Abstract

Magnetooptic (MO) effects in magnetic multilayers with periodically stratified regions are analyzed for the case of normal light wave incidence and polar magnetization (Faraday and polar Kerr effects). From the universal 4×4-matrix formalism simplified analytical representations restricted to terms linear in the off-diagonal permittivity tensor elements are obtained with no loss in accuracy. The MO effects are expressed as weighted sums of contributions from individual layers. Approximate expressions useful for the evaluation of trends in MO effects are given for periodic multilayers consisting of blocks with ultrathin magnetic films. The procedure is illustrated on periodic systems built of symmetric units. Limits on the ultrathin approximation are discussed.

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References

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  1. Heinrich and J. A. C. Bland, eds., Ultrathin Magnetic Structures, (Springer Verlag, Berlin Heidelberg, 1994).
  2. M. Schubert, T. E. Tiwald and J. A. Woollam, "Explicit solutions for the optical properties of arbitrary magneto-optic materials in generalized ellipsometry," Appl. Opt. 38, 177-187 (1999).
    [CrossRef]
  3. P. Yeh, "Optics of anisotropic layered media: a new 4 x 4 matrix algebra," Surf. Sci. 96, 41-53 (1980).
    [CrossRef]
  4. S. Visnovsky, "Magneto-optical ellipsometry," Czech. J. Phys. B 36, 625-650 (1986).
    [CrossRef]
  5. K. Balasubramian, A. Marathay and H. A. Macleod, "Modelling magneto-optical thin-film media for optical data storage," Thin Solid Films 164, 122-128 (1988).
  6. F. Abeles, "Recherches sur la propagation des ondes electromagnetiques sinusoidales dans les milieux stratifies. Application aux couches minces," Ann. Phys. Paris 5, 596-640 (1950).
  7. M. Born and E. Wolf, Principles of Optics, (Pergamon Press, Oxford, 1959).
  8. J. Lafait, T. Yamaguchi, J. M. Frigerio, A. Bichri and K. Driss-Khodja, "Effective medium equivalent to a symmetric multilayer at oblique incidence," Appl. Opt. 29, 2460-2465 (1990).
    [CrossRef] [PubMed]
  9. S. Visnovsky, M. Nyvlt, V. Prosser, R. Lopusnik and R. Urban, J. Ferre and G. Penissard, D. Renard, R. Krishnan, "Polar magneto-optics in simple ultrathin-magnetic-film structures," Phys. Rev. B 52, 1090-1106 (1995).
    [CrossRef]
  10. M. Nyvlt, J. Ferre, J.-P. Jamet, P. Houdy, P. Boher, S. Visnovsky, R. Urban and R. Lopusnik "MO Kerr and Faraday studies of Au/Co ultrathin film sandwiches," J. Magn. Magn. Mater. 148, 281-282 (1995).
    [CrossRef]
  11. Z. Q. Qiu and S. D. Bader, "Surface magneto--optic Kerr effect (SMOKE)," J. Magn. Magn. Mater. 200, 664-78 (1999).
    [CrossRef]
  12. J. Ferre, M. Nyvlt, G. Penissard, V. Prosser, D. Renard, S. Visnovsky, "MO Kerr and Faraday studies of Au/Co ultrathin film sandwiches, J. Magn. Magn. Mater. 148, 281-282 (1995).
    [CrossRef]
  13. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light. (North Holland, Elsevier, Amsterdam, 1987).
  14. J. Badoz, M. Billardon, J. C. Canit, M. F. Russel, "Sensitive devices to determine the state and degree of polarization of a light using a birefringent modulator," J. Opt. 8, 373-384 (1977).
    [CrossRef]
  15. G. E. Jellison, Jr. and F. A. Modine, "Two channel polarization modulation ellipsometer," Appl. Opt. 29, 959-974 (1990).
    [CrossRef] [PubMed]
  16. P. B. Johnson and R. W. Christy, "Optical constants of transition metals: Ti, V, Cr, Mn, Fe, Co, Ni, and Pd," Phys. Rev. B 9, 5056-5070 (1974).
    [CrossRef]
  17. S. Visnovsky, M. Nyvlt, V. Parizek, P. Kielar, V. Prosser and R. Krishnan, "Magneto-Optical Studies of Pt/Co Multilayers and Pt-Co Alloy Thin Films," IEEE Trans. Magn. 29, 3390-3392 (1993).
    [CrossRef]
  18. J. H. Weaver, "Optical properties of Rh, Pd, Ir and Pt," Phys. Rev. B 11, 1416-1425 (1975).
    [CrossRef]
  19. D. W. Lynch and W. R. Hunter, "Comments on the optical constants of metals and an introduction to the several metals," in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic Press, Inc., Orlando, 1985) pp. 275-367.
  20. Zak, E. R. Moog, C. Liu, and S. D. Bader, "Universal approach to magneto-optics," J. Magn. Magn. Mater. 89, 107-123 (1990).
    [CrossRef]

Other (20)

Heinrich and J. A. C. Bland, eds., Ultrathin Magnetic Structures, (Springer Verlag, Berlin Heidelberg, 1994).

M. Schubert, T. E. Tiwald and J. A. Woollam, "Explicit solutions for the optical properties of arbitrary magneto-optic materials in generalized ellipsometry," Appl. Opt. 38, 177-187 (1999).
[CrossRef]

P. Yeh, "Optics of anisotropic layered media: a new 4 x 4 matrix algebra," Surf. Sci. 96, 41-53 (1980).
[CrossRef]

S. Visnovsky, "Magneto-optical ellipsometry," Czech. J. Phys. B 36, 625-650 (1986).
[CrossRef]

K. Balasubramian, A. Marathay and H. A. Macleod, "Modelling magneto-optical thin-film media for optical data storage," Thin Solid Films 164, 122-128 (1988).

F. Abeles, "Recherches sur la propagation des ondes electromagnetiques sinusoidales dans les milieux stratifies. Application aux couches minces," Ann. Phys. Paris 5, 596-640 (1950).

M. Born and E. Wolf, Principles of Optics, (Pergamon Press, Oxford, 1959).

J. Lafait, T. Yamaguchi, J. M. Frigerio, A. Bichri and K. Driss-Khodja, "Effective medium equivalent to a symmetric multilayer at oblique incidence," Appl. Opt. 29, 2460-2465 (1990).
[CrossRef] [PubMed]

S. Visnovsky, M. Nyvlt, V. Prosser, R. Lopusnik and R. Urban, J. Ferre and G. Penissard, D. Renard, R. Krishnan, "Polar magneto-optics in simple ultrathin-magnetic-film structures," Phys. Rev. B 52, 1090-1106 (1995).
[CrossRef]

M. Nyvlt, J. Ferre, J.-P. Jamet, P. Houdy, P. Boher, S. Visnovsky, R. Urban and R. Lopusnik "MO Kerr and Faraday studies of Au/Co ultrathin film sandwiches," J. Magn. Magn. Mater. 148, 281-282 (1995).
[CrossRef]

Z. Q. Qiu and S. D. Bader, "Surface magneto--optic Kerr effect (SMOKE)," J. Magn. Magn. Mater. 200, 664-78 (1999).
[CrossRef]

J. Ferre, M. Nyvlt, G. Penissard, V. Prosser, D. Renard, S. Visnovsky, "MO Kerr and Faraday studies of Au/Co ultrathin film sandwiches, J. Magn. Magn. Mater. 148, 281-282 (1995).
[CrossRef]

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light. (North Holland, Elsevier, Amsterdam, 1987).

J. Badoz, M. Billardon, J. C. Canit, M. F. Russel, "Sensitive devices to determine the state and degree of polarization of a light using a birefringent modulator," J. Opt. 8, 373-384 (1977).
[CrossRef]

G. E. Jellison, Jr. and F. A. Modine, "Two channel polarization modulation ellipsometer," Appl. Opt. 29, 959-974 (1990).
[CrossRef] [PubMed]

P. B. Johnson and R. W. Christy, "Optical constants of transition metals: Ti, V, Cr, Mn, Fe, Co, Ni, and Pd," Phys. Rev. B 9, 5056-5070 (1974).
[CrossRef]

S. Visnovsky, M. Nyvlt, V. Parizek, P. Kielar, V. Prosser and R. Krishnan, "Magneto-Optical Studies of Pt/Co Multilayers and Pt-Co Alloy Thin Films," IEEE Trans. Magn. 29, 3390-3392 (1993).
[CrossRef]

J. H. Weaver, "Optical properties of Rh, Pd, Ir and Pt," Phys. Rev. B 11, 1416-1425 (1975).
[CrossRef]

D. W. Lynch and W. R. Hunter, "Comments on the optical constants of metals and an introduction to the several metals," in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic Press, Inc., Orlando, 1985) pp. 275-367.

Zak, E. R. Moog, C. Liu, and S. D. Bader, "Universal approach to magneto-optics," J. Magn. Magn. Mater. 89, 107-123 (1990).
[CrossRef]

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Figures (3)

Fig. 1.
Fig. 1.

Symmetric A/B/A unit.

Fig. 2.
Fig. 2.

Magnetooptic azimuth rotation (full lines), θK , and ellipticity (dashed), K , in a periodic multilayer consistingo f symmetric A/B/A blocks, Pt(0.4 nm)/Co(0.4 nm)/Pt(0.4 nm), Pt(1.2 nm)/Co(0.4 nm)/Pt(1.2 nm), and Cu(1.2 nm)/Co(0.4 nm)/Cu(1.2 nm), as a function of number of the blocks. The curves 1 correspond to the ultrathin film approximation applied to all layers (Eq. (46)), the curves 2 were obtained with the ultrathin film approximation applied selectively to the magnetic layers using Eq. (45). The curves 3 were computed with Eq. (31).

Fig. 3.
Fig. 3.

The effect of the Co film thickness, d (Co), on the reflection MO azimuth rotation, θK (full lines), and ellipticity, K , (dashed) at the wavelength λ=632.8 nm. The Co film is deposited on Pt or Cu substrate. The curves 1, 2 and, 3 correspond to the approximations limited to first, second (Eq.(48)), and third orders in β (Co)=(2π/λ)N (Co) d (Co). The curves 4 were obtained without restriction on the magnetic layer thickness (Eq.(47)).

Equations (71)

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ε ( n ) = ( ε xx ( n ) ε xy ( n ) 0 ε xy ( n ) ε xx ( n ) 0 0 0 ε zz ( n ) ) .
[ ( c ω γ z ( n ) ) 2 ε xx ( n ) ε xy ( n ) 0 ε xy ( n ) ( c ω γ z ( n ) ) 2 ε xx ( n ) 0 0 0 ε zz ( n ) ] [ E 0 x ( n ) E 0 y ( n ) E 0 z ( n ) ] = 0 ,
E 0 ( 0 ) = [ E 01 ( 0 ) E 02 ( 0 ) E 03 ( 0 ) E 04 ( 0 ) ] = [ M 11 M 12 0 0 M 21 M 22 0 0 0 0 M 33 M 34 0 0 M 43 M 44 ] [ E 01 ( 𝓝 + 1 ) E 02 ( 𝓝 + 1 ) E 03 ( 𝓝 + 1 ) E 04 ( 𝓝 + 1 ) ] = M E 0 ( 𝓝 + 1 ) .
M = [ D ( 0 ) ] i n = 1 𝓝 S ( n ) D ( 𝓝 + 1 ) ,
S ( n ) = D ( n ) P ( n ) [ D ( n ) ] 1 =
= [ cos β + ( n ) i N + ( n ) 1 sin β + ( n ) 0 0 i N + ( n ) sin β + ( n ) cos β + ( n ) 0 0 0 0 cos β ( n ) i N ( n ) 1 sin β ( n ) 0 0 i N ( n ) sin β ( n ) cos β ( n ) ]
β ± ( n ) = ω c N ± ( n ) d n ,
D ( n ) = [ 1 1 0 0 N + ( n ) N + ( n ) 0 0 0 0 1 1 0 0 N ( n ) N ( n ) ]
P ( n ) = [ exp ( i β + ( n ) ) 0 0 0 0 exp ( i β + ( n ) ) 0 0 0 0 exp ( i β ( n ) ) 0 0 0 0 exp ( i β ( n ) ) ] ,
[ E + ( t ) E ( t ) ] = [ t + 0 0 t ] [ E + ( i ) E ( i ) ] ,
[ E + ( r ) E ( r ) ] = [ r + 0 0 r ] [ E + ( i ) E ( i ) ] ,
[ t ] = [ t xx t xy t xy t xx ] = 1 2 [ ( t + t + ) i ( t t + ) i ( t t + ) ( t + t + ) ] ,
[ r ] = [ r xx r xy r xy r xx ] = 1 2 [ ( r + r + ) i ( r r + ) i ( r r + ) ( r + r + ) ] .
[ E x ( t ) E y ( t ) ] = [ t ] [ E x ( i ) E y ( i ) ] ,
[ E x ( r ) E y ( r ) ] = [ r ] [ E x ( i ) E y ( i ) ]
χ t = t xy t xx = i t + t t + + t = i M 11 M 33 M 11 + M 33 ,
χ r = r xy r xx = i r + r r + + r = i M 43 M 11 M 21 M 33 M 43 M 11 + M 21 M 33 .
t xy ( 0 , 𝓝 + 1 ) = i 2 ( t + ( 0 , 𝓝 + 1 ) t ( 0 , 𝓝 + 1 ) ) ,
r xy ( 0 , 𝓝 + 1 ) = i 2 ( r + ( 0 , 𝓝 + 1 ) r ( 0 , 𝓝 + 1 ) ) .
Δ N ( n ) = 1 2 ( N + ( n ) N ( n ) ) = i ε xy ( n ) 2 N ( n ) ,
t xx ( 0 , 𝓝 + 1 ) = 1 2 ( t + ( 0 , 𝓝 + 1 ) + t ( 0 , 𝓝 + 1 ) ) ,
r xx ( 0 , 𝓝 + 1 ) = 1 2 ( r + ( 0 , 𝓝 + 1 ) + r ( 0 , 𝓝 + 1 ) ) ,
t xy ( 0 , 𝓝 + 1 ) = i Δ t xx ( 0 , 𝓝 + 1 ) = i n = 1 𝓝 t xx ( 0 , 𝓝 + 1 ) N n Δ N n ,
r xy ( 0 , 𝓝 + 1 ) = i Δ r xx ( 0 , 𝓝 + 1 ) = i n = 1 𝓝 r xx ( 0 , 𝓝 + 1 ) N n Δ N n ,
t xx ( 0 , 𝓝 + 1 ) = 1 M 11 , t xy ( 0 , 𝓝 + 1 ) = i Δ ( M 11 ) M 11 2 = i M 11 2 n = 1 𝓝 M 11 N n Δ N n ,
r xx ( 0 , 𝓝 + 1 ) = M 21 M 11 ,
r xy ( 0 , 𝓝 + 1 ) = i Δ ( M 21 M 11 ) = i M 11 2 ( M 21 n = 1 𝓝 + 1 M 11 N n M 11 n = 1 𝓝 + 1 M 21 N n ) Δ N n
χ t ( 0 , 𝓝 + 1 ) = t xy t xx = i Δ ( M 11 ) M 11 = i n = 1 𝓝 + 1 ln ( M 11 ) N n Δ N n
χ r ( 0 , 𝓝 + 1 ) = r xy r xx = i M 11 Δ ( M 21 ) M 21 Δ ( M 11 ) M 11 M 21 = i n = 1 𝓝 + 1 N n [ ln ( M 21 M 11 ) ] Δ N n .
M = C L q W ,
L = [ m 11 + m 12 + 0 0 m 21 + m 22 + 0 0 0 0 m 11 m 12 0 0 m 21 m 22 ] .
L q = [ m 11 + p q + p q 1 + m 12 + p q + 0 0 m 21 + p q + m 11 + p q + p q 1 + 0 0 0 0 m 11 p q p q 1 m 12 p q 0 0 m 21 p q m 11 p q p q 1 ] .
L = S ( A ) S ( B ) S ( A ) ,
m 11 ± = cos β ± ( A ) cos β ± ( B ) 1 2 ( N ± ( A ) N ± ( B ) + N ± ( B ) N ± ( A ) ) sin β ± ( A ) sin β ± ( B ) ,
m 12 ( ± ) = i N ± ( A ) [ sin β ± ( A ) cos β ± ( B ) + 1 2 ( N ± ( A ) N ± ( B ) + N ± ( B ) N ± ( A ) ) cos β ± ( A ) sin β ± ( B )
+ 1 2 ( N ± ( A ) N ± ( B ) N ± ( B ) N ± ( A ) = sin β ± ( B ) ] ,
m 21 ± = i N ± ( A ) [ sin β ± ( A ) cos β ± ( B ) + 1 2 ( N ± ( A ) N ± ( B ) + N ± ( B ) N ± ( A ) ) cos β ± ( A ) sin β ± ( B )
1 2 ( N ± ( A ) N ± ( B ) N ± ( B ) N ± ( A ) ) sin β ± ( B ) ] .
M = C L q W = 1 2 N 0 ( N 0 1 N 0 1 ) ( L 11 ( q ) L 12 ( q ) L 21 ( q ) L 11 ( q ) ) ( 1 1 N 2 N 2 ) .
χ t ( 0 , q L , 2 ) = i { p q [ N 0 ( m 11 + N 2 m 12 ) + ( N 2 m 11 + m 21 ) ] p q 1 ( N 0 + N 2 ) } 1
× { [ ( p q p q 1 ) ( N 0 + N 2 ) + p q N 0 ( m 11 + N 2 m 12 )
+ p q ( N 2 m 11 + m 21 ) ] Δ m 11 + p q ( N 0 N 2 Δ m 12 + Δ m 21 ) } ,
χ r ( 0 , q L , 2 ) = 2 i N 0 { p q [ N 0 ( m 11 + N 2 m 12 ) + ( N 2 m 11 + m 21 ) ] p q 1 ( N 0 + N 2 ) } 1
× { p q [ N 0 ( m 11 + N 2 m 12 ) ( N 2 m 11 + m 21 ) ] p q 1 ( N 0 N 2 ) } 1
× { Δ m 11 ( N 2 2 m 12 m 21 ) ( p q 2 + p q p q 1 p q p q 1 ) Δ m 12 N 2 p q
× [ ( N 2 m 11 + m 21 ) p q N 2 p q 1 ] + Δ m 21 p q [ p q ( m 11 + N 2 m 12 ) p q 1 ] } .
m 11 = U AB [ 1 r AB 2 e 2 i β A + e 2 i β B ( e 2 i β A r AB 2 ) ] ,
m 12 = U AB N A [ ( 1 + r AB e i β A ) 2 e 2 i β B ( r AB + e i β A ) 2 ] ,
m 21 = U AB N A [ ( 1 r AB e i β A ) 2 e 2 i β B ( r AB e i β A ) 2 ] .
r n 1 , n = N n 1 N n N n 1 + N n , and t n 1 , n = 2 N n 1 N n 1 + N n .
Δ m 11 = i Δ N B N B U AB { β B [ 1 r AB 2 e 2 i β A e 2 i β B ( e 2 i β A r AB 2 ) ]
+ i r AB ( 1 e 2 i β A ) ( 1 e 2 i β B ) } ,
Δ m 12 = i Δ N B N B U AB 1 N A { β B [ ( 1 + r AB e i β A ) 2 + e 2 i β B ( r AB + e i β A ) 2 ]
+ i ( 1 + r AB e i β A ) ( r AB + e i β A ) ( 1 e 2 i β B ) } ,
Δ m 21 = i Δ N B N B U AB N A { β B [ ( 1 r AB e i β A ) 2 + e 2 i β B ( r AB e i β A ) 2 ]
+ i ( 1 r AB e i β A ) ( r AB e i β A ) ( 1 e 2 i β B ) } .
m 11 cos β A , m 12 i N A sin β A , m 21 i N A sin β A .
Δ m 11 β B sin β A Δ N B N A ,
Δ m 12 i β B ( 1 cos β A ) Δ N B N A 2 ,
Δ m 21 i β B ( 1 + cos β A ) Δ N B .
χ r ( 0 , q L , A ) 4 N 0 β B Δ N B p q ( N 0 2 N A 2 ) ( p q e i β A p q 1 ) ,
e i q β A = p q e i β A p q 1
p q = e i ( q 1 ) β A ( 1 + e 2 i β A + e 4 i β A + + e 2 i ( q 1 ) β A ) ,
χ r ( 0 , qL , A ) 4 N 0 β B Δ N B e i β A ( 1 e 2 i q β A ) ( N 0 2 N A 2 ) ( 1 e 2 i β A ) .
χ r ( 0 , qL , 2 ) 4 q β B N 0 Δ N B N 0 2 N 2 2 .
χ r ( 0 , 1 , 2 ) = Δ N 1 2 N 1 ( 1 + e 2 i β 1 r 01 r 12 ) 1 ( r 01 + e 2 i β 1 r 12 ) 1
× ( 1 r 01 2 ) [ 4 β 1 r 12 e 2 i β 1 i ( 1 e 2 i β 1 ) ( 1 + r 12 2 e 2 i β 1 ) ] .
χ r ( 0 , 1 , 2 ) 4 ( Δ N 1 ) N 0 N 0 2 N 2 2 [ β 1 + N 2 N 1 ( 1 2 N 0 2 N 1 2 N 0 2 N 2 2 ) β 1 2 ] .
p 0 = 0 , p 6 = 32 m 11 5 32 m 11 3 + 6 m 11 , p 1 = 1 , p 7 = 64 m 11 6 80 m 11 4 + 24 m 11 2 1 , p 2 = 2 m 11 , p 8 = 128 m 11 7 192 m 11 5 + 80 m 11 3 8 m 11 , p 3 = 4 m 11 2 1 , p 9 = 256 m 11 8 448 m 11 6 + 240 m 11 4 40 m 11 2 + 1 , p 4 = 8 m 11 3 4 m 11 , p 10 = 512 m 11 9 1024 m 11 7 + 672 m 11 5 p 5 = 16 m 11 4 12 m 11 2 + 1 , 160 m 11 3 + 10 m 11 ,
[ p 0 ] = 0 , [ p 4 ] = 6 p 3 + 2 p 1 , [ p 8 ] = 14 p 7 + 10 p 5 + 6 p 3 + 2 p 1 , [ p 1 ] = 0 , [ p 5 ] = 8 p 4 + 4 p 2 , [ p 9 ] = 16 p 8 + 12 p 6 + 8 p 4 + 4 p 2 , [ p 2 ] = 2 p 1 = 2 , [ p 6 ] = 10 p 5 + 6 p 3 + 2 p 1 , [ p 10 ] = 18 p 9 + 14 p 7 + 10 p 5 + [ p 3 ] = 4 p 2 , [ p 7 ] = 12 p 6 + 8 p 4 + 4 p 2 , + 6 p 3 + 2 p 1 .
p q + 1 = 2 m 11 p q p q 1 , p q + 1 2 2 m 11 p q p q + 1 + p q 2 = 1 , p q + 2 = 2 ( q + 1 ) p q + 1 + p q , p q 2 p q + 1 p q 1 1 = 0 , p q + 1 = p q [ m 11 p q + 1 p q ] + p q p q + 1 p q + 1 m 11 p q .

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