Polaron & Self-Trapping

Electron-phonon coupling, Fröhlich model, and the self-trapping transition

Lattice Distortion & Electron Wavefunction

Polaron Properties vs Coupling α

Energy landscape

Theory — Fröhlich Polaron

H = p²/2m + Σ_k ℏω_LO(a†_k a_k + ½) + Σ_k V_k(a_k e^{ik·r} + h.c.)
E_polaron ≈ −α ℏω_LO (weak coupling α≪1)
m*/m = 1/(1 − α/6) (valid α < 6)
E_polaron ≈ −0.109 α² ℏω_LO (strong coupling, Pekar)

Fröhlich coupling α describes interaction strength between electron and longitudinal optical (LO) phonons in a polar crystal. Small α: perturbative polaron with slightly enhanced mass. Large α > ~6: self-trapped "large polaron" with m* → ∞.


Self-trapping: at strong coupling, the electron localizes itself in the potential well it creates — a nonlinear feedback. The wavefunction shrinks as α increases. Materials: SrTiO₃ (α≈2), ionic crystals (α=4-6), bipolarons in cuprates.


The Feynman path-integral variational method (1955) gives the best analytical result across all α, bridging weak and strong coupling regimes.