Measuring the Berry phase of graphene from wavefront dislocations in Friedel oscillations

Electronic band structures dictate the mechanical, optical and electrical properties of crystalline solids. Their experimental determination is therefore crucial for technological applications. Although the spectral distribution in energy bands is routinely measured by various techniques¹, it is more difficult to access the topological properties of band structures such as the quantized Berry phase, γ, which is a gauge-invariant geometrical phase accumulated by the wavefunction along an adiabatic cycle². In graphene, the quantized Berry phase γ = π accumulated by massless relativistic electrons along cyclotron orbits is evidenced by the anomalous quantum Hall effect^4,5. It is usually thought that measuring the Berry phase requires the application of external electromagnetic fields to force the charged particles along closed trajectories³. Contradicting this belief, here we demonstrate that the Berry phase of graphene can be measured in the absence of any external magnetic field. We observe edge dislocations in oscillations of the charge density ρ (Friedel oscillations) that are formed at hydrogen atoms chemisorbed on graphene. Following Nye and Berry⁶ in describing these topological defects as phase singularities of complex fields, we show that the number of additional wavefronts in the dislocation is a real-space measure of the Berry phase of graphene. Because the electronic dispersion relation can also be determined from Friedel oscillations⁷, our study establishes the charge density as a powerful observable with which to determine both the dispersion relation and topological properties of wavefunctions. This could have profound consequences for the study of the band-structure topology of relativistic and gapped phases in solids.