Multiferroic Magnetoelectric Coupling
Ferroelectric P · Magnetic M · Cross-Coupling · Landau Free Energy
Landau Parameters
α (FE quadratic, <0 = ordered):
-1.0
β (mag quadratic, <0 = ordered):
-0.5
ME coupling γ:
0.4
Electric field E:
0.0
Magnetic field H:
0.0
F = α/2·P² + β/4·P⁴ + α_m/2·M² + β_m/4·M⁴
+ γ·P²M² - E·P - H·M
∂F/∂P=0, ∂F/∂M=0 → equilibrium
ME effect: ∂P/∂H = -2γPM/α_eff
Multiferroics simultaneously exhibit ferroelectric and magnetic order. The cross-coupling term γP²M² enables electric control of magnetism and magnetic control of polarization — key for spintronics.