Concurrent influence of magnetic field and temperature on the quantum capacitance of silicene and germanene

We investigate the quantum capacitance (C_Q) of silicene and germanene under perpendicular magnetic and electric fields, incorporating temperature effects and Gaussian broadening of Landau levels. When the perpendicular electric field is switched off (∆_z=0), the spectra display sharp Landau oscillations and a pronounced peak at the charge neutrality point, characteristic of massless Dirac fermions. Increasing the magnetic field strength increases the number of oscillation peaks and their separation, reflecting the √B dependence of Landau level energies in two-dimensional Dirac systems. When an external electric field is applied (∆_z=λ_so and 〖2λ〗_so), the oscillation peaks shift away from the Fermi level, indicating band gap opening due to inversion symmetry breaking. A comparison between silicene and germanene reveals distinct behaviors arising from their different intrinsic spin-orbit couplings, with germanene exhibiting larger energy gaps and peaks at higher Fermi energies. Importantly, our finite temperature formulation demonstrates that, in the presence of a perpendicular magnetic field, magneto-quantum capacitance exhibits remarkable thermal robustness, establishing the finite-temperature validity regime of zero temperature magneto-quantum capacitance theories. These results provide a physically grounded framework for interpreting capacitance measurements in buckled Dirac materials under realistic experimental conditions.