Quantum theory of the complex dielectric constant of free carriers in polar semiconductors

The optical constants and reflectivity of a semiconductor are known as functions of the real and imaginary parts of the complex dielectric constant. The imaginary part of the complex dielectric constant e
2
is proportional to the optical conductivity, which has recently been calculated from the quantum density matrix equation of motion. The expression obtained for e
2
reduces to the Drude result, as obtained from the quasi-classical Boltzmann transport equation, in the limit of low frequencies and elastic scattering mechanisms, and to the quantum result found using time dependent perturbation theory in the limit of high frequencies. This paper derives the real part of the complex dielectric constant e
1
for a III-V or II-VI semiconductor with the band structure of the Kane theory, using the quantum density matrix method. The relation of e
1
to the second order perturbation energy of the system is shown, and the reflectivity is a minimum when the second order perturbation energy vanishes. The quantum calculation for e
1
gives approximately the same result as the Drude theory, except near the fundamental absorption edge, and reduces to the Drude result at low frequencies. Using the complex dielectric constant, the real and imaginary parts of the complex refractive index, the skin depth, the surface impedance, and the reflectivity are found. The plasma resonance is examined. The surface impedance and the skin depth are shown to reduce to the usual classical result in the limit that
e_{1} = 0
and
w\tau \ll 1
, where
w
is the angular frequency of the applied field and τ is the electron scattering time.