Figure 1. Bandedge profile of heterojunction before contact.
If we look at the energy band diagram before contact as shown in figure 1. The built voltage will be
The difference has been taken as F p -F P , so that forward bias is for the positive potential connected to the P-side. In this case, V 0 will be
For positive or zero built in voltage, it will generally be near zero, the energy band diagram is then given by expressions similar to the nP case, simply replacing all the donor concentration, which is N d , to acceptor concentration that is N a . However, if built in voltage were too negative, the energy band diagram is given by expressions similar to that of pN heterojunction case with N D replaced by N A .
From equation (17), we and obtain an expression for P side, which is
at P side.
we obtain at x= 0
Thus from equations (16) and (17)
, and , and hence
The solution using Matlab is given by
where LambertW( x ) is the solution to w *exp( w ) = x and LambertW( x ) is linear when x <<1. B is usually very large, so that we can ignore the lambertW and hence
For nN heterojunction the built in Voltage will be
ΔE c is generally larger than the difference (E cn - F n ) (E cN F N ), the built in voltage will be positive. The expressions for the energy band diagrams are similar to the p-N heterojunction case with Na replaced by N d.
The energy diagram before contact is shown at figure 2
Figure 2. Bandedge profile before contact in p-P heterojunction.
Following the similar steps as for the p-P heterojunction, we can find an equation at x =0 for electric fields at both sides. The displacement vector must continuous at zero, therefore we obtain:
the only unknown V on is obtained by solving equation (27)