= Solution
At equilibrium, majority <Electrons> undergo <diffusion> from n to p, while majority <electron holes> diffuse from p to n. Their departure exposes positive donors on the n side and negative acceptors on the p side. The resulting <electric field> points from n to p and opposes further diffusion. Equilibrium is reached when the <chemical potential> is constant across the junction. The <built-in potential> is this electrostatic adjustment, not a discontinuity of the material's <band gap>.
Since electronic band energies shift by $-eV(x)$, both the <conduction band> and <valence band> undergo <band bending> down from p to n by $eV_b$ at zero applied bias. Their separation remains the same gap. In the nondegenerate <thermal equilibrium> limit the built-in voltage can also be expressed using neutral-edge densities as $eV_b=k_BT\log(n_n/n_p)=k_BT\log(n_np_p/n_i^2)$.
Take the n-contact potential and its <electrochemical potential> as reference. Applying $V_{\rm ext}$ to the p side changes the total depletion-layer drop to
$$
\boxed{U=V_b-V_{\rm ext},\qquad
E_{c,p}-E_{c,n}=E_{v,p}-E_{v,n}=eU.}
$$
Thus $V_{\rm ext}<0$ is <reverse bias>: it increases the barrier and widens the <depletion region>. The bands have horizontal tangents at the neutral edges in this approximation and a smooth cubic bend within the linearly graded layer.
A biased junction is not in global thermal equilibrium. Its contact electron <electrochemical potentials> obey
$$
\boxed{\mu_p-\mu_n=-eV_{\rm ext}.}
$$
Under reverse bias the p-contact level is higher by $e|V_{\rm ext}|$. A single horizontal equilibrium <chemical potential> cannot describe this state; electron and hole <quasi-Fermi levels> describe the nonequilibrium populations. The sketch marks the contact levels. Detailed spatial quasi-Fermi profiles require transport and recombination information beyond the electrostatic doping profile.
\Image[/past-exam-of-the-quantum-condensed-matter-course-of-the-university-of-cambridge/2021-q6-graded-junction.png]
{title=Band bending, electric field and capacitance of a linearly graded junction}
{description=Original depletion-model sketches with the p region on the left. The top panels compare equilibrium with reverse bias; under bias only the contact electrochemical levels are specified. The bottom panels show the negative electric field and the normalized positive differential capacitance. Energy and length scales are illustrative.}
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