Dielectric functions and collective excitations in MgB_2

The frequency- and momentum-dependent dielectric function $\epsilon{(\bf q,\omega)}$ as well as the energy loss function Im[-$\epsilon^{-1}{(\bf q,\omega)}$\protect{]} are calculated for intermetallic superconductor $MgB_2$ by using two {\it ab initio} methods: the plane-wave pseudopotential method and the tight-binding version of the LMTO method. We find two plasmon modes dispersing at energies $\sim 2$-8 eV and $\sim 18$-22 eV. The high energy plasmon results from a free electron like plasmon mode while the low energy collective excitation has its origin in a peculiar character of the band structure. Both plasmon modes demonstrate clearly anisotropic behaviour of both the peak position and the peak width. In particular, the low energy collective excitation has practically zero width in the direction perpendicular to boron layers and broadens in other directions.

The frequency-and momentum-dependent dielectric function ǫ(q, ω) as well as the energy loss function Im[-ǫ −1 (q, ω)] are calculated for intermetallic superconductor M gB2 by using two ab initio methods: the plane-wave pseudopotential method and the tight-binding version of the LMTO method. We find two plasmon modes dispersing at energies ∼ 2-8 eV and ∼ 18-22 eV. The high energy plasmon results from a free electron like plasmon mode while the low energy collective excitation has its origin in a peculiar character of the band structure. Both plasmon modes demonstrate clearly anisotropic behaviour of both the peak position and the peak width. In particular, the low energy collective excitation has practically zero width in the direction perpendicular to boron layers and broadens in other directions. After the discovery of superconductivity in the M gB 2 compound with the transition temperature T c ∼ 39K [1] much effort has been devoted to understanding the mechanism of the superconductivity [2][3][4][5][6][7][8][9][10][11] as well as to studying different electronic and atomic characteristics of this compound. Among these characteristics are the superconducting gap [3,12,13], the crystal structure and its influence on T c [1,2,[14][15][16], band structure [4,5,[17][18][19] and the Fermi surface [4], lattice vibrations [6,9,20,21] as well as thermodynamic and transport properties [22]. Recently Voelker et al [8] explored collective excitations very near the Fermi level by using a simple band structure model and found the plasmon acoustic mode at very small momenta (q ≃ 0.01 a.u. −1 ) and low energies (ω ≃ 0.01 eV). Here we study collective excitations in M gB 2 for different momenta and energies, namely, for q ≥0.1 a.u. −1 and ω ≥ 2 eV. We report first-principle calculations for the real, ǫ 1 (q, ω), and imaginary, ǫ 2 (q, ω), part of the dielectric function as well as the energy loss function Im[-ǫ −1 (q, ω)]. As a result of the calculation two plasmon modes and a few features arising from interband transitions are obtained.
M gB 2 crystallizes in the so-called AlB 2 structure in which B atoms form graphitelike honeycomb layers that alternate with hexagonal layers of M g atoms. The magnesium atoms are located at the center of hexagons formed by borons and donate their electrons to the boron planes. Similar to graphite M gB 2 exhibits a strong anisotropy in the B-B lengths: the distance between the boron planes is significantly longer than in-plane B-B distance. We use this resemblance between graphite and M gB 2 in order to clear up the origin of plasmon peaks in M gB 2 by comparing the calculated energy loss function features with those obtained from EELS measurements for graphite and single-wall carbon nanotubes (SWCN) [26,27]. To shed more light on the problem we also evaluate the dielectric functions and the energy loss spectra for the M gB 2 crystal structure with the M g atoms re-moved (this hypothetical crystal structure is designated as B 2 ).
Information on the energy lost by electrons in their interactions with metals is carried by the dynamical structure factor S(q, ω) which is related by the fluctuationdissipation theorem to the energy loss function Im[ǫ −1 00 (q, ω)]. To calculate the inverse dielectric function we invoke the random phase approximation (RPA) where where υ c is the bare Coulomb potential and χ 0 is the density response function of the noninteracting electron system. The dielectric function is related to χ 0 as ǫ = 1 − υ c χ 0 . The energy loss function may be obtained by inverting the first matrix element of ǫ that leads to neglecting short-range exchange and correlation effects or directly from ǫ −1 when these effects are included. We have computed the energy loss function by using both of these approaches and found that the inclusion of the local field effects leads to negligible changes of both the width and energy of the plasmon peaks. In the calculation of the density response matrix χ 0 GG ′ (q, ω) we have used two different first-principle methods: the plane wave pseudopotential method [23] and the tight-binding version of the LMTO method [24,25].
In Figs. 1a and 1b we show the evaluated band structure of M gB 2 and B 2 along the symmetry directions. In general, these band structures are quite similar. The distinctions between them in the vicinity of the Fermi level (E F ) are due to the lower position of E F relative to the σ band in the ΓA direction for B 2 . The states of this band, which are of p x,y symmetry, are degenerate in ΓA and their charge density is located in B layer showing a clear 2D character. This character leads to weaker interactions between the B layers and to smaller dispersion of the σ band along ΓA in B 2 . The p z band which is occupied at Γ in M gB 2 becomes unoccupied at Γ in B 2 . In Figs. 2a and 2b we present the momentum dependence of the energy loss function in the ΓA, ΓK and ΓM directions. In M gB 2 we have found two plasmon modes. The higher collective excitation mode originates from the free electron like excitation mode with energy ω p1 =19.1 eV that corresponds to the electron density parameter r s =1.82 a.u. of M gB 2 . This free electron like mode is transformed into two separated submodes ω xy p1 and ω z p1 in a real crystal. One of them is very isotropic in the hexagonal plane (the ΓK and ΓM directions) and disperses linearly up for momenta q ≥ 0.2 a.u., while another one, in the ΓA direction, has smaller energy and is nearly constant. Because of limitations of the calculation methods and a large width of the energy loss peaks at small momenta we could not determine accurately the plasmon peak position in this region. Therefore we define the plasmon energy at the Γ point by extrapolation of the computed plasmon dispersions at q≥ 0.2 a.u. This extrapolation results in ω xy p1 =19.4 eV and ω z p1 =18.8 eV in good agreement with the free electron gas value ω p1 . The width, △ p1 , of both these energy loss peaks decreases with the increasing momentum, and the △ z p1 width deacreases faster than △ xy p1 . A different behavior is shown by the low-energy loss function peak which disperses linearly up in both the ΓA direction and in the hexagonal plane. In the ΓA direction, at q ⊥ ∼ 0.1 a.u. −1 , the peak is very narrow, △ z p2 ∼ 0.01 eV, that can be seen in the very small value of ǫ 2 in the energy interval around the peak position where ǫ 1 =0 (Figs. 3a and 3b). In particular, for q ⊥ =0.12 a.u. −1 this interval is between 1 and 5 eV (Fig. 3a) and the energy loss peak is located at 2.9 eV. On changing the momentum to the A point this interval becomes more narrow and moves to higher energies (Fig. 3c). At small momenta the first peak of Im[-ǫ 00 (q, ω)] placed in the energy interval 0-1 eV is determined by intraband transitions within the two σ bands in the x, y plane around the ΓA direction, while the second peak located at 5.4 eV (Fig. 3a) is determined by the interband σ−π transitions in the KM , AH, and AL directions. So, the low energy plasmon excitation corresponds to electron excitations in the σ bands and one can define it as the σ plasmon. In the hexagonal plane, in the ΓK direction, the low energy EELS peak broadens (Fig. 4a) and disperses linearly up on going from the Γ point to K. In the ΓM direction the plasmon peak disperses similar to that in the ΓK one showing a nearly ideal isotropy in the hexagonal plane. However, it becomes smaller and wider on going from Γ to M and vanishes finally at q ≃ 0.8| ΓM |. Comparing the plasmon energies obtained from the LMTO and pseudopotential calculations one can find only a small difference of ∼ 0.1 eV between them. For instance, at q=0.2 a.u. −1 the LMTO ω z p2 is slightly smaller than the pseudopotential one and vice versa for larger momenta. This slight difference results in different energy loss peak positions at Γ: the extrapolation of both plasmon energies ω xy p2 and ω z p2 calculated for q≥ 0.1 a.u. −1 to the Γ point gives ω z p2 =1.8 eV, ω xy p2 =2.0 eV (LMTO) and ω z p2 =2.2 eV, ω xy xy =2.4 (pseudopotential). We estimate the accuracy of these values to be better than 0.2 eV.
Besides two plasmon modes we have obtained four small features in Im[-ǫ −1 00 (q, ω)] that correspond to interband excitations. It is difficult to find out what transitions are responsible for these features, nevertheless we show them in Fig. 2a. One of them arises at q ≃0.1a.u. −1 at an energy of ≈2 eV in the hexagonal plane, another one occurs at q ≥ 0.4 a.u. −1 in the ΓM direction at an energy of ≃ 4 eV and the other two small features arise in the ΓA direction for q=0.1-0.25 a.u. −1 at energies of 10 eV and 13 eV, respectively.
In Fig. 2b we show the momentum dependence of the energy loss function calculated for the hypothetical crystal structure B 2 . In general, the energy loss function in B 2 shows relatively similar features to those in M gB 2 , though there are some important distinctions. In particular, all collective excitations including two plasmon modes manifest a smaller dispersion in the ΓA direction. This effect is a direct consequence of a weaker interactions between adjacent layers of boron in B 2 compared to M gB 2 . Another distinction is that all features in the energy loss function in B 2 are much clearer than those in M gB 2 (Figs. 3a-3c and 4a-4c). One exception is the high energy plasmon mode. The third distinction is that B 2 has more features in Im[-ǫ −1 00 (q, ω)] than does M gB 2 . The low energy plasmon mode extrapolation to the Γ point gives ω z p2 =4.1 eV which is ∼ 2 eV higher than that in M gB 2 . This shift in energy is due to the higher energy position of the second maximum of Im[ǫ 00 ] (Fig. 3a). While the position of the first peak of Imǫ in B 2 nearly coincides with that in M gB 2 the second peak is moved by 2 eV to higher energies. Via the Hilbert transform (Kramers-Kronig relation) it also moves the node of ǫ 1 to higher energy. Despite some quantitative distinctions between the energy loss functions in M gB 2 and B 2 one can conclude that mostly the features of the excitation spectrum of M gB 2 can be derived, with the relevant corrections, from those of the hypothetical crystal B 2 .
The two plasmon modes similar to those obtained in M gB 2 were also observed in EELS experiments for graphite and SWCN [26,27] which are even more anisotropic than M gB 2 . The upper plasmon mode which has larger energy in graphite and SWCN than in M gB 2 (near the Γ point ω xy p1 ≃21 eV (SWCN) and ω xy p1 ≃26 eV (graphite)) [27] also results from excitations of all valence electrons. The lower plasmon mode ω xy p2 shows a linear dependence on momentum, like that in M gB 2 , with energies ω xy p2 ≃5 eV (SWCN) and ω xy p2 ≃6.5.eV (graphite) [27] at Γ. But in contrast to SWCN and graphite where ω xy p2 represents the collective excitation of the π-electron system [26][27][28] in M gB 2 this mode is a result of the collective excitation of the σ-electron system. The different origin of the low-energy plasmon mode in M gB 2 /B 2 and graphite/SWCN can be qualitatively understood from the Fermi energy (E F ) position. In graphite the Fermi level is pinned by π-electrons in the KH direction. In M gB 2 and B 2 with a smaller number of electrons per atom the Fermi level is pinned by σ-electrons. So, lowenergy excitations in M gB 2 and in B 2 are expected to be derived from the σ-band electrons.
In conclusion, we have performed the first-principle calculations of the dielectric functions ǫ 1 (q, ω) and ǫ 2 (q, ω) as well as the energy loss function Im[-ǫ −1 (q, ω)]. The calculations reveal the two plasmon modes in M gB 2 and B 2 and a few interband collective excitations. The low energy plasmon mode corresponding to the excitations of electrons in the σ bands shows a very anisotropic behavior of the peak width. The energy loss spectrum of M gB 2 can be derived, with the relevant corrections, from that of the hypothetical crystal structure B 2 .
We thank R.H.Ritchie and A.Bergara for helpful discussions. This work was partially supported by the Basque Country University, Basque Hezkuntza Saila, and Iberdrola S.A.
After the submission of this paper first-principles calculations of collective excitations in M gB 2 have also been presented by Wei Ku et al. [29].