Imaging the nanoscale phase separation in vanadium dioxide thin films at terahertz frequencies

Vanadium dioxide (VO2) is a material that undergoes an insulator–metal transition upon heating above 340 K. It remains debated as to whether this electronic transition is driven by a corresponding structural transition or by strong electron–electron correlations. Here, we use apertureless scattering near-field optical microscopy to compare nanoscale images of the transition in VO2 thin films acquired at both mid-infrared and terahertz frequencies, using a home-built terahertz near-field microscope. We observe a much more gradual transition when THz frequencies are utilized as a probe, in contrast to the assumptions of a classical first-order phase transition. We discuss these results in light of dynamical mean-field theory calculations of the dimer Hubbard model recently applied to VO2, which account for a continuous temperature dependence of the optical response of the VO2 in the insulating state.


Suplementary Note 1: Gaussian fits to near-field image histograms
First we obtain histograms of the near-field signal distribution in each THz and MIR image. We crop each image to a 210x125 pixel region that does not contain any of the reference gold. We As mentioned in the main text, the THz histograms are Gaussian at every temperature, while the MIR histograms are bimodal for intermediate temperatures.
To extract an average signal level at each temperature we fit the THz histograms to the skewed Gaussian function where the four free parameters are the amplitude A, the standard deviation σ, the skew parameter α, and the mean µ. The fitted functions are plotted as solid lines for three representative temperatures in Supplementary Figure 1a. This last parameter µ is what is plotted as a function of temperature in Fig. 3c (open circles) of the main text.
Because the MIR images display phase separation, the histograms near the transition temperature are bimodal. Therefore we fit the MIR histograms to a function for the sum of two skewed Gaussians: The fitted functions are plotted as solid lines in Supplementary Figure 1b  THz near-field signal at each temperature is extracted from histogram fitting as described in Supplementary Note 1. Although the data for decreasing temperatures does not fully close the hysteresis loop, the hysteretic behavior in the THz near-field signal upon cooling is evident. We were unable to obtain a full set of THz data for decreasing temperatures due to degradation of the AFM tip used in the experiment. Because the formation of real-space patterns in the phase transition is spontaneous, a full imaging series must be performed in a single shot to compare images at different temperatures. Unfortunately, our setup prohibits the exchange of AFM tips without altering the sample temperature and thus disrupting a cooling or heating cycle. For this reason, we were not able to replace the AFM tip and complete the full hysteresis loop.
We overlay the average THz near-field signal with DC conductance measured in the same apparatus on the same film (green). The several orders of magnitude jump in DC conductance, signifying the insulator-metal transition, occurs to within 1 K of the same temperatures as the factor of 2 jump in the THz near-field signal. The slight offset in the THz near-field signal transition temperature compared to the transport transition temperature can be understood as due to several effects. First, we would expect local imaging experiments to record higher average near-field signal at temperatures below the DC transport transition because the formation of unconnected metallic domains precedes the percolative transition forming a continuous conducting path through the sample. Second, we would expect the THz near-field signal to increase at temperatures below the DC transport transition as our theory predicts: the slight increase in the insulating state THz conductivity leads to a rather large THz near-field signal.

Supplementary Note 3: Autocorrelation analysis of AFM resolution
The custom AFM tips used for THz-SNOM limit the AFM resolution in THz measurements.
Supplementary Figure 3a

Supplementary Note 4: Width of the IMT in MIR near-field images
In Supplementary Figure 4 we plot the difference in the MIR signal levels between the insulating and metallic states at each temperature for which there is coexistence. The coexistence region spans from 342 K to 350 K and is a good indication of the width of the transition in temperature.

Supplementary Note 5: Low-frequency reflectivity of an incoherent metal
The Drude model provides a consistent basis for discussing the frequency dependence of material optical response. The dielectric function˜ = 1 + i 2 is given by the Drude model as where ω P and τ are the plasma frequency and scattering time, respectively. For simplicity we consider the reflectivity of a material in vacuum at normal incidence, which is whereñ = n + ik is the complex index of refraction of the material. When the magnetic permiability of the material is equal to 1, the index of refraction is related to the optical permitivitty as follows: Consider the case of an incoherent metal where ω P τ ≈ 1 in the low frequency limit ωτ 1.
This represents a material with a low but finite conductivity, as for the high t ⊥ /U case in the insulating state (main text). In this limit we obtain the following approximations: Substituting Eqs. 8 and 9 into Eq. 5, we obtain This is equivalent to having a material where T c is homogenous, but there are local variations in the temperature of the film. We use the calculated near-field signal as a function of temperature to model the near-field signal distribution that would be acquired in an imaging experiment. For each temperature T j we generate an N -element vector V of temperatures that are normally distributed about T j with a standard deviation of 1K. In Figs. 5a and 5c of the main text, we plot the histogram of S(V ) + δS, where S(x) is the near-field signal calculated at temperature x, and δS is a random white noise term. The white noise is added at each pixel to simulate the experimental error in detected near-field signal due to electronic and environmental background.