The current across a metal-semiconductor junction is mainly due to majority carriers. Three distinctly different mechanisms exist: diffusion of carriers from the semiconductor into the metal, thermionic emission of carriers across the Schottky barrier and quantum-mechanical tunneling through the barrier. The diffusion theory assumes that the driving force is distributed over the length of the depletion layer. The thermionic emission theory on the other hand postulates that only energetic carriers, those, which have an energy equal to or larger than the conduction band energy at the metal-semiconductor interface, contribute to the current flow. Quantum-mechanical tunneling through the barrier takes into account the wave-nature of the electrons, allowing them to penetrate through thin barriers. In a given junction, a combination of all three mechanisms could exist. However, typically one finds that only one current mechanism dominates.

The analysis reveals that the diffusion and thermionic emission currents can be written in the following form:

This expression states that the current is the product of the electronic charge, q, a velocity, v, and the density of available carriers in the semiconductor located next to the interface. The velocity equals the mobility multiplied with the field at the interface for the diffusion current and the Richardson velocity (see section 3.4.2) for the thermionic emission current. The minus one term ensures that the current is zero if no voltage is applied as in thermal equilibrium any motion of carriers is balanced by a motion of carriers in the opposite direction.

The tunneling current is of a similar form, namely:

where v_R is the Richardson velocity and n is the density of carriers in the semiconductor. The tunneling probability term, Θ, is added since the total current depends on the carrier flux arriving at the tunnel barrier multiplied with the probability, Θ, that they tunnel through the barrier.

3.4.1. Diffusion current

This analysis assumes that the depletion layer is large compared to the mean free path, so that the concepts of drift and diffusion are valid. The resulting current density equals:

The current therefore depends exponentially on the applied voltage, V_a, and the barrier height, φ_B. The prefactor can more easily be understood if one rewrites it as a function of the electric field at the metal-semiconductor interface, _max:

yielding:

so that the prefactor equals the drift current at the metal-semiconductor interface, which for zero applied voltage exactly balances the diffusion current.

3.4.2 Thermionic emission

The thermionic emission theory assumes that electrons, with an energy larger than the top of the barrier, will cross the barrier provided they move towards the barrier. The actual shape of the barrier is hereby ignored. The current can be expressed as:

where is the Richardson constant and φ_B is the Schottky barrier height.

The expression for the current due to thermionic emission can also be written as a function of the average velocity with which the electrons at the interface approach the barrier. This velocity is referred to as the Richardson velocity given by:

So that the current density becomes:

3.4.3. Tunneling

The tunneling current is obtained from the product of the carrier charge, velocity and density. The velocity equals the Richardson velocity, the velocity with which on average the carriers approach the barrier. The carrier density equals the density of available electrons, n, multiplied with the tunneling probability, Θ, yielding:

Where the tunneling probability is obtained from:

and the electric field equals = φ_B/L.

The tunneling current therefore depends exponentially on the barrier height, φ_B, to the 3/2 power.

3.4.4. Derivation of the Metal-Semiconductor Junction Current

We start from the expression for the total current and then integrate it over the width of the depletion region:

which can be rewritten by using = -dφ/dx and multiplying both sides of the equation with exp(-φ/V_t), yielding:

Integration of both sides of the equation over the depletion region yields:

Where the values listed in Table 3.4.1 were used for the electron density and the potential:

and φ^* = φ + φ_I - V_a. The integral in the denominator can be solved using the potential obtained from the full depletion approximation solution, or:

so that φ^* can be written as:

where the second term is dropped since the linear term is dominant if x << xsub>d. Using this approximation one can solve the integral as:

for (φ_i – V_a) > V_t. This yields the final expression for the current due to diffusion:

This expression indicates that the current depends exponentially on the applied voltage, V_a, and the barrier height, φ_B. The prefactor can be understood physically if one rewrites that term as a function of the electric field at the metal-semiconductor interface, _max:

yielding:

so that the prefactor equals the drift current at the metal-semiconductor interface, which for zero applied voltage exactly balances the diffusion current.

The thermionic emission theory assumes that electrons, which have an energy larger than the top of the barrier will cross the barrier, provided they move towards the barrier. The actual shape of the barrier is hereby ignored. The current can be expressed as:

For non-degenerately doped material, the density of electrons between E and E + dE is given by: (using (2.4.7) and assuming E_F,n < E_c - 3kT)

Assuming a parabolic conduction band (with constant effective mass m^*), the carrier energy, E, can be related to its velocity, v, by:

Combining (3.4.21) with (3.4.22) yields:

when replacing v² by v_x² + v_y² +v_z² and 4<π v²dv by dv_xdv_ydv_z the current becomes:

using

The velocity v_ox is obtained by setting the kinetic energy equal to the potential across the n-type region:

so that v_ox is the minimal velocity of an electron in the quasi-neutral n-type region, needed to cross the barrier. Using

which is valid for a metal-semiconductor junction, one obtains:

where is the Richardson constant and φ_B is the Schottky barrier height which equals the difference between the Fermi level in the metal, E_F,M and the conduction band edge, E_c, evaluated at the interface between the metal and the semiconductor. The -1 term is added to account for the current flowing from right to left. The current flow from right to left is independent of the applied voltage since the barrier is independent of the band bending in the semiconductor and equal to φ_B. Therefore it can be evaluated at any voltage. For V_a = 0 the total current must be zero, yielding the -1 term.

The expression for the current due to thermionic emission can also be written as a function of the average velocity with which the electrons at the interface approach the barrier. This velocity is referred to as the Richardson velocity given by:

So that the current density becomes:

To derive the tunnel current, we start from the time independent Schrödinger equation:

which can be rewritten as

Assuming that V(x) - E is independent of position in a section between x and x+dx this equation can be solved yielding:

The minus sign is chosen since we assume that the particle moves from left to right. For a slowly varying potential the amplitude of the wave function at x = L can be related to the wave function at x = 0 :

This equation is referred to as the WKB approximation. From this the tunneling probability, Θ, can be calculated for a triangular barrier for which V(x)-E = q<φ_B (1- x/L)

The tunneling probability then becomes:

where the electric field equals = φ_B/L.

The tunneling current is obtained from the product of the carrier charge, velocity and density. The velocity equals the Richardson velocity, the velocity with which on average the carriers approach the barrier while the carrier density equals the density of available electrons multiplied with the tunneling probability, yielding:

The tunneling current therefore depends exponentially on the barrier height to the 3/2 power.

Boulder, 2022