Comparison of microscopic models for disorder in bilayer graphene: Implications for density of states and optical conductivity

We study the effects of disorder on bilayer graphene using four different microscopic models and directly compare their results. We compute the self-energy, density of states, and optical conductivity in the presence of short-ranged scatterers and screened Coulomb impurities, using both the Born approximation and self-consistent Born approximation for the self-energy. We also include a finite interlayer potential asymmetry, which generates a gap between the valence and conduction bands. We find that the qualitative behavior of the two scattering potentials is similar, but that the choice of approximation for the self-energy leads to important differences near the band edge in the gapped case. Finally, we describe how these differences manifest in the measurement of the band gap in optical and transport experimental techniques.

Figure 1

(a) Sketch of the low-energy band structure of bilayer graphene in the presence of an external electric field, which creates a band gap at low energy. The bands are labeled with the indices $c,v$ for conduction and valence bands, and $l,s$ for low-energy and split bands, respectively. Diagrams corresponding to (b) the Born approximation and (c) the self-consistent Born approximation for the self-energy, and the definition of the Green's function in terms of the self-energy. (d) The Thomas-Fermi screening wave vector $q_{\mathrm{TF}}$ as a function of the Fermi energy. The band edge in the $u=0.2$-eV case is at $E_{\mathrm{F}}=0.083$ eV. (e) The screening wave vector in the RPA as a function of wave vector in the intrinsic case.

Figure 2

The real and imaginary parts of the self-energy in the low-energy conduction band (solid lines) and split conduction band (dashed lines) for $n_{\mathrm{i}}={10}^{11}{\mathrm{cm}}^{-2}$ (thin lines) and $n_{\mathrm{i}}={10}^{12}{\mathrm{cm}}^{-2}$ (thick lines) at $\left|k\right|a=0.01$. The lines are labeled in panel (o). The scattering potential (CI or SRI), approximation for the self-energy (BA or SCBA) and the band gap ($u=0$ or 0.2 eV) are labeled in each panel. The self-energy is trivially zero in the clean limit.

Figure 3

The density of states for $n_{\mathrm{i}}={10}^{11}{\mathrm{cm}}^{-2}$ (thin solid lines), $n_{\mathrm{i}}={10}^{12}{\mathrm{cm}}^{-2}$ (thick solid lines), and (near the band edge) for $n_{\mathrm{i}}=5\times {10}^{11}{\mathrm{cm}}^{-2}$ (dot-dashed lines). The scattering potential (CI or SRI), approximation for the self-energy (BA or SCBA), and gap ($u=0$ or 0.2 eV) are labeled in each panel. The dotted lines give the DOS in the clean limit.

Figure 4

(a)–(h) The optical conductivity of bilayer graphene in units of ${\sigma }_0=\frac{\pi e^2}{2h}$ (the conductivity of monolayer graphene in the clean limit) with $u=0.2\mathrm{eV}$. The type of scatterer (CI or SRI), approximation to the self-energy (BA or SCBA), and the carrier density are labeled in each panel. In all plots, the dotted line represents the contribution from the interband conductivity in the clean system, the thick solid line is for $n_{\mathrm{i}}={10}^{12}{\mathrm{cm}}^{-2}$, and the thin solid line is $n_{\mathrm{i}}=5\times {10}^{11}{\mathrm{cm}}^{-2}$. Throughout we have assumed a ${\mathrm{SiO}}_2$ substrate with no top gate when defining the dielectric environment (see text for full definition of parameters). (i), (j) Show the transitions that give rise to each feature in the nondisordered conductivity.

Figure 5

The band gaps extracted from the optical conductivity of intrinsic bilayer graphene and the DOS. (a) SCBA approximation for the CI on an SiO${}_2$ substrate; (b) the same approximation but for a high-quality h-BN substrate (note the difference in the range of the horizontal axis); (c) SRI for BA and SCBA; (d), (e) definition of optical and DOS gaps.

Figure 6

(a), (b) The SRI strength set by the effective CI and renormalized to approximate the TF screening. (c), (d) The direct comparison of the DOS for CI and SRI for $u=0.2\mathrm{eV}$. The impurity density is labeled in each plot. (c) $V^{\prime }=2\mathrm{eV}{\mathrm{nm}}^2$ and (d) $V^{\prime }=1.6\mathrm{eV}{\mathrm{nm}}^2$.