  The theory of perfect crystals, founded upon the Bloch theorem, gives an
understanding of extended quantum states grouped into energy bands, and permits
the derivation of the dynamics of electrons in those states. The semiconductor
physics used to explain the operation of electronic devices treats the
(imperfect) semiconductor crystal as a uniform effective medium in which
positively and negatively charged quasi-particles mostly obey Newtonian
dynamics, and in which the chemistry of impurity atoms is far different from
that of those same atoms in free space. The connection between these two
pictures can be made by made by invoking a mathematical transformation that
takes the finite-temperature, impure device structure and algebraically
subtracts from it a perfect crystal, leaving only the residual differences to
be analyzed. This notion of the residual difference offers a conceptual basis
for understanding many aspects of semiconductor physics, including the
properties of impurity states and heterogeneous interfaces. The mesoscopic
transformation that underlies the residual-difference picture provides the
systematic way to define a concept that is essential to the understanding of
semiconductor devices: a position-dependent band structure.
