Abstract
Manybody phenomena are paramount in physics. In condensed matter, their hallmark is considerable on a wide range of material characteristics spanning electronic, magnetic, thermodynamic and transport properties. They potentially imprint nontrivial signatures in spectroscopic measurements, such as those assigned to Kondo, excitonic and polaronic features, whose emergence depends on the involved degrees of freedom. Here, we address systematically zerobias anomalies detected by scanning tunneling spectroscopy on Co atoms deposited on Cu, Ag and Au(111) substrates, which remarkably are almost identical to those obtained from firstprinciples. These features originate from gaped spinexcitations induced by a finite magnetic anisotropy energy, in contrast to the usual widespread interpretation relating them to Kondo resonances. Resting on relativistic timedependent density functional and manybody perturbation theories, we furthermore unveil a new manybody feature, the spinaron, resulting from the interaction of electrons and spinexcitations localizing electronic states in a well defined energy.
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Introduction
Signatures of manybody phenomena in solid state physics are diverse^{1,2,3,4,5}. One of them is the Kondo effect emerging from the interaction between the sea of electrons in a metal and the magnetic moment of an atom^{6,7}, whose signature is expected below a characteristic Kondo temperature T_{K}. One of its manifestations is a resistivity minimum followed by a strong increase upon reducing the temperature, as initially observed in metals doped with a low concentration of magnetic impurities^{8}. When the latter are deposited on surfaces, they can develop Kondo resonances evinced by zerobias anomalies, with various Fanoshapes^{9,10} that are detectable by scanning tunneling spectroscopy (STS), as shown schematically in Fig. 1a. The discovery of such lowenergy spectroscopic features by pioneering STS measurements^{5,11,12,13} opened an active research field striving to address and learn about manybody physics at the subnanoscale. A seminal example is Co adatoms deposited on Cu, Ag and Au(111) surfaces^{5,11,13,14,15,16,17,18}, which develop a dip in the transport spectra, with a minimum located at a positive bias voltage surrounded by steps from either sides (Fig. 1b). Although being commonly called Kondo resonances, the hallmarks of the Kondo effect have so far not been established for those particular Co atoms, i.e. the disappearance of the Kondo resonance at temperatures above T_{K} and the splitting of the feature after applying a magnetic field^{7,12,19,20,21}. A huge progress was made in advanced simulations combining quantum impurity solvers or even GW with density functional theory (DFT) addressing Kondo phenomena for various impurities (see e.g. refs. ^{22,23,24,25,26}), often neglecting spinorbit interaction. The electronic structure spectra of realistic systems do not reproduce, in general, the experimental ones.
In the current work, we provide an alternative interpretation for the observed zerobias anomalies in Co adatoms deposited on Cu, Ag and Au(111) surfaces, utilizing a recently developed framework resting on relativistic timedependent DFT (TDDFT) in conjunction with manybody perturbation theory (MBPT). Similar results were found for Co adatoms on Cu, Ag(001) surfaces and Ti adatom on Ag(001), which are shown in Supplementary Figure 1 for the sake of brevity. Our firstprinciples simulations indicate that the observed features find their origin in inelastic spinexcitations (SE), as known for other systems^{12,27,28,29,30,31,32,33,34,35}, which are gaped SE owing to the magnetic anisotropy energy that favors the outofplane orientation of the Co moment. Therefore, the physics is dictated by relativistic effects introduced by the spinorbit interaction. As illustrated in Fig. 1b, the resulting theoretical transport spectra are nearly identical to the experimental ones, advocating for a nonKondo origin of the features. This effect induces two steps, asymmetric in their height, originating from intrinsic spinexcitations, and leads to the typically observed shape in the differential conductance, thanks to the emergence in one side of the bias voltage of a new type of manybody feature: a bound state that we name spinaron, emanating from the interaction of the spinexcitation and electrons. Finally, we propose possible experiments that enable the verification of the origin of the investigated zerobias anomalies.
Results
We compare our theoretical data to measurements obtained with lowtemperature STS and proceed with a threepronged approach for the firstprinciples simulations. We start from regular DFT calculations based on the fullelectron KorringaKohnRostoker (KKR) Green function^{36,37} method, which is ideal to treat Co adatoms on metallic substrates. We continue by building the tensor of relativistic dynamical magnetic susceptibilities for the adatomsubstrate complex, \(\underline{\chi }(\omega )\), encoding the spectrum of SEs^{29,38,39}. Finally, the manybody selfenergy, \(\underline{\Sigma }(\varepsilon )\), is computed accounting for the SEelectron interaction including the spinorbit coupling. The TersoffHammann approach^{40} allows the access to the differential conductance via the groundstate LDOS decaying from the substrate to the vacuum, where the STStip is located, here assumed to be located at 6.3 Å above the adatom for the Cu(111) surface and 7.1 Å for the Ag and Au(111) ones. This is then used to evaluate the renormalization of the differential conductance because of the SEs. More details are given in the Methods section and Supplementary Note 1.
Zerobias anomaly of Co adatom on Au(111)
We discuss here the different ingredients leading to the spectrum shown in Fig. 1b, which was found to be in a remarkable agreement with the data of Ref. ^{17}, in particular. The adatom on Au(111) surface carries a spin moment of 2.22 μ_{B} and a relatively large orbital moment of 0.43 μ_{B}. The easy axis of the Co magnetic moment is outofplane favored by a substantial magnetic anisotropy energy (MAE) of 4.46 meV (Table 1). This opens a gap in the SE spectrum, as illustrated in Fig. 2a, which shows the density of transversal SEs describing spinflip processes, \(\frac{1}{\pi }\ {\rm{Im}}\ {\chi }^{+}(\omega )\). The SE arises at 6.8 meV, which is shifted from the expected ideal location, 8 meV, as obtained from \(4\frac{{\rm{MAE}}}{{M}_{{\rm{spin}}}}\) because of dynamical corrections^{39}. As a result of electronhole excitations of opposite spins^{41,42}, the lifetime τ of the SE is reduced down to 0.29 ps (\(\tau =\frac{\hslash }{\Gamma }\), Γ being the resonance width at half maximum). A simplified theory indicates that this effective damping is enhanced by the finite LDOS at the Fermi energy, which settles the density of electronhole excitations^{38}. The interaction of electrons and spinexcitations is incorporated in the socalled selfenergy. It is represented by a complex quantity, with the real part shifting the energy of the electrons, and the imaginary part describing their inverse lifetimes. The significant components of the selfenergy are spin diagonal, considering that the contribution of the offdiagonal elements is negligible for the investigated C_{3v}symmetric adatomsubstrate systems (see Supplementary Note 1). These quantities are computed from the dynamical susceptibilities \(\underline{\chi }(\omega )\) and the groundstate density n_{0}(ε). For instance, the imaginary part for a given spin channel σ after taking the trace over angular momentum indices, \({\rm{Im}}\ {\Sigma }^{\sigma \sigma }({\varepsilon }_{{\rm{F}}}+V)\), is proportional to \(\mathop{\int}\nolimits_{0}^{V}d\omega \ {n}_{0}^{\bar{\sigma }}({\varepsilon }_{{\rm{F}}}+V+\omega )\ {\rm{Im}}\ {\chi }^{\sigma \bar{\sigma },\bar{\sigma }\sigma }(\omega )\), i.e. it is a convolution of the groundstate density, n_{0}(ε), of the opposite spincharacter and the SE density integrated over the bias voltage of interest. This quantity is plotted in Fig. 2b. Two steps are present, one for each spin channel, located at positive (negative) bias voltage for the minority (majority) selfenergy. This is expected from the integration of the SE density over a resonance. As the selfenergy of a given spinchannel is proportional to the LDOS of the opposite spincharacter, the majorityspin selfenergy has a higher intensity than the minority one, as expected from adatom LDOS illustrated in Fig. 2c, which decreases substantially the lifetime of the majorityspin electrons compared to that of minorityspin type.
The groundstate LDOS, \({n}_{0}^{\sigma }(\varepsilon )\), of Co adatom (depicted in Fig. 2c as black lines) varies very weakly for a bias voltage range of ~60 meV at the vicinity of the Fermi energy. The LDOS is then renormalized by the SEelectron interaction upon solving the Dyson equation
from which the renormalized LDOS is obtained by tracing over site, spin and angular momenta of the Green function: \(n(\varepsilon )=\frac{1}{\pi }\ {\rm{Im}}\ {\rm{Tr}}\ {\underline{G}}_{{\rm{R}}}(\varepsilon )\).
At the adatom site (Fig. 2c), steplike features arise in the LDOS at the SE energy. The minorityspin LDOS hosts one single feature above the Fermi energy as expected from the corresponding selfenergy. In contrast, the majorityspin LDOS is marked with an additional feature at positive voltage, which we identify as a manybody bound state — a spinaron. One can recognize it (spinaron) either from a onetoone comparison between the spinresolved LDOS and the selfenergy, as being a feature not present in the latter one (see Supplementary Figs. 3 and 4) or from tracking the intersections of Green functions and selfenergies leading to the vanishing of the denominator of Eq. (1). The presence of spinfluctuations affect the electronic behavior in terms of the electronSE interaction encoded in the selfenergy. This additional interaction can act as an attractive potential permitting the localization of electrons in a finite energy window, giving rise to a bound state. The spinaron emerges then when the denominator of the Dyson equation, Eq. (1), cancels out, i.e. when \({\rm{Re}}(\underline{G}\underline{\Sigma })=1\), which occurs for the \({d}_{{z}^{2}}\) orbital having the ideal symmetry to be detected by STS, as illustrated in Supplementary Figs. 5 and 6. The spinaron is characterized by an energy and a lifetime (settled by \({\rm{Im}}(\underline{G}\underline{\Sigma })\)), both affected by the spinorbit interaction, since it dictates the magnitude of the SEgap defining the selfenergy, and the electronhole excitations.
The adatom electronic features decay into the vacuum, which are probed by the STStip in terms of the differential conductance. The signature of the SE is better seen in the change of the vacuum LDOS, Δn = n − n_{0}, illustrated in Fig. 1b and being spindecomposed in Fig. 2d. One sees that the origin of the two steps and their asymmetry observed experimentally and theoretically is the concomitant contribution of the spinexcitation features and the spinaron. The signal is mainly emanating from the majorityspin LDOS, with the spinaron showing up as a step being higher than the one corresponding to the intrinsic SE below the Fermi energy. We note that the spinaron bears similarities to the spin polaron suggested to exist in halfmetallic ferromagnets^{43}.
Systematic study of Co adatoms on Cu, Ag, Au(111) surfaces
We performed a systematic comparison between simulated and experimental data and evidenced that the the spinexcitations combined with the spinaron are generic features for Co adatoms deposited on Cu(111), Ag(111) and Au(111). The agreement shown in Fig. 3a is staggering, certifying that the result obtained for Au(111) surface is not accidental. Similarly to Au, the spinaron originates from the \({d}_{{z}^{2}}\) on Ag and Cu, conferring the right symmetry to be detected efficiently with STS (more details are provided in Supplementary Figs. 5 and 6). This enforces our view that the experimentally observed zerobias features for Co adatoms are captured by gaped SEs. The energies and lifetimes (ε_{spinaron}, τ_{spinaron}) of the spinarons as obtained from the theoretical spectra in vacuum are: (4.42 meV, 0.34 ps), (12.6 meV, 0.20 ps) and (9.41 meV, 0.20 ps) for Cu, Ag, and Au(111), respectively, which are of the same order of magnitude than those of the intrinsic spinexcitations listed in Table 1. Interestingly, the lifetimes of the latter excitations increase slightly on Cu and Ag surfaces when compared to that obtained on Au. Interestingly, the spectra obtained for Co/Cu(111) are in line with those reported in ref. ^{48} based on a simplified theoretical approach (more details are provided in Supplementary Note 2).
Merino and Gunnarsson^{49} suggested that the surface states of the investigated substrates give rise to the particular shape of the lowenergy excitations. In the case of Ag(111), STS experiments showed the possible alteration of the tunneling signal depending on the scattering of the Ag surface state at surrounding defects and step edges^{50}. When compared to other surfaces (see Supplementary Fig. 1), our theory indicates that the surface states are important to enhance the overall signal in the vacuum while the main origin of the spectral anomalies of the isolated adatoms is a combination of the intrinsic spinexcitations signatures and the spinaron. The weight and shape of each of the features depend on the substrate and interference effects induced by decay of the electronic states of both the adatom and surface. Moreover Eq. (1), shows that both the steplike and peaklike features of the respective imaginary and real parts of the selfenergy are mixed up, contributing to the signature of the observed lowenergy anomalies (see Supplementary Fig. 3). One sees in Fig. 3b, that the main difference between the three surfaces originates from intrinsic spinexcitation occurring in the minorityspin channel (positive bias voltage), which is for Ag more enhanced than the signal coming from the majorityspin channel carrying both the intrinsic spinexcitation (negative bias voltage) and the spinaron (positive bias voltage). When deposited on Cu and Au, the asymmetry between majority and minorityspin channels switches. This is induced by the electronic structure of the adatom on the three surfaces (see Supplementary Fig. 2). The LDOS at the Fermi energy of Co on Ag hosts a larger minorityspin DOS than on Cu and Au. The reason is the weaker hybridization strength between the electronic states of Co and the substrate, when compared to Cu or Au(111), which reduces the broadening of the minorityspin resonance on the former. The positions of the steps pertaining to the intrinsic SEs correlate with the magnitude of the MAE, which favors the outofplane orientation of the Co magnetic moment on the three substrates as listed in Table 1.
For the quantitative validation of the agreement between the theoretical and experimental data, we fit our data with the commonly used Fanoresonance formula for the differential conductance of Kondo resonances^{9,10,15}:
with \({\mathcal{A}}\) being the amplitude of the signal and q the coupling parameter. The latter plays an important role in the Fano formalism as it determines the shape and asymmetry of the STSsignal. \(\varepsilon =({\mathrm{eV}}{E}_{0})/{k}_{{\rm{B}}}{T}_{{\rm{K}}}^{{\rm{eff}}}\) encodes the information regarding the effective Kondo temperature \({T}_{{\rm{K}}}^{{\rm{eff}}}\), as well as the bias voltage V (with k_{B} being the Boltzmann constant and E_{0} the position of the investigated resonance). The fitted Fanoparameters are listed in Table 2. Astonishingly, the recovered effective temperatures are in perfect agreement with the ones obtained from experimental data.
Experimental proposals: impact of spinpolarized tip, magnetic field, and proximityeffects
As mentioned before, the Kondo origin of the lowenergy spectral features has so far not been evidenced for the systems investigated in the current work. This is usually realized by performing temperaturedependent measurements and/or upon the application of a magnetic field, which would respectively result in a broadening of the anomalies and/or their splitting. However, the large broadening of the diplike structure require improved energy resolutions than currently available, preventing the realization of such experimental studies. Here we address possible experiments that can further verify our predictions.
Kondo resonances should not change when probed by a spinpolarized tip. Our spectra are however spindependent and thus we expect the alteration of their shape depending on the spinpolarization of the tip, \({P}_{{\rm{tip}}}=\frac{{n}_{{\rm{tip}}}^{\uparrow }{n}_{{\rm{tip}}}^{\downarrow }}{{n}_{{\rm{tip}}}^{\uparrow }+{n}_{{\rm{tip}}}^{\downarrow }}\), since the differential conductance is approximately proportional to \((1+{P}_{{\rm{tip}}}){n}_{{\rm{adatom}}}^{\uparrow }+(1{P}_{{\rm{tip}}}){n}_{{\rm{adatom}}}^{\downarrow }\)^{40,51}. Ultimately, manipulating the spinpolarization of the tip (see e.g. Ref. ^{52}) would help spinresolving the LDOS as depicted in Fig. 3b for the three investigated substrates.
Furthermore, the zerobias dip is expected to split into two features for a traditional Kondo resonance once an external magnetic field is applied^{7}. Figure 3c shows a completely different behavior. The field applied along the easy axis of the Co atoms yields an increase of the excitation gap, as expected, and of the spinaron energy (see the spinresolved spectra in Fig. S1). The interplay of the various features gives the impression that the observed dip drifts to energetically higher unoccupied states, which occurs because of the presence of the spinaron. We note that applying a magnetic field in the direction perpendicular to the magnetic moment, would affect the excitation gap in a nontrivial way^{32}. A field of 14 Tesla is available in some STS setups^{45} and can even reach 38 Tesla^{46,47}. Larger fields can be accessed effectively via magneticexchangemediated proximity effect by either (i) bringing another magnetic atom to the vicinity of the probed adatom or (ii) depositing the probed adatom on a magnetic surface with a nonmagnetic spacer inbetween (see Fig. 3d). If the adjacent atom is nonmagnetic, it can modify the MAE, which dictates the magnitude of the SE gap. If the MAE is reduced, the lifetime of the spinexcitations is expected to increase, since the amount of electronhole excitations available in the respective energy range would decrease. This can then favor the monitoring of the impact of temperature and magnetic field on the zerobias anomalies, helping to distinguish a Kondo behavior from the one emerging from spinexcitations.
Discussion
The zerobias anomalies probed by lowtemperature scanning tunneling spectroscopy on Co atoms deposited on Cu, Ag, and Au (111) surfaces, usually identified as Kondo resonances, are shown to be the hallmarks of gaped spinexcitations enhanced by the presence of spinarons. We note that there are other examples of materials, such as quantum wires, where zerobias anomalies have been challenged to be Kondo features^{53,54}. However, the adatoms investigated in the current work represent the most traditional systems, where the surface science community converges to the Kondorelated interpretation. The gap of the spinexcitations is induced by the magnetic anisotropy energy of the Co adatom, defining the meV energy scale requested to excite the magnetic moment, and therefore its magnitude can be extracted from the position of the observed steps. Considering that the large magnetic moments of the Co adatoms are characterized by an outofplane easy axis, Kondoscreening is unlikely to occur^{21}, and enforces the view that the zerobias anomalies result from spinexcitations. Additional simulations performed on Co adatoms on Cu and Ag(001) surfaces as well as Ti adatom on Ag(001), shown in Supplementary Fig. 1, provide additional evidence that spinexcitations are potentially present on other materials, giving rise to the experimentally observed zerobias anomalies.
Grounding on a powerful theoretical framework based on relativistic timedependent density functional and manybody perturbation theories, we obtain differential conductance spectra reproducing extremely well the measured data. We systematically demonstrate the presence of spinarons, which are manybody boundstates emerging from the interaction of electrons and spinexcitations. While the selfenergies quantifying the interaction of the electrons and spinexcitations are dynamical in nature and account for various correlation effects, it would be interesting to prospect in the future the impact of correlations (in the spirit of DFT + U^{55}) on the ground state properties, such as the magnetic anisotropy energy and subsequently on the excitation behavior of the investigated materials. In general, our findings call for a profound change of our understanding of measured zerobias anomalies of various nanostructures, which stimulates further theoretical developments permitting the abinitio investigation of Kondo features, spinarons, spinexcitations, and spinorbit driven physics on equal footing.
The onetoone agreement between our firstprinciples spectra and the available experimental ones strongly advocates for the importance of the spinexcitations in the interpretation of the origin of the zerobias anomalies. Xray magnetic circular dichroism (XMCD) experiments should help to confort our findings by unveiling the magnetic nature as well as the magnetic anisotropy energy of the investigated adatoms as done for Co adatoms on Pt(111)^{56}. Surprisingly, this was, so far, not performed. Temperaturedependent and magneticfield STMbased measurements were, to our knowledge, not reported, which is explained by the extreme difficulty to probe with enough resolution modifications induced in the rather broad spectral features. We conjecture that this might change in the near future, for example with electronspinresonance STM (ESRSTM)^{57,58} if realized on metallic substrates. In this work, various experimental setups were proposed, which would permit to further confirm our predictions. For instance, the theoretical spectra are spindependent and therefore the weight of each spinchannel to the total STM spectrum should depend on the spinpolarization of the tip. Furthermore, the application of a magnetic field is expected to increase the gap of the intrinsic spinexcitations, while a splitting is expected for Kondo features. However, the presence of the spinaron leads to an unconventional behavior, that is the excitation gap increases but the effective dip is not fixed and shifts to larger bias voltages. Currently, a few STM setup allow to reach large magnetic fields (e.g., 14 T and even 38 T), which would be enough to check our predictions. But even if those fields are not available, a reasonable alternative would be to use the proximityinduced effective magnetic field emerging from an adjacent magnetic adatom. Finally, one could tune down the magnetic anisotropy energy in order to reduce the amount of electronhole excitations that are responsible for the broadening of the spinexcitations. This could be realized by attaching a nonmagnetic atom such as Cu, for example, to Co adatom, after which the experimental investigation of the impact of temperature and magnetic fields would become more amenable.
By opening a new perspective on lowenergy spectroscopic features characterizing subnanoscale structures deposited on substrates, built upon the pioneering work of the STS community (see e.g. refs. ^{5,11,12,13}), our findings motivate new experiments exploring the interplay of temperature, proximity effects, and response to an external magnetic field, which can help identifying the real nature of the observed excitations and unravel the complexity and richness of the physics behind the spinaron.
Methods
Our firstprinciples approach is implemented in the framework of the scalarrelativistic fullelectron KorringaKohnRostoker (KKR) Green function augmented selfconsistently with spinorbit interaction^{36,37}, where spinexcitations are described in a formalism based on timedependent density functional theory (TDDFT)^{39,41,59,60} including spinorbit interaction. Manybody effects triggered by the presence of spinexcitations are approached via manybody perturbation theory^{48,61,62} extended to account for relativistic effects. The method is based on multiplescattering theory allowing an embedding scheme, which is versatile for the treatment of nanostructures in real space. The full charge density is computed within the atomicsphere approximation (ASA) and local spin density approximation (LSDA) is employed for the evaluation of the exchangecorrelation potential^{63}. We assume an angular momentum cutoff at \({l}_{\max }=3\) for the orbital expansion of the Green function and when extracting the local density of states a kmesh of 300 × 300 is considered. The Co adatoms sit on the fcc stacking site relaxed towards the surface by 20% (14%) of the lattice parameter of the underlying Au and Ag (Cu) substrates.
After obtaining the groundstate electronic structure properties, the singleparticle Green functions are then employed for the construction of the tensor of dynamical magnetic susceptibilities, \(\underline{\chi }(\omega )\), within timedependent density functional theory (TDDFT)^{29,38,39} including spinorbit interaction. The susceptibility is obtained from a Dysonlike equation, which renormalizes the bare KohnSham susceptibility, \({\underline{\chi }}_{{\rm{KS}}}(\omega )\) as
\({\underline{\chi }}_{{\rm{KS}}}(\omega )\) describes uncorrelated electronhole excitations, while \(\underline{{\mathcal{K}}}\) represents the exchangecorrelation kernel, taken in adiabatic LSDA (such that this quantity is local in space and frequencyindependent^{64}). The energy gap in the spin excitation spectrum is accurately evaluated using a magnetization sum rule^{29,38,39}.
Data availability
All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Information. Additional data related to this paper may be requested from the authors.
Code availability
The KKR Green function code that supports the findings of this study is available from the corresponding author on reasonable request.
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Acknowledgements
We thank Markus Ternes, Alexander Weismann, Nicolas Lorente and WolfDieter Schneider for fruitful discussions. We are grateful to Michael Crommie, Lars Diekhöner, Peter Wahl, Alexander Schneider, Markus Ternes, Klaus Kern for sharing with us their original data measured with scanning tunneling microscopy. This work is supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (ERCconsolidator grant 681405  DYNASORE). We acknowledge the computing time granted by the JARAHPC Vergabegremium and VSR commission on the supercomputer JURECA at Forschungszentrum Jülich.
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S.L. initiated, designed and supervised the project. J.B. developed the theoretical abinitio scheme accounting for spinorbit interaction in the calculation of the manybody selfenergies. J.B. performed the simulations and F.S.M.G. contributed to data postprocessing. All authors discussed the results and helped writing the paper.
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Bouaziz, J., Mendes Guimarães, F.S. & Lounis, S. A new view on the origin of zerobias anomalies of Co atoms atop noble metal surfaces. Nat Commun 11, 6112 (2020). https://doi.org/10.1038/s41467020197461
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DOI: https://doi.org/10.1038/s41467020197461
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