Magnetooptics of Exciton Rydberg States in a Monolayer Semiconductor

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Magnetooptics of Exciton Rydberg States in a Monolayer Semiconductor

A. V. Stier, N. P. Wilson, K. A. Velizhanin, J. Kono, X. Xu, and S. A. Crooker

Phys. Rev. Lett. 120, 057405 – Published 1 February 2018

Abstract

We report 65 T magnetoabsorption spectroscopy of exciton Rydberg states in the archetypal monolayer semiconductor ${WSe}_{2}$ . The strongly field-dependent and distinct energy shifts of the $2 s$ , $3 s$ , and $4 s$ excited neutral excitons permits their unambiguous identification and allows for quantitative comparison with leading theoretical models. Both the sizes (via low-field diamagnetic shifts) and the energies of the $n s$ exciton states agree remarkably well with detailed numerical simulations using the nonhydrogenic screened Keldysh potential for 2D semiconductors. Moreover, at the highest magnetic fields, the nearly linear diamagnetic shifts of the weakly bound $3 s$ and $4 s$ excitons provide a direct experimental measure of the exciton’s reduced mass $m_{r} = 0.20 \pm 0.01 m_{0}$ .

Received 14 August 2017

DOI:https://doi.org/10.1103/PhysRevLett.120.057405

Physics Subject Headings (PhySH)

Diamagnetism Excitons Magneto-optical spectra Zeeman effect

2-dimensional systems Transition metal dichalcogenides

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

A. V. Stier¹, N. P. Wilson², K. A. Velizhanin³, J. Kono⁴, X. Xu², and S. A. Crooker¹

¹National High Magnetic Field Laboratory, Los Alamos, New Mexico 87545, USA
²Department of Physics, University of Washington, Seattle, Washington 98195, USA
³Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
⁴Departments of Electrical and Computer Engineering, Physics and Astronomy, and Materials Science and NanoEngineering, Rice University, Houston, Texas 77005, USA

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Vol. 120, Iss. 5 — 2 February 2018

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Figure 1
(a) Experimental schematic: A ${WSe}_{2}$ monolayer sandwiched between hBN slabs is positioned over the $3.5 μ m$ diameter core of a single-mode optical fiber. Circularly polarized $σ^{\pm}$ transmission spectra are acquired to 65 T at low temperature (4 K). (b) Normalized transmission spectra $T / T_{0}$ at selected magnetic fields $B$ from 0 to 65 T. The $1 s$ ground state of the neutral $A$ exciton $X^{0}$ appears at 1.723 eV. Its $2 s$ excited state is also clearly visible at 1.853 eV (130 meV higher in energy); it exhibits a much larger diamagnetic blueshift in accordance with its much larger spatial extent (vertical dashed lines indicate their zero-field energies). Inset: The $3 s$ excited state is faintly visible even at $B = 0$ . (c) Intensity plots showing all the $σ^{\pm}$ spectra from 0–65 T. The large shifts of the weak $3 s$ and $4 s$ states of $X^{0}$ are readily apparent. (A small amount of $σ^{+}$ polarization leaks into the $σ^{-}$ spectra, especially for the strong $1 s$ feature at large $B$ ).
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Figure 2
(a) Measured $1 s$ , $2 s$ , $3 s$ , and $4 s$ exciton energies versus $B$ , for both $σ^{+}$ and $σ^{-}$ polarizations. (b) The average energy of the $σ^{\pm}$ transitions for all the $n s$ states (note $10 \times$ vertical scale for the $1 s$ state). Dotted red lines show purely quadratic fits to $B^{2}$ . The quadratic shifts of the $2 s$ and $3 s$ states are $15 \times$ and $71 \times$ larger than that of $1 s$ , quantitatively consistent with their larger radii computed from theory. The $3 s$ and $4 s$ states evolve toward a more linear shift at large $B$ , which can be calculated numerically in this intermediate-field regime [40]. Blue lines show the numerically calculated $3 s / 4 s$ energies using $m_{r} = 0.2 m_{0}$ . (c) The $σ^{\pm}$ energy difference reveals a similar valley Zeeman splitting for all $n s$ states. The dotted straight line has slope $- 245 μ eV / T$ ( $g_{v} \approx - 4.2$ ). (d) The blue lines show numerically calculated energies for all $n s$ Rydberg excitons to very large $B$ . The straight dashed lines show $⟨ E_{N} ⟩ = (N + \frac{1}{2}) ℏ ω_{c}^{*}$ , the valley-averaged energies of interband transitions between free electron and hole LLs [40]. At very large $B$ , the slope of the $n s$ exciton shift approaches that of $⟨ E_{N} ⟩$ , where $N = n - 1$ .
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Figure 3
(a) Plots of $ψ_{n s} (r)$ , the $1 s$ , $2 s$ , $3 s$ , and $4 s$ wave functions of $X^{0}$ in an hBN-encapsulated ${WSe}_{2}$ monolayer, computed using $V_{K} (r)$ (black line), using $r_{0} = 4.5 nm$ , $κ_{hBN} = 4.5$ , and $m_{r} = 0.20 m_{0}$ . The calculated rms exciton radii $r_{n s} = \sqrt{⟨ ψ_{n s} | r_{⊥}^{2} | ψ_{n s} ⟩}$ are close to experimental values. Crucially, we note that a rms radius is not the same as a “Bohr radius”; in the context of 3D (2D) hydrogenic models, rms radii are larger by a factor of $\sqrt{2}$ ( $\sqrt{1.5}$ ) [30]. Thus, the $1 s$ rms exciton radius in bulk ${WSe}_{2}$ was reported [51] to be $\sqrt{2} \times 1.8 nm = 2.54 nm$ . (b) Calculated $n s$ exciton binding energies versus $κ$ . When $κ = 4.5$ , the calculated $1 s - 2 s$ and $2 s - 3 s$ separations are 124 and 21.3 meV, very close to the experimental values of 130 and 22 meV. Correlating the model with the data (dark blue points) indicates a free-particle gap of $\sim 1.890 eV$ for hBN/ ${WSe}_{2}$ /hBN.
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