Chiral-phonon-induced current in helical crystals

In this paper, we theoretically show that in a helical crystal, a current is induced by chiral phonons representing the microscopic local rotation of atoms. By treating the rotational motion as a perturbation, we calculate the time-dependent current by using the adiabatic Berry phase method. The time average of the current along the helical axis becomes finite in the metallic phase but it vanishes in the insulating phase. On the other hand, the current in the hexagonal plane changes with time, but its time average vanishes due to the threefold rotation space-time symmetry. We show that the time evolutions of the current follow the space-time symmetries of the helical systems. Moreover, we explain the reason for the vanishing of the time average of the current in the insulating phase from the aspect of the Chern number in the parameter space.

Figure 1

Helical structure with honeycomb-lattice layers. (a) One layer of the helical lattice, forming a honeycomb lattice. (b) Hopping pattern in the left-handed helical crystal. (c) Hopping pattern in the right-handed helical crystal.

Figure 2

Two modes of microscopic rotation for the left-handed and right-handed helices related by the mirror reflection $M_x$ with respect to the $yz$ plane. (a-1) The left-handed helix with the CW (clockwise) phonon mode. (a-2) The right-handed helix with the CCW (counterclockwise) phonon mode. (b-1) The left-handed helix with the CCW phonon mode. (b-2) The right-handed helix with the CW phonon mode.

Figure 3

Band structure for the helical model of a stacked honeycomb lattice without phonons. The lattice constants are $a=1, c=2$ and the hopping parameters are set to be $t_A=0.5t_1, t_B=0.8t_1$. (a) The first Brillouin zone of the model. (b) The metallic band dispersion with the on-site energy value ${\lambda }_{\nu }=3t_1$. (c) The insulating band dispersion with the on-site energy value ${\lambda }_{\nu }=5t_1$.

Figure 4

Results of the calculation of the geometric current for the metallic phases with $t_A=0.5, t_B=0.8, \delta t=0.1t_1$, and ${\lambda }_{\nu }=3t_1$. For the left-handed helix with CW phonons, (a-1) is the geometric current in $x,y,z$ directions and (a-2) represents the evolution of the geometric current for one period in the $xy$ plane. For the right-handed helix with CCW phonons, (b-1) is the geometric current in $x,y,z$ directions and (b-2) represents the evolution of the geometric current for one period in the $xy$ plane.

Figure 5

Results of the calculation of the geometric current for insulating phases with $t_A=0.5, t_B=0.8, \delta t=0.1t_1$, and ${\lambda }_{\nu }=5t_1$. For the left-handed helix with CW phonons, (a-1) is the geometric current in $x,y,z$ directions and (a-2) represents the evolution of the geometric current for one period in the $xy$ plane. For the right-handed helix with CCW phonons, (b-1) is the geometric current in $x,y,z$ directions and (b-2) represents the evolution of the geometric current for one period in the $xy$ plane.

Figure 6

Dependence of the geometric current on the on-site energy ${\lambda }_{\nu }$ with the parameter values $t_A=0.5, t_B=0.8$, and $\delta t=0.1t_1$ for the right-handed helix with CCW phonons. The on-site energies are set to be (a-1), (a-2): ${\lambda }_{\nu }=5t_1$, (b-1), (b-2): ${\lambda }_{\nu }=4t_1$, (c-1), (c-2): ${\lambda }_{\nu }=3t_1$, and (d-1), (d-2): ${\lambda }_{\nu }=2t_1$. (a-1)–(d-1) are the results of the geometric current induced by the chiral phonon. (a-2)–(d-2) are the trajectories of the current vector ${\mathit{j}}^{\mathrm{geom},\mathrm{R}}$ within the $xy$ plane. The arrows show the direction of time evolution and these trajectories form the trianglelike cycles.

Figure 7

Contribution to the $z$ component of the geometric current from the respective time-averaged energy band with the parameter values $t_A=0.5, t_B=0.8$, and $\delta t=0.1t_1$ for the right-handed helix with CCW phonons. (a) Dependence on the on-site energy of the $z$ component of time-averaged geometric current. (b)–(e) Time-averaged energy bands with the color map showing the contribution to the time-averaged geometric current. The on-site energies are set to be (b) ${\lambda }_{\nu }=2t_1$, (c) ${\lambda }_{\nu }=3t_1$, (d) ${\lambda }_{\nu }=4t_1$, and (e) ${\lambda }_{\nu }=5t_1$.

Figure 8

Toy model for topological charge pumping. (a) Monopole inside the cylinder at the $k_z=0$ plane in the three-dimensional parameter space $(t_c,t_s,k_z)$. (b) Gapless area in the five-dimensional parameter space $(k_x,k_y,k_z,t_c,t_s)$. (c) Trajectory formed by $t_c$ and $t_s$ encircles the gapless area to realize the charge pumping. (d) Distribution of Weyl points on the $k_z=0$ plane and their monopole charges with $t_c=t_s=1$.