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Equilibrium properties of transition metal aluminides
Solid State Communications 150 (2010) 189–194
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Equilibrium properties of transition metal aluminidesI C. Paduani ∗ DF-UFSC, Florianópolis, CEP 88040-900, SC, Brazil
article
info
Article history: Received 7 July 2009 Received in revised form 5 October 2009 Accepted 25 October 2009 by D.D. Sarma Available online 30 October 2009 PACS: 71.15.Ap 71.15.Mb 71.15.Nc 71.18.+y 71.20.Be
abstract In this work are studied the band structure and Fermi surfaces of the transition metal aluminides MnAl, FeAl, CoAl, NiAl and CuAl, in the ordered B2 (CsCl) structure, by means of first-principles spin-polarized scalar relativistic calculations. The ground state of MnAl is controversial, and it has been reported as the L10 (fct) structure, with a moment of 2.0 µB on the Mn atom [2]. The present calculations indicate that ferromagnetic MnAl and FeAl are stable in the ordered B2 (CsCl) structure. The magnetic moments on the Mn and Fe atoms are 1.87 µB and 0.80 µB , respectively, and increase with lattice expansion. CoAl, NiAl and CuAl are nonmagnetic, and CoAl has the smallest unit cell volume and the largest bulk modulus in this series. © 2009 Elsevier Ltd. All rights reserved.
Keywords: A. Aluminides A. Transition metals D. Electronic structure D. Fermi surface
1. Introduction In the Mn–Al system from 40–100 at.% Mn forms a metastable ferromagnetic (FM) phase obtained by special heat treatment [1], the τ phase, which has a face-centered-tetragonal (L10 ) structure with lattice constants a = 2.77 Å and c = 3.57 Å (and lattice positions 000 and 12 21 12 ). First-principles calculations performed to study the phase stability of 3d transition-metal aluminides with equiatomic composition showed that the ordered tetragonal L10 structure is preferred for early transition metal aluminides, which evolves to the cubic B2 (CsCl-type) structure for late transition metal aluminides [2]. The ordered B2 phase thus becomes more stable for FeAl, CoAl and NiAl. As has been pointed out [2], the reasoning is that, in addition to the size effect, the charge transfer from Al to 3d metals and the filling of d bands requires that a late 3d metal have as many Al atom as its nearest neighbors as possible to facilitate the charge transfer and bonding hybridization. Spinpolarized calculations revealed a FM L10 phase for MnAl, with a
I This work was supported by CNPq and Finep.
∗ Corresponding address: Universidade Federal de Santa Catarina Florianopolis, Departamento de Fisica, CEP 88040-900 Florianopolis, SC, Brazil Tel.: +55 48 3721 9234; fax: +55 48 3721 9946. E-mail address: [email protected]. 0038-1098/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2009.10.031
magnetic moment of 2.0 µB on the Mn atoms; for MnAl the ordered B2 phase has the lowest energy in the paramagnetic state, whereas in the FM state (µMn = 1.7 µB ) this structure is no longer stable, and the L10 phase is preferred. However, the obtained equilibrium cell volume yields a = 3.79 Å, and a ratio c /a = 0.87 [2], which is well above the reported experimental values for the fct τ phase [1]. It is worth mentioning that transition metal aluminides Mn–Al–T, T = Fe, Co, Mn and Cu have been experimentally investigated by Tsuboya et al. [3], and are stable in the B2 structure (CsCl-type, the so-called κ phase), and within a wide compositional range the lattice constants are below 3.0 Å. Calculations of ordered (B2) FeAl have shown that it is stable in a FM phase, with a moment of 0.6 µB on the Fe atoms. Experimentally, ordered FeAl is nonmagnetic. However, electronic structure calculations performed by different methods confirm a stable FM state for FeAl, CoAl and NiAl [4–6]. Calculations of the Stoner criterion [7] for the stability of ferromagnetism in 3d transition metal aluminides in the B2 structure have shown a FM ground state for both MnAl and FeAl [8]. The conventional wisdom is that the magnetic interaction between Mn atoms is considered to be antiferromagnetic if the distance between nearest-neighbor Mn atoms is shorter than 2.8 Å, and to be ferromagnetic if it is larger than 2.9 Å. From experimental results it has been observed that in the Mn–Al–Cu system the sign of the exchange interaction
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Fig. 1. Top: Total energy E0 against average atomic volume for MnAl and FeAl in the B2 structure (FM and NM indicates ferromagnetic and nonmagnetic calculations). Bottom: Calculated E0 vs. c /a ratio curves for FM MnAl in the L10 structure (s.g. P4/mmm) in two different cell volumes. The equilibrium lattice constant and c /a ratio are indicated.
between Mn atoms at the same sites is positive, thus indicating parallel moments, and it is negative between Mn atoms in different sites, thus indicating antiparallel moments [3]. From a study of the bonding mechanism in later transition metal aluminides in the B2 structure a depletion of electron density at the Al sites has been observed, and a buildup of d-bonding charge at the transition metal sites, and as a consequence, there is no directional bonding between nearest-neighbor atoms [2]. Since in ordered B2-type systems there are no nearest-neighbor atoms of the same kind, the short-range band mixing is expected to play a leading rule in the bonding mechanism in these compounds. In this contribution is studied the electronic structure of the transition metal aluminides MnAl, FeAl, CoAl, NiAl and CuAl in the B2 structure in spin-polarized scalar relativistic calculations, employing the full-potential linearized augmented-plane wave (FP–LAPW) method [9]. A description of the band structure and Fermi surfaces is provided for both ferromagnetic (FM) and nonmagnetic (NM) states at the equilibrium volumes. Since the ground state of MnAl is still controversial, it is included in the series of late transition metal aluminides in order to shed more light on this subject. 2. Method and results The FP–LAPW method and the density functional theory are used with the Perdew–Burke–Ernzerhof (1996) parameterization [10] for exchange and correlation effects to investigate the electronic structure of the studied compounds. The calculations were performed at the equilibrium cell volumes obtained by fitting the total energy curves to the Murnaghan equation of states [11].
For both transition metal and Al atoms were adopted equal muffintin radii of 2.20 a.u. The integration in the Brillouin zone was performed using 6000 k-points. A criterion of energy was used for convergence, taken to 10−2 mRy, and the atomic wave function expansion was set up to l = 10. The charge density Fourier expansion cutoff was made with Gmax = 14. The cutoff parameter is Rmt Kmax = 9.0, where Kmax is the maximum value of the reciprocal lattice vector used in the plane wave expansion and Rmt is the smallest atomic sphere radius in the unit cell. According to the calculations of Zou and Fu [2] a FM state exists in the L10 fct phase of MnAl with c /a = 0.87 (a = 3.79 Å), with a moment of 2.03 µB on the Mn atoms; in the B2 phase no local minimum was observed in the total energy vs c /a curve, which at c /a = 0.707 corresponds to the B2 phase. However, the region investigated in the minimization of the total energy corresponds to a large cell volume with a = 3.79 Å. This is far above the value for the unit cell volume of MnAl-based compounds which are stable in the B2 structure [3]. In Fig. 1 (top) are shown plots of the calculated total energy Eo versus equilibrium cell volume in the B2 phase for both MnAl and FeAl, as well as a plot (bottom) of the total energy vs. c /a ratio for MnAl in the L10 structure. The calculations were performed in spin-polarized and non-spin-polarized cases. Since for the other aluminides a null moment was found from the FM calculations they are not included in Fig. 1. As one sees, the FM state has the lowest energy for both MnAl and FeAl. In the former the Eo curves merge at smaller volumes, whereas in the latter this does not happen. The equilibrium lattice constants are listed in Table 1. For MnAl a = 5.571 a.u. (2.947 Å), which is much smaller than the result of Zou and Fu, and is larger than that value where ferromagnetism is expected (2.9 Å), and is very close to the lattice constants of the MnAl-based compounds.
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Table 1 Equilibrium properties of aluminides in the B2 structure: calculated lattice constant, equilibrium volume Vo , Bulk Modulus, Fermi energy (EF ), calculated DOS at the Fermi level (N (EF )), calculated electronic heat-capacity coefficient γ and atomic magnetic moment. Experimental values wherever available are given in parentheses [12,13] (FM = ferromagneticcalculations, NM = nonmagnetic).
MnAl MnAl FeAl FeAl CoAl NiAl CuAl
FM NM FM NM FM FM FM
Lattice constant (a.u.)
Vo (a.u.)3
B (GPa)
EF (Ry)
N(EF )
5.571 5.498 5.504 5.506 (5.50) 5.488 (5.42) 5.633 (5.46) 5.733
173.79 167.09 167.57 167.81 166.12 179.68 189.42
124.32 174.58 112.81 197.25 335.33 320.60 60.96
0.648 0.649 0.622 0.606 0.659 0.581 0.602
33.06 59.79 33.72 42.16 10.58 12.73 9.14
Fig. 2. Magnetic moment of Mn and Fe atoms versus atomic volume in MnAl and FeAl, respectively. The arrows indicate equilibrium volumes.
In the plot at the bottom of Fig. 1 are shown the results of FM calculations of the total energy as function of the c /a ratio for MnAl in the L10 structure, with two different unit cell volumes: the minimization of the c/a ratio was performed with the s.g. P4/mmm of the L10 structure, first by using the equilibrium cell volume of the B2 structure, and after by using that of the L10 structure. A minimum is found at −2802.812 Ry (solid curve), for a = 7.234 a.u. (3.827 Å) and c /a = 0.87, which agrees well with the earlier results of Zou and Fu. However, the minimum in the other (dotted) curve shown at the bottom of Fig. 1 is seen at −2802.940 Ry, for a = 5.500 a.u. and c /a = 1.0, which corresponds to the B2 structure, and is 128 mRy below that (solid curve) for the L10 structure. Hence, one may conclude that the B2 structure indeed is the stable phase for MnAl, with a lattice constant a = 5.571 a.u. (2.947 Å), as obtained initially by performing the calculations with s.g. Pm-3m of the B2 structure. In the L10 structure a secondary minimum is observed at a smaller c /a ratio, which corresponds to a metastable state which can be reached by lattice compression. With a null moment the lattice constant of MnAl is 5.498 a.u. (2.908 Å). Interesting enough is the fact that, FeAl has practically the same value for the lattice constant in both FM and NM states, and the present calculated equilibrium lattice constant in this phase agrees well with the experimental result as well as with earlier calculations [5]. Furthermore, the unit cell volume is the smallest for CoAl, where the Co atom carries a null moment. For CuAl, a similar procedure as performed for MnAl in the L10 structure, by using different volumes, also shows that the B2 structure is preferred for this compound (the difference between the minima is 120 mRy).
states Ryatom
γ
mJ molcellK2
5.26 10.37 5.85 7.31 1.83 2.21 1.59
µ ( µB ) 1.87 0.80 0 0 0
Fig. 3. Band structure along symmetry directions around the Fermi level and total DOS for MnAl (the t2g and eg characters of d-states are indicated only in the majority spin sub-band).
Fig. 2 shows the behavior of the atomic magnetic moment versus cell volume on the Mn and Fe atoms in MnAl and FeAl. Starting with the equilibrium volume (indicated by an arrow in Fig. 2), by lattice compression the magnetic moment on the Mn atom decreases continuously, whereas by lattice expansion an abrupt increase is observed above ≈10% of increase in the unit cell volume. It is noteworthy that in larger cells of superstructures the Mn atoms in FM phases carry a moment larger than 3 µB , as has been observed in Heusler alloys [14]. On the other hand, the calculated Fe moment in FeAl increases steadily by expanding the lattice from smaller volumes, but still remains about half of that value found on the Mn atom in MnAl. In Figs. 3–5 are shown the total DOS and band structures for the studied aluminides. For MnAl in Fig. 3 are shown the results of FM calculations. The character of Mn d-states is indicated only in the spin-up DOS diagram. The Fermi level EF lies in a gap in the majority spin sub-band, which separates t2g states from a sharp eg peak in the antibonding region, while in the minority spin subband EF is located on a shoulder of a broad peak which consists primarily of Mn d(t2g )-states. In both spin sub-bands a tendency to form degenerate bands around Γ and M points is observed, which leads to the peaky structure near EF in the DOS diagrams. For FM FeAl, in Fig. 4 one sees similar features, as observed in FM MnAl. In spite of the shift of EF to the right of the t2g peak in the minority spin sub-band (accounting for the additional electron of Fe), the density of states at EF (N(EF )) is nearly equal to that of FM
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Fig. 4. Band structure and total DOS for FeAl.
Fig. 6. Fermi surfaces for MnAl. From top to bottom: Hole sheets of majority spin from bands 4–6 (left); hole sheets from bands 4–5 and electron pockets from band 6 of minority spin (right).
Fig. 5. Band structure and total DOS for CoAl (left), NiAl (center) and CuAl (right).
MnAl, which yields close results for the calculated electronic heatcapacity coefficient given by γ = 13 π 2 k2B N (EF ), shown in Table 1. Nevertheless, results of nonmagnetic calculations reveal much larger discrepancies between these results. In Fig. 5, apart from the shift of the Fermi level with the additional electrons of Co and Ni, the band structure around EF keeps the same characteristics, despite these aluminides being nonmagnetic. However, a band narrowing is observed, from CoAl to CuAl; in this latter the structure of peaks appears well below EF , and an enhanced degeneracy is seen around M points. Figs. 6–8 display plots of the Fermi surfaces (FS) from the bands crossing EF for these aluminides. In MnAl, on the left in Fig. 3 are seen hole pockets for majority spin from the three FS crossing EF , which exhibit features around R points. On the right, for minority spin are shown mixed electron and hole sheets from bands 4–5 and electron pockets from band 6. The holes are centered at Γ and the electron pockets are seen mostly around M points. For FeAl, in Fig. 7 the hole sheets (on the left) in the majority spin
Fig. 7. FS for FeAl: Hole sheets from bands 4–6 of majority spin (left); hole sheets from bands 4–5 and electron pockets from band 6 of minority spin (right).
C. Paduani / Solid State Communications 150 (2010) 189–194
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Fig. 8. Top: FS for CoAl displaying a hole sheet from band 6 (left) and electron pocket from band 7 (right); middle: FS for NiAl showing hole sheet from band 6 (left) and electron pocket from band 7 (right); bottom: Hole sheet from band 6 (left) and electron pockets from bands 7–9 in CuAl.
sub-band appear more ‘‘unfolded’’ around R points, whereas in the minority spin sub-band, with the additional d electron of Fe, the hole sheets of the FS from bands 4–5 are reduced, and the filling of more electron states leads to the multiple connected structure seen in Fig. 7 for the electron pockets built up by the FS from band 6. In Fig. 8 similar features are seen for the FS of holes from band 6 for CoAl−→CuAl. The filling of electron pockets from band 7 in these aluminides occurs predominantly around the M points, which leads to the structures shown in Fig. 8. The poor conductivity of these compounds and the absence of extended or open orbits is also depicted in these plots. 3. Discussion In Table 1 one sees that the unit cell volume changes drastically between the NM and FM phases of MnAl. The bulk modulus B is smaller for the phase with larger cell volume. In FeAl, despite nearly equal cell volumes, a large difference is observed for the bulk modulus between the FM and NM phases. In Table 1 the smallest B belongs to CuAl (which has also the largest cell volume), while the largest B belongs to CoAl. From Table 1 one sees that only MnAl and FeAl fulfill the Stoner criterion for ferromagnetism, N(EF ) I ≥ 1, where I is an exchange-correlation enhancement parameter, and is an atomic-like property, insensitive to crystal structure [7,15]. Interesting enough is that close values are observed for N(EF ) (and γ ) in both CoAl and CuAl, despite the large discrepancies in the cell volumes and bulk moduli, as listed in Table 1. In the FM phase, the magnetic moment on the Mn atoms in MnAl is about twice that of the Fe atoms in FeAl.
In Fig. 9 are plotted the l-projected DOS for d(eg )- and d(t2g )states of 3d metals and for s- and p-states of Al atoms. In the first two panels are shown the partial DOS plots for both spin sub-bands in MnAl. One can see that the Al p-states (light-gray area in the panel) participate more actively in the bonding mechanism and are strongly hybridized with the t2g d-states of Mn within the energy range 0.1–0.3 Ry. Although some admixture also occurs with eg states, a directional characteristic of the bonding mechanism is inferred from this plot, which has been also mentioned in the work of Zou and Fu. Moreover, as has been pointed out before, the existence of short Al–Al bonds is energetically costly and unfavorable in the B2 lattice, and is prohibited by preferably creating vacancies at the 3d metal sites. This is exactly what is depicted by the 3D plots of the Fermi surfaces shown in Figs. 6–8. The point is, at smaller unit cell volumes, the B2 structure indeed is the stable phase, where the magnetic moment on the Mn atoms is 1.87 µB , and is more preferable than the L10 structure. For FeAl the calculated equilibrium volume is in very good agreement with experimental results, and the FM state is stable, with a magnetic moment on the Fe atoms of 0.8 µB . These results also agree well with the results of Moruzzi and Marcus [5]. Moreover, the formation of open orbits in FeAl is also shown by the multiple connected structure of the electron pockets built up by the Fermi surfaces from band 6 in the minority spin sub-band. This introduces an important effect on the transport properties of this compound. CoAl has the largest bulk modulus and the smallest cell volume. In Fig. 7 one can see that there is an stronger admixture between bonding t2g and eg d-states of Co atoms and between these and Al bonding p-states, which represents a strengthening
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decreases the bandwidth, which causes weaker interactions and a volume expansion. Finally, in CuAl the electronic states are almost filled up, and the narrowest bandwidth takes place, which in turn implies weaker interactions and a larger cell volume, and as a necessary accompaniment, the smallest bulk modulus in this series. Summarizing, the present calculations indicate that FM MnAl, FeAl and CuAl are stable in the ordered B2 (CsCl) structure, with a magnetic moment of 1.87 µB and 0.80 µB on the Mn and Fe atoms, respectively, whereas CoAl, NiAl and CuAl are nonmagnetic (NM). The separation in energy between the minima of the FM and NM states are 24.6 mRy for MnAl and 2.5 mRy for FeAl. By shrinking the lattice the NM state can be reached from the FM state in MnAl; this does not occur in FeAl. In both FM states, the magnetic moment on the Mn and Fe atoms increases with a lattice expansion. A band narrowing is observed throughout this series, which indicates a weakening of interactions. The cell volume decreases from MnAl to CoAl, where it is the smallest, and has a steep increase for NiAl and CuAl, where it is the largest. What happens in CoAl is that, while the antibonding (eg ) orbitals are still not completely filled up, there is an enhanced hybridization between the bonding d(eg )and d(t2g )-states of Co atoms with the bonding p-states of Al atoms. This leads to a strengthening of interactions which results in the smallest unit cell volume as well as in the largest bulk modulus. From the calculated Fermi surfaces it is observed that, among these aluminides, only in FeAl are formed open orbits. References
Fig. 9. L-projected DOS for d-states of 3d atoms and for s- and p-states from Al atoms. The d(t2g )-states of the 3d metals are indicated by a dotted line, and the eg states are showed as a solid line. The light-gray (dark) area corresponds to the Al p-(s-)states.
of interactions, yielding therein a smaller cell volume. To a lesser extent this also happens in NiAl, but the filling of electron states
[1] A.J. Koch, P. Hokkeling, M.G.v.d. Steeg, K.J. de Vos, J. Appl. Phys. 31 (5) (1960) 75S. [2] J. Zou, C.L. Fu, Phys. Rev. B 51 (1995) 2115. [3] I. Tsuboya, M. Sugihara, J. Phys. Soc. Japan 16 (1961) 1257; J. Phys. Soc. Japan 17 (1962) 410; J. Phys. Soc. Japan 16 (1961) 571; J. Phys. Soc. Japan 15 (1960) 1534; I. Tsuboya, J. Phys. Soc. Japan 16 (1961) 1875. [4] D.J. Singh, in: J.H. Westbrook, R.L. Fleisher (Eds.), Intermetallic Compounds: Principles, vol. 1, Wiley, New York, 1944, pp. 127–147. [5] V.L. Moruzzi, P.M. Marcus, Phys. Rev. B 47 (1993) 7878. [6] I. Dasgupta, T. Saha-Dasgupta, A. Mookerjee, G.P. Das, J. Phys. Condens. Matter 9 (1997) 3529. [7] E.C. Stoner, Proc. R. Soc. London Ser. A 169 (1939) 339. [8] D.A. Papaconstantopoulos, K.B. Hathaway, J. Appl. Phys. 87 (2000) 5872. [9] P. Blaha, K. Schwarz, G.K.H. Madsen, D. Kvasnicka, J. Luitz, WIEN2k, Vienna University of Technology, 2002, A Full-Potential Linearized Augmented-Plane Wave Package for Calculating Crystal Properties; K. Schwarz, P. Sorantin, S.B. Trickey, Comput. Phys. Commun. 59 (1990) 399; K. Schwarz, P. Blaha, G.K.H. Madsen, Comput. Phys. Commun. 147 (2002) 71. [10] J.P. Perdew, S. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865–3868. [11] F.D. Murnaghan, Proc. Natl. Acad. Sci. USA 30 (1944) 244. [12] P.A. Beck (Ed.), Electronic Structure and Alloy Chemistry of the Transition Elements, Interscience Publishers, New York, 1963. [13] M.V. Nevit, in: J.H. Westbrook (Ed.), Intermetallics Compounds, R.E. Krieger Publishing Co, Huntington, NY, 1977. [14] A. Ayuela, J. Enkovaara, K. Ullakko, R.M. Nieminen, J. Phys. Condens. Matter 11 (1999) 2017. [15] J.F. Janak, Phys. Rev. B 16 (1977) 255.
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Equilibrium properties of transition metal aluminidesI C. Paduani ∗ DF-UFSC, Florianópolis, CEP 88040-900, SC, Brazil
article
info
Article history: Received 7 July 2009 Received in revised form 5 October 2009 Accepted 25 October 2009 by D.D. Sarma Available online 30 October 2009 PACS: 71.15.Ap 71.15.Mb 71.15.Nc 71.18.+y 71.20.Be
abstract In this work are studied the band structure and Fermi surfaces of the transition metal aluminides MnAl, FeAl, CoAl, NiAl and CuAl, in the ordered B2 (CsCl) structure, by means of first-principles spin-polarized scalar relativistic calculations. The ground state of MnAl is controversial, and it has been reported as the L10 (fct) structure, with a moment of 2.0 µB on the Mn atom [2]. The present calculations indicate that ferromagnetic MnAl and FeAl are stable in the ordered B2 (CsCl) structure. The magnetic moments on the Mn and Fe atoms are 1.87 µB and 0.80 µB , respectively, and increase with lattice expansion. CoAl, NiAl and CuAl are nonmagnetic, and CoAl has the smallest unit cell volume and the largest bulk modulus in this series. © 2009 Elsevier Ltd. All rights reserved.
Keywords: A. Aluminides A. Transition metals D. Electronic structure D. Fermi surface
1. Introduction In the Mn–Al system from 40–100 at.% Mn forms a metastable ferromagnetic (FM) phase obtained by special heat treatment [1], the τ phase, which has a face-centered-tetragonal (L10 ) structure with lattice constants a = 2.77 Å and c = 3.57 Å (and lattice positions 000 and 12 21 12 ). First-principles calculations performed to study the phase stability of 3d transition-metal aluminides with equiatomic composition showed that the ordered tetragonal L10 structure is preferred for early transition metal aluminides, which evolves to the cubic B2 (CsCl-type) structure for late transition metal aluminides [2]. The ordered B2 phase thus becomes more stable for FeAl, CoAl and NiAl. As has been pointed out [2], the reasoning is that, in addition to the size effect, the charge transfer from Al to 3d metals and the filling of d bands requires that a late 3d metal have as many Al atom as its nearest neighbors as possible to facilitate the charge transfer and bonding hybridization. Spinpolarized calculations revealed a FM L10 phase for MnAl, with a
I This work was supported by CNPq and Finep.
∗ Corresponding address: Universidade Federal de Santa Catarina Florianopolis, Departamento de Fisica, CEP 88040-900 Florianopolis, SC, Brazil Tel.: +55 48 3721 9234; fax: +55 48 3721 9946. E-mail address: [email protected]. 0038-1098/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2009.10.031
magnetic moment of 2.0 µB on the Mn atoms; for MnAl the ordered B2 phase has the lowest energy in the paramagnetic state, whereas in the FM state (µMn = 1.7 µB ) this structure is no longer stable, and the L10 phase is preferred. However, the obtained equilibrium cell volume yields a = 3.79 Å, and a ratio c /a = 0.87 [2], which is well above the reported experimental values for the fct τ phase [1]. It is worth mentioning that transition metal aluminides Mn–Al–T, T = Fe, Co, Mn and Cu have been experimentally investigated by Tsuboya et al. [3], and are stable in the B2 structure (CsCl-type, the so-called κ phase), and within a wide compositional range the lattice constants are below 3.0 Å. Calculations of ordered (B2) FeAl have shown that it is stable in a FM phase, with a moment of 0.6 µB on the Fe atoms. Experimentally, ordered FeAl is nonmagnetic. However, electronic structure calculations performed by different methods confirm a stable FM state for FeAl, CoAl and NiAl [4–6]. Calculations of the Stoner criterion [7] for the stability of ferromagnetism in 3d transition metal aluminides in the B2 structure have shown a FM ground state for both MnAl and FeAl [8]. The conventional wisdom is that the magnetic interaction between Mn atoms is considered to be antiferromagnetic if the distance between nearest-neighbor Mn atoms is shorter than 2.8 Å, and to be ferromagnetic if it is larger than 2.9 Å. From experimental results it has been observed that in the Mn–Al–Cu system the sign of the exchange interaction
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Fig. 1. Top: Total energy E0 against average atomic volume for MnAl and FeAl in the B2 structure (FM and NM indicates ferromagnetic and nonmagnetic calculations). Bottom: Calculated E0 vs. c /a ratio curves for FM MnAl in the L10 structure (s.g. P4/mmm) in two different cell volumes. The equilibrium lattice constant and c /a ratio are indicated.
between Mn atoms at the same sites is positive, thus indicating parallel moments, and it is negative between Mn atoms in different sites, thus indicating antiparallel moments [3]. From a study of the bonding mechanism in later transition metal aluminides in the B2 structure a depletion of electron density at the Al sites has been observed, and a buildup of d-bonding charge at the transition metal sites, and as a consequence, there is no directional bonding between nearest-neighbor atoms [2]. Since in ordered B2-type systems there are no nearest-neighbor atoms of the same kind, the short-range band mixing is expected to play a leading rule in the bonding mechanism in these compounds. In this contribution is studied the electronic structure of the transition metal aluminides MnAl, FeAl, CoAl, NiAl and CuAl in the B2 structure in spin-polarized scalar relativistic calculations, employing the full-potential linearized augmented-plane wave (FP–LAPW) method [9]. A description of the band structure and Fermi surfaces is provided for both ferromagnetic (FM) and nonmagnetic (NM) states at the equilibrium volumes. Since the ground state of MnAl is still controversial, it is included in the series of late transition metal aluminides in order to shed more light on this subject. 2. Method and results The FP–LAPW method and the density functional theory are used with the Perdew–Burke–Ernzerhof (1996) parameterization [10] for exchange and correlation effects to investigate the electronic structure of the studied compounds. The calculations were performed at the equilibrium cell volumes obtained by fitting the total energy curves to the Murnaghan equation of states [11].
For both transition metal and Al atoms were adopted equal muffintin radii of 2.20 a.u. The integration in the Brillouin zone was performed using 6000 k-points. A criterion of energy was used for convergence, taken to 10−2 mRy, and the atomic wave function expansion was set up to l = 10. The charge density Fourier expansion cutoff was made with Gmax = 14. The cutoff parameter is Rmt Kmax = 9.0, where Kmax is the maximum value of the reciprocal lattice vector used in the plane wave expansion and Rmt is the smallest atomic sphere radius in the unit cell. According to the calculations of Zou and Fu [2] a FM state exists in the L10 fct phase of MnAl with c /a = 0.87 (a = 3.79 Å), with a moment of 2.03 µB on the Mn atoms; in the B2 phase no local minimum was observed in the total energy vs c /a curve, which at c /a = 0.707 corresponds to the B2 phase. However, the region investigated in the minimization of the total energy corresponds to a large cell volume with a = 3.79 Å. This is far above the value for the unit cell volume of MnAl-based compounds which are stable in the B2 structure [3]. In Fig. 1 (top) are shown plots of the calculated total energy Eo versus equilibrium cell volume in the B2 phase for both MnAl and FeAl, as well as a plot (bottom) of the total energy vs. c /a ratio for MnAl in the L10 structure. The calculations were performed in spin-polarized and non-spin-polarized cases. Since for the other aluminides a null moment was found from the FM calculations they are not included in Fig. 1. As one sees, the FM state has the lowest energy for both MnAl and FeAl. In the former the Eo curves merge at smaller volumes, whereas in the latter this does not happen. The equilibrium lattice constants are listed in Table 1. For MnAl a = 5.571 a.u. (2.947 Å), which is much smaller than the result of Zou and Fu, and is larger than that value where ferromagnetism is expected (2.9 Å), and is very close to the lattice constants of the MnAl-based compounds.
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Table 1 Equilibrium properties of aluminides in the B2 structure: calculated lattice constant, equilibrium volume Vo , Bulk Modulus, Fermi energy (EF ), calculated DOS at the Fermi level (N (EF )), calculated electronic heat-capacity coefficient γ and atomic magnetic moment. Experimental values wherever available are given in parentheses [12,13] (FM = ferromagneticcalculations, NM = nonmagnetic).
MnAl MnAl FeAl FeAl CoAl NiAl CuAl
FM NM FM NM FM FM FM
Lattice constant (a.u.)
Vo (a.u.)3
B (GPa)
EF (Ry)
N(EF )
5.571 5.498 5.504 5.506 (5.50) 5.488 (5.42) 5.633 (5.46) 5.733
173.79 167.09 167.57 167.81 166.12 179.68 189.42
124.32 174.58 112.81 197.25 335.33 320.60 60.96
0.648 0.649 0.622 0.606 0.659 0.581 0.602
33.06 59.79 33.72 42.16 10.58 12.73 9.14
Fig. 2. Magnetic moment of Mn and Fe atoms versus atomic volume in MnAl and FeAl, respectively. The arrows indicate equilibrium volumes.
In the plot at the bottom of Fig. 1 are shown the results of FM calculations of the total energy as function of the c /a ratio for MnAl in the L10 structure, with two different unit cell volumes: the minimization of the c/a ratio was performed with the s.g. P4/mmm of the L10 structure, first by using the equilibrium cell volume of the B2 structure, and after by using that of the L10 structure. A minimum is found at −2802.812 Ry (solid curve), for a = 7.234 a.u. (3.827 Å) and c /a = 0.87, which agrees well with the earlier results of Zou and Fu. However, the minimum in the other (dotted) curve shown at the bottom of Fig. 1 is seen at −2802.940 Ry, for a = 5.500 a.u. and c /a = 1.0, which corresponds to the B2 structure, and is 128 mRy below that (solid curve) for the L10 structure. Hence, one may conclude that the B2 structure indeed is the stable phase for MnAl, with a lattice constant a = 5.571 a.u. (2.947 Å), as obtained initially by performing the calculations with s.g. Pm-3m of the B2 structure. In the L10 structure a secondary minimum is observed at a smaller c /a ratio, which corresponds to a metastable state which can be reached by lattice compression. With a null moment the lattice constant of MnAl is 5.498 a.u. (2.908 Å). Interesting enough is the fact that, FeAl has practically the same value for the lattice constant in both FM and NM states, and the present calculated equilibrium lattice constant in this phase agrees well with the experimental result as well as with earlier calculations [5]. Furthermore, the unit cell volume is the smallest for CoAl, where the Co atom carries a null moment. For CuAl, a similar procedure as performed for MnAl in the L10 structure, by using different volumes, also shows that the B2 structure is preferred for this compound (the difference between the minima is 120 mRy).
states Ryatom
γ
mJ molcellK2
5.26 10.37 5.85 7.31 1.83 2.21 1.59
µ ( µB ) 1.87 0.80 0 0 0
Fig. 3. Band structure along symmetry directions around the Fermi level and total DOS for MnAl (the t2g and eg characters of d-states are indicated only in the majority spin sub-band).
Fig. 2 shows the behavior of the atomic magnetic moment versus cell volume on the Mn and Fe atoms in MnAl and FeAl. Starting with the equilibrium volume (indicated by an arrow in Fig. 2), by lattice compression the magnetic moment on the Mn atom decreases continuously, whereas by lattice expansion an abrupt increase is observed above ≈10% of increase in the unit cell volume. It is noteworthy that in larger cells of superstructures the Mn atoms in FM phases carry a moment larger than 3 µB , as has been observed in Heusler alloys [14]. On the other hand, the calculated Fe moment in FeAl increases steadily by expanding the lattice from smaller volumes, but still remains about half of that value found on the Mn atom in MnAl. In Figs. 3–5 are shown the total DOS and band structures for the studied aluminides. For MnAl in Fig. 3 are shown the results of FM calculations. The character of Mn d-states is indicated only in the spin-up DOS diagram. The Fermi level EF lies in a gap in the majority spin sub-band, which separates t2g states from a sharp eg peak in the antibonding region, while in the minority spin subband EF is located on a shoulder of a broad peak which consists primarily of Mn d(t2g )-states. In both spin sub-bands a tendency to form degenerate bands around Γ and M points is observed, which leads to the peaky structure near EF in the DOS diagrams. For FM FeAl, in Fig. 4 one sees similar features, as observed in FM MnAl. In spite of the shift of EF to the right of the t2g peak in the minority spin sub-band (accounting for the additional electron of Fe), the density of states at EF (N(EF )) is nearly equal to that of FM
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Fig. 4. Band structure and total DOS for FeAl.
Fig. 6. Fermi surfaces for MnAl. From top to bottom: Hole sheets of majority spin from bands 4–6 (left); hole sheets from bands 4–5 and electron pockets from band 6 of minority spin (right).
Fig. 5. Band structure and total DOS for CoAl (left), NiAl (center) and CuAl (right).
MnAl, which yields close results for the calculated electronic heatcapacity coefficient given by γ = 13 π 2 k2B N (EF ), shown in Table 1. Nevertheless, results of nonmagnetic calculations reveal much larger discrepancies between these results. In Fig. 5, apart from the shift of the Fermi level with the additional electrons of Co and Ni, the band structure around EF keeps the same characteristics, despite these aluminides being nonmagnetic. However, a band narrowing is observed, from CoAl to CuAl; in this latter the structure of peaks appears well below EF , and an enhanced degeneracy is seen around M points. Figs. 6–8 display plots of the Fermi surfaces (FS) from the bands crossing EF for these aluminides. In MnAl, on the left in Fig. 3 are seen hole pockets for majority spin from the three FS crossing EF , which exhibit features around R points. On the right, for minority spin are shown mixed electron and hole sheets from bands 4–5 and electron pockets from band 6. The holes are centered at Γ and the electron pockets are seen mostly around M points. For FeAl, in Fig. 7 the hole sheets (on the left) in the majority spin
Fig. 7. FS for FeAl: Hole sheets from bands 4–6 of majority spin (left); hole sheets from bands 4–5 and electron pockets from band 6 of minority spin (right).
C. Paduani / Solid State Communications 150 (2010) 189–194
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Fig. 8. Top: FS for CoAl displaying a hole sheet from band 6 (left) and electron pocket from band 7 (right); middle: FS for NiAl showing hole sheet from band 6 (left) and electron pocket from band 7 (right); bottom: Hole sheet from band 6 (left) and electron pockets from bands 7–9 in CuAl.
sub-band appear more ‘‘unfolded’’ around R points, whereas in the minority spin sub-band, with the additional d electron of Fe, the hole sheets of the FS from bands 4–5 are reduced, and the filling of more electron states leads to the multiple connected structure seen in Fig. 7 for the electron pockets built up by the FS from band 6. In Fig. 8 similar features are seen for the FS of holes from band 6 for CoAl−→CuAl. The filling of electron pockets from band 7 in these aluminides occurs predominantly around the M points, which leads to the structures shown in Fig. 8. The poor conductivity of these compounds and the absence of extended or open orbits is also depicted in these plots. 3. Discussion In Table 1 one sees that the unit cell volume changes drastically between the NM and FM phases of MnAl. The bulk modulus B is smaller for the phase with larger cell volume. In FeAl, despite nearly equal cell volumes, a large difference is observed for the bulk modulus between the FM and NM phases. In Table 1 the smallest B belongs to CuAl (which has also the largest cell volume), while the largest B belongs to CoAl. From Table 1 one sees that only MnAl and FeAl fulfill the Stoner criterion for ferromagnetism, N(EF ) I ≥ 1, where I is an exchange-correlation enhancement parameter, and is an atomic-like property, insensitive to crystal structure [7,15]. Interesting enough is that close values are observed for N(EF ) (and γ ) in both CoAl and CuAl, despite the large discrepancies in the cell volumes and bulk moduli, as listed in Table 1. In the FM phase, the magnetic moment on the Mn atoms in MnAl is about twice that of the Fe atoms in FeAl.
In Fig. 9 are plotted the l-projected DOS for d(eg )- and d(t2g )states of 3d metals and for s- and p-states of Al atoms. In the first two panels are shown the partial DOS plots for both spin sub-bands in MnAl. One can see that the Al p-states (light-gray area in the panel) participate more actively in the bonding mechanism and are strongly hybridized with the t2g d-states of Mn within the energy range 0.1–0.3 Ry. Although some admixture also occurs with eg states, a directional characteristic of the bonding mechanism is inferred from this plot, which has been also mentioned in the work of Zou and Fu. Moreover, as has been pointed out before, the existence of short Al–Al bonds is energetically costly and unfavorable in the B2 lattice, and is prohibited by preferably creating vacancies at the 3d metal sites. This is exactly what is depicted by the 3D plots of the Fermi surfaces shown in Figs. 6–8. The point is, at smaller unit cell volumes, the B2 structure indeed is the stable phase, where the magnetic moment on the Mn atoms is 1.87 µB , and is more preferable than the L10 structure. For FeAl the calculated equilibrium volume is in very good agreement with experimental results, and the FM state is stable, with a magnetic moment on the Fe atoms of 0.8 µB . These results also agree well with the results of Moruzzi and Marcus [5]. Moreover, the formation of open orbits in FeAl is also shown by the multiple connected structure of the electron pockets built up by the Fermi surfaces from band 6 in the minority spin sub-band. This introduces an important effect on the transport properties of this compound. CoAl has the largest bulk modulus and the smallest cell volume. In Fig. 7 one can see that there is an stronger admixture between bonding t2g and eg d-states of Co atoms and between these and Al bonding p-states, which represents a strengthening
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decreases the bandwidth, which causes weaker interactions and a volume expansion. Finally, in CuAl the electronic states are almost filled up, and the narrowest bandwidth takes place, which in turn implies weaker interactions and a larger cell volume, and as a necessary accompaniment, the smallest bulk modulus in this series. Summarizing, the present calculations indicate that FM MnAl, FeAl and CuAl are stable in the ordered B2 (CsCl) structure, with a magnetic moment of 1.87 µB and 0.80 µB on the Mn and Fe atoms, respectively, whereas CoAl, NiAl and CuAl are nonmagnetic (NM). The separation in energy between the minima of the FM and NM states are 24.6 mRy for MnAl and 2.5 mRy for FeAl. By shrinking the lattice the NM state can be reached from the FM state in MnAl; this does not occur in FeAl. In both FM states, the magnetic moment on the Mn and Fe atoms increases with a lattice expansion. A band narrowing is observed throughout this series, which indicates a weakening of interactions. The cell volume decreases from MnAl to CoAl, where it is the smallest, and has a steep increase for NiAl and CuAl, where it is the largest. What happens in CoAl is that, while the antibonding (eg ) orbitals are still not completely filled up, there is an enhanced hybridization between the bonding d(eg )and d(t2g )-states of Co atoms with the bonding p-states of Al atoms. This leads to a strengthening of interactions which results in the smallest unit cell volume as well as in the largest bulk modulus. From the calculated Fermi surfaces it is observed that, among these aluminides, only in FeAl are formed open orbits. References
Fig. 9. L-projected DOS for d-states of 3d atoms and for s- and p-states from Al atoms. The d(t2g )-states of the 3d metals are indicated by a dotted line, and the eg states are showed as a solid line. The light-gray (dark) area corresponds to the Al p-(s-)states.
of interactions, yielding therein a smaller cell volume. To a lesser extent this also happens in NiAl, but the filling of electron states
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