Simplified bond-hyperpolarizability model of second harmonic generation, group theory, and Neumann’s principle

Abstract

We discuss the susceptibility third-rank tensor for second harmonic and sumfrequency generation, associated with low index surfaces of silicon (Si(001), Si(011) and Si(111)) from two different approaches: the Simplified Bond-Hyperpolarizablility Model (SBHM) and Group Theory (GT). We show that 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. OCIS codes: (240.4350) Nonlinear optics at surfaces; (240.6700) Surfaces; (190.4350) Nonlinear optics at surfaces R .In this work, we are going to represent a general third rank tensor as a 9 × 3 matrix divided into three matrix of dimensions 3 × 3. Hence, an explicit representation in this notation of a general third rank tensor ijk d (i, j, k = 1, 2, 3) is given in Eq. (3):

Model of third harmonic generation and electric field induced optical second harmonic using simplified bond-hyperpolarizability model

“…The main difference is that for the Si bulk two neighboring tetrahedral elements are required to model the response of the complete conventional cell whereas only one tetrahedral is required to describe the surface. Therefore, the point group for this surface is C 2v , as we have discussed in our previous work [16]. Based on this, we can contract the general EFISH tensor in Eq.…”

“…[18,19] is defined in this way and their representation changes when the sample is rotated. Therefore we apply the following bond definition [16]:…”

“…The SBHM advantage is that the model can describe SHG generated by a Silicon (Si) surface with only two fitting parameter [13] which is rather different from the standard tensorial approach [15]. In our recent work [16] we have also demonstrated how SBHM and GT are related for several crystal orientations such as Si(111), Si(001), and Si(011) and here we expand SBHM to include THG and EFISH.…”

Bond Model of Second‐ and Third‐Harmonic Generation in Body‐ and Face‐Centered Crystal Structures

“…Furthermore, it can also be inferred from the location of the nonzero elements that all the tensors are symmetric. In other words, they obey the Kleinman symmetry [14]:…”

“…Another important step towards the understanding of SHG in diamond and zincblende lattices was the investigation of the third-rank susceptibility tensor that was obtained from the simplified bond-hyperpolarizability model (SBHM) and group theory (GT) [14,15], where it was shown that one can derive from GT the SBHM tensor. e work was later extended to show that the model can fit electric-field-induced second-harmonic (EFISH) experimental results in metal-Oxide semiconductor (MOS) with good accuracy [16].…”