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Insights into the design of thermoelectric Mg3Sb2 and its analogs by combining theory and experiment

Insights into the design of thermoelectric Mg3Sb2 and its analogs by combining theory and experiment www.nature.com/npjcompumats REVIEW ARTICLE OPEN Insights into the design of thermoelectric Mg Sb and its 3 2 analogs by combining theory and experiment 1 1 1 Jiawei Zhang , Lirong Song and Bo Brummerstedt Iversen Over the past two decades, we have witnessed a strong interest in developing Mg Sb and related CaAl Si -type materials for low- 3 2 2 2 and intermediate-temperature thermoelectric applications. In this review, we discuss how computations coupled with experiments provide insights for understanding chemical bonding, electronic transport, point defects, thermal transport, and transport anisotropy in these materials. Based on the underlying insights, we examine design strategies to guide the further optimization and development of thermoelectric Mg Sb -based materials and their analogs. We begin with a general introduction of the Zintl 3 2 concept for understanding bonding and properties and then reveal the breakdown of this concept in AMg X with a nearly 2 2 isotropic three-dimensional chemical bonding network. For electronic transport, we start from a simple yet powerful atomic orbital scheme of tuning orbital degeneracy for optimizing p-type electrical properties, then discuss the complex Fermi surface aided by high valley degeneracy, carrier pocket anisotropy, and light conductivity effective mass responsible for the exceptional n-type transport properties, and finally address the defect-controlled carrier density in relation to the electronegativity and bonding character. Regarding thermal transport, we discuss the insight into the origin of the intrinsically low lattice thermal conductivity in Mg Sb . Furthermore, the anisotropies in electronic and thermal transport properties are discussed in relation to crystal orbitals and 3 2 chemical bonding. Finally, some specific challenges and perspectives on how to make further developments are presented. npj Computational Materials (2019) 5:76 ; https://doi.org/10.1038/s41524-019-0215-y INTRODUCTION properties. During the past two decades, both theoretical and experimental aspects in CaAl Si -type TE materials have under- The increasing energy consumption worldwide pushes significant 2 2 13–52 gone a rapid development, which results in a significant efforts in developing alternative energy technologies. Thermo- improvement of zT above unity (Fig. 1). Experimentally, strategies electric (TE) technology, capable of converting waste heat into 45–47,49,50 15,16,18,19,21,23,25,29,35,36 such as doping, alloying, and con- electrical energy, shows potential for waste heat-harvesting trolling vacancy concentrations have been used to optimize applications with a great advantage that TE devices are entirely electrical transport properties and to reduce thermal conductivity in solid state with no moving parts, compact, quiet, and 1,2 through point defect scattering. As nearly all these compounds maintenance free. One key bottleneck of this promising are persistently p-type, it is a remarkable breakthrough that the technology is its low conversion efficiency, which is essentially low-cost Te-doped Mg Sb -based compounds were recently limited by low-performing materials. The performance of a TE 3 2 discovered to show exceptional n-type TE properties at low and material can be characterized by the dimensionless figure of merit, 53–55 intermediate temperatures (Fig. 1), comparable or even zT = α σT/(κ + κ ), where α is the Seebeck coefficient, σ is the L e superior to the commercial n-type TE materials, such as Bi Te electrical conductivity, T is the absolute temperature, κ is the 2 3 56–73 and PbTe. Hence, significant research efforts on this promis- lattice thermal conductivity, and the electronic thermal conduc- tivity κ is related to σ through the Wiedemann–Franz law κ = LσT ing material system are currently ongoing. Theoretically, electronic e e 20,74–77 11,12,78,79 55,60,61,80 structures, chemical bonding, defects, and (L denotes the Lorenz number). The interdependent correlation 81–84 3–10 phonon-related properties have been extensively studied to among the TE transport parameters invokes numerous efforts worldwide on decoupling these parameters so as to improve zT.In understand the transport properties. The interplay between theory general, achieving a superior zT requires a combined effort of and experiment has given rise to many important guidelines based on the underlying physical and chemical insights for improving the power factor α σ and reducing the lattice thermal conductivity. optimizing CaAl Si -type TEs. 2 2 AB X compounds with the CaAl Si -type structure, usually In this review, we summarize some of the successful guiding 2 2 2 2 described as Zintl phases, are well recognized due to their principles for understanding and rationalizing the electrical and promising TE properties. In these compounds, A is an alkaline thermal transport in Mg Sb and its derivatives with the CaAl Si - 3 2 2 2 type structure. We show how computational efforts integrated earth or a divalent rare earth element, B is a transition metal or a main group element, and X usually comes from group 14, 15, or with experiments lead to additional physical and chemical insights 11,12 occasionally group 13. The rich variety of compositions for the profound understanding of chemical bonding, electronic covered by AB X (CaAl Si -type) compounds enables consider- transport, defect chemistry, phonon transport, and transport 2 2 2 2 able chemical tunability of the electronic and thermal transport anisotropy. We start from general introductions of crystal structure Center for Materials Crystallography, Department of Chemistry and iNANO, Aarhus University, 8000 Aarhus, Denmark Correspondence: Bo Brummerstedt Iversen (bo@chem.au.dk) Received: 19 February 2019 Accepted: 26 June 2019 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences J. Zhang et al. ab n-type Mg Sb 1.4 3 2 2.0 p-type Mg Sb 3 2 1.2 Other p-type AB X 2 2 1.6 1.0 1.2 0.8 0.6 0.8 0.4 0.4 0.2 0.0 0.0 2000 2004 2008 2012 2016 2020 2000 2004 2008 2012 2016 2020 Year Year Fig. 1 a, b Timelines of a the maximum zT and b the average zT values of the reported AB X thermoelectric materials with the CaAl Si -type 2 2 2 2 structure. The average zT is calculated by the direct integration method within the entire measurement temperature range. The experimental 14–71 data are taken from refs. and the Zintl concept that has been widely applied to understand 1933, much earlier than those of ternary derivatives. From the 88,89 the structure, bonding, and electronic transport in CaAl Si -type reported phase diagram, Mg Sb and Mg Bi show a phase 2 2 3 2 3 2 compounds. After that, we reveal the nearly isotropic three- transition from the low-temperature α phase to the high- dimensional (3D) chemical bonding network in AMg X , where the 2 2 temperature β phase at ~900 and ~700 °C, respectively. The Zintl formalism is no longer applicable. For p-type electronic high-temperature β phase is superionic with liquid-like magne- transport, we discuss how electronic transport can be optimized sium ions and it is assumed to have a cubic structure, but the by minimizing the crystal orbital splitting energy via forming solid exact crystal structure remains unclear. solutions and tuning biaxial strains, whereas for n-type transport we reveal the multi-valley conduction bands and complex Fermi surface as the electronic origin of the extraordinary n-type TE ZINTL FORMALISM 91,92 properties. Then we review the defect chemistry of the intrinsic p- The Zintl concept plays a vital role in understanding structure, type behavior and the surprising n-type behavior under different chemical bonding, and properties of a wide range of solid-state thermodynamic states, followed by highlighting the defect- materials. Within this simple but powerful concept, the mostly controlled carrier transport and its correlation with the electro- ionic cations are considered merely as electron donors, donating negativity and bonding character. For thermal transport, we their electrons to the covalently bonded anionic substructures. review the studies on exploring the origin of the intrinsically low The covalent bonding in the anionic substructures ensures a lattice thermal conductivity in Mg Sb from first principles 3 2 significant orbital overlap, a light band mass, and thereby a high calculations. Moreover, we discuss the anisotropy in electrical carrier mobility, whereas the ionic cations are suitable for doping and thermal transport properties with respect to crystal orbitals with the aim to tune the carrier density without affecting the and chemical bonding. Finally, we conclude with some current 93,94 covalent anionic networks. Such bonding features, as well as challenges and prospects for future development. ideal band gaps, make charge-balanced Zintl compounds very promising for TE applications. 11,12 CRYSTAL STRUCTURE As described by Hoffmann et al., AB X compounds with 2 2 the CaAl Si -type structure can easily be understood as Zintl AB X compounds with the trigonal CaAl Si -type structure 2 2 2 2 2 2 δ− δ+ phases, where the covalently bonded [B X ] layers receive (space group: P3m1) can be viewed as the A cationic layers 2 2 δ+ δ− electrons from the ionic A layers. Based on the aforementioned intercalated between the tightly bound [B X ] anionic layers 2 2 Zintl concept, considerable efforts have been made to explore (Fig. 2a). The three types of atoms A, B, and X occupy three AB X -type TE compounds and optimize them via doping, different crystallographic sites 1a (0, 0, 0), 2d (1/3, 2/3, z ), and 2d 2 2 substituting, or creating vacancies on the cationic A sites. One (1/3, 2/3, z ) with point symmetries of 3m,3m, and 3m, δ− notable example can be seen in Ca Yb Zn Sb alloys reported respectively. In the [B X ] networks, B has a normal tetrahedral 1−x x 2 2 2 2 by Gascoin et al., which show the fine tunability of the carrier coordination, while X shows a unique distorted umbrella-like local concentration through doping or substituting Ca with Yb. The environment. As a result, the vertical B-X bond (d ) along the c alloying of Ca and Yb indeed has a negligible effect on the axis is usually longer than the three symmetry-equivalent tilted B- 2− δ+ covalent anionic [Zn Sb ] networks, so that the good carrier X bonds (d ). Taking into account the cationic A layers, the 2 2 mobility and ideal energy gap are maintained. In addition, this interlayer A-X bonds (d ) hold together the alternating cationic and anionic layers. Unlike the B atoms being tetrahedrally alloying creates disorder in the cationic layers, which results in a reduction in thermal conductivity and an enhancement in zT. coordinated by the X atoms, each A atom occupies the octahedral Ever since this early work, the Zintl formalism has been broadly site with six equal adjacent bonds. Each X atom, surrounded by applied in developing and optimizing TE AZn Sb , ACd Sb , and four B atoms and three A atoms, has seven adjacent bonds 2 2 2 2 20,23,27,32 their alloys. In these compounds, however, it was found including one vertical B-X bond, three tilted B-X bonds, and three interlayer A-X bonds (Fig. 2c–e). that the hole mobility may be greatly affected by changing the Mg X (X = As, Sb, and Bi) crystallizing in the inverse α-La O cation A from alkaline earth elements to rare earth elements. 3 2 2 3 structure is a special case of the CaAl Si (AB X ) structure with A Different synthesis methods were also found to result in a great 2 2 2 2 86 95 and B being, respectively, Mg1 and Mg2. The crystal structures of change in carrier mobility of AZn Sb . These results are likely 2 2 Mg Sb and Mg Bi were reported by Zintl and Husemann in induced by the change in carrier scattering time τ because the 3 2 3 2 npj Computational Materials (2019) 76 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences 1234567890():,; zT max zT average J. Zhang et al. Fig. 2 a Crystal structure of layered Zintl phases A(Zn,Cd) Sb with the CaAl Si -type structure, showing a clear distinction between the 2 2 2 2 δ+ δ− ionic A layer and the covalent [B X ] layer. b A more accurate structural view of AMg X with the CaAl Si -type structure, which shows 2 2 2 2 2 2 comparable interlayer and intralayer bonds being mostly ionic with partial covalent nature (high polarity). c–e Coordination environments of the three nonequivalent atoms c A, d B, and e XinAB X compounds with the CaAl Si -type structure. f Static deformation electron density 2 2 2 2 −3 map on (110) plane of Mg Sb with a nearly isotropic three-dimensional bonding network (ρ =ρ  1). The contour interval is 0.006 e Å . 3 2 intra inter Positive (negative) contours are plotted with red solid (blue dotted) lines. g Noncovalent interaction analysis with reduced density gradient as a function of sign(λ )ρ for the interlayer and intralayer interactions in Mg Sb . sign(λ ) is the sign of the second eigenvalue of the electron 2 3 2 2 101 79 density Hessian matrix. Figure is reproduced with permission from ref. , CC-BY-4.0 effective mass of the valence band is shown to be unaffected by rational explanation for the unusual umbrella-like local coordina- varying the cation A. tion environment of the anionic X atom and the bond length δ− difference in the [B X ] networks of the CaAl Si (AB X ) 2 2 2 2 2 2 structure through making and breaking chemical bonds. In NEARLY ISOTROPIC 3D BONDING NETWORK IN AMG X : 2 2 addition, they compared the local and extended constructions BREAKDOWN OF THE ZINTL FORMALISM of the bonding in transition metal CaAl Si -type compounds with 2 2 Despite being very successful in rationalizing the electronic those in ThCr Si -type compounds and explained why com- 2 2 transport, the Zintl formalism and electron counting are not pounds with the B-site elements being occupied by transition 0 5 10 universal and can only provide a qualitative insight. It does not metals with the d ,d , and d configurations only form the give us any indications of the site preference, bond length CaAl Si -type structure. Burdett and Miller also conducted 2 2 difference in anionic networks, or the reason why intrinsic p-type extended Hückel calculations to reveal the formalism of the Al Si 2 2 Mg Sb shows very poor carrier mobility. More accurate theore- fragment in the CaAl Si -type structure and extended the 3 2 2 2 tical calculations based on quantum mechanics or density fragment formalism to several other main group compounds. functional theory (DFT) are required to understand the structure The above studies were conducted only based on the chemical δ− and chemical bonding. bonds in the [B X ] networks, neglecting the effect of the 2 2 δ+ 2+ Using the crystal orbital overlap population approach based on cationic A layer. The contribution of the cationic Ca layer to the extended Hückel calculations, Hoffmann et al. gave a the chemical bonding and electronic structure of CaAl Si was 2 2 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences npj Computational Materials (2019) 76 J. Zhang et al. ionic nature in both cationic and anionic layers. This indicates that Table 1. Bader atomic charge Q, the degree of ionicity, and ρ =ρ intra inter any description of AMg X , especially Mg Sb , as layered Zintl 2 2 3 2 of several AB X compounds with the CaAl Si -type structure 2 2 2 2 phases in all previous literature is incorrect. Chemical bonds in AB X (CaAl Si -type) compounds can be 2 2 2 2 Compounds ρ =ρ Q (e) Degree of intra inter described as polar bonds based on the topological properties of ionicity (%) the bond critical points (BCPs) using the classification scheme by A B X A layer B X layer 2 2 Gatti. The covalency of the intralayer bonds in the B X 2 2 networks, however, may vary a lot among different compounds. Mg Sb 1.47 1.51 1.47 −2.23 75.5 73.9 3 2 The intralayer B-X bonds are much more covalent and stronger Mg Bi 1.38 1.43 1.40 −2.11 71.5 70.2 3 2 than the mostly ionic interlayer A-X bonds in AZn Sb and 2 2 CaMg Sb 1.39 1.39 1.48 −2.17 69.5 73.2 2 2 ACd Sb , whereas the intralayer bonds in AMg X are only slightly 2 2 2 2 stronger than the largely ionic interlayer bonds (see Table 1 and CaMg Bi 1.37 1.37 1.42 −2.10 68.5 70.5 2 2 Fig. 2a, b, f). This is further confirmed by the similar reduced CaZn Sb 2.34 1.37 0.31 −0.99 68.5 24.3 2 2 density gradient distributions of the three nonequivalent bonds CaCd Sb 2.20 1.38 0.28 −0.97 69.0 23.2 2 2 101 from the non-covalent interaction analysis in Mg Sb (Fig. 2g). 3 2 The data of Bader atomic charge and ρ =ρ are adapted from ref. . The chemical bonding analysis may be conducive to under- intra inter The degree of ionicity is estimated from the charge transfer index, which standing transport properties. The mostly ionic feature with partial is calculated by the average of the atomic charge divided by the nominal covalency (high polarity) of chemical bonds explains the 36,102 oxidation charge of the atoms in unit cell. ρ denotes the electron inter intrinsically low carrier mobility in p-type Mg Sb and 3 2 density value at bond critical point (BCP) of the interlayer A-X bond, while CaMg Sb . However, the high carrier mobility in AMg Bi (A = 2 2 2 2 ρ represents the average electron density value at BCPs of the two 26,41 intra Mg, Ca, Eu, and Yb) cannot be understood from chemical intralayer B-X bonds. The data of CaCd Sb are calculated in this work 2 2 bonding, though it could be attributed to the bipolar effect or the using the previous computational methods change in carrier scattering time induced by defects. The electron density ratio ρ =ρ is devised as a simplified intra inter parameter measuring the degree of anisotropy of the chemical 95 79 investigated by Alemany et al. using DFT calculations. They bonding network in the Mg Sb -related structures, where ρ 3 2 intra found that the cationic layers show an important contribution to and ρ represent the electron density values at BCPs of the inter the covalent bonding of CaAl Si , although the cationic layers do intralayer and interlayer bonds, respectively. Unlike AZn Sb and 2 2 2 2 not play a dominant role in the electronic structure and the origin ACd Sb with clear anisotropic chemical bonding networks ( 2 2 of the conductivity behavior. In addition to these significant ρ =ρ >2), the AMg X compounds show nearly isotropic 3D 2 2 intra inter 11,12,78,96 studies on rationalizing structural formation and electro- chemical bonding networks with ρ =ρ typically being intra inter nic structure using the molecular orbital approaches, Grin et al. smaller than 1.5 and often close to unity (Table 1). The nearly studied the chemical bonding in YbCd Sb using the electron isotropic 3D bonding networks can be used to understand the 2 2 localizability indicator and showed that four-center bonding exists nearly isotropic structural and thermal properties especially lattice in this compound and the whole bonding picture can be thermal conductivities in AMg X , which will be discussed later. 2 2 2+ 2+ 3− described by the closed-shell configuration [Yb ][Cd ] [Sb ] . 2 2 Based on the analysis of the electron density difference, Toberer CRYSTAL FIELD ORBITAL SPLITTING, ORBITAL DEGENERACY, et al. discussed the chemical bonding in AZn Sb (A = Sr, Ca, 2 2 AND P-TYPE ELECTRONIC TRANSPORT and Yb) and revealed the largely covalent nature of the Zn-Sb bonds as well as the charge accumulation from Sb to A. The A-Sb In principle, the intrinsic electronic transport of a material is bond was found to be very asymmetric and slightly larger in determined almost exclusively by the degeneracy and curvature YbZn Sb than in SrZn Sb or CaZn Sb , which might be of the electronic bands at the band edges. The Seebeck coefficient 2 2 2 2 2 2 attributed to the less electron transfer from the cation A to the is typically determined by the density of states effective mass m * 2/3 1,103 anionic Zn Sb slabs induced by the larger electronegativity of Yb = N m *, where N represents the valley degeneracy of the 2 2 v s v in comparison to Sr or Ca. Using DFT calculations, Li et al. electronic bands and m * denotes the effective mass of a single 1,103 revealed a much lower shear strength in Mg Sb compared to valley. As proposed by Goldsmid, the optimum electrical 3 2 CaMg Sb and CaZn Sb , which was explained by the weaker transport performance can be expressed as being proportional to 2 2 2 2 97 3/2 interlayer Mg-Sb bond in Mg Sb . Despite significant efforts on the weighted mobility μ(m */m ) , where μ and m represent the 3 2 d e e understanding chemical bonding, most of these efforts are carrier mobility and the mass of an electron, respectively. qualitative and there is clearly a lack of a quantitative chemical Considering the case with predominant acoustic phonon scatter- bonding description of CaAl Si -type compounds based on ing or alloy scattering, the weighted mobility can be simplified 2 2 topological analysis of the electron density. and written as N /m *(N /m * for an isotropic band), where m *is v c v s c Recent theoretical calculations extend our knowledge of the conductivity effective mass. It is clear that excellent electrical chemical bonding in CaAl Si -type compounds. Quantitative performance requires a high valley degeneracy as well as a light 2 2 analysis of full DFT electron density by Zhang et al. elucidated conductivity effective mass. Increasing valley degeneracy (includ- that the Zintl formalism is perfectly applicable in AZn Sb and ing orbital degeneracy) has long been recognized as an efficient 2 2 ACd Sb but not in AMg X (including Mg X ). The analysis was way to improve electrical transport performance through enhan- 2 2 2 2 3 2 conducted using the quantum theory of atoms in molecules cing the Seebeck coefficient without explicitly decreasing the (QTAIM) developed by Bader. QTAIM is based on the analysis of carrier mobility when the intervalley scattering is insignifi- 4,6,9,103,104 critical points of the electron density ρ, which are the points cant. Below we will introduce how electrical transport satisfying ∇ρ = 0. The readers are referred to ref. for the details can be rationalized through enhancing orbital degeneracy via of the method. The idea is clearly proved by the result of Bader minimizing the orbital splitting energy in p-type AB X 2 2 atomic charges (see Table 1). In CaZn Sb and CaCd Sb , the compounds with the CaAl Si -type structure. 2 2 2 2 2 2 cationic and anionic layers show a remarkable difference in the In CaAl Si -type compounds, the valence band maximum (VBM) 2 2 degree of ionicity estimated from atomic charges, i.e., the located at the Г point shows p orbital characteristics of the anions cationic layers are largely ionic while the anionic layers are mostly (Fig. 3a, b). Unlike the triply degenerate p orbitals at the Γ point in covalent. In contrast, nearly complete charge transfers are cubic structures protected by the point symmetry, the p orbital, observed for all atoms in AMg X , which elucidates the mostly due to the crystal field effect, is usually well separated from the p 2 2 x, npj Computational Materials (2019) 76 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences J. Zhang et al. 2.0 ab Mg1 CaAl Si -type structure 2 2 1.5 Mg2 CC C 1.0 Sb =E(Γ(p ))-E(Γ(p )) x,y z 0.5 0.0 p d Γ( ) x y −0.5 Γ( ) −1.0 Δ Δ ≈0 >0 Δ <0 −1.5 Γ Γ MK A L H A ef optimal p 20 -3 p=5×10 cm 25 20 -3 p=10 cm 19 -3 p=5×10 cm 20 19 -3 15 p=10 cm -0.4 -0.2 0.0 0.2 -0.4 -0.2 0.0 0.2 Δ (eV) (eV) Fig. 3 a Schematic diagram of orbital engineering to realize three-fold degenerate p orbitals in CaAl Si -type compounds. Nondegenerate 2 2 band Γ(p ) and doubly degenerate band Γ(p ) are mainly composed of p and p orbitals from the anions, respectively. Δ denotes the crystal z x,y z x,y field splitting energy between p and p orbitals at the Γ point. b Band structure of Mg Sb by TB-mBJ potential without spin–orbit coupling. x,y z 3 2 −3 The p orbitals of Sb are projected on the band structure. c, d Partial charge densities of the valence bands c Γ(p ) (isovalue: 0.06 e Å ) and d Γ −3 (p ) (isovalue: 0.13 e Å ) at the Γ point in Mg Sb . e The theoretical power factor at 300 K versus Δ of Mg Sb at various hole concentrations x,y 3 2 3 2 p. The red curve shows the best values corresponding to the optimum carrier concentrations. Data points for unoptimized carrier 14,15,17–27,39–41 concentrations fill up the pink area right below the red curve. f The experimental power factors of reported CaAl Si -type 2 2 18,21 compounds at 600 K as a function of Δ. Red points represent the alloys with zT larger than unity at high temperatures. a, b, e, f are adapted with permission from ref. , CC-BY-4.0 orbitals in the trigonal CaAl Si -type structure. Accordingly, the The zero-Δ selection rule may be combined with the band gap y 2 2 three-fold degenerate valence band at the Γ point splits into a criterion E < 1.5 eV with the aim to search for promising TE nondegenerate band Γ(p ) and a doubly degenerate band Γ(p ), candidates from a rich variety of CaAl Si (AB X )-type compounds. z x,y 2 2 2 2 where Γ(p ) and Γ(p ) are typically a heavy and light hole band, The band gaps of these AB X materials show a clear decreasing x,y z 2 2 9,105 respectively (Fig. 3a–d). The crystal field splitting energy Δ trend when the electronegativity difference between B and X between them is defined as Δ = E(Γ(p )) − E(Γ(p )), where the decreases, which may be rationalized by a decrease of the energy x,y z orbital degeneracy is effectively increased when the splitting between the atomic orbitals of B and X and an increase in band 106,107 energy approaches zero. The basic idea of the orbital engineering widths using the molecular orbital scheme. However, it approach proposed by Zhang et al. is to minimize the splitting should be noted that SOC also plays an important role in energy Δ with the aim to maximize orbital degeneracy and decreasing energy gaps of Bi-based compounds. Excellent TE thereby optimize electrical transport performance. This concept is properties are well confirmed in Sb-based and Bi-based com- 17 29 15 well confirmed by the Boltzmann transport calculations as well as pounds such as EuZn Sb , EuMg Bi , and YbZn Sb with 2 2 2 2 2 2 the experimental observations, which show peak values in power small Δ values close to zero and ideal band gaps. In particular, factor as Δ ≈ 0 (see Fig. 3e, f). As the thermal conductivities of most Shuai et al. reported strongly enhanced power factors in EuMg Bi 2 2 promising CaAl Si -type TE compounds are comparable especially with a nearly zero Δ value compared with those of CaMg Bi and 2 2 2 2 at high temperatures, the strongly enhanced power factor results YbMg Bi with Δ values largely deviating from zero, which 2 2 in peak zT values at Δ ≈ 0. The strong correlation between provides a solid confirmation of the zero-Δ rule. Though simple experimental power factors and Δ also indicates the minimal and powerful, the zero-Δ rule only has resulted in the discovery of intervalley scattering in different AB X compounds, which a few potential TE candidates. 2 2 ensures the success of this approach. Spin–orbit coupling (SOC) Materials design and optimization requires effective approaches is found to lift the degeneracy of the p band, but will not change for manipulating crystal field orbital splitting energy. The x,y the main idea of the approach. Accordingly, a simple selection hybridizations or overlap integrals of p orbitals control their rule, i.e., maintaining the valence band splitting energy around splitting energy at the VBM. In general, tuning structural zero (zero-Δ rule), is proposed to optimize the electrical transport parameters (for instance, the interlayer distance, bond lengths performance. and angles, and c/a) by the crystal deformation is able to Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences npj Computational Materials (2019) 76 2 14 -1 -2 -1 Energy (eV) ασ/τ (10 μWcm K s ) 2 -1 -2 Energy (eV) α σ (μWcm K ) Substrate J. Zhang et al. a b Thin Film 0.8 0.2 SrZn Sb 2 2 3.0 EuCd Sb 2 2 CaZn Sb 2 2 CaCd Sb 0.1 2 2 SrCd Sb 2 2 YbCd Sb BaMg Sb 2 2 2 2 0.4 YbZn Sb 2 2 2.4 ε 0.0 SrMg Sb 2 2 YbCd Zn Sb EuZn Sb 1.5 0.5 2 2 2 EuMg Sb -0.1 0.0 2 2 1.8 EuZn Cd Sb CaMg Sb 1.75 0.25 2 2 2 YbMg Sb -0.2 2 2 -0.4 1.2 -0.3 CaZn Sb 2 2 Mg Sb 3 2 Mg Sb 3 2 -0.4 -0.8 0.6 4.4 4.5 4.6 4.7 4.8 -4 -2 0 2 4 a (Å) ε (%) Fig. 4 Solid solution map and biaxial strain engineering for materials design. a Calculated Δ versus the lattice constant a in CaAl Si -type 2 2 14,15,17–27,39–41 compounds with E < 1.5 eV. Reported thermal conductivities at 500 K are shown in color bar. The stars correspond to solid solutions YbCd Zn Sb and EuZn Cd Sb with nearly zero Δ values. b Δ as a function of biaxial strain ε in two representative CaAl Si - 1.5 0.5 2 1.75 0.25 2 2 2 type compounds Mg Sb and CaZn Sb . Biaxial strain ε is defined as (a − a )/a × 100%, where a and a are the in-plane lattice parameters 3 2 2 2 0 0 0 with unstrained and strained states, respectively. Figure is adapted from ref. , CC-BY-4.0 effectively manipulate the orbital interactions and thereby the accelerate the screening and design of new TE materials from orbital splitting energy. In principle, crystal deformation can be layered or noncubic compounds. induced by both external and internal forces. External forces include physical pressure and strain effect, while internal forces MULTI-VALLEY CONDUCTION BANDS, COMPLEX FERMI involve chemical doping or forming solid solutions. Two SURFACE, AND EXCEPTIONAL N-TYPE ELECTRONIC efficient approaches, solid solution map and biaxial strain TRANSPORT engineering, can be used to realize the manipulation of Δ. The valley degeneracy is defined as the number of different carrier Using the solid solution map with calculated Δ values versus pockets (for the same type of carriers) existing at a given energy lattice parameter or band gap E , one can conveniently choose level. In general, the valley degeneracy of an individual electronic two or more compounds with positive and negative Δ values to band can be defined as N = N N , where N is the form a solid solution with the desirable Δ value of zero, which v v,sym v,band v,sym number of symmetry equivalent positions in the Brillouin zone for leads to excellent electronic transport performance (see Fig. 4a). a given k point at which the electronic band occurs and N Since alloying is also conducive to reducing the thermal v,band represents the number of electronic bands degenerate at the conductivity owing to the point defect scattering, strongly same k point and energy level. For the high-symmetry Γ point enhanced zT values can generally be achieved using this located at the center of the Brillouin zone, N = 1. The orbital approach. This powerful strategy is confirmed in several alloys v,sym 18 21 engineering approach discussed in the previous section is such including YbCd Zn Sb , EuZn Cd Sb , Eu Yb - 1.6 0.4 2 1.8 0.2 2 0.2 0.2 29 35 36 a case. Although the band degeneracy N of 3 at the Γ point Ca Mg Bi , YbCd Zn Sb , and Ca Na MgZnSb with v,band 0.6 2 2 1.5 0.5 2 0.99 0.01 2 (at the VBM) can be achieved via tuning the splitting energy of p superior TE power factors and zTs. In particular, Wood et al. orbitals, N of 1 at the Γ point limits the overall valley revealed the valence band crossing in CaMg Sb -CaZn Sb alloys, v,sym 2 2 2 2 which combines with reducing thermal conductivities by the alloy degeneracy to be N ≤ 3. It is thereby clear that a high valley degeneracy requires not only a high band degeneracy N but scattering to result in a peak zT of 0.87 at 850 K. As shown by v,band also a high N . In order to obtain a high N value, we need a Wang et al., optimizing electrical transport performance induced v,sym v,sym high-symmetry Brillouin zone as well as a band extremum that by minimizing the p orbital splitting energy as well as minimizing 1,2 the lattice thermal conductivity by the point defect scattering occurs at a low-symmetry k point. Below we will introduce the leads to an optimal zT of ~1.3 at 700 K in the YbCd Zn Sb multi-valley band behavior as the electronic origin of the 2−x x 2 extraordinary electrical transport performance in n-type Mg Sb - alloys. In addition to the approach of forming solid solutions, 3 2 based TE materials, which exhibit a unique near-edge conduction biaxial strain engineering can be applied to continuously tune Δ band minimum (CBM) at a low-symmetry k point with a high N so as to optimize electrical transport performance in thin-film TE v, materials (see Fig. 4b). A general principle for optimizing TE of 6. sym performance via biaxial strain is that for compounds with negative The electronic structure of Mg Sb has been studied using 3 2 (positive) Δ value the compressive (tensile) biaxial strain is more different theoretical methods in various reports. Traditional 108 109 functionals such as GGA or LDA are known to underestimate effective. 54,74 band gaps while recent calculations with the TB-mBJ Since most semiconductors show p-orbital characteristics at the potential gives more accurate band gaps close to the valence band edges, the orbital engineering approach can easily be extended to other structures with the same orbital splitting experimental values. However, band structures in all earlier features. One notable extension is the earlier reported pseudo- reports are typically calculated along the high-symmetry k 9 74–77 cubic approach in chalcopyrite structures, where the crystal field paths, which often overlook the actual CBM at the low- splitting energy can be directly linked to the structural deforma- symmetry k point. As revealed in the recent calculations including tion parameter η = c/2a. Moreover, the approach can be extended SOC by Zhang et al., the accurate CBM in Mg Sb is located at 3 2 * * to layered metal dichalcogenides and lithium intercalated metal the CB point (0, 0.417, 0.333), which is along the M –L line inside dichalcogenides. Therefore, the orbital engineering approach the first Brillouin zone (see Fig. 5a). Mg Sb shows an indirect 3 2 with physical and chemical insights based on the underlying band gap of ~0.6 eV with the VBM at the Γ point and the CBM at atomic orbitals enriches band engineering and may substantially the CB point. Moreover, there is a secondary band minimum npj Computational Materials (2019) 76 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences Δ (eV) -1 -1 κ (W m K ) Δ (eV) J. Zhang et al. With SOC A L L a 2.5 b CB 2.0 M* K M 1.5 1.0 n-type Total CB c 0.5 1 Mg p-type Sb 0.0 n-type -0.5 -1.0 -1.5 p-type -2.0 M K A L H A ** Γ Γ ML024 DOS Fig. 5 a Band structure and partial density of states of Mg Sb by TB-mBJ potential with SOC. b High-symmetry k points and k paths in the 3 2 Brillouin zone. c Multiple carrier pockets shown in Fermi surface of n-type Mg Sb for an energy level at 0.1 eV above the conduction band 3 2 minimum CB . d One highly anisotropic carrier pocket at the Г point shown in Fermi surface of p-type Mg Sb corresponding to an energy 1 3 2 level at 0.1 eV below the valence band maximum. a, b are adapted with permission from ref. , American Chemical Society. c, d are adapted with permission from ref. , CC-BY-4.0 located just above the CB extremum at the K point. The small surface complexity, it is crucial to take into account the energy difference ΔE of 0.078 eV between the two band contribution of the anisotropic feature of the single carrier pocket KCB minima CB and K suggests that they might even be treated as at the CB , which shows a moderate anisotropy parameter K = 1 1 * * * 1/2 effectively converged at elevated temperatures because of the m /(m m ) ≈ 2.3. || ⊥,1 ⊥,2 * * thermal broadening of Fermi function. Inspired by the biaxial The Fermi surface complexity factor N K can be calculated by 105 * * 3/2 112,114 * strain engineering in p-type compounds, Li et al. revealed that (m /m ) , where m is evaluated using the carrier density d c c n-type electrical performance of Mg Sb can also be optimized by n and theoretical electrical conductivity σ/τ from a BoltzTraP 3 2 111 * 2 tuning ΔE towards zero via the biaxial strain. calculation by m = ne /(σ/τ) under the constant carrier’s scatter- KCB c 116 * The iso-energy Fermi surface provides an intuitive shortcut to ing time approximation. m is estimated by fitting the Seebeck understand the multi-valley conduction bands in Mg Sb (see coefficient from Boltzmann transport calculations applied to the 3 2 Fig. 5b–d). In contrast to only one highly anisotropic hole pocket full DFT band structure using the single band model. The * * of the VBM at the Γ point, the Fermi surface of n-type Mg Sb , theoretical calculations reveal a peak N K of ~19 in n-type 3 2 v corresponding to an energy level 0.1 eV above the conduction Mg Sb , much higher than that of p-type Mg Sb (see Fig. 6a). 3 2 3 2 band minimum CB , shows 6 isolated full electron pockets inside This is consistent with both the theoretical and experimental the Brillouin zone and 6 one-third pockets at the K point (see results (see Fig. 6b, c), displaying significantly enhanced power Fig. 5c). As a result, the valley degeneracies for the conduction factors for n-type doping in comparison with those for p-type band minima CB and K are, respectively, 6 and 2, which may be doping in Mg Sb -based materials. The high Fermi surface 1 3 2 added up to 8 when they are effectively converged. Such a high complexity, as well as the superior n-type electrical transport valley degeneracy is comparable to many state-of-the-art TE performance, can be attributed to a combined effect of the large 2,103 * materials such as (Bi,Sb) Te and PbTe. m induced by the high valley degeneracy, the light conductivity 2 3 d The complex Fermi surface is conducive to good TE perfor- effective mass, and the carrier pocket anisotropy (see Fig. 6a). The mance. As proposed by Gibbs et al., the complexity of Fermi n-type transport of Mg Sb shows a primary contribution from the 3 2 * * surface may be characterized by the simplified parameter N K , nontrivial band minimum CB as well as a certain contribution v 1 * * where N and K represent the effective valley degeneracy and from the secondary band minimum at the K point. This multiple anisotropy parameter of a single carrier pocket, respectively. In band behavior in n-type Mg Sb is confirmed by the result from 3 2 * * general, the higher the N K , the better the electrical transport the BoltzTraP calculation that Mg SbBi with a much larger ΔE v 3 KCB performance. In addition to the high valley degeneracy, the of ~0.18 eV shows lower Seebeck coefficients in comparison with complexity of Fermi surface in n-type Mg Sb is clearly reflected in those of Mg Sb (see Fig. 6d). 3 2 3 2 the ellipsoidal-like carrier pockets at the CB , indicating a clear Though the multiple conduction band behavior was also found anisotropic feature. For every individual carrier pocket at the CB in ternary AMg X and AZn Sb , the unique conduction band 1 2 2 2 2 point, the effective mass is anisotropic along one longitudinal CB with a six-fold valley degeneracy only exists in binary Mg X 1 3 2 (elongated) and two transverse directions in k space with m = (X = As, Sb, and Bi). As shown by the theoretical calculations of || * * 0.55m , m = 0.21m , and m = 0.28m , although when Sun et al., the unique CBM at the CB point in Mg Sb might e ⊥,1 e ⊥,2 e 1 3 2 averaging over the six equivalent carrier pockets the overall possibly be explained by the bonding states between Mg1 and average effective mass tensor follows the crystal symmetry and Mg2 atoms. However, the very large interatomic distance * * shows a nearly isotropic feature with m = m = (0.21 × 4 + between Mg1 and Mg2 (~3.7 Å) indicates that the orbital kx ky * 113 0.55 × 2)/6 = 0.32m and m = 0.28m . Regarding the Fermi interaction between Mg1 and Mg2, if any, should be very weak. e kz e Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences npj Computational Materials (2019) 76 Energy (eV) J. Zhang et al. ab 20 m * m * DFT, n-type Mg Sb N *K * 20 c d 3 2 2.4 n-type DFT, p-type Mg Sb 3 2 p-type 2.0 1.6 1.2 0.8 0.4 0.0 0 0 0.1 1 10 0.1 1 20 -3 20 -3 n (10 cm ) n (10 cm ) cd n-type 400 DFT, Mg Sb 3 2 DFT, Mg SbBi SPB, 1.05me p-type DFT, Mg3Sb2 -1 0 0 20 40 60 80 100 120 140 160 10 10 2 -1 -1 20 -3 μ (cm V s ) n (10 cm ) Expt., n-type Mg3+xSb1.5Bi0.5 Te-doped, Zhang et al. Te-doped, Kanno et al. Te-doped, Mao et al. Se-doped, Zhang et al. S-doped, Zhang et al.]:1123/1073K-pressed [ Mn,Te-codoped,Chen et al.]:1073K-pressed Te-doped, Tamaki et al. Te-doped, Imasato et al. Te-doped, Shuai et al.] :873/923K-pressed Mn,Te-codoped, Chen et al. Fe/Co/Hf/Ta,Te-codoped, Mao et al. Nb,Te-codoped, Shuai et al. La-doped, Imasato et al.]:873/923K-pressed Expt., p-type doping in Mg Sb 3 2 Fig. 6 a Conductivity effective mass m *, density of states (DOS) effective mass m *, and Fermi surface complexity factor N *K* estimated from c d v 115 113 BoltzTraP as a function of Hall carrier concentration (n ) in p-type and n-type Mg Sb . n is estimated by 1/eR , where R is Hall H 3 2 H H H coefficient. b Theoretical power factor at 300 K from BoltzTraP versus n in Mg Sb . c Experimental power factor versus Hall mobility at room H 3 2 temperature for reported p-type and n-type Mg Sb -based materials. d Seebeck coefficient values (|α|) as a function of n at 300 K. The black 3 2 H and red solid lines correspond to the prediction of p-type and n-type Mg Sb from BoltzTraP, taken from ref. . The orange dash-dotted line 3 2 shows the prediction by a single parabolic band (SPB) model with a DOS effective mass equal to that of the CB band at the CBM of Mg Sb . 1 3 2 The red dashed line represents the theoretical prediction of n-type Mg SbBi taken from ref. .In c, d, experimental data of p-doped Mg Sb 3 3 2 41,45–47,49,50 54–60,63,64,66,67,71 and n-doped Mg Sb Bi are taken from refs. and refs. , respectively 3+x 1.5 0.5 Moreover, the calculations were only conducted for Mg Sb which makes a correction to the simple schematic plot by Imasato 3 2 without a systematic comparison with many other ternary et al. The Mg Bi alloying results in a moderate increase in the 3 2 compounds. The underlying origin of the unique six-fold CB energy separation between conduction band minima K and CB , 1 1 band minimum in Mg X still requires further investigation. In making the contribution of the secondary band minimum K to 3 2 Mg X , the band gaps decrease from Mg As (1.6 eV) to Mg Sb electronic transport insignificant. The conduction band minimum Г 3 2 3 2 3 2 (0.6 eV) to Mg Bi (semimetal). It is found that doping or (N = 1) shows a noticeable downward shift with the Mg Bi 3 2 v 3 2 substituting on the anion sites will not destroy the favorable alloying. However, the band minimum Г is too far from the band conduction band minimum CB since the conduction bands are minimum CB to play a significant role in n-type electronic transport 1 1 54,57 mainly contributed by the electronic states of Mg (Fig. 5a). before the band-gap closure (x ≈ 1.7). As the composition of Mg Bi 3 2 Therefore, n-doped Mg Sb ,Mg As , as well as the solid solutions increases in Mg Sb Bi solid solutions, the band gap is linearly 3 2 3 2 3 2−x x Mg Sb Bi ,Mg Sb As , and Mg As Bi with suitable band reduced from ~0.6 eV (x= 0) to ~0.24 eV (x= 1) to zero (semimetal, 3 2−x x 3 2−x x 3 2−x x gaps are predicted to show promising electrical transport x ≈ 1.7), indicating an increasing detrimental bipolar contribution. In performance if they are properly doped on the anion sites. addition, the narrowing of the band gap, accompanied by the Experimentally, we are especially interested in n-type Mg Sb Bi increasing near-edge band widths, results in the decreasing band 3 2−x x 62,111 with the energy gaps E ≤ ~0.6 eV within ~10k T (T= 300–725 K), effective masses. This is confirmed in several reports as well as g B 119 113 which is generally the suitable value for a good TE material. DFT the theoretical result by BoltzTraP (Fig. 7b), which shows a clear 113 * calculations with the TB-mBJ potential reveal the accurate decreasing trend in m of near-edge conduction bands from conduction band alignments of Mg Sb Bi alloys (see Fig. 7a), Mg Sb to Mg SbBi to Mg Bi . 3 2−x x 3 2 3 3 2 npj Computational Materials (2019) 76 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences 2 -1 -2 m*(m ) ασ (μWcm K ) N *K * -1 |α| (μVK ) 2 14 -1 -2 -1 ασ/τ (10 μWcm K s ) J. Zhang et al. ab SOC noSOC N =1 Mg3Sb2 0.35 1.6 Mg3SbBi Mg3Bi2 0.30 1.2 N =2 v 0.25 0.8 N =6 CB 0.4 0.20 0.0 0.15 -0.4 0.10 0.0 0.5 1.0 1.5 2.0 0.1 1 10 20 -3 n (10 cm ) x in Mg Sb Bi 3 2-x x Fig. 7 a The conduction band alignments in Mg Sb Bi (x = 0, 1, and 2) solid solutions by the TB-mBJ method with SOC. It should be 3 2−x x noted that the three conduction band minima are located at different k points (i.e., CB , K, and Г). The band alignment is calculated by aligning the Mg-1s core levels of different compounds. The valence band maximum of Mg Sb at the Г point is set to 0 eV. b Conductivity 3 2 effective mass versus Hall carrier concentration simulated from BoltzTraP for n-type Mg Sb Bi . SOC and noSOC denote the cases with 3 2−x x and without SOC, respectively. It is clear that SOC has a negligible effect on near-edge conduction bands in Mg Sb Bi because the CBM is 3 2−x x dominated by electronic states of the light element Mg (see Fig. 5a). Figure is adapted from ref. 58 66 Although the Mg Bi alloying lowers the Seebeck coefficient Mao et al., and Chen et al. Following the earlier work of Zhang 3 2 due to the lighter conduction band mass and the weaker et al., the other approach is to increase the pressing temperature contribution from the secondary band minimum K, it is conducive to 1123 or 1073 K, which was also confirmed in the studies by 64 59 to increasing the carrier mobility and weighted mobility as the Kanno et al. and Mao et al. Combining the above two conductivity effective mass m is decreased (see Fig. 7). An c approaches, Chen et al. was able to further improve the room- appropriate amount of the Mg Bi alloying (x ≤ 1) would be ideal 3 2 temperature zT and average zT in n-type porosity-mediated for n-doped Mg Sb Bi to have an enhanced weighted mobility 3 2−x x Mg Mn Sb Bi Te with an SPS pressing temperature 3.225 0.025 1.5 0.49 0.01 for superior TE performance without a noticeable bipolar effect at of 1073 K. Regarding the underlying mechanism, Mao et al. elevated temperatures. This is consistent with the widely reported suggested that the increasing mobility might be due to reduction compositions of n-doped Mg Sb Bi (x = 0, 0.5, and 1) with 3 2−x x of the number of Mg vacancies, whereas Kuo et al. proposed an excellent high-temperature TE performance. An increasing bipolar illuminating two-phase model that explains the experimental effect with increasing Bi content in Mg Sb Bi is expected to 3 2−x x observation by the larger grain sizes reducing the grain boundary shift the peak zT to lower temperatures, but an improved zT at low electrical resistance under a higher pressing temperature. In temperatures may be achieved owing to the lighter conductivity addition, enhanced low-temperature TE performance was shown effective mass. Thus more efforts should be made to investigate n- in n-type Mg SbBi Te and Te-doped Mg Sb Bi by 3.2 0.99 0.01 3 0.6 1.4 type Mg Sb Bi with a wider range of compositions (x = 0–1.7). 62,70 3 2−x x Imasato et al. and Mg Sb Bi Te (y = 1.1–1.5) by Shu 3.02 y 1.99−y 0.01 It will also be interesting to design a functionally graded n-doped 68 et al. with increasing Bi content, which can be attributed to the Mg Sb Bi (x = 0–1.7) system for TE applications. Below we give 3 2−x x lighter conductivity effective mass. Regarding n-type dopants a brief overview of the recent experimental reports on n-doped 57 60 other than the Te element, Se and S on the anion site as well Mg Sb Bi . 67 69 3 2−x x as La and Y on the cation site have also been experimentally 53 53 As shown originally by Pedersen in 2012 (see the link in ref. explored in Mg Sb Bi . With continuous experimental efforts, 55 54 3+x 1.5 0.5 for details) and later reported by Tamaki et al. and Zhang et al. , a low-temperature zT of ~0.9 at 323 K and a peak zT of ~1.85 at an exceptionally high n-type TE performance can be achieved in 723 K have been achieved so far in n-type (Mn, Te)-codoped Mg Sb Bi through Te doping on the anion site with or without 3 1.5 0.5 66,71 54 Mg Sb Bi . In addition to the intensive developments in TE 3 1.5 0.5 excess Mg. Zhang et al. reported n-type Mg Sb Bi Te 3 1.5–0.5x 0.5–0.5x x performance, the thermal stability of n-type Te-doped with varying Te content and no excess Mg synthesized by Mg Sb Bi was investigated by Jørgensen et al. using 3 1.5 0.5 combining arc melting and spark plasma sintering (SPS) at 1123 K synchrotron powder X-ray diffraction and X-ray total scattering, based on the reproduction and improvement of the original work where a clear evolution of bismuth as a secondary phase was by Pedersen. The samples with an optimal zT of 0.56–1.65 at observed in the powdered sample during thermal cycling. 300–725 K reported by Zhang et al. show the intrinsic 54,60,61,63,65,121,122 Several experimental and theoretical efforts temperature-dependent mobility with the dominant acoustic reveal that a large amount of excess Mg is not needed for phonon scattering at low temperatures. Tamaki et al. reported achieving n-type properties as claimed by Tamaki et al. . A tiny n-type Mg Sb Bi Te with a large nominal excess Mg 3.2 1.5 0.49 0.01 amount of excess Mg or even no excess Mg is required to realize prepared using ball milling followed by SPS pressing at 873 K, n-type properties as long as great care is taken to avoid the Mg which exhibits low carrier mobility and TE performance below loss during the synthesis, consistent with the earlier report 500 K due to impurity scattering. Later, two approaches were without excess Mg by Zhang et al. The samples with smaller applied to improve the mobility and power factor at low amounts of excess or no excess Mg even show better temperatures (Fig. 6c) by tuning the carrier scattering mechanism performance and stability than those with large amounts of from the ionized impurity scattering to the acoustic-phonon- 63,122 excess Mg. Various reports with different amounts of excess dominated mixed scattering. One approach is to dope Mg Sb Bi with the transition metals (i.e., Nb, Fe, Co, Hf, Ta, Mg may arise from the difficulty in preparing these materials 3.2 1.5 0.5 and Mn) on the Mg sites and the Te dopant on the anion sites without the Mg loss due to the high reactivity, easy oxidation, and using a pressing temperature of 927 K as reported by Shuai et al., high vapor pressure of Mg. Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences npj Computational Materials (2019) 76 Energy (eV) m * (m ) c e Te doping limit CBM CBM Te doping limit J. Zhang et al. DEFECT-CONTROLLED CARRIER TRANSPORT, electronegativity might be related to the relative energy cost of ELECTRONEGATIVITY, AND BONDING CHARACTER the electron transfer influenced by the bond strength between the cation A and the anion, which in turn affects the vacancy Nearly all CaAl Si -type compounds, except CaAl Si and 2 2 2 2 formation energies. In general, the formation of the vacancy is SrAl Si , have been reported as intrinsically p-type, which is 2 2 easier when the bonding is weak. However, in order to explain the attributed to the intrinsic point defects pinning the chemical deduction, accurate calculations should be conducted to char- potential close to the valence bands. According to the intrinsic 80,125 acterize the bond strength between A and Sb. Though the defect calculations, the cation vacancy at the A site is the underlying mechanism regarding the bond strength is compli- most energetically stable point defect in all AZn Sb (A = Ca, Sr, 2 2 cated, the results provide insight on tuning carrier density by Eu, and Yb) compounds, which leads to the persistent p-type alloying on the A site with elements of varying electronegativities. behavior in these compounds. Pure Mg Sb is intrinsically p-type, 3 2 15,19,23,32 This idea is well confirmed in many significant reports on and many attempts at doping or substituting with a series of 38–50,52,102,126–130 optimizing p-type AB X TEs, where excellent zTs were achieved 2 2 elements result in p-type behaviors as well. This is owing to the carrier density optimization and thermal conductivity because of the low formation energies of the negatively charged reduction via alloying at the A site. 2 2 Mg vacancies (V and V ) (Fig. 8), which prevent the chemical Mg1 Mg2 Defect-controlled carrier transport with the insight from the potential from moving close to the conduction bands for the Sb- 55,61 electronegativity difference was explored in n-type rich condition. The n-type doping in Mg Sb -based materials 3 2 Mg Sb Bi with the chalcogens Q (S, Se, and Te) as electron 51,53– 3 1.5 0.5 is challenging and only recently discovered to be successful. dopants. Experimentally, it is found that both the maximum 55 55,61,121 Several defect calculations were conducted to explain attainable carrier concentration and mobility increase with the n-type behavior in Mg Sb under the Mg-excess condition. 3 2 decreasing electronegativity difference χ  χ between the Q Mg The results vary from different methods applied in defect energy chalcogen dopants Q and Mg (Fig. 9c). Using DFT calculations, calculations. Ohno et al. concluded from calculations using the 60 Zhang et al. revealed that the improving carrier concentration is PBE functional with finite-size and band-gap corrections that caused by the increasing doping limit induced by the lower under the Mg-excess condition the Mg vacancy suppression rather extrinsic defect-formation energy (Fig. 9d), which may be than the Mg interstitial is responsible for n-type properties explained by the stronger bonding between the dopant and the (Fig. 8), while Chong et al. based on more accurate calculations matrix with decreasing χ  χ . Moreover, the enhanced 131 Q Mg with the HSE06 functional revealed that the role of the excess mobility is attributed to the smaller effective mass of conduction Mg for achieving n-type behavior is to compensate the electronic bands originating from the enhanced bond covalency with 1− 134,135 charge of the defect complex (V + Mg ) . It is clear that defect Mg2 i decreasing electronegativity difference, which is supported calculations are usually sensitive to the applied theoretical by the decreasing theoretical density of states. According to the methods; however, the robust relative trend of defect formation above trends, using dopants with a small electronegativity energies is supposed to render insightful guidance for the carrier difference compared to its bonding partner was proposed as a transport. Below we introduce guidelines for the defect-controlled guiding rule for efficient n-type doping in Mg Sb -based 3 2 carrier transport with respect to the electronegativity in p-type compounds. As confirmed in extrinsic defect calculations for n- AZn Sb - and n-type Mg Sb -based materials. 2 2 3 2 type Mg Sb by Gorai et al., this guideline was found to work 3 2 In p-type AZn Sb (A = Ca, Sr, Eu, and Yb), the experimental 2 2 pretty well for the anion site doping but might not work well for carrier density increases with increasing electronegativity of A the cation site doping due to the strong charge compensation and (Fig. 9a). As revealed by Pomrehn et al. using DFT calculations, complex competing phases involved. Moreover, theoretical 136,137 the increasing carrier density can be rationalized by the calculations by Gorai et al. suggested the group-3 elements decreasing trend in the vacancy formation energy at the cation including La, Y, and Sc as effective n-type cation dopants for 67,69 A site with increasing electronegativity (Fig. 9b). The relationship Mg Sb , which was confirmed by the recent experiments of 3 2 between the vacancy defect-formation energy and La-doped and Y-doped Mg Sb Bi . 3+x 1.5 0.5 E range F E range with Te doping with Te doping Mg-excess Sb-excess ab 1.5 1.5 2+ Mg 1+ i Sb Mg2 3+ Mg 1+ Sb 1.0 1.0 Sb Mg1 2- 1+ Mg2 Sb 2- 1+ Te Mg1 0.5 Sb 0.5 2+ Mg 2- 1+ Mg2 2- Te V Sb Mg1 0 0 -0.2 -0.2 0 0.2 0.4 0.6 0 0.2 0.4 0.6 E - E (eV) E - E (eV) F VBM F VBM Fig. 8 a, b Defect formation energies of intrinsic point defects as well as the extrinsic defect Te under a Sb-excess (Δμ ¼ 0) and b Mg- Sb Sb excess (Δμ ¼ 0) conditions. Figure is adapted with permission from ref. , Elsevier Mg npj Computational Materials (2019) 76 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences Defect Formation Energy (eV) Defect Formation Energy (eV) J. Zhang et al. ab 0.6 p-type AZn Sb 2 2 Yb Sr 0.4 1.0 Eu 0.2 Ca Ca Eu 0.0 Yb Sr -0.2 VBM 0.1 -0.4 0.98 1.00 1.02 1.04 1.06 -0.2 0.0 0.2 0.4 0.6 Electronegativity E−E (eV) F V cd n-type Mg Sb Bi Te VBM VBM 1.6 1.6 3 1.5 0.5 Sb Se Se Sb Te Sb 1.2 1.2 Mg1 Mg2 0.8 0.8 0.1 0.4 0.4 0.0 0.0 Mg rich Sb rich 0.01 -0.4 0.8 1.0 1.2 1.4 0.0 0.2 0.4 0.0 0.2 0.4 |χ -χ | Fermi energy (eV) Fermi energy (eV) Q Mg 15–17,20,95 Fig. 9 Defect-controlled carrier density. a The experimental Hall carrier concentration at room temperature versus the electronegativity of the cation A (A = Sr, Eu, Ca, and Yb) in p-type AZn Sb . The Allred–Rochow scale of the electronegativity is used. b 2 2 Defect formation energy of the cation vacancy in AZn Sb . The respective conduction band minima of different compounds are shown in the 2 2 54,57,60 colored dashed lines. c Hall carrier concentration at 300 K as a function of the electronegativity difference χ  χ (Q = S, Se, and Te) Q Mg in n-type chalcogen-doped Mg Sb Bi . The Pauling scale of the electronegativity is adopted. d Defect formation energies of extrinsic 3 1.5 0.5 defects Q as well as the intrinsic defects V and V under Mg-rich and Sb-rich conditions. a, b and c, d are adapted with permissions, Sb Mg1 Mg2 80 60 respectively, from refs. and , Wiley-VCH ANHARMONIC PHONON MODES AND LOW LATTICE THERMAL The lattice thermal conductivity of Mg Sb and the two ternary 3 2 CONDUCTIVITY analogs (CaMg Sb and CaMg Bi ) was calculated by Zhang 2 2 2 2 14–69,71 et al. using first-principles Boltzmann transport calculations Reported lattice thermal conductivities κ at room tem- considering the anharmonic phonon–phonon scattering with the perature in CaAl Si -type TE compounds span a relatively wide 2 2 −1 −1 −1 −1 ShengBTE code. Importantly, the theoretical result shows a range from ~0.6 W m K in CaCd Sb to ~4.5 W m K in 2 2 noticeable decrease in κ from CaMg Sb and CaMg Bi to CaMg Sb . Among them, Mg Sb with a low density of ~4.0 g L 2 2 2 2 2 2 3 2 −3 Mg Sb , which is in good agreement with experiments (see 3 2 cm shows an intrinsically low room-temperature κ of about −1 −1 40,45,46 Fig. 10a). It is found that the phonon lifetime plays a key role in 1.0–1.5 W m K , comparable to those of the state-of-the- 82,84 1,138,139 reducing κ in Mg Sb . In general, the phonon lifetime for L 3 2 art TE materials PbTe and Bi Te . A small difference (~15%) 2 3 intrinsic crystalline materials is dominated by the three-phonon in κ is found between single crystalline and polycrystalline 40,102,130 anharmonic scattering process. As revealed in Fig. 10b, the Mg Sb samples, suggesting that the grain boundary 3 2 calculated anharmonic scattering rates of Mg Sb are considerably 3 2 scattering does not play a significant role in phonon transport. larger than those of CaMg Sb and CaMg Bi in the low- and mid- 2 2 2 2 Though alloying is often applied to further reducing lattice frequency regions (<5.5 THz). In particular, the anharmonic thermal conductivity through point defect scattering, the intrinsi- scattering rates of Mg Sb display one peak at 0.7–1.04 THz and 3 2 cally low thermal conductivity induced by the phonon–phonon two peaks in the mid-frequency region of 3.3–4.8 THz, which are scattering is indispensable to the exceptionally high zT in n-type about (or even more than) one order of magnitude larger than Mg Sb -based materials. Below we will discuss the origin of the 3 2 those of ternary compounds within the same frequency regions. intrinsically low lattice thermal conductivity in Mg Sb from first 3 2 The peak in anharmonic scattering rates of Mg Sb at 0.7–1.04 THz 3 2 principles. is typically induced by a unique feature in phonon dispersion in Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences npj Computational Materials (2019) 76 20 -3 20 -3 n (10 cm ) n (10 cm ) Defect Formation Energy (eV) Defect Formation Energy (eV) J. Zhang et al. ab Theory Expt. CaMg Sb 2 2 CaMg Bi 2 2 Mg Sb 3 2 Nano-Mg Sb 3 2 Mg Sb 3 2 CaMg Sb 2 2 CaMg Bi 2 2 200 300 400 500 600 700 800 T (K) cd 2 Mg Sb 3 2 Total CaMg Sb 2 2 Mg1 Mg2 CaMg Bi 2 2 Sb Γ MK Γ A L H A PDOS Wave vectors 79,84 Fig. 10 a The experimental and theoretical lattice thermal conductivities. The theoretical and experimental data are taken from refs. 26,36,40,43 43 and , respectively. Nano-Mg Sb denotes the nanostructured Mg Sb sample prepared by the high-energy ball milling method. b 3 2 3 2 The calculated anharmonic scattering rates of Mg Sb , CaMg Sb , and CaMg Bi . c Phonon band structure and density of states of Mg Sb . 3 2 2 2 2 2 3 2 79 84 Adapted with permission from ref. , CC-BY-4.0. The mode Grüneisen parameters are projected on the phonon band structure. Curve width indicates the relative magnitude of the mode Grüneisen parameter. The positive and negative mode Grüneisen parameters are colored in red and blue, respectively. d Mode Grüneisen parameters of Mg Sb , CaMg Sb , and CaMg Bi . For the theoretical results shown here, the 3 2 2 2 2 2 longitudinal optical/transverse optical (LO/TO) splitting is not considered since the LO/TO splitting only has a minor effect on the results. The readers are referred to ref. for the results considering the LO/TO splitting comparison with ternary compounds as pointed out earlier by modes in Mg Sb and Mg Bi may be attributed to the unstable 3 2 3 2 Peng et al., i.e., much softer transverse acoustic phonon modes Mg1 in the octahedral sites as judged by the small cation-to-anion at the M (~0.71 THz), L (~0.81 THz), and A (~1.04 THz) points ionic radius ratio below the stability limit (0.414), which results characterized by very large mode Grüneisen parameters (Fig. 10c, from the small ionic radius of Mg (Fig. 11b). Although the insight d). In particular, the soft transverse acoustic phonon mode at the L proposed by Peng et al. is illuminating, it cannot explain why point exhibits the largest negative mode Grüneisen parameter. A Mg As , which also exhibits a cation-to-anion ionic radius ratio 3 2 similar behavior was also found in Mg Bi , where the phonon below the stability limit, does not show soft transverse acoustic 3 2 calculations often show negative (imaginary) frequencies for phonon modes. Clearly, the ionic radius ratio rule is somewhat transverse acoustic branches at the L point. In addition, the peak oversimplified by assuming purely ionic bonding with a complete anharmonic scattering rates of Mg Sb at 3.3–4.8 THz in Mg Sb charge transfer. In reality, chemical bonds in Mg X show certain 3 2 3 2 3 2 are induced by the anharmonic optical phonon modes within this degrees of covalency with incomplete charge transfers (see frequency region with much larger mode Grüneisen parameters Table 1), which will result in a larger effective ionic radius of than those of CaMg Sb and CaMg Bi (Fig. 10d). Therefore, the Mg1 and probably push the cation-to-anion ionic radius ratio 2 2 2 2 soft transverse acoustic phonon modes within 0.7–1.04 THz as well above the stability limit. as the anharmonic optical modes within 3.3–4.8 THz contribute to The instability of Mg1 is well confirmed in the relatively larger the large peak values in the anharmonic scattering rates and atomic displacement parameter and flatter potential energy thereby intrinsically low lattice thermal conductivity in Mg Sb . landscape of Mg1 compared to those of Mg2 in Mg Sb (Fig. 11c, 3 2 3 2 The unique soft transverse phonon modes are related to the d), which might be attributed to the relatively weaker Mg1-Sb structural instability in Mg Sb and Mg Bi , which is well bond as well as the larger void space of the octahedral site of Mg1. 3 2 3 2 141,142 confirmed in their less negative formation enthalpies However, the unstable Mg1 at the octahedral site better explains compared with other CaAl Si -type compounds (Fig. 11a). As the anharmonic optical modes at 3.3–4.8 THz rather than the soft 2 2 claimed by Peng et al., the soft transverse acoustic phonon transverse acoustic modes since the anharmonic optical modes npj Computational Materials (2019) 76 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences -1 -1 κ (W m K ) Frequency (THz) L J. Zhang et al. ab AB X 2 2 Mg Bi Ba 3 2 -0.2 X=P 0.7 X=As Mg Sb X=Sb 3 2 0.6 Eu -0.4 X=Bi Sr Ca 0.5 Yb -0.6 0.4 Stability limit Mg -0.8 0.3 -1.0 0.2 0.70.8 0.91.0 1.1 0.6 0.8 1.0 1.2 1.4 Electronegativity difference ΔEN r (Å) cation 0.12 cd a c Mg1 0.06 Mg1 Mg2 Mg2 Sb Sb 0.05 0.08 0.04 0.03 0.04 0.02 0.01 0.00 0.00 200 400 600 800 -0.2 -0.1 0.0 0.1 0.2 T (K) Displacement (Å) Fig. 11 a Formation enthalpy versus the electronegativity difference ΔEN in AB X (CaAl Si -type) compounds. ΔEN is calculated by the 2 2 2 2 difference between the electronegativity of the anion X and the average electronegativity of the atoms A and B. Formation enthalpy values 141,142 are taken from MaterialsProject.org. b The cation-to-anion ionic radius ratio r /r versus the ionic radius of the cation A at the cation anion octahedral site in AMg X . The minimum stability limit of r /r for the octahedral site is 0.414. The data points for Mg Sb and Mg Bi 2 2 cation anion 3 2 3 2 are colored in pink. Adapted with permissions from ref. , Elsevier. c Experimental isotropic atomic displacement parameters of Mg Sb . d The 3 2 potential energy curves for the nonequivalent atoms in Mg Sb displacing along the a and c directions. c, d are reproduced with permissions 3 2 from ref. , CC-BY-4.0 are mainly contributed by the motion of Mg1 while the acoustic since it is sensitive to the factors such as the sample quality, crystal phonon modes are dominated by the vibration of Sb (Fig. 10b, c). alignment, and preferred orientations. Therefore, here we will It is more likely that the unstable or weak chemical bonds lead to focus on the anisotropy of electrical and thermal transport based the soft transverse acoustic phonon modes in Mg Sb and Mg Bi . on the perfect crystal structure from the theoretical point of view. 3 2 3 2 The soft transverse acoustic phonon modes in Mg Sb and In general, only electronic states near the Fermi level contribute to 3 2 Mg Bi involve the interlayer shearing motion, which results in the electronic transport. Under the constant and isotropic carrier 3 2 82,97 their reported soft shear moduli. However, whether the soft scattering time assumption, the anisotropy of electrical conduc- shear modes are mainly induced by the weak interlayer Mg1-Sb tivity is completely determined by the anisotropy in carrier (or Mg1-Bi) bond requires further confirmation. Quantitative mobility and thereby the anisotropy in the inverse effective mass chemical bonding analysis based on the electron density is tensor, which can be intuitively understood from the anisotropy of reliable in comparing different chemical bonds in the same the atomic orbitals interactions. For p-type transport in Mg Sb , 3 2 compound, but it cannot be used for reliably comparing the the electronic states at the VBM are mainly comprised of the p relative strength of chemical bonds in different compounds. orbitals of the Sb atoms. The anisotropic p orbitals show much Therefore, other computational methods should be applied to stronger orbital overlaps along the c axis, which results in much quantitatively characterize and compare the relative strength of larger band dispersion, smaller effective mass, and larger electrical the interlayer A-X bonds in different compounds. Furthermore, conductivity in comparison to those along the a axis. This is other factors such as the difference in atomic masses should also confirmed by the calculated strong anisotropy in the electrical be considered. Overall, more comprehensive work should be conductivity for p-type Mg Sb with σ /σ being much smaller 3 2 xx zz conducted in the future to reveal the origin of these unique soft than unity (see Fig. 12a), consistent with the experimental Sb and Mg Bi . reports. In contrast, for n-type transport in Mg Sb , the transverse modes in Mg 3 2 3 2 3 2 electronic states at the CBM are dominated by spherical s orbitals 55,57 of the Mg atoms, which generally result in nearly isotropic ELECTRONIC AND THERMAL TRANSPORT ANISOTROPY features in orbital interactions, average effective mass tensor, and Experimentally, the anisotropy of the transport properties in bulk thereby electrical conductivity (Fig. 12a). For the Seebeck 48,102,130,143 TE materials can vary a lot between different reports coefficient, both p-type and n-type Mg Sb show a nearly isotropic 3 2 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences npj Computational Materials (2019) 76 U (Å ) Formation energy (eV per atom) iso Potential Energy (eV) r /r cation anion J. Zhang et al. 2.0 ab p-type n-type SnS σ /σ 1.8 MoS TiS xx zz α /α 1.6 xx zz MoSe 1.4 1.2 1.0 CaZn Sb 2 2 0.8 SrZn Sb 0.6 2 2 CaMg Sb 2 2 0.4 Mg Sb 3 2 0.2 CaMg Bi 2 2 0.0 0.01 0.1 1 ρ /ρ 20 -3 intra inter n (10 cm ) Anisotropy of the chemical bonding network Fig. 12 a Anisotropy of the calculated electronic transport properties including Seebeck coefficient (α /α ) and electrical conductivity (σ / xx zz xx σ ) as a function of Hall carrier concentration in p-type and n-type Mg Sb . The data are calculated in this work using the previous zz 3 2 computational methods . b Anisotropy of the theoretical lattice thermal conductivity κ /κ as a function of the anisotropy of the chemical a c bonding network characterized by ρ =ρ . b is reproduced with permissions from ref. , CC-BY-4.0 intra inter feature even though the effective mass tensor is anisotropic development. Below we present some challenges and possible (Fig. 12a). This may be attributed to the notion, which was future directions in both theoretical and experimental aspects. Theoretically, critical challenges include the difficulties for current proposed by Parker et al., that the anisotropic effective mass computations to describe the carrier scattering time, doping, results in isotropic Seebeck coefficient as long as the electronic temperature effects, defects, and disorder in experiments. One band is assumed parabolic, the carrier scattering time is assumed typical example of these challenges can be seen in electrical only dependent on energy, and the bipolar effect is insignificant. transport calculations under the rigid band and constant carrier’s Moreover, as investigated by Sun et al., the Seebeck coefficients scattering time approximation, which assumes electronic bands and of several ternary compounds AMg X and AZn Sb (A = Ca, Sr, 2 2 2 2 scattering time being insensitive to temperature and doping. In Ba; X = Sb, Bi) also show isotropic features when there is no reality, doping on the cation sites and varying carrier scattering bipolar effect, whereas the n-type electrical conductivities in these mechanisms result in the deviation from theory and the scatter of compounds show anisotropic features due to the anisotropic experimental data in the Pisarenko plot of the Seebeck coefficient average effective mass tensors induced by the anisotropic versus carrier density (see Fig. 6d). Another example can be found in electron pockets, which may be understood from the anisotropic the situation where the high carrier mobility cannot be understood orbital interactions induced by the increased contributions of from the band structure. In several compounds containing rare earth anisotropic p and d orbitals to the CBM. elements, the carrier’s scattering time is responsible for the In general, the thermal transport anisotropy is related to the 79,145 intrinsically high carrier mobility, whereas the detailed mechanism anisotropy of the chemical bonding network. In Mg Sb and 3 2 requires further investigation. In n-type Mg Sb Bi ,changing 3 2−x x related structures, the anisotropy of the chemical bonding carrier scattering behavior can improve carrier mobility by doping network can be quantified by the intralayer-to-interlayer bond- with transition metals on the Mg sites or increasing the pressing strength ratio ρ =ρ based on electron density. A nearly intra inter temperature, but the underlying mechanism is still under debate. linear correlation between the anisotropy ratio κ /κ of lattice a c In addition, the insights into the origin of valence and thermal conductivity and ρ =ρ indicates that ρ =ρ can intra inter intra inter conduction band alignments, how thermal expansion affects be adopted as an indicator measuring the anisotropy of lattice band alignments, and the origin of carrier pocket anisotropy thermal conductivity in Mg Sb -related materials (see Fig. 12b). 3 2 remain unclear. The underlying mechanism on why the favorable For AMg X compounds, the nearly isotropic 3D bonding network 2 2 six-fold CBM at the low-symmetry CB point only exists in binary with ρ =ρ ≈ 1 results in the nearly isotropic features in intra inter Mg X requires further investigation. For defect calculations, more 3 2 phonon dispersion, group velocity, Grüneisen parameter, and potential n-type dopants need to be predicted for the experi- ultimately lattice thermal conductivity. However, the lattice mental validations. Regarding thermal transport, further efforts are thermal conductivity in AZn Sb is relatively anisotropic due to the 2 2 required to elucidate the origin of the soft transverse acoustic anisotropic chemical bonding network with ρ =ρ > 2. phonon modes in Mg Sb and Mg Bi that remains obscurely intra inter 3 2 3 2 understood to date. Whether the soft transverse acoustic modes are induced by the weak interlayer interaction should be carefully SUMMARY AND OUTLOOK examined. The critical challenge here for computations is how to In this review, we have explored many illuminating insights such quantitatively and reliably compare the strength of interlayer as orbital overlap, orbital degeneracy, orbital splitting energy, interactions in different compounds. In addition to this possible valley degeneracy, effective mass, carrier pocket anisotropy, Fermi notion, the impact of other factors such as the atomic mass surface complexity, point defects, electronegativity, and bond difference on phonon transport should be examined. covalency for understanding electronic and thermal transport of Experimentally, the reported compositions are mainly limited to Mg Sb and related CaAl Si -type TEs. Although current insights Sb- and Bi-based compounds, while As- and P-based compounds 3 2 2 2 for the materials design have been very helpful to date, many remain largely unexplored. According to the solid solution challenges need to be addressed in order to achieve further compound map, there are still many unexplored alloying npj Computational Materials (2019) 76 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences Electronic transport anisotropy Anisotropy of lattice thermal conductivity κ /κ a c J. Zhang et al. combinations showing the potential to achieve superior p-type 17. Zhang, H. et al. A new type of thermoelectric material, EuZn Sb . J. Chem. Phys. 2 2 129, 164713 (2008). electrical properties with minimal crystal orbital splitting energy 18. Wang, X.-J. et al. Synthesis and high thermoelectric efficiency of Zintl phase values. Moreover, excellent n-type properties have only been YbCd Zn Sb . Appl. Phys. Lett. 94, 092106 (2009). 2−x x 2 discovered in Mg Sb Bi with few effective electron dopants 3 2−x x 19. Cao, Q.-G. et al. Zintl phase Yb Ca Cd Sb with tunable thermoelectric prop- 1-x x 2 2 such as the chalcogens, Y, and La. Hence, other effective n-type erties induced by cation substitution. J. Appl. Phys. 107, 053714 (2010). dopants for Mg Sb Bi as well as other potential material 3 2−x x 20. Toberer, E. S., May, A. F., Melot, B. C., Flage-Larsen, E. & Snyder, G. J. Electronic systems such as Mg Sb As and Mg Bi As for n-type doping 3 2−x x 3 2−x x structure and transport in thermoelectric compounds AZn Sb (A=Sr, Ca, Yb, 2 2 need to be explored in the future. Despite the outstanding TE Eu). Dalton Trans. 39, 1046–1054 (2010). performance shown in n-type Mg Sb -based materials, the long- 3 2 21. Zhang, H. et al. Thermoelectric properties of Eu(Zn Cd ) Sb . Dalton Trans. 39, 1-x x 2 2 term thermal stability under a temperature gradient and an 1101–1104 (2010). electrical current should be addressed as an essential step toward 22. Zhang, H. et al. Synthesis and properties of CaCd Sb and EuCd Sb . Inter- 2 2 2 2 metallics 18, 193–198 (2010). the practical applications. 23. Zhang, H. et al. Thermoelectric properties of Yb Eu Cd Sb . J. Chem. Phys. 133, x 1−x 2 2 194701 (2010). 24. Zhang, H. et al. Thermoelectric properties of polycrystalline SrZn Sb prepared 2 2 ACKNOWLEDGEMENTS by spark plasma sintering. J. Electron. Mater. 39, 1772–1776 (2010). We thank K.F.F. Fischer for discussions and comments. This work was supported by 25. Guo, K. et al. Enhanced thermoelectric figure of merit of Zintl phase YbCd 2- the Danish National Research Foundation (Center for Materials Crystallography, Mn Sb by chemical substitution. Eur. J. Inorg. Chem. 2011,4043–4048 x x 2 DNRF93) and the Danish Center for Scientific Computing. The numerical results (2011). presented in this work were obtained at the Center for Scientific Computing, Aarhus. 26. May, A. F. et al. Thermoelectric transport properties of CaMg Bi , EuMg Bi , and 2 2 2 2 Affiliation with the Center for Integrated Materials Research (iMAT) at Aarhus YbMg Bi . Phys. Rev. B 85, 035202 (2012). 2 2 University is gratefully acknowledged. 27. Zevalkink, A. et al. Nonstoichiometry in the Zintl phase Yb Zn Sb as a route to 1-δ 2 2 thermoelectric optimization. Chem. Mater. 26, 5710–5717 (2014). 28. Min, W., Guo, K., Wang, J. & Zhao, J. Effect of manganese doping on the ther- AUTHOR CONTRIBUTIONS moelectric properties of Zintl phase EuCd Sb . J. Rare Earth. 33, 1093–1097 2 2 J.Z. wrote the manuscript with the inputs and suggestions from B.B.I. L.S. collected (2015). the reported experimental data and edited the manuscript. 29. Shuai, J. et al. Higher thermoelectric performance of Zintl phases (Eu Yb ) 0.5 0.5 1- Ca Mg Bi by band engineering and strain fluctuation. Proc. Natl Acad. Sci. USA x x 2 2 113, E4125–E4132 (2016). ADDITIONAL INFORMATION 30. Shuai, J. et al. Thermoelectric properties of Zintl compound Ca Na Mg Bi . 1-x x 2 1.98 Appl. Phys. Lett. 108, 183901 (2016). Competing interests: The authors declare no competing interests. 31. Shuai, J. et al. Thermoelectric properties of Bi-based Zintl compounds Ca 1- Yb Mg Bi . J. Mater. Chem. A 4, 4312–4320 (2016). x x 2 2 Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims 32. Wubieneh, T. 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Bjerg, L., Madsen, G. K. H. & Iversen, B. B. Ab initio calculations of intrinsic point from the copyright holder. To view a copy of this license, visit http://creativecommons. defects in ZnSb. Chem. Mater. 24, 2111–2116 (2012). org/licenses/by/4.0/. 126. Xin, H. X. & Qin, X. Y. Electrical and thermoelectric properties of nanocrystal substitutional semiconductor alloys Mg (Bi Sb ) prepared by mechanical 3 x 1-x 2 © The Author(s) 2019 alloying. J. Phys. D Appl. Phys. 39, 5331 (2006). Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences npj Computational Materials (2019) 76 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png npj Computational Materials Springer Journals

Insights into the design of thermoelectric Mg3Sb2 and its analogs by combining theory and experiment

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www.nature.com/npjcompumats REVIEW ARTICLE OPEN Insights into the design of thermoelectric Mg Sb and its 3 2 analogs by combining theory and experiment 1 1 1 Jiawei Zhang , Lirong Song and Bo Brummerstedt Iversen Over the past two decades, we have witnessed a strong interest in developing Mg Sb and related CaAl Si -type materials for low- 3 2 2 2 and intermediate-temperature thermoelectric applications. In this review, we discuss how computations coupled with experiments provide insights for understanding chemical bonding, electronic transport, point defects, thermal transport, and transport anisotropy in these materials. Based on the underlying insights, we examine design strategies to guide the further optimization and development of thermoelectric Mg Sb -based materials and their analogs. We begin with a general introduction of the Zintl 3 2 concept for understanding bonding and properties and then reveal the breakdown of this concept in AMg X with a nearly 2 2 isotropic three-dimensional chemical bonding network. For electronic transport, we start from a simple yet powerful atomic orbital scheme of tuning orbital degeneracy for optimizing p-type electrical properties, then discuss the complex Fermi surface aided by high valley degeneracy, carrier pocket anisotropy, and light conductivity effective mass responsible for the exceptional n-type transport properties, and finally address the defect-controlled carrier density in relation to the electronegativity and bonding character. Regarding thermal transport, we discuss the insight into the origin of the intrinsically low lattice thermal conductivity in Mg Sb . Furthermore, the anisotropies in electronic and thermal transport properties are discussed in relation to crystal orbitals and 3 2 chemical bonding. Finally, some specific challenges and perspectives on how to make further developments are presented. npj Computational Materials (2019) 5:76 ; https://doi.org/10.1038/s41524-019-0215-y INTRODUCTION properties. During the past two decades, both theoretical and experimental aspects in CaAl Si -type TE materials have under- The increasing energy consumption worldwide pushes significant 2 2 13–52 gone a rapid development, which results in a significant efforts in developing alternative energy technologies. Thermo- improvement of zT above unity (Fig. 1). Experimentally, strategies electric (TE) technology, capable of converting waste heat into 45–47,49,50 15,16,18,19,21,23,25,29,35,36 such as doping, alloying, and con- electrical energy, shows potential for waste heat-harvesting trolling vacancy concentrations have been used to optimize applications with a great advantage that TE devices are entirely electrical transport properties and to reduce thermal conductivity in solid state with no moving parts, compact, quiet, and 1,2 through point defect scattering. As nearly all these compounds maintenance free. One key bottleneck of this promising are persistently p-type, it is a remarkable breakthrough that the technology is its low conversion efficiency, which is essentially low-cost Te-doped Mg Sb -based compounds were recently limited by low-performing materials. The performance of a TE 3 2 discovered to show exceptional n-type TE properties at low and material can be characterized by the dimensionless figure of merit, 53–55 intermediate temperatures (Fig. 1), comparable or even zT = α σT/(κ + κ ), where α is the Seebeck coefficient, σ is the L e superior to the commercial n-type TE materials, such as Bi Te electrical conductivity, T is the absolute temperature, κ is the 2 3 56–73 and PbTe. Hence, significant research efforts on this promis- lattice thermal conductivity, and the electronic thermal conduc- tivity κ is related to σ through the Wiedemann–Franz law κ = LσT ing material system are currently ongoing. Theoretically, electronic e e 20,74–77 11,12,78,79 55,60,61,80 structures, chemical bonding, defects, and (L denotes the Lorenz number). The interdependent correlation 81–84 3–10 phonon-related properties have been extensively studied to among the TE transport parameters invokes numerous efforts worldwide on decoupling these parameters so as to improve zT.In understand the transport properties. The interplay between theory general, achieving a superior zT requires a combined effort of and experiment has given rise to many important guidelines based on the underlying physical and chemical insights for improving the power factor α σ and reducing the lattice thermal conductivity. optimizing CaAl Si -type TEs. 2 2 AB X compounds with the CaAl Si -type structure, usually In this review, we summarize some of the successful guiding 2 2 2 2 described as Zintl phases, are well recognized due to their principles for understanding and rationalizing the electrical and promising TE properties. In these compounds, A is an alkaline thermal transport in Mg Sb and its derivatives with the CaAl Si - 3 2 2 2 type structure. We show how computational efforts integrated earth or a divalent rare earth element, B is a transition metal or a main group element, and X usually comes from group 14, 15, or with experiments lead to additional physical and chemical insights 11,12 occasionally group 13. The rich variety of compositions for the profound understanding of chemical bonding, electronic covered by AB X (CaAl Si -type) compounds enables consider- transport, defect chemistry, phonon transport, and transport 2 2 2 2 able chemical tunability of the electronic and thermal transport anisotropy. We start from general introductions of crystal structure Center for Materials Crystallography, Department of Chemistry and iNANO, Aarhus University, 8000 Aarhus, Denmark Correspondence: Bo Brummerstedt Iversen (bo@chem.au.dk) Received: 19 February 2019 Accepted: 26 June 2019 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences J. Zhang et al. ab n-type Mg Sb 1.4 3 2 2.0 p-type Mg Sb 3 2 1.2 Other p-type AB X 2 2 1.6 1.0 1.2 0.8 0.6 0.8 0.4 0.4 0.2 0.0 0.0 2000 2004 2008 2012 2016 2020 2000 2004 2008 2012 2016 2020 Year Year Fig. 1 a, b Timelines of a the maximum zT and b the average zT values of the reported AB X thermoelectric materials with the CaAl Si -type 2 2 2 2 structure. The average zT is calculated by the direct integration method within the entire measurement temperature range. The experimental 14–71 data are taken from refs. and the Zintl concept that has been widely applied to understand 1933, much earlier than those of ternary derivatives. From the 88,89 the structure, bonding, and electronic transport in CaAl Si -type reported phase diagram, Mg Sb and Mg Bi show a phase 2 2 3 2 3 2 compounds. After that, we reveal the nearly isotropic three- transition from the low-temperature α phase to the high- dimensional (3D) chemical bonding network in AMg X , where the 2 2 temperature β phase at ~900 and ~700 °C, respectively. The Zintl formalism is no longer applicable. For p-type electronic high-temperature β phase is superionic with liquid-like magne- transport, we discuss how electronic transport can be optimized sium ions and it is assumed to have a cubic structure, but the by minimizing the crystal orbital splitting energy via forming solid exact crystal structure remains unclear. solutions and tuning biaxial strains, whereas for n-type transport we reveal the multi-valley conduction bands and complex Fermi surface as the electronic origin of the extraordinary n-type TE ZINTL FORMALISM 91,92 properties. Then we review the defect chemistry of the intrinsic p- The Zintl concept plays a vital role in understanding structure, type behavior and the surprising n-type behavior under different chemical bonding, and properties of a wide range of solid-state thermodynamic states, followed by highlighting the defect- materials. Within this simple but powerful concept, the mostly controlled carrier transport and its correlation with the electro- ionic cations are considered merely as electron donors, donating negativity and bonding character. For thermal transport, we their electrons to the covalently bonded anionic substructures. review the studies on exploring the origin of the intrinsically low The covalent bonding in the anionic substructures ensures a lattice thermal conductivity in Mg Sb from first principles 3 2 significant orbital overlap, a light band mass, and thereby a high calculations. Moreover, we discuss the anisotropy in electrical carrier mobility, whereas the ionic cations are suitable for doping and thermal transport properties with respect to crystal orbitals with the aim to tune the carrier density without affecting the and chemical bonding. Finally, we conclude with some current 93,94 covalent anionic networks. Such bonding features, as well as challenges and prospects for future development. ideal band gaps, make charge-balanced Zintl compounds very promising for TE applications. 11,12 CRYSTAL STRUCTURE As described by Hoffmann et al., AB X compounds with 2 2 the CaAl Si -type structure can easily be understood as Zintl AB X compounds with the trigonal CaAl Si -type structure 2 2 2 2 2 2 δ− δ+ phases, where the covalently bonded [B X ] layers receive (space group: P3m1) can be viewed as the A cationic layers 2 2 δ+ δ− electrons from the ionic A layers. Based on the aforementioned intercalated between the tightly bound [B X ] anionic layers 2 2 Zintl concept, considerable efforts have been made to explore (Fig. 2a). The three types of atoms A, B, and X occupy three AB X -type TE compounds and optimize them via doping, different crystallographic sites 1a (0, 0, 0), 2d (1/3, 2/3, z ), and 2d 2 2 substituting, or creating vacancies on the cationic A sites. One (1/3, 2/3, z ) with point symmetries of 3m,3m, and 3m, δ− notable example can be seen in Ca Yb Zn Sb alloys reported respectively. In the [B X ] networks, B has a normal tetrahedral 1−x x 2 2 2 2 by Gascoin et al., which show the fine tunability of the carrier coordination, while X shows a unique distorted umbrella-like local concentration through doping or substituting Ca with Yb. The environment. As a result, the vertical B-X bond (d ) along the c alloying of Ca and Yb indeed has a negligible effect on the axis is usually longer than the three symmetry-equivalent tilted B- 2− δ+ covalent anionic [Zn Sb ] networks, so that the good carrier X bonds (d ). Taking into account the cationic A layers, the 2 2 mobility and ideal energy gap are maintained. In addition, this interlayer A-X bonds (d ) hold together the alternating cationic and anionic layers. Unlike the B atoms being tetrahedrally alloying creates disorder in the cationic layers, which results in a reduction in thermal conductivity and an enhancement in zT. coordinated by the X atoms, each A atom occupies the octahedral Ever since this early work, the Zintl formalism has been broadly site with six equal adjacent bonds. Each X atom, surrounded by applied in developing and optimizing TE AZn Sb , ACd Sb , and four B atoms and three A atoms, has seven adjacent bonds 2 2 2 2 20,23,27,32 their alloys. In these compounds, however, it was found including one vertical B-X bond, three tilted B-X bonds, and three interlayer A-X bonds (Fig. 2c–e). that the hole mobility may be greatly affected by changing the Mg X (X = As, Sb, and Bi) crystallizing in the inverse α-La O cation A from alkaline earth elements to rare earth elements. 3 2 2 3 structure is a special case of the CaAl Si (AB X ) structure with A Different synthesis methods were also found to result in a great 2 2 2 2 86 95 and B being, respectively, Mg1 and Mg2. The crystal structures of change in carrier mobility of AZn Sb . These results are likely 2 2 Mg Sb and Mg Bi were reported by Zintl and Husemann in induced by the change in carrier scattering time τ because the 3 2 3 2 npj Computational Materials (2019) 76 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences 1234567890():,; zT max zT average J. Zhang et al. Fig. 2 a Crystal structure of layered Zintl phases A(Zn,Cd) Sb with the CaAl Si -type structure, showing a clear distinction between the 2 2 2 2 δ+ δ− ionic A layer and the covalent [B X ] layer. b A more accurate structural view of AMg X with the CaAl Si -type structure, which shows 2 2 2 2 2 2 comparable interlayer and intralayer bonds being mostly ionic with partial covalent nature (high polarity). c–e Coordination environments of the three nonequivalent atoms c A, d B, and e XinAB X compounds with the CaAl Si -type structure. f Static deformation electron density 2 2 2 2 −3 map on (110) plane of Mg Sb with a nearly isotropic three-dimensional bonding network (ρ =ρ  1). The contour interval is 0.006 e Å . 3 2 intra inter Positive (negative) contours are plotted with red solid (blue dotted) lines. g Noncovalent interaction analysis with reduced density gradient as a function of sign(λ )ρ for the interlayer and intralayer interactions in Mg Sb . sign(λ ) is the sign of the second eigenvalue of the electron 2 3 2 2 101 79 density Hessian matrix. Figure is reproduced with permission from ref. , CC-BY-4.0 effective mass of the valence band is shown to be unaffected by rational explanation for the unusual umbrella-like local coordina- varying the cation A. tion environment of the anionic X atom and the bond length δ− difference in the [B X ] networks of the CaAl Si (AB X ) 2 2 2 2 2 2 structure through making and breaking chemical bonds. In NEARLY ISOTROPIC 3D BONDING NETWORK IN AMG X : 2 2 addition, they compared the local and extended constructions BREAKDOWN OF THE ZINTL FORMALISM of the bonding in transition metal CaAl Si -type compounds with 2 2 Despite being very successful in rationalizing the electronic those in ThCr Si -type compounds and explained why com- 2 2 transport, the Zintl formalism and electron counting are not pounds with the B-site elements being occupied by transition 0 5 10 universal and can only provide a qualitative insight. It does not metals with the d ,d , and d configurations only form the give us any indications of the site preference, bond length CaAl Si -type structure. Burdett and Miller also conducted 2 2 difference in anionic networks, or the reason why intrinsic p-type extended Hückel calculations to reveal the formalism of the Al Si 2 2 Mg Sb shows very poor carrier mobility. More accurate theore- fragment in the CaAl Si -type structure and extended the 3 2 2 2 tical calculations based on quantum mechanics or density fragment formalism to several other main group compounds. functional theory (DFT) are required to understand the structure The above studies were conducted only based on the chemical δ− and chemical bonding. bonds in the [B X ] networks, neglecting the effect of the 2 2 δ+ 2+ Using the crystal orbital overlap population approach based on cationic A layer. The contribution of the cationic Ca layer to the extended Hückel calculations, Hoffmann et al. gave a the chemical bonding and electronic structure of CaAl Si was 2 2 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences npj Computational Materials (2019) 76 J. Zhang et al. ionic nature in both cationic and anionic layers. This indicates that Table 1. Bader atomic charge Q, the degree of ionicity, and ρ =ρ intra inter any description of AMg X , especially Mg Sb , as layered Zintl 2 2 3 2 of several AB X compounds with the CaAl Si -type structure 2 2 2 2 phases in all previous literature is incorrect. Chemical bonds in AB X (CaAl Si -type) compounds can be 2 2 2 2 Compounds ρ =ρ Q (e) Degree of intra inter described as polar bonds based on the topological properties of ionicity (%) the bond critical points (BCPs) using the classification scheme by A B X A layer B X layer 2 2 Gatti. The covalency of the intralayer bonds in the B X 2 2 networks, however, may vary a lot among different compounds. Mg Sb 1.47 1.51 1.47 −2.23 75.5 73.9 3 2 The intralayer B-X bonds are much more covalent and stronger Mg Bi 1.38 1.43 1.40 −2.11 71.5 70.2 3 2 than the mostly ionic interlayer A-X bonds in AZn Sb and 2 2 CaMg Sb 1.39 1.39 1.48 −2.17 69.5 73.2 2 2 ACd Sb , whereas the intralayer bonds in AMg X are only slightly 2 2 2 2 stronger than the largely ionic interlayer bonds (see Table 1 and CaMg Bi 1.37 1.37 1.42 −2.10 68.5 70.5 2 2 Fig. 2a, b, f). This is further confirmed by the similar reduced CaZn Sb 2.34 1.37 0.31 −0.99 68.5 24.3 2 2 density gradient distributions of the three nonequivalent bonds CaCd Sb 2.20 1.38 0.28 −0.97 69.0 23.2 2 2 101 from the non-covalent interaction analysis in Mg Sb (Fig. 2g). 3 2 The data of Bader atomic charge and ρ =ρ are adapted from ref. . The chemical bonding analysis may be conducive to under- intra inter The degree of ionicity is estimated from the charge transfer index, which standing transport properties. The mostly ionic feature with partial is calculated by the average of the atomic charge divided by the nominal covalency (high polarity) of chemical bonds explains the 36,102 oxidation charge of the atoms in unit cell. ρ denotes the electron inter intrinsically low carrier mobility in p-type Mg Sb and 3 2 density value at bond critical point (BCP) of the interlayer A-X bond, while CaMg Sb . However, the high carrier mobility in AMg Bi (A = 2 2 2 2 ρ represents the average electron density value at BCPs of the two 26,41 intra Mg, Ca, Eu, and Yb) cannot be understood from chemical intralayer B-X bonds. The data of CaCd Sb are calculated in this work 2 2 bonding, though it could be attributed to the bipolar effect or the using the previous computational methods change in carrier scattering time induced by defects. The electron density ratio ρ =ρ is devised as a simplified intra inter parameter measuring the degree of anisotropy of the chemical 95 79 investigated by Alemany et al. using DFT calculations. They bonding network in the Mg Sb -related structures, where ρ 3 2 intra found that the cationic layers show an important contribution to and ρ represent the electron density values at BCPs of the inter the covalent bonding of CaAl Si , although the cationic layers do intralayer and interlayer bonds, respectively. Unlike AZn Sb and 2 2 2 2 not play a dominant role in the electronic structure and the origin ACd Sb with clear anisotropic chemical bonding networks ( 2 2 of the conductivity behavior. In addition to these significant ρ =ρ >2), the AMg X compounds show nearly isotropic 3D 2 2 intra inter 11,12,78,96 studies on rationalizing structural formation and electro- chemical bonding networks with ρ =ρ typically being intra inter nic structure using the molecular orbital approaches, Grin et al. smaller than 1.5 and often close to unity (Table 1). The nearly studied the chemical bonding in YbCd Sb using the electron isotropic 3D bonding networks can be used to understand the 2 2 localizability indicator and showed that four-center bonding exists nearly isotropic structural and thermal properties especially lattice in this compound and the whole bonding picture can be thermal conductivities in AMg X , which will be discussed later. 2 2 2+ 2+ 3− described by the closed-shell configuration [Yb ][Cd ] [Sb ] . 2 2 Based on the analysis of the electron density difference, Toberer CRYSTAL FIELD ORBITAL SPLITTING, ORBITAL DEGENERACY, et al. discussed the chemical bonding in AZn Sb (A = Sr, Ca, 2 2 AND P-TYPE ELECTRONIC TRANSPORT and Yb) and revealed the largely covalent nature of the Zn-Sb bonds as well as the charge accumulation from Sb to A. The A-Sb In principle, the intrinsic electronic transport of a material is bond was found to be very asymmetric and slightly larger in determined almost exclusively by the degeneracy and curvature YbZn Sb than in SrZn Sb or CaZn Sb , which might be of the electronic bands at the band edges. The Seebeck coefficient 2 2 2 2 2 2 attributed to the less electron transfer from the cation A to the is typically determined by the density of states effective mass m * 2/3 1,103 anionic Zn Sb slabs induced by the larger electronegativity of Yb = N m *, where N represents the valley degeneracy of the 2 2 v s v in comparison to Sr or Ca. Using DFT calculations, Li et al. electronic bands and m * denotes the effective mass of a single 1,103 revealed a much lower shear strength in Mg Sb compared to valley. As proposed by Goldsmid, the optimum electrical 3 2 CaMg Sb and CaZn Sb , which was explained by the weaker transport performance can be expressed as being proportional to 2 2 2 2 97 3/2 interlayer Mg-Sb bond in Mg Sb . Despite significant efforts on the weighted mobility μ(m */m ) , where μ and m represent the 3 2 d e e understanding chemical bonding, most of these efforts are carrier mobility and the mass of an electron, respectively. qualitative and there is clearly a lack of a quantitative chemical Considering the case with predominant acoustic phonon scatter- bonding description of CaAl Si -type compounds based on ing or alloy scattering, the weighted mobility can be simplified 2 2 topological analysis of the electron density. and written as N /m *(N /m * for an isotropic band), where m *is v c v s c Recent theoretical calculations extend our knowledge of the conductivity effective mass. It is clear that excellent electrical chemical bonding in CaAl Si -type compounds. Quantitative performance requires a high valley degeneracy as well as a light 2 2 analysis of full DFT electron density by Zhang et al. elucidated conductivity effective mass. Increasing valley degeneracy (includ- that the Zintl formalism is perfectly applicable in AZn Sb and ing orbital degeneracy) has long been recognized as an efficient 2 2 ACd Sb but not in AMg X (including Mg X ). The analysis was way to improve electrical transport performance through enhan- 2 2 2 2 3 2 conducted using the quantum theory of atoms in molecules cing the Seebeck coefficient without explicitly decreasing the (QTAIM) developed by Bader. QTAIM is based on the analysis of carrier mobility when the intervalley scattering is insignifi- 4,6,9,103,104 critical points of the electron density ρ, which are the points cant. Below we will introduce how electrical transport satisfying ∇ρ = 0. The readers are referred to ref. for the details can be rationalized through enhancing orbital degeneracy via of the method. The idea is clearly proved by the result of Bader minimizing the orbital splitting energy in p-type AB X 2 2 atomic charges (see Table 1). In CaZn Sb and CaCd Sb , the compounds with the CaAl Si -type structure. 2 2 2 2 2 2 cationic and anionic layers show a remarkable difference in the In CaAl Si -type compounds, the valence band maximum (VBM) 2 2 degree of ionicity estimated from atomic charges, i.e., the located at the Г point shows p orbital characteristics of the anions cationic layers are largely ionic while the anionic layers are mostly (Fig. 3a, b). Unlike the triply degenerate p orbitals at the Γ point in covalent. In contrast, nearly complete charge transfers are cubic structures protected by the point symmetry, the p orbital, observed for all atoms in AMg X , which elucidates the mostly due to the crystal field effect, is usually well separated from the p 2 2 x, npj Computational Materials (2019) 76 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences J. Zhang et al. 2.0 ab Mg1 CaAl Si -type structure 2 2 1.5 Mg2 CC C 1.0 Sb =E(Γ(p ))-E(Γ(p )) x,y z 0.5 0.0 p d Γ( ) x y −0.5 Γ( ) −1.0 Δ Δ ≈0 >0 Δ <0 −1.5 Γ Γ MK A L H A ef optimal p 20 -3 p=5×10 cm 25 20 -3 p=10 cm 19 -3 p=5×10 cm 20 19 -3 15 p=10 cm -0.4 -0.2 0.0 0.2 -0.4 -0.2 0.0 0.2 Δ (eV) (eV) Fig. 3 a Schematic diagram of orbital engineering to realize three-fold degenerate p orbitals in CaAl Si -type compounds. Nondegenerate 2 2 band Γ(p ) and doubly degenerate band Γ(p ) are mainly composed of p and p orbitals from the anions, respectively. Δ denotes the crystal z x,y z x,y field splitting energy between p and p orbitals at the Γ point. b Band structure of Mg Sb by TB-mBJ potential without spin–orbit coupling. x,y z 3 2 −3 The p orbitals of Sb are projected on the band structure. c, d Partial charge densities of the valence bands c Γ(p ) (isovalue: 0.06 e Å ) and d Γ −3 (p ) (isovalue: 0.13 e Å ) at the Γ point in Mg Sb . e The theoretical power factor at 300 K versus Δ of Mg Sb at various hole concentrations x,y 3 2 3 2 p. The red curve shows the best values corresponding to the optimum carrier concentrations. Data points for unoptimized carrier 14,15,17–27,39–41 concentrations fill up the pink area right below the red curve. f The experimental power factors of reported CaAl Si -type 2 2 18,21 compounds at 600 K as a function of Δ. Red points represent the alloys with zT larger than unity at high temperatures. a, b, e, f are adapted with permission from ref. , CC-BY-4.0 orbitals in the trigonal CaAl Si -type structure. Accordingly, the The zero-Δ selection rule may be combined with the band gap y 2 2 three-fold degenerate valence band at the Γ point splits into a criterion E < 1.5 eV with the aim to search for promising TE nondegenerate band Γ(p ) and a doubly degenerate band Γ(p ), candidates from a rich variety of CaAl Si (AB X )-type compounds. z x,y 2 2 2 2 where Γ(p ) and Γ(p ) are typically a heavy and light hole band, The band gaps of these AB X materials show a clear decreasing x,y z 2 2 9,105 respectively (Fig. 3a–d). The crystal field splitting energy Δ trend when the electronegativity difference between B and X between them is defined as Δ = E(Γ(p )) − E(Γ(p )), where the decreases, which may be rationalized by a decrease of the energy x,y z orbital degeneracy is effectively increased when the splitting between the atomic orbitals of B and X and an increase in band 106,107 energy approaches zero. The basic idea of the orbital engineering widths using the molecular orbital scheme. However, it approach proposed by Zhang et al. is to minimize the splitting should be noted that SOC also plays an important role in energy Δ with the aim to maximize orbital degeneracy and decreasing energy gaps of Bi-based compounds. Excellent TE thereby optimize electrical transport performance. This concept is properties are well confirmed in Sb-based and Bi-based com- 17 29 15 well confirmed by the Boltzmann transport calculations as well as pounds such as EuZn Sb , EuMg Bi , and YbZn Sb with 2 2 2 2 2 2 the experimental observations, which show peak values in power small Δ values close to zero and ideal band gaps. In particular, factor as Δ ≈ 0 (see Fig. 3e, f). As the thermal conductivities of most Shuai et al. reported strongly enhanced power factors in EuMg Bi 2 2 promising CaAl Si -type TE compounds are comparable especially with a nearly zero Δ value compared with those of CaMg Bi and 2 2 2 2 at high temperatures, the strongly enhanced power factor results YbMg Bi with Δ values largely deviating from zero, which 2 2 in peak zT values at Δ ≈ 0. The strong correlation between provides a solid confirmation of the zero-Δ rule. Though simple experimental power factors and Δ also indicates the minimal and powerful, the zero-Δ rule only has resulted in the discovery of intervalley scattering in different AB X compounds, which a few potential TE candidates. 2 2 ensures the success of this approach. Spin–orbit coupling (SOC) Materials design and optimization requires effective approaches is found to lift the degeneracy of the p band, but will not change for manipulating crystal field orbital splitting energy. The x,y the main idea of the approach. Accordingly, a simple selection hybridizations or overlap integrals of p orbitals control their rule, i.e., maintaining the valence band splitting energy around splitting energy at the VBM. In general, tuning structural zero (zero-Δ rule), is proposed to optimize the electrical transport parameters (for instance, the interlayer distance, bond lengths performance. and angles, and c/a) by the crystal deformation is able to Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences npj Computational Materials (2019) 76 2 14 -1 -2 -1 Energy (eV) ασ/τ (10 μWcm K s ) 2 -1 -2 Energy (eV) α σ (μWcm K ) Substrate J. Zhang et al. a b Thin Film 0.8 0.2 SrZn Sb 2 2 3.0 EuCd Sb 2 2 CaZn Sb 2 2 CaCd Sb 0.1 2 2 SrCd Sb 2 2 YbCd Sb BaMg Sb 2 2 2 2 0.4 YbZn Sb 2 2 2.4 ε 0.0 SrMg Sb 2 2 YbCd Zn Sb EuZn Sb 1.5 0.5 2 2 2 EuMg Sb -0.1 0.0 2 2 1.8 EuZn Cd Sb CaMg Sb 1.75 0.25 2 2 2 YbMg Sb -0.2 2 2 -0.4 1.2 -0.3 CaZn Sb 2 2 Mg Sb 3 2 Mg Sb 3 2 -0.4 -0.8 0.6 4.4 4.5 4.6 4.7 4.8 -4 -2 0 2 4 a (Å) ε (%) Fig. 4 Solid solution map and biaxial strain engineering for materials design. a Calculated Δ versus the lattice constant a in CaAl Si -type 2 2 14,15,17–27,39–41 compounds with E < 1.5 eV. Reported thermal conductivities at 500 K are shown in color bar. The stars correspond to solid solutions YbCd Zn Sb and EuZn Cd Sb with nearly zero Δ values. b Δ as a function of biaxial strain ε in two representative CaAl Si - 1.5 0.5 2 1.75 0.25 2 2 2 type compounds Mg Sb and CaZn Sb . Biaxial strain ε is defined as (a − a )/a × 100%, where a and a are the in-plane lattice parameters 3 2 2 2 0 0 0 with unstrained and strained states, respectively. Figure is adapted from ref. , CC-BY-4.0 effectively manipulate the orbital interactions and thereby the accelerate the screening and design of new TE materials from orbital splitting energy. In principle, crystal deformation can be layered or noncubic compounds. induced by both external and internal forces. External forces include physical pressure and strain effect, while internal forces MULTI-VALLEY CONDUCTION BANDS, COMPLEX FERMI involve chemical doping or forming solid solutions. Two SURFACE, AND EXCEPTIONAL N-TYPE ELECTRONIC efficient approaches, solid solution map and biaxial strain TRANSPORT engineering, can be used to realize the manipulation of Δ. The valley degeneracy is defined as the number of different carrier Using the solid solution map with calculated Δ values versus pockets (for the same type of carriers) existing at a given energy lattice parameter or band gap E , one can conveniently choose level. In general, the valley degeneracy of an individual electronic two or more compounds with positive and negative Δ values to band can be defined as N = N N , where N is the form a solid solution with the desirable Δ value of zero, which v v,sym v,band v,sym number of symmetry equivalent positions in the Brillouin zone for leads to excellent electronic transport performance (see Fig. 4a). a given k point at which the electronic band occurs and N Since alloying is also conducive to reducing the thermal v,band represents the number of electronic bands degenerate at the conductivity owing to the point defect scattering, strongly same k point and energy level. For the high-symmetry Γ point enhanced zT values can generally be achieved using this located at the center of the Brillouin zone, N = 1. The orbital approach. This powerful strategy is confirmed in several alloys v,sym 18 21 engineering approach discussed in the previous section is such including YbCd Zn Sb , EuZn Cd Sb , Eu Yb - 1.6 0.4 2 1.8 0.2 2 0.2 0.2 29 35 36 a case. Although the band degeneracy N of 3 at the Γ point Ca Mg Bi , YbCd Zn Sb , and Ca Na MgZnSb with v,band 0.6 2 2 1.5 0.5 2 0.99 0.01 2 (at the VBM) can be achieved via tuning the splitting energy of p superior TE power factors and zTs. In particular, Wood et al. orbitals, N of 1 at the Γ point limits the overall valley revealed the valence band crossing in CaMg Sb -CaZn Sb alloys, v,sym 2 2 2 2 which combines with reducing thermal conductivities by the alloy degeneracy to be N ≤ 3. It is thereby clear that a high valley degeneracy requires not only a high band degeneracy N but scattering to result in a peak zT of 0.87 at 850 K. As shown by v,band also a high N . In order to obtain a high N value, we need a Wang et al., optimizing electrical transport performance induced v,sym v,sym high-symmetry Brillouin zone as well as a band extremum that by minimizing the p orbital splitting energy as well as minimizing 1,2 the lattice thermal conductivity by the point defect scattering occurs at a low-symmetry k point. Below we will introduce the leads to an optimal zT of ~1.3 at 700 K in the YbCd Zn Sb multi-valley band behavior as the electronic origin of the 2−x x 2 extraordinary electrical transport performance in n-type Mg Sb - alloys. In addition to the approach of forming solid solutions, 3 2 based TE materials, which exhibit a unique near-edge conduction biaxial strain engineering can be applied to continuously tune Δ band minimum (CBM) at a low-symmetry k point with a high N so as to optimize electrical transport performance in thin-film TE v, materials (see Fig. 4b). A general principle for optimizing TE of 6. sym performance via biaxial strain is that for compounds with negative The electronic structure of Mg Sb has been studied using 3 2 (positive) Δ value the compressive (tensile) biaxial strain is more different theoretical methods in various reports. Traditional 108 109 functionals such as GGA or LDA are known to underestimate effective. 54,74 band gaps while recent calculations with the TB-mBJ Since most semiconductors show p-orbital characteristics at the potential gives more accurate band gaps close to the valence band edges, the orbital engineering approach can easily be extended to other structures with the same orbital splitting experimental values. However, band structures in all earlier features. One notable extension is the earlier reported pseudo- reports are typically calculated along the high-symmetry k 9 74–77 cubic approach in chalcopyrite structures, where the crystal field paths, which often overlook the actual CBM at the low- splitting energy can be directly linked to the structural deforma- symmetry k point. As revealed in the recent calculations including tion parameter η = c/2a. Moreover, the approach can be extended SOC by Zhang et al., the accurate CBM in Mg Sb is located at 3 2 * * to layered metal dichalcogenides and lithium intercalated metal the CB point (0, 0.417, 0.333), which is along the M –L line inside dichalcogenides. Therefore, the orbital engineering approach the first Brillouin zone (see Fig. 5a). Mg Sb shows an indirect 3 2 with physical and chemical insights based on the underlying band gap of ~0.6 eV with the VBM at the Γ point and the CBM at atomic orbitals enriches band engineering and may substantially the CB point. Moreover, there is a secondary band minimum npj Computational Materials (2019) 76 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences Δ (eV) -1 -1 κ (W m K ) Δ (eV) J. Zhang et al. With SOC A L L a 2.5 b CB 2.0 M* K M 1.5 1.0 n-type Total CB c 0.5 1 Mg p-type Sb 0.0 n-type -0.5 -1.0 -1.5 p-type -2.0 M K A L H A ** Γ Γ ML024 DOS Fig. 5 a Band structure and partial density of states of Mg Sb by TB-mBJ potential with SOC. b High-symmetry k points and k paths in the 3 2 Brillouin zone. c Multiple carrier pockets shown in Fermi surface of n-type Mg Sb for an energy level at 0.1 eV above the conduction band 3 2 minimum CB . d One highly anisotropic carrier pocket at the Г point shown in Fermi surface of p-type Mg Sb corresponding to an energy 1 3 2 level at 0.1 eV below the valence band maximum. a, b are adapted with permission from ref. , American Chemical Society. c, d are adapted with permission from ref. , CC-BY-4.0 located just above the CB extremum at the K point. The small surface complexity, it is crucial to take into account the energy difference ΔE of 0.078 eV between the two band contribution of the anisotropic feature of the single carrier pocket KCB minima CB and K suggests that they might even be treated as at the CB , which shows a moderate anisotropy parameter K = 1 1 * * * 1/2 effectively converged at elevated temperatures because of the m /(m m ) ≈ 2.3. || ⊥,1 ⊥,2 * * thermal broadening of Fermi function. Inspired by the biaxial The Fermi surface complexity factor N K can be calculated by 105 * * 3/2 112,114 * strain engineering in p-type compounds, Li et al. revealed that (m /m ) , where m is evaluated using the carrier density d c c n-type electrical performance of Mg Sb can also be optimized by n and theoretical electrical conductivity σ/τ from a BoltzTraP 3 2 111 * 2 tuning ΔE towards zero via the biaxial strain. calculation by m = ne /(σ/τ) under the constant carrier’s scatter- KCB c 116 * The iso-energy Fermi surface provides an intuitive shortcut to ing time approximation. m is estimated by fitting the Seebeck understand the multi-valley conduction bands in Mg Sb (see coefficient from Boltzmann transport calculations applied to the 3 2 Fig. 5b–d). In contrast to only one highly anisotropic hole pocket full DFT band structure using the single band model. The * * of the VBM at the Γ point, the Fermi surface of n-type Mg Sb , theoretical calculations reveal a peak N K of ~19 in n-type 3 2 v corresponding to an energy level 0.1 eV above the conduction Mg Sb , much higher than that of p-type Mg Sb (see Fig. 6a). 3 2 3 2 band minimum CB , shows 6 isolated full electron pockets inside This is consistent with both the theoretical and experimental the Brillouin zone and 6 one-third pockets at the K point (see results (see Fig. 6b, c), displaying significantly enhanced power Fig. 5c). As a result, the valley degeneracies for the conduction factors for n-type doping in comparison with those for p-type band minima CB and K are, respectively, 6 and 2, which may be doping in Mg Sb -based materials. The high Fermi surface 1 3 2 added up to 8 when they are effectively converged. Such a high complexity, as well as the superior n-type electrical transport valley degeneracy is comparable to many state-of-the-art TE performance, can be attributed to a combined effect of the large 2,103 * materials such as (Bi,Sb) Te and PbTe. m induced by the high valley degeneracy, the light conductivity 2 3 d The complex Fermi surface is conducive to good TE perfor- effective mass, and the carrier pocket anisotropy (see Fig. 6a). The mance. As proposed by Gibbs et al., the complexity of Fermi n-type transport of Mg Sb shows a primary contribution from the 3 2 * * surface may be characterized by the simplified parameter N K , nontrivial band minimum CB as well as a certain contribution v 1 * * where N and K represent the effective valley degeneracy and from the secondary band minimum at the K point. This multiple anisotropy parameter of a single carrier pocket, respectively. In band behavior in n-type Mg Sb is confirmed by the result from 3 2 * * general, the higher the N K , the better the electrical transport the BoltzTraP calculation that Mg SbBi with a much larger ΔE v 3 KCB performance. In addition to the high valley degeneracy, the of ~0.18 eV shows lower Seebeck coefficients in comparison with complexity of Fermi surface in n-type Mg Sb is clearly reflected in those of Mg Sb (see Fig. 6d). 3 2 3 2 the ellipsoidal-like carrier pockets at the CB , indicating a clear Though the multiple conduction band behavior was also found anisotropic feature. For every individual carrier pocket at the CB in ternary AMg X and AZn Sb , the unique conduction band 1 2 2 2 2 point, the effective mass is anisotropic along one longitudinal CB with a six-fold valley degeneracy only exists in binary Mg X 1 3 2 (elongated) and two transverse directions in k space with m = (X = As, Sb, and Bi). As shown by the theoretical calculations of || * * 0.55m , m = 0.21m , and m = 0.28m , although when Sun et al., the unique CBM at the CB point in Mg Sb might e ⊥,1 e ⊥,2 e 1 3 2 averaging over the six equivalent carrier pockets the overall possibly be explained by the bonding states between Mg1 and average effective mass tensor follows the crystal symmetry and Mg2 atoms. However, the very large interatomic distance * * shows a nearly isotropic feature with m = m = (0.21 × 4 + between Mg1 and Mg2 (~3.7 Å) indicates that the orbital kx ky * 113 0.55 × 2)/6 = 0.32m and m = 0.28m . Regarding the Fermi interaction between Mg1 and Mg2, if any, should be very weak. e kz e Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences npj Computational Materials (2019) 76 Energy (eV) J. Zhang et al. ab 20 m * m * DFT, n-type Mg Sb N *K * 20 c d 3 2 2.4 n-type DFT, p-type Mg Sb 3 2 p-type 2.0 1.6 1.2 0.8 0.4 0.0 0 0 0.1 1 10 0.1 1 20 -3 20 -3 n (10 cm ) n (10 cm ) cd n-type 400 DFT, Mg Sb 3 2 DFT, Mg SbBi SPB, 1.05me p-type DFT, Mg3Sb2 -1 0 0 20 40 60 80 100 120 140 160 10 10 2 -1 -1 20 -3 μ (cm V s ) n (10 cm ) Expt., n-type Mg3+xSb1.5Bi0.5 Te-doped, Zhang et al. Te-doped, Kanno et al. Te-doped, Mao et al. Se-doped, Zhang et al. S-doped, Zhang et al.]:1123/1073K-pressed [ Mn,Te-codoped,Chen et al.]:1073K-pressed Te-doped, Tamaki et al. Te-doped, Imasato et al. Te-doped, Shuai et al.] :873/923K-pressed Mn,Te-codoped, Chen et al. Fe/Co/Hf/Ta,Te-codoped, Mao et al. Nb,Te-codoped, Shuai et al. La-doped, Imasato et al.]:873/923K-pressed Expt., p-type doping in Mg Sb 3 2 Fig. 6 a Conductivity effective mass m *, density of states (DOS) effective mass m *, and Fermi surface complexity factor N *K* estimated from c d v 115 113 BoltzTraP as a function of Hall carrier concentration (n ) in p-type and n-type Mg Sb . n is estimated by 1/eR , where R is Hall H 3 2 H H H coefficient. b Theoretical power factor at 300 K from BoltzTraP versus n in Mg Sb . c Experimental power factor versus Hall mobility at room H 3 2 temperature for reported p-type and n-type Mg Sb -based materials. d Seebeck coefficient values (|α|) as a function of n at 300 K. The black 3 2 H and red solid lines correspond to the prediction of p-type and n-type Mg Sb from BoltzTraP, taken from ref. . The orange dash-dotted line 3 2 shows the prediction by a single parabolic band (SPB) model with a DOS effective mass equal to that of the CB band at the CBM of Mg Sb . 1 3 2 The red dashed line represents the theoretical prediction of n-type Mg SbBi taken from ref. .In c, d, experimental data of p-doped Mg Sb 3 3 2 41,45–47,49,50 54–60,63,64,66,67,71 and n-doped Mg Sb Bi are taken from refs. and refs. , respectively 3+x 1.5 0.5 Moreover, the calculations were only conducted for Mg Sb which makes a correction to the simple schematic plot by Imasato 3 2 without a systematic comparison with many other ternary et al. The Mg Bi alloying results in a moderate increase in the 3 2 compounds. The underlying origin of the unique six-fold CB energy separation between conduction band minima K and CB , 1 1 band minimum in Mg X still requires further investigation. In making the contribution of the secondary band minimum K to 3 2 Mg X , the band gaps decrease from Mg As (1.6 eV) to Mg Sb electronic transport insignificant. The conduction band minimum Г 3 2 3 2 3 2 (0.6 eV) to Mg Bi (semimetal). It is found that doping or (N = 1) shows a noticeable downward shift with the Mg Bi 3 2 v 3 2 substituting on the anion sites will not destroy the favorable alloying. However, the band minimum Г is too far from the band conduction band minimum CB since the conduction bands are minimum CB to play a significant role in n-type electronic transport 1 1 54,57 mainly contributed by the electronic states of Mg (Fig. 5a). before the band-gap closure (x ≈ 1.7). As the composition of Mg Bi 3 2 Therefore, n-doped Mg Sb ,Mg As , as well as the solid solutions increases in Mg Sb Bi solid solutions, the band gap is linearly 3 2 3 2 3 2−x x Mg Sb Bi ,Mg Sb As , and Mg As Bi with suitable band reduced from ~0.6 eV (x= 0) to ~0.24 eV (x= 1) to zero (semimetal, 3 2−x x 3 2−x x 3 2−x x gaps are predicted to show promising electrical transport x ≈ 1.7), indicating an increasing detrimental bipolar contribution. In performance if they are properly doped on the anion sites. addition, the narrowing of the band gap, accompanied by the Experimentally, we are especially interested in n-type Mg Sb Bi increasing near-edge band widths, results in the decreasing band 3 2−x x 62,111 with the energy gaps E ≤ ~0.6 eV within ~10k T (T= 300–725 K), effective masses. This is confirmed in several reports as well as g B 119 113 which is generally the suitable value for a good TE material. DFT the theoretical result by BoltzTraP (Fig. 7b), which shows a clear 113 * calculations with the TB-mBJ potential reveal the accurate decreasing trend in m of near-edge conduction bands from conduction band alignments of Mg Sb Bi alloys (see Fig. 7a), Mg Sb to Mg SbBi to Mg Bi . 3 2−x x 3 2 3 3 2 npj Computational Materials (2019) 76 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences 2 -1 -2 m*(m ) ασ (μWcm K ) N *K * -1 |α| (μVK ) 2 14 -1 -2 -1 ασ/τ (10 μWcm K s ) J. Zhang et al. ab SOC noSOC N =1 Mg3Sb2 0.35 1.6 Mg3SbBi Mg3Bi2 0.30 1.2 N =2 v 0.25 0.8 N =6 CB 0.4 0.20 0.0 0.15 -0.4 0.10 0.0 0.5 1.0 1.5 2.0 0.1 1 10 20 -3 n (10 cm ) x in Mg Sb Bi 3 2-x x Fig. 7 a The conduction band alignments in Mg Sb Bi (x = 0, 1, and 2) solid solutions by the TB-mBJ method with SOC. It should be 3 2−x x noted that the three conduction band minima are located at different k points (i.e., CB , K, and Г). The band alignment is calculated by aligning the Mg-1s core levels of different compounds. The valence band maximum of Mg Sb at the Г point is set to 0 eV. b Conductivity 3 2 effective mass versus Hall carrier concentration simulated from BoltzTraP for n-type Mg Sb Bi . SOC and noSOC denote the cases with 3 2−x x and without SOC, respectively. It is clear that SOC has a negligible effect on near-edge conduction bands in Mg Sb Bi because the CBM is 3 2−x x dominated by electronic states of the light element Mg (see Fig. 5a). Figure is adapted from ref. 58 66 Although the Mg Bi alloying lowers the Seebeck coefficient Mao et al., and Chen et al. Following the earlier work of Zhang 3 2 due to the lighter conduction band mass and the weaker et al., the other approach is to increase the pressing temperature contribution from the secondary band minimum K, it is conducive to 1123 or 1073 K, which was also confirmed in the studies by 64 59 to increasing the carrier mobility and weighted mobility as the Kanno et al. and Mao et al. Combining the above two conductivity effective mass m is decreased (see Fig. 7). An c approaches, Chen et al. was able to further improve the room- appropriate amount of the Mg Bi alloying (x ≤ 1) would be ideal 3 2 temperature zT and average zT in n-type porosity-mediated for n-doped Mg Sb Bi to have an enhanced weighted mobility 3 2−x x Mg Mn Sb Bi Te with an SPS pressing temperature 3.225 0.025 1.5 0.49 0.01 for superior TE performance without a noticeable bipolar effect at of 1073 K. Regarding the underlying mechanism, Mao et al. elevated temperatures. This is consistent with the widely reported suggested that the increasing mobility might be due to reduction compositions of n-doped Mg Sb Bi (x = 0, 0.5, and 1) with 3 2−x x of the number of Mg vacancies, whereas Kuo et al. proposed an excellent high-temperature TE performance. An increasing bipolar illuminating two-phase model that explains the experimental effect with increasing Bi content in Mg Sb Bi is expected to 3 2−x x observation by the larger grain sizes reducing the grain boundary shift the peak zT to lower temperatures, but an improved zT at low electrical resistance under a higher pressing temperature. In temperatures may be achieved owing to the lighter conductivity addition, enhanced low-temperature TE performance was shown effective mass. Thus more efforts should be made to investigate n- in n-type Mg SbBi Te and Te-doped Mg Sb Bi by 3.2 0.99 0.01 3 0.6 1.4 type Mg Sb Bi with a wider range of compositions (x = 0–1.7). 62,70 3 2−x x Imasato et al. and Mg Sb Bi Te (y = 1.1–1.5) by Shu 3.02 y 1.99−y 0.01 It will also be interesting to design a functionally graded n-doped 68 et al. with increasing Bi content, which can be attributed to the Mg Sb Bi (x = 0–1.7) system for TE applications. Below we give 3 2−x x lighter conductivity effective mass. Regarding n-type dopants a brief overview of the recent experimental reports on n-doped 57 60 other than the Te element, Se and S on the anion site as well Mg Sb Bi . 67 69 3 2−x x as La and Y on the cation site have also been experimentally 53 53 As shown originally by Pedersen in 2012 (see the link in ref. explored in Mg Sb Bi . With continuous experimental efforts, 55 54 3+x 1.5 0.5 for details) and later reported by Tamaki et al. and Zhang et al. , a low-temperature zT of ~0.9 at 323 K and a peak zT of ~1.85 at an exceptionally high n-type TE performance can be achieved in 723 K have been achieved so far in n-type (Mn, Te)-codoped Mg Sb Bi through Te doping on the anion site with or without 3 1.5 0.5 66,71 54 Mg Sb Bi . In addition to the intensive developments in TE 3 1.5 0.5 excess Mg. Zhang et al. reported n-type Mg Sb Bi Te 3 1.5–0.5x 0.5–0.5x x performance, the thermal stability of n-type Te-doped with varying Te content and no excess Mg synthesized by Mg Sb Bi was investigated by Jørgensen et al. using 3 1.5 0.5 combining arc melting and spark plasma sintering (SPS) at 1123 K synchrotron powder X-ray diffraction and X-ray total scattering, based on the reproduction and improvement of the original work where a clear evolution of bismuth as a secondary phase was by Pedersen. The samples with an optimal zT of 0.56–1.65 at observed in the powdered sample during thermal cycling. 300–725 K reported by Zhang et al. show the intrinsic 54,60,61,63,65,121,122 Several experimental and theoretical efforts temperature-dependent mobility with the dominant acoustic reveal that a large amount of excess Mg is not needed for phonon scattering at low temperatures. Tamaki et al. reported achieving n-type properties as claimed by Tamaki et al. . A tiny n-type Mg Sb Bi Te with a large nominal excess Mg 3.2 1.5 0.49 0.01 amount of excess Mg or even no excess Mg is required to realize prepared using ball milling followed by SPS pressing at 873 K, n-type properties as long as great care is taken to avoid the Mg which exhibits low carrier mobility and TE performance below loss during the synthesis, consistent with the earlier report 500 K due to impurity scattering. Later, two approaches were without excess Mg by Zhang et al. The samples with smaller applied to improve the mobility and power factor at low amounts of excess or no excess Mg even show better temperatures (Fig. 6c) by tuning the carrier scattering mechanism performance and stability than those with large amounts of from the ionized impurity scattering to the acoustic-phonon- 63,122 excess Mg. Various reports with different amounts of excess dominated mixed scattering. One approach is to dope Mg Sb Bi with the transition metals (i.e., Nb, Fe, Co, Hf, Ta, Mg may arise from the difficulty in preparing these materials 3.2 1.5 0.5 and Mn) on the Mg sites and the Te dopant on the anion sites without the Mg loss due to the high reactivity, easy oxidation, and using a pressing temperature of 927 K as reported by Shuai et al., high vapor pressure of Mg. Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences npj Computational Materials (2019) 76 Energy (eV) m * (m ) c e Te doping limit CBM CBM Te doping limit J. Zhang et al. DEFECT-CONTROLLED CARRIER TRANSPORT, electronegativity might be related to the relative energy cost of ELECTRONEGATIVITY, AND BONDING CHARACTER the electron transfer influenced by the bond strength between the cation A and the anion, which in turn affects the vacancy Nearly all CaAl Si -type compounds, except CaAl Si and 2 2 2 2 formation energies. In general, the formation of the vacancy is SrAl Si , have been reported as intrinsically p-type, which is 2 2 easier when the bonding is weak. However, in order to explain the attributed to the intrinsic point defects pinning the chemical deduction, accurate calculations should be conducted to char- potential close to the valence bands. According to the intrinsic 80,125 acterize the bond strength between A and Sb. Though the defect calculations, the cation vacancy at the A site is the underlying mechanism regarding the bond strength is compli- most energetically stable point defect in all AZn Sb (A = Ca, Sr, 2 2 cated, the results provide insight on tuning carrier density by Eu, and Yb) compounds, which leads to the persistent p-type alloying on the A site with elements of varying electronegativities. behavior in these compounds. Pure Mg Sb is intrinsically p-type, 3 2 15,19,23,32 This idea is well confirmed in many significant reports on and many attempts at doping or substituting with a series of 38–50,52,102,126–130 optimizing p-type AB X TEs, where excellent zTs were achieved 2 2 elements result in p-type behaviors as well. This is owing to the carrier density optimization and thermal conductivity because of the low formation energies of the negatively charged reduction via alloying at the A site. 2 2 Mg vacancies (V and V ) (Fig. 8), which prevent the chemical Mg1 Mg2 Defect-controlled carrier transport with the insight from the potential from moving close to the conduction bands for the Sb- 55,61 electronegativity difference was explored in n-type rich condition. The n-type doping in Mg Sb -based materials 3 2 Mg Sb Bi with the chalcogens Q (S, Se, and Te) as electron 51,53– 3 1.5 0.5 is challenging and only recently discovered to be successful. dopants. Experimentally, it is found that both the maximum 55 55,61,121 Several defect calculations were conducted to explain attainable carrier concentration and mobility increase with the n-type behavior in Mg Sb under the Mg-excess condition. 3 2 decreasing electronegativity difference χ  χ between the Q Mg The results vary from different methods applied in defect energy chalcogen dopants Q and Mg (Fig. 9c). Using DFT calculations, calculations. Ohno et al. concluded from calculations using the 60 Zhang et al. revealed that the improving carrier concentration is PBE functional with finite-size and band-gap corrections that caused by the increasing doping limit induced by the lower under the Mg-excess condition the Mg vacancy suppression rather extrinsic defect-formation energy (Fig. 9d), which may be than the Mg interstitial is responsible for n-type properties explained by the stronger bonding between the dopant and the (Fig. 8), while Chong et al. based on more accurate calculations matrix with decreasing χ  χ . Moreover, the enhanced 131 Q Mg with the HSE06 functional revealed that the role of the excess mobility is attributed to the smaller effective mass of conduction Mg for achieving n-type behavior is to compensate the electronic bands originating from the enhanced bond covalency with 1− 134,135 charge of the defect complex (V + Mg ) . It is clear that defect Mg2 i decreasing electronegativity difference, which is supported calculations are usually sensitive to the applied theoretical by the decreasing theoretical density of states. According to the methods; however, the robust relative trend of defect formation above trends, using dopants with a small electronegativity energies is supposed to render insightful guidance for the carrier difference compared to its bonding partner was proposed as a transport. Below we introduce guidelines for the defect-controlled guiding rule for efficient n-type doping in Mg Sb -based 3 2 carrier transport with respect to the electronegativity in p-type compounds. As confirmed in extrinsic defect calculations for n- AZn Sb - and n-type Mg Sb -based materials. 2 2 3 2 type Mg Sb by Gorai et al., this guideline was found to work 3 2 In p-type AZn Sb (A = Ca, Sr, Eu, and Yb), the experimental 2 2 pretty well for the anion site doping but might not work well for carrier density increases with increasing electronegativity of A the cation site doping due to the strong charge compensation and (Fig. 9a). As revealed by Pomrehn et al. using DFT calculations, complex competing phases involved. Moreover, theoretical 136,137 the increasing carrier density can be rationalized by the calculations by Gorai et al. suggested the group-3 elements decreasing trend in the vacancy formation energy at the cation including La, Y, and Sc as effective n-type cation dopants for 67,69 A site with increasing electronegativity (Fig. 9b). The relationship Mg Sb , which was confirmed by the recent experiments of 3 2 between the vacancy defect-formation energy and La-doped and Y-doped Mg Sb Bi . 3+x 1.5 0.5 E range F E range with Te doping with Te doping Mg-excess Sb-excess ab 1.5 1.5 2+ Mg 1+ i Sb Mg2 3+ Mg 1+ Sb 1.0 1.0 Sb Mg1 2- 1+ Mg2 Sb 2- 1+ Te Mg1 0.5 Sb 0.5 2+ Mg 2- 1+ Mg2 2- Te V Sb Mg1 0 0 -0.2 -0.2 0 0.2 0.4 0.6 0 0.2 0.4 0.6 E - E (eV) E - E (eV) F VBM F VBM Fig. 8 a, b Defect formation energies of intrinsic point defects as well as the extrinsic defect Te under a Sb-excess (Δμ ¼ 0) and b Mg- Sb Sb excess (Δμ ¼ 0) conditions. Figure is adapted with permission from ref. , Elsevier Mg npj Computational Materials (2019) 76 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences Defect Formation Energy (eV) Defect Formation Energy (eV) J. Zhang et al. ab 0.6 p-type AZn Sb 2 2 Yb Sr 0.4 1.0 Eu 0.2 Ca Ca Eu 0.0 Yb Sr -0.2 VBM 0.1 -0.4 0.98 1.00 1.02 1.04 1.06 -0.2 0.0 0.2 0.4 0.6 Electronegativity E−E (eV) F V cd n-type Mg Sb Bi Te VBM VBM 1.6 1.6 3 1.5 0.5 Sb Se Se Sb Te Sb 1.2 1.2 Mg1 Mg2 0.8 0.8 0.1 0.4 0.4 0.0 0.0 Mg rich Sb rich 0.01 -0.4 0.8 1.0 1.2 1.4 0.0 0.2 0.4 0.0 0.2 0.4 |χ -χ | Fermi energy (eV) Fermi energy (eV) Q Mg 15–17,20,95 Fig. 9 Defect-controlled carrier density. a The experimental Hall carrier concentration at room temperature versus the electronegativity of the cation A (A = Sr, Eu, Ca, and Yb) in p-type AZn Sb . The Allred–Rochow scale of the electronegativity is used. b 2 2 Defect formation energy of the cation vacancy in AZn Sb . The respective conduction band minima of different compounds are shown in the 2 2 54,57,60 colored dashed lines. c Hall carrier concentration at 300 K as a function of the electronegativity difference χ  χ (Q = S, Se, and Te) Q Mg in n-type chalcogen-doped Mg Sb Bi . The Pauling scale of the electronegativity is adopted. d Defect formation energies of extrinsic 3 1.5 0.5 defects Q as well as the intrinsic defects V and V under Mg-rich and Sb-rich conditions. a, b and c, d are adapted with permissions, Sb Mg1 Mg2 80 60 respectively, from refs. and , Wiley-VCH ANHARMONIC PHONON MODES AND LOW LATTICE THERMAL The lattice thermal conductivity of Mg Sb and the two ternary 3 2 CONDUCTIVITY analogs (CaMg Sb and CaMg Bi ) was calculated by Zhang 2 2 2 2 14–69,71 et al. using first-principles Boltzmann transport calculations Reported lattice thermal conductivities κ at room tem- considering the anharmonic phonon–phonon scattering with the perature in CaAl Si -type TE compounds span a relatively wide 2 2 −1 −1 −1 −1 ShengBTE code. Importantly, the theoretical result shows a range from ~0.6 W m K in CaCd Sb to ~4.5 W m K in 2 2 noticeable decrease in κ from CaMg Sb and CaMg Bi to CaMg Sb . Among them, Mg Sb with a low density of ~4.0 g L 2 2 2 2 2 2 3 2 −3 Mg Sb , which is in good agreement with experiments (see 3 2 cm shows an intrinsically low room-temperature κ of about −1 −1 40,45,46 Fig. 10a). It is found that the phonon lifetime plays a key role in 1.0–1.5 W m K , comparable to those of the state-of-the- 82,84 1,138,139 reducing κ in Mg Sb . In general, the phonon lifetime for L 3 2 art TE materials PbTe and Bi Te . A small difference (~15%) 2 3 intrinsic crystalline materials is dominated by the three-phonon in κ is found between single crystalline and polycrystalline 40,102,130 anharmonic scattering process. As revealed in Fig. 10b, the Mg Sb samples, suggesting that the grain boundary 3 2 calculated anharmonic scattering rates of Mg Sb are considerably 3 2 scattering does not play a significant role in phonon transport. larger than those of CaMg Sb and CaMg Bi in the low- and mid- 2 2 2 2 Though alloying is often applied to further reducing lattice frequency regions (<5.5 THz). In particular, the anharmonic thermal conductivity through point defect scattering, the intrinsi- scattering rates of Mg Sb display one peak at 0.7–1.04 THz and 3 2 cally low thermal conductivity induced by the phonon–phonon two peaks in the mid-frequency region of 3.3–4.8 THz, which are scattering is indispensable to the exceptionally high zT in n-type about (or even more than) one order of magnitude larger than Mg Sb -based materials. Below we will discuss the origin of the 3 2 those of ternary compounds within the same frequency regions. intrinsically low lattice thermal conductivity in Mg Sb from first 3 2 The peak in anharmonic scattering rates of Mg Sb at 0.7–1.04 THz 3 2 principles. is typically induced by a unique feature in phonon dispersion in Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences npj Computational Materials (2019) 76 20 -3 20 -3 n (10 cm ) n (10 cm ) Defect Formation Energy (eV) Defect Formation Energy (eV) J. Zhang et al. ab Theory Expt. CaMg Sb 2 2 CaMg Bi 2 2 Mg Sb 3 2 Nano-Mg Sb 3 2 Mg Sb 3 2 CaMg Sb 2 2 CaMg Bi 2 2 200 300 400 500 600 700 800 T (K) cd 2 Mg Sb 3 2 Total CaMg Sb 2 2 Mg1 Mg2 CaMg Bi 2 2 Sb Γ MK Γ A L H A PDOS Wave vectors 79,84 Fig. 10 a The experimental and theoretical lattice thermal conductivities. The theoretical and experimental data are taken from refs. 26,36,40,43 43 and , respectively. Nano-Mg Sb denotes the nanostructured Mg Sb sample prepared by the high-energy ball milling method. b 3 2 3 2 The calculated anharmonic scattering rates of Mg Sb , CaMg Sb , and CaMg Bi . c Phonon band structure and density of states of Mg Sb . 3 2 2 2 2 2 3 2 79 84 Adapted with permission from ref. , CC-BY-4.0. The mode Grüneisen parameters are projected on the phonon band structure. Curve width indicates the relative magnitude of the mode Grüneisen parameter. The positive and negative mode Grüneisen parameters are colored in red and blue, respectively. d Mode Grüneisen parameters of Mg Sb , CaMg Sb , and CaMg Bi . For the theoretical results shown here, the 3 2 2 2 2 2 longitudinal optical/transverse optical (LO/TO) splitting is not considered since the LO/TO splitting only has a minor effect on the results. The readers are referred to ref. for the results considering the LO/TO splitting comparison with ternary compounds as pointed out earlier by modes in Mg Sb and Mg Bi may be attributed to the unstable 3 2 3 2 Peng et al., i.e., much softer transverse acoustic phonon modes Mg1 in the octahedral sites as judged by the small cation-to-anion at the M (~0.71 THz), L (~0.81 THz), and A (~1.04 THz) points ionic radius ratio below the stability limit (0.414), which results characterized by very large mode Grüneisen parameters (Fig. 10c, from the small ionic radius of Mg (Fig. 11b). Although the insight d). In particular, the soft transverse acoustic phonon mode at the L proposed by Peng et al. is illuminating, it cannot explain why point exhibits the largest negative mode Grüneisen parameter. A Mg As , which also exhibits a cation-to-anion ionic radius ratio 3 2 similar behavior was also found in Mg Bi , where the phonon below the stability limit, does not show soft transverse acoustic 3 2 calculations often show negative (imaginary) frequencies for phonon modes. Clearly, the ionic radius ratio rule is somewhat transverse acoustic branches at the L point. In addition, the peak oversimplified by assuming purely ionic bonding with a complete anharmonic scattering rates of Mg Sb at 3.3–4.8 THz in Mg Sb charge transfer. In reality, chemical bonds in Mg X show certain 3 2 3 2 3 2 are induced by the anharmonic optical phonon modes within this degrees of covalency with incomplete charge transfers (see frequency region with much larger mode Grüneisen parameters Table 1), which will result in a larger effective ionic radius of than those of CaMg Sb and CaMg Bi (Fig. 10d). Therefore, the Mg1 and probably push the cation-to-anion ionic radius ratio 2 2 2 2 soft transverse acoustic phonon modes within 0.7–1.04 THz as well above the stability limit. as the anharmonic optical modes within 3.3–4.8 THz contribute to The instability of Mg1 is well confirmed in the relatively larger the large peak values in the anharmonic scattering rates and atomic displacement parameter and flatter potential energy thereby intrinsically low lattice thermal conductivity in Mg Sb . landscape of Mg1 compared to those of Mg2 in Mg Sb (Fig. 11c, 3 2 3 2 The unique soft transverse phonon modes are related to the d), which might be attributed to the relatively weaker Mg1-Sb structural instability in Mg Sb and Mg Bi , which is well bond as well as the larger void space of the octahedral site of Mg1. 3 2 3 2 141,142 confirmed in their less negative formation enthalpies However, the unstable Mg1 at the octahedral site better explains compared with other CaAl Si -type compounds (Fig. 11a). As the anharmonic optical modes at 3.3–4.8 THz rather than the soft 2 2 claimed by Peng et al., the soft transverse acoustic phonon transverse acoustic modes since the anharmonic optical modes npj Computational Materials (2019) 76 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences -1 -1 κ (W m K ) Frequency (THz) L J. Zhang et al. ab AB X 2 2 Mg Bi Ba 3 2 -0.2 X=P 0.7 X=As Mg Sb X=Sb 3 2 0.6 Eu -0.4 X=Bi Sr Ca 0.5 Yb -0.6 0.4 Stability limit Mg -0.8 0.3 -1.0 0.2 0.70.8 0.91.0 1.1 0.6 0.8 1.0 1.2 1.4 Electronegativity difference ΔEN r (Å) cation 0.12 cd a c Mg1 0.06 Mg1 Mg2 Mg2 Sb Sb 0.05 0.08 0.04 0.03 0.04 0.02 0.01 0.00 0.00 200 400 600 800 -0.2 -0.1 0.0 0.1 0.2 T (K) Displacement (Å) Fig. 11 a Formation enthalpy versus the electronegativity difference ΔEN in AB X (CaAl Si -type) compounds. ΔEN is calculated by the 2 2 2 2 difference between the electronegativity of the anion X and the average electronegativity of the atoms A and B. Formation enthalpy values 141,142 are taken from MaterialsProject.org. b The cation-to-anion ionic radius ratio r /r versus the ionic radius of the cation A at the cation anion octahedral site in AMg X . The minimum stability limit of r /r for the octahedral site is 0.414. The data points for Mg Sb and Mg Bi 2 2 cation anion 3 2 3 2 are colored in pink. Adapted with permissions from ref. , Elsevier. c Experimental isotropic atomic displacement parameters of Mg Sb . d The 3 2 potential energy curves for the nonequivalent atoms in Mg Sb displacing along the a and c directions. c, d are reproduced with permissions 3 2 from ref. , CC-BY-4.0 are mainly contributed by the motion of Mg1 while the acoustic since it is sensitive to the factors such as the sample quality, crystal phonon modes are dominated by the vibration of Sb (Fig. 10b, c). alignment, and preferred orientations. Therefore, here we will It is more likely that the unstable or weak chemical bonds lead to focus on the anisotropy of electrical and thermal transport based the soft transverse acoustic phonon modes in Mg Sb and Mg Bi . on the perfect crystal structure from the theoretical point of view. 3 2 3 2 The soft transverse acoustic phonon modes in Mg Sb and In general, only electronic states near the Fermi level contribute to 3 2 Mg Bi involve the interlayer shearing motion, which results in the electronic transport. Under the constant and isotropic carrier 3 2 82,97 their reported soft shear moduli. However, whether the soft scattering time assumption, the anisotropy of electrical conduc- shear modes are mainly induced by the weak interlayer Mg1-Sb tivity is completely determined by the anisotropy in carrier (or Mg1-Bi) bond requires further confirmation. Quantitative mobility and thereby the anisotropy in the inverse effective mass chemical bonding analysis based on the electron density is tensor, which can be intuitively understood from the anisotropy of reliable in comparing different chemical bonds in the same the atomic orbitals interactions. For p-type transport in Mg Sb , 3 2 compound, but it cannot be used for reliably comparing the the electronic states at the VBM are mainly comprised of the p relative strength of chemical bonds in different compounds. orbitals of the Sb atoms. The anisotropic p orbitals show much Therefore, other computational methods should be applied to stronger orbital overlaps along the c axis, which results in much quantitatively characterize and compare the relative strength of larger band dispersion, smaller effective mass, and larger electrical the interlayer A-X bonds in different compounds. Furthermore, conductivity in comparison to those along the a axis. This is other factors such as the difference in atomic masses should also confirmed by the calculated strong anisotropy in the electrical be considered. Overall, more comprehensive work should be conductivity for p-type Mg Sb with σ /σ being much smaller 3 2 xx zz conducted in the future to reveal the origin of these unique soft than unity (see Fig. 12a), consistent with the experimental Sb and Mg Bi . reports. In contrast, for n-type transport in Mg Sb , the transverse modes in Mg 3 2 3 2 3 2 electronic states at the CBM are dominated by spherical s orbitals 55,57 of the Mg atoms, which generally result in nearly isotropic ELECTRONIC AND THERMAL TRANSPORT ANISOTROPY features in orbital interactions, average effective mass tensor, and Experimentally, the anisotropy of the transport properties in bulk thereby electrical conductivity (Fig. 12a). For the Seebeck 48,102,130,143 TE materials can vary a lot between different reports coefficient, both p-type and n-type Mg Sb show a nearly isotropic 3 2 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences npj Computational Materials (2019) 76 U (Å ) Formation energy (eV per atom) iso Potential Energy (eV) r /r cation anion J. Zhang et al. 2.0 ab p-type n-type SnS σ /σ 1.8 MoS TiS xx zz α /α 1.6 xx zz MoSe 1.4 1.2 1.0 CaZn Sb 2 2 0.8 SrZn Sb 0.6 2 2 CaMg Sb 2 2 0.4 Mg Sb 3 2 0.2 CaMg Bi 2 2 0.0 0.01 0.1 1 ρ /ρ 20 -3 intra inter n (10 cm ) Anisotropy of the chemical bonding network Fig. 12 a Anisotropy of the calculated electronic transport properties including Seebeck coefficient (α /α ) and electrical conductivity (σ / xx zz xx σ ) as a function of Hall carrier concentration in p-type and n-type Mg Sb . The data are calculated in this work using the previous zz 3 2 computational methods . b Anisotropy of the theoretical lattice thermal conductivity κ /κ as a function of the anisotropy of the chemical a c bonding network characterized by ρ =ρ . b is reproduced with permissions from ref. , CC-BY-4.0 intra inter feature even though the effective mass tensor is anisotropic development. Below we present some challenges and possible (Fig. 12a). This may be attributed to the notion, which was future directions in both theoretical and experimental aspects. Theoretically, critical challenges include the difficulties for current proposed by Parker et al., that the anisotropic effective mass computations to describe the carrier scattering time, doping, results in isotropic Seebeck coefficient as long as the electronic temperature effects, defects, and disorder in experiments. One band is assumed parabolic, the carrier scattering time is assumed typical example of these challenges can be seen in electrical only dependent on energy, and the bipolar effect is insignificant. transport calculations under the rigid band and constant carrier’s Moreover, as investigated by Sun et al., the Seebeck coefficients scattering time approximation, which assumes electronic bands and of several ternary compounds AMg X and AZn Sb (A = Ca, Sr, 2 2 2 2 scattering time being insensitive to temperature and doping. In Ba; X = Sb, Bi) also show isotropic features when there is no reality, doping on the cation sites and varying carrier scattering bipolar effect, whereas the n-type electrical conductivities in these mechanisms result in the deviation from theory and the scatter of compounds show anisotropic features due to the anisotropic experimental data in the Pisarenko plot of the Seebeck coefficient average effective mass tensors induced by the anisotropic versus carrier density (see Fig. 6d). Another example can be found in electron pockets, which may be understood from the anisotropic the situation where the high carrier mobility cannot be understood orbital interactions induced by the increased contributions of from the band structure. In several compounds containing rare earth anisotropic p and d orbitals to the CBM. elements, the carrier’s scattering time is responsible for the In general, the thermal transport anisotropy is related to the 79,145 intrinsically high carrier mobility, whereas the detailed mechanism anisotropy of the chemical bonding network. In Mg Sb and 3 2 requires further investigation. In n-type Mg Sb Bi ,changing 3 2−x x related structures, the anisotropy of the chemical bonding carrier scattering behavior can improve carrier mobility by doping network can be quantified by the intralayer-to-interlayer bond- with transition metals on the Mg sites or increasing the pressing strength ratio ρ =ρ based on electron density. A nearly intra inter temperature, but the underlying mechanism is still under debate. linear correlation between the anisotropy ratio κ /κ of lattice a c In addition, the insights into the origin of valence and thermal conductivity and ρ =ρ indicates that ρ =ρ can intra inter intra inter conduction band alignments, how thermal expansion affects be adopted as an indicator measuring the anisotropy of lattice band alignments, and the origin of carrier pocket anisotropy thermal conductivity in Mg Sb -related materials (see Fig. 12b). 3 2 remain unclear. The underlying mechanism on why the favorable For AMg X compounds, the nearly isotropic 3D bonding network 2 2 six-fold CBM at the low-symmetry CB point only exists in binary with ρ =ρ ≈ 1 results in the nearly isotropic features in intra inter Mg X requires further investigation. For defect calculations, more 3 2 phonon dispersion, group velocity, Grüneisen parameter, and potential n-type dopants need to be predicted for the experi- ultimately lattice thermal conductivity. However, the lattice mental validations. Regarding thermal transport, further efforts are thermal conductivity in AZn Sb is relatively anisotropic due to the 2 2 required to elucidate the origin of the soft transverse acoustic anisotropic chemical bonding network with ρ =ρ > 2. phonon modes in Mg Sb and Mg Bi that remains obscurely intra inter 3 2 3 2 understood to date. Whether the soft transverse acoustic modes are induced by the weak interlayer interaction should be carefully SUMMARY AND OUTLOOK examined. The critical challenge here for computations is how to In this review, we have explored many illuminating insights such quantitatively and reliably compare the strength of interlayer as orbital overlap, orbital degeneracy, orbital splitting energy, interactions in different compounds. In addition to this possible valley degeneracy, effective mass, carrier pocket anisotropy, Fermi notion, the impact of other factors such as the atomic mass surface complexity, point defects, electronegativity, and bond difference on phonon transport should be examined. covalency for understanding electronic and thermal transport of Experimentally, the reported compositions are mainly limited to Mg Sb and related CaAl Si -type TEs. Although current insights Sb- and Bi-based compounds, while As- and P-based compounds 3 2 2 2 for the materials design have been very helpful to date, many remain largely unexplored. According to the solid solution challenges need to be addressed in order to achieve further compound map, there are still many unexplored alloying npj Computational Materials (2019) 76 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences Electronic transport anisotropy Anisotropy of lattice thermal conductivity κ /κ a c J. 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Electronic systems such as Mg Sb As and Mg Bi As for n-type doping 3 2−x x 3 2−x x structure and transport in thermoelectric compounds AZn Sb (A=Sr, Ca, Yb, 2 2 need to be explored in the future. Despite the outstanding TE Eu). Dalton Trans. 39, 1046–1054 (2010). performance shown in n-type Mg Sb -based materials, the long- 3 2 21. Zhang, H. et al. Thermoelectric properties of Eu(Zn Cd ) Sb . Dalton Trans. 39, 1-x x 2 2 term thermal stability under a temperature gradient and an 1101–1104 (2010). electrical current should be addressed as an essential step toward 22. Zhang, H. et al. Synthesis and properties of CaCd Sb and EuCd Sb . Inter- 2 2 2 2 metallics 18, 193–198 (2010). the practical applications. 23. Zhang, H. et al. Thermoelectric properties of Yb Eu Cd Sb . J. Chem. Phys. 133, x 1−x 2 2 194701 (2010). 24. Zhang, H. et al. Thermoelectric properties of polycrystalline SrZn Sb prepared 2 2 ACKNOWLEDGEMENTS by spark plasma sintering. J. Electron. Mater. 39, 1772–1776 (2010). We thank K.F.F. Fischer for discussions and comments. This work was supported by 25. Guo, K. et al. Enhanced thermoelectric figure of merit of Zintl phase YbCd 2- the Danish National Research Foundation (Center for Materials Crystallography, Mn Sb by chemical substitution. Eur. J. Inorg. Chem. 2011,4043–4048 x x 2 DNRF93) and the Danish Center for Scientific Computing. The numerical results (2011). presented in this work were obtained at the Center for Scientific Computing, Aarhus. 26. May, A. F. et al. Thermoelectric transport properties of CaMg Bi , EuMg Bi , and 2 2 2 2 Affiliation with the Center for Integrated Materials Research (iMAT) at Aarhus YbMg Bi . Phys. Rev. B 85, 035202 (2012). 2 2 University is gratefully acknowledged. 27. Zevalkink, A. et al. Nonstoichiometry in the Zintl phase Yb Zn Sb as a route to 1-δ 2 2 thermoelectric optimization. Chem. Mater. 26, 5710–5717 (2014). 28. Min, W., Guo, K., Wang, J. & Zhao, J. Effect of manganese doping on the ther- AUTHOR CONTRIBUTIONS moelectric properties of Zintl phase EuCd Sb . J. Rare Earth. 33, 1093–1097 2 2 J.Z. wrote the manuscript with the inputs and suggestions from B.B.I. L.S. collected (2015). the reported experimental data and edited the manuscript. 29. Shuai, J. et al. Higher thermoelectric performance of Zintl phases (Eu Yb ) 0.5 0.5 1- Ca Mg Bi by band engineering and strain fluctuation. Proc. Natl Acad. Sci. USA x x 2 2 113, E4125–E4132 (2016). ADDITIONAL INFORMATION 30. Shuai, J. et al. Thermoelectric properties of Zintl compound Ca Na Mg Bi . 1-x x 2 1.98 Appl. Phys. Lett. 108, 183901 (2016). Competing interests: The authors declare no competing interests. 31. Shuai, J. et al. Thermoelectric properties of Bi-based Zintl compounds Ca 1- Yb Mg Bi . J. Mater. Chem. A 4, 4312–4320 (2016). x x 2 2 Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims 32. Wubieneh, T. 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