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Research Papers

Phonon Scattering in One-Dimensional Anharmonic Crystals and Superlattices: Analytical and Numerical Study

[+] Author and Article Information
Nichlas Z. Swinteck

Postdoctoral Research Associate
e-mail: swinteck@email.arizona.edu

Krishna Muralidharan

Assistant Professor

Pierre A. Deymier

Professor
Department of Materials Science and Engineering,
University of Arizona,
Tucson, AZ 85721

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received April 29, 2012; final manuscript received October 22, 2013; published online June 6, 2013. Assoc. Editor: Michael Leamy.

J. Vib. Acoust 135(4), 041016 (Jun 06, 2013) (13 pages) Paper No: VIB-12-1125; doi: 10.1115/1.4023824 History: Received April 29, 2012; Revised October 22, 2013

Second-order perturbation theory based on multiple time scale analysis is used to illuminate three-phonon scattering processes in the one-dimensional anharmonic monoatomic crystal. Molecular dynamics simulation techniques in conjunction with spectral energy density analyses are used to quantify phonon mode lifetime in (1) the monoatomic crystal and (2) a series of superlattice configurations. It is found that the lifetime of vibrational modes in the monoatomic crystal is inherently long, because the conditions for conservation of wave vector and frequency are pathologically difficult to satisfy. Superlattice configurations, however, offer band-folding effects, whereby the availability of phonon decay channels decreases the lifetime of the vibrational modes supported by the medium.

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Figures

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Fig. 1

(a) Schematic representation of 1D crystal with linear stiffness β and quadratic nonlinearity parameter ε. (b) The potential energy function describing the 1D crystal.

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Fig. 2

(Left) Band structure of 1D harmonic monoatomic crystal. (Right) SED-frequency plot showing wave vector mode k = π/a.

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Fig. 3

(Left) Band structure (constant SED contours) for 1D harmonic monoatomic crystal near k = π/10 a. (Right) SED-frequency plots for four MD simulations differing in their initial random configurations.

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Fig. 4

(Left) Band structure (constant SED contours) for 1D anharmonic monoatomic crystal near k = π/10a. (Right) SED-frequency plots for four MD simulations differing in their initial random configurations.

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Fig. 5

(Left) Band structure for 1D anharmonic monoatomic crystal. (Right) SED-frequency plot showing wave vector mode k = π/a. ε = 3.0 and initial random displacement does not exceed 10% of a.

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Fig. 6

(Top, left) SED-frequency plots for wave vector modes k′ and k″, corresponding to Case I. (Bottom, left) SED-frequency plots for wave vector modes k′ and k″, corresponding to Case II. (Right) SED-frequency plot corresponding to k* = π/a. For Cases I and II, wave vectors k′ and k″ satisfy wave vector conservation for mode k*. The frequencies of modes k′ and k″ add (or subtract) to yield near-resonance peaks near ω*(k*).

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Fig. 7

SED-frequency plots for 1D anharmonic monoatomic crystal at k = π/a for MD systems of varying sizes. The parameter characterizing the degree of anharmonicity in the 1D crystal is ε = 3.0.

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Fig. 8

Frequency-shift evaluated at k = π/a for 1D anharmonic crystal relative to harmonic case. Symbols represent different magnitudes for the maximum initial random displacement imposed upon the masses in the 1D crystal in terms of percentage of the lattice spacing. Circle, square, and triangle symbols represent small, intermediate, and large initial displacements, respectively.

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Fig. 9

(Top) Unit cell for diatomic crystal. (Center) Band structure for 1D anharmonic diatomic crystal. (Left) SED-frequency plot at k = π/20a with peaks for harmonic (dashed line) and anharmonic (solid line) cases. (Right) SED-frequency plot at k = π/a with peaks for harmonic (dashed line) and anharmonic (solid line) cases.

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Fig. 10

(Top) Four-atom unit cell. (Left) Anharmonic band structure corresponding to the four-atom unit cell. (Right) SED-frequency plot at k = π/2a. Dashed line represents the harmonic case, whereas the solid line represents the anharmonic case.

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Fig. 11

(Top) Eight-atom unit cell. (Left) Anharmonic band structure corresponding to the eight-atom unit cell. (Right) SED-frequency plot at k = π/4a. Dashed line represents the harmonic case, whereas the solid line represents the anharmonic case.

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Fig. 12

(Top) Sixteen-atom unit cell. (Left) Anharmonic band structure corresponding to the sixteen-atom unit cell. (Right) SED-frequency plot at k = π/8a. Dashed line represents the harmonic case, whereas the solid line represents the anharmonic case.

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Fig. 13

(a) (Left to right) 1:1, 2:2, 4:4, and 8:8 SED-frequency plot with peaks respectively corresponding to the superlattice configurations depicted in Figs. 9–12. (b) Lorentzian function fits to the SED-frequency spectra in (a). Lorentzian peaks are labeled with half-width at half-maximum values in units of 10–6 Hz.

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