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Journal of Physics Condensed Matter

Large polarons with quadratic electron-phonon interaction

T. Altanhan, B. Ünal

Published May 15, 1989• 0 citations
Publisher

Abstract

The Frohlich Hamiltonian of large polarons with quadratic electron-phonon interaction is obtained and the ground-state energy is calculated by using second-order perturbation theory.
DOI
10.1088/0953-8984/1/19/010

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The Fr\"ohlich polaron confined to move in one space dimension and interacting with a three-dimensional lattice is studied. We calculate exact and approximate results for the ground-state energy and the effective mass. The following exact results were obtained. In (a) the weak-coupling limit for the energy E=-\ensuremath{\alpha}'-[(3/2 \ensuremath{\surd}2 )-1]\ensuremath{\alpha}' $^{2}$, the mass ${\mathit{m}}^{\mathrm{*}}$/${\mathit{m}}_{\mathit{b}}$=1/2\ensuremath{\alpha}'+[(5/2 \ensuremath{\surd}2 )-1]/4, and the average number of virtual phonons N=1/2\ensuremath{\alpha}'+[(3/ \ensuremath{\surd}2 )-2]\ensuremath{\alpha}' $^{2}$, and in (b) the strong-coupling limit E=-0.333 088\ensuremath{\alpha}' $^{2}$,${\mathit{m}}^{\mathrm{*}}$/${\mathit{m}}_{\mathit{b}}$=2.1254\ensuremath{\alpha}' $^{4}$, and N=0.666 176\ensuremath{\alpha}' $^{2}$. Here, \ensuremath{\alpha}' is the normalized electron-phonon coupling constant, which was chosen such that E=-\ensuremath{\alpha}' within second-order perturbation theory.

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CRC Press eBooks•Oct 9, 2018•0 citations
The Fr\"ohlich Hamiltonian is generalized to the case of an electron moving in n space dimensions. For n=2 and n=3 the familiar Fr\"ohlich Hamiltonian is reobtained. The polaron ground-state energy is calculated up to fourth order in perturbation theory. We found that within the Feynman two-particle polaron model approximation the polaron ground-state energy satisfies the scaling relation ${E}_{n\mathrm{D}(\mathrm{\ensuremath{\alpha}})}$=(n/3)E3D{\mathrm{E}}_{3\mathrm{D}}E3D​({\ensuremath{\Gamma}[ (n-1)/2]3 \ensuremath{\surd}\ensuremath{\pi} /2n\ensuremath{\Gamma}(n/2)}\ensuremath{\alpha}), where EnD{E}_{n\mathrm{D}}EnD​ is the Feynman polaron ground-state energy for the polaron in n dimensions and E3D{E}_{3\mathrm{D}}E3D​ the energy in three dimensions.
The ground-state energy of a two-dimensional (2D) Fr\"ohlich polaron is calculated to second order in the coupling constant (\ensuremath{\alpha}) and gives E/\ensuremath{\Elzxh}${\ensuremath{\omega}}_{s}$=-(\ensuremath{\pi}/2)\ensuremath{\alpha}-0.063 97${\ensuremath{\alpha}}^{2}$ with \ensuremath{\Elzxh}${\ensuremath{\omega}}_{s}$ the surface optical-phonon energy. In the strong-coupling limit the adiabatic approximation is used and E/\ensuremath{\Elzxh}${\ensuremath{\omega}}_{s}$=-0.4047${\ensuremath{\alpha}}^{2}$ is found to leading order in \ensuremath{\alpha}. The Feynman path-integral approximation, the Gaussian approximation, and the modified Lee-Low-Pines unitary transformation approximation to the polaron ground-state energy all satisfy the scaling relation E2D{E}_{2\mathrm{D}}E2D​(\ensuremath{\alpha})=(2/3)E3D{E}_{3\mathrm{D}}E3D​(( 3\ensuremath{\pi}/4)\ensuremath{\alpha}), where E2D{E}_{2\mathrm{D}}E2D​ (E3D{E}_{3\mathrm{D}}E3D​) is the ground-state energy of the 2D (3D) polaron.
The Fr\"ohlich polaron confined to move in one space dimension and interacting with a three-dimensional lattice is studied. We calculate exact and approximate results for the ground-state energy and the effective mass. The following exact results were obtained. In (a) the weak-coupling limit for the energy E=-\ensuremath{\alpha}'-[(3/2 \ensuremath{\surd}2 )-1]\ensuremath{\alpha}' 2^{2}2, the mass m∗{\mathit{m}}^{\mathrm{*}}m∗/mb{\mathit{m}}_{\mathit{b}}mb​=1/2\ensuremath{\alpha}'+[(5/2 \ensuremath{\surd}2 )-1]/4, and the average number of virtual phonons N=1/2\ensuremath{\alpha}'+[(3/ \ensuremath{\surd}2 )-2]\ensuremath{\alpha}' 2^{2}2, and in (b) the strong-coupling limit E=-0.333 088\ensuremath{\alpha}' 2^{2}2,m∗{\mathit{m}}^{\mathrm{*}}m∗/mb{\mathit{m}}_{\mathit{b}}mb​=2.1254\ensuremath{\alpha}' 4^{4}4, and N=0.666 176\ensuremath{\alpha}' 2^{2}2. Here, \ensuremath{\alpha}' is the normalized electron-phonon coupling constant, which was chosen such that E=-\ensuremath{\alpha}' within second-order perturbation theory.
A refined approach where the wave function is gradually improved is developed in the one-dimensional (1D) large-Frohlich-polaron problem. The results for the ground-state energy, the effective mass, and the average number of virtual phonons are obtained for a wide range of the coupling constant It, and are in agreement with those of the Feynman path-integral formalism. In addition, in the weak-coupling limit, the expansions of these observables in powers of the coupling constant are exactly calculated up to the alpha 3 -term. They are E=- alpha -0.06066 alpha 2 -0.00844 alpha 3 , m*=1+0.5 alpha +0.19194 alpha 2 +0.06912 alpha 3 , and N=0.5 alpha +0.12132 alpha 2 +0.029540 alpha 3 , which agree with the known results of the fourth-order perturbation theory.
After deriving the quantum mechanical model polaron Hamiltonian, Frohlich, Pelzar and Zineau presented a perturbative solution to the problem. This chapter examines the perturbation theory from Frohlich et al. and presents other important weak and intermediate coupling polaron theories. It discusses several polaronic properties like average number of virtual phonons in the polaron cloud, electron life-time, lattice polarization potential using the first-order perturbative wave function and obtained the polaron energy and the polaron mass using the second-order Rayleigh-Schrodinger perturbation theory (RSPT). The second-order RSPT is known to have a limited range of validity. In the one-phonon Tamm-Dancoff approximation, the exact ground state energy appears in the denominator instead of the unperturbed energy that appears in RSPT. The chapter also discusses the work of Lee and Pines who have used the Tomonaga's approximation made in the formulation of the meson-theoretic problem to the case of polaron.