Abstract
The transport and optical properties of solids are due primarily to the electron system and to a lesser extent to the ions. These properties were discussed in Chapters 24 and 25 without taking the interaction between electrons into account, although its role may be important in some cases. In this chapter we will study the response of the interacting electron system to external perturbations, to an applied electromagnetic field. We will consider first the effect of an external scalar potential and will derive general expressions that relate the dielectric function to the density–density response function and the dynamical structure factor introduced in the previous chapter. This will then allow us to get approximate expressions for the frequency and wave number dependence of the dielectric function. The study of the redistribution of electrons induced by an external charge will lead to a proper description of screening in metals. It will be shown that the optical conductivity can be calculated from the current–current correlation function. Finally, by studying the response of the electron system to an external magnetic field, we will be able to derive an approximate expression for the wave number- and frequency-dependent susceptibility which is the magnetic analog of the dielectric function.
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Notes
- 1.
For this reason \(\widetilde{{\mathnormal{\Pi}}}\) is sometimes called irreducible polarization function or irreducible polarization insertion.
- 2.
l. h. Thomas, Thomas, L. H. 1927 and e. Fermi, Fermi, E. 1928.
- 3.
As we will see, the \(1/q^2\) singularity of the Fourier transform of the Coulomb interaction vanishes when the screened interaction is considered.
- 4.
Acronym for random phase approximation.
- 5.
j. Lindhard, 1954.
- 6.
n. d. Mermin, 1970.
- 7.
The results will be given by replacing the effective mass with the electron mass.
- 8.
r. Kubo, Kubo, R. 1957. The name Kubo formula refers more generally to the formula that expresses the generalized susceptibility as a retarded correlation function. This is discussed in more detail in Appendix J. The expression for transport coefficients is often referred to as Green–Kubo formula (m. s. Green, 1952, 1954), Green, M. S. while the expression for the conductivity is sometimes referred to as Kubo–Nakano formula (H. Nakano. 1956). Nakano, H.
- 9.
We note that the order of the limits is different when the static dielectric constant or the static magnetic susceptibility is calculated. The limit \(\omega \rightarrow 0\) has to be performed first for finite \({\boldsymbol{q}}\), and only after that we can take the limit \({\boldsymbol{q}} \rightarrow 0\). Otherwise the Lindhard function gives zero, because the number of particles and the magnetization are conserved quantities.
- 10.
D. A. Greenwood, Greenwood, D. A. 1958.
- 11.
E. C. Stoner, 1938.
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Sólyom, J. (2010). Electronic Response to External Perturbations. In: Fundamentals of the Physics of Solids. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04518-9_2
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DOI: https://doi.org/10.1007/978-3-642-04518-9_2
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