Work function seen with sub-meV precision through laser photoemission

The work function (φ) is a fundamental surface electronic property, defined as the minimum energy required to remove an electron from a solid to vacuum at 0 K. Accurate values of φ test theories of surface electronic structure and are central to understanding device junction behavior, carrier injection, and surface catalysis. Conventional thermionic, field-, and photoemission measurements yield relatively smooth emission thresholds due to kinematic constraints in electron emission, typically limiting φ to two or three significant digits. This low precision has hampered detailed investigations of φ-dependence on temperature, strain, and other perturbations. The key challenge is that, near threshold, only electrons emitted along the surface normal can surmount the barrier, causing angle-integrated spectra to display broad slopes rather than sharp steps. The present work asks whether angle-resolved photoemission spectroscopy (ARPES), by selecting emission direction, can isolate the truly slowest photoelectrons and measure φ with substantially higher precision. By optimizing a laser-based ARPES setup to detect slow electrons, the authors demonstrate that the slow-end cutoff becomes an intrinsically sharp step with a parabolic angular dispersion, enabling sub-meV precision in determining φ.

Prior literature established that emission thresholds are smoothed by kinematic effects (Fowler analysis) and by angle integration in ultraviolet/X-ray PES, resulting in ~0.1 eV-wide slopes and limiting precision of φ to 2–3 significant digits, occasionally four. Work function compilations (Kawano; Derry et al.) highlight variability across techniques and sensitivity to surface conditions. Theoretical and experimental studies address temperature dependence of φ and strain effects, but progress is constrained by measurement precision. Previous ARPES and PES works characterized band dispersions and Fermi cutoffs with sub-meV precision on the fast end, but not the slow-end threshold. The authors build on laser-based low-hν ARPES advances that allow controlled detection of ≤1 eV kinetic energy electrons and small beam spots, prerequisites for resolving slow-electron trajectories. Earlier work did not delineate the two key features necessary for reliable slow-end ARPES—parabolic angular dispersion and step-like cutoff independent of Fermi-Dirac and photon bandwidth broadening—criteria explicitly demonstrated here.

Samples: Two Au(111) single-crystal surfaces (samples 1 and 2) were prepared via cycles of 1.8-keV Ar-ion sputtering and 550 K annealing. Final annealing pressures were <7×10^−10 Torr (sample 1) and 4×10^−10 Torr (sample 2). Samples were cooled (~15 K/min) to 400 K and transferred to an ARPES chamber at 3.7×10^−11 Torr. Shockley surface states were characterized with helium-lamp ARPES. ARPES setup: A homemade Yb-doped fiber-laser light source with photon energy hν = 5.988 eV was coupled to a hemispherical analyzer (Scienta-Omicron DA30-L) equipped with a one-dimensional entrance slit, retractable He lamp (He Iα = 21.2180 eV), six-axis manipulator, and cryostat. The laser oscillator was stably mode-locked over months; beam diameter at the e-lens focal point was ~0.1 mm (set using a pinhole). Pass energy was 2 eV, and spectra were recorded by sweeping the retardation voltage in the e-lens. The wattage of the 5.988 eV probe was ~1 μW. The emission angle θ and photoelectron energy EPE−EF were acquired simultaneously, yielding E–θ maps. Temperatures were controlled from 30 to 90 K. Energy calibration: Absolute EPE−EF referenced to the sample EF in electrical contact with the analyzer was calibrated with He-lamp ARPES on evaporated Au at ~10 K by locating the Fermi cutoff at 21.2180 eV (He Iα). With this calibration and the known laser hν, the slowest-end cutoff energy equals the absolute work function φ of the sample, independent of the analyzer’s absolute work function (requiring only its stability). Measurement procedure: Fiber-laser ARPES mapped the slow-end cutoff across sample rotations R = 0°, 2.5°, 5.0°, 7.5°, observing the Shockley surface state truncated by the slow-end cutoff. The angular dependence of the cutoff was analyzed in E–θ space. Fitting and analysis: At each θ, energy distribution curves (EDCs) across the slow-end cutoff were fit with a cutoff function CF defined as a step-edge (Θ of a linear slope) convolved with a Gaussian G(ε; γ). The fit returns the cutoff energy εth(θ) and Gaussian FWHM γ. The θ-dependence of εth was compared to an analytical model of threshold photoelectrons. Model of threshold photoelectrons and dispersion: Electrons near threshold undergoing refraction at the surface experience kinematic constraints; at a critical angle they travel tangentially along the surface (threshold photoelectrons). Due to the work-function difference ΔΦ = Φs − Φe between sample (Φs) and analyzer entrance/e-lens coating (Φe), an electric field exists in vacuum that drags threshold electrons off the surface. Assuming a field predominantly along the surface normal (z), the parallel momentum is conserved and the z-momentum gained satisfies (ħkz)^2/2m = ΔΦ. The analyzer observes a nominal emission angle tanθ = kth/kz, giving the angle-resolved cutoff dispersion: Eth − EF = Φs + ΔΦ tan^2θ. Thus, the cutoff bottom at θ = 0° gives φ = Φs, and the curvature determines ΔΦ. Under applied negative bias −v/e to the sample, the dispersion generalizes to E−EF = v + (ΔΦ + v) tan^2θ, predicting increased curvature with bias. The model implies that the slow-end cutoff sharpness is unaffected by Fermi-Dirac broadening or photon bandwidth. Controls and additional considerations: The analyzer and sample were commonly grounded; Schottky lowering Δφs due to residual fields, estimated from dispersion curvature (ΔΦ ≈ 0.9 eV) and geometry (za−zs ≈ 32 mm), was ~0.2 meV, below measurement scatter. Under large applied bias (v ≥ 25 eV), Schottky lowering can exceed 1 meV. Background signals (bulk bands and scattered electrons) were also truncated by the same cutoff, indicating a homogeneous surface characterized by a single φ; multiple edges would indicate patchiness. Temperature-dependent measurements (30–90 K) assessed the stability and temperature dependence of φ.

- ARPES resolves an intrinsically sharp, step-like slow-end cutoff with a parabolic angular dispersion that bottoms at normal emission (θ = 0°), in agreement with the threshold-photoelectron model E−EF = Φ + ΔΦ tan^2θ. - Sample 1 (Au(111)) at 30 K: The four rotational datasets (R = 0°, 2.5°, 5.0°, 7.5°) yield overlapping parabolic dispersions. The cutoff energy near θ = ±0.5° (115 points) averaged to φ = 5.5034 eV with a standard deviation σ = 1.2 meV; the curvature corresponds to ΔΦ ≈ 0.9 eV. The fitted cutoff width FWHM γ = 10.8 ± 3.4 meV. - Surface aging: After 10 h in the spectrometer, the slow-end cutoff shifted downward by 5.5 meV, consistent with reduced φ due to weak physisorption by residual gas; the Fermi cutoff remained stable, indicating analyzer stability. - Sample 2 (Au(111)) temperature series (30–90 K): The slow-end cutoff remained sharp (γ ≈ 8.3 ± 1.0 meV near θ = ±0.5°). The work function at the slowest end remained at φ = 5.5553 eV with σ = 0.4 meV over 30–90 K, i.e., no detectable temperature dependence within ±0.4 meV over 60 K (|dφ/dT| < ±0.08 kB). This value is higher than sample 1 and comparable to the highest reported Au(111) φ ≈ 5.6 ± 0.1 eV from Fowler plots. - Independence from Fermi-Dirac and photon bandwidth: The slow-end cutoff’s sharpness is not broadened by temperature or by the light bandwidth; the observed width is set by analyzer resolution and surface/analyzer work-function inhomogeneity (γ = (ψs + ψa)/2). The slightly larger γ for sample 1 suggests greater surface inhomogeneity. - Bias-dependent demonstration (supplementary): On HOPG at room temperature and 6×10^−10 Torr, with −v/e = 1.62 V, the slowest-end cutoff width was ~8.0 meV and the slowest-end energy near θ = ±0.5° was leveled within σ ≈ 0.15 meV as v was varied (1–4 batteries), demonstrating sub-meV precision under bias.

By resolving the emission direction, ARPES isolates threshold photoelectrons that define the true slow-end cutoff, overcoming the Fowler-type smoothing inherent to angle-integrated PES. The observed parabolic angular dispersion and sharp step-like cutoff directly reflect the kinematics of photoelectron refraction at the surface and acceleration in the vacuum field set by the work-function difference between sample and analyzer entrance. The cutoff bottom gives the absolute φ, while the curvature encodes ΔΦ. Since the slow-end cutoff is unaffected by Fermi-Dirac broadening and photon bandwidth, φ can be determined with sub-meV precision limited mainly by analyzer resolution, voltage stability, and surface/analyzer work-function homogeneity. The negligible temperature dependence of φ in Au(111) from 30–90 K (|dφ/dT| < ±0.08 kB) imposes strong constraints on theories that typically predict O(kB) behavior and suggests compensation between bulk and surface contributions to dφ/dT. The sensitivity also enables detection of subtle surface aging effects at the meV level.

The study demonstrates that laser-based low-hν ARPES can measure the work function with unprecedented precision by directly observing the slowest-end cutoff: a sharp, step-like feature with a parabolic angular dispersion bottomed at normal emission. Applying a kinematic model connects the cutoff bottom to φ and the curvature to the vacuum-level slope (ΔΦ). For Au(111), φ = 5.5034 ± 0.0012 eV (sample 1) and 5.5553 ± 0.0004 eV (sample 2) were obtained; temperature dependence from 30–90 K was undetectable within ±0.4 meV, and a 5.5 meV aging-induced shift was resolved. These results open precise investigations of the fifth significant digit of φ and enable monitoring of subtle surface phenomena. Future work could deploy this approach under ambient/near-ambient pressures, during controlled strain, across phase transitions/crossovers, and under femtosecond optical excitation, as well as extend to diverse materials and engineered interfaces.

- The absolute φ pertains to the specific prepared Au(111) surfaces; achieving truly ideal, contamination-free surfaces is practically impossible, and φ is highly sensitive to atomic-scale cleanliness and adsorbates. - The method assumes surface homogeneity within the illuminated area; patchiness would produce multiple edges and broaden the cutoff. - Precision depends on analyzer stability, voltage stability, and beam size/focusing to ensure controlled electron trajectories; slow electrons are sensitive to stray fields. - Under large applied biases, Schottky lowering can exceed 1 meV and must be accounted for. - Demonstrations focused on Au(111) (and supplementary HOPG); generalization to other materials may require surface-specific optimizations.