A theory of sound propagation in disordered one-dimensional metals

Abstract

This paper presents a consistent quantum mechanical derivation of the Hamiltonian to describe the interaction of electrons with long-wavelength phonons in one-dimensional conductors. Due to the nearly total mutual cancellation of the common deformational and Coulomb electron-phonon interaction mechanisms, the sound field happens to be much less coupled to electrons than in three-dimensional metals. In 1d metals a major part is played by a new mechanism, called “cross-deformational”, and by the inertial interaction due to the Stewart-Tolman effect. The cross-deformational interaction results from the modulation by the sound wave of the potential of the interaction between the electrons and the randomly located impurities. It is nondiagonal with respect to the electron momenta on the Fermi surface. Equations of motion are derived for the 1d metal in the continuum approximation consistently taking into account the electron elasticity. A new technique is suggested for the calculation of dynamic electron correlation functions, permitting one to take into account directly and automatically the multiple electron scattering by impurities and the effect of electron localization in a disordered 1d conductor. The sound velocity and absorption are analysed as functions of frequency for all possible frequencies and wavelengths, both for weak and strong spatial dispersion. We found that the sound absorption due to the cross-dimensional interaction is an essentially localized effect occuring at the moment when the electron is scattered by a single impurity. A geometric resonance type oscillatory behavior is predicted for the frequency dependence of the absorption. It occurs due to the hopping character of the electron motion, with a fixed hop length, in the nonuniform sound field. Since Landau damping is absent, the frequency dependence of the absorption in the range of strong spatial dispersion proves to be quadratic rather than linear as in 3d metals. The effects of temperature on conductivity, sound absorption and sound velocity are analysed. The scattering frequency of electrons by long-wavelength thermal phonons is established. It is controlled by the cross-deformational interaction and increases quadratically with temperature. The role of the Fermi liquid interaction between conduction electrons is discussed and the criteria of three-dimensionality are established. In contrast to 3d metals, there is a wide range of low frequencies where the single-particle gas approximation is valid.