Spin-Controlled Superconductivity and Tunable Triplet Correlations in Graphene Nanostructures

We study graphene ferromagnet/superconductor/ferromagnet ($F/S/F$) nanostructures via a microscopic self-consistent Dirac Bogoliubov–de Gennes formalism. We show that as a result of proximity effects, experimentally accessible spin switching phenomena can occur as one tunes the Fermi level ${\mu }_F$ of the $F$ regions or varies the angle $\theta$ between exchange field orientations. Superconductivity can then be switched on and off by varying either $\theta$ or ${\mu }_F$ (a spin-controlled superconducting graphene switch). The induced equal-spin triplet correlations in $S$ can be controlled by tuning ${\mu }_F$, effectively making a graphene based two-dimensional spin-triplet valve.

Figure 1

$F/S/F$ geometry investigated. The magnetic structure is described by exchange fields ${\overrightarrow{\mathit{h}}}_L$ and ${\overrightarrow{\mathit{h}}}_R$ in the left and right magnets, respectively. Their orientation is given by angles ${\theta }_{L,R}$ and ${\phi }_{L,R}$: ${\overrightarrow{\mathit{h}}}_{L,R}=h_{L,R}(\cos {\theta }_{L,R},\sin {\theta }_{L,R}\sin {\phi }_{L,R},\sin {\theta }_{L,R}\cos {\phi }_{L,R})$. The Fermi level in the $F$ regions can be tuned via the gate electrodes (G.E.). The system is infinite in the $x$-$y$ plane. $x$ is normal to the interfaces.

Figure 2

(a): Critical temperature $T_c$ (normalized by its bulk value $T_{c0}$) vs the magnet doping parameter ${\tilde{\mu }}_F$ for various values of $h/{\mu }_S$ with $\theta =90°$ (orthogonal exchange fields). In (b), the singlet pair amplitude at $T/T_{c0}=0.29$ is shown as a function of position, for an exchange field $h/{\mu }_S=0.2$, and with $\theta =180°$ (antiparallel exchange fields). Results for several ${\tilde{\mu }}_F$ are shown.

Figure 3

(a): Critical temperature vs the relative field angle $\theta$ at several exchange fields, $h/{\mu }_S$. The doping parameter ${\tilde{\mu }}_F$ is fixed at ${\tilde{\mu }}_F=0.5$. In (b), we show the singlet pair amplitude normalized to its bulk value for different field orientations, $\theta$ (see legend). $T$ is set at $T/T_{C0}=0.12$, and ${\tilde{\mu }}_F=0.5$.

Figure 4

Triplet spin valve effect. In panels (a) and (b), respectively, we plot the real and imaginary parts of the equal-spin triplet correlations, $f_1$, in the $S$ region. The curves in each of the panels represents a different doping level ${\tilde{\mu }}_F$ in the $F$ regions: by tuning ${\tilde{\mu }}_F$, the degree of equal-spin correlations in the superconductor can be controlled. The two relative exchange field orientations studied correspond to when the internal exchange fields are parallel ($\theta =0°$) or perpendicular ($\theta =90°$).

Figure 5

The imaginary part of $f_1$ in $F$ [panel (a)] and the real part of $f_0$ in $S$ [panel (b)] are plotted for multiple values of $\theta$ and ${\tilde{\mu }}_F=-0.2$. (The corresponding imaginary and real parts vanish). The long range nature of $f_1$ in $F$ is seen in panel (a) while the obvious contrast between the long- and short-ranged nature of the equal- and opposite-spin triplets ($f_1$ and $f_0$) in $S$ is seen by comparing panel (b) with Fig. 4.