Ferroelectric Domain Switching & the P-E Hysteresis Loop

Applied Field

Field E 0.00 MV/m

Sweep amplitude 1.5 × E꜀

Sweep speed 1.0×

Grain disorder (σ) 0.25

Live Readouts

Applied field E

0.00 MV/m

Polarization P

+35.0 µC/cm²

Domains at +P

100%

Remnant Pᵣ (E=0)

— µC/cm²

Coercive field E꜀ (this material)

1.20 MV/m

How it works

Piezoelectric ceramics like lead zirconate titanate (PZT) are ferroelectric: below the Curie temperature their perovskite (ABO₃) unit cell is non-centrosymmetric, so the central B-site cation sits off-center and each unit cell carries a permanent electric dipole. Neighboring cells lock into domains that all point the same way. A strong enough electric field can reverse a domain's dipole — this is what "poling" a piezo ceramic during manufacturing actually does, and it is why the P-E curve is a loop, not a line.

This sim models the ceramic as an ensemble of independent domains (a Preisach-type hysteron model, the standard textbook description of ferroelectric/ferromagnetic hysteresis). Each domain i has its own local coercive field E꜀ᵢ, drawn from a Gaussian centered on the material's E꜀ with spread σ (real ceramics are polycrystalline, so grain orientation and defects spread the switching threshold from grain to grain):

domain flips +P → -P  when  E < -E꜀ᵢ
domain flips -P → +P  when  E > +E꜀ᵢ
otherwise the domain keeps its current state (memory ⇒ hysteresis)

macroscopic polarization:  P(E) = Pₛ · (1/N) Σᵢ sᵢ,   sᵢ ∈ {-1,+1}

Field slider — drive the applied field E by hand and watch domains (cones) flip red↔blue once |E| crosses each grain's own threshold.
Auto-Sweep — cycles E through a full triangle wave, tracing the closed P-E hysteresis loop in the corner plot in real time.
Grain disorder σ — 0 gives an idealized square loop (every domain flips at exactly E꜀); higher σ rounds the loop's shoulders, matching real polycrystalline PZT.
Remnant polarization Pᵣ — the polarization left over at E = 0 after saturating the loop; this residual dipole alignment is exactly what makes a poled piezo element respond to strain afterward.

Real-world relevance: this loop is measured on every batch of PZT, PVDF and KNN piezo material during production — its area is the electrical energy dissipated per switching cycle, and Pᵣ/Pₛ tells manufacturers how well a part was poled before it goes into a sonar transducer, fuel injector, or ultrasound probe.

P–E hysteresis loop

Domain dipole +P (up) Domain dipole -P (down)

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