Interplay of spin–orbit and exchange interaction in a ferromagnet/heavy-metal hybrid system: Ni on W(110)

In a combined experimental and theoretical study, we investigate the interplay of spin–orbit interaction (SOI) and exchange interaction (XI) in the electronic structure of ultrathin Ni films on W(110). Using spin- and angle-resolved inverse photoemission, we observe that the size of the spin splitting of Ni-related exchange-split states differs for opposite magnetization directions. A quenched spin splitting for one of the magnetization directions reveals a contribution of SOI on an equal footing with XI. Using density-functional theory calculations, we explore the underlying mechanisms responsible for the experimentally observed coupling of SOI and XI. We find that a hybridization between adsorbate and substrate states, along with a high probability density of the respective states at the heavy W nuclei, cause the strong influence of SOI on the Ni-related exchange-split states.


Introduction
The interplay of spin-orbit interaction (SOI) and exchange interaction (XI) provides the foundation for a wide range of prospective spin-based applications.One research branch of this field focuses on the development of efficient non-volatile data storage for information technology.The high coercivity needed for the stability of non-volatile magnetic bits concurrently causes data to be difficult to write.Therefore, an efficient writing of nanosized magnetic bits, or in other words, an efficient magnetization switching of magnetic nanostructures or ultrathin magnetic films is necessary.Due to their prospective high efficiency, spin-orbit torques (SOTs) consisting of a thin ferromagnetic (FM) film on a heavy metal (HM) substrate have emerged as a promising solution to this challenge [1].Thereby, the interplay of SOI, induced by the HM substrate, and XI, intrinsic in the ultrathin FM adlayer, is suggested to play a crucial role in the efficiency of the current-induced magnetization switching [2].Studying the electronic structure of a FM/HM hybrid system provides access to explore the mechanisms causing the interplay of SOI and XI.
Ferromagnetic 3d transition metals on a heavy metal substrate provide a playground to explore various phenomena related to the interplay between SOI and XI.For Fe/W(110), for example, SOI-induced topological modifications of Fe-related states are observed [3].Vice versa, XI-induced topological phase transitions of W-related states with non-trivial topology are debated [4,5].In thin Co films on W(110) [6], exchange-split quantum-well states alter their energy vs momentum dispersion upon magnetization reversal.This magnetization-dependent band dispersion is caused by the interplay of SOI and XI.Surprisingly, (spin-integrated) angle-resolved photoemission measurements of ultrathin Ni films on W(110) provide no indication of an interplay between SOI and XI [7].
In this paper, we present spin-and angle-resolved inverse photoemission (IPE) data for ultrathin Ni films on W(110).In contrast to [7], our data unambiguously reveals a strong interplay of SOI and XI.The influence of the HM substrate leads to an SOI-induced modified spin splitting of adlayer-related exchange-split states.In the study of 10 ML (monolayer) Co on W(110) [6], the spin splitting is dominated by XI.SOI introduced by the HM substrate acts as a small perturbation to the XI-induced spin splitting.By replacing Co with Ni, the reduced magnetic moment per atom is expected to result in a reduced exchange splitting.By thinning the film to about 4 ML, the influence of the substrate and thus of the SOI on the Ni adlayer is expected to increase.Thereby, we achieve an equally sized contribution of SOI and XI in Ni-related states.With density-functional theory (DFT) calculations, we reveal that the origin of the large interplay is caused by the hybridization of W and Ni states.This hybridization is accompanied by a pronounced probability density of the respective states at the heavy W nuclei.
For the interplay to affect the size of the spin splitting, the magnetization direction has to be aligned with the Rashba spin direction.The Rashba spin component is in the surface plane and perpendicular to the in-plane wave vector k ∥ of the electron.Therefore, the magnetization axis of the sample has to be perpendicular to k ∥ .For a visualization of that scenario, a sketch of a free-electron-like band influenced by SOI and XI is displayed in figures 1(a) and (b) for M↑ and M↓, respectively.For M↑ in figure 1(a), the size of the spin splitting is large (small) for +k ∥ (−k ∥ ).The opposite applies if the magnetization is reversed to M↓ (figure 1(b)).This in contrast to a ferromagnetic system with negligible SOC where the size of the spin splitting is identical upon sign reversal of k ∥ (or M).For convenience, we refer in this paper to the different sized spin splitting upon magnetization reversal as spin splitting asymmetry (SSA).A more detailed description of this well-established effect can be found, e.g. in [8][9][10].Our approach to observe the influence of the interplay of SOI and XI within the bandstructure of Ni/W(110) is sketched in figure 1(c).Red and black hexagons depict the surface Brillouin zones of Ni(111) and W(110), respectively.In that measurement geometry, the interplay of SOI and XI manifests in a different-sized spin splitting for opposite magnetization directions, i.e. in an SSA for M↑ and M↓.

Experimental details
A W(110) single crystal was cleaned by cycles of annealing in an oxygen atmosphere of 10 −8 mbar at 1500 K and subsequent flashing up to 2300 K [11].A sharp (1 × 1) low-energy electron diffraction (LEED) pattern with low background intensity and the absence of O and C signals in Auger electron spectroscopy confirmed a good surface quality.Ni films were deposited on the clean W(110) surface by electron beam evaporation.The well-studied epitaxial growth of Ni(111) on W(110) is described by the Nishiyama-Wassermann mode [12,13], where the W[001] direction is aligned with the Ni[110] direction.This results in a large lattice mismatch of about ≈27% along the W[001] direction and only ≈3% along the W[110] direction.Assuming a reconstruction of the Ni film in such a way that 9 Ni atoms coincide with 7 W along the W[001]/Ni [110] direction reduces the lattice mismatch from 27% to 1.3%.In line with this expectation, a more or less developed (7 × 1) LEED superstructure is observed experimentally for the first few monolayers.Details of this complex reconstruction are given in [14,15].
With LEED the coverage was calibrated by the appearance of the (7 × 1) superstructure [12,13].From this, we estimated our coverage to be between 3 ML and 4 ML.After deposition, the Ni film was annealed to 420 K to improve the film quality.The magnetic anisotropy of the Ni film is influenced by the 2-fold symmetry of the W(110) substrate.The resulting easy axis is along the distinguished Γ K direction, which coincides with the Γ H direction of W(110).Therefore, the film was remanently magnetized along the easy axis Γ K [16].Due to the limited magnetic field of our magnetization coil at the sample position (≈25 mT), we were unable to magnetize the thin film at room temperature.To address this limitation, we employed multiple magnetization pulses during the cooling phase post annealing.Due to a reduced coercivity at elevated temperatures, we were able to remanently magnetize our thin film.The Curie temperature T C for 4ML Ni/W(110) is approximately 450 K [17].
Spin-and angle-resolved IPE experiments were performed along Γ M [Γ N] with respect to Ni(111) [W(110)] by varying the angle of electron incidence θ.The direction of the spin polarization of the electron beam was chosen to be in-plane and perpendicular to the measurement direction Γ M. Thus, the spin direction of our incoming electron beam is aligned to both the magnetization and the Rashba spin directions.This measurement geometry allows us to explore the combined influence of XI and SOI on the unoccupied electronic structure.The apparatus offers the possibility to detect photons at different angles [18] as displayed in figure 1(d) to explore the angular distribution of the emitted photons.IPE data presented in this paper for θ = 0 • and for positive values θ were recorded with C4 and for negative values of θ with C1.Additional data from C2 and C3 (not shown) were used for qualitatively analyzing the angular photon emission characteristics.The measurements were carried out at room temperature unless otherwise stated.The total energy and angular resolutions are approximately 350 meV and ±1.5 • , respectively.The IPE apparatus is detailed in [19,20].

Theoretical details
The calculations have been performed within the local spin-density approximation [21] of DFT [22] employing nonlocal norm-conserving pseudopotentials [23] in the separable Kleinman-Bylander form [24] including scalar-relativistic corrections and spin-orbit coupling (SOI) [25].The electronic wave functions are represented by Gaussian orbitals with s, p, and d symmetries and several decay constants [26].For details of the method, see [27].
We performed calculations for W(110), for freestanding Ni layers, as well as for 4 ML Ni on W(110).To account for the substantial lattice mismatch between the Ni adlayers and the underlying W substrate, we use a superstructure with 9 Ni atoms and 7 W atoms per layer in the unit cell.The calculations have been carried out for four Ni layers on top of six W layers.The bottom W layer has been saturated by H atoms to reduce artificial interaction between the two surfaces inside the slab.In total, there are 85 atoms in each unit cell.Relaxations of the outermost eight layers have been taken into account in the structural optimizations.As a constraint, the orientation of the magnetization has been aligned with the Γ K direction during the self-consistent computations.
We simulate the IPE process by assuming a plane wave as the initial state and compute the dipole transition matrix elements employing the final states resulting from our DFT calculations.An exponential damping term with a decay length of 5 Å has been used to consider the finite penetration depth of the incoming electrons.

Results and discussion
In this section, we present results obtained by spin-and angle-resolved IPE measurements and DFT calculations for ultrathin Ni films on W(110).Thereby, we focus on the Γ M direction, as depicted in figure 1, such that the magnetization direction is aligned with the Rashba spin direction.Yellow and purple colors indicate the magnetization direction along ±Γ K as specified in figure 1(c).

Magnetization-dependent IPE spectra
We start with a series of IPE spectra for various angles of electron incidence θ along Γ M. Red (green) triangles represent data points of minority (majority) spin.The pointing direction of the triangular-shaped data points resembles the opposite spatial alignment of the electron spin polarization upon magnetization reversal.Figures 2(a 111) surface [28].(d) Spectra for θ = 0 • of pristine W(110) (solid line) and Ni(111) (dashed line) [28].(e) and (f) display the spin-dependent E vs k ∥ dispersion of α2 with peak positions obtained from the spectra for θ = 25 • , 29 • , and 35 • .The blue area serves as a guide to the eye for the theoretically obtained dispersion of the α2 band (cf figure 5).
respectively.For θ = 0 • , two maxima dominate the spectra, located close to E F and at 2.2 eV above E F .The intensity located close to E F resembles the Ni surface state (NiSS), revealed by a comparison with the spectrum of pristine Ni(111) shown in figure 2(d).The high density of Ni d states contributes to the higher minority spin intensity within the feature located close to E F .The intensity difference for the two different magnetization directions M↓ and M↑ at E F is due to aging of the sample.The M↑ spectra were acquired immediately after the sample preparation and the M↓ spectra after approximately 12 h.
The second feature at 2.2 eV above E F is also present on the pristine W(110) surface (figure 2(d)), and therefore denoted as W1.Considering the spectra for θ = 6 • , a dispersion of W1 towards lower energies upon increasing k ∥ is evident.As we will see in section 4.3, the W1 band is crucial for the hybridization with Ni-related exchange-split states.Apart from this, the intensities of W1 in the spectra for θ = 0 • show a peculiar spin dependence.This effect is known from the pristine W(110) surface.Due to the mixed orbital symmetry of W1 and the symmetry breaking by the experimental geometry (off-normal light detection with C4), spin-dependent intensities appear in the IPE spectra [18].Please note, the sign of the spin-dependent intensity differences is independent of the magnetization directions M↑ and M↓, as indicated by the pointing direction of the triangular-shaped data points.With respect to the colors indicating minority and majority, however, the differences appear reversed upon magnetization reversal.Additionally, spin-dependent intensities of W1 in the FM/HM system may result due to spin-dependent electron attenuation within the ferromagnetic adlayer [29,30].However, due to the rather small difference in the inelastic mean free path for majority and minority electrons in Ni, only a minor influence of this effect on the intensities of W1 is expected.
In the spectra for θ = 6 • , the asymmetric shape of the intensity feature located close to E F indicates contributions from more than one state.Furthermore, a small but significant qualitative difference in the spin signal is apparent with respect to the different magnetization directions.For M↑, a higher minority spin intensity is observed on the high-energy side of the feature, while for M↓ a higher minority spin intensity appears at the low-energy side.
In the spectra for θ = 13 • , two intensity maxima are identified, labeled α and β.The former is cut off by the Fermi energy, while the latter splits off towards higher energies.As those intensities are not observed on the pristine W(110) surface, we attribute them to Ni-related exchange-split states.Comparing the spectra for opposite magnetization directions, a clear spin-dependent intensity difference of the α feature is apparent.While for M↑ no intensity difference can be observed in the spin signal, the minority intensity exceeds the majority intensity for M↓.For β, a spin-dependent energy splitting is observed, whose size is larger for M↓ than for M↑ as indicated by green and red tick marks.
The spectra for θ = 19 • reveal again a larger spin splitting of β for M↓ compared with M↑.The maximum of the α feature is now shifted towards higher energies and a splitting becomes apparent as well.The differences of the spin splitting follow the trend of the β feature: A larger spin splitting for M↑ compared with M↓.Within the α feature a double-peak structure is observed.Its origin will be discussed in more detail below.
The overall trend of the SSA continues for larger θ.In the majority spectra for θ = 25 • , α appears again as a double-peak feature.For electronic states in ultrathin films, we expect quantum-size effects.This results in quantum-well-like states whose energy changes with the film thickness, as observed, e.g. in [31].We attribute the appearance of the double-peak structure to areas of different coverages sampled by the IPE measurement.The energy difference of these two states within the α feature is in the order of 0.4 eV, compatible with calculations of the α state for a freestanding Ni film of 3 ML (not shown) and 4 ML (see figure 4) thickness.We conclude that the maxima at lower energies, α 1 , correspond to a coverage of 3 ML, while the maxima at higher energies, α 2 , are associated with a coverage of 4 ML.
The SSA as well as the double-peak structure can also be traced in the spectra for θ = 29 • and θ = 35 • with a similar manner.For the sake of clarity, we now restrict our discussion to the α 2 state.For θ = 25 • , 29 • , and 35 • , the peak positions observed for α 2 are transferred to E(k ∥ ) plots in figures 2(e) and (f).The blue area serves as a guide to the eye for the theoretically obtained dispersion of the α 2 state (cf figure 5).The E(k ∥ ) plots unambiguously illustrate the different-sized spin splittings for M↓ and M↑.Additional verification of the magnetization-dependent spin splitting is gained from the line width of the spin-integrated spectra in figure 2(c).Due to the enhanced spin splitting, the FWHM (full width at half maximum) for M↓ (1 eV) is significantly larger than for M↑ (0.8 eV).
Comparing our spin-integrated spectra for θ = 35 • with corresponding spectra of a Ni(111) surface [28] (dashed lines in figure 2(c)), the α feature is reminiscent of transitions into so-called sp bands of Ni.The dispersion of the β feature, on the other hand, is reminiscent of NiSS [28].A complete congruence cannot be expected due to the quantum size effects, the influence of the substrate, and a slightly different photon detection energy of 9.4 eV used in [28].However, we assume that for larger coverages, the α band evolves into the so-called sp band and the β band into the NiSS.

Experimental proof of SOI contribution
To experimentally prove the SOI contribution, we performed two experiments: (i) measurement above T C and (ii) measurements for opposite angles θ, i.e. reversed k ∥ .
Firstly, we exclude a pure magnetic origin leading to the observed SSA, e.g.incomplete remanent magnetizations of the Ni film.A measurement above the Curie temperature at T ≈ 470 K [17] was performed (figure 3(a), upper panel).Above T C , the long-range magnetic order breaks down.Thus, no spin splitting is expected above T C [32,33].Here, the absence of long range magnetic order is evident from the disappearance of the spin signal just above E F .In comparison, below T C (figures 2(a) and (b)), a dominant intensity of the minority spin component is apparent.For the α band, however, a significant spin splitting remains above T C (figure 3(a)).We conclude that the spin splitting above T C originates from SOI.The spin effects, apparent at room temperature, i.e. below T C (see figure 2), are thus not of pure magnetic origin.
Secondly, to cross-check the contribution of SOI leading to the observed SSA, we validate the symmetry predicted by the simple Rashba plus XI model: A reversal of the SSA from +θ to −θ [8,34].Figure 3(b) displays the IPE spectra for M↑ and M↓ and ±θ.The upper spectra for +θ are identical to those displayed in , the expected symmetry becomes apparent: While in the spectra for +θ a large (small) spin splitting is observed for M↓ (M↑), the spectra for −θ spectra disclose a large (small) spin splitting for M↑ (M↓).The different photon detection angles used in the spectra for −θ (C1) and +θ (C4) with respect to the plane of incidence (cf figure 1(d)) causes intensity differences of the α feature.In addition, due to the measurements not being taken at the precise opposite angle, the state energies differ slightly.
From the non-vanishing spin splitting above T C as well as the observed Rashba-type behavior, we conclude that the observed SSA is a result of the interplay between XI and SOI.Although the bands discussed above deviate from the free-electron model illustrated in figure 1, the observed SSA still adheres to the basic model governing the interaction between SOI and XI [8].Furthermore, the almost complete quenching of the spin splitting for M↑ (figure 2(e)) provides evidence of a similar strength of SOI and XI within the α 2 band.In comparison with the Co film on W(110) used in [6], the Ni film used here exhibits bands with intrinsically smaller exchange splitting.Additionally, the influence of the interface, and thus the SOI, is assumed to decrease with increasing film thickness.In our case, a balanced ratio of SOI and XI, tuned by a suitable choice of constituents and dimensions of our system, results in a magnetization-and momentum-dependent quenching of the exchange splitting of α 2 .
However, reducing the thickness of the ferromagnetic adlayer itself is not sufficient to guarantee strong influence of SOI on the adlayer states.It is necessary that the electrons feel the strong potential gradient of the substrate's nuclei [35].Therefore, a coupling of adlayer to substrate states is necessary.The precondition comprises a spatial overlap of the wave functions of substrate and adlayer states as well as energetic proximity in k space [6].

Origin of SOI contribution
To elucidate the influence of SOI on the FM adlayer, we compare the bandstructure of the pristine W(110) surface with the bandstructure of a freestanding 4 ML Ni film, as shown in figure 4. Thin gray lines represent a slab calculation of the electronic bands for 19 layers of W with (110) surface along Γ N. The size of the turquoise dots represents the surface contribution.The red and green lines are the minority and majority bands of a freestanding 4 ML Ni film along Γ M, respectively.The W1 band crosses two quantized Ni-bands.For higher k ∥ , the slope of W1 changes sign and W1 disperses upwards, nearly parallel to the α2 band of the freestanding Ni film.The requirement of energetic proximity in k space is thus fulfilled.Our DFT calculations reveal that the W1 state consists of d x 2 −y 2 , d z 2 and p z orbitals.Up to k ∥ ≈ 0.5 Å −1 , the orbital contribution of W1 is dominated by d x 2 −y 2 .The change of the slope is caused by a hybridization with an upwards dispersing band of W d z 2 states.The α2 state of the freestanding Ni layer consists mainly of d xz with a minor contribution from p z and d z 2 orbitals.Thus, the orbitals of α2 and W1 extend along the z-direction, providing an overlap of these orbitals.
To further examine the possible hybridization of W and Ni states and the resulting SSA, we performed DFT calculations for a 4 ML Ni film on W(110) in the described superstructure.Figures 5(a) and (b) display a photocurrent calculation based on the bandstructure provided by the DFT calculation for M↑ and M↓, respectively.The size of the red (green) dots represents the photocurrent intensities for the minority (majority) spin component.The calculation shows distinct photocurrent intensities for the α 2 (blue background) and β bands.The photocurrent intensity highlighted by the turquoise-shaded area is related to the W1 state.This assignment results from an analysis of the orbital composition as well as from a comparison with the bandstructure of the pristine W(110) surface.
The α 2 state of the hybrid system does not only contain the above-mentioned orbitals of α2 and W1 but contains an additional admixture of tungsten d xz and d yz orbitals.This contribution is essential for the resulting SOI and thus an SSA.In the dominant on-site terms, there is no SOI-induced coupling between the d z 2 and d x 2 −y 2 orbitals (due to ∆m l = 2) of the W1 state.The d xz and d yz orbitals, however, couple with these orbitals via SOC.
Comparing the spin splittings for M↑ and M↓, a significantly smaller spin splitting is obtained for M↑ compared with M↓, consistent with our measurements.The difference in size of the spin splittings is slightly smaller in the calculations compared with the experiment.In particular, the spin splitting within the α 2 band for M↑ is not completely quenched.Nonetheless, the distinctive aspects, the hybridization of the α 2 band with W1 and the resulting SSA, are confirmed by the photocurrent calculation.
The hybridization effects are reflected in the real-space probability density for the wave function, cut along the surface normal and the [110] direction, as displayed in figure 6.In addition, the layer-dependent probability densities averaged over the unit cell |Ψ|    Clearly, the wave function is confined within the Ni film and, therefore, the electrons do not feel the strong potential gradient.On the basis of this calculation, the absence of an SSA reported in [7] for the occupied bands may be understood.As a further result, we note that the lobes of the orbitals point mainly along the surface normal.This is in concordance with the observed reduced photon intensity for the spectra of 35 • (not shown) in the counter C2 oriented along the surface normal.

Conclusion
We observed a different-sized spin splitting for opposite magnetization directions in the unoccupied electronic structure of an ultrathin Ni film on W(110).This result is attributed to an interplay of SOI and XI at the interface.With the support of DFT calculations, the origin of this interplay is traced back to hybridization between Ni-related adlayer and W-related substrate states, as evident by the extension of the respective wave function into the substrate.Due to the choice of the constituents of our FM/HM system and its low dimension, we were able to tune SOI and XI interaction to the same scale, resulting in a k-and magnetization-dependent quenching of the exchange splitting.

Figure 1 .
Figure 1.(a) and (b) Sketch of a free-electron-like band in the presence of SOI and XI for M↑ and M↓.Red and green colors denote the minority and majority character of the band, respectively.The red and green (gray) filled Gaussian curves on the left and right side of the sketch represent the resulting spin-resolved IPE spectra for minority and majority states, with an increased (quenched) spin splitting.(c) Sample and measurement geometry.Red and black hexagons depict the surface Brillouin zones of Ni(111) and W(110), respectively.The magnetization direction (purple and yellow arrow) is perpendicular to k ∥ .(d) Experimental geometry of the IPE set-up.Counter C1 (α = −70 • ) and C2 (α = 35 • ) are in the plane of incidence (xz-plane), while C3 (α = 0 • ) and C4 (α = 65 • ) are tilted out of the xz-plane at angles of β = 35 • and 32 • , respectively.
) and (b) display spin-resolved IPE spectra of 4 ML Ni/W(110) for M↑ and M↓,

Figure 2 .
Figure 2. Spin-and angle-resolved inverse photoemission spectra along Γ M measured with C4 for (a) M↑, and (b) M↓.Red and green symbols denote the minority and majority spin components.The pointing direction of the triangular-shaped symbols indicates the absolute spatial alignment of the spins along Γ K. (c) Spin-integrated spectra for θ = 35 • for M↑ (yellow) and M↑ (purple).For comparison, the dashed line represents a spectrum of the pristine Ni(111) surface[28].(d) Spectra for θ = 0 • of pristine W(110) (solid line) and Ni(111) (dashed line)[28].(e) and (f) display the spin-dependent E vs k ∥ dispersion of α2 with peak positions obtained from the spectra for θ = 25 • , 29 • , and 35 • .The blue area serves as a guide to the eye for the theoretically obtained dispersion of the α2 band (cf figure5).

Figure 3 .
Figure 3. Spin-and angle-resolved inverse photoemission spectra of the α feature along Γ M (a) measured at T = 470 K > TC, (b) measured at room temperature, for almost opposite angle θ for M↑ (left panel) and M↓ (right panel).

Figure 4 .
Figure 4. DFT calculations for a freestanding 4 ML Ni film along Γ N (minority bands in red, majority bands in green).Bandstructure of W(110) for a slab of 19 W layers along Γ M [gray lines].The size of the turquoise dots denotes the surface contribution of the respective W band.
2 average are shown.The probability density in figure 6(a) refers to the α 2 band at E − E F = 1.6 eV and k ∥ = 0.77 Å −1 .At this E(k ∥ ) point the α 2 band is influenced by SOI as revealed by the observed SSA in figure 5.It is obvious from figure 6(a) that the wave function of the Ni-related α 2 band extends well into the first W layers.The electrons, therefore, feel the strong potential

Figure 5 .
Figure 5. Photocurrent calculations for 4 ML Ni/W(110) on a grid of 43 k ∥ points along Γ M in the superstructure.The size of the dots denotes the intensity of the IPE process for electrons with minority (red) and majority (green) spin components.The blue-shaded area serves as a guide to the eye for the intensities related to the α2 band.Areas with white background relate to the E(k ∥ ) areas of figures 2(e) and (f).The turquoise areas highlight the intensities related to the W1 state.

Figure 6 .
Figure 6.Probability density of the α2 state for 4 ML Ni/W(110), cut perpendicular to the surface and along the crystallographic direction [110] of W (a) at E − EF = 1.6 eV and k ∥ = 0.77 Å −1 and (b) at E − EF = −0.01eV and k ∥ = 0.38 Å −1 .The left-and rightmost panels display the layer-averaged probability densities.The four topmost layers consist of Ni atoms, the layers below of W atoms. Black (red) and white (orange) dots denote the position of the Ni (W) atoms within and outside the drawing plane, respectively.