Large Tunable Thermophase in Superconductor – Quantum Dot – Superconductor Josephson Junctions

In spite of extended efforts, detecting thermoelectric effects in superconductors has proven to be a challenging task, due to the inherent superconducting particle-hole symmetry. Here we present a theoretical study of an experimentally attainable Superconductor – Quantum Dot – Superconductor (SC-QD-SC) Josephson Junction. Using Keldysh Green’s functions we derive the exact thermo-phase 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 particle-hole symmetry in a gradual manner, allowing, in the presence of a temperature gradient, for gate controlled appearance of a superconducting thermo-phase. This thermo-phase 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.

where measurable thermo-phase 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 p-h 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 s-type 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 SC-QD- where σ σ † d d ( ) 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 non-equilibrium 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: quasi-particle current, I QP , Josephson (pair) current, I sc , and a term involving pair-QP transition, I  I  T T  I  T T   I  T T   ( , , ,  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 thermo-phase 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 thermo-phase, the thermally induced current is completely canceled by the appearance of a thermo-phase across the junction 8 . This serves as the definition for the thermo-phase φ th :

Linear Response
In the linear response regime (linear in Δ T), 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 G r 11 and A, whose contributions to the current are plotted in Fig. 2, is given by r r 0 1 ( 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 p-h 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 pair-QP transition term I pair−QP vanishes identically, since this thermal transport process is perfectly particle-hole 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 = σΔ φ + S φ σΔ T, in analogy to the Seebeck coefficient, we can define the thermo-phase Seebeck coefficient (TPSC) in a similar manner, I QP sc 0 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 p-h 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  (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 Δ T, 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, Δ T. 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 pair-QP 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 Δ T c (the red curve in Fig. 3, corresponding to Δ T = 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 Δ T = Δ T c the thermo-phase 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 time-dependent AC response (as in the AC Josephson effect), an effect which has in fact been measured in tunnel junctions 26 . We leave the time-dependent thermal Josephson effect for a future study, and concentrate here on Δ T below the critical value Δ T c .
Solving the condition (4) for vanishing current, we plot in Fig. 4 the thermo-phase 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 Δ T ≥ Δ T 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 Δ T c can be directly measured in experiments, by applying a temperature difference and monitoring for which Δ T a finite current (or voltage) begins to appear.

Coulomb Interaction
So far we have ignored the on-site 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 on-site Coulomb interaction, represented by a term H U = Un ↑ n ↓ , where ≡ σ σ σ † n d d . Within the Hartree-Fock (HF) approximation, the dot levels are renormalized according to ε ε = + σ σ U n , where the dot occupations 〈 n σ 〉 are calculated self-consistently (σ is the spin opposite to σ). Once the dot levels and occupations are determined (inset in Fig. 5), the thermo-phase can be calculated using Eq. 4.
Compared to the non-interacting 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 π-junction transition 29,30 . It is important to note that the HF approximation, while found to generally describe the SC-QD-SC 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 spin-asymmetric solutions are doubly-degenerate 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 thermo-phase 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 thermo-phase is symmetric not around ε = 0, but around the new (and only) point of particle-hole symmetry, ε − U/2 = 0. There are other points, inside the singly occupied region, where the QP term (and consequently also the thermo-phase) vanish, but this is due to cancellation of contributions and not because of p-h 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 π-junction transition, where any thermal gradient will produce the maximal thermo-phase 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 non-monotonous 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 p-h 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 thermo-phase, 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 dot-electrode 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 is the Nambu particle-hole spinor. We find the relevant self energies using the equations of motion, in Nambu space: , 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.