Capacitance of graphene bilayer as a probe of layer-specific properties

The unique capabilities of capacitance measurements in bilayer graphene enable probing of layer-specific properties that are normally out of reach in transport measurements. Furthermore, capacitance measurements in the top-gate and penetration field geometries are sensitive to different physical quantities: The penetration field capacitance probes the two layers equally, whereas the top-gate capacitance preferentially samples the near layer, resulting in the “near-layer capacitance enhancement” effect observed in recent top-gate capacitance measurements. We present a detailed theoretical description of this effect and show that capacitance can be used to determine the equilibrium layer polarization, a potentially useful tool in the study of broken symmetry states in graphene, stemming from the interplay between interlayer screening, disorder, and the inverse-square-root van Hove singularity particular to the bilayer graphene band structure. We show how capacitance experiments can be used to probe the ground-state layer polarization, a potentially useful tool in the study of broken symmetry states in graphene.

Figure 1

(a) Bilayer graphene capacitor schematic. Layer densities ($n_1$ and $n_2$) and electrostatic potentials ($v_1$ and $v_2$) are controlled by voltages on external gates ($v_t$ and $v_b$), which couple to the bilayer through the fixed geometric capacitances $C_t^0$ and $C_b^0$. Capacitance measurements[12] are performed by measuring the current flowing through both layers in the presence of an ac driving potential on one of the gates. (b) Top-gate capacitance as a function of external gate potentials for a clean bilayer, calculated using the self-consistent approach of Sec. 4 [see Eq. (26) as well as Eqs. (17, 18, 19, 20) and (13, 14, 15)]. The capacitance, which is small in the insulating regime and high in the metallic regime, is enhanced at the edges of the metallic region due to the presence of van Hove singularities in the density of states at the band edge. The enhancement is asymmetric, reflecting the asymmetric population of the layers, Eq. (2).

Figure 2

Energy dependence of the interlayer compressibility matrix elements ${\nu }_{ij}$ in the 1/2 [left panel, Eqs. (13, 14, 15)] and ${\nu }_{\pm }$ [right panel, Eqs. (9, 10, 11)] bases for fixed interlayer asymmetry $v_{-}=50$ meV and $\Lambda =5$ eV. In the left panel, single layer charge compressibilities ${\nu }_{11}$ and ${\nu }_{22}$ are divergent only on one side of the charge gap, allowing the interlayer asymmetry to be probed by single side capacitance measurements. In the $+/-$ basis, this asymmetry is reflected by the charge-flavor response, ${\nu }_{+-}$.

Figure 3

Calculated $C_t$ (a) and $C_p$ (b) for the clean bilayer. Different color traces correspond to different values of the top-gate capacitance, measured relative to a fixed $C_b^0$ (taken to be 120 aF$/\mu {\mathrm{m}}^2$ corresponding to the standard 285 nm SiO${}_2$). Penetration field traces are normalized by the geometric value corresponding to full penetration, $C_p^0={(1/C_b^0+1/C_{\mathrm{BLG}}^0+1/C_t^0)}^{-1}$.

Figure 4

The effect of disorder on the density of states. Partial density of states ${\rho }_i$, Eq. (38) for layers $i=1$ (solid lines) and $i=2$ (dashed lines) of a graphene bilayer, obtained from the self-consistent Born approximation, Eqs. (36) and (37). Increasing the disorder strength leads to smearing of van Hove singularities and, eventually, to a closing of the energy gap.

Figure 5

Top-gate (left panel) and penetration field (right panel) capacitance for different values of the short-range disorder parameter $\gamma$, here measured in millielectron volts. Interlayer asymmetry parameter $v_{-}=50$ meV and the cutoff $\Lambda =5$ eV. Geometric parameters are chosen to match experiment reported in Ref. 12, $C_t^0/C_b^0=30$, $C_b^0=120$ aF$/\mu$m${}^2$. Color scheme corresponds to varying values of $\gamma$ as in Fig. 4.