Origin of the MBFSs

By definition, a FE-like final state in the crystal is one single plane wave *e**i*(**k**+**G)r** which matches the outgoing photoelectron plane wave. In the whole multitude of bands, formally available under *E*k and **K**// conservation, this plane wave corresponds to one single band that we will refer to as primary, relaying Mahan's primary photoemission cones 41. All other bands in the multitude give strictly zero contribution to the photocurrent. We will be calling them secondary, relaying Mahan's secondary cones. The MBFS effects, observed in our ARPES data, indicate that the corresponding final states may include, for given *E*k and **K**//, several bands with different *k*zs giving comparable contributions to the ARPES intensity. These effects obviously fall beyond the FE-like picture. As the first-principles calculations can not yet exhaustively describe our experimental results, we will analyse the MBFS effects based on insightful model calculations.

The non-FE effects in the final states, in particular their multiband composition, is certainly a phenomenon not new for low-energy ARPES. They have been studied experimentally and theoretically for 3D bulk band dispersions in various materials including Cu31,32, Mg34 and even Al the paradigm FE metal14,42, semiconductors35, various transition metal dichalcogenides36–38 as well as surface states, in particular for the Al(100) and (111) surfaces33. However, it is intriguing to observe such effects in our soft-X-ray energy range. Why do they appear in spite of the fact that the photoelectron *E*k is overwhelmingly large compared to the *V*(**r**) modulations?

We will now build a physically appealing picture of the non-FE effects in the photoemission final states using their standard treatment as the time-reversed LEED states43. They are superpositions of damped Bloch waves *ф***k**(**r**) with complex *k*z, whose imaginary part Im*k*z represents the (1) inelastic electron scattering, described by a constant optical potential *V*i (imaginary part of the self-energy), and (2) elastic scattering off the crystal potential44–47. The amplitudes *A***k** of these *ф***k**(**r**), determining their contribution to the total ARPES signal, were determined within the matching approach of the dynamic theory of LEED17,38,44,45,48,49 where the electron wavefunction in the vacuum half-space (superposition of the incident plane wave *e**i***K**0**r** and all diffracted ones *e**i*(**K**+**g**)**r**, **g** being the surface reciprocal vectors) is matched, at the crystal surface, to that in the crystal half-space (superposition of *ф***k**(**r**) satisfying the surface-parallel momentum conservation **k**//=**K**//+**g**). The underlying complex bandstructure calculations utilised the empirical-pseudopotential scheme, where *ф***k**(**r**) are formed by hybridization of plane waves *e**i*(**k**+**G**)**r**, **G** being 3D reciprocal-lattice vectors. The Fourier components *V**Δ***K** = < *e**i*(**k**+**G**)**r**|*V*(**r**)|*e**i*(**k**+**G'**)**r**> of the local pseudopotential *V*(**r**) were adjustable parameters.

We start from the ideal FE case, where *V*(**r**) is constant and equal to *V*0 (so-called empty lattice). The corresponding calculations are plotted in Fig. 3 (*a*) as the *E*(Re*k*z) bands (the corresponding *E*(Im*k*z) bands are not shown here for brevity). Due to the absence of hybridization between the plane waves in the empty-lattice case, each *ф***k**(**r**) contains one single plane wave corresponding to a certain **G** vector. Typical of high energies, we observe a dense multitude of bands brought in by an immense number of all **G** vectors falling into our energy region. Starting from the ultimate V0 = 0 case, when the vacuum half-space is identical to the crystal one, it is obvious that only one band will couple to the photoelectron plane wave in vacuum *e**i***Kr** and thus be effective in the ARPES final state, specifically, only the primary band whose plane wave – in the context of LEED often called conducting plane wave – has **k** + **G** equal to the photoelectron **K**. The whole multitude of the secondary bands, whose plane wave's **k** + **G** is different from **K**, will give no contribution to the photocurrent. In our more general case *V*(**r**) = *V*0, the *k*z component of the photoelectron distorts upon its escape to vacuum, and the above momentum-equality condition to identify the conducting plane wave should be cast in terms of the in-plane components as **k**// + **G**// = **K**//. In a formal language, these intuitive considerations can be expressed through the partial contributions of each *ф***k**(**r**) into the total current absorbed in the sample in the LEED process, which are the so-called partial absorbed currents *T***k** ∝ *V*i⋅\({\int }_{0}^{\infty }{\left|{A}_{k}{\varphi }_{k}\left(z\right)\right|}^{2}dz\), with the integration extending from the crystal surface into its depth31,32,37. Importantly in the ARPES context, the *T*k values multiplied by the photoemission matrix elements define the partial photocurrents emanating from the individual *ф***k**(**r**) in the MBFS31. In Fig. 3(a) the calculated *T*k are marked in blue colorscale. As expected for the empty-lattice case, *T***k** is equal to 1 for the primary (in the LEED context often called conducting) band and strictly zero for all other ones, realising the ideal FE final state containing one single plane wave. In Mahan's language, only the primary-cone photoemission is active in our ideal FE case.

We will now introduce spatial modulations of *V*(**r**) as expressed by *V**Δ***K** for non-zero *Δ***K**. The plane waves start to hybridise through the *V**Δ***K** matrix elements, and each *ф***k**(**r**) becomes a superposition of a few plane waves as *ф***k**(**r**) = Σ**G***C***G***e**i*(**k**+**G**)**r**. In this case not only one but several *ф***k**(**r**) can acquire a certain admixture of the **k**// + **G**// = **K**// conducting plane wave – in the formal language, their *T***k** becomes non-zero – and give a certain contribution to the total photocurrent. Our model calculations for this case are sketched in Fig. 3 (*b*). The ARPES final state appears multiband in a sense that it consists of several *ф***k**(**r**) with different *k*zs (typically alongside the primary band) which give comparable contributions to the total ARPES signal as quantified by the corresponding *T***k**. In Mahan's language, the qualitative distinction between the primary- and secondary-cone photoemission dissolves. Correspondingly, the ARPES spectra will show up several peaks corresponding to different *k*z or, if the separation of these *k*zs is smaller than the intrinsic Δ*k*z, excessive broadening of the spectral peaks. This is exactly what we have just seen in our ARPES data on Ag(100). We note in passing that on the qualitative level the bands contributing to the photocurrent can be easily identified based on the Fourier expansion of their *ф***k**(**r**) which should have a substantial weight of the **k**// + **G**// = **K**// conducting plane wave50.

Whereas for the sake of physical insight we have intentionally simplified the above picture, the exact treatment of the MBFSs based on the matching approach of LEED has been developed in a series of previous works albeit limited to relatively low final-state energies31,34,37,38. Finally, we note that the MBFS phenomenon can also be understood within the simplified three-step model of photoemission, where the whole quantum-mechanical photoemission process is splitted into the photoexcitation of a photoelectron, its transport out of the crystal, and escape to vacuum. In this framework, the MBFSs can be viewed as resulting from multiple scattering of photoelectrons on their way out of the crystal that creates multiple Bloch-wave modes of the scattered wavefield.

Whereas the effects of MBFSs have already been established at low excitation energies, their survival in high-energy ARPES might seem puzzling. In a naive way of thinking, photoelectrons with energies much higher than the modulations of *V*(**r**) should not feel them, recovering the FE case with one single *ф***k**(**r**). However, *V**Δ***K** as the strength of hybridization between two plane waves depends, somewhat counter-intuitively, not on energy but rather on *Δ***K** between them. As sketched in the insert in Fig. 3 (b), *V**Δ***K** typically has its maximal negative value at *ΔK* = 0 (which is the *V*0), and with increase of *ΔK* sharply rises and then asymptotically vanishes. Importantly, however high the energy is, the multitude of the plane waves always contains pairs of those whose *ΔK* is small. The corresponding bands can be identified by close dispersions. For such pairs *V**Δ***K** is large, giving rise to their strong hybridization. Importantly, all bands hybridising with the **k**// + **G**// = **K**// plane wave will receive non-zero *T***k** and thus contribute to the total photocurrent, as shown in Fig. 3 (b). This forms the MBFSs that should survive even at high energies.

Effect of MBFSs on the spectral structure

We will now follow in more detail how the MBFSs affect the ARPES spectra. As an example, we will analyse the experimental *k*z-MDC from Fig. 2(d) in the region of the X point at *hv* ~ 1100 eV, reproduced in Fig. 4 (with the linear background subtracted). Within the FE approximation, we might expect to observe here two Lorentzian peaks, placed symmetrically around the X point and broadened by the same intrinsic Δ*k*z. However, the *k*z-MDC shows three distinct peaks *A*-*C*, with the peak *B* coming from a final-state band falling beyond the FE approximation. Moreover, Lorentzian fitting of the peaks finds that whereas the peak *C* has a relatively small width of 0.11 Å-1, the widths of the peaks *A* and *B* are more than twice larger, 0.30 and 0.32 Å-1, respectively. The picture of MBFSs neatly explains this observation, suggesting that whereas the peak *C* is formed by a final state having one dominant *k*z contribution, and the peaks *A* and *B* by final states incorporating a multitude of *k*zs separated less than Δ*k*z. Whereas it is generally believed that the intrinsic broadening of the ARPES peaks in *k*z is determined exclusively by finite λPE the photoelectron mean free path, our example demonstrates that the multiband final-state composition may not only create additional spectral peaks but also be an important factor of their broadening additional to λPE.

Intriguingly, however, we note that even the narrowest peak *C* is almost twice broader than Δ*k*z ~ 0.065 Å-1 expected from λPE ~ 15.5 Å suggested by the TPP-2M formula 51 well-established in XPS and Auger electron spectroscopy. One explanation might be that already the peak *C* would incorporate multiple final-state bands with smaller *k*z separation compared to other two peaks. Another explanation would trace back to quasielastic electron-electron or electron-phonon scattering, which would increase with energy owing to the increase of the phase-space volume available for such scattering. Altering **k** of photoelectrons, it should destroy the coherence of photoelectrons and thus reduce λPE as reflected in the observed Δ*k*z. At the same time, the quasielastic scattering should have only a little effect on attenuation of the **k**-integrated signal of the core-level or intrinsically incoherent Auger electrons. In other words, the effective λPE in ARPES should be smaller than that in XPS/Auger spectroscopy, described by the TPP-2M and related formalism. Such intriguing fundamental physics certainly deserves further investigation.

MBFS phenomena through various materials

The phenomenon of MBFSs surviving at high excitation energies is certainly not restricted to Ag only and, strengthening with the strength of *V*(**r**) modulations, should be fairly general over various materials. Even for Al the paradigm FE metal, astonishingly, such MBFSs can be detected at least up to excitation energies of a few hundreds of eV14,42. Quite commonly the MBFS effects at high energies are observed in van-der-Waals materials such as MoTe252, which should be connected with a large modulation of *V*(**r**) across the van-der-Waals gap.

Another vivid example of the MBFS effects is the soft-X-ray ARPES data for GaN presented in Fig. 5, compiled from the previously published results on AlN/GaN(1000) heterostructures13. The panel (a) shows the ARPES spectral structure plot expected from the DFT valence bands and FE final states with *V*0 = 5 eV. With the non-symmorphic space group of bulk GaN, the ARPES dispersions allowed by the dipole selection rules (though in our case somewhat relaxed due to the band bending in GaN) are marked bold. The panels (b,c) present the experimental out-of-plane ARPES dispersions measured at *k*x in two formally equivalent \(\underset{\_}{Г}\) points of the surface BZ, \(\underset{\_}{Г}\)0 in the first and \(\underset{\_}{Г}\)1 in the second zone. As expected because of weaker electron screening of the atomic potential and thus sharper modulations of *V*(**r**) in the covalent GaN compared to the metallic Ag, the deviations of experimental dispersions from the predictions of the FE approximation are much stronger than for Ag. One can clearly see the MBFSs where the individual bands (marked by arrows at their top) are separated in *k*z more than the intrinsic Δ*k*z broadening. In the multitude of the experimental ARPES dispersions, one can identify the one which can be associated with the primary-cone photoemission (bold arrows) although in the \(\underset{\_}{Г}\)0 data this band cannot be traced below 1000 eV. Remarkably, for the same initial-state *E*(**k**) the ARPES dispersions measured at the \(\underset{\_}{Г}\)0 and \(\underset{\_}{Г}\)1 points appear completely different, identifying different final-state bands selected from the continuum of all unoccupied states available for given final-state energy and **K**//. These bands are identified by their leading plane-wave component to have **k**// + **G**// = **K**//, where **K**// of the photoelectron changes between the surface BZs32.

The high-energy final states in Si are a counter-example though. Figure 6 presents soft-X-ray ARPES data on a few-nm thick layer of Si(100) n-doped with As53 as the out-of-plane band dispersions (b) and iso-*E*B contours (c), respectively. The panel (a) shows the ARPES spectral structure plot expected from the DFT-GGA calculated valence bands and FE final states with *V*0 = 10 eV, with the bold lines indicating the dispersions allowed by the selection rules (for in-depth discussion see Ref. 54). Because of the covalent character of Si, one might again expect that the non-FE effects here would be comparable to those for GaN and in any case stronger than for the metallic Ag. Contrary to such expectations, however, the experimental data in (b,c) does not show any clear signatures of the MBFSs in Si in the shown (*E*k,**k**) region, although at low excitation energies they are profound35. At the moment we can not decipher any simple arguments that would relate the strength of the non-FE effects in the high-energy electron states to any obvious electronic-structure parameters of various materials.

Non-FE effects beyond ARPES

The non-FE effects in high-energy electron states such as MBFS manifest themselves not only in the ARPES dispersions. Another manifestation will be the circular dichroism in the angular distribution of photoelectrons (CDAD) that necessitates that the final-state wavefunctions deviate from the free-electron plane waves55,56. The CDAD has indeed been observed already in the early soft-X-ray ARPES study on Ag(100)39. Another example is the orbital tomography of adsorbed molecules (see, for example, Refs. 57–59) which takes advantage of the Fourier relation between the angle distribution of photoelectrons and electron density of the valence electron orbitals. The non-FE effects introduce additional plane-wave components in the final states, calling for refinement of the straightforward Fourier-transform processing of the experimental data59. Beyond ARPES, the very fact of electron diffraction at crystalline surfaces identifies non-FE effects in the electron states in the crystal, because otherwise the incident electrons would upon entering the crystal follow the same FE wavefunction and thus would not reflect. The Reflection High-Energy Electron Diffraction (RHEED) evidences that the non-FE effects survive even in the energy range of a few tens of keV, when *Δ***K** between the incident and diffracted plane waves is small and thus the corresponding *V**Δ***K** large. These considerations suggest that the MBFSs should survive even in hard-X-ray ARPES, waiting for a direct experimental observation.

Finally, we should point out that the coherent photoemission process underlying the ARPES experiment discussed above (as well as the orbital tomography) is fundamentally different to the essentially incoherent process of X-ray photoelectron diffraction (XPD) (see, for example, the reviews60–62. In the first case, all photoelectron emitters (atoms) throughout the crystal surface region within the depth λPE are coherent – or entangled, in the modern quantum mechanics discourse – and emit a coherent photoelectron wavefield characterised by a well-defined **k**. The resulting ARPES intensity as a function of *E*k and *θ* bears sharp structures reflecting, through the momentum conservation, the **k**-resolved band structure of the valence states. In the XPD, other way around, the coherence between the emitters throughout the surface region is lost. This takes place, for example, for isolated impurity atoms or adsorbed molecules, localised core levels, where the initial-state wavefunctions at different atoms are decoupled from each other, or when the coherence of photoelectrons is broken by thermal or defect scattering, or when the signal from certain valence-band states, like *d*-states, is integrated in energy63,64. The result is that each photoelectron emitter creates scattered waves within a sphere of the radius λPE, which interfere with each other incoherently with the waves emanating from another emitter. Typical of diffraction with a few interfering rays, the resulting XPD intensity distribution as a function of *E*k and *θ* is fairly smooth, and reflects the local atomic structure. With *E*k increasing into the hard-X-ray energy range, λPE and thereby the number of coherently scattered waves increases. This forms sharp Kikuchi-like structures in the XPD angular distribution, reflecting the long-range atomic structure62. In any case, the XPD stays incoherent between the emitters. This fundamental difference between the coherent photoemission and incoherent XPD processes is stressed, for example, by the fact that in the first case the photoelectron angular distribution follows **p**ph, shifting with *hv*, and in the second case it is insensitive to **p**ph19.