CERMIT protocol

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; [3] 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]

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

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)

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]

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].