Dark excitons in transition metal dichalcogenides

Ermin Malic, Malte Selig, Maja Feierabend, Samuel Brem, Dominik Christiansen, Florian Wendler, Andreas Knorr, and Gunnar Berghäuser

Phys. Rev. Materials 2, 014002 – Published 17 January 2018

Abstract

Monolayer transition metal dichalcogenides (TMDs) exhibit a remarkably strong Coulomb interaction that manifests in tightly bound excitons. Due to the complex electronic band structure exhibiting several spin-split valleys in the conduction and valence band, dark excitonic states can be formed. They are inaccessibly by light due to the required spin-flip and/or momentum transfer. The relative position of these dark states with respect to the optically accessible bright excitons has a crucial impact on the emission efficiency of these materials and thus on their technological potential. Based on the solution of the Wannier equation, we present the excitonic landscape of the most studied TMD materials including the spectral position of momentum- and spin-forbidden excitonic states. We show that the knowledge of the electronic dispersion does not allow to conclude about the nature of the material's band gap since excitonic effects can give rise to significant changes. Furthermore, we reveal that an exponentially reduced photoluminescence yield does not necessarily reflect a transition from a direct to a nondirect gap material, but can be ascribed in most cases to a change of the relative spectral distance between bright and dark excitonic states.

Received 1 September 2017

DOI:https://doi.org/10.1103/PhysRevMaterials.2.014002

Physics Subject Headings (PhySH)

Excitons Luminescence

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Ermin Malic¹, Malte Selig², Maja Feierabend¹, Samuel Brem¹, Dominik Christiansen², Florian Wendler², Andreas Knorr², and Gunnar Berghäuser^1,*

¹Department of Physics, Chalmers University of Technology, Gothenburg, Sweden
²Institut für Theoretische Physik, Technische Universität Berlin, Berlin, Germany

^*gunbergh@chalmers.se

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Vol. 2, Iss. 1 — January 2018

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Figure 1
Dark and bright excitons with the hole located at the $K$ valley. Schematic electronic dispersions around the $K$ and $Λ$ valley for (a) tungsten-based TMDs $({WS}_{2}$ and ${WSe}_{2})$ and (b) molybdenum-based TMDs $({MoS}_{2}$ and ${MoSe}_{2})$ . In the first case, the lowest conduction and the highest valence band have the opposite spin. Spin-up and spin-down bands are denoted by red and blue lines, respectively. The spin-down valence band is not shown since it is a few hundreds of meV away contributing to B excitons that are not considered here. The yellow arrow describes the lowest optically induced transition between the bands of the same spin at the $K$ point. The correlated electron-hole pairs are enclosed by a yellow (bright $A_{1 s}$ exciton), red, and orange (momentum-forbidden dark $K − Λ$ and $K − K^{'}$ exciton, respectively) and purple ovals (spin-forbidden dark $K − K$ exciton). Exciton dispersion in (c) ${WS}_{2}$ , (d) ${MoS}_{2}$ , (e) ${WSe}_{2}$ , and (f) ${MoSe}_{2}$ calculated by solving the Wannier equation. Dashed lines reflect the relative band ordering in a free-particle picture without taking into account excitonic binding energies. While in molybdenum-based TMDs, the bright exciton is the energetically lowest state (yellow line), tungsten-based TMDs exhibit lower lying dark excitonic states. Note that one finds for every spin-like state an energetically degenerated spin-unlike state in the corresponding opposite valley.
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Figure 2
Dark and bright excitons with the hole located at the $Γ$ valley. The same as in Fig. 1, however, now considering $Γ - K$ and $Γ - Λ$ excitons, where the hole is located at the $Γ$ valley. Here, bright excitons do not appear since there is no direct band gap at the $Γ$ point. Thus, we only find spin- and momentum-forbidden dark exciton states. Since there is no spin-orbit-induced splitting of the valence band at the $Γ$ point, we find for every spin-forbidden state a degenerate momentum-forbidden state (as in the case of $K$ -hole excitons). We find that for TMDs including selen atoms [(e, f)], $Γ$ -hole excitons are located far above the lowest bright exciton transition (not shown), while TMDs including sulfur atoms [(c, d)] exhibit excitons close to the bright state (dashed orange line). Note that the dashed lines correspond to the relative band ordering in the free-particle picture.
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Figure 3
Impact of dark-bright separation and layer number on quantum yield. Quantum yield $Y$ as defined in Eq. (4) is shown in dependence of (a) the relative spectral position of the bright and the lowest lying dark state $Δ_{d b} = E_{bright} - E_{dark}$ and (b) the number of TMD layers $l$ in the case of ${MoS}_{2}$ . The quantum yield is extremely sensitive to $Δ_{d b}$ and shows an exponential dependence (except for the region around $Δ_{d b} \approx 0)$ reflecting the Boltzmann distribution of excitonic states. Grey circles in (b) show experimental data on the layer dependence taken from Ref. [25]. Using the experimental results we can estimate the dark-bright separation $Δ_{d b}^{l}$ for different layer numbers $l$ . We find that monolayer ${MoS}_{2}$ is a slightly indirect-gap semiconductor with the $Γ − K$ exciton located 10 meV below the bright $A_{1 s}$ transition, i.e., $Δ_{d b}^{1} \approx 10 meV$ . Note that the theoretical prediction based on the input from ab initio density functional theory calculations [20] for the electronic band structure gives a larger value of approximately 60 meV.
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Figure 4
Temperature dependence of the quantum yield. Quantum yield for monolayer (a) ${MoS}_{2}, {WS}_{2}$ and (b) ${MoSe}_{2}, {WSe}_{2}$ as a function of temperature. The relative position of dark and bright excitonic states in a TMD material can be directly read off by the temperature dependence of the quantum yield. An increasing (decreasing) quantum yield indicates a relatively higher (lower) occupation of the bright exciton state suggesting that the dark (bright) exciton is energetically lowest. While the solid lines are based on DFT input parameters on the electronic band structure [20], the dashed line for ${MoS}_{2}$ shows the results obtained by assuming a dark-bright distance of $Δ_{db} \approx 10 meV$ according to the estimation from Fig. 3. In this case we find a nonmonotonous temperature behavior, where after an initial increase the quantum yield starts to decrease again after a turning-point temperature of approximately 100 K. This can be ascribed to the enhanced filling of dark states, which lie slightly above the bright $A_{1s}$ exciton [cf. Fig. 1].
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