Polarizable-bond model for second-harmonic generation Bernardo S. Mendoza Centro de Investigaciones en Optica, A.C. Apartado Postal 1-948, 37 000 Leo ´n, GTO, Mexico W. Luis Mocha ´ n Laboratorio de Cuernavaca, Instituto de Fı ´sica, Universidad Nacional Auto ´noma de Mexico, Apartado Postal 48-3, 62 251 Cuernavaca, Morelos, Mexico Received 18 April 1996 We develop a theory for the calculation of the optical second-harmonic generation spectra of Si incorporat- ing the nonlinear surface local field effect. Our model consists of four interpenetrated fcc lattices of nonlinearly polarizable bonds. Each of them is anisotropic and although they are centrosymmetric, they respond quadrati- cally to the spatial inhomogeneities of the polarizing local field. The large gradient of the field induced at a bond due to the dipole moment of a neighbor leads to a second order polarization. In the bulk, each bond lies within a centrosymmetric environment, so this contribution is canceled out after summing over all other bonds. However, at the surface it is not compensated and it leads to a large nonlinear macroscopic response. Our model parameters are fitted to the nonlinear anisotropy measured at 1.17 and 2.34 eV. We calculate a linear anisotropy spectra for the 110 surface in agreement with previous measurements. Our nonlinear spectra show peaks at 1.65 eV for a strained 001 surface and at 1.75 eV for a 111 surface, in agreement with some recent experimental results. S0163-18299708104-6 I. INTRODUCTION The electric-dipolar quadratic susceptibility is a third rank tensor, and therefore it must be null within the bulk of any centrosymmetric system. For this reason, a large portion of the light with frequency 2  reflected from an interface illu- minated with monochromatic radiation at  is surface origi- nated, making second-harmonic generation SHG a sensi- tive optical surface probe for this class of systems. Besides being nondestructive and noninvasive, SHG has the added advantage of accessing surfaces such as buried interfaces, out of ultrahigh vacuum conditions and within arbitrary transparent ambients. However, the efficiency of the surface SHG is extremely low, of the order of 1 1/c ( a B 3 /  e ) 3 10 -20 cm 2 /W, where a B is the Bohr radius,  the wavelength, e the electronic charge, and c the speed of light, and very powerful laser systems are required for its observa- tion. Most experiments have been performed only at a few selected frequencies, emphasizing the polar and azimuthal angular dependence of the signal for different crystal sur- faces and combinations of incoming and outgoing polarizations. 2–12 The possible angular dependence of SHG is well understood from a phenomenological point of view, in terms of the independent components of the bulk and sur- face nonlinear susceptibilities and their symmetry originated constraints. 13–16 The recent development of high power tunable lasers with a wide spectral range has stimulated experiments in nonlin- ear surface spectroscopy. In particular, SHG spectra have recently been measured for different clean, oxidized, and ad- sorbate covered surfaces of Si. 17,18 These spectra show a well developed peak close to 2  =3.3 eV. Its position and its relative insensitivity to surface conditions suggest that it is originated from a bulk transition between the valence and conduction bands, which becomes SH electric-dipolarly ac- tive close to the surface. More recently, nonlinear anisotropy and electroreflectance spectroscopy experiments have shown that different components of the nonlinear susceptibility peak at slightly different frequencies. 19–21 These peaks have been associated to particular interband bulk transitions frequency- shifted at the surface. There are different theoretical approaches in the literature to calculate SHG. The nonlinear surface response of simple metals was estimated 22,23 and later calculated 24,25 within the hydrodynamic model, and microscopic calculations for simple metals have been performed using self-consistent jel- lium models. 26–28 A peak in the SHG spectrum has been predicted at the subharmonic of the ionization threshold 28 and giant resonances were obtained at the frequencies of the multipolar surface plasmon and its subharmonic. 29 The an- isotropy due to lattice effects has been incorporated using a Boltzmann equation approach for systems with a nearly spherical Fermi surface 30 and within the ‘‘Swiss cheese’’ model 31 for noble metals. 32 On the other side, there are a few calculations of the SHG spectra of semiconductors. Simple analytical expressions for model semiconductors made up of a continuous distribution of polarizable entities 1 were ob- tained by neglecting crystallinity effects. The latter were incorporated 33–35 within a dipolium model that also accounts for local field effects. A more microscopic approach has been employed to calculate SHG from As terminated Si111 slabs using a tight binding formalism. 36 The purpose of the present paper is the development of a simple quantitative theory for the SHG spectra of semicon- ductor surfaces accounting in an approximate way for the bulk transitions and the crystalline symmetry. A previous successful theory for the surface linear response of natural Si incorporated the geometrical arrangement of the atoms at the surface through the surface local field effect. 37 In this paper we extend that theory to the nonlinear response. We expect PHYSICAL REVIEW B 15 JANUARY 1997-II VOLUME 55, NUMBER 4 55 0163-1829/97/554/248914/$10.00 2489 © 1997 The American Physical Society