Domain-wall magnetoresistance and low-frequency noise have been studied in epitaxial antiferromagnetically-coupled [Fe/Cr(001)]_10 multilayers and ferromagnetic Co line structures as a function of DC current intensity. In [Fe/Cr(001)]_10 multilayers a transition from excess to suppressed domain-wall induced 1/f noise above current densities of j_c ~ 2*10^5 A/cm^2 has been observed. In ferromagnetic Co line structures the domain wall related noise remains qualitatively unchanged up to current densities exceeding 10^6A/cm^2. Theoretical estimates of the critical current density for a synthetic Fe/Cr antiferromagnet suggest that this effect may be attributed to current-induced domain-wall motion that occurs via spin transfer torques.
We theoretically investigate dynamics of antiferromagnetic domain walls driven by spin-orbit torques in antiferromagnet/heavy metal bilayers. We show that spin-orbit torques drive antiferromagnetic domain walls much faster than ferromagnetic domain walls. As the domain wall velocity approaches the maximum spin-wave group velocity, the domain wall undergoes Lorentz contraction and emits spin-waves in the terahertz frequency range. The interplay between spin orbit torques and the relativistic dynamics of antiferromagnetic domain walls leads to the efficient manipulation of antiferromagnetic spin…
The control of domain walls or spin textures is crucial for spintronic applications of antiferromagnets. Despite many efforts, it has been challenging to directly visualize antiferromagnetic domains or domain walls with nanoscale resolution, especially in magnetic field. Here, we report magnetic imaging of domain walls in several uniaxial antiferromagnets, the topological insulator MnBi$_2$Te$_4$ family and the Dirac semimetal EuMnBi$_2$, using cryogenic magnetic force microscopy (MFM). Our MFM results reveal higher magnetic susceptibility or net moments inside the domain walls than in domains…
Magnetotransport measurements on small single crystals of Cr, the elemental antiferromagnet, reveal the hysteretic thermodynamics of the domain structure. The temperature dependence of the transport coefficients is directly correlated with the real-space evolution of the domain configuration as recorded by x-ray microprobe imaging, revealing the effect of antiferromagnetic domain walls on electron transport. A single antiferromagnetic domain wall interface resistance is deduced to be of order $5\times10^{-5}\mathrm{μΩ\cdot cm^{2}}$ at a temperature of 100 K.
For antiferromagnetically coupled Fe/Cr multilayers the low field contribution to the resistivity, which is caused by the domain walls, is strongly enhanced at low temperatures. The low temperature resistivity varies according to a power law with the exponent about 0.7 to 1. This behavior can not be explained assuming ballistic electron transport through the domain walls. It is necessary to invoke the suppression of anti-localization effects (positive quantum correction to conductivity) by the nonuniform gauge fields caused by the domain walls.
Antiferromagnetic (AF) domain walls have recently attracted revived attention, not only in the emerging field of AF spintronics, but also more specifically for offering fast domain wall velocities and dynamic excitations up to the terahertz frequency regime. Here we introduce an approach to nucleate and stabilize an AF domain wall in a synthetic antiferromagnet (SAF). We present experimental and micromagnetic studies of the magnetization reversal in [(Co/Pt)$_{X-1}$/Co/Ir]$_{N-1}$(Co/Pt)$_X$ SAFs, where interface induced perpendicular magnetic anisotropy (PMA) and AF interlayer exchange coupli…
The antiferromagnetic domain structure of a multiferroic has been investigated in the presence of a ferroelectric domain structure. It has been demonstrated that an inhomogeneous magnetoelectric (flexomagnetoelectric) interaction leads to pinning of antiferromagnetic domain walls at the walls of the ferroelectric domains and to a change in the structure of antiferromagnetic domain walls.
Deterministic control of antiferromagnetic domain walls by circular phonons
Radboud University（拉德堡德大学）；Italian Institute of Technology（意大利理工学院）；Nanyang Technological University（南洋理工大学）；Scuola Normale Superiore（比萨高等师范学院）；University of Genoa（热那亚大学）
The motion of a Néel-like ${180}^{\circ}$ domain wall induced by a time-dependent staggered spin-orbit field in the layered collinear antiferromagnet Mn$_2$Au is explored. Through an effective version of the two sublattice nonlinear $σ$-model which does not take into account the antiferromagnetic exchange interaction directed along the tetragonal c-axis, it is possible to replicate accurately the relativistic and inertial traces intrinsic to the magnetic texture dynamics obtained through atomistic spin dynamics simulations for quasistatic processes. In the case in which the steady-state magnet…
Spin canting and complex spin textures in antiferromagnetic materials can often be described in terms of Dzyaloshinskii-Moriya interactions (DMI). Values for DMI parameters are not easily measurable directly, and often inferred from other quantities. In this work, we examine how domain wall dynamics in an antiferromagnetic optomagnonic system can display unique features directly related to the existence of DMI. Our results indicate that the presence of DMI enables spin interactions with cavity photons in a geometry which otherwise allows no magneto-optical coupling, and on the other hand modul…
The interest in understanding scaling limits of magnetic textures such as domain walls spans the entire field of magnetism from its relativistic quantum fundamentals to applications in information technologies. The traditional focus of the field on ferromagnets has recently started to shift towards antiferromagnets which offer a rich materials landscape and utility in ultra-fast and neuromorphic devices insensitive to magnetic field perturbations. Here we report the observation that domain walls in an epitaxial crystal of antiferromagnetic CuMnAs can be atomically sharp. We reveal this ultimat…
We numerically compute current-induced spin-transfer torques for antiferromagnetic domain walls, based on a linear response theory in a tight-binding model. We find that, unlike for ferromagnetic domain wall motion, the contribution of adiabatic spin torque to antiferromagnetic domain wall motion is negligible, consistent with previous theories. As a result, the non-adiabatic spin-transfer torque is a main driving torque for antiferromagnetic domain wall motion. Moreover, the non-adiabatic spin-transfer torque for narrower antiferromagnetic domain walls increases more rapidly than that for fer…
Antiferromagnetic spintronics is an emerging research field which aims to utilize antiferromagnets as core elements in spintronic devices. A central motivation toward this direction is that antiferromagnetic spin dynamics is expected to be much faster than ferromagnetic counterpart because antiferromagnets have higher resonance frequencies than ferromagnets. Recent theories indeed predicted faster dynamics of antiferromagnetic domain walls (DWs) than ferromagnetic DWs. However, experimental investigations of antiferromagnetic spin dynamics have remained unexplored mainly because of the immunit…
A storage scheme based on racetrack memory, where the information can be coded in a domain or a skyrmion, seems to be an alternative to conventional hard disk drive for high density storage. Here, we perform a full micromagnetic study of the performance of synthetic antiferromagnetic (SAF) racetrack memory in terms of velocity and sensitivity to defects by using experimental parameters. We find that to stabilize a SAF skyrmion, the Dzyaloshinskii-Moriya Interaction in the top and the bottom ferromagnet should have an opposite sign. The velocity of SAF skyrmion and SAF Neel domain wall are of t…
In antiferromagnets, the Dzyaloshinskii-Moriya interaction lifts the degeneracy of left- and right-circularly polarized spin waves. This relativistic coupling increases the efficiency of spin-wave-induced domain wall motion and leads to higher drift velocities. We show that in biaxial antiferromagnets, the spin-wave helicity controls both the direction and magnitude of the magnonic force on chiral domain walls. By contrast, in uniaxial antiferromagnets, the magnonic force is propulsive with a helicity dependent strength.
Future applications of antiferromagnets (AFs) in many spintronics devices rely on the precise manipulation of domain walls. The conventional approach using static magnetic fields is inefficient due to the low susceptibility of AFs. Recently proposed electrical manipulation with spin-orbit torques is restricted to metals with a specific crystal structure. Here we propose an alternative, broadly applicable approach: using asymmetric magnetic field pulses to induce controlled ratchet motion of AF domain walls. The efficiency of this approach is based on three peculiarities of AF dynamics. First, …
In contrast with rich investigations about the translation of an antiferromagnetic (AFM) texture, spin precession in an AFM texture is seldom concerned for lacking an effective driving method. In this work, however, we show that under an alternating spin-polarized current with spin along the AFM anisotropy axis, spin precession can be excited in an AFM DW. Especially, chaotic spin precession occurs at moderate interfacial Dzyaloshinskii-Moriya interaction (DMI), which contributes to a nonlinear term in the dynamic equation of DW precession. Also, crisis-induced intermittent chaos appears when …
We predict a mechanism to controllably manipulate domain walls in kagome antiferromagnets via a single linearly polarized spin-wave source. We show by means of atomistic spin dynamics simulations of antiferromagnets with kagome structure that the speed and direction of the domain wall motion can be regulated by only tuning the frequency of the applied spin-wave. Starting from microscopics, we establish an effective action and derive the corresponding equations of motion for the spin-wave-driven domain wall. Our analytical calculations reveal that the coupling of two spin-wave modes inside the …
The control of magnetic domain walls is essential for the magnetic-based memory and logic applications. As an elementary excitation of magnetic order, spin wave is capable of moving magnetic domain walls just as the conducting electric current. Ferromagnetic spin waves can only be right-circularly polarized. In contrast, antiferromagnetic spin waves have full polarization degree of freedom, including both left- and right-circular polarizations, as well as all possible linear or elliptical ones. Here we demonstrate that, due to the Dzyaloshinskii-Moriya interaction, the spin wave driven domain …
Polarization, denoting the precession direction with respect to the background magnetization, is an intrinsic degree of freedom of spin wave. Using magnetic textures to control the spin wave polarization is fundamental and indispensable toward reprogrammable polarization-based magnonics. Here, we show that due to the intrinsic cubic anisotropy, a $90^\circ$ antiferromagnetic domain wall naturally acts as a spin wave retarder (wave-plate). Moreover, for a $90^\circ$ domain wall pair developed by introducing a second domain in a homogenous antiferromagnetic wire, the sign of retarding effect can…
We demonstrate the possibility to drive an antiferromagnet domain-wall at high velocities by field-like Néel spin-orbit torques. Such torques arise from current-induced local fields that alternate their orientation on each sub-lattice of the antiferromagnet and whose orientation depend primarily on the current direction, giving them their field-like character. The domain-wall velocities that can be achieved by this mechanism are two orders of magnitude greater than the ones in ferromagnets. This arises from the efficiency of the staggered spin-orbit fields to couple to the order parameter and …
It is shown that magnetic states and field-driven reorientation transitions in synthetic antiferromagnets crucially depend on contributions of higher-order anisotropies. A phenomenological macrospin model is derived to describe the magnetic states of two antiferromagnetically coupled magnetic thin film elements. The calculated phase diagrams show that magnetic states with out-of-plane magnetization, symmetric escaped spin-flop phases, exist in a broad range of the applied magnetic field. Due to the formation of such states and concomitant multidomain patterns, the switching processes in toggle…
We resolve the domain-wall structure of the model antiferromagnet $\text{Cr}_2\text{O}_3$ using nanoscale scanning diamond magnetometry and second-harmonic-generation microscopy. We find that the 180$^\circ$ domain walls are predominantly Bloch-like, and can co-exist with Néel walls in crystals with significant in-plane anisotropy. In the latter case, Néel walls that run perpendicular to a magnetic easy axis acquire a well-defined chirality. We further report quantitative measurement of the domain-wall width and surface magnetization. Our results provide fundamental input and an experimental m…
We experimentally study the structure and dynamics of magnetic domains in synthetic antiferromagnets based on Co/Ru/Co films. Dramatic effects arise from the interaction among the topological defects comprising the dual domain walls in these structures. Under applied magnetic fields, the dual domain walls propagate following the dynamics of bi-meronic (bi-vortex/bi-antivortex) topological defects built in the walls. Application of an external field triggers a rich dynamical response: The propagation depends on mutual orientation and chirality of bi-vortices and bi-antivortices in the domain wa…
Antiferromagnets offer remarkable promise for future spintronics devices, where antiferromagnetic order is exploited to encode information. The control and understanding of antiferromagnetic domain walls (DWs) - the interfaces between domains with differing order parameter orientations - is a key ingredient for advancing such antiferromagnetic spintronics technologies. However, studies of the intrinsic mechanics of individual antiferromagnetic DWs remain elusive since they require sufficiently pure materials and suitable experimental approaches to address DWs on the nanoscale. Here we nucleate…
Magnetic nanowires supporting field- and current-driven domain wall motion are envisioned for new methods of information storage and processing. A major obstacle for their practical use is the domainwall velocity, which is traditionally limited due to the Walker breakdown occurring when the forcing field or current reaches a critical threshold value. We show through numerical and analytical modeling that the Walker breakdown limit can be extended or completely eliminated in antiferromagnetically coupled magnetic nanowires. These coupled nanowires allow for giant domain-wall velocities driven b…
In this work, we derive the Landau-Lifshitz-Bloch equation accounting for the multi-domain antiferromagnetic (AFM) lattice at finite temperature, in order to investigate the domain wall (DW) motion, the core issue for AFM spintronics. The continuity equation of the staggered magnetization is obtained using the continuum approximation, allowing an analytical calculation on the domain wall dynamics. The influence of temperature on the static domain wall profile is investigated, and the analytical calculations reproduce well earlier numerical results on temperature gradient driven saturation velo…
We develop a theory to compute the domain-wall magnetoresistance (DWMR) in antiferromagnetic (AFM) metals with different spin structures. In the diffusive transport regime, the DWMR can be either {\it negative} or positive depending on the domain-wall orientation and spin structure. In contrast, when the transport is in the ballistic regime, the DWMR is always positive, and the magnitude depends on the width and orientation of the domain wall. Our results pave the way of using electrical measurements for probing the internal spin structure in antiferromagnetic metals.
The magnetization dynamics of a synthetic antiferromagnet subject to a short magnetic field pulse, has been studied by using a combination of first-principles and atomistic spin dynamics simulations. We observe switching phenomena on the time scale of tens of picoseconds, and inertia-like behavior in the magnetization dynamics. We explain the latter in terms of a dynamic redistribution of magnetic energy from the applied field pulse to other possible energy terms, such as the exchange interaction and the magnetic anisotropy, without invoking concepts such as inertia of an antiferromagnetic vec…
Antiferromagnets have a number of favourable properties as active elements in spintronic devices, including ultra-fast dynamics, zero stray fields and insensitivity to external magnetic fields . Tetragonal CuMnAs is a testbed system in which the antiferromagnetic order parameter can be switched reversibly at ambient conditions using electrical currents . In previous experiments, orthogonal in-plane current pulses were used to induce 90 degree rotations of antiferromagnetic domains and demonstrate the operation of all-electrical memory bits in a multi-terminal geometry . Here, we demonstrate th…
The magnetic microstructure and domain wall distribution of antiferromagnetic $α$-Fe$_2$O$_3$ epitaxial layers is determined by statistical image analyses. Using dichroic spectro-microscopy images, we demonstrate that the domain structure is statistically invariant with thickness and that the antiferromagnetic domain structure of the thin films is inherited from the ferrimagnetic precursor layer one, even after complete transformation into antiferromagnetic $α$-Fe$_2$O$_3$. We show that modifying the magnetic domain structure of the precursor layer is a genuine way to tune the magnetic domain …
The dynamics of a 90$^{\circ }$ domain wall in an antiferromagnetic nanostrip driven by the current-induced spin-orbital torque are theoretically examined in the presence of random thermal fluctuations. A soliton-type equation of motion is developed on the basis of energy balance between the driving forces and dissipative processes in terms of the domain wall velocity. Comparison with micromagnetic simulations in the deterministic conditions shows good agreement in both the transient and steady-state transport. When the effects of random thermal fluctuations are included via a stochastic treat…
In Mn$_3$X (X=Sn, Ge) antiferromagnets domain walls are thick and remarkably complex because of the non-collinear arrangement of spins in each domain. A planar Hall effect (PHE), an electric field perpendicular to the applied current but parallel to the applied magnetic field, was recently observed inside the hysteresis loop of Mn$_3$Sn. The sign of the PHE displayed a memory tuned by the prior orientation of the magnetic field and its history. We present a study of PHE in Mn$_3$Ge extended from room temperature down to 2 K and show that this memory effect can be manipulated by either magnetic…
In ferromagnetic materials, the rich dynamics of magnetic domain walls (DWs) under magnetic field or current have been successfully described using the well-known q-ϕ analytical model. We demonstrate here that this simple unidimensional model holds for multiple-sublattice materials such as ferrimagnetic alloys or synthetic antiferromagnets (SAF) by using effective parameters, and is in excellent agreement with double-lattice micromagnetic simulations. We obtain analytical laws for the DW velocity and internal precession angle as a function of net magnetisation for different driving forces (mag…
The discovery of magnetism in two-dimensional (2D) van der Waals (vdW) materials has flourished a new endeavour of fundamental problems in magnetism as well as potential applications in computing, sensing and storage technologies. Of particular interest are antiferromagnets, which due to their intrinsic antiferromagnetic exchange coupling show several advantages in relation to ferromagnets such as robustness against external magnetic perturbations. This property is one of the cornerstones of antiferromagnets and implies that information stored in antiferromagnetic domains is invisible to appli…
We describe the dynamics of magnetic skyrmions in synthetic antiferromagnets (SAF), with a finite interlayer coupling. Due to the opposite gyrovector of the skyrmions in the two SAF layers, the coupled skyrmions reach a stationary regime with a spatial separation, in a direction orthogonal to their velocity. As a consequence, and contrary to the ferromagnetic situation, a transient regime necessarily occurs, with a finite acceleration, related to an inertia, and that limits its time response. The formalism developed here, based on two coupled Thiele equations, allows a quantitative description…
Heusler alloy Co2FeAl was employed as ferromagnetic layers in Co2FeAl/Ru/Co2FeAl synthetic antiferromagnet structures. The experimental results show that the structure with a Ru thickness of 0.45 nm takes on strongly antiferromagnetic coupling, which maintains up to 150 oC annealing for 1 hour. The structure has a very low saturation magnetization Ms of 425 emu/cc, a low switching field Hsw of 4.3 Oe and a high saturation field Hs of 5257 Oe at room temperature, which are favorable for application in ultrahigh density magnetic read heads or other magnetic memory devices. XRD study testifies th…
Chemically ordered B2 FeRh exhibits a remarkable antiferromagnetic-ferromagnetic phase transition that is first order. It thus shows phase coexistence, usually by proceeding though nucleation at random defect sites followed by propagation of phase boundary domain walls. The transition occurs at a temperature that can be varied by doping other metals onto the Rh site. We have taken advantage of this to yield control over the transition process by preparing an epilayer with oppositely directed doping gradients of Pd and Ir throughout its height, yielding a gradual transition that occurs between …
We investigate ferrimagnetic domain wall dynamics induced by circularly polarized spin waves theoretically and numerically. We find that the direction of domain wall motion depends on both the circular polarization of spin waves and the sign of net spin density of ferrimagnet. Below the angular momentum compensation point, left- (right-) circularly polarized spin waves push a domain wall towards (away from) the spin-wave source. Above the angular momentum compensation point, on the other hand, the direction of domain wall motion is reversed. This bidirectional motion originates from the fact t…
Hybrid Graphene/magnetic structures offer a unique playground for fundamental research, and opportunities for emerging technologies. Graphene-spaced ultrathin structures with antiferromagnetic exchange-coupling (AFC) seem a relevant scenario, analogous to that of conventional metallic multilayer devices. Unfortunately, the AFC found so far between bulk magnetic single crystals and Graphene-spaced adatoms, clusters or molecules either requires low temperatures, is too weak, or of complex nature, for realistic exploitation. Here we show theoretically and experimentally that a strong perpendicula…
Antiferromagnetic materials are outstanding candidates for next generation spintronic applications, because their ultrafast spin dynamics makes it possible to realize several orders of magnitude higher-speed devices than conventional ferromagnetic materials1. Though spin-transfer torque (STT) is a key for electrical control of spins as successfully demonstrated in ferromagnetic spintronics, experimental understanding of STT in antiferromagnets has been still lacking despite a number of pertinent theoretical studies2-5. Here, we report experimental results on the effects of STT on domain-wall (…
This paper addressses the kinetics and dynamics of a family of domain wall solutions along classical antiferromagnetic Heisenberg spin chains at low energies. The equation of motion is derived and found to have long range position- and velocity-dependent two-body forces. A 'quiescent' regime is identified where the forces between walls are all repelling. Outside this regime some of the interactions are attractive, giving rise to wall collisions whereupon the colliding walls annihilate. The momentum of the system is found to be conserved in the quiescent regime and to suffer discontinuous jumps…
Due to the difficulty in detecting and manipulating magnetic states of antiferromagnetic materials, studying their switching dynamics using electrical methods remains a challenging task. In this work, by employing heavy metal/rare earth-transition metal alloy bilayers, we experimentally studied current-induced domain wall dynamics in an antiferromagnetically coupled system. We show that the current-induced domain wall mobility reaches a maximum close to the angular momentum compensation. With experiment and modelling, we further reveal the internal structures of domain walls and the underlying…
Efficient manipulation of antiferromagnetic (AF) domains and domain walls has opened up new avenues of research towards ultrafast, high-density spintronic devices. AF domain structures are known to be sensitive to magnetoelastic effects, but the microscopic interplay of crystalline defects, strain and magnetic ordering remains largely unknown. Here, we reveal, using photoemission electron microscopy combined with scanning X-ray diffraction imaging and micromagnetic simulations, that the AF domain structure in CuMnAs thin films is dominated by nanoscale structural twin defects. We demonstrate t…
Magnetic materials that possess large bulk perpendicular magnetic anisotropy (PMA) are essential for the development of magnetic tunnel junctions (MTJs) used in future spintronic memory and logic devices. The addition of an antiferromagnetic layer to these MTJs was recently predicted to facilitate ultra-fast magnetization switching. Here, we report a demonstration of a bulk perpendicular synthetic antiferromagnetic (P-SAFM) structure comprised of a (001) textured FePd/Ru/FePd trilayer with a face-centered-cubic (fcc) phase Ru spacer. The L10 FePd P-SAFM structure shows a large bulk PMA (~10.2 …
We examine the combined effects of interlayer exchange coupling (IEC) and the interfacial Dzyaloshinskii-Moriya Interaction (DMI) on the structure of magnetic domain walls in fully compensated synthetic anti-ferromagnets (SAFs). Ir-based SAFs with ferromagnetic (FM) layers based on [Pt/(Co/Ni)M]N were characterized by Lorentz transmission electron microscopy (LTEM). The multi-layer design of the individual ferromagnetic layers enables control of the interfacial Dzyaloshinskii-Moriya interaction (via 'M') and, in turn, the structure and chirality of domain walls (DWs). We compare the Fresnel-mo…
We derive a phenomenological theory of current-induced staggered magnetization dynamics in antiferromagnets. The theory captures the reactive and dissipative current-induced torques and the conventional effects of magnetic fields and damping. A Walker ansatz describes the dc current-induced domain-wall motion when there is no dissipation. If magnetic damping and dissipative torques are included, the Walker ansatz remains robust when the domain-wall moves slowly. As in ferromagnets, the domain-wall velocity is proportional to the ratio between the dissipative-torque and the magnetization dampin…
A flat band in fermionic system is a dispersionless single-particle state with a diverging effective mass and nearly zero group velocity. These flat bands are expected to support exotic properties in the ground state, which might be important for a wide range of promising physical phenomena. For many applications it is highly desirable to have such states in Dirac materials, but so far they have been reported only in non-magnetic Dirac systems. In this work we propose a realization of topologically protected spin-polarized flat bands generated by domain walls in planar magnetic topological ins…
Topological insulators, in contrast to ordinary semiconductors, accompany protected metallic surfaces described by Dirac-type fermions. Here, we theoretically show another emergent two-dimensional metal embedded in the bulk insulator is realized at a magnetic domain wall. The domain wall has long been studied as ingredients of both old-fashioned and leading-edge spintronics. The domain wall here, as an interface of seemingly trivial antiferromagnetic insulators, emergently realizes a functional interface preserved by zero modes with robust two-dimensional Fermi surfaces, where pyrochlore iridi…
Polarization is a key ingredient of all waves, including the electromagnetic wave, the acoustic wave, as well as the spin wave. Due to the fixed ferromagnetic order, the spin wave in ferromagnet is limited to the right circular polarization. The spin wave in antiferromagnet, however, is endowed with the full polarization degree of freedom because of the two identical magnetic sublattices. In the synthetic antiferromagnet, the two magnetic sublattices are spatially separated into two sublayers. The circular polarization of spin wave is partially locked to the magnetic sublattice of the antiferr…
We explore the emergence of spin-polarised flat-bands at head-to-head domain walls in a recently predicted class of antiferromagnetic topological insulators hosting planar magnetisation. We show, in the framework of quantum well physics, that by tuning the width of a domain wall one can control the functional form of the bound states appearing across it. Furthermore, we demonstrate the effect that the parity of the number of layers in a multilayer sample has on the electronic dispersion. In particular, the alignment of the magnetisation vectors on the terminating surfaces of odd layer samples …
