Ultralow Lattice Thermal Conductivity and Thermoelectric Properties of

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Cite This: ACS Appl. Energy Mater. 2019, 2, 3004−3008

Ultralow Lattice Thermal Conductivity and Thermoelectric Properties of Monolayer Tl2O Muhammad Sajjad, Nirpendra Singh, Shahid Sattar, Stefaan De Wolf, and Udo Schwingenschlögl* Physical Science and Engineering Division (PSE), King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia

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S Supporting Information *

ABSTRACT: We report first-principles results on the thermal and thermoelectric properties of monolayer Tl2O. The lattice thermal conductivity and electronic transport coefficients are obtained by semiclassical Boltzmann transport theory. Monolayer Tl2O is found to be a semiconductor with a direct band gap of 1.62 eV. The lattice thermal conductivity turns out to be ultralow, for example, 0.17 W/mK at 300 K. Combined with a high power factor, this results in excellent thermoelectric performance. For example, at 500 K the p-type and ntype thermoelectric figures of merit reach peak values of 0.96 and 0.94 at hole and electron concentrations of 1.2 × 1011 and 0.8 × 1011 cm−2, respectively. KEYWORDS: lattice thermal conductivity, thermoelectric properties, monolayer, Tl2O, first principles

T

We employ the Vienna ab initio simulation package.14 The long-range van der Waals interaction is taken into account by means of the DFT-D3 method,15 and a cutoff energy of 700 eV is adopted for the plane wave basis set. To construct monolayer Tl2O, we start from the unit cell of bulk Tl2O with optimized in-plane lattice parameter (3.58 Å), keep one of the six layers (three atoms), and set the out-of-plane lattice parameter of the simulation cell to 23 Å. The generalized gradient approximation (Perdew−Burke−Ernzerhof flavor) of the exchange correlation functional and a Γ-centered 10 × 10 × 1 k-mesh are used for the structure relaxation, which is assumed to be converged when the Hellmann−Feynman forces have declined below 10−4 eV/Å for all atoms. Selfconsistent and non-self-consistent calculations are performed on 20 × 20 × 1 and 40 × 40 × 1 k-meshes, respectively, employing the Heyd−Scuseria−Ernzerhof hybrid exchange− correlation functional. The electronic transport coefficients are calculated by Boltzmann transport theory within the constant relaxation time and rigid band approximations, as implemented in the BoltzTraP code.16 This approach has shown reliability in determining the transport properties of a variety of materials.17−19 In order to evaluate the heat conduction by lattice vibrations, κl is obtained from the phonon Boltzmann transport equation using the ShengBTE code,20 which requires the second and third order force constants as input. The second

he efficiency of a thermoelectric material in converting heat into electricity (and vice versa) can be measured by the dimensionless figure of merit ZT = S2σT/(κe + κl), where S is the Seebeck coefficient (thermopower), σ the electrical conductivity, and κe/l the electronic/lattice contribution to the thermal conductivity.1 Efficient materials, therefore, combine a high power factor (S2σ) with a low thermal conductivity (κe + κl). Finding materials with large ZT, however, is a challenging task, because S, σ, κe, and κl are interdependent quantities. In many cases, the thermoelectric performance can be enhanced by disorder,2 resonant doping,3 nanostructuring,4 and rattling modes,5 for example. Low-dimensional materials with high density of states at the Fermi energy are also interesting for thermoelectric applications, as phonon scattering at the surface lowers the lattice thermal conductivity6,7 and thus enhances the figure of merit as compared to their bulk counterparts. Thallium oxide appears in three modifications: trigonal Tl2O,8 cubic Tl2O3,9 and monoclinic Tl4O3.10 Tl2O has a layered structure and can be prepared by thermal decomposition of Tl2CO3.8 It is used as Tl source for highly refractive optical glasses and synthetic gemstones.11 The cleavage energy of 0.43 J m−1 is similar to that of graphene (0.37 J m−1) and MoS2 (0.42 J m−1),12,13 indicating that metal-shrouded monolayer Tl2O can be fabricated by mechanical exfoliation. In the present study, we use first-principles calculations and Boltzmann transport theory to determine the thermal and thermoelectric properties of monolayer Tl2O, which combines the advantage of low dimensionality with the high atomic weight of Tl. Neither experimental nor theoretical investigations are available in the literature. © 2019 American Chemical Society

Received: February 6, 2019 Accepted: April 23, 2019 Published: April 30, 2019 3004

DOI: 10.1021/acsaem.9b00249 ACS Appl. Energy Mater. 2019, 2, 3004−3008

Letter

ACS Applied Energy Materials order force constants are determined using the Phonopy code21 with a 5 × 5 × 1 supercell and 3 × 3 × 1 k-mesh. We have checked that there are no relevant differences in the results when we use a 6 × 6 × 1 supercell and 5 × 5 × 1 kmesh. The third order force constants are obtained by a finite displacement scheme22 in that two atoms up to eighth nearest neighbors are displaced simultaneously, corresponding to an interaction range of 8.1 Å and a total of 432 displacements. A 60 × 60 × 1 q-mesh is used, which is well converged, as the value of κl at 300 K deviates by less than 1% from the result obtained with an alternative 80 × 80 × 1 q-mesh. In the case of bulk materials the BoltzTraP and ShengBTE codes normalize the results automatically correctly, while in the case of a monolayer an additional scaling is required. The nominal thickness of monolayer Tl2O for this scaling (multiplication with the out-of-plane lattice parameter of the simulation cell and division by the nominal thickness) is set to 6.34 Å (optimized out-of-plane lattice parameter of bulk Tl2O, 38.06 Å, divided by the number of layers per bulk unit cell, six). Trigonal bulk Tl2O has space group R3̅m (No. 166).8 Referring to Figure 1, monolayer Tl2O is derived from bulk

Figure 2. (a) Electronic band structure, (b) phonon band structure, and (c) partial electronic densities of states of monolayer Tl2O. The full and dotted lines in panel b represent results for 5 × 5 × 1 and 6 × 6 × 1 supercells, showing full convergence of the acoustic phonons.

functions of the carrier concentration (for both holes and electrons), where τ is the relaxation time. As expected, both σ/τ and κe/τ are enhanced when the carrier concentration increases. Because the hole effective mass is larger than the electron effective mass (see Figure 2a), |S| is higher in the case of hole carriers than in the case of electron carriers. It turns out that hole carriers also result in higher S2σ/τ than electron carriers (see Figure 3d). At 300, 400, and 500 K, S2σ/τ reaches very high peak values of 10.7 × 1011, 13.0 × 1011, and 15.0 × 1011 W/(mK2 s) (5.2 × 1011, 6.9 × 1011, and 8.6 × 1011 W/ (mK2 s)) at hole (electron) concentrations of 2.8 × 1013, 3.3 × 1013, and 3.9 × 1013 cm−2 (0.8 × 1013, 1.0 × 1013, and 1.2 × 1013 cm−2), respectively. At 300 K, for example, the peak value is 14 (6) times higher than in the case of monolayer MoSe2 (0.8 × 1011 W/(mK2 s) at a hole concentration of 1.6 × 1013 cm−2 and 0.9 × 1011 W/(mK2 s) at an electron concentration of 1.5 × 1013 cm−2) and 4 (3) times higher than in the case of monolayer SnSe2 (2.8 × 1011 W/(mK2 s) at a hole concentration of 3.4 × 10 13 cm −2 and 1.7 × 10 11 W/(mK2 s) at an electron concentration of 1.4 × 1013 cm−2).24,25 Constituting the main aspect of our analysis of monolayer Tl2O, we next address the heat conduction by lattice vibrations. The phonon band structure in Figure 2b, obtained with nonanalytical corrections to account for the ionicity of the bonds, shows nine phonon branches, with A1′(R) + 2A2″(IR) + 2E′(R IR) + E″(R) decomposition at the Γ point (R, Raman activity; IR, infrared activity).26 While the heat conduction is mainly due to the acoustic phonons, the low-energy optical phonons are important as they increase the scattering crosssection of the acoustic phonons.27 Figure 4a demonstrates low values of κl (in-plane) with a significant temperature dependence. We obtain at 300 K, for example, a value of 0.17 W/mK (0.18 W/mK when only seventh instead of eighth nearest neighbors are considered), which is much smaller than in conventional thermoelectric materials such as PbTe (2.00 W/mK) and Bi2Te3 (1.24 W/mK).28,29 It is dramatically

Figure 1. (a) Top and (b) side views of the crystal structure of monolayer Tl2O. Tl atoms appear in gray (large spheres) and O atoms in red (small spheres). The circle indicates the considered interaction range.

Tl2O by cleaving the (001) surface to obtain a structure with a layer of O atoms sandwiched between layers of Tl atoms. The O atoms therefore are octahedrally coordinated by six Tl atoms. Optimization of the structure results in a lattice parameter of 3.60 Å and a Tl−O bond length of 2.58 Å, in agreement with ref 13 (3.58 and 2.57 Å). We obtain a direct band gap of 1.62 eV (see Figure 2a), which is slightly larger than the value reported in ref 13 (1.56 eV). According to the partial densities of states shown in Figure 2c, the top of the valence band is composed of Tl s,p and O p states, whereas the Tl s,p states dominate at the bottom of the conduction band. As a consequence, effects of spin−orbit coupling on the results presented in the following are negligible. Turning to the thermoelectric performance of monolayer Tl2O, we study carrier concentrations from 108 to 5 × 1013 cm−2 and temperatures from 300 to 500 K, noting that the melting point of bulk Tl2O is 852 K.23 The calculated in-plane electronic transport coefficients are given in Figure 3a−c as 3005

DOI: 10.1021/acsaem.9b00249 ACS Appl. Energy Mater. 2019, 2, 3004−3008

Letter

ACS Applied Energy Materials

Figure 4. (a) Lattice thermal conductivity as a function of the temperature, (b) phonon scattering rate at 300 K as a function of the frequency, and (c) cumulative lattice thermal conductivity as a function of the mean free path.

Figure 3. Electronic and thermoelectric properties of monolayer Tl2O as functions of the carrier concentration (left, holes; right, electrons).

Table 1. Comparison of Lattice Thermal Conductivities and Spring Constantsa

smaller than predicted for monolayer transition metal dichalcogenides30 (see Table 1). We notice that a value of κl < 2 W/mK is usually set as target to be achieved by highperformance thermoelectric materials.31 The low κl of monolayer Tl2O can be explained by the low cutoff frequency of the acoustic phonon branches (53 cm−1 (see Figure 2b; Debye temperature, 76.3 K), which reflects low group velocities. Additionally, the mentioned coupling between the acoustic and optical phonons results in high phonon scattering rates (see Figure 4b). With a spring constant of only 1.6 eV/Å2 (trace of the harmonic force constant tensor) the Tl−O bonds are much less stiff than the metal−chalcogen bonds in monolayer transition metal dichalcogenides30 (see Table 1) because of the large atomic size of Tl and the involvement of Tl 6p electrons in the bonding. We next study the average mode Grüneisen parameter (∑i γiCV,i/∑i CV,i , where γi and CV,i are the Grüneisen parameter and specific heat

TlO2 MoS2 MoSe2 WS2 WSe2 ZrS2 ZrSe2 HfS2 HfSe2

κl (W/mK)

spring constant (eV/Å2)

0.17 103 54 142 53 14 10 17 11

1.6 11.2 9.8 11.7 10.2 4.6 3.7 5.2 4.2

a

The values of the monolayer transition metal dichalcogenides are predictions from ref 30.

capacity of mode i, respectively) to address the lattice anharmonicity of monolayer Tl2O. The value obtained at 3006

DOI: 10.1021/acsaem.9b00249 ACS Appl. Energy Mater. 2019, 2, 3004−3008

ACS Applied Energy Materials

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300 K is 2.69, which is higher than in PbTe (2.18), Bi2Te3 (1.5), and monolayers MoS2 (1.22), MoSe2 (1.20), and WS2 (1.15).32−34 The Tl 6s2 lone pair (compare Figure 3a of ref 13) contributes to the high lattice anharmonicity (strong phonon scattering; low κl) by deforming the electron density due to electrostatic repulsion. Figure 4c addresses the cumulative lattice thermal conductivity (κcl ) as a function of the phonon mean free path. At 300 K, for example, half of κl is due to phonons with a mean free path below 1.3 nm. This threshold is significantly lower than in the case of PbTe (5.6 nm) but comparable to the case of Bi2Te3 (1.5 nm).28,29 Higher thresholds are predicted for monolayer transition metal dichalcogenides (MoSe2, 247 nm; WSe2, 122 nm; HfSe2, 31 nm; ZrSe2, 12 nm).24,35 Since no experimental values of τ are given in the existing literature, we use the relation τ = μm*/e to derive from the mobilities (μ = 4.30 × 103 and 3.34 × 103 cm2/(V s)) and effective masses (m* = 0.14m0 and 0.10m0) of the holes and electrons (Γ−M direction) at 300 K, as obtained in ref 13 by deformation potential theory, values of 3.53 × 10−13 and 1.83 × 10−13 s, respectively (τ300). Introduction of a temperature 300 K dependence τ(T ) = τ300 T and evaluation of ZT then leads to the results in Figure 3e. We note that a carrier concentration of 1011 cm−2, for example, corresponds to a defect density of 0.03% for Tl and of 0.01% for O. As the weight of κe/τ (and therefore the influence of its dependence on the carrier concentration) in the denominator increases for decreasing κl/τ, the maximum of ZT (as a function of the carrier concentration) shifts away from the maximum of S2σ/τ for increasing τ. While Table 2 already demonstrates high peak

carrier type

ZT

300

hole electron hole electron hole electron

0.95 0.91 0.96 0.93 0.97 0.94

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsaem.9b00249. Out-of-plane lattice thermal conductivity as a function of the temperature; cumulative lattice thermal conductivity as a function of the phonon frequency; POSCAR file (PDF)

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400 500

× × × × × ×

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Nirpendra Singh: 0000-0001-8043-0403 Shahid Sattar: 0000-0003-4409-0100 Stefaan De Wolf: 0000-0003-1619-9061 Udo Schwingenschlögl: 0000-0003-4179-7231 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The research reported in this publication was supported by funding from King Abdullah University of Science and Technology (KAUST).

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REFERENCES

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carrier concentration (cm−2) 2.4 1.6 1.8 1.1 1.2 0.8

ASSOCIATED CONTENT

S Supporting Information *

Table 2. Peak Values of the Thermoelectric Figure of Merit at Different Temperatures with Corresponding Carrier Concentrations temperature (K)

Letter

1011 1011 1011 1011 1011 1011

values of ZT, it should be realized that there will be additional scattering channels in real samples that are not taken into account in our calculations, such as boundary and impurity scattering, and will further reduce κl. The experimental values of ZT, therefore, will be higher than predicted here. In conclusion, the thermal and thermoelectric properties of monolayer Tl2O have been investigated by first-principles calculations, including the lattice contribution to the thermal conductivity to obtain an accurate prediction. The phonon band structure demonstrates stability with low group velocities and strong coupling between the acoustic and optical phonons. Accordingly, ultralow values are found for κl, much lower than in the class of monolayer transition metal dichalcogenides, for example. Since additionally S2σ/τ turns out to be high, excellent thermoelectric performance is predicted for monolayer Tl2O. 3007

DOI: 10.1021/acsaem.9b00249 ACS Appl. Energy Mater. 2019, 2, 3004−3008

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DOI: 10.1021/acsaem.9b00249 ACS Appl. Energy Mater. 2019, 2, 3004−3008