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Studies on magnetic photonic band-gap material at microwave frequency
Solid State Communications 130 (2004) 451–454 www.elsevier.com/locate/ssc
Studies on magnetic photonic band-gap material at microwave frequency Ping Xub,*, Tian-Yi Caib, Zhen-Ya Lia,b b
a CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, China Department of Physics, Suzhou University 172, Suzhou 215006, China
Received 18 January 2004; accepted 26 February 2004 by H. Eschrig
Abstract The microwave transmission characteristics of magnetic photonic band-gap (MPBG) materials, in which the periodic structures are composed of alternating layers of polycarbonate with and without ferromagnetic nanowires, are studied. We present a theoretical method to investigate the transmission spectra of the MPBG material. The band-gap effect varies with the periodic parameters of the MPBG structure. Our calculation can describe well the experimental measurement value. The influence of the applied static magnetic field on the MPBG and the ferromagnetic resonance phenomenon has been studied. q 2004 Elsevier Ltd. All rights reserved. PACS: 76.50. þ g; 42.70.Qs; 41.20.Jb; 42.25.Bs Keywords: A. Magnetic photonic band-gap materials; C. Transmission spectra; E. Ferromagnetic resonance
1. Introduction Photonic band-gap (PBG) materials have attracted much attention in recent years for theoretical and practical importance in fundamental science and application [1 –5]. These artificially fabricated materials are periodically ordered structures made of various dielectric or metal inclusions with different refractive indices. The interest of these PBG materials comes from their specific properties that the propagation of electromagnetic waves in periodic composite materials can be controlled. The frequency spectrum of these materials has the band-gap structure. There is a range of frequencies at which the light propagation is strictly forbidden. For the last decade, numerous studies have devoted to PBG materials composed of isotropic dielectric materials [6 –8]. Then the constitutive components were diversified to conductors which also show * Corresponding author. Tel.: þ86-512-651-123-25; fax: þ 86512-651-125-97. E-mail addresses: [email protected] (P. Xu), [email protected] (Z.Y. Li). 0038-1098/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2004.02.049
PBG effects in the microwave and the optical regions of the electromagnetic wave spectrum [9 –11]. Rather recently, some publications have emerged in which magnetic photonic crystals (MPC) are considered [12 – 14]. Wider applications are possible with PBG materials utilizing the magnetic media, for example, an enhancement of the Kerr effect and Faraday rotation can be achieved in one-dimensional MPC. Only a few of papers were devoted to the studies of PBG materials composed of magnetic materials. In the optical range the relative permeability m of magnetic material is equal to 1. But for most ferrites m is quite different from 1 in the microwave range and thus can be exploited for microwave PBG materials. The value of m of ferrites in the microwave range depends on the saturation magnetization, the microwave frequency, and the external static magnetic field. Therefore, ferrites make tunable PBG materials possible with applied magnetic field [15]. The magnetic PBG materials can operate at microwave frequencies having strong spectral non-reciprocity. Recent developments in the manufacturing of materials permit the fabrication of magnetic nanowires of small diameter in a background. An experimental investigation of
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P. Xu et al. / Solid State Communications 130 (2004) 451–454
one kind of magnetic photonic band-gap (MPBG) materials is presented in Ref. [16]. It consists of an array of parallel ferromagnetic nanowires electrodeposited into an insulating polycarbonate membrane. Ferromagnetic materials have higher resonance frequencies. The MPBG effect is also available accompanied by the existence of the ferromagnetic resonance (FMR) in the vicinity of the band-gap frequency. Our study has been motivated by the works of A. Saib et al. [16]. In this paper, we undertake theoretical investigation on MPBG materials. We present a theoretical method to study plane-wave propagation in MPBG materials. The transmission characteristics of multilayer film composed of periodic insulating polycarbonate with and without ferromagnetic nanowires are investigated. The calculation results obtained for the MPBG are compared with the experimental results. They show agreement well to the experimental data.
2. Film structures In this paper, we consider a model system composed of an array of alternating layers of insulating polycarbonate (P) pffiffiffi with the refractive index np ¼ e p and ferromagnetic composite (F) with the effective refractive index nF made of ferromagnetic nanowires embedded in a polycarbonate matrix. Fig. 1 shows a three-dimensional view of the MPBG structure. The periodicity parameters d and a are the thickness of F layer and the distance between two F layers, respectively. A schematic description of F layer is shown in Fig. 2. Compared to the skin depth, the diameter of nanowires is small, resulting in the full penetration of the electromagnetic fields inside the nanowires. The relative permittivity and magnetic permeability of insulating polycarbonate e p ¼ 2:89; mp ¼ 1; respectively. In this configuration, the waves propagate along the OZ-axis and the direction of the electric field is along OX-axis for the transverse electric (TE) mode. The applied magnetic field Hex is parallel to OX-axis. The magnetic field of the TE
Fig. 2. Top-view picture of F layer of the MPBG sample, an insulating polycarbonate membrane filled with ferromagnetic nanowires.
mode is perpendicular to the applied magnetic field Hex meeting the requirements for a FMR. The relative magnetic permeability of ferromagnetic nanowires m~f is a tensor and assumed to have the form 1 0 mf ikf 0 C B C m~f ¼ B @ 2ikf mf 0 A 0
0
1
where mf and kf are, respectively, the diagonal and offdiagonal components of the magnetic permeability tensor of one nanowire.
3. Formulation An example of multilayer film made of alternating layers of F and P is shown in Fig. 1. The transmission characteristics of this periodic structure can be evaluated by the transfer matrix method [17]. According to this method, the electromagnetic fields in adjacent layers are linked by a matrix Mj which depends on the interface and on the properties of the material. 0 1 i 2 sin dj C B cos dj hj Mj ¼ @ A 2ihj sin dj cos dj where 2p nd l j j rffiffiffiffiffi 10 hj ¼ n m0 j
dj ¼
nj ; refractive index; dj ; thickness of the j layer; l; wavelength, respectively. When the number of the layers is N M ¼ M1 M2 …Mj …MN Fig. 1. Schematic description of the MPBG structure.
The transmission can be given by the matrix elements of M: In order to obtain the effective refractive index of F layer
P. Xu et al. / Solid State Communications 130 (2004) 451–454
nF ; the polycarbonate membrane with ferromagnetic nanowires can be regarded as a composite with nF ¼ pffiffiffiffipffiffiffiffiffi e eF meF where e eF ; meF are the effective relative permittivity and magnetic permeability of the composite polymer þ nanowires. For the TE mode (s-polarized light), the electric field is along the nanowires. The effective relative permittivity of the composite can be evaluated by 1eF ¼ 1f f þ 1p ð1 2 f Þ where f and 1f are the filling fraction and the relative permittivity of ferromagnetic nanowires and 1p ; the relative permittivity of polycarbonate matrix. The effective relative magnetic permeability of F layer meF can be written in the form
m2 2 k2 m
453
magnetic permeability tensor of one nanowire is given by the classical FMR theory.
mf ¼ 1 þ kf ¼
vm ðvr þ ivaÞ ðvr þ ivaÞ2 2 v2
vm v ðvr þ ivaÞ2 2 v2
where vm ¼ gMs and vr ¼ gHex þ 1=2vm ; g; Ms and Hex are the gyromagnetic ratio, the saturation magnetization and the applied static magnetic field. a is the damping factor.
4. Results and discussion
For the TE mode, the relative magnetic permeability of the ferromagnetic nanowire is a function of frequency v; the saturation magnetization Ms and the strength of the applied field Hex : The diagonal and off-diagonal components of the
Fig. 3 shows the calculated spectra of transmission T of periodic structure without applied magnetic field as on Fig. 1. The numerical calculations are performed using the parameters as shown in Ref. [16]. A forbidden PBG exists in the microwave range at 20.2 GHz. In this sample, the periodic parameters are a ¼ 3:636 mm; d ¼ 1:818 mm: In the experiment the MPBG peak occurs at 21 GHz. The circles are experimental data. Our results seem to match the experimental data in magnitude. In Fig. 4, a ¼ 3:076 mm; d ¼ 1:538 mm; the transmission curves reach the minimum due to the PBG effect at 25.6 GHz, while the experimental result is 26 GHz. Around the band-gap frequency, the electromagnetic waves are reflected. In the vicinity of the band-gap frequency, corresponding to the FMR frequency of the ferromagnetic material at the remanent state, there exhibits another peak in the transmission spectra. Around the FMR frequency, the electromagnetic waves are absorbed
Fig. 3. Theoretical and experimental microwave transmission of a MPBG with a ¼ 3:636 mm; d ¼ 1:818 mm at zero dc magnetic field. The circles represent the experimental data in Ref. [16]. Solid line is calculation results.
Fig. 4. Theoretical and experimental microwave transmission of a MPBG with a ¼ 3:076 mm; d ¼ 1:538 mm at zero dc magnetic field. The circles represent the experimental data in Ref. [16]. Solid line is calculation results.
meF ¼
m and k are the diagonal and off-diagonal components of the effective magnetic permeability tensor of the composite. The calculation can be conducted approximately by the Maxwell – Garnett approximation [18]. m ¼ mp þ 2f mp k¼
ðmf 2 mp Þðmf þ mp Þ 2 kf2 ðmf þ mp þ kf Þðmf þ mp 2 kf Þ
4f m2p kf ðmf þ mp þ kf Þðmf þ mp 2 kf Þ
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P. Xu et al. / Solid State Communications 130 (2004) 451–454
the magnetic permeability of the MPBG materials, there appears a band-gap. The influences of the periodic parameters a and d; and of the applied static magnetic field on the MPBG effect and the FMR effect are different. The periodic parameters play a more important role in the MPBG effect. The FMR effect is significantly influenced by the applied static magnetic field. Differed from the classical PBG structures, of particular interest of the MPBG materials is that the exploiting of dc magnetic fields can give an additional tuning tool to control the optical properties of the PBG materials. The using of magnetic materials in combination with dielectrics and metals can lead to a new growing for photonics applications.
Acknowledgements
Fig. 5. Theoretical microwave transmission of a MPBG with a ¼ 2:666 mm; d ¼ 1:333 mm at different values of dc magnetic field applied parallel to the nanowires of a MPBG.
by the magnetic materials. Compared with the experimental results, our calculation approximately agrees with them. Fig. 5 shows the transmission spectra of a MPBG where a ¼ 2:666 mm and d ¼ 1:333 mm when the static magnetic field is applied. The MPBG peak is observed at 27.8 GHz. According to the experimental report, there is a band-gap at 28 GHz. With decreasing thickness of each layer of the MPBG structures, the MPBG peak moves toward the higher frequency region. The position of the FMR peak varies with the static magnetic field. On the other hand, near the FMR frequency, the MPBG peak shifts a little with the static magnetic field applied. The effects of the applied magnetic field on the FMR peak and the MPBG peak are different.
5. Conclusion To conclude, we have studied the transmission characteristics of MPBG structures composed of alternating layers of polycarbonate with and without ferromagnetic nanowires. We have proposed a theoretical method to investigate the MPBG effect in periodic structure with ferromagnetic composite. In F layer of the MPBG sample, the ferromagnetic nanowires are randomly distributed inside the polycarbonate membrane. When the filling fraction of the nanowires is small, it is still insulating. The nanowires are dominantly surrounded by the polycarbonate host. It is reasonable to take this layer as a composite with one kind of cylinders being surrounded by the other host component. The Maxwell – Garnett theory has morphological consistency with such composite. From the calculation, we have found that by using our theoretical method, the results can describe the experimental data. Due to a periodic change of
This work was supported by the National Natural Science Foundation of China under Grant No. 10174049 and was partially supported by the National Natural Science Foundation of China under Grant No. 10204017 and by the Natural Science of Jiangsu Province for financial support under Grant No. BK2002038.
References [1] E. Yablonovitch, Phys. Rev. Lett. 58 (1987) 2059. [2] S. John, Phys. Rev. Lett. 58 (1987) 2486. [3] J.D. Joannopoulos, P.R. Villeneuve, S. Fan, Nature 386 (1997) 143. [4] O. Painter, R.K. Lee, A. Scherer, et al., Science 284 (1999) 1819. [5] I.L. Lyubchanskii, N.N. Dadoenkova, M.I. Lyubchanskii, et al., J. Phys. D: Appl. Phys. 36 (2003) R277. [6] E. Yablonovitch, T.J. Gmitter, K.M. Leung, Phys. Rev. Lett. 67 (1991) 2295. [7] J.C. Knight, J. Broeng, T.A. Birks, et al., Science 282 (1998) 1476. [8] Y. Fink, J.N. Winn, S. Fan, et al., Science 282 (1998) 1679. [9] M.J. Bloemer, M. Scalora, Appl. Phys. Lett. 72 (1998) 1676. [10] A. Suarez-Garcia, R. del Coso, R. Serna, et al., Appl. Phys. Lett. 83 (2003) 1842. [11] M.C. Larciprete, C. Sibilia, S. Paoloni, et al., J. Appl. Phys. 93 (2003) 5013. [12] M.M. Sigalas, C.M. Soukoulis, R. Biswas, K.M. Ho, Phys. Rev. B 56 (1997) 959. [13] M. Inoue, K.I. Arai, T. Fujii, M. Abe, J. Appl. Phys. 85 (1999) 5768. [14] H. Kato, T. Matsushita, A. Takayama, M. Egawa, J. Appl. Phys. 93 (2003) 3906. [15] C.S. Kee, J.E. Kim, H.Y. Park, Phys. Rev. E 57 (1998) 2327. [16] A. Saib, D. Vanhoenacker-Janvier, I. Huynen, et al., Appl. Phys. Lett. 83 (2003) 2378. [17] M. Born, E. Wolf, Principles of optics, Pergamon, New York, 1980. [18] P.M. Hui, D. Stroud, Appl. Phys. Lett. 50 (1987) 950.
Studies on magnetic photonic band-gap material at microwave frequency Ping Xub,*, Tian-Yi Caib, Zhen-Ya Lia,b b
a CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, China Department of Physics, Suzhou University 172, Suzhou 215006, China
Received 18 January 2004; accepted 26 February 2004 by H. Eschrig
Abstract The microwave transmission characteristics of magnetic photonic band-gap (MPBG) materials, in which the periodic structures are composed of alternating layers of polycarbonate with and without ferromagnetic nanowires, are studied. We present a theoretical method to investigate the transmission spectra of the MPBG material. The band-gap effect varies with the periodic parameters of the MPBG structure. Our calculation can describe well the experimental measurement value. The influence of the applied static magnetic field on the MPBG and the ferromagnetic resonance phenomenon has been studied. q 2004 Elsevier Ltd. All rights reserved. PACS: 76.50. þ g; 42.70.Qs; 41.20.Jb; 42.25.Bs Keywords: A. Magnetic photonic band-gap materials; C. Transmission spectra; E. Ferromagnetic resonance
1. Introduction Photonic band-gap (PBG) materials have attracted much attention in recent years for theoretical and practical importance in fundamental science and application [1 –5]. These artificially fabricated materials are periodically ordered structures made of various dielectric or metal inclusions with different refractive indices. The interest of these PBG materials comes from their specific properties that the propagation of electromagnetic waves in periodic composite materials can be controlled. The frequency spectrum of these materials has the band-gap structure. There is a range of frequencies at which the light propagation is strictly forbidden. For the last decade, numerous studies have devoted to PBG materials composed of isotropic dielectric materials [6 –8]. Then the constitutive components were diversified to conductors which also show * Corresponding author. Tel.: þ86-512-651-123-25; fax: þ 86512-651-125-97. E-mail addresses: [email protected] (P. Xu), [email protected] (Z.Y. Li). 0038-1098/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2004.02.049
PBG effects in the microwave and the optical regions of the electromagnetic wave spectrum [9 –11]. Rather recently, some publications have emerged in which magnetic photonic crystals (MPC) are considered [12 – 14]. Wider applications are possible with PBG materials utilizing the magnetic media, for example, an enhancement of the Kerr effect and Faraday rotation can be achieved in one-dimensional MPC. Only a few of papers were devoted to the studies of PBG materials composed of magnetic materials. In the optical range the relative permeability m of magnetic material is equal to 1. But for most ferrites m is quite different from 1 in the microwave range and thus can be exploited for microwave PBG materials. The value of m of ferrites in the microwave range depends on the saturation magnetization, the microwave frequency, and the external static magnetic field. Therefore, ferrites make tunable PBG materials possible with applied magnetic field [15]. The magnetic PBG materials can operate at microwave frequencies having strong spectral non-reciprocity. Recent developments in the manufacturing of materials permit the fabrication of magnetic nanowires of small diameter in a background. An experimental investigation of
452
P. Xu et al. / Solid State Communications 130 (2004) 451–454
one kind of magnetic photonic band-gap (MPBG) materials is presented in Ref. [16]. It consists of an array of parallel ferromagnetic nanowires electrodeposited into an insulating polycarbonate membrane. Ferromagnetic materials have higher resonance frequencies. The MPBG effect is also available accompanied by the existence of the ferromagnetic resonance (FMR) in the vicinity of the band-gap frequency. Our study has been motivated by the works of A. Saib et al. [16]. In this paper, we undertake theoretical investigation on MPBG materials. We present a theoretical method to study plane-wave propagation in MPBG materials. The transmission characteristics of multilayer film composed of periodic insulating polycarbonate with and without ferromagnetic nanowires are investigated. The calculation results obtained for the MPBG are compared with the experimental results. They show agreement well to the experimental data.
2. Film structures In this paper, we consider a model system composed of an array of alternating layers of insulating polycarbonate (P) pffiffiffi with the refractive index np ¼ e p and ferromagnetic composite (F) with the effective refractive index nF made of ferromagnetic nanowires embedded in a polycarbonate matrix. Fig. 1 shows a three-dimensional view of the MPBG structure. The periodicity parameters d and a are the thickness of F layer and the distance between two F layers, respectively. A schematic description of F layer is shown in Fig. 2. Compared to the skin depth, the diameter of nanowires is small, resulting in the full penetration of the electromagnetic fields inside the nanowires. The relative permittivity and magnetic permeability of insulating polycarbonate e p ¼ 2:89; mp ¼ 1; respectively. In this configuration, the waves propagate along the OZ-axis and the direction of the electric field is along OX-axis for the transverse electric (TE) mode. The applied magnetic field Hex is parallel to OX-axis. The magnetic field of the TE
Fig. 2. Top-view picture of F layer of the MPBG sample, an insulating polycarbonate membrane filled with ferromagnetic nanowires.
mode is perpendicular to the applied magnetic field Hex meeting the requirements for a FMR. The relative magnetic permeability of ferromagnetic nanowires m~f is a tensor and assumed to have the form 1 0 mf ikf 0 C B C m~f ¼ B @ 2ikf mf 0 A 0
0
1
where mf and kf are, respectively, the diagonal and offdiagonal components of the magnetic permeability tensor of one nanowire.
3. Formulation An example of multilayer film made of alternating layers of F and P is shown in Fig. 1. The transmission characteristics of this periodic structure can be evaluated by the transfer matrix method [17]. According to this method, the electromagnetic fields in adjacent layers are linked by a matrix Mj which depends on the interface and on the properties of the material. 0 1 i 2 sin dj C B cos dj hj Mj ¼ @ A 2ihj sin dj cos dj where 2p nd l j j rffiffiffiffiffi 10 hj ¼ n m0 j
dj ¼
nj ; refractive index; dj ; thickness of the j layer; l; wavelength, respectively. When the number of the layers is N M ¼ M1 M2 …Mj …MN Fig. 1. Schematic description of the MPBG structure.
The transmission can be given by the matrix elements of M: In order to obtain the effective refractive index of F layer
P. Xu et al. / Solid State Communications 130 (2004) 451–454
nF ; the polycarbonate membrane with ferromagnetic nanowires can be regarded as a composite with nF ¼ pffiffiffiffipffiffiffiffiffi e eF meF where e eF ; meF are the effective relative permittivity and magnetic permeability of the composite polymer þ nanowires. For the TE mode (s-polarized light), the electric field is along the nanowires. The effective relative permittivity of the composite can be evaluated by 1eF ¼ 1f f þ 1p ð1 2 f Þ where f and 1f are the filling fraction and the relative permittivity of ferromagnetic nanowires and 1p ; the relative permittivity of polycarbonate matrix. The effective relative magnetic permeability of F layer meF can be written in the form
m2 2 k2 m
453
magnetic permeability tensor of one nanowire is given by the classical FMR theory.
mf ¼ 1 þ kf ¼
vm ðvr þ ivaÞ ðvr þ ivaÞ2 2 v2
vm v ðvr þ ivaÞ2 2 v2
where vm ¼ gMs and vr ¼ gHex þ 1=2vm ; g; Ms and Hex are the gyromagnetic ratio, the saturation magnetization and the applied static magnetic field. a is the damping factor.
4. Results and discussion
For the TE mode, the relative magnetic permeability of the ferromagnetic nanowire is a function of frequency v; the saturation magnetization Ms and the strength of the applied field Hex : The diagonal and off-diagonal components of the
Fig. 3 shows the calculated spectra of transmission T of periodic structure without applied magnetic field as on Fig. 1. The numerical calculations are performed using the parameters as shown in Ref. [16]. A forbidden PBG exists in the microwave range at 20.2 GHz. In this sample, the periodic parameters are a ¼ 3:636 mm; d ¼ 1:818 mm: In the experiment the MPBG peak occurs at 21 GHz. The circles are experimental data. Our results seem to match the experimental data in magnitude. In Fig. 4, a ¼ 3:076 mm; d ¼ 1:538 mm; the transmission curves reach the minimum due to the PBG effect at 25.6 GHz, while the experimental result is 26 GHz. Around the band-gap frequency, the electromagnetic waves are reflected. In the vicinity of the band-gap frequency, corresponding to the FMR frequency of the ferromagnetic material at the remanent state, there exhibits another peak in the transmission spectra. Around the FMR frequency, the electromagnetic waves are absorbed
Fig. 3. Theoretical and experimental microwave transmission of a MPBG with a ¼ 3:636 mm; d ¼ 1:818 mm at zero dc magnetic field. The circles represent the experimental data in Ref. [16]. Solid line is calculation results.
Fig. 4. Theoretical and experimental microwave transmission of a MPBG with a ¼ 3:076 mm; d ¼ 1:538 mm at zero dc magnetic field. The circles represent the experimental data in Ref. [16]. Solid line is calculation results.
meF ¼
m and k are the diagonal and off-diagonal components of the effective magnetic permeability tensor of the composite. The calculation can be conducted approximately by the Maxwell – Garnett approximation [18]. m ¼ mp þ 2f mp k¼
ðmf 2 mp Þðmf þ mp Þ 2 kf2 ðmf þ mp þ kf Þðmf þ mp 2 kf Þ
4f m2p kf ðmf þ mp þ kf Þðmf þ mp 2 kf Þ
454
P. Xu et al. / Solid State Communications 130 (2004) 451–454
the magnetic permeability of the MPBG materials, there appears a band-gap. The influences of the periodic parameters a and d; and of the applied static magnetic field on the MPBG effect and the FMR effect are different. The periodic parameters play a more important role in the MPBG effect. The FMR effect is significantly influenced by the applied static magnetic field. Differed from the classical PBG structures, of particular interest of the MPBG materials is that the exploiting of dc magnetic fields can give an additional tuning tool to control the optical properties of the PBG materials. The using of magnetic materials in combination with dielectrics and metals can lead to a new growing for photonics applications.
Acknowledgements
Fig. 5. Theoretical microwave transmission of a MPBG with a ¼ 2:666 mm; d ¼ 1:333 mm at different values of dc magnetic field applied parallel to the nanowires of a MPBG.
by the magnetic materials. Compared with the experimental results, our calculation approximately agrees with them. Fig. 5 shows the transmission spectra of a MPBG where a ¼ 2:666 mm and d ¼ 1:333 mm when the static magnetic field is applied. The MPBG peak is observed at 27.8 GHz. According to the experimental report, there is a band-gap at 28 GHz. With decreasing thickness of each layer of the MPBG structures, the MPBG peak moves toward the higher frequency region. The position of the FMR peak varies with the static magnetic field. On the other hand, near the FMR frequency, the MPBG peak shifts a little with the static magnetic field applied. The effects of the applied magnetic field on the FMR peak and the MPBG peak are different.
5. Conclusion To conclude, we have studied the transmission characteristics of MPBG structures composed of alternating layers of polycarbonate with and without ferromagnetic nanowires. We have proposed a theoretical method to investigate the MPBG effect in periodic structure with ferromagnetic composite. In F layer of the MPBG sample, the ferromagnetic nanowires are randomly distributed inside the polycarbonate membrane. When the filling fraction of the nanowires is small, it is still insulating. The nanowires are dominantly surrounded by the polycarbonate host. It is reasonable to take this layer as a composite with one kind of cylinders being surrounded by the other host component. The Maxwell – Garnett theory has morphological consistency with such composite. From the calculation, we have found that by using our theoretical method, the results can describe the experimental data. Due to a periodic change of
This work was supported by the National Natural Science Foundation of China under Grant No. 10174049 and was partially supported by the National Natural Science Foundation of China under Grant No. 10204017 and by the Natural Science of Jiangsu Province for financial support under Grant No. BK2002038.
References [1] E. Yablonovitch, Phys. Rev. Lett. 58 (1987) 2059. [2] S. John, Phys. Rev. Lett. 58 (1987) 2486. [3] J.D. Joannopoulos, P.R. Villeneuve, S. Fan, Nature 386 (1997) 143. [4] O. Painter, R.K. Lee, A. Scherer, et al., Science 284 (1999) 1819. [5] I.L. Lyubchanskii, N.N. Dadoenkova, M.I. Lyubchanskii, et al., J. Phys. D: Appl. Phys. 36 (2003) R277. [6] E. Yablonovitch, T.J. Gmitter, K.M. Leung, Phys. Rev. Lett. 67 (1991) 2295. [7] J.C. Knight, J. Broeng, T.A. Birks, et al., Science 282 (1998) 1476. [8] Y. Fink, J.N. Winn, S. Fan, et al., Science 282 (1998) 1679. [9] M.J. Bloemer, M. Scalora, Appl. Phys. Lett. 72 (1998) 1676. [10] A. Suarez-Garcia, R. del Coso, R. Serna, et al., Appl. Phys. Lett. 83 (2003) 1842. [11] M.C. Larciprete, C. Sibilia, S. Paoloni, et al., J. Appl. Phys. 93 (2003) 5013. [12] M.M. Sigalas, C.M. Soukoulis, R. Biswas, K.M. Ho, Phys. Rev. B 56 (1997) 959. [13] M. Inoue, K.I. Arai, T. Fujii, M. Abe, J. Appl. Phys. 85 (1999) 5768. [14] H. Kato, T. Matsushita, A. Takayama, M. Egawa, J. Appl. Phys. 93 (2003) 3906. [15] C.S. Kee, J.E. Kim, H.Y. Park, Phys. Rev. E 57 (1998) 2327. [16] A. Saib, D. Vanhoenacker-Janvier, I. Huynen, et al., Appl. Phys. Lett. 83 (2003) 2378. [17] M. Born, E. Wolf, Principles of optics, Pergamon, New York, 1980. [18] P.M. Hui, D. Stroud, Appl. Phys. Lett. 50 (1987) 950.