Abstract

Ba2EuFeNb4O15-based epitaxial ferroelectric thin films with tetragonal tungsten bronze crystal structures are shown to have nonlinear optical properties at room temperature and are promising candidates for integrated optical frequency convertors and related applications. High quality epitaxial ferroelectric thin films of tetragonal tungsten bronze Ba2EuFeNb4O15 have been synthesized on MgO(100) by pulsed laser deposition. Structural investigation reveals that the c-oriented films are rotated in plane by ±18.4° and ±31° with respect to the substrate structure. Stable ferroelectric properties were obtained by microelectromechanical characterization. Second harmonic generation related to the spontaneous polarization in the films was studied and the independent components of the nonlinear susceptibility were determined. Ferroelectric films of Ba2EuFeNb4O15 having a tetragonal tungsten bronze structure are new candidate for room temperature nonlinear optical applications.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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    [Crossref]
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2020 (1)

T. Hajlaoui, N. Émond, C. Quirouette, B. Le Drogoff, J. Margot, and M. Chaker, “Metal–insulator transition temperature of boron-doped VO2 thin films grown by reactive pulsed laser deposition,” Scr. Mater. 177, 32–37 (2020).
[Crossref]

2017 (5)

H. Kalhori, M. Coey, I. Abdolhosseini Sarsari, K. Borisov, S. B. Porter, G. Atcheson, M. Ranjbar, H. Salamati, and P. Stamenov, “Oxygen Vacancy in WO3 Film-based FET with Ionic Liquid Gating,” Sci. Rep. 7(1), 12253 (2017).
[Crossref]

T. Hajlaoui, C. Chabanier, C. Harnagea, and A. Pignolet, “Epitaxial Ba2LnFeNb4O15-based multiferroic nanocomposite thin films with tetragonal tungsten bronze structure,” Scr. Mater. 136, 1–5 (2017).
[Crossref]

T. Hajlaoui, C. Harnagea, and A. Pignolet, “Influence of lanthanide ions on multiferroic properties of Ba2LnFeNb4O15 (Ln = Eu3+, Sm3+ and Nd3+) thin films grown on silicon by pulsed laser deposition,” Mater. Lett. 198, 136–139 (2017).
[Crossref]

T. Hajlaoui, L. Corbellini, C. Harnagea, M. Josse, and A. Pignolet, “Enhanced ferroelectric properties in multiferroic epitaxial Ba2EuFeNb4O15 thin films grown by pulsed laser deposition,” Mater. Res. Bull. 87, 186–192 (2017).
[Crossref]

T. Hajlaoui, C. Harnagea, D. Michau, M. Josse, and A. Pignolet, “Highly oriented multiferroic Ba2LnFeNb4O5-based composite thin films with tetragonal tungsten bronze structure on silicon substrates,” J. Alloys Compd. 711, 480–487 (2017).
[Crossref]

2016 (2)

W. J. Hu, Z. Wang, W. Yu, and T. Wu, “Optically controlled electroresistance and electrically controlled photovoltage in ferroelectric tunnel junctions,” Nat. Commun. 7(1), 1–9 (2016).
[Crossref]

M. İlhan, M. K. Ekmekçi, A. Mergen, and C. Yaman, “Synthesis and Optical Characterization of Red-Emitting BaTa2O6:Eu3+ Phosphors,” J. Fluoresc. 26(5), 1671–1678 (2016).
[Crossref]

2015 (4)

P. Castera, D. Tulli, A. M. Gutierrez, and P. Sanchis, “Influence of BaTiO3 ferroelectric orientation for electro-optic modulation on silicon,” Opt. Express 23(12), 15332 (2015).
[Crossref]

X. Chen, P. Karpinski, V. Shvedov, K. Koynov, B. Wang, J. Trull, C. Cojocaru, W. Krolikowski, and Y. Sheng, “Ferroelectric domain engineering by focused infrared femtosecond pulses,” Appl. Phys. Lett. 107(14), 1–4 (2015).
[Crossref]

D. Chen, I. Harward, J. Baptist, S. Goldman, and Z. Celinski, “Curie temperature and magnetic properties of aluminum doped barium ferrite particles prepared by ball mill method,” J. Magn. Magn. Mater. 395, 350–353 (2015).
[Crossref]

M. A. Houle, C. A. Couture, S. Bancelin, J. Van der Kolk, E. Auger, C. Brown, K. Popov, L. Ramunno, and F. Légaré, “Analysis of forward and backward Second Harmonic Generation images to probe the nanoscale structure of collagen within bone and cartilage,” J. Biophotonics 8(11-12), 993–1001 (2015).
[Crossref]

2014 (1)

M. Sharma, S. C. Kashyap, H. C. Gupta, M. C. Dimri, and K. Asokan, “Enhancement of Curie temperature of barium hexaferrite by dense electronic excitations,” AIP Adv. 4(7), 077129 (2014).
[Crossref]

2013 (1)

T. Miyamoto, H. Yada, H. Yamakawa, and H. Okamoto, “Ultrafast modulation of polarization amplitude by terahertz fields in electronic-type organic ferroelectrics,” Nat. Commun. 4(1), 1–9 (2013).
[Crossref]

2011 (1)

S. A. Denev, T. T. A. Lummen, E. Barnes, A. Kumar, and V. Gopalan, “Probing ferroelectrics using optical second harmonic generation,” J. Am. Ceram. Soc. 94(9), 2699–2727 (2011).
[Crossref]

2009 (4)

Y. Zhang, J. Martinez-Perdiguero, U. Baumeister, C. Walker, J. Etxebarria, M. Prehm, J. Ortega, C. Tschierske, M. J. O’Callaghan, A. Harant, and M. Handschy, “Laterally azo-bridged H-shaped ferroelectric dimesogens for second-order nonlinear optics: Ferroelectricity and second harmonic generation,” J. Am. Chem. Soc. 131(51), 18386–18392 (2009).
[Crossref]

M. Josse, O. Bidault, F. Roulland, E. Castel, A. Simon, D. Michau, R. Von der Muhll, O. Nguyen, and M. Maglione, “The Ba2LnFeNb4O5 “tetragonal tungsten bronze”: Towards RT composite multiferroics,” Solid State Sci. 11(6), 1118–1123 (2009).
[Crossref]

F. Roulland, M. Josse, E. Castel, and M. Maglione, “Influence of ceramic process and Eu content on the composite multiferroic properties of the Ba6-2xLn2xFe1 + xNb9-xO30 TTB system,” Solid State Sci. 11(9), 1709–1716 (2009).
[Crossref]

E. Castel, M. Josse, D. Michau, and M. Maglione, “Flexible relaxor materials: Ba2PrxNd1−xFeNb4O15 tetragonal tungsten bronze solid solution,” J. Phys.: Condens. Matter 21(45), 452201 (2009).
[Crossref]

2008 (1)

S. Alkoy, C. Duran, and D. A. Hall, “Electrical properties of textured potassium strontium niobate (KSr2Nb5O15) ceramics fabricated by reactive templated grain growth,” J. Am. Ceram. Soc. 91(5), 1597–1602 (2008).
[Crossref]

2007 (1)

A. Guarino, G. Poberaj, D. Rezzonico, R. Degl’Innocenti, and P. Günter, “Electro-optically tunable microring resonators in lithium niobate,” Nat. Photonics 1(7), 407–410 (2007).
[Crossref]

2006 (2)

I. Levin, M. C. Stennett, G. C. Miles, D. I. Woodward, A. R. West, and I. M. Reaney, “Coupling between octahedral tilting and ferroelectric order in tetragonal tungsten bronze-structured dielectrics,” Appl. Phys. Lett. 89(12), 2–5 (2006).
[Crossref]

K. M. Ok, E. O. Chi, and P. S. Halasyamani, “Bulk characterization methods for non-centrosymmetric materials: Second-harmonic generation, piezoelectricity, pyroelectricity, and ferroelectricity,” Chem. Soc. Rev. 35(8), 710–717 (2006).
[Crossref]

2005 (1)

Y. Xiao, V. B. Shenoy, and K. Bhattacharya, “Depletion layers and domain walls in semiconducting ferroelectric thin films,” Phys. Rev. Lett. 95(24), 247603 (2005).
[Crossref]

2004 (3)

K. J. Choi, M. Biegalski, Y. L. Li, A. Sharan, J. Schubert, R. Uecker, P. Reiche, Y. B. Chen, X. Q. Pan, V. Gopalan, L. Q. Che, D. C. Schlom, and C. B. Eom, “Enhancement of ferroelectricity in strained BaTiO3 thin films,” Science 306(5698), 1005–1009 (2004).
[Crossref]

E. O. Chi, A. Gandini, K. M. Ok, L. Zhang, and P. S. Halasyamani, “Syntheses, Structures, Second-Harmonic Generating, and Ferroelectric Properties of Tungsten Bronzes: A6M2M′8O30 (A) Sr2+, Ba2+, or Pb2+ M) Ti4+, Zr4+, or Hf4+ M′) Nb5+or Ta5+),” Chem. Mater. 16(19), 3616–3622 (2004).
[Crossref]

J. Li, B. Nagaraj, H. Liang, W. Cao, C. H. Lee, and R. Ramesh, “Ultrafast polarization switching in thin-film ferroelectrics,” Appl. Phys. Lett. 84(7), 1174–1176 (2004).
[Crossref]

2003 (1)

M. Eßer, M. Burianek, P. Held, J. Stade, S. Bulut, C. Wickleder, and M. Mühlberg, “Optical characterization and crystal structure of the novel bronze type CaxBa1−xNb2O6 (x = 0.28; CBN-28),” Cryst. Res. Technol. 38(6), 457–464 (2003).
[Crossref]

2000 (1)

C. W. Teng, J. F. Muth, U. Ozgur, M. J. Bergmann, H. O. Everitt, A. K. Sharma, C. Jin, and J. Narayan, “Refractive indices and absorption coefficients of Mg x Zn 1− x O alloys,” Appl. Phys. Lett. 76(8), 979–981 (2000).
[Crossref]

1999 (1)

G. Ghosh, “Dispersion-equation coefficients for the refractive index and birefringence of calcite and quartz crystals,” Opt. Commun. 163(1-3), 95–102 (1999).
[Crossref]

1998 (2)

C. Z. Tan, “Determination of refractive index of silica glass for infrared wavelengths by IR spectroscopy,” J. Non-Cryst. Solids 223(1-2), 158–163 (1998).
[Crossref]

D. Damjanovic, “Ferroelectric, dielectric and piezoelectric properties of ferroelectric thin films and ceramics,” Rep. Prog. Phys. 61(9), 1267–1324 (1998).
[Crossref]

1996 (1)

K. Kintaka, M. Fujirnura, T. Suhara, and H. Nishihara, “High-Efficiency LiNbO3 Waveguide Second-Harmonic Generation Devices with Ferroelectric-Domain-Inverted Gratings Fabricate by Applying Voltage,” J. Lightwave Technol. 14(3), 462–468 (1996).
[Crossref]

1995 (1)

Y. Uesu, S. Kurimura, and Y. Yamamoto, “Optical second harmonic images of 90° domain structure in BaTiO3 and periodically inverted antiparallel domains in LiTaO3,” Appl. Phys. Lett. 66(17), 2165–2167 (1995).
[Crossref]

1993 (1)

J. J. E. Reid, “Resonantly enhanced, frequency doubling of an 820 nm GaAlAs diode laser in a potassium lithium niobate crystal,” Appl. Phys. Lett. 62(1), 19–21 (1993).
[Crossref]

1967 (1)

P. V. Lenzo, E. G. Spencer, and A. A. Ballman, “Electro-optic coefficients of ferroelectric strontium barium niobate,” Appl. Phys. Lett. 11(1), 23–24 (1967).
[Crossref]

1952 (1)

R. E. Stephens and I. H. Malitson, “Index of refraction of magnesium oxide,” J. Res. Natl. Bur. Stand. (1934). 49, 2360 (1952).

Abdolhosseini Sarsari, I.

H. Kalhori, M. Coey, I. Abdolhosseini Sarsari, K. Borisov, S. B. Porter, G. Atcheson, M. Ranjbar, H. Salamati, and P. Stamenov, “Oxygen Vacancy in WO3 Film-based FET with Ionic Liquid Gating,” Sci. Rep. 7(1), 12253 (2017).
[Crossref]

Alkoy, S.

S. Alkoy, C. Duran, and D. A. Hall, “Electrical properties of textured potassium strontium niobate (KSr2Nb5O15) ceramics fabricated by reactive templated grain growth,” J. Am. Ceram. Soc. 91(5), 1597–1602 (2008).
[Crossref]

Asokan, K.

M. Sharma, S. C. Kashyap, H. C. Gupta, M. C. Dimri, and K. Asokan, “Enhancement of Curie temperature of barium hexaferrite by dense electronic excitations,” AIP Adv. 4(7), 077129 (2014).
[Crossref]

Atcheson, G.

H. Kalhori, M. Coey, I. Abdolhosseini Sarsari, K. Borisov, S. B. Porter, G. Atcheson, M. Ranjbar, H. Salamati, and P. Stamenov, “Oxygen Vacancy in WO3 Film-based FET with Ionic Liquid Gating,” Sci. Rep. 7(1), 12253 (2017).
[Crossref]

Auger, E.

M. A. Houle, C. A. Couture, S. Bancelin, J. Van der Kolk, E. Auger, C. Brown, K. Popov, L. Ramunno, and F. Légaré, “Analysis of forward and backward Second Harmonic Generation images to probe the nanoscale structure of collagen within bone and cartilage,” J. Biophotonics 8(11-12), 993–1001 (2015).
[Crossref]

Ballman, A. A.

P. V. Lenzo, E. G. Spencer, and A. A. Ballman, “Electro-optic coefficients of ferroelectric strontium barium niobate,” Appl. Phys. Lett. 11(1), 23–24 (1967).
[Crossref]

Bancelin, S.

M. A. Houle, C. A. Couture, S. Bancelin, J. Van der Kolk, E. Auger, C. Brown, K. Popov, L. Ramunno, and F. Légaré, “Analysis of forward and backward Second Harmonic Generation images to probe the nanoscale structure of collagen within bone and cartilage,” J. Biophotonics 8(11-12), 993–1001 (2015).
[Crossref]

Baptist, J.

D. Chen, I. Harward, J. Baptist, S. Goldman, and Z. Celinski, “Curie temperature and magnetic properties of aluminum doped barium ferrite particles prepared by ball mill method,” J. Magn. Magn. Mater. 395, 350–353 (2015).
[Crossref]

Barnes, E.

S. A. Denev, T. T. A. Lummen, E. Barnes, A. Kumar, and V. Gopalan, “Probing ferroelectrics using optical second harmonic generation,” J. Am. Ceram. Soc. 94(9), 2699–2727 (2011).
[Crossref]

Baumeister, U.

Y. Zhang, J. Martinez-Perdiguero, U. Baumeister, C. Walker, J. Etxebarria, M. Prehm, J. Ortega, C. Tschierske, M. J. O’Callaghan, A. Harant, and M. Handschy, “Laterally azo-bridged H-shaped ferroelectric dimesogens for second-order nonlinear optics: Ferroelectricity and second harmonic generation,” J. Am. Chem. Soc. 131(51), 18386–18392 (2009).
[Crossref]

Bergmann, M. J.

C. W. Teng, J. F. Muth, U. Ozgur, M. J. Bergmann, H. O. Everitt, A. K. Sharma, C. Jin, and J. Narayan, “Refractive indices and absorption coefficients of Mg x Zn 1− x O alloys,” Appl. Phys. Lett. 76(8), 979–981 (2000).
[Crossref]

Bhattacharya, K.

Y. Xiao, V. B. Shenoy, and K. Bhattacharya, “Depletion layers and domain walls in semiconducting ferroelectric thin films,” Phys. Rev. Lett. 95(24), 247603 (2005).
[Crossref]

Bidault, O.

M. Josse, O. Bidault, F. Roulland, E. Castel, A. Simon, D. Michau, R. Von der Muhll, O. Nguyen, and M. Maglione, “The Ba2LnFeNb4O5 “tetragonal tungsten bronze”: Towards RT composite multiferroics,” Solid State Sci. 11(6), 1118–1123 (2009).
[Crossref]

Biegalski, M.

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K. J. Choi, M. Biegalski, Y. L. Li, A. Sharan, J. Schubert, R. Uecker, P. Reiche, Y. B. Chen, X. Q. Pan, V. Gopalan, L. Q. Che, D. C. Schlom, and C. B. Eom, “Enhancement of ferroelectricity in strained BaTiO3 thin films,” Science 306(5698), 1005–1009 (2004).
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S. A. Denev, T. T. A. Lummen, E. Barnes, A. Kumar, and V. Gopalan, “Probing ferroelectrics using optical second harmonic generation,” J. Am. Ceram. Soc. 94(9), 2699–2727 (2011).
[Crossref]

K. J. Choi, M. Biegalski, Y. L. Li, A. Sharan, J. Schubert, R. Uecker, P. Reiche, Y. B. Chen, X. Q. Pan, V. Gopalan, L. Q. Che, D. C. Schlom, and C. B. Eom, “Enhancement of ferroelectricity in strained BaTiO3 thin films,” Science 306(5698), 1005–1009 (2004).
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[Crossref]

T. Hajlaoui, C. Harnagea, D. Michau, M. Josse, and A. Pignolet, “Highly oriented multiferroic Ba2LnFeNb4O5-based composite thin films with tetragonal tungsten bronze structure on silicon substrates,” J. Alloys Compd. 711, 480–487 (2017).
[Crossref]

T. Hajlaoui, C. Chabanier, C. Harnagea, and A. Pignolet, “Epitaxial Ba2LnFeNb4O15-based multiferroic nanocomposite thin films with tetragonal tungsten bronze structure,” Scr. Mater. 136, 1–5 (2017).
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T. Hajlaoui, L. Corbellini, C. Harnagea, M. Josse, and A. Pignolet, “Enhanced ferroelectric properties in multiferroic epitaxial Ba2EuFeNb4O15 thin films grown by pulsed laser deposition,” Mater. Res. Bull. 87, 186–192 (2017).
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T. Hajlaoui, C. Harnagea, and A. Pignolet, “Influence of lanthanide ions on multiferroic properties of Ba2LnFeNb4O15 (Ln = Eu3+, Sm3+ and Nd3+) thin films grown on silicon by pulsed laser deposition,” Mater. Lett. 198, 136–139 (2017).
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[Crossref]

E. O. Chi, A. Gandini, K. M. Ok, L. Zhang, and P. S. Halasyamani, “Syntheses, Structures, Second-Harmonic Generating, and Ferroelectric Properties of Tungsten Bronzes: A6M2M′8O30 (A) Sr2+, Ba2+, or Pb2+ M) Ti4+, Zr4+, or Hf4+ M′) Nb5+or Ta5+),” Chem. Mater. 16(19), 3616–3622 (2004).
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S. Alkoy, C. Duran, and D. A. Hall, “Electrical properties of textured potassium strontium niobate (KSr2Nb5O15) ceramics fabricated by reactive templated grain growth,” J. Am. Ceram. Soc. 91(5), 1597–1602 (2008).
[Crossref]

Handschy, M.

Y. Zhang, J. Martinez-Perdiguero, U. Baumeister, C. Walker, J. Etxebarria, M. Prehm, J. Ortega, C. Tschierske, M. J. O’Callaghan, A. Harant, and M. Handschy, “Laterally azo-bridged H-shaped ferroelectric dimesogens for second-order nonlinear optics: Ferroelectricity and second harmonic generation,” J. Am. Chem. Soc. 131(51), 18386–18392 (2009).
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T. Hajlaoui, C. Harnagea, and A. Pignolet, “Influence of lanthanide ions on multiferroic properties of Ba2LnFeNb4O15 (Ln = Eu3+, Sm3+ and Nd3+) thin films grown on silicon by pulsed laser deposition,” Mater. Lett. 198, 136–139 (2017).
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T. Hajlaoui, L. Corbellini, C. Harnagea, M. Josse, and A. Pignolet, “Enhanced ferroelectric properties in multiferroic epitaxial Ba2EuFeNb4O15 thin films grown by pulsed laser deposition,” Mater. Res. Bull. 87, 186–192 (2017).
[Crossref]

T. Hajlaoui, C. Harnagea, D. Michau, M. Josse, and A. Pignolet, “Highly oriented multiferroic Ba2LnFeNb4O5-based composite thin films with tetragonal tungsten bronze structure on silicon substrates,” J. Alloys Compd. 711, 480–487 (2017).
[Crossref]

T. Hajlaoui, C. Chabanier, C. Harnagea, and A. Pignolet, “Epitaxial Ba2LnFeNb4O15-based multiferroic nanocomposite thin films with tetragonal tungsten bronze structure,” Scr. Mater. 136, 1–5 (2017).
[Crossref]

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D. Chen, I. Harward, J. Baptist, S. Goldman, and Z. Celinski, “Curie temperature and magnetic properties of aluminum doped barium ferrite particles prepared by ball mill method,” J. Magn. Magn. Mater. 395, 350–353 (2015).
[Crossref]

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M. Eßer, M. Burianek, P. Held, J. Stade, S. Bulut, C. Wickleder, and M. Mühlberg, “Optical characterization and crystal structure of the novel bronze type CaxBa1−xNb2O6 (x = 0.28; CBN-28),” Cryst. Res. Technol. 38(6), 457–464 (2003).
[Crossref]

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M. A. Houle, C. A. Couture, S. Bancelin, J. Van der Kolk, E. Auger, C. Brown, K. Popov, L. Ramunno, and F. Légaré, “Analysis of forward and backward Second Harmonic Generation images to probe the nanoscale structure of collagen within bone and cartilage,” J. Biophotonics 8(11-12), 993–1001 (2015).
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T. Hajlaoui, C. Harnagea, D. Michau, M. Josse, and A. Pignolet, “Highly oriented multiferroic Ba2LnFeNb4O5-based composite thin films with tetragonal tungsten bronze structure on silicon substrates,” J. Alloys Compd. 711, 480–487 (2017).
[Crossref]

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[Crossref]

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[Crossref]

M. Josse, O. Bidault, F. Roulland, E. Castel, A. Simon, D. Michau, R. Von der Muhll, O. Nguyen, and M. Maglione, “The Ba2LnFeNb4O5 “tetragonal tungsten bronze”: Towards RT composite multiferroics,” Solid State Sci. 11(6), 1118–1123 (2009).
[Crossref]

F. Roulland, M. Josse, E. Castel, and M. Maglione, “Influence of ceramic process and Eu content on the composite multiferroic properties of the Ba6-2xLn2xFe1 + xNb9-xO30 TTB system,” Solid State Sci. 11(9), 1709–1716 (2009).
[Crossref]

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[Crossref]

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[Crossref]

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Figures (5)

Fig. 1.
Fig. 1. Schematic of the experimental set-up. λ/2: half-wave plate, L: +5 cm lens, F: 405 ± 10 nm filter, A: analyser, S: sample (TTB-Eu).
Fig. 2.
Fig. 2. (a) X-ray diffractogram obtained in Bragg-Brentano geometry demonstrating the synthesis of TTB-Eu films oriented with their c-axis normal to the substrate surface.
Fig. 3.
Fig. 3. (a) Longitudinal piezoelectric coefficient (ξZZ), and (b) transverse piezoelectric coefficient (ξXZ) versus an applied voltage. The in-plane piezoelectric coefficient ξXZ represents roughly 15% of the out-of-plane piezoelectric coefficient ξZZ.
Fig. 4.
Fig. 4. (a) SHG microscopy image of the TTB-Eu thin film (forward direction). (b) Square root of the detected SHG intensity as a function of the excitation power (forward and backward). The linear behavior shows a second order process.
Fig. 5.
Fig. 5. (a) Measured SHG counts as a function of the angle θ (the angle between the plane of the sample and the XY plane as explained in the Fig. 1) for a p-polarization. (b) Measured SHG counts as a function of the direction of the polarization with respect to the vertical axis (X). Experimental points are black symbols, and fitting curves are in red.

Tables (4)

Tables Icon

Table 1. Ordinary refractive index [39,40] and coherence lengths for the TTB-Eu thin film, MgO substrate and reference quartz (S, M and q superscript, respectively), along with transmission coefficient of the sample (at θ = 32°).

Tables Icon

Table 2. Nonlinear susceptibility components of TTB-Eu thin film: while d15 and d33 are determined by curve fitting, the d32 is calculated using Eq. (6).

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Table 3. Induced SHG polarization in specific directions, excited by a specific polarization (first line), and the component of the nonlinear tensor it allows to retrieve

Tables Icon

Table 4. Verification of the nonlinear susceptibility components of TTB-Eu thin film by direct measurement.

Equations (11)

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d = [ 0 0 0 0 d 15 0 0 0 0 d 15 0 0 d 32 d 32 d 33 0 0 0 ]
[ P x ( 2 ) P y ( 2 ) P z ( 2 ) ] [ d 15 C φ S φ S θ [ ( d 15 + d 32 ) C θ 2 S φ 2 + d 32 C φ 2 + d 33 S θ 2 S φ 2 ] S θ [ d 32 ( C φ 2 + C θ 2 S φ 2 ) + d 33 S θ 2 S φ 2 + d 15 S φ 2 S θ 2 ] C θ ]
I y ( φ , θ 0 ) [ ( C φ 2 + ( 1 + d 15 d 32 ) C θ 0 2 S φ 2 + d 33 d 32 S θ 0 2 S φ 2 ) S θ 0 ] 2
I y ( φ 0 = 90 , θ ) [ ( ( 1 + d 15 d 32 ) C θ 2 + d 33 d 32 S θ 2 ) S θ ] 2
2 ω X ( φ = 0 , θ 0 ) = K ω 2 ε 0 c n 2 ω n ω 2 λ 2 w 2 d e f f 2 ( L s i n c ( L L c ) ) 2
d 32 S = d 11 q sin θ n 2 ω S ( n ω S ) 2 n 2 ω q ( n ω q ) 2 T q T S 2 ω S 2 ω q ( L q sinc ( L q / L c q ) L S sinc ( L θ S / L c S ) ) 2
P x ( 2 ) = P X ( 2 ) P y ( 2 ) = C θ P Y ( 2 ) S θ P Z ( 2 ) P z ( 2 ) = C θ P Z ( 2 ) + S θ P Y ( 2 )
R P = | n 1 1 ( n 1 n 2 sin θ ) 2 n 2 cos θ / n 1 1 ( n 1 n 2 sin θ ) 2 + n 2 cos θ | 2
2 ω S ( φ = 0 , θ ) 2 ω q ( φ = 0 , θ ) = T S T q ( d 32 S sin θ d 11 q ) 2 n 2 ω q ( n ω q ) 2 n 2 ω S ( n ω S ) 2 ( L S sinc ( L θ S / L c S ) L q sinc ( L q / L c q ) ) 2
d eff = ( d 15 + d 32 ) S θ C θ 2 + d 33 S θ 3 d 33 = 1 si n 3 θ [ d eff ( d 15  +  d 32 ) sin θ cos 2 θ ]
d 33 = 1 s i n 3 θ [ d 11 q n 2 ω S ( n ω S ) 2 n 2 ω q ( n ω q ) 2 T q T S 2 ω S 2 ω q ( L q s i n c ( L q / L c q ) L S s i n c ( L θ S / L c S ) ) 2 ( d 15  +  d 32 ) s i n θ c o s 2 θ ]