For a classical capacitor the capacitance is determined by only the geometry, where the electrons distribute themselves such a way that it minimizes the energy. Thus, energy and capacitance are inversely proportional. In quantum mechanics, addition of electron requires kinetic energy and therefore, contributes to the capacitance, which is known as the quantum capacitance, which directly probes the electronic density of states of the material. The Quantum capacitance also contains the fundamental information about the ground state of the system, for example electron-electron interaction and quantum correlation contributes to the negative capacitance value. The thermodynamic compressibility factor near the Fermi energy has been measured by using quantum capacitance measurements at zero and high magnetic field. Recently, Van-Hove singularities and Luttinger parameter have been measured in one-dimensional system by this technique. The linear density of states of single layer graphene and topological insulator (Bi2Se3) have also been shown by quantum capacitance measurements. We have developed the quantum capacitance measurement technique combined with transport measurements from low-temperature to room temperature with magnetic fields in order to study the electronic structures, correlation effect etc in two-dimensional novel materials like single, bilayer, twisted double layer graphene. We have achieved to measure below femtofarad capacitance.

References:

- Energetics of the complex phase diagram of a tunable bilayer graphene probed by quantum capacitance, Phys. Rev. B 98, 035418 (2018).
- Large Landau-level splitting in a tunable one-dimensional graphene superlattice probed by magnetocapacitance measurements, Physical Review B 98 (3), 035418 (2018).

(Top-left): Optical image of the device. (Top-right): Schematic of the device architecture and measurement scheme. (Bottom-left): Cut lines showing Ct as a function of top gate voltage Vtg for several values of backgate voltage (Vbg). (Bottom-right): C_{q} as a function of E_{F}, for small value of displacement field showing different insulating pahses.

(Top-left): Schematic of one-dimensional periodic potential. Schematic of the device architecture. Bottom inset shows the scanning electron microscope image of the grids. Derivative of the quantum capacitance as a function of top-gate and grid voltage. Measured total capacitance (Ct ) as a function of top-gate voltage (Vtg) at B = 9.8 T (blue line). The red line shows the corresponding change in chemical potential as a function of Vtg using the charge conservation relation.