Half‐Metallic Full‐Heusler and Half‐Heusler Compounds with Perpendicular Magnetic Anisotropy

Herein, a mechanism is proposed as to how thin films formed from a Heusler compound can simultaneously have both perpendicular magnetic anisotropy (PMA) and be half‐metallic. It is proposed that a thin film formed from a half‐metallic full‐Heusler or half‐Heusler compound that is cubic in the bulk can undergo a tetragonal distortion by adopting the lattice constant of the underlayer material during the thin film deposition process. The value of this distortion can be tuned by using underlayer materials with different in‐plane lattice constants. The distortion can thereby be optimized so that it is large enough to give rise to sufficient PMA, while, simultaneously, small enough to retain the half‐metallic properties (and, therefore, high tunneling magnetoresistance properties) of the Heusler compound. Density functional theory (DFT) calculations that are carried out for 90 full‐Heuslers and 147 half‐Heuslers that have been identified in the literature as half‐metals show that of these, 14 full‐Heusler and 59 half‐Heusler compounds display both half‐metallicity and PMA for optimal tetragonal distortions.


Introduction
Spin-transfer torque magnetic random-access memory (STT-MRAM) is considered today as the most promising contender for a "universal memory" that combines all the strengths and none of the weaknesses of existing memory types. [1] STT-MRAM combines nonvolatility, high read and write speeds, unlimited endurance, reconfigurable logic functions, and ultralow power consumption. Key to the successful development of this technology is the discovery of new magnetic materials for the magnetic tunnel junction (MTJ) memory elements that have sufficient stability against thermal fluctuations to sustain deeply scaled devices.
The magnetic electrodes that form the MTJ must possess sufficient perpendicular magnetic anisotropy (PMA) that their magnetizations lie perpendicular to the plane of the MTJ device because this allows for lower currents to switch the magnetization of the electrode that forms the memory layer of the device using spin torque. [2,3] The most promising magnetic materials to date are considered to be alloys formed from Co, Fe, and B, in conjunction with MgO tunnel barriers. [3][4][5] Unfortunately, the PMA of the Co-Fe-B layers arises from the interfaces between these layers and the tunnel barrier and/or under-or overlayers and is too weak to overcome thermal fluctuations when the diameter of the MTJ has a critical size of less than %20 nm to allow for the required 10 year memory stability.
Magnetic materials in which the PMA is derived from volume magnetocrystalline anisotropy (MCA) are thus needed to allow for increased PMA values, robust to scaling. Two of the most promising classes of such materials are the full-Heusler compounds that have the chemical formula X 2 YZ and half-Heusler compounds that have the chemical formula XYZ, wherein X and Y are transition metals or lanthanides (rare-earth metals), and Z is the main group element. [6] In the case of half-Heuslers, X and Y can also be alkali or alkaline earth metals. While the parent Heusler compounds are cubic and do not exhibit significant magnetic anisotropy, the structure of some of these compounds is found to be tetragonally distorted and thus can potentially display large PMA. (In this article, we will use the term "Heuslers" as a common name for both full-Heuslers and half-Heuslers. ) Some examples of tetragonal Heusler compounds are Mn 3Àx Ga [7] and Mn 3 Ge. [8] Thin films of these materials have been shown to exhibit large PMA for films grown epitaxially on single-crystalline substrates such as STiO 3 (001) or MgO(001), [8][9][10][11][12] as well as on amorphous substrates formed from Si(001)/SiO 2 . [13,14] Unfortunately, the experimental values of the tunneling magnetoresistance (TMR) for MTJs with Mn 3Àx Ga or Mn 3 Ge electrodes and MgO tunneling barriers were found to be very small, far below the values needed for MRAM applications. [13] Tetragonal Heuslers with theoretically predicted large PMA that could potentially show high TMR ether due to the large spin polarization or due to so-called Brillouin zone spin filtering effect have been recently suggested in ref. [15]. While tetragonal ground state and high PMA have been experimentally verified for some of suggested compounds, [15] there is no experimental conformation yet of high TMR in MTJs with these electrodes.
On the other hand, very high TMR values have been reported for MTJs with electrodes formed from cubic full-Heuslers, as high as 354% at room temperature (RT) for Co 2 MnSi. [16] Such a high TMR is attributed to the half-metallicity of these compounds. [16,17] Unfortunately, all half-metallic full-Heusler and half-Heusler compounds known to date are cubic, and thus have negligible bulk magnetocrystalline anisotropy.
In order to have both PMA and half-metallicity, we propose to create a tetragonal distortion in thin films formed from nominally cubic, half-metallic Heuslers by preparing these films on underlayer materials that have corresponding (smaller or larger) in-plane lattice constants. The distortion should be large enough to make the Heusler compound tetragonal with sufficient PMA, while, simultaneously, small enough to retain the half-metallic properties and, therefore, high TMR of the compound. In this article, we systematically study the properties (PMA and halfmetallicity) of Heusler compounds that have been identified in the literature as being half-metallic, as a function of their in-plane lattice constant that we vary in the vicinity of the equilibrium cubic lattice constant.

Crystal Structure
Cubic full-Heusler compounds X 2 YZ can have a regular structure or an inverse structure. These two crystal structures are shown schematically in Figure 1a,c. Four sites form four fcc sublattices: site Z (occupied by atom Z); site II, octahedrally coordinated by Z; and two equivalent sites I that are tetrahedrally coordinated by Z. In the regular Heusler structure shown in Figure 1a, two X atoms [red, labeled as X(I)] have an identical environment-they are located on sites I in the same xy-plane. In this structure, the Y atom (cyan) on site II and Z atom (grey) are located in a second xy-plane. In the inverse Heusler structure shown in Figure 1c, two X atoms have different environments─one X atom [red, labeled as X(I)] is located on site I in one xy-plane together with a Y atom (cyan), while another X atom [orange, labeled as X(II)] is located on site II in a second xy-plane with the Z atom (gray). Regular ( Figure 1b) and inverse ( Figure 1d) tetragonal Heusler structures can be obtained from regular and inverse cubic structures, correspondingly, by stretching (or compressing) the parent cubic structure along the z-axis (note that the z-axis is along the film normal). Tetragonal unit cells shown in Figure 1b,d are rotated by 45°around the z-axis relative to the parent cubic structures shown in Figure 1a,c, respectively. (Note that only a subset of the atoms from Figure 1a,c are shown in Figure 1b,d.) The lattice constant a cub of the cubic Heusler is indicated in Figure 1a and the lattice constants a and c of the tetragonal Heusler are shown in Figure 1b. For characterization of the tetragonal unit cell we use the dimensionless parameter c 0 = c/(2a) that is equal to 1/ ffiffi ffi 2 p % 0.707 for the cubic structure, and varies between 0.6 and 0.9 for most of the tetragonal Heuslers we consider here (see Table 3, 6, 7). Note that for c 0 = 1 the tetragonal structure of a full-Heusler becomes the fcc structure if all four atoms of the compound are considered as equivalent.

Computational Details
We performed DFT calculations for both the regular and inverse structures with various initial magnetic configurations (four possible relative magnetic orientations of X, X, and Y atoms) and various cubic lattice constants for 90 cubic full-Heusler compounds that have been identified as half-metals in the literature. For selected compounds (see Table 3), we also performed calculations for tetragonally distorted structures. For the DFT calculations, we used the generalized gradient approximation of the density functional theory (GGA/DFT) implemented within the VASP program [18] with projector augmented wave potentials [19,20] and the PBE GGA/DFT functional. [21] The convergence of the results presented below was verified for selected compounds by varying the number of divisions in reciprocal space from 10 Â 10 Â 8 to 18 Â 18 Â 14 and by varying the energy cutoff that defines the plane waves basis in VASP code from 400 to 520 eV. The final density of states (DOS) calculations for all compounds were performed with 12 Â 12 Â 12 divisions in reciprocal space and 520 eV energy cutoff that provided better than 0.01 eV accuracy for the bandgap values. Structure optimization and DOS calculations were performed without inclusion of the spin-orbit interaction. The MCA energy per formula unit, K mc , of the tetragonally distorted Heusler compounds is calculated as the difference between the total energies of the magnetic states with magnetization directed along the x-axis and the z-axis, i.e., K mc = E(100)À E(001), where positive K mc corresponds to out-of-plane magnetization. For MCA calculations, we used finer 20 Â 20 Â 20 mesh in reciprocal space and included spin-orbit interactions selfconsistently in the DFT cycle. We also calculated the volume magnetic anisotropy, K v = K mc ÀK sh , where K sh = μ 0 M 2 s V/2 is the shape anisotropy energy of a thin film per unit cell volume V, M s is the saturation magnetization, and μ 0 is the vacuum permeability.
The Curie temperature, T C , was calculated within the standard mean-field approximation (MFA) [22] using the exchange constants, J ij , of the effective Heisenberg Hamiltonian (i and j are the site indexes). In this approach, T C can be estimated as [23] k B T C = 2/3J max , where J max is the maximal eigenvalue of the (4 Â 4) J μν matrix, with J μν ¼ P j∈ν J 0j . Here, 0 is fixed index for the sublattice μ and the sum is taken over sites in the sublattice ν. The exchange constants J ij were calculated with the QUESTAAL code (www.questaal.org) using a Green's function approach implemented within the LMTO-ASA framework. [24,25]

Results and Discussion
The results of calculations for full-Heuslers compounds are summarized in Table 1-3. We found that out of the 90 considered full-Heusler cubic compounds, 35 compounds are either half-metals or near half-metals (these compounds are included in Table 1) and 55 compounds are neither half-metals nor near half-metals (these are compounds included in Table 2). Here, we define a compound as being a half-metal if it has a bandgap in one of the spin channels and the Fermi energy is located within this bandgap (lower edge of the bandgap, E min , is located below the Fermi energy, E F , while the upper edge of the bandgap, E max , is located above the Fermi energy). We define a compound as being a near-half-metal if it has a bandgap in one of the spin channels and either E min is located not too far above E F such that 0 < E min À E F ≤ 0.1 eV, or E max is located not too far below E F so that 0 < E F À E max ≤ 0.1 eV. Calculated values of E min , E max (where we set E F = 0), and the bandgap, E gap = E max À E min , are presented in Table 1. The next-to-last column of Table 1 also indicates if a compound is a half-metal (HM) or a near-half-metal (near-HM). The bandgaps of all compounds presented in Table 1 are in the minority spin channel, except for Mn 2 VAl and Cr 2 MnAl where the bandgap is in the majority spin channel (which corresponds to a negative magnetic moment m (in μ B ) for these compounds according to the Slater-Pauling rule, [26][27][28] m = N À 24, where N is the total number of valence electrons per formula unit). Table 1 and 2 show the calculated values of the cubic lattice constants in the regular structure, a r , and inverse structure, a i , the total magnetic moment in the regular structure, m r , and the inverse structure, m i , the difference between the total energy of the regular and inverse structures, E r ÀE i , and an indicator if the ground state is a regular or inverse structure (column "ground state"). The ground state has an inverse structure if E r À E i > 0 and a regular structure if E r À E i < 0. The values of E max , E min , E gap , and the HM/near-HM indicator that are shown in Table 1 correspond to the ground state of the compound.
References to published papers where a specific compound was identified as being a half-metal are presented in Table 1 and 2. We have found that compounds presented in Table 2 are not half-metallic in the ground state, but many compounds in Table 2 that have the regular structure ground state are half-metallic in the inverse structure (this can be seen from the nonfractional values of m i in Table 2, which are in agreement with the Slater-Pauling rule). Some of the compounds in Table 2 have a bandgap in one spin channel, but either with E min located too far above E F , with E min À E F > 0.1 eV, or E max located too far below E F , with E F À E max > 0.1 eV. Also, some of the compounds in Table 2 do not have a bandgap but have a very small density of states (DOS) in one of the spin channels at or near E F . We attribute different conclusions regarding the half-metallicity of compounds presented in Table 2 obtained in the present work and in the literature to the sensitivity of the bandgap (or even existence of the bandgap) of these compounds to the details of the computational method used (e.g., the chosen DFT functional). Also, the authors of ref. [29] only considered inverse structures, while we found that for many of the Heuslers considered in ref. [29] the ground state is the regular structure. Table 3 presents results of calculations of the PMA and halfmetallic properties of 22 half-metals from Table 1 when these compounds undergo a tetragonal distortion. The list of compounds in Table 3 is that of Table 1 with exclusion of 11 near-half-metals and two half-metallic compounds, Cr 2 MnAl and Cr 2 NiAl, whose bandgap, E gap ≤ 0.05 eV, is too small. As mentioned in the introduction, the tetragonal distortion of a thin film of Heusler compound is expected to be a result of the Heusler compound adopting (via thin film epitaxy) the lattice constant of the underlayer material (that could be larger or smaller than the cubic lattice constant of the Heusler) during the thin film deposition process. Table 3 shows the indicator of the ground state structure (regular or inverse), the lattice constant, a c , the total magnetic moment, m c , the bandgap, E gap , and the Curie temperature, T C , calculated for the ground state cubic structure. (Note that the lattice constant a c corresponds to a shown in Figure 1b and is related to a cub shown in Figure 1a as a c = a cub / ffiffi ffi 2 p ). Table 3 also shows the minimum, a min , and the maximum, a max , values of the in-plane lattice constant for which the tetragonally distorted compound retains its half-metallicity. The width of the half-metallic range, w hm , defined as w hm = a max À a min serves as a measure of the robustness of the half-metallic properties of the compound. Larger values of w hm and larger values of the bandgap, E gap , result in more stable half-metallic properties (stable, nonvanishing bandgap less susceptible to details of calculations or other effects). Effects that can affect half-metallic properties include change in the in-plane and out-of-plane lattice constants, details of the computational method (the chosen DFT functional, inclusion of the spin-orbit coupling, use of beyond DFT methods, such as LDA þ U, GGA þ U, or GW), finite temperature effects, effects of surfaces and interfaces, and effects of defects such as impurities or dislocations.
The a opt , shown in Table 3, are the in-plane lattice constants that provide the largest (optimal) MCA energy, K mc , for a in the half-metallicity range (a min ≤a opt ≤ a max ). The values, c 0 opt , are the out-of-plane dimensionless constants that correspond to a opt .
For cubic phase c 0 = 1/ ffiffi ffi 2 p % 0.707. Therefore, the deviation of the c 0 opt from 1/ ffiffi ffi 2 p provides the measure of deviation from cubic structure (the measure of how large is the tetragonal distortion) at a opt . We found that for all the considered compounds in Table 3, K mc is a nearly linear function of a that crosses zero at a = a c due to the cubic symmetry at this value. Therefore, a opt for all the compounds presented in Table 3 coincide either with a min or a max . Thus, the range of lattice constants for which considered full Heusler compounds are simultaneously half-metallic and have PMA is from a c to a opt . The values of K mc , K sh , and K v that correspond to the optimal values of the lattice constants a opt and c 0 opt are shown in Table 3. Table 3 is divided into two parts. In the upper part of the table, we show compounds that can be simultaneously half-metals and have PMA (K v > 0) under optimal tetragonal distortions. In the lower part of the table, we show compounds that are not simultaneously half-metals and have PMA for any tetragonal distortion that we considered. While for compounds in the lower part of the table, the largest K mc for a in the half-metallicity range is still positive (K mc > 0); nevertheless, K mc is smaller than the shape Table 1. Calculated cubic lattice constant of full-Heuslers that we have identified as half-metals or near-half-metals (see definition of near-half-metals in the text) in the regular structure, a r , and in the inverse structure, a i , total magnetic moment in the regular structure, m r , and in the inverse structure, m i , the difference between the total energy in the regular and inverse structures, E r À E i , bottom edge of the bandgap, E min (where we set E F = 0), top edge of the bandgap, E max , and the value of the bandgap, E gap = E max À E min in the ground state (as shown in the "Ground state" column). www.advancedsciencenews.com www.pss-b.com anisotropy constant, K sh , so that K v = K mc À K sh is negative. Unfortunately, one of the most well-known half-metallic Heusler compounds, Co 2 MnSi, belongs to the lower list of compounds that cannot be simultaneously half-metallic and have PMA. Compounds in the upper and lower parts of Table 3 are listed in order of the magnitude of the width of the half-metallicity range, w hm . Six compounds from the top of Table 3, namely, Mn 2 CoSi, Mn 2 CoAl, Co 2 CrSi, Mn 2 CuSi, Mn 2 CoGe, and Co 2 CrGe, have a wide half-metallicity range, w hm ≥ 0.24 Å, a relatively large bandgap, E gap ≥ 0.30 eV, a large Curie temperature, T C ≥ 490 K, and a relatively large PMA under the optimal tetragonal distortion, K v ≥ 0.18 MJ m À3 . These six compounds therefore form our list of the most promising full-Heusler compounds for MTJ devices.
Note that Mn 2 FeSb is listed in Table 3 twice. The reason is that Mn 2 FeSb has two ranges of a where it is half-metallic. One half-metallicity range has a min = 4.17 Å and a max = 4.39 Å (in this range the magnetic moment is 3.0 μ B per formula unit). The second half-metallicity range corresponds to a min = 4.03 Å and a max = 4.05 Å (in this range the magnetic moment is 1.0 μ B per formula unit). While the width of the second half-metallicity range is only w hm = 0.02 Å, we nevertheless include the second entry for Mn 2 FeSb in Table 3 for two reasons. The first reason is that due to a stronger tetragonal distortion the optimal K v increases from K v = 0.04 MJ m À3 for the first range to K v = 1.15 MJ m À3 for the second range (this is the largest value of K v presented in Table 3). The second reason is that a opt = 4.03 Å ideally matches the lattice constant of a CoAl underlayer material (a = 4.04 Å). It has been shown recently that using a novel chemical templating technique, the use of a chemical template underlayer formed from an atomically ordered CoAl layer allows for near-bulk-like magnetic properties in tetragonally distorted Heusler films grown on top of it, even for film deposition at room temperature, and that films only 1 or 2 unit cells thick display excellent PMA properties. [14,30] Table 2. Calculated cubic lattice constant of full-Heuslers that we have identified as not half-metals or near-half-metals in the regular structure, a r , and in the inverse structure, a i , total magnetic moment in the regular structure, m r , and in the inverse structure, m i , the difference between the total energy in the regular and inverse structures, www.advancedsciencenews.com www.pss-b.com

Crystal Structure
The crystal structure of the cubic half-Heusler XYZ compounds can be obtained from the crystal structure of the full-Heusler X 2 YZ compounds shown in Figure 1c as follows: the fcc sublattice of Z-atoms stays intact, but one of the remaining three fcc sublattices of atoms represented by either blue, red, or orange spheres in the figure is removed and the remaining two fcc sublattices are occupied by X and Y atoms. There are three possible resulting structures: we will use notation indicators 1, 2, and 3 for these three possible crystal structures of half-Heusler compounds as follows: 1) 1 denotes the crystal structure when the X atoms are located in one xy-plane and Y and Z atoms are located in a second xy-plane; 2) 2 denotes the crystal structure when Y atoms are located in one xy-plane and X and Z atoms are located in a second xy-plane; and 3) 3 denotes the crystal structure when Z atoms are located in one xy-plane and X and Y atoms are located in a second xy-plane.
Analogous to the case of full-Heuslers a tetragonal half-Heusler structure can be obtained from the cubic structure by stretching (or compressing) the parent cubic structure along the z-axis.
We performed DFT calculations for all three possible crystal structures with various initial magnetic configurations and various cubic lattice constants for 147 cubic half-Heusler compounds that were identified as half-metals in the literature. For DFT calculations we used the computational approach described in Section 2.2. For selected compounds (see Table 6 and 7), we also performed calculations for tetragonally distorted structures.

Results and Discussion
The results of calculations for half-Heuslers compounds are summarized in Table 4-7. We found that out of 147 considered Table 3. Lattice constant, a c , total magnetic moment, m c , the bandgap, E gap , and the Curie temperature, T C , calculated for the ground state cubic structure of half-metallic full-Heuslers, the minimum, a min , and the maximum, a max , values of the in-plane lattice constant for which the tetragonally distorted compound is staying half-metallic, the width of the half-metallic range, w hm = a max À a min , the optimal in-plane lattice constant, a opt , and corresponding dimensionless out-of-plane lattice constant, c 0 opt , that provide the largest (optimal) MCA energy for a in the half-metallicity range (a min ≤a opt ≤a max ), the MCA energy, K mc , shape anisotropy energy, K sh , and the volume magnetic anisotropy, K v = K mc À K sh , calculated for the optimal lattice constant, a opt (PMA corresponds to positive K v ). Top part of the table contains compounds with PMA (K v > 0) and bottom part of the table contain compounds without PMA (K v < 0). In both parts of the table, compounds are ordered by the width of the half-metallic range, w hm . www.advancedsciencenews.com www.pss-b.com Table 4. Ground state structure indicator (see definition of the three possible structures of half-Heuslers in the text) of half-Heuslers that we identified as half-metals or near-half-metals, calculated cubic lattice constant, a c , and total magnetic moment, m c , in the ground state, the indicator of the crystal structure with the second lowest total energy, the difference, E 1 -E 2 , between the total energy of the ground state crystal structure, E 1 , and the total energy of the crystal structure with the second lowest total energy, E 2 , the bottom edge of the bands gap, E min (where we set E F = 0), top edge of the bandgap, E max , and the value of the bandgap, E gap = E max -E min , calculated for the ground state structure, the spin channel which has the bandgap. Note that CoTiSb has bandgaps in both spin channels, so it is a semiconductor, not a half-metal. www.advancedsciencenews.com www.pss-b.com  Table 5. Ground state structure indicator of half-Heuslers that we identified as not half-metals or near-half-metals, calculated cubic lattice constant, a c , and total magnetic moment, m c , in the ground state, the indicator of the crystal structure with the second lowest total energy, the difference, E 1 -E 2 , between the total energy of the ground state crystal structure, E 1 , and the total energy of the crystal structure with the second lowest total energy, E 2 . www.advancedsciencenews.com www.pss-b.com Table 6. Ground state structure indicator, lattice constant, a c , total magnetic moment, m c , the bandgap, E gap , and the Curie temperature, T C , calculated for the ground state cubic structure of half-metallic half-Heuslers, the minimum, a min , and the maximum, a max , values of the in-plane lattice constant for which the tetragonally distorted compound stays half-metallic, the width of the half-metallic range, w hm = a max À a min , the optimal in-plane lattice constant, a opt , and corresponding dimensionless out-of-plane lattice constant, c 0 opt , that provide the largest (optimal) MCA energy for a in the half-metallicity range (a min ≤a opt ≤a max ), the MCA energy, K mc , shape anisotropy energy, K sh , and the volume magnetic anisotropy, K v = K mc À K sh , calculated for the optimal lattice constant, a opt . This table includes compounds that have PMA (K v > 0). Compounds are ordered by the width of the half-metallic range, w hm . The bold font for some values of a min and a max indicates that the true value of a min is smaller and the true value of a max is larger than the values shown in table (see text for details). compounds 111 compounds are half-metals or near-half-metals (these compounds are included in Table 4) and 36 compounds are neither half-metals nor near-half-metals (these compounds are included in Table 5). (For definition of half-metals and near-half-metals, see Section 2.3.) In the second and fifth columns of Table 4 and 5, we denote the cubic crystal structure with the lowest total energy (ground state) and the cubic crystal structure with the second lowest total energy. The difference, E 1 À E 2 , between the total energy of the ground state crystal structure, E 1 , and the total energy of the crystal structure with the second lowest total energy, E 2 , is shown in the sixth column of Table 4 and 5. The third and fourth columns of Table 4 and 5 show the lattice constant, a c , and the total magnetic moment, m c , calculated for the ground state cubic crystal structure. Calculated values of the bottom edge of the bandgap, E min , top edge of the bandgap, E max (we set E F = 0), the bandgap, E gap = E max À E min , the spin channel in which the bandgap is located, and the (HM/near-HM) indicator are shown in Table 4 in columns 7-11, respectively.
References to published papers where a specific half-Heusler compound was identified as a half-metal are shown in the last column of Table 4 and 5. Note that the ratio of the number of compounds that we have identified as neither half-metals or near-half-metals (these compounds are listed in Table 5) to the number of compounds that we have identified as half-metals or near-half-metals (these compounds are listed in Table 4) is significantly smaller for half-Heuslers as compared to full-Heuslers (compare the ratio of 36 compounds in Table 5 to the 111 compounds in Table 4 and analogous ratio of 55 compounds in Table 2 to 35 compounds in Table 1). This can be explained by the fact that half-Heusler compounds have significantly larger bandgaps as compared to the full-Heusler compounds (the typical bandgap for half-Heuslers is %1 eV, while the typical bandgap for full-Heuslers is %0.3 eV), so the effect of the computational method used or the chosen DFT functional on the conclusion as to whether a given compound is a half-metal or not is much smaller for half-Heuslers as compared to full-Heuslers, so leading to a better agreement between our calculations and previous papers.
The larger values of the bandgaps for half-Heuslers as compared to full-Heuslers can be explained by presence of the s-character alkali or alkaline earth elements in most of the considered half-Heusler compounds. The bands with large contributions from s-character are typically wider (and, therefore, bandgaps are also wider) as compared to less dispersive p-or d-character bands. Note that half-Heusler compounds without alkali or alkaline earth elements have smaller bandgaps (see Table 6 and 7).
Analogous to the case of full-Heuslers, some of the compounds in Table 5 have a bandgap in one spin channel, but either with E min located too far above E F , with E min À E F > 0.1 eV, or E max located too far below E F , with E F À E max > 0.1 eV. Also, some of the compounds in Table 5 do not have the bandgap but have a very small density of states in one of the spin channels at or near E F . Table 6 and 7 present results of calculations of the PMA and half-metallic properties of 100 half-metals and near-half-metals from Table 4 when these compounds undergo a tetragonal distortion (the list of compounds in Table 6 and 7 is that of Table 4 with the exclusion of 10 near-half-metals that do not become half-metals under any tetragonal distortion, and CoTiSb that is a nonmagnetic semiconductor). Table 6 shows the ground state crystal structure indicator (1, 2, or 3), the lattice constant, a c , the total magnetic moment, m c , the bandgap, E gap , and the Curie temperature, T C , calculated for the ground state cubic structure. Table 6 also shows the minimum, a min , and the maximum, a max , values of the in-plane lattice constant for which the tetragonally distorted compound remains half-metallic (note that for near-half-metals either a min > a c or a max < a c ). The values of some a min and a max are shown in bold. The bold font means that the true value of a min is smaller and the true value of a max is larger than the values shown in Table 6. The reason for this is that we have not calculated tetragonal distortions which correspond to a too large deviation from the cubic structure (typically, we stopped further calculations when a c À a min or a max À a c exceeded 0.6 Å). Thus, the width of the half-metallic range, w hm = a max À a min , that is presented in Table 6 is a lower bound of the true w hm for the compounds that have a min or a max shown in bold font. Note that due to larger values of the bandgaps w hm is significantly wider for the half-Heuslers as compared to the full-Heuslers. As we discussed above, larger calculated values of E gap and w hm make the half-metallic properties of compounds more stable and less susceptible to changes in the in-plane and out-of-plane lattice constants, details of the computational method, chosen DFT functional, finite temperature effects, and effects of defects such as impurities and dislocations. The values of a opt , shown in Table 6, are the in-plane lattice constant that provides the largest (optimal) MCA energy, K mc , for a in the half-metallicity range (a min ≤a opt ≤a max ). The values of c 0 opt are the out-of-plane dimensionless constant that corresponds to a opt . We found that due to the wide width of the half-metallic range of typical half-metallic half-Heusler compounds, K mc as a function of a in the range a min ≤a ≤a max often does not follow a simple linear function (as opposed to the case of full-Heuslers), while it still crosses zero at a = a c due to the cubic symmetry at this value. Therefore, a opt in Table 6 often do not coincide with a min or a max . The values of K mc , K sh , and K v that correspond to the optimal values of the lattice constants a opt and c 0 opt are shown in Table 6. Note that in Table 6 we show only the compounds with PMA (K v > 0). Thus, the compounds shown in Table 6 are compounds that can simultaneously be half-metals and have PMA under an optimal tetragonal distortion.
In order to understand the origin of the large Curie temperature for half-Heusler compounds presented in Table 6, we studied the contribution to the J μν matrix of exchange constants J ij for various sites i and j for compounds with T C > 400 K. We found that for compounds XYZ where both X and Y elements are the alkali or alkaline earth elements the main contribution to J μν comes from J ij with i and j being the nearest-neighbor Z (main group element) sites. For compounds XYZ where X is alkali or alkaline earth element and Y is the transitional metal, the main contribution to J μν comes from J ij with i and j being the nearestneighbor Y (transitional metal) sites. For compounds XYZ where both X and Y are transitional metals, the main contribution to J μν comes from J ij with i and j being the nearest-neighbor transitional metal sites (X and Y). Table 7 contains the same information as Table 6 (except that we have not calculated the Curie temperature) but for compounds with K v ≤ 0 at the optimal values of the lattice constants a opt and c 0 opt . Thus, compounds shown in Table 7 cannot simultaneously be half-metals and have PMA even under optimal tetragonal distortions. Note that the compounds listed in Table 6 and 7 are ordered by the values of the width of the half-metallicity range, w hm .

Conclusion
In this work, we have proposed a mechanism by which a Heusler compound can simultaneously display both PMA and halfmetallic properties, which is highly desirable for STT-MRAM devices. In particular, we propose that a tetragonal distortion of a thin film of a half-metallic cubic full-Heusler or half-Heusler compound can be set in the film by having the film adopt the in-plane lattice constant of an underlayer material during the thin film deposition process. The value of the distortion can be tuned by using an underlayer material with a suitable in-plane lattice constant. The optimal distortion should be large enough to make the Heusler compound tetragonal with sufficient PMA, while, simultaneously, small enough to maintain the half-metallic properties (and, therefore, high TMR properties) of the compound.
We performed DFT calculations for 90 full-Heuslers that were identified in the literature as half-metals. We found that 24 of them are half metals and that 14 of these half-metals (see Table 3) can simultaneously keep their half-metallicity and have PMA under an optimal tetragonal distortion (optimal in-plane lattice constant). Our most promising list of full-Heusler compounds for MTJ devices includes six compounds, namely, Mn 2 CoSi, Mn 2 CoAl, Co 2 CrSi, Mn 2 CuSi, Mn 2 CoGe, and Co 2 CrGe, that simultaneously have a wide half-metallicity range, w hm ≥ 0.24 Å, a relatively large bandgap, E gap ≥ 0.30 eV, a large Curie temperature, T C ≥ 490 K, and a relatively large PMA under an optimal tetragonal distortion, K v ≥ 0.18 MJ m À3 .
We also performed DFT calculations for 147 half-Heuslers that were identified in the literature as half-metals. We found that 94 of them are half metals and that 59 of these half-metals (see Table 6) can simultaneously keep their half-metallicity and have PMA under an optimal tetragonal distortion (optimal in-plane lattice constant). Our most promising list of half-Heusler compounds for MTJ devices includes seven compounds CsBaC, RbTaGe, CsSrC, RbTaSi, RbNbSi, CoCrAs, and RhFeSn that simultaneously display a wide half-metallicity range, w hm ≥ 0.40 Å, a relatively large bandgap, E gap ≥ 0.40 eV, a large Curie temperature, T C ≥ 410 K, and a relatively large PMA under an optimal tetragonal distortion, K v ≥ 0.25 MJ m À3 .