Dielectric fluctuations in force microscopy: noncontact friction and frequency jitter

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 poly(methylmethacrylate) 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.