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3.5  Recombination and Generation

In Section 2.2.2 recombination/generation has been discussed for a single two-state defect located energetically within the bandgap, better known as the SRH model  [31]. This model has been incorporated into the bipolar spherical harmonics expansion of the BTE  [62]. Thus the electron and hole BTE are now coupled by a recombination term Γ and the solution fνT(t) of Equation (2.32) per trap level ET. The recombination operators Γp∕n{fn,fp} for electrons and holes are

Γn{f n,fp} = NT ----3 (2π) ∫ B Gνn(k)fν T(t)(1 - fn) - Rn(k)(1 - fν T(t))fnd3k, (3.34)
Γp{f n,fp} = -NT-- (2π)3 ∫ B Gνp(k)(1 - fν T(t))(1 - fp) - Rp(k)fν T(t)fpd3k, (3.35)
where NT is the trap concentration. The expansion into spherical harmonics using Equation (3.3), Equation (3.4) as well as the H-transfrom gives
Γn{f n,fp}l,m = NTZn(H)( n n n 1 n ) G (H)fT (1 - fl,m) - R (H )(Y---- fT)fl,m 0,0δl,0δl,m, (3.36)
Γp{f n,fp}l,m = NTZp(H)( ) Gp(H )(-1--- f )(1 - fp )- Rp (H )f fp Y0,0 T l,m T l,mδl,0δl,m. (3.37)
To provide an example of the implementation of the above equations, a 2D NPN-transistor in steady state has been simulated with a trap density of 1019cm-3 and a single trap level at 0.1eV away from the center of the bandgap. The domain has been discretized using an orthonormal grid with 3600 vertices. The doping was set to ND = 1016cm-3 in both n-doped regions and NA = 1016cm-3 in the p-doped center. The gate potential was set to 0.7V and V cc to 0.1V. The resulting trap occupancy is shown in Figure 3.4.

PIC

Figure 3.4: Trap occupancy in a NPN-transistor, where the gate potential is 0.7V and V cc = 0.1V. Note that the base of the transistor has been named ‘Gate’.

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