Imaging heat transport in suspended diamond nanostructures with integrated spin defect thermometers

TL;DR

This work addresses non-diffusive heat transport in ultrahigh-κ diamond nanostructures by employing embedded NV centers as nanoscale thermometers to image temperature fields in suspended diamond cantilevers. It combines in-situ temperature imaging with first-principles multiscale modeling based on the linearized Boltzmann transport equation and viscous heat equations to predict heat flow in sub-micrometer channels. The experiments reveal a pronounced width-dependent reduction of the effective thermal conductivity that cannot be explained by Fourier diffusion alone, and the reduction is quantitatively captured by the VHE, with ballistic phonon transport dominating over viscous hydrodynamic effects. This approach establishes a versatile platform for probing non-Fourier heat transport in complex geometries and provides a rigorous framework to dissect intrinsic and extrinsic scattering mechanisms in high-κ materials.

Abstract

Among all materials, mono-crystalline diamond has one of the highest measured thermal conductivities, with values above 2000 W/m/K at room temperature. This stems from momentum-conserving `normal' phonon-phonon scattering processes dominating over momentum-dissipating `Umklapp' processes, a feature that also suggests diamond as an ideal platform to experimentally investigate phonon heat transport phenomena that violate Fourier's law. Here, we introduce dilute nitrogen-vacancy color centers as in-situ, highly precise spin defect thermometers to image temperature inhomogeneities in single-crystal diamond microstructures heated from ambient conditions. We analyze cantilevers with cross-sections in the range from about 0.2 to 2.6 $μ$m$^2$, observing a strong reduction of the cantilevers' conductivity as the width decreases. We use first-principles simulations based on the linearized phonon Boltzmann transport equation and viscous heat equations to quantitatively predict the cantilevers' thermal transport properties, rationalizing how the interplay between intrinsic and extrinsic phonon scattering mechanisms determines the observed non-diffusive behavior. Our temperature-imaging method paves the way for the exploration of unconventional, non-diffusive heat transport phenomena in devices and nanostructures of arbitrary geometries.

Figures (14)

• Figure 1: (a) Schematic representation of the experiment. Right inset: atomic structure of an NV center in the diamond lattice. (b) Scanning electron microscope (SEM) image with false colors for the diamond cantilever (blue) and Cr patch (yellow). Scale bar is 2 µ m. Right inset: triangular cantilever cross-section, with width $w$ and angle $\theta$. (c) ZFS shift $\Delta D (T) = D(T) - D(22 \mathrm{^\circ C})$ versus bath temperature, measured in bulk diamond (circles) and on a cantilever (squares). Lower inset: Energy levels of the NV center ground state. The "bright" $m_s=0$ state shows higher photoluminescence than the degenerate, "dark" $m_s=\pm1$ states, allowing to measure $D$ through ODMR. Top inset: lock-in ODMR spectra measured at $22~\mathrm{^\circ C}$ (solid line) and $62~\mathrm{^\circ C}$ (dashed).
• Figure 2: (a) Temperature shift $\Delta T$ at a fixed readout position on a cantilever versus heating laser power $P_h$. Dashed line is a linear fit. (b) SEM image (top view) of a 1.53 µ m-wide cantilever (blue) with a narrow section (0.73 µ m) between the Cr patch (yellow) and the bulk. Vertical dashed lines highlight the extremities of the narrow section. The black cross indicates the heating laser position (c) Temperature profile measured along the cantilever at $P_h = 3.0$ mW, by sweeping the position $x$ of the readout laser. The dashed blue line is a linear fit of $\Delta T(x)$ in the narrow section.
• Figure 3: (a) SEM images, top view of cantilevers with width: $w = 0.95$ µ m (A), 1.43 µ m (B) and 1.98 µ m (C). The black cross indicates the heating laser position, corresponding to $x=0$ in (b). The white cross in II indicates the readout position for the power scan in Fig. \ref{['fig:Fig2']}a. (b) Temperature profile measured in corresponding cantilevers, normalized by the absorbed heating power $P_{abs} = \alpha P_h$. (c) Normalized temperature gradient extracted from the profiles in (b) versus cantilever width $w$. Black lines correspond to the $w^{-2}$ scaling predicted by Fourier's law (solid line) and the $w^{-3}$ scaling (dashed line) expected in the ballistic limit; green line is the first-principle prediction from the VHE.
• Figure 4: Effective thermal conductivity of diamond cantilevers as a function of width. Solid line is the prediction based on the VHE with parameters computed from first principles. Scatter points are experimental data, measured in two distinct diamond samples and fabricated either with or without FIB milling SupMat. The dashed line in the inset is the bulk conductivity.
• Figure S1: (a) Influence of readout laser power on readout error, for a microwave power +41 dBm. The procedure for estimation of readout error is detailed in Fig. \ref{['fig:AllanDev']}. (b) Influence microwave power on readout error, for a readout laser power of 0.15 mW. The reported microwave power values correspond to the output of the microwave generator, before +43 dBm amplification.