Charge and spin edge currents in two-dimensional Floquet topological superconductors

A time periodic driving on a topologically trivial system induces edge modes and topological properties. In this work we consider triplet and singlet superconductors subject to periodic variations of the chemical potential, spin-orbit coupling, and magnetization in both topologically trivial and nontrivial phases, and study their influence on the charge and spin currents that propagate along the edges of the two-dimensional system, for moderate to large driving frequencies. Currents associated with the edge modes are induced in the trivial phases and enhanced in the topological phases. In some cases there is a sign reversal of the currents as a consequence of the periodic driving. The edge states associated with the finite quasienergy states at the edge of the Floquet zone are in general robust, while the stability of the zero quasienergy states depends on the parameters. Also, the spin polarization of the Floquet spectrum quasienergies is strong as for the unperturbed topological phases. It is found that in some cases the unperturbed edge states are immersed in a continuum of states due to the perturbation, particularly if the driving frequency is not large enough. However, their contribution to the edge currents and spin polarization is still significant.

Figure 1

Phase diagram for the spin triplet superconductor with $\mathit{d}=(d_x,d_y,0)$. The phase diagram for an $s$-wave superconductor is very similar except in the vicinity of zero magnetization (limited approximately by the dotted lines), where the system is topologically trivial. In the case of a triplet superconductor with $\mathit{d}=(0,0,d_z)$ with a complex $z$ component, the phase diagram is also similar with the exception of a region around zero magnetic field (limited by dashed lines) that is topological, with Chern number 2.

Figure 2

Floquet spectra for $l=1$ for a $d_x,d_y$ triplet superconductor in a topologically trivial phase with $\alpha =0,M_z=0,\mu =-5,{\mathrm{\Delta }}_s=0,d_z=0,d=0.6$, where the hopping is changed with time with different frequencies $w=4,6,8,12$. The periodic driving is $t_{d}coswt$ with $t_d=2$. Note that the spectrum is symmetric around zero quasienergy.

Figure 3

Floquet spectra for $l=1$ for a $d_x,d_y$ triplet superconductor in a topologically trivial phase with $\alpha =0,M_z=0,\mu =-5,{\mathrm{\Delta }}_s=0,d_z=0,d=0.6$, where the magnetization is changed with time with different frequencies $w=4,6,8,12$. The periodic driving is $M_{zd}coswt$ with $M_{zd}=2$.

Figure 4

Evolution of Floquet spectra for various values of $l,l=0,1,2,3$ for a $d_x,d_y$ triplet superconductor in a topologically trivial phase $\alpha =0.1,M_z=0,\mu =-5,{\mathrm{\Delta }}_s=0.1,d_z=0,d=0.6$, where the magnetization is changed with time with frequency $w=6$. The periodic driving is $M_{zd}coswt$ with $M_{zd}=4$. The case $l=0$ is the unperturbed Hamiltonian. The results show the stability of two sets of Majorana modes both at zero energy and at the limit of the Floquet zone.

Figure 5

Evolution of Floquet spectra for $l=1$ for a $d_x,d_y$ triplet superconductor in a topologically nontrivial phase $\alpha =0,M_z=2,\mu =-5,{\mathrm{\Delta }}_s=0,d_z=0,d=0.6$, where the hopping is changed with time with frequencies $w=6,12$. The periodic driving is ${\tilde{t}}_{d}coswt$ with ${\tilde{t}}_d=0.5$ (top row) and ${\tilde{t}}_d=2$ (bottom row).

Figure 6

Charge current profile for $l=2$ for a $d_x,d_y$ superconductor for $P_1,P_3,P_4$ for (a) ${\mu }_d=1$ and (b) $M_{zd}=1$. Only one half of the system is shown since the current profile is antisymmetric around the middle point. The system size is $N_y=100$. Charge current profile for $l=2$ for a $d_z$ superconductor for $P_5,P_6,P_7,P_8$ for (c) ${\mu }_d=1$. Note that the current profile has no symmetry around the middle point in some cases. The results for the $d_x,d_y$ pairing consider three points in the phase diagram: $\mu =-5,M_z=0$ (case $P_1$ with $C=0$), $P_3$ with $\mu =-5,M_z=2$ (with $C=1$), and $P_4$ with $\mu =-1,M_z=2$ (with $C=-2$). The results for the $p+ip$ pairing consider $P_5$ with $\mu =-5,M_z=0$ (with $C=0$), $P_6$ with $\mu -2,M_z=0$ (with $C=2$), $P_7$ with $\mu =-3,M_z=2$ (with $C=1$), and $P_8$ with $\mu =-1,M_z=2$ (with $C=-2$). These points are indicated in Fig. 1.

Figure 7

Influence of the spin-orbit coupling on the spin current in the unperturbed case. The parameters are $d=0.6,{\mathrm{\Delta }}_s=0.1,d_z=0,M_z=0,\mu =-3,-5$. Increasing the spin-orbit in the topologically nontrivial phase is detrimental while in the topologically trivial phase, increasing the spin-orbit coupling considerably increases the spin current.

Figure 8

Influence of intensity of periodic driving for the cases of ${\mu }_d,{\alpha }_d,M_{zd}$ on the spin current $\left(l=2\right)$. The parameters are $\mu =-5,d=0.6,{\mathrm{\Delta }}_s=0.1,d_z=0,M_z=0,\alpha =0.6$ and the frequency of the periodic driving is $w=6$. Note the reversal of the direction of the spin current at small couplings. At weak coupling the increase of the spin current is linear. As the transition from weak to strong coupling occurs the spin current becomes nonlinear. Changing the spin-orbit coupling has a small effect on the edge spin current while both changing the chemical potential and the magnetization have strong effects including a change of the current direction.

Figure 9

Spin current for the triplet superconductors $d_x,d_y$ [(a) and (b)] and $d_z$ [(c) and (d)] for the periodic drivings ${\mu }_d=1$ [(a) and (c)] and $M_{zd}=1$ [(b) and (d)].

Figure 10

Time evolution of the spin current for a $d_x,d_y$ triplet superconductor for the periodic drivings (a) ${\mu }_d$, (b) ${\alpha }_d$, and (c) $M_{zd}=1$ for $P_1,P_2,P_3,P_4$, respectively.

Figure 11

Spin polarization of an unperturbed $\left(l=0\right)$ and perturbed $\left(l=2\right)$ system with ${\mu }_d=1$ for (a) and (b) $P_1$ (trivial case) and (c) and (d) $P_2$ ($Z_2$ case). In these cases $C=0$ and $M_z=0$. The maximal values of the spin polarizations are (a) $7.3\times {10}^{-5}$, (b) $1.9\times {10}^{-3}$, (c) $2.5\times {10}^{-2}$, and (d) $5.8\times {10}^{-3}$. The maximal value displayed in (c) has been reduced to $0.007$ for better visualization.

Figure 12

Spin polarization of unperturbed $\left(l=0\right)$ and perturbed $\left(l=2\right)$ system with ${\mu }_d=1$ for (a) and (b) $P_3 \left(C=1\right)$ and (c) and (d) $P_4 \left(C=-2\right)$. In these cases $C\ne 0$ and $M_z\ne 0$. The maximal values of the spin polarizations are (a) $3.8\times {10}^{-2}$, (b) $7.8\times {10}^{-3}$, (c) $4.7\times {10}^{-2}$, and (d) $8.3\times {10}^{-3}$. The maximal values of the panels have been reduced to $0.005$.