- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Universal low-density power laws of the dc conductivity and Hall constant in the self-consistent Born approximation

Giacomo Morpurgo, Christophe Berthod, and Thierry Giamarchi

PDF

Phys. Rev. Research 7, 033038 – Published 8 July, 2025

DOI: https://doi.org/10.1103/nzrk-yfqk

Abstract

The dc conductivity tensor of two-dimensional one-band metals with weak pointlike disorder and magnetic field is studied in the self-consistent Born approximation, with special emphasis on the regime of low carrier density. In this theory, the Kubo conductivity is a functional of the electron dispersion and local (momentum-independent) electron self-energy, which is itself a causal functional of the dispersion and disorder strength. We obtain closed expressions for the asymptotic low-density conductivities at zero temperature in the form of power laws of the density and disorder strength with universal exponents. The crossover to the semiclassical regime of high density is studied numerically, as well as the temperature dependence. Our model and results may be relevant to interpret linear magnetotransport experiments performed in the metallic regime of gated two-dimensional semiconductors.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (56)

  1. T. Ando, A. B. Fowler, and F. Stern, Electronic properties of two-dimensional systems, Rev. Mod. Phys. 54, 437 (1982).
  2. A. Tsumura, H. Koezuka, and T. Ando, Macromolecular electronic device: Field-effect transistor with a polythiophene thin film, Appl. Phys. Lett. 49, 1210 (1986).
  3. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Electric field effect in atomically thin carbon films, Science 306, 666 (2004).
  4. S. Manzeli, D. Ovchinnikov, D. Pasquier, O. V. Yazyev, and A. Kis, 2D transition metal dichalcogenides, Nat. Rev. Mater. 2, 17033 (2017).
  5. Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras, and P. Jarillo-Herrero, Unconventional superconductivity in magic-angle graphene superlattices, Nature (London) 556, 43 (2018).
  6. E. Y. Andrei, D. K. Efetov, P. Jarillo-Herrero, A. H. MacDonald, K. F. Mak, T. Senthil, E. Tutuc, A. Yazdani, and A. F. Young, The marvels of moiré materials, Nat. Rev. Mater. 6, 201 (2021).
  7. T. Kopp and J. Mannhart, Calculation of the capacitances of conductors: Perspectives for the optimization of electronic devices, J. Appl. Phys. 106, 064504 (2009).
  8. J. V. Li, A. M. Nardes, Z. Liang, S. E. Shaheen, B. A. Gregg, and D. H. Levi, Simultaneous measurement of carrier density and mobility of organic semiconductors using capacitance techniques, Org. Electron. 12, 1879 (2011).
  9. T. Fukuzawa, M. Koshino, and T. Ando, Weak-field Hall effect in graphene calculated within self-consistent Born approximation, J. Phys. Soc. Jpn. 78, 094714 (2009).
  10. M. Noro and T. Ando, Weak-field Hall effect in graphene with long-range scatterers, J. Phys. Soc. Jpn. 85, 014708 (2016).
  11. G. Morpurgo, L. Rademaker, C. Berthod, and T. Giamarchi, Hall response of locally correlated two-dimensional electrons at low density, Phys. Rev. Res. 6, 013112 (2024).
  12. H. Zhang, C. Berthod, H. Berger, T. Giamarchi, and A. F. Morpurgo, Band filling and cross quantum capacitance in ion-gated semiconducting transition metal dichalcogenide monolayers, Nano Lett. 19, 8836 (2019).
  13. C. Berthod, H. Zhang, A. F. Morpurgo, and T. Giamarchi, Theory of cross quantum capacitance, Phys. Rev. Res. 3, 043036 (2021).
  14. H. H. Choi, Y. I. Rodionov, A. F. Paterson, J. Panidi, D. Saranin, N. Kharlamov, S. I. Didenko, T. D. Anthopoulos, K. Cho, and V. Podzorov, Accurate extraction of charge carrier mobility in 4-probe field-effect transistors, Adv. Funct. Mater. 28, 1707105 (2018).
  15. J. Ziman, Electrons and Phonons: The Theory of Transport Phenomena in Solids (Oxford University Press, Oxford, 2001).
  16. H. Fukuyama, H. Ebisawa, and Y. Wada, Theory of Hall effect. I: Nearly free electron, Prog. Theor. Phys. 42, 494 (1969).
  17. H. Fukuyama, Theory of Hall effect. II: Bloch electrons, Prog. Theor. Phys. 42, 1284 (1969).
  18. M. Itoh, The Hall effect in the ladder approximation, J. Phys.: Condens. Matter 2, 5357 (1990).
  19. M. Khodas and A. M. Finkel'stein, Hall coefficient in an interacting electron gas, Phys. Rev. B 68, 155114 (2003).
  20. J. Kovačević, M. Ferrero, and J. Vučičević, Towards numerically exact computation of conductivity in the thermodynamic limit of interacting lattice models, arXiv:2501.19118 [Phys. Rev. Lett. (to be published)].
  21. P. Prelovšek and X. Zotos, Reactive Hall constant of strongly correlated electrons, Phys. Rev. B 64, 235114 (2001).
  22. A. Auerbach, Hall number of strongly correlated metals, Phys. Rev. Lett. 121, 066601 (2018).
  23. H. E. Castillo and C. A. Balseiro, Hall conductivity and Fermi surface in highly correlated systems, Phys. Rev. Lett. 68, 121 (1992).
  24. F. F. Assaad and M. Imada, Hall coefficient for the two-dimensional Hubbard model, Phys. Rev. Lett. 74, 3868 (1995).
  25. W. O. Wang, J. K. Ding, B. Moritz, E. W. Huang, and T. P. Devereaux, DC Hall coefficient of the strongly correlated Hubbard model, npj Quantum Mater. 5, 51 (2020).
  26. T. Pruschke, M. Jarrell, and J. Freericks, Anomalous normal-state properties of high-Tc superconductors: intrinsic properties of strongly correlated electron systems? Adv. Phys. 44, 187 (1995).
  27. A. A. Markov, G. Rohringer, and A. N. Rubtsov, Robustness of the topological quantization of the Hall conductivity for correlated lattice electrons at finite temperatures, Phys. Rev. B 100, 115102 (2019).
  28. J. Vučičević and R. Žitko, Electrical conductivity in the Hubbard model: Orbital effects of magnetic field, Phys. Rev. B 104, 205101 (2021).
  29. S. M. Manning and Y. Chen, The Hall effect in the normal state of the t–J model, J. Phys.: Condens. Matter 5, L23 (1993).
  30. H. Kontani, K. Kanki, and K. Ueda, Hall effect and resistivity in high-Tc superconductors: The conserving approximation, Phys. Rev. B 59, 14723 (1999).
  31. H. Kontani, Theory of infrared Hall conductivity based on the Fermi liquid theory: Analysis of high-Tc superconductors, J. Phys. Soc. Jpn. 76, 074707 (2007).
  32. J. Mitscherling and W. Metzner, Longitudinal conductivity and Hall coefficient in two-dimensional metals with spiral magnetic order, Phys. Rev. B 98, 195126 (2018).
  33. G. León and T. Giamarchi, Hall effect in quasi one-dimensional organic conductors, J. Phys. F 142, 315 (2006).
  34. G. León, C. Berthod, and T. Giamarchi, Hall effect in strongly correlated low-dimensional systems, Phys. Rev. B 75, 195123 (2007).
  35. G. León, C. Berthod, T. Giamarchi, and A. J. Millis, Hall effect on the triangular lattice, Phys. Rev. B 78, 085105 (2008).
  36. M. Filippone, C.-E. Bardyn, S. Greschner, and T. Giamarchi, Vanishing Hall response of charged fermions in a transverse magnetic field, Phys. Rev. Lett. 123, 086803 (2019).
  37. S. Greschner, M. Filippone, and T. Giamarchi, Universal Hall response in interacting quantum systems, Phys. Rev. Lett. 122, 083402 (2019).
  38. M. Buser, S. Greschner, U. Schollwöck, and T. Giamarchi, Probing the Hall voltage in synthetic quantum systems, Phys. Rev. Lett. 126, 030501 (2021).
  39. A. Khurana, Electrical conductivity in the infinite-dimensional Hubbard model, Phys. Rev. Lett. 64, 1990 (1990).
  40. S. Liu, Nonlocal conductivities and dielectric functions of free electron gas with impurity scattering, Ann. Phys. (NY) 59, 165 (1970).
  41. H. Fukuyama, Hall effect in two-dimensional disordered systems, J. Phys. Soc. Jpn. 49, 644 (1980).
  42. T. Ando, Bilayer graphene with long-range scatterers: Diamagnetism and weak-field Hall effect, Physica E 58, 6 (2014).
  43. M. Pickem, E. Maggio, and J. M. Tomczak, Prototypical many-body signatures in transport properties of semiconductors, Phys. Rev. B 105, 085139 (2022).
  44. H. Bruus and K. Flensberg, Many-Body Quantum Theory in Condensed Matter Physics: An Introduction (Oxford University Press, Oxford, 2016).
  45. C. Berthod, Spectroscopic Probes of Quantum Matter (IOP, Bristol, UK, 2018).
  46. C. Berthod, Piecewise: Flexible piecewise functions for fast integral transforms in Julia (2025), https://github.com/ChristopheBerthod/Piecewise.jl.
  47. C. Berthod, Linear magneto-transport with a local self-energy in Julia (2025), https://github.com/ChristopheBerthod/MagnetoTransport.jl.
  48. As −ImΣ(ɛ) must remain strictly positive for causality reasons, an infinitesimal shift i0 must in principle be inserted into Eq. (11), such that ImΣSCBA(ɛ) is never strictly zero.
  49. M. Yamagishi, J. Soeda, T. Uemura, Y. Okada, Y. Takatsuki, T. Nishikawa, Y. Nakazawa, I. Doi, K. Takimiya, and J. Takeya, Free-electron-like Hall effect in high-mobility organic thin-film transistors, Phys. Rev. B 81, 161306 (2010).
  50. T. Uemura, M. Yamagishi, J. Soeda, Y. Takatsuki, Y. Okada, Y. Nakazawa, and J. Takeya, Temperature dependence of the Hall effect in pentacene field-effect transistors: Possibility of charge decoherence induced by molecular fluctuations, Phys. Rev. B 85, 035313 (2012).
  51. N. R. Pradhan, D. Rhodes, S. Memaran, J. M. Poumirol, D. Smirnov, S. Talapatra, S. Feng, N. Perea-Lopez, A. L. Elias, M. Terrones, P. M. Ajayan, and L. Balicas, Hall and field-effect mobilities in few layered p−WSe2 field-effect transistors, Sci. Rep. UK 5, 8979 (2015).
  52. M. Masseroni, M. Gull, A. Panigrahi, N. Jacobsen, F. Fischer, C. Tong, J. D. Gerber, M. Niese, T. Taniguchi, K. Watanabe, L. Levitov, T. Ihn, K. Ensslin, and H. Duprez, Spin-orbit proximity in MoS2/bilayer graphene heterostructures, Nat. Commun. 15, 9251 (2024).
  53. I. Gutiérrez-Lezama, N. Ubrig, E. Ponomarev, and A. F. Morpurgo, Ionic gate spectroscopy of 2D semiconductors, Nat. Rev. Phys. 3, 508 (2021).
  54. P. W. Anderson, Absence of diffusion in certain random lattices, Phys. Rev. 109, 1492 (1958).
  55. P. A. Lee and T. V. Ramakrishnan, Disordered electronic systems, Rev. Mod. Phys. 57, 287 (1985).
  56. G. Morpurgo, C. Berthod, and T. Giamarchi, Open data to “Universal low-density power laws of the dc conductivity and Hall constant in the self-consistent Born approximation”, Yareta, University of Geneva (2025), 10.26037/yareta:gl44nnfso5enhohdbfruvfnvge.

Outline

Information

Phys. Rev. Research 7, 033038– Published 8 July, 2025
  • Received 7 April 2025
  • Accepted 5 June 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation