[Paper Review] Longitudinal and spin/valley Hall optical conductivity in single layer $MoS_{2}$

Zhou Li, Jules Carbotte|Nov 13, 2012

2D Materials and Applications4 citations

TL;DR

This paper presents a theoretical study of the longitudinal and transverse (spin/valley Hall) optical conductivity in monolayer MoS₂ using a low-energy Hamiltonian incorporating spin-orbit coupling and valley degrees of freedom. It reveals that circularly polarized light selectively excites carriers in one valley with high efficiency, and temperature significantly modulates spin admixture near the absorption threshold, enabling control over spin polarization in valleytronic applications.

ABSTRACT

A monolayer of $MoS_{2}$ has a non-centrosymmetric crystal structure, with spin polarized bands. It is a two valley semiconductor with direct gap falling in the visible range of the electromagnetic spectrum. Its optical properties are of particular interest in relation to valleytronics and possible device applications. We study the longitudinal and the transverse Hall dynamical conductivity which is decomposed into charge, spin and valley contributions. Circular polarized light associated with each of the two valleys separately is considered and results are filtered according to spin polarization. Temperature can greatly change the spin admixture seen in the frequency window where they are not closely in balance.

Motivation & Objective

• To understand the optical response of monolayer MoS₂, particularly its longitudinal and transverse (spin/valley Hall) conductivity.
• To investigate how circularly polarized light selectively excites carriers in one valley due to valley-specific selection rules.
• To examine the role of temperature in altering spin admixture in the optical conductivity near the absorption edge.
• To decompose the transverse conductivity into charge, spin, and valley Hall contributions using the Kubo formula.
• To provide quantitative predictions for optical conductivity relevant to device applications in valleytronics and 2D optoelectronics.

Proposed method

• Uses a low-energy effective Hamiltonian for MoS₂ near the K and -K points, including spin-orbit coupling and valley index τ.
• Applies the Kubo formula to compute dynamic conductivity, decomposing it into charge, spin, and valley Hall components.
• Evaluates the conductivity using Green's functions and momentum-space integrals centered at the K and -K valleys.
• Incorporates temperature effects via Fermi-Dirac distribution, which smears the occupation and affects spin admixture.
• Computes the real part of the optical conductivity for right (σ₊) and left (σ₋) circular polarization to model valley-selective excitation.
• Derives analytical expressions in the infinite band limit and performs numerical calculations with a momentum cutoff (ka = 3.0).

Experimental results

Research questions

• RQ1 How does circularly polarized light selectively excite carriers in one valley of monolayer MoS₂?
• RQ2 What is the relative contribution of charge, spin, and valley Hall conductivity to the total transverse optical response?
• RQ3 How does temperature affect the spin admixture in the optical conductivity near the absorption threshold?
• RQ4 What is the magnitude and sign of the valley Hall conductivity compared to the spin Hall conductivity in MoS₂?
• RQ5 To what extent does the second valley contribute to optical absorption under circularly polarized excitation?

Key findings

• The real part of the valley Hall conductivity is positive and significantly larger in magnitude than the spin Hall conductivity, which is negative for certain chemical potential values.
• For circularly polarized light, the second valley contributes less than 20% to absorption at 4 eV photon energy, confirming strong valley selectivity.
• The main absorption peak occurs at approximately 1.7 eV, with a sharp onset near the band gap energy (1.66 eV).
• Temperature strongly affects the spin admixture in the conductivity: at 300 K, the spin-up contribution is more strongly smeared than spin-down, shifting and broadening the spin polarization peak.
• Near the absorption threshold, the spin polarization ratio P(ω) can drop below 0.5 and approach 0 due to a delay in the onset of spin-up transitions, indicating a transient imbalance.
• The DC limit of the valley Hall conductivity is analytically derived and found to be large and positive, supporting its potential for valleytronic applications.