CERMIT time dependent offset

Overview

The force-gradient detection protocol was used to detect magnetic resonance in samples with spin-lattice relaxation times longer than a cantilever period (of order 200 µs). Reference [1] presented the CERMIT (Cantilever-Enabled Readout of Magnetization Inversion Transients) protocol used to detect electron spin resonance modulation with microwave (MW) pulses. Reference [2] introduced a signal model accounting for both adiabatic losses and \(T_2\) relaxation from cantilever motion during the MW pulses. The signal model is valid in the limit of \(T_2 \ll T_1\), where the transverse magnetization reaches a pseudo-equilibrium with the slowly evolving longitudinal magnetization.

Following the derivation from CERMIT steady-state saturation, the unitless time variable \(\tau\) and relaxation time variables \(\alpha\) and \(\beta\) are defined as,

\[\tau = \gamma B_1 t,\]

\[\alpha = \frac{1}{\gamma B_1 T_1},\]

and

\[\beta = \frac{1}{\gamma B_1 T_2},\]

where \(\gamma\) is the electron gyromagnetic ratio, and \(B_1\) is the irradiation intensity in the rotating frame. At low-\(B_1\) region, we can determine numerically the final magnetization \(M_z^\mathrm{final}\) during a cantilever sweep from \(\tau_i\) to \(\tau_f\), and sweep time \(\tau_f - \tau_i \ll T_1\) as follows,

(1)\[M_z^\mathrm{finial} \approx e^{-R(\tau_i, \tau_f)} M_z^\mathrm{initial}\]

and

(2)\[R(\tau_i, \tau_f) = \frac{\arctan{(\pi \alpha_1 \tau_f / \beta)} - \arctan{(\pi \alpha_1 \tau_i / \beta)}}{\pi \alpha_1},\]

where

\[\alpha_1 = \frac{1}{\pi \gamma B_1^2}\frac{d\Delta B_0}{dt}\]

is a unitless sweep-rate parameter, assuming the resonance offset changes linearly in time.

At the time \(\tau = 0\), the system is at the resonance condition. The argument of the arctan function can be written as

(3)\[\frac{\pi \alpha_1 \tau}{\beta} = \gamma T_2 \frac{d \Delta B_0}{d t} t = \gamma T_2 \Delta B_0 (t).\]

Additionally, since we assume that the cantilever velocity remains the same during the sweep,

\[\pi \alpha_1 =\frac{1}{\gamma B_1^2}\frac{d \Delta B_0}{d t} = \frac{1}{\gamma B_1^2} \frac{\partial \Delta B_0}{\partial x} \frac{\partial x}{\partial t} = \frac{1}{\gamma B_1^2} \frac{\partial \Delta B_0}{\partial x} v_\mathrm{tip} = \frac{B_{zx}v_\mathrm{tip}}{\gamma B_1^2}.\]

Finally,

(4)\[\frac{M_z^\mathrm{final}}{M_z^\mathrm{initial}} \approx e^{-R(\tau_i, \tau_f)},\]

where

\[R(\tau_i, \tau_f) = \frac{\gamma B_1^2}{B_{zx}v_\mathrm{tip}} (\arctan{(\gamma T_2 \Delta B_0(\tau_f)}) - \arctan{(\gamma T_2 \Delta B_0(\tau_i))}).\]

The above derivation accounts for the magnetization change after a single pulse. For the multi-pulse CERMIT experiment, we account for the \(T_1\) relaxation between pulses. Given the system is at equilibrium, we use the simplified notation that before and after the pulse \(p\), the magnetization is \(M^{-} = M_z^\mathrm{eq}(\tau_p^-)\) and \(M^{+} = M_z^\mathrm{eq}(\tau_p^+)\); and the magnetization ratio \(r\) is \(r = M^{+}/M^{-} =e^{-R(\tau_p^+, \tau_p^-)}\). The magnetization relaxes towards the initial magnetization \(M_0\),

\[M^{-} - M_0 = (M^{+}- M_0)e^{-\frac{t}{T_1}}\]

where \(t\) is the time between pulses. The microwave pulse occurs at the same position during the cantilever cycle, and \(R\) remains the same regardless of the system magnetization state. With the pulse interval \(\Delta t\), the change in magnetization before the relaxation is

\[\Delta M^{+} = M^{+} - M_0 = \frac{r - 1}{1 - r e^{-\frac{\Delta t}{T_1}}} M_0.\]

Therefore, the average change in magnetization \(\Delta M_z^\mathrm{avg}\) is

(5)\[\Delta M_z^\mathrm{avg} = \frac{\int_{0}^{\Delta t}{\Delta M^{+}e^{-\frac{t}{T_1}}} \,dt}{\Delta t} = \frac{(r - 1) (1 - e^{-\frac{t}{T_1}})}{1-re^{-\frac{t}{T_1}}}\frac{T_1}{\Delta t}.\]

The final signal sums over spin at location \(\boldsymbol{r_j}\)

(6)\[\delta f = \ \frac{\sqrt{2}f_\mathrm{c}}{2\pi k_\mathrm{c}}\sum_j \Delta M_z^\mathrm{avg}(\boldsymbol{r_j}) \frac{\partial ^2 B^\mathrm{tip}_z (\boldsymbol{r_j})}{\partial x^2}\]

where \(f_\mathrm{c}\) is the cantilever frequency, and \(k_\mathrm{c}\) is the cantilever spring constant.

[1]

Moore, E. W.; Lee, S.-G.; Hickman, S. A.; Wright, S. J.; Harrell, L. E.; Borbat, P. P.; Freed, J. H. & Marohn, J. A. “Scanned-Probe Detection of Electron Spin Resonance from a Nitroxide Spin Probe”, Proc. Natl. Acad. Sci. U.S.A., 2009, 106, 22251 - 22256 [10.1073/pnas.0908120106].

[2]

Boucher, E.; Sun, P.; Keresztes, I.; Harrell, H. L.; Marohn, J. A. “The Landau–Zener–Stückelberg–Majorana transition in the T2 \(\ll\) T1 limit”, J. Magn. Reson., 2023, 354, 107523 [10.1016/j.jmr.2023.107523].

Experiment Summary

mrfmsim.experiment.CermitTDGroup	Simulates a Cornell-style frequency shift magnetic resonance force microscope experiment considering the time-dependent nature of the saturation, averaged over multiple pulses and with small-step approximation.
mrfmsim.formula.polarization.rel_dpol_sat_td(...)	Relative change in polarization for time-dependent saturation.
mrfmsim.formula.polarization.rel_dpol_sat_td_smallsteps(B1, ...)	Small step approximation of the time-dependent relative change in polarization.

mrfmsim.experiment.CermitTDGroup = <mrfmsim.group.ExperimentGroup 'CermitTDGroup'>

Simulates a Cornell-style frequency shift magnetic resonance force microscope experiment considering the time-dependent nature of the saturation, averaged over multiple pulses and with small-step approximation.

• CermitTD

  CermitTD(B0, B1, cantilever, dt_pulse, f_rf, grid, h, magnet, mw_x_0p, sample,
  tip_v)
  returns: df_spin
  group: CermitTDGroup
  graph: CermitTD_graph
  handler: MemHandler
  
  Time-dependent CERMIT experiment for a large tip.
  

• CermitTDSmallTip

  CermitTDSmallTip(B0, B1, cantilever, dt_pulse, f_rf, grid, h, magnet, mw_x_0p,
  sample, tip_v, trapz_pts, x_0p)
  returns: df_spin
  group: CermitTDGroup
  graph: CermitTDSmallTip_graph
  handler: MemHandler
  
  Time-dependent CERMIT experiment for a small tip.