MOS Charge

Contacts

Source, Drain and substrate (Body/Bulk) are ohmic contacts and at equilibrium (no voltage applied and thermal equilibrium) the Fermi energy is uniquely defined:

When an external voltage () is applied to said contacts, the Fermi energy is fixed at the external potential (this means that the voltage applied “sets” the level) The Fermi energy at thermal equilibrium is set in all the semiconductor connected to the metal contact (source, drain or substrate).

Instead if the voltage is applied between two ohmic contacts, the two Fermi energy are shifted one from each other proportionally to the voltage applied:

where contact 1 has the lower voltage. For example if a voltage is applied between source and drain, with the difference in Fermi energies is given by:

Any potential energy of the electrons can be derived from a unique potential as all levels are parallel. The electrostatic potential is associated with the intrinsic Fermi level of the substrate where the semiconductor is neutral (, far from the interface.)

(in a n-MOS p-doped , in n-doped substrate )

In general:

and in the bulk reference system where

Channel potential

Is defined as:

so it is the Fermi energy of the substrate minus the channel electrostatic potential by the charge (electrons, with holes there are sign changes) NOTICE: this is not a potential, is an energy! Being the current along the x axis negligible it is assumed that the channel potential does not depend on x

and if

Channel potential: the meaning

The channel potential does not represent the voltage drop between source and drain, but describes the behaviour of the electron quasi-Fermi energy

Only if the current is mainly drift (no diffusion), the electrostatic potential is parallel to the channel potential

Electron concentration

Starting from the Shockley equation

and in a p-semiconductor (nMOS) the is given by:

so the Shockley equation becomes:

where is the electron charge in a stand-alone p-doped sample with doping.

Hole concentration

Again from the Shockley equation:

we can notice that the exponential is negative with respect to the potential, on the contrary in the electron concentration was positive. We make the strong approximation that the hole current is negligible overall, this means that we make the approximation of constant and equal to (bulk referred models) The hole concentration is now:

where is the hole charge in a stand-alone p-doped sample with doping

While a stand-alone p-type semiconductor is neutral, in the MOS system the charge density is not 0.

and substituting the previous expressions for the hole and electron concentration we obtain:

In static conditions Gauss law can be converted into the Poisson equation:

that can be simplified with the gradual channel hypothesis (GCA), so named because we assume that the voltages vary gradually along the channel from drain to source but at the same time they vary quickly perpendicularly to the channel moving from gate to the bulk. with this model we assume that we can separate the problem in two simple one-dimensional problems.

with the boundary conditions:

as means on the bulk contact, where the semiconductor is neutral With GCA the Poisson equation becomes one-dimensional and with some calculations the solution for the electric field is found

The electrostatic potential cannot be found by integrating the electric field The sign holds fo the working condition, ”-” for accumulation and ”+” for depletion or invertion Defining the surface potential as: we find the electric field at the surface:

and it is possible to determine the total semiconductor charge per unit area, by means of the Gauss law

Charges

The total charge per unit area is defined as:

The electron charge per unit area is similar, but only the electron concentration is integrated:

and the same goes for the depletion charge

with that is the depletion depth.

The sum between the depletion charge and the electron charge is equal to the total charge in the semiconductor

The MOS charge for changes with respect to the voltage, in the figure is possible to see the charge at the surface for the four operating conditions #todo add image with the four op. cond

Surface Potential Equation (SPE)

Now that is known the relation between the charge (per unit area) and the surface potential, it is possible to obtain the Surface Potential Equation.

Starting from the voltage balance equation:

and being the the SPE is:

Is interesting to see the behaviour of both the charge and the surface voltage in the different working condition

Depletion

When we are in depletion, the charge in the semiconductor is only the depletion charge:

the voltages then are:

Where is the body coefficient, we can see that is linear with and independent from the channel potential

Weak inversion

In weak inversion is not possible to do approximations on the charges in the substrate, so is not taken into consideration, instead we will evaluate the charge and the surface potential in strong inversion

Strong inversion

In strong inversion we have the accumulation of charges of the opposite kind with respect to the type of substrate. The total charge is:

In this case is independent from but only depends on

In strong inversion we obtain the charge control model where depends on and through linearization of the MOS charge with respect to around and we obtain:

Where is the bulk charge linearization coefficient (not reported in these notes, check the slides)

The model becomes linear with and . From the definition of we can rewrite the electron charge as:

todo add graph