Ordering tendencies and electronic properties in quaternary Heusler derivatives
The phase stabilities and ordering tendencies in the quaternary full-Heusler alloys NiCoMnAl and NiCoMnGa have been investigated by in-situ neutron diffraction, calorimetry and magnetization measurements. NiCoMnGa was found to adopt the L2 structure, with distinct Mn and Ga sublattices but a common Ni-Co sublattice. A second-order phase transition to the B2 phase with disorder also between Mn and Ga was observed at . In contrast, in NiCoMnAl slow cooling or low-temperature annealing treatments are required to induce incipient L2 ordering, otherwise the system displays only B2 order. Linked to this L2 ordering, a drastic increase in the magnetic transition temperature was observed in NiCoMnAl, while annealing affected the magnetic behavior of NiCoMnGa only weakly due to the low degree of quenched-in disorder. First principles calculations were employed to study the thermodynamics as well as order-dependent electronic properties of both compounds. It was found that a near half-metallic pseudo-gap emerges in the minority spin channel only for the completely ordered Y structure, which however is energetically unstable compared to the predicted ground state of a tetragonal structure with alternating layers of Ni and Co. The experimental inaccessibility of the totally ordered structures is explained by kinetic limitations due to the low ordering energies.
i.1 Motivation and Scope
The class of Heusler alloys, with the ternary system CuMnAl as the prototypical representative,Heusler (1903) hosts a variety of systems displaying intriguing properties.Graf et al. (2011) For instance, the latent structural instability in the magnetic NiMn-based compounds gives rise to significant magnetic shape memorySozinov et al. (2002) and magnetocaloricKrenke et al. (2005) effects. On the other hand, they can also display attractive properties that are directly related to their electronic configuration, with the proposal of spintronics, which relies on the detection and manipulation of spin currents, as an example. In a magnetic tunnel junction for instance, the achievable tunneling magnetoresistive effect and thereby the miniaturization of components depends on the spin polarization of the conduction electrons in the electrodesJullière (1975). As a consequence, half-metallic materials, which have a 100% spin polarization due to a band gap at the Fermi level in one spin channel, are highly sought after and currently the focus of both theoretical and experimental investigations.
While the first half-metal identified by theoretical calculations in 1983 by Groot et al.de Groot et al. (1983) was the half-Heusler compound NiMnSb of C1 structure, also in full-Heusler alloys with L2 structure half-metallic properties have been predictedIshida et al. (1982); Fujii et al. (1990) and experimentally observedJourdan et al. (2014). Recently, also a large number of quaternary Heusler derivatives, among them NiCoMnAlHalder et al. (2015) and NiCoMnGaAlijani et al. (2011a), have been suggested by ab initio calculations to be half-metals in their fully ordered Y structure Özdoğan et al. (2013). Half-metallic properties in the NiCoMnAlOkubo et al. (2011) and NiCoMnGaKanomata et al. (2009) systems have additionally been proposed for the Co-rich side of the respective phase diagrams on the basis of magnetization measurements via the Generalized Slater-Pauling rule. It is obvious that the degree of chemical order will have direct consequences for the half-metallic properties of these systems. However, the connection between atomic order, segregation and functional properties has also been established for the magnetocaloric and metamagnetic shape memory effects,Recarte et al. (2012); Barandiaran et al. (2013), ferroic glassesMonroe et al. (2015) and the recently reported shell-ferromagnetism in off-stoichiometric Heusler compounds.Çakır et al. (2016)
In assessing the potential of a given material for application following from its electronic structure, theoretical and experimental investigations have contrasting characteristics: in ab initio calculations, the distribution of electronic charge is the fundamental quantity that is considered, which depends in principle only on the positions of the ions and their atomic numbers. From this, other important properties can be derived, like total energies, magnetic moments and forces on the ions. Different structures can be compared in terms of their total energies, but chemical disorder must be taken into account appropriately. This can be handled very efficiently in terms of the coherent potential approximation (see Ref. Ruban and Abrikosov (2008) for a recent review), which, however, does not provide an easy way to account for ionic relaxations. On the other hand, explicit calculations of disordered structures with randomly distributed atoms in larger super-cells are much more involved and numerically intensive. Thus, for practical reasons often an ordered configuration is assumed to be representative. On the other hand, in experiments the state of order in the sample is relevant for the potential application, while the determination of aspects of the electronic structure is often quite hard, which especially applies for the spin polarization. Thus, it seems indicated to combine the respective strengths of experiment and theory, which is what we set out to do in this paper. Specifically, in the systems of NiCoMnAl and NiCoMnGa we study the degrees of equilibrium long-range order and the associated order/disorder phase transitions by in situ neutron diffraction, and the kinetics of order relaxation during isothermal annealing by way of its effect on magnetization and Curie temperature. Further, we perform ab initio calculations on different ordered and disordered structures to determine the associated electronic structures as well as ordering energies. As we will show, these calculations imply that among the realistic candidates only the hitherto assumed Y ordering displays half-metallicity, but does not correspond to the actual ground state. In addition, the associated ordering energies are small, which explains the experimentally observed stability of disorder among Ni and Co.
i.2 States of order in quaternary Heusler derivatives
To facilitate the discussion of the different ordered quaternary structures and their relations later in this article, we enumerate here the structures, define the nomenclature and summarize the pertinent knowledge on their ternary parent compounds.
Heusler alloys in the strict sense of the word are ternary systems of composition XYZ displaying L2 order, which is defined by the space group 225 (Fmm) with inequivalent occupations of the Wyckoff positions 4a, 4b and 8c. Typically, X is a late transition metal occupying preferentially 8c, while an early transition metal Y and a main-group metal Z occupy the other two sites,Graf et al. (2011) with CuMnAl the prototypical representative.
NiMnGa conforms to above definition and displays a stable L2 phase at intermediate temperatures.111We neglect here the martensitic transitions below room temperature. Around it shows a second-order disordering transition to the B2 (CsCl) structure,Sánchez-Alarcos et al. (2007) corresponding to a mixing of Mn and Ga, that is, it acquires space group 221 (Pmm) with Wyckoff position 1a occupied preferentially by Ni, while Mn and Ga share position 1b. This partial disordering can be understood by the observation that B2 order, i.e., the distinction between Ni on the one hand and Mn and Ga on the other hand, is stabilized by nearest-neighbor interactions on the common bcc lattice, while the ordering between Mn and Ga corresponding to full L2 order can only be effected by the presumably weaker next-nearest-neighbor interactions. Indeed, in NiMnAl only the B2 state or at the most very weak L2 order can experimentally be observed.Acet et al. (2002) In both systems the B2 state is stable up to the melting point, that is, there is no transition to the fully disordered bcc state. In the Co-based systems, the situation is remarkably similar, with well-developed L2 order in CoMnGa and only B2 order in CoMnAl.Webster (1971)
It seems probable, and is indeed corroborated by our experimental observations to be reported below, that the behavior of the quaternary systems NiCoMnGa and NiCoMnAl can be traced back to their ternary parent compounds. The most plausible candidates of ordered structures following this reasoning are illustrated Fig. 1. Given that both Ni- and Co-based ternary parents display the B2 structure at high temperatures, it is natural to assume this to be also the case for NiCoMnZ, with site 1a shared by Ni and Co and site 1b by Mn and Z. We will denote this as (NiCo)(MnZ), where the parentheses denote mixing between the enclosed elements.222We do not consider the other two B2 possibilities (NiMn)(CoZ) and (NiZ)(CoMn).
As temperature is decreased, transitions to states of higher order can appear. For the Mn-Z sublattice, a NaCl-like ordering of Mn and Z is most likely by analogy with the ternary parents. Assuming the same kind of interaction favoring unlike pairs also between Ni and Co, the realized structures depend on the relative strengths: for dominating Mn-Z interactions, the B2 phase would transform to an L2 structure of type (NiCo)MnZ, where Ni and Co are randomly arranged over the 8c sites, and in the converse case to L2 NiCo(MnZ) with Mn and Z on 8c. In either case, the ordering of the other sublattice at some lower temperature would transform the system to the so-called Y structurePauly et al. (1968); Bacon and Plant (1971) of prototype LiMgPdSnEberz et al. (1980) with space group 216 (F3m) and Wyckoff positions 4a, 4b, 4c, and 4d being occupied by Ni, Co, Mn and Z, respectively.
However, as the kind of chemical interaction within the Ni-Co sublattice is as yet unknown, also other possibilities have to be considered. In principle, there is an unlimited number of superstructures on the L2 (NiCo)MnZ structure, corresponding to different Ni/Co orderings. In particular, apart from the above-mentioned cubic Y structure (with NaCl-type Ni/Co ordering) there are two other structures with a four-atom primitive cell, making them appear a priori equally likely to be realized as the Y structure. These are tetragonal structures characterized by either alternating columns or planes of Ni and Co atoms, which we denote by T and T. Specifically, the T structure has space group 131 (P4/mmc), with Ni on Wyckoff position 2e, Co on 2f, Mn 2c, and Z on 2d, while T has space group 129 (P4/nmm) with Ni on 2a and Co on 2b, while Mn and Z reside on two inequivalent 2c positions, with prototype ZrCuSiAsJohnson and Jeitschko (1974). Note that the Ni/Co ordering in these three fully-ordered structures can equally be understood as alternating planes in different crystallographic orientations, with T corresponding to planes, T to , and Y to planes. Finally, of course the possibility of phase separation into L2 NiMnZ and CoMnZ has to be considered.
Ii Macroscopic properties
ii.1 Sample preparation and thermal treatments
Nominally stoichiometric NiCoMnAl and NiCoMnGa alloys have been prepared by induction melting and tilt casting of high-purity elements under argon atmosphere. After casting, the samples have been subjected to a solution-annealing treatment at followed by quenching in room-temperature water. In this state, the samples have been checked for their actual composition using wavelength-dispersive X-ray spectroscopy (WDS). For each alloy, eight independent positions have been measured. The average over the retrieved values are given in Table 1, showing satisfactory agreement with the nominal compositions. Additionally, sample homogeneity was confirmed by microstructure observation using backscattered electrons.
In order to track the ordering processes of the alloys upon low-temperatures isothermal aging, samples have been annealed at for different times and water-quenched. Thus, for both systems we consider four states, corresponding to the as-quenched state and after annealings for , , and , respectively, denoted in the following by aq, ann6h, ann24h, and ann72h. Previous results Neibecker et al. (2014) have proven this low-temperature annealing protocol to be successful for increasing the achievable state of order in structurally similar alloys of the NiMnAl system.
ii.2 Magnetization measurements
Magnetization measurements corresponding to the different annealing conditions were performed, specifically the Curie temperatures and spontaneous magnetization values have been determined. Temperature-dependent magnetization measurements were done in a TOEI Vibrating Sample Magnetometer (VSM) applying an external magnetic field of in a temperature range from room temperature to . The spontaneous magnetization for NiCoMnGa has been determined with a Superconducting Quantum Interference Device (SQUID) based Quantum Design MPMS system at employing external magnetic fields up to . Since for the ductile NiCoMnAl alloy sample preparation turned out to have an effect on sample properties, presumably due to introduced mechanical stresses, in this alloy system temperature-dependent magnetization measurements have been performed by VSM on samples of larger size in an external field of .
Figure 2 a) and b) shows field-dependent magnetization curves () of NiCoMnGa in four different annealing conditions measured at and . The spontaneous magnetization has been retrieved via constructing Arrott plots. The obtained values are given in Table 2. While, as expected, the spontaneous magnetization increases with decreasing measurement temperature, no apparent effects on due to annealing are visible, with the values of scattering around and at and , respectively. These values show reasonable agreement to previous studies with being stated as at a temperature of .Kanomata et al. (2009)
Figure 2 c) and d) show curves of NiCoMnAl in four different annealing conditions measured at and . Additionally, curves of NiCoMnAl in the four annealing conditions at an external magnetic field of that were used to extrapolate the value of the spontaneous magnetization are given in the Supporting Information. The determined values for are listed in Table 2 and range from in the aq state to in the ann72h state. This increase of during annealing in NiCoMnAl is significantly larger than the corresponding effect in NiCoMnGa. Clearly, the difference in with annealing further increases at higher measurement temperatures due to lower magnetic transition temperatures in the shorter annealed samples. The values obtained for are in good agreement with two previous studies where in the B2 ordered state was reported as Halder et al. (2015) and Okubo et al. (2011)
Figure 3 shows the corresponding temperature-dependent magnetization measurements () of NiCoMnGa and NiCoMnAl under an external field of . The magnetization curves are normalized since absolute magnetization values are, due to sample shape-specific demagnetization fields, not meaningful under low external magnetic fields. In the following discussion, we define the apparent Curie temperature as the locus of the maximal slope of the curves.
In NiCoMnAl, the magnetic transition temperature increases from to with annealing of the samples, reflecting a corresponding increase of L order. The specific transition temperatures are given in Table 2. This compares satisfactorily with the value of quoted by Okubo et al.Okubo et al. (2011) for samples quenched from the B2 region. An important point to note is that, in order to probe the high Curie temperatures in these systems, during the measurements the sample is subjected to temperatures where the ordering kinetics become appreciable. Specifically, with ordering kinetics at on the order of hours, the Curie temperatures below measured on heating at a rate of can safely be assumed to correspond to the degree of order imposed by the isothermal annealing treatments. However, at the maximum temperature of the degree of order will relax during the measurement towards the corresponding equilibrium value, leading to an increase of order for the aq sample and a decrease when starting from a high degree of order. This difference between heating and cooling curves is well discernible.
NiCoMnGa shows a magnetic transition from the ferromagnetic to the paramagnetic state between 636.0 and . Since the degree of L order in NiCoMnGa is high in all annealing conditions, annealing has a much smaller effect on than in NiCoMnAl. The determined transition temperatures are given in Table 2. Apparently, still a small increase of order with annealing exists in this alloy system. We interpret the constant offset of about between heating and cooling to effects of thermal inertia.
ii.3 Differential scanning calorimetry
Differential Scanning Calorimetry (DSC) has been employed to analyze both alloys with respect to magnetic and structural phase transitions on a Netzsch DSC 404 C Pegasus. All measurements have been performed at a heating rate of over a temperature range from 300 K to 1273 K. Figure 4 shows DSC results for the NiCoMnGa and NiCoMnAl alloys that have been subject to solution annealing at , quenching to room temperature, followed by a low-temperature annealing at applied with the intention to adjust a large degree of L2 order. Taking into account different ordering kinetics, the NiCoMnGa alloy was annealed for K, while the NiCoMnAl alloy was annealed for .
Both alloys show a clear magnetic transition from the ferromagnetic to the paramagnetic state at and for NiCoMnAl and NiCoMnGa, respectively. Those values show good agreement to the values obtained by magnetization measurements (Table 2), justifying our approach of defining the Curie temperatures via the position of maximal slope in the magnetization under constant field. NiCoMnGa shows additionally an order-disorder phase transition at higher temperatures that can be assigned according to our neutron diffraction measurements (Sec. III) to the transition from the L2-(NiCo)MnGa to the B2-(NiCo)(MnGa) structure, which is in accordance with the behavior of the structurally similar NiMnGa compoundSánchez-Alarcos et al. (2007) and with previous results from Kanomata et al.Kanomata et al. (2009) The phase transition temperature was determined as , a value in excellent agreement to the reported in Ref. Kanomata et al., 2009. In contrast, NiCoMnAl does not show any further apparent peaks in the calorimetric signal besides the magnetic transition.
Iii Neutron diffraction
Neutron diffraction measurements have been performed at the SPODIHoelzel et al. (2015) high-resolution neutron powder diffractometer at the Heinz Maier-Leibnitz Zentrum (MLZ) in Garching, Germany. Polycrystalline samples were measured continuously on heating and cooling between room temperature and , employing rates of approximately and a recording frequency of approximately one pattern per 15 minutes. Measurements have been done using Nb sample holders and employing a neutron wavelength of . Temperature-dependent lattice constants, peak widths and structure factors corresponding to the different degrees of long-range order have been refined. Additionally, for the depiction of the waterfall plots, data treatment as described in Ref. Hoelzel et al., 2012 has been applied.
Figure 5 shows waterfall plots of the neutron diffraction patterns of NiCoMnGa/Al upon heating and cooling on a logarithmic pseudocolor scale. All reflection families, namely L2, B2 and A2, as well as the peaks due to the Nb sample holder, are labeled in the figure. Their presence correspond to the symmetry breaking into inequivalent sublattices as discussed in Sec. I, and their strength indicates the quantitative degree of long-range order. The A2 peaks are not influenced by any disorder in the system, since here all lattice sites contribute in phase. The presence of the B2 peak family indicates different average scattering lengths on the Ni-Co and the Mn-Z sublattices. Finally, L2 peaks are due to a further symmetry breaking between either the 4a and 4b and/or 4c and 4d sublattices. Note that such a qualitative reasoning cannot distinguish whether the system has the Y structure or one of the two possible L2 structures, which can only be decided by a quantitative analysis (as will be done below). Similar as for the A2 peak family, the intensity of the B2 peak family is not influenced by the degree of L2 order.
In the waterfall depiction, the evolution of peak position (in qualitative terms) peak intensity with temperature can be followed nicely. Initially, the samples correspond to the state quenched from . Already in this state, NiCoMnGa exhibits L2 order as evidenced by the presence of the corresponding diffraction peaks. Upon heating, first of all the thermal expansion of the lattice is observed with the peak positions shifting to smaller scattering angles. Simultaneously, the peaks stemming from the Nb sample holder can clearly be distinguished from the sample peaks due to their lower rate of thermal expansion. At approximately , a disordering phase transition from the L2 phase to the B2 phase is observed. This is reversed on cooling at nearly the same temperature, which shows that at these high temperatures the equilibrium states of order are followed closely. The observed value of is in good agreement to the determined by calorimetry.
In contrast to NiCoMnGa, NiCoMnAl is found to have a B2 state in the as-quenched condition with no L2 reflections visible. On cooling, the peaks are slightly narrower than on heating, indicating the release of internal stresses in the sample remaining from quenching. Interestingly, upon slow cooling the sample down from , at approximately very diffuse maxima are appearing at the positions where L2 reflections would be expected. Numerical analysis of the corresponding regions on heating suggests that also already here a very weak intensity is found as soon as temperature regions are reached that are sufficient to facilitate a relaxation of order via diffusion. Arguably, the diffuse intensity observed is the manifestation of L2 short-range order or incipient L2 long-range order with very small anti-phase domains. Such anti-phase domains have previously been observed in NiMnAlGa alloys,Ishikawa et al. (2008); Umetsu et al. (2011) where the phase transition temperature implies ordering kinetics on experimentally accessible time scales. The absence of well-defined L2 order as well as the pronouncedly lower B2-L2 transition temperature in NiCoMnAl compared to the NiCoMnGa alloy is consistent with the behavior observed in the related NiMnAl and NiMnGa compounds where transition temperatures of, respectively, Kainuma et al. (2000) and Sánchez-Alarcos et al. (2007) have been reported.
While confirming a state of B2 order, in neutron diffraction no magnetic superstructure peaks are observed. Thus, in contrast to NiMnAl, where Ziebeck et al. Ziebeck and Webster (1975) discovered a helical magnetic structure manifesting itself in form of antiferromagnetic superstructure reflections and satellite peaks at the (200) and (220) reflections, NiCoMnAl is entirely ferromagnetic even under B2 order. This goes along with measurements (Sec. II) showing prototypical ferromagnetic properties. In contrast, for NiMnAl, antiferromagnetic properties haven been reported.Acet et al. (2002) Presumably, the drastic difference in magnetic structure results from strong ferromagnetic interactions in the system introduced by Co, overcoming the antiparallel coupling between neighboring Mn atoms.
Figure 6 shows the temperature-dependent lattice constants retrieved from fitting the in-situ neutron diffraction data as well as the corresponding temperature-dependent thermal expansion coefficients. At , the lattice constant of approximately in as quenched NiCoMnGa is only slightly larger than the one of NiCoMnAl with approximately , while the thermal expansion is similar in both alloys with a value of approximately . In the case of NiCoMnGa, the heating and cooling curves coincide, indicating little effect of the applied quenching treatment. The B2-L2 transition is clearly mirrored in the lattice constant, with a maximum in the thermal expansion coefficient around , in agreement with the calorimetric transition temperature and the vanishing of L2 intensities in neutron diffraction.
In the case of NiCoMnAl, the B2–L2 ordering transition is neither visible directly in the lattice constant nor in the thermal expansion coefficient. However, this system displays another striking effect with the divergence of the lattice constants on heating and cooling at intermediate temperatures. The absence of an analogous effect in the determined Nb lattice constants proves that this deviation is real as opposed to, e.g., an error in the determination of the sample temperature. We interpret it to be due to a superposition of a lattice expansion due to quenched-in disorder with a lattice contraction due to a quenched-in vacancy supersaturation. On heating, around ordering kinetics become active, leading to a relaxation of the lattice expansion, while only at temperatures above vacancies become mobile enough to equilibrate their concentrations at vacancy sinks such as surfaces or grain boundaries. Thus, in this interpretation the agreement in the lattice constants of the slow-cooled and quenched states at low temperatures is just a coincidence.
Figures 7 show the temperature-dependent structure factors of NiCoMnGa and NiCoMnAl, i.e., essentially the ratio of the intensities of the B2 and L2 peaks to the A2 peak families after taking into account Lorentz factors and Debye-Waller factors. The theoretical structure factors for different kinds of disorder are depicted in the figures as stroked lines. In the case of NiCoMnGa, the second-order B2–L2 transition at is clearly visible. Additionally, this evaluation gives credence to the scenario of the observed L2 intensity being due to solely Mn/Ga order as opposed to Y ordering or only Ni/Co ordering. Interestingly, with increasing temperature also the degree of B2 order decreases somewhat in both systems. Also, the degrees of order on cooling are always higher than on heating, which is indicative of some amount of disorder after quenching. These qualitative conclusions seem valid even though quantitative interpretations of the data have to be treated with caution considering the limited number of crystallite grains fulfilling the Bragg condition that defines the statistical precision.
Iv First-principles calculations
We performed ab initio calculations for the structures proposed in Sect. I.2. Specifically, we computed ordering energies and electronic densities of states (DOS) by plane-wave density functional theory as implemented in VASP (Vienna Ab-initio Simulation Package),Kresse and Furthmüller (1996) and magnetic interactions in the Liechtenstein approachLiechstenstein et al. (1987) as implemented by Ebert et al. Ebert et al. (2011) in their Korringa-Kohn-Rostoker Green’s function code (SPR-KKR).
iv.0.1 Computational details
In the VASP calculations, the disordered structures were realized by 432 atom supercells (corresponding to 666 bcc cells) with random occupations, taking advantage of the efficient parallelization in VASP for massively parallel computer hardware. Here, the wavefunctions of the valence electrons are described by a plane wave basis set, with the projector augmented wave approach taking care of the interaction with the core electrons.Kresse and Joubert (1999) Exchange and correlation was treated in the generalized gradient approximation using the formulation of Perdew, Burke and Ernzerhof.Perdew et al. (1996) We converged unit cell dimensions and atomic positions by a conjugate gradient scheme until forces and pressures reached values around and , respectively. For the structural relaxations of the disordered systems, we used a 222 Monkhorst-Pack -mesh with the 432 atom supercells in combination with Methfessel-PaxtonMethfessel and Paxton (1989) Fermi surface smearing (), while total energies and densities of states were calculated by the tetrahedron method with Blöchl correctionsBlöchl et al. (1994) using a 444 -mesh. A 171717 -mesh was employed for the ordered structures represented in a cubic 16 atom unit cell. In all our calculations we allowed for a spontaneous spin polarization, always resulting in stable ferromagnetism.
In the SPR-KKR calculations, the ferromagnetic ground state was chosen as reference and disorder was treated analytically in the framework of the coherent potential approximation. The electronic density of states obtained for the different disordered structures agreed very well with the results obtained from the plane wave calculations, which corroborates our explicit supercell-based description.
iv.0.2 Formation energies and stable structures
The results of our total-energy calculations are shown in Fig. 8 and given in Tab. 3. In addition to the quaternary systems, we also computed the ternary full Heusler systems for use as reference energies, specifically cubic L CoMnAl, NiMnAl, and CoMnGa, as well as tetragonal L NiMnGa, according to the martensitic transition occurring in the latter case. The energy differences are always specified with respect to the four-atom Heusler formula unit in the fully relaxed states. As expected for isoelectronic systems, the energy differences of the different phases behave similar in NiCoMnAl and NiCoMnGa. In both cases, we observe a significant gain in energy by ordering the main group element Z and Mn. As one would expect, the B2 phase is among the least favorable ones in terms of total energy, and thus its observed thermodynamic stability at high temperatures is due to its large configurational entropy. The fully disordered bcc phases turned out to be significantly higher in energy, at 1.00 eV/f.u. for NiCoMnGa and 1.07 eV/f.u. for NiCoMnAl (both without relaxation), and are therefore not included in Fig. 8.
In contrast, the fully ordered Y structure, which has previously been proposed as a new candidate for a half metal, appears significantly more stable, not only against B2 disorder but also against decomposition into the ternary phases. However, a surprising result of our calculations is that NaCl-type ordering of Ni and Co is always disfavored compared to random disorder: this pertains both to L2 NiCo(MnZ), which is about higher in energy than B2 (NiCo)(MnZ), as well as the energetical gain of about when Y NiCoMnZ is disordered to L2 (NiCo)MnZ. Thus, considering only structural thermodynamics, the Y structure will not be thermodynamically stable at any temperature, as it has both higher internal energy as well as lower configurational entropy compared to L2 (NiCo)MnZ.
However, the partially disordered L2 structure should not be the ground state. Indeed, in both NiCoMnAl and NiCoMnGa the two tetragonal structures with a four-atom unit cell T and T have lower energies than all structures considered up to now. Thus, our calculations identify T with the alternation of Ni and Co planes as the ground state structure. We are confident that, at least among the superstructures on the bcc lattice, there should be no structures with significantly lower energies, as the NaCl-type Mn/Z order with its large energy gain seems quite stable, while any Ni/Co order different from the three kinds considered here would need to rely on quite long-range interactions.
We observe that the relaxation procedure yields a considerable energy gain for the disordered structures. An analysis of the corresponding atomic displacements is given in the supplementary information, evidencing an expansion of -coordinated pairs made up of equal atoms due to Pauli repulsion as common characteristic of the relaxations. Specifically for Mn/Z disorder, the mean bond lengths show an asymmetry, reflecting the larger size of the Z atom, particularly in the case for Z Ga. The relaxation energies of the disordered structures as given in Tab. 3 can be satisfactorily reproduced by assuming independent contributions of due Ni/Co disorder, due Mn/Al disorder, and due Mn/Ga disorder, with the prominence of the latter value again due to the larger size of Ga.
Due to the tetragonal arrangement of the Ni and Co atoms in T and T, the cubic symmetry is reduced to tetragonal, which is reflected also in the lattice parameters. Specifically, as reported in Tab. 3, , the lattice constant along the fourfold tetragonal axis, is 3–4% larger than for the T structure, while it is about 2% smaller for T. Indeed, this behavior is expected due to above-reported tendency of -coordinated equal elements in L2 (NiCo)MnZ to be pushed apart due to Pauli repulsion, while Ni-Co pairs are contracted. Further, the Wyckoff positions 2c in space group 129, which are occupied by Mn and Z in the T structure, have an internal degree of freedom correspond to a translation along the tetragonal axis. For NiCoMnAl, the parameters are and , and for NiCoMnGa and , being practically the same in both compounds. With Ni in 2a at and Co in 2b at , this means that Mn and Z are slightly shifted away from the Ni planes. Again, this is mirrored in the increased bond lengths of -coordinated Ni-Mn and Ni-Z pairs compared to Co-Mn and Co-Z pairs under disorder as given in the supplementary information. Thus, with these small tetragonal distortions and deviations of the internal degrees of freedom from the ideal values, it is clearly appropriate to consider also the tetragonal phases as superstructures on the bcc lattice.
While the tetragonal distortions as mentioned above are on the order of a few percent, the differences in the unit cell volumes between the cubic and the tetragonal structures is much smaller. Indeed, we observe that there is a nearly perfect monotonic decrease of unit cell volume with internal energy of the structures: while the volume contraction with Mn/Z ordering by values of about 0.2% for NiCoMnAl and 0.6% for NiCoMnGa was expected, and also the bigger effect in the latter case can be rationalized by the larger Ga atoms, NaCl-type Ni/Co order, which was already found to be energetically unfavorable, leads to a lattice expansion by about 0.5% in both systems. In contrast, the energy gains with T and T are reflected in a corresponding volume contraction.
Our theoretical results explain the experimental observations: as reported above, experimentally NiCoMnGa displays the B2 phase at high temperatures with a well-defined ordering transition to the L2 (NiCo)MnGa phase at lower temperatures, while for NiCoMnAl the transition temperature is reduced and only barely kinetically accessible. This is reproduced by our calculations, with L2 (NiCo)MnZ being the lowest-energy cubic phase, while B2 can be stabilized by entropy, with indeed a larger energy gain and thus a higher expected transition temperature for Z = Ga. Ni/Co ordering always increases the internal energy and decreases configurational entropy, thus L2 NiCo(MnZ) and Y NiCoMnZ are not predicted to be existing. Also, as the energy cost of disordering B2 to A2 is about five times larger than the gain of ordering to L2 (NiCo)MnZ, with the latter happening around , we do not expect a transition to A2 in the stability range of the solid.
On the other hand, also the L2 (NiCo)MnZ phase is only stabilized by entropy, and thus should transform at some temperature to T. With an argumentation as above, where the B2–L2 and the L2–T transitions have the same entropy difference, but the latter’s energy difference of is about a factor of 4–5 lower, we predict a transition temperature around (see the supplementary material for a more detailed discussion of these issues). As already the B2–L2 transition in NiCoMnAl at around is only barely progressing, a bulk transition to T is therefore not to be expected on accessible timescales.
Extrapolating the lattice constants measured on cooling to gives for NiCoMnGa and for NiCoMnAl. The deviation of about to the calculated values for L2 (NiCo)MnZ is quite satisfactory, corresponding to a relative error of 0.7%. Of course, the difference in lattice constants between the two alloys should be predicted even much more accurately, giving to be compared to the value of determined experimentally. Further, the predicted contraction at the B2–L2 transition in NiCoMnGa of agrees perfectly with the experimental value as obtained by integrating the excess thermal expansion coefficient between and , while in NiCoMnAl the contraction with ordering is estimated as by the differences in the heating and cooling curves evaluated at and to be compared with the predicted value of . These two small discrepancies imply that the experimental lattice constant of NiCoMnAl at low temperatures on cooling is increased compared to the theoretical predictions, which is consistent with a reduced degree of L2 long-range order in NiCoMnAl due to kinetical reasons.
From the non-integer values of the total magnetic moment per formula unit listed in Tab. 3, one can already conclude that neither of the structures yields the desired half-metallic properties. For the fully ordered Y structure, our calculated values for are slightly lower than the values of Halder et al. (2015) and Alijani et al. (2011a) previously reported for NiCoMnAl and NiCoMnGa, and the integer moment of 5 , which follows from the generalized Slater-Pauling rule for half-metallic full-Heusler compounds with valence electrons per unit cell. The calculated is even smaller for the other structures. Experimentally, we measured values between 4.47 and for L2 (NiCo)MnGa, while for Z Al an increase from in the as quenched state to after the longest annealing was observed. Thus, it seems that the dependence of on the state of order in the intermediate states is more complicated than the situation captured by our calculation of the respective extremes, corresponding to a decrease from perfect Mn/Z disorder in the B2 case to perfect order in the L2 case.
The values in Tab. 3 imply that the magnetic moment per formula unit depends primarily on the order on the Ni/Co sublattice, with values around for NaCl-type order, for columnar order or disorder, and for planar order. The equilibrium unit cell volume grows with increasing , with an additional lattice expansion in the cases of Mn/Z disorder.
The induced Ni moments show the largest variation between the different structures, in absolute and relative numbers. The Co moments follow the behavior of the Ni moments with a smaller variation. This is a consequence of the hybridization of Co and Ni in the minority spin density of states, which is responsible for the formation of a gap-like feature at as discussed in detail in the next section. The Mn moments appear well localized with values slightly above and vary only by a tenth of a Bohr magneton.
The ferromagnetic ground state of the compounds arises from the strong ferromagnetic coupling between nearest-neighbor Mn-Ni (coupling constant approximately ) and, in particular, Mn-Co (coupling constant approximately ) pairs. On the other hand, Mn pairs in coordination, which randomly occur in the B2 case, exhibit large frustrated antiferromagnetic coupling (coupling constant approximately ). This behavior is well known from ternary stoichiometric and off-stoichiometric Mn-based Heusler systems.Şaşıoğlu et al. (2004); Kurtulus et al. (2005); Şaşıoğlu et al. (2005); Rusz et al. (2006); Buchelnikov et al. (2008); Şaşıoğlu et al. (2008); Sokolovskiy et al. (2012); Comtesse et al. (2014); Entel et al. (2014) A more detailed account of the coupling constants for NiCoMnZ is given in the supplementary material. Thus for the low-energy structures, which do not exhibit Mn pairs with negative coupling constant, we expect the magnetic ordering temperature to be significantly higher than in the B2 case. This agrees nicely with the significant increase under annealing observed for NiCoMnAl, while NiCoMnGa is already L2-ordered in the as quenched state and thus has still a higher transition temperature.
iv.0.4 Electronic Structure
The shape of the electronic density of states (DOS) of ternary L2 Heusler compounds of the type XYZ, including the appearance of a half-metallic gap, has been explained convincingly by Galanakis et al.Galanakis et al. (2002, 2006) in terms of a molecular orbital picture. First, we consider the formation of molecular orbitals on the simple cubic sublattice occupied by atoms of type X. Here, the , and orbitals hybridize forming a pair of and molecular orbitals, while the and the states form and molecular orbitals. The molecular orbitals of and symmetry can hybridize with the respective orbitals of the nearest neighbor on the Y-position (in the present case Mn), splitting up in pairs of bonding and anti-bonding hybrid orbitals. However, due to their symmetry, no partner for hybridization is available for the and orbitals, which therefore remain sharp. Accordingly, these orbitals are dubbed “non-bonding”.
If the band filling is adjusted such that the Fermi level is located between the and states in one spin channel, the compound can become half-metallic. This is for instance the case for CoMnGe with , which has according to the generalized Slater–Pauling rule .Picozzi et al. (2002); Galanakis et al. (2002) If additional valence electrons are made available, also the states may become occupied. This is the case for NiMnGa and NiMnAl (), which do not possess half-metallic properties. Here, the Ni- states form a sharp peak just below the Fermi energy and gives rise to a band-Jahn-Teller mechanism leading to a martensitic transformation and modulated phases arising from strong electron-phonon coupling due to nesting features of the Fermi surface Brown et al. (1999); Lee et al. (2002); Bungaro et al. (2003); Zayak et al. (2003); Opeil et al. (2008); Haynes et al. (2012). Consequently, the magnetic moments of these compounds are significantly smaller. First principles calculations report values of 3.97 – 4.22 Ayuela et al. (1999); Godlevsky and Rabe (2001); Galanakis et al. (2002) and 4.02 – 4.22Fujii et al. (1989); Ayuela et al. (1999); Godlevsky and Rabe (2001); Ayuela et al. (2002); Şaşıoğlu et al. (2004); Gruner et al. (2008) for NiMnAl and NiMnGa, respectively.
Figure 9 shows the total and element-resolved electronic densities of states (DOS) of NiCoMnGa for the most relevant structures, which have the same valence electron concentration as the half metal CoMnGe (NiCoMnAl shows an analogous picture and can be found in the supporting material). Here, the perfectly alternating NaCl-type order of the elements on the Ni-Co sublattice in the Y-structure enforces a complete hybridization of the Ni- and Co-states, since the atoms find only neighbors of the other species. This becomes apparent from the pertinent illustration, as essentially the same features are present in the partial density of states of both elements. The magnitude of a specific peak may, however, be larger for one or the other species. This can be understood from the concept of covalent magnetismWilliams et al. (1981); Schwarz et al. (1984); Mohn (2003) which has been applied to Heusler alloys recently.Dannenberg et al. (2010) The molecular orbitals are occupied by each species with a weight scaling inversely with the energy difference to the constituting atomic levels. In the minority spin channel, the bonding molecular orbitals are dominated by Ni-states, while the non-bonding states around and the anti-bonding orbitals above are dominated by the Co-states. The non-bonding orbitals directly above are equally shared by Co and Ni states.
As expected, with decreasing order the features of the DOS smear out and become less sharp. Specifically the pseudo-gap at the Fermi level in one spin channel, which corresponds to the near half-metallic behavior suggested first by Entel et al. Entel et al. (2010), and subsequently by Alijani et al. Alijani et al. (2011a), Singh et al. Singh et al. (2012) and Halder et al. Halder et al. (2015), is in particular sensitive to ordering on the Ni-Co sublattice, and only encountered for the NaCl-type ordering of the fully ordered Y and the partially ordered L2 NiCo(MnZ) structures.
In fact, the minority spin gap is not complete. A close inspection of the band structure (see supporting material) of Y NiCoMnGa/Al clearly shows several bands crossing the Fermi level. Since this occurs in the immediate vicinity of the point, the weight of the respective states in the Brillouin zone is small and a gap-like feature appears in the DOS. Thus, in this configuration the compound should be classified as a half-semimetal rather than a half-metal. Nearly perfect gaps are observed if Z is a group IV element with a half-filled -shell. In our case, the missing electron of the main group element has to be compensated by the additional valence electron from one of the transition metals. These are only available on parts of the X-sites, which can lead to a distribution of -states between the sharp and states of the transition metals below and above .
In all other structures, the Ni-Co sublattice contains neighboring pairs of the same element. In this case, the respective -orbitals can hybridize independently at different levels. As a consequence, the and molecular orbitals split up. This is best seen in the DOS of the T structure. Here, the Ni-dominated part of the former peak moves to below the Fermi level (where we expect it in NiMnGa/Al), which creates considerable DOS right at and fully destroys the half-metallic character. An analogous argument can be applied to the cubic L2 (NiCo)MnZ and B2 structures with disorder on the Ni-Co sublattice (Fig. 9c and d). The disorder on the Mn-Z sublattice in the B2 phase causes only minor changes in the electronic structure and manifests mainly in a larger band width of the valence states and the disappearance of a pronounced peak at in the minority channel, which originates from the hybridization between the Ni-Co and Mn-Z sublattice. In contrast, the distribution of the Ni and Co states near the Fermi level, which are decisive for the functional properties of this compound family, is not significantly changed compared to the L2 case.
Employing in-situ neutron diffraction, magnetic measurements and calorimetry, we studied the ordering tendencies in the quaternary Heusler derivatives NiCoMnAl and NiCoMnGa. NiCoMnGa was found to display an L2 (NiCo)MnGa structure with strong Mn/Ga order and no to minor Ni/Co ordering tendencies, where the degree of order achieved upon slow cooling was higher than in quenched samples. The B2–L2 second-order phase transition was observed at . NiCoMnAl after quenching was found to adopt the B2 structure, while on slow cooling from high temperatures broadened L2 reflections were observed to emerge at temperatures below in neutron diffractometry. Yet, kinetics at these temperatures are so slow that the adjustment of large degrees of L2 order in this compound is kinetically hindered. Still, low-temperature annealing treatments at in samples quenched from showed a strong effect on the magnetic transition temperatures, proving that this parameter probes sensitively the state of order in the sample.
Density functional theory reproduces the experimentally observed trends of the order-dependent magnetic behavior and of the ordering tendencies between the two systems. Our calculations reveal that the fully ordered Y structure with F3m symmetry is thermodynamically not accessible, since the partially disordered L2 phase is lower in energy. Instead, we propose as the ground state a tetragonal structure with a planar arrangement of Ni and Co. This structure is stable against decomposition into the ternary Heusler compounds, but we expect the energetic advantage to be too small to compensate for the larger entropy of the L2 phase at reasonable annealing conditions. However, the fabrication of this structure by layered epitaxial growth on appropriately matching substrates, which favor the slight tetragonal distortion, could be possible. From the electronic density of states and band structure, we could conclude that neither of the structures is half-metallic in the strict definition. This specifically pertains also to the hypothetical Y structure, which exhibits several bands crossing the Fermi level close to the point in the minority spin channel, and is thus a half-semimetal.
Since the first quaternary Heusler derivatives adopting the Y structure have been proposed to possess half-metallic propertiesDai et al. (2009), the interest in these materials has developed rapidly with numerous publications dealing with the topic.Entel et al. (2010); Alijani et al. (2011a); Gökoğlu (2012); Singh et al. (2012); Al-zyadi et al. (2015); Wei et al. (2015); Halder et al. (2015); Mukadam et al. (2016); Alijani et al. (2011b); Gao et al. (2013); Özdoğan et al. (2013); Zhang et al. (2014); Bainsla et al. (2014); Xiong et al. (2014); Enamullah et al. (2015); Feng et al. (2015); Gao et al. (2015); Elahmar et al. (2015); Enamullah et al. (2016); Berri et al. (2014) Density-functional theory calculations have been used to identify promising systems among the NiCo-,Entel et al. (2010); Alijani et al. (2011a); Gökoğlu (2012); Singh et al. (2012); Al-zyadi et al. (2015); Wei et al. (2015); Halder et al. (2015) NiFe-Alijani et al. (2011a); Wei et al. (2015); Mukadam et al. (2016) and CoFe-basedDai et al. (2009); Alijani et al. (2011b); Gao et al. (2013); Özdoğan et al. (2013); Berri et al. (2014); Zhang et al. (2014); Bainsla et al. (2014); Xiong et al. (2014); Enamullah et al. (2015); Feng et al. (2015); Gao et al. (2015); Elahmar et al. (2015); Enamullah et al. (2016) compounds. However, in most cases the phase stability of the Y structure is tested, if at all, only against stacking order variations of this Y structure (see, for instance, Refs. Alijani et al., 2011b; Dai et al., 2009) but rarely against disorderEnamullah et al. (2016) or other states of order. Simultaneously, experimental investigations as a rule either point towards disordered structures Alijani et al. (2011b); Bainsla et al. (2014); Enamullah et al. (2015); Halder et al. (2015); Mukadam et al. (2016) or, specifically for the case of X-ray diffraction on ordering between transition metal elements, cannot decide these issues.Alijani et al. (2011a, b); Enamullah et al. (2015)
Based on our findings, we conclude that at least in the NiCo-based, but probably also in the NiFe- and CoFe-based alloys, the stability of the Y-structure is doubtful and, even if it was thermodynamically stable, might still not be kinetically accessible in most quaternary Heusler derivatives. Indeed, preliminary first-principles results show that also for the NiFeMnGa and the CoFeMnGa alloys, the tetragonal T order is lower in energy than the Y structure by 62 meV/f.u and 80 meV/f.u., respectively. This underlines that a detailed analysis of phase stabilities in those systems that have been identified as promising half-metals, especially with respect to the tetragonal structures and/or L2 type disorder, is essential in order to evaluate their actual potential. More generally, the comparatively small energetical differences between the various possible types of order along with small disordering energies specifically with respect to the late transition metal constituents as obtained here suggest that in these quaternary Heusler derivatives disorder could be the norm rather than the exception in physical reality.
This work was funded by the Deutsche Forschungsgemeinschaft (DFG) within the Transregional Collaborative Research Center TRR 80 “From electronic correlations to functionality”. P.N. acknowledges additional support from the Japanese Society for the Promotion of Science (JSPS) via a short-term doctoral scholarship for research in Japan. We thank O. Dolotko and A. Senyshyn of the MLZ for facilitating the neutron diffraction measurements. SQUID measurements were performed at the Center for Low Temperature Science, Institute for Materials Research, Tohoku University. Computing resources for the supercell calculations were kindly provided by the Center for Computational Sciences and Simulation (CCSS) at University of Duisburg-Essen on the supercomputer magnitUDE (DFG grants INST 20876/209-1 FUGG and INST 20876/243-1 FUGG).
- Heusler (1903) F. Heusler, in Verhandlungen der Deutschen Physikalischen Gesellschaft, Vol. 5 (1903) p. 219.
- Graf et al. (2011) T. Graf, C. Felser, and S. S. Parkin, Prog. Solid State Chem. 39, 1 (2011).
- Sozinov et al. (2002) A. Sozinov, A. A. Likhachev, N. Lanska, and K. Ullakko, Appl. Phys. Lett. 80, 1746 (2002).
- Krenke et al. (2005) T. Krenke, E. Duman, M. Acet, E. F. Wassermann, X. Moya, L. Mañosa, and A. Planes, Nature Mater. 4, 450 (2005).
- Jullière (1975) M. Jullière, Phys. Lett. A 54, 225 (1975).
- de Groot et al. (1983) R. A. de Groot, F. M. Mueller, P. G. van Engen, and K. H. J. Buschow, Phys. Rev. Lett. 50, 2024 (1983).
- Ishida et al. (1982) S. Ishida, S. Akazawa, Y. Kubo, and J. Ishida, J. Phys. F: Met. Phys. 12, 1111 (1982).
- Fujii et al. (1990) S. Fujii, S. Sugimura, Ishida, and S. Asano, J. Phys.: Condens. Matter 2, 8583 (1990).
- Jourdan et al. (2014) M. Jourdan, J. Minár, J. Braun, A. Kronenberg, S. Chadov, B. Balke, A. Gloskovskii, M. Kolbe, H. J. Elmers, G. Schönhense, H. Ebert, C. Felser, and M. Kläui, Nat. Commun. 5 (2014), 10.1038/ncomms4974.
- Halder et al. (2015) M. Halder, M. D. Mukadam, K. G. Suresh, and S. M. Yusuf, J. Magn. Magn. Mater. 377, 220 (2015).
- Alijani et al. (2011a) V. Alijani, J. Winterlik, G. H. Fecher, S. S. Naghavi, and C. Felser, Phys. Rev. B 83, 184428 (2011a).
- Özdoğan et al. (2013) K. Özdoğan, E. Şaşıoğlu, and I. Galanakis, J. Appl. Phys. 113, 193903 (2013).
- Okubo et al. (2011) A. Okubo, X. Xu, R. Y. Umetsu, T. Kanomata, K. Ishida, and R. Kainuma, J. Appl. Phys. 109, 07B114 (2011).
- Kanomata et al. (2009) T. Kanomata, Y. Kitsunai, K. Sano, Y. Furutani, H. Nishihara, R. Y. Umetsu, R. Kainuma, Y. Miura, and M. Shirai, Phys. Rev. B 80, 214402 (2009).
- Recarte et al. (2012) V. Recarte, J. I. Pérez-Landazábal, and V. Sánchez-Alarcos, J. Alloys Comp. 536, S5308 (2012).
- Barandiaran et al. (2013) J. M. Barandiaran, V. A. Chernenko., E. Cesari, D. Salas, J. Gutierrez, and P. Lazpitza, J. Phys.: Condens. Matter 25, 484005 (2013).
- Monroe et al. (2015) J. A. Monroe, J. E. Raymond, X. Xu, N. Nagasako, R. Kainuma, Y. I. Chumlyakov, R. Arroyave, and I. Karaman, Acta Mater. 101, 107 (2015).
- Çakır et al. (2016) A. Çakır, M. Acet, and M. Farle, Sci. Rep. 6, 28931 (2016).
- Ruban and Abrikosov (2008) A. V. Ruban and I. A. Abrikosov, Rep. Prog. Phys. 71, 046501 (2008).
- (20) We neglect here the martensitic transitions below room temperature.
- Sánchez-Alarcos et al. (2007) V. Sánchez-Alarcos, V. Recarte, J. I. Pérez-Landazábal, and G. J. Cuello, Acta Mater. 55, 3883 (2007).
- Acet et al. (2002) M. Acet, E. Duman, E. F. Wassermann, L. Mañosa, and A. Planes, J. Appl. Phys. 92, 3867 (2002).
- Webster (1971) P. J. Webster, J. Phys. Chem. Solids 32, 1221 (1971).
- (24) We do not consider the other two B2 possibilities (NiMn)(CoZ) and (NiZ)(CoMn).
- Pauly et al. (1968) H. Pauly, A. Weiss, and H. Witte, Z. Metallkd. 59, 47 (1968).
- Bacon and Plant (1971) G. E. Bacon and J. S. Plant, J. Phys. F: Met. Phys. 1, 524 (1971).
- Eberz et al. (1980) U. Eberz, W. Seelentag, and H. Schuster, Z. Naturforsch. B 35, 1341 (1980).
- Johnson and Jeitschko (1974) V. Johnson and W. Jeitschko, J. Solid State Chem. 11, 161 (1974).
- Neibecker et al. (2014) P. Neibecker, M. Leitner, G. Benka, and W. Petry, Appl. Phys. Lett. 105, 261904 (2014).
- Hoelzel et al. (2015) M. Hoelzel, A. Senyshyn, and O. Dolotko, J. Large-Scale Res. Facil. 1, A5 (2015).
- Hoelzel et al. (2012) M. Hoelzel, A. Senyshyn, N. Juenke, H. Boysen, W. Schmahl, and H. Fuess, Nucl. Instrum. Methods A 667, 32 (2012).
- Ishikawa et al. (2008) H. Ishikawa, R. Y. Umetsu, K. Kobayashi, A. Fujita, R. Kainuma, and K. Ishida, Acta Mater. 56, 4789 (2008).
- Umetsu et al. (2011) R. Y. Umetsu, H. Ishikawa, K. Kobayashi, A. Fujita, K. Ishida, and R. Kainuma, Scripta Mater. 65, 41 (2011).
- Kainuma et al. (2000) R. Kainuma, F. Gejima, Y. Sutou, I. Ohnuma, and K. Ishida, Mat. Trans. JIM 41, 943 (2000).
- Ziebeck and Webster (1975) K. R. A. Ziebeck and P. J. Webster, J. Phys. F: Met. Phys. 5, 1756 (1975).
- Kresse and Furthmüller (1996) G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996).
- Liechstenstein et al. (1987) A. I. Liechstenstein, M. I. Katsnelson, V. P. Antropov, and V. A. Gubanov, J. Magn. Magn. Mater. 67, 65 (1987).
- Ebert et al. (2011) H. Ebert, D. Ködderitzsch, and J. Minár, Rep. Prog. Phys. 74, 096501 (2011).
- Kresse and Joubert (1999) G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).
- Perdew et al. (1996) J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).
- Methfessel and Paxton (1989) M. Methfessel and A. T. Paxton, Phys. Rev. B 40, 3616 (1989).
- Blöchl et al. (1994) P. E. Blöchl, O. Jepsen, and O. K. Andersen, Phys. Rev. B 49, 16223 (1994).
- Şaşıoğlu et al. (2004) E. Şaşıoğlu, L. M. Sandratskii, and P. Bruno, Phys. Rev. B 70, 024427 (2004).
- Kurtulus et al. (2005) Y. Kurtulus, R. Dronskowski, G. D. Samolyuk, and V. P. Antropov, Phys. Rev. B 71, 014425 (2005).
- Şaşıoğlu et al. (2005) E. Şaşıoğlu, L. M. Sandratskii, P. Bruno, and I. Galanakis, Phys. Rev. B 72, 184415 (2005).
- Rusz et al. (2006) J. Rusz, L. Bergqvist, J. Kudrnovský, and I. Turek, Phys. Rev. B 73, 214412 (2006).
- Buchelnikov et al. (2008) V. D. Buchelnikov, P. Entel, S. V. Taskaev, V. V. Sokolovskiy, A. Hucht, M. Ogura, H. Akai, M. E. Gruner, and S. K. Nayak, Phys. Rev. B 78, 184427 (2008).
- Şaşıoğlu et al. (2008) E. Şaşıoğlu, L. M. Sandratskii, and P. Bruno, Phys. Rev. B 77, 064417 (2008).
- Sokolovskiy et al. (2012) V. V. Sokolovskiy, V. D. Buchelnikov, M. A. Zagrebin, P. Entel, S. Sahoo, and M. Ogura, Phys. Rev. B 86, 134418 (2012).
- Comtesse et al. (2014) D. Comtesse, M. E. Gruner, M. Ogura, V. V. Sokolovskiy, V. D. Buchelnikov, A. Grünebohm, R. Arróyave, N. Singh, T. Gottschall, O. Gutfleisch, V. A. Chernenko, F. Albertini, S. Fähler, and P. Entel, Phys. Rev. B 89, 184403 (2014).
- Entel et al. (2014) P. Entel, M. E. Gruner, D. Comtesse, V. V. Sokolovskiy, and V. D. Buchelnikov, phys. stat. sol. (b) 251, 2135 (2014).
- Galanakis et al. (2002) I. Galanakis, P. H. Dederichs, and N. Papanikolaou, Phys. Rev. B 66, 174429 (2002).
- Galanakis et al. (2006) I. Galanakis, P. Mavropoulos, and P. H. Dederichs, J. Phys. D: Appl. Phys. 39, 765 (2006).
- Picozzi et al. (2002) S. Picozzi, A. Continenza, and A. J. Freeman, Phys. Rev. B 66, 094421 (2002).
- Brown et al. (1999) P. J. Brown, A. Y. Bargawi, J. Crangle, K.-U. Neumann, and K. R. A. Ziebeck, J. Phys.: Condens. Matter 11, 4715 (1999).
- Lee et al. (2002) Y. Lee, J. Y. Rhee, and B. N. Harmon, Phys. Rev. B 66, 054424 (2002).
- Bungaro et al. (2003) C. Bungaro, K. M. Rabe, and A. Dal Corso, Phys. Rev. B 68, 134104 (2003).
- Zayak et al. (2003) A. T. Zayak, P. Entel, J. Enkovaara, A. Ayuela, and R. M. Nieminen, Phys. Rev. B 68, 132402 (2003).
- Opeil et al. (2008) C. P. Opeil, B. Mihaila, R. K. Schulze, L. Mañosa, A. Planes, W. L. Hults, R. A. Fisher, P. S. Riseborough, P. B. Littlewood, J. L. Smith, and J. C. Lashley, Phys. Rev. Lett. 100, 165703 (2008).
- Haynes et al. (2012) T. D. Haynes, R. J. Watts, J. Laverock, Z. Major, M. A. Alam, J. W. Taylor, J. A. Duffy, and S. B. Dugdale, New J. Phys. 14, 035020 (2012).
- Ayuela et al. (1999) A. Ayuela, J. Enkovaara, K. Ullakko, and R. Nieminen, J. Phys.: Condens. Matter 11, 2017 (1999).
- Godlevsky and Rabe (2001) V. V. Godlevsky and K. M. Rabe, Phys. Rev. B 63, 134407 (2001).
- Fujii et al. (1989) S. Fujii, S. Ishida, and S. Asano, J. Phys. Soc. Japan 58, 3657 (1989).
- Ayuela et al. (2002) A. Ayuela, J. Enkovaara, and R. Nieminen, J. Phys.: Condens. Matter 14, 5325 (2002).
- Gruner et al. (2008) M. E. Gruner, W. A. Adeagbo, A. T. Zayak, A. Hucht, S. Buschmann, and P. Entel, Eur. Phys. J. Special Topics 158, 193 (2008).
- Williams et al. (1981) A. R. Williams, R. Zeller, V. L. Moruzzi, C. D. Gelatt, Jr., and J. Kübler, J. Appl. Phys. 52, 2067 (1981).
- Schwarz et al. (1984) K. Schwarz, P. Mohn, P. Blaha, and J. Kübler, J. Phys. F: Met. Phys. 14, 2659 (1984).
- Mohn (2003) P. Mohn, Magnetism in the Solid State, Springer Series in Solid-State Sciences, Vol. 134 (Springer, Berlin, Heidelberg, 2003).
- Dannenberg et al. (2010) A. Dannenberg, M. E. G. M. Siewert, M. Wuttig, and P. Entel, Phys. Rev. B 82, 214421 (2010).
- Entel et al. (2010) P. Entel, M. E. Gruner, A. Dannenberg, M. Siewert, S. K. Nayak, H. C. Herper, and V. D. Buchelnikov, Mater. Sci. Forum 635, 3 (2010).
- Singh et al. (2012) M. Singh, H. S. Saini, and M. K. Kashyap, Adv. Mater. Res. 585, 270 (2012).
- Dai et al. (2009) X. Dai, G. Liu, G. H. Fecher, C. Felser, Y. Li, and H. Liu, J. Appl. Phys. 105, 07E901 (2009).
- Gökoğlu (2012) G. Gökoğlu, Solid State Sci. 14, 1273 (2012).
- Al-zyadi et al. (2015) J. M. K. Al-zyadi, G. Y. Gao, and K.-L. Yao, J. Magn. Magn. Mater. 378, 1 (2015).
- Wei et al. (2015) X.-P. Wei, Y.-L. Zhang, Y.-D. Chu, X.-W. Sun, T. Sun, P. Guo, and J.-B. Deng, J. Phys. Chem. Solids 82, 28 (2015).
- Mukadam et al. (2016) M. D. Mukadam, S. Roy, S. S. Meena, P. Bhatt, and S. M. Yusuf, Phys. Rev. B 94, 214423 (2016).
- Alijani et al. (2011b) V. Alijani, S. Ouardi, G. H. Fecher, J. Winterlik, S. S. Naghavi, X. Kozina, G. Stryganyuk, C. Felser, E. Ikenaga, Y. Yamashita, S. Ueda, and K. Kobayashi, Phys. Rev. B 84, 224416 (2011b).
- Gao et al. (2013) G. Y. Gao, L. Hu, K. L. Yao, B. Luo, and N. Liu, J. Alloys Comp. 551, 539 (2013).
- Zhang et al. (2014) Y. J. Zhang, Z. H. Liu, G. T. Li, X. Q. Ma, and G. D. Liu, J. Alloys Comp. 616, 449 (2014).
- Bainsla et al. (2014) L. Bainsla, K. G. Suresh, A. K. Nigam, M. Manivel Raja, B. S. D. C. S. Varaprasad, Y. K. Takahashi, and K. Hono, J. Appl. Phys. 116, 203902 (2014).
- Xiong et al. (2014) L. Xiong, L. Yi, and G. Y. Gao, J. Magn. Magn. Mater. 360, 98 (2014).
- Enamullah et al. (2015) Enamullah, Y. Venkateswara, S. Gupta, M. R. Varma, P. Singh, K. G. Suresh, and A. Alam, Phys. Rev. B 92, 224413 (2015).
- Feng et al. (2015) Y. Feng, H. Chen, H. Yuan, Y. Zhou, and X. Chen, J. Magn. Magn. Mater. 378, 7 (2015).
- Gao et al. (2015) Q. Gao, L. Li, G. Lei, J.-B. Deng, and X.-R. Hu, J. Magn. Magn. Mater. 379, 288 (2015).
- Elahmar et al. (2015) M. H. Elahmar, H. Rached, D. Rached, R. Khenata, R. Murtaza, S. Bin Omran, and W. K. Ahmed, J. Magn. Magn. Mater. 393, 165 (2015).
- Enamullah et al. (2016) Enamullah, D. D. Johnson, K. G. Suresh, and A. Alam, Phys. Rev. B 94, 184102 (2016).
- Berri et al. (2014) S. Berri, D. Maouche, M. Ibrir, and F. Zerarga, J. Magn. Magn. Mater. 354, 65 (2014).