Abstract

A sensor capable of mapping microwave (mw) currents in semiconductor circuits can be realized by exploiting magneto-optic effects (MO) at transverse magnetization (M) in ultrathin ferromagnetic or ferrimagnetic films. In the sensor, Mwould be induced in the magnetic film by the fringing fields of mw currents flowing in the semiconductor circuit along the plane of incidence. In this work, an evaluation of MO sensor performance was made for nanostructures consisting of ultrathin Fe layers sandwiched between AlN dielectric layers. The multilayer thin film stacks were grown on Si wafer substrates. The performance of the sensor systems is characterized in terms of magnetization-induced changes in the MO multilayer reflection coefficients, expressed analytically. Sensor configurations which optimize the operation at the laser wavelength of 410 nm, and which are still easy to fabricate, are proposed. Modeling predicts the strongest MO enhancement in a sensor incorporating two Fe nanolayers, each of a different thickness, formed by the layer sequence AlN/Fe/AlN/Fe/AlN/Au/Si. The use of ferrimagnetic hexagonal ferrite films with the in-plane c-axis as an alternative sensor material is also discussed.

© 2017 Optical Society of America

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References

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2016 (4)

J. A. Girón-Sedas, J. R. Mejía-Salazar, E. Moncada-Villa, and N. Porras-Montenegro, “Enhancement of the transverse magneto-optical Kerr effect via resonant tunneling in trilayers containing magneto-optical metals,” Appl. Phys. Lett. 109(3), 033106 (2016).
[Crossref]

J. H. Liang, Y. L. Chen, L. Sun, C. Zhou, Y. Yang, and Y. Z. Wu, “The anisotropic linear and quadratic magneto-optical Kerr effects in epitaxial Fe/GaAs(110) film,” Appl. Phys. Lett. 108(8), 082404 (2016).
[Crossref]

A. S. Sokolov, M. Geiler, and V. G. Harris, “Broadband ferromagnetic resonance linewidth measurement by a microstripline transmission resonator,” Appl. Phys. Lett. 108(17), 172408 (2016).
[Crossref]

T. Kaihara, H. Shimizu, A. Cebollada, and G. Armelles, “Magnetic field control and wavelength tunability of SPP excitations using Al2O3/SiO2/Fe structures,” Appl. Phys. Lett. 109(11), 111102 (2016).
[Crossref]

2015 (4)

J. McCord, “Progress in magnetic domain observation by advanced magneto-optical microscopy,” J. Phys. D Appl. Phys. 48(33), 333001 (2015).
[Crossref]

C. Briley, D. Schmidt, T. Hofmann, E. Schubert, and M. Schubert, “Anisotropic magneto-optical hysteresis of permalloy slanted columnar thin films determined by vector magneto-optical generalized ellipsometry,” Appl. Phys. Lett. 106(13), 133104 (2015).
[Crossref]

C. A. Herreño-Fierro and E. J. Patiño, “Maximization of surface-enhanced transversal magneto-optic Kerr effect in Au/Co/Au thin films,” Phys. Status Solidi, B Basic Res. 252(2), 316–322 (2015).
[Crossref]

M. Moradi, S. M. Mohseni, S. Mahmoodi, D. Rezvani, N. Ansari, S. Chung, and J. Åkerman, “Au/NiFe Magnetoplasmonics: Large Enhancement of Magneto-Optical Kerr,” Electron. Mater. Lett. 11(3), 440–446 (2015).
[Crossref]

2013 (4)

M. Pohl, L. E. Kreilkamp, V. I. Belotelov, I. A. Akimov, A. N. Kalish, N. E. Khokhlov, V. J. Yallapragada, A. V. Gopal, M. Nur-E-Alam, M. Vasiliev, D. R. Yakovlev, K. Alameh, A. K. Zvezdin, and M. Bayer, “Tuning of the transverse magneto-optical effect in magneto-plasmonic crystals,” New J. Phys. 15, 075024 (2013).
[Crossref]

L. Halagačka, M. Vanwolleghem, K. Postava, B. Dagens, and J. Pištora, “Coupled mode enhanced giant magnetoplasmonics transverse Kerr effect,” Opt. Express 21(19), 21741–21755 (2013).
[Crossref] [PubMed]

Š. Višňovský, E. Lišková-Jakubisová, I. Harward, and Z. Celinski, “Analytical analysis of a multilayer structure with ultrathin Fe film for magneto-optical sensing,” Opt. Express 21(3), 3400–3416 (2013).
[Crossref] [PubMed]

I. Harward, Y. Nie, D. Chen, J. Baptist, J. M. Shaw, E. Jakubisová-Lišková, Š. Višňovský, P. Široký, M. Lesňák, J. Pištora, and Z. Celinski, “Physical properties of Al doped Ba hexagonal ferrite thin films,” J. Appl. Phys. 113(4), 043903 (2013).
[Crossref]

2012 (2)

V. Zayets, H. Saito, S. Yuasa, and K. Ando, “Enhancement of the transverse non-reciprocal magneto-optical effect,” J. Appl. Phys. 111(2), 023103 (2012).
[Crossref]

J. A. Arregi, J. B. Gonzalez-Diaz, E. Bergaretxe, O. Idigoras, T. Unsal, and A. Berger, “Study of generalized magneto-optical ellipsometry measurement reliability,” J. Appl. Phys. 111(10), 103912 (2012).
[Crossref]

2011 (3)

V. I. Belotelov, I. A. Akimov, M. Pohl, V. A. Kotov, S. Kasture, A. S. Vengurlekar, A. V. Gopal, D. R. Yakovlev, A. K. Zvezdin, and M. Bayer, “Enhanced magneto-optical effects in magnetoplasmonic crystals,” Nat. Nanotechnol. 6(6), 370–376 (2011).
[Crossref] [PubMed]

I. Harward, T. O’Keevan, A. Hutchison, V. Zagorodnii, and Z. Celinski, “A broadband ferromagnetic resonance spectrometer to measure thin films up to 70 GHz,” Rev. Sci. Instrum. 82(9), 095115 (2011).
[Crossref] [PubMed]

A. Boltasseva and H. A. Atwater, “Materials science. Low-loss plasmonic metamaterials,” Science 331(6015), 290–291 (2011).
[Crossref] [PubMed]

2010 (1)

J. Pištora, M. Lesňák, E. Lišková, Š. Višňovský, I. Harward, P. Maslankiewicz, K. Balin, Z. Celinski, J. Mistrík, T. Yamaguchi, R. Lopusnik, and J. Vlček, “The effect of FeF2 on the magneto-optic response in FeF2/Fe/FeF2 sandwiches,” J. Phys. D Appl. Phys. 43(15), 155301 (2010).
[Crossref]

2009 (1)

Y. Dong and X. Zhang, “Enhanced magneto-optical Kerr effect in magnetic multilayers containing double-negative metamaterials,” J. Appl. Phys. 105(5), 054105 (2009).
[Crossref]

2008 (2)

Y.-Y. Song, J. Das, Z. Wang, W. Tong, and C. E. Patton, “In-plane oriented barium ferrite films with self-bias and low microwave loss,” Appl. Phys. Lett. 93(17), 172503 (2008).
[Crossref]

E. Liskova, S. Visnovsky, R. Lopusnik, I. Harward, M. Wenger, T. Christensen, and Z. Celinski, “Magneto-optical AlN/Fe/AlN structures optimized for operation in the violet spectral region,” J. Phys. D Appl. Phys. 41(15), 155007 (2008).
[Crossref]

2006 (1)

Y. Shi, J. Zukrowski, M. Przybylski, M. Nyvlt, A. Winkelmann, J. Barthel, and J. Kirschner, “A mode-of-growth-dependent magneto-optical response from ultrathin Co film on Pd surfaces,” Surf. Sci. 600(18), 4180–4184 (2006).
[Crossref]

2005 (1)

V. S. Merkulov, “Matrix of light reflection from a crystal with an arbitrary permittivity tensor,” J. Appl. Spectrosc. 72(4), 610–613 (2005).
[Crossref]

2003 (1)

K.-S. Lee, S.-K. Kim, and J. B. Kortright, “Atomic-scale depth selectivity of soft x-ray resonant Kerr effect,” Appl. Phys. Lett. 83(18), 3764–3766 (2003).
[Crossref]

2002 (1)

2001 (1)

2000 (1)

M. R. Pufall and A. Berger, “Studying the reversal mode of the magnetization vector versus applied field angle using generalized magneto-optical ellipsometry,” J. Appl. Phys. 87(9), 5834–5836 (2000).
[Crossref]

1999 (3)

A. Berger and M. R. Pufall, “Quantitative vector magnetometry using generalized magneto-optical ellipsometry,” J. Appl. Phys. 85(8), 4583–4585 (1999).
[Crossref]

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(1), 177–187 (1999).
[Crossref] [PubMed]

K. Postava, J. Pištora, and Š. Višňovský, “Magneto-optical effects in ultrathin structures at transversal magnetization,” Czech. J. Phys. B 49(8), 1185–1204 (1999).
[Crossref]

1996 (2)

Š. Višňovský, R. Krishnan, M. Nývlt, and V. Prosser, “Optical behaviour of Fe in magnetic multilayers,” J. Magn. Soc. Jpn. 20(S1), 41–46 (1996).
[Crossref]

A. Y. Elezzabi and M. R. Freeman, “Ultrafast magneto-optic sampling of picosecond current pulses,” Appl. Phys. Lett. 68(25), 3546–3548 (1996).
[Crossref]

1994 (1)

R. Atkinson, P. Papakonstantinou, I. W. Salter, and R. Gerber, “Optical and magneto-optical properties of Co-Ti-substituted barium hexaferrite single crystals and thin films produced by laser ablation deposition,” J. Magn. Magn. Mater. 138(1-2), 222–231 (1994).
[Crossref]

1990 (3)

1986 (2)

Š. Višňovský, “Magneto-optical ellipsometry,” Czech. J. Phys. B 36(5), 625–650 (1986).
[Crossref]

Š. Višňovský, “Magneto-optical transverse Kerr effect in a film-substrate system,” Czech. J. Phys. B 36(10), 1203–1208 (1986).
[Crossref]

1980 (2)

P. Yeh, “Optics of anisotropic layered media: A new 4×4 matrix algebra,” Surf. Sci. 96(1-3), 41–53 (1980).
[Crossref]

R. E. McClure, “An electrical equivalent circuit for the transverse magneto-optic effect in thin films,” IEEE Trans. Magn. 16(5), 1185–1187 (1980).
[Crossref]

1974 (1)

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(12), 5056–5070 (1974).
[Crossref]

1969 (1)

G. S. Krinchik and V. S. Gushchin, “Investigation of interband transitions in ferromagnetic metals by the magneto-optical method,” Sov. Phys. JETP 29(6), 984–988 (1969).

1968 (1)

G. S. Krinchik and V. A. Artemjev, “Magneto-optic properties of nickel, iron and cobalt,” J. Appl. Phys. 39(2), 1276–1278 (1968).
[Crossref]

1950 (1)

F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. (Paris) 12(5), 596–640, 706–782 (1950).
[Crossref]

1948 (1)

C. Kittel, “On the theory of ferromagnetic resonance absorption,” Phys. Rev. 73(2), 155–161 (1948).
[Crossref]

Abelès, F.

F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. (Paris) 12(5), 596–640, 706–782 (1950).
[Crossref]

Åkerman, J.

M. Moradi, S. M. Mohseni, S. Mahmoodi, D. Rezvani, N. Ansari, S. Chung, and J. Åkerman, “Au/NiFe Magnetoplasmonics: Large Enhancement of Magneto-Optical Kerr,” Electron. Mater. Lett. 11(3), 440–446 (2015).
[Crossref]

Akimov, I. A.

M. Pohl, L. E. Kreilkamp, V. I. Belotelov, I. A. Akimov, A. N. Kalish, N. E. Khokhlov, V. J. Yallapragada, A. V. Gopal, M. Nur-E-Alam, M. Vasiliev, D. R. Yakovlev, K. Alameh, A. K. Zvezdin, and M. Bayer, “Tuning of the transverse magneto-optical effect in magneto-plasmonic crystals,” New J. Phys. 15, 075024 (2013).
[Crossref]

V. I. Belotelov, I. A. Akimov, M. Pohl, V. A. Kotov, S. Kasture, A. S. Vengurlekar, A. V. Gopal, D. R. Yakovlev, A. K. Zvezdin, and M. Bayer, “Enhanced magneto-optical effects in magnetoplasmonic crystals,” Nat. Nanotechnol. 6(6), 370–376 (2011).
[Crossref] [PubMed]

Alameh, K.

M. Pohl, L. E. Kreilkamp, V. I. Belotelov, I. A. Akimov, A. N. Kalish, N. E. Khokhlov, V. J. Yallapragada, A. V. Gopal, M. Nur-E-Alam, M. Vasiliev, D. R. Yakovlev, K. Alameh, A. K. Zvezdin, and M. Bayer, “Tuning of the transverse magneto-optical effect in magneto-plasmonic crystals,” New J. Phys. 15, 075024 (2013).
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Ando, K.

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T. Kaihara, H. Shimizu, A. Cebollada, and G. Armelles, “Magnetic field control and wavelength tunability of SPP excitations using Al2O3/SiO2/Fe structures,” Appl. Phys. Lett. 109(11), 111102 (2016).
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E. Liskova, S. Visnovsky, R. Lopusnik, I. Harward, M. Wenger, T. Christensen, and Z. Celinski, “Magneto-optical AlN/Fe/AlN structures optimized for operation in the violet spectral region,” J. Phys. D Appl. Phys. 41(15), 155007 (2008).
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J. H. Liang, Y. L. Chen, L. Sun, C. Zhou, Y. Yang, and Y. Z. Wu, “The anisotropic linear and quadratic magneto-optical Kerr effects in epitaxial Fe/GaAs(110) film,” Appl. Phys. Lett. 108(8), 082404 (2016).
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E. Liskova, S. Visnovsky, R. Lopusnik, I. Harward, M. Wenger, T. Christensen, and Z. Celinski, “Magneto-optical AlN/Fe/AlN structures optimized for operation in the violet spectral region,” J. Phys. D Appl. Phys. 41(15), 155007 (2008).
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[Crossref] [PubMed]

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J. Pištora, M. Lesňák, E. Lišková, Š. Višňovský, I. Harward, P. Maslankiewicz, K. Balin, Z. Celinski, J. Mistrík, T. Yamaguchi, R. Lopusnik, and J. Vlček, “The effect of FeF2 on the magneto-optic response in FeF2/Fe/FeF2 sandwiches,” J. Phys. D Appl. Phys. 43(15), 155301 (2010).
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I. Harward, T. O’Keevan, A. Hutchison, V. Zagorodnii, and Z. Celinski, “A broadband ferromagnetic resonance spectrometer to measure thin films up to 70 GHz,” Rev. Sci. Instrum. 82(9), 095115 (2011).
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J. A. Arregi, J. B. Gonzalez-Diaz, E. Bergaretxe, O. Idigoras, T. Unsal, and A. Berger, “Study of generalized magneto-optical ellipsometry measurement reliability,” J. Appl. Phys. 111(10), 103912 (2012).
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I. Harward, Y. Nie, D. Chen, J. Baptist, J. M. Shaw, E. Jakubisová-Lišková, Š. Višňovský, P. Široký, M. Lesňák, J. Pištora, and Z. Celinski, “Physical properties of Al doped Ba hexagonal ferrite thin films,” J. Appl. Phys. 113(4), 043903 (2013).
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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(12), 5056–5070 (1974).
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T. Kaihara, H. Shimizu, A. Cebollada, and G. Armelles, “Magnetic field control and wavelength tunability of SPP excitations using Al2O3/SiO2/Fe structures,” Appl. Phys. Lett. 109(11), 111102 (2016).
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M. Pohl, L. E. Kreilkamp, V. I. Belotelov, I. A. Akimov, A. N. Kalish, N. E. Khokhlov, V. J. Yallapragada, A. V. Gopal, M. Nur-E-Alam, M. Vasiliev, D. R. Yakovlev, K. Alameh, A. K. Zvezdin, and M. Bayer, “Tuning of the transverse magneto-optical effect in magneto-plasmonic crystals,” New J. Phys. 15, 075024 (2013).
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V. I. Belotelov, I. A. Akimov, M. Pohl, V. A. Kotov, S. Kasture, A. S. Vengurlekar, A. V. Gopal, D. R. Yakovlev, A. K. Zvezdin, and M. Bayer, “Enhanced magneto-optical effects in magnetoplasmonic crystals,” Nat. Nanotechnol. 6(6), 370–376 (2011).
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G. S. Krinchik and V. S. Gushchin, “Investigation of interband transitions in ferromagnetic metals by the magneto-optical method,” Sov. Phys. JETP 29(6), 984–988 (1969).

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I. Harward, Y. Nie, D. Chen, J. Baptist, J. M. Shaw, E. Jakubisová-Lišková, Š. Višňovský, P. Široký, M. Lesňák, J. Pištora, and Z. Celinski, “Physical properties of Al doped Ba hexagonal ferrite thin films,” J. Appl. Phys. 113(4), 043903 (2013).
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J. Pištora, M. Lesňák, E. Lišková, Š. Višňovský, I. Harward, P. Maslankiewicz, K. Balin, Z. Celinski, J. Mistrík, T. Yamaguchi, R. Lopusnik, and J. Vlček, “The effect of FeF2 on the magneto-optic response in FeF2/Fe/FeF2 sandwiches,” J. Phys. D Appl. Phys. 43(15), 155301 (2010).
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J. H. Liang, Y. L. Chen, L. Sun, C. Zhou, Y. Yang, and Y. Z. Wu, “The anisotropic linear and quadratic magneto-optical Kerr effects in epitaxial Fe/GaAs(110) film,” Appl. Phys. Lett. 108(8), 082404 (2016).
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E. Liskova, S. Visnovsky, R. Lopusnik, I. Harward, M. Wenger, T. Christensen, and Z. Celinski, “Magneto-optical AlN/Fe/AlN structures optimized for operation in the violet spectral region,” J. Phys. D Appl. Phys. 41(15), 155007 (2008).
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J. Pištora, M. Lesňák, E. Lišková, Š. Višňovský, I. Harward, P. Maslankiewicz, K. Balin, Z. Celinski, J. Mistrík, T. Yamaguchi, R. Lopusnik, and J. Vlček, “The effect of FeF2 on the magneto-optic response in FeF2/Fe/FeF2 sandwiches,” J. Phys. D Appl. Phys. 43(15), 155301 (2010).
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Lopusnik, R.

J. Pištora, M. Lesňák, E. Lišková, Š. Višňovský, I. Harward, P. Maslankiewicz, K. Balin, Z. Celinski, J. Mistrík, T. Yamaguchi, R. Lopusnik, and J. Vlček, “The effect of FeF2 on the magneto-optic response in FeF2/Fe/FeF2 sandwiches,” J. Phys. D Appl. Phys. 43(15), 155301 (2010).
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J. Pištora, M. Lesňák, E. Lišková, Š. Višňovský, I. Harward, P. Maslankiewicz, K. Balin, Z. Celinski, J. Mistrík, T. Yamaguchi, R. Lopusnik, and J. Vlček, “The effect of FeF2 on the magneto-optic response in FeF2/Fe/FeF2 sandwiches,” J. Phys. D Appl. Phys. 43(15), 155301 (2010).
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M. Moradi, S. M. Mohseni, S. Mahmoodi, D. Rezvani, N. Ansari, S. Chung, and J. Åkerman, “Au/NiFe Magnetoplasmonics: Large Enhancement of Magneto-Optical Kerr,” Electron. Mater. Lett. 11(3), 440–446 (2015).
[Crossref]

Moncada-Villa, E.

J. A. Girón-Sedas, J. R. Mejía-Salazar, E. Moncada-Villa, and N. Porras-Montenegro, “Enhancement of the transverse magneto-optical Kerr effect via resonant tunneling in trilayers containing magneto-optical metals,” Appl. Phys. Lett. 109(3), 033106 (2016).
[Crossref]

Moradi, M.

M. Moradi, S. M. Mohseni, S. Mahmoodi, D. Rezvani, N. Ansari, S. Chung, and J. Åkerman, “Au/NiFe Magnetoplasmonics: Large Enhancement of Magneto-Optical Kerr,” Electron. Mater. Lett. 11(3), 440–446 (2015).
[Crossref]

Nie, Y.

I. Harward, Y. Nie, D. Chen, J. Baptist, J. M. Shaw, E. Jakubisová-Lišková, Š. Višňovský, P. Široký, M. Lesňák, J. Pištora, and Z. Celinski, “Physical properties of Al doped Ba hexagonal ferrite thin films,” J. Appl. Phys. 113(4), 043903 (2013).
[Crossref]

Nur-E-Alam, M.

M. Pohl, L. E. Kreilkamp, V. I. Belotelov, I. A. Akimov, A. N. Kalish, N. E. Khokhlov, V. J. Yallapragada, A. V. Gopal, M. Nur-E-Alam, M. Vasiliev, D. R. Yakovlev, K. Alameh, A. K. Zvezdin, and M. Bayer, “Tuning of the transverse magneto-optical effect in magneto-plasmonic crystals,” New J. Phys. 15, 075024 (2013).
[Crossref]

Nyvlt, M.

Y. Shi, J. Zukrowski, M. Przybylski, M. Nyvlt, A. Winkelmann, J. Barthel, and J. Kirschner, “A mode-of-growth-dependent magneto-optical response from ultrathin Co film on Pd surfaces,” Surf. Sci. 600(18), 4180–4184 (2006).
[Crossref]

Nývlt, M.

Š. Višňovský, R. Krishnan, M. Nývlt, and V. Prosser, “Optical behaviour of Fe in magnetic multilayers,” J. Magn. Soc. Jpn. 20(S1), 41–46 (1996).
[Crossref]

O’Keevan, T.

I. Harward, T. O’Keevan, A. Hutchison, V. Zagorodnii, and Z. Celinski, “A broadband ferromagnetic resonance spectrometer to measure thin films up to 70 GHz,” Rev. Sci. Instrum. 82(9), 095115 (2011).
[Crossref] [PubMed]

Papakonstantinou, P.

R. Atkinson, P. Papakonstantinou, I. W. Salter, and R. Gerber, “Optical and magneto-optical properties of Co-Ti-substituted barium hexaferrite single crystals and thin films produced by laser ablation deposition,” J. Magn. Magn. Mater. 138(1-2), 222–231 (1994).
[Crossref]

Patiño, E. J.

C. A. Herreño-Fierro and E. J. Patiño, “Maximization of surface-enhanced transversal magneto-optic Kerr effect in Au/Co/Au thin films,” Phys. Status Solidi, B Basic Res. 252(2), 316–322 (2015).
[Crossref]

Patton, C. E.

Y.-Y. Song, J. Das, Z. Wang, W. Tong, and C. E. Patton, “In-plane oriented barium ferrite films with self-bias and low microwave loss,” Appl. Phys. Lett. 93(17), 172503 (2008).
[Crossref]

Pištora, J.

I. Harward, Y. Nie, D. Chen, J. Baptist, J. M. Shaw, E. Jakubisová-Lišková, Š. Višňovský, P. Široký, M. Lesňák, J. Pištora, and Z. Celinski, “Physical properties of Al doped Ba hexagonal ferrite thin films,” J. Appl. Phys. 113(4), 043903 (2013).
[Crossref]

L. Halagačka, M. Vanwolleghem, K. Postava, B. Dagens, and J. Pištora, “Coupled mode enhanced giant magnetoplasmonics transverse Kerr effect,” Opt. Express 21(19), 21741–21755 (2013).
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J. Pištora, M. Lesňák, E. Lišková, Š. Višňovský, I. Harward, P. Maslankiewicz, K. Balin, Z. Celinski, J. Mistrík, T. Yamaguchi, R. Lopusnik, and J. Vlček, “The effect of FeF2 on the magneto-optic response in FeF2/Fe/FeF2 sandwiches,” J. Phys. D Appl. Phys. 43(15), 155301 (2010).
[Crossref]

K. Postava, J. Pištora, and Š. Višňovský, “Magneto-optical effects in ultrathin structures at transversal magnetization,” Czech. J. Phys. B 49(8), 1185–1204 (1999).
[Crossref]

Pohl, M.

M. Pohl, L. E. Kreilkamp, V. I. Belotelov, I. A. Akimov, A. N. Kalish, N. E. Khokhlov, V. J. Yallapragada, A. V. Gopal, M. Nur-E-Alam, M. Vasiliev, D. R. Yakovlev, K. Alameh, A. K. Zvezdin, and M. Bayer, “Tuning of the transverse magneto-optical effect in magneto-plasmonic crystals,” New J. Phys. 15, 075024 (2013).
[Crossref]

V. I. Belotelov, I. A. Akimov, M. Pohl, V. A. Kotov, S. Kasture, A. S. Vengurlekar, A. V. Gopal, D. R. Yakovlev, A. K. Zvezdin, and M. Bayer, “Enhanced magneto-optical effects in magnetoplasmonic crystals,” Nat. Nanotechnol. 6(6), 370–376 (2011).
[Crossref] [PubMed]

Porras-Montenegro, N.

J. A. Girón-Sedas, J. R. Mejía-Salazar, E. Moncada-Villa, and N. Porras-Montenegro, “Enhancement of the transverse magneto-optical Kerr effect via resonant tunneling in trilayers containing magneto-optical metals,” Appl. Phys. Lett. 109(3), 033106 (2016).
[Crossref]

Postava, K.

Prosser, V.

Š. Višňovský, R. Krishnan, M. Nývlt, and V. Prosser, “Optical behaviour of Fe in magnetic multilayers,” J. Magn. Soc. Jpn. 20(S1), 41–46 (1996).
[Crossref]

Przybylski, M.

Y. Shi, J. Zukrowski, M. Przybylski, M. Nyvlt, A. Winkelmann, J. Barthel, and J. Kirschner, “A mode-of-growth-dependent magneto-optical response from ultrathin Co film on Pd surfaces,” Surf. Sci. 600(18), 4180–4184 (2006).
[Crossref]

Pufall, M. R.

M. R. Pufall and A. Berger, “Studying the reversal mode of the magnetization vector versus applied field angle using generalized magneto-optical ellipsometry,” J. Appl. Phys. 87(9), 5834–5836 (2000).
[Crossref]

A. Berger and M. R. Pufall, “Quantitative vector magnetometry using generalized magneto-optical ellipsometry,” J. Appl. Phys. 85(8), 4583–4585 (1999).
[Crossref]

Rezvani, D.

M. Moradi, S. M. Mohseni, S. Mahmoodi, D. Rezvani, N. Ansari, S. Chung, and J. Åkerman, “Au/NiFe Magnetoplasmonics: Large Enhancement of Magneto-Optical Kerr,” Electron. Mater. Lett. 11(3), 440–446 (2015).
[Crossref]

Saito, H.

V. Zayets, H. Saito, S. Yuasa, and K. Ando, “Enhancement of the transverse non-reciprocal magneto-optical effect,” J. Appl. Phys. 111(2), 023103 (2012).
[Crossref]

Salter, I. W.

R. Atkinson, P. Papakonstantinou, I. W. Salter, and R. Gerber, “Optical and magneto-optical properties of Co-Ti-substituted barium hexaferrite single crystals and thin films produced by laser ablation deposition,” J. Magn. Magn. Mater. 138(1-2), 222–231 (1994).
[Crossref]

Schmidt, D.

C. Briley, D. Schmidt, T. Hofmann, E. Schubert, and M. Schubert, “Anisotropic magneto-optical hysteresis of permalloy slanted columnar thin films determined by vector magneto-optical generalized ellipsometry,” Appl. Phys. Lett. 106(13), 133104 (2015).
[Crossref]

Schubert, E.

C. Briley, D. Schmidt, T. Hofmann, E. Schubert, and M. Schubert, “Anisotropic magneto-optical hysteresis of permalloy slanted columnar thin films determined by vector magneto-optical generalized ellipsometry,” Appl. Phys. Lett. 106(13), 133104 (2015).
[Crossref]

Schubert, M.

C. Briley, D. Schmidt, T. Hofmann, E. Schubert, and M. Schubert, “Anisotropic magneto-optical hysteresis of permalloy slanted columnar thin films determined by vector magneto-optical generalized ellipsometry,” Appl. Phys. Lett. 106(13), 133104 (2015).
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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(1), 177–187 (1999).
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I. Harward, Y. Nie, D. Chen, J. Baptist, J. M. Shaw, E. Jakubisová-Lišková, Š. Višňovský, P. Široký, M. Lesňák, J. Pištora, and Z. Celinski, “Physical properties of Al doped Ba hexagonal ferrite thin films,” J. Appl. Phys. 113(4), 043903 (2013).
[Crossref]

Shi, Y.

Y. Shi, J. Zukrowski, M. Przybylski, M. Nyvlt, A. Winkelmann, J. Barthel, and J. Kirschner, “A mode-of-growth-dependent magneto-optical response from ultrathin Co film on Pd surfaces,” Surf. Sci. 600(18), 4180–4184 (2006).
[Crossref]

Shimizu, H.

T. Kaihara, H. Shimizu, A. Cebollada, and G. Armelles, “Magnetic field control and wavelength tunability of SPP excitations using Al2O3/SiO2/Fe structures,” Appl. Phys. Lett. 109(11), 111102 (2016).
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Široký, P.

I. Harward, Y. Nie, D. Chen, J. Baptist, J. M. Shaw, E. Jakubisová-Lišková, Š. Višňovský, P. Široký, M. Lesňák, J. Pištora, and Z. Celinski, “Physical properties of Al doped Ba hexagonal ferrite thin films,” J. Appl. Phys. 113(4), 043903 (2013).
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Sokolov, A. S.

A. S. Sokolov, M. Geiler, and V. G. Harris, “Broadband ferromagnetic resonance linewidth measurement by a microstripline transmission resonator,” Appl. Phys. Lett. 108(17), 172408 (2016).
[Crossref]

Song, Y.-Y.

Y.-Y. Song, J. Das, Z. Wang, W. Tong, and C. E. Patton, “In-plane oriented barium ferrite films with self-bias and low microwave loss,” Appl. Phys. Lett. 93(17), 172503 (2008).
[Crossref]

Sun, L.

J. H. Liang, Y. L. Chen, L. Sun, C. Zhou, Y. Yang, and Y. Z. Wu, “The anisotropic linear and quadratic magneto-optical Kerr effects in epitaxial Fe/GaAs(110) film,” Appl. Phys. Lett. 108(8), 082404 (2016).
[Crossref]

Tiwald, T. E.

Tong, W.

Y.-Y. Song, J. Das, Z. Wang, W. Tong, and C. E. Patton, “In-plane oriented barium ferrite films with self-bias and low microwave loss,” Appl. Phys. Lett. 93(17), 172503 (2008).
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Unsal, T.

J. A. Arregi, J. B. Gonzalez-Diaz, E. Bergaretxe, O. Idigoras, T. Unsal, and A. Berger, “Study of generalized magneto-optical ellipsometry measurement reliability,” J. Appl. Phys. 111(10), 103912 (2012).
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van Exter, M.

Vanwolleghem, M.

Vasiliev, M.

M. Pohl, L. E. Kreilkamp, V. I. Belotelov, I. A. Akimov, A. N. Kalish, N. E. Khokhlov, V. J. Yallapragada, A. V. Gopal, M. Nur-E-Alam, M. Vasiliev, D. R. Yakovlev, K. Alameh, A. K. Zvezdin, and M. Bayer, “Tuning of the transverse magneto-optical effect in magneto-plasmonic crystals,” New J. Phys. 15, 075024 (2013).
[Crossref]

Vengurlekar, A. S.

V. I. Belotelov, I. A. Akimov, M. Pohl, V. A. Kotov, S. Kasture, A. S. Vengurlekar, A. V. Gopal, D. R. Yakovlev, A. K. Zvezdin, and M. Bayer, “Enhanced magneto-optical effects in magnetoplasmonic crystals,” Nat. Nanotechnol. 6(6), 370–376 (2011).
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Visnovsky, S.

E. Liskova, S. Visnovsky, R. Lopusnik, I. Harward, M. Wenger, T. Christensen, and Z. Celinski, “Magneto-optical AlN/Fe/AlN structures optimized for operation in the violet spectral region,” J. Phys. D Appl. Phys. 41(15), 155007 (2008).
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Višnovský, Š.

Š. Višňovský, E. Lišková-Jakubisová, I. Harward, and Z. Celinski, “Analytical analysis of a multilayer structure with ultrathin Fe film for magneto-optical sensing,” Opt. Express 21(3), 3400–3416 (2013).
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I. Harward, Y. Nie, D. Chen, J. Baptist, J. M. Shaw, E. Jakubisová-Lišková, Š. Višňovský, P. Široký, M. Lesňák, J. Pištora, and Z. Celinski, “Physical properties of Al doped Ba hexagonal ferrite thin films,” J. Appl. Phys. 113(4), 043903 (2013).
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J. Pištora, M. Lesňák, E. Lišková, Š. Višňovský, I. Harward, P. Maslankiewicz, K. Balin, Z. Celinski, J. Mistrík, T. Yamaguchi, R. Lopusnik, and J. Vlček, “The effect of FeF2 on the magneto-optic response in FeF2/Fe/FeF2 sandwiches,” J. Phys. D Appl. Phys. 43(15), 155301 (2010).
[Crossref]

K. Postava, J. Pištora, and Š. Višňovský, “Magneto-optical effects in ultrathin structures at transversal magnetization,” Czech. J. Phys. B 49(8), 1185–1204 (1999).
[Crossref]

Š. Višňovský, R. Krishnan, M. Nývlt, and V. Prosser, “Optical behaviour of Fe in magnetic multilayers,” J. Magn. Soc. Jpn. 20(S1), 41–46 (1996).
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Š. Višňovský, “Magneto-optical ellipsometry,” Czech. J. Phys. B 36(5), 625–650 (1986).
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Š. Višňovský, “Magneto-optical transverse Kerr effect in a film-substrate system,” Czech. J. Phys. B 36(10), 1203–1208 (1986).
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Vlcek, J.

J. Pištora, M. Lesňák, E. Lišková, Š. Višňovský, I. Harward, P. Maslankiewicz, K. Balin, Z. Celinski, J. Mistrík, T. Yamaguchi, R. Lopusnik, and J. Vlček, “The effect of FeF2 on the magneto-optic response in FeF2/Fe/FeF2 sandwiches,” J. Phys. D Appl. Phys. 43(15), 155301 (2010).
[Crossref]

Wang, Z.

Y.-Y. Song, J. Das, Z. Wang, W. Tong, and C. E. Patton, “In-plane oriented barium ferrite films with self-bias and low microwave loss,” Appl. Phys. Lett. 93(17), 172503 (2008).
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Wenger, M.

E. Liskova, S. Visnovsky, R. Lopusnik, I. Harward, M. Wenger, T. Christensen, and Z. Celinski, “Magneto-optical AlN/Fe/AlN structures optimized for operation in the violet spectral region,” J. Phys. D Appl. Phys. 41(15), 155007 (2008).
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Winkelmann, A.

Y. Shi, J. Zukrowski, M. Przybylski, M. Nyvlt, A. Winkelmann, J. Barthel, and J. Kirschner, “A mode-of-growth-dependent magneto-optical response from ultrathin Co film on Pd surfaces,” Surf. Sci. 600(18), 4180–4184 (2006).
[Crossref]

Woollam, J. A.

Wu, Y. Z.

J. H. Liang, Y. L. Chen, L. Sun, C. Zhou, Y. Yang, and Y. Z. Wu, “The anisotropic linear and quadratic magneto-optical Kerr effects in epitaxial Fe/GaAs(110) film,” Appl. Phys. Lett. 108(8), 082404 (2016).
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Yakovlev, D. R.

M. Pohl, L. E. Kreilkamp, V. I. Belotelov, I. A. Akimov, A. N. Kalish, N. E. Khokhlov, V. J. Yallapragada, A. V. Gopal, M. Nur-E-Alam, M. Vasiliev, D. R. Yakovlev, K. Alameh, A. K. Zvezdin, and M. Bayer, “Tuning of the transverse magneto-optical effect in magneto-plasmonic crystals,” New J. Phys. 15, 075024 (2013).
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V. I. Belotelov, I. A. Akimov, M. Pohl, V. A. Kotov, S. Kasture, A. S. Vengurlekar, A. V. Gopal, D. R. Yakovlev, A. K. Zvezdin, and M. Bayer, “Enhanced magneto-optical effects in magnetoplasmonic crystals,” Nat. Nanotechnol. 6(6), 370–376 (2011).
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Yallapragada, V. J.

M. Pohl, L. E. Kreilkamp, V. I. Belotelov, I. A. Akimov, A. N. Kalish, N. E. Khokhlov, V. J. Yallapragada, A. V. Gopal, M. Nur-E-Alam, M. Vasiliev, D. R. Yakovlev, K. Alameh, A. K. Zvezdin, and M. Bayer, “Tuning of the transverse magneto-optical effect in magneto-plasmonic crystals,” New J. Phys. 15, 075024 (2013).
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Yamaguchi, T.

J. Pištora, M. Lesňák, E. Lišková, Š. Višňovský, I. Harward, P. Maslankiewicz, K. Balin, Z. Celinski, J. Mistrík, T. Yamaguchi, R. Lopusnik, and J. Vlček, “The effect of FeF2 on the magneto-optic response in FeF2/Fe/FeF2 sandwiches,” J. Phys. D Appl. Phys. 43(15), 155301 (2010).
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S. Višňovský, K. Postava, T. Yamaguchi, and R. Lopusník, “Magneto-optic ellipsometry in exchange-coupled films,” Appl. Opt. 41(19), 3950–3960 (2002).
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J. H. Liang, Y. L. Chen, L. Sun, C. Zhou, Y. Yang, and Y. Z. Wu, “The anisotropic linear and quadratic magneto-optical Kerr effects in epitaxial Fe/GaAs(110) film,” Appl. Phys. Lett. 108(8), 082404 (2016).
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V. Zayets, H. Saito, S. Yuasa, and K. Ando, “Enhancement of the transverse non-reciprocal magneto-optical effect,” J. Appl. Phys. 111(2), 023103 (2012).
[Crossref]

Zagorodnii, V.

I. Harward, T. O’Keevan, A. Hutchison, V. Zagorodnii, and Z. Celinski, “A broadband ferromagnetic resonance spectrometer to measure thin films up to 70 GHz,” Rev. Sci. Instrum. 82(9), 095115 (2011).
[Crossref] [PubMed]

Zayets, V.

V. Zayets, H. Saito, S. Yuasa, and K. Ando, “Enhancement of the transverse non-reciprocal magneto-optical effect,” J. Appl. Phys. 111(2), 023103 (2012).
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Y. Dong and X. Zhang, “Enhanced magneto-optical Kerr effect in magnetic multilayers containing double-negative metamaterials,” J. Appl. Phys. 105(5), 054105 (2009).
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Zhou, C.

J. H. Liang, Y. L. Chen, L. Sun, C. Zhou, Y. Yang, and Y. Z. Wu, “The anisotropic linear and quadratic magneto-optical Kerr effects in epitaxial Fe/GaAs(110) film,” Appl. Phys. Lett. 108(8), 082404 (2016).
[Crossref]

Zukrowski, J.

Y. Shi, J. Zukrowski, M. Przybylski, M. Nyvlt, A. Winkelmann, J. Barthel, and J. Kirschner, “A mode-of-growth-dependent magneto-optical response from ultrathin Co film on Pd surfaces,” Surf. Sci. 600(18), 4180–4184 (2006).
[Crossref]

Zvezdin, A. K.

M. Pohl, L. E. Kreilkamp, V. I. Belotelov, I. A. Akimov, A. N. Kalish, N. E. Khokhlov, V. J. Yallapragada, A. V. Gopal, M. Nur-E-Alam, M. Vasiliev, D. R. Yakovlev, K. Alameh, A. K. Zvezdin, and M. Bayer, “Tuning of the transverse magneto-optical effect in magneto-plasmonic crystals,” New J. Phys. 15, 075024 (2013).
[Crossref]

V. I. Belotelov, I. A. Akimov, M. Pohl, V. A. Kotov, S. Kasture, A. S. Vengurlekar, A. V. Gopal, D. R. Yakovlev, A. K. Zvezdin, and M. Bayer, “Enhanced magneto-optical effects in magnetoplasmonic crystals,” Nat. Nanotechnol. 6(6), 370–376 (2011).
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Appl. Phys. Lett. (8)

T. Kaihara, H. Shimizu, A. Cebollada, and G. Armelles, “Magnetic field control and wavelength tunability of SPP excitations using Al2O3/SiO2/Fe structures,” Appl. Phys. Lett. 109(11), 111102 (2016).
[Crossref]

A. S. Sokolov, M. Geiler, and V. G. Harris, “Broadband ferromagnetic resonance linewidth measurement by a microstripline transmission resonator,” Appl. Phys. Lett. 108(17), 172408 (2016).
[Crossref]

Y.-Y. Song, J. Das, Z. Wang, W. Tong, and C. E. Patton, “In-plane oriented barium ferrite films with self-bias and low microwave loss,” Appl. Phys. Lett. 93(17), 172503 (2008).
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[Crossref]

J. H. Liang, Y. L. Chen, L. Sun, C. Zhou, Y. Yang, and Y. Z. Wu, “The anisotropic linear and quadratic magneto-optical Kerr effects in epitaxial Fe/GaAs(110) film,” Appl. Phys. Lett. 108(8), 082404 (2016).
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Š. Višňovský, “Magneto-optical ellipsometry,” Czech. J. Phys. B 36(5), 625–650 (1986).
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Electron. Mater. Lett. (1)

M. Moradi, S. M. Mohseni, S. Mahmoodi, D. Rezvani, N. Ansari, S. Chung, and J. Åkerman, “Au/NiFe Magnetoplasmonics: Large Enhancement of Magneto-Optical Kerr,” Electron. Mater. Lett. 11(3), 440–446 (2015).
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J. A. Arregi, J. B. Gonzalez-Diaz, E. Bergaretxe, O. Idigoras, T. Unsal, and A. Berger, “Study of generalized magneto-optical ellipsometry measurement reliability,” J. Appl. Phys. 111(10), 103912 (2012).
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Figures (10)

Fig. 1
Fig. 1 Ferromagnetic resonance frequency, f, as a function of the applied magnetic flux density, Bappl, in thin films of iron (Fe) and hexagonal ferrite (BaM).
Fig. 2
Fig. 2 Interpretation of geometry for transverse magnetization magnetooptic probe. The microwave current, Imw, the applied magnetic flux density field, Bappl, and the magnetization, M, are parallel to the y-axis. M and Bmw denote the microwave magnetization and field components parallel to the x-axis and normal to the plane yz of light incidence.
Fig. 3
Fig. 3 Effect of the thickness, d(Fe), of the Fe layer on a Si substrate on the reflectance, Rpp, and the magneto-optic parameters, |∆rpp|, ∆Rpp, ∆Rpp/Rpp (See Table 2).
Fig. 4
Fig. 4 Absolute values of the magnetoooptic amplitude reflection coefficient, |∆rpp| = |∆rpp(2) + ∆rpp(4)|, and its components, |∆rpp(2)| and |∆rpp(4)| as functions of the thickness, d(3), of the AlN(d(3)) layer in the AlN(45 nm)Fe(17 nm)AlN(d(3))Fe(31 nm) Au(2 nm) multilayer on a Si substrate. See Table 10.
Fig. 5
Fig. 5 Effect of the thickness, d(3), of the AlN(d(3)) layer, on |∆rpp(2)| + |∆rpp(4)|, |∆rpp(2) + ∆rpp(4)|, and their difference in the AlN(45 nm)Fe(17 nm)AlN(d(3))Fe(31 nm)Au(2 nm) multilayer on a Si substrate. See Table 10.
Fig. 6
Fig. 6 Transverse magneto-optic reflectance difference, ∆Rpp = ∆Rpp(2) + ∆Rpp(4), and its components as functions of the thickness, d(3), of the AlN(d(3)) layer in the AlN(45 nm) Fe(17 nm) AlN(d(3)) Fe(31 nm) Au(2 nm) multilayer on a Si substrate. See Table 10.
Fig. 7
Fig. 7 Transverse magneto-optic reflectance difference ratio, ∆Rpp/Rpp, reflectance, Rpp, and the power transmitted into the Si substrate, Tpp, in the AlN(45 nm) Fe(17 nm) AlN(d(3)) Fe(31 nm) Au(2 nm) multilayer. See Table 10.
Fig. 8
Fig. 8 Power, App, absorbed in the AlN(45 nm) Fe(17 nm) AlN(d(3)) Fe(31 nm) Au(2 nm) multilayer on Si substrate and its distribution among the absorbing layers, Fe(d(2)), App(2), and Fe(d(4)), App(4), as functions of the thickness, d(3), of the AlN(d(3)). The power absorbed in the Au reflector, Au(d(5)) was below 0.0023 in the whole d(3) range. See Table 10.
Fig. 9
Fig. 9 Transverse magneto-optic reflectance difference, ∆Rpp = ∆Rpp(2) + ∆Rpp(4), and its components as functions of the thickness, d(3), of the AlN(d(3)) layer in the AlN(26 nm) Fe(18 nm) AlN(d(3)) Fe(29 nm) Au(2 nm) multilayer on a Si substrate. See Table 10.
Fig. 10
Fig. 10 Absolute value of the transverse magneto-optic parameter, q, for iron (Fe) and hexagonal ferrite (BaF) as a function of photon energy in eV at an angle of incidence of 45 deg.

Tables (10)

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Table 1 Optical and magneto-optical parameters of Fe

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Table 3 Thick Fe film, dFe = 104 nm, with AlN cover, dAlN denotes the thickness of AlN cover

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Table 4 AlN(d1)Fe(d2)Si structure

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Table 5 AlN(d1)Fe(d2)AlN(d3)Si structure.

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Table 6 AlN(d1)Fe(d2)Au(d3)Si structure, d3 = 2 nm

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Table 7 AlN(d1)Fe(d2) AlN(d3)Au(d4)Si structure, d4 = 2 nm

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Table 8 Fe(d1)AlN(d2)Fe(d3)AlN(d4)Fe(d5) structure on Si

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Table 9 AlN(d1)Fe(d2)AlN(d3)Fe(d4)AlN(d5) structure on Si

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Table 10 AlN(d1)Fe(d2)AlN(d3)Fe(d4)Au(d5)Si structure, d5 = 2 nm

Equations (38)

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ε (m) ( ω )=( ε xx (m) ( ω ) 0 0 0 ε yy (m) ( ω ) ε yz (m) ( ω ) 0 ε zy (m) ( ω ) ε zz (m) ( ω ) ).
( ε 0 ( m ) N y 2 N z ( m )2 ) E x ( m ) x ^ +( ε 0 ( m ) j ε 1 ( m ) + N y N z ( m )2 N z ( m )2 ) E y ( m ) y ^ +( ε 0 ( m ) +j ε 1 ( m ) N y 2 + N y N z ( m )2 ) E z ( m ) z ^ =0.
N z3 (m) = N z0 (m) ( 1 ε 1 (m)2 2 ε 0 (m) N z0 (m)2 ) N z0 (m) , N z4 (m) = N z0 (m) ( 1 ε 1 (m)2 2 ε 0 (m) N z0 (m)2 ) N z0 (m) ,
D ( m _ ) =( N z0 (m) ε 0 (m)1/2 N z0 (m) ε 0 (m)1/2 ε 0 (m)1/2 +j ε 1 (m) N y N z0 (m)1 ε 0 (m)1/2 ε 0 (m)1/2 +j ε 1 (m) N y N z0 (m)1 ε 0 (m)1/2 )( C 3 (m) 0 0 C 4 (m) ),
( E 0y ( m _ ) ( z ( m _ 1 ) ) H 0x ( m _ ) ( z ( m _ 1 ) ) )= D ( m _ ) ( E 03 ( m _ ) ( z ( m _ 1 ) ) E 04 ( m _ ) ( z ( m _ 1 ) ) ).
( E 0y ( m _ ) ( z ( m _ 1 ) ) H 0x ( m _ ) ( z ( m _ 1 ) ) )= S ( m _ ) ( E 0y ( m _ ) ( z ( m _ ) ) H 0x ( m _ ) ( z ( m _ ) ) ).
S ( m _ ) = D ( m _ ) P (m) D ( m _ )1 =( cos β (m) j N z0 (m) ε 0 (m) sin β (m) j ε 0 (m) N z0 sin β (m) cos β (m) )+ q ( m ) sin β (m) ( 1 0 0 1 ),
q (m) ( M x )= ε 1 (m) ( M x ) N y ε 0 (m) N z0 (m) = q (m) ( M x ).
P ( m _ ) = P (m) =( e j β ( m ) 0 0 e j β ( m ) ),
β ( m _ ) = β ( m ) = ω c N z0 ( m ) d ( m ) .
M=( M 33 M 34 M 43 M 44 )= D (0)1 m _ =1 N S ( m _ ) D ( N+1 _ ) .
( E p ( 0 ) E p ( 0 ) )=( M 33 M 34 M 43 M 44 )( E p ( N+1 ) E p ( N+1 ) ),
D (0)1 = ( 2 N (0) α (0) ) 1 ( N (0) α (0) N (0) α (0) )
D (N+1) =( α ( N+1 ) α ( N+1 ) N ( N+1 ) N ( N+1 ) ),
r p p ( 0 1 _ N , N + 1 _ ) = E p ( 0 ) E p ( 0 ) = [ M ( 0 1 _ N , N + 1 _ ) ] 43 [ M ( 0 1 _ N , N + 1 _ ) ] 33 1 .
M ( 0 123456 _ ) = D (0)1 S ( 1 _ ) S ( 2 _ ) S ( 3 _ ) S ( 4 _ ) S ( 5 _ ) D ( 6 _ ) .
r pp ( 0 123456 _ ) = r pp ( 0123456 ) +Δ r pp ( 0 1 _ 23456 ) +Δ r pp ( 01 2 _ 3456 ) +...+Δ r pp ( 012345 6 _ ) ,
Δ r pp ( 0 1 _ 23456 ) = 1 2 [ r pp ( 0 1 _ 23456 ) ( M x ) r pp ( 0 1 _ 23456 ) ( M x ) ], Δ r pp ( 01 2 _ 3456 ) = 1 2 [ r pp ( 01 2 _ 3456 ) ( M x ) r pp ( 01 2 _ 3456 ) ( M x ) ], Δ r pp ( 012345 6 _ ) = 1 2 [ r pp ( 012345 6 _ ) ( M x ) r pp ( 012345 6 _ ) ( M x ) ],
Δ r pp ( 0 1 _ 23456 ) = j 2 q (1) ( 1 e j2 β (1) )( 1 r pp (123456)2 e 2j β (1) ) t pp (01) t pp (10) ( 1 r pp (10) r pp (123456) e 2j β (1) ) 2 , Δ r pp ( 01 2 _ 3456 ) = j 2 q (2) ( 1 e j2 β (2) )( 1 r pp (23456)2 e 2j β (2) ) t pp (012) t pp (210) ( 1 r pp (210) r pp (23456) e 2j β (2) ) 2 , Δ r pp ( 012 3 _ 456 ) = j 2 q (3) ( 1 e j2 β (3) )( 1 r pp (3456)2 e 2j β (3) ) t pp (0123) t pp (3210) ( 1 r pp (3210) r pp (3456) e 2j β (3) ) 2 ,
Δ r pp ( 0123 4 _ 56 ) = j 2 q (4) ( 1 e j2 β (4) )( 1 r pp (456)2 e 2j β (4) ) t pp (01234) t pp (43210) ( 1 r pp (43210) r pp (456) e 2j β (4) ) 2 , Δ r pp ( 01234 5 _ 6 ) = j 2 q (5) ( 1 e j2 β (5) )( 1 r pp ( 56 )2 e 2j β (5) ) t pp (012345) t pp (543210) ( 1 r pp ( 543210 ) r pp (56) e 2j β (5) ) 2 , Δ r pp ( 012345 6 _ ) =j 1 2 q (6) t pp (0123456) t pp (6543210) .
Δ R pp ( 0 123456 _ ) = 1 2 [ R pp ( + M ) R pp ( M ) ]= 1 2 [ r pp ( 0 123456 _ ) r pp ( 0 123456 _ )* ] + 1 2 [ r pp ( 0 123456 _ ) r pp ( 0 123456 _ )* ]
R pp ( 0 123456 _ ) = 1 2 [ r pp ( 0 123456 _ ) r pp ( 0 123456 _ )* ] + + 1 2 [ r pp ( 0 123456 _ ) r pp ( 0 123456 _ )* ] R pp ( 0123456 )
Δ R p p ( 0 123456 _ ) 1 2 r p p ( 0123456 ) ( Δ r p p ( 0 1 _ 23456 ) * + Δ r p p ( 01 2 _ 3456 ) * + ... + Δ r p p ( 012345 6 _ ) * ) + 1 2 r p p ( 0123456 ) * ( Δ r p p ( 0 1 _ 23456 ) + Δ r p p ( 01 2 _ 3456 ) + ... + Δ r p p ( 012345 6 _ ) ) .
A 01 =1, B 01 = r pp (01) , C 01 = r pp (01) , D 01 =1,
A 02 =1+ r pp (01) r pp (12) e 2j β (1) , B 02 = r pp (01) e 2j β (1) + r pp (12) , C 02 = r pp (01) + r pp (12) e 2j β (1) , D 02 = r pp (01) r pp (12) + e 2j β (1) .
A 03 =[ ( 1+ r pp (01) r pp (12) e 2j β (1) )+( r pp (01) e 2j β (1) + r pp (12) ) r pp (23) e 2j β (2) ]= A 02 + B 03 r pp (23) e 2j β (2) , B 03 =[ ( 1+ r pp (01) r pp (12) e 2j β (1) ) r pp (23) +( r pp (01) e 2j β (1) + r pp (12) ) e 2j β (2) ]= A 02 r pp (23) + B 03 e 2j β (2) ,
C 03 =[ ( r pp (01) + r pp (12) e 2j β (1) )+( r pp (01) r pp (12) + e 2j β (1) ) r pp (23) e 2j β (2) ]= C 02 + D 02 r pp (23) e 2j β (2) , D 03 =[ ( r pp (01) + r pp (12) e 2j β (1) ) r pp (23) +( r pp (01) r pp (12) + e 2j β (1) ) e 2j β (2) ]= C 02 r pp (23) + D 02 e 2j β (2) ,
A 04 = A 03 + B 03 r pp (34) e 2j β (3) , B 04 = A 03 r pp (34) + B 03 e 2j β (3) , C 04 = C 03 + D 03 r pp (34) e 2j β (3) , D 04 = D 03 e 2j β (3) + C 03 r pp (34) ,
A 05 = A 04 + B 04 r pp (45) e 2j β (4) , B 05 = A 04 r pp (45) + B 04 e 2j β (4) , C 05 = C 04 + D 04 r pp (45) e 2j β (4) , D 05 = D 04 e 2j β (4) + C 04 r pp (45) , A 06 = A 05 + B 05 r pp (56) e 2j β (5) , B 06 = A 05 r pp (56) + B 05 e 2j β (5) , C 06 = C 05 + D 05 r pp (56) e 2j β (5) , D 06 = D 05 e 2j β (5) + C 05 r pp (56) ,
r pp (i1,i) = r pp (i,i1) = N ( i1 ) α ( i ) N ( i ) α ( i1 ) N ( i1 ) α ( i ) + N ( i ) α ( i1 ) , i=1,...,6.
r pp ( 0,1,...n,n+1 ) = C 0,n+1 A 0,n+1 1 ,
r pp ( n+1,n,...,1,0 ) = B 0,n+1 A 0,n+1 1 .
t pp ( 0,1,...n,n+1 ) = t pp (01) e j β (1) t pp (12) e j β (2) ... t pp (n1,n) e j β (n1) t pp (n,n+1) A 0,n+1 1 ,
t pp ( n+1,n,...,1,0 ) = t pp (10) e j β (1) t pp (21) e j β (2) ... t pp (n,n1) e j β (n1) t pp (n+1,n) A 0,n+1 1 ,
t pp (i1,i) = 2 N ( i1 ) α ( i1 ) N ( i1 ) α ( i ) + N ( i ) α ( i1 ) , i=1,...,6, and t pp (i,i1) = 2 N ( i ) α ( i ) N ( i1 ) α ( i ) + N ( i ) α ( i1 ) , i=1,...,6.
r ss (i1,i) = r ss (i,i1) = N ( i1 ) α ( i1 ) N ( i ) α ( i ) N ( i1 ) α ( i1 ) + N ( i ) α ( i ) , i=1,...,6,
t ss (i1,i) = 2 N ( i1 ) α ( i1 ) N ( i1 ) α ( i1 ) + N ( i ) α ( i ) , i=1,...,6,
t s s ( i , i 1 ) = 2 N ( i ) α ( i ) N ( i 1 ) α ( i 1 ) + N ( i ) α ( i ) , i = 1 , ... , 6.

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