IBM cyclic

Overview

A number of methods have been devised for detecting spin magnetic resonance using a cantilever. The methods are different enough that numerically calculating the effect of the spins on the cantilever requires a distinct approach for each method. We are most interested in simulating the signal from Degen et al. [1] and Longenecker et al. [2] experiments.

In these experiments, adiabatic rapid passages were used to repeatedly invert the sample’s spin magnetization in time with the natural oscillation period of the cantilever. The modulated spin magnetization interacted with a magnetic field gradient to produce a resonant force that excited the cantilever. The cantilever position was observed with a lock-in detector; spin resonance was registered as a change in the amplitude of the cantilever oscillation. In the experiments cited above, the number of spins in resonance was so small that the spin fluctuations exceeded the average thermal spin polarization. In this small-ensemble limit, nuclear magnetic resonance (NMR) was detected as a change in the variance of the cantilever position fluctuations observed in the in-phase channel of the lock-in detector.

The net polarization between spin-up and spin-down fluctuates on the same time scale as the random spin-flip rate. For a random ensemble of spin-1/2 nuclei with a small mean polarization, the variance of the net polarization \(\Delta N\) is \(\sigma^2_{\Delta N} = N\), where \(N\) is the total number of spins. The standard deviation of such statistical polarization far exceeds the Boltzmann polarization in a small detection volume. In this limit, the variance can be used as the MRFM signal. For \(n\) independent configurations of the spin ensemble, the sample variance \(s^2_{\Delta N}\) is

\[s^2_{\Delta N} = \frac{1}{n-1} \sum^n_{j=1} (\Delta N_j - \overline{\Delta N})\]

where \(\overline{\Delta N}\) is the Boltzmann polarization. The standard error of the variance is

\[\sigma_{s^2_{\Delta N}} = \sqrt{\frac{2}{n-1}} \sigma^2_{\Delta N} \approx \sqrt{\frac{2}{n-1}} N.\]

The signal-to-noise ratio of the variance signal, in the approximation that no other noise is present, is

\[\mathrm{SNR} = \frac{s^2_{\Delta N}}{\sigma_{s^2_{\Delta N}}} = \sqrt{\frac{n-1}{2}}.\]

Numerically, we can simulate the signal from the three-dimensional convolution integral,

\[\sigma^2_\mathrm{spin}(\boldsymbol{r}_s) = \int_\mathrm{sv} d^3(\boldsymbol{r})K(\boldsymbol{r}_s - \boldsymbol{r})\rho(\boldsymbol{r})\]

\[K(\boldsymbol{r}) = A \mu_\mathrm{p}^2\left(\frac{\partial B_z^\mathrm{tip}}{\partial x}\right)^2\eta (\Delta B_0(\boldsymbol{r}))\]

\[\Delta B_0(\boldsymbol{r}) = B_0 + B^\mathrm{tip}(\boldsymbol{r}) - 2\pi \frac{f_\mathrm{rf}}{\gamma_\mathrm{p}}\]

where \(\sigma^2_\mathrm{spin}\) is the force variance, \(\boldsymbol{r}_s\) is the tip position, \(K\) is the point spread function related to the resonance slice, \(A\) is a scaling factor, \(\rho\) is the proton density of the sample, \(\mu_\mathrm{p} = 1.4 \times 10^{26}\) J/T is the proton magnetic moment, \(\eta (\Delta B_0(\boldsymbol{r}))\) is a function that characterizes the off-resonance spin response, \(\gamma_\mathrm{p}/2 \pi = 42.56\) MHz/T is the proton gyromagnetic ratio, \(B_0\) is an external magnetic field, and \(B^\mathrm{tip}(\boldsymbol{r})\) is the tip field. For the cyclic inversion protocol with the triangle-wave frequency, the off-resonance response is well approximated by [2]

\[\begin{split} \eta (\Delta B_0(\boldsymbol{r}))= \begin{cases} \cos^2{\left(\dfrac{\gamma_\mathrm{p}\Delta B_0(\boldsymbol{r})}{2\Delta f_\mathrm{FM}}\right)} & \mathrm{for}\; \Delta B_0(\boldsymbol{r}) \leq \pi \Delta f_\mathrm{FM}/\gamma_\mathrm{p}\\ 0 & \mathrm{otherwise}. \\ \end{cases}\end{split}\]

where \(\Delta f_\mathrm{FM}\) is the peak-to-peak frequency modulation deviation.

Reference

[1]

Degen, C. L.; Poggio, M.; Mamin, H. J.; Rettner, C. T. & Rugar, D. “Nanoscale Magnetic Resonance Imaging”, Proc. Natl. Acad. Sci. U.S.A., 2009, 106, 1313 - 1317 [10.1073/pnas.0812068106].

[2] (1,2)

Longenecker, J. G.; Mamin, H. J.; Senko, A. W.; Chen, L.; Rettner, C. T.; Rugar, D. & Marohn, J. A. “High-Gradient Nanomagnets on Cantilevers for Sensitive Detection of Nuclear Magnetic Resonance”, ACS Nano, 2012, 6, 9637 - 9645 [10.1021/nn3030628].

Experiment Summary

mrfmsim.experiment.IBMCyclic	Simulate an IBM-style cyclic-inversion magnetic resonance force microscope experiment.
mrfmsim.formula.polarization.rel_dpol_ibm_cyclic(...)	Relative change in polarization for IBM adiabatic rapid passage.

mrfmsim.experiment.IBMCyclic = <mrfmsim.model.Experiment 'IBMCyclic'>

Simulate an IBM-style cyclic-inversion magnetic resonance force microscope experiment.

IBMCyclic(B0, df_fm, f_rf, grid, h, magnet, sample)
returns: (dF2_spin, dF_spin)
graph: ibm_cyclic_graph
handler: MemHandler

Simulate an IBM-style cyclic-inversion magnetic resonance force microscope
experiment.