General theory of intraband relaxation processes in heavily doped graphene

The frequency and wave-vector-dependent memory function in the longitudinal conductivity tensor of weakly interacting electronic systems is calculated by using an approach based on quantum transport equations. In this paper, we show that there is a close relation between the single-electron self-energy, the electron-hole pair self-energy, and the memory function. It is also shown in which way singular long-range Coulomb interactions, together with other $\mathbf{q}\approx \mathbf{0}$ scattering processes, drop out of both the memory function and the related transport equations. The theory is illustrated on heavily doped graphene, which is the prototype of weakly interacting single-band electron-phonon systems. A steplike increase of the width of the quasiparticle peak in angle-resolved photoemission spectra at frequencies of the order of the frequency of in-plane optical phonons is shown to be consistent with the behavior of an intraband plasmon peak in the energy loss spectroscopy spectra. Both anomalies can be understood as a direct consequence of weak electron scattering from in-plane optical phonons.

Figure 1

The skeleton series for the single-electron self-energy $\mathrm{\Sigma }(\mathbf{k},\mathit{i}{\omega }_n)$. $U(\mathbf{k},{\mathbf{k}}^{\prime },\mathit{i}{\omega }_n,\mathit{i}{\omega }_m)$ (white rectangle) and $W(\mathbf{k},{\mathbf{k}}^{\prime },\mathit{i}{\nu }_m)=\phi \left({\mathbf{q}}^{\prime }\right)/V+\mathcal{F}(\mathbf{k},{\mathbf{k}}^{\prime },\mathit{i}{\nu }_m)$ (bold dashed line) are the completely irreducible four-point interaction and the renormalized two-point interaction.

Figure 2

The real part (solid line) and the imaginary part (dashed line) of $-\hslash {\mathrm{\Sigma }}_{\mathrm{op}}(\mathbf{k},\omega )$ in the ${\pi }^{*}$ band, for $\mathbf{k}=(k_x,0),k_x=1.95{\AA }^{-1},t=2.52$ eV, $E_{\mathrm{F}}=1.35$ eV, $|G_{\mathrm{op}}{|}^2=0.2{\mathrm{eV}}^2,\hslash {\omega }_{\mathrm{op}}=0.2$ eV, $\eta =\hslash {\mathrm{\Sigma }}^i=20$ meV, and $T=30$ K. For these values of parameters, we obtain $-\mathrm{Re}\left\{\hslash {\mathrm{\Sigma }}_{\mathrm{op}}(\mathbf{k},0)\right\}=81.5$ meV, which corresponds to the change of the chemical potential, $\delta \mu$.

Figure 3

The spectral function $(1/2\pi )\mathcal{A}(\mathbf{k},\varepsilon )$ in the ${\pi }^{*}$ band for $\mathbf{k}=(k_x,0),k_x=1.8,1.85,1.9$, and 1.95 ${\AA }^{-1}$. Dotted line: the damping function $-\hslash {\mathrm{\Sigma }}^i(\mathbf{k},\omega )$. The parameters are the same as in Fig. 2, with $\hslash \delta {\mathrm{\Sigma }}^i(\mathbf{k},\omega )=a+\left|\hslash \omega \right|b$, and $a=0.03$ eV, $b=0.05$.

Figure 4

The Bethe-Salpeter equation for the auxiliary electron-hole propagator ${\mathrm{\Phi }}_{\nu }^0(\mathbf{k},{\mathbf{k}}_{+},\mathit{i}{\omega }_n,\mathit{i}{\omega }_{n+})$. The white rectangle represents the irreducible four-point interaction $U^0({\mathbf{k}}_{+},{\mathbf{k}}^{\prime },{\mathbf{k}}_{+}^{\prime },\mathbf{k},\mathit{i}{\omega }_{n+},\mathit{i}{\omega }_m,\mathit{i}{\omega }_{m+},\mathit{i}{\omega }_n)$.

Figure 5

The diagrammatic illustration of the ${\lambda }^2$ contribution to ${\sigma }_{\alpha \alpha }(\mathbf{q},\omega )$ from Eqs. (37), (26), and (28).

Figure 6

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

Figure 7

The first set of four mutually related contributions to ${\sigma }_{\alpha \alpha }^{\left(2\right)}\left(\omega \right)$ that are proportional to ${\left(H_2^{\prime }\right)}^2$ [or ${\left(H_1^{\prime }\right)}^4]$.

Figure 8

The diagrammatic illustration of the $\mathrm{\Delta }{\mathrm{\Pi }}_{\alpha }^{\left(CD\right)}(\mathbf{k},\omega ,\varepsilon )$ contributions to ${\sigma }_{\alpha \alpha }^{\left(2\right)}\left(\omega \right)$.

Figure 9

The first three ${\left(H_1^{\prime }\right)}^4$ contributions to ${\mathrm{\Sigma }}^{\left[4\right]}(\mathbf{k},\omega )$.

Figure 10

The real part (solid lines) and the imaginary part (dashed lines) of the memory function $M_{\alpha }^{\mathrm{op}}({\mathbf{k}}_{\mathrm{F}},\omega )$, Eq. (49), for $t=2.52$ eV, $E_{\mathrm{F}}=0.45,1.35$ eV, $|{\tilde{G}}_{\mathrm{op}}{|}^2=0.2{\mathrm{eV}}^2,\hslash {\omega }_{\mathrm{op}}=0.2$ eV, $\eta =\hslash {\mathrm{\Sigma }}^i=20$ meV, and $T=30$ K. Here, $\mathrm{Re}\left\{M_{\alpha }^{\mathrm{op}}({\mathbf{k}}_{\mathrm{F}},0)\right\}=0$.

Figure 11

Solid line: the real part of the intraband conductivity (50) for $E_{\mathrm{F}}=0.45$ eV. The parameters are the same as in Fig. 10, with $\delta M_{\alpha }^i\left(\omega \right)=\tilde{a}+\left|\hslash \omega \right|\tilde{b}$ and $\tilde{a}=0.01$ eV, $\tilde{b}=0.015$. Dashed line: the result of the relaxation-time approximation, Eq. (51), for $\hslash {\mathrm{\Gamma }}_{\alpha }=14$ meV.

Figure 12

The energy loss function $-\mathrm{Im}\{1/\varepsilon (\mathbf{q},\omega \left)\right\}$ as a function of the wave vector $\mathbf{q}=(q_x,0)$, for $q_x{a}_0=0.001,E_{\mathrm{F}}=0.4$ eV, ${\tilde{\varepsilon }}_{\infty }=1$, and for $\hslash \omega$ close to the energy of the in-plane optical phonons. The conductivity is given by Eq. (50). The parameters are the same as in Fig. 11. The wave vector $q_x$ increases with the step $\mathrm{\Delta }{q}_x{a}_0=0.00025$ up to $q_x{a}_0=0.004$. $a_0$ is the Bohr radius.

Figure 13

The relevant vectors in the honeycomb lattice in graphene. The $t_j\equiv \tilde{t}(\overline{\mathbf{R}},{\mathbf{r}}_j)$ represent three hopping integrals between neighboring $2p_z$ orbitals. Two different carbon sites in the unit cell are labeled by $A$ and $B$.