Interacting electrons confined in one dimension are generally described by the Luttinger liquid formalism, where the low-energy electronic dispersion is assumed to be linear and the resulting plasmonic excitations are non-interacting. Instead, a Luttinger liquid in one-dimensional materials with nonlinear electronic bands is expected to show strong plasmon–plasmon interactions, but an experimental demonstration of this behaviour has been lacking. Here, we combine infrared nano-imaging and electronic transport to investigate the behaviour of plasmonic excitations in semiconducting single-walled carbon nanotubes with carrier density controlled by electrostatic gating. We show that both the propagation velocity and the dynamic damping of plasmons can be tuned continuously, which is well captured by the nonlinear Luttinger liquid theory. These results contrast with the gate-independent plasmons observed in metallic nanotubes, as expected for a linear Luttinger liquid. Our findings provide an experimental demonstration of one-dimensional electron dynamics beyond the conventional linear Luttinger liquid paradigm and are important for understanding excited-state properties in one dimension.
Subscribe to Journal
Get full journal access for 1 year
only $16.58 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.
The numerical data represented in Fig. 4b–d are provided with the paper as source data. All other data that support results in this Article are available from the corresponding author upon reasonable request.
Matlab codes for nonlinear theory calculation are available from the corresponding author upon reasonable request.
Tomonaga, S. Remarks on Bloch’s method of sound waves applied to many-fermion problems. Prog. Theor. Phys. 5, 544–569 (1950).
Luttinger, J. M. An exactly soluble model of a many-fermion system. J. Math. Phys. 4, 1154–1162 (1963).
Haldane, F. D. M. ‘Luttinger liquid theory’ of one-dimensional quantum fluids. I. Properties of the Luttinger model and their extension to the general 1D interacting spinless Fermi gas. J. Phys. C Solid State 14, 2585–2609 (1981).
Giamarchi, T. Quantum Physics in One Dimension Ch. 3 (Oxford Univ. Press, 2004).
Giuliani, G. & Vignale, G. Quantum Theory of the Electron Liquid Ch. 9 (Cambridge Univ. Press, 2005).
Deshpande, V. V., Bockrath, M., Glazman, L. I. & Yacoby, A. Electron liquids and solids in one dimension. Nature 464, 209–216 (2010).
Voit, J. One-dimensional Fermi liquids. Rep. Prog. Phys. 58, 977–1116 (1995).
Bockrath, M. et al. Luttinger-liquid behaviour in carbon nanotubes. Nature 397, 598–601 (1999).
Yao, Z., Postma, H. W. C., Balents, L. & Dekker, C. Carbon nanotube intramolecular junctions. Nature 402, 273–276 (1999).
Ishii, H. et al. Direct observation of Tomonaga–Luttinger-liquid state in carbon nanotubes at low temperatures. Nature 426, 540–544 (2003).
Jompol, Y. et al. Probing spin-charge separation in a Tomonaga-Luttinger liquid. Science 325, 597–601 (2009).
Zhao, S. et al. Correlation of electron tunneling and plasmon propagation in a Luttinger liquid. Phys. Rev. Lett. 121, 047702 (2018).
Kane, C., Balents, L. & Fisher, M. P. A. Coulomb interactions and mesoscopic effects in carbon nanotubes. Phys. Rev. Lett. 79, 5086–5089 (1997).
Egger, R. & Gogolin, A. O. Effective low-energy theory for correlated carbon nanotubes. Phys. Rev. Lett. 79, 5082–5085 (1997).
Williams, P. F. & Bloch, A. N. Self-consistent dielectric response of a quasi-one-dimensional metal at high frequencies. Phys. Rev. B 10, 1097–1108 (1974).
Pustilnik, M., Khodas, M., Kamenev, A. & Glazman, L. I. Dynamic response of one-dimensional interacting fermions. Phys. Rev. Lett. 96, 196405 (2006).
Imambekov, A. & Glazman, L. I. Universal theory of nonlinear Luttinger liquids. Science 323, 228–231 (2009).
Imambekov, A. & Glazman, L. I. Phenomenology of one-dimensional quantum liquids beyond the low-energy limit. Phys. Rev. Lett. 102, 126405 (2009).
Barak, G. et al. Interacting electrons in one dimension beyond the Luttinger-liquid limit. Nat. Phys. 6, 489–493 (2010).
Schmidt, T. L., Imambekov, A. & Glazman, L. I. Spin-charge separation in one-dimensional fermion systems beyond Luttinger liquid theory. Phys. Rev. B 82, 245104 (2010).
Imambekov, A., Schmidt, T. L. & Glazman, L. I. One-dimensional quantum liquids: beyond the Luttinger liquid paradigm. Rev. Mod. Phys. 84, 1253–1306 (2012).
Jin, Y. et al. Momentum-dependent power law measured in an interacting quantum wire beyond the Luttinger limit. Nat. Commun. 10, 2821 (2019).
Wang, S. et al. Logarithm diameter scaling and carrier density independence of one-dimensional Luttinger liquid plasmon. Nano Lett. 19, 2360–2365 (2019).
Shi, Z. et al. Observation of a Luttinger-liquid plasmon in metallic single-walled carbon nanotubes. Nat. Photonics 9, 515–519 (2015).
Liu, M., Sternbach, A. J. & Basov, D. N. Nanoscale electrodynamics of strongly correlated quantum materials. Rep. Prog. Phys. 80, 014501 (2016).
Saito, R., Dresselhaus, G. & Dresselhaus, M. S. Physical Properties of Carbon Nanotubes Ch. 4 (Imperial College Press, 1998).
Wang, S. et al. Nonlinear Luttinger liquid plasmons in semiconducting single walled carbon nanotubes. Figshare https://doi.org/10.6084/m9.figshare.11900307.v2 (2020).
das Sarma, S. & Hwang, E. H. Dynamical response of a one-dimensional quantum-wire electron system. Phys. Rev. B 54, 1936–1946 (1996).
Jorio, A., Dresselhaus, G., & Dresselhaus, M. S. Carbon Nanotubes: Advanced Topics in the Synthesis, Structure, Properties and Applications Ch. 15 (Springer, 2008).
Zhou, X., Park, J.-Y., Huang, S., Liu, J. & McEuen, P. L. Band structure, phonon scattering, and the performance limit of single-walled carbon nanotube transistors. Phys. Rev. Lett. 95, 146805 (2005).
Purewal, M. S. et al. Scaling of resistance and electron mean free path of single-walled carbon nanotubes. Phys. Rev. Lett. 98, 186808 (2007).
He, X. et al. Carbon nanotubes as emerging quantum-light sources. Nat. Mater. 17, 663–670 (2018).
Ozbay, E. Plasmonics: merging photonics and electronics at nanoscale dimensions. Science 311, 189–193 (2006).
We thank N. Yao, R. Vasseur, J. Kang and H. B. Balch for helpful discussions. This work was mainly supported by the Director, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division of the US Department of Energy under contract no. DE-AC02-05-CH11231 (sp2-Bonded Materials Program KC2207). The device fabrication and electrical measurement were supported by the Office of Naval Research (MURI award N00014-16-1-2921). The data analysis was supported by the NSF award 1808635. Z.S. acknowledges support from the National Natural Science Foundation of China (11774224 and 11574204). F. Wu, Z.Z. and C.Z. acknowledge the National Science Foundation for financial support under grant no. 769K521. K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, Japan and the CREST (JPMJCR15F3), JST.
The authors declare no competing interests.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Wang, S., Zhao, S., Shi, Z. et al. Nonlinear Luttinger liquid plasmons in semiconducting single-walled carbon nanotubes. Nat. Mater. (2020). https://doi.org/10.1038/s41563-020-0652-5
Nature Materials (2020)