Abstract
The concept of a phononic crystal can in principle be realized at the nanoscale whenever the conditions for coherent phonon transport exist. Under such conditions, the dispersion characteristics of both the constitutive material lattice (defined by a primitive cell) and the phononic crystal lattice (defined by a supercell) contribute to the value of the thermal conductivity. It is therefore necessary in this emerging class of phononic materials to treat the lattice dynamics at both periodicity levels. Here we demonstrate the utility of using supercell lattice dynamics to investigate the thermal transport behavior of three-dimensional nanoscale phononic crystals formed from silicon and cubic voids of vacuum. The periodicity of the voids follows a simple cubic arrangement with a lattice constant that is around an order of magnitude larger than that of the bulk crystalline silicon primitive cell. We consider an atomic-scale supercell which incorporates all the details of the silicon atomic locations and the void geometry. For this supercell, we compute the phonon band structure and subsequently predict the thermal conductivity following the Callaway-Holland model. Our findings dictate that for an analysis based on supercell lattice dynamics to be representative of the properties of the underlying lattice model, a minimum supercell size is needed along with a minimum wave vector sampling resolution. Below these minimum values, a thermal conductivity prediction of a bulk material based on a supercell will not adequately recover the value obtained based on a primitive cell. Furthermore, our results show that for the relatively small voids and void spacings we consider (where boundary scattering is dominant), dispersion at the phononic crystal unit cell level plays a noticeable role in determining the thermal conductivity.
Highlights
This study focuses on silicon-based nanoscale phononic crystals, to understand the underlying physics of wave dispersion and scattering we must first consider the most basic lattice forming the material’s atomic structure, that is, the lattice described by a primitive cell
For our lattice dynamics and subsequent thermal conductivity calculations we focus on the -X wavevector path bordering the edge of the irreducible face-centered cubic (FCC) Brillouin zone (Fig. 1(d))
For our bulk silicon model, we observe that a minimum supercell size of Nmin = 6 and a minimum κ-space sampling resolution of nκ,min = 1024 steps yield a prediction that is within 1% of the nominal value of 142 W/mK
Summary
Influence atomic-scale phonons is opening up a new direction in nanoscale heat transfer and material design.[17,18,19,20,21,22,23,24,25,26,27,28] For example, among the configurations considered is the insertion of periodic holes in silicon slabs.[24,25,26,28] Should the conditions of coherent transport exist in such configurations, phonon waves can in principal linearly interfere while simultaneously experience nonlinear scattering events (see Ref. 21 for an analysis on the interplay between the effects of dispersion and anharmonic scattering in a one-dimensional nanoscale phononic crystal). 24, 26) may exhibit a reduction in thermal conductivity without a significant impact on the scattering of electrons – a favorable outcome for increasing the figure of merit of thermoelectric materials.[29,30] It is conceivable that a phononic crystal with a relatively large lattice spacing (as considered in Refs. 24, 26) may exhibit a reduction in thermal conductivity without a significant impact on the scattering of electrons – a favorable outcome for increasing the figure of merit of thermoelectric materials.[29,30]
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