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Effective-medium theory for nonlinear magneto-optics in magnetic granular alloys: cubic nonlinearity
Journal of Magnetism and Magnetic Materials 258–259 (2003) 103–105
Effective-medium theory for nonlinear magneto-optics in magnetic granular alloys: cubic nonlinearity Alexander B. Granovskya,*, Michail V. Kuzmichova, Jean-Pierre Clercb, Mitsuteru Inouec,d a
Faculty of Physics, Lomonosov Moscow State University, 119992 Moscow, Russia b IUSTI, Universite de Provence, 13453 Marseille, France c Toyohashi University of Technology, Toyohashi 441-8580, Japan d CREST, Japan Science & Technology Corporation, Kawaguchi 332-0012, Japan
Abstract We propose a simple effective-medium approach for calculating the effective dielectric function of a magnetic metal– insulator granular alloy in which there is a weakly nonlinear relation between electric displacement D and electric field ð3Þ ð0Þ ð3Þ 2 E for both constituent materials of the form Di ¼ eð0Þ i Ei þwi jE i j E i : We assume that linear ei and cubic nonlinear wi dielectric functions are diagonal and linear with magnetization non-diagonal components. For such metal–insulator composite magneto-optical effects depend on a light intensity and the effective cubic dielectric function wð3Þ eff can be significantly greater (up to 103 times) than that for constituent materials. The calculation scheme is based on the Bergman and Stroud-Hui theory of nonlinear optical properties of granular matter. The giant cubic magneto-optical nonlinearity is found for composites with metallic volume fraction close to the percolation threshold and at a resonance of optical conductivity. It is shown that a composite may exhibit nonlinear magneto-optics even when both constituent materials have no cubic magneto-optical nonlinearity. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Nonlinear magneto-optics; Metal–insulator composites; Effective-medium approximation
1. Introduction Granular metal–insulator alloys, consisting of nanoscale magnetic particles in an insulating matrix are of unique consideration for the technological application and fundamental physics. They display a wide variety of both unusual linear and nonlinear electric-transport, optical and magneto-optical properties, especially for compositions close to the percolation threshold. At finite frequencies, the nonlinear dependence of electrical displacement D on electric field E is the basis of nonlinear optics and magneto-optics. Nonlinear optical and magneto-optical phenomena can be observed at the frequency of incident light and in the case of high harmonic generation. Magneto-induced second harmonic generation and the so-called nonlinear *Corresponding author. Fax: +7-095-939-47-87. E-mail address: [email protected] (A.B. Granovsky).
magneto-optical Kerr effect [1,2] are the well-known examples of the second type nonlinear magnetooptics, and these effects require inversion symmetry breaking. In this paper, we are discussing nonlinear magnetooptics in granular metal–insulator alloys at the frequency of incident light, which is due to a weakly nonlinear relation between D and E of the form: ð3Þ 2 Di ¼ ei Ei ¼ eð0Þ i Ei þ wi jEi j Ei ;
ð0Þ 2 wð3Þ i jEi j 5ei :
ð1Þ
for both constituent materials (i ¼ 1 for metal and i ¼ 2 and cubic for insulator). We assume that linear eð0Þ i nonlinear wð3Þ dielectric functions are diagonal and linear i with magnetization non-diagonal components 1 1 0 d 0 d 0 0 ei eod wi wod i i C C B B od eð0Þ ¼ @ eod edi 0 Awð3Þ wdi 0 A: ð2Þ i i ¼ @ wi i 0
0304-8853/03/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 2 ) 0 1 0 7 8 - 8
0
edi
0
0
wdi
A.B. Granovsky et al. / Journal of Magnetism and Magnetic Materials 258–259 (2003) 103–105
104
Since non-diagonal components of dielectric function are responsible for magneto-optical effects, one can expect that magneto-optical effects in such materials and composites depend on light intensity. It is quite difficult to observe this nonlinear effect for homogeneous ferromagnets because of rather small magnitude of cubic non-diagonal term wod i . In this paper, within the effective-medium approximation (EMA) it is shown that magneto-optical cubic nonlinearity in nano-composites can be at least three orders of magnitude greater than that for constituent materials. Moreover, a composite may exhibit nonlinear magneto-optics even when both constituent materials have no cubic magneto-optical nonlinearity.
3. Results and discussion The Eq. (6) describes cubic nonlinear optics and is the same as the result of Refs. [3–5]. The Eq. (7) describes cubic nonlinear magneto-optics and is our principal result. The coefficients K; Q and M are enhancement factors. They strongly depend on metallic volume fraction f ¼ f1 and frequency w: We calculated these coefficients using three different EMA schemes developed for linear magneto-optics [6], namely, Maxwell– Garnett approximation, Bruggeman approximation (EMA), and symmetrised Maxwell–Garnett approximation [6]. The details of these tedious but straightforward calculations will be published elsewhere. Some of results are shown in Figs. 1–3 and can be summarised as follows:
2. Theory The space-averaged fields and displacements for a composite with metallic volume fraction f1 are related by an equation of the same form as Eq. (1): ð0Þ 2 /DS ¼ eeff /ES ¼ eeff /ES þ wð3Þ eff j/ESj /ES:
The diagonal and non-diagonal elements of the effective dielectric function eeff can be written as d d od od edeff ¼ F ðed1 ; ed2 ; f1 Þ; eod eff ¼ Fðe1 ; e2 ; e1 ; e2 ; f1 Þ;
ð3Þ
where F and F are well-known functions in the linear limit of EMA [3–6]. We expand these functions in a dð0Þ odð0Þ od ð0) Taylor series about eeff and eeff eeff taking into ð0Þ 2 account that op > ei ¼ ei þ wð3Þ ojE j >; where i i 2 ojEi j > is the mean square of the electric field in the ith component in the linear limit [3–5]. Since within the linear limit [3–5]
1. The coefficients K; Q; M are large only at some charactereristic frequencies for every composition. The characteristic frequencies are the same for both optics (K) and magneto-optics (Q; M) are connected with plasmon-type resonances in metallic grains or near metal-insulator interfaces, as well as with resonances of optical conductivity [7]. Therefore, as a rule KðwÞ; QðwÞ; MðwÞ spectra have at least two peaks for every composition (Fig. 1). 2. The spectra KðwÞ; QðwÞ; MðwÞ depend on the type of EMA approach used for calculation (Fig. 2). It means that nonlinear effects are more sensitive to granular alloy microstructure in comparison with linear phenomena. 3. Nonlinear magneto-optics in composites can be observed even when both components have no cubic
fi ojE i j2 > =E20 ¼ ðqedeff =qedi Þ0 ¼ Fi0 ðed1 ð0Þ; ed2 ð0Þ; f1 Þ ¼ Fi0 ;
ð4Þ
where E 0 is the external field, we have as a result dð0Þ þ wdeff E 20 ; edeff ¼ eeff
odð0Þ 2 edeff ¼ eeff þ wod eff E 0 ;
ð5Þ
where wdeff ¼ f1 K1 wd1 þ f2 K2 wd2 ;
ð6Þ
d d od od wod eff ¼ f1 Q1 w1 þ f2 Q2 w2 þ f1 M1 w1 þ f2 M2 w2 ;
ð7Þ
! dð0Þ dð0Þ 1 0 0 1 qeeff qeeff ; Ki ¼ 2 Fi jFi j ¼ 2 dð0Þ fi fi qei qeidð0Þ
ð8Þ
1 qF Qi ¼ 2 dð0Þ jFi0 j; fi qei Mi ¼
1 qF jF 0 j: fi2 qeiodð0Þ i
ð9Þ
ð10Þ
Fig. 1. Enhancement factor of optical nonlinearity ð1 f ÞK2 vs. frequency w for a composite with metallic volume fraction f1 ¼ f ¼ 0:1 (dashed line) and f ¼ 0:2 (solid line) calculated in the Maxwell–Garnett approximation. Optical and magnetooptical parameters for constituent materials were taken from Refs. [6,8,9] corresponding to Co and Al2O3; edð0Þ ¼ 2; plasmon 2 frequency wpl ¼ 8:95 1015 c1.
A.B. Granovsky et al. / Journal of Magnetism and Magnetic Materials 258–259 (2003) 103–105
od magneto-optical nonlinearity (wod 1 ¼ w2 ¼ 0), but at least one of them exhibits nonlinear optics (Eq. (7)). 4. The enhancement factor for cubic magneto-optical nonlinearity can be as large as 103 (Fig. 3) and is larger than that for cubic optical nonlinearity (Fig. 2).
80 60 f K1
105
40 20 0 0
0.2
0.4
0.6
0.8
1
0.6
0.8
1
f 160
f K1
120 80 40 0 0
0.2
0.4 f
Fig. 2. Enhancement factor of optical nonlinearity fK1 vs. metallic volume fraction f calculated in the framework of Bruggeman (top) and symmetrised Maxwell–Garnett approximation (bottom) at w=wpl ¼ 1 for the same set of parameters as in Fig. 1.
Acknowledgements
20 00
The work was supported by the Russian Foundation for Basic Research (grant 02-02-17797), and by the Scientific Program ‘‘Universities of Russia’’ (grant 01.03.003). A.G. is grateful to University of Provence for hospitality.
15 00 f Q1
Two other features of cubic magneto-optical nonlinearity should also be mentioned. First of all, within the EMA, the enhancement factor of magneto-optics at visible wavelengths cannot exceed 104. By analogy with nonlinear optics, it is quite reasonable to expect that fractal structure of composites close to the percolation threshold can provide both an additional enhancement and a shift of resonance frequency. Second, since magneto-optical effects, for example transversal Kerr effect [6], are linear with non-diagonal dielectric function but also depend on its diagonal component, the total enhancement factor of nonlinearity for every Kerr or Faraday magneto-optical effect is a rather complicated function of both linear and nonlinear optical and magneto-optical parameters, light frequency and incident angle. We hope that our calculations will stimulate experimental study of cubic nonlinear magneto-optics in magnetic nano-composites.
10 00 50 0 0
References 0
0.2
0.4
0.6
0.8
1
f 20
f M1
15 10 5 0 0
0.2
0.4
0.6
0.8
1
f Fig. 3. Enhancement factors of magneto-optical nonlinearity fQ1 (top) and fM1 (bottom) vs. metallic volume fraction f in the framework of symmetrised Maxwell–Garnett approximation at w=wpl ¼ 1 for the same set of parameters as in Fig. 1.
[1] K.H. Bennemann, J. Magn. Magn. Mater. 200 (1999) 679. [2] O.A. Aktsipetrov, E.A. Ganshina, V.S. Guschin, T.V. Misuryaev, T.V. Murzina, J. Magn. Magn. Mater. 196– 197 (1999) 80. [3] D.J. Bergman, Phys. Rev. B 39 (1989) 4598. [4] D. Stroud, P.M. Hui, Phys. Rev. B 37 (1988) 8719. [5] X.C. Zeng, D.J. Bergman, P.M. Hui, D. Stroud, Phys. Rev. B 38 (1988) 10970. [6] A. Granovsky, M. Kuzmichev, J.P. Clerc, J. Magn. Soc. Jpn. 23 (1999) 382. [7] A. Petrov, Physics of small particles, Nauka, Moscow, 1982, p. 427 (in Russian). [8] E. Ganshina, A. Granovsky, V. Guschin, M. Kuzmichov, P. Podrugin, A. Kravetz, E. Shipel, Physica A 241 (1997) 45. [9] T.S. Eriksson, A. Hjortsberg, G.A. Nikolasson, C.G. Granqvist, Appl. Opt. 20 (1981) 2742.
Effective-medium theory for nonlinear magneto-optics in magnetic granular alloys: cubic nonlinearity Alexander B. Granovskya,*, Michail V. Kuzmichova, Jean-Pierre Clercb, Mitsuteru Inouec,d a
Faculty of Physics, Lomonosov Moscow State University, 119992 Moscow, Russia b IUSTI, Universite de Provence, 13453 Marseille, France c Toyohashi University of Technology, Toyohashi 441-8580, Japan d CREST, Japan Science & Technology Corporation, Kawaguchi 332-0012, Japan
Abstract We propose a simple effective-medium approach for calculating the effective dielectric function of a magnetic metal– insulator granular alloy in which there is a weakly nonlinear relation between electric displacement D and electric field ð3Þ ð0Þ ð3Þ 2 E for both constituent materials of the form Di ¼ eð0Þ i Ei þwi jE i j E i : We assume that linear ei and cubic nonlinear wi dielectric functions are diagonal and linear with magnetization non-diagonal components. For such metal–insulator composite magneto-optical effects depend on a light intensity and the effective cubic dielectric function wð3Þ eff can be significantly greater (up to 103 times) than that for constituent materials. The calculation scheme is based on the Bergman and Stroud-Hui theory of nonlinear optical properties of granular matter. The giant cubic magneto-optical nonlinearity is found for composites with metallic volume fraction close to the percolation threshold and at a resonance of optical conductivity. It is shown that a composite may exhibit nonlinear magneto-optics even when both constituent materials have no cubic magneto-optical nonlinearity. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Nonlinear magneto-optics; Metal–insulator composites; Effective-medium approximation
1. Introduction Granular metal–insulator alloys, consisting of nanoscale magnetic particles in an insulating matrix are of unique consideration for the technological application and fundamental physics. They display a wide variety of both unusual linear and nonlinear electric-transport, optical and magneto-optical properties, especially for compositions close to the percolation threshold. At finite frequencies, the nonlinear dependence of electrical displacement D on electric field E is the basis of nonlinear optics and magneto-optics. Nonlinear optical and magneto-optical phenomena can be observed at the frequency of incident light and in the case of high harmonic generation. Magneto-induced second harmonic generation and the so-called nonlinear *Corresponding author. Fax: +7-095-939-47-87. E-mail address: [email protected] (A.B. Granovsky).
magneto-optical Kerr effect [1,2] are the well-known examples of the second type nonlinear magnetooptics, and these effects require inversion symmetry breaking. In this paper, we are discussing nonlinear magnetooptics in granular metal–insulator alloys at the frequency of incident light, which is due to a weakly nonlinear relation between D and E of the form: ð3Þ 2 Di ¼ ei Ei ¼ eð0Þ i Ei þ wi jEi j Ei ;
ð0Þ 2 wð3Þ i jEi j 5ei :
ð1Þ
for both constituent materials (i ¼ 1 for metal and i ¼ 2 and cubic for insulator). We assume that linear eð0Þ i nonlinear wð3Þ dielectric functions are diagonal and linear i with magnetization non-diagonal components 1 1 0 d 0 d 0 0 ei eod wi wod i i C C B B od eð0Þ ¼ @ eod edi 0 Awð3Þ wdi 0 A: ð2Þ i i ¼ @ wi i 0
0304-8853/03/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 2 ) 0 1 0 7 8 - 8
0
edi
0
0
wdi
A.B. Granovsky et al. / Journal of Magnetism and Magnetic Materials 258–259 (2003) 103–105
104
Since non-diagonal components of dielectric function are responsible for magneto-optical effects, one can expect that magneto-optical effects in such materials and composites depend on light intensity. It is quite difficult to observe this nonlinear effect for homogeneous ferromagnets because of rather small magnitude of cubic non-diagonal term wod i . In this paper, within the effective-medium approximation (EMA) it is shown that magneto-optical cubic nonlinearity in nano-composites can be at least three orders of magnitude greater than that for constituent materials. Moreover, a composite may exhibit nonlinear magneto-optics even when both constituent materials have no cubic magneto-optical nonlinearity.
3. Results and discussion The Eq. (6) describes cubic nonlinear optics and is the same as the result of Refs. [3–5]. The Eq. (7) describes cubic nonlinear magneto-optics and is our principal result. The coefficients K; Q and M are enhancement factors. They strongly depend on metallic volume fraction f ¼ f1 and frequency w: We calculated these coefficients using three different EMA schemes developed for linear magneto-optics [6], namely, Maxwell– Garnett approximation, Bruggeman approximation (EMA), and symmetrised Maxwell–Garnett approximation [6]. The details of these tedious but straightforward calculations will be published elsewhere. Some of results are shown in Figs. 1–3 and can be summarised as follows:
2. Theory The space-averaged fields and displacements for a composite with metallic volume fraction f1 are related by an equation of the same form as Eq. (1): ð0Þ 2 /DS ¼ eeff /ES ¼ eeff /ES þ wð3Þ eff j/ESj /ES:
The diagonal and non-diagonal elements of the effective dielectric function eeff can be written as d d od od edeff ¼ F ðed1 ; ed2 ; f1 Þ; eod eff ¼ Fðe1 ; e2 ; e1 ; e2 ; f1 Þ;
ð3Þ
where F and F are well-known functions in the linear limit of EMA [3–6]. We expand these functions in a dð0Þ odð0Þ od ð0) Taylor series about eeff and eeff eeff taking into ð0Þ 2 account that op > ei ¼ ei þ wð3Þ ojE j >; where i i 2 ojEi j > is the mean square of the electric field in the ith component in the linear limit [3–5]. Since within the linear limit [3–5]
1. The coefficients K; Q; M are large only at some charactereristic frequencies for every composition. The characteristic frequencies are the same for both optics (K) and magneto-optics (Q; M) are connected with plasmon-type resonances in metallic grains or near metal-insulator interfaces, as well as with resonances of optical conductivity [7]. Therefore, as a rule KðwÞ; QðwÞ; MðwÞ spectra have at least two peaks for every composition (Fig. 1). 2. The spectra KðwÞ; QðwÞ; MðwÞ depend on the type of EMA approach used for calculation (Fig. 2). It means that nonlinear effects are more sensitive to granular alloy microstructure in comparison with linear phenomena. 3. Nonlinear magneto-optics in composites can be observed even when both components have no cubic
fi ojE i j2 > =E20 ¼ ðqedeff =qedi Þ0 ¼ Fi0 ðed1 ð0Þ; ed2 ð0Þ; f1 Þ ¼ Fi0 ;
ð4Þ
where E 0 is the external field, we have as a result dð0Þ þ wdeff E 20 ; edeff ¼ eeff
odð0Þ 2 edeff ¼ eeff þ wod eff E 0 ;
ð5Þ
where wdeff ¼ f1 K1 wd1 þ f2 K2 wd2 ;
ð6Þ
d d od od wod eff ¼ f1 Q1 w1 þ f2 Q2 w2 þ f1 M1 w1 þ f2 M2 w2 ;
ð7Þ
! dð0Þ dð0Þ 1 0 0 1 qeeff qeeff ; Ki ¼ 2 Fi jFi j ¼ 2 dð0Þ fi fi qei qeidð0Þ
ð8Þ
1 qF Qi ¼ 2 dð0Þ jFi0 j; fi qei Mi ¼
1 qF jF 0 j: fi2 qeiodð0Þ i
ð9Þ
ð10Þ
Fig. 1. Enhancement factor of optical nonlinearity ð1 f ÞK2 vs. frequency w for a composite with metallic volume fraction f1 ¼ f ¼ 0:1 (dashed line) and f ¼ 0:2 (solid line) calculated in the Maxwell–Garnett approximation. Optical and magnetooptical parameters for constituent materials were taken from Refs. [6,8,9] corresponding to Co and Al2O3; edð0Þ ¼ 2; plasmon 2 frequency wpl ¼ 8:95 1015 c1.
A.B. Granovsky et al. / Journal of Magnetism and Magnetic Materials 258–259 (2003) 103–105
od magneto-optical nonlinearity (wod 1 ¼ w2 ¼ 0), but at least one of them exhibits nonlinear optics (Eq. (7)). 4. The enhancement factor for cubic magneto-optical nonlinearity can be as large as 103 (Fig. 3) and is larger than that for cubic optical nonlinearity (Fig. 2).
80 60 f K1
105
40 20 0 0
0.2
0.4
0.6
0.8
1
0.6
0.8
1
f 160
f K1
120 80 40 0 0
0.2
0.4 f
Fig. 2. Enhancement factor of optical nonlinearity fK1 vs. metallic volume fraction f calculated in the framework of Bruggeman (top) and symmetrised Maxwell–Garnett approximation (bottom) at w=wpl ¼ 1 for the same set of parameters as in Fig. 1.
Acknowledgements
20 00
The work was supported by the Russian Foundation for Basic Research (grant 02-02-17797), and by the Scientific Program ‘‘Universities of Russia’’ (grant 01.03.003). A.G. is grateful to University of Provence for hospitality.
15 00 f Q1
Two other features of cubic magneto-optical nonlinearity should also be mentioned. First of all, within the EMA, the enhancement factor of magneto-optics at visible wavelengths cannot exceed 104. By analogy with nonlinear optics, it is quite reasonable to expect that fractal structure of composites close to the percolation threshold can provide both an additional enhancement and a shift of resonance frequency. Second, since magneto-optical effects, for example transversal Kerr effect [6], are linear with non-diagonal dielectric function but also depend on its diagonal component, the total enhancement factor of nonlinearity for every Kerr or Faraday magneto-optical effect is a rather complicated function of both linear and nonlinear optical and magneto-optical parameters, light frequency and incident angle. We hope that our calculations will stimulate experimental study of cubic nonlinear magneto-optics in magnetic nano-composites.
10 00 50 0 0
References 0
0.2
0.4
0.6
0.8
1
f 20
f M1
15 10 5 0 0
0.2
0.4
0.6
0.8
1
f Fig. 3. Enhancement factors of magneto-optical nonlinearity fQ1 (top) and fM1 (bottom) vs. metallic volume fraction f in the framework of symmetrised Maxwell–Garnett approximation at w=wpl ¼ 1 for the same set of parameters as in Fig. 1.
[1] K.H. Bennemann, J. Magn. Magn. Mater. 200 (1999) 679. [2] O.A. Aktsipetrov, E.A. Ganshina, V.S. Guschin, T.V. Misuryaev, T.V. Murzina, J. Magn. Magn. Mater. 196– 197 (1999) 80. [3] D.J. Bergman, Phys. Rev. B 39 (1989) 4598. [4] D. Stroud, P.M. Hui, Phys. Rev. B 37 (1988) 8719. [5] X.C. Zeng, D.J. Bergman, P.M. Hui, D. Stroud, Phys. Rev. B 38 (1988) 10970. [6] A. Granovsky, M. Kuzmichev, J.P. Clerc, J. Magn. Soc. Jpn. 23 (1999) 382. [7] A. Petrov, Physics of small particles, Nauka, Moscow, 1982, p. 427 (in Russian). [8] E. Ganshina, A. Granovsky, V. Guschin, M. Kuzmichov, P. Podrugin, A. Kravetz, E. Shipel, Physica A 241 (1997) 45. [9] T.S. Eriksson, A. Hjortsberg, G.A. Nikolasson, C.G. Granqvist, Appl. Opt. 20 (1981) 2742.