AB INITIO STUDY OF ELECTRONIC AND MAGNETIC STRUCTURE OF INTERMETALLIC RE 5 Ge 3 COMPOUNDS

DOI: 10.1142/S0217984911027649

November 1, 2011 8:54 WSPC/147-MPLB S0217984911027649

Modern Physics Letters B, Vol. 25, No. 32 (2011) 2427–2438c© World Scientific Publishing Company

AB INITIO STUDY OF ELECTRONIC AND MAGNETIC

STRUCTURE OF INTERMETALLIC RE5Ge3 COMPOUNDS

M. DJERMOUNI, M. BELHADJ, S. KACIMI and A. ZAOUI∗,†

Modelling and Simulation in Materials Science Laboratory,

Djillali Liabes University of Sidi Bel-Abbes, Sidi Bel-Abbes 22000, Algeria∗ali [email protected]†[email protected]

Received 2 May 2011Revised 11 September 2011

A series of binary rare-earth metal germanides RE5Ge3 (RE=La,Ce,Pr and Nd)adopting the Mn5Si3-type hexagonal structure is studied. These intermetallic phasesshow a complex magnetic behavior. Using a modification of the local densityapproximation (LSDA+U) the magnetic and electronic properties of these compoundsare calculated. The spin-orbit coupling (SOC) was included using a full relativistic basis.Besides the structural parameters, bonding characters, total and partial densities of stateand band structures are analyzed and compared with the experimental findings.

Keywords: Intermetallic compounds; magnetic properties; FP-LAPW method; LDA+U +SOC.

1. Introduction

Recently, there is a considerable interest in understanding the microstructure of

new magnetic materials. Current technology is dependent on advances made in

the search for new compounds that may contain the properties of the materials

superior than those of traditional ones. The performance in such a search is

significantly dependent on the theoretical foundations, which allow one to envisage

new properties of the materials with new crystal structures. The formation of

materials based on rare-earth metals is of particular significance because of the

unique combination of properties of the rare-earth metals themselves and of

their compounds. The most of rare-earth intermetallic compounds RE5Ge3 (RE=

rare-earth elements) crystallize in the Mn5Si3-type hexagonal structure with space

group P63/mcm in which RE occupies two non-equivalent crystallographic sites.1,2

They present both particular challenges in synthesis and novel opportunities for

a new kind of chemistry. They have been receiving much attention in recent

years due to the interesting properties such as the magnetization and electrical

∗Corresponding author.

2427

http://dx.doi.org/10.1142/S0217984911027649

mailto:[email protected]

mailto:[email protected]


2428 M. Djermouni et al.

transport magnetic ordering, magnetoelastic phenomena, thermal capacity and

enthalpy and complicated magnetism.3–7 Numerous experimental works have been

reported.8–19 The magnetic and electrical properties of these compounds were

studied by Buschow and coworkers1,2 using polycrystalline samples. The Kondo

effect was observed in Ce5Ge3 (see Ref. 20) with complication in magnetization

orders. The discovery of giant magnetocaloric effect in Gd5(SiGe)4 (see Ref. 21)

and a strong interplay between a magnetic and structural behavior in RE5Ge4compounds, give opportunity to study the RE-Ge system.22,23 This phenomena is

confirmed in RE5Ge3 (RE = Nd and Gd)24 with a new magnetoelastic irreversibility

character. The magnetic transition in these compounds is also studied by Tsutaoka

et al.25 Very recently Joshi and co-workers26 give magnetic behavior of Pr5Ge3

and Tb5Ge3. These intermetallic compounds at low temperature show different

magnetic transitions.

So, it is clear that the magnetic structure is still an open question. Analyzing

the experimental investigations above, it is worthy to note that these interesting

physical properties mainly result from the 4f electrons whose theoretical research

is still a challenge due to the more local and complicated features than 3d electrons.

A number of questions remain unanswered in relation to the electronic structure

and magnetic behavior of RE5Ge3 series of compounds (RE = La, Ce, Pr and Nd)

that are well suited to an investigation by modern ab initio techniques. So, most of

interests are limited to the synthesization and magnetic analysis and the absence

in literature of any information about the ab initio electronic structure of these

compounds have lead us to study these new materials. The main aim of this work

is to investigate the electronic structure of these intermetallics. The plan of the

present paper is as follows. Section 2 gives a description of the method as well as

some details of the calculations. The calculated structural and electronic properties

of RE5Ge3 are presented and discussed in Sec. 3. Finally, the conclusion is given in

Sec. 3.2.

2. Methodology

The calculations have been performed within DFT implemented in the WIEN2K

code.27 Atoms were represented by hybrid full-potential (linear) augmented

plane-wave plus local orbitals (L/APW+ lo) method.28 In this method, wave

functions, charge density, and potential are expanded in spherical harmonics within

no overlapping muffin-tin spheres, and plane waves are used in the remaining

interstitial region of the unit cell. In the code, the core and valence states are

treated differently. Core states are treated within a multi-configuration relativistic

Dirac–Fock approach, while valence states are treated in a scalar relativistic

approach. The exchange-correlation energy was calculated using the Perdew–Wang

local spin density approximations (LSDA)29 and (LSDA+U)30. Very careful step

analysis is done to ensure convergence of the total energy in terms of the variational

cutoff-energy parameter. At the same time we have used an appropriate set of


Electronic and Magnetic Structure of Intermetallic RE5Ge3 Compounds 2429

Fig. 1. (Color online) Crystal structure of RE5Ge3 compounds (a) tripled along the c-axis and(b) along the ab-plane: blue atoms for rare earth elements and smaller green for germaniumelements.

k-points to compute the total energy. We compute equilibrium lattice constants and

bulk moduli by fitting the total energy versus volume to the Murnaghan31 equation.

The standard built-in basis functions were applied with the valence configurations

of (RE: 5s2 5p6 6s2 5d1 4fn: 0 ≤ n ≤ 3), and (Ge: 3d10 4s2 4p2). The total energy

was minimized using a set of 60 k-points in the irreducible sector of Brillouin zone,

equivalent to an 8× 8× 10 Monkhorst–Pack32 grid in the unit cell, and the value of

8 Ry for the cutoff energy were used. We have adopted the values of 2.75 bohr for

rare earth elements and 2.5 bohr for germanium, as MT radii. Most of the RE5M3

crystallize in the Mn5Si3-type hexagonal structure (P63/mcm, No. 193) (Fig. 1).

In this structure prototype, the RE atoms occupy the two inequivalent 4(d) and

6(g) crystal sites [i.e. in the special position (1/3, 2/3, 0) and in (xRE, 0, 1/4),

respectively], whilst the M atoms occupy the 6(g) site (xGe, 0, 1/4).

We note that spin-orbit coupling can also be included in the calculations.

Spin-orbit coupling does not very much affect the overall band structure except

for some minor details. However, spin-up and spin-down states are then mixed.

Therefore, we present all band structures with and without spin-orbit coupling.

We perform a single non-self-consistent calculation including spin-orbit coupling in

order to obtain the orbital magnetic moments.

3. Results and Discussion

3.1. Structural properties of RE5Ge3

For hexagonal RE5Ge3, our calculated results at 0 GPa for the lattice parameter,

bulk moduli and their first derivative obtained by LSDA and LSDA+U

approximations, compared with other experimental results are illustrated in



Table 1. The calculated equilibrium lattice parameters a (A), the bulk modulus B (GPa) andHubbard U parameter (eV) of the RE5Ge3 (RE = La, Ce, Pr and Nd) intermetallic compoundsaccording to FLAPW-LSDA and (LSDA+U) calculations in comparison to the data for alreadysynthesized compounds.

Position:

RE5Ge3 Exc xRE, xGe V (A3) a (A) c (A) c/a B (GPa) B′

This work LDA 0.2427, 0.6086 456.446 8.772 6.85 0.781 60.971 3.833

LDA+U ,

La5Ge3 Ud = 0.531 0.2456, 0.6041 461.708 8.872 6.774 0.763 60.035 3.588

Ref. 19 — 476.19 8.941 6.878 0.769 — —

Ref. 10 — 478.644 8.95 6.9 0.771 — —

This work LDA 0.2627, 0.6171 386.456 8.679 5.923 0.682 79.617 4.674

LDA+U ,

Ce5Ge3 Uf = 1.619 0.2613, 0.6203 399.286 8.726 6.055 0.694 63.076 4.493

Ref. 10 — 454.77 8.84 6.72 0.76 — —

This work LDA 0.2324, 0.5966 388.155 8.521 6.173 0.72 64.217 4.5

LDA+U ,

Uf = 2.332 0.2304, 0.6031 422.598 8.917 6.137 0.688 59.821 3.449

Pr5Ge3 Ref. 26 0.2300, 0.6040 442.226 8.804 6.588 0.748 — —

Ref. 10 — 445.63 8.79 6.66 0.758 — —

Ref. 1 — 450.233 8.818 6.686 0.758 — —

This work LDA 0.2499, 0.6073 393.075 8.632 6.092 0.706 58.206 3.929

LDA+U ,

Uf = 2.403 0.2499, 0.6072 420.112 8.718 6.382 0.688 62.764 3.558

Ref. 16 0.238, 0.606 433.909 8.703 6.615 0.76 — —

Ref. 18 — 439.981 8.7508 6.6345 0.757 — —

Nd5Ge3 Ref. 25 — 434.609 8.712 6.612 0.759 — —

Ref. 17 0.2415, 0.6047 438.6 8.743 6.625 0.758 — —

Ref. 2 0.2435, 0.6069 439.68 8.758 6.619 0.756 — —

Ref. 14 — 438.241 8.747 6.614 0.756 — —

Ref. 10 0.24, 0.6 436.62 8.74 6.6 0.755 — —

Table 1. The LSDA+U method30,33 is employed to calculate the on-site Coulomb

correlation for the RE-4f electrons, where the Coulomb repulsion energy Ueff is

calculated using Madsen, Novak approach34 and presented in the same table. From

Table 1, one notices that the determined volume by the LSDA+U is higher than

that calculated by LSDA for all studied compounds. This over-estimate is caused by

the potential U which plays the part of a repulsive potential. We find equilibrium

volumes of all systems in agreement with the experimental values.1,8–19,25,26 The

principal effect of Hubbard correction is to prevent the participation of the 4f

electrons in the bond: the 4f states are pushed towards lower and/or higher



energies away from the Fermi level. Thus the LSDA+U give better results while

by comparing with the experimental data. Optimization of the atomic positions

within LSDA is found to be substantially reduced compared to the experimental

estimate10,12,16,17,26 In order to check the influence of electron–electron correlation

on the structural optimization, which has been found to be important, we have

further optimized the atomic positions within the LSDA+U approach using the

calculated value of U for RE. Remarkably, we observe that with the consideration

of U , these parameters become comparable with experiment (see Table 1). These

results indicate the importance of correlation effects for the description of the

correct internal parameters in these compounds.

3.2. Electronic and magnetic properties

We show only the band structure, DOS and charge density of only one compound,

because they are very similar to those of the other compounds with only a few

corrections in the energy levels. As a first step to understand the electronic and

magnetic properties of RE5Ge3 compounds, the electronic structure calculations are

carried out. We note that the present calculations were performed using standard

density functional theory. As mentioned, there are problems in describing the

magnetism of the rare earth metallic in this approach. There has been considerable

discussion of the nature and role of electronic correlations in these materials. As

shown in Fig. 2, the band structure along the high symmetry point in the first

Brillouin zone of all compounds was calculated with different approaches. First, we

show the full band structures of Pr5Ge3 within LSDA scheme in Fig. 2 (LDA). The

Ge 4s bands lie between −9 and −7 eV. Between −3.4 and 3.1 eV, there are very

Fig. 2. FPLAPW-calculated band structure of spin polarized Pr5Ge3 intermetallic compoundalong several high-symmetry axes using LSDA and LSDA+U approach by taking account of thespin-orbit coupling.



flat bands which come from the Pr-4f electrons. From 3.2 eV, there is a mixture

of Pr-5d, Ge-4s and 4p bands. Those very flat bands below and above the Fermi

level are mainly of the Pr-4f characters. A prominent feature of the band structure

near EF , besides the 4f bands, is the Pr-5d character which hybridizes with the

Pr-4f bands. We also study the on-site atomic-like correlation effects beyond

LSDA by using LSDA+U approach in a rotationally invariant, full potential

implementation. Minimizing the LSDA+U total energy functional with spin-orbit

coupling (SOC) treated self-consistently generates not only the ground state energy

and spin densities, but also effective one-electron states and energies that provide

the orbital contribution to the moment and Fermi surfaces. The basic difference of

LSDA+U calculations from the LSDA is its explicit dependence on the on-site spin

and orbitally resolved occupation matrices. The Coulomb potential U for the Pr-4f

orbitals have been calculated (Table 1). The resulting band structure calculated

within the LSDA+U scheme is shown in middle panel of Fig. 2 (LDA+U). We

observe that the crystal field splitting of Pr-4f bands within LSDA is quite small

and in fact difficult to identify due to hybridization with itinerant bands. From

LSDA+U the 4f bands are still very flat but are split into some manifolds by

some combination of the crystal field and the anisotropy of the U interaction. The

spin-orbit effect has been observed to play a significant role and was proposed

to be important for the magnetic and orbital physics of RE5Ge3. We performed

LSDA+U +SO calculations with the same U values as mentioned above, where

the spin-orbit effects have been introduced as a second variation using the scalar

relativistic approximation. We also calculated fully relativistic band structure to

see the SOC effects, which is shown in Fig. 2 (LDA+U +SO). As expected, SOC

splits the 4f electrons into several band located in the vicinity of the Fermi level.

The total and partial densities of state for the Nd5Ge3 are shown in Figs. 3

and 4, respectively. It is clearly that the states, which are approximately located

between −3.5 eV and the Fermi level, originate from the bonding of Ge-p and

Nd-d-f states. The bonding states, as seen in Fig. 4, are dominated by Ge-s, Ge-p,

and Nd-f , while the antibonding states are dominated by Nd-f states. Below the

Fermi level, Ge-p, Nd-d and Nd-f peaks show evidence for strong hybridization,

while the other contributions are small. The figures indicate clearly that, for both

spin channels, there is a slight shift between the spin-up and spin-down states, which

can be attributed to the exchange interaction of the spin-down Ge-4p states with

the corresponding RE-5d-4f levels. As mentioned above, the band structure and

density of state results prove that the coulomb potential and spin-orbit coupling are

key factors to understand the electronic and magnetic properties of this series of

materials. The 4f electrons are splitting into several states located near the Fermi

level. The SOC effect is very weak on electronic structure of our materials; this is

due to the hexagonal environment contrary to the cubic structures (see Fig. 5).

The figures also indicate that the magnetism in RE5Ge3 originates mainly from

RE atoms. This is due to a large exchange splitting between the spin-up and

spin-down RE-4f states so that, the former are fully occupied and deeply located



Fig. 3. Calculated projected total density of state (TDOS) plot for spin polarized Nd5Ge3intermetallic compounds using: LSDA, LSDA+U and LSDA+U +SO approaches.

below EF . Such a large exchange splitting results in a dominant localization of the

magnetic moment on the rare earth atoms with a subsequently induced moments

on the Ge atom.



Spin down

Spin up

Fig. 4. (Color online) Calculated projected partial density of state (PDOS) plots for spinpolarized Nd5Ge3 intermetallic compound using LSDA+U +SO approach.

Table 2 summarizes the total and local spin magnetic moments for RE5Ge3compounds. For these compounds, our spin polarized calculation shows that the

spin magnetic moments on RE (Ce, Pr and Nd) atoms are very important. Of

course, for RE5Ge3 the spin magnetic moments are found to be due to the rare earth



Fig. 5. (Color online) Calculated projected total density of state plots for RE5Ge3 intermetalliccompounds using LSDA+U +SO approach.

Table 2. The calculated local and total magnetic moments of RE5Ge3 (RE = La, Ce, Pr and Nd)

intermetallic compounds using LSDA, LSDA+U and LSDA+U +SO approaches.

RE5Ge3 Exc MSpin (RE1) MSpin (RE2) MSpin (Ge) M (Total)

LDA 0.042 0.221 −0.031 1.217

Ce5Ge3 This work LDA+U 0.128 0.392 −0.017 2.973

LDA+U +SO 0.111 0.367 −0.016 2.742

Ref. 20 2.660 2.660 — —

LDA

This work LDA+U 2.183 2.143 −0.0200 23.233

Pr5Ge3 LDA+U +SO 2.336 2.170 −0.022 24.378

Ref. 26 3.580 3.580 — —

Ref. 16 3.800 3.800 — —

LDA 3.358 3.391 −0.107 36.207

This work LDA+U 3.492 3.218 −0.049 35.299

Nd5Ge3 LDA+U +SO 3.262 3.228 −0.046 34.793

Ref. 14 (Theory) 3.620 3.620 — —

Ref. 14 3.320 3.320 — —

Ref. 25 3.640 3.640 — —

atom [(Ce1 = 0.11, Ce2 = 0.36 µB/atom), (Pr1 = 2.336, Pr2 = 2.170 µB/atom)

and (Nd1 = 3.262, Nd2 = 3.228 µB/atom)] with almost no contribution from

the Ge sites using LSDA+U +SO approach. It is clear that the total magnetic

moment increase along the series going from Ce to Nd. The calculated spin magnetic



(a)

(b)

Fig. 6. (a) Contour plots of the valence–charge distribution of nonmagnetic La5Ge3 intermetalliccompound in different planes using LSDA+U +SO approach. (b) Spin density and total electrondensity contours of Pr5Ge3 in (110) plane using LSDA+U +SO approach.

moments at the RE sites are found to be in good agreement with the experimental

estimate.14,16,20,25,26

The total valence charge density is displayed by LSDA+U +SO approximation

for La5Ge3 compound (Fig. 6(a)) in different planes containing the different atoms.



The bonding charge density shows a depletion of the electronic density at the lattice

sites together with an increase in the electronic density in the interstitial region.

The covalent character of the bonding is dominant and the build-up of bonding

charge along the RE–Ge bond direction is very strong. This feature is consistent

with the PDOS plots, in Fig. 4, showing the importance of the RE-d-f and Ge-p

hybridization, associated with the RE–Ge bonding. A significant redistribution of

the bonding charge in the interstitial region is also seen. The bonding between RE

and Ge is not so strong than the RE–RE bonding. From Fig. 6(a), the Ge–Ge

bonding is also stronger than the RE–Ge and the RE–RE bonding.

Figure 6(b) shows the total electron density and the spin density (the difference

between the electron density of spin-up and spin-down) in plane perpendicular to

the symmetry axis containing Pr and Ge atoms. Germanium spin density in the

vicinity of the spin density of the Pr atoms is of opposite sign. From this figure, it

becomes clear why the Germanium magnetic moments are so small and negative:

they result from the combination of positive and negative 4p-orbital spin densities

on each Ge atom. It is also clear that the origin of magnetism comes from the rare

earth element, since the spin density presents a spin magnetic moment which is

mainly due to the 4f orbitals.

4. Conclusion

To conclude, we have carried out DFT-based first-principles calculations to

investigate the electronic and magnetic structure of RE5Ge3. Within the

LSDA+U +SO, the structural, electronic, and magnetic properties were studied for

the hexagonal phase of RE5Ge3 (RE = La, Ce, Pr and Nd) intermetallic compounds.

The calculated lattice constants and magnetic moments turned out to be in good

agreement with available experimental data. The effect of the Hubbard correction

and the spin-orbit interaction are investigated and are found to be a necessary

requirement for the accurate description of correct state of these compounds. The

electronic band structure, density of states and the bonding charge density have

also been calculated to elucidate the bonding mechanism in these compounds and

the results indicate that bonding is mostly of covalent nature. The magnetism

originated mainly from the RE atoms, in all magnetic compounds.

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AB INITIO STUDY OF ELECTRONIC AND MAGNETIC STRUCTURE OF INTERMETALLIC RE 5 Ge 3 COMPOUNDS

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