Ultrafast nonequilibrium evolution of excitonic modes in semiconductors

We study the time evolution of excitonic states after photoexcitation in the one-dimensional spinless extended Falicov-Kimball model. Several numerical methods are employed and benchmarked against each other: time-dependent mean-field simulations, the second-Born approximation (2BA) within the Kadanoff-Baym formalism, the generalized Kadanoff-Baym ansatz (GKBA) implemented with the 2BA, and the infinite time-evolving block decimation (iTEBD) method. It is found that the GKBA gives the best agreement with iTEBD and captures the relevant physics. Excitations to the particle-hole continuum and resonant excitations of the equilibrium exciton result in a qualitatively different dynamics. In the former case, the exciton binding energy remains positive and the frequency of the corresponding coherent oscillations is smaller than the band gap. On the other hand, resonant excitations trigger a collective mode whose frequency is larger than the band gap. We discuss the origin of these different behaviors by evaluating the nonequilibrium susceptibility using the nonthermal distribution and a random phase approximation. The peculiar mode with frequency larger than the band gap is associated with a partial population inversion with a sharp energy cutoff. We also discuss the effects of the cooling by a phonon bath. We demonstrate the real-time development of coherence in the polarization, which indicates excitonic condensation out of equilibrium.

Figure 1

Dispersion of the conduction band and valence band for $J_c=1,J_v=-1,{\mathrm{\Delta }}_v=-3.2,{\mathrm{\Delta }}_c=1.2$, and $U=2.0$ at $T=0$. The green (blue) arrows indicate the above-gap (resonant) excitation with frequency $\mathrm{\Omega }=3.0$ ($\mathrm{\Omega }=1.9$).

Figure 2

(a) Comparison of the exciton binding energy ${\mathcal{E}}_b$ estimated from the oscillations after a short pulse using different numerical methods. The dashed line indicates $\frac{U^2}{8}$. (b) Results of ${\gamma }_{0,22}$ [Eq. (18b)] for $T=0$. Here, $0^{+}=0.02$ is used. The horizontal dotted line indicates $-\frac{2}{U}$. The shaded area indicates the particle-hole continuum. (c) The spectrum of the linear response function $-\mathrm{Im}{\chi }_{11}(\omega ;q)$ in equilibrium evaluated by the TEBD for ${\mathrm{\Delta }}_v=-3.2,{\mathrm{\Delta }}_c=1.2$, and $U=2.0$ at $T=0$. Here, we take $\sigma =(t_1-t_0)/\left(2\sqrt{2}\right)$ with $t_0=0$ and $t_1=40$. The green (blue) arrows indicate the above-gap (resonant) excitation with excitation frequency $\mathrm{\Omega }=3.0$ ($\mathrm{\Omega }=1.9$).

Figure 3

$\mathrm{GKBA}+\mathrm{s}2\mathrm{BA}$ time evolution of the excited charge (a), (d), the dipole moment (b), (e), and the total energy (c), (f) during and after the photoexcitation with $\mathrm{\Omega }=3.0$ (a)–(c) and $\mathrm{\Omega }=1.9$ (d)–(f), pulse parameters defined in Eq. (32), and different field strengths $E_0$.

Figure 4

Comparison of the density of conduction band electrons and the polarization among s2BA, $\mathrm{GKBA}+\mathrm{s}2\mathrm{BA}$, iTEBD, and MF for $\mathrm{\Omega }=1.9$. (a), (b) are for $E_0=0.1$, (c), (d) are for $E_0=0.2$, and (e), (f) are for $E_0=0.3$.

Figure 5

(a), (c), (e), (g) Difference in the dipole moment $\mathrm{\Delta }P(t;t_{\mathrm{probe}}$) between the cases with and without the probe pulse for different delay times. (b), (d), (f), (h) The Fourier transformation of $\mathrm{\Delta }P$ with respect to $t$ $\left[\right|\mathrm{\Delta }P(\omega ;t_{\mathrm{probe}})|$ defined in Eq. (35)] is plotted in the space of $\omega$ and $t_{\mathrm{probe}}$. Panels (a)–(d) show the result for pump-pulse frequency $\mathrm{\Omega }=3.0$ and (e)–(h) for $\mathrm{\Omega }=1.9$. The other pulse parameters are defined in Eq. (32), and the field strength of the pump pulse is $E_0=0.2$ and 0.3 for (a), (b), (e), (f) and (c), (d), (g), (h), respectively. Black solid lines indicate ${\mathcal{E}}_{\mathrm{ex}}^{\mathrm{eq}}$ and back dashed lines show the renormalized band gap ${\mathcal{E}}_{\mathrm{gap}}^{\mathrm{ren}}$ after the excitation.

Figure 6

GKBA+s2BA time evolution of the momentum distribution of the conduction band electrons $\left[n_c\left(k\right)\right]$ for different pump-pulse excitations and amplitudes.

Figure 7

Results for ${\gamma }_{0,22}$ [Eq. (18b)] and ${\gamma }_{22}$ [Eq. (18c)] for different field strengths $E_0$ and $\mathrm{\Omega }=3.0$. The momentum distribution is obtained from the $\mathrm{GKBA}+\mathrm{s}2\mathrm{BA}$ simulation at $t=150$. The vertical dotted lines indicate the band gap estimated by the MF Hamiltonian (12). Here, we set $0^{+}=0.02$ in Eq. (18) for the numerical evaluation, which explains the finite weight in $\mathrm{Im}{\gamma }_{0,22}$ below the band gap. “Eq.” indicates the results in equilibrium.

Figure 8

Results for ${\gamma }_{0,22}$ [Eq. (18b)] and ${\gamma }_{22}$ [Eq. (18c)] for different field strengths and $\mathrm{\Omega }=1.9$. The momentum distribution is obtained from the GKBA+s2BA simulation at $t=150$. The vertical dotted lines indicate the band gap estimated by the MF Hamiltonian (12). Here, we set $0^{+}=0.02$ in Eq. (18) for the numerical evaluation, which explains the finite weight in $\mathrm{Im}{\gamma }_{0,22}$ below the band gap. “Eq.” indicates the results in equilibrium.

Figure 9

(a), (b) Real and imaginary parts of the susceptibility estimated by the GKBA+s2BA simulation averaged around $t_{\mathrm{probe}}=150$ $\left[{\tilde{\chi }}_{\mathrm{GKBA}}\left(\omega \right)\right]$. See the main text for detailed explanations. (c) Summary of the frequency of the oscillation (${\omega }_{\mathrm{coh}}^{*}$), the band gap, and the phase of the susceptibility at $\omega ={\omega }_{\mathrm{coh}}^{*}$. Here, ${\omega }_{\mathrm{coh}}^{*}$ is estimated by the peak position in $|{\tilde{\chi }}_{\mathrm{GKBA}}\left(\omega \right)|$ or ${\chi }_{11}\left(\omega \right)$ and the phase is defined as the argument of $-{\tilde{\chi }}_{\mathrm{GKBA}}\left({\omega }_{\mathrm{coh}}^{*}\right)$ or $-{\chi }_{11}\left({\omega }_{\mathrm{coh}}^{*}\right)$. The gap size is estimated by the GKBA analysis at $t=150$.

Figure 10

Imaginary part of the momentum-resolved correlation function $-\mathrm{Im}{\underline{\chi }}^{>}(\omega ;\mathbf{q},t_0)$ after the pump with $\mathrm{\Omega }=1.9$ and $E_0=0.1$ for $t_0=33.08$ and $t_1=80$. Here, $\sigma =(t_1-t_0)/\left(2\sqrt{2}\right)$ is used for the window function.

Figure 11

(a), (c) Time evolution of the momentum distribution of the conduction band electrons $\left[n_c\left(k\right)\right]$ within GKBA+s2BA for nonzero electron-phonon couplings. (b), (d) $\left|\mathrm{\Delta }P\right(\omega ;t_{\mathrm{probe}}\left)\right|$ obtained by the pump-probe simulation [Eq. (35)] plotted in the space of $\omega$ and $t_{\mathrm{probe}}$. The solid black lines indicate the frequency of the exciton in equilibrium ${\mathcal{E}}_{\mathrm{ex},\mathrm{eq}}$, while the dashed black lines indicate the renormalized band gap ${\mathcal{E}}_{\mathrm{gap}}^{\mathrm{ren}}$ after the pulse measured at $t=150$. Panels (a), (b) show results for $E_0=0.2$ and $\mathrm{\Omega }=3.0$, while (c), (d) is for $E_0=0.35$ and $\mathrm{\Omega }=3.0$. Here, ${\omega }_c=0.2$ and $g=0.25$ are used.

Figure 12

(a)–(c) Comparison between the evolution after a strong resonant excitation with and without phonon bath using $\mathrm{GKBA}+\mathrm{s}2\mathrm{BA}$. Panel (a) shows the evolution of the excited charge, (b) shows the dipole moment, and (c) shows the total energy (solid line) and the kinetic energy (dashed line). (d) Evolution of the momentum distribution of the conduction band electrons within GKBA+s2BA. Here, ${\omega }_c=0.2$, $g=0.25,0.0$, $\mathrm{\Omega }=1.9,E_0=0.3$. The other pulse parameters are defined in Eq. (32).

Figure 13

(a) An example of a ladder diagram for ${\chi }_{\mu \nu }(t,t^{\prime };\mathbf{q})$ represented as a Feynman diagram. (b) An example of a ring contribution, which can appear in the diagrams for ${\chi }_{\mu \nu }(t,t^{\prime };\mathbf{q})$. (c) An example of a ladder-type diagram with crossed interaction lines for ${\chi }_{\mu \nu }(t,t^{\prime };\mathbf{q})$. Double lines with arrows indicate the full electron Green's function, while the dashed lines represent the Coulomb interaction.

Figure 14

Time evolution of the momentum distribution of the conduction band electrons $\left[n_c\left(k\right)\right]$ for various indicated conditions within tdMF.

Figure 15

Comparison of the density of conduction band electrons and the polarization among s2BA, 2BA, $\mathrm{GKBA}+\mathrm{s}2\mathrm{BA}$, GKBA+2BA, and iTEBD for $\mathrm{\Omega }=1.9$. Panels (a) and (b) are for $E_0=0.1$ and (c) and (d) are for $E_0=0.2$.