LLG equation with Slonczewski spin-transfer torque auto-oscillation
Frequency: — GHz | Precession angle: — °
A DC current drives precession of a free magnetic layer via spin-transfer torque (Slonczewski 1996). The LLG equation: dM/dt = −γ(M×H) − αγ(M×(M×H))/Ms + aJ(M×(M×p̂)), where aJ ∝ I is the spin-torque amplitude. Above a threshold current I_c = αγMsH/aJ₀, spin torque overcomes damping, driving sustained auto-oscillation at microwave frequencies (1–40 GHz).