Low-energy collective electronic excitations in ${\mathrm{LiC}}_6,{\mathrm{SrC}}_6$, and ${\mathrm{BaC}}_6$

We present a first-principles study of the electronic band structure and low-energy dielectric response properties of a representative group of graphite intercalated compounds—${\mathrm{LiC}}_6,{\mathrm{SrC}}_6$, and ${\mathrm{BaC}}_6$. Our results obtained from time-dependent density functional theory calculations reveal the presence of different plasmons in these compounds at energies below 12 eV. The presence of these collective electronic excitations is discussed in terms of intra- and interband transitions. In addition to the bulk plasmon, we find the $\pi$- and intraband plasmons. However, their properties vary greatly from one material to another. In ${\mathrm{LiC}}_6$ and ${\mathrm{BaC}}_6$, the $\pi$ plasmon is a two-dimensional excitation, whereas in ${\mathrm{SrC}}_6$ it has a three-dimensional character. As for the intraband plasmon, it is observed in all momentum-transfer symmetry directions, with rather strong anisotropy in the energy position between momentum transfers in the basal carbon plane and perpendicular to it. Also, we discuss the appearance in ${\mathrm{SrC}}_6$ and ${\mathrm{BaC}}_6$ of a low-energy collective electronic mode with the characteristic soundlike dispersion in terms of the energy bands crossing the Fermi level with different group velocities. We find a correlation between the appearance of such low-energy electronic modes and the occupation of the graphite interlayer band.

Figure 1

(a),(b) Hexagonal unit cell and stacking sequence of GICs. All carbon atomic layers denoted by $A$ have the same arrangement in the basal plane as shown by small gray circles. Large red circles represent intercalant atoms located in the interlayer positions above the centers of carbon hexagons in $\alpha$ or $\beta$ or $\gamma$ site as schematized in (b). (c) Projection of the first Brillouin zone (1BZ) onto the hexagonal carbon basal plane. In (c) the $\mathrm{\Gamma }KM$ and AHL planes cross the $c^{*}$ axis, directed in the direction perpendicular to the basal plane, at $k_z=0$ and $k_z=\pi /c$, respectively. In the reciprocal space, three main symmetry directions of the momentum transfer—$a^{*},b^{*}$, and $c^{*}$—are directed along the $\mathrm{\Gamma }K,\mathrm{\Gamma }M$, and $\mathrm{\Gamma }A$ directions of the 1BZ, respectively.

Figure 2

Calculated band structure of (a) ${\mathrm{LiC}}_6$, (b) ${\mathrm{SrC}}_6$, and (c) ${\mathrm{BaC}}_6$ along high-symmetry directions of the 1BZ. The interlayer and hybridized bands [31] are highlighted with blue and green, respectively. The Fermi level located at the zero energy is shown by a horizontal dashed line.

Figure 3

Loss function, $L(\mathbf{Q},\omega )$, of ${\mathrm{LiC}}_6$ at momentum transfers $\mathbf{Q}$ along the $a^{*}$ axis calculated (a) with and (b) without inclusion of the local-field effects. The respective ${\log }_{10}\left\{\mathrm{Im}\left[\epsilon \right]\right\}$ is shown in (c). In (d), (e), and (f) and in (g), (h), and (i) the same quantities are presented for $\mathbf{Q}$'s directed along the $b^{*}$ and $c^{*}$ axes, respectively. Peaks in the loss function corresponding to the $\pi$ and intraband plasmons are labeled by $\pi \mathrm{P}$ and IP, respectively. In (a) the peak corresponding to an intraband plasmon reappearing at large momentum transfers is denoted as ${\mathrm{IP}}^{\prime }$. Peak $M$ in (d) and (e) is discussed in the text. In (c), (f), and (i) the peaks in $\mathrm{Im}\left[\epsilon \right]$ generated by the intraband transitions are labeled as IB, IB1, and IB2. A clear peak in $\mathrm{Im}\left[\epsilon \right]$ produced by the interband transitions is denoted as $T$. The dispersion of all these peaks is highlighted by white and black dotted lines. In (g) vertical arrows point to regions where there is enhanced hybridization of the IP mode with the $T$ peak in $\mathrm{Im}\left[\epsilon \right]$.

Figure 4

Upper panel: Real $\mathrm{Re}\left[\epsilon \right]$ (blue solid line) and imaginary $\mathrm{Im}\left[\epsilon \right]$ (red dashed line) parts of the dielectric function in ${\mathrm{LiC}}_6$ at $Q_{\parallel a^{*}}=0.03{\AA }^{-1}$. Lower panel: Corresponding loss function evaluated with the inclusion of LFEs (black solid line). Peaks in the loss function corresponding to the intraband and $\pi$ plasmons are marked as IP and $\pi \mathrm{P}$, respectively.

Figure 5

Density of states distribution in ${\mathrm{LiC}}_6$ vs energy and group velocity components along (a) $a^{*}$, (b) $b^{*}$, and (c) $c^{*}$ symmetry directions. Peaks at the Fermi level are labeled as P1 and P2. The energies are relative to the Fermi level set to zero.

Figure 6

Loss function, $L(\mathbf{Q},\omega )$, of ${\mathrm{SrC}}_6$ at momentum transfers $\mathbf{Q}$ along the $a^{*}$ axis calculated (a) with and (b) without inclusion of the local-field effects. The respective ${\log }_{10}\left\{\mathrm{Im}\left[\epsilon \right]\right\}$ is shown in (c). In (d), (e), and (f) and in (g), (h), and (i) the same quantities are presented for $\mathbf{Q}$'s directed along the $b^{*}$ and $c^{*}$ axes, respectively. Peaks in the loss function corresponding to the bulk, $\pi$, intraband, and acoustic plasmons are labeled by symbols BP, $\pi \mathrm{P}$, IP, and AP, respectively. The peaks $K,M1,M2,\pi {\mathrm{P}}^{\prime },{\mathrm{IP}}^{\prime }$, and $A$ are discussed in the main text. Dispersion of weak peaks in the loss function is highlighted by white dotted lines. In (c), (f), and (i) the peaks in $\mathrm{Im}\left[\epsilon \right]$ generated by the intraband transitions are labeled as IB, IB1, and IB2, whose dispersion as well as that of the interband $T$ peak is highlighted by black dotted lines.

Figure 7

Upper panel: Real (blue solid line) and imaginary (red dashed line) parts of the dielectric function in ${\mathrm{SrC}}_6$ evaluated at momentum transfer $Q_{\parallel a^{*}}=0.12{\AA }^{-1}$. The intraband peaks in $\mathrm{Im}\left[\epsilon \right]$ generated by the slow and fast carriers at the Fermi level are labeled as IB1 and IB2, respectively. The second zero-crossing of $\mathrm{Re}\left[\epsilon \right]$ caused by the presence of two intraband peaks in $\mathrm{Im}\left[\epsilon \right]$ is highlighted by an open circle. Lower panel: Loss function with (black solid line) and without (orange long-dashed line) inclusion of the LFEs evaluated with the RPA kernel. The TDDFT result is shown by a green dotted line. The peak in the loss function labeled as AP corresponds to the acoustic plasmon.

Figure 8

Density of states distribution in ${\mathrm{SrC}}_6$ vs energy and group velocity components along (a) $a^{*}$, (b) $b^{*}$, and (c) $c^{*}$ symmetry directions. Peaks at the Fermi level are labeled as P1 and P2. The energies are relative to the Fermi level set to zero.

Figure 9

Upper panel: Real (blue solid line) and imaginary (red dashed line) parts of the dielectric function in ${\mathrm{SrC}}_6$ evaluated at momentum transfer $Q_{\parallel c^{*}}=0.02{\AA }^{-1}$ with the inclusion of the LFEs. The same quantities obtained without LFEs are shown by cyan and orange dotted lines, respectively. The zero-crossings of $\mathrm{Re}\left[\epsilon \right]$ confirming the collective nature of the $\pi \mathrm{P}$ and IP loss peaks are highlighted by open circles. Lower panel: Loss function with (black solid line) and without (black dotted line) inclusion of the LFEs. The $\pi$ and intraband plasmon peaks are labeled as $\pi \mathrm{P}$ and IP, respectively.

Figure 10

Upper panel: Real (blue solid line) and imaginary (red dashed line) parts of the dielectric function in ${\mathrm{SrC}}_6$ evaluated with the inclusion of the LFEs at momentum transfer $Q_{\parallel c^{*}}=0.61{\AA }^{-1}$. The same quantities obtained without LFEs are shown by cyan and orange dotted lines, respectively. Lower panel: Loss function obtained with (black solid line) and without (black dotted line) inclusion of the LFEs. The peaks $\pi \mathrm{P}$, IP, and $A$ in the loss function are discussed in the text.

Figure 11

Upper panel: Real (blue solid line) and imaginary (red dashed line) parts of the dielectric function in ${\mathrm{SrC}}_6$ evaluated with the inclusion of the LFEs at momentum transfer $Q_{\parallel c^{*}}=1.25{\AA }^{-1}$. The same quantities obtained without LFEs are shown by cyan and orange dotted lines, respectively. Lower panel: Loss function obtained with (black solid line) and without (black dotted line) inclusion of the LFEs. The peaks $\pi {\mathrm{P}}^{\prime }$ and ${\mathrm{IP}}^{\prime }$ in the loss function are discussed in the text.

Figure 12

Loss function, $L(\mathbf{Q},\omega )$, of ${\mathrm{BaC}}_6$ at momentum transfers $\mathbf{Q}$ along the $a^{*}$ axis calculated (a) with and (b) without inclusion of the local-field effects. The respective ${\log }_{10}\left\{\mathrm{Im}\left[\epsilon \right]\right\}$ is shown in (c). In (d), (e), and (f) and in (g), (h), and (i) the same quantities are presented for $\mathbf{Q}$'s directed along the $b^{*}$ and $c^{*}$ axes, respectively. Peaks in the loss function corresponding to the bulk, $\pi$, intraband, and acoustic plasmons are labeled by BP, $\pi \mathrm{P}$, IP, and AP, respectively. The peaks $K,M$, and ${\mathrm{IP}}^{\prime }$ are discussed in the main text. The dispersion of the acoustic plasmon peaks AP1 and AP2 is highlighted by white dotted lines. In (c), (f), and (i) the peaks in $\mathrm{Im}\left[\epsilon \right]$ generated by the intraband transitions are labeled as IB, IB1, IB2, and IB3, whose dispersion as well as that of the interband $T$ peak is highlighted by black dotted lines.

Figure 13

Upper panel: Real (blue solid line) and imaginary (red dashed line) parts of the dielectric function in ${\mathrm{BaC}}_6$ evaluated at momentum transfer $Q_{\parallel b^{*}}=0.02{\AA }^{-1}$. Lower panel: The loss function obtained at the same $Q_{\parallel b^{*}}$ is shown by a black solid line. The interband peak $T$ in $\mathrm{Im}\left[\epsilon \right]$ produces a sharp drop in $\mathrm{Re}\left[\epsilon \right]$ at the adjacent energy region via the Kramers-Kronig relation. The zero-crossings in $\mathrm{Re}\left[\epsilon \right]$ producing the plasmon peaks IP and $\pi \mathrm{P}$ in the loss function are highlighted by open circles.

Figure 14

Upper panel: Real (blue solid line) and imaginary (red dashed line) parts of the dielectric function in ${\mathrm{BaC}}_6$ evaluated at momentum transfer $Q_{\parallel a^{*}}=0.09{\AA }^{-1}$. The intraband peaks in $\mathrm{Im}\left[\epsilon \right]$ generated by the states at the Fermi level moving with different group velocities are labeled as IB1, IB2, and IB3. The zero-crossing of $\mathrm{Re}\left[\epsilon \right]$ producing a peak AP1 in the loss function is highlighted by an open circle. Lower panel: Loss function (black solid line). Peaks in the loss function labeled as AP1 and AP2 correspond to the acoustic plasmons.

Figure 15

Density of states distribution in ${\mathrm{BaC}}_6$ vs energy and group velocity components along (a) $a^{*}$, (b) $b^{*}$, and (c) $c^{*}$ symmetry directions. Peaks at the Fermi level are labeled as P1, P2, and P3. The energies are relative to the Fermi level set to zero.

Figure 16

Upper panel: Real (blue solid line) and imaginary (red dashed line) parts of the dielectric function in ${\mathrm{BaC}}_6$ evaluated with the inclusion of the LFE's at momentum transfer $Q_{\parallel c^{*}}=0.02{\AA }^{-1}$. The same quantities obtained without the LFE's are shown by cyan and orange dotted lines, respectively. Lower panel: Loss function with (black solid line) and without (black dotted line) inclusion of the LFE's. The zero-crossings in $\mathrm{Re}\left[\epsilon \right]$ producing the IP and BP peaks in the loss function are highlighted by open circles.