Thus the surface electron density has grown to the bulk majority-carrier density. The chemical potential need not actually enter the conduction band. The phrase “just forms” can also refer to the earlier weak-inversion crossover; the diagram uses the conventional strong-inversion threshold and labels it explicitly.

Within the depletion approximation the negative space charge is approximately

- e N_{A}

over a width

W_{d} = \frac{2 ϵ _{s} ψ _{s}}{e N _{A}}, W_{d, th} = \frac{4 ϵ _{s} ϕ _{F}}{e N _{A}},

(4)

where

ϵ_{s}

is the absolute semiconductor permittivity. Its outer edge is where the bands become flat and mobile electron holes recover their bulk density. The electrical insulator blocks steady carrier transport while sustaining an electric field; the metal gate supplies positive surface charge balancing depletion and inversion charge. Under gate bias, the gate and substrate electrochemical potentials are different, so a single equilibrium Fermi level should not be drawn through the whole biased stack.

b

 0  0

Solution

 0  0

For a hydrogenic donor state, dielectric screening replaces

ϵ_{0}

ϵ_{r} ϵ_{0}

and the electron mass by the band effective mass. The characteristic orbital radius is therefore

a^{*} = \frac{4 π ϵ _{0} ϵ _{r} ℏ ^{2}}{m ^{*} e ^{2}} = ϵ_{r} \frac{m _{e}}{m ^{*}} a_{0} .

(1)

Using

a_{0} = 0.0529177 nm

gives

a^{*} = 120 a_{0} = 6.35 nm .

(2)

A donor-state overlap estimate compares this radius with the mean dopant spacing

d \sim N_{D}^{- 1/3}

. Taking

d \sim a^{*}

yields

N_{D} \sim (a^{*})^{- 3} = 3.91 \times 1 0^{24} m^{- 3} = 3.91 \times 1 0^{18} c m^{- 3} .

(3)

There is no sharp geometrical onset: hydrogenic wavefunctions have exponential tails. If “overlap” is defined by neighbouring orbital diameters touching,

d \sim 2 a^{*}

, the corresponding estimate is

4.88 \times 1 0^{23} m^{- 3}

, smaller by a factor of eight. These are conventions for the onset of appreciable orbital overlap, not an exact critical density for a metal-insulator transition.