Spectrum of exciton states in monolayer transition metal dichalcogenides: Angular momentum and Landau levels

M. Van der Donck and F. M. Peeters

Phys. Rev. B 99, 115439 – Published 28 March 2019

Abstract

A four-band exciton Hamiltonian is constructed starting from the single-particle Dirac Hamiltonian for charge carriers in monolayer transition metal dichalcogenides (TMDs). The angular part of the exciton wave function can be separated from the radial part, in the case of zero center of mass momentum excitons, by exploiting the eigenstates of the total exciton angular momentum operator with which the Hamiltonian commutes. We explain why this approach fails for excitons with finite center of mass momentum or in the presence of a perpendicular magnetic field and present an approximation to resolve this issue. We calculate the (binding) energy and average interparticle distance of different excited exciton states in different TMDs and compare these with results available in the literature. Remarkably, we find that the intervalley exciton ground state in the $\mp K$ valley has angular momentum $j = \pm 1$ , which is due to the pseudospin of the separate particles. The exciton mass and the exciton Landau levels are calculated and we find that the degeneracy of exciton states with opposite relative angular momentum is altered by a magnetic field.

Received 14 December 2018

DOI:https://doi.org/10.1103/PhysRevB.99.115439

Physics Subject Headings (PhySH)

Excitons Landau levels

2-dimensional systems Transition metal dichalcogenides

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

M. Van der Donck^* and F. M. Peeters^†

Department of Physics, University of Antwerp, Groenenborgerlaan 171, B-2020 Antwerp, Belgium

^*matthias.vanderdonck@uantwerpen.be
^†francois.peeters@uantwerpen.be

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Issue

Vol. 99, Iss. 11 — 15 March 2019

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Images

Figure 1
Schematic representation of the low-energy band structure of 2D TMDs and different kinds of excitons. Blue and red bands are spin up and spin down bands, respectively. The open and closed circles indicate holes and electrons, respectively. The blue solid ellipse and the red dotted ellipse indicate intravalley $A$ excitons in the $K$ and $K^{'}$ valley, respectively. The red dashed ellipse indicates an intravalley $B$ exciton in the $K$ valley. The large purple dot-dashed ellipse indicates an intervalley $A$ exciton.
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Figure 2
Intravalley $(τ_{e} = - τ_{h} = 1$ , left) and intervalley $(τ_{e} = τ_{h} = - 1$ , right) exciton energy levels for ${MoS}_{2}$ suspended in vacuum (a) and placed on a ${SiO}_{2}$ substrate $(ɛ_{b} = 3.8)$ (b) for $K = 0$ . The results of the present work (P. W.) are compared with results of the literature (Lit.) from Ref. [30] (a) and Ref. [31] (b). Our results are labeled on the figure according to $n {[L]}_{j}$ , where $[L]$ represents $s, p, d$ depending on the orbital angular momentum of the dominant component of the wave function, with $s, p$ , and $d$ excitons indicated in blue, red, and black, respectively. No labeling was given in Ref. [31] for intervalley excitons. Exciton states with negative orbital angular momentum are indicated with dashed lines. When two states with opposite orbital angular momentum are degenerate only the solid line is shown. $B$ exciton states are indicated with the letter $B$ . To facilitate comparison, the energy levels are uniformly shifted downwards in energy such that the ground state (which can be either intravalley or intervalley) has zero energy.
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Figure 3
Different components $(β = c, γ = v$ : blue curve, $β = c, γ = c$ and $β = v, γ = v$ : red curve, $β = v, γ = c$ : black curve) of the $K = 0$ intravalley $1 s$ -state wave function for excitons in ${MoS}_{2}$ suspended in vacuum obtained from Eq. (13) (solid) and Eq. (10) (dashed). The blue and black curves are $s$ -like and have zero pseudospin, the red curves are $p$ -like and have pseudospin $\pm 1$ . The total angular momentum of this state is $j = 0$ . The inset shows the corresponding total radial probability distribution for the two cases.
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Figure 4
Six lowest Landau levels of the intravalley $(τ_{e} = - τ_{h} = 1) 2 p$ (a) and $3 d$ (b) states for excitons in ${MoS}_{2}$ suspended in vacuum. Solid blue (red dashed) lines indicate positive (negative) relative angular momentum states.
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Figure 5
Dominant component of the intravalley $(τ_{e} = - τ_{h} = 1) 2 p$ -state wave function for the degenerate states with $(j_{r}, j_{R}) = (1, 0)$ (a) and $(j_{r}, j_{R}) = (- 1, 1)$ (b) for excitons in ${MoS}_{2}$ suspended in vacuum in the presence of a magnetic field of $B = 50$ T.
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