Strong-Field Bloch Electron Interferometry for Band-Structure Retrieval

Tobias Weitz, Christian Heide, and Peter Hommelhoff

Phys. Rev. Lett. 132, 206901 – Published 14 May 2024

Abstract

When Bloch electrons in a solid are exposed to a strong optical field, they are coherently driven in their respective bands where they acquire a quantum phase as the imprint of the band shape. If an electron approaches an avoided crossing formed by two bands, it may be split by undergoing a Landau-Zener transition. We here employ subsequent Landau-Zener transitions to realize strong-field Bloch electron interferometry, allowing us to reveal band structure information. In particular, we measure the Fermi velocity (band slope) of graphene in the vicinity of the K points as $(1.07 \pm 0.04) nm {fs}^{- 1}$ . We expect strong-field Bloch electron interferometry for band structure retrieval to apply to a wide range of material systems and experimental conditions, making it suitable for studying transient changes in band structure with femtosecond temporal resolution at ambient conditions.

Received 26 September 2023
Revised 11 April 2024
Accepted 20 April 2024

DOI:https://doi.org/10.1103/PhysRevLett.132.206901

Physics Subject Headings (PhySH)

Ballistic transport Band gap Density of states Electrical conductivity Optical interactions with surfaces Photocurrent Photogalvanic effect Stark effect

Graphene

Band structure methods Bloch wave theory

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Tobias Weitz ^1,*, Christian Heide ^1,2, and Peter Hommelhoff ^1,†

¹Department of Physics, Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Staudtstrasse 1, D-91058 Erlangen, Germany
²Stanford PULSE Institute, SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA

^*Corresponding author: tobias.weitz@fau.de
^†Corresponding author: peter.hommelhoff@fau.de

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Vol. 132, Iss. 20 — 17 May 2024

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Images

Figure 1
Field-driven electron dynamics in graphene and the influence of the hopping parameter $γ$ . (a) With increasing $| γ |$ , the linear slope, i.e., the Fermi velocity $v_{F} = \frac{3}{2} ℏ^{- 1} | γ | a$ , increases. For instance, at $γ = - 2.8 eV$ , $v_{F} = 1.03 {nm fs}^{- 1}$ whereas at $γ = - 3.0 eV$ , $v_{F} = 1.11 {nm fs}^{- 1}$ . (b) Dirac conelike dispersion around the K point with valence (blue) and conduction band (orange). The intraband trajectory of an exemplary electron-hole pair (red and blue circles) starting at $k_{0} = [14.1, 6.6] {nm}^{- 1}$ and driven by the vector potential shown in (c) is shown as double arrows (blue in valence band, red in conduction band). (c) Electric field $E (t)$ (red line) and its associated vector potential $A (t)$ of a few-cycle light pulse. (d) An electron starting in the valence band (blue circle) can undergo two different pathways within one optical cycle: a Landau-Zener transition after a quarter of an optical cycle followed by three-quarters of intraband motion, or vice versa. The final state occupation depends on the interference of both pathways with conduction (valence) band occupation for constructive (destructive) interference.
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Figure 2
Subcycle dynamics as a function of the hopping parameter $γ$ . For this showcase, a 1.2 cycle waveform at 800 nm (1 cycle $= 2.7 fs$ ) is employed with a peak optical field strength $E_{0} = 3 {V nm}^{- 1}$ and a carrier-envelope phase (CEP) $φ_{CE} = - π / 2$ [see Fig. 1]. (a) Instantaneous band gap $ϵ^{cv} (t)$ experienced by the electron-hole pair indicated in (a). We show three cases for a choice of $γ = [- 3.0, - 2.9, - 2.8] eV$ [dark to light green, compare to Fig. 1]. Time integration over the green shaded area yields the dynamic phase $ϕ^{cv} (t)$ (modulo $ℏ$ ), which is shown on the right axis, starting at $ϕ^{cv} (- 2 cycles) = 0$ . (b) Absolute value of the associated transient interband dipole coupling $V_{k}^{cv} = E (t) \cdot d^{cv} (k)$ (blue line) induced by $E (t)$ (red line). We note that $d^{cv}$ is independent of $γ$ . Regions of likely Landau-Zener tunneling events refer to the maxima in $| V_{k}^{cv} |$ , which are indicated as gray dashed lines through all panels. (c) Conduction band population $ρ^{c}$ resulting from the analytical model (full lines) and the TDSE (dashed lines). Crucially, the residual $ρ^{c}$ (circles) differ significantly even for slight changes in $γ$ due to the interferometrically high sensitivity of the electron’s dynamic phase on band modifications.
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Figure 3
Carrier-envelope phase-dependent current in monolayer graphene. (a) Experimentally obtained CEP-dependent current (red circles with error bars) as a function of the optical peak field strength $E_{0}$ . Error bars indicate the standard deviation of the mean of 33 independent samples. The three green lines are obtained from numerical TDSE computations with the absolute current magnitude as only free parameter, with hopping parameters $γ$ as indicated. Inset: $χ^{2}$ distribution from fitting the simulated data to the experimentally obtained data points. The dashed white ellipse indicates the $1 σ$ interval around the global minimum marked by the full white lines. (b) Sketch of the experimental setup. Epitaxially grown monolayer graphene on a SiC substrate is illuminated with tightly focused CEP-stable 2-cycle laser pulses. Induced CEP-dependent currents are measured via two gold electrodes attached to the graphene.
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