APPLIED PHYSICS LETTERS 107, 251904 (2015)

Anisotropic lattice thermal conductivity in chiral tellurium from first principles Hua Peng,1,2 Nicholas Kioussis,1,a) and Derek A. Stewart3,4 1

Department of Physics, California State University Northridge, Northridge, California 91330-8268, USA College of Physics and Optoelectronics, Taiyuan University of Technology, Taiyuan 030024, China 3 Cornell Nanoscale Facility, Cornell University, Ithaca, New York 14853, USA 4 San Jose Research Center, HGST, a Western Digital company, 3403 Yerba Buena Rd., San Jose, California 95135, USA 2

(Received 21 August 2015; accepted 2 December 2015; published online 22 December 2015) Using ab initio based calculations, we have calculated the intrinsic lattice thermal conductivity of chiral tellurium. We show that the interplay between the strong covalent intrachain and weak van der Waals interchain interactions gives rise to the phonon band gap between the lower and higher optical phonon branches. The underlying mechanism of the large anisotropy of the thermal conductivity is the anisotropy of the phonon group velocities and of the anharmonic interatomic force constants (IFCs), where large interchain anharmonic IFCs are associated with the lone electron pairs. We predict that tellurium has a large three-phonon scattering phase space that results in low thermal conductivity. The thermal conductivity anisotropy decreases under applied hydroC 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4938203] static pressure. V

Understanding heat transfer in materials is critical for many industries, ranging from microelectronics to thermoelectric devices. Since heat in semiconductors is primarily carried by lattice vibrations (phonons),1,2 developing robust theoretical approaches to predict the lattice thermal conductivity is crucial for optimizing materials properties for industrial applications. In the last few years, ab initio techniques have been developed and successfully applied to predict the thermal conductivity in common semiconductors (Si [Refs. 3 and 4], Ge [Ref. 4]), diamond,5 cubic III-V compounds,6 and other typical thermoelectric materials (PbTe [Ref. 7], SiGe [Ref. 8], skutterudites [Ref. 9], and Mg2Si [Ref. 10]). Most previous works have focused on cubic crystal structures with isotropic thermal conductivity. However, reduced symmetry crystals with anisotropic chemical bond configurations represent an interesting class of materials that can be used to engineer heat flow. At ambient pressure, elemental tellurium is a chiral material with trigonal crystal structure composed of helical chains arranged in a hexagonal array. This trigonal crystal structure is associated with an intriguing chemical-bond-hierarchy of strong intra-chain covalent bonding and weak inter-chain van der Waals bonding. Since each tellurium atom forms covalent bonding with its two nearestneighbors (NNs) along the chain, the remaining nonbonding p electron pairs form lone-pair configurations normal to the chain axis. This bonding geometry and lone-pair electrons give rise to several intriguing physical properties in tellurium. Recently, we predicted11 that tellurium undergoes a trivial insulator to strong topological insulator transition under shear strain. The strain depopulates the lone-pair valence orbital leading to a band inversion. Spin-orbit coupling (SOC) also causes considerable band structure changes, resulting in a camel-back shape of the valence band maximum that leads to large electrical conductivity and Seebeck coefficient.12 The

average room-temperature Seebeck coefficient (250 lV/K) for tellurium is comparable to Bi2Te3, making elemental tellurium a potential p-type thermoelectric. Structures with anisotropic chemical bonding or lone pair electrons play important roles in heat transport. For example, silicon spiral chains in higher manganese silicides lead to multiple low lying optical modes that increase the phase space for phonon scattering.13 Moreover, the hingelike layered SnSe crystal structure exhibits exceptionally low lattice thermal conductivity of 0.23 W/mK at 973 K,1 which was attributed to the large lattice anharmonicity arising from lone-pair electrons.14 Elemental tellurium is one of the simplest examples of anisotropic quasi-one dimensional structures16 and can serve as an important test case for understanding thermal transport in chiral materials. While the tellurium phonon spectra have been calculated using semiempirical17,18 and ab-initio approaches,19 first principles studies of the lattice thermal conductivity are still lacking. In this letter, we calculate the thermal conductivity of tellurium by iteratively solving the phonon Boltzmann transport equation that incorporates harmonic and anharmonic interatomic force constants (IFC) calculated from first principles. The results show that heat transport along the chain is larger than that along the perpendicular direction in agreement with experiment. The underlying mechanism is the anisotropy of the phonon velocity and of the anharmonic IFCs associated with lone-pair electrons which enhance the scattering processes normal to the chains. We predict that the three phonon scattering phase space is very large, comparable to that in the extremely low thermal conductivity material SnSe. Methodology—The lattice thermal conductivity for heat current in the ath direction from a temperature gradient along b can be expressed as4,5 jab ¼

a)

E-mail: [email protected]

0003-6951/2015/107(25)/251904/4/$30.00

107, 251904-1

1X Ck tka tkb skb ; V k

(1)

C 2015 AIP Publishing LLC V

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where V is the crystal volume, k  (j, q) is shorthand for phonon mode branch index j with wave vector q, Ck is the mode specific heat, and tkb and skb are the velocity component and the phonon transport lifetime along b. The specific heat and group velocities are obtained from the phonon dispersion calculated using the harmonic IFCs, while third order anharmonic IFCs are required to calculate the scattering lifetimes (skb) by iteratively solving the linearized phonon BTE.20 We consider only three-phonon scattering rates based on Fermi’s golden rule since they are the dominant contributions to the intrinsic lattice thermal conductivity in the temperature range considered.5 The details of the theoretical approach have been discussed fully in previous works.5,20,21 The IFC calculations are carried out by the supercell method using the Vienna Ab-initio Simulation Package (VASP)22 with the projector augmented wave formalism. The generalized gradient approximation (GGA) to the exchange-correlation potential in the PBE form is used.23 A plane wave cut-off energy of 500 eV is used. The harmonic IFCs including SOC are calculated using a real space supercell approach with a 6  6  4 supercell with a displacement ˚ . Tests using smaller supercells showed artifisize of 0.01 A cial imaginary frequencies and nonlinear behavior for acoustic phonon modes near C. This issue was also seen in a previous ab initio calculation of the Te phonon dispersion.19 The phonon frequencies and eigenmodes are calculated by diagonalizing the dynamical matrix, which is constructed from harmonic IFCs using the Phonopy code.24 The thirdorder anharmonic IFCs are calculated with a interaction ˚ . The ShengBTE21 package is used to range cutoff of 5.6 A solve the BTE using a 20  20  20 q mesh. The crystal structure is fully optimized before the IFC calculation. The ˚ , c ¼ 5.96 A ˚ , and inrelaxed unit cell parameters, a ¼ 4.51 A ternal atomic position parameter u ¼ 0.269 are in good agree˚ , 5.93 A ˚ , and 0.263, ment with experimental values, 4.45 A respectively. Phonon dispersion—In Fig. 1(a), we show the calculated phonon dispersion with and without SOC and compare them with room temperature neutron scattering data.25,26 Overall, the calculated phonon dispersion of the acoustic and three lower optical branches agree well with experiment. There are some discrepancies for the three higher optical branches where the theoretical values are consistently lower than the measured values. This is likely due to the fact that GGA underbinds and overestimates the lattice constants. However, since the predicted thermal conductivity is dominated by acoustic phonon modes, this will have little overall effect. The phonon dispersion exhibits a band gap of 20 cm1 between the lower and higher optical branches which results from the interplay between covalent and van der Waals bonding in tellurium.17 For example, the two lowfrequency optical branches (Figs. 1(d) and 1(e)) at C of E symmetry correspond to bond bending vibrations; the A1 branch is a chain rotating mode (Fig. 1(f)); the highfrequency A2 symmetry optical mode is a bond breathing mode (Fig. 1(g)); and the two high-frequency E modes are bond stretching modes (Figs. 1(h) and 1(i)). The bond bending and chain rotational modes are linked to interchain van der Waals interactions while the breathing and stretching modes are related to intrachain covalent bonding.

Appl. Phys. Lett. 107, 251904 (2015)

FIG. 1. (a) Phonon dispersion with (solid blue curves) and without (dashed purple) SOC. Black circles denote room-temperature experimental data.25,26 Branch index and optical mode symmetries at C are labeled. (b) Trigonal structure of tellurium containing helical chain and (c) Brillouin zone. (d)–(i) Top view of atomic displacements (blue arrows) of Te atoms in the unit cell for the six optical modes (labeled 4–9) at C.

Effect of SOC—We also examined the effect of SOC on the tellurium phonon dispersion. Although our recent ab initio calculations show that the SOC effect of the 5p electrons plays a crucial role in the electronic band structure,12 Figure 1(a) shows that it has little effect on the phonon dispersion. Therefore, to reduce computational cost, the anharmonic IFCs are determined without SOC. In trigonal tellurium, there is an additional contribution to the restoring force due to the dipole moment induced from the electronic shell deformation associated with the atomic vibrations. As shown in Fig. 1(a), the zone-center phonon frequencies of Te differ along the A-C and K-C directions. Our calculations show that the longitudinal optical-transverse optical (LO-TO) splitting of the lower optical mode near C arises from the unusually high polarizability of tellurium in sharp contrast to other elemental semiconductors. We have calculated the  ¼ 0 and mode effective Born charge,27 and found that Z?  Zk ¼ 4:2 for the rotational mode normal and parallel to the chains. The non-zero z-component of the Born effective charge gives non-analytic contributions to the dynamical matrix which causes the LO-TO splitting. Lattice Thermal Conductivity—The lattice thermal conductivity parallel and perpendicular to the chains of the converged and zeroth-order solutions to the linearized BTE as a function of temperature are shown in Fig. 2(a). The zerothorder solution is equivalent to the single-mode relaxation time approximation. The zeroth-order result is very close to

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the converged result and underestimates the full iterative solution on the average by 3%, indicating that Umklapp phonon scattering dominates three-phonon interactions.5 As the temperature increases, the thermal conductivity follows approximately a 1/T relation. Analysis of the contributions from different branches shows that the lattice thermal conductivity of tellurium is predominantly due to heat carried by acoustic phonons, which account for up to 95% (85%) of the total lattice thermal conductivity at 50 K (300 K). The lattice thermal conductivity exhibits large directional anisotropy with larger thermal transport parallel to the j chain direction. For example, at room temperature, j?k  2:3. This anisotropy can occur due to differences in group velocity, anharmonicity, and scattering phase space along the different directions. We find that the anisotropy ratio of the group velocity is t 2k ¼ 1:84t 2? (Table S115). The lattice anharmonicity can be estimated from the Gruneisen parameter, which characterizes the change of phonon frequencies with changes in crystal volume. The calculated average Gruneisen parameters at room temperature including all phonon modes along and perpendicular to the chains are 0.92 and 1.37, respectively, and in reasonably good agreement with measured values,16 0.89 and 1.05. These values are much greater than the average high temperature Gruneisen parameter found in Si [Ref. 29] (0.56) and SnSe [Ref. 30] (0.63), indicating a high degree of anharmonicity in tellurium. They also show that anharmonicity is greater in the plane perpendicular to Te chains. The third-order anharmonic IFCs show intriguing radial and angular behavior which support this. As shown in Fig. 2(b), the first NN Te pairs belong to the same chain while the second and third NN pairs lie on nearest-neighbor chains. We find that the 3rd-order IFCs between the third NN atoms (perpendicular

FIG. 2. The calculated lattice thermal conductivity parallel (red curves) and perpendicular (black curves) to the chains is shown in comparison with experimental results28 (red and black circles). The solid (dashed) curves show the converged (zeroth order) solution to the linearized BTE. (b) Top view of two nearest-neighbor chains where the interatomic bond lengths are labeled and atoms with the same z-coordinate are shown in the same color. (c) Lone-pair charge density associated with the highest valence band.

Appl. Phys. Lett. 107, 251904 (2015)

to the chains) are about eight times larger than that between the second-nearest NN atoms. This large anharmonic IFC acts to increase the scattering rates, reducing j?. Charge density analysis shows that while the p-bonding state is along the chain, the lone pair electrons (Fig. 2(c)) extend perpendicular to the chains, indicating that the high scattering rates perpendicular to the chains arise from the lone pair electrons. Minor differences between the theoretical and measured values of jL are due to the fact that the experimental data28 includes contributions from the electronic thermal conductivity. At higher temperatures, the measured thermal conductivity also shows evidence of contributions from the bipolar effect which is expected in small bandgap semiconductors10 like tellurium. Scattering Phase Space—Fig. 3 shows the three-phonon X emission and absorption scattering phase space,31 P6 3 ¼ ð6npÞ3 ÐÐ P 0 0 3 0 3 0 00 j;j0 ;j00 BZ d½xj ðqÞ6xj ðq Þ þ xj ðq þ q  QÞd q d q, for the nine phonon modes. Here, X is the unit cell volume, n is the number of phonon branches, Q is a reciprocal lattice vector, and the þ() sign refers to absorption (emission) processes. We find that P3 ¼ 1.67  102 eV1 in Te is rather large, i.e., about five time larger than that in silicon and comparable to the lowest conductivity SnSe compound30 (1.9  102 eV1). This high phonon scattering probability results in low thermal conductivity. P 3 increases with increasing frequency since more phonons can take part in threephonon emission scattering processes. The large P 3 is due to significant acoustic-acoustic-optical scattering of the lowest frequency optical phonons around 1.8 THz (60 cm1). Fig. 3(b) clearly shows that the acoustic and three lower optical branches have large three-phonon scattering phase space for absorption over a wide frequency range. In order to study the thermal conductivity in nanocrystalline tellurium, we show in P Fig. 4 the cumulative thermal K
FIG. 3. Room temperature (a) emission and (b) absorption three-phonon scattering phase space. Colors correspond to modes.

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Appl. Phys. Lett. 107, 251904 (2015)

conductivity materials, is the cause of its small lattice thermal conductivity. The lattice thermal conductivity increases with pressure with a larger effect on the heat flow perpendicular to the chains which in turn reduces the anisotropy. The research at CSUN was supported by NSF-PREM Grant No. DMR-1205734 and by Grant No. HDTRA1-10-10113. H.P. acknowledges the travel support by Natural Foundation of Shanxi Grant. No. 2013021010-2. 1

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FIG. 4. Normalized cumulative thermal conductivity of tellurium parallel (red curves) and perpendicular (black curves) to the chains at 100 K and room temperature. TABLE I. Effect of hydrostatic pressure (in GPa) on the lattice constants a ˚ ) and on the lattice thermal conductivities (in W/mK) at room and c (in A temperature. P 0 0.4 0.9

a

c

j?

jk

jk =j?

4.509 4.462 4.415

5.959 5.964 5.967

1.22 1.44 1.68

2.77 3.01 3.26

2.27 2.09 1.94

phonon branch with wave vector q and dj is the differential thermal conductivity. At 300 K, j? is primarily due to the phonons with MFPs from 1nm to 110nm, while for jk the MFP range expands to 1 lm. As temperature decreases, the mean free path increases. These results suggest that introducing nanoparticles with size less than 100nm in bulk tellurium can reduce the total lattice thermal conductivity by up to 30%. Pressure Effects—The effect of hydrostatic pressure on the lattice constants and the parallel (jk ) or perpendicular (j?) lattice thermal conductivity is shown in Table I. Applied pressure leads to a strong decrease (weak increase) in the interchain (intrachain) distances which in turn tune the interchain and intrachain IFCs, the latter being about 10% of the former (0.32  104 dyn/cm). We find that both room-temperature jk and j? increase with increasing pressure while the heat transport anisotropy is reduced. As the chains come closer together under pressure, this increases the inter-chain IFC and the thermal conductivity perpendicular to the chains. In conclusion, using an ab initio based phonon Boltzmann transport approach we have investigated the lattice thermal conductivity of trigonal tellurium. The large anisotropy of the thermal conductivity parallel and perpendicular to the chains arises from the anisotropy of the phonon velocities and of the anharmonic IFCs associated with the lone-pair electrons perpendicular to the chains which gives rise to high scattering rates. The extremely large three-phonon scattering phase space in tellurium, comparable with other known low-

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