Abstract
I construct a simple model to demonstrate that when the many-electron quantum state of a material is near a quantum phase transition and the vibrational motion of a phonon explores the potential energy surface near the transition point, then an impenetrable barrier appears at the potential energy surface which restricts the phonon from crossing the transition point and abnormally increases the phonon frequency. The ensuing anomalous enhancement of the electron phonon coupling is general and independent of the specific nature of the electronic quantum phase transition. Understanding and modelling this strong electron–phonon coupling may potentially lead to the design of phonon superconductors with high critical temperatures by choosing their parameters appropriately near an electronic quantum phase transition.
Highlights
The coupling between electrons and phonons is investigated when a phonon drives an electronic system towards a transition of the many-electron state
In the limit α → 0, the diagonal Born-Oppenheimer correction (DBOC) term has become an infinitely thin but impenetrable, delta-function barrier, confining perfectly the phonon wave functions on the left of the transition point. This picture is physically intuitive, especially compared with the prediction when the DBOC is ignored, that phonons near e.g. a metal to Mottinsulator transition can tunnel between the two regions and induce recurring metal-insulator transitions, or perhaps bring the electronic state in a superposition of entirely different macroscopic states
I note the critical temperature in phonon-mediated superconductivity increases quasilinearly with phonon frequency[10]
Summary
The coupling between electrons and phonons is investigated when a phonon drives an electronic system towards a transition of the many-electron state. To analyse this effect, it is useful to review briefly the adiabatic separation of the electronic and nuclear degrees of freedom[1, 2, 8]. The corresponding electronic energy, dependent on the nuclear positions, provides the potential energy surface (PES) where the nuclei move. As the nuclei slowly vibrate around their equilibrium positions, the electronic wave function evolves adiabatically with them. The evolved wave function is expanded in terms of the electronic eigenfunctions for the periodic structure and in the deviations of the phonons from equilibrium.
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