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 effect4,5. 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.

The datasets generated and/or analysed during the current study are available from the corresponding author on reasonable request.

We thank P. Mallet, J.-Y. Veuillen and J. M. Gómez Rodriguez for experimental support. H.G.-H. and I.B. were supported by AEI and FEDER under project MAT2016-80907-P (AEI/FEDER, UE), by the

Fundación Ramón Areces and by the Comunidad de Madrid NMAT2D-CM programme under grant S2018/NMT-4511. M.I.K. acknowledges the support of NWO via the Spinoza Prize.

Laboratoire Ondes et Matière d’Aquitaine, Université de Bordeaux, CNRS UMR 5798, Talence, France

Departamento de Física de la Materia Condensada, Universidad Autónoma de Madrid, Madrid, Spain

Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, Madrid, Spain

Instituto Nicolás Cabrera, Universidad Autónoma de Madrid, Madrid, Spain

Institute for Molecules and Materials, Radboud University, Nijmegen, The Netherlands

Université Grenoble Alpes, CEA, IRIG, PHELIQS, Grenoble, France

H.G.-H. and I.B. performed the experiments. V.T.R. discovered the dislocations, which were explained with the theory derived by C.D. M.I.K. and C.C. gave technical support and conceptual

advice. C.D. and V.T.R. wrote the manuscript with the input of all authors. V.T.R. coordinated the collaboration.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Peer review information Nature thanks An-Ping Li and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

This video illustrates the pseudospin rotation in intervalley back-scattering and its winding as the STM tip circles around a H adatom. The STM tip is symbolized by the purple dot. Momentums

are symbolized by grey arrows, the pseudospin of the incident electron in valley K is symbolized by a blue arrow and the pseudospin of the reflected electron in the K´ valley is symbolized

by a red arrow.

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