Abstract

We discuss the susceptibility third-rank tensor for second harmonic and sum-frequency generation, associated with low index surfaces of silicon (Si(001), Si(011), and Si(111)), from two different approaches: the simplified bond-hyperpolarizability model (SBHM) and group theory (GT). We show that the SBHM agrees very well with the experimental results for simple surfaces because the definitions of the bond vectors implicitly include the geometry of the crystal and therefore the symmetry group. However, for more complex surfaces it is shown that one can derive from GT the SBHM tensor, if Kleinman symmetry is allowed.

© 2014 Optical Society of America

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  1. E. J. Adles and D. E. Aspnes, “Application of the anisotropic bond model to second-harmonic generation from amorphous media,” Phys. Rev. B 77, 165102 (2008).
    [CrossRef]
  2. J. Kwon, M. C. Downer, and B. S. Mendoza, “Second-harmonic and reflectance-anisotropy spectroscopy of vicinal Si(001)/SiO2 interfaces: experiment and simplified microscopic model,” Phys. Rev. B 73, 195330 (2006).
    [CrossRef]
  3. J. T. Madden, V. J. Hall, and G. J. Simpson, “Mining the polarization-dependence of nonlinear optical measurements,” Analyst 136, 652–662 (2011), and references therein.
    [CrossRef]
  4. C. J. Dehen, R. M. Everly, R. M. Plocinik, H. G. Hedderich, and G. J. Simpson, “Discrete retardance second harmonic generation ellipsometry,” Rev. Sci. Instrum. 78, 013106 (2007).
    [CrossRef]
  5. N. J. Begue, R. M. Everly, V. J. Hall, L. Haupert, and G. J. Simpson, “Nonlinear optical Stokes ellipsometry. 2. Experimental demonstration,” J. Phys. Chem. C 113, 10166–10175 (2009).
    [CrossRef]
  6. M. A. van der Veen, V. K. Valev, T. Verbiest, and D. E. De Vos, “In situ orientation-sensitive observation of molecular adsorption on a liquid/zeolite interface by second-harmonic generation,” Langmuir 25, 4256–4261 (2009).
    [CrossRef]
  7. B. S. Mendoza and W. L. Mochán, “Polarizable-bond model for second-harmonic generation,” Phys. Rev. B 55, 2489–2502 (1997).
    [CrossRef]
  8. N. Arzate and B. S. Mendoza, “Polarizable bond model for optical spectra of Si(100) reconstructed surfaces,” Phys. Rev. B 63, 113303 (2001).
    [CrossRef]
  9. G. D. Powell, J. F. Wang, and D. E. Aspnes, “Simplified bond-hyperpolarizability model of second harmonic generation,” Phys. Rev. B 65, 205320 (2002).
  10. J.-F. T. Wang, G. D. Powell, R. S. Johnson, G. Lucovsky, and D. E. Aspnes, “Simplified bond-hyperpolarizability model of second harmonic generation: application to Si-dielectric interfaces,” J. Vac. Sci. Technol. B 20, 1699–1705 (2002).
    [CrossRef]
  11. H. Hardhienata, A. Prylepa, D. Stifter, and K. Hingerl, “Simplified bond-hyperpolarizability model of second-harmonic-generation in Si(111): theory and experiment,” J. Phys. 423, 012046 (2013).
    [CrossRef]
  12. J. F. McGilp, “Using steps at the Si–SiO2 interface to test simple bond models of the optical second-harmonic response,” J. Phys. 19, 016006 (2007).
    [CrossRef]
  13. E. J. Adles and D. E. Aspnes, “The anisotropic bond model of nonlinear optics,” Phys. Stat. Sol. A 205, 728–731 (2008).
    [CrossRef]
  14. J. F. Nye, Physical Properties of Crystals, Their Representations by Tensors and Matrices (Clarendon, 1957).
  15. R. C. Powell, Symmetry, Group Theory, and the Physical Properties of Crystals, Lecture Notes in Physics (Springer, 2010).
  16. F. A. Cotton, Chemical Applications of Group Theory, 3rd ed. (Wiley, 1990).
  17. D. C. Harris and M. D. Bertolucci, Symmetry and Spectroscopy—An Introduction to Vibrational and Electronic Spectroscopy (Dover, 1989).
  18. R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic, 2003).
  19. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 1984).
  20. T. A. Driscoll and D. Guidotti, “Symmetry analysis of second-harmonic generation in silicon,” Phys. Rev. B 28, 1171–1173 (1983).
    [CrossRef]
  21. J. E. Sipe, D. J. Moss, and H. M. van Driel, “Phenomenological theory of optical second- and third-harmonic generation from cubic centrosymmetric crystals,” Phys. Rev. B 35, 1129–1141 (1987).
    [CrossRef]
  22. Factor 2 in all the tensors in Table 3.5 from Ref. [15] is a mistake. Prof. R. C. Powell, (private communication, September 2013).
  23. G. G. Malliaras, H. A. Wierenga, and T. Rasing, “Study of the step structure of vicinal Si(110) surfaces using optical second harmonic generation,” Surf. Sci. 287, 703–707 (1993).
    [CrossRef]
  24. G. Lüpke, D. J. Bottomley, and H. M. van Driel, “Second- and third-harmonic generation from cubic centrosymmetric crystals with vicinal faces: phenomenological theory and experiment,” J. Opt. Soc. Am. B 11, 33–44 (1994).
    [CrossRef]

2013 (1)

H. Hardhienata, A. Prylepa, D. Stifter, and K. Hingerl, “Simplified bond-hyperpolarizability model of second-harmonic-generation in Si(111): theory and experiment,” J. Phys. 423, 012046 (2013).
[CrossRef]

2011 (1)

J. T. Madden, V. J. Hall, and G. J. Simpson, “Mining the polarization-dependence of nonlinear optical measurements,” Analyst 136, 652–662 (2011), and references therein.
[CrossRef]

2009 (2)

N. J. Begue, R. M. Everly, V. J. Hall, L. Haupert, and G. J. Simpson, “Nonlinear optical Stokes ellipsometry. 2. Experimental demonstration,” J. Phys. Chem. C 113, 10166–10175 (2009).
[CrossRef]

M. A. van der Veen, V. K. Valev, T. Verbiest, and D. E. De Vos, “In situ orientation-sensitive observation of molecular adsorption on a liquid/zeolite interface by second-harmonic generation,” Langmuir 25, 4256–4261 (2009).
[CrossRef]

2008 (2)

E. J. Adles and D. E. Aspnes, “Application of the anisotropic bond model to second-harmonic generation from amorphous media,” Phys. Rev. B 77, 165102 (2008).
[CrossRef]

E. J. Adles and D. E. Aspnes, “The anisotropic bond model of nonlinear optics,” Phys. Stat. Sol. A 205, 728–731 (2008).
[CrossRef]

2007 (2)

J. F. McGilp, “Using steps at the Si–SiO2 interface to test simple bond models of the optical second-harmonic response,” J. Phys. 19, 016006 (2007).
[CrossRef]

C. J. Dehen, R. M. Everly, R. M. Plocinik, H. G. Hedderich, and G. J. Simpson, “Discrete retardance second harmonic generation ellipsometry,” Rev. Sci. Instrum. 78, 013106 (2007).
[CrossRef]

2006 (1)

J. Kwon, M. C. Downer, and B. S. Mendoza, “Second-harmonic and reflectance-anisotropy spectroscopy of vicinal Si(001)/SiO2 interfaces: experiment and simplified microscopic model,” Phys. Rev. B 73, 195330 (2006).
[CrossRef]

2002 (2)

G. D. Powell, J. F. Wang, and D. E. Aspnes, “Simplified bond-hyperpolarizability model of second harmonic generation,” Phys. Rev. B 65, 205320 (2002).

J.-F. T. Wang, G. D. Powell, R. S. Johnson, G. Lucovsky, and D. E. Aspnes, “Simplified bond-hyperpolarizability model of second harmonic generation: application to Si-dielectric interfaces,” J. Vac. Sci. Technol. B 20, 1699–1705 (2002).
[CrossRef]

2001 (1)

N. Arzate and B. S. Mendoza, “Polarizable bond model for optical spectra of Si(100) reconstructed surfaces,” Phys. Rev. B 63, 113303 (2001).
[CrossRef]

1997 (1)

B. S. Mendoza and W. L. Mochán, “Polarizable-bond model for second-harmonic generation,” Phys. Rev. B 55, 2489–2502 (1997).
[CrossRef]

1994 (1)

1993 (1)

G. G. Malliaras, H. A. Wierenga, and T. Rasing, “Study of the step structure of vicinal Si(110) surfaces using optical second harmonic generation,” Surf. Sci. 287, 703–707 (1993).
[CrossRef]

1987 (1)

J. E. Sipe, D. J. Moss, and H. M. van Driel, “Phenomenological theory of optical second- and third-harmonic generation from cubic centrosymmetric crystals,” Phys. Rev. B 35, 1129–1141 (1987).
[CrossRef]

1983 (1)

T. A. Driscoll and D. Guidotti, “Symmetry analysis of second-harmonic generation in silicon,” Phys. Rev. B 28, 1171–1173 (1983).
[CrossRef]

Adles, E. J.

E. J. Adles and D. E. Aspnes, “Application of the anisotropic bond model to second-harmonic generation from amorphous media,” Phys. Rev. B 77, 165102 (2008).
[CrossRef]

E. J. Adles and D. E. Aspnes, “The anisotropic bond model of nonlinear optics,” Phys. Stat. Sol. A 205, 728–731 (2008).
[CrossRef]

Arzate, N.

N. Arzate and B. S. Mendoza, “Polarizable bond model for optical spectra of Si(100) reconstructed surfaces,” Phys. Rev. B 63, 113303 (2001).
[CrossRef]

Aspnes, D. E.

E. J. Adles and D. E. Aspnes, “Application of the anisotropic bond model to second-harmonic generation from amorphous media,” Phys. Rev. B 77, 165102 (2008).
[CrossRef]

E. J. Adles and D. E. Aspnes, “The anisotropic bond model of nonlinear optics,” Phys. Stat. Sol. A 205, 728–731 (2008).
[CrossRef]

G. D. Powell, J. F. Wang, and D. E. Aspnes, “Simplified bond-hyperpolarizability model of second harmonic generation,” Phys. Rev. B 65, 205320 (2002).

J.-F. T. Wang, G. D. Powell, R. S. Johnson, G. Lucovsky, and D. E. Aspnes, “Simplified bond-hyperpolarizability model of second harmonic generation: application to Si-dielectric interfaces,” J. Vac. Sci. Technol. B 20, 1699–1705 (2002).
[CrossRef]

Begue, N. J.

N. J. Begue, R. M. Everly, V. J. Hall, L. Haupert, and G. J. Simpson, “Nonlinear optical Stokes ellipsometry. 2. Experimental demonstration,” J. Phys. Chem. C 113, 10166–10175 (2009).
[CrossRef]

Bertolucci, M. D.

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

Bottomley, D. J.

Boyd, R. W.

R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic, 2003).

Cotton, F. A.

F. A. Cotton, Chemical Applications of Group Theory, 3rd ed. (Wiley, 1990).

De Vos, D. E.

M. A. van der Veen, V. K. Valev, T. Verbiest, and D. E. De Vos, “In situ orientation-sensitive observation of molecular adsorption on a liquid/zeolite interface by second-harmonic generation,” Langmuir 25, 4256–4261 (2009).
[CrossRef]

Dehen, C. J.

C. J. Dehen, R. M. Everly, R. M. Plocinik, H. G. Hedderich, and G. J. Simpson, “Discrete retardance second harmonic generation ellipsometry,” Rev. Sci. Instrum. 78, 013106 (2007).
[CrossRef]

Downer, M. C.

J. Kwon, M. C. Downer, and B. S. Mendoza, “Second-harmonic and reflectance-anisotropy spectroscopy of vicinal Si(001)/SiO2 interfaces: experiment and simplified microscopic model,” Phys. Rev. B 73, 195330 (2006).
[CrossRef]

Driscoll, T. A.

T. A. Driscoll and D. Guidotti, “Symmetry analysis of second-harmonic generation in silicon,” Phys. Rev. B 28, 1171–1173 (1983).
[CrossRef]

Everly, R. M.

N. J. Begue, R. M. Everly, V. J. Hall, L. Haupert, and G. J. Simpson, “Nonlinear optical Stokes ellipsometry. 2. Experimental demonstration,” J. Phys. Chem. C 113, 10166–10175 (2009).
[CrossRef]

C. J. Dehen, R. M. Everly, R. M. Plocinik, H. G. Hedderich, and G. J. Simpson, “Discrete retardance second harmonic generation ellipsometry,” Rev. Sci. Instrum. 78, 013106 (2007).
[CrossRef]

Guidotti, D.

T. A. Driscoll and D. Guidotti, “Symmetry analysis of second-harmonic generation in silicon,” Phys. Rev. B 28, 1171–1173 (1983).
[CrossRef]

Hall, V. J.

J. T. Madden, V. J. Hall, and G. J. Simpson, “Mining the polarization-dependence of nonlinear optical measurements,” Analyst 136, 652–662 (2011), and references therein.
[CrossRef]

N. J. Begue, R. M. Everly, V. J. Hall, L. Haupert, and G. J. Simpson, “Nonlinear optical Stokes ellipsometry. 2. Experimental demonstration,” J. Phys. Chem. C 113, 10166–10175 (2009).
[CrossRef]

Hardhienata, H.

H. Hardhienata, A. Prylepa, D. Stifter, and K. Hingerl, “Simplified bond-hyperpolarizability model of second-harmonic-generation in Si(111): theory and experiment,” J. Phys. 423, 012046 (2013).
[CrossRef]

Harris, D. C.

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

Haupert, L.

N. J. Begue, R. M. Everly, V. J. Hall, L. Haupert, and G. J. Simpson, “Nonlinear optical Stokes ellipsometry. 2. Experimental demonstration,” J. Phys. Chem. C 113, 10166–10175 (2009).
[CrossRef]

Hedderich, H. G.

C. J. Dehen, R. M. Everly, R. M. Plocinik, H. G. Hedderich, and G. J. Simpson, “Discrete retardance second harmonic generation ellipsometry,” Rev. Sci. Instrum. 78, 013106 (2007).
[CrossRef]

Hingerl, K.

H. Hardhienata, A. Prylepa, D. Stifter, and K. Hingerl, “Simplified bond-hyperpolarizability model of second-harmonic-generation in Si(111): theory and experiment,” J. Phys. 423, 012046 (2013).
[CrossRef]

Johnson, R. S.

J.-F. T. Wang, G. D. Powell, R. S. Johnson, G. Lucovsky, and D. E. Aspnes, “Simplified bond-hyperpolarizability model of second harmonic generation: application to Si-dielectric interfaces,” J. Vac. Sci. Technol. B 20, 1699–1705 (2002).
[CrossRef]

Kwon, J.

J. Kwon, M. C. Downer, and B. S. Mendoza, “Second-harmonic and reflectance-anisotropy spectroscopy of vicinal Si(001)/SiO2 interfaces: experiment and simplified microscopic model,” Phys. Rev. B 73, 195330 (2006).
[CrossRef]

Lucovsky, G.

J.-F. T. Wang, G. D. Powell, R. S. Johnson, G. Lucovsky, and D. E. Aspnes, “Simplified bond-hyperpolarizability model of second harmonic generation: application to Si-dielectric interfaces,” J. Vac. Sci. Technol. B 20, 1699–1705 (2002).
[CrossRef]

Lüpke, G.

Madden, J. T.

J. T. Madden, V. J. Hall, and G. J. Simpson, “Mining the polarization-dependence of nonlinear optical measurements,” Analyst 136, 652–662 (2011), and references therein.
[CrossRef]

Malliaras, G. G.

G. G. Malliaras, H. A. Wierenga, and T. Rasing, “Study of the step structure of vicinal Si(110) surfaces using optical second harmonic generation,” Surf. Sci. 287, 703–707 (1993).
[CrossRef]

McGilp, J. F.

J. F. McGilp, “Using steps at the Si–SiO2 interface to test simple bond models of the optical second-harmonic response,” J. Phys. 19, 016006 (2007).
[CrossRef]

Mendoza, B. S.

J. Kwon, M. C. Downer, and B. S. Mendoza, “Second-harmonic and reflectance-anisotropy spectroscopy of vicinal Si(001)/SiO2 interfaces: experiment and simplified microscopic model,” Phys. Rev. B 73, 195330 (2006).
[CrossRef]

N. Arzate and B. S. Mendoza, “Polarizable bond model for optical spectra of Si(100) reconstructed surfaces,” Phys. Rev. B 63, 113303 (2001).
[CrossRef]

B. S. Mendoza and W. L. Mochán, “Polarizable-bond model for second-harmonic generation,” Phys. Rev. B 55, 2489–2502 (1997).
[CrossRef]

Mochán, W. L.

B. S. Mendoza and W. L. Mochán, “Polarizable-bond model for second-harmonic generation,” Phys. Rev. B 55, 2489–2502 (1997).
[CrossRef]

Moss, D. J.

J. E. Sipe, D. J. Moss, and H. M. van Driel, “Phenomenological theory of optical second- and third-harmonic generation from cubic centrosymmetric crystals,” Phys. Rev. B 35, 1129–1141 (1987).
[CrossRef]

Nye, J. F.

J. F. Nye, Physical Properties of Crystals, Their Representations by Tensors and Matrices (Clarendon, 1957).

Plocinik, R. M.

C. J. Dehen, R. M. Everly, R. M. Plocinik, H. G. Hedderich, and G. J. Simpson, “Discrete retardance second harmonic generation ellipsometry,” Rev. Sci. Instrum. 78, 013106 (2007).
[CrossRef]

Powell, G. D.

J.-F. T. Wang, G. D. Powell, R. S. Johnson, G. Lucovsky, and D. E. Aspnes, “Simplified bond-hyperpolarizability model of second harmonic generation: application to Si-dielectric interfaces,” J. Vac. Sci. Technol. B 20, 1699–1705 (2002).
[CrossRef]

G. D. Powell, J. F. Wang, and D. E. Aspnes, “Simplified bond-hyperpolarizability model of second harmonic generation,” Phys. Rev. B 65, 205320 (2002).

Powell, R. C.

Factor 2 in all the tensors in Table 3.5 from Ref. [15] is a mistake. Prof. R. C. Powell, (private communication, September 2013).

R. C. Powell, Symmetry, Group Theory, and the Physical Properties of Crystals, Lecture Notes in Physics (Springer, 2010).

Prylepa, A.

H. Hardhienata, A. Prylepa, D. Stifter, and K. Hingerl, “Simplified bond-hyperpolarizability model of second-harmonic-generation in Si(111): theory and experiment,” J. Phys. 423, 012046 (2013).
[CrossRef]

Rasing, T.

G. G. Malliaras, H. A. Wierenga, and T. Rasing, “Study of the step structure of vicinal Si(110) surfaces using optical second harmonic generation,” Surf. Sci. 287, 703–707 (1993).
[CrossRef]

Shen, Y. R.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 1984).

Simpson, G. J.

J. T. Madden, V. J. Hall, and G. J. Simpson, “Mining the polarization-dependence of nonlinear optical measurements,” Analyst 136, 652–662 (2011), and references therein.
[CrossRef]

N. J. Begue, R. M. Everly, V. J. Hall, L. Haupert, and G. J. Simpson, “Nonlinear optical Stokes ellipsometry. 2. Experimental demonstration,” J. Phys. Chem. C 113, 10166–10175 (2009).
[CrossRef]

C. J. Dehen, R. M. Everly, R. M. Plocinik, H. G. Hedderich, and G. J. Simpson, “Discrete retardance second harmonic generation ellipsometry,” Rev. Sci. Instrum. 78, 013106 (2007).
[CrossRef]

Sipe, J. E.

J. E. Sipe, D. J. Moss, and H. M. van Driel, “Phenomenological theory of optical second- and third-harmonic generation from cubic centrosymmetric crystals,” Phys. Rev. B 35, 1129–1141 (1987).
[CrossRef]

Stifter, D.

H. Hardhienata, A. Prylepa, D. Stifter, and K. Hingerl, “Simplified bond-hyperpolarizability model of second-harmonic-generation in Si(111): theory and experiment,” J. Phys. 423, 012046 (2013).
[CrossRef]

Valev, V. K.

M. A. van der Veen, V. K. Valev, T. Verbiest, and D. E. De Vos, “In situ orientation-sensitive observation of molecular adsorption on a liquid/zeolite interface by second-harmonic generation,” Langmuir 25, 4256–4261 (2009).
[CrossRef]

van der Veen, M. A.

M. A. van der Veen, V. K. Valev, T. Verbiest, and D. E. De Vos, “In situ orientation-sensitive observation of molecular adsorption on a liquid/zeolite interface by second-harmonic generation,” Langmuir 25, 4256–4261 (2009).
[CrossRef]

van Driel, H. M.

G. Lüpke, D. J. Bottomley, and H. M. van Driel, “Second- and third-harmonic generation from cubic centrosymmetric crystals with vicinal faces: phenomenological theory and experiment,” J. Opt. Soc. Am. B 11, 33–44 (1994).
[CrossRef]

J. E. Sipe, D. J. Moss, and H. M. van Driel, “Phenomenological theory of optical second- and third-harmonic generation from cubic centrosymmetric crystals,” Phys. Rev. B 35, 1129–1141 (1987).
[CrossRef]

Verbiest, T.

M. A. van der Veen, V. K. Valev, T. Verbiest, and D. E. De Vos, “In situ orientation-sensitive observation of molecular adsorption on a liquid/zeolite interface by second-harmonic generation,” Langmuir 25, 4256–4261 (2009).
[CrossRef]

Wang, J. F.

G. D. Powell, J. F. Wang, and D. E. Aspnes, “Simplified bond-hyperpolarizability model of second harmonic generation,” Phys. Rev. B 65, 205320 (2002).

Wang, J.-F. T.

J.-F. T. Wang, G. D. Powell, R. S. Johnson, G. Lucovsky, and D. E. Aspnes, “Simplified bond-hyperpolarizability model of second harmonic generation: application to Si-dielectric interfaces,” J. Vac. Sci. Technol. B 20, 1699–1705 (2002).
[CrossRef]

Wierenga, H. A.

G. G. Malliaras, H. A. Wierenga, and T. Rasing, “Study of the step structure of vicinal Si(110) surfaces using optical second harmonic generation,” Surf. Sci. 287, 703–707 (1993).
[CrossRef]

Analyst (1)

J. T. Madden, V. J. Hall, and G. J. Simpson, “Mining the polarization-dependence of nonlinear optical measurements,” Analyst 136, 652–662 (2011), and references therein.
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Phys. (2)

H. Hardhienata, A. Prylepa, D. Stifter, and K. Hingerl, “Simplified bond-hyperpolarizability model of second-harmonic-generation in Si(111): theory and experiment,” J. Phys. 423, 012046 (2013).
[CrossRef]

J. F. McGilp, “Using steps at the Si–SiO2 interface to test simple bond models of the optical second-harmonic response,” J. Phys. 19, 016006 (2007).
[CrossRef]

J. Phys. Chem. C (1)

N. J. Begue, R. M. Everly, V. J. Hall, L. Haupert, and G. J. Simpson, “Nonlinear optical Stokes ellipsometry. 2. Experimental demonstration,” J. Phys. Chem. C 113, 10166–10175 (2009).
[CrossRef]

J. Vac. Sci. Technol. B (1)

J.-F. T. Wang, G. D. Powell, R. S. Johnson, G. Lucovsky, and D. E. Aspnes, “Simplified bond-hyperpolarizability model of second harmonic generation: application to Si-dielectric interfaces,” J. Vac. Sci. Technol. B 20, 1699–1705 (2002).
[CrossRef]

Langmuir (1)

M. A. van der Veen, V. K. Valev, T. Verbiest, and D. E. De Vos, “In situ orientation-sensitive observation of molecular adsorption on a liquid/zeolite interface by second-harmonic generation,” Langmuir 25, 4256–4261 (2009).
[CrossRef]

Phys. Rev. B (7)

B. S. Mendoza and W. L. Mochán, “Polarizable-bond model for second-harmonic generation,” Phys. Rev. B 55, 2489–2502 (1997).
[CrossRef]

N. Arzate and B. S. Mendoza, “Polarizable bond model for optical spectra of Si(100) reconstructed surfaces,” Phys. Rev. B 63, 113303 (2001).
[CrossRef]

G. D. Powell, J. F. Wang, and D. E. Aspnes, “Simplified bond-hyperpolarizability model of second harmonic generation,” Phys. Rev. B 65, 205320 (2002).

E. J. Adles and D. E. Aspnes, “Application of the anisotropic bond model to second-harmonic generation from amorphous media,” Phys. Rev. B 77, 165102 (2008).
[CrossRef]

J. Kwon, M. C. Downer, and B. S. Mendoza, “Second-harmonic and reflectance-anisotropy spectroscopy of vicinal Si(001)/SiO2 interfaces: experiment and simplified microscopic model,” Phys. Rev. B 73, 195330 (2006).
[CrossRef]

T. A. Driscoll and D. Guidotti, “Symmetry analysis of second-harmonic generation in silicon,” Phys. Rev. B 28, 1171–1173 (1983).
[CrossRef]

J. E. Sipe, D. J. Moss, and H. M. van Driel, “Phenomenological theory of optical second- and third-harmonic generation from cubic centrosymmetric crystals,” Phys. Rev. B 35, 1129–1141 (1987).
[CrossRef]

Phys. Stat. Sol. A (1)

E. J. Adles and D. E. Aspnes, “The anisotropic bond model of nonlinear optics,” Phys. Stat. Sol. A 205, 728–731 (2008).
[CrossRef]

Rev. Sci. Instrum. (1)

C. J. Dehen, R. M. Everly, R. M. Plocinik, H. G. Hedderich, and G. J. Simpson, “Discrete retardance second harmonic generation ellipsometry,” Rev. Sci. Instrum. 78, 013106 (2007).
[CrossRef]

Surf. Sci. (1)

G. G. Malliaras, H. A. Wierenga, and T. Rasing, “Study of the step structure of vicinal Si(110) surfaces using optical second harmonic generation,” Surf. Sci. 287, 703–707 (1993).
[CrossRef]

Other (7)

Factor 2 in all the tensors in Table 3.5 from Ref. [15] is a mistake. Prof. R. C. Powell, (private communication, September 2013).

J. F. Nye, Physical Properties of Crystals, Their Representations by Tensors and Matrices (Clarendon, 1957).

R. C. Powell, Symmetry, Group Theory, and the Physical Properties of Crystals, Lecture Notes in Physics (Springer, 2010).

F. A. Cotton, Chemical Applications of Group Theory, 3rd ed. (Wiley, 1990).

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

R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic, 2003).

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 1984).

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Figures (3)

Fig. 1.
Fig. 1.

(a) GaAs conventional cell, Ga (red) and As (blue). (b) Tetrahedral element, Ga in the center. (c) Tetrahedral element, As in the center.

Fig. 2.
Fig. 2.

Bond orientation according to the surface for Si: (a) (111), (b) (001), and (c) (011).

Fig. 3.
Fig. 3.

Top view of the Si(011) surface; the dashed line is a glide plane. (a) Surface, (b) mirror plane and its image, and (c) translation by a quarter of the lattice constant generates the original crystal again.

Equations (26)

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P⃗=1Vj[α1jR(z)(ϕ)·b^j]·E⃗+1Vj[α2j(R(z)(ϕ)·b^j)(R(z)(ϕ)·b^j)(R(z)(ϕ)·b^j)]·E⃗E⃗=χ(1)E⃗+χ(2)E⃗E⃗.
χ(2)=1Vjα2j(R(z)(ϕ)·b^j)(R(z)(ϕ)·b^j)(R(z)(ϕ)·b^j),
d=((d111d121d131d112d122d132d113d123d133)--------(d211d221d231d212d222d232d213d223d233)--------(d311d321d331d312d322d332d313d323d333)),
b1=(12sinβ212sinβ2cosβ2)b2=(12sinβ212sinβ2cosβ2)b3=(12sinβ212sinβ2cosβ2)b4=(12sinβ212sinβ2cosβ2),
b5=b1b6=b2b7=b3b8=b4,
χ(2)=1Vj=18α2j(R(z)(ϕ)·b^j)(R(z)(ϕ)·b^j)(R(z)(ϕ)·b^j),
((00αGaAsSsin2ϕ00αGaAsScos2ϕαGaAsSsin2ϕαGaAsScos2ϕ0)(00αGaAsScos2ϕ00αGaAsSsin2ϕαGaAsScos2ϕαGaAsSsin2ϕ0)(αGaAsSsin2ϕαGaAsScos2ϕ0αGaAsScos2ϕαGaAsSsin2ϕ0000)),
((00000αGaAsS0αGaAsS0)(00αGaAsS000αGaAsS00)(0αGaAsS0αGaAsS00000)),
((00αGaAsS000αGaAsS00)(00000αGaAsS0αGaAsS0)(αGaAsS000αGaAsS0000)),
b1=(001)b2=(sinβ0cosβ)b3=(12sinβ32sinβcosβ)b4=(12sinβ32sinβcosβ).
((3αl4sin3βcos3ϕ3αl4sin3βsin3ϕ3αl2sin2βcosβ3αl4sin3βsin3ϕ3αl4sin3βcos3ϕ03αl2sin2βcosβ00)(3αl4sin3βsin3ϕ3αl4sin3βcos3ϕ03αl4sin3βcos3ϕ3αl4sin3βsin3ϕ3αl2sin2βcosβ03αl2sin2βcosβ0)(3αl2sin2βcosβ0003αl2sin2βcosβ000αu+3αlcos3β)),
((00S[αucos2ϕ+αlsin2ϕ]0012(αuαl)Ssin2ϕS[αucos2ϕ+αlsin2ϕ]12(αuαl)Ssin2ϕ0)(0012(αuαl)Ssin2ϕ00S[αlcos2ϕ+αusin2ϕ]12(αuαl)Ssin2ϕS[αlcos2ϕ+αusin2ϕ]0)(S[αucos2ϕ+αlsin2ϕ]12(αuαl)Ssin2ϕ012(αuαl)Ssin2ϕS[αusin2ϕ+αlcos2ϕ]0002(αu+αl)cos2(β2))),
((002αeffcos2ϕ00αeffsin2ϕ2αeffcos2ϕαeffsin2ϕ0)(00αeffsin2ϕ002αeffsin2ϕαeffsin2ϕ2αeffsin2ϕ0)(2αeffcos2ϕαeffsin2ϕ0αeffsin2ϕ2αeffsin2ϕ0004αeff)),
((00000d1320d1320)(00d132000d13200)(0d1320d13200000)),
((02d222d1312d22200d13100)(d222000d222d1310d1310)(d311000d311000d333)).
((00d131000d13100)(00000d2320d2320)(d311000d322000d333)).
χijk(ϕ)=Ril(ϕ)Rjm(ϕ)Rkn(ϕ)χlmn.
((d111cos3ϕd111sin3ϕd131d111sin3ϕd111cos3ϕ0d13100)(d111sin3ϕd111cos3ϕ0d111cos3ϕd111sin3ϕd1310d1310)(d311000d311000d333))
((00d131cos2ϕ+d232sin2ϕ0012(d131d232)sin2ϕd131cos2ϕ+d232sin2ϕ12(d131d232)sin2ϕ0)(0012(d131d232)sin2ϕ00d131sin2ϕ+d232cos2ϕ12(d131d232)sin2ϕd131sin2ϕ+d232cos2ϕ0)(d311cos2ϕ+d322sin2ϕ12(d311d322)sin2ϕ012(d311d322)sin2ϕd311sin2ϕ+d322cos2ϕ000d333)),
d11134αlsin3β,d131=d31132αlcosβsin2β,andd333αu+3αlcos3β.
dijk=djki=dkij=dikj=djik=dkji.
d111d131=12tanβ.
d131=d311αuS,d232=d322αlSandd3332(αu+αl)cos2(β2),
d232=d322=0,
d311=d131=12d333=2αeff,
d333d311=2.

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