Very large thermophase in ferromagnetic Josephson junctions

The concept of thermophase refers to the appearance of a phase gradient inside a superconductor originating from the presence of an applied temperature bias across it. The resulting supercurrent flow may, in suitable conditions, fully counterbalance the temperature-bias-induced quasiparticle current therefore preventing the formation of any voltage drop, i.e., a thermovoltage, across the superconductor. Yet, the appearance of a thermophase is expected to occur in Josephson-coupled superconductors as well. Here we theoretically investigate the thermoelectric response of a thermally-biased Josephson junction based on a ferromagnetic insulator. In particular, we predict the occurrence of a very large thermophase which can reach $\pi/2$ across the contact for suitable temperatures and structure parameters, i.e., the quasiparticle thermal current can reach the critical current. Such a thermophase can be several orders of magnitude larger than that predicted to occur in conventional Josephson tunnel junctions. In order to assess experimentally the predicted very large thermophase we propose a realistic setup realizable with state-of-the-art nano fabrication techniques and well-established materials which is based on a superconducting quantum interference device. This effect could be of strong relevance in several low-temperature applications, for example, for revealing tiny temperature differences generated by coupling the electromagnetic radiation to one of the superconductors forming the junction.

Thermoelectric currents in superconductors are often shorted by supercurrents which generate a phase gradient, a thermophase, inside the superconductor.As suggested a long time ago by Ginzburg [1], a bimetallic superconducting loop constrains the possible phase gradients, and allows observation of thermoelectric effects via magnetic fields arising from circulating currents manipulated by temperature differences.Measurements of such circulating currents [2] were larger than that predicted by theory by several orders of magnitude, a discrepancy that is yet to be explained [3,4].Here we propose an alternative way to produce a huge thermophase in a SQUID loop consisting of a conventional superconductor in a spin-polarized contact with a superconductor-ferromagnet bilayer.The resulting thermophase can be close to π/2 across the contact, several orders of magnitude larger than in conventional Josephson junctions [5,6].Such a giant thermophase could be used for detecting tiny temperature differences, for instance, generated by radiation coupling to one of the superconductors.
We consider a Josephson junction (see Fig. 1a) consisting of two superconductors S L and S R tunnel coupled through a ferromagnetic insulator (FI) and a non-magnetic (I) barrier.The FI has different transmissivities for spin-up and spin down electrons and therefore acts as a spin-filter.[7] The interaction between the conduction electrons in S L with the localized magnetic moments of the FI leads to an effective spin-splitting field h in the left electrode that decays away from the interface over the superconducting coherence length ξ 0 .[8] The thin I layer placed on the right side of the FI prevents such a spinsplitting field to be induced in S R [9,10].We assume that the thickness t L of S L is smaller than ξ 0 so that the induced h is spatially uniform across the entire S L layer [11].The junction is temperature biased, so that T L,R is the temperature in S L,R , respectively, and ϕ th denotes the phase difference between the superconducting order parameters induced by such a temperature difference.We focus on the static (i.e., time-independent) regime so that a dc Josepson current can flow in response to the applied thermal gradient but no thermovoltage develops across the junction.
In order to analyze the setup, we generalize the calculation of [12] to the case of two superconductors.The total electric current I flowing through the junction is given by the sum of the quasiparticle (I qp ) and the Josephson contribution (I J ) where I c is the critical supercurrent.The current contribution proportional to cos ϕ th [13] does not contribute, since it does not possess any thermoelectric response, and it would require a finite voltage.The explicit forms for I qp and I c in Eq. ( 1) can be obtained from the expressions for the current through arXiv:1403.1231v1[cond-mat.mes-hall]5 Mar 2014 a spin-filter barrier with polarization P [14][15][16], and In Eqs. ( 2) and (3), ), and ∆ L,R is the energy gap in S L,R which has to be determined self-consistently due to the presence of h and finite T L/R .For ideal superconductors Γ → 0 + [17]; we have checked that the value Γ = 10 −4 ∆ 0 chosen in our numerics negligibly affect the results.Above, ∆ 0 denotes the zerotemperature, zero-exchange field superconducting energy gap.Furthermore, e is the electron charge, k B is the Boltzmann constant, and R T is the normal-state resistance of the junction.Equation (2) shows that for a non-vanishing I qp to exist (i) a finite h should be induced in one of the superconductors, and (ii) P has to be finite.[12] In an electrically-open configuration the total current has to vanish.In order to ensure a vanishing thermovoltage across the junction, the quasiparticle current I qp induced by the temperature gradient has to be canceled by an opposite dc supercurrent I J .This cancelation is the origin of the thermophase, which is defined as This thermophase is thus a measure of the amplitude of the thermoelectric effect at the contact between the superconductors.
Let us analyze the behavior of I qp and of I c under thermal bias.Figures 2a and 2b show I qp vs T L and T R , respectively, when the temperature of the other electrode is fixed to 0.1T c .Here T c is the superconducting critical temperature in the absence of h which we assume, for simplicity, to be the same for both superconductors.By varying T L , and depending on the temperature range, the thermocurrent can be either positive, i.e., flowing according to the thermal gradient set across the junction, or negative.The sign of the current can be ascribed to a more electron-or hole-like contribution to thermoelectric transport, respectively.The amplitude of I qp is, in general, larger for larger h, and drops eventually to zero at the temperature for which ∆ L vanishes.The value of such a critical temperature depends on the value of h.By contrast, Fig. 2b shows that the thermocurrent does not change sign if T L is held constant and T R (> T L ) is varied.The amplitude of I qp in this case is larger than that obtained by varying T L > T R , and is finite at T c .Notably, the quasiparticle characteristics exhibits sharp dips positioned at the temperatures satisfying the condition The large values obtained by I qp are the origin of a giant thermophase achievable in ferromagnetic Josephson junctions.Similarly, the lower panels of Fig. 2 display the behavior of I c vs T R and T L by holding the other electrode at 0.1T c .The I c (T ) curves differ drastically from those obtained for h = 0 at T L = T R (dash-dotted curves).In particular, at a low enough temperature I c gets larger by increasing h.This remarkable effect corresponds to the supercurrent enhancement discussed in Ref. [18] which occurs even for P = 0. We stress that the I c strenghtening joined with the sharp jumps appearing at those temperatures where Eq. ( 5) holds are a manifestation of an exchange field induced in S L and of a nonequilibrium condition stemming from the thermal bias [18].
The thermophase ϕ th is obtained from Eq. ( 4). Figure 3a  and b shows the dependence of ϕ th on T L and T R , respectively, when the other electrode is kept at 0.1T c .We have chosen a reasonable polarization of the barrier (P = 90%) and moderate h values easily achievable with present-day experiments.[7,10] We find that ϕ th can be vary large, close to π/2 for h 0.3∆ 0 .This substantial effect has to be compared to the minute one achievable in conventional nonferromagnetic Josephson tunnel junctions where ϕ th ∼ 10 −4 is expected.[5,6] We also notice the presence of temperature regions where ϕ th is not defined since I qp may exceed I c (see Fig. 3b).In this case a finite dc voltage is induced across the contact.
The impact of P on ϕ th is shown in panel c and d of Fig. 3 0.0 0.5 where we set h = 0.2∆ 0 , and the temperature in the superconductors is varied similarly to panel a and b, respectively.The increase of P leads to a sizable thermophase enhancement, as From Fig. 3 it becomes clear that large values of ϕ th can be obtained more easily by increasing T R while keeping S L at a low temperature, consistently with the I qp (T R ) dependence shown in Fig. 2b.The full behavior of ϕ th vs T R for T L = 0.1T c is displayed in the color plots of Fig. 4. Specifically, we set P = 96% and varied h in panel a, whereas in panel b we set h = 0.3∆ 0 and varied P. The figures show that a sizeable ϕ th can be obtained in a rather large range of parameters, and may provide a valuable tool for tailoring optimized junctions where ϕ th is maximized.
To assess experimentally the predicted giant thermophase we propose the setup depicted in Fig. 1b.It consists of a superconducting quantum interference device (SQUID) including a FI, and comprising a number of superconducting tunnel junctions which can either heat or perform accurate electron thermometry [19].From the materials side, FIs such as EuO or EuS (providing P up to ∼ 98%) [20] in contact with superconducting Al appear as ideal candidates for the implementation of the structure which can be realized with standard lithographic techniques.The ratio h/∆ 0 in such structures depends on the thickness of the Al layer and quality of the contact.In the superconducting state of Al, values ranging from h/∆ 0 ≈ 0.2 up to 0.6 have been reported.[10,[21][22][23] Alternatively, GdN barriers in combination with Nb or NbN could be used as well.[24,25] In the SQUID, the thermophase ϕ th 1/2 developed across the two junctions results into a nonzero circulating supercurrent I circ .In the absence of an external flux, the amplitude of the circulating current is given by where I ci is the critical current for contact i = 1, 2. Since ϕ th i depends on the ratio of the quasiparticle and critical supercurrents, an asymmetry of the resistances between the contacts would not cause a circulating current.However, replacing one of the junctions, say 2, by a conventional SIS junction would set ϕ th 2 ≈ 0 and therefore would lead to a large I circ even without an external flux.
An alternative way to measure the thermophase is to consider the case of a finite external flux Φ ext .If the SQUID junctions are identical, I circ can be written as I circ = I c (sin ϕ 1 + sin ϕ th ) = −I c (sin ϕ 2 + sin ϕ th ), and can be analytically determined for negligible ring inductance by imposing fluxoid quantization, ϕ 2 − ϕ 1 = 2πΦ ext /Φ 0 , where Φ 0 = 2.067 × 10 −15 Wb is the flux quantum.The result for I circ is thus which holds for I qp ≤ I c [see Eq. ( 4)].Equation (7) reduces to I circ = I c sin(πΦ ext /Φ 0 ) for ϕ th = 0.As shown in Fig. 4(c), the presence of a finite thermophase gives rise to regions of flux close to Φ 0 /2 and 3Φ 0 /2 where I circ vanishes because the thermoelectric current becomes larger than the effective critical current of the SQUID.The presence of these regions is a direct evidence of the thermophase.The circulating current can be detected through a conventional dc SQUID inductively-coupled to the first loop (see Fig. 1b) so that Φ SQUID ∼ MI circ , where Φ SQUID is the magnetic flux induced in the read-out SQUID and M is the mutual inductance coefficient.For typical values of I c = 1 µA and M = 10 pH, Φ SQUID up to ∼ 10 −17 Wb ≈ 5 × 10 −3 Φ 0 can be generated with a proper temperature bias.This can be well detected with standard SQUIDs which provide routinely magnetic flux sensitivities ∼ 10 −6 Φ 0 / √ Hz [26].In conclusions, we have predicted the occurrence of a giant thermophase in thermally-biased Josephson junctions based on FIs.This sizeable effect can be detected in a structure realizable with current state-of-the-art nanofabrication techniques and well-established materials.Besides shedding light onto fundamental problems related to the thermoelectric response of superconductors and exotic weak links in the Josephson regime, the very sharp thermophase response (see Fig. 3) combined with the low heat capacity of superconductors could al-low realizing ultrasensitive radiation detectors [19] where radiation induced heating of one of the superconductors is detected via the thermophase.The presence of magnetic material also allows for adding a new control parameter to the experimental investigation of coherent manipulation of heat flow at the nanoscale [27][28][29][30].

FIG. 1 .
FIG. 1. Thermally-biased ferromagnetic Josephson junction and the proposed experimental setup.(a) Sketch of a generic S-FI-I-S Josephson junction discussed in the text.It consists of two identical superconductors, S L and S R , tunnel-coupled by a ferromagnetic insulator FI and a non-magnetic barrier I.The direct contact between FI and S L leads to an induced exchange field in the latter, while the non-magnetic barrier prevents such a field to appear in S R .T L and T R are the temperature in S L and S R , respectively, whereas ϕ th denotes the thermophase originated from thermally biasing the Josephson junction.t L is the thickness of S L .(b) Scheme of a detection setup consisting of a temperature-biased superconducting quantum interference device (SQUID) based on the previous junction.Superconducting electrodes tunnel-coupled to S L and S R serve either as heaters (h) or thermometers (th), and allow one to impose and detect a temperature gradient across the SQUID.The magnitude of the induced ϕ th can be determined by measuring variations of the supercurrent (I circ ) circulating in the interferometer through a conventional dc SQUID inductively coupled to the first ring.M denotes the mutual inductance between the loops, and Φ ext is the external applied magnetic flux.

FIG. 2 .
FIG. 2. Quasiparticle and Josephson critical currents under thermal-bias conditions for different values of the exchange field.Quasiparticle current I qp (a) vs T L at T R = 0.1T c and (b) vs T R at T L = 0.1T c ; Josephson critical current I c (c) vs T L at T R = 0.1T c and (d) vs T R at T L = 0.1T c .Dash-dotted curves in panels c and d are calculated for h = 0 and T L = T R .∆ 0 denotes the zero-exchange field, zero-temperature superconducting energy gap corresponding to the critical temperature T c ≈ ∆ 0 /1.764kB whereas I 0 is the zeroexchange field, zero-temperature Josephson critical current.

FIG. 3 .
FIG. 3. Behavior of the thermophase in a ferromagnetic Josephson junction.(a) Thermophase ϕ th vs T L calculated for several values of the exchange field h at T R = 0.1T c and P = 90%.(b) ϕ th vs T R calculated at T L = 0.1T c and P = 90% for the same exchange field values as in panel a.(c) ϕ th vs T L calculated for several values of the polarization P of the spin-filter barrier at T R = 0.1T c and h = 0.2∆ 0 .(d) ϕ th vs T R calculated at T L = 0.1T c and h = 0.2∆ 0 for the same P values as in panel c. ϕ th is not defined in the temperature regions where |I qp | exceeds I c .

FIG. 4 .
FIG. 4. Full behavior of the thermophase and response of the circulating current in a temperature-biased FI SQUID.(a) Color plot of the thermophase ϕ th vs T R and h calculated at T L = 0.1T c for P = 96%.(b) Color plot of ϕ th vs T R and P calculated at T L = 0.1T c for h = 0.3∆ 0 .(c) Circulating current vs. flux for a few values of the thermophase in a symmetric SQUID.