This example describes the simulation of a metal-oxide-semiconductor (MOS) Capacitor. In the first part we will perform steady-state simulations to calculate the static capacitance of the MOSCap. In the second part of this example we will perform a small-signal analysis to investigate the frequency dependence of the MOS capacitance.

Two project files are provided to assist with the application example. The project files contain the material properties, geometry, and simulation region required to run the example.

A MOS Capacitor can be created by placing an insulator between a metal plate and a semiconductor. Voltages applied on the metal induces positive (holes) or negative (electrons) charges on the surface of the semiconductor. Charges of opposite polarity accumulates on the metal place and the the structure works as a capacitor. Depending on the type (doping) of the semiconductor and the voltage applied on the metal gate, the MOS Capacitor has three modes of operation. In one mode, the voltage applied on the metal contact accumulates majority carriers on the surface of the semiconductor and this is called "Accumulation." In the other case, the applied voltage induces minority carriers on the semiconductor surface. This initially creates a depletion region at the surface (the "Depletion" mode of operation) and eventually the majority carrier type at the surface of the semiconductor gets inverted. This final mode is called the "Inversion." Both in accumulation and strong inversion, the MOS Capacitor can be approximated by a conventional parallel plate capacitor and assumes constant values. In depletion mode however, the width of the depletion region affects the capacitance of the MOSCap and the capacitance becomes bias dependent.

In accumulation and strong inversion, the capacitance of the MOSCap can be approximated to be,

$$ C_{a / i}=C_{o x}=\frac{\varepsilon_{i n s} A}{d} $$

where, eps_ins is the dielectric constant of the insulator, A is surface area, and d is the thickness of the insulator. In depletion, the capaciatnce of the MOSCap depends on the thickness of the depletion region and becomes smallest at the maximum width of the depletion region. The minimum capacitance of the MOSCap, just before it switches from depletion to inversion can be approximated by,

$$ C_{d}=\frac{\varepsilon_{i n s} A}{W_{m}} $$

$$ W_{m}=2\left[\frac{2 \varepsilon_{\text {ins}} \phi_{\text {s,inv}}}{q N_{A / D}}\right]^{1 / 2} $$

where, W_{m} is the maximum width of the depletion region, phi_s_inv is the surface potential at the beginning of inversion and N_{A/D} is the doping concentration of the semiconductor. In static case and low frequency of operation, the capacitance values can be expressed by these analytic expressions. For the case of depletion and accumulation, the movement of charge in the semiconductor is due to the majority carriers which is restricted by the dielectric relaxation time of the semiconductor and therefore is extremely fast. The high frequency capacitance of the MOSCap in accumulation and depletion region therefore follows their static values. In inversion however, the inversion layer is separated from the bulk semiconductor by the depletion region and the movement of charge at the surface is restricted by the generation rate of the carriers. Thus the inversion capacitance becomes smaller at high frequency and becomes almost equal to the minimum capacitance of the MOSCap.

## Steady State Simulation

In this part of the example, we will perform a steady state simulation of a MOS Capacitor and calculate its static capacitance. Download the moscap.ldev project file and the moscap_static_C.lsf script file in the same folder. Run the script file. It will perform two voltage sweeps, one from 0 to 2.5 V and another from 0 to -2.5 V and will calculate the capacitance of the MOSCAP. The script will generate the following plot and save the capacitance value in a .mat file. The different regions of the capacitance curve has been labeled.

MOSCap static capacitance

## Small Signal Simulation

Next, we are going to perform a small-signal ac analysis of the same MOSCap. To do this, download the ssac_moscap.ldev project file and the moscap_ac_Cap.lsf script file in the same folder. The script will load the project file and perform a small-signal ac analysis at each DC bias point from -2.5 V to 2.5 V and calculate the admittance of the MOSCap. The imaginary part of the admittance will give the value of the capacitance. The small-signal ac analysis will be done for a frequency range of 0.001 Hz to 1 MHz. After calculating the ac capacitance at different frequencies, the script will plot the low frequency (0.001 Hz) and high frequency (1 MHz) ac capacitance of the MOSFET along with the static capacitance calculated from the steady state simulation as well as the analytic values for static inversion/accumulation capacitance and minimum capacitance of the MOSCap as given by the equations above.

It can be seen from the plot that the results from both the steady state and the small signal ac simulations agree with the theoretical maximum and minimum values of the MOS Capacitor.