Many-body theory of trion absorption features in two-dimensional semiconductors

Recent optical studies of monolayer transition-metal dichalcogenides have demonstrated that their excitonic absorption feature splits into two widely separated peaks at finite carrier densities. The additional peak is usually attributed to the presence of trions, bound states of two electrons and a hole or an electron and two holes. Here we argue that in the density range over which the trion peak is well resolved, it cannot be interpreted in terms of weakly coupled three-body systems and that the appropriate picture is instead one in which excitons are dressed by interactions with a Fermi sea of excess carriers. This coupling splits the exciton spectrum into a lower-energy attractive exciton-polaron branch, normally identified as a trion branch, and a higher-energy repulsive exciton-polaron branch, normally identified as an exciton branch. We have calculated the frequency and doping dependence of the optical conductivity and found that (i) the splitting varies linearly with the Fermi energy of the excess quasiparticles, (ii) the trion peak is dominant at high carrier densities, and (iii) the trion peak width is considerably smaller than that of the excitonic peak. Our results are in good agreement with recent experiments.

Figure 1

Optical conductivity $\sigma \left(\omega \right)/{\sigma }_0$, where ${\sigma }_0=e^2/h$ is the quantum unit of conductance. (a) Theoretical conductivity when Fermi sea dressing of excitons is neglected and (b) full conductivity including interactions between excitons and a fluctuating Fermi sea. We refer to the two peaks in (b), often interpreted as trion and exciton peaks, as the attractive and repulsive exciton-polaron branches. Energies are measured from the bare semiconductor band gap and measured in units of the exciton binding energy. The dashed lines show the bare interband absorption threshold renormalized by interactions.

Figure 2

Dependence on carrier Fermi energy ${\epsilon }_{\mathrm{F}}$ of (a) the excitonic ground-state energy ${\epsilon }_{\mathrm{X}}$ and (b) its squared optical matrix element $D^2$. The ground-state energy approaches the renormalized interband absorption threshold $2{\epsilon }_{\mathrm{F}}+{\mathrm{\Sigma }}_{\mathrm{g}}$ (dashed line) in the high-carrier-density limit.

Figure 3

(a) Excitonic contribution to the optical conductivity. The triangle vertexes correspond to the optical matrix elements $D$, defined in Eq. (5). The paired solid and dashed lines represent excitons, bound states of conduction band electrons, and valence-band holes described by summation of all scattering ladder diagrams. (b) Exciton self-energy due to interactions $\mathrm{\Gamma }$ with Fermi sea fluctuations. (c) Bethe-Salpeter equation for the exciton/Fermi sea interaction $\mathrm{\Gamma }$ vertex. The value of these diagrams depends on the total and relative motion momenta $\mathbf{q}$ and $\mathbf{p}$. For a given $\mathbf{q}$ and $\mathbf{p}$ the exciton momentum is $2\mathbf{q}/3-\mathbf{p}$, and the electron momentum is $\mathbf{q}/3+\mathbf{p}$.

Figure 4

The spectral function $A_{\mathrm{\Gamma }}(\omega ,\mathbf{q})=-2\mathrm{Im}\left[{\mathrm{\Gamma }}^{\mathrm{R}}(\omega ,\mathbf{q})\right]$ for the many-body vertex function ${\mathrm{\Gamma }}^{\mathrm{R}}(\omega ,\mathbf{q})$. The dashed line follows ${\omega }_{\mathbf{q}}$, given by (8) and corresponding to the bound state of an exciton with a Fermi sea of electrons. The behavior evolves from the two-particle one at ${\epsilon }_{\mathrm{F}}\ll {\epsilon }_{\mathrm{T}}$ to the polaronic one ${\epsilon }_{\mathrm{F}}\sim {\epsilon }_{\mathrm{T}}$, where the dispersion ${\omega }_{\mathbf{q}}$ achieves minimum at finite momentum $q_{*}$ and can be expanded in its vicinity according to (9) and (10).

Figure 5

(a)–(f) Frequency dependence of the optical conductivity $\sigma \left(\omega \right)$ for different values of the Fermi energy of electrons ${\epsilon }_{\mathrm{F}}$. Two peaks represent attractive and repulsive exciton-polaron branches.

Figure 6

(a) Dependence of the energy splitting $\mathrm{\Delta }{\epsilon }^{*}={\epsilon }_{\mathrm{X}}^{*}-{\epsilon }_{\mathrm{T}}^{*}$ between exciton $X$ and trion $T$ absorption features on carrier Fermi energy. Dependence of absorption feature (b) peaks $\sigma \left({\epsilon }_{\mathrm{X}\left(\mathrm{T}\right)}^{*}\right)/{\sigma }_0$ and (c) widths $\delta {\epsilon }_{\mathrm{X}\left(\mathrm{T}\right)}/{\overline{\epsilon }}_{\mathrm{X}}$ on carrier Fermi energy ${\epsilon }_{\mathrm{F}}$. The splitting interpolates between two linear behaviors, $\mathrm{\Delta }{\epsilon }^{*}={\epsilon }_{\mathrm{T}}+m{\epsilon }_{\mathrm{F}}/{\mu }_{\mathrm{T}}$ at ${\epsilon }_{\mathrm{F}}\lesssim {\epsilon }_{\mathrm{T}}$ and $\mathrm{\Delta }{\epsilon }^{*}=m{\epsilon }_{\mathrm{T}}/{\mu }_{\mathrm{T}}+{\epsilon }_{\mathrm{F}}$ at ${\epsilon }_{\mathrm{F}}\gtrsim {\epsilon }_{\mathrm{T}}$.

Figure 7

(a) Diagrammatic representation for the current-current correlation function $\chi \left(i{\omega }_n\right)$. Solid and dashed lines correspond to electrons from conduction and valence bands. (b) Excitons correspond to the ladder series of scattering diagrams, and their summation can be reduced to the renormalization of current vertex $W_{\mathbf{p}}^0\to W_{\mathbf{p}}$. (c) and (d) Self-energies of electrons in conduction and valence bands in the Hartree-Fock approximation. Hartree contributions (the first two terms) in ${\mathrm{\Sigma }}_{\mathbf{p}}^{\mathrm{c}}$ and ${\mathrm{\Sigma }}_{\mathbf{p}}^{\mathrm{v}}$ are equal to each other, renormalize the chemical potential, and do not influence the gap ${\epsilon }_{\mathrm{g}}$ between conduction and valence bands. The gap renormalization ${\mathrm{\Sigma }}_{\mathrm{g}}={\mathrm{\Sigma }}_{\mathbf{p}=0}^{\mathrm{c}}-{\mathrm{\Sigma }}_{\mathbf{p}=0}^{\mathrm{v}}\equiv {\mathrm{\Sigma }}^{\mathrm{c}}-{\mathrm{\Sigma }}^{\mathrm{v}}$ is governed by the difference of Fock terms representing exchange interactions between electrons.

Figure 8

(a)–(f) Frequency dependence of the optical conductivity $\sigma \left(\omega \right)$ for different values of electron density $n$. (g) Density dependence of the energy splitting $\mathrm{\Delta }{\epsilon }^{*}={\epsilon }_{\mathrm{X}}^{*}-{\epsilon }_{\mathrm{T}}^{*}$ between exciton $X$ and trion $T$ absorption features. Density dependence of absorption feature (h) peaks $\sigma \left({\epsilon }_{\mathrm{X}\left(\mathrm{T}\right)}^{*}\right)/{\sigma }_0$ and (i) widths $\delta {\epsilon }_{\mathrm{X}\left(\mathrm{T}\right)}/{\overline{\epsilon }}_{\mathrm{X}}$. The splitting interpolates between two linear behaviors, $\mathrm{\Delta }{\epsilon }^{*}={\epsilon }_{\mathrm{T}}+m{\epsilon }_{\mathrm{F}}/{\mu }_{\mathrm{T}}$ at ${\epsilon }_{\mathrm{F}}\lesssim {\epsilon }_{\mathrm{T}}$ and $\mathrm{\Delta }{\epsilon }^{*}=m{\epsilon }_{\mathrm{T}}/{\mu }_{\mathrm{T}}+{\epsilon }_{\mathrm{F}}$ at ${\epsilon }_{\mathrm{F}}\gtrsim {\epsilon }_{\mathrm{T}}$.