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Figure of merit for transverse magneto-optical Kerr effect
Journal of Magnetism and Magnetic Materials 226}230 (2001) 1686}1687
Figure of merit for transverse magneto-optical Kerr e!ect Y. Souche *, A.P.B. Tufaile, C.E. Santi, V. Novosad , A.D. Santos Laboratoire Louis Ne& el-CNRS/UJF, BP 166, 38042 Grenoble Cedex 9, France Laborato& rio de Materiais Magne& ticos, Universidade de SaJ o Paulo, CP 66.318, CEP 05315-970 SaJ o Paulo, Brazil
Abstract An expression for the Fresnel coe$cient r is established in the general case of an interface between two semi-in"nite media and is extended to multilayers. By analogy with the well-known (R for polar Kerr e!ect, a "gure of merit is ) proposed for the transverse geometry. It is an alternative characterizing quantity to the relative change of re#ectivity for p polarization, R /R . 2001 Elsevier Science B.V. All rights reserved. Keywords: Magneto-optics; Kerr e!ect; Thin "lms
In the present study, the "gure of merit allows the characterization of pure magnetic contribution to magneto-optical e!ect. It is an objective means of comparison between materials, particularly for the industrial applications. Many authors [1}4] calculated the Fresnel coe$cients r (i, j"s, p) at the dielectric-magnetic GH interface. Moreover, some of them only considered the polar and longitudinal Kerr e!ects. Hereafter, the main paths for calculating r in the transverse geometry are described following Yeh method's [5] with uncoupled modes. Let us consider the interface between two magnetic media, 1 and 2, lying in the plane xOy, with magnetization vectors M and M along Oy, perpendicular to the plane of incidence xOz. Positive z corresponds to medium 2. According to the Maxwell equations, the wave equation is kG (kG E)# GE"0, c
(1)
where kG is the wave vector in medium i (i"1, 2), such that kG"(2/G)(G, 0, G) and G is the permittivity tensor of
* Corresponding author. Tel.: #33-4-76-88-79-22; fax: #334-76-88-11-91. E-mail address: [email protected] (Y. Souche).
the medium:
G"
G
0
0
G
jm GQG V
0
!jm GQG V 0 , G
(2)
G is independent of the direction of magnetization. m "M /M and QG is the complex magneto-optic coefV V "cient. Assuming that EG"EG exp(jt!G ) r) and taking into account that ", two eigenmodes can be deduced from Eq. (2): (G)"G!(G) and (G)"G(1!Q)!(G).
(3)
The corresponding electric "elds are the s and p polarizations. In the latter case, for E "1, V G!(G) jm GQG " . (4) E "! V X G! jm GQG# V The electric and magnetic "elds can be written in terms of polarization eigenvectors as E " AG EG exp j(x#G z!t) G N N N N and H " AG HG exp j(x#G z!t), G N N N N
0304-8853/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 1 1 5 6 - 2
(5)
Y. Souche et al. / Journal of Magnetism and Magnetic Materials 226}230 (2001) 1686}1687
1687
where "1, 2 correspond to the incident s and p polarizations and "3, 4 to the p and s re#ected ones. The continuity of the tangential components, x and y, of E and H at the interface can be written as the product: MA"MA where
EG V EG MGAG" W EG V EG W
EG V EG W EG V EG W
EG V EG W EG V EG W
EG V EG W EG V EG W
AG AG . AG AG
(6)
The re#ected modes do not exist in the semi-in"nite medium 2, so that r "A /A and r "A /A . By taking into account that "! and that QG;1, it follows from Eq. (6) and for p polarization that
Fig. 1. F , R /R and R for a sputtered "lm of CoZr, 83 nm 2 thick, as a function of the angle of incidence.
r " ( ! )!jW( )mV Q!( )mV QX . (7) ( # )#jW( )mQ!( )mQX V V
At the "rst order in QG, ! r " # !j [mQ( )(1 ( # ) V !( ! ))#mQ( )(1 V !( ! ))]. (8) A similar form was obtained for a non magneticmagnetic bilayer. r can be rewritten as the sum of two complex terms corresponding to non-magnetic and magnetic contributions r " exp( j )# exp( j ). (9) The commonly used "gure of merit of the polar e!ect is (R, with the Kerr rotation and R the re#ectivity ) ) at the interface. Depending on the incident polarization, p or s, "r /r and "!r /r . At the normal ) ) incidence, p and s are equivalent and (R " . The "gure of merit is F " (R " cos( ! ). (10) ) In the case of the TMOKE, the saturated states correspond to a change of sign of r , i.e. a change by of . The resulting change of re#ectivity R obtained from the quadratic module of Eq. (9) is R "4 cos( ! ) +4(R cos( ! ). (11) Finally, the "gure of merit for the TMOKE is proposed as R . F " 2 4(R
(12)
Fig. 2. F for a sputtered "lm of Co, 80 nm thick, as a function 2 of the angle of incidence.
Figs. 1 and 2 represent plots of F , R /R and R , 2 respectively, for a sputtered "lm of CoZr, 83 nm thick, and F for a sputtered "lm of Co, 80 nm thick, as a func2 tion of the angle of incidence. It can be noted that R /R and F show an identical behaviour demonstrat 2 ing that both of them are a satisfactory criterium of the magneto-optical properties, at least in this case of a rather thick "lm of a ferromagnetic metal.
References [1] A.V. Sokolov, Optical Properties of Metals, Blackie and Son Ltd, London, 1967. [2] G. Metzger, P. Pluvinage, R. Torguet, Ann. Phys. 10 (1965) 5. [3] Z.J. Yang, M.R. Scheinfein, J. Appl. Phys. 74 (1993) 6810. [4] J. Zak, E.R. Moog, C. Liu, S.D. Bader, J. Magn. Magn. Mater. 89 (1990) 107. [5] P. Yeh, Surf. Sci. 96 (1980) 41.
Figure of merit for transverse magneto-optical Kerr e!ect Y. Souche *, A.P.B. Tufaile, C.E. Santi, V. Novosad , A.D. Santos Laboratoire Louis Ne& el-CNRS/UJF, BP 166, 38042 Grenoble Cedex 9, France Laborato& rio de Materiais Magne& ticos, Universidade de SaJ o Paulo, CP 66.318, CEP 05315-970 SaJ o Paulo, Brazil
Abstract An expression for the Fresnel coe$cient r is established in the general case of an interface between two semi-in"nite media and is extended to multilayers. By analogy with the well-known (R for polar Kerr e!ect, a "gure of merit is ) proposed for the transverse geometry. It is an alternative characterizing quantity to the relative change of re#ectivity for p polarization, R /R . 2001 Elsevier Science B.V. All rights reserved. Keywords: Magneto-optics; Kerr e!ect; Thin "lms
In the present study, the "gure of merit allows the characterization of pure magnetic contribution to magneto-optical e!ect. It is an objective means of comparison between materials, particularly for the industrial applications. Many authors [1}4] calculated the Fresnel coe$cients r (i, j"s, p) at the dielectric-magnetic GH interface. Moreover, some of them only considered the polar and longitudinal Kerr e!ects. Hereafter, the main paths for calculating r in the transverse geometry are described following Yeh method's [5] with uncoupled modes. Let us consider the interface between two magnetic media, 1 and 2, lying in the plane xOy, with magnetization vectors M and M along Oy, perpendicular to the plane of incidence xOz. Positive z corresponds to medium 2. According to the Maxwell equations, the wave equation is kG (kG E)# GE"0, c
(1)
where kG is the wave vector in medium i (i"1, 2), such that kG"(2/G)(G, 0, G) and G is the permittivity tensor of
* Corresponding author. Tel.: #33-4-76-88-79-22; fax: #334-76-88-11-91. E-mail address: [email protected] (Y. Souche).
the medium:
G"
G
0
0
G
jm GQG V
0
!jm GQG V 0 , G
(2)
G is independent of the direction of magnetization. m "M /M and QG is the complex magneto-optic coefV V "cient. Assuming that EG"EG exp(jt!G ) r) and taking into account that ", two eigenmodes can be deduced from Eq. (2): (G)"G!(G) and (G)"G(1!Q)!(G).
(3)
The corresponding electric "elds are the s and p polarizations. In the latter case, for E "1, V G!(G) jm GQG " . (4) E "! V X G! jm GQG# V The electric and magnetic "elds can be written in terms of polarization eigenvectors as E " AG EG exp j(x#G z!t) G N N N N and H " AG HG exp j(x#G z!t), G N N N N
0304-8853/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 1 1 5 6 - 2
(5)
Y. Souche et al. / Journal of Magnetism and Magnetic Materials 226}230 (2001) 1686}1687
1687
where "1, 2 correspond to the incident s and p polarizations and "3, 4 to the p and s re#ected ones. The continuity of the tangential components, x and y, of E and H at the interface can be written as the product: MA"MA where
EG V EG MGAG" W EG V EG W
EG V EG W EG V EG W
EG V EG W EG V EG W
EG V EG W EG V EG W
AG AG . AG AG
(6)
The re#ected modes do not exist in the semi-in"nite medium 2, so that r "A /A and r "A /A . By taking into account that "! and that QG;1, it follows from Eq. (6) and for p polarization that
Fig. 1. F , R /R and R for a sputtered "lm of CoZr, 83 nm 2 thick, as a function of the angle of incidence.
r " ( ! )!jW( )mV Q!( )mV QX . (7) ( # )#jW( )mQ!( )mQX V V
At the "rst order in QG, ! r " # !j [mQ( )(1 ( # ) V !( ! ))#mQ( )(1 V !( ! ))]. (8) A similar form was obtained for a non magneticmagnetic bilayer. r can be rewritten as the sum of two complex terms corresponding to non-magnetic and magnetic contributions r " exp( j )# exp( j ). (9) The commonly used "gure of merit of the polar e!ect is (R, with the Kerr rotation and R the re#ectivity ) ) at the interface. Depending on the incident polarization, p or s, "r /r and "!r /r . At the normal ) ) incidence, p and s are equivalent and (R " . The "gure of merit is F " (R " cos( ! ). (10) ) In the case of the TMOKE, the saturated states correspond to a change of sign of r , i.e. a change by of . The resulting change of re#ectivity R obtained from the quadratic module of Eq. (9) is R "4 cos( ! ) +4(R cos( ! ). (11) Finally, the "gure of merit for the TMOKE is proposed as R . F " 2 4(R
(12)
Fig. 2. F for a sputtered "lm of Co, 80 nm thick, as a function 2 of the angle of incidence.
Figs. 1 and 2 represent plots of F , R /R and R , 2 respectively, for a sputtered "lm of CoZr, 83 nm thick, and F for a sputtered "lm of Co, 80 nm thick, as a func2 tion of the angle of incidence. It can be noted that R /R and F show an identical behaviour demonstrat 2 ing that both of them are a satisfactory criterium of the magneto-optical properties, at least in this case of a rather thick "lm of a ferromagnetic metal.
References [1] A.V. Sokolov, Optical Properties of Metals, Blackie and Son Ltd, London, 1967. [2] G. Metzger, P. Pluvinage, R. Torguet, Ann. Phys. 10 (1965) 5. [3] Z.J. Yang, M.R. Scheinfein, J. Appl. Phys. 74 (1993) 6810. [4] J. Zak, E.R. Moog, C. Liu, S.D. Bader, J. Magn. Magn. Mater. 89 (1990) 107. [5] P. Yeh, Surf. Sci. 96 (1980) 41.