Figure 3(b) depicts a schematic two-band approximation to a semiconductor crystal. The band rather than discrete energy level picture arises because of the periodicity of the crystalline lattice. Solution to the quantum Schrodinger equation with a periodic potential lead to so-called Bloch states akin to the solutions to Mathieu's equation. An extra complication in the semiconductor is that carriers (electrons in the conduction and holes in the valence band) obey Fermi-Dirac statistics and display strong many-body Coulomb interactions (repulsion and attraction). These many-body phenomena strongly modify the interaction of optical pulses with semiconductor materials. Further details on light-semiconductor interactions are given in the article in this proceedings by Andreas Knorr and Stephan Koch.
A set of Semiconductor-Bloch equations (SBE) can be derived from the microscopic physics which, in form at least, mimic the two-level atom optical Bloch equations above[12, 13]. The SBE in the envelope approximation have the following form,
Here, q is the momentum resolved carrier wavenumber,
is the renormalized Rabi frequency,
denote the distribution functions for
electrons or holes
and
is the renormalized energy dispersion for a parabolic two-band semiconductor
with unrenormalized transition frequency
.
The Coulomb potential, 
,
is treated in a quasi-statical screening model; 
is
the bare potential. The Rabi frequency
is determined by the dipole matrix
element 
and the amplitude of the external electrical field
. The collisional relaxation
terms appearing above simulate
carrier-carrier (c-c) and carrier-phonon (c-ph) scattering.
In the 
-equation
we include c-c and c-ph scattering at the level of a
relaxation rate approximation (no memory)
with uniform time constants
(a=e,h) for c-c collisions and
for c-ph collisions:
where i=c,p and f denotes Fermi functions with chemical potentials
and temperature 
.
Memory effects are included by employing a pole approximation,
where the damping constants 
are chosen to
to remove a spurious absorption tail below the renormalized bandgap energy.
Another unique feature of the semiconductor material response is the occurrence of sharp spectral features below the renormalized bandgap. These ``exciton'' features arise as a consequence of the discrete quantum energy levels associated with a bound ``electron-hole'' pair. Thus the details of the nonequilibrium carrier distribution within the valence and conduction bands can significantly modify the characteristics of an ultrashort pulse propagating in an absorbing or amplifying (inverted) semiconductor medium. An illustration of pulse distortion in a semiconductor amplifier will be given in Section 3.4.