= Solution
A positive gate voltage attracts <Electrons> towards the oxide interface of the <p-type semiconductor> and repels mobile <electron holes>. The uncovered ionized acceptors form a negatively charged <depletion region>. Increasing the voltage further creates a thin electron-rich <semiconductor inversion layer> at the surface. The <conduction band> and <valence band> both bend downwards towards the positively biased gate, because an electron's electrostatic energy is $-e\psi$.
In an ideal nondegenerate substrate with acceptor density $N_A$, write $\phi_F=(k_BT/e)\ln(N_A/n_i)>0$ and surface potential $\psi_s$ relative to the neutral bulk. The electron and hole surface densities are
$$
n_s=\frac{n_i^2}{N_A}e^{e\psi_s/(k_BT)},\qquad p_s=N_Ae^{-e\psi_s/(k_BT)}.
$$
The first change of majority-carrier type occurs at $\psi_s=\phi_F$, where $n_s=p_s=n_i$. The conventional <MOS inversion threshold> for appreciable strong inversion is
$$
\boxed{\psi_s\simeq2\phi_F,\qquad n_s\simeq N_A.}
$$
At threshold the inversion sheet is just developing, while a much thicker <depletion region> lies between it and the neutral p-type bulk. Inversion does not require the <conduction band> to cross the <Fermi level>: the surface can remain nondegenerate.
\Image[/past-exam-of-the-quantum-condensed-matter-course-of-the-university-of-cambridge/2018-q1b-mos-inversion.png]
{title=Downward band bending and an incipient electron inversion layer in a MOS structure}
{description=Original schematic with a positively biased metal gate, insulating oxide, a thin surface inversion region, and a deeper depletion region. The semiconductor band gap is constant. The separate gate and substrate Fermi levels illustrate applied bias; band offsets and widths are schematic.}
\b[The metal and semiconductor Fermi levels need not be equal.] The ideal oxide blocks particle exchange, so the gate and substrate can maintain different <electrochemical potentials>. In a biased <MOSFET>, the <gate-to-substrate Fermi-level difference> is $\widetilde\mu_g-\widetilde\mu_s=-eV_g$, when the applied voltage is defined in the electron-reservoir convention. Each region can separately be in internal equilibrium. If an external zero-voltage connection instead equilibrates the reservoirs, their Fermi levels coincide, but different <work functions> can still cause <band bending>. Here the chemical potential shown in a band diagram is the electron electrochemical potential; the purely chemical contribution alone is not the equilibrium criterion for charged particles.
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