Dielectric fluctuations in force microscopy: Noncontact friction and frequency jitter

Abstract

Electric force microscopy, in which a charged probe oscillates tens to hundreds of nanometers above a sample surface, provides direct mechanical detection of relaxation in molecular materials. Noncontact friction, the damping of the probe's motions, reflects the dielectric function at the resonant frequency of the probe, while fluctuations in the probe frequency are induced by slower molecular motions. We present a unified theoretical picture of both measurements, which relates the noncontact friction and the power spectrum of the frequency jitter to dielectric properties of the sample and to experimental geometry. Each observable is related to an equilibrium correlation function associated with electric field fluctuations, which is determined by two alternative, complementary strategies for a dielectric continuum model of the sample. The first method is based on the calculation of a response function associated with the polarization of the dielectric by a time-varying external charge distribution. The second approach employs a stochastic form of Maxwell's equations, which incorporate a fluctuating electric polarization, to compute directly the equilibrium correlation function in the absence of an external charge distribution. This approach includes effects associated with the propagation of radiation. In the experimentally relevant limit that the tip-sample distance is small compared to pertinent wavelengths of radiation, the two methods yield identical results. Measurements of the power spectrum of frequency fluctuations of an ultrasensitive cantilever together with measurements of the noncontact friction over a poly͑methylmethacrylate͒ film are used to estimate the minimum experimentally detectable frequency jitter. The predicted jitter for this polymer is shown to exceed this threshold, demonstrating the feasibility of the measurement.

Scanned-probe detection of electron spin resonance from a nitroxide spin probe

“…The measured frequency noise at height h = 120 nm, drive amplitude x rms = 233 nm, and frequency offset f mod = 6.48 Hz is approximately P δf = 1 × 10 −6 Hz 2 /Hz (see SI Appendix), 100 times the noise expected from thermomechanical cantilever motion (28,31). Spins directly below the tip -those giving rise to the signal at field B c in Fig.…”

“…A small voltage was applied between the cantilever and the gold coating to minimize noncontact friction (25)(26)(27) and surface frequency noise (10,28). In parallel, samples were prepared in the same way (minus the gold layer), manually removed from the substrate, and inserted into a quartz tube for characterization by inductively detected pulsed ESR at low temperatures.…”

“…The cantilever was excited into self-oscillation via a piezo mounted at the cantilever base by using a fixed-gain positive feedback loop (31). Cantilever position was detected by using a fiberoptic interferometer (λ = 1, 310 nm, incident power 3 μW), digitized, and sent to a high-bandwidth software frequency demodulation algorithm (28).…”

Unified picture of cantilever frequency shift measurements of magnetic resonance

“…13. Let us assume that the frequency noise is thermally limited, with a power spectrum given by 44,46 …”

Switching through intermediate states seen in a single nickel nanorod by cantilever magnetometry

“…A software frequency demodulator (supporting information of Ref. 22) was used to determine the instantaneous cantilever frequency from a record of the cantilever deflection versus time, from which the average cantilever frequency was obtained and the frequency-fluctuation power spectrum was computed. Spring constant shifts were calculated from observed frequency shifts using Dk ¼ 2 kDf/f c , with k and f c the cantilever spring constant and resonance frequency, respectively.…”