Ferromagnetic insulator-based superconducting junctions as sensitive electron thermometers

We present an exhaustive theoretical analysis of charge and thermoelectric transport in a normal metal-ferromagnetic insulator-superconductor (NFIS) junction, and explore the possibility of its use as a sensitive thermometer. We investigated the transfer functions and the intrinsic noise performance for different measurement configurations. A common feature of all configurations is that the best temperature noise performance is obtained in the non-linear temperature regime for a structure based on an europium chalcogenide ferromagnetic insulator in contact with a superconducting Al film structure. For an open-circuit configuration, although the maximal intrinsic temperature sensitivity can achieve $10$nKHz$^{-1/2}$, a realistic amplifying chain will reduce the sensitivity up to $10$$\mu$KHz$^{-1/2}$. To overcome this limitation we propose a measurement scheme in a closed-circuit configuration based on state-of-art SQUID detection technology in an inductive setup. In such a case we show that temperature noise can be as low as $35$nKHz$^{-1/2}$. We also discuss a temperature-to-frequency converter where the obtained thermo-voltage developed over a Josephson junction operated in the dissipative regime is converted into a high-frequency signal. We predict that the structure can generate frequencies up to $\sim 120$GHz, and transfer functions up to $200$GHz/K at around $\sim 1$K. If operated as electron thermometer, the device may provide temperature noise lower than $35$nKHz$^{-1/2}$ thereby being potentially attractive for radiation sensing applications.

We propose and analyze a temperature-to-frequency converter based on a normal metal-ferromagnetic insulatorsuperconductor (NFIS) junction embedded in a superconducting loop which contains a superconducting quantum interference device (SQUID). By setting a temperature difference across the NFIS junction a thermovoltage will be generated across the circuit if the SQUID is in the resistive regime. This thermovoltage depends on both the magnitude and sign of the temperature difference, and will generate radiation at the Josephson frequency. In Eu-based FIs joined with superconducting Al the structure is in principle capable to generate frequencies up to ∼ 120GHz, and transfer functions up to 200GHz/K at around ∼ 1K. Yet, if operated as electron thermometer, the device may provide temperature noise better than 35nK Hz −1/2 thereby being potentially attractive for radiation sensing applications.
It has been suggested recently that the spin-splitting induced in a superconductor (S) placed in contact with a ferromagnetic insulator (FI) can be exploited in different kinds of spin caloritronic devices such as heat valves 1,2 or thermoelectric elements 3,4 . They can be used as building blocks in phase-coherent thermoelectric transistors 5 , and for the creation of magnetic fields induced by a temperature gradient in Josephson junctions (JJs) due to the thermophase effect 6 . Yet, NFIS junctions have been proposed as well for efficient electron cooling of a normal metal 7 .
Here we propose a prototype structure based on the FIS building block acting as an efficient temperature-tofrequency converter that can be used, for example, for sensitive electron thermometry as well as for radiation sensing applications [8][9][10][11][12] . Our device consists of a normal metalferromagnetic insulator-superconductor (NFIS) junction, denoted here as the thermoelectric element (TE), which is connected, via the superconducting wires S 1 , to a dc superconducting quantum interference device (SQUID), as shown in Fig. 1(a). A temperature difference localized between the N and S side of TE induces thermoelectricity 4 . By properly tuning the magnetic flux (Φ) piercing the SQUID one can set the latter to operate in two regimes: i) Peltier regime, where the SQUID is in the Josephson regime, and a circulating thermocurrent is generated. This regime can be probed by an inductive measurement of the current. ii) Seebeck regime where the SQUID is in the resistive regime thereby allowing the generation of a Seebeck thermovoltage (V ) across the TE element.
In what follows we focus on the latter regime regime in which the generated thermovoltage will induce an ac-Josephson effect with oscillatory supercurrent at frequency ν = |V |/Φ 0 14 , where Φ 0 2.067 × 10 −15 Wb is the flux quantum. The frequency ν can be measured with great accuracy providing accurate information about temperature difference across the TE. a) Electronic mail: francesco.giazotto@sns.it b) Electronic mail: sebastian bergeret@ehu.es It is instructive to start with the description of the TE. We assume the S layer to be thinner than the superconducting coherence length ξ 0 , so that the exchange field (h exc ) induced in S by FI is spatially homogenous 15 . In such a case the superconductor density of the states (DoSs) is given by the sum of the densities for spin-up and spin-down quasiparticles, is the pairing potential that depends both on temperature T S in S and h exc , and it is computed self-consistently in a standard way 15,16 . We are interested in the current through the NFIS junction which is given by 4 Here R T is the normal-state resistance of the tunneling junction and N ± = (N ↑ ± N ↓ ). We assume thermalization on both the S and the N layer neglecting any deviation of the distribution functions from their equilibrium form 13 Here T N is the temperature in the N layer, −e is the electron charge and k B is the Boltzmann constant. The role of the FI layer is twofold: it acts as a spin filter with polarization P 17 and correspondly it is at the origin of the spin-split DoSs in the S layer. These two features have been experimentally demonstrated [18][19][20][21][22][23] . Finally, according to Eq. (1), even in the absence of a voltage bias across the junction a finite current I T E can flow provided T N = T S , as demonstrated in Ref. 4 .
Before analyzing the role of a temperature bias across TE, we first determine the current-voltage characteristics (IVCs) and differential conductance G of the NFIS junction. We set a low temperature, is the critical temperature of the superconductor, and ∆ 0 is the zero-temperature, zero-exchange field energy-gap. The results obtained from Eq. We now assume a finite temperature difference between the electrodes, δ T = T S − T N , and re-calculate the IVCs from   Fig. 2(a,b) reveal two main properties of the IVC. First, the IVC strongly depends on the amplitude of the temperature difference δ T : the larger the temperature difference, the larger is the current at low voltages. In the case that the S electrode is heated [see Fig. 2(a)], this trend is limited by the reduced critical temperature T * c < T c of the superconductor originating from the presence of a finite h exc which suppresses the ∆(T S , h exc ) calculated selfconsistently 15,16 . When T S → T * c , the TE goes into the normal state with ohmic characteristic [red curve in Fig. 2(a)]. There is another interesting feature of the IVCs: they strongly depend on the sign of δ T . For the same value of |δ T |, the current at V = 0 is larger when the N electrode is colder than the S one, i.e., when δ T > 0. In other words, the thermoelectric effect in the TE depends on the sign of the temperature difference. This feature was not realized in previous works 3,4 where only the linear response regime was analyzed.
If   pression of the superconducting energy gap. This explains the suppression of V 0 at large T S until it vanishes when superconductivity is fully destroyed. We note that V 0 reaches zero continuously owing to the fact that we have chosen values of h exc for which the superconducting-normal state transition is of the second order. A different temperature behavior of V 0 is obtained when S is kept at low temperature T S = 0.01T c and T N is varied, as shown in Fig. 2(d). In particular, besides the obvious change of sign, V 0 grows monotonically by increasing T N until it reaches an asymptotic value. In contrast to the V 0 (T S ) dependence shown in Fig. 2(c), heating of the N electrode does not affect the superconducting gap ∆, and therefore V 0 is not suppressed by increasing T N . It is also important to stress that the curves V 0 (T N ) depends strongly on the polarization P of the barrier [see Fig. 2(d)]. In particular, the larger P the larger is the thermovoltage V 0 (T N ) developed across the TE. By contrast, the V 0 (T S ) amplitude turns out to be almost unaffected by the value of P. The different behaviors with respect to the sign of δ T allow one to reconstruct both the amplitude and direction of the thermal gradient in the TE element even without directly addressing the sign of V 0 . This further information could be eventually exploited to reconstruct the spatial position of a heating event, thereby opening interesting possibilities to build detector-like devices.
Having analyzed the electronic transport in the TE we now focus on the temperature-to-frequency conversion process. This conversion is achieved with the device sketched in Fig. 1(a). The TE is connected via two superconducting arms S 1 to a dc-SQUID formed by two identical JJs. We assume to place a tunnel barrier between S and S 1 to isolate the S element ensuring its description as a thermally homogeneous superconductor with a spin-split DoSs, reducing any influence of S 1 arms and therefore, guaranteeing that the cur-rent through the TE is described by Eq. (1). Superconductors S and S 1 are Josephson coupled through the barrier and no additional voltage drop will occur. Furthermore, we assume the NS 1 junction to be a clean metallic contact, thereby contributing negligibly to the total resistance of the system but, for simplicity, we disregard the proximity effect induced into the N layer by the nearby superconductor S 1 16 . The electric current through the SQUID (I SQUID ) depends both on the voltage V and on the magnetic flux Φ piercing the loop. Within the RSJmodel (neglecting the loop inductance) the current is given by 14 where R J and I c are the shunting resistance and the Josephson critical current of each junction of the SQUID, respectively. Thus, if one applies a magnetic flux such that Φ/Φ 0 = 1/2 ± n, where n is an integer, the SQUID is operated in the resistive regime with an ohmic IVC, I SQUID = 2V /R J . In this case there is a time oscillating current through the interferometer with a frequency equal to the Josephson frequency, ν = |V |/Φ 0 . As discussed above, the value of V depends on the temperature difference δ T across the TE, and therefore the frequency emitted by the SQUID is a measure of δ T . In order to quantify the temperature-to-frequency conversion effect, one has to determine the voltage V developed for any given δ T imposed across the TE which satisfies the following equation where I T E is defined in Eq. (1). The solution to the above equation is given by the point in which the dashed line in Figs. 2(a) and 2(b) intersects the IVCs. Before analyzing the exact solution of Eq. (2), we discuss the linear response regime of the TE analyzed first in Refs 3,4 .
In this regime the voltage V and the temperature difference δ T T ≡ (T S + T N )/2 across the NFIS junction are small so that the current through the TE is given by ] −1 is the electric conductance, and α is thermoelectric coefficient 4 By imposing the condition (2) we obtain the voltage (V lin ) across the SQUID within the linear response V lin −PαR J δ T /[T (R J σ + 2)]. It is obvious that |V lin |, and therefore the frequency, does not depend on the sign of δ T . As we show below, this symmetric behavior of ν(δ T ) only holds in the linear response regime, i.e, for small values of δ T . Furthermore, from the expression for V lin , it clearly follows that the larger the polarization P the larger the achievable frequency ν, and that large R J values allow to maximize the achievable voltage drop across the structure. In the limit of open-circuit configuration, i.e., for R J → ∞, we get V lin −Pαδ T /(σ T ).
The situation drastically changes if one goes beyond the linear response. The results for the non-linear regime are summarized in Fig. 3. Panels 3(a) and 3(b) show the frequency generated by the SQUID, for positive and negative δ T , respectively. We have assumed a spin-filter efficiency P = 0.98 which is representative for EuO or EuS barriers 24 , a superconductor with T c = 3K which would be implementable with ultra-thin Al films [19][20][21][22] , and R T /R J = 0.1. If T N is kept at 0.01T c the maximum frequency is achieved around h exc ≈ 0.2∆ 0 for T S ≈ 0.75T c , and obtains values as large as ∼ 120GHz. By contrast, if T S is kept at low temperature, ν increases monotonically by increasing both T N and/or h exc but reaches smaller frequencies than in the previous case. A relevant figure of merit of the structure is represented by the temperature-to-frequency transfer function (τ), defined as τ = ∂ ν/∂ T . Figures 3(c) and 3(d) display the transfer functions corresponding to panels (a) and (b), respectively. In particular, τ exceeding 200 GHz/K around T S ∼ 1K can be achieved for h exc = 0.5∆ 0 by heating S, while τ up to ∼ 55 GHz/K can be achieved with the same values by heating N.
We now turn on addressing the noise performance of the temperature-to-frequency conversion operation when operating the NFIS junction as an electron thermometer 25 .
We identify the main source of noise in the current shot noise generated in the TE 26 , T N )] and the bias V is given by the solution of (2). The bias fluctuations are generated from the current noise via the load resistance seen from the TE, i.e., the parallel of SQUID resistance R J /2 and the TE resistance R d . Note that the differential resistance R d = ∂V /∂ I T E is calculated over the solutions of Eq. (2). The important quantity is represented by the frequency noise spectral density (S ν ) which can be expressed as S ν = S I R 2 /Φ 2 0 where R = R d R J /(2R d + R J ) describes the total load resistance as seen by TE. Finally, the intrinsic temperature noise (temperature sensitivity) per unit bandwidth of the thermometer (s T ) is related to the frequency noise spectral density as s T = √ S ν |τ| −1 . Figure 4(a) and (b) show the calculated square root of the frequency noise spectral density S ν for positive and negative δ T , respectively, calculated for the same parameters as in Fig. 3, and for R T = 1Ω. In particular, for positive δ T , the noise spectrum S 1/2 ν shows a non-monotonic behavior with a maximum at intermediate temperatures, and suppression at higher δ T . By contrast, for δ T < 0 the noise spectrum grows monotonically, and it is less influenced by h exc .
The behavior of s T for positive and negative δ T is displayed in Fig. 4(c) and (d), respectively. At small |δ T | we can see the noise sensitivity in the linear regime, which is independent of the sign of δ T . By increasing |δ T | the growth of S . When the maximum of |τ| is reached the best noise figure is roughly reached. The best performance of s T ∼ 35 nK Hz −1/2 is obtained around 1K for h exc = 0.5∆ 0 . After the minimum of s T the nonlinear behavior will dominate. For the δ T > 0 we see the peak of the divergency of |τ| −1 , i.e., the annihilation of the transfer function, as shown in Fig. 3(c). Differently, for δ T < 0 one observes a smooth increase of s T determined by the progressive reduction of the transfer function [see Fig. 3(d)]. We conclude by noting that the best noise performance is obtained in the nonlinear regime for |δ T | 1K where s T is almost independent of the sign of δ T . Therefore, the non-linearities that are useful to measure the temperature difference in the TE are important as well to maximize the temperature sensitivity.
In summary, we have theoretically investigated a temperature-to-frequency converter based on a normal metal-ferromagnetic insulator-superconductor (NFIS) thermoelectric element integrated with a SQUID. In particular, we have shown that with suitable structural material parameters the device allows for the generation of Josephson radiation at a frequency that depends on both the amplitude and sign of the temperature difference across the NFIS junction. Frequencies up to ∼ 120GHz and large transfer functions (i.e., up to 200GHz/K) around ∼ 1 − 2K can be obtained in a structure achievable with prototype FIs such as EuS or EuO providing P up to ∼ 98% in combination with superconducting Al thin films. If operated as a thermometer, the device is capable to provide intrinsic temperature noise down to ∼ 35nKHz −1/2 around 1K for a sufficiently large h exc . The proposed structure has the potential for the realization of effective on-demand on-chip temperature-to-frequency converters as well as sensitive electron thermometers or radiation sensors easily integrable with current superconducting electronics. We