Thermoelectric properties of semimetals
Maxime Markov, Emad Rezaei, Safoura Nayeb Sadeghi, Keivan Esfarjani, Mona Zebarjadi
TThermoelectric properties of semimetals
Maxime Markov , Emad Rezaei , Safoura Nayeb Sadeghi , Keivan Esfarjani , , and Mona Zebarjadi , ∗ Department of Electrical and Computer Engineering,University of Virginia, Charlottesville, Virginia 22904, USA Department of Mechanical and Aerospace Engineering,University of Virginia, Charlottesville, Virginia 22904, USA Department of Physics, University of Virginia, Charlottesville, Virginia 22904, USA and Department of Materials Science and Engineering,University of Virginia, Charlottesville, Virginia 22904, USA
Heavily doped semiconductors are by far the most studied class of materials for thermoelectricapplications in the past several decades. They have Seebeck coefficient values which are 2-3 orders ofmagnitude higher than metals, making them attractive for thermoelectric applications. Semimetalsgenerally demonstrate smaller Seebeck coefficient values due to lack of an energy bandgap. However,when there is a large asymmetry in their electron and hole effective masses, semimetals could havelarge Seebeck coefficient values. In this work, we study the band structure of a class of 18 semimet-als using first principles calculations and calculate their Seebeck coefficient using the linearizedBoltzmann equation within the constant relaxation time approximation. We conclude, despite theabsence of the band gap, that some semimetals are good thermoelectrics with Seebeck coefficientvalues reaching up to 200 µ V/K. We analyze the metrics often used to describe thermoelectricproperties of materials, and show that the effective mass ratio is a key parameter resulting in highSeebeck coefficient values in semimetals.
PACS numbers: 63.20.kg
I. INTRODUCTION.
Material thermoelectric figure of merit, zT is defined as zT = σS Tκ , wherein σ is the electrical conductivity, S isthe Seebeck coefficient, κ is the thermal conductivity andT is the temperature. A thermoelectric module is madeout of n and p legs electrically in series and thermallyin parallel. The efficiency of a thermoelectric modulein power generation mode and in refrigeration cycle ,and the thermal switching ratio in switching mode areincreasing functions of the n and p materials’ figure ofmerit. Hence, finding thermoelectric materials with largefigure of merit is of high interest.Metals were the first class of materials studied for ther-moelectric applications. While they have large electricalconductivity, they usually have small Seebeck coefficientvalues and large thermal conductivity values, makingthem non ideal candidates for traditional thermoelectricapplications. Semiconductors usually own Seebeck coef-ficient values that are orders of magnitude larger thanmetals. The large Seebeck coefficient is the result ofthe presence of the bandgap which breaks the symme-try between electrons and holes. There are two majorcompeting factors in optimization of the figure of meritin semiconductors. First, when the Fermi level is insidethe bandgap, the Seebeck coefficient is large. As theFermi level moves into the valence or conduction bands,the difference between the density of states (DOS) of hotelectrons (above the Fermi level) and cold electrons (be-low the Fermi level) becomes small, and so does the See-beck coefficient. On the contrary, the electrical conduc-tivity increases since there are more available electronicstates. As a result, one needs to adjust the position of the Fermi level to optimize the thermoelectric power fac-tor, P = σS . Second, as the Fermi level moves insidethe band, similar to the electrical conductivity, the elec-tronic part of the thermal conductivity also increases. Itis therefore difficult to design a material with very largefigure of merit although no theoretical upper limit hasbeen found for zT .Semimetals are a class of materials with properties inbetween semiconductors and metals. They usually havevery small overlap of bands and therefore while they donot have an energy gap, their intrinsic carrier densitycan vary in a large range, between 10 − cm − , de-pending on the band overlap and the size of the carrierpockets. For example, the intrinsic concentrations at liq-uid helium temperature 4.2 K are about 5 . · cm − for HgTe , 3 . · cm − for HgSe , 2 . · cm − for Bi , 3 . · cm − for Sb , and 2 . · cm − forAs . These values are much smaller than in metals, whichare typically around 10 cm − , and are comparable withand in some cases smaller than in heavily-doped semicon-ductors used for thermoelectric applications, 10 − cm − . However, semimetals generally have much largercarrier mobility values compared to metals and heavilydoped semiconductors. For example, electron mobilitiesat 4.2 K are 6 . · cm V − s − in HgTe , 1 . · cm V − s − in HgSe and 11 · cm V − s − in Bi . Asa result, the electrical conductivity of semimetals is com-parable to those of heavily-doped semiconductors. Notethat the carrier mobility is much lower in heavily-dopedsemiconductors due to ionized impurity doping and inmetals due to electron-electron and electron-phonon in-teractions. The thermal conductivity values in semimet-als could be also small, especially if they consist of heavyelements. For example, the thermal conductivity at room a r X i v : . [ c ond - m a t . m t r l - s c i ] M a y FIG. 1. Schematic illustration of different types of semimet-als: a) direct semi-metal with parabolic bands b) indirectsemimetal with parabolic bands c) Dirac or Weyl semimetalwith linear dispersion. The Fermi level is denoted by thedashed line. In each case, the band structure could be sym-metric as shown by black curves or asymmetric as shown byred curves. Semi-metals with asymmetric bands are the focusof this work temperature is about 1.7
W m − K − in HgSe , 1.9-2.9 W m − K − in HgTe , 6.0 W m − K − in the trigonaldirection in pure bismuth and could be as low as 1.6 W m − K − in Bi-Sb alloys .Semimetallic and zero gap materials show many in-teresting properties. They have attracted interests astopologically non trivial materials . Many of them havestrong spin-orbit coupling and comprise of heavy ele-ments. As a result, they possess a low thermal conduc-tivity. Inversion of bands happens in many of the zero-gap alloys such as Bi x Sb − x and Hg x Cd − x Te, leadingto interesting transport properties. While many of thesematerials have been studied in other fields, there has notbeen a systematic study of their thermoelectric proper-ties due to their lack of band gap.If one is to avoid doping and only choose to work withintrinsic materials, semimetals would be the best poten-tial candidate for having a large thermoelectric powerfactor . In our recent publication, we studied thermo-electric properties of HgTe as a well-known semimetal.One of the interesting features observed was that the See-beck coefficient in intrinsic HgTe was not sensitive to thenumber of defects and impurities inside the sample. Thismeans one can change the carrier concentration by or-ders of magnitude while keeping the Seebeck coefficientconstant. This is because these large changes in carrierconcentration did not result in a considerable shift in thechemical potential, so that Seebeck was not changed. Ifthis is a general trend in semimetals, then the interplaybetween electrical conductivity and Seebeck coefficientis much weaker in semimetals compared to semiconduc-tors and therefore it is easier to increase the figure ofmerit in semimetallic samples especially at low tempera-tures where the dominant source of scattering is impurityscattering.In this work, the thermoelectric response of severalsemimetallic elements, i.e. their Seebeck coefficient val-ues, are studied using first principles calculations withproper corrections for the energy levels. We restrict our-selves to room temperature where the diffusive part ofthe Seebeck coefficient is known to be dominant. The Irr. k-points cut-off energy BTE solver
PBE, HSE/ mBJ PBE, HSE/ mBJ
HgTe 1661, 752 350, 350 BoltzTraP1HgSe 1661, 752 420, 420 BoltzTraP1 α − HgS 1661, 752 420, 420 BoltzTraP1TlP 413, 321 420, 420 BoltzTraP1TlAs 413, 321 420, 420 BoltzTraP1Li AgSb 413, 104 420, 420 BoltzTraP1Na AgSb 413, 104 420, 420 BoltzTraP1Rb AgSb 413, 104 420, 420 BoltzTraP1 α − Sn 294, 294 600, 560 BoltzTraP2Sb 868, 868 600, 500 BoltzTraP2Bi 868, 868 600, 560 BoltzTraP2TaAs 512, 512 600, 500 BoltzTraP2TaP 512, 512 600, 500 BoltzTraP2NbP 512, 512 600, 500 BoltzTraP2Mg Pb 294, 294 600, 560 BoltzTraP2PtSb rationale to focus only on the Seebeck coefficient is thefollowing: As was discussed, the carrier mobility is ex-pected to be large in semimetals. If semimetals consistof heavy elements, then their thermal conductivity is alsoexpected to be low. The biggest concern with semimet-als is therefore the Seebeck coefficient and thus the pro-cess of searching for good semimetals for thermoelectricapplications should start with the scan for the Seebeckcoefficient. From a computational point of view, amongthe three transport properties determining the figure ofmerit, the Seebeck coefficient is the least sensitive oneto the scattering rates. Therefore, the only propertythat could be reliably calculated under constant relax-ation time approximation and still be of value, is theSeebeck coefficient. We should acknowledge that evenSeebeck coefficient values can be modified when energydependent relaxation times are introduced . In-cluding energy dependent relaxation times would be avery difficult task when scanning many materials. Here,as the first step towards finding promising semimetalliccandidates, we limit ourselves to the constant relaxationtime approximation. II. COMPUTATIONAL METHODS
We preselect 18 materials which were mentioned in theliterature as semimetals. Information about their crystalstructure, space group number and number of atoms perunit cell are summarized in Table II. Vienna Ab-initioSimulation Package (VASP) is used to perform first-principles calculations. Pseudopotentials based on the (a) (b) (c)
FIG. 2. The band structures (left panels) and density of states (right panels) of HgTe, Mg Pb and Bi representing the threetypes of semi-metals described in Figure 1. Black curves - PBE, red curves - HSE06. projector augmented wave method from VASP librarywith the generalized gradient approximation by Perdew,Burke and Ernzehof (GGA-PBE) as well as hybridHeyd-Scuseria-Ernzehof (HSE06) exchange-correlation(XC) functionals are employed to calculate band struc-ture and density of states. The details about the cut-off energy and number of k points may be found inTable I. We used relaxed PBE lattice parameters forall materials but for HgTe, HgSe and HgS. For thelatter three materials the experimental lattice parame-ters were considered. The summary of the latticeparameters can be found in the Supplementary Mate-rial. Spin-orbit coupling is included in all calculations(except for TiS and TiSe ) and transport calculationsare performed within the Constant Relaxation Time Ap-proximation (CRTA) as implemented in BoltzTraP and BoltzWann codes to obtain the diffusive part ofthe Seebeck coefficient (see Table I). The interpolating kpoint grid was taken to be at least 30 times denser thanthe initial DFT grid. III. RESULTS
We divide all semimetals into three separate groups.These are shown schematically in Fig. 1. The first grouppossesses a distinct feature in the band structure: thelowest conduction band has a deep minimum at the cen-ter of the Brillouin zone (BZ) where it overlaps with thehighest valence bands. When the two bands are sym-metric (shown by black curves), the intrinsic chemicalpotential is expected to be at the midpoint between theto band extrema, and the intrinsic Seebeck coefficient isexpected to be very small. However, it is possible to havea band structure similar to the red curve in Fig 1a, wherethe low degeneracy of the conduction band in vicinity ofthe Γ point results in a small density-of-states (DOS), themagnitude of which is essentially defined by the electron’seffective mass ( i.e. the curvature of the band). On theother hand, valence bands have heavier effective massesand higher degeneracy with contributions from elsewhere in the BZ. As a result, the DOS is asymmetric aboutthe chemical potential. This is known to be beneficialfor the material’s electronic properties in general and, inparticular, leads to a high Seebeck coefficient. A typicalexample of such material is HgTe which has been studiedin our recent publication both theoretically and experi-mentally . Other (predominantly cubic) materials areHgSe and HgS, TlAs and TlP , α -Sn as well as inverseHeusler materials (Li AgSb, Na AgSb, Rb AgSb) .The band structures of these materials along with theirDOS are shown in the Supplementary Material. Here, asthe representative of this class of materials, we show theband structure and the DOS of HgTe as shown in Fig. 2a.Black curves are used to show PBE results for the bandstructure and the DOS of all materials reported in thiswork. Red curves show the HSE results.Among the materials studied within the first class, thehybrid functional calculations (red curves) reveal thatHgS and Li AgSb are in fact semiconductors with bandgaps of 0.33 eV and 0.67 eV respectively. In almost allcases, we note that the effective masses of the conduc-tion band significantly decreases in HSE comparing withPBE calculations. A possible explanation for this effecthas been given in Ref. where the small effective masseswere attributed to the strong level repulsion betweenthe s-like conduction band and p-like valence band atΓ. This repulsion is inversely proportional to the squareof the difference between these two levels which reducesfrom − .
93 eV for PBE to -0.27 eV for HSE06 in case ofHgTe
The second group (Fig. 3b) includes other semimetal-lic materials without any distinct feature in their bandstructure but possessing a low density of states at theFermi level. The top of the valence band and the bottomof the conduction band are at different k points as shownschematically in Fig. 1b. Electron and hole pockets co-exist. This class includes, for instance, Mg Pb, cubicpyrite structures (PtSb and PtBi ) , TiS , TiSe ,TaP, NbP and α − Zn Sb . We note TiS gap opens upwhen HSE functional is used and therefore this materialis a semiconductor with the bandgap of 0.4 eV. Despite FIG. 3. The band structure, the density of states and the Seebeck coefficient of Na AgSb are shown in panels a, b, and crespectively. Similarly those of Rb AgSb are shown in the lower panels of d,e, and f. The black curves in the band structureand DOS plots are PBE results and the red curves are HSE results. The Seebeck coefficients are only reported for HSE results.Red curves are Seebeck coefficient values vs. doping concentration and black curves are vs. chemical potential. Left side ofeach Seebeck plot refers to p-type doping and right side to n-type doing. its large Seebeck coefficient which is expected for a mate-rial with a bandgap, the intrinsic carrier concentration islow and therefore it does not fall in the class of materialswe are interested in this work. On the other hand, TiSe remains semimetallic under HSE, with overlapping con-duction (L and M points) and valence bands (Γ point).Its Seebeck coefficient is however found to be small dueto the small asymmetry in the bands. In another workwhere properties of the monolayer TiSe were studied ,we found that the bandgap can be opened under tensilestrain, leading to a metal-insulator transition and corre-sponding non-linear effects. As for Mg Pb, the overlap ofbands is relatively large and the ratio of the DOS effec-tive mass of the conduction band to that of the valenceband is close to one. (see Fig. 3b) Therefore this mate-rial exhibits a small intrinsic Seebeck coefficient value ofabout -10 µV /K .The third class of materials includes relativistic (Diracand Weyl) semimetals with linear bands close to theFermi level. These are schematically shown in Fig. 1c.The examples include Bi, Sb, Na Bi and TaAs-family andinverse Heusler materials Na AgSb and Rb AgSb. Ther-moelectric properties of the latter family as well as someother topologically non-trivial semimetals have been re-cently investigated in Ref. . The band structure of Bias the representative of this class of materials is shown inFig. 3c. Most samples in this group demonstrate rathersmall Seebeck coefficient values. This is expected becausethere is an inherent symmetry in the band structure atthe Dirac point. The symmetry can breakdown only if additional bands exist close to the Dirac point as shownschematically in Fig. 1c.Two examples are Na AgSb and Rb AgSb which show,within HSE, a Dirac dispersion at Γ point, in addition
FIG. 4. Computational Seebeck coefficient values calculatedin this work using HSE band structures versus experimentalSeebeck coefficient values from literature. When available,single crystals in [001] (trigonal) and [100] (binary) direc-tions are compared. Most experimental samples are intrinsicincluding Bi , Sb ,Sn , HgTe , PtSb , and TaAs .Other samples are n-doped including: α -HgS (10 cm − ),TiS (8 · cm − ), and HgSe ( 10 cm − ). FIG. 5. (a) Absolute value of intrinsic Seebeck coefficient (color bar) as a function of band gap E g (x-axis) and effective massratio γ (y-axis). Negative bandgap refers to overlapping bands. (b) Intrinsic Seebeck coefficient as a function of asymmetryparameter Ξ defined in eq. 3. Black circles - PBE data, red squares - HSE06 data, black dotted line is a linear dependenceassumed from Eq. 2 to a parabolic valence band. The band structure, theDOS and the corresponding Seebeck coefficient of thesetwo materials are shown in Fig.3. The Seebeck is onlyreported for HSE calculations. For PBE results, wherebands were parabolic instead of linear, we refer the readerto the supplementary materials. Both materials showlarge intrinsic Seebeck coefficient values and large in-trinsic carrier concentrations. The Seebeck coefficient ofNa AgSb is larger than 200 µV /K and interestingly it isinsensitive to the changes in the carrier concentration upto ± cm − . The large value can be associated withthe extra parabolic valence band, and the flat Seebeckto the constancy of the slope of the DOS and group ve-locities in this region. Rb AgSb is similar. We see thistrend more or less for all our calculated materials, indi-cating that the coupling between electrical conductivityand the Seebeck coefficient is weaker in the semimetal-lic samples compared to heavily doped semiconductors.We also emphasize that these large Seebeck values areobtained at relatively large carrier concentrations. Notethe carrier concentration reported in the plots, are theHall type carrier concentration, i.e., the difference be-tween free electron, n, and free holes, p, densities. Theactual carrier concentration that determines the electri-cal conductivity is larger and is the sum of n and p.Bismuth and antimony are well-known materials andhave been the subject of studies for many years. The ex-perimentally measured values for Bi and Sb can providea good comparison to the theoretical calculations. In ad-dition to Bi and Sb, in Fig. 4 our computational resultsare compared to reported experimental values of α -Sn, α -HgS, HgSe, HgTe, TiS , TaAs and PtSb .As shown inFig. 4, and considering there are no fitting parameters,the agreement between theory and experiment is satis-factory.Several of the samples that we have studied in this work have Seebeck coefficient values larger than100 µV /K . We expect the Seebeck coefficient to be largeonly when there is a band gap or when there is asym-metry between electron and hole effective masses. Todemonstrate this, we extract an effective mass from thedensity of states estimated as the slope of density ofstates of the electrons (conduction band) and the holes(valence band) with respect to the square root of energy.The absolute value of intrinsic Seebeck coefficient of dif-ferent materials with respect to the effective mass ratio(effective mass of the holes to that of the electrons) andbandgap energy is plotted in Fig. 5a. We see an increas-ing trend in the Seebeck coefficient values with respectto the mass ratio for semimetals where the band gap iszero or close to zero. There are a few exceptions, namelyNa AgSb and Rb AgSb. In these materials, due to thepresence of the Dirac point, the parabolic assumption en-abling the extraction of an effective mass is not accurate.There are two parameters to which Seebeck is sensi-tive: one is the bandgap and the position of the chemicalpotential, and the other is the mass ratio. For this rea-son, we would like to define a single parameter whichwe call ”asymmetry parameter” to characterize Seebeck,and show their correlation.We start by using the equation for bipolar Seebeck co-efficient under constant relaxation time approximation S = − k B q (cid:20) σ e − σ h σ e + σ h ( βE g + 5) + β ( ε c + ε v − µ ) (cid:21) (1)where k B is the Boltzmann constant, q is the elemen-tary charge, β = ( k B T ) − , σ e and σ h are electron andhole conductivities. The bandgap E g is defined similarto semiconductors as a difference between the bottom ofthe conduction band ε c and the top of the valence band ε v with the the chemical potential µ somewhere in be-tween. The bandgap E g is positive for semiconductorsand negative for semimetals where there is band overlap.Its values for different materials studied in this work arelisted in Table II.Assuming non-degenerate statistics, constant relax-ation time approximation and intrinsic conditions ( n = p ), one can simplify Eq. 1 to S = − k B q (cid:20) γ − γ + 1 ( βE g + 5) −
32 ln( γ ) (cid:21) (2)where γ is defined as effective mass ratio of holes to elec-trons (listed in Table II). Note that the condition n = p automatically places the chemical potential at the rightplace, and we do not need to specify it. Since we are inter-ested in the absolute value of the Seebeck coefficient, wedefine the following parameter as the indicator of asym-metry: Ξ = (cid:12)(cid:12)(cid:12)(cid:12) γ − γ + 1 (cid:12)(cid:12)(cid:12)(cid:12) ( βE g + 5) + (cid:12)(cid:12)(cid:12)(cid:12) ln ( γ ) (cid:12)(cid:12)(cid:12)(cid:12) (3)In Figure 5b, we show the dependence of the calculatedintrinsic Seebeck coefficient to the asymmetry parameterΞ. According to Eq. 2, a linear dependence to asymmetryparameter is expected. The obtained results show a noisylinear trend. Therefore, we conclude that this parame-ter can be used to estimate the Seebeck coefficient fromthe band structure information. The fact that there is alarge level of noise is attributed to several factors. First,many of the studied materials do not obey the parabolicband dispersion and therefore it is not possible to definea proper effective mass for them. Second, the model usesnon-degenerate statistics that is not accurate when thereis an overlap between the bands. Despite these, thereis clear increasing trend of the Seebeck coefficient withrespect to the defined asymmetric parameter. ACKNOWLEDGEMENTS
This work is support by National Science Foundation,grant number 1653268. Calculations were performed us-ing Rivanna cluster of UVA.
IV. CONCLUSIONS.
First principle DFT calculations were employed to scanamong semimetallic materials potential candidates withhigh Seebeck coefficient. Our calculated Seebeck coef-ficient values are found to be in agreement with ex-perimental results when the latter were available. Ageneral increase in the intrinsic Seebeck coefficient asa function of materials’ bandgap and the ratio of holemass to electron mass is observed. It is shown thatmaterials with no bandgap but with large mass ratio,can have large Seebeck coefficient values comparable tothose of heavily doped semiconductors. We observedthat the Seebeck coefficient values of semimetals werein many cases insensitive to carrier concentration in awide range around the intrinsic density (see supplemen-tary materials for details). Therefore the coupling be-tween the Seebeck coefficient and the electrical conduc-tivity is weaker in semimetals compared to semiconduc-tors, allowing for simpler optimization of thermoelectricproperties. Many of the studied semimetals includingNa AgSb, Rb AgSb,TIP, TaP, and HgSe showed Seebeckcoefficient values close to or larger than 100 µV /K .Due to relatively high intrinsic carrier concentrationand, simultaneously, high mobility, in the absence ofdoping, these semimetals may show high thermoelectricpower factor values. The ones with heavy atoms are goodcandidates for high zT materials. ∗ [email protected] H. J. Goldsmid, in Semiconductors and Semimetals (ed.Tritt, T.) (Academic Press, 2001). A. F. Ioffe,
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