Tip size: small tip vs. large tip

Currently, we have two different sizes of magnets — the large tips are micron-sized spherical magnets (1 - 4 µm in diameter); the small tips are nanometer-sized rectangular prism (~100 nm in cross-section length).

For large tips, the cantilever amplitude (100 - 300 nm) is small compared to the tip size, and for small tips, the cantilever amplitude is comparable to the tip size. They result in different approximations for the CERMIT protocol.

Large tip - small amplitude limit

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. [3]

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.

In the NMR-CERMIT experiment of Garner et al., [6] 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.

Small tip - large amplitude limit

The small-amplitude approximation used to derive the above equations may not be valid in a small-tip ESR-CERMIT experiment.[#Lee2012apra]_ In this case, we must calculate the signal using Equation 20: [3] [4]

(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 synchrony with the cantilever oscillation. In the i-OSCAR experiment of Rugar and coworkers, [5] the resulting position-dependent change in magnetization led to a measurable frequency 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 in the SPAM geometry and the “hangdown” geometry. The geometries are shown in doc Experiment geometry. In both cases, the first term vanishes: the sum over sample spins is zero since the gradient is 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 is not 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 equivalent spring constant shift,

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

We showed in Ref. [3] 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.

Eric Moore and co-workers previously implemented Eqs. (3) and (6) to calculate the ESR-MRFM signal from a single spin. [3] [2]

And to simulate the amplitude dependence of the signal from a single slice whose magnetization has been inverted via an adiabatic rapid passage. [1]

References

[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]

Moore, E. W. & Marohn, J. A. Unpublished calculation, 2009.

[3] (1,2,3,4)

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].

[4]

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

[5]

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].

[6]

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].