Three-particle states and brightening of intervalley excitons in a doped ${\mathrm{MoS}}_{2}$ monolayer

Three-particle states and brightening of intervalley excitons in a doped ${MoS}_{2}$ monolayer

Y. V. Zhumagulov, A. Vagov, N. Yu. Senkevich, D. R. Gulevich, and V. Perebeinos

Phys. Rev. B 101, 245433 – Published 22 June 2020

Abstract

Optical spectra of two-dimensional transition-metal dichalcogenides (TMDC) are influenced by complex multiparticle excitonic states. Their theoretical analysis requires solving the many-body problem, which in most cases, is prohibitively complicated. In this work, we calculate the optical spectra by exact diagonalization of the three-particle Hamiltonian within the Tamm-Dancoff approximation where the doping effects are accounted for via the Pauli blocking mechanism, modelled by a discretized mesh in the momentum space. The single-particle basis is extracted from the ab initio calculations. Obtained three-particle eigenstates and the corresponding transition dipole matrix elements are used to calculate the linear absorption spectra as a function of the doping level. Results for negatively doped ${MoS}_{2}$ monolayer (ML) are in excellent quantitative agreement with the available experimental data, validating our approach. The results predict additional spectral features due to the intervalley exciton that is optically dark in an undoped ML but is brightened by the doping. Our approach can be applied to a plethora of other atomically thin semiconductors, where the doping induced brightening of the many-particle states is also anticipated.

Received 6 March 2020
Revised 25 May 2020
Accepted 28 May 2020

DOI:https://doi.org/10.1103/PhysRevB.101.245433

Physics Subject Headings (PhySH)

Biexcitons Excitons Trions

Quantum wells Semiconductor compounds

Density functional calculations Optical absorption spectroscopy Photoluminescence

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Y. V. Zhumagulov^1,2, A. Vagov^1,3, N. Yu. Senkevich¹, D. R. Gulevich¹, and V. Perebeinos ^4,1,*

¹ITMO University, St. Petersburg 197101, Russia
²University of Regensburg, Regensburg 93040, Germany
³Institute for Theoretical Physics III, University of Bayreuth, Bayreuth 95440, Germany
⁴Department of Electrical Engineering, University at Buffalo, The State University of New York, Buffalo, New York 14260, USA

^*vasilipe@buffalo.edu

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Vol. 101, Iss. 24 — 15 June 2020

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Figure 1
(a) Schematic discretization of the Brillouin zone with the basis vectors ${\vec{b}}_{1, 2}$ and the $6 \times 6$ $k$ -point mesh. A magenta filled polygon shows an elementary area $Δ k^{2} \sqrt{3} / 2$ per $k$ point. Points $Γ$ , $K$ , $K^{'}$ , and $M$ are shown for reference. (b) Relation between the Fermi momentum ( $x$ axis), the Fermi energy of the doping electrons (left $y$ axis) and the number of $k$ points of the mesh (right $y$ axis).
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Figure 2
Band structure of a ${MoS}_{2}$ ML near points $K$ and $- K$ of the Brillouin zone. The single particle density of states (DOS) for trion wave functions is illustrated by circles: Their centers point to the contributing states and the radii give the contribution weights. Panels (a), (b), and (c) correspond to different three-particle states $X^{-}$ , $i X^{0} e$ , and $X^{0} e$ , schematically illustrated, where longer orbitals depict weakly bound electrons. Three-particle states $X^{0} e$ and $i X^{0} e$ are related, respectively, to the intervalley $X^{0}$ and intravalley $i X^{0}$ two-particle excitons in the undoped ML. The calculations are shown for the Fermi level $E_{F} = 3.04$ meV.
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Figure 3
(a) Absorption spectrum calculated for different values of the doping density $n$ (right) corresponding to the Fermi levels $E_{F}$ (left); the lines are offset vertically for clarity. The three peaks correspond to three-particle states $X^{-}$ , $i X^{0} e$ , and $X^{0} e$ shown in Fig. 2. (b) Absorption spectra of the undoped (red) and doped (blue) ML's at the Fermi level $E_{F} = 3.04$ meV. (c) The oscillator strength (OS) of the spectral peaks and the total OS (the sum for the three peaks) as a function of the Fermi level. Dashed line gives the OS of the $X$ exciton peak of the undoped ML. (d) The energy position of the peaks as a function of the Fermi level, experimental data [25] for the two peaks energies $X^{-}$ and $X^{0} e$ are shown for comparison. Dashed lines show the intervalley $i X^{0}$ and intravalley $X^{0}$ exciton energies from the BSE solution in the undoped ML.
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Figure 4
(a) Energy differences $Δ E$ (splitting) for pairs of states: $X^{0} e - X^{-}$ and $X^{0} e - i X^{0} e$ , are plotted as a function of the Fermi energy $E_{F}$ . Dashed lines illustrate the linear dependence $Δ E = Δ_{0} + α E_{F}$ with $α = 1$ . Available experimental data [25] for the energy difference of states $X^{0} e - X^{-}$ are also shown for comparison. (b) The doping dependence of the potential energy contribution to the splitting that yields the deviations from the linear dependence in panel (a) (see text). The horizontal dashed lines in (b) are a guide to the eye.
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Figure 5
(a) Binding energy of the trion state relative to the energy of the $X^{0}$ exciton in the undoped ML as a function of the substrate dielectric constant $ɛ$ , calculated at $E_{F} = 10.89$ meV and $E_{F} = 4.87$ meV. Results for the variational calculations (magenta circles) for the lowest $X^{-}$ states are given for comparison. (b) and (c) OS for the spectral peaks, calculated at $E_{F} = 10.89$ meV and $E_{F} = 4.87$ meV (in the last case the $i X^{0} e$ peak is not visible).
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Physical Review B

covering condensed matter and materials physics

Three-particle states and brightening of intervalley excitons in a doped ${MoS}_{2}$ monolayer

Y. V. Zhumagulov, A. Vagov, N. Yu. Senkevich, D. R. Gulevich, and V. Perebeinos

Phys. Rev. B 101, 245433 – Published 22 June 2020

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Physical Review B

covering condensed matter and materials physics

Three-particle states and brightening of intervalley excitons in a doped MoS2 monolayer

Y. V. Zhumagulov, A. Vagov, N. Yu. Senkevich, D. R. Gulevich, and V. Perebeinos

Phys. Rev. B 101, 245433 – Published 22 June 2020

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Three-particle states and brightening of intervalley excitons in a doped ${MoS}_{2}$ monolayer