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Classical and quantum capacitances calculated locally considering a two-dimensional Hall bar

In this work we investigate the electrostatic properties of two dimensional electron system (2DES) in the integer quantum Hall regime. The alternating screening properties of compressible and incompressible strips are formed due to edge effects together with electron-electron interactions. As it is well known, the Landau quantization emanates from strong perpendicular magnetic fields. The (Landau) energy levels are broadened due to impurities, which we embedded their effects in density of states (DOS). In a basic level DOS has two different forms: the Gaussian and semi-elliptic descriptions. The second form is calculated within the self consistent Born approximation (SCBA). Having in hand the density of states, we obtain both the longitudinal and Hall (transversal) conductivities ($σ_{l}, σ_{H}$) utilizing Thomas-Fermi-Poisson approximation to calculate position dependent charge density profile and use Drude formalism to obtain transport coefficients. Since, the definition of capacitance is closely related with compressibility via DOS, (local) screening properties of 2DES is extremely important to understand local capacitances. Here we numerically simulate a translational invariant Hall bar subject to high magnetic fields which is perpendicular to the plane of the 2DES using realistic parameters extracted from the related experiments. Using the above mentioned approaches the local capacitances are calculated, numerically. Our findings are in perfect agreement with related experimental results which are based on a dynamic scanning capacitance microscopy technique.