Dynamical polarizability of graphene beyond the Dirac cone approximation

We compute the dynamical polarizability of graphene beyond the usual Dirac cone approximation, integrating over the full Brillouin zone. We find deviations at $\hslash \omega =2t$ (where $t$ is the hopping parameter) which amount to a logarithmic singularity due to the Van Hove singularity and derive an approximate analytical expression. Also at low energies, we find deviations from the results obtained from the Dirac cone approximation which manifest themselves in a peak spitting at arbitrary direction of the incoming wave vector $\mathbf{q}$. Consequences for the plasmon spectrum are discussed.

Figure 1

(a) The hexagonal and (b) rhombical Brillouin zone. The symmetrized rhombical Brillouin zone and its segmentation. The inner square refers to $j=-$, the outer triangles refer to $j=+$; additionally the values of $s$ and $s^{\prime }$ are given.

Figure 2

(Color online) Left-hand side: the imaginary part of the polarizability $\text{Im}{P}^{\left(1\right)}\left(\left|\mathbf{q}\right|,\phi ,\omega \right)$ as a function of the energy $\omega$ at $k_{B}T/t=0.01$ for different angles $\phi$ with $\left|\mathbf{q}\right|a=0.1$. The result obtained from the Dirac cone approximation is also shown (dashed line). Inset: energy region around the Van Hove singularity of the same curves. Right-hand side: the imaginary part of the polarizability $\text{Im}{P}^{\left(1\right)}\left(\mathbf{q},\omega \right)$ as a function of the energy $\omega$ at $k_{B}T/t=0.01$ for various wave vectors $\mathbf{q}$ defined in the text.

Figure 3

(Color online) The imaginary part of the polarizability $\text{Im}{P}^{\left(1\right)}\left(\left|\mathbf{q}\right|,\phi ,\omega \right)$ with $\left|\mathbf{q}\right|a=0.01$ as a function of the energy $\omega$ for various angles $\phi$ and two chemical potentials $\mu /t=0.05$ (left) and $\mu /t=0.5$ (right) at $k_{B}T/t=0.01$. Also shown is the analytical result coming from the Dirac cone approximation at zero temperature (dashed line).

Figure 4

(Color online) The imaginary part of the polarizability $\text{Im}{P}^{\left(1\right)}\left(\left|\mathbf{q}\right|,\phi ,\omega \right)$ with $\left|\mathbf{q}\right|a=0.01$ as a function of the energy $\omega$ for various angles $\phi$ and two chemical potentials $\mu /t=1$ (left) and $\mu /t=1.5$ (right) at $k_{B}T/t=0.01$. Also shown is the analytical result coming from the Dirac cone approximation at zero temperature (dashed line).

Figure 5

(Color online) The real part of the polarizability $\text{Re}{P}^{\left(1\right)}\left(\left|\mathbf{q}\right|,\phi ,\omega \right)$ as a function of the energy $\omega$ for various angles $\phi$ at $k_{B}T/t=0.01$. Left: for energies close to the Van Hove singularity with $\left|\mathbf{q}\right|a=0.1$ and $\mu =0$. Right: for low energies with $\left|\mathbf{q}\right|a=0.01$ and $\mu /t=0.05$. Also shown is the analytical result coming from the Dirac cone approximation at zero temperature (dashed line).

Figure 6

(Color online) The real part of the polarizability $\text{Re}{P}^{\left(1\right)}\left(\left|\mathbf{q}\right|,\phi ,\omega \right)$ for $\left|\mathbf{q}\right|a=0.01$ as a function of the energy $\omega$ for various angles $\phi$ at $k_{B}T/t=0.01$. Left: for the chemical potential $\mu /t=1$. Right: for the chemical potential $\mu /t=1.5$. Also shown is the analytical result coming from the Dirac cone approximation at zero temperature (dashed line).