Surface Space Charge Region of an n-Type MOS Capacitor

B.1 Surface Space Charge Region of an
n-Type MOS Capacitor

To be able to calculate and interpret the $C (V )$ -characteristics of pMOSFETs the understanding of the semiconductor charge $Qs$ as a function of the surface potential $ψs$ is inevitable. The nomenclature of these calacultions is based on [10] but an n-type instead of a p-type semiconductor is used.

The measure for the local band bending in the semiconductor is given by $ψn (x ) = Ei(x)∕q$ , which determines the local electron concentration $nn(x)$ and hole concentration $pn(x)$ given by

( ) qψn nn(x) = nn0exp k-T- = nn0exp (β ψn) (B.1 ) ( B ) pn(x) = pn0exp −-qψn = pn0exp(− βψn) (B.2 ) kBT

where $nn0$ and $pn0$ denote the equilibrium densities (flatband case), $kB$ the Boltzmann constant and $1∕β$ the thermoelectrical potential at $T$ . Starting from the 1d-Poisson equation

d2ψ (x ) ρ(x) ----n2---= − ---- dx ϵrϵ0

(B.3)

with the local space charge density $ρ(x) = q(ND+ − NA − + pn − nn )$ the potential is obtained. Here, $N + D$ and $N − A$ are the densities of the ionized donors and acceptors. Deep in the substrate (and under flatband conditions in the whole semiconductor) charge neutrality can be assumed $ρ(x ) = 0$ due to $ψn(x → ∞ ) = 0$ . Inserting and evaluating (B.1) and (B.2) yields $ND+ − NA − = nn0 − pn0$ and finally

d2ψn(x) -q-- dx2 = − ϵrϵ0(nn0 − pn0 + pn − nn) -q-- = − ϵrϵ0(pn0(exp(− βψn) − 1)− nn0(exp(βψn )− 1)). (B.4 )

To integrate (B.4) the following integration trick is necessary:

( ) 2 2 2 d ddψx- = d-ψdx-−-d2-ψd-x-= d-ψ- ( ) dx dx dψd dψ- = dψ-d2ψ- dx dx dx dx d2ψ = dψ dx2- ∫ ( ) ∫ 2 ddψxd ddψx- = d-ψdψ (B.5 ) dx2

With (B.5) the Poisson equation (B.4) can be rewritten as

∫ψs −E∫sdψn (dψn) -q-- 0 dx d dx = − ϵrϵ0(pn0(exp(− βψn) − 1)− nn0(exp(βψn )− 1))dψn 0 ( )2||−Es q [ ( exp(− βψn) ) (exp (β ψn) )] ||ψs 12 ddψxn || = − ---- pn0 -----------− ψn − nn0 ---------− ψn || (B.6) 0 ϵrϵ0 [ ( − β ) ( β 0 ) ] 1E2 = − -q-- p exp(− β-ψs)− ψ + 1- − n exp(βψs)-− ψ − 1- 2 s ϵrϵ0 n0 − β s β n0 β s β qn [( p ) ] = ---n0 -n0- (exp(− βψs)+ β ψs − 1) + (exp(βψs)− β ψs − 1) (B.7.) ϵrϵ0β nn0

The integration boundaries (B.6) range from the substrate, where $ψn = 0$ and $dψn ∕dx = 0$ to the surface with $ψn = ψs$ and $d ψn∕dx = − Es$ .

For non-degenerate semiconductors the Fermi level is far enough away from $E c$ and $E v$ and Fermi-Dirac statistics can be approximated by Boltzmann statistics. The carrier concentrations for an n-type semiconductor with $ND+ > NA −$ can be approximated [10]. Then,

(E − Ei ) ( qψBn ) nn0 ≈ ND = niexp --f---- = niexp ----- (B.8 ) kBT kBT -n2i- n2i-- pn0 = nn0 ≈ ND , (B.9 )

which yields

pn0-= exp(− 2 βψBn). nn0

(B.10)

The electric field at the surface in (B.7) is now simplified to

∘ ------ 2qnn0- Es = ± ϵrϵ0β F(βψBn, ψs)

(B.11)

with

∘ -------------------------------------------------------- F(β ψBn,ψs) = + exp (− 2βψBn )(exp (− βψs) + βψs − 1)+ (exp(βψs) − βψs − 1).

(B.12)

By applying Gauss’ law $∇ ⋅E = ρ (x )∕(ϵrϵ0)$ the space-charge-density per area is finally obtained as

∘ --------- Qs = − ϵrϵ0Es = ∓ 2qϵrϵ0nn0F (βψBn,ψs). β

(B.13)

For n-type (pMOS) semiconductors (B.13) can now be approximated for the different regimes of the surface potential $ψs ≡ V$ with the contact voltage $V$ , defined in the beginning of the chapter. For accumulation with $ψs > 0$ , the term $exp(βψs)$ dominates in (B.12), making

Qs ∝ − exp(β ψs∕2).

For $ψs < 0$ , depletion and successively weak inversion set in till $ψs = − 2ψBn$ is fulfilled. Here

∘ ------ Qs ∝ + − β ψs.

Finally, beyond $ψs > − |2ψBn |$ the first term in (B.12) starts to dominate, which yields

Qs ∝ + exp (− βψs∕2 ).

Figure B.2: The surface charge density $Qs$ compared for both p-type and n-type semiconductors depending on $ψs$ . While the solid lines stand for positive $Qs$ , the dashed lines symbolize a negative $Qs$ . The approximations very well fit the exact solutions (B.13) and (B.16) drawn in black. Deviations from the latter only exist at the transitions of the operating regimes. 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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B.1 Surface Space Charge Region of ann-Type MOS Capacitor

B.1 Surface Space Charge Region of an
n-Type MOS Capacitor