first principles calculations of al-rich re (re = ho, er, tm and yb) intermetallic compounds

© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Phys. Status Solidi B 246, No. 6, 1206–1214 (2009) / DOI 10.1002/pssb.200844352 p s sbasic solid state physics

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First principles calculations of Al-rich RE (RE = Ho, Er, Tm and Yb) intermetallic compounds

Vipul Srivastava*, 1, Gitanjali Pagare2, Sankar P. Sanyal1, and M. Rajagopalan3

1 Department of Physics, Barkatullah University, Hoshangabad Road, Bhopal, Madhya Pradesh 462 026, India 2 Department of Physics, M. L. B. Girls Govt. P. G. College, Bhopal 462 011, India 3 Crystal Growth Centre, Anna University, Chennai 600 025, India

Received 1 September 2008, revised 9 December 2008, accepted 9 January 2009

Published online 27 February 2009

PACS 62.20.–x, 65.40.–b, 71.15.Mb, 71.15.Nc, 71.20.Lp

* Corresponding author: e-mail [email protected], Phone: +91 755 2491835, Fax: +91 755 2491823

© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction Intermetallic compounds are among the longest studied human-made materials, being the sub-ject of constant interest for inorganic chemists, physicists and material scientists [1–3]. Intermetallic compounds consist of two or more metals combined with specific stoichiometries by mixed metallic, covalent and ionic bonding. Their chemical, physical, electrical, magnetic and mechanical properties are often superior to those of ordi-nary metals, but their enormous potential for improving engineering performance remains largely unused because they are brittle and fracture easily at room temperature. These compounds have attracted considerable attention due to their unique mechanical properties such as high ten-sile strength, good ductility, high corrosion resistance and thermal stability [4]. The Al-rich rare earth (RE) intermetallics, namely AlRE (RE = Ho, Er, Tm, Yb and Lu), crystallize in the cu-bic CsCl-type structure (B2, Pm3m, space group no. 221). The CsCl-type AlRE intermetallics are very significant

with respect to their thermal and electronic properties [5, 6]. These intermetallics are of significant importance to novel material design and further scientific and technical investigations. A first principles study of the phase stability of Ti–Al intermetallic compounds was performed by Asta et al. [7]. There are a number of intermetallics [8] that have similar CsCl-type structures and have been studied ex-perimentally. The structure of binary alloys of rare earth metals with Ag and Au was investigated by X-ray diffrac-tion and was reported to be a CsCl-type structure [8]. Some experimental information regarding the structure of AlRE equiatomic intermetallics can be found in the Ref. [9]. In AlRE the rare earths have different occupation numbers for the shallow inner 4f shell, ranging from 0 to 14 through the series from La to Lu. This changing 4f occupation means the RE elements and their compounds have a wide range of different magnetic and electronic properties. The mixing of Al with RE elements has made a new class of highly or-dered and ductile compounds. It is evident from the litera-

The electronic structure of nonmagnetic intermetallic AlRE

(RE = Ho, Er, Tm and Yb) compounds, which crystallize in

the CsCl-type (B2) structure, is studied by means of the self-

consistent tight binding linear muffin tin orbital method at

ambient as well as at high pressure. The total energies are

computed as a function of volume and fitted to the Birch

equation of state. The ground-state properties such as the

equilibrium lattice parameter, the bulk modulus and the cohe-

sive properties of these compounds are calculated. Interest-

ingly, in AlTm and AlYb, f-like electrons may get delocal-

ized and show fluctuations in the total density of states.

Thermal properties like Debye temperature and Grüneisen

constants are estimated within the Debye–Grüneisen model.

The results obtained are compared with the available experi-

mental values and other theoretical results. We find good

agreement between theory and experiment. The variation of

Debye temperature with pressure is also studied. We also pre-

sent electronic charge density plots to reveal bonding proper-

ties in the intermetallics under investigation.

Phys. Status Solidi B 246, No. 6 (2009) 1207

www.pss-b.com © 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Original

Paper

ture that the role of f-electrons in RE ions with Al in struc-tural, electronic and elastic properties at both normal and high pressures has not been extensively studied and is yet to be explored in detail. The mechanical and thermody-namic properties of some CsCl-type AlRE intermetallics were explored by Tao et al. [6]. They used the projector augmented wave (PAW) method within the generalized gradient approximation (GGA). Very recently, some of the lighter AlRE intermetallics have been investigated by us [10] with respect to their electronic and thermal properties, where we have predicted unusual behaviour of AlLa, AlCe and AlPr under compression. This has motivated us to in-vestigate whether heavier AlRE intermetallics fall into this category or not. In this paper we present a first principles tight binding linear muffin tin orbital (TB-LMTO) study of a number of AlRE (RE = Ho, Er, Tm and Yb) intermetallics aiming to build further confidence in this approach and to provide important crystal chemical information for a wide variety of AlRE compounds. Theoretical data in this area can use-fully supplement experimental data in various cases where X-ray diffraction is not possible. The thermal properties are evaluated using the Debye–Grüneisen (DG) model. The calculated lattice parameters, bulk modulus, Debye temperature and Grüneisen constants are in good agree-ment with previously reported values [5, 6].

The rest of the paper is arranged as follows. The meth-odology that was used to calculate the electronic properties of the AlRE intermetallics and the first principles computa-tional details are briefly provided in Section 2.1. The De-bye temperature and Grüneisen constants, by incorporating first principles theory in the DG model, are briefly de-scribed in Section 2.2. The results and a discussion of the electronic and thermal properties of the AlRE intermetal-lics are then presented in Section 3, followed by some con-cluding remarks in Section 4.

2 Method of calculation 2.1 TB-LMTO method The self-consistent non-spin polarized TB-LMTO [11, 12] method within the atomic sphere approximation (ASA) [12] was used to calculate to-tal energy, band structure and density of states for the AlRE compounds. In the ASA, a crystal is divided into space-filling spheres centred on each of the atomic sites. Combined corrections are also induced which also account for the non-spherical shapes of the atomic spheres and the truncation of the higher partial waves inside the sphere to minimize the errors in the LMTO method. The AlRE in-termetallics crystallize in the B2-type structure, and posi-tioned at Al: (0, 0, 0); and RE: (0.5, 0.5, 0.5). In the present calculation this method does not include empty spheres because the AlRE intermetallics have the B2-type close-

Figure 1 Variation of total energy with

relative volume for AlRE intermetallics.

1208 V. Srivastava et al.: First principles calculations of Al-rich RE intermetallic compounds

© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.pss-b.com

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packed structure. As is already known [10, 13–15], the TB-LMTO method is best suited for close-packed struc-tures. For the exchange–correlation potential within the local density approximation (LDA), the parameterization scheme of van Barth and Hedin [16] was used. The Wigner–Seitz sphere radius was chosen in such a way that the sphere boundary potential is a minimum and the charge flow between the two atoms is in accordance with the elec-tronegativity criteria. The E and k convergence was conse-quently checked. The tetrahedron method [12] of Brillouin zone integration was used to calculate the total density of states. The total energy was computed by reducing the vol-ume from 1.05V0 to 0.65V0, where V0 is the equilibrium cell volume. The calculated total energy was fitted to the Birch equation of state [17] to obtain the pressure–volume (P–V) relation. The pressure was obtained by taking the volume derivative of the total energy. The bulk modulus B = −V0 dP/dV was also calculated from the P–V relation. 2.2 DG model To calculate the important thermal properties of a vibrating Debye lattice we used the DG model [18]. The Debye temperature θD was calculated us-ing the following expression:

D67.48 ,

rB

Mθ = (1)

where r is in a.u., B is the bulk modulus in kbar and M is the average atomic weight, which is the arithmetic average of the masses of the species in the compound. In Eq. (1) if we substitute the calculated (or experimental) value of bulk modulus at r = r0 (at equilibrium) the calculated value of (θD)0 deviates from the experimental value. To overcome this problem a scaling factor was introduced. In the modi-fied expression, the theoretical value of r at r = r0 and bulk modulus derived from the above first principles calculation

Table 1 Calculated lattice parameter a, bulk modulus B0, Wigner–Seitz radius r0, Debye temperature (θD)0 and Grüneisen constant γ0 for CsCl-type AlRE (RE = Ho, Er, Tm and Yb) under ambient conditions.

solid a (Å) B0 (GPa) r0 (Å) (θD)0 (K) γ0

this work 3.4758 65.85 1.7114 196.16 1.0655AlHo previous work

3.580a 65.50a – 233b 1.209b

this work 3.4799 65.43 1.7129 194.44 1.0632AlEr previous work

3.565a 66.43a – 232b 1.132b

this work 3.4826 65.57 1.7598 1.0569AlTm previous work

3.554a 66.79a – 231b 1.120b

this work 3.5000 62.54 1.7228 189.53 1.0526AlYb previous work

3.697a 39.11a – 192b 1.319b

a Ref. [5];

b Ref. [6].

Figure 2 Non-spin polarized band structure under ambient con-ditions for AlRE intermetallics.



Original

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at r0 were substituted and the Debye temperature (θD)0 can be calculated using the expression

0

D41.63 .

r B

Mθ = (2)

Details of the prefactor 41.63 used in Eq. (2) can be found elsewhere [19]. The Grüneisen constant γ was calculated [19] using

Dln

,lnV

θγ

∂=

∂ (3)

where V is the atomic volume. 3 Results and discussion Modern ab initio simula-tion provides a powerful tool to study structural and elec-tronic properties with reasonable confidence. Therefore in order to investigate the electronic and thermodynamic properties of the AlRE intermetallics we computed the band structure, density of states, bulk modulus, cohesive energy, etc. The non-spin polarized electronic band struc-ture calculations were carried out to estimate the total en-ergy of the AlRE intermetallics using the first principles TB-LMTO method. The total energy was plotted against different compressions, as shown in Fig. 1. The minimum of the curves defines the equilibrium volume V0 (or equi-librium separation r0), which is found to be 41.9947 Å3 for AlHo and the corresponding lattice parameter a = 3.4758 Å. Similarly, for the other AlRE intermetallics the equilibrium volumes are 42.1405 Å3, 42.2387 Å3 and 42.875 Å3 and the corresponding lattice parameter values are 3.4799 Å, 3.4826 Å and 3.5 Å for AlEr, AlTm and AlYb, respectively (Table 1). We compared our results with those of previous theoretical studies (also given in Table 1). A good agreement is established. The self-consistent band structures along the high-symmetry directions for all the intermetallics were obtained and are shown in Fig. 2. On the whole, the band profiles are seen to be almost the same for all the AlRE compounds except movement in f-like states. It is seen from Fig. 2 that the lowest lying band in these compounds is mainly due to the Al s-like state, lying at around −0.58 Ry relative to the Fermi level. It is well separated from the other states. The bands lying above this and just below the Fermi level (approximately −0.2 Ry) are mainly due to Al p-like states, which are hybridized with the RE p-like states. The d-bands of the RE atoms generally lie above the Fermi level and are hybridized with Al p-like states near the Fermi level. Some d-bands just touch the Fermi level. The cluster of bands that are situated at the Fermi level are the RE f-like states hybridized with Al p-states and are responsible for the strong metallic behav-iour. The density of states (DOS) plot provides an even more comprehensive picture of the elemental contributions to the electronic structure of the AlRE intermetallics. Fig-ure 3 shows the partial DOS for AlHo, AlEr, AlTm and

AlYb at ambient pressure. Al s-like states start at around −0.58 Ry for all the AlRE intermetallics, while at −0.2 Ry the peak is predominantly due to Al p-like states, which hybridize with RE d-like states. The RE f-like state is rep-resented by a sharp peak near EF for all the AlRE intermet-allics. The hybridization of RE f-like states with d-states and Al p-states at the Fermi level can be seen. For AlHo the sharp peak (f-like states) corresponds to a DOS value of ~800 states/Ry cell, and for AlEr, AlTm and AlYb these values are ~1800, ~1700 and ~900 states/Ry cell, respec-tively. One can understand the reason for this increase and then decrease in DOS for these four intermetallics from the fact that in the periodic table as one moves from Ho to Yb the number of f-electrons increases. The DOS value of ~800 states/Ry determined for AlHo increases to ~1800 states/Ry for AlEr, and then in the case of AlTm starts decreasing to a value of ~1700 states/Ry. But in the case of AlYb a sharp decrease in the DOS value to ~900 states/Ry is calculated. This sharp decrease in DOS could be the reason that f-electrons in Yb get delocalized even at ambient pressure. The delocalization of f-electrons will be discussed later in this section. We also calculated the total DOS at the Fermi level under compression for all the AlRE intermetallics. Figure 4 shows the total DOS at EF under compression for AlHo, AlEr, AlTm and AlYb. One notices a linear decrease in DOS as we compress the first two of these intermetallics. The values of DOS are calculated to be from 575.02 to 337.91 states/Ry cell for AlHo and 598.29 to 256.07 states/Ry cell for AlEr as V/V0 changes from 1 to 0.70, respectively. In contrast, an increase in DOS is calcu-lated for the other two AlRE intermetallics, as also shown in Fig. 4. For AlTm an increase in DOS is seen for a com-pression value of V/V0 between 0.80 and 0.75. In the case of AlYb such an increase in DOS occurs twice, firstly for compression V/V0 = 1.00 to 0.95 and secondly for V/V0 = 0.85 to 0.75. In order to understand such increase and decrease in the DOS value we have plotted only Yb f-states for various compression values in Fig. 5. From an analysis of this figure we could conclude that the fluctua-tion in total DOS is due to unstable f-states. A similar study is reported for the lighter AlRE intermetallics in our very recent paper [10]. The anomalous behaviour of the DOS for AlTm and AlYb can be compared with the f-electron system of lighter AlRE (RE = La, Ce and Pr) in-termetallics. Such anomalous behaviour can be related to the delocalization of f-electrons, which seems to be greater in Yb than in Tm. In another way this delocalization of f-states may also be due to the possibility of structural phase transformations in AlTm and AlYb under compres-sion. We suggest a future extension of the present study at high pressure for consideration by experimentalists. To reveal the bonding properties in the AlRE intermet-allics the electronic charge density plots under ambient as well as under compression for the [110] direction for AlHo, AlEr, AlTm and AlYb are shown in Fig. 6. The corner at-oms are Al, while the central atoms represent the RE (Ho,



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Figure 3 Partial density of states for AlRE intermetallics.



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Figure 5 Variation of Yb f-states under compression for AlYb.

Er, Tm and Yb) atoms. From the plots one observes that the bonding in these materials is metallic in nature at am-bient conditions. Under compression there is a delocaliza-tion of f-like electrons, which can be seen from the band structure plots in Fig. 7. The extent of delocalization in-creases one moves from Ho to Yb. This can be seen also from the charge density plots. The delocalization of f-like electrons is responsible for conductivity, ductility and hardness properties in these intermetallics. In condensed matter f-electrons play a crucial role in comparison with spd valence electrons. The bonding properties of conven-tional materials are well described in the framework of density functional theory (DFT). But the intricate nature of f-electrons makes their quantum mechanical behaviour much more complicated than that of ‘normal’ electrons in solids. The LDA functional used in the present calculation within DFT has some limitations [20]. The results obtained from the LDA functional may differ from experimental values. The following is a brief overview of LDA limita-tions at a glance, most of them not being relevant in the present context. (i) Over binding in LDA: small lattice constant, large cohesive energies and high bulk modulus. The over binding can largely be corrected by the introduc-tion of the GGA. (ii) Neglect of strong correlations: ex-change splitting underestimated for narrow d- and f-bands. This can be solved by using LDA + U [21, 22], self-interaction corrected scheme [23]. (iii) Neglect of van der Waals interactions: van der Waals forces arise from mutual dynamical polarization of the interacting atoms not included in the LDA functional. (iv) Band gap problem:

Figure 4 Variation of density of states at the Fermi

level with relative volume for AlRE intermetallics.



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Figure 6 Electronic charge density plots

for AlRE intermetallics in the [110] direc-

tion a) under ambient conditions and b) un-

der compression (V/V0 = 0.65).



Original

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Figure 7 Non-spin polarized band structure for AlRE intermetal-

lics under compression (V/V0 = 0.65).

Figure 8 Variation of Debye temperature with pressure for AlRE

intermetallics.

band gaps in semiconductors and insulators are always un-derestimated. However, various schemes [21–25] have been presented to overcome these problems, but in the pre-sent paper we have tried to emphasize more the physics to explore precise structural, electronic, mechanical and thermal properties of the AlRE intermetallics under inves-tigation using the LDA functional. In the present work we also calculated the thermal properties of AlRE compounds using the DG model [18]. This method has gained wide use because of its simplicity. In this method, the Debye temperature θD [19] can be ob-tained from the calculated values of bulk modulus and the Grüneisen constant γ can be obtained from Eq. (3). The bulk modulus (B0), Wigner–Seitz radius (r0), Debye tem-perature (θD)0 and Grüneisen constant (γ 0) at absolute tem-perature were calculated for all AlRE intermetallics and compared with other theoretical calculations [5, 6] in Ta-ble 1. Our results show good agreement. We varied θD with respect to pressure for all the AlRE intermetallics, as shown in Fig. 8. It can be seen that as pressure increases, θD also increases for all cases up to 500 kbar, which is quite natural. No decrease in θD is noticed up to the very high pressure region of 500 kbar. In the literature several relations have been derived that correlate θD and bulk modulus by a square root law. An early attempt in this di-rection was made by Madelung [26] and Einstein [27]. It was found that by averaging the elastic constants of the transverse acoustic phonon modes as the elastic modulus instead of the bulk modulus, the square root law is estab-lished with high precession. In future work, we will incor-porate this model to get more accurate results. 4 Conclusions A theoretical study of the electronic and thermal properties of some heavier AlRE (RE = Ho, Er,



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Tm and Yb) intermetallics has been presented using com-bined first principles TB-LMTO and DG model, respec-tively. All the AlRE intermetallics crystallize in the B2 structure, which is in agreement with previous theoretical and experimental work. It is found from the present study that the AlRE intermetallics are metallic in nature. The pressure-induced variation in DOS at EF was calculated. The value of DOS decreases almost linearly for AlHo and AlEr; conversely, it increases for AlTm and AlYb. We tried to determine the reason for such behaviour in these AlRE intermetallics and concluded that delocalization of f-electrons under pressure could be the reason. One can also correlate this delocalization of f-electrons with the possibility of some kind of structural phase transformations in the latter two intermetallics. We also investigated the nature of the bonding in the AlRE intermetallics under am-bient and under compression. We found metallic bonding between Al and RE atoms under both conditions. We used the DG model to calculate the Debye temperature and Grüneisen constant using the bulk modulus and the sphere radius obtained from the TB-LMTO study. A linear in-crease in θD with pressure is noticed up to 400 kbar. No experimental study has been reported in the literature on the thermal properties and structural phase transitions in such intermetallics. We have therefore presented results at ambient as well as at high pressure and open the way for experimental studies. Our results are in good agreement with previous theoretical results.

Acknowledgements V.S. is grateful to DST, New Delhi,

for a Young Scientist award under the Fast Track scheme (DST

project no. SR/FTP/PS-30/2005) and for financial support. G.P. is

grateful to MPCST, Bhopal, for a Research Project award and fi-

nancial support.

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first principles calculations of al-rich re (re = ho, er, tm and yb) intermetallic compounds

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