Determination of the gate-tunable band gap and tight-binding parameters in bilayer graphene using infrared spectroscopy

We present a compelling evidence for the opening of a bandgap in exfoliated bottom-gated bilayer graphene by fitting the gate-voltage-modulated infrared reflectivity spectra in a large range of doping levels with a tight-binding model and the Kubo formula. A close quantitative agreement between the experimental and calculated spectra is achieved, allowing us to determine self-consistently the full set of Slonczewski-Weiss-McClure tight-binding parameters together with the gate-voltage-dependent bandgap. The doping dependence of the bandgap shows a good agreement with the existing calculations that take the effects of self-screening into account. We also identify certain mismatches between the tight-binding model and the data, which can be related to electron-electron and electron-phonon interactions.

Figure 1

(Color online) (a) Absolute (gold-normalized) reflectivity of bare oxide and graphene at $V_g=+100$, $-20$, and $-100\text{V}$. The substrate temperature is 10 K. (b) Reflectivity of graphene normalized to bare oxide. (c) Self-normalized reflectivity of graphene $R_{gr}\left(V_g\right)/R_{gr}\left(-20\text{V}\right)$.

Figure 2

(Color online) Crystal structure of Bernal-stacked bilayer graphene and the considered tight-binding parameters.

Figure 3

(Color online) (a) Self-normalized and (b) difference reflectivity spectra as a function of $V_g$. Dots are the experimental data points, solid and dashed lines represent model fits 1 and 2 (without and with a bandgap), respectively. (c) The real part of the optical conductance. Curves—the result of the reflectivity fits (the colors and the line types correspond to panels a and b), symbols—the refined conductance obtained using a KK-consistent inversion as described in the text. In all panels the curves are vertically displaced for clarity.

Figure 4

(Color online) The extracted values of $\left|U\right|$ (top panel) and the chemical potential (bottom panel) as functions of the gate voltage and doping. Dash-dotted line—the calculated “unscreened” value of $\left|U\right|$, solid line—an ab initio DFT calculation from Ref. 42, which takes screening effects into account.

Figure 5

Electronic bands (calculated using parameters extracted from optical spectra) and chemical potential at selected gate voltages.

Figure 6

(Color online) (a) The difference reflectivity at $V_g=-45\text{V}$ at three temperatures of the substrate: 300, 150, and 10 K. (b) Model curves of the same quantity calculated by varying the effective temperature of graphene and keeping other parameters unchanged (shown in Table ).

Figure 7

(Color online) Model demonstration of the effect of the bandgap on optical spectra in the case of finite doping and the case of zero doping. Top panel: the chemical potential is larger than the bandgap; bottom panel: $\mu$ is zero. The inset shows the corresponding bands (energy, in eV, vs momentum near the K point). A simplified set of SWMcC parameters is used: ${\gamma }_0=3\text{eV}$, ${\gamma }_1=0.4\text{eV}$, and ${\gamma }_3={\gamma }_4=\Delta =0$. The broadening $\Gamma$ is 0.01 eV.

Figure 8

(Color online) Color maps of $\left(\Delta R/R\right)\left(\omega ,V_g\right)$. (a)—experiment, (b)—model curves corresponding to Fit 1 (zero bandgap), (c)—model curves corresponding to Fit 2 (including bandgap), and (d)—model curves in the absence of the bandgap and electron-hole asymmetry. The color scheme is the same in all graphs. The dashed and dash-dotted lines in panel (a) are the dependencies of ${\omega }_A=2\left|\mu \right|$ and ${\omega }_B={\gamma }_1+2\left|\mu \right|$, respectively.

Figure 9

(Color online) The experimental curves $\left(\Delta R/R\right)\left(\omega \right)$ (solid lines) at $V_g=-95\text{V}$ (left panel) and $+65\text{V}$ (right panel), together with the best fitting curves (dash-dotted lines).

Figure 10

(Color online) The functions ${\beta }_1\left(\omega \right)$ and ${\beta }_2\left(\omega \right)$, describing the sensitivity of $\left(\Delta R/R\right)\left(\omega \right)$ to $\Delta G_1\left(\omega \right)$ and $\Delta G_2\left(\omega \right)$, respectively (obtained at 10 K).