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Unformatted text preview: Physics 927 E.Y.Tsymbal 1 Section 13: Optical properties of solids Optical methods are very useful for the quantitative determination of the electronic band structure of solids. Experiments on optical reflectivity, transmission and refraction provide the way to determine the dielectric constant of the solid, which is related to the band structure. The dielectric constant is related to the optical conductivity. The term “optical conductivity” means the electrical conductivity in the presence of an alternating electric field. The term “optical” here covers the entire frequency range, and is not restricted only to the visible region of the spectrum. In order to derive the relation between the dielectric constant and the optical conductivity, we assume that electric field is oscillating with angular frequency ω : ( ) ( ) i t t e ω ω − = E E . (1) This wave propagates through the medium with conductivity ( ) σ ω and the dielectric constant ( ) L ε ω , both being the function of ω . Index L for the dielectric constant here reflects the fact that this is the dielectric constant of the lattice and does not include the conduction electrons. This implies the polarization of the medium occurs only due to the bound charges (polarization due to ions). The electric current and the electrical displacement are related to the electric field by ( ) ( ) ( ) ω σ ω ω = j E , (2) ( ) ( ) ( ) L ω ε ω ω = D E . (3) In general the conductivity and the dielectric function are tensors. Here we assume for simplicity that the medium is isotropic, so that ( ) σ ω and ( ) L ε ω are scalars, and therefore D and j are parallel to E . Now let me show that the dielectric constant and conductivity enter into a determination of the optical properties of a solid only in the combination. This can be seen from the Maxwell equation (CGS units) 1 4 c t c π ∂ ∇× = + ∂ D H j . (4) The first term on the right hand side of this equation corresponds to the displacement current associated with the polarization of the ion cores. The second term which is proportional to j is the convective current of the conduction electrons. Using Eq.(1) this formula can be rewritten in the following form: 1 4 1 4 1 L L i c t c c t c t π π σ ε σ ε ε ω ∂ ∂ ∂ ∇× = + = + = ∂ ∂ ∂ E E E H E , (5) where ε is a complex dielectric function, 4 L i π σ ε ε ω = + . (6) In this representation the conduction electrons are considered as a part of the dielectric medium. This consideration is plausible because in the presence of alternating electric field the conduction Physics 927 E.Y.Tsymbal 2 electrons oscillate around their equilibrium positions without net transnational motion. This is different from the DC conductivity when ( ) σ ω and ( ) L ε ω describe distinguishable physical processes, where ( ) σ ω describes the "free charges" (those that can move freely over arbitrary distances in response to the DC field) and ( ) L...
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- Fall '11