Quantum Paraelectric — Soft Phonon Mode

SrTiO₃-like: Quantum Fluctuations Suppress Ferroelectric Order
Phonon dispersion ω²(k) — soft mode at Γ
Effective potential V(Q) — polarization landscape
T = 80 K
Δ_QF = 0.55
T_c0 = 80 K
E = 0.00
β = 0.50
Soft mode dispersion:
ω²(T) = A(T² + T_Q²)
Quantum limit:
T_Q = Δ_QF · T_c0
Dielectric const:
ε(T) ∝ 1/ω²(T)
Barrett formula:
ε⁻¹ ~ T_1/2·coth(T₁/2T)−T_0
QP condition:
T_Q > T_c0 → no FE order
SrTiO₃ parameters:
T_c0 ≈ 36 K (suppressed)
T_Q ≈ 40 K → quantum PE
SrTiO₃ is a quantum paraelectric: classical Landau theory predicts ferroelectric order at ~36 K, but quantum zero-point fluctuations of the soft transverse optical phonon mode prevent condensation. The soft mode frequency ω²(T) → 0 but stops at a finite value set by quantum fluctuations T_Q. Applying pressure or isotope substitution (¹⁶O→¹⁸O) can tip it into a FE state.