Regions of existence of surface spin-waves in surface Brillouin zones of hexagonal simple ferromagnetic thin films

Journal Pre-proof Regions of existence of surface spin-waves in surface Brillouin zones of hexagonal simple ferromagnetic thin films

K.L. Yang, R.K. Qiu, D.Z. Li, J. Li PII:

S1386-9477(19)30416-3

DOI:

https://doi.org/10.1016/j.physe.2019.113773

Reference:

PHYSE 113773

To appear in:

Physica E: Low-dimensional Systems and Nanostructures

Received Date:

15 March 2019

Accepted Date:

11 October 2019

Please cite this article as: K.L. Yang, R.K. Qiu, D.Z. Li, J. Li, Regions of existence of surface spinwaves in surface Brillouin zones of hexagonal simple ferromagnetic thin films, Physica E: Lowdimensional Systems and Nanostructures (2019), https://doi.org/10.1016/j.physe.2019.113773

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Journal Pre-proof

Regions of existence of surface spin-waves in surface Brillouin zones of hexagonal simple ferromagnetic thin films K. L. Yang, R. K. Qiu*, D. Z. Li and J. Li Shenyang University of Technology, Shenyang 110870, P.R. China

Abstract A quantum statistic Green’s-function method is used to study the surface in-plane propagating spin-waves (k//  0) in a symmetric magnetic film consisting of twelve atomic layers with damping. The effects of temperature, external magnetic field, surface exchange coupling, surface anisotropy and thickness on the regions of existence of low- and high-frequency surface spin-waves in the surface two-dimensional Brillouin zone have been investigated. The regions of existence of surface spin-waves increase first and then decrease with increasing temperature or external magnetic field. There is a special temperature or external magnetic field, at which the regions of existence of surface spin-waves exhibit a maximum. Additionally, the spatial distribution of spin-waves in different spin-wave frequency range is also studied. For a film with low-frequency surface spin-waves, there is a special spin-wave frequency range, in which no spin-waves propagate in surfaces, the spin-waves propagate only in internal layers of the film. Namely, in this frequency range, the film is a " spin-waves surface insulator ". For a film with high-frequency surface spin-waves, there is also a special spin-wave frequency range, in which the spin-waves propagate only in surfaces. Namely, in this frequency range, the film is a " spin-waves internal insulator ". In the present work, a quantum statistic approach is developed to study the surface spin-wave of magnetic film. The results show the method to adjust the regions of existence of surface spin-waves in the surface two-dimensional Brillouin zone and are beneficial for building future microwave device based upon magnetic film.

*Corresponding author. E-mail: [email protected] ( R. K. Qiu)

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Journal Pre-proof 1. Introduction Spin-waves, being propagating collective excitations of the transverse component of the magnetization, are regarded as information carriers[1-6]. Ferromagnetic films play important roles in modern information technology ranging ferromagnetic data storage to sensors or magnetic random access memory. Spin-waves are also excited during the switching of magnetic thin film memory elements[7] and contribute significantly to noise associated with high-frequency applications. The magnetic properties of spin located at surfaces can be considerably different from those of spin within the bulk of the material and the interesting physical phenomena such as surface spin-waves can emerge[8-11]. Surface spin-waves are localized at the surface of a magnetic thin films and characterized by a wave vector k// parallel to the surface. Understanding the surface spin-waves of magnetic thin film is essential for the development of these modern information devices that are based on magnetic thin film technology. Multi-layer spin-waves are accessible to investigation by different experimental techniques. The ferromagnetic resonance (FMR)[12] is restricted to excitations not propagating in the plane of the film (k// = 0), whereas Brillouin light scattering (BLS)[13] serves for studying spin-wave excitations with non-zero in-plane wave-vector (k//  0). The spin polarized electron energy loss spectroscopy (SPEELS) gives possibility to scan the entire surface Brillouin zone (SBZ)，thus it allows for the direct observation of spin-waves localized dynamically at the surface[14,15]. A considerable number of experimental studies concerning large wave-vector surface spin-waves of ultrathin cobalt and iron films have emerged lately[16-23]. The large wave-vector surface spin-waves are entirely determined by the microscopic exchange coupling. Most of the theory of surface spin-waves have been concerned with one of two types of geometry, a semi-infinite Heisenberg magnets[24-31] or a Heisenberg magnetic thin film [11,32-38]. Mills and Maradudin first discussed the existence of surface states near the (001) surface of a Heisenberg semi-infinite ferromagnet with a simple cubic lattice. They also studied the effects of the surface state on thermodynamical properties of the magnet[24]. Compared with semi-infinite magnets, surface spin-wave of magnetic thin film has great 2

Journal Pre-proof effect on magnetic properties because of the large surface-to-volume ratio of thin film. The theory of surface spin-waves of magnetic thin film mainly includes the Hamiltonian diagonalization procedure[32,33], the effective-medium formulation[34], the spin operator formalism[11], the Green’s-function method[35-38], etc. The properties of magnetic excitations at surface of a geometrically frustrated lattice thin film are investigated using a spin operator formalism, where the influence of surface exchange and anisotropy parameters on surface and bulk spin-wave energies is discussed[11]. Most of the previous theory mainly uses a semiclassical approach for low temperature region, while the quantum methods have seldom been employed[36]. The method of retarded Green's-function was introduced by Diep-The-Hung et al. in their study of spin-waves and other magnetic properties of ferromagnetic and antiferromagnetic films[36]. It is valuable to develop some quantum methods to study surface spin-waves of magnetic films for entire temperature region below Tc. In the previous theoretical work, great attention has been given to a display of surface spin-waves dispersion relation, and surface spin-waves can be significantly affected by magnetic surface anisotropy[11,35-37] and surface exchange coupling[11,35,37,38]. Puszkarski et al.[32,33] discussed the regions of existence of interface/surface spin-waves in the two-dimensional (in-plane) Brillouin zone for bilayer/single layer film by using a Hamiltonian diagonalization procedure based on the Tyablikov-Bogolyubov scheme. So far the effect of temperature on the regions of existence of surface spin-waves in the surface Brillouin zone for magnetic thin film has not been investigated. In the present work, we attempt to study the effect of temperature, external magnetic field, surface exchange coupling, surface anisotropy and thickness on the regions of existence of surface spin-waves in the surface Brillouin zone for a symmetric magnetic thin film with damping, using a quantum statistic method for the entire temperature region below Tc. The spatial distribution of spin-waves in different frequency range has also been studied. We will use Callen’s Green’s-function method[39], Tyablikov decoupling approximation and Anderson-Callen decoupling approximation[40,41] to deal with the Heisenberg Hamiltonian with single-ion anisotropy. The spatial distribution of spin-wave is obtained by the imaginary of the Green’s-function. The method used here extends the Callen’s Green’s-function method employed for magnetization in ferromagnetism[39]. The motivation of the present paper is to 3

Journal Pre-proof provide a new quantum statistic method to study surface spin-waves and spin-waves spatial distributions of magnetic thin film and to show how to adjust the regions of existence of surface spin-waves in the surface Brillouin zone for a magnetic thin film. The outline of the paper is as follows: In Section 2, we describe the model and the calculation procedure. Section 3 presents spatial distributions of spin-waves and the effect of temperature, external magnetic field, surface exchange coupling, surface anisotropy and thickness on the regions of existence of surface spin-waves in the surface Brillouin zone. In Section 4, conclusions are presented.

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2. Model and calculation procedure

Fig. 1. Geometries: (a) magnetic film and choice of X, Y and Z axes. (b) schematic representation of a film with twelve atomic layers. Only the surface exchange couplings J1 ( within layer 1) and J12 ( within layer 12 ) are illustrated. (c) Planar view of the magnetic atomic layer (d) Brillouin zone in x-y plane.

We consider a Heisenberg modes for a magnetic film. The magnetic film is composed of twelve infinitely extended FM atomic layers. The twelve magnetic atomic layer lie in the X-Y plane, the Z-direction is normal to the X-Y plane. The structure of the film is hexagonal simple. A schematic diagram of the magnetic film, planar view of the magnetic atomic layer and Brillouin zone in the x-y plane are given in Fig. 1(a)-(d). The Hamiltonian is : H=   J ljl ' j ' S lj S l ' j '  2 Dl ( S ljz ) 2   g B B0 S ljz ljl ' j '

lj

lj

5

(2.1)

Journal Pre-proof In Eq. (2.1), the magnetic ions are specified by the set of indices lj, where l is an integer labeling the atomic layers and j is a two-dimensional lattice vector in the X-Y plane. The summation in the first term is over the nearest-neighbor (NN) spins only. The second (anisotropy) term of Eq. (2.1) comprises the surface and bulk anisotropy effects. The assumptions for the NN exchange terms and the definition of the surface and bulk anisotropy terms involved in the Hamiltonian are shown in Table 1. The bulk coupling (J) and surface couplings (J1 and J12) all are ferromagnetic. The direction of the spins in the initial state in a ferromagnetic film is along the positive z-axis. The external magnetic field B0 is also along the positive z-axis. The frequency of the exchange spin-waves lies in the THz range, which makes them attractive for use in the emerging THz wave technology.

Table 1 The assumptions for the NN exchange terms and the definition of the surface and bulk anisotropy terms involved in the Hamiltonian of a ferromagnetic twelve-layer film.

J ljl ' j ' 

J 1 (if both spin are in lower surface layer) J 12 (if both spin are in upper surface layer) J (if both spin are in internal layers)

or (if a spin is in surface layer, another spin is in internal layer) D (for bulk spins) Dl 

D1 (for lower surface spins) D12 (for upper surface spins)

We introduce the Green’s-functions according to Callen[39]:

Gll ' ( , lj , l ' j ' )  Slj | exp(al ' Slz' j ' ) Sl' j ' 

(l=1-12; l'=1-12)

(2)

Where al ' (l'=1-12) are parameters. For the interlayer and the intralayer exchange coupling terms, we adopt the Tyablikov decoupling approximation. For the single-ion anisotropy term, 6

Journal Pre-proof the Anderson-Callen decoupling approximation[40,41] is applied. Using the technique of the equation of motion for Green’s-functions, we obtain the Fourier components of the Green’s-functions of the magnetic film: D * ( ) G  det( D( ))

(3)

Where the matrix D* is adjoint matrix of matrix D,  represents the energy spectrum of the system. And  is diagonal matrix with matrix elements:

 l ,l (al )  [ Sl , exp(al Slz ) Sl ]

(l=1-12)

D( )  Wl ,l ' 1212 - H l ,l ' 1212

(4) (5)

Here, in the twelve-order matrix W, only the diagonal matrix elements Wl,l (l=1-12) are non-zero and equal to  . The matrix H is a three-diagonal matrix with matrix elements Hl,l' (l, l'=1-12):

H 1,1  12 J 1 S1z (1   k // )  2 J S 2z  2 D1 S1z C1  g B B0 H 12,12  12 J 12 S12z (1   k // )  2 J S11z  2 D12 S12z C12  g B B0

H l ,l  12 J Slz (1   k // )  2 J Slz1  2 J Slz1  2 D Slz Cl  g B B0 H l ,l 1  -2 J Slz

(l =1-11)

H l ,l 1  -2 J Slz

(l =2-12)

Here  S lz  1 6  //

 k   eik //

(l =2-11)

(6)

are the spin components in the Z-direction per site in the layers l, and //  //

.  // represents that only the exchanges between the nearest neighbors in xy -

planes are taken into account.

Cl  1  [ S l ( S l  1)  ( S lz ) 2 ] / 2 S l2

(7)

( Slz ) 2  Sl ( Sl  1)  Slz (1  2nl )

(8)

nl 

1 N

12

 (e  k

i 1

i

M ll (i )  1) (i   m )

(9)

m i

Here   1

k BT

with T the temperature. The parameter M ll (l=1-12) is the algebraic 7

Journal Pre-proof cofactor of matrix element H ll in the matrix H. After using the spectral theorem and Callen’s[39] technique, we finally obtain the magnetization of each layer of the film with twelve layers:

( Sl  1  nl )nl2 Sl 1  ( Sl  nl )(1  nl ) 2 Sl 1 S   (1  nl ) 2 Sl 1  nl2 Sl 1 z l

(10)

where l = 1-12. The spectra have been calculated using: det( D( ))  0

(11)

After carrying out the numerical calculations to solve self-consistently the fundamental Eqs. (9), (10) and (11), we obtain solutions for the spin-wave spectra. The spin-wave density of states can be obtained from

 ( )  (1 /  )Tr Im G

(12)

where G is the matrix Green’s function. Next, we will investigate the surface spin-wave in a magnetic film with twelve atomic layers. For the sake of simplicity, in the calculations, we always take the spin quantum number S=1 for each layers. The bulk exchange couplings are set to unity (J =1). If damping is considered in the system, then the spin-wave frequency will be a complex quantity[42]. Therefore, we will introduce a low imaginary frequency in the Green’s-function. For the sake of simplicity, the damping in the twelve magnetic atomic layers has been assumed to be the same and the imaginary frequency has also been taken the same for different frequency, anisotropy, exchange coupling, temperature and magnetic field. The imaginary frequency is

 ，  , with  Gilbert’s damping coefficient and  the frequency. Like in Ref. [43], we take a small Gilbert’s damping coefficient  = 0.01 and  = 10 (units of J) (the frequency is in the range of our numerical results). The low imaginary frequency is taken  ，= 0.1 (units of J). With Eq. (12), the spin-wave density of states can be calculated. The surface exchange parameters used in this paper refer to Ref. [11].

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Journal Pre-proof 3. Results and discussion

Fig. 2. (Color on-line) Dependence of the spin-wave spectra of a symmetric magnetic film with (a) surface exchange coupling J1 = J12 = 0.75 and (b) J1 = J12 = 2 on the wave vector in the Brillouin zone

M  K 

at reduced temperatures τ = T/Tc = 0.5, surface

anisotropies D1 = D12 = 0.05, bulk anisotropy D = 0 and external magnetic field B (= g B B0 ) = 0.15. The Tc is the Curie temperature of the respective system. These curves correspond to the spin-wave spectra 1 , 2 …, 12 , starting from the energetically lowest spectrum.

First, we will present the spin-wave spectra of a symmetric magnetic film. Fig. 2(a) and (b) show the spin-wave spectra of a magnetic film with surface exchange coupling J1 = J12 = 0.75 and J1 = J12 = 2, respectively, in the Brillouin zone   M  K   . In Fig. 2, one can see that there are twelve spin wave spectra. As indicated in the figure caption, the spin-wave spectra 1 , 2 …, 12 , start from the energetically lowest spectrum. The spin-wave spectra increase rapidly with increasing wave vector k in the Brillouin zone   M and K -  . In the Brillouin zone M  K , when the wave vector k changes from the M point to the K point, the twelve spin-wave frequencies increase slowly. From Fig.2(a), when surface exchange coupling J1 = J12 = 0.75, there is a degeneracy of the lower spin-wave spectra 1 and 2 (or 3 and  4 ) for larger wave vector k. When surface intralayer coupling J1 = J12 = 2 (see Fig.2(b)), there is also a degeneracy of the higher spin-wave spectra 11 and 12 (or 9 and 10 ) (or 7 and 8 ) in the Brillouin zone   M  K   . 9

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Fig. 3. Dependence of the reduced spin-wave density of states (  /  0 ) of each layer in a symmetric magnetic film with J1 = J12 = 0.75 and D1 = D12 = 0.05 on the spin-wave frequency in the Brillouin zone   M , at reduced temperatures τ = T/Tc = 0.5, external magnetic field B (= g B B0 ) = 0.15 and at different wave vector (a) K X  0 , (b) K X  1.2542 , (c) K X  2.5048 and (d) K X  2 / 3 . The Tc is the Curie temperature of the system.

Next, we will study the spatial distribution of the spin-waves 1 , 2 ..., 12 of a 10

Journal Pre-proof magnetic film for different wave vector and different layers. Fig. 3 shows the dependence of the reduced spin-wave density of states (  /  0 ) of twelve layers of a magnetic film with J1 = J12 = 0.75 on the spin-wave frequency in the Brillouin zone   M . The  is the spin-wave density of states for different spin-wave frequencies, and the  0 the spin-wave density of states of layer 1 of a magnetic film with D1 = D12 = 0.08, D = 0.01 and J1 = J12 = 1 at τ = T/Tc = 0.125 and B (= g B B0 ) = 0.5 in the  point of the Brillouin zone and in frequency  =1.0575. From Fig.3(a), when kx=0, low frequency spin-waves distribute on all layers of the film (i.e. layers 1-12), while high frequency spin-waves distribute on internal layers of the film. With increasing wave vector kx (from Fig.3(a)-(d)), the distribution of the spin-wave

1 (or 2 ) increases gradually in surface layers (i.e. layer 1 and layer 12) and decreases gradually in internal layers, while the distribution of other spin-waves decreases gradually in surface layers. When the wave vector kx is equal to or greater than 2.5048 (from Fig.3(c)-(d)), the spin-wave 1 (or 2 ) propagates only in surface layers, and other spin-waves propagate only in internal layers. This means that there is a special wave vector in the Brillouin zone   M , when the wave vector kx is equal to or greater than the special wave vector, spin-wave 1 (or 2 ) propagates only in surface, i.e. surface spin-wave appears in the magnetic film. We refer to the special wave vector as critical wave vector kc. In the calculation, when the wave vector k is equal to the critical wave vector kc, the density of states of spin wave 1 (or 2 ) at surface is 18 times higher than that of spin wave 1 (or

2 ) at sub-surface (2nd and 11th layer).

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Fig. 4. Dependence of the reduced spin-wave density of states (  /  0 ) of each layer in a symmetric magnetic film with J1 = J12 = 2 and D1 = D12 = 0.05 on the spin-wave frequency in the Brillouin zone   M , at reduced temperatures τ = T/Tc = 0.5, external magnetic field B (= g B B0 ) = 0.15 and at different wave vector (a) K X =0, (b) K X =0.6864, (c) K X =1.3728 and (d) K X = 2 / 3 . The Tc is the Curie temperature of the system.

Fig.4 shows the spatial distribution of the spin waves 1 , 2 ..., 12 of a magnetic film with J1 = J12 = 2 in the Brillouin zone   M . With increasing wave vector kx (from 12

Journal Pre-proof Fig.4(a)-(d)), the distribution of the high frequency spin-wave 12 (or 11 ) in surface layers (internal layers) gradually increases (decreases), while the distribution of other spin-waves in surface layers gradually decreases. From Fig.4(c)-(d)), when the wave vector kx is equal to or greater than 1.3728 (i.e. critical wave vector kc ), the spin-wave 12 (or 11 ) propagates only in surface layers, and other spin-waves propagate only in internal layers. This means that surface spin-waves appear in the magnetic film as the wave vector kx is equal to or greater than the critical wave vector kc. In addition, in Fig.4d, there is the subsurface localization, i.e. spin-wave 9 and 10 are localized at 2nd and 11th layers. In the case of ferromagnetic thin films with natural surface, the occurrence of the surface/subsurface localization is caused by exchange interactions oblique to in-plane propagation directions, which was shown first by Wallis et al. [25] and then studied in details by H. Puszkarski [44,45]. In the thin films discussed in our work, there are no oblique-to-surface neighbors, the reason for surface/subsurface localization is that the parameters of surface layer are different from those of bulk.

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Fig. 5. The regions of existence of surface spin-waves in the two-dimensional (in-plane) Brillouin zone for a symmetric magnetic film with twelve atomic layers. The surface spin-waves exist in the shaded regions. Note that the boundaries of the regions of existence of surface spin-waves correspond to the critical wave vector kc. In the calculation, surface anisotropies D1 = D12 = 0.05, bulk anisotropy D = 0, external magnetic field B (= g B B0 ) = 0.15 and surface exchange coupling (a)-(d) J1 = J12 = 0.75 (e)-(h) J1 = J12 = 2. In Fig.5(a)-(d), the reduced temperature t=0.321, 0.333, 0.5 and 0.56, respectively. In Fig.5(e)-(h), the reduced temperature t= 0.036, 0.357, 0.571 and 0.93, respectively. The Tc is the Curie temperature of the respective systems.

Next, we will study the existence regions of surface spin-waves in the hexagonal two-dimensional surface Brillouin zone. Fig.5 shows the Brillouin zone with, marked thereon, the existence regions of surface spin-waves determined by the critical wave vector kc for different temperatures and two different surface exchange coupling films (J1 = J12 = 0.75 and J1 = J12 = 2). From Fig.5, it can be said that the emergence of surface spin-waves occurs much more easily on the edges of the Brillouin zone ( and especially on its apices ) than at its centre. It agrees with the results of interface spin-waves in literature[33]. Moreover, the regions of existence of low-frequency (Fig.5(a)-(d)) and high-frequency (Fig.5(e)-(h)) surface spin-waves increase first (specifically, towards the centre of the Brillouin zone) and then decrease with increasing temperature. The regions of existence of low- (high-) frequency surface spin-waves exhibit a maximum at τ =0.5 (0.571). It can be explained that the 14

Journal Pre-proof difference of spins in surface and bulk is the largest at this temperature. The low-frequency surface spin-waves only exist in a finite temperature range, while high-frequency surface spin-waves almost exist in the whole temperature range (below the Curie temperature). The regions of existence of high-frequency surface spin-waves of a film with J1 = J12 = 2 are larger than that of low-frequency surface spin-waves of a film with J1 = J12 = 0.75 if other parameters are the same. Namely, surface spin-waves are generated more easily in a film with surface exchange couplings J1 = J12 = 2 than in a film with surface exchange couplings J1 = J12 = 0.75. The above differences of the regions of existence of low- and high-frequency surface spin-waves are related to the surface exchange coupling parameters.

Fig. 6. The regions of existence of surface spin-waves in the two-dimensional (in-plane) Brillouin zone for a symmetric magnetic film with twelve atomic layers. The surface spin-waves exist in the shaded regions. The boundaries of the regions of existence of surface spin-waves correspond to the critical wave vector kc. In the calculation, surface anisotropies D1 = D12 = 0.05, bulk anisotropy D = 0, reduced temperature τ = T/Tc = 0.5 and surface exchange coupling (a)-(d) J1 = J12 = 0.75 (e)-(h) J1 = J12 = 2. In Fig.6(a)-(d), the external magnetic field B = 0.07, 0.15, 1 and 2, respectively. In Fig.6(e)-(h), the external magnetic field B = 0.02, 0.15, 2 and 4, respectively. The value of Tc of the respective systems is the value at magnetic field B = 0.15.

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Journal Pre-proof Fig.6 shows the Brillouin zone with, marked thereon, the existence regions of surface spin-waves determined by the critical wave vector kc for different external magnetic fields and two different surface exchange coupling films (J1 = J12 = 0.75 and J1 = J12 = 2). From Fig.6, the regions of existence of surface spin-waves increase first and then decrease with increasing external magnetic field. The regions of existence of low- or high-frequency surface spin-waves exhibits a maximum at B =0.15. It can be also explained that the difference of spins in surface and bulk is the largest at this external magnetic field. Similar to Fig.5, when other parameters are the same, the regions of existence of surface spin-waves of the system (J1 = J12 = 2) are larger than that of the system (J1 = J12 = 0.75).

Fig. 7. The regions of existence of surface spin-waves in the two-dimensional (in-plane) Brillouin zone for a symmetric magnetic film with twelve atomic layers. The surface spin-waves exist in the shaded regions. Note that the boundaries of the regions of existence of surface spin-waves correspond to the critical wave vector kc. In the calculation, surface anisotropies D1 = D12 = 0.05, bulk anisotropy D = 0, reduced temperature τ = T/Tc = 0.5 and external magnetic field B (= g B B0 ) = 0.15. In Fig.7(a)-(c), the surface exchange couplings J1 = J12 = 0.8, 0.87 and 0.89, respectively. In Fig.7(d)-(f), the surface exchange couplings J1 = J12 = 1.41, 1.42 and 1.75, respectively. The value of Tc is the value at surface exchange couplings J1 = J12 =0.75 (Fig.7(a)-(c)) or J1 = J12 =2 (Fig.7(d)-(f)). 16

Journal Pre-proof Fig.7 shows the Brillouin zone with, marked thereon, the existence regions of surface spin-waves determined by the critical wave vector kc for different surface exchange couplings. From Fig.7, it can be seen that the regions of existence of surface spin-waves are considerably broadened with decreasing (increasing) surface exchange couplings for the film with J1 = J12 < 1 (J1 = J12 > 1). It is because for the films with J1 = J12 < 1 (J1 = J12 > 1), when surface exchange couplings decrease (increase), the difference between surface and bulk of the film becomes larger, as a consequence, the region of existence of surface spin-waves also becomes larger.

Fig.8. (Color on-line) The regions of existence of surface spin-waves in the two-dimensional (in-plane) Brillouin zone for a symmetric magnetic film with twelve atomic layers. The surface spin-waves exist in the shaded regions. Note that the boundaries of the regions of existence of surface spin-waves correspond to the critical wave vector kc. In the calculation, surface anisotropies D1 = D12 = 0.1 and -0.1 (blue and red shaded regions, respectively), bulk anisotropy D = 0, reduced temperature τ = T/Tc = 0.5, external magnetic field B (= g B B0 ) = 0.15 and surface exchange coupling (a) J1 = J12 = 0.75 (b) J1 = J12 = 2. The value of Tc of the respective systems is the value at surface anisotropies D1 = D12 =0.05.

Fig.8 shows the Brillouin zone with, marked thereon, the existence regions of surface spin-waves determined by the critical wave vector kc for two different surface anisotropies and two different surface exchange coupling films (J1 = J12 = 0.75 and J1 = J12 = 2). From 17

Journal Pre-proof Fig.8(a), it can be seen that the regions of existence of low-frequency surface spin-waves of a film with an easy-plane surface anisotropy (i.e. D1 = D12 < 0) are slightly larger than that of a film with an easy-axis surface anisotropy (i.e. D1 = D12 > 0). But the regions of existence of high-frequency surface spin-waves of a film with an easy-plane surface anisotropy are slightly smaller than that of a film with an easy-axis surface anisotropy (see Fig.8(b)). It agrees with the results of literature[33], in which the regions of existence of low-frequency interface spin-waves of a bilayer film with easy-plane interface anisotropy are larger than those of a bilayer film with easy-axis interface anisotropy.

Fig. 9. (Color on-line) The regions of existence of acoustic surface spin-waves in the two-dimensional (in-plane) Brillouin zone for a symmetric magnetic film with twelve and four atomic layers (red and blue shaded regions, respectively). Note that the boundaries of the regions of existence of surface spin-waves correspond to the critical wave vector kc. In the calculation, surface anisotropies D1 = D12 = 0.05, bulk anisotropy D = 0, external magnetic field B (= g B B0 ) = 0.15 and surface exchange coupling J1 = J12 = 0.75. In Fig.9(a)-(f), the reduced temperature t = 0.32, 0.36, 0.417, 0.5, 0.53 and 0.56, respectively. The value of Tc is the value for the magnetic film with twelve atomic layers.

Fig.9 shows the dependence of the regions of existence of surface spin-waves on 18

Journal Pre-proof temperature for two magnetic films with twelve and four atomic layers. From Fig.9(b)-(e), there are surface spin-waves in both magnetic films with twelve and four atomic layers, and the regions of existence of surface spin-waves in the twelve atomic layers are larger than that in the four atomic layers. From Fig.9(a) and (f), surface spin-waves exist only in the twelve atomic layers. Therefore, the temperature region of existence of surface spin-waves in the twelve atomic layers are larger than that in the four atomic layers. By our calculation, the regions of existence of surface spin-waves in the two-dimensional Brillouin zone for eight or six atomic layers smaller slightly than that of twelve atomic layers, and the temperature region of the existence of surface spin-waves in eight or six atomic layers is almost the same as that in twelve atomic layers. Our results are agree with that in literature[46], where the experimental data of Etzkorn et al. show that the surface spin-wave is practically independent of the number of layers beyond five monolayers.

Fig. 10. (Color on-line) The spin-wave spectra in a symmetric magnetic film with twelve atomic layers in the Brillouin zone   M  K   . Several special spin-wave frequency ranges and the distributions of spin-waves are illustrated. Here the kc is critical wave vector and the  c is the frequency of the spin-wave spectra 1 (or  2 ) with critical wave vector kc. The k4 (k3), k5, k6 , k7 and k8 is the wave vector of spin-wave spectra  4 (or 3 ), 5 ,

6 , 7 and 8 with frequency c , respectively. In the calculation, J1 = J12 = 0.75, D1 = D12 = 0.05, D = 0, reduced temperatures τ = T/Tc = 0.5 and B (= g B B0 ) = 0.15. 19

Journal Pre-proof Next, we will study the spatial distribution of spin-waves in different spin-wave frequency range. From Fig. 10, there are two special frequency ranges 1 and 2 in the Brillouin zone   M for a symmetric magnetic film with surface exchange coupling J1 = J12 = 0.75. Here, the 1 is from the highest frequency of the spin wave 1 (or  2 ) to the highest frequency of the spin-wave 12 , the 2 from the spin-wave frequency  c (the

c is the frequency of the spin-wave 1 (or  2 ) with the critical wave vector kc) to the highest frequency of the spin wave 1 (or  2 ). By the calculation of the spatial distribution of spin-waves, it can be obtained that, when wave vector k is larger than or equal to k4 (or k3)/k5/k6/k7/k8, the spin-waves  4 (or 3 )/ 5 /  6 /  7 / 8 does not propagate in surface layers (i.e. when the frequency is larger than  c , the spin-waves  4 (or 3 ), 5 ,  6 ,  7 and 8 propagate only in internal layers of the film). When wave vector k is larger than or equal to zero, the spin-waves 9 , 10 , 11 and 12 also do not propagate in surface layers (i.e. when the frequency is larger than  c , the spin-waves 9 , 10 , 11 and 12 also propagate only in internal layers). When wave vector k is larger than or equal to the critical wave vector kc, the spin-wave 1 (or 2 ) propagates only in surface layers (i.e. when the frequency is larger than  c , the spin-wave 1 (or 2 ) propagate only in surface). Therefore, in the range of spin-wave frequency 1 ( there is not the spin-wave 1 (or

2 )), the spin-waves can propagate only in internal layers. However, in the range of spin-wave frequency 2 , the lowest mode spin-waves 1 and 2 propagate only in surfaces layers, other spin-waves propagate in internal layers. Similarly, there are also the ranges of spin-wave frequency 1 and 2 in the Brillouin zone K   , which is slightly wider than that in the Brillouin zone   M , respectively.

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Fig. 11. (Color on-line) The spin-wave spectra in a symmetric magnetic film with twelve atomic layers in the Brillouin zone   M  K   . Several special spin-wave frequency range and the distribution of spin-waves are illustrated. Here the kc is critical wave vector and the  c is the frequency of spin-wave spectra

11 (or 12 ) with critical wave vector kc. In

the calculation, J1 = J12 = 2, D1 = D12 = 0.05, D=0, reduced temperatures τ = T/Tc = 0.5 and B (= g B B0 ) = 0.15.

The spatial distribution of spin-waves in different spin-wave frequency range of a symmetric magnetic film with J1 = J12 = 2 is given in Fig.11. Form Fig.11, in the Brillouin zone   M , there is a special frequency range 1 , which is from the spin-wave frequency

c to the highest frequency of the spin-wave 11 (or 12 ). When wave vector k is larger than or equal to the critical wave vector kc, the spin-wave

11 (or 12 ) propagates only in

surface layers. Namely, when the frequency is larger than  c , the spin-waves

11 (or 12 )

propagate only in surface layers and there are no other modes spin-waves. Therefore, in the range of spin-wave frequency 1 , the spin-waves propagate only in surface layers. In the Brillouin zone K   , in addition to the range of spin-wave frequency 1 , there is the range 21

Journal Pre-proof of spin-wave frequency 2 , which is from the spin-wave frequency  c to the highest frequency of the spin wave 9 (or 10 ). By the calculation of the spatial distribution of spin-waves, it can be obtained that in the range of spin-wave frequency 2 , the highest mode spin-waves

11 (or 12 ) propagate only in surfaces layers, the spin-waves 9 (or

10 ) propagate in internal layers. The spatial distribution of spin-waves in the range of spin-wave frequency 1 is the same as that in the Brillouin zone   M . From above discussion, we find some interesting spatial distributions of spin-wave in a symmetric magnetic film. For a film with low- frequency surface spin-waves, in a special frequency range, no spin waves propagate in surfaces, the excited spin wave propagate only in internal layers of the film. We refer to the film as " spin-waves surface insulator ". For a film with high-frequency surface spin-waves, there is also a special spin-wave frequency range, in which the excited spin waves propagate only in surfaces. We refer to the film as " spin-waves internal insulator ". The distribution of spin-wave of the "spin-waves internal insulator" is similar to that of electron of the topological insulators[47], where the electrons travelling only on surfaces. In our study, the surface spin-waves and the special spatial distributions of spin-wave originates from the asymmetry of surface and bulk. In magnetic thin films, some interesting phenomena occur just due to the spin-wave propagation. For exchange spin waves, there 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[48]. In our work, the structure of the film is hexagonal simple. In this structure, there are no oblique-to-surface neighbors and no the vanishing of the effective coupling between adjacent layers of spins. Therefore, there is no the reversal of the mode order in the spin-wave spectrum. 4. Conclusions In conclusion, a quantum statistic Green’s-function method has been used to study the surface in-plane propagating spin-waves in a symmetric magnetic film. There are low- or high-frequency surface spin-waves in the film. We have studied the effect of temperature, 22

Journal Pre-proof external magnetic field, surface exchange coupling, surface anisotropy and thickness on the regions of existence of surface spin-waves in the surface two-dimensional Brillouin zone. The distribution of spin-waves in different spin-wave frequency range has also been investigated. The emergence of surface spin-waves occurs much more easily on the edges of the two-dimensional Brillouin zone (and especially on its apices) than at its centre. The regions of existence of low- and high-frequency surface spin-waves increase first and then decrease with increasing temperature and external magnetic field. However, the effects of surface exchange coupling or surface anisotropy on the regions of existence of low-frequency surface spin-waves are different from that on the regions of existence of high-frequency surface spin-waves. The regions of existence of low-/high-frequency surface spin-waves are considerably broadened with decreasing/increasing surface exchange couplings. The regions of existence of low-/high-frequency surface spin-waves of a film with easy-plane surface anisotropy are slightly larger/smaller than that of a film with easy-axis surface anisotropy. Additionally, the regions of existence of surface spin-waves increases with increasing thickness of the film, but it is almost independent of the number of layers beyond five monolayers. The distribution of spin-waves in different spin-wave frequency range in a film with low-frequency surface spin-waves is different from that in a film with high-frequency surface spin-waves. In the present work, a quantum statistic approach is developed to study the surface spin-wave of a magnetic film. The results reported in this work give the fundamental knowledge on the surface in-plane propagating spin-waves in a symmetric magnetic film and are beneficial for building future microwave device based upon magnetic film.

Acknowledgments The work has been supported by the Doctoral Scientific Research Foundation of Liaoning Province of China under Grant No. 20170520405.

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Journal Pre-proof Conflict of interest statement We declared that we have no conflicts of interest to this work.

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Highlights (1) A quantum approach is developed to study the surface spin-waves (SSW) of magnetic thin film. (2) The effects of temperature and external magnetic field on the regions of existence of SSW have been investigated. (3) The spatial distribution of spin-waves in different spin-waves frequency range is studied. (4) The present results direct the method to adjust the surface spin-waves of the magnetic thin film.