Thermoelectric Response in Single Quintuple Layer Bi2Te3 S. Sharma and U. Schwingenschlögl* Physical Science and Engineering Division (PSE), King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia ABSTRACT: Because Bi2Te3 belongs to the most important thermoelectric materials, the successful exfoliation of a single quintuple layer has opened access to an interesting two-dimensional material. For this reason, we study the thermoelectric properties of single quintuple layer Bi2Te3 by considering both the electron and phonon transport. On the basis of first-principles density functional theory, the electronic and phononic contributions are calculated by solving Boltzmann transport equations. The dependence of the lattice thermal conductivity on the phonon mean free path is evaluated along with the contributions of the acoustic and optical branches. We find that the thermoelectric response is significantly better for p- than for n-doping. By optimizing the carrier concentration, at 300 K, a ZT value of 0.77 is achieved, which increases to 2.42 at 700 K.
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the c-axis, referred to as quintuple layers (QLs). Weak van der Waals (vdW) interaction between the QLs makes it possible to obtain single-QL samples. Mechanically exfoliated single-QL samples and ultrathin films show enhanced thermoelectric response.36,37 Density functional theory has been used to describe the behavior of single-QL Bi2Te3,38 and the thermal conductivities of perfect and nanoporous few-QL films have been studied by molecular dynamics simulations,39 finding that nanoporousity reduces the thermal conductivity. The thermoelectric parameters of single-QL Bi2Te3 have been calculated by the Landauer approach using ab initio band structure data.40 Employing the experimental value of κlat (1.5 W K−1 m−1), the authors have predicted an increase of ZT at room temperature by a factor of 10 as compared to bulk Bi2Te3. The thermoelectric behavior of single-QL Bi2Te3 also has been investigated by means of Boltzmann equations for the electronic and phononic transport, based on molecular dynamics simulations,41 predicting ZT = 0.6 at room temperature. Recently, the same approach has been used for studying the electronic transport in few-QL Bi2Te3 nanofilms,42 employing κlat values from ref 39. Single-QL Bi2Te3 (ZT ≈ 1) turns out to be superior to thicker films. All previous theoretical studies have predicted κlat using molecular dynamics simulations, though the accuracy depends critically on the accuracy of the interatomic potentials.30,39 The problem can be avoided in density functional theory, as applied in the present work in combination with Boltzmann transport theory. We obtain κlat by solving the Boltzmann transport
hile the consumption of energy is increasing, the amount of available natural resources is depleting quickly. Search for clean and renewable energy resources thus is becoming more and more critical. One very promising approach is the recovery of waste heat with the help of thermoelectric materials.1−6 The conversion efficiency of such materials depends on the dimensionless figure of merit, ZT = S2σT/κ, where S, σ, κ, and T are the Seebeck coefficient, electrical conductivity, thermal conductivity (sum of electronic (κele) and lattice (κlat) contributions), and temperature, respectively. Achieving a high ZT value requires maximizing the power factor S2σ and minimizing κ. However, the interdependence between these transport coefficients makes optimization a challenging task. Hicks and Dresselhaus have theoretically demonstrated that the reduction of the dimensionality of a material can enhance ZT strongly.7,8 In recent years, rapid progress in nanotechnology has provided a large variety of low-dimensional materials, such as graphene,9 phosphorene,10,11 silicene,12 stanene, 13 transition metal dichalcogenides, 14,15 and MXenes,16,17 which are currently attracting growing interest from the thermoelectric point of view. For example, monolayer transition metal dichalcogenides have been predicted to improve ZT strongly as compared to the corresponding bulk materials,14,15 and a superior thermoelectric response has been reported for SnSe nanosheets.18 The thermoelectric properties of Bi2Te3 and its alloys have been studied extensively both experimentally19−28 and theoretically.7,29−34 The material has a layered rhombohedral structure with lattice parameters a = 0.438 nm and c = 3.045 nm (space group R3̅m),35 consisting of layers of five atomic planes with the sequence Te(1)−Bi−Te(2)−Bi−Te(3) along © XXXX American Chemical Society
Received: July 20, 2016 Accepted: October 3, 2016
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DOI: 10.1021/acsenergylett.6b00289 ACS Energy Lett. 2016, 1, 875−879
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ACS Energy Letters equation for phonons (without any assumption about the phonon lifetimes) iteratively by means of the ShengBTE code.43 This approach is parameter-free and requires only crystal structure information. It has been used successfully for predicting κlat of GeSb2Te4 (experimental value of 1.3 W K−1 m−1, theoretical value of 1.5 W K−1 m−1).44 For describing the electronic transport, we employ also Boltzmann transport theory. Different density functionals are compared, in particular, the Perdew−Burke−Ernzerhof (PBE) functional45 and the computationally expensive hybrid Heyd−Scuseria− Ernzerhof (HSE06) functional.46 We use density functional theory as implemented in the Vienna ab initio simulation package (projector augment wave approach).47 A vacuum slab of more than 15 Å thickness is used to build a two-dimensional model and to prevent artificial interaction because of the periodic boundary conditions. In order to account for the vdW interaction, the optB86b-vdW functional is employed for structure optimization48 because previous results on bulk Bi2Te3 indicate best performance under the vdW functionals with respect to the description of the lattice parameters.34 The energy cutoff of the plane wave expansion is set to 450 eV, and the Brillouin zone is sampled on a Monkhorst−Pack 11 × 11 × 1 k-mesh. Both the lattice constants and atomic positions are relaxed until the forces on the atoms have declined to 0.01 eV/Å and the total energy changed to 0.001 meV. The transport coefficients are obtained by Boltzmann transport theory within the constant relaxation time approximation. For this purpose, the electronic band structure is calculated at the PBE level on a 45 × 45 × 1 k-mesh (following ref 42, while we find that a 21 × 21 × 1 k-mesh is already well converged) and at the HSE06 level on a 21 × 21 × 1 k-mesh, taking into account the spin−orbit coupling. The effect of doping is studied within the rigid band approach as implemented in the BoltzTrap code.49 Moreover, the phonon dispersion relation is calculated using the phonopy code.50 For evaluating κlat, the required harmonic interatomic force constants are obtained by density functional perturbation theory51 and the third-order interatomic force constants by a finite difference method, using a 4 × 4 × 1 supercell and a 3 × 3 × 1 k-mesh. For the third-order force constants, the interaction up to fourth nearest neighbors is considered. Using both the harmonic and third-order force constants as input, the Boltzmann transport equation for phonons is solved selfconsistently by means of the ShengBTE code43 in order to determine the phonon relaxation times and κlat. A 95 × 95 × 1 q-mesh is employed to obtain the phonon lifetimes. Long-range electrostatic interactions are included by evaluating the dielectric tensor and Born effective charges using density functional perturbation theory. The fully optimized structure of single-QL Bi2Te3 is shown in Figure 1. The obtained lattice constants are a = b = 4.38 Å, with a QL thickness of 7.60 Å, in agreement with previous calculations.38,41 The present work is the first to fully optimize the structure of single-QL Bi2Te3 using a vdW functional. The modifications with respect to the PBE results (a = b = 4.43 Å, with a QL thickness of 7.56 Å) amount to about 1%. Because a very accurate description of the electronic band structure is required to predict transport coefficients, we compare results obtained by the PBE and HSE06 functionals. Figure 2 shows for the PBE functional a band gap of 310 meV, while the HSE06 functional yields a value of 570 meV.
Figure 1. Side (left) and top (right) views of single-QL Bi2Te3. Red and blue spheres represent Bi and Te atoms, respectively.
Figure 2. Electronic band structures (hexagonal Brillouin zone) and densities of states at the PBE and HSE06 levels.
For temperatures of 300, 500, and 700 K, Figure 3 shows the variations of S, σ, S2σ, and ZT with the electron and hole concentrations. Because σ is obtained with respect to the electronic relaxation time τ, τ in principle can be determined by fitting to the experimental value of σ at a particular carrier concentration. However, due to a lack of experimental data for single-QL Bi2Te3, earlier studies have used τ of bulk Bi2Te3.41,42 In order to enable comparison to these studies, we employ the same values of 2.13 × 10−14 s (300 K), 1.23 × 10−14 s (500 K), and 1.12 × 10−14 s (700 K). Furthermore, κele is obtained from σ using the Wiedemann−Franz relation, κele = LσT (L being the Lorenz number), which is a good approximation for Bi-based bulk and low-dimensional materials,22,26,52,53 while ref 30 has raised doubts about the accuracy of the Boltztrap code in the metallic limit. It is clear from a comparison of the PBE and HSE06 results in Figure 3 that the accurate description of the band structure by the HSE06 functionals has important effects on the predicted transport coefficients. According to Figure 2, the band forming the valence band edge is much flatter than that forming the conduction band edge, that is, holes have larger effective masses than electrons. Hence, p-doping leads to higher S values than n-doping. At high temperature and low carrier concentration, the minority carriers lower S and enhance σ due to the small band gap.54 For the HSE06 functional at 300 K, a 876
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ACS Energy Letters
Figure 3. Transport coefficients as functions of the carrier concentration for p-doping (left) and n-doping (right) at the PBE and HSE06 levels.
Figure 4. (a) Harmonic phonon dispersion relation in the hexagonal Brillouin zone and (b) lattice thermal conductivity as a function of the temperature along with data from previous molecular dynamics studies.39,41
Figure 5. (a) Lattice thermal conductivity as a function of the temperature and (b) cumulative lattice thermal conductivity as a function of the mean free path.
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ZT value of 0.77 is obtained for a hole concentration of 4.0 × 1020 cm−3, while at 700 K, the maximal ZT value of 2.42 appears for a hole concentration of 3.2 × 1020 cm−3. Employing κlat of ref 41 would result in ZT values of 0.66 at 300 K and 1.96 at 700 K, whereas κlat of ref 39 would yield ZT values of 1.06 at 300 K and 3.01 at 700 K. The phonon dispersion of single-QL Bi2Te3 is shown in Figure 4a, agreeing well with previous results.41 Absence of negative frequencies confirms structural stability. The group velocities of the acoustic branches near the Γ-point turn out to be 1192, 1859, and 2920 m/s, while those of the three lowest optical branches are less than 5 m/s. In Figure 4b, the obtained values of κlat are compared to previous results based on molecular dynamics simulations. These results show a large variation, which is probably due to the employed interatomic potentials. In any case, the description of anharmonic force constants by empirical interatomic potentials is less accurate than our ab initio treatment. The fact that three low-frequency optical branches appear in close proximity to the acoustic branches (see Figure 4) makes significant contributions to the heat conduction possible. However, Figure 5a shows that the contributions of the acoustic branches clearly dominate, as a consequence of the small group velocities of the optical branches. In order to understand the effect of the sample size on the thermal transport, we analyze the cumulative lattice thermal conductivity (κlat c ) as a function of the phonon mean free path for temperatures of 300 and 700 K (see Figure 5b). The values are obtained by cumulating the thermal conductivities of all phonons up to a given mean free path. At 300 K, for example, it turns out that half of the contributions to κlat are from phonons with mean free paths below 320 nm, indicating that appropriate nanostructuring can be used to lower κlat. We note that the maximal electron/hole mean free path in the studied temperature range is smaller than 13 nm, which shows that the electron transport is not affected by the nanostructuring. The thermoelectric performance of single-QL Bi2Te3 has been investigated by means of the Boltzmann transport equations for the electrons and phonons. The electronic band structure, phonon dispersion relation, and interatomic force constants have been calculated using density functional theory. Results have been obtained and compared for the PBE and HSE06 functionals. The improved description of the band structure by the HSE06 hybrid functional is found to be of key importance for an accurate prediction of the thermoelectric properties, which demonstrates that electronic exchange effects play an unexpectedly large role in Bi2Te3. The lattice contribution to the thermal conductivity is dominated by the acoustic phonon branches despite the fact that low-frequency optical phonon branches appear. Beyond, the calculated dependence of the lattice thermal conductivity on the phonon mean free path suggests that the thermoelectric performance of single-QL Bi2Te3 can be enhanced efficiently by nanostructuring. Predictions of the lattice thermal conductivity of single-QL Bi2Te3 by means of molecular dynamics simulations, as reported in the literature, show large variations (for example, from 1.63 to 2.97 W K−1 m−1 at 300 K), which are likely due to issues with the employed interatomic potentials. The method used in the present work, on the other hand, does not suffer from this limitation.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. 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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