Current-phase relation and flux-dependent thermoelectricity in Andreev interferometers
We predict a novel -junction state of multi-terminal Andreev interferometers that emerges from an interplay between long-range quantum coherence and non-equilibrium effects. Under non-zero bias the current-phase relation resembles that of a -junction differing from the latter due to a non-zero average . The flux-dependent thermopower of the system exhibits features similar to those of a -junction and in certain limits it can reduce to either odd or even function of in the agreement with a number of experimental observations.
Multi-terminal heterostructures composed of interconnected superconducting (S) and normal (N) terminals (frequently called Andreev interferometers) are known to exhibit non-trivial behavior provided the quasiparticle distribution function inside the system is driven out of equilibrium. For instance, it was demonstrated both theoretically V (); WSZ (); Yip () and experimentally Teun () that biasing two N-terminals in a four-terminal NS configuration by an external voltage one can control both the magnitude and the phase dependence of the supercurrent flowing between two S-terminals and – in particular – provide switching between zero- and -junction states at certain values of . In other words, a -junction state in SNS structures can be induced simply by driving electrons in the N-metal out of equilibrium.
Another way to generate non-equilibrium electron states in Andreev interferometers is to expose the system to a temperature gradient. As a result, an electric current (and/or voltage) response occurs in the system which is the essence of the thermoelectric effect Gi (). Usually the magnitude of this effect in both normal metals and superconductors is small in the ratio between temperature and the Fermi energy , however, it can increase dramatically in the presence of electron-hole asymmetry. The symmetry between electrons and holes in superconducting structures can be lifted for a number of reasons, such as, e.g., spin-dependent electron scattering (for instance, at magnetic impurities Kalenkov12 (), spin-active interfaces KZ15 () or superconductor-ferromagnet boundaries Beckmann ()) or Andreev reflection at different NS-interfaces in an SNS structure with a non-zero phase difference between two superconductors V2 (); KZ17 () (see also VH ()). The latter mechanism could be responsible for large thermoelectric signal observed in various types of Andreev interferometers Venkat1 (); Venkat2 (); Petrashov03 (); Venkat3 (); Petrashov16 ().
Yet another important feature of some of the above observations is that the detected thermopower was found to oscillate as a function of the applied magnetic flux with the period equal to the flux quantum , thus indicating that the thermoelectric effect essentially depends on the phase of electrons in the interferometer. The symmetry of such thermopower oscillations was observed to be either odd or even in depending on the sample topology Venkat1 (). Also, with increasing bias voltage these oscillations were found to vanish and then re-appear at yet higher voltages with the phase shifted by Petrashov03 (). Despite subsequent attempts to attribute the results Venkat1 () to charge imbalance effects Titov2008 () or mesoscopic fluctuations JW () no unified and consistent explanation for the observations Venkat1 (); Venkat2 (); Petrashov03 (); Venkat3 (); Petrashov16 () has been offered so far.
In this paper we address the properties of SNS junctions embedded in multi-terminal configurations with both bias voltage and thermal gradient applied to different normal terminals. For the configuration depicted in Fig. 1 we will demonstrate that at low enough temperatures and with no thermal gradient the corresponding SNS structure exhibits characteristic features of what we will denote as -junction state: The current flowing through the superconducting contour of our setup (as shown in Fig. 1) is predicted to have the form
where and is a -periodic function of the superconducting phase difference across our SNS junction. At zero bias both and vanish and the term reduces to the equilibrium supercurrent in diffusive SNS structures ZZh (); GreKa (). At low enough the contribution essentially coincides with the voltage-controlled Josephson current WSZ () (with jumping from 0 to with increasing ), while at higher voltages with a good accuracy we have with non-zero phase shift which tends to in the limit of large . This behavior resembles that of an equilibrium -junction which develops nonvanishing supercurrent at . In contrast to the latter situation, however, here we drive electrons out of equilibrium, thereby generating extra current along with the phase shift . Remarkably, also a thermoelectric signal does not vanish at for non-zero , as it will be demonstrated below.
The article is organized as follows. In Section II we briefly describe the quasiclassical Green function formalism employed in our further analysis. The general current-phase relation for our Andreev interferometer summarized in Eq. (1) is derived and analyzed in Section III. In Section IV we elaborate on the implications of this relation for the flux-dependent thermopower in multi-terminal Andreev intererometers thereby proposing an interpretation for long-standing experimental puzzles Venkat1 (); Petrashov03 (). We close with a brief summary of our key observations in Section V.
Ii Quasiclassical formalism
In what follows we will employ the quasiclassical Usadel equations which can be written in the form belzig1999quasiclassical ()
Here matrix represent the Green function in the Keldysh-Nambu space
is the diffusion constant, is the electric potential, is the quasiparticle energy, is the superconducting order parameter equal to in the first (second) S-terminal and to zero otherwise. The retarded, advanced, and Keldysh components of the matrix are matrices in the Nambu space
where is the distribution function matrix and is the Pauli matrix. The current density is related to the matrix by means of the formula
where is the Drude conductivity of a normal metal.
which enter the kinetic equations as
Equation (5) for the current density can then be cast to the form
Analogously one can define the heat current density
Eqs. (2) should be supplemented by proper boundary conditions. Here we only address the limit of transparent interfaces and continuously match the normal wires Green functions to those in the normal terminals
and in the superconducting ones
The spectral currents obey the Kirchhoff-like equations in all nodes of our structure.
We first consider a symmetric four-terminal setup of Fig. 1 with wire lengths , equal cross sections and voltagesFN () . The spectral part of the Usadel equation (2) is solved numerically in a straightforward manner (cf, e.g., Ref. WSZ, ). This solution enables us to find the retarded and advanced Green functions and to evaluate the spectral current (6) as well as the kinetic coefficients (7)-(9). In order to resolve the kinetic equations and to determine the current-phase relation for our setup we will adopt the following strategy. We first obtain a simple approximate analytic solution and then verify it by a rigorous numerical analysis.
Let us for a moment assume that the phase difference is small as compared to unity and relax this assumption in the very end of our calculation. In this case one can proceed perturbatively and resolve the kinetic equations in the first order in . Within the same accuracy, one can drop the small terms and neglect the energy dependence of . With the aid of Eq. (12) we arrive at the expressions for the spectral currents flowing in the superconducting (normal) contours of our circuit FNC (), see Fig. 1. We obtain
where we defined
and the spectral resistances (which reduce to that for a normal wire of length in the normal state with ). The distribution functions are given by Eq. (15) with and . Integrating Eqs. (18) and (19) over energy we obtain approximate expressions for the currents and .
In addition to the above perturbative analysis we carried out a rigorous numerical calculation of both and involving no approximations. In the low temperature limit the corresponding results are displayed in Figs. 2 and 3 along with approximate results derived from Eqs. (18) and (19) in the same limit. It is satisfactory to observe that our simple perturbative procedure yields very accurate result for the current not only for small phases but for all values of , see Fig. 2. This current is an even -periodic function of and . Likewise, for the system under consideration we have .
Below in this section we will mainly concentrate on the phase dependence of the current . Fig. 3 demonstrates that – in the agreement with our expectations – our simple analytic result for derived from Eq. (18) is quantitatively accurate at sufficiently small phase values or, more generally, at all phases in the vicinity of the points . Moreover, even away from these points Eq. (18) remains qualitatively correct capturing all essential features obtained within our rigorous numerical analysis. These considerations yield Eq. (1) which represents the first key result of our work.
It is instructive to analyze the above expressions in more details. The first term in the right-hand side of Eq. (18) is a familiar one. In equilibrium it accounts for dc Josephson current ZZh (); GreKa (), while at non-zero bias and in the limit (in which the last term in Eq. (18) vanishes) it reduces to the results WSZ (); Yip () demonstrating voltage-controlled transitions in SNS junctions. In contrast, the last term in Eq. (18) is a new one being responsible for both and parts. This term is controlled by the combination , where is an even function of . Hence, the net current is no longer an odd function of .
The physics behind this result is transparent. In the presence of a non-zero bias a dissipative current component, which we will further label as , is induced in the normal wire segments and . At NS interfaces this current gets converted into extra (-dependent) supercurrent flowing across a superconducting loop. Since at low temperatures and energies electrons in normal wires attached to a superconductor remain coherent keeping information about the phase , dissipative currents in such wires also become phase (or flux) dependent demonstrating even in Aharonov-Bohm-like (AB) oscillations nakano1991quasiparticle (); SN96 (); GWZ97 (); Grenoble (), i.e. , where . Combining this contribution to the current with an (odd in ) Josephson current we immediately arrive at Eq. (1) with .
The behavior of the phase shift displayed in the inset of Fig. 3 is the result of a trade-off between Josephson and Aharonov-Bohm contributions to . At low bias voltages dominates over , and we have . Increasing the bias to values , in full agreement with previous results WSZ () we observe the transition to the -junction state implying the sigh change of . Here we defined the Thouless energy , where is the total length of three wire segments between two S-terminals (see Fig. 1). At even higher bias voltages both terms and eventually become of the same order. For and at we have WSZ () , where for our geometry
We also approximate FN1 () , where and is the normal resistance of the wire with length . Hence, for we obtain
The function (restricted to the interval ) shows damped oscillations and saturates to the value in the limit of large , as it is also illustrated in the inset of Fig. 3.
At higher the Josephson current decays exponentially with increasing whereas the Aharonov-Bohm term shows a much weaker power-law dependence GWZ97 (); Grenoble () , thus dominating the expression for and implying that at such values of .
For completeness, we point out that a -junction state is also realized in a cross-like geometry with provided we set and (see, e.g., Fig. 4 below). Under these conditions the distribution function at the wire crossing point differs from zero resulting in a non-vanishing even in contribution to containing . However, if either or this even in contribution vanishes and we get back to the results WSZ (); Yip () describing 0- and -junction states.
Iv Flux-dependent thermopower
We now turn to the thermoelectric effect. It was argued V2 (); KZ17 (); VH () that in Andreev interferometers this effect may become large provided the phase difference between superconducting electrodes differs from . Below we will demonstrate that a large thermopower can be induced by a temperature gradient even if .
To this end let us somewhat modify the setup in Fig. 1 by setting and attaching two extra normal terminals N and N as shown in Fig. 4. These terminals are disconnected from the external circuit and are maintained at different temperatures and , while the temperature of the remaining four terminals equals to .
We first set and evaluate the thermoelectric voltage between N and N induced by a thermal gradient . For simplicity, below we consider the configuration with . As no current can flow into the terminals N and N, we obtain
where is the spectral conductance. Eq. (22) defines the relation between and the induced voltages . In the first order in it yields the thermoelectric voltage in the form
Here is the induced electric potential of the terminals N and N evaluated at . For any nonzero bias the voltage differs from zero as long as . In this case the thermovoltage (23) also remains nonzero as the spectral conductance explicitly depends on energy due to the superconducting proximity effect. On the other hand, in the absence of superconductivity the latter dependence disappears and the expression (23) vanishes identically even for nonzero . This observation emphasizes a non-trivial interplay between superconductivity, quantum coherence and thermoelectricity in hybrid metallic nanostructures.
In order to recover the phase dependence of the thermoelectric voltage we treated the problem numerically. The corresponding results are displayed in Figs. 4 and 5. Fig. 4 demonstrates the temperature dependence of the thermopower at different values of . In Fig. 5 we present the thermopower as a function of at together with the current-phase relation evaluated for the same setup. We observe that both functions and demonstrate essentially the same behavior and, hence, in complete analogy with Eq. (1) we have
where and is a -periodic function of , which at high enough voltages only slightly deviates from a simple form (cf. Fig. 5).
Eqs. (23), (24) represent the second key result of our work. It allows to conclude that in general the periodic dependence of the thermopower on the magnetic flux in Andreev interferometers is neither even nor odd in , but it can reduce to either one of them depending on the system topology or, more specifically, on the relation between , and the relevant Thouless energy . The phase shift in Eq. (24) is not strictly identical to in Eq. (1) FN2 (), however, both these functions behave similarly. In fact, only slightly deviates from (cf., e.g, Fig. 5). With increasing , the phase also experiences an abrupt transition from 0 to and then tends to in the limit of large voltages and/or temperatures.
Our findings allow to naturally interpret the experimental results Venkat1 () where both odd and even dependencies of on were detected depending on the system topology. Indeed, while at small enough and we have and remains an odd function, at larger voltages and/or temperatures the phase shift approaches and the flux dependence of the thermopower (24) turns even, just as it was observed for some of the structures Venkat1 (). Furthermore, as we already discussed, with increasing bias the phase jumps from 0 to which is fully consistent with the observations Petrashov03 (). Thus, we believe the transition for the flux-dependent thermopower detected in experiments Petrashov03 () has the same physical origin as that predicted V (); WSZ (); Yip () and observed Teun () earlier for dc Josephson current.
In this work we have elucidated a non-trivial interplay between proximity-induced quantum coherence and non-equilibrium effects in multi-terminal hybrid normal-superconducting nanostructures. We have demonstrated that applying an external bias one drives the system to a -junction state in Eq. (1) determined by a trade-off between non-equilibrium Josephson and Aharonov-Bohm-like contributions. We have also analyzed the phase-coherent thermopower in such nanostructures which exhibits periodic dependence on the magnetic flux being in general neither even nor odd in . Our results allow to formulate a clear physical picture explaining a number of existing experimental observations and calling for further experimental analysis of the issue.
We would like to thank A.G. Semenov for fruitful discussions. This work is a part of joint Russian-Greek Projects No. RFMEFI61717X0001 and No. T4P-00031 Experimental and theoretical studies of physical properties of low-dimensional quantum nanoelectronic systems. One of us (P.E.D.) also acknowledges support by Skoltech as a part of Skoltech NGP program and the hospitality of KIT during November 2017.
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