Small Polaron Transformation Research Articles

Analytical derivation of equilibrium state for open quantum system.

Calculation of the equilibrium state of an open quantum system interacting with a bath remains a challenge to this day, mostly due to a huge number of bath degrees of freedom. Here, we present an analytical expression for the reduced density operator in terms of an effective Hamiltonian for a high temperature case. Comparing with numerically exact results, we show that our theory is accurate for slow baths and up to intermediate system-bath coupling strengths. Our results demonstrate that the equilibrium state does not depend on the shape of spectral density in the slow bath regime. The key quantity in our theory is the effective coupling between the states, which depends exponentially on the ratio of the reorganization energy to temperature and, thus, has opposite temperature dependence than could be expected from the small polaron transformation.

The current-induced heat generation in a spin-flip quantum dot sandwiched between a ferromagnetic and a superconducting electrode

Using non-equilibrium Green's functions' theory based on extended Nambu representation and small polaron transformation, we studied the current-induced heat generation in a spin-flip quantum dot sandwiched between a ferromagnetic and a superconducting electrode. We focused on moderate dot-leads coupling and relative small phonon energy, and derived the detailed expression of heat generation. The numerical results show (i) the heat generation decreases with polarization degree increasing, (ii) the intradot spin-flip can have a great effect on the heat generation at both zero temperature and finite temperature and (iii) at finite temperature an optimal workspace of keeping spin current and tuning heat generation by modulating the spin-flip intensity can be found.

Magnetoresistance effect of heat generation in a single-molecular spin-valve

Based on non-equilibrium Green's functions' theory and small polaron transformation's technology, we study the heat generation by current through a single-molecular spin-valve. Numerical results indicate that the variation of spin polarization degree can change heat generation effectively, the spin-valve effect happens not only in electrical current but also in heat generation when Coulomb repulsion in quantum dot is smaller than phonon frequency and interestingly, when Coulomb repulsion is larger than phonon frequency, the inverse spin-valve effect appears by sweeping gate voltage and is enlarged with bias increasing. The inverse spin-valve effect will induce the unique heat magnetoresistance effect, which can be modulated from heat-resistance to heat-gain by gate voltage easily.

Dissipative time-dependent quantum transport theory: Quantum interference and phonon induced decoherence dynamics.

A time-dependent inelastic electron transport theory for strong electron-phonon interaction is established via the equations of motion method combined with the small polaron transformation. In this work, the dissipation via electron-phonon coupling is taken into account in the strong coupling regime, which validates the small polaron transformation. The corresponding equations of motion are developed, which are used to study the quantum interference effect and phonon-induced decoherence dynamics in molecular junctions. Numerical studies show clearly quantum interference effect of the transport electrons through two quasi-degenerate states with different couplings to the leads. We also found that the quantum interference can be suppressed by the electron-phonon interaction where the phase coherence is destroyed by phonon scattering. This indicates the importance of electron-phonon interaction in systems with prominent quantum interference effect.

The inelastic spin Seebeck effects in a controllable Aharonov–Bohm interferometer based on a molecular quantum dot

Using non-equilibrium Green's functions' technology combined with small polaron transformation, we theoretically study the inelastic spin Seebeck effects in a controllable Aharonov–Bohm interferometer based on a degenerate molecular quantum dot level. The numerical results show that when the direct coupling between electrodes is switched on, the spin-dependent thermoelectric efficiency will be suppressed, but the magnetic flux can modulate the thermoelectric line-shapes regularly and the pure spin current generation can be achieved by the temperature gradient. When the direct coupling between electrodes is closed, the spin-dependent thermoelectric efficiency will be raised greatly and the heating resulting from the electron–phonon coupling can improve it effectively. • We develop a new numerical method to implement the calculation for inelastic Fano-resonance thermoelectric transport. • When the electrodes' direct coupling is open, the magnetic flux can modulate the line-shape of thermoelectric functions regularly. • The pure spin current generation can be achieved by the temperature gradient and the electrodes' direct coupling. • When the electrodes' coupling is closed, the spin-dependent thermoelectric efficiency will be raised greatly and the heating to the phonon bath can improve it effectively.

Inelastic electron transport through mesoscopic systems: Heating versus cooling and sequential tunneling versus cotunneling processes

Inelastic electron transport in quantum dot systems is studied via the hierarchical equations of motion combining a small polaron transformation approach, with differential conductance $dI/dV\ensuremath{\sim}V$ characteristics being evaluated accurately at the cotunneling level. We observe (i) the peak feature of phonon emission Franck-Condon sidebands to the zero-phonon peaks of both polaron and bipolaron in sequential electron transport; (ii) phonon absorption peaks occurring if the phonon temperature is sufficiently higher than that of the carrier electron; and (iii) the step feature of Raman sidebands in the cotunneling transport regime. We also evaluate the polaron transport response to a continuous-wave irradiation that induces bias-voltage oscillation. We observe, consistent with experimental results, that (iv) the photon-phonon-assisted tunneling enhances phonon absorptions while suppressing emissions. As the phonon absorption (emission) is associated with the process of absorbing (emitting) energy from (to) the phonon environment, an alternating or tailored field applied to contacts could be a practical means of cooling the mesoscopic quantum-transport device.

Transport through a mixed-valence molecular transistor in the sequential-tunneling regime: Theoretical insight from the two-site Peierls–Hubbard model

Transport through a mixed-valence system in the sequential-tunneling region is investigated using the master equation method and a simple two-site Peierls-Hubbard model that includes electron-phonon (e-p) coupling, electron hopping, and electron-electron (e-e) repulsion. The characteristics of Coulomb diamonds in the conductance spectra under three regimes are discussed. In the regime of zero e-p coupling, we found that the widths of Coulomb diamonds are dominated by the competition of electron-hopping and Coulomb repulsion. In the regime of weak and intermediate e-p coupling, by virtue of the normal-mode transformation we found that coupling to the symmetric-mode decreases the widths of Coulomb diamonds. In the regime of strong e-p coupling, an analytical expression for the widths of Coulomb diamonds can be derived using the small polaron transformation. The derived formula provides a new way to estimate e-e interactions and e-p couplings experimentally.

Effectiveness of Perturbation Theory Approaches for Computing Non-Condon Electron Transfer Dynamics in Condensed Phases

A description of electron transfer in condensed-phase media requires models that adequately describe the coupling of the electronic degrees of freedom to the surrounding nuclear coordinates. The spin-boson model has been the canonical model used to understand quantum dynamic processes in condensed-phase media over the last 25 years. Inherent in the standard model of a two-state quantum system coupled to a bosonic bath is the assumption that the Condon approximation is valid. In this context, the Condon approximation assumes that the bath configurations (coordinates) have no effect on the nonadiabatic coupling matrix element. While this is a useful model for electron transfer in small molecular systems, the validity of this approximation is less likely when large-scale motions of solvent molecules are strongly coupled to the electron transfer event, e.g., in molecular clamps and long-range electron transfer in biopolymers. In the present paper a general model for two-state electron transfer which allows for system-bath coupling in both the diagonal and off-diagonal (nonadiabatic) terms is studied. Time-dependent perturbation theory for this Hamiltonian is developed using a small polaron transformation. As noted in several recent studies, in a certain regime of parameter space, the relevant Hamiltonian admits an exact solution, termed the exactly solvable non-Condon Hamiltonian (or NCE). This limit, for which exact solutions are available, is used to benchmark the short- and long-time accuracy of various perturbative approaches. The validated perturbation equations are subsequently used to explore the role of non-Condon effects on electron transfer by systematically increasing the strength of the non-Condon coupling term from zero (i.e., the canonical spin-boson model) to the value that pertains to the exactly solvable non-Condon model (where non-Condon effects are significant).

Ab initioelectron propagators in molecules with strong electron-phonon interaction: II. Electron Green’s function

Ab initio electron propagator methods are developed to study electronic properties of molecular systems with strong electron-electron and electron-phonon interactions. For the calculation of electron Green's functions we apply a canonical small polaron transformation that intrinsically contains strong electron-phonon effects. In the transformed Hamiltonian, the energy levels for the noninteracting particles are shifted down by the relaxation (solvation) energies. The Coulomb integrals are also renormalized by the electron-phonon interaction. For certain values of the electron-phonon coupling constants, the renormalized Coulomb integrals can be negative which implies the attraction between two electrons. Within the small polaron transformation we develop a diagrammatic technique for the calculation of electron Green's function in which the electron-phonon interaction is already included into the multiple phonon correlation functions. Since the decoupling of the phonon correlation functions is impossible, and therefore, a Wick's theorem for such correlation functions is invalid, there is no Dyson equation for the electron Green's function. To find the electron Green's function, we use different approximations. One of them is a link-cluster approximation that includes diagonal transitions for the renormalized zeroth Green's function. In the linked-cluster approach the Dyson equation is derived in the most general case, where the self-energy operator is an arbitrary functional (not only in the Hartree-Fock approximation). It is shown that even a Hartree-Fock electron (hole) is not a particle any longer. It is a quasiparticle with a finite lifetime that depends on energy of particle and hole states in different ways. As a consequence of this, a standard description of a Hartree-Fock approximation in terms of wave functions becomes inappropriate in this problem. To challenge the linked-cluster approximation we develop a different approach: a sequential propagation approximation where scattering events occur only for sequential transitions. A self-consistent Hartree-Fock equation for a four-index Green's function matrix is derived. In conclusion, the proposed schemes can be considered for future method developments for quantum chemical calculations for large molecules with strong nonadiabatic effects, e-e correlated electron transfer reactions, and electron transport in molecular transport junctions.

Conduction in molecular junctions: inelastic effects

The effect of a thermal environment on electron (or hole) transfer through molecular bridges and on the electron conduction properties of such bridges is studied. Our steady state formalism based on an extension of the Redfield theory [J. Phys. Chem. B 104 (2000) 3817; Chem. Phys. 268 (2001) 315] is extended in two ways: First, a better description of the weak-coupling limit, which accounts for the asymmetry of the energy dependence of the quasi-elastic component of the transmission is employed. Secondly, for strong coupling to the thermal bath the small polaron transformation is employed prior to the Redfield expansion. It is shown that the thermal coupling is mainly characterized by two physical parameters: the reorganization energy that measures the coupling strength and the correlation time (or its inverse – the spectral width) of the thermal bath. Implications for the observed dependence of the bridge-length dependence of the transmissions are discussed. It is argued that in the intermediate regime between tunneling behavior and site-to-site thermally induced hopping, the transport properties may depend on the interplay between the local relaxation rate and the transmission dynamics.

Interplay of exciton or electron transfer and relaxation

Mori theory together with canonical transformation but with quantities of interest remaining untransformed is applied to the exciton- or electron-phonon system with site-diagonal linear coupling and full exciton or electron density matrix; finite concentrations of the carriers are admitted. Partial small polaron transformation is used which allows to investigate relaxation of initially relaxed systems without initial-condition terms. The result is applied, in the lowest order in the coupling, to a particle on a symmetric dimer. Coefficients suggested in a recent generalization of the Haken-Strobl-Reineker model are identified. The theory surprisingly, is found to be, in the above order, insufficient to yield a loss of coherence of carrier propagation with increasing time, though standard Markovian treatment suggests the opposite. A new type of a so far unnoticed drawback, typical of finite-order time-convulution theories, which is connected with the infinite memory of the initial lattice relaxation becomes conspicuous.

On the calculation of transfer rates between impurity states in solids

We comment on the calculation of energy transfer in solids, showing that the calculation can be done to infinite order in the phonon-impurity interaction using the small-polaron transformation. When the final result is expanded, the results of low order perturbation theory are recovered, and, in addition, new terms are found which are due to cross products.