CERMIT steady-state saturation

CERMIT steady-state saturation#

Overview#

Spins are subjected to a static magnetic field \(\boldsymbol{B}_0 = B_0 \hat{\boldsymbol{z}}\) and irradiated with an oscillating magnetic field

\[B(t) = B_1 \left(\hat{\boldsymbol{x}} \, \cos{\omega t} + \hat{\boldsymbol{y}} \, \sin{\omega t}\right).\]

In a frame of reference rotating about the \(z\) axis at angular frequency \(\omega\), the rotating-frame components of spin magnetization will evolve according to the Bloch equation

\[\frac{d}{d t} \boldsymbol{M}_\text{R} = \gamma \boldsymbol{M}_{\text{R}} \times \boldsymbol{B}_{\text{eff}} + \text{relaxation terms}\]

with \(\times\) the vector cross product, \(\gamma\) the spins’s gyromagnetic ratio, and

\[\boldsymbol{B}_{\text{eff}} = \underbrace{\left( B_0 - \frac{\omega}{\gamma} \right)}_{\Delta B} \hat{\boldsymbol{z}} + B_1 \, \hat{\boldsymbol{x}}_{\text{R}}\]

an effective magnetic field in the rotating frame. Dropping the subscript “R” and adding in relaxation terms, the Bloch equations in the rotating frame are

\[\begin{split}\begin{align} \dot{M_x} & = + \gamma \Delta B \, M_y - \frac{M_x}{T_2} \\ \dot{M_y} & = - \gamma \Delta B \, M_x + \gamma B_1 \, M_z - \frac{M_y}{T_2} \\ \dot{M_z} & = - \gamma B_1 \, M_y + \frac{M_0 - M_z}{T_1} \end{align}\end{split}\]

with \(M_0\) the equilibrium magnetization, \(T_1\) the spin-lattice relaxation time, \(T_2\) the spin dephasing time. These equations can be written as a set of three coupled differential equations of the form

(1)#\[\dot{x} = A x + b\]

with

(2)#\[\begin{split}x = \begin{pmatrix} M_x \\ M_y \\ M_z \end{pmatrix}, \: A = \begin{pmatrix} - r_2 & + \Delta & 0 \\ - \Delta & -r_2 & + \omega_1 \\ 0 & - \omega_1 & -r_1 \end{pmatrix}, \text{ and } b = \begin{pmatrix} 0 \\ 0 \\ r_1 M_0 \end{pmatrix}.\end{split}\]

Here \(x\) is a vector describing the magnetization in the rotating frame. The magnetic field \(B_0\) is applied along the \(z\) axis and so the equilibrium magnetization \(M_0\) is also along the \(z\) axis: \(M_{\mathrm{eq}} = (0, 0, M_0)^{\text{T}}\). We will assume the initial condition to be

\[\begin{split}x(0) = \begin{pmatrix} 0 \\ 0 \\ M_{z}(0) \end{pmatrix}.\end{split}\]

That is, the magnetization is initially assumed to lie parallel to the \(z\) axis. The variables in the above equations are summarized in the Table below.

variable

description

\(x\)

magnetization vector in the rotating frame

\(r_2\)

spin dephasing rate

\(r_1\)

spin-lattice relaxation rate

\(B_0\)

longitudinal magnetic field intensity

\(M_0\)

longitudinal equilibrium magnetization

\(B_1\)

transverse oscillating magnetic field intensity

\(\omega\)

transverse oscillating magnetic field frequency

\(\gamma\)

electron spin gyromagnetic ratio

\(T_2 = 1/r_2\)

spin dephasing time

\(T_1 = 1/r_1\)

spin-lattice relaxation time

\(\omega_0 = \gamma B_0\)

Larmor frequency

\(\omega_1 = \gamma B_1\)

Rabi frequency

\(\Delta B = B_0 - \omega/\gamma\)

resonance offset in field units

\(\Delta = \omega_0 - \omega\)

resonance offset in frequency units

Variables appearing in eqs. (1), (2), and subsequent equations. he magnetic field \(B_0\) is applied along the \(z\) axis, and the equilibrium magnetization \(M_0\) also lies along the \(z\) axis.

Steady-state solution#

The steady-state solution to eq. (1) is obtained by setting \(\dot{x} = 0\) and solving for \(x\). The result is \(x_{\mathrm{ss}} = - A^{-1} \, b\) Computing the \(A^{-1}\) matrix analytically, substituting this result and \(b\) into the above equation, and writing the result in terms of the variables \(T_1 = 1/r_1\) and \(T_2 = 1/r_2\), we obtain the following three components of the steady-state magnetization. Let us write the components in a number of equivalent, useful forms.

(3)#\[\begin{split}\begin{aligned} \frac{M_x^{\mathrm{ss}}}{M_0} & = \frac{r_1 \omega_1 \Delta}{r_1 r_2^2 + r_2 \omega_1^2 + r_1 \Delta^2} = \frac{T_2^2 \, \omega_1 \Delta}{1 + T_1 T_2 \, \omega_1^2 + T_2^2 \Delta^2} = \sqrt{\frac{T_2}{T_1}} \frac{S \, \Omega}{1 + S^2 + \Omega^2} \\ \frac{M_y^{\mathrm{ss}}}{M_0} &= \frac{r_1 r_2 \omega_1}{r_1 r_2^2 + r_2 \omega_1^2 + r_1 \Delta^2} = \frac{T_2 \, \omega_1}{1 + T_1 T_2 \, \omega_1^2 + T_2^2\Delta^2} = \sqrt{\frac{T_2}{T_1}} \frac{S}{1 + S^2 + \Omega^2} \\ \frac{M_z^{\mathrm{ss}}}{M_0} &= \frac{r_1 r_2^2 + r_1 \Delta^2}{r_1 r_2^2 + r_2 \omega_1^2 + r_1 \Delta^2} = \frac{1 + T_2^2 \, \Delta^2}{1 + T_1 T_2 \, \omega_1^2 + T_2^2 \, \Delta^2} = \frac{1 + \Omega^2}{1 + S^2 + \Omega^2} \end{aligned}\end{split}\]

These equations are exact and valid both on and off-resonance. In the last form, we have written the magnetization in terms of a unitless saturation parameter

\[S \equiv \omega_1 \sqrt{T_1 T_2} = \gamma B_1 \sqrt{T_1 T_2}\]

and unitless resonance offset

(4)#\[\Omega \equiv T_2 \Delta = T_2 (\gamma B_0 - \omega).\]

Our laboratory’s magnetic resonance force microscope (MRFM) experiments detect \(M_z\). The expression for the \(z\) component of the steady-state magnetization is especially simple on-resonance:

\[M_z^{\mathrm{ss}}(\Omega = 0) = \frac{1}{1 + S^2} M_0.\]

In our experiments we detect the change in the \(z\) magnetization \(\Delta M_z = M_z(\text{final}) - M_z(\text{initial})\) with \(M_z(\text{initial}) = M_0\) the thermal-equilibrium magnetization and \(M_z(\text{final}) = M_z^{\mathrm{ss}}\) the steady-state magnetization in the presence of irradiation. Using the above expressions, we compute

\[\Delta M_z = - M_0 \frac{S^2}{1 + S^2 + \Omega^2} = - M_0 \frac{S^2}{1 + S^2} \frac{1}{1 + \left( \Omega \big/ \sqrt{1+S^2} \right)^2}\]

The signal on resonance approaches \(-M_0\) for \(S \gg 1\), i.e., in the saturation limit. The signal is largest on resonance and has a Lorentzian dependence on the resonance offset parameter \(\Omega\). Expressed in terms of the magnetic field, the width of this Lorentzian is

\[\Delta B = \frac{1}{\gamma T_2} \sqrt{1+S^2} \approx \frac{S}{\gamma T_2}\]

with the approximation valid when \(S \gg 1\). When the irradiation intensity is low, \(S \ll 1\) and \(\Delta B \approx 1/(\gamma T_2) = B_{\mathrm{homog}}\), the homogeneous linewidth. When the irradiation intensity is high, then the signal linewidth becomes \(\Delta B \approx S/(\gamma T_2)\), \(S\) times larger than the homogenous linewidth. To see an appreciable signal requires \(S > 1\), which necessarily broadens the linewidth.

In a magnetic resonance force microscope (MRFM) experiment, the resonance offset is spatially dependent due to the presence of the magnetic field gradient provided by the cantilever’s magnetic tip. Consider spins at a distance \(z_0\) from the center of the magnetic tip. Expand the magnetic field in a Taylor series in \(z\) about the point \(z_0\) and plug the resulting expression into eq. (4) to obtain

\[\Omega \approx\gamma T_2 \left( B_0(z_0) - \frac{\omega}{\gamma} + G_{zz} (z - z_0) \right)\]

with \(G_{zz} \equiv \frac{\partial B_z}{\partial z}(z_0)\). For simplicity, we are neglecting the dependence of the field \(B_0\) on \(x\) and \(y\). We are also implicitly assuming that the cantilever is not moving appreciably. Suppose that we have set the irradiation frequency so that spins at a distance \(z_0\) are in resonance; then \(B_0(z_0) - \omega \big/ \gamma = 0\) and \(\Omega \approx \gamma T_2 G_{zz} (z - z_0)\). With these assumptions, the change in magnetization is

(5)#\[\Delta M_z = - M_0 \frac{S^2}{1 + S^2} \frac{1}{1 + \left( z^{\prime} \big/ \Delta_z \right)^2}\]

where we have introduced the variables \(z^{\prime} = z - z_0\) and

(6)#\[\Delta_z = \frac{\sqrt{1+S^2}} {\gamma \, T_2 G_{zz}} \approx \frac{S}{\gamma \, T_2 G_{zz}},\]

with the approximation valid when \(S \gg 1\), i.e., in the saturation limit. In this limit \(\Delta_z\) can be written simply as \(S B_{\mathrm{homog}}/G_{zz}\). We can see from the above equations that the magnetization will vary over a resonant slice of width \(\Delta_z\). This width increases as the irradiation intensity increases and is \(\propto B_1\) in the saturation limit.

To obtain the force-gradient magnetic resonance signal \(\Delta k_{\mathrm{spin}}\) we should multiply \(\Delta M_z\) by the tip field’s second derivative \(G_{zxx}(\boldsymbol{r})\) and the spin density \(\rho_{\mathrm{s}}\) and integrate the result over space. For simplicity, let us treat the resonant slice as a cylinder of thickness \(\Delta_z\) and cross-sectional area \(A\) and let us assume that \(G_{zxx}(\boldsymbol{r})\) is constant over the resonance slice. Making these approximations and integrating the signal over space gives

\[\begin{split}\begin{align} \Delta k_{\mathrm{spin}} & = -M_0 \frac{S^2}{1 + S^2} \, G_{zxx} \, \rho_{\mathrm{s}} \, A \int dz^{\prime} \frac{1} {1 + (z^{\prime} \big/ \Delta_z)^2} \\ & = -M_0 \frac{S^2}{1 + S^2} \, G_{zxx} \, \rho_{\mathrm{s}} \, A \, \pi \Delta_z \end{align}\end{split}\]

where in the second line we have substituted the value of the integral, \(\pi \, \Delta_z\). Substituting eq. (6) for \(\Delta z\) gives

\[\Delta k_{\mathrm{spin}} = - \frac{\pi M_0 A \rho_{\mathrm{s}}}{\gamma T_2} \frac{G_{zxx}}{G_{zz}} \frac{S^2}{\sqrt{1 + S^2}}\]

This equation predicts that \(\Delta k_{\mathrm{spin}} \propto S\) in the \(S \gg 1\) limit. In this picture, the magnetization saturates yet the signal continues to grow because the width of the sensitive slice continues to increase due to power broadening.

Experiment Summary#

mrfmsim.experiment.CermitESRGroup

Simulates a Cornell-style frequency shift magnetic resonance force microscope experiment in which microwaves are applied for half a cantilever cyclic to saturate electron spin resonance in a bowl-shaped region swept out by the cantilever motion.

mrfmsim.formula.polarization.rel_dpol_sat_steadystate(...)

Relative change in polarization for steady-state.
mrfmsim.experiment.CermitESRGroup = <mrfmsim.group.ExperimentGroup 'CermitESRGroup'>#

Simulates a Cornell-style frequency shift magnetic resonance force microscope experiment in which microwaves are applied for half a cantilever cyclic to saturate electron spin resonance in a bowl-shaped region swept out by the cantilever motion.

  • CermitESR
    CermitESR(B0, B1, cantilever, f_rf, grid, h, magnet, mw_x_0p, sample)
returns: df_spin
group: CermitESRGroup
graph: CermitESR_graph
handler: MemHandler

CERMIT ESR experiment for a large tip.
  • CermitESRStationaryTip
    CermitESRStationaryTip(B0, B1, cantilever, f_rf, grid, h, magnet, sample)
returns: df_spin
group: CermitESRGroup
graph: CermitESRStationaryTip_graph
handler: MemHandler

CERMIT ESR experiment for a stationary tip.
  • CermitESRSmallTip
    CermitESRSmallTip(B0, B1, cantilever, f_rf, grid, h, magnet, mw_x_0p, sample,
trapz_pts, x_0p)
returns: df_spin
group: CermitESRGroup
graph: CermitESRSmallTip_graph
handler: MemHandler

CERMIT ESR experiment for a small tip.
  • CermitESRStationaryTipPulsed
    CermitESRStationaryTipPulsed(B0, B1, cantilever, f_rf, grid, h, magnet, sample,
t_off, t_on)
returns: df_spin
group: CermitESRGroup
graph: CermitESRStationaryTipPulsed_graph
handler: MemHandler

CERMIT ESR experiment for a stationary tip with a pulsed microwave.