Article pubs.acs.org/cm
Cite This: Chem. Mater. 2018, 30, 1540−1546
Inverse Band Structure Design via Materials Database Screening: Application to Square Planar Thermoelectrics Eric B. Isaacs and Chris Wolverton* Department of Materials Science and Engineering, Northwestern University, Evanston, Illinois 60208, United States S Supporting Information *
ABSTRACT: Electronic band structure contains a wealth of information on the electronic properties of a solid and is routinely computed. However, the more difficult problem of designing a solid with a desired band structure is an outstanding challenge. In order to address this inverse band structure design problem, we devise an approach using materials database screening with materials attributes based on the constituent elements, nominal electron count, crystal structure, and thermodynamics. Our strategy is tested in the context of thermoelectric materials, for which a targeted band structure containing both flat and dispersive components with respect to crystal momentum is highly desirable. We screen for thermodynamically stable or metastable compounds containing d8 transition metals coordinated by anions in a square planar geometry in order to mimic the properties of recently identified oxide thermoelectrics with such a band structure. In doing so, we identify 157 compounds out of a total of more than half a million candidates. After further screening based on electronic band gap and structural anisotropy, we explicitly compute the band structures for the several of the candidates in order to validate the approach. We successfully find two new oxide systems that achieve the targeted band structure. Electronic transport calculations on these two compounds, Ba2PdO3 and La4PdO7, confirm promising thermoelectric power factor behavior for the compounds. This methodology is easily adapted to other targeted band structures and should be widely applicable to a variety of design problems.
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INTRODUCTION
In this work, we take a different approach to the inverse band structure design problem via a materials database screening based on several materials attributes. Whereas the search space in the approach of Franceschetti and Zunger is all the possible substitutional decorations of a given structure type for a particular chemistry, we search over different stoichiometries, chemistries, and a wide variety of known crystal structures. Our chemical and structural space is the more than half a million inorganic crystalline solids in the Open Quantum Materials Database (OQMD),7,8 a database of electronic structure calculations. In addition to ∼38 0009 of the known crystalline solids from the Inorganic Crystal Structure Database (ICSD),10,11 the OQMD currently contains electronic structure calculations of ∼493 000 hypothetical crystalline compounds based on decorating known binary and ternary prototype crystal structures.12 Our approach takes advantage of this existing, extensive collection of materials data. Though this database contains the electronic structures of all compounds, the band structures along high-symmetry directions in k-space are not computed or stored in the database. In other words, we cannot simply “search” the database directly for the targeted band structure.
Electronic band structure, the energy ϵ as a function of crystal momentum k for each electron band, encapsulates a wealth of fundamental information on the electronic properties of a crystalline material. Simple examples include the nature and magnitude of the fundamental band gap and the carrier effective masses (for a semiconductor) and the Fermi surface (for a metal). Band structure impacts virtually all aspects of material properties, including electronic, thermal, optical, magnetic, and mechanical behavior. It has now become a routine task to compute the band structure for a given crystal structure, in particular using Kohn− Sham density functional theory (DFT).1,2 However, the inverse problem of finding or designing a compound with a targeted band structure is a much more difficult challenge. Franceschetti and Zunger devised an approach to this inverse problem relying on the rapid, approximate evaluation of the forward problem.3 In this work, an optimization algorithm (simulated annealing) was used to achieve a targeted band structure property (maximum band gap) within the configurational space of a specific superlattice or alloy system. Variants of this approach involve other optimization algorithms (e.g., genetic algorithms, particle swarm optimization), other targeted properties (e.g., minimum band gap, direct band gap, large optical transition oscillator strength), and expanded structural spaces.4−6 © 2018 American Chemical Society
Received: October 26, 2017 Revised: January 4, 2018 Published: February 26, 2018 1540
DOI: 10.1021/acs.chemmater.7b04496 Chem. Mater. 2018, 30, 1540−1546
Chemistry of Materials
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Article
COMPUTATIONAL DETAILS Using a Boltzmann transport approach, Kuroki and Arita showed that a band with both flat (small ∇kϵ) and dispersive (large ∇kϵ) parts along a direction in k-space can lead to high power factor if the electronic chemical potential μ lies at an energy separating the two.17 For this special “pudding-mold” band structure, the band velocity difference across μ (proportional to S) is large, and the high band velocity of the dispersive part and large Fermi surface enable a large σ. In addition to leading to the exceptionally high power factor of NaxCoO2,17 the pudding-mold band structure has been suggested to contribute to the record-breaking thermoelectric performance of SnSe18−20 and to the large computed power factors of recently proposed Fe-based Heusler compounds.21 Although the original pudding-mold band structure concept corresponds to regions of small and large ∇kϵ along the same direction in k-space, one can consider a broader definition in which the flat and dispersive regions can occur along in different directions in k-space. Such a band structure naturally emerges from a low-dimensional (e.g., one- or two-dimensional) crystal structure, which can be thought of as analogous to the low-dimensional nanostructures proposed by Hicks and Dresselhaus to exhibit high thermoelectric performance.22,23 Usui and Kuroki showed that one-dimensional crystal structures in particular can lead to a large power factor due to the enhanced density of states at the band edge.24 A previous computational study proposed Bi2PdO4 as a possibly promising thermoelectric oxide due in part to large power factor.13 A key component to the promising electrical properties was the pudding-mold band structure, which relates to the square planar coordination (illustrated in Figure 1a) of d8 Pd by oxygen. As shown in Figure 1b, a d8 (as well as d4 or d6, in principle) electronic configuration of the metallic element in this coordination can lead to a semiconducting compound according to crystal field theory,25 as is the case for Bi2PdO4. In this particular compound, the stacked arrangement of square planar PdO4 units with d3z2−r2 highest-occupied orbitals (whose z axis is aligned with the stacking direction) leads to a valence band that is dispersive in the stacking direction and flat in the other directions (a quasi-one-dimensional crystal structure). Although it has a distinct structure, PbPdO2 also has a d8 Pd square planar coordination, pudding-mold-like band structure, and promising thermoelectric properties,15,16 which is suggestive of a connection between square planar coordination and pudding-mold band structure. We note that Co-doped PbPdO2 has also been studied as a possible spin-gapless semiconductor.26−28 Therefore, we choose our materials attributes to mimic those of Bi2PdO4 and PbPdO2 and screen the OQMD based on the following criteria: • Chemistry − Compound must contain TM and anion elements, and (for practicality) no radioactive elements. • Structure − In the compound, TMs must be coordinated by anions in square planar coordination (4 nearest neighbors and bond angles 4 × 85−95°, 2 × ≥175°).29 • Electron count − All TM in the compound must have d4, d6, or d8 configuration. As shown in Figure 1b, for a TM in square planar coordination, such nominal electron counts can lead to a semiconductor based on crystal field theory. • Thermodynamics − Compound must be thermodynamically stable, metastable (within 25 meV/atom of the
The materials attributes that we use in our design/screening strategy include (1) the elements contained in the material, (2) nominal valence electron count for each element, (3) the local coordination geometry of atoms in the crystal structure, and (4) thermodynamic stability. For example, here we focus (for reasons described below) on (1) compounds containing transition metals and anions, (2) transition metals in a d8 electronic configuration, (3) transition metals coordinated by anions in a square planar arrangement, and (4) compounds that are thermodynamically stable or metastable. We design the targeted band structure by querying for any crystal that simultaneously possesses all the specified materials attributes. In our approach, full computation of the band structure (the forward problem) is only necessary for a small number of candidate compounds as a validation. This is highly advantageous since band structures require additional DFT calculations and a quantitative figure of merit for a candidate band structure is not always easily computed. We test our design strategy in the context of thermoelectric materials, which can enable waste heat capture via the conversion of a temperature gradient into a electrical current.14 The thermoelectric figure of merit is proportional to the power factor σS2, where σ is the electronic conductivity and S is the Seebeck coefficient. Because the power factor is strongly dependent on band structure, thermoelectrics provides an excellent context to study band structure design. In particular, as discussed below, a band structure containing both flat and dispersive parts with respect to k leads to large power factor. In our case, the materials attributes specified above related to constituent elements, nominal electron count, and structure (d8 transition metals coordinated by anions in square planar geometry, shown in Figure 1) are chosen to mimic those of
Figure 1. (a) Geometry and (b) corresponding crystal field splitting of d orbitals of a transition metal in square planar coordination (D4h point group). The d8 electronic configuration is shown as an example. Note that the order of the A1g and B2g levels is sometimes reversed, as in the case of Bi2PdO4 in ref 13.
Bi2PdO413 and PbPdO2,15,16 two oxide thermoelectrics which were recently found to exhibit such a band structure. Screening the entire OQMD, we find 157 candidates for this targeted band structure. After further screening based on electronic band gap and structural anisotropy, we compute the band structures for several compounds to verify our approach. We identify two existing oxide compounds (Ba2PdO3 and La4PdO7) that successfully achieve the targeted band structure, providing validation to our inverse band structure design approach. Additionally, electronic transport calculations indicate promising power factor behavior in these compounds, particularly for n-type doping. 1541
DOI: 10.1021/acs.chemmater.7b04496 Chem. Mater. 2018, 30, 1540−1546
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Chemistry of Materials
namics criterion (reduction by a factor of 7) and the square planar structural criterion (reduction by a factor of 21). The vast majority of the compounds identified (150/157) turn out to be present in ICSD (including stable and metastable compounds). 68 of the these compounds have one of 32 structural prototypes listed in the ICSD, which are included in the Supporting Information. The structural prototypes corresponding to the most compounds are KBrF4-type (9, e.g., PbPdF4), La2Sb-type (8, e.g., GdZrSb), Rb3PdF5-type (7, e.g., BaLa2PtO5), and K2PtCl4-type (5, e.g., Tl2PdCl4). The lack of non-ICSD compounds found might be explained by the fact that few of the structural prototypes used to generate hypothetical compounds in the OQMD contain square planar bonding. Only two of the prototype structures used to systematically and comprehensively (i.e., considering nearly all periodic table elements) generate hypothetical compounds contain any square planar coordination. The only one currently in the OQMD in which all sites for an element are in square planar coordination is the L12 structure (e.g., Cu in CuAu3). The other, the D022 prototype, has square planar coordination for 1/3 of the sites of one of the elements (e.g., Al in Al3Ti). The 7 non-ICSD compounds found by our query, (Ce/Pr/Nd/Sm/Gd/Tb/Dy)2PdO4, actually correspond to neither the L12 nor the D022 prototype. These compounds have the distinct layered Ca2RuO4 prototype structure, which was previously explored for a very small number (∼25) of Pd oxide compounds to search for and assess the stability of materials similar to Bi2PdO4.13 The chemistry of the 157 identified compounds is summarized in Figure 3a. Oxides are the most prevalent chemistry, with 70 compounds. Note that we include compounds with polyatomic anions in this category if the oxygen of the polyatomic anion is bonded to the transition metal. Examples include iodates, nitrates, sulfates, selenates, and pyrophosphates. There are also a significant number of halides (56). The remaining 31 compounds include sulfides, selenides, phosphides, nitrides, carbonyls, and cyanides. Our query is able to find arbitrarily complex stoichiometries and crystal structures, as exemplified by H8CrSiO4F6. Each Cr is bonded to 2 hydroxide groups and 2 F in this compound, which we choose to count as an oxide in Figure 3a. The crystal structure also contains H2, H2O2, and SiF6 units. Figure 3b illustrates the particular elements involved in the square planar bonding. Pd oxides are dominant with 37 occurrences, and Pd halides also occur significantly (29 times) in the group of compounds. This is consistent with the prevalence of Pd2+ in square planar environments as has been found previously.40 Au oxides and chlorides also feature prominently. Despite the dominance of Pd oxides, there is still overall a diverse range of transition metals and anions in the 157 compounds. We also observe significant diversity in structures. There are compounds of all dimensionalities (e.g., 0D PdCl2, 1D PdBr2, 2D PdS2, 3D Mn3Sb). Here dimensionality refers to the connectivity of the square planar units. For example, in PdBr2 there are 1D channels of edge-sharing PdBr4 square planar units. Even within a particular dimensionality of the square planar connectivity, there is significant diversity in structure. For example, for 0D (unconnected) square planar compounds, there are distinct relative orientations of the square planar units (e.g., perpendicular in PbPdF4 vs parallel in Tl2PdCl4). Further Screening of Oxides Based on Band Gap. The band gap, in addition to the dispersion of the bands, is critically
convex hull), or unstable but in ICSD. This attribute relates the desired synthesizability of the compound, rather than the band structure. Because the OQMD and the associated qmpy framework30 do not intrinsically compute or store the coordination environments of different atoms in a crystal structure, we wrote a separate code to enable the assessment of our structural criterion. To query for a particular coordination environment, this code first computes all the distances between TMs and anions (taking into account translation vectors) to check if the coordination number is a desired value, e.g., 4 for square planar. If so, the bond lengths and bond angles for the nearestneighbor shell are computed to enable any local structural query. Details on the parameters we employ in this code, including how the coordination number is computed, are included in the Supporting Information. Additional DFT calculations are performed for the most promising candidate materials using VASP.31−34 The generalized gradient approximation35 is employed and the Kohn− Sham equations are solved using a 500 eV plane wave basis with the projector augmented wave (PAW) method36,37 and uniform k-point meshes of k-point density ≥700/Å−3. The total energy and ionic forces are converged to within 10−6 eV and 0.001 eV/Å, respectively. Electronic band structures are computed for high-symmetry k-paths based on the conventions of Setwayan et al. 38 Semiclassical electronic transport calculations within the constant relaxation time approximation are performed in BOLTZTRAP with k-point density ≥2300/ Å−3.39
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RESULTS AND DISCUSSION Characterization of Identified Compounds. Figure 2 summarizes the number of compounds found in the OQMD to
Figure 2. Total number of compounds (thick bars) as a function of consecutive screening filters on a logarithmic scale, with breakdown in terms of thermodynamic categories (thin bars). Unstable compounds are included only if they are reported in ICSD.
satisfy the screening criteria. As different criteria are applied successively, the number of candidates decreases from the total of more than half a million to just 157. The largest fractional decreases in candidate compounds come from the thermody1542
DOI: 10.1021/acs.chemmater.7b04496 Chem. Mater. 2018, 30, 1540−1546
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Chemistry of Materials
known square planar thermoelectric candidate materials Bi2PdO4 and PbPdO2 are included in these 33, which is a validation of the screening strategy. One of the 33 compounds is CaPd3O4, which has been studied as a possible excitonic insulator43,44 and topological Dirac semimetal.45 The electronic band structures are computed for five of the compounds in which the connectivity of the square planar units is not the same in all spatial directions: NaBi2AuO5, Ba2TlCuO5, Ba2PdO3, LiCuO2, and La4PdO7. We focus on such “nonisotropic” compounds since the difference in bonding in different directions should lead in principle to different ∇kϵ in different directions of k-space as is targeted for the puddingmold band structure. Compounds Achieving the Targeted Band Structure. Two of the compounds for which the band structure is computed, Ba2PdO3 and La4PdO7, successfully achieve the targeted band structure. Ba2PdO3 (space group Immm) and La4PdO7 (space group C2/m) both contain one-dimensional chains of corner-sharing PdO4 square planar units (shown going into the plane for La4PdO7 in Figure 4). For both of these materials, the number of d electrons (roughly estimated via projection within the PAW sphere) on the transition metal (Pd) site is 8.4, consistent with the nominal d8 configuration. As shown in Figure 4, Ba2PdO3 and La4PdO7 possess the pudding-mold band structure. Both show flat and dispersive parts of the low-energy spectrum both below and above the Fermi energy. For example, for Ba2PdO3 the conduction band disperses strongly (weakly) along Γ−Y (Γ−Z). The valence band disperses strongly (weakly) along W−R (W−T). La4PdO7 similarly shows both dispersive and flat components, though it is seen most strongly in the conduction band. We note that La4PdO7 appears to have a smaller valence bandwidth than Ba2PdO3 as the highest-occupied band is flat for much of the k-path shown in Figure 4b. The density of states in Figure 4 shows strong Pd d character in the band edges. The fact that we find the targeted band structure in these two compounds is a strong success of our inverse design strategy. Although the square planar unit itself is two-dimensional, the dimensionality of the band structure for the extended system depends strongly on the connectivity of the square planar units. For example, one-dimensional chains of PdO4 square planar units (e.g., stacked in the case of Bi2PdO4, corner-sharing in the case of Ba2PdO3) lead to a one-dimensional band structure with dispersion primarily in one direction, whereas a two-dimensional layer of square planar PdO4 units (e.g., La2PdO4) leads to a two-dimensional band structure with dispersion primarily in two directions. This suggests our type of inverse band structure design for pudding-mold thermoelectrics might also have success for other coordinations, in addition to square planar, that lend themselves to low-dimensional crystal structures. We note that our design strategy did not uncover any candidate compounds with a pudding-mold band structure that contain a lone pair cation, which would also be desirable for a thermoelectric by lowering lattice thermal conductivity.46,47 There are several compounds with Zr bonded to Sb in square planar coordination (e.g., GdZrSb), but here Sb is nominally the anion. In nearly all of the other compounds identified, either a metal is found (e.g., Ba2TlCuO5, Li8Bi2PdO10) or the electronic band gap is too large (e.g., 1.4 eV for Pb2PdCl6, 2.3 eV for Tl2Ni(CN)4). For NaBi2AuO5, the valence band edge does not primarily corresponding to the TM d orbitals, though it does contain some pudding-mold-like features.
Figure 3. Characterization of the 157 candidate compounds identified. (a) Distribution of stabilities split up in terms of oxides, halides, and other chemistries. (b) Network of chemical bonding of the square planar environment in which node size represents elemental occurrence and edge thickness is proportional to bonding frequency. (c) Histogram of DFT band gaps (greyed out for values larger than 1.1 eV).
important for thermoelectric materials.41 In particular, highefficiency thermoelectric materials such as PbTe, SnSe, and Bi2Te3 have band gaps no larger than around 1 eV. As such, to further screen the 157 candidate compounds, we further require a DFT band gap no larger than 1.1 eV. We choose not to put a lower limit on the band gap since semilocal DFT is well-known to underestimate the electronic band gap.42 As illustrated in the band gap distribution in Figure 3c, this leaves 72 remaining compounds. We focus on the 33 of these compounds that are pure oxides (i.e., containing no other anion element). These candidate compounds are listed in the Supporting Information. The 1543
DOI: 10.1021/acs.chemmater.7b04496 Chem. Mater. 2018, 30, 1540−1546
Article
Chemistry of Materials
Figure 4. Electronic band structure and total and projected density of states of (a) Ba2PdO3 and (b) La4PdO7. The dashed red line marks the valence band maximum. In the crystal structures shown in the insets, Pd, O, and Ba/La atoms are shown in blue, red, and green, respectively.
Electronic Transport Properties of Ba2PdO3 and La4PdO7. Finally, as a confirmation that the designed electronic band structure leads to promising thermoelectric behavior, we perform electronic transport calculations via the constant relaxation time approximation. To isolate the effect of band structure shape, rather than the magnitude of the band gap, we compare the transport properties of Ba2PdO3 and La4PdO7 to Bi2PdO4 at a fixed band gap of 1.41 eV. This value corresponds to the band gap of Bi2PdO4 from hybrid DFT calculations and was used in ref 13. Figure 5a shows the behavior of the power factor (divided by the relaxation time) for these 3 materials as a function of doping at 700 K. Typical doping magnitudes leading to peak thermoelectric performance that can be experimentally achieved are 1 × 1019 to 1 × 1021 cm−3.48 For p-type doping, the power factor magnitude of Ba2PdO3 and La4PdO7 is similar to that of Bi2PdO4 (∼1 × 1012 W/(m K2 s)). Bi2PdO4 achieves a peak in S2 σ/τ of 1.36 × 1012 W/(m K2 s) for a carrier concentration of 5.3 × 1020 cm−3. One of the new compounds (Ba2PdO3) achieves a higher peak value at a higher doping (3.5 × 1021 cm−3), but for more reasonable dopings (
