^{1}, John A. Marohn

^{1}and Roger F. Loring

^{1,a)}

### 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’sequations, 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.

We acknowledge Seppe Kuehn for assistance in fabricating the cantilevers and in building the friction microscope used here. S.M.Y. acknowledges primary support from the National Science Foundation via the Cornell Center for Nanoscale Systems (EEC-0117770 and EEC-0646547) and supplementary support from National Science Foundation Grant Nos. DMR-0134956 and DMR-0706508. This work was performed in part at the Cornell NanoScale Science and Technology Facility, a member of the National Nanotechnology Infrastructure Network, which is supported by the National Science Foundation through Grant No. ECS-0335765. J.A.M. acknowledges supplemental support from the National Institute of Health (5R01GM-070012), the Army Research Office Multi-University Research Initiative (W911NF-05-1-0403), and the National Science Foundation (DMR-0134956). R.F.L. acknowledges support from the National Science Foundation through Grant Nos. CHE0413992 and CHE0743299.

I. INTRODUCTION

II. EXPERIMENTAL OBSERVABLES

III. FIELD CORRELATION FUNCTIONS FROM RESPONSE THEORY AND THE QUASISTATIC APPROXIMATION

IV. FIELD CORRELATION FUNCTIONS FROM STOCHASTIC ELECTRODYNAMICS

V. EXPERIMENTAL PROCEDURE

VI. FEASIBILITY OF JITTER MEASUREMENT ON A POLYMER FILM

### Key Topics

- Friction
- 50.0
- Dielectrics
- 39.0
- Maxwell equations
- 36.0
- Electric fields
- 35.0
- Correlation functions
- 32.0

## Figures

To probe electric field fluctuations, a charged cantilever tip oscillates in the direction at height above the surface of a dielectric sample of thickness .

To probe electric field fluctuations, a charged cantilever tip oscillates in the direction at height above the surface of a dielectric sample of thickness .

Measured power spectrum of cantilever frequency fluctuations without a sample. Both axes are logarithmic. The solid line is a fit to Eq. (6.7). At the lowest frequencies, the power spectrum is a constant determined by thermomechanical fluctuations [dash-dot line; Eq. (6.8)]. At higher frequency, the spectral density of cantilever frequency fluctuations increases quadratically due to detector noise [dashed line; Eqs. (6.7)–(6.9)].

Measured power spectrum of cantilever frequency fluctuations without a sample. Both axes are logarithmic. The solid line is a fit to Eq. (6.7). At the lowest frequencies, the power spectrum is a constant determined by thermomechanical fluctuations [dash-dot line; Eq. (6.8)]. At higher frequency, the spectral density of cantilever frequency fluctuations increases quadratically due to detector noise [dashed line; Eqs. (6.7)–(6.9)].

Predicted cantilever frequency jitter vs probe height. Note the logarithmic scale on both axes. The dashed line shows the calculated root-mean-squared frequency jitter from Eq. (6.3) for a cantilever having a tip radius of 30 nm held at 0.5 V over a 280 nm thick film of PMMA at room temperature. The solid line shows the minimum detectable frequency jitter calculated from Eq. (6.11) in a 100 Hz bandwidth, using known cantilever parameters and the measured height-dependent ringdown time.

Predicted cantilever frequency jitter vs probe height. Note the logarithmic scale on both axes. The dashed line shows the calculated root-mean-squared frequency jitter from Eq. (6.3) for a cantilever having a tip radius of 30 nm held at 0.5 V over a 280 nm thick film of PMMA at room temperature. The solid line shows the minimum detectable frequency jitter calculated from Eq. (6.11) in a 100 Hz bandwidth, using known cantilever parameters and the measured height-dependent ringdown time.

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