3.2.1 Full depletion analysis of an M-S junction

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3.2.1 Full depletion analysis of an M-S junction

The simple analytic model of the M-S junction is based on the full depletion approximation which assumes that the semiconductor is depleted over a distance x_d, called the depletion region. While this assumption does not provide an accurate charge distribution, it does provide very reasonable approximate expressions for the electric field and potential throughout the semiconductor as shown in the figure below.

Fig.3.2.1

Charge density, electric field, potential and energy as obtained for an n-type metal-semiconductor junction (N_d = 10¹⁷ cm^-3 , F_M = 4.75 V, c = 4.05 V and V_a = -0.5 V) using the full depletion model.

Active figure

A comparison of this approximate analytical solution and a numeric calculation can be found in section 3.2.4.

We now apply the full depletion approximation to an M-S junction containing an n-type semiconductor. We define the depletion region to be between the metal-semiconductor interface (x = 0) and the edge of the depletion region (x = x_d). The depletion layer width, x_d, is unknown at this point but will later be expressed as a function of the applied voltage.

As the semiconductor is depleted of mobile carriers within the depletion region, the charge density is due to the ionized donors. Outside the depletion region the semiconductor is assumed to be neutral. This yields the following expressions for the charge density, r:

and

where we assumed full ionization so that the ionized donor density equals the donor density, N_d. Using Gauss's law we obtain electric field as a function of position:

and

where e_s is the dielectric constant of the semiconductor. We also assumed that the electric field is zero outside the depletion region. It is expected to be zero there since a non-zero field would cause the mobile carriers to redistribute untill there is no field. The depletion region does not contain mobile carriers so that there can be a non-zero field. The largest (absolute) value of the electric field is obtained at the interface and is given by:

where the electric field was also related to the total charge (per unit area), Q_d, in the depletion layer. Since the electric field is minus the gradient of the potential, one obtains the potential by integrating the expression for the electric field, yielding:

and

The total potential difference across the semiconductor equals the built-in potential, f_i in thermal equilibrium and is further reduced by the applied voltage when a positive voltage is applied to the metal. This boundary condition provides the following relation between the potential at the surface, the applied voltage and the depletion layer width:

Solving this expression for the depletion layer width, x_d, yields:

In addition one can also obtain the capacitance as a function of the applied voltage by taking the derivative of the charge with respect to the applied voltage yielding:

The last term in the equation indicates that the expression of a parallel plate capacitor still applies. One can understand this once one realizes that the charge added (removed) from the depletion layer as one decreases (increases) the applied voltage is added (removed) only at the edge of the depletion region. While the parallel plate capacitor expression seems to imply that the capacitance is constant, the M-S junction capacitance is not constant since the depletion layer width, x_d, varies with the applied voltage.

3.2 Ź ® 3.2.2