Non-Interacting Models of Inhomogeneous SB

The junction current of an inhomogeneous SB has been proposed to be described by a parallel conduction model. In such a model, the junction current is a linear sum of the contribution from every individual area, namely,

I(V_a) = A^* T² [ exp(b V_a ) - 1 ] S_i A_i exp( b F_i )

where A_i and F_i are, respectively, the area and the SBH of the i-th "patch". Here the overall transport mechanism has been assumed to be thermionic emission, as evidenced by the unity ideality factor assumed in the equation. In general, the transport mechanism can be assumed to be any other mechanism with only minor modifications. The main assumption behind the parallel conduction model is the independence, as far as the electrical conduction is concerned, of the different segments of an interface to each other. Under this condition, transport theories for homogeneous SBH can be simply extended to cover inhomogeneous SBs. Such a model has been applied to describe mixed-phase diodes and also to analyze SB diodes with an assumed continuous range of SBH.

Interactions at Inhomogeneous SBs

What has been missing from all models based on the parallel conduction concept is the interaction between neighboring sections of the same interface. When the actual band bending is considered, not only the probabilistic distribution of the magnitude of the SBH is found to be important, but also the length scale with which the SBH varies is found to be very important. As first revealed by numerical simulations at MS interfaces and semiconductor-liquid interfaces, the parallel conduction model is in significant error when the SBH varies spatially on a scale less than, or comparable to, the width of the space charge region. The error arises because Eq. (1) fails to take into account of the potential "pinch-off" effect, as shown in Fig. 1, which is a three-dimensional schematic of a saddle point near a low-SBH patch in a high-SBH background.

From earlier discussion, it is clear that the shape of the inhomogeneous SBH has a strong effect on the current transport. For example, small-amplitude variation of the SBH and isolated high-SBH patches lead to a junction current which obeys the parallel conduction model. However, when the SBH varies with large amplitudes on a small lateral length scale potential pinch-off occurs and the current transport deviates considerably from the parallel conduction model. Numerical simulations gave excellent support to expressing the total current as a modified sum of currents flowing in each patch,

I(V_a) = A^* T² [ exp(b V_a ) - 1 ] S_i A_i,eff exp( b F_i,eff )

where the effective SBH, F_i,eff, is simply the local SBH for patches which are not pinched off, but F_i,eff assumes the value of the saddle-point potential in front of the i-th low-SBH patch when pinch-off occurs. Likewise, the effective area of a patch, A_i,eff, is given in Table 1 for pinched-off patches, but reverts to the size of the patch, A_i, when it is not pinched off. The most important difference between Eqs. (2) and (1) is that the F_i,eff's and the A_i,eff 's depend on bias, temperature, doping level, geometry, etc., while the F_i 's and the A_i 's do not. It is the various dependencies of the parameters in Eq. (2) which leads to a wealth of interesting I-V characteristics. For example, the increase of the saddle point potential with forward bias leads to an ideality factor greater than 1 for the component of current which flows through this saddle-point.