Colloquium: Excitons in atomically thin transition metal dichalcogenides

Atomically thin materials such as graphene and monolayer transition metal dichalcogenides (TMDs) exhibit remarkable physical properties resulting from their reduced dimensionality and crystal symmetry. The family of semiconducting transition metal dichalcogenides is an especially promising platform for fundamental studies of two-dimensional (2D) systems, with potential applications in optoelectronics and valleytronics due to their direct band gap in the monolayer limit and highly efficient light-matter coupling. A crystal lattice with broken inversion symmetry combined with strong spin-orbit interactions leads to a unique combination of the spin and valley degrees of freedom. In addition, the 2D character of the monolayers and weak dielectric screening from the environment yield a significant enhancement of the Coulomb interaction. The resulting formation of bound electron-hole pairs, or excitons, dominates the optical and spin properties of the material. Here recent progress in understanding of the excitonic properties in monolayer TMDs is reviewed and future challenges are laid out. Discussed are the consequences of the strong direct and exchange Coulomb interaction, exciton light-matter coupling, and influence of finite carrier and electron-hole pair densities on the exciton properties in TMDs. Finally, the impact on valley polarization is described and the tuning of the energies and polarization observed in applied electric and magnetic fields is summarized.

Figure 1

(a) Monolayer transition metal dichalcogenide crystal structure. The transition metal atoms appear in black, the chalcogen atoms in yellow. (b) Typical band structure for ${MX}_2$ monolayers calculated using density functional theory and showing the quasiparticle band gap $E_g$ at the $K$ points and the spin-orbit splitting in the valence band ([165]). (c), (d) Schematic illustrations in a single-particle picture show that the order of the conduction bands is opposite in $\mathrm{M}\mathrm{o}{X}_2$ (c) and $\mathrm{W}{X}_2$ (d) monolayers ([99]). The contribution from Coulomb-exchange effects that has to be added to calculate the separation between optically active (bright—spin-allowed) and optically inactive (dark—spin-forbidden) excitons is not shown ([60]).

Figure 2

(a) Schematic real-space representation of the electron-hole pair bound in a Wannier-Mott exciton showing the strong spatial correlation of the two constituents. The arrow indicates the center-of-mass wave vector responsible for the motion of the exciton as a whole. (b) A typical exciton wave function calculated for monolayer ${\mathrm{MoS}}_2$. The modulus squared of the electron wave function is plotted in color scale for the hole position fixed at the origin. The inset shows the corresponding wave function in momentum space across the Brillouin zone, including both $K^{+}$ and $K^{-}$ exciton states. From [161]. (c) Representation of the exciton in reciprocal space with the contributions of the electron and hole quasiparticles in the conduction (CB) and valence (VB) bands, respectively, shown schematically by the widths of the shaded areas. (d) Schematic illustration of the optical absorption of an ideal 2D semiconductor including the series of bright exciton transitions below the renormalized quasiparticle band gap. In addition, the Coulomb interaction leads to the enhancement of the continuum absorption in the energy range exceeding $E_B$, the exciton binding energy. The inset shows the atomlike energy level scheme of the exciton states, designated by their principal quantum number $n$, with the binding energy of the exciton ground state ($n=1$) denoted by $E_B$ below the free-particle (FP) band gap.

Figure 3

Presentation of commonly used experimental techniques to determine exciton binding energies in TMD monolayers. (a) Direct measurement of the free-particle band-gap energy using scanning tunneling spectroscopy of ML ${\mathrm{MoSe}}_2$ on bilayer graphene (right panel) combined with a measurement of the absolute energy of the exciton ground state from photoluminescence (left panel). From [216]. (b) Exciton states of ML ${\mathrm{WS}}_2$ on a ${\mathrm{SiO}}_2/\mathrm{Si}$ substrate from reflectance contrast measurements. The extracted transition energies of the states and the inferred band-gap position are presented in the right panel. From [39]. (c) The linear absorption spectrum and the third-order susceptibility extracted from two-photon photoluminescence excitation spectra of ML ${\mathrm{WSe}}_2$ on fused silica substrate with exciton resonances of the ground and excited states. From [78]. (d) Exciton states as measured by second-harmonic spectroscopy of the $A$ and $B$ transitions in ML ${\mathrm{WSe}}_2$. From [227]. (e) One-photon photoluminescence excitation spectra and the degree of linear polarization of the luminescence of ML ${\mathrm{WSe}}_2$ with features of the excited $2s$ state of the $A$ and the ground state of the $B$ exciton. From [227].

Figure 4

(a) A schematic overview of typical allowed and forbidden electronic transitions for the respective bright and dark exciton states. The underlying band structure is simplified for clarity, including only the upper valence band at $K^{+}$ and the high-symmetry points $K^{\pm }$ and $Q$ in the conduction band. The order of the spin states in the conduction band corresponds to W-based TMD MLs; see [115], [70], and [99] for details. (b) Schematic illustration of the exciton ground-state dispersion in the two-particle representation. The light cone for bright excitons is marked by the free-space photon dispersion $c{q}_{\parallel }$, where $c$ is the speed of light and the excitons outside of the cone are essentially dark.

Figure 5

(a) Brightening of the dark exciton transition observed in ML ${\mathrm{WSe}}_2$ by photoluminescence experiments within an in-plane magnetic field. From [269]. (b) Schematic of the brightening of the dark exciton transitions involving the spin states in the conduction bands 1 and 2. For simplicity we do not show the Coulomb-exchange energy term that also contributes to the dark-bright splitting. From [60]. (c), (d) Using in-plane optical excitation and detection, the dark ($X^D$) and bright ($X^0$) excitons can be distinguished by polarization dependent measurements. The ${\mathrm{WSe}}_2$ ML is encapsulated in $h\mathrm{BN}$ for improved optical quality. Adapted from [231].

Figure 6

(a) Absorbance and photoluminescence experiments exhibiting signatures of neutral ($A$) and charged ($A^{-}$) excitons in a charge tunable $\mathrm{M}\mathrm{o}{\mathrm{S}}_2$ monolayer. From [127]. (b) Color contour plot of PL from an electrically gated $\mathrm{M}\mathrm{o}\mathrm{S}{\mathrm{e}}_2$ monolayer that can be tuned to show emission from positively charged ($X^{+}$) to negatively charged ($X^{-}$) trion species. From [172]. (c) Contour plot of the first derivative of the differential reflectivity in a charge tunable $\mathrm{W}\mathrm{S}{\mathrm{e}}_2$ monolayer. The $n$- and $p$-type regimes are manifested by the presence of $X^{+}$ and $X^{-}$ transitions. Around charge neutrality, the neutral exciton $X^0$ and an excited state $X^{0*}$ are visible. From [48].

Figure 7

(a) Exciton PL emission time of the order of 2 ps measured in time-resolved photoluminescence for ML ${\mathrm{WSe}}_2$ at $T=7\mathrm{K}$. From [170]. (b) Schematic showing that $|+1⟩$ and $|-1⟩$ neutral excitons are coupled by the electron-hole Coulomb-exchange interaction. From [70]. (c) Decay of the neutral exciton polarization in ${\mathrm{WSe}}_2$ monolayers on ps time scales as measured by Kerr rotation. From [280]. (d) Decay of resident electron polarization as measured by Kerr rotation in monolayer ${\mathrm{WS}}_2$ with a typical time constant of 5 ns for $T=8\mathrm{K}$. From [27]. (e) Decay of hole polarization in a charge tunable ${\mathrm{WSe}}_2$ monolayer with a time constant of $2\mu \mathrm{s}$ for $T=4\mathrm{K}$, where $B_y$ is the magnetic field applied in the sample plane. From [57].

Figure 8

(a) Schematic of Zeeman shifts in magnetic field $B$ perpendicular to the monolayer plane. (b) Measurements on $\mathrm{M}\mathrm{o}\mathrm{S}{\mathrm{e}}_2$ MLs that show a clear Zeeman splitting. From [122]. (c) Reflectivity measurements on $\mathrm{W}{\mathrm{S}}_2$ MLs in high magnetic fields and (d) the Zeeman splitting extracted for $A$ and $B$ excitons. From [205].