Efficient phase-tunable Josephson thermal rectifier

Josephson tunnel junctions are proposed as efficient phase-tunable thermal rectifiers. The latter exploit the strong temperature dependence of the superconducting density of states and phase-dependence of heat currents flowing through Josephson junctions to operate. Remarkably, large heat rectification coefficients up to 800% can potentially be achieved using conventional materials and standard fabrication methods. In addition, these devices allow for the in-situ fine tuning of the thermal rectification magnitude and direction.

leading to critical temperatures T c1 and T c2 , respectively. The electronic temperature in both S 1 and S 2 is kept at fixed T 1 and T 2 , respectively, and the voltage drop across the junction is set to zero. Additionally, ϕ denotes the macroscopic phase difference across the junction with normal-state resistance R J . In the forward thermal bias configuration, a thermal gradient is created by setting T 1 = T hot > T 2 = T cold which leads to a total heat current J + flowing from S 1 to S 2 [see Fig. 1(a)]. In the reverse thermal bias configuration, the thermal gradient is inverted so that T 1 = T cold < T 2 = T hot leading to a heat current J − flowing from S 2 to S 1 [see Fig. 1(b)]. Under these hypothesis we define the rectification coefficient as We describe now the equations governing heat transport in the Josephson thermal rectifier. We focus on the electronic contribution to heat transport only, and neglect eventual heat currents carried by lattice phonons. The forward and reverse total heat currents flowing through the Josephson junction read 6 ϕ ϕ ϕ ϕ = 0 0 0 0 ϕ ϕ ϕ ϕ = π/2 π/2 π/2 π/2 ϕ ϕ ϕ ϕ = π π π π (%) and k, l =hot,cold.
In Eq.
is the normalized smeared (by non zero γ) BCS quasiparticle DOS in S α with α = 1, 2. 24 In the following we will assume γ ∼ 10 −5 , which describes realistic SIS' junctions. [25][26][27] Additionally, f (T k ) = (1 + e ε/k B T k ) −1 is the Fermi energy distribution, ∆ α (T k ) is the temperaturedependent energy gap of S α , k B is the Boltzmann constant and e is the electron charge. On the other hand, the term J int in Eq. (2) is the amplitude of the phase-dependent component of the heat current, [21][22][23] 28 This component is peculiar to the Josephson effect and arises as a consequence of tunneling processes through the junction involving both quasiparticles and Cooper pairs. [21][22][23] It is worthwhile to emphasize that, depending on the value of ϕ, the second term in Eq. (2) may switch its sign. The existence and sign of this component was experimentally demonstrated in Ref. 5 It is illustrative to start by analyzing the case in which one of the two electrodes is a normal metal, i.e., a NIS junction.  For this purpose we can simply set ∆ 2 = 0 which leads to the complete suppression of J int in Eq. (2). We calculate R as a function of T hot for different values of T cold , i.e., for different temperature gradients established across the weak link. As shown in Fig. 1(c) a maximum positive rectification of ∼ 26% is obtained for T hot 0.85T c1 at T cold = 0.01T c1 . As T hot increases, heat rectification starts to decrease eventually inverting its sign which implies that heat flux from the normal metal to S 1 becomes preferred. Furthermore, by increasing T cold leads to a reduction of R which reaches its maximum for larger values of T hot . We note than, a bare NIS junction offers, in this way, an already quite rich response in terms of heat rectification.
In the following analysis we shall focus on the case in which both junction electrodes are superconductors, i.e., a SIS' junction. We define for clarity r = ∆ 1 /∆ 2 ≤ 1. By fixing the temperature of the second electrode to T cold = 0.01T c1 , we calculate R as a function of T hot and as a function of r. The result is plotted in Fig. 2(a), (b) and (c) for three representative cases, corresponding to ϕ = 0, ϕ = π/2 and ϕ = π, respectively. Three selected profiles of R as a function of r for different T hot values are shown as well in Fig. 2(d), (e) and (f) for the same values of ϕ. R depends strongly on r reaching its maximum at r 0.75 and T hot 0.77T c1 for ϕ = π, dropping then to zero at r = 1. The inspection of these graphs also reveals how phase biasing across the junction does make a substantial difference. In particular, the heat rectification coefficient does not only change by almost two orders of magnitude from ϕ = 0 to ϕ = π but it also switches its sign.
The dependence of R on T hot for ϕ = 0 is shown in Fig.  3(a) for different values of T cold and r = 0.75. In particular, R is negative as long as T hot < T c1 , i.e., as long as one of the two electrodes remains in the superconducting state, reaching values of R −25% at T hot 0.76T c1 and T cold = 0.01T c1 . Above this point, the sign of R and, therefore, the rectification direction depends on the thermal gradient being however slightly reduced. On the other hand, curves corresponding to ϕ = π are shown in Fig. 3(b) vs. T hot for the same values of T cold . Remarkably, large values of R ∼ 800% can be obtained at T hot 0.77T c1 and T cold = 0.01T c1 . If T cold is increased, R is in general reduced reaching its maximum for larger values of T hot . Above T c1 , the behavior is identical to that calculated for ϕ = 0 since one electrode remains always in the normal state and the phase plays no role any more. In Fig. 3(c), the periodicity of R with ϕ is shown for the same values of T cold and the corresponding T hot that maximize R. We stress how ϕ-dependence provides the Josephson thermal diode with an unique tunability.
It is worthwhile to emphasize that the SIS' junction rectifies heat only if ∆ 1 = ∆ 2 . As for the case of the NIS diode, heat rectification demands the combination of two different DOS being (at least one of them) strongly temperature-dependent. 10 Yet, a fairly large thermal gradient is required as well. Indeed, heat rectification is absent in the linear-response regime, that is, for small temperature differences δ T = T hot −T cold << T = T hot +T cold 2 . In this case, the total electron heat current flowing through the Josephson junction reduces to 28 which depends on the average temperature of the two electrodes only.
We discuss in the following some possible experimental realizations faced to make the most of the phase-dependence of the Josephson thermal rectifier. On the very first place we are interested in playing with the macroscopic phase difference across the Josephson junction during operation. Phase biasing of a Josephson junction can be achieved, in general, through supercurrent injection or by applying an external magnetic flux. In the former case, schematized in Fig.  4(a), a Josephson current i J is forced to flow through the junction via two extra control superconducting wires connected to the diode's core through clean or tunnel contacts. The control wires can be made of a third superconductor S 3 with energy gap ∆ 3 >> ∆ 1 , ∆ 2 so to suppress heat losses. The phase-current relation under such circumstances is given by sin(ϕ) = i J /i c J . 29 Provided that i J ≤ i c J , i c J being the junction critical current, a phase gradient contained within the sections −π/2 ≤ ϕ ≤ π/2 can be established. An analogous phase gradient can be obtained using a direct current superconducting quantum interference device (DC SQUID) pierced by an external control magnetic flux Φ as shown in Fig.  4(b). If both junctions are identical and neglecting the loop's geometrical inductance, the phase-flux relation must satisfy cos(ϕ a ) = cos(ϕ b ) = [1 − cos(2πΦ/Φ 0 )]/2 where ϕ a and ϕ b are the phase drops across each junction 6,7 and Φ 0 is the flux quantum. The optimum phase configuration in terms of heat rectification, i.e., ϕ = π, can be reached by using a rf SQUID as shown in Fig. 4(c). Fur such a purpose, the thermal diode can be enclosed through clean contacts within a superconducting ring S 3 pierced by a control flux Φ. Neglecting again the loop's inductance, the phase-flux relation is given in this case by ϕ = 2πΦ/Φ 0 29 enabling the phase drop across the junction to vary within the whole phase space, i.e., −π ≤ ϕ ≤ π. Figure 4(d) shows a device envisioned to probe experimentally the effects discussed above. Two identical normal metal electrodes, source and drain, are weakly connected one each via a resistance R N to both S 1 and S 2 , respectively. Superconducting probes can be tunnel-coupled to these electrodes so to implement SINIS thermometers and heaters. 1 Yet, the forward thermal bias configuration can be realized by intentionally increasing the electronic temperature in source electrode up to T + src = T h and probing the temperature in drain electrode T + dr . On the reverse configuration, we set T − dr = T h and T − src is measured in a similar way. The difference δ T e = T + dr − T − src for a given T h can be used to assess experimentally heat rectification. δ T e can be computed numerically by solving a system of thermal equations accounting for the heat exchange mechanisms present in our device [see Fig. 5(a)]. On the forward configuration, electrons in S 1 exchange heat with electrons in the source at power J src→S 1 and, at power J + , with electrons in S 2 . On the other hand, electrons in S 2 exchange heat with drain's electrons at power J S 2 →dr . Finally, electrons in the whole structure exchange heat at power J e-ph with lattice phonons that we assume to reside at bath temperature T bath . Under such circumstances, the three unknown quantities, i.e., T hot , T cold and T + dr can be calculated for given initial conditions by solving the following system of thermal balance equations 30 In the above expressions, J e-ph,src(dr) = Σ N V N (T 5 src(dr) − T 5 bath ) 1 , V N and Σ N being the volume of the normal metal electrode and the electron-phonon coupling constant, respectively. Furthermore, we assume T bath = 10 mK << T k << ∆(T k )/k B so that J e-ph,S α 0.95Σ S V S T 5 k e −∆α (T k ) k B T k , 31 with V S and Σ S being the volume of each superconducting electrode and the electron-phonon coupling constant, respectively. As representative parameters we set V N = V S = 2 × 10 −20 m 3 , R J = 10 kΩ and R N = 100 Ω. Source and drain electrodes can be made, for instance, of Cu for which Σ N 3 × 10 9 WK −5 m −31 whereas the diode's core can be made of Al and Mn-doped Al with Σ S 0.3 × 10 9 WK −5 m −3 , 1 since the latter allows for fine tuning of the aluminum superconducting gap. 32,33 In this way, we set T c1 = ∆ 1 1.764k B = 1.4 K and r = 0.75. With this set of parameters, R can be determined as a function of T h for given values of ϕ. The resulting curves are plotted in Fig.  4(b) together with the computed values of δ T e vs. T h which are plotted in Fig. 4(d). Remarkably, in the present setup, a maximum R ∼ 340% can be reached. The latter corresponds to a temperature difference as large as δ T e ∼ 140 mK which is easily measurable with standard SINIS or SNS thermometry techniques. 1,5 Even more interesting, phase-coherence fingerprints are clearly observable as well. Notably, R and δ T e show the expected 2π-periodicity as shown in Fig. 4(c) and (e). In closing, we emphasize that, at such low bath temperatures, both superconducting electrodes are only marginally coupled to phonons. 1 Indeed, neglecting the contribution of J e-ph,S i leads to differences less than ∼ 5% of the values presented here.
To summarize, we have proposed and analyzed the concept of a Josephson thermal rectifier. Under appropriate conditions, a remarkably large rectification coefficient of R ∼ 800% can be obtained. In addition, the Josephson thermal diode is phase-tunable. This latter property allows to maximize heat rectification in-situ or, even, to switch its sign. Such a device might find a straightforward application, e.g., in the field of electronic refrigeration enabling magnetic-flux dependent heat management and thermal isolation at the nanoscale. The operation principle which is at the basis of this heat rectifier will likely contribute, on the other hand, to improve the performance of other different coherent thermal components such as heat transistors or splitters. [5][6][7] These thermal devices might potentially lead to the emergence of coherent caloritronic nanocircuits.
We acknowledge R. Aguado and C. Altimiras for comments, and the FP7 program No. 228464 MICROKELVIN, the Italian Ministry of Defense through the PNRM project TERASUPER, and the Marie Curie Initial Training Action (ITN) Q-NET 264034 for partial financial support.