We use first-principles calculations to reveal the effects of divalent Pb, Ca, and Sn doping of Bi2Te3 on the band structure and transport properties, including the Seebeck coefficient, α, and the reduced power factor, α2σ/τ, where σ is the electrical conductivity and τ is the effective relaxation time. Pb and Ca additions exhibit up to 60%–75% higher peak α2σ/τ than that of intrinsic Bi2Te3 with Bi antisite defects. Pb occupancy and Ca occupancy of Bi sites increase σ/τ by activating high-degeneracy low-effective-mass bands near the valence band edge, unlike Bi antisite occupancy of Te sites that eliminates near-edge valence states in intrinsic Bi2Te3. Neither Pb doping nor subatomic-percent Ca doping increases α significantly, due to band averaging. Higher Ca levels increase α and diminish σ, due to the emergence of a corrugated band structure underpinned by high-effective-mass bands, attributable to Ca–Te bond ionicity. Sn doping results in a distortion of the bands with a higher density of states that may be characterized as a resonant state but decreases α2σ by up to 30% due to increases in the charge carrier effective mass and decreases in both spin–orbit coupling and valence band quasidegeneracy. These results, and thermal conductivity calculations for nanostructured Bi2Te3, suggest that Pb or Ca doping can enhance the thermoelectric figure of merit ZT to values up to ZT ∼ 1.7, based on an experimentally determined τ. Our findings suggest that divalent doping can be attractive for realizing large ZT enhancements in pnictogen chalcogenides.
Realizing thermoelectric devices with >20% device energy conversion efficiencies1–3 requires materials with simultaneously high Seebeck coefficient α, high electrical conductivity σ, and low thermal conductivity κ. This criterion is met when the material has a thermoelectric figure of merit ZT > 3, where , and T is the absolute temperature. Much work has focused on lowering κ by suppressing phonon transport through crystal design, nanostructuring, and alloying while preserving a high bulklike power factor α2σ.1,4,5 However, it is widely recognized that further ZT increases require realizing α2σ values beyond limits imposed by natural bulk properties.
Our recent theoretical work has shown that manipulating the electronic structure is an attractive means to obtain α2σ values multifold higher than those of intrinsic single crystals.6 For example, we showed that manipulating lattice strain in Bi2–xSbxTe3–ySey alloys can increase α2σ by ∼250% due to favorable distortions of the band structure and resulting density of states (DOS). This is corroborated by experiments showing up to eightfold greater power factors compared to bulk alloys.7,8 These exciting results underscore the importance of correlations between the electronic band topology and power factor, motivating us to explore subatomic-percent doping as a low-pressure means to realize such beneficial band topologies.
Here, we reveal the effects of three divalent p-type dopants, namely, Pb, Ca, and Sn, on the electronic structure and power factor of Bi2Te3. We chose these dopants because subatomic-percent additions of Pb or Sn have been shown9,10 to result in 10%–50% higher power factor α2σ, but the underlying mechanisms are not yet clear. Ca doping has been shown to allow tuning of the Fermi energy EF to the Dirac point,11 but the consequent effects on band structure, α, and σ are not known. Accordingly, for each dopant addition, we computed the Bi2Te3 electronic band structures using first-principles density functional theory (DFT) calculations. We extracted power factors using Boltzmann transport equations (BTE). Baseline calculations were carried out for intrinsic Bi2Te3 to separate the effects of antisite point defects.
Our results show that subatomic-percent Pb doping leads to a 75% α2σ/τ enhancement because Pb occupancy of Bi sites () preserves the high-mobility bands that are otherwise suppressed in intrinsic Bi2Te3 due to Bi occupancy of Te sites (i.e., antisite defects). Ca occupancy of Bi sites () creates up to 60% higher α2σ/τ by forming a corrugated band structure from a mix of high and low-effective-mass bands. In contrast, Sn occupancy of Bi sites () results in a 30% lower α2σ/τ due to decreased spin–orbit coupling and increased charge carrier effective mass arising from resonant state formation. These insights along with thermal conductivity calculations for nanostructured Bi2Te3 suggest that ZT ≥ 1.7 could be attained by Pb or Ca doping of nanostructured Bi2Te3.
II. FIRST-PRINCIPLES COMPUTATIONAL METHODS
Our DFT calculations used the Vienna ab initio simulation package (VASP v5.3.5)12,13 with projector-augmented wave (PAW) pseudopotentials and the modified Perdew–Burke–Ernzerhof generalized gradient approximation (PBEsol) exchange-correlation functional.14 The plane wave energy cutoff was set to 300 eV. Except ionic relaxation, all our calculations included spin–orbit coupling. Ionic relaxations ceased when interatomic forces fell to < 0.1 eV/nm, and electronic relaxations completed when the difference in ground state energy between successive steps was <10−5 eV.
Ionic and electronic relaxations were performed using Γ-centered, 12 × 12 × 12, 4 × 4 × 4, and 2 × 2 × 2 k-meshes for the rhombohedral Bi2Te3 unit cell [Fig. 1(a)], 135-atom 3 × 3 × 3 supercell [Fig. 1(b)], and 320-atom 4 × 4 × 4 supercell, respectively. A single defect in the 320-atom supercell allowed the simulation of ∼0.3 at. % doping. Doping levels of ∼0.75 and ∼2.2 at. % were achieved in the 135-atom supercell by introducing 1 and 3 dopant atoms, respectively. For the 2.2 at. % concentration, we varied the separation distances of the 3 defects (see Sec. IV E), and the configuration with the lowest ground state energy was chosen for further analysis.
Any nondefective supercell was otherwise identical to a primitive unit cell (see band structure comparison in the supplemental material). We sampled the electronic structure with 100 k-points between each high-symmetry point along the L-Z-a-Γ-Z-F path [Fig. 1(c)] in the first Brillouin zone (IBZ) to obtain band structure cross sections from ellipsoidal energy surfaces near the valence band edge (VBE). The reciprocal lattice coordinates of these high-symmetry k-points6 scale linearly with supercell size. For clarity in comparing the effects of different dopants, we plotted only the band closest to the VBE, since bands within ∼3kBT of EF dominate transport.15 We determined the electron localization function (ELF) and density of states (DOS) using the linear tetrahedron method to sample the converged charge density over the irreducible Brillouin zone (IBZ). We used symmetrized 36 × 36 × 36, 8 × 8 × 8, and 6 × 6 × 6 Γ-centered k-meshes to sample the unit cell, 3 × 3 × 3 supercells, and 4 × 4 × 4 supercells, respectively.
We used BoltzTraP,16 the version available in 2016, to extract α, σ/τ, and κe—the electronic contribution to thermal conductivity, at room temperature (300 K) along the high-ZT direction, from the band structure data used for DOS calculations, but interpolated onto a k-mesh 5 times as dense as the mesh used in our DFT calculations. We applied a 0.055 eV scissors bandgap shift to nondefective Bi2Te3 to match the experimental value EG = 0.15 eV.17 Scissors shifts were applied to each of our doped systems to reach the same EG, to enable us to separate the effects of band topology from bandgap. Hole concentrations were obtained by dividing band occupancy at the calculated Fermi level EF by the respective supercell volume.
We extracted α and στ−1 from the band structures of Bi2Te3 with defects (see Sec. IV A). We used a charge scattering relaxation time τ = 3.4 × 10−14 s to match our calculated α2σ values with intrinsic, single-crystal experimental data.9,18–20 This τ value was subsequently used to determine σ values for all systems. For the hole concentration range 1019 ≤ h ≤ 1020 cm−3, our α and σ values determined from each nondefective supercell were within 2% of those obtained from a unit cell. We report thermoelectric transport properties over a range of carrier concentrations for each dopant. Our aim is to mainly capture the overall trends of how doping and antisite defects separately impact the electronic band structure and hence the thermoelectric properties. Our results could serve as a basis for interpreting and understanding experimental results that can be impacted by a variety of factors.9,10,18 We note that the effects of codoping, chemical stability, or formation energy, or doping efficiency are beyond the scope of this work.
III. BISMUTH TELLURIDE PROPERTIES
A. Crystal structure
Bi2Te3 has a rhombohedral structure [Fig. 1(a)] specified by the space group with Te6c-Bi6c-Te3a-Bi6c-Te6c quintuple layer stacking, wherein Bi6c sites are given by fractional coordinates (±0.400, ±0.400, ±0.400), Te3a sites by (0,0,0), and Te6c sites by (±0.209, ±0.209, ±0.209). Our calculated lattice parameters = 1.042 nm and = 24.322° are within 1% of experimentally determined values.21 Since Pb,9,22 Ca,11,23 and Sn10,23 are known to occupy pnictogen sites as acceptors in V2VI3 materials, we placed the dopants at Bi sites in the supercell.
B. Orbital interaction
In Bi2Te3, experimental charge carrier effective mass ranges between 0.06m0 ≤ m* ≤ 0.08m0 for electrons and 0.08m0 ≤ m* ≤ 0.11m0 for holes,24,25 where m0 is the free electron mass and is the geometric mean of the Cartesian surface curvatures of energy surfaces in the IBZ. Bohr ionization energies are, therefore, ≤ 0.44 meV for donors and ≤ 0.6 meV for acceptors,26 both of which are <0.5% of the bandgap 0.13 ≤ Eg ≤ 0.15 eV.17 These shallow dopant levels are easily ionizable and strongly interact with the host bands, consistent with experimentally observed low-temperature charge carrier saturation in intrinsic p-Bi2Te3.19 The Bohr radii 26 are >33 nm for donors and >24 nm for acceptors, indicating delocalized effects for defect concentrations ≤ 10−4 at. %. Despite a factor-of-two uncertainty,26 our estimates of EI and rB indicate that Bi2Te3 should have nearly completely ionized dopants at all concentrations investigated here.
IV. RESULTS AND DISCUSSION
A. Intrinsic defects
Our DFT calculations reveal that Bi occupancy of Te sites (i.e., antisite defect) alters the electronic structure and, therefore, the transport properties in Bi2Te3. Our calculated electron localization function (ELF) map (Fig. 2) shows an increased electron localization near the center of Bi– bonds, indicating increased covalency compared to ionic Bi–Te6c bonds in nondefective Bi2Te3,27,28 where the ELF is high near the atoms. The increased covalency due to Bi– bonds should promote high hole mobility because of increased overlaps in ppσ bonds.29–31
The relatively low value of our calculated 200 ≤ ≤ 300 cm2 V−1 s−1 suggests the primacy of a factor other than bonding. Bi antisite occupancy of Te sites decreases the near-edge valence state population built mostly from Te6c p-orbitals.15 Increasing , therefore, tends to retract valence band states farther from the edge in nondefective Bi2Te3 (Fig. 3). For example, = 2.2 at. % causes electronic bands along Z-a, a-Γ, Z-F, and to retract by 30–70 meV (1.2kBT to 2.7kBT at 300 K), leading to a 50%–75% lower density of states (DOS) near the VBE compared to that in nondefective Bi2Te3 (Fig. 4).
Increasing from 0.3 to 2.2 at. % decreases σ by 30% [Fig. 5(a)] due to an equivalent decrease in [Fig. 5(b)] and decreases α by 20% [Fig. 5(c)]. This result can be understood in terms of the doping-induced movement of the electronic bands near EF. Recalling that individual band degeneracy g represents the number of equivalent energy surfaces in the IBZ near EF32 and recognizing multiple inequivalent bands with 2 ≤ g ≤ 6 near the VBE of Bi2Te3, we note that movement of the bands alters the quasidegeneracy, i.e., the total number of bands within ∼3kBT of the VBE. In particular, doping-induced retraction of bands decreases quasidegeneracy, which in turn decreases both α and σ because ,33 where ,6 and 34 where σi is the contribution from each band. We obtain the highest α 2σ ∼ 3.9 mW m−1 K−2 at h = 3 × 1019 cm−3 with = 0.3 at. % [Fig. 5(d)]. Increasing [ to 2.2 at. % reduces α2σ to 1.7 mW m−1 K−2.
B. Pb doping
The band structure and bonding of Bi2Te3 with substitutional doping strongly resembles that of nondefective Bi2Te3, and correlates with up to 25% increase in σ over intrinsic Bi2Te3 with . In particular, we find that Pb-doped and nondefective Bi2Te3 exhibit similar band convergence and quasidegeneracy near the VBE (Fig. 6). For example, for = 0.75 at. %, the high-degeneracy (g = 6) low-effective-mass Z-F band resides 0.65kBT from the VBE, which is the same as in nondefective Bi2Te3. In contrast, this same low-m* Z-F band is 0.8kBT farther from the VBE for intrinsic Bi2Te3 with = 0.75 at. %. This similarity can be understood in terms of the and defect chemistries.
Contrary to the effect of as discussed above, doping does not suppress Te p-orbital states. The similar atomic masses of Pb and Bi ensure that doping does not significantly alter the spin–orbit coupling effect observed in nondefective Bi2Te3.32 However, the higher electronegativity difference between Pb and Te (0.3) compared to Bi and Te (0.2)35 suggests that Pb doping increases bond polarity. Indeed, we observe increased electron localization on both Pb and Te atoms (Fig. 7), compared with Bi and Te atoms in nondefective Bi2Te3. This Pb-doping-induced ionicity increase thereby decreases electron wave function overlap, which decreases intraband energy level splitting and thus, . The consequent increase in m* for = 2.2 at. % manifests as a 0.4kBT retraction of the low- Z-F band, and a negligible change in the density of states near EF (Fig. 8).
Preservation of the low-m* Z-F band in -doped Bi2Te3 increases α2σ by enhancing relative to intrinsic Bi2Te3 with equivalent doping. For example, at = 0.3 at. %, this band contributes to a 25% higher σ ∼ 1.9 × 105 Ω−1 m−1 [Fig. 9(a)] due to an equivalent increase in mobility to ∼ 400 cm2 V−1 s−1 [Fig. 9(b)], at h ∼ 3 × 1019 cm−3. Since band contributions are averaged as ,6 α only increases by ∼5% [Fig. 9(c)], suggesting that the contribution of the high- bands to α dominates over that of the low- Z-F band. Therefore, doping of 0.3 at. % provides a ∼30% higher α2σ ∼ 5.1 mW m−1 K−2 [Fig. 9(d)]. Increasing Pb doping to = 0.75 at. % does not further alter α2σ, in contrast to a 25% α2σ decrease seen upon a similar increase in in intrinsic Bi2Te3.
At the highest studied doping level of  = 2.2 at. %, the charge carrier mobility decreases by 30% to ∼ 275 cm2 V−1 s−1 [Fig. 9(b)] due to Z-F band retraction from the VBE, while α decreases by <5% [Fig. 9(c)]. Thus, the power factor decreases to α2σ ∼ 3.3 mW m−1 K−2 [Fig. 9(d)]. Our results agree well with experimentally measured α2σ ∼ 4.5 mW m−1 K−2 in Bi2Te3 with = 0.03 at. %, which is 50% higher than that measured in Bi2Te3 with .9 No power factor enhancements over intrinsic Bi2Te3 with defects were experimentally reported for > 0.2 at. %.9
C. Ca doping
We find that substitutional doping in Bi2Te3 activates new bands, whose topology is sensitive to the Ca doping concentration. Controlling the doping level leads to up to 20% higher σ or 30% higher α than in intrinsic Bi2Te3 with . Subatomic-percent doping shifts multiple secondary low-m* bands toward the VBE. For example, for = 0.75 at. %, the low-m* Z-F energy ellipsoid transforms such that an alternate cross section labeled as the band (Fig. 10), aligns with the a-Γ band at the VBE at ∼0.8kBT higher than in nondefective Bi2Te3. This ellipsoid transformation also manifests as a movement of the left portion of the Z-F band 1.2kBT closer to the VBE. In addition, two low-m* bands along L-Z and Z-a shift to within 2kBT of the VBE. Collectively, this results in an up to threefold increase in DOS within 3kBT from the VBE (Fig. 11) compared to Bi2Te3 with equivalent .
Such a convergence of low-m* bands at = 0.3 at. % yields a 20% higher σ ∼ 1.8 × 105 Ω−1 m−1 at h ∼ 3 × 1019 cm−3 [Fig. 12(a)] compared to that with equivalent , due to a high ∼ 375 cm2 V−1 s−1 [Fig. 12(b)]. The 10% increase in α [Fig. 12(c)] is twofold greater than that observed for equivalent doping. The net effect is a ∼33% increase in the power factor to α2σ ∼ 5.3 mW m−1 K−2 [Fig. 12(d)]. At = 0.75 at. %, α2σ ∼ 4.7 mW m−1 K−2, which is 8% lower than that obtained with equivalent , but still 60% higher than in Bi2Te3 with equivalent
Increasing Ca doping to = 2.2 at. % creates bands with lower curvatures (Fig. 10) and thus higher effective-mass than at = 0.75 at. % near the VBE. In particular, the highest-effective-mass Γ-Z band advances to within 1.6kBT of the band edge of intrinsic Bi2Te3 (Fig. 10). The valence band maxima (VBM) along L-Z and a-Γ retract ≤ 0.8kBT along with both the Z-F and bands, which transform into flatter, higher-effective-mass bands. The Z-a band flattens and advances to the VBE. The collective effect is a more than tenfold increase in DOS near the VBE (Fig. 11), created by a fivefold increase in , if we assume that DOS .15 The higher m* (lower ) is consistent with the higher ionicity35 of –Te6c and –Te3a bonds (Fig. 13) than the Bi–Te counterparts in nondefective Bi2Te3. The defects form equal-length bonds of high ionic character with both Te6c and Te3a involving electron transfer from Ca. In nondefective Bi2Te3, however, Bi–Te6c bonds formed by electron transfer from Te are 5% shorter than the covalent Bi–Te3a bonds.28 Since Ca atomic mass is only ∼20% of that of Bi, doping decreases spin–orbit coupling, which usually inhibits convergence of bands involving 5p–6p orbitals.32 Our observation of increased band convergence at all suggests that the interaction of 4s–5p orbitals in Ca-doped Bi2Te3 is fundamentally different than the 5p–6p orbital interactions for , , and doping.
The increased m* with = 2.2 at. % manifests as a 20% higher α ∼ 200 μV K−1 at h ∼ 3 × 1019 cm−3, which is 30% higher than Bi2Te3 with optimum = 0.3 at. %, and 65% higher than with equivalent = 2.2 at. %. The σ decreases by 30% due to a μh decrease to 160 cm2 V−1 s−1 (Fig. 12), leading to a net α2σ ∼ 3.2 mW m−1 K−2. This power factor is only 40% of the maximum achieved with subatomic-percent doping, indicating that enhancements in α due to high-effective-mass bands cannot fully counteract the concomitant diminution of μh. These results suggest that highly degenerate, low-effective-mass bands, generated by covalent bonds, are perhaps more important for realizing high α2σ.
D. Sn doping
Experimentally, subatomic-percent Sn doping yields 40%–75% higher α than undoped Bi2Te3 at room temperature because of a 25%–100% increase in m*.10,23 However, our first-principles calculations indicate that doping of Bi2Te3 produces lower α2σ than equivalent doping of intrinsic Bi2Te3, despite the formation of an Sn-induced resonant state, i.e., a DOS peak within the valence band. Although doping does not suppress Te p-orbital states, we observe that all secondary bands retract away from the VBE, reminiscent of the decreased quasidegeneracy in intrinsic Bi2Te3 with doping (Fig. 14). Decreased spin–orbit coupling resulting from the ∼43% lower Sn atomic mass than Bi32 is the likely reason for retraction of the bands. The –Te6c bond has a higher covalent character than the Bi–Te6c bond, as seen from the slightly higher electron localization half-way between the Sn–Te bond (Fig. 15). This result is consistent with the 30% lower electronegativity difference of Sn–Te (0.14) than Bi–Te (0.2).35
The possibility of charge carrier mobility enhancement due to increased covalency, however, is counteracted by increases in band effective mass. All secondary VBMs in -doped Bi2Te3 have lower curvatures (higher m*) than those in intrinsic Bi2Te3 with equivalent levels (Fig. 14). For example, at = 0.75%, Z-a, Z-F, and band edges have ∼15% higher m* than in intrinsic Bi2Te3. At = 2.2 at. %, these three bands have ∼40% higher m*. These near-edge band topology changes are consistent with the 50%–200% higher DOS than that of nondefective Bi2Te3, attributable to distortion of the band structure, termed a resonant state deeper in the valence band [Fig. 16(a)]. Assuming DOS ,15 the resonant state corresponds to 30%–110% higher than intrinsic Bi2Te3 in agreement with an experiment.10
Decreasing doping shifts the resonant state toward the VBE, e.g., from 200 meV below at = 2.2 at. %, to 95 meV below the VBM at = 0.3 at. % [Fig. 16(b)]. The reported experimental value is 15 meV at [Sn] ∼ 0.1 at. %.36 Our calculations reveal two α2σ peaks: a primary peak around h ∼ 1019–1020 cm−3 is followed by a smaller secondary peak at h ∼ 1020–1021 cm−3 range [Figs. 16(c) and 17(d)]. These results mirror experimental results10 showing a primary α2σ peak at = 0.05 at. % (h = 1 × 1019 cm−3), followed by a secondary peak at = 0.3 at. % (h = 5 × 1019 cm−3). Additionally, our calculations show that decreasing the Sn doping level increases the primary peak magnitude and shifts the secondary peak to lower hole concentrations. Thus, if we were able to model dopant concentrations more similar to experiment (0.05–0.1 at. %, requiring ∼ to ∼ supercell size), we expect that the secondary α2σ peak would shift into the range of h ∼ 1–5 × 1019 cm−3, and increase the primary peak magnitude, in agreement with the experimentally observed α2σ enhancement.
Sn-doping-induced power factor behavior can be understood in terms of the nature and relative magnitude of the effects of Sn doping on σ and α. In particular, doping increases m* due to resonant state formation, but decreases α2σ due to a vastly greater decrease in σ. For example, at = 0.3 at. %, σ ∼ 9 × 104 Ω−1 m−1 [Fig. 17(a)] is 35% lower than Bi2Te3 with equivalent concentration, due to a decreased ∼ 185 cm2 V−1 s−1 [Fig. 17(b)] at h ∼ 3 × 1019 cm−3. A 10% increase in α ∼ 175 μV K−1 [Fig. 17(c)] slightly offsets the decreased mobility to produce a peak α2σ ∼ 2.75 mW m−1 K−2 [Fig. 17(d)] which is ∼30% lower than the value obtained from intrinsic Bi2Te3 with equivalent ]. Increasing Sn doping to = 2.2 at. % decreases α2σ to ∼1.4 mW m−1 −2, which is ∼20% lower than the value obtained from intrinsic Bi2Te3 with ] = 2.2 at. %.
E. Defect configuration
The α2σ of Bi2Te3 is sensitive to both defect type and configuration. defects show the strongest α2σ disparity between clustered and distributed configurations. At = 2.2 at. %, distributed defects have a separation distance of 0.75 ≤ dsep ≤ 1.5 nm. However, for the same defect concentration, if we force defects to cluster with dsep ∼ 0.4 nm, we get a 0.4 meV/atom lower total energy. This result indicates that clustered antisite defects, which yield a lower α2σ, are energetically favored. For example, for dsep ∼ 0.4 nm, α2σ ∼ 1.7 mW m−1 K−2 is more than twofold lower than α2σ ∼ 4 mW m−1 K−2 obtained at higher dsep.
If we decrease dsep by increasing , we increase the Bi– repulsion as well as Te6c– attraction for dsep ≥ 0.75 nm (see Fig. 18). In contrast, if we force dsep ∼ 0.4 nm for constant = 2.2 at. %, both bond lengths decrease Thus, –Bi and/or – bonds are responsible for degrading α2σ due to band retraction from the VBE, as shown in Fig. 19. Such differences in defect clustering (inset of Fig. 19) may explain experimental outliers, e.g., α2σ > 4 mW m−1 K−2 at h > 1020 cm−3.20
Forcing defects to cluster with dsep ∼ 0.4 nm at = 2.2 at. % yields a 0.1 meV/atom lower ground state energy (Fig. 20) than sparsely distributed defects with 0.75 ≤ dsep ≤ 1.5 nm. This result suggests a weaker preference for clustering than the above-discussed clustering. The distributed configuration yields only 5% lower α2σ due to the similarity of the valence 6p orbitals of Pb and Bi. Given the similar energetics of defect clustering, and the opposing effects of and on band structure and α2σ, it is conceivable that doping suppresses formation by altering the defect energetics.
Contrary to Pb doping, defects with separations in the 0.75 ≤ dsep ≤ 1.5 nm range exhibit a 0.3 meV/atom lower total energy (Fig. 20) than when clustered such that dsep ∼ 0.4 nm, indicating that distributed defects are energetically preferred. When defects are forced to cluster, α2σ decreases by 20%. Similar to Ca doping, distributed defects yield a ∼0.2 meV/atom lower total energy (Fig. 20) at 2.2 at. % concentration, suggesting that Sn does not prefer to cluster in Bi2Te3. When defects are forced to cluster, α2σ decreases by 25%.
F. Maximum ZT in nanostructured Bi2Te3
Results of our first-principles calculations described above clearly demonstrate and doping of Bi2Te3 can result in ≥30% higher α2σ/τ than that in intrinsic Bi2Te3 with antisite defects (Fig. 21). In addition, we show that subatomic-percent dopant concentrations yield the highest maximum α2σ, providing up to ≥40% higher α2σ compared to the respective highly doped systems. Since τ increases with decreasing dopant concentration, our use of a constant τ implies that our calculated increases in α2σ at subatomic-percent dopant concentration are a conservative estimate. We show below that we can potentially realize room-temperature ZT ∼ 1.7 in Bi2Te3 by combining such doping-induced electronic property enhancements with low thermal conductivity achieved in bulk pellets with an average grain size of dG ∼ 10 nm (Fig. 22). This represents a nearly twofold increase over the highest reported ZT = 0.87 (Refs. 37–39) from high-quality intrinsic single-crystal Bi2Te3, and a 240% increase over single-crystal p-Bi2Te3 with a more typical ZT ∼ 0.5,9,18,20 which we used to benchmark our DFT-based transport calculations (Sec. IV A). Nanostructuring with 10 nm grains enables a 60% decreased lattice thermal conductivity κL, accompanied by only a 20% decrease in α2σ, due to a 35% decrease in σ and a 10% increase in α (Fig. 21).
The key question raised by the above results is the extent to which the benefits of dopant induced band structure distortions can be isolated from possible changes in relaxation time. Since BoltzTraP cannot model the scattering, we analytically solved Boltzmann transport equations for α , σ, and κe, using the nonparabolic two-band Kane model.15 We baselined the models40 against experimental single-crystal transport data19 by implementing experimental carrier effective masses and bandgap (see Sec. III B). Energy-dependent optical, acoustic, and polar-optical phonon–electron scattering were included using standard expressions,15 and electron-defect scattering with the Brooks–Herring model.41 We have experimentally and theoretically shown that κL in nanograined Bi2Te3 is unaffected by the subatomic-percent doping and is lower than either the single-crystal or direction.42 Thus, our κL modeling methodology42 using modified Debye–Callaway equations43 started with κL ∼ 0.85 W m−1 K−1 for polycrystalline Bi2Te3 with randomly oriented grains, previously employed for the large-grain limit.42 Grain scattering was subsequently added to both electronic and thermal models as an additional scattering mechanism with mean-free-path equal to grain diameter dG. At dG = 10 nm, this additional scattering lowers κL more than σ, due to the grain size being higher than the mean-free-path of holes, but lower than that of phonons.42 The slight α enhancement is due to the energy filtering effect, i.e., increased scattering of high-energy carriers.44,45
Using our analytical models, we calculated each parameter in for 5 nm ≤ dG ≤ 1 μm at h = 3 × 1019 cm−3 (Fig. 22) and normalized all values to a grain size of dG = 1 μm, beyond which transport values are constant with grain size. Our DFT-based BoltzTraP transport values for α, σ, and κe (h = 3 × 1019 cm−3) as well as the experimental κL = 0.85 Wm−1 K−1 were subsequently scaled using these normalized transport trends to calculate nanostructured ZT for each dopant at 0.3 at. % concentration.
Our first-principles calculations on Bi2Te3 reveal band structure alterations created by Pb, Ca, and Sn doping, as well as by intrinsic Bi antisite defects, and their correlation to thermoelectric power factor. Subatomic-percent Pb and Ca doping enhance the power factor by 30%–75% over Bi2Te3 with intrinsic antisite defects, due to the retention and creation of highly degenerate, low-effective-mass bands near the valence band edge, which enhance carrier mobility. Concentrated Ca doping fosters high α, due to the ionic-bonding-induced actuation of high-effective-mass bands, despite the loss in spin–orbit coupling strength. Sn doping also increases the carrier effective mass and reduces the spin–orbit coupling but provides no enhancement in power factor compared with intrinsic p-Bi2Te3. Sn resonant states underlying the increased effective mass do, however, show the capability of sustaining α and μh, thereby inducing a secondary power factor peak at higher hole concentrations.
We show that higher-than-bulk ZT is best achieved in Bi2Te3 nanocrystals doped with subatomic-percent quantities of Pb or Ca. Although a high DOS near band edges is considered beneficial for high thermoelectric efficiency, our work illustrates the importance of the underlying band topology. Only the confluence of many, highly degenerate, low-effective-mass bands, rather than high-effective-mass bands or resonant states, can create the electronic landscape necessary to realize higher-than-bulk power factor. With nanostructuring-induced suppression of κ, ZT ≥ 1.7 can be achieved. We suggest exploring additional elements outside of periodic table groups 4–7 to find dopants that have different orbital characters, can occupy different Wyckoff positions, and thereby introduce new bands along different momenta.
See the supplementary material for reference data for band structure, effective mass, bonding, transport, Fermi levels and band gaps for each dopant concentration. It also includes experimental comparison data for the correlation of hole and doping concentrations..
For our calculations, we used the IBM Blue Gene/Q supercomputer at the Rensselaer Center for Computational Innovations. We gratefully acknowledge funding from the Department of Energy Basic Energy Sciences program through the S3TEC EFRC, Award No. DE-SC0001299/DE-FG02-09ER4657.