uˆ , (D.1) β B D α A C kτˆ − kτˆ = Φˆ Uˆ (kτˆ + 2iz ) , (D.2) 3 β 3 B 3 0 D u sv eiφ/2 0 Uˆ = , Φˆ = . (D.3) v u 0 e−iφ/2 Here we have assumed that the incoming waves from a superconductor are absent (i.e., γ = δ = 0). A sign factor s = ±1 is introduced for the latter use. From the first condition, we obtain, α A C (kτˆ + 2iz )Uˆ −1 Φˆ ∗ + = (kτˆ + 2iz ) . (D.4) 3 0 β B 3 0 D Together with the second boundary condition, it is possible to eliminate C and D, h i A τˆ k Uˆ −1 Φˆ ∗ + kUˆ −1 Φˆ ∗τˆ + 2iz Uˆ −1 Φˆ ∗ 3 3 0 B h i α = − τˆ k Uˆ −1 Φˆ ∗ − kUˆ −1 Φˆ ∗τˆ + 2iz Uˆ −1 Φˆ ∗ . (D.5) 3 3 0 β The elements of the matrices are represented as ∗ ∗ (k + iz0)u χ −i z0 s v χ A iz0 u χ −(k + iz0)s v χ α ∗ = − ∗ , (D.6) −i z0 v χ −(k − iz0)u χ B (k − iz0)v χ iz0 u χ β with χ = eiφ/2. By calculating the matrix inversion, we reach at A 1 −(k − iz0)u χ i z0 s v χ = ∗ ∗ B (k2 + z2)[|t |2u2 + |r |2(u2 − sv2)] i z v χ (k + iz )u χ 0 n n 0 0 ∗ iz0 u χ −(k + iz0)s v χ α × ∗ . (D.7) (k − iz0)v χ iz0 u χ β

86 APPENDIX D. REFLECTION COEFFICIENTS IN ONE-DIMENSIONAL JUNCTIONS 87

The reflection coefficients are calculated as −iz (k − iz )(u2 − sv2) r [(1 + s)Ω + (1 − s)E] r = 0 0 = n , (D.8) ee 2 2 | |2 2 | |2 2 − 2 − | |2 − | |2 (k + z0)[ tn u + rn (u sv )] (1 + s s tn )Ω + (1 s + s tn )E k2u v e−iφ |t |2∆ e−iφ r = = n . (D.9) he 2 2 | |2 2 | |2 2 − 2 − | |2 − | |2 (k + z0)[ tn u + rn (u sv )] (1 + s s tn )Ω + (1 s + s tn )E The results with s = 1 are the reflection coefficients for an s-wave NS junction discussed in Chap. 4. On the other hand, the results with s = −1 plus ∆ → ∆k represent the reflection coefficients for a px-wave NS junction discussed in Chap. 5. 2 2 Since GN = (2e /h)|tn| in one dimension, the normalized conductance for s = 1 is calculated as

2 2 GNS 2∆ |tn| = 2 2 2 2 4 2 (D.10) GN (2 − |tn| ) (∆ − E ) + |tn| E for E < ∆, and √ G 2[|t |2E2 + (2 − |t |2) E2 − ∆2E] NS = n √ n , (D.11) 2 2 2 2 2 GN [(2 − |tn| ) E − ∆ + |tn| E] for E ≥ ∆. In the tunnel limit |tn| ≪ 1, the normalized conductance becomes 2| |2 GNS ∆ tn ≪ = 2 2 1, (D.12) GN 2(∆ − E ) for E < ∆. Thus the conductance spectra have U-shaped gap structure. For E ≥ ∆, on the other hand, the spectra G E NS = √ , (D.13) GN E2 − ∆2 is proportional to the density of states in a uniform superconductor. In the case of s = −1, the normalized conductance results in for E < ∆,

2 2 2 GNS 2|tn| (2E + ∆ ) = 2 2 4 2 . (D.14) GN 4(1 − |tn| )E + |tn| ∆

2 At E = 0, the conductance ratio 2/|tn| is much larger than unity. As a result, the tunnel conductance spectra have a peak at the zero energy. The limiting behavior

| |2 2 GNS ≈ tn ∆ /2 2 2 2 , (D.15) GN E + (|tn| ∆/2)

2 2 for E ≪ ∆ and |tn| ≪ 1 suggests that the peak width is about |tn| ∆/2.

√ √ G 2[|t |2E2 + (2 − |t |2) E2 − ∆2E] E2 − ∆2 NS = n n √ → . (D.16) 2 2 2 2 2 GN [(2 − |tn| )E + |tn| E − ∆ ] E for E ≥ ∆. In the last line, we consider tunnel limit. APPENDIX D. REFLECTION COEFFICIENTS IN ONE-DIMENSIONAL JUNCTIONS 88

In an SIS junction, the boundary conditions are given by α A C Φˆ Uˆ + Uˆ = Uˆ , (D.17) 1 β 2 B 1 D

α A C Φˆ Uˆ k τˆ − Uˆ k τˆ = Uˆ (k τˆ + 2i z ) , (D.18) 1 3 β 2 3 B 1 3 0 D u sv u v eiφ/2 0 Uˆ = , Uˆ = , Φˆ = , (D.19) 1 v u 2 sv u 0 e−iφ/2 with φ = φL − φR. We introduced a sign factor s = ±1 for the latter convenience. They are transformed as α A C (k τˆ + 2i z )Pˆ + (k τˆ + 2i z )Pˆ =(k τˆ + 2i z ) , (D.20) 3 0 11 β 3 0 12 B 3 0 D α A C Pˆ k τˆ − Pˆ k τˆ =(k τˆ + 2i z ) , (D.21) 11 3 β 12 3 B 3 0 D

Pˆ11 = Uˆ1 Φˆ Uˆ1, Pˆ12 = Uˆ1 Φˆ Uˆ2. (D.22)

By eliminating C and D, we obtain the relation, h i h i α A k τˆ Pˆ − Pˆ k τˆ + 2iz Pˆ = − k τˆ Pˆ + Pˆ k τˆ + 2iz Pˆ . (D.23) 3 11 11 3 0 11 β 3 12 12 3 0 12 B

The elements of the matrices are represented explicitly, 2 2 ∗ ∗ (k + iz0)(u χ − v χ ) iz0 uv(χ − sχ ) A −iz uv(χ − sχ∗) −(k − iz )(u2 χ∗ − v2 χ) B 0 0 2 − 2 ∗ − ∗ − iz0 (u χ sv χ )(k + iz0) s uv(χ χ ) α = ∗ 2 ∗ 2 . (D.24) (k − iz0) uv(χ − χ ) iz0 (u χ − sv χ) β

By calculating the matrix inversion, we obtain, A 1 1 = B k2 + z2 (u2 − sv2)2 + |t |2u2v2(cos φ − 1) 0 n − 2 ∗ − 2 − ∗ × (k iz0)(u χ v χ) iz0 uv(χ sχ ) −iz uv(χ − sχ∗) −(k + iz )(u2 χ − v2 χ∗) 0 0 2 − 2 ∗ − ∗ × iz0 (u χ sv χ )(k + iz0) s uv(χ χ ) α ∗ 2 ∗ 2 . (D.25) (k − iz0) uv(χ − χ ) iz0 (u χ − sv χ) β

[1] H. K. Onnes, Comm. Phys. Lab. Univ. Leiden Supple. 29 (1911).

[2] J. Bardeen, L. N. Cooper, and J. R. Schrieﬀer, Phys. Rev. 108, 1175 (1957).

[3] N. N. Bogoliubov, JETP 7, 41 (1958); JETP 7, 51 (1958).

[4] M. Tinkham, ’Introduction to Superconductivity’, 2nd Ed. McGraw-Hill (1996).

[5] S. Nakajima, ’Introduction to Superconductivity’, (Japanese) Baifukan (1971).

[6] A. A. Abrikosov, Fundamentals of the Theory of Metals, North-Holland (1988).

[7] P. G. de Gennes, ’Superconductivity of Metals and Alloys’, Benjamin, New York, 1966.

[8] G. Mahan, ’Many-particle Physics’ 3rd edition, Plenum Publishers (2000).

[9] J. R. Schrieﬀer, ’Theory of Superconductivity’, Addison-Wesley, (1983).

[10] A. A. Abrikosov, L. P. Gor’kov, and I. E. Dzyaloshinskii, ’Methods of Quantum Field Theory in Statistical Physics’, Dover Pubns (1975).

[11] A. L. Fetter and J. D. Walecka, ’Quantum Theorey of Many-Particle Systems’ McGraw-Hill (1971).

[12] B. D. Josephson, Phys. Lett. 1, 251 (1962).

[13] A. F. Andreev, Zh. Eksp. Theor. Fiz, 46, 1823 (1964) [Sov. Phys. JETP 19, 1228 (1964)].

[14] Y. Asano, Phys. Rev. B 61, 1732 (2000).

[15] Y. Tanaka and S. Kashiwaya, Phys. Rev. Lett. 74, 3451 (1995).

[16] Y. Asano, Y. Tanaka, and S. Kashiwaya, Phys. Rev. B 69, 134501 (2004).

[17] A. Furusaki and M. Tsukada, Solid State Commun. 78, 299 (1991).

[18] V. Ambegaokar and A. Baratoﬀ, Phys. Rev. Lett. 10, 486 (1963).

[19] I. O. Kulik and A. N. Omel’yanchuk, Sov. J. Low. Temp. Phys. 3, 459 (1977). [Fiz. Nizk. Temp. 3, 945 (1977)].

[20] Y. Tanaka and S. Kashiwaya, Phys. Rev. B 53, 11957R (1996).

[21] Y. Asano, Phys. Rev. B 64, 224515 (2001).

[22] Y. Tanaka, and S. Kashiwaya, Phys. Rev. B 70, 012507 (2004).

89 BIBLIOGRAPHY 90

[23] Y. Asano, Y. Tanaka, and S. Kashiwaya, Phys. Rev. Lett. 96, 097007 (2006).

[24] S. Ikegaya, S.-I. Suzuki, Y. Tanaka, and Y. Asano, Phys. Rev. B 94, 054512 (2016).

[25] M. Sato, Y. Tanaka, K. Yada, and T. Yokoyama, Phys. Rev. B 83, 224511(2011).

[26] Y. Tanaka, Phys. Rev. Lett. 72, 3871 (1994).

[27] Y. Tanaka and S. Kashiwaya, Phys. Rev. B 56, 892 (1997).

[28] Y. Asano, Phys. Rev. B 72, 092508 (2005).

[29] Y. Asano, Phys. Rev. B 74, 220501R (2006).

[30] P. M. R. Brydon, W. Chen, Y. Asano, and D. Manske, Phys. Rev. B 88, 054509 (2013).

[31] A. Millis, D. Rainer, and J. A. Sauls, Phys. Rev. B 38, 4504 (1988).