Study of radiative heat transfer in Ångström- and nanometre-sized gaps

Radiative heat transfer in Ångström- and nanometre-sized gaps is of great interest because of both its technological importance and open questions regarding the physics of energy transfer in this regime. Here we report studies of radiative heat transfer in few Å to 5 nm gap sizes, performed under ultrahigh vacuum conditions between a Au-coated probe featuring embedded nanoscale thermocouples and a heated planar Au substrate that were both subjected to various surface-cleaning procedures. By drawing on the apparent tunnelling barrier height as a signature of cleanliness, we found that upon systematically cleaning via a plasma or locally pushing the tip into the substrate by a few nanometres, the observed radiative conductances decreased from unexpectedly large values to extremely small ones—below the detection limit of our probe—as expected from our computational results. Our results show that it is possible to avoid the confounding effects of surface contamination and systematically study thermal radiation in Ångström- and nanometre-sized gaps.


Supplementary Note 2: Estimation of the Stiffness of the Scanning Probes via Modeling
The stiffness of the probes was estimated by employing finite element analysis using COMSOL TM . In this computation, we included a 500 µm thick silicon (Si) block, which is the cantilevered portion of our probe, to which a 8 µm tall tip made of silicon oxide (SiO 2 ) is added as shown in Fig. S2. The values of Young's modulus (E) and Poisson's ratio (ν) assumed in these calculations are as follows: Si (E = 170 GPa, ν = 0.28), SiO 2 (E = 70 GPa, ν = 0.17). Further, in order to estimate the stiffness of the probe, the following boundary conditions were assigned: a 100 nN of either a normal or a shear force was applied at the apex of SiO 2 tip, while the opposite end of the Si block was fixed (see Supplementary Fig. 2). Note that we evaluated three sets of deflections, where the normal deflection (i.e. deflection in the z-direction, see Supplementary Fig. 2) is determined by the cantilever stiffness, whereas the shear deflections (x-or y-direction) are related to the transverse stiffness of the tip. From the computed deflections, the stiffness of our probe was estimated to be ~10700 Nm -1 in the normal direction and ~5300 Nm -1 in the lateral directions (x and y directions labeled in Supplementary Fig. 2).

Supplementary Note 3: Characterization of the Thermal Resistance of Scanning Probes
To characterize the thermal resistance of the SThM probes, we followed an experimental procedure developed by some of us recently 1 . The first step in this process was to determine the heat flux Q into the probe when it contacted a hot surface and measure the temperature increase of the probe ( P T Δ ) via the embedded thermocouple. The resistance of the probe can thus be found to be P probe . To accomplish this procedure, a suspended calorimeter 1 with an integrated Pt resistance heater-thermometer was employed. If an AC current at a frequency ω and amplitude I ω is supplied to the Pt heater, temperature oscillations at a frequency 2ω and amplitude T 2ω are induced. When the SThM probe was placed in contact with the heated calorimeter (see the inset of Supplementary Fig. 3), an additional conduction path (via the probe) was established resulting in a heat current through the probe. This additional conduction path also reduced the amplitude of temperature oscillations by ΔT 2ω . The heat flux into the probe where sus G is the thermal conductance of the suspended calorimeter. By measuring the temperature increase of the probe ( p 2 T ω Δ ) we obtained the probe resistance as Supplementary Fig. 3, the measured temperature increase of the probe at frequency 2ω is plotted against the heat input into the probe.
The slope of the plot gives the probe resistance, which we determined to be 4 -1 probe 9 10 KW R = × .
Supplementary Figure 3: Amplitude of temperature oscillations vs. heat current input to the tip. The slope of the line is used to determine the thermal resistance of the probe. Inset shows a schematic of the experiment where the scanning probe is placed in contact with the suspended calorimeter.

Supplementary Note 4: Surface Characterization
The surface topography of template-stripped Au samples and Au-coated scanning probes were characterized by scanning tunneling microscopy (STM) and scanning electron microscopy (SEM), respectively. We note that template-stripped Au surfaces have been widely used in scanning probe microscopy studies due to their high quality in terms of the ultra-small surface roughness. 2 As shown in Supplementary Fig. 4a, the RMS roughness of the Au sample (100 nm thick) obtained from STM studies was found to be <0.1 nm within a scanning area of 150 nm x 150 nm. On the other hand, the probe's (as shown in Supplementary Fig. 4b) surface roughness is found to be much larger (RMS of 2 -3 nm) than that of the Au sample. This is mostly because that the tip of the scanning probe is a layered structure comprising of metallic and dielectric materials the fabrication (Supplementary Fig. 1) of which involves multiple deposition and etching steps during which the layers tend to become progressively rougher. Since the surface roughness of the Au substrates used in our experiments was substantially smaller than that of the scanning probes, the roughness of the substrates was considered negligible in our computational analysis. It can be seen that the measured noise power spectral density increases rapidly at low frequencies.

Supplementary
The measured PSD at high frequencies agrees reasonably well with the expected Johnson noise (PSD [V/Hz 1/2 ]= 4k B TR ) and is estimated to be ~10 nVHz -1/2 for our scanning probes whose thermocouple resistance is ~5 kΩ. At lower frequencies there are significant contributions due to ambient temperature drift. To demonstrate this point, we recorded the fluctuation of thermal conductance for a period of ~1 hour when the scanning probe was placed at a constant distance of ~100 nm from the substrate (Supplementary Fig. 6). Under these conditions the thermal conductance between the tip and the sample is expected to be invariant with time. It can be seen that there is an apparent thermal conductance change of ~15 nWK -1 (peak-to-peak), similar to the noise level shown in Fig. 4 of the main manuscript which is likely due to temperature drift of a few 10s of mK. Furthermore, a comparison of the measured frequency-dependent noise spectral densities in Supplementary Fig. 6 and Fig. 4, suggests that the low frequency noise present in the data of Fig. 4 is most likely due to the temperature drift of the ambient.

Supplementary Figure 6: Fluctuations in temperature and radiative thermal conductances.
Temperature drift of the scanning probe or the ambient temperature leads to fluctuations of the thermoelectric voltage output of the scanning probe, which in turn manifests itself as apparent thermal conductance fluctuations. The above data, which illustrate the level of noise in the thermal conductance data, were obtained from an experiment where the probe and the sample were separated by 100 nm and a temperature differential of 40 K was applied.

Experiment
In the "controlled crashing" experiments the tip of the scanning probe is indented into the substrate by a very short distance (a few nanometers at most). To verify that there is no significant change in the tip diameter during this process we obtained SEM images for a new probe and of a probe after the crashing experiment. As shown in Supplementary Fig. 7, we found no gross/observable change in the tip shape or curvature. Figure 7: SEM images of a pristine probe and a probe subjected to "controlled" crashing. Analysis of the SEM images of the tips of scanning thermal probes before and after subjecting to controlled crashing suggests that there is nor observable difference in the tip geometry. The diameter of the dashed circles is ~300 nm.

Supplementary Note 7: Determination of the Gap-size
Unambiguous determination of the gap-size between the scanning probe and the sample is key to successful interpretation of the data obtained in our experiments. We estimated the minimum gap-size from the conductance vs. gap-size curve (Fig. 2) by extrapolating the curve to a conductance of 1G 0 (corresponding to a single-atom Au-Au contact) and estimating the additional distance the probe would have to be moved to achieve this conductance. This additional distance is an indicator of the minimal gap-size in our experiments. Since the measured tunneling barrier was found to be ~1 to 2.5 eV, the extrapolated gap-size was estimated to be ~1.3 to 2.2 Å.
This estimate was also experimentally validated by displacing our probes (in control experiments) until the conductance was increased to 1G 0 from 0.1G 0 . Data from such experiments is shown in Supplementary Fig. 8 and it can be seen that an additional displacement of ~1 to 2.5 Å is required to increase the conductance to 1G 0 .