Phonon Interference and Thermal Conductance Reduction in Atomic-Scale Metamaterials
We introduce and model a three-dimensional (3D) atomic-scale phononic metamaterial producing two-path phonon interference antiresonances to control the heat flux spectrum. We show that a crystal plane partially embedded with defect-atom arrays can completely reflect phonons at the frequency prescribed by masses and interaction forces. We emphasize the predominant role of the second phonon path and destructive interference in the origin of the total phonon reflection and thermal conductance reduction in comparison with the Fano-resonance concept. The random defect distribution in the plane and the anharmonicity of atom bonds do not deteriorate the antiresonance. The width of antiresonance dip can provide a measure of the coherence length of the phonon wave packet. All our conclusions are confirmed both by analytical studies of the equivalent quasi-1D lattice models and by numerical molecular dynamics (MD) simulations of realistic 3D lattices.
Two-photon interference can result in a total cancellation of the photon output because of the coalescence of the two single photons, which was first observed by Hong et al.(1). This interference effect occurs because two possible photon paths interfere destructively, which produces the famous “Hong-Ou-Mandel (HOM) dip” in the detection probability of the output photons. The HOM dip has since then been demonstrated in both the optical(2); (3) and microwave(4) regime. In particular, two-photon destructive interference was recently demonstrated in a 3D optical metamaterial(5). Similar destructive interference effect which results in a total reflection can be also realized in a phonon system. For the sound waves, the two-path phonon interference antiresonance was first described in Ref.[(6)], where the anomalous zero-transmission and total absorption of long-wavelength acoustic waves in a crystal with 2D (planar) defect were related with the between the two possible phonon paths: through the nearest-neighbor bonds and through the non-nearest-neighbor bonds which couple directly crystal layers adjacent to the defect atomic plane.
Constant endeavour has been devoted to the precisely control of heat conduction. Recent efforts concentrated on reducing the thermal conductivity via nanostructured materials with superlattices(7); (8) and embedded nanoparticles(9); (10); (11). Most works attributed the reduction of to the decreased phonon life time and thus the mean free path (MFP), which belong to the particle description of heat conduction. However, the role of destructive phonon interference in the reduction of is much less understood in the wave picture of thermal transport.
In this Letter we introduce and model a realistic 3D atomic-scale phononic metamaterial that allows for manipulating the flow of thermal energy. Two-path phonon interference is generated by exploiting the phonon reflection on internal interfaces embedded with defect-atom arrays. The 2D planar defects force phonons to propagate through two paths: through unperturbed (matrix) and perturbed (defect) interatomic bonds(6); (12). The resulting phonon interference yields transmission antiresonance (zero-transmission dip) in the spectrum of short-wavelength phonons that can be controlled by the masses, force constants and 2D concentration of the defect atoms. Our results show that the patterning of the defect-atom arrays can lead to a new departure in thermal energy management(13), offering potential applications in thermal filters(14), thermal diodes(15) and thermal cloaking(18); (16); (17).
An atomic presentation of the 3D phononic metamaterial with a face-centered cubic (FCC) lattice including a 2D array of heavy defect atoms is depicted in Fig. 1(a). The defect-atom arrays are distributed periodically or randomly in the defect crystal plane with different filling fractions . When the defects do not fill entirely the defect plane, phonons have two paths to cross such an atom array as shown in Fig. 1(a), whereas the phonon path through the host atoms is blocked when the defect layer is constituted by a uniform impurity-atom array, packed with impurity atoms. Two types of atomic metamaterials were studied using realistic interatomic potentials: a FCC lattice of Argon (Ar) where the defects are heavy isotopes and a diamond lattice of Si with Germanium (Ge) atoms as the defects. The interactions between Ar atoms are described by the Lennard-Jones potential(19). The covalent Si:Si/Ge:Ge/Si:Ge interactions are modeled by the Stillinger-Weber (SW) potential(20). To probe the phonon transmission, MD-based phonon wave packet (WP) method(21) was used to provide the per-phonon-mode energy transmission coefficient. The spatial width (coherence length) of the WP is taken much larger than the wavelength of the WP central frequency, corresponding to the plane-wave approximation. All the MD simulations were performed with the LAMMPS code package(23).
The transmission coefficient of the WP with , retrieved from MD simulations of an Ar metamaterial is presented in Fig. 2. The incident phonons undergo a total reflection on the defect layer at the antiresonance frequency . Phonon transmission spectra displays an interference antiresonance profile since the two phonon paths interfere destructively at , analogous to the two-photon interference which results in the HOM dip(1); (5). A total transmission at follows the interference antiresonance, which is reminiscent of the Fano resonances(24). For a uniform defect-atom array, the zero-transmission antiresonance profile will be totally suppressed and replaced by a monotonous decay of transmission with frequency. In the later case, only the phonon path through the defect atoms is accessible.
We emphasize that the second phonon path is indispensable to the emergence of the zero-transmission dip, which cannot be sufficiently described by the Fano resonance. We clarify this by studying the phonon transmission through two successive uniform defect-array, where a local resonant minimum is observed instead of a zero-transmission dip in Fig. 2. This minimum satisfies well the Fano-resonance condition(24) of a discrete state resonating with its continuum background, but no zero-transmission dip occurs because of the absence of the second phonon path(22); (6). This minimum can be considered as phonon analogy of the Fabry-Pérot resonance in optics, which requires only a single phonon path. Therefore this clearly corroborates the two-path destructive interference nature of the zero-transmission dip in .
To further understand the phonon antiresonances caused by the interference between two phonon channels, we use an equivalent model of monatomic quasi-1D lattice of coupled harmonic oscillators(12), depicted in the inset in Fig. 2. In model (a), phonons propagate through two paths: through the host atom bonds, and through those of the impurity atoms, whereas in model (b) and (c), only the second channel remains open. The model (a) gives the energy transmission coefficient for plane wave:
where are the frequencies of the reflection and transmission resonances, is the maximal phonon frequency for a given polarization, . is a real positive coefficient given by the atomic masses, force constants and , for . The frequency exists only in the presence of an additional channel which is open for wave propagation through the bypath around the defect atom, see inset (a) in Fig. 2. As follows from Eq. (1) and Fig. 2, and . In the transmission of a narrow WP with , given by the convolution of for plane wave from Eq. (1) with a gaussian WP in frequency domain with , the interference effect is weakened by more frequency components when the plane-wave approximation () is broken and the transmission at is not zero any more, i.e. , which is the case also in . As depicted in Fig. 2, an excellent agreement in transmission coefficients is demonstrated between the equivalent quasi-1D model provided by Eq. (1) and the MD simulations of the 3D atomic-scale phononic metamaterial with the use of realistic interatomic potential.
In a lattice with atomic impurities, the substituent atoms scatter phonons due to differences in mass and/or bond stiffness. Since no bond defect was introduced, the loci of the resonances are only determined by the mass of the isotope atoms. As the defect atoms get heavier, the two-path phonon interference antiresonance becomes more pronounced in terms of phonon attenuation depth and width, and demonstrates a red-shift in the phonon transmission spectrum thus impeding the long-wavelength phonons, as shown in Fig. 3(a). The equivalent quasi-1D lattice model gives the following expression for the frequency of the transmission dip:
where and refer to the atomic mass of the defect and host atom, . The transmission resonance at is much less sensitive to the defect mass since it is largely determined by the mass of the host atom. As depicted in Fig. 3(b), the spectral positions of the interference resonances are again in an excellent agreement with the analytical prediction of the equivalent quasi-1D lattice model given by Eq. (2).
In Fig. 3(a), the transmission spectra for longitudinal phonons across the uniform defect-atom array is plotted to be compared with that of the -filled defect-atom array. At the two-path interference antiresonance frequency , an array of defect atoms has a transmittance two orders of magnitude smaller than that of a uniform defect-atom array. The difference between the very strong phonon reflection on a -filled defect array and the high phonon transmission across a uniform defect array can result in a counter-intuitive effect: an array of segregated impurity atoms can scatter more thermal phonons than an array with a uniform distribution of heavy isotopes. This anomalous phonon transmission phenomenon in molecular systems can find its acoustic conterpart in macroscopic structures(12); (25); (27). In Ref.[(25)], perforated plates were proved to shield ultrasonic acoustic waves in water much more effectively than uniform plates. Liu et al.(27) managed to break the mass-density law for sonic transmission by embedding high-density spheres coated with a soft material in a single layer of a stiff matrix.
We calculate the interfacial thermal conductance by following the Landauer-like formalism(26):
where is phonon group velocity in the cross-plane direction, is the Bose-Einstein distribution of phonons at temperature . The integral is carried out over the whole Brillouin zone and the sum is over the phonon branches. By embedding defect atoms in a monolayer, we manage to reduce the thermal conductance by in respect to the case of no defects, as shown in Fig 4(a). This destructive-interference-induced effect can be used to explain the remarkable decrease of the thermal conductivity of SiGe alloy with very small amount of Ge, with respect to pristine Si(28). can be further reduced by considering the second-nearest-neighbor bonds between the host atoms on the two sides of the uniform defect layer in addition to the nearest-neighbor bonds linking the host and adjacent defect atoms, see also Ref.[(6)]. This reduction comes from the suppression of phonon transmission at high frequencies, shown in Fig. 4(b), which is due to the opening of the second phonon path through the host atom bonds interfering destructively with the first path through the nearest-neighbor bonds . The emergence of the second phonon path substantially reduces , by , despite the weakness of the corresponding bond: in Ar lattice(19). This demonstrates another advantage of the application of the two-path destructive phonon interference for the thermal conductivity reduction: more heat is blocked by the opening of the additional phonon paths.
In Fig. 5, we report the two-path phonon antiresonance in Si crystal as the metamaterial incorporating 2D planar impurity array of Ge atoms. Ge and Si atom has a mass ratio of 2.57 and thus the Ge-atom array introduces both heavy mass and bond defects due to a weaker coupling between Ge:Si than the Si:Si interaction(20).
The nonlinear effects on the two-path phonon interference antiresonance was investigated by increasing the amplitude of the incident phonon WP, as shown in Fig. 5(b). As increases, the reflection becomes less pronounced with more heat flux passing through, which provides direct evidence of inelastic phonon scattering at the defect plane. The antiresonances demonstrate a red-shift in frequency due to the higher-order (cubic) terms in the interatomic potiential. We also note in this connection that our computation of a quasi-1D atomic chain containing an impurity atom characterized by non-parabolic (nonlinear) interaction potential with neighboring host atoms agree with our MD results. The interference antiresonance remains pronounced even when the interaction nonlinearity becomes fairly strong. Therefore the two-path phonon interference antiresonance in the proposed phononic metamaterial makes it possible to control thermal energy transport in the case of huge lattice distortions, for instance at very high temperatures.
In contrast to light(29), even a single defect atom in a lattice plane produces interference antiresonance for phonons because of the presence of the two phonon paths. Therefore, phonon reflection should be apparent even for defect-atom arrays in the absence of periodicity because of the localized nature of the resonance. This is supported by further investigating the phonon transmission through the arrays of Ge atoms, distributed in a plane in Si phononic metamaterials with different and randomness. Strong transmission dip, similar to that in periodic arrays, remains pronounced in both cases, as shown in Fig. 5(c). This was shown experimentally to be equally valid in macroscopic acoustic metamaterials(27).
Chen et al. reduced below the alloy limit by distributing random Ge segregates in Si superlattices(11). Their ab initio calculations showed that phonon MFP was substantially reduced in the low frequencies(11). We note that the Ge segregates can be regarded as randomly dispersed heavy resonators which scatter low-frequency thermal phonons by the interference antiresonances, red-shifted according to the isotopic-shift law, cf. Eq. (2). With this destructive interference, we can also explain the extremely low found in the InGaAs alloy randomly embedded with heavy ErAs nanoparticles(8).
The decreased 2D defect filling fraction narrows the antiresonance width because of weakening of the relative strength of the “defect-bond” phonon path through the crystal plane, see Fig. 1 and Eq. (1). In general, the width of the antiresonance dip for our two-path phonon interference is determined by both the and the finite coherence length of the phonon WP. As follows from Fig. 2, for the large , is not sensitive to . In the limit of small and for , is narrow and proportional to , as shown in Fig. 5(c) and 5(d). In this limit, for the WP with short , , will be determined mainly by (1). From Fig. 5(d), the full-width-at-half-minimum of the dip for WP with is THz. Then from the inequality , we get the WP width in the time domain ps and the WP spatial width (coherence length) nm, where km/s is the longitudinal phonon group velocity in Si at . This length coincides with the WP coherence length of 3.2nm, which was used in the MD simulation shown in Fig. 5(d). The width of the antiresonance dip for the WP with a shorter coherence length is larger than that of the WP with , see Fig. 5(d). Therefore the width of the two-path phonon interference dip in the transmission spectrum can provide a measure of the coherence length of the phonon WP, similar to the width of the HOM dip in the two-photon interference(1).
In conclusion, we provide comprehensive modeling of atomic-scale phononic metamaterial
for the control of heat transport
by exploiting two-path phonon interference antiresonances.
Thermal phonons crossing the defect plane with the two paths of different nature
undergo destructive interference, which results in the reduction of thermal conductance.
Interference antiresonances are not deteriorated by the aperiodicity in the defect arrays and by the
anharmonicity of atom bonds.
The width of antiresonance dip provides a measure of the coherence length of the phonon wave packet.
Such patterned atomic planes can be considered as high-finesse atomic-scale phononic mirrors.
And, at last, we would emphasize that strong optical reflections observed in 3D
stereometamaterials(30) can also be interpreted as photon interference
antiresonances in optically transparent plane, embedded with plasmonic nanostructures.
- C.K. Hong, Z.Y. Ou and L. Mandel, Phys. Rev. Lett. 59, 2044 (1987).
- C. Santori, D. Fattal, J. Vuckovic, G. Solomon, and Y. Yamamoto, Nature (London) 419, 594 (2002).
- J. Beugnon, M.P.A. Jones, J. Dingjan, B. Darquie, G. Messin, A. Browaeys, and P. Grangier, Nature (London) 440, 779 (2006).
- C. Lang, C. Eichler, L. Steffen, J. M. Fink, M. J. Woolley, A. Blais, and A. Wallraff, Nat. Phys. 9, 345 (2013).
- S. M. Wang, S. Y. Mu, C. Zhu, Y. X. Gong, P. Xu, H. Liu, T. Li, S. N. Zhu, and X. Zhang, Opt. Express 20, 5213 (2012).
- Yu. A. Kosevich, Prog. Surf. Sci. 55, 1 (1997).
- I. Chowdhury, R. Prasher, K. Lofgreen, G. Chrysler, S. Narasimhan, R. Mahajan, D. Koester, R. Alley, and R. Venkatasubramanian, Nat. Nano. 4, 235 (2009)
- W. Kim, J. Zide, A. Gossard, D. Klenov, S. Stemmer, A. Shakouri, and A. Majumdar, Phys. Rev. Lett. 96, 045901 (2006)
- N. Mingo, D. Hauser, N. P. Kobayashi, M. Plissonnier, and A. Shakouri, Nano Lett. 2, 711 (2009)
- G. Pernot, M. Stoffel, I. Savic, F. Pezzoli, P. Chen, G. Savelli, A. Jacquot, J. Schumann, U. Denker, I. Mönch, Ch. Deneke, O. G. Schmidt, J. M. Rampnoux, S. Wang, M. Plissonnier, A. Rastelli, S. Dilhaire, and N. Mingo, Nat. Mat. 6, 491 (2009)
- P. Chen, N. A. Katcho, J. P. Feser, W. Li, M. Glaser, O. G. Schmidt, D. G. Cahill, N. Mingo, and A. Rastelli, Phys. Rev. Lett. 111, 115901 (2013)
- Yu. A. Kosevich, Physics-Uspekhi 51, 848 (2008).
- M. Maldovan, Phys. Rev. Lett. 110, 025902 (2013).
- L. Zhang, P. Keblinski, J.S. Wang, and B. Li, Phys. Rev. B. 83, 064303 (2011).
- N. Li, J. Ren, L. Wang, G. Zhang, P. Hänggi, and B. Li, Rev. Mod. Phys. 84, 1045 (2012).
- H. Xu, X. Shi, F. Gao, H. Sun, and B. Zhang, Phys. Rev. Lett. 112, 054301 (2014).
- T. Han, X. Bai, D. Gao, J.T.L. Thong, B. Li, and C.W. Qiu, Phys. Rev. Lett. 112, 054302 (2014).
- S. Narayana and Y. Sato, Phys. Rev. Lett. 108, 214303 (2012).
- H. Kaburaki, J Li, S. Yip and H. Kimizuka, J. Appl. Phys. 102, 043514 (2007)
- F. H. Stillinger and T. A. Weber, Phys. Rev. B 31, 5262 (1985).
- P. Schelling, S. Phillpot, and P. Keblinski, Appl. Phys. Lett. 80, 2484 (2002).
- Yu. A. Kosevich and E. S. Syrkin, Sov. Phys. Solid State 33, 1156 (1991).
- S. Plimpton, J. Comput. Phys. 117, 1 (1995).
- U. Fano, Phys. Rev. 124, 1866 (1961).
- H. Estrada, P. Candelas, A. Uris, F. Belmar, F. J. GarcÃa de Abajo, and F. Meseguer, Phys. Rev. Lett. 101, 084302 (2008).
- I. M. Khalatnikov, An introduction to the theory of superfluidity (Benjamin, New York, 1965).
- Z. Liu, X. Zhang, Y. Mao, Y. Y. Zhu, Z. Yang, C. T. Chan, and P. Sheng, Science 289, 1734 (2000).
- J. Garg, N. Bonini, B. Kozinsky, and N. Marzari, Phys. Rev. Lett. 106, 045901 (2011)
- A. Degiron, H. Lezec, N. Yamamoto, and T. Ebbesen, Opt. Commun. 239, 61 (2004).
- N. Liu, H. Liu, S. Zhu, and H. Giessen, Nature Photon. 3, 157 (2009)