Recently, black phosphorus and other materials with biaxial symmetry, namely, the orthorhombic, monoclinic, or triclinic symmetries, have attracted significant attentions [1, 2].&nbsp; On fundamental side, the thermal conductivity tensor of these materials has three distinct diagonal elements (principal components), referred to its principal axes.&nbsp; On application side, black phosphorus has been widely seen as a promising candidate for electronic, optoelectronic, and thermoelectric applications, due to its unique properties , including high hole mobility, tunable direct bandgap, and its opposite anisotropy in thermal and electrical conductivities.Though there are existing detailed first principle calculations [3, 4] on the anisotropic thermal properties of black phosphorus, a general Debye-type model with compact algebraic expressions, at least in limiting cases, is still lack.&nbsp; This general model should be easily applied to other materials with biaxial symmetry, and more importantly the simple algebraic expressions should offer more physical insights in a more straightforward manner.Building upon our anisotropic Debye model for materials with uniaxial symmetry [5, 6], namely, the tetragonal, trigonal, or hexagonal symmetries, we develop a general framework to model heat transfer in materials with biaxial symmetry.&nbsp; Here we generalize both the first Brillouin zone (FBZ) and the isoenergy surfaces from standard ellipsoids, in which only one of the three principal directions is anisotropic, to index ellipsoids, , in which all three principal directions are anisotropic.&nbsp; We obtain compact algebraic expressions in limiting temperature regimes for specific heat (C), phonon irradiation (Hs) and thermal conductivity (ks) along different principal direction, s.&nbsp; The temperature dependent specific heat shows power law transitions from T3 – T2 – T1 – T0, which indicates dimensionality crossovers from 3D – 2D – 1D and finally recovers the Dulong and Petit limit.&nbsp; These dimensionality crossovers originate from the anisotropy of the phonon velocities along the principal axes and the FBZ truncation effect.&nbsp; The analytical expressions of Hs, and ks explicitly re-iterate the phonon focusing effect: in many cases the heat transfer along one specific principal direction can be increased by reducing a phonon velocity component in either of the other two principal directions.&nbsp; This model is checked by comparison with the experimental results of C of black phosphorus, thermal boundary conductance (Gs) between aluminum and black phosphorus, and ks of black phosphorus oriented in every principal direction.&nbsp;References:[1] S. Lee, F. Yang, J. Suh, S. Yang, Y. Lee, G. Li, H. Sung Choe, A. Suslu, Y. Chen, C. Ko, J. Park, K. Liu, J. Li, K. Hippalgaonkar, J. J. Urban, S. Tongay and J. Wu, Nature Communications 6 (1), 8573 (2015).[2] Z. Luo, J. Maassen, Y. Deng, Y. Du, R. P. Garrelts, M. S. Lundstrom, P. D. Ye and X. Xu, Nature Communications 6 (1), 8572 (2015).[3] L. Zhu, G. Zhang and B. Li, Physical Review B 90 (21), 214302 (2014).[4]&nbsp; M. Li, J. S. Kang, H. D. Nguyen, H. Wu, T. Aoki and Y. Hu, Advanced Materials 31 (33), 1901021 (2019).[5]&nbsp;Z. Chen, Z. Wei, Y. Chen and C. Dames, Physical Review B 87 (12) (2013).[6]&nbsp;Z. Chen and C. Dames, Applied Physics Letters 107 (19), 193104 (2015).

Modeling Anisotropic Thermal Transport in Black-Phosphorus-Like Materials

Paper Proceedings: https://asmedigitalcollection.asme.org/IMECE/IMECE2021/volume/85543

ASME 2021 International Mechanical Engineering Congress and Exposition (IMECE2021)

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