[Paper Review] Resonant Enhancement of Second Harmonic Generation by Edge States in Transition Metal Dichalcogenide Monolayers

V. V. Enaldiev|arXiv (Cornell University)|Sep 28, 2016

Strong Light-Matter Interactions3 citations

TL;DR

This paper develops a low-energy effective theory for edge states in transition metal dichalcogenide (TMD) monolayers using a two-band k·p Hamiltonian, showing that edge states with linear dispersion—governed by a single phenomenological parameter—affect second harmonic generation (SHG). When edge states cross the band gap, they induce resonant enhancement of SHG at half the band gap frequency, with the parameter extracted from experimental resonance data.

ABSTRACT

We derive a low-energy theory for edge states in transition metal dichalcogenide monolayers for a two-band $\bm{kp}$-Hamiltonian in case of uncoupled valleys. In the absence of spin-orbit interaction at the edge, these states possess a linear dispersion described by a single phenomenological parameter characterizing the edge structure. Depending on the sign of the parameter, the edge state spectrum can either cross the band gap or lie outside of it. In the first case, the presence of edge states leads to resonant enhancement of the second harmonic generation at frequencies about the half of the band gap, in agreement with recent experiments. The value of the phenomenological boundary parameter is extracted from the resonance frequency position.

Motivation & Objective

• To develop a general analytical framework for edge states in TMD monolayers using the k·p method, independent of microscopic edge details.
• To investigate how edge states influence non-linear optical responses, particularly second harmonic generation (SHG).
• To explain the experimentally observed resonant enhancement of SHG near half the band gap frequency in MoS2 monolayers.
• To extract the phenomenological boundary parameter from experimental resonance frequencies, linking theory to measurable quantities.
• To establish a unified description of edge states across TMD materials by focusing on boundary conditions rather than atomic-scale structure.

Proposed method

• Formulate a two-band k·p Hamiltonian for electrons in the K and K' valleys of TMD monolayers, incorporating spin and valley degrees of freedom.
• Introduce a boundary condition (BC) with a single real phenomenological parameter a to describe edge structure, derived from vanishing normal probability current.
• Assume no valley coupling or spin-orbit interaction at the edge, allowing independent treatment of spin and valley indices.
• Use the Sokhotski–Plemelj formula to analytically evaluate the non-linear conductivity tensor components in the presence of edge states.
• Derive expressions for the real and imaginary parts of the non-linear conductivity (δσ) in terms of energy integrals involving edge state dispersion and Fermi functions.
• Identify power-law singularities (ω − ω₀)^{-3/2} in conductivity components that signal resonant behavior when edge states cross the band gap.

Experimental results

Research questions

• RQ1 How do edge states in TMD monolayers influence the second harmonic generation (SHG) response in the presence of a band gap?
• RQ2 What role does the sign and magnitude of the phenomenological boundary parameter play in determining whether edge states lie within or outside the band gap?
• RQ3 Can the observed resonant enhancement of SHG at half the band gap frequency in MoS2 be explained by the existence of edge states with linear dispersion?
• RQ4 How can the phenomenological boundary parameter be extracted from experimental data on SHG resonance frequency?
• RQ5 To what extent can a k·p-based effective theory describe edge states in TMDs without relying on detailed DFT or tight-binding calculations?

Key findings

• Edge states in TMD monolayers exhibit linear dispersion and are described by a single phenomenological parameter a, which determines whether the edge state spectrum lies within or outside the band gap.
• When the edge state spectrum crosses the band gap (for |a| < 1), a resonant enhancement of second harmonic generation occurs at frequencies near half the band gap energy.
• The resonance frequency in the SHG spectrum is directly linked to the edge state energy, allowing the phenomenological parameter a to be extracted from experimental data.
• The non-linear conductivity components δσ_{xxx} and δσ_{yxx} exhibit a power-law singularity (ω − ω₀)^{-3/2} at the resonance frequency ω₀, confirming the presence of a sharp enhancement.
• The analytical expressions for the non-linear conductivity include a universal functional form dependent on the Fermi distribution difference f(ε₀ − ħω) − 2f(ε₀) + f(ε₀ + ħω), which modulates the resonance strength.
• The theory successfully explains the experimentally observed resonant SHG in MoS2 monolayers without requiring detailed atomic-level modeling of the edge structure.