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

We investigate the quantum capacitance (CQ) 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 (∆ᵦ=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 (∆ᵦ=λₛo 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.