Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

Observation of electron–hole puddles in graphene using a scanning single-electron transistor

Abstract

The electronic structure of graphene causes its charge carriers to behave like relativistic particles. For a perfect graphene sheet free from impurities and disorder, the Fermi energy lies at the so-called ‘Dirac point’, where the density of electronic states vanishes. But in the inevitable presence of disorder, theory predicts that equally probable regions of electron-rich and hole-rich puddles will arise. These puddles could explain graphene’s anomalous non-zero minimal conductivity at zero average carrier density. Here, we use a scanning single-electron transistor to map the local density of states and the carrier density landscape in the vicinity of the neutrality point. Our results confirm the existence of electron–hole puddles, and rule out extrinsic substrate effects as explanations for their emergence and topology. Moreover, we find that, unlike non-relativistic particles the density of states can be quantitatively accounted for by considering non-interacting electrons and holes.

This is a preview of subscription content, access via your institution

Access options

Buy this article

Prices may be subject to local taxes which are calculated during checkout

Figure 1: Characterization of a graphene monolayer with transport and inverse compressibility measurements.
Figure 2: The Dirac point measured as a function of position.
Figure 3: Spatial density fluctuations and electron/hole puddles.
Figure 4: Potential variations on the substrate.
Figure 5: Estimate of the disorder from high magnetic field measurements.

Similar content being viewed by others

References

  1. Brandt, N. B., Chudinov, S. M. & Ponomarev, Y. G. Semimetals 1, Graphite and Its Compounds (North-Holland, Amsterdam, 1988).

    Google Scholar 

  2. Semenoff, G. W. Condensed-matter simulation of a three dimensional anomaly. Phys. Rev. Lett. 53, 2449–2452 (1984).

    Article  ADS  Google Scholar 

  3. Haldane, F. D. M. Model for a quantum Hall effect without Landau levels: Condensed-matter realization of the “parity anomaly”. Phys. Rev. Lett. 61, 2015–2018 (1988).

    Article  ADS  MathSciNet  Google Scholar 

  4. Zhou, S. Y. et al. First direct observation of Dirac fermions in graphite. Nature Phys. 2, 595–599 (2006).

    Article  ADS  Google Scholar 

  5. Bostwick, A., Ohta, T., Seyller, T., Horn, K. & Rotenberg, E. Quasiparticle dynamics in graphene. Nature Phys. 3, 36–40 (2007).

    Article  ADS  Google Scholar 

  6. Fetter, A. L. & Walecka, J. D. Quantum Theory of Many Particle Systems (McGraw-Hill, New York, 1971).

    Google Scholar 

  7. Bello, M. S., Levin, E. I., Shklovskii, B. I. & Efros, A. L. Density of localized states in the surface impurity band of a metal-insulator-semiconductor structure. Sov. Phys. JETP 53, 822–829 (1981).

    Google Scholar 

  8. Eisenstein, J. P., Pfeiffer, L. N. & West, K. W. Compressibility of the two-dimensional electron gas: Measurements of the zero-field exchange energy and fractional quantum Hall gap. Phys. Rev. B 50, 1760–1778 (1994).

    Article  ADS  Google Scholar 

  9. Peres, N. M. R., Guinea, F. & Castro Neto, A. H. Coulomb interactions and ferromagnetism in pure and doped graphene. Phys. Rev. B 72, 174406 (2005).

    Article  ADS  Google Scholar 

  10. Barlas, Y., Pereg-Barnea, T., Polini, M., Asgari, R. & MacDonald, A. H. Chirality and correlations in graphene. Phys. Rev. Lett. 98, 236601–236604 (2007).

    Article  ADS  Google Scholar 

  11. Hwang, E. H., Hu, B. Y.-K. & Das Sarma, S. Density dependent exchange contribution to δμ/δn in extrinsic graphene. Preprint at <http://www.arxiv.org/cond-mat/0703499v1> (2007).

  12. Eisenstein, J. P., Pfeiffer, L. N. & West, K. W. Negative compressibility of interacting two-dimensional electron and quasiparticle gases. Phys. Rev. Lett. 68, 674–677 (1992).

    Article  ADS  Google Scholar 

  13. Millard, S. et al. Effect of finite quantum well width on the compressibility of a two-dimensional electron gas. Phys. Rev. B 55, 6715–1618 (1997).

    Article  ADS  Google Scholar 

  14. Shapira, S. et al. Thermodynamics of a charged fermion layer at high r s values. Phys. Rev. Lett. 77, 3181–3184 (1996).

    Article  ADS  Google Scholar 

  15. Kravchenko, S. V., Ringberg, D. A., Semenchinsky, S. G. & Pudalov, V. M. Evidence for the influence of electron–electron interaction on the chemical potential of the two-dimensional electron gas. Phys. Rev. B 42, 3741–3744 (1990).

    Article  ADS  Google Scholar 

  16. Novoselov, K. S. et al. Electric field effect in atomically thin carbon films. Science 306, 666–669 (2004).

    Article  ADS  Google Scholar 

  17. Novoselov, K. S. et al. Two dimensional gas of massless Dirac fermions in graphene. Nature 438, 197–200 (2005).

    Article  ADS  Google Scholar 

  18. Zhang, Y., Tan, Y.-W., Stormer, H. L. & Kim, P. Experimental observation of the quantum Hall effect and Berry’s phase in graphene. Nature 438, 201–204 (2005).

    Article  ADS  Google Scholar 

  19. Peres, N. M. R., Guinea, F. & Castro Neto, A. H. Electronic properties of disordered two-dimensional carbon. Phys. Rev. B 73, 125411 (2006).

    Article  ADS  Google Scholar 

  20. Sheng, D. N., Sheng, L. & Sheng, Z. Y. Quantum Hall effect in graphene: disorder effect and phase diagram. Phys. Rev. B 73, 233406 (2006).

    Article  ADS  Google Scholar 

  21. Khveschenko, D. V. Electron localization properties in graphene. Phys. Rev. Lett. 97, 036802–036805 (2006).

    Article  ADS  Google Scholar 

  22. Ziegler, K. Robust transport properties in graphene. Phys. Rev. Lett. 97, 266802–266805 (2006).

    Article  ADS  Google Scholar 

  23. Ostrovsky, P. M., Gornyi, I. V. & Mirlin, A. D. Electron transport in disordered graphene. Phys. Rev. B 74, 235443 (2006).

    Article  ADS  Google Scholar 

  24. Aleiner, I. L. & Efetov, K. B. Effect of disorder on transport in graphene. Phys. Rev. Lett. 97, 236801–236804 (2006).

    Article  ADS  Google Scholar 

  25. Hwang, E. H., Adam, S. & Das Sarma, S. Carrier transport in two-dimensional graphene layers. Phys. Rev. Lett. 98, 186806–186809 (2007).

    Article  ADS  Google Scholar 

  26. Nomura, K. & MacDonald, A. H. Quantum transport of massless Dirac fermions in graphene. Phys. Rev. Lett. 98, 076602–076605 (2007).

    Article  ADS  Google Scholar 

  27. Cheianov, V. & Fal’ko, V. I. Friedel oscillations, impurity scattering and temperature dependence of resistivity in graphene. Phys. Rev. Lett. 97, 226801–226804 (2006).

    Article  ADS  Google Scholar 

  28. McCann, E. et al. Weak-localization magnetoresistance and valley symmetry in graphene. Phys. Rev. Lett. 97, 146805–146809 (2006).

    Article  ADS  Google Scholar 

  29. Gonzales, J., Guinea, F. & Vozmediano, M. A. H. Electron–electron interactions in graphene sheets. Phys. Rev. B. 63, 134421 (2001).

    Article  ADS  Google Scholar 

  30. Castro Neto, A. H. & Kim, E.-A. Charge inhomogeneity and the structure of graphene sheets. Preprint at <http://www.arxiv.org/cond-mat/0702562v1> (2007).

  31. Novoselov, K. S. et al. Two-dimensional atomic crystals. Proc. Natl Acad. Sci. USA 102, 10451–10453 (2005).

    Article  ADS  Google Scholar 

  32. Schedin, F. et al. Detection of individual gas molecules adsorbed on graphene. Nature Mater. 6, 652–655 (2007).

    Article  ADS  Google Scholar 

  33. Matsui, T. et al. STS observations of Landau levels at graphite surfaces. Phys. Rev. Lett. 94, 226403–226406 (2005).

    Article  ADS  Google Scholar 

  34. Niimi, Y., Kambara, H., Matsui, T., Yoshioka, D. & Fukuyama, H. Real-space imaging of alternate localization and extension of quasi-two-dimensional electronic states at graphite surfaces in magnetic fields. Phys. Rev. Lett. 97, 236804–236807 (2006).

    Article  ADS  Google Scholar 

  35. Niimi, Y. et al. Scanning tunneling microscopy and spectroscopy of the electronic local density of states of graphite surfaces near monoatomic step edges. Phys. Rev. B 73, 085421 (2006).

    Article  ADS  Google Scholar 

  36. Ishigami, M., Chen, J. H., Cullen, W. G., Fuhrer, M. S. & Williams, E. D. Atomic structure of graphene on SiO2 . Nano Lett. 7, 1643–1648 (2007).

    Article  ADS  Google Scholar 

  37. Rutter, G. M. et al. Scattering and interference in epitaxial graphene. Science 317, 219–222 (2007).

    Article  ADS  Google Scholar 

  38. Yoo, M. J. et al. Scanning single-electron transistor microscopy: Imaging individual charges. Science 276, 579–582 (1997).

    Article  Google Scholar 

  39. Yacoby, A., Hess, H. F., Fulton, T. A., Pfeiffer, L. N. & West, K. W. Electrical imaging of the quantum Hall state. Solid State Commun. 111, 1–13 (1999).

    Article  ADS  Google Scholar 

  40. Jiang, Z. et al. Infrared spectroscopy of Landau levels of graphene. Phys. Rev. Lett. 98, 197403–197407 (2007).

    Article  ADS  Google Scholar 

  41. Cheianov, V. V., Fal’ko, V. I., Altschuler, B. L. & Aleiner, I. L. Random resistor network model of minimal conductivity in graphene. Phys. Rev. Lett. 99, 176801 (2007).

    Article  ADS  Google Scholar 

  42. Geim, A. K. & Novoselov, K. S. The rise of graphene. Nature Mater. 6, 183–191 (2007).

    Article  ADS  Google Scholar 

  43. Katnelson, M. I., Novoselov, K. S. & Geim, A. K. Chiral tunnelling and the Klein paradox in graphene. Nature Phys. 2, 620–625 (2006).

    Article  ADS  Google Scholar 

  44. Cho, S. & Fuhrer, M. S. Charge transport and inhomogeneity near the charge neutrality point in graphene. Preprint at <http://www.arxiv.org/abs/0705.3239> (2007).

  45. Prange, R. E. & Girvin, S. M. The Quantum Hall Effect (Springer, New York, 1990).

    Book  Google Scholar 

  46. Ilani, S. et al. The microscopic nature of localization in the quantum Hall effect. Nature 427, 328–332 (2004).

    Article  ADS  Google Scholar 

Download references

Acknowledgements

We acknowledge helpful discussions on the preparation of graphene flakes with K. Novoselov and A. Geim. We would also like to acknowledge fruitful discussions with F. von Oppen.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to A. Yacoby.

Supplementary information

Supplementary Information

Supplementary Material (PDF 58 kb)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Martin, J., Akerman, N., Ulbricht, G. et al. Observation of electron–hole puddles in graphene using a scanning single-electron transistor. Nature Phys 4, 144–148 (2008). https://doi.org/10.1038/nphys781

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nphys781

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing