Abstract

The electronic band structure governs the electron dynamics in solids. It defines a group velocity and an effective mass of the electronic wave packet. Recent experimental and theoretical studies suggest that an electron acquires the effective mass of its excited state over distances much larger than the lattice period of the solid. Therefore, electron propagation on atomic length scales was typically considered to be free-electron-like. Here, we test this hypothesis by probing attosecond photoemission from a Cu(111) surface. We use attosecond pulse trains in the extreme-ultraviolet (21–33 eV) to excite electrons from two initial bands within the 3d-valence band of copper. We timed their arrival at the crystal surface with a probing femtosecond infrared pulse, and found an upper limit of 350±40 as (1  as=10−18  s) for the propagation time an electron requires to assume the effective mass of its excited state. This observation implies that a final-state Bloch wave packet forms within a travel distance of 5–7 Å, which is at most two atomic layers. Using well-established theory, our measurements demonstrate the importance of the band structure even for atomic-scale electron transport.

© 2017 Optical Society of America

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  1. M. Chhowalla, D. Jena, and H. Zhang, “Two-dimensional semiconductors for transistors,” Nat. Rev. 1, 1–15 (2016).
  2. S. B. Desai, S. R. Madhvapathy, A. B. Sachid, J. P. Llinas, Q. Wang, G. H. Ahn, G. Pitner, M. J. Kim, J. Bokor, C. Hu, H. P. Wong, and A. Javey, “MoS2 transistors with 1-nanometer gate lengths,” Science 354, 99–102 (2016).
    [Crossref]
  3. N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, 1976).
  4. M. Gertsvolf, M. Spanner, D. M. Rayner, and P. B. Corkum, “Demonstration of attosecond ionization dynamics inside transparent solids,” J. Phys. B 43, 131002 (2010).
    [Crossref]
  5. R. Chang, S. Potnis, R. Ramos, C. Zhuang, M. Hallaji, A. Hayat, F. Duque-Gomez, J. E. Sipe, and A. M. Steinberg, “Observing the onset of effective mass,” Phys. Rev. Lett. 112, 170404 (2014).
    [Crossref]
  6. F. Duque-Gomez and J. E. Sipe, “Response of a particle in a one-dimensional lattice to an applied force: dynamics of the effective mass,” Phys. Rev. A 85, 053412 (2012).
    [Crossref]
  7. I. Bloch, J. Dalibard, and W. Zwerger, “Many-body physics with ultracold gases,” Rev. Mod. Phys. 80, 885–964 (2008).
    [Crossref]
  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, 138–141 (2010).
    [Crossref]
  9. A. L. Cavalieri, N. Muller, T. Uphues, V. S. Yakovlev, A. Baltuska, B. Horvath, B. Schmidt, L. Blumel, R. Holzwarth, S. Hendel, M. Drescher, U. Kleineberg, P. M. Echenique, R. Kienberger, F. Krausz, and U. Heinzmann, “Attosecond spectroscopy in condensed matter,” Nature 449, 1029–1032 (2007).
    [Crossref]
  10. R. Locher, L. Castiglioni, M. Lucchini, M. Greif, L. Gallmann, J. Osterwalder, M. Hengsberger, and U. Keller, “Energy-dependent photoemission delays from noble metal surfaces by attosecond interferometry,” Optica 2, 405–410 (2015).
    [Crossref]
  11. S. Neppl, R. Ernstorfer, A. L. Cavalieri, C. Lemell, G. Wachter, E. Magerl, E. M. Bothschafter, M. Jobst, M. Hofstetter, U. Kleineberg, J. V. Barth, D. Menzel, J. Burgdorfer, P. Feulner, F. Krausz, and R. Kienberger, “Direct observation of electron propagation and dielectric screening on the atomic length scale,” Nature 517, 342–346 (2015).
    [Crossref]
  12. C. Chen, Z. Tao, A. Carr, P. Matyba, T. Szilvasi, S. Emmerich, M. Piecuch, M. Keller, D. Zusin, S. Eich, M. Rollinger, W. You, S. Mathias, U. Thumm, M. Mavrikakis, M. Aeschlimann, P. M. Oppeneer, H. Kapteyn, and M. Murnane, “Distinguishing attosecond electron-electron scattering and screening in transition metals,” Proc. Natl. Acad. Sci. USA 114, E5300–E5307 (2017).
    [Crossref]
  13. F. Siek, S. Neb, P. Bartz, M. Hensen, C. Struber, S. Fiechter, M. Torrent-Sucarrat, V. M. Silkin, E. E. Krasovskii, N. M. Kabachnik, S. Fritzsche, R. D. Muino, P. M. Echenique, A. K. Kazansky, N. Muller, W. Pfeiffer, and U. Heinzmann, “Angular momentum-induced delays in solid-state photoemission enhanced by intra-atomic interactions,” Science 357, 1274–1277 (2017).
    [Crossref]
  14. C. Lemell, B. Solleder, K. Tőkési, and J. Burgdörfer, “Simulation of attosecond streaking of electrons emitted from a tungsten surface,” Phys. Rev. A 79, 062901 (2009).
    [Crossref]
  15. A. K. Kazansky and P. M. Echenique, “One-electron model for the electronic response of metal surfaces to subfemtosecond photoexcitation,” Phys. Rev. Lett. 102, 177401 (2009).
    [Crossref]
  16. E. E. Krasovskii, “Attosecond spectroscopy of solids: streaking phase shift due to lattice scattering,” Phys. Rev. B 84, 195106 (2011).
    [Crossref]
  17. Z. Tao, C. Chen, T. Szilvasi, M. Keller, M. Mavrikakis, H. Kapteyn, and M. Murnane, “Direct time-domain observation of attosecond final-state lifetimes in photoemission from solids,” Science 353, 62–67 (2016).
    [Crossref]
  18. R. Locher, M. Lucchini, J. Herrmann, M. Sabbar, M. Weger, A. Ludwig, L. Castiglioni, M. Greif, M. Hengsberger, L. Gallmann, and U. Keller, “Versatile attosecond beamline in a two-foci configuration for simultaneous time-resolved measurements,” Rev. Sci. Instrum. 85, 013113 (2014).
    [Crossref]
  19. M. Lucchini, L. Castiglioni, L. Kasmi, P. Kliuiev, A. Ludwig, M. Greif, J. Osterwalder, M. Hengsberger, L. Gallmann, and U. Keller, “Light-matter interaction at surfaces in the spatiotemporal limit of macroscopic models,” Phys. Rev. Lett. 115, 137401 (2015).
    [Crossref]
  20. F. Schlaepfer, A. Ludwig, M. Lucchini, L. Kasmi, M. Volkov, L. Gallmann, and U. Keller, “Gouy phase shift for annular beam profiles in attosecond experiments,” Opt. Express 25, 3646–3655 (2017).
    [Crossref]
  21. C. H. Zhang and U. Thumm, “Streaking and Wigner time delays in photoemission from atoms and surfaces,” Phys. Rev. A 84, 033401 (2011).
    [Crossref]
  22. C. N. Berglund and W. E. Spicer, “Photoemission studies of copper and silver: theory,” Phys. Rev. 136, A1030–A1044 (1964).
    [Crossref]
  23. A. Damascelli, “Probing the electronic structure of complex systems by ARPES,” Phys. Scripta T109, 61–74 (2004).
    [Crossref]
  24. H. G. Muller, “Reconstruction of attosecond harmonic beating by interference of two-photon transitions,” Appl. Phys. B 74, s17–s21 (2014).
    [Crossref]
  25. P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Auge, P. Balcou, H. G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science 292, 1689–1692 (2001).
    [Crossref]
  26. A. S. Kheifets, “Time delay in valence-shell photoionization of noble-gas atoms,” Phys. Rev. A 87, 063404 (2013).
    [Crossref]
  27. L. Cattaneo, J. Vos, M. Lucchini, L. Gallmann, C. Cirelli, and U. Keller, “Comparison of attosecond streaking and RABBITT,” Opt. Express 24, 29060–29076 (2016).
    [Crossref]
  28. D. R. Penn, “Electron mean-free-path calculations using a model dielectric function,” Phys. Rev. B 35, 482–486 (1987).
    [Crossref]
  29. S. Hüfner, Photoelectron Spectroscopy: Principles and Applications (Springer, 2003).
  30. D. J. Spanjaard, D. W. Jepsen, and P. M. Marcus, “Effects of transmission factors and matrix elements on angular distribution of photoemission from Ag(111),” Phys. Rev. B 15, 1728–1737 (1977).
    [Crossref]
  31. L. Ilver and P. O. Nilsson, “Angular-resolved u.v. photoemission from single crystals of copper,” Solid State Commun. 18, 677–680 (1976).
    [Crossref]
  32. I. Jordan, M. Huppert, S. Pabst, A. S. Kheifets, D. Baykusheva, and H. J. Wörner, “Spin-orbit delays in photoemission,” Phys. Rev. A 95, 013404 (2017).
    [Crossref]

2017 (4)

C. Chen, Z. Tao, A. Carr, P. Matyba, T. Szilvasi, S. Emmerich, M. Piecuch, M. Keller, D. Zusin, S. Eich, M. Rollinger, W. You, S. Mathias, U. Thumm, M. Mavrikakis, M. Aeschlimann, P. M. Oppeneer, H. Kapteyn, and M. Murnane, “Distinguishing attosecond electron-electron scattering and screening in transition metals,” Proc. Natl. Acad. Sci. USA 114, E5300–E5307 (2017).
[Crossref]

F. Siek, S. Neb, P. Bartz, M. Hensen, C. Struber, S. Fiechter, M. Torrent-Sucarrat, V. M. Silkin, E. E. Krasovskii, N. M. Kabachnik, S. Fritzsche, R. D. Muino, P. M. Echenique, A. K. Kazansky, N. Muller, W. Pfeiffer, and U. Heinzmann, “Angular momentum-induced delays in solid-state photoemission enhanced by intra-atomic interactions,” Science 357, 1274–1277 (2017).
[Crossref]

F. Schlaepfer, A. Ludwig, M. Lucchini, L. Kasmi, M. Volkov, L. Gallmann, and U. Keller, “Gouy phase shift for annular beam profiles in attosecond experiments,” Opt. Express 25, 3646–3655 (2017).
[Crossref]

I. Jordan, M. Huppert, S. Pabst, A. S. Kheifets, D. Baykusheva, and H. J. Wörner, “Spin-orbit delays in photoemission,” Phys. Rev. A 95, 013404 (2017).
[Crossref]

2016 (4)

L. Cattaneo, J. Vos, M. Lucchini, L. Gallmann, C. Cirelli, and U. Keller, “Comparison of attosecond streaking and RABBITT,” Opt. Express 24, 29060–29076 (2016).
[Crossref]

Z. Tao, C. Chen, T. Szilvasi, M. Keller, M. Mavrikakis, H. Kapteyn, and M. Murnane, “Direct time-domain observation of attosecond final-state lifetimes in photoemission from solids,” Science 353, 62–67 (2016).
[Crossref]

M. Chhowalla, D. Jena, and H. Zhang, “Two-dimensional semiconductors for transistors,” Nat. Rev. 1, 1–15 (2016).

S. B. Desai, S. R. Madhvapathy, A. B. Sachid, J. P. Llinas, Q. Wang, G. H. Ahn, G. Pitner, M. J. Kim, J. Bokor, C. Hu, H. P. Wong, and A. Javey, “MoS2 transistors with 1-nanometer gate lengths,” Science 354, 99–102 (2016).
[Crossref]

2015 (3)

R. Locher, L. Castiglioni, M. Lucchini, M. Greif, L. Gallmann, J. Osterwalder, M. Hengsberger, and U. Keller, “Energy-dependent photoemission delays from noble metal surfaces by attosecond interferometry,” Optica 2, 405–410 (2015).
[Crossref]

S. Neppl, R. Ernstorfer, A. L. Cavalieri, C. Lemell, G. Wachter, E. Magerl, E. M. Bothschafter, M. Jobst, M. Hofstetter, U. Kleineberg, J. V. Barth, D. Menzel, J. Burgdorfer, P. Feulner, F. Krausz, and R. Kienberger, “Direct observation of electron propagation and dielectric screening on the atomic length scale,” Nature 517, 342–346 (2015).
[Crossref]

M. Lucchini, L. Castiglioni, L. Kasmi, P. Kliuiev, A. Ludwig, M. Greif, J. Osterwalder, M. Hengsberger, L. Gallmann, and U. Keller, “Light-matter interaction at surfaces in the spatiotemporal limit of macroscopic models,” Phys. Rev. Lett. 115, 137401 (2015).
[Crossref]

2014 (3)

R. Locher, M. Lucchini, J. Herrmann, M. Sabbar, M. Weger, A. Ludwig, L. Castiglioni, M. Greif, M. Hengsberger, L. Gallmann, and U. Keller, “Versatile attosecond beamline in a two-foci configuration for simultaneous time-resolved measurements,” Rev. Sci. Instrum. 85, 013113 (2014).
[Crossref]

R. Chang, S. Potnis, R. Ramos, C. Zhuang, M. Hallaji, A. Hayat, F. Duque-Gomez, J. E. Sipe, and A. M. Steinberg, “Observing the onset of effective mass,” Phys. Rev. Lett. 112, 170404 (2014).
[Crossref]

H. G. Muller, “Reconstruction of attosecond harmonic beating by interference of two-photon transitions,” Appl. Phys. B 74, s17–s21 (2014).
[Crossref]

2013 (1)

A. S. Kheifets, “Time delay in valence-shell photoionization of noble-gas atoms,” Phys. Rev. A 87, 063404 (2013).
[Crossref]

2012 (1)

F. Duque-Gomez and J. E. Sipe, “Response of a particle in a one-dimensional lattice to an applied force: dynamics of the effective mass,” Phys. Rev. A 85, 053412 (2012).
[Crossref]

2011 (2)

E. E. Krasovskii, “Attosecond spectroscopy of solids: streaking phase shift due to lattice scattering,” Phys. Rev. B 84, 195106 (2011).
[Crossref]

C. H. Zhang and U. Thumm, “Streaking and Wigner time delays in photoemission from atoms and surfaces,” Phys. Rev. A 84, 033401 (2011).
[Crossref]

2010 (2)

M. Gertsvolf, M. Spanner, D. M. Rayner, and P. B. Corkum, “Demonstration of attosecond ionization dynamics inside transparent solids,” J. Phys. B 43, 131002 (2010).
[Crossref]

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, 138–141 (2010).
[Crossref]

2009 (2)

C. Lemell, B. Solleder, K. Tőkési, and J. Burgdörfer, “Simulation of attosecond streaking of electrons emitted from a tungsten surface,” Phys. Rev. A 79, 062901 (2009).
[Crossref]

A. K. Kazansky and P. M. Echenique, “One-electron model for the electronic response of metal surfaces to subfemtosecond photoexcitation,” Phys. Rev. Lett. 102, 177401 (2009).
[Crossref]

2008 (1)

I. Bloch, J. Dalibard, and W. Zwerger, “Many-body physics with ultracold gases,” Rev. Mod. Phys. 80, 885–964 (2008).
[Crossref]

2007 (1)

A. L. Cavalieri, N. Muller, T. Uphues, V. S. Yakovlev, A. Baltuska, B. Horvath, B. Schmidt, L. Blumel, R. Holzwarth, S. Hendel, M. Drescher, U. Kleineberg, P. M. Echenique, R. Kienberger, F. Krausz, and U. Heinzmann, “Attosecond spectroscopy in condensed matter,” Nature 449, 1029–1032 (2007).
[Crossref]

2004 (1)

A. Damascelli, “Probing the electronic structure of complex systems by ARPES,” Phys. Scripta T109, 61–74 (2004).
[Crossref]

2001 (1)

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Auge, P. Balcou, H. G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science 292, 1689–1692 (2001).
[Crossref]

1987 (1)

D. R. Penn, “Electron mean-free-path calculations using a model dielectric function,” Phys. Rev. B 35, 482–486 (1987).
[Crossref]

1977 (1)

D. J. Spanjaard, D. W. Jepsen, and P. M. Marcus, “Effects of transmission factors and matrix elements on angular distribution of photoemission from Ag(111),” Phys. Rev. B 15, 1728–1737 (1977).
[Crossref]

1976 (1)

L. Ilver and P. O. Nilsson, “Angular-resolved u.v. photoemission from single crystals of copper,” Solid State Commun. 18, 677–680 (1976).
[Crossref]

1964 (1)

C. N. Berglund and W. E. Spicer, “Photoemission studies of copper and silver: theory,” Phys. Rev. 136, A1030–A1044 (1964).
[Crossref]

Aeschlimann, M.

C. Chen, Z. Tao, A. Carr, P. Matyba, T. Szilvasi, S. Emmerich, M. Piecuch, M. Keller, D. Zusin, S. Eich, M. Rollinger, W. You, S. Mathias, U. Thumm, M. Mavrikakis, M. Aeschlimann, P. M. Oppeneer, H. Kapteyn, and M. Murnane, “Distinguishing attosecond electron-electron scattering and screening in transition metals,” Proc. Natl. Acad. Sci. USA 114, E5300–E5307 (2017).
[Crossref]

Agostini, P.

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, 138–141 (2010).
[Crossref]

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Auge, P. Balcou, H. G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science 292, 1689–1692 (2001).
[Crossref]

Ahn, G. H.

S. B. Desai, S. R. Madhvapathy, A. B. Sachid, J. P. Llinas, Q. Wang, G. H. Ahn, G. Pitner, M. J. Kim, J. Bokor, C. Hu, H. P. Wong, and A. Javey, “MoS2 transistors with 1-nanometer gate lengths,” Science 354, 99–102 (2016).
[Crossref]

Ashcroft, N. W.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, 1976).

Auge, F.

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Auge, P. Balcou, H. G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science 292, 1689–1692 (2001).
[Crossref]

Balcou, P.

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Auge, P. Balcou, H. G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science 292, 1689–1692 (2001).
[Crossref]

Baltuska, A.

A. L. Cavalieri, N. Muller, T. Uphues, V. S. Yakovlev, A. Baltuska, B. Horvath, B. Schmidt, L. Blumel, R. Holzwarth, S. Hendel, M. Drescher, U. Kleineberg, P. M. Echenique, R. Kienberger, F. Krausz, and U. Heinzmann, “Attosecond spectroscopy in condensed matter,” Nature 449, 1029–1032 (2007).
[Crossref]

Barth, J. V.

S. Neppl, R. Ernstorfer, A. L. Cavalieri, C. Lemell, G. Wachter, E. Magerl, E. M. Bothschafter, M. Jobst, M. Hofstetter, U. Kleineberg, J. V. Barth, D. Menzel, J. Burgdorfer, P. Feulner, F. Krausz, and R. Kienberger, “Direct observation of electron propagation and dielectric screening on the atomic length scale,” Nature 517, 342–346 (2015).
[Crossref]

Bartz, P.

F. Siek, S. Neb, P. Bartz, M. Hensen, C. Struber, S. Fiechter, M. Torrent-Sucarrat, V. M. Silkin, E. E. Krasovskii, N. M. Kabachnik, S. Fritzsche, R. D. Muino, P. M. Echenique, A. K. Kazansky, N. Muller, W. Pfeiffer, and U. Heinzmann, “Angular momentum-induced delays in solid-state photoemission enhanced by intra-atomic interactions,” Science 357, 1274–1277 (2017).
[Crossref]

Baykusheva, D.

I. Jordan, M. Huppert, S. Pabst, A. S. Kheifets, D. Baykusheva, and H. J. Wörner, “Spin-orbit delays in photoemission,” Phys. Rev. A 95, 013404 (2017).
[Crossref]

Berglund, C. N.

C. N. Berglund and W. E. Spicer, “Photoemission studies of copper and silver: theory,” Phys. Rev. 136, A1030–A1044 (1964).
[Crossref]

Bloch, I.

I. Bloch, J. Dalibard, and W. Zwerger, “Many-body physics with ultracold gases,” Rev. Mod. Phys. 80, 885–964 (2008).
[Crossref]

Blumel, L.

A. L. Cavalieri, N. Muller, T. Uphues, V. S. Yakovlev, A. Baltuska, B. Horvath, B. Schmidt, L. Blumel, R. Holzwarth, S. Hendel, M. Drescher, U. Kleineberg, P. M. Echenique, R. Kienberger, F. Krausz, and U. Heinzmann, “Attosecond spectroscopy in condensed matter,” Nature 449, 1029–1032 (2007).
[Crossref]

Bokor, J.

S. B. Desai, S. R. Madhvapathy, A. B. Sachid, J. P. Llinas, Q. Wang, G. H. Ahn, G. Pitner, M. J. Kim, J. Bokor, C. Hu, H. P. Wong, and A. Javey, “MoS2 transistors with 1-nanometer gate lengths,” Science 354, 99–102 (2016).
[Crossref]

Bothschafter, E. M.

S. Neppl, R. Ernstorfer, A. L. Cavalieri, C. Lemell, G. Wachter, E. Magerl, E. M. Bothschafter, M. Jobst, M. Hofstetter, U. Kleineberg, J. V. Barth, D. Menzel, J. Burgdorfer, P. Feulner, F. Krausz, and R. Kienberger, “Direct observation of electron propagation and dielectric screening on the atomic length scale,” Nature 517, 342–346 (2015).
[Crossref]

Breger, P.

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Auge, P. Balcou, H. G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science 292, 1689–1692 (2001).
[Crossref]

Burgdorfer, J.

S. Neppl, R. Ernstorfer, A. L. Cavalieri, C. Lemell, G. Wachter, E. Magerl, E. M. Bothschafter, M. Jobst, M. Hofstetter, U. Kleineberg, J. V. Barth, D. Menzel, J. Burgdorfer, P. Feulner, F. Krausz, and R. Kienberger, “Direct observation of electron propagation and dielectric screening on the atomic length scale,” Nature 517, 342–346 (2015).
[Crossref]

Burgdörfer, J.

C. Lemell, B. Solleder, K. Tőkési, and J. Burgdörfer, “Simulation of attosecond streaking of electrons emitted from a tungsten surface,” Phys. Rev. A 79, 062901 (2009).
[Crossref]

Carr, A.

C. Chen, Z. Tao, A. Carr, P. Matyba, T. Szilvasi, S. Emmerich, M. Piecuch, M. Keller, D. Zusin, S. Eich, M. Rollinger, W. You, S. Mathias, U. Thumm, M. Mavrikakis, M. Aeschlimann, P. M. Oppeneer, H. Kapteyn, and M. Murnane, “Distinguishing attosecond electron-electron scattering and screening in transition metals,” Proc. Natl. Acad. Sci. USA 114, E5300–E5307 (2017).
[Crossref]

Castiglioni, L.

M. Lucchini, L. Castiglioni, L. Kasmi, P. Kliuiev, A. Ludwig, M. Greif, J. Osterwalder, M. Hengsberger, L. Gallmann, and U. Keller, “Light-matter interaction at surfaces in the spatiotemporal limit of macroscopic models,” Phys. Rev. Lett. 115, 137401 (2015).
[Crossref]

R. Locher, L. Castiglioni, M. Lucchini, M. Greif, L. Gallmann, J. Osterwalder, M. Hengsberger, and U. Keller, “Energy-dependent photoemission delays from noble metal surfaces by attosecond interferometry,” Optica 2, 405–410 (2015).
[Crossref]

R. Locher, M. Lucchini, J. Herrmann, M. Sabbar, M. Weger, A. Ludwig, L. Castiglioni, M. Greif, M. Hengsberger, L. Gallmann, and U. Keller, “Versatile attosecond beamline in a two-foci configuration for simultaneous time-resolved measurements,” Rev. Sci. Instrum. 85, 013113 (2014).
[Crossref]

Cattaneo, L.

Cavalieri, A. L.

S. Neppl, R. Ernstorfer, A. L. Cavalieri, C. Lemell, G. Wachter, E. Magerl, E. M. Bothschafter, M. Jobst, M. Hofstetter, U. Kleineberg, J. V. Barth, D. Menzel, J. Burgdorfer, P. Feulner, F. Krausz, and R. Kienberger, “Direct observation of electron propagation and dielectric screening on the atomic length scale,” Nature 517, 342–346 (2015).
[Crossref]

A. L. Cavalieri, N. Muller, T. Uphues, V. S. Yakovlev, A. Baltuska, B. Horvath, B. Schmidt, L. Blumel, R. Holzwarth, S. Hendel, M. Drescher, U. Kleineberg, P. M. Echenique, R. Kienberger, F. Krausz, and U. Heinzmann, “Attosecond spectroscopy in condensed matter,” Nature 449, 1029–1032 (2007).
[Crossref]

Chang, R.

R. Chang, S. Potnis, R. Ramos, C. Zhuang, M. Hallaji, A. Hayat, F. Duque-Gomez, J. E. Sipe, and A. M. Steinberg, “Observing the onset of effective mass,” Phys. Rev. Lett. 112, 170404 (2014).
[Crossref]

Chen, C.

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M. Lucchini, L. Castiglioni, L. Kasmi, P. Kliuiev, A. Ludwig, M. Greif, J. Osterwalder, M. Hengsberger, L. Gallmann, and U. Keller, “Light-matter interaction at surfaces in the spatiotemporal limit of macroscopic models,” Phys. Rev. Lett. 115, 137401 (2015).
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R. Locher, L. Castiglioni, M. Lucchini, M. Greif, L. Gallmann, J. Osterwalder, M. Hengsberger, and U. Keller, “Energy-dependent photoemission delays from noble metal surfaces by attosecond interferometry,” Optica 2, 405–410 (2015).
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F. Siek, S. Neb, P. Bartz, M. Hensen, C. Struber, S. Fiechter, M. Torrent-Sucarrat, V. M. Silkin, E. E. Krasovskii, N. M. Kabachnik, S. Fritzsche, R. D. Muino, P. M. Echenique, A. K. Kazansky, N. Muller, W. Pfeiffer, and U. Heinzmann, “Angular momentum-induced delays in solid-state photoemission enhanced by intra-atomic interactions,” Science 357, 1274–1277 (2017).
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C. Chen, Z. Tao, A. Carr, P. Matyba, T. Szilvasi, S. Emmerich, M. Piecuch, M. Keller, D. Zusin, S. Eich, M. Rollinger, W. You, S. Mathias, U. Thumm, M. Mavrikakis, M. Aeschlimann, P. M. Oppeneer, H. Kapteyn, and M. Murnane, “Distinguishing attosecond electron-electron scattering and screening in transition metals,” Proc. Natl. Acad. Sci. USA 114, E5300–E5307 (2017).
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Z. Tao, C. Chen, T. Szilvasi, M. Keller, M. Mavrikakis, H. Kapteyn, and M. Murnane, “Direct time-domain observation of attosecond final-state lifetimes in photoemission from solids,” Science 353, 62–67 (2016).
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F. Schlaepfer, A. Ludwig, M. Lucchini, L. Kasmi, M. Volkov, L. Gallmann, and U. Keller, “Gouy phase shift for annular beam profiles in attosecond experiments,” Opt. Express 25, 3646–3655 (2017).
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M. Lucchini, L. Castiglioni, L. Kasmi, P. Kliuiev, A. Ludwig, M. Greif, J. Osterwalder, M. Hengsberger, L. Gallmann, and U. Keller, “Light-matter interaction at surfaces in the spatiotemporal limit of macroscopic models,” Phys. Rev. Lett. 115, 137401 (2015).
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F. Siek, S. Neb, P. Bartz, M. Hensen, C. Struber, S. Fiechter, M. Torrent-Sucarrat, V. M. Silkin, E. E. Krasovskii, N. M. Kabachnik, S. Fritzsche, R. D. Muino, P. M. Echenique, A. K. Kazansky, N. Muller, W. Pfeiffer, and U. Heinzmann, “Angular momentum-induced delays in solid-state photoemission enhanced by intra-atomic interactions,” Science 357, 1274–1277 (2017).
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A. K. Kazansky and P. M. Echenique, “One-electron model for the electronic response of metal surfaces to subfemtosecond photoexcitation,” Phys. Rev. Lett. 102, 177401 (2009).
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C. Chen, Z. Tao, A. Carr, P. Matyba, T. Szilvasi, S. Emmerich, M. Piecuch, M. Keller, D. Zusin, S. Eich, M. Rollinger, W. You, S. Mathias, U. Thumm, M. Mavrikakis, M. Aeschlimann, P. M. Oppeneer, H. Kapteyn, and M. Murnane, “Distinguishing attosecond electron-electron scattering and screening in transition metals,” Proc. Natl. Acad. Sci. USA 114, E5300–E5307 (2017).
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Z. Tao, C. Chen, T. Szilvasi, M. Keller, M. Mavrikakis, H. Kapteyn, and M. Murnane, “Direct time-domain observation of attosecond final-state lifetimes in photoemission from solids,” Science 353, 62–67 (2016).
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Keller, U.

F. Schlaepfer, A. Ludwig, M. Lucchini, L. Kasmi, M. Volkov, L. Gallmann, and U. Keller, “Gouy phase shift for annular beam profiles in attosecond experiments,” Opt. Express 25, 3646–3655 (2017).
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L. Cattaneo, J. Vos, M. Lucchini, L. Gallmann, C. Cirelli, and U. Keller, “Comparison of attosecond streaking and RABBITT,” Opt. Express 24, 29060–29076 (2016).
[Crossref]

M. Lucchini, L. Castiglioni, L. Kasmi, P. Kliuiev, A. Ludwig, M. Greif, J. Osterwalder, M. Hengsberger, L. Gallmann, and U. Keller, “Light-matter interaction at surfaces in the spatiotemporal limit of macroscopic models,” Phys. Rev. Lett. 115, 137401 (2015).
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R. Locher, L. Castiglioni, M. Lucchini, M. Greif, L. Gallmann, J. Osterwalder, M. Hengsberger, and U. Keller, “Energy-dependent photoemission delays from noble metal surfaces by attosecond interferometry,” Optica 2, 405–410 (2015).
[Crossref]

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[Crossref]

Kheifets, A. S.

I. Jordan, M. Huppert, S. Pabst, A. S. Kheifets, D. Baykusheva, and H. J. Wörner, “Spin-orbit delays in photoemission,” Phys. Rev. A 95, 013404 (2017).
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A. L. Cavalieri, N. Muller, T. Uphues, V. S. Yakovlev, A. Baltuska, B. Horvath, B. Schmidt, L. Blumel, R. Holzwarth, S. Hendel, M. Drescher, U. Kleineberg, P. M. Echenique, R. Kienberger, F. Krausz, and U. Heinzmann, “Attosecond spectroscopy in condensed matter,” Nature 449, 1029–1032 (2007).
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Kim, M. J.

S. B. Desai, S. R. Madhvapathy, A. B. Sachid, J. P. Llinas, Q. Wang, G. H. Ahn, G. Pitner, M. J. Kim, J. Bokor, C. Hu, H. P. Wong, and A. Javey, “MoS2 transistors with 1-nanometer gate lengths,” Science 354, 99–102 (2016).
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Kleineberg, U.

S. Neppl, R. Ernstorfer, A. L. Cavalieri, C. Lemell, G. Wachter, E. Magerl, E. M. Bothschafter, M. Jobst, M. Hofstetter, U. Kleineberg, J. V. Barth, D. Menzel, J. Burgdorfer, P. Feulner, F. Krausz, and R. Kienberger, “Direct observation of electron propagation and dielectric screening on the atomic length scale,” Nature 517, 342–346 (2015).
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A. L. Cavalieri, N. Muller, T. Uphues, V. S. Yakovlev, A. Baltuska, B. Horvath, B. Schmidt, L. Blumel, R. Holzwarth, S. Hendel, M. Drescher, U. Kleineberg, P. M. Echenique, R. Kienberger, F. Krausz, and U. Heinzmann, “Attosecond spectroscopy in condensed matter,” Nature 449, 1029–1032 (2007).
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Kliuiev, P.

M. Lucchini, L. Castiglioni, L. Kasmi, P. Kliuiev, A. Ludwig, M. Greif, J. Osterwalder, M. Hengsberger, L. Gallmann, and U. Keller, “Light-matter interaction at surfaces in the spatiotemporal limit of macroscopic models,” Phys. Rev. Lett. 115, 137401 (2015).
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F. Siek, S. Neb, P. Bartz, M. Hensen, C. Struber, S. Fiechter, M. Torrent-Sucarrat, V. M. Silkin, E. E. Krasovskii, N. M. Kabachnik, S. Fritzsche, R. D. Muino, P. M. Echenique, A. K. Kazansky, N. Muller, W. Pfeiffer, and U. Heinzmann, “Angular momentum-induced delays in solid-state photoemission enhanced by intra-atomic interactions,” Science 357, 1274–1277 (2017).
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E. E. Krasovskii, “Attosecond spectroscopy of solids: streaking phase shift due to lattice scattering,” Phys. Rev. B 84, 195106 (2011).
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Krausz, F.

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Supplementary Material (1)

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Figures (4)

Fig. 1.
Fig. 1. (a) Experimental setup. The p-polarized XUV and IR beams are focused first into an argon gas target to obtain a reference RABBITT measurement. Subsequently, the pulses are refocused by a gold-coated toroidal mirror onto the copper sample at an incidence angle of 75°, and the photoelectrons are detected by a hemispherical analyzer 30° from the surface normal. A second RABBITT from copper is recorded. (b), (c), Sketch of the three-step model of photoemission. (1) Upon excitation by light, an electron is promoted from an occupied band in the valence band below the Fermi level into an unoccupied band. (2) The excited electron propagates in real space towards the solid surface. (3) The electron passes the surface barrier potential and escapes the solid, where it will then interact with the IR. With the given XUV photon energies, the typical photoelectron escape depths λ range between 5 and 7 Å.

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Fig. 2.
Fig. 2. Extraction of attosecond photoemission delays from split 3d-valence band. (a) XUV photon spectrum used in our experiment. (b) Each XUV harmonic ionizes the 3d-valence band of copper and gives a photoelectron signal from which the band structure can be retrieved [23]. The photoemission spectra are obtained at a detection angle of 30° from the surface normal and by rotating the copper sample around the (111) direction, hence varying the azimuthal angle ϕ. In black, the 3d-band structure of copper calculated with DFT is superimposed over the data measured with harmonic order 17. Thick lines highlight the two bands yielding high photoemission intensity; they are labeled with “upper” and “lower” according to their binding energy. (c) Sketch of the surface Brillouin zone with main crystallographic directions. The red dashed lines in (b) and (c), corresponding to ϕ=46°, indicate the position at which the RABBITT measurements were taken. ϕ=0° corresponds to the ΓM¯ direction. (d), (e) Portions of RABBITT traces measured from the copper sample at the second focus and from argon gas at the first focus, respectively. (f) Sideband 20, integrated over the respective split peaks and separated into an upper and lower part corresponding to the bands highlighted in (b). (g) Integrated sideband 20 in the argon RABBITT data that serves as the reference in the photoemission delay measurement.

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Fig. 3.
Fig. 3. Experimental results for photoemission delays (black) are plotted for upper and lower parts of the 3d-valence band as a function of the electron kinetic energy. The free-electron propagation time associated with mechanism (i) is plotted in blue. The propagation delays, based on group velocity derivation from the electronic band structure obtained from DFT calculations and associated with mechanism (ii), are plotted for the lower and upper parts of the 3d-valence band (in green and orange, respectively). The error bar around the zero level shows the uncertainty of our delay calibration.

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Fig. 4.
Fig. 4. (a) Scheme showing the two types of transitions that can occur: (i) transition into a free-electron final state and (ii) transition to a bulk final state. The different final band curvature for these two transitions determines the velocity at which the excited electron propagates towards the surface. (b) Difference between the experimental delays from the upper and lower parts of the 3d-valence band (in black), extracted from the experimental results shown in Fig. 3 together with the difference between the delays for transitions from the upper and lower parts of the valence band obtained with DFT calculations (in brown). The brown shaded area accounts for the standard deviation of the individual points.

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Equations (10)

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τ2qCu=τW+τtransp+τcc,
Ekin,s=ℏ2k22me−E0,
v→fe(k→)=ℏk→me.
τfe(Ekin)=λ(Ekin)|v→fe|
v→g(k→)=1ℏ∇→E(k→).
τi(Ekin)=λ(Ekin)|v→g(k∥,k⊥(i))|.
τ(Ekin,avg)=∑iwiτi(Ekin)∑iwi.
N(E,ℏω)∝1ω2∑i,fTf×kf×Θ(vg,f,⊥)×vg,f,⊥vg,i,⊥×|Mi,fB|2×δ(ki→−kf→)×δ(p∥/ℏ−kf,∥+G∥)×δ(Ef−Ei−ℏω)×δ(E−Ef−Wf).
k∥=p∥ℏ+G∥,p∥=2meEkin,vsin θ,
Ekin,v=ℏω−Eb−Wf,

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