Prediction of New Stable Compounds and Promising

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Prediction of New Stable Compounds and Promising Thermoelectrics in the Cu−Sb−Se System Yongsheng Zhang,*,† Vidvuds Ozoliņs,̌ ‡ Donald Morelli,§ and C. Wolverton† †

Department of Materials Science & Engineering, Northwestern University, Evanston, Illinois 60208, United States Department of Materials Science & Engineering, University of California, Los Angeles, California 90095-1595, United States § Department of Chemical Engineering & Materials Science, Michigan State University, East Lansing, Michigan 48824, United States ‡

S Supporting Information *

ABSTRACT: We study the phase stability and predict as-yet-unreported compounds in the thermoelectric Cu−Sb−Se ternary system. We use a combination of total energies obtained from density-functional-theory-based (DFT) calculations with vibrational entropies from phonon calculations (within the harmonic approximation) and configurational entropies, treated with cluster expansions (CE). The Cu−Sb−Se ternary phase diagram is determined (treating all phases as line compounds) using the grandcanonical linear programming method. We find the following results: (1) we predict the stability of a new previously unknown, zinc blende-based Cu4SbSe5 compound but find that it is thermodynamically stable up to only ∼300 K; (2) we also predict that a Cu12Sb4Se13 phase (isostructural with Cu12Sb4S13, but unreported in the Cu−Sb−Se system) appears in the phase diagram at high temperatures (but below the temperatures where the observed Cu3SbSe3 phase is stable); (3) based on quasi-harmonic phonon and band structure calculations, we find that Cu12Sb4Se13 has thermal conductivity and an electronic structure that suggests it as a promising thermoelectric material.

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INTRODUCTION Thermoelectric materials play an important role in addressing energy efficiency needs, such as converting waste heat generated by power plants, vehicles or even living beings into electricity.1−4 The conversion efficiency is characterized by the thermoelectric figure of merit ZT = Sσ2T/κ, where S is the Seebeck coefficient, σ is the electrical conductivity, and κ is the thermal conductivity. Enhancing the figure of merit can be achieved by increasing S or σ, or decreasing κ. However, the energy-conversion efficiencies of current thermoelectric materials are not high enough to make their widespread use economical. Recent advances in thermoelectric materials have relied on embedding nanostructures within the bulk material, which reduce thermal conductivity,4−8 but which can also potentially reduce electronic conductivity. An alternative tactic is to search for new materials with ordered crystal structures having intrinsically low thermal conductivity due to strong lattice anharmonicity.9−16 Ternary Cu−Sb−Se compounds are attractive materials for numerous applications. Compounds from this family are candidates for optical absorbers, optical materials, 2D electronic materials and thermoelectric materials. Skoug et al.12 experimentally investigated the thermoelectric behavior of two different ternary Cu−Sb−Se compounds with different nominal Sb valence states: Cu3SbSe4 and Cu3SbSe3. They found that Cu3SbSe3 exhibits anomalously low and nearly temperature-independent lattice thermal conductivity, whereas Cu3SbSe4 does not exhibit this anomalous behavior. This interesting behavior of the two seemingly similar compounds © XXXX American Chemical Society

has recently been investigated (ref 13) via a combination of density-functional theory (DFT) and a Debye−Callaway model17 of thermal conductivity. These DFT calculations have shown that the average of the square of the Grüneisen parameter for the acoustic modes in Cu3SbSe3 is larger than that of Cu3SbSe4, which theoretically confirms that Cu3SbSe3 has stronger lattice anharmonicity than Cu3SbSe4. The soft phonon frequencies and high Grüneisen parameters in Cu3SbSe3 arise from the electrostatic repulsion between the lone s2 electron pair at Sb sites and the bonding charge in Sb− Se bonds. A combination of first-principles determined longitudinal and transverse acoustic mode Grüneisen parameters, zone-boundary frequencies, and phonon group velocities, produces a theoretically calculated lattice thermal conductivity in good agreement with the experimental measurements. Hence, the DFT calculations are able to accurately model the relationship between crystal structure and anharmonicity in this system. Due to this interesting hebavior of Cu3SbSe3 and Cu3SbSe4, one might wonder if the Cu−Sb−Se system contains other instances of strong anharmonicity or promising thermoelectric compounds. To understand the properties of all phases in the Cu−Sb−Se ternary system, the full ternary phase diagram is required. However, this phase diagram is not well-established either by experimental measurements or by theoretical Received: February 25, 2014 Revised: May 8, 2014

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dx.doi.org/10.1021/cm5006828 | Chem. Mater. XXXX, XXX, XXX−XXX

Chemistry of Materials

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Figure 1. Experimental crystal structures of Cu3SbSe419 (I4̅2m, aexp = bexp = 5.66 Å, cexp = 11.28 Å), Cu3SbSe320 (Pnma, aexp = 7.99 Å, bexp = 10.61 Å, cexp = 6.84 Å), and CuSbSe221 (Pnma, aexp = 6.40 Å, bexp = 3.95 Å, cexp = 13.33 Å). The cluster expansion predicted zinc-blende Cu−Sb−Se crystal structures of Cu6SbSe7, Cu5SbSe6, and Cu4SbSe5 (Table 2). The Cu12Sb4Se13 structure using the Cu12Sb4S13 prototype and the corresponding phonon-stabilized Cu12Sb4Se13 structure (Table 2). The Cu atom at the center of the CuSe3 plane shifts along one of the blue arrows after phonon stabilizations. The red arrows on the Sb atoms in Cu12Sb4Se13 represent atomic displacements responsible for the anomalously high Grüneisen parameters of the transverse acoustic phonon branch in Figure 4. Cu = blue, Sb = brown, Se = green. scheme,22 and the generalized gradient approximation of Perdew, Burke, and Ernzerhof23 (GGA-PBE) for the electronic exchange− correlation functional. The energy cutoff for the plane wave expansion is 800 eV. The Brillouin zones are sampled by Monkhorst−Pack24 kpoint meshes for all compounds with meshes chosen to give a roughly constant density of k-points (30 Å3 for all compounds). Atomic positions and unit cell vectors are relaxed until all the forces and components of the stress tensor are below 0.01 eV/Å and 0.2 kbar, respectively. Phonons are calculated using the supercell force constant method25,26 implemented in the Alloy Theoretical Automated Toolkit (ATAT).27 We performed phonon calculations on various compounds with various symmetries and unit cell sizes. In each case, the supercell was constructed to make sure each superlattice vector is larger than 8 Å. In the quasi-harmonic DFT phonon calculations, the system volume is isotropically expanded by +6% from the DFT relaxed volume. The Grüneisen parameter is defined as49

calculations. To date, only a partial phase diagram of Cu−Sb− Se is known, e.g., a small Cu composition region from Cu23.85Sb25.69Se50.46 to Cu26.10Sb24.34Se49.56.18 The experimentally determined melting temperatures (Tm) of Cu3SbSe4 and CuSbSe2 are 698 and 733 K, respectively.48 A more complete understanding the Cu−Sb−Se ternary phase diagram is required. Because of the lack of experimental information about the Cu−Sb−Se phase diagram, we suspect there could be new, undiscovered phases in this system. Hence, we use computational techniques to search for new compounds and calculate their stability. Specifically, we (1) use the cluster expansion (CE) method28,30 to predict new possible Cu−Sb−Se ternary phases and (2) use prototype structures from the Cu−Sb−S system as candidates structures in Cu−Sb−Se. From these calculations, we predict the stability of the experimentally observed phases as well as two previously unreported phases in this system. We predict a Cu4SbSe5 phase that is stable below T ∼ 300 K. We also predict the existence of Cu12Sb4Se13, which becomes stable at high temperatures (but below temperatures where the Cu3SbSe3 phase has been observed). Using our firstprinciples determined longitudinal and transverse acoustic mode Grüneisen parameters, zone-boundary frequencies, and phonon group velocities, we calculate the lattice thermal conductivity of Cu4SbSe5 and Cu12Sb4Se13 using the Debye− Callaway model. The theoretical thermal conductivity of Cu12SbSe13 is lower than all other Cu−Sb−Se ternary phases. Combined with favorable electronic band structure properties, we suggest Cu12Sb4Se13 is a promising new thermoelectric material.

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γi = −

V ∂ωi ωi ∂V

(1)

where γi, V, and ωi are the Grüneisen parameter, the volume of the primitive cell, and the phonon frequency, respectively. The Grüneisen parameter characterizes the relationship between phonon frequency and volume change. The Grüneisen parameters provide an estimate of the strength of the phonon−phonon interactions in a compound. Cluster Expansion. The cluster expansion (CE) method28−33 has been long used to identify unknown structures in bulk materials,34−37 surfaces38−40 and surface adsorptions.41−43 The crystal structure is represented by a three-dimensional lattice, in which any system state is defined by the occupation of sites in the lattice and the total energy of any configuration (σ) is expanded in series over discrete interactions (the effective cluster interactions, or ECI) between the lattice sites.

E(σ ) = J0 +

∑ Ji Si + i

METHODOLOGY

1 2

∑ Jij SiSj + i≠j

1 3!

∑

Jijk SiSjSk + ...

i≠j≠k

(2) where in a binary alloy (A1−xBx), Si = 1 or −1 indicate whether site i in the lattice is occupied by an atom of type A or B, respectively. Jα are effective cluster interactions, for a cluster of sites α (e.g., pairs, triplets,

Density-Functional Theory and Phonon Calculations. We perform DFT calculations using the Vienna Ab Initio Simulation Package (VASP) with the projector augmented wave (PAW) B



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etc). Formally, higher and higher order interaction terms follow in the infinite expansion of eq 2. In practice, the expansion must be truncated after a finite number of terms. The judicious choice of which interactions to consider and which ones to neglect critically affects the reliability of the entire approach.27,44,45 To quantify the impact of this choice on accuracy, we rely on the concept of leave-one-out cross validation (LOO−CV)27,30,42 to identify the most important interactions out of a larger pool of possible interactions. The mixing energy [ΔE(σ)] of a configuration (σ) can be calculated as

ΔE(σ ) = [E(σ ) − xEA − (1 − x)EB]/Nmix

(3)

where E(σ), EA, and EB are the total energies of the new configuration (σ), the pure constituents A and B, respectively. x is the composition of A in the configuration σ. Nmix is the total number of mixed atoms. We calculate many A1−xBx structures using DFT, and use ATAT27 to construct and fit the cluster expansion. The ground states can be determined by enumerating all possible A1−xBx configurations.

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GROUND-STATE STABILITY AND PREDICTED TERNARY CU−SB−SE CRYSTAL STRUCTURES To search for new possible Cu−Sb−Se ternary phases, we use the cluster expansion method. The CE is a lattice-based method to describe configurational decorations of atoms on a given lattice type. Because the crystal structures of bulk Cu (facecentered cubic), Sb (rhombohedral), and Se (hexagonal) are very different, direct application of CE to the full Cu−Sb−Se ternary system is difficult. Thus, we focus on an isoplethal section of the ternary phase diagram: CuSe-SbSe. Experiments have determined three stable Cu−Sb−Se ternary phases, Cu3SbSe4,19 Cu3SbSe3,20 and CuSbSe221 (Figure 1). Cu3SbSe4 is a zinc blende (ZB)-based structure. There are two sets of face-centered cubic (fcc) sublattices in ZB and Cu/Sb and Se occupy these sublattices, respectively, in Cu3SbSe4 and each atom is 4-fold coordinated. Although Cu3SbSe3 and CuSbSe2 are orthorhombic structures rather than the ZB structure, we still find that some atoms such as Se in Cu3SbSe3 and Se and Cu in CuSbSe2 sit in the middle of a tetrahedron. Because several of the observed phases are related to orderings on the ZB lattice, we use the ZB structure as the lattice in our cluster expansion to mix cation Cu and Sb in the pseudobinary CuSe and SbSe systems. We calculate 22 Cu− Sb−Se ZB structures using DFT, and use ATAT to construct and fit the cluster expansion. We use leave-one-out crossvalidation (LOO−CV) to assess the predictive ability of the cluster expansion fit. The cluster expansion is terminated when all structures predicted to be ground states from the CE are also ground states from the set of DFT energies and the LOO−CV score is ∼50 meV/cation.46 Our results for ZB-ordered structures from the cluster expansion are shown in Figure 2a (black line). Considering only ZB orderings, our CE successfully predicts the experimentally determined ZB Cu3SbSe4 structure (but not Cu3SbSe3 or CuSbSe2 because these are not ZB-based structures). Additionally, we find several other ZB Cu−Sb−Se ternary phases, which fall on this ”ZB-only” convex hull (Figure 1, Figure 2a, Table 1, and Table 2): Cu6SbSe7, Cu5SbSe6, Cu4SbSe5, and CuSb5Se6. (We do not list the ZB crystal structures of CuSbSe2 and CuSb5Se6 in Table 2 or Figure 1 because although these energies lie on the ”ZB-only” convex hull, they lie above the true convex hull, as shown below). Our CE only determines substitutional ordered arrangements on the ZB lattice. A true ground state must be lower in energy than all other compounds (or combinations of compounds), not merely those on a fixed lattice. Therefore, we supplement

Figure 2. (a) Mixing energies of CuSe−SbSe pseudobinary phases using cluster expansion predictions based on the ZB structure. DFT ground-state line (black circle line). (b) Using the energies based on the experimentally determined CuSe and (Sb + Sb2Se3)/3 as the end points, the convex hull is replotted (black line). The experimentally determined Pnma CuSbSe2 is included as well, and the ZB CuSbSe2 and CuSb5Se6 are not shown because they have large positive mixing energies. After phonon stabilizations, the DFT energies of experimental CuSe and Sb2Se3 structures shift down the two end points by ∼20 meV/fu and ∼13 meV/fu, respectively, which leaves the Cu6SbSe7 and Cu5SbSe6 phases above the convex hull (blue line).

Table 1. Collections of Important Information of All Phases Presented in Figure 2: Stoichiometry, Space Group, on ZB Convex Hull, on “Full” Convex Hull, Experimentally Observed or Theoretically Predicted stoichiometry

space group

on ZB convex hull

Cu6SbSe7 Cu5SbSe6 Cu4SbSe5 Cu3SbSe4 CuSbSe2-ZB CuSb5Se6 CuSbSe2-exp.

R3 Cm I4̅ I4̅2m Pmn21 Cm Pnma

yes yes yes yes yes yes no

on the full experimentally observed convex hull or theoretically predicted yes yes yes yes no no yes

theor theor theor exp/theor theor theor exp

the CE search with experimentally known, non-ZB structures. The experimentally determined CuSe is not in the ZB structure, but a hexagonal structure (P63/mmc).47 The DFT energy of the experimentally determined hexagonal CuSe structure is only ∼7 meV/fu lower than that of the ZB structure. SbSe is not an experimentally observed compound. For the SbSe composition, we give a reference energy from a combination of the energies of the experimentally determined Sb and Sb2Se3 structures: (ESb + ESb2Se3)/3, which is 900 meV/fu lower than the fictitious ZB SbSe structure. Therefore, we use the experimental CuSe and (Sb + Sb2Se3)/3 structures as the two end points to replot the ground-state energies (Figure 2b, the black line). Additionally, we add the energy of the experimentally determined orthorhombic CuSbSe2 phase (Pnma).21 (We cannot show the energy of the experimentally determined C



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predicted ZB CuSbSe2 and CuSb5Se6 phases are now clearly above the convex hull (with significantly positive mixing energies), which indicates that for SbSe rich compositions, simple ZB orderings are unstable. Cu12Sb4Se13 Structure using the Cu12Sb4S13 Prototype. To increase the likelihood of finding new stable compounds, we can expand our search over a wider set of candidate crystal structures. A simple method of expanding the search50,51 is to consider experimentally observed crystal structures for systems that are isoelectronic with Cu−Sb−Se. Hence, we look for crystal structures of stable compounds in the Cu−Sb−S system. From the inorganic crystal structure database (ICSD), we find that the Cu−Sb−Se system has many crystal structures in common with the Cu−Sb−S system: the ZB-based structure of Cu3SbSe4 is analogous to Cu3SbS4; the orthorhombic (Pnma) structure of Cu3SbSe3 and Cu3SbS3; the orthorhombic (Pnma) structure of CuSbSe2 and CuSbS2. However, in the Cu−Sb−S ternary system, a cubic Cu12Sb4S13 structure has been experimentally observed, but the corresponding Cu12Sb4Se13 compound has not yet been observed. We suspect that a Cu12Sb4Se13 phase could be stable, and therefore, we take the cubic Cu12Sb4S13 structure as a prototype and replace all S by Se and then relax the unit cell and atomic positions within DFT. In the Cu12Sb4Se13 structure (Figure 1), there are two inequivalent Cu atoms and Se atoms: for Cu, one kind of Cu has four Se neighbors with 2.43 Å bond lengths, the other kind of Cu is surrounded by three Se atoms forming a plane with ∼2.40 Å bond lengths; for Se, one Se is tetrahedrally surrounded by three Cu (∼2.40 Å) and one Sb (2.63 Å), the other Se at the center of the structure is octahedrally coordinated with six Cu (2.42 Å). Sb only has three neighbors, which leaves a dangling bond (or lone-pair electron density) as in Cu3SbSe3 discussed in ref 13. We next perform phonon calculations to check the local structural stability of the hypothetical Cu12Sb4Se13 (the top left panel in Figure 4). We find the cubic Cu12Sb4Se13 structure to be dynamically unstable, with several unstable optical phonon branches involving out-of-plane vibrations of the 3-fold coordinated Cu atoms (Figure 1). Interestingly, similar structural instabilities are also found in Cu12Sb4S1315 (suggesting that this phase is not a T = 0 K ground state but is stabilized at finite temperatures). But unlike Cu12Sb4S13, in which transverse acoustic (TA) branches are harmonically unstable near the Brillouin Zone (BZ) boundary, the TA branches of Cu12Sb4Se13 are stable in the whole BZ. Following the eigenvectors of the imaginary optical frequencies, we obtain a lower-energy, distorted Cu12Sb4Se13 crystal structure (with a P21 space group, Table 2) that can be used to represent the T = 0 K energetics of this hypothetical compound (Figure 1). We refer to this P21 compound as “phonon stabilized Cu12Sb4Se13”. We recompute the phonons of the distorted phase and find that it is stable. In this stabilized structure (Figure 1), the Se-6Cu octahedron is significantly distorted and Cu is displaced in the perpendicular direction out of the Cu-3Se plane (the blue arrows in Figure 1). The Se−Cu−Se bond angles in the plane significantly change from 131.3° to 88.6−134.0°. This distortion suggests that instabilities in Cu12Sb4Se13 are caused by the (unfavorable) 3-fold planar bonding environment of Cu as in Cu12Sb4S13.15 Cu−Sb−Se Ternary Phase Diagram. The discussion above only treats a pseudobinary, isoplethal section of the full ternary phase diagram, and only at T = 0 K. To fully understand the thermodynamic stability of Cu−Sb−Se phases (and to

Table 2. CE predicted Cu6SbSe7, Cu5SbSe6, and Cu4SbSe5 Using the ZB Structure, and the Phonon Stabilized Cu12Sb4Se13 Structure system Cu6SbSe7 (R3) a = 10.553 Å b = 10.553 Å c = 9.704 Å γ = 120.0° Cu5SbSe6 (Cm) a = 6.898 Å b = 11.966 Å c = 6.933 Å β = 109.19° Cu4SbSe5 (I4̅) a = 8.985 Å b = 8.985 Å c = 5.672 Å Cu12Sb4Se13 (P21) a = 10.923 Å b = 10.850 Å c = 10.843 Å β = 89.42°

x

y

z

atom

Wyckoff

Cu Cu Sb Se Se Se Cu Cu Cu Sb Se Se Se Se Cu Sb Se Se

9b 9b 3a 9b 9b 3a 4b 4b 2a 2a 2a 4b 4b 2a 8g 2d 8g 2a

0.2299 0.3772 0.0000 −0.0763 0.0473 0.0000 −0.1707 −0.1581 −0.1745 0.3323 −0.2640 0.1998 0.2081 0.1980 −0.3055 0.0000 0.1070 0.0000

0.1837 −0.0914 0.0000 −0.4601 0.2383 0.0000 0.1672 0.3285 0.0000 0.0000 0.0000 0.1793 0.3331 0.0000 0.4021 0.5000 −0.2845 0.0000

0.1960 0.1859 −0.1428 0.2809 0.2729 −0.4147 −0.0129 −0.4902 −0.4912 −0.0025 0.1353 0.1373 −0.3766 −0.4064 0.2583 0.7500 0.4810 0.0000

Cu Cu Cu Cu Cu Cu Cu Cu Cu Cu Cu Cu Sb Sb Sb Sb Se Se Se Se Se Se Se Se Se Se Se Se Se

2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a

−0.000 58 0.4996 0.2535 −0.2493 −0.2433 −0.2582 0.0156 −0.4578 0.2930 −0.2891 0.2971 0.2926 −0.0027 0.4651 −0.0149 0.4736 0.1430 −0.3678 0.1427 −0.3704 −0.1146 0.3818 −0.1146 0.3823 0.1467 −0.3729 0.3649 −0.1358 0.2315

0.001 11 0.4954 −0.4945 −0.0044 0.2502 −0.2525 −0.0321 −0.4714 0.0345 0.4603 0.2135 −0.2112 0.2791 −0.2331 −0.2746 0.2295 0.3629 −0.1382 −0.3646 0.1417 0.1221 −0.3887 −0.1177 0.3926 0.1128 −0.3885 −0.1137 0.3879 0.0017

0.247 91 −0.2478 −0.0096 −0.4978 −0.2463 −0.2572 −0.2069 0.2145 −0.0391 0.4699 −0.2126 −0.2908 0.0312 −0.4840 0.4817 −0.0211 −0.1415 0.3666 −0.3628 0.1430 −0.1194 0.3538 −0.3615 0.1335 0.1165 −0.3897 0.1130 −0.3857 −0.2562

orthorhombic Cu3SbSe3 phase in the convex hull because its stoichiometry lies outside of the pseudobinary CuSe-SbSe line.) The phase stability trend at the CuSe rich side (xSbSe ≤ 0.25, the black line in Figure 2b) is the same as the previous convex hull using ZB structures, which indicates that for CuSe rich compositions, the Cu−Sb−Se ternary compounds have an energetic preference for ZB-type structures. However, with increasing SbSe composition (xSbSe > 0.25), the convex hull significantly deviates from the ZB convex hull, and the CE D



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Figure 3. Ternary phase diagrams of Cu−Sb−Se: before phonon calculations at T = 0 K without zero-point energy (ZPE) (a), after structural stabilizations based on phonon calculations but still no ZPE (b), T = 0 K with ZPE (c), T = 190 K (d), T = 270 K (e), T = 910 K (f), T = 1100K (g), and T = 1570K (h). The scale bar is used to show the magnitude of formation free energies (unit: meV/atom).

phonon calculations as imaginary optical frequencies. These structures are stabilized by moving atoms along the eigenvectors of those imaginary frequencies until a new energy minimum is found. The phonons of those distorted structures are recomputed. The DFT energies of the phonon stabilized experimental CuSe and Sb2Se3 structures shift down by ∼20 meV/fu and ∼13 meV/fu (the blue dots at the end points in Figure 2b), respectively, which leave the Cu6SbSe7 and Cu5SbSe6 phases above the convex hull (the blue line in Figure 2b). Correcting for these dynamic instabilities significantly alters the Cu−Sb−Se ternary phase diagram (Figure 3b) compared to the simple T = 0 K static calculations (Figure 3a). Even though the CE predicted Cu6SbSe7 and Cu5SbSe6 ternary phases become thermodynamically unstable, the theoretically predicted Cu4SbSe5 phase is still a stable phase in the ternary phase diagram. Including zero-point energies at T = 0 K (Figure 3c) does not change the stable phases from those in the phase diagram without ZPE (Figure 3b). However, the inclusion of finitetemperature vibrational thermodynamics changes the stabilities of many phases: (1) At 190 K (Figure 3d), the experimentally observed CuSbSe2 appears as a stable phase. (2) As we increase the temperature to 270 K, we find our theoretically predicted Cu4SbSe5 is no longer a stable phase. Such low-T transformations suggest that the energetic competition between these phases is very subtle, with nearly degenerate energies. (3) At 910 K, the experimentally determined Cu3SbSe4 becomes unstable and disappears from the phase diagram as well. (4) Interestingly, at 1100 K, our hypothetical Cu12Sb4Se13 phase becomes stable. (5) At 1570 K, another experimentally wellknown phase, Cu3SbSe3 appears as a stable phase. From the phase diagram evolution with increasing temperature, the theoretically predicted Cu4SbSe5 phase is thermodynamically stable up to around room temperature (∼300 K); the exprimentally determined CuSbSe2 and Cu3SbSe3 are stable compounds at high temperatures, suggesting entropy contributions are important in the stabilization of these

check the stability of the theoretically predicted ternary phases), we need to calculate the full Cu−Sb−Se ternary phase diagram as a function of temperature. We use DFT to calculate the total energies of 18 Cu−Sb−Se phases from the ICSD, as well as our theoretically predicted structures: Cu, Sb, Se, Cu2Sb, Cu3Sb, Cu11Sb3, CuSe, CuSe2, Cu2Se, Cu3Se2, Sb2Se3, Cu3SbSe3, Cu 3 SbSe 4 , CuSbSe 2 , Cu 6 SbSe 7 , Cu 5 SbSe 6 , Cu 4 SbSe 5 , Cu12Sb4Se13. Only solid-state phase equilibria are considered. The observed melting temperatures (Tm) of Cu3SbSe4 and CuSbSe2 are ∼700 K.48 Because simulating liquid phase free energies using DFT is very difficult, we neglect the liquid phase in the current Cu−Sb−Se phase diagram. Also we neglect offstoichiometry in all compounds (all compounds are treated as purely stoichiometric line compounds). Nevertheless, we can still capture useful information regarding the phase stability of solid compounds. Using the grand-canonical linear programming method (GCLP52), we compute the Cu−Sb−Se ternary phase diagram (Figure 3) on the basis of the DFT calculated energies for different contributions to the thermodynamics: (1) T = 0 K (static energies) without phonon stabilization, (2) T = 0 K (static energies), after phonon stabilization of unstable phases, (3) T = 0 K, including zero-point energy (ZPE), and (4) T > 0 K including vibrational enthalpies and entropies. Without phonon calculations, the T = 0 K Cu−Sb−Se phase diagram (Figure 3a) shows that Sb2Se3 has the most negative formation energy and the experimentally determined Cu3SbSe4 and theoretically predicted (CE) Cu6 SbSe 7, Cu 5SbSe6 , and Cu4SbSe5 are stable compounds in the full ternary Cu−Sb− Se phase diagram, not only in the CuSe−SbSe pseudobinary phase diagram. However, the experimentally determined Cu3SbSe3 is not stable in the ground-state (T = 0 K) phase diagram, which suggests that at finite-temperature free energy contribution plays an important role in the stabilization of the experimental compound. Several Cu−Sb−Se structures (CuSe, Cu2Se, Sb2Se3, and Cu12Sb4Se13) possess dynamic instablities, which appear in our E



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Figure 4. Phonon dispersions (top panels) and corresponding acoustic Grüneisen parameters (bottom panels) of Cu4SbSe5 (left panels) and Cu12Sb4Se13 (right panels). Because the acoustic modes play an important role in thermal conductivity, we highlight these modes with different colors in the plot: The red, green and blue lines highlight TA, TA′, and LA modes, respectively. The inset figures are the first Brillouin zones of the two compounds with high symmetry points (red points) we considered in our phonon dispersion calculations.

Table 3. Average Transverse (TA/TA′) and Longitudinal (LA) Grüneisen Parameters (γT̅ A/TA′/LA), Debye Temperatures (ΘTA/TA′/LA), and Phonon Velocities (vTA/TA′/LA) in Cu3SbSe4, Cu3SbSe3, Cu4SbSe5, and Cu12Sb4Se13a system

γT̅ A

γT̅ A′

γL̅ A

ΘTA (K)

ΘTA′ (K)

ΘLA (K)

vTA (m/s)

vTA′ (m/s)

vLA (m/s)

Cu3SbSe4 Cu3SbSe3 Cu4SbSe5 Cu12Sb4Se13

1.27 3.03 0.92 9.48

1.14 2.92 0.74 8.50

1.26 1.26 0.83 4.65

60 33 62 30

65 34 66 30

78 36 76 39

1485 1072 1342 1256

1699 1344 1707 1398

3643 3014 3718 2961

a

The values of Cu3SbSe4 and Cu3SbSe3 are taken from ref 13, and the values of Cu4SbSe5 and Cu12Sb4Se13 are calculated using their phonon dispersions (Figure 4). The Debye temperate is calculated using Θ = ωD/kB (ωD is the largest acoustic frequency in each direction); the phonon velocity is the slope of the acoustic phonon dispersion around the Γ point. The Grüneisen parameters, Debye temperatures, and phonon velocities are averaged by the weight of the high symmetry points.

efficient thermoelectric materials. To this end, we investigate the lattice thermal conductivity of these compounds. One method to calculate thermal conductivity55 involves solving the Peierls−Boltzmann equation.54 Another approach, which we use here, is the Debye−Callaway model, which has recently been demonstrated to be successful in applications to semiconductors17 and thermoelectric materials.13 (The model has been further improved by Allen in a very recently published paper.56) In our previous work,12,13 we demonstrated that Cu3SbSe3 has a much lower thermal conductivity than Cu3SbSe4 due to the strong anharmonicity in Cu 3SbSe3 , which can be characterized via the Grüneisen parameters. The lone s2 electron pair at Sb site plays an import role in the strong anharmonicity in Cu3SbSe3. Thus, we wish to describe the extent of anharmonic behavior of Cu4SbSe5 and Cu12Sb4Se13. In Figure 4, we plot the calculated phonon dispersion curves53 along the high symmetry directions in their respective Brillouin Zones (BZ) (insets in Figure 4), and highlight the acoustic modes with different colors in the plot because these phonons contribute most to the lattice thermal conductivity. The

compounds; and Cu12Sb4Se13 and Cu3SbSe3 appear as stable phases above the melting temperature (∼700 K48). The stability at high temperatures (above melting temperatures) for Cu3SbSe3 is consistent with the experimental synthesis conditions for the compound:12 The Cu3SbSe3 phase is experimentally synthesized by melting (1123 K) Cu, Sb, and Se mixture at a 3:1:3 mixing ratio and then quenching to the room temperature. From our thermodynamical Cu−Sb−Se ternary phase diagram, the Cu12Sb4Se13 phase appears at even lower (∼500 K lower) temperature than the Cu3SbSe3 phase. Therefore, it is possible that the theoretically predicted Cu12Sb4Se13 compound could be synthesized using a similar procedure; that is, by melting a 12:4:13 mixture of Cu, Sb, and Se followed by a quench to room temperature. Anharmonicity and Lattice Thermal Conductivity of Cu4SbSe5 and Cu12Sb4Se13. We have examined the phase stability in the Cu−Sb−Se system, and found several candidates for new, unexplored compounds in this system. Specifically, from the last section, we have predicted Cu4SbSe5 and Cu12Sb4Se13 as stable phases. We wish to further investigate if these new ternary compounds are promising candidates for F



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corresponding acoustic Grüneisen parameters (eq 1) of the two compounds are given in Figure 4, and the average Grüneisen parameter (γ)̅ of each acoustic branch (TA, TA,′ and LA in Table 3) is calculated using the method described in refs 13 and17: γ ̅ = (⟨γi2⟩)1/2. The acoustic phonon dispersions and Grüneisen parameters of ZB-Cu4SbSe5 (Figure 4) are similar to those of ZB-Cu3SbSe4 (ref 13), and hence, these compounds have similar average Debye temperatures (Θ) and phonon velocities (v) (Table 3). These similarities can be attributed to similar bonding environments in Cu 3 SbSe 4 and Cu 4 SbSe 5 , each atom tetrahedrally coordinated by four nearest-neighbors. However, Cu12Sb4Se13, which has lone-pair electrons at the Sb sites, has acoustic phonon modes that are softer than those of Cu4SbSe5; the zone-boundary frequencies are ∼20−30 cm −1 in Cu12Sb4Se13 compared to ∼40−50 cm−1 in Cu4SbSe5. Thus, the Debye temperatures and phonon velocities of Cu12Sb4Se13 are much lower than those of the ZB-based compounds, Cu4SbSe5 and Cu3SbSe4 (Table 3). We find that the zoneboundary frequencies of Cu12Sb4Se13 and Cu3SbSe3 are similar, which results in nearly identical Debye temperatures and phonon group velocities (Table 3). However, the average Grüneisen parameters of Cu12Sb4Se13, especially of the transverse modes (TA and TA′), are much larger than those of Cu3SbSe3 and other Cu−Sb−Se compounds (Table 3). From the Grüneisen parameter dispersions of Cu12Sb4Se13, we can see that along high symmetry directions (Z-Γ and T-Γ) in the BZ, the acoustic modes have very large Grüneisen parameters (>10), indicating that the branches in Cu12Sb4Se13 are very anharmonic and interact strongly with other acoustic phonons. Analysis of the eigenvectors of these modes shows that they involves Sb vibrations away from Sb−Se bonds, such as along the [110] direction (red arrows in Figure 1), roughly along which the Sb lone-pair electrons are also oriented. We hypothesize that the interactions between the lone pair and the three Sb−Se bonds (Figure 1) are sensitive to the Sb−Se bond angles and result in large Grüneisen parameters of the TA′ modes in Cu12Sb4Se13. The Debye−Callaway model (details in refs 13 and 17) of lattice thermal conductivity is a function of Debye temperatures, phonon velocities, and Grüneisen parameters. All of these quantities may be computed from DFT, leading to a firstprinciples description of lattice thermal conductivity with no adjustable parameters. We have shown previously that the theoretically calculated lattice thermal conductivities of Cu3SbSe4 and Cu3SbSe3 using this model are in good agreement with the experimental measurements:13 the theoretically calculated thermal conductivity captures the trends of the experimental measurements and the deviation for both cases is at most only about 20%, which is quite satisfactory given the approximations inherent in the Debye−Callaway formalism. We therefore use the longitudinal (LA) and transverse (TA/TA′) Debye temperatures (ΘTA/TA′/LA), phonon velocities (vTA/TA′/LA), and Grüneisen parameters (γ ̅T A/TA′/LA ) of Cu 3 SbSe 4 , Cu 3 SbSe 3 , Cu 4 SbSe 5 , and Cu12Sb4Se13 (Table 3) to parametrize the Debye−Callaway model and calculate the lattice thermal conductivity of each of these phases. The resulting thermal conductivities are shown in Figure 5. Our predicted ZB-Cu4SbSe5 has higher lattice thermal conductivity than the ZB-Cu3SbSe4 compound due to small Grüneisen parameters. However, Cu12Sb4Se13 has an extremely low lattice thermal conductivity, lower than all other Cu−Sb− Se compounds. The theoretically calculated lattice thermal

Figure 5. Theoretical lattice thermal conductivity of Cu3SbSe4 (black line), Cu3SbSe3 (red line), Cu4SbSe5 (green line), and Cu12Sb4Se13 (blue line) using their corresponding Grüneisen parameters (γ), Debye temperatures (Θ), and phonon velocities (v) in Table 3.

conductivity of Cu12Sb4Se13 at 300 K is ∼0.1 W m−1 K−1, which is lower than the minimum lattice thermal conductivity of the phonon glass limit (∼0.25 W m−1 K−1). Such an extremely low theoretical lattice thermal conductivity could be due to approximations in the Debye−Callaway model, which does not take into account contributions from optical modes. From our previous work,13 we found that the theoretically calculated thermal conductivities using the Debye−Callaway model are typically smaller than the experimental observed values. Considering the smallest error bar (∼0.28 W m−1 K−1) in the previous Cu−Sb−Se calculations, we can estimate the lattice theormal conductivity of Cu12Sb4Se13 at 300 K should be less than ∼0.38 W m−1 K−1, which is still an extremely small value. Such low lattice thermal conductivity indicates that Cu12Sb4Se13 could be a promising thermoelectric material. Electronic Properties of Cu4SbSe5 and Cu12Sb4Se13. To further determine whether Cu4SbSe5 and Cu12Sb4Se13 are promising thermoelectric materials, we now investigate their electronic structure. Figure 6 shows the calculated electronic band structures of Cu4SbSe5 and Cu12Sb4Se13. Even though the two stoichiometric compounds are metallic (the Fermi level crosses valence bands), the two band structures have the same kind of unusual features as found in Cu12Sb4S13;15 i.e., there are unoccupied hole states in the valence bands. By doping with Zn or Fe, as was done in Cu12Sb4S13,15 we can introduce additional electrons in the system to fill the holes and move the Fermi level higher, where the thermoelectric power factor reaches a maximum. We have investigated this possibility using DFT. We substitute Zn for Cu in various sites, finding an energetic preference for Zn in tetrahedral positions. We find that 75% Zn in Cu4SbSe5 (Zn3CuSbSe5) and 16.6% Zn in Cu12Sb4Se13 (Zn2Cu10Sb4Se13) are enough to push the Fermi level up within the band gap (Figure 6). The band gap of Zn2Cu10Sb4Se13 is 0.66 eV, which is smaller than that of the Zn substituted Cu12Sb4S13 system (∼0.8 eV).15 Moreover, from the band structures of the pure, undoped compounds (Figure 6), we note the favorable band structure features for good thermoelectric performance, such as valence band degeneracy, band extrema at off-Γ points, and high effective masses. Lu et al.15 recently investigated Cu12Sb4S13 doped with Zn or Fe and obtained high ZT values (around 1) over a large range of substitution level. Because Cu12Sb4Se13 bears resemblance to G



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in this phase. Band structure calculations provide further evidence that Cu12Sb4Se13 could be a promising thermoelectric compound, on the basis of valence band characteristics, provided it is sufficiently doped (e.g., with Zn) to move the Fermi level into the band gap.

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ASSOCIATED CONTENT

S Supporting Information *

Theoretically calculated phonon density of states of Cu4SbSe5 and Cu12Sb4Se13. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*Y. Zhang: e-mail, [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Collaborative work among NU, MSU, and UCLA is supported as part of the Center for Revolutionary Materials for Solid State Energy Conversion, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Award Number DESC0001054. The authors gratefully acknowledge the use of the high performance computing system at Northwestern University and computing resources at the National Energy Research Scientific Computing Center, which is supported by the US DOE under Contract No. DE-AC02-05CH11231.

Figure 6. Electronic band structures of Cu4SbSe5 (a) and Cu12Sb4Se13 (b) with and without Zn doping. The high symmetry points in the first Brillouin zones (BZ) of the two compounds can be found in Figure 4. The red dashed lines indicate the position of the Fermi level.

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Cu12Sb4S13 in crystal structure, band structure, and low lattice thermal conductivity, we suggest that this theoretically predicted compound is a promising thermoelectric material as well.

REFERENCES

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SUMMARY We have investigated phase stability, predicted new compounds, and determined promising thermoelectric behavior of compounds in the Cu−Sb−Se ternary system via a computational DFT approach. Our theoretically computed Cu−Sb−Se ternary phase diagram provides important information for future experimental investigation of this system, such as temperature-dependent phase stability and possible new compounds and stoichiometries. We predict the stability of a new previously unknown, zinc blende-based Cu 4 SbSe 5 compound but find that it is thermodynamically stable up to only ∼300 K. However, we also predict the existence of a new, previously unreported Cu12Sb4Se13 phase (isostructural with Cu12Sb4S13), which appears in the phase diagram at high temperatures (but below the temperatures where the observed Cu3SbSe3 phase is stable). Thus, we hypothesize that similar procedures previously used to synthesize Cu3SbSe3 could be successful in producing this new Cu12Sb4Se13 compound. Using quasi-harmonic phonon calculations, we compute the Debye temperatures, phonon velocities, and Grüneisen parameters for our predicted phases, and use these quantities in the Debye− Calloway model to determine lattice thermal conductivity. Our calculations suggest that strong anharmonicity is present in Cu12Sb4Se13 and leads to an extremely low thermal conductivity H



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Prediction of New Stable Compounds and Promising

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