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Robust strong-field theory model for ultrafast electron transport through metal-insulator-metal tunneling nanojunctions

Boyang Ma and Michael Krüger*

  • *Contact author: krueger@technion.ac.il
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Phys. Rev. A 112, 033104 – Published 11 September, 2025

DOI: https://doi.org/10.1103/h2zq-7l2n

Abstract

Ultrafast science studies the dynamics of electrons in matter with extreme temporal precision, typically in the attosecond and femtosecond time domain. Recent experimental and theoretical progress has put metal-insulator-metal (MIM) tunneling nanojunctions in the spotlight of ultrafast science. Waveform-controlled laser fields can induce ultrafast currents in these junctions, opening the door to petahertz electronic operation and attosecond-scale scanning tunneling microscopy. Inspired by our strong-field model for attosecond tunneling microscopy [B. Ma and M. Krüger, Phys. Rev. Lett. 133, 236901 (2024)], here we generalize our model to MIM nanojunctions. We introduce several refinements and corrections, accounting for the image potential inside the gap and boundary effects. Moreover, we also find that the Keldysh parameter, which is a hallmark parameter for ultrafast light-matter interactions, alone is insufficient for describing the physics in thin MIM nanojunctions. We introduce a new parameter ζ that accounts for the effects of the limited size of the junction and provide several interpretations of the parameter that shed new light on the complex physics in the light-driven junction. Most strikingly, we find that for ζ>1 photon-assisted tunneling is dominating the ultrafast electron transport across the junction, regardless of the value of the Keldysh parameter. Here one or more photons are absorbed and the electron undergoes static tunneling through the barrier. ζ<1 is required for a regime in which the laser field and the Keldysh parameter dominate the transport. We also discuss the three-step model of electron transport across the nanojunction in the adiabatic regime, the energy cutoff of the transported electrons, and the carrier-envelope phase control of the net current. Our theory model provides a rich toolbox for understanding and predicting the physics of ultrafast MIM nanojunctions.

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References (78)

  1. F. Krausz and M. Ivanov, Attosecond physics, Rev. Mod. Phys. 81, 163 (2009).
  2. P. B. Corkum and F. Krausz, Attosecond science, Nat. Phys. 3, 381 (2007).
  3. F. Lépine, M. Y. Ivanov, and M. J. J. Vrakking, Attosecond molecular dynamics: Fact or fiction? Nat. Photonics 8, 195 (2014).
  4. A. Palacios and F. Martín, The quantum chemistry of attosecond molecular science, WIREs Comput. Mol. Sci. 10, e1430 (2020).
  5. M. Nisoli, P. Decleva, F. Calegari, A. Palacios, and F. Martín, Attosecond electron dynamics in molecules, Chem. Rev. 117, 10760 (2017).
  6. A. L. Cavalieri, N. Müller, T. Uphues, V. S. Yakovlev, A. Baltŭska, B. Horvath, B. Schmidt, L. Blümel, R. Holzwarth, S. Hendel, M. Drescher, U. Kleineberg, P. M. Echenique, R. Kienberger, F. Krausz, and U. Heinzmann, Attosecond spectroscopy in condensed matter, Nature (London) 449, 1029 (2007).
  7. G. Vampa, C. R. McDonald, G. Orlando, P. B. Corkum, and T. Brabec, Semiclassical analysis of high harmonic generation in bulk crystals, Phys. Rev. B 91, 064302 (2015).
  8. S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, Observation of high-order harmonic generation in a bulk crystal, Nat. Phys. 7, 138 (2011).
  9. P. B. Corkum, Plasma perspective on strong field multiphoton ionization, Phys. Rev. Lett. 71, 1994 (1993).
  10. G. Vampa, C. R. McDonald, G. Orlando, D. D. Klug, P. B. Corkum, and T. Brabec, Theoretical analysis of high-harmonic generation in solids, Phys. Rev. Lett. 113, 073901 (2014).
  11. R. Weissenbilder, S. Carlström, L. Rego, C. Guo, C. M. Heyl, P. Smorenburg, E. Constant, C. L. Arnold, and A. L'Huillier, How to optimize high-order harmonic generation in gases, Nat. Rev. Phys. 4, 713 (2022).
  12. F. Krausz and M. I. Stockman, Attosecond metrology: From electron capture to future signal processing, Nat. Photonics 8, 205 (2014).
  13. A. Feist, K. E. Echternkamp, J. Schauss, S. V. Yalunin, S. Schäfer, and C. Ropers, Quantum coherent optical phase modulation in an ultrafast transmission electron microscope, Nature (London) 521, 200 (2015).
  14. D. Nabben, J. Kuttruff, L. Stolz, A. Ryabov, and P. Baum, Attosecond electron microscopy of sub-cycle optical dynamics, Nature (London) 619, 63 (2023).
  15. M. Eldar, S. Abo-Toame, and M. Krüger, Sub-optical-cycle electron pulse trains from metal nanotips, J. Phys. B: At. Mol. Opt. Phys. 55, 074001 (2022).
  16. I. Giaever and K. Megerle, Study of superconductors by electron tunneling, Phys. Rev. 122, 1101 (1961).
  17. G. Binnig, H. Rohrer, C. Gerber, and E. Weibel, Surface studies by scanning tunneling microscopy, Phys. Rev. Lett. 49, 57 (1982).
  18. C. J. Chen, Introduction to Scanning Tunneling Microscopy (Oxford University Press, Oxford, UK, 2021).
  19. T. A. Jung, R. R. Schlittler, and J. K. Gimzewski, Conformational identification of individual adsorbed molecules with the STM, Nature (London) 386, 696 (1997).
  20. O. Marti and M. Amrein, STM and SFM in Biology (Academic Press, New York, 2012).
  21. T. Rybka, M. Ludwig, M. F. Schmalz, V. Knittel, D. Brida, and A. Leitenstorfer, Sub-cycle optical phase control of nanotunnelling in the single-electron regime, Nat. Photonics 10, 667 (2016).
  22. M. Ludwig, G. Aguirregabiria, F. Ritzkowsky, T. Rybka, D. C. Marinica, J. Aizpurua, A. G. Borisov, A. Leitenstorfer, and D. Brida, Sub-femtosecond electron transport in a nanoscale gap, Nat. Phys. 16, 341 (2020).
  23. M. R. Bionta, F. Ritzkowsky, M. Turchetti, Y. Yang, D. C. Mor, W. P. Putnam, F. X. Kärtner, K. K. Berggren, and P. D. Keathley, On-chip sampling of optical fields with attosecond resolution, Nat. Photonics 15, 456 (2021).
  24. M. Garg and K. Kern, Attosecond coherent manipulation of electrons in tunneling microscopy, Science 367, 411 (2020).
  25. J. Schötz, Z. Wang, E. Pisanty, M. Lewenstein, M. F. Kling, and M. F. Ciappina, Perspective on petahertz electronics and attosecond nanoscopy, ACS Photonics 6, 3057 (2019).
  26. M. F. Ciappina, J. A. Pérez-Hernández, A. S. Landsman, W. A. Okell, S. Zherebtsov, B. Förg, J. Schötz, L. Seiffert, T. Fennel, T. Shaaran, T. Zimmermann, A. Chacón, R. Guichard, A. Zaïr, J. W. G. Tisch, J. P. Marangos, T. Witting, A. Braun, S. A. Maier, L. Roso et al., Attosecond physics at the nanoscale, Rep. Prog. Phys. 80, 054401 (2017).
  27. P. Dombi, Z. Pápa, J. Vogelsang, S. V. Yalunin, M. Sivis, G. Herink, S. Schäfer, P. Groß, C. Ropers, and C. Lienau, Strong-field nano-optics, Rev. Mod. Phys. 92, 025003 (2020).
  28. D. Ehberger, J. Hammer, M. Eisele, M. Krüger, J. Noe, A. Högele, and P. Hommelhoff, Highly coherent electron beam from a laser-triggered tungsten needle tip, Phys. Rev. Lett. 114, 227601 (2015).
  29. S. Meier, T. Higuchi, M. Nutz, A. Högele, and P. Hommelhoff, High spatial coherence in multiphoton-photoemitted electron beams, Appl. Phys. Lett. 113, 143101 (2018).
  30. T. L. Cocker, V. Jelic, M. Gupta, S. J. Molesky, J. A. J. Burgess, G. D. L. Reyes, L. V. Titova, Y. Y. Tsui, M. R. Freeman, and F. A. Hegmann, An ultrafast terahertz scanning tunnelling microscope, Nat. Photonics 7, 620 (2013).
  31. T. L. Cocker, D. Peller, P. Yu, J. Repp, and R. Huber, Tracking the ultrafast motion of a single molecule by femtosecond orbital imaging, Nature (London) 539, 263 (2016).
  32. V. Jelic, K. Iwaszczuk, P. H. Nguyen, C. Rathje, G. J. Hornig, H. M. Sharum, J. R. Hoffman, M. R. Freeman, and F. A. Hegmann, Ultrafast terahertz control of extreme tunnel currents through single atoms on a silicon surface, Nat. Phys. 13, 591 (2017).
  33. K. Yoshioka, I. Katayama, Y. Minami, M. Kitajima, S. Yoshida, H. Shigekawa, and J. Takeda, Real-space coherent manipulation of electrons in a single tunnel junction by single-cycle terahertz electric fields, Nat. Photonics 10, 762 (2016).
  34. M. Garg, A. Martin-Jimenez, M. Pisarra, Y. Luo, F. Martín, and K. Kern, Real-space subfemtosecond imaging of quantum electronic coherences in molecules, Nat. Photonics 16, 196 (2022).
  35. W. Zhu, R. Esteban, A. G. Borisov, J. J. Baumberg, P. Nordlander, H. J. Lezec, J. Aizpurua, and K. B. Crozier, Quantum mechanical effects in plasmonic structures with subnanometre gaps, Nat. Commun. 7, 11495 (2016).
  36. K. J. Savage, M. M. Hawkeye, R. Esteban, A. G. Borisov, J. Aizpurua, and J. J. Baumberg, Revealing the quantum regime in tunnelling plasmonics, Nature (London) 491, 574 (2012).
  37. U. S. Sainadh, R. T. Sang, and I. V. Litvinyuk, Attoclock and the quest for tunnelling time in strong-field physics, J. Phys. Photonics 2, 042002 (2020).
  38. B. Ma and M. Krüger, Strong-field theory of attosecond tunneling microscopy, Phys. Rev. Lett. 133, 236901 (2024).
  39. D. J. Tannor, Introduction to Quantum Mechanics: A Time-Dependent Perspective (University Science Books, Melville, NY, 2007).
  40. M. Krüger, M. Schenk, and P. Hommelhoff, Attosecond control of electrons emitted from a nanoscale metal tip, Nature (London) 475, 78 (2011).
  41. M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L'Huillier, and P. B. Corkum, Theory of high-harmonic generation by low-frequency laser fields, Phys. Rev. A 49, 2117 (1994).
  42. J. G. Simmons, Generalized formula for the electric tunnel effect between similar electrodes separated by a thin insulating film, J. Appl. Phys. 34, 1793 (1963).
  43. J. A. Scholl, A. García-Etxarri, A. L. Koh, and J. A. Dionne, Observation of quantum tunneling between two plasmonic nanoparticles, Nano Lett. 13, 564 (2013).
  44. S. V. Yalunin, M. Gulde, and C. Ropers, Strong-field photoemission from surfaces: Theoretical approaches, Phys. Rev. B 84, 195426 (2011).
  45. J. G. Simmons, Potential barriers and emission-limited current flow between closely spaced parallel metal electrodes, J. Appl. Phys. 35, 2472 (1964).
  46. M. Garg, A. Martin-Jimenez, Y. Luo, and K. Kern, Ultrafast photon-induced tunneling microscopy, ACS Nano 15, 18071 (2021).
  47. W. Schottky, Über den Einfluß von Strukturwirkungen, besonders der Thomsonschen Bildkraft, auf die Elektronenemission der Metalle, Phys. Z. 15, 872 (1914).
  48. M. Schenk, M. Krüger, and P. Hommelhoff, Strong-field above-threshold photoemission from sharp metal tips, Phys. Rev. Lett. 105, 257601 (2010).
  49. Y. Luo and P. Zhang, Ultrafast optical-field-induced photoelectron emission in a vacuum nanoscale gap: An exact analytical formulation, Appl. Phys. Lett. 119, 194101 (2021).
  50. D. B. Milošević, G. G. Paulus, D. Bauer, and W. Becker, Above-threshold ionization by few-cycle pulses, J. Phys. B: At. Mol. Opt. Phys. 39, R203 (2006).
  51. M. Y. Ivanov, M. Spanner, and O. Smirnova, Anatomy of strong field ionization, J. Mod. Opt. 52, 165 (2005).
  52. J. Bardeen, Tunnelling from a many-particle point of view, Phys. Rev. Lett. 6, 57 (1961).
  53. M. Krüger, M. Schenk, P. Hommelhoff, G. Wachter, C. Lemell, and J. Burgdörfer, Interaction of ultrashort laser pulses with metal nanotips: A model system for strong-field phenomena, New J. Phys. 14, 085019 (2012).
  54. K. Amini, J. Biegert, F. Calegari, A. Chacón, M. F. Ciappina, A. Dauphin, D. K. Efimov, C. F. de Morisson Faria, K. Giergiel, P. Gniewek, A. S. Landsman, M. Lesiuk, M. Mandrysz, A. S. Maxwell, R. Moszyński, L. Ortmann, J. A. Pérez-Hernández, A. Picón, E. Pisanty, J. Prauzner-Bechcicki et al., Symphony on strong field approximation, Rep. Prog. Phys. 82, 116001 (2019).
  55. D. B. Milošević, G. G. Paulus, and W. Becker, Phase-dependent effects of a few-cycle laser pulse, Phys. Rev. Lett. 89, 153001 (2002).
  56. W. Becker, S. P. Goreslavski, D. B. Milošević, and G. G. Paulus, The plateau in above-threshold ionization: The keystone of rescattering physics, J. Phys. B: At. Mol. Opt. Phys. 51, 162002 (2018).
  57. D. M. Volkov, Über eine Klasse von Lösungen der Diracschen Gleichung, Z. Phys. 94, 250 (1935).
  58. D. Shafir, H. Soifer, B. D. Bruner, M. Dagan, Y. Mairesse, S. Patchkovskii, M. Y. Ivanov, O. Smirnova, and N. Dudovich, Resolving the time when an electron exits a tunnelling barrier, Nature (London) 485, 343 (2012).
  59. H. R. Reiss and V. P. Krainov, Approximation for a Coulomb-Volkov solution in strong fields, Phys. Rev. A 50, R910 (1994).
  60. O. Smirnova, M. Spanner, and M. Ivanov, Analytical solutions for strong field-driven atomic and molecular one- and two-electron continua and applications to strong-field problems, Phys. Rev. A 77, 033407 (2008).
  61. K. G. Kay, Time-dependent semiclassical tunneling through barriers, Phys. Rev. A 88, 012122 (2013).
  62. M. V. Berry, Quantum scars of classical closed orbits in phase space, Proc. R. Soc. London A 423, 219 (1989).
  63. A. M. O. de Almeida, The Weyl representation in classical and quantum mechanics, Phys. Rep. 295, 265 (1998).
  64. O. Smirnova, M. Spanner, and M. Ivanov, Coulomb and polarization effects in sub-cycle dynamics of strong-field ionization, J. Phys. B: At. Mol. Opt. Phys. 39, S307 (2006).
  65. G. Wachter, C. Lemell, J. Burgdörfer, M. Schenk, M. Krüger, and P. Hommelhoff, Electron rescattering at metal nanotips induced by ultrashort laser pulses, Phys. Rev. B 86, 035402 (2012).
  66. F. H. M. Faisal, J. Z. Kamiński, and E. Saczuk, Photoemission from solid surfaces in intense laser fields, Laser Phys. 16, 284 (2006).
  67. F. H. M. Faisal, J. Z. Kamiński, and E. Saczuk, Photoemission and high-order harmonic generation from solid surfaces in intense laser fields, Phys. Rev. A 72, 023412 (2005).
  68. O. Pedatzur, G. Orenstein, V. Serbinenko, H. Soifer, B. D. Bruner, A. J. Uzan, D. S. Brambila, A. G. Harvey, L. Torlina, F. Morales, O. Smirnova, and N. Dudovich, Attosecond tunnelling interferometry, Nat. Phys. 11, 815 (2015).
  69. P. Salières, B. Carré, L. Le Déroff, F. Grasbon, G. G. Paulus, H. Walther, R. Kopold, W. Becker, D. B. Milošević, A. Sanpera, and M. Lewenstein, Feynman's path-integral approach for intense-laser-atom interactions, Science 292, 902 (2001).
  70. G. G. Paulus, W. Nicklich, H. Xu, P. Lambropoulos, and H. Walther, Plateau in above threshold ionization spectra, Phys. Rev. Lett. 72, 2851 (1994).
  71. A. M. Zheltikov, Keldysh parameter, photoionization adiabaticity, and the tunneling time, Phys. Rev. A 94, 043412 (2016).
  72. L. V. Keldysh, Ionization in the field of a strong electromagnetic wave, Sov. Phys. JETP 20, 1307 (1965).
  73. S. V. Popruzhenko, Keldysh theory of strong field ionization: History, applications, difficulties and perspectives, J. Phys. B: At. Mol. Opt. Phys. 47, 204001 (2014).
  74. G. Herink, D. R. Solli, M. Gulde, and C. Ropers, Field-driven photoemission from nanostructures quenches the quiver motion, Nature (London) 483, 190 (2012).
  75. R. G. Forbes, Field electron and ion emission from charged surfaces: A strategic historical review of theoretical concepts, Ultramicroscopy 95, 1 (2003).
  76. R. H. Fowler and L. Nordheim, Electron emission in intense electric fields, Proc. R. Soc. London A 119, 173 (1928).
  77. A. G. Borisov, B. Ma, M. Zapata-Herrera, A. Babaze, M. Krüger, and J. Aizpurua, Femtosecond optical-field-driven currents in few-nanometer-size gaps with hot electron injection into metallic leads, ACS Photonics 12, 2137 (2025).
  78. A. Wirth, M. T. Hassan, I. Grguraš, J. Gagnon, A. Moulet, T. Luu, S. Pabst, R. Santra, Z. A. Alahmed, A. M. Azzeer, V. S. Yakovlev, V. Pervak, F. Krausz, and E. Goulielmakis, Synthesized light transients, Science 334, 195 (2011).

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Phys. Rev. A 112, 033104– Published 11 September, 2025
  • Received 16 March 2025
  • Accepted 4 August 2025
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