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Laser picoscopy of valence electrons in solids

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Valence electrons contribute a small fraction of the total electron density of materials, but they determine their essential chemical, electronic and optical properties. Strong laser fields can probe electrons in valence orbitals1,2,3 and their dynamics4,5,6 in the gas phase. Previous laser studies of solids have associated high-harmonic emission7,8,9,10,11,12 with the spatial arrangement of atoms in the crystal lattice13,14 and have used terahertz fields to probe interatomic potential forces15. Yet the direct, picometre-scale imaging of valence electrons in solids has remained challenging. Here we show that intense optical fields interacting with crystalline solids could enable the imaging of valence electrons at the picometre scale. An intense laser field with a strength that is comparable to the fields keeping the valence electrons bound in crystals can induce quasi-free electron motion. The harmonics of the laser field emerging from the nonlinear scattering of the valence electrons by the crystal potential contain the critical information that enables picometre-scale, real-space mapping of the valence electron structure. We used high harmonics to reconstruct images of the valence potential and electron density in crystalline magnesium fluoride and calcium fluoride with a spatial resolution of about 26 picometres. Picometre-scale imaging of valence electrons could enable direct probing of the chemical, electronic, optical and topological properties of materials.

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Fig. 1: Strong-field quasi-free electron motion in a crystal.
Fig. 2: Probing of the ionic/covalent radius of atoms in solids.
Fig. 3: Measurement of the Fourier coefficients of the crystal potential in MgF2.
Fig. 4: Reconstruction of the valence electron potential and density of MgF2.

Data availability

The datasets generated and/or analysed during this study are available from the corresponding authors on reasonable request.

Code availability

The analysis codes that support the findings of the study are available from the corresponding authors on reasonable request.

Change history

  • 31 July 2020

    This Article was amended to correct the Peer review information, which was originally incorrect.


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This work was supported by a European Research Council grant (Attoelectronics-258501), the Deutsche Forschungsgemeinschaft Cluster of Excellence, the Munich Centre for Advanced Photonics and the Max Planck Society.

Author information

Authors and Affiliations



E.G. conceived and supervised the project. H.L., H.Y.K. and M.Z. performed the experiments and analysed the experimental data. H.Y.K. and H.L. performed the theoretical modelling and calculations. S.H and S.M. conducted the DFT and TDDFT modelling. E.G., H.L. and H.Y.K. interpreted the experimental data and contributed to the preparation of the manuscript. These authors contributed equally: H. Lakhotia, H. Y. Kim.

Corresponding author

Correspondence to E. Goulielmakis.

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Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature thanks Michael Sentef, Andre Staudte, Marco Taucer and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data figures and tables

Extended Data Fig. 1 Strong field-driven electron dynamics in MgF2 (ħωL = 2eV).

ac, Comparison of crystal (νc; blue curves) and free (νfree; red dashed curves) electron velocities along the [100] direction of an MgF2 crystal as calculated by TDDFT for laser field strengths F0 of 0.1 V Å−1 (a), 0.9 V Å−1 (b) and 2.0 V Å−1, and carrier at an energy of ħωL = 2eV.

Extended Data Fig. 2 High-harmonic generation in MgF2 (theory).

High-harmonic spectra calculated by TDDFT simulations (red curve) and by use of the scattering model (blue curve) for laser parameters (ħωL = 2eV and F0 = 0.9 V Å−1) and crystal orientation settings as quoted in Fig. 1d.

Extended Data Fig. 3 Crystal orientation dependence of high-harmonic generation in MgF2.

The intensity of the third, ninth and thirteenth harmonics measured as a function of the crystal angle at field strengths (F0 = 0.58, 0.65 and 0.7 V Å−1) of the driving pulse. The rotation of the crystal is performed with respect to the c axis. The azimuthal angle represents the orientation of the crystal with respect to the laser polarization and the radius represents the harmonic yield. The four-fold symmetry of the crystal suggests a square lattice. Error bars in the measured data indicate the standard deviation of the mean value from four measurements acquired under identical conditions.

Extended Data Fig. 4 Laser picoscopy in CaF2.

a, Intensity yields of representative harmonics (N = 9, 11 and 13) in CaF2 measured as a function of the crystal rotation angle with respect to the c axis and for three representative driving field strengths (F0 = 0.58, 0.65 and 0.7 V Å−1). b, c, Intensity yields (black dots) of harmonics versus field strengths measured along the [110] (b) and [100] (c) axes of the crystal. The red and blue curves are the fitting of the intensity yields according to equation (18) or equation (3). Error bars in ac indicate the standard deviation of the mean value from three measurements acquired under identical conditions. d, e, Retrieved amplitudes \({\tilde{V}}_{{k}_{{\rm{l}}}}\,\) and their relative phases (0 rad in blue and π rad in red) along the [110] (d) and [100] (e) axes of the crystal.

Extended Data Fig. 5 Reconstruction of the valence electron potential and density of CaF2.

a, Crystal structure of CaF2. The laser pulse (orange curve) impinges on the crystal along the c axis. The potential is probed along lines determined by laser polarization vectors (orange arrows) and the symmetry point C. b, c, Reconstructed 1D slices of the valence potential (blue curves) when the laser polarization vector is aligned with the [110] (b) and [100] (c) axes. Grey and cyan spheres represent F and Ca2+, respectively, as aligned along the measurement line. d, Reconstructed 2D slice of the valence electron potential of CaF2 on the (002) plane. Bright spots represent Ca+2 ions and the light broad spots represent F ions. e, Valence electron density evaluated from the data in d. f, DFT-calculated valence electron density of CaF2 on the (002) plane.

Extended Data Fig. 6 Electron density of CaF2 extended over multiple unit cells.

Bright dots correspond to Ca+2 ions centred on (002) plane while the light dots correspond to F ions centred on (004) plane but penetrating into the (002) plane.

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Lakhotia, H., Kim, H.Y., Zhan, M. et al. Laser picoscopy of valence electrons in solids. Nature 583, 55–59 (2020).

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