Abstract
Utilizing state-dependent projection operators and the Kang-Choi reduction identities, we derive the linear, first, and second-order nonlinear optical conductivities for an electron system interacting with phonons. The lineshape functions included in the conductivity tensors satisfy “the population criterion” saying that the Fermi distribution functions for electrons and Planck distribution functions for phonons should be combined in multiplicative forms. The results also contain energy denominator factors enforcing the energy conservation as well as interaction factors describing electron-phonon interaction properly. Therefore, the phonon absorption and emission processes as well as photon absorption and emission processes in all electron transition processes can be presented in an organized manner and the results can be represented in diagrams that can model the quantum dynamics of electrons in a solid.
Highlights
Studies of the optical transitions for electron systems interacting with a background are useful for examining the electronic properties of solids because the absorption lineshapes are sensitive to the type of scattering mechanisms affecting the carrier behavior
“many-body criteria,” one of the most difficult being “the population criterion,” meaning that the Fermi distribution functions for electrons and the Planck distribution functions for phonons should be combined in multiplicative forms because the electrons and phonons belong to different categories in quantum-statistical physics
We presented a method for deriving the nonlinear optical conductivity tensors for a system of electrons interacting with phonons using the PR method
Summary
Studies of the optical transitions for electron systems interacting with a background are useful for examining the electronic properties of solids because the absorption lineshapes are sensitive to the type of scattering mechanisms affecting the carrier behavior. The second kind (see [13, equation (4.18)]) was used along with current-dependent projectors, which is called the state-independent projectors, to derive the first-order nonlinear optical conductivity [13]. This satisfies the population criterion but has limited applications because it does not satisfy the projection criterion. The corrected extended version of the nonlinear optical conductivity formalism, including the second-order nonlinear optical conductivity, was presented recently by applying the projection-reduction (PR) method by combining the generalized state-dependent projection operator (SDPO) and the KCRI of the 3rd kind (see [14, equation (2.5)]). The aim of this study is to generalize the KCRI to cover higher-order nonlinear optical conductivities and show how physically acceptable forms of the optical conductivities can be obtained by PR method. A method is introduced to represent the optical conductivity formulae using diagrams, through which a physical intuition to the quantum dynamics of the electrons in a solid can be achieved
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