Effective numbers of charge carriers in doped graphene: Generalized Fermi liquid approach

The single-band current-dipole Kubo formula for the dynamical conductivity of heavily doped graphene from Kupčić [Phys. Rev. B 91, 205428 (2015)] is extended to a two-band model for conduction $\pi$ electrons in lightly doped graphene. Using a posteriori relaxation-time approximation in the two-band quantum transport equations, with two different relaxation rates and one quasiparticle lifetime, we explain a seemingly inconsistent dependence of the dc conductivity of ultraclean and dirty lightly doped graphene samples on electron doping, in a way consistent with the charge continuity equation. It is also shown that the intraband contribution to the effective number of conduction electrons in the dc conductivity vanishes at $T=0$ K in the ultraclean regime, but it remains finite in the dirty regime. The present model is shown to be consistent with a picture in which the intraband and interband contributions to the dc conductivity are characterized by two different mobilities of conduction electrons, the values of which are well below the widely accepted value of mobility in ultraclean graphene. The dispersions of Dirac and $\pi$ plasmon resonances are reexamined to show that the present, relatively simple expression for the dynamical conductivity tensor can be used to study simultaneously single-particle excitations in the dc and optical conductivity and collective excitations in energy loss spectroscopy experiments.

Figure 1

(a) The Bethe-Salpeter equations for the auxiliary electron-hole propagators ${\mathrm{\Phi }}_{\nu }^{L{L}^{\prime }}(\mathbf{k},{\mathbf{k}}_{+},\mathit{i}{\omega }_n,\mathit{i}{\omega }_{n+})$ in the Hartree-Fock approximation. The dashed line represents the force-force correlation function ${\mathcal{F}}_{\lambda }({\mathbf{k}}^{\prime }-\mathbf{k},\mathit{i}{\nu }_m)$ and the bold solid lines are the single-electron propagators ${\mathcal{G}}_L(\mathbf{k},\mathit{i}{\omega }_n)$. (b) The Dyson equation for ${\mathcal{G}}_L(\mathbf{k},\mathit{i}{\omega }_n)$ in the same approximation.

Figure 2

The Bethe-Salpeter expression for the $4\times 4$ current-current correlation function ${\pi }_{\mu \nu }(\mathbf{q},\mathit{i}{\nu }_n)$ [15, 21].

Figure 3

The effective numbers of charge carriers $n_{\alpha \alpha }^{\mathrm{intra}}\left(\mathbf{q}\right)$ and $n_{\alpha \alpha }^{\mathrm{tot}}\left(\mathbf{q}\right)$ calculated by using Eq. (26), for $n_L\left(\mathbf{k}\right)=f_L\left(\mathbf{k}\right),t=2.52$ eV, $T=100$ K, and $\mathbf{q}\approx \mathbf{0}$. $V_{0}n=(1/N){\sum }_{L\mathbf{k}\sigma }{f}_L\left(\mathbf{k}\right)$ is the concentration of conduction electrons ($V_{0}n=2$ in pristine graphene) and $V_0$ is the primitive cell volume.

Figure 4

Solid, dashed, and dotted lines: the real part of the dynamical conductivity of doped graphene obtained by Eq. (35) beyond the Dirac cone approximation, for $n_L\left(\mathbf{k}\right)=f_L\left(\mathbf{k}\right),E_{\mathrm{F}}=-0.105$ eV, $T=150$ K, $q_x{a}_0=0.0001$, and for realistic values of the damping energies, $\hslash {\mathrm{\Gamma }}_1=4$ meV and $\hslash {\mathrm{\Gamma }}_2=20$ meV. The interband part calculated by using ${\pi }_{00}^{\mathrm{inter}}(\mathbf{q},\omega )$ and ${\pi }_{\alpha 0}^{\mathrm{inter}}(\mathbf{q},\omega )$ from Eq. (34) is also shown (dot-dashed and dot-dot-dashed lines). Experimental data (full triangles) are from Ref. [14]. $a_0$ is the Bohr radius.

Figure 5

One ${\left(H_1^{\prime }\right)}^0$ and three ${\left(H_1^{\prime }\right)}^2$ contributions to ${\pi }_{\mu \nu }(\mathbf{q},\mathit{i}{\nu }_n)$, labeled by 0 (bare contribution), $2A_1$ (electron self-energy term), $2A_2$ (hole self-energy term), and $2B=2B_1+2B_2$ (vertex correction).

Figure 6

The doping dependence of ${\sigma }_{\alpha \alpha }^{\mathrm{dc}}$ in ultraclean graphene calculated by using the current-dipole conductivity formula (35), for $n_L\left(\mathbf{k}\right)=f_L\left(\mathbf{k}\right)$, and for realistic values of $\hslash \mathrm{\Gamma },\hslash \mathrm{\Gamma }\left(T\right)=a+bT,a=0.5$ meV, and $b=0.5/200$ meV/K. Experimental data, taken at $T=40$ K, are from Ref. [13].

Figure 7

The doping dependence of the effective number $n_{\alpha \alpha }^{\mathrm{dc},\mathrm{tot}}$ and the dc conductivity ${\sigma }_{\alpha \alpha }^{\mathrm{dc}}$ for different values of the ratio ${\mathrm{\Sigma }}^i/\mathrm{\Gamma }$ at $T=50$ K and $\hslash \mathrm{\Gamma }=5$ meV. Experimental data (for $n_e>0$ and $T=40$ K) are from Ref. [13].

Figure 8

The real part of the dynamical conductivity of pristine graphene calculated by using Eq. (35), for $n_L\left(\mathbf{k}\right)=f_L\left(\mathbf{k}\right),\hslash {\mathrm{\Gamma }}_1=\hslash {\mathrm{\Gamma }}_2=5$ meV, and $T=50$, 40, 30, 20, 15 K. The intraband and interband contributions are also shown.

Figure 9

The real part of $\varepsilon (\mathbf{q},\omega )$ calculated by using three versions of ${\pi }_{\mu \nu }^{\mathrm{inter}}(\mathbf{q},\omega )$ from Eq. (34), for $E_{\mathrm{F}}=-0.5$ eV and $q_x{a}_0=0.02,q_y=0$. ${\sigma }_{\alpha \alpha }^{\mathrm{intra}}(\mathbf{q},\omega )$ is given by the intraband term in Eq. (35). The dot-dot-dashed line is the prediction of the charge-charge conductivity formula. The parameters of the model are $\hslash {\mathrm{\Gamma }}_1=10$ meV, $\hslash {\mathrm{\Gamma }}_2=50$ meV, and $T=150$ K.

Figure 10

The two-dimensional plot of the energy loss function $-\mathrm{Im}\{1/\varepsilon (\mathbf{q},\omega \left)\right\}$, for $q_x=q$ and $q_y=0$. Main figure: the interband plasmon resonance ($\pi$ plasmon, in common language). Inset: the intraband (Dirac) plasmon resonance. The dashed lines show the frequencies ${\omega }_{\mathrm{pl}}^0\left(\mathbf{q}\right)=\sqrt{(2\pi e^{2}q/m)n_{\alpha \alpha }^{\mathrm{intra}}}$, with $V_0{n}_{\alpha \alpha }^{\mathrm{intra}}=0.109$ (inset) and ${\omega }_{\mathrm{pl}}^{\mathrm{tot},0}\left(\mathbf{q}\right)=\sqrt{(2\pi e^{2}q/m)n_{\alpha \alpha }^{\mathrm{tot}}}$, with $V_0{n}_{\alpha \alpha }^{\mathrm{tot}}=1.045$ (main figure). The parameters are the same as in Fig. 9.

Figure 11

The doping dependence of ${\sigma }_{\alpha \alpha }^{\mathrm{dc}}$ at $T=50$, 100, and 150 K. The solid, dashed, and dot-dashed lines represent the results of the current-dipole conductivity formula (35) for $\hslash {\mathrm{\Gamma }}_1=\hslash {\mathrm{\Gamma }}_2=5$ meV and $n_L\left(\mathbf{k}\right)=f_L\left(\mathbf{k}\right)$. The full circles, squares, and diamonds are the results of the current-current conductivity formula (A2) for $\hslash {\mathrm{\Sigma }}^i=2.5$ meV.