University of Pennsylvania ScholarlyCommons Department of Mechanical Engineering & Departmental Papers (MEAM) Applied Mechanics March 2008 Acoustic phonon scattering from particles embedded in an anisotropic medium: A molecular dynamics study Neil Zuckerman University of Pennsylvania Jennifer R. Lukes University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/meam_papers Recommended Citation Zuckerman, Neil and Lukes, Jennifer R., "Acoustic phonon scattering from particles embedded in an anisotropic medium: A molecular dynamics study" (2008). Departmental Papers (MEAM). 145. https://repository.upenn.edu/meam_papers/145 Reprinted from Physical Review B, Volume 77, Article 094302, March 2008, 20 pages. Publisher URL: http://dx.doi.org/10.1103/PhysRevB.77.094302 This paper is posted at ScholarlyCommons. https://repository.upenn.edu/meam_papers/145 For more information, please contact [email protected]. Acoustic phonon scattering from particles embedded in an anisotropic medium: A molecular dynamics study Abstract Acoustic phonon scattering from isolated nanometer-scale impurity particles embedded in anisotropic media is investigated using molecular dynamics simulation. The spectral-directional dependence of the scattering, for both longitudinal and transverse modes, is found through calculation of scattering cross sections and three dimensional scattering phase functions for inclusions of varying sizes, shapes, and stiffnesses and for waves of different wave numbers. The technique enables direct observation of the effects of mode conversion, lattice mismatch strain, elastic anisotropy, and atomistic granularity on acoustic phonon scattering from nanoparticles. The results will be useful for the design of nanoparticle- based thermal insulating materials, for example, quantum dot superlattices for thermoelectric energy conversion. Keywords acousto-optical effects, molecular dynamics method, nanoparticles, phonons Comments Reprinted from Physical Review B, Volume 77, Article 094302, March 2008, 20 pages. Publisher URL: http://dx.doi.org/10.1103/PhysRevB.77.094302 This journal article is available at ScholarlyCommons: https://repository.upenn.edu/meam_papers/145 PHYSICAL REVIEW B 77, 094302 ͑2008͒ Acoustic phonon scattering from particles embedded in an anisotropic medium: A molecular dynamics study Neil Zuckerman and Jennifer R. Lukes Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA ͑Received 24 August 2007; revised manuscript received 6 November 2007; published 5 March 2008͒ Acoustic phonon scattering from isolated nanometer-scale impurity particles embedded in anisotropic media is investigated using molecular dynamics simulation. The spectral-directional dependence of the scattering, for both longitudinal and transverse modes, is found through calculation of scattering cross sections and three- dimensional scattering phase functions for inclusions of varying sizes, shapes, and stiffnesses and for waves of different wave numbers. The technique enables direct observation of the effects of mode conversion, lattice mismatch strain, elastic anisotropy, and atomistic granularity on acoustic phonon scattering from nanoparticles. The results will be useful for the design of nanoparticle-based thermal insulating materials, for example, quantum dot superlattices for thermoelectric energy conversion. DOI: 10.1103/PhysRevB.77.094302 PACS number͑s͒: 63.22.Ϫm, 66.70.Ϫf, 63.20.kp, 46.40.Cd I. INTRODUCTION on thermal conductivity.1,17 Recent experiments indicate that including particles of different sizes may be more effective at Nanostructured thermoelectric materials based on super- 1,2 3–5 reducing conductivity by increasing scattering for a wide lattice arrays of thin films, nanowires, and quantum 1 dots5–8 have attracted increasing research attention in recent spectral range of phonons. Also, for a given mass fraction of years because of their potential to enable dramatic efficiency particles, a larger quantity of smaller particles may yield k .19 increases9–11 in solid-state cooling, heating, and direct energy lower c conversion applications as compared to conventional ther- The scattering of phonons during these events is not spa- moelectric materials. Increased efficiency is related to in- tially uniform but instead occurs preferentially in certain di- creased thermoelectric figure of merit ZT,12 rections. The exact spatial distribution depends on many variables, including phonon wavelength, particle size, and S2 ␴ T particle arrangement. The ability to quantify the spectral- ZT = TE e TE , ͑1͒ k directional nature of scattering from embedded nanoparticles c for different phonon polarizations is critically important for ␴ where STE is the Seebeck coefficient, e is the electrical con- understanding thermal transport in quantum dot-based mate- ductivity, TTE is the temperature, and kc is the thermal con- rials. With this knowledge, thermoelectric materials can be ductivity. The goal of thermoelectric materials research has engineered to minimize forward scattering, and thus mini- thus been to increase ZT by engineering materials with high mize thermal conductivity and maximize ZT, over the entire ␴ STE, high e, and low kc. spectral range of phonons present in the material. The inves- Quantum dot superlattices, which are regular arrays of tigation described herein uses a molecular dynamics ͑MD͒ semiconductor nanoparticles embedded in a semiconducting simulation to quantify this scattering in anisotropic host ma- host material, are potentially the most promising materials terials and to examine intermediate spectral ranges well re- for ZT enhancement.6,13,14 These materials have recently moved from the analytically straightforward limiting cases of demonstrated ZT values higher than 1, and this enhancement small and large wavelengths. Section II summarizes the key 6,15,16 arises primarily from the reduction of kc. The funda- background information on previous work on acoustic scat- mental reasons for thermal conductivity reduction in quan- tering and the importance of crystal anisotropy on phonon tum dot superlattices are still not fully understood,17 but pho- scattering. Sections III and VII discuss the MD modeling non scattering from the embedded particles, changes in the approach developed for the present calculations and the ad- phonon dispersion relation due to nanostructuring, and quan- vantages and limitations of the technique. Section IV covers tum confinement are all likely mechanisms.18 Phonons are the method applied to resolve mode polarizations from the quantized traveling elastic waves associated with the dis- simulation data. Section V presents the methodology used to placement of atoms from their equilibrium lattice positions, calculate scattering phase functions and cross sections, Sec. and their scattering from “obstacles” in their paths governs VI discusses the results, and Sec. VIII summarizes the con- the thermal conductivity of nonmetallic solids. These ob- clusions. stacles include other phonons, grain boundaries, impurity at- oms, structural defects such as vacancies and dislocations, II. BACKGROUND changes in atomic mass ͑isotopes͒, and, in general, any fea- tures that change the bond stiffness, bond orientation, or The scattering of phonons due to impurities in crystals is mass of adjacent atoms from those of the host lattice. The a fundamentally three-dimensional problem due to the aniso- scattering of phonons from particles is strongly affected by tropic nature of elastic waves in crystals. For a given energy, the ratio of particle size to phonon wavelength and by par- changes in phonon wave vector relative to the crystal lattice ticle arrangement, both of which have a significant influence orientation will result in change in speed of phonon propa- 1098-0121/2008/77͑9͒/094302͑20͒ 094302-1 ©2008 The American Physical Society NEIL ZUCKERMAN AND JENNIFER R. LUKES PHYSICAL REVIEW B 77, 094302 ͑2008͒ kz in the geometric regime. Prasher also explained via scatter- ing equations how wave-interference effects, dependent scat- tering, and multiple scattering events yield anisotropic phase functions, which can make large changes in the rate of pho- vg1 non transport. These publications focused on the ultimate change in phonon propagation rate due to scattering events, k1 described as a change in intensity, and considered calcula- tions for conversion between two distinct modes in an iso- tropic solid.22 Well-established phonon transport models ͑ ͒ ky e.g., Ref. 25 also generally assume elastic isotropy and iso- k ≠ k tropic scattering. Existing mesoscale approaches use a cross 1 2 v v section based on phonon wavelength and particle size, but g1= g2 k2 the model equations in their final form do not track the de- v 26 g2 tails of the directional nature of scattering. Other calcula- tions use a more complex anisotropic phase function but do not include the details of mode conversion or the effects of contour of constant ω material anisotropy in order to reduce the complexity of the calculations.27 Such calculations may use directionally de- pendent dispersion curves for phonons, but the phase func- FIG. 1. ͑Color online͒ Slowness surface for quasitransverse tions do not explicitly account for the relative alignment of mode in the ͑100͒ plane, plotted in k space with various wave the three-component group velocity vector and the crystal vectors and group velocity vectors, based on linear