The Classical Surface in Modern Theories of Solid Surfaces: II. Subband Formalism and Dimensionality Theory

Der-Ruenn Su

《Chinese Journal of Physics》 39卷5期 (2001/10) Pp. 498-528

並列摘要

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 treat the surface or interface as a classical surface for semiconductors, etc., we find some dimensionality discriminations in our theory. We study the one-dimensional energy to incorporate subbands, reduced from the three-dimensional space. Because the surface condition μ(z), as a one-dimensional-boundary condition is not mathematically compatible with the one-dimensional Schrödinger equation in 3D, the plane subband theory thus obtained is not the 2D Bloch theory of electrons. There are different structural formulation influences when reduced from 3D. They imply that we can have, for any one crystal energy, three different appearances and nature: a three-dimensional E3D and two one-dimensional E^¤zn and E^(sl) n . The latter two provide 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 energies: from E3D, E^(3D) F ; from E^¤zn, E^(1D) F1 ; and from E^(sl) n , E^(1D) F3 , for running waves. This also implies that the physical contents of the crystal momentum include (a) a complex nature to indicate the mean free path, and (b) an extension to other (extra or deviate) dimensionalities. The origins of the mobilities are mentioned and related to the electron scatterings with 2D structure from the k^(2D) Z , the k^(2D) Z pz-perturbations. 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 the envelope function; and (b) the k^(2D) Z pz-perturbation for this E^¤zn for all kinds of perfect crystals with a plane-surface, has a peculiar k^(2D) Z coming from the other dimensions (2D). This reveals a new dimension and is a dimensionality-breakthrough from 2D into 3D. One obvious reason for this dimensionality-breakthrough is the existence of lattice lateral-vibration dynamics on the static surface μ(z). This resolves the problem that subband theories are 2D, e.g., in the xy-plane while the surface equation is expressed in one-variable with one degree of freedom in variable-variations, such as μ(z). Consequently a two-dimensional plane, μ(z), is physically measurable in three-dimensional space, by a certain periodic potential condition in terms of the primitive vectors in an (xy)-plane. Contrary 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.