$\frac{d\mu }{dn}$ in suspended bilayer graphene: The interplay of disorder and band gap

We present an interpretation of recent experimental measurements of $d\mu /dn$ in suspended bilayer graphene samples. We demonstrate that the data may be quantitatively described by assuming a spatially varying interlayer potential asymmetry (which generates a band gap) induced by local electric fields resulting from charged-impurity disorder in the graphene environment. We demonstrate that the fluctuations in the interlayer potential asymmetry and density vary between different samples, and that the potential asymmetry fluctuations increase in magnitude as the density fluctuations increase. This indicates that the mechanism causing this effect is likely to be extrinsic. We also provide predictions for the optical conductivity and mobility of suspended bilayer graphene samples with small band gaps in the presence of disorder.

Figure 1

(a) Fits of the density averaged $\overline{K}$ from Eq. (2) to the experimental data. Note that this theory includes a spatially uniform band gap $u_c$. The black lines are for published data[1] and the red lines for the unpublished data.[2] (b) Fits of the double averaged $\overline{\overline{K}}$ from Eq. (4) with $u_{\mathrm{c}}=0$ to the experimental data. Note that in this theory, the band gap $u=\tilde{u}\left(r\right)$ fluctuates in space, and the distribution of these fluctuations is parameterized by $\delta u$. The table shows a comparison of the fitting parameters for the two samples and two procedures.

Figure 2

Plots of distributions $P$ and $Q$ for (a) $x_0<\delta x$, (b) $x_0=\delta x$, and (c) $x_0>\delta x$. In all cases, we have $\delta x=1$.

Figure 3

Optical conductivity for (a) $u_{\mathrm{c}}=5\mathrm{meV}$ and (b) $u_{\mathrm{c}}=2\mathrm{meV}$ for various degrees of disorder broadening. The color of the lines is the same in both panels. Dashed lines indicate the clean case.

Figure 4

(a) Calculated mobility of suspended bilayer graphene as a function of impurity density for different strengths of the short-ranged impurities. (b) Conductivity calculated within effective medium theory as a function of gate voltage $V_g$ for a charged impurity density, $n_i=2\times {10}^{10}{\mathrm{cm}}^{-2}$, and a fixed short-range potential $n_d{V}_0^2=0.3$ (eVÅ)${}^2$. Different curves correspond to the band gap $E_g=0$, 0.15, 0.3, 0.33 eV (from top to bottom). Inset shows the calculated conductivities as a function of carrier density for two different charged impurity densities $n_i=2\times {10}^{10}{\mathrm{cm}}^{-2}$ (dashed line) and $n_i=4\times {10}^{10}{\mathrm{cm}}^{-2}$ (solid line) with zero band gap.

Figure 5

(a) and (b) The band structure of clean bilayer graphene with a band gap. The dashed lines correspond to the electron-hole symmetric band structure found with ${\gamma }_4=\Delta =0$ [see Eq. (A1)] and the solid line to the full band structure with ${\gamma }_4=150\mathrm{meV}$ and $\Delta =18\mathrm{meV}$. Black lines are for $u=20\mathrm{meV}$ and red lines for $u=60\mathrm{meV}$. (c) The bottom of the conduction band for $u=20\mathrm{meV}$ showing the absence of sombrero structure for finite ${\gamma }_4,\Delta$. (d) The bottom of the conduction band for $u=60\mathrm{meV}$ showing the restoration of the sombrero structure for finite ${\gamma }_4,\Delta$. (e) $d\mu /dn$ for clean bilayer graphene. The lines correspond to the same as in (a).