Effects of the phase coherence on the local density of states in superconducting proximity structures

TL;DR

This paper investigates how phase coherence influences the local density of states in superconducting proximity structures, highlighting differences between s-wave and p-wave junctions and the effects of interface transparency.

Contribution

It provides a theoretical analysis of the energy spectrum in superconducting proximity structures with both s-wave and p-wave pairing, emphasizing the role of interface transparency and depairing effects.

Findings

Decay length limited by inelastic scattering at zero temperature

Decay length in p-wave junctions depends on interface transparency

Anomalous proximity effect occurs in odd-parity superconductors

Abstract

We theoretically study the local density of states in superconducting proximity structure where two superconducting terminals are attached to a side surface of a normal-metal wire. Using the quasiclassical Green's function method, the energy spectrum is obtained for both of spin-singlet $s$-wave and spin-triplet $p$-wave junctions. In both of the cases, the decay length of the proximity effect at the zero temperature is limited by a depairing effect due to inelastic scatterings. In addition to the depairing effect, in $p$-wave junctions, the decay length depends sensitively on the transparency at the junction interfaces, which is a unique property to odd-parity superconductors where the anomalous proximity effect occurs.

Equations

$$\mathcal{D}\nabla \left(\mathbb{G}\nabla \mathbb{G}\right)+i{\left[\mathbb{H},\mathbb{G}\right]}_{-}=0,$$

$$\displaystyle\mathbb{G}(\boldsymbol{r},\varepsilon)=\left(\begin{array}[]{cc}\check{g}^{R}(\boldsymbol{r},\varepsilon)&\check{g}^{K}(\boldsymbol{r},\varepsilon)\\[2.84526pt] 0&\check{g}^{A}(\boldsymbol{r},\varepsilon)\\ \end{array}\right),$$

$$\mathcal{D}{\partial }_x\left(\mathbb{G}{\partial }_x\mathbb{G}\right)+i{\left[\mathbb{H},\mathbb{G}\right]}_{-}+\mathbb{S}{\Theta }_S(x)=0,$$

$${\check{g}}^K={\check{g}}^R\check{F}-\check{F}{\check{g}}^A,$$

$$\check{F}={\check{\tau }}_0{f}_L+{\check{\tau }}_3{f}_T,$$

$$\mathcal{D}{\partial }_x\left({\check{g}}^X{\partial }_x{\check{g}}^X\right)+i{\left[{\check{H}}^X,{\check{g}}^X\right]}_{-}+{\check{S}}^X{\Theta }_S(x)=0,$$

$$\displaystyle\check{g}^{X}(x,\varepsilon)=\left(\begin{array}[]{rr}\hat{g}^{X}&\hat{f}^{X}\\[2.84526pt] -\undertilde{\hat{f}}^{X}&-\hat{g}^{X}\\[2.84526pt] \end{array}\right).$$

$${\hat{g}}^X={\hat{\sigma }}_0{g}^X,$$

$${\hat{f}}^X=f_{\mu }^X(i{\hat{\sigma }}_{\mu }{\hat{\sigma }}_2),\text{undertilde}{\hat{f}}^X=f_{\mu }^X(i{\hat{\sigma }}_2{\hat{\sigma }}_{\mu }),$$

$$\mathcal{D}{\partial }_x\left({\tilde{g}}^X{\partial }_x{\tilde{g}}^X\right)+i{\left[{\epsilon }^X{\tilde{\tau }}_3,{\tilde{g}}^X\right]}_{-}+{\tilde{S}}^X{\Theta }_S(x)=0,$$

$$\displaystyle\hskip 17.07164pt\tilde{g}^{X}(x,\varepsilon)=\left(\begin{array}[]{rr}g^{X}&f^{X}\\[2.84526pt] -f^{X}&-g^{X}\\[2.84526pt] \end{array}\right),$$

$${\tilde{g}}^X$$

$$\displaystyle=\left[\begin{array}[]{rr}\cosh\theta&\sinh\theta\\[2.84526pt] -\sinh\theta&-\cosh\theta\\ \end{array}\right],$$

$$\mathcal{D}\frac{{\partial }^2\theta }{\partial x^2}+2i\bar{\epsilon }\sinh \theta +{\Theta }_S(x)S(x,\epsilon )=0.$$

$${\frac{d\theta }{dy}|}_{y=0}={\gamma }_B^{-1}⟨F{⟩}_{\varphi },$$

$$F=\frac{-2T_N(f_S\cosh {\theta }_0-g_S\sinh {\theta }_0)}{(2-T_N)\Xi +T_N(g_S\cosh {\theta }_0-f_S\sinh {\theta }_0)},$$

$$g_S=g_{S+}+g_{S-},$$

$$\displaystyle f_{S}=\left\{\begin{array}[]{ll}f_{S+}+f_{S-}&\text{~{}for singlet SCs, }\\[2.84526pt] i(g_{S-}f_{S+}-g_{S+}f_{S-})&\text{~{}for triplet SCs, }\\ \end{array}\right.$$

$$g_{S\pm }(\varphi )=\frac{\epsilon }{\sqrt{{\epsilon }^2-|{\Delta }_{\pm }{|}^2}},f_{S\pm }(\varphi )=\frac{{\Delta }_{\pm }}{\sqrt{{\epsilon }^2-|{\Delta }_{\pm }{|}^2}},$$

$$\Xi =1+g_{S+}{g}_{S-}-f_{S+}{f}_{S-}$$

$$\displaystyle\Delta_{+}(\phi)=\left\{\begin{array}[]{ll}\Delta_{0}&\text{~{}for an $s$\,-wave, }\\[2.84526pt] \Delta_{0}\cos[\phi-\alpha_{\phi}]&\text{~{}for a $p$\,-wave,}\\[2.84526pt] \end{array}\right.$$

$$S(x,\epsilon )={\gamma }_B^{-1}⟨F(x,\epsilon ,\varphi ){⟩}_{\varphi }.$$

$$\displaystyle\theta(x,\varepsilon)\bigg{|}_{x=\pm L}=0,\hskip 22.76219pt\frac{d\theta(x,\varepsilon)}{dx}\bigg{|}_{x=0}=0.$$

$$\displaystyle\theta(x,\varepsilon)\bigg{|}_{x=\pm L}=0,\hskip 22.76219pt\theta(x,\varepsilon)\bigg{|}_{x=0}=0.$$

$$\nu (x,\epsilon )=\frac{{\nu }_0}{8}\mathrm{Tr}\left[{\check{\tau }}_3\left({\check{g}}^R-{\check{g}}^A\right)\right].$$

$$\delta \nu (x,\epsilon )=\frac{\nu (x,\epsilon )-{\nu }_0}{{\nu }_0}.$$

$$\overline{\delta \nu }(\gamma )=\frac{\delta \nu (x=0,\epsilon =0;\gamma )}{\delta \nu (x=0,\epsilon =0;\gamma =0.001{\Delta }_0)}.$$

$$\displaystyle D\boldsymbol{\nabla}_{\boldsymbol{r}}\big{(}\check{g}_{o}^{X}\boldsymbol{\nabla}_{\boldsymbol{r}}\check{g}^{X}_{o}\big{)}+i\big{[}\check{H}^{X}_{o},\check{g}^{X}_{o}\big{]}_{-}=0,$$

$$\displaystyle\check{H}^{X}_{o}=\left[\begin{array}[]{cc}\varepsilon^{X}\hat{\sigma}_{0}&\hat{\Delta}(\boldsymbol{r},\varepsilon)\\[5.69054pt] \hat{\Delta}^{*}(\boldsymbol{r},-\varepsilon)&-\varepsilon^{X}\hat{\sigma}_{0}\\ \end{array}\right],$$

$$\displaystyle\check{g}^{X}_{o}(\boldsymbol{r},\varepsilon)=\left[\begin{array}[]{cc}\hat{g}^{X}(\boldsymbol{r},\varepsilon)&\hat{f}^{X}(\boldsymbol{r},\varepsilon)\\[5.69054pt] -\undertilde{\hat{f}}^{X}(\boldsymbol{r},\varepsilon)&-\undertilde{\hat{g}}^{X}(\boldsymbol{r},\varepsilon)\\ \end{array}\right],$$

$${\check{H}}_o^X$$

$$\displaystyle=\left[\begin{array}[]{cc}\varepsilon^{X}\hat{\sigma}_{0}&\Delta_{\mu}(\boldsymbol{r})(i\hat{\sigma}_{\mu}\hat{\sigma}_{2})\\[5.69054pt] \Delta_{\mu}^{*}(\boldsymbol{r})(i\hat{\sigma}_{2}\hat{\sigma}_{\mu})&-\varepsilon^{X}\hat{\sigma}_{0}\\ \end{array}\right].$$

$$\displaystyle\check{g}^{X}_{o}=\left[\begin{array}[]{cc}g^{X}\hat{\sigma}_{0}&f^{X}_{\mu}(i\hat{\sigma}_{\mu}\hat{\sigma}_{2})\\[5.69054pt] -\undertilde{f}^{X}_{\mu}(i\hat{\sigma}_{2}\hat{\sigma}_{\mu})&-{g}^{X}\hat{\sigma}_{0}\\ \end{array}\right].$$

$$\displaystyle D\boldsymbol{\nabla}_{\boldsymbol{r}}\big{(}\check{g}^{X}\boldsymbol{\nabla}_{\boldsymbol{r}}\check{g}^{X}\big{)}+i\big{[}\check{H}^{X},\check{g}^{X}\big{]}_{-}=0,$$

$$\displaystyle\check{g}^{X}(\boldsymbol{r},\varepsilon)=\left[\begin{array}[]{rr}g^{X}&f^{X}\\[5.69054pt] \undertilde{f}^{X}&-g^{X}\\ \end{array}\right]\otimes\hat{\sigma}_{0},\hskip 17.07164pt$$

$$\displaystyle\check{H}^{X}{(\boldsymbol{r},\varepsilon)}=\left[\begin{array}[]{cc}\varepsilon^{X}&\Delta(\boldsymbol{r})\\[5.69054pt] \Delta^{*}(\boldsymbol{r})&-\varepsilon^{X}\\ \end{array}\right]\otimes\hat{\sigma}_{0},$$

$${\check{H}}^X(x,\epsilon )={\check{\tau }}_3{\epsilon }^X+i{\check{\tau }}_2{\Delta }_{\mathrm{R}}(x)+i{\check{\tau }}_1{\Delta }_{\mathrm{I}}(x),$$

$${\check{H}}^R(x,\epsilon )=-{\check{\tau }}_1{\left[{\check{H}}^A(x,\epsilon )\right]}^{\ast }{\check{\tau }}_1,$$

Full text

††preprint: APS/123-QED

I Introduction

The proximity effect is a phenomenon observed in a normal metal (N) attached to a superconductor (SC)Deutscher and Gennes (1969). Cooper pairs penetrating into an N causes superconducting-like phenomena such as the screening of magnetic fields and the suppression of the local density of states (LDOS) at the Fermi level (zero energy). The penetration length of Cooper pairs is limited by the thermal coherence length $\xi_{T}=\sqrt{\mathscr{D}/2\pi T}$, where $\mathscr{D}$ is the diffusion constant in the N and $T$ is the temperature. Indeed, the Josephson current is present only when the spacing between two SCs $L_{1}$ is shorter than $\xi_{T}$.DE GENNES (1964) Although $\xi_{T}$ is the typical length scale of the proximity effect, Volkov and Takayanagi have shown that the characteristic length depends on observablesVolkov and Takayanagi (1996, 1997). They studied the conductance of a normal-metal wire whose side surface is connected to two superconducting terminals [See Fig. 1(b).]. The conductance depends on the phase difference of the two SCs even when $L_{1}\gg\xi_{T}$Volkov and Takayanagi (1996, 1997). Thus this phenomenon is named the long-range phase-coherent effect.

The analysis by Volkov and Takayanagi is unfortunately restricted to the weak-proximity-effect regime, where the solutions of the linearized Usadel equation describe the long-range phase-coherent effect. However, the magnitude of the proximity effect is generally sensitive to the transparency of an N/SC interface and the pairing symmetry of the superconductor. The strong proximity effect leads a gap-like energy spectrum at low energy in the LDOS Volkov et al. (1993); Golubov and Kupriyanov (1988, 1989); Golubov et al. (1997). The boundary condition for the quasiclassical Green’s functionKuprianov and Lukichev (1988); Nazarov (1994, 1999) enables these analysis.

Taking the essence of the circuit theoryNazarov (1994, 1999) into account, a boundary condition for the quasiclassical Usadel Green’s function at an N/SC interface has been derivedTanaka and Kashiwaya (2004); Tanaka et al. (2005a, 2003, 2004). This boundary condition enables to describe junctions of unconventional SCs such as high-$T_{c}$ cuprate, spin-triplet SCs, and topological SCs. It has been well established that the Andreev bound states (ABSs) due to the unconventional pairingTanaka and Kashiwaya (1995) modifies the proximity effect in various ways. In an N/$d$ -wave junction, the proximity effect can not contribute to ensemble-averaged values over random-impurity configurationsTanaka et al. (2003, 2004). However, the amplitude of the Josephson current in each $d$ -wave/N/$d$ -wave junction can exceed the ensemble-averaged Josephson current for the $s$ -wave/N/$s$ -wave junctionsAsano (2001); Asano et al. (2006a). Spin-triplet paringsShivaram et al. (1986); Choi and Sauls (1991); Maeno et al. (1994); Rice and Sigrist (1995); Ishida et al. (1998); Graf et al. (2000); Saxena et al. (2000); Aoki et al. (2001); Mackenzie and Maeno (2003); Huy et al. (2007); Kashiwaya et al. (2011); Machida et al. (2012) cause several anomalies (i.e., the anomalous proximity effect) such as large zero-energy peaks in the LDOS in the NTanaka and Kashiwaya (2004); Tanaka et al. (2005a, b) and resonant charge transport through a dirty N Tanaka and Kashiwaya (2004); Tanaka et al. (2005a, b); Asano et al. (2006b); Asano and Tanaka (2013); Ikegaya et al. (2015, 2016). The anomalous proximity effect is a result of the penetration of the ABSsBuchholtz and Zwicknagl (1981); Hara and Nagai (1986); Hu (1994); Tanaka and Kashiwaya (1995); Kashiwaya and Tanaka (2000) into the normal metal or equivalently the appearance of odd-frequency Cooper pairs in the normal metalTanaka and Golubov (2007); Tanaka et al. (2007a, b); Eschrig et al. (2007); Asano et al. (2007a, 2011); Asano and Tanaka (2013); Higashitani et al. (2013); Suzuki and Asano (2014, 2015); Ikegaya and Asano (2016); Suzuki and Asano (2016). Such unusual phenomena have attracted much attention these days because they are equivalent to the physics of Majorana Fermions appearing topologically nontrivial SCsRead and Green (2000); Ivanov (2001); Qi and Zhang (2011); L.Fu and Kane (2008); Tanaka et al. (2009); Fu and Kane (2009); Akhmerov et al. (2009); Sau et al. (2010); Tanaka et al. (2012); Mizushima et al. (2015); Sato and Fujimoto (2016); Mizushima et al. (2016); Sato and Ando (2017); van Weperen et al. (2015); Shabani et al. (2016); Lutchyn et al. ; Deacon et al. (2017); Gazibegovic et al. (2017); Gül et al. (2018); Sestoft et al. (2017); Chen et al. (2016); Suzuki et al. (2018); Olde Olthof et al. (2018). At present, however, we have never known how the anomalous proximity effect modifies the long-range phase-coherent phenomena.

In this paper, we study the local density of state (LDOS) in a wire of a diffusive normal metal (DN) by solving numerically the quasiclassical Usadel equation in the regime of the strong proximity effect. We consider two types of proximity structures: T-shaped junction shown in Fig. 1(a) and Volkov-Takayanagi (VT) junction shown in Fig. 1(b). We found in the T-shaped junction that the quasiparticle density of states depends strongly on the barrier potential at the junction interface. In the VT junction, the LDOS between the two superconducting electrodes depends sensitively on the phase difference of the two superconducting electrode. In an in-phase junction, the LDOS in DN between the $s$ -wave ($p$ -wave) superconducting electrodes shows the zero-energy dip (peak), whereas such dip and peak structures vanish in an out-of-phase junction because of the destructive interference of Cooper pairs. In an $s$-wave junction, the phase-coherent effect is spatially limited by a decay length due to depairing of Cooper pairs. In a $p$-wave junction, in addition to the depairing effect, the low transparency at the junction interface limits the long-range phase-coherent effect as well.

This paper is organized as following. In Sec. II, the Keldysh-Usadel formalism and the system we consider are explained. In Sec. III, we discuss the calculated LDOS for the T-shaped junction. In Sec. IV, we shown the LDOS in the Volkov-Takayanagi junction and discuss the long-range coherence. In particular, we focus on the junction-length and depairing-ratio dependences of the LDOS. We summarize this study in Sec. V.

II Keldysh-Usadel formalism

II.1 Usadel equation

In this paper, we consider the junctions of a diffusive normal metal (DN) where superconducting (S) wires are attached to a side surface of the DN as shown in Fig. 1. We refer to the junction shown in Figs. 1(a) and 1(b) as T-shaped and Volkov-Takayanagi (VT) junctions, respectively. In the T-shaped junction, a narrow S wire is attached to a wire of the DN at $|x|<w/2$ and $y=0$ with a finite interface resistance $R_{b}$, where $w$ is the width of the S arm which is much shorter than the superconducting coherence length in the diffusive system $\xi_{0}=\sqrt{\mathscr{D}/2\pi T_{c}}$. In the VT junction, narrow S wires are attached at $|x\mp L_{1}|<w/2$. The DN is connected to lead wires of clean normal metal at $x=\pm L$, but sufficiently narrow and thin in the $y$ and $z$ directions (i.e., $L_{y(z)}\ll\xi_{0}$).

The Green’s function in the DN obeys the Usadel equationUsadel (1970):

[TABLE]

where $\mathscr{D}$ is the diffusion constant in the DN, $\check{g}^{X}$ with $X=K$, $R$, and $A$ are the Keyldysh, retarded, and advanced components of the Usadel Green’s function, and $\mathbb{H}=\mathrm{diag}[\check{H}^{R},\check{H}^{A}]$. Assuming the width of the DN is much narrower than $\xi_{0}$, we can ignore the spatial variation of the Green’s function in the $y$ direction in the DN. Namely, one need to consider a one-dimensional diffusive system where the Usadel equation is reduced to

[TABLE]

where the last term $\mathbb{S}(x,\varepsilon)$ represents effects of the S wires. The source term $\mathbb{S}(x,\varepsilon)$ is reduced from the boundary condition in the $y$ directionVolkov and Takayanagi (1996, 1997). The step-like function is unity only at the place where the S wires are attached: $\Theta_{S}(x)=\Theta(w/2-|x|)$ for the T-shaped junction and $\Theta_{S}(x)=\Theta(w/2-|x-L_{1}|)+\Theta(w/2-|x+L_{1}|)$ for the VT junction. In this paper, the symbols written in a bold mean matrices in the Keldysh space, and the accents $\check{\cdot}$ and $\hat{\cdot}$ means matrices in particle-hole space and spin space. The identity matrices in particle-hole and spin space are respectively denoted by $\check{\tau}_{0}$ and $\hat{\sigma}_{0}$. The Pauli matrices are denoted by $\check{\tau}_{j}$ and $\hat{\sigma}_{j}$ with $j\in[1,3]$. The Keldysh-Usadel equation is supplemented by the so-called normalization condition: $\mathbb{G}\mathbb{G}=\mathbbm{1}$. The Keldysh Green’s function can be obtained from the following relation:

[TABLE]

where $f_{L}=\tanh(\varepsilon/2T)$ and $f_{T}=f_{T}(x,\varepsilon)$ describes the derivation from equilibrium.

The LDOS is related to the retarded and advanced components of the Usadel Green’s function. The Usadel equation for $X=R$ and $A$ in one dimension is given by

[TABLE]

where $\check{H}^{X}=\tilde{\varepsilon}^{X}\check{\tau}_{3}$. The factor $\bar{\varepsilon}^{X}$ depends on $X$: $\bar{\varepsilon}^{R}=\varepsilon+i\gamma$ and $\bar{\varepsilon}^{A}=\varepsilon-i\gamma$, where $\varepsilon$ and $\gamma$ being the energy and the depairing ratio due to inelastic scatterings. In this paper, we assumed that there is no spin-dependent potential, that the Cooper pairs has one single spin component (i.e., $\hat{\Delta}=\Delta_{\mu}i\hat{\sigma}_{\mu}\hat{\sigma}_{2}$ with $\Delta_{\mu}$ being the scalar pair potential), and the phase difference between two SCs is $\delta\Phi=0$ or $\pi$ (i.e., no-current states). In this case, one can parametrize the matrix structure of the Green’s functions as follows:

[TABLE]

where $\mu$ is related to the direction of the synthetic spin of Cooper pairs: $\mu=0$ and $\mu=1$-$3$ correspond to the spin-singlet and spin-triplet parings. The Usadel equation can be simplified by this parametrization:

[TABLE]

where we have introduced the symbol $\tilde{\cdot}$ meaning a $2\times 2$ matrix in spin-reduced particle-hole space [e.g., $\check{g}^{X}(x,\varepsilon)=\tilde{g}^{X}(x,\varepsilon)\otimes\hat{\sigma}_{0}$]. Here we assumed the phase difference between two SCs is [math] or $\pi$ which simplifies the relation between $f^{X}$ and $\undertilde{f}^{X}$ as discussed in Appendix.

The standard angular parametrization makes the Usadel equation much simplerVolkov et al. (1993); Golubov and Kupriyanov (1988); Golubov et al. (1997). The Green’s function can be well parametrized by the following parameterization:

[TABLE]

where we omit the index $X$ from $\theta=\theta^{X}(x,\varepsilon)$. This parametrization always satisfies the normalization condition: $\tilde{g}^{X}\tilde{g}^{X}=\tilde{\tau}_{0}$. The Usadel equation is reduced by this parametrization:

[TABLE]

II.2 Effects of superconducting terminals

The last term in the left hand side of Eq. (22) [i.e., $S(x,\varepsilon)$] represents the effect of the S arms attached to the side surface of the DN Volkov et al. (1993); Golubov et al. (1997). The typical boundary conditionsKuprianov and Lukichev (1988) are no longer available for junctions of unconventional SCs. In order to discuss the proximity effect by unconventional pairings, one must employ the so-called Tanaka-Nazarov conditionTanaka and Kashiwaya (2004); Tanaka et al. (2005a), which is an extension of the circuit theoryNazarov (1994, 1999). The source term ${S}$ is derived from the boundary condition in the $y$ direction. We employ the Tanaka-Nazarov boundary condition discussed in Refs. [Tanaka et al., 2003, 2004; Tanaka and Kashiwaya, 2004; Tanaka et al., 2005a]:

[TABLE]

where $\gamma_{B}=R_{b}/R_{N}\xi_{0}$ is the barrier parameter with $R_{b}$ and $R_{N}$ being the interface resistance per unit area and the specific resistance of the DN, $T_{N}(\phi)=\cos^{2}\phi/(\cos^{2}\phi+z_{0}^{2})$ is the transmission coefficient of an N/N interface with a barrier potential $\hbar v_{F}z_{0}$, $\phi$ is the angle of the momentum measured from the $k_{y}$ axis, and $\theta_{0}(x)=\theta(x)|_{y=0}$. The angle $\phi$ is measured from the $y$-axis. The angular bracket means angle average: \langle\cdots\rangle_{\phi}\equiv\big{(}\int_{-\pi/2}^{\pi/2}\cdots\cos\phi~{}d\phi\big{)}\big{(}\int_{-\pi/2}^{\pi/2}T_{N}\cos\phi~{}d\phi\big{)}^{-1}. The functions $g_{S}$ and $f_{S}$ can be obtained from the Green’s functions in a homogeneous ballistic superconductor:

[TABLE]

where $\Delta_{+}(\phi)=\Delta_{-}(\pi-\phi)$. The pair potential depends on the pairing symmetry of the superconductor:

[TABLE]

where $\Delta_{0}\in\mathbb{R}$ is the amplitude of the pair potential in a homogeneous superconductor and $\alpha_{\phi}$ parameterizes the direction of the anisotropic superconductor. The boundary condition (23) is transformed into the source term in the present case. The source term is given by

[TABLE]

The diffusivity changes the symmetry of Cooper pairs because only the isotropic $s$ -wave pairs can survive in diffusive systems. In the present case, the symmetry of S wires determines The symmetry of the Cooper pairs induced in the DN. In the $s$ -wave junction, spin-singlet $s$-wave Cooper pairs are induced, whereas spin-triplet $s$ -wave Cooper pairs are induced in the $p$ -wave junctionTanaka and Golubov (2007); Tanaka et al. (2007a). In order to satisfy the Fermi-Dirac statistics, the spin-triplet Cooper pairs must belong to the odd-frequency pairing symmetryBerezinskii (1974).

II.3 Boundary conditions

The Usadel equation (22) is supplemented by the boundary conditions. The boundary conditions for the T-shaped junction and the VT junction without a phase difference are given by

[TABLE]

The boundary conditions for the VT junction with the $\pi$-phase difference is given by

[TABLE]

The details are written in Appendix.

The LDOS $\nu(x,\varepsilon)$ can be obtained from the Green’s function:

[TABLE]

where $\nu_{0}$ is the density of states per spin at the Fermi level in the normal states. In proximity structures, it is convenient to introduce the deviation of the LDOS:

[TABLE]

We solve numerically Eq. (22) using the so-called “forward elimination, backward substitution method”.

III T-shaped junctions

We first discuss the roles of the important interface parameters (i.e., $z_{0}$ and $\gamma_{B}$) in a junction where a SC is attached to a side surface of the DN. The deviation of the LDOS $\delta\nu(x,\varepsilon)$, which is given in Eq. (38), in the T-shaped junction with an $s$ -wave SC are shown in Fig. 2. The deviation $\delta\nu$ is obtained at $x=0$ (beneath the S wire), $2\xi_{0}$, $4\xi_{0}$, $6\xi_{0}$, $8\xi_{0}$. The length of the DN and the width of the S arm is set to $L=10\xi_{0}$ and $w=0.3\xi_{0}$, respectively. The barrier parameter is set to $\gamma_{B}=1$ in (a), (b), and (c), $\gamma_{B}=3.33$ in (d). The interface-potential parameter is set to $z_{0}=0.1$ in (a), $1.0$ in (b) and (d), and $3.0$ in (c).

In an $s$ -wave junction, the coherence peak appears beneath the S arm at the energy $\varepsilon\sim\Delta_{0}$ because of the proximity effectGolubov et al. (1997). Simultaneously, at the low energy, an energy dip appears reflecting the energy gap in the S arm 111We have confirmed that the anomalous Green’s function becomes pure imaginary, meaning that the conventional even-frequency Cooper pairs are injected from the SC to the diffusive normal metal.. The peak height and dip depth monotonically decrease with increasing the distance from the S terminal.

Comparing Figs. 2(a), 2(b), and 2(c), we can see that the coherence peak around $\varepsilon=\Delta_{0}$ becomes sharper and higher as $z_{0}$ increases. On the other hand, the dip width in the energy and real space does not strongly depends on $z_{0}$. The dip width and depth are mainly determined by the spacing between normal lead wires (i.e., $2L$). We have confirmed that the low-energy dip becomes narrower with increasing system sizeGolubov et al. (1997). Comparing Figs. 2(d) with 2(b), we can see that the amplitude of $\delta\nu$ becomes smaller with increasing the interface resistance (i.e., increasing of $\gamma_{B}$).

Contrary to the $s$ -wave case, in the T-shaped junction with a $p$ -wave SC, the so-called zero-energy peak appears due to the anomalous proximity effect by odd-frequency spin-triplet $s$ -wave Cooper pairs Tanaka and Kashiwaya (2004); Tanaka et al. (2005a); Tanaka and Golubov (2007); Asano et al. (2007b) where topologically protected zero-energy states penetrate into the DN Ikegaya et al. (2015, 2016). 222We have confirmed that the anomalous Green’s function becomes a real function, meaning that the odd-frequency Cooper pairs are induced. Differing from the $d$-wave case (not shown), the zero energy peak can survive in a $p$ -wave junction even in a diffusive system reflecting the orbital symmetry of odd-frequency pairing Tanaka and Golubov (2007) and the topological nature of a $p$ -wave SC Asano and Sasaki (2015); Suzuki and Asano (2015); Ikegaya et al. (2016); Suzuki and Asano (2016). The peak becomes higher but narrower in energy space with increasing $z_{0}$. The peak width in real space, on the other hand, does not strongly depend on the $z_{0}$. As happened in the $s$ -wave junctions, $\gamma_{B}$ changes basically only the amplitude of the deviation $|\delta\nu|$. The coherence peak around $\varepsilon=\Delta_{0}$ does not appear in the $p$ -wave case. The zero-energy anomaly in $p$ -wave T-shaped junctions can be observed by the charge transport measurementsAsano et al. (2007b).

IV Volkov-Takayanagi junctions

IV.1 quasiparticle spectrum

In a two-superconductor system such as Josephson junctions, the phase difference between the two S wires affects significantly on the quasiparticle spectrum in the junction. The LDOS in the VT junction with $s$ -wave SCs are shown in Fig. 4(a) and 4(b), where the phase difference is set to $\delta\Phi=0$ and $\pi$, respectively. The parameters are set to $L=6\xi_{0}$, $L_{1}=5\xi_{0}$, $w=0.3\xi_{0}$, $\gamma=0.01\Delta_{0}$, $\gamma_{B}=1$, and $z_{0}=1$. When there is no phase difference, there is an energy dip whose size is about $0.2\Delta_{0}$ at the zero energy. This energy dip spreads between the S arms even though the spacing between the two arms is set to $2L_{1}=10\xi_{0}$.

When the phase difference is $\delta\Phi=\pi$ the LDOS at the center of the junction becomes completely flat as shown in Fig. 4(b). In addition, even at intermediate points, the kink around $0.2\Delta_{0}$, which exists when $\delta\Phi=0$, vanishes and $\delta\nu$ is more insensitive to $\varepsilon$. As a result, the energy dip is no longer prominent in Fig. 4(b). These behavior can be interpreted in terms of the destructive interference of Cooper pairs injected from the S arms. The phase of the anomalous Green’s function describing the Cooper pairs is related to the sign of the pair potential. In the $\delta\Phi=\pi$ junction, the Cooper pairs from each arm have an opposite phase. In other words, the pair amplitude of Cooper pairs cancel perfectly each other at the center of a junction. As a consequence, the LDOS at the center becomes completely flat. Reflecting this behavior, the Green’s function has an additional symmetry in real space $f^{X}(x,\varepsilon)=-f^{X}(x,\varepsilon)$ (See Appendix for details).

The LDOS in the $p$ -wave VT junction are shown in Fig. 5(a) and 5(b), where the phase difference is set to $\delta\Phi=0$ and $\pi$, respectively. When $\delta\Phi=0$, the zero-energy peak spreads between the two S wires (i.e., $|x|\leq L_{1}$). The peak is the highest beneath the S wires and the lowest at the center of the junction. The low-energy dip at the center of the junction is more prominent than that beneath the S wire. The dip width at $x=0$ is about $0.2\Delta_{0}$ which is comparable with that for the $s$-wave case shown in Fig. 4(a). When $\delta\Phi=\pi$, as happened in the $s$-wave VT junction, the LDOS is completely flat at $x=0$. Moreover, the height of the zero energy peak is lower than the $\delta\Phi=0$ case due to the destructive interference of the Cooper pairs injected from each SC.

Differing from the typical $p$ -wave Josephson junctionAsano et al. (2006b), in the VT junction, the most constructive and destructive interferences occur when $\delta\Phi=0$ and $\delta\Phi=\pi$, respectively. As shown in Fig. 1(b), the S wires are attached to the side surface which is normal to the $y$ axis. On the contrary, in the typical Josephson junction, $p_{x}$ -wave SCs attached in the $x$ direction. In the $p$ -wave VT junction without a phase difference, the anomalous Green’s functions injected from both of the S wires have the same sign. When the phase difference is $\pi$, however, Cooper pairs from each S wire have opposite phase, which leads the destructive interference.

IV.2 Junction-length dependence

The coherence is diminished with increasing the junction length. The junction-length dependence of the LDOS at the center of the VT junction with the $s$ - and $p$ -wave SCs are plotted in Figs. 6(a) and 6(b), respectively. In the calculations, we set the phase difference $\delta\Phi=0$, $z_{0}=1$, and $L-L_{1}=\xi_{0}$. In the $s$ -wave VT junction, the LDOS shows a dip structure at low energy even in a sufficiently long junction. This energy dip becomes wider with decreasing the junction length. The height of the coherence peak around $\varepsilon=\Delta_{0}$ strongly depends on the junction length. With decreasing the junction length, $\delta\nu|_{\varepsilon=\Delta}$ is almost the unity for $L>4\xi_{0}$, is negative for $L=4\xi_{0}$, and becomes positive for $L=2\xi_{0}$. In the short-junction limit, $\delta\nu$ becomes qualitatively the same as that in the T-shaped junction.

The coherence in a $p$ -wave junction modifies the LDOS as happened in the $s$ -wave case. As shown in Fig. 6(b), the zero-energy peak and the energy dip can be seen even when $L=10\xi_{0}$. The width of the zero-energy peak in energy space decreases monotonically with increasing the junction length. The peak height at $x=0$ and $\varepsilon=0$ decreases monotonically with increasing the junction length.

The junction-length dependence of the correction at $\varepsilon=0$ and $x=0$ (i.e., $\delta\nu|_{x=\varepsilon=0}$) in the $s$ -wave VT junction is shown in Fig. 7(a), where the barrier parameter at the interface is set to $z_{0}=0.1$, $1.0$, and $3.0$, and the depairing ratio is set to $\gamma=0.01\Delta_{0}$. The amplitude of the correction $|\delta\nu|$ decreases with increasing the junction length where the curvature of $|\delta\nu|$ as a function of $L$ is positive. We have confirmed that the curvature changes at a certain length. In the long-junction limit (i.e., $L_{1}\gg\xi_{0}$), $\delta\nu|_{x=\varepsilon=0}$ approaches to $\nu_{0}$ (i.e., normal state) where the VT junction can be regarded as a pair of two T-shaped junctions. In the $p$ -wave junction, the amplitude of the correction $|\delta\nu|$ decreases with increasing $L_{1}$ as seen in the $s$ -wave case. However, contrary to the $s$ -wave case, the degree of correction decreases with increasing $L$ more rapidly when the magnitude of $z_{0}$ is large. This behavior is unique to the spin-triplet $p$ -wave junction.

The junction-length dependences with a larger depairing ratio $\gamma=0.1\Delta_{0}$ are shown in Figs. 7(c) and 7(d). Both of the $s$ - and $p$ -wave cases, the amplitudes of $\delta\nu$ are smaller and decrease more rapidly compared with the results for $\gamma=0.01\Delta_{0}$. When $L=20\xi_{0}$, the correction $\delta\nu$ is almost zero in all of the cases. Therefore, the decay length for $|\delta\nu(x,\varepsilon=0)|$ in the strong-proximity-effect regime would be mainly determined by $\sqrt{\mathscr{D}/\gamma}$, which is consistent with the $s$ -wave results with weak-proximity effectVolkov and Takayanagi (1996).

IV.3 Depairing-ratio dependence

In real samples, the deparing effects such as inelastic scatterings are inevitably present. We lastly discuss the $\gamma$ dependence of $\delta\nu$. The $\gamma$ dependence of $\delta\nu|_{x=\varepsilon=0}$ for $s$ - and $p$ -wave junctions are shown in Figs. 8(a) and 8(b), respectively. The junction length and the interface barrier are fixed at $L/\xi_{0}=4$ or $8$ and $z_{0}=0.1$ or $1.0$. The corrections for $L/\xi_{0}=4$ and $8$ approach to a certain value even though the distance between the two S electrodes are different. We therefore can conclude that the decay length of $\delta\nu$ in the VT junction is determined by $\gamma$. This behavior is consistent with that demonstrated within the weak-proximity-effect approximationVolkov and Takayanagi (1996, 1997). In the $s$ -wave case, the slopes of $\delta\nu|_{x=0,\varepsilon=0}$ curves do not strongly depends on $z_{0}$.

As shown in Fig. 8(b), the decay length of $\delta\nu$ is determined by $\gamma$ in the $p$ -wave junction as well. The corrections at $\gamma=0.001\Delta_{0}$ is almost independent of the junction length, meaning which the decay length for $p$ -wave junction is determined by the depairing ratio $\gamma$ as well. Contrary to the $s$ -wave case, however, the slopes for the $p$ -wave junctions strongly depends on $z_{0}$.

We show the $\gamma$ dependence of $\overline{\delta\nu}(\gamma)$ in Fig. 9, where $\overline{\delta\nu}(\gamma)$ is a function of $\gamma$ normalised by its value at $\gamma=0.001\Delta_{0}$;

[TABLE]

We compare the following four cases: the $p$ -wave junctions with $z_{0}=0.1$, $1.0$, and $3.0$ and the $s$ -wave junction with $z_{0}=0.1$. Figure 9 clearly shows that the decay length for the $p$ -wave junction strongly depends on $z_{0}$. The $p$ -wave result with $z_{0}=0.1$ and the $s$ -wave results with $z_{0}=0.1$ are not qualitatively different. Therefore, we conclude that the decay length for the $p$ -wave junction depends on the amplitude of Cooper pairs injected by the proximity effect.

Differing from the N/DN/$p$ -wave junctionTanaka et al. (2005a) where the zero-energy LDOS at the DN/$p$ -wave interface diverges as $\propto 1/\sqrt{\gamma}$, the zero-energy correction $\delta\nu(x,\varepsilon=0)$ does not diverge even when $\gamma\to 0$ everywhere in the DN because our system is essentially different from the system where a $p$ -wave SC is used as an electrodeTanaka et al. (2005a, b).

V Conclusion

We have theoretically studied the quasiparticle spectrum in a junction of a diffusive normal metal where superconductors are attached to its side surface. We have considered two types of junctions: the T-shaped junction where one superconductor is attached to the diffusive normal metal and the Volkov-Takayanagi junction where two superconductors are attached to it. In the T-shaped junction, when the superconductor is spin-singlet $s$-wave, the local density of states which can be measured by scanning tunneling spectroscopy (STS) measurements has a dip structure which is consistent with the standard proximity effect. On the other hand, in the spin-triplet $p$-wave case, there is a zero-energy peak in the local density of states due to the anomalous proximity effect by odd-frequency pairing. The amplitude of the correction in the local density of states is strongly depends on the interface barrier. In the $p$ -wave case, in particular, the larger barrier results in the larger density of states at the zero energy.

In the Volkov-Takayanagi junction, the phase difference between the two superconductors affects significantly on the energy spectrum. In the $s$ -wave junction without a phase difference, the low-energy dip appears at the center of the junction. On the contrary, when the phase difference is $\pi$, such a low-energy dip vanishes and the local density of state at the center becomes one in the normal state because of the destructive interference of Cooper pairs. When spin-triplet $p$ -wave superconductors are employed instead of spin-singlet $s$ -wave superconductors, the zero-energy resonant states appear. When there is no phase difference, the zero-energy peak spreads spatially between the two superconductors, whereas the peak vanishes at the center of the junction when the phases differ by $\pi$.

We have also studied the characteristic length scale of the phase coherence. We have shown that, in both of the $s$ -wave and $p$ -wave cases, the decay length of the zero-energy state is mainly characterized by the depairing ratio $\gamma$ by, for example, inelastic scatterings. We have demonstrated that the decay length is not simply determined by $\gamma$ for spin-triplet $p$ -wave junctions. The decay length for a $p$ -wave junction depends also on the quality of the interface because the strength of the resonance depends strongly on the interface barrier potential.

Acknowledgements.

The authors would like to thank T. Yokoyama and S. Tamura for useful discussions. This work was supported by Grants-in-Aid from JSPS for Scientific Research on Innovative Areas “Topological Materials Science” (KAKENHI Grant Numbers JP15H05851, JP15H05852, JP15H05853 and JP15K21717), Scientific Research (B) (KAKENHI Grant Number JP18H01176), Japan-RFBR Bilateral Joint Research Projects/Seminars number 19-52-50026, JSPS Core-to-Core Program (A. Advanced Research Networks). A. A. G. acknowledges supports by the European Union H2020-WIDESPREAD-05-2017-Twinning project “SPINTECH” under grant agreement Nr. 810144.

Appendix A Additional symmetry of the Usadel equation

In the quasiclassical formalism, the anomalous Green’s functions $f$ and $\undertilde{f}$ are related by several symmetry relations. In a diffusive system (i.e., Usadel formalism), the Green’s functions can have additional symmetry compared with the ballistic case.

A.1 General symmetry

The Usadel equation for the retarded and advanced component is given by

[TABLE]

where $\check{g}^{X}_{o}$ with $X=R$ $(A)$ means retarded (advanced) Green’s function. Assuming the single-component pair potential (i.e., either of the even-frequency spin-singlet or odd-frequency spin-triplet SCs), the matrix $\check{H}$ becomes

[TABLE]

where $\Delta_{\mu}(\boldsymbol{r})\in\mathbb{C}$ is the scalar pair potential with $\mu\in[1,3]$. The factor $s_{\varepsilon}$, which is defined as $s_{\varepsilon}=+1$ $(-1)$ for even-frequency (odd-frequency) SCs, stems from the frequency symmetry of the pair potential. We have used $s_{\varepsilon}s_{\mu}=-1$ (i.e., Pauli rule) and $\hat{\sigma}_{\mu}^{*}\hat{\sigma}_{2}=-s_{\mu}\hat{\sigma}_{2}\hat{\sigma}_{\mu}$. In this case, it is convenient to parametrise the spin structure of the Green’s function as following:

[TABLE]

We can simplify the Usadel equation by the unitary transform. We first define the unitary matrix: $[\check{U}_{1}]^{-1}=\mathrm{diag}\left[\hat{\sigma}_{0},-i\hat{\sigma}_{2}\hat{\sigma}_{\mu}\right]$. Multiplying $\check{U}_{1}$ and $\check{U}_{1}^{-1}$ from the left and right side of the Usadel equation (40), we have the simplified Usadel equation:

[TABLE]

where we redefine the Greens functions and the matrix $\check{H}^{X}$ as following: $\check{g}^{X}{(\boldsymbol{r},\varepsilon)}=\check{U}_{1}\check{g}^{X}_{0}{(\boldsymbol{r},\varepsilon)}\check{U}_{1}^{-1}$ and $\check{H}^{X}{(\boldsymbol{r},\varepsilon)}=\check{U}_{1}\check{H}^{X}_{0}{(\boldsymbol{r},\varepsilon)}\check{U}_{1}^{-1}$, and the subscript $\mu$ is omitted.

The matrix $\check{H}^{X}(x,\varepsilon)$ satisfies several symmetric relations. Hereafter, we consider the one-dimensional system. Using the Pauli matrices in the particle-hole space, we can express the matrix $\check{H}^{X}(x,\varepsilon)$ with a simpler form:

[TABLE]

where $\Delta_{\mathrm{R}(\mathrm{I})}\in\mathbb{R}$ is the real (imaginary) part of the pair potential. The first symmetry is given by

[TABLE]

where we have used $\varepsilon^{R}=[\varepsilon^{A}]^{*}$. The relations above connect the retarded and advanced Green’s functions. The second symmetry is given by

[TABLE]

The third symmetry is given by

[TABLE]

where $\phi(x)$ is the local phase defined as $\phi=\mathrm{arctan}\left(\Delta_{\mathrm{I}}/\Delta_{\mathrm{R}}\right)$. We can reduce the following relations from Eq. (70):

[TABLE]

When the pair potential is a real function, we can parametrise the Green’s function as

[TABLE]

A.2 Symmetry in Josephson(-ish) junctions

In Josephson(-ish) junctions, the Green’s functions have additional symmetry. In this paper, we refer to the junctions in which the relation $\phi(x)=-\phi(x)$ is satisfied as the Josephson-ish junctions (e.g., Volkov-Takayanagi junctions). In other words, the real and imaginary parts of the pair potential are even and odd function of $x$:

[TABLE]

In this case, the matrix $\check{H}^{X}(x,\varepsilon)$ and the Green’s functions satisfy the symmetry relations related to the real space:

[TABLE]

Combining Eqs. (74) and (82), we have

[TABLE]

In particular, the relation above can further be reduced when the phase difference is either $\delta\Phi=0$ or $\pi$:

[TABLE]