Excitons without effective mass: Biased bilayer graphene

Understanding the dynamics of excitons in two-dimensional semiconductors requires a theory that incorporates the essential physics distinct from their three-dimensional counterparts. In addition to the modified dielectric environment, single-particle states with strongly nonparabolic dispersion appear in many two-dimensional band structures, so that “effective mass” is ill-defined. Focusing on electrostatically biased bilayer graphene as an example where quartic (and higher) dispersion terms are necessary, we present a semianalytic theory used to investigate the properties of ground and excited excitonic states. This includes determination of relative oscillator strengths and magnetic moments (valley $g$-factors) which can be directly compared to recent experimental measurements.

Figure 1

Single-particle electronic structure of biased bilayer graphene near the $K\left(K^{\prime }\right)$-point under bias $2V=100$ meV shown in black. Red line is least-squares eighth-order polynomial fit within the vertical dashed lines. Magnified region (circled in green) emphasizes the nonparabolic dispersion and energetic depth of band-edge extrema. Inset above shows schematic optical absorption spectrum close to the interband transition threshold, with a manifold of discrete exciton states at lower energy.

Figure 2

(a) Energy landscape of a single $m=0$ Gaussian trial function for the ground-state exciton envelope function of BBG, using Eqs. (4) and (5). Kinetic energy polynomial coefficients $A_{\xi }$, where $\xi =1,2,3,4$, are determined by least-squares fitting the exact dispersion between our choice of $\pm k_{\text{fit}}$, giving a real-space length scale shown as a dashed white line. The energetic minima, indicating variational optimum length scale $b_0>\pi /k_{\text{fit}}$, is shown as a solid black line. Black dashed line presents half of the 2D screening length $r_0>b_0$. (b) Variational optimum length scale shown in blue (left axis) for both ground-state ($b_0$) and $m=1$ excited state ($b_1$). Binding energy in green (right axis) accounts for reduction in bound-continuum energy due to $E_{\text{min}}$ (see Fig. 1). Dashed lines are energies of the single trial function, which are improved by the lowest generalized eigenvalue of the problem with an optimized five-function basis (see text).

Figure 3

(a) Relative oscillator strengths of the $m=0$ and $m=1$ excitons (blue curves) and their ratio (red dashed curve), as a function of the gate bias energy $2V$. (b) Valley $g$-factors of the $m=0$ and $m=1$ excitons. Inset: $k$-dependent $g$-factor differences of the conduction and valence bands, under bias conditions $V=$ 10, 30, and 100 meV. Here, the full scale of $k$ is normalized according to $\frac{\sqrt{3}}{2}a{\gamma }_0{k}_{\text{max}}=\frac{{\gamma }_1}{2}$, so that $k_{\text{max}}$ is less than 2% from the $K$ point to the $\mathrm{\Gamma }$ point.