The classical surface in modern theories of solid surfaces: II. Subband formalism and dimensionality theory

In a previous paper [Chin. J. Phys. 29, 49 (1991)], we found that in front of a classical surface, the dimensionality has some ambiguity. Because we t reat the surface or interface as a classical surface for semiconductors, et c., we find some dimensionality discriminations in our theory. We study the one-dimensional energy to incorporate subbands, reduced from the three-dim ensional space. Because the surface condition theta (z), as a one-dimension al-boundary condition is not mathematically compatible with the one-dimensi onal Schrodinger equation in 3D, the plane subband theory thus obtained is not the 2D Bloch theory of electrons. There are different structural formul ation influences when reduced from 3D. They imply that we can have, for any one crystal energy, three different appearances and nature: a three-dimens ional E-3D and two one-dimensional E-zn* and E-n((sl)). The latter two prov ide envelope functions, with reductions in dimensionality. This Suggests a new picture having various dimensionalities for energy quanta packets. This implies that in the statistical method we have three different Fermi energ ies: from E-3D, E-F((3D)); from E-zn*, E-F1((1D)); and from E-n((sl)), E-F3 ((1D)) for running waves. This also implies that the physical contents of t he crystal momentum include (a) a complex nature to indicate the mean free path, and (b) an extension to other (extra or deviate) dimensionalities. Th e origins of the mobilities are mentioned and related to the electron scatt erings with 2D structure from the k(Z)((2D)), the k(Z)((2D)) p(z)-perturbat ions. They are related to some energy eigen-equation with energy value E-zn * which has the peculiarities: (a) it is not exactly one-dimensional for th e envelope function; and (b) the k(Z)((2D)) p(z)-perturbation for this E-zn * for all kinds of perfect crystals with a plane-surface, has a peculiar k (2D) coming from z the other dimensions (2D). This reveals a new dimension and is a dimensionality-breakthrough from 2D into 3D. One obvious reason fo r this dimensionality-breakthrough is the existence of lattice lateral-vibr ation dynamics on the static surface theta (z). This resolves the problem t hat subband theories are 2D, e.g., in the xy-plane while the surface equati on is expressed in one-variable with one degree of freedom in variable-vari ations, such as theta (z). Consequently a two-dimensional plane, theta (z), is physically measurable in three-dimensional space, by a certain periodic potential condition in terms of the primitive vectors in an (xy)-plane. Co ntrary to mathematics, the area integral is immeasurable in the formulation of volume integrals in Riemann integral theory due to the lattice dynamics vibrating into new dimensions.