Spectrally Resolved Specular Reflections of Thermal Phonons from Atomically Rough Surfaces

The reflection of waves from rough surfaces is a fundamental process that plays a role in diverse fields such as optics, acoustics, and seismology. While a quantitative understanding of the reflection process has long been established for many types of waves, the precise manner in which thermal phonons of specific wavelengths reflect from atomically rough surfaces remains unclear owing to limited control over terahertz-frequency phonon generation and detection. Knowledge of these processes is critical for many applications, however, and is particularly important for recent attempts to create novel materials by coherently interfering thermal phonons. Here, we report measurements of a key property for these efforts, the phonon-wavelength-dependent specularity parameter, which describes the probability of specular reflections of thermal phonons at a surface. Our experiments show evidence of specular surface reflections of terahertz thermal phonons in our samples around room temperature and indicate a sensitivity of these reflections to surface imperfections on the scale of just 2–3 atomic planes. Our work demonstrates a general route to probe the microscopic interactions of thermal phonons with surfaces that are typically inaccessible with traditional experiments.

Popular Summary

Heat flow can often be thought of as a flow of particles. These particles, known as phonons, are actually collective vibrations of atoms or molecules. The ability to manipulate phonons, and hence heat flow, partly depends on understanding how these quasiparticles reflect from surfaces. Here, we describe experiments that allow us to resolve the nature of these reflections over the phonon spectrum for the first time. Focusing on terahertz thermal phonons, we obtain the first measurements of their specularity parameter, which quantifies to what degree phonons of different frequencies undergo mirrorlike (or specular) reflections.

Our approach exploits the breakdown of heat-diffusion theory at length scales comparable to the mean free paths of phonons, along with a first-principles interpretation of the experimental observables. The combination of these approaches allows us to place tight constraints on whether thermal phonons with wavelengths on the angstrom scale preserve their phase as they reflect from an atomically rough surface.

Our results show that phonons are exquisitely sensitive to roughness on the scale of 10 Å but can indeed specularly reflect from surfaces with sufficiently small roughness on the scale of 2.5 Å, providing key knowledge required for the coherent manipulation of heat.

Our findings will drive novel applications such as the synthesis of artificial, heterogeneous materials with exceptional thermal transport properties that do not occur naturally.

Figure 1

(a) Schematic of the TG experiment. Two pump laser pulses (green) are crossed at the sample to form a sinusoidal temperature grating. A probe laser beam (bright orange) diffracts from the temperature grating and measures its transient decay. To improve the signal-to-noise ratio, the diffracted probe (signal) beam is optically heterodyned with a reference local oscillator beam (dull orange) and directed to a photodetector. (b) Excited thermal phonons reflect from the rough membrane boundaries either specularly or diffusely, depending on their wavelength.

Figure 2

(a) Representative transient signals from TG on thin silicon membranes at different grating periods and the corresponding biexponential fits (dashed black lines). (b) Measured thermal conductivity of membrane $M1$ versus the grating period at 350 K obtained by fitting the data in (a). The thermal conductivity lies close to the specular limit.

Figure 3

(a) Candidate specularity parameters versus the phonon wavelength. (b) Thermal conductivity versus grating period at 90 K from the experiment (symbols) and the profiles in (a) (lines). (c) Thermal conductivity versus temperature for a grating period of $30\mu \mathrm{m}$ from the experiment (symbols) and the profiles in (a) (lines). Only the increasing profile (red solid curve) matches all of the data.

Figure 4

Measured thermal conductivity of membrane $M1$ at different grating periods for (a) 80, (b) 175, (c) 250, and (d) 400 K, compared with specular and diffuse limits. The gray region represents the thermal conductivity corresponding to the specularity parameter in (e). (e) Specularity parameter versus phonon wavelength obtained from Bayesian inference. The gray region is an intensity plot of the posterior probability distribution for the specularity parameter, with the dashed lines indicating a 95% credible interval. Phonons with wavelengths less than 20 Å are nearly entirely diffusely reflected, while phonons with wavelengths longer than 60 Å are reflected nearly completely specularly.

Figure 5

Cross-sectional TEM images and AFM scans of the surface of membranes $M1$ [(a) and (b), respectively] and $M2$ [(c) and (d), respectively]. The rms roughness on the crystalline silicon membrane surface is estimated to be $2.5\pm 0.5\AA$ for membrane $M1$ and $7\pm 0.5\AA$ for membrane $M2$. Scale bar, 20 Å. The AFM measurement on both membranes yielded a rms roughness of $8\pm 2\AA$. Additional TEM images on different locations of $M1$ and $M2$ are provided in Supplemental Material, Sec. I [49].

Figure 6

Measured thermal conductivity of membrane $M2$ versus the grating period at various temperatures along with calculated specular and diffuse limits. The thermal conductivity lies nearly on the diffuse limit, indicating primarily diffuse reflections at the surface shown in Figs. 5 and 5.

Figure 7

Calculated thermal-conductivity accumulation versus phonon wavelength for membranes with thicknesses of 1145 nm [(a),(b)] corresponding to $M1$ and 515 nm [(c),(d)] corresponding to $M2$ at 80 and 350 K. The solid (dashed) black curve represents the specular (diffuse) scattering limit, and the gray region is calculated using the measured specularity profile in Fig. 4. Partially specular reflections as in Fig. 4 enhance the thermal conductivity of $M1$ by 40%–500% at 80 K and 15%–50% at 350 K compared to the diffuse limit. Since $M2$ is thinner, a specularity profile as in Fig. 4 could enhance the thermal conductivity by 100%–600% at 80 K and 20%–60% at 350 K compared to the observed diffuse limit.