Abstract

Despite an intense research effort, the physical mechanism underlying the formation of a quasi-two-dimensional electron gas at the interface between the band insulators LaAlO3 and SrTiO3 is still not fully understood. Interface-sensitive optical second-harmonic spectroscopy can shed light on this mechanism, by accessing specific information on the orbital and structural reconstruction taking place at the interface that is not accessible by other techniques, and in particular by transport measurements. Here we present a detailed theoretical analysis of the spectral transitions that are most relevant in the second-order nonlinear optical response of oxide interfaces with a square symmetry, in general. In particular, we discuss the case of LaAlO3/SrTiO3 interfaces, using symmetry arguments to derive specific selection rules, which have strong consequences on the second-harmonic spectra recorded with different input/output polarization combinations of light. These selection rules may in particular explain recent experimental findings.

© 2013 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  16. T. Higuchi, Y. Hotta, T. Susaki, A. Fujimori, and H. Y. Hwang, “Modulation doping of a Mott quantum well by a proximate polar discontinuity,” Phys. Rev. B 79, 075415 (2009).
    [CrossRef]
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    [CrossRef]
  19. M. Salluzzo, J. Cezar, N. Brookes, V. Bisogni, G. De Luca, C. Richter, S. Thiel, J. Mannhart, M. Huijben, A. Brinkman, G. Rijnders, and G. Ghiringhelli, “Orbital reconstruction and the two-dimensional electron gas at the LaAlO3/SrTiO3 interface,” Phys. Rev. Lett. 102, 166804 (2009).
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    [CrossRef]
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  24. N. Ogawa, K. Miyano, M. Hosoda, T. Higuchi, C. Bell, Y. Hikita, and H. Hwang, “Enhanced lattice polarization in SrTiO3/LaAlO3 superlattices measured using optical second-harmonic generation,” Phys. Rev. B 80, 081106 (2009).
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  27. A. Rubano, T. Güenter, T. Fink, D. Paparo, L. Marrucci, C. Cancellieri, S. Gariglio, J.-M. Triscone, and M. Fiebig, “Influence of atomic termination on the LaAlO3/SrTiO3 interfacial polar rearrangement,” Phys. Rev. B 88, 035405 (2013).
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    [CrossRef]
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    [CrossRef]
  30. M. Bertocchi, E. Luppi, E. Degoli, V. Véniard, and S. Ossicini, “Large crystal local-field effects in second-harmonic generation of a Si/CaF2 interface: an ab initio study,” Phys. Rev. B 86, 035309 (2012).
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    [CrossRef]
  33. D. C. Harris and M. D. Bertolucci, Symmetry and Spectroscopy: An Introduction to Vibrational and Electronic Spectroscopy (Dover, 1989).
  34. A. H. Kahn and A. J. Leyendecker, “Electronic energy bands in strontium titanate,” Phys. Rev. 135, A1321–A1325 (1964).
    [CrossRef]
  35. L. F. Mattheiss, “Energy bands for KNiF3, SrTiO3, KMoO3, and KTaO3,” Phys. Rev. B 6, 4718–4740 (1972).
  36. K. van Benthem, C. Elsässer, and R. H. French, “Bulk electronic structure of SrTiO3: experiment and theory,” J. Appl. Phys. 90, 6156–6164 (2001).
    [CrossRef]
  37. W. Luo, W. Duan, S. G. Louie, and M. L. Cohen, “Structural and electronic properties of n-doped and p-doped SrTiO3,” Phys. Rev. B 70, 214109 (2004).
    [CrossRef]
  38. A. F. Santander-Syro, O. Copie, T. Kondo, F. Fortuna, S. Pailhes, R. Weht, X. G. Qiu, F. Bertran, A. Nicolaou, A. Taleb-Ibrahimi, P. Le Fevre, G. Herranz, M. Bibes, N. Reyren, Y. Apertet, P. Lecoeur, A. Barthelemy, and M. J. Rozenberg, “Two-dimensional electron gas with universal subbands at the surface of SrTiO3,” Nature 469, 189–193 (2011).
    [CrossRef]
  39. C. Cantoni, J. Gazquez, F. M. Granozio, M. P. Oxley, M. Varela, A. R. Lupini, S. J. Pennycook, C. Aruta, U. Scotti di Uccio, P. Perna, and D. Maccariello, “Electronic transfer and ionic displacements as the origin of the 2D electron gas at the LAO/STO interface: direct measurements with atomic-column spatial resolution,” Adv. Mater. 24, 3952–3957 (2012).
    [CrossRef]

2013

A. Rubano, T. Güenter, T. Fink, D. Paparo, L. Marrucci, C. Cancellieri, S. Gariglio, J.-M. Triscone, and M. Fiebig, “Influence of atomic termination on the LaAlO3/SrTiO3 interfacial polar rearrangement,” Phys. Rev. B 88, 035405 (2013).

2012

T. Güenter, A. Rubano, D. Paparo, M. Lilienblum, L. Marrucci, F. Miletto Granozio, U. Scotti di Uccio, R. Jany, C. Richter, J. Mannhart, and M. Fiebig, “Spatial inhomogeneities at the LaAlO3/SrTiO3 interface: evidence from second harmonic generation,” Phys. Rev. B 86, 235418 (2012).

M. Bertocchi, E. Luppi, E. Degoli, V. Véniard, and S. Ossicini, “Large crystal local-field effects in second-harmonic generation of a Si/CaF2 interface: an ab initio study,” Phys. Rev. B 86, 035309 (2012).

C. Cantoni, J. Gazquez, F. M. Granozio, M. P. Oxley, M. Varela, A. R. Lupini, S. J. Pennycook, C. Aruta, U. Scotti di Uccio, P. Perna, and D. Maccariello, “Electronic transfer and ionic displacements as the origin of the 2D electron gas at the LAO/STO interface: direct measurements with atomic-column spatial resolution,” Adv. Mater. 24, 3952–3957 (2012).
[CrossRef]

2011

A. F. Santander-Syro, O. Copie, T. Kondo, F. Fortuna, S. Pailhes, R. Weht, X. G. Qiu, F. Bertran, A. Nicolaou, A. Taleb-Ibrahimi, P. Le Fevre, G. Herranz, M. Bibes, N. Reyren, Y. Apertet, P. Lecoeur, A. Barthelemy, and M. J. Rozenberg, “Two-dimensional electron gas with universal subbands at the surface of SrTiO3,” Nature 469, 189–193 (2011).
[CrossRef]

A. Rubano, M. Fiebig, D. Paparo, A. Marino, D. Maccariello, U. Scotti di Uccio, F. Miletto Granozio, L. Marrucci, C. Richter, S. Paetel, and J. Mannhart, “Spectral and spatial distribution of polarization at the LaAlO3/SrTiO3 interface,” Phys. Rev. B 83, 155405 (2011).

P. Zubko, S. Gariglio, M. Gabay, P. Ghosez, and J.-M. Triscone, “Interface physics in complex oxide heterostructures,” Annu. Rev. Condens. Matter Phys. 2, 141–165 (2011).
[CrossRef]

2010

S. Seo, M. Han, G. Hassink, W. Choi, S. Moon, J. Kim, T. Susaki, Y. Lee, J. Yu, C. Bernhard, H. Hwang, G. Rijnders, D. Blank, B. Keimer, and T. Noh, “Two-dimensional confinement of 3d1 electrons in LaTiO3/LaAlO3 multilayers,” Phys. Rev. Lett. 104, 036401 (2010).
[CrossRef]

E. Luppi, H. Hübener, and V. Véniard, “Ab initio second-order nonlinear optics in solids: second-harmonic generation spectroscopy from time-dependent density-functional theory,” Phys. Rev. B 82, 235201 (2010).
[CrossRef]

2009

C. Cen, S. Thiel, J. Mannhart, and J. Levy, “Oxide nanoelectronics on demand,” Science 323, 1026–1030 (2009).
[CrossRef]

M. Salluzzo, J. Cezar, N. Brookes, V. Bisogni, G. De Luca, C. Richter, S. Thiel, J. Mannhart, M. Huijben, A. Brinkman, G. Rijnders, and G. Ghiringhelli, “Orbital reconstruction and the two-dimensional electron gas at the LaAlO3/SrTiO3 interface,” Phys. Rev. Lett. 102, 166804 (2009).
[CrossRef]

M. Sing, G. Berner, K. Groß, A. Müller, A. Ruff, A. Wetscherek, S. Thiel, J. Mannhart, A. Pauli, C. Schneider, P. Willmott, M. Gorgoi, F. Schäfers, and R. Claessen, “Profiling the interface electron gas of LaAlO3/SrTiO3 heterostructures with hard x-ray photoelectron spectroscopy,” Phys. Rev. Lett. 102, 176805 (2009).
[CrossRef]

M. Huijben, A. Brinkman, G. Koster, G. Rijnders, H. Hilgenkamp, and D. H. A. Blank, “Structure-property relation of SrTiO3/LaAlO3 interfaces,” Adv. Mater. 21, 1665–1677 (2009).
[CrossRef]

A. Savoia, D. Paparo, P. Perna, Z. Ristic, M. Salluzzo, F. Miletto Granozio, U. Scotti di Uccio, C. Richter, S. Thiel, J. Mannhart, and L. Marrucci, “Polar catastrophe and electronic reconstructions at the LaAlO3/SrTiO3 interface: evidence from optical second harmonic generation,” Phys. Rev. B 80, 075110 (2009).

N. Ogawa, K. Miyano, M. Hosoda, T. Higuchi, C. Bell, Y. Hikita, and H. Hwang, “Enhanced lattice polarization in SrTiO3/LaAlO3 superlattices measured using optical second-harmonic generation,” Phys. Rev. B 80, 081106 (2009).

T. Higuchi, Y. Hotta, T. Susaki, A. Fujimori, and H. Y. Hwang, “Modulation doping of a Mott quantum well by a proximate polar discontinuity,” Phys. Rev. B 79, 075415 (2009).
[CrossRef]

2008

R. Ramesh and D. G. Schlom, “Whither oxide electronics?” MRS Bull. 33, 1006–1014 (2008).
[CrossRef]

C. Cen, S. Thiel, G. Hammerl, C. Schneider, K. Andersen, C. Hellberg, J. Mannhart, and J. Levy, “Nanoscale control of an interfacial metal-insulator transition at room temperature,” Nat. Mater. 7, 298–302 (2008).
[CrossRef]

M. Basletic, J.-L. Maurice, C. Carrétéro, G. Herranz, O. Copie, M. Bibes, E. Jacquet, K. Bouzehouane, S. Fusil, and A. Barthélémy, “Mapping the spatial distribution of charge carriers in LaAlO3/SrTiO3 heterostructures,” Nat. Mater. 7, 621–625 (2008).
[CrossRef]

K. Yoshimatsu, R. Yasuhara, H. Kumigashira, and M. Oshima, “Origin of metallic states at the heterointerface between the band insulators LaAlO3 and SrTiO3,” Phys. Rev. Lett. 101, 026802 (2008).
[CrossRef]

A. Caviglia, S. Gariglio, N. Reyren, D. Jaccard, T. Schneider, M. Gabay, S. Thiel, G. Hammerl, J. Mannhart, and J.-M. Triscone, “Electric field control of the LaAlO3/SrTiO3 interface ground state,” Nature 456, 624–627 (2008).
[CrossRef]

2007

A. Tsukazaki, A. Ohtomo, T. Kita, Y. Ohno, H. Ohno, and M. Kawasaki, “Quantum Hall effect in polar oxide heterostructures,” Science 315, 1388–1391 (2007).
[CrossRef]

Y. Hotta, T. Susaki, and H. Y. Hwang, “Polar discontinuity doping of the LaVO3/SrTiO3 interface,” Phys. Rev. Lett. 99, 236805 (2007).
[CrossRef]

A. Brinkman, M. Huijben, M. V. Zalk, J. Huijben, U. Zeitler, J. C. Maan, W. G. V. der Wiel, G. Rijnders, D. H. A. Blank, and H. Hilgenkamp, “Magnetic effects at the interface between non-magnetic oxides,” Nat. Mater. 6, 493–496 (2007).
[CrossRef]

N. Reyren, S. Thiel, A. Caviglia, L. Fitting Kourkoutis, G. Hammerl, C. Richter, C. Schneider, T. Kopp, A.-S. Rüetschi, D. Jaccard, M. Gabay, D. Muller, J.-M. Triscone, and J. Mannhart, “Superconducting interfaces between insulating oxides,” Science 317, 1196–1199 (2007).
[CrossRef]

2006

M. Huijben, G. Rijnders, D. H. A. Blank, S. Bals, S. V. Aert, J. Verbeeck, G. V. Tendeloo, A. Brinkman, and H. Hilgenkamp, “Electronically coupled complementary interfaces between perovskite band insulators,” Nat. Mater. 5, 556–560 (2006).
[CrossRef]

H. Hwang, “Tuning interface states,” Science 313, 1895–1896 (2006).
[CrossRef]

N. Nakagawa, H. Y. Hwang, and D. A. Muller, “Why some interfaces cannot be sharp,” Nat. Mater. 5, 204–209 (2006).
[CrossRef]

S. Thiel, G. Hammerl, A. Schmehl, C. Schneider, and J. Mannhart, “Tunable quasi-two-dimensional electron gases in oxide heterostructures,” Science 313, 1942–1945 (2006).
[CrossRef]

2004

A. Ohtomo and H. Hwang, “A high-mobility electron gas at the LaAlO3/SrTiO3 heterointerface,” Nature 427, 423–426 (2004).
[CrossRef]

W. Luo, W. Duan, S. G. Louie, and M. L. Cohen, “Structural and electronic properties of n-doped and p-doped SrTiO3,” Phys. Rev. B 70, 214109 (2004).
[CrossRef]

2002

L. Marrucci, D. Paparo, G. Cerrone, C. de Lisio, E. Santamato, S. Solimeno, S. Ardizzone, and P. Quagliotto, “Probing interfacial properties by optical second-harmonic generation,” Opt. Lasers Eng. 37, 601–610 (2002).
[CrossRef]

2001

K. van Benthem, C. Elsässer, and R. H. French, “Bulk electronic structure of SrTiO3: experiment and theory,” J. Appl. Phys. 90, 6156–6164 (2001).
[CrossRef]

1997

D. C. Hutchings and J. M. Arnold, “Determination of second-order nonlinear coefficients in semiconductors using pseudospin equations for three-level systems,” Phys. Rev. B 56, 4056–4067 (1997).
[CrossRef]

1972

L. F. Mattheiss, “Energy bands for KNiF3, SrTiO3, KMoO3, and KTaO3,” Phys. Rev. B 6, 4718–4740 (1972).

1965

J. F. Ward, “Calculation of nonlinear optical susceptibilities using diagrammatic perturbation theory,” Rev. Mod. Phys. 37, 1–18 (1965).
[CrossRef]

1964

A. H. Kahn and A. J. Leyendecker, “Electronic energy bands in strontium titanate,” Phys. Rev. 135, A1321–A1325 (1964).
[CrossRef]

Aert, S. V.

M. Huijben, G. Rijnders, D. H. A. Blank, S. Bals, S. V. Aert, J. Verbeeck, G. V. Tendeloo, A. Brinkman, and H. Hilgenkamp, “Electronically coupled complementary interfaces between perovskite band insulators,” Nat. Mater. 5, 556–560 (2006).
[CrossRef]

Andersen, K.

C. Cen, S. Thiel, G. Hammerl, C. Schneider, K. Andersen, C. Hellberg, J. Mannhart, and J. Levy, “Nanoscale control of an interfacial metal-insulator transition at room temperature,” Nat. Mater. 7, 298–302 (2008).
[CrossRef]

Apertet, Y.

A. F. Santander-Syro, O. Copie, T. Kondo, F. Fortuna, S. Pailhes, R. Weht, X. G. Qiu, F. Bertran, A. Nicolaou, A. Taleb-Ibrahimi, P. Le Fevre, G. Herranz, M. Bibes, N. Reyren, Y. Apertet, P. Lecoeur, A. Barthelemy, and M. J. Rozenberg, “Two-dimensional electron gas with universal subbands at the surface of SrTiO3,” Nature 469, 189–193 (2011).
[CrossRef]

Ardizzone, S.

L. Marrucci, D. Paparo, G. Cerrone, C. de Lisio, E. Santamato, S. Solimeno, S. Ardizzone, and P. Quagliotto, “Probing interfacial properties by optical second-harmonic generation,” Opt. Lasers Eng. 37, 601–610 (2002).
[CrossRef]

Arnold, J. M.

D. C. Hutchings and J. M. Arnold, “Determination of second-order nonlinear coefficients in semiconductors using pseudospin equations for three-level systems,” Phys. Rev. B 56, 4056–4067 (1997).
[CrossRef]

Aruta, C.

C. Cantoni, J. Gazquez, F. M. Granozio, M. P. Oxley, M. Varela, A. R. Lupini, S. J. Pennycook, C. Aruta, U. Scotti di Uccio, P. Perna, and D. Maccariello, “Electronic transfer and ionic displacements as the origin of the 2D electron gas at the LAO/STO interface: direct measurements with atomic-column spatial resolution,” Adv. Mater. 24, 3952–3957 (2012).
[CrossRef]

Bals, S.

M. Huijben, G. Rijnders, D. H. A. Blank, S. Bals, S. V. Aert, J. Verbeeck, G. V. Tendeloo, A. Brinkman, and H. Hilgenkamp, “Electronically coupled complementary interfaces between perovskite band insulators,” Nat. Mater. 5, 556–560 (2006).
[CrossRef]

Barthelemy, A.

A. F. Santander-Syro, O. Copie, T. Kondo, F. Fortuna, S. Pailhes, R. Weht, X. G. Qiu, F. Bertran, A. Nicolaou, A. Taleb-Ibrahimi, P. Le Fevre, G. Herranz, M. Bibes, N. Reyren, Y. Apertet, P. Lecoeur, A. Barthelemy, and M. J. Rozenberg, “Two-dimensional electron gas with universal subbands at the surface of SrTiO3,” Nature 469, 189–193 (2011).
[CrossRef]

Barthélémy, A.

M. Basletic, J.-L. Maurice, C. Carrétéro, G. Herranz, O. Copie, M. Bibes, E. Jacquet, K. Bouzehouane, S. Fusil, and A. Barthélémy, “Mapping the spatial distribution of charge carriers in LaAlO3/SrTiO3 heterostructures,” Nat. Mater. 7, 621–625 (2008).
[CrossRef]

Basletic, M.

M. Basletic, J.-L. Maurice, C. Carrétéro, G. Herranz, O. Copie, M. Bibes, E. Jacquet, K. Bouzehouane, S. Fusil, and A. Barthélémy, “Mapping the spatial distribution of charge carriers in LaAlO3/SrTiO3 heterostructures,” Nat. Mater. 7, 621–625 (2008).
[CrossRef]

Bell, C.

N. Ogawa, K. Miyano, M. Hosoda, T. Higuchi, C. Bell, Y. Hikita, and H. Hwang, “Enhanced lattice polarization in SrTiO3/LaAlO3 superlattices measured using optical second-harmonic generation,” Phys. Rev. B 80, 081106 (2009).

Berner, G.

M. Sing, G. Berner, K. Groß, A. Müller, A. Ruff, A. Wetscherek, S. Thiel, J. Mannhart, A. Pauli, C. Schneider, P. Willmott, M. Gorgoi, F. Schäfers, and R. Claessen, “Profiling the interface electron gas of LaAlO3/SrTiO3 heterostructures with hard x-ray photoelectron spectroscopy,” Phys. Rev. Lett. 102, 176805 (2009).
[CrossRef]

Bernhard, C.

S. Seo, M. Han, G. Hassink, W. Choi, S. Moon, J. Kim, T. Susaki, Y. Lee, J. Yu, C. Bernhard, H. Hwang, G. Rijnders, D. Blank, B. Keimer, and T. Noh, “Two-dimensional confinement of 3d1 electrons in LaTiO3/LaAlO3 multilayers,” Phys. Rev. Lett. 104, 036401 (2010).
[CrossRef]

Bertocchi, M.

M. Bertocchi, E. Luppi, E. Degoli, V. Véniard, and S. Ossicini, “Large crystal local-field effects in second-harmonic generation of a Si/CaF2 interface: an ab initio study,” Phys. Rev. B 86, 035309 (2012).

Bertolucci, M. D.

D. C. Harris and M. D. Bertolucci, Symmetry and Spectroscopy: An Introduction to Vibrational and Electronic Spectroscopy (Dover, 1989).

Bertran, F.

A. F. Santander-Syro, O. Copie, T. Kondo, F. Fortuna, S. Pailhes, R. Weht, X. G. Qiu, F. Bertran, A. Nicolaou, A. Taleb-Ibrahimi, P. Le Fevre, G. Herranz, M. Bibes, N. Reyren, Y. Apertet, P. Lecoeur, A. Barthelemy, and M. J. Rozenberg, “Two-dimensional electron gas with universal subbands at the surface of SrTiO3,” Nature 469, 189–193 (2011).
[CrossRef]

Bibes, M.

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A. Brinkman, M. Huijben, M. V. Zalk, J. Huijben, U. Zeitler, J. C. Maan, W. G. V. der Wiel, G. Rijnders, D. H. A. Blank, and H. Hilgenkamp, “Magnetic effects at the interface between non-magnetic oxides,” Nat. Mater. 6, 493–496 (2007).
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A. Brinkman, M. Huijben, M. V. Zalk, J. Huijben, U. Zeitler, J. C. Maan, W. G. V. der Wiel, G. Rijnders, D. H. A. Blank, and H. Hilgenkamp, “Magnetic effects at the interface between non-magnetic oxides,” Nat. Mater. 6, 493–496 (2007).
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N. Reyren, S. Thiel, A. Caviglia, L. Fitting Kourkoutis, G. Hammerl, C. Richter, C. Schneider, T. Kopp, A.-S. Rüetschi, D. Jaccard, M. Gabay, D. Muller, J.-M. Triscone, and J. Mannhart, “Superconducting interfaces between insulating oxides,” Science 317, 1196–1199 (2007).
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M. Basletic, J.-L. Maurice, C. Carrétéro, G. Herranz, O. Copie, M. Bibes, E. Jacquet, K. Bouzehouane, S. Fusil, and A. Barthélémy, “Mapping the spatial distribution of charge carriers in LaAlO3/SrTiO3 heterostructures,” Nat. Mater. 7, 621–625 (2008).
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T. Güenter, A. Rubano, D. Paparo, M. Lilienblum, L. Marrucci, F. Miletto Granozio, U. Scotti di Uccio, R. Jany, C. Richter, J. Mannhart, and M. Fiebig, “Spatial inhomogeneities at the LaAlO3/SrTiO3 interface: evidence from second harmonic generation,” Phys. Rev. B 86, 235418 (2012).

A. Rubano, M. Fiebig, D. Paparo, A. Marino, D. Maccariello, U. Scotti di Uccio, F. Miletto Granozio, L. Marrucci, C. Richter, S. Paetel, and J. Mannhart, “Spectral and spatial distribution of polarization at the LaAlO3/SrTiO3 interface,” Phys. Rev. B 83, 155405 (2011).

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N. Reyren, S. Thiel, A. Caviglia, L. Fitting Kourkoutis, G. Hammerl, C. Richter, C. Schneider, T. Kopp, A.-S. Rüetschi, D. Jaccard, M. Gabay, D. Muller, J.-M. Triscone, and J. Mannhart, “Superconducting interfaces between insulating oxides,” Science 317, 1196–1199 (2007).
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A. Rubano, T. Güenter, T. Fink, D. Paparo, L. Marrucci, C. Cancellieri, S. Gariglio, J.-M. Triscone, and M. Fiebig, “Influence of atomic termination on the LaAlO3/SrTiO3 interfacial polar rearrangement,” Phys. Rev. B 88, 035405 (2013).

P. Zubko, S. Gariglio, M. Gabay, P. Ghosez, and J.-M. Triscone, “Interface physics in complex oxide heterostructures,” Annu. Rev. Condens. Matter Phys. 2, 141–165 (2011).
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A. Caviglia, S. Gariglio, N. Reyren, D. Jaccard, T. Schneider, M. Gabay, S. Thiel, G. Hammerl, J. Mannhart, and J.-M. Triscone, “Electric field control of the LaAlO3/SrTiO3 interface ground state,” Nature 456, 624–627 (2008).
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M. Salluzzo, J. Cezar, N. Brookes, V. Bisogni, G. De Luca, C. Richter, S. Thiel, J. Mannhart, M. Huijben, A. Brinkman, G. Rijnders, and G. Ghiringhelli, “Orbital reconstruction and the two-dimensional electron gas at the LaAlO3/SrTiO3 interface,” Phys. Rev. Lett. 102, 166804 (2009).
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T. Güenter, A. Rubano, D. Paparo, M. Lilienblum, L. Marrucci, F. Miletto Granozio, U. Scotti di Uccio, R. Jany, C. Richter, J. Mannhart, and M. Fiebig, “Spatial inhomogeneities at the LaAlO3/SrTiO3 interface: evidence from second harmonic generation,” Phys. Rev. B 86, 235418 (2012).

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C. Cen, S. Thiel, G. Hammerl, C. Schneider, K. Andersen, C. Hellberg, J. Mannhart, and J. Levy, “Nanoscale control of an interfacial metal-insulator transition at room temperature,” Nat. Mater. 7, 298–302 (2008).
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A. Caviglia, S. Gariglio, N. Reyren, D. Jaccard, T. Schneider, M. Gabay, S. Thiel, G. Hammerl, J. Mannhart, and J.-M. Triscone, “Electric field control of the LaAlO3/SrTiO3 interface ground state,” Nature 456, 624–627 (2008).
[CrossRef]

N. Reyren, S. Thiel, A. Caviglia, L. Fitting Kourkoutis, G. Hammerl, C. Richter, C. Schneider, T. Kopp, A.-S. Rüetschi, D. Jaccard, M. Gabay, D. Muller, J.-M. Triscone, and J. Mannhart, “Superconducting interfaces between insulating oxides,” Science 317, 1196–1199 (2007).
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C. Cen, S. Thiel, G. Hammerl, C. Schneider, K. Andersen, C. Hellberg, J. Mannhart, and J. Levy, “Nanoscale control of an interfacial metal-insulator transition at room temperature,” Nat. Mater. 7, 298–302 (2008).
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A. F. Santander-Syro, O. Copie, T. Kondo, F. Fortuna, S. Pailhes, R. Weht, X. G. Qiu, F. Bertran, A. Nicolaou, A. Taleb-Ibrahimi, P. Le Fevre, G. Herranz, M. Bibes, N. Reyren, Y. Apertet, P. Lecoeur, A. Barthelemy, and M. J. Rozenberg, “Two-dimensional electron gas with universal subbands at the surface of SrTiO3,” Nature 469, 189–193 (2011).
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M. Basletic, J.-L. Maurice, C. Carrétéro, G. Herranz, O. Copie, M. Bibes, E. Jacquet, K. Bouzehouane, S. Fusil, and A. Barthélémy, “Mapping the spatial distribution of charge carriers in LaAlO3/SrTiO3 heterostructures,” Nat. Mater. 7, 621–625 (2008).
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M. Huijben, A. Brinkman, G. Koster, G. Rijnders, H. Hilgenkamp, and D. H. A. Blank, “Structure-property relation of SrTiO3/LaAlO3 interfaces,” Adv. Mater. 21, 1665–1677 (2009).
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T. Higuchi, Y. Hotta, T. Susaki, A. Fujimori, and H. Y. Hwang, “Modulation doping of a Mott quantum well by a proximate polar discontinuity,” Phys. Rev. B 79, 075415 (2009).
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M. Huijben, A. Brinkman, G. Koster, G. Rijnders, H. Hilgenkamp, and D. H. A. Blank, “Structure-property relation of SrTiO3/LaAlO3 interfaces,” Adv. Mater. 21, 1665–1677 (2009).
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M. Salluzzo, J. Cezar, N. Brookes, V. Bisogni, G. De Luca, C. Richter, S. Thiel, J. Mannhart, M. Huijben, A. Brinkman, G. Rijnders, and G. Ghiringhelli, “Orbital reconstruction and the two-dimensional electron gas at the LaAlO3/SrTiO3 interface,” Phys. Rev. Lett. 102, 166804 (2009).
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A. F. Santander-Syro, O. Copie, T. Kondo, F. Fortuna, S. Pailhes, R. Weht, X. G. Qiu, F. Bertran, A. Nicolaou, A. Taleb-Ibrahimi, P. Le Fevre, G. Herranz, M. Bibes, N. Reyren, Y. Apertet, P. Lecoeur, A. Barthelemy, and M. J. Rozenberg, “Two-dimensional electron gas with universal subbands at the surface of SrTiO3,” Nature 469, 189–193 (2011).
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C. Cantoni, J. Gazquez, F. M. Granozio, M. P. Oxley, M. Varela, A. R. Lupini, S. J. Pennycook, C. Aruta, U. Scotti di Uccio, P. Perna, and D. Maccariello, “Electronic transfer and ionic displacements as the origin of the 2D electron gas at the LAO/STO interface: direct measurements with atomic-column spatial resolution,” Adv. Mater. 24, 3952–3957 (2012).
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A. Rubano, M. Fiebig, D. Paparo, A. Marino, D. Maccariello, U. Scotti di Uccio, F. Miletto Granozio, L. Marrucci, C. Richter, S. Paetel, and J. Mannhart, “Spectral and spatial distribution of polarization at the LaAlO3/SrTiO3 interface,” Phys. Rev. B 83, 155405 (2011).

Mannhart, J.

T. Güenter, A. Rubano, D. Paparo, M. Lilienblum, L. Marrucci, F. Miletto Granozio, U. Scotti di Uccio, R. Jany, C. Richter, J. Mannhart, and M. Fiebig, “Spatial inhomogeneities at the LaAlO3/SrTiO3 interface: evidence from second harmonic generation,” Phys. Rev. B 86, 235418 (2012).

A. Rubano, M. Fiebig, D. Paparo, A. Marino, D. Maccariello, U. Scotti di Uccio, F. Miletto Granozio, L. Marrucci, C. Richter, S. Paetel, and J. Mannhart, “Spectral and spatial distribution of polarization at the LaAlO3/SrTiO3 interface,” Phys. Rev. B 83, 155405 (2011).

A. Savoia, D. Paparo, P. Perna, Z. Ristic, M. Salluzzo, F. Miletto Granozio, U. Scotti di Uccio, C. Richter, S. Thiel, J. Mannhart, and L. Marrucci, “Polar catastrophe and electronic reconstructions at the LaAlO3/SrTiO3 interface: evidence from optical second harmonic generation,” Phys. Rev. B 80, 075110 (2009).

M. Salluzzo, J. Cezar, N. Brookes, V. Bisogni, G. De Luca, C. Richter, S. Thiel, J. Mannhart, M. Huijben, A. Brinkman, G. Rijnders, and G. Ghiringhelli, “Orbital reconstruction and the two-dimensional electron gas at the LaAlO3/SrTiO3 interface,” Phys. Rev. Lett. 102, 166804 (2009).
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A. Rubano, T. Güenter, T. Fink, D. Paparo, L. Marrucci, C. Cancellieri, S. Gariglio, J.-M. Triscone, and M. Fiebig, “Influence of atomic termination on the LaAlO3/SrTiO3 interfacial polar rearrangement,” Phys. Rev. B 88, 035405 (2013).

T. Güenter, A. Rubano, D. Paparo, M. Lilienblum, L. Marrucci, F. Miletto Granozio, U. Scotti di Uccio, R. Jany, C. Richter, J. Mannhart, and M. Fiebig, “Spatial inhomogeneities at the LaAlO3/SrTiO3 interface: evidence from second harmonic generation,” Phys. Rev. B 86, 235418 (2012).

A. Rubano, M. Fiebig, D. Paparo, A. Marino, D. Maccariello, U. Scotti di Uccio, F. Miletto Granozio, L. Marrucci, C. Richter, S. Paetel, and J. Mannhart, “Spectral and spatial distribution of polarization at the LaAlO3/SrTiO3 interface,” Phys. Rev. B 83, 155405 (2011).

A. Savoia, D. Paparo, P. Perna, Z. Ristic, M. Salluzzo, F. Miletto Granozio, U. Scotti di Uccio, C. Richter, S. Thiel, J. Mannhart, and L. Marrucci, “Polar catastrophe and electronic reconstructions at the LaAlO3/SrTiO3 interface: evidence from optical second harmonic generation,” Phys. Rev. B 80, 075110 (2009).

L. Marrucci, D. Paparo, G. Cerrone, C. de Lisio, E. Santamato, S. Solimeno, S. Ardizzone, and P. Quagliotto, “Probing interfacial properties by optical second-harmonic generation,” Opt. Lasers Eng. 37, 601–610 (2002).
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Miletto Granozio, F.

T. Güenter, A. Rubano, D. Paparo, M. Lilienblum, L. Marrucci, F. Miletto Granozio, U. Scotti di Uccio, R. Jany, C. Richter, J. Mannhart, and M. Fiebig, “Spatial inhomogeneities at the LaAlO3/SrTiO3 interface: evidence from second harmonic generation,” Phys. Rev. B 86, 235418 (2012).

A. Rubano, M. Fiebig, D. Paparo, A. Marino, D. Maccariello, U. Scotti di Uccio, F. Miletto Granozio, L. Marrucci, C. Richter, S. Paetel, and J. Mannhart, “Spectral and spatial distribution of polarization at the LaAlO3/SrTiO3 interface,” Phys. Rev. B 83, 155405 (2011).

A. Savoia, D. Paparo, P. Perna, Z. Ristic, M. Salluzzo, F. Miletto Granozio, U. Scotti di Uccio, C. Richter, S. Thiel, J. Mannhart, and L. Marrucci, “Polar catastrophe and electronic reconstructions at the LaAlO3/SrTiO3 interface: evidence from optical second harmonic generation,” Phys. Rev. B 80, 075110 (2009).

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N. Ogawa, K. Miyano, M. Hosoda, T. Higuchi, C. Bell, Y. Hikita, and H. Hwang, “Enhanced lattice polarization in SrTiO3/LaAlO3 superlattices measured using optical second-harmonic generation,” Phys. Rev. B 80, 081106 (2009).

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S. Seo, M. Han, G. Hassink, W. Choi, S. Moon, J. Kim, T. Susaki, Y. Lee, J. Yu, C. Bernhard, H. Hwang, G. Rijnders, D. Blank, B. Keimer, and T. Noh, “Two-dimensional confinement of 3d1 electrons in LaTiO3/LaAlO3 multilayers,” Phys. Rev. Lett. 104, 036401 (2010).
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N. Reyren, S. Thiel, A. Caviglia, L. Fitting Kourkoutis, G. Hammerl, C. Richter, C. Schneider, T. Kopp, A.-S. Rüetschi, D. Jaccard, M. Gabay, D. Muller, J.-M. Triscone, and J. Mannhart, “Superconducting interfaces between insulating oxides,” Science 317, 1196–1199 (2007).
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Muller, D. A.

N. Nakagawa, H. Y. Hwang, and D. A. Muller, “Why some interfaces cannot be sharp,” Nat. Mater. 5, 204–209 (2006).
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M. Sing, G. Berner, K. Groß, A. Müller, A. Ruff, A. Wetscherek, S. Thiel, J. Mannhart, A. Pauli, C. Schneider, P. Willmott, M. Gorgoi, F. Schäfers, and R. Claessen, “Profiling the interface electron gas of LaAlO3/SrTiO3 heterostructures with hard x-ray photoelectron spectroscopy,” Phys. Rev. Lett. 102, 176805 (2009).
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N. Nakagawa, H. Y. Hwang, and D. A. Muller, “Why some interfaces cannot be sharp,” Nat. Mater. 5, 204–209 (2006).
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A. F. Santander-Syro, O. Copie, T. Kondo, F. Fortuna, S. Pailhes, R. Weht, X. G. Qiu, F. Bertran, A. Nicolaou, A. Taleb-Ibrahimi, P. Le Fevre, G. Herranz, M. Bibes, N. Reyren, Y. Apertet, P. Lecoeur, A. Barthelemy, and M. J. Rozenberg, “Two-dimensional electron gas with universal subbands at the surface of SrTiO3,” Nature 469, 189–193 (2011).
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S. Seo, M. Han, G. Hassink, W. Choi, S. Moon, J. Kim, T. Susaki, Y. Lee, J. Yu, C. Bernhard, H. Hwang, G. Rijnders, D. Blank, B. Keimer, and T. Noh, “Two-dimensional confinement of 3d1 electrons in LaTiO3/LaAlO3 multilayers,” Phys. Rev. Lett. 104, 036401 (2010).
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N. Ogawa, K. Miyano, M. Hosoda, T. Higuchi, C. Bell, Y. Hikita, and H. Hwang, “Enhanced lattice polarization in SrTiO3/LaAlO3 superlattices measured using optical second-harmonic generation,” Phys. Rev. B 80, 081106 (2009).

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A. Tsukazaki, A. Ohtomo, T. Kita, Y. Ohno, H. Ohno, and M. Kawasaki, “Quantum Hall effect in polar oxide heterostructures,” Science 315, 1388–1391 (2007).
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A. Tsukazaki, A. Ohtomo, T. Kita, Y. Ohno, H. Ohno, and M. Kawasaki, “Quantum Hall effect in polar oxide heterostructures,” Science 315, 1388–1391 (2007).
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A. Tsukazaki, A. Ohtomo, T. Kita, Y. Ohno, H. Ohno, and M. Kawasaki, “Quantum Hall effect in polar oxide heterostructures,” Science 315, 1388–1391 (2007).
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A. Ohtomo and H. Hwang, “A high-mobility electron gas at the LaAlO3/SrTiO3 heterointerface,” Nature 427, 423–426 (2004).
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K. Yoshimatsu, R. Yasuhara, H. Kumigashira, and M. Oshima, “Origin of metallic states at the heterointerface between the band insulators LaAlO3 and SrTiO3,” Phys. Rev. Lett. 101, 026802 (2008).
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M. Bertocchi, E. Luppi, E. Degoli, V. Véniard, and S. Ossicini, “Large crystal local-field effects in second-harmonic generation of a Si/CaF2 interface: an ab initio study,” Phys. Rev. B 86, 035309 (2012).

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C. Cantoni, J. Gazquez, F. M. Granozio, M. P. Oxley, M. Varela, A. R. Lupini, S. J. Pennycook, C. Aruta, U. Scotti di Uccio, P. Perna, and D. Maccariello, “Electronic transfer and ionic displacements as the origin of the 2D electron gas at the LAO/STO interface: direct measurements with atomic-column spatial resolution,” Adv. Mater. 24, 3952–3957 (2012).
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A. Rubano, M. Fiebig, D. Paparo, A. Marino, D. Maccariello, U. Scotti di Uccio, F. Miletto Granozio, L. Marrucci, C. Richter, S. Paetel, and J. Mannhart, “Spectral and spatial distribution of polarization at the LaAlO3/SrTiO3 interface,” Phys. Rev. B 83, 155405 (2011).

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A. F. Santander-Syro, O. Copie, T. Kondo, F. Fortuna, S. Pailhes, R. Weht, X. G. Qiu, F. Bertran, A. Nicolaou, A. Taleb-Ibrahimi, P. Le Fevre, G. Herranz, M. Bibes, N. Reyren, Y. Apertet, P. Lecoeur, A. Barthelemy, and M. J. Rozenberg, “Two-dimensional electron gas with universal subbands at the surface of SrTiO3,” Nature 469, 189–193 (2011).
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Paparo, D.

A. Rubano, T. Güenter, T. Fink, D. Paparo, L. Marrucci, C. Cancellieri, S. Gariglio, J.-M. Triscone, and M. Fiebig, “Influence of atomic termination on the LaAlO3/SrTiO3 interfacial polar rearrangement,” Phys. Rev. B 88, 035405 (2013).

T. Güenter, A. Rubano, D. Paparo, M. Lilienblum, L. Marrucci, F. Miletto Granozio, U. Scotti di Uccio, R. Jany, C. Richter, J. Mannhart, and M. Fiebig, “Spatial inhomogeneities at the LaAlO3/SrTiO3 interface: evidence from second harmonic generation,” Phys. Rev. B 86, 235418 (2012).

A. Rubano, M. Fiebig, D. Paparo, A. Marino, D. Maccariello, U. Scotti di Uccio, F. Miletto Granozio, L. Marrucci, C. Richter, S. Paetel, and J. Mannhart, “Spectral and spatial distribution of polarization at the LaAlO3/SrTiO3 interface,” Phys. Rev. B 83, 155405 (2011).

A. Savoia, D. Paparo, P. Perna, Z. Ristic, M. Salluzzo, F. Miletto Granozio, U. Scotti di Uccio, C. Richter, S. Thiel, J. Mannhart, and L. Marrucci, “Polar catastrophe and electronic reconstructions at the LaAlO3/SrTiO3 interface: evidence from optical second harmonic generation,” Phys. Rev. B 80, 075110 (2009).

L. Marrucci, D. Paparo, G. Cerrone, C. de Lisio, E. Santamato, S. Solimeno, S. Ardizzone, and P. Quagliotto, “Probing interfacial properties by optical second-harmonic generation,” Opt. Lasers Eng. 37, 601–610 (2002).
[CrossRef]

Pauli, A.

M. Sing, G. Berner, K. Groß, A. Müller, A. Ruff, A. Wetscherek, S. Thiel, J. Mannhart, A. Pauli, C. Schneider, P. Willmott, M. Gorgoi, F. Schäfers, and R. Claessen, “Profiling the interface electron gas of LaAlO3/SrTiO3 heterostructures with hard x-ray photoelectron spectroscopy,” Phys. Rev. Lett. 102, 176805 (2009).
[CrossRef]

Pennycook, S. J.

C. Cantoni, J. Gazquez, F. M. Granozio, M. P. Oxley, M. Varela, A. R. Lupini, S. J. Pennycook, C. Aruta, U. Scotti di Uccio, P. Perna, and D. Maccariello, “Electronic transfer and ionic displacements as the origin of the 2D electron gas at the LAO/STO interface: direct measurements with atomic-column spatial resolution,” Adv. Mater. 24, 3952–3957 (2012).
[CrossRef]

Perna, P.

C. Cantoni, J. Gazquez, F. M. Granozio, M. P. Oxley, M. Varela, A. R. Lupini, S. J. Pennycook, C. Aruta, U. Scotti di Uccio, P. Perna, and D. Maccariello, “Electronic transfer and ionic displacements as the origin of the 2D electron gas at the LAO/STO interface: direct measurements with atomic-column spatial resolution,” Adv. Mater. 24, 3952–3957 (2012).
[CrossRef]

A. Savoia, D. Paparo, P. Perna, Z. Ristic, M. Salluzzo, F. Miletto Granozio, U. Scotti di Uccio, C. Richter, S. Thiel, J. Mannhart, and L. Marrucci, “Polar catastrophe and electronic reconstructions at the LaAlO3/SrTiO3 interface: evidence from optical second harmonic generation,” Phys. Rev. B 80, 075110 (2009).

Qiu, X. G.

A. F. Santander-Syro, O. Copie, T. Kondo, F. Fortuna, S. Pailhes, R. Weht, X. G. Qiu, F. Bertran, A. Nicolaou, A. Taleb-Ibrahimi, P. Le Fevre, G. Herranz, M. Bibes, N. Reyren, Y. Apertet, P. Lecoeur, A. Barthelemy, and M. J. Rozenberg, “Two-dimensional electron gas with universal subbands at the surface of SrTiO3,” Nature 469, 189–193 (2011).
[CrossRef]

Quagliotto, P.

L. Marrucci, D. Paparo, G. Cerrone, C. de Lisio, E. Santamato, S. Solimeno, S. Ardizzone, and P. Quagliotto, “Probing interfacial properties by optical second-harmonic generation,” Opt. Lasers Eng. 37, 601–610 (2002).
[CrossRef]

Ramesh, R.

R. Ramesh and D. G. Schlom, “Whither oxide electronics?” MRS Bull. 33, 1006–1014 (2008).
[CrossRef]

Reyren, N.

A. F. Santander-Syro, O. Copie, T. Kondo, F. Fortuna, S. Pailhes, R. Weht, X. G. Qiu, F. Bertran, A. Nicolaou, A. Taleb-Ibrahimi, P. Le Fevre, G. Herranz, M. Bibes, N. Reyren, Y. Apertet, P. Lecoeur, A. Barthelemy, and M. J. Rozenberg, “Two-dimensional electron gas with universal subbands at the surface of SrTiO3,” Nature 469, 189–193 (2011).
[CrossRef]

A. Caviglia, S. Gariglio, N. Reyren, D. Jaccard, T. Schneider, M. Gabay, S. Thiel, G. Hammerl, J. Mannhart, and J.-M. Triscone, “Electric field control of the LaAlO3/SrTiO3 interface ground state,” Nature 456, 624–627 (2008).
[CrossRef]

N. Reyren, S. Thiel, A. Caviglia, L. Fitting Kourkoutis, G. Hammerl, C. Richter, C. Schneider, T. Kopp, A.-S. Rüetschi, D. Jaccard, M. Gabay, D. Muller, J.-M. Triscone, and J. Mannhart, “Superconducting interfaces between insulating oxides,” Science 317, 1196–1199 (2007).
[CrossRef]

Richter, C.

T. Güenter, A. Rubano, D. Paparo, M. Lilienblum, L. Marrucci, F. Miletto Granozio, U. Scotti di Uccio, R. Jany, C. Richter, J. Mannhart, and M. Fiebig, “Spatial inhomogeneities at the LaAlO3/SrTiO3 interface: evidence from second harmonic generation,” Phys. Rev. B 86, 235418 (2012).

A. Rubano, M. Fiebig, D. Paparo, A. Marino, D. Maccariello, U. Scotti di Uccio, F. Miletto Granozio, L. Marrucci, C. Richter, S. Paetel, and J. Mannhart, “Spectral and spatial distribution of polarization at the LaAlO3/SrTiO3 interface,” Phys. Rev. B 83, 155405 (2011).

A. Savoia, D. Paparo, P. Perna, Z. Ristic, M. Salluzzo, F. Miletto Granozio, U. Scotti di Uccio, C. Richter, S. Thiel, J. Mannhart, and L. Marrucci, “Polar catastrophe and electronic reconstructions at the LaAlO3/SrTiO3 interface: evidence from optical second harmonic generation,” Phys. Rev. B 80, 075110 (2009).

M. Salluzzo, J. Cezar, N. Brookes, V. Bisogni, G. De Luca, C. Richter, S. Thiel, J. Mannhart, M. Huijben, A. Brinkman, G. Rijnders, and G. Ghiringhelli, “Orbital reconstruction and the two-dimensional electron gas at the LaAlO3/SrTiO3 interface,” Phys. Rev. Lett. 102, 166804 (2009).
[CrossRef]

N. Reyren, S. Thiel, A. Caviglia, L. Fitting Kourkoutis, G. Hammerl, C. Richter, C. Schneider, T. Kopp, A.-S. Rüetschi, D. Jaccard, M. Gabay, D. Muller, J.-M. Triscone, and J. Mannhart, “Superconducting interfaces between insulating oxides,” Science 317, 1196–1199 (2007).
[CrossRef]

Rijnders, G.

S. Seo, M. Han, G. Hassink, W. Choi, S. Moon, J. Kim, T. Susaki, Y. Lee, J. Yu, C. Bernhard, H. Hwang, G. Rijnders, D. Blank, B. Keimer, and T. Noh, “Two-dimensional confinement of 3d1 electrons in LaTiO3/LaAlO3 multilayers,” Phys. Rev. Lett. 104, 036401 (2010).
[CrossRef]

M. Salluzzo, J. Cezar, N. Brookes, V. Bisogni, G. De Luca, C. Richter, S. Thiel, J. Mannhart, M. Huijben, A. Brinkman, G. Rijnders, and G. Ghiringhelli, “Orbital reconstruction and the two-dimensional electron gas at the LaAlO3/SrTiO3 interface,” Phys. Rev. Lett. 102, 166804 (2009).
[CrossRef]

M. Huijben, A. Brinkman, G. Koster, G. Rijnders, H. Hilgenkamp, and D. H. A. Blank, “Structure-property relation of SrTiO3/LaAlO3 interfaces,” Adv. Mater. 21, 1665–1677 (2009).
[CrossRef]

A. Brinkman, M. Huijben, M. V. Zalk, J. Huijben, U. Zeitler, J. C. Maan, W. G. V. der Wiel, G. Rijnders, D. H. A. Blank, and H. Hilgenkamp, “Magnetic effects at the interface between non-magnetic oxides,” Nat. Mater. 6, 493–496 (2007).
[CrossRef]

M. Huijben, G. Rijnders, D. H. A. Blank, S. Bals, S. V. Aert, J. Verbeeck, G. V. Tendeloo, A. Brinkman, and H. Hilgenkamp, “Electronically coupled complementary interfaces between perovskite band insulators,” Nat. Mater. 5, 556–560 (2006).
[CrossRef]

Ristic, Z.

A. Savoia, D. Paparo, P. Perna, Z. Ristic, M. Salluzzo, F. Miletto Granozio, U. Scotti di Uccio, C. Richter, S. Thiel, J. Mannhart, and L. Marrucci, “Polar catastrophe and electronic reconstructions at the LaAlO3/SrTiO3 interface: evidence from optical second harmonic generation,” Phys. Rev. B 80, 075110 (2009).

Rozenberg, M. J.

A. F. Santander-Syro, O. Copie, T. Kondo, F. Fortuna, S. Pailhes, R. Weht, X. G. Qiu, F. Bertran, A. Nicolaou, A. Taleb-Ibrahimi, P. Le Fevre, G. Herranz, M. Bibes, N. Reyren, Y. Apertet, P. Lecoeur, A. Barthelemy, and M. J. Rozenberg, “Two-dimensional electron gas with universal subbands at the surface of SrTiO3,” Nature 469, 189–193 (2011).
[CrossRef]

Rubano, A.

A. Rubano, T. Güenter, T. Fink, D. Paparo, L. Marrucci, C. Cancellieri, S. Gariglio, J.-M. Triscone, and M. Fiebig, “Influence of atomic termination on the LaAlO3/SrTiO3 interfacial polar rearrangement,” Phys. Rev. B 88, 035405 (2013).

T. Güenter, A. Rubano, D. Paparo, M. Lilienblum, L. Marrucci, F. Miletto Granozio, U. Scotti di Uccio, R. Jany, C. Richter, J. Mannhart, and M. Fiebig, “Spatial inhomogeneities at the LaAlO3/SrTiO3 interface: evidence from second harmonic generation,” Phys. Rev. B 86, 235418 (2012).

A. Rubano, M. Fiebig, D. Paparo, A. Marino, D. Maccariello, U. Scotti di Uccio, F. Miletto Granozio, L. Marrucci, C. Richter, S. Paetel, and J. Mannhart, “Spectral and spatial distribution of polarization at the LaAlO3/SrTiO3 interface,” Phys. Rev. B 83, 155405 (2011).

Rüetschi, A.-S.

N. Reyren, S. Thiel, A. Caviglia, L. Fitting Kourkoutis, G. Hammerl, C. Richter, C. Schneider, T. Kopp, A.-S. Rüetschi, D. Jaccard, M. Gabay, D. Muller, J.-M. Triscone, and J. Mannhart, “Superconducting interfaces between insulating oxides,” Science 317, 1196–1199 (2007).
[CrossRef]

Ruff, A.

M. Sing, G. Berner, K. Groß, A. Müller, A. Ruff, A. Wetscherek, S. Thiel, J. Mannhart, A. Pauli, C. Schneider, P. Willmott, M. Gorgoi, F. Schäfers, and R. Claessen, “Profiling the interface electron gas of LaAlO3/SrTiO3 heterostructures with hard x-ray photoelectron spectroscopy,” Phys. Rev. Lett. 102, 176805 (2009).
[CrossRef]

Salluzzo, M.

A. Savoia, D. Paparo, P. Perna, Z. Ristic, M. Salluzzo, F. Miletto Granozio, U. Scotti di Uccio, C. Richter, S. Thiel, J. Mannhart, and L. Marrucci, “Polar catastrophe and electronic reconstructions at the LaAlO3/SrTiO3 interface: evidence from optical second harmonic generation,” Phys. Rev. B 80, 075110 (2009).

M. Salluzzo, J. Cezar, N. Brookes, V. Bisogni, G. De Luca, C. Richter, S. Thiel, J. Mannhart, M. Huijben, A. Brinkman, G. Rijnders, and G. Ghiringhelli, “Orbital reconstruction and the two-dimensional electron gas at the LaAlO3/SrTiO3 interface,” Phys. Rev. Lett. 102, 166804 (2009).
[CrossRef]

Santamato, E.

L. Marrucci, D. Paparo, G. Cerrone, C. de Lisio, E. Santamato, S. Solimeno, S. Ardizzone, and P. Quagliotto, “Probing interfacial properties by optical second-harmonic generation,” Opt. Lasers Eng. 37, 601–610 (2002).
[CrossRef]

Santander-Syro, A. F.

A. F. Santander-Syro, O. Copie, T. Kondo, F. Fortuna, S. Pailhes, R. Weht, X. G. Qiu, F. Bertran, A. Nicolaou, A. Taleb-Ibrahimi, P. Le Fevre, G. Herranz, M. Bibes, N. Reyren, Y. Apertet, P. Lecoeur, A. Barthelemy, and M. J. Rozenberg, “Two-dimensional electron gas with universal subbands at the surface of SrTiO3,” Nature 469, 189–193 (2011).
[CrossRef]

Savoia, A.

A. Savoia, D. Paparo, P. Perna, Z. Ristic, M. Salluzzo, F. Miletto Granozio, U. Scotti di Uccio, C. Richter, S. Thiel, J. Mannhart, and L. Marrucci, “Polar catastrophe and electronic reconstructions at the LaAlO3/SrTiO3 interface: evidence from optical second harmonic generation,” Phys. Rev. B 80, 075110 (2009).

Schäfers, F.

M. Sing, G. Berner, K. Groß, A. Müller, A. Ruff, A. Wetscherek, S. Thiel, J. Mannhart, A. Pauli, C. Schneider, P. Willmott, M. Gorgoi, F. Schäfers, and R. Claessen, “Profiling the interface electron gas of LaAlO3/SrTiO3 heterostructures with hard x-ray photoelectron spectroscopy,” Phys. Rev. Lett. 102, 176805 (2009).
[CrossRef]

Schlom, D. G.

R. Ramesh and D. G. Schlom, “Whither oxide electronics?” MRS Bull. 33, 1006–1014 (2008).
[CrossRef]

Schmehl, A.

S. Thiel, G. Hammerl, A. Schmehl, C. Schneider, and J. Mannhart, “Tunable quasi-two-dimensional electron gases in oxide heterostructures,” Science 313, 1942–1945 (2006).
[CrossRef]

Schneider, C.

M. Sing, G. Berner, K. Groß, A. Müller, A. Ruff, A. Wetscherek, S. Thiel, J. Mannhart, A. Pauli, C. Schneider, P. Willmott, M. Gorgoi, F. Schäfers, and R. Claessen, “Profiling the interface electron gas of LaAlO3/SrTiO3 heterostructures with hard x-ray photoelectron spectroscopy,” Phys. Rev. Lett. 102, 176805 (2009).
[CrossRef]

C. Cen, S. Thiel, G. Hammerl, C. Schneider, K. Andersen, C. Hellberg, J. Mannhart, and J. Levy, “Nanoscale control of an interfacial metal-insulator transition at room temperature,” Nat. Mater. 7, 298–302 (2008).
[CrossRef]

N. Reyren, S. Thiel, A. Caviglia, L. Fitting Kourkoutis, G. Hammerl, C. Richter, C. Schneider, T. Kopp, A.-S. Rüetschi, D. Jaccard, M. Gabay, D. Muller, J.-M. Triscone, and J. Mannhart, “Superconducting interfaces between insulating oxides,” Science 317, 1196–1199 (2007).
[CrossRef]

S. Thiel, G. Hammerl, A. Schmehl, C. Schneider, and J. Mannhart, “Tunable quasi-two-dimensional electron gases in oxide heterostructures,” Science 313, 1942–1945 (2006).
[CrossRef]

Schneider, T.

A. Caviglia, S. Gariglio, N. Reyren, D. Jaccard, T. Schneider, M. Gabay, S. Thiel, G. Hammerl, J. Mannhart, and J.-M. Triscone, “Electric field control of the LaAlO3/SrTiO3 interface ground state,” Nature 456, 624–627 (2008).
[CrossRef]

Scotti di Uccio, U.

C. Cantoni, J. Gazquez, F. M. Granozio, M. P. Oxley, M. Varela, A. R. Lupini, S. J. Pennycook, C. Aruta, U. Scotti di Uccio, P. Perna, and D. Maccariello, “Electronic transfer and ionic displacements as the origin of the 2D electron gas at the LAO/STO interface: direct measurements with atomic-column spatial resolution,” Adv. Mater. 24, 3952–3957 (2012).
[CrossRef]

T. Güenter, A. Rubano, D. Paparo, M. Lilienblum, L. Marrucci, F. Miletto Granozio, U. Scotti di Uccio, R. Jany, C. Richter, J. Mannhart, and M. Fiebig, “Spatial inhomogeneities at the LaAlO3/SrTiO3 interface: evidence from second harmonic generation,” Phys. Rev. B 86, 235418 (2012).

A. Rubano, M. Fiebig, D. Paparo, A. Marino, D. Maccariello, U. Scotti di Uccio, F. Miletto Granozio, L. Marrucci, C. Richter, S. Paetel, and J. Mannhart, “Spectral and spatial distribution of polarization at the LaAlO3/SrTiO3 interface,” Phys. Rev. B 83, 155405 (2011).

A. Savoia, D. Paparo, P. Perna, Z. Ristic, M. Salluzzo, F. Miletto Granozio, U. Scotti di Uccio, C. Richter, S. Thiel, J. Mannhart, and L. Marrucci, “Polar catastrophe and electronic reconstructions at the LaAlO3/SrTiO3 interface: evidence from optical second harmonic generation,” Phys. Rev. B 80, 075110 (2009).

Seo, S.

S. Seo, M. Han, G. Hassink, W. Choi, S. Moon, J. Kim, T. Susaki, Y. Lee, J. Yu, C. Bernhard, H. Hwang, G. Rijnders, D. Blank, B. Keimer, and T. Noh, “Two-dimensional confinement of 3d1 electrons in LaTiO3/LaAlO3 multilayers,” Phys. Rev. Lett. 104, 036401 (2010).
[CrossRef]

Sing, M.

M. Sing, G. Berner, K. Groß, A. Müller, A. Ruff, A. Wetscherek, S. Thiel, J. Mannhart, A. Pauli, C. Schneider, P. Willmott, M. Gorgoi, F. Schäfers, and R. Claessen, “Profiling the interface electron gas of LaAlO3/SrTiO3 heterostructures with hard x-ray photoelectron spectroscopy,” Phys. Rev. Lett. 102, 176805 (2009).
[CrossRef]

Solimeno, S.

L. Marrucci, D. Paparo, G. Cerrone, C. de Lisio, E. Santamato, S. Solimeno, S. Ardizzone, and P. Quagliotto, “Probing interfacial properties by optical second-harmonic generation,” Opt. Lasers Eng. 37, 601–610 (2002).
[CrossRef]

Susaki, T.

S. Seo, M. Han, G. Hassink, W. Choi, S. Moon, J. Kim, T. Susaki, Y. Lee, J. Yu, C. Bernhard, H. Hwang, G. Rijnders, D. Blank, B. Keimer, and T. Noh, “Two-dimensional confinement of 3d1 electrons in LaTiO3/LaAlO3 multilayers,” Phys. Rev. Lett. 104, 036401 (2010).
[CrossRef]

T. Higuchi, Y. Hotta, T. Susaki, A. Fujimori, and H. Y. Hwang, “Modulation doping of a Mott quantum well by a proximate polar discontinuity,” Phys. Rev. B 79, 075415 (2009).
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Y. Hotta, T. Susaki, and H. Y. Hwang, “Polar discontinuity doping of the LaVO3/SrTiO3 interface,” Phys. Rev. Lett. 99, 236805 (2007).
[CrossRef]

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A. F. Santander-Syro, O. Copie, T. Kondo, F. Fortuna, S. Pailhes, R. Weht, X. G. Qiu, F. Bertran, A. Nicolaou, A. Taleb-Ibrahimi, P. Le Fevre, G. Herranz, M. Bibes, N. Reyren, Y. Apertet, P. Lecoeur, A. Barthelemy, and M. J. Rozenberg, “Two-dimensional electron gas with universal subbands at the surface of SrTiO3,” Nature 469, 189–193 (2011).
[CrossRef]

Tendeloo, G. V.

M. Huijben, G. Rijnders, D. H. A. Blank, S. Bals, S. V. Aert, J. Verbeeck, G. V. Tendeloo, A. Brinkman, and H. Hilgenkamp, “Electronically coupled complementary interfaces between perovskite band insulators,” Nat. Mater. 5, 556–560 (2006).
[CrossRef]

Thiel, S.

M. Sing, G. Berner, K. Groß, A. Müller, A. Ruff, A. Wetscherek, S. Thiel, J. Mannhart, A. Pauli, C. Schneider, P. Willmott, M. Gorgoi, F. Schäfers, and R. Claessen, “Profiling the interface electron gas of LaAlO3/SrTiO3 heterostructures with hard x-ray photoelectron spectroscopy,” Phys. Rev. Lett. 102, 176805 (2009).
[CrossRef]

C. Cen, S. Thiel, J. Mannhart, and J. Levy, “Oxide nanoelectronics on demand,” Science 323, 1026–1030 (2009).
[CrossRef]

M. Salluzzo, J. Cezar, N. Brookes, V. Bisogni, G. De Luca, C. Richter, S. Thiel, J. Mannhart, M. Huijben, A. Brinkman, G. Rijnders, and G. Ghiringhelli, “Orbital reconstruction and the two-dimensional electron gas at the LaAlO3/SrTiO3 interface,” Phys. Rev. Lett. 102, 166804 (2009).
[CrossRef]

A. Savoia, D. Paparo, P. Perna, Z. Ristic, M. Salluzzo, F. Miletto Granozio, U. Scotti di Uccio, C. Richter, S. Thiel, J. Mannhart, and L. Marrucci, “Polar catastrophe and electronic reconstructions at the LaAlO3/SrTiO3 interface: evidence from optical second harmonic generation,” Phys. Rev. B 80, 075110 (2009).

A. Caviglia, S. Gariglio, N. Reyren, D. Jaccard, T. Schneider, M. Gabay, S. Thiel, G. Hammerl, J. Mannhart, and J.-M. Triscone, “Electric field control of the LaAlO3/SrTiO3 interface ground state,” Nature 456, 624–627 (2008).
[CrossRef]

C. Cen, S. Thiel, G. Hammerl, C. Schneider, K. Andersen, C. Hellberg, J. Mannhart, and J. Levy, “Nanoscale control of an interfacial metal-insulator transition at room temperature,” Nat. Mater. 7, 298–302 (2008).
[CrossRef]

N. Reyren, S. Thiel, A. Caviglia, L. Fitting Kourkoutis, G. Hammerl, C. Richter, C. Schneider, T. Kopp, A.-S. Rüetschi, D. Jaccard, M. Gabay, D. Muller, J.-M. Triscone, and J. Mannhart, “Superconducting interfaces between insulating oxides,” Science 317, 1196–1199 (2007).
[CrossRef]

S. Thiel, G. Hammerl, A. Schmehl, C. Schneider, and J. Mannhart, “Tunable quasi-two-dimensional electron gases in oxide heterostructures,” Science 313, 1942–1945 (2006).
[CrossRef]

Triscone, J.-M.

A. Rubano, T. Güenter, T. Fink, D. Paparo, L. Marrucci, C. Cancellieri, S. Gariglio, J.-M. Triscone, and M. Fiebig, “Influence of atomic termination on the LaAlO3/SrTiO3 interfacial polar rearrangement,” Phys. Rev. B 88, 035405 (2013).

P. Zubko, S. Gariglio, M. Gabay, P. Ghosez, and J.-M. Triscone, “Interface physics in complex oxide heterostructures,” Annu. Rev. Condens. Matter Phys. 2, 141–165 (2011).
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A. Caviglia, S. Gariglio, N. Reyren, D. Jaccard, T. Schneider, M. Gabay, S. Thiel, G. Hammerl, J. Mannhart, and J.-M. Triscone, “Electric field control of the LaAlO3/SrTiO3 interface ground state,” Nature 456, 624–627 (2008).
[CrossRef]

N. Reyren, S. Thiel, A. Caviglia, L. Fitting Kourkoutis, G. Hammerl, C. Richter, C. Schneider, T. Kopp, A.-S. Rüetschi, D. Jaccard, M. Gabay, D. Muller, J.-M. Triscone, and J. Mannhart, “Superconducting interfaces between insulating oxides,” Science 317, 1196–1199 (2007).
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A. Tsukazaki, A. Ohtomo, T. Kita, Y. Ohno, H. Ohno, and M. Kawasaki, “Quantum Hall effect in polar oxide heterostructures,” Science 315, 1388–1391 (2007).
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K. van Benthem, C. Elsässer, and R. H. French, “Bulk electronic structure of SrTiO3: experiment and theory,” J. Appl. Phys. 90, 6156–6164 (2001).
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C. Cantoni, J. Gazquez, F. M. Granozio, M. P. Oxley, M. Varela, A. R. Lupini, S. J. Pennycook, C. Aruta, U. Scotti di Uccio, P. Perna, and D. Maccariello, “Electronic transfer and ionic displacements as the origin of the 2D electron gas at the LAO/STO interface: direct measurements with atomic-column spatial resolution,” Adv. Mater. 24, 3952–3957 (2012).
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M. Bertocchi, E. Luppi, E. Degoli, V. Véniard, and S. Ossicini, “Large crystal local-field effects in second-harmonic generation of a Si/CaF2 interface: an ab initio study,” Phys. Rev. B 86, 035309 (2012).

E. Luppi, H. Hübener, and V. Véniard, “Ab initio second-order nonlinear optics in solids: second-harmonic generation spectroscopy from time-dependent density-functional theory,” Phys. Rev. B 82, 235201 (2010).
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M. Huijben, G. Rijnders, D. H. A. Blank, S. Bals, S. V. Aert, J. Verbeeck, G. V. Tendeloo, A. Brinkman, and H. Hilgenkamp, “Electronically coupled complementary interfaces between perovskite band insulators,” Nat. Mater. 5, 556–560 (2006).
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J. F. Ward, “Calculation of nonlinear optical susceptibilities using diagrammatic perturbation theory,” Rev. Mod. Phys. 37, 1–18 (1965).
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A. F. Santander-Syro, O. Copie, T. Kondo, F. Fortuna, S. Pailhes, R. Weht, X. G. Qiu, F. Bertran, A. Nicolaou, A. Taleb-Ibrahimi, P. Le Fevre, G. Herranz, M. Bibes, N. Reyren, Y. Apertet, P. Lecoeur, A. Barthelemy, and M. J. Rozenberg, “Two-dimensional electron gas with universal subbands at the surface of SrTiO3,” Nature 469, 189–193 (2011).
[CrossRef]

Wetscherek, A.

M. Sing, G. Berner, K. Groß, A. Müller, A. Ruff, A. Wetscherek, S. Thiel, J. Mannhart, A. Pauli, C. Schneider, P. Willmott, M. Gorgoi, F. Schäfers, and R. Claessen, “Profiling the interface electron gas of LaAlO3/SrTiO3 heterostructures with hard x-ray photoelectron spectroscopy,” Phys. Rev. Lett. 102, 176805 (2009).
[CrossRef]

Willmott, P.

M. Sing, G. Berner, K. Groß, A. Müller, A. Ruff, A. Wetscherek, S. Thiel, J. Mannhart, A. Pauli, C. Schneider, P. Willmott, M. Gorgoi, F. Schäfers, and R. Claessen, “Profiling the interface electron gas of LaAlO3/SrTiO3 heterostructures with hard x-ray photoelectron spectroscopy,” Phys. Rev. Lett. 102, 176805 (2009).
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K. Yoshimatsu, R. Yasuhara, H. Kumigashira, and M. Oshima, “Origin of metallic states at the heterointerface between the band insulators LaAlO3 and SrTiO3,” Phys. Rev. Lett. 101, 026802 (2008).
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K. Yoshimatsu, R. Yasuhara, H. Kumigashira, and M. Oshima, “Origin of metallic states at the heterointerface between the band insulators LaAlO3 and SrTiO3,” Phys. Rev. Lett. 101, 026802 (2008).
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Figures (3)

Fig. 1.
Fig. 1.

Feynman’s diagrams showing all possible electronic transition paths for SHG. The solid-line arrows going from left to right stand for an initially empty electron state, those going from right to left stand for an initially empty hole state (i.e., occupied electron state), the dots (graph vertices indicated with italic letters) stand for the transition dipoles with the specified Cartesian index. (A) and (B) refer to the nondegenerate cases, occurring when all three states are different, and the eight possible permutations of the vertices correspond to the eight terms appearing in Eq. (7). (C)–(E) correspond to the degenerate cases in which two or all three states coincide.

Fig. 2.
Fig. 2.

Possible arrangements of O (2p) orbitals forming the nine uppermost valence bands are indicated. Note that in the bulk, at the Γ point in the BZ, these orbitals are grouped in three threefold-degenerate levels (here separated by vertical lines) according to the cubic symmetry. At surface this degeneracy is partly lifted with the separation among singlet and doublet states. It should be also noted that orbitals sharing the same symmetry character (e.g., bonding and antibonding orbitals) can be partly hybridized in the actual energy–eigenvalue states and the exact weight of their contributions cannot be determined by symmetry and can only be computed by numerical methods.

Fig. 3.
Fig. 3.

Symmetry-allowed O(2p)-Ti(3d) 2ω transitions for all three symmetry-allowed χijk components within the 4 mm symmetry group are shown. Note that the allowed transitions for χzzz (contributing to the pp signal together with the other χ^ elements) and χzxx (sp signal) are the same, whereas two different transitions are allowed for χxxz (ds signal).

Tables (2)

Tables Icon

Table 1. List of the Selection Rules that Apply for Each χ^ Element and for Each Kind of Transition (Fundamental or Second-Harmonic Frequency)a

Tables Icon

Table 2. List of all the Nonvanishing Tensor Components (Left Column) Associated to the Most Likely 2ω-Transitions between Resonant States a and b, Through Nonresonant State n (Rows) at the Γ Point of the BZa

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

Pi(2ω)=ε0χijk(2ω)Ej(ω)Ek(ω),
ESHG=iωE02ϵ0ccosβχeff,
χeff=eioutLiioutχijkLjjinLkkinejinekindz,
ein=(cosαcosβ,sinα,cosαsinβ),eout=(cosαcosβ,sinα,cosαsinβ).
Lxx=2cosβncosβ+cosβ,Lyy=2cosβcosβ+ncosβ,Lzz=2cosβn(ncosβ+cosβ),
χzzzχzxx=χzyyχxxz=χyyz=χxzx=χyzy.
χspeff=χzxxtLzzout(Lyyin)2sinβ,χdseff=χxxztLyyoutLyyinLzzinsinβ,χppeff=χzzztLzzout(Lzzin)2sin3β+(χzxxdLzzoutLxxin)2χxxztLxxoutLzzin)Lxxinsinβcos2β,
χijk(2)(2ω,ω,ω)=N22lmnρll0[μlni(μnmjμmlk+μnmkμmlj)(ωnl2ωiγnl)(ωmlωiγml)+μnmi(μlnjμmlk+μlnkμmlj)(ωmn2ωiγmn)(ωnl+ω+iγnl)+μnmi(μlnjμmlk+μlnkμmlj)(ωnm+2ω+iγnm)(ωmlωiγml)+μmli(μlnjμnmk+μlnkμnmj)(ωml+2ω+iγml)(ωnl+ω+iγnl)],
μnmi=allel.jψn|(e)xj,i|ψm=pqp|(e)x1,i|qψn|a^pa^q|ψm=pqμpqiδψn,ψmq+p,
χijkA(2ω,ω,ω)=12V2qpr{f(Eq)[1f(Ep)][1f(Er)]×[μqri(μrpjμpqk+μrpkμpqj)(ωrq2ωiγrq)(ωpqωiγpq)+μrpi(μqrjμpqk+μqrkμpqj)(ωpr2ωiγpr)(ωrq+ω+iγrq)+μrpi(μqrjμpqk+μqrkμpqj)(ωrp+2ω+iγrp)(ωpqωiγpq)+μpqi(μqrjμrpk+μqrkμrpj)(ωpq+2ω+iγpq)(ωrq+ω+iγrq)]},
χijkB(2ω,ω,ω)=12V2qpr{f(Eq)f(Er)[1f(Ep)]×[μrpi(μqrjμpqk+μqrkμpqj)(ωpr2ωiγpr)(ωpqωiγpq)+μqri(μrpjμpqk+μrpkμpqj)(ωrq2ωiγrq)(ωpr+ω+iγpr)+μqri(μrpjμpqk+μrpkμpqj)(ωqr+2ω+iγqr)(ωpqωiγpq)+μpqi(μrpjμqrk+μrpkμqrj)(ωpq+2ω+iγpq)(ωpr+ω+iγpr)]}.
χijkCD(2ω,ω,ω)=122pr{f(Er)[1f(Ep)]×[μpri(μrpjPk+μrpkPj)(1(ωrp2ωiγrp)(ωpr+ω+iγpr)+1(ωrp+2ω+iγrp)(ωprωiγpr))+Pi0(μrpjμprk+μrpkμprj)2ω(1ωpr+ω+iγpr+1ωprωiγpr)]},
χijk(2ω,ω,ω)=f(Ea)[1f(Eb)]22(ωbaωiγba)×{1Vnanb[(12f(En))(μani(μnbjμbak+μnbkμbaj)ωna2ωiγna+μnbi(μanjμbak+μankμbaj)ωnb+2ω+iγnb)]+Pi0(μabjμbak+μabkμbaj)2ω+μabi(Pj0μbak+Pk0μbaj)ωab+2ω+iγab}.
a,must be an initially full state;b,must be an initially empty state;χzzz0,χzxx=0,χxxz0,ifa,bhave the same symmetry;χzzz=0,χzxx0,χxxz0,ifa,bhave different symmetry,one beingE;χzzz=0,χzxx=0,χxxz=0,ifa,bhave different symmetry,notE.
χijk(2ω,ω,ω)=μabi2V2(ωba2ωiγba)·{f(Ea)[1f(Eb)]n[1f(En)ωnaωiγna+f(En)ωbnωiγbn](μbnjμnak+μbnkμnaj)+[1f(Ea)][1f(Eb)]nf(En)(μnajμbnk+μnakμbnj)(1ωan+ω+iγan1ωbnωiγbn)+f(Ea)f(Eb)n[1f(En)](μnajμbnk+μnakμbnj)(1ωan+ω+iγan1ωbnωiγbn)}μbai22(ωba2ωiγba)f(Ea)[1f(Eb)]μabjPk0+μabkPj0ωbaωiγba.
χzzz0,χzxx0,χxxz=0,ifa,bhave the same symmetryχzzz=0,χzxx=0,χxxz0,ifa,bhave different symmetry,one beingEχzzz=0,χzxx=0,χxxz=0,ifa,bhave different symmetry,notE.

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