Supercurrent-induced charge-spin conversion in spin-split superconductors

Faluke Aikebaier, Mihail A. Silaev, and T. T. Heikkilä

Phys. Rev. B 98, 024516 – Published 25 July 2018

Abstract

We study spin-polarized quasiparticle transport in a mesoscopic superconductor with a spin-splitting field in the presence of coflowing supercurrent. In such a system, the nonequilibrium state is characterized by charge, spin, energy, and spin-energy modes. Here we show that in the presence of both spin splitting and supercurrent, all these modes are mutually coupled. As a result, the supercurrent can convert charge imbalance, which in the presence of spin splitting decays on a relatively short scale, to a long-range spin accumulation decaying only via inelastic scattering. This effect enables coherent charge-spin conversion controllable by a magnetic flux, and it can be detected by studying different symmetry components of the nonlocal conductance signal.

Received 22 December 2017
Revised 27 March 2018

DOI:https://doi.org/10.1103/PhysRevB.98.024516

Physics Subject Headings (PhySH)

Spin accumulation Spin current Spin relaxation Superfluid density Transport phenomena

Ferromagnetic superconductors Thermoelectrics

Nonequilibrium Green's function

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Faluke Aikebaier^*, Mihail A. Silaev^†, and T. T. Heikkilä^‡

Department of Physics and Nanoscience Center, University of Jyvaskyla, P.O. Box 35 (YFL), FI-40014 University of Jyväskylä, Finland

^*faluke.aikebaier@jyu.fi
^†mikesilaev@gmail.com
^‡tero.t.heikkila@jyu.fi

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Issue

Vol. 98, Iss. 2 — 1 July 2018

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Images

Figure 1
Schematic pictures illustrating the couplings between different types of nonequilibrium states in a superconductor in the presence of the phase gradient driving the condensate to the velocity $v_{s}$ . (a) Generation of charge imbalance by the temperature gradient. (b) Generation of spin accumulation by the charge imbalance gradient $\nabla μ$ under the restriction that energy current is absent $I_{e} = 0$ . Shown in the plots are the quasiparticle electronlike (el) and holelike (hl) spectral branches in the superconductor in the presence of Doppler shifted energy $\pm p_{F} v_{s}$ . The solid and open circles show the extra occupied and empty states compared to the equilibrium distribution, respectively, and the circles with crosses show the states which become depopulated due to the Doppler shift.
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Figure 2
Schematic picture illustrating the nonzero spin energy in the ground state of a spin-singlet superconductor with spin splitting. $N_{↑, ↓} (ɛ)$ are the spin-up and -down densities of states as functions of the energy $ɛ$ . The relative Zeeman shift of the electronic bands is $2 h$ . The case of $T = 0$ is shown, so that all states below the Fermi level $ɛ_{F}$ are occupied.
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Figure 3
(a) Schematic view of the setup. Here the spin-splitting field is induced from either the ferromagnetic insulator or external magnetic field $B$ . (b)–(d) Spin accumulation as a function of parameters $μ_{z} = μ_{z} (h, T, τ_{sn})$ at the detector position $L_{\det} = L / 8$ in the linear-response regime (small $V_{inj})$ . (b) The dependence on the spin-relaxation rate for $k_{B} T = 0.15 Δ_{0}$ and $h = 0.3 Δ_{0}$ . (c) Temperature and (d) spin-splitting field dependence. The orbital depairing rate is $τ_{orb}^{- 1} = 0.176 h^{2} / Δ_{0}$ . Here we normalize the induced spin signal by the supercurrent amplitude $I_{s}$ .
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Figure 4
Spin accumulation and nonlocal conductance. (a) Position dependence of heat-induced (red) and supercurrent-induced (blue) charge and spin imbalances. Here the results are calculated for $T = 0.15 Δ_{0}, h = 0.3 Δ_{0}$ at $V_{inj} = 0.1 Δ_{0}$ . The solid curves are odd in injection voltage, and dashed curves are even. (b) Injection voltage dependence on spin accumulation for $T = 0.25 Δ_{0}$ . (c) Nonlocal conductance as a function of injection voltage in separate scales for heat- and supercurrent-induced effects (with $ξ_{0} \partial_{x} φ = 0.1)$ for $T = 0.02 Δ_{0}$ . (d) Heat-induced and total conductance as a function of injection voltage for $T = 0.25 Δ_{0}$ (with $ξ_{0} \partial_{x} φ = 0.5)$ . The parameters $τ_{so}^{- 1} = 0.0475 Δ_{0}, τ_{sf}^{- 1} = 0.0025 Δ_{0}, h = 0.05 Δ_{0}$ , and $L = 20 ξ_{0}$ are the same in (b)–(d).
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Figure 5
Spin accumulation with and without relaxation in the linear-response regime. (a) The dependence on orbital depairing rate in the case without spin relaxation. Position dependence in the case of (b) pure spin-flip relaxation and (c) pure spin-orbit relaxation. An exchange field $h = 0.3 Δ_{0}$ is the same for all panels, and a temperature $T = 0.15 Δ_{0}$ is used in (b) and (c). The red curves describe the charge imbalance. The spin-relaxation length is defined as $λ_{sn} = \sqrt{τ_{sn} D}$ .
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