Phase coherent electron transport in asymmetric cross-like Andreev interferometers

TL;DR

This paper provides a theoretical analysis of quantum coherent electron transport in asymmetric cross-like Andreev interferometers, highlighting how geometric and electron-hole asymmetries influence phase-dependent currents and their controllability.

Contribution

It reveals the crucial role of asymmetries in enabling Aharonov-Bohm-like effects and phase-coherent currents in these interferometers, advancing understanding of their transport properties.

Findings

Asymmetries induce Aharonov-Bohm-like contributions to supercurrent.

Phase-coherent currents depend on voltage, temperature, and topology.

Electron-hole asymmetry causes odd-in-phase current components.

Abstract

We present a detailed theoretical description of quantum coherent electron transport in voltage-biased cross-like Andreev interferometers. Making use of the charge conjugation symmetry encoded in the quasiclassical formalism, we elucidate a crucial role played by geometric and electron-hole asymmetries in these structures. We argue that a non-vanishing Aharonov-Bohm-like contribution to the current $I_S$ flowing in the superconducting contour may develop only in geometrically asymmetric interferometers making their behavior qualitatively different from that of symmetric devices. The current $I_N$ in the normal contour -- along with $I_S$ -- is found to be sensitive to phase-coherent effects thereby also acquiring a $2\pi$-periodic dependence on the Josephson phase. In asymmetric structures this current develops an odd-in-phase contribution originating from electron-hole asymmetry. We demonstrate that both phase dependent currents $I_S$ and $I_N$ can be controlled and manipulated by tuning the applied voltage, temperature and system topology, thus rendering Andreev interferometers particularly important for future applications in modern electronics.

Equations

$$\check{G}(ϵ,\mathbf{r})=\left(\begin{array}{cc}{\hat{G}}^R& {\hat{G}}^K\\ \hat{0}& {\hat{G}}^A\end{array}\right).$$

$$D\nabla \left(\check{G}\nabla \check{G}\right)=\left[-iϵ{\hat{\tau }}_z,\check{G}\right],$$

$${\hat{G}}^R=\frac{1}{1+\gamma \tilde{\gamma }}\left(\begin{array}{cc}1-\gamma \tilde{\gamma }& 2\gamma \\ 2\tilde{\gamma }& \gamma \tilde{\gamma }-1\end{array}\right).$$

$$\Delta \gamma -\frac{2\tilde{\gamma }}{1+\gamma \tilde{\gamma }}(\nabla \gamma {)}^2+2iϵ\gamma =0,$$

$$\Delta \tilde{\gamma }-\frac{2\gamma }{1+\gamma \tilde{\gamma }}(\nabla \tilde{\gamma }{)}^2+2iϵ\tilde{\gamma }=0.$$

$${\hat{G}}^K={\hat{G}}^R\hat{F}-\hat{F}{\hat{G}}^A,\hat{F}=f_L+{\hat{\tau }}_z{f}_T,$$

$$\nabla j_L$$

$$\nabla j_T$$

$$j_s=\frac{1}{4}\mathrm{tr}{\hat{\tau }}_z\left({\hat{G}}^R\nabla {\hat{G}}^R-{\hat{G}}^A\nabla {\hat{G}}^A\right)$$

$$D_L$$

$$D_T$$

$$\mathcal{Y}$$

$$j=-\frac{{\sigma }_N}{2e}\int j_T(ϵ)dϵ,$$

$$f_{L/T}^N$$

$$\displaystyle L\check{G}\partial_{x}\check{G}=\pm\frac{1}{2r}\Big{[}\check{G}_{\rm SC},\check{G}\Big{]},$$

$${\hat{G}}_{\mathrm{SC}}^R=\frac{\left(\begin{array}{cc}ϵ& \Delta e^{i\chi }\\ -\Delta e^{-i\chi }& -ϵ\end{array}\right)}{\sqrt{(ϵ+i\delta {)}^2-{\Delta }^2}},$$

$$\displaystyle\hat{G}^{K}_{\rm SC}=\tanh\frac{\epsilon}{2T}\Big{(}\hat{G}^{R}_{\rm SC}-\hat{G}^{A}_{\rm SC}\Big{)}$$

$$r=\frac{\mathcal{A}{\sigma }_N}{L\mathcal{G}},$$

$${\check{G}}_c(ϵ,\mathbf{r})=-{\hat{\tau }}_1\check{G}(ϵ,\mathbf{r}){\hat{\tau }}_1.$$

$$I_i(-\varphi ,-V_1,-V_2)=-I_i(\varphi ,V_1,V_2),$$

$$I_{N1}(\varphi ,V_1,V_2)=I_{N2}(\varphi ,V_1,V_2)=I_N,$$

$$I_{S1}(\varphi ,V_1,V_2)=I_{S2}(\varphi ,V_1,V_2)=I_S,$$

$$I_S(-\varphi )=-I_S(\varphi ),$$

$$I_N(-\varphi )=I_N(\varphi ).$$

$${\gamma }_i^{\prime \prime }+{\lambda }^2{\gamma }_i=0,$$

$${\gamma }_{S1}(0)={\gamma }_{S2}(0)={\gamma }_{N1}(0)={\gamma }_{N2}(0)={\gamma }_0,$$

$${\mathcal{A}}_{S1}{\gamma }_{S1}^{\prime }(0)+{\mathcal{A}}_{N2}{\gamma }_{N2}^{\prime }(0)={\mathcal{A}}_{S2}{\gamma }_{S2}^{\prime }(0)+{\mathcal{A}}_{N1}{\gamma }_{N1}^{\prime }(0),$$

$${\gamma }_{N1}(l_{N1})={\gamma }_{N2}(-l_{N2})=0,$$

$$L{\gamma }_{S1}^{\prime }(-l_{S1})=-\frac{{\mathcal{F}}_L}{2r_L},L{\gamma }_{S2}^{\prime }(l_{S2})=\frac{{\mathcal{F}}_R}{2r_R}.$$

$${\mathcal{F}}_R=\frac{-i\Delta e^{i{\varphi }_R}}{\sqrt{{\Delta }^2-(ϵ+i\delta {)}^2}},{\mathcal{F}}_L=\frac{-i\Delta e^{i{\varphi }_L}}{\sqrt{{\Delta }^2-(ϵ+i\delta {)}^2}}.$$

$${\gamma }_{N1}={\gamma }_0\frac{\sin \lambda (l_{N1}-x)}{\sin \lambda l_{N1}},{\gamma }_{N2}={\gamma }_0\frac{\sin \lambda (l_{N2}+x)}{\sin \lambda l_{N2}},$$

$${\gamma }_{S1}={\gamma }_0\frac{\cos \lambda (l_{S1}+x)}{\cos \lambda l_{S1}}-\frac{{\mathcal{F}}_L}{2L\lambda r_L}\frac{\sin \lambda x}{\cos \lambda l_{S1}},$$

$${\gamma }_{S2}={\gamma }_0\frac{\cos \lambda (l_{S2}-x)}{\cos \lambda l_{S2}}+\frac{{\mathcal{F}}_R}{2L\lambda r_R}\frac{\sin \lambda x}{\cos \lambda l_{S2}},$$

$${\gamma }_0$$

$$\mathcal{N}$$

$$j_{s,S1}{\mathcal{A}}_{S1}=j_{s,S2}{\mathcal{A}}_{S2}=\frac{{\mathcal{A}}_{S1}{\mathcal{A}}_{S2}}{L^2{r}_R{r}_L}\sin \varphi \times$$

Full text

I Introduction

An interplay between quantum coherence and non-equilibrium phenomena is an intriguing topic in condensed matter physics. Hybrid metallic heterostructures composed of superconducting (S) and normal (N) terminals constitute an important playground to realize and investigate rich physics associated with the above phenomena. In these systems – frequently called Andreev interferometers – long-range quantum coherence is induced due to the superconducting proximity effect, while non-equilibrium conditions can be created by virtue of biasing different terminals with external voltages and/or temperature gradients Belzig et al. (1999); Fornieri and Giazotto (2017); Giazotto et al. (2006). Distinctive electrical and thermal properties of such systems – including, e.g., large phase-dependent thermoelectric effects Eom et al. (1998); Dikin et al. (2001); Parsons et al. (2003); Cadden-Zimansky et al. (2007); Shelly et al. (2016), conductance re-entrance Stoof and Nazarov (1996); Golubov et al. (1997), Aharonov-Bohm-like behavior of SN-rings Stoof and Nazarov (1996); Golubov et al. (1997); Nakano and Takayanagi (1991); Petrashov et al. (1995); Courtois et al. (1996) and non-local (or crossed) Andreev reflection Byers and Flatté (1995); Deutscher and Feinberg (2000); Beckmann et al. (2004); Russo et al. (2005); Cadden-Zimansky and Chandrasekhar (2006); Golubev and Zaikin (2007); Kalenkov and Zaikin (2007); Golubev et al. (2009) – render them a promising platform for modern electronics and caloritronics.

Yet another remarkable effect is the so-called $\pi$-junction state that can occur in systems with two normal and two superconducting terminals interconnected by normal metallic wires forming a cross. Applying a phase twist $\phi$ to two superconducting terminals of this cross-like Andreev interferometer one induces dc Josephson current $I_{S}(\phi)$ between these terminals just like in usual SNS junctions ZZh ; Dubos et al. (2001); Golubov et al. (2004). Simultaneously biasing two normal terminals with an external voltage $V$ one can modify the electron distribution function in the system, and thereby control the magnitude of $I_{S}\equiv I_{S}(\phi,V)$. At some values of $V$ the supercurrent flowing between S-terminals becomes negative signaling the $\pi$-junction state Volkov (1995); Wilhelm et al. (1998); Yip (1998); Baselmans et al. (1999).

Recently three of us demonstrated Dolgirev et al. (2018); Dolgirev et al. (2019a) that the above scenario – being appropriate for symmetric cross-like Andreev interferometers – becomes by far incomplete as soon as the system topology is made asymmetric. It turns out that in the latter situation the underlying physics becomes much richer being essentially determined by a competition between voltage-dependent (odd in $\phi$) Josephson and (even in $\phi$) Aharonov-Bohm-like currents flowing in the superconducting contour. This trade-off may have a drastic impact on the current-phase relation $I_{S}(\phi)$ in voltage-biased Andreev interferometers resulting in a novel $(I_{0},\phi_{0})$-junction state Dolgirev et al. (2018), predicted to occur at low $T$ and high enough $eV$ exceeding an effective Thouless energy of our device. This state is characterized by coherent $2\pi$-periodic oscillations of $I_{S}$ as a function of $\phi$ shifted from the origin by the phase $\phi_{0}(V)$ that can take any value, thus being in general different from zero or $\pi$.

It should be emphasized that asymmetric topology of cross-like Andreev interferometers plays a crucial role for this effect: With the aid of simple charge-conjugation symmetry arguments to be outlined below one can demonstrate that by making the interferometer in Fig. 1 symmetric in at least one of the two contours (either normal or superconducting) one totally suppresses the Aharonov-Bohm contribution to $I_{S}$, hence, getting back to the physical picture Volkov (1995); Wilhelm et al. (1998); Yip (1998); Baselmans et al. (1999) with $\phi_{0}(V)=0,\pi$.

Here we will argue that the physics of asymmetric cross-like interferometers is actually even richer than that discussed in our previous studies Dolgirev et al. (2018); Dolgirev et al. (2019a). In particular, it turns out that the current $I_{N}$ flowing between the two normal terminals of such interferometer – similarly to $I_{S}$ – exhibits proximity induced coherent $2\pi$-periodic oscillations as a function of the superconducting phase difference $\phi$. The function $I_{N}(\phi)$ is in general neither even nor odd, i.e. it consists of both even and odd in $\phi$ harmonics. While the first of these contributions ($\propto\cos\phi$) to the current $I_{N}$ can be interpreted in terms of the Aharonov-Bohm-like effect, the second one ($\propto\sin\phi$) is much more tricky as it obviously cannot have anything to do with the Josephson current. Below we will demonstrate that the physical origin of the latter contribution to $I_{N}$ is directly related to electron-hole asymmetry generated due to the mechanism of sequential Andreev reflections at different NS-interfaces Kalenkov and Zaikin (2017).

The structure of the paper is as follows. In Sec. II, we introduce the system under consideration, describe the quasiclassical formalism used throughout our paper and elucidate the charge-conjugation symmetry properties of this formalism important for our further considerations. In Sec. III we focus our analysis on the limit of highly resistive NS interfaces, in which case it is possible to obtain a full analytic solution of our problem. Our key results and the corresponding discussion are formulated in Sec. IV. In Sec. V, we briefly summarize our findings. Further technical details are relegated to appendices.

II The model and basic formalism

Below we will consider cross-like Andreev interferometers schematically depicted in Fig. 1. The system consists of two normal-metal diffusive wires of total lengths $l_{N1}+l_{N2}$ and $l_{S1}+l_{S2}=L$ connected between each other in the form of a cross, and attached respectively to two normal and two superconducting terminals. We will address a general case of asymmetric Andreev interferometers with $l_{N1}\neq l_{N2}$ and $l_{S1}\neq l_{S2}$ which demonstrate a variety of quantum coherent effects some of which do not occur in symmetric configurations. Electrostatic potentials of both S-terminals are set equal to zero $V=0$, while the potentials of the normal terminals are denoted as $V_{1}$ and $V_{2}$. These N-terminals are biased by an external voltage $V$ implying $V_{2}=V_{1}+V$. The superconducting order parameter of the left and right S-terminals is chosen to be respectively $\Delta\exp(i\phi_{L})$ and $\Delta\exp(i\phi_{R})$. The value of the phase difference between these terminals $\phi=\phi_{L}-\phi_{R}$ can easily be controlled by an external magnetic flux inserted inside a superconducting loop.

Obviously, electric current can flow between S-terminals (superconducting contour) as well as between N-terminals (normal contour) being dependent on external bias $V$, temperature $T$ and phase difference $\phi$. The task at hand is to determine the distribution of voltages and electric currents in our structure in the presence of long-range quantum coherent effects, and to demonstrate the importance of geometric and electron-hole asymmetries in our problem.

II.1 Quasiclassical formalism

We will adopt the standard quasiclassical formalism Belzig et al. (1999) aimed at describing non-equilibrium quantum properties of hybrid metallic structures like the one in Fig. 1. The quasiclassical Green’s functions in each metallic wire are represented with the aid of a $4\times 4$-matrices in the Keldysh-Nambu space composed of retarded ($\hat{G}^{R}$), advanced ($\hat{G}^{A}$) and Keldysh ($\hat{G}^{K}$) functions

[TABLE]

This matrix Green’s function obeys the normalization condition $\check{G}^{2}=\check{1}$ and satisfies the Usadel equation

[TABLE]

where $D$ stands for a diffusion coefficient and $\hat{\tau}_{z}$ is the Pauli matrix in the Nambu space.

In what follows it will be convenient for us to employ the so-called Riccati parameterization Schopohl and Maki (1995); Schopohl (1998). For the retarded Green’s function it reads

[TABLE]

A similar representation holds for the advanced Green’s function since $\hat{G}^{A}=-\hat{\tau}_{z}(\hat{G}^{R})^{+}\hat{\tau}_{z}$. The spectral part of the Usadel equation then becomes

[TABLE]

With the aid of the standard representation for the Keldysh Green’s function

[TABLE]

the kinetic part of the Usadel equation can be cast to the form

[TABLE]

Here $j_{T}=\operatorname{tr}(\check{G}\nabla\check{G}\hat{\tau}_{z})^{K}$ and $j_{L}=\operatorname{tr}(\check{G}\nabla\check{G})^{K}$ represent the spectral densities of respectively electric and thermal currents, $f_{L}$ and $f_{T}$ are respectively symmetric and anisymmetric parts of the electron distribution function,

[TABLE]

stands for the supercurrent density, and the kinetic coefficients $D_{L}$, $D_{T}$ and ${\cal Y}$ are defined as

[TABLE]

Note that the kinetic coefficient (12) explicitly accounts for the presence of the electron-hole asymmetry in our system.

Resolving the Usadel equations one can evaluate the electric current density $j$ in our system defined as

[TABLE]

where $\sigma_{N}$ is the Drude conductivity of a normal metal.

II.2 Boundary conditions

As usually, the Usadel equation (2) should be supplemented by proper boundary conditions allowing to match the Green’s functions at all inter-metallic interfaces. Below we will assume that the central node – the contact between the two normal wires – is characterized by perfect transmission, meaning that the Green’s functions are continuous and that the spectral currents associated with them are conserved. The same applies to the boundaries with the N-terminals: The Green’s functions inside the normal-metal wire are continuously matched to the corresponding bulk values $\hat{G}^{R/A}_{N}=\pm\hat{\tau}_{z}$ and

[TABLE]

What remains is to define the boundary conditions at two NS interfaces. Here, we will restrict our analysis to the tunneling limit, i.e. we assume that the transmission of both NS interfaces is small compared to unity. This limit is accounted for by the well-known Kupriyanov-Lukichev (KL) boundary conditions Kuprianov and Lukichev (1988)

[TABLE]

where $\check{G}$ is the Green’s function in the normal wire, $\check{G}_{\rm SC}$ denotes the bulk Green’s function of the corresponding S-terminal with

[TABLE]

and phase $\chi$ equals to either $\phi_{L}$ or $\phi_{R}$ depending on the terminal. The parameter $r$ is defined as

[TABLE]

where $\mathcal{A}$ is the interface cross section and ${\mathcal{G}}$ is the normal-state conductance of the interface. Note that within the applicability range of KL boundary conditions (15) and depending on the relation between ${\mathcal{G}}$ and the conductance of the normal wire of length $L$, the parameter $r$ can in general take any value both smaller and larger than unity.

II.3 Symmetry considerations

Let us define charge-conjugated Green’s function as

[TABLE]

It is straightforward to verify that the function (19) represents a solution of the Usadel equation (2) with inverted signs of both electric and magnetic fields as well as of that of the superconducting phase. This symmetry has important consequences for the charge transport properties of the system under consideration.

Resolving the Usadel equation (2) we determine the charge currents in all four metallic wires as functions of the phase difference $\phi$ and the applied voltages $V_{1}$ and $V_{2}$. Making use of Eq. (19) one can demonstrate that all currents invert their signs under the transformation $V_{1}\leftrightarrow-V_{1}$, $V_{2}\leftrightarrow-V_{2}$, $\phi\leftrightarrow-\phi$, i.e. we have

[TABLE]

where the index $i$ labels the wires $N1$, $N2$, $S1$ and $S2$.

The electrostatic potentials $V_{1}$ and $V_{2}$ as functions of both the phase $\phi$ and the bias voltage $V$ are determined from the current conservation conditions

[TABLE]

combined with the condition $V_{2}-V_{1}=V$. Likewise, the currents $I_{S1}$, $I_{S2}$, $I_{N1}$, $I_{N2}$ can also be expressed as functions of $\phi$ and $V$.

In general all these currents are $2\pi$-periodic functions of $\phi$. Extra geometric symmetries of our structure may enforce higher symmetries for the above currents rendering them, e.g., either purely even or purely odd functions of $\phi$. In particular, it is instructive to distinguish two special cases: (i) symmetric connectors to S-terminals (implying that $l_{S1}=l_{S2}$ and $r_{L}=r_{R}$) and (ii) symmetric connectors to N-terminals ($l_{N1}=l_{N2}$). It follows immediately (see also Appendix A for more details) that in both cases (i) and (ii) the current $I_{S}$ turns out to be an odd function of $\phi$, i.e.

[TABLE]

whereas the current $I_{N}$ is even in $\phi$,

[TABLE]

Hence, for partially symmetric cross-like Andreev interferometers (in both cases (i) and (ii)) the Aharonov-Bohm-like contribution to the current $I_{S}$ vanishes and we are back to the situation of only 0- or $\pi$-junction states considered in Refs. Volkov (1995); Wilhelm et al. (1998); Yip (1998). On top of that, no odd-in-$\phi$ contribution to $I_{N}$ can occur in such structures.

In what follows we will, therefore, address the most general case of fully asymmetric interferometers with $l_{N1}\neq l_{N2}$ and $l_{S1}\neq l_{S2}$.

III Highly resistive interfaces: analytic solution

Let us now employ the above equations and evaluate the Green’s functions for the structure depicted in Fig. 1. As usually, one can split the problem into spectral, Eqs. (4–5), and kinetic, Eqs. (7–8), parts which can be treated separately. Below in this section, we will stick to the limit of sufficiently large values of the parameter $r$ at both NS interfaces and construct a full analytic solution of the problem.

III.1 Spectral part

Let us assume that tunnel barriers at both NS interfaces are sufficiently large and, hence, anomalous correlations penetrating into the normal-metal wires from the superconducting terminals are strongly suppressed. In this case, one can linearize the spectral part of the Usadel equation and get

[TABLE]

where $\lambda^{2}=2i\epsilon/(\mathcal{E}_{\rm Th}L^{2})$ and $\mathcal{E}_{\rm Th}\equiv D/L^{2}$ (with $L=l_{\rm S1}+l_{\rm S2}$) is an effective Thouless energy of our setup. The same equation also holds for $\tilde{\gamma}$. Here and below, we also assume $\mathcal{E}_{\rm Th}\ll\Delta$ enabling us to restrict our analysis to subgap energies $|\epsilon|<\Delta$.

The boundary conditions take the form

[TABLE]

Equations in the first two lines follow directly from the continuity of $\gamma$ and from the spectral current conservation at the central node (with coordinate set equal to zero). The equation in the third line implies that anomalous correlations vanish at the boundaries with both N-terminals. Finally, the two equations in the last line just represent KL boundary conditions at the left and right NS interfaces characterized by parameters $r_{L}$ and $r_{R}$ respectively (defined in Eq. (18) with ${\mathcal{G}}\equiv{\mathcal{G}}_{L/R}$). We also choose

[TABLE]

Resolving the linearized Usadel equations with the above boundary conditions, we obtain

[TABLE]

where

[TABLE]

Then, for the spectral supercurrent density, one readily finds

[TABLE]

The above analytic solution of the spectral part of the problem enables one to easily derive the applicability condition for the linearized Usadel equation (25). Setting functions $\gamma$ to be much smaller than unity within the normal wires and making use of Eqs. (26)–(30), we arrive at the following conditions

[TABLE]

Note that depending on the system parameters these conditions may substantially deviate from simple inequalities $r_{R,L}\gg 1$ which one could naively expect to be sufficient in order to linearize the Usadel equations.

III.2 Kinetic part

Below, we proceed similarly to the above subsection, and resolve the kinetic equations perturbatively by formally expanding them in $1/(r_{i}r_{j})$, where $i,j=L,R$. In the zeroth order, we have

[TABLE]

with the boundary conditions

[TABLE]

Equations in the first line account for boundaries with both N-terminals, the second line represents KL boundary conditions at both NS interfaces, whereas the last equation just reflects both electric and energy currents conservation and, hence, it remains valid to all orders. In fact, the condition $j_{L,\rm S1/S2}=0$ is also valid to all orders at energies $|\epsilon|<\Delta$, since subgap excitations do not contribute to the energy current flowing into the S-terminals.

We observe that – to the zeroth order – functions $f^{(0)}_{L}$ and $f^{(0)}_{T}$ depend linearly on the coordinate along the wire in the normal contour, whereas in the wire that belongs to the superconducting contour these functions remain constant equal to

[TABLE]

Here, $R_{\mathrm{N1}}$ and $R_{\mathrm{N2}}$ are normal-state Drude resistances of the normal wires connected to terminals $N_{1}$ and $N_{2}$

[TABLE]

Note that with the aid of the above zeroth order solution combined with KL boundary conditions one can establish the spectral electric current in the superconducting contour to the next order in parameter $\sim 1/(r_{L}r_{R})$. Indeed, the latter conditions can be written in the form $j^{(1)}_{T}=\pm\alpha_{T}/(Lr_{L/R})$ with

[TABLE]

With this in mind, the electric current conservation condition yields

[TABLE]

which after some algebra can further be cast to the form

[TABLE]

This equation together with the condition $V_{2}-V_{1}=V$ defines electrostatic potentials of both normal terminals $V_{1}$ and $V_{2}$ demonstrating that these potentials depend not only on $V$ and $T$, but also on phase difference $\phi$ between the superconducting terminals. The latter dependence clearly illustrates the importance of long-range proximity induced quantum coherence effects spreading not only into the superconducting contour but also into the normal contour, thereby influencing the potentials of both normal terminals. It follows from Eq. (38) that both electrostatic potentials $V_{1}$ and $V_{2}$ depend on $\cos\phi$, thus being even functions of $\phi$.

The above perturbative analysis of the kinetic equations can be justified if the interface resistances are much larger than the resistances of the corresponding attached normal wires

[TABLE]

Having determined $V_{1}$ and $V_{2}$, we are ready to find the electric current in the superconducting contour. It reads

[TABLE]

where

[TABLE]

The first term in the right-hand side of Eq. (41) represents the Josephson contribution, the second term (proportional to both $1/{(r_{L}r_{R})}$ and $\cos\phi$) defines the coherent Aharonov-Bohm-like current, while the last term has to do with the Andreev conductance of SN interfaces.

As already mentioned above, Eq. (38) contains only terms depending on $\cos\phi$, while the Josephson contribution proportional to $\sin\phi$ drops out from this equation. In other words, the terms entering the electric current conservation condition, cf. Eq. (37), represent the combination of $\cos\phi$-dependent (Aharonov-Bohm) and $\phi$-independent (Andreev) contributions. This observation appears to be specific to the chosen cross-like geometry (as suggested, e.g., by Eqs. (31) and (34)) and, furthermore, it only holds in the leading order in $1/(r_{i}r_{j})$. A more detailed numerical analysis indicates that for smaller values of $r_{L,R}$ the $\sin\phi$-harmonic is present and might even play an important role. We also note that Eq. (38) and Eq. (41) can be combined in a way that allows to expel an explicit dependence on $\cos\phi$ from the expression for $I_{S}$. In this case, the even in $\phi$ contribution to $I_{S}$ appears implicitly due to the dependencies of potentials $V_{1}$ and $V_{2}$ on $\phi$.

IV Results and discussion

Let us now explicitly evaluate the distribution of voltages and currents in our cross-like Andreev interferometer. We first determine electrostatic potentials of the two normal terminals, $V_{1}$ and $V_{2}$, and then evaluate the currents in both superconducting and normal contours which depend on these potentials.

IV.1 Electrostatic potentials

According to Eq. (38) the corresponding Aharonov-Bohm term is proportional to $\sim 1/(r_{L}r_{R})$, i.e. it has the same order as the other two terms $\sim 1/r_{L}^{2}$ and $\sim 1/r_{R}^{2}$. On the other hand, in the limit $l_{\mathrm{S1}},l_{\mathrm{S2}}\gg\sqrt{D/\Delta}$ considered here and at high enough voltages $\mathcal{E}_{\mathrm{Th}}\ll e|V_{1,2}|<|\Delta|$ the Aharonov-Bohm contribution becomes exponentially suppressed as $\propto e^{-\sqrt{e|V_{1,2}|/\mathcal{E}_{\mathrm{Th}}}}$ and, hence, it can be treated as a small perturbation. Then we obtain

[TABLE]

where

[TABLE]

and

[TABLE]

The presence of the $\phi$-dependent term in Eq. (42) indicates that electrostatic potentials $V_{1/2}$ are sensitive to proximity-induced long-range quantum coherence in our structure. The magnitude of this coherent contribution to $V_{1,2}$ is controlled by parameter $V_{\rm AB}$ defined in Eq. (44). Note that this approximate analytic expression for $V_{\rm AB}$ turns out to be very accurate, as it is demonstrated in Fig. 2.

IV.2 Current $I_{S}$ in the superconducting contour

Following our previous analysis Dolgirev et al. (2018); Dolgirev et al. (2019a) we can express the current $I_{S}$ in the superconducting contour in the form

[TABLE]

Here $I_{0}(V)$ defines a dissipative Andreev-like term entering Eq. (41), while $I_{J}$ and $I_{\rm AB}$ represent odd in $\phi$ Josephson and even in $\phi$ Aharonov-Bohm-like currents. It follows directly from Eq. (38) that the last (Aharonov-Bohm) term differs from zero only for asymmetric structures with both $l_{\rm S1}\neq l_{\rm S2}$ and $l_{\rm N1}\neq l_{\rm N2}$, in accordance with our general symmetry analysis in Sec IIC.

The results derived in the previous subsection imply that in the particular case of identical NS boundaries with $r_{L}=r_{R}=r\gg 1$ we have $I_{S}(r)\propto 1/r^{2}$. In this limit, the amplitudes of both odd and even in $\phi$ oscillations as functions of $V$ are shown in Fig. 3. We observe that $I_{J}(V)$ experiences zero-to-$\pi$-junction switching Volkov (1995); Wilhelm et al. (1998); Yip (1998); Baselmans et al. (1999) at around $eV\simeq 17{\cal E}_{\rm Th}$ and becomes exponentially suppressed at higher voltages, similarly to the case of fully transparent NS boundaries Dolgirev et al. (2018); Dolgirev et al. (2019a). Integrating the supercurrent density in Eq. (41) over energies, at sufficiently high bias voltages $eV\gg{\cal E}_{\rm Th}$ we get

[TABLE]

This formula turns out to be in a good agreement with the numerical solution of Eq. (38) at $eV\gtrsim 5{\cal E}_{\rm Th}$, cf. Fig. 3a.

Just like in the case of transparent SN-boundaries Dolgirev et al. (2018); Dolgirev et al. (2019a), here one could expect $I_{\rm AB}(r,V)$ to saturate to some non-zero value $I_{\rm AB}^{\infty}(r)$ at sufficiently high voltages $eV\gg{\cal E}_{\rm Th}$. In contrast to such expectations, in the limit $r\gg 1$ one finds $I_{\rm AB}^{\infty}=0$, cf. also Fig. 3b. The latter result applies in the leading order in $1/r^{2}$ and has the same origin as a similar behavior of the coherent contribution to $V_{1/2}$ at large voltages, cf. Fig. 2. Hence, one can expect that $I_{\rm AB}^{\infty}(r)\propto 1/r^{4}$ for $r\gg 1$.

For amplitude $I_{\rm AB}(V)$, within the voltage interval $\Delta\gg eV\gg{\cal E}_{\rm Th}$, one can derive an expression similar to the one in Eq. (46). Under the condition $l_{\rm S1},\,l_{\rm S2}\gg\sqrt{D/(eV)}$ we obtain

[TABLE]

Note that in the particular case $r_{R}=r_{L}=r$, the expression (47) vanishes identically implying that a more accurate treatment is required in this case. The corresponding analysis can be worked out and yields

[TABLE]

As far as the temperature dependence of $I_{S}$ is concerned, we point out that, while the Aharonov-Bohm-like contribution $I_{\rm AB}$ decays as a power law with increasing $T>eV$, the Josephson term $I_{J}$ decays exponentially, thus becoming negligible as compared to $I_{\rm AB}$ in the high-temperature limit $T\gg{\cal E}_{\rm Th}$. In the case of fully transparent NS-interfaces the temperature dependencies of both even and odd in $\phi$ components of $I_{S}$ have already been studied elsewhere Dolgirev et al. (2018); Dolgirev et al. (2019a), therefore we can avoid further details here.

IV.3 Current $I_{N}$ in the normal contour

Let us now turn to the electric current $I_{N}$ flowing between the two normal terminals. This current also demonstrates a $2\pi$-periodic dependence on phase $\phi$ and can be represented as a sum of even and odd in $\phi$ contributions:

[TABLE]

The first – even in $\phi$ – term again describes the Aharonov-Bohm-like contribution C (1) and is by no means surprising. At the same time, the presence of the odd periodic in $\phi$ contribution to current $I_{N}$ is curious. In contrast to the superconducting contour, here term $I^{\rm odd}_{N}(\phi,V)$ obviously cannot be attributed to the Josephson effect, and its physical nature requires further analysis.

For simplicity, let us assume that all cross sections are equal ${\cal A}_{i}={\cal A}$. Then, we obtain

[TABLE]

where $j^{(0)}_{L}=(f_{L}^{N}(V_{1})-f_{L}^{N}(V_{2}))/(l_{\rm N1}+l_{\rm N2})$ is the corresponding spectral current in the normal contour. Since the current $I^{\rm odd}_{N}$ in Eq. (50) is controlled by the kinetic coefficient ${\cal Y}$ we conclude that this current should be attributed to the electron-hole asymmetry in our system generated due to the phase-sensitive mechanism of sequential Andreev reflections at different NS interfaces Kalenkov and Zaikin (2017). This conclusion is further supported by observing that $I^{\rm odd}_{N}\propto\sin\phi/(r_{L}r_{R})$.

At sufficiently large voltages $e|V|\gg{\cal E}_{\rm Th}$ the integrals in Eq. (50) can be handled explicitly, and we get

[TABLE]

where $R_{\textrm{S1}}=l_{\textrm{S1}}/(\mathcal{A}_{\textrm{S1}}\sigma_{N})$ and $R_{\textrm{S2}}=l_{\textrm{S2}}/(\mathcal{A}_{\textrm{S2}}\sigma_{N})$ are the normal state Drude resistances of the wires connected to the superconducting terminals. We also note that the current in Eq. (51) vanishes identically for $l_{\rm S1}=l_{\rm S2}$ (and $r_{\rm S1}=r_{\rm S2}$) and/or $l_{\rm N1}=l_{\rm N2}$, in full agreement with our symmetry considerations in Sec. IIC.

It is also interesting to study current $I_{N}(\phi,V)$ at different values of $r$. This can be done by numerically solving the Usadel equation (2). In Fig. 4 we display the corresponding results for the amplitudes of both odd and even oscillations of the current $I_{N}$ as functions of $V$ at $T\mapsto 0$ for different values of $r$. We observe that the odd in $\phi$ harmonics persists at all values of $r$ becoming progressively more pronounced with decreasing $r$, cf. also the inset in Fig. 4 where we present the result obtained in the limit $r=0$. This amplitude first increases with increasing $V$ reaching its maximum at around $eV\simeq 10{\cal E}_{\rm Th}$ and then falls off being exponentially suppressed already at $eV\simeq 50{\cal E}_{\rm Th}$ in accordance with Eq. (51). Let us mention that in contrast to the current $I_{J}(V)$ which exhibits the transition to the $\pi$-junction state Dolgirev et al. (2018); Dolgirev et al. (2019a) at around $eV\gtrsim 15{\cal E}_{\rm Th}$ (cf. also Fig 3a), the amplitude of $I_{N}^{\rm odd}(V)$ demonstrates similar switching at much higher voltages $eV\gtrsim 60{\cal E}_{\rm Th}$. This behavior of $I_{N}^{\rm odd}(V)$ is related to the presence of an extra parameter $(l_{\rm S1}-l_{\rm S2})/L<1$ in the argument of the $\sin$-term, cf. Eq. (51).

The even in $\phi$ current amplitude $I_{N}^{\rm even}(V)$ saturates to a non-zero value at large voltages (see Fig. 4), just as one would expect for the Aharonov-Bohm-like contribution. Notably, the value of the plateau scales as $1/r^{2}$ for $r\gg 1$. This is in contrast to $I_{\rm AB}^{\infty}(r)$ which scales as $1/r^{4}$.

In order to complete our analysis of the current $I_{N}$, we note that with increasing temperature $T>eV$ the even in $\phi$ contribution to this current decays as a power law similarly to $I_{\rm AB}(T)$, which could serve as a signature of the Aharonov-Bohm-like effect. In contrast, the odd in $\phi$ contribution $I_{N}^{\rm odd}(T)$ behaves qualitatively similarly to the Josephson term $I_{J}(T)$ decaying much faster than $I_{N}^{\rm even}(T)$ and becoming invisibly small already at temperatures of order several ${\cal E}_{\rm Th}$.

V Conclusions

In this work we performed a detailed analysis of a non-trivial interplay between proximity induced long-range quantum coherence and non-equilibrium effects in cross-like Andreev interferometers as well as of its impact on electron transport properties of such devices.

We emphasized a crucial role of various symmetries in our problem. The charge conjugation symmetry encoded in the Usadel equations allowed us to establish an important general relation (20) which, in turn, helps to demonstrate that topology of cross-like Andreev interferometers is essential for determining charge transport properties of these devices.

We showed that in symmetric interferometers, the current $I_{S}$ in the superconducting contour is an odd function of the superconducting phase difference $\phi$. In other words, the even in $\phi$ Aharonov-Bohm-like contribution vanishes identically in such structures. These setups can only support the voltage-controlled Josephson current, and demonstrate switching between 0- and $\pi$-states depending on the applied voltage bias. In contrast, non-vanishing Aharonov-Bohm-like currents do survive in asymmetric structures. The physics of such devices is dominated by a trade-off between Josephson and Aharonov-Bohm-like quantum coherent contributions to the current $I_{S}$, leading to the $(I_{0},\phi_{0})$-junction state at sufficiently high bias voltages Dolgirev et al. (2018); Dolgirev et al. (2019a). Hence, the current-phase relation $I_{S}(\phi)$ can be manipulated by external voltage bias, temperature and topology of the setup.

The current $I_{N}(\phi)$ flowing in the normal contour is also $2\pi$-periodic function of the superconducting phase difference $\phi$, i.e. it is directly affected by the proximity induced long-range quantum coherence. With the aid of our symmetry arguments we demonstrated that in symmetric Andreev interferometers, the current $I_{N}(\phi,V)$ is an even function of $\phi$ associated with the Aharonov-Bohm-like contribution.

A non-trivial effect discovered here is that in asymmetric cross-like interferometers the current $I_{N}$ develops an odd harmonics $I^{\rm odd}_{N}$, cf. Eq. (50). The appearance of this contribution is particularly interesting because it can be attributed neither to the Aharonov-Bohm effect nor to the Josephson physics. In fact, our analysis demonstrates that the origin of the term $I^{\rm odd}_{N}$ is linked to violation of yet one more – electron-hole – symmetry that occurs under non-zero phase bias due to sequential Andreev reflections at different NS interfaces Kalenkov and Zaikin (2017). In the tunneling limit the magnitude of this effect is controlled by $\sin\phi$ thereby resulting in a “Josephson-like” contribution $I^{\rm odd}_{N}\propto\sin\phi$ to the current $I_{N}(\phi,V)$ between normal terminals. Similarly to $I_{S}(\phi)$, the current-phase relation $I_{N}(\phi)$ can also be manipulated by external voltage bias, temperature and topology of the interferometer.

Finally, we note that electron-hole symmetry violation is believed to also be responsible for large thermoelectric effects in Andreev interferometers Kalenkov and Zaikin (2017); Dolgirev et al. (2019b). Our work, therefore, establishes an intimate relation between the current $I_{N}$ and thermoelectricity in hybrid superconducting nanostructures KDZ .

To conclude, we developed a detailed theory of quantum coherent charge transport in phase-and-voltage-biased asymmetric cross-like Andreev interferometers. The electron currents in both superconducting and normal contours demonstrate the presence of both even and odd $2\pi$-periodic in $\phi$ contributions. We identified key physical mechanisms responsible for different contributions to these currents, and described their non-trivial behavior depending on the applied voltage, temperature and the system topology. Our findings allow for full control of the current pattern in biased Andreev interferometers, thus rendering them particularly promising for future applications in modern electronics.

Acknowledgements

The authors would like to thank A. Radkevich and A.G. Semenov for fruitful discussions. P.E.D. acknowledges the hospitality of Skoltech during the fall of 2018. P.E.D. and A.E.T. were supported by the Skoltech NGP program (Skoltech-MIT joint project). This work is partially supported by RFBR Grant No. 18-02-00586 for M.S.K. and A.D.Z.

Appendix A Symmetric Andreev interferometers

Let us focus our attention on the two special cases: (i) symmetric connectors to S-terminals with $l_{S1}=l_{S2}$ and $r_{L}=r_{R}$ and (ii) symmetric connectors to N-terminals with $l_{N1}=l_{N2}$.

In the case (i), we observe an extra symmetry related to the possibility of interchanging the terminals $S_{1}\leftrightarrow S_{2}$ with simultaneous inversion of the phase $\phi\rightarrow-\phi$ implying that all the functions $V_{1}(\phi,V)$, $V_{2}(\phi,V)$, $I_{N1}(\phi,V_{1},V_{2})$, $I_{N2}(\phi,V_{1},V_{2})$ are even in $\phi$, cf. Eq. (24). Besides that, we have

[TABLE]

Combining these equations with Eq. (22) we arrive at the relation (23).

In the case (ii), our system is symmetric with respect to interchanging the normal terminals $N_{1}\leftrightarrow N_{2}$. Then we have

[TABLE]

Let us define $\delta V=V_{1}+V/2=V_{2}-V/2$. By symmetry in the case (ii) the function $\delta V$ is even in $V$. It follows from Eq. (19) that for $\phi\rightarrow-\phi,\,V\rightarrow-V$ we get

[TABLE]

Here we also employed Eq. (54). Note that the currents $I_{S1},\,I_{S2}$ are even functions of $V$ and, hence, we obtain

[TABLE]

i.e. $\delta V$ turns out to be an odd function of the superconducting phase $\phi$.

Applying the relations (54), (59) and (19) to the current $I_{S1}$ we get

[TABLE]

implying that the current $I_{S}$ again turns out to be an odd function of the phase $\phi$, i.e. just like in the case (i) it obeys the relation (23).

Furthermore, making use of the relations (21), (56), (57), and (59) we recover the following properties of the current $I_{N1}$:

[TABLE]

implying that the current $I_{N}$ obeys the relation (24), thus being an even function of the phase $\phi$.

Obviously, the relations (23) and (24) are also obeyed in a special case of fully symmetric Andreev interferometers with $l_{S1}=l_{S2}$, $r_{L}=r_{R}$ and $l_{N1}=l_{N2}$. In this case we have $V_{2}=-V_{1}=V/2$, i.e. the potentials $V_{1}$ and $V_{2}$ do not depend on $\phi$.

Appendix B Transparent SN interfaces

While the main part of our paper is devoted to the tunneling limit described by KL boundary conditions (15), it is useful to extend our analysis to the case of fully transparent SN interfaces corresponding to the limit $r=0$. In particular, we performed a numerical analysis of the current $I_{N}$ (the current $I_{S}$ was investigated in Refs. Dolgirev et al. (2018); Dolgirev et al. (2019a)). The corresponding results are displayed in Fig. 4 at different values of $r$, including $r=0$ in the inset of Fig. 4.

In the case of fully transparent SN interfaces and in the limit of high voltages $\mathcal{E}_{\mathrm{Th}}\ll e|V_{1,2}|<\Delta$ one can also derive an explicit expression for the odd harmonic of the current $I_{N}$. Employing the approach developed in Ref. Dolgirev et al. (2019a) one can easily find the anomalous Green’s function in the normal wires connected to the normal terminals. We obtain

[TABLE]

where $x$ is the distance from the crossing point, and $F_{c}^{R}$ is the anomalous Green’s function evaluated at this crossing point

[TABLE]

Here we defined

[TABLE]

The function $\tilde{F}_{c}$ can be recovered from Eq. (63) by replacing $F_{c}$ by $-\tilde{F}_{c}$ and $\chi_{1,2}$ by $-\chi_{1,2}$. Then it is straightforward to derive the function $\mathcal{Y}$ and evaluate the integral in Eq. (50). In the case of equal cross sections we obtain

[TABLE]

As shown in the inset of Fig. 4, this analytic expression perfectly matches with our numerical result at $eV\gtrsim 20{\cal E}_{\rm Th}$. In addition, Eq. (65) can be used to estimate the maximum value of the current $I^{\rm odd}_{N}$.