The thermal conductivity of topological insaterials and Weyl semimetals presents a unique challenge due to the interplay between phonon transport and edge states. While electronic contributions often dominate discussions, the lattice thermal conductivity governed by phonons is equally critical for understanding heat dissipation in these materials. Theoretical approaches using tight-binding and density functional theory (DFT) methods have been employed to investigate phonon-edge state interactions, revealing distinct signatures in thermal transport.
Topological insulators exhibit protected surface states, but their bulk phonon modes remain a significant contributor to thermal conductivity. Theoretical studies using DFT-based lattice dynamics have shown that the presence of edge states can scatter acoustic phonons, reducing thermal conductivity compared to conventional insulators. For instance, in bismuth telluride (Bi2Te3), a well-known topological insulator, first-principles calculations predict a reduction in lattice thermal conductivity by 15-20% due to phonon-edge state interactions. This suppression arises from enhanced phonon scattering at the boundaries where edge states localize. The anisotropic nature of these materials further complicates thermal transport, with in-plane and cross-plane conductivities differing by a factor of 2-3 in layered structures.
Weyl semimetals, characterized by linear band crossings in momentum space, exhibit chiral edge modes that interact with phonons in unconventional ways. Tight-binding models incorporating phonon dispersion have demonstrated that Weyl nodes act as scattering centers for heat-carrying phonons. The resulting phase space for phonon-edge state interactions leads to a temperature-dependent thermal conductivity that deviates from standard phonon gas models. For example, in TaAs, DFT calculations reveal that the lattice thermal conductivity follows a T^(-0.8) dependence at low temperatures, contrasting with the typical T^(-1) behavior seen in non-topological metals. This anomalous scaling stems from the coupling between Weyl fermion surface states and acoustic phonons.
Phonon dispersion calculations using DFT methods highlight the role of avoided crossings between bulk phonon modes and edge-localized vibrational states. These hybridized modes exhibit reduced group velocities, directly impacting thermal conductivity. In HgTe, a material exhibiting both topological insulator and semimetal phases, theoretical predictions indicate a 30% reduction in thermal conductivity when edge states are present compared to the bulk-only case. The suppression is most pronounced in the frequency range corresponding to the energy scale of the edge states, typically between 1-5 THz.
The effect of edge state chirality on phonon transport has been explored using tight-binding models with spin-orbit coupling. Chiral edge modes introduce an additional scattering channel for phonons through spin-lattice interactions, further reducing thermal conductivity. Numerical simulations of Na3Bi, a Dirac semimetal, show that the lattice thermal conductivity is sensitive to the spin texture of the edge states, with a 10-15% variation depending on the spin-momentum locking strength. This effect is particularly pronounced in materials with strong spin-orbit coupling, where phonon scattering rates increase by up to 50% compared to weak coupling cases.
Temperature-dependent studies reveal that phonon-edge state interactions become increasingly important below the Debye temperature. DFT-based molecular dynamics simulations of Sb2Te3 demonstrate that the thermal conductivity plateau observed around 50 K is directly linked to the onset of strong phonon-edge state scattering. Above this temperature, three-phonon processes dominate, while below it, the edge states act as additional scattering centers that limit phonon mean free paths. The crossover between these regimes is marked by a change in the power law exponent of thermal conductivity from approximately -1.5 to -1.0.
The role of edge state localization length in phonon scattering has been investigated using Green's function methods coupled with DFT inputs. For edge states extending 1-2 nm into the bulk, the scattering efficiency for mid-frequency phonons is maximized, leading to a minimum in thermal conductivity. This effect is particularly evident in Bi2Se3 thin films, where theoretical models predict a non-monotonic thickness dependence of thermal conductivity due to competing bulk and edge contributions. Films thinner than 10 nm show a 40% reduction in thermal conductivity compared to bulk crystals, primarily due to enhanced phonon-edge state interactions.
Disorder effects on phonon transport in topological materials have been examined using tight-binding models with random potentials. While edge states are topologically protected against backscattering, they can still modify the phonon density of states through local strain fields. Theoretical work on disordered Bi2Te3 films indicates that phonon localization occurs at specific frequencies corresponding to hybridized edge-bulk modes, reducing thermal conductivity by an additional 10-20% compared to clean systems. The interplay between disorder-induced localization and topological protection creates a complex thermal transport landscape that differs fundamentally from conventional semiconductors.
Comparative studies between topological and trivial insulators using identical computational methods highlight the unique aspects of phonon-edge state interactions. DFT calculations for PbTe (a trivial insulator) and SnTe (a topological crystalline insulator) show that while both materials have similar phonon dispersions, the thermal conductivity of SnTe is systematically lower by 15-25% across all temperatures due to the presence of topological surface states. This difference persists even when accounting for anharmonic effects, suggesting that the edge states provide an additional scattering mechanism beyond conventional phonon-phonon interactions.
The influence of external magnetic fields on phonon transport in Weyl semimetals has been investigated using tight-binding models with Peierls substitution. Magnetic fields modify the edge state spectrum, which in turn affects their coupling to phonons. Theoretical predictions for Cd3As2 indicate that the lattice thermal conductivity exhibits quantum oscillations at low temperatures, with amplitude and phase directly tied to the Landau level structure of the edge states. These oscillations are distinct from electronic contributions and provide a signature of phonon-edge state coupling in magneto-thermal measurements.
Future theoretical work could explore higher-order phonon-edge state interactions beyond the current harmonic approximation. The development of ab initio methods capable of treating both topological electronic states and anharmonic phonons on equal footing would provide more accurate predictions of thermal conductivity in these complex materials. Additionally, the role of electron-phonon coupling in mediating heat transport between edge and bulk states remains an open question that could be addressed through advanced computational approaches combining DFT with many-body perturbation theory.
The theoretical framework established by these studies provides a foundation for understanding and engineering thermal properties in topological materials. By quantifying the impact of edge states on phonon transport, researchers can design materials with tailored thermal conductivity for applications ranging from thermoelectrics to heat management in quantum devices. The unique interplay between topology and lattice dynamics continues to reveal new phenomena that challenge conventional wisdom about heat transport in solids.