Renormalization of the optical band gap through an effective Thirring interaction for massive Dirac-like electrons

TL;DR

This paper models the temperature-dependent renormalization of the optical band gap in 2D Dirac-like materials using an effective Thirring interaction, aligning well with experimental data for transition metal dichalcogenides.

Contribution

It introduces a theoretical framework combining Dirac-like electrons and phonon interactions to explain band gap renormalization at finite temperatures in 2D materials.

Findings

The renormalized band gap remains nearly constant at low temperatures.

At higher temperatures, the band gap decreases linearly with temperature.

The model's predictions agree with experimental data for TMD monolayers.

Abstract

We analyze mass renormalization in massive Dirac-like systems in (2+1) dimensions arising from electron-phonon interactions at finite temperatures, employing the large-$N$ expansion. Our model combines the low-energy description of charge carriers in a buckled honeycomb lattice with the low-energy approximation for phonons and electron-phonon interactions in two-dimensional materials. Consequently, the system is modeled as a massive Dirac-like field coupled to a two-component vector field $\mathcal{A}_i$, representing the phonon modes. This framework allows us to compute the one-loop electron self-energy at finite temperature, from which we derive the renormalized band gap, $m^R$. The effective model is subsequently applied to describe the renormalized optical band gap in monolayers of transition metal dichalcogenides (TMDs), including MoS$_2$, MoSe$_2$, WS$_2$, and WSe$_2$. A good agreement is observed with experimental data for reasonable values of the ultraviolet cutoff, $\Lambda \approx 1$ eV. Our main findings indicate that $m^R$ remains nearly constant at low temperatures, whereas at higher temperatures it decreases linearly with the temperature $T$. Specifically, we find that $m^R$ reduces by approximately $\approx [0.1,0.2]$ eV as the temperature increases from $\approx 4$ K to $500$ K, consistent with recent experimental observations. Furthermore, we estimate the temperature range at which the transition to the linear regime occurs, obtaining typical values within $\approx [110,150]$ K for the four materials under consideration.