Individual electrons in graphene behave as massless quasiparticles1,2,3,4,5,6,7,8. Unexpectedly, it is inferred from plasmonic investigations9,10,11,12 that electrons in graphene must exhibit a non-zero mass when collectively excited. The inertial acceleration of the electron collective mass is essential to explain the behaviour of plasmons in this material, and may be directly measured by accelerating it with a time-varying voltage and quantifying the phase delay of the resulting current. This voltage–current phase relation would manifest as a kinetic inductance, representing the reluctance of the collective mass to accelerate. However, at optical (infrared) frequencies, phase measurements of current are generally difficult, and, at microwave frequencies, the inertial phase delay has been buried under electron scattering13,14,15. Therefore, to date, the collective mass in graphene has defied unequivocal measurement. Here, we directly and precisely measure the kinetic inductance, and therefore the collective mass, by combining device engineering that reduces electron scattering and sensitive microwave phase measurements. Specifically, the encapsulation of graphene between hexagonal boron nitride layers16, one-dimensional edge contacts17 and a proximate top gate configured as microwave ground18,19 together enable the inertial phase delay to be resolved from the electron scattering. Beside its fundamental importance, the kinetic inductance is found to be orders of magnitude larger than the magnetic inductance, which may be utilized to miniaturize radiofrequency integrated circuits. Moreover, its bias dependency heralds a solid-state voltage-controlled inductor to complement the prevalent voltage-controlled capacitor.
Subscribe to Journal
Get full journal access for 1 year
only $15.58 per issue
All prices are NET prices.
VAT will be added later in the checkout.
Rent or Buy article
Get time limited or full article access on ReadCube.
All prices are NET prices.
Novoselov, K. S. et al. Two-dimensional gas of massless Dirac fermions in graphene. Nature 438, 197–200 (2005).
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).
Geim, A. K. & Novoselov, K. S. The rise of graphene. Nature Mater. 6, 183–191 (2007).
Castro Neto, A. H., Guinea, F., Peres, N. M. R., Novoselov, K. S. & Geim, A. K. The electronic properties of graphene. Rev. Mod. Phys. 81, 109–162 (2009).
Das Sarma, S., Adam, S., Hwang, E. H. & Rossi, E. Electronic transport in two-dimensional graphene. Rev. Mod. Phys. 83, 407–470 (2011).
Koppens, F. H. L., Chang, D. E. & García de Abajo, F. J. Graphene plasmonics: a platform for strong light–matter interactions. Nano Lett. 11, 3370–3377 (2011).
Chen, J. et al. Optical nano-imaging of gate-tunable graphene plasmons. Nature 487, 77–81 (2012).
Yeung, K. Y. M. et al. Far-infrared graphene plasmonic crystals for plasmonic band engineering. Nano Lett. 14, 2479–2484 (2014).
Hwang, E. H. & Das Sarma, S. Dielectric function, screening, and plasmons in two-dimensional graphene. Phys. Rev. B 75, 205418 (2007).
Grigorenko, A. N., Polini, M. & Novoselov, K. S. Graphene plasmonics. Nature Photon. 6, 749–758 (2012).
Ju, L. et al. Graphene plasmonics for tunable terahertz metamaterials. Nature Nanotech. 6, 630–634 (2011).
Yan, H. et al. Tunable infrared plasmonic devices using graphene/insulator stacks. Nature Nanotech. 7, 330–334 (2012).
Deligeorgis, G. et al. Microwave propagation in graphene. Appl. Phys. Lett. 95, 073107 (2009).
Lee, H-J., Kim, E., Yook, J-G. & Jung, J. Intrinsic characteristics of transmission line of graphenes at microwave frequencies. Appl. Phys. Lett. 100, 223102 (2012).
Jeon, D-Y. et al. Radio-frequency electrical characteristics of single layer graphene. Jpn. J. Appl. Phys. 48, 091601 (2009).
Dean, C. R. et al. Boron nitride substrates for high-quality graphene electronics. Nature Nanotech. 5, 722–726 (2010).
Wang, L. et al. One-dimensional electrical contact to a two-dimensional material. Science 342, 614–617 (2013).
Andress, W. F. et al. Ultra-subwavelength two-dimensional plasmonic circuits. Nano Lett. 12, 2272–2277 (2012).
Yeung, K. Y. M. et al. Two-path solid-state interferometry using ultra-subwavelength two-dimensional plasmonic waves. Appl. Phys. Lett. 102, 021104 (2013).
Abedinpour, S. H. et al. Drude weight, plasmon dispersion, and ac conductivity in doped graphene sheets. Phys. Rev. B 84, 045429 (2011).
Burke, P. J., Spielman, I. B., Eisenstein, J. P., Pfeiffer, L. N. & West, K. W. High frequency conductivity of the high-mobility two-dimensional electron gas. Appl. Phys. Lett. 76, 745–747 (2000).
Rana, F. Graphene terahertz plasmon oscillators. IEEE Trans. Nanotechnol. 7, 91–99 (2008).
Yoon, H., Yeung, K. Y. M., Umansky, V. & Ham, D. A Newtonian approach to extraordinarily strong negative refraction. Nature 488, 65–69 (2012).
Xia, J., Chen, F., Li, J. & Tao, N. Measurement of the quantum capacitance of graphene. Nature Nanotech. 4, 505–509 (2009).
Chauhan, J. & Guo, J. Assessment of high-frequency performance limits of graphene field-effect transistors. Nano Res. 4, 571–579 (2011).
Yoon, H., Yeung, K. Y. M., Kim, P. & Ham, D. Plasmonics with two-dimensional conductors. Phil. Trans. R. Soc. Lond. A 372, 20130104 (2014).
Jang, C. et al. Tuning the effective fine structure constant in graphene: opposing effects of dielectric screening on short- and long-range potential scattering. Phys. Rev. Lett. 101, 146805 (2008).
Hwang, C. et al. Fermi velocity engineering in graphene by substrate modification. Sci. Rep. 2, 590 (2012).
Elias, D. C. et al. Dirac cones reshaped by interaction effects in suspended graphene. Nature Phys. 7, 701–704 (2011).
Marks, R. B. A multiline method of network analyzer calibration. IEEE Trans. Microw. Theory Tech. 39, 1205–1215 (1991).
Ohba, N., Miwa, K., Nagasako, N. & Fukumoto, A. First-principles study on structural, dielectric, and dynamical properties for three BN polytypes. Phys. Rev. B 63, 115207 (2001).
Maex, K. et al. Low dielectric constant materials for microelectronics. J. Appl. Phys. 93, 8793–8841 (2003).
D.H. and H.Y. acknowledge support from the Air Force Office of Scientific Research (contract no. FA9550-13-1-0211), from the Office of Naval Research (contract no. N00014-13-1-0806), from the National Science Foundation (NSF; contract no. DMR-1231319), from the Samsung Advanced Institute of Technology and its Global Research Opportunity programme (contract no. A18960). P.K. acknowledges support from the Nano Material Technology Development Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (2012M3A7B4049966). J.H. and L.W. acknowledge support from the NSF (contract no. DMR-1124894) and the Office of Naval Research (award no. N000141310662). C.F. acknowledges support of the Columbia Optics and Quantum Electronics IGERT under NSF grant DGE-1069420. N.T. acknowledges support from the Netherlands Organisation for Scientific Research Device fabrication was performed in part at the Center for Nanoscale Systems at Harvard University.
The authors declare no competing financial interests.
About this article
Cite this article
Yoon, H., Forsythe, C., Wang, L. et al. Measurement of collective dynamical mass of Dirac fermions in graphene. Nature Nanotech 9, 594–599 (2014). https://doi.org/10.1038/nnano.2014.112
ACS Nano (2019)
Journal of Physics Communications (2019)
Graphene Antidot Terahertz Plasmonic Metasurfaces Employing Self-Aligned Metal Cores for Sensing Applications
ACS Applied Nano Materials (2019)
ACS Applied Electronic Materials (2019)
Fullerenes, Nanotubes and Carbon Nanostructures (2019)