Quantum Capacitance of 2D Material - I¶
Now, we will derive an expression for the quantum capacitance of a 2D material. Consider a neutral isolated sheet of degenerately doped 2D semiconductor of area \(A\). Let the geometric capacitance of the sheet per unit area be \(C_0\). The equilibrium condition of the sheet is shown on the left side of the figure below. The density of states \(D(E)\) is plotted as a function of energy \(E\). The area under the blue curve in the figure represents the total number of free electrons per unit area, \(N_0\).
As we have seen in the chapter on carrier statistics, for degenerately doped 2D semiconductors, the number of free electrons can be written as
where \(D_0\) is the density of states, \(E_F\) is the Fermi energy and \(E_0\) is the first subband energy. Now, suppose we add \(\delta N\) electrons per unit area to the sheet. This means we add an additional charge of \(\delta \sigma = e \delta N\) to the sheet. This will lead to a change in electrostatic potential \(\delta \phi_e = eC_0\delta N\). A change in electrostatic potential is depicted as an upward shift in the bands by \(e\delta \phi_e\) on the right side of the figure above. Therefore the first subband \(E_0\) is at \(E_0 + e\delta \phi_e\) now.
What about the change in the Fermi level ? Let the Fermi level shift by \(\delta E_F\). The new Fermi level is at \(E_{F1} = E_F + \delta E_F\). The new carrier concentration \(N_1 = N_0 + \delta N\) is shown as the blue shaded area in the figure above. We can write it as
From this equation, it is very easy to conclude that -
This equation shows that the shift in Fermi energy (\(\delta E_F\)) is not equal to the shift electrostatic potential energy (\(e\phi_e\)). There is an additional shift of \(\delta N / D_0\). However, in the limit that the density of states is very high, \(D_0 \to \infty\), the shift in Fermi level is nearly equal to the shift in electrostatic potential energy.
We have already seen the physics behind this previously. A high density of states at a given energy implies that we can pump more electrons to a given energy. In contrast, a low density of states implies we need to go to higher energy levels to add more electrons.
Classical physics assumes a very high density of states. In such a situation, we can see that the system’s capacitance is nearly the same as that of its geometrical/electrostatic capacitance.