Graphene/hBN–based valley transistor: Dynamic control of valley current in synchronized nonzero voltages within the time-dependent regime

Graphene/hexagonal boron nitride (hBN) heterostructures represent a promising class of metal--insulator--semiconductor systems widely explored for multifunctional digital device applications. In this work, we demonstrate that graphene, when influenced by carrier-dependent trapping in the hBN spacer, triggered by a localized potential from Kelvin probe force microscopy (KPFM), can exhibit the behavior of a valley transistor under specific conditions. We employ a tight-binding model that self-consistently incorporates a Gaussian-shaped potential to represent the effect of the tip gate. Crucially, we show that the heterostructure can function as a field-effect transistor (FET), with its operation governed by the bias gate (which shifts the Fermi level) and the tip-induced potential (which breaks electron--hole symmetry by selectively trapping electron or hole quasiparticles). Our results reveal that, under specific conditions involving lattice geometry, pulse frequency, and gate voltages, the device exhibits valley transistor functionality. The valley current $({I}_{{K}_{1}=\ensuremath{-}K} \text{or} {I}_{{K}_{2}=+K})$ can be selectively controlled by synchronizing the frequencies and polarities of the tip and bias gate voltages. Notably, when both gates are driven with the same polarity, the graphene channel outputs a periodically modulated, pure valley-polarized current. This enables the device to switch between distinct ON/OFF valley current states even at finite bias. Interestingly, when the ${I}_{{K}_{1}=\ensuremath{-}K}$ current is in the ON (forward current) state, the ${I}_{{K}_{2}=+K}$ current is OFF. Reversing the gate polarity inverts this behavior: ${I}_{{K}_{1}=\ensuremath{-}K}$ becomes OFF, while ${I}_{{K}_{2}=+K}$) turns ON (reversed-current). These findings pave the way toward realizing low-voltage valley transistors within metal--insulator--semiconductor architectures, offering avenues for multifunctional applications in valleytronics and advanced gating technologies.