The varied relation of effective mass and mean number of phonon of polaron with coupling strength and magnetic field was derived by using the linear combination operator and unitary transformation methods.The numerical calculation indicates that the vibration frequency λ firstly falls and late increase with increasing coupling constant α,λ monotonous increases with increasing magnetic field strength;effective mass of polaron monotonous increases with increasing coupling strength and occurs resonance with varing magnetic field strength.In resonance point,because coupling between electron and crystal vibration is highest strength,effective mass of polaron achieves maximum.The numerical calculation also shows that mean number of phonon of polaron monotonous increases with increasing coupling strength, but decreases with increasing magnetic field strength when magnetic field strength is larger than magnetic strength of resonance point,since the farther distance from resonance point is,the weaker the interaction between electron and phonons is,conclusion is contrary when magnetic field strength is smaller than magnetic strength of resonance point.
The varied relation of effective mass and mean number of phonon of polaron with coupling strength and magnetic field was derived by using the linear combination operator and unitary transformation methods.The numerical calculation indicates that the vibration frequency λ firstly falls and late increase with increasing coupling constant α,λ monotonous increases with increasing magnetic field strength;effective mass of polaron monotonous increases with increasing coupling strength and occurs resonance with varing magnetic field strength.In resonance point,because coupling between electron and crystal vibration is highest strength,effective mass of polaron achieves maximum.The numerical calculation also shows that mean number of phonon of polaron monotonous increases with increasing coupling strength, but decreases with increasing magnetic field strength when magnetic field strength is larger than magnetic strength of resonance point,since the farther distance from resonance point is,the weaker the interaction between electron and phonons is,conclusion is contrary when magnetic field strength is smaller than magnetic strength of resonance point.
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. LeeLowPines calculated the groundstate energy of polaron by a variational technique. Huybrechts calculated the energy of the groundstate and the first internal excited state of the optical polaron for different values of the electronphonon coupling. N.Tokuda studied the dependence of groundstate 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 groundstate energy of the twodimension magnetoplaron by using forthorder perturbationtheoretic method. Wei et al. investigated the cyclotronresonance mass and frequency of the interface polaron by using Feyman path integration method. Recently, Catandella et al. studied the properties of polaron in onedimension 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 groundstate 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 groundstate energy, selftrapping energy and Landau energy with coupling constant and cyclotronresonance frequency are derived by using linearcombination 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 cyclotronresonance frequency ωc;the groundstate energy E0 decrease with increasing α and monotonous rises with increasing ωc;the selftrapping energy E0tr increases with increasing α and ωc;Landau energy E0L firstly increases to a maximum and then fall with increasing α;E0L firstly rises to maximum also and then decreases with increasing ωc.
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 groundstate energy, the excitedstate energies with coupling constant; Xiao Jinglin, 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 linearcombination 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 electronphonon 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 weakcoupling,the effective mass of slowmoving polaron has nothing to do with temperature because a electron is approximate a freeelectron at limit of weakcoupling 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.
In most cases, as the strength of electron-phonon coupling increases, the effective mass of polarons typically increases. However, in this paper, we uncover a fascinating phenomenon: the presence of light polarons even within the strong coupling regime, where electron-phonon coupling includes both Holstein and bond types (electron-phonon coupling occurring on the hopping bonds). To investigate this, we employ a diagrammatic Monte Carlo method based on the path-integral formulation of the particle sector and the Fock-state representation of the phonon sector. Our study centers on analyzing the impact of bond electron-phonon coupling on the Holstein polaron's essential properties, including its effective mass, ground-state energy, and the average number of phonons. We examine two distinct scenarios: one where the phonon frequency of both Holstein and bond types is the same, and another where the phonon frequency of the Holstein type is twice as large as that of the bond type. In both of these cases, when the Holstein coupling falls within the light mass regime, we observe minimal changes in the effective mass as a function of bond coupling, compared to the case of the bare bond polaron. However, the behavior of the effective mass undergoes a significant change in the heavy mass regime. Most intriguingly, we discover a nonmonotonic dependence of the effective mass on the bond coupling gB when the Holstein coupling gH falls within the heavy mass regime. This finding holds promise, particularly in the context of bipolarons, where achieving a light effective mass and a large binding energy to get a compact size is crucial.
In this paper,the properties of the mean number of optical phonons of the polaron in a polyatomic crystals are studied.The mean number of optical phonons of strong and weak coupling polaron in a polyatomic crystals were derived respectively,by using linear combination operator and unitary transformation methods.For weak-coupling,we obtain the usual result that the mean number of phonons is the sum of the contribution of different electron-phonon branches coupling.For strong-coupling,there is an extra contribution due to crossed terms between the different phonon branches.
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.