Fractal Nature of Metallic and Insulating Domain Configurations in a VO2 Thin Film Revealed by Kelvin Probe Force Microscopy

We investigated the surface work function (WS) and its spatial distribution for epitaxial VO2/TiO2 thin films using Kelvin probe force microscopy (KPFM). Nearly grain-boundary-free samples allowed observation of metallic and insulating domains with distinct WS values, throughout the metal–insulator transition. The metallic fraction, estimated from WS maps, describes the evolution of the resistance based on a two-dimensional percolation model. The KPFM measurements also revealed the fractal nature of the domain configuration.

All of these considerations raise the following question: How can we directly observe the metallic and insulating domain configurations of VO 2 thin films and their evolution throughout the transition? Conventional x-ray diffraction (XRD), photoemission spectroscopy, and optical spectroscopies can be used to investigate the physical properties of VO 2 during the MIT 21 ; however, these techniques provide only the physical quantities averaged at the macroscopic scale. Nanoscale probe-based tools, such as scanning tunneling microscopy (STM) and scanning probe microscopy (SPM), have superior spatial resolution. Various SPM-based techniques have been developed to probe the electrical properties of samples. In particular, Kelvin probe force microscopy (KPFM) is a versatile tool for investigating the local surface potential under ambient conditions 31,33 .
In this work, we demonstrated that KPFM measurements of the local surface work function (W S ) can be used to reveal the intriguing phase transition behaviors of the VO 2 /TiO 2 thin films. The W S maps clearly show the spatial distribution of metallic and insulating domains during the transition. The evolution of the metallic domain fraction well explains the temperature dependence of the resistivity based on the 2D percolation model. The domain size is tens of nanometers at intermediate temperatures. The perimeter and area of the metallic domains follow power-law behaviors: a power exponent larger than 1/2 suggests that the metallic domains are fractal objects. Figure 1 shows the XRD pattern for the 15-nm-thick VO 2 thin film grown on a TiO 2 (001); two significant peaks corresponding to tetragonal VO 2 (002) at the higher angle and TiO 2 (002) at the lower angle were evident. The cross-sectional high-resolution transmission electron microscopy (HR-TEM) image shown in the inset of Fig. 1 confirms the epitaxial crystallinity. Previously, we reported details of the sample preparation procedures and characterization results 7 . Figure 2a shows a schematic diagram of the measurement setup. Transport and KPFM measurements (XE-100, Park Systems) can be simultaneously performed in a glove box filled with N 2 gas. The sample temperature was varied from 285 to 355 K using a sample stage equipped with a Peltier device. The VO 2 Figure 1. An x-ray diffraction (XRD) pattern of a 15-nm-thick VO 2 thin film grown on a TiO 2 (001) substrate. The inset shows a cross-sectional high-resolution transmission electron microscopy (HR-TEM) image of the VO 2 /TiO 2 thin film. thin film was patterned by photolithography and reactive ion etching with SF 6 gas. The width and length of the stripe-shaped pattern were 4 and 30 μ m, respectively. Figure 2b shows the resistivity (ρ) and surface work function (W S ) measured during a heating and cooling cycle. W S was averaged over a 2 × 1-μ m 2 scanned area on the VO 2 surface. ρ undergoes an abrupt change around 300 K. The transition temperature (T C ) is less than that of a single crystal, due to the shorter c-axis length caused by epitaxial strain 28 . W S gradually decreases (increases) while raising (lowering) the sample temperature, indicating variation in the electronic structure. The temperature dependences of both ρ and W S were reproducible under repeated heating and cooling cycles. Thus, although surface redox must be accounted for under ultra-high-vacuum conditions 29 , it was not a consideration in this study. Figure 3a-d show the W S maps obtained from an identical region, whose morphology is shown in Fig. 3e, while heating the sample from 285 to 355 K. The sample surface was very flat; the root-mean-square roughness was only 0.42 nm. Hence, topographic artifacts had little effect on the W S images. From Fig. 2b, two values (i.e., 5.06 and 4.96 eV) can be chosen as representative values for low-and high-temperature W S , respectively. The two values correspond to blue and red colors in the W S maps shown in Fig. 3a-d; as the temperature increased, the area of the red (blue) region decreased (increased). Note that the VO 2 surface regions were either red or blue in color only, with the exception of the boundary region between the two colors. In contrast, our earlier study showed that the VO 2 /Al 2 O 3 films exhibited a gradual change of W S over the entire surface area with variations in the sample temperature 31 . The VO 2 / Al 2 O 3 films suffer from rather large strain due to the lattice mismatch, generating high-density GBs and tens-of-nm-sized grains. The domain size should be limited by the GBs and the width of the space charge region (SCR) formed at the boundaries of the metallic and insulating domains, which can cover large portions of the sample surface area during the transition 31 . Consequently, we expect that band bending at the domain boundaries would dominate the spatial distribution and temperature evolution of W S in the VO 2 /Al 2 O 3 film with small grains. A comparison of VO 2 /TiO 2 (this work) and VO 2 /Al 2 O 3 (Ref. 31) thin films showed that distinct strain states and resulting microstructures can significantly influence the evolution of the W S maps in the VO 2 thin films.
Kelvin probe force microscopy is based on the electrostatic interaction between the probe tip and sample, and has a spatial resolution of tens of nm 34 Fig. 3a-d) are significantly larger than the spatial resolution limit and red spots can be seen at the center of such green regions. The limited resolution alone may not explain such results. Therefore, the intermediate regions in the work function maps should be explained by the SCR formation at the domain boundaries as well as the resolution limit of KPFM.
Near-field scanning optical microscopy (NSOM) 3 and hard x-ray nanoprobe (HXN) measurements 30 revealed metallic and insulating domains in the VO 2 thin films as they underwent the MIT. In both NSOM and HXN data, the spectra and diffraction patterns near T C were broader than those far from T C , implying the coexistence of the two phases; however, extraction of a physical parameter from individual domains is not straightforward. The dielectric constant from optical spectra and the lattice constant from diffraction patterns can be obtained only after model-based fittings, in which the associated sample states and fitting parameters are chosen subjectively. In contrast, KPFM results provide the local surface work function of a specific area without the need for numerical analysis.
From the map for each temperature, the metallic region can be identified as the region with the high-temperature W S (i.e., the blue-colored region in Fig. 3a-d). The typical size (tens of nm) and the shape of the metallic domain were similar to those obtained from the NSOM and HXN results 3,30 . The area with the high-temperature W S enables us to estimate the metallic fraction, P M . P M increases linearly with the sample temperature. Interestingly, the metallic domains (the blue region) were present even at 285 K, far below T C (Fig. 3a). Ramirez et al. suggested the existence of persistent metallic domains in VO 2 from analysis of the resistance hysteresis during the MIT 27 . Additionally, a small part of the sample surface had a low-temperature W S even at 355 K, far above T C (the red region in Fig. 3d); this may correspond to an area where stain relaxation occurs, stabilizing the insulating phase at high temperatures 26 . Figure 4a and 4b show the difference in W S between high and low temperatures. The average W S decreased as the temperature increased (see Fig. 2b), and hence we would expect local W S to be smaller at high temperatures than at low temperatures (see the dark color in Fig. 4a and 4b). From Fig. 4a and b, however, in some localized regions (see light colored regions) we can see that W S increased with temperature. This suggests that an inverse phase transition (i.e., from insulating to metallic phases) occurred in these local areas. Qazilbash et al. observed a similar inverse phase transition at the nanoscale in HXN experiments 30 . Our data appear to show relevant experimental results, although a clear understanding of the physical origin is currently lacking. Figure 5a shows the full-width-half-maximum (FWHM) of the W S distributions as a function of temperature. Figure 5b shows histograms of W S at 285, 305, and 355 K; P M gradually increased with the sample temperature, as expected from the progressive decrease in W S (see Fig. 1c). The FWHM exhibited a broad single peak centered at 305 K, corresponding to a P M value of nearly 1/2. When the metallic and  insulating domains have nearly the same area, W S will experience its highest standard deviation, and the FWHM was a maximum. Figure 6 shows the sample conductivity, σ, as a function of P M obtained from the W S maps. σ, increases abruptly at P M ~ 0.3. The percolation model can be applied to describe the relationship between σ, and P M , as reported previouly 7,21 . According to the percolation model, σ, should have a power-law dependence as a function of P M , i.e., σ, ∝ (P M -P C ) t , where P C is the percolation threshold and t is the critical exponent 32 . The universal values are P C = 0.45 and t = 1.4, for the 2D percolation conduction model, and 0.15 < P C < 0.17 and t = 2.0, for the 3D percolation model 7,21,32 . The P M dependence of σ, can is described well by the 2D percolation model with P C = 0.36 and t = 1.46, as shown in the inset of Fig. 6. These values are similar to those reported from optical microscopy measurements 7 of epitaxial VO 2 /TiO 2 thin films. These results suggest that the value of P M from our KPFM measurements accurately describes the metallic fraction in the VO 2 /TiO 2 thin films.
Chang et al. reported that P M , estimated from scanning tunneling spectroscopy (STS) experiments deviated significantly from that obtained from XRD data 21 . STS is a technique used to study the surface density of states; atomic resolution can be achieved with this technique for well-prepared surfaces. VO 2 has very strong electron-lattice coupling and correlation effects; hence, the surface physical properties can be altered significantly from the bulk properties 21 . However, the surface preparation procedures to obtain atomically ordered oxide surfaces have not been well established 35 . In particular, it can be difficult to achieve oxygen stoichiometry at the oxide surface in ultrahigh vacuum 29 , thus, inherent features of metal oxides can limit observation of intrinsic MIT behavior in VO 2 using STS analysis. Figure 7 shows that the relationship between the perimeter (Σ) and area (A) of metallic domains at 285, 305, and 335 K follows a power law; specifically, log(Σ) ∝ log(A). More interestingly, the data for the three temperatures overlapped a single line. The power exponent, i.e., the slope in the log(Σ)-log(A) plot, was estimated to be 0.74. Objects with Euclidian shapes should have a slope of 0.5, because A ∝ Σ 2 .  Thus, the larger exponent suggests that the metallic domains are fractal objects, having a fractal dimension, D = 1.48 (i.e., Σ ∝ A 1.48/2 ). The percolation cluster is known to be an example of a random fractal 32 , and our KPFM measurements provide direct evidence for the fractal nature of the domain shapes in VO 2 .
Simultaneous measurements of W S and the resistivity of VO 2 /TiO 2 thin films in this study provided information on the temperature-dependent domain configurations and their influence on MIT behaviors. The local surface W S of a specific area was measured directly using KPFM, without the need for additional numerical analysis. W S maps, obtained by KPFM, showed that the nearly GB-free VO 2 /TiO 2 thin films had tens-of-nm-sized metallic and insulating domains with clearly distinct W S values, throughout the MIT transition. The 2D percolation model well explains the relationship between the metallic domain fraction and the sample resistivity. Real-space domain maps also suggest that the domains form a fractal surface, which is a well-known feature of percolation clusters.

Methods
VO 2 thin films that were 15-nm-thick were deposited on rutile TiO 2 (001) substrates using pulsed laser deposition using an ArF Excimer laser with a wavelength of λ = 193 nm, a repetition rate of 2 Hz, and a fluence of 10 mJ cm −2 at 430 °C in an 1.0-Pa oxygen atmosphere. A V 2 O 5 pellet was used as the target, and the deposition rate was approximately 0.3 nm min −1 . KPFM measurements were made using an atomic force microscope (XE-100, Park Systems) with a glove box. The glove box was purged using N 2 for more than 3 hours, and the sample was then heated to 100 °C for 30 min to remove any adsorbed water. The KPFM and transport measurements were then carried out while varying the sample temperature using a Peltier device. Conductive Pt-coated Si cantilevers with a resonance frequency of ~240 kHz (NSG10/Pt, NT-MDT) were used to characterize the work function and topography. Immediately following each measurement, the work function of the tip was calibrated using highly ordered pyrolytic graphite (HOPG) (SPI Supplies) as a reference sample. The measurements were repeated several times and no noticeable differences were observed, confirming the reproducibility of the data. For the transport experiments, Al wires were bonded at the ends of a striped pattern using a wire bonder (7476D, West Bond). Two-probe electrical measurements were performed using a semiconductor parameter analyzer (4156B, Hewlett Packard) simultaneously with the work function measurements.