Frequency in Middle of Magnon Band Gap in a Layered Ferromagnetic Superlattice

Commun. Theor. Phys. (Beijing, China) 54 (2010) pp. 1144 1150 c Chinese Physical Society and IOP Publishing Ltd Vol. 54, No. 6, December 15, 2010 Frequency in Middle of Magnon Band Gap in a Layered Ferromagnetic Superlattice QIU Rong-Ke ( Â ), ZHAO Jian ( ), and YING Cai-Hong ( ) Shenyang University of Technology, Shenyang 110870, China (Received February 23, 2010; revised manuscript received April 19, 2010) Abstract The frequency in middle of magnon energy band in a five-layer ferromagnetic superlattice is studied by using the linear spin-wave approach and Green s function technique. It is found that four energy gaps and corresponding four frequencie in middle of energy gaps exist in the magnon band along K x direction perpendicular to the superlattice plane. The spin quantum numbers and the interlayer exchange couplings all affect the four frequencies in middle of the energy gaps. When all interlayer exchange couplings are same, the effect of spin quantum numbers on the frequency ω g1 in middle of the energy gap ω 12 is complicated, and the frequency ω g1 depends on the match of spin quantum numbers in each layer. Meanwhile, the frequencies ω g2, ω g3, and ω g4 in middle of other energy gaps increase monotonously with increasing spin quantum numbers. When the spin quantum numbers in each layer are same, the frequencies ω g1, ω g2, ω g3, and ω g4 all increase monotonously with increasing interlayer exchange couplings. PACS numbers: 75.70.Cn, 75.10.Jm, 75.30.Ds Key words: layered ferromagnetic superlattice, magnon energy band, frequency in middle of the magnon energy gap, spin quantum number, interlayer exchange couplings 1 Introduction The band spectrum, which is the signature of periodicity, has been intensively studied in photonic crystals, [1 2] plasmonic crystals, [3] phononic crystals, [4] periodic ferroelectric media. [5] Especially, attention has been focused on searching photonic and phononic crystals. Yablonovitch [1] and John [2] anticipated the existence of complete energy gaps in electromagnetic wave spectra of three-dimensional (3D) periodic composites. Subsequently, the photonic band structure was calculated for electromagnetic waves in the periodic dielectric materials. [6 7] Kushwaha et al. [8 9] studied the acoustic band structure of periodic elastic composites. In decade, attention has been focused on the search of photonic and phononic crystals in which both the position and the width of the energy gap could be controlled by external factors, such as applied voltage or magnetic field. The materials with periodically magnetic structure can be refereed to as magnonic crystals in which the magnon acting as information carrier. A theory of 3D magnonic crystals was developed to be applicable for explaining the existence of the spin-wave frequency gap. A confrontation of the theory with experimental results proves that certain manganites can be regarded as 3D magnonic crystals existing in the nanoscale in nature. [10] The planewave method was used to determine spin-wave spectra of 3D magnonic crystals composed of two different ferromagnetic materials. It was demonstrated that magnonic gaps occur at spontaneous magnetization contrast and/or exchange contrast values above a certain critical level, which depends on the lattice type, and the reduced width of the gap (i.e., the width referred to the gap center) is, in good approximation, a linear function of both the two contrasts. [11] Further research was focused on magnetic multilayer/superlattice systems, which can be regarded as one-dimensional magnonic crystals. [12 23] It was found that the magnon energy gap strongly affects physical properties of layered magnetic systems. [14] Temperature dependence of magnetization and optical magnon gap in bilayer antiferromagnetic YBa 2 Cu 3 O 6 was obtained by employing the Green s function technique and the Callen decoupling approximation. [15] Deng et al. [16] studied the magnon energy gap of a two-layer magnetic superlattice with a periodic anisotropy, by using local coordinates and a spin-bose transformation quantum approach. The magnon dispersion relation with two ferromagnetic layers in the pure exchange limit, the pure dipolar limit, and both exchange and dipolar interactions was calculated, and it was presented that the magnon energy gap derives mainly from the exchange interaction. [17 18] In our previous work, [19 21] the width of magnon energy gap was studied for magnetic superlattices. The results showed that there are modulated energy gaps, and the absence of the energy gap at a certain condition means that the system has a high magnetically structural symmetry, and there is a competition effect of the anisotropy and the spin quantum number on the magnon energy gap. [19 21] In order to understand further the magnon band of multilayered superlattices, in this report, we study the positions of the magnon energy gaps in a superlattice system constructed by five kinds of ferromagnetic materials that are coupled ferromagnetically. The scientific significance is that, the positions of the magnon energy gap should Supported by the Natural Science Foundation of Liaoning Province of China under Grant No. 20062040 E-mail: rkqiu@163.com

No. 6 Frequency in Middle of Magnon Band Gap in a Layered Ferromagnetic Superlattice 1145 correlate with forbidden band frequency of absorbing electromagnetic wave in the magnetic thin films or layered superllatice. The outline of this paper is as follows: In Sec. 2, we describe the model, Hamiltonian of the system and calculation procedure. Section 3 represents the effect of spin quantum numbers and interlayer exchange couplings on the frequency in middle of the magnon energy gaps of the system. Section 4 gives a conclusion. 2 Model and Calculation Procedure We consider a Heisenberg model for a five-layer ferromagnetic superlattice on a simple cubic lattice. A schematic model of the five-layer ferromagnetic superlattice is illustrated in Fig. 1. state in the five sublayers is along the positive z-direction. Therefore, all the interlayer exchange couplings J 12, J 23, J 34, J 45, and J 51 and the intralayer exchange couplings J 1, J 2, J 3, J 4, and J 5 are positive. There are N sites on each layer of the superlattice, and total 5N sites for the system. By use of the Holstein Primakoff transform [22] and the linear spin-wave approximation, [23] introducing the spin-wave operators b lk (b lk ) (l = 1, 2, 3, 4, 5), we can rewrite Eq. (1) as follows: ( Zyz 5 5 ) H = N J l Sl 2 2 + J l,l+1 S l S l+1 + 5 (J l S l Z yz + J l,l+1 S l+1 + J l 1,l S l 1 ) k 5 Sl S l+1 J l,l+1 k (γ kx b l,k b l+1,k + γ kxb l,k b l+1,k) b lk b lk 1 2 Z yz 5 J l S l γ k (2b lk b lk + 1). (2) k Here if l = 1, l 1 = 5; if l = 5, l+1 = 1. And Z yz = 4 represents the number of nearest neighbors in the yz-planes that are the same. Note, γ k = 1 e ik δ, (3a) Z yz δ γ ±kx = δ x e ±ik δx. (3b) Fig. 1 A schematic model for the five-layer ferromagnetic superlattice. The interlayer exchange couplings and spin quantum numbers are illustrated. A unit cell of the superlattice consists of five layers, 1, 2, 3, 4, and 5, where spins are denoted by S l (l = 1, 2, 3, 4, 5) for each layer. The nearest neighboring spins within each sublayer are coupled ferromagnetically by the intralayer exchange couplings J l (l = 1, 2, 3, 4, 5), respectively. The ferromagnetic interlayer exchange couplings J l,l+1 between the spins at the nearest neighboring layers exist. The superlattice is stacked periodically along x- direction that is normal to layers (yz-planes). The Hamiltonian is: H = 1 5 J l S l,ρ S l,ρ+δ 2 ρ,δ 5 J l,l+1 S l,ρ S l+1,ρ, (1) ρ where l is the number of sublayers, δ represents that only the exchanges between the nearest neighbors in yz-planes are taken into account. The direction of spins of the initial Here γ kx γ kx. γ kx and γ kx are complex because δ x has only one value, but γ k is real because δ can take four different values within the layer. We first define the five-order matrix retarded Green s functions: G(k, ω) = [G i,j ] 5 5, (4) here G i,j = b ik ; b jk ω (i=1 5; j=1 5). By using the equation of the Green s functions, we obtain the solution of the Green s function as follows: 1 G(k, ω) = det(d(ω)) [M i,j] 5 5, (5) where D(ω) = [W i,j ] 5 5 + [H i,j ] 5 5. (6) Here in the five-order matrix W, only the diagonal matrix elements W ii (i=1 5) are non-zero and equal to ω. ω represents the energy spectrum of the system. The matrix elements H ij (i, j = 1, 2, 3, 4, 5) in the five-order matrix, which are given in Appendix. Setting the determinant to zero, i.e., det(d(ω)) = 0, we obtain five positive numerical solutions for the spin wave spectra of the ferromagnetic five-layer superlattice.

1146 QIU Rong-Ke, ZHAO Jian, and YING Cai-Hong Vol. 54 In the following, we shall discuss whether the energy gaps exist in the energy band structure along the K x direction (perpendicular to the superlattice plane) of the present system, and how the spin quantum number, the interlayer exchange couplings affect the frequency in middle of the energy gap along the K x direction. In this paper, setting the intralayer exchange couplings a unit (J 1 = J 2 = J 3 = J 4 = J 5 = 1.0). 3 Results and Discussion Figure 2 shows the K x dependence of the energy spectra of the five-layer ferromagnetic superlattice. From Fig. 2, five energy spectra branches ω l (l = 1, 2, 3, 4, 5) exist in the energy band, ω 1 is an acoustic branch and ω 2, ω 3, ω 4, and ω 5 are optic. Furthermore, there are four magnon energy gaps ω ll+1 (l = 1, 2, 3, 4) in the energy band. The magnon energy gap ω 12 is defined as the gap between the top of the branch ω 1 and the bottom of the branch ω 2, while ω g1 is the frequency in the middle of this gap; the magnon energy gap ω ll+1 (l = 2, 3, 4) is between the bottom of the branch ω l+1 and the top of the branch ω l, while ω gl is the frequency in the middle of the corresponding gap. According to numerical calculations, the relation ω g1 < ω g2 < ω g3 < ω g4 is always satisfied. As the cases of photon, electron, and phonon, the magnon energy gap should strongly affect the physical properties of the magnetic system. We shall focus on the effects of the spin quantum numbers and interlayer exchange couplings on the frequency in the middle of the magnon energy gaps of the system. numbers S 1 and S 2 of the five-layer ferromagnetic superlattice, as J 12 = J 23 = J 34 = J 45 = J 51 = 1.0. From Fig. 3(a), the frequency ω g1 increases with increasing the spin quantum numbers S 1 and S 2, as S 3 = S 4 = S 5 = 0.5. Fig. 3 (a) and (b) S 1 and S 2 dependence of the frequencies ω g1 in middle of the magnon energy gap ω 12 of the five-layer ferromagnetic superlattice. The parameters J 1 = J 2 = J 3 = J 4 = J 5 = 1.0, (a) S 3 = S 4 = S 5 = 0.5 (b) S 3 = S 4 = S 5 = 2.5. Fig. 2 K x dependence of the energy spectra ωof the fivelayer ferromagnetic superlattice with S 1 = 1.5, S 2 = 1.0, S 3 = 0.5, S 4 = 2.5, S 5 = 1.5, J 12 = 0.5, J 23 = 1.0, J 34 = 0.6, J 45 = 0.8, J 51 = 0.7, J 1 = J 2 = J 3 = J 4 = J 5 = 1.0 and K y = K z = 0. Here, ω ll+1 is the energy gap between the energy spectraω 1 and ω l+1 (l = 1, 2, 3, 4), The frequencies ω g1, ω g2, ω g3, and ω g4 are the frequencies in middle of the magnon energy gaps ω 12, ω 23, ω 34 and ω 45, respectively. Figures 3(a) and 3(b) represent the dependence of the frequency ω g1 in the middle of the magnon energy gap ω 12 on the first- and the second-layer spin quantum If the spin quantum numbers S 3, S 4 and S 5 are bigger [see Fig. 3(b) for S 3 = S 4 = S 5 = 2.5], the change of the frequency ω g1 is richer. When S 1 = 0.5, ω g1 decreases with increasing S 2 ; when S 1 = 1.0, ω g1 decreases first and then increases with increasing S 2, there is minimum ω g1 at S 2 = 1.0; when S 1 > 1.0, ω g1 increases with increasing S 2. Figures 4(a) and 4(b) represent the dependence of the frequency ω g1 in the middle of the magnon energy gap ω 12 on the first- and the third-layer spin quantum numbers S 1 and S 3 of the five-layer ferromagnetic superlattice, as J 12 = J 23 = J 34 = J 45 = J 51 = 1.0. As S 2 = S 4 = S 5 = 1.0 (see Fig. 4(a)), the frequency ω g1 in the middle of the energy gap ω 12 increases with increasing S 1 and S 3. As the difference between S 2 and S 4 (= S 5 ) is bigger [see Fig. 4(b) with S 2 = 0.5, S 4 = S 5 = 3.5)], the variety of the frequency ω g1 is plenty. When S 1 = 0.5, 1.0, 1.5, and 2.0, the ω g1 increases with increasing S 3 ; when S 1 = 2.5, the ω g1 increases first and then decreases with increasing S 3, and there is a maximum for ω g1 at

No. 6 Frequency in Middle of Magnon Band Gap in a Layered Ferromagnetic Superlattice 1147 S 3 = 2.0; when S 1 > 2.5, the ω g1 decreases monotonously with increasing S 3. if they are small (0.5 or 1.0), the frequency ω g1 increases monotonously with increasing other two spin quantum numbers S 1 and S 2. For the bigger spin quantum numbers (S 3 = S 4 = S 5 2.0), when the spin quantum number S 1 is small/large, the frequency ω g1 decreases/increases with increasing a spin quantum number S 2. When the spin quantum numbers of two neighboring layers and a next neighboring layer are same, i.e. S 2 = S 4 = S 5, the frequency ω g1 increases with increasing S 1 and S 3. For the large difference between S 2 and S 4 (=S 5 ), when the spin quantum number S 1 is small/large, the frequency ω g1 increases/decreases with increasing a spin quantum number S 3. Namely, the tendency of variation of the frequency ω g1 in the middle of the magnon energy gap ω 12 is correlated with the match between the spin quantum numbers at each layer. Fig. 4 (a) and (b) S 1 and S 3 dependence of the frequencies ω g1 in middle of the magnon energy gap ω 12 of the five-layer ferromagnetic superlattice. The parameters J 1 = J 2 = J 3 = J 4 = J 5 = 1.0, (a) S 2 = S 4 = S 5 = 1.0 (b) S 2 = 0.5, S 4 = S 5 = 3.5. Fig. 5 S 1 and S 2(S 3) dependence of the frequencies ω g2 in middle of the magnon energy gap ω 23 of the five-layer ferromagnetic superlattice. Here, the solid lines are corresponding to the curves of S 1 and S 2, the dash lines corresponding to the curves of S 1 and S 3. The solid lines and the dash lines are same. The parameters J 1 = J 2 = J 3 = J 4 = J 5 = 1.0. For solid lines, S 3 = S 4 = S 5 = 3.5, for dash lines, S 2 = S 4 = S 5 = 3.5. From results above, it is found that, when three adjacent spin quantum numbers are same, i.e. S 3 = S 4 = S 5, Fig. 6 S 1 and S 2 (S 3) dependence of the frequencies ω g4 in middle of the magnon energy gap ω 45 of the five-layer ferromagnetic superlattice. Here, the solid lines are corresponding to the curves of S 1 and S 2, the dash lines corresponding to the curves of S 1 and S 3. The solid lines and the dash lines are same. The parameters J 1 = J 2 = J 3 = J 4 = J 5 = 1.0. For solid lines, S 3 = S 4 = S 5= 3.5, for dash lines, S 2 = S 4 = S 5 = 3.5. Figure 5 represents the dependence of the frequency ω g2 in the middle of the magnon energy gap ω 23 on the first- and the second (or third)-layer spin quantum numbers S 1 and S 2 (or S 3 ) of the five-layer ferromagnetic superlattice, as J 12 = J 23 = J 34 = J 45 = J 51 = 1.0. It is found that the frequency ω g2 increases monotonously with increasing spin quantum numbers S 1, S 2 (or S 3 ), while the curves of ω g2 -S 1 and S 2 are consistent with the curves of ω g2 -S 1 and S 3. It explains that, the effects of the second- and third- layer spin quantum numbers S 2 and S 3 on the frequency ω g2 in the middle of the magnon energy gap ω 23 of five-layer ferromagnetic superlattice are same. Namely, the spin quantum numbers S 2 and S 3 are symmetry for the frequency ω g2. The effect of spin quantum numbers on the frequency ω g4 in the middle of the magnon energy gap ω 45 is similar as that on the frequency ω g2 (see Fig. 6). It may be due to that the frequencies ω g2 and ω g4 all are in K x = 0. Figure 7 demonstrates

1148 QIU Rong-Ke, ZHAO Jian, and YING Cai-Hong Vol. 54 the dependence of the frequency ω g3 in the middle of the magnon energy gap ω 34 on the first- and the second (or third)-layer spin quantum numbers S 1 and S 2 (or S 3) of the five-layer ferromagnetic superlattice, as J 12 = J 23 = J 34 = J 45 = J 51 = 1.0, and S 3 = S 4 = S 5 = 0.5 (for solid lines), and S 2 = S 4 = S 5 = 0.5 (for dotted lines). From Fig. 7, the frequency ω g3 becomes bigger with increasing the spin quantum numbers S 1, S 2 (or S 3 ). As spin quantum number S 1 = 0.5, the dot lines are consistent with the solid lines. Meanwhile, if the spin quantum numbers S 1 and S 2 (or S 3 ) are not equal to 0.5, the dot lines are always higher than the solid lines for same spin quantum number S 1. It indicates that, when the spin quantum number S 1 is larger than 0.5, the effect of the spin quantum number S 3 on the frequency ω g3 is stronger than that of the spin quantum number S 2. Fig. 8 J 12 and J 23 (J 34) dependence of the frequencies ω g1 in middle of the magnon energy gap ω 12 of J 45 = J 51 = 0.5, for dash lines, J 23 =J 45 = J 51 = 0.5. Fig. 7 S 1 and S 2 (S 3) dependence of the frequencies ω g3 in middle of the magnon energy gap ω 34 of the five-layer ferromagnetic superlattice. Here, the solid lines are corresponding to the curves of S 1 and S 2, the dash lines corresponding to the curves of S 1 and S 3. The parameters are J 12 = J 23 = J 34 = J 45 = J 51 = 1.0 and J 1 = J 2 = J 3 = J 4 = J 5 = 1.0. For solid lines, S 3 = S 4 = S 5 = 0.5, for dash lines, S 2 = S 4 = S 5 = 0.5. Fig. 9 J 12 and J 23 (J 34) dependence of the frequencies ω g2 in middle of the magnon energy gap ω 23 of J 45 = J 51 = 0.5, for dash lines, J 23 = J 45 = J 51 = 0.5. Next we will study the effects of the interlayer exchange couplings on the frequency in middle of the magnon energy gap. The effects of the interlayer exchange couplings J 12 and J 23 (or J 34 ) on the frequencies ω g1, ω g2, ω g3, and ω g4 are shown in Figs. 8 11, respectively. From Fig. 8, the frequency ω g1 is higher, as the interlayer exchange couplings J 12, J 23, and J 34 increase. Except J 12 = 0.5, the sold curves are not consistent with the dot curves for same interlayer exchange coupling J 12. It indicates that, when the interlayer exchange coupling J 12 is not equal to other interlayer exchange couplings, the effects of the interlayer exchange couplings J 23 and J 34 on the frequency ω g1 are not symmetry. The effects of the interlayer exchange couplings J 12 and J 23 (or J 34 ) on the frequencies ω g2, ω g3, and ω g4 are same with that on the frequency ω g1 (see Figs. 9, 10, and 11). Fig. 10 J 12 and J 23 (J 34) dependence of the frequencies ω g3 in middle of the magnon energy gap ω 34 of J 45 = J 51 = 0.5, for dash lines, J 23 = J 45 = J 51 = 0.5.

No. 6 Frequency in Middle of Magnon Band Gap in a Layered Ferromagnetic Superlattice 1149 Fig. 11 J 12 and J 23 (J 34) dependence of the frequencies ω g4 in middle of the magnon energy gap ω 45 of J 45 = J 51 = 0.5, for dash lines, J 23 = J 45 = J 51 = 0.5. Our conclusion can be compared with experiment result. Hillebrands et al. [24] reported Brillouin-lightscattering experiments in Co/Pd multilayers with individual layer thicknesses between 1 and 64 atomic layers. Evidence is found for exchange-dominated collective spinwave excitations when interlayer exchange coupling is active. As the Pd spacer layer thickness was decreased, the frequencies of a band of collective spin-waves was observed to increase. This means that, the frequencies of a band of collective spin-waves is higher, as the interlayer exchange couplings increase. It was in agreement with our results. 4 Conclusion In conclusion, we have studied systematically the frequencies in the middle of the magnon energy gaps in the five-layer ferromagnetic superlattice, by using the linearspin wave approach and Green s function technique. We have discussed the effects of the spin quantum number and the interlayer exchange couplings on the frequencies in the middle of the magnon energy band gaps of the system. The following remarks have been derived. There are five energy spectra branches along K x direction in the five-layer ferromagnetic superlattice. The four energy gaps ω 12, ω 23, ω 34, and ω 45 exist in the magnon energy band of the system, and there are four frequencies ω g1, ω g2, ω g3, and ω g4 in the middle of the magnon energy gaps ω 12, ω 23, ω 34, and ω 45, respectively. When all interlayer exchange couplings are same, the effect of spin quantum numbers on the frequency ω g1 in middle of the energy gap ω 12 is very complicated. Take three adjacent spin quantum numbers are same, i.e. S 3 = S 4 = S 5, if they are small, the frequency ω g1 increases monotonously with increasing other two spin quantum numbers (S 1 and S 2 ). If they are bigger (S 3 = S 4 = S 5 2.0), when the spin quantum S 1 is small/large, the frequency ω g1 decreases/increases with increasing a spin quantum number S 2. Take spin quantum numbers of two neighboring layers and a next neighboring layer are same, i.e. S 2 = S 4 = S 5, the frequency ω g1 increases with increasing S 1 and S 3. Take the difference between S 2 and S 4 (= S 5 ) bigger, when the spin quantum number S 1 is small/large, the frequency ω g1 increases/decreases with increasing a spin quantum number S 3. Namely, the frequency ω g1 depends on the match in each layer spin quantum numbers. Meanwhile, the frequencies ω g2, ω g3, and ω g4 in middle of the energy gaps increase monotonously with increasing spin quantum numbers. When the spin quantum numbers at each layer are same, the frequencies ω g1, ω g2, ω g3, and ω g4 all increase monotonously with increasing interlayer exchange couplings. The scientific significance of the study is that, the positions of the magnon energy gap should correlate with forbidden band frequency of absorbing electromagnetic wave in the magnetic thin films or layered superllatice. Appendix The parameters H ij are described as: H 11 = J 1 S 1 Z yz (1 γ k ) + S 2 J 12 + S 5 J 51, H 12 = S 1 S 2 J 12 γ kx, H 13 = H 31 = H 14 = H 41 = H 24 = H 42 = H 25 = H 52 = H 35 = H 53 = 0, H 15 = S 1 S 5 J 51 γ kx, H 21 = S 1 S 2 J 12 γ kx, H 22 = S 2 J 2 Z yz (1 γ k ) + S 1 J 12 + S 3 J 23, H 23 = S 2 S 3 J 23 γ kx, H 32 = S 2 S 3 J 23 γ kx, H 33 = S 3 J 3 Z yz (1 γ k ) + S 2 J 23 + S 4 J 34, H 34 = S 3 S 4 J 34 γ kx, H 45 = S 4 S 5 J 45 γ kx, H 43 = S 3 S 4 J 34 γ kx, H 44 = S 4 J 4 Z yz (1 γ k ) + S 3 J 34 + S 5 J 45, H 51 = S 5 S 1 J 51 γ kx, H 54 = S 5 S 4 J 45 γ kx, H 55 = S 5 J 5 Z yz (1 γ k ) + S 4 J 45 + S 1 J 51.

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