Pushing the space-charge limit in electron momentum microscopy

To paraphrase a biological mantra [1]: in condensed-matter physics, if seeking to understand function, it is usually a good idea to study electronic structure. To do this, the most attractive experimental method is angle-resolved photoelectron spectroscopy (ARPES) as it provides a direct view of the momentum (i.e., direction)-dependent propagation behavior of the active electrons near the Fermi edge. ARPES, in fact, is nowadays an industry fueled by the widespread availability of powerful synchrotron-radiation sources, the commercial availability of high-throughput high-resolution spectrometers, and the continuing emergence of novel materials with intriguing electronic properties. This stream of discovery includes copper- and iron-based high-temperature superconductors, graphene, as well as topological insulators and semimetals. Yet, significant technological advances are also still possible as demonstrated by the recent development of a novel type of spectrometer: the time-of-flight (ToF) momentum microscope [2]. The paper by Schönhense et al [3] presents a correction scheme for resolution-limiting Coulomb interaction effects between photoelectrons inside this instrument, thus enabling its efficient use at pulsed high-brilliance photon sources.

The experimental task in ARPES is to image the E(k_∣∣) band structure of photoelectrons by measuring the energy (E) and direction (ϑ, φ or k_∣∣x, k_∣∣y) dependence of the photocurrent that is driven into the hemisphere above the sample surface by a monochromatic photon beam. It is the job of the spectrometer to provide an efficient, precise, invertible, and distortion-free mapping of the three initial parameters at the sample (E, ϑ, φ) onto measurable coordinates at the detector, i.e., typically two spatial coordinates (x, y). When photoemission is triggered by sufficiently short photon pulses, another viable coordinate is the arrival time of the photoelectrons at the detector (t). Thus, an ultimate simultaneous 'three-to-three' mapping (E, ϑ, φ) → (t, x, y) is in principle possible, however not with the common hemispherical analyzer whose entrance slit limits angular detection to one dimension [4] and at the expense of a pulse repetition rate limited to a few MHz for flight-time measurement.

In the ToF momentum microscope, the complete mapping is realized by directly projecting the (linearly scaled) momentum (k_∣∣x, k_∣∣y) image from the back focal plane of the objective lens of a photoelectron emission microscope through an energy-dispersing field-free drift section onto a two-dimensional delay-line detector [2]. Compared to angle-resolving ToF spectrometers [4, 5], elaborate transformations from the detector coordinates (t, x, y) via angles (ϑ, φ) to momenta (k_∣∣x, k_∣∣y) are not required. And since it is based on a 'real' microscope, the momentum microscope offers the additional advantage of spatially resolved and confined measurements [2, 6]. Resolution-degrading Coulomb interaction effects inside the instrument, however, can become stronger because all photoelectrons, the few primary ones from near the Fermi edge as well as the many secondary ones at low energies, are collected by the accelerating field between the sample and the objective lens. So, counterintuitively, high collection efficiency can effectively limit the information flow at a given resolution.

Figures 1 and 2 of the paper by Schönhense et al [3] vividly illustrate how significant space-charge-induced spectral distortions near the Fermi edge emerge when the number of emitted electrons per pulse is increased to about 10⁶. In particular, two effects are observed: a momentum-dependent energy shift of up to 10 eV on axis with a Lorentzian-shaped profile around the optical axis, and an increase in diffuse background intensity.

Schönhense et al [3] use ray-tracing simulations and a simple analytical mean-field model to rationalize the spatio-temporal space-charge dynamics in the accelerating and decelerating fields of the momentum microscope. Immediately after pulsed emission, all photoelectrons are confined to a small volume. The probability for individual scattering events is high, increasing with the number of emitted electrons. Thus, an increasing fraction of the fast electrons will be scattered to lower energies and different momenta. This can explain the diffuse background, which is similar to the one produced by indirect transitions in the photoemission process. Later on, slow and fast electrons become separated. The slow electrons stay near the optical axis and behind the fast electrons pushing them to higher kinetic energies, while their constant-energy shells expand to form the momentum image. In the decelerating field section of the microscope, the energy push becomes anisotropic, strongest for on-axis electrons, and the result is the observed two-dimensional Lorentzian-like energy profile centered on the optical axis.

An intriguing observation is that the measured bands and Fermi edge remain remarkably sharp despite the space-charge-induced energy shifts of several eV. This opens the way to space-charge correction. Based on the results of their analytical model, Schönhense et al [3] have developed an algorithm that can restore the energy distribution near the Fermi edge. When tested on the model system W(110), the algorithm indeed manages to recover fine details of the measured three-dimensional Fermi surface.

The common way to deal with space-charge effects has been to limit the number of emitted electrons per pulse by reducing the photon pulse intensity and thus increasing measurement time. The correction algorithm developed by Schönhense et al [3] can give us back some of that time, and it may inspire us to think about further creative ways to push the space-charge limit.

On a more general note, with the exploitation of parallel energy and dual momentum-component imaging, photoemission spectroscopists are joining the 'big-data' club. The flow of high-quality data is of the order of gigabytes per hour. On the one hand, this enables qualitatively new types of studies, such as complete E(k_∣∣, k_⊥) bulk band structure tomography by k_⊥ cross-sectioning, i.e., scanning the photon energy at a tunable high-brilliance soft x-ray source (see figure 5 and the 'k-space movies' in the supplementary material of the paper by Schönhense et al [3]), or in situ combined time-resolved pump–probe ARPES and x-ray photoelectron diffraction at emerging x-ray free-electron lasers with high repetition rates. On the other hand, we will face the common big data challenges of data acquisition, storage, analysis, and visualization, and of how to transform large data sets into research answers. The correction algorithm and automated data processing pipeline described by Schönhense et al [3] are a taste of things to come.

Acknowledgments