Coulomb effects in the absorbance spectra of two-dimensional Dirac materials

A wide range of materials like graphene, topological insulators, and transition-metal dichalcogenides (TMDs) share an interesting property: the low-energy excitations behave as Dirac particles. This emergent behavior of Dirac quasiparticles defines a large class of media that are usually called Dirac materials. In this work we build the foundations of a way to study the linear optical properties of these two-dimensional media. Our approach is based on a Dirac-like formulation of the standard semiconductor Bloch equations used in semiconductor physics. We provide an explicit expression of the linear absorbance, which we call the relativistic Elliott formula, and use it to quantify the variation of the continuum absorbance spectrum with the strength of the Coulomb interaction (the Sommerfeld factor). Our calculations also show how the Coulomb enhancements scales with the band gap and vanishes for zero band gap, shedding light on the behavior of graphene for low-light intensities. The results presented are in good quantitative agreement with published experimental results. Our theory can help researchers to explore the nonlinear interactions of intense, ultrashort pulses with TMDs, and the framework is flexible enough to be adapted to different experimental situations, such as cavities, multilayers, heterostructures, and microresonators.

Figure 1

Plot of the absorbance [Eq. (13)] using parameters for a ${\text{MoS}}_2$ TMD on a fused-silica substrate: energy gap $\mathrm{\Delta }=2.82\mathrm{eV}$ and background refractive index $n_b=1.5$. We use a $\delta =0.1\mathrm{\Delta }$ Lorentzian broadening on both exciton and continuum transitions. $E_b$ is the binding energy for the $1s$ Dirac exciton.

Figure 2

Plot of the Elliott formula reducing the strength of the Coulomb coupling constant ${\alpha }_c$. Dark blue, ${\alpha }_c=0.45$; blue, ${\alpha }_c=0.30$; cyan, ${\alpha }_c=0.20$). Free-carrier limit, dash-dotted gray line; zero-gap graphene limit, solid black line; nonrelativistic free-carrier limit, dashed green line. Inset: Continuum absorbance at the band edge ($\mathrm{\Delta }=2.18\mathrm{eV}$) when increasing the strength of the Coulomb interactions.

Figure 3

Plot of the continuum absorbance contribution while reducing the band gap $\mathrm{\Delta }$ (${\alpha }_c=0.35$) with a finite Lorentzian energy broadening $\delta =10\mathrm{meV}$. Blue, $\mathrm{\Delta }=100$ meV; cyan, $\mathrm{\Delta }=50$ meV; green, $\mathrm{\Delta }=25$ meV; violet, $\mathrm{\Delta }=5$ meV. Inset: Continuum absorbance while reducing the band gap for zero-energy broadening (${\alpha }_c=0.35$). The axis labels are the same as in the main plot.

Figure 4

(a) Real part of the dielectric function of ${\mathrm{MoSe}}_2$. (b) Imaginary part of the dielectric function of ${\mathrm{MoSe}}_2$. (c) Real part of the dielectric function of ${\mathrm{MoS}}_2$. (d) Imaginary part of the dielectric function of ${\mathrm{MoS}}_2$. This figure shows good agreement with the results in Figs. 1(e), 1(i), 1(g), and 1(k) of [19].

Figure 5

Fit of the radial components of the real and imaginary parts of the dipole moment in units of area $L^2$ and electric charge $e$.