Strength of Effective Coulomb Interactions in Graphene and Graphite

To obtain an effective many-body model of graphene and related materials from first principles we calculate the partially screened frequency dependent Coulomb interaction. In graphene, the effective on-site (Hubbard) interaction is $U_{00}=9.3\mathrm{eV}$ in close vicinity to the critical value separating conducting graphene from an insulating phase emphasizing the importance of nonlocal Coulomb terms. The nearest-neighbor Coulomb interaction strength is computed to $U_{01}=5.5\mathrm{eV}$. In the long-wavelength limit, we find the effective background dielectric constant of graphite to be $ϵ=2.5$ in very good agreement with experiment.

Figure 1

Effect of lattice expansion on the strength of Coulomb interactions obtained with $h=21.2\AA$. (a) On-site $U_{00}(a)/U_{00}(a_0)$ and nearest-neighbor $U_{01}(a)/U_{01}(a_0)$ Coulomb interaction as well as nearest-neighbor hopping $t(a)/t(a_0)$ as a function of isotropic strain ($1-a/a_0$). $a$ is the lattice constant. The parameters are given relative to their values at the equilibrium lattice constant $a_0=2.47\AA$. The solid lines are linear fits serving as a guide to the eye. (b) Ratios of on-site $U_{00}(a)/t(a)$ and nearest-neighbor $U_{01}(a)/t(a)$ Coulomb interaction to the nearest-neighbor hopping. The dashed line indicates the phase boundary at $U_{\mathrm{SL}}/t=3.5$ separating the zero gap phase from a gapped spin-liquid one [8].

Figure 2

Static cRPA dielectric functions $ϵ(\mathbf{k})$ of graphene and graphite as function of (in-plane) momentum transfer $\mathbf{k}=(k_x,k_y)$. (a) Color coded (gray scale) $ϵ(\mathbf{k})$ for graphene. The most pronounced effect is the decrease of $ϵ(\mathbf{k})$ for $\mathbf{k}\to 0$. There is a small directional modulation at intermediate momentum transfer $k=|\mathbf{k}|\gtrsim 1{\AA }^{-1}$. (b) cRPA dielectric functions $ϵ(k)$ of graphene and graphite as function of $k=|\mathbf{k}|$. Equation (3) fits well the background dielectric screening for freestanding graphene in the limit of $k\to 0$. For graphite, two values of the perpendicular momentum transfer are considered: $k_z=0$ and $k_z=0.5(2\pi /c)$ with $c=3.3\AA$ being the graphite interlayer spacing.

Figure 3

Frequency dependence of the on-site and nearest-neighbor interaction obtained from cRPA for graphene ($h=21.2\AA$) and graphite. For graphite $U_{00}(\omega )=U_{00}^A(\omega )$ is shown, which is virtually the same as $U_{00}^B(\omega )$. $|U_{00}^A(\omega )-U_{00}^B(\omega )|<0.15\mathrm{eV}$ for $\omega <20\mathrm{eV}$.