The work function (Φ), the minimum energy required to remove an electron from a crystal surface at 0 K, is a fundamental electronic property. Its precise determination is crucial for validating theories of surface electronic structures and understanding a wide range of surface and interfacial phenomena, including junction device behavior, charge-carrier injection, and surface catalytic interactions. While several techniques exist to measure Φ, including thermionic, field, and photoemission, these methods typically yield values with only two or three significant digits, limiting our understanding of its dependence on temperature, strain, or other factors. This low precision is partly due to the kinematic effect on electron emission, where only electrons directed normal to the surface can overcome the potential barrier. This effect broadens the threshold of electron emission, creating a smooth slope instead of a sharp step edge in photoelectron spectra, making precise Φ determination challenging. This research aims to address these limitations by applying angle-resolved photoemission spectroscopy (ARPES) with a laser-based source to improve the precision of work function measurements.

Extensive research has been dedicated to the measurement of the work function, leading to the tabulation of over 1000 values obtained through various methods (Kawano, 2008; Derry et al., 2015). However, these values generally lack high precision. The challenges in achieving high precision have been discussed in several studies (Cardona & Ley, 1978; Krolikowski & Spicer, 1969), focusing on the effects of surface contamination and the kinematic limitations of traditional photoemission techniques. Fowler's work (1931) on photoelectric sensitivity curves and later refinements (DuBridge, 1933; Jensen, 2007) have explored the smoothing of the emission threshold due to the angular distribution of emitted electrons. Previous studies using ultraviolet and X-ray photoemission spectroscopy (PES) have observed this smoothing effect, resulting in broad slopes (approximately 0.1 eV wide) instead of sharp step edges at the slow end of the photoelectron spectrum (Park et al., 1996; Helander et al., 2010; Akaike et al., 2014; Koitaya et al., 2012). These studies, while providing valuable insights, generally report work function values with two to three significant figures (Park et al., 1996; Koitaya et al., 2012) or, in some cases, four (Cardona & Ley, 1978; Pescia & Meier, 1982). The need for techniques capable of sub-meV precision has been highlighted by studies investigating the temperature (Kiejna, 1986) and strain dependence (Sekiba et al., 2007; Wang et al., 2010; Peng et al., 2012; Lanzillo et al., 2015; Wu et al., 2016) of the work function. Atomic-level resolved studies (Paggel et al., 2002) have also underscored the sensitivity of the work function to surface quality.

This study uses angle-resolved photoemission spectroscopy (ARPES) with a fiber-laser-based light source (hv = 5.988 eV) to measure the work function of Au(111). The laser's beam is precisely aligned and focused to control the trajectory of slow photoelectrons, a critical aspect for achieving high precision. The experimental setup consists of a fiber-laser source, a hemispherical electron analyzer, and a helium lamp for calibration. The Au(111) sample was prepared through cycles of Ar-ion bombardment and annealing, resulting in two samples with varying degrees of surface cleanliness. Angle-resolved photoemission measurements were performed with the sample at various temperatures (30-90 K) and emission angles. The data was acquired as a two-dimensional matrix of energy and emission angle. The slowest photoelectrons, which are those emitted normal to the surface, are the focus of the analysis. The energy distribution curves (EDCs) were fitted using a cutoff function (CF) consisting of a step function, a linear slope, and a Gaussian to extract the cutoff energy and width as functions of emission angle. This procedure allows for a precise determination of the work function. The effect of surface aging was assessed by observing changes in the cutoff energy over a prolonged period (10 hours). The absolute value of the work function was determined by referencing the photoelectron energy to the Fermi level of the sample, calibrated using helium-lamp ARPES. The researchers also used a kinematic model to explain the sharp parabolic distribution of the slow-side cutoff. This model considers the trajectory of the threshold photoelectrons and incorporates the effect of electric fields between the sample and electron lens. The model allows for deriving an analytical relationship between the cutoff energy, emission angle and the sample and analyzer work functions.

The key findings of this study include: 1. **Sub-meV Precision:** ARPES with the laser-based source achieved exceptionally high precision in work function measurements. The work function of the cleaner sample (sample 2) was determined to be 5.5553 eV with a standard deviation of ±0.4 meV. This represents a significant improvement over the precision of traditional methods. 2. **Sharp Parabolic Cutoff:** The slow-end cutoff in the ARPES spectra showed a sharp parabolic angular distribution, unlike the smooth slopes observed in conventional PES. The bottom of this parabolic distribution corresponds to the slowest photoelectrons emitted normal to the surface. 3. **Temperature Independence:** The work function of sample 2 exhibited minimal temperature dependence between 30 K and 90 K, with variations smaller than ±0.4 meV, corresponding to ±0.08k_B. This small temperature dependence contrasts with theoretical predictions that often estimate larger variations. 4. **Surface Aging Detection:** Surface aging effects were detected as a 5.5 meV downward shift in the cutoff energy after 10 hours, indicating the sensitivity of this method to subtle surface changes. 5. **Kinematic Model:** A kinematic model was developed to explain the observed sharp parabolic cutoff, considering the trajectory of the threshold photoelectrons and the electric field between the sample and the electron lens. This model shows that the curvature of the dispersion depends on the electric field between the sample and the analyzer while the bottom of the dispersion reflects the sample work function. 6. **Role of Monochromaticity:** The study clarifies the role of light monochromaticity. While precise determination of the Fermi level requires a monochromatic source, observing the parabolic cutoff itself and monitoring relative changes in work function can be achieved using non-monochromatic sources.

The results significantly advance our ability to measure work functions with unprecedented precision. The sub-meV accuracy achieved using the optimized ARPES technique allows for a much more detailed investigation of the factors influencing work function, including subtle surface effects, temperature variations, and strain. The minimal temperature dependence observed for Au(111) presents a stringent constraint for theoretical models that often predict more substantial temperature-dependent variations, suggesting a possible compensation between bulk and surface contributions. The detection of surface aging as a small meV shift demonstrates the method's sensitivity to environmental factors. The developed kinematic model provides a sound theoretical framework for understanding the unique features of the ARPES spectra and highlights the importance of considering the trajectory of photoelectrons in accurate work function determination. This improved precision in work function measurement opens up possibilities for investigating its behaviour in various complex scenarios that previously escaped accurate measurements.

This research has demonstrated a significant advancement in the precise measurement of the work function using laser-based ARPES. The sub-meV precision achieved is a major improvement over existing techniques, allowing for the detection of subtle changes in work function due to temperature variations and surface aging. The development of a kinematic model explaining the observed parabolic cutoff further strengthens the understanding of the underlying physics. Future research directions could involve applying this technique to study the work function under various conditions (ambient pressure, strain, phase transitions) and to investigate other material systems. The improved precision opens new avenues for studies that were previously hindered by the limitations of conventional measurement techniques.

While this study demonstrates exceptional precision for Au(111), the generalizability of this method to other materials might require careful optimization of experimental parameters, including the selection of the photon energy and the sample preparation procedures. The high vacuum conditions required for the measurements limit the direct study of surfaces under ambient conditions. The surface sensitivity of the technique also needs to be taken into consideration when interpreting the results as any inhomogeneity in work function will contribute to the width of the slow-end cutoff. Moreover, the interpretation of the results relies on the accuracy of the kinematic model, which may need further refinement for complex systems or under different experimental conditions.