Trion liquid and its photoemission signatures

Abstract

We study the formation of a trion liquid in doped low-dimensional semiconductors with strong electron-hole interactions and analyze its signatures in angle-resolved photoemission spectroscopy (ARPES). We show that this strongly correlated state of matter forms naturally in the vicinity of the phase boundary between a normal band insulator and an excitonic insulator upon doping. By studying the photoemission spectrum, we show that a partially occupied trion band gives rise to an in-gap feature in the ARPES spectrum with vanishing spectral weight at the Fermi energy. We demonstrate our findings using a 1D microscopic model employing exact, unbiased, matrix product state (MPS)-based calculations.

Figures (6)

• Figure 1: Phase diagram obtained for the model in Eq. \ref{['eq:H2c1f']}. The light blue and grey regions correspond to the excitonic insulator (EI) and band insulator (BI) phases obtained at charge neutrality, respectively. The dashed region, indicates the parameter regime in which a trion liquid forms upon doping. Model parameters used are $t_f/t_c=1,U_1/U_0=1/3$. The inset shows a cut across the phase diagram at fixed $U_0$ varying $D$, along the dashed line. The trion liquid phase is obtained in the regime $E^b_{\rm ex}<E_{\rm gap}<E^b_{\rm tr}$ upon doping. The yellow star marks the point $D=3.4$, $U=3.0$ at which Figs. \ref{['fig:energetics']}(b), \ref{['fig:model']}(b), and \ref{['fig:ARPES']} are obtained. At this point, the binding energies are equal to $E^b_{\rm ex}=1.00$, $E^b_{\rm ex-e}=0.71$ and the gap is $E_{\rm gap}=1.40$. All energies are given in units of $t_c$.
• Figure 2: (a) Schematic representation of the semiconducting system coupled to a charge reservoir, which fixes the Fermi energy, $\varepsilon_{\rm F}$. In the trion liquid regime, due to the energy gain upon trion formation, tunneling of electrons into the system is possible for $\mu>\mu^*=\varepsilon_c-(E^b_{\rm tr}-E_{\rm gap})$ (see text). (b) Density of electrons and holes as a function of the chemical potential obtained numerically for the model in Eq. \ref{['eq:H2c1f']} with parameters corresponding to the point marked by a star in Fig. \ref{['fig:PD']} and $V/U_0=1/6$ for an $N=51$-long chain. When the chemical potential crosses $\mu^*$ the density of electrons and holes starts increasing in a ratio of 2:1 as expected for a formation of trions.
• Figure 3: (a) Schematic illustration of the model in Eq. \ref{['eq:H2c1f']}. The operator $\tau_x^\dagger=c^\dagger_{1,x} f_x c^\dagger_{2,x}$ is a trion creation operator at position $x$ along the chain. (b) Single electron and trion static correlation functions in the trion liquid phase with parameters corresponding to the point marked by a star in Fig. \ref{['fig:PD']} and $V/U_0=1/6$, in presence of 5 trions in a system of $N=101$ sites ($\sim5\%$ electron doping). While single electrons are gapped, trions exhibit power-law correlations.
• Figure 4: Photoemission spectrum, $A^-(k,\omega)=A^-_c(k,\omega)+A^-_f(k,\omega)$, obtained numerically using model parameters corresponding to the point marked by a star in Fig. \ref{['fig:PD']} and $V/U_0=1/6$, in presence of 3 trions in a system of size $N=101$ sites ($\sim3\%$ electron doping). Insets schematically depict the processes contributing to the spectral weight, with the emitted electron shown in red (blue) for processes in the $c$ ($f$) channel. The red (purple) dashed line corresponds to the non-interacting $c$ ($f$) band. Exciton dispersion as well as the boundary of the 2-exciton continuum obtained using ED are plotted as orange dashed and cyan dotted lines respectively.
• Figure S1: Band-resolved photoemission spectrum at $3\%$ and $10\%$ electron doping. With increasing trion density the in-gap feature in the $c$-channel evolves to have the trion band dispersion (vertical grey dashed lines indicate the Fermi momenta of the trions). In the $f$-channel a $\pm k_{\rm F}$ splitting of the valence band is observed, origination from the fermionic commutation relations between the trions and the $f$-electrons. The intensity of the in-gap two-exciton feature also increases.