- Email: [email protected]
Exchange interactions and crystal-field effects in HoX (X=Ag , Cd, Cu, Mg, Rh, Zn) intermetallic compounds
ARTICLE IN PRESS
Physica B 381 (2006) 265–270 www.elsevier.com/locate/physb
Exchange interactions and crystal-field effects in HoX (X ¼ Ag, Cd, Cu, Mg, Rh, Zn) intermetallic compounds Ja´n Rusz, Ilja Turek, Martin Divisˇ Faculty of Mathematics and Physics, Department of Electronic Structures, Charles University, Ke Karlovu 5, 12116 Prague 2, Czech Republic Received 6 January 2006; accepted 12 January 2006
Abstract We present an ab initio treatment of the exchange interactions and crystal-field effects in HoX (X ¼ Ag, Cd, Cu, Mg, Rh, Zn) intermetallic compounds with the CsCl crystal structure. The exchange interactions are determined using the magnetic-force theorem starting from both a ferromagnetically ordered state (zero temperature) and a disordered-local-moments (effectively high temperature) state up to an interatomic distance of 10a, where a is the lattice parameter. The exchange interactions are used for construction of the Heisenberg Hamiltonian, from which we determine the magnetic-transition temperatures using the mean-field approximation and the Green’s-function based random-phase approximation. The crystal-field parameters are derived from an ab initio electronic structure of the non-magnetic state. A combined Hamiltonian is treated within the mean-field approximation to study the effect of the crystal-field interaction on the critical temperatures and the magnetization. r 2006 Elsevier B.V. All rights reserved. PACS: 75.10.Hk; 71.70.Gm; 75.10.Jm Keywords: Critical temperature; Exchange interactions; Crystal field; Heisenberg model; HoX compounds
1. Introduction The first principles calculation of non-zero magnetic properties of solids is an active field of current solid-statetheory research. There are several approaches to this problem with different degrees of accuracy and demands on computational power. A large part of recent calculations is based on a combined approach, which makes use of the density functional theory (DFT, [1]) as its base. A suitable model Hamiltonian is then constructed and its parameters are calculated ab initio within the (zero-temperature) DFT. The non-zero-temperature properties are studied within the model Hamiltonian using methods of statistical physics. For spherically symmetric magnetic ions, like Gd or 3d transition elements (where the orbital momentum is frozen), it was found that low-energy magnetic excitations Corresponding author.
E-mail address: [email protected] (J. Rusz). 0921-4526/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2006.01.509
(magnons) are well described by an effective Heisenberg model [2]. This approach has been applied to many systems involving transition metals [3] or to rare-earth-based systems [4]. Recently, we employed this method to evaluate propagation vectors and critical temperatures in GdX systems (X is a non-magnetic element) [5,6] obtaining fair agreement with experiment, but the situation becomes more complicated when the magnetic ion is not spherically symmetric. It is necessary then to take into account its interaction with the crystal field (CF). Therefore, we decided to study Ho-based intermetallics with the CsCl structure, to understand to what extent the CF does influence the critical temperatures and the magnetization. This work has the following structure: in Section 2 we describe the applied methods of calculations—electronicstructure calculations and evaluation of the exchange interactions and CF parameters. Section 3 describes the calculation of the magnetization and the critical temperatures based on constructed model Hamiltonians and Section 4 contains a discussion of obtained results.
ARTICLE IN PRESS 266
J. Rusz et al. / Physica B 381 (2006) 265–270
2. Calculation of the parameters of model Hamiltonians
ing Stevens operators O^ m l [12]
2.1. Exchange interactions
H^ CF ¼ bA04 hr4 iðO^ 04 þ 5O^ 44 Þ þ gA0 hr6 iðO^ 0 21O^ 4 Þ,
The exchange interactions are determined using the magnetic force theorem [2,7], which is based on infinitesimal spin rotations on particular atomic sites starting from some collinear magnetic structure—‘a reference state’. Because of the strong spin–orbit interaction in HoX systems, a feasible assumption is that the Ho orbital momentum follows the rotation of the spin momentum. Therefore, the obtained exchange interactions describe the dynamics of the total angular momenta, not just the spin part. The underlying electronic structure was calculated by the tight binding linearized muffin-tin orbitals method (TB-LMTO) within the atomic spheres approximation (ASA) [8]. The radii of the atomic spheres in the ASA are proportional to the atomic radii of the respective elements. The holmium 4f states were treated within the open-core approach [9] and the spd basis was used for the valence states. The lattice parameters were taken from experiment [10]. The mapping of the exchange interactions was done in two reference states: (i) the ferromagnetic state, and (ii) the disordered-local-moments state (DLM). DLM calculation is an application of the coherent potential approximation (CPA) [8]. It corresponds to a hypothetical high-temperature state where magnetic moments are completely randomly oriented without any short-range order. Within this method, magnetic moments of the X-element completely vanish, so that only Ho–Ho interactions are to be considered. For further details about the applied method of calculation of the exchange parameters see Refs. [3,4].
2.2. Crystal-field parameters Calculation of CF parameters requires a full-potential method because the non-spherical components of the potential around the particular atom are necessary. Therefore, we repeated the calculation of the electronic structure using the WIEN2k code [11], which implements the full-potential linearized-augmented-plane-waves (FP LAPW) method of solution of the Kohn–Sham equations. Band structures calculated within both methods agreed very well. Within the WIEN2k, we used up to 150 basis functions per atom. Brillouin-zone (BZ) integration was performed by an improved tetrahedron method using 20 000 k-points in the full BZ. The holmium 4f states were treated within the open-core approach. The CsCl structure has a simple cubic Bravais cell and both the Ho and the X sites have full cubic symmetry. Therefore, the crystal-field Hamiltonian is described by only two parameters A04 hr4 i and A06 hr6 i and the correspond-
6
6
ð1Þ
6
where b and g are the reduced matrix elements. In the paramagnetic state the CF parameters are determined by the following integral [13]: m l Am l hr i ¼ C l
Z
1 0
2 R24f ðrÞV m l ðrÞr dr,
(2)
where V m l ðrÞ are the l; m components of the holmium-site potential, R4f ðrÞ is the radial part of the wave-function of the localized 4f states. The constants C m l appear due to different conventions in the definition of the Stevens’ operators and the spherical harmonics used in the WIEN2k code. The radial 4f wavefunction of the Ho atom was calculated using the self-interaction-corrected optimizedlinear-combination-of-atomic-orbitals (OLCAO) method [14,15]. The integration in Eq. (2) up to the muffin-tin radius RMT is quite straightforward. However, for the parameters of lower order this is usually not sufficient, because the non-spherical components of the potential V m l ðrÞ grow quickly with distance from the core. To enlarge the integration domain beyond RMT , the plane-wave representation of the potential in the interstitial region is expanded into spherical harmonics using the Rayleigh formula, as described in Ref. [16]. The resulting CF parameters are summarized in Table 1 together with the known experimental values [17], which are based on the results of a direct spectroscopic method (inelastic neutron scattering). Measurements of the CF parameters of HoCd are, to the best of our knowledge, not available, but one can expect them to be similar to ones for TmCd [18]. We performed numerical tests by enlarging l max in the calculation of non-spherical contributions to the Hamiltonian matrix and found our results to be well converged within the precision shown in Table 1.
Table 1 Calculated and available experimental values [17] of CF parameters of HoX (X ¼ Ag, Cd, Cu, Mg, Rh, Zn) compounds (in kelvins) Compound
HoAg HoCd HoCu HoMg HoRh HoZn
Theory
Experiment
A04 hr4 i
A06 hr6 i
A04 hr4 i
A06 hr6 i
52 20 55 þ11 35 20
3.1 3.1 3.7 2.1 8.3 3.2
67 7 32% 68 8 þ42 9 125 19 14 4
12 1 15% 15 1 13 1 18 3 18 1
The stars denote experimental CF parameters of TmCd [18].
ARTICLE IN PRESS J. Rusz et al. / Physica B 381 (2006) 265–270
3. Treatment of model Hamiltonians 3.1. Heisenberg model We write the Heisenberg Hamiltonian in the following form: X ^A ^B H^ ¼ J AB (3) ij J i J j , ij;AB
where i; j are indices labeling the unit cells and A; B are indices of the sublattice (basis indices). J^ A i are operators of the total angular momentum in unit cell i at sublattice A. The exchange interactions are denoted by J AB ij . We tried to extract from the Heisenberg model the propagation vector of the magnetic structure and the corresponding magnetic-transition temperature T C . The propagation vector Q corresponds to the latticeFourier-transformed matrix of the exchange interactions JðQÞ with the maximal eigenvalue. The elements of JðQÞ are given by X AB J AB ðQÞ ¼ eiQR0i J AB (4) 0i , i
RAB 0i
where is a vector connecting the sites ð0; AÞ and ði; BÞ. We scanned the irreducible wedge of the first BZ on a mesh corresponding to 107 points in the whole first BZ. The resulting Q vectors are summarized in Table 2. Within the mean-field approximation (MFA) the corresponding critical temperatures are given by maximal eigenvalue (T MFA ) of the generalized eigenvalue problem C X
J AB ðQÞhJ^ zB i ¼
B
3kB T MFA C hJ^ z i, 2J A ðJ A þ 1Þ A
(5)
where kB is the Boltzmann constant, J A the total angular momentum of the atoms at sublattice A and hJ^ zA i the mean value of J^ zA . The Ho ground-state multiplet (GSM) has total angular momentum J Ho ¼ 8. The second lowest energy multiplet of the Ho ion (with J Ho ¼ 7) is due to strong spin–orbital interaction roughly 7000 K higher in energy [19]. This energy difference is much larger than the exchange and CF splitting of the GSM and around T C this excited level is practically unoccupied. This allows us to restrict our
267
treatment to the GSM only. The spin polarization of valence electrons in the Ho sphere contributes to the total Ho spin moment by less than 10% and can safely be neglected. The small X-sublattice magnetic moments are of itinerant nature and, therefore, we treated them as classical moments—i.e. unit vectors of size one parallel with onsite magnetization. Such combination of ‘classical’ and ‘quantum’ magnetic moments is discussed in Section 2 of Ref. [5]. To go beyond the simple MFA we employed also the Green’s-function-based technique—random-phase approximation (RPA). The derivation of the multiple sublattice RPA formalism is presented in detail in Ref. [5]. Here, we only show key equations for collinear magnetic structures, which are solved by iterative methods: Z 1 1 2J A ðJ A þ 1Þ 1 dq N ðqÞ , ð6Þ hJ^ zA i ¼ AA O 3kB T RPA C N AB ðqÞ ¼ dAB
X
J AC ð0ÞhJ^ zC i hJ^ zA iJ AB ðqÞ,
ð7Þ
C
where O is the volume of the unit cell. MFA Generally, for collinear magnetic structures T RPA C oT C [5]. It is known that the MFA overestimates T C because neglecting spin fluctuations makes the system more rigid. On the other hand, for purely ferromagnetic exchange interactions it has been shown that the RPA underestimates T C [20]. From our studies of Heusler-alloy systems using several approaches to calculate exchange interactions [21], MFA we observed that T RPA holds also for more C oT C oT C general cases (by T C we mean the exact critical temperature given by the model, which can be calculated by Monte Carlo techniques). However, there is no general proof for this statement yet. Above-mentioned RPA formulas hold only for collinear magnetic structures. Therefore, they are not applicable to treatment of magnetic structures with general propagation vectors. In this case (HoCu within the FM reference state), the multi-sublattice RPA value of T C is missing in Table 2. In the DLM reference state, the magnetic moments of the X sublattice vanish, so that the system is effectively singlesublattice. In such a case, we can employ the approach
Table 2 Propagation vectors Q and critical temperatures T C (in kelvins) of studied HoX compounds with the CsCl structure calculated using MFA and RPA within two different reference states for mapping the exchange interactions: a ferromagnetic (FM) state and a disordered-local-moments (DLM) state. Experimental values are taken from Ref. [10] System
HoAg HoCd HoCu HoMg HoRh HoZn
DLM
FM
Experiment
Q
T MFA C
T RPA C
Q
T MFA C
T RPA C
Q
TC
ð0; 1=2; 1=2Þ ð0; 0; 0Þ ð0; 0:45; 1=2Þ ð0; 0; 0Þ ð0:11; 0:12; 0:15Þ ð0; 0; 0Þ
19.8 45.3 13.5 44.9 4.3 88.9
16 39 9.3 43 1.5 82
ð0; 1=2; 1=2Þ ð0; 0; 0Þ ð0; 0:45; 1=2Þ ð0; 0; 0Þ ð0; 0; 0Þ ð0; 0; 0Þ
19.9 57.2 13.5 61.5 10.2 130
14 48 — 57 6.7 120
ð0; 1=2; 1=2Þ ð0; 0; 0Þ ð0; 1=2; 1=2Þ ð0; 0; 0Þ AFM ð0; 0; 0Þ
32–33 19–30 28 21–25 3.2 75–80
ARTICLE IN PRESS J. Rusz et al. / Physica B 381 (2006) 265–270
268
described in Ref. [4] for calculation of T RPA of the spinC spiral magnetic structure.
80 60
3.2. Inclusion of crystal-field effects
AB
For calculation of magnetization curves it is necessary to determine the easy axis of the magnetization. In a cubic CF there are three possible orientations—(0 0 1), (0 1 1) or (1 1 1). For particular directions, some terms in H^ exch will vanish. Note, that for non-collinear magnetic structures this Hamiltonian would contain unit-cell-dependent terms and, consequently, would introduce a complication into the Hamiltonian, because the local effective magnetic field will have a different orientation with respect to the crystallographic axes in different unit cells. Therefore, we apply this formalism only for collinear magnetic structures. The combined Hamiltonian can be diagonalized exactly in the jJJ z i basis, leading to the splitting of the 4f GSM. However, below the critical temperature, evaluation of hJ^ aA i requires knowledge of the eigenfunctions. Therefore, selfconsistent techniques are applied. Above T C , all hJ^ aA i are identically zero, so that there is no dependence of the Hamiltonian on its eigenvectors. The calculated CF splitting in the paramagnetic state is shown in Fig. 1. For direct evaluation of the T C , one can use a simpler approach. Because hJ^ a i ! 0 as T ! T C , the exchange interaction can be treated as a small perturbation to the unperturbed CF Hamiltonian. The derivation of T C goes along same lines as the MFA for the Heisenberg model (see e.g. Ref. [23]). Different is the treatment of the Ho
CF splitting (K)
In a crystal, the Ho ions are non-spherical, so that the (spherically symmetric) Heisenberg model cannot give a complete picture of the magnetic properties of the Ho sublattice. The non-sphericity of the Ho ions leads to a splitting of the 4f energy level due to the CF interaction. The combination of the CF Hamiltonian with the Heisenberg model gives a more appropriate description. The treatment of this combined model Hamiltonian is much more complex than that of pure Heisenberg model. There are several publications in which a simple axial CF together with exchange interactions is treated within the RPA for some particular (rather low) total angular momenta [22]. But such formalism has not yet been developed for a fully cubic (or lower symmetry) CF Hamiltonian and this task goes beyond the scope of present paper. Within the MFA, the exchange interactions enter the calculation only as an effective magnetic field acting on the Ho sublattice. An additional Zeeman-like term is added to the CF Hamiltonian (for collinear magnetic structures only; note that in CsCl structure thanks to inversion symmetry all elements of JðQÞ are real, so that J AB ðQÞ ¼ J BA ðQÞ) X H^ exch ¼ 2 J AB ðQÞðhJ^ xB iJ^ xA þ hJ^ yB iJ^ yA þ hJ^ zB iJ^ zA Þ. (8)
40 20 0 -20 -40 -60 -80
HoAg
HoCd
HoCu
HoMg
HoRh
HoZn
Fig. 1. Calculated CF splitting of the Ho 4f ground-state multiplet in the paramagnetic state. The energies are plotted relative to an unperturbed 4f level.
sublattice, where we employ the differential susceptibility at zero magnetic field ! @hJ^ a iðTÞ , (9) wab ðTÞ ¼ @Bb B¼0
which describes the response of the Ho sublattice (M a ¼ gL mB hJ^ aHo i) to infinitesimal exchange field acting P as a perturbation term a Ba J^ a . Using the perturbation theory it is straightforward to derive the following expression for the susceptibility wab ðTÞ ¼
1 X a b eE n =kB T eE m =kB T J J , Z mn mn nm En Em
(10)
where m; n label eigenstates of unperturbed CF Hamiltonian, E m and E n are corresponding eigenenergies, Z ¼ P E m =kB T is the statistical sum and J amn are matrix me a elements hmjJ^ jni. For degenerated energy levels E m ¼ E n , the fraction ðeE m =kB T eE n =kB T Þ=ðE m E n Þ is replaced by its l’Hospital limit eE n =kB T =kB T. In the derivation we took an advantage of hJ^ a i ¼ 0 in the unperturbed state. For a two-sublattice HoX system, we obtain the set of equations X wab ðT C ÞfJ HoHo ðQÞhJ^ bHo i þ J HoX ðQÞhJ^ bX ig, hJ^ aHo i ¼ b
hJ^ aX i ¼
2 fJ HoHo ðQÞhJ^ aHo i þ J HoX ðQÞhJ^ aX ig. 3kB T C
ð11Þ
Because of the temperature dependence of the susceptibility, this leads to a non-linear eigenvalue problem. For vanishing CF parameters this set of equations reduces to MFA of the pure Heisenberg model, Eq. (5). In cubic symmetry at zero magnetic field, the symmetry requires that the differential susceptibility is scalar, i.e. wab ¼ wdab . This simplifies the equations above. As a corollary, in the MFA the critical temperature does not
ARTICLE IN PRESS J. Rusz et al. / Physica B 381 (2006) 265–270
4.1. Magnetic structure and critical temperatures within the Heisenberg model The results of the calculation of propagation vectors and critical temperatures are summarized in Table 2. The magnetic structure is predicted correctly in most cases. A small deviation from the observed antiferromagnetic structure of HoCu occurs in both the DLM and the FM approach. While the propagation vector of HoRh is not known experimentally (there are indications for some kind of antiferromagnetic ordering [10]), only the DLM approach predicts a non-ferromagnetic solution. The calculated critical temperatures agree with experiment on a semi-quantitative level. They reproduce the trends and they agree with experiment within the factor of three in the worst case. 4.2. Effect of the crystal-field interaction on magnetic properties Comparison of the calculated parameters of the CF Hamiltonian with experimental values also reveals semiquantitative agreement. Signs, orders and trends are reproduced. The magnitudes of the higher-order parameters A06 hr6 i are systematically underestimated in the calculations, the same occurs in the case of A04 hr4 i for HoRh and HoMg. For other systems, the predicted A04 hr4 i parameters agree fairly well with experimental data. The calculated CF splitting in the paramagnetic regime is, however, underestimated by factors from 2 to 5, see Fig. 1 and compare with Fig. 6 of Ref. [17]. Interesting questions arise, when both Hamiltonians are combined. How will this influence the calculated critical temperatures? How strongly do exchange interactions alter the energy-level scheme? To address these questions we performed a set of calculations (within the MFA) of the temperature dependence of the Ho-atom magnetizations and of the splitting of the 4f GSM for three orientations of the exchange field—(0 0 1), (0 1 1) and (1 1 1). Here, we present results based on exchange interactions calculated within the DLM state. Qualitatively different behavior is observed for different systems. The HoRh compound has the lowest critical temperature among the studied compounds and largest CF splitting (see Fig. 1). This indicates that HoRh will be dominated by the CF interaction also below the critical temperature. This is indeed the case, as can be seen from Fig. 2 (left panel). The exchange interactions remove the remaining degeneracy after the 4f GSM is split due to the CF interaction. This
120
-1
10x mHo (µB. atom )
100 50
80 Splitting of GS multiplet (K)
4. Results and discussion
75
Splitting of GS multiplet (K)
(in cubic symmetry) depend on the direction of the exchange field. The anisotropy of the magnetization at finite magnetic field is an effect of the breaking of the cubic symmetry.
269
-1
10x mHo (µB. atom )
25
0
-25
60 40 20 0 -20 -40 -60 -80
-50
-100 -75
-120 0
0.5
1
1.5 2 T (K)
2.5
3
0
10
20
30
40
50
T (K)
Fig. 2. (Color online) Temperature dependence of the splitting of ground state multiplet in HoRh (left panel) and HoMg (right panel). Temperature dependence of magnetization of the Ho sublattice is plotted here in different style (see legend). For HoRh, different line styles are used for triplets (solid line), doublets (dashed line) and singlet (dash-dotted line), respectively.
further splitting is of a size of a few kelvins, while the CF splitting is of order of several tens kelvins. There is no crossing of levels. HoRh is dominated by the CF interaction which strongly influences its magnetic properties. The easy axis of magnetization is found to be (1 1 1) for this compound. The low-temperature ordered Ho magnetic moment is reduced from the free ion value of 10mB to a much lower value of 6:18mB per atom (see Fig. 2 (left panel)). Also the critical temperature derived within the MFA is considerably lowered from 4.3 to only 2.6 K. HoMg on the other hand displays the opposite limit— smallest CF splitting (Fig. 1) and relatively high critical temperature (Table 2). Both the calculated and the experimental CF parameters are reduced in comparison to HoRh, while the critical temperature is one order higher. As a result, this system is dominated by the exchange interactions. From Fig. 2 (right panel) we can deduce, that the CF interaction is a rather small effect. The temperature dependence of the energy levels below T C resembles a pure Zeeman splitting of a degenerate 4f GSM in an external magnetic field (with exception of the near vicinity of the critical temperature, where the exchange interaction looses its dominance due to diminishing hJ^ zHo i). The ordered magnetic moment 9:90mB (along (0 0 1) easy axis) is only slightly reduced from the free ion value and the same holds for the critical temperature 45 K, which is very similar to T C in the pure Heisenberg model (within the MFA). The behavior of HoZn is similar to that of HoMg. Its CF parameters are also smaller than for HoRh, while T C is even larger than for HoMg. The maximal Ho magnetic moment is nearly 10mB (along (0 1 1) easy axis). Its critical temperature of 89 K is practically unchanged by the presence of the CF interaction. HoCd has very similar theoretical CF parameters as HoZn. As a result, the splitting of its 4f GSM in the
ARTICLE IN PRESS 270
J. Rusz et al. / Physica B 381 (2006) 265–270
paramagnetic state is nearly identical to the HoZn splitting (Fig. 1). On the other hand, it has a lower critical temperature than HoZn, i.e. its exchange interactions are weaker. As a result, the exchange splitting of the GSM is smaller and the Ho moment is a bit reduced to 9:88mB (along the (0 1 1) easy axis). Anyway, the critical temperature remains slightly above 45 K, as without the CF interaction. The antiferromagnets HoAg and HoCu both display larger CF parameters than HoMg or HoZn and therefore the CF interaction plays a more important role here. Their magnetic moments are slightly more reduced to 9:56mB and 9:47mB (both along the (011) low-temperature easy axis) for HoAg and HoCu, respectively. Their critical temperatures are, however, only weakly reduced with respect to the CF-free calculation: 20.5 and 12.7 K (CF and exchange interactions included) should be compared to 21.0 and 13.5 K (Heisenberg Hamiltonian only, MFA). 5. Conclusions In conclusion, we have performed systematic ab initio calculations of exchange interactions and CF parameters in the HoX series of compounds (X ¼ Ag, Cd, Cu, Mg, Rh, Zn). The calculated CF parameters agree with experiment on a semiquantitative level. The critical temperatures calculated from the pure Heisenberg model agree reasonably well with experiment, also the predicted propagation vectors give rather reliable results. Within an approach, which combines both the exchange interactions and the CF interaction, it was found that with the exception of HoRh, the calculated critical temperatures are only weakly reduced by the CF interaction. In the presence of the CF, the Ho magnetic moments are smaller than the free ion values. Acknowledgment We thank Pavel Nova´k for valuable discussions. This work is a part of the research project MSM 0021620834 financed by the Ministry of Education of the Czech Republic.
References [1] P. Hohenberg, W. Kohn, Phys. Rev. 136 (1964) B864. [2] V.P. Antropov, B.N. Harmon, A.N. Smirnov, J. Magn. Magn. Mater. 200 (1999) 148. [3] M. Pajda, J. Kudrnovsky´, I. Turek, V. Drchal, P. Bruno, Phys. Rev. B 64 (2001) 174402. [4] I. Turek, J. Kudrnovsky´, M. Divisˇ , P. Franek, G. Bihlmayer, S. Blu¨gel, Phys. Rev. B 68 (2003) 224431. [5] J. Rusz, I. Turek, M. Divisˇ , Phys. Rev. B 71 (2005) 174408. [6] J. Rusz, I. Turek, M. Divisˇ , Physica B, accepted for publication, doi:10.1016/j.physb.2006.01.427. [7] A.I. Liechtenstein, M.I. Katsnelson, V.A. Gubanov, J. Phys. F 14 (1984) L125. [8] I. Turek, V. Drchal, J. Kudrnovsky´, M. Sˇob, P. Weinberger, Electronic Structure of Disordered Alloys, Surfaces and Interfaces, Kluwer, Boston, 1997. [9] M. Richter, J. Phys. D 31 (1998) 1017. [10] K.H.J. Buschow, Rep. Prog. Phys. 42 (1979) 1373. [11] P. Blaha, K. Schwarz, G.K.H. Madsen, D. Kvasnicka, J. Luitz, WIEN2k: An Augmented PlaneWave + Local Orbitals Program for Calculating Crystal Properties, Technical University Vienna, 2001. [12] M.T. Hutchings, Sol. State Phys. 16 (1964) 227. [13] P. Nova´k, J. Kuriplach, J. Magn. Magn. Mater. 104 (1992) 1499. [14] H. Eschrig, Optimized LCAO Method and the Electronic Structure of Extended Systems, Akademie-Verlag, Springer, Berlin, 1988. [15] M. Richter, H. Eschrig, Solid State Commun. 72 (1989) 263. [16] M. Divisˇ , J. Rusz, H. Michor, G. Hilscher, P. Blaha, K. Schwarz, J. Alloys Compd. 403 (2005) 29. [17] D. Schmitt, P. Morin, J. Pierre, Phys. Rev. B 15 (1977) 1698. [18] M. Giraud, P. Morin, J. Rouchy, D. Schmitt, J. Magn. Magn. Mater. 59 (1986) 255. [19] G.H. Dieke, Spectral and Energy Levels of Rare Earth Ions in Crystals, Interscience, New York, 1968. [20] Y. Sinai, Theory of Phase Transitions: Rigorous Results, Pergamon, Oxford, 1982. [21] J. Rusz, L. Bergqvist, J. Kudrnovsky´, I. Turek, Phys. Rev. B, submitted for publication. [22] G.J. Bowden, J.P.D. Martin, J. Phys.: Condens. Matter 2 (1990) 10435; G.J. Bowden, J.P.D. Martin, J. Phys.: Condens. Matter 2 (1990) 10451; G.J. Bowden, J.P.D. Martin, J. Phys.: Condens. Matter 2 (1990) 10461. [23] N. Majlis, The Quantum Theory of Magnetism, World Scientific, Singapore, 2001.
Physica B 381 (2006) 265–270 www.elsevier.com/locate/physb
Exchange interactions and crystal-field effects in HoX (X ¼ Ag, Cd, Cu, Mg, Rh, Zn) intermetallic compounds Ja´n Rusz, Ilja Turek, Martin Divisˇ Faculty of Mathematics and Physics, Department of Electronic Structures, Charles University, Ke Karlovu 5, 12116 Prague 2, Czech Republic Received 6 January 2006; accepted 12 January 2006
Abstract We present an ab initio treatment of the exchange interactions and crystal-field effects in HoX (X ¼ Ag, Cd, Cu, Mg, Rh, Zn) intermetallic compounds with the CsCl crystal structure. The exchange interactions are determined using the magnetic-force theorem starting from both a ferromagnetically ordered state (zero temperature) and a disordered-local-moments (effectively high temperature) state up to an interatomic distance of 10a, where a is the lattice parameter. The exchange interactions are used for construction of the Heisenberg Hamiltonian, from which we determine the magnetic-transition temperatures using the mean-field approximation and the Green’s-function based random-phase approximation. The crystal-field parameters are derived from an ab initio electronic structure of the non-magnetic state. A combined Hamiltonian is treated within the mean-field approximation to study the effect of the crystal-field interaction on the critical temperatures and the magnetization. r 2006 Elsevier B.V. All rights reserved. PACS: 75.10.Hk; 71.70.Gm; 75.10.Jm Keywords: Critical temperature; Exchange interactions; Crystal field; Heisenberg model; HoX compounds
1. Introduction The first principles calculation of non-zero magnetic properties of solids is an active field of current solid-statetheory research. There are several approaches to this problem with different degrees of accuracy and demands on computational power. A large part of recent calculations is based on a combined approach, which makes use of the density functional theory (DFT, [1]) as its base. A suitable model Hamiltonian is then constructed and its parameters are calculated ab initio within the (zero-temperature) DFT. The non-zero-temperature properties are studied within the model Hamiltonian using methods of statistical physics. For spherically symmetric magnetic ions, like Gd or 3d transition elements (where the orbital momentum is frozen), it was found that low-energy magnetic excitations Corresponding author.
E-mail address: [email protected] (J. Rusz). 0921-4526/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2006.01.509
(magnons) are well described by an effective Heisenberg model [2]. This approach has been applied to many systems involving transition metals [3] or to rare-earth-based systems [4]. Recently, we employed this method to evaluate propagation vectors and critical temperatures in GdX systems (X is a non-magnetic element) [5,6] obtaining fair agreement with experiment, but the situation becomes more complicated when the magnetic ion is not spherically symmetric. It is necessary then to take into account its interaction with the crystal field (CF). Therefore, we decided to study Ho-based intermetallics with the CsCl structure, to understand to what extent the CF does influence the critical temperatures and the magnetization. This work has the following structure: in Section 2 we describe the applied methods of calculations—electronicstructure calculations and evaluation of the exchange interactions and CF parameters. Section 3 describes the calculation of the magnetization and the critical temperatures based on constructed model Hamiltonians and Section 4 contains a discussion of obtained results.
ARTICLE IN PRESS 266
J. Rusz et al. / Physica B 381 (2006) 265–270
2. Calculation of the parameters of model Hamiltonians
ing Stevens operators O^ m l [12]
2.1. Exchange interactions
H^ CF ¼ bA04 hr4 iðO^ 04 þ 5O^ 44 Þ þ gA0 hr6 iðO^ 0 21O^ 4 Þ,
The exchange interactions are determined using the magnetic force theorem [2,7], which is based on infinitesimal spin rotations on particular atomic sites starting from some collinear magnetic structure—‘a reference state’. Because of the strong spin–orbit interaction in HoX systems, a feasible assumption is that the Ho orbital momentum follows the rotation of the spin momentum. Therefore, the obtained exchange interactions describe the dynamics of the total angular momenta, not just the spin part. The underlying electronic structure was calculated by the tight binding linearized muffin-tin orbitals method (TB-LMTO) within the atomic spheres approximation (ASA) [8]. The radii of the atomic spheres in the ASA are proportional to the atomic radii of the respective elements. The holmium 4f states were treated within the open-core approach [9] and the spd basis was used for the valence states. The lattice parameters were taken from experiment [10]. The mapping of the exchange interactions was done in two reference states: (i) the ferromagnetic state, and (ii) the disordered-local-moments state (DLM). DLM calculation is an application of the coherent potential approximation (CPA) [8]. It corresponds to a hypothetical high-temperature state where magnetic moments are completely randomly oriented without any short-range order. Within this method, magnetic moments of the X-element completely vanish, so that only Ho–Ho interactions are to be considered. For further details about the applied method of calculation of the exchange parameters see Refs. [3,4].
2.2. Crystal-field parameters Calculation of CF parameters requires a full-potential method because the non-spherical components of the potential around the particular atom are necessary. Therefore, we repeated the calculation of the electronic structure using the WIEN2k code [11], which implements the full-potential linearized-augmented-plane-waves (FP LAPW) method of solution of the Kohn–Sham equations. Band structures calculated within both methods agreed very well. Within the WIEN2k, we used up to 150 basis functions per atom. Brillouin-zone (BZ) integration was performed by an improved tetrahedron method using 20 000 k-points in the full BZ. The holmium 4f states were treated within the open-core approach. The CsCl structure has a simple cubic Bravais cell and both the Ho and the X sites have full cubic symmetry. Therefore, the crystal-field Hamiltonian is described by only two parameters A04 hr4 i and A06 hr6 i and the correspond-
6
6
ð1Þ
6
where b and g are the reduced matrix elements. In the paramagnetic state the CF parameters are determined by the following integral [13]: m l Am l hr i ¼ C l
Z
1 0
2 R24f ðrÞV m l ðrÞr dr,
(2)
where V m l ðrÞ are the l; m components of the holmium-site potential, R4f ðrÞ is the radial part of the wave-function of the localized 4f states. The constants C m l appear due to different conventions in the definition of the Stevens’ operators and the spherical harmonics used in the WIEN2k code. The radial 4f wavefunction of the Ho atom was calculated using the self-interaction-corrected optimizedlinear-combination-of-atomic-orbitals (OLCAO) method [14,15]. The integration in Eq. (2) up to the muffin-tin radius RMT is quite straightforward. However, for the parameters of lower order this is usually not sufficient, because the non-spherical components of the potential V m l ðrÞ grow quickly with distance from the core. To enlarge the integration domain beyond RMT , the plane-wave representation of the potential in the interstitial region is expanded into spherical harmonics using the Rayleigh formula, as described in Ref. [16]. The resulting CF parameters are summarized in Table 1 together with the known experimental values [17], which are based on the results of a direct spectroscopic method (inelastic neutron scattering). Measurements of the CF parameters of HoCd are, to the best of our knowledge, not available, but one can expect them to be similar to ones for TmCd [18]. We performed numerical tests by enlarging l max in the calculation of non-spherical contributions to the Hamiltonian matrix and found our results to be well converged within the precision shown in Table 1.
Table 1 Calculated and available experimental values [17] of CF parameters of HoX (X ¼ Ag, Cd, Cu, Mg, Rh, Zn) compounds (in kelvins) Compound
HoAg HoCd HoCu HoMg HoRh HoZn
Theory
Experiment
A04 hr4 i
A06 hr6 i
A04 hr4 i
A06 hr6 i
52 20 55 þ11 35 20
3.1 3.1 3.7 2.1 8.3 3.2
67 7 32% 68 8 þ42 9 125 19 14 4
12 1 15% 15 1 13 1 18 3 18 1
The stars denote experimental CF parameters of TmCd [18].
ARTICLE IN PRESS J. Rusz et al. / Physica B 381 (2006) 265–270
3. Treatment of model Hamiltonians 3.1. Heisenberg model We write the Heisenberg Hamiltonian in the following form: X ^A ^B H^ ¼ J AB (3) ij J i J j , ij;AB
where i; j are indices labeling the unit cells and A; B are indices of the sublattice (basis indices). J^ A i are operators of the total angular momentum in unit cell i at sublattice A. The exchange interactions are denoted by J AB ij . We tried to extract from the Heisenberg model the propagation vector of the magnetic structure and the corresponding magnetic-transition temperature T C . The propagation vector Q corresponds to the latticeFourier-transformed matrix of the exchange interactions JðQÞ with the maximal eigenvalue. The elements of JðQÞ are given by X AB J AB ðQÞ ¼ eiQR0i J AB (4) 0i , i
RAB 0i
where is a vector connecting the sites ð0; AÞ and ði; BÞ. We scanned the irreducible wedge of the first BZ on a mesh corresponding to 107 points in the whole first BZ. The resulting Q vectors are summarized in Table 2. Within the mean-field approximation (MFA) the corresponding critical temperatures are given by maximal eigenvalue (T MFA ) of the generalized eigenvalue problem C X
J AB ðQÞhJ^ zB i ¼
B
3kB T MFA C hJ^ z i, 2J A ðJ A þ 1Þ A
(5)
where kB is the Boltzmann constant, J A the total angular momentum of the atoms at sublattice A and hJ^ zA i the mean value of J^ zA . The Ho ground-state multiplet (GSM) has total angular momentum J Ho ¼ 8. The second lowest energy multiplet of the Ho ion (with J Ho ¼ 7) is due to strong spin–orbital interaction roughly 7000 K higher in energy [19]. This energy difference is much larger than the exchange and CF splitting of the GSM and around T C this excited level is practically unoccupied. This allows us to restrict our
267
treatment to the GSM only. The spin polarization of valence electrons in the Ho sphere contributes to the total Ho spin moment by less than 10% and can safely be neglected. The small X-sublattice magnetic moments are of itinerant nature and, therefore, we treated them as classical moments—i.e. unit vectors of size one parallel with onsite magnetization. Such combination of ‘classical’ and ‘quantum’ magnetic moments is discussed in Section 2 of Ref. [5]. To go beyond the simple MFA we employed also the Green’s-function-based technique—random-phase approximation (RPA). The derivation of the multiple sublattice RPA formalism is presented in detail in Ref. [5]. Here, we only show key equations for collinear magnetic structures, which are solved by iterative methods: Z 1 1 2J A ðJ A þ 1Þ 1 dq N ðqÞ , ð6Þ hJ^ zA i ¼ AA O 3kB T RPA C N AB ðqÞ ¼ dAB
X
J AC ð0ÞhJ^ zC i hJ^ zA iJ AB ðqÞ,
ð7Þ
C
where O is the volume of the unit cell. MFA Generally, for collinear magnetic structures T RPA C oT C [5]. It is known that the MFA overestimates T C because neglecting spin fluctuations makes the system more rigid. On the other hand, for purely ferromagnetic exchange interactions it has been shown that the RPA underestimates T C [20]. From our studies of Heusler-alloy systems using several approaches to calculate exchange interactions [21], MFA we observed that T RPA holds also for more C oT C oT C general cases (by T C we mean the exact critical temperature given by the model, which can be calculated by Monte Carlo techniques). However, there is no general proof for this statement yet. Above-mentioned RPA formulas hold only for collinear magnetic structures. Therefore, they are not applicable to treatment of magnetic structures with general propagation vectors. In this case (HoCu within the FM reference state), the multi-sublattice RPA value of T C is missing in Table 2. In the DLM reference state, the magnetic moments of the X sublattice vanish, so that the system is effectively singlesublattice. In such a case, we can employ the approach
Table 2 Propagation vectors Q and critical temperatures T C (in kelvins) of studied HoX compounds with the CsCl structure calculated using MFA and RPA within two different reference states for mapping the exchange interactions: a ferromagnetic (FM) state and a disordered-local-moments (DLM) state. Experimental values are taken from Ref. [10] System
HoAg HoCd HoCu HoMg HoRh HoZn
DLM
FM
Experiment
Q
T MFA C
T RPA C
Q
T MFA C
T RPA C
Q
TC
ð0; 1=2; 1=2Þ ð0; 0; 0Þ ð0; 0:45; 1=2Þ ð0; 0; 0Þ ð0:11; 0:12; 0:15Þ ð0; 0; 0Þ
19.8 45.3 13.5 44.9 4.3 88.9
16 39 9.3 43 1.5 82
ð0; 1=2; 1=2Þ ð0; 0; 0Þ ð0; 0:45; 1=2Þ ð0; 0; 0Þ ð0; 0; 0Þ ð0; 0; 0Þ
19.9 57.2 13.5 61.5 10.2 130
14 48 — 57 6.7 120
ð0; 1=2; 1=2Þ ð0; 0; 0Þ ð0; 1=2; 1=2Þ ð0; 0; 0Þ AFM ð0; 0; 0Þ
32–33 19–30 28 21–25 3.2 75–80
ARTICLE IN PRESS J. Rusz et al. / Physica B 381 (2006) 265–270
268
described in Ref. [4] for calculation of T RPA of the spinC spiral magnetic structure.
80 60
3.2. Inclusion of crystal-field effects
AB
For calculation of magnetization curves it is necessary to determine the easy axis of the magnetization. In a cubic CF there are three possible orientations—(0 0 1), (0 1 1) or (1 1 1). For particular directions, some terms in H^ exch will vanish. Note, that for non-collinear magnetic structures this Hamiltonian would contain unit-cell-dependent terms and, consequently, would introduce a complication into the Hamiltonian, because the local effective magnetic field will have a different orientation with respect to the crystallographic axes in different unit cells. Therefore, we apply this formalism only for collinear magnetic structures. The combined Hamiltonian can be diagonalized exactly in the jJJ z i basis, leading to the splitting of the 4f GSM. However, below the critical temperature, evaluation of hJ^ aA i requires knowledge of the eigenfunctions. Therefore, selfconsistent techniques are applied. Above T C , all hJ^ aA i are identically zero, so that there is no dependence of the Hamiltonian on its eigenvectors. The calculated CF splitting in the paramagnetic state is shown in Fig. 1. For direct evaluation of the T C , one can use a simpler approach. Because hJ^ a i ! 0 as T ! T C , the exchange interaction can be treated as a small perturbation to the unperturbed CF Hamiltonian. The derivation of T C goes along same lines as the MFA for the Heisenberg model (see e.g. Ref. [23]). Different is the treatment of the Ho
CF splitting (K)
In a crystal, the Ho ions are non-spherical, so that the (spherically symmetric) Heisenberg model cannot give a complete picture of the magnetic properties of the Ho sublattice. The non-sphericity of the Ho ions leads to a splitting of the 4f energy level due to the CF interaction. The combination of the CF Hamiltonian with the Heisenberg model gives a more appropriate description. The treatment of this combined model Hamiltonian is much more complex than that of pure Heisenberg model. There are several publications in which a simple axial CF together with exchange interactions is treated within the RPA for some particular (rather low) total angular momenta [22]. But such formalism has not yet been developed for a fully cubic (or lower symmetry) CF Hamiltonian and this task goes beyond the scope of present paper. Within the MFA, the exchange interactions enter the calculation only as an effective magnetic field acting on the Ho sublattice. An additional Zeeman-like term is added to the CF Hamiltonian (for collinear magnetic structures only; note that in CsCl structure thanks to inversion symmetry all elements of JðQÞ are real, so that J AB ðQÞ ¼ J BA ðQÞ) X H^ exch ¼ 2 J AB ðQÞðhJ^ xB iJ^ xA þ hJ^ yB iJ^ yA þ hJ^ zB iJ^ zA Þ. (8)
40 20 0 -20 -40 -60 -80
HoAg
HoCd
HoCu
HoMg
HoRh
HoZn
Fig. 1. Calculated CF splitting of the Ho 4f ground-state multiplet in the paramagnetic state. The energies are plotted relative to an unperturbed 4f level.
sublattice, where we employ the differential susceptibility at zero magnetic field ! @hJ^ a iðTÞ , (9) wab ðTÞ ¼ @Bb B¼0
which describes the response of the Ho sublattice (M a ¼ gL mB hJ^ aHo i) to infinitesimal exchange field acting P as a perturbation term a Ba J^ a . Using the perturbation theory it is straightforward to derive the following expression for the susceptibility wab ðTÞ ¼
1 X a b eE n =kB T eE m =kB T J J , Z mn mn nm En Em
(10)
where m; n label eigenstates of unperturbed CF Hamiltonian, E m and E n are corresponding eigenenergies, Z ¼ P E m =kB T is the statistical sum and J amn are matrix me a elements hmjJ^ jni. For degenerated energy levels E m ¼ E n , the fraction ðeE m =kB T eE n =kB T Þ=ðE m E n Þ is replaced by its l’Hospital limit eE n =kB T =kB T. In the derivation we took an advantage of hJ^ a i ¼ 0 in the unperturbed state. For a two-sublattice HoX system, we obtain the set of equations X wab ðT C ÞfJ HoHo ðQÞhJ^ bHo i þ J HoX ðQÞhJ^ bX ig, hJ^ aHo i ¼ b
hJ^ aX i ¼
2 fJ HoHo ðQÞhJ^ aHo i þ J HoX ðQÞhJ^ aX ig. 3kB T C
ð11Þ
Because of the temperature dependence of the susceptibility, this leads to a non-linear eigenvalue problem. For vanishing CF parameters this set of equations reduces to MFA of the pure Heisenberg model, Eq. (5). In cubic symmetry at zero magnetic field, the symmetry requires that the differential susceptibility is scalar, i.e. wab ¼ wdab . This simplifies the equations above. As a corollary, in the MFA the critical temperature does not
ARTICLE IN PRESS J. Rusz et al. / Physica B 381 (2006) 265–270
4.1. Magnetic structure and critical temperatures within the Heisenberg model The results of the calculation of propagation vectors and critical temperatures are summarized in Table 2. The magnetic structure is predicted correctly in most cases. A small deviation from the observed antiferromagnetic structure of HoCu occurs in both the DLM and the FM approach. While the propagation vector of HoRh is not known experimentally (there are indications for some kind of antiferromagnetic ordering [10]), only the DLM approach predicts a non-ferromagnetic solution. The calculated critical temperatures agree with experiment on a semi-quantitative level. They reproduce the trends and they agree with experiment within the factor of three in the worst case. 4.2. Effect of the crystal-field interaction on magnetic properties Comparison of the calculated parameters of the CF Hamiltonian with experimental values also reveals semiquantitative agreement. Signs, orders and trends are reproduced. The magnitudes of the higher-order parameters A06 hr6 i are systematically underestimated in the calculations, the same occurs in the case of A04 hr4 i for HoRh and HoMg. For other systems, the predicted A04 hr4 i parameters agree fairly well with experimental data. The calculated CF splitting in the paramagnetic regime is, however, underestimated by factors from 2 to 5, see Fig. 1 and compare with Fig. 6 of Ref. [17]. Interesting questions arise, when both Hamiltonians are combined. How will this influence the calculated critical temperatures? How strongly do exchange interactions alter the energy-level scheme? To address these questions we performed a set of calculations (within the MFA) of the temperature dependence of the Ho-atom magnetizations and of the splitting of the 4f GSM for three orientations of the exchange field—(0 0 1), (0 1 1) and (1 1 1). Here, we present results based on exchange interactions calculated within the DLM state. Qualitatively different behavior is observed for different systems. The HoRh compound has the lowest critical temperature among the studied compounds and largest CF splitting (see Fig. 1). This indicates that HoRh will be dominated by the CF interaction also below the critical temperature. This is indeed the case, as can be seen from Fig. 2 (left panel). The exchange interactions remove the remaining degeneracy after the 4f GSM is split due to the CF interaction. This
120
-1
10x mHo (µB. atom )
100 50
80 Splitting of GS multiplet (K)
4. Results and discussion
75
Splitting of GS multiplet (K)
(in cubic symmetry) depend on the direction of the exchange field. The anisotropy of the magnetization at finite magnetic field is an effect of the breaking of the cubic symmetry.
269
-1
10x mHo (µB. atom )
25
0
-25
60 40 20 0 -20 -40 -60 -80
-50
-100 -75
-120 0
0.5
1
1.5 2 T (K)
2.5
3
0
10
20
30
40
50
T (K)
Fig. 2. (Color online) Temperature dependence of the splitting of ground state multiplet in HoRh (left panel) and HoMg (right panel). Temperature dependence of magnetization of the Ho sublattice is plotted here in different style (see legend). For HoRh, different line styles are used for triplets (solid line), doublets (dashed line) and singlet (dash-dotted line), respectively.
further splitting is of a size of a few kelvins, while the CF splitting is of order of several tens kelvins. There is no crossing of levels. HoRh is dominated by the CF interaction which strongly influences its magnetic properties. The easy axis of magnetization is found to be (1 1 1) for this compound. The low-temperature ordered Ho magnetic moment is reduced from the free ion value of 10mB to a much lower value of 6:18mB per atom (see Fig. 2 (left panel)). Also the critical temperature derived within the MFA is considerably lowered from 4.3 to only 2.6 K. HoMg on the other hand displays the opposite limit— smallest CF splitting (Fig. 1) and relatively high critical temperature (Table 2). Both the calculated and the experimental CF parameters are reduced in comparison to HoRh, while the critical temperature is one order higher. As a result, this system is dominated by the exchange interactions. From Fig. 2 (right panel) we can deduce, that the CF interaction is a rather small effect. The temperature dependence of the energy levels below T C resembles a pure Zeeman splitting of a degenerate 4f GSM in an external magnetic field (with exception of the near vicinity of the critical temperature, where the exchange interaction looses its dominance due to diminishing hJ^ zHo i). The ordered magnetic moment 9:90mB (along (0 0 1) easy axis) is only slightly reduced from the free ion value and the same holds for the critical temperature 45 K, which is very similar to T C in the pure Heisenberg model (within the MFA). The behavior of HoZn is similar to that of HoMg. Its CF parameters are also smaller than for HoRh, while T C is even larger than for HoMg. The maximal Ho magnetic moment is nearly 10mB (along (0 1 1) easy axis). Its critical temperature of 89 K is practically unchanged by the presence of the CF interaction. HoCd has very similar theoretical CF parameters as HoZn. As a result, the splitting of its 4f GSM in the
ARTICLE IN PRESS 270
J. Rusz et al. / Physica B 381 (2006) 265–270
paramagnetic state is nearly identical to the HoZn splitting (Fig. 1). On the other hand, it has a lower critical temperature than HoZn, i.e. its exchange interactions are weaker. As a result, the exchange splitting of the GSM is smaller and the Ho moment is a bit reduced to 9:88mB (along the (0 1 1) easy axis). Anyway, the critical temperature remains slightly above 45 K, as without the CF interaction. The antiferromagnets HoAg and HoCu both display larger CF parameters than HoMg or HoZn and therefore the CF interaction plays a more important role here. Their magnetic moments are slightly more reduced to 9:56mB and 9:47mB (both along the (011) low-temperature easy axis) for HoAg and HoCu, respectively. Their critical temperatures are, however, only weakly reduced with respect to the CF-free calculation: 20.5 and 12.7 K (CF and exchange interactions included) should be compared to 21.0 and 13.5 K (Heisenberg Hamiltonian only, MFA). 5. Conclusions In conclusion, we have performed systematic ab initio calculations of exchange interactions and CF parameters in the HoX series of compounds (X ¼ Ag, Cd, Cu, Mg, Rh, Zn). The calculated CF parameters agree with experiment on a semiquantitative level. The critical temperatures calculated from the pure Heisenberg model agree reasonably well with experiment, also the predicted propagation vectors give rather reliable results. Within an approach, which combines both the exchange interactions and the CF interaction, it was found that with the exception of HoRh, the calculated critical temperatures are only weakly reduced by the CF interaction. In the presence of the CF, the Ho magnetic moments are smaller than the free ion values. Acknowledgment We thank Pavel Nova´k for valuable discussions. This work is a part of the research project MSM 0021620834 financed by the Ministry of Education of the Czech Republic.
References [1] P. Hohenberg, W. Kohn, Phys. Rev. 136 (1964) B864. [2] V.P. Antropov, B.N. Harmon, A.N. Smirnov, J. Magn. Magn. Mater. 200 (1999) 148. [3] M. Pajda, J. Kudrnovsky´, I. Turek, V. Drchal, P. Bruno, Phys. Rev. B 64 (2001) 174402. [4] I. Turek, J. Kudrnovsky´, M. Divisˇ , P. Franek, G. Bihlmayer, S. Blu¨gel, Phys. Rev. B 68 (2003) 224431. [5] J. Rusz, I. Turek, M. Divisˇ , Phys. Rev. B 71 (2005) 174408. [6] J. Rusz, I. Turek, M. Divisˇ , Physica B, accepted for publication, doi:10.1016/j.physb.2006.01.427. [7] A.I. Liechtenstein, M.I. Katsnelson, V.A. Gubanov, J. Phys. F 14 (1984) L125. [8] I. Turek, V. Drchal, J. Kudrnovsky´, M. Sˇob, P. Weinberger, Electronic Structure of Disordered Alloys, Surfaces and Interfaces, Kluwer, Boston, 1997. [9] M. Richter, J. Phys. D 31 (1998) 1017. [10] K.H.J. Buschow, Rep. Prog. Phys. 42 (1979) 1373. [11] P. Blaha, K. Schwarz, G.K.H. Madsen, D. Kvasnicka, J. Luitz, WIEN2k: An Augmented PlaneWave + Local Orbitals Program for Calculating Crystal Properties, Technical University Vienna, 2001. [12] M.T. Hutchings, Sol. State Phys. 16 (1964) 227. [13] P. Nova´k, J. Kuriplach, J. Magn. Magn. Mater. 104 (1992) 1499. [14] H. Eschrig, Optimized LCAO Method and the Electronic Structure of Extended Systems, Akademie-Verlag, Springer, Berlin, 1988. [15] M. Richter, H. Eschrig, Solid State Commun. 72 (1989) 263. [16] M. Divisˇ , J. Rusz, H. Michor, G. Hilscher, P. Blaha, K. Schwarz, J. Alloys Compd. 403 (2005) 29. [17] D. Schmitt, P. Morin, J. Pierre, Phys. Rev. B 15 (1977) 1698. [18] M. Giraud, P. Morin, J. Rouchy, D. Schmitt, J. Magn. Magn. Mater. 59 (1986) 255. [19] G.H. Dieke, Spectral and Energy Levels of Rare Earth Ions in Crystals, Interscience, New York, 1968. [20] Y. Sinai, Theory of Phase Transitions: Rigorous Results, Pergamon, Oxford, 1982. [21] J. Rusz, L. Bergqvist, J. Kudrnovsky´, I. Turek, Phys. Rev. B, submitted for publication. [22] G.J. Bowden, J.P.D. Martin, J. Phys.: Condens. Matter 2 (1990) 10435; G.J. Bowden, J.P.D. Martin, J. Phys.: Condens. Matter 2 (1990) 10451; G.J. Bowden, J.P.D. Martin, J. Phys.: Condens. Matter 2 (1990) 10461. [23] N. Majlis, The Quantum Theory of Magnetism, World Scientific, Singapore, 2001.