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Physical review. B, Condensed matter

Polaron effective mass

Dinkar C. Khandekar, K.V. Bhagwat, S. V. Lawande

Published Feb 15, 1988• 19 citations
Publisher

Abstract

Adequacy of various definitions of polaron effective mass ${m}^{\mathrm{*}}$ is examined. The expression for ${m}^{\mathrm{*}}$ obtained by using the zero-temperature kernel is compared with the zero-temperature limit of the temperature-dependent effective mass obtained by employing the finite-temperature kernel. We find that consistent results are obtained when Saitoh's definition is employed. Our calculations are illustrated for a simple case of weak electron-phonon interaction.

DOI
10.1103/physrevb.37.3085

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The Properties of Effective Mass of Polaron

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With the improvement of experimental techniques and the replacement of equipment, composition and application of solid material have been gained a series new development. So there is continuing interests in polaron. N.Tokuda discussed that the variational law of the groundstate energy, the excitedstate energies with coupling constant; Xiao Jinglin, et al. investigated that the properties of polaron in various coupling strength; Xiao Wei, et al. studied that the affact of temperature on polaron; We discussed that the properties of acoustic deformation potential polaron. This paper was going to illustrate the variational law of the effective mass of polaron with temperature and coupling strength.In this paper, the variational relation of the effective mass of polaron in crystals with temperature and coupling strength was derived by using linearcombination operator and unitary transformation method. The results indicated that the effective mass increases with increasing coupling strength when temperature is constant, resulting in the interaction of electronphonon strengthens with increasing coupling strength. The results also illustrate that the effective mass varies with variation of temperature when coupling strength is constant, the stronger coupling strength is, the more obvious the affect of temperature on effective mass. At limit of weakcoupling,the effective mass of slowmoving polaron has nothing to do with temperature because a electron is approximate a freeelectron at limit of weakcoupling and it's motion is not affected by crystals vibration.In sum, the effective mass of polaron increases with increasing coupling strength when temperature is constant, and affect of temperature on effective mass also increases with increasing coupling strength.

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Abstract Exact perturbational results up to the second order in the coupling constant or the fourth order in the electron‐phonon interactions for the free energy, the inertial mass, and the magnetic mass of a polaron are calculated by use of the path‐integral method. New explicit analytical results are obtained for a Fröhlich polaron at finite temperatures, which in the zero temperature limit reduce to the known results.

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Ashok Sethia, Fumio Hirata, Yoshitaka Tanimura +1 more

We calculate the density matrix elements of the polaron using Feynman's variational method. The density matrix is diagonalized, and the eigenvalues and eigenfunctions are derived. These results and the calculated density matrix are used to evaluate the root-mean-square (rms) displacements ${R}_{\ensuremath{\beta}}$ and the effective mass (EM) ${m}^{*}$ for various values of coupling strengths between electron and medium and of the temperature. We find that EM is related to the rms displacement. The temperature dependence of the EM is in qualitative agreement with experiment.

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

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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

Temperature Dependence of Energy-band Width and Effective Mass for the Holstein Polaron

Dejun Li, Mi Xian-Wu

As ones know,the polaron is a quasiparticle formed by an electron and phonons.Due to Coulomb interaction,the electron will make the lattice deformation take place,meanwhile,the deformation potential(induced) by the lattice deformation will exert on the electron,the electron-phonon interaction influence doubtless on the state of motion of an electron,for example,the energy band and effective mass of an electron could change by the interaction,the electron can also be trapped by the deformable lattice,therefore,the research on the(polaron) is very important in understanding optical and transport properties of a solid. Studying electron-phonon interaction has been a main subject in condensed matter physics;in experiments,the influences of the temperature on the forbidden-band width of semiconductors have been observed very early,with the increase of the temperature,the forbidden-band width of most semiconductors will narrow,but a few will widen,these situations are closely relevant to the electron-phonon interaction.In particular,(recently),a large amount of experimental results,ranging from infrared spectroscopy to transport properties(involving) the colossal magnetoresistance and high-T_c superconductivity,has pointed out the importance of the electron-phonon interaction,which has renewed the interest in studying the models of the electron-phonon(coupled) system,especially,the Holstein polaron,for its relative simplicity,is the most considered model for the interaction of a single tight-binding electron coupled to phonons. In the past several decades,the Holstein poloron has been studied in detail in one-dimensional lattices,various theoretical methods,such as variational approaches,the Global-Local variational method,the(coherent)-state expansion method,have been developed to study features of the Holstein polaron including ground-state energy,polaron energy band,effective mass and so on.However,so far,the work considering the influences of the temperature on features of the Holstein polaron is little still. The main aim of this paper is to investigate the influences of the temperature on the energy-band width and the effective mass of the Holstein polaron by using the canonical transformation method.It is shown that,at certain coupling strength,the energy-band width of the Holstein polaron will narrow and the effective mass will augment with the increase of the temperature,in particular,when electron-phonon coupling strength is big enough,the energy-band width will become zero rapidly in small region of temperature,in fact,the situation is rapid transition of the polaron from the energy-band state to the self-trapping local state,this is an(interesting) result obtained in the paper.It is obvious that these features of the Holstein polaron are very useful in explaining optical and transport properties of a solid.

Chinese Journal of Luminescence•Jan 1, 2006•0 citations
Adequacy of various definitions of polaron effective mass m∗{m}^{\mathrm{*}}m∗ is examined. The expression for m∗{m}^{\mathrm{*}}m∗ obtained by using the zero-temperature kernel is compared with the zero-temperature limit of the temperature-dependent effective mass obtained by employing the finite-temperature kernel. We find that consistent results are obtained when Saitoh's definition is employed. Our calculations are illustrated for a simple case of weak electron-phonon interaction.
With the improvement of experimental techniques and the replacement of equipment, composition and application of solid material have been gained a series new development. So there is continuing interests in polaron. N.Tokuda discussed that the variational law of the groundstate energy, the excitedstate energies with coupling constant; Xiao Jinglin, et al. investigated that the properties of polaron in various coupling strength; Xiao Wei, et al. studied that the affact of temperature on polaron; We discussed that the properties of acoustic deformation potential polaron. This paper was going to illustrate the variational law of the effective mass of polaron with temperature and coupling strength.In this paper, the variational relation of the effective mass of polaron in crystals with temperature and coupling strength was derived by using linearcombination operator and unitary transformation method. The results indicated that the effective mass increases with increasing coupling strength when temperature is constant, resulting in the interaction of electronphonon strengthens with increasing coupling strength. The results also illustrate that the effective mass varies with variation of temperature when coupling strength is constant, the stronger coupling strength is, the more obvious the affect of temperature on effective mass. At limit of weakcoupling,the effective mass of slowmoving polaron has nothing to do with temperature because a electron is approximate a freeelectron at limit of weakcoupling and it's motion is not affected by crystals vibration.In sum, the effective mass of polaron increases with increasing coupling strength when temperature is constant, and affect of temperature on effective mass also increases with increasing coupling strength.
Using the exactly soluble Feynman polaron as a point of departure, I discuss and compare the properties of three projected, or momentum dependent, partition functions of a particle coupled to an environment and reconsider the anomalies that appear in some definitions of the effective mass. Although the momentum-dependent free energy associated with the projected partition function at fixed total momentum has exponentially suppressed finite-temperature corrections and yields the correct zero-temperature effective mass, the same is not necessarily true of the free energies associated with the simpler momentum-dependent partition functions derived from either the reduced density matrix of the particle or the density matrixlike quantity obtained by replacing in the particle's nonlocal effective action the phonon Green's function with its zero-temperature form.
We calculate the density matrix elements of the polaron using Feynman's variational method. The density matrix is diagonalized, and the eigenvalues and eigenfunctions are derived. These results and the calculated density matrix are used to evaluate the root-mean-square (rms) displacements ${R}_{\ensuremath{\beta}}$ and the effective mass (EM) m∗{m}^{*}m∗ for various values of coupling strengths between electron and medium and of the temperature. We find that EM is related to the rms displacement. The temperature dependence of the EM is in qualitative agreement with experiment.
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)
As ones know,the polaron is a quasiparticle formed by an electron and phonons.Due to Coulomb interaction,the electron will make the lattice deformation take place,meanwhile,the deformation potential(induced) by the lattice deformation will exert on the electron,the electron-phonon interaction influence doubtless on the state of motion of an electron,for example,the energy band and effective mass of an electron could change by the interaction,the electron can also be trapped by the deformable lattice,therefore,the research on the(polaron) is very important in understanding optical and transport properties of a solid. Studying electron-phonon interaction has been a main subject in condensed matter physics;in experiments,the influences of the temperature on the forbidden-band width of semiconductors have been observed very early,with the increase of the temperature,the forbidden-band width of most semiconductors will narrow,but a few will widen,these situations are closely relevant to the electron-phonon interaction.In particular,(recently),a large amount of experimental results,ranging from infrared spectroscopy to transport properties(involving) the colossal magnetoresistance and high-T_c superconductivity,has pointed out the importance of the electron-phonon interaction,which has renewed the interest in studying the models of the electron-phonon(coupled) system,especially,the Holstein polaron,for its relative simplicity,is the most considered model for the interaction of a single tight-binding electron coupled to phonons. In the past several decades,the Holstein poloron has been studied in detail in one-dimensional lattices,various theoretical methods,such as variational approaches,the Global-Local variational method,the(coherent)-state expansion method,have been developed to study features of the Holstein polaron including ground-state energy,polaron energy band,effective mass and so on.However,so far,the work considering the influences of the temperature on features of the Holstein polaron is little still. The main aim of this paper is to investigate the influences of the temperature on the energy-band width and the effective mass of the Holstein polaron by using the canonical transformation method.It is shown that,at certain coupling strength,the energy-band width of the Holstein polaron will narrow and the effective mass will augment with the increase of the temperature,in particular,when electron-phonon coupling strength is big enough,the energy-band width will become zero rapidly in small region of temperature,in fact,the situation is rapid transition of the polaron from the energy-band state to the self-trapping local state,this is an(interesting) result obtained in the paper.It is obvious that these features of the Holstein polaron are very useful in explaining optical and transport properties of a solid.