Abstract
The thermoelectric transport properties of ptype Bi_{2}Te_{3} nanofilms with various quintuple layers (QL) were systematically investigated based on ab initio electronic structure calculations and Boltzmann transport equations. Our results demonstrated that ptype fewquintuple Bi_{2}Te_{3} nanofilms could exhibit high thermoelectric performance. It was found out that the 1QL Bi_{2}Te_{3} nanofilm had the highest ZT value as compared with other nanofilms, which is mainly attributed to the significant enhancement of the density of states near the edge of the valence band resulting from the strong coupling between the top and bottom electronic states and the quantum confinement effect. The dependence of the thermoelectric transport properties on carrier concentration and temperature was also discussed in detail, which can be useful for searching highefficiency fewquintuple Bi_{2}Te_{3} thermoelectric nanofilms.
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Introduction
Thermoelectric materials have attracted intense interest due to their potential applications in cooling and power generation^{1,2,3,4,5}. The performance of thermoelectric devices depends on the dimensionless figure of merit, ZT, which is defined as
where S is the Seebeck coefficient, T is the temperature, σ is the electrical conductivity, κ_{l} is the lattice thermal conductivity and κ_{e} is the thermal conductivity of the electronic carriers. To achieve a high ZT, it requires maximizing the Seebeck coefficient and electrical conductivity but minimizing the thermal conductivity. In general, the thermal conductivity can be minimized by introducing nanostructures into the thermoelectric materials to increase phonon scattering^{6,7,8}, the Seebeck coefficient can be enhanced by quantum confinement effects^{9,10} and the electrical conductivity can be improved by adjusting the carrier concentration through doping^{11,12}, however, the interdependence among these parameters makes the maximization of ZT a challenge so far.
Among various thermoelectric materials, Bi_{2}Te_{3} and its solid solutions have been investigated intensely because they possess outstanding thermoelectric properties at room temperature, which is due to the high degeneracy at the edges of their energy bands. The ZT value of the Bi_{2}Te_{3}based thermoelectric materials has been improved by many approaches such as the synthesis of nanostructured bulk materials or nanocomposites with a precise control of the size, structure, composition and carrier concentration as well as the deposition of various thin films by meltspinning, sputtering, evaporation and molecular beam epitaxy^{13,14,15,16}. Recently, Bi_{2}Te_{3} has been demonstrated as a threedimensional (3D) topological insulator, which further stimulates a strong research interest in this material^{17}. Bulk Bi_{2}Te_{3} crystallizes in a rhombohedral unit cell with the space group () and has a layered structure with five atomic layers referred to as a quintuple layer (QL)^{18,19}. The interaction between two adjacent QL is of the weak van der Waals type, which allows one to disassemble Bi_{2}Te_{3} crystal into its quintuple building block^{20}. The experimental results have shown that the exfoliated quintuples and ultrathin fewquintuple films have a low thermal conductivity, which is favorable to improve the thermoelectric performance^{20}. Very recently, an efficient thermoelectric conversion was observed in the atomic monolayer steps of Bi_{2}Te_{3} (ntype) and Sb_{2}Te_{3} (ptype) QL by scanning photoinduced thermoelectric current probe^{21}. Theoretically, P. Ghaemi et al. have discovered that nanometerscale Bi_{2}Te_{3} thin film generates a hybridizationinduced band gap of the unconventional surface states, which can lead to an increased thermoelectric performance at low temperatures^{22}. Recent theoretical researches show that Bi_{2}Te_{3} nanofilms have enhanced thermoelectric properties^{18,23}. B. Qiu et al. have studied the thermal conductivity of the fewquintuple Bi_{2}Te_{3} nanofilms by using molecular dynamics simulation and have shown that nanoporous Bi_{2}Te_{3} nanofilms display significantly reduced thermal conductivity as compared to the bulk couterparts^{24}. All these results indicate that the fewquintuple Bi_{2}Te_{3} nanofilms can be considered as promising thermoelectric materials.
In this work, we report an effective scheme of combining density functional theory and Boltzmann transport equations to provide a microscopic description of the thermoelectric properties of fewquintuple Bi_{2}Te_{3} nanofilms. Our results demonstrate that ptype fewquintuple Bi_{2}Te_{3} nanofilms can exhibit high thermoelectric performance and thus they can stand as a promising candidate for nanoscale thermoelectric applications.
Computational methods
Firstprinciples calculation
The calculation of the energy band structure has been performed by using the projector augmented wave (PAW) method within the framework of density functional theory as implemented in the Vienna Ab initio Simulation Package (VASP)^{25,26,27}. The exchangecorrelation function is in the form of the PerdewBurkeErnzerh (PBE) with generalized gradient approximation (GGA)^{28,29}. The MonkhorstPack scheme is adopted for the integration of Brillouin zone with a k mesh of 11 × 11 × 11 for the bulk Bi_{2}Te_{3} and a k mesh of 11 × 11 × 1 for the Bi_{2}Te_{3} fewquintuple films. In order to investigate the transport properties of the Bi_{2}Te_{3} nanofilms, a denser k mesh of 45 × 45 × 1 is used. A cutoff energy of 450 eV is used in the plane wave basis. For the lattice constants one uses experimental value^{30} and interlayer separations for Bi_{2}Te_{3} fewquintuple nanofilms are fully relaxed until the magnitude of the force acting on all atoms is less than 0.01 eV/Å and the total energy converges within 0.001 meV. The Bi_{2}Te_{3} nanofilms with various thicknesses are modeled as a slab containing a few quintuples and a vacuum layer and Fig. 1 shows the models used in this simulation. For the five slabs with different thicknesses considered, a vacuum layer of 28 Å is added. Our tests revealed that this is sufficient for the prevention of spurious interactions between image slabs.
Calculation of the transport properties
In this work, the semiclassical Boltzmann equations with the relaxation time approximation were used to evaluate the thermoelectric transport coefficients^{31}. Previous calculations confirmed that it is an effective approach to predict the thermoelectric transport properties of bulk materials and nanosystems^{9,32,33}. By solving the Boltzmann transport equations, σ, S and κ_{e} can be obtained as^{9,34}
where e is the charge of the carriers, T is the temperature, E_{F} is the Fermi level, f_{0} is the FermiDirac distribution function and ∑(ε) is the transport distribution function and can be expressed as^{31,35}
where v_{k} is the group velocity of the carriers associated with wave vector k, τ_{k} is the relaxation time, N_{k} is the number of the sampled k points in the Brillouin Zone (BZ) and Ω is the volume of unit cell. For a quasitwodimensional (2D) system, Ω should be substituted by the product of Γ and h, where Γ is the area of a 2D unit cell and h is the thicknesses of the quasi2D system. As a result, for the quasi2D system Eq. (6) can be expressed as
Because the current version of the BoltzTrap program^{36} is restricted to analyze 3D materials, according to the discussion above we implemented the codes (modified from the BoltzTrap codes^{36}) for quasi2D systems to obtain the transport coefficients.
The relaxation time τ depends on the temperature, electronic energy and carrier concentration and generally is treated as a constant for convenience. Due to lack of available data for the relaxation time of Bi_{2}Te_{3} nanofilms, we have adopted the bulk value as a reliable approximation by noting that the nanofilms possess similar covalent bonds with bulk Bi_{2}Te_{3}^{23}. Similar treatments have been done for nanoscale systems such as Si nanowires^{33}, Bi_{2}Te_{3} nanowires^{37}, Bi_{2}Te_{3} nanofilms^{23} and so on. For the relaxation time at 300 K, we used the value of 2.2 × 10^{−14} s, which T. J. Scheidemantel et al^{32} have adopted to study the thermoelectric properties of bulk Bi_{2}Te_{3} and obtained consistent results with the experimental data^{38}. The temperature dependence of the relaxation time of bulk Be_{2}Te_{3} has been summarized in Table 1^{37}. The phonon thermal conductivity is another important parameter to determine the thermoelectric figure of merit. The previous theoretical study showed that phonon thermal conductivity of the perfect 1QL Bi_{2}Te_{3} nanofilm decreases monotonically with temperature and exhibits a T^{−1} dependence^{23,24} and this inverse temperature dependence has also been found in bulk Bi_{2}Te_{3} both theoretically^{39} and experimentally^{40}. For the perfect Bi_{2}Te_{3} nanofilms with a few QL, the inverse temperature dependence^{24} will be adopted in our calculations and the coefficient b was determined by the value of the phonon thermal conductivity of Bi_{2}Te_{3} nanofilms at 300 K^{24}. Previous results showed that the thermal conductivity of fewquintuple Bi_{2}Te_{3} nanofilms could be significantly reduced by introducing structure defects such as nanopore^{24}. In the following, we also investigated the thermoelectric figure of merit of nanoporous Bi_{2}Te_{3} nanofilms with few quintuples by assuming the defects don't significantly influence the electronic structure. Since the molecular dynamics simulation showed the thermal conductivity of nanoporous nanofilms depends weakly on the temperature and film thickness, we adopted a constant value of 0.3 W/mK in our calculation^{24}.
Results and discussion
Band structure
The ab initio electronic band structures of Bi_{2}Te_{3} bulk material and nanofilms with various thicknesses are shown in Fig. 2. The spinorbit interaction was taken into account in the calculation, which was proved to be essential for calculating the electronic structures of Bi_{2}Te_{3} and our results for bulk Bi_{2}Te_{3} (Fig. 2(a)) are in good agreement with the data reported in previous work^{41}. As shown in Figs. 2(b–f), band structures of the Bi_{2}Te_{3} nanofilms are similar to each other except for the significant difference in band gap and energy spacing between adjacent bands. Firstly, the indirect band gap decreases with increasing the film thickness. For example, the band gap is about 320 meV for 1QL film (Fig. 2(b)) and it disappears when the film thickness increases to 3QL (Fig. 2(d)). Secondly, the energy spacing between the adjacent bands decreases with increasing the number of QL. The reasons are that with the thickness increasing, the coupling between the top and bottom electronic states and the quantum confinement effect on the electrons in the nanofilm are weakened. It is also noteworthy to point out that the conduction bands of all five nanofilms are much more dispersive than the valence bands.
The density of states (DOS) for the Bi_{2}Te_{3} nanofilms with various thicknesses is shown in Fig. 3. The DOS near the edge of the valence band has a sharper and larger peak than that near the edge of the conduction band due to flatter valence bands (as shown in Fig. 2). It can be observed from Fig. 3 that the total DOS for 1 QL Bi_{2}Te_{3} nanofilm displays staircaselike behavior near the edges of conduction bands and valence bands, induced by the quantum confinement effect in quasi2D systems. With the number of QL decreasing, the spacing between the peaks (singularity) of the DOS significantly increases, which is consistent with the change of the energy spacing between the adjacent bands.
Transport properties
Based on the calculated electronic band structures, the thermoelectric transport coefficients of the fewquintuple Bi_{2}Te_{3} nanofilms can be evaluated by the methods presented in Sec. II. It is well known that the standard density functional theory usually underestimates the band gaps of semiconductors. In order to achieve a better agreement with the experimental results, the band gaps of the nanofilms were adjusted to match the experimental data^{42} by applying the socalled scissor operator in our calculations. Similar treatments have been widely adopted in previous publications^{43,44}.
Fig. 4 shows the calculated electrical conductivity, Seebeck coefficient, power factor and figureofmerit of the perfect ptype Bi_{2}Te_{3} films with different QL and ptype bulk Bi_{2}Te_{3} as a function of carrier concentration at 300 K. As shown in Fig. 4(a), the electrical conductivity of all the ptype Bi_{2}Te_{3} nanofilms increases with increasing carrier concentration. In contrast, the Seebeck coefficient decreases with increasing carrier concentration and in general the Seebeck coefficient for the thinner nanofilm is larger than that for the thicker nanofilm at the same carrier concentration (Fig. 4(b)). This can be explained by the following discussion. Taking into account that MaxwellBoltzmann approximation is justified in the region of E_{F} − E_{V} ≫ k_{B}T, S can be expressed as a function of n:
where N_{V} is the effective density of states at the edge of the highest valence band and A_{V} represents a scattering factor of the semiconductor^{8}. N_{V} can be expressed as
where g_{V}(ε) is the density of states. Eq. (8) directly reveals that the magnitude of S decreases with increasing n, as shown in Fig. 4(b). In addition, with the number of QL decreasing, the DOS near the edge of the valence band increases due to the quantum confinement effect (Fig. 3), leading to an increase of N_{V} and hence an enhancement of S. It is worth noting that S of 1QL Bi_{2}Te_{3} nanofilm is significantly larger than those of other nanofilms, which results from the significantly enhanced DOS near the edge of the valence band induced by the strong coupling between the top and bottom electronic states and the quantum confinement effect. The electrical conductivity σ increases with increasing carrier concentration n, whereas the Seebeck coefficient S decreases, therefore, there is an optimized carrier concentration, n_{opt}, at which the power factor is maximized (as shown in Fig. 4(c)). The n_{opt} to obtain the maximum power factor for Bi_{2}Te_{3} nanofilms with various QL occurs in the range from 0.3 × 10^{20} cm^{−3} to 5 × 10^{20} cm^{−3}. In general, the n_{opt} increases with decreasing the number of QL. The n_{opt} can be derived as^{8}
Eq. (10) reveals that the n_{opt} for the power factor is proportional to N_{V}. With the number of QL decreasing, the DOS near the edge of the valence band increases, thus, N_{V} rises and hence n_{opt} increases (as shown in Fig. 4(c)). Figure 4(d) shows that the dependence of ZT on the carrier concentration is similar to that of the power factor, indicating that the carrier concentration is a key parameter to adjust the thermoelectric performance of the fewquintuple films. The n_{opt} to obtain the highest ZT value is in the range from 0.1 × 10^{20} cm^{−3} to 1.5 × 10^{20} cm^{−3}. The perfect 1QL film has the largest ZT value of 1.0 at an n_{opt} of 1.5 × 10^{20} cm^{−3}. The n_{opt} for the ZT is different from that for the power factor because of the effect of the electronic thermal conductivity that is related to the carrier concentration. We have calculated the thermoelectric transport properties of bulk Bi_{2}Te_{3} by using the value of 2.2 × 10^{−14} s at 300 K^{32} for the relaxation time and the value of 1.5 W/mK at 300 K^{32} for the phonon thermal conductivity. Our calculated results are in excellent agreement with the previous theoretical^{32} and experimental data^{38}. It can be seen from Fig. 4 that the electrical conductivity of bulk Bi_{2}Te_{3} is larger while the seebeck coefficient is smaller in most range of carrier concentrations, as compared with those of Bi_{2}Te_{3} nanofilms at a fixed carrier concentration. The optimized ZT value of bulk Bi_{2}Te_{3} is about 0.6, which is smaller than that of 1QL Bi_{2}Te_{3} nanofilm (Fig. 4(d)).
Fig. 5 shows the calculated electrical conductivity, Seebeck coefficient, power factor and figure of merit of the ptype perfect 1QL Bi_{2}Te_{3} nanofilm as a function of carrier concentration at different temperatures. As shown in Fig. 5(a), at a fixed carrier concentration the electrical conductivity decreases with increasing temperature, mainly because the scattering rate of the carriers increases and the relaxation time decreases with increasing temperature (as shown in Table 1). In contrast, the S slightly increases with increasing the temperature at a fixed carrier concentration, as shown in Fig. 5(b). This is mainly because when the temperature rises, the term, exp[−(E_{V} − ε)/k_{B}T] with ε < E_{V}, in Eq. (9) increases and then it leads to an increasing N_{V}. Due to the reduction of σ, the power factor decreases with increasing temperature and the n_{opt} to obtain the maximum power factor shifts to a larger value (Fig. 5(c)). Fig. 5(d) shows that the dependence of ZT on the temperature is reversed as compared with that of the power factor. Although the power factor decreases with raising temperature, ZT value increases because both phonon and electron thermal conductivity decrease. The peak values of ZT for the ptype perfect 1QL Bi_{2}Te_{3} film are 1.0 and 1.9 at 300 K and 800 K, respectively.
Figs. 6(a) and 6(b) summarize the dependence of the optimized ZT (at n_{opt}) of the ptype perfect and nanoporous Bi_{2}Te_{3} nanofilms with various QL on temperature, respectively. As shown in Fig. 6(a), for all of the perfect nanofilms, their optimized ZT values increase with increasing temperature and 1QL nanofilm has the highest optimized ZT value in the entire temperature range as compared with other nanofilms. The temperature dependence of optimized ZT value has also been found in the previous study of thermoelectric properties of 1QL Bi_{2}Te_{3} nanofilm^{23}. For the nanofilms with a thickness larger than 1QL, their ZT values are close to each other because they possess approximately similar DOS near the edge of the valence band. Previous results showed that the perfect nanofilms have high thermal conductivity, e.g. 1.65 W/mK for perfect 1QL nanofilm at 300 K^{24}, which limits further improvement of ZT. Fortunately, the molecular dynamics simulations showed that the phonon thermal conductivity can be significantly reduced by introducing structure defects such as nanopore^{24}. Considering the fact that phonon thermal conductivity can be significantly reduced by introducing defects without significantly sacrificing the thermoelectric power factor S^{2}σ^{45}, which has been seen as a main strategy^{2} to enhance the thermoelectric properties of materials and has been confirmed by a lot of experimental works^{2,45,46}, we give the calculated results of thermoelectric properties of nanoporous Bi_{2}Te_{3} nanofilms by assuming that introduction of nanopores don't influence the electronic structure and power factor of nanofilms. Due to the reduction of phonon thermal conductivity, the ptype Bi_{2}Te_{3} nanofilms with nanopore defects exhibit a significantly improved ZT value as compared with the perfect nanofilms. Particularly, the 1QL nanoporous Bi_{2}Te_{3} nanofilms can achieve an optimized ZT value of 2.65 at room temperature, as shown in Fig. 6(b). The temperature dependence of the optimized ZT values of the ptype Bi_{2}Te_{3} nanofilms with nanopore defects becomes complicated in comparison to that of the perfect nanofilms. This can be explained by the following discussions. By using the Wiedemann–Franz law: κ_{e} = LσT, the ZT may be written as follows:^{4}
For perfect Bi_{2}Te_{3} nanofilms, the phonon dominates thermal transport (κ_{l} ≫ κ_{e}) and ZT increases with temperature mainly due to that κ_{l} decreases with temperature. For nanoporous Bi_{2}Te_{3} nanofilms, κ_{l} remains constant over the entire temperature range and ZT depends on temperature through S and κ_{e} according to Eq. (11). In this case, the values of κ_{l} and κ_{e} at optimized carrier concentration are comparable to each other. The temperature dependence of the optimized ZT values of nanoporous Bi_{2}Te_{3} nanofilms is less prominent than that of perfect nanofilms and several factors including S and κ_{e} as well as its magnitude relative to κ_{l} come into effect. We hope that these predictions could provide a clue for experimental researchers to optimize the thermoelectric performance of Bi_{2}Te_{3} nanofilms.
Conclusions
In summary, we have employed a combination of firstprinciples electronic structure and Boltzmann transport calculations to predict the thermoelectric transport properties of fewquintuple Bi_{2}Te_{3} nanofilms. Our results show that ptype Bi_{2}Te_{3} nanofilms could be good candidates for thermoelectric nanomaterials, provided that the phonon thermal conductivity can be decreased by introducing structure defects without significantly sacrificing the thermoelectric power factor and ptype 1QL nanofilm with nanopore defects can achieve an upperlimit ZT value of 2.65 at room temperature. It was found out that the ptype 1QL Bi_{2}Te_{3} nanofilm as compared with other nanofilms has the highest ZT value, which is mainly attributed to the significantly enhanced density of states resulting from the strong coupling between the top and bottom electronic states and the quantum confinement effect. The dependence of thermoelectric transport properties on doping level and temperature was investigated in detail, which provides guidance for future experiments to optimize ZT value of ptype Bi_{2}Te_{3} nanofilms with few quintuple layers. The type of calculation reported in this work represents a valuable investigation tool to study the thermoelectric properties of quasi2D nanomaterials.
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Acknowledgements
This work was supported by China Postdoctoral Science Foundation (Grant No. 2012M520236) and National Natural Science Foundation of China (Grant Nos. 21273124 and 21290190). Computational resources were provided by Tsinghua National Laboratory for Information Science and Technology of China.
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G.Z. performed the calculations and modified the code. G.Z. and D.W. analyzed the data and discussed the results. All the authors reviewed the manuscript.
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Zhou, G., Wang, D. Fewquintuple Bi_{2}Te_{3} nanofilms as potential thermoelectric materials. Sci Rep 5, 8099 (2015). https://doi.org/10.1038/srep08099
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DOI: https://doi.org/10.1038/srep08099
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