# Interplay between many-body effects and charge transfers in Cr2AlC bulk plasmon excitation

Vincent Mauchamp, M. Bugnet, P. Chartier, T. Cabioc'H, M. Jaouen, J.

Vinson, K. Jorissen, J. J. Rehr

► To cite this version:

Vincent Mauchamp, M. Bugnet, P. Chartier, T. Cabioc'H, M. Jaouen, et al. Interplay between many-body effects and charge transfers in Cr2AlC bulk plasmon excitation. Physical Review B, 2012, 86 (12), pp.125109. 10.1103/PhysRevB.86.125109. hal-05487978

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Interplay between many-body effects and charge transfers in Cr₂AlC bulk plasmon excitation.

V. Mauchamp,¹,* M. Bugnet,¹,² P. Chartier,¹ T. Cabioc'h,¹ M. Jaouen,¹ J. Vinson,³ K. Jorissen,³ and J. J. Rehr¹,³

¹Institut Pprime, UPR 3346 CNRS - Université de Poitiers - ISAE-ENSMA,
Département Physique et Mécanique des Matériaux, SP2MI Téléport 2,
BP 30179, 86962 Futuroscope-Chasseneuil Cedex, France

²Canadian Centre for Electron Microscopy, Mc Master University,
1280 Main street West, Hamilton, Ontario L8S 4L7, Canada

³Department of Physics, Box 351560, University of Washington, Seattle, Washington 98195-1560, USA

(Dated: August 30, 2012)

The bulk plasmon excitation anisotropy of Cr₂AlC, a nanolaminated ternary carbide, is investi-
gated by Electron Energy-Loss Spectroscopy (EELS) and ab initio calculations. Depending on the
crystallographic orientation, the valence EELS signal is dominated either by a single plasmon or split
into a superposition of two independent plasmons with mutually orthogonal momentum transfers.
This splitting arises from the electron-hole interaction anisotropy, important along the hexagonal
structure c-axis and negligible within the basal plane. It is shown that within the basal plane, the
plasmon behavior can be reproduced from an effective medium theory considering Cr₂AlC as an
atomic scale superlattice built from pure aluminum and CrC planes.

Keywords: 61.10.Ht,68.37.Lp,79.20.Uv

# I. INTRODUCTION

Plasmon excitations in metallic systems are collective
oscillations of the electronic states when screening an ex-
ternal electromagnetic perturbation.¹ Depending on the
material, these longitudinal modes can be understood
from a simple free electron picture or more complex quan-
tum mechanical frameworks.²,³ When combined to the
adapted level of theory, the probe of plasmons leads to
information concerning the intrinsic electronic response
in solids or nanostructured materials such as trans-
port properties, quantum confinement, modifications of
electron-ion interactions, spatial confinement etc.⁴⁻⁷ In
this context, Electron Energy-Loss Spectroscopy (EELS)
in the Transmission Electron Microscope (TEM) is a
powerful tool to probe these excitations, with unchal-
lenged spatial resolution.⁸ EELS is especially suited for
the investigation of anisotropic materials since it permits
probing the dielectric response of single grains in well de-
fined crystallographic orientations and is thus relevant in
the investigation of the electronic properties of nanolam-
inated compounds such as MAX phases.⁹,¹⁰

Cr₂AlC is one of these MAX phases, which are ternary
carbides or nitrides with general formula M<sub>n+1</sub>AX<sub>n</sub> (n =
1-3) where M is an early transition metal, A is an A-
group element and X is either C or N. These compounds
crystallize in the hexagonal P6<sub>3</sub>/mmc space group. They
are built on M<sub>6</sub>X octahedra layers interleaved with pure
A element layers (see Fig. 1 (a) for Cr<sub>2</sub>AlC). Due to this
stacking sequence, which makes these materials good pre-
cursors for the synthesis of a new class of two dimen-
sional nanocrystals,<sup>11</sup> MAX phases combine the most
important properties of metals and ceramics.<sup>12</sup> More-
over, their electronic structure involves complex charge
transfers with metallic, ionic and covalent bonds.<sup>13</sup> Ow-
ing to this peculiar bonding scheme, these materials ex-

<figure><img src="image_2.png" /><figcaption>FIG. 1. (Color online) (a) Cr₂AlC crystal structure. Bulk plasmon recorded along the (b) [0001] and the (c) [1 ¯2 10] zone axis. The shaded area should not be considered because it is subjected to zero-loss extraction artifacts.</figcaption></figure>

hibit very interesting mechanical and electronic proper-
ties, from both fundamental and technological points of
view.¹⁴,¹⁵ Of special interest are the titanium-based com-
pounds, investigated as ohmic contacts in the semicon-
ductor industry,¹⁶,¹⁷ and chromium-based ones which ex-
hibit outstanding high-temperature corrosion resistance
properties.¹⁸ In all cases, the anisotropy resulting from
the nanolaminated structure is key to the understanding
of their electronic properties. In this work we focus on
the plasmon excitation anisotropy of the Cr₂AlC valence
electron, chosen as a model case for the 211 MAX phase
compounds.

The paper is divided into three parts. In the first, the
material processing and TEM experiments are described.
The second part gives an overview of the theory used to
solve the BSE equation together with the computational
details. Finally experiments are compared to calculations
and the anisotropy of Cr₂AlC bulk plasmon excitation is
discussed in terms of a combination between many body
and charge transfer effects.

## II. EXPERIMENTAL DETAILS

Cr₂AlC samples were synthesized from chromium, aluminum and carbon graphite highly pure powder mixture, weighted according to the atomic composition with Cr:Al:C = 2:1.1:1 ratio.¹⁹ The powders were then cold-compacted and sintered using hot isostatic pressing (HIP) at 1450°C under 100MPa during 24 hours. The x-ray diffraction pattern exhibits single phase Cr₂AlC. The average grain size of the such polycrystalline sample is about 50µm.

For the EELS experiments the samples were cut by a wire saw, mechanically thinned by tripod polishing prior to Ar⁺ (2.5 keV) milling in order to reach electron transparency. Given the sample large grain size compared to the electron probe, single grains with definite crystallographic orientations, determined using the electron diffraction patterns, were selected. In order to investigate the anisotropy of the dielectric tensor, two types of grain were chosen: one in [0001] zone axis (the hexagonal c axis parallel to the electron beam) and the other in [11210] zone axis (basal plane of the hexagonal structure parallel to the electron beam). The spectra were recorded in image mode using parallel illumination with a collection semiangle (fixed by the spectrometer aperture) of 13.5 mRad, in a JEOL 2200FS TEM. The energy resolution was about 1 eV as measured by the zero loss peak (full width-half maximum) recorded in vacuum. The spectra were deconvoluted from plural scattering using the Fourier-Log method.⁸

## III. THEORY AND COMPUTATIONAL DETAILS

Experimental spectra are interpreted using ab initio calculations of the dielectric tensor using a Bethe-Salpeter equation (BSE) approach in order to treat the electron hole interaction resulting from the hole created by the probe. The AI2NBSE code was used: this approach combines the NIST BSE solver with ABINIT, a pseudopotential Density Functional Theory electronic structure code that calculates the Kohn-Sham wave functions which form a basis for the BSE.²⁰,²¹

### A. The Bethe-Salpeter Equation

We give here a brief summary of the BSE treatment which is described in details in reference 20 and references therein. The excited state, $|f\rangle$, results from the promotion of a valence electron to the conduction band due to the interaction with the electron beam. Provided some approximations,²⁰ the BSE is reduced to an eigenvalue problem of an effective particle-hole Hamiltonian

acting on the subspace of singly excited states (we adopt here the notations of reference 20):²⁰

$$[H_{1e} + H_{2e}] |f\rangle = \Omega_f |f\rangle \quad (1)$$

where the eigenstates $|f\rangle$ are described as the superposition of particle-hole basis states $|nn'k(q)\rangle$:

$$|f\rangle = \sum_{nn'k} \psi_f [nn'k(q)] |nn'k(q)\rangle \quad (2)$$

k and q are respectively the Bloch crystal momentum index and momentum transfer of a given particle-hole state. n and n' are the indices running over all occupied and unoccupied bands respectively.

In Eq. 1, $H_{1e}$ is the single-particle contribution to the Hamiltonian. It is diagonal in the quasiparticle basis and corresponds to independent transitions. $H_{2e}$ describes the electron-hole interactions which include both a direct and an exchange part: $\hat{H}_{2e} = V_D + V_X$. The matrix elements of these two terms are given, in terms of two-particle integrals between electron and hole Kohn-Sham orbitals $\psi_{nk}(\mathbf{x})$ and $\psi_{n'k+q}(\mathbf{x})$, by:

$$V_D(n n' \mathbf{k}, n'' n''' \mathbf{k}'; \mathbf{q}) \approx \\ - \int d^3 \mathbf{x} \psi_{n'' \mathbf{k}'+\mathbf{q}}^*(\mathbf{x}) \psi_{n' \mathbf{k}+\mathbf{q}}(\mathbf{x}) \\ \times \int d^3 \mathbf{x}' \psi_{n \mathbf{k}}^*(\mathbf{x}') \psi_{n'' \mathbf{k}'}(\mathbf{x}') W(\mathbf{x}, \mathbf{x}'; \omega = 0).$$

for the direct term, and

$$V_X(n n' \mathbf{k}, n'' n''' \mathbf{k}'; \mathbf{q}) \approx \\ 2 \int d^3 \mathbf{x} \psi_{n'' \mathbf{k}'+\mathbf{q}}^*(\mathbf{x}) \psi_{n'' \mathbf{k}'}(\mathbf{x}) \\ \times \int d^3 \mathbf{x}' \psi_{n \mathbf{k}}^*(\mathbf{x}') \psi_{n' \mathbf{k}+\mathbf{q}}(\mathbf{x}') \frac{1}{|\mathbf{x}-\mathbf{x}'|}.$$

for the exchange part.

In AI2NBSE, the screened Coulomb interaction $W = \epsilon^{-1}(\omega)V$ entering the direct term is computed using the static, spatially dependent Hybertsen-Levine-Louie dielectric function.²²,²³ Although the screening should in principle be frequency dependent, this static approximation gives good results in many systems.²⁰,²⁴

The imaginary part of the frequency dependent dielectric function, $\epsilon_2(\mathbf{q}, \omega)$ could then be obtained using a Fermi golden rule that couples the ground state of the system to the excited states $|f\rangle$, through the macroscopic current operator. In the present calculations a slightly different approach is used: $\epsilon_2(\mathbf{q}, \omega)$ is computed using resolvent techniques the derivation of which from the Fermi golden rule is detailed in reference 24.
## B. Computational details

ABINIT calculations were performed within the Perdew-Burke-Ernzerhof Generalized Gradient Approximation (GGA) using Trouilliers-Martins pseudopotentials from the ABINIT data base.²⁵ A 40 Hartree cut-off was used and the unit-cell was described using optimized parameters ($a = 2.85$ Å and $c = 12.71$ Å). The chromium $d$ band correlation effects were not accounted for,²⁶ but given the good agreement between calculations and experiments (see Fig. 2), their impact on plasmon excitation is expected to be small.

In addition to the BSE calculations, band structure calculations were performed using the WIEN2k code. WIEN2k is an all electron Full Potential Linearized Augmented Plane Wave (FLAPW) method.²⁷ An augmented plane wave (APW) basis set combined with local orbitals (APW+lo) was used with a converged basis set obtained for a RK<sub>max</sub> value of 7.5.²⁸ Calculations were performed considering the Perdew-Burke-Ernzerhof GGA and 1000 k-points were used in the full Brillouin zone for the self-consistent field determination of the potentials.

## IV. RESULTS AND DISCUSSION

The experimental bulk plasmons recorded for grains oriented along the [0001] and [12̅10] zone axes are sketched in Fig. 1 (b) and (c). As evidenced by this figure, the bulk plasmon excitations in Cr₂AlC are complex structures with several modes and reveal a strong anisotropy when recorded with the *c*-axis or basal plane parallel to the electron beam. The former exhibits two distinct structures at 14.3 and 21.5 eV and a shoulder at 18.1 eV whereas the plasmon in the other orientation shows two distinct modes around 18.8 and 22.2 eV.

Due to the scattering geometry, the experimental EELS signal mixes the two components of the dielectric tensor $\epsilon_{ab}$ and $\epsilon_c$, which correspond to the basal-plane and *c*-axis response respectively. However, given the collection angles used in the present experiments, the plasmon is largely dominated by the contribution from the response perpendicular to the beam axis. The inelastic cross section corresponding to the [0001] zone axis is thus approximated to the basal plane loss function: $\sigma_{[0001]}(\mathbf{q}, \omega) \approx -\mathrm{Im}(\frac{1}{\epsilon_{ab}(\mathbf{0}, \omega)})$. The EELS signal along the [12̅10] zone axis is simulated by taking the average of the basal plane and *c*-axis loss functions: $\sigma_{[12̅10]}(\mathbf{q}, \omega) \approx \frac{1}{2}[-\mathrm{Im}(\frac{1}{\epsilon_{ab}(\mathbf{0}, \omega)}) - \mathrm{Im}(\frac{1}{\epsilon_c(\mathbf{0}, \omega)})]$. Comparison between theoretical and experimental spectra is sketched in Fig. 2: BSE calculations are in good agreement with the experiments [see Fig. 2 (a) and (b)]. Except for the shoulder around 18.1 eV (which energy is slightly overestimated) the major features of the [0001] bulk plasmon are well reproduced with good intensity ratios. Similarly, the two structures of the [12̅10] bulk plasmon are also reproduced at the right energy positions

<figure><img src="image_3.png" /></figure>

FIG. 2. (Color online) Experimental bulk plasmons (black lines) compared to calculations considering the full effective Hamiltonian (grey thick lines) or the single particle term (red dashed line) for the (a) [0001] or (b) [12̅10] zone axis. (c) Decomposition of the BSE calculation for the [12̅10] orientation between the basal-plane (green dashed line) and *c*-axis (blue line) loss functions. (d) BSE calculation of $\mathrm{Re}\epsilon(\mathbf{0}, \omega)$ for the basal plane (green dashed line) and *c*-axis (blue line) orientations. Only the energy range above the shaded area is plotted.

by the BSE calculation. The plasmon width is underestimated, which can be attributed to dispersion effects which are ignored in the present calculation.

Comparison of BSE and single particle calculations for both orientations reveal a strong anisotropy of the electron-hole interaction. This anisotropy results from the strong electron-hole interaction effect in the *c*-axis dielectric response and much weaker one within the basal plane. The [0001] bulk plasmon, mainly driven by the basal plane response, can thus be interpreted in a single particle picture. However, the double structure observed for the [12̅10] plasmon can only be reproduced when including two particle effects which redshift the *c*-axis plasmon component giving the first mode at 18.8 eV. The structure at 22.2 eV is due to the basal plane component. The plasmonic character of these two peaks is shown in Fig. 2 (d): they both correspond to zeros of the real part of the dielectric function. As a consequence, this double structure can be interpreted as the superposition of two independent bulk plasmons with two mutually orthogonal momenta transfers along the *c*-axis and within the basal plane of the hexagonal structure. Similar superposition of bulk plasmons is observed in various lamellar compounds such as graphite.²⁹ However, in Cr₂AlC, the splitting between these two modes results mainly from electron/hole interactions whereas for graphite it is already obtained from the ground state band structure (i.e. considering independent transitions).

The Cr₂AlC basal plane plasmon excitation ([0001] zone axis) is thus determined by the ground state elec-
<figure><img src="image_4.png" /><figcaption>FIG. 3. (Color online) FLAPW calculations of (a) Cr₂AlC projected Density of States (DOS); the shaded peak shows the coupling between Al and CrC layers in Cr₂AlC. (b) Superposition of CrC and pure Al DOS. (c) Cr₂AlC band structure calculated along the Γ → K and Γ → M directions of the first Brillouin zone. These directions correspond to crystal momenta along the unit cell basal plane and c-axis orientations respectively. Al-s, $p_x p_y$, and $p_z$ character in the orbital are emphasized as open circles.</figcaption></figure>

tronic structure of the material which, as shown by the FLAPW calculations in Fig. 3 (a),²⁷ can be schematically divided into four contributions: C-s states between -15 and -11 eV, C-p/ Cr-d hybridization between -9 and -3.7 eV, Al-p/Cr-d hybridization between -3.7 and -1.6 eV and the Cr-d band close to the Fermi level. Among these hybridizations, one can observe the widely dispersed Al states which exhibit a “quasi-free” electron behavior. This DOS compares well with that computed for an hypothetical face centered cubic CrC with a lattice parameter corresponding to the Cr-C bond length in the MAX phase, superimposed to that of pure aluminum (Fig. 3 (b)). The same hybridizations within the same energy ranges are observed excepted for Al-p/Cr-d one (shaded area in Fig. 3 (a) and (b)) which describes the coupling between the CrC and Al layers. As seen in Fig. 3 (c), this peak arises from the weakly dispersed Al-$p_z$ bands which are responsible for the charge transfer to the CrC layers. Compared to bulk aluminum, the confinement of the Al layers in the Cr₂AlC structure results in the localization of these Al-$p_z$ states directed along the c-axis of the hexagonal structure. On the contrary both Al-s and $p_x p_y$ states keep their free electron behavior within

the basal plane as shown by the parabolic dispersion of the corresponding bands. Such an orbital reconstruction involving both charge transfer and symmetry breaking is also reported for the silicon atom in Ti₃SiC₂.³⁰ This phenomenon is similar to that observed in the transition metal d-bands at interfaces in oxides heterostructure.³¹

Following this picture and since the characteristic plasmon length scale, given by $2\pi/(k_0\theta_E)$ where $k_0$ is the wave vector of the probe electron and $\theta_E$ the characteristic angle at the plasmon energy, is 60 times larger than the stacking periodicity $c/2$, one can use an effective medium (E.M.) approach to describe the basal plane plasmon excitation of Cr₂AlC.³² In this approach, the dielectric behavior of Cr₂AlC, considered as a stacking of uncoupled CrC and Al layers, can be calculated as: $\epsilon_{a,b}(\omega) = \frac{d}{c/2} \cdot \epsilon_{Al}(\omega) + \frac{c/2-d}{c/2} \cdot \epsilon_{CrC}(\omega)$ and $\frac{1}{\epsilon_c(\omega)} = \frac{d}{c/2} \cdot \frac{1}{\epsilon_{Al}(\omega)} + \frac{c/2-d}{c/2} \cdot \frac{1}{\epsilon_{CrC}(\omega)}$, $d$ and $c$ being defined in Fig. 1 while $\epsilon_{Al}$ and $\epsilon_{CrC}$ are the bulk dielectric response of aluminum and the fcc CrC structure. We used a Drude model to describe the aluminum dielectric response, the plasma frequency and relaxation time being deduced from the free electron picture. Although not rigorous from a microscopic point of view, it gives a reasonable description for the aluminum plasmon.⁸ The CrC dielectric function was computed with Al2NBSE considering only the single particle term of the Hamiltonian. The corresponding calculations are plotted in Fig. 4.

As expected, the *ab initio* calculation of the basal plane plasmon can be nicely reproduced by the E.M. approach: the three modes labeled 1 to 3 are reproduced at the correct energy positions with relative intensities in qualitative agreement with the *ab initio* calculation. It should be mentioned that this bulk plasmon cannot be reproduced considering only the CrC layer as shown by the green dotted line in Fig. 4 (a). On the contrary, the [0001] plasmon (Fig. 4 (b)) cannot be understood from the E.M. approach due to charge transfer between Al and CrC layers along the c-axis: both the plasmon energy and the global shape are off when compared to the *ab initio* calculation.

This behavior is opposite to that observed for the description of GaAs/AlAs superlattices optical properties where the E.M. approach gives a good description along the stacking sequence.³³ In both cases, the crucial point is the competition between confinement effects, linked to the intrinsic structure of the material, and electron-hole interactions. In GaAs/AlAs superlattices, the negative contribution of the Local Field Effects (LFE) in the growth direction cancels the confinement effects that tend to increase the dielectric constant: the E.M. approach gives a reasonable picture in the growth direction whereas it is less accurate for the basal plane dielectric response. For Cr₂AlC, there is the same competition, but confinement effects and electron-hole interactions are important only along the c-axis. The basal plane dielectric response can thus be simply understood from the ground state electronic structure which, in a first approximation
<figure><img src="image_5.png" /></figure>

FIG. 4. (Color online) Effective medium approach (black) compared to AI2NBSE calculations in the independent particle approximation (red dashed line) for the calculation of the (a) basal plane plasmon and the (b) c-axis plasmon. In (a) the aluminum (blue dashed dotted line) and CrC (green dots) plasmons are also given for comparison.

tion, is a superimposition of Cr<sub>6</sub>C octahedra electronic states with a 2D quasi-free electron gas resulting from the aluminum layers. An E.M. approach describing Cr<sub>2</sub>AlC dielectric behavior from CrC and Al bulk dielectric functions is thus valid in this orientation.

V. CONCLUSION

In conclusion, the plasmon excitation anisotropy in
Cr₂AlC has been investigated using EELS experiments

and DFT/BSE calculations. It has been shown that the
plasmon response in the [12\overline{1}0] zone axis can be inter-
preted as a superposition of two independent plasmons
corresponding to the basal plane and c-axis dielectric
response. This double mode can only be reproduced
when including electron-hole interactions in the calcu-
lation. These two-particles effects, important along the
stacking sequence and weak within the basal plane, are
responsible for the experimentally observed anisotropy.

As in semiconductors superlattices, the dielectric re-
sponse of the MAX phase is driven by the competition
between electron-hole interactions and confinement ef-
fects, both being weak within the basal plane. In the
latter case, the Cr<sub>2</sub>AlC dielectric response can be well
understood from an E.M. approach considering the MAX
phase as an atomic scale CrC/Al superlattice built from
pure Cr<sub>6</sub>C octahedra and aluminum layers. This simple
picture, the validity of which should be investigated in
other energy ranges, brings new insights into the basal
plane electronic properties of these complex materials.

* vincent.mauchamp@univ-poitiers.fr

1 D. Pines and P. Nozieres, *The Theory of Quantum Liquids*, W. A. Benjamin, Inc. (1966).

2 K. Sturm, Adv. Phys. 31, 1 (1982).

3 G. Onida, L. Reining and A. Rubio, Rev. Mod. Phys. **74**, 601 (2002).

4 I. Loa, K. Syassen, G. Monaco, G. Vankó, M. Krisch, and M. Hanfland, Phys. Rev. Lett **107**, 086402 (2011).

5 M. M. Özer, E.J. Moon, A.G. Eguiluz, and H.H. Weitering, Phys. Rev. Lett **106**, 197601 (2011).

6 C. Kramberger, R. Hambach, C. Giorgetti, M.H. Rümmeli, M. Knupfer, J. Fink, B. Büchner, L. Reining, E. Einarsson, S. Maruyama, F. Sottile, K. Hannewald, V. Olevano, A.G. Marinopoulos, and T. Pichler, Phys. Rev. Lett **100**, 196803 (2008).

7 A. Goldoni, L. Sangaletti, F. Parmigiani, G. Comelli, and G. Paolucci, Phys. Rev. Lett **87**, 076401 (2001).

8 R.F. Egerton *Electron Energy-Loss Spectroscopy in the Electron Microscope*, Plenum, New York, 1996.

9 G. Hug, P. Eklund, and A. Orchowski, Ultramicroscopy **110**, 1054 (2010).

10 V. Mauchamp, G. Hug, M. Bugnet, T. Cabioc'h, and M. Jaouen, Phys. Rev. B **81**, 035109 (2010).

11 M. Naguib, M. Kurtoglu, V. Presser, J. Lu, J. Niu, M. Heon, L. Hultman, Y. Gogotsi, and M. Barsoum, Adv. Mater. **23**, 4248 (2011).

12 P. Eklund, M. Beckers, U. Jansson, H. Högberg, L. Hult-

man, Thin Solid Films **518**, 1851 (2010).

13 G. Hug, M. Jaouen, and M. W. Barsoum, Phys. Rev. B **71**, 024105 (2005).

14 M. W. Barsoum, A. Murugaiah, S.R. Kalidindi, and T. Zhen, Phys. Rev. Lett **92**, 255508 (2004).

15 H.-I. Yoo, M. Barsoum, and T. El-Raghy, Nature **407**, 581 (2000).

16 K. Buchholt, R. Ghandi, M. Domeij, C.M. Zetterling, J. Lu, P. Eklund, L. Hultman, and A. Lloyd Spetz, App. Phys. Lett. **98**, 042108 (2011).

17 Z. Wang, S. Tsukimoto, M. Saito, K. Ito, M. Murakami, and Y. Ikuhara, Phys. Rev. B **80**, 245303 (2009).

18 Z.J. Lin, M.S. Li, J.Y. Wang and Y.C. Zhou, Acta Materialia **55**, 6182 (2007).

19 Z.J. Lin, Y.C. Zhou, M.S. Li and J.Y. Wang, Z. Metallkd. **96**, 291 (2005).

20 H.M. Lawler, J.J. Rehr, F. Vila, S.D. Dalosto, E.L. Shirley, and Z.H. Levine, Phys. Rev. B **78**, 205108 (2008).

21 X. Gonze, J.-C. Beuken, R. Caracas, F. Detraux, M. Fuchs, G.M. Rignanese, L. Sindic, M. Verstraete, G. Zerah, F. Jollet, M. Torrent, A. Roy, M. Mikami, P. Ghosez, J.Y. Raty, and D.C. Allan, Comput. Mater. Sci. **25**, 478 (2002).

22 Z.H. Levine and S.G. Louie, Phys. Rev. B **25**, 6310 (1982).

23 M.S. Hybertsen and S.G. Louie, Phys. Rev. B **37**, 2733 (1988).

24 L.X. Benedict and E.L. Shirley, Phys. Rev. B **59**, 5441 (1999).
25 J.P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. **77**, 3865 (1996).

26 M. Ramzan, S. Lebègue, R. Ahuja, Phys. Stat. Solidi (RRL) **5**, 122 (2011).

27 P. Blaha, K. Schwarz, G.K.H. Madsen, D. Kvaniscka, and J. Luitz, *WIEN2k*, an augmented plane wave+local orbitals program for calculating crystal properties, Technische Universität WIEN, Austria, 2001.

28 E. Sjöstedt, L. Nordström, and D.J. Singh, Solid State Commun. **114**, 15 (2000).

29 A.G. Marinopoulos, L. Reining, V. Olevano, A. Rubio, T. Pichler, X. Liu, M. Knupfer, and J. Fink, Phys. Rev. Lett. **89**, 076402 (2002).

30 M. Magnuson, M. Mattesini, N. Van Nong, P. Eklund, and L. Hultman Phys. Rev. B **85**, 195134 (2012).

31 J. Chakhalian, J.W. Freeland, H.U. Habermeier, G. Cristiani, G. Khaliullin, M. van Veenendaal, and B. Keimer, Science **318**, 1114 (2007).

32 D.J. Bergman, Phys. Rep. **9**, 377 (1978).

33 S. Botti, N. Vast, L. Reining, V. Olevano, and L. C. Andreani, Phys. Rev. Lett. **89**, 216803 (2002).