Crystal field calculations for alloys

ARTICLE IN PRESS Physica B 404 (2009) 2091–2093

Contents lists available at ScienceDirect

Physica B journal homepage: www.elsevier.com/locate/physb

Crystal field calculations for alloys M. Divisˇ Department of Condensed Matter Physics, Faculty of Mathematics and Physics, Charles University, Ke Karlovu 5, 121 16 Prague 2, Czech Republic

a r t i c l e in fo

abstract

Article history: Received 31 March 2009 Accepted 31 March 2009

A new method for calculations of crystal field (CF) interaction in alloys containing rare earths (4f electron CF) has been developed. The method is based on density-functional theory (DFT) and uses the full-potential local-orbitals (FPLO) method combined with the coherent-potential approximation (CPA). The method is successfully applied to calculate the CF interaction in RENi4Al alloys. & 2009 Elsevier B.V. All rights reserved.

PACS: 71.15.Mb 71.20.Eh 71.70.Ch Keywords: Electronic-structure calculations Crystal field Rare-earth alloys

1. Introduction For description of the properties of compounds and alloys with strongly correlated localized electrons, the effective-Hamiltonian approach is still often used [1]. The effective Hamiltonian contains a number of parameters which must be provided as an input. Although the values of most of the parameters are known with sufficient accuracy, this is not true, however, for the parameters describing the crystal field (CF) in rare-earth (RE) compounds. Usually, they are determined by fitting experimental data and in some cases it is even difficult to estimate a reasonable interval in which the CF parameters should lie. The calculation of CF parameters is therefore desirable and several papers addressing this problem have appeared recently [2–8]. As the 4f states are well localized, the total (electrostatic plus exchange correlation) potential usually dominates and a reasonable agreement with the experiment was achieved for e.g. RECo5 compounds [3,4,6]. On the other hand, the calculation of the CF interaction for alloys has not been performed up to now to our best knowledge. There only exists calculation of rare-earth impurities in noble metals but this was not a full-potential calculation without semicore RE-5p states and the supercell technique was used [9]. This is certainly a drawback since some RE alloys have large application potential and knowledge of the CF interaction is very important for such systems. To fill this gap, we propose a firstprinciples method, based on the coherent-potential approximation (CPA), which is able to treat effectively and accurately the

E-mail address: [email protected] 0921-4526/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2009.03.045

effects of atom disorder in alloys. To demonstrate the applicability of the new method, we have calculated the CF parameters for RENi4Al alloys. The microscopic CF Hamiltonian was diagonalized and the resulting CF energies are compared with experimental results available [10].

2. Method of calculations Our first-principles calculations of the electronic structure and the CF interaction are based on the density-functional theory (DFT). We use the local-density approximation [11] and computercode full-potential local orbitals (FPLO) [12], which contains the CPA for alloys [13]. The crystal structure of RENi4Al alloys is hexagonal of the CaCu5 type. The RE atoms (RE ¼ Pr, Nd, Dy) occupy the 1a ¼ (0,0,0) site and Ni the two crystallographic sites 2c ¼ (1/3,2/3,0) and 3g ¼ (1/2,0,1/2). Substituted Al atoms randomly occupy the 3g position [10]. The RENi4Al system can thus be classified as a partially random alloy suitable for application of the CPA method. The first principles FPLO-CPA calculations were performed in the scalar relativistic mode to obtain the normal paramagnetic ground state. We have used experimental a- and c-axis data [10]. In the FPLO calculations, the RE (6s6p5d), Ni (4s4p3d) and Al (3s3p3d) states were treated as valence Bloch states. To improve the accuracy of DFT calculations, the fully occupied RE (5s5p4d), Ni (3s3p) and Al (2s2p) states were treated as semicore Bloch states. The localized RE3+ 4f states have negligible hybridization with the valence states [10], so that they were treated in the open-core approximation using integer electron occupations n ¼ 2 (Pr), 3

ARTICLE IN PRESS M. Divisˇ / Physica B 404 (2009) 2091–2093

(Nd) and 9 (Dy). The open-core treatment removes the selfinteraction for aspherical components of the total potential which is very important for our CF calculations. The k-mesh contained 133 k-points in the irreducible wedge of the Brillouin zone (BZ) which corresponds to 1728 k-points in the full BZ. To check the accuracy the additional calculations with 793 irreducible k-points (13824 in the full BZ) were performed. 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 the RENi4Al alloys. The interaction with the CF produced by the neighboring core charges and the anisotropic valence electronic charge density is the strongest perturbation of the localized atomic-like 4f states of the trivalent RE ions in RENi4Al. In the hexagonal symmetry, the corresponding CF Hamiltonian can be written as

20 EF

15 DOS (states/eV)

2092

a

DyNi4Al

10 5 0

b

2 Dy

1

^ CF ¼ a  A0 hr 2 i  O0 þ b  A0 hr 4 iO0 þ g  ðA0 hr 6 iO0 H 2 4 6 2 4 6 þ A66 hr 6 iO66 Þ,

0

where OM L are the Stevens ‘‘operator equivalents’’ and a, b, g the L reduced matrix elements [14]. To calculate AM L hr i, we use a firstprinciples method based on DFT. Within this method, the electronic structure and the corresponding distribution of the ground-state charge density is obtained using the FPLO code with L the CPA solver. The CF parameters AM L hr i originating from the aspherical part of the total single-particle DFT potential in the crystal, can be obtained from

10

L AM L hr i ¼ aLM 

Z

RC 0

2 jR4f ðrÞj2 V M L ðrÞr dr;

(2)

where V M L ðrÞ are the components of the total (Coulomb and exchange-correlation) potential. The term V M L ðrÞ is readily obtained with the FPLO-CPA code. The integration is done from zero to an upper radial limit Rc beyond which the 4f-charge density can be neglected. The R4f ðrÞ describes the radial shape of the localized 4f-charge density of the RE atom. In the present study, we have used a self-interaction corrected (SIC) wave function from atomic calculations. This approach (see Ref. [3]) was found to provide a 4f-charge density being very close to that obtained from more rigorous SIC-DFT band calculations [5]. The conversion factors aLM establish the relation between the symmetrized spherical harmonics used within FPLO and the real tesseral harmonics which transform in the same way as the tensor operators OM L [14].

PDOS (states/eV; atom)

(1)

c Ni(3g), Ni(2c)

5

0 2

d

Al

1

0 -10

-8

-6

-4

-2

0

2

4

E (eV) Fig. 1. Total density of electronic states (DOS) as a function of energy (a) for DyNi4Al. Projected DOS (PDOS) of Dy (b), Ni(3 g) (c, full line), Ni(2c) (c, dashed line) and Al (d). The Fermi level is set at zero energy.

Table 1 CF parameters (in kelvin) of RENi4Al (RE ¼ Pr, Nd, Dy), calculated by the FPLO-CPA methodology.

3. Results and discussion

RENi4Al

A02 hr 2 i(K)

A04 hr 4 i(K)

A06 hr6 i(K)

A66 hr 6 i(K)

The calculated electronic density of states (DOS) of DyNi4Al is shown in Fig. 1a. The occupied part of the DOS has a width of 8.9 eV. The first region, from 8.9 to 4.2 eV, consists mainly of free-electron-like states Pr-6s, Ni-4s, Al-3s and Al-3p (see Fig. 1(b)–(d)). The states from 4.2 eV to the Fermi level are mainly Ni-3d states, hybridized with Dy-5d and Al-3p states. The unoccupied states above the Fermi level have mixed Dy-5d, Ni-3d and Al-3p character. The DOS curves for RE ¼ Pr, Nd are very similar and therefore not shown. The Fermi level of DyNi4Al is situated at a local minimum of the DOS, N(EF) ¼ 3.57 states eV1 f. u.1. Orbital analysis of the DOS shows that mainly Ni-3d, Dy-5d and Al-3p contribute to the total DOS at EF. The value of the DOS at EF is too small to cause spontaneous magnetic polarization of the Ni-3d states. The above value of the DOS of DyNi4Al at EF corresponds to an electronicspecific-heat coefficient g ¼ 8.4 mJ mol1 K1, which is lower than the g value of 16 mJ mol1 K1 derived from specific-heat data [10]. This points to a medium value of the mass-enhancement

Pr Nd Dy

553 549 656

156 147 55

34 25 8.5

813 515 142

coefficient l ¼ 0.9 for DyNi4Al (gexp ¼ gband (1+l)), indicating a medium electron-phonon interaction in RENi4Al alloys. The calculated CF parameters for RENi4Al alloys are summarized in Table 1. The CF parameters have negligible dependence on the number of k-points (1728 or 13 824 in the full BZ). We also tested the dependence on 4f core wave function confinement and only the second-order CF parameters were influenced less than five percent. The CF parameters were used to diagonalize the CF Hamiltonian (1). The CF eigenenergies are plotted in Fig. 2. A ground-state singlet and an energy gap larger than 150 K were found for PrNi4Al. The exchange energy is not sufficient to induce magnetic ordering which is in the nice agreement with the absence of anomaly in the specific heat corresponding to

ARTICLE IN PRESS M. Divisˇ / Physica B 404 (2009) 2091–2093

RNi4 Al, CF energy levels 800 2

ECF (K)

600 2

400

200

2

0

Pr

Nd

Dy

Fig. 2. CF energy levels of RENi4Al (RE ¼ Pr, Nd, Dy) in kelvin. The J ¼ 4 groundstate multiplet of Pr is split into three singlets and three doublets (marked by ‘‘2’’). The J ¼ 9/2 and J ¼ 15/2 ground-state multiplets of Nd and Dy are split into five and eight doublets, respectively.

magnetic ordering [10]. The pronounced anomalies found in the NdNi4Al and DyNi4Al specific heat indicate magnetic-ordering temperatures of 5.7 and 13.5 K, respectively [10]. The total CF splittings of the ground-state J ¼ 9/2 (Nd) and J ¼ 15/2 (Dy) multiplets, as calculated with our FPLO-CPA method, are more than 343 K (Nd) and 700 K (Dy), respectively. Therefore, the CF interaction must be much larger than the exchange interaction in RENi4Al. This is in agreement with the fact that substantial Schottky contributions are observed in the specific heat of RENi4Al in the paramagnetic region above the ordering temperatures for RE ¼ Nd and Dy [10]. From the analysis of the Schottky contribution, it follows that the first excited state is at 30 and 23 K for NdNi4Al and DyNi4Al, respectively [10]. Results of our first-principles calculations provide 15 and 42 K, respectively, showing semiquantitative agreement between the results of our calculations and the experimental data available [10]. The phonon contribution dominates at temperatures above 50 K so that the energies of other CF states are rather uncertain. Moreover, the experience with first-principles CF calculations is that usually only semiquantitative agreement with experimental data is obtained [7]. The effect of alloying Al is mainly pronounced for

2093

A04 hr 4 iCF parameter since it changes the sign from negative for RENi5 [3] compounds to positive for RENi4Al alloys. In summary, first-principles calculations of the electronic structure of partially disordered RENi4Al (RE ¼ Pr, Nd, Dy) alloys have been performed. The crystal-field interaction for 4f electrons in RENi4Al was calculated for the first time from first principles, using the FPLO-CPA methodology. We have found semiquantitative agreement between theory and experiment. For further refinement of the CF parameters, single-crystal data (susceptibility, inelastic neutron scattering) would be desirable. We like to emphasize that the our new method of CF calculations, based on FPLO-CPA methodology, has general applicability for more complex alloys.

Acknowledgment This work is a part of the research program MSM 0021620834 financed by the Ministry of Education of the Czech Republic. The work of M.D. was also supported by the Grant Agency of Czech Republic (Project 202/09/1027). M.D. wishes to thank Manuel Richter and Klaus Koepernik for the help with the FPLO code (open-core treatment) and to Vladimı´r Sechovsky´ for critical discussion of the manuscript.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

F. de Groot, Coord. Chem. Rev. 249 (2005) 31. M. Divisˇ, J. Kuriplach, Physica B 205 (1995) 353. P. Nova´k, Phys. Stat. Sol. (b) 198 (1996) 729. K. Hummler, M. Fa¨hnle, Phys. Rev. B 53 (1996) 3272. M. Richter, J. Phys. D Appl. Phys. 31 (1998) 1017. M.D. Kuzmin, L. Steinbeck, M. Richter, Phys. Rev. B 65 (2002) 064409. M. Divisˇ, J. Rusz, H. Michor, G. Hilscher, P. Blaha, K. Schwarz, J. Alloys Comp. 403 (2005) 29. P. Nova´k, M. Divisˇ, Phys. Stat. Sol. (b) 244 (2007) 3168. L. Steinbeck, M. Richter, H. Eschrig, U. Nitzsche, Phys. Rev. B 49 (1994) 16289. P. Svoboda, J. Vejpravova´, S. Danisˇ, M. Mihalik, Physica B 378–380 (2006) 1107. J.P. Perdew, Y. Wang, Phys. Rev. B 45 (1992) 13244. K. Koepernik, H. Eschrig, Phys. Rev. B 59 (1999) 1743 /http://www.fplo.de/S. K. Koepernik, B. Velicky´, R. Hayn, H. Eschrig, Phys. Rev. B 55 (1997) 5717. M.T. Hutchings, Sol. State Phys. 16 (1964) 227.