TlAgTe: Glass-like Heat Transport in Crystals

• TlAgTe is a crystalline semiconductor defined by a distorted AgTe4 framework and isolated Tl chains that induce low, glass-like thermal conductivity.
• The anomalous heat transport arises from combined factors including localized rattling of heavy Tl atoms, strong anharmonicity, and significant four-phonon Umklapp scattering.
• First-principles studies using SCPH and iterative phonon transport models reveal a particle-to-wave crossover, reconciling theory with the experimental thermal conductivity of around 0.44 W m⁻¹K⁻¹.

TlAgTe is an ordered crystalline semiconductor distinguished in the recent literature by an ultralow lattice thermal conductivity with a weak, glass-like temperature dependence, despite its crystalline atomic order (Semwal et al., 26 Aug 2025). In the first-principles account presently available, its anomalous heat transport is not attributed to a single mechanism but to the combined action of a distorted three-dimensional $\mathrm{AgTe_4}$ tetrahedral framework, disconnected Tl chains in the hollow space of that framework, localized rattling-like Tl vibrations, strong anharmonicity, substantial four-phonon Umklapp scattering, and a crossover from particle-like Peierls transport to wave-like coherent transport. TlAgTe is therefore notable less for electronic topology than for a phonon transport regime in which crystalline order coexists with partially glass-like thermal conduction (Semwal et al., 26 Aug 2025).

1. Crystal chemistry and structural identity

TlAgTe crystallizes as a three-dimensional framework of distorted $\mathrm{AgTe_4}$ tetrahedra, with disconnected chains of Tl atoms residing in the hollow space of that framework (Semwal et al., 26 Aug 2025). The bonding is described as mixed covalent–ionic, with disparate bond strengths. This bonding hierarchy is central to the material’s physical behavior because it creates a relatively rigid Ag–Te network together with more weakly bound heavy Tl cations.

The structural distortion is not incidental. The $\mathrm{AgTe_4}$ tetrahedra are explicitly described as distorted rather than ideal, and that local distortion is identified as one of the microscopic ingredients that amplifies anharmonicity. The Tl sublattice is especially unusual: despite Tl being heavy, its calculated mean-square displacement is as large as that of the much lighter Ag atoms. The interpretation given is that Tl is loosely bound and undergoes rattling-like motion inside the structural voids. Electronic density of states and electron localization function calculations further show localized lone-pair electrons on Tl, and those lone pairs, together with weak Tl-centered restoring forces and framework distortion, are identified as major microscopic sources of strong anharmonicity and reduced thermal transport.

Within this structural picture, the atomic roles are differentiated. Tl dominates the low-frequency localized modes and acts as a heavy-cation rattler, more precisely as a concerted rattling subsystem because the relevant vibrations involve collective Tl motion rather than isolated independent Einstein oscillators. Ag and Te form the tetrahedral backbone that supports the overall crystalline phonon spectrum, although Ag also shows a relatively large mean-square displacement and is therefore not entirely rigid. Te contributes strongly to higher-frequency vibrational regions and, after renormalization, contributes localized spectral weight around $\sim 95\ \mathrm{cm^{-1}}$.

2. Vibrational spectrum and anharmonic renormalization

The harmonic phonon spectrum of TlAgTe is divided into three frequency groups: GR1 $=30$–$55\ \mathrm{cm^{-1}}$, GR2 $=55$–$112\ \mathrm{cm^{-1}}$, and GR3 $=112$–$150\ \mathrm{cm^{-1}}$ (Semwal et al., 26 Aug 2025). The most remarkable feature is GR1, which contains several nearly dispersionless localized branches mainly due to Tl vibrations. These are described as rattling-like, and eigenvector visualizations show concerted Tl motion at the zone center. Such flat branches hinder heat flow because they add many scattering channels while contributing little group velocity.

Temperature-dependent renormalization is pronounced. The low-energy spectrum hardens appreciably with temperature: the first transverse optical mode shifts from $\mathrm{AgTe_4}$0 at $\mathrm{AgTe_4}$1 to $\mathrm{AgTe_4}$2 at $\mathrm{AgTe_4}$3 at $\mathrm{AgTe_4}$4, and from $\mathrm{AgTe_4}$5 to $\mathrm{AgTe_4}$6 at the S point. The strongest hardening occurs in GR2, followed by GR1. At the same time, the renormalized phonon density of states shows increased localization near $\mathrm{AgTe_4}$7 and $\mathrm{AgTe_4}$8, mainly from Tl, and near $\mathrm{AgTe_4}$9, mainly from Te.

This behavior is taken as strong evidence of pronounced anharmonicity. The analysis attributes it to three intertwined microscopic factors: local structural distortion of the $\mathrm{AgTe_4}$0 tetrahedral network, localized Tl lone-pair electrons, and weakly bound heavy Tl cations undergoing rattling-like motion. Quartic anharmonicity is emphasized as especially important because it drives the self-consistent phonon renormalization, whereas cubic renormalization is acknowledged but treated as secondary because the cubic bubble correction is perturbatively smaller than the quartic effect.

3. Thermal-transport formalism

The theoretical description adopts a unified transport framework in which the total lattice thermal conductivity is decomposed into particle-like and coherent terms (Semwal et al., 26 Aug 2025): $\mathrm{AgTe_4}$1 Here $\mathrm{AgTe_4}$2 is the standard phonon-gas or Peierls contribution, governed by intraband group velocities and phonon lifetimes, whereas $\mathrm{AgTe_4}$3 is the coherent or wave-like contribution, arising from off-diagonal matrix elements of the velocity operator between different phonon branches.

The computational workflow is first-principles throughout. Density functional theory with the PBEsol exchange-correlation functional, implemented in VASP, is used to obtain the electronic structure and interatomic force constants. Harmonic phonons are computed using finite displacements with Phonopy. Three-phonon transport is treated by solving the Peierls-Boltzmann transport equation iteratively using ShengBTE, and four-phonon scattering is added using FourPhonon within the relaxation-time approximation. Strong anharmonic frequency shifts are treated with the self-consistent phonon method based on many-body Green’s-function theory.

In this formalism, anharmonicity renormalizes the harmonic frequencies $\mathrm{AgTe_4}$4 into temperature-dependent frequencies $\mathrm{AgTe_4}$5,

$\mathrm{AgTe_4}$6

with linewidth

$\mathrm{AgTe_4}$7

The Peierls contribution is written as

$\mathrm{AgTe_4}$8

and the paper emphasizes the approximate scaling

$\mathrm{AgTe_4}$9

When both three-phonon and four-phonon scattering are included, the total scattering rate follows Matthiessen’s rule,

$\sim 95\ \mathrm{cm^{-1}}$0

The four-phonon Umklapp channels are explicitly classified as splitting,

$\sim 95\ \mathrm{cm^{-1}}$1

redistribution,

$\sim 95\ \mathrm{cm^{-1}}$2

and recombination,

$\sim 95\ \mathrm{cm^{-1}}$3

This decomposition is essential because TlAgTe does not conform to a purely quasiparticle picture. The particle-like term captures the remnant crystalline phonon-gas transport, whereas the coherent term captures interbranch tunneling or mode-mixing transport that becomes important when branches are dense, flat, and strongly broadened.

4. Quantitative thermal conductivity and scattering hierarchy

Experiment gives about $\sim 95\ \mathrm{cm^{-1}}$4 at $\sim 95\ \mathrm{cm^{-1}}$5 with a glass-like temperature dependence $\sim 95\ \mathrm{cm^{-1}}$6 (Semwal et al., 26 Aug 2025). The paper compares this with successive levels of theory and shows that neither harmonic phonons nor three-phonon scattering alone are sufficient.

Model	$\sim 95\ \mathrm{cm^{-1}}$7	Temperature dependence
HA+3ph	$\sim 95\ \mathrm{cm^{-1}}$8	nearly $\sim 95\ \mathrm{cm^{-1}}$9
SCPH+3ph	$=30$0	$=30$1
SCPH+3ph+4ph	$=30$2	$=30$3
Experiment	$=30$4	$=30$5

At $=30$6, HA+3ph predicts $=30$7, only $=30$8 of the experimental $=30$9, and gives an approximately $55\ \mathrm{cm^{-1}}$0 temperature dependence. When SCPH-renormalized frequencies are used in the three-phonon calculation, $55\ \mathrm{cm^{-1}}$1 rises to $55\ \mathrm{cm^{-1}}$2 because hardening reduces the three-phonon scattering rates and increases phonon lifetimes. This improves the temperature trend but overshoots the magnitude, showing that higher-order scattering is indispensable.

Including four-phonon scattering resolves much of the discrepancy. The four-phonon rates are found to be comparable in importance to three-phonon rates, and below $55\ \mathrm{cm^{-1}}$3 there are especially many allowed four-phonon processes because of the increased number of mode combinations satisfying energy and momentum conservation. Four-phonon scattering is dominated by Umklapp rather than normal processes. At $55\ \mathrm{cm^{-1}}$4, adding four-phonon scattering on top of SCPH+3ph reduces $55\ \mathrm{cm^{-1}}$5 by $55\ \mathrm{cm^{-1}}$6, from $55\ \mathrm{cm^{-1}}$7 to $55\ \mathrm{cm^{-1}}$8, and changes the temperature dependence to $55\ \mathrm{cm^{-1}}$9, very close to the experimental $=55$0.

The paper further notes the scaling

$=55$1

so that

$=55$2

with $=55$3. This mixed temperature dependence explains why TlAgTe does not follow the ordinary crystal-like $=55$4 law.

Mode by mode, the particle-like conductivity comes mainly from acoustic phonons below $=55$5 and low-frequency optical modes in GR1 up to about $=55$6. The cumulative $=55$7 saturates around $=55$8, indicating that higher-frequency modes contribute little through the Peierls channel.

5. Particle-to-wave crossover and coherent transport

One of the defining results for TlAgTe is the crossover from particle-like to wave-like thermal transport (Semwal et al., 26 Aug 2025). In ordinary crystals, heat is dominated by intraband propagation of spectrally separated phonons. In TlAgTe, by contrast, many branches are flat and closely spaced, while anharmonicity broadens them strongly. In that regime, off-diagonal velocity matrix elements linking different branches become important, and heat can be carried by interband tunneling or coherence between modes.

The crossover is analyzed using the Wigner formulation and the Wigner limit in time,

$=55$9

Modes with

$112\ \mathrm{cm^{-1}}$0

behave predominantly particle-like, whereas modes with

$112\ \mathrm{cm^{-1}}$1

enter the interband wave-like tunneling regime. In TlAgTe this crossover occurs prominently for highly localized modes above about $112\ \mathrm{cm^{-1}}$2. Mean-free-path analysis gives a consistent picture, with the average bond length serving as a spatial crossover scale analogous to an Ioffe-Regel-like criterion.

Importantly, all scattering lifetimes remain above the strict amorphous/Ioffe-Regel limit, so the perturbative phonon picture remains formally meaningful. Nonetheless, many modes already behave in a strongly non-Peierls, coherence-dominated way. This is why $112\ \mathrm{cm^{-1}}$3 becomes significant and increases with temperature: as temperature rises, occupations increase and linewidths broaden, enhancing spectral overlap between nearby branches.

At $112\ \mathrm{cm^{-1}}$4, for the highest-level theory, the coherent contribution is about $112\ \mathrm{cm^{-1}}$5 of the particle-like one,

$112\ \mathrm{cm^{-1}}$6

Given $112\ \mathrm{cm^{-1}}$7, this places $112\ \mathrm{cm^{-1}}$8 at roughly $112\ \mathrm{cm^{-1}}$9, bringing the total close to the measured $=112$0. The coherent spectral decomposition shows strong peaks in the GR1 and GR2 regions, confirming that the same localized modes that suppress phonon-gas transport also enable wave-like heat transfer.

TlAgTe thus occupies an intermediate transport regime: crystalline enough that phonons remain identifiable, but anharmonic and locally distorted enough that glass-like wave transport coexists with phonon-gas transport.

6. Microscopic interpretation and broader research context

The microscopic origin proposed for TlAgTe’s glass-like thermal conductivity is a coupled hierarchy rather than a single dominant effect (Semwal et al., 26 Aug 2025). The distorted $=112$1 tetrahedral framework creates chemical and structural heterogeneity with hollow regions. Heavy Tl atoms with localized lone-pair electrons and unusually large thermal displacements occupy those hollows and support concerted rattling-like localized modes in the low-frequency spectrum. Those modes flatten the dispersion, reduce group velocities, and expand the scattering phase space. Strong quartic anharmonicity hardens and further localizes parts of the spectrum as temperature rises, while cubic and quartic interactions generate substantial linewidths. Four-phonon Umklapp processes then strongly diminish $=112$2, and the resulting dense, broadened spectrum enables a sizable $=112$3 that grows with temperature.

This analysis also defines what TlAgTe should not be conflated with. The binary AgTe monolayer reported as a flat honeycomb sheet on Ag(111) is a different material: it is a planar 2D interfacial Ag–Te compound whose free-standing limit hosts mirror-protected Dirac node lines and SOC-gapped edge-state physics, whereas TlAgTe is treated as a three-dimensional crystalline semiconductor with anomalous lattice thermal transport (Liu et al., 2019). Likewise, direct spectroscopic evidence for mixed-valence Tl has been established for Pb$=112$4Tl$=112$5Te, where Tl$=112$6 and Tl$=112$7 are resolved in Tl $=112$8 core levels, but that result is a spectroscopic and chemical analogue rather than a demonstration of mixed valency in TlAgTe itself (Walmsley et al., 2018). Other Tl–Te ternaries such as CdTl$=112$9Te$150\ \mathrm{cm^{-1}}$0 and HgTl$150\ \mathrm{cm^{-1}}$1Te$150\ \mathrm{cm^{-1}}$2 have been proposed as $150\ \mathrm{cm^{-1}}$3 strong topological insulators with SOC-induced nontrivial gaps, yet those calculations concern a distinct structural family and do not establish a corresponding topology for TlAgTe (Li et al., 2022). Even elemental tellurene is relevant only indirectly, as an illustration of Te bonding flexibility and low-dimensional structural chemistry rather than as a structural model for TlAgTe (Zhu et al., 2017).

A plausible implication is that TlAgTe has become a model system for an ordered crystal near a particle-to-wave crossover in phonon transport. In the present literature, its significance lies in the mechanism-resolved demonstration that ultralow and weakly temperature-dependent $150\ \mathrm{cm^{-1}}$4 can emerge in a crystalline semiconductor when local structural distortions, bonding hierarchy, lone-pair-active heavy cations, localized rattling-like branches, strong quartic renormalization, and dominant four-phonon Umklapp scattering occur simultaneously (Semwal et al., 26 Aug 2025).