The origin of a flat band

The origin of a flat band as defined in the flat band-steep band model for superconductivity has been studied. A local inversion center is derived mathematically as the necessary and sufficient condition for an occurrence of a flat band at an arbitrary point. From the (k) over right arrow -dependent part of E((k) over right arrowj), a pseudo-inversion center is demonstrated to be a sufficient condition based on the Patterson symmetry. The chemical bonding origin of a flat band is also studied analytically with a tight-binding approach. (C) 2003 Elsevier Inc. All rights reserved.