A theoretical and experimental assessment of adiabatic losses in force-gradient-detected magnetic resonance of nitroxide spin labels

Abstract

We recently introduced a new theoretical description of Landau--Zener--Stückelberg--Majorana (LZSM) transitions that accounts for both adiabatic and spin-dephasing losses during sweeps through resonance. Here, we use this new description to assess signal loss due to cantilever tip motion in magnetic resonance force microscopy experiments on electron spins. We derive equations for spin-induced cantilever frequency shifts that account for the time-dependent magnetization present during cantilever-synchronized periods of irradiation and relaxation. We show that a frequency shift can be created by either a force- or force-gradient coupling mechanism, depending on the periodicity and timing of the microwave irradiation; the frequency shift decreases when the spin-lattice relaxation time becomes shorter than the cantilever oscillation period. Equations were validated by comparing with the magnetization computed by numerically integrating the time-dependent Bloch equations. Numerical simulations incorporating the new equations were compared to frequency-shift electron-spin signals collected as a function of magnetic field, tip-sample separation, microwave power, and microwave timing. The simulations quantitatively describe the observed signals with essentially no free parameters. Finally, motivated by our new frequency-shift equations, we present a new experimental spin-excitation protocol that eliminates spurious signals arising from direct microwave excitation of the cantilever in a magnetic resonance force microscope experiment.

Figures (11)

• Figure 1: Magnetic resonance force microscope apparatus and microwave irradiation scheme. (a) A 40mM sample of 4-amino-TEMPO in polystyrene was spin-coated onto a quartz substrate and placed over a microstripline resonator, the source of an oscillating magnetic field. The cantilever was oscillated in the $x$-direction, with a zero-to-peak amplitude of $x_\mathrm{pk} = 164nm$. Note that the $x$ direction is orthogonal to both the external field and the long axis of the cantilever Marohn1998dec. (b) Bursts of microwave (MW) irradiation were applied in synchrony with the cantilever oscillation. These bursts lasted for a half-cantilever-cycle duration and were typically applied every third cantilever cycle. In (b), we depict the resulting $z$ component of spin magnetization $\mu_z$ versus time at a single sample location.
• Figure 2: Time-delayed intermittent microwave bursts applied every $n$ cantilever cycles. Cantilever position (upper) and microwave intensity (lower) are plotted versus time, with the microwave phase delay indicated as $\phi$.
• Figure 3: Intermittent microwave bursts applied every $(n - \frac{1}{2})$ cantilever cycles, with $n \geq 1$. Cantilever position (upper) and microwave intensity (lower) are plotted versus time.
• Figure 4: Electron spin resonance signal vs. external field $B_0$ at four tip-sample separations (colored circles), and signal simulations calculated using two different methods (black lines). (a) Saturation calculated using the steady-state Bloch equations Moore2009dec with $B_1 = 2.9µT$ and $k_0 = 3.9mN\per m$. (b) Saturation calculated according to eq. \ref{['eq:DeltaMz-strong-B1']} with $B_1 = 2.9µT$, $k_0 = 0.8mN\per m$, and velocity assumed to be the maximum velocity during the 164nm amplitude motion of the cantilever. Inset: Tip field inferred from the data at each height (circles) and the best fit to eq. \ref{['eq:Btip-fit']}.
• Figure 5: Calculated (solid line) and measured (circles) spin signal vs. microwave power at $B_0 = 642mT$ and $h = 150nm$. The upper $x$-axis is the estimated total power into the microwave resonator. The lower $x$-axis is derived from the $B_1$ used in the simulation. A coil constant of 28.9µT W^-1/2 was inferred by requiring the calculated signal plotted versus both the upper and lower $x$ axes to agree. This coil constant was used for subsequent simulations. We note that the $B_1 \ll B_1^{\mathrm{crit}}$ approximation may no longer hold beyond $B_1 = 6.3µT$ (dashed line).