Curie Transition in a Ferroelectric — Landau Free-Energy Landscape

Drive

Temperature (T − T_c) -0.60

Field amplitude E₀ 0.70

Sweep frequency 0.18

Live State

Ferroelectric (double well)

Polarization P

0.00

Field E

0.00

Coercive field E꜀

—

Spontaneous P꜀

—

How it works

The polarization P of a ferroelectric crystal is treated as an order parameter minimizing a Landau free-energy density (Landau–Devonshire theory):

F(P) = a(T−Tc)·P² + b·P⁴ − E·P
a > 0 → single well (paraelectric, no spontaneous P)
a < 0 → double well (ferroelectric, ±P₀ stable)

The ball on the left is P itself, rolling under damped inertial dynamics driven by that landscape — a visual form of the Landau–Khalatnikov equation γṖ = −∂F/∂P:

m·P̈ + γ·Ṗ = −∂F/∂P = −(2aP + 4bP³ − E)
E(t) = E₀·sin(2π·f·t)

Below the Curie point (a < 0) the two wells give a spontaneous polarization P₀ = √(−a/2b) even at E = 0. As the field sweeps back and forth the ball is pushed to switch wells only once the field exceeds the coercive value — the saddle-node point where one well disappears — tracing the closed P–E loop on the right:

P₀ (remnant)  = √(−a / 2b)
E꜀ (coercive) = (4/3)|a|·√(|a| / 6b)

Temperature slider — sets a = T−Tc directly. Above 0 the double well merges into one minimum: the loop collapses to a single line (linear, non-hysteretic dielectric response), exactly what happens physically above the Curie temperature.
Field amplitude — must exceed the coercive field to fully switch the polarization each half-cycle; too small and the loop pinches into a thin lens instead of a full saturated loop.
Sweep frequency — how fast E(t) oscillates; the instanced arrows at the back show the crystal's dipoles rotating in lock-step with the macroscopic P.

Real-world relevance: this P–E loop is the standard characterization measurement for PZT, BaTiO₃ and other ferroelectric ceramics used in capacitors, FeRAM memory and piezoelectric actuators.

Free-energy landscape (left) P–E hysteresis loop (right)

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