Damping effects in doped graphene: The relaxation-time approximation

The dynamical conductivity of interacting multiband electronic systems derived by Kupčić et al. [J. Phys.: Condens. Matter 90, 145602 (2013)] is shown to be consistent with the general form of the Ward identity. Using the semiphenomenological form of this conductivity formula, we have demonstrated that the relaxation-time approximation can be used to describe the damping effects in weakly interacting multiband systems only if local charge conservation in the system and gauge invariance of the response theory are properly treated. Such a gauge-invariant response theory is illustrated on the common tight-binding model for conduction electrons in doped graphene. The model predicts two distinctly resolved maxima in the energy-loss-function spectra. The first one corresponds to the intraband plasmons (usually called the Dirac plasmons). On the other hand, the second maximum $(\pi$ plasmon structure) is simply a consequence of the Van Hove singularity in the single-electron density of states. The dc resistivity and the real part of the dynamical conductivity are found to be well described by the relaxation-time approximation, but only in the parametric space in which the damping is dominated by the direct scattering processes. The ballistic transport and the damping of Dirac plasmons are thus the problems that require abandoning the relaxation-time approximation.

Figure 1

The solid lines represent the electron dispersions (6) along the $K^{\prime }\text{−}\Gamma \text{−}K$ line in the Brillouin zone, for $t=2.52$ eV. The dashed lines are the asymmetric dispersions corresponding to the finite overlap integral $s=0.07$ and $t=3$ eV. The diamonds are the dispersions obtained by solving the local density approximation Kohn-Sham equations [21]. The dot-dashed line labels the position of the Fermi level $E_{\mathrm{F}}$ in a typical hole-doped case $(E_{\mathrm{F}}=-0.5$ eV) [5, 6, 7, 15].

Figure 2

The Bethe-Salpeter expression for the current-current correlation function ${\pi }_{\mu \nu }^{\mathrm{C}}\left(q\right)$ [3, 13, 23].

Figure 3

The real part of the dynamical conductivity (61) in hole-doped graphene obtained by means of the relaxation-time approximation. The parameters of the model are $t=2.52$ eV, $E_{\mathrm{F}}=-0.5$ eV, $\hslash {\Gamma }_1=10$ meV, $\hslash {\Gamma }_2=50$ meV, $T=150$ K, and $\mathbf{q}=(q_x,0)$. The solid line in the inset of the figure represents the ordinary Drude formula (53). $a_0$ is the Bohr radius.

Figure 4

Elementary excitations in hole-doped graphene in the random-phase approximation [for $E_{\mathrm{F}}=\hslash {\omega }_{\mathrm{F}}=-0.5$ eV and $\mathbf{q}=(q_x,0)\text{]}$. The solid and dashed lines represent the intraband plasmon dispersions calculated by using, respectively, ${\varepsilon }_{\infty }(\mathbf{q},\omega )=1$ and ${\tilde{\varepsilon }}_{\infty }(\mathbf{q},\omega )=1$ in Eqs. (1) and (67).

Figure 5

The dependence of the effective number $V_0{n}_{\alpha \alpha }^{\mathrm{intra},0}$, Eq. (35) with $n_L\left(\mathbf{k}\right)$ replaced by $f_L\left(\mathbf{k}\right)$, and the density of states $V_0{\rho }_0\left(E_{\mathrm{F}}\right)$, Eq. (58), on the electron doping in hole-doped graphene. Notice that $n_{\alpha \alpha }^{\mathrm{intra},0}\approx n$ for $V_{0}n\ll 2$ and that $n_{\alpha \alpha }^{\mathrm{intra},0}\approx (3t/4){\rho }_0\left(E_{\mathrm{F}}\right)\propto \sqrt{n_h}$ for $V_0{n}_h\ll 2$.

Figure 6

The dependence of the real part of the dielectric function (67) on $\hslash \omega$, for $E_{\mathrm{F}}=-0.5$ eV and for two values of the wave vector $\mathbf{q}=(q_x,0),q_x{a}_0=0.005$, and 0.02. The solid and dashed lines correspond, respectively, to ${\varepsilon }_{\infty }(\mathbf{q},\omega )=1$ and ${\tilde{\varepsilon }}_{\infty }(\mathbf{q},\omega )=1$.

Figure 7

Solid line: the real part of the dynamical conductivity, Eq. (61), in hole-doped graphene, for $E_{\mathrm{F}}=-0.5$ eV, $\hslash {\Gamma }_1=10$ meV, $\hslash {\Gamma }_2=50$ meV, $T=150$ K, and $q_x{a}_0=0.02$. Dashed and dot-dashed lines: the corresponding real and imaginary parts of the dielectric function.

Figure 8

The Dirac plasmon peak in the energy loss function for different values of $q_x{a}_0$. The parameters are the same as in Fig. 7.

Figure 9

The $\pi$ plasmon peak in the energy loss function for different values of $\mathbf{q}=(q_x,0)$. The spectra are divided by $q{a}_0$ for clarity. The imaginary part of the screened long-range Coulomb interaction is $\mathrm{Im}\left\{\tilde{v}(\mathbf{q},\omega )\right\}=2\pi a_0\mathrm{Im}\{1/q{a}_0\varepsilon (\mathbf{q},\omega )\}$.

Figure 10

Main figure: the dispersion of the Dirac plasmons for $E_{\mathrm{F}}=-0.4$ eV. Inset: the energy loss function $-\mathrm{Im}\{1/\varepsilon (\mathbf{q},\omega \}$ calculated by means of the relaxation-time approximation, for $\hslash \omega$ close to the energy of the in-plane optical phonons and for $\hslash {\Gamma }_1=0.02$ eV.

Figure 11

Three ${\left(H_1^{\prime }\right)}^2$ contributions to ${\pi }_{\mu \nu }^{\mathrm{intra}}(\mathbf{q},\omega )$, labeled by $2A_1$ (electron self-energy term), $2A_2$ (hole self-energy term), and $2B=2B_1+2B_2$ (vertex correction).

Figure 12

Four contributions to ${\pi }_{\mu \nu }^{\mathrm{intra}}(\mathbf{q},\omega )$ out of eight irreducible contributions that are proportional to ${\left(H_2^{\prime }\right)}^2$ [or ${\left(H_1^{\prime }\right)}^4\text{]}$.