High-frequency damping of collective excitations in fermion systems. I. Plasmon damping and frequency-dependent local-field factor in a two-dimensional electron gas

Starting from a general expression of Glick and Long [Phys. Rev. B 4, 3455 (1971)] for the imaginary part of the frequency- and wave-number-dependent dielectric function ε(k,ω) of an electron gas in a two-particle–hole pair excitation approximation, an asymptotic expression for ${\epsilon }_2$(k,ω) is derived, which is valid for any wave number k provided ω is much higher than a certain characteristic k-dependent frequency. The formula for ${\epsilon }_2$(k,ω) is cast in a form suitable for application to any arbitrary potential v(k) in three-, two-, and one-dimensional space. In the case of D=3 and the Coulomb potential v(k) our result is the same as that of Glick and Long, although its region of validity is now larger. As a specific example, the damping of plasmons in a 2D electron gas has been calculated. Also, an interpolation formula for the complex local-field factor G(k,ω) in a 2D electron gas has been derived. The latter immediately leads to an expression for the ω-dependent exchange-correlation potential—a result analogous to the one derived by Gross and Kohn in the 3D case. The asymptotic expression for ${\epsilon }_2$(k,ω) is used in paper II to calculate the damping of zero sound in normal liquid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$.