Designing High-Efficiency Nanostructured Two-Phase Heusler

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Designing high-efficiency nanostructured two-phase Heusler thermoelectrics Vancho Kocevski, and Chris Wolverton Chem. Mater., Just Accepted Manuscript • DOI: 10.1021/acs.chemmater.7b03379 • Publication Date (Web): 13 Oct 2017 Downloaded from http://pubs.acs.org on October 13, 2017

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First-principles theory

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Chemistry of Materials

Single phase Thermoelectric

DFT mixing energies / favorable solvus

screen for stability/tie-line between Matrix/NS

Host Matrix

T T1

S

L+S solvus fit to:

promising two-phase thermoelectrics

Nanostructured phase

Solid solution

T2

matrix + nanostructured phase

e

_

e _ e _

Synthesis

e

_

Nanostructured material phonon

h

h+ h

+

+

h

+

ACS Paragon Plus Environment

xi


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Designing high-efficiency nanostructured two-phase Heusler thermoelectrics Vancho Kocevski∗,†,‡ and Chris Wolverton† †Department of Materials Science and Engineering, Northwestern University, Evanston, IL 60208, United States ‡Current address: Department of Mechanical Engineering, University of South Carolina, Columbia, SC 29208 E-mail: [email protected],[email protected] Abstract Nanostructured systems formed by two distinct phases are particularly promising for high efficiency thermoelectrics, due to the reduction in thermal conductivity afforded by the nanostructured phase. However, the choice of the matrix and nanostructured phases represent a challenging materials discovery problem, due to the large compositional space involved. Heusler phase thermoelectrics are particularly promising candidates for nanostructuring, since these compounds often possess favorable electronic thermoelectric properties, but relatively high thermal conductivity. Here, we have developed a high-throughput screening strategy to predict promising candidates for nanostructuring systems based on two Heusler phases. Our search includes all two-phase systems involving full, half, and inverse Heusler in the Open Quantum Materials Database, in total a search space of ∼ 1011 possible combinations of two Heusler compounds. To reduce this space, our screening approach starts with a set of known thermoelectrics as matrix phases, and screens for all second phase compounds that are stable and form a two-phase equilibrium with the matrix. We compute mixing energies for the resulting

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combinations of a matrix and a nanostructured phase, and find systems that have a moderately large positive mixing energy, and hence show an appropriate balance between tendency for nanostructuring and solubility of the second phase. Our screening approach gives 31 pairs, 2 of which have been explored experimentally (thus validating our screening strategy), and 29 of which represent new predictions of systems awaiting experimental synthesis. In addition, our results show that matrix/nanostructure pairs consisting of distinct crystal structures (e.g., mixing of half Heusler with full Heusler) typically have low mutual solubility, whereas isostructural matrix/nanostructured phase pairs (where both matrix and nanostructure have the same structure type, half Heusler or full Heusler) more often have energetics suitable for forming nanostructures or solid solutions.

1

Introduction

Utilizing the thermoelectric effect to scavenge waste heat into usable electricity has been of great interest for global energy sustainability. Thermoelectric materials could, in principle be used in power generation from industrial processes, home heating, automotive exhaust and other heat generating sources, with materials that have large conversion efficiency being of great importance. However, identifying and developing novel thermoelectric materials with high conversion efficiency faces many challenges mainly due to the difficulties in independent manipulation of their properties. The efficiency of thermoelectric materials is determined by the thermoelectric figure of merit, ZT , defined as:

ZT =

σS 2 T κ

(1)

where S is the Seebeck coefficient, σ is the electrical conductivity, T is the thermodynamic temperature, and κ is the thermal conductivity of the material [the sum of the electrical (κel ) and lattice (κlat ) thermal conductivity]. Evidently, ZT can be increased by decreasing

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the denominator in Eq. 1, κ, and/or by increasing the numerator, σS 2 , known as the thermoelectric power factor (PF) of a material (PF = σS 2 ). Unfortunately, the S, σ and κel are strongly correlated with the electronic structure of the material and they often cannot be independently optimized. One method for improving ZT of bulk materials that has received a lot of attention involves lowering the lattice thermal conductivity by introducing nanostructures into a host matrix, 1,2 i.e., nanostructuring. For example forming two-phase nanostructured (NS) material can significantly increase the ZT of Bi2 Te3 3 and PbTe 4–8 based thermoelectric materials. These materials have moderate PFs (1−3 mW m−1 K−2 ) and the increase in ZT comes largely from the significant lowering of the thermal conductivity. Thus, having materials with even higher PF, where the thermal conductivity alone can be decreased by nanostructuring, would be of considerable interest. A class of materials that are shown to have these particular properties of interest are Heusler compounds: full Heusler (FH) and half Heusler (HH). This adds to the widespread applicability of the Heusler compounds 9,10 in areas such as optoelectronic applications, 11,12 spintronics, 13–15 superconductivity 16–19 and topological insulators. 20–22 Many thermoelectric studies of NS HH based materials focus on MNiSn 23–39 and MCoSb 40–50 (M = Ti, Zr, Hf) as host matrices, where the NS phase can be either a FH MNi2 Sn or other M (M = Ti, Zr, Hf) rich HH materials. Similarly, there have been studies of the thermoelectric performance of other HHs, namely MFeSb (M = V, Nb), 51–54 and of FHs such as Fe2 TiAl 55 and Fe2 VAl. 56–58 One should note that, all of these previous studies have focused on nanostructuring in systems where the host matrix is a reasonably good thermoelectric. Interestingly, some studies show that mixtures of HH phases, with no obvious NS phase formation, can also have high ZT, 59–62 showing the broad usability of the HH compounds as thermoelectrics. Expanding the field of research of single phase Heusler compounds as thermoelectrics can be accelerated by the use of first-principles calculations. In recent years there have been theoretical studies aimed at finding better single phase Heusler thermoelectrics, based on

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screening strategy to determine Heusler thermoelectric compounds that can form two-phase NS materials with other Heusler compounds. The main goal is to utilize the predictive power of the DFT to determine suitable combination of matrix and nanostructured phase, termed matrix/NS pairs, that can be synthesized as two-phase NS materials, and hence should possess improved thermoelectric properties. The main objective of our study is schematically outlined in Fig. 1. We consider two-phase systems involving the ∼ 105 full, half, and inverse Heusler compounds in the Open Quantum Materials Database (OQMD), 70,71 in total a search space of ∼ 1011 possible combination of two Heusler compounds. We reduce this space significantly by starting from host matrices which are known to possess favorable thermoelectric properties and screen for all second phase systems that are stable, and form a two-phase equilibrium with the matrix. This gives us ∼ 104 possible pairs, which we reduce further to ∼ 102 pairs by considering systems with mixing of only one element, and eliminating radioactive/actinide elements. Using this screening approach, we find 106 matrix/NS phase pairs. We use DFT to investigate the mixing energy of the matrix/NS phase pairs, and we estimate the solvus boundary between the matrix and the NS phase, from which we determine a mixing energy interval in which formation of two-phase NS materials is favorable. We found 31 pairs with mixing energy within the energy interval that promotes formation of twophase NS materials. Two of the systems have been experimentally reported in the literature as NS Heusler systems (thus validating our approach), leaving 29 promising themoelectric matrix/NS phase pairs which are predictions of this work, and await experimental synthesis. Several of the 29 matrix/NS pairs are very promising candidates for NS thermoelectrics due to the large mass difference between the mixing atoms (e.g. Fe2 TiSn/Fe2 HfSn pair) or because the mixing atom serves as dopant (e.g. Fe2 TiSi/Mn2 TiSi pair).

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ionic positions. Note that the calculated energies are at zero temperature and pressure. In addition, for better estimate of the solubility between the two compounds we also calculated the vibrational entropy of mixing (see Sec. 3.3), for which the phonon dispersion is required. The phonon dispersion was calculated using finite displacement method in 2×2×2 supercells with the Phonopy code. 76 The mixing of the host matrix with the NS phase is done such that the stoichiometry of the mixed system always lies on the tie-line between the two constituent compounds. A tie-line connecting two phases implies a stable two-phase equilibrium exists between those phases. In this paper, we generally look for the existence of a tie-line on the T = 0 K convex hull, to determine stable two-phase equilibria. Mixing on the Y or Z site is performed by respectively substituting one of the Y or Z atoms with the mixing atom. However, keeping in mind the difference in number of atoms on X site in HH and FH compounds and the possibility of mixing between HH and FH, mixing on the X site requires explanation. The case of HH/HH mixing (or FH/FH) mixing is simpler, where one (or two) X atoms are substituted by the mixing atom. On the other hand, in the case of HH matrix and FH NS phase (HH/FH mixing), one atom on X1 site is substituted by the mixing atom and another mixing atom is placed in the neighboring X2 (vacant) site, thus keeping the stoichiometry of the system still on the tie-line between the two compounds. Similarly, in the case of FH matrix and HH NS phase (FH/HH mixing) the X1 atoms is substituted by the mixing atom and a neighboring X2 atom is removed leaving a vacancy, as this sublattice is vacant in the HH structure. For clearer picture of the mixing on the X site, in Fig. 3 we have shown an atomistic representation of the supercells depicting the mixing between different types of Heusler compounds. Having in mind the large number of calculations that need to be performed, identifying the smallest possible supercell that can produce reliable results is important. Therefore, we considered 5 different sizes of supercells; for the case of HH structure there are: 12-atoms (conventional cell), 24-atoms (2×2×2 of the primitive cell), 48-atoms (rhombohedral bcc-

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XY(A)Z

where Etot

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is the DFT total energy of a supercell of XYZ with 1 mixing atom and NY

XYZ is the number of atoms in a particular sublattice (in this case the Y sublattice). Etot and XAZ Etot are the DFT total energies of the supercells of pure XYZ and XAZ, respectively. The

calculated mixing energies of the matrix/NS phase pairs using different supercell sizes are summarized in Table 1. Table 1: Mixing energies (in eV) between half Heusler matrices and different Heusler compounds as NS phase, as a function of the supercell size. System (mixing atom) NiTi(Zr)Sn NiZr(Ti)Sn NiTiSn(Sb) Ni(Ni2 )TiSn Ni(Co2 )TiSn

12-atoms 0.16 0.15 −0.17 0.37 0.44

24-atoms 0.15 0.14 −0.23 0.40 0.33

Supercell size 48-atoms 81-atoms 0.15 0.15 0.14 0.14 −0.27 −0.29 0.39 0.39 0.34 0.33

96-atoms 0.16 0.14 −0.29 0.39 0.32

Evidently, for each of the studied systems the changes of the mixing energy when using 81atom or 48-atom supercell is very small (< 5 %) compared to the mixing energy for 96-atom supercell. In the case of the 24-atom supercell, except for the NiTiSn/NiTiSb system that exhibits significant change in the mixing energy(∼ 20 %), the mixing energies for all other systems show small difference with respect to the largest supercell. Further decreasing of the supercell size influences only the mixing energy in the NiTiSn/NiTiSb and NiTiSn/Co2 TiSb systems. The other 3 studied systems show that even at the smallest supercell, 12 atom, the mixing energy are converged, and the mixing energies change insignificantly compare to the largest supercell. This demonstrates that although using the 24-atom supercell can yield good results in some cases, the 48-atom supercell would give reliable results in most of the cases. Therefore, for the rest of the study we used the 48-atom supercell.

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3

Results and discussion

3.1

Host matrices and nanostructured phase screening

In this study, we use the OQMD. 70,71 As of October 2016, 77 the OQMD contains ∼265,000 FH, HH and IH compounds. 78 Exploring every possible combination of two-phase matrix/NS pair would entail a space of (265, 000)2 ∼ 1011 possible systems. Evidently, this is an impractically large space of possible systems, and hence, we need to significantly narrow the search space. We start with the host matrices, and to narrow the composition space we are considering the HH compounds that have been shown to exhibit favorable thermoelectric properties, 23–54,59 NiM1 Sn and CoM1 Sb (M1 = Ti, Zr, Hf) and FeM2 Sb (M2 = V, Nb). As FH host matrices, we are considering compounds that are shown to have high power factor, 63 Fe2 M1Sn and Fe2 M1Si (M1 = Ti, Zr, Hf), or have been previously experimentally studied as thermoelectrics, 55–58,79 Fe2 TiAl and Co2 TiSn. See Table 2 for the full list of host matrices and their respective references. In addition, because we are are going to screen for stable, two-phase mixtures, the host matrices should be thermodynamically stable, i.e., to be on the convex hull according to the OQMD. After introducing the latter criteria we were left with 5 HH and 9 FH stable matrices, the onces specified with S in Table 2. As mentioned in the previous section, the main goal of the study is to find compounds that can form NS within the host matrices. Therefore, it is crucial to find compounds that can form a two-phase equilibrium, where both the matrix and NS phase are stable, i.e., compounds that have convex hull tie-lines with the host matrices. From the initial screen for possible tie-lines between the considered host matrices and any Heusler compound (half, inverse Heusler and full Heusler) in the OQMD, we obtained 15350 possible matrix/NS phase pairs. Considering that we have 14 host matrices, this means that, on average, each one of the host matrices has ∼1000 tie-lines to other Heusler phases in the OQMD. This is a substantial number of pairs, and by screening out the NS phases that have 2 common elements with the host matrices, we were left with 189 pairs. We further screened out the 10



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Table 2: List of considered host matrices and the respective known experimental and theoretical studies. Y and N indicates if the compounds were previous experimentally and/or theoretically studied or not, respectively. The OQMD column shows if the compound is stable (S) or unstable (U) in the OQMD (using the data set as of October 2016 77 ), where in parenthesis are shown the the distance from the convex hull of the unstable compounds. Half Heusler matrices Comp. Expr. OQMD a NiTiSn Y S NiZrSn Ya S a NiHfSn Y U (0.640 eV) 77 b S CoTiSb Y CoZrSb Yb U (0.887 eV) 77 U (0.925 eV) 77 CoHfSb Yb c FeVSb Y S c FeNbSb Y S

a b c d e f g

Comp. Fe2 TiSi Fe2 ZrSi Fe2 HfSi Fe2 TiSn Fe2 ZrSn Fe2 HfSn Fe2 VAl Fe2 NbAl Fe2 TaAl Fe2 VGa Fe2 NbGa Fe2 TaGa Fe2 TiAl Co2 TiSn

Full Heusler matrices Expr. Theory OQMD g N Y S N Yg U (0.129 eV) N Yg U (0.064 eV) g N Y S N Yg U (0.048 eV) N Yg S d Y Yg S g N Y S g N Y S N Yg U (0.054 eV) N Yg U (0.076 eV) N Yg S e Y S f Y S

see Refs. 23–39 see Refs. 40–50 see Refs. 51–54,59 see Refs. 56–58 see Ref. 55 see Ref. 79 see Ref. 63

compounds having actinides and lanthanides, arriving at 138 pairs. Having in mind the high PF of the Heusler compounds, our interest is having matrix/NS phase pairs that can form a coherent interface that might facilitate electron transport without much scattering and hence have an insignificant influence on the PF. In previous studies has been argued that although the influence of the coherent interface on the lattice thermal conductivity is smaller compared to the influence of the incoherent interface, the coherent interface has a greater influence on the increasing ZT of NS thermoelectrics. 34,35,80,81 A coherent interface typically requires a relatively small lattice mismatch between the matrix and NS phase. Therefore, we screened out the NS phases that have a lattice parameter 11


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nanostructured phase.

3.2

Alloying of Heusler compounds

To facilitate the calculations of the mixing energies for the whole set of matrix/NS phase pairs, we considered various alloying sites and different number of alloying elements to verify that the method of mixing the host matrix and NS phase gives the lowest mixing energy. We took NiTiSn as an illustrative host matrix and we alloyed the matrix with one atom, taking Sb and Zr as alloying atoms, which corresponds to HH/HH mixing. In addition, we are interested in the mixing between HH and FH compounds, therefore we alloyed NiTiSn with: two Fe atoms, corresponding to NiTiSn/Fe2 TiSn (HH/FH) mixing, and one Ni atom that corresponds to NiTiSn/Ni2 TiSn (another type of HH/FH mixing). The defect formation energy of a element A, ∆EfA , is calculated using the equation:

∆EfA

A

0

=E +E −

N X

∆ni (µ0i + ∆µi )

(3)

i=1

where E A and E 0 are the DFT total energies of a supercell with and without the alloying element, respectively. The summation is done over all (N ) atomic species i in the supercell (e.g. X, Y, Z and A). ∆ni is the change in the number of atom type i in the supercell, where ∆ni > 0 and ∆ni < 0 show that atom is being added and removed, respectively. µ0i and ∆µi are the chemical potential of the atomic species i in their elemental state (DFT ground state crystal structure) and the change in the chemical potential relative to µ0i as a result of the N -phase equilibrium, respectively. The N -phase equilibrium is obtained from the OQMD, by calculating the stable phases in equilibrium at the particular composition. For example, alloying TiNiSn with Sb can be done on Ti, Ni, Sn or vacancy site, yielding four different compositions, Ti15 Ni16 Sn16 Sb, Ti16 Ni15 Sn16 Sb, Ti16 Ni16 Sn15 Sb, and Ti16 Ni16 Sn16 Sb, respectively. To obtain the 4-phase equilibrium, for each of the four composition we calculated the phases in equilibrium ac13


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cording to the OQMD. For example, in the case of Ti16 Ni16 Sn16 Sb those four phases are TiNiSn, TiSb2 , NiSb and Ni3 Sn4 . After the N -phase equilibrium has been defined, the set of chemical potentials, ∆µi can be calculated solving the system of linear equations:

∆Ek =

N X

cik ∆µi

(4)

i=1

where ∆Hk is the formation energy of phase k (k = 1, ..., N ), taken from the OQMD, and cik is the composition of element i in phase k. In the case of a single alloying element, the NiTiSn 48-atom supercell was alloyed separately on of the four sublattices: vacancy (Vac), Ti, Ni or Sn (see Fig. 2 for more detail on the different sublattices), with one Sb, one Zr or one Ni atom. Our calculations show that Sb alloying on the Sn site (NiTiSn/NiTiSb mixing) and Zr alloying on Ti site (NiTiSn/NiZrSn mixing) have the lowest defect formation energies: -0.27 eV and 0.13 eV, respectively. On the other hand the defect formation energies for any of the other three sites are larger than 1.0 eV, showing that the energetically favorable site is consistent with the one that is intuitively expected from the chemical similarity and structures of the constituent Heusler compounds. In the case of Ni alloying, the configuration with a Ni on a vacancy site, corresponding to the tie-line between NiTiSn and Ni2 TiSn, has defect formation energy of 0.23 eV, which is significantly lower compared to the defect formation energy on Ti or Sn site. Having two atoms as alloying elements is more complicated because of the added extra degree of freedom where the alloying elements can be placed. For example, one of the Fe atoms can be placed on a vacancy site of the NiTiSn. The second Fe atom can go on one of the four sites, Ni, Ti, Sn or vacancy, that are nearest neighbors (NNs) to the first Fe atom, or on one of the four sites that are second NNs, and so on. For the purpose of the study, we considered the distance between two alloying elements only up to the second NN. Summarized in Table 3 are the calculated defect formation energies for the NiTiSn with two Fe atoms as alloying elements. In Table 3 the sublattices column notes the alloyed sublattices

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Table 3: Defect formation energies (in eV) for two Fe atoms in NiTiSn matrix. Shown in bold are the lowest defect formation energy, and the corresponding alloying sublattices and the NN position of the Fe atoms. Sublattices Ni Ni Ni Ni Ni Sn Ni Sn Ni Ti Ni Ti Ni Vac Ni Vac Sn Vac Sn Vac Sn Sn Sn Sn Ti Sn Ti Sn Ti Ti Ti Ti Ti Vac Ti Vac Vac Vac Vac Vac

NN ∆EfA 1st 2.34 2nd 2.30 1st 3.13 2nd 3.91 1st 2.44 2nd 2.72 1st 0.71 2nd 0.87 1st 2.50 2nd 3.94 1st 6.54 2nd 6.63 1st 4.28 2nd 5.59 1st 3.47 2nd 3.52 1st 2.11 2nd 3.48 1st 1.23 2nd 1.31

(e.g. Ni and Ni, Ni and Sn etc.), the NN column shows if the alloying Fe atoms are placed as first or second NN neighbors, and the third column is the defect formation energy. Evidently, the energy of the alloying on Ni and Vac sublattices is significantly lower compared to energy of alloying at any other combination of sublattices, showing that it is the most energetically favorable way of HH/FH mixing. Moreover, substituting Ni atom with Fe and having one Fe atom on the 1st NN vacancy site is the most favorable configuration (shown in bold in Table 3). Interestingly, the structure with one Fe atom on Ni site and the other Fe atom on vacancy site is non magnetic, in contrast to the structures alloyed on two vacancy sites and on two Ni sites, with the latter two being magnetic. Based on COHP analysis we see that the magnetic configuration significantly weakens the Fe–Ti and Fe–Ni bonds in the Fe alloyed NiTiSn,

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compared to the non magnetic configuration, and hence increases the energy of the whole system. This decrease in the bonding strength between Fe–Ni and Fe–Ti is the main reason for the considerably higher defect formation energy of NiTiSn alloyed with two Fe atoms on Ni and vacancy sites compared to the NiTiSn alloyed on two vacancy sites or two Ni sites.

3.3

Mixing energy intervals

Because the main interest in this study is to find matrix/NS phase pairs that can phase separate at typical processing temperatures, having a very high or very low solubility is impractical. If the solubility is too high, there will be little driving force for phase separation at reasonable temperatures, and if the solubility is too low, it will be difficult to incorporate the alloying elements in the material to begin with. Therefore, it is meaningful to find a mixing energy interval in which the solubility of the two compounds at elevated temperatures (e.g. typical processing temperatures between the solvus boundary and melting point of the host matrix) is high enough for the two compounds to mix, but at lower temperatures the solubility is low enough to facilitate phase separation. To estimate this interval of mixing energies, ∆Emix , that suggests a favorable formation of two-phase NS material (NSing energy interval), we calculated the solvus of a solute i, xi (T ), using the following expression: 82 ∆Emix ∆Svib exp − xi (T ) = exp kb kb T

(5)

where kb and T are the Boltzmann constant and the thermodynamic temperature, respectively. ∆Svib is the vibrational entropy, defined as:

∆Svib = kb

Z

νmax

ln 0

16

kb T hν

∆D(ν)dν


(6)


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where h is the Planck’s constant and ν is the phonon frequency. ∆D(ν) is the composition weighted phonon density of states (DOS), given by: ∆D(ν) = DXY(A)Z (ν) −

NY − 1 XYZ 1 XAZ D (ν) − D (ν) NY NY

where DXY(A)Z (ν), DXYZ (ν) and DXAZ (ν) are the DFT phonon DOS of XYZ supercell with 1 mixing atom and the supercells of pure XYZ and XAZ, respectively, and NY is the number of atoms in a particular sublattice. Calculating the vibrational entropy for every matrix/NS phase pair is very computationally demanding and time consuming. Thus, we considered 5 matrix/NS phase pairs: Co2 TiSn/Fe2 TiSn, NiTiSn/NiZrSn, NiZrSn/NiTiSn, Fe2 TiAl/Fe2 TiSn and Fe2 TiSn/Fe2 TiAl, representing the mixing on X, Y and Z sites, for which we calculated the vibrational entropy. It has been shown that the experimental solvus in various Al alloys can be very well represented when the vibrational entropy is considered when calculating the solvus. 83–85 In the case of mixing between two Heusler compounds we were able to find only two studies detailing the phase diagram of NiTiSn with excess Ni. 86,87 However, our DFT calculations show that the Ni2 TiSn is dynamically unstable at 0 K (has negative phonon frequencies), hence it is difficult to compare the calculated vibrational entropy of the mixing between NiTiSn and Ni2 TiSn. Dynamical instability of the Ni2 TiSn has been previously reported by Page et al., 67 and its origin was proposed to be the small size of the Ti 3d orbitals. Nevertheless, the calculated vibrational entropies for all 5 cases considered are within ±0.2kb interval, which has rather small effect on the solubility, especially at lower temperatures – see the yellow shaded area in Fig. 5A. Thus, for estimating the NSing energy interval we ignore the contribution from vibrational entropy and calculate the solubility using: ∆Emix xi ∼ exp − . kb T

(7)

The lower limit for the NSing energy interval was obtained by specifying that the solubil17


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peratures at which the Heusler two-phase NS materials are synthesized. It is also important to note that in experiments of mixing between NiTiSn and Ni2 TiSn, an obvious formation of Ni2 TiSn NS in the NiTiSn matrix has been reported, which experimentalists argue is indicative of sufficient mutual solubility of Ni2 TiSn in NiTiSn. 32,34,87,88 Our calculations show a defect formation energy of 0.23 eV for alloying of Ni on a vacancy site in NiTiSn (which corresponds to a composition on the tie-line between NiTiSn and Ni2 TiSn), see Sec. 3.2. The defect formation energy is within the proposed NS energy interval, indicating that a two-phase system with mixing energy within the proposed NS energy interval can probably form nanostructured material. However, we note that the choice of the solubility criteria is somewhat arbitrary, but the idea is quite general, and the mixing energies we report can be utilized for a screening using different temperature and concentration conditions. Nevertheless, indicating an energy interval in which nanostructuring is expected to occur will help guide our screening strategy toward systems that experimentally should form two-phase NS materials. In this way the matrix/NS phase pairs can be divided in four distinct categories: I a stable quaternary compound exists (negative mixing energy) II compounds that have a very high solubility (mixing energy below the lower NSing limit) III possibility for forming two-phase NS materials (mixing energy within the NSing interval) IV compounds that form stable interfaces (mixing energy higher the the upper NSing limit)

3.4

Mixing in half Heusler host matrices

We begin the study of the nanostructuring in Heusler compounds by analyzing the mixing between the HH matrices: NiTiSn, NiZrSn, CoTiSb, FeVSb and FeNbSb, and compounds that were found to have tie-lines with the host matrices. For each of the matrix/NS phase pairs we calculated the mixing energies, as in Eq. 2, using a 48-atom supercell, where the NS phases is mixed in the host matrix as detailed in Sec. 2. The calculated mixing energies

19


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Table 4: Mixing energies (in eV) of half Heusler matrices and various NS phases, arranged in ascending order of mixing energies (increasingly unfavorable for mixing). Shown in bold are the compounds with a mixing energy within the NSing energy interval, defined based on the solubility model (see discussion in Sec. 3.3). Host matrix

NiZrSn

NS compound ∆Emix NiTiSb 0.07 Co2 TiSn 0.80 Co2 TiIn 1.28 Co2 TiGa 1.55 Co2 TiAl 1.82 Co2 TiZn 1.83 Co2 TiMn 2.42 FeNbSb 0.13 Fe2 VGa 1.30 Fe2 VAl 1.59

FeNbSb

Host matrix CoTiSb

NS compound ∆Emix NiTiSb -0.27 NiZrSn 0.15 PtTiSn 0.17 Co2 TiSn 0.34 Ni2 TiSn 0.39 Fe2 TiSn 0.71 Ni2 TiIn 0.80 Ni2 MgSn 0.83 Ni2 TiGa 0.91 Ni2 TiZn 1.29

FeVSb

Host matrix

NiTiSn

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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NS compound ∆Emix NiTiSn 0.14 Ni2 HfSn 0.56 Ni2 TmSn 0.56 Ni2 ScSn 0.65 Ni2 ZrAl 1.14 Ni2 MgSn 1.24 Ni2 LiSn 1.60 FeVSb 0.11 RuNbSb 0.17 Fe2 NbGa 1.48 Fe2 NbAl 1.74

only one of the pairs, NiTiSn/NiTiSb, is within the category I. Also, only one pair, CoTiSb/NiTiSb, can be placed in the category II, suggesting that these compounds have a very low-temperature miscibility gap, and hence extremely high solubility. Lastly, 6 matrix/NS pairs have mixing energy within the NSing energy interval – the category III. Two of those pairs, NiTiSn/NiZrSn and NiZrSn/NiTiSn, have been previously shown to form two-phase NS materials. 36–39 Bhattacharya et al. 36 show that NiTi0.75 Zr0.25 Sn forms distinct Zr-rich regions ∼ 1 µm in size, and in the case NiTix Zry Hf1−x−y Sn (with 0.3 < x < 0.5 and 0.25 < y < 0.37) a clear separation between Ti-rich and (Zr,Hf)-rich regions has been reported. 37–39 The other four pairs with mixing energy within the NS energy interval, NiTiSn/PtTiSn, FeVSb/FeNbSb, FeNbSb/FeVSb and FeNbSb/RuNbSb, have not been hitherto studied, are promising candidates as NS thermoelectrics, and we call for imminent experimental testing and validation of these predictions.

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3.5

Mixing in full Heusler host matrices

The study is further extended to the mixing of the FH matrices: Co2 TiSn, Fe2 HfSn, Fe2 NbAl, Fe2 TaAl, Fe2 TiAl, Fe2 TiSi, Fe2 TaGa, Fe2 TiSn, and Fe2 VAl, with different compounds as possible NS phases. For each of the matrix/NS phase pair the mixing energies were calculated using Eq. 2 and the calculated mixing energies are shown in Table 5. As in the case of the HH matrices, for a better overview in which of the 4 categories the matrix/NS phase pair can be placed, the mixing energies are plotted as a function of mixing site and type of NS phase – see Fig. 7. It is noticeable that the mixing energies of the FH/FH systems are typically lower, compared to the mixing energies between FH/HH pairs, which is similar to the mixing in HH matrices, where HH/HH mixing has lower energy. However, there is larger number of pairs that have mixing energy in the NSing energy interval (category III) and in high solubility interval (category II). From the whole list of pairs, 25 FH matrix/NS phase pairs have mixing energy within the NSing energy interval, showing the possibility of forming two-phase NS materials and are good candidates for potential future use in thermoelectric applications. For a more clear separation from the other matrix/NS pars, the 25 FH matrix/NS phase pairs that have mixing energy within the NSing energy interval are given in Table 6, together with the 6 HH matrix/NS phase pairs. Table 5: Mixing energies (in eV) of full Heusler host matrix and various Heusler compounds as NS phase, arranged in ascending order of mixing energies (increasingly unfavorable for mixing). Shown in bold are the compounds with a mixing energy within the NSing energy interval, defined based on the solubility model (see discussion in Sec. 3.3).

22


NS compound ∆Emix Fe2 TiAl -0.11 Fe2 TaGe -0.10 Fe2 WAl -0.03 Fe2 TaGa -0.01 Fe2 NbAl 0.00 Fe2 MnAl 0.08 Fe2 VAl 0.18 Co2 TaAl 0.25 Fe2 TiGa -0.22 Fe2 TaGe -0.08 Fe2 TaAl -0.01 Fe2 NbGa 0.00 Fe2 VGa 0.15 Co2 TaGa 0.23 Ru2 TaGa 0.44 PtTaGa 2.00

Host matrix

Fe2 TiSn

Host matrix

Fe2 VAl

∆Emix -0.12 -0.04 -0.01 0.00 0.10 0.19 0.22 0.33 1.59 -0.11 -0.06 0.03 0.20 1.39

Fe2 TaAl

NS compound Fe2 TiAl Fe2 WAl Fe2 NbGa Fe2 TaAl Fe2 MnAl Fe2 VAl Co2 NbAl Mn2 NbAl FeNbSb Fe2 TiAl Fe2 TiGa Fe2 TiGe Mn2 TiSi Fe2 CoSi

Fe2 TaGa

Host matrix

Fe2 NbAl

NS compound ∆Emix Fe2 WAl -0.72 Cu2 TiAl -0.33 Fe2 TiGe -0.07 Fe2 MnAl -0.02 Fe2 VAl -0.01 Fe2 TiGa -0.01 Fe2 NbAl 0.00 Fe2 TaAl 0.01 Fe2 TiSi 0.04 Fe2 TiSn 0.10 Ru2 TiAl 0.24 Mn2 TiAl 0.64 Fe2 TiSn 0.10 Ru2 HfSn 0.27 PdHfSn 0.79 PtHfSn 1.25

Fe2 TiSi

Host matrix

Fe2 TiAl

NS compound ∆Emix Co2 ScSn -0.08 Co2 TiIn -0.07 Fe2 TiSn 0.02 Co2 LuSn 0.07 Co2 HfSn 0.10 Co2 MnSn 0.12 Rh2 TiSn 0.12 Co2 ZrSn 0.14 Co2 TiGa 0.14 Co2 TiAl 0.15 NiTiSn 0.38 Ru2 TiSn 0.50 PtTiSn 0.92

Fe2 HfSn

Host matrix

Co2 TiSn

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


NS compound ∆Emix Fe2 TiAl -0.18 Fe2 TiGa -0.09 Fe2 HfSn 0.12 Co2 TiSn 0.16 Fe2 TiGe 0.21 Ru2 TiSn 0.29 NiTiSn 0.51 PtTiSn 1.01 Fe2 TiAl -0.16 Fe2 MnAl -0.12 Fe2 WAl -0.12 Fe2 VGa -0.02 Co2 VAl 0.09 Fe2 VBe 0.10 Fe2 TaAl 0.22 Fe2 NbAl 0.25



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Table 6: Matrix/NS phase pairs that are predicted to favor formation of two-phase NS materials (pairs with mixing energy within the NSing energy interval, defined based on the solubility model – see discussion in Sec. 3.3). HH matrices Matrix

NS comp. NiZrSnh NiTiSn PtTiSn NiZrSn NiTiSnh FeVSb FeNbSb FeVSb FeNbSb RuNbSb h

4

FH matrices Matrix

Co2 TiSn

NS comp. Co2 HfSn Co2 MnSn Rh2 TiSn Co2 ZrSn Co2 TiGa Co2 TiAl

Matrix

NS comp. Fe2 TiSn Fe2 TiAl Ru2 TiAl Fe2 TiSn Fe2 HfSn Ru2 HfSn Fe2 MnAl Fe2 NbAl Fe2 VAl Co2 NbAl

Matrix Fe2 TiSi Fe2 TaAl Fe2 VAl

NS comp. Mn2 TiSi Fe2 VAl Co2 TaAl Fe2 VBe Fe2 TaAl Fe2 NbAl

Matrix Fe2 TiSn

Fe2 TaGa

NS comp. Fe2 HfSn Co2 TiSn Fe2 TiGe Ru2 TiSn Fe2 VGa Co2 TaGa

Matrix/NS phase pairs that have been previously studied experimentally

Conclusions

Heusler compounds have been emerging as vital thermoelectric materials, whose efficiency can be significantly increased by nanostructuring. Utilizing the predictive power of firstprinciples calculations can considerably speed up the process of finding new compounds that can be precipitated as nanostructures within a Heusler matrix. In the presented study, first we screened for convex hull tie-lines, indicating a stable two-phase equilibrium between HH and FH compounds that are known to exhibit favorable thermoelectric properties and every HH, FH and IH in the OQMD, arriving at 106 unique matrix/NS phase pair. For each of these matrix/NS phase pairs we calculated the mixing energies and estimated the solubility window in which nanostructures are expected to form. We show that the solubility between matrix/NS pairs consisting of distinct crystal structures (e.g. mixing of HH with FH) typically have low mutual solubility, thus favoring the formation of stable interfaces with very low intermixing. On the other hand, the Heusler compounds of the same type (HH mixing with HH or FH mixing with FH) can either form two-phase NS materials, have very high solubility or can form stable quaternary compounds. Moreover, we found 31 matrix/NS phase pairs that have mixing energy within the proposed mixing energy interval that favors formation of two-phase NS materials, out of which 29 pairs have not been considered previously for thermoelectric applications. Out of these 29 pairs, 24



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there are several pairs that we believe will be of immediate interest for the experimentalists, such as the FeVSb/FeNbSb and Fe2 TiSn/Fe2 HfSn mixtures, with a large mass difference between the mixing atoms, and the Fe2 TiSi/Mn2 TiSi pair, where the Mn atoms can serve as dopant. Additionally, the lattice mismatch between the matrix and the NS phases in the predicted 31 pairs is low (< 3 %), thus indicating possible formation of a coherent interface, mostly preserving the favorable electrical properties of the host matrices.

Acknowledgement This work was supported by the DARPA grant N66001-15-C-4036 (DFT calculations) and by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, under Award No. DE-SC0014520 (design of the high-throughput screening strategy). This research used computational resources provided by the Quest high performance computing facility at Northwestern University which is jointly supported by the Office of the Provost, the Office for Research, and Northwestern University Information Technology, as well as the National Energy Research Scientific Computing Center.

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(61) Rogl, G.; Sauerschnig, P.; Rykavets, Z.; Romaka, V.; Heinrich, P.; Hinterleitner, B.; Grytsiv, A.; Bauer, E.; Rogl, P. (V,Nb)-doped half Heusler alloys based on Ti,Zr,HfNiSn with high ZT. Acta Mater. 2017, 131, 336 – 348. (62) Chen, L.; Liu, Y.; He, J.; Tritt, T. M.; Poon, S. J. High thermoelectric figure of merit by resonant dopant in half-Heusler alloys. AIP Adv. 2017, 7, 065208 – 065203. (63) Bilc, D. I.; Hautier, G.; Waroquiers, D.; Rignanese, G.-M.; Ghosez, P. Low-Dimensional Transport and Large Thermoelectric Power Factors in Bulk Semiconductors by Band Engineering of Highly Directional Electronic States. Phys. Rev. Lett. 2015, 114, 136601 – 136605. (64) Page, A.; Poudeu, P.; Uher, C. A first-principles approach to half-Heusler thermoelectrics: Accelerated prediction and understanding of material properties. J. Materiomics 2016, 2, 104 – 113, Special Issue on Adv. in Thermoelectric Research. (65) Carrete, J.; Mingo, N.; Wang, S.; Curtarolo, S. Nanograined Half-Heusler Semiconductors as Advanced Thermoelectrics: An Ab Initio High-Throughput Statistical Study. Adv. Funct. Mater. 2014, 24, 7427 – 7432. (66) Carrete, J.; Li, W.; Mingo, N.; Wang, S.; Curtarolo, S. Finding Unprecedentedly LowThermal-Conductivity Half-Heusler Semiconductors via High-Throughput Materials Modeling. Phys. Rev. X 2014, 4, 011019 – 011017. (67) Page, A.; Uher, C.; Poudeu, P. F.; Van der Ven, A. Phase separation of full-Heusler nanostructures in half-Heusler thermoelectrics and vibrational properties from firstprinciples calculations. Phys. Rev. B 2015, 92, 174102 – 17412. (68) Do, D. T.; Mahanti, S. D.; Pulikkoti, J. J. Electronic structure of Zr-Ni-Sn systems: role of clustering and nanostructures in half-Heusler and Heusler limits. J. Phys. Condens. Matter 2014, 26, 275501 – 275511.

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in the newer versions of OQMD, or phases might become stable in the re-evaluated version of OQMD. (78) The stated total number of Heusler compounds is taken from our group’s internal version of the OQMD, and information for some of the listed compounds might not be available to the users of the public-facing version. The additional compounds will be included in the forthcoming public release of the OQMD. (79) Barth, J.; Fecher, G. H.; Balke, B.; Graf, T.; Shkabko, A.; Weidenkaff, A.; Klaer, P.; Kallmayer, M.; Elmers, H.-J.; Yoshikawa, H.; Ueda, S.; Kobayashi, K.; Felser, C. Anomalous transport properties of the half-metallic ferromagnets Co2 TiSi, Co2 TiGe and Co2 TiSn. Philos. Trans. R. Soc., A 2011, 369, 3588 – 3601. (80) Liu, W.; Yan, X.; Chen, G.; Ren, Z. Recent advances in thermoelectric nanocomposites. Nano Energy 2012, 1, 42 – 56. (81) Yang, J.; Yip, H.-L.; Jen, A. K.-Y. Rational Design of Advanced Thermoelectric Materials. Adv. Energy Mater. 2013, 3, 549 – 565. (82) Asta, M.; Ozolin, š, V. Structural, vibrational, and thermodynamic properties of Al-Sc alloys and intermetallic compounds. Phys. Rev. B 2001, 64, 094104 – 094107. (83) Ozolin, š, V.; Asta, M. Large Vibrational Effects upon Calculated Phase Boundaries in Al-Sc. Phys. Rev. Lett. 2001, 86, 448 – 451. (84) Ozolin, š, V.; Sadigh, B.; Asta, M. Effects of vibrational entropy on the Al-Si phase diagram. J. Phys. Condens. Matter 2005, 17, 2197 – 2210. (85) Ravi, C.; Wolverton, C.; Ozolin, š, V. Predicting metastable phase boundaries in Al-Cu alloys from first-principles calculations of free energies: The role of atomic vibrations. EPL (Europhysics Lett.) 2006, 73, 719 – 725.

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Designing High-Efficiency Nanostructured Two-Phase Heusler

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