Nanostructured Thermoelectric: Phonon Glass, Electron Crystal

Nanostructuring

Grain size 60 nm

Smaller grains → more phonon-boundary scattering. Below ~20 nm the grains start scattering electrons too, so σ falls.

Rattler atoms (filled skutterudite) 4%

Heavy guest ions (e.g. La, Ce, Yb) rattling in CoSb₃ cages resonantly scatter low-frequency phonons.

Temperature 500 K

Higher T strengthens intrinsic phonon-phonon Umklapp scattering (mean free path ∝ 1/T).

Carrier density 2.0 ×10¹⁹ cm⁻³

Doping trade-off (Pisarenko relation): more carriers → higher σ but lower Seebeck S.

Live Readouts

Lattice κ_l— W/m·K

Figure of merit ZT—

η (vs. Carnot)—

Legend

Phonon (heat carrier) — scatters often Electron (charge carrier) — scatters rarely Rattler guest atom Host lattice atom Grain boundary

How it works

A good thermoelectric wants to conduct electricity like a crystal but heat like a glass (Slack's "phonon-glass, electron-crystal" idea). Phonons that carry heat have long wavelengths, so they scatter off grain boundaries and heavy rattler atoms; electrons that carry charge have much shorter wavelengths, so the same defects barely touch them — if the grains stay above ~10-20 nm.

Matthiessen's rule (combine scattering mechanisms):
  1/Λ = 1/Λ_boundary + 1/Λ_rattler + 1/Λ_Umklapp(T)

Kinetic theory of heat conduction:
  κ_l = (1/3)·C_v·v_s·Λ

Wiedemann-Franz law (electronic heat vs. charge):
  κ_e = L·σ·T           (L = 2.44×10⁻⁸ WΩ/K²)

Pisarenko relation (doping trade-off):
  S ∝ n^(-2/3)           σ = n·e·μ

Figure of merit and device efficiency:
  ZT = S²σT / (κ_l + κ_e)
  η = η_Carnot · (√(1+ZT) − 1) / (√(1+ZT) + T_c/T_h)

Grain size — sets the boundary mean free path Λ_boundary directly; drag it down and watch phonons (orange) bounce far more than electrons (cyan).
Rattler atoms — heavy guest ions vibrating loosely in cage voids (the mechanism behind filled skutterudites like La-CoSb₃, ZT≈1.7) resonantly scatter phonons without touching the electron gas.
Temperature — raises intrinsic Umklapp (phonon-phonon) scattering, which shortens Λ even with no nanostructuring at all.
Carrier density — the Pisarenko trade-off: more free carriers raise electrical conductivity σ but suppress the Seebeck coefficient S, so ZT peaks at an intermediate doping level, not at the extremes.

Real materials: bulk-nanostructured BiSbTe reaches κ_l ≈ 0.5 W/m·K via ~20 nm grains (Poudel et al.); filled skutterudites reach κ_l ≈ 1.5 W/m·K via rattling (vs. 10 W/m·K unfilled); record SnSe single crystals hit ZT = 2.6 through extreme lattice anharmonicity instead of nanostructuring at all.

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