Experiment

single spin:

mrfmsim_marohn.experiment.cermitesr_singlespin

Experiment class for mrfmsim.

See how experiments are tested: experiment tests

Experiment Protocol: Spin Noise

In this package we implement in Python the protocol outlined in the supporting information of Longenecker 12 for determining the small-ensemble force variance signal in a cyclic modulation NMR-MRFM experiment.

mrfmsim_marohn.experiment.ibmcyclic	Experiment class for mrfmsim.
mrfmsim_marohn.formula.polarization.rel_dpol_arp_ibm(...)	Relative change in polarization for IBM adiabatic rapid passage.

Experiment Protocol: Frequency shift

In Garner et al. 1 and Moore et al. 2 experiments, spin magnetization interacted with the second derivative of a magnetic field to produce a change in the cantilever’s frequency of oscillation. This approach to detecting magnetic resonance was termed the CERMIT protocol, which stands for Cantilever-Enabled Readout of Magnetization Inversion Transients.

In the Garner experiment 1, nuclear spin magnetization was inverted once using a swept-frequency adiabatic rapid passage, and the resulting step-change in the cantilever frequency indicated nuclear spin resonance (NMR). In the Moore experiment 2, electron spin magnetization was modulated slowly, by switching the spin-saturating microwaves on and off periodically. The cantilever oscillation was digitized and sent to a (software) frequency demodulator. The resulting frequency-versus-time data was fed to a (software) lock-in detector, operated with the microwave modulation trigger as the reference signal. A change in the lock-in output indicated electron spin resonance (ESR).

To observe a change in the cantilever frequency, the cantilever in these experiments were driven into self-oscillation. In the presence of the tip field gradient, the motion of the cantilever led to a dithering of the resonance frequency of the spins in the sample. In the Moore experiment 2, microwave irradiation was applied at a fixed frequency and this dithering was used to sweep out a region of saturated electron spins below the tip. In the Garner experiment 1, in contrast, the region of inverted magnetization swept out by the dithering of the tip was much smaller than the region of inverted spin magnetization created by sweeping the frequency of the applied radio frequency field.

In experiments like these involving a driven cantilever, the observed frequency shift depends on the amplitude of the cantilever oscillation and different equations are needed to calculate the spin signal in small-amplitude and large-amplitude limits. A unified set of equations describing frequency-shift experiments were derived from first principles 4; those results are summarized below.

In this package we implement in Python the protocol for calculating the dc-NMR-CERMIT signal outlined in the Garner et al. manuscript 1 and the protocol for calculating the ac-ESR-CERMIT signal outlined in the supporting information of the Moore et al. manuscript 2.

Small amplitude limit (large tip)

In the small-amplitude limit, we can calculate the spin-dependent frequency shift experienced by the cantilever using

(1)\[\Delta f = - \frac{f}{2 k} \sum_j \mu_z(\vec{r}_j) \frac{\partial^{\, 2} B_z^{\mathrm{tip}}( \vec{r}_j )}{\partial x^2}\]

where \(f \: [\mathrm{Hz}]\) is the cantilever resonance frequency and \(k \: [\mathrm{N} \: \mathrm{m}^{-1}]\) is the cantilever spring constant. The direction of the applied magnetic field is the \(z\) and the direction of the cantilever motion is \(x\). In equation (1), \(\mu_z \: [\mathrm{N} \: \mathrm{m} \: \mathrm{T}^{-1}]\) is the \(z\) component of the spin magnetic moment, and \(B_z^{\mathrm{tip}} \: [\mathrm{T}]\) is the \(z\) component of the magnetic field produced by the cantilever’s magnetic tip. The sum represents a sum over all spins in resonance (discussed below). The frequency shift arises from a spring constant shift of

(2)\[\Delta k = - \sum_j \mu_z(\vec{r}_j) \frac{\partial^{\, 2} B_z^{\mathrm{tip}}( \vec{r}_j )}{\partial x^2}\]

Equations (1) and (2) are valid when the zero-to-peak amplitude of the cantilever oscillation is much smaller than the distance between the center of the (spherical) magnet and the sample spins 4.

In the ESR-CERMIT experiment of Moore et al., the magnetization distribution \(\mu_z (\vec{r})\) depends, according to the steady-state Bloch equations, on the frequency \(f_{\mathrm{rf}}\) and strength \(B_1\) of the microwave field, the sample relaxation times \(T_1\) and \(T_2\) , the sample spin density, the applied magnetic field \(B_0\), the tip magnetic field \(B_z^{\mathrm{tip}}\), and cantilever position. In the Moore experiment, the cantilever sweeps out a region of saturated magnetization as it moves.

mrfmsim_marohn.experiment.cermitesr	Experiment class for mrfmsim.
mrfmsim_marohn.formula.polarization.rel_dpol_sat_steadystate(...)	Relative change in polarization for steady-state.

Assuming a stationary tip, an approximation can be made for the ESR experiment (Michael Boucher):

mrfmsim_marohn.experiment.cermitesr_stationarytip	Experiment class for mrfmsim.
mrfmsim_marohn.formula.polarization.rel_dpol_sat_steadystate(...)	Relative change in polarization for steady-state.

In the NMR-CERMIT experiment of Garner et al., the frequency of the applied radio frequency field \(f_{\mathrm{rf}}\) is swept. The initial magnetization follows the effective field at each location in the sample, resulting in a region of inverted magnetization below the tip.

Adiabatic Rapid Passage experiment:

mrfmsim_marohn.experiment.cermitarp	Experiment class for mrfmsim.
mrfmsim_marohn.formula.polarization.rel_dpol_sat_steadystate(...)	Relative change in polarization for steady-state.

Nutation experiments also implemented:

mrfmsim_marohn.experiment.cermitnut	Experiment class for mrfmsim.
mrfmsim_marohn.formula.polarization.rel_dpol_nut(...)	Relative change in polarization under the evolution of irradiation.

Large amplitude limit (small tip)

The small-amplitude approximation used to derive the above equations may not be valid in a small-tip ESR-CERMIT experiment 4. In this case we must calculate the signal using Equation 20 4 5:

(3)\[\Delta f = \frac{f}{2 \pi k x_{\mathrm{pk}}^2} \sum_j \int_{-\pi}^{\pi} \mu_z(\vec{r}_j,\theta) \frac{\partial B_z^{\mathrm{tip}}(x - x_{\mathrm{pk}} \cos{\theta},y,z)}{\partial x} x_{\mathrm{pk}} \cos{\theta} d\theta\]

where \(x_{\mathrm{pk}}\) is the zero-to-peak amplitude of the cantilever oscillation. We write \(\mu_z(\vec{r}_j,\theta)\) to indicate that, if the microwaves are left on during cantilever motion, then the magnetization may vary in syncrony with the cantilever oscillation. In the i-OSCAR experiment of Rugar and coworkers 6, the resulting position-dependent change in magnetization led to a measurable freqency shift.

Equation (3) is exact. To understand the nature of the large-tip approximation, Eq. (1), let us expand the Eq. (3) gradient in the \(x\) variable:

(4)\[\frac{\partial B_z^{\mathrm{tip}}(x - x_{\mathrm{pk}} \cos{\theta},y,z)} {\partial x} \approx \frac{\partial B_z^{\mathrm{tip}}(x,y,z)}{\partial x} - x_{\mathrm{pk}} \cos{\theta} \frac{\partial^2 B_z^{\mathrm{tip}}(x,y,z)} {\partial x^2} + {\cal O}(x_{\mathrm{pk}}^2)\]

In calculating the signal from our ESR-CERMIT experiment we will assume for simplicity that the spin distribution \(\mu_z(\vec{r}_j)\) has reached steady-state; we neglect any change in the magnetization during the cantilever motion. In this approximation

(5)\[\Delta f = \frac{f}{2 \pi k x_{\mathrm{pk}}^2} \sum_j \int_{-\pi}^{\pi} \mu_z(\vec{r}_j) \left( \frac{\partial B_z^{\mathrm{tip}}(x,y,z)}{\partial x} - x_{\mathrm{pk}} \cos{\theta} \: \frac{\partial^2 B_z^{\mathrm{tip}}(x,y,z)}{\partial x^2} \right) \: x_{\mathrm{pk}} \cos{\theta} \: d\theta\]

There are two terms. The first term is

\[\Delta f^{(1)} = \frac{f}{2 \pi k x_{\mathrm{pk}}} \sum_j \mu_z(\vec{r}_j) \frac{\partial B_z^{\mathrm{tip}}(x,y,z)}{\partial x} \int_{-\pi}^{\pi} \cos{\theta} \: d\theta\]

We are interested in experiments carried out in the SPAM geometry and the hangdown geometry. In both of these cases the first term vanishes: the sum over sample spins is zero since the gradient is both positive and negative over the sensitive slice. Moreover, the integral over \(\theta\) is zero. The second term in Eq. (5) is

\[\Delta f^{(2)} = - \frac{f}{2 k} \sum_j \mu_z(\vec{r}_j) \frac{\partial^2 B_z^{\mathrm{tip}}(x,y,z)}{\partial x^2} \underbrace{\frac{1}{\pi} \int_{-\pi}^{\pi} \cos^2{\theta} \: d\theta}_{= 1}\]

which simplifies to the large-tip result, Eq. (1),

\[\Delta f^{(2)} = - \frac{f}{2 k} \sum_j \mu_z(\vec{r}_j) \frac{\partial^2 B_z^{\mathrm{tip}}(x,y,z)}{\partial x^2}\]

We see from this derivation that the validity of Eq. (1) rests on the validity of the approximation in Eq. (4). According to Eq. (4), for Eq. (1) to be valid, the change in the gradient experienced by any spin in the sample should be strictly linear in the cantilever amplitude. This will not be true for a large-amplitude motion of the cantilever.

Let us rewrite Eq. (3) by

1. assuming that the magnetization distribution is in steady-state,

2. writing the frequency shift in terms of an eqivalent spring constant shift,

3. expressing the result in terms of an equivalent force.

We showed in Reference 4 that maximizing this equivalent force will maximize the signal-to-noise ratio in a frequency-shift experiment. In terms of a force, the ESR-CERMIT signal is

(6)\[\Delta F = \Delta k \: x_{\mathrm{pk}} = \frac{2}{\pi} \sum_j \int_{0}^{\pi} \mu_z(\vec{r}_j) \: \frac{\partial B_z^{\mathrm{tip}}(x - x_{\mathrm{pk}} \cos{\theta},y,z)}{\partial x} \: \cos{\theta} \: d\theta\]

In writing Eq. (6) we have condensed the integral to a half cycle of the cantilever oscillation. In the integrand, the position variable \(x(\theta) = x - x_{\mathrm{pk}} \cos{\theta}\) runs from \(x - x_{\mathrm{pk}}\) to \(x + x_{\mathrm{pk}}\) as \(\theta\) runs from \(0\) to \(\pi\). In the steady-state approximation, the spin distribution \(\mu_z(\vec{r}_j)\) in Eq. (6) is determined in the same way as in the large-tip experiment.

mrfmsim_marohn.experiment.cermitesr_smalltip	Experiment class for mrfmsim.
mrfmsim_marohn.formula.polarization.rel_dpol_sat_steadystate(...)	Relative change in polarization for steady-state.
mrfmsim_marohn.formula.field.xtrapz_field_gradient(...)	Calculate CERMIT integral using Trapezoidal summation.

Eric Moore and co-workers previously implemented Eqs. (3) and (6) to calculate the ESR-MRFM signal from a single spin 4 3

mrfmsim_marohn.experiment.cermitesr_singlespin	Experiment class for mrfmsim.
mrfmsim_marohn.experiment.cermitesr_singlespin_approx	Experiment class for mrfmsim.
mrfmsim_marohn.formula.field.xtrapz_field_gradient(...)	Calculate CERMIT integral using Trapezoidal summation.

And to simulate the amplitude dependence of the signal from a single slice whose magnetization has been inverted via an adiabatic rapid passage 2.

mrfmsim_marohn.experiment.cermitarp_smalltip	Experiment class for mrfmsim.
mrfmsim_marohn.formula.field.xtrapz_field_gradient(...)	Calculate CERMIT integral using Trapezoidal summation.

The single-slice simulation is quite slow because, essentially, a full magnetic field and field gradient simulation must be done for each \(\theta\). If you approximate the \(\theta\) integral using (only!) 32 points, then the simulation will take 32 times longer to run than single simulation. Since the \(x(\theta)\) values are not equally spaced, you cannot simplify the integral by translating the \(x\) coordinates.

John’s intermittent irradiation experiments also implemented:

Intermittent irradiation

mrfmsim_marohn.experiment.cermitesr_periodirrad_stationarytip

Experiment class for mrfmsim.

Cornell-style frequency-shift 1

mrfmsim_marohn.experiment.cermitarp	Experiment class for mrfmsim.
mrfmsim_marohn.experiment.cermitnut	Experiment class for mrfmsim.

IBM-style cyclic-inversion 11 12

mrfmsim_marohn.experiment.ibmcyclic

Experiment class for mrfmsim.

Cornell-style cyclic saturation ESR 2 13 14

There are two different versions of this experiment to simulate:

1. an experiment employing a large spherical magnetic tip (radius \(r \sim 2 \: \mu \mathrm{m}\)).

2. an experiment employing a small spherical magnetic tip (radius \(r = 100 \; \mathrm{nm}\)).

The cyclic-saturation experiment has a few important differences from the NMR experiments previously described include:

• the cantilever motion cannot be neglected and actually (partially) determines the volume of spins in resonance as the cantilever motion sweeps out the resonant slice

• the magnetization is determined by the Bloch equations

Large tip approximation

mrfmsim_marohn.experiment.cermitesr

Experiment class for mrfmsim.

• In practice, spins are saturated with a microwave pulse every n cantilever cycles. This saturation is modulated with a time of no microwave pulses to generate a modulated cantilever frequency shift detected via lock-in detection.

• For this simulation, we can calculate the spin-dependent frequency shift experiments by the cantilever using

\[ \begin{align}\begin{aligned} \Delta f = - \frac{f_c}{2 k} \sum_j \mu_z(\vec{r}_j) \frac{\partial^2 B_z^{\mathrm{tip}}( \vec{r}_j )}{\partial x^2}\\which is valid when the tip radius is much smaller than the zero-to-peak the amplitude of the cantilever oscillation. Where :math:`f_c` is the cantilever resonance\end{aligned}\end{align} \]

• At a fixed cantilever position in this ESR-CERMIT experiment, the magnetization distribution \(\mu_z( \vec{r} )\) depends on the frequency \(f_{\mathrm{rf}}\) and strength \(B_1\) of the microwave field, the sample relaxation times (\(T_1\) and \(T_2\)), the sample spin density, the applied magnetic field \(B_0\), and the tip magnetic field \(B_z^{\mathrm{tip}}\).

• As the cantilever moves it sweeps over a region of saturated spins that we must sum over to obtain the signal. We must include these cantilever motion effects in our simulations.

Small tip approximation

In the small tip experiment, the cantilever amplitude may not be assumed small when compared to the tip radius. We therefore must evaluate the full integral given by Lee, et al..

mrfmsim_marohn.experiment.cermitarp_smalltip	Experiment class for mrfmsim.
mrfmsim_marohn.experiment.cermitesr_smalltip	Experiment class for mrfmsim.

Cornell Single Spin ESR

mrfmsim_marohn.experiment.cermitesr_singlespin

Experiment class for mrfmsim.

Additional Experiments

mrfmsim_marohn.experiment.cermitesr_stationarytip	Experiment class for mrfmsim.
mrfmsim_marohn.experiment.cermitesr_periodirrad_stationarytip	Experiment class for mrfmsim.

Reference

1(1,2,3,4,5)

Garner, S. R.; Kuehn, S.; Dawlaty, J. M.; Jenkins, N. E. & Marohn, J. A. “Force-Gradient Detected Nuclear Magnetic Resonance” Appl. Phys. Lett., 2004, 84, 5091 - 5093 [10.1063/1.1762700].

2(1,2,3,4,5,6)

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].

3

Moore, E. W. & Marohn, J. A. Unpublished calculation, 2009.

4(1,2,3,4,5,6)

Lee, S.-G.; Moore, E. W. & Marohn, J. A. “A Unified Picture of Cantilever Frequency-Shift Measurements of Magnetic Resonance”, Phys. Rev. B, 2012, 85, 165447 [10.1103/PhysRevB.85.165447].

5

Equation 20 in Lee et al. 2012 is off by a factor of \(-1\). We give the correct equation above.

6

Rugar, D.; Budakian, R.; Mamin, H. J. & Chui, B. W. “Single Spin Detection by Magnetic Resonance Force Microscopy”, Nature, 2004 , 430, 329 - 332 [10.1038/nature02658].

7

Wago, K.; Botkin, D.; Yannoni, C. & Rugar, D. “Force-detected Electron-spin Resonance: Adiabatic Inversion, Nutation, and Spin Echo”, Phys. Rev. B, 1998, 57, 1108 - 1114 [10.1103/PhysRevB.57.1108].

8

Klein, O.; Naletov, V. & Alloul, H. “Mechanical Detection of Nuclear Spin Relaxation in a Micron-size Crystal”, Eur. Phys. J. B, 2000, 17, 57 - 68 [10.1007/s100510070160].

9

Baum, J.; Tycko, R. & Pines, A. “Broadband and Adiabatic Inversion of a Two-level System by Phase-modulated Pulses”, Phys. Rev. A , 1985, 32, 3435 - 3447 [10.1103/PhysRevA.32.3435].

10

Kupce, E. & Freeman, R. “Optimized Adiabatic Pulses for Wideband Spin Inversion”, Journal of Magnetic Resonance, Series A, 1996, 118, 299 - 303 [10.1006/jmra.1996.0042].

11

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].

12(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].

13

Isaac, C. E.; Gleave, C. M.; Nasr, P. T.; Nguyen, H.L.; Curley, E. A.; Yoder, J.L.; Moore, E. W.; Chen, L & Marohn, J. A. “Dynamic nuclear polarization in a magnetic resonance force microscope experiment”, Phys. Chem. Chem. Phys., 2016, 18, 8806 [10.1039/c6cp00084c].

14

Lee, S. G.; Moore, E. W.; & Marohn, J. A. “Unified picture of cantilever frequency shift measurements of magnetic resonance”, Phys. Rev. B, 2012, 85, 165447 [10.1103/PhysRevB.85.165447].

experiment module

Import the experiments dynamically to the mrfmsim_marohn.experiment module.