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Possible existence of a new type of left-handed materials in coupled ferromagnetic bilayer films
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Journal of Magnetism and Magnetic Materials 302 (2006) 368–374 www.elsevier.com/locate/jmmm
Possible existence of a new type of left-handed materials in coupled ferromagnetic bilayer films Jiangwei Chena,b,, Baoshan Zhanga, Dongming Tanga, Yi Yanga, Weidong Xua,c, Huaixian Lua a
National Laboratory of Solid State Microstructures, Department of Physics, Nanjing University, Nanjing 210093, PR China b Nanjing Artillery Academy, Nanjing 211132, PR China c Engineering Institute of Engineer Corps., PLA University of Science and Technology, Nanjing 210007, PR China Received 26 July 2005; received in revised form 4 September 2005 Available online 24 October 2005
Abstract On the basis of Landau–Lifshitz–Gilbert (LLG) equation, an anomalous ferromagnetic resonance behavior is demonstrated in detail. Coupling between the magnetic moments produces a 3p/2 phase delay for one of the moments ferromagnetic resonance unusually, thus leads the sign of magnetic susceptibility w~ ¼ w0 jw00 to be opposite to that induced by the usual ferromagnetic resonance. Consequently, a left-handed material (LHM) may be formed near the low-frequency side of the resonance. Particularly, a LHM with negative value of real part of permeability only is predicted. r 2005 Elsevier B.V. All rights reserved. PACS: 73.20.Mf; 75.70.Ak; 42.70.a; 75.75.+a; 76.20.+q Keywords: Ferromagnetic resonance; Left-handed material
1. Introduction In 1960s, Veselago [1] predicted a strong effect of sign of both the permittivity and the permeability on electromagnetic wave propagation properties, i.e., if the real part of both the permittivity e~ and the magnetic permeability m~ are negative, the phase velocity direction of an electromagnetic wave will be opposite to its energy flow direction. In these * * media, *the electric field E , magnetic field H , and wave vector k form a left-handed triplet of vectors, instead of a right-handed triplet observed in conventional materials. Such media are called left-handed materials (LHM). Based upon the early ideas of Pendry et al. [2,3], Smith et al. [4] found negative real part of both effective m~ eff and e~eff simultaneously in a system consisting of spilt ring resonators and metal wires, and Shelby et al. [5] have observed negative values of the index of refraction n in this
Corresponding author. Tel.: +0253724979; fax: +0253593011.
E-mail address: [email protected] (J. Chen). 0304-8853/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2005.09.040
system. This discovery reinvigorated the search for LHM [6–12]. Recently, an increased attention has been paid to the microwave performance of thin ferromagnetic films [7–15]. The interest is also rooted on possible application of such films as LHM [7–12]. A number of novel ferromagnetic resonance properties are shown in the nanostructured systems, which may be due to the shape effect, surface effect, interface effect, and the exchange coupling, etc. [13–15]. The frequency dependence of magnetic permeability is typically due to the ferromagnetic resonance. As it is well known, the phase delay of the forced oscillation with respect to the pumping field is usually less than p; thus, the imaginary part of the magnetic permeability is positive. It is also noted that, for all LHMs mentioned above, the real part of both the permittivity e~ and the magnetic permeability m~ are negative, and the frequency range is near the high-frequency side of the ferromagnetic resonance [4–12]. In this paper, we first present the magnetic susceptibility properties of the exchange-coupled bilayer ferromagnetic films. We have investigated the effects of either exchange
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coupling between the moments or the inhomogenous anisotropy effective magnetic field in the ferromagnetic films. It has been shown that an ‘acoustic’ mode and an ‘optic’ mode may exist in the ferromagnetic bilayers [14,15]. However, the previous works focused mainly on the resonance frequency and the total magnetic susceptibility of the system. Here, we shall pay more attention to the magnetic susceptibility of the single layer of the bilayer film. For simplicity, we choose a parallel coupled ferromagnetic bilayer film to demonstrate in detail that, for the ‘optic’ mode, a 3p=2 phase delay exists in one of the moments ferromagnetic resonance. (Similar results can also be obtained in the anti-parallel coupled systems.) Since phase delay of the resonance is usually equal to p/2, the formerly mentioned ferromagnetic resonance is abnormal. We call this phenomenon as an anomalous ferromagnetic resonance. It is interesting to note that, here, the sign of the corresponding magnetic susceptibility w~ ¼ w0 jw00 is significantly different from that induced by the usual ferromagnetic resonance. Consequently, we predict the possible existence of a new type of LHM due to the unique sign properties of the magnetic susceptibility, in which the left-handed propagation properties exist near the low-frequency side of the ferromagnetic resonance, and also, only the real part of the magnetic permeability is negative.
(2) Effective anisotropy energy, " # 2 X 1 M i;jj H eff , 2 u2;i M s i
2. Model
H eff1 ¼ H 0 þ H u2;1 bM 2 ¼ ðH 0 þ H eff u2;1 Þ e z bM 2 ,
A parallel coupled system to be discussed in this paper consists of two ferromagnetic layers and is shown in Fig. 1, in which the magnetization is in the direction perpendicular to the film plane, defined as z-direction, and the film is infinite along x- and y-axes. Under certain conditions, the lowest-order spin wave mode within each layer is close to the uniform precession mode. Thus, for simplicity, two * * single magnetization vectors M 1 (for layer 1) and M 2 (for layer 2) are used to represent the magnetic moment in the layers [14,15]. Usually, the energies involved in the exchange-coupled system are the following [15]:
H eff2 ¼ H 0 þ H u2;2 bM 1 ¼ ðH 0 þ H eff u2;2 Þ e z bM 1 .
(2)
where the z-axis is normal to the film plane and M i;jj is the in-plane component of the magnetization of the ith * eff
layer. H u2;i are effective fields due to the shape, surface, magnetocrystalline, stress, or other anisotropy. Here, * eff
we assume the direction of H u2;i is in the positive direction of the z-axis. (3) Interlayer exchange energy, *
*
* * M1 M2 bM 1 M 2 , A12 M 1M 2
(3)
where A12 is the bilinear exchange energy constant per unit surface area. Here, the sign of A12 (i.e., the sign of b) is chosen as negative for a parallel coupled system. *
Apparently, when H 0 is equal to zero or its direction is in the positive direction of z-axis, the equilibrium position of the system is such that both the two moments stay parallel to the positive direction of z-axis as shown in Fig. 1. The effective static magnetic fields applied in the two magnetic moments are as follows: *
*
* eff
*
*
*
*
*
* eff
*
*
*
(4) (5)
When the magnetic moments are perturbed from their equilibrium orientation, the response of * the two moments to the alternating magnetic fields h 1 exp ðjotÞ and * h 2 exp ðjotÞ are governed by [14–17] *
* * * dM 1 ¼ g1 M 1 ðH eff1 þ h 1 exp ðjotÞÞ dt *
*
M 1 dM 1 þ a1 , M1 dt
ð6Þ
*
(1) Zeeman energy, X * * ðH 0 M i Þ,
(1)
i *
where H 0 is the external field, and i ¼ 1 (or 2) refers to the first (or the second) ferromagnetic layer.
Z Y Layer 1
M1
Layer 2
M2
X
Fig. 1. Bilayer structure and coordinate system used in the model.
* * * dM 2 ¼ g2 M 2 ðH eff2 þ h 2 exp ðjotÞÞ dt *
þ a2
*
M 2 dM 2 , M2 dt
ð7Þ
where g1 and g2 are the gyromagnetic constants, a1 and a2 the Gilbert damping coefficients of the two moments, * * * * respectively. Let M 1 ¼ M 01 e z þ m1 exp ðjotÞ and M 2 ¼ * * * * M 02 e z þ m2 exp ðjotÞ (here, M 01 bjm1 j, M 02 bjm2 j, and in the calculation, we assume M 01 ¼ M 02 ). To obtain the magnetic susceptibility, Eqs. (6) and (7) are linearized as [16] jom1x þ ½g1 ðH 0 þ H eff u2;1 bM 02 Þ þ jo a1 m1y þ 0m2x þ g1 bM 01 m2y ¼ g1 M 01 h1y ,
ð8Þ
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½g1 ðH 0 þ H eff u2;1 bM 02 Þ þ jo a1 m1x þ jom1y
3. Results and discussion
g1 bM 01 m2x þ 0m2y ¼ g1 M 01 h1x ,
ð9Þ 3.1. Magnetic susceptibility
0m1x þ g2 bM 02 m1y þ jom2x þ ½g2 ðH 0 þ H eff u2;2 bM 01 Þ þ jo a2 m2y ¼ g2 M 02 h2y ,
ð10Þ
* eff
g2 bM 02 m1x þ 0m1y ½g2 ðH 0 þ H eff u2;2 bM 01 Þ þ jo a2 m2x þ jom2y ¼ g2 M 02 h2x ,
ð11Þ
and m1z ¼ m2z ¼ 0.
(12)
The solution of the Eqs. (8)–(11) yields the dynamic * * magnetization m1 and m2 . Then the scalar extrinsic dynamic susceptibility of ith layer is defined by *
*
w~ i ¼
w0i
jw00i
¼
mi h i *
,
(13)
j h i j2
and the corresponding magnetic permeability is *
m~ i ¼ 1 þ w~ i ¼ 1 þ
*
mi h i *
.
(14)
j h i j2
The oscillation curves of the magnetic moments are calculated according to the following linearized equations: dm1x 1 ¼ ½g1 a1 ðH 0 þ H eff u2;1 Þm1x dt 1 þ a21 g1 ðH 0 þ H eff u2;1 bM 02 Þm1y g1 bM 01 m2y þ g1 a1 M 01 h1x exp ðjotÞ þ g1 M 01 h1y exp ðjotÞ,
ð15Þ
dm1y 1 ¼ ½g ðH 0 þ H eff u2;1 bM 02 Þm1x dt 1 þ a21 1 g1 a1 ðH 0 þ H eff u2;1 Þm1y þ g1 bM 01 m2x g1 M 01 h1x exp ðjotÞ þ g1 a1 M 01 h1y exp ðjotÞ,
ð16Þ
dm2x 1 ¼ ½g2 bM 02 m1y g2 a2 ðH 0 þ H eff u2;2 Þm2x dt 1 þ a22 g2 ðH 0 þ H eff u2;2 bM 01 Þm2y þ g2 a2 M 02 h2x exp ðjotÞ þ g2 M 02 h2y exp ðjotÞ, ð17Þ dm2y 1 ¼ ½g bM 02 m1x þ g2 ðH 0 þ H eff u2;2 bM 01 Þm2x dt 1 þ a22 2 g2 a2 ðH 0 þ H eff u2;2 Þm2y g2 M 02 h2x exp ðjotÞ þ g2 a2 M 02 h2y exp ðjotÞ, and dm1z dm2z ¼ ¼ 0. dt dt
First, we consider the case in which the two moments have same value of ferromagnetic resonance frequency in
ð18Þ
(19)
their uncoupled state. Since the magnitude of H u2;i and the effective field of the exchange coupling may be in the order of 1–10 kG s [15], we assume that g1 ðH 0 þ H eff u2;1 Þ ¼ eff g2 ðH 0 þ H u2;2 Þ ¼ 1:000 GHz, and the coupling parameter b is taken as g1 bM 02 ¼ g2 bM 01 ¼ 0:9 GHz arbitrarily. The two Gilbert damping coefficients are taken as a1 ¼ a2 ¼ 0:01 for commonality and conventionally. Thus, the system has two characteristic frequencies of 1.000 and 2.800 GHz, respectively. Furthermore, we assume that the electromagnetic wave propagates in the positive direction * of the z-axis, considering absorption of*the moment M 2 , * and the amplitude j h 1 j is smaller than j h 2 j. The obtained complex magnetic susceptibilities w~ 1xx ¼ w01xx jw001xx for layer 1 and w~ 2xx ¼ w02xx jw002xx for layer 2 are shown in Fig. 2(a) and (b), respectively. It is interesting to find that the imaginary part w001xx (dashed line in Fig. 2(a)) near 2.800 GHz is negative, which is apparently opposite to the positive w001xx and w002xx near 1.00 GHz and positive w002xx near 2.800 GHz as shown in Fig. 2(a) and (b), respectively. In addition, the change of the real part w01xx (solid line in Fig. 2(a)) near 2.800 GHz with increasing frequency is also different from that of the other three, i.e., w01xx and w02xx near 1.00 GHz, and w02xx near 2.800 GHz. The anomalous characters mentioned above are related to the * fact that phase delay of the oscillation of moment M 1 near 2.800 GHz is larger than p with respect to the alternating magnetic fields, which can be directly seen from Fig. 2(c)–(f). Fig. 2(c) and (d) demonstrates the case in which the system is driven by the magnetic field (c) with frequency of 2.785 GHz, which is slightly smaller than the high characteristic * frequency 2.800 GHz. It is shown that oscillation of M 2 (dotted line in Fig. 2(d)) has a phase delay with a value less than p/2; however, the phase delay * of the M 1 oscillation (dot-dashed line in Fig. 2(d)) is in the range from p to 3p/2. The phase delay of both the magnetic moments oscillations increases with increasing frequency of the alternating magnetic field. When the frequency of the driven field is equal to the high characteristic frequency 2.800 GHz, the amplitudes of the two forced oscillations increase with increasing of time, which is similar to the case in the usual resonance. However, the phase delays of the * * two moments oscillation are 3p/2 for M 1 , and p/2 for M 2 . Furthermore, when the frequency of the alternating field is slightly larger than 2.800 GHz, from Fig. 2(e) and (f), we * can see that the phase delay of the M 1 oscillation (dotdashed line in Fig. 2(f)) is in the range 3p/2–2p. We have also investigated the effects of the value of either b or * * (j h 2 j j h 1 j). It is found that increasing the value of b may highly shift the high characteristic frequency of the system,
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magnetic
susceptibility
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(a)
3 0 −3
(b)
3 0 −3
1.5
0
1
2 frequency (GHz) 1.5
(c)
3
4
(e)
amplitude (a.u.)
0.5 0 −1.5 2
−0.5 −1.5
(d)
5 0
0 −2 −4 390 392
(f )
−5 394 396 time (ns)
398
400
−10 390
392
394 396 time (ns)
398
400
Fig. 2. (a) and (b) Calculated complex magnetic susceptibilities w~ 1xx for layer 1 and w~ 2xx for layer 2. Here, the solid lines indicate the real*part w0 and the of the M 1 (dot-dashed dashed lines refer to the imaginary part, w00 . (c) Alternating magnetic field with frequency of 2.785 GHz, (d) Oscillation curves * * * line) and M 2 (dotted line). (e) Alternating magnetic field with frequency of 2.810 GHz, and (f) Oscillation curves of the M 1 (dot-dashed line) and M 2 (dotted line).
but not alter the anomalous forced oscillation characters; * * and increasing the difference of (j h 2 j j h 1 j) may enhance the value of the complex magnetic susceptibility near the frequency of the anomalous resonance. Then, we demonstrate similar phenomena exist in the systems formed by two moments which have different ferromagnetic resonance frequency in their uncoupled state. The typical results obtained are shown in Fig. 3. In the calculation, the ferromagnetic resonance frequencies of * * M 1 and M 2 are arbitrarily taken as g1 ðH 0 þ H eff u2;1 Þ ¼ 1:000 GHz and g2 ðH 0 þ H eff u2;2 Þ ¼ 1:500 GHz, respectively. g1 bM 02 and g2 bM 01 are taken as g1 bM 02 ¼ g2 bM 01 ¼ 0:2 GHz. The system has two characteristic frequencies of 1.122 and 1.762 GHz, respectively. It is also found that the imaginary part w001xx (dashed line in Fig. 3(a)) of magnetic susceptibility w~ 1xx for layer 1 near high characteristic frequency 1.762 GHz is negative unusually. The anomalous characters are also related to the fact that phase delay of the anomalous oscillation is larger than p, which can be directly seen from Fig. 3(c)–(f). Fig. 3(c) and (d) demonstrate the case that the system is driven by the magnetic field (c) with frequency of 1.750 GHz, which is slightly smaller than the high characteristic frequency of * 1.762 GHz. It is shown that oscillation of M 2 (dotted line in Fig. 3(d)) has a phase delay with value smaller than p/2; * however, the phase delay of the M 1 oscillation (dot-dashed line in Fig. 3(d)) is in the range p–3p/2. When the frequency of the driven field is slightly larger than 1.762 GHz, from
*
Fig. 3(e) and (f), we can see that the phase delay of the M 1 oscillation (dot-dashed in Fig. 3(f)) is in the range 3p/2–2p. Physical mechanism of the anomalous forced oscillation may be explained in a two-step driven model. In both the cases mentioned above, the high characteristic frequency is larger than the ferromagnetic resonance frequencies of the two magnetic moments. When frequency of the alternating magnetic *field is near the high characteristic frequency, moment M 2 oscillation driven by the alternating magnetic * field h 2 exp ðjotÞ has a phase delay in the range p/2–p. Meanwhile, the coupling between the * two moments decreases the value of the phase delay* of M 2 oscillation, which induces the phase delay of the M 2 oscillation to be less than p/2 even when the frequency of the alternating magnetic field is slightly smaller than the high characteristic frequency (see dotted lines, Fig. 2(d) and Fig. 3(d)). On the * other hand, the oscillation amplitude of M is larger than 2 * that of the moment M 1 when only taking into account the effect of the alternating magnetic fields; thus, the moment * * M 1 is driven*by both the magnetic field h 1 exp ðjotÞ and the moment M 2 . However, it is noted that the phase delay * of the M 1 oscillation is dominated by the driving of the * magnetic moment M , from the 2 which can be seen directly * * case in which j h j is equal to zero, thus M is driven by 1 2 * * * h 2 exp ðjotÞ, and then M 2 forcing M 1 to oscillate and the * two steps driven make the phase delay of the M 1 oscillation to be in the range p–2p. In addition, the negative imaginary part of the magnetic susceptibility means outputting the
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susceptibility
372
(a)
4.5 0 −4.5
magnetic
−9 9 (b) 4.5 0 −4.5 −9 0
1
2
3
frequency (GHz)
amplitude (a. u.)
1.5
1.5
(c)
0
0 −1.5 (d) 10 0 −10 −20 390
(e)
392
394 396 time (ns)
398
400
−1.5 15 (f ) 5 −5 −15 −25 390 392
394 396 time (ns)
398
400
Fig. 3. (a) and (b) are similar to the cases shown in Fig. 2(a) and (b), respectively, the only change is that, here, the two moments’ ferromagnetic resonance * frequencies are 1.000 and 1.500 GHz, respectively, (c) Alternating magnetic field with frequency of 1.750 GHz, (d) Oscillation * curves of the M 1 (dot-dashed * * line) and M 2 (dotted line). (e) The alternating magnetic field with frequency of 1.780 GHz, and (f) Oscillation curves of the M 1 (dot-dashed line) and M 2 (dotted line).
energy; we can*see that the energy comes from the driving by the moment M 2 . On the other hand, the sum of imaginary part of the two magnetic susceptibilities is not negative. So, here, no thermodynamic principles are being violated. 3.2. Effects on the electromagnetic wave propagation properties We only consider incident electromagnetic wave propagating in the direction of the magnetization. According to Maxwell’s equations, here, electromagnetic wave is either right circularly polarized (RCP) or left circularly polarized (LCP). For the wave with propagation wave vector k~ ¼ k0 jk00 , its electric and magnetic fields can be expressed as * *
*ðÞ
E ð r ; tÞ ¼ E 0 * *
* ðÞ
H ð r ; tÞ ¼ H 0 *ðÞ
exp ðjot jk0 z k00 zÞ, exp ðjot jk0 z k00 zÞ, * ðÞ
(20) (21)
^ In Eqs. (20) and where E 0 ¼ x^ jy^ and H 0 ¼ x^ jy. (21) the signs of k0 and k00 can both be positive or negative depending on the directions of the wave vector and the energy flow. Here, we assume that the direction of the wave vector is in the positive direction of the z-axis, i.e., we assume k0 40 in Eqs. (20) and (21), but the sign of k00 still can be positive or negative. In this case, if k00 40, the phase velocity and energy flow are in the same directions, this is the usual case for right-handed materials. In contrast, if k00 o0, the phase velocity and energy flow are in opposite
*
*
*
directions, and E , H and k will form a left-handed triplet of vectors. This is the case for LHMs. So, for incident waves of a given frequency o, we can determine whether wave propagations in the materials is right-handed or lefthanded through the relative sign changes of k0 and k00 [7,9]. We shall investigate unique effects of the magnetic susceptibility induced by anomalous ferromagnetic resonance on electromagnetic wave propagation properties by ~ As an example, calculation of propagation wave vector k. the magnetic susceptibility obtained in the second case (shown in Fig. 3) are adopted. On the other hand, we assume that the studied film is metallic. According to the Drude–Lorentz model, the permittivity can be written as [7,9,18] e~eff ¼ e0eff je00eff ¼ 1
o2p , o2 jo=t
(22)
where op is the plasma frequency and t is the relaxation time. We only consider incident electromagnetic waves propagating in the direction perpendicular to the film plane. Therefore, the complex propagation wave vectors for RCP in layer 1 and 2 of the film are [7,9,18] o pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jðae þ am;i Þ 0 00 ~ ki ¼ ki jki ¼ j~eeff jjm~ iRCP j exp c 2 ~ jki j exp ðjak;i Þ, ð23Þ where m~ iRCP ¼ m~ ixx m~ ixy is the permeability corresponding to the RCP waves, i ¼ 1 (or 2), ae , am;i are the damping
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J. Chen et al. / Journal of Magnetism and Magnetic Materials 302 (2006) 368–374
5e+05 4e+05 3e+05 2e+05 1e+05 0 −1e+05
1.5
(a)
(d)
1 0.5 0 −0.5
K1(a.u.)
(b)
(e)
2000
5
0
0
−2000
−5
−4000 K2(a.u.)
373
−10
(c)
(f )
6
2000 0
2
−2000 −4000
0
2 Frequency (GHz)
−2
0
2 Frequency (GHz)
Fig. 4. (a) Effective permittivity e~eff of the metallic film with op ¼ 1000 GHz, (b) and (c) corresponding wave vectors k~1 in layer 1 and k~2 in layer 2. The solid lines indicate the real part e0eff , k01 and k02 , and the dashed lines refer the imaginary part e00eff , k001 and k002 , respectively. (d), (e) and (f) are similar to the cases of (a), (b) and (c), respectively, only the op reduced down to 0.5 GHz.
angles of the permittivity and permeability, respectively. ak;i ¼ ðae þ am;i Þ=2. It is seen that while p=2oak op or 3p=2oak o2p, sign of the k0 is opposite to that of the k00 , and the electromagnetic wave propagating in this medium is left-handed. For usual LHM, p=2oae op, p=2oam op, thus p=2oak op. It is interesting to note that for the case discussed in Ref. [9] by Wu et al., where e0 5e00 , ae p=2, it is easy to get p=2oak op when p=2oam op; thus, even when e0 is not negative, the corresponding medium may still be a LHM. Since the value of op may be modified by controlling the density of the electron, etc., we first assume that op ¼ 1000 GHz and op t ¼ 100, the corresponding e~eff and complex wave vectors k~1 and k~2 are shown in Fig. 4(a), (b) and (c), respectively. It is seen that, here, the real part e0 eff of the e~eff is less than zero. The sign change of the imaginary part of the vectors with frequency of 1.122 GHz results in the change in direction of the energy flow, i.e., LHM properties. It is also noted that, unusually, near the low-frequency side of 1.762 GHz, where ferromagnetic * resonance of the M 1 is abnormal, layer 1 is still a LHM; however, layer 2 becomes LHM near the high-frequency side of 1.762 GHz. These phenomena can be understood by considering the different value of the ak near frequency of 1.762 GHz. For layer 1, in the low-frequency side of 1.762 GHz, ae 4p=2ð p=2Þ, poam o3p=2 and p=2oak op; and for layer 2, in the high-frequency side of 1.762 GHz, ae 4p=2ð p=2Þ, p=2oam op and p=2oak op. We are also interested in the case where 0oae op=2 (i.e., e0eff 40). However, as poam o2p leads p=2oak op, the electromagnetic wave propagating in this medium is possible to be
left-handed. As a supposed model, we assume that op may be reduced down to 0.5 GHz (e.g., by using the way described by Pendry in Ref. [2]), and do not change the value of relaxation time t, then the real part e0eff of e~eff is positive (see Fig. 4(d)). It is interesting to find from Fig. 4(e) and (f) that in a narrow range near the lowfrequency side of 1.762 GHz, a passband exists for the film, and layer 1 is a LHM. It is stressed that, in contrast to the usual LHM, here, only the real part of permeability is negative.
4. Conclusion In summary, a theoretical basis for an anomalous ferromagnetic resonance behavior has been provided, which is induced by the coupling between the magnetic moments. Due to the unique sign properties of the magnetic susceptibility, a LHM may be formed near the low-frequency side of the resonance. Furthermore, a LHM, in which only the real part of permeability is less than zero, is predicted. Experimental quantifications of the predictions of this paper should also motivate further theoretical progress.
Acknowledgments J. Chen acknowledges support in this work from the Postdoctoral Support Program in Scientific Research of Jiangsu Province, No. 0204003409.
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Journal of Magnetism and Magnetic Materials 302 (2006) 368–374 www.elsevier.com/locate/jmmm
Possible existence of a new type of left-handed materials in coupled ferromagnetic bilayer films Jiangwei Chena,b,, Baoshan Zhanga, Dongming Tanga, Yi Yanga, Weidong Xua,c, Huaixian Lua a
National Laboratory of Solid State Microstructures, Department of Physics, Nanjing University, Nanjing 210093, PR China b Nanjing Artillery Academy, Nanjing 211132, PR China c Engineering Institute of Engineer Corps., PLA University of Science and Technology, Nanjing 210007, PR China Received 26 July 2005; received in revised form 4 September 2005 Available online 24 October 2005
Abstract On the basis of Landau–Lifshitz–Gilbert (LLG) equation, an anomalous ferromagnetic resonance behavior is demonstrated in detail. Coupling between the magnetic moments produces a 3p/2 phase delay for one of the moments ferromagnetic resonance unusually, thus leads the sign of magnetic susceptibility w~ ¼ w0 jw00 to be opposite to that induced by the usual ferromagnetic resonance. Consequently, a left-handed material (LHM) may be formed near the low-frequency side of the resonance. Particularly, a LHM with negative value of real part of permeability only is predicted. r 2005 Elsevier B.V. All rights reserved. PACS: 73.20.Mf; 75.70.Ak; 42.70.a; 75.75.+a; 76.20.+q Keywords: Ferromagnetic resonance; Left-handed material
1. Introduction In 1960s, Veselago [1] predicted a strong effect of sign of both the permittivity and the permeability on electromagnetic wave propagation properties, i.e., if the real part of both the permittivity e~ and the magnetic permeability m~ are negative, the phase velocity direction of an electromagnetic wave will be opposite to its energy flow direction. In these * * media, *the electric field E , magnetic field H , and wave vector k form a left-handed triplet of vectors, instead of a right-handed triplet observed in conventional materials. Such media are called left-handed materials (LHM). Based upon the early ideas of Pendry et al. [2,3], Smith et al. [4] found negative real part of both effective m~ eff and e~eff simultaneously in a system consisting of spilt ring resonators and metal wires, and Shelby et al. [5] have observed negative values of the index of refraction n in this
Corresponding author. Tel.: +0253724979; fax: +0253593011.
E-mail address: [email protected] (J. Chen). 0304-8853/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2005.09.040
system. This discovery reinvigorated the search for LHM [6–12]. Recently, an increased attention has been paid to the microwave performance of thin ferromagnetic films [7–15]. The interest is also rooted on possible application of such films as LHM [7–12]. A number of novel ferromagnetic resonance properties are shown in the nanostructured systems, which may be due to the shape effect, surface effect, interface effect, and the exchange coupling, etc. [13–15]. The frequency dependence of magnetic permeability is typically due to the ferromagnetic resonance. As it is well known, the phase delay of the forced oscillation with respect to the pumping field is usually less than p; thus, the imaginary part of the magnetic permeability is positive. It is also noted that, for all LHMs mentioned above, the real part of both the permittivity e~ and the magnetic permeability m~ are negative, and the frequency range is near the high-frequency side of the ferromagnetic resonance [4–12]. In this paper, we first present the magnetic susceptibility properties of the exchange-coupled bilayer ferromagnetic films. We have investigated the effects of either exchange
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coupling between the moments or the inhomogenous anisotropy effective magnetic field in the ferromagnetic films. It has been shown that an ‘acoustic’ mode and an ‘optic’ mode may exist in the ferromagnetic bilayers [14,15]. However, the previous works focused mainly on the resonance frequency and the total magnetic susceptibility of the system. Here, we shall pay more attention to the magnetic susceptibility of the single layer of the bilayer film. For simplicity, we choose a parallel coupled ferromagnetic bilayer film to demonstrate in detail that, for the ‘optic’ mode, a 3p=2 phase delay exists in one of the moments ferromagnetic resonance. (Similar results can also be obtained in the anti-parallel coupled systems.) Since phase delay of the resonance is usually equal to p/2, the formerly mentioned ferromagnetic resonance is abnormal. We call this phenomenon as an anomalous ferromagnetic resonance. It is interesting to note that, here, the sign of the corresponding magnetic susceptibility w~ ¼ w0 jw00 is significantly different from that induced by the usual ferromagnetic resonance. Consequently, we predict the possible existence of a new type of LHM due to the unique sign properties of the magnetic susceptibility, in which the left-handed propagation properties exist near the low-frequency side of the ferromagnetic resonance, and also, only the real part of the magnetic permeability is negative.
(2) Effective anisotropy energy, " # 2 X 1 M i;jj H eff , 2 u2;i M s i
2. Model
H eff1 ¼ H 0 þ H u2;1 bM 2 ¼ ðH 0 þ H eff u2;1 Þ e z bM 2 ,
A parallel coupled system to be discussed in this paper consists of two ferromagnetic layers and is shown in Fig. 1, in which the magnetization is in the direction perpendicular to the film plane, defined as z-direction, and the film is infinite along x- and y-axes. Under certain conditions, the lowest-order spin wave mode within each layer is close to the uniform precession mode. Thus, for simplicity, two * * single magnetization vectors M 1 (for layer 1) and M 2 (for layer 2) are used to represent the magnetic moment in the layers [14,15]. Usually, the energies involved in the exchange-coupled system are the following [15]:
H eff2 ¼ H 0 þ H u2;2 bM 1 ¼ ðH 0 þ H eff u2;2 Þ e z bM 1 .
(2)
where the z-axis is normal to the film plane and M i;jj is the in-plane component of the magnetization of the ith * eff
layer. H u2;i are effective fields due to the shape, surface, magnetocrystalline, stress, or other anisotropy. Here, * eff
we assume the direction of H u2;i is in the positive direction of the z-axis. (3) Interlayer exchange energy, *
*
* * M1 M2 bM 1 M 2 , A12 M 1M 2
(3)
where A12 is the bilinear exchange energy constant per unit surface area. Here, the sign of A12 (i.e., the sign of b) is chosen as negative for a parallel coupled system. *
Apparently, when H 0 is equal to zero or its direction is in the positive direction of z-axis, the equilibrium position of the system is such that both the two moments stay parallel to the positive direction of z-axis as shown in Fig. 1. The effective static magnetic fields applied in the two magnetic moments are as follows: *
*
* eff
*
*
*
*
*
* eff
*
*
*
(4) (5)
When the magnetic moments are perturbed from their equilibrium orientation, the response of * the two moments to the alternating magnetic fields h 1 exp ðjotÞ and * h 2 exp ðjotÞ are governed by [14–17] *
* * * dM 1 ¼ g1 M 1 ðH eff1 þ h 1 exp ðjotÞÞ dt *
*
M 1 dM 1 þ a1 , M1 dt
ð6Þ
*
(1) Zeeman energy, X * * ðH 0 M i Þ,
(1)
i *
where H 0 is the external field, and i ¼ 1 (or 2) refers to the first (or the second) ferromagnetic layer.
Z Y Layer 1
M1
Layer 2
M2
X
Fig. 1. Bilayer structure and coordinate system used in the model.
* * * dM 2 ¼ g2 M 2 ðH eff2 þ h 2 exp ðjotÞÞ dt *
þ a2
*
M 2 dM 2 , M2 dt
ð7Þ
where g1 and g2 are the gyromagnetic constants, a1 and a2 the Gilbert damping coefficients of the two moments, * * * * respectively. Let M 1 ¼ M 01 e z þ m1 exp ðjotÞ and M 2 ¼ * * * * M 02 e z þ m2 exp ðjotÞ (here, M 01 bjm1 j, M 02 bjm2 j, and in the calculation, we assume M 01 ¼ M 02 ). To obtain the magnetic susceptibility, Eqs. (6) and (7) are linearized as [16] jom1x þ ½g1 ðH 0 þ H eff u2;1 bM 02 Þ þ jo a1 m1y þ 0m2x þ g1 bM 01 m2y ¼ g1 M 01 h1y ,
ð8Þ
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½g1 ðH 0 þ H eff u2;1 bM 02 Þ þ jo a1 m1x þ jom1y
3. Results and discussion
g1 bM 01 m2x þ 0m2y ¼ g1 M 01 h1x ,
ð9Þ 3.1. Magnetic susceptibility
0m1x þ g2 bM 02 m1y þ jom2x þ ½g2 ðH 0 þ H eff u2;2 bM 01 Þ þ jo a2 m2y ¼ g2 M 02 h2y ,
ð10Þ
* eff
g2 bM 02 m1x þ 0m1y ½g2 ðH 0 þ H eff u2;2 bM 01 Þ þ jo a2 m2x þ jom2y ¼ g2 M 02 h2x ,
ð11Þ
and m1z ¼ m2z ¼ 0.
(12)
The solution of the Eqs. (8)–(11) yields the dynamic * * magnetization m1 and m2 . Then the scalar extrinsic dynamic susceptibility of ith layer is defined by *
*
w~ i ¼
w0i
jw00i
¼
mi h i *
,
(13)
j h i j2
and the corresponding magnetic permeability is *
m~ i ¼ 1 þ w~ i ¼ 1 þ
*
mi h i *
.
(14)
j h i j2
The oscillation curves of the magnetic moments are calculated according to the following linearized equations: dm1x 1 ¼ ½g1 a1 ðH 0 þ H eff u2;1 Þm1x dt 1 þ a21 g1 ðH 0 þ H eff u2;1 bM 02 Þm1y g1 bM 01 m2y þ g1 a1 M 01 h1x exp ðjotÞ þ g1 M 01 h1y exp ðjotÞ,
ð15Þ
dm1y 1 ¼ ½g ðH 0 þ H eff u2;1 bM 02 Þm1x dt 1 þ a21 1 g1 a1 ðH 0 þ H eff u2;1 Þm1y þ g1 bM 01 m2x g1 M 01 h1x exp ðjotÞ þ g1 a1 M 01 h1y exp ðjotÞ,
ð16Þ
dm2x 1 ¼ ½g2 bM 02 m1y g2 a2 ðH 0 þ H eff u2;2 Þm2x dt 1 þ a22 g2 ðH 0 þ H eff u2;2 bM 01 Þm2y þ g2 a2 M 02 h2x exp ðjotÞ þ g2 M 02 h2y exp ðjotÞ, ð17Þ dm2y 1 ¼ ½g bM 02 m1x þ g2 ðH 0 þ H eff u2;2 bM 01 Þm2x dt 1 þ a22 2 g2 a2 ðH 0 þ H eff u2;2 Þm2y g2 M 02 h2x exp ðjotÞ þ g2 a2 M 02 h2y exp ðjotÞ, and dm1z dm2z ¼ ¼ 0. dt dt
First, we consider the case in which the two moments have same value of ferromagnetic resonance frequency in
ð18Þ
(19)
their uncoupled state. Since the magnitude of H u2;i and the effective field of the exchange coupling may be in the order of 1–10 kG s [15], we assume that g1 ðH 0 þ H eff u2;1 Þ ¼ eff g2 ðH 0 þ H u2;2 Þ ¼ 1:000 GHz, and the coupling parameter b is taken as g1 bM 02 ¼ g2 bM 01 ¼ 0:9 GHz arbitrarily. The two Gilbert damping coefficients are taken as a1 ¼ a2 ¼ 0:01 for commonality and conventionally. Thus, the system has two characteristic frequencies of 1.000 and 2.800 GHz, respectively. Furthermore, we assume that the electromagnetic wave propagates in the positive direction * of the z-axis, considering absorption of*the moment M 2 , * and the amplitude j h 1 j is smaller than j h 2 j. The obtained complex magnetic susceptibilities w~ 1xx ¼ w01xx jw001xx for layer 1 and w~ 2xx ¼ w02xx jw002xx for layer 2 are shown in Fig. 2(a) and (b), respectively. It is interesting to find that the imaginary part w001xx (dashed line in Fig. 2(a)) near 2.800 GHz is negative, which is apparently opposite to the positive w001xx and w002xx near 1.00 GHz and positive w002xx near 2.800 GHz as shown in Fig. 2(a) and (b), respectively. In addition, the change of the real part w01xx (solid line in Fig. 2(a)) near 2.800 GHz with increasing frequency is also different from that of the other three, i.e., w01xx and w02xx near 1.00 GHz, and w02xx near 2.800 GHz. The anomalous characters mentioned above are related to the * fact that phase delay of the oscillation of moment M 1 near 2.800 GHz is larger than p with respect to the alternating magnetic fields, which can be directly seen from Fig. 2(c)–(f). Fig. 2(c) and (d) demonstrates the case in which the system is driven by the magnetic field (c) with frequency of 2.785 GHz, which is slightly smaller than the high characteristic * frequency 2.800 GHz. It is shown that oscillation of M 2 (dotted line in Fig. 2(d)) has a phase delay with a value less than p/2; however, the phase delay * of the M 1 oscillation (dot-dashed line in Fig. 2(d)) is in the range from p to 3p/2. The phase delay of both the magnetic moments oscillations increases with increasing frequency of the alternating magnetic field. When the frequency of the driven field is equal to the high characteristic frequency 2.800 GHz, the amplitudes of the two forced oscillations increase with increasing of time, which is similar to the case in the usual resonance. However, the phase delays of the * * two moments oscillation are 3p/2 for M 1 , and p/2 for M 2 . Furthermore, when the frequency of the alternating field is slightly larger than 2.800 GHz, from Fig. 2(e) and (f), we * can see that the phase delay of the M 1 oscillation (dotdashed line in Fig. 2(f)) is in the range 3p/2–2p. We have also investigated the effects of the value of either b or * * (j h 2 j j h 1 j). It is found that increasing the value of b may highly shift the high characteristic frequency of the system,
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susceptibility
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(a)
3 0 −3
(b)
3 0 −3
1.5
0
1
2 frequency (GHz) 1.5
(c)
3
4
(e)
amplitude (a.u.)
0.5 0 −1.5 2
−0.5 −1.5
(d)
5 0
0 −2 −4 390 392
(f )
−5 394 396 time (ns)
398
400
−10 390
392
394 396 time (ns)
398
400
Fig. 2. (a) and (b) Calculated complex magnetic susceptibilities w~ 1xx for layer 1 and w~ 2xx for layer 2. Here, the solid lines indicate the real*part w0 and the of the M 1 (dot-dashed dashed lines refer to the imaginary part, w00 . (c) Alternating magnetic field with frequency of 2.785 GHz, (d) Oscillation curves * * * line) and M 2 (dotted line). (e) Alternating magnetic field with frequency of 2.810 GHz, and (f) Oscillation curves of the M 1 (dot-dashed line) and M 2 (dotted line).
but not alter the anomalous forced oscillation characters; * * and increasing the difference of (j h 2 j j h 1 j) may enhance the value of the complex magnetic susceptibility near the frequency of the anomalous resonance. Then, we demonstrate similar phenomena exist in the systems formed by two moments which have different ferromagnetic resonance frequency in their uncoupled state. The typical results obtained are shown in Fig. 3. In the calculation, the ferromagnetic resonance frequencies of * * M 1 and M 2 are arbitrarily taken as g1 ðH 0 þ H eff u2;1 Þ ¼ 1:000 GHz and g2 ðH 0 þ H eff u2;2 Þ ¼ 1:500 GHz, respectively. g1 bM 02 and g2 bM 01 are taken as g1 bM 02 ¼ g2 bM 01 ¼ 0:2 GHz. The system has two characteristic frequencies of 1.122 and 1.762 GHz, respectively. It is also found that the imaginary part w001xx (dashed line in Fig. 3(a)) of magnetic susceptibility w~ 1xx for layer 1 near high characteristic frequency 1.762 GHz is negative unusually. The anomalous characters are also related to the fact that phase delay of the anomalous oscillation is larger than p, which can be directly seen from Fig. 3(c)–(f). Fig. 3(c) and (d) demonstrate the case that the system is driven by the magnetic field (c) with frequency of 1.750 GHz, which is slightly smaller than the high characteristic frequency of * 1.762 GHz. It is shown that oscillation of M 2 (dotted line in Fig. 3(d)) has a phase delay with value smaller than p/2; * however, the phase delay of the M 1 oscillation (dot-dashed line in Fig. 3(d)) is in the range p–3p/2. When the frequency of the driven field is slightly larger than 1.762 GHz, from
*
Fig. 3(e) and (f), we can see that the phase delay of the M 1 oscillation (dot-dashed in Fig. 3(f)) is in the range 3p/2–2p. Physical mechanism of the anomalous forced oscillation may be explained in a two-step driven model. In both the cases mentioned above, the high characteristic frequency is larger than the ferromagnetic resonance frequencies of the two magnetic moments. When frequency of the alternating magnetic *field is near the high characteristic frequency, moment M 2 oscillation driven by the alternating magnetic * field h 2 exp ðjotÞ has a phase delay in the range p/2–p. Meanwhile, the coupling between the * two moments decreases the value of the phase delay* of M 2 oscillation, which induces the phase delay of the M 2 oscillation to be less than p/2 even when the frequency of the alternating magnetic field is slightly smaller than the high characteristic frequency (see dotted lines, Fig. 2(d) and Fig. 3(d)). On the * other hand, the oscillation amplitude of M is larger than 2 * that of the moment M 1 when only taking into account the effect of the alternating magnetic fields; thus, the moment * * M 1 is driven*by both the magnetic field h 1 exp ðjotÞ and the moment M 2 . However, it is noted that the phase delay * of the M 1 oscillation is dominated by the driving of the * magnetic moment M , from the 2 which can be seen directly * * case in which j h j is equal to zero, thus M is driven by 1 2 * * * h 2 exp ðjotÞ, and then M 2 forcing M 1 to oscillate and the * two steps driven make the phase delay of the M 1 oscillation to be in the range p–2p. In addition, the negative imaginary part of the magnetic susceptibility means outputting the
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susceptibility
372
(a)
4.5 0 −4.5
magnetic
−9 9 (b) 4.5 0 −4.5 −9 0
1
2
3
frequency (GHz)
amplitude (a. u.)
1.5
1.5
(c)
0
0 −1.5 (d) 10 0 −10 −20 390
(e)
392
394 396 time (ns)
398
400
−1.5 15 (f ) 5 −5 −15 −25 390 392
394 396 time (ns)
398
400
Fig. 3. (a) and (b) are similar to the cases shown in Fig. 2(a) and (b), respectively, the only change is that, here, the two moments’ ferromagnetic resonance * frequencies are 1.000 and 1.500 GHz, respectively, (c) Alternating magnetic field with frequency of 1.750 GHz, (d) Oscillation * curves of the M 1 (dot-dashed * * line) and M 2 (dotted line). (e) The alternating magnetic field with frequency of 1.780 GHz, and (f) Oscillation curves of the M 1 (dot-dashed line) and M 2 (dotted line).
energy; we can*see that the energy comes from the driving by the moment M 2 . On the other hand, the sum of imaginary part of the two magnetic susceptibilities is not negative. So, here, no thermodynamic principles are being violated. 3.2. Effects on the electromagnetic wave propagation properties We only consider incident electromagnetic wave propagating in the direction of the magnetization. According to Maxwell’s equations, here, electromagnetic wave is either right circularly polarized (RCP) or left circularly polarized (LCP). For the wave with propagation wave vector k~ ¼ k0 jk00 , its electric and magnetic fields can be expressed as * *
*ðÞ
E ð r ; tÞ ¼ E 0 * *
* ðÞ
H ð r ; tÞ ¼ H 0 *ðÞ
exp ðjot jk0 z k00 zÞ, exp ðjot jk0 z k00 zÞ, * ðÞ
(20) (21)
^ In Eqs. (20) and where E 0 ¼ x^ jy^ and H 0 ¼ x^ jy. (21) the signs of k0 and k00 can both be positive or negative depending on the directions of the wave vector and the energy flow. Here, we assume that the direction of the wave vector is in the positive direction of the z-axis, i.e., we assume k0 40 in Eqs. (20) and (21), but the sign of k00 still can be positive or negative. In this case, if k00 40, the phase velocity and energy flow are in the same directions, this is the usual case for right-handed materials. In contrast, if k00 o0, the phase velocity and energy flow are in opposite
*
*
*
directions, and E , H and k will form a left-handed triplet of vectors. This is the case for LHMs. So, for incident waves of a given frequency o, we can determine whether wave propagations in the materials is right-handed or lefthanded through the relative sign changes of k0 and k00 [7,9]. We shall investigate unique effects of the magnetic susceptibility induced by anomalous ferromagnetic resonance on electromagnetic wave propagation properties by ~ As an example, calculation of propagation wave vector k. the magnetic susceptibility obtained in the second case (shown in Fig. 3) are adopted. On the other hand, we assume that the studied film is metallic. According to the Drude–Lorentz model, the permittivity can be written as [7,9,18] e~eff ¼ e0eff je00eff ¼ 1
o2p , o2 jo=t
(22)
where op is the plasma frequency and t is the relaxation time. We only consider incident electromagnetic waves propagating in the direction perpendicular to the film plane. Therefore, the complex propagation wave vectors for RCP in layer 1 and 2 of the film are [7,9,18] o pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jðae þ am;i Þ 0 00 ~ ki ¼ ki jki ¼ j~eeff jjm~ iRCP j exp c 2 ~ jki j exp ðjak;i Þ, ð23Þ where m~ iRCP ¼ m~ ixx m~ ixy is the permeability corresponding to the RCP waves, i ¼ 1 (or 2), ae , am;i are the damping
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5e+05 4e+05 3e+05 2e+05 1e+05 0 −1e+05
1.5
(a)
(d)
1 0.5 0 −0.5
K1(a.u.)
(b)
(e)
2000
5
0
0
−2000
−5
−4000 K2(a.u.)
373
−10
(c)
(f )
6
2000 0
2
−2000 −4000
0
2 Frequency (GHz)
−2
0
2 Frequency (GHz)
Fig. 4. (a) Effective permittivity e~eff of the metallic film with op ¼ 1000 GHz, (b) and (c) corresponding wave vectors k~1 in layer 1 and k~2 in layer 2. The solid lines indicate the real part e0eff , k01 and k02 , and the dashed lines refer the imaginary part e00eff , k001 and k002 , respectively. (d), (e) and (f) are similar to the cases of (a), (b) and (c), respectively, only the op reduced down to 0.5 GHz.
angles of the permittivity and permeability, respectively. ak;i ¼ ðae þ am;i Þ=2. It is seen that while p=2oak op or 3p=2oak o2p, sign of the k0 is opposite to that of the k00 , and the electromagnetic wave propagating in this medium is left-handed. For usual LHM, p=2oae op, p=2oam op, thus p=2oak op. It is interesting to note that for the case discussed in Ref. [9] by Wu et al., where e0 5e00 , ae p=2, it is easy to get p=2oak op when p=2oam op; thus, even when e0 is not negative, the corresponding medium may still be a LHM. Since the value of op may be modified by controlling the density of the electron, etc., we first assume that op ¼ 1000 GHz and op t ¼ 100, the corresponding e~eff and complex wave vectors k~1 and k~2 are shown in Fig. 4(a), (b) and (c), respectively. It is seen that, here, the real part e0 eff of the e~eff is less than zero. The sign change of the imaginary part of the vectors with frequency of 1.122 GHz results in the change in direction of the energy flow, i.e., LHM properties. It is also noted that, unusually, near the low-frequency side of 1.762 GHz, where ferromagnetic * resonance of the M 1 is abnormal, layer 1 is still a LHM; however, layer 2 becomes LHM near the high-frequency side of 1.762 GHz. These phenomena can be understood by considering the different value of the ak near frequency of 1.762 GHz. For layer 1, in the low-frequency side of 1.762 GHz, ae 4p=2ð p=2Þ, poam o3p=2 and p=2oak op; and for layer 2, in the high-frequency side of 1.762 GHz, ae 4p=2ð p=2Þ, p=2oam op and p=2oak op. We are also interested in the case where 0oae op=2 (i.e., e0eff 40). However, as poam o2p leads p=2oak op, the electromagnetic wave propagating in this medium is possible to be
left-handed. As a supposed model, we assume that op may be reduced down to 0.5 GHz (e.g., by using the way described by Pendry in Ref. [2]), and do not change the value of relaxation time t, then the real part e0eff of e~eff is positive (see Fig. 4(d)). It is interesting to find from Fig. 4(e) and (f) that in a narrow range near the lowfrequency side of 1.762 GHz, a passband exists for the film, and layer 1 is a LHM. It is stressed that, in contrast to the usual LHM, here, only the real part of permeability is negative.
4. Conclusion In summary, a theoretical basis for an anomalous ferromagnetic resonance behavior has been provided, which is induced by the coupling between the magnetic moments. Due to the unique sign properties of the magnetic susceptibility, a LHM may be formed near the low-frequency side of the resonance. Furthermore, a LHM, in which only the real part of permeability is less than zero, is predicted. Experimental quantifications of the predictions of this paper should also motivate further theoretical progress.
Acknowledgments J. Chen acknowledges support in this work from the Postdoctoral Support Program in Scientific Research of Jiangsu Province, No. 0204003409.
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