Session A2: Compressibility and Transport in Bilayer Graphene

Electronic compressibility of bilayer graphene

We have recently measured the electronic compressibility of bilayer graphene [1], allowing exploration of the thermodynamic density of states as a function of applied electric and magnetic fields. Utilizing dual-gated field-effect devices, we can independently vary both the carrier density and the size of the tunable band gap. An oscillating voltage applied to a back gate generates corresponding signals in the top gate via electric fields lines which penetrate the graphene, thereby allowing a direct measurement of the inverse compressibility, $K^{-1}$, of the bilayer [2]. We have mapped $K^{-1}$, which is proportional to the inverse density of states, as a function of the top and back gate voltages in zero and finite magnetic field. A sharp increase in $K^{-1}$ near zero density is observed with increasing electric field strength, signaling the controlled opening of a band gap. At high magnetic fields, broad Landau level (LL) oscillations are observed, directly revealing the doubled degeneracy of the lowest LL and allowing for a determination of the disorder broadening of the levels. We compare our results to tight-binding calculations of the bilayer band structure, and to recent theoretical studies of the compressibility of bilayer graphene. Together, these clearly illustrate the unusual hyperbolic nature of the low energy band structure, reveal a sizeable electron-hole asymmetry, and suggest that many-body interactions play only a small role in bilayer-on-substrate devices. This work is a collaboration with J. P. Eisenstein of Caltech, and is supported by the NSF under Grant No. DMR-0552270 and the DOE under Grant No. DE-FG03-99ER45766. \\[4pt] [1] E. A. Henriksen and J. P Eisenstein, Phys. Rev. B {\bf82}, 041412(R) (2010). \\[0pt] [2] J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. {\bf 68}, 674 (1992); Phys. Rev. B {\bf 50}, 1760 (1994).   

Electronic Structure and Carrier Transport in Graphene Bilayers

Graphene bilayers come in different varieties ranging from the micro-mechanically exfoliated Bernal stacked sheets where the strongly coupled layers act like a single electronic material, to the essentially decoupled turbostratic graphene bilayers observed in both epitaxial and CVD grown graphene. In this talk I will first review the experimental evidence and early theoretical understanding for the band structure of bilayer graphene. I will then discuss electrical transport measurements and present the semi-classical theory for carrier transport in bilayer graphene. I will show that close to the Dirac point, the co-existence of electron and hole carriers gives rise to an interesting interplay between disorder and temperature [1-3]. For example, we predict that knowing the strength of the disorder potential from low temperature conductivity measurements completely determines the temperature dependence of the conductivity. Detailed comparison with recent experiments highlights both the successes and the shortcomings of this theoretical model. Finally, I will examine the different factors influencing the transport in twisted graphene bilayers. For example, the breaking of inversion symmetry results in a charge imbalance between the two layers giving rise to unexpected features in magneto-transport. \\[4pt] [1] S. Adam and S. Das Sarma, ``Boltzmann transport and residual conductivity in bilayer graphene," {\it Phys. Rev. B}, {\bf 77}, 115436, (2008). \\[0pt] [2] S. Adam and M. D. Stiles,``Temperature dependence of the diffusive conductivity of bilayer graphene,'' {\it Phys. Rev. B}, {\bf 82}, 075423, (2010). \\[0pt] [3] S. Xiao, J. Chen, S. Adam, E. D. Williams, and M. S. Fuhrer, ``Charged impurity scattering in bilayer graphene,'' {\it Phys. Rev. B}, {\bf 82}, 041406, (2010).   

Probing layer imbalance in bilayer graphene with electrostatic capacitance measurements

In bilayer graphene, application of an external electric field modulates both the charge carrier density and the band structure itself. In particular, application of an electric field perpendicular to the sample plane opens up a band gap in the bilayer graphene energy spectrum, leading to insulating behavior at charge neutrality. Using capacitance measurements, we extract the electronic compressibility as a function of density, applied bias, and temperature. We find that the compressibility remains high even in the region in which a gap is expected, confirming that the insulating behavior observed in transport is due to transport via localized states. Temperature dependent capacitance measurements allow us to estimate the gap in the spectrum, which we find to be in qualitative agreement with that measured by optics. Away from charge neutrality, the density dependence of the compressibility is consistent with hyperbolic electronic bands. Features identified with the $1/\sqrt{\epsilon}$ van Hove singularity---expected for the nearly quartic dispersion of gapped bilayer graphene---are observed near the band edge. These features show a polarization dependent asymmetry, appearing only where the near layer is at lower energy layer for the corresponding carrier type. Using a model of bilayer graphene that incorporates the finite interlayer separation, we show that capacitance measurements in bilayer graphene are sensitive to $\textit{layer indexed}$ compressibilities, in addition to the total charge compressibility. This allows an unambiguous determination of the layer polarization of the ground state, a particularly useful tool in the study of the broken symmetry states observed at high magnetic field.   

Compressibility of bilayer graphene: the role of disorder

We discuss the role of disorder caused by charged impurities on the compressibility of bilayer graphene. In doing so, we take into account the full hyperbolic dispersion relation and the presence of a gap between the valence and conduction bands to produce an exact calculation of $\frac{d\mu}{dn}$ for the non-disordered case. We then introduce two methods for including the disorder in a statistical way and evaluate the effectiveness of each by comparing their predictions with recent experiments. We find that averaging is best done at the level of the observable quantity: in this case the compressibility. This work is done in collaboration with Sankar Das Sarma and Euyheon Hwang, and supported by US-ONR, NRI-SWAN, and UMD-CNAM.   

Local Compressibility Measurements of Correlated States in Suspended Bilayer Graphene

Bilayer graphene has attracted considerable interest due to the important role played by many-body effects, particularly at low energies. The exceptional quality of suspended devices has enabled the observation of interaction-driven broken-symmetry states and the fractional quantum Hall effect. Here we report local compressibility measurements of a suspended graphene bilayer. We find that the energy gaps at filling factors +/- 4 do not vanish at low fields, but instead merge into an incompressible region near the charge neutrality point at zero electric and magnetic field. These results indicate the existence of a zero-field ordered state and are consistent with the formation of either an anomalous quantum Hall state or a nematic phase with broken rotational symmetry. At higher fields, we measure the intrinsic energy gaps of broken-symmetry states at filling factor 0, +/- 1 and +/-2, and find that they scale linearly with magnetic field, yet another manifestation of the strong Coulomb interactions in bilayers.