Band Structure Mapping of Bilayer Graphene via Quasiparticle Scattering

A perpendicular electric field breaks the layer symmetry of Bernal-stacked bilayer graphene, resulting in the opening of a band gap and a modification of the effective mass of the charge carriers. Using scanning tunneling microscopy and spectroscopy, we examine standing waves in the local density of states of bilayer graphene formed by scattering from a bilayer/trilayer boundary. The quasiparticle interference properties are controlled by the bilayer graphene band structure, allowing a direct local probe of the evolution of the band structure of bilayer graphene as a function of electric field. We extract the Slonczewski-Weiss-McClure model tight binding parameters as $\gamma_0 = 3.1$ eV, $\gamma_1 = 0.39$ eV, and $\gamma_4 = 0.22$ eV.

Quasiparticle interference in metals results in standing waves in the local density of states (LDOS) whose properties reflect both the nature of the defect scatterer and the underlying electronic properties of the material itself. For the case of scattering from a straight edge, these Friedel oscillations have been studied at length for metals such as Cu 1 and Au 2 . They have recently been examined in monolayer graphene as well, where a step in a hexagonal boron nitride (hBN) substrate acted as the scatterer 3 . These measurements allow direct reconstruction of the Fermi surface properties of the material carrying the LDOS waves. Peculiar properties of monolayer graphene, such as its small Fermi surface at low energy and the pseudospin of its charge carriers, result in Friedel oscillations with unusually long wavelength and faster decay than those seen in noble metals. The Friedel oscillations in Bernal-stacked bilayer graphene are expected to be of similarly unusual nature as those observed in monolayer graphene. Differences in the exact properties of these Friedel oscillations should arise due to the hyperbolic band structure of bilayer graphene, as opposed to the linear band structure of monolayer graphene 4-6 . The ability to tune the band gap of bilayer graphene with an electric field 7 adds an additional degree of freedom to these Friedel oscillations, whereas the band structures of metals and monolayer graphene are independent of electric field.
In this letter, we present scanning tunneling microscopy (STM) and spectroscopy (STS) measurements of exfoliated few-layer graphene on hBN. Specifically, we examine interfaces between bilayer and trilayer graphene, where the layer-change region acts as an edge scatterer for carriers in the bilayer graphene. By tracking the behavior of Friedel oscillations in bilayer graphene, we are able to map its band structure as a function of electric field. Prior measurements have indirectly mapped the band structure of bilayer graphene via optical spectroscopy and transport measurement techniques [8][9][10][11][12][13][14][15] . In the former case, features of the band structure were deduced from optical transition energies, and in the latter from cyclotron mass and chemical potential measurements. The band structure of bilayer graphene has also been examined locally with STM 4,6 , but without the ability to probe both bands or tune the electric field. This work provides the first direct probe of the full energy versus momentum dispersion of bilayer graphene as a function of electric field.  acquired by turning off the feedback circuit and adding a small (5 mV) a.c. voltage at 563 Hz to the sample voltage. The current was measured by lock-in detection. Fig. 1

(b) shows
Raman spectroscopy mapping of a representative graphene on hBN flake measured in this study. A region of bilayer graphene neighbors trilayer graphene in a continuous sheet. These regions are identified by a change in the width of the Raman 2D peak 16 .  To confirm our results, we have simulated the LDOS in bilayer graphene resulting from electron scattering off a trilayer graphene edge. We performed the simulation using density functional theory (DFT) implemented in ATK 19 , with the density matrix calculated using non-equilibrium Green's functions. As shown in Fig. 3(d) for E = +100 meV, the simulation also exhibits a standing wave in the LDOS of bilayer graphene 20 . We chose the ABC stacking for the trilayer graphene to match our data in Fig. 4, although we find virtually no quantitative difference in the LDOS wavelength between the two stacking configurations of trilayer graphene 21 . We performed the simulation at varying energies, and the resulting energy versus momentum dispersion is plotted in the black star symbols of Fig. 4. We note that this data should be compared with the experimental data points at V g = +15 V due to an experimental offset in the zero of the electric field. We find relatively good quantitative agreement between our experimental data points and the DFT simulations, especially in the valence band. We see less electron-hole asymmetry in our simulations than in our experimental data, and as a result the simulated conduction band is slightly wider than our experimental results. However, the overall agreement between the two suggests that we can safely describe the bilayer/trilayer interface scattering (which is complicated by different pseudospins in each layer and an unknown atomic edge configuration) with a simple 2k scattering model 3 .
To analyze the dispersion in Fig. 4, we take a simple tight-binding model Hamiltonian Similarly, by tracking the approximate band edges we extract the band gap ( Fig. 1(d) inset) as a function of gate voltage. The direct gapŨ = U √ 1+(U/γ 1 ) 2 differs from the K-point gap by less than 1% at our maximum gap values, therefore we ignore this distinction. We find We fit the electron and hole data at all electric fields simultaneously following Eq. 1, taking γ 1 = 0.39 eV and ∆ = 0.018 eV as known from prior measurements 8,10-14 , and leaving γ 0 and γ 4 as fitting parameters. The best fit 27 to the data gives γ 0 = 3.1 eV (which corresponds to v F = 1.0 × 10 6 m/s) and γ 4 = 0.22 eV. This value of γ 4 falls just above the high end of prior results obtained via optical spectroscopy [9][10][11]13 and magnetoresistance measurements 14 , and represents pronounced electron-hole asymmetry. The resulting bands are plotted on top of the experimental data points in Fig. 4, and are well fit to the data for all electric fields.
In conclusion, we have examined quasiparticle scattering in bilayer graphene and observe coherent standing waves in the local density of states. By concurrently tracking the band gap and Fermi surface profile extracted from these standing waves, we can construct a complete picture of the bilayer graphene band structure and its evolution with electric field. We observe pronounced electron-hole asymmetry in bilayer graphene, which can be captured by the tight-binding model with a large dimer to non-dimer hopping parameter γ 4 . This