Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review

Polaron Energy Spectrum

David M. Larsen

Published Apr 15, 1966• 83 citations
Publisher

Abstract

The energy spectrum of a polaron---an electron interacting with the longitudinal-optical phonons of a polar crystal---is studied with particular attention given to the region where the polaron excitation energy closely approaches the energy of a longitudinal-optical phonon. In this region, existing theories of the polaron spectrum are inadequate; in particular, the usual Rayleigh-Schr\"odinger perturbation theory is shown to be inconsistent. A self-consistent weak-coupling theory is developed, and a variational theory of the polaron spectrum which, for small coupling, reduces to this weak-coupling theory is presented. Whitfield and Puff, and Schultz have conjectured that the polaron energy $E\mathrm{E}p\mathrm{R}$, bends over and becomes horizontal as the polaron momentum $p$ approaches from below the value at which the polaron excitation energy, $E(p)\ensuremath{-}E(0)$, becomes equal to the optical-phonon energy. Using the weak-coupling method of the present paper, this conjecture has been verified to lowest order and next higher order in the coupling constant.

DOI
10.1103/physrev.144.697

Related papers

Weak-Coupling Theory of the Polaron Energy-Momentum Relation

George Whitfield, Robert D. Puff

We present arguments to show that the energy-momentum relation for a polaron is quadratic for small $k$, but bends over and becomes horizontal when the energy approaches the optical-phonon energy above the ground state. We present a calculation for zero temperature based on a thermodynamic Green's-function theory which is especially designed to give a reliable polaron energy-momentum relation.

Physical Review•Jul 5, 1965•96 citations

Green's function theory of the polaron energy‐momentum relation

R. v. Baltz

Abstract The energy E(p) of a polaron is studied as a function of temperature and momentum p with particular attention focused on the region where the excitation energy E(p) – E(0) closely approaches the energy of a longitudinal optical phonon. It is shown that in this region a quasi‐particle description is not adequate, e.g., the horizontal tangent in E(p) does not give rise to a singularity in the density of electronic states. The shape of the spectral weight function indicates that even for small coupling strength there is an internal excited state of the polaron.

physica status solidi (b)•Mar 1, 1972•7 citations

Polaron inndimensions

Bo E. Sernelius

The energy dispersion for arbitrary momentum of a polaron in n dimensions is calculated by the Rayleigh-Schr\"odinger perturbation theory and by a modified Brillouin-Wigner perturbation theory. For the energy and effective mass of the polaron ground state, both methods give results that agree with the results of V. V. Paranjape and P. V. Panjat [Phys. Rev. B 35, 2942 (1987)] to linear order in the polaron coupling constant \ensuremath{\alpha}. The self-energy shifts are progressively weakened with increasing dimension. In particular, this weakening occurs for the structure in the energy dispersion near the momentum value at which the polaron starts decaying via the emission of phonons.

Physical review. B, Condensed matter•Dec 15, 1987•13 citations

Energy of a polaron bound to a Coulomb potential

Ward Huybrechts, G. Remaut

A variational calculation of the ground state and of the first excited state of a bound polaron is given for all values of the electron-phonon coupling and of the Coulomb potential. The applied unitary transformation formalism is adapted in order to take account of correlation between the longitudinal optical phonons.

physica status solidi (a)•Jun 16, 1980•6 citations

A Variational Calculation of the Polaron Self-Energy and Effective Mass.

Benjamin U. Stewart

An approximate calculation is made of the polaron self-energy and effective mass as defined by Frohlich.The approach taken is a variational one based on an ex tended version of Hohler's ansatz in which the trial state function employed is an exact eigenfunction of the total wave vector of the polaron.The effective mass obtained simulates the Feynmann-Schultz result while the corresponding polaron self-energy is fairly accurate but inferior to the result of Feynman and Schultz.CONTENTS Chapters « -« T -| > £ ( l ^' (1'2) n in which n is the optical refractive index, k is the static dielectric constant, e is the electronic charge and 2 t\& is Plank's constant.The limit S->°° is to be taken with lim I = -~ S-*oo v 8tt d 3v .(1,3) A natural unit system is employed in which 2^=o)=2m=l.The first term in the Hamiltonian is the quantum mechanical operator corresponding to the kinetic energy of the electron.The second term represents the energy of the polarization field resulting from the long wave length longitudinal optical crystal modes in the absence of interaction with the electron.The last term represents the energy of interaction between the electron and the phonons.The name "polaron" has been given to the entity consisting of the Bloch conduction electron together with its accompanying nonradiative polarization field.Accordingly Frohlich defines the polaron self-energy, EQ (a), and the (ground-state) polaron effective mass, y(a), by the expansion EQ (a,ic) = Ec (a) + k 2/y(a) + 0(k4 ), (1,4)

Civil War Book Review•Jan 1, 1971•0 citations

The Ground State of the Optical Polaron in the Strong-Coupling Case

Satoru J. Miyake

The next lowest order term of the ground state energy has its origin in lowering of frequencies of phonons localized around the polaron. An eigenvalue equation is derived, the solution of which gives phonon eigenmodes and frequencies in the presence of the polaron. Several eigenvalues are obtained by numerical integration. From the results, the ground state energy is estimated as \(E_{0}/\hbar\omega_{0}{=}-0.108513\alpha^{2}-2.836+O(1/\alpha^{2})\), where ω 0 is the frequency of the optical phonon and α is the electron-phonon coupling constant. The numerical value of the effective mass is 0.0227019α 4 .

Journal of the Physical Society of Japan•Sep 15, 1976•69 citations
The energy spectrum of a polaron---an electron interacting with the longitudinal-optical phonons of a polar crystal---is studied with particular attention given to the region where the polaron excitation energy closely approaches the energy of a longitudinal-optical phonon. In this region, existing theories of the polaron spectrum are inadequate; in particular, the usual Rayleigh-Schr"odinger perturbation theory is shown to be inconsistent. A self-consistent weak-coupling theory is developed, and a variational theory of the polaron spectrum which, for small coupling, reduces to this weak-coupling theory is presented. Whitfield and Puff, and Schultz have conjectured that the polaron energy EEpRE\mathrm{E}p\mathrm{R}EEpR, bends over and becomes horizontal as the polaron momentum ppp approaches from below the value at which the polaron excitation energy, E(p)\ensuremath−E(0)E(p)\ensuremath{-}E(0)E(p)\ensuremath−E(0), becomes equal to the optical-phonon energy. Using the weak-coupling method of the present paper, this conjecture has been verified to lowest order and next higher order in the coupling constant.
We present arguments to show that the energy-momentum relation for a polaron is quadratic for small kkk, but bends over and becomes horizontal when the energy approaches the optical-phonon energy above the ground state. We present a calculation for zero temperature based on a thermodynamic Green's-function theory which is especially designed to give a reliable polaron energy-momentum relation.
Abstract The energy E(p) of a polaron is studied as a function of temperature and momentum p with particular attention focused on the region where the excitation energy E(p) – E(0) closely approaches the energy of a longitudinal optical phonon. It is shown that in this region a quasi‐particle description is not adequate, e.g., the horizontal tangent in E(p) does not give rise to a singularity in the density of electronic states. The shape of the spectral weight function indicates that even for small coupling strength there is an internal excited state of the polaron.
The energy dispersion for arbitrary momentum of a polaron in n dimensions is calculated by the Rayleigh-Schr\"odinger perturbation theory and by a modified Brillouin-Wigner perturbation theory. For the energy and effective mass of the polaron ground state, both methods give results that agree with the results of V. V. Paranjape and P. V. Panjat [Phys. Rev. B 35, 2942 (1987)] to linear order in the polaron coupling constant \ensuremath{\alpha}. The self-energy shifts are progressively weakened with increasing dimension. In particular, this weakening occurs for the structure in the energy dispersion near the momentum value at which the polaron starts decaying via the emission of phonons.
An approximate calculation is made of the polaron self-energy and effective mass as defined by Frohlich.The approach taken is a variational one based on an ex tended version of Hohler's ansatz in which the trial state function employed is an exact eigenfunction of the total wave vector of the polaron.The effective mass obtained simulates the Feynmann-Schultz result while the corresponding polaron self-energy is fairly accurate but inferior to the result of Feynman and Schultz.CONTENTS Chapters « -« T -| > £ ( l ^' (1'2) n in which n is the optical refractive index, k is the static dielectric constant, e is the electronic charge and 2 t\& is Plank's constant.The limit S->°° is to be taken with lim I = -~ S-*oo v 8tt d 3v .(1,3) A natural unit system is employed in which 2^=o)=2m=l.The first term in the Hamiltonian is the quantum mechanical operator corresponding to the kinetic energy of the electron.The second term represents the energy of the polarization field resulting from the long wave length longitudinal optical crystal modes in the absence of interaction with the electron.The last term represents the energy of interaction between the electron and the phonons.The name "polaron" has been given to the entity consisting of the Bloch conduction electron together with its accompanying nonradiative polarization field.Accordingly Frohlich defines the polaron self-energy, EQ (a), and the (ground-state) polaron effective mass, y(a), by the expansion EQ (a,ic) = Ec (a) + k 2/y(a) + 0(k4 ), (1,4)
The next lowest order term of the ground state energy has its origin in lowering of frequencies of phonons localized around the polaron. An eigenvalue equation is derived, the solution of which gives phonon eigenmodes and frequencies in the presence of the polaron. Several eigenvalues are obtained by numerical integration. From the results, the ground state energy is estimated as E0/ℏω0=−0.108513α2−2.836+O(1/α2)E_{0}/\hbar\omega_{0}{=}-0.108513\alpha^{2}-2.836+O(1/\alpha^{2})E0​/ℏω0​=−0.108513α2−2.836+O(1/α2), where ω 0 is the frequency of the optical phonon and α is the electron-phonon coupling constant. The numerical value of the effective mass is 0.0227019α 4 .