PHYSICAL REVIE%' B VOLUME 45, NUMBER 24 15 JUNE 1992-II Simple model of second-harmonic generation W. L. Schaich and Bernardo S. Mendoza* Physics Department and Materials Research Institute, Indiana Uniuersity, Bloomington, Indiana 47405 (Received 5 August 1991) A model for second-harmonic generation from materials with inversion symmetry is developed. The reAecting medium is treated as an ordered array of localized, polarizable entities and the self-consistent relations between induced multipole moments and local fields are derived. The resulting equations are evaluated for a simple model of an entity's response, yielding microscopic estimates of both surface and bulk susceptibility components. The difficulties of applying the model to describe real materials are dis- cussed. I. INTRODUCTION Although second-harmonic generation by centrosym- metric materials has been extensively studied as an exper- imental phenomenon over the past 30 years, ' there has been much less work on the theoretical side. ' In the 1960s estimates of various contributing mechanisms were made, but no systematic calculations were correctly completed. The 1970s began with the important work of Rudnick and Stern, who pointed out the need for a more careful analysis of surface effects. However, it proved difficult to fully evaluate microscopic theories. This situ- ation began to change in the 1980s. Sipe et al. formulat- ed a hydrodynamic model (relevant for metals) which, with some slight modifications, was evaluated in detail by Corvi and Schaich. More sophisticated computations soon followed, done first in the static limit ' and then at finite frequencies. ' ' These calculations, however, only obtain the surface optical response to a normal elec- tric field (i e , yj ~. ~). . For the jellium model that is usually used by those doing calculations, this is the only nontrivi- al case; but experimentally, matters are complicated by additional responses (e.g. , y~) ~~) and anisotropies that arise from crystallinity effects. Consequently, there have also been theoretical efforts to formally retain all contri- butions consistent with symmetry. ' Each distinct term is described with a different phenomenological pa- rameter and one has now developed catalogues of the complete range of possible induced polarizations for arbi- trary incident fields' ' and of the resulting radia- tion. ' In this process it was discovered that it is sometimes experimentally impossible to separate certain surface and bulk contributions, an ambiguity that might be reduced if one had theoretical estimates of the (relative) size of the various parameters. In this paper we develop a microscopic model for second-harmonic generation in reAection from cen- trosymmetric crystals. It is based on viewing the materi- al as an ordered array of pointlike, polarizable entities, which we shall refer to as "dipolium. " In an applied field each entity develops various multipole moments which, in turn, produce fields that inhuence its neighbors. The self-consistent solution for the net polarization of the sys- tern allows one to calculate the various parameters intro- duced by the phenomenological theories. We acknowledge at the outset that a dipolium model has definite limitations, but assert that it also has several attractive features. We will demonstrate that it is compu- tationally tractable and will note how it can be refined to a certain extent. Its greatest advantage over presently used models is that it allows an explicit, microscopic treatment of crystallinity effects. Although the specific numerical results obtained here are of uncertain relevance to any real material, they may provide helpful qualitative insights. Even the derivation of the basic equations, which requires a consistent treatment of local fields and multipolar response, is a useful conceptual ex- ercise. There certainly have been previous attempts to esti- mate the contribution of bound charges, ' ' but these were done before the realization of the need to treat sur- face and bulk response in fairly distinct ways. Our devel- opment is a natural extension of earlier work on linear optical response, where the use of point-dipole models has a long tradition, going back in the previous century to the Clausius-Mossotti or Lorentz-Lorenz relations. The issue of separate treatment of surface and bulk response arises there, too, and we will apply the under- standing that has been developed. This involves exploit- ing the large difference in length scales between typical lattice constants and optical wavelengths. Similar ideas have proven useful for jellium calculations. ' The formal development of the model is presented in Sec. II. The required polarization fields near the surface are shown to be obtainable from the solution of various matrix equations of small dimension, while in the bulk explicit results can be found. Every phenomenological parameter that has been introduced can be evaluated and usually has a nonzero value. In Sec. III we describe some model calculations for the simple case of harmonic oscil- lators on a semi-infinite fcc lattice. In some cases the answers are nearly analytic and the structure in the fre- quency dependence of the parameters is fairly easy to un- derstand. We then discuss the range of experimental relevance of the dipolium model. The most severe con- straint on extending this range lies in describing the non- linear response of a single entity. 45 14 279 1992 The American Physical Society