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Alloying bismuth telluride with antimony telluride and bismuth selenide for _{2}Te_{3-}_{2}Te_{2}Se due to chalcogen site occupancy preference. The parametrization of the electronic and thermal transports presented can be used in future optimization efforts.

Bismuth telluride is the dominant thermoelectric material for applications near room temperature due to its inherently low lattice thermal conductivity and high electronic weighted mobility [_{2}Te_{3} for _{2}Se_{3} for

The _{2}Te_{3}-Sb_{2}Te_{3} alloy has been well characterized and modeled. The thermal conductivity variation with composition is well described using a mass contrast alloy scattering model where composition varies on a single cation site [_{0.5}Sb_{1.5}Te_{3} resulting in an enhancement of the thermoelectric power factor [

The _{2}Te_{3}-Bi_{2}Se_{3} alloy is less straightforward. Mass contrast alloy scattering modeling does not match thermal conductivity values reported in the literature, and there is wide variability in qualitative trends reported [_{2}Te_{3} is comprised of conduction and valence band extrema with high valley degeneracy while the structure of Bi_{2}Se_{3} is far simpler with a direct gap between singly degenerate extrema at the _{2}Te_{2}Se suggests that complex band dynamics are in play [_{2}Te_{3} and Bi_{2}Se_{3} have been extensively studied, a comprehensive description of the dynamics in the alloy has not been reported.

In this report, we aim to explain long standing unresolved issues in the

Bismuth telluride, bismuth selenide, and all intermediate alloys _{2}Te_{3}-Sb_{2}Te_{3} alloy system where Bi and Sb occupy equivalent sites without preference.

The crystal structure of _{2}Te_{2}Se composition.

The difference in bonding at each chalcogen site and the occupancy preference has significant consequences for the electrical and thermal properties of the system. The preference results in the formation of an ordered compound at Bi_{2}Te_{2}Se which acts as a demarcation point between which the chalcogen site is being changed during alloying. There is some disagreement in the literature as to whether the ordered compound is stable or if solid state phase separation occurs as observed in Bi_{2}Te_{2}S [_{2}Te_{2}S, this separation occurs to relieve bond angle strains that would otherwise occur in the ordered Te^{(1)}–Bi–S^{(2)}–Bi–Te^{(1)} compound. Ordering of the Bi_{2}Te_{2}Se phase has been observed in numerous studies [

The layered structure of the tetradymites results in significant anisotropy in transport properties. In the _{2}Te_{3}-Sb_{2}Te_{3} alloy system where the electrical and thermal conductivity ratios are both near 2-3 resulting in a nearly isotropic _{0.5}Sb_{1.5}Te_{3}, but only in the range of 1-1.2 for

The electronic transport properties vary significantly with alloy composition as shown in Figure _{e} for Bi_{2}Te_{3} to 0.25 _{e} for Bi_{2}Se_{3} [_{w}^{2}V^{-1} s^{-1} going from Bi_{2}Te_{3} to Bi_{2}Se_{3} [

The band gap and electronic transport in

In the Bi_{2}Te_{3}-Sb_{2}Te_{3} alloy system, a peak in the effective mass and an abrupt change in the band gap slope with composition is observed near Bi_{0.5}Sb_{1.5}Te_{3} [_{2}Te_{2}Se and also a peak in Seebeck effective mass of ~1.30 _{e} near Bi_{2}Te_{2.5}Se_{0.5} (Figure

The electronic structure of the

Experimental characterization of the Bi_{2}Te_{3} conduction band Fermi surface has found the conduction band minimum to be sixfold degenerate and nonparabolic [

Schematic illustration of the impact of the spin-orbit interaction induced band overlap in the _{2}Se_{3} and Bi_{2}Te_{3}, an anticrossing occurs at the overlap creating a band gap and change of character for states at _{2}Te_{3}, the inversion is significant enough to invert the curvature at _{2}Se_{3} with varying strengths of spin-orbit interaction. Doubling the strength of the interaction causes the CB extremum at _{2}Te_{3} (Figure

ARPES and SdH measurements have found the CBM in Bi_{2}Se_{3} to be a single ellipsoidal valley centered at _{2}Se_{3} (Figures _{2}Te_{3} at

Fermi surface maps in the bisectrix plane illustrate the variation in positions and energies of the conduction band extrema with alloy composition. In Bi_{2}Se_{3} (a), the CBM is singly degenerate at _{2}Te_{3} (c), the CBM is at the sixfold f point slightly displaced from the _{2}Te_{2.5}Se_{0.5} (b), the pockets at f and z are at almost the same energy level. Varying the alloy composition manipulates the magnitude of the spin-orbit interaction and shifts the energy and positions of various extrema. This is illustrated in (d) where doubling the spin-orbit interaction in Bi_{2}Se_{3} calculations results in a CBM at f as in Bi_{2}Te_{3}.

Köhler et al. performed Shubnikov-de Haas measurements on _{2}Te_{3} side, the energy separation between the CBM (_{2}Te_{1.5}Se_{1.5} does not clearly show a conduction band pocket near

Band structure calculations on the ordered Bi_{2}Te_{2}Se compound disagree regarding the order and location of the conduction band extrema. Reports can be found with the CBM occurring along the _{2}Te_{3} to be near but not on the _{2}Te_{2}Se; however, that study focused on the Dirac point in the surface states very near the VBM and did not capture the bulk conduction band [

In the absence of SOI, both the conduction band minimum (CBM) and valence band maximum (VBM) would be single valleys located at _{2}Te_{3}, the overlap is significant enough to invert the curvature of the bands at _{2}Se_{3}, it is not enough to invert the curvature at

The magnitude of the SOI induced overlap increases monotonically from Bi_{2}Se_{3} to Bi_{2}Te_{3} when alloying [_{2}Te_{3} and Bi_{2}Se_{3} can be qualitatively understood as modulating the magnitude of the SOI-induced band overlap. This is illustrated by electronic structure calculations in Figures _{2}Se_{3} with varying degrees of SOI. Doubling the SOI for Bi_{2}Se_{3} inverts the curvature at _{2}Te_{3} where the SOI effect is greater (Figures

Knowing the effects of varying SOI allows for understanding of trends in the band gap with alloy composition. The opening of the band gap when substituting Te into Bi_{2}Se_{3} results from the CBM at _{2}Te_{2}Se is not caused by the crossing of the CBM of Bi_{2}Te_{3} and the CBM of Bi_{2}Se_{3} as has been previously suggested [_{2}Te_{3}, the VBM is located near the _{2}Se_{3}, the VBM is found at _{2}Te_{2}Se, there is a change in the location of the lowest energy direct transition [

The shifts in band edge energies discussed in the previous section have been summarized schematically in Figure

(a) Four band edges are primarily responsible for the electronic transport in the _{2}Te_{2.7}Se_{0.3} occurs due to the crossing of the sixfold _{2}Te_{2}Se can be understood as a crossing between the

The Seebeck coefficient,

Here,

A more complete effective mass model for this system would contain the variation with composition for each band of the Seebeck and conductivity masses, the deformation potentials, and the band degeneracies in addition to changes in lattice stiffness and alloy scattering energies. Unfortunately, most of these parameters have not been reported in the literature, and the large number of unknown variables leads to fit solutions which are not unique. Despite this difficulty, some conclusions may still be drawn regarding the band parameters.

Kohler reported that the lowest conduction band in Bi_{2}Te_{3} was sixfold degenerate with a density-of-states mass of 0.27 _{e} and a second conduction band was present only 25 meV higher having a much higher mass estimated to be ~3 _{e} [_{2}Te_{3} found the relationship between conductivity and Seebeck could be well described by a weighted mobility of _{e} behaving like a single band with a mass of 1.06 _{e} necessitates the second band having both a higher mass and weighted mobility. This result is contrary to expectation considering the _{2}Te_{2.7}Se_{0.3}, Konstantinov et al. found that the deformation potential of the sixfold pocket must be 6-8 times larger than the second, higher mass band edge [_{2}Te_{3} is attributable to the

In Bi_{2}Se_{3}, the lowest band is singly degenerate at _{e}; however, the nonparabolicity of the band makes this a poor descriptor of the density-of-states beyond _{e} accurately predicts the Seebeck data for Hall carrier concentrations between _{2}Se_{3}. The weighted mobility of any second band must be significantly lower than that of the

We present in Figure _{2}Se_{3} and _{2}Te_{3}. Similarly, predicting the conductivity mass variations requires more significant knowledge than is available of pocket anisotropy changes with alloy composition. The values presented here are meant to serve as reasonable estimates based upon available data to be used for further experimental and theoretical verification.

Linear variations are assumed for fitting parameters between the binary compositions. Wherever possible, data from experimental results and band structure calculations are used to set parameters of the model. It must be noted that many of these experimental values were measured at temperatures lower than 300 K where the transport data was taken. Band offsets and masses could shift with temperature, and the degree to which this occurs is not clear. The values solved for in fitting are given in Table

Fitting parameters used in a two-conduction band effective mass model. Linear variation was assumed between the binary compositions.

Seebeck mass, _{e}) |
Deformation potential, |
_{e}^{-1}) | ||||
---|---|---|---|---|---|---|

Bi_{2}Se_{3} |
0.97 | 0.25 | 62.4 | 62.4 | 44.3 | 8.2 |

Bi_{2}Te_{3} |
0.27 | 1.19 | 5.0 | 16.4 | 124.7 | 2.7 |

We attempted to include in our model the effects of alloy scattering of charge carriers as described by Harrison and Hauser [_{2}Te_{3} due to the relatively soft lattice and large static dielectric permittivity [

Shown in Figure _{2}Te_{3} to 1 in Bi_{2}Se_{3}; however, its mass decreases substantially such that the weighted mobility would still be expected to increase. This suggests that the deformation potential of the _{2}Te_{3} to Bi_{2}Se_{3}. Our model fits this increase as from 5 to 16 eV. An improved fit could be obtained by not assuming a linear variation in parameters across the entire alloy composition; however, experimental and theoretical data do not provide sufficient guidance to avoid overfitting. Future studies could refine the understanding of this important thermoelectric and topological insulator material system.

Much of thermoelectric material engineering pertains to reducing a material’s lattice thermal conductivity while still maintaining a high weighted mobility. This is possible because of the difference in order of magnitudes of the mean free paths of charge carriers (small) and phonons (small to large). Nevertheless, most methods of reducing thermal conductivity also reduce carrier mobility somewhat, and evaluation of the success in balancing electrical and thermal engineering can be performed using the quality factor [

The effect on lattice thermal conductivity of alloying Bi_{2}Te_{3} and Bi_{2}Se_{3} is shown in Figure _{2}Te_{2.5}Se_{0.5}. To gain further insight, the alloy scattering model of Callaway and von Baeyer and Klemens was fit to the data of each study as shown in Figure

Alloying Bi_{2}Te_{3} and Bi_{2}Se_{3} significantly reduces the lattice thermal conductivity in comparison to the pure binaries and the ordered Bi_{2}Te_{2}Se phase [_{2}Te_{3} and Bi_{2}Te_{2}Se and between Bi_{2}Te_{2}Se and Bi_{2}Se_{3} (b). Mass contrast alone nearly describes the trend between Bi_{2}Te_{2}Se and Bi_{2}Se_{3}; however, the lattice thermal conductivity Bi_{2}Te_{3} and Bi_{2}Te_{2}Se requires considerations of strain or bonding changes at least an order of magnitude higher.

The lattice thermal conductivity of the alloy (Equation (_{2}Te_{2}Se, the Bi_{2}Te_{3-}_{2}Te_{3}-Bi_{2}Te_{2}Se and Bi_{2}Te_{2}Se and Bi_{2}Se_{3}. In order to account for alloying on two different chalcogen sublattices, we use the scattering parameter form of Yang et al. which was used to describe the effects of alloying on each site in ZrNiSn [

Here, the scattering parameter (Equation (

While there is discrepancy in the reported endpoint lattice thermal conductivities of Bi_{2}Te_{3}, Bi_{2}Te_{2}Se, and Bi_{2}Se_{3}, the least-squared-error fit _{2}Te_{1.5}Se_{1.5} [_{2}Te_{3}-Sb_{2}Te_{3} _{2}Te_{2}S, the more electronegative sulfur atom preferentially occupies the (2) site. At this composition, large bond angle strains prevent the formation of the ordered compound, and instead, some of the (2) site are occupied by Te [_{2}Te_{3} and Bi_{2}Te_{2}Se.

Summary of “endpoint” lattice thermal conductivity data from various authors and the

Reference | Lattice thermal conductivity (W/mK) | Best fit | |||
---|---|---|---|---|---|

Bi_{2}Te_{3} |
Bi_{2}Te_{2}Se |
Bi_{2}Se_{3} |
Te/Se^{(1)} |
Te/Se^{(2)} | |

Birkholz [ |
1.65 | 1.26 | 2.39 | 7.7 | 75.2 |

Rosi et al. [ |
1.30 | 1.24 | 2.00 | 0 | 72.8 |

Goldsmid [ |
1.54 | 1.23 | 82.7 | ||

Champness et al. [ |
1.77 | 1.35 | 1.74 | 13.7 | 169.3 |

Average | 1.56 | 1.27 | 2.04 | 7.1 | 100.0 |

Transport modeling of the _{2}Te_{2}Se compound provides clarity and identifies bond strain as a key factor between Bi_{2}Te_{3} and Bi_{2}Te_{2}Se. It is our hope that the analyses provided here will aid future efforts to characterize and engineer

The Bi_{2}Te_{3} (_{2}Se_{3}, and Bi_{2}Te_{2}Se (

Experimental cell parameters used for electronic structure calculations [

Compound | Volume^{3}) | ||
---|---|---|---|

Bi_{2}Te_{3} |
10.468 | 24.164 | 168.933 |

Bi_{2}Se_{3} |
9.841 | 24.304 | 141.890 |

Sb_{2}Te_{3} |
10.284 | 23.851 | 156.217 |

For the composition

The electronic band structures were calculated with Density Functional Theory (DFT), using the Vienna Ab initio Simulation Package (VASP) [_{2}Se_{3}. In order to get a band structure that fits better the experimental band structure, we set a SOC weight equal to 0.7, corresponding to 100% in Figure

The band structure on the bisectrix plane for the Bi_{2}Te_{3} and Bi_{2}Se_{3}, shown in Figure

The band structure on the high symmetry path, shown in Figure

The band structure of the Bi_{2}Te_{2.5}Se_{0.5} compound was calculated using BandUP software [

All data analyses, such as band structure plotting and supercell generation, were carried out using the pymatgen python package [

In a single band system with a parabolic energy dependence on

The authors declare no conflicts of interest.

This research was carried out under a contract with the National Aeronautics and Space Administration and was supported by the NASA Science Missions Directorate’s Radioisotope Power Systems Technology Advancement Program. The authors also gratefully acknowledge thermoelectrics research at Northwestern University through the award 70NANB19H005 from U.S. Department of Commerce, National Institute of Standards and Technology as part of the Center for Hierarchical Materials Design (CHiMaD). FR acknowledges support from the LOCOTED (Low Cost ThermoElectric Devices) project funded by the Walloon Region (Programmes FEDER).

_{2}Te

_{3}and Bi

_{2}Te

_{2}Se from first principles

_{2}Te

_{3}-Bi

_{2}Se

_{3}alloys

_{2}Te

_{3}-Sb

_{2}Te

_{3}alloys between 2-15 microns

_{2}Te

_{3}sowie der festen lösungen Bi

_{2-x}Sb

_{x}Te

_{3}und Bi

_{2}Te

_{3-x}Se

_{x}hinsichtlich ihrer eignung als material für halbleiter-Thermoelemente

_{0.25}Sb

_{0.75})

_{2}Te

_{3}due to band convergence and improved by carrier concentration control

_{2}Te

_{3}-Sb

_{2}Te

_{3}system

_{2}Te

_{3}-related materials

_{2}Se

_{3}, Bi

_{2}Te

_{3}and Sb

_{2}Te

_{3}with a single Dirac cone on the surface

_{2}Te

_{3}

_{2}Te

_{3}und Bi

_{2}Te

_{2}S

_{2}Te

_{3−x}Se

_{x}

_{2}Te

_{2}Se

_{2}V

_{I3}thermoelectric materials

_{14}Te

_{13}Se

_{8}and the phase Bi

_{14}Te

_{15}S

_{6}

_{2}Te

_{2}Se

_{2}Te

_{3}-Bi

_{2}Se

_{3}system

_{2}Se

_{3}-Bi

_{2}Te

_{3}and Bi

_{2}Se

_{3}-In

_{2}Se

_{3}phase diagrams

_{2}(Te

_{1−x}Se

_{x})

_{3}single crystal solid solutions (

_{2}Se

_{3}single crystals

_{2}Te

_{2.7}Se

_{0.3}

_{x}Bi

_{2}Te

_{2.7}Se

_{0.3}nanocomposites

_{2}Te

_{3}single crystals

_{2−x}In

_{x}Se

_{3}single crystals

_{2−x}Sb

_{x}Se

_{3}mixed crystals by Shubnikov–de Haas and cyclotron resonance measurements in high magnetic fields

_{2}Se

_{3}

_{1−x}Sb

_{x})

_{2}Se

_{3}single crystals

_{1−x}Sb

_{x})

_{2}Te

_{3}single crystals

_{2}Te

_{3}-Bi

_{2}Se

_{3}

_{2}(Te

_{1−x}Se

_{x})

_{3}-based compounds as n-type legs for low-temperature power generation

_{2}(Te

_{1−x}Se

_{x}

_{3}

_{2}Te

_{3}valence band structure

_{2}Te

_{3}

_{2}Te

_{3}

_{2}Te

_{3}

_{2}Te

_{3}from Shubnikov-de Haas effect

_{2}Te

_{3}

_{2}Se

_{3}due to the three-dimensional bulk Fermi surface

_{Bi2See}from Shubnikov-de Haas investigations

_{2}(Se

_{x}Te

_{1−x})

_{3}

_{2}Te

_{2}Se topological insulator

_{2}

_{3}(

_{2}Te

_{2}Se thin films: a first-principles study

_{2}Te

_{2}X (X = Te, Se, S) topological insulators

_{2}Te

_{3}

_{1/2}corrections in the electronic structure of Bi

_{2}Te

_{3}compounds

_{2}Te

_{3}

_{2}Te

_{3}and Sb

_{2}Te

_{3}from first-principles calculations

_{2}Se

_{3}and Bi

_{2}Te

_{3}topological insulator surfaces

_{2}Te

_{3}and Sb

_{2}

_{ofBi2Te3andBaBiTe3}

_{2}Te

_{3}

_{2}Se

_{3}crystals

_{2}Te

_{2.7}Se

_{0.3}in a two-band model of the electron spectrum

_{2}Te

_{3}–Sb

_{2}Te

_{3}alloys