Results for p-Type Semiconductors

B.2 Results for p-Type Semiconductors

When compared to n-type semiconductors, some of the equations from (B.1) to (B.13) differ for p-type semiconductors. Due to their exchanged concentrations of holes $pp$ and electrons $np$

pp(x) = pp0exp(− βψp ) pp0 ≈ NA = ni exp(βψBp) np(x) = np0exp (β ψp) np0 = n2∕pp0 ≈ n2∕NA i i

some of the equations change their signs depending on the applied bias conditions. The calculations are derived in [10] and for that reason only the differences between n-type and p-type are summarized below. Starting with the electric field at the interface

1 qp [ (n ) ] -E2s = --p0- (exp(− βψs)+ β ψs − 1) + --p0 (exp (β ψs)− βψs − 1) 2 ϵrϵ0β pp0

∘ ------ E = ± 2qpp0F(βψ ,ψ ) s ϵrϵ0β Bp s

(B.14)

with

∘ -------------------------------------------------------- F(β ψ ,ψ ) = + (exp(− βψ )+ βψ − 1 )+ exp(− 2βψ )(exp(βψ ) − βψ − 1), Bp s s s Bp s s

(B.15)

the space-charge-density becomes

∘ --------- Qs = − ϵrϵ0Es = ∓ 2qϵrϵ0pp0F (βψBp,ψs). β

(B.16)

Again, the charge at the surface (B.16) can be approximated for certain surface potentials $ψs$ .

For accumulation with $ψs < 0$ , the term $exp(− βψs)$ dominates the root in (B.15), making $Qs ∝ exp(− β ψs∕2)$ .

Starting from the flatband condition at $ψs = 0$ , first depletion of holes and afterwards weak inversion set in till $ψs = 2ψBp$ is fulfilled. In these two regimes $Qs ∝ − √ βψs-$ .

Finally, beyond $ψ > 2ψ s Bn$ the first term in (B.15), starts to dominate by outbalancing the negative exponent in $exp (− 2βψBp )$ which yields $Qs ∝ − exp (β ψs∕2)$ .

In Fig. B.2 the different operating conditions with its resulting surface charge density $Qs$ at the interface side of the semiconductor are opposite for both p-type and n-type semiconductors. The above mentioned approximations very well fit the exact solutions (B.13) and (B.16), as deviations are only present at the intersections of the different regimes. Furthermore, it is shown that $Q (ψ ) ≡ − Q (− ψ ) s,n s s,p s$ , where the subscripts $n$ and $p$ denote the type of semiconductor.

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