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arXiv:cond-mat/9704237 · Physical review. B, Condensed matter

Spectral properties of the small polaron

J. M. Robin

Published Dec 1, 1997• 34 citations
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Abstract

We compute exactly both the spectral function of the electron and of the small polaron for the two-site Holstein model in the case where phonon and electron energies are of the same order. We find that for intermediary coupling, the small polaron is a better fundamental excitation of the system than the electron. However, the Lang-Firsov approximation fails to predict accurately the dispersion relation for the small polaron.

DOI
10.1103/physrevb.56.13634

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We compute exactly both the spectral function of the electron and of the small polaron for the two-site Holstein model in the case where phonon and electron energies are of the same order. We find that for intermediary coupling, the small polaron is a better fundamental excitation of the system than the electron. However, the Lang-Firsov approximation fails to predict accurately the dispersion relation for the small polaron.
The phonon spectral function of the one-dimensional Holstein model is obtained within weak and strong-coupling approximations based on analytical self-energy calculations. The characteristic excitations found in the limit of small charge-carrier density are related to the known (electronic) spectral properties of Holstein polarons such as the polaron band dispersion. Particular emphasis is laid on the different physics occurring in the adiabatic and anti-adiabatic regimes, respectively. Comparison is made with a cluster approach exploiting exact numerical results on small systems to yield an approximation for the thermodynamic limit. This method, similar to cluster perturbation theory, confirms the analytical findings, and yields accurate results also in the intermediate-coupling regime.
The phonon spectral function of the one-dimensional Holstein model is obtained within weak and strong-coupling approximations based on analytical self-energy calculations.The characteristic excitations found in the limit of small charge-carrier density are related to the known (electronic) spectral properties of Holstein polarons such as the polaron band dispersion.Particular emphasis is laid on the different physics occurring in the adiabatic and antiadiabatic regimes, respectively.Comparison is made with a cluster approach exploiting exact numerical results on small systems to yield an approximation for the thermodynamic limit.This method, similar to cluster perturbation theory, confirms the analytical findings, and yields accurate results also in the intermediate-coupling regime.
We calculate the spectral properties of the one-dimensional Holstein and breathing polarons using the self-consistent Born approximation. The Holstein model electron-phonon coupling is momentum independent while the breathing coupling increases monotonically with the phonon momentum. We find that for a linear or tight binding electron dispersion: (i) for the same value of the dimensionless coupling the quasiparticle renormalization at small momentum in the breathing polaron is much smaller, (ii) the quasiparticle renormalization at small momentum in the breathing polaron increases with phonon frequency unlike in the Holstein model where it decreases, and (iii) in the Holstein model the quasiparticle dispersion displays a kink and a small gap at an excitation energy equal to the phonon frequency ${\ensuremath{\omega}}_{0}$ while in the breathing model it displays two gaps, one at excitation energy ${\ensuremath{\omega}}_{0}$ and another one at $2{\ensuremath{\omega}}_{0}$. These differences have two reasons: first, the momentum of the relevant scattered phonons increases with increasing polaron momentum and second, the breathing bare coupling is an increasing function of the phonon momentum. These result in an effective electron-phonon coupling for the breathing model which is an increasing function of the total polaron momentum, such that the small momentum polaron is in the weak coupling regime while the large momentum one is in the strong coupling regime. However, the first reason does not hold if the free electron dispersion has low energy states separated by large momentum, as in a higher dimensional system, for example, in this situation the difference between the two models becomes less significant.
According to recent quantum Monte Carlo simulations, the small polaron theory is practically exact in a wide range of the long-range (Fröhlich) electron-phonon coupling and adiabatic ratio. Applying Lang-Firsov transformation, we derive a single-particle Green's function describing the propagation of small Fröhlich polaron. One- and two-dimensional spectral functions are studied by expanding Green's function perturbatively. Remarkably, the coherent spectral weight ( Z ) and effective mass ( Z ′) renormalisation exponents are found to be different with Z ′ ≫ Z , which can explain a small coherent spectral weight and a relatively moderate mass enhancement in oxides.
According to recent Quantum Monte Carlo simulations the small polaron theory is practically exact in a wide range of the long-range (Frohlich) electron-phonon coupling and adiabatic ratio. We apply the Lang-Firsov transformation to convert the strong-coupling term in the Hamiltonian into the form of an effective hopping integral and derive the single-particle Green's function describing propagation of the small Frohlich polaron. One and two dimensional spectral functions are studied by expanding the Green's function perturbatively. Numerical calculations of the spectral functions are produced. Remarkably, the coherent spectral weight (Z) and effective mass (Z') renormalisation exponents are found to be different with Z'>>Z, which can explain a small coherent spectral weight and a relatively moderate mass enhancement in oxides.
An analytical approach to the one-dimensional spinless Holstein model is proposed, which is valid at finite charge-carrier concentrations. Spectral functions of charge carriers are computed on the basis of self-energy calculations. A generalization of the Lang–Firsov canonical transformation method is shown to provide an interpolation scheme between the extreme weak-coupling and strong-coupling cases. The transformation depends on a variationally determined parameter that characterizes the charge distribution across the polaron volume. The relation between the spectral functions of the polaron and electron, the latter corresponding to the photoemission spectrum, is derived. Particular attention is paid to the distinction between the coherent and incoherent parts of the spectra, and their evolution as a function of band filling and model parameters. Results are discussed and compared with recent numerical calculations for the many-polaron problem.