Complete magnetic control over the superconducting thermoelectric effect

Jabir Ali Ouassou, César González-Ruano, Diego Caso, Farkhad G. Aliev, and Jacob Linder

Phys. Rev. B 106, 094514 – Published 21 September 2022

Abstract

Giant thermoelectric effects are known to arise at the interface between superconductors and strongly polarized ferromagnets, enabling the construction of efficient thermoelectric generators. We predict that the thermopower of such a generator can be completely controlled by a magnetic input signal: Not only can the thermopower be toggled on and off by rotating a magnet, but it can even be entirely reversed. This in situ control diverges from conventional thermoelectrics, where the thermopower is usually fixed by the device design.

Received 29 April 2022
Revised 29 July 2022
Accepted 9 September 2022

DOI:https://doi.org/10.1103/PhysRevB.106.094514

Physics Subject Headings (PhySH)

Proximity effect Seebeck effect Spin caloritronics Spin filtering Superconductivity

Ferromagnets Magnetic insulators Spin valves Superconductors Thermoelectric generators

Nonequilibrium Green's function Usadel equation

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Jabir Ali Ouassou ¹, César González-Ruano ², Diego Caso², Farkhad G. Aliev², and Jacob Linder¹

¹Center for Quantum Spintronics, Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway
²Departamento Física de la Materia Condensada C-III, INC and IFIMAC, Universidad Autónoma de Madrid, Madrid 28049, Spain

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Issue

Vol. 106, Iss. 9 — 1 September 2022

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Images

Figure 1
Proposed experiment. The junction's thermopower $S (φ)$ is determined by the angle $φ$ between the magnetic orientations $m$ and $m^{'}$ of the two ferromagnets. The latter is rotated using an external magnetic field, providing in situ control over the thermopower. When the normal metal is heated using, e.g., a light-emitting diode, one can measure a magnetically controlled thermoelectric voltage $Δ V_{OC} (φ)$ (open circuit) or current $I_{SC} (φ)$ (short circuit).
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Figure 2
Numerical results for the experiment proposed in Fig. 1. The polar angle is the magnetic misalignment $φ$ between the two ferromagnets, while the radius corresponds to $| Δ V_{OC} |$ and $| I_{SC} |$ , respectively. The color shows the thermoelectric polarity, i.e., whether the predicted values for $Δ V_{OC}$ and $I_{SC}$ are positive (blue) or negative (red). The total radius shown is (a) $1.1 \times 10^{- 2} V_{0}$ and (b) $6.5 \times 10^{- 5} I_{0}$ .
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Figure 3
Cartoon picture of the mechanism behind the proposed effect. (a) On the left we sketch the spin-split density of states in an S/FM bilayer at $T = 0$ . All states below the Fermi level are filled and all states above it are empty. On the right is an N layer at $T > 0$ . Spin-up electrons and spin-down holes can tunnel into the lowest-energy states of the S/FM subsystem, creating a net heat and spin flow to the left. However, the electron flow (blue) and hole flow (red) have opposite charges and equal magnitude, so there is no net electric current. (b) A fully spin-up polarized FI is inserted between the S/FM and N subsystems. The quasiparticles in N can now only “see” the spin-up bands of the S/FM subsystem. Electrons can thus tunnel like before, but the lowest-energy hole states are no longer accessible, which strongly suppresses the hole flow. Thus, net-negative charge flows to the left due to the thermal excitations. (c) The FI is now polarized in the spin-down direction. Since the excitations in N only see the spin-down bands in S/FM, mainly holes can tunnel to the left, and we get a net-positive charge flow. (d) When the FI is polarized perpendicularly to the FM, half of each spin band is visible. The resulting electric currents cancel as in panel (a).
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Figure 4
Cartoon picture explaining the sign reversal of the thermoelectric effect. (a) Let us first consider $φ = 0^{\circ}$ as in Fig. 3. As long as the spin-splitting field $m^{'} ≲ Δ_{0}$ , there is a low-energy coherence peak in the electron band and a gap in the hole band, resulting in an electron-dominated thermoelectric effect. (b) If $m^{'} ≳ Δ_{0}$ , then the spin splitting is sufficiently large to push the coherence peak down into the hole band. This is an oversimplification of the actual spin-dependent density of states in such a junction, but the explanation is qualitatively correct and yields the correct physical intuition. In this case, both electrons and holes can tunnel from the hot N layer, but the coherence peak makes the hole current dominant at low temperatures. This reverses the sign of the thermopower compared to panel (a). (c) If the temperature of the normal metal is increased, then tunneling contributions at moderate energies become increasingly important. This increases the electron current more than the hole current due to the gap in the hole band, and at sufficiently high temperatures the sign of the thermopower therefore flips again. The effect is exacerbated if the S is heated as well, since more of the states in the coherence peak will then already be occupied, which again increases the importance of tunneling processes at higher energies. (d) This shows the equivalent to panel (b) for $φ = 180^{\circ}$ . The thermopower $S (φ)$ remains antisymmetric for $m^{'} ≳ Δ_{0}$ since the spin-up and spin-down bands are still shifted equally in opposite directions.
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Figure 5
Evolution of the thermoelectric response with the strength of the magnetic exchange field $m^{'}$ in the FM. Panels (a–h) show the short-circuit current $I_{SC} (φ)$ , where the plot radius is $1.2 \times 10^{- 4} I_{0}$ . Panels (i–p) show the open-circuit voltage $Δ V_{OC} (φ)$ , where the plot radius is $3 \times 10^{- 2} V_{0}$ . Except for the exchange field and radial scale, all parameters used for the simulations and visualization are the same as in Fig. 2.
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Figure 6
Thermoelectric current $I_{SC} (φ)$ as function of the temperatures $T_{S}$ in S and $T_{N}$ in N for $m^{'} = 3 Δ_{0}$ . Blue regions correspond to $I_{SC} > + I_{min}$ and red regions to $I_{SC} < - I_{min}$ , where $I_{min} \equiv 10^{- 5} I_{0}$ .
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