= Solution
A positive <electric potential difference> between gate and substrate attracts <Electrons> towards the insulating interface and repels the mobile <electron holes> of the <p-type semiconductor> substrate. The exposed, negatively charged <acceptor impurities> form a <depletion region>; the thin mobile <Electron> layer at its surface is the <semiconductor inversion layer>. The <conduction band> and <valence band> bend \i[downwards] towards the positively biased gate, because the <Electron> <potential energy> is $-e\psi$.
\Image[/past-exam-of-the-quantum-condensed-matter-course-of-the-university-of-cambridge/2012-mos-inversion.png]
{title=<Band bending> and charge separation at the conventional <MOS inversion threshold>}
To specify the conventional <MOS inversion threshold>, let $N_A$ be the ionized acceptor density, $n_i$ the <intrinsic carrier concentration>, and
$$
\phi_F=\frac{k_BT}{e}\log\frac{N_A}{n_i}>0.
$$
In the neutral bulk the intrinsic level lies $e\phi_F$ above the <semiconductor> <chemical potential>. A positive surface potential $\psi_s$ lowers both band edges by $e\psi_s$, giving, in the nondegenerate approximation,
$$
p_s=N_Ae^{-e\psi_s/(k_BT)},\qquad
n_s=\frac{n_i^2}{N_A}e^{e\psi_s/(k_BT)}.
$$
At $\psi_s=\phi_F$, the surface is intrinsic: $n_s=p_s=n_i$. The usual onset of \i[strong inversion] instead means
$$
\boxed{\psi_s\simeq2\phi_F,\qquad n_s\simeq N_A.}
$$
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 $-eN_A$ over a width
$$
W_d=\sqrt{\frac{2\epsilon_s\psi_s}{eN_A}},\qquad
W_{d,\mathrm{th}}=\sqrt{\frac{4\epsilon_s\phi_F}{eN_A}},
$$
where $\epsilon_s$ 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.
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