Rayleigh waves, surface disorder, and phonon localization in nanostructures

We introduce a technique to calculate thermal conductivity in disordered nanostructures: a finite-difference time-domain solution of the elastic-wave equation combined with the Green-Kubo formula. The technique captures phonon wave behavior and scales well to nanostructures that are too large or too surface disordered to simulate with many other techniques. We investigate the role of Rayleigh waves and surface disorder on thermal transport by studying graphenelike nanoribbons with free edges (allowing Rayleigh waves) and fixed edges (prohibiting Rayleigh waves). We find that free edges result in a significantly lower thermal conductivity than fixed ones. Free edges both introduce Rayleigh waves and cause all low-frequency modes (bulk and surface) to become more localized. Increasing surface disorder on free edges draws energy away from the center of the ribbon and toward the disordered edges, where it gets trapped in localized surface modes. These effects are not seen in ribbons with fixed boundary conditions and illustrate the importance of phonon-surface modes in nanostructures.

Figure 1

Illustration of a Rayleigh wave propagating along the free top surface. The wave amplitude decays exponentially with increasing distance from the surface.

Figure 2

Phonon dispersions and thermal transport in graphene and in our graphenelike model system simulated via FDTD. All temperature-dependent plots are at 300 K. (a1),(a2) The contribution to thermal conductivity $\kappa$ from phonons with wave numbers around $k$ (a1) or frequencies around $f$ (a2). Thermal conductivity is calculated from the RTA and standard umklapp and isotope scattering rates, with scattering lifetime having an upper bound due to the finite size of the sample. The maximum contributions for transverse acoustic (TA) and longitudinal acoustic (LA) phonons occur at wave numbers 3.37 and $5.28 {\text{nm}}^{-1}$ (a1) (wavelengths of 1.86 and 1.19 nm) and frequencies 9.33 and 11.0 THz (a2), respectively. Dashed black-blue and black-orange lines in all panels indicate the phonon wave number or frequency that contribute the most to thermal conductivity from TA and LA modes, respectively. (b1)–(b4) Contour plots of the phonon dispersion relations for graphene TA (b1), graphene LA (b2) [55], FDTD transverse (b3), and FDTD longitudinal (b4) modes [Eq. (11)]. Dashed black-blue and black-orange circles are the constant-wavenumber curves for the peak wave numbers in (a1). (c1) The density of states (DOS) for TA (blue) and LA (orange) modes in graphene, calculated based on full phonon dispersions [55]. (c2) The number of phonons per unit frequency for graphene, calculated based on the full-dispersion DOS from (c1) and Bose-Einstein statistics. A large fraction of the graphene phonon population falls in the continuum limit. (d1) The DOS for transverse (blue dashed) and longitudinal (orange dashed) modes in our graphenelike model system. (d2) The number of phonons per unit frequency in our graphenelike model system based on the DOS from (d1). The system is classical, so the phonon number per unit frequency is calculated with Maxwell-Boltzmann statistics and tends to zero at low frequencies. However, a large fraction of the phonon population is still in the continuum limit, and the population is sizable at the frequencies of maximum contribution to thermal conductivity.

Figure 3

The staggered grid used for the FDTD solution to the elastic (top) and scalar (bottom) wave equations. $(i,j)$ enumerate the grid cells along the $x$ and $y$ directions. The symbols (squares, triangles, and circles) show where on the grid the different components of $\mathbf{v},\sigma$, and $\tau$ are defined.

Figure 4

Representative heat current autocorrelation function versus time. The curve is similar to those obtained for equilibrium molecular-dynamics simulations [33].

Figure 5

(Left) Visualization of elastic-wave mode conversion at a smooth surface. Color represents the spatial profile of the energy density (arbitrary units: red, high; blue, low). A longitudinal wave packet is incident on a free, smooth top surface at ${60}^{\circ }$ from the surface normal. One longitudinal and one transverse wave packet are reflected. The transverse wave can be identified by its shorter wavelength and slower group velocity. The plot to the right shows the relative energy in the scattered longitudinal (L) and transverse (T) wave packets as a function of the angle of incidence. The material parameters have a strong effect on the angular dependence seen in the plot.

Figure 6

Snapshot of a Rayleigh wave that scattered from a rough surface. Color represents the energy-density profile (log scale, arbitrary units: red, high; blue, low). A Rayleigh wave packet was launched from left to right, moving first along a smooth bottom surface (left) and then along a rough bottom surface (right). Once the packet reaches the roughness (snapshot was taken shortly thereafter), it starts radiating energy into bulk modes. However, the conversion from Rayleigh into bulk modes is relatively weak. The energy leaves the packet slowly, and the packet continues to the right with little distortion.

Figure 7

Snapshots of bulk elastic waves scattered from rough surfaces (top of each panel). Color represents the energy-density profile (log scale, arbitrary units: red, high; blue, low). A longitudinal wave packet was incident on a surface that is nearly smooth and free (top left panel), very rough and free (top right), nearly smooth and fixed (bottom left), and very rough and fixed (bottom right). Surface Rayleigh modes are visible only for the free surfaces (i.e., energy is localized near the free surfaces), because fixed surfaces do not support surface modes. (Inset to top left panel) Magnification of the energy-density profile for a free, nearly smooth surface (region inside the dashed box on the main panel) reveals a wave packet localized near the surface: a Rayleigh mode. (Compare with the Rayleigh wave packet in Fig. 6.) The relative size of the rms roughness $\mathrm{\Delta }$ and packet wavelength $\lambda$ are $\mathrm{\Delta }/\lambda =0.025$ for the left two figures, $\mathrm{\Delta }/\lambda =0.25$ for the right two figures. $\xi /\lambda =0.5$ for all figures ($\xi$ is the correlation length).

Figure 8

The participation ratio versus frequency for all modes of a $70h$-wide graphenelike nanoribbon with $\mathrm{\Delta }=2h,\xi =5h$, and free (top) and fixed (bottom) BCs. Free BC results in significant localization at low frequencies, where we expect to find Rayleigh waves. Traveling waves in a smooth ribbon will have a participation ratio of $2/3$. Indeed, all the calculated participation ratios are below $2/3$ except for one spurious point. $4/9$ is the smallest participation ratio possible in a smooth ribbon. Almost all the low-frequency modes in the fixed ribbon have participation ratios above $4/9$, while almost all the low-frequency modes in the free ribbon have participation ratios below $4/9$. This clearly shows how free surfaces can increase the localization of all low-frequency modes; nearly all surface Rayleigh modes are in this range.

Figure 9

Illustration of surface modes calculated with VEA for a $50h$-wide ribbon with $\mathrm{\Delta }=3h$ and $\xi =5h$. Color is the relative amplitude of the velocity field (proportional to the square root of kinetic energy) throughout the ribbon (color bar to the right). (Left) A wag mode (participation ratio $p=0.089)$: The energy is concentrated in a protuberance that “wags” side to side, as illustrated by (center) a close-up of the wag mode with a superimposed snapshot of the velocity field. (Right) A mode that is less localized than the wag mode but still concentrates energy at the surface.

Figure 10

(Top) Thermal conductivity $\kappa$ of a nanoribbon as a function of rms roughness $\mathrm{\Delta }$ (given in the units of $h$, the grid-cell size), as obtained based on the FDTD solution to elastic (solid lines) and scalar (dashed lines) wave equations with free (black circles) and fixed (red squares) boundary conditions. The correlation length $\xi$ is fixed at $9h$ and the nanoribbon width is $100h$ (grid-cell size is $h=0.246\mathrm{nm}$). (Bottom) Ratio of thermal conductivities resulting from free and fixed boundary conditions with the elastic-wave equation. The thermal conductivities appear to converge in the limits of no surface disorder and very high surface disorder.

Figure 11

Thermal conductivity $\kappa$ of a $100h$-wide wire (grid-cell size $h=0.246\mathrm{nm}$) versus wire length. The correlation length is $9h$, and the rms roughness is $3h$. There are length effects at small lengths, but the length effects are mostly gone by a length of $300h$. Most importantly, $\kappa$ does not appear to diverge like $\kappa \propto L^{\alpha }$, where $L$ is the system length and $\alpha$ is a nonzero, system-dependent constant.

Figure 12

Spatial distributions of the energy density across the ribbon, normalized with respect to the average energy density $\overline{u}$. The position is given in the units of $W$, the ribbon width. Because of roughness, the distance of the edge from the nanoribbon axis varies; stars denote the minimal while squares denote the maximal distance of the edge from the axis; the star-to-star region can be considered the nanoribbon bulk. (Top) The energy densities for identical very rough ribbons ($\mathrm{\Delta }/W=0.05$, with $\xi /W=0.09$) with free BCs (supporting Rayleigh waves) and fixed BCs (Rayleigh waves absent) versus position with respect to the ribbon axis. The free surface siphons energy while the fixed surface has little effect on the energy profile. (Bottom) Difference in the energy densities between free and fixed BCs for very rough (purple, $\mathrm{\Delta }/W=0.05$) and slightly rough (orange, $\mathrm{\Delta }/W=0.02$) ribbons; $\xi /W=0.09$ for both. The free-minus-fixed energy-density profile (i.e., the difference between the curves from the top panel) represents the energy redistribution that stems from the presence of Rayleigh surface waves. It is notable that the energy gets moved away from the bulk and into the Rayleigh surface waves, an effect that becomes more pronounced with increasing roughness.