Skip to main content

Thank you for visiting 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.

Simultaneous voltage and current density imaging of flowing electrons in two dimensions


A variety of physical phenomena associated with nanoscale electron transport often results in non-trivial spatial voltage and current patterns, particularly in nonlocal transport regimes. While numerous techniques have been devised to image electron flows, the need remains for a nanoscale probe capable of simultaneously imaging current and voltage distributions with high sensitivity and minimal invasiveness, in a magnetic field, across a broad range of temperatures and beneath an insulating surface. Here we present a technique for spatially mapping electron flows based on a nanotube single-electron transistor, which achieves high sensitivity for both voltage and current imaging. In a series of experiments using high-mobility graphene devices, we demonstrate the ability of our technique to visualize local aspects of intrinsically nonlocal transport, as in ballistic flows, which are not easily resolvable via existing methods. This technique should aid in understanding the physics of two-dimensional electronic devices and enable new classes of experiments that image electron flow through buried nanostructures in the quantum and interaction-dominated regimes.

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

Relevant articles

Open Access articles citing this article.

Access options

Rent or buy this article

Prices vary by article type



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

Fig. 1: Overview of the nanoscale voltage and current imaging technique.
Fig. 2: Spatial imaging of the voltage drop of flowing electrons in the diffusive and ballistic regimes.
Fig. 3: Extracting local quantities from electrostatic potential maps.
Fig. 4: Imaging the local current density in a graphene device with a bend.
Fig. 5: Imaging nonlocal ballistic voltage drops.

Data availability

The data that support the plots and other analysis in this work are available from the corresponding author upon request.


  1. Beenakker, C. W. J. & van Houten, H. Quantum transport in semiconductor nanostructures. Solid State Phys. Adv. Res. Appl. 44, 1–228 (1991).

    Google Scholar 

  2. Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).

    Article  CAS  Google Scholar 

  3. Qi, X.-L. & Zhang, S.-C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011).

    Article  CAS  Google Scholar 

  4. Lucas, A. & Fong, K. C. Hydrodynamics of electrons in graphene. J. Phys. Condens. Matter 30, 053001 (2018).

    Article  Google Scholar 

  5. Landauer, R. Conductance determined by transmission: probes and quantised constriction resistance. J. Phys. Condens. Matter 1, 8099–8110 (1989).

    Article  Google Scholar 

  6. McLennan, M. J., Lee, Y. & Datta, S. Voltage drop in mesoscopic systems: a numerical study using a quantum kinetic equation. Phys. Rev. B 43, 13846–13884 (1991).

    Article  CAS  Google Scholar 

  7. Datta, S. Electron Transp ort Mesoscopic System (Cambridge Univ. Press, 1997).

  8. Muralt, P. & Pohl, D. W. Scanning tunneling potentiometry. Appl. Phys. Lett. 48, 514–516 (1986).

    Article  CAS  Google Scholar 

  9. McCormick, K. L. et al. Scanned potential microscopy of edge and bulk currents in the quantum Hall regime. Phys. Rev. B 59, 4654–4657 (1999).

    Article  CAS  Google Scholar 

  10. Bachtold, A. et al. Scanned probe microscopy of electronic transport in carbon nanotubes. Phys. Rev. Lett. 84, 6082–6085 (2000).

    Article  CAS  Google Scholar 

  11. Weitz, P., Ahlswede, E., Weis, J., Klitzing, K. V. & Eberl, K. Hall-potential investigations under quantum Hall conditions using scanning force microscopy. Phys. E 6, 247–250 (2000).

    Article  CAS  Google Scholar 

  12. Melitz, W., Shen, J., Kummel, A. C. & Lee, S. Kelvin probe force microscopy and its application. Surf. Sci. Rep. 66, 1–27 (2011).

    Article  CAS  Google Scholar 

  13. Roth, B. J., Sepulveda, N. G. & Wikswo, J. P. Using a magnetometer to image a two-dimensional current distribution. J. Appl. Phys. 65, 361–372 (1989).

    Article  Google Scholar 

  14. Tokura, Y., Honda, T., Tsubaki, K. & Tarucha, S. Noninvasive determination of the ballistic-electron current distribution. Phys. Rev. B 54, 1947–1952 (1996).

    Article  CAS  Google Scholar 

  15. Huber, M. E. et al. Gradiometric micro-SQUID susceptometer for scanning measurements of mesoscopic samples. Rev. Sci. Instrum. 79, 053704 (2008).

    Article  Google Scholar 

  16. Rondin, L. et al. Magnetometry with nitrogen-vacancy defects in diamond. Rep. Prog. Phys. 77, 056503 (2014).

    Article  CAS  Google Scholar 

  17. Tetienne, J.-P. et al. Quantum imaging of current flow in graphene. Sci. Adv. 3, e1602429 (2017).

    Article  Google Scholar 

  18. Chang, K., Eichler, A., Rhensius, J., Lorenzelli, L. & Degen, C. L. Nanoscale imaging of current density with a single-spin magnetometer. Nano Lett. 17, 2367–2373 (2017).

    Article  CAS  Google Scholar 

  19. Vasyukov, D. et al. A scanning superconducting quantum interference device with single electron spin sensitivity. Nat. Nanotechnol. 8, 639–644 (2013).

    Article  CAS  Google Scholar 

  20. Eriksson, M. A. et al. Cryogenic scanning probe characterization of semiconductor nanostructures. Appl. Phys. Lett. 69, 671–673 (1996).

    Article  CAS  Google Scholar 

  21. Bhandari, S. et al. Imaging cyclotron orbits of electrons in graphene. Nano Lett. 16, 1690–1694 (2016).

    Article  CAS  Google Scholar 

  22. Fei, Z. et al. Gate-tuning of graphene plasmons revealed by infrared nano-imaging. Nature 487, 82–85 (2012).

    Article  CAS  Google Scholar 

  23. Chen, J. et al. Optical nano-imaging of gate-tunable graphene plasmons. Nature 487, 77–81 (2012).

    Article  CAS  Google Scholar 

  24. Mueller, T., Xia, F., Freitag, M., Tsang, J. & Avouris, P. Role of contacts in graphene transistors: a scanning photocurrent study. Phys. Rev. B 79, 245430 (2009).

    Article  Google Scholar 

  25. Fontein, P. F. et al. Spatial potential distribution in GaAs/AlxGa1-xAs heterostructures under quantum Hall conditions studied with the linear electro-optic effect. Phys. Rev. B 43, 12090–12093 (1991).

    Article  CAS  Google Scholar 

  26. Knott, R., Dietsche, W., v. Klitzing, K., Eberl, K. & Ploog, K. Inside a 2D electron system: images of potential and dissipation. Solid. State. Electron. 37, 689–692 (1994).

    Article  CAS  Google Scholar 

  27. Gao, C., Wei, T., Duewer, F., Lu, Y. & Xiang, X. D. High spatial resolution quantitative microwave impedance microscopy by a scanning tip microwave near-field microscope. Appl. Phys. Lett. 71, 1872–1874 (1997).

    Article  CAS  Google Scholar 

  28. Wang, Z. et al. Quantitative measurement of sheet resistance by evanescent microwave probe. Appl. Phys. Lett. 86, 153118 (2005).

    Article  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

  31. Honig, M. et al. Local electrostatic imaging of striped domain order in LaAlO3/SrTiO3. Nat. Mater. 12, 1112–1118 (2013).

    Article  CAS  Google Scholar 

  32. 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  CAS  Google Scholar 

  33. Waissman, J. et al. Realization of pristine and locally tunable one-dimensional electron systems in carbon nanotubes. Nat. Nanotechnol. 8, 569–574 (2013).

    Article  CAS  Google Scholar 

  34. Ben Shalom, M. et al. Quantum oscillations of the critical current and high-field superconducting proximity in ballistic graphene. Nat. Phys. 12, 318–322 (2016).

    Article  Google Scholar 

  35. Levinson, I. B. Potential distribution in a quantum point contact. Sov. Phys. JETP 68, 1257–1265 (1989).

    Google Scholar 

  36. Xia, F., Perebeinos, V., Lin, Y. M., Wu, Y. & Avouris, P. The origins and limits of metal-graphene junction resistance. Nat. Nanotechnol. 6, 179–184 (2011).

    Article  CAS  Google Scholar 

  37. Yu, Y. J. et al. Tuning the graphene work function by electric field effect. Nano Lett. 9, 3430–3434 (2009).

    Article  CAS  Google Scholar 

  38. Morikawa, S. et al. Imaging ballistic carrier trajectories in graphene using scanning gate microscopy. Appl. Phys. Lett. 107, 243102 (2015).

    Article  Google Scholar 

  39. Bhandari, S. et al. Imaging electron flow from collimating contacts in graphene. 2D Mater. 5, 021003 (2018).

    Article  Google Scholar 

  40. Tarucha, S., Saku, T., Tokura, Y. & Hirayama, Y. Sharvin resistance and its breakdown observed in long ballistic channels. Phys. Rev. B 47, 4064–4067 (1993).

    Article  CAS  Google Scholar 

  41. Falkovich, G. & Levitov, L. Linking spatial distributions of potential and current in viscous electronics. Phys. Rev. Lett. 119, 066601 (2017).

    Article  Google Scholar 

Download references


We thank G. Falkovich, L. Levitov, A. Shytov and A. Stern for discussions and D. Mahalu for electron-beam lithography. We further acknowledge support from the Helmsley Charitable Trust grant, the ISF (grant no. 712539), WIS-UK collaboration grant and the ERC-Cog (See-1D-Qmatter, no. 647413).

Author information

Authors and Affiliations



L.E., S.I. and J.A.S. conceived the technique. L.E., A.R., S.I. and J.A.S. created the SETs, performed the measurements and analysed the data. J.B., D.P., J.Z. and M.B.-S. fabricated the graphene devices. K.W. and T.T. supplied the hBN crystals. L.E., S.I. and J.A.S. wrote the manuscript with input from all authors.

Corresponding author

Correspondence to Joseph A. Sulpizio.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Journal peer review informationNature Nanotechnology thanks Klaus Ensslin and other anonymous reviewer(s) for their contribution to the peer review of this work.

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

Supplementary information

Supplementary Information for

“Simultaneous voltage and current density imaging of flowing electrons in two

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ella, L., Rozen, A., Birkbeck, J. et al. Simultaneous voltage and current density imaging of flowing electrons in two dimensions. Nat. Nanotechnol. 14, 480–487 (2019).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

This article is cited by


Quick links

Find nanotechnology articles, nanomaterial data and patents all in one place. Visit Nano by Nature Research