The Impact of Electric and Magnetic Fields on the Quantum Capacitance of Monolayer Germanene

In this work, we present the theory of quantum capacitance in two-dimensional buckled germanene. Germanene is a material with a natural bandgap and strong spin–orbit interaction, offering significant advantages over graphene. The results of this study reveal the behavior of the quantum capacitance in response to variations in the spin–orbit interaction and electric field. The capacitance in each valley can be adjusted with the electric field, shifting from a minimum to a maximum as the external gate voltage is applied. The total capacitance, derived from both the K and K ′ valleys, shows a tunable band gap at $${{E}_{{\text{F}}}} = 0$$ . When the electric field energy $${{\Delta }_{z}} = 2{{{{\lambda }}}_{{{\text{so}}}}}$$ , the capacitance is zero at $${{E}_{{\text{F}}}} = 0$$ , whereas the electric field energy $${{\Delta }_{z}} = {{{{\lambda }}}_{{{\text{so}}}}}$$ , the capacitance reaches maximum with a single peak at $${{E}_{{\text{F}}}} = 0$$ and electron–hole symmetric spectrum. As the electric field affects the band structure, the spin-up states retain their energy gap, whereas the spin-down states converge. Shubnikov–de Haas oscillations in quantum capacitance are also observed, showing a be-ating pattern at low magnetic fields that disappears as the electric field dominates. At higher magnetic fields, the splitting of the oscillations becomes evident. The disappearance of beating pattern and subsequent oscillation splitting are attributed to the interplay between the spin–orbit interaction, electric field, and magnetic field.