Negative differential transmission in graphene

By using the Kubo linear response theory with the Keldysh Green function approach, we investigate the mechanism leading to the negative differential transmission in a system with the equilibrium electron density much smaller than the photon-excited one. It is shown that the negative differential transmission can appear at low probe-photon energy (in the order of the scattering rate) or at high energy (much larger than the scattering rate). For the low probe-photon energy case, the negative differential transmission is found to come from the increase of the intraband conductivity due to the large variation of electron distribution after the pumping. As for the high probe-photon energy case, the negative differential transmission is shown to tend to appear with the hot-electron temperature being closer to the equilibrium one and the chemical potential higher than the equilibrium one but considerably smaller than half of the probe-photon energy. We also show that this negative differential transmission can come from both the inter- and the intraband components of the conductivity. Especially, for the interband component, its contribution to the negative differential transmission is shown to come from both the Hartree-Fock self-energy and the scattering. Furthermore, the influence of the Coulomb-hole self-energy is also addressed.

Figure 1

(a) Chemical potential and (b) electron temperature dependencies of $\Delta {\Sigma }_{\mathrm{Re}}$ and $\Delta {\left(2\tau \right)}^{-1}$ with and without the CH self-energy. Here the probe-photon energy $\omega$ is chosen to be 1500 meV. For the chemical potential ${\mu }_1$ dependence, the electron temperature $T_e=700$ K and for the electron temperature dependence, the chemical potential ${\mu }_1=300$ meV. Moreover, to make the results clearer, the curves with the CH self-energy plotted in the figure are $\Delta {\Sigma }_{\mathrm{Re}}/100$. Note that the scale of $\Delta {\left(2\tau \right)}^{-1}$ is on the right-hand side of the frame.

Figure 2

Interband as well as total differential conductivities with and without the CH self-energy as function of the probe-photon energy $\omega$. The states after the pumping are chosen to be (a) $T_e=500$ K and ${\mu }_1=-{\mu }_{-1}=300$ meV, and (b) $T_e=1000$ K and ${\mu }_1=-{\mu }_{-1}=300$ meV. The total conductivity without the CH self-energy in (a) is fitted for $\omega \le 50$ meV by Eq. (16) with the fitted results plotted as the green dotted curve. The inset zooms the range at high $\omega$ with the interband differential conductivity without the CH and HF self-energies plotted as brown solid triangles. The black chain line in the figure marks $\Delta \sigma /{\sigma }_0=0$.

Figure 3

(a) Total differential conductivity in $\omega$-${\mu }_1$ space with $T_{\mathrm{ph}}=T_e=700$ K. (b) Chemical potential dependence of the total differential conductivity at $\omega =1500$ meV (blue solid dots) and 1700 meV (blue solid curve) as well as the interband one at $\omega =1500$ meV (red open squares) and 1700 meV (red dashed curve). The chemical potential dependence of the interband differential conductivity without the HF self-energy at $\omega =1700$ meV is also plotted as the green dashed curve. Here $T_e=T_{\mathrm{ph}}=700$ K. The red arrows mark the peak of the interband differential conductivities. (c) Total differential conductivity in the $\omega$-${\mu }_1$ parameter space with $T_{\mathrm{ph}}$ being 200 K lower than $T_e$. (d) Chemical potential dependence of the total differential conductivity at $\omega =1700$ meV as well as that for the interband one with and without the HF self-energy. Here $T_e=700$ K and $T_{\mathrm{ph}}=500$ K. (e) Total differential conductivities in the $T_e$-${\mu }_1$ parameter space. Here $\omega =1500$ meV and $T_{\mathrm{ph}}=T_e$. (f) Temperature dependence of the total differential conductivity as well as that for the interband one with and without the HF self-energy. Here ${\mu }_1$ is chosen to be 320 meV. For clarity, only the positive differential conductivity is plotted in (a), (c), and (e). The chain lines in (b), (d), and (f) mark $\Delta \sigma /{\sigma }_0=0$.

Figure 4

Total differential conductivity with the CH self-energy for graphene on SiO${}_2$ (a) and SiC (b) in $T_e$-${\mu }_1$ parameter space. For clarity, only the positive differential conductivity is plotted.