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Journal of Physics C Solid State Physics

On the use of variational models involving a magnetic field to calculate the ground-state energy of the polaron

Yves Lépine

Published May 30, 1987• 2 citations
Publisher

Abstract

The ground-state energy of the free polaron is calculated, using the Fock approximation. A new model Hamiltonian involving an electron in a harmonic potential of cylindrical symmetry in the presence of a variational magnetic field, internal in origin, is used. This magnetic field, which has no physical existence, introduces the possibility of symmetry breaking. This approach is shown to be equivalent to that of Peters and Devreese (1982), who used path integrals to study the properties of the polaron in a magnetic field, if the variational parameters of their model action are chosen to match those of the author's model and if the magnetic field in the model action is used as a variational parameter whose origin is internal. Using that approach, a strong-coupling ground-state energy much lower than the usual one is found (E 0 =-0.1658 alpha 2 ). It is shown that such a low value cannot be obtained from the adiabatic approximation because a model spectrum having a degenerate ground state has been used. His results call into question the assumption that the energies obtained from the Fock approximation or from the Feynman integrals are upper bounds to the polaron ground-state energy in the case of model actions or Hamiltonians involving magnetic fields.

DOI
10.1088/0022-3719/20/15/007

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With the development of technological and experimental techniques , crystals with all kinds of dimension and shape have been made by experiment, which has brought out a large market for the apply of new materials. So there is continuing interests in polaron. LeeLowPines calculated the groundstate energy of polaron by a variational technique. Huybrechts calculated the energy of the groundstate and the first internal excited state of the optical polaron for different values of the electronphonon coupling. N.Tokuda studied the dependence of groundstate energy, effective mass and the mean number of polaron on the coupling constant by using the unitary transformation and the method of a lagrange multiplier. Larsen got the groundstate energy of the twodimension magnetoplaron by using forthorder perturbationtheoretic method. Wei et al. investigated the cyclotronresonance mass and frequency of the interface polaron by using Feyman path integration method. Recently, Catandella et al. studied the properties of polaron in onedimension Holstein molecular crystals by using a new variational method, which chosen the linear overlap of Bloch wave of large polaron and small polaron as a trial wavefunction, obtained the variational regularities of the groundstate energy, mass and the mean number of polaron with coupling constant. We also did some works on polaron by using Huybrechts' method. But all works were based on only one considering either magnetic field or coupling strength,respectively. This paper is going to illustrate the common influence of magnetic field and coupling strength on the properties of polaron.The variational relations of the groundstate energy, selftrapping energy and Landau energy with coupling constant and cyclotronresonance frequency are derived by using linearcombination operator method. Numerical calculation indicates that the vibration frequency λ firstly falls, arriving to a minimum value, late increases with increasing coupling constant α; λ monotonously rises with increasing cyclotronresonance frequency ωc;the groundstate energy E0 decrease with increasing α and monotonous rises with increasing ωc;the selftrapping energy E0tr increases with increasing α and ωc;Landau energy E0L firstly increases to a maximum and then fall with increasing α;E0L firstly rises to maximum also and then decreases with increasing ωc.

Chinese Journal of Luminescence•Jan 1, 2003•0 citations
The ground-state energy of the free polaron is calculated, using the Fock approximation. A new model Hamiltonian involving an electron in a harmonic potential of cylindrical symmetry in the presence of a variational magnetic field, internal in origin, is used. This magnetic field, which has no physical existence, introduces the possibility of symmetry breaking. This approach is shown to be equivalent to that of Peters and Devreese (1982), who used path integrals to study the properties of the polaron in a magnetic field, if the variational parameters of their model action are chosen to match those of the author's model and if the magnetic field in the model action is used as a variational parameter whose origin is internal. Using that approach, a strong-coupling ground-state energy much lower than the usual one is found (E 0 =-0.1658 alpha 2 ). It is shown that such a low value cannot be obtained from the adiabatic approximation because a model spectrum having a degenerate ground state has been used. His results call into question the assumption that the energies obtained from the Fock approximation or from the Feynman integrals are upper bounds to the polaron ground-state energy in the case of model actions or Hamiltonians involving magnetic fields.
New results for the free energy of a polaron in a strong magnetic field are presented. The method used here is the extended version of Feynman's pathintegral formalism in which the most general quadratic form containing an infinite number of variational parameters is employed to simulate the variational trial action. A set of nonlinear integral equations is established to determine the upper bound for the free energy in the entire range of electron-phonon coupling constants, temperatures and magnetic fields. Analytical solutions are provided for some limiting cases of interest, and are compared critically with existing calculations of the ground-state energy at zero temperature. The free energy thus obtained is exact to the lowest order in the coupling constant and provides the good interpolation up to the strong coupling regime.
Abstract The properties of the ground state of the bound large polaron within the Fock approximation are investigated by using a new model spectrum which is known to describe better the polaron at low external fields. The results are compared with those of Matsuura who used the same approximation, obtained from a path integral formulation, with the usual Gaussian and Coulombic trial spectra, Both formulations are variational. It is found that the new spectrum gives better results (lower variational energies) than those of Matsuura for intermediate or large values of electron–phonon coupling and small values of the electrostatic coupling of the electron to the defect. This is related to that fact that the polaron seems to prefer a cylindrical symmetry to a spherical one in the absence of external fields.
The free energy of an ideal polaron gas in a static magnetic field is calculated using Feynman's path-integral formalism. The trial action, used in Feynman's polaron theory, is extended to take into account the anisotropy of the effective electron-phonon interaction. This results in a free-energy expression with four variational parameters. According to Feynman's conjecture, the resulting free energy is an upper bound to the exact result. The approximate free energy is expected to provide accurate results for arbitrary electron-phonon coupling strength ($\ensuremath{\alpha}$), temperature (TTT), and magnetic field strength (H\mathcal{H}H). The free energy per polaron is evaluated for limiting values of $\ensuremath{\alpha}$, TTT, and H\mathcal{H}H.
With the development of technological and experimental techniques , crystals with all kinds of dimension and shape have been made by experiment, which has brought out a large market for the apply of new materials. So there is continuing interests in polaron. LeeLowPines calculated the groundstate energy of polaron by a variational technique. Huybrechts calculated the energy of the groundstate and the first internal excited state of the optical polaron for different values of the electronphonon coupling. N.Tokuda studied the dependence of groundstate energy, effective mass and the mean number of polaron on the coupling constant by using the unitary transformation and the method of a lagrange multiplier. Larsen got the groundstate energy of the twodimension magnetoplaron by using forthorder perturbationtheoretic method. Wei et al. investigated the cyclotronresonance mass and frequency of the interface polaron by using Feyman path integration method. Recently, Catandella et al. studied the properties of polaron in onedimension Holstein molecular crystals by using a new variational method, which chosen the linear overlap of Bloch wave of large polaron and small polaron as a trial wavefunction, obtained the variational regularities of the groundstate energy, mass and the mean number of polaron with coupling constant. We also did some works on polaron by using Huybrechts' method. But all works were based on only one considering either magnetic field or coupling strength,respectively. This paper is going to illustrate the common influence of magnetic field and coupling strength on the properties of polaron.The variational relations of the groundstate energy, selftrapping energy and Landau energy with coupling constant and cyclotronresonance frequency are derived by using linearcombination operator method. Numerical calculation indicates that the vibration frequency λ firstly falls, arriving to a minimum value, late increases with increasing coupling constant α; λ monotonously rises with increasing cyclotronresonance frequency ωc;the groundstate energy E0 decrease with increasing α and monotonous rises with increasing ωc;the selftrapping energy E0tr increases with increasing α and ωc;Landau energy E0L firstly increases to a maximum and then fall with increasing α;E0L firstly rises to maximum also and then decreases with increasing ωc.