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Effect of antiferromagnetic interfacial coupling on spin-wave resonance frequency of multi-layer film
Accepted Manuscript Effect of antiferromagnetic interfacial coupling on spin-wave resonance frequency of multi-layer film Rong-ke Qiu, Wei Cai PII: DOI: Reference:
S0304-8853(17)30065-3 http://dx.doi.org/10.1016/j.jmmm.2017.03.038 MAGMA 62561
To appear in:
Journal of Magnetism and Magnetic Materials
Received Date: Revised Date: Accepted Date:
16 January 2017 10 March 2017 21 March 2017
Please cite this article as: R-k. Qiu, W. Cai, Effect of antiferromagnetic interfacial coupling on spin-wave resonance frequency of multi-layer film, Journal of Magnetism and Magnetic Materials (2017), doi: http://dx.doi.org/10.1016/ j.jmmm.2017.03.038
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Effect of antiferromagnetic interfacial coupling on spin-wave resonance frequency of multi-layer film Rong-ke Qiu1, and Wei Cai1 1
Shenyang University of Technology, Shenyang 110870, P.R. China
Abstract We investigate the spin-wave resonance (SWR) frequency in a bicomponent bilayer and triple-layer films with antiferromagnetic or ferromagnetic interfacial couplings, as function of interfacial coupling, surface anisotropy, interface anisotropy, thickness and external magnetic field, using the linear spin-wave approximation and Green’s function technique.
The
microwave
properties
for
multi-layer
magnetic
film
with
antiferromagnetic interfacial coupling is different from those for multi-layer magnetic film with ferromagnetic interfacial coupling. For the bilayer film with antiferromagnetic interfacial couplings, as the lower (upper) surface anisotropy increases, only the SWR frequencies of the odd (even) number modes increase. The lower (upper) surface anisotropy does not affect the SWR frequencies of the even (odd) number modes. For the multi-layer film with antiferromagnetic interfacial coupling, the SWR frequency of modes m=1, 3 and 4 decreases while that of mode m=2 increases with increasing thickness of the film within a proper parameter region. The present results could be useful in enhancing our fundamental understanding and show the method to enhance and adjust the SWR frequency of bicomponent multi-layer magnetic films with antiferromagnetic or ferromagnetic interfacial coupling.
1
Key words: Ferromagnetic multi-layer film; Spin-wave resonance frequency; Surface and interface anisotropies; Antiferromagnetic and ferromagnetic interfacial couplings; Thickness of multi-layer film Corresponding author: Rong-ke Qiu, Shenyang University of Technology, Shenyang 110870, P.R. China, Tel: 13591494026, e-mail: [email protected]
1. Introduction Ferromagnetic resonance (FMR) and spin-wave resonance (SWR) are a power tool for studying dynamic microwave properties of magnetic thin film, multi-layers and nanostructrue[1-5]. These materials are important for magnonics[6], magnetic memory units [7], microwave filters [8], and for other low power consumption spintronic devices [9–10]. The magnetic field dependence of resonant spin-wave modes has been studied in layered NiFe (30nm)/Cu (10nm)/NiFe (15nm)/Cu (10nm)/NiFe (30nm) nanowires of rectangular cross section, encompassing both the parallel and antiparallel alignments of the middle stripe with respect to the magnetization direction of the outermost ones[2]. Under zero external magnetic fields, single-layer FeCo thin films exhibit no FMR peaks, while multiple FMR peaks were obtained by growing FeCo thin films on NiFe underlayers with various thicknesses up to 50 nm[3]. In addition, a high permeability of magnetic materials at high frequencies is a requirement for electromagnetic interference suppressors, microtransformers, etc.[11-14]. However, the enhancement in the resonance frequency of bulk magnetic materials is limited in accordance with the Snoek’s limit [15]. Magnetic thin films or multi-layer materials are expected to extend Snoek’s limit and have higher resonance frequencies and wide resonance band[16-20]. The resonance
2
frequency of FeNiCo/Teflon film was enhanced to the 3.0-4.7 GHz range [16]. The static and high-frequency magnetic properties of multi-layers (Co90Nb10/Ta)n have been investigated [17]. The results show that the in-plane uniaxial magnetic anisotropy fields can be adjusted from 12 to 520 Oe only by decreasing the thickness of the Ta interlayers from 8.0 to 1.8 nm. As a consequence, the resonance frequencies of the multi-layers continuously increase from 1.4 to 6.5 GHz [17]. The SWR and FMR effect observed in multi-layer films has been investigated by different theoretical method. The classical method of the Landau-Lifshitz equation (or based on the Landau-Lifshitz equation) has mainly been used[21-26], while microscopic[27-29] and quantum theory [30,31,34] have seldom been used. Puszkarski [27-29] presented a microscopic theory of standing SWRs in exchange-coupled bilayer films, in which the analysis is based on the exact solution of the eigenvalue problem of a bilayer film achieved with the interface rescaling approach. Here, the effects of interfacial coupling, interface and surface anisotropies of a symmetric bilayer film on SWR were studied. Perchmin [31] presented a quantum theory of spin-wave resonance in cubic ferromagnets, in which Dyson’s formalism[32] rather than that of Holstein-Primakoff [33] has been used in a manner analogous to the quantum theory of ferromagnetic resonance[34]. It is valuable to develop some quantum methods to study the microwave properties of magnetic multi-layer films. According to earlier work, the microwave properties of a magnetic multi-layer thin film are significantly affected by magnetic surface anisotropies [28, 30], interface anisotropies [27-29], interfacial couplings [27-29], external magnetic field [3,28,30] and thickness of magnetic layer[3,35], but the effect of thickness of
3
magnetic layer of a bicomponent multi-layer magnetic film with antiferromagnetic interfacial coupling on the SWR frequency have seldom been studied. Therefore, in the present work, we will utilize the linear spin-wave approach and Green’s function technique to study the SWR frequency (including FMR frequency) in a bicomponent multi-layer thin film consisting of two different magnetic materials with single-ion surface and interface anisotropies, and with antiferromagnetic or ferromagnetic interfacial coupling. We found that the multi-layer magnetic film with antiferromagnetic interfacial coupling has some novel microwave properties. The SWR frequencies for multi-layer magnetic film with ferromagnetic interfacial coupling all decrease with increasing the film thickness. While for multi-layer magnetic film with antiferromagnetic interfacial coupling, the SWR frequency in mode m=2 increases with increasing the film thickness within a proper parameter region. Our present work presents a natural extension of our previous results [36], derived for single component multi-layer films with ferromagnetic interfacial exchange coupling. Our present work is beneficial to fundamental understanding of microwave properties of a bicomponent multi-layer magnetic film and shows how to obtain high and controllable SWR frequencies (including FMR frequencies) of a bicomponent multi-layer film with antiferromagnetic or ferromagnetic interfacial coupling with a wide band. The outline of this paper is as follows: In Section 2, we describe the model, Hamiltonian of the system and calculation procedure. The SWR frequencies in a bilayer and triple-layer films are discussed in Section 3. In Section 4 conclusions are presented.
4
2 Model and calculation procedure We consider a Heisenberg model with a single-ion anisotropy on a simple cubic lattice for multi-layer ferromagnetic films with antiferromagnetic or ferromagnetic interfacial coupling built up by N monolayers parallel to two infinitely extended surfaces. A schematic diagram of the bilayer (or triple-layer) film is given in Fig. 1(a) and (b) (or Fig. 1(c) and (d)). For the bilayer film, the first layer A consists of N1 monolayers and the second layer B of N2 monolayers. For the triple-layer film, the first, second and third layer (A, B and A) consist of N1, N2 and N1 monolayers, respectively. The Hamiltonian is:
H = − ∑ J ll ' S lj S l ' j ' − ∑ Dl ( S ljz ) 2 − ∑ gµ B B S ljz ljl ' j '
lj
(2.1)
lj
In Eq. (2.1), the magnetic ions are specified by the set of indices lj, where l is an integer labelling the monolayers and j is a two-dimensional lattice vector in the yz-plane. The summation in the first term is over the nearest-neighbour (NN) spins only. The second (anisotropy) term of Eq. (2.1) comprises the surface, interface and bulk anisotropy effects. The assumptions for the NN exchange terms and the definition of the surface, interface and bulk anisotropy terms involved in the Hamiltonian are shown in Table. The bulk exchange coupling (J) is ferromagnetic and the interfacial coupling (JAB) is antiferromagnetic or ferromagnetic. The direction of the spins in the initial state in film A is along the positive z-axis, while those in film B are along the negative z-axis (for the film with antiferromagnetic intrefacial coupling) or the positive z-axis (for the film with ferromagnetic intrefacial coupling). The external magnetic field B is also along the positive z-axis. In the following, we shall introduce only the calculation procedure for the triple-layer film, because of the similarity of the calculation procedure for the bilayer and 5
triple-layer films. By use of the Holstein–Primakoff transform [33] and the linear spinwave approximation [37], introducing the spin-wave operators blk ( blk+ // ) (l = 1-N), we //
can rewrite Eq. (2.1) as follows:
H = −n (2 N1 − 4) DS A2 + ( N 2 − 2) DS B2 + ∑ Dl S A2 + 2 Dint ( S A2 + S B2 ) l =1, N 2 2 − 4S A nJ (3 N1 − 1) − 2 S B nJ (3N 2 − 1) − 4 S A S B nJ AB − gµ B Bn( 2 N1S A + N 2 S B ) N −1
∑
+
l = 2 ( l ≠ N1 ~ N1 + N 2 +1)
+
(12 JS A + 2 S A D + gµ B B) ∑ bl+, k// bl ,k // k //
N1 + N 2 −1
+ + ∑ (12 JS B + 2 S B D + gµ B B ) ∑ bl ,k // bl ,k // + ∑ (10 JS A + 2S A Dl + gµ B B) ∑ bl ,k // bl ,k //
l = N1 + 2
+
l =1, N
k //
∑ l = N1 , N1 + N 2 +1
k //
(10 JS A + 2 J AB S B + 2 S A Dint + gµ B B )∑ bl+,k // bl ,k // k //
+ ∑ (10 JS B + 2 J AB S A + 2 S B Dint + gµ B B ) ∑ bl+, k // bl ,k // l = N1 +1, N1 + N 2
−
k //
N −1
+ + ∑ 2S A J l ,l +1 ∑ (bl ,k // bl +1, k // + bl , k // bl +1,k // ) −
l =1( l ≠ N1 ~ N1 + N 2 )
−
∑ l = N1 , N1 + N 2
− 4S A J
k //
N1 + N 2 −1
+ + ∑ 2S B J l ,l +1 ∑ (bl ,k // bl +1,k // + bl ,k // bl +1,k // )
l = N1 +1
k //
2 S A S B J AB ∑ (bl ,k // bl++1, k // + bl+, k// bl +1,k // ) k //
N
∑
N1 + N 2
+ + ∑ γ k // (2bl , k// bl ,k // − 1) − 4 S B J ∑ ∑ γ k// (2bl , k // bl ,k // − 1)
l =1( l ≠ N1 +1~ N1 + N 2 ) k //
l = N1 +1 k //
(2.2a)
6
H = − n (2 N1 − 4) DS A2 + ( N 2 − 2) DS B2 + ∑ Dl S A2 + 2 Dint ( S A2 + S B2 ) l =1, N − 4S A2 nJ (3 N1 − 1) − 2S B2 nJ (3 N 2 − 1) + 4S A S B nJ AB − gµ B B n(2 N1S A − N 2 S B ) N −1
∑
+
l = 2 ( l ≠ N1 ~ N1 + N 2 +1)
+
(12 JS A + 2S A D + gµ B B)∑ bl+,k // bl ,k // k //
N1 + N 2 −1
+ + ∑ (12 JS B + 2 S B D - gµ B B )∑ bl ,k // bl ,k // + ∑ (10 JS A + 2S A Dl + gµ B B )∑ bl ,k // bl ,k //
l = N1 + 2
+
l =1, N
k //
∑ l = N1 , N1 + N 2 +1
k //
(10 JS A - 2 J AB S B + 2S A Dint + gµ B B )∑ bl+,k // bl ,k // k //
+ ∑ (10 JS B - 2 J AB S A + 2S B Dint - gµ B B )∑ bl+,k // bl , k // l = N1 +1, N1 + N 2
−
k //
N −1
+ + ∑ 2S A J ∑ (bl , k// bl +1,k // + bl ,k // bl +1,k // ) −
l =1( l ≠ N1 ~ N1 + N 2 )
−
∑ l = N1 , N1 + N 2
− 4S A J
k //
N1 + N 2 −1
+ + ∑ 2S B J ∑ (bl ,k // bl +1,k // + bl , k// bl +1,k // )
l = N1 +1
k //
2 S A S B J AB ∑ (bl , k// bl +1,k // + bl+,k // bl++1,k // ) k //
N
∑
N1 + N 2
+ + ∑ γ k // ( 2bl ,k // bl ,k // − 1) − 4S B J ∑ ∑ γ k// (2bl ,k // bl , k// − 1)
l =1( l ≠ N1 +1~ N1 + N 2 ) k //
l = N1 +1 k //
(2.2b) The equations (2.2a) and (2.2b) correspond to the case of the triple-layer film with ferromagnetic and antiferromagnetic interfacial coupling, respectively. Note that,
γk = //
1 eik // ⋅δ // ∑ 4 δ //
(2.3)
where δ // represents that only the exchanges between the nearest neighbors in yz – planes are taken into account. There are n sites on each layer-lattice, and, in total, nN sites in the system. We define the N-order matrix retarded Green’s functions: 7
G (k // , f ) = [Gi , j ] N × N
(2.4)
here Gi , j =<< bik // ; b +jk // >> f （i = 1 ~ N; j = 1 ~ N)
(2.5a)
Gi , j =<< bik // ; b +jk // >> f (i = 1 ~ N1; j = 1 ~ N1 or i = N1+N2+1 ~ N; j = N1+N2+1 ~ N) Gi , j =<< bik+// ; b jk // >> f （i = N1+1 ~ N1+N2; j = N1+1 ~ N1+N2) Gi , j =<< bik// ; b jk // >> f (i = 1 ~ N1; j = N1+1 ~ N1+N2 or i = N1+N2+1 ~ N; j = N1+1 ~
N1+N2) Gi , j =<< bik+// ; b +jk // >> f (i = N1+1 ~ N1+N2; j = 1 ~ N1 or i = N1+1 ~ N1+N2; j = N1+N2+1 ~
N)
(2.5b)
The equations (2.5a) and (2.5b) are the matrix elements of the N-order matrix retarded Green’s functions of the triple-layer film with ferromagnetic and antiferromagnetic interfacial coupling, respectively. By using the equation of the Green’s functions, we obtain the solution of the Green’s function as follows:
D* ( f ) G (k // , f ) = det( D( f ))
(2.6)
Where the matrix D* is adjoint matrix of matrix D. And
[ ]
D( f ) = Wi, j
N×N
[ ]
+ Hi, j
(2.7)
N ×N
8
Here, in the N-order matrix W, only the diagonal matrix elements Wii (i = 1 ~ N) are nonzero and equal to f . f represents the energy spectrum of the system. The matrix H is a three-diagonal matrix with matrix elements H ij (i, j =1~N): H i ,i =（ - 8 JS A (1 − γ k // ) + 4 JS A + 2 DS A + gµ B B ）
(i = 2~ (N-1), i ≠ N1~N1+N2+1)
H i ,i = ∓（8 JS B (1 − γ k // ) + 4 JS B + 2 DS B ± gµ B B ）
(i = N1+2~ N1+N2-1)
H1,1 =（ - 8 JS A (1 − γ k// ) + 2 JS A + 2 D1 S A + gµ B B ） H N , N =（ - 8 JS A (1 − γ k // ) + 2 JS A + 2 DN S A + gµ B B ） H N1 , N1 = N N1 + N 2 +1, N1 + N 2 +1 =（ - 8 JS A (1 − γ k // ) + 2 JS A ± 2 J AB S B + 2 Dint S A + gµ B B） H N1 + N 2 , N1 + N 2 = H N1 +1, N1 +1 = ∓（8 JS B (1 − γ k // ) + 2 JS B ± 2 J AB S A + 2 Dint S B ± gµ B B ） H i ,i +1 = 2 JS A
(i = 1~ (N1-1); i = (N1+N2+1) ~ (N-1))
H i,i +1 = ±2 JS B
(i = (N1+1)~ (N1+N2-1))
H i ,i −1 = 2 JS A
(i= 2 ~ N1; i =(N1+N2+1)~N)
H i,i −1 = ±2 JS B
(i= (N1+2) ~(N1+N2))
H N1 , N1 +1 = H N1 + N 2 +1, N1 + N 2 = 2 J AB S A S B H N1 +1, N1 = H N1 + N 2 , N1 + N 2 +1 = ±2 J AB S A S B
(2.8)
The upper/lower sign above corresponds to the case of the triple-layer film with ferromagnetic/antiferromagnetic interfacial coupling. Setting the determinant to zero, i.e. det (D( f ))=0, we obtain N numerical solutions for the spin-wave spectra of the triple-layer film with antiferromagnetic or ferromagnetic interfacial coupling.
9
We now discuss the effect of the interfacial coupling between the sublayer films, the magnetic surface anisotropy, the interface anisotropy, the thickness and the external magnetic field on the SWR frequencies (including the FMR frequencies) in a bicomponent bilayer and triple-layer films with antiferromagnetic or ferromagnetic interfacial coupling, and compare the SWR frequencies of films with antiferromagnetic and ferromagnetic interfacial coupling. Because k// = 0 , the dipole-dipole interaction vanishes [38,39] and is neglected in our model. In the calculations, the exchange coupling between nearest neighboring spins in the sublayer films A and B is set to unity (J = 1.0). For the bicomponent bilayer or triple-layer films, both the sublayer film A and B consist of six monolayers (i.e. N1=N2=6), and the spin quantum numbers in the sublayer film A and B are SA=1.0 and SB=1.5, respectively. 3. Results and discussion
3.1. SWR frequency in a bicomponent bilayer film For a bilayer film AB (N1=N2=6), there are twelve SWR frequencies. The spin-wave modes with mode numbers m= 1, 2…, N, start from the energetically lowest mode. Next we will study only the lowest four SWR frequencies (i.e. modes m=1, 2, 3 and 4), because the resonance of low energy has larger excitation probability. Figure 2 presents the dependence of the SWR frequencies in a bicomponent bilayer film with D1 = 0.08,
D12 = 0.08, D = 0.02, Dint = 0.02 and B = 0 on the ferromagnetic or antiferromagnetic interfacial couplings JAB. From Fig.2, the SWR frequencies of mode m=1 are almost unchanged by ferromagnetic interfacial coupling. The other three SWR frequencies increase with increasing ferromagnetic interfacial coupling. Moreover, the change of the SWR frequencies of mode m=4 is the biggest. While four SWR frequencies all increase
10
with enhancing antiferromagnetic interfacial coupling, and the effect of antiferromagnetic interfacial coupling on the SWR frequencies of mode m=3 is the biggest. When the interfacial coupling is absent, the SWR frequencies of the bilayer films with ferromagnetic and antiferromagnetic interfacial couplings are same, because these systems are same. When the interfacial coupling is stronger, the SWR frequencies of modes m=1 and 3 (or m=2 and 4) of the bilayer films with antiferromagnetic interfacial couplings are larger (or smaller) than those of the bilayer films with ferromagnetic interfacial couplings. Our results on the effect of the ferromagnetic interfacial coupling of bicomponent bilayer film on the resonance frequencies agree with the calculation result of single component asymmetric bilayer film in Ref. 36, and also agree with the experimental and calculation result in Ref. 5. Figure 3 shows the dependence of the SWR frequencies on the lower surface anisotropy D1 and upper surface anisotropy D12 of a bilayer films with ferromagnetic or antiferromagnetic interfacial couplings. For the bilayer films with ferromagnetic interfacial coupling (from Fig.3(a)), when the surface anisotropy D1 or D12 increases, the SWR frequencies all increase, and the effects of the lower surface anisotropy D1 and the upper surface anisotropy D12 on the SWR frequencies are almost the same. For the bilayer film with antiferromagnetic interfacial coupling (from Fig.3(b)), when the surface anisotropy D1 (or D12) increases, only the SWR frequencies of modes m=1 and 3 (or modes m=2 and 4) increase, the SWR frequencies of modes m=2 and 4 (or modes m=1 and 3) are unchanged by the surface anisotropy D1 (or D12). Namely, for the bilayer films with antiferromagnetic interfacial coupling, the surface anisotropy D1 (or D12) only affects the SWR frequencies of odd (or even) number modes. From Figs.3(c) and (d), the
11
effects of the surface anisotropy D1 (or D12) of the bilayer films with antiferromagnetic interfacial couplings on the SWR frequencies of odd (or even) number modes are larger than those of the bilayer films with ferromagnetic interfacial couplings. The effects of the interface anisotropy on the SWR frequency of a bilayer films with ferromagnetic or antiferromagnetic interfacial couplings are shown in Figs 4(a) and 4(b), respectively. For the bilayer film with ferromagnetic or antiferromagnetic interfacial coupling, when the interface anisotropy Dint increases, all SWR frequencies increase. For the film with ferromagnetic interfacial coupling (see Fig. 4(a)), the effect of the interface anisotropy Dint on mode m=3 is larger. When JAB=0.2, the effect of the interface anisotropy Dint on mode m=4 is also larger. This agrees with the result of single component symmetric bilayer film in Ref. 36. For the film with antiferromagnetic interfacial coupling (see Fig. 4(b)), the effect of Dint on modes m= 4 is larger. From Fig.4(a) and (b), the different ferromagnetic interfacial coupling can lead to different effects of the interface anisotropy on the SWR frequency, while the antiferromagnetic interfacial coupling almost does not affect the effects of the interface anisotropy on the SWR frequency. 3.2. SWR frequency in a bicomponent triple-layer film How will the SWR frequency be affected if we add a sublayer film A to the bilayer film AB (forming a symmetry triple-layer film ABA)? In next, we will study the SWR frequencies of a triple-layer film. Figure 5 shows the effects of the interfacial couplings on the SWR frequencies of triple-layer films with ferromagnetic or antiferromagnetic interfacial couplings. In the process of changing the ferromagnetic or antiferromagnetic interfacial couplings, the SWR frequencies increase with increasing interfacial couplings,
12
only the lowest SWR frequency is not affected by ferromagnetic interfacial coupling. The effect of ferromagnetic (or antiferromagnetic) interfacial coupling on the SWR frequencies in mode m=3 (or m=4) is larger. When the interfacial coupling is stronger, the SWR frequencies of modes m=1, 2 and 4 (or m=3) of the triple-layer films with antiferromagnetic interfacial couplings are larger (or smaller) than those of the triplelayer films with ferromagnetic interfacial couplings. Our results on the effect of the ferromagnetic interfacial coupling of a bicomponent triple-layer film on the resonance frequencies agree with the calculation result of single component symmetric triple-layer film in Ref. 36 The dependence of the SWR frequencies on the surface anisotropies D1 and D18 of a triple-layer films with ferromagnetic or antiferromagnetic interfacial couplings is shown in Fig. 6. For triple-layer films with ferromagnetic interfacial couplings (see Fig.6(a)), when the surface anisotropy D1 or D18 increases, all SWR frequencies increase. For triple-layer films with antiferromagnetic interfacial couplings (see Fig.6(b)), when the surface anisotropy D1 or D18 increases, only the SWR frequencies in mode m=1 and 4 increase, the SWR frequency in mode m=2 and 3 is almost independent of the surface anisotropy D1 or D18. For both triple-layer films with ferromagnetic and antiferromagnetic interfacial couplings, the effects of the surface anisotropies D1 and D18 on the SWR frequencies are the same, because of the symmetry of the triple-layer films with ferromagnetic or antiferromagnetic interfacial couplings. From Fig.6(c), the effects of the surface anisotropies D1 on the SWR frequencies with modes m=1 and 4 (or modes m=2 and 3) of the triple-layer films with antiferromagnetic interfacial couplings are
13
slightly larger (or smaller) than those of the triple-layer films with ferromagnetic interfacial couplings. Figures 7(a) and 7(b) present the effects of the interface anisotropies on the SWR frequency in triple-layer films with ferromagnetic and antiferromagnetic interfacial couplings, respectively. For these films, when the interface anisotropy Dint increases, all SWR frequencies increase. For the triple-layer films with ferromagnetic interfacial couplings, as seen in Fig. 7(a), the effect of the interface anisotropy Dint on mode m=1 and 4 is larger. For mode m=3 or m=4, the different interfacial coupling JAB can lead to different effect of the interface anisotropy Dint on the SWR frequencies. For the triplelayer films with antiferromagnetic interfacial couplings ( see Fig. 7(b) ), the effect of Dint on modes m=3 and 4 (the mode number is defined at Dint=0.1) is larger. The antiferromagnetic interfacial coupling almost does not affect the effects of the interface anisotropy on the SWR frequency. Figs 7(a) and 7(b) show that, for the lowest mode m=1, the effect of Dint of the film with ferromagnetic interfacial coupling on the SWR frequency is larger than those of the film with antiferromagnetic interfacial coupling. Figure 8 shows the comparison of the SWR frequencies in bilayer and triple-layer films, these curves are chosen from Figs.2 and 5. For the films with ferromagnetic interfacial coupling (from Fig.8(a)), we can see that the SWR frequencies of four modes decrease with increasing thickness of the film. This agrees with the experimental and theoretical results reported in Refs. 35 and 36, respectively. For the films with antiferromagnetic interfacial coupling (from Fig.8(b)), the SWR frequencies of modes m=1, 3 and 4 decrease with increasing thickness of the film. While for the mode m=2, when JAB is stronger, the SWR frequency increases with increasing thickness of the film
14
(see the inset of Fig.8(b)). It means that, the SWR frequency of mode m=2 can increase with increasing thickness of the film with antiferromagnetic interfacial coupling within a proper parameter region. It is also calculated that the SWR frequency increases with increasing external magnetic field for bilayer and triple-layer films with ferromagnetic interfacial coupling. This agrees with the experimental results reported in Refs. 3 and 20. As the external magnetic field increases, the SWR frequencies of two (or three) modes increase while that of the other two (or one) modes decrease for bilayer (or triple-layer) film with antiferromagnetic interfacial coupling. This is because of the antiparallel alignments of spins in the initial state in film with antiferromagnetic interfacial coupling. 4. Conclusions
In conclusion, we have discussed the effect of the interfacial couplings, the surface anisotropies, the interface anisotropies, the thickness and the external magnetic field on the lowest four SWR frequencies (i.e. modes m=1, 2, 3, and 4) in a bicomponent bilayer and triple-layer films with ferromagnetic or antiferromagnetic interfacial couplings by means of the linear spin-wave approximation and Green’s function technique. The comparison of the SWR frequencies in multi-layer magnetic films with ferromagnetic and antiferromagnetic interfacial coupling has been made. The main results are concluded as following. The SWR frequency of a triple-layer film is different from those of a bilayer film, and the SWR frequency of a film with antiferromagnetic interfacial couplings is also different from those of a film with ferromagnetic interfacial couplings. For a bicomponent bilayer or triple-layer film with ferromagnetic interfacial couplings, the SWR frequencies all increase with increasing the surface anisotropy, the interface
15
anisotropy and the external magnetic field, respectively, and with decreasing the film thickness. As the ferromagnetic interfacial coupling increases, only the SWR frequencies of modes m=2, 3 and 4 increase. The ferromagnetic interfacial coupling almost does not affect the lowest mode. For a bicomponent bilayer or triple-layer film with antiferromagnetic interfacial couplings, the SWR frequencies all increase with increasing the antiferromagnetic interfacial couplings and the interface anisotropy, respectively. As the surface anisotropy
D1 (or D12) of a bilayer film increases, only the SWR frequencies of the odd (or even) number modes increase. The surface anisotropy D1 (or D12) does not affect the SWR frequencies of the even (or odd) number modes. When the surface anisotropy D1 or D18 of a triple-layer film increases, only the SWR frequencies in modes m=1 and 4 increase, the SWR frequencies in modes m=2 and 3 are almost unchanged. When the external magnetic field increases, two (or three) SWR frequencies increase while the other two (or one) decrease for the bilayer (or triple-layer) film. For the films with antiferromagnetic interfacial coupling, the SWR frequencies of modes m=1, 3 and 4 decrease while the SWR frequency of mode m=2 can increase with increasing thickness of the film within a proper parameter region. Theoretically, this work provides a quantum method to study the SWR frequency of bicomponent multi-layer magnetic films with ferromagnetic or antiferromagnetic interfacial couplings. The present results could be used to understand the microwave properties of multi-layer magnetic film and direct the method to enhance and adjust the SWR frequency of the bicomponent multi-layer thin film with ferromagnetic or antiferromagnetic interfacial couplings.
16
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[1] X. Zhou and A. O. Adeyeye, Phys. Rev. B 94 (2016) 054410. [2] G. Gubbiotti, H.T. Nguyen, R. Hiramatsu, S. Tacchi, M.G. Cottam, T. Ono, J. Magn. Magn. Mater. 384 (2015) 45. [3] X. Zhong, W. T. Soh, N. N. Phuoc, Y. Liu, and C. K. Ong, J. Appl. Phys. 117 (2015) 013906. [4] K. Chalapat, J.V.I. Timonen, M. Huuppola, L. Koponen, C. Johans, R. H. A. Ras, O. Ikkala, M. A. Oksanen, E. Seppälä and G. S. Paraoanu, Nanotech. 25 (2014) 485707. [5] J. Lindner and K. Baberschke, J. Phys.: Condens. Matter 15 (2003) R193. [6] V. V. Kruglyak, S. O. Demokritov, and D. Grundler, J. Phys.D: Appl. Phys. 43 (2010) 264001. [7] F. G. Aliev, J. F. Sierra, A. A. Awad, G. N. Kakazei, D. S. Han, S. K. Kim, V. Metlushko, B. Ilic, and K. Y. Guslienko, Phys. Rev. B 79 (2009) 174433. [8] S. Choi, K. S. Lee, K. Y. Guslienko, and S. K. Kim, Phys. Rev. Lett. 98 (2007) 087205. [9] A. A. Nikitin, A. B. Ustinov, A. A. Semenov, A. V. Chumak, A. A. Serga, V. I. Vasyuchka, E. Lähderanta, B. A. Kalinikos, and B. Hillebrands, Appl. Phys. Lett. 106 (2015) 102405. [10] A. Haldar, D. Kumar, and A. O. Adeyeye, Nat. Nano. 11 (2016) 437. [11] Y. W. Zhao, X. K. Zhang and J. Q. Xiao, Adv. Mater. 17 (2005) 915 . [12] D. S. Gardner, G. Schrom, P. Hazucha, F. Paillet, T. Karnik, S. Borkar, R. Hallstein, T. Dambrauskas, C. Hill, C. Linde, W. Worwag, R. Baresel and S. Muthukumar, J. Appl. Phys. 103 (2008) 07E927.
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[13] N. N. Phuoc, F. Xu and C. K. Ong, Appl. Phys. Lett. 94 (2009) 092505. [14] S. A. Ramakrishna, Rep. Prog. Phys. 68 (2005) 449. [15] J. L. Snoek, Nature (London) 160 (1947) 90. [16] H. Greve, C. Pochstein, H. Takele, V. Zaporojtchenko, F. Faupel, A. Gerber, M. Frommberger and E. Quandt, Appl. Phys. Lett. 89 (2006) 242501. [17] G. Z. Chai, Y. C. Yang, J. Y. Zhu, M. Lin, W. B. Sui, D. W. Guo, X. L. Li and D. S. Xue, Appl. Phys. Lett. 96 (2010) 012505. [18] X. H. Wang, G. Z. Chai and D. S. Xue, J. Alloys Comp. 584 (2014) 171. [19] C. Wu, A. N. Khalfan, C. Pettiford, N. X. Sun, S. Greenbaum and Y. H. Ren, J. Appl. Phys. 103 (2008) 07B525. [20] M. A. de Sousa, F. Pelegrini, W. Alayo, J. Quispe-Marcatoma, E. Baggio-Saitovitch, Physica B 450 (2014) 167. [21] C. H. Wilts and O. G. Ramer, J. Appl. Phys. 47 (1976) 1151. [22] B. Heinrich, S. T. Purcell, J. R. Dutcher, J. F. Cochran and A. S. Arrot, Phys. Rev. B 38 (1988) 12879.
[23] B. Hillebrands, Phys. Rev. B 37 (1988) 9885. [24] M. G. Cottam (ed) 1994 Linear and Nonlinear Spin Waves in Magnetic Films and Superlattices (Singapore: World Scientific). [25] A. M. Zyuzin, M. A. Bakulin, D. A. Zyuzin, V. V. Radaikin, and S. N. Sabaev, Tech. Phys. 56 (2011) 809. [26] K. Seemann, H. Leiste, K. Krüger, J. Magn. Magn. Mater. 345 (2013) 36. [27] H. Puszkarski, Phys. Rev. B 46 (1992) 8926. [28] H. Puszkarski, Phys. Rev. B 49 (1994) 6718.
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[29] H. Puszkarski, Surf. Sci. Rep. 20 (1994) 45. [30] X. J. Hou and G. H. Yun, Chin. J. Mater. Res. 21 (2007) 389. [31] A. R. Ferchmin, Phys. Lett. 1 (1962) 281. [32] F. J. Dyson, Phys. Rev. 102 (1956) 1217; 102 (1956) 1230. [33] T. Holstein and H. Primakoff, Phys. Rev. 58 (1940) 1098. [34] D. Polder, Phil. Mag. 40 (1949) 99. [35] Y.V. Khivintsev, L. Reisman, J. Lovejoy, R. Adam, C. M. Schneider, R. E. Camley and Z. J. Celinski, J. Appl. Phys. 108 (2010) 023907. [36] R. K. Qiu, F. J. Ma and Z. D. Zhang, J. Magn. Magn. Mater. 394 (2015) 494. [37] P. W. Anderson, Phys. Rev. 86 (1952) 694. [38] P. Grünberg, J. Barnas, F. Saurenbach, J. A. Fuss, A. Wolf and M. Vohl, J. Magn. Magn. Mater. 93 (1991) 58. [39] M. Vohl, J. A. Wolf, P. Grünberg, K. Spörl, D. Weller and B. Zeper, J. Magn. Magn. Mater. 93 (1991) 403.
19
Figure 1. Geometries: (a) bilayer system and choice of X, Y and Z axes; (b) schematic
representation of a bilayer film, showing sublayer A with N1 monolayers and sublayer B with N2 monolayers; (c) triple-layer system and choice of X, Y and Z axes; (d) schematic representation of a triple-layer film, showing sublayers A, B and A with N1, N2 and N1 monolayers, respectively.
20
Figure 2. (Color on-line) SWR frequency as a function of the ferromagnetic (solid black
lines) and antiferromagnetic (dashed red lines) interfacial coupling JAB for a bilayer film with D = Dint = 0.02, D1 = D12= 0.08, B = 0, SA = 1.0 and SB=1.5. These curves correspond to the spin-wave modes m= 1, 2, 3 and 4, starting from the energetically lowest mode.
21
Figure 3. (Color on-line) SWR frequency as a function of the lower surface anisotropy
D1 or upper surface anisotropy D12 for a bilayer film with (a) JAB = 0.5 (bilayer film with ferromagnetic interfacial coupling), and (b) JAB = -0.5 (bilayer film with antiferromagnetic interfacial coupling). The comparisons of Fig. (a) and Fig.(b) are shown in Fig.(c) and (d). The parameters are: D = D12 (or D1) =Dint=0.08, and B = 0.
22
Figure 4. (Color on-line) SWR frequency as a function of the interface anisotropy with
D1 = D12 = 0.08, D = 0.02, and B = 0. The solid black ,the dotted green, and the dashed red lines correspond to |JAB| = 0.2, 0.5 and 0.8, respectively. (a) for a bilayer film with ferromagnetic interfacial coupling and (b) for a bilayer film with antiferromagnetic interfacial coupling.
23
Figure 5. (Color on-line) SWR frequency as a function of the interfacial couplings JAB
for a triple-layer film with ferromagnetic (the solid black lines) or antiferromagnetic (the dashed red lines) interfacial couplings. The parameters are: D = Dint = 0.02, D1 = D18= 0.08, and B= 0.
24
Figure 6. (Color on-line) SWR frequency as a function of the surface anisotropies D1 and
D18 for a triple-layer film with (a) JAB = 0.5, and (b) JAB = -0.5. The comparison of Fig. (a) and Fig.(b) is shown in Fig.(c). The parameters are: D = D18 (or D1) = Dint = 0.08, and
B = 0.
25
Fig. 7. (Color on-line) Dependence of interface anisotropies Dint on the SWR frequency
for a triple-layer film with D1 =D18 = 0.08, D = 0.02 and B = 0. The solid black ,the dashed green, and the dotted red lines correspond to |JAB| = 0.2, 0.5 and 0.8, respectively. (a) for a triple-layer film with ferromagnetic interfacial coupling and (b) for a triple-layer film with antiferromagnetic interfacial coupling.
26
Fig. 8. (Color on-line) Dependence of interlayer coupling on the SWR frequency for a
bilayer (solid black lines) and triple-layer (dashed red lines) film with (a) ferromagnetic and (b) antiferromagnetic interfacial couplings. The parameters are: D1=D18 (or
D12)=0.08, D = Dint = 0.02 and B = 0.
27
Table.
The assumptions for the NN exchange terms and the definition of the surface, interface and bulk anisotropy terms involved in the Hamiltonian of a bilayer and triple-layer films.
28
Highlights
(1) A quantum approach is developed to study the SWR frequency of a bicomponent multi-layer films with antiferromagnetic or ferromagnetic interfacial couplings. (2) The comparison of the SWR frequencies in multi-layer magnetic films with ferromagnetic and antiferromagnetic interfacial coupling has been made. (3) The present results could be useful in enhancing our fundamental understanding and show the method to enhance and adjust the SWR frequency of bicomponent multi-layer magnetic films with antiferromagnetic or ferromagnetic interfacial coupling.
29
S0304-8853(17)30065-3 http://dx.doi.org/10.1016/j.jmmm.2017.03.038 MAGMA 62561
To appear in:
Journal of Magnetism and Magnetic Materials
Received Date: Revised Date: Accepted Date:
16 January 2017 10 March 2017 21 March 2017
Please cite this article as: R-k. Qiu, W. Cai, Effect of antiferromagnetic interfacial coupling on spin-wave resonance frequency of multi-layer film, Journal of Magnetism and Magnetic Materials (2017), doi: http://dx.doi.org/10.1016/ j.jmmm.2017.03.038
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Effect of antiferromagnetic interfacial coupling on spin-wave resonance frequency of multi-layer film Rong-ke Qiu1, and Wei Cai1 1
Shenyang University of Technology, Shenyang 110870, P.R. China
Abstract We investigate the spin-wave resonance (SWR) frequency in a bicomponent bilayer and triple-layer films with antiferromagnetic or ferromagnetic interfacial couplings, as function of interfacial coupling, surface anisotropy, interface anisotropy, thickness and external magnetic field, using the linear spin-wave approximation and Green’s function technique.
The
microwave
properties
for
multi-layer
magnetic
film
with
antiferromagnetic interfacial coupling is different from those for multi-layer magnetic film with ferromagnetic interfacial coupling. For the bilayer film with antiferromagnetic interfacial couplings, as the lower (upper) surface anisotropy increases, only the SWR frequencies of the odd (even) number modes increase. The lower (upper) surface anisotropy does not affect the SWR frequencies of the even (odd) number modes. For the multi-layer film with antiferromagnetic interfacial coupling, the SWR frequency of modes m=1, 3 and 4 decreases while that of mode m=2 increases with increasing thickness of the film within a proper parameter region. The present results could be useful in enhancing our fundamental understanding and show the method to enhance and adjust the SWR frequency of bicomponent multi-layer magnetic films with antiferromagnetic or ferromagnetic interfacial coupling.
1
Key words: Ferromagnetic multi-layer film; Spin-wave resonance frequency; Surface and interface anisotropies; Antiferromagnetic and ferromagnetic interfacial couplings; Thickness of multi-layer film Corresponding author: Rong-ke Qiu, Shenyang University of Technology, Shenyang 110870, P.R. China, Tel: 13591494026, e-mail: [email protected]
1. Introduction Ferromagnetic resonance (FMR) and spin-wave resonance (SWR) are a power tool for studying dynamic microwave properties of magnetic thin film, multi-layers and nanostructrue[1-5]. These materials are important for magnonics[6], magnetic memory units [7], microwave filters [8], and for other low power consumption spintronic devices [9–10]. The magnetic field dependence of resonant spin-wave modes has been studied in layered NiFe (30nm)/Cu (10nm)/NiFe (15nm)/Cu (10nm)/NiFe (30nm) nanowires of rectangular cross section, encompassing both the parallel and antiparallel alignments of the middle stripe with respect to the magnetization direction of the outermost ones[2]. Under zero external magnetic fields, single-layer FeCo thin films exhibit no FMR peaks, while multiple FMR peaks were obtained by growing FeCo thin films on NiFe underlayers with various thicknesses up to 50 nm[3]. In addition, a high permeability of magnetic materials at high frequencies is a requirement for electromagnetic interference suppressors, microtransformers, etc.[11-14]. However, the enhancement in the resonance frequency of bulk magnetic materials is limited in accordance with the Snoek’s limit [15]. Magnetic thin films or multi-layer materials are expected to extend Snoek’s limit and have higher resonance frequencies and wide resonance band[16-20]. The resonance
2
frequency of FeNiCo/Teflon film was enhanced to the 3.0-4.7 GHz range [16]. The static and high-frequency magnetic properties of multi-layers (Co90Nb10/Ta)n have been investigated [17]. The results show that the in-plane uniaxial magnetic anisotropy fields can be adjusted from 12 to 520 Oe only by decreasing the thickness of the Ta interlayers from 8.0 to 1.8 nm. As a consequence, the resonance frequencies of the multi-layers continuously increase from 1.4 to 6.5 GHz [17]. The SWR and FMR effect observed in multi-layer films has been investigated by different theoretical method. The classical method of the Landau-Lifshitz equation (or based on the Landau-Lifshitz equation) has mainly been used[21-26], while microscopic[27-29] and quantum theory [30,31,34] have seldom been used. Puszkarski [27-29] presented a microscopic theory of standing SWRs in exchange-coupled bilayer films, in which the analysis is based on the exact solution of the eigenvalue problem of a bilayer film achieved with the interface rescaling approach. Here, the effects of interfacial coupling, interface and surface anisotropies of a symmetric bilayer film on SWR were studied. Perchmin [31] presented a quantum theory of spin-wave resonance in cubic ferromagnets, in which Dyson’s formalism[32] rather than that of Holstein-Primakoff [33] has been used in a manner analogous to the quantum theory of ferromagnetic resonance[34]. It is valuable to develop some quantum methods to study the microwave properties of magnetic multi-layer films. According to earlier work, the microwave properties of a magnetic multi-layer thin film are significantly affected by magnetic surface anisotropies [28, 30], interface anisotropies [27-29], interfacial couplings [27-29], external magnetic field [3,28,30] and thickness of magnetic layer[3,35], but the effect of thickness of
3
magnetic layer of a bicomponent multi-layer magnetic film with antiferromagnetic interfacial coupling on the SWR frequency have seldom been studied. Therefore, in the present work, we will utilize the linear spin-wave approach and Green’s function technique to study the SWR frequency (including FMR frequency) in a bicomponent multi-layer thin film consisting of two different magnetic materials with single-ion surface and interface anisotropies, and with antiferromagnetic or ferromagnetic interfacial coupling. We found that the multi-layer magnetic film with antiferromagnetic interfacial coupling has some novel microwave properties. The SWR frequencies for multi-layer magnetic film with ferromagnetic interfacial coupling all decrease with increasing the film thickness. While for multi-layer magnetic film with antiferromagnetic interfacial coupling, the SWR frequency in mode m=2 increases with increasing the film thickness within a proper parameter region. Our present work presents a natural extension of our previous results [36], derived for single component multi-layer films with ferromagnetic interfacial exchange coupling. Our present work is beneficial to fundamental understanding of microwave properties of a bicomponent multi-layer magnetic film and shows how to obtain high and controllable SWR frequencies (including FMR frequencies) of a bicomponent multi-layer film with antiferromagnetic or ferromagnetic interfacial coupling with a wide band. The outline of this paper is as follows: In Section 2, we describe the model, Hamiltonian of the system and calculation procedure. The SWR frequencies in a bilayer and triple-layer films are discussed in Section 3. In Section 4 conclusions are presented.
4
2 Model and calculation procedure We consider a Heisenberg model with a single-ion anisotropy on a simple cubic lattice for multi-layer ferromagnetic films with antiferromagnetic or ferromagnetic interfacial coupling built up by N monolayers parallel to two infinitely extended surfaces. A schematic diagram of the bilayer (or triple-layer) film is given in Fig. 1(a) and (b) (or Fig. 1(c) and (d)). For the bilayer film, the first layer A consists of N1 monolayers and the second layer B of N2 monolayers. For the triple-layer film, the first, second and third layer (A, B and A) consist of N1, N2 and N1 monolayers, respectively. The Hamiltonian is:
H = − ∑ J ll ' S lj S l ' j ' − ∑ Dl ( S ljz ) 2 − ∑ gµ B B S ljz ljl ' j '
lj
(2.1)
lj
In Eq. (2.1), the magnetic ions are specified by the set of indices lj, where l is an integer labelling the monolayers and j is a two-dimensional lattice vector in the yz-plane. The summation in the first term is over the nearest-neighbour (NN) spins only. The second (anisotropy) term of Eq. (2.1) comprises the surface, interface and bulk anisotropy effects. The assumptions for the NN exchange terms and the definition of the surface, interface and bulk anisotropy terms involved in the Hamiltonian are shown in Table. The bulk exchange coupling (J) is ferromagnetic and the interfacial coupling (JAB) is antiferromagnetic or ferromagnetic. The direction of the spins in the initial state in film A is along the positive z-axis, while those in film B are along the negative z-axis (for the film with antiferromagnetic intrefacial coupling) or the positive z-axis (for the film with ferromagnetic intrefacial coupling). The external magnetic field B is also along the positive z-axis. In the following, we shall introduce only the calculation procedure for the triple-layer film, because of the similarity of the calculation procedure for the bilayer and 5
triple-layer films. By use of the Holstein–Primakoff transform [33] and the linear spinwave approximation [37], introducing the spin-wave operators blk ( blk+ // ) (l = 1-N), we //
can rewrite Eq. (2.1) as follows:
H = −n (2 N1 − 4) DS A2 + ( N 2 − 2) DS B2 + ∑ Dl S A2 + 2 Dint ( S A2 + S B2 ) l =1, N 2 2 − 4S A nJ (3 N1 − 1) − 2 S B nJ (3N 2 − 1) − 4 S A S B nJ AB − gµ B Bn( 2 N1S A + N 2 S B ) N −1
∑
+
l = 2 ( l ≠ N1 ~ N1 + N 2 +1)
+
(12 JS A + 2 S A D + gµ B B) ∑ bl+, k// bl ,k // k //
N1 + N 2 −1
+ + ∑ (12 JS B + 2 S B D + gµ B B ) ∑ bl ,k // bl ,k // + ∑ (10 JS A + 2S A Dl + gµ B B) ∑ bl ,k // bl ,k //
l = N1 + 2
+
l =1, N
k //
∑ l = N1 , N1 + N 2 +1
k //
(10 JS A + 2 J AB S B + 2 S A Dint + gµ B B )∑ bl+,k // bl ,k // k //
+ ∑ (10 JS B + 2 J AB S A + 2 S B Dint + gµ B B ) ∑ bl+, k // bl ,k // l = N1 +1, N1 + N 2
−
k //
N −1
+ + ∑ 2S A J l ,l +1 ∑ (bl ,k // bl +1, k // + bl , k // bl +1,k // ) −
l =1( l ≠ N1 ~ N1 + N 2 )
−
∑ l = N1 , N1 + N 2
− 4S A J
k //
N1 + N 2 −1
+ + ∑ 2S B J l ,l +1 ∑ (bl ,k // bl +1,k // + bl ,k // bl +1,k // )
l = N1 +1
k //
2 S A S B J AB ∑ (bl ,k // bl++1, k // + bl+, k// bl +1,k // ) k //
N
∑
N1 + N 2
+ + ∑ γ k // (2bl , k// bl ,k // − 1) − 4 S B J ∑ ∑ γ k// (2bl , k // bl ,k // − 1)
l =1( l ≠ N1 +1~ N1 + N 2 ) k //
l = N1 +1 k //
(2.2a)
6
H = − n (2 N1 − 4) DS A2 + ( N 2 − 2) DS B2 + ∑ Dl S A2 + 2 Dint ( S A2 + S B2 ) l =1, N − 4S A2 nJ (3 N1 − 1) − 2S B2 nJ (3 N 2 − 1) + 4S A S B nJ AB − gµ B B n(2 N1S A − N 2 S B ) N −1
∑
+
l = 2 ( l ≠ N1 ~ N1 + N 2 +1)
+
(12 JS A + 2S A D + gµ B B)∑ bl+,k // bl ,k // k //
N1 + N 2 −1
+ + ∑ (12 JS B + 2 S B D - gµ B B )∑ bl ,k // bl ,k // + ∑ (10 JS A + 2S A Dl + gµ B B )∑ bl ,k // bl ,k //
l = N1 + 2
+
l =1, N
k //
∑ l = N1 , N1 + N 2 +1
k //
(10 JS A - 2 J AB S B + 2S A Dint + gµ B B )∑ bl+,k // bl ,k // k //
+ ∑ (10 JS B - 2 J AB S A + 2S B Dint - gµ B B )∑ bl+,k // bl , k // l = N1 +1, N1 + N 2
−
k //
N −1
+ + ∑ 2S A J ∑ (bl , k// bl +1,k // + bl ,k // bl +1,k // ) −
l =1( l ≠ N1 ~ N1 + N 2 )
−
∑ l = N1 , N1 + N 2
− 4S A J
k //
N1 + N 2 −1
+ + ∑ 2S B J ∑ (bl ,k // bl +1,k // + bl , k// bl +1,k // )
l = N1 +1
k //
2 S A S B J AB ∑ (bl , k// bl +1,k // + bl+,k // bl++1,k // ) k //
N
∑
N1 + N 2
+ + ∑ γ k // ( 2bl ,k // bl ,k // − 1) − 4S B J ∑ ∑ γ k// (2bl ,k // bl , k// − 1)
l =1( l ≠ N1 +1~ N1 + N 2 ) k //
l = N1 +1 k //
(2.2b) The equations (2.2a) and (2.2b) correspond to the case of the triple-layer film with ferromagnetic and antiferromagnetic interfacial coupling, respectively. Note that,
γk = //
1 eik // ⋅δ // ∑ 4 δ //
(2.3)
where δ // represents that only the exchanges between the nearest neighbors in yz – planes are taken into account. There are n sites on each layer-lattice, and, in total, nN sites in the system. We define the N-order matrix retarded Green’s functions: 7
G (k // , f ) = [Gi , j ] N × N
(2.4)
here Gi , j =<< bik // ; b +jk // >> f （i = 1 ~ N; j = 1 ~ N)
(2.5a)
Gi , j =<< bik // ; b +jk // >> f (i = 1 ~ N1; j = 1 ~ N1 or i = N1+N2+1 ~ N; j = N1+N2+1 ~ N) Gi , j =<< bik+// ; b jk // >> f （i = N1+1 ~ N1+N2; j = N1+1 ~ N1+N2) Gi , j =<< bik// ; b jk // >> f (i = 1 ~ N1; j = N1+1 ~ N1+N2 or i = N1+N2+1 ~ N; j = N1+1 ~
N1+N2) Gi , j =<< bik+// ; b +jk // >> f (i = N1+1 ~ N1+N2; j = 1 ~ N1 or i = N1+1 ~ N1+N2; j = N1+N2+1 ~
N)
(2.5b)
The equations (2.5a) and (2.5b) are the matrix elements of the N-order matrix retarded Green’s functions of the triple-layer film with ferromagnetic and antiferromagnetic interfacial coupling, respectively. By using the equation of the Green’s functions, we obtain the solution of the Green’s function as follows:
D* ( f ) G (k // , f ) = det( D( f ))
(2.6)
Where the matrix D* is adjoint matrix of matrix D. And
[ ]
D( f ) = Wi, j
N×N
[ ]
+ Hi, j
(2.7)
N ×N
8
Here, in the N-order matrix W, only the diagonal matrix elements Wii (i = 1 ~ N) are nonzero and equal to f . f represents the energy spectrum of the system. The matrix H is a three-diagonal matrix with matrix elements H ij (i, j =1~N): H i ,i =（ - 8 JS A (1 − γ k // ) + 4 JS A + 2 DS A + gµ B B ）
(i = 2~ (N-1), i ≠ N1~N1+N2+1)
H i ,i = ∓（8 JS B (1 − γ k // ) + 4 JS B + 2 DS B ± gµ B B ）
(i = N1+2~ N1+N2-1)
H1,1 =（ - 8 JS A (1 − γ k// ) + 2 JS A + 2 D1 S A + gµ B B ） H N , N =（ - 8 JS A (1 − γ k // ) + 2 JS A + 2 DN S A + gµ B B ） H N1 , N1 = N N1 + N 2 +1, N1 + N 2 +1 =（ - 8 JS A (1 − γ k // ) + 2 JS A ± 2 J AB S B + 2 Dint S A + gµ B B） H N1 + N 2 , N1 + N 2 = H N1 +1, N1 +1 = ∓（8 JS B (1 − γ k // ) + 2 JS B ± 2 J AB S A + 2 Dint S B ± gµ B B ） H i ,i +1 = 2 JS A
(i = 1~ (N1-1); i = (N1+N2+1) ~ (N-1))
H i,i +1 = ±2 JS B
(i = (N1+1)~ (N1+N2-1))
H i ,i −1 = 2 JS A
(i= 2 ~ N1; i =(N1+N2+1)~N)
H i,i −1 = ±2 JS B
(i= (N1+2) ~(N1+N2))
H N1 , N1 +1 = H N1 + N 2 +1, N1 + N 2 = 2 J AB S A S B H N1 +1, N1 = H N1 + N 2 , N1 + N 2 +1 = ±2 J AB S A S B
(2.8)
The upper/lower sign above corresponds to the case of the triple-layer film with ferromagnetic/antiferromagnetic interfacial coupling. Setting the determinant to zero, i.e. det (D( f ))=0, we obtain N numerical solutions for the spin-wave spectra of the triple-layer film with antiferromagnetic or ferromagnetic interfacial coupling.
9
We now discuss the effect of the interfacial coupling between the sublayer films, the magnetic surface anisotropy, the interface anisotropy, the thickness and the external magnetic field on the SWR frequencies (including the FMR frequencies) in a bicomponent bilayer and triple-layer films with antiferromagnetic or ferromagnetic interfacial coupling, and compare the SWR frequencies of films with antiferromagnetic and ferromagnetic interfacial coupling. Because k// = 0 , the dipole-dipole interaction vanishes [38,39] and is neglected in our model. In the calculations, the exchange coupling between nearest neighboring spins in the sublayer films A and B is set to unity (J = 1.0). For the bicomponent bilayer or triple-layer films, both the sublayer film A and B consist of six monolayers (i.e. N1=N2=6), and the spin quantum numbers in the sublayer film A and B are SA=1.0 and SB=1.5, respectively. 3. Results and discussion
3.1. SWR frequency in a bicomponent bilayer film For a bilayer film AB (N1=N2=6), there are twelve SWR frequencies. The spin-wave modes with mode numbers m= 1, 2…, N, start from the energetically lowest mode. Next we will study only the lowest four SWR frequencies (i.e. modes m=1, 2, 3 and 4), because the resonance of low energy has larger excitation probability. Figure 2 presents the dependence of the SWR frequencies in a bicomponent bilayer film with D1 = 0.08,
D12 = 0.08, D = 0.02, Dint = 0.02 and B = 0 on the ferromagnetic or antiferromagnetic interfacial couplings JAB. From Fig.2, the SWR frequencies of mode m=1 are almost unchanged by ferromagnetic interfacial coupling. The other three SWR frequencies increase with increasing ferromagnetic interfacial coupling. Moreover, the change of the SWR frequencies of mode m=4 is the biggest. While four SWR frequencies all increase
10
with enhancing antiferromagnetic interfacial coupling, and the effect of antiferromagnetic interfacial coupling on the SWR frequencies of mode m=3 is the biggest. When the interfacial coupling is absent, the SWR frequencies of the bilayer films with ferromagnetic and antiferromagnetic interfacial couplings are same, because these systems are same. When the interfacial coupling is stronger, the SWR frequencies of modes m=1 and 3 (or m=2 and 4) of the bilayer films with antiferromagnetic interfacial couplings are larger (or smaller) than those of the bilayer films with ferromagnetic interfacial couplings. Our results on the effect of the ferromagnetic interfacial coupling of bicomponent bilayer film on the resonance frequencies agree with the calculation result of single component asymmetric bilayer film in Ref. 36, and also agree with the experimental and calculation result in Ref. 5. Figure 3 shows the dependence of the SWR frequencies on the lower surface anisotropy D1 and upper surface anisotropy D12 of a bilayer films with ferromagnetic or antiferromagnetic interfacial couplings. For the bilayer films with ferromagnetic interfacial coupling (from Fig.3(a)), when the surface anisotropy D1 or D12 increases, the SWR frequencies all increase, and the effects of the lower surface anisotropy D1 and the upper surface anisotropy D12 on the SWR frequencies are almost the same. For the bilayer film with antiferromagnetic interfacial coupling (from Fig.3(b)), when the surface anisotropy D1 (or D12) increases, only the SWR frequencies of modes m=1 and 3 (or modes m=2 and 4) increase, the SWR frequencies of modes m=2 and 4 (or modes m=1 and 3) are unchanged by the surface anisotropy D1 (or D12). Namely, for the bilayer films with antiferromagnetic interfacial coupling, the surface anisotropy D1 (or D12) only affects the SWR frequencies of odd (or even) number modes. From Figs.3(c) and (d), the
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effects of the surface anisotropy D1 (or D12) of the bilayer films with antiferromagnetic interfacial couplings on the SWR frequencies of odd (or even) number modes are larger than those of the bilayer films with ferromagnetic interfacial couplings. The effects of the interface anisotropy on the SWR frequency of a bilayer films with ferromagnetic or antiferromagnetic interfacial couplings are shown in Figs 4(a) and 4(b), respectively. For the bilayer film with ferromagnetic or antiferromagnetic interfacial coupling, when the interface anisotropy Dint increases, all SWR frequencies increase. For the film with ferromagnetic interfacial coupling (see Fig. 4(a)), the effect of the interface anisotropy Dint on mode m=3 is larger. When JAB=0.2, the effect of the interface anisotropy Dint on mode m=4 is also larger. This agrees with the result of single component symmetric bilayer film in Ref. 36. For the film with antiferromagnetic interfacial coupling (see Fig. 4(b)), the effect of Dint on modes m= 4 is larger. From Fig.4(a) and (b), the different ferromagnetic interfacial coupling can lead to different effects of the interface anisotropy on the SWR frequency, while the antiferromagnetic interfacial coupling almost does not affect the effects of the interface anisotropy on the SWR frequency. 3.2. SWR frequency in a bicomponent triple-layer film How will the SWR frequency be affected if we add a sublayer film A to the bilayer film AB (forming a symmetry triple-layer film ABA)? In next, we will study the SWR frequencies of a triple-layer film. Figure 5 shows the effects of the interfacial couplings on the SWR frequencies of triple-layer films with ferromagnetic or antiferromagnetic interfacial couplings. In the process of changing the ferromagnetic or antiferromagnetic interfacial couplings, the SWR frequencies increase with increasing interfacial couplings,
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only the lowest SWR frequency is not affected by ferromagnetic interfacial coupling. The effect of ferromagnetic (or antiferromagnetic) interfacial coupling on the SWR frequencies in mode m=3 (or m=4) is larger. When the interfacial coupling is stronger, the SWR frequencies of modes m=1, 2 and 4 (or m=3) of the triple-layer films with antiferromagnetic interfacial couplings are larger (or smaller) than those of the triplelayer films with ferromagnetic interfacial couplings. Our results on the effect of the ferromagnetic interfacial coupling of a bicomponent triple-layer film on the resonance frequencies agree with the calculation result of single component symmetric triple-layer film in Ref. 36 The dependence of the SWR frequencies on the surface anisotropies D1 and D18 of a triple-layer films with ferromagnetic or antiferromagnetic interfacial couplings is shown in Fig. 6. For triple-layer films with ferromagnetic interfacial couplings (see Fig.6(a)), when the surface anisotropy D1 or D18 increases, all SWR frequencies increase. For triple-layer films with antiferromagnetic interfacial couplings (see Fig.6(b)), when the surface anisotropy D1 or D18 increases, only the SWR frequencies in mode m=1 and 4 increase, the SWR frequency in mode m=2 and 3 is almost independent of the surface anisotropy D1 or D18. For both triple-layer films with ferromagnetic and antiferromagnetic interfacial couplings, the effects of the surface anisotropies D1 and D18 on the SWR frequencies are the same, because of the symmetry of the triple-layer films with ferromagnetic or antiferromagnetic interfacial couplings. From Fig.6(c), the effects of the surface anisotropies D1 on the SWR frequencies with modes m=1 and 4 (or modes m=2 and 3) of the triple-layer films with antiferromagnetic interfacial couplings are
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slightly larger (or smaller) than those of the triple-layer films with ferromagnetic interfacial couplings. Figures 7(a) and 7(b) present the effects of the interface anisotropies on the SWR frequency in triple-layer films with ferromagnetic and antiferromagnetic interfacial couplings, respectively. For these films, when the interface anisotropy Dint increases, all SWR frequencies increase. For the triple-layer films with ferromagnetic interfacial couplings, as seen in Fig. 7(a), the effect of the interface anisotropy Dint on mode m=1 and 4 is larger. For mode m=3 or m=4, the different interfacial coupling JAB can lead to different effect of the interface anisotropy Dint on the SWR frequencies. For the triplelayer films with antiferromagnetic interfacial couplings ( see Fig. 7(b) ), the effect of Dint on modes m=3 and 4 (the mode number is defined at Dint=0.1) is larger. The antiferromagnetic interfacial coupling almost does not affect the effects of the interface anisotropy on the SWR frequency. Figs 7(a) and 7(b) show that, for the lowest mode m=1, the effect of Dint of the film with ferromagnetic interfacial coupling on the SWR frequency is larger than those of the film with antiferromagnetic interfacial coupling. Figure 8 shows the comparison of the SWR frequencies in bilayer and triple-layer films, these curves are chosen from Figs.2 and 5. For the films with ferromagnetic interfacial coupling (from Fig.8(a)), we can see that the SWR frequencies of four modes decrease with increasing thickness of the film. This agrees with the experimental and theoretical results reported in Refs. 35 and 36, respectively. For the films with antiferromagnetic interfacial coupling (from Fig.8(b)), the SWR frequencies of modes m=1, 3 and 4 decrease with increasing thickness of the film. While for the mode m=2, when JAB is stronger, the SWR frequency increases with increasing thickness of the film
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(see the inset of Fig.8(b)). It means that, the SWR frequency of mode m=2 can increase with increasing thickness of the film with antiferromagnetic interfacial coupling within a proper parameter region. It is also calculated that the SWR frequency increases with increasing external magnetic field for bilayer and triple-layer films with ferromagnetic interfacial coupling. This agrees with the experimental results reported in Refs. 3 and 20. As the external magnetic field increases, the SWR frequencies of two (or three) modes increase while that of the other two (or one) modes decrease for bilayer (or triple-layer) film with antiferromagnetic interfacial coupling. This is because of the antiparallel alignments of spins in the initial state in film with antiferromagnetic interfacial coupling. 4. Conclusions
In conclusion, we have discussed the effect of the interfacial couplings, the surface anisotropies, the interface anisotropies, the thickness and the external magnetic field on the lowest four SWR frequencies (i.e. modes m=1, 2, 3, and 4) in a bicomponent bilayer and triple-layer films with ferromagnetic or antiferromagnetic interfacial couplings by means of the linear spin-wave approximation and Green’s function technique. The comparison of the SWR frequencies in multi-layer magnetic films with ferromagnetic and antiferromagnetic interfacial coupling has been made. The main results are concluded as following. The SWR frequency of a triple-layer film is different from those of a bilayer film, and the SWR frequency of a film with antiferromagnetic interfacial couplings is also different from those of a film with ferromagnetic interfacial couplings. For a bicomponent bilayer or triple-layer film with ferromagnetic interfacial couplings, the SWR frequencies all increase with increasing the surface anisotropy, the interface
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anisotropy and the external magnetic field, respectively, and with decreasing the film thickness. As the ferromagnetic interfacial coupling increases, only the SWR frequencies of modes m=2, 3 and 4 increase. The ferromagnetic interfacial coupling almost does not affect the lowest mode. For a bicomponent bilayer or triple-layer film with antiferromagnetic interfacial couplings, the SWR frequencies all increase with increasing the antiferromagnetic interfacial couplings and the interface anisotropy, respectively. As the surface anisotropy
D1 (or D12) of a bilayer film increases, only the SWR frequencies of the odd (or even) number modes increase. The surface anisotropy D1 (or D12) does not affect the SWR frequencies of the even (or odd) number modes. When the surface anisotropy D1 or D18 of a triple-layer film increases, only the SWR frequencies in modes m=1 and 4 increase, the SWR frequencies in modes m=2 and 3 are almost unchanged. When the external magnetic field increases, two (or three) SWR frequencies increase while the other two (or one) decrease for the bilayer (or triple-layer) film. For the films with antiferromagnetic interfacial coupling, the SWR frequencies of modes m=1, 3 and 4 decrease while the SWR frequency of mode m=2 can increase with increasing thickness of the film within a proper parameter region. Theoretically, this work provides a quantum method to study the SWR frequency of bicomponent multi-layer magnetic films with ferromagnetic or antiferromagnetic interfacial couplings. The present results could be used to understand the microwave properties of multi-layer magnetic film and direct the method to enhance and adjust the SWR frequency of the bicomponent multi-layer thin film with ferromagnetic or antiferromagnetic interfacial couplings.
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References
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Figure 1. Geometries: (a) bilayer system and choice of X, Y and Z axes; (b) schematic
representation of a bilayer film, showing sublayer A with N1 monolayers and sublayer B with N2 monolayers; (c) triple-layer system and choice of X, Y and Z axes; (d) schematic representation of a triple-layer film, showing sublayers A, B and A with N1, N2 and N1 monolayers, respectively.
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Figure 2. (Color on-line) SWR frequency as a function of the ferromagnetic (solid black
lines) and antiferromagnetic (dashed red lines) interfacial coupling JAB for a bilayer film with D = Dint = 0.02, D1 = D12= 0.08, B = 0, SA = 1.0 and SB=1.5. These curves correspond to the spin-wave modes m= 1, 2, 3 and 4, starting from the energetically lowest mode.
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Figure 3. (Color on-line) SWR frequency as a function of the lower surface anisotropy
D1 or upper surface anisotropy D12 for a bilayer film with (a) JAB = 0.5 (bilayer film with ferromagnetic interfacial coupling), and (b) JAB = -0.5 (bilayer film with antiferromagnetic interfacial coupling). The comparisons of Fig. (a) and Fig.(b) are shown in Fig.(c) and (d). The parameters are: D = D12 (or D1) =Dint=0.08, and B = 0.
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Figure 4. (Color on-line) SWR frequency as a function of the interface anisotropy with
D1 = D12 = 0.08, D = 0.02, and B = 0. The solid black ,the dotted green, and the dashed red lines correspond to |JAB| = 0.2, 0.5 and 0.8, respectively. (a) for a bilayer film with ferromagnetic interfacial coupling and (b) for a bilayer film with antiferromagnetic interfacial coupling.
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Figure 5. (Color on-line) SWR frequency as a function of the interfacial couplings JAB
for a triple-layer film with ferromagnetic (the solid black lines) or antiferromagnetic (the dashed red lines) interfacial couplings. The parameters are: D = Dint = 0.02, D1 = D18= 0.08, and B= 0.
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Figure 6. (Color on-line) SWR frequency as a function of the surface anisotropies D1 and
D18 for a triple-layer film with (a) JAB = 0.5, and (b) JAB = -0.5. The comparison of Fig. (a) and Fig.(b) is shown in Fig.(c). The parameters are: D = D18 (or D1) = Dint = 0.08, and
B = 0.
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Fig. 7. (Color on-line) Dependence of interface anisotropies Dint on the SWR frequency
for a triple-layer film with D1 =D18 = 0.08, D = 0.02 and B = 0. The solid black ,the dashed green, and the dotted red lines correspond to |JAB| = 0.2, 0.5 and 0.8, respectively. (a) for a triple-layer film with ferromagnetic interfacial coupling and (b) for a triple-layer film with antiferromagnetic interfacial coupling.
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Fig. 8. (Color on-line) Dependence of interlayer coupling on the SWR frequency for a
bilayer (solid black lines) and triple-layer (dashed red lines) film with (a) ferromagnetic and (b) antiferromagnetic interfacial couplings. The parameters are: D1=D18 (or
D12)=0.08, D = Dint = 0.02 and B = 0.
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Table.
The assumptions for the NN exchange terms and the definition of the surface, interface and bulk anisotropy terms involved in the Hamiltonian of a bilayer and triple-layer films.
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Highlights
(1) A quantum approach is developed to study the SWR frequency of a bicomponent multi-layer films with antiferromagnetic or ferromagnetic interfacial couplings. (2) The comparison of the SWR frequencies in multi-layer magnetic films with ferromagnetic and antiferromagnetic interfacial coupling has been made. (3) The present results could be useful in enhancing our fundamental understanding and show the method to enhance and adjust the SWR frequency of bicomponent multi-layer magnetic films with antiferromagnetic or ferromagnetic interfacial coupling.
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