Spin injection and spin relaxation in odd-frequency superconductors

The spin transport inside an odd-frequency spin-triplet superconductor differs from that of a conventional superconductor due to its distinct symmetry properties. We study spin transport inside an emergent odd-frequency superconductor by replacing the spin-singlet gap matrix in the Usadel equation with a matrix representing spin-triplet pairing that is odd under inversion of energy. We show that the peculiar nature of the density of states allows for an even larger spin injection than in the normal state. Moreover, when the odd-frequency pairing inherits its temperature dependence from a conventional superconductor through the proximity effect, the density of states can transition from gapless to gapped as the temperature decreases. At the transition point, the spin accumulation inside the odd-frequency superconductor is peaked and larger than in the normal state. While the spin-flip scattering time is known to decrease below the superconducting transition temperature in conventional superconductors, we find that the same is true for the spin-orbit scattering time in odd-frequency superconductors. This renormalization is particularly large for energies close to the gap edge, if such a gap is present.

Figure 1

(a) We study the nonequilibrium spin accumulation in a conventional spin-singlet $(\uparrow \downarrow -{\downarrow \uparrow )}_z$ and odd-frequency spin-triplet $(\uparrow \downarrow +{\downarrow \uparrow )}_z$ superconductor (SC) upon applying a spin-dependent voltage $\left|e\right|V$ to an adjacent normal metal (NM). The spin-dependent voltage has opposite sign for spin-up and spin-down electrons, and can be induced from an electric voltage $\left|e\right|V_{\text{ch}}$ applied between two oppositely oriented ferromagnets (FM). The applied electric voltage $\left|e\right|V_{\text{ch}}$ is in general not equal to the induced spin-dependent voltage $\left|e\right|V$ in the NM contact. The FMs are polarized along the $z$ axis so that the spins injected into the SC cannot be carried by the Cooper pairs. The injected spins are relaxed by spin-flip and spin-orbit scattering until equilibrium is reached a distance $L$ from the NM contact. (b) We suggest inducing odd-frequency superconductivity through proximity to a conventional superconductor. Spin-singlet Cooper pairs are partially converted into triplets as they leak from a conventional superconductor into a ferromagnet. Upon leaking through a second sufficiently thick ferromagnet magnetized perpendicularly to the first one, only spin triplets survive [22, 23, 31]. In the highly disordered materials considered here, only $s$-wave pairing can be present [1, 8]. The remaining triplet pairing then transforms the adjacent normal metal into an emergent odd-frequency superconductor.

Figure 2

While the DOS for a singlet superconductor (a) is gapped, the DOS of an odd-frequency superconductor (b) can either be peaked at zero temperature (blue) or gapped (purple and red). The nonequilibrium spin accumulation corresponding to the DOS in (b) is shown for spin-flip scattering with $l_{\text{sf}}=0.15L$ (c) and for spin-orbit scattering with $l_{\text{so}}=0.15L$ (d). All plots correspond to a spin voltage of $\left|e\right|V=0.5{\mathrm{\Delta }}_0$ and temperature $T=0.5T_c$. The above corresponds to the pairing type described in Eq. (12). The second pairing type described in Eq. (13) gives similar results.

Figure 3

The nonequilibrium spin accumulation is plotted as a function of temperature in the presence of spin-flip scattering for $l_{\text{sf}}=0.15L$. (a) A gapless DOS where $0\le Cf\left(T\right)\le 0.90$ for all temperatures. (b) A DOS that is gapped at low temperatures $[1.05>Cf(T)>1]$, and gapless at higher temperatures $[0\le Cf(T)\le 1]$. Both are measured at a distance $0.25L$ away from the normal-metal contact and correspond to an applied spin voltage of $\left|e\right|V=0.1{\mathrm{\Delta }}_0$. The triplet pairing follows the model described by Eq. (12), however the model described by Eq. (13) gives similar results.

Figure 4

The nonequilibrium spin accumulation is plotted as a function of temperature in the presence of spin-orbit scattering for $l_{\text{so}}=0.15L$. (a) and (b) A gapless DOS where $0\le Cf\left(T\right)\le 0.90$ for all temperatures. (c) and (d) A DOS that is gapped at low temperatures $[1<Cf(T)<1.05]$, and gapless at higher temperatures $[0\le Cf(T)\le 1]$. (a) and (c) Measured at a distance $0.05L$ away from the normal-metal contact, while (b) and (d) are measured at a distance $0.25L$ away from the normal-metal contact. The applied spin voltage is $\left|e\right|V=0.1{\mathrm{\Delta }}_0$ for all panels. The triplet pairing follows the model described by Eq. (12), however the model described by Eq. (13) gives similar results.