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First-principles study of magnetism in NpRhAl
ARTICLE IN PRESS
Physica B 371 (2006) 332–336 www.elsevier.com/locate/physb
First-principles study of magnetism in NpRhAl M. Divisˇ Department of Electronic Structures, Faculty of Mathematics and Physics, Charles University, Ke Karlovu 5, 121 16 Prague 2, Czech Republic Received 14 October 2005; accepted 15 October 2005
Abstract First-principles calculations based on density-functional theory were performed for the compound NpRhAl for the first time. The electronic density of states, bonding properties and equilibrium volume were studied using relativistic full-potential APW plus local orbitals calculations. The magnetocrystalline anisotropy energy was estimated from total-energy calculations and the c-axis was predicted to be the easy axis of magnetization. Finally, we employed the LSDA+U method to mimic orbital polarization and to obtain the correct total magnetic moments in experimental equilibrium. r 2005 Elsevier B.V. All rights reserved. PACS: 71.15.Mb; 71.20.Lp; 71.27.+a Keywords: Neptunium intermetallics; Magnetic properties; Band-structure calculations
1. Introduction The magnetic properties of actinide compounds are determined by the character of the 5f-electron states ranging from itinerant to almost localized. Systematic studies of large isostructural families of ternary compounds have helped to understand the main tendencies for U compounds [1]. The much more limited set of experimental data for other light actinides, namely Np and Pu, leaves considerably more uncertainty about the general rules and characteristics of magnetism in transuranium systems. From the few known examples of Np compounds forming in the same structure as the U counterparts, the An2T2X (An ¼ U or Np, T ¼ transition metal, X ¼ p metal) compounds can be seen to exhibit a very similar pattern in which the presence of the T and X components leads to suppression of actinide magnetic moments. This makes the clear impression that 5f-ligand hybridization is the principal mechanism determining the magnetic order, and that the Np compounds, in close analogy with their U cousins, have a narrow 5f band at the Fermi level (EF). On the other hand, the higher 5f count in Np leads to the Tel.: +42 0221911367; fax: +42 0224911051.
E-mail address: [email protected]. 0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.10.129
general belief that electron–electron correlations should have a more prominent role and that Np systems tend to be less itinerant than the corresponding U ones. Indeed, the discovery of unconventional superconductivity in PuTGa5 compounds initiated an effort to compare UCoGa5, which is a weak Pauli paramagnet, with the antiferromagnet NpCoGa5 with the heavy-fermion superconductor PuCoGa5 [2]. This dramatic development of physical properties led to conjectures about other type of physics dominating the magnetic properties in the transuranium systems [3]. But such variations can be generally explained also within simple band theory, as the higher f-count can simply move the Fermi level up to a higher density of the 5f states. Equiatomic ternaries of the NpTX type, crystallizing in the ZrNiAl structure type, do not exhibit any exotic ground state. They are mainly interesting as counterparts of UTX compounds, which represent one of the most thoroughly studied groups with a well-mapped boundary between magnetic and non-magnetic ground states. Moreover, magnetic and other electronic properties are well accounted for by the existing theoretical approaches. It is therefore tempting to extend the present calculations successfully for example from URhAl to NpRhAl and to investigate whether a tendency to 5f localization, which can be
ARTICLE IN PRESS M. Divisˇ / Physica B 371 (2006) 332–336
associated with a possible narrowing of the 5f band, appears. The ternary Np compounds with composition NpTX, where T is a transition metal (Fe–Ni and 4d, 5d equivalents) and X is a p element (Al, Ga, Sn) were studied by Kergadallan, who proposed magnetic ordering of Np moments in NpRhAl [4]. The magnetic-ordering temperature (TN ¼ 16.5 K) was established on the basis of specificheat data [5]. 237Np Mo¨ssbauer spectroscopy reveals two different values of the magnetic hyperfine field, with onethird of the Np moments strongly reduced in magnitude [4]. The heavy-fermion character deduced from the Sommerfeld coefficient g ¼ 200 mJ/mol K2 is understood in the context of a low value of the magnetic entropy [5] and Np moments much smaller than expected from, e.g. the 5f4 free-ion value (2.25mB). Using the Dunlap–Lander relation, the magnetic hyperfine fields of 229 and 170 T, give Np moments m ¼ 1.065mB and 0.791mB, respectively. Such a situation implies a complicated antiferromagnetic structure, perhaps frustrated [5], which is markedly different from the simple ferromagnetism in URhAl (TC ¼ 27 K). The distinct tendency to antiferromagnetism in NpTX compounds is generally contrasting with UTX compounds, where the tendency to ferromagnetism dominates at least in the range close to the onset of magnetism [1]. Here, we present results of relativistic band–structure calculations for NpRhAl in the simplified ferromagnetic structure including spin–orbit coupling in the framework of density-functional theory (DFT). In the study of the magnetic moments, we have also gone beyond local-spindensity approximation (LSDA) and employed the static limit of the dynamical mean-field theory, the LSDA+U approximation in particular. 2. Theoretical aspects NpRhAl crystallizes in the hexagonal structure (ZrNiAl type), which contains three formula units per unit cell. The ZrNiAl structure has a layered structure, consisting of planes of Np atoms admixed with one-third of the Rh atoms (Rh1), that are stacked along the c-axis, while two adjacent Np planes are separated from one another by a layer consisting of the remaining Rh (Rh2) and Al atoms. The Np atoms have Rh atoms as nearest neighbors and vice versa, so that both the Np and the Rh atoms are well separated from atoms of the same type. There are two internal parameters for the Np (xNp, 0, 1/2) and the Al (xAl, 0, 0) atomic positions in the NpRhAl structure. To obtain direct information about the ground-state electronic structure and related properties, we applied firstprinciples theoretical methods. The ground-state electronic structure was calculated on the basis of the DFT within the LSDA [6] and the generalized gradient (GGA) approximation [7]. For this purpose, we used the full-potential augmented-plane-wave plus local-orbitals method (APWlo) as implemented in the latest version (WIEN2k) of the
333
original WIEN code [8]. The spin–orbit coupling (SOC) is treated by a second variational step within this implementation [9]. The calculations were performed with the following parameters. The non-overlapping atomic sphere radii of 2.7, 2.3 and 2.1 a.u. (1 a.u. ¼ 52.9177 pm) were taken for Np, Rh, and Al, respectively. The basis for expansion of the valence states (higher than 6 Ry below the Fermi energy) consisted of more than 1000 basis functions (more than 100 APW/atom) plus Np (6s, 6p), Pt (5p), and Al (2p) local orbitals. Relativistic local orbitals for the description of Np 6p1/2 states were included [10]. The Np 5f states were also treated as valence Bloch states and thus Np is characterized by a non-integer occupation number. The Brillouin-zone integrations were performed with the tetrahedron method [8] on a 35–105 special k-points mesh. The unknown values of the internal parameters for Np and Al were obtained by minimizing the forces at experimental values of a ¼ 697.6 pm and c ¼ 402.5 pm [11]. The values xNp ¼ 0.58017291 and the xAl ¼ 0.23685063 were found. We carefully tested the convergence of the results presented with respect to the parameters mentioned and found them to be fully sufficient for all presented characteristics of NpRhAl. We also tried the around-mean-field (AMF) LDA+U method [8], because by construction it is better suited for systems with a lower degree of localization, which should be the case for NpRhAl. The LDA+U potential is implemented in a rotationally invariant way. In these calculations, we varied the parameters U and J, which were used to describe the onsite Coulomb (direct and exchange) interaction among correlated 5f electrons at the Np-site. The effective U was varied in the range from 0.2 to 0.5 eV with several different J values, namely 0.22, 0.33 eV and an atomic value 0.55 eV [12]. Assuming that the LSDA+U method is appropriate for the description of the magnetism of NpRhAl, we heuristically expect that the value of J does not deviate dramatically from its atomic value, and that the value of U should not be lower than the value of J. We are fully aware that on this level such a calculation loses its first-principles character but, on the other hand, we will show that these heuristically derived values allow us to obtain valuable results. As a figure of merit for the calculations we take the saturated magnetic moment, which has been determined by 237Np Mo¨ssbauer spectroscopy [4]. We also tested to which extent the results of the LSDA+U calculations depend on the starting density matrix. Experience has shown that the converged self-consistent electronic structure does not depend on the starting density matrices in the cases where U and J are smaller than the bandwidth of 5f states. In the treated compound NpRhAl, this bandwidth exceeds 3 eV, which is well above the upper limit of the interval of U parameters used (0.5 eV). 3. Results and discussion The total density of electronic states (DOS) and the siteprojected DOS from LSDA calculations including SOC at
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334
experimental equilibrium are shown in Fig. 1. The lowest band, from about 8 to 5 eV, originates from Al 3s states. There is a pseudo-gap around 5 eV. The Rh 4d states form the main contribution to the occupied energy range in the energy range 5 to 2 eV (‘‘4d band’’) but they show an admixture of the Np 7s, 6d states, Rh 5s states, and Al 3s, 3p states. The highest occupied bands (between 2 eV and the Fermi level) originate mainly from the hybridized Np 5f states and Rh 4d states but all remaining (Np 7s, 6d states; Al 3p states) are also present. Finally, we see that the Fermi level is situated inside the Np 5f band with a rather large DOS. The total occupied bandwidth of Np 5f states is roughly 1.5 eV. The important results concerning the Np-f and Rh-d bonding in NpRhAl are contained in Table 1, which shows the charge distribution over the atomic spheres and its orbital decomposition. Although the Rh 4d band is filled with the Fermi energy lying in the region of the Np 5f band, the Rh-4d and Np-5f occupation numbers are 6.737 (Rh1), 6.716 (Rh2), and 3.580 (Np) electrons, respectively. This is a consequence of the hybridization between the Rh4d and Np-5f states. The region of the rather low DOS (2 eVoEo1.5 eV) between the Np 5f and the Rh 4d
20
(a) total
15 10 5 0
(b) 12
Np
DOS (states/ev)
8 4 0 0.8
(c)
Al
0.6 0.4 0.2 0.0
(d)
EF
Rh2 4 Rh1 2 0 -10
-8
-6
-4
-2 E (eV)
0
2
4
Fig. 1. Calculated DOS of NpRhAl: (a) the total DOS and (b,c,d) the Np-, Al-, Rh1-, Rh2 atom-projected DOS, respectively.
Table 1 Angular decomposition of the APW valence-charge density in the atomic spheres (electron charge) for NpRhAl NpRhAl
Qs
Qp
Qd
Qf
Qt(GS)
Qt(S)
Np Rh1 Rh2 Al INT
2.167 0.337 0.367 0.532
5.664 6.205 6.175 6.489
0.912 6.737 6.716 0.068
3.580 0.010 0.013 0.007
90.333 43.285 43.271 11.096 6.294
90.037 42.585 42.585 10.920 7.458
INT is the electron charge in the interstitial region per formula unit of NpRhAl. Qt(GS) is the total electron charge for the fully self-consistent calculations. Qt(S) is the total electron charge obtained from superposition of neutral atomic densities.
bands is determined mainly by the energy difference between the corresponding atomic levels and the strength of the Np-5f and Rh-4d hybridization. When we compare the converged ground-state density with that of the simple superposition of neutral atomic charge densities, which is the initial charge density in the Kohn–Sham equations, we find a small charge transfer from the interstitial region to both rhodium sites (see Table 1). The DOS value at the Fermi level is N(EF) ¼ 16 states/ eV which corresponds to gband ¼ 13 mJ/mol K2. The experimental specific-heat value is gexp ¼ 200 mJ/mol K2, that leads to an enhancement factor l ¼ 14.4 with l defined by gexp ¼ gband(1+l). This total enhancement is most likely due to the electron–phonon coupling and a rather strong many-body enhancement. This indicates strong correlations within the narrow 5f band of NpRhAl. Including SOC and assuming for simplicity a ferromagnetic alignment of the Np magnetic moments along the c-axis, the spin moments MS(Np) ¼ 2.69mB and the orbital moment ML(Np) ¼ 2.73mB were obtained. The Rh and Al moments are due to 5f(Np)-4d(Rh) and 5f(Np)-3p(Al) hybridization, respectively, and they are small (mo0.15mB). The calculated total Np magnetic moment Mt(Np) ¼ 0.04mB is dramatically smaller than the values 1.06mB and 0.79mB obtained from analysis of the Mo¨ssbauer-spectroscopy data [4]. This discrepancy points to the importance of orbital polarization effects [13]. A very interesting feature of U intermetallic compounds is the giant magnetic anisotropy as one of products of the strong spin–orbit coupling and orbital polarization. Because of the lack of single-crystal studies, very few data exist for Np systems. In order to estimate the magnetic anisotropy at least in the hypothetical ferromagnetic state, we used a well-known property of the DFT, namely that different orientations of the magnetic moments with respect to the crystal axes provide different Kohn–Sham eigenvalues in the presence of the SOC. The magnetocrystalline anisotropy energy (MAE) was calculated as the difference of the total energies corresponding to the moment directions along the a- and the c-axis, respectively. The calculated MAE is 10 meV/f.u. and the c-axis is the
ARTICLE IN PRESS M. Divisˇ / Physica B 371 (2006) 332–336
easy axis. The MAE value of 34 meV/f.u. found for URhAl is larger [9]. We would like to compare the performance of LSDA and GGA with respect to the equilibrium volume of NpRhAl. In Fig. 2a, we show the variation of the total energy with the relative volume V/V0 (V0 is the experimental equilibrium volume). The experimental c/a ratio and symmetry-free structure parameters obtained from minimization of the forces were used and kept constant in the calculations. The LSDA value of the equilibrium volume deviates from the experimental value by about 5.5% (see Fig. 2a). This is a typical deviation usually obtained in LSDA calculations. The corresponding bulk modulus is B ¼ 141 GPa. The GGA (see Fig. 2b), on the other hand, overestimated V0 by only 1%, so the GGA (bulk modulus B ¼ 116 GPa) provides a better equilibrium volume than LSDA. To estimate the effects of orbital polarization inside the correlated Np 5f bands and to improve the values of the magnetic moments, we employed the LSDA+U method. We succeeded in reproducing the magnetic moments of
-0.275
-0.280
E (V)
E - E0 (Ry)
Fit -0.285
-0.290
-0.295
(a)
-0.300 0.84
0.92
0.96
1.00
1.04
M.D. wishes to thank to L. Havela and V. Sechovsky´ for fruitful discussions and for their help with the preparation of manuscript. This work is a part of the research plan MSM0021620834, financed by the Ministry of Education of the Czech Republic. The work of M.D. was also partially supported by Grant Agency of Czech Republic (Grant No. 202/04/1103).
E (V) E - E0 (Ry)
NpRhAl with U ¼ 0.36 eV and J ¼ 0.33 eV (mexp ¼ 1.065mB) and U ¼ 0.26 eV and J ¼ 0.22 eV (mexp ¼ 0.791mB). The corresponding spin and orbital moments are MS(Np) ¼ 2.613mB and ML(Np) ¼ 3.678mB for the larger total moment mexp ¼ 1.065mB and MS(Np) ¼ 2.639mB and ML(Np) ¼ 3.430mB for the smaller total moment mexp ¼ 0.791mB. These values of U and J are in the same range as the values found for UTX compounds, for instance for the isostructural URhAl compound we have found U ¼ 0.5 eV and J ¼ 0.33 eV with total U magnetic moment mexp ¼ 0.94mB [14]. The rather strong many-body enhancement found in specific-heat measurements is a fingerprint of dynamical effects which cannot be captured by the LSDA+U method, which is merely the static limit of the dynamical mean-field theory. In summary, first-principles electronic-structure calculations were performed for the compound NpRhAl for the first time. The Np-5f states were treated as itinerant Bloch states. We studied DOS, hybridization between the Np-5f and Rh-4d states and bonding properties of NpRhAl and have found a small charge transfer from the interstitial region to the Rh atoms. Comparison of the experimental and calculated Sommerfeld coefficient g points to strong mass enhancement effects. Relativistic DFT calculations including SOC strongly underestimate the values of the experimental magnetic moments. The calculated MAE value predicts the c-axis to be the easy axis of magnetization. While the LSDA method underestimates the equlibrium volume of NpRhAl by 5.5%, the GGA method reduces the disagreement to 1% in comparison with experiment. To reproduce the measured magnetic moments, we employed the LSDA+U method and have found moderate electron correlations described by the U and J values between Np-5f electrons at the Np site. Acknowledgments
0.88
-0.45
Fit -0.46
References -0.47
-0.48 0.92 (b)
335
0.96
1.00
1.04
1.08
1.12
V/V0
Fig. 2. Calculated total energy as a function of relative volume V/V0, where V0 denotes the experimental volume. (a) and (b) correspond to LSDA (E0 ¼ 203328 Ry) and GGA (E0 ¼ 203447 Ry) calculations, respectively. The lines correspond to polynomial fits.
[1] V. Sechovsky´, L. Havela, Magnetism of ternary intermetallic compounds of uranium, Handbook of Magnetic Materials, vol. 11, Elsevier Science B.V., Amsterdam, 1998. [2] I. Opahle, P.M. Oppeneer, Phys. Rev. Lett. 90 (2003) 157001. [3] H. Onishi, T. Hotta, New J. Phys. 6 (2005) 193. [4] Y. Kergadallan, Ph.D. Thesis, University of Paris XI, Orsay, 1993. [5] P. Javorsky´, P. Boulet, J. Rebizant, F. Wastin, E. Colineau, Physica B 359–361 (2005) 1018. [6] J.P. Perdew, Y. Wang, Phys. Rev. B 45 (1992) 13244. [7] J.P. Perdew, S. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. [8] P. Blaha, K. Schwarz, G. Madsen, D. Kvasnicka, J. Luitz, WIEN2k, an Augmented Plane Wave+Local Orbitals Program for Calculating
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Crystal Properties, Karlheinz Schwarz, TU Wien, Austria, 2001, ISBN 3-9501031-1-2. [9] J. Kunesˇ , P. Nova´k, M. Divisˇ , P.M. Oppeneer, Phys. Rev. B 63 (2001) 205111. [10] J. Kunesˇ , P. Nova´k, R. Schmid, P. Blaha, K. Schwarz, Phys. Rev. B 64 (2001) 153102.
[11] F. Wastin, J.C. Spirlet, J. Rebizant, J. Alloys Compounds 219 (1995) 232. [12] A.B. Shick, W.E. Pickett, Phys. Rev. Lett. 86 (2001) 300. [13] O. Eriksson, M.S.S. Brooks, B. Johansson, Phys. Rev. B 41 (1990) 7311. [14] J. Rusz, M. Divisˇ , J. Phys.: Condens. Matter 16 (2004) 6675.
Physica B 371 (2006) 332–336 www.elsevier.com/locate/physb
First-principles study of magnetism in NpRhAl M. Divisˇ Department of Electronic Structures, Faculty of Mathematics and Physics, Charles University, Ke Karlovu 5, 121 16 Prague 2, Czech Republic Received 14 October 2005; accepted 15 October 2005
Abstract First-principles calculations based on density-functional theory were performed for the compound NpRhAl for the first time. The electronic density of states, bonding properties and equilibrium volume were studied using relativistic full-potential APW plus local orbitals calculations. The magnetocrystalline anisotropy energy was estimated from total-energy calculations and the c-axis was predicted to be the easy axis of magnetization. Finally, we employed the LSDA+U method to mimic orbital polarization and to obtain the correct total magnetic moments in experimental equilibrium. r 2005 Elsevier B.V. All rights reserved. PACS: 71.15.Mb; 71.20.Lp; 71.27.+a Keywords: Neptunium intermetallics; Magnetic properties; Band-structure calculations
1. Introduction The magnetic properties of actinide compounds are determined by the character of the 5f-electron states ranging from itinerant to almost localized. Systematic studies of large isostructural families of ternary compounds have helped to understand the main tendencies for U compounds [1]. The much more limited set of experimental data for other light actinides, namely Np and Pu, leaves considerably more uncertainty about the general rules and characteristics of magnetism in transuranium systems. From the few known examples of Np compounds forming in the same structure as the U counterparts, the An2T2X (An ¼ U or Np, T ¼ transition metal, X ¼ p metal) compounds can be seen to exhibit a very similar pattern in which the presence of the T and X components leads to suppression of actinide magnetic moments. This makes the clear impression that 5f-ligand hybridization is the principal mechanism determining the magnetic order, and that the Np compounds, in close analogy with their U cousins, have a narrow 5f band at the Fermi level (EF). On the other hand, the higher 5f count in Np leads to the Tel.: +42 0221911367; fax: +42 0224911051.
E-mail address: [email protected]. 0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.10.129
general belief that electron–electron correlations should have a more prominent role and that Np systems tend to be less itinerant than the corresponding U ones. Indeed, the discovery of unconventional superconductivity in PuTGa5 compounds initiated an effort to compare UCoGa5, which is a weak Pauli paramagnet, with the antiferromagnet NpCoGa5 with the heavy-fermion superconductor PuCoGa5 [2]. This dramatic development of physical properties led to conjectures about other type of physics dominating the magnetic properties in the transuranium systems [3]. But such variations can be generally explained also within simple band theory, as the higher f-count can simply move the Fermi level up to a higher density of the 5f states. Equiatomic ternaries of the NpTX type, crystallizing in the ZrNiAl structure type, do not exhibit any exotic ground state. They are mainly interesting as counterparts of UTX compounds, which represent one of the most thoroughly studied groups with a well-mapped boundary between magnetic and non-magnetic ground states. Moreover, magnetic and other electronic properties are well accounted for by the existing theoretical approaches. It is therefore tempting to extend the present calculations successfully for example from URhAl to NpRhAl and to investigate whether a tendency to 5f localization, which can be
ARTICLE IN PRESS M. Divisˇ / Physica B 371 (2006) 332–336
associated with a possible narrowing of the 5f band, appears. The ternary Np compounds with composition NpTX, where T is a transition metal (Fe–Ni and 4d, 5d equivalents) and X is a p element (Al, Ga, Sn) were studied by Kergadallan, who proposed magnetic ordering of Np moments in NpRhAl [4]. The magnetic-ordering temperature (TN ¼ 16.5 K) was established on the basis of specificheat data [5]. 237Np Mo¨ssbauer spectroscopy reveals two different values of the magnetic hyperfine field, with onethird of the Np moments strongly reduced in magnitude [4]. The heavy-fermion character deduced from the Sommerfeld coefficient g ¼ 200 mJ/mol K2 is understood in the context of a low value of the magnetic entropy [5] and Np moments much smaller than expected from, e.g. the 5f4 free-ion value (2.25mB). Using the Dunlap–Lander relation, the magnetic hyperfine fields of 229 and 170 T, give Np moments m ¼ 1.065mB and 0.791mB, respectively. Such a situation implies a complicated antiferromagnetic structure, perhaps frustrated [5], which is markedly different from the simple ferromagnetism in URhAl (TC ¼ 27 K). The distinct tendency to antiferromagnetism in NpTX compounds is generally contrasting with UTX compounds, where the tendency to ferromagnetism dominates at least in the range close to the onset of magnetism [1]. Here, we present results of relativistic band–structure calculations for NpRhAl in the simplified ferromagnetic structure including spin–orbit coupling in the framework of density-functional theory (DFT). In the study of the magnetic moments, we have also gone beyond local-spindensity approximation (LSDA) and employed the static limit of the dynamical mean-field theory, the LSDA+U approximation in particular. 2. Theoretical aspects NpRhAl crystallizes in the hexagonal structure (ZrNiAl type), which contains three formula units per unit cell. The ZrNiAl structure has a layered structure, consisting of planes of Np atoms admixed with one-third of the Rh atoms (Rh1), that are stacked along the c-axis, while two adjacent Np planes are separated from one another by a layer consisting of the remaining Rh (Rh2) and Al atoms. The Np atoms have Rh atoms as nearest neighbors and vice versa, so that both the Np and the Rh atoms are well separated from atoms of the same type. There are two internal parameters for the Np (xNp, 0, 1/2) and the Al (xAl, 0, 0) atomic positions in the NpRhAl structure. To obtain direct information about the ground-state electronic structure and related properties, we applied firstprinciples theoretical methods. The ground-state electronic structure was calculated on the basis of the DFT within the LSDA [6] and the generalized gradient (GGA) approximation [7]. For this purpose, we used the full-potential augmented-plane-wave plus local-orbitals method (APWlo) as implemented in the latest version (WIEN2k) of the
333
original WIEN code [8]. The spin–orbit coupling (SOC) is treated by a second variational step within this implementation [9]. The calculations were performed with the following parameters. The non-overlapping atomic sphere radii of 2.7, 2.3 and 2.1 a.u. (1 a.u. ¼ 52.9177 pm) were taken for Np, Rh, and Al, respectively. The basis for expansion of the valence states (higher than 6 Ry below the Fermi energy) consisted of more than 1000 basis functions (more than 100 APW/atom) plus Np (6s, 6p), Pt (5p), and Al (2p) local orbitals. Relativistic local orbitals for the description of Np 6p1/2 states were included [10]. The Np 5f states were also treated as valence Bloch states and thus Np is characterized by a non-integer occupation number. The Brillouin-zone integrations were performed with the tetrahedron method [8] on a 35–105 special k-points mesh. The unknown values of the internal parameters for Np and Al were obtained by minimizing the forces at experimental values of a ¼ 697.6 pm and c ¼ 402.5 pm [11]. The values xNp ¼ 0.58017291 and the xAl ¼ 0.23685063 were found. We carefully tested the convergence of the results presented with respect to the parameters mentioned and found them to be fully sufficient for all presented characteristics of NpRhAl. We also tried the around-mean-field (AMF) LDA+U method [8], because by construction it is better suited for systems with a lower degree of localization, which should be the case for NpRhAl. The LDA+U potential is implemented in a rotationally invariant way. In these calculations, we varied the parameters U and J, which were used to describe the onsite Coulomb (direct and exchange) interaction among correlated 5f electrons at the Np-site. The effective U was varied in the range from 0.2 to 0.5 eV with several different J values, namely 0.22, 0.33 eV and an atomic value 0.55 eV [12]. Assuming that the LSDA+U method is appropriate for the description of the magnetism of NpRhAl, we heuristically expect that the value of J does not deviate dramatically from its atomic value, and that the value of U should not be lower than the value of J. We are fully aware that on this level such a calculation loses its first-principles character but, on the other hand, we will show that these heuristically derived values allow us to obtain valuable results. As a figure of merit for the calculations we take the saturated magnetic moment, which has been determined by 237Np Mo¨ssbauer spectroscopy [4]. We also tested to which extent the results of the LSDA+U calculations depend on the starting density matrix. Experience has shown that the converged self-consistent electronic structure does not depend on the starting density matrices in the cases where U and J are smaller than the bandwidth of 5f states. In the treated compound NpRhAl, this bandwidth exceeds 3 eV, which is well above the upper limit of the interval of U parameters used (0.5 eV). 3. Results and discussion The total density of electronic states (DOS) and the siteprojected DOS from LSDA calculations including SOC at
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334
experimental equilibrium are shown in Fig. 1. The lowest band, from about 8 to 5 eV, originates from Al 3s states. There is a pseudo-gap around 5 eV. The Rh 4d states form the main contribution to the occupied energy range in the energy range 5 to 2 eV (‘‘4d band’’) but they show an admixture of the Np 7s, 6d states, Rh 5s states, and Al 3s, 3p states. The highest occupied bands (between 2 eV and the Fermi level) originate mainly from the hybridized Np 5f states and Rh 4d states but all remaining (Np 7s, 6d states; Al 3p states) are also present. Finally, we see that the Fermi level is situated inside the Np 5f band with a rather large DOS. The total occupied bandwidth of Np 5f states is roughly 1.5 eV. The important results concerning the Np-f and Rh-d bonding in NpRhAl are contained in Table 1, which shows the charge distribution over the atomic spheres and its orbital decomposition. Although the Rh 4d band is filled with the Fermi energy lying in the region of the Np 5f band, the Rh-4d and Np-5f occupation numbers are 6.737 (Rh1), 6.716 (Rh2), and 3.580 (Np) electrons, respectively. This is a consequence of the hybridization between the Rh4d and Np-5f states. The region of the rather low DOS (2 eVoEo1.5 eV) between the Np 5f and the Rh 4d
20
(a) total
15 10 5 0
(b) 12
Np
DOS (states/ev)
8 4 0 0.8
(c)
Al
0.6 0.4 0.2 0.0
(d)
EF
Rh2 4 Rh1 2 0 -10
-8
-6
-4
-2 E (eV)
0
2
4
Fig. 1. Calculated DOS of NpRhAl: (a) the total DOS and (b,c,d) the Np-, Al-, Rh1-, Rh2 atom-projected DOS, respectively.
Table 1 Angular decomposition of the APW valence-charge density in the atomic spheres (electron charge) for NpRhAl NpRhAl
Qs
Qp
Qd
Qf
Qt(GS)
Qt(S)
Np Rh1 Rh2 Al INT
2.167 0.337 0.367 0.532
5.664 6.205 6.175 6.489
0.912 6.737 6.716 0.068
3.580 0.010 0.013 0.007
90.333 43.285 43.271 11.096 6.294
90.037 42.585 42.585 10.920 7.458
INT is the electron charge in the interstitial region per formula unit of NpRhAl. Qt(GS) is the total electron charge for the fully self-consistent calculations. Qt(S) is the total electron charge obtained from superposition of neutral atomic densities.
bands is determined mainly by the energy difference between the corresponding atomic levels and the strength of the Np-5f and Rh-4d hybridization. When we compare the converged ground-state density with that of the simple superposition of neutral atomic charge densities, which is the initial charge density in the Kohn–Sham equations, we find a small charge transfer from the interstitial region to both rhodium sites (see Table 1). The DOS value at the Fermi level is N(EF) ¼ 16 states/ eV which corresponds to gband ¼ 13 mJ/mol K2. The experimental specific-heat value is gexp ¼ 200 mJ/mol K2, that leads to an enhancement factor l ¼ 14.4 with l defined by gexp ¼ gband(1+l). This total enhancement is most likely due to the electron–phonon coupling and a rather strong many-body enhancement. This indicates strong correlations within the narrow 5f band of NpRhAl. Including SOC and assuming for simplicity a ferromagnetic alignment of the Np magnetic moments along the c-axis, the spin moments MS(Np) ¼ 2.69mB and the orbital moment ML(Np) ¼ 2.73mB were obtained. The Rh and Al moments are due to 5f(Np)-4d(Rh) and 5f(Np)-3p(Al) hybridization, respectively, and they are small (mo0.15mB). The calculated total Np magnetic moment Mt(Np) ¼ 0.04mB is dramatically smaller than the values 1.06mB and 0.79mB obtained from analysis of the Mo¨ssbauer-spectroscopy data [4]. This discrepancy points to the importance of orbital polarization effects [13]. A very interesting feature of U intermetallic compounds is the giant magnetic anisotropy as one of products of the strong spin–orbit coupling and orbital polarization. Because of the lack of single-crystal studies, very few data exist for Np systems. In order to estimate the magnetic anisotropy at least in the hypothetical ferromagnetic state, we used a well-known property of the DFT, namely that different orientations of the magnetic moments with respect to the crystal axes provide different Kohn–Sham eigenvalues in the presence of the SOC. The magnetocrystalline anisotropy energy (MAE) was calculated as the difference of the total energies corresponding to the moment directions along the a- and the c-axis, respectively. The calculated MAE is 10 meV/f.u. and the c-axis is the
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easy axis. The MAE value of 34 meV/f.u. found for URhAl is larger [9]. We would like to compare the performance of LSDA and GGA with respect to the equilibrium volume of NpRhAl. In Fig. 2a, we show the variation of the total energy with the relative volume V/V0 (V0 is the experimental equilibrium volume). The experimental c/a ratio and symmetry-free structure parameters obtained from minimization of the forces were used and kept constant in the calculations. The LSDA value of the equilibrium volume deviates from the experimental value by about 5.5% (see Fig. 2a). This is a typical deviation usually obtained in LSDA calculations. The corresponding bulk modulus is B ¼ 141 GPa. The GGA (see Fig. 2b), on the other hand, overestimated V0 by only 1%, so the GGA (bulk modulus B ¼ 116 GPa) provides a better equilibrium volume than LSDA. To estimate the effects of orbital polarization inside the correlated Np 5f bands and to improve the values of the magnetic moments, we employed the LSDA+U method. We succeeded in reproducing the magnetic moments of
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E (V)
E - E0 (Ry)
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(a)
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1.04
M.D. wishes to thank to L. Havela and V. Sechovsky´ for fruitful discussions and for their help with the preparation of manuscript. This work is a part of the research plan MSM0021620834, financed by the Ministry of Education of the Czech Republic. The work of M.D. was also partially supported by Grant Agency of Czech Republic (Grant No. 202/04/1103).
E (V) E - E0 (Ry)
NpRhAl with U ¼ 0.36 eV and J ¼ 0.33 eV (mexp ¼ 1.065mB) and U ¼ 0.26 eV and J ¼ 0.22 eV (mexp ¼ 0.791mB). The corresponding spin and orbital moments are MS(Np) ¼ 2.613mB and ML(Np) ¼ 3.678mB for the larger total moment mexp ¼ 1.065mB and MS(Np) ¼ 2.639mB and ML(Np) ¼ 3.430mB for the smaller total moment mexp ¼ 0.791mB. These values of U and J are in the same range as the values found for UTX compounds, for instance for the isostructural URhAl compound we have found U ¼ 0.5 eV and J ¼ 0.33 eV with total U magnetic moment mexp ¼ 0.94mB [14]. The rather strong many-body enhancement found in specific-heat measurements is a fingerprint of dynamical effects which cannot be captured by the LSDA+U method, which is merely the static limit of the dynamical mean-field theory. In summary, first-principles electronic-structure calculations were performed for the compound NpRhAl for the first time. The Np-5f states were treated as itinerant Bloch states. We studied DOS, hybridization between the Np-5f and Rh-4d states and bonding properties of NpRhAl and have found a small charge transfer from the interstitial region to the Rh atoms. Comparison of the experimental and calculated Sommerfeld coefficient g points to strong mass enhancement effects. Relativistic DFT calculations including SOC strongly underestimate the values of the experimental magnetic moments. The calculated MAE value predicts the c-axis to be the easy axis of magnetization. While the LSDA method underestimates the equlibrium volume of NpRhAl by 5.5%, the GGA method reduces the disagreement to 1% in comparison with experiment. To reproduce the measured magnetic moments, we employed the LSDA+U method and have found moderate electron correlations described by the U and J values between Np-5f electrons at the Np site. Acknowledgments
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335
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Fig. 2. Calculated total energy as a function of relative volume V/V0, where V0 denotes the experimental volume. (a) and (b) correspond to LSDA (E0 ¼ 203328 Ry) and GGA (E0 ¼ 203447 Ry) calculations, respectively. The lines correspond to polynomial fits.
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