Propagation Effects in the Spin-Wave Spectrum of the Ferromagnetic Thin Film

Exchange spin waves propagating in magnetic thin films exhibit some dynamic effects, that is, the effects caused solely by the wave propagation. In this paper we put our attention in four phenomena of such kind: the surface (and subsurface) localization, the collapse of the bulk band into a single energy level, the reversal of the mode order in the spin-wave spectrum, and the dynamic separation of the thin film into two subsystems. We link these effects to properties of Hamiltonian matrix elements inherited from the spatial distribution of neighboring spinswith respect to the film surface.We also provide necessary conditions for the occurrence of these effects.


Introduction
Magnetic thin films are the subject of intensive studies, both experimental [1][2][3][4][5] and theoretical [6][7][8][9][10], for more than half a century.This interest is stimulated by two main reasons: the huge progress in the fabrication technology of such structures [11] and the rich physics they exhibit.The latest is also inherited in more complicated systems such as magnetic multilayers [12][13][14] or thin-film magnonic crystals [15][16][17][18].A particular class of phenomena is related to the parallel to the surface propagation of spin waves (in-plane propagation).
The propagation of spin waves in thin magnetic films leads to several effects including surface localization.For example, if magnetostatic interactions are taken into account famous Damon-Eshbach modes occur [19].These modes are localized only at one surface, bottom or top, depending on the propagation direction.As a consequence nonreciprocity effect appears in the spin-wave spectrum for thin films with one surface differing from another [20].Both effects are dynamic and disappear for standing spin waves, that is, for in-plane wave vector equal to zero.The dynamic localization appears also for purely exchange spin waves.Even in very simple model such as ferromagnetic thin film with natural surface and only nearest neighbor (NN) exchange interactions taken into account the surface localization occurs for propagating spin waves [21,22].In this case also another dynamic effect was observed, namely, the collapse of the bulk band, where all bulk modes are degenerated having the same energy [23].
Dynamic surface localization in thin magnetic films was studied theoretically in last decades [24][25][26][27] but it strongly gained in importance since spin polarized electron energy loss spectroscopy (SPEELS) was used to investigate propagating spin waves [28][29][30].In contrast to the ferromagnetic resonance (FMR), which investigates standing spin waves, SPEELS gives possibility to scan the entire surface Brillouin zone (SBZ); thus it proves to be the powerful tool for the experimental study of propagation effects.The method allows experimental studies of the exchange interactions [31], magnetic anisotropy [32], or thermal properties [33].For example, latest SPEELS researches show the interaction type in iron thin films to be strongly dependent on the substrate: from antiferromagnetic exchange [34] to Dzyaloshinskii-Moriya one [35].It allows also for the direct observation of spin waves localized dynamically at the surface [36].
In the planar model used in our work the thin film is treated as a set of layers representing lattice planes parallel to the surface.This allows us to reduce the problem to a onedimensional finite chain [7] with effective coupling between its elements (lattice planes).Propagation effects are strongly influenced by this effective interlayer coupling which reflects the total influence of exchange interactions from neighboring Advances in Condensed Matter Physics spins.The interlayer coupling was studied for standing spin waves in both dipolar and exchange coupled multilayers [37][38][39][40][41][42].For exchange coupled layers the effective coupling depends on the wave vector of the propagating spin wave which may lead to its vanishing and the changing of its sign (between ferro-and antiferromagnetic) [43].
The spin-wave spectrum of different magnetic systems is strongly influenced by the range of considered interactions.For example, concurrence of the short range exchange and long range dipolar interactions generates menagerie of (meta)stable magnetic configurations [44][45][46][47][48][49][50] and causes rich spectrum of magnetic excitations [51][52][53][54][55] in magnetic dots, leads to the complete bandgap opening in magnonic crystals [56], or induces the spin-wave pinning in strongly curveted wires [57].In the case of thin films it is responsible for the occurrence of negative group velocity [58] and the splitting of the spin-wave spectrum into subbands [14].Also purely exchange spin waves exhibit new features if next nearest neighbors (NNN) interactions are considered, for example, subsurface localization [59] or separation of the film into subsystems [43].
These dynamic effects and their dependence on NNN interactions as well as on the crystallographic structure of thin film are the main objective of the present paper.We show that some of these effects could be useful for the determination of the NNN exchange interactions which is often a rather difficult task.One of more spectacular examples is the case of europium monochalcogenides (EuX).For EuS the literature data differ by the factor 2, and for EuO even the sign of   remains in question [60][61][62].The bulk band collapse, followed by the dynamic mode order reversal with changing wave vector, seems especially useful for this purpose.
In our work we use the planar model of the thin film with spin waves described by the Heisenberg Hamiltonian which contains NN and NNN interactions.The model has been known since the late sixties and its derivation is performed in several papers (see, e.g., [7]); thus we shall only recall its main assumptions in Section 2. In the next sections we use this model to explain dynamic (propagation) effects in the spin-wave spectrum of thin films.In Section 3 we consider a surface localization of spin waves caused solely by their propagation and provide necessary conditions for this type of localization to occur.In Section 4 we study the dynamic collapse of the bulk band caused by the vanishing of the effective coupling between adjacent layers.For certain in-plane wave vectors all bulk modes in the spin-wave spectrum have equal energy, which makes the whole bulk band collapse into one energy level.The effect is usually followed by the reversal of the order of modes in two regions of the SBZ.In Section 5 we show that the thin film can separate into subsystems in which spin waves propagate independently for wave vectors from the border of the SBZ.Major results are summarized in Section 6.The paper is completed with two appendixes.In Appendix A we provide the neighbors distribution in the thin films considered in the paper.Appendix B contains the explicit dependence on the in-plane wave vector of structural sums introduced in Section 2.
The planar model of thin film: the film can be treated as a set of  crystallographic planes (layers) parallel to the surface.The wave vector is decomposed into two components, k ‖ = (  ,   ) and k ⊥ , parallel and perpendicular to the surface, respectively.

The Model
The schematic plot of a model is shown in Figure 1.The system under consideration is a thin magnetic film that consists of spins  arranged in sites of a crystal lattice.The film can be treated as a set of crystallographic planes (layers) parallel to its surface (- plane).An external magnetic field strong enough to stabilize the homogenous ferromagnetic ground state is applied perpendicularly to the surface (along the  direction).Under this assumption the Zeeman energy can be neglected as it only results in a uniform shift of the spin-wave spectrum.The wave vector can be decomposed into a parallel to the film surface in-plane component k ‖ = (  ,   ) and perpendicular component k ⊥ .The first component is related to the in-plane propagation of spin waves while the second one describes their quantization along the film thickness.
Considering uniform NN and NNN exchange interactions the Hamiltonian of the film takes the following form: where  and   are the NN and NNN exchange integrals, respectively.In the planar model of thin film the position of each spin is given by the layer number, , and the position vector within the layer, j.The symbols   j  in (1) refer to the neighbors of the spin j, nearest in the first summation and next nearest in the second one.The spin from the plane  can have neighbors in the same lattice plane (j  ∈   = ), in neighboring planes (j  ∈   =  + 1), or in further planes.The detailed description of the diagonalization procedure of Hamiltonian (1) is given in several papers (see, e.g., [7]).The first step is the Holstein-Primakoff transformation, that is, the mapping to boson operators.Then the Fourier transformation in the plane of the film is performed and, subsequently, the Bogolyubov-Tyablikov transformation in the direction perpendicular to it (mapping to the wave-vector space).The assumptions the method is based on make it valid for low concentration of the spin waves (low temperatures in the case of thermal magnons) and for thin films large in the directions parallel to the surface.
The final outcome is an  ×  band matrix, where  is the number of layers forming the thin film, in the following form: The number of nonzero elements depends on the spatial distribution of the NN and NNN (provided in Appendix A) and their explicit forms depend on the exchange integrals  and   , as well as on the in-plane wave vector.Matrix ( 2) is typical for one-dimensional chain where the interaction between its links is described by the off-diagonal elements of the matrix.Other entries of the Hamiltonian matrix have the following meanings:  represents the onsite energy;  1-2 are surface (subsurface) parameters (see, e.g., [7] for more details).In other words, the method used transforms thin film into a one-dimensional chain, where the role of chain elements is played by lattice planes parallel to the surface.This means the off-diagonal entries of the matrix describe the effective coupling between lattice planes:  1 refers to the interaction between nearest layers,  2 to the interaction between next nearest ones, and so forth.In our microscopic model this effective interlayer coupling reflects the total influence of exchange interactions from neighboring spins taking into account NN and NNN.The coupling depends not only on exchange integrals and the surface cut (NN and NNN distribution) but also on the inplane wave vector which is the reason for the occurrence of propagation (dynamic) effects in the spin-wave spectrum.Thus the knowledge of the dependence of off-diagonal elements on the in-plane wave vector allows forecasting the propagation effects.It is also worth noticing that the Zeeman energy, as an on-site energy, only affects the diagonal terms, shifting the whole spectrum of eigenvalues in a uniform way.
During the diagonalization of the Hamiltonian it is useful to introduce the so-called structural sums, which describe the spatial distribution of the NN and NNN [7], defined as follows: where summation over j  runs over the projections of (next) nearest neighbors of the spin in the plane  belonging to the plane  + .The explicit forms of structural sums for surface cuts considered in this paper are given in Appendix B.
Elements of the Hamiltonian matrix expressed in terms of structural sums are the following: where  = 1, 2. Please notice that dependence of Hamiltonian matrix elements on the in-plane wave vector is introduced only by structural sums.Therefore the evolution of these sums with k ‖ determines propagation effects.

Dynamic Surface Localization
Surface localization of spin waves in thin magnetic films is widely studied in literature both experimentally and theoretically.For exchange spin waves it was found that the key factor for the surface localization is geometric disposition of interacting neighbors with respect to the surface rather than the interaction range: the emergence of surface states is due to interactions with neighbors the position vectors of which are oblique to the film surface, irrespectively of whether these "oblique" interactions are NN or NNN type [21,22,63].Here we systemize the results and link them to the properties of Hamiltonian matrix elements.We also show that the range of interactions is important as well, however not the direct spinspin interactions but the effective interlayer ones.The geometric disposition of neighboring spins is particularly important in the case of purely dynamical localization, that is, for the thin film with natural surfaces (no pinning).In such system there is no surface localization for the SBZ center but it could occur for nonzero in-plane wave vectors (propagating spin waves).To obtain surface localization for k ‖ = 0 (which can be detected by FMR experiments) one needs to introduce surface anisotropy to generate additional pinning at the surface.The anisotropy has large influence on the surface localization and can cover the dynamical localization effect.Therefore it is not taken into account in this paper.(The influence of the anisotropy on the spin-wave localization is studied, e.g., in [64].) The simplest case is sc(001) surface cut for which the Hamiltonian matrix is three-diagonal only.This is a consequence of the NN and NNN disposition.Every spin has 4 NN and 4 NNN at the same lattice plane; at the adjacent plane it has 1 NN and 4 NNN (see Appendix A).This means the effective coupling is restricted only to the neighboring lattice planes even if we include NNN interactions.In such case, that is, if the Hamiltonian matrix is three-diagonal, conditions for surface localization are [7]                 < 1, |/| ( − 1) for symmetric and antisymmetric mode, respectively.Here  is off-diagonal element and  is the surface parameter.Please notice that the surface parameter according to (6) depends only on the number of NN and NNN at the adjacent plane so the dynamic surface localization can be induced solely by  through its dependence on structural sums (and then on in-plane wave vector).
According to the definition (3) if the position vector of the neighboring spin from adjacent layer is perpendicular to the surface (we will call it perpendicular neighbor) then the corresponding structural sum is equal to 1 (the number of adequate neighbors) independently of k ‖ .In this case  = − and neither condition ( 7) nor ( 8) is fulfilled and thus surface states cannot occur.In a consequence the necessary condition for the dynamic localization is an oblique disposition of the neighbors from adjacent lattice plane, which makes offdiagonal elements of the Hamiltonian matrix functions of the in-plane wave vector.For example, for sc(001) surface cut there is only one "perpendicular" nearest neighbor at the layer  + 1 and thus corresponding structural sum Γ 1 = 1.The structural sum for NNN is given by Γ  1 = 2(cos   + cos   ) which means these 4 "oblique" neighbors form layer  + 1 are responsible for eventual surface localization.Thus, the offdiagonal element  is given by where   and   are components of the in-plane wave vector k ‖ .
In Figure 2 we show exemplary spin-wave spectra for sc(001) thin films consisting of  = 7 lattice planes.The spectra are plotted along the high-symmetry path marked as a green solid line in the surface Brillouin zone (SBZ) shown in the inset in Figure 2(a).In the first case NNN interactions are assumed to be ferromagnetic (FM) with the ratio of NNN to NN exchange integral set to   / = 0.5 (Figure 2(a)).The second spectrum (Figure 2(b)) is obtained for antiferromagnetic (AFM) NNN interactions with   / = −0.5.In both graphs the shaded area represents the bulk band.Below each spectrum we provide a graph with (k ‖ ) plotted as a green dotted line accompanied with horizontal blue lines for ± (dashed lines) and ±( − 1)/( + 1) (solid lines).In Figures 2(c)-2(h) we show spin-wave profiles for some particular wave vectors marked with purple lines in the corresponding spectra.
For   / = 0.5 case surface parameter  = 6 and condition (7) means that || < 6 ((k ‖ ) should be between dashed lines in Figure 2(a)) which is fulfilled for any k ‖ ̸ = 0; that is, there is a symmetric surface state within the whole SBZ except point Γ.In Figure 2(c) we show spin-wave profiles for the wave vector between points Γ and Δ in the SBZ.The lowest mode is localized at both surfaces while all other modes are bulk modes.The second surface mode, antisymmetric, occurs for wave vector large enough to fulfill condition (8) which in the case in question means || < 4.5 ((k ‖ ) between solid lines in Figure 2(a)).The points at which this mode becomes localized are marked by red arrows in Figure 2(a).In Figure 2(d) we show spin-wave profiles for point  in the SBZ where two lowest modes are localized.(The third profile given in panel (e) shows the mode order reversal whose effect is explained in Section 4.) For   / = −0.5 case  = −2 and the surface localization of the symmetric mode appears in some region in the SBZ around its center (again, except point Γ), that is, if (k ‖ ) is between dashed lines in Figure 2(b).For larger k ‖ there is no surface state in the spectrum.Condition (6) for antisymmetric mode localization boils down to || < 1.5 ((k ‖ ) between solid lines in Figure 2(b)) which is fulfilled for wave vectors from some ring around point Γ (between points marked by red arrows, cf.profiles in Figures 2(f) and 2(g)).
Other surface cuts for which the Hamiltonian matrix is three-diagonal are bcc(110) and fcc(111) but now neighbors of both types, NN and NNN, which are located on the adjacent layer are disposed obliquely with respect to the surface plane.Therefore, both interactions are responsible for the dynamic surface localization.
Concerning the influence of the NN and NNN on the surface localization very instructive are surface cuts bcc(001) and fcc(001).In both cases there are 4 NN located "obliquely" at the adjacent plane ( + 1) and one "perpendicular" NNN at next-neighboring plane ( + 2).This means only NN are responsible for spin-wave localization.In other words, for the hypothetical situation when NN interactions are switched off ( = 0) and only NNN interactions are taken into account (  ̸ = 0) there will be no surface states in the spinwave spectrum.Exemplary spectra are shown in Figure 3 for bcc(001) thin film consisting of 11 lattice planes ( = 11) with only one type of interactions taken into account: NN in panel (a) and NNN in panel (b).For the first case  1 = 8 and the localization condition for symmetric mode is fulfilled for any in-plane wave vector except 0 ( 1 (k ‖ ) above dashed blue line in Figure 3(a)).For antisymmetric mode the condition is | 1 | < 20/3 = 6.67; thus the second surface state occurs if  1 (k ‖ ) is above solid blue line (between points marked by red arrows).In Figure 3(b), that is, for the hypothetical case of only NNN interactions switched on, there is no surface localization due to the perpendicular disposition of the NNN from the plane  + 2.
In the very last case the Hamiltonian matrix is fivediagonal; thus we should take into account both surface parameters  1 and  2 , and simple conditions (7) and (8) for the surface localization are not valid anymore.This also means that the effective coupling between layers now extends up to the second-neighboring layer which is described by offdiagonal element  2 .For bcc(001) surface cut  2 does not depend on k ‖ because at the layer  + 2 there is only one "perpendicular" neighbor.Different situation holds for fcc(110) surface cut for which there are two "oblique" NNN at the layer  + 2 (and four "oblique" NN at  + 1).Hence, we expect surface localization caused by NN and additional localization   It is also worth noticing that each surface mode is localized at both surfaces.Moreover, in the case of symmetric thin film studied here the localization of particular mode is the same at both surfaces (the spin-wave profile is symmetric or antisymmetric).To differ one surface from the other one should introduce additional pinning.It could be done by taking into account the surface anisotropy [64].However, Step: 0.2 J/J N = 0.9 J/J N = 0.3 anisotropy terms in the elements of the Hamiltonian matrix do not depend on the wave vector (see (19) and ( 20) in [64]).Therefore we do not observe such effects like nonreciprocity found in the case of Damon-Eshbach modes [20].

Band Collapse and Mode Order Reversal
Let us go back to the case for which the Hamiltonian matrix is three-diagonal only.Since off-diagonal element  describes the effective coupling between neighboring layers only nearest layers are coupled.For  < 0 the coupling is of ferromagnetic nature, that is, it favors the in-phase precession of spins belonging to neighboring lattice planes.By analogy with phonons [65], this type of excitation is referred to as acoustic.On the other hand, if  > 0 the coupling has antiferromagnetic nature energetically favoring antiphase precession (if the external field saturates the sample; if not the situation is opposite).This type of excitation is called optic.In the case of phonons the acoustic mode has a lower energy than the optic mode and the same pattern is considered the normal order of modes in the spin-wave spectrum.If the energy of the optic mode is lower than that of the acoustic one, the spin-wave spectrum has a reversed mode order, which occurs as a consequence of AFM interaction between adjacent layers.(This problem is widely discussed for standing spin waves for both types of interlayer coupling, exchange and dipolar; see, e.g., [37,[39][40][41][42].)However, for propagating spin waves  depends on in-plane wave vector which implies evolution of effective coupling within the SBZ.This means the character of effective coupling may change from FM to AFM for different wave vectors and, consequently, the order of modes in the spin-wave spectrum may change as well.
Such dynamic (due to the wave propagation) mode order reversal we observe in spin-wave spectra shown in Figure 2, that is, for sc(001) thin film composed of  = 7 layers.In Figure 2(a) we show the spectrum for FM NNN interactions (  / = 0.5) accompanied by the dependence of (k ‖ ).Going along high-symmetry path in the SBZ (green line in the inset in Figure 2(a)) at point Γ the effective interlayer coupling is FM ( < 0) until point  is reached.In Figures 2(c) and 2(d) we show spin-wave profiles for two points from this section of the high-symmetry path marked in panel (a) with purple lines.For both of them we observe normal order of modes with lowest mode acoustic and highest one optic.After crossing point  off-diagonal element  becomes greater than zero and effective coupling is AFM.This results in reversed mode order shown in Figure 2(e) where spinwave profiles for point  in the SBZ are plotted.Going further along the high-symmetry path  restores its negative value after some point between  and Σ.In other words, the reversed mode order occurs in some region of SBZ around point .
A similar feature we observe for AFM NNN interactions for which case the spin-wave spectrum is presented in Figure 2(b).The main difference is that reversed mode order appears around point Γ now (see the profiles provided in Figures 2(f) and 2(g)) because in this region  > 0. After crossing point  effective coupling is FM ( < 0) and the order of modes is normal (see spin-wave profiles for point  in Figure 2(h)).
These two regions, of normal and reversed order of modes, are separated by line at which  = 0 (see (k ‖ ) dependence on Figures 2(a) and 2(b)).In such situation Hamiltonian matrix has nonzero elements only at its diagonal.In a consequence there are  − 2 eigenvalues equal to  (bulk modes) and two eigenvalues equal to  −  (surface modes).This is what we call band collapse: whole bulk band is collapsed into one energy level.The collapse was discovered by Lévy et al. [23] in a semi-infinite ferromagnet of sc structure and (001) surface.Originally this effect was described as caused by NNN interactions.In [66] we studied this effect in thin films with different surface cuts and found it to occur even without NNN interactions.In [43] we show that the collapse may be followed by the reversal of the order of modes in the spin-wave spectrum.
Since  depends on exchange integrals for NN and NNN also two effects described above should.For sc(001) surface cut element  is given by ( 9) and the equation  = 0 has real solutions only if |  /| ≥ 0.25.Therefore, if NNN interactions are too weak there is no bulk band collapse nor dynamical change of the mode order in the spin-wave spectrum of sc(001) films.For weak FM NNN interactions  is negative and the order of modes in the spectrum is normal within the whole SBZ.On the other hand, for weak AFM NNN interactions the situation is opposite:  > 0 and the order of modes is reversed for any k ‖ .For stronger NNN interactions the solutions for different   / ratios are plotted in Figure 5 in a quarter of SBZ.For the critical value   / ± 0.25 the equation  = 0 is satisfied only by one in-plane wave vector, corresponding to the high-symmetry point  or Γ for FM (Figure 5(a)) or AFM (Figure 5(b)) NNN interactions, respectively.This means that only at this one point does collapse occur and there is no dynamical change of the mode order.For NNN interactions stronger than the critical value solutions of the equation  = 0 form a line in the SBZ which shifts from the corresponding highsymmetry point toward the center of the SBZ quarter.Thus, as we already mentioned, the region of the reversed mode order appears around the point  for FM NNN interactions and around point Γ for AFM ones under the condition that NNN interactions are strong enough.
A similar situation we have for the bcc(110) surface cut; the Hamiltonian matrix is also three-diagonal.In this case the SBZ has the shape of an irregular hexagon which results in rather complicated formulas for the wave vector k ‖ within the SBZ and, consequently, for the elements of the Hamiltonian matrix.To simplify the description of the problem the socalled uniform zone can be introduced [43], based on new coordinates,  1 and  2 , assumed to range from −0.5 to 0.5 within the SBZ.With such definition the uniform zone has the shape of a square regardless of the original SBZ geometry.For bcc(110) case the one-to-one correspondence that maps   Reduced energy [E/SJ] Figure 9: Spin-wave spectrum of an 11-layer thick fcc(110) thin film for   / = 0.5, that is, the special case for which the bulk band collapse occurs even for NNN interactions switched on.Solid lines represent the energy of individual modes; the shaded area is the bulk band for in-plane spin waves.The spectrum is plotted along the high-symmetry path shown in (a), inset, and accompanied with the evolution of off-diagonal elements of the Hamiltonian matrix  1 (green dotted line) and  2 (purple dotted line).
each point in the SBZ to a point in the uniform zone is given by [43] The off-diagonal element of the Hamiltonian matrix expressed in coordinates  1 and  2 is given by Solutions of the equation  = 0 for FM NNN interactions are shown in Figure 6.In contrast to sc(001) case there is no threshold of   / ratio and the collapse occurs even without NNN interactions.For   / = 0 it happens at the border of the SBZ (segment -) and is not followed by the mode order reversal.While FM NNN interactions increase the line  = 0 is moving towards point  and the region of reversed mode order appears.For AFM NNN interactions a necessary condition for the mode order reversal is   / < −1, which means the NNN interactions must be stronger than NN ones.The effect of band collapse followed by mode order reversal should be observable in experiments which gives possibility to look at spin waves over the whole SBZ, for example, by spin polarized electron energy loss spectroscopy (SPEELS) [30].Observation of this effect gives possibility to determine precisely the   / ratio.For example, in sc(001) thin film the eventual collapse appears for some point in between Γ and .From relation (9) for the wave vector from the segment Γ- the condition for the collapse is fulfilled if where  0 is the length of the wave vector along the Γ- segment for which the collapse occurs and  is the lattice constant.For bcc(110) surface cut and FM NNN interactions if the wave vector changes along the line  1 = 0.5(1 −  2 ) (the green dashed line in Figure 6) the condition for the collapse is   / = cos( 2 ).Thus the observation of the collapse for some particular directions in the SBZ gives automatically the ratio of NNN to NN interactions.
For other surface cuts the Hamiltonian matrix is threediagonal if NNN interactions are absent and then the collapse occurs only at the border of the SBZ.For example, for sc(110) thin films with only NN interactions taken into account the off-diagonal element of the Hamiltonian matrix is given by which means  = 0 if   = ±/ √ 2, which is satisfied by wave vectors from the two borders of the SBZ (bold lines in Figure 7(a)).For fcc(001) surface cut and   / = 0 we have and thus  = 0 for the whole SBZ border (  ±  = ±2, bold lines in Figure 7(b)).The same situation, that is, the collapse of the bulk band at the SBZ border, exists also for bcc(001) surface cut (see segment - in Figure 3(a)) for which

Separation into Subsystems
If the Hamiltonian matrix is five-diagonal and the offdiagonal element  1 depends on k ‖ it could happen that  1 = 0 and  2 ̸ = 0 for some in-plane wave vectors.In such case, according to the physical meaning of the off-diagonal elements, there is no effective coupling between neighboring layers while next-neighboring layers are coupled.In other words, from the point of view of propagating spin waves the   system is separated into two subsystems consisting of the layers with odd and even numbers, respectively.
In Figure 8 we show the spin-wave spectrum of bcc(001) thin film with FM (panel (a)) and AFM (panel (b)) NNN interactions.For bcc(001) surface cut there are 4 "oblique" NN at the adjacent layer (+1) and one "perpendicular" NNN at layer +2.This means that  2 depends on NNN interactions only and does not depend on the spin-wave propagation.It has FM character for FM NNN interactions ( 2 < 0, Figure 8(a)) and AFM character for AFM NNN interactions ( 2 > 0, Figure 8(b)).Element  1 depends only on NN and k ‖ ; thus in both cases shown in Figure 8 its evolution with k ‖ is the same:  1 = 0 at the SBZ border, that is, between points  and .For both cases we also show spin-wave profiles (Figures 8(c)-8(f)) for two points in the SBZ: Δ and  (marked with red lines in the corresponding spectrum).For point Δ both spectra are typical for FM thin film but for point  we can clearly see the separation of the thin film into two subsystems.For two lowest modes, localized at surfaces, the nonzero dynamical magnetization (the amplitude of the spin

Conclusion
In magnetic thin films some interesting phenomena occur just due to the spin-wave propagation.In this paper we collect such dynamic effects in the case of exchange spin waves and link them to properties of Hamiltonian matrix elements.First of all, we study the dynamically induced (due to the wave propagation) surface localization which appears only if neighboring spins are located obliquely with respect to the surface of the film regardless of the circumstance of the interactions.We show that the same rule holds for the subsurface localization as well.
Another effect is the collapse of the bulk band into one energy level for some certain wave vectors caused by the vanishing of the effective coupling between adjacent layers of spins.The collapse, usually followed by the reversal of the mode order in the spin-wave spectrum, can be utilized for precise determination of the next nearest neighbors exchange interactions.If the effective interlayer coupling ranges up to the next-neighboring layers the spin waves from the border of  the surface Brillouin zone may exhibit another phenomenon: they can be excited in only every second lattice plane; that is, the thin film acts like that divided into two subsystems: one consists of odd-numbered layers and the second one consists of even-numbered ones.

A. Nearest and Next Nearest Neighbors Distribution
Here we present an analysis of the nearest (NN) and next nearest (NNN) neighbors distribution in thin films discussed in this paper.In Figures 10-12 the 3D distribution is provided as well as 2D mapping onto the lattice plane parallel to the surface of the film.In each figure an arbitrary chosen spin is marked as the central black circle and the shaded plane represents the plane  parallel to the film surface which contains the chosen spin.Below each 3D distribution the 2D mapping of respective neighbors is shown, that is, their projection onto the plane , where shaded circles (rings) represent NN (NNN) from the same plane  while empty dashed ones represent the projections of respective neighbors from adjacent planes ( + ).
The numbers of neighbors within the individual lattice planes parallel to the surface are summarized in Table 1.

B. Structural Sums
In this appendix we provide the explicit dependence on the in-plane wave vector k ‖ = (  ,   ) of structural sums for thin films considered in the paper for NN (Table 2) and NNN (Table 3).For the detailed calculations please see [67].

Figure 2 : 6 AdvancesFigure 3 :
Figure 2: ((a), (b)) Spin-wave spectra of a 7-layer thick sc(001) thin film with NNN exchange interactions (a) ferromagnetic and (b) antiferromagnetic.Solid lines represent the energy of individual modes; the shaded area is the bulk band for in-plane spin waves.Each spectrum is plotted along the high-symmetry path shown in (a), inset.Below each spectrum the evolution of the off-diagonal Hamiltonian matrix element  is shown as a green dotted line.Blue lines represent conditions for the surface localization of symmetric (solid lines) and antisymmetric (dashed lines) modes.Wave vectors for which the antisymmetric mode starts to localize are marked with red arrows.((c)-(h)) Mode profiles corresponding to points in the SBZ marked with purple lines in ((a), (b)).

Figure 4 :
Figure 4: ((a), (b)) Spin-wave spectra of an 11-layer thick fcc(110) thin film (a) without and (b) with NNN exchange interactions.Solid lines represent the energy of individual modes; the shaded area is the bulk band for in-plane spin waves.Each spectrum is plotted along the highsymmetry path shown in (a), inset, and accompanied with the evolution of off-diagonal elements  1 (green dotted line) and  2 (purple dotted line) of the Hamiltonian matrix.(c) Mode profiles of the four lowest modes corresponding to the point  in the SBZ, marked with a red line in (b).

Figure 5 :
Figure 5: Two regions in the SBZ for sc(001) surface cut, with normal () and reverse () mode order in the spin-wave spectrum, for (a) ferromagnetic and (b) antiferromagnetic NNN interactions.The boundary between the regions shifts with growing |  |/ towards the SBZ center as shown by blue curves.For the critical value |  |/ = 0.25 the reversal domain does not occur (its boundary amounts to points  and Γ in (a) and (b), resp.).

Figure 6 :
Figure 6: Two regions in the uniform zone for bcc(110) surface cut and ferromagnetic NNN interactions, with normal () and reverse () mode order in the spin-wave spectrum.The boundary between the regions shifts with growing |  |/ as shown by the arrow.For no NNN interactions (|  |/ = 0) the reversal domain does not occur (its boundary amounts to the segment -).

Figure 7 :
Figure 7: SBZ for (a) sc(110) and (b) fcc(001) surface cuts with the segments of the SBZ border at which the collapse of the bulk band occurs marked with bold blue lines.

Figure 8 :C 2
Figure 8: ((a), (b)) Spin-wave spectra of an 11-layer thick bcc(001) thin film with NNN exchange interactions (a) ferromagnetic and (b) antiferromagnetic.Solid lines represent the energy of individual modes; the shaded area is the bulk band for in-plane spin waves.Each spectrum is plotted along the high-symmetry path shown in (b), inset, and accompanied with the evolution of the off-diagonal Hamiltonian matrix elements  1 and  2 shown as a green and purple dotted line, respectively.((c)-(f)) Mode profiles corresponding to points in the SBZ marked with red lines in ((a), (b)).

Figure 12 :
Figure 12: The same distribution and mapping of neighbors as in Figure 10 but for fcc thin films.Due to the large number of lattice sites in this case circles are shown only for spins from the plane  and for respective neighbors.

Table 1 :
The numbers of nearest neighbors (NN) and next nearest (NNN) neighbors in adjacent lattice planes parallel to the surface in cubic thin films and three surface cuts.Individual planes are numbered  ± , where  = 0, . . ., 3 for considered structures.

Table 2 :
Structural sums Γ  for nearest neighbors disposed in adjacent lattice planes  ± ,  = 0, . . ., 3, parallel to the surface in thin films considered in the paper.

Table 3 :
Structural sums Γ   for next nearest neighbors disposed in adjacent lattice planes  ± ,  = 0, . . ., 2, parallel to the surface in thin films considered in the paper.