This is a 2D energy-diagram companion to the 3D nanowire viewer: instead of orbiting a rendered wire, it draws the actual electron energy ladder. Confining a conductor to a nanowire (1D) or thin film (2D) turns its smooth bulk density of states g(E) ∝ √E into a staircase of subbands, each opening with a van Hove singularity g(E) ∝ 1/√(E−En) in 1D — real physical energies here, computed from particle-in-a-box confinement, not a rescaled placeholder.
Subband energies: E_n = n²·E1, E1 = ħ²π²/(2m*d²) ≈ 0.376/(m*_rel·d[nm]²) eV
3D bulk DOS: g(E) ∝ √E
2D well DOS: g(E) = Σ_n step(E − E_n) (a staircase)
1D wire DOS: g(E) = Σ_n 1/√(E − E_n) (van Hove spikes)
Occupation: f(E) = 1 / (exp((E−E_F)/k_BT) + 1) (Fermi–Dirac, shown as fill)
Mott formula: S(E_F) = -(π²/3)(k_B²T/e)·d[ln g(E)]/dE |_E=E_F
Power factor: PF = S² · g(E_F)
- Dimensionality — switches which g(E) shape drives the Mott formula; the energy ladder redraws its subbands and the wire lane redraws its drift accordingly.
- Wire cross-section d — smaller d pushes the subbands E_n = n²E1 further apart (E1 ∝ 1/d²), sharpening the DOS steps/spikes that boost S.
- Fermi level EF — where the Fermi–Dirac fill line sits on the ladder; placing it right where g(E) rises steeply maximizes |d ln g/dE|, hence |S|.
- Temperature gradient ΔT — sets both the open-circuit Seebeck voltage V ≈ S·ΔT and how thermally smeared the Fermi–Dirac fill line looks (width ∝ kBT).
Real-world relevance: this is the Hicks–Dresselhaus mechanism used to design nanostructured thermoelectric materials (nanowires, superlattices, quantum-dot arrays) that convert waste heat into electricity or run solid-state Peltier coolers with a higher figure of merit ZT than any bulk crystal of the same composition. Because the Mott formula differentiates the density of states, the Seebeck coefficient genuinely diverges right at a van Hove edge in this idealized (unbroadened) model — real devices see a large but finite peak once scattering broadens the step.