Ab initio study of the electron energy loss function in a graphene-sapphire-graphene composite system

The propagator of a dynamically screened Coulomb interaction $W$ in a sandwichlike structure consisting of two graphene layers separated by a slab of ${\mathrm{Al}}_2{\mathrm{O}}_3$ (or vacuum) is derived from single-layer graphene response functions and by using a local dielectric function for the bulk ${\mathrm{Al}}_2{\mathrm{O}}_3$. The response function of graphene is obtained using two approaches within the random phase approximation (RPA): an ab initio method that includes all electronic bands in graphene and a computationally less demanding method based on the massless Dirac fermion (MDF) approximation for the low-energy excitations of electrons in the $\pi$ bands. The propagator $W$ is used to derive an expression for the effective dielectric function of our sandwich structure, which is relevant for the reflection electron energy loss spectroscopy of its surface. Focusing on the range of frequencies from THz to mid-infrared, special attention is paid to finding an accurate optical limit in the ab initio method, where the response function is expressed in terms of a frequency-dependent conductivity of graphene. It was shown that the optical limit suffices for describing hybridization between the Dirac plasmons in graphene layers and the Fuchs-Kliewer phonons in both surfaces of the ${\mathrm{Al}}_2{\mathrm{O}}_3$ slab, and that the spectra obtained from both the ab initio method and the MDF approximation in the optical limit agree perfectly well for wave numbers up to about 0.1 ${\mathrm{nm}}^{-1}$. Going beyond the optical limit, the agreement between the full ab initio method and the MDF approximation was found to extend to wave numbers up to about 0.3 ${\mathrm{nm}}^{-1}$ for doped graphene layers with the Fermi energy of 0.2 eV.

Figure 1

(a) Crystal structure of a gr-${\mathrm{Al}}_2{\mathrm{O}}_3$-gr composite. (b) Simplified model where the sapphire layer is approximated by a homogeneous dielectric slab described by local dielectric function ${\epsilon }_S\left(\omega \right)$, whereas graphene sheets are described by 2D response functions ${\chi }_{1,2}(\mathbf{Q},\omega )$.

Figure 2

The Feynman diagrams for interaction between charge density fluctuations in two graphene layers described by response functions ${\chi }_1(\mathbf{Q},\omega )$ and ${\chi }_2(\mathbf{Q},\omega )$ in two cases, when the region between the layers is (a) vacuum and (b) dielectric slab described by the local dielectric function ${\epsilon }_S\left(\omega \right)$. The black dashed lines represent the bare “intrasystem” $v_{11}$ and “intersystem” $v_{12}$ Coulomb interactions, whereas the blue wavy lines represent the screened intrasystem ${\tilde{v}}_{11}$ and intersystem ${\tilde{v}}_{12}$ Coulomb interactions in the presence of dielectric slab.

Figure 3

The EELS in single-layer doped graphene with $E_F=1\mathrm{eV}$. The black vertical line denotes the boundary wave number $Q_b=0.34{\mathrm{nm}}^{-1}$, which separates the EELS obtained using the optical ab initio RPA method (left region) and the full ab initio RPA method (right region). The thin white lines show the lower, $\omega =v_F(Q-2k_F)$, and the upper, $\omega =v_{F}Q$, edges of the intraband ${\pi }^{*}\leftrightarrow {\pi }^{*}$ electron-hole excitations, as well as the lower edge, $\omega =2E_F-v_{F}Q$, of the interband $\pi \leftrightarrow {\pi }^{*}$ electron-hole excitations in the Dirac cone approximation.

Figure 4

The low-energy EELS in single-layer doped graphene ($E_F=1\mathrm{eV}$) where the optical ab initio RPA spectrum is calculated in the region $Q<Q_b$ by using (a) only the Drude optical conductivity $\sigma ={\sigma }^{\mathrm{intra}}$ (8) and (b) the “full” optical conductivity $\sigma ={\sigma }^{\mathrm{intra}}+{\sigma }^{\mathrm{inter}}$ (8),(10). Vertical black lines denote the boundary wave number $Q_b=0.34{\mathrm{nm}}^{-1}$. The thin white lines represent the electron-hole excitations edges in $\mathrm{gr}(1\mathrm{eV}$), as described in Fig. 3.

Figure 5

The EELS for two graphene layers calculated using: (a) the optical ab initio RPA method in the whole range of wave numbers, (b) the optical ab initio RPA method for $Q<Q_b$, and the full ab initio RPA method for $Q>Q_b$. The solid magenta and white lines show the dispersion relations of odd (${\omega }_{-}$) and even (${\omega }_{+}$) Dirac plasmons, respectively, calculated using the optical MDF-RPA model. The dotted magenta and white lines show the dispersion relations of odd (${\omega }_{-}^0$) and even (${\omega }_{+}^0$) Dirac plasmons, respectively, calculated using only the Drude model (8). The boundary wave number is $Q_b=0.098{\mathrm{nm}}^{-1}$, the separation between graphene layers is $a=5$ nm and both graphene layers are doped such that $E_F=200\mathrm{meV}$. The thin white lines represent the electron-hole excitations edges in $\mathrm{gr}(200\mathrm{meV}$), as described in Fig. 3.

Figure 6

The EELS in reduced $(Q,\omega )$ space in a symmetric gr-${\mathrm{Al}}_2{\mathrm{O}}_3$-gr sandwich calculated using (a) the optical ab initio RPA method and (b) the optical MDF-RPA method. The separation between graphene layers (or the ${\mathrm{Al}}_2{\mathrm{O}}_3$ slab thickness) is $a=5$ nm and both layers are doped such that $E_F=200\mathrm{meV}$. The black lines show dispersion relations of six hybridized modes (${\omega }_1^{\pm }, {\omega }_2^{\pm }$ and ${\omega }_3^{\pm }$) calculated using the optical MDF-RPA method, and the red dotted lines show dispersion relations of the two highest modes (${\omega }_3^{0\pm }$) calculated using only the Drude model. The thin white lines represent the upper edge, $\omega =v_{F}Q$, of the intraband ${\pi }^{*}\leftrightarrow {\pi }^{*}$ electron-hole excitations in Dirac cone approximation.

Figure 7

The EELS in a symmetric gr-${\mathrm{Al}}_2{\mathrm{O}}_3$-gr sandwich calculated using (a) the MDF-RPA method and (b) a combination of the optical ab initio RPA and the full ab initio RPA methods with the boundary wave number $Q_b=0.049{\mathrm{nm}}^{-1}$. The separation between graphene layers (or the ${\mathrm{Al}}_2{\mathrm{O}}_3$ slab thickness) is $a=5$ nm and they are both doped with equal densities, such that $E_F=200$ meV. The magenta dotted lines show dispersion relations of three odd modes (${\omega }_1^{-}, {\omega }_2^{-}$ and ${\omega }_3^{-}$), while the white dotted lines show dispersion relations of three even modes (${\omega }_1^{+}, {\omega }_2^{+}$ and ${\omega }_3^{+}$) calculated using the optical MDF-RPA method. The thin white lines represent the electron-hole excitations edges in $\mathrm{gr}(200\mathrm{meV}$), as described in Fig. 3.

Figure 8

The EELS in four asymmetric structures: (a) vacuum-${\mathrm{Al}}_2{\mathrm{O}}_3$-gr(200 meV), (b) gr(200 meV)-${\mathrm{Al}}_2{\mathrm{O}}_3$-vacuum, (c) gr(0 meV)-${\mathrm{Al}}_2{\mathrm{O}}_3$-gr(200 meV), and (d) gr(200 meV)-${\mathrm{Al}}_2{\mathrm{O}}_3$-gr(0 meV), calculated using the optical-RPA method. The ${\mathrm{Al}}_2{\mathrm{O}}_3$ slab thickness is $a=5$ nm. The black lines show dispersion relations of five modes (${\omega }_1, {\omega }_2, {\omega }_3, {\omega }_4$, and ${\omega }_5$) calculated using the optical MDF-RPA method, while the red dotted lines show the dispersion relations of the two highest (${\omega }_4^0$ and ${\omega }_5^0$) modes calculated using only the Drude approximation (8). The thin white lines represent the upper edge, $\omega =v_{F}Q$, of the intraband ${\pi }^{*}\leftrightarrow {\pi }^{*}$ electron-hole excitations in $\mathrm{gr}(200\mathrm{meV}$) in the Dirac cone approximation.

Figure 9

The EELS in four asymmetric structures: (a) vacuum-${\mathrm{Al}}_2{\mathrm{O}}_3$-gr(200 meV), (b) gr(200 meV)-${\mathrm{Al}}_2{\mathrm{O}}_3$-vacuum, (c) gr(0 meV)-${\mathrm{Al}}_2{\mathrm{O}}_3$-gr(200 meV), and (d) gr(200 meV)-${\mathrm{Al}}_2{\mathrm{O}}_3$-gr(0 meV), calculated using the MDF-RPA model. The ${\mathrm{Al}}_2{\mathrm{O}}_3$ slab thickness is $a=5\mathrm{nm}$. The white dotted lines show dispersion relations of five modes (${\omega }_1, {\omega }_2, {\omega }_3, {\omega }_4$, and ${\omega }_5$) calculated using the optical MDF-RPA model. The thin white lines represent the electron-hole excitation edges in $\mathrm{gr}(200\mathrm{meV}$), as described in Fig. 3.