Strain-driven phonon topological phase transition impedes thermal transport in titanium monoxide

TL;DR

This study demonstrates how strain-induced topological phonon phase transitions in titanium monoxide can significantly reduce thermal conductivity by breaking symmetry and increasing phonon scattering, offering new ways to control heat transport in materials.

Contribution

It reveals the impact of topological phonon phase transitions on thermal transport and shows how uniaxial strain can effectively tune thermal conductivity in TiO.

Findings

Applying 10% tension reduces thermal conductivity by up to 77%.

Symmetry breaking lifts degeneracy of acoustic phonon branches.

Enhanced phonon scattering increases thermal resistance.

Abstract

Topological phonon states in crystalline materials have attracted significant research interests due to their importance for fundamental physical phenomena, yet their implication on phonon thermal transport remains largely unexplored. Here, we use density functional theory calculations and symmetry analyses to explore topological phonon phase transitions under uniaxial strains and their tuning effects on thermal transport in titanium monoxide (TiO). Our calculation shows that application of 10% tension significantly diminishes lattice thermal conductivity of TiO by 77% and 66% along the a and c axes, respectively, at room temperature. This suppression is found to result largely from the breaking of symmetry protected degeneracy of acoustic branches, which induces a substantial enhancement of phonon scattering phase space due to the easier fulfillment of scattering selection rules. Our study provides evidence for the importance of phononic band topology in modulating thermal conductivity and offers a promising route towards controlling solid-state heat transport.

Full text

The emergence of topological quantum states is one of the most prominent advancements in condensed matter physics Qi and Zhang (2011); Stern and Lindner (2013); Haldane (2017); Hao and Hu (2018); Li et al. (2021a). The concepts of topological electronic states have been generalized to bosonic systems, leading to the birth of topological phonons Stenull et al. (2016a); Liu et al. (2020a); Süsstrunk and Huber (2016); Yang et al. (2017); Li et al. (2012); Huber (2016); Liu et al. (2017); Xia et al. (2017); Zhang et al. (2018a). Recently, topological phonons have received increasing research interest in both experiments and theories Stenull et al. (2016b); Zhang et al. (2018b); Lv et al. (2021); Osterhoudt et al. (2021); Ren et al. (2022), since it is believed that the phonon is an ideal platform for realizing topological quantum states due to its bosonic nature and hard-to-break time-reversal symmetry Liu et al. (2020b); Wang et al. (2022). In addition, phonons, energy quanta of lattice vibration, also play an essential role in fundamental physical phenomena including heat conduction, light-matter interaction, thermoelectrics, and superconductivity Liu et al. (2018). In this context, it is of essential importance to realize and tune topological nontrivial phonon states in certain materials, which may result in novel physical properties and promising application prospects in a wide range of fields. Hitherto, various types of symmetry-enforced topological phonons have been identified theoretically in thousands of materials Chen et al. (2021); Li et al. (2021b); Xu et al. (2022) , and some of them have been confirmed experimentally Zhang et al. (2018b). However, the physical consequences underlying topological states of phonons in these materials remain largely unexplored. Particularly, as one of the most exciting consequences, tuning of thermal transport by topological phase transition (TPT) in these systems is rarely studied Yue et al. (2020); Tang and Cao (2021).

It is well known that phonons usually govern thermal transport in most materials, especially in nonmetals. In the kinetic theory of phonon gases Ziman (2001); Srivastava (1990), the lattice thermal conductivity ($\kappa_{L}$) can be expressed as the sum of contributions from all phonon modes: $\kappa_{L}=\sum_{\lambda}C_{\lambda}v_{\lambda}^{2}\tau_{\lambda}$, where $C_{\lambda}$ is the heat capacity of a phonon mode $\lambda$, $v_{\lambda}$ is the group velocity, and $\tau_{\lambda}$ is the phonon lifetime. All three factors that determine $\kappa_{L}$ are closely associated with the phonon dispersion of a material: $v_{\lambda}$ and $\tau_{\lambda}$ are completely determined from the phonon dispersion, while $\tau_{\lambda}$ relies largely on the phonon dispersion due to the restriction of phonon-phonon scattering selection rules Lindsay et al. (2018); Qian et al. (2021). In this scenario, one can utilize the phononic TPT to alter the phonon dispersion relation of a material and thus effectively regulate its heat conduction. However, limited by the charge neutral and spinless nature, the TPT of phonons generally cannot be achieved by conventional approaches of treating electrons Liu et al. (2020b); Yue et al. (2020). It is noted that previous studies have demonstrated TPTs of phonons induced in different approaches, including spin–lattice interactions for magnetic lattices Zhang et al. (2010); Ioselevich and Capellmann (1995), magnetic field for ionic lattices Zhang et al. (2010), Coriolis field Wang et al. (2015), and optomechanical interactions Peano et al. (2015). In contrast, applying external pressure or strain to break the symmetry of crystals provides an alternative and universal method for achieving TPTs of phonons Jiang and Park (2019); Rostami et al. (2022); Katailiha et al. (2021).

In this work, we perform first-principles calculations to examine the effect of phononic TPT on the thermal transport in hexagonal TiO, a Weyl semimetal that has been synthesized experimentally Möhr and Müller-Buschbaum (1994). With the aid of symmetry analyses, we identify the existence of phononic triple degenerated point (TDP) and the nodal line formed doubly degenerated acoustic phonon branches along the high-symmetric $\Gamma\mathrm{A}$ direction in the Brillouin zone (BZ). We show that the application of uniaxial strain enables the TPT of phonons, leading to significant suppression of $\kappa_{L}$. By further phonon transport analyses, we attribute this giant reduction in $\kappa_{L}$ to the increased phonon-phonon scattering channels, enabled by the breaking of degenerate states of topological phonons. Our work reveals the strong relevance of topological phonons with thermal conductivity, and opens up a new pathway towards controlling heat conduction in topological materials.

All first-principles calculations are implemented in the Vienna Ab initio Simulation Package (VASP) Kresse and Hafner (1993); Kresse and Furthmüller (1996) with the projector augmented wave potential Blöchl (1994), under the generalized gradient approximation with the Perdew-Burke-Ernzerhof functional for the exchange-correlation functional Perdew et al. (2008); Le et al. (2012). The topological surface states are calculated by the Wanniertools package Wu et al. (2018), and the irreducible representations of phonon branches are obtained by the PhononIrep package Zhang et al. (2022). To obtain accurate $\kappa_{L}$, we exactly solve the Boltzmann transport equation within an iterative scheme, by considering phonon lifetimes due to the isotope and three-phonon scattering Li et al. (2014). More computational details are given in Supplemental Material SM .

TiO crystallizes in the tungsten carbide type (WC-type) hexagonal structure with space group of $P\overline{6}m2$ (No. 187), and has recently been predicted to have nontrivial topological phonon states Xie et al. (2019). The crystal structure of WC-type TiO is shown in Fig. 1(a), where the Ti atom and the O atom occupy the wyckoff position $1b$ (0, 0, 1/2) and $1c$ (1/3, 2/3, 0), respectively. The topological electronic states in TiO have been well studied Ullah et al. (2022). As is seen in Fig. 1(b), in the electronic band of TiO, there exists a TDP along the high-symmetry line $\Gamma\mathrm{A}$ that brings in the vanishing of the density of states near the Fermi level, revealing the nature of its semimetal. Fig. 1(c) shows the calculated phonon dispersion of TiO along the high-symmetry lines along with corresponding phonon density of states. The missing of imaginary frequency modes indicates the dynamical stability of TiO. From the phonon density of states, it is also observed that the low-frequency phonon modes below 12 THz are dominated by Ti atoms, while those above 12 THz mainly originate from the lighter O atoms. Notably, we observe rich types of degeneracy of phonon branches in the phonon spectrum of TiO. Specifically, two transverse acoustic (TA) branches degenerate along the $\Gamma\mathrm{A}$ line (marked as $\Delta$), forming a nodal line running through the BZ, and the longitudinal acoustic (LA) branch crosses with a double-degenerated optical branches along $\Delta$ forming a TDP at $\sim$12 THz. The emergence of these types of phonon band degeneracy is explained as follows.

In terms of symmetry analysis, the generating elements of the little group at an arbitrary point in $\Delta$ are $C_{3z}^{+}$, $M_{100}$ associate with joint symmetry $M_{001}T$. For the degenerated TA branches, the eigenvalues of $C_{3z}^{+}$ symmetry are $\left[e^{-2\pi i/3},e^{2\pi i/3}\right]$. Due to the existence of the joint symmetry $M_{001}T$ at any arbitrary $k$ point on $\Delta$, these two symmetry eigenvalues that are complex conjugates of each other must be degenerated forming a two-dimensional irreducible representation (Irrep) $\Delta_{3}$ written under the BCS convention Elcoro et al. (2017); Aroyo et al. (2006). Thus, the double-fold degenerated nodal line is protected by $M_{001}T$ and $C_{3z}^{+}$. As for the LA phonon branch, its Irrep is $\Delta_{1}$ and the eigenvalue of $C_{3z}^{+}$ symmetry is 1. The similar symmetry analysis is also applied to the three optical branches, and corresponding Irreps are labeled in Fig. 1(c). Notably, the TDP at 12 THz is formed by two sets of bands ($\Delta_{1}$ and $\Delta_{3}$) with different $C_{3z}^{+}$ eigenvalues, indicating that this degeneracy is protected by $C_{3z}^{+}$ and $M_{001}T$. Since there does not exist a fully gapped sphere surrounding the TDP, this TDP does not have a well-defined topological charge. Even though, this TDP can be regarded as two Weyl points with opposite topological charges $\pm 1$ and formed by band pairs whose $C_{3z}^{+}$ eigenvalues are $\left[e^{2\pi i/3},1\right]$ and $\left[e^{-2\pi i/3},1\right]$ degenerate together. As a result, when projected onto surface, one can observe the Fermi arc whose origin and terminal are located at the same projected TDP. To verify this point, the density of states of (100) surface of TiO is calculated and shown in Fig.1(e). One notices that the surface state is clearly visible. To understand the connection of the surface state clearer, we also plot the Fermi surface of the phonon dispersion of the semi-infinite (100) surface in Fig.1(f). As expected, the Fermi arc origins and terminate exist at the same point in the projected BZ.

The presence of these phonon band topologies should have non-negligible impact on thermal conductivity, since they can profoundly affect the phase space for phonon-phonon scatterings. To explore their effect on $\kappa_{L}$, we attempt to achieve topological phonon phase transitions by applying strain. By symmetry analysis, we find that applying an uniaxial strain along (100) orientation can break the phonon band degeneracy protected by the $C_{3z}^{+}$ and $M_{001}T$ symmetries. The phonon dispersion of TiO under the tension strain of 10$\%$ is compared with that at zero strain in Fig.2(a). It is seen that different from the original acoustic bunching under ambient condition, with the presence of strain, the significant softening of the low-lying TA modes along the $\Gamma\mathrm{A}$ direction leads to the large separation of three acoustic branches, which allows for increased phonon-phonon scattering channels due to the easier fulfillment of energy and momentum conservation rules Yang et al. (2019, 2020); Ravichandran and Broido (2019). It is also found that the softening of the LA branch makes the TDP vanish when the strain is applied, resulting in the avoided crossing between LA and TO phonon branches, which facilitates to enlarge the phase space for available phonon-phonon scatterings as mentioned previously Delaire et al. (2011); Lu et al. (2018). These prominent changes in the phonon dispersion induced by the TPT are expected to enhance phonon-phonon scattering due to the increased phase space and thus suppress the thermal conductivity. In parallel, we also explore the strain effect on the electronic band structure. As plotted in blue in Fig.2(b), in the presence of strain, the TDP electronic state is split into two double degenerated points, and the system remains the semimetal phase. Given the vanishing electronic density of states at the Fermi level, the phonon scattering by electrons across the TPT should thus be insignificant Yang et al. (2021a, b).

To examine the lattice thermal conductivity across the TPT, we present the calculated temperature-dependent $\kappa_{L}$ under 0 and 10$\%$ strain in Fig.2(c). In the absence of strain, the thermal conductivities along the $a$ and $b$ axes are the same and larger than those along the $c$ axis. Nevertheless, the thermal conductivities along three axes are all different when TiO is subjected to 10$\%$ strain, with the one along $b$ axis being largest, followed by the $a$ axis and then the $c$ axis. Importantly, the thermal conductivities along three axes under 10$\%$ strain are all significantly declined over the entire temperature range, compared with the intrinsic case. Specifically, at 300 K, $\kappa_{L}$ along the $a$, $b$, and $c$ axes decreases by 78$\%$ (from 83.19 to 18.51 W/mK), 64$\%$ (from 83.19 to 29.56 W/mK), and 68$\%$ (from 57.71 to 18.34 W/mK), respectively. Note that the effect of four-phonon scattering on $\kappa_{L}$ is not strong in TiO, with the less than 7$\%$ reduction in $\kappa_{L}$ at room temperature as confirmed in Fig.S(1) SM , and it is thus not considered in the present calculation. To illustrate which frequency ranges are affected across the TPT, the spectral lattice thermal conductivity along the $a$ axis and its accumulation with frequency at different strains are provided in Fig.2(d). It is evident that phonons below 10 THz, corresponding to acoustic branches, are a major contributor to thermal conductivity and their contributions are largely diminished across the TPT.

Further decomposing $\kappa_{L}$ into different branches, we can see from Fig.3(a) that under 10$\%$ strain the contributions to $\kappa_{L}$ from all three acoustic branches are substantially decreased, e.g., at 300 K, the $\kappa_{L}$ contributed from the TA1, TA2, and LA phonons is reduced by 80$\%$, 74$\%$, and 80$\%$, respectively. The thermal conductivity of a material is jointly determined by group velocities $v_{\lambda}$, heat capacity $C_{v}$, and phonon lifetimes $\tau_{\lambda}$. Therefore, to elucidate the microscopic origins behind lowering $\kappa_{L}$, it is useful to conduct a detailed analysis on these vibrational properties. In Fig.3(b, c) we separately plot these quantities for different strains and find all to contribute to reducing $\kappa_{L}$ in varying degrees. First, we notice that the overall phonon group velocities are lowered when the strain is applied, stemming from the acoustic phonon softening effect. Second, we observe a minor change in the heat capacity shown in the inset of Fig.3(b), with the decrease less than 2$\%$ under the strain. Third, we find that through the TPT, the phonon scattering rates, inverse of phonon lifetimes, are significantly increased in the whole frequency range. Particularly, the original phonon scattering rates have a very marked dip in the frequency range of 10-12 THz, while the scattering rates of these phonons increase by almost an order of magnitude when the strain is applied. This prominent feature is closely related to the enlargement of phase space for three-phonon scattering as discussed later. In contrast, the phonon group velocities and heat capacity do not change much, while the phonon scattering rates increase by a large margin. Hence, we conclude that the significant suppression in $\kappa_{L}$ by strain is mainly due to the increased phonon scattering rates.

For more insights, we compare the phonon scattering rates of each branch at different strains in Fig.3(d). It is showed that the phonon scattering rates of all the heat-carrying acoustic branches under the strain are substantially larger than those without the strain, especially for the low-lying TA modes below 6 THz. The phonon scattering strength depends on the phase space and anharmonicity, both of which can be altered by strain. To find out which one dominates the increase in phonon scattering rates, we detect these two quantities separately. Fig.3(e) shows the mode-resolved Gr$\ddot{\rm u}$neisen parameter, which can be used to measure the phonon anharmonicity Jin et al. (2022a); Zhou et al. (2022); Jin et al. (2022b). It can be seen that the magnitudes of Gr$\ddot{\rm u}$neisen parameter under the strain are obviously larger than those without the strain, implying the stronger phonon anharmonicity in the former. The phonon phase space characterizes the number of all available phonon scattering channels for simultaneously satisfying the energy and momentum conservation. As seen in Fig.3(f), the weighted phase space of all phonons is significantly increased when the strain is applied, demonstrating that strain-induced TPT substantially increases the available phonon scattering processes. These results reveal that both the stronger phonon anharmonicity and enlarged phase space are responsible for the increased phonon scattering rates.

In the formalism of three-phonon scattering rates given in Eq.S2-S5 SM , the phonon anharmonicity is reflected in the scattering matrix elements, which depends both on the normalized eigenfunctions and the anharmonic force constants (FCs). Hence, to quantitatively single out the topological effects of phonons, it is instructive to calculate the thermal conductivity using the mixed FCs. By combining the harmonic FCs under 10$\%$ strain and anharmonic FCs at zero strain, the resultant $\kappa_{L}$ along the $a$ axis is 7.26 W/mK, which is lower 91$\%$ than that at zero strain, more than the actual reduction of 78$\%$ across the TPT, as given in the inset of Fig.3(f). This implies that the anharmonic FCs is decreased after imposing the strain, and that the enhanced phonon anharmonicity should be rooted in a large increase in eigenfunctions resulting from the change of phonon dispersion induced by the TPT. From this analysis, we can conclude that the increased phonon scattering rates under the strain originates fundamentally from the topological phonon effects, which is enabled by simultaneously increasing the phase space and anharmonicity.

To gain a deeper understanding of connection between the phonon band topology and phonon scattering rates, we further analyze the contributions of different processes to phonon scattering rates. As shown in Fig.4(a-c), the phonon scattering rates below 10 THz are governed by three-phonon processes involving either three acoustic phonons ($aaa$) or two acoustic and one optical modes ($aao$), while processes among one acoustic and two optical phonons ($aoo$) have a minor contribution. Under the ambient condition, the double-fold degenerated nodal line between TA branches allows the acoustic phonons to bunch together, which prevents the $aaa$ processes due to the constraint of phonon scattering selection rules. As a consequence, the phonon scattering rates arising from $aaa$ processes exhibit a large dip in a wide frequency range, corresponding to the weak phase space of $aaa$ processes shown in Fig.4 (d). Upon the uniaxial tension, the opening of the degenerated TA branches largely weakens the acoustic bunching effect, resulting in the greatly increased phonon scattering rates by increasing the phase space of $aaa$ processes as illustrated in Fig.4 (d). Besides, the three optical phonon branches also show softening with strain, accompanied by the avoiding crossing between the LA an optical phonons, which makes it easier for $aao$ processes to satisfy the energy conservation rules. As a result, the phase space of $aao$ processes is considerably enlarged under the strain as seen in Fig.4 (b), thereby giving rise to the enhancement of phonon scattering involving $aao$ processes. To highlight the importance of topological effects of phonons, we also plot the phonon phase space along the $\Gamma$A line in the inset of Fig.4 (f). It is apparent that across the TPT, the phase spaces of three acoustic phonon branches are all increased significantly, providing strong evidence that topological effects of phonons strength phonon scattering by increasing available scattering channels and thus suppress $\kappa_{L}$.

In summary, we have disclosed significant impact of topological phonon phase transition in suppressing the lattice thermal transport of TiO by first-principles calculations. The systematic symmetry analyses explicitly show the coexistence of the nodal line formed by two TA branches and a TDP among the LA and optical branches. By applying an uniaxial tension, we demonstrate the TPT of these nontrivial phonons and find that the thermal conductivity is significantly suppressed, with a 78$\%$ reduction at room temperature. Further analyses on vibrational properties reveal that the giant reduction in $\kappa_{L}$ arises from the increased phonon-phonon scattering rates, which is largely due to the enlarged phase space for $aaa$ and $aao$ processes across the TPT. This finding establishes the correlation between the phononic TPT and thermal conductivity, and provides a new fundamental basis for regulating heat conduction in solids.

I Acknowledgments

X.Y. acknowledges support from the Natural Science Foundation of China (NSFC) (Grant No. 12004254) and the Chongqing Natural Science Foundation (Grant No. CSTB2022NSCQ-MSX0834). D.-S. Ma. acknowledges support from the NSFC (Grants No. 12204074) and the China National Postdoctoral Program for Innovative Talent (Grant No. BX20220367). X.L. and X.J. acknowledge support from the National Key R$\&$D Program of China (Grant No. 2018YFC1900500), the Foundation Research Fund for NSFC (Grant No. U1902217), and Chongqing Outstanding Youth Project (Grant No. CSTC2019JCYJJQX0024). Simulations have been performed on Hefei advanced computing center.