Abstract
In spite of extended efforts, detecting thermoelectric effects in superconductors has proven to be a challenging task, due to the inherent superconducting particlehole symmetry. Here we present a theoretical study of an experimentally attainable Superconductor – Quantum Dot – Superconductor (SCQDSC) Josephson Junction. Using Keldysh Green’s functions we derive the exact thermophase and thermal response of the junction, and demonstrate that such a junction has highly tunable thermoelectric properties and a significant thermal response. The origin of these effects is the QD energy level placed between the SCs, which breaks particlehole symmetry in a gradual manner, allowing, in the presence of a temperature gradient, for gate controlled appearance of a superconducting thermophase. This thermophase increases up to a maximal value of ±π/2 after which thermovoltage is expected to develop. Our calculations are performed in realistic parameter regimes, and we suggest an experimental setup which could be used to verify our predictions.
Introduction
Thermoelectric (TE) effects correspond to the response of electrical charge (via induced current or voltage) when a thermal bias is applied across a junction. Since the warmer side has an equal excess of both particles and holes, the direction and magnitude of the TE response are determined by the asymmetry between particles and holes. Consequently, TE effects have proven to be a powerful tool in probing the density of states near the Fermi energy, particularly in materials with strong electronelectron interactions^{1,2,3}. However, in superconductors (SCs), which are a paradigmatic example of interacting electron systems, the TE response is both small in magnitude and hard to control. This is because SCs are inherently particlehole (ph) symmetric, and the ph asymmetry stems primarily from impurity scattering^{4,5,6}. Measuring a substantial and controllable TE response in SCs is therefore a major challenge.
Early experiments searching for thermocurrent in superconductors found that even the expected small thermocurrent was generally absent^{7}. An explanation for the absence of thermoelectric response was proposed by Ginzburg^{8}, who suggested, within the two fluid scheme, that the superfluid is expected, under certain conditions, to counterbalance the quasiparticle (QP) current with a nondissipative supercurrent^{9}. The existence of such a supercurrent is accompanied by an induced gradient in the phase of the SC order parameter^{10,11}.
To overcome the absence of current, experiments in which the setup comprises a bimetallic loop (taking advantage of the fact that the SC phase has to be geometrically quantized), were proposed and performed^{11,12}. However, different experiments^{13} disagreed with each other and with theory^{14,15}, a discrepancy which only recently may have been resolved^{16}. Suggestions for increasing the thermal response and ph asymmetry include using magnetic impurities^{17} or a ferromagnetic junction setup^{18}, leading to a thermophase of greater magnitude. However, using a magnetic field for tuning the system parameters^{19} leads to substantial experimental limitations.
In spite of all these efforts, the challenge of devising a SC system which exhibits substantial TE effects and with a large degree of control is yet to be met. Here, we demonstrate that in a SCquantum dot (QD)SC setup (schematically depicted in Fig. 1(a)), the TE response can be considerably larger than in SC tunnel junctions^{10}, where measurable thermophase can only arise around the transition temperature. Control over its magnitude can be achieved by a gate voltage, which shifts the energy levels of the QD, allowing for breaking of the ph symmetry even for ideal SC electrodes, thus enabling experimental control of the magnitude and direction of the thermal response. It is important to note, that such a setup is within current experimental capabilities^{20,21,22,23}, making our predictions experimentally verifiable.
Model
Our model consists of bulk stype superconductors as leads, with individual gap energies and arbitrary phases (taken symmetrically for convenience), and a single QD level in between. The Hamiltonian for the SCQDSC junction is given by H = H_{L} + H_{R} + H_{QD} + H_{V}, with the lead Hamiltonians H_{s} (s = L, R) given by
where is the creation (annihilation) of an electron on side s with momentum k, spin σ. The order parameter is complex, and the phase difference is taken, without loss of generality, as ϕ_{L} = −ϕ_{R} = ϕ/2. The chemical potential in the SC leads is defined as the zero of energy. We first start with a noninteracting, singlelevel QD. In this case, the QD Hamiltonian H_{QD} and the hopping Hamiltonian between the QD and the SCs, H_{V}, are
where is the creation (annihilation) of an electron on the dot with spin σ. While the calculation is quite general, in the present context we assume spin degeneracy, ε_{↑} = ε_{↓} ≡ ε, and uniform tunneling V_{ks} ≡ V_{s}. From this Hamiltonian, the currents and other quantities are calculated using the nonequilibrium Green’s function method, as described in the Methods section.
Results
We start by addressing the general form of the current. Substituting the Green’s function (Eq. 10 in the Methods section) into the expression for the current, we find that the current can be generally divided into three terms: quasiparticle current, I_{QP}, Josephson (pair) current, I_{sc}, and a term involving pairQP transition, I_{pair−QP},
A temperature dependence exists in all the terms through the temperature dependence of the superconducting order parameter. The usefulness of this form for the current stems from the fact that within the relevant parameter range discussed here, the phase dependence of the amplitude of the various current terms is negligible. This phase dependence originates from multiple reflections between the QD and the leads, giving rise to higher harmonic processes with a non trivial phase function. These reflections diminish as a function of Γ_{s}/ε, i.e. as the energy level in the QD moves away from the Fermi level, and as a result Cooper pairs have smaller probability of tunneling across the junction. In the parameter range for which the thermophase is appreciable – the tunnel junction regime – the ratio Γ_{s}/ε is small and thus multiple reflections can practically be neglected.
In an open junction setup (Fig. 1(c)), with no externally imposed constrains over the thermophase, the thermally induced current is completely canceled by the appearance of a thermophase across the junction^{8}. This serves as the definition for the thermophase ϕ_{th}:
Linear Response
In the linear response regime (linear in ΔT), assuming a symmetric junction, one can write the different terms in Eq. 3 explicitly:
where the parameters Γ and Δ were taken equal on both sides of the junction. In these equations, the expression for and A, whose contributions to the current are plotted in Fig. 2, is given by (Fig. 2(a)) and (Fig. 2(b)). The transmission channel for the QPs, , demonstrates the asymmetry of transmission as a function of energy, required to generate a thermoelectric response. On the other hand, the transmission channel for pairs, 1/A(ω), is driven by a superconducting phase difference, generated to compensate for the QP contribution, and thus does not require ph asymmetry. As can be seen in Fig. 2(a,b), both the QP channel and the pair channel contain sharp resonances which are Andreev bound states (ABS) (though the ABS do not participate in the QP transport due to Re[ρ(ω)] term in Eq. 5, as ρ has a real part only outside the gap). The pairQP transition term I_{pair−QP} vanishes identically, since this thermal transport process is perfectly particlehole symmetric (mathematically, writing I_{pair−QP} as an integral similar to Eq. 5, the integrand is an odd function of ω, as a result of a symmetric transmission channel in this process).
Since in linear response one can define I = σΔϕ + S_{ϕ}σΔT, in analogy to the Seebeck coefficient, we can define the thermophase Seebeck coefficient (TPSC) in a similar manner,
In Fig. 2(c), the TPSC S_{ϕ} is plotted as a function of dot level energy ε, for various temperatures T and couplings Γ. S_{ϕ} consistently peaks around ε = Δ + aΓ, with the factor a being typically a ~ 1–2.5 for the relevant parameters (Γ > 0.05). The TPSC peak occurs when the dot energy is slightly above the SC coherence peaks in the BCS DOS, at the point that maximizes the interplay of ph asymmetry and transmission. This is similar in nature to the Seebeck coefficient peak through a QD between normal leads^{24}, which resides a distance Γ above (or below) the QD energy level resonance. In the inset of Fig. 2(c) we plot the inverse temperature dependence of the TPSC on a log scale, for various level energies. For T ≪ Δ the leading contribution to the temperature dependence of the TPSC stems from the activated form of the the Fermi function in the QP term (5), which can be approximated by , where E_{g} is an activation energy. Indeed, the logarithmic slopes of S_{ϕ}, depicted in the inset of Fig. 2(c), are linear with an activation gap Δ_{0}, as expected for QPs. This description works rather well for most of the relevant temperature range. In the opposite limit, for T approaching the SC transition temperature T_{c}, the TPSC diverges due to vanishing of the Josephson term, as 1/Δ^{2} ~ (T − T_{c})^{−1}.
Beyond Linear Response
The formulation described in the previous section applies, in fact, also beyond the linear response regime in ΔT, where the main deviation from linear response stems from the difference in order parameters on both sides due to thermal difference. The full analytical expression, including all contributions, is quite long and thus will not be shown here. Figure 3 depicts the total current as a function of phase for several values of temperature difference, ΔT. The general division of the current into the three terms (Eq. 3) holds also beyond the linear response regime, as can be seen in the inset of Fig. 3, which shows the QP and the Josephson contributions to the total current (the contribution from the pairQP transition term I_{pair−QP} still vanishes). The Josephson term is modified due the difference in Δ between the two sides^{25}. As can be clearly seen from the figure, the QP term is almost insensitive to phase difference, but sensitive to changes in temperature difference, while the Josephson term oscillates with the phase difference, but weakly sensitive to temperatures far from the SC transition temperature.
As the temperature difference increases beyond a critical value ΔT_{c} (the red curve in Fig. 3, corresponding to ΔT = 0.181 for the depicted set of parameters) the QP current reaches a value such that the Josephson current can no longer compensate for it (for ΔT = ΔT_{c} the thermophase is exactly ±π/2). If the total current is kept at zero, an effective voltage will develop in this regime, which will give rise to a timedependent AC response (as in the AC Josephson effect), an effect which has in fact been measured in tunnel junctions^{26}. We leave the timedependent thermal Josephson effect for a future study, and concentrate here on ΔT below the critical value ΔT_{c}.
Solving the condition (4) for vanishing current, we plot in Fig. 4 the thermophase ϕ_{th} as a function of the left lead temperature T_{L} (for fixed T_{R}) and QD level energy ε. The region of ϕ_{th} = ±π/2 (red or blue plateau in Fig. 4) corresponds to the regime for which ΔT ≥ ΔT_{c}, and is not covered in this work. The value of the critical temperature difference as a function of dot energy can be read from Fig. 4 as the contour of the ±π/2 plateau. The value of the critical ΔT_{c} can be directly measured in experiments, by applying a temperature difference and monitoring for which ΔT a finite current (or voltage) begins to appear.
Coulomb Interaction
So far we have ignored the onsite interaction on the QD, which may be important, for example, in the Coulomb blockade regime^{27}. In order to address this, we add to the Hamiltonian an onsite Coulomb interaction, represented by a term H_{U} = Un_{↑}n_{↓}, where . Within the HartreeFock (HF) approximation, the dot levels are renormalized according to , where the dot occupations 〈n_{σ}〉 are calculated selfconsistently ( is the spin opposite to σ). Once the dot levels and occupations are determined (inset in Fig. 5), the thermophase can be calculated using Eq. 4.
Compared to the noninteracting problem, the interaction introduces a new regime where the dot is singly occupied, the effective spin energy levels split^{28}, and a magnetic moment forms. This splitting suppresses the pair tunneling amplitude, where eventually the Josephson term becomes smaller and, unless smeared by temperature, changes sign, leading to a πjunction transition^{29,30}. It is important to note that the HF approximation, while found to generally describe the SCQDSC physics very well^{29,31}, will not be valid in the Kondo regime, where T_{k} > Δ^{32}. In addition, it may give qualitatively inaccurate values for the boundaries of the singly occupied regime^{33}. These failures of the HF approximation are rather limited in the small Γ/Δ limit^{33}, which is the regime of interest in the present work, and therefore we can safely proceed with its usage. Furthermore, we note that although there seems to be an apparent spin symmetry breaking from the form of the HF solution, these spinasymmetric solutions are doublydegenerate with the degenerate solutions having opposite spins. It is thus important to take both solutions into account to preserve spin symmetry. In Fig. 5 we show the thermophase ϕ_{th} as a function of the bare dot energy (shifted by half the Coulomb interaction), for various values of U. We first note that the thermophase is symmetric not around ε = 0, but around the new (and only) point of particlehole symmetry, ε − U/2 = 0. There are other points, inside the singly occupied region, where the QP term (and consequently also the thermophase) vanish, but this is due to cancellation of contributions and not because of ph symmetry. Inside this region, we also see sharp ±π/2 peaks (points A, B in Fig. 5) which correspond to a vanishing Josephson term during the πjunction transition, where any thermal gradient will produce the maximal thermophase of ±π/2. Other new features (such as a small peak at point C and a tiny peak at point D in Fig. 5) emerge from the nonmonotonous behavior of the QP term, in the singly occupied regime. In this regime, the contributions from the two spin levels have opposite signs and their magnitude difference also changes sign as a function of average dot energy. The features outside this region, however, are unaffected by the interaction except for the trivial shift away from the ph symmetry point by U/2.
Discussion
All the results presented in this paper can be directly tested experimentally. To measure the thermoelectric effect and the thermophase, we suggest the experimental setup depicted in Fig. 1(b). It consists of a SC ring with one branch including a QD while the other branch including a thin insulating barrier. One side of the ring is heated in order to create a temperature gradient, and as a result, a unidirectional circulating thermocurrent arises. This setup makes use of the geometrical constraint on the gauge invariant phase, and of the fact that the phase drop occurs primarily at the point of most resistance^{34} (which is the QD as opposed to the insulating barrier). This implies that the phase difference across the QD junction is ϕ = 2π(Φ/Φ_{0} + n), where Φ is the magnetic flux penetrating the ring and Φ_{0} = hc/2e is the flux quantum. Since there is no external magnetic flux, the phase difference across the QD, necessary to produce the supercurrent that cancels the thermal current in the bulk of the SC, is accompanied by a generation of a magnetic flux through the ring. This flux, in fact, arises from supercurrents running on the surface of the ring^{12}. This experimentally measurable flux can be continuously modified by the applied temperature gradient, or the position of the dot level energy. In order to measure the critical temperature difference, an open setup (Fig. 1(c)) can also be utilized (where no phase detection is necessary). The temperature difference for which effective thermovoltage begins to appear, is the critical temperature difference. Ref. 22 has already applied setups that involve SCs and a QD, while ref. 21 has already demonstrated applying a temperature bias in SC^{21}, but these two approaches have yet to be experimentally explored together.
In summary, we have demonstrated that a superconductor  quantum dot  superconductor junction can serve as a model system to study thermoelectric effects in SC systems, as it exhibits a large and controllable TE response. The current response to a temperature difference has been studied as a function of the most important control parameters, namely temperature, gate voltage and dotelectrode couplings. Specific experimental realizations to test our predictions have been suggested, and we believe that they are well within current experimental capabilities. Further studies that examine the AC thermal Josephson effect (beyond the critical temperature difference) are currently under way.
Methods
In order to find the current across the junction we calculate the Green’s function in Nambu space^{30}, , where is the Nambu particlehole spinor. We find the relevant self energies using the equations of motion, in Nambu space:
where , N_{s}(0) being the normal metal density of states (DOS). ρ_{s} can be regarded as the generalized DOS in the superconductor, normalized by the normal metal value, where there is an imaginary (ω < Δ_{s}) contribution from inside the gap. Applying the self energies to the Dyson equation^{30} we find the retarded Green’s Function in Fourier space , where
and . Using , we express the current in terms of the Green’s functions on the dot^{35}, , where the lesser Green’s functions , and we can calculate the term in the square brackets in the expression for J using the Langreth relation^{36} [A(ω)B(ω)]^{<} = A^{r}(ω)B^{<} (ω) + A^{<} (ω)B^{a}(ω).
From a numerical perspective, a broadening of the superconducting gap energy is required to avoid divergence of the superconducting DOS. A suitable Dynes Broadening^{37,38} is required, and if done carefully (the broadening should be mutually conjugate for particles and holes, namely Δ(ω) = Δ_{0} − i sign(ω)η), it enables one to directly see the contribution from the Andreev bound states^{28,30}, which are usually numerically elusive (being ideally a delta function contribution to the local DOS), as can be seen in Fig. 2(a,b). In all the calculations the zero temperature SC order parameter on both sides was set as the unit energy, Δ_{s}(T = 0) ≡ Δ_{0} = 1, and all other energy values are measured in units of Δ_{0}. The value of the Dynes broadening parameter used in our calculations is η = 10^{−4}, but the results are largely independent of this value.
Additional Information
How to cite this article: Kleeorin, Y. et al. Large Tunable Thermophase in Superconductor–Quantum Dot–Superconductor Josephson Junctions. Sci. Rep. 6, 35116; doi: 10.1038/srep35116 (2016).
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Acknowledgements
We thank M. Di Ventra and O. EntinWohlman for illuminating discussions. YM acknowledges support from ISF under grants 11/11 and 292/15. FG acknowledges the European Research Council under the European Union’s Seventh Framework Program (FP7/20072013)/ERC Grant agreement No. 615187COMANCHE for partial financial support.
Author information
Affiliations

Department of Physics, BenGurion University of the Negev, Beer Sheva, 84105, Israel
 Yaakov Kleeorin
 & Yigal Meir

The Ilse Katz Institute for Nanoscale Science and Technology, BenGurion University of the Negev, Beer Sheva, 84105, Israel
 Yigal Meir
 & Yonatan Dubi

NEST Istituto NanoscienzeCNR and Scuola Normale Superiore, I56127 Pisa, Italy
 Francesco Giazotto

Department of Chemistry, BenGurion University of the Negev, Beer Sheva, 84105, Israel
 Yonatan Dubi
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Contributions
F.G. and Y.D. initiated the project. Y.K. performed the analytical and numerical calculations. All authors discussed the results and participated in writing the manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding authors
Correspondence to Yaakov Kleeorin or Yonatan Dubi.
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