Atomistic Simulation Centre School of Mathematics and Physics Queen's University Belfast University Road Belfast BT7 1NN Northern Ireland
"If we take the view that quantisation of energy levels, tunnelling and interference are where quantum mechanics departs most violently from classical notions, we may ask where do the two come closest? Nowhere is this proximity more tangible than in the realm of particle interactions." Undated, Anon
This we can put to a simple test. In a random array of barriers, a combination
of quantisation, tunnelling and interference generates
trapped quantum states (left); but couple this to another set of degrees of freedom, and the picture washes out (right):
Without electron-phonon interactions life would crash. For starters we’d lose our sense of smell (PRL 98 (2007) 038101) (not always a bad thing). Electrical and electronic equipment would seize up. Even if we could keep appliances going, we wouldn't be getting the energy in the forms and places needed. E-phonon scattering is what turns the laptop armies at meetings into one of today's biggest energy wasters. E-phonon scattering is how electrons and atomic vibrations exchange energy and momentum. This controls charge and energy transport in molecular and condensed matter systems, and enforces thermal equilibration. Despite their fundamental role e-phonon interactions remain a challenge. Stepping beyond the Born-Oppenheimer approximation (the basis for the vast majority of MD simulations) opens a many-body problem (even ignoring e-e interactions). The hardest case are problems involving the simultaneous coupled dynamics of the two subsystems. We work on this problem in molecular electronics and systems under irradiation. Something these two very different contexts have in common is that in each case it is possible to drive the electrons and the ionic motion very far out of equilibrium, resulting in exceptionally violent momentum and energy exchange. We've developed a scalable method for real-time quantum e-phonon dynamics that can reach significant size/time scales [eceid] . Take say a 2d quantum dot on a substrate, following electronic excitation. We can then track the subsequent relaxation back towards equilibrium, as monitored say via the evolving dot state occupancies. Notice the interplay between transitions below. A vital process here is spontaneous phonon emission. It's often the hardest process to capture, and is where mean-field electron-nuclear dynamics (aka Ehrenfest dynamics) usually breaks down.
[eceid] Valerio Rizzi, Tchavdar Todorov, Jorge Kohanoff and Alfredo Correa, PRB 93 (2016) 024306
Quantum correlated e-phonon simulations are never easy though. A short-cut which throws out some physics but makes the problem a lot easier is to keep just the 'diagonal part' of the dynamics, resulting in a set of coupled kinetic equations for the electronic and phonon occupancies [eceid] . As an example, consider a narrow-gap doped semiconducting system. Initially a population of electrons from the valence band have been excited into impurity levels just below the conduction band. Below we track the e-phonon relaxation from there. The electronic occupancies eventually relax into a thermal distribution but notice how the process is split between two very different time-scales: fast initial thermalization with the nearby conduction band, followed by much slower overall thermalization involving the far valence band.