Momentum resolution in inverse photoemission A. Zumbülte, A. B. Schmidt, and M. Donatha) Physikalisches Institut, Westfälische Wilhelms-Universität Münster, Wilhelm-Klemm-Str. 10, 48149 Münster, Germany

(Received 8 December 2014; accepted 12 January 2015; published online 29 January 2015) We present a method to determine the electron beam divergence, and thus the momentum resolution, of an inverse-photoemission setup directly from a series of spectra measured on Cu(111). Simulating these spectra with different beam divergences shows a distinct influence of the divergence on the appearance of the Shockley surface state. Upon crossing the Fermi level, its rise in intensity can be directly linked with the beam divergence. A comparison of measurement and simulation enables us to quantify the momentum resolution independent of surface quality, energy resolution, and experimental geometry. With spin resolution, a single spectrum taken around the Fermi momentum of a spin-split surface state, e.g., on Au(111), is sufficient to derive the momentum resolution of an inverse-photoemission setup. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4906508]

I. INTRODUCTION

Experimental access to the electronic structure of solids is most directly provided by photoelectron spectroscopy (PES) for the occupied1 and bremsstrahlung spectroscopy for the unoccupied states.2–5 The first analyzes the energy distribution of the electrons, which are emitted after photon irradiation. The latter detects the radiation emitted after electron bombardment of the sample. Early experiments revealed the density of states D(E), without resolving the dependence of the electronic structure on the wave vector k, i.e., the energymomentum dispersion relation E(k). In photoelectron spectroscopy, k resolution is realized by analyzing the emission angle of the electrons in the detection channel. In order to gain k resolution in bremsstrahlung spectroscopy, a defined electron incidence angle in the excitation channel is needed, which demands an electron beam with small divergence. Since E and k are intimately connected in a solid, a finite momentum resolution influences the energy resolution of the measured spectra and vice versa. The momentum resolution is directly linked to the angular resolution, yet it is a function of the electron energy. The higher the electron energy, the higher are the requirements on the angular resolution to keep the same momentum resolution. The great success story of angle-resolved photoemission (ARPES) for the determination of the valence band structure of solids, which is based on direct optical interband transitions, was made possible by the use of vacuum ultraviolet (VUV) light. This led to sufficient k resolution, with respect to the size of the Brillouin zone, at reasonable angular resolution of about 1◦ for the emitted electrons. The counterpart, bremsstrahlung spectroscopy in the VUV region, was introduced in the late 1970s by using a bandpass-type Geiger-Müller counter sensitive to radiation of about 10 eV.6 It was first called bremsstrahlung isochromat spectroscopy (BIS), as is

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still the case for high electron energies, but later renamed as inverse photoemission (IPE) for radiation in the VUV range.7,8 Evidence of direct interband transitions in IPE was found in experiments, where the mean angle and the angular range of the incident electrons were varied.9,10 This was the start of k-resolved inverse photoemission. The accessible momentum range typically covers the full Brillouin zone. The energy resolution of ARPES was pushed from some tenth of an eV into the µeV range, the angular resolution to below 0.2◦.1 With these parameters, the study of systems such as high-TC superconductors11 with ARPES became feasible. The use of display analyzers, i.e., the parallel detection of kinetic energy and emission angle of the photoelectrons over a wide range, now allows for “snap shots” of a large part of the band structure E(k). Beautiful measurements of, e.g., Dirac cones on graphene12,13 and topological insulators,14,15 as well as spinorbit-split surface states16,17 are the result of this development. Although IPE is just the time-reversed counterpart of PES, technical improvements in IPE face two fundamental problems: (i) Due to phase space arguments, the cross section for IPE in the VUV range is reduced by a factor of 105 compared with PES.7 (ii) The simultaneous measurement of different electron energies and angles is not possible in IPE because these important parameters are in the excitation channel. This requires subsequent measurements of different electron energies and angles in IPE. In addition, due to the surface sensitivity of the technique, the measurement time is limited to minutes or hours, before the sample surface becomes contaminated by adsorption of residual gas. Therefore, all technical advances in IPE have to be evaluated with respect to their practical benefit, i.e., the detectable photon count rate. Nevertheless, the energy resolution of state-of-the-art IPE was improved to 165 meV at still reasonable count rates.18 The experimental investigation of spin-dependent electronic states due to exchange or spin-orbit interaction called for spin-polarization analysis in PES and spin-polarized electron sources in IPE. The intensity loss accompanied with spin resolution is in the order of 10−2–10−4 in PES, while spinpolarized electron sources provide equivalent sample currents

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as in the spin-integrated case. Only the need for two independent measurements for the two spin directions and the noncomplete electron-beam spin polarization leads to somewhat longer data acquisition times. This considerably smaller intensity loss in IPE makes spin-resolved IPE attractive despite its limitations.19–21 While there are numerous papers in the literature on energy and spin resolution in IPE, there is much less information on momentum resolution and how to determine it for a specific experimental setup. In this article, we give an overview on momentum resolution in IPE and how it is connected to the angular resolution, i.e., the electron beam divergence. We evaluate measuring the beam divergence via a Faraday cup. Then, we present a method to determine the electron-beam divergence directly from a series of IPE spectra for Cu(111). We simulate IPE spectra with varying angular resolution and compare them with measured data. We develop criteria for determining the beam divergence from this comparison. Employing our method, we find a total beam divergence of ∆θ = 3.9◦ ± 0.5◦ for our IPE setup, which relates to a momentum −1 resolution of ∆k ∥ ≈ 0.04 Å at the Fermi energy EF for typical values of the sample work function.

II. ANGULAR VS. MOMENTUM RESOLUTION

For measurements of E(k), the momentum resolution is equally as important as the energy resolution. The angular resolution, which in ARPES is typically given by the diameter of the entrance aperture into the photoelectron spectrometer (or, more recently, the capability of the display analyzer), defines the momentum resolution for a fixed photon energy. The influence of the momentum resolution on ARPES spectra was analyzed in detail, e.g., by Matzdorf:22 Measurements of the Shockley-type surface state on Cu(111) show a broadening of the peak for increasing aperture width. In addition, at the band minimum, a shift of the energetic peak position is detected. It was shown how the influence of the angular resolution depends on the gradient of the E(k) dispersion relation. An analogous influence is expected in IPE. A typical experimental setup for IPE consists of an electron gun and a photon detector. With these, two operation modes are possible. The spectrometer mode uses a fixed electron energy, while the energy of the detected photons is varied. This requires a grating spectrometer in the VUV range with high sensitivity. The energy resolution is connected with the width of the entrance slit, which limits the solid angle for the detected photons. This severe limitation cannot be compensated by the gain in intensity by simultaneous detection of different photon energies. Therefore, IPE in the isochromat mode has become more widespread. Here, the kinetic energy of the electrons is varied, while the emitted photons are detected at a fixed photon energy, typically with a highly sensitive Geiger-Müller counter serving as a bandpass filter. As in ARPES, only the component of the electron wave vector, which is parallel to the surface, is conserved during traversing the vacuum-solid interface (apart from energetically restricted surface-umklapp processes). With the angle of incidence θ, the work function of the sample Φ, the photon energy

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~ω, and the final state energy E f , the parallel momentum k ∥ is given by k∥ =

2m (~ω + E f − Φ) sin θ, ~2

(1)

which is analogous to the formula used in ARPES for the timereversed process. This formula shows the relationship between angle and momentum. As a consequence, the angular resolution caused by the angular spread of the incident electron beam determines the momentum resolution. Improving the k resolution in IPE is considerably more challenging than in ARPES: A parallel electron beam with low kinetic energy (7 to 15 eV) and high current (µA range) is needed. To produce a parallel beam, the emission spot of the electron source should be minimized to represent a point source as closely as possible. Any deviation from a point source limits the angular resolution, not to mention space charge effects. Additionally, the subsequent electron optics has to be designed in such a way that it preserves a parallel electron beam for varying kinetic energies. In general, the angular resolution is, therefore, a function of the electron energy. Equation (1) contains the photon energy ~ω. Low photon energies result in higher momentum resolutions at a given angular resolution, yet they limit the accessible momentum range. For a given final state energy E f in IPE (equivalent to the initial state energy Ei in ARPES), different photon energies ~ω1 and ~ω2 correspond to different k1 and k2, indicated by the different lengths of the corresponding vectors in Fig. 1(a). The same angular resolution ∆θ in both cases leads to quite different momentum resolutions ∆k1 and ∆k2. Low photon energies of about 10 eV, as used in IPE, are favorable in comparison with higher photon energies commonly used in ARPES experiments with gas discharge lamps or synchrotron radiation. This important relation is demonstrated in Fig. 1(b), which illustrates the momentum resolution ∆k ∥ at the Fermi level for common photon energies used in IPE (9.9 eV), ARPES (HeI: 21.2 eV and HeII: 40.8 eV), and hard X-ray photoelectron spectroscopy (HAXPES) (5 keV). The work function of Cu(111) was used as an example. The shaded areas cover the momentum resolutions at the Fermi level for incidence (emission) angles between 0◦and 45◦. To achieve a −1 momentum resolution of, e.g., 0.1 Å at the Fermi level, an ◦ angular resolution of 6 is sufficient at ~ω = 9.9 eV, while a few tenths of a degree are required at ~ω = 5 keV.

III. ANGULAR RESOLUTION IN INVERSE PHOTOEMISSION A. Definition of angular resolution ∆θ

Unfortunately, there is no clear definition of angular resolution ∆θ in the literature. In some cases (including work from our group), it is not even clear, whether the full or the half-angular width is stated. Table I summarizes a number of specifications from the literature. The given values vary between 2 and 10◦. In many cases, it is not described how these values are defined or how they have been determined.

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momentum resolution resulting from it will be characterized by the FWHM of an assumed Gaussian distribution. Please note that, throughout this paper, we use angular resolution and beam divergence interchangeably.

B. ∆θ from Faraday-cup measurements of the electron beam

FIG. 1. Influence of the photon energy on the momentum resolution: (a) For a given angular resolution ∆θ, an increasing photon energy results in an increasing kinetic energy, illustrated by the different lengths of the k vectors. This leads to different momentum resolutions ∆k 1 and ∆k 2. (b) Momentum resolutions at the Fermi level for common photon energies (IPE: 9.9 eV, ARPES: 21.2 eV (HeI), 40.8 eV (HeII), and HAXPES: 5 keV) as function of angular resolution ∆θ for incidence/emission angles θ between 0 and 45◦. The values were calculated with the work function of Cu(111). ∆k ∥ for higher angles is always smaller, so the momentum resolution for normal incidence/emission (0◦) gives always an upper limit.

The beam divergence ∆θ of an electron gun can be determined by measuring beam profiles via a Faraday cup. The experimental setup is schematically shown in Fig. 2(a). Beam profiles are measured by recording the beam current through a small entrance hole of the cup, while varying the x and y positions of the cup (see Fig. 2(b)). Changes of the beam profile as a function of distance z from the electron gun permit an estimate of the beam divergence. This method has some disadvantages. First, it requires a (possibly additional) Faraday cup that can be placed at the sample position. Measuring the electron beam profile in this configuration, i.e., Faraday cup at the sample position and sample somewhere else, could also distort the result due to modified electric and/or magnetic field configurations. Moreover, for reliable results, the Faraday cup requires a large range of linear motion in z direction. Measuring the beam diameter in a too small range can easily lead to the situation depicted in Figures 2(b) and 2(c). Despite a considerable beam divergence, beam-profile measurements at positions A and B suggest a much smaller divergence, while measurements at positions A and C even imply a parallel beam.

Where the angular distribution can be approximated with a Gaussian function, the standard deviation σ or the full width at half-maximum (FWHM) is a useful specification. Therefore, in the following, the beam divergence ∆θ and the

TABLE I. Selection of angular resolutions in IPE as found in the literature. Angular spread Angular resolution Full angular width Total angular divergence Angular divergence ∆θ (FWHM) Angular divergence Beam divergence Angular spread Beam divergence Well collimated electron beam (∆θ) Full angular divergence Angular divergence Full beam divergence Angular spread Beam divergence Full angular convergence

±2.6◦ ≤6◦ 5◦ 3◦ 7◦