peierls boltzmann equation

We first sketch the derivation of the 2D G-K equation45,46. Commun. The relaxation from local to global equilibrium or steady state is governed by hydrodynamic equations describing the conserved quantities during the N-process, including energy and crystal momentum. The negative sign means heat flows opposite to the temperature gradient. 3) passing the center of the ribbon in y (Fig. Information about registration may be found here. We have checked that, the maximum relative difference, defined as |(qq0)/q|, locates at the upper and lower boundaries. (52), and denotes the center of the nonsharp particle-wave crossover for phonons. Here, we take the \({\tau }_{N}={10}^{-10}\) s. The green dotted line corresponds to the 3D Debye model with group velocity \({v}_{g}=1.6\times {10}^{4}\) m/s, the blue dashed line is the 2D Debye model with the same group velocity, while the red solid line stands for the 2D case with one quadratic ZA mode and two degenerate linear acoustic modes (longitudinal and transverse) with the same vg. Ackerman, C. C., Bertman, B., Fairbank, H. A. Solutions of the Peierls-Boltzmann transport equation using inputs from density functional theory calculations have been successful in predicting the thermal conductivity in a wide range of materials. Bulk thermal conductivity of CsPbBr3 in the smooth phase convention. Phys. [108], Wan etal. [109], Zhang etal. Regime diagram for thermal transport. Use of the American Physical Society websites and journals implies that Lindsay, L.; Broido, D. A.; Reinecke, T. L. Li, Wu; Carrete, Jess; Madsen, Georg K. H. Berman, R.; Hudson, P. R. W.; Martinez, M. Landon, Colin D.; Hadjiconstantinou, Nicolas G. Souvatzis, P.; Eriksson, O.; Katsnelson, M. I. Jo, Insun; Pettes, Michael T.; Lindsay, Lucas, Carrete, Jess; Mingo, Natalio; Curtarolo, Stefano, Giannozzi, Paolo; Baroni, Stefano; Bonini, Nicola, Yates, Jonathan R.; Wang, Xinjie; Vanderbilt, David, Qin, Guangzhao; Yan, Qing-Bo; Qin, Zhenzhen, Fugallo, Giorgia; Cepellotti, Andrea; Paulatto, Lorenzo, Lindsay, L.; Broido, D. A.; Mingo, Natalio, Skelton, Jonathan M.; Parker, Stephen C.; Togo, Atsushi, Garg, Jivtesh; Bonini, Nicola; Kozinsky, Boris, Wu, Xufei; Lee, Jonghoon; Varshney, Vikas, Giannozzi, Paolo; de Gironcoli, Stefano; Pavone, Pasquale, Balandin, Alexander A.; Ghosh, Suchismita; Bao, Wenzhong, Lee, Sangyeop; Broido, David; Esfarjani, Keivan, Debernardi, Alberto; Baroni, Stefano; Molinari, Elisa. Turkyilmazoglu, M. Latitudinally deforming rotating sphere. For linear dispersion, we have \({{\boldsymbol{q}}}_{0L}={v}_{g}^{2}{{\boldsymbol{p}}}_{0L}\). The area of each circle is proportional to the contribution to the total conductivity and colored according to the origin of the contribution: green for particlelike propagation of populations; blue for wavelike tunneling of coherences; intermediate colors represents phonons contributing to both mechanisms, with red corresponding to 50% of each [see Eq. Wang, J.-S., Wang, J. 90, 041002, https://doi.org/10.1103/RevModPhys.90.041002 (2018). (33) reduces to a one-dimension form \({\partial }^{2}q/\partial {y}^{2}=A\). WebThe reduced Boltzmann-Peierls equation ( 1) induces an infinite number of balance equations. Two-dimensional density of states for the thermal conductivity, C,12, which resolves how much a Zener-like coupling between two vibrational modes having frequencies 1, 2 contributes to the coherences conductivity. 025901, Materials Research Letters, Vol. Here non-local thermal conductance/resistance means the temperature difference and the induced heat current (or vice versa) are separated in real space. We get \(\beta \approx 0.08\) at \(T=100\) K, smaller than value obtained from the 2D Debye model \(\beta =\mathrm{1/4}\). is a kinetic equation for the phase density of phonons. It is not necessary to obtain permission to reuse this Adv 4, eaat3374 https://doi.org/10.1126/sciadv.aat3374 (2018). Numerical results of the model applied to a system of phonons and photo-excited electrons in gallium arsenide (GaAs) are compared with data obtained by an ensemble Monte Carlo technique. The region above it represents well-defined phonons; the Wigner transport equation(37) describes all these phonons (blue, red, green), while the semiclassical Peierls-Boltzmann equation accounts only for phonons that propagate particlelike (green). Physical Review Physics Education Research, Log in with individual APS Journal Account , Log in with a username/password provided by your institution , Get access through a U.S. public or high school library . Recently, it has been shown that these two conduction mechanisms emerge from a Wigner transport equation, which unifies and extends the Peierls-Boltzmann and Allen-Feldman formulations, allowing one to describe also complex crystals where particlelike and wavelike conduction mechanisms coexist. We find that, in the presence of quadratic ZA modes, vss increases with temperature and is much smaller than results from the Debye model in the relevant temperature ~100 K. The physical mechanism is the following. We need to use the full form of \({f}_{N}^{eq}\) instead of Eq. 2004 Society for Industrial and Applied Mathematics The unit cell, and top and side views of supercells of (a)graphene, (b)silicene, and (c)-NP. The horizontal black like is the Wigner limit in time, =1/av (i.e., the inverse average interband spacing, see Eq. 33,52. Google Scholar. simplified kinetic equation, The fields e, pi, Qi, and Nij denote the energy article or its components as it is available under the terms of In this work, we discuss the theoretical foundations of a Wigner heat-transport equation (named after the Wigner formulation of quantum mechanics used to derive it) that naturally encompasses the coexistence of particlelike and wavelike conduction mechanisms, unifying and extending the Peierls and Allen-Feldman formulations. Thank you for visiting nature.com. The authors thank Nuo Yang, Jin-Hua Gao for discussions. & Yang, R. Colloquium: Phononic thermal properties of two-dimensional materials. The vertical lines have the same length of the horizontal lines below them; they show that in the excited state |Rb=a^(R)b|0 (where |0 is the ground state), atoms at position R+b have a mean square displacement that becomes smaller as the distance |R+bRb| increases. We illustrate the general method of Chapman-Enskog projection construction and selection of the subclass of correct boundary conditions determining the attractive manifold. Macroscopic collective behavior emerges from microscopic many-body interactions between individual degrees of freedom comprising the system. We give a positive existence result for the maximum entropy principle if the underlying kinetic equation is the Boltzmann-Peierls equation. Lett. Scatter points represent measurements from Suresh etal. As an example, we have plotted typical heat current flow patterns (lines) and the resulting temperature distribution (color) with \(\chi =0.5\) and 2104 in Fig. The quadratic dispersion of graphene ZA acoustic phonon mode is argued to play an important role in widening the temperature range11. (54) and related discussion for details]. The plot in Fig. This is the wave equation describing propagation of second sound with velocity vss and damping coefficient \({\tau }_{R}^{-1}\). 114, 30683073, https://doi.org/10.1073/pnas.1612181114 (2017). Rev. Comparative study of phonon spectrum and thermal expansion of graphene, silicene, germanene, and blue phosphorene. 29, Issue 12, Journal of Physics and Chemistry of Solids, Vol. Article The yellow region below the Ioffe-Regel limit represents overdamped phonons, which require full spectral-function approaches [78, 79] to be described correctly. CAS (37) analytically with the help of the streaming function (see Sec. Due to these limitations, studies on the hydrodynamic transport of quasi-particles in solid state system are scarce. Provided by the Springer Nature SharedIt content-sharing initiative. We take into account the out of plane quadratic phonon dispersion of the ZA mode, normally present in 2D materials. description of our journals and our newly announced SIAM Journals Online, Phys. [37]). obtain the proper permission from the rights holder directly for WebThese data can be used with the Peierls- Boltzmann transport equation to calculate a spectral TC by 2469-9950/2021/103(20)/205421(11) 205421-1 2021 American Physical Society Heat conduction in crystalline semiconductor films occurs by lattice vibrations that result in the propagation of quanta of energy called phonons. The classical PBE is based on the GouyChapman theory VanGessel, Francis; Peng, Jie; Chung, Peter W. Yang, Xiuxian; Dai, Zhenhong; Zhao, Yinchang, Smith, Brandon; Lindsay, Lucas; Kim, Jaehyun, Pandey, Tribhuwan; Polanco, Carlos A.; Lindsay, Lucas, Simoncelli, Michele; Marzari, Nicola; Cepellotti, Andrea, DeAngelis, Freddy; Muraleedharan, Murali Gopal; Moon, Jaeyun, Plata, Jose J.; Nath, Pinku; Usanmaz, Demet, Yuan, Kunpeng; Sun, Zhehao; Zhang, Xiaoliang, - Physical Review. Observation of the Dirac fluid and the breakdown of the Wiedemann-Franz law in graphene. We recall that the populations and coherences conductivity tensors of La2Zr2O7 are proportional to the identity (see Sec7a); thus populations and coherences conductivities are represented here as scalars. Phonon-defect scattering rates are computed from a Green's function methodology that is nonperturbative and includes interatomic force constant variance induced near the defects. 1(c), we have performed fully numerical calculation by solving the Boltzmann equation under Callaway approximation using the discrete ordinate method33,52. The other was to use Non-Equilibrium Molecular Dynamics (NEMD). 54, Issue 2, Physical Review, Vol. Length and width dependent thermal conductivity in suspended single layer graphene has been reported experimentally49,50. Phys. Observation of Poiseuille flow of phonons in black phosphorus. II. The Exploration of Hot Nuclear Matter. & Fong, K. C. Hydrodynamics of electrons in graphene. In order to formulate better heat removal strategies and designs, it is first necessary to understand the fundamental mechanisms of heat transport in semiconductor thin films. Phonon Hydrodynamic Heat Conduction and Knudsen Minimum in Graphite. A. 91, Issue 23, Applied Physics Letters, Vol. This chapter presents an overview of the theoretical underpinnings of typical numerical methods for solving the Peierls-Boltzmann equation (PBE) for phonon transport when coupled with density functional theory (DFT). The purpose of this note is to clarify the solution of the non-local Peierls Boltzmann equation found by Hua and Lindsay (Phys. Nat. 16, 789791, https://doi.org/10.1103/PhysRevLett.16.789 (1966). For 3D materials with the Debye model, the second sound velocity is \({v}_{ss}={v}_{g}/\sqrt{3}\), similar model for 2D material gives \({v}_{ss}={v}_{g}/\sqrt{2}\). At low temperature, phonons with small k contribute dominantly to the propagation of second sound. We can see that, the form of the G-K equation is the same as the 3D case. Following ref. The parameters are the same as ref. We rationalize the conditions determining the crossover from particlelike to wavelike heat conduction, showing that phonons below the Ioffe-Regel limit (i.e., with a mean free path shorter than the interatomic spacing) contribute to heat transport due to their wavelike capability to interfere and tunnel. The analytical and numerical results show good agreements in both cases. Jacak, B. V. & Mller, B. The coherences conductivities (Cxx) are blue. [114], and Sedmidubsk etal. 12, 672676, https://doi.org/10.1038/nphys3667 (2016). & Mackenzie, A. P. Evidence for hydrodynamic electron flow in pdcoo2. The magnitude of the linear group velocity is vg, and the magnitude of the wave vector is k=|k|. (7) is zero because the scattering processes conserve energy. : Condens. de Tomas, C., Cantarero, A., Lopeandia, A. F. & Alvarez, F. X. In J. Web1. Solid lines are temperature-dependent simulations performed relying on the same perturbative treatment of anharmonicity that we employed in our thermal-conductivity calculations. Turkyilmazoglu, M. MHD natural convection in saturated porous media with heat generation/absorption and thermal radiation: closed-form solutions. --Magazines for Libraries, Eighth Edition, 1995, R. R. It can be checked that, our results reduce to that of Debye model if we ignore the quadratic phonon mode. Using the steplike phase convention yields a total conductivity larger than that obtained using the smooth phase convention. Even negative thermal resistance can be observed, where the heat current flows are from the low to the high temperature regime. Its effect on the hydrodynamic transport is analyzed. objectives of this organization is to make the flow of information between Here in the hydrodynamic case, vortex formation is due to frequent momentum-conserving N-processes together with the boundary conditions imposed here, i.e., heat current injection and collection at local positions. Publishers note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This paper presents new transport equations to describe the kinetics of coupled electron-phonon systems in semiconductors. We highlight how the nonsharp crossover from dominant particlelike conduction to dominant wavelike conduction occurs around this value, in agreement with the theoretical predictions from the Wigner framework [Eq. In the limit of \(\chi \to +\,\infty \), the R-process is dominant, Eq. Here, this work elucidates important properties of phonon-defect scattering in thermal transport and demonstrates the predictive power of the coupling of Peierls-Boltzmann transport, Green's function methods, and density functional theory. It is a result of frequent momentum exchange between different phonons, akin to the vortex formation in classical gas or liquid flow. Phys. Here, a flow of heat current from a point source \(I(x)=I\delta (x)\) is injected into the ribbon and collected at the opposite side. 110, Journal of Applied Physics, Vol. We show the phonon lifetimes (q)=[(q)s]1 as a function of the energy (q)s for La2Zr2O7 [at a temperature of 200K (a), 800K (b), and 1300K (c)], and for CsPbBr3 [at a temperature of 50K (d), 225K (e), and 350K (f)]. 117, 166601, https://doi.org/10.1103/PhysRevLett.117.166601 (2016). Importantly, we consider both linear and quadratic acoustic phonon dispersions, which is critical to 2D materials. Erhaltungsgleichungen'' (ANumE - Analysis and numerics for (b)Zoomed-in dispersion close to along M. The calculated phonon dispersions, heat capacity, and thermal expansion coefficient are found to be in good agreement with measured data. Bowker, New Providence, New Jersey The resulting equation reads, Here, the presence of last term at the right side of the equation is due to the fact that normal scattering process conserves crystal momentum, but not the heat flux, Here, q0 is defined similarly by replacing fsk by \({f}_{N,s{\boldsymbol{k}}}^{eq}\). R. Soc. Colloquium: Phononics: Manipulating heat flow with electronic analogs and beyond. Physical Review X is a trademark of the American Physical Society, registered in the United States, Canada, European Union, and Japan. The N-processes describe phonon-phonon interactions Science 351, 10611064, https://doi.org/10.1126/science.aac8385 (2016). All rights reserved. The coordinate of the ribbon center is (x=10, y=5) m. Use of the American Physical Society websites and journals implies that The results are plotted in Fig. Dashed black lines are experiments, taken from Paul etal. When the temperature increases, more linear phonons contribute and vss increases correspondingly. We also get the dimensionless parameter \(\chi \approx 5\times {10}^{9}{w}^{2}\) m2. In the meantime, to ensure continued support, we are displaying the site without styles B 95, 184304, https://doi.org/10.1103/PhysRevB.95.184304 (2017). This has not been considered before. This is compared to Fig. B 99, 085202, https://doi.org/10.1103/PhysRevB.99.085202 (2019). MathSciNet corresponding vector of densities , fluxes ,, and productions P, according to, With these definitions, the kinetic equation implies, The Maximum Entropy Principle (MEP) can be formulated as follows. Prior to solution of the BTE, it is necessary to compute the lifetimes (or scattering rates) for phonons of all wave-vector and polarization. Blue, coherences conductivity (C). If the mean free, The coupling of lattice dynamics and phonon transport methodologies with density functional theory has become a powerful tool for calculating lattice thermal conductivity () with demonstrated quantitative accuracy and applicability to a wide range of materials. 212, Issue 2, Physical Review Letters, Vol. Article The calculation of p0N needs special care. Considering the universal behaviours of hydrodynamics, we expect similar transport behaviours may exist for other quasi-particles in solid. (b) Schematic of Poiseuille flow generated by the temperature difference along the nano-ribbon. The advent of coupled thermal transport calculations with interatomic forces derived from density functional theory has ushered in a new era of fundamental microscopic insight into lattice thermal conductivity. 103, Issue 25, Physical Review, Vol. The newly developed AIMD approach allows determination of harmonic and anharmonic interatomic forces at each temperature, which is particularly appropriate for highly anharmonic materials such as Bi 2 Te 3 . As examples, we have plotted the distribution of heat flux along two line cuts (green dashed lines in Fig. Experimental Studies of KBr and NaI, Method to extract anharmonic force constants from first principles calculations, Thermal conductivity of isotopically modifiedgraphene, Distributions of phonon lifetimes in Brillouin zones, Isotope effect in the thermal conductivity of germanium single crystals, Direct Measurement of Room-Temperature Nondiffusive Thermal Transport Over Micron Distances in a Silicon Membrane, Diameter Dependence of Lattice Thermal Conductivity of Single-Walled Carbon Nanotubes: Study from Ab Initio, Phonon Dispersion Curves in Wurtzite-Structure GaN Determined by Inelastic X-Ray Scattering, Nonmetallic crystals with high thermal conductivity, Phonon Self-Energy and Origin of Anomalous Neutron Scattering Spectra in SnTe and PbTe Thermoelectrics, Anharmonic force constants extracted from first-principles molecular dynamics: applications to heat transfer simulations, Phonons in single-layer and few-layer MoS, Direct Solution to the Linearized Phonon Boltzmann Equation, Isotope scattering of dispersive phonons in Ge, First-Principles Theory of Anharmonicity and the Inverse Isotope Effect in Superconducting Palladium-Hydride Compounds, Coherent Phonon Heat Conduction in Superlattices, Calculated transport properties of CdO: Thermal conductivity and thermoelectric power factor, Phonons and related crystal properties from density-functional perturbation theory, Thermal conductivity of half-Heusler compounds from first-principles calculations, High Thermal Conductivity in Short-Period Superlattices, Nanoscale thermal transport. Mater. In the year 2001 we have studied the following problems: Kinetic solutions of the Boltzmann-Peierls equation and access http://www.siam.org/. However, for 2D materials, when we include the quadratic phonon mode, the simple relation does not hold any more, qq0. Phys. The equilibrium layer distance d=3.35 is used to convert qy into the standard unit. High-resolution X-ray luminescence extension imaging, Raman tensor calculated from the 2n+1 theorem in density-functional theory, Physically founded phonon dispersions of few-layer materials and the case of borophene [Supplemental information], Tensile Strains Give Rise to Strong Size Effects for Thermal Conductivities of Silicene, Germanene and Stanene, Ab initio theory of the lattice thermal conductivity in diamond, Dislocation-induced thermal transport anisotropy in single-crystal group-III nitride films, Tutorial: Time-domain thermoreflectance (TDTR) for thermal property characterization of bulk and thin film materials, Phonon properties and thermal conductivity from first principles, lattice dynamics, and the Boltzmann transport equation, A scattering rate model for accelerated evaluation of lattice thermal conductivity bypassing anharmonic force constants, A Review of Thermal Transport in Low-Dimensional Materials Under External Perturbation: Effect of Strain, Substrate, and Clustering, A lower bound to the thermal diffusivity of insulators, Glass-like thermal conductivity in nanostructures of a complex anisotropic crystal, Phonon hydrodynamics for nanoscale heat transport at ordinary temperatures, Infrared reflectance, transmittance, and emittance spectra of MgO from first principles, Quartic Anharmonicity of Rattlers and Its Effect on Lattice Thermal Conductivity of Clathrates from First Principles, Thermal Expansion for Charring Ablative Materials, A review of computational phononics: the bulk, interfaces, and surfaces, Pressure induced excellent thermoelectric behavior in skutterudites CoSb, Phonon interaction with ripples and defects in thin layered molybdenum disulfide, Generalization of Fouriers Law into Viscous Heat Equations, Thermal conductivity of InN with point defects from first principles, Modulating the thermal conductivity in hexagonal boron nitride via controlled boron isotope concentration, Thermal Transport in Disordered Materials, An efficient and accurate framework for calculating lattice thermal conductivity of solids: AFLOWAAPL Automatic Anharmonic Phonon Library, Deducing Phonon Scattering from Normal Mode Excitations, Tailoring phononic, electronic, and thermoelectric properties of orthorhombic GeSe through hydrostatic pressure, https://doi.org/10.1103/PhysRevB.90.134309, First Principles Modeling of Phonon Heat Conduction in Nanoscale Crystalline Structures, Survey of ab initio phonon thermal transport, https://doi.org/10.1016/j.mtphys.2018.11.008, Thermal Transport by First-Principles Anharmonic Lattice Dynamics, https://doi.org/10.1007/978-3-319-50257-1_10-1, https://doi.org/10.1103/PhysRevB.98.014306, Oak Ridge National Lab.
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