Resolving thermoelectric “paradox” in superconductors

Researchers bring the theory of thermoelectricity in superconductors and experiment into agreement.

Abstract

For almost a century, thermoelectricity in superconductors has been one of the most intriguing topics in physics. During its early stages in the 1920s, the mere existence of thermoelectric effects in superconductors was questioned. In 1944, it was demonstrated that the effects may occur in inhomogeneous superconductors. Theoretical breakthrough followed in the 1970s, when the generation of a measurable thermoelectric magnetic flux in superconducting loops was predicted; however, a major crisis developed when experiments showed puzzling discrepancies with the theory. Moreover, different experiments were inconsistent with each other. This led to a stalemate in bringing theory and experiment into agreement. With this work, we resolve this stalemate, thus solving this long-standing “paradox,” and open prospects for exploration of novel thermoelectric phenomena predicted recently.

INTRODUCTION

A temperature gradient applied to a superconductor, S, will yield an electric current ${\overrightarrow{\mathit{j}}}_{\text{q}}=-{\mathrm{\eta }}_{\text{q}}\nabla \mathit{T}$, where η_q is the thermoelectric coefficient. It is carried by quasiparticles, the unpaired “normal” charge carriers that exist at finite temperatures in any superconductor and behave similarly to conduction electrons in a normal metal. However, in contrast to a normal metal, the total current in the bulk of a homogeneous superconductor vanishes because the thermoelectric current, ${\overrightarrow{\mathit{j}}}_{\text{q}}$, is canceled by a superconducting current ${\overrightarrow{\mathit{j}}}_{\text{s}}$, ensuring that the total current in the bulk is zero: $\overrightarrow{\mathit{j}}={\overrightarrow{\mathit{j}}}_{\text{s}}+{\overrightarrow{\mathit{j}}}_{\text{q}}=0$ to obey the Meissner effect (1, 2). What renders the thermoelectric supercurrent measurable is its relation to the gradient of the macroscopic phase, $\mathrm{\theta }(\overrightarrow{\mathit{r}})$, of the superconducting condensate wave function ${\mathrm{\psi }}_{\text{s}}(\overrightarrow{\mathit{r}})={({\mathit{n}}_{\text{s}}/2)}^{1/2}\text{exp}(\mathit{i}\mathrm{\theta }(\overrightarrow{\mathit{r}}))$, where n_s is the concentration of “superconducting” electrons (3). Combining ${\overrightarrow{\mathit{j}}}_{\text{s}}=(\mathit{e}\mathrm{\hslash }{\mathit{n}}_{\text{s}}/2\mathit{m})\nabla \mathrm{\theta }(\overrightarrow{\mathit{r}})$ and ${\overrightarrow{\mathit{j}}}_{\text{s}}=-{\overrightarrow{\mathit{j}}}_{\text{q}}={\mathrm{\eta }}_{\text{q}}\nabla \mathit{T}$, we find that the application of a temperature gradient to the superconductor creates a phase difference Δθ = (2mη_q/eℏn_s)ΔT across the superconductor. The phase difference is analogous to the difference in electrochemical potential across the ends of a normal metal placed in a temperature gradient (4). When the ends of the superconductor, S, are connected by a dissimilar superconductor, S′, making a closed loop, as in Fig. 1A, a circulating supercurrent, I_cs, flowing within penetration depth, Λ, is induced, generating a thermoelectric magnetic field, B_Th, and a corresponding thermoelectric magnetic flux Φ_Th.

Fig. 1. Bimetallic superconducting loop and experimental setup for thermoelectric flux measurements.

(A) Bimetallic superconducting loop in the temperature gradient. Thermoelectric quasiparticle current, I_q, in the bulk of the superconductor with a smaller gap is opposed by counterflowing supercurrent I_s with superconducting phase difference, Δθ, created and circulating current I_cs induced within the penetration depth, Λ; with an increase in average temperature, the penetration depth acquires an increment, δΛ, that results in an increase in the effective area of the loop; the screening current I₂ keeps the total magnetic flux through the bimetallic loop constant. (B) Diagram of experimental setup. A bimetallic loop made of different superconductors is placed within a loop of a hybrid quantum interferometer a-b-c-d. The total flux through the bimetallic loop, Φ₂, is created by the external magnetic field B and the fields B_Th and B₂ induced by thermoelectric circulating current I_cs and screening current I₂. The interferometer measures the superconducting phase difference φ between c and d that is proportional to the total magnetic flux through the interferometer loop. This includes the flux through the bimetallic loop and externally induced flux that is partially screened by the current I₁. (C) False-colored scanning electron micrograph of a bimetallic loop coupled to a heater and hybrid quantum interference device (HyQUID). (D) A fluxless heater f generating a temperature gradient in the bimetallic loop by local spot heating of the contact e. The temperature T₁ at e is measured by superconductor/normal/superconductor (SNS) thermometer g. (E) The HyQUID measuring a superconducting phase difference between c and d with folded normal wires insulated by a spacer [see (B) for details].

Practical measurements of thermoelectric magnetic flux became feasible with the emergence of extremely sensitive magnetometers based on superconducting quantum interference devices (SQUIDs) (5). A SQUID contains a superconducting loop interrupted by “weak links” (5), making it sensitive to the magnetic flux Φ₁ threading the loop. When the loop is coupled to the bimetallic superconducting loop via mutual inductance, M, the flux Φ₁ becomes dependent on the supercurrent I₂ circulating in the bimetallic loop: Φ₁ = Φ_1e − MI₂, where Φ_1e is the flux through the interferometer loop area generated by sources other than the bimetallic loop. The circulating supercurrent I_cs and the associated thermoelectric flux Φ_Th can be measured when one of the contacts of the bimetallic loop is heated, thus creating a temperature gradient. The first thermoelectric flux measurement (6) was in a reasonable agreement with the existing theory. However, further experiments (4, 7, 8) showed temperature-dependent magnetic fluxes up to five orders of magnitude larger than predicted by the theory (3) with unexpected dependence on the temperature. From an experimental viewpoint, the genuine thermoelectric effect could be masked by concomitant effects that may occur in the presence of a background field due to the temperature dependence of the penetration depth, Λ, of the superconductor (9). To minimize spurious Λ effects, an ingenious experiment was undertaken using a bimetallic superconducting torus (4) to minimize contributions of the background magnetic field. Nevertheless, an unexpectedly large temperature-dependent flux was still observed. Such a large thermoelectric flux would imply “giant” values for the superconducting component of the thermoelectric current and similar values for the quasiparticle component, ${\overrightarrow{\mathit{j}}}_{\text{q}}$; yet, the experiments that followed revealed a paradoxical mismatch between the values of ${\overrightarrow{\mathit{j}}}_{\text{s}}$ and ${\overrightarrow{\mathit{j}}}_{\text{q}}$. The measured quasiparticle current (10) was orders of magnitude smaller than the supercurrent and agreed well with the theory (11). No plausible explanations for the observations and no experimentally supported analysis have been offered to date, and the paradox remained unresolved [see Galperin et al. (12) and Gurevich et al. (13) and references therein].

Here, we resolve this paradox. We explain the reason why the masking Λ effects were not excluded completely in previous experiments and present experimental results of the separation of genuine thermoelectric flux from the masking effects. Theoretically, we deduce a new formula connecting the thermoelectric magnetic flux and the current circulating in the loop in the presence of a temperature gradient. For calculations, we use a method suggested recently (13), taking into account the energy balance in the system. Our proof-of-principle experiments agree well with the new theory.

In the presence of background external field, B, the current circulating in the bimetallic loop, I₂, is present even in the absence of the temperature gradient. It keeps the number, k, of flux quanta trapped in the bimetallic loop constant, in accordance with the quantization rule for the flux, Φ₂, through the bimetallic loop: Φ₂ = Φ_2e − L₂I₂ = kΦ₀, where Φ_2e = A₂B is the flux generated by external field, B, through the bimetallic loop area, A₂; L₂ is the self-inductance of the loop; and k = 0, ±1, ±2 and so on. As a result, the total flux through the interferometer loop is Φ₁ = Φ_1e + (M/L₂)(A₂B + kΦ₀). Heating of one of the contacts of the bimetallic loop not only creates a temperature gradient but also increases the average temperature of the loop. Although the temperature gradient induces a thermoelectric flux Φ_Th, the rise in the temperature results in an increase of the penetration depth Λ → Λ + δΛ (Fig. 1A) with the corresponding changes of the effective area of the loop δA₂ = (dA₂/dΛ)δΛ and inductances δL₂ = (dL₂/dΛ)δΛ and δM = (dM/dΛ)δΛ. Taking into account that usually the changes are relatively small, δA₂ < < A₂, δL₂ < < L₂, and δM < < M, the flux increment induced by heating is given as δΦ₁ = δ(MA₂/L₂)B + δ(M/L₂)kΦ₀ + (M/L₂)Φ_Th.

The signal measured by the quantum interferometer is a periodic function of Φ₁ with the period equal to the flux quantum Φ₀ = h/2e. In practice, the interferometer signal oscillations are measured as a function of magnetic field, B, with the period and phase of the oscillations depending on M, L₂, A₂, and the k number. The flux increment, δΦ₁, results in the phase shifts of the oscillations. They include the thermoelectric flux contribution (the third term) and contributions due to the changes in the penetration depth, the Λ effect (the first and second terms). The first Λ effect phase shift corresponds to the changes in the oscillations’ period and depends on the ambient magnetic field. The second shift is k-dependent. The Λ effect contributions may exceed the contribution of thermoelectric flux by several orders of magnitude (4, 8, 9). Although the first phase shift can be calculated using precise enough measurements of the period of oscillations at different temperature gradient values, the second shift is practically indistinguishable from the thermoflux contribution when the control of the number of flux quanta, k, trapped in the bimetallic loops is lacking. Moreover, fluctuations in the k values from one experiment to another may lead to scatter in the measured values. The previous experiments used macroscopic loops of several millimeters in diameter that corresponded to about 10⁶ trapped flux quanta due to the geomagnetic field. To precisely determine and control the k value in such large loops as well as measure the period of oscillations, one had to control the absolute magnitude of magnetic field with a precision of about 10⁻⁹ T, an extremely difficult task that was never done. Consequently, separation of the genuine thermoelectric flux has not been completed, leaving the puzzling discrepancy between the theory and the experiment unexplained (1, 12, 13). The conceptual and technological advance reported here is based on highly sensitive hybrid quantum interferometers (14, 15) coupled to the bimetallic loops fabricated using modern nanolithography with areas up to five orders of magnitude smaller than those measured previously. This allowed both precise control of the number of trapped flux quanta and measurements of the changes in the oscillations’ period under the temperature gradient. The thermoflux is separated using characteristic phase shifts in the quantum interference oscillations that are independent of the applied magnetic field and k numbers. The measured thermoelectric flux changes its sign upon reversal of the direction of the temperature gradient, a distinctive feature of a genuine thermoelectric effect.

RESULTS

Theory of thermoelectric magnetic flux

The thermoelectric magnetic flux, Φ_Th, and the circulating current, I_cs, are connected by the formula

$${\mathrm{\Phi }}_{\text{Th}}={\mathit{L}}_2{\mathit{I}}_{\text{cs}}$$ (1)

We consider a superconducting thermocouple consisting of a superconductor, S, and a superconductor, S′, with a much larger gap and, hence, negligible quasiparticle current. This situation corresponds with our experiments; however, it is simple to generalize it to an arbitrary superconductor S′.

By the current conservation, the circulating current is independent of the coordinate, x, along the bimetallic loop, I_q(x) + I_s(x) = I_cs = const, where I_q and I_s are the bulk thermoelectric quasiparticle and superconducting currents, respectively. The theory of thermoelectric quasiparticle current, I_q, was developed by Galperin et al. (3), Guénault and Webster (16), and Van Harlingen et al. (4); however, calculation of the superconducting circulating current, I_cs, which generates the thermoelectric magnetic flux, was not attempted. Here, we proceed with the calculation using minimization of the total energy of the superconducting loop, W, with respect to I_cs, as suggested by Gurevich et al. (13); see the Supplementary Materials for details.

The total energy consists of the energy of the magnetic field created by the circulating current, ${\mathit{W}}_{\text{m}}={\mathit{L}}_2{\mathit{I}}_{\text{cs}}^2/2$, and the kinetic energy of superconducting electrons, ${\mathit{W}}_{\text{k}}={\mathit{L}}_{\text{k}}{\mathit{I}}_{\text{s}}^2/2$, where L_k is the kinetic inductance. We find that the circulating current is defined by the quasiparticle thermoelectric current, I_q(0), and the kinetic inductance, L_k(0), of the weaker superconductor at the hot contact: I_cs = I_q(0)L_k(0)/(L₂ + L_k(0)).

Next, we calculate the quasiparticle current at the hot contact, I_q(0) = − η_q(0)(dT_q/dx)s, where s is the cross-sectional area of the superconductor. We use the heat equation that relates the rate at which the quasiparticles accumulate the energy from the heat flow, ${\dot{\mathit{Q}}}_{\text{q}}=-{\mathrm{\kappa }}_{\text{q}}\mathit{s}(\mathit{d}{\mathit{T}}_{\text{q}}/\mathit{d}\mathit{x})$, and the rate of quasiparticle energy transfer to the “local” phonons in the wire, ${\dot{\mathit{Q}}}_{\text{qp}}=\mathrm{\Sigma }({\mathit{T}}_{\text{q}}^{\mathit{n}}-{\mathit{T}}_{\text{p}}^{\mathit{n}})\mathit{s}\mathit{d}\mathit{x}$; T_p is the phonon temperature, κ_q is the quasiparticle thermal conductivity, and Σ is a material electron-phonon interaction parameter (17). We obtain ${\mathit{I}}_{\text{q}}(0)=({\mathrm{\eta }}_{\text{q}}(0)/{\mathrm{\kappa }}_{\text{q}}(0))\langle \mathrm{\Sigma }({\mathit{T}}_{\text{q}}^{\mathit{n}}-{\mathit{T}}_{\text{p}}^{\mathit{n}})\rangle \mathit{s}{\mathit{l}}_0$, where κ_q(0) is the thermal conductivity at the hot contact and the brackets 〈 … 〉 mean an average over the length l₀ of the weaker superconductor, with n = 5 or 6 depending on the relationship between the electron mean free path, l, and the phonon wavelength, λ_p (18, 19). Using connections of the coefficients η_q(0), κ_q(0) and Σ in the superconducting state to their values in the normal state, η_N(0), κ_N(0) (3, 20) and Σ_N (21), and assuming that the normal state quasiparticle thermal conductivity κ_N(0) obeys the Wiedemann-Franz law ${\mathrm{\kappa }}_{\text{N}}(0)=\mathcal{L}{\mathrm{\sigma }}_{\text{N}}{\mathit{T}}_{\text{q}}(0)$, where $\mathcal{L}=2.4\times {10}^{-8}$ V²/K² is the Lorenz number and σ_N is conductivity of the smaller gap superconductor in the normal state, we arrive at the following formula for the thermoelectric flux

$${\mathrm{\Phi }}_{\text{Th}}=-{\mathrm{\alpha }}_{\text{N}}{\mathrm{\Sigma }}_{\text{N}}\langle \mathit{G}({\mathit{T}}_{\text{q}})({\mathit{T}}_{\text{q}}^{\mathit{n}}-{\mathit{T}}_{\text{p}}^{\mathit{n}})\rangle {\mathit{L}}_{\text{eff}}\mathcal{V}/\mathcal{L}$$ (2)

where α_N = η_N(0)/σ_NT_q(0) (V/K²) is the gradient in the dependence of thermopower on temperature that is constant in the investigated temperature range [see, for example, Mamin et al. (10)], L_eff = L₂L_k(0)/(L₂ + L_k(0)) is the effective inductance, $\mathcal{V}$ is the volume of the small-gap superconductor, and the function G(T_q) describes the drop in electron-phonon interaction upon the temperature decrease below the critical temperature. The value of the gap at the hot contact is the smallest within the bimetallic loop, making the contact essentially a weak link. The kinetic inductance of such a link is given by L_k = Φ₀/2πI_c, where I_c is the critical current (22). At temperatures close to critical, the kinetic inductance diverges as the critical current goes to zero. The value of the circulating current approaches that of the quasiparticle current, I_cs → I_q(0). According to Eq. 2, the thermoflux originates from the temperature difference between the heated electron system and the phonon system. It is a continuous function at the critical temperature and above it. We started with finite supercurrent I_s; however, no singularities in the thermoflux are expected at the critical temperature. When the bulk supercurrent fades, the value of circulating current I_cs continuously approaches the value of quasiparticle current, I_q. This is a result of current conservation. Simultaneously, kinetic inductance diverges at the critical temperature and the effective inductance approaches the value of geometric inductance, L₂.

The formula gives several conditions to maximize the thermoflux in the superconducting state. Besides an obvious condition of the large thermopower, the formula requires large effective inductance and strong electron-phonon interaction as well as coupling of the “local” phonon system to the environment of the bimetallic loop to ensure difference in the electron and phonon temperatures. The formula is valid when one of the superconductors enters the normal state. In this case, by the current conservation, the circulating current is equal to the thermoelectric current in the normal wire, I_cs = I_N(0); in addition, in this case, G(T_q) = 1 and L_eff = L₂.

Experiments

Our experimental setup is shown in Fig. 1. The bimetallic loop is placed inside the measuring loop of a quantum magnetometer (Fig. 1, B and C). The characteristic magnetic field B = Φ₀/A₂ ≈ 10⁻⁵ T was high enough to allow precise control of the k number.

We used a magnetometer based on a HyQUID (14, 23, 24). Similar to a standard SQUID, the magnetometer consisted of a superconducting loop interrupted by a weak link. The weak link consisted of a normal conductor connected to the superconductor at c and d (Fig. 1, B to E). The electrical resistance R between points a and b is an oscillating function of the total magnetic flux Φ₁ threading the interferometer loop

$$\mathit{R}={\mathit{R}}_0-\mathit{r} \text{cos}(2\mathrm{\pi }{\mathrm{\Phi }}_1/{\mathrm{\Phi }}_0)$$ (3)

where R₀ is independent of magnetic field and r is the amplitude of oscillations. To determine the thermoelectric flux, we measured several resistance oscillations as a function of applied magnetic field. The values B_n,k of the field at the extrema of the resistance correspond to Φ₁ = nΦ₀, with n = (2m + 1)/2 for maxima and n = m for minima; m = 0, ±1, ±2, ±3 and so on. In the absence of the temperature gradient, the measured values B_n,k can be found using the formula

$${\mathit{B}}_{\mathit{n},\mathit{k}}\mathit{A}+(\mathit{M}/{\mathit{L}}_2)\mathit{k}{\mathrm{\Phi }}_0=\mathit{n}{\mathrm{\Phi }}_0$$ (4)

where A = A₁ − (M/L₂)A₂ is the effective area of the measuring interferometer loop and A₁ is its actual area (see Materials and Methods for details).

Examples of oscillations as a function of applied magnetic field are shown in Fig. 2. Figure 2A shows the oscillations in the absence of the temperature gradient at different k numbers. The k numbers were manipulated by thermocycling of the bimetallic loop between the normal state and the superconducting state in the precalculated applied magnetic field. When the k number was changed to k + Δk, the positions of the extrema were shifted by ΔB, which was constant for all maxima and equal to ΔB = (M/L₂)(Φ₀/A)Δk with (M/L₂) = 0.2, which is in agreement with our design value. When the temperature gradient is established, the total flux Φ₁ = nΦ₀ corresponding to a particular extremum of the resistance is intact, whereas the measured positions of the extrema are shifted to the new values ${\mathit{B}}_{\mathit{n},\mathit{k}}^{\prime }$ satisfying the equation ${\mathit{B}}_{\mathit{n},\mathit{k}}^{\prime }{\mathit{A}}^{\prime }+({\mathit{M}}^{\prime }/{\mathit{L}}_2^{\prime })(\mathit{k}{\mathrm{\Phi }}_0-{\mathrm{\Phi }}_{\text{Th}})=\mathit{n}{\mathrm{\Phi }}_0$, where A′ = A + δA and $({\mathit{M}}^{\prime }/{\mathit{L}}_2^{\prime })=(\mathit{M}/{\mathit{L}}_2)+\mathrm{\delta }(\mathit{M}/{\mathit{L}}_2)$ are the values modified by the Λ effect. We used the difference $\mathrm{\Delta }{\mathit{B}}_{\mathit{n},\mathit{k}}={\mathit{B}}_{\mathit{n},\mathit{k}}-{\mathit{B}}_{\mathit{n},\mathit{k}}^{\prime }$, the phase shift in oscillations, to determine the thermoelectric flux

$$\frac{\mathrm{\Delta }{\mathit{B}}_{\mathit{n},\mathit{k}}}{{\mathit{B}}_0}=\mathit{a}\mathit{n}-\mathit{b}\mathit{k}-\mathit{c}$$ (5)

Fig. 2. Phase shifts in oscillations at different k numbers and heater currents.

(A) Oscillations at different k numbers of magnetic flux quanta trapped in the bimetallic loop with the heater current OFF. Left maximum corresponds to k = 4; middle, k = 5; right, k = 6; 1, n = 8.5; 2, n = 9.5; 3, n = 10.5. A change in k number introduces constant phase shift, leaving the period intact. (B) Oscillations with the heater current OFF (red line) and with the heater current ON (black line). (C) The shifts ΔB_n,k [as in (B)] of the values B_n,k with the heater current ON versus the total flux Φ₁ = nΦ₀ at different fixed k numbers of the trapped flux quanta. (D) The shifts ΔB_n,k versus k number at a fixed total flux. (E) Three-dimensional (3D) representation of the shifts ΔB_n,k versus k number and the total flux Φ₁ = nΦ₀ through the measurement loop. The planes correspond to different fixed heater currents I_h = 10, 15, 20, and 25 μA.

The coefficients a = δA/A and b = δ(M/L₂) describe the phase shifts due to the Λ effect, and the phase shift c is associated with the thermoelectric flux, Φ_Th, and is independent of n, k and magnetic field; B₀ = Φ₀/A is the period of oscillations. To create a temperature gradient, we used a heater made of a normal metallic silver wire (Fig. 1, C and D) connected to one of the contacts of the bimetallic loop. Figure 2B shows the oscillations at a fixed k number at different heater currents. The heater produced two effects described in Eq. 5: (i) the phase shift associated with the thermoelectric flux, Φ_Th, and (ii) the change in the period of oscillations due to the Λ effect.

The dependencies ΔB_n,k/B₀ versus n and k at fixed heater currents are shown in Fig. 2 (C and D). They are linear, in agreement with Eq. 5. The slopes give the values a and b.

Equation 5 at a constant heater current is an equation for a plane in 3D space with x, y, and z coordinates corresponding to n, k and ΔB_n,k/B₀. The planes plotted using measurement results are shown in Fig. 2E. They allow visualization of the Λ effect contribution (an − bk) at different k and n values at a glance. The Λ effect is minimal along the line where the planes intersect. The n and k numbers along the line satisfy the relation an ≈ bk; hence, keeping ratio k/n constant and changing p = Integer(bk/an), we move along the line. For this particular sample, b/a ≈ 5/2 and k/n = 2/5. To decrease error in the value of c, we averaged measurements at different combinations of n and k. The measured values of c as a function of p at several values n and k are shown in Fig. 3A. As expected, the value of c that is directly related to thermoelectric flux was independent of k and n and depended on the heater current. Here, we emphasize that to obtain the value of the thermoflux, one has to know the absolute values of n and k. Thermocycling adds integers to the k numbers: k = k₀ + m, where m = ±1, ±2, etc. and k₀ = Integer(A₂B_r/Φ₀) is the number of flux quanta trapped because of the residual magnetic field, B_r, in the solenoid. We have measured B_r using sensitive Hall probes and found it to be close to the geomagnetic field B_g = 41 μT in our case. With Φ₀/A₂ = 10 μT, we find k₀ = 4. In previous experiments, this number was up to five orders of magnitude larger, leading to a much larger Λ effect (8, 9). Next, we have undertaken an experiment to verify that the measured thermoflux shows the signature of the genuine thermoflux with a change of the sign when the temperature gradient is reversed. We have nanofabricated a mirror-reflected bimetallic loop with the aluminum part replacing lead and vice versa. This is equivalent to a change in the direction of the temperature gradient and circulation of the thermoelectric current generating thermoelectric flux. The values of c versus p for the two samples are plotted in Fig. 3A. The sign of c changes when the temperature gradient is reversed, as expected for the genuine thermoflux.

Fig. 3. Thermoelectric phase shifts and magnetic flux at different heater currents and hot contact temperatures.

(A) The values of ΔB_n,k versus p at different k and n values close to the line ΔB_n,k = 0 in Fig. 2E with Λ effect excluded. (B) The measured thermoelectric flux values at different hot spot temperatures for two samples with opposite direction of the temperature gradient (symbols). Curves are theory predictions according to Eq. 2.

DISCUSSION

To compare the measured values of the thermoflux with theory, we take into account the following aspects of our experiments. The thermal resistance of the wire/substrate interface (the Kapitza resistance) for our aluminum films was negligible because the thickness, h = 80 nm, was much less than the phonon wavelength, λ_p ≈ hv_s/k_BT ≥ 200 nm at T ≤ 1.2 K, where v_s ≈ 5 × 10³ ms⁻¹ is the sound velocity. The phonon temperature in such films is close to the substrate temperature, T_p ≈ T_sub (21), and the phonon term, ${\mathit{T}}_{\text{p}}^{\mathit{n}}$, in the difference, ${\mathit{T}}_{\text{q}}^{\mathit{n}}-{\mathit{T}}_{\text{p}}^{\mathit{n}}$, as seen in Eq. 2, can be neglected because the substrate temperature is estimated to be lower than the quasiparticle temperature T_sub < T_q; therefore, ${\mathit{T}}_{\text{q}}^{\mathit{n}}-{\mathit{T}}_{\text{p}}^{\mathit{n}}={\mathit{T}}_{\text{q}}^{\mathit{n}}(1-{({\mathit{T}}_{\text{sub}}/{\mathit{T}}_{\text{q}})}^{\mathit{n}})\approx {\mathit{T}}_{\text{q}}^{\mathit{n}}$ (17). Second, to calculate the effective inductance L_eff, we consider the aluminum at the hot contact as essentially a weak link within the bimetallic loop. This is justified because the gap in aluminum is much smaller than that in lead. The kinetic inductance of such a link is given by L_k = Φ₀/2πI_c, where I_c is the critical current (22). The absolute value of thermoelectric flux at temperatures close to critical is ${\mathrm{\Phi }}_{\text{Th}}({\mathit{T}}_{\text{c}})\approx {\mathrm{\alpha }}_{\text{N}}{\mathrm{\Sigma }}_{\text{N}}\langle {\mathit{T}}_{\text{c}}^{\mathit{n}}\rangle {\mathit{L}}_{\text{eff}}\mathcal{V}/\mathcal{L}$, with L_eff(T_c) ≈ L₂, because the kinetic inductance at the hot end acquires large values L_k(0) >> L₂ when the critical current approaches zero at T ≈ T_c. At temperatures below critical, the value of the thermoflux can be written as Φ_Th(T_q) = Φ_Th(T_c)〈G(T_q/T_c) ⋅ (T_q/T_c)ⁿ〉(L_eff/L₂) with scaling factors describing the decrease in quasiparticle concentration, the electron-phonon interaction, and the decrease in the kinetic inductance (see the Supplementary Materials for details).

The experimentally measured values fitted with the theoretical curves are shown in Fig. 3B. We used the calculated value L₂ ≈ 10⁻¹⁰ H for the geometrical inductance of our bimetallic loops, the cross section s = 6 × 10⁻¹⁴ m² and the length l₀ = 5 × 10⁻⁵ m, and critical temperature T_c ≈ 1.2 K. Using a typical value of the electron mean free path l ≈ 50 nm and the wavelength of phonons in our aluminum films calculated above, one would expect the films to belong to the strong disorder limit l << λ_p with the electron-phonon interaction described by n = 6 (18). However, as shown in the Supplementary Materials, the results of fit with n = 5 do not differ much from that with n = 6 in the investigated temperature range. Both of the fits give the value of the product of the thermopower, α_N, and the electron-phonon interaction parameter in the normal state, Σ_N: α_NΣ_N ≈ 10³ VW/Kⁿ⁺²m³. Using the sample-dependent values of α_N in a realistic range, 2 × 10⁻⁸ to 6 × 10⁻⁷ V/K², we obtain the sample-dependent values of Σ_N in the range 2 × 10⁹ to 5 × 10¹⁰ W/m³Kⁿ, which is in agreement with previous measurements (25). To fit the dependence of the thermoflux on temperature, we have taken the dependence of critical current at the hot contact in the form I_c = I₀(1 − T_q(0)/T_c) (26) with a fitting parameter value of I₀ = 30 μA, which is realistic for our wires (see the Supplementary Materials for more comments on the temperature dependence).

In conclusion, we have developed a method allowing the separation of the genuine thermoelectric flux from spurious Λ effects. Theoretically, we deduce the formula for the circulating current by minimizing the total energy of the system. We solve the heat equation for the loop and calculate the thermoelectric flux using the relation of the circulating current to the heat flow and the energy transfer rate from quasiparticles to phonons. The measured value of the thermoelectric flux agrees well with the theory. The technological and theoretical advance reported here opens a new avenue for exploration of thermal phenomena in superconducting devices (27–30), which is critical for energy management in nanoscale structures.

MATERIALS AND METHODS

Magnetometry with HyQUID

The HyQUID is especially suitable for this experiment. Its fabrication technology and composition are compatible with the other elements, which allows the whole structure to be placed on the same chip. This avoids the use of macroscopic leads to connect the bimetallic loop and the magnetometer, thus preventing stray signal pickup and the introduction of parasitic inductance (5). The device is designed to reduce coupling of the flux-sensitive loop with the measuring electronics: the normal segment of the flux-sensitive loop makes contact with the measuring wire a-b at a single point. As a result, the current noise is not transferred to the interferometer loop from the measuring circuit. To reduce inductive coupling of the flux-sensitive loop with the measuring circuit, the a-b wire was folded (see Fig. 1, B, C, and E, for details). Furthermore, the device was designed to make the voltage between the N/S interfaces negligible, thus reducing the Josephson radiation and extra noises that are typical in standard SQUID-based magnetometers. The interferometer proved to be a sensitive probe for minute supercurrents (23).

The total magnetic flux Φ₁ in Eq. 3 threading the interferometer loop was Φ₁ = BA₁ − L₁I₁ − MI₂, where B is the applied magnetic field; A₁ and L₁ are the area and inductance of the interferometer loop, respectively; and I₁ is the Josephson screening current in the interferometer loop I₁ = I_c1 sin(2πΦ₁/Φ₀). To avoid hysteresis, the HyQUID was designed so that the screening factor was less than unity: L₁I_c1 < 1. The measured values B_n,k in this regime were independent of the screening current because it completely vanishes at the extrema: I₁ ∝ sin(2πn) = 0. Calculating the current I₂ using quantization of the flux through the bimetallic loop Φ₂ = B_n,kA₂ − L₂I₂ = kΦ₀, we arrived at Eq. 5 for the measured values B_n,k.

The bimetallic loop was made of lead and aluminum with critical temperatures 7.2 and 1.2 K, respectively. Thus, the concentration of quasiparticles in lead was negligible in the investigated temperature range 0.25 to 1.1 K. The structure consisted of five layers. The first layer was silver; the second, spacers made of silicon monoxide; the third, upper silver layer; the fourth, aluminum; and the fifth, lead. The deposition was performed in medium vacuum of 10⁻⁶ Torr. To provide a clean interface for good electrical contact between the layers and good thermal contact with the substrate, we used in situ argon plasma etch. The thickness of silver, aluminum, silicon monoxide, and lead layers was 50, 80, 30, and 150 nm, respectively. The areas of measurement and bimetallic loops were A₁ = 380 μm² and A₂ = 170 μm², respectively. To apply magnetic field in a wide range, we used a superconducting solenoid. For fine field tuning, a wire made of lead was fabricated on the same chip (Fig. 1C). The measurements of the resistance R were made using a standard low-frequency lock-in technique. To provide optimal operation of the magnetometer, the length of the c-d wire, L_cd, was made smaller than the electron phase breaking length, ${\mathit{L}}_{\mathrm{\phi }}=\sqrt{\mathit{D}{\mathrm{\tau }}_{\mathrm{\phi }}}$, where D is the diffusion coefficient and τ_Φ is the electron phase breaking time; however, L_cd was large enough to suppress the Josephson current so that L₁I₁ < Φ₀ to avoid hysteresis (23, 24). These requirements defined the value of the resistance of the L_cd wire, R_cd. The resistivity of silver was ρ = 4 × 10⁻⁸ Ω ⋅ m, which gave the typical values of resistance R_cd ≈ 10 Ω for a given material and wire dimensions. The resistance of the L_ab wire, R₀, in our interferometer was close to R_cd. The value of r/R₀ was in the range 2 to 15%, depending on the dimensions of the wires, in accordance with theory and previous experiments (14, 24, 31).

Thermometry

The temperature gradient was created by a heater made of a normal metallic silver wire (Fig. 1, C and D). The heater was made bifilar to minimize magnetic flux generated by the heating current. The residual flux induced by the heater was canceled by averaging the measurements with opposite heater currents. The heater was connected to one of the contacts of the bimetallic loop by a silver wire. To measure the temperature at the hot end, T_q(0), we used the superconducting/normal/superconducting junction (32). The thermometer was placed away from the hot contact at point g to avoid coupling of the thermometer wires to the bimetallic and interferometer loops (Fig. 1, C and D). To make the temperatures at g and e (Fig. 1D) as close to each other as possible, the heater was thermally coupled to the thermometer and to the loop in a similar way. Points g and e were connected to the heater at point f by normal silver wires of the same length and cross-sectional area. Both wires had similar thermal boundary conditions at the substrate and the superconductors; hence, the temperature distribution in the wires obeyed similar heat equations. To calibrate the thermometer, we used the transition of aluminum in the bimetallic loop to the normal state that was measured using the dependence of penetration depth, Λ, on temperature (33); for details, see the Supplementary Materials. The substrate temperature T₀ close to the cold end was measured using the sensitivity of the amplitude of oscillations to the temperature; an increase in T₀ was negligible at the heater currents investigated.

SUPPLEMENTARY MATERIALS

Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/2/2/e1501250/DC1

Calculating thermoelectric flux

Thermometry

Fig. S1. Conversion of currents at the interface between two superconductors, S and S′, with different energy gaps, Δ and Δ′.

Fig. S2. Calculated dependence of the thermoelectric flux scaling factors on the quasiparticle temperature normalized to critical temperature, T_c.

Fig. S3. Calculated temperature dependence of thermoelectric flux, Φ_th, normalized to its value, Φ_th(T_c), at critical temperature with different values of critical current, I₀, and electron-phonon parameter, n.

Fig. S4. Dependence of critical current in the SNS thermometer on the length of the normal element at base temperature T = 245 mK.

Fig. S5. Differential resistance versus bias current curves for an SNS thermometer at different bath temperatures.

Fig. S6. Differential resistance versus bias current curves for an SNS thermometer at different heater currents.

Fig. S7. Temperature calibration curve.

Fig. S8. Determination of the heater current corresponding to the onset of superconductivity in the bimetallic loop.

Reference (34)