Tunable valley filtering in dynamically strained $\alpha$-$\mathcal{T}_3$ lattices

Mechanical deformations in $\alpha$-$\mathcal{T}_3$ lattices induce local pseudomagnetic fields of opposite directionality for different valleys. When this strain is equipped with a dynamical drive, it generates a complementary valley-asymmetric pseudoelectric field which is expected to accelerate electrons. We propose that by combining these effects by a time-dependent nonuniform strain, tunable valley filtering devices can be engineered that extend beyond the static capabilities. We demonstrate this by implementing an oscillating Gaussian bump centered in a four-terminal Hall bar $\alpha$-$\mathcal{T}_3$ setup and calculating the induced pseudoelectromagnetic fields analytically. Within a recursive Floquet Green-function scheme, we determine the time-averaged transmission and valley polarization, as well as the spatial distributions of the local density of states and current density. As a result of the periodic drive, we detect novel energy regimes with highly valley-polarized transmission, depending on $\alpha$. Analyzing the spatial profiles of the time-averaged local density of states and current density we can relate these regimes to the pseudoelectromagnetic fields in the setup.By means of the driving frequency, we can manipulate the valley-polarized states, which might be advantageous for future device applications.