ST-FMR Mixing Voltage Derivation
The total voltage measured involves the mixing of the RF current and the oscillating resistance due to AMR:
$$ V_{total} = I_{rf} \times R_{AMR} $$
$$ I_{rf} = I_{0} \cos(\omega t) $$
$$ R_{AMR} = R_{0} + \Delta R \cos^{2}(\theta(t)) $$
where the angle \(\theta(t)\) is defined as:
$$ \theta(t) = \theta_{H} + \theta_{c} \cdot \cos(\omega t +\delta) $$
So the term \( \cos(\theta (t)) \) can be expanded. Using the small angle approximation:
$$
\begin{aligned}
\cos(\theta (t)) &= \cos[\theta_{H}+\theta_{c}\cdot \cos(\omega t +\delta)],\hspace{10pt} \text{let } \theta'_{c} \equiv \theta_{c} \cdot \cos(\omega t +\delta)\\
&= \cos(\theta_{H}+\theta'_{c}) \\
&= \cos\theta_{H}\cdot \cos\theta'_{c} - \sin\theta_{H}\cdot \sin\theta'_{c}\\
&\approx \cos\theta_{H} - \sin\theta_{H}\cdot \theta'_{c}\\
&=\cos\theta_{H} - \sin\theta_{H}\cdot\theta_{c}\cdot \cos(\omega t+\delta)
\end{aligned}
$$
Therefore, \( \cos^2(\theta(t)) \) is approximately:
$$
\begin{aligned}
\cos^{2}(\theta (t)) \approx \cos^{2} (\theta_{H}) - 2\cos\theta_{H}\cdot \sin\theta_{H}\cdot \theta_{c} \cdot \cos(\omega t + \delta)
\end{aligned}
$$
Substituting this back into \( V_{total} \):
$$
\begin{aligned}
V_{total} &= I_{rf} \times R_{AMR} \\
&= I_{0} \cos(\omega t) \times [R_{0} + \Delta R \cos^{2}(\theta(t))] \\
&= I_0 R_0 \cos(\omega t) + I_0 \Delta R \cos(\omega t) [\cos^{2} (\theta_{H}) - 2\cos\theta_{H}\cdot \sin\theta_{H}\cdot \theta_{c} \cos(\omega t + \delta)] \\
&= (I_0 R_0 + I_0 \Delta R\cos^{2} \theta_{H})\cdot \cos(\omega t) - I_0 \Delta R \sin(2\theta_{H})\cdot \theta_{c} \cos(\omega t)\cos(\omega t + \delta) \\
&=(I_0 R_0 + I_0 \Delta R\cos^{2} \theta_{H})\cdot \cos(\omega t) - I_0 \Delta R \sin(2\theta_{H})\cdot \theta_{c}\cdot \frac{1}{2}[\cos(2\omega t + \delta)+ \cos(\delta)] \\
&= \underbrace{(I_0 R_0 + I_0 \Delta R\cos^{2} \theta_{H})\cdot \cos(\omega t)}_{\text{1st harmonic term}} - \underbrace{[\frac{I_0}{2} \Delta R \sin(2\theta_{H})\cdot \theta_{c}]\cdot \cos(2\omega t +\delta)}_{\text{2nd harmonic term}} \\
&\quad - \underbrace{[\frac{I_0}{2} \Delta R \sin(2\theta_{H})\cdot \theta_{c}]\cos \delta}_{\text{DC term } (V_{mix})}
\end{aligned}
$$
So the ST-FMR voltage (DC component) measured from the bias tee is:
$$
\begin{aligned}
V_{mix} &= \frac{I_0}{2} \Delta R \sin(2\theta_{H})\cdot (-\theta_{c})\cdot \cos \delta \\
& = \frac{I_0}{2} \Delta R \sin(2\theta_{H})\cdot [\Re(m_y)\cdot \cos\delta]
\end{aligned}
$$
LLG Equation Analysis
The Landau-Lifshitz-Gilbert equation with SOT terms can be expressed as:
$$
\dot{\hat{\mathbf{m}}} = -\gamma(\hat{\mathbf{m}}\times\vec{H}_{eff}) +\alpha \hat{\mathbf{m}} \times \dot{\hat{\mathbf{m}}} + \tau_{DL} (\hat{\mathbf{m}}\times\vec{\sigma}\times\hat{\mathbf{m}}) + \tau_{FL} (\hat{\mathbf{m}}\times\vec{\sigma})
$$
Assuming small precession approximations:
$$
\left(
\begin{aligned}
m_{x'} &\approx 1 \\
m_{y'} &\approx \theta_{c} e^{i\omega t} \\
m_{z'} &\approx \theta_{c} e^{i\omega t}
\end{aligned}
\right)
\quad \text{and} \quad
\left(
\begin{aligned}
\dot{m}_{x'} &= 0 \\
\dot{m}_{y'} &= i \omega m_{y'} \\
\dot{m}_{z} &= i \omega m_{z}
\end{aligned}
\right)
$$
With polarization \( \vec{\sigma} = (\sin\theta,\cos\theta,0) \).
1. Precession Term
$$
\begin{aligned}
-\gamma(\hat{\mathbf{m}}\times\vec{H}_{eff}) &= -\gamma\begin{pmatrix} 1 \\ m_{y'} \\ m_{z} \end{pmatrix}\times\begin{pmatrix} H \\ 0 \\ -(M_s -\frac{2K_{\perp}}{M_s})m_{z} \end{pmatrix} \\
&= -\gamma \begin{pmatrix}-m_{y'}M_{eff}m_z \\ (H+M_{eff})m_z \\ -H\cdot m_{y'} \end{pmatrix}
\end{aligned}
$$
2. Damping Term
$$
\begin{aligned}
\alpha \hat{\mathbf{m}} \times \dot{\hat{\mathbf{m}}} &= \alpha \begin{pmatrix} 1 \\ m_{y'} \\ m_z \end{pmatrix} \times \begin{pmatrix} 0 \\ i\omega m_{y'} \\ i \omega m_z \end{pmatrix} \\
&= \alpha \begin{pmatrix} 0 \\ -i\omega m_{z} \\ i \omega m_{y'} \end{pmatrix}
\end{aligned}
$$
3 & 4. SOT Terms
$$ \tau_{DL} (\hat{\mathbf{m}}\times\vec{\sigma}\times\hat{\mathbf{m}}) \approx \tau_{DL}\cdot \begin{pmatrix} 0 \\ -\cos\theta \\ 0 \end{pmatrix} $$
$$ \tau_{FL} (\hat{\mathbf{m}}\times\vec{\sigma}) \approx \tau_{FL}\cdot \begin{pmatrix} 0 \\ 0 \\ \cos\theta \end{pmatrix} $$
Solving for \( m_{y'} \)
$$
\left(
\begin{aligned}
i\omega m_{y'} &=-\gamma(H+M_{eff})m_{z}-\alpha i \omega m_{z} -\tau_{DL} \cos\theta \\
i \omega m_{z} &= \gamma Hm_{y'} + \alpha i \omega m_{y'}+\tau_{FL}\cos\theta
\end{aligned}
\right)
$$
Let \( \gamma (H+M_{eff})\equiv \omega_{1} \) and \( \gamma{H}\equiv \omega_{2} \). The solution is:
$$ m_{y'} = \frac{(\omega_{1}+i \alpha\omega)\tau_{FL}+i\omega \tau_{DL}}{(1+\alpha^2)\omega^2-\omega_1\omega_2-i\omega\alpha(\omega_1+\omega_2)}\cos\theta $$
Considering the phase difference \( \delta \), we set \( m_{y'}^0 e^{-i\delta} \) to represent the response relative to \( \cos\theta \):
$$
\Re[m_{y'}^0]\cos\delta = \frac{(\omega^2-\omega_0^2)\omega_1\tau_{FL}+\omega^2\alpha\omega^+(\tau_{DL}+\alpha\tau_{FL})}{(\omega^2-\omega_0^2)^2+\omega^2\alpha^2(\omega^+)^2}
$$
Using the Kittel formula \( \omega_0^2 \approx \gamma \omega_{H_0}^+ (H-H_0) \) near resonance:
$$
\begin{aligned}
\Re[m_{y'}^0]\cos\delta &= \frac{-\gamma \omega_{H_0}^+ (H-H_0)\omega_1\tau_{FL}+\alpha\omega^2\omega_{H_0}^+(\tau_{DL}+\alpha\tau_{FL})}{\gamma^2(\omega_{H_0}^+)^2(H-H_0)^2+\omega^2\alpha^2(\omega_{H_0}^+)^2} \\
&= \frac{\omega}{\gamma \Delta H \omega_{H_0}^+} \left[ \frac{\Delta H(H-H_0)\frac{\omega_1}{\omega}\tau_{FL}}{(H-H_0)^2+\Delta H^2} + \frac{\Delta H^2(\tau_{DL})}{(H-H_0)^2+\Delta H^2} \right]
\end{aligned}
$$
This leads to the final ST-FMR line shape components:
$$
\begin{aligned}
V_{mix} &= V_S \frac{\Delta H^2}{(H-H_0)^2+\Delta H^2} + V_A \frac{\Delta H (H-H_0)}{(H-H_0)^2+\Delta H^2}
\end{aligned}
$$
where the first term is the Symmetric Lorentzian (proportional to \( \tau_{DL} \)) and the second is the Antisymmetric Lorentzian (proportional to \( \tau_{FL} \)).