Intrinsically low lattice thermal conductivity in layered Mn3Si2Te6

The ferrimagnetic nodal-line semiconductor Mn3Si2Te6 has recently received much attention due to its colossal angular magnetoresistance (Seo et al 2021 Nature 599 581). The magnetic and electronic properties of Mn3Si2Te6 have been extensively studied. Meanwhile, a recent experiment showed that Mn3Si2Te6 has a low in-plane lattice thermal conductivity, which implies its potential applications in thermoelectricity. Here, we have investigated phonon dispersion and lattice thermal conductivity of Mn3Si2Te6 by the first-principles calculations and the Peierls–Boltzmann transport equation. It is found that the lattice thermal conductivities of Mn3Si2Te6 are quite low, which are 1.33 and 0.96 Wm−1K−1 along the a and c axes at 300 K, respectively. A significant contribution (>90%) to the thermal conductivity comes from the acoustic phonons and low-frequency optical phonons linked to the vibration of Te atoms. Meanwhile, it is found that such low thermal conductivities of Mn3Si2Te6 are a consequence of the low group velocities and relatively short phonon lifetimes, which are intrinsically derived from the quite complex crystal structure, heavy Te atoms, and relatively weak chemical bonding. Our work not only explains the origin of the intrinsically low thermal conductivity of Mn3Si2Te6 but also could be helpful to the study on the thermal conductivity of other similar layered magnetic materials.


Introduction
Layered magnetic materials have received extensive attention due to their exotic physical properties, including colossal magnetoresistance (CMR) [1,2], long-range magnetic order [3], high-temperature superconductivity [4], quantum anomalous Hall effect [5], and electronic phase separation [6], which are induced by their spin, orbital, charge, and lattice degrees of freedom.The magnetic cations in these materials are often arranged in either a honeycomb or triangular lattice of each monolayer.One of the important features in layered magnetic materials is the anisotropic interaction resulting from the interlayer van der Waals gap, that is, the weak interlayer interaction and strong intralayer interaction.Moreover, some layered magnetic materials have been reported as promising candidates in thermoelectric applications because of their low thermal conductivities [7,8].
Recently, Mn 3 Si 2 Te 6 , a layered ferrimagnetic nodal-line semiconductor [9,10], has attracted increasing interest since the discovery of its special angular-dependent CMR [10,11].Unlike other CMR materials, the Mn 3 Si 2 Te 6 only manifests CMR when the magnetic field is applied along the magnetically hard c-axis.The in-plane resistivity of Mn 3 Si 2 Te 6 drops by seven orders of magnitude if the magnetic field is above 9 T [10,11].It was also reported that there is an exotic quantum state driven by ab plane chiral orbital currents in Mn 3 Si 2 Te 6 [12].These intriguing features have inspired a lot of research works to unearth the electronic and magnetic properties of Mn 3 Si 2 Te 6 [13][14][15][16].
Besides the study of electrical transport properties, a recent experiment shows that the thermal conductivity of Mn 3 Si 2 Te 6 is rather low, which can be further suppressed by applying the magnetic field [17].At 300 K, the in-plane thermal conductivity of Mn 3 Si 2 Te 6 is about 1.76 WK −1 m −1 and the electronic thermal conductivity is negligible due to its high resistivity [17].Such a low lattice thermal conductivity (κ) suggests that Mn 3 Si 2 Te 6 has great potential applications in various fields, such as thermal barrier coatings and thermoelectrics.Compared with the extensive investigations on the magnetic properties of Mn 3 Si 2 Te 6 , a comprehensive and quantitative understanding of its lattice dynamics is lacking, which is crucial to identify the origin of low lattice thermal conductivity in Mn 3 Si 2 Te 6 .
In this work, we performed first-principles calculations to explore the lattice dynamics and thermal conductivity of Mn 3 Si 2 Te 6 .We discovered that the low lattice thermal conductivity is predominantly attributed to the low phonon group velocities which are connected with the optical phonon branches intersecting the acoustic branches caused by the vibration of Te atoms.This is distinct from many thermoelectric compounds with low lattice thermal conductivity which manifest themselves in strong anharmonicity.

Methodology
The first principles calculations are performed based on the density functional theory (DFT) by the Vienna Ab-initio Simulation Package [18,19].We use a plane wave cutoff energy of 350 eV within the exchange and correlation functional of Perdew-Burke-Ernzerhof for solid (PBEsol) form [20].The cell and atomic positions are fully optimized until the total energies and maximal Hellmann-Feynman force are converged within 10 −8 eV and 10 −4 eV Å −1 , respectively.The DFT-D3 method of Grimme [21] is employed to obtain the van der Waals corrections.The experimental ferrimagnetic configuration is used in our calculations.An effective Hubbard U of 3 eV is set for Mn's 3d orbitals to consider the electron correlation.
The harmonic interatomic force constants (IFCs) are calculated by the finite-displacement approach [22], which is implemented in the ALAMODE package [23,24].The 2 × 2 × 2 supercells containing 176 atoms are used.The 3 × 3 × 1 k-mesh is adopted to sample the Brillouin zone.The cubic and quartic IFCs are obtained by compressive sensing lattice dynamics [25].To further account for the anharmonic renormalization resulting from quartic anharmonicity, the self-consistent phonon (SCPH) theory [26] is used.The SCPH equations are solved on a 14 × 14 × 7 q mesh, which is confirmed to be sufficient.The lattice thermal conductivities are calculated by the ALAMODE code [23,24].

Phonon dispersions and Raman-active modes
The crystal structure of Mn 3 Si 2 Te 6 is similar to that of the quasi-two-dimensional ferromagnet CrSiTe 3 .In the ab plane, it is composed of the edge-sharing MnTe 6 (Mn 1 site) octahedra and Si-Si dimers, which form the layers of MnSiTe 3 .In addition, other Mn atoms (Mn 2 site) are intercalated between the MnSiTe 3 layers and Mn 1 -Mn 2 is linked through the face-sharing octahedral along the c axis.The side and top views of the crystal structure of Mn 3 Si 2 Te 6 are shown in figures 1(a) and (b), respectively.Our optimized lattice constants are a = b = 6.93 Å and c = 13.87Å, which are in good agreement with the low-temperature experimental values (a = b = 7.017 and c = 14.172Å) [11].
Based on the optimized structure, we calculated the phonon dispersion of Mn 3 Si 2 Te 6 , which is shown in figure 1(c).It is clear that the phonon dispersion has no imaginary modes in the whole Brillouin zone, indicating the dynamical stability of Mn 3 Si 2 Te 6 .There are a total of 66 phonon branches (3 acoustic and 63 optical ones) due to the 22 atoms in the primitive cell.Intuitively, the whole phonon dispersions are separated into four frequency ranges isolated by the full gaps.The highest frequency of three acoustic branches is only ∼1.4 THz (see the enlarged plot in figure S1), which is even lower than that (∼2.0 THz) [27] of SnSe with low lattice thermal conductivity (κ = 0.62 Wm −1 K −1 at 300 K) [27].The almost dispersionless phonon bands at ∼6.5, 10.9, and 14.3 THz are observed.We can infer that these high-frequency optical phonons have low phonon group velocities, thus these phonon branches only have a negligible contribution to the lattice thermal conductivity.At the same time, it is found that there are many low-frequency optical branches in the phonon dispersion.It is noted that the low-frequency optical branches are strongly overlapped with the acoustic branches.Such a significant entanglement of acoustic and optical modes would invoke intensive phonon scattering.Hence, the low-frequency phonon lifetimes will be dramatically suppressed.The suppression of phonon lifetimes is harmful to the phonon transport.Furthermore, these low-frequency phonons play the most important role in the thermal transport.Therefore, the significant overlap between the optical modes and acoustic modes leads to the low thermal conductivity.These results are consistent with the subsequent conclusions related to the frequency-dependent thermal conductivity.
The projected density of states (PDOS) in figure 1(d) shows that the high-frequency (above 10.9 THz) phonon modes are governed by the vibration of Si atoms due to the light Si atoms and strong Si-Si bonds.A sharp PDOS peak appears at about 6.5 THz, which is composed of the phonon branches linked to the Si and Mn atoms.In the frequency range of 3.5-5.2THz, the large mixing modes caused by the vibrations of Mn and Te atoms can be observed.The Te atoms' vibrations predominate below 3.5 THz due to the heavy atom mass, which implies that the lattice thermal conductivity of Mn 3 Si 2 Te 6 is mainly related to the phonons associated with Te atoms.This is confirmed by the subsequent calculations.
Based on the phonon dispersion of Mn 3 Si 2 Te 6 , the Raman modes can be easily obtained.The Mn 3 Si 2 Te 6 belongs to the trigonal crystal system (space group P 31c, No. 163), and the corresponding irreducible representation at Γ point is 5A 1g ⊕11E g ⊕6A 2u ⊕11E u ⊕5A 2g ⊕6A 2u , in which the A 1g and E g modes are Raman-active.Therefore, there are a total of 16 Raman-active modes in Mn 3 Si 2 Te 6 which are listed in table 1.For comparison, some other theoretical calculations and experimental data [15,28] are also given in table 1.We have converted the unit of the calculated Raman frequency from THz to cm −1 for the convenience of comparison.It is found that our calculated results are in good agreement with the recent Raman measurement [15,28], confirming the accuracy of the PBEsol + D3 + U calculations in Mn 3 Si 2 Te 6 .
We further calculated the sound velocities of Mn 3 Si 2 Te 6 by the slope of acoustic phonon branches, which are listed in table 2. The average sound velocity is calculated by the formula [27,29]: where i represents the direction of x, y, or z.The average sound velocity of Mn 3 Si 2 Te 6 along the x (y) axis is slightly larger than that in the z-direction.Specifically, the maximum sound velocities are no more than 3.78  [27], and BiOCuSe (v LA = 4.095 km s −1 , v TA = 2.394 km s −1 , κ = 0.9 Wm −1 K −1 at 300 K) [31].
Elastic properties are often used to qualitatively evaluate the lattice thermal conductivity.Generally, low Young's modulus (E) and large Grüneisen parameter (γ) signify a small lattice thermal conductivity [27,32].The calculated elastic modulus (85.13 GPa) of Mn 3 Si 2 Te 6 along the x and y directions is also higher than that (52.60 GPa) along the z direction.This indicates that the in-plane chemical bonds of Mn 3 Si 2 Te 6 are stronger than those between the layers, but their difference is not very large.Our calculated E, G, Poisson's ratio (ν), Debye temperature (Θ D ), and γ of Mn 3 Si 2 Te 6 are 61.97GPa, 25.05 GPa, 0.24, 233 K, and 1.45, respectively, which are also presented in table S1.It is interesting to compare the above elastic parameters with those of some well-known thermoelectric materials with low κ, as illustrated in table S1.The E (61.97 GPa) of Mn 3 Si 2 Te 6 is lower than that of Bi 2 Se 3 (70.3GPa) [30] and BiCuOSe (76.5 GPa) [31], but higher than those of Sb 2 Se 3 (45.3GPa) [30] and PbTe (53.7 GPa) [27].The Grüneisen parameter γ of Mn 3 Si 2 Te 6 is comparable to those of Sb 2 Se 3 (γ = 1.3) [30] and PbTe (γ = 1.49) [27].The low acoustic phonon velocities, Young's modulus, and moderate Grüneisen parameter are expected to lead to the low κ in Mn 3 Si 2 Te 6 .

Lattice thermal conductivity
We now turn to the calculations of the lattice thermal conductivity of Mn 3 Si 2 Te 6 .We employ the CLSD method [25] to calculate κ based on the Peierls-Boltzmann transport equation.In the framework, the lattice thermal conductivity tensor κ µν is given below: where N q and V p is the total number of q points and the volume of the primitive unit cell.C jq represents the heat capacity.µ and ν represent the direction of x, y, or z. v µ jq and v ν jq are the group velocity tensor for the phonon mode with the wavevector of q and band index of j. τ jq is the phonon lifetime for the phonon mode of jq.In the calculations, different approximations can be used.For example, if the lattice thermal conductivity is calculated by using the phonon dispersion based on the harmonic approximation (HA) and the phonon lifetimes based on the three-phonon (3ph) scattering, it can be named as HA + 3ph method.If the phonon dispersion is corrected by the SCPH theory, it is called the SCPH + 3 ph method.Figure 2(a) shows the lattice thermal conductivities of the Mn 3 Si 2 Te 6 in the range of 300-800 K obtained by the HA + 3 ph and SCPH + 3 ph methods.It is found that the κ by using HA + 3 ph method is about 0.3 Wm −1 K −1 lower than the corresponding value by the SCPH + 3 ph method, which is similar to the case of Tl 3 VSe 4 [33].The calculated lattice thermal conductivities by using the HA + 3 ph method follow the power law κ ∝ T −1 , while the ones by the SCPH + 3 ph method obey the power law κ ∝ T −0.78 .The combination of 3ph interactions and anharmonic renormalization stemmed from quartic anharmonicity results in the unusual temperature dependence.Because the HA + 3 ph method often underestimates κ [33], we will only discuss the results based on the SCPH + 3 ph method in the following analysis.
Figure 2(a) indicates that Mn 3 Si 2 Te 6 indeed has low lattice thermal conductivities, which are 1.33 and 0.96 Wm −1 K −1 along the a-axis and c-axis directions at 300 K.The values are comparable with those of some well-known thermoelectric materials, such as PbTe (2.3 Wm −1 K −1 ) [27], Sb 2 Se 3 (1.30Wm −1 K −1 ) [30] and BiOCuSe (0.9 Wm −1 K −1 ) [31].When the temperature is raised to 800 K, the a-axis and c-axis lattice thermal conductivities of Mn 3 Si 2 Te 6 decrease to 0.62 and 0.46 Wm −1 K −1 , respectively.It is noticed that the thermal conductivity of Mn 3 Si 2 Te 6 shows a small anisotropy, which is related to its special layered structure.
In order to analyze the size effects on the lattice thermal transport, we further calculated the cumulative thermal conductivity of Mn 3 Si 2 Te 6 at room temperature, as displayed in figure 2(b), which is linked to the accumulated contributions of the mean-free path (MFP) to the thermal conductivity [27].The cumulative κ values first increase as MFP, then gradually approach a plateau, and the maximum values of MFP in Mn 3 Si 2 Te 6 along the a-and c-axes are about 342 and 811 nm, respectively.The contribution of phonons within MFPs less than 50 nm contributes more than 70% to the lattice thermal conductivity.It implies that the nanostructuring with a length of 50 nm would only potentially decrease the lattice thermal conductivity by 30% at the most.Therefore, we think nanostructuring may be challenging in the experiment to lower the lattice thermal conductivity of Mn 3 Si 2 Te 6 .The representative MFP (rMFP) is defined as the size of MFP and all phonons with an MFP smaller than this value contribute half of the thermal conductivity.The rMFP of Mn 3 Si 2 Te 6 is about 20.8 and 22.0 nm along the a-and c-axes which are much longer than that (∼5 nm) of SnSe [34].
We also calculated the frequency-dependent thermal conductivity at room temperature, as shown in figure 3. The width of each column in figure 3 is 0.7 THz, and the total thermal conductivity is the summation of all columns.Obviously, most of the κ results from the phonon modes below 3.5 THz, i.e. the vibrations of Te atoms.The thermal conductivity contributed by phonons below 3.5 THz accounts for about 92% and 91% of the total thermal conductivity along the a-and c-axes, respectively.Noticeably, the phonon modes ranging from 0.7 to 1.4 THz contribute significantly to lattice thermal conductivity, especially along the a-axis direction, the corresponding contribution ratio is circa 46%.As seen in figure S1, there is a large overlap between the acoustic and low-frequency optical phonon modes in this range.These results imply the lattice thermal conductivities of Mn 3 Si 2 Te 6 mainly stem from the vibration of Te atoms.

Group velocities and phonon lifetimes
To further gain insight into the thermal conductivity of Mn 3 Si 2 Te 6 , we also calculated its phonon group velocities, lifetimes, and Grüneisen, which are given in figure 4. According to the equation ( 2), the phonon group velocities and lifetimes have important influences on the lattice thermal conductivity.At 300 K, Both phonon group velocities and lifetimes below 3.5 THz exhibit large values, which indicates that these  low-frequency phonons have a large contribution to the lattice thermal conductivity.These findings are consistent with the above conclusion from the frequency-dependent thermal conductivity.Furthermore, it is also found that the phonon group velocities along the a-and c-axes exhibit a little anisotropy.Especially between 1 and 2 THz, the group velocities along the a-axis are slightly larger than that along the c-axis, which is the main reason for the anisotropic thermal conductivity of Mn 3 Si 2 Te 6 .It is found that the group velocities of Mn 3 Si 2 Te 6 are excessively low, and most of the group velocities are not more than 1 km s −1 .The low phonon group velocity is the main reason for the low lattice thermal conductivity of Mn 3 Si 2 Te 6 .
As shown in figure 4(b), it is found that the lifetimes of most phonons in Mn 3 Si 2 Te 6 are less than 15 ps but the ones of low-frequency phonons can exceed 100 ps.The phonon lifetimes are comparable with that of BiCuOSe (0-10 ps) [35].Finally, we also calculated the Grüneisen parameters γ of Mn 3 Si 2 Te 6 at 300 K, which mostly lie between −1.5 and 2.2, as shown in figure 4(c).The absolute value of γ is slightly lower than that of PbTe (γ ≈ −2.5-3.0,κ = 2.3 at 300 K) [27], and much smaller than those of some other materials such as Sb 2 Se 3 (γ ≈ 0-10) [30] and BiOCuSe (γ ≈ 0-8) [31].Compared with the well-known materials with low κ, the phonon lifetimes of Mn 3 Si 2 Te 6 are moderately low, and the Grüneisen parameters are not outstanding.Therefore, Mn 3 Si 2 Te 6 does not exhibit very strong anharmonicity.

Discussion
The calculated thermal conductivities of Mn 3 Si 2 Te 6 based on SCPH + 3 ph are 1.33 and 0.96 Wm −1 K −1 along the a-and c-axes, respectively.The measured in-plane thermal conductivity is about 1.76 Wm −1 K −1 at 300 K [17].Our calculated results are somewhat lower than the experimental values.The possible reason is that we neglect the off-diagonal terms of the heat-flux operator which correspond to the loss of coherence between different vibrational eigenstates and heat conduction attached to the wave-like tunneling [36].We did not calculate it because the off-diagonal terms are too computationally expensive for Mn 3 Si 2 Te 6 .Considering the off-diagonal term will appropriately improve the lattice thermal conductivity [33].
Layered materials often exhibit strong anisotropy in thermal conductivity between in-plane and out-of-plane directions [37,38] due to their weak van der Waals interactions.The difference between in-plane and out-of-plane κ in the transition metal dichalcogenides such as WSe 2 and WTe 2 is even circa an order of magnitude [37].However, the in-plane and out-of-plane κ of the layered ferrimagnet Mn 3 Si 2 Te 6 only exhibit weak anisotropy with the anisotropic ratio κ a /κ c ≈ 1.39.Compared with other layered materials, the interlayers of Mn 3 Si 2 Te 6 are additionally filled with one-third of Mn atoms, so the interlayer interactions are greatly enhanced.
The low thermal conductivity of Mn 3 Si 2 Te 6 essentially comes from its quite complex crystal structure and heavy Te elements by the analysis of the crystal structure.Meanwhile, a small elastic modulus indicates that the chemical bonds in Mn 3 Si 2 Te 6 are weak, which also usually leads to low lattice thermal conductivity.In addition to low thermal conductivity, Mn 3 Si 2 Te 6 has a quite large Seebeck coefficient, which is 507 µVK −1 at 300 K [17].However, the experiment also showed that the electric conductivity of Mn 3 Si 2 Te 6 is quite small, resulting in a low thermoelectric figure of merit (about 0.01 at 300 K) [17].The electric conductivity of Mn 3 Si 2 Te 6 can be expected to be improved by doping in experiments, thereby enhancing its thermoelectric efficiency.
In addition, we also checked the effect of Hubbard U on the above results.As shown in table S2, the lattice constants of Mn 3 Si 2 Te 6 increase as the Hubbard U increases from 3 to 6 eV.When the U is 6 eV, the difference between calculated lattice constants and experimental values [11] are 0.68% and 0.54% for the in-plane and out-of-plane directions.However, the phonon dispersions between U = 3 and 6 eV are almost the same, as shown in figure S2.The Raman frequencies with U = 3 and 6 eV are given in table S3, in which we find that they differ by at most a few cm −1 and more Raman mode frequencies at U = 3 eV match better with the experiments [15,28].The band structures of Mn 3 Si 2 Te 6 with U = 3 and 6 eV are shown in figure S3.It can be seen the semiconductor characteristics with indirect band gap are not affected by the value of U, and the band gaps increase as U increases.The calculated local magnetic moments and band gaps for U = 3 and 6 eV are presented in table S4.The corresponding local magnetic moments of Mn slightly increase with the increase of U, and the magnetic moments and band gap with U = 3 are better consistent with previous theoretical results [39].We also calculated the lattice thermal conductivities at T = 300 K for U = 6 eV.The calculated in-plane and out-of-plane lattice thermal conductivities obtained by SCPH + 3 ph (HA + 3 ph) method are 1.24 (1.11) Wm −1 K −1 and 0.96 (0.83) Wm −1 K −1 , respectively.Therefore, the difference of lattice thermal conductivity between U = 3 and 6 eV is less than 0.1 Wm −1 K −1 at 300 K.

Conclusion
We systematically investigated the phonon, elastic properties, and intrinsic lattice thermal properties of Mn 3 Si 2 Te 6 based on the first-principle three-phonon calculations.First, we confirmed the reliability of the calculation parameters by comparing the theoretical and experimental Raman frequencies.Then, we find that the thermal conductivities of Mn 3 Si 2 Te 6 are quite low, which are 1.33 and 0.96 Wm −1 K −1 along the aand c-axes at 300 K, respectively.The small anisotropy of the thermal conductivity between the a-and c-axes is due to the additional intercalated Mn atoms between the layers.From the frequency point of view, the phonons between 0 and 3.5 THz, that is, the acoustic and low-frequency optical ones, contribute most (>90%) of the thermal conductivity.Further calculations revealed that the low thermal conductivity of Mn 3 Si 2 Te 6 comes from the low group velocities and relatively short phonon lifetimes, which is closely related to its quite complex crystal structure, heavy Te elements, and weak chemical bonds.Our research not only provides an insightful view to understand the thermal transport characteristics of Mn 3 Si 2 Te 6 but also could promote the application of the materials.

Figure 1 .
Figure 1.(a) Side and (b) top views of the crystal structure of Mn3Si2Te6.The purple, blue, and orange balls represent Mn, Si, and Te atoms, respectively.The primitive unit cell is indicated by the black line.(c) Phonon dispersions and (d) projected density of states (PDOS) in Mn3Si2Te6.The red, purple, and green dash highlight one longitudinal and two transverse acoustic (LA, TA1, TA2) branches.

Figure 3 .
Figure 3. Calculated frequency-dependent thermal conductivities at 300 K along the (a) a-axis and (b) c-axis, respectively.

Table 2 .
Calculated sound velocities of the longitudinal acoustic (LA) and two transverse acoustic (TA1, TA2) modes in Mn3Te2Te6.
and 2.56 km s −1 for the LA and TA branches, respectively.These values are comparable to those of good thermoelectric materials with low lattice thermal conductivities, such as Bi 2 Se 3