Electronic structure and crystal field in REMg and RERh (RE = rare earth) intermetallics

PHYSICA rn

Physica B 183 (1993) 25-32 North-Holland

Electronic structure and crystal field In REMg and RERh eRE = rare earth) intermetallics M. Divis a and 1. Kuriplach b aDepartment of Metal Physics, Charles University, Prague, Czechoslovakia bDepartment of Low Temperature Physics, Charles University, Prague, Czechoslovakia Received 25 September 1992

Electronic structures of the rare earth intermetallics YRh, YMg, GdRh and GdMg were calculated using the full potential LAPW method at LDA approximation. The crystal field splittings of RE 3 + energy levels were obtained from the aspherical components of the self-consistent potential. Our method is shown to describe the trends in experimental values of the crystal field parameters for RERh and REMg systems correctly.

1. Introduction

The anisotropic properties of rare earth intermetallic compounds are the object of wide scientific interest. It is currently believed that the principal source of this anisotropy is the crystal field (CF) which acts as a perturbation on localized 4f states [1]. Several ab initio calculations of CF interaction for rare earth metals [2,3] and few intermetallics [4] have appeared recently. These calculations are based on the local density approximation (LDA) to the treatment of the total Coulomb and exchange-correlation potential in crystals. This potential [2,4] or charge density [3] obtained self-consistently was used to derive the CF splitting of RE 3 + energy levels. In some cases such calculations provide reasonable agreement with experimental data of CF parameters [2]. Otherwise a rather large discrepancy between experimental and theoretical results was found [4]. Especially the validity and accuracy of such model treatment for the fourth- and sixth-order terms in the CF hamiltonian is still in question. Correspondence to: Martin DiviS, Department of Metal Physics, Charles University, Ke Karlovu 5, 121 16 Prague 2, Czechoslovakia.

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To proceed further we decided to study such systems in which significant trends in values and even signs of CF parameters were found and reliable experimental data are available. REMg and RERh manifest themselves as such desired examples because the fourth-order CF parameter A 4 • is positive for REMg compounds and negative for RERh compounds, respectively [5]. Moreover the cubic symmetry of CF in RE sites excludes the second-order terms in the CF hamiltonian which dominate in many intermetallics with lower crystal symmetry. This fact has enabled us to test our model assumptions for higher-order terms directly. The CF parameters for REMg and RERh compounds were also earlier analysed iJ;l terms of the superposition model (SM) by Newman [6]. This phenomenological model assumes that the CF potential is the superposition of two-body potentials due to the nearest-neighbour atoms alone. The SM is rather attractive for experimentalists owing to its simplicity and recently it was successfully used to reduce the number of adjustable CF parameters for intermetallic RECu 2 [7] and RECu 2 Si 2 [8] compounds with low orthorhombic and tetragonal symmetry, respectively. From a theoretical point of view it was shown in ref. [9] that the more-distant-atom contribution

1993 - Elsevier Science Publishers B.Y. All rights reserved

26

M. Divis, 1. Kuriplach / Electronic structure and crystal field in RE compounds

to the CF parameters in metals is very small. It was also found using first-principle calculations 6 6 that the ratio A~ .
2. Method

For above-mentioned purposes we have chosen to study YRh, YMg, GdRh and GdMg compounds in particular. All these compounds crystallize in the cubic CsCI structure (space group Pm3m) [11]. Both YRh and YMg are Pauli paramagnets while a ferromagnetic ground state was reported for GdRh and GdMg intermetallic compounds [11]. The values of the lattice constants and the muffin-tin (MT) radius of spheres used in our calculations are given in table 1. We have performed electronic structure calculations of the above-mentioned compounds using the full potential linearized augmented plane wave (FLAPW) package of programs developed by Blaha et al. [12]. The spin-polarized calculations were performed with the exchange-correlation potential [13] in the case of GdRh and GdMg, respectively. The intermetallics YRh and YMg were treated as non-spin-polarized and we Table 1 Values of the parameters involved in the FLAPW calculations for REX compounds (X = Rh, Mg). Compound

Lattice parameter [a.u.] RE muffin-tin radius X muffin-tin radius

YRh

YMg

GdRh

GdMg

6.442 2.600 2.800

7.173 3.213 2.269

6.491 3.213 2.269

7.196 3.213 2.269

have used the exchange-correlation term in the form described by Hedin and Lundqvist [14]. In FLAPW calculations it is common practice to distinguish the electronic states into the core, semicore and valence states, respectively [12]. The inner shells for YRh and YMg were treated as core states with exception of the 4s, 4p states for Y (Rh) and 2s, 2p states for Mg which were treated as semicore states. The remaining states (Y, Rh-5s, 4d, 5p; Mg-3s, 3p) were assumed to be the valence states. Two different types of calculation, which will be referred to as A and B, were performed for GdRh and GdMg, respectively: A - Gd - core == Kr, 4d, 4f; semicore == 5s, 5p; valence == 6s, 5d, 6p. Rh - core == Ar, 3d; semi core == 4s, 4p; valence == 5s, 4d, 5p. Mg - core == He; semicore == 2s, 2p; valence == 3s, 3p. B - Gd - core == Xe; semicore == 4f; valence == as in A. Rh - core == Kr; valence == as in A. Mg - core == Ne; valence == as in A. The reliability of these model calculations concerning the treatment of the localized 4f electrons in RE metals was discussed by one of us recently [2]. The possibility of treating the hybridization effects between the 4f and other-band electrons is included in case B simply by the choice of energy parameters for states with different angular momentum (see ref. [12]). Scalar relativistic corrections were included in calculating energies and wave functions of valence and semicore electrons. We have also improved the band structure code [12] to take into account matrix elements of the FLAPW potential with L = 6 in the LAPW basis [15]. The total number of nonspherical matrix elements was 131 in comparison with 18 terms in the original version of the FLAPW code. The core electrons were treated fully relativistically and the corre-

M. Divis, J. Kuriplach / Electronic structure and crystal field in RE compounds

sponding charge densities were relaxed and updated at each iteration. The starting crystal potentials were obtained by superimposing atomic potentials for neutral atomic configurations Y(Kr, 5s 2, 4d 1 ), Rh(Kr, 2 5s\ 4d 8 ), Mg(Ne, 3s 2 ) and Gd(Xe, 6s , 5d\ 4e), respectively. Spin-polarized potentials were used in the case of GdRh and GdMg. At each iteration cycle the Brillouin zone was sampled using 84 and 20 k points in the irreducible wedge of the zone for valence and semicore states, respectively. Within muffin-tin spheres lattice harmonics with angular momentum I up to 12 were employed to expand the wave functions. We have also used a basis set cutoff RMT Kmax = 8.0 (more than 200 wave functions) and 7.5 for valence and semicore states, respectively. In the case where the 4f electrons were treated as semicore, the RMT Kmax = 10.5 was employed and we have calculated the semicore bands in the r point only [2]. The various numerical sensitive quantities (total energy, Fermi levels for valence and semicore electrons, density integrals) were monitored during the self-consistent cycles. Convergence is assumed when these quantities obtained from subsequent iterations differ less than 10- 4 Ry. We have also checked the reability of our numerical results using 455 and 84 k points in the irreducible wedge of the Brillouin zone (RMT Kmax = 10.5) for valence and semicore states, respectively. The total densities of states (DOS) and integrated density of states (NOS) for plotting purposes were calculated from FLAPW eigenenergies using the Green function method described in ref. [16]. The CF hamiltonian for a single RE 3 + ion in CsCI structure compounds can be written in the form

(/{CF

= a 4 A~ (r

4 )

[O~(J)

+ 5 O:(J)]

+ a 6 A~ (r 6 ) [O~(J) - 21 O:(J)] where O~ (J) are the Stevens operators, a L the reduced matrix elements and A~ (r L ) the two independent CF parameters which scribe the strength of the CF interaction

(1) are are de[1].

27

Within the approximation that the interaction between the 4f electrons and valence electrons may be treated perturbatively, the values of the CF parameters may be calculated within the framework of LDA (LSDA) in a simple and natural way. We have expanded the total (Coulomb) potential vCr) within the muffin-tin sphere on the RE site using the cubic harmonics

eM. L·

vCr) = ~ V~ (r) e~ (it,

4» .

(2)

L,M

The CF parameters were calculated using the following relation: RMT

A~ (rL) = a~

f IR (rW V~(r) 4f

r2 dr

(3)

o

where a~ are the conversion factors [17] and R 4f (r) is the radial part of the 4f wave function. The contribution to the CF parameters from the region outside the RE muffin-tin sphere is neglected here. The corresponding correction of 1-2% of the final value can be estimated from the 4f electron density located outside this sphere.

3. Results

3.1. Electronic structures In fig. 1 we show the band structure and total DOS for paramagnetic YRh. The bottom of the spectrum (I'r symmetry) is predominantly of stype. At higher energy one can find bands with mainly d-character. These narrow bands strongly contribute to the DOS which exhibits a doublepeak structure around 0.4 Ry. The first peak and second peak are connected with bands of e g and t 2g cubic symmetry, respectively. On the other hand the Fermi level EF = 596.6 mRy is situated in the region of rather low density of states (N(FF) = 8.3 states/Ry). Thus the low values of the linear coefficient of specific heat 'Y and the paramagnetic susceptibility X [11] can be attributed to the presence of such a minimum in the

M. Divis, 1. Kuriplach I Electronic structure and crystal field in RE compounds

28

NOS [states/unit cell]

o

20

40

NOS [states / 20

o

60

unit cell] 40 60

1.0

YRh

0.8

0.8

>:

>:

~ 0.6

~

>, OIl

H Cll <=1

~ ~

0.4

YMg 0.6 EF

04 ~~==~~----~~~~--------~

<=1

Oil

Oil

0.2

0.2

r

M

R

o.0

r DOS [st./Ry unit cell]

'-------"=""-----'------'------'''-----~'---------'--------"

r

X

M

r

R

DOS [st./ Ry unit cell]

Fig. 1. Energy bands and total DOS, NOS for YRh.

Fig. 2. Energy bands and total DOS, NOS for YMg.

DOS at the Fermi surface. The partial electron charges in the MT spheres as well as the ratio p = n(t2g)/n(e g) of the number of t 2g - and e g type electrons are given in table 2. The band energies along the main symmetry directions for YMg are drawn in fig. 2. The bottom of the spectrum (Tl symmetry) has again dominantly s-character. The d-type bands are well split through cubic symmetry into two bands e g (doubly degenerated at Tr2) and three bands t 2g (triply degenerated at I;s). In comparison with YRh the Fermi level EF = 421.6 mRy is now situated at the peak of DOS (N(EF) = 15.8 states/Ry) which originates from e g electrons of yttrium mainly. This difference is also manifested through the different ratio p (see table 2). Schmitt et al. have calculated the band structure of YMg by a self-consistent symmetrized APW

method earlier [18]. They have used a muffin-tin potential without reference to relativistic corrections, up to 35 k points and smaller APW basis (33-57 basis functions). Nevertheless these results are qualitatively consistent with those presented in this work and shown in table 3. It is to be noted that Schmitt's different ratio p = 1.30 can influence the resulting values of CF parameters (see below). The electronic structures of the valence states of GdRh (see fig. 3) and GdMg (see fig. 4) are rather similar to those of YRh and YMg, respectively. The main difference comes from the exchange splitting of spin-up and spin-down subbands. The DOS at the Fermi level is higher for GdMg than for GdRh which reflects the fact that the magnetic-ordering temperature for GdMg (Tc = 119-120 K) is higher than for GdRh

Table 2 Fermi energy of valence electrons, electron character in RE muffin-tin spheres, ratio p of the number of t 2g - and eg-type electrons in the RE sphere, spin magnetic moments M, inside spheres and experimental values of magnetic moments [11]. YRh

YMg

GdRh(B) Gd!

EF

-

Erl

s P

d

eg t 2g Total

f p

M,APW Mexp

0.431 0.199 0.171 0.220 0.572 0.792 0.033 2.60

0.413 0.400 0.310 0.464 0.673 1.137 0.016 1.45

0.442 0.243 0.220 0.226 0.566 0.792 0.068

Gd"

0.411 0.221 0.186 0.165 0.438 0.603 0.067 2.57 7.25 6.45-6.6

GdMg (B)

GdMg (A) Gd!

0.434 0.197 0.150 0.422 0.411 0.833 0.010

Gd"

0.394 0.184 0.098 0.101 0.273 0.374 0.006 1.31 7.51 5.6-7.55

Gd!

0.434 0.198 0.149 0.425 0.414 0.840 0.010

Gd" 0.394

0.184 0.097 0.101 0.274

0.375 0.007 1.31 7.47

M. Divis, f. Kuriplach I Electronic structure and crystal field in RE compounds

29

Table 3 Comparison between the results of the FLAPW and symmetrized APW method [18] given for YMg at selected symmetry points.

x.' FLAPW SAPW

o

o

391,9 355,2

195,5 184,8

0.8

>::

377,1 363,6

414,3 393,7

320,8 293,3

3.2. Crystal field parameters

t>II

Q)

223,2 199,5

0,6

>.

...

445,9 401,6

seven spin-up 4f electrons in the core for GdRh so the model calculation B is presented for GdRh only.

1.0

~

M s'

M3 2lO,3 196,9

0.4

~

"'"

spin down

spin up

0.2

0.0 "------~----'---~-----' 40 20 a 20 40

DOS [states/Ry unit cell]

Fig, 3, Total spin-up and spin-down DOS for GdRh,

(Tc = 24-29.5 K) [11]. We have calculated the spin magnetic moments per Gd atom MJ GdMg) > Ms (GdRh) in agreement with the observed trend in the measured magnetic moments [11] (see table 2). A more precise comparison should require the analysis of experimental data taken from polarized neutron diffraction. The other significant results of our calculations summarized in table 2 are rather independent of the way the 4f electrons are treated (model A or B). We were not able to stabilize all 0.8

GdMg 0.6

>::

EF

~ 0.4 >.

...t>II Q)

0.2

~

"'"

spin

up

0.0

-0.2 "------~----'----~-40 20 a 20 40

DOS [states/Ry unit cell]

Fig. 4. Total spin-up and spin-down DOS for GdMg.

We have used the nonspherical terms of the self-consistent Coulomb and total (Coulomb +exchange-correlation) potential for the YRh and YMg intermetallics to calculate the CF parameters according to eq. (3). In this case the R 4f (r) wave functions for Gd were approximated by the atomic ones obtained from the atomic code [12]. The values reported in table 4 are not significantly influenced by the choice of the R4tCr)(J = ~) or R4tCr)(J = ~) relativistic wave function. For comparison we have also included the experimental results for RERh and REMg compounds. The calculated values of A~ (r 4 ) account for the observed change of sign from negative (RERh) to positive (REMg). In our model this phenomenon is related to the different shape of the Coulomb (total) LDA potential within the muffin-tin APW sphere (see fig. 5). On the other hand a rather large discrepancy between experiment and our calculation for the RERh system can be seen. The value derived from Coulomb and total potential is 1.4 and 2.5 times smaller, respectively. The similar discrepancy was found for A~ (r 4 ) in the RECo s compounds recently [4]. The calculated sixthorder CF parameter also correctly reproduces experimental trend A~ (r 6 )RERh < A~ (r 6 )REMg but its magnitude is again smaller. In the case of gadolinium compounds we have used both spin-up and spin-down nonspherical components of the potential to calculate the CF and we have obtained similar results. In this case the R 4f (r) for Gd were the self-consistently calcu-

30

M. Divis, 1. Kuriplach / Electronic structure and crystal field in RE compounds

Table 4 CF parameters calculated for YRh, YMg, GdRh and GdMg as well as a comparison with results of previous calculations [20, 21] and with experiment [5], respectively. The values of the CF parameters for GdRh and GdMg were obtained using the spin-up potentials.

4

A~ (r >[K]

YRh Coulomb Total YMg Coulomb Total GdRh Model B Coulomb Total GdMg Model A Coulomb Total Model B Coulomb Total DyRh cal. [20] Coulomb cal. [21] Total YMg cal. [20] Coulomb cal. [21] Total HoRh expo [5] ErRh expo [5] HoMg expo [5] ErMg expo [5]

A~

(r6) [K]

-93.3 -41.6

-2.8 -3.0

19.5 12.6

-2.1 -1.9

0.04 (a)

>: ~

YMg

0.02 YRh

0

....

:>

:

0.00 0

2 radius [a.u.]

3

0.00

(b)

-86.7 -52.2

-1.5 -7.7

>: ~

-2.3 -2.0

50.4 0.4

-0.6 -2.7

-98.1 14.7

-0.1 1.3

44.1 -4.5 -125 ± 19 -123 ± 10 +42±9 +4±3

-0.1 0.4 -18±3 -19± 1 -13 ± 1 -11 ± 1

lated core wave functions. To be noted are the large enhancement (5 times) and slight reduction of A~ from the exchange-correlation part of the potential for GdRh and GdMg, respectively (see table 4). If one compares our calculated CF for gadolinium and yttrium compounds (table 4) one can find no significant differences. This fact manifests: - first, the chemical similarity of Gd and Y which is reflected in the shape of nonspherical components of the crystal potential; - second, gadolinium atomic wave functions are almost not influenced by the crystal environment. In addition the effect of hybridization (see section 2) on the CF seems to be small.

-

YMg

0

:>'"

40.5 0.3

-0.01

-0.02

-0.03

--- YRh

~--~-~--~

o

2 radius [a.u.]

3

Fig. 5. Radial dependence of (a) V~(r) and (b) V~(r) components of the total potential for YRh and YMg.

4. Discussion and conclusions

The CF splitting of 4f localized states is due to the interaction between the aspherical 4f charge density and the aspherical potential of the crystal lattice. We have treated this problem in a perturbative way. It means that the aspherical potential was calculated for a system with a spherical 4f shell (GdRh, GdMg) or without the 4f shell itself (YRh, YMg). This approach is supported by the experimental observation that the CF has predominantly one-electron character [6,19] and the CF parameters do not significantly depend on the particular RE atom in the isostructural series of intermetallic compounds (see e.g. ref. [6,10]). On the other hand it would also be possible to modify our model calculations A, B in order to incorporate the contribution to the CF owing to the aspherical 4f charge density. We did not do this because it was shown recently that meaningless values for the CF parameters for Tb metal were obtained in this way [2]. From this it follows that the calculation of CF parameters may serve as a very sensitive probe for future im-

M. Divis, f. Kuriplach / Electronic structure and crystal field in RE compounds

provements of exchange-correlation potentials which go beyond LDA in electronic structure calculations. The CF parameters were calculated for selected compounds with CsCI structure (DyRh, YMg) by Schmitt recently [20]. He has used the electronic density obtained from the symmetrized APW method [20] to derive the 'direct' part of the CF potential. These results are quoted in table 4 and can be compared with our results obtained using the Coulomb term only. From table 4 one can see that our results for 4 A~ (r ) are similar to his in the case of RERh only. In contrast our calculated A~ (r 6 ) values differ by an order of magnitude matching closer to the experimental data. Nevertheless the sign change of A~ (r 4 ) between RERh and REMg compounds can be ascribed to the different ratio p of the number of t Zg - and e g-type electrons localized inside the RE muffin-tin sphere in our calculation as well as in ref. [20]. Thus we can confirm the original idea of Schmitt about opposite contributions of t Zg - and eg-type electrons to the value of the CF parameters. In his following paper [21] Schmitt also attempted to include the exchange contribution and he has found a large reduction of resulting values of the CF parameters A~ (r 4 ) and in the case of DyRh he even obtained a change of sign of the parameter 4 A~ (r ). This trend is in contradiction with experiment [5]. On the other hand we did not find 4 such a large reduction of the A~ (r ) parameter using the total (Coulomb plus exchange-correlation) potential (see table 4). We have also derived a large value of A~ (r6) in comparison with ref. [21] and our resulting sixth-order parameters are in much better agreement with experimental data (see table 4). In conclusion the better overall agreement with experimental data of the CF parameters obtained by us can be ascribed to using a more precise FLAPW method with LDA approximation while Schmitt employed a muffintin symmetrized APW method with ambiguous X-a treatment of the exchange-correlation potential [18,21]. In the case of CsCI structure compounds the 4 6 CF parameters A~ (r ) and A~ (r ) can be e~­ pressed in terms of the intrinsic parameters A4

31

and A6 of SM, respectively [6]. These intrinsic parameters represent the assumption of a cylindrically symmetric CF potential between the RE ion and each ligand. The nearest-neighbour atoms in CsCI structure systems are eight-fold coordinated giving the very simple relations 6 4 A~ (r ) = -28/9 A4 and A~ (r ) = 16/9 A 6 , respectively [6]. Analysing the experimental data of the CF parameters Newman [6] has found that A4 > 0 and A6 < 0 with the exception of REMg compounds. We would like to note at this point that the intrinsic parameters satisfy the relations A z ~ 4 A4 ~ 8 A6 ~ 0 in ionic crystals and hightemperature superconductors [19]. The positive intrinsic parameter A 4 was also found in RE metals [6] and many RE intermetallic compounds studied so far [6,10]. Thus the REMg system with A4 negative would exhibi! an exception and so Newman argued that A4 remains positive even for REMg compounds because the second-neighbour RE atom (six-fold) contributions are relatively more important in this case [6]. On the other hand we have unambiguously 4 shown that the positive A~ (r ) for the REMg 4 system and the negative A~ (r ) for the RERh system come from the shape of the LDA (LSDA) potential inside the RE muffin-tin sphere entirely. Therefore in terms of SM the intrinsic parameter A4 should be negative for the REMg system which was confirmed by our calculations. We can conclude that our concept used to calculate the CF splitting is reliable to provide the trends found in experiment.

Acknowledgements

The authors wish to thank Dr. P. Novak for many stimulating discussions and critical reading of the manuscript.

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M. Divis, J. Kuriplach / Electronic structure and crystal field in RE compounds

32

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