Semi-metals as potential thermoelectric materials: case of HgTe
Maxime Markov, Xixiao Hu, Han-Chun Liu, Naiming Liu, Joseph Poon, Keivan Esfarjani, Mona Zebarjadi
SSemi-metals as potential thermoelectric materials: case ofHgTe
Maxime Markov , Xixiao Hu , Han-Chun Liu , Naiming Liu , JosephPoon , Keivan Esfarjani , and Mona Zebarjadi ∗ Department of Electrical and Computer Engineering, University of Virginia, Charlottesville,Virginia 22904, USA Department of Physics, University of Virginia, Charlottesville, Virginia 22904, USA Department of Materials Science and Engineering, University of Virginia, Charlottesville,Virginia 22904, USA Department of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville,Virginia 22904, USA
Abstract
The best thermoelectric materials are believed to be heavily dopedsemiconductors. The presence of a bandgap is assumed to be essentialto achieve large thermoelectric power factor and figure of merit. In thiswork, we study HgTe as an example semimetal with competitive ther-moelectric properties. We employ ab initio calculations with hybridexchange-correlation functional to accurately describe the electronicband structure in conjunction with the Boltzmann Transport theoryto investigate the electronic transport properties. We show that in-trinsic HgTe, a semimetal with large disparity in its electron and holemasses, has a high thermoelectric power factor that is comparable tothe best known thermoelectric materials. We also calculate the latticethermal conductivity using first principles calculations and evaluatethe overall figure of merit. Finally, we prepare semi-metallic HgTesamples and we characterize their transport properties. We show thatour theoretical calculations agree well with the experimental data. ∗ Corresponding author.
E-mail: [email protected] a r X i v : . [ c ond - m a t . m t r l - s c i ] J a n Introduction
Since its discovery in 1821, thermoelectricity remains in the center of interestsof the scientific community. Thermoelectric effect (Seebeck effect) refers todirect conversion of thermal to electrical energy in solids and can be used forpower generation and waste heat recovery. [1, 2, 3, 4]. Despite their clean,environmentally friendly and reliable performances, thermoelectric modulesare only used in niche applications such as in powering space probes. Themain obstacle preventing thermoelectric technology to be widely used on amass market today is its relatively low efficiency [5].The thermoelectric efficiency is an increasing function of the material’sdimensionless figure of merit ZT = S σκ T where S is the Seebeck coefficient, σ is the electrical conductivity, κ is the thermal conductivity, and T is theabsolute temperature. The first two quantities can be combined togetherinto the thermoelectric power factor P F = S σ describing electronic trans-port, in contrast to the thermal conductivity, κ , related to thermal transport.The power factor is often used as a guide to preselect the class of potentialthermoelectric materials. Indeed, metals have highest electrical conductivitybut suffer from a low Seebeck coefficient. The reason for their low Seebeckcoefficient is the symmetry of the density of states around the chemical po-tential. The number of hot electrons above the chemical potential in a metalis roughly the same as the number of cold empty states below the chemicalpotential. As a result under a temperature gradient, the number of elec-trons diffusing from the hot side to the cold side, is approximately equal tothe number of cold electrons diffusing from the cold side to the hot side.The same problem does not exist in semiconductors due to the presence of aband gap allowing only one type of the carriers to diffuse. Typical Seebeckcoefficient of semiconductors is two orders of magnitude larger than metals.Ioffe first noticed this advantage of semiconductors [6] and paved the wayfor many successful demonstration of doped semiconductors with high ZTvalues. Later, several research groups including Chasmar & Stratton [7] andSofo & Mahan [8] studied the effect of band gap on thermoelectric prop-erties of materials employing two-band toy models for electronic structureand reached the conclusion that best thermoelectrics must have band gapgreater than at least 6 k B T. Today, this criteria has become a golden rule andheavily doped semiconductors are the main focus of the thermoelectric soci-ety [9]. While opening a band gap is a proven way of increasing the Seebeckcoefficient, in this article we show that to have a large Seebeck coefficient,a band gap is not a must. What needed is an asymmetric density of stateswhich could be achieved also in semi-metals with slight overlap of electronsand holes bands but with large asymmetry in the electron and hole effective2asses.We turn our attention to semi-metallic HgTe whose properties are in thetransition region between semiconductors and metals. HgTe has a very highelectron/hole effective mass ratio m e /m h (cid:39) . µV /K [11] and -135 µV /K [12]at room temperatures which is similar to the Seebeck coefficient of heavilydoped semiconductors with a bandgap. The carrier concentration of intrinsicHgTe is only 10 − cm − which is much smaller than a metal or a typ-ical good heavily-doped semiconductor thermoelectric. However, the largeelectron mobility in HgTe ( µ > cm /V.s ) [10] makes up for its low carrierconcentration and as a result, the electrical conductivity of an intrinsic sam-ple is relatively large and is about σ = 1700 S/cm [12, 11] at room temper-atures. The large electron mobility is partly due to the small effective massof the electrons and partly because of the absence of dopants. The mobilityof a heavily doped semiconductor is limited by ionized impurity scatteringwhich is not the case in an intrinsic semi-metal. The experiment reveals thatintrinsic HgTe is a high power factor material with P F = 14 − µ W cm − K − at T = 300 K [12, 11] that is comparable to well-known thermoelectricmaterials such as SnSe ( P F (cid:39) µ W cm − K − ), PbTe − x Se x ( P F (cid:39) µ Wcm − K − ) and Bi Te ( P F (cid:39) µ W cm − K − ) at their ZT maximum [13].Apart from having a good electrical transport properties, mercury tellurideis a good thermal insulator with κ = 2 . p -type samples of HgTe [11].In this work, we perform a combined theoretical and experimental studyof thermoelectric properties of HgTe at high temperatures. To address theabove mentioned issues, we employ ab initio calculations with hybrid exchange-correlation functional in conjunction with the Boltzmann Transport theory3ith energy dependent relaxation times obtained from the fitting of experi-mental electrical conductivity. We do not attempt to optimize the thermo-electric properties of HgTe using nanostructuring, alloying or slight doping.Instead, we attempt to develop a platform based on first principles calcula-tions to study its transport properties and to make a case for semi-metals aspotential candidates for thermoelectric applications. The electronic band structure of zinc-blende HgTe has been extensively stud-ied over the past decade. [17, 18, 19, 20] It has been shown that ab initio calculations with standard LDA and GGA exchange-correlation functionalscan not accurately describe the band structure of HgTe. To achieve a goodagreement with experiment, one must perform either GW calculations [18, 19]or use a hybrid functional [17, 20] where a portion of exact Fock exchangeinteraction is introduced into a standard exchange-correlation functional.Figure 1: Electronic band structure (panel a), density of states g ( E ) (panel b)and differential conductivity σ xx ( E ) (panel c) calculated using PBE (blackcurves) and HSE06 (red curves) exchange-correlation functionals. Energylevels from the latter calculation are labeled according to their symmetries.4GA-PBE HSE06 Expt. E Γ = E(Γ ) - E(Γ ) -0.93 -0.27 -0.29 [21],-0.30 [22]∆ Γ = E(Γ ) - E(Γ ) 0.76 0.89 0.91 [21] E L = E( L c ) - E( L , ) 1.45 2.19 2.25 [22]∆ L = E( L , ) - E( L v ) 0.54 0.56 0.62 [22], 0.75 [23] E X = E( X c ) - E( X ) 4.15 5.02 5.00 [23]∆ X = E( X ) - E( X v ) 0.19 0.22 0.1-0.2 [23]Table 1: Energy band edges, E , and spin-orbit splittings, ∆, at Γ, L andX high symmetry points calculated with the GGA-PBE and hybrid-HSE06functionals. Experimental results from the literature are shown.In Fig. 1 (a), we compare the electronic band structures calculated usingGGA-PBE [24] (black curves) and hybrid-HSE06 [25] (red curves) exchange-correlation functionals and summarize the theoretical and experimental bandedges, E, and spin-orbit splittings, ∆, at Γ, L and X high symmetry pointsin Table 1. First, we note that the HSE06 calculation predicts the correctlevel ordering Γ , Γ , Γ [18, 20] that is consistent with experiment [21] incontrast to the GGA-PBE calculation where the Γ and Γ bands are re-versed. Second, the band energies obtained with the hybrid functional arein excellent agreement with experiment. For instance, the inverted band gap E g = E Γ − E Γ = − .
27 eV and spin-orbit splitting ∆ = E Γ − E Γ = 0 . m e = 0.18 m in GGA-PBE to m e = 0.04 m in HSE06 in the [100] direction,whereas the effective mass of the top valence bands remains essentially un-changed m h = 0.29 m in GGA-PBE to m h = 0.33 m in HSE06 . Thus,HgTe is a material with a very high electron-hole effective mass ratio.Finally, the electronic properties of HgTe near the Fermi level are definedby the region of the Brillouin zone close to the Γ point, where the bands havea low degeneracy. This low degeneracy in combination with a small electroneffective mass in HSE06 calculation results in a small density of states of con-duction bands. The asymmetry between the conduction and valence bandsis clearly seen in both, the density of states g ( E ) and the differential con-ductivity σ xx ( E ), as can be seen in Fig. 1 (b) and (c) respectively.In Fig. 2 (a), we show the Seebeck coefficient as a function of dopingconcentration for p - and n -types of doping at T = 290 K calculated usingthe constant relaxation time approximation. Our results with the GGA-PBE functional agree well with the previous calculation of Chen et al. [15]done with the same exchange-correlation potential. As it is expected from theband structure calculations, one can see a noticeable change in the magnitude5igure 2: The Seebeck coefficient (panel (a)) and the electrical conductiv-ity (panel (b)) as a function of carrier concentration for p -type and n -typesamples at T = 290 K calculated with GGA-PBE (black dashed line) andhybrid-HSE06 (solid red line) functionals. Experimental data from Whitsett et al. [11] is shown by a blue circle and experimental data measured in thiswork is shown by black square.of the Seebeck coefficient due to the increase of the electron-hole effectivemass ratio in HSE06 calculation. For instance, the maximum of the Seebeckcoefficient is increased from 142 µ V/K to 202 µ V/K and is slightly shiftedtowards the lower doping concentrations from 2 · cm − to 9 · cm − .In intrinsic and low doped samples (up to 10 cm − ), the Seebeck coefficientremains constant but also has a sufficiently higher magnitude of -81 µ V/Kwith HSE06 instead of − µ V/K with GGA-PBE. Our HSE06 result is ingood agreement with experimental result -91 µ V/K (blue circle) reported byWhitsett et al [11] for p-type sample. However, our measurements in n-typeHgTe sample with n = 3 . · cm − doping concentration show much largervalues of the Seebeck coefficient of -136 µ V/K (black square).The constant relaxation time theory, does not allow to compute the elec-trical conductivity but only its ratio to the unknown relaxation time στ . Ascan be seen in Fig. 2 (b), this ratio varies slowly at low doping concentra-tions and grows rapidly at high doping concentrations. However, one wouldexpect a different behavior for the electrical conductivity at least in the highdoping concentration region where a strong charged carrier scattering lim-its the mobilities. Thus, to further investigate the behavior of the electricalconductivity and the Seebeck coefficient, we introduce the phenomenolog-ical scattering rates and fit them to reproduce our experimental electricalconductivity data in n -type sample.Fig. 3 (a) show our experimental data obtained using the four-terminal6igure 3: Panel (a) : Temperature variation of the electrical conductivity σ measured in the experiment in n -type samples before (violet circles) and after(red and green squares) annealing. The fitting curves are shown by dashedblue and solid black lines respectively. Panel (b) : Temperature variation ofthe Seebeck coefficient S for n -type samples measured in experiment (violetcircles and green squares). The theoretical Seebeck coefficients calculatedin the CRTA and in the ERTA are shown by black solid and red dasheddotted lines respectively. We have used ZEM and PPMS systems for themeasurements.probe method [26] in the samples prepared using the spark plasma sintering(SPS) technique. Two sets of measurements before (violet circles) and after(red and green squares) annealing have been performed. As expected, anneal-ing improves the electrical conductivity [27, 11] which reaches its maximumvalue of σ = 1036 (Ω cm) − at T = 350 K and then starts monotonicallydecreasing at higher temperatures. We notice that our results are much lowerthan the electrical conductivity σ = 1700 Ω − cm − measured in the intrin-sic samples at T = 300 K [12, 11]. These intrinsic samples were preparedby multiple annealing of the originally p -type samples in the presence of Hggas [12, 11]. However, in this work we do not follow this procedure due tothe extreme toxicity of mercury.We fit the measured electrical conductivity using ab initio data for thedifferential conductivity σ xx ( E ) and the density of states g ( E ) obtained withthe hybrid-HSE06 functional and phenomenological energy dependent scat-tering rates accounting for the acoustic deformation potential, polar opticaland ionized impurity scattering rates. [3] Details of the considered scatter-ing rates are described in Supplementary information. We then recalculatethe Seebeck coefficient using the obtained scattering rates and find that itsmagnitude is increased about 2 times with respect to the constant relax-ation time approximation (CRTA). The energy dependent relaxation time7pproximation (ERTA) results in Seebeck coefficient values that are closerto the experimentally measured ones. Therefore we conclude that the differ-ence between the CRTA calculations (Fig. 2a) and experimental values is aresult of the energy dependence of the scattering rates. Although the See-beck coefficient is not as sensitive as the conductivity to the relaxation times,this example demonstrates that CRTA results could be misleading even incalculation of the Seebeck coefficient.The temperature variation of the Seebeck coefficient calculated in theCRTA (black solid lines), the ERTA (red dashed dotted line) and measured inexperiment are shown in Fig. 3 (b). As one can see, both the theoretical andexperimental Seebeck coefficients remain almost temperature independent inthe studied temperature range between 300 and 500 K.Our study reveals that for the accurate description of the electrical trans-port properties of HgTe, one needs to accurately reproduce the electron-holeeffective mass ratio that can not be achieved using standard LDA or GGAexchange-correlation functionals. Moreover, we find that the inclusion ofenergy dependent scattering rates changes the magnitude of the Seebeck co-efficient drastically. The latter has been unexpected since, according to thecommon believe [29], the CRTA reproduces well the behavior of the diffusionpart of the Seebeck coefficient. The magnitude of the Seebeck coefficient ofHgTe is an order of magnitude higher than the one in typical metals andclose to the typical values of narrow-gap semiconductors. That is explainedby the the low effective mass and low degeneracy of the conduction bandnear the Fermi level. We then conclude that the presence of a bandgap isnot essential for obtaining large Seebeck coefficient values. Now, we turn our attention to the thermal transport properties of HgTe.First, we investigate the lattice dynamics by calculating the phonon spectrumalong the high symmetry directions. The phonon dispersion is shown inFig. 4 and is in an excellent agreement with previous theoretical results [30,16, 31] as well as with available data from the inelastic neutron scatteringexperiments [32, 33] (green circles). In our calculations we do not take intoaccount the non-analytical correction to split the optical phonons at Γ point.However, this correction should not strongly affect the thermal conductivitysince the contribution is usually small due to the low group velocities ofoptical phonons. Our theoretical frequencies for optical phonons ω O (Γ) = 118cm − agree well with the Raman spectroscopy data for the transverse opticalphonons ω T O (Γ) = 116 cm − [34].To further validate the vibration spectrum, we calculated the elastic con-8igure 4: Theoretical phonon dispersion calculated using DFPT in this work(black curves) compared to the inelastic neutron scattering data (green cir-cles) [32, 33]. C , GPa C , GPa C , GPaPresent 57.3 41.0 22.0Experiment 59.7 [35] 41.5 [35] 22.6 [35]Other 56.3 [36] 37.9 [36] 21.2 [36]67.4 [37] 45.7 [37] 30.0 [37]Table 2: Elastic constants C ij (GPa) calculated in the present work andcompared with other theoretical calculations [36, 37] and experiment [35].stants C ij . As shown in Table 2, the difference between our theoretical resultsand experiment does not exceed 4%. Then, we compare the sound veloci-ties in [100] direction obtained from the elastic constants, from the slopes ofacoustic branches near the Γ point and experimental data in Table 3. Thelargest differences with the experiment, 7.1% and 2.5% for the transverse(TA) and longitudinal (LA) sound velocities respectively, are found for theevaluation of sound velocities from the slopes of acoustic phonons.Figure 5 summarizes the theoretical and experimental thermal conduc-tivity obtained in this work as well as those reported by other groups. Weperform the lattice thermal conductivity calculations by exactly solving theBoltzmann Transport Equation (BTE). First, we include only the intrinsicthree-phonon anharmonic scattering (dotted black curve). We obtain thelattice thermal conductivity that is much lower than the previous ab initio L , m/s v T , m/sElastic constant 2655 1645Slope 2747 1504Experiment 2680 1620Table 3: The longitudinal v L and transverse v T sound velocities (m/s) in [100]direction calculated in the present work from the elastic constants, slopes ofacoustic phonons and experiment.Figure 5: Panal (a):
Temperature dependence of the thermal conductivitycalculated with account for anharmonic three-phonon processes only (blackdashed line ) and with addition of isotopic disorder scattering (black solidline); green squares - experimental data from Whitsett et al [11]; blue circles -our experimental data; dashed red curve - previous computational result fromRefs [16, 31].
Panal (b):
Accumulated thermal conductivity as a function ofphonon mean free path Λ at T = 100 K (blue curve), 300K (black curve) and500K (red curve). Horizontal dotted line denotes 50 % thermal conductivityreduction.calculations (red dashed curve) [16]. For instance, we get κ L = 5 .
48 W/mKinstead of κ L = 10 .
46 W/mK in Ref. [16]. Our theoretical values are stillhigher than ones measured in experiment κ L = 2 . κ L = 2 .
14 W/mK (Ref. [11]). This discrepancy can not be attributed tothe extrinsic sources of scattering such as the impurity scattering since theexperimental data for the p -type samples with doping concentration between10 − cm − show essentially the same thermal conductivity [11]. Theaddition of isotopic disorder scattering significantly decreases the thermalconductivity mainly at low temperatures (black solid curve) whereas at hightemperatures the isotopic scattering plays a minor role. At room temperaturewe get κ L = 4 .
68 W/mK that is still higher than experimental values.10igure 6: Temperature variation of the power factor
P F T = S σT (panel a)and thermoelectric figure of merit ZT (panel b) measured in the experiment(green squares) and calculated in the CRTA (black solid line) and in theERTA (red dashed dotted line).While we capture the low temperature trend, we attribute the disagree-ment between experiment and theory at higher temperatures to some intrinsicscattering mechanism which has not been taken into account in our calcu-lations. We assume that four-phonon anharmonic processes or higher orderthree phonons are important because of the deviations of κ ( T ) from the 1/Tbehavior. Thus, the lattice thermal conductivity of HgTe should be subjectto further investigation.In Fig. 5 b, we analyze the accumulated lattice thermal conductivity κ L (Λ) as a function of phonon mean free path Λ (see supplementary materialfor details) at three different temperatures T = 100 K (blue curve), 300 K(black curve), 500 K (red curve). As one can see, the thermal conductivity ismainly cumulated below 1 micron and the mean free paths become shorterwhen temperature is increased. The accumulated function can be used topredict the effective size L of a nanostrucure necessary to reduce the thermalconductivity and, thus, increase the thermoelectric performance of a material.Indeed, phonons with mean free paths larger than L are scattered by sampleboundaries and their contribution to the thermal conductivity is suppressed.The horizontal dotted line denotes a 50% reduction of thermal conductivity.It is found to be L = 136 nm at T = 100 K, L = 42 nm at T = 300 K and L = 25 nm at T = 500 K. Finally, we evaluate the overall thermoelectric power factor
P F T = S σT based on our experimental and theoretical data in Fig. 6 (a). As one can see,11gTe possess a high power factor which grows with temperature linearly from0.8 W m − K − at T = 310 K to 0.9 W m − K − at T = 475 K. Our theoreticalvalues obtained in the ERTA slightly overestimate the experimental powerfactor but show the same temperature dependence reaching 1.0 W m − K − at T = 500 K. The CRTA underestimates the magnitude of the Seebeckcoefficient and results in a low power factor around 0.2 W m − K − .Thefigure of merit also increases linearly since thermal conductivity is relativelyunchanged in this temperature range.While ZT values reported here are small. We would like to emphasize thatthis is not an optimized sample. One can increase the ZT values by manydifferent techniques. For example, further increase in the electrical conduc-tivity (a factor of two) is expected after annealing in Hg gas with relativelyunchanged Seebeck coefficient and thermal conductivity values [12, 11]. Asmentioned earlier we avoid this process due to both toxicity of Hg gas and thefact that optimization of the thermoelectric properties of HgTe is not the sub-ject of this work. One can also implement nanostructuring to further reducethe thermal conductivity, a technique that is routinely performed to opti-mize the thermoelectric figure of merit. Similarly, slight doping (tunning ofthe chemical potential) and slight alloying could be used to further optimizethe performance of semimetallic HgTe. For example, alloying with cadmiumcould lower the thermal conductivity and still preserves the semimetallic na-ture of the HgTe for small molar fractions of cadmium ( x < . Our theoretical calculations are based on density functional theory (DFT).For the electrical transport calculations, we use Vienna Ab-initio SimulationPackage (VASP) [38, 39] combined with Boltzmann Transport Theory as im-plemented in Boltztrap code [29]. We use pseudopotentials based on the pro-jector augmented wave method [40] from VASP library with the generalizedgradient approximation by Perdew, Burke and Ernzehof (GGA-PBE) [24]and with a hybrid Heyd-Scuseria-Ernzehof (HSE06) [25] exchange-correlationfunctionals. A plane wave kinetic cut-off of E cut = 350 eV and Γ-centeredk-point mesh of 8x8x8 were found to be enough to converge the total energyup to 5 meV [20, 41]. We use a tetrahedron method for the Brillouin zoneintegration and the experimental lattice parameter a = 6.460 ˚ A in both cal-culations. In our calculations, we take into account the spin-orbit couplingwhich is important to accurately reproduce the electronic band structure of12gTe. To ensure the convergence of transport integrals in Boltztrap, we use20 times denser interpolated grid than we do in our ab initio calculations.For the thermal transport calculations, we use Quantum Espresso [42]package combined with D3Q code to calculate third-order anharmonic forceconstants using ”2n+1” theorem [43] and to solve the Boltzmann Transportequation for phonons variationally [2]. We use the norm-conserving pseu-dopotentials with the exchange-correlation part treated in the local densityapproximation by Perdew and Zunger (LDA-PZ) [45]. We use a cut-off en-ergy of E cut = 1360 eV (100 Ry), 8x8x8 k-points mesh to sample the Brillouinzone with Methfessel-Paxton smearing of σ = 0.068 eV (0.005 Ry). The equi-librium lattice parameter is found to be 6 .
431 ˚ A . Spin-orbit coupling is notincluded in the calculations since it has a weak effect on vibrational prop-erties of HgTe as has been pointed out by M. Cardona et al. [30]. Phononfrequencies and group velocities are calculated using the density functionalperturbation theory (DFPT) [46] on a 8x8x8 q -point grid centered at Γ. Thethird-order anharmonic constants are calculated on a 4x4x4 q -point grid inthe Brillouin zone that amounts to 42 irreducible triplets. Both phonon har-monic and anharmonic constants are then interpolated on a dense 24x24x24 q -point grid necessary to converge the thermal conductivity calculations.The detailed information about the charged carrier scattering rates ob-tained from the electrical conductivity fit and about the isotopic disorderscattering rates used in the lattice thermal conductivity calculation is re-ported in the supplementary material. A 99.99% purity of HgTe ingot was purchased from 1717 CheMall Corpo-ration for HgTe sample preparation and the density of the ingot was 7.82 ± obtained by Archimedes (cid:48) principle. We crashed the ingot andmilled it with a mortar and pestle for about 10 minutes to obtain fine pow-ders. Later, they were consolidated into a 0.5 (cid:48)(cid:48) -diameter compact disk byusing spark plasma sintering (SPS) method at 783K, 50MPa for 15 minutes.After SPS process, the density of the HgTe disk is increased to 7.98 ± , which is quite close to the theoretical fully-dense value of the HgTedensity 8.13 g/cm . For annealing preparation, the compact HgTe samplewas sealed in an evacuated capsule, and it was situated in the middle of afurnace at 523 K for 5 days.For the ingot samples, we used the machine to cut it into a rectangular-bar-shaped sample with the dimension of 2x4x8 mm . For the SPS samples,due to their fragality, we hand-polished the disk into a rectangular-shaped barof the same size as the ingot one instead of cutting them in the machine. The13igure 7: The comparison of the x-ray diffraction (XRD) results between (a)ingot, (b) SPS, and (c) SPS-annealing samples. The excess Te peaks in theingot samples are highlighted by red color.14our-probe electrical conductivity and Seebeck coefficient measurements wereperformed in the helium atmosphere with a ZEM-3 equipment from UlvacTech., Inc. The Hall coefficient measurements were conducted in QuantumDesign Versa-Lab. The thermal diffusivity experiments were carried out witha LFA 467 HyperFlash equipment from NETZSCH. The measured thermaldiffusivity was then multiplied by the theoretical heat capacity [1] C p ( T ) = C V ( T ) + 1 . · − T where C V ( T ) was obtained from the Debye model.X-ray diffraction data are shown in fig. 7. Figure. 7(a) shows the originalingot contains single-phase HgTe with excess tellurium. After the SPS pro-cess, in additional to the original HgTe phase, a new crystal phase, Hg x Te z ,emerges (see panel (b)), and the excess Te peaks, i.e., Te (101) and Te (102),disappear. Panel (c) shows that the SPS-annealing sample is a single-phaseHgTe crystal without the excess of Te. Note that the phase of Hg x Te z van-ishes after annealing.Additional information about the Hall coefficient measurements is pro-vided with the supplementary material. In conclusion, we have investigated, both experimentally and theoretically,the electrical and thermal transport properties of HgTe at high-temperaturesbetween 300 and 500 K. We have found that HgTe is a good thermoelec-tric material in a low pressure semi-metallic zinc-blende phase as it has ahigh Seebeck coefficient and a low thermal conductivity. To explain theexperimental data for the Seebeck coefficient, we accurately reproduce theelectron-hole effective mass ratio by performing ab initio calculations withthe hybrid-HSE06 exchange-correlation functional and take into account thephenomenological scattering rates extracted from a fit to electrical conduc-tivity. Finally, we perform the lattice thermal conductivity calculations by exactly solving the Boltzmann Transport Equation (BTE). We include three-phonon anharmonic scattering and isotopic disorder scattering processes. Weattribute the disagreement between experiment and theory to some intrinsicscattering mechanism which has not been taken into account in our calcula-tions. Our work demonstrates that large thermoelectric power factors couldbe achieved even in the absence of an energy bandgap.15 cknowledgements
M.M., H.L and M.Z acknowledges support from Air Force Young InvestigatorAward (Grant FA9550-14-1-0316). M.Z. acknowledges support from NationalSpace Grant College and Fellowship Program (SPACE Grant) Training Grant2015-2018, grant number NNX15A120H. We acknowledge the SEAS for thecomputational time on Rivanna HPC cluster.
Conflict of interest
There are no conflicts to declare.
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In this supplementary material, we provide the supporting information aboutthe Hall coefficient measurements, the electrical conductivity fitting, the lat-tice thermal conductivity calculations and measurements.
In Fig. S1, we show the experimental data for the Hall effect resistance R xy asa function of an applied external magnetic field B in the temperature range200 K ≤ T ≤
400 K. The Hall coefficient R H can be extracted from the slopeof the R xy ( B ) curve as R H ( T ) = R xy ( T ) B l (S.1)where l is the sample thickness. The net carrier concentration and electronmobilities can be found from the Hall coefficient data as n ( T ) = 1 eR H ( T ) (S.2) µ e ( T ) = R H ( T ) Rl (S.3)and are shown in Fig. S2 and Fig. S3 respectively. Here e is an elementarycharge, R is the resistance of the sample without magnetic field. The electrical conductivity can be found using the following expression σ ( T, µ ) = 1 V cell (cid:90) σ ( E ) (cid:34) − ∂f µ ( T, E ) ∂E (cid:35) dE (S.4)where V cell is a unit cell volume, E is energy, µ is the chemical potential, f µ is the Fermi-Dirac distribution function and σ ( E ) is the differential conduc-tivity σ ( E ) = e τ ( E ) g ( E ) v g ( E ) (S.5)21here g ( E ) is the density of states, v g ( E ) is a group velocity and τ ( E ) =1 / Γ( E ) is the total relaxation time that is inversely proportional to the to-tal scattering rate Γ( E ). In our calculations we use g ( E ) and v g obtainedwith the HSE06 exchange-correlation functional. In the constant relaxationtime approximation (CRTA), one assumes that τ ( E ) is constant and energyindependent.In this work, we consider the energy dependent scattering rates. Weconsider 3 types of carrier scattering including acoustic deformation poten-tial scattering Γ ac ( E ), ionized impurity scattering Γ imp ( E ) and polar opticalscattering Γ pop ( E ) [3]. As follows from the Matthiessen’s rule, the total scat-tering rate Γ( E ) is a sum of all three contributions. Overall, we have 4 fittingparameters A , A , A and phonon energy ¯ hω .Acoustic deformation potential scattering rate isΓ ac ( E ) = A g ( E ) (S.6)Ionized impurity scattering rate isΓ imp ( E ) = A n C T E − / (S.7)where n C is the net carrier concentration obtained from the Hall coefficientmeasurements (see Fig. S2) n C ( T ) = n exp( − T d /T ) (S.8)where T d = 450 . n = 16 . · cm − .Polar optical scattering rate isΓ pop ( E ) = A ¯ hωv g n BE (cid:115) hωE − n BE ¯ hωE sinh − (cid:115) E ¯ hω ++( n BE + 1) (cid:115) − ¯ hωE + ( n BE + 1) ¯ hωE sinh − (cid:115) E ¯ hω − (S.9)where n BE is the Bose-Einstein distribution function, v g is the group velocity.The first two terms represent the polar-optical absorption while the last twoterms describe the emission.The energy dependent scattering rates obtained from the fitting to ex-perimental electrical conductivity for the samples before and after annealingare shown in Fig. S4. 22 .3 Thermal conductivity measurements. To obtain the thermal conductivity κ , we use the following formula κ ( T ) = ρc p ( T ) D ( T ) (S.10)where ρ is the measured density of a sample, D ( T ) is the measured thermaldiffusivity and c p ( T ) is the theoretical specific heat capacity. The measuredthermal diffusivity for the original ingot sample and the sample after theSPS is shown in Fig. S5. The ingot sample has an excess of Te atoms, anda lower density, ρ =7.82 ± , comparing to ρ = 7.98 ± after the SPS. The thermal diffusivity is higher for the ingot samples (blackcircles) than in the SPS samples (blue triangles), but does not change afterthe annealing of the SPS sample. The theoretical heat capacity is c p = c v + V α β T T (S.11)where α is the coefficient of thermal expansion, β T is the isothermal diffusiv-ity, c V can be found from the Debye model c v = 9 N A k B (cid:18) TT D (cid:19) (cid:90) x D dx x e x ( e x − (S.12)where T D = 140 K is the Debye temperature. For the second term in Eq. S.11,we use the experimental values from Ref. [1] and get the following expressionfor the specific heat c p ( T ) = c V ( T ) + 1 . · − T (S.13)The obtained heat capacity c p ( T ) linearly changes from 0.158 JK − g − at T= 250 K to 0.171 JK − g − at T = 700 K In this work, we perform ab initio calculations solving the Boltzmann Trans-port Equation (BTE). The algorithm we use is described in details in Ref. [2].Apart from the intrinsic three-phonon scattering processes, we include theisotopic disorder scattering processes with rates given by P iso q j = π N q ω q j ω q (cid:48) j (cid:48) δ (¯ hω q j − ¯ hω q (cid:48) j (cid:48) ) (cid:20) n q j n q (cid:48) j (cid:48) + n q j + n q (cid:48) j (cid:48) (cid:21) (cid:88) s g s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) α z sα q j z sα q (cid:48) j (cid:48) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (S.14)23here q - phonon wave vector, j - phonon branch index, ω q j - frequencyof phonon ( q , j ), n q j - Bose-Einstein distribution function, α - Cartesiancoordinate, s - atom type, z sα q j - phonon eigenmode, g s - isotopic fluctuationparameter g s = (cid:80) i c i M i − ( (cid:80) i c i M i ) ( (cid:80) i c i M i ) (S.15)We use the natural isotopic composition of Hg and Te as summarized inTable 4. The resulting isotopic fluctuation parameters are g s = 6 . · − forHg and g s = 28 . · − for Te. M Hg , amu % M T e , amu %195.966 0.15 119.904 0.09197.967 9.97 121.903 2.55198.968 16.87 122.904 0.89199.968 23.10 123.903 4.74200.970 13.18 124.904 7.07201.971 29.86 125.903 18.84203.973 6.87 127.904 31.74129.906 34.08Table 4: List of natural isotopes of Hg and Te.
The lattice thermal conductivity can be written as κ L = 1 k B T V cell N q (cid:88) ν n ν (1 + n ν ) ω ν c ν F ν (S.16)where V cell is the unit cell volume, ν = { q , j } , c ν is the group velocity, F ν isthe linear deviation of the out-of-equilibrium phonon distribution n outν fromits equilibrium value n ν n outν = n ν − F ν · ∇ T ∂n ν ∂T (S.17)It can be found from the solution of the Boltzmann Trasport Equation. Inthe relaxation time approximation (RTA) F RT Aν = Λ
RT Aν = τ ν c ν . In the exactsolution it plays a role of a vectorial mean free-path dispacement. To find ascalar mean-free path Λ exactν , one needs to project it onto velocity directionΛ exactν = F ν · c ν | c ν | (S.18)24he lattice thermal conductivity can be rewritten as a function of one singlevariable Λ as κ L = (cid:88) ν κ L (Λ ν ) = (cid:90) d Λ κ accL (Λ) (S.19)where the accumulated thermal conductivity is defined as κ accL (Λ) = (cid:88) ν κ L (Λ) δ (Λ − Λ ν ) (S.20)In Fig. S6 we show the difference in the accumulated thermal conductivitiesin the two approaches discussed above. As one can see, the mean free pathdistribution in the exact approach is shifted toward the longer values. References [S1] V. M. Glazov and L. M. Pavlova.
Rus. J. Phys. Chem. , 70:441, 1996.[S2] G. Fugallo, M. Lazzeri, L. Paulatto, and F. Mauri.
Ab initio variationalapproach for evaluating lattice thermal conductivity.
Phys. Rev. B ,88:045430, 2013.[S3] M. Lundstrom.
Fundamentals of carrier transport . Cambridge Univer-sity Press, New York, 2000. 25igure S1: The Hall effect resistance R xy measured as a function of magneticfield B at different temperatures. 26igure S2: The net carrier concentration obtained from the Hall coefficientmeasurements as a function of temperature for the samples before (blue di-amonds) and after (red diamonds) annealing. The samples are found to be nn