Electronic compressibility of layer-polarized bilayer graphene

We report on a capacitance study of dual gated bilayer graphene. The measured capacitance allows us to probe the electronic compressibility as a function of carrier density, temperature, and applied perpendicular electrical displacement $\overline{D}$. As a band gap is induced with increasing $\overline{D}$, the compressibility minimum at charge neutrality becomes deeper but remains finite, suggesting the presence of localized states within the energy gap. Temperature dependent capacitance measurements show that compressibility is sensitive to the intrinsic band gap. For large displacements, an additional peak appears in the compressibility as a function of density, corresponding to the presence of a one-dimensional van Hove singularity (vHs) at the band edge arising from the quartic bilayer graphene band structure. For $\overline{D}>0$, the additional peak is observed only for electrons, while for $\overline{D}<0$ the peak appears only for holes. This asymmetry can be understood in terms of the finite interlayer separation and may be useful as a direct probe of the layer polarization.

Figure 1

(a)–(c) Optical microscope images. (a) A thin hBN single crystal is transferred onto a (b) mono- or bilayer graphene flake and (c) contacts and gate electrodes deposited by electron beam lithography. Scale bars are 10 $\mu$m. (d) Cross section schematic of resulting dual gated device. (e) Schematic circuit diagram of the capacitance bridge. A reference capacitor (Johanson Technology R14S) is mounted on the probe, and a reference voltage is chosen to balance the capacitance bridge. A small AC excitation signal is added to the DC gate bias through a transformer (Triad Magnetics SP67).

Figure 2

Measured conductance and capacitance of monolayer graphene at 2 K and $B=0$ (a) and $B=9$ T (b). In (a), capacitance and conductance resemble each other closely due to the fact that both contain a spurious contribution adding “in parallel”; $G={(1/R+1/R_C)}^{-1}$ and $C_T={(1/C_Q+1/C_T^0)}^{-1}$ where $R$ is the device resistance, $C_Q$ is the compressibility, or “quantum capacitance,” $C_T^0$ is the geometric capacitance, and $R_C$ is the contact resistance of our two-terminal devices.

Figure 3

Capacitance at $B=0$ and 1.5 K as a function of V${}_T$ and V${}_B$. Colored traces in (b) are taken at 30 volt intervals in V${}_B$, corresponding to the colored lines in (a). (c) Traces at constant $\overline{D}$, extracted from the data set shown in (a). Data is plotted as a function of $C_T^0\Delta V_T+C_B^0\Delta V_B$, which would correspond to the density if the bilayer were perfectly two dimensional and perfectly compressible. Curves in (c) are offset for clarity by 2 fF per V/nm in $\overline{D}$.

Figure 4

Bottom panel: Temperature dependence of the inverse compressibility ${\left(A{e}^2\nu \right)}^{-1}$ for $V_B=80$, 0, and $-$80 V. A single value of $C_{\mathrm{para}}$ is chosen for all gate voltages at a given temperature, but a different $C_{\mathrm{para}}$ is chosen for each temperature so that the curves match at high density. We attribute the variation in $C_{\mathrm{para}}$ to thermal expansion of the bonding wires, the capacitance of which constitute the majority of $C_{\mathrm{para}}$. Top panel: comparison between the temperature dependence of the minimal compressibility and minimal conductivity at $V_B=-60$ V. The dashed line is a guide for the eye.

Figure 5

Left panel: Subtracted capacitance, $\Delta C_T=C_T\left(\overline{D}\right)-C_T(\overline{D}=0)$, as a function of approximate density for different applied displacements. Curves are offset by 1 fF per V/nm in $\overline{D}$. Right panel: Calculated top gate capacitance for disordered bilayer graphene, following Ref. 17. The colors correspond to the displacements in the left panel; the disorder parameter is $\gamma =4$ for all curves. Curves are offset by 3 fF per V/nm in $\overline{D}$.