We study coupled electron-vibrational condensed-matter systems. Of primary interest are many-electron effects in two types of systems: nano- and micromechanical semiconductor resonators and electrons on the surface of liquid helium. Both types of systems are well characterized and display rich and unusual behavior, which makes them particularly attractive for revealing manifestations of the electron-vibrational coupling. We show that, in nano- and micro-mechanical resonators, the coupling leads to strong nonlinearity of the modes and a specific temperature dependence of their frequencies. The mechanism is the lifting of the degeneracy of the multi-valley electron energy spectrum by the vibration-induced strain. The redistribution of the electrons between the valleys is controlled by a large ratio of the electron-phonon coupling constant to the electron chemical potential or temperature. We find unusually large quartic in the strain terms in the electron free energy, which result in an unusually strong amplitude dependence of the mode frequencies. This dependence is calculated for silicon micro-systems, which at present are most broadly used in applications. It is significantly different for different modes and crystal orientations, and can vary nonmonotonously with the electron density and temperature. The proposed mechanism leads also to a very strong temperature dependence of the mode frequencies. These results explain the experimental observations and suggest ways of improving the stability and sensitivity of modern micromechanical devices. The other system studied, the electrons floating above the helium surface and coupled to the vibrations in the liquid helium, is special at least in two respects. First, this system is free from defects, which makes it the best known conductor and a perfect system for studying electron-vibrational coupling unmasked by disorder. Second, the electron-electron interaction is strong, so that the electrons form a strongly correlated liquid or a solid. We have developed an algorithm that allowed us to carry out extensive molecular dynamics simulations of the system with the account taken of the microscopic mechanisms of the electron scattering by the excitations in helium. The emphasis was made on calculating the experimentally observable characteristic, the many-electron mobility. The mobility displays particularly interesting features when the electrons are placed in a periodic potential. These features are a direct consequence of the strongly correlated electron motion. As we found, they enable a direct characterization of the correlations in the electron liquid and a direct measurement of the correlation length through its effect on the activated transport. They also provide a means for studying commensurate-incommensurate transitions through mobility measurements. We show that the crystallization in the electron system can be identified from the disappearance of self-diffusion. The way the diffusion coefficient vanishes is qualitatively different in the cases of incommensurate and commensurate electron systems. The mobility transverse to the periodic potential displays an activation dependence on the potential strength, with the exponent that strongly depends on the extent of the incommensurability.
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