Microwave impedance microscopy (MIM) is a scanning probe technique to measure local changes in tip-sample admittance. The imaginary part of the reported change is calibrated with finite element simulations and physical measurements of a standard capacitive sample, and thereafter the output $\Delta Y$ is given a reference value in siemens. Simulations also provide a means of extracting sample conductivity and permittivity from admittance, a procedure verified by comparing the estimated permittivity of polytetrafluoroethlyene (PTFE) to the accepted value. Simulations published by others have investigated the tip-sample system for permittivity at a given conductivity, or conversely conductivity and a given permittivity; here we supply the full behavior for multiple values of both parameters. Finally, the well-known effective medium approximation of Bruggeman is considered as a means of estimating the volume fractions of the constituents in inhomogeneous two-phase systems. Specifically, we consider the estimation of porosity in carbide-derived carbon, a nanostructured material known for its use in energy storage devices.

## I. INTRODUCTION

Microwave impedance microscopy (MIM), a scanning probe technique used to measure the change in admittance local to the tip, has been the subject of a number of recent studies. ^{1–7} The technique has been used, for example, to image the highly conductive domain walls in ferromagnetics,^{2–4} and identify metallic and semiconducting carbon nanotubes individually and non-destructively.^{6} Many of the existing studies using MIM are somewhat qualitative in nature and the technique is used generally to measure the contrast in the tip-sample admittance across an image; however, some effort has been given to quantification of results and analyzing the tip-sample system in detail. Wei *et al*. have produced a Green’s theorem approach to quantify the tip-sample capacitance and verified this approach by comparison of the MIM signal with an electrostatic force microscopy (EFM) approach curve,^{8} and have also modeled effects of the height of the probe on the measurement.^{9} The EFM approach curve method is also used in the calibration of a very similar instrument, the scanning microwave microscope (SMM) of Keysight Technologies (formerly Agilent Technologies).^{10} Furthermore, Lai *et al*. have demonstrated a quantitative calibration scheme for MIM by imaging samples of known bulk permittivity, comparing the average output (in arbitrary units) of the instrument with finite element simulations.^{7} Lai *et al*.^{11} and Tselev *et al*.^{4} have also offered estimates of the ratio of capacitance to the signal output. Their estimates differ somewhat, Tselev *et al*. citing 1.6 aF/mV (the ratio of the tip-sample capacitance to the output voltage of the imaginary sMIM-C or MIM-Im channel), and Lai *et al*. reporting 0.1 aF/mV. The instruments used in the two studies are slightly different, however, and employ different operating conditions. The instrument of Lai *et al*. used a −20 dBm, 1 GHz input to the tip; the instrument of Tselev *et al*. is the commercialized version of MIM, the scanning microwave microscope (sMIM), the default input of which is a −10 dBm, 3 GHz wave. The tip radius is also known to play a role in the amount of output contrast per unit of admittance.^{12}

Using simulations and MIM measurements, we create a quantification scheme very similar to that of Lai *et al*^{7} with only the use of the capacitive calibration die provided with the instrument and verify the prediction with a sample of polytetrafluoroethylene (PTFE). We furthermore provide finite element simulations mapping the full behavior of the tip-sample admittance as a function of both permittivity and conductivity. Prior published simulation results only give the simulated response as a function of conductivity at a fixed value of permittivity,^{12} or permittivity at a given conductivity.^{7} The full results suggest constraints on the prediction of conductivity or permittivity of a sample where these values are not well known. Finally, we also use an effective medium approximation to estimate the porosity of a nanostructured carbon material, carbide-derived carbon (CDC).

## II. MATERIALS

Carbide-derived carbon is a highly porous carbon structure synthesized by the high-temperature chlorination of almost any carbide.^{13} The chlorination strips the metal atom from the carbide, leaving a tortuous, disordered pore network. Synthesis conditions and the choice of precursor allow the tuning of porosity, pore size distribution, and electrical conductivity.^{14} The temperature of chlorination and a subsequent annealing in hydrogen tends to affect the reorganization of the carbon and the conversion of *sp*^{3} hybridization to graphitic *sp*^{2} carbon.^{15} The lattice structure of the precursor also affects the order in the pore network.

Bulk porosity of this and other porous materials is typically measured macroscopically by interpreting gas adsorption isotherms with the Brunauer-Emmett-Teller (BET) theory for multilayer molecular formation. By using local admittance measurements with MIM, simulation-assisted permittivity prediction and effective medium approximations, we investigate local changes in porosity of a CDC sample, as opposed to the more traditional macroscopic methods.

## III. SIMULATIONS

The tip-sample system can be simulated with a simplified two-dimensional geometry in the finite element method software COMSOL. The basic geometry is shown in Figure 1. A small separation between the tip and the sample is included since the contact is not Ohmic, usually a surface passivation layer or Schottky barrier exists between the tip and the sample. The tip is actually pyramidal in shape,^{16} though we instead approximate the tip as a cone so that the geometry can be reduced to a two-dimensional axisymmetric simulation to avoid a more computationally intense three-dimensional model. A similar simulation geometry has already been demonstrated for a non-contact mode,^{11} though the sMIM mode is used only in the contact mode. Furthermore, it is informative to perform the simulation over a range of conductivities and permittivities to observe the complete response of the system. The real and imaginary parts of the simulated admittance between the tip and the sample are shown in Figure 2.

An important observation from the simulated admittance surfaces is that the in the high-conductivity region, the imaginary part of the admittance is relatively insensitive to changes in permittivity. The MIM-Im channel (the instrument output corresponding to $Im(\Delta Y)$ is sometimes generally assumed to be an indicator of permittivity, but this is not always the case, as the simulations show. Similarly, a change in the imaginary part of the admittance and thus MIM-Im could reflect a change in conductivity rather than permittivity.

From the simulations, we also produce and equivalent circuit, shown in Figure 3. Choosing a value of *ϵ* and fitting with respect to *σ*, the goodness of fit is high for both the real and imaginary parts, with $R2>0.99$. Previous studies interpret the results of simulations qualitatively with a two-element equivalent circuit^{11} or with a more complicated circuit of arbitrary size,^{17} though the three element circuit is the simplest to quantitatively fit the results of the non-Ohmic contact-mode simulation.

## IV. INSTRUMENT CALIBRATION

In this section we distinguish between two calibrations. One calibration procedure is performed on the instrument itself and serves to align the orthogonal output axes of the instrument—the MIM-Re and MIM-Im signals—to the real and imaginary parts of the admittance change. Otherwise, MIM-Re might be a linear combination of $Re(\Delta Y)$ and $Im(\Delta Y)$, and likewise for MIM-Im. The other procedure utilizes the results of the simulations, similar to those presented above, of the calibration die to set a reference value for $\Delta Ytip-sample$, the quantity to which the MIM-Re and MIM-Im signals are proportional. The result is a value for admittance in siemens for any given point in the scan, rather than the unreferenced admittance contrast using only the first calibration procedure.

The procedure to find a reference scale is as follows. First, we perform simulations of the calibration die with a similar geometry to that of Figure 1, but including the dielectric and conducting layers of the calibration die. The die consists of an SiO_{2}-covered silicon wafer patterned with circles of Al followed by Al_{2}O_{3}. The instrument will have essentially two different responses for the die, one when the tip is in contact with the Al/Al_{2}O_{3} circles, another for contact with only the SiO_{2}-covered substrate. We simulate both cases to obtain two admittances, then perform a measurement with the instrument. A line is fit through the two points to reference the contrast map:

where *Y*_{2} and *V*_{2} are the simulated admittance and instrument output voltage for one region of the sample, and *Y*_{1}, *V*_{1} for the other region. The instrument calibration—the alignment of the output axes—remains valid for all new samples so long as the probe and waveguides are not changed, so in principle this linear extrapolation is able to predict the admittance for any sample.

Verification of this step itself is difficult, as we have not yet measured an intrinsic parameter of the material. We can, however, use the simulated admittance surfaces in Figure 2 as lookup tables for the conductivity and permittivity given an estimate of measured admittance. There is a notable caveat in this prediction. The simulation shows that different combinations of *ϵ* and *σ* can produce the same value for the imaginary part of admittance; in other words, constant-admittance contours may be drawn that contain points of various *ϵ* and *σ* values. It is also possible that for intermediate conductivities, one may see no change in the MIM-Im channel even for changes in permittivity over a large range. This means that when extracting the parameters *ϵ* and *σ* from the estimated admittance *Y*, one must know something about the sample in advance. Fortunately, there are limiting behaviors. At low conductivities, the admittance does not vary with permittivity.

Therefore, we may verify the simulation-assisted estimate of *ϵ* with a dielectric sample. Here we have imaged a PTFE sheet, where the MIM-Im channel is shown in Figure 4. The MIM-Im voltage is then averaged over the whole image, then used with the simulation results to predict a value of $Im(Y)$ as above. The contour on the simulated $Im(Y)$-surface corresponding to this admittance is shown in Figure 4. PTFE has a very low conductivity, lower even than the lower bound in the plot, there is no ambiguity in the prediction of *ϵ*. The predicted value of the relative permittivity is approximately 2.2, the accepted value usually tabulated as 2.1. The weak contrast in Figure 4 is a reflection of topographical changes in the sample, and in general the cross-talk between MIM-Im and topography or surface roughness is high.^{18}

The entirety of the prediction process is summarized in Figure 5. The MIM-Im voltage of the calibration die is averaged over the each of the two regions (on and off the capacitive stack), and compared with the simulated admittances of the die to establish the admittance-voltage map. In the Figure, the resulting correspondence between an average MIM-Im voltage and permittivity for the PTFE sample is highlighted with a blue line. The same approach is used for a CDC sample described below, but under a different calibration—a different admittance-voltage map. The calibration remains valid so long as the tip and instrument cabling are not exchanged, but comparing MIM-Im voltages for different calibrations is not valid.

A different calibration scheme to assign values of capacitance to the MIM-Im signal has been demonstrated wherein approach curves for MIM and EFM are compared for the same point on a sample.^{8} The authors use a Green’s function approach for calculating analytically the capacitance between the probe and a small region beneath the tip.

## V. RESULTS

We can use the same predictive model for other samples. It is even possible to estimate volume fractions in two-phase systems with the use of effective medium theories. The approximation of Bruggeman, for example, relates the effective permittivity, volume fractions, and constituent permittivities of a multiphase system:

where *f*_{i} is the volume fraction of the *i*th constituent, $\u03f5i$ the permittivity, and $\u03f5eff$ the effective permittivity of the composite. If for a two-phase system we can estimate the local effective permittivity with MIM, and if we have knowledge of the constituent permittivities, we can deduce the volume fractions.

To demonstrate the volume fraction prediction, we consider the nanostructured carbon described in Section I, carbide-derived carbon.^{13} This material is highly porous, so the two-phase system in this case is composed of air and carbon, where the volume fraction of air is called the porosity. Carbide-derived carbon has several published reports on its use as an electrode in energy storage devices called electrochemical double layer capacitors. For our purposes, this material is useful for its tunability in both structure and conductivity, two features which strongly influence the sample response in a technique such as MIM. Using the same procedure as mentioned above, we can make predictions of the local permittivity in a sample of CDC. Admittance contours for a CDC sample, synthesized at 1200 °C from an SiC precursor with particle sizes on the order of tens of nanometers, are shown in Figure 6.

The Bruggeman approximation relates the effective permittivity of the inhomogeneous medium to the volume fractions and permittivities of the constituents. The effective permittivity is plotted in Figure 7 for different possible values of the permittivity of the carbon alone. If we then map the predicted values of permittivity for the CDC as in Figure 6, we can produce estimates of the porosity. The results shown in Figure 8, except for the lowest carbon permittivity give ($\u03f5r=5.0$) are in general agreement with bulk characterization of CDC, which is known to exhibit porosities of over 50%. To our knowledge, no published study correlating the bulk permittivity and bulk porosity of carbide-derived carbon exists, and such a study would aid in the calibration of local porosity measurements The bulk permittivity is generally near 10 for amorphous carbons.^{19}

## VI. CONCLUSIONS

Microwave impedance microscopy (MIM) is a valuable technique that can image admittance contrast with minimal sample preparation. We have provided with simulations the full tip-sample behavior as a function of both permittivity and conductivity, and demonstrated a method for extracting values of permittivity from the contrast map under certain constraints for the conductivity. Values of conductivity could likewise be extracted with a similar calibration with a resistive standard sample. We have also used the predicted permittivity and the Bruggeman effective medium approximation to generate estimates of local porosity, though an additional bulk characterization of permittivity and porosity is required to obtain more precise estimates.

## ACKNOWLEDGMENTS

We acknowledge PrimeNano and K. Jones of Asylum Research for instrument time and technical support. We also thank J. McDonough and B. Dyatkin, students of Y. Gogotsi at Drexel University, for samples. This work was partially supported by the Department of Energy Office of Basic Science grant DE-FG02-00ER45813-A000, and the Nano/Bio Interface Center of the University of Pennsylvania, via the National Science Foundation NSEC DMR08-32802 fund.

## REFERENCES

^{2}-bonded carbon networks through the metal-insulator transition