Orbital-separation approach for consideration of finite electric bias within density-functional total-energy formalism

We present a simple approach for the consideration of bias voltage within the Kohn-Sham formalism of density-functional theory. To be specific, the electronic charging of a metal-insulator-metal capacitor under bias voltage is considered explicitly. This is achieved by separating the Kohn-Sham orbitals around the Fermi level into anode or cathode parts and applying different Fermi levels in the determination of occupation numbers. The formal basis of the present approach is discussed in detail. We test this method against Au-vacuum-Au and Au-MgO-Au capacitors with various dielectric thicknesses. It is shown that the bulk optical and static dielectric constants can be obtained accurately. We also demonstrate that interface effects on capacitance can be investigated straightforwardly. Furthermore, we apply this method to the graphene capacitor and identify the quantum effects in the capacitance, which is well explained by the contribution of the kinetic energy of electrons to the capacitance. The present method can be readily implemented in conventional first-principles codes and provides a unified approach to evaluate capacitance of nanodevices.

Figure 1

(a) Schematic of the simulation models considered in this work. The box indicates the boundaries of the periodic boundary condition employed in the calculations. (b) Schematic of the separation procedure within a preset window around the Fermi level.

Figure 2

(a) The Au-vacuum-Au capacitor model used in this work. The box represents the periodic boundary condition. (b) $xy$-plane average of the electron density obtained by integrating the states between $\pm 1$ eV around the Fermi level under zero bias, (c) $xy$-plane average of the induced charge density at various bias voltages, and (d) $xy$-plane average of the electrostatic potential difference ($\Delta V_{\mathrm{H}}$) with respect to zero bias results. In (b)–(d), the results for $d^{\prime }=2$ nm are presented. The vertical dashed lines indicate the position of the surface Au nuclei, while the dotted lines indicate the position of the surface corresponding to the effective vacuum thickness $d_{\mathrm{eff}}$ (see text).

Figure 3

(a) The accumulated surface charge density (left $y$ axis) and total energy change (right $y$ axis) as functions of bias voltage. (b) The capacitance evaluated from the total energy (open circle) and from the accumulated charge (solid square). The results are presented for the Au-vacuum-Au capacitor with a vacuum thickness of $d^{\prime }=2$ nm.

Figure 4

The inverse capacitance of the Au-vacuum-Au capacitor with respect to $d^{\prime }$.

Figure 5

(a) Schematic of the Au-MgO-Au capacitor model under study, (b) $xy$-plane average of the electron density obtained by integrating the states between $\pm 1$ eV around the Fermi level under zero bias, and (c) $xy$-plane average of the electrostatic potential with respect to that at zero bias. The results are given for the capacitor with 9 MgO layers whose structure was fixed (solid line) or relaxed (dashed line) at $V=0.6$ V. The vertical dashed lines indicate the positions of the surface nuclei.

Figure 6

(a) The total energy with respect to that at zero bias and (b) capacitance as functions of bias voltage. The results are given for the capacitor with 9 MgO layers whose structure was fixed at the zero bias structure (solid squares) and that relaxed at each bias voltage (open circles).

Figure 7

Inverse capacitances of the Au-MgO-Au capacitor under bias voltage of 0.6 V are plotted with respect to the nominal dielectric thickness for fixed (solid squares) and relaxed (open circles) conditions.

Figure 8

(a) Schematic of the MgO capacitor with monolayer Au electrodes, (b) electron density corresponding to the states at $\pm 1$ eV around the Fermi level under zero bias, and (c) the $xy$-plane average of the change in the electrostatic potential due to bias application for fixed (solid line) and relaxed (dashed line) structures.

Figure 9

Comparison of the inverse capacitance of MgO capacitors with 8-layer (solid squares) and monolayer (open circles) Au electrodes.

Figure 10

(a) Schematic of the graphene-vacuum-graphene capacitor, (b) $xy$-plane average of the electron density obtained by integrating the states between $\pm 1$ eV around the Fermi level under zero bias, (c) $xy$-plane average of the induced charge density at various values of applied bias, and (d) $xy$-plane average of the electrostatic potential difference with respect to zero bias. The dashed vertical lines correspond to the position of the graphene sheets.

Figure 11

(a) The accumulated charge of the graphene-1 nm vacuum-graphene capacitor (black squares) and total-energy change (white circles) as a function of bias voltage. (b) The capacitance calculated from voltage vs energy (white circles) and voltage vs induced charge (black squares). The dashed line indicates the geometric capacitance obtained from the $V$–$Q$ curve.