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 ).


PIC


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.