User:Luman2009/sandbox

Since Tamm published his first paper on surface states, the investigations on various properties and/or problems of surface states have been a very active and important area in condensed matter physics.

A naturally simple but fundamental question is how many surface states are in a band gap in a one-dimensional crystal of length $$ N a $$ ($$ a $$ is the potential period, and $$ N $$ is a positive integer)? A well-accepted concept proposed by Fowler first in 1933, then written in Seitz's classic book that "in a finite one-dimensional crystal the surface states occur in pairs, one state being associated with each end of the crystal." Such a concept seemly was never doubted since then for nearly a century, as shown, for example, in. However, a recent new investigation gives an entirely different answer.

The investigation tries to understand electronic states in ideal crystals of finite size based on the mathematical theory of periodic differential equations. This theory provides some fundamental new understandings of those electronic states, including surface states.

The theory found that a one-dimensional finite crystal with two ends at $$ \tau $$ and $$ N a + \tau $$ always has one and only one state whose energy and properties depend on $$ \tau $$ but not $$ N $$ for each band gap. This state is either a band-edge state or a surface state in the band gap. Numerical calculations have confirmed such findings. Further, these behaviors have been seen in different one-dimensional systems, such as in.

Therefore:
 * The fundamental property of a surface state is that its existence and properties depend on the location of the periodicity truncation.
 * Truncation of the lattice's periodic potential may or may not lead to a surface state in a band gap.
 * An ideal one-dimensional crystal of finite length $$ L = N a $$ with two ends can have, at most, only one surface state at one end in each band gap.

Further investigations extended to multi-dimensional cases found that
 * An ideal simple three-dimensional finite crystal may have vertex-like, edge-like, surface-like, and bulk-like states.
 * A surface state is always in a band gap is only valid for one-dimensional cases.