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Structure and physical properties of quaternary Heusler alloy NiMnCuSb
Accepted Manuscript Structure and physical properties of quaternary Heusler alloy NiMnCuSb S.K. Bose, J. Kudrnovský, Y. Liu PII: DOI: Reference:
S0304-8853(17)31965-0 http://dx.doi.org/10.1016/j.jmmm.2017.08.049 MAGMA 63083
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Journal of Magnetism and Magnetic Materials
Please cite this article as: S.K. Bose, J. Kudrnovský, Y. Liu, Structure and physical properties of quaternary Heusler alloy NiMnCuSb, Journal of Magnetism and Magnetic Materials (2017), doi: http://dx.doi.org/10.1016/j.jmmm. 2017.08.049
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Structure and physical properties of quaternary Heusler alloy NiMnCuSb S.K. Bosea,∗, J. Kudrnovsk´yb , and Y. Liuc a
Physics Department, Brock University, St. Catharines, ON, L2S 3A1, Canada Institute of Physics, Academy of the Sciences of the Czech Republic, Na Slovance 2, 182 21 Prague 8, Czech Republic c State Key Laboratory of Metastable Materials Science and Technology, and Hebei Key Laboratory of Microstructural Material Physics, School of Sciences, Yanshan University, Qinhuangdao, 066004, China
b
Abstract The relationship between structure and physical properties such as magnetic moment and Curie temperature is studied theoretically for the recently synthesized equiatomic quaternary Heusler alloy CuNiMnSb, which is found to crystallize in the LiMgPdSb structure. Calculations are performed for various different possible structural phases within the framework of the D03 -type structure (space group F¯43m) by changing the sequence of the atoms along the body diagonal of fcc lattice that characterizes the structure of this alloy. Total energy, magnetic moment and the Curie temperature results, in particular the latter two, suggest that the correct order of the atoms along the body diagonal is most likely Ni(0,0,0), Mn(0.25,0.25,0.25), Cu(0.5,0.5,0.5) and Sb(0.75,0.75,0.75) in contrast to (disagreement with) the conclusions of a recent experimental study. Neither the sequence Ni-Cu-Mn-Sb nor Ni-MnCu-Sb shows a large spin polarization at the Fermi level. Keywords: Heusler alloys, FP-LAPW, TB-LMTO, exchange interaction, Curie temperature
∗
Corresponding author: S.K. Bose Email address: [email protected] (S.K. Bose)
Preprint submitted to JMMM
August 14, 2017
1. Introduction Lattice structure is one of the most fundamental properties dictating the physical properties of crystalline solids. For solids with a multisublattice structure with different atoms occupying the sublattices, the importance of the lattice structure becomes paramount. Important examples of such solids are Heusler alloys with a variety of interesting physical properties. Many of such alloys have a cubic structure consisting of four interpenetrating fcc-sublattices A-B-C-D aligned along the body diagonal and centered about equidistant positions (0,0,0)a, (0.25,0.25,0,25)a, (0.5,0.5.0,5)a, and (0.75,0.75,0.75)a, where a is the lattice constant. Regular ternary Heusler alloys formed by atoms X,Y,and Z have formula (in terms of the fcc-sublattice occupations) X-Y-X-Z as, e.g., the Cu2 MnAl system. Closely related alloys with the anti-Heusler structure follow the formula X-X-Y-Z (e.g. Mn2 CuAl). Another class of Heusler alloys is the recently studied quaternary compounds (system) with the so-called LiMgPdSb(Sn) type structure [1], where the sequence of the atoms along the body diagonal is Li, Mg, Pd, and Sb(Sn). Special cases of such alloys are semi-Heusler alloys such as NiMnSb where the ’missing’ element is a vacancy. There are numerous articles on Heusler alloys encompassing both theoretical and experimental results. For review purposes readers may consult [2, 3, 4, 5] and references therein. The typical experimental tool for the determination of lattice structure of Heusler alloys, and the order of occupation of sublattices A to D by atomic species, is the powder X-ray diffraction (XRD) method. However, if constituent atoms have similar electronic form factors, such as Cu, Ni, or Mn, the reliability of the XRD to determine the correct atomic order becomes questionable. In such cases neutron diffraction studies would be more reliable, since the magnetic scattering factors should differ greatly among Mn, Ni and Cu. In the absence of neutron scattering results guidance from first-principles theoretical calculations can be very useful, as the calculated physical properties such as total energy, magnetic moments and the critical temperatures (for magnetic alloys) are sensitive to the chemical order. We illustrate this point by considering the newly synthesized quaternary NiMnCuSb Heusler alloy [6]. This alloy is interesting in that it is one of the few known ordered and cubic non-Fe Heusler alloys that is ferromagnetic with a relatively high Curie temperature (∼ 690 K), as determined experimentally by Haque et al. [6] . The total magnetic moment per formula unit (f.u.) is ∼ 3.85 µB . The XRD experiments carried out by the authors of the 2
paper [6] have led to the conclusion that the crystallographic order is given by the formula Ni-Cu-Mn-Sb. However, in the words of the authors themselves, ’it is difficult to confirm the ordering in structure based on the X-ray diffraction studies as the diffraction from Cu, Ni and Mn atoms is identical due to identical scattering factor.’ Based on their magnetic measurements the authors conclude that the moment primarily resides on the Mn atoms, the Ni atoms do not carry any moment. In this work we present theoretical results for the total energy, magnetic moments, and Curie temperatures Tc for the ideal NiCuMnSb compound and also for the Cu- and Ni-rich counterparts studied by Haque et al. in [6]. While we confirm that magnetic moment resides predominantly on the Mn atoms, we question the suggested formula and propose a new one, namely the formula NiMnCuSb (with the sequence Ni-Mn-Cu-Sb along the diagonal), the results for which agree better with measured total magnetic moment and the Curie temperature. Note also that the analogy of NiMnCuSb to Ni2 MnSb, and in particular to NiMnSb or (Cu,Ni)MnSb may be useful in this context: in none of these alloys do Mn and Sb occupy nearest neighbor sublattices. It should be noted that the structure of the quaternary NiMnCuSb alloy can be visually related to the structure of the known semi-Heusler alloy NiMnSb: Cu atoms of the former occupy the empty fcc sublattice of the latter. This simple view is further corroborated by the fact that the experimental lattice parameter of NiMnCuSb is almost the same as that of NiMnSb[7]. This indicates that there is enough space in the open NiMnSb structure to accommodate Cu atoms without noticeable change in volume per atom. 2. Formalism We have studied the electronic structure and magnetic properties of this alloy system using two complementary methods: the full-potential linear augmented plane wave (FP-LAPW) method, as implemented in WIEN2k [8] code, and the Green’s function formulation [9, 10] of the tight-binding linear muffin-tin orbital (TB-LMTO) method in the atomic sphere approximation (ASA) [11]. Both approaches are based on the density functional theory (DFT) [12]. While the more accurate WIEN2k code is used for the phase stability study, we employ the TB-LMTO method in conjunction with the coherent potential approximation (CPA) [9, 10] for the study of cases involving disorder between sublattices, e.g., for disordered Cu- and Ni-rich phases. We employ the TB-LMTO-CPA method primarily for the determination of ex3
change integrals [13] which are needed for the study of the magnetic stability of the alloys and the Curie temperatures. Three different forms of the generalized gradient approximation (GGA), which is considered to be optimal for the structural determination, were tested: PBE-96 [14], Wu-Cohen [15], and PBE-sol [16]. TB-LMTO calculations generally used the exchange-correlation potential of Vosko, Wilk and Nussair (VWN)[17]. In some cases we have also used the LDA+U method with the Hubbard U parameter in the fully-localized limit [18]. The LDA+U approach is sometimes used for the fine-tuning of band gaps, magnetic moments, critical temperatures, etc. (see, e.g., a recent study [19]). In both methods we have neglected the spin-orbit coupling because of its marginally small effect on the reported results. Finally, for both the WIEN2k and TB-LMTO calculations, careful tests were done concerning various methoddependent parameters to warrant proper convergence of the results. Values of various parameters for the WIEN2k method, such as the number of wave vectors in the Brillouin zone (BZ), BZ sampling scheme, convergence criteria etc. were all the same as discussed in our recent publications[20, 21]. 3. Results and discussion 3.1. Total energies and magnetic moments We have calculated the total energies and magnetic moments for three different realizations of the structure by placing the atoms Ni, Cu, and Mn in different sequences at the A(0,0,0), B(0.25,0.25,02.5), and C(0.5,0.5,0.5) sites, while the Sb atom was fixed at the D(0.75,0.75,0.75) site. In Table 1 we show the WIEN2k results for the total energies and magnetic moments for three different GGA versions of the exchange-correlation potential. The results are shown for the experimental lattice parameter of 5.92382 ˚ A. Similarly, in Table 2 we summarize related results for the energy optimization with respect to the sample volume or the lattice constant. The following conclusions can be drawn: (i) The sequence Cu(A)-Ni(B)-Mn(C) can readily be discarded on the basis of energy consideration for both the experimental and optimized lattice constants; (ii) On the other hand, the total energies of the sequences Ni(A)-Cu(B)-Mn(C) (denoted below as NiCuMnSb) and Ni(A)-Mn(B)-Cu(C) (denoted as NiMnCuSb) are very close in both cases. The calculated lattice constants, magnetic moments, or relative energy differences among various phases are not sensitive to the choice of the type of the GGA functional. While for the experimental volume the total energy of 4
the Ni-Cu-Mn-Sb sequence is slightly lower than that of the Ni-Mn-Cu-Sb sequence, for the optimized volumes the order of total energies is reversed. It should be noted that the optimized lattice constants are overestimated by about 3-4% with respect to the experiment; (iii) The most important part of the total magnetic moment Mtot is the contribution from local Mn-moments as assumed in the paper by Haque et al. [6]. Moments for the optimized structures are larger than the experimental values due their larger lattice constants. The local moments on the Ni-atoms are small (about 0.1−0.25 µB ) and those on Cu/Sb atoms are negligible. Most important, however, is the fact that Mtot is generally too small for the sequence Ni-Cu-Mn-Sb, while reasonable agreement is obtained for the proposed Ni-Mn-Cu-Sb sequence; (iv) Because of the higher magnetic moment for the sequence Ni-Mn-Cu-Sb, the solid will have a higher magnetic entropy, causing its free energy to be lower than that of the sequence Ni-Cu-Mn-Sb at room temperatures. Furthermore, it is not just the total energy (or free energy) that dictates the phase of the experimentally prepared sample, as the latter depends also on the annealing and related atomic kinetics; (v) Although we have estimated the total energies using the WIEN2k, corresponding results from the TB-LMTO method at the experimental lattice constant come out to be reasonably close. For example, the energy difference between sequences Ni-Cu-Mn-Sb and Ni-Mn-Cu-Sb are -0.0018 Ry and -0.002 Ry to -0.003 Ry for TB-LMTO and WIEN2k, respectively. Thus total energy calculations and, more importantly, agreement between the calculated and measured magnetic moment values support the preference for the proposed sequence Ni-Mn-Cu-Sb. Further support, as discussed in the next section, comes from the study of magnetic stability and calculated Curie temperatures. Additional discussion and results related to magnetic moments (calculated via TB-LMTO) appear in sec. 3.2. We conclude this Section by comparing the total and atom-resolved densities of states (DOS) for NiMnCuSb and NiCuMnSb, respectively, in Fig. 1. The following observations can be made: (i) There is a significantly larger splitting of majority and minority Mn-local DOS for NiMnCuSb, resulting in its larger total magnetic moment; (ii) The majority states for NiMnCuSb are separated in energy into two parts, one with low binding energies (0.1,-0.3) Ry, originating predominantly from Ni-and Mn-states, and another with higher binding energies (-0.3,-0.5) Ry, with dominant contribution from Cu-states. These two parts of the energy spectra are separated by a gap. The minority bands behave differently: minority Mn-states are shifted upwards above the Fermi energy, while the majority and minority Ni-states oc5
cupy similar (overlapping) positions due to the small Ni-local moment (about 0.17 µB ); (iii) The DOS’s for the Ni-Cu-Mn-Sb sequence behave differently, with the energy gaps around -0.3 Ry now vanishing due to a strong Cu-Ni hybridization on the neighboring sublattices; (iv) Neither Ni-Mn-Cu-Sb nor Ni-Cu-Mn-Sb shows large spin polarization at the Fermi level. 3.2. Exchange integrals and the Curie temperature We determine the Curie temperature Tc from the classical Heisenberg Hamiltonian H X Mn,Mn H =− Jij ei · ej . (1) i,j
Here i, j are site indices, ei is the unit vector pointing along the direction of the local magnetic moment at site i, JijMn,Mn is the exchange interaction between Mn moments at sites i and j, and the positive/negative values of J Mn,Mn correspond to the ferromagnetic (FM)/antiferromagnetic (AFM) coupling of two spins, respectively. We have estimated J Mn,Mn on the Mn-sublattice using the Lichtenstein mapping [22] procedure as formulated in the framework of the TB-LMTO method [13]. In this approach values of corresponding magnetic moments are included in the definition of J Mn,Mn . Exchange integrals depend, in general, on the reference state from which they are extracted. In the present case we employ the disordered local moment (DLM) state [23] as the reference state. The DLM state describes the paramagnetic state above the Curie temperature with fluctuating Mn-moments and assumes no prescribed magnetic phase. It is thus the appropriate choice for estimating the Curie temperature, which describes the transition between the ferromagnetic and paramagnetic states. Alternatively, one can employ the FM reference state [22], which describes the system better at low temperatures (T = 0, theoretically). The DLM state is formally treated as a 50-50 random binary AB alloy with Mn-moments pointing randomly in opposite directions, allowing for the labeling Mn[+] (A) and Mn[-] (B). Such a state can be naturally treated using the CPA [23]. Because the DLM state is described by the CPA, the spatial extent of J Mn,Mn is smaller as compared to the FM reference state and a smaller number of J Mn,Mn is needed to obtain converged values for the Curie temperature. Note that local moments on Ni-, Cu- and Sb-atoms are strictly zero in the DLM state, which simplifies the estimate of the Curie temperature based on the Heisenberg Hamiltonian (1), since only exchange interactions among the Mn-atoms need to be considered. 6
We determine J Mn,Mn ’s not only for the ideal NiMnCuSb/NiCuMnSb alloys, but also for the Ni- and Cu-rich compounds, NiMn(Cu0.9 ,Ni0.1 )Sb and (Ni0.9 ,Cu0.1 )MnCuSb, respectively, discussed in the experimental study of Haque et al. [6]. A detailed analysis of the site occupations by Haque et al. [6] reveals that both Cu- and Ni-rich compounds are in fact slightly Sbpoor containing about 5% vacancies on the Sb-lattice. Similarly, the ideal NiMnCuSb compound is in fact slightly Sb-rich, containing a few per cent of Sb[Cu]-antisites on the Cu-sublattice. It should be noted, however, that in all of these cases relevant exchange interactions are only those among the Mn-atoms on the unperturbed Mn-sublattice. The disorder on Cu-/Ni/Sb-sublattices influences such Mn-Mn interactions only indirectly via the hybridization of Mn-orbitals with those on other atoms. We refer the reader to [13] for details. Once the classical Heisenberg Hamiltonian (1) is constructed, the Curie temperature Tc is determined from it in the framework of the random phase approximation (RPA) [13]. RPA values of Tc are more reliable than those obtained from the mean field approximation (MFA), which is known to often overestimate the Curie temperature. One can obtain the MFA estimate of the Curie temperature from 2 X Mn,Mn kB TcMFA = J0i , (2) 3 i6=0
where the sum extends over all the neighboring shells. The RPA expression for Tc is given by 3 1 X Mn,Mn (kB TcRPA )−1 = [J (0) − J Mn,Mn(q)]−1 . (3) 2N q Here J Mn,Mn (q) is the lattice Fourier transform of the real-space exchange integrals JijMn,Mn, and N is the number of wave vectors q considered in the irreducible Brillouin zone of the fcc lattice used in the sum. The RPA calculations were done for 62 shells of J Mn,Mn with distances up to about 4.0 a. The dependence of the calculated Curie temperature Tc on the number of shells of exchange integrals used in the calculation is an interesting issue. The MFA Tc values are robust with respect to the cut-off distance. The RPA Tc , as shown in Eq.(3), depends on the difference J Mn,Mn(0)−J Mn,Mn (q). Obviously the values of J Mn,Mn(q) for q ∼ 0 dominate the contribution. This, in turn, implies that the long range variations in the real-space exchange 7
integrals J Mn,Mn(d) are important and cannot be easily ignored. The RPA Tc values thus show large fluctuations with respect to the cut-off distance or the number of shells. The sensitivity of Tc to the cut-off distance has been illustrated in detail in [24] with the example of Fe3 Al. We have verified that the use of 229 shells (distances up to about 6.4 a) changes the calculated Curie temperature by about 2%. We also mention that exchange integrals in the Heisenberg Hamiltonian can be used for the study of the magnetic phase stability by performing their lattice Fourier transformation [25]. The exchange integrals J Mn,Mn(d) as a function of distance d between two Mn-spins are shown in Fig. 2 for the two ideal cases: NiMnCuSb and NiCuMnSb. One observes the dominating FM interactions for the Ni-Mn-Cu-Sb sequence, in contrast with the Ni-Cu-Mn-Sb sequence which shows a complex behavior with competing FM and AFM interactions. We therefore present in Fig. 3 the lattice Fourier transformation of the real-space exchange integrals J Mn,Mn(d) along the lines of the high symmetry in the corresponding BZ. Such a plot, in fact, characterizes the stability of the system with respect to periodic magnetic excitations with the wave vector q. The maximum characterizes the magnetic phase, as the system is cooled below the critical temperature. For example, excitations with the wave-vector q=0 (the Γ-point) correspond to the FM state, which is compatible with the experiment (for the Ni-Mn-Cu-Sb sequence). On the contrary, the Ni-Cu-Mn-Sb sequence shows a maximum at the X-point giving an AFM ground state, in striking disagreement with the experiment. One can thus clearly exclude the Ni-Cu-Mn-Sb sequence proposed in [6]. Total and local (Mn,Ni) magnetic moments at experimental volumes obtained by the TB-LMTO-CPA method are shown in Table 3. Results for the ideal NiMnCuSb, and Cu- and Ni-rich alloys as well as alloys containing defects are shown (cf. Table 1 for corresponding values obtained using WIENGGA for the ideal NiMnCuSb alloy). Local moments on Cu- and Sb-atoms are negligible (few hundredth of µB ) and are not presented. The following conclusions can be drawn: (i) There is good agreement between Mtot values obtained in the TB-LMTO LDA+U method with U[Mn]=0.05 Ry and the WIEN2k GGA values. Note that the GGA represents an approximate treatment of electron correlations beyond the LDA; (ii) The local Ni-moment on the fcc-Ni sublattice (about 0.17 µB ) is smaller than the bulk fcc-Ni moment value (0.6 µB ) despite a slightly larger nearest-neighbor distance in NiMnCuSb as compared to fcc-Ni. This is due to the effect of the environment: hybridization of the Ni-sublattice with other sublattices; (iii) The lowest 8
value of Mtot is obtained for the Cu-rich case, in agreement with the experiment. While the values for NiMnCuSb and Ni-rich alloys agree with the experimental results quantitatively, the calculated total moment for the Curich case is too large; and (iv) The relative order of Mtot values is reproduced correctly, i.e., in agreement with the the experiment. One could, of course, speculate about differently annealed Ni- and Cu-rich samples. On the other hand, the differences among the three cases are small and total moments depend weakly on the alloy composition. The effect of alloy composition on the Curie temperature is stronger, as shown below. Estimates of the Curie temperature Tc for the above alloys were obtained using the random phase approximation (RPA) as well as the simple meanfield approximation (MFA). The latter is known to significantly overestimate the Curie temperature, and is usually less reliable. A key ingredient for the RPA/MFA estimates of the Curie temperature is the lattice Fourier transformation (J Mn,Mn (q)) [13]. The results are summarized in Fig. 4. We have the following comments: (i) The MFA values overestimate Tc s, while the RPA values underestimate, when compared with the measured values. This is particularly valid for smaller values of U; (ii) The experimentally observed order Tc [Cu-rich] > Tc [ideal] > Tc [Ni-rich] is obtained only for models including Sb-vacancies/Cu[Sb]-antisites (Models B and C); (iii) The model C also gives a correct order for the MFA values; and (iv) The best agreement between calculated and measured Tc -values is obtained for the Model C with larger effective U[Mn]-values, while Model B with smaller U gives acceptable results as well. Note that the calculations for NiMnCuSb and the corresponding Ni-rich and Cu-rich cases were all done for corresponding experimental lattice parameters, as reported in Haque et al. [6]. 4. Conclusions We have presented a theoretical study of the structural and magnetic properties of the Heusler alloy CuNiMnSb (NiMnCuSb), by combining the FP-LAPW method (WIEN2k code) for the determination of the lattice structure/magnetic moments and the TB-LMTO approach to estimate exchange integrals and corresponding Curie temperatures. Our motivation was to present arguments in favor of the Ni(A)-Mn(B)-Cu(C)-Sb(D) sequence rather than the sequence Ni(A)-Cu(B)-Mn(C)-Sb(D) along the body diagonal of the cube characterizing the DO3 -type (space group F¯43m) structure. The authors of the experimental study of this system reported the structure as 9
NiCuMnSb, based on their XRD measurements, while admitting that uncertainties related to the closeness in the values of the X-ray form factors of Cu, Ni, and Mn could not be ruled out [6]. Theoretical total energy values and especially the nature of magnetic interactions strongly support the Ni(A)Mn(B)-Cu(C)-Sb(D) sequence. Calculated exchange interactions show that the sequence Ni(A)-Mn(B)-Cu(C)-Sb(D) yields a strong ferromagnet with high local moment on Mn and high Curie temperature. Calculations also reveal that were the sequence to be Ni(A)-Cu(B)-Mn(C)-Sb(D), the substance would be either an antiferromagnet or, at best, a weak ferromagnet, with an Mn moment noticeably lower than the measured value. In addition, we obtain the correct (i.e, in agreement with experiment) relative order among the RPA Curie temperatures for the ideal NiMnCuSb and its Cu- and Ni-rich counterparts, but only if experimentally realistic scenarios involving Sb-vacancies and/or Sb[Cu]-antisites are considered. The calculated values of Curie temperatures are in fair agreement with measured data. Density of states calculations reveal that neither NiMnCuSb nor NiCuMnSb has large spin polarization at the Fermi level. 5. Acknowledgments The work of Y.L. is supported by the Natural Science Foundation of Hebei Province (No. A2015203021), and also by Key Project of Heibei Educational Department, China (No. ZD2014015). The work of J.K. was supported by a Grant from the Czech Science Foundation (No. 15-13436S). The work of S.K.B. was funded by Brock University, Canada. References [1] K. Ozdogan,E.Sasioglu, I. Galanakis, J. Appl. Phys. 113 (2013) 193903. [2] J. Ma, V.I. Hedge, K. Munira, Yu. Xie, S. Keshavarz, D.T. Mildebrath, C. Wolverton, A.W. Ghosh, W.H. Butler, Phys. Rev. B 95 (2017) 024411. [3] L. Wollmann, A.K. Nayak, S.S.P. Parkin, C. Felser, cond-mat arXiv:1612.05947. [4] L. Bainsla and K.G. Suresh, App. Phys. Rev. 3 (2016) 031101.
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[5] S.E. Kulkova, S.V. Eremeev. T. Kakeshita, S.S. Kulkova, G.E. Rudenski, Materials Transactions 47 (2006) 599. [6] Z. Haque, G.S. Thakur, S. Ghara, L.C. Gupta, A. Sundaresan, and A.K. Ganguli, J. Magn. Magn. Mat. 397 (2016) 315. [7] P. J. Webster and K. R. A. Ziebeck, in Alloys and Compounds of dElements with Main Group Elements, Part 2, edited by H. R. J. Wijn, Landolt-B¨ornstein, New Series, Group III, Vol. 19, Pt. C (SpringerVerlag, Berlin, 1988). [8] P. Blaha, K. Schwarz, G.K.H. Madsen, D. Kvasnicka and J. Luitz, WIEN2k, An Augmented Plane Wave plus Local Orbitals Program for Calculating Crystal Properties, (ISBN 3-9501031-1-2), Vienna University of Technology, Vienna, Austria (2016). [9] J. Kudrnovsk´y and V. Drchal, Phys. Rev. B 41 (1990) 7515. ˇ [10] I. Turek, V. Drchal, J. Kudrnovsk´y, M. Sob, and P. Weinberger, Electronic Structure of Disordered Alloys, Surfaces and Interfaces (Kluwer, Boston-London-Dordrecht, 1997). [11] O.K. Andersen, O. Jepsen and D. Gl¨otzel, in Highlights of CondensedMatter Theory, edited by F. Bassani, F. Fumi, and M.P. Tosi (NorthHolland, Amsterdam, 1985), 59. [12] P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964); W. Kohn and 14 L. J. Sham, Phys. Rev. 140 (1965) A1133. [13] I. Turek, J. Kudrnovsk´y, V. Drchal, P. Bruno, Phil. Mag. 86 (2006) 1713. [14] J.P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3685. [15] Z. Wu, and R.E. Cohen, Phys. Rev. B 73 (2006) 235116. [16] J.P. Perdew, A. Ruzsinszky, G.I. Csonka, O.A. Vydrov, G.E. Scuseria, L.A. Constantin, X. Zhou, K. Burke, Phys. Rev. Lett. 100 (2008)136406. [17] S.H.Vosko, L. Wilk, and M. Nusair, Can. J. Phys. 58 (1980) 1200. 11
[18] S.L. Dudarev, G.A. Botton, S.Y. Savrasov, C.J. Humphreys, A. P. Sutton, Phys. Rev. B 57 (1990) 1505. [19] M. Krause and F. Bechstedt, J. Supercond. Nov. Magn. 26 (2013) 1963. [20] Y.Liu, S.K. Bose, and J. Kudrnovsk´y, J. Magn. Magn. Mat. 423 (2017) 12. [21] Y.Liu, S.K. Bose, and J. Kudrnovsk´y, AIP Advances 6 (2016) 125005. [22] A.I. Liechtenstein, M.I. Katsnelson, V.P. Antropov, V.A. Gubanov, J. Magn. Magn. Mater. 67 (1987) 65. [23] B.L. Gyorffy, A.J. Pindor, J. Staunton, G.M. Stocks, and H. Winter, J. Phys. F: Met. Phys. 15 (1985) 1337. ´ [24] J. Kudrnovsk´y, V. Drchal, L. Bergqvist, J. Rusz, I.Turek, B. Ujfalussy, I. Vincze, Phys. Rev. B 90 (2014) 134408. [25] J. Kudrnovsk´y, F. M´aca, I. Turek, J. Redinger, Phys. Rev. B 80 (2009) 064405.
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Table 1: Magnetic moments and total energies per formula unit as obtained from WIEN2k calculations for different occupations of the sublattices A(0,0,0); B(0.25,0.25,0.25); C(0.5,0.5,0.5); and D-Sb(0.75,0.75,0.75) at the experimental lattice parameter a=5.924 ˚ A. Results are shown for three different exchange-correlation potentials: PBE-96 [14], WuCohen [15], and PBE-sol [16].
A(Cu) PBE-96 Wu-Cohen PBE-sol A(Ni) PBE-96 Wu-Cohen PBE-sol A(Ni) PBE-96 Wu-Cohen PBE-sol
B(Ni)
C(Mn)
B(Cu)
C(Mn)
B(Mn)
Mtot (µB ) 3.090 3.032 3.081
mM n (µB ) 2.703 2.651 2.689
E (Ry) -21636.25275 -21632.34196 -21623.21040
2.950 2.878 2.927
2.743 2.674 2.772
-21636.27389 -21632.36321 -21623.23197
3.922 3.904 3.923
3.449 3.427 3.436
-21636.27177 -21632.35994 -21623.22897
C(Cu)
13
Table 2: Magnetic moments and total energies as obtained from WIEN2k calculations for different occupations of the sublattices A(0,0,0); B(0.25,0.25,0.25); C(0.5,0.5,0.5); and DSb(0.75,0.75,0.75) for the optimized lattice parameter a (˚ A). Other labels and quantities have the same meaning as in Table 1.
A(Cu) PBE-96 Wu-Cohen PBE-sol A(Ni) PBE-96 Wu-Cohen PBE-sol A(Ni) PBE-96 Wu-Cohen PBE-sol
B(Ni)
C(Mn)
B(Cu)
C(Mn)
B(Mn)
˚) Mtot (µB ) mMn (µB ) E (Ry) a (A 3.583 3.167 -21636.27624 6.1665 3.408 2.989 -21632.34990 6.0609 3.405 2.983 -21623.21720 6.0493 3.401 3.233 3.228
3.187 3.001 2.990
-21636.29344 6.1394 -21632.36894 6.0400 -21623.23665 6.0274
4.182 4.092 4.099
3.681 3.589 3.587
-21636.30137 6.1790 -21632.37286 6.0888 -21623.24015 6.0776
C(Cu)
14
Table 3: Calculated total (Mtot ) and local magnetic moments (mQ ) on Q=Mn- and Nisites (in µB ) for the NiMnCuSb alloy, and its Ni-rich and Cu-rich counterparts assuming U[Mn]=0.05 Ry. The second row for NiMnCuSb shows the values for the case with 2% of Sb[Cu] antisites. For Ni- and Cu-rich cases, the second row shows the values for alloys containing 5% Sb-vacancies. The last column shows the experimental total moments in each case [6].
Alloy NiMnCuSb NiMn(Cu0.9 ,Ni0.1 )Sb (Ni0.9 ,Cu0.1 )MnCuSb
Mtot 3.870 3.843 3.886 3.898 3.863 3.883
15
mMn 3.660 3.639 3.675 3.691 3.670 3.691
exp mNi Mtot 0.170 3.85 0.167 0.166 3.87 0.171 0.168 3.57 0.175
100 80
(a)
maj
LDOS (states/spin/Ry)
60 40 20 0 -20 -40 tot -60
Ni Mn
-80
min
Cu
-100 100 80
(b)
maj
LDOS (states/spin/Ry)
60 40 20 0 -20 -40 tot -60
Ni
-80
Mn
min
Cu
-100 -0.6
-0.4
-0.2
0
0.2
E - EF (Ry)
Figure 1: The total and local densities of states (DOS) for the ferromagnetic NiMnCuSb alloys as calculated by the TB-LMTO-CPA method: (a) the sequence Ni-Mn-Cu-Sb, and (b) the sequence Ni-Cu-Mn-Sb. Both the total DOS (tot) and atom-resolved DOS’s on Ni, Mn, and Cu-atoms are shown. The alloy Fermi energy is shifted to zero. In both cases we have used a small effective Hubbard U[Mn]=0.05 Ry to reproduce the total magnetic moment (3.87 µB ) for the sequence Ni-Mn-Cu-Sb.
16
0.8 0.7
Ni-Mn-Cu-Sb
JMn,Mn(d)(mRy)
0.6 0.5
Ni-Cu-Mn-Sb
0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 0.5 0.75
1
1.25 1.5 1.75
2
2.25 2.5
d/a
Figure 2: The exchange interactions J Mn,Mn between Mn-atoms for the sequences NiMn-Cu-Sb (full circles) and Ni-Cu-Mn-Sb (empty circles) are plotted as a function of the distance d between Mn-atoms (in units of the lattice constant a). In both cases the interactions are derived from the corresponding DLM reference state, and we have added a small effective Hubbard U[Mn]=0.05 Ry as in Fig. 1. The positive/negative values of JMn,Mn correspond to the ferromagnetic/antiferromagnetic couplings, respectively.
17
10
6 4 2 0
J
Mn,Mn
(q) (mRy)
8
-2 -4 -6
L
Γ
X
W
K
Γ
Figure 3: The lattice Fourier transformation of exchange interactions J Mn,Mn (q), along the lines of the high symmetry in the fcc BZ corresponding to the Mn-sublattice is shown for the sequences Ni-Mn-Cu-Sb (full lines) and Ni-Cu-Mn-Sb (dashed lines). The same model as in Fig. 2 is also employed here. The magnetic ground state corresponds to the maximum of each curve due to the convention in the definition of the Heisenberg Hamiltonian (1) [13].
18
1000 (a)
MFA
Tc [K]
800 600 400 200 0 1000
Mod A Mod B Mod C Ideal Ni-rich Cu-rich
(b) RPA
800
Tc [K]
Exp
600 400 200 0 Mod A Mod B Mod C
Exp
Figure 4: Calculated Curie temperatures for NiMnCuSb (sequence Ni-Mn-Cu-Sb along the diagonal) and its Ni-rich (NiMn(Cu0.9 ,Ni0.1 )Sb) and Cu-rich ((Ni0.9 ,Cu0.1 )MnCuSb) counterparts. Three models were studied: (i) Model A (Mod A): the ideal alloys without Sb-vacancies or Sb[Cu]-antisites. A small effective Hubbard U[Mn]=0.05 Ry is added; (ii) Model B (Mod B): the same as Model A but for alloys with 0.5% antisites for NiMnCuSb and 5% Sb-vacancies for Ni-/Cu-rich alloys; and (iii) Model C (Mod C): the same as for Model B but with effective Hubbard U[Mn]=0.13 Ry and Sb[Cu] concentration of 2%. We show both the MFA (a) and RPA (b) values. For convenience, experimental results (Exp) for each model are also shown.
19
Research Highlights We present a theoretical study of magnetism of the Quaternary Heusler alloy NiMnCuSb. Our theoretical study identifies the structure to be NiMnCuSb as opposed to NiCuMnSb. The identification of the true structure as being NiMnCuSb is in contradiction to a recently published work. We show that the measured values of magnetic moments and Curie temperatures support our results.
S0304-8853(17)31965-0 http://dx.doi.org/10.1016/j.jmmm.2017.08.049 MAGMA 63083
To appear in:
Journal of Magnetism and Magnetic Materials
Please cite this article as: S.K. Bose, J. Kudrnovský, Y. Liu, Structure and physical properties of quaternary Heusler alloy NiMnCuSb, Journal of Magnetism and Magnetic Materials (2017), doi: http://dx.doi.org/10.1016/j.jmmm. 2017.08.049
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Structure and physical properties of quaternary Heusler alloy NiMnCuSb S.K. Bosea,∗, J. Kudrnovsk´yb , and Y. Liuc a
Physics Department, Brock University, St. Catharines, ON, L2S 3A1, Canada Institute of Physics, Academy of the Sciences of the Czech Republic, Na Slovance 2, 182 21 Prague 8, Czech Republic c State Key Laboratory of Metastable Materials Science and Technology, and Hebei Key Laboratory of Microstructural Material Physics, School of Sciences, Yanshan University, Qinhuangdao, 066004, China
b
Abstract The relationship between structure and physical properties such as magnetic moment and Curie temperature is studied theoretically for the recently synthesized equiatomic quaternary Heusler alloy CuNiMnSb, which is found to crystallize in the LiMgPdSb structure. Calculations are performed for various different possible structural phases within the framework of the D03 -type structure (space group F¯43m) by changing the sequence of the atoms along the body diagonal of fcc lattice that characterizes the structure of this alloy. Total energy, magnetic moment and the Curie temperature results, in particular the latter two, suggest that the correct order of the atoms along the body diagonal is most likely Ni(0,0,0), Mn(0.25,0.25,0.25), Cu(0.5,0.5,0.5) and Sb(0.75,0.75,0.75) in contrast to (disagreement with) the conclusions of a recent experimental study. Neither the sequence Ni-Cu-Mn-Sb nor Ni-MnCu-Sb shows a large spin polarization at the Fermi level. Keywords: Heusler alloys, FP-LAPW, TB-LMTO, exchange interaction, Curie temperature
∗
Corresponding author: S.K. Bose Email address: [email protected] (S.K. Bose)
Preprint submitted to JMMM
August 14, 2017
1. Introduction Lattice structure is one of the most fundamental properties dictating the physical properties of crystalline solids. For solids with a multisublattice structure with different atoms occupying the sublattices, the importance of the lattice structure becomes paramount. Important examples of such solids are Heusler alloys with a variety of interesting physical properties. Many of such alloys have a cubic structure consisting of four interpenetrating fcc-sublattices A-B-C-D aligned along the body diagonal and centered about equidistant positions (0,0,0)a, (0.25,0.25,0,25)a, (0.5,0.5.0,5)a, and (0.75,0.75,0.75)a, where a is the lattice constant. Regular ternary Heusler alloys formed by atoms X,Y,and Z have formula (in terms of the fcc-sublattice occupations) X-Y-X-Z as, e.g., the Cu2 MnAl system. Closely related alloys with the anti-Heusler structure follow the formula X-X-Y-Z (e.g. Mn2 CuAl). Another class of Heusler alloys is the recently studied quaternary compounds (system) with the so-called LiMgPdSb(Sn) type structure [1], where the sequence of the atoms along the body diagonal is Li, Mg, Pd, and Sb(Sn). Special cases of such alloys are semi-Heusler alloys such as NiMnSb where the ’missing’ element is a vacancy. There are numerous articles on Heusler alloys encompassing both theoretical and experimental results. For review purposes readers may consult [2, 3, 4, 5] and references therein. The typical experimental tool for the determination of lattice structure of Heusler alloys, and the order of occupation of sublattices A to D by atomic species, is the powder X-ray diffraction (XRD) method. However, if constituent atoms have similar electronic form factors, such as Cu, Ni, or Mn, the reliability of the XRD to determine the correct atomic order becomes questionable. In such cases neutron diffraction studies would be more reliable, since the magnetic scattering factors should differ greatly among Mn, Ni and Cu. In the absence of neutron scattering results guidance from first-principles theoretical calculations can be very useful, as the calculated physical properties such as total energy, magnetic moments and the critical temperatures (for magnetic alloys) are sensitive to the chemical order. We illustrate this point by considering the newly synthesized quaternary NiMnCuSb Heusler alloy [6]. This alloy is interesting in that it is one of the few known ordered and cubic non-Fe Heusler alloys that is ferromagnetic with a relatively high Curie temperature (∼ 690 K), as determined experimentally by Haque et al. [6] . The total magnetic moment per formula unit (f.u.) is ∼ 3.85 µB . The XRD experiments carried out by the authors of the 2
paper [6] have led to the conclusion that the crystallographic order is given by the formula Ni-Cu-Mn-Sb. However, in the words of the authors themselves, ’it is difficult to confirm the ordering in structure based on the X-ray diffraction studies as the diffraction from Cu, Ni and Mn atoms is identical due to identical scattering factor.’ Based on their magnetic measurements the authors conclude that the moment primarily resides on the Mn atoms, the Ni atoms do not carry any moment. In this work we present theoretical results for the total energy, magnetic moments, and Curie temperatures Tc for the ideal NiCuMnSb compound and also for the Cu- and Ni-rich counterparts studied by Haque et al. in [6]. While we confirm that magnetic moment resides predominantly on the Mn atoms, we question the suggested formula and propose a new one, namely the formula NiMnCuSb (with the sequence Ni-Mn-Cu-Sb along the diagonal), the results for which agree better with measured total magnetic moment and the Curie temperature. Note also that the analogy of NiMnCuSb to Ni2 MnSb, and in particular to NiMnSb or (Cu,Ni)MnSb may be useful in this context: in none of these alloys do Mn and Sb occupy nearest neighbor sublattices. It should be noted that the structure of the quaternary NiMnCuSb alloy can be visually related to the structure of the known semi-Heusler alloy NiMnSb: Cu atoms of the former occupy the empty fcc sublattice of the latter. This simple view is further corroborated by the fact that the experimental lattice parameter of NiMnCuSb is almost the same as that of NiMnSb[7]. This indicates that there is enough space in the open NiMnSb structure to accommodate Cu atoms without noticeable change in volume per atom. 2. Formalism We have studied the electronic structure and magnetic properties of this alloy system using two complementary methods: the full-potential linear augmented plane wave (FP-LAPW) method, as implemented in WIEN2k [8] code, and the Green’s function formulation [9, 10] of the tight-binding linear muffin-tin orbital (TB-LMTO) method in the atomic sphere approximation (ASA) [11]. Both approaches are based on the density functional theory (DFT) [12]. While the more accurate WIEN2k code is used for the phase stability study, we employ the TB-LMTO method in conjunction with the coherent potential approximation (CPA) [9, 10] for the study of cases involving disorder between sublattices, e.g., for disordered Cu- and Ni-rich phases. We employ the TB-LMTO-CPA method primarily for the determination of ex3
change integrals [13] which are needed for the study of the magnetic stability of the alloys and the Curie temperatures. Three different forms of the generalized gradient approximation (GGA), which is considered to be optimal for the structural determination, were tested: PBE-96 [14], Wu-Cohen [15], and PBE-sol [16]. TB-LMTO calculations generally used the exchange-correlation potential of Vosko, Wilk and Nussair (VWN)[17]. In some cases we have also used the LDA+U method with the Hubbard U parameter in the fully-localized limit [18]. The LDA+U approach is sometimes used for the fine-tuning of band gaps, magnetic moments, critical temperatures, etc. (see, e.g., a recent study [19]). In both methods we have neglected the spin-orbit coupling because of its marginally small effect on the reported results. Finally, for both the WIEN2k and TB-LMTO calculations, careful tests were done concerning various methoddependent parameters to warrant proper convergence of the results. Values of various parameters for the WIEN2k method, such as the number of wave vectors in the Brillouin zone (BZ), BZ sampling scheme, convergence criteria etc. were all the same as discussed in our recent publications[20, 21]. 3. Results and discussion 3.1. Total energies and magnetic moments We have calculated the total energies and magnetic moments for three different realizations of the structure by placing the atoms Ni, Cu, and Mn in different sequences at the A(0,0,0), B(0.25,0.25,02.5), and C(0.5,0.5,0.5) sites, while the Sb atom was fixed at the D(0.75,0.75,0.75) site. In Table 1 we show the WIEN2k results for the total energies and magnetic moments for three different GGA versions of the exchange-correlation potential. The results are shown for the experimental lattice parameter of 5.92382 ˚ A. Similarly, in Table 2 we summarize related results for the energy optimization with respect to the sample volume or the lattice constant. The following conclusions can be drawn: (i) The sequence Cu(A)-Ni(B)-Mn(C) can readily be discarded on the basis of energy consideration for both the experimental and optimized lattice constants; (ii) On the other hand, the total energies of the sequences Ni(A)-Cu(B)-Mn(C) (denoted below as NiCuMnSb) and Ni(A)-Mn(B)-Cu(C) (denoted as NiMnCuSb) are very close in both cases. The calculated lattice constants, magnetic moments, or relative energy differences among various phases are not sensitive to the choice of the type of the GGA functional. While for the experimental volume the total energy of 4
the Ni-Cu-Mn-Sb sequence is slightly lower than that of the Ni-Mn-Cu-Sb sequence, for the optimized volumes the order of total energies is reversed. It should be noted that the optimized lattice constants are overestimated by about 3-4% with respect to the experiment; (iii) The most important part of the total magnetic moment Mtot is the contribution from local Mn-moments as assumed in the paper by Haque et al. [6]. Moments for the optimized structures are larger than the experimental values due their larger lattice constants. The local moments on the Ni-atoms are small (about 0.1−0.25 µB ) and those on Cu/Sb atoms are negligible. Most important, however, is the fact that Mtot is generally too small for the sequence Ni-Cu-Mn-Sb, while reasonable agreement is obtained for the proposed Ni-Mn-Cu-Sb sequence; (iv) Because of the higher magnetic moment for the sequence Ni-Mn-Cu-Sb, the solid will have a higher magnetic entropy, causing its free energy to be lower than that of the sequence Ni-Cu-Mn-Sb at room temperatures. Furthermore, it is not just the total energy (or free energy) that dictates the phase of the experimentally prepared sample, as the latter depends also on the annealing and related atomic kinetics; (v) Although we have estimated the total energies using the WIEN2k, corresponding results from the TB-LMTO method at the experimental lattice constant come out to be reasonably close. For example, the energy difference between sequences Ni-Cu-Mn-Sb and Ni-Mn-Cu-Sb are -0.0018 Ry and -0.002 Ry to -0.003 Ry for TB-LMTO and WIEN2k, respectively. Thus total energy calculations and, more importantly, agreement between the calculated and measured magnetic moment values support the preference for the proposed sequence Ni-Mn-Cu-Sb. Further support, as discussed in the next section, comes from the study of magnetic stability and calculated Curie temperatures. Additional discussion and results related to magnetic moments (calculated via TB-LMTO) appear in sec. 3.2. We conclude this Section by comparing the total and atom-resolved densities of states (DOS) for NiMnCuSb and NiCuMnSb, respectively, in Fig. 1. The following observations can be made: (i) There is a significantly larger splitting of majority and minority Mn-local DOS for NiMnCuSb, resulting in its larger total magnetic moment; (ii) The majority states for NiMnCuSb are separated in energy into two parts, one with low binding energies (0.1,-0.3) Ry, originating predominantly from Ni-and Mn-states, and another with higher binding energies (-0.3,-0.5) Ry, with dominant contribution from Cu-states. These two parts of the energy spectra are separated by a gap. The minority bands behave differently: minority Mn-states are shifted upwards above the Fermi energy, while the majority and minority Ni-states oc5
cupy similar (overlapping) positions due to the small Ni-local moment (about 0.17 µB ); (iii) The DOS’s for the Ni-Cu-Mn-Sb sequence behave differently, with the energy gaps around -0.3 Ry now vanishing due to a strong Cu-Ni hybridization on the neighboring sublattices; (iv) Neither Ni-Mn-Cu-Sb nor Ni-Cu-Mn-Sb shows large spin polarization at the Fermi level. 3.2. Exchange integrals and the Curie temperature We determine the Curie temperature Tc from the classical Heisenberg Hamiltonian H X Mn,Mn H =− Jij ei · ej . (1) i,j
Here i, j are site indices, ei is the unit vector pointing along the direction of the local magnetic moment at site i, JijMn,Mn is the exchange interaction between Mn moments at sites i and j, and the positive/negative values of J Mn,Mn correspond to the ferromagnetic (FM)/antiferromagnetic (AFM) coupling of two spins, respectively. We have estimated J Mn,Mn on the Mn-sublattice using the Lichtenstein mapping [22] procedure as formulated in the framework of the TB-LMTO method [13]. In this approach values of corresponding magnetic moments are included in the definition of J Mn,Mn . Exchange integrals depend, in general, on the reference state from which they are extracted. In the present case we employ the disordered local moment (DLM) state [23] as the reference state. The DLM state describes the paramagnetic state above the Curie temperature with fluctuating Mn-moments and assumes no prescribed magnetic phase. It is thus the appropriate choice for estimating the Curie temperature, which describes the transition between the ferromagnetic and paramagnetic states. Alternatively, one can employ the FM reference state [22], which describes the system better at low temperatures (T = 0, theoretically). The DLM state is formally treated as a 50-50 random binary AB alloy with Mn-moments pointing randomly in opposite directions, allowing for the labeling Mn[+] (A) and Mn[-] (B). Such a state can be naturally treated using the CPA [23]. Because the DLM state is described by the CPA, the spatial extent of J Mn,Mn is smaller as compared to the FM reference state and a smaller number of J Mn,Mn is needed to obtain converged values for the Curie temperature. Note that local moments on Ni-, Cu- and Sb-atoms are strictly zero in the DLM state, which simplifies the estimate of the Curie temperature based on the Heisenberg Hamiltonian (1), since only exchange interactions among the Mn-atoms need to be considered. 6
We determine J Mn,Mn ’s not only for the ideal NiMnCuSb/NiCuMnSb alloys, but also for the Ni- and Cu-rich compounds, NiMn(Cu0.9 ,Ni0.1 )Sb and (Ni0.9 ,Cu0.1 )MnCuSb, respectively, discussed in the experimental study of Haque et al. [6]. A detailed analysis of the site occupations by Haque et al. [6] reveals that both Cu- and Ni-rich compounds are in fact slightly Sbpoor containing about 5% vacancies on the Sb-lattice. Similarly, the ideal NiMnCuSb compound is in fact slightly Sb-rich, containing a few per cent of Sb[Cu]-antisites on the Cu-sublattice. It should be noted, however, that in all of these cases relevant exchange interactions are only those among the Mn-atoms on the unperturbed Mn-sublattice. The disorder on Cu-/Ni/Sb-sublattices influences such Mn-Mn interactions only indirectly via the hybridization of Mn-orbitals with those on other atoms. We refer the reader to [13] for details. Once the classical Heisenberg Hamiltonian (1) is constructed, the Curie temperature Tc is determined from it in the framework of the random phase approximation (RPA) [13]. RPA values of Tc are more reliable than those obtained from the mean field approximation (MFA), which is known to often overestimate the Curie temperature. One can obtain the MFA estimate of the Curie temperature from 2 X Mn,Mn kB TcMFA = J0i , (2) 3 i6=0
where the sum extends over all the neighboring shells. The RPA expression for Tc is given by 3 1 X Mn,Mn (kB TcRPA )−1 = [J (0) − J Mn,Mn(q)]−1 . (3) 2N q Here J Mn,Mn (q) is the lattice Fourier transform of the real-space exchange integrals JijMn,Mn, and N is the number of wave vectors q considered in the irreducible Brillouin zone of the fcc lattice used in the sum. The RPA calculations were done for 62 shells of J Mn,Mn with distances up to about 4.0 a. The dependence of the calculated Curie temperature Tc on the number of shells of exchange integrals used in the calculation is an interesting issue. The MFA Tc values are robust with respect to the cut-off distance. The RPA Tc , as shown in Eq.(3), depends on the difference J Mn,Mn(0)−J Mn,Mn (q). Obviously the values of J Mn,Mn(q) for q ∼ 0 dominate the contribution. This, in turn, implies that the long range variations in the real-space exchange 7
integrals J Mn,Mn(d) are important and cannot be easily ignored. The RPA Tc values thus show large fluctuations with respect to the cut-off distance or the number of shells. The sensitivity of Tc to the cut-off distance has been illustrated in detail in [24] with the example of Fe3 Al. We have verified that the use of 229 shells (distances up to about 6.4 a) changes the calculated Curie temperature by about 2%. We also mention that exchange integrals in the Heisenberg Hamiltonian can be used for the study of the magnetic phase stability by performing their lattice Fourier transformation [25]. The exchange integrals J Mn,Mn(d) as a function of distance d between two Mn-spins are shown in Fig. 2 for the two ideal cases: NiMnCuSb and NiCuMnSb. One observes the dominating FM interactions for the Ni-Mn-Cu-Sb sequence, in contrast with the Ni-Cu-Mn-Sb sequence which shows a complex behavior with competing FM and AFM interactions. We therefore present in Fig. 3 the lattice Fourier transformation of the real-space exchange integrals J Mn,Mn(d) along the lines of the high symmetry in the corresponding BZ. Such a plot, in fact, characterizes the stability of the system with respect to periodic magnetic excitations with the wave vector q. The maximum characterizes the magnetic phase, as the system is cooled below the critical temperature. For example, excitations with the wave-vector q=0 (the Γ-point) correspond to the FM state, which is compatible with the experiment (for the Ni-Mn-Cu-Sb sequence). On the contrary, the Ni-Cu-Mn-Sb sequence shows a maximum at the X-point giving an AFM ground state, in striking disagreement with the experiment. One can thus clearly exclude the Ni-Cu-Mn-Sb sequence proposed in [6]. Total and local (Mn,Ni) magnetic moments at experimental volumes obtained by the TB-LMTO-CPA method are shown in Table 3. Results for the ideal NiMnCuSb, and Cu- and Ni-rich alloys as well as alloys containing defects are shown (cf. Table 1 for corresponding values obtained using WIENGGA for the ideal NiMnCuSb alloy). Local moments on Cu- and Sb-atoms are negligible (few hundredth of µB ) and are not presented. The following conclusions can be drawn: (i) There is good agreement between Mtot values obtained in the TB-LMTO LDA+U method with U[Mn]=0.05 Ry and the WIEN2k GGA values. Note that the GGA represents an approximate treatment of electron correlations beyond the LDA; (ii) The local Ni-moment on the fcc-Ni sublattice (about 0.17 µB ) is smaller than the bulk fcc-Ni moment value (0.6 µB ) despite a slightly larger nearest-neighbor distance in NiMnCuSb as compared to fcc-Ni. This is due to the effect of the environment: hybridization of the Ni-sublattice with other sublattices; (iii) The lowest 8
value of Mtot is obtained for the Cu-rich case, in agreement with the experiment. While the values for NiMnCuSb and Ni-rich alloys agree with the experimental results quantitatively, the calculated total moment for the Curich case is too large; and (iv) The relative order of Mtot values is reproduced correctly, i.e., in agreement with the the experiment. One could, of course, speculate about differently annealed Ni- and Cu-rich samples. On the other hand, the differences among the three cases are small and total moments depend weakly on the alloy composition. The effect of alloy composition on the Curie temperature is stronger, as shown below. Estimates of the Curie temperature Tc for the above alloys were obtained using the random phase approximation (RPA) as well as the simple meanfield approximation (MFA). The latter is known to significantly overestimate the Curie temperature, and is usually less reliable. A key ingredient for the RPA/MFA estimates of the Curie temperature is the lattice Fourier transformation (J Mn,Mn (q)) [13]. The results are summarized in Fig. 4. We have the following comments: (i) The MFA values overestimate Tc s, while the RPA values underestimate, when compared with the measured values. This is particularly valid for smaller values of U; (ii) The experimentally observed order Tc [Cu-rich] > Tc [ideal] > Tc [Ni-rich] is obtained only for models including Sb-vacancies/Cu[Sb]-antisites (Models B and C); (iii) The model C also gives a correct order for the MFA values; and (iv) The best agreement between calculated and measured Tc -values is obtained for the Model C with larger effective U[Mn]-values, while Model B with smaller U gives acceptable results as well. Note that the calculations for NiMnCuSb and the corresponding Ni-rich and Cu-rich cases were all done for corresponding experimental lattice parameters, as reported in Haque et al. [6]. 4. Conclusions We have presented a theoretical study of the structural and magnetic properties of the Heusler alloy CuNiMnSb (NiMnCuSb), by combining the FP-LAPW method (WIEN2k code) for the determination of the lattice structure/magnetic moments and the TB-LMTO approach to estimate exchange integrals and corresponding Curie temperatures. Our motivation was to present arguments in favor of the Ni(A)-Mn(B)-Cu(C)-Sb(D) sequence rather than the sequence Ni(A)-Cu(B)-Mn(C)-Sb(D) along the body diagonal of the cube characterizing the DO3 -type (space group F¯43m) structure. The authors of the experimental study of this system reported the structure as 9
NiCuMnSb, based on their XRD measurements, while admitting that uncertainties related to the closeness in the values of the X-ray form factors of Cu, Ni, and Mn could not be ruled out [6]. Theoretical total energy values and especially the nature of magnetic interactions strongly support the Ni(A)Mn(B)-Cu(C)-Sb(D) sequence. Calculated exchange interactions show that the sequence Ni(A)-Mn(B)-Cu(C)-Sb(D) yields a strong ferromagnet with high local moment on Mn and high Curie temperature. Calculations also reveal that were the sequence to be Ni(A)-Cu(B)-Mn(C)-Sb(D), the substance would be either an antiferromagnet or, at best, a weak ferromagnet, with an Mn moment noticeably lower than the measured value. In addition, we obtain the correct (i.e, in agreement with experiment) relative order among the RPA Curie temperatures for the ideal NiMnCuSb and its Cu- and Ni-rich counterparts, but only if experimentally realistic scenarios involving Sb-vacancies and/or Sb[Cu]-antisites are considered. The calculated values of Curie temperatures are in fair agreement with measured data. Density of states calculations reveal that neither NiMnCuSb nor NiCuMnSb has large spin polarization at the Fermi level. 5. Acknowledgments The work of Y.L. is supported by the Natural Science Foundation of Hebei Province (No. A2015203021), and also by Key Project of Heibei Educational Department, China (No. ZD2014015). The work of J.K. was supported by a Grant from the Czech Science Foundation (No. 15-13436S). The work of S.K.B. was funded by Brock University, Canada. References [1] K. Ozdogan,E.Sasioglu, I. Galanakis, J. Appl. Phys. 113 (2013) 193903. [2] J. Ma, V.I. Hedge, K. Munira, Yu. Xie, S. Keshavarz, D.T. Mildebrath, C. Wolverton, A.W. Ghosh, W.H. Butler, Phys. Rev. B 95 (2017) 024411. [3] L. Wollmann, A.K. Nayak, S.S.P. Parkin, C. Felser, cond-mat arXiv:1612.05947. [4] L. Bainsla and K.G. Suresh, App. Phys. Rev. 3 (2016) 031101.
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[5] S.E. Kulkova, S.V. Eremeev. T. Kakeshita, S.S. Kulkova, G.E. Rudenski, Materials Transactions 47 (2006) 599. [6] Z. Haque, G.S. Thakur, S. Ghara, L.C. Gupta, A. Sundaresan, and A.K. Ganguli, J. Magn. Magn. Mat. 397 (2016) 315. [7] P. J. Webster and K. R. A. Ziebeck, in Alloys and Compounds of dElements with Main Group Elements, Part 2, edited by H. R. J. Wijn, Landolt-B¨ornstein, New Series, Group III, Vol. 19, Pt. C (SpringerVerlag, Berlin, 1988). [8] P. Blaha, K. Schwarz, G.K.H. Madsen, D. Kvasnicka and J. Luitz, WIEN2k, An Augmented Plane Wave plus Local Orbitals Program for Calculating Crystal Properties, (ISBN 3-9501031-1-2), Vienna University of Technology, Vienna, Austria (2016). [9] J. Kudrnovsk´y and V. Drchal, Phys. Rev. B 41 (1990) 7515. ˇ [10] I. Turek, V. Drchal, J. Kudrnovsk´y, M. Sob, and P. Weinberger, Electronic Structure of Disordered Alloys, Surfaces and Interfaces (Kluwer, Boston-London-Dordrecht, 1997). [11] O.K. Andersen, O. Jepsen and D. Gl¨otzel, in Highlights of CondensedMatter Theory, edited by F. Bassani, F. Fumi, and M.P. Tosi (NorthHolland, Amsterdam, 1985), 59. [12] P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964); W. Kohn and 14 L. J. Sham, Phys. Rev. 140 (1965) A1133. [13] I. Turek, J. Kudrnovsk´y, V. Drchal, P. Bruno, Phil. Mag. 86 (2006) 1713. [14] J.P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3685. [15] Z. Wu, and R.E. Cohen, Phys. Rev. B 73 (2006) 235116. [16] J.P. Perdew, A. Ruzsinszky, G.I. Csonka, O.A. Vydrov, G.E. Scuseria, L.A. Constantin, X. Zhou, K. Burke, Phys. Rev. Lett. 100 (2008)136406. [17] S.H.Vosko, L. Wilk, and M. Nusair, Can. J. Phys. 58 (1980) 1200. 11
[18] S.L. Dudarev, G.A. Botton, S.Y. Savrasov, C.J. Humphreys, A. P. Sutton, Phys. Rev. B 57 (1990) 1505. [19] M. Krause and F. Bechstedt, J. Supercond. Nov. Magn. 26 (2013) 1963. [20] Y.Liu, S.K. Bose, and J. Kudrnovsk´y, J. Magn. Magn. Mat. 423 (2017) 12. [21] Y.Liu, S.K. Bose, and J. Kudrnovsk´y, AIP Advances 6 (2016) 125005. [22] A.I. Liechtenstein, M.I. Katsnelson, V.P. Antropov, V.A. Gubanov, J. Magn. Magn. Mater. 67 (1987) 65. [23] B.L. Gyorffy, A.J. Pindor, J. Staunton, G.M. Stocks, and H. Winter, J. Phys. F: Met. Phys. 15 (1985) 1337. ´ [24] J. Kudrnovsk´y, V. Drchal, L. Bergqvist, J. Rusz, I.Turek, B. Ujfalussy, I. Vincze, Phys. Rev. B 90 (2014) 134408. [25] J. Kudrnovsk´y, F. M´aca, I. Turek, J. Redinger, Phys. Rev. B 80 (2009) 064405.
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Table 1: Magnetic moments and total energies per formula unit as obtained from WIEN2k calculations for different occupations of the sublattices A(0,0,0); B(0.25,0.25,0.25); C(0.5,0.5,0.5); and D-Sb(0.75,0.75,0.75) at the experimental lattice parameter a=5.924 ˚ A. Results are shown for three different exchange-correlation potentials: PBE-96 [14], WuCohen [15], and PBE-sol [16].
A(Cu) PBE-96 Wu-Cohen PBE-sol A(Ni) PBE-96 Wu-Cohen PBE-sol A(Ni) PBE-96 Wu-Cohen PBE-sol
B(Ni)
C(Mn)
B(Cu)
C(Mn)
B(Mn)
Mtot (µB ) 3.090 3.032 3.081
mM n (µB ) 2.703 2.651 2.689
E (Ry) -21636.25275 -21632.34196 -21623.21040
2.950 2.878 2.927
2.743 2.674 2.772
-21636.27389 -21632.36321 -21623.23197
3.922 3.904 3.923
3.449 3.427 3.436
-21636.27177 -21632.35994 -21623.22897
C(Cu)
13
Table 2: Magnetic moments and total energies as obtained from WIEN2k calculations for different occupations of the sublattices A(0,0,0); B(0.25,0.25,0.25); C(0.5,0.5,0.5); and DSb(0.75,0.75,0.75) for the optimized lattice parameter a (˚ A). Other labels and quantities have the same meaning as in Table 1.
A(Cu) PBE-96 Wu-Cohen PBE-sol A(Ni) PBE-96 Wu-Cohen PBE-sol A(Ni) PBE-96 Wu-Cohen PBE-sol
B(Ni)
C(Mn)
B(Cu)
C(Mn)
B(Mn)
˚) Mtot (µB ) mMn (µB ) E (Ry) a (A 3.583 3.167 -21636.27624 6.1665 3.408 2.989 -21632.34990 6.0609 3.405 2.983 -21623.21720 6.0493 3.401 3.233 3.228
3.187 3.001 2.990
-21636.29344 6.1394 -21632.36894 6.0400 -21623.23665 6.0274
4.182 4.092 4.099
3.681 3.589 3.587
-21636.30137 6.1790 -21632.37286 6.0888 -21623.24015 6.0776
C(Cu)
14
Table 3: Calculated total (Mtot ) and local magnetic moments (mQ ) on Q=Mn- and Nisites (in µB ) for the NiMnCuSb alloy, and its Ni-rich and Cu-rich counterparts assuming U[Mn]=0.05 Ry. The second row for NiMnCuSb shows the values for the case with 2% of Sb[Cu] antisites. For Ni- and Cu-rich cases, the second row shows the values for alloys containing 5% Sb-vacancies. The last column shows the experimental total moments in each case [6].
Alloy NiMnCuSb NiMn(Cu0.9 ,Ni0.1 )Sb (Ni0.9 ,Cu0.1 )MnCuSb
Mtot 3.870 3.843 3.886 3.898 3.863 3.883
15
mMn 3.660 3.639 3.675 3.691 3.670 3.691
exp mNi Mtot 0.170 3.85 0.167 0.166 3.87 0.171 0.168 3.57 0.175
100 80
(a)
maj
LDOS (states/spin/Ry)
60 40 20 0 -20 -40 tot -60
Ni Mn
-80
min
Cu
-100 100 80
(b)
maj
LDOS (states/spin/Ry)
60 40 20 0 -20 -40 tot -60
Ni
-80
Mn
min
Cu
-100 -0.6
-0.4
-0.2
0
0.2
E - EF (Ry)
Figure 1: The total and local densities of states (DOS) for the ferromagnetic NiMnCuSb alloys as calculated by the TB-LMTO-CPA method: (a) the sequence Ni-Mn-Cu-Sb, and (b) the sequence Ni-Cu-Mn-Sb. Both the total DOS (tot) and atom-resolved DOS’s on Ni, Mn, and Cu-atoms are shown. The alloy Fermi energy is shifted to zero. In both cases we have used a small effective Hubbard U[Mn]=0.05 Ry to reproduce the total magnetic moment (3.87 µB ) for the sequence Ni-Mn-Cu-Sb.
16
0.8 0.7
Ni-Mn-Cu-Sb
JMn,Mn(d)(mRy)
0.6 0.5
Ni-Cu-Mn-Sb
0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 0.5 0.75
1
1.25 1.5 1.75
2
2.25 2.5
d/a
Figure 2: The exchange interactions J Mn,Mn between Mn-atoms for the sequences NiMn-Cu-Sb (full circles) and Ni-Cu-Mn-Sb (empty circles) are plotted as a function of the distance d between Mn-atoms (in units of the lattice constant a). In both cases the interactions are derived from the corresponding DLM reference state, and we have added a small effective Hubbard U[Mn]=0.05 Ry as in Fig. 1. The positive/negative values of JMn,Mn correspond to the ferromagnetic/antiferromagnetic couplings, respectively.
17
10
6 4 2 0
J
Mn,Mn
(q) (mRy)
8
-2 -4 -6
L
Γ
X
W
K
Γ
Figure 3: The lattice Fourier transformation of exchange interactions J Mn,Mn (q), along the lines of the high symmetry in the fcc BZ corresponding to the Mn-sublattice is shown for the sequences Ni-Mn-Cu-Sb (full lines) and Ni-Cu-Mn-Sb (dashed lines). The same model as in Fig. 2 is also employed here. The magnetic ground state corresponds to the maximum of each curve due to the convention in the definition of the Heisenberg Hamiltonian (1) [13].
18
1000 (a)
MFA
Tc [K]
800 600 400 200 0 1000
Mod A Mod B Mod C Ideal Ni-rich Cu-rich
(b) RPA
800
Tc [K]
Exp
600 400 200 0 Mod A Mod B Mod C
Exp
Figure 4: Calculated Curie temperatures for NiMnCuSb (sequence Ni-Mn-Cu-Sb along the diagonal) and its Ni-rich (NiMn(Cu0.9 ,Ni0.1 )Sb) and Cu-rich ((Ni0.9 ,Cu0.1 )MnCuSb) counterparts. Three models were studied: (i) Model A (Mod A): the ideal alloys without Sb-vacancies or Sb[Cu]-antisites. A small effective Hubbard U[Mn]=0.05 Ry is added; (ii) Model B (Mod B): the same as Model A but for alloys with 0.5% antisites for NiMnCuSb and 5% Sb-vacancies for Ni-/Cu-rich alloys; and (iii) Model C (Mod C): the same as for Model B but with effective Hubbard U[Mn]=0.13 Ry and Sb[Cu] concentration of 2%. We show both the MFA (a) and RPA (b) values. For convenience, experimental results (Exp) for each model are also shown.
19
Research Highlights We present a theoretical study of magnetism of the Quaternary Heusler alloy NiMnCuSb. Our theoretical study identifies the structure to be NiMnCuSb as opposed to NiCuMnSb. The identification of the true structure as being NiMnCuSb is in contradiction to a recently published work. We show that the measured values of magnetic moments and Curie temperatures support our results.