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Giant orbital magnetic moments and paramagnetic shift in artificial relativistic atoms and molecules

Nature Nanotechnology volume 18, pages 250–256 (2023)Cite this article

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Abstract

Materials such as graphene and topological insulators host massless Dirac fermions that enable the study of relativistic quantum phenomena. Single quantum dots and coupled quantum dots formed with massless Dirac fermions can be viewed as artificial relativistic atoms and molecules, respectively. Such structures offer a unique testbed to study atomic and molecular physics in the ultrarelativistic regime (particle speed close to the speed of light). Here we use a scanning tunnelling microscope to create and probe single and coupled electrostatically defined graphene quantum dots to unravel the magnetic-field responses of artificial relativistic nanostructures. We observe a giant orbital Zeeman splitting and orbital magnetic moment up to ~70 meV T–1 and ~600μB (μB, Bohr magneton) in single graphene quantum dots. For coupled graphene quantum dots, Aharonov–Bohm oscillations and a strong Van Vleck paramagnetic shift of ~20 meV T–2 are observed. Our findings provide fundamental insights into relativistic quantum dot states, which can be potentially leveraged for use in quantum information science.

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Fig. 1: Experimental setup and orbital Zeeman splitting of GQD states.
Fig. 2: Experimental observation of linear orbital Zeeman splitting.
Fig. 3: Quantum number and gate dependence of magnetic moments of GQD states.
Fig. 4: Paramagnetic shift and AB effect in coupled double GQDs.

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Source data are provided with this paper. Any additional material is available from the corresponding authors upon reasonable request.

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All the codes used in this Article are available from the corresponding authors upon request.

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Acknowledgements

We thank the Hummingbird Computational Cluster team at University of California Santa Cruz for providing computational resources for the numerical TB calculations performed in this work, and M. Hance for providing insight into the accelerator physics considerations related to our experimental findings. J.V.J. and Z.G. acknowledge support from the National Science Foundation under award DMR-1753367. J.V.J. acknowledges support from the Army Research Office under contract W911NF-17-1-0473. V.I.F. and S.S. acknowledge support from the European Graphene Flagship Core 3 Project. V.I.F. acknowledges support from the Lloyd Register Foundation Nanotechnology Grant and EPSRC grants EP/V007033/1, EP/S030719/1 and EP/N010345/1. K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, Japan, via grant no. JPMXP0112101001 and JSPS KAKENHI via grant no. JP20H00354.

Author information

Author notes

  1. These authors contributed equally: Zhehao Ge, Sergey Slizovskiy.

Authors and Affiliations

  1. Department of Physics, University of California Santa Cruz, Santa Cruz, CA, USA

    Zhehao Ge, Peter Polizogopoulos, Toyanath Joshi, David Lederman & Jairo Velasco Jr

  2. Department of Physics and Astronomy, University of Manchester, Manchester, UK

    Sergey Slizovskiy & Vladimir I. Fal’ko

  3. National Graphene Institute, University of Manchester, Booth Street East, Manchester, UK

    Sergey Slizovskiy & Vladimir I. Fal’ko

  4. International Center for Materials Nanoarchitectonics and National Institute for Materials Science, Tsukuba, Japan

    Takashi Taniguchi

  5. Research Center for Functional Materials, National Institute for Materials Science, Tsukuba, Japan

    Kenji Watanabe

  6. Henry Royce Institute for Advanced Materials, Manchester, UK

    Vladimir I. Fal’ko

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  1. Zhehao Ge
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Contributions

Z.G. and J.V.J. conceived the work and designed the research strategy. Z.G. fabricated the samples and performed the data analysis under the supervision of J.V.J. K.W. and T.T. provided the hBN crystals. Z.G. carried out the tunnelling spectroscopy measurements with assistance from P.P. and T.J. under the supervision of D.L. and J.V.J. S.S. developed the interpretation for experimental findings and performed the continuum model calculations under the supervision of V.I.F. Z.G. performed the numerical TB calculations with input from S.S. under the supervision of V.I.F. and J.VJ. Z.G. and J.V.J. wrote the paper. All the authors discussed the paper and commented on the manuscript.

Corresponding authors

Correspondence to Vladimir I. Fal’ko or Jairo Velasco Jr.

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Extended data

Extended Data Fig. 1 Raw dI/dVS(VS, B) used to get \({\boldsymbol{d}}^{\boldsymbol{3}}{\boldsymbol{I/dV}}_{\mathbf{S}}^{\boldsymbol{3}}{\boldsymbol{(}} {\boldsymbol{V}}_{\boldsymbol{S}},\,{\boldsymbol{B}} {\boldsymbol{)}}\) plot in Fig. 2d.

dI/dVS (VS, B) taken at d = 40 nm for the same GQD shown in Fig. 2a,b at VG = –16 V. The set point used to acquire the tunneling spectra was I = 1 nA, VS = –60 mV, with a 2 mV ac modulation.

Source data

Extended Data Fig. 2 Deviation between experimental potential well and parabolic potential well.

a, Experimentally measured dI/dVS (VS, d) for the same QD shown in Fig. 2a at VG = −16 V along a line across the center of a circular graphene pn junction. Colored lines are quadratic potential wells with different κ values. The set point used to acquire the tunneling spectra was I = 1 nA, VS = −200 mV, with a 2 mV ac modulation. b, Schematic of the deviation between experimental potential well and parabolic potential well at more negative energies. The experimental potential well deviates from parabolic potential well at more negative energies, and the actual QD radius will be larger than the parabolic potential well. This explains the faster increase of experimentally measured μ than the theoretical values for graphene QDs with a quadratic potential well.

Source data

Extended Data Fig. 3 \({\boldsymbol{d}}^{\boldsymbol{3}}{\boldsymbol{I/dV}}_{\boldsymbol{S}}^{\boldsymbol{3}}{\boldsymbol{(}} {\boldsymbol{V}}_{\boldsymbol{S,}}\,{\boldsymbol{B}} {\boldsymbol{)}}\) plot at different VG.

a, \(d^3I/dV_S^3\left( {V_S,B} \right)\) at VG = −24 V and at d = 36 nm. b, \(d^3I/dV_S^3\left( {V_S,B} \right)\) at VG = −20 V and at d = 40 nm. c, \(d^3I/dV_S^3\left( {V_S,B} \right)\) at VG = −10 V and at d = 36 nm. d, \(d^3I/dV_S^3(V_S,B)\) at VG = 0 V and at d = 25 nm. The quantum number (n,m) in (a–d) corresponds to radial and angular quantum number, respectively.

Source data

Supplementary information

Supplementary Information

Supplementary Figs. 1–15 and Sections 1–14.

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Source Data Extended Data Fig. 1

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Ge, Z., Slizovskiy, S., Polizogopoulos, P. et al. Giant orbital magnetic moments and paramagnetic shift in artificial relativistic atoms and molecules. Nat. Nanotechnol. 18, 250–256 (2023). https://doi.org/10.1038/s41565-023-01327-0

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