Electronic Structure and Properties of the Alkaline Earth Monosilicides

ARTICLE pubs.acs.org/JPCC

Electronic Structure and Properties of the Alkaline Earth Monosilicides Eduardo Cuervo Reyes and Reinhard Nesper* Laboratory of Inorganic Chemistry, ETH-Zurich, Switzerland ABSTRACT: In continuation of a previous work, the electronic structure of the alkaline earth monosilicides (ÆSi; Æ = Ca, Sr, and Ba) is thoroughly investigated here. We employ several first principles codes at the density functional theory level. ÆSi are found to be metallic with a low density of states at the Fermi level. The conductivity takes place over a network of metal d- and silicon p-states. We discuss how increasing cation size causes not only a flattening of the bands but also a depopulation of silicon antibonding states and the shrinking of the Fermi surface. The hole-like branch of the Fermi surface is more anisotropic than the electron-like; the anisotropy of the former seems to grow, approaching two-dimensional metallic features, as the cation size increases. Estimated heats of formation are in fairly good agreement with previous experimental results.

’ INTRODUCTION The study of the electronic structure of silicides has been of great interest; on one hand, because metal
silicon interaction plays a relevant role on the properties of interfaces in electronic devices, and on the other hand, because they are genuine examples for the fundamental understanding of Zintl phases and intermetallics. Around a decade ago, alkaline earth monosilicides (ÆSi) regained attention because of their capability to reversibly absorb hydrogen, revitalizing the studies on their structure
properties relationships and those of the hydrogenated phases.1,2 In a previous work,3 we showed that ÆSi also have an intriguing property; they are weakly magnetic due to the polarization of low density of charge carriers. This magnetism is very sensitive to impurities. Æ atoms can be partially replaced by rare earth metals. It has been reported4 that Ba1
xEuxSi exists as a solid solution for any x ∈ [0;1] and that the magnetic response of these diluted magnetic phases depends on x in a nontrivial way. For most x values, Eu atoms may be statistically distributed, and therefore, modeling their electronic structure is a very challenging task. Being ÆSi is at the edge of this solid solution, mastering their properties constitutes a pillar for the understanding of the more complex ternary family. ÆSi, with Æ = Ca, Sr, and Ba, crystallize in the CrB structure; space group Cmcm, No. 63.5
7 Silicon ions (formally Si2
) fulfill the geometrical conditions for a six valence electron species (Figure 1). They form polyanions in the form of zigzag chains, 2
1 ∞[Si ], which run along the c axis, and are ecliptically stacked in the a direction. From Zintl’s viewpoint,8
19 Æ2+ cations are isolated; meaning that silicon derived bands may be considered to be full. To date, it is known20
23 that compounds having ecliptically stacked silicon polyanions have high electric resistivity and metallic behavior. This, if doping is excluded, has been related to the depopulation of silicon-derived antibonding states π*. r 2011 American Chemical Society

Figure 1. Primitive cell of ÆSi structure. Æ atoms are represented by black spheres; silicon is in gray.

First principles studies of the electronic structure of silicides have not been well explored, to our knowledge. A detailed discussion has only been presented for CaSi,23 where the trends in bonding character and electron distribution in the CaxSiy family, were investigated; reporting that for low silicon content, the compound is a semicontuctor and for high silicon content, metallic. This has been explained by the fact that materials containing more silicon tend to form polyanions, leaving less orbitals avilable for lone pairs and pushing the antibonding states up above the calcium orbitals. The effect of the cation size on the bandwidth of the monosilicides have been briefly commented by Watanabe et al.,24 where they argued the reduction of the bandwidth with the increase of the cation weight. Their discussion was limited to compare the separation between the principal peaks in the density of states. As we indicated in a former article,3 we dedicate the present manuscript to discuss the electronic structures of CaSi, SrSi, and BaSi. We present here our estimates for the charge distribution, band structure, Fermi surface, density of states (DOS), cohesive Received: June 21, 2011 Revised: December 5, 2011 Published: December 13, 2011 2536

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The Journal of Physical Chemistry C energy (εcoh), and heat of formation (ΔHf°), and we comment on the dependence on the cation size.

’ SETTINGS OF THE CALCULATIONS The calculations presented in this article were performed using three different DFT codes; CASTEP25 and DMol3 packages from Materials Studio, and the code based on linear-muffintin-orbitals (LMTO) from O. K. Andersen’s group.26
31 The reasons for using more than one algorithm are manifold. First, it may be interesting to see how the charge partitioning looks when the bands are projected onto the basis sets employed in each method. Second, every code has pros and cons, and we can chose to use the best modules from each of them. Third, different features are available in each program; therefore, a more complete description can be achieved by combining these programs in a suitable way. Within the framwork of CASTEP, we have chosen ultrasoft pseudopotentials and the exchange-correlation functional from Perdew
Burke
Erzenhof (PBE)32 in the generalized gradient approximation (GGA). Pseudoatomic calculations were done by employing the reference configuration Si 3s23p2 and Æ ns2np6(n + 1)s2 with n = 3, 4, and 5 for Ca, Sr, and Ba, respectively. The plane wave cutoff was set to 310 eV and the k-space was sampled over a 24  24  26 mesh following the Monkhorst
Pack procedure.33 Convergence of the total energy was tested as a function of the number of k-points (already shown for BaSi in ref 3); 0.1 eV were applied for the Gaussian smearing of the states; systems were treated as metallic. The self-consistency (SCF) tolerance was set to 0.5  10
6 eV for the total energy per atom, 10
7 eV for the eigenvalues, and 10
8 eV for the Fermi level. Optimization of lattice constants and atomic positions reveals that the optimized positions differ by less than 10
3 Å from the experimental ones; the differences in total energies being of the order of 10
3 eV. The charge spilling due to the use of the default projector operator was larger that 1% for SrSi and BaSi; thus, calculations generating the pseudopotentials on the fly (OTF) were repeated, with a larger number of states in the projector. With this, the spilling became less than 1% (see Table 1), which is a good reference. The charge transferred from the metal to the silicon was reduced. That can be understood by the need of including excited atomic states in order to properly project delocalized electrons. Table 2 shows the identity difference for each of the atoms involved. Values in the order of 10
3 are already acceptablel and thereforel we did not check for possible improvement including higher excited states. The DMol3 uses spherical harmonics, which are well suited for the partitioning of atomic charges onto partial waves. Our calculations were performed using the radial basis set doublenumerical (DNP) 3.5, which has proven to be very efficient.34 Two GGA exchange correlation potentials were tested: the PBE and the one due to Perdew and Wang.35 The number of k-points was chosen to give a separation of less than 10
2 Å
1 between them. The SCF tolerance was set to 10
6 Ha, and in order to ease the convergence, the orbital smearing was set to 5  10
4 Ha. DMol3 also features relativistic corrections. Including relativistic corrections allocates a larger amount of charge in the valence s orbitals, as expected due to the relativistic contraction, with a reduction of the charge in p and d states and a smaller charge transfer toward silicon. The influence of this correction on the cohesive energy is negligible for Ca and CaSi but important for

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Table 1. Charge Spilling and Transfered Charge (Mulliken) Obtained with Pseudopotentials from the Database (DB) and Generated on the Fly (OTF) within CASTEP carge spill. (10
2)

ion charge (e)

material

DB

OTF

DB

OTF

CaSi

0.60

0.60

0.82

0.82

SrSi

2.36

0.37

0.99

0.65

BaSi

1.48

0.26

0.77

0.57

Table 2. Projector-Identity Difference (ΔI) atom

Si

Ca

Sr

Ba

ΔI (10
3)

0.3

1.5

3.3

7.8

the heavier elements. Consequently, results that include relativistic corrections are presented. LMTO calculations were performed using the atomic sphere approximation (ASA), version LMASA-46, and in order to minimize the overlap, empty spheres were introduced with radii between 1.0 and 4.5 au. The SCF tolerance was set to 10
5 Ry for the total energy and 10
5 e for the rms of the change of atomic charges. For silicon atoms, orbitals (nlm) with n < 3 were treated as core states and, for Æ atoms, orbitals with energies below the valence s orbital were considered as core states; l = 2 was the highest partial wave considered. Broyden mixing was employed in the self-consistency algorithm. LMTO makes affordable the k-space integration over very fine mesh; grids of 32  32  32 points were chosen, and integrals were calculated using the tetrahedral method.36,37 Systems were treated as metallic and the Fermi level was determined with an accuracy of 10
6 Ry. The results obtained by all different methods are consistent. Just for an overview, the total DOS are compared in Figure 2; a closer look will be taken during the discussion in the next section. In Figure 2, DM stands for density mixing algorithm, and All, for the standard DFT, both within the CASTEP code. PW-UPS means that graphs denoted as such correspond to calculations using plane waves and ultrasoft pseudopotentials. The label TBLMTO-ASA, in the right panels, is self-explanatory. In the left panel, the black line corresponds to DOS obtained with the pseudopotentials (PS) from the database, while the gray line shows those obtained generating the PS on the fly. We chose to present the energy bands and DOS in an energy window of approximately 1 Ry (from
12 eV to 2 eV with respect to the Fermi level), excluding the core (or semicore) states and excited states well above EF. DOS are reported in units of states per eV per primitive cell (two formula units). In the charge partitioning, only one entry per atom type is presented since there is only one unequivalent crystallographic position per atomic species, the other positions being symmetry-related and thus, electronically equivalent.

’ RESULTS AND DISCUSSION The band structures and the projected DOS from CASTEP calculations are shown in Figure 3. In average, there are six bands below EF, which account for the allocation of the 12 active electrons; two per Æ- and four per silicon-atom. The bands are 2-fold degenerate at kz = 1/2 because there are two equivalent formula units along the z direction, and this doubling of the cell 2537

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Figure 2. Density of states of CaSi, SrSi, and BaSi by different methods. Description is given in the text.

content results in the folding of the Brillouin zone at kz = 1/2. The bands are grouped into two blocks. Starting from lower energies, the first feature in the DOS appears between
10 eV and
5 eV for the three compounds. These two bands are mainly silicon s-states with a small contribution of silicon p-states. This small p contribution arises from the nonorthogonality of the s orbital with respect to two of the p orbitals of neighboring silicon atoms. The dispersion of the s-bands (from Γ to Z, T to Y, and S to R) corresponds to the strong bonding along the chains; being rather flat in the other directions. The bandwidth decreases from CaSi to BaSi. This can be explained by the reduction of the orbital overlap with the lengthening of the silicon
silicon nearest and next nearest neighbor distances along the chain, due to the increase of the cation size. However, the relative contribution of Si-s, Si-p, and Æ orbitals does not change. This rigidity or stability of the structure is consistent with the broad range of cation substitution supported by these silicides. The gap around
8 eV (between the

Figure 3. Band structure of ÆSi. The labels for the symmetry points in the Brillouin zone represent: Γ =(000), Z =(00(1)/(2)), T = ((1)/ (2)(1)/(2)(1)/(2)), Y =((1)/(2)(1)/(2)0), S =(0(1)/(2)0), R =(0(1)/ (2)(1)/(2)). Partial DOS: s-, p-, and d-states in dotted, dashed, and solid lines, respectively.

two s-bands), observed in the DOS calculated with CASTEP, is just an artifact (or a bug) of the k-integration implemented in this code. One can see from the band structure that this gap does not really exist. These two s bands are degenerate at kz = 1/2, or in other words, they come from a single Si band, after folding the Brillouin zone. At this point, it may be worth making a connection between bands and the chemical bond pictures. The two silicon-s bands are full; still, they should have a bonding character. In order to understand this, we must admit that there is a small contribution of cation orbitals in these bands; the bond is not just silicon
silicon but a combination of Si and Æ orbitals. Of course, because of the large energy difference between Si and Æ states, the bonding 2538

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Table 3. Mulliken and Hirshfeld Charge, Si
Si Overlap (ovl) and DOS at the Fermi Level from CASTEP CaSi

SrSi

BaSi

cation s electrons

0.32

0.30

0.23

cation p electrons cation d electrons

0.00 0.86

0.00 1.06

0.00 1.23

silicon s electrons

1.55

1.54

1.56

silicon p electrons

3.26

3.11

3.01

Mulliken charge

0.82

0.65

0.57

Hirshfeld charge

0.11

0.14

0.14

Si
Si ovl population

0.91

1.03

1.09

DOSEF (states/eV cell)

1.45

1.01

0.67

Table 4. Mulliken and Hirshfeld Charge and DOS at the Fermi Level from DMol3

Figure 4. DOS of ÆSi from LMTO. Area under black line, total DOS; under gray border, Æ DOS; black area, Æ-d DOS; in red, DOS in empty spheres.

bands are almost pure silicon states, and the antibonding states have a dominant contribution from Æ orbitals. This goes back to

CaSi

SrSi

BaSi

cation s electrons

0.50

0.42

0.35

cation p electrons

0.21

0.15

0.03

cation d electrons

0.25

0.61

0.87

silicon s electrons

1.67

1.67

1.71

silicon p electrons

3.30

3.10

2.98

silicon d electrons Mulliken charge

0.06 1.04

0.05 0.82

0.05 0.75

Hirshfeld charge

0.15

0.15

0.14

DOSEF (states/eV cell)

1.43

1.40

1.40

the well-known fact that most anions do not exist without the cation environment. The second group of states, from
4 eV to EF, is mostly a mix of silicon p and Æ d and s valence orbitals. These bands account for the binding in the a and b directions and the metallic conductivity. As one moves down in the Æ group, from lighter to heavier metal, we observe (Figures 3 and 4) a reduction of the interband distances and a narrowing of the peaks in the DOS. This trend can be explained by the decrease of the distance between Æ atomic levels and the reduction of Si
Si and Æ
Si overlaps. On this same line, as we move from Ca to Ba, there is a transfer of the charge from the Si-p toward Æ-d orbitals (Tables 3 and 4), implying a further depopulation of π* states. Si
Si overlap population is also shown in Table 3; (negative) positive values indicate (anti)bonding. There is an increase in the bonding character of the Si
Si overlap as one goes from Ca to Ba. Since an increase in the cation size results in a reduction of the overlap between silicon atoms, the larger positive values of the overlap population can be only achieved with a depopulation of the antibonding states, i.e., the aforementioned transfer of electrons from the π* toward Æ-d states. ÆSi are compounds for which the Zintl
Klemm concept seems to be adequate, predicting a closed shell configuration (silicon atoms, with two single bonds and two lone pairs, fill their shells with the two electrons donated by every divalent cation). At the same time, the ÆSi are bad metals, and as such, they adhere to the (almost a rule) general phenomenon of metallic behavior observed in silicides, which contain ecliptically stacked polyanions. Looking at the intersects with the Fermi level, we could infer an anisotropic conductivity preferential in the a
c plane. Of this, however, we have no experimental evidence. Only pellets from 2539

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Fermi level laying in a region of low DOS; i.e., small Fermi surface. The actual values of the calculated DOSEF are slightly method-dependent but in all cases between 0.7 eV
1 and 1.5 eV
1 per unit cell (Tables 3 and 4 and Figure 4). This is in good agreement with the results obtained by Bisi et al.23 for CaSi. The Fermi surfaces (ΣF) are shown in Figure 5. The red contours are the paths in reciprocal space used for the drawing of the band structures in Figure 3; the Brillouin zones are shown in gray, and the real space primitive cells are shown in black. A Fermi surface contains as many branches as there are bands crossing the Fermi level. Every branch devides the k-space into empty and occupied regions. In Figure 5, the (gray) blue side faces the (un)occupied states. In CaSi, ΣF has three branches, one holelike and two electron-like. In SrSi and BaSi, ΣF has two branches, one electron-like and one hole-like. The Fermi surface shrinks as one moves down in the group. The reduction of the Fermi surface affects mostly those regions where the normal vector has a nonvanishing b component; this must cause a reduction of the mobility of the charge carriers in the b direction. The topology and small size of the Fermi surfaces make the electronic properties of these materials highly sensitive to impurities. Either magnetic polarization or charge localization can be triggered. However, the expected small size of these effects makes them difficult to measure and to engage in further applications. The partial substitution of the Æ atoms by magnetic rare earths cations may lead to a variety of interesting effects, which is the subject of current investigations that will be addressed in future publications.4 Coming back to the current subject, we cannot predict in a straightforward way how the conductivity should change from CaSi to BaSi since it also depends on electron scattering processes. From the Fermi surfaces, we infer that CaSi has the highest conductivity. Although this is not conclusive, it is in agreement with four-point measurements performed on pellets.1 We have also estimated the cohesive energy and the heat of formation of the monosilicides. The former is defined as the difference between the energy of the solid and the sum of the energies of the constituent atoms infinitely apart. It is implemented in DMol3. There are several approximate ways of calculating ΔHf°. Since ΔHf° is, in general, several orders of magnitude smaller than the total energy, it is difficult to obtain good estimates. The standard heats of formation per atom were computed as the weighted differences in cohesive energies between the compound and the elemental solids (in their standard structure modification: Si diamond, Ca and Sr fcc, and Ba bcc), ΔHf oðAx By Þ ¼

Figure 5. Fermi surface of CaSi (top), SrSi (middle), and BaSi (bottom) with blue sides toward occupied states. In gray, the Brillouin zone. In red lines, the path used for drawing the band structures. In black, the edges of the primitive cell.

powder have been measured,1 showing an overall low conductivity and metallic behavior with very small temperature coefficients. Small crystal size and their high reactivity have prohibited performing higher quality single crystal measurements. Our calculations predict CaSi, SrSi, and BaSi to be metallic, with the

xεcoh ðAÞ þ yεcoh ðBÞ
εcoh ðAx By Þ x þ y

ð1Þ

We observed that the calculated cohesive energies for the elemental solids (Si, Ca, Sr, and Ba) were in all cases smaller than the experimental values reported in the literature.38 Expecting that the cohesive energy of the compounds are also underestimated by using the GGA approximation, we employed our theoretical values for both, elements and compounds, for the calculation of the formation enthalpies with eq 1. These figures are shown in Table 5. We have also included results from some previous works for comparison. ΔHf° values from ref 39 were obtained from the relationship between the formation enthalpies and the atomization enthalpies of the compound and the components. Balducci et al.40
43 obtained the formation enthalpies of CaSi, SrSi, and BaSi from the reaction enthalpy changes at room temperature, 2540

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Table 5. Cohesive Energy (εcoh) and Standard Heat of Formation at 0 K (ΔHf°) εcoh (eV/atom)


ΔHf° (eV/atom)

previous experimental and theoretical results. There is no clear dependence between the sizes of the cations and the formation enthalpies.

Si

this work

4.54

’ AUTHOR INFORMATION

Ca

expt38 this work

4.66 1.61

Corresponding Author

38

expt

1.84

this work

1.49

expt38

1.72

this work

1.75

expt38

1.90

this work

3.72

0.63

LMTO-ASA23 DFT-GGA24

3.88

0.64 0.53

Sr Ba CaSi

SrSi

BaSi

*Phone: +41 (0)44 632 30 69. Fax: +41 (0)44 632 11 49. E-mail: [email protected].

expt39

0.72

expt40

0.51

this work

3.48

0.45

DFT-GGA24

0.45

DFT-GGA42

0.48

expt39

0.69

expt43 this work

3.59

0.53 0.44

DFT-GGA41

0.50

DFT-GGA24

0.37

expt39

0.72

expt41

0.52

derived from vapor pressure measurements by the second and third law methods of analysis. Their theoretical estimates were calculated within the DFT approach in the GGA-PBE approximation, using the plane wave SCF package in Quantum-Espresso, with ultrasoft pseudopotentials. Theoretical values from ref 24 were evaluated from the total energies calculated with the CASTEP package (DFT-GGA). The data reported in literature are in general scattered. Overall, our results are in the right order of magnitude. As the difference in ΔHf° between the monosilides are smaller than the scattering of the data, it is not possible to define a relationship between the size of the cation, or any other property, with the heat of formation.

’ SUMMARY We present the electronic structures and properties of alkaline earth monosilicides CaSi, SrSi, and BaSi from a theoretical perspective, and we have compared our results with available experimental and theoretical data. To our opinion, most important results are the following. (1) The high stability of the electronic structure of the zigzag chains. (2) The narrowing of the features in the density of states with the increase of the cation size. This can be explained by the reduction of the inter atomic overlap and the reduction of the energy difference between the valence states of the metal atoms. (3) The lowering (with respect to silicon states) of the d orbitals of the Æ metals causes a depopulation of the silicon antibonding states. (4) The reduction of the dispersion of the bands has major effects at the Fermi level. The Fermi surface shrinks. Its features (mainly those of the holelike branch) become more two-dimensional. (5) ÆSi are metallic with a low DOS at the Fermi level. (6) Our estimates of cohesive energy and heat of formation are in good agreement with

’ ACKNOWLEDGMENT This work has been supported by the Swiss National Science Foundation under project no. 2-77937-10. ’ REFERENCES (1) Armbruster, M. M. No. 17553. Ph.D. Dissertation, ETH, Z€urich, 2008. (2) Armbruster, M. M.; W€orle, M.; Krumeich, F.; Nesper, R. Z. Anorg. Allg. Chem. 2009, 635, 1758. (3) Cuervo Reyes, E; Stalder, E. D.; Mensing, C.; Budnyk, S.; Nesper, R. J. Phys. Chem. C 2010, 115, 1090. (4) Spahr, M. No. 12281. Ph.D. Dissertation, ETH, Z€urich, 1997. (5) Rieger, W.; Parthe, E. Acta Crystallogr. 1967, 22, 919. (6) Eisenmann, B.; Sch€afer, H.; Turban, K. Z. Naturforsch. B 1974, 29, 464. (7) Merlo, F.; Fornasini, M. L. J. Less-Common Metals 1967, 13, 603. (8) Zintl, E.; Goubeau, J.; Dullenkopf, W. Z. Phys. Chem. A 1931, 154, 1. (9) Zintl, E.; Harder, A. Z. Phys. Chem. A 1931, 154, 47. (10) Zintl, E.; Dullenkopf, W. Z. Phys. Chem. B 1932, 16, 195. (11) Zintl, E. Angew. Chem. 1939, 52, 1. (12) Klemm, W. Proc. Chem. Soc. London 1959, 329. (13) Mooser, E.; Pearson, W. B. Progress in Semiconductors; Wiley & Sons Inc.: New York, 1960; Vol. 5. (14) Klemm, W. Festk€orperprobleme; Vieweg: Braunschweig, Germany, 1963. (15) Busmann, E. Z. Anorg. Allg. Chem. 1961, 313, 90. (16) Klemm, W.; Busmann, E. Z. Anorg. Allg. Chem. 1963, 319, 297. (17) Nesper, R. Prog. Solid State Chem. 1990, 20, 1. (18) Kautzlarich, S. M., Ed. Chemistry, Structure, and Bonding of Zintl Phases and Ions: Selected Topics and Recent Advances; Wiley-VCH: Weinheim, Germany, 1996. (19) Nesper, R. Zintl Phases Revisited. In Silicon Chemistry; Jutzi, P., Schubert, U., Eds.; Wiley VCH: Weinheim, Germany, 2004. (20) Evers, J.; Weiss, A. Solid State Commun. 1975, 17, 41. (21) Wengert, S. No. 12070. Ph.D. Dissertation, ETH, Z€urich, 1997. (22) Savin, A.; Nesper, R.; Wengert, S.; F€assler, T. F. Angew. Chem., Int. Ed. 1997, 36, 1808. (23) Bisi, O.; Braicovich, L.; Carbone, C.; Lindau, I.; Iandelli, A.; Olcese, G. L.; Palenzona, A. Phys. Rev. B 1989, 40, 10194. (24) Imai, Y.; Watanabe, A. Intermetallics 2002, 10, 333. (25) Segall, M. D.; Lindan, P. J. D.; Probert, M. J.; Pickard, C. J.; Hasnip, P. J.; Clark, S. J.; Payne, M. C. J. Phys.: Condens. Matter. 2002, 14 (11), 2717. (26) Andersen, O. K. Phys. Rev. B 1975, 12, 3060. (27) Jepsen, O.; Andersen, O. K.; Machintosh, A. R. Phys. Rev. B 1975, 12, 3084. (28) Andersen, O. K.; Jepsen, O. Phys. Rev. Lett. 1984, 53, 2571. (29) Andersen, O. K.; Pawlowska, Z.; Jepsen, O. Phys. Rev. B 1986, 34, 5253. (30) Nowak, H. J.; Andersen, O. K.; Fujiwara, T.; Jepsen, O.; Vargas, P. Phys. Rev. B 1991, 44, 3577. (31) Lambrecht, W. R. L.; Andersen, O. K. Phys. Rev. B 1986, 34, 2439. (32) Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. Rev. Lett. 1996, 77, 3865. 2541

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