A Kelvin probe force microscopy (KPFM) image is sometimes difficult to interpret because it is a blurred representation of the true surface potential (SP) distribution of the materials under test. The reason for the blurring is that KPFM relies on the detection of electrostatic force, which is a long-range force compared to other surface forces. Usually, KPFM imaging model is described as the convolution of the true SP distribution of the sample with an intrinsic point spread function (PSF) of the measurement system. To restore the true SP signals from the blurred ones, the intrinsic PSF of the system is needed. In this work, we present a way to experimentally calibrate the PSF of the KPFM system. Taking the actual probe shape and experimental parameters into consideration, this calibration method leads to a more accurate PSF than the ones obtained from simulations. Moreover, a nonlinear reconstruction algorithm based on total variation (TV) regularization is applied to KPFM measurement to reverse the blurring caused by PSF during KPFM imaging process; as a result, noises are reduced and the fidelity of SP signals is improved.

## I. INTRODUCTION

Kelvin probe force microscopy (KPFM), as an advanced mode of atomic force microscopy (AFM), has been proven to be capable of detecting surface potential (SP) distribution of various materials with a resolution in nano-scales.^{1–5} Such technique provides quantitative basis for the analysis of electronic properties such as band alignment of semiconductors,^{5} surface potential of organic solar cells,^{6,7} work function of various materials,^{8–10} etc. Ideally, each pixel of the KPFM image should represent the SP of the material at the specific area corresponding to the pixel. However, due to the fact that electrostatic force is a long-range force, SP mapping obtained by KPFM is usually a distorted and blurred reproduction of the real potential distribution.^{11–15} More specifically, KPFM measurement at a specific point is a weighted result of SP signals of a much larger area. The weight matrix is an intrinsic parameter of the KPFM system, which is usually denoted as a point spread function (PSF) and is the point source response of the KPFM imaging system.

A widely accepted model describing the correlation between measured SP signals and the actual SP distribution was derived by Jacobs *et al.* in 1998.^{16} As illustrated in their work, the interaction between the probe and the sample surface was modeled as a set of ideal conductors with mutual capacitances, and the calculation was based on a strong assumption of uniform electric field along field lines.^{16} Strassburg *et al.*, on the other hand, developed an electrostatic force model based on a boundary element method without assuming a uniform electric field.^{17} As demonstrated in their report, the PSF of a specific probe shape and tip-sample distance could be derived by modeling semiconductor as equivalent dipole layers.

Several other works have also reported on the simulation of PSF and its application in the deconvolution of KPFM signals.^{17–19} To make the problem solvable, most of the simulations assume a regular probe with symmetric shape.^{17–20} Based on such assumption, the resultant PSF will be an estimation of the actual system PSF but not the accurate one. In this work, we present a new approach to experimentally calibrate PSF of a KPFM system. The calibration is based on the COMSOL simulation of the electrical field and the actual KPFM measurement of a biased electrode pair. This calibration method not only provides a direct way to obtain the PSF but also improves the accuracy of the PSF by taking into account the real probe shape. The calibrated PSF was then applied to a non-linear total variation (TV) regularization deconvolution algorithm for the reconstruction of SP images. This iterative algorithm restores the accuracy of KPFM measurement while suppressing noises, which greatly helps the interpretation of KPFM data.

## II. IMAGING MODEL AND POINT SPREAD FUNCTION

Due to the long-range electrostatic force, the KPFM imaging system is modeled as the convolution of the SP distribution and the PSF of the imaging system. As expressed in the following:

where *g*(*x*,*y*) is the KPFM measurement, *f*(*x*,*y*) is the true SP distribution of the sample, *h*(*x*,*y*) is the PSF of the imaging system, and *n*(*x*,*y*) is the additive noise. The convolution process often results in the blurring of acquired image, leading to the decrease in resolution and loss of fidelity of the measurement. We can demonstrate such effect with a simulated PSF based on Strassburg’s electrostatic force model.^{17} Shown in Figure 1(a) is the simulated PSF with a tip-sample distance of 5 nm. The profile of the probe used in the calculation is composed of two parts—a conical shaped body and a hemispherical shaped tip end. Tip length is chosen to be 10 $\mu m$ and tip radius is 15 nm based on typical manufacture’s data.^{21} As reported in our previous work, if the feature size under test is too small, there will be a large discrepancy between sample’s true SP and SP acquired by KPFM.^{21} Results are recreated in Figure 1(b) where the simulated KPFM measurements of circular electrodes are shown. In the simulation, we assumed a uniform SP of 1 V on electrode and 0 V on the rest of the sample. Also, the radiuses of the electrodes were varied to demonstrate the correlation between the feature size and KPFM accuracy. From Figure 1(b), it can be concluded that a smaller feature size results in a larger discrepancy between the KPFM measurement and actual SP due to the averaging effect of PSF. From the simulation, with this specific tip-sample system, a feature radius of at least 50 nm is needed for an accurate measurement. Figure 1(c) demonstrates the degradation of resolution caused by the KPFM imaging process. When two small features are closely located, only one blurred and broaden feature can be detected in the KPFM measurement due to the convolution with the PSF. Such blurring caused by PSF often leads to the misinterpretation and misunderstanding of the measurement data. In this work, to improve the fidelity of KPFM measurement, a novel experimental approach to accurately calibrate the PSF of our KPFM system is presented. Then, with the calibrated PSF, a non-linear reconstruction algorithm based on TV regularization is utilized to reverse the blurring process and reconstruct SP signals.

## III. CALIBRATION OF PSF WITH KPFM MEASUREMENT

Taking into consideration that the PSF of a specific tip-sample system is reflected in the KPFM measurement, it can be extracted through experimental calibration. In this work, to calibrate the PSF of our KPFM system, KPFM measurement was conducted on a pair of calibration electrodes. By comparing the KPFM measurement and simulating true SP distribution with COMSOL, the PSF of this tip-sample system was derived. The calibration electrode pair was fabricated with standard photolithography and e-beam lithography process on a silicon substrate with a 150 nm thick silicon oxide layer. A pair of electrodes was first fabricated through photolithography, then, more delicate features were patterned by e-beam lithography. The metal used for our electrode pair is palladium (Pd) and the electrode pair has a thickness of 50 nm deposited by an e-beam evaporator.

KPFM measurement was carried out with Agilent SPM 5500 system using SCM-PIT KPFM probe fabricated by Bruker. The electric oscillation frequency used was 13.311 KHz and the tip-sample distance calculated from force distance curve is 31 nm. In the KPFM measurement, the two electrodes were biased with specific voltages supplied by a voltage source, as shown in Figure 2, and the electrode pair was placed perpendicular to the KPFM probe. The relative position between the sample and KPFM probe is important because that, unlike the results obtained from simulations, the real PSF of a tip-sample system is not rotationally symmetric.

The topography of the calibration electrode is shown in Figure 3(a). Because silicon oxide is non-conductive, charges could be trapped on silicon oxide surface, especially at edges of the fabricated patterns. However, since the trapped charges will not migrate with different biases, the subtraction of SP signals with different biases can help to remove the signals caused by trapped charges. To do so, the KPFM measurement was first performed when both electrodes were biased at 0 V, the corresponding SP image is shown in Figure 3(b). Then the KPFM measurement was taken when the left electrode was biased at −1 V while the right electrode was biased at +1 V; the corresponding SP measurement is shown in Figure 3(c). The subtraction of these two measurements removes trapped charges and gives the corrected SP distribution of the calibration sample with −1 V bias on the left electrode and +1 V bias on the right electrode. After noise removal using a median filter, the corrected KPFM result is shown in Figure 3(d).

After acquiring the KPFM measurement of the calibration electrodes, the ideal SP distribution of the calibration electrodes was simulated in COMSOL. A topography image of the calibration electrode was imported into COMSOL and the electrode model was built based on the measured electrode shape. Other parameters such as metal thickness and substrate dielectric were also incorporated into COMSOL software. Initial voltages applied to electrodes were again −1 V on the left electrode and +1 V on the right electrode, respectively. Zero charge was used as the boundary condition to compute SP distribution with no surface charge. And a charge conservation node was added to include equations for charge conservation based on Gauss’ law. Finally, with coarse mesh at locations far from electrode center and a much finer mesh compared to electrode size in electrode center, the simulation of SP distribution was obtained. The simulation was performed with a much larger area of 200 $\mu m$ by 100 $\mu m$ than the size of KPFM measurement, which is 1.5 $\mu m$ by 1.5 $\mu m$ to ensure its fidelity. The simulation result is shown in Figure 4. The inset figure is a zoomed-in image with the same size to our KPFM measurement, which is also the ideal SP distribution of our calibration electrode with this specific bias.

With the acquired KPFM measurement and simulated ideal SP distribution, Lucy-Richardson (L-R) algorithm was applied to calibrate the PSF of our system. L-R algorithm is an iterative algorithm with the iteration process being described as^{22}

where $hi(x,y)$ is the estimation of the PSF of our system at *i*th iteration; $f(x,y)$ is the ideal SP distribution of the calibration electrode with $f(\u2212x,\u2212y)$ being its corresponding adjoint matrix; and $g(x,y)$ is the measured SP distribution. The basic idea of this iteration process is to calculate the most likely $hi(x,y)$ with known $f(x,y)$ and observed $g(x,y)$. The iteration process was implemented with functions available in MATLAB.

The calibrated PSF is shown in Figure 5. Figure 5(a) is the calibrated PSF, Figure 5(b) is its top view, and Figure 5(c) is the SEM image of a SCM-PIT probe provided by Bruker.^{23} As can be seen from the calibrated PSF, different from the Gaussian-like, symmetric PSF typically obtained from simulation,^{17} the real PSF is asymmetric. Also, the top view of the calibrated PSF mimics the projection of probe tip on the surface; this agrees with our understanding that the shape of a PSF is largely dependent on the profile of the probe being used. Later, with this calibrated PSF, we apply the reconstruction algorithm to reverse the blurring effect of PSF and reconstruct SP images.

## IV. NONLINEAR TOTAL VARIATION REGULARIZATION BASED SIGNAL RECONSTRUCTION

With the calibrated PSF, a nonlinear TV regularization based signal reconstruction algorithm was applied to KPFM images to reconstruct the true SP images. The reason for the choice of nonlinear algorithm instead of the linear ones is that the nonlinear reconstruction method is much better at conserving the resolution of original signals.^{24} We can formulate an optimization problem to retrieve SP signals from the blurred KPFM measurement *g* and the calibrated PSF *h*. The optimization problem is equivalent to the minimization of the following function:

where ||·|| denotes vector 2-norm and ||·||_{1} is the vector 1-norm. The first term of this function is to force the reconstructed result to be the most likely estimation of the undistorted signal. The second term is the TV regularization which helps to smooth reconstructed signals and eliminate noise. The balance between these two terms is achieved with parameter *λ* which is a positive real number. Intuitively, it can be concluded from Equation (3) that with a larger *λ*, more stress will be on the first term and the reconstructed result will be more constrained to resemble the estimation of the undistorted image, while a smaller *λ* emphasizes on the suppressing of noise level but could lead to the loss of fidelity in reconstructed signals. Usually, the optimal value of *λ* is chosen based on specific images under processing and might vary greatly with different images. In this work, we demonstrate the application of this algorithm in the deconvolution of SP image of nanorods (1-D material) and SP image of graphene oxide (2-D material).

To solve the optimization problem, the augmented Lagrangian method was used. Since convolution is a linear operation, it can be expressed with matrices. Now denoting that the convolution with PSF $h(x,y)$ is associated with an operator H and introducing *u* as the intermediate variable, the optimization problem is equivalent to^{25,26}

and the corresponding augmented Lagrangian problem can be written as

where *z* is the Lagrange multiplier associated with constraint $u=\u2207f$ and $\beta $ is the regularization parameter associated with penalty term $||u\u2212\u2207f||2$. To find the saddle point of this problem, the alternating direction method was used to solve the following sub-problems iteratively:^{27}

To solve the f-subproblem, we find the optimality condition of Equation (6),

The solution to u-subproblem has a closed form of^{28,29}

where $wx,y=\u2207x,yf+1\beta zx,y$ and represents the *x* and *y* components of the matrix *w*, respectively. With solutions to the subproblems and initial values of $f0=g,u0=\u2207f0$, *f* and *u* can be updated iteratively. The Lagrange multiplier *z* is also calculated in each iteration using Equation (8). Convergence was checked at the end of each iteration using $\epsilon =\u2225fk+1\u2212fk\u22252/\u2225fk\u22252$: if ε is less than the predefined tolerance, the iteration ends. The tolerance we used in this work is 10^{−6}. Using the TV regularization algorithm, we demonstrate the successful deconvolution of SP images of a zinc oxide (ZnO) nanorod (1-D material) and a graphene oxide flake (2-D material) on a semiconductor polymer surface. They are chosen simply because of their availability in the lab.

We first demonstrate the deconvolution of SP image of a nanorod using the TV regularization algorithm. Figure 6 shows the deconvolution results. Figure 6(a) is the topography image of a ZnO nanorod on poly(3-hexylthiophene-2,5-diyl) (P3HT) and phenyl-C61-butyric acid methyl ester (PCBM) blend and Figure 6(b) is the corresponding SP image. Due to the band alignment between ZnO and P3HT/PCBM polymer blend, electrons were accumulating at ZnO/polymer interfaces. This can be observed from Figure 6(b) where the interfaces of ZnO and P3HT/PCBM are much darker than the rest of the image, indicating the accumulation of electrons. To deconvolute the measured SP image, we first reconstructed the SP image in MATLAB, as shown in Figure 6(c). The TV regularization algorithm was then applied to the SP data and the deconvoluted image is shown in Figure 6(d). Figures 6(c) and 6(d) have the same colormap and color scale, providing a direct comparison between the original measurement and the deconvoluted image. Compared to Figure 6(c), noises in Figure 6(d) are reduced; also, the electron accumulation at the interfaces between ZnO and P3HT/PCBM is much more obvious and clearer. Cross sections (indicated by the black dashed lines drawn on Figures 6(c) and 6(d)) are shown in Figure 6(e) for a closer observation. In Figure 6(e), the blue dashed curve is the cross section of the original SP image and the orange solid curve is the cross section of the deconvoluted SP image. Compared to the original SP signal, the deconvoluted signal is much smoother and the charge accumulation is more evident due to the enhanced signal intensity. In this deconvolution, since the features (the nanorod) are relatively small, emphasis is on the restoration of the accuracy of SP signals. To do so, the regularization parameter $\lambda $ was chosen to be 100. This enforces the deconvoluted image to represent the maximum likelihood of the undistorted image while loosens the stress on the removal of noises.

Next, we demonstrate the TV regularization algorithms on a 2-D material graphene oxide, where the emphasis is on the removal of noises. Shown in Figure 7 are the measurement and deconvolution result of a graphene oxide flake on P3HT/PCBM blend. Figure 7(a) is the topography image and Figure 7(b) is its SP image. From the SP image, it can be observed that the whole image is buried in noise. For the purpose of comparing with the deconvoluted image, the SP image was recreated in MATLAB and is shown in Figure 7(c). The TV regularization algorithm was again applied and the corresponding result is shown in Figure 7(d). Compared with the raw image, the noise in the deconvoluted image is greatly suppressed. This can also be observed from the cross section signals shown in Figure 7(e). Cross sections indicated by black dashed lines drawn on Figures 7(c) and 7(d) are shown in Figure 7(e), where the dashed blue curve is the cross section of the original SP image and the orange solid curve is the cross section of the deconvoluted SP image. Since the feature (the graphene oxide flake) in this measurement is relatively large, the value of the measured SP should maintain a relatively high fidelity as we discussed previously. Hence, the focus is on the removal of noises. To do so, the regularization parameter $\lambda $ was chosen to be 10, much smaller than the value we had for the ZnO nanorod. As a result, the SP image of the graphene oxide flake was significantly smoothed.

## V. CONCLUSION

To sum up, in this work, we demonstrated an innovative way to experimentally calibrate the PSF of a KPFM system. This calibration method not only provides a more direct way to obtain the PSF than regular simulation approach but also improves the accuracy of the PSF by taking into account the real probe shape. Moreover, a reconstruction algorithm based on nonlinear TV regularization is applied to the deconvolution of KPFM signals. This iterative algorithm suppresses noise and restores the accuracy of KPFM measurement which greatly helps the interpretation of KPFM data.

## ACKNOWLEDGMENTS

This research is supported in part by the NSF Grant Nos. CBET 1132819 and CBET 1404591.