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arXiv:1012.1716 · Physical Review B

Nonequilibrium dynamics of the Holstein polaron driven by an external electric field

Lev Vidmar, J. Bonča, Marcin Mierzejewski, Peter Prelovšek, S. A. Trugman

Published Apr 5, 2011• 79 citations
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Abstract

This work represents a fundamental study of a Holstein polaron in one dimension driven away from the ground state by a constant electric field. Taking fully into account quantum effects, we follow the time evolution of the system from its ground state as the constant electric field is switched on at $t=0$ until it reaches a steady state. At weak electron-phonon coupling (EP), the system experiences damped Bloch oscillations (BO's) characteristic for a noninteracting electron band. An analytic expression of the steady-state current is proposed in terms of weak EP coupling and large electric field. For moderate values of EP coupling, the oscillations are almost critically damped and the system reaches the steady state after a short time. In the strong-coupling limit, weakly damped BO's, consistent with nearly adiabatic evolution within the polaron band, persist up to extremely large electric fields. A traveling polaron under the influence of the electric field leaves behind a trail of phonon excitations absorbing the excess energy gained from the electric field. The shape of the traveling polaron is investigated in detail.

DOI
10.1103/physrevb.83.134301

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This work represents a fundamental study of a Holstein polaron in one dimension driven away from the ground state by a constant electric field. Taking fully into account quantum effects, we follow the time evolution of the system from its ground state as the constant electric field is switched on at t=0t=0t=0 until it reaches a steady state. At weak electron-phonon coupling (EP), the system experiences damped Bloch oscillations (BO's) characteristic for a noninteracting electron band. An analytic expression of the steady-state current is proposed in terms of weak EP coupling and large electric field. For moderate values of EP coupling, the oscillations are almost critically damped and the system reaches the steady state after a short time. In the strong-coupling limit, weakly damped BO's, consistent with nearly adiabatic evolution within the polaron band, persist up to extremely large electric fields. A traveling polaron under the influence of the electric field leaves behind a trail of phonon excitations absorbing the excess energy gained from the electric field. The shape of the traveling polaron is investigated in detail.
Using a variational numerical method, we compute the time evolution of the Hubbard-Holstein bipolaron from its ground state when at t=0t=0t=0 the constant electric field is switched on. The system is evolved taking into account full quantum effects until it reaches a quasistationary state. In the zero-field limit, the current shows Bloch oscillations characteristic for the adiabatic regime where the electric field causes the bipolaron to evolve along the quasiparticle band. The bipolaron remains bound and the net current remains zero in this regime. At larger electric fields, the system enters the dissipative regime with a finite quasistationary current. Concomitantly, the bipolaron dissociates into two separate polarons. By examining different parameter regimes, we show that the appearance of a finite quasistationary current is inevitably followed by the dissociation of the bipolaron.
We investigate polaron formation in a many-electron system in the presence of a local repulsion sufficiently strong to prevent local-bipolaron formation. Specifically, we consider a Hubbard-Holstein model of interacting electrons coupled to dispersionless phonons of frequency ω 0 . Numerically solving the model in a small one-dimensional cluster, we find that in the nearly adiabatic case ω 0 < t , the necessary and sufficient condition for the polaronic regime to occur is that the energy gain in the atomic ( i.e. , extremely localized) regime ε pol overcomes the energy of the purely electronic system ε el . In the antiadiabatic case, ω 0 > t , polaron formation is instead driven by the condition of a large ionic displacement g / ω 0 < 1 ( g being the electron-phonon coupling). Dynamical properties of the model in the weak and moderately strong coupling regimes are also analyzed.
We study the quantum dynamics of small polaron formation and polaron transport through finite quantum structures in the framework of the one-dimensional Holstein model with site-dependent potentials and interactions. Combining Lanczos diagonalization with Chebyshev moment expansion of the time evolution operator, we determine how different initial states, representing stationary ground states or injected wave packets, after an electron-phonon interaction quench, develop in real space and time. Thereby, the full quantum nature and dynamics of electrons and phonons is preserved. We find that the decay out of the initial state sensitively depends on the energy and momentum of the incoming particle, the electron-phonon coupling strength, and the phonon frequency, whereupon bound polaron-phonon excited states may emerge in the strong-coupling regime. The tunneling of a Holstein polaron through a quantum wall or dot is generally accompanied by strong phonon number fluctuations due to phonon emission and reabsorption processes.
We present a detailed numerical study of the one-dimensional Holstein model with a view to understanding the self-trapping process of electrons or excitons in crystals with short-range particle-lattice interactions. Applying a very efficient variational Lanczos method, we are able to analyze the ground-state properties of the system in the weak- and strong-coupling, adiabatic and nonadiabatic regimes on lattices large enough to eliminate finite-size effects. In particular, we obtain the complete phase diagram and comment on the existence of a critical length for self-trapping in finite (closed) one-dimensional systems. In order to characterize large and small polaron states we calculate self-consistently the lattice distortions and the particle-phonon correlation functions. In the strong-coupling case, two distinct types of small polaron states are shown to be possible according to the relative importance of static displacement field and dynamic polaron effects. Special emphasis is on the intermediate-coupling regime, which we also study by means of direct diagonalization, preserving the full dynamics and quantum nature of phonons. The crossover from large to small polarons shows up in a strong decrease of the kinetic energy accompanied by a substantial change in the optical absorption spectra. We show that our numerical results in all important limiting cases reveal excellent agreement with both analytical perturbation theory predictions and very recent density matrix renormalization group data.
Keeping the full quantum nature of the problem, we compute the relaxation time of the Holstein polaron in one dimension after it was driven far from the equilibrium by a strong oscillatory pulse. Just after the pulse, the polaron's kinetic energy increases and subsequently exhibits a relaxation-type decrease with simultaneous emission of phonons. In the weak coupling regime, partial tunneling of the electron from the polaron self-potential is observed. The inverse relaxation time is for small values of electron-phonon coupling λ linear with λ, while it deviates downwards from the linear regime at λ>/~0.1/ω(0). The imaginary part of the equilibrium self-energy shows good agreement with the inverse relaxation time obtained from nonequilibrium simulations.
We calculate the real space and time formation of a polaron quasiparticle from a bare electron. The time-dependent Schr\"odinger equation for the Holstein model of electron-phonon coupling is numerically integrated in a large Hilbert space to obtain the time evolution of the electron and phonon densities and the electron-phonon (el-ph) correlation functions. The quantum dynamical nature of the phonons is preserved. As the el-ph coupling increases, qualitative changes in polaron formation occur when the one-phonon polaron bound state forms. A potential barrier between the quasifree and heavy polaron states exists in dimensions $D\ensuremath{\geqslant}2$, consistent with earlier adiabatic theory. We compare to recent experiments.