Interaction effects on electronic Floquet spectra: Excitonic effects

Abstract

Floquet engineering of electronic states by light is a central topic in modern experiments. However, the impact of many-body interactions on the single-electron properties remains unclear in this non-equilibrium situation. We propose that interaction effects could be reasonably understood by performing perturbative expansion in both the pump field and the electron-electron interaction when computing physical quantities. As an example, we apply this approach to semiconductors and show analytically that excitonic effects, i.e., effects of electron-hole interaction, lead to dramatic corrections to the single-electron Floquet spectra even when the excitons are only virtually excited by the pump light. We compute these effects in phosphorene and monolayer MoS$_2$ for time- and angle-resolved photoemission spectroscopy and ultrafast optical experiments.

Figures (2)

• Figure 1: (a) Schematic of a semiconductor with conduction band $c$ and valence band $v$ represented by black lines. Driven by pump light at frequency $\omega_{\text{P}}$, the electron is virtually pumped to a Floquet side band $v^{+1}$ (red curve), which is dramatically corrected by the virtual excitation of the zero-momentum excitonic mode on the blue dashed curve. (b), (c) The Feynman diagrams of the self-energy $\Sigma$ and the light-matter vertex $\Gamma$. The red wavy lines represent the electric field of the pump light connected in pairs to the "disorder" vertices. The black wavy lines are the electron-hole interaction.
• Figure 2: (a)-(c) Floquet electronic spectra of phosphorene driven by pump light at frequencies $\omega_{\text{P}}=1.200 \,\mathrm{eV}$, $1.300\,\mathrm{eV}$, and $1.490\,\mathrm{eV}$, respectively. The colored dashed curves labeled by $c^0$, $c^{-1}$, $v^{+1}$, and $v^0$ are the four low-energy Floquet electronic bands predicted by free theory without interaction corrections. The color map shows the photoemission intensity of the interaction corrected Floquet bands. The equilibrium conduction-band minimum (valence-band maximum) is indicated by the orange (green) horizontal line. The gray dashed line marks the energy of the $1s$ exciton ($E_{1s}=1.331$ eV). The brown dashed line in (c) marks the energy of the $2s$ exciton ($E_{2s}=1.499$ eV). Here, $m_c=0.17m_e$, $m_v=0.18m_e$, $E_{\text{g}}=1.52 \,\mathrm{eV}$, $\epsilon=5$, $\gamma_{1s}=\gamma_{2s}=15 \,\mathrm{meV}$Tian.2020Yoon.2021, and $M_k=\mathrm{i}5.25 \,\mathrm{eV \text{\AA}}$Pereira_PRB_2015. The pump electric field is $E_0=5\times 10^5 \,\mathrm{V/cm}$ and along the armchair ($x$) direction of phosphorene. The probe matrix elements are constrained by the symmetry of phosphorene: $M_{{fc}}=b_1$ and $M_{{fv}}=-b_1 M_k k/E_{\text{g}}$, where we set $b_1=1$, the probe incident plane is the $z-y$ plane, and the probe field is in the zigzag ($y$) direction Bao2024Fan2025SA. (d) Similar to (a)-(c), but for the lowest conduction band and highest valence band of monolayer $1$H-MoS$_2$ with momenta measured from the $K$ point. The Floquet bands with excitonic corrections are instead shown by solid lines. The parameters are $m_c=0.40 m_e$, $m_v=0.48 m_e$, $E_{\text{g}}=1.66 \,\mathrm{eV}$, $\epsilon=5$, $E_{1s}=1.18 \,\mathrm{eV}$, $\gamma_{1s}=\gamma_{2s}=5 \,\mathrm{meV}$, $M_k=3.51 \,\mathrm{eV \text{\AA}}$, $\omega_{\text{P}}=1.16 \,\mathrm{eV}$ and $E_0= 2.9\times 10^5 \,\mathrm{V/cm}$.