STRUCTURAL AND MAGNETIC PROPERTIES OF HEUSLER ALLOYS Pd2MnZ (Z=Ga, Ge, As): AB INITIO STUDY

In this work, we report results of ab initio and Monte Carlo investigations of structural and magnetic properties in a series of Heusler compositions Pd2MnZ (Z = Ga, Ge, As). It was found that for Pd2MnGa and Pd2MnAs, the stable martensitic state is realized on the contrast with Pd2MnGe. The equilibrium lattice parameters for the series of Pd2MnZ (Z = Ga, Ge, As) compounds increase with increasing the number of valence electrons per atom (e/a ratio). Having calculated total magnetic moments and magnetic exchange parameters from ab initio methods, the Curie temperature for Pd2Mn-based alloys has been estimated in the framework of Monte Carlo simulations of Heisenberg model.


Introduction
Heusler alloys are of great interest to the scientific community due to their unique properties such as shape memory effect, effects of superelasticity and superplasticity, giant magnetocaloric effect, giant magnetoresistance and magnetostrain, etc.These effects are associated with martensitic transformations [1][2][3][4].
Full-Heusler alloys are ternary intermetallic systems with L2 1 crystal structure and are derived from a parent compound with the generic formula X 2 YZ.Here, X and Y are 3d transition metals, and Z are sp elements of III-V group of periodic table of elements.
At present, a small number of works is devoted to the study of Pd 2 Mn-based alloys.For instance, Natera et al. found that Pd 2 MnGe shows a preferentially disordered L2 1 structure with 0.98 Pd + 0.02 Mn on the A-and C-sites and 0.96 Mn + 0.04 Pd on the B-sites [5].
The objective of this paper is the theoretical investigation of the structural and magnetic properties of Pd 2 MnZ (Z = Ga, Ge, As) Heusler alloys and comparison with available experimental data.
In the first step, the structural ground state calculations were performed using the VASP package.In order to determine the presence of a stable martensitic state, calculations of the total energy as a function of the tetragonal distortion (с/a, where a and c are the crystallographic axis) of a cubic crystal are performed.Subsequently, the optimized structural configurations from the VASP calculations were used for the calculations of magnetic exchange constants by means of the SPR-KKR package.
Computational parameters used for the VASP package are following.The pseudopotentials were taken with the electronic configurations as follows: Pd (4p 6 4d 9 5s 1 ), Mn (3p 6 3d 6 4s 1 ), Ga (3d 1 4s 2 4p 1 ), Ge (3d 1 4s 2 4p 2 ), and As (4s 2 4p 3 ).The interaction between ions and electrons was described by the projector augmented wave (PAW) method [7,8].For the kinetic energy cut-off was 400 eV and the kinetic energy cut-off for the augmentation charges was 800 eV.Brillouin zone sampling was performed according to the Monkhorst-Pack grid [10].The k-points for self-consistent field cycles in the Brillouin zone were generated with 12 3 meshes.All considered structures were relaxed using the conjugate gradient algorithm and both the atomic position and the volume were optimized.
As for the SPR-KKR package, the calculations of exchange parameters were performed using of the spinpolarized scalar-relativistic mode.
To obtain the temperature dependences of the magnetization of the Pd 2 MnZ (Z = Ga, Ge, As) compounds, well-working and well-tested Heisenberg model was applied [11,12,13,14].With the use of longrange exchange interactions, it allows obtaining Curie temperatures close to the experiment.We carried out MC simulations of the three-dimensional Heisenberg model in the absence of anisotropy and magnetic fields where S i is the spin of the unit length (|S i | = 1) placed at the lattice site, and J ij are the exchange coupling parameters calculated using an expression proposed by Liechtenstein et al. [15].The magnetic exchange parameters clearly show the damping oscillatory behavior, changing the type of interaction from ferromagnetic (FM) to antiferromagnetic (AFM) and vice versa.We included the exchange interactions up to eight coordination shell for all interaction pairs.The MC simulations were performed for a simulation cell with periodic boundary conditions using the Metropolis algorithm [16].Therefore, for the Pd 2 MnZ (Z = Ga, Ge, As) alloys, we used the simulation cell containing 1098 Pd, 1099 Mn, and 1728 Z atoms.Changes in the independent spin variables {S i x , S i y , S i z } are accepted or rejected in accordance with the single-site transition probability, W = min{1; exp(−∆H/k B T)}.As a time unit, we used one MC step consisting of N attempts to change the spin variables.For each temperature, the properties (internal energy of the system H and magnetic order parameter m) were analyzed, allowing 5×10 5 MC steps and 10 4 thermalization steps.
To calculate the total average magnetization M at a certain temperature point, we used the following equation: where µ Pd and µ Mn are the magnetic moments of Pd and Mn atoms, which were obtained from ab initio calculations; m Pd and m Mn are the normalized magnetizations calculated as follows: 3 Results and discussion

Structural properties
Initially, we determined the equilibrium lattice constant of a crystal of Pd 2 MnZ compounds using the geometry optimization in VASP, focusing on the equilibrium volume of the cell and the tetragonal distortion parameterized by the c/a ratio.In this case, a negligibly small change in volume between the cubic and tetragonal phases was assumed: V a a c = = .
The calculated equilibrium lattice parameters and bulk modulus of the Pd 2 MnZ austenitic compounds are summarized in Table 1.Noted parameters were estimated according to the Birch-Murnagahan equation of state.As one can see from Table 1, the equilibrium lattice parameters for a series of Pd 2 MnZ (Z = Ga, Ge, As) compounds increase with increasing e/a due to increasing Z element's radius.Experimental values of Pd 2 MnGa and Pd 2 MnGe are 6.180 Å and 6.174 Å, respectively [17].In order to investigate the possibility of the martensitic transformation in alloys studied, we performed calculations of the total energy relative to the cubic phase as a function of the tetragonality ratio c/a.The total energy differences between the tetragonally distorted and the cubic phases as functions of c/a ratio for Pd 2 MnZ (Z = Ga, Ge, As) alloys are shown in Fig. 1.In this case, ∆E = 0 corresponds to the equilibrium austenitic phase for each compound.
It is seen that for Pd 2 MnGa and Pd 2 MnAs, the stable martensite can be realized at c/a ratio of 1.30 and 1.35, respectively.While in the case of the stoichiometric Pd 2 MnGe alloy, the austenite-martensite transformation does not occur due to the absence of any energy minima in the E(c/a) curve for c/a ≠ 1, as shown in Fig. 1.Fig. 1.The total energy differences with respect to equilibrium value for a series of Pd 2 MnZ (Z = Ga, Ge, As) alloys as functions of tetragonal distortion, c/a.

Electronic and magnetic properties
The total densities of the electronic states are shown in Fig. 2(a-c).The analysis of the densities of electron states shows the relative shift of the Fermi level to higher energy regions in the Ga-Ge-As sequence, which arises from the increase in the number of valence electrons in this series of alloys.The character of the densities of electronic states is similar to those obtained for other Pd 2 Mn-containing alloys [18,19].
Magnetic properties of Pd 2 Mn-based series of alloys are also of significant interest, because they allow us to predict compositional trends of the Curie temperature.
Firstly, the total magnetic moment of Pd 2 MnZ (Z = Ga, Ge, As) compounds was calculated using the SPR-KKR package.We found that Pd 2 MnGa possesses the lowest magnetic moment (4.12 µB/f.u.) and it increases slightly with the increasing number of valence electrons per atom e/a.Whereat we evaluated the magnetic exchange parameters.
The exchange coupling constants, J ij , for Pd 2 MnZ as a function of the distance between interacting atoms i and j are shown in Fig. 2 (d-f).It is worth mentioning that positive values of J ij correspond to the FM exchange, whereas the negative ones mark the AFM exchange.It should be noticed that exchange interactions between Pd-Z (Z = Ga, Ge, As) and Pd-Pd atoms are very small due to the lowest magnetic moments and omitted from the figures.It is seen that for all compounds studied the Mn-Pd pair exhibits a similar behavior, where the nearest-neighbor interactions provide the maximum contribution to the exchange energy than subsequent interactions.The latter ones are close to zero.
As for the Mn-Mn exchange interactions, they have an AFM character between nearest-neighbor atoms within the first coordination shell for Pd 2 MnGa, then it becomes FM and increases slightly with increasing e/a ratio.The magnetic exchange parameters clearly show the damping oscillatory behavior for all compositions, which can be associated with various exchange mechanisms such as the magnetic interaction between localized d and collectivized s electrons, magnetic exchange interaction between localized and collectivized d electrons, and super-exchange interaction via sp-conduction electrons contributing to the RKKY-type indirect exchange interaction [18].In addition, for Pd 2 MnGa and Pd 2 MnGe, the AFM contributions to the exchange energy are accounted for Mn-Mn interactions within the sixth and fourth coordination shells, respectively.On the other side, for Pd 2 MnAs, the AFM contributions are significantly less than for other alloys of studying series.
Obtained exchange coupling parameters and partial magnetic moments for Pd 2 MnZ (Z = Ga, Ge, As) alloys were used for further MC simulations of the thermomagnetization curves to obtain corresponding Curie temperatures.The set of magnetization curves as a function of temperature for alloys studied is presented in Fig. 3.As can be seen from Fig. 3, for all cases, the ferromagnetic -paramagnetic phase transition in austenite occurs.In a case of Pd 2 MnGa, the highest Curie temperature can be observed due to Mn-Mn FM interactions within second, third, fourth, and fifth coordination shells jointly with nearest-neighbor Mn-Pd FM interaction.On the other hand, for Pd 2 MnGe, the lowest value of Curie temperature is found because of the largest AFM contribution between Mn atoms interacting within the fourth shell.We note that the experimental value of Curie temperature of Pd 2 MnGe is 170 K [17].The calculated total magnetic moments and Curie temperatures for alloys studied are summarized in Table 2.

Conclusion
The optimized lattice parameters for austenite and martensite, total magnetic moments as well as thermomagnetization curves and Curie temperatures for a series of Pd 2 MnZ (Z = Ga, Ge, As) compounds are successfully obtained using the first-principles ground state calculations in association with the finitetemperature Monte Carlo simulations.Ab initio calculations were performed using VASP and SPR-KKR packages.It was shown that the optimized lattice parameters of considered alloys increase along the sequence of Pd 2 MnGa→Pd 2 MnGe→Pd 2 MnAs.This has to do with the fact that Z element's radius increases along the respective sequence.The lattice energy calculations for the distorted cubic structure have shown that the stable martensitic phase can be realized only for Pd 2 MnGa and Pd 2 MnAs compounds with c/a ratio of 1.3 and 1.35, respectively.The total magnetic moments for the cubic austenitic state of Pd 2 MnZ (Z = Ga, Ge, As) were found to be increasing with the increasing e/a ratio.Using the Heisenberg Hamiltonian and exchange coupling constants, the Curie temperatures for compounds studied have been obtained.It was shown that Pd 2 MnGa possesses the Curie temperature close to room temperature, while for remaining compositions, it found to be around 190 K.We consider that discussed alloys could be interesting for further investigations.

Fig. 2 .
Fig. 2. The total densities of the electronic states and the exchange coupling parameters for (a) Pd 2 MnGa, (b) Pd 2 MnGe, and (c) Pd 2 MnAs alloys as a function of the distance (d/a) between atoms i and j.Here a is the optimized lattice parameter.

Table 1 .
Calculated lattice constants a 0 (in Å) and bulk modulus B (in GPa) for Pd 2 MnZ alloys

Table 2 .
Calculated total magnetic moment µ tot (in µ B /f.u.) and Curie temperature T C (in K) for Pd 2 MnZ (Z=Ga, Ge, As) alloys