The paper under discussion promises a spin- and angle-resolved inverse-photoemission (IPE) setup, where the spin-polarization direction of the electron beam used for excitation “can be tuned to any preferred direction” while “preserving the parallel beam condition.” We support the idea to improve IPE setups by introducing a three-dimensional spin-polarization rotator, but we put the presented results to the test by comparing them with the literature results obtained by existing setups. Based on this comparison, we conclude that the presented proof-of-principle experiments miss the target in several aspects. Most importantly, the key experiment of tuning the spin-polarization direction under otherwise allegedly identical experimental conditions causes changes in the IPE spectra that are in conflict with existing experimental results and basic quantum-mechanical considerations. We propose experimental test measurements to identify and overcome the shortcomings.

Spin-resolved inverse photoemission (IPE) is a well established technique for probing the unoccupied electronic structure of solid surfaces, in particular their spin dependence caused by exchange interaction in ferromagnets and/or by spin–orbit coupling (SOC) in heavy elements. Complex wavevector-dependent spin textures in SOC-influenced systems demand tuning the spin polarization direction of the exciting electron beam in order to follow the spin orientation of the electrons in particular states.1 While spin rotation via an external magnetic field was successfully applied for spin-resolved appearance-potential spectroscopy2 at rather high kinetic energies of the exciting electrons, this approach is not easily applicable to IPE with low kinetic energies of typically 5–20 eV. For IPE, a mechanically rotable spin-polarized electron source was constructed, which provides an electron beam with transversal spin polarization in any direction in the plane perpendicular to the propagation direction of the electrons.3,4 Furthermore, a three-dimensional (3D) spin rotator using magnetic and electrical fields was developed and successfully used in SPLEEM (spin-polarized low-energy-electron microscopy) experiments5,6 The promising and rewarding idea of the authors of Ref. 7 is to combine the 3D spin rotator, so far used for SPLEEM, with a state-of-the-art IPE experiment. We strongly support the conceptual idea of the authors but question the performance test of the setup achieved so far.

In Fig. 4(a) of Ref. 7, the authors present a series of angle-resolved IPE spectra for Cu(001). Compared with literature data, the intensities of the sp-derived bulk transition Bsp and the image-potential state IS are remarkably low. While the data show some similarity with the spectra from 1982,10 much more pronounced spectral intensities were already obtained in 1984.11 In Fig. 1(a), a comparison of the normal-incidence spectrum of Ref. 7 with state-of-the-art data8 reveals a remarkable difference.

In Fig. 5(c) of Ref. 7, a series of IPE spectra of Au(111) for various angles of electron incidence θ is presented, intended to show the dispersion of the L-gap surface state SS. This state is occupied around Γ̄, and therefore, no intensity is expected in IPE for θ = 0°. Due to its parabolic E(k) dispersion, it crosses the Fermi energy upon increasing θ. With an electron beam of low angular divergence, this Fermi-level crossing can be observed: smallest intensity of SS around θ = 0° and rising intensity for larger θ. In contrast to literature data,9 this expected intensity behavior is not reflected in the data in Fig. 5(c) of Ref. 7: small intensity for θ = 1° and almost vanishing intensities for higher θ. In Fig. 1(b), we present a direct comparison of spin-integrated data from the two publications. The spectra have been normalized to equal background intensities. It becomes very clear that almost no intensity of SS is observed in Ref. 7. Despite these obvious discrepancies, it is claimed that “the spectra are in qualitative agreement with earlier studies though with a lower signal-to-noise ratio.” (Note that the IPE spectra show signals from direct transitions and background intensity from secondary processes, which is not noise.)

FIG. 1.

(a) Comparison of normal-incidence IPE spectra of Cu(001) from Ref. 7 (black dots) and Ref. 8 (green dots), normalized to equal background intensities. The spectrum of Ref. 7 is duplicated in the upper part of the figure with enhanced intensity (2×). (b) Comparison of the spin-integrated IPE spectra of Au(111) from Ref. 7 (black dots) and Ref. 9 (green dots).

FIG. 1.

(a) Comparison of normal-incidence IPE spectra of Cu(001) from Ref. 7 (black dots) and Ref. 8 (green dots), normalized to equal background intensities. The spectrum of Ref. 7 is duplicated in the upper part of the figure with enhanced intensity (2×). (b) Comparison of the spin-integrated IPE spectra of Au(111) from Ref. 7 (black dots) and Ref. 9 (green dots).

Close modal

The reasons for the discrepancies can be manyfold, e.g., sample condition, energy resolution, and angular resolution. The latter is of crucial importance for measurements of the Au(111) SS at the Fermi wavevector kF. The authors of Ref. 7 estimate the electron beam divergence to ±3.0° based on a recipe from the literature.12 However, the use of the recipe is only applicable “provided the peak-to-background ratio of SS is high enough (in our case about 10:1).”12 This criterion is far missed, as it is less than 1.5:1 [see Fig. S4(a) in Ref. 7].

In Figs. 2(a) and 2(b), we compare the spin-resolved IPE data of Au(111) for three selected θ from Refs. 7 and 9, respectively. The substantial intensity differences between the two datasets cannot be explained by differences in the photon-detection angles because they are almost identical (75° vs 70°). While in (b), the IPE data have been normalized to 100% spin polarization of the incoming beam, the data in (a) are shown as raw data as in the original publication, except for θ = 8°, where the spin-normalized data are added as solid lines [taken from Figs. 5(c), 6(a), and S4(a) in Ref. 7]. Apart from the intensity differences, additional questions arise: Why are the spin differences in (a) so small? Why do the spin-dependent intensity differences in (a) not change sign upon sign reversal of k as expected for a Rashba-type state and observed in (b)? Why is the intensity of SS quenched in (a) for θ = 13°, while the data in (b) show a clear spin-dependent energy splitting of SS as expected for this k value?

FIG. 2.

Comparison of the spin-resolved IPE spectra of Au(111) for spin-up (red) and spin-down (blue) electrons from Ref. 7 [Figs. 5(c), 6(a), and S4(a)] in (a) and from Ref. 9 in (b). In (b), the data are normalized to 100% spin polarization of the incoming electron beam, while raw data are shown in (a), except for θ = 8°, where the spin-normalized data are added as solid lines. The background intensities are scaled by a factor of nine between (a) and (b).

FIG. 2.

Comparison of the spin-resolved IPE spectra of Au(111) for spin-up (red) and spin-down (blue) electrons from Ref. 7 [Figs. 5(c), 6(a), and S4(a)] in (a) and from Ref. 9 in (b). In (b), the data are normalized to 100% spin polarization of the incoming electron beam, while raw data are shown in (a), except for θ = 8°, where the spin-normalized data are added as solid lines. The background intensities are scaled by a factor of nine between (a) and (b).

Close modal

In Fig. 6 of Ref. 7, spin-resolved data are presented as raw data (a) and as spin-normalized data with additional steplike background subtraction (b). This steplike function, adapted from the literature,13 where it was used to model the fundamental gap edge of a semiconductor, is not justified in the present case of a conductor, such as Au, with a Fermi edge due to an almost constant density of states at the Fermi level. In addition, the data with background subtraction are then compared with literature data without background subtraction to “fairly reproduce the spin asymmetry” of the literature data. This is, however, an inappropriate approach.

Note also the inconsistent use of uncertainty intervals in Ref. 7: large and almost constant bars independent of count numbers are used in Fig. 7, even below the Fermi level, where only dark counts are present. In particular, an identical dataset for θ = 8° is shown with very different uncertainty intervals in Figs. 6(a) and 7(a).

The key experiment of the paper is the spin rotation under otherwise identical measurement conditions. Before looking at the experimental results, we discuss the influence of the spin rotation on the intensity of the Au(111) surface state from a theoretical point of view. Within Fermi’s golden rule, the photocurrent due to transitions between an initial state ψi of the IPE process with energy Ei, i.e., an incoming electron, into a final state ψf with energy Ef, i.e., the Au(111) surface state, by emission of a photon with energy ℏω is given by

Ifiψi|Ap̂|ψf2δ(EiEfω)
(1)

with the vector potential A of the emitted light and the electron momentum operator p̂. To illustrate the behavior of Ifi, we use in the following discussion the coordinate system employed in Ref. 7 with a Rashba spin polarization of the spin-split Au surface state along the z-axis. The corresponding basic spinors |χ10 and |χ01 give rise to the spin-expectation values Ŝ=/2(0,0,1) and Ŝ=/2(0,0,1), respectively. Due to SOC, the final state has, in general, the form

r|ψfαf(r)10+βf(r)01
(2)

with different spatial components for the spin-up and spin-down parts. We note in passing that in the case of the spin-split Au(111) surface state (f = 1, 2), density-functional calculations find that its spin polarization exceeds 90%.14 This corresponds to |α1(r)| ≫ |β1(r)| for the spin-up part of the state (f = 1) and |β2(r)| ≫ |α2(r)| for the corresponding spin-down part (f = 2).

First, we consider incoming electrons with spin polarization along the ±z direction. To ease the discussion, we neglect SOC for the incident electron states

r|ψi(z)=γ(r)|χandr|ψi(z)=γ(r)|χ.
(3)

The corresponding transition intensities are

If(+z)Mαf1010+Mβf10012
(4)
=|Mαf|2and analogIf(z)|Mβf|2
(5)

with Mηf=γ*(r)Ap̂ηf(r)d3r for ηα, β.

Second, we examine transitions for incoming electrons with spin orientations orthogonal to those of the final state. Electrons with spin polarization along the ±x direction have states15 

r|ψi(±x)=γ(r)2(|χ±|χ)γ(r)21±1.
(6)

Note that these states are not orthogonal to ψf from Eq. (2). Thus, the transition intensities do not vanish,

If(±x)Mαf21110+Mβf21±1012,
If(±x)12Mαf±Mβf2.
(7)

Our calculation shows that a spin rotation of the incoming electrons [Eqs. (3) and (6)] does not change the spin-integrated intensities for the Au(111) surface state,

If(+z)+If(z)=If(+x)+If(x)Mαf2+Mβf2.
(8)

This theoretically expected behavior has been convincingly confirmed by IPE experiments for magnetic films3 and SOC-influenced systems, such as Tl/Si(111)3,4 and Au(111).4 However, it is at variance with the key experiment of Ref. 7. In Figs. 3(a) and 3(b), we compare the results of a 90°-spin-rotation experiment from Refs. 7 and 4, respectively. In striking contrast to (b), the overall surface-state intensity is quenched upon spin rotation in (a). Even the spin difference does not totally disappear after rotation in (a). The decrease in the intensity, also observed for spin rotation from transversal to longitudinal in Fig. 7(c) of Ref. 7, is even taken as evidence of a successful spin rotation, allegedly supported by literature results.16,17 A close look to the cited literature, however, shows that in all cases, the spin-integrated intensities do not change upon spin rotation (see Fig. 8 in Ref. 16, Fig. 6 in Ref. 3, and Fig. 3.7 in Ref. 4).

FIG. 3.

Spin-resolved IPE spectra of the Au(111) surface state for two different orientations of the electron spin polarization: collinear to the Rashba direction (red and blue triangles) and perpendicular to it (beige and violet triangles). (a) and (b) Data of Refs. 7 and 4, respectively.

FIG. 3.

Spin-resolved IPE spectra of the Au(111) surface state for two different orientations of the electron spin polarization: collinear to the Rashba direction (red and blue triangles) and perpendicular to it (beige and violet triangles). (a) and (b) Data of Refs. 7 and 4, respectively.

Close modal

To clarify the described controversies between the new experimental results and literature data as well as expectations from theoretical considerations, we propose the following experiments: (i) characterization of the incident electron beam during spin manipulation with respect to changes of energy broadening, angle of incidence, and beam divergence via a Faraday cup and, even more sensitively, with IPE measurements of Cu(111) (angle-dependent Fermi-level crossing of the L-gap surface state, width and dispersion of image-potential state).12 (ii) Measuring IPE spectra of spin-independent spectral features, e.g., the bulk-derived transition Bsp of Cu(001), during spin manipulation. No changes in the spectra must be observed. (iii) Measuring spin-dependent spectral features of ferromagnetic and/or SOC-influenced systems during spin manipulation. The spin asymmetries are expected to change, but the spin-integrated intensities must not change.3,4

The authors have no conflicts to disclose.

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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