Scattering effects from neighboring atoms in core-level WSe2 photoemission

PHYSICAL REVIEW B 105, 125405 (2022) Scattering effects from neighboring atoms in core-level WSe2photoemission M. J. Ambrosio,1E. Plésiat,2P. Decleva ,3P. M. Echenique,1,4R. Díez Muiño ,1,5and F. Martín2,6 1Donostia International Physics Center, 20018 San Sebastián, Spain 2Instituto Madrileño de Estudios Avanzados de Nanociencia, Campus de Cantoblanco, 28049 Madrid, Spain 3Dipartamento di Scienze Chimiche e Farmaceutiche, Universitá di Trieste, 34127 Trieste, Italy 4Universidad del País Vasco, Apdo. 1072, 20018 San Sebastián, Spain 5Centro de Física de Materiales (CSIC/UPV-EHU), 20018 San Sebastián, Spain 6Departamento de Química, Universidad Autónoma de Madrid, 28049 Madrid, Spain (Received 17 November 2021; revised 31 January 2022; accepted 10 February 2022; published 8 March 2022) Methods of attosecond science originally developed to investigate systems in the gas phase are currently being adapted to obtain temporal information on the electron dynamics that takes place in condensed-matter systems. In particular, streaking measurements have recently been performed to determine photoemission time delays from the WSe2dichalcogenide. In this work we present a fully atomistic description of the photoemission process in WSe2and provide angularly resolved photoemission cross sections and time delays from the W 4 f,Se3dand Se 4score states of the system. Since these states are spatially localized, we propose a cluster approach in which we build up from smaller to larger clusters, so that we can assess the importance of scattering effects by each new layer of neighboring atoms. We use a static-exchange density functional theory method with B-spline functions, where a one-center angular-momentum expansion is supplemented by off-center expansions with fewer partial waves. This enhances convergence in comparison with a one-center expansion, which would require very high angular momenta to characterize the localized fast oscillations near each off-center atomic core. We find that the photoemission delays and fully differential cross sections are strongly affected by scattering events that take place off the neighboring atoms, implying the need to consider their effects for quantitative descriptions of the photoemission process. DOI: 10.1103/PhysRevB.105.125405 I. INTRODUCTION Transition metal dichalcogenides (TMDCs) are a family of two-dimensional materials, whose composition follows the formula MX2, with Ma group IV, V, or VI transition metal while Xdenotes a chalcogen, examples of which are S, Se, and Te. WSe2belongs to the TMDC family and is a semiconductor, which can be both, nand p-doped, making it suitable for electronic devices requiring p-njunctions. Internally, each 2H-WSe2layer is held together by covalent-ionic bonds between a W atom and the six closest Se atoms. Relatively weak van der Waals bonds between layers mean that fewor singlelayer structures can be produced by an exfoliation process [1]. Its most stable form is the 2Hstructure, an A-B-A-B stacking [2], the B layers having an in-plane reflection with respect to A layers. Other less common or stable forms for TMDCs, which have different interlayer displacements and coordinations, are 3R,1T, and 1T[3]. In its bulk form, WSe2possesses an indirect band gap (1.25 eV), which becomes direct and wider (1.9 eV) reaching the monolayer configuration, making it favorable for photonic applications [4]. Band structure [2] and dielectric function tuning can be achieved by mechanical strain, which results in varying electrical and optical properties [5–7]. Doping and defects are also means of modifying the TMDC’s characteristics [1]. A review on the possible applications of WSe2in the field of electronics, photoelectronics, and gas sensors can be found in [1]. The high potential of WSe2for applications has drawn recent interest towards a better understanding of the electronsubstrate and light-electron interactions via angle-resolved photoelectron spectroscopy (ARPES) or timeand angleresolved photoelectron spectroscopy (TRARPES). Tanabe et al. [8] probed the symmetries of the valence band states using ARPES, while in a recent application of TRARPES by Liu et al. [9], the authors apply a 28 eV probe and a 1.55 eV pump pulse to excite electrons from the valence band. Taking advantage of the spatial localization of core states, in comparison with the delocalized nature of the valence-band states probed in the above-mentioned experiments, TRARPES has also been used to get insight into the photoemission time delays from the deeply bound W 4 f,Se3d, and Se 4sstates of WSe2[10]. In this experiment, attosecond-streaked spectra from the core states were measured and, using the streaking traces from the Se 4sstate as a reference, relative photoemission time delays were determined. Attosecond streaking generates a spectrum that can be understood from a classical picture, where the electron is alternatively slowed down and sped up by an IR dressing field after being promoted to the continuum by an XUV pulse. The measured delays thus contain dynamical information of the escaping photoelectron from its birth to its final destination 2469-9950/2022/105(12)/125405(13) 125405-1 ©2022 American Physical Society M. J. AMBROSIO et al. PHYSICAL REVIEW B 105, 125405 (2022) outside the material. Hence, in addition to the initial state characteristics (e.g., the initial angular momentum), the rich dynamics that the photoelectron undergoes on its way to the continuum is also imprinted in the measured signal. Due to the inherent complexity of a periodic solid system such as WSe2, previous quantum mechanical models separate the contribution of the different effects and do not describe the surrounding atoms on the same footing as the atoms from which the electron is ejected [10]. Purely classical Monte Carlo calculations can more easily handle the latter effect and, in fact, have been successfully used to characterize the physics of the escaping electron in metal nanoparticles [11,12], but logically they are not fully appropriate when quantum interferences are expected to play a prominent role. Thus, a full quantum mechanical treatment that treats all atomic centers at a similar level of accuracy and describes the expected interferences between different electronic paths associated with multiple scattering events is highly desirable to better understand the physics of the photoemission process in TMDCs. In this work we present a fully atomistic quantum mechanical description of the photoemission process in WSe2and provide angularly resolved photoemission cross sections and time delays from the W 4 f,Se3d, and Se 4score states of the system. Since these states are spatially localized, we propose a cluster approach in which we build up from smaller to larger clusters, so that we can assess the importance of scattering effects by each new layer of neighboring atoms. Our method consists of a static-exchange density functional theory (DFT) partial-wave expansion with supplemental off-center terms. The radial coordinates are discretized by sets of Bspline functions. The method has a proven track record when applied to single-photon electron spectroscopy from small molecules [13–16], while more recently it has been extended to time-resolved spectroscopy [17]. In this contribution we are extending the method to the realm of solid state physics. We show that the photoemission delays and fully differential cross sections are strongly affected by scattering events that take place off the neighboring atoms, implying the need to consider such effects for a quantitative description of the photoemission process. The paper is organized as follows. First we present the necessary theory for our model, then we examine how the fully differential cross section evolves for increasingly larger clusters, assessing whether convergence is achieved with respect to the cluster size, and then we focus on a more detailed crosssection-and-Wigner-delay joint analysis. This more detailed study is performed mainly for the few-atom clusters, where the nature of the photoelectron interactions with the cluster can be inferred. We start the analysis by the small clusters, namely, Se2, W3Se, W3Se2,WSe 3, and WSe6, and use the acquired information to tackle the more complex clusters: W7Se6,W 7Se12,W 7Se24,W 3Se14, and W6Se14. All clusters are depicted in Fig. 1. II. THEORY The objective of this work is to evaluate transition amplitudes, and from them, extract the physically measurable fully differential cross sections (FDCS) and Wigner time FIG. 1. Top left panel: Diagram indicating the W-centric model clusters within the periodic crystal. WSe3: white long-dashed line. W7Se6: blue short-dashed line. W7Se12: black solid line. W7Se24: maroon dot-dashed line. Bottom left panel: Se-centric model clusters. W3Se W3Se2: white long-dashed line. W3Se14: blue shortdashed line. W6Se14: black solid line. Center column: top view of every model cluster in this work. Right column: side view of every model cluster in this work. delays for WSe2one-photon ionization. Our technique incorporates effects introduced by the neighboring atoms into the photoemission process as in state-of-the-art theoretical studies of molecular photoionization. In order to calculate transition-matrix elements we first obtain the initial and final (continuum) states for the system in question by diagonalizing the Kohn-Sham Hamiltonian HKS =− 1 2∇2− N ZN |r−RN|+n(r) |r−r|+VXC[n(r)], (1) where VXC is the LB94 exchange-correlation functional [18], RNare the nuclei positions, and ZNtheir charges. The LB94 functional has been chosen to guarantee that the photoelectron sees the correct asymptotic charge in its escape from the atomic center. The ground state density n(r) has been evaluated with the Amsterdam density functional (ADF) commercial software [19–22]. Bound states, with negative eigenvalues, are directly provided by diagonalization of the Kohn-Sham Hamiltonian in Eq. (1), while continuum asymptotic-momentum eigenstates at a chosen positive energy are obtained by solving the scattering K-matrix equations in the basis of KS orbitals through an inverse iteration procedure [13,15,23]. This procedure, which is widely used to describe the electronic continuum of atoms and molecules in the gas phase, ensures that the correct asymptotic boundary conditions of scattering states are correctly imposed. Continuum states have been evaluated for energies between 0 and 110 eV, with particular attention to 91 eV, which is the energy chosen in Ref. [10] for their streaking measurements. The wave functions are represented in a basis of Bspline functions located at the center of mass of the cluster [one-center expansion (OCE)] for different symmetry-adapted partial waves, complemented by smaller off-center angularmomentum expansions located at the nuclei’s positions. We adopt a notation similar to Toffoli et al. [15], and denote the OCE basis elements at the origin Oby χO nlhλμ =1 rO Bn(rO)Xlhλμ(θO,φ O),(2) 125405-2 SCATTERING EFFECTS FROM NEIGHBORING ATOMS … PHYSICAL REVIEW B 105, 125405 (2022) TABLE I. One-center angular momentum LMAX: semiclassical estimate and the values used in this report to ensure convergence. W-centric W WSe3WSe6W7Se6W7Se12 W7Se24 Semiclassical 12 12 16 20 25 Converged 10 24 24 26 32 40 Se-centric Se Se2 W3Se W3Se2W3Se14 W6Se14 Semiclassical 8 9 9 18 19 Converged 10 24 20 26 28 28 and define the angular functions Xas Xlhλμ(θ,φ)= m YR lm(θ,φ)βlmhλμ,(3) with YR lm being real spherical harmonics and the coefficients βlmhλμ give the symmetry-dictated weights. The off-center basis elements are symmetrized combinations of functions localized at each augmentation sphere j: χi nlhλμ = j∈Qi 1 rj Bn(rj) m β(j) lmhλμYR lm(θj,φj).(4) Index iruns over the nonequivalent nuclei sets, denoted by Qi, while jenumerates the equivalent centers within each Qi. Each center has its set of off-center coordinates rj,θj,φj. λindicates the irreducible representation, while μdenotes the degeneracy, if present, and hidentifies elements within a set fixed by l,λ,μ. Turning to the radial coordinates, Bn stands for the nth spline basis element. The B-spline bases span the radial interval [0,RO MAX] for the main expansion, and [0,Ri MAX] for the off-center ones. Indices l,mare the usual angular-momentum quantum numbers. For the off centers we chose a maximum angular momentum Lj MAX one unit larger than the highest bound-state angular momentum, in order for the basis to properly describe continuum states accounting for dipole emission near each core. The B-spline basis set is truncated at the outermost radius by excluding the three outermost Bsplines from each off-center basis in order to achieve continuity up to the second derivative at rj=Rj MAX. The B-spline knot grid is tuned to have a finer step near the cores, reaching an asymptotic value far away from the main center. A simple semiclassical estimation provides a starting point, a baseline, for the main angular-momentum expansion. We calculate the classical angular momentum of an electron with maximum linear momentum placed at the farthest atomic core. This acts as a bare minimum of the angular momentum LMAX of the main partial-wave expansion. Table Ishows the bare minimum LMAX by the semiclassical requirement for an asymptotic kinetic energy of 3 a.u., as well as the final values with which numerical convergence was reached. In all cases the semiclassical baseline was indeed surpassed by the angular-momentum values that ensured convergence. A quantum mechanical interpretation surmises the minimum LMAX requirement as related to the number of partial waves necessary to characterize the wave function oscillations in the transverse direction at the cluster’s farthest atomic positions. The off-center complementary expansions, which take care of the very tightly localized oscillations at each atomic center, account for the expected increase in the kinetic energy of the electron due to the deep atomic-core potential wells. We evaluate the photoemission transition matrix for - polarized light, starting from a bound (initial, i) molecular orbital (MO) i Jindexed by Jto a (final, f)kf-asymptoticmomentum state f kf[24]: Tkf,J=f kf·ri J.(5) The fully differential cross section is subsequently derived [24], as dσJ d=8πω|Tkf,J|2 3c,(6) with cbeing the speed of light (≈137 a.u.) and ωthe photon energy. The Wigner photoemission delay is given by τWig =dϕJ dEf ,(7a) ϕJ=arg(Tkf,J),(7b) i.e., the derivative of the transition-element phase with respect to the kinetic energy Ef(see Refs. [25–28]). Emission from W 4 fwas modeled by considering clusters which had a central W atom, namely, WSe3,WSe 6,W 7Se6, W7Se12, and W7Se24. For Se 3dand Se 4semission we chose arrays with one or two atoms at their central axis: Se2,W 3Se, W3Se2,W 3Se14, and W6Se14. Figure 1(left panels) shows how the model clusters progressively incorporate neighboring atoms from the full periodic system. Photoemission from localized cores, especially nonzeroangular-momentum orbitals, implies the aggregation of photoelectrons from a number of MOs. Specifically, we are interested in the emission from the central atoms, since they are the ones this cluster approach intends to model, by progressively surrounding them with additional neighbors. Due to the symmetry properties of the clusters, we find that, to a very good approximation, the probability density associated with all degenerate localized orbitals (LOs) coincides with that resulting from an incoherent sum over the MOs given in Table II. The information in this table has been obtained by observing the molecular orbital construction in terms of symmetry-adapted atomic orbitals after the ADF-calculation stage. An example of how to read the information in Table II is as follows: MOs 35A1,22A2,54E(doubly degenerate), 21A2, and 31E (doubly degenerate) are the mutually orthogonal W 4 fLOs (i.e., 4 f−3,4f−2,4f−1,4f0,4f+1,4f+2, and 4 f+3) of the central W atom in W7Se6. Therefore, to calculate the Wigner delay and cross section corresponding to the LO, we need to gather those indexes for the said MOs. Following a recipe similar to that proposed in Ref. [29], we use the following cross-section-weighted average to evaluate the Wigner delay associated with all degenerate LOs: ˜τWig,I(θ,φ)=J dσJ d(θ,φ)τWig,J(θ,φ) J dσJ d(θ,φ),(8) where Jindexes the MOs (e.g., 22A2), and Iidentifies the set of degenerate LOs (e.g., 4f−3,4f−2,4f−1,4f0,4f+1,4f+2, 125405-3 M. J. AMBROSIO et al. PHYSICAL REVIEW B 105, 125405 (2022) TABLE II. Localized orbitals (LO) composition in terms of molecular orbital (MO) of point groups C3vor D3h, indexed by growing principal quantum number within their subspecies A1,A2,Eand A1,A2,E,A1,A2,E. LO A1A2E W4f@ W 11,12 1 8,9 Se 3d@ Se 6 3,4 Se 4s@Se 7 LO A1A2EA1 A2 E W4f@WSe 618 5 20 14 17 W4f@W 7Se635 22 54 21 31 W4f@W 7Se12 45 26 68 31 45 W4f@W 7Se24 63 44 104 48 79 Se 3d@W 3Se218 20,21 11 10,11 Se 3d@W 3Se14 27 55,56 25 35,36 Se 3d@W 6Se14 39 38,39 20 28,29 Se 4s@W 3Se222 15 Se 4s@W 3Se14 36 29 Se 4s@W 6Se14 51 37 and 4 f+3for W 4 f). The corresponding FDCS is just the incoherent sum of the FDCSs for each degenerate state. The FDCS and Wigner delay ˜τWig,Iin Eq. (8) are defined for each emission direction and energy. For finite collection angles we define an integrated Wigner time delay ˜τWig,Ias follows [29]: ˜τWig,I=dσ d(θ,φ)τWig(θ,φ)d dσ d(θ,φ)d.(9) Our symmetry-adapted partial wave expansion makes use of point-group theory, leading to transition matrices for each molecular orbital. Since we are interested in the emission from the innermost atoms, as opposed to emission from the whole cluster, we need a way to separate the center-specific contributions. To that end we observe the symmetry-adapted orbital composition (symmetry combinations of fragment orbitals, or SFO in ADF [19]) in terms of individual-atom orbitals from ADF, and in turn how these SFOs are combined to create the molecular orbitals. As a by-product of point-group symmetry, the photoelectron yield from the core levels of interest is, to good approximation, made up of incoherent MO contributions. This is due to some of the MOs being completely localized by construction, meaning they are dominantly constituted by one ADF SFO which in turn is dominated by orbitals belonging to an innermost atom or atomic pair. Therefore, in order to obtain observables like the cross section or Wigner delay, we are able to combine the yields incoherently. As a consequence, the probability contribution from the central W 4 forbitals is to within 99% constructed as the incoherent summation of a specific MO’s yield. The same applies for the Se 3dorbitals from the innermost Se atoms, to about 99%, and lowering for Se 4sto about 60%. Table II specifically shows which MOs the localized-orbital photocurrent stems from. All the model clusters belong to either the C3vor the D3h point groups with the Se2and isolated atoms being the exceptions and belonging to higher-symmetry groups (C∞and R3). This means that all the systems will present a minimum of three vertical symmetry planes. III. RESULTS We arrange the results as follows. In Sec. III A we show polar FDCS plots as a function of the emission directions for a fixed photon energy of 91 eV, as in the experiment [10], where the growing emission-pattern complexity becomes evident as we consider bigger model clusters. Section III C explores the scattering mechanisms that emerge with the gradual addition of atoms to the clusters, outlining the processes that lead to observable cross section and Wigner delay structures. We stress that this procedure relies on comparing calculations for progressively larger clusters, in order to disentangle the information by comparison. The convergence with respect to cluster size is addressed in Sec. IV. A. Fully differential cross section Based on the experimental conditions from Ref. [10], we consider a photon energy of 91 eV and we present the detailed FDCS structure with respect to the outgoing direction in Figs. 2–4. The addition of neighbor-atom layers plays a role in the FDCS shape more significantly than initially expected. Given that the core levels are strongly localized, we expect the effects to stem from the final state, which contains every possible scattering and confinement process. Figures 2(a) and 2(h) show that the presence of the top Se layer draws the photoemission from the central W atom, deflecting it in the direction of the Se cores. Panel (c) incorporates the bottom three-Se-atom layer. Besides the three-lobed emission being tilted towards the Se atoms, each lobe is distorted, presenting small shoulder structures, which are due to a reflection effect. This will become more apparent when we jointly analyze the FDCSs and Wigner delays in Sec. III C. A subsequent incorporation of a W ring seems to focus the lobes towards normal emission, as shown in panels (d) and (j). The addition of an outer perimeter of Se atoms [see panels (e) and (k) in Fig. 2] introduces higher complexity in the FDCS. The simple three-lobe upwards and downwards structures give way to a number of fine protrusions but also a focused normal-emission 125405-4 SCATTERING EFFECTS FROM NEIGHBORING ATOMS … PHYSICAL REVIEW B 105, 125405 (2022) FIG. 2. Top row: Side view of the W 4 fphotoemission FDCS at 91 eV photon energy, with light polarization along the zaxis (indicated by the leftmost arrow and circle). Bottom row: top view of the W 4 fFDCS. peak [see panels (e), (f), (k), and (l) from Fig. 2]. When comparing with the results of larger clusters, W7Se24 to W7Se12, we observe that the W 4 fFDCS has not reached complete convergence. At the same time, the comparison shows that the effects are mostly rounding up the existing shape, in addition to an upward-focusing effect. From a purely mathematical perspective, the addition of more atomic perimeters introduces higher angular-momentum partial waves, which translates into higher LMAX needed to achieve numerical convergence. In turn, higher angular momenta enables the finer photoemission lobes [cf. left to right halves in Figs. 2,3, and 4]. We now turn to Se-centric systems, namely, Se, Se2,W 3Se, W3Se2,W 3Se14, and W6Se14. The study evidences a better cluster-size converged picture. We start with the Se 3dstate, building up from the isolated atom. The addition of a second Se atom as emitter creates interference, as attested by the middle rings in the FDCS, and increases the photoelectron yield [cf. Fig. 3, panels (a) and (b)]. The yield increase is, as expected, present in the comparison between W3Se and W3Se2; however, these systems incorporate the threefold vertical symmetry. Turning to the more sophisticated clusters in panels (e) and (f) [and (k) and (l)] we see that the FDCS exhibits only minor changes while keeping the overall pattern in place. The cluster model for Se 3demission shows a better degree of convergence than the models for W 4 f. The scenario for the Se 4sphotoemission (see Fig. 4)is much more simplistic than for the Se 3dand W 4 forbitals. The dipole nature of the s-type-orbital photoemission shape dominates all the cases but W3Se. We observe that the cluster produces a focusing effect towards normal emission at photon energies near 91 eV, with only a minor flux scattered away from the zaxis. The FDCS shape largely stabilizes even with a relatively small cluster, W3Se2, varying by less than a factor 2 towards W3Se14 and W6Se14. The shape change between W3Se14 to W6Se14 is a minor off-axis lobe narrowing. B. Wigner delays Here, we present the Wigner time delays resulting from each cluster model for each of the orbitals of interest. The Wigner delays correspond solely to the ionization process induced by XUV radiation [30], and they—-or their differences between selected orbitals—are not expected to match the values obtained from streaking measurements, which are affected by the so-called continuum-continuum delays introduced by the accompanying IR pulse. The experimental measurement from Siek et al. [10] collected photoelectrons emitted perpendicular to the surface, therefore we adjust the collection angle around the normal axis, setting the collection width to 5◦. Table III shows the calculated Wigner delays in this angular range. While for W emission the Wigner delay ˜τWig,Ishows appreciable changes even for the largest clusters, the opposite is true for both Se 3dand Se 4sorbitals, where the delay stabilizes. Figure 5shows that the same is true for other photon energies. The underlying mechanisms that give rise to the Wigner delay structures will be discussed in the next section where we jointly examine Wigner delays and cross sections in terms of photon energies and polar emission angles. C. Lowest-order mechanisms on small clusters In this section we examine the mechanisms responsible for the cross section and time delay structures. Figures 7–12 present photoemission FDCSs and Wigner delays at φ=0◦ and φ=60◦, as functions of polar emission angle and photon FIG. 3. Top row: Side view of the Se 3dphotoemission FDCS at 91 eV photon energy, with light polarization along the zaxis (indicated by the leftmost arrow and circle). Bottom row: top view of the Se 3dphotoemission FDCS. 125405-5 M. J. AMBROSIO et al. PHYSICAL REVIEW B 105, 125405 (2022) FIG. 4. Top row: Side view of the Se 4sphotoemission FDCS at 91 eV photon energy, with light polarization along the zaxis (indicated by the leftmost arrow and circle). Bottom row: top view of the Se 4sphotoemission FDCS. energy, in order to probe the effects of having or not having a neighboring atom in the emission plane. The specifics of scattering processes can be inferred by comparison, building up from smallest clusters to the more complex ones. We overlay the FDCS as contours over the Wigner time delays as a device to help us better distinguish the underlying processes. The arguments in this section originate from three main points: (i) whether the cluster possesses one or two emitting atoms, (ii) scattering events, whether low angle scattering events (deflections) or sharp angle scattering effects (collisions or rebounds), and (iii) interference of two or more of the above. See Fig. 6for a pictorial representation of these phenomena. Before delving into the specific mechanisms present on each system, we have to point out that W 4fphotoemission differs from Se 3dphotoemission regarding point (i): while the central W atom is a single-atom emitter, the axial Se atoms make up a pair of emitters that can interfere with each other. If we analyze normal emission from any Se orbital, regardless of how the emission from the atom below is being scattered by the Se atom on top, the latter is always producing an unobstructed photocurrent. In order to discuss the structures present in the FDCS and Wigner delays we will use the following naming convention. We start by the orbital in question, which in this work is one of 4s,3d,or 4 f, with no room for confusion regarding from which atomic species they originate. We denote the model cluster by a letter from A to F, as indicated by Table IV. The azimuthal emission angle φis added next to the cluster identifier, and lastly, we add an integer counter for peaks belonging to a given orbital, cluster, and φ. By default we refer to peaks in the FDCS, and we preappend a “T” to the label if pointing to Wigner time delays. TABLE III. Angularly integrated Wigner time delays ˜τWig,I with an acceptance cone of 5◦around the normal axis, in attoseconds, for the 91 eV experimental XUV photon energy in Ref. [10]. Orbital W WSe3WSe6W7Se6W7Se12 W7Se24 W4f82 65 77 156 137 171 Orbital Se Se2W3Se W3Se2W3Se14 W6Se14 Se 3d71 88 120 103 185 198 Se 4s15 48 29 70 95 100 We start with W 4 fphotoemission at φ=0◦in Fig. 7. We observe that the top Se layer addition to the isolated W introduces the most prominent feature in the FDCS, peak FIG. 5. Wigner time delays along the normal direction with a 5◦ acceptance cone. (a) W 4 f,(b)Se3d,and(c)Se4s. The polarization direction is indicated by the leftmost arrow and circle. 125405-6 SCATTERING EFFECTS FROM NEIGHBORING ATOMS … PHYSICAL REVIEW B 105, 125405 (2022) FIG. 6. Top panel: Azimuthal angle configurations and depiction of nearest neighbors in W-centric (left) and Se-centric clusters. To the right we present the line coding for the figures below, where we indicate the azimuthal distance to atoms lying at the given polar angle in said figures. Bottom panels: Pictorial representation of (a) a deflection and (b) two emitters leading to scattering plus interference. 4fB00|1. This is a constructive interference process with the scattered flux at the in-plane Se atom, marked by the yellow dashed line, right below the emission direction. The emission through the Se atoms, on the contrary, is inhibited. At near grazing emission there is another structure 4fB00|2 that stems from a similar process; however, for such emission polar angle our cluster model is not expected to match the periodic system, as in the latter case the electron would encounter other atoms. Both structures appear, albeit slightly reshaped, in panel (c): 4fC00|1 and 4fC00|2. Turning to φ=60◦,we note that no analogs to 4fB00|1 and 4fB00|2 are present in Fig. 8(b), meaning that it is a direct collision with the top Se atom, which takes place at φ=0◦but not at φ=60◦, that gives rise to said structures. An interesting picture emerges when comparing Figs. 7(b) and 8(b). There are no dominant features in Fig. 8(b), evidencing that the Se atoms lying away from the emission direction are only marginally affecting the electron flux. However, WSe6presents two peaks at φ=60◦, 4fC60|1 and 4fC60|2 [Fig. 8(c)], that manifest from a photoelectron probability rebound off the bottom Se layer, given that no similar structures TABLE IV. Peak labeling scheme. Label A B C D E F W centric W WSe3WSe6W7Se6W7Se12 W7Se24 Se centric Se Se2W3Se W3Se2W3Se14 W6Se14 FIG. 7. FDCS and Wigner delays for W 4 fphotoemission fromW,WSe 3,WSe 6,W 7Se6,W 7Se12,andW 7Se24 at φ=0◦. The CS and TW labels indicate FDCS and Wigner time delays, respectively. The number between brackets specifies φfor quick visual reference. 125405-7 M. J. AMBROSIO et al. PHYSICAL REVIEW B 105, 125405 (2022) FIG. 8. FDCS and Wigner delays for W 4 fphotoemission fromW,WSe 3,WSe 6,W 7Se6,W 7Se12,andW 7Se24 at φ=60◦. The CS and TW labels indicate FDCS and Wigner time delays, respectively. The number between brackets specifies φfor quick visual reference. are present in the WSe3case [Fig. 8(h)]. This off-Se rebound flux contributes some yield in the normal emission which is present for WSe6but not for WSe3. Looking at the W, WSe3, and WSe6time-delay structures supports the explanation, as the WSe3delay structure in both Figs. 7(h) and 8(h) closely resemble the isolated atom Wigner delays in Fig. 7(g). However, WSe6shows a more complex Wigner-delay structure. There are delay increases of 50–100 as corresponding to Wigner delay peaks T4fC60|1 and T4fC60|2. A careful comparison of peaks 4fB00|1 and 4fC00|1 shows that the latter has embedded structures analogous to 4fC60|1 and 4fC60|1, further supporting that they stem from flux rebounding off the bottom three Se atoms, as said structures are visible in Fig. 8(b) because there is no prominent low-angle deflection interference like 4fB00|1 along a φmissing a Se atom. Noting that the addition of the six perimetric W atoms in W7Se6introduces peaks 4fD00|3, 4fD00|4, and 4fD00|5 [Fig. 7(d)], none with an analog in smaller clusters, suggests therefore a direct participation of these W atoms. Peak 4fD00|3 is the same structure 4fD60|1 in Fig. 8(d), both lying at normal emission, and we see coincident Wigner delay increases T4fD00|1 and T4fD60|1, attesting to the travel time the photoelectron experiences inside the substrate. At φ=60◦ we expect to see the interplay of many-center scattering, given that there is no Se atom lying in the emission plane that would otherwise yield the most dominant features. The W7Se12 cross section is very complex to disentangle extensively, but 4fE00|2, which lays at 20◦, appears to be 4fD00|3 distorted in the same way 4fA00|1 becomes 4fB00|1 by a low-angle deflection against a Se atom in the emission plane. Four of the six W7Se12 perimetral W atoms have a Se atom at their φ=0◦, ending up with a 20◦deflection of otherwise normally emitted (as would be the case for W7Se6) probability flux, for which only two W atoms have a Se neighbor to their right. Further backing the analogy is the fact that there is no significant change in delay between T4fD00|3 and the delay level (there is no peak) corresponding to 4fE00|2. Less prominent structures appear as analogs to 4fE00|2 in the φ=60◦case [Fig. 8(e)], which account for only three Se atoms lying in the plane of secondary emission at φ=60◦ from the perimeter W atoms (see Fig. 1for W7Se12). This supports the idea that photoelectron dynamics gains in complexity as we add more neighboring atoms in the pursuit of capturing its behavior inside the periodic system, considering that higher-order processes (i.e., beyond a single scattering event, appear in the larger model clusters. There is a practical limit to the ability to disentangle every mechanism involved: namely, the finite 4πsolid angle. As more and more processes produce overlapping yield, the FDCS and Wigner-delay structures become inextricably entangled. The step-by-step cluster approach, however, allows us to explain the most prominent effects, and to an extent, determine how they get distorted and others are added when we study successively larger systems. Some FDCS structures like 4fD60|1 to 4 all have a matching Wigner-delay increase, indicating that the processes correspond to time consuming scattering events with large 125405-8 SCATTERING EFFECTS FROM NEIGHBORING ATOMS … PHYSICAL REVIEW B 105, 125405 (2022) deflections, instead of minor ones (e.g., 4fB00|1). The distortion from 4fB00|1 and 2 to 4fC00|1 and 2, corresponding mostly to reflections on the bottom Se layer, do incur in Wigner-delay increases that are not present for WSe3.However, the deflection and interference leading to the large FDCS structures 4fB00|1 and 2 do not seem to significantly delay the photoelectrons. In Figs. 9and 10 we turn to photoemission from the Se 3d orbital. The isolated-atom Se 3demission is mostly isotropic, without strong angular dependencies within the energy ranges explored. The dimer exhibits four FDCS peaks, 3dB00|1 to 3dB00|4, that are produced by the dual emitters through mutual interference and scattering [see Fig. 6(b)]. These structures translate to the W3Se2emission, which we will explain shortly. For Se-centric clusters, φ=0◦means the outgoing photoelectron does not make an in-plane close-up to a W atom, which allows for the exploration of less prominent but more intricate collisions. In Fig. 9(c) the Se 3dphotoemission FDCS from W3Se evidences three peaks: 3dC00|1, 3dC00|2, and 3dC00|3 that originate from hard collisions with the W atoms [see Fig. 6(a)], given the Se atom is above the three W atoms, and the lowest order collision mechanism requires a downwards emission plus a sharp-angle scattering event with the W atoms below. It is worth noting that 3dB00|1 matches a 100 as Wigner-delay increase with respect to the isolated atom emission, supporting the interpretation of a rebound process being involved. The W3Se2FDCS in Fig. 9(c) can be well described as a direct combination of the Se and W3Se processes. There is a peak that appears in the W3Se2FDCS, 3dD00|5, which does not have a clear parent on either Se2’s or W3Se’s FDCS. This leaves its interpretation to scattering events by the W atoms of the photocurrent emitted from the lower Se atom. We turn to analyzing the φ=60◦scenario (Fig. 10), where there is a W atom in the emission plane. The same type of low-angle collision type as seen in W 4 femission from WSe3is enabled from the bottom Se atom (W3Se2) as well as more direct in-plane photoelectron rebounds with a W atom [see Fig. 6(a)] for both W3Se and W3Se2.The W3Se system exhibits two notable peaks, labeled in Fig. 10(c) as 3dC60|1 and 3dC60|2. The latter, by comparison with isolated-Se emission, can be explained by a rebound process with the W layer, while the former corresponds to interference of direct emission [cf. Fig. 10(a), top right] and a rebound process. Having a W atom in the emission plane enables the detection of more direct collision processes that require lower momentum transfers than out-of-plane processes, and therefore, their amplitude is proportionally more significant. Peak 3dD60|4’s origin is explained by looking at 3dC60|2, which implies a rebound process by the electron flux emitted by the top Se atom, although the reader can recall the same structure is observed for W3Se2for φ=0◦but not for W3Se, leading to the interpretation that 3dD60|4 collects amplitude from two different processes: a reflection from the top photocurrent and a scattering process by the bottom one. The expected delay increases are similar, considering the path lengths are equivalently long, therefore the time-delay FIG. 9. FDCS and Wigner delays for Se 3dphotoemission from Se, Se2,W 3Se, W3Se2,W 3Se14,andW 6Se14 at φ=0◦.TheCS and TW labels indicate FDCS and Wigner time delays, respectively, and the number between brackets specifies φfor quick visual reference. 125405-9