advanced Semiconductor Physics

Semiconductor Band Diagrams

Visualise energy band diagrams for PN junctions, MOS capacitors, and MOSFET channels under applied bias using the Poisson equation.

Applied bias0.00 V

— Ec (conduction band)— Ev (valence band)- - Ei (intrinsic level)- - Ef (Fermi level)

Select 'PN Junction' mode. The diagram shows the conduction band Ec (blue), valence band Ev (red), intrinsic level Ei (dashed grey), and Fermi level Ef (green).

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Energy increases upward. Electrons 'fall' toward lower Ec; holes 'rise' toward higher Ev — they behave like bubbles.
At zero bias (Vbias = 0), the Fermi level is flat — thermal equilibrium. The band bending at the junction shows the built-in potential Vbi as the difference in Ec from p-side to n-side.

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A flat Fermi level is the hallmark of thermal equilibrium. No net current flows despite the built-in field.
Apply forward bias (Vbias → +0.5 V). The bands flatten — the barrier is reduced. The two quasi-Fermi levels (EFn, EFp) split; this split equals the applied voltage.

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Quasi-Fermi levels (EFn for electrons, EFp for holes) represent the non-equilibrium carrier distributions. Current flows when they split.
Switch to 'MOS Cap.' mode. At Vg = 0, bands are flat. Apply positive Vg — the bands bend downward at the surface (n-type inversion). Apply negative Vg — bands bend up (depletion/accumulation).

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The MOS capacitor is the basis of the MOSFET gate. Band bending by Vg = 2φF achieves strong inversion.
Switch to 'MOSFET' mode and vary Vg. The band diagram shows the channel potential along the source-to-drain direction. This potential profile determines where carriers flow.

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In a real MOSFET, the channel potential varies from 0 at the source to Vds at the drain, causing the varying carrier density that leads to pinch-off in saturation.