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Magnetostatic modes of a ferromagnetic superlattice in an oblique applied field
ARTICLE IN PRESS Physica B 404 (2009) 2086–2090
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Magnetostatic modes of a ferromagnetic superlattice in an oblique applied field Rui-Hua Zhu , Hong-Yan Peng, Mei-Heng Zhang, Yu-Qiang Chen Department of Physics, Mudanjiang Teachers College, Mudanjiang 157012, People’s Republic of China
a r t i c l e in f o
a b s t r a c t
Article history: Received 18 November 2008 Received in revised form 15 March 2009 Accepted 30 March 2009
The dispersion relations for both bulk and surface magnetostatic waves are obtained in a case where the applied field makes an arbitrary angle with the interfaces of the structure and the static magnetization is tipped out of the interfaces. With the transfer matrix treatment, we find infinite sets of bulk mode bands and infinite branches of surface modes. Both the bulk modes and the surface modes are sensitive to the direction of the static magnetization. Many new properties of the magnetostatic modes are thus presented. & 2009 Elsevier B.V. All rights reserved.
PACS: 75.70.Cn 75.40.Gb 75.30.Ds Keywords: Ferromagnetic superlattice Magnetostatic modes Oblique applied field
1. Introduction The long-wavelength collective excitations in a magnetic superlattice, including the magnetostatic modes or retarded modes, are intriguing subjects. Early works on magnetostatic modes in a ferromagnetic superlattice were carried out by Camley et al. [1], and Grunberg and Mike [2]. They calculated the collective magnetostatic modes in a ferromagnetic/nonmagnetic superlattice with the magnetization and applied field parallel to the interfaces (the so-called Voigt configuration). Experimental results from such a system were derived using Brillouin light scattering by Grimsditch et al. [3,4]. After that, there were many theoretical studies on collective excitations in various superlattices using different techniques, such as the transfer matrix formalism [5–7], effective-medium theory [8–12] and recurrent relations [13,14]. As shown in previous discussions, the spin configuration is an important factor for magnetostatic modes in magnetic superlattices. For the Voigt configuration, a well-known finding is that the ferromagnetic/nonmagnetic superlattice supports magnetostatic surface modes only if the volume fraction of magnetic materials exceeds 50%. However, this conclusion is not applicable for the transverse field configuration, since there are no magnetostatic modes in a perfect semi-infinite ferromagnetic/ nonmagnetic superlattice [15,16]. Corresponding author. Tel.: +86 0453 6486595.
E-mail addresses: [email protected], [email protected] (R.-H. Zhu). 0921-4526/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2009.03.046
The oblique applied field (OAF) configuration receives less consideration, and original work on this configuration includes the magnetostatic surface modes for a ferromagnetic slab [17] and surface polaritons for an antiferromagnet [18]. However, due to the low symmetry of this configuration, the magnetostaic modes of superlattice structures are usually addressed with an effectivemedium approximation. As an example, Lu¨ et al. calculated the magnetostatic modes in a lateral ferromagnetic superlattice where the applied field makes an arbitrary angle with the interfaces by deriving an effective-medium permeability tensor [19]. Subsequently, this method was extended to a metallic magnetic superlattice including the existence of eddy currents [20]. In this paper, we examine the problem of collective magnetostatic modes in a ferromagnetic/nonmagnetic superlattice, where the superlattice is in an OAF. The dispersion relations for the magnetostatic modes are calculated by applying Bloch’s theorem in spite of the low symmetry configuration. As we will see, the direction of the applied field has great impact on the magnetostatic surface modes and bulk modes, and many new properties of the magnetostatic modes are revealed.
2. Dispersion relations for bulk and surface modes The aim of this section is to derive the magnetostatic surface modes and bulk modes in the superlattice structure. For this
ARTICLE IN PRESS R.-H. Zhu et al. / Physica B 404 (2009) 2086–2090
2087
purpose, we employ a macroscopic theory, which is usually used to study the interface effects of multilayer structures [21,22]. However, this problem also can be treated with microscope theory, as demonstrated previously [23–25]. The waves are modeled in the magnetostatic region using Maxwell’s equations:
mxz ¼ mzx ¼ om oi sin yM cos yM =ðo2i o2 Þ, myz ¼ mzy ¼ om o sin yM =ðo2i o2 Þ. The related parameters are om ¼ 4pgM s and oi ¼ gHi .
r B ¼ 0;
"
r h ¼ 0.
(1)
The geometry of the superlattice is shown in Fig. 1. It consists of alternating ferromagnetic and nonmagnetic layers. The thickness of a ferromagnetic layer is d1 , while d2 represents the thickness of a nonmagnetic layer. Thus, the periodic length of the superlattice is L, which equals d1 plus d2. The magnetic volume ratio f1 is defined by f 1 ¼ d1 =L. The external magnetic field is in the x–z plane and makes an angle y with z-axis. We assume that the superlattice is quite large in the y and z directions, such that the demagnetizing field is directed along the x-axis. We also suppose the superlattice is well magnetized such that the static magnetizations of all layers are parallel to the internal field Hi. Thus, Hi can be calculated by a vector addition between the external field H0 and the demagnetizing field Hd while angle yM can be derived using the following: tan yM ¼ ðHo sin y 4pMs sin yM Þ=ðHo cos yÞ:
(2)
If y ¼ yM ¼ 0, the static magnetizations are parallel to the z-axis, which is the case in the Voigt configuration and the corresponding Voigt direction is along the y-axis. The magnitude of the internal field Hi is a function of the angle yM, with the expression Hi ¼ ½ðHo sin y 4pM s sin yM Þ2 þ ðHo sin yÞ2 1=2 .
(3)
The magnetic material’s long-wave dynamics is governed by the frequency-dependent magnetic permeability tensor l(o), which is calculated by linearizing the magnetization’s torque equation, and it is given by 0 mxx imxy mxz 1 B im myy imyz C (4) lðoÞ ¼ @ yx A, mzx imzy mzz
mxx ¼ 1 þ om oi cos2 yM =ðo2i o2 Þ, mzz ¼ 1 þ om oi sin2 yM =ðo2i o2 Þ, mxy ¼ myx ¼ om o cos yM =ðo2i o2 Þ,
myy ¼ 1 þ om oi =ðo2i o2 Þ,
From Eqs. (1) and (4), we can derive an equation that the scalar potential must satisfy:
mxx
# @2 @2 @2 @2 þ m þ m þ 2 m F ¼ 0. yy zz xz @x@z @x2 @y2 @z2
(5)
For the bulk magnetostatic modes, we assume a plane-wave solution:
F ¼ CðxÞ expbiðqk rk otÞc.
(6)
The vector rk is the projection of position vector r on y–z plane and qk is parallel to the y–z plane, so qk rk ¼ qy y þ qz z ¼ qjj ðcos fy þ sin fzÞ. The function cðxÞ in Eq. (6) is different between magnetic layers and nonmagnetic layers. For the region nLoxonL þ d1 , attributed to a magnetic layer, it is set by
cðxÞ ¼ eiq? nd1 eiq? ðxnLÞ ðAn eaðxnLÞ þ Bn eaðxnLd1 Þ Þ.
(7)
The factor exp½iq? ðx nLÞ in Eq. (7) is for OAF, which is after a method proposed by Rahman and Mills [17]. The factor expðiq? nd1 Þ in Eq. (7) serves to ensure that the determinant of the transfer matrix equals unity. However, this factor results in a phase shift from one period to the next. For the neighboring nonmagnetic layer nL þ d1 oxoðn þ 1ÞL; the scalar potential is ordinary, and is expressed as
cðxÞ ¼ eiq? nd1 fC n eqjj ðxnLd1 Þ þ Dn eqjj ½xðnþ1ÞL g,
(8)
where q? and a in Eq. (7) are solved from Eq. (5), where q? ¼ qjj ðmxz sin f=mxx Þ,
(9)
and
a ¼ qjj f½m2xz sin2 f þ mxx ðmyy cos2 f þ mzz sin2 fÞ=m2xx g1=2 ¼ qjj ½ðo2 O2þ Þðo2 O2 Þ=ðo2 O2 Þ2 1=2 , 2 i
1=2
2 i
(10) 2
1=2
where Oþ ¼ ðo þ om oi Þ , O ¼ ðo þ om oi cos yM Þ and O ¼ ðo2i þ om oi cos2 yM cos2 fÞ1=2 . It is clear that Oþ and O are the solutions for myy ðoÞ ¼ 0 and mxx ðoÞ ¼ 0, respectively. If a is a real number, we determine that o4Oþ or ooO, and that the magnetostatic modes in the layer are of the surface type. Otherwise, if a is a pure imaginary number, while o is in the range from O to Oþ , the modes are bulk waves. With the continuity of the magnetic potential F and the normal component Bx at the interfaces x ¼ nL þ d1 and x ¼ ðn þ 1ÞL, one may find a matrix equation for the amplitudes of waves in neighbor periods, ! ! Anþ1 An ¼T , (11) Bnþ1 Bn and T¼
1
Gþ Gþ Gþþ G expðad1 ÞðGþ Gþ Eþ Gþþ G E Þ 2 sinhðqjj d2 ÞGþþ Gþ
2 sinhðqjj d2 ÞG Gþ
!
expðad1 ÞðGþ Gþ E Gþþ G Eþ Þ
,
(12) with the notations defined by G ¼ qjj cos fmxy mxx a qjj and E ¼ expðqjj d2 Þ. The dispersion relations may then be acquired by applying Bloch’s theorem, where Fig. 1. A superlattice consists of alternating ferromagnetic layers and nonmagnetic layers, which are numbered by integer n. The x-axis is normal to the interfaces, and the y–z plane is parallel to the interfaces. The OAF is parallel to the x–z plane and has an angle y with z-axis.
cosðQLÞ ¼ 12 trðTÞ ¼ 12ðT 11 þ T 22 Þ. Q is the Bloch wave vector.
(13)
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To analyze the surface modes, we consider the truncation of the superlattice at the plane x ¼ 0, so that the superlattice occupies the region x40. Bloch’s theorem still holds, but Q becomes complex (Q ¼ ib). The dispersion relation for surface modes is
Gþ G sinhðad1 Þ ¼ 0
(14)
Eq. (14) is thus an extension of the results for the Voigt configuration [26]. Note that for the transverse field configuration, the derivation for surface modes is slightly different [27,28], and 1 the result T 11 þ l T 12 ¼ lT 21 þ T 22 and l ¼ ða þ kÞ=ða kÞ is void for OAF configuration. The physical solution for Eq. (14) is G ¼ 0 and sinhðad1 Þ ¼ 0. For the case G ¼ 0, the attenuated constant of surface modes is
b ¼ af 1 qjj ð1 f 1 Þ,
(15)
which must be greater than zero for surface modes. Hence, one has the following restriction for surface modes
a4qjj ð1 f 1 Þ=f 1 .
(16)
The mode frequencies are calculated by G ¼ 0, and are given by 1 oi (17) o¼ þ ðom þ oi Þ cos yM cos f . 2 cos yM cos f The restriction for the magnetic ratio is determined by applying Eqs. (10), (16) and (17), with the expression ð1 f 1 Þ a4 , f1 0
(18)
bðom þ oi Þcos2 yM cos2 f oi c bðom oi Þ cos2 yM cos2 f þ oi c ½ðom þ oi
Þcos2
yM
cos2
2
f oi þ 4om oi
cos2
2
yM sin yM
cos2
f
These kinds of surface modes may vanish if yM exceeds arccos ½oi =ðom þ oi Þ1=2 . For ordinary cases f 1 o1, the surface modes will exist while fC X0. Thus, the condition for the existence of the surface modes is ðom oi Þ oi þ . 2om 2om cos2 yM
(20)
For yM ¼ 0, Eq. (20) can be simplified to f 1 X1=2, which is the classical result stated in Ref. [1]. Another possible solution can be derived from sinhðad1 Þ ¼ 0, which results in
2
2O
o2 ¼
mp qjj d1
2
!2 þ
ðO2þ
þ
O2 Þ
44o2m o2i 2 241 þ
mp qjj d1
om oi sin2 f 1þ
.
A necessary condition for Eq. (18) is that a0 is greater than zero, which is also the sufficient solution for f 1 ¼ 1. For this case, the surface modes are propagating in a wedge for fC ofofC and the critical angles are determined by 1=2 oi . (19) cos fC cos yM ¼ om þ oi
f 1X
modes of the layers. The Voigt configuration is covered as yM ¼ 0, while the mode frequencies are given by
o2 ¼ O2þ
where
a0 ¼
Fig. 2. Mode frequencies versus the propagating angle f for a constant magnitude of wave vector qjj d1 ¼ 2. The magnetic ratio for the superlattice is f 1 ¼ 0:8. The applied field is 6.5 kg, and has an angle y ¼ 551. The shaded areas represent the bulk modes; solid curves represent the surface modes. The five highest frequency and five lowest frequency bulk modes together with the corresponding surface modes are shown in this and after figures.
mp qjj d1
!2 .
(22)
3. Numerical calculation and discussions Our numerical calculations are explored with a Ni/Mo superlattice. Ni is a ferromagnetic material with saturated magnetization of M s ¼ 480 G and a gyromagnetic ratio of g ¼ 1:93 107 rad=G, while Mo is taken as a ‘nonmagnetic spacer’. With the conversion of 1 kg ¼ 3:072 GHz, the frequencies are quoted in Gauss, where om ¼ 6:032 kg, and oi equals to Hi in magnitude. First, we calculate the mode frequencies versus the propagating angles. These results are shown in Fig. 2. A general feature of the magnetostatic modes for the OAF configuration is that there are infinite sets of bulk mode bands and in the gap between the neighboring bulk bands, there is a branch of surface modes calculated from Eq. (21). In Fig. 2, we see that except for the highest frequency bulk mode bands and the highest frequency branch of surface modes, bulk modes and surface modes are in the frequency region from O to O+. With increasing m, both o+ and 3
!2 2
2
2
cos yM sin yM sin f þ
!2 3 mp 5 qjj d1
m ¼ 0; 1; 2; 3 . . . . If m ¼ 0, Eq. (21) states that o ¼ O , which provides the boundaries of those surface modes. Since the mode frequencies are between O and Oþ , these modes are composed by bulk
ðO2þ
O2 Þ2 5 ,
(21)
o approach O. In the vicinity of frequency O, the bulk mode bands and the surface modes become very dense. With decreasing
f, the surface mode frequency between O and O+ decreases, but
ARTICLE IN PRESS R.-H. Zhu et al. / Physica B 404 (2009) 2086–2090
these surface mode frequencies increase when the frequency is between O and O. For f ¼ 0, the difference between O and O vanishes, meaning these surface modes and bulk modes are equal at O, and no magnetostatic modes are below O. The highest frequency bulk mode band is still above O. With an increase in f, the highest frequency bulk mode band narrows and its frequencies decrease. The highest frequency branch of the surface modes can be solved using Eq. (17). Its frequencies are still above O+ and decrease with increasing f. It is terminated by touching the highest frequency bulk mode bands. In Fig. 2, we see that the highest frequency branch propagates in an allowed angle region. Since there is a gap between the low-edge cut-off frequency and O+, this region is smaller than the surface modes in a semi-infinite ferromagnet. The results for mode frequencies as a function of the magnetic ratio f1 for different applied fields are shown in Fig. 3. As seen in Fig. 3, with an increase in f 1 , the bulk mode bands are enlarged, except for the highest frequency bulk mode band, so the gaps
Fig. 3. Mode frequency versus the magnetic ratio f 1 . Shaded areas represent the bulk modes, solid lines represent the surface modes. O and O are shown by dashed lines.
2089
between the bulk bands narrow. If f1 approaches 1, the superlattice becomes a bulk ferromagnet and the gaps between the bands may disappear. On the contrary, for very small f1, the bulk bands narrow to a single frequency. The effects of the direction of static magnetization can be seen by comparing Fig. 3(a) with Fig. 3(b). In Fig. 3(a), the applied field is y ¼ 55 and H0 ¼ 6:5 kg, so yM ¼ 30:9 ; however, in Fig. 3(b), the applied field is y ¼ 80 and H0 ¼ 7:65 kg, which results in yM ¼ 60 . The highest frequency branch of the surface modes may vanish for the larger yM. Note that with increasing yM , the difference between O+ and O is increased, and in some extreme conditions, the results may be valid for the Voigt configuration and the transverse field configuration. For y ¼ yM ¼ 0, O is equivalent to O+, so the bulk mode bands and the surface modes between O and O+ may vanish, however, the highest frequency bulk mode band still exists and the low-edge frequency restriction turns to O+. For y ¼ yM ¼ p=2, the difference between O+ and O is enlarged, and O converges with, O so the modes between O and O disappear, whereas the bulk mode bands whose frequencies are in the region from O to O+ still exist. The mode frequencies versus qjj d1 are plotted in Fig. 4. We observe that the width of the highest and the lowest frequency bulk mode bands are continuously decreasing with the increasing qjj d1 , whereas the widths of other bulk mode bands turn narrow both at very large or very small qjj d1 . We see that as qjj increases, with the exception of the highest branch of surface modes, the surface mode frequencies above O increase, but surface mode frequencies below O decrease. As we know, the group velocity can be defined by do dqjj, which is inverse near O. For ooO, the group velocity of the surface modes is negative, but for o4O, the group velocity is positive. However, it is easy to understand. These surface modes are composed of bulk waves in each layer. The modes below O still exist after changing to the Voigt configuration. For this configuration, the magnetostatic bulk modes in a single magnetic layer were discussed by Damon and Eshbach [29]. These modes are nondispersive for the Voigt direction, but for the direction parallel to the applied field, they decrease with increases in the wave vector. The magnetostatic modes that correspond to frequencies above O are in the upper region, are similar to the transverse field
Fig. 4. Mode frequency versus the wave vectors qjj . The magnitude and direction of the applied field is the same as in Fig. 2. The magnetic ratio f 1 ¼ 0:8 is held constant. The propagating angle f is 351. Shaded areas represent bulk modes and solid lines represent the surface modes. The highest branch of the surface modes is not shown, since it is irrelevant with respect to qjj d1 .
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configuration. For this configuration, while magnetostatic surface modes exist, as in quasiperiodic superlattices, etc. [15,27], the magnetostatic surface modes increase with increases in the wave vector. This is consistent with our results.
4. Conclusions We discuss the magnetostatic surface modes and bulk modes for a ferromagnetic superlattice in an oblique applied field. For this system, the scalar potentials are set by Eqs. (6)–(8), and the dispersion equations are derived by applying the transfer matrix. The results are shown as numerical calculations. The surface modes and bulk mode bands (with the exception of the highest bulk mode band and the highest branch of surface modes) are separated into the upper and lower region by the frequency O, together with O+ and O. The properties of the modes in the lower region are similar to the Voigt configuration, and the properties of the modes in the upper region are similar to the transverse field configuration. The direction and the magnitude of the applied field affect the direction of the static magnetization, which determines the widths of the regions according to O+, O and O. The group velocity of the surface modes scales inversely with the proximity of O. For the mode frequencies below O, the velocity is negative, and for the mode frequencies above O, it is positive. A powerful technique for detecting magnetostatic modes is the Brillouin scattering. We do not believe there should be any unusual difficulties in applying this technique.
Acknowledgments This work was supported by Educational Department of Heilongjiang Province under Grant no. 11521298. The authors
thank the Key Laboratory of Superhard Materials of Heilongjiang Province for support. Reference [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]
R.E. Camley, T.S. Rahman, D.L. Mills, Phys. Rev. B 27 (1983) 261. P. Gru¨nberg, K. Mika, Phys. Rev. B 27 (1983) 2995. M. Grimsdith, M.R. Khan, A. Kueny, I.K. Schuller, Phys. Rev. Lett. 51 (1983) 498. A. Kueny, M.R. Khan, I.K. Schuller, M. Grimsditch, Phys. Rev. B 29 (1984) 2879. J. Barnas´, Solid State Commun. 61 (1987) 405. J. Barnas´, Phys. C Solid State Phys. 21 (1988) 1021. J. Barnas´, Phys. C Solid State Phys. 21 (1988) 4097. N. Raj, D.R. Tilley, Phys. Rev. B 36 (1987) 7003. M.C. Oliverost, N.S. Almeida, D.R. Tilley, J. Thomas, R.E. Camley, J. Phys. Condens. Matter 4 (1992) 8497. X.-Z. Wang, D.R. Tilley, Phys. Lett. A 187 (1994) 325. X.-Z. Wang, D.R. Tilley, J. Phys. Condens. Matter 9 (1997) 5777. R.-H. Zhu, C. Jia, X.-Z. Wang, Phys. Lett. A 355 (2006) 77. H.K. Sy, F. Chen, Phys. Rev. B 50 (1994) 3411. H.K. Sy, F. Chen, J. Magn. Magn. Mater 152 (1996) L282. R.E. Camley, M.G. Cottam, Phys. Rev. B 35 (1987) 189. F.G. Elmzughi, N.C. Constantinou, D.R. Tilley, Phys. Rev. B 51 (1995) 11515. T.S. Rahman, D.L. Mills, J. Appl. Phys. 53 (1982) 2084. N.S. Almeida, D.L. Mills, Phys. Rev. B 37 (1988) 3400. S.-C. Lu¨, X.-Z. Wang, D.R. Tilley, Phys. Rev. B 55 (1997) 12402. C. Jia, X.-Z. Wang, S.-C. Lu¨, Phys. Rev. B 61 (2000) 9494. M.A.A. Monteiro, G.A. Farias, R.N. Costa Filho, N.S. Almeida, Eur. Phys. J. B 61 (2008) 121. C.A.A. Arau´jo, E.L. Albuquerque, D.H.A.L. Anselmo, M.S. Vasconcelos, Phys. Lett. A 372 (2008) 1135. J. Milton Pereira Jr., R.N. Costa Filho, M.G. Cottam, J. Magn. Magn. Mater 272 (2004) 1235. J. Milton Pereira Jr., R.N. Costa Filho, Eur. Phys. J. B 40 (2004) 137. J.M. de Souza, E.L. Albuquerque, D.H.A.L. Anselmo, G.A. Farias, Physica B 324 (2002) 217. R.E. Camley, R.L. Stamps, J. Phys. Condens. Matter 5 (1993) 3727. D.H.A.L. Anselmo, M.G. Cottam, E.L. Albuquerque, J. Appl. Phys. 85 (1999) 5774. M.S. Vasconcelos, D.H.A.L. Anselmo, C.G. Bezerra, Solid State Commun. 135 (2005) 673. R.W. Damon, J.R. Eshbach, J. Phys. Chem. Solids 19 (1961) 308.
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Magnetostatic modes of a ferromagnetic superlattice in an oblique applied field Rui-Hua Zhu , Hong-Yan Peng, Mei-Heng Zhang, Yu-Qiang Chen Department of Physics, Mudanjiang Teachers College, Mudanjiang 157012, People’s Republic of China
a r t i c l e in f o
a b s t r a c t
Article history: Received 18 November 2008 Received in revised form 15 March 2009 Accepted 30 March 2009
The dispersion relations for both bulk and surface magnetostatic waves are obtained in a case where the applied field makes an arbitrary angle with the interfaces of the structure and the static magnetization is tipped out of the interfaces. With the transfer matrix treatment, we find infinite sets of bulk mode bands and infinite branches of surface modes. Both the bulk modes and the surface modes are sensitive to the direction of the static magnetization. Many new properties of the magnetostatic modes are thus presented. & 2009 Elsevier B.V. All rights reserved.
PACS: 75.70.Cn 75.40.Gb 75.30.Ds Keywords: Ferromagnetic superlattice Magnetostatic modes Oblique applied field
1. Introduction The long-wavelength collective excitations in a magnetic superlattice, including the magnetostatic modes or retarded modes, are intriguing subjects. Early works on magnetostatic modes in a ferromagnetic superlattice were carried out by Camley et al. [1], and Grunberg and Mike [2]. They calculated the collective magnetostatic modes in a ferromagnetic/nonmagnetic superlattice with the magnetization and applied field parallel to the interfaces (the so-called Voigt configuration). Experimental results from such a system were derived using Brillouin light scattering by Grimsditch et al. [3,4]. After that, there were many theoretical studies on collective excitations in various superlattices using different techniques, such as the transfer matrix formalism [5–7], effective-medium theory [8–12] and recurrent relations [13,14]. As shown in previous discussions, the spin configuration is an important factor for magnetostatic modes in magnetic superlattices. For the Voigt configuration, a well-known finding is that the ferromagnetic/nonmagnetic superlattice supports magnetostatic surface modes only if the volume fraction of magnetic materials exceeds 50%. However, this conclusion is not applicable for the transverse field configuration, since there are no magnetostatic modes in a perfect semi-infinite ferromagnetic/ nonmagnetic superlattice [15,16]. Corresponding author. Tel.: +86 0453 6486595.
E-mail addresses: [email protected], [email protected] (R.-H. Zhu). 0921-4526/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2009.03.046
The oblique applied field (OAF) configuration receives less consideration, and original work on this configuration includes the magnetostatic surface modes for a ferromagnetic slab [17] and surface polaritons for an antiferromagnet [18]. However, due to the low symmetry of this configuration, the magnetostaic modes of superlattice structures are usually addressed with an effectivemedium approximation. As an example, Lu¨ et al. calculated the magnetostatic modes in a lateral ferromagnetic superlattice where the applied field makes an arbitrary angle with the interfaces by deriving an effective-medium permeability tensor [19]. Subsequently, this method was extended to a metallic magnetic superlattice including the existence of eddy currents [20]. In this paper, we examine the problem of collective magnetostatic modes in a ferromagnetic/nonmagnetic superlattice, where the superlattice is in an OAF. The dispersion relations for the magnetostatic modes are calculated by applying Bloch’s theorem in spite of the low symmetry configuration. As we will see, the direction of the applied field has great impact on the magnetostatic surface modes and bulk modes, and many new properties of the magnetostatic modes are revealed.
2. Dispersion relations for bulk and surface modes The aim of this section is to derive the magnetostatic surface modes and bulk modes in the superlattice structure. For this
ARTICLE IN PRESS R.-H. Zhu et al. / Physica B 404 (2009) 2086–2090
2087
purpose, we employ a macroscopic theory, which is usually used to study the interface effects of multilayer structures [21,22]. However, this problem also can be treated with microscope theory, as demonstrated previously [23–25]. The waves are modeled in the magnetostatic region using Maxwell’s equations:
mxz ¼ mzx ¼ om oi sin yM cos yM =ðo2i o2 Þ, myz ¼ mzy ¼ om o sin yM =ðo2i o2 Þ. The related parameters are om ¼ 4pgM s and oi ¼ gHi .
r B ¼ 0;
"
r h ¼ 0.
(1)
The geometry of the superlattice is shown in Fig. 1. It consists of alternating ferromagnetic and nonmagnetic layers. The thickness of a ferromagnetic layer is d1 , while d2 represents the thickness of a nonmagnetic layer. Thus, the periodic length of the superlattice is L, which equals d1 plus d2. The magnetic volume ratio f1 is defined by f 1 ¼ d1 =L. The external magnetic field is in the x–z plane and makes an angle y with z-axis. We assume that the superlattice is quite large in the y and z directions, such that the demagnetizing field is directed along the x-axis. We also suppose the superlattice is well magnetized such that the static magnetizations of all layers are parallel to the internal field Hi. Thus, Hi can be calculated by a vector addition between the external field H0 and the demagnetizing field Hd while angle yM can be derived using the following: tan yM ¼ ðHo sin y 4pMs sin yM Þ=ðHo cos yÞ:
(2)
If y ¼ yM ¼ 0, the static magnetizations are parallel to the z-axis, which is the case in the Voigt configuration and the corresponding Voigt direction is along the y-axis. The magnitude of the internal field Hi is a function of the angle yM, with the expression Hi ¼ ½ðHo sin y 4pM s sin yM Þ2 þ ðHo sin yÞ2 1=2 .
(3)
The magnetic material’s long-wave dynamics is governed by the frequency-dependent magnetic permeability tensor l(o), which is calculated by linearizing the magnetization’s torque equation, and it is given by 0 mxx imxy mxz 1 B im myy imyz C (4) lðoÞ ¼ @ yx A, mzx imzy mzz
mxx ¼ 1 þ om oi cos2 yM =ðo2i o2 Þ, mzz ¼ 1 þ om oi sin2 yM =ðo2i o2 Þ, mxy ¼ myx ¼ om o cos yM =ðo2i o2 Þ,
myy ¼ 1 þ om oi =ðo2i o2 Þ,
From Eqs. (1) and (4), we can derive an equation that the scalar potential must satisfy:
mxx
# @2 @2 @2 @2 þ m þ m þ 2 m F ¼ 0. yy zz xz @x@z @x2 @y2 @z2
(5)
For the bulk magnetostatic modes, we assume a plane-wave solution:
F ¼ CðxÞ expbiðqk rk otÞc.
(6)
The vector rk is the projection of position vector r on y–z plane and qk is parallel to the y–z plane, so qk rk ¼ qy y þ qz z ¼ qjj ðcos fy þ sin fzÞ. The function cðxÞ in Eq. (6) is different between magnetic layers and nonmagnetic layers. For the region nLoxonL þ d1 , attributed to a magnetic layer, it is set by
cðxÞ ¼ eiq? nd1 eiq? ðxnLÞ ðAn eaðxnLÞ þ Bn eaðxnLd1 Þ Þ.
(7)
The factor exp½iq? ðx nLÞ in Eq. (7) is for OAF, which is after a method proposed by Rahman and Mills [17]. The factor expðiq? nd1 Þ in Eq. (7) serves to ensure that the determinant of the transfer matrix equals unity. However, this factor results in a phase shift from one period to the next. For the neighboring nonmagnetic layer nL þ d1 oxoðn þ 1ÞL; the scalar potential is ordinary, and is expressed as
cðxÞ ¼ eiq? nd1 fC n eqjj ðxnLd1 Þ þ Dn eqjj ½xðnþ1ÞL g,
(8)
where q? and a in Eq. (7) are solved from Eq. (5), where q? ¼ qjj ðmxz sin f=mxx Þ,
(9)
and
a ¼ qjj f½m2xz sin2 f þ mxx ðmyy cos2 f þ mzz sin2 fÞ=m2xx g1=2 ¼ qjj ½ðo2 O2þ Þðo2 O2 Þ=ðo2 O2 Þ2 1=2 , 2 i
1=2
2 i
(10) 2
1=2
where Oþ ¼ ðo þ om oi Þ , O ¼ ðo þ om oi cos yM Þ and O ¼ ðo2i þ om oi cos2 yM cos2 fÞ1=2 . It is clear that Oþ and O are the solutions for myy ðoÞ ¼ 0 and mxx ðoÞ ¼ 0, respectively. If a is a real number, we determine that o4Oþ or ooO, and that the magnetostatic modes in the layer are of the surface type. Otherwise, if a is a pure imaginary number, while o is in the range from O to Oþ , the modes are bulk waves. With the continuity of the magnetic potential F and the normal component Bx at the interfaces x ¼ nL þ d1 and x ¼ ðn þ 1ÞL, one may find a matrix equation for the amplitudes of waves in neighbor periods, ! ! Anþ1 An ¼T , (11) Bnþ1 Bn and T¼
1
Gþ Gþ Gþþ G expðad1 ÞðGþ Gþ Eþ Gþþ G E Þ 2 sinhðqjj d2 ÞGþþ Gþ
2 sinhðqjj d2 ÞG Gþ
!
expðad1 ÞðGþ Gþ E Gþþ G Eþ Þ
,
(12) with the notations defined by G ¼ qjj cos fmxy mxx a qjj and E ¼ expðqjj d2 Þ. The dispersion relations may then be acquired by applying Bloch’s theorem, where Fig. 1. A superlattice consists of alternating ferromagnetic layers and nonmagnetic layers, which are numbered by integer n. The x-axis is normal to the interfaces, and the y–z plane is parallel to the interfaces. The OAF is parallel to the x–z plane and has an angle y with z-axis.
cosðQLÞ ¼ 12 trðTÞ ¼ 12ðT 11 þ T 22 Þ. Q is the Bloch wave vector.
(13)
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To analyze the surface modes, we consider the truncation of the superlattice at the plane x ¼ 0, so that the superlattice occupies the region x40. Bloch’s theorem still holds, but Q becomes complex (Q ¼ ib). The dispersion relation for surface modes is
Gþ G sinhðad1 Þ ¼ 0
(14)
Eq. (14) is thus an extension of the results for the Voigt configuration [26]. Note that for the transverse field configuration, the derivation for surface modes is slightly different [27,28], and 1 the result T 11 þ l T 12 ¼ lT 21 þ T 22 and l ¼ ða þ kÞ=ða kÞ is void for OAF configuration. The physical solution for Eq. (14) is G ¼ 0 and sinhðad1 Þ ¼ 0. For the case G ¼ 0, the attenuated constant of surface modes is
b ¼ af 1 qjj ð1 f 1 Þ,
(15)
which must be greater than zero for surface modes. Hence, one has the following restriction for surface modes
a4qjj ð1 f 1 Þ=f 1 .
(16)
The mode frequencies are calculated by G ¼ 0, and are given by 1 oi (17) o¼ þ ðom þ oi Þ cos yM cos f . 2 cos yM cos f The restriction for the magnetic ratio is determined by applying Eqs. (10), (16) and (17), with the expression ð1 f 1 Þ a4 , f1 0
(18)
bðom þ oi Þcos2 yM cos2 f oi c bðom oi Þ cos2 yM cos2 f þ oi c ½ðom þ oi
Þcos2
yM
cos2
2
f oi þ 4om oi
cos2
2
yM sin yM
cos2
f
These kinds of surface modes may vanish if yM exceeds arccos ½oi =ðom þ oi Þ1=2 . For ordinary cases f 1 o1, the surface modes will exist while fC X0. Thus, the condition for the existence of the surface modes is ðom oi Þ oi þ . 2om 2om cos2 yM
(20)
For yM ¼ 0, Eq. (20) can be simplified to f 1 X1=2, which is the classical result stated in Ref. [1]. Another possible solution can be derived from sinhðad1 Þ ¼ 0, which results in
2
2O
o2 ¼
mp qjj d1
2
!2 þ
ðO2þ
þ
O2 Þ
44o2m o2i 2 241 þ
mp qjj d1
om oi sin2 f 1þ
.
A necessary condition for Eq. (18) is that a0 is greater than zero, which is also the sufficient solution for f 1 ¼ 1. For this case, the surface modes are propagating in a wedge for fC ofofC and the critical angles are determined by 1=2 oi . (19) cos fC cos yM ¼ om þ oi
f 1X
modes of the layers. The Voigt configuration is covered as yM ¼ 0, while the mode frequencies are given by
o2 ¼ O2þ
where
a0 ¼
Fig. 2. Mode frequencies versus the propagating angle f for a constant magnitude of wave vector qjj d1 ¼ 2. The magnetic ratio for the superlattice is f 1 ¼ 0:8. The applied field is 6.5 kg, and has an angle y ¼ 551. The shaded areas represent the bulk modes; solid curves represent the surface modes. The five highest frequency and five lowest frequency bulk modes together with the corresponding surface modes are shown in this and after figures.
mp qjj d1
!2 .
(22)
3. Numerical calculation and discussions Our numerical calculations are explored with a Ni/Mo superlattice. Ni is a ferromagnetic material with saturated magnetization of M s ¼ 480 G and a gyromagnetic ratio of g ¼ 1:93 107 rad=G, while Mo is taken as a ‘nonmagnetic spacer’. With the conversion of 1 kg ¼ 3:072 GHz, the frequencies are quoted in Gauss, where om ¼ 6:032 kg, and oi equals to Hi in magnitude. First, we calculate the mode frequencies versus the propagating angles. These results are shown in Fig. 2. A general feature of the magnetostatic modes for the OAF configuration is that there are infinite sets of bulk mode bands and in the gap between the neighboring bulk bands, there is a branch of surface modes calculated from Eq. (21). In Fig. 2, we see that except for the highest frequency bulk mode bands and the highest frequency branch of surface modes, bulk modes and surface modes are in the frequency region from O to O+. With increasing m, both o+ and 3
!2 2
2
2
cos yM sin yM sin f þ
!2 3 mp 5 qjj d1
m ¼ 0; 1; 2; 3 . . . . If m ¼ 0, Eq. (21) states that o ¼ O , which provides the boundaries of those surface modes. Since the mode frequencies are between O and Oþ , these modes are composed by bulk
ðO2þ
O2 Þ2 5 ,
(21)
o approach O. In the vicinity of frequency O, the bulk mode bands and the surface modes become very dense. With decreasing
f, the surface mode frequency between O and O+ decreases, but
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these surface mode frequencies increase when the frequency is between O and O. For f ¼ 0, the difference between O and O vanishes, meaning these surface modes and bulk modes are equal at O, and no magnetostatic modes are below O. The highest frequency bulk mode band is still above O. With an increase in f, the highest frequency bulk mode band narrows and its frequencies decrease. The highest frequency branch of the surface modes can be solved using Eq. (17). Its frequencies are still above O+ and decrease with increasing f. It is terminated by touching the highest frequency bulk mode bands. In Fig. 2, we see that the highest frequency branch propagates in an allowed angle region. Since there is a gap between the low-edge cut-off frequency and O+, this region is smaller than the surface modes in a semi-infinite ferromagnet. The results for mode frequencies as a function of the magnetic ratio f1 for different applied fields are shown in Fig. 3. As seen in Fig. 3, with an increase in f 1 , the bulk mode bands are enlarged, except for the highest frequency bulk mode band, so the gaps
Fig. 3. Mode frequency versus the magnetic ratio f 1 . Shaded areas represent the bulk modes, solid lines represent the surface modes. O and O are shown by dashed lines.
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between the bulk bands narrow. If f1 approaches 1, the superlattice becomes a bulk ferromagnet and the gaps between the bands may disappear. On the contrary, for very small f1, the bulk bands narrow to a single frequency. The effects of the direction of static magnetization can be seen by comparing Fig. 3(a) with Fig. 3(b). In Fig. 3(a), the applied field is y ¼ 55 and H0 ¼ 6:5 kg, so yM ¼ 30:9 ; however, in Fig. 3(b), the applied field is y ¼ 80 and H0 ¼ 7:65 kg, which results in yM ¼ 60 . The highest frequency branch of the surface modes may vanish for the larger yM. Note that with increasing yM , the difference between O+ and O is increased, and in some extreme conditions, the results may be valid for the Voigt configuration and the transverse field configuration. For y ¼ yM ¼ 0, O is equivalent to O+, so the bulk mode bands and the surface modes between O and O+ may vanish, however, the highest frequency bulk mode band still exists and the low-edge frequency restriction turns to O+. For y ¼ yM ¼ p=2, the difference between O+ and O is enlarged, and O converges with, O so the modes between O and O disappear, whereas the bulk mode bands whose frequencies are in the region from O to O+ still exist. The mode frequencies versus qjj d1 are plotted in Fig. 4. We observe that the width of the highest and the lowest frequency bulk mode bands are continuously decreasing with the increasing qjj d1 , whereas the widths of other bulk mode bands turn narrow both at very large or very small qjj d1 . We see that as qjj increases, with the exception of the highest branch of surface modes, the surface mode frequencies above O increase, but surface mode frequencies below O decrease. As we know, the group velocity can be defined by do dqjj, which is inverse near O. For ooO, the group velocity of the surface modes is negative, but for o4O, the group velocity is positive. However, it is easy to understand. These surface modes are composed of bulk waves in each layer. The modes below O still exist after changing to the Voigt configuration. For this configuration, the magnetostatic bulk modes in a single magnetic layer were discussed by Damon and Eshbach [29]. These modes are nondispersive for the Voigt direction, but for the direction parallel to the applied field, they decrease with increases in the wave vector. The magnetostatic modes that correspond to frequencies above O are in the upper region, are similar to the transverse field
Fig. 4. Mode frequency versus the wave vectors qjj . The magnitude and direction of the applied field is the same as in Fig. 2. The magnetic ratio f 1 ¼ 0:8 is held constant. The propagating angle f is 351. Shaded areas represent bulk modes and solid lines represent the surface modes. The highest branch of the surface modes is not shown, since it is irrelevant with respect to qjj d1 .
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configuration. For this configuration, while magnetostatic surface modes exist, as in quasiperiodic superlattices, etc. [15,27], the magnetostatic surface modes increase with increases in the wave vector. This is consistent with our results.
4. Conclusions We discuss the magnetostatic surface modes and bulk modes for a ferromagnetic superlattice in an oblique applied field. For this system, the scalar potentials are set by Eqs. (6)–(8), and the dispersion equations are derived by applying the transfer matrix. The results are shown as numerical calculations. The surface modes and bulk mode bands (with the exception of the highest bulk mode band and the highest branch of surface modes) are separated into the upper and lower region by the frequency O, together with O+ and O. The properties of the modes in the lower region are similar to the Voigt configuration, and the properties of the modes in the upper region are similar to the transverse field configuration. The direction and the magnitude of the applied field affect the direction of the static magnetization, which determines the widths of the regions according to O+, O and O. The group velocity of the surface modes scales inversely with the proximity of O. For the mode frequencies below O, the velocity is negative, and for the mode frequencies above O, it is positive. A powerful technique for detecting magnetostatic modes is the Brillouin scattering. We do not believe there should be any unusual difficulties in applying this technique.
Acknowledgments This work was supported by Educational Department of Heilongjiang Province under Grant no. 11521298. The authors
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