# Work function seen with sub-meV precision through laser photoemission

Sep 11, 2020

### Laser-ARPES setup for slow photoelectrons

Slow photoelectrons are vulnerable to fields and their trajectories can easily be bent by the electric and magnetic fields remaining in the spectrometer24. Particularly, the slowest photoelectrons are those that have once been at rest on the outer surface, and taking control of their trajectory from the surface to the electron analyser is the most difficult. As will be explained later, the genuine cutoff of the slowest end seen in ARPES should exhibit the following two features (Fig. 1a). (1) The slow-side cutoff shows a parabolic angular dispersion. (2) The cutoff is a step edge whose width is not broadened by Fermi-Dirac distribution nor by the bandwidth of light. We set these two features, which were not described in the previous studies of the work function1,32, as the criteria for the successful ARPES on the slow photoelectrons.

The technical difficulty to perform reliable ARPES on slow photoelectrons is gradually being removed, and currently, slow photoelectrons of 1 eV kinetic energy can be passed through modern electron analysers in a controlled trajectory. This advance has partly been propelled by the advent of low-hν sources of 10 eV based on lasers27,30,33,34,35, wherein the detection of slow photoelectrons becomes a mandatory. To attain a fair angular resolution particularly when the photoelectrons are slow, it is requested to keep the beam diameter less than 0.3 mm at the focal point of the electron lens (e-lens) attached to the analyser, and the laser-based sources can easily meet this demand.

We set up a fibre-laser-based source of hν = 5.988 eV29 and docked it to an ARPES spectrometer equipped with a helium lamp (Fig. 1b; also see Supplementary Note 1 and Supplementary Fig. S1). The beam diameter at the focal point of the e-lens was set to ~0.1 mm through a procedure that utilises a pin hole30. The oscillator of the fibre-laser source was stably mode locked for at least three months, which ensures that the profile of the laser such as the photon energy and band width was locked during the measurements. While fibre-laser ARPES was used to detect the slow photoelectrons, helium-lamp ARPES was also used to characterise the band dispersions and to calibrate the photoelectron energy EPE − EF referenced to the Fermi level of the sample in electrical contact to the analyser; see ‘Methods’.

### Preparation of Au(111)

As a model system, we investigated the (111) surface of gold. Gold is a noble metal and is an exemplary metal for electrodes. Studies of the work function of gold begun partly in pursuit of a reference standard36,37, and the values for Au(111) have been found to be in 5.2–5.6 eV10,11,25,38. It is well known that the work function is sensitive to the surface quality at the atomic level, and there is consensus that the cleaner the surface is, the higher the work function is, especially for materials with the values greater than ~4 eV39. We prepared two samples of Au(111) (samples 1 and 2) through cycles of Ar-ion bombardment and annealing (‘Methods’). The dispersion of the Shockley surface state formed on Au(111)40,41,42 was observed by using helium-lamp ARPES (Fig. 1c). The band dispersion for sample 2 was slightly sharper than that for sample 1 (Supplementary Fig. S2 and Supplementary Note 2). As we shall see later, sample 2 indeed exhibited higher ϕs than that of sample 1.

### Slow end seen with fibre-laser ARPES

The panels in Fig. 2a–d show the photoelectron distributions obtained when Au(111) of sample 1 was illuminated by the fibre-laser source. The fastest photoelectrons formed the Fermi cutoff at the known photon energy 5.988 eV of the source, which ensures the accuracy of EPE − EF calibrated by using the He Iα line. As the sample was rotated from R = 0 (Fig. 2a; normal-emission configuration) to 7.5° (Fig. 2d), the Shockley surface band became fully visible up to the Fermi cutoff. However, the bottom of the band was truncated by the cutoff on the slow side because the electrons excited from the bottom could not overcome the work function. The visibility of the Shockley band down to the cutoff provides a credit that our ARPES setup was operating agreeably for the slowest photoelectrons even without applying any bias voltage on the sample; also see Supplementary Fig. S2 and Supplementary Note 2.

Our main observations for Fig. 2a–d are of the slow side cutoff. First, the cutoff energy depended on the emission angle of the photoelectrons (θ) and exhibited a parabolic dispersion bottomed at the surface normal, θ = 0°. Second, the slow-side cutoff was a sharp step edge in contrast to the 0.1-eV-wide slopes observed in ultraviolet and X-ray PES23,24,25,26. The parabolic cutoff is distinct from the parabolic boundary that occurs in the photoelectron distribution mapped onto energy–momentum (Ek) plane. As is well known, the mapping function43 has a range that sets a parabolic boundary in Ek plane whose bottom occurs not in the surface normal but in the direction towards the e-lens. Such a boundary in Ek plane can be seen for example in Bovet et al.44 The cutoff herein is observed in Eθ plane and its bottom followed the surface normal when the sample was rotated from R = 0 to 7.5°. Therefore, the parabolic cutoff is intrinsic to the emission from the surface.

In order to quantify the sharp-and-parabolic cutoff, we performed a fitting procedure and extracted the energy and width of the cutoff as functions of θ. In the procedure, we first set a cutoff function (CF), which was a step distribution function of a linear slope convoluted with a Gaussian; for the explicit form of CF, see ‘Methods’. Then, with the CF, we fitted the energy distribution curves (EDCs) of the data shown in Fig. 2a–d. The case for the R = 0° distribution is presented in Fig. 2e. The four parabolic curves nicely overlapped each other (Fig. 2f) and the full width at half maximum of the Gaussian (FWHM: γ) was averaged as γ = 10.8 ± 3.4 meV (Fig. 2g). The cutoff energy around θ = ±0.5°, in which there are 115 data points, was averaged as 5.5034 eV with one standard deviation of σ = 1.2 meV (inset to Fig. 2f). The energy at the bottom of the parabolic cutoff will subsequently be identified as the absolute value of the work function. Note, the overlap of the four dispersions seen in Fig. 2f shows that the dispersion shifted with the known interval of 2.5° along the emission angle axis and ensures the angular scaling of θ.

With the sharp cutoff as a measure, we were able to discern the aging of the surface. After keeping Au(111) of sample 1 in the spectrometer for 10 h, the sample surface became less clean and the cutoff shifted downwards; see, Supplementary Fig. S3 and Supplementary Note 3. The shift was as small as 5.5 meV and was attributed to the reduction in the work function due to residual gas weakly physisorbed on the surface. The reduction is consistent with the trend that the work function lowers as the surface becomes less clean39. There was no discernible shift of the Fermi cutoff during the 10 h, which indicated that the analyser condition was stable during the measurement; see ‘Methods’.

In a separate run of the measurement on sample 2, we varied the temperature from 30 to 90 K and monitored the cutoff (Fig. 3a–c). In contrast to the Fermi cutoff, the slow-end cutoff remained sharp (Fig. 3d). The width of the slowest end around θ = ±0.5°, in which there are 45 data points, was γ = 8.3 ± 1.0 meV and the cutoff energy of the slowest end stayed at 5.5553 eV with one standard deviation of σ = 0.4 meV (Fig. 3e). That is to say, there was no temperature dependence in the work function with the precision of  ±0.4 meV/60 K = ±0.08kB, where kB is the Boltzmann constant. The absolute value 5.5553 eV was higher than the value 5.5034 eV for sample 1 and is comparable to the highest reported work function of 5.6 ± 0.1 eV on Au(111) obtained through a Fowler plot38. Thus, with the work function as the measure39, the surface quality of sample 2 was better than that of sample 1 and was comparable to that studied in Pescia and Meier38.

Before explaining why the cutoff on the slow side appears sharp and parabolic, we point out that the cutoff is not only truncating the Shockley band but also the background signal, as clearly seen at θ > 10° in Fig. 2d. The background signal could originate from bulk bands as well as photoelectrons that have encountered some scattering45. This observation indicates that, when explaining the features of the cutoff, the photoelectrons forming the Shockley band and background signal have to be treated on equal footings. We thus consider a model for whatever photoelectrons that pass through a homogeneous surface characterised by a single work function ϕs. If there had been multiple edges in the spectrum24, the surface would have been judged as non-uniform, or patchy12,24.

### Trajectory of the threshold photoelectrons

The sharp-and-parabolic appearance of the cutoff can be understood by considering the trajectory of ‘threshold photoelectrons’. Below, we first explain the photoelectron refraction, or the kinematic constraints upon the emission across the surface1,19,20, and define the threshold photoelectrons. Then, we consider their trajectories from the surface to the entrance of the e-lens that collects the photoelectrons into the hemispherical analyser.

When passing through the surface, photoelectrons are refracted because the work function acts as a potential barrier. As shown in Fig. 4a, the angle of refraction becomes large as the angle of incidence increases, and at a critical angle, the photoelectrons travel tangential to surface. We call these tangential photoelectrons the threshold photoelectrons. Their kinetic energy on the surface is ({varepsilon }_{{rm{s}}}^{{rm{th}}} = {(hslash {{bf{k}}}_{{rm{s}}}^{{rm{th}}})}^{2}/2m ge 0), where m is the electron mass and (hslash {{bf{k}}}_{{rm{s}}}^{{rm{th}}}) is the momentum that is parallel to the surface by definition. The energy level of a threshold photoelectron ({E}_{{rm{PE}}}^{{rm{th}}}) can be described as (Fig. 4b),

$${E}_{{rm{PE}}}^{{rm{th}}}={E}_{{rm{F}}}+{phi }_{{rm{s}}}+{varepsilon }_{{rm{s}}}^{{rm{th}}}ge {E}_{{rm{F}}}+{phi }_{{rm{s}}}equiv {V}_{{rm{vac}}}^{{z}_{{rm{s}}}},$$

(1)

where ({V}_{{rm{vac}}}^{{z}_{{rm{s}}}}) is the vacuum level just outside the surface.

The threshold photoelectrons cannot reach the e-lens as long as they are travelling tangential to the surface. Here, we are reminded that, even when the sample and e-lens are electrically connected, electric fields can exist between the two; see Fig. 4b. The vacuum level is a solution to the Poisson equation, ({nabla }^{2}{V}_{{rm{vac}}}({bf{r}})=0), with the boundary condition set by the work function on the vacuum boundary. Thus, Vvac differs by Δϕ = ϕs − ϕa between the sample and entrance of the e-lens, where ϕa is the work function of the material that coats the interior of the e-lens and analyser. When Δϕ > 0, the threshold photoelectrons can take off the surface and be dragged towards the e-lens. Their kinetic energy at the e-lens entrance becomes ({varepsilon }_{{rm{a}}}^{{rm{th}}}={varepsilon }_{{rm{s}}}^{{rm{th}}}+Delta phi) (Fig. 4b). For the case when Δϕ < 0, see later.

Analytic solutions for the trajectory exist when we can regard the electric field to be directed along the surface normal (z) (Fig. 4c). This arrangement is similar to an infinitely large parallel-plate capacitor, but each plate is made of different materials. Then, while dragged, the momentum parallel to the surface is unchanged. At the e-lens entrance, the momentum along z ((hslash {k}_{{rm{a}}}^{perp })) can be obtained through ({(hslash {k}_{{rm{a}}}^{perp })}^{2}/2m=Delta phi), and the nominal emission angle (θ) seen by the analyser becomes (tan theta =| {{bf{k}}}_{{rm{s}}}^{{rm{th}}}| /{k}_{{rm{a}}}^{perp }) (Fig. 4c). Thus, equation (1) can be described as

$${E}_{{rm{PE}}}^{{rm{th}}}-{E}_{{rm{F}}}={phi }_{{rm{s}}}+Delta phi {tan }^{2}theta .$$

(2)

Equation (2) illustrates that, when entering the e-lens, the energy of the threshold photoelectron exhibits a parabolic angular dispersion, and this is the dispersion detected by the analyser. In Fig. 2f, we overlaid the curve of equation (2) with Δϕ = 0.9 eV, which determines the curvature, and the bottom of the dispersion ϕs = 5.5034 ± 0.0012 eV is identified as the absolute value of the work function.

For completeness, we present the dispersion of the angle-resolved cutoff when the sample is applied with negative bias voltage  −v/e with respect to the analyser:

$${E}_{{rm{PE}}}^{{rm{th}}}-{E}_{{rm{F}}}={phi }_{{rm{s}}}+v+(Delta phi +v){tan }^{2}theta .$$

(3)

Here, EF + v becomes the Fermi level of the sample. When Δϕ + v > 0, the threshold photoelectrons are dragged towards the e-lens and the intrinsic cutoff becomes visible. With increasing v, the dispersion shifts upwards in energy and its curvature (prefactor of ({tan }^{2}theta)) is increased. The increase of the curvature is in accordance to the photoelectron acceptance cone being tunable with v46,47. At v 25 eV, the lowering of the work function due to the Schottky effect12 can exceed 1 meV and may prevail when seen with the sub-meV precision; see ‘Methods’. When integrated over a certain anglular cone about the normal emission, the cutoff is smoothed into a slope (Fig. 1a right schematic) as seen in ultraviolet and X-ray PES23,24,25,26. This slope seen in the normal-emission configuration is the lineshape formulated in  Cardona and Ley1 and in Krolikowski and Spicer20 that took the Fowler effect into account, although the role of Δϕ was not explicated. According to Eq. (3), the width of the slope in the integrated spectrum becomes wider as v is increased, while the angle-resolved cutoff remains sharp; see Fig. 1a for the relationship between the curvature seen in ARPES and width of the sloped region seen in PES.

The demonstration of the sub-meV precision measurement under the biased condition is presented in Supplementary Fig. S4: We performed the measurement at room temperature and at the pressure of 6 × 10−10 Torr on the surface of an exfoliated highly-oriented pyrolitic graphite29. The width of the slowest-end cutoff was as narrow as γ = 8.0 meV when one battery of v/e = 1.62 V was attached, and the energy of the slowest end around θ = ±0.5° was leveled within one standard deviation of σ = 0.15 meV when the number of the attached batteries was varied from one to four. For the details of the demonstration, see Supplementary Note 4.

### On the role of the monochromaticity

It was shown in the previous section that the parabolic dispersion of the slow-end cutoff depends solely on how the threshold photoelectrons were dragged from the outer surface to the e-lens; it does not depend on how the threshold photoelectrons were generated. Whatever the value of hν may be, the threshold photoelectrons emerged on the outer surface line up on the identical dispersion when entering the e-lens. This point is implicit in Eqs. (1)–(3) as well as the definition of the work function ϕs = Es − EF being independent of hν. Thus, the cutoff on the slow end is not blurred by the bandwidth of light (γhν) besides not being affected by the temperature dependence of the Fermi-Dirac function. Therefore, the cutoff can be observed with the resolution (γa) set by the analyser and the stability of the bias voltage. This point is in strong contrast to the bands and Fermi cutoffs seen with the convoluted resolution ({({gamma }_{{rm{a}}}^{2}+{gamma }_{hnu }^{2})}^{1/2}). The width of the slow-end cutoff being slightly larger for sample 1 (10.8 ± 3.4 meV) than that for sample 2 (8.3 ± 1.0 meV) can be attributed to the degree of inhomogeneity of the work function (γϕ) within the probed area set by the  ~0.1-mm beam size. That is to say, the width of the cutoff seen in ARPES is (gamma ={({gamma }_{{rm{a}}}^{2}+{gamma }_{phi }^{2})}^{1/2}). When γϕ → 0, the width γ becomes the direct measure of γa.

The only but important role for the light to be monochromatic was to precisely locate the energy level of the Fermi cutoff Ef, which was the requisite to refer to the absolute value of the work function1,23; see ‘Methods’. If the purpose was only to observe the parabolic cutoff and monitor the relative value of the work function, the monochromaticity of the source is not needed; whatever sources that can generate an ensemble of excited electrons in the crystal, or an ensemble of threshold photoelectrons travelling along the outer surface, can be utilised. For example, synchrotron light can be used32 even when its photon energy is drifting; intense femtosecond infra-red pulses that can generate muti-photon photoelectrons can be used48,49 provided that the intense field of the pulse does not significantly alter the work function12; deuterium lamps and electron guns, the latter in the setup of momentum-resolved electron-energy loss spectroscopy50,51, may also be used if the beam size can be reduced sufficiently.