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

## Abstract

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.

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## Data availability

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

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## Acknowledgements

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.

## Author information

Authors

### Contributions

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.

### Corresponding authors

Correspondence to C. Dutreix or V. T. Renard.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

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

## Supplementary information

### Supplementary Information

This file contains Supplementary Figures S1 to S10

### Video 1: Locking of pseudospin rotation on STM tip position.

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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Dutreix, C., González-Herrero, H., Brihuega, I. et al. Measuring the Berry phase of graphene from wavefront dislocations in Friedel oscillations. Nature 574, 219–222 (2019). https://doi.org/10.1038/s41586-019-1613-5

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