Imaging thermal conductivity with nanoscale resolution using a scanning spin probe

The ability to probe nanoscale heat flow in a material is often limited by lack of spatial resolution. Here, we use a diamond-nanocrystal-hosted nitrogen-vacancy centre attached to the apex of a silicon thermal tip as a local temperature sensor. We apply an electrical current to heat up the tip and rely on the nitrogen vacancy to monitor the thermal changes the tip experiences as it is brought into contact with surfaces of varying thermal conductivity. By combining atomic force and confocal microscopy, we image phantom microstructures with nanoscale resolution, and attain excellent agreement between the thermal conductivity and topographic maps. The small mass and high thermal conductivity of the diamond host make the time response of our technique short, which we demonstrate by monitoring the tip temperature upon application of a heat pulse. Our approach promises multiple applications, from the investigation of phonon dynamics in nanostructures to the characterization of heterogeneous phase transitions and chemical reactions in various solid-state systems.

we determine as a function of the applied voltage .

Supplementary Note 1 | Sensitivity to the substrate thermal conductivity
The sensitivity to the substrate thermal conductivity is given by where is the total measurement time and ( ) is the minimum detectable change of thermal conductivity. Assuming a shot-noise-limited signal, we recast Supplementary where ( ) is the average number of photons collected during an individual measurement interval (~300 ns) when the NV spin is in the ( ) level of the ground state triplet, and is the total number of repeats.
Unlike the experiments herein (where the full NV spectrum is recorded to determine , Fig. 2a), an optimized thermal sensing protocol would monitor the NV spin signal after a time of coherent evolution 2,3 . In this case, the NV signal is given by is the NV resonance frequency shift relative to a reference, is the NV spin coherence lifetime, and is an integer in the range from 1 to 4. Since , , and are functions of the nanocrystal temperature ( ), Supplementary Eq. (2) takes the form In deriving Supplementary Eq. (6), we are neglecting the temperature dependence of , comparatively week within the observed range, and we use . 4 Choosing the near-optimum evolution interval ( ) ⁄ ⁄ , and using Supplementary Eqs.
(3) and (6) Fig. 1b). Using the Ramsey coherence lifetime ( ) ns ( Supplementary Fig. 4), we find K -1 at K, implying that contributions to originating from a thermallyinduced resonance shift or a fluorescence change are comparable for short lived NV spins evolving under the simplest sensing protocol (see Supplementary Eq. (7)). This is not the case, however, if echo sequences are exploited to extend the NV spin coherence lifetime.
In the regime ns we find and Supplementary Eq. (7) simplifies to where we ignore contributions of order ⁄ .
In general, the tip temperature is a function of the substrate thermal conductivity and the heater temperature . From our experiments, we find  Fig. 1a) and the concomitant reduction of NV fluorescence ( Supplementary Fig. 1b) Finally, we note that in Supplementary Eq. (11) and are also functions of the heater material, implying that different sensitivities are to be expected for cantilevers of different composition.

Measurements
To quantitatively understand the thermal transport and temperature gradient of the tipsample system, we6use a one-dimensional thermal transport model as shown in Supplementary Fig. 7a The contact radius a is approximately 10 nm. The resulting fitting curves (red) to the experimental data are shown in Supplementary Fig. 6 with the corresponding raw data points. The fitting is not ideal, but it shows that Supplementary Eq. (14) can capture part of the thermal transport physics of our system. Furthermore, it provides an approximate relationship between the measured ratio and the thermal conductivities of different materials, which can be used to estimate the thermal conductivity of an unknown material. Also worth noting is that the assumption in Supplementary Eq. (14) that R int is constant for different materials is clearly a crude approximation. In general, R int depends on the material properties (particularly on its thermal conductivity) so a reasonable hypothesis is (14). Not only the agreement is better but, more remarkably, the exponent n has a very small range 0.2 to 0.3 for the different temperatures and materials, with n average ~ 0.25.
The fitting also gives R tip = 5×10 6 K/W which is in good agreement with previous results.
Supplementary Eq. (16) with n ~ 0.25, R tip = 5×10 6 K/W and C = 2×10 7 is therefore a relationship which can be used with good accuracy to determine the thermal conductivity of an unknown material at the nanoscale from a measured temperature ratio (we assume SI units for ).