Polaron effective mass

Formulae similar to (6) have wide application to the movement of a system through an activated intermediate state, and will be applied to a simplified description of the Kondo effect in Chapter 3. For polarons the effect is to allow band motion at low temperatures with well-defined wavenumber k but enhanced effective mass mp given by... 

The prefactor is about 3 m is the effective mass in the undistorted lattice. The small polaron, for T<% will be scattered by phonons its mobility when plotted against T l will be as in Fig. 2.1. 

Up to this point we have discussed the formation of polarons in ionic crystals. Polarons of another type can also form in elements and other systems, such as the valence bands of alkali and silver halides, where the polarizability is not the relevant factor. In fact Holstein s (1959) original discussion of the small polaron was of this form. This kind of polaron is sometimes called a molecular polaron, and is illustrated in Fig, 2.3(a), and in Fig. 2.3(b) in the activated configuration of the atoms when the electron can move from one site to another. There is nothing analogous to the large polaron in this case in three-dimensional systems either a small polaron is formed or there is little effect on the effective mass from interaction with phonons. 

The energy E will necessarily have this minimum, but its value at this point can be positive or negative only in the latter case will a stable self-trapped particle (i.e. a small polaron) form. This is most likely to occur for large effective mass, and thus for holes in a narrow valence band or for carriers in d-bands. If the polaron is unstable then there is practically no change in the effective mass of an electron or hole in equilibrium in the conduction or valence band. 

A spin polaron should move at low temperatures with a fixed wave vector k, like any other pseudoparticle, and be scattered by phonons and magnons. The effective mass is expected to be of the form mey /0, where y l. To obtain this result, we compute the transfer integral when the polaron moves through one atomic distance. The spin will contribute a term proportional to... 

An alternative treatment of the effective mass of an antiferromagnetic spin polaron is given by Kasuya (1970), who also obtained in a one-dimensional model a high effective mass. The theory was also discussed by Nagaev (1971). 

Very direct evidence for the existence of bound spin polarons is provided by the work of Torrance et al (1972) on the metal-insulator transition in Eu-rich EuO At low temperatures, when the moments on the Eu ions are ferromagnetically aligned, the electrons in the oxygen vacancies cannot form spin polarons and are present in sufficient concentration to give metallic conduction. Above the Curie temperature the conductivity drops by a factor of order 10 , because the electrons now polarize the surrounding moments, forming spin polarons with higher effective mass. 

It is of course possible that a carrier in the conduction band or a hole in the valence band will form a spin polaron, giving considerable mass enhancement. The arguments of Chapter 3, Section 4 suggest that the effective mass of a spin polaron will depend little on whether the spins are ordered or disordered (as they are above the Neel temperature TN). This may perhaps be a clue to why the gap is little affected when T passes through TN. If the gap is U —%Bt -f B2 and Bt and B2 are small because of polaron formation and little affected by spin disorder, the insensitivity of the gap to spin disorder is to be expected. 

We have no direct evidence for the formation of spin polarons in any conductor, apart from gadolinium sulphide. The best evidence would be a decrease in the effective mass and increase in the conductivity of nonsuperconductors with magnetic field. 

In Kemeny and Rosenberg s model to explain the compensation rule (Kemeny and Rosenberg, 1970a, b) the tunnelling of small polarons (effective mass < I00me, where me is the free electron mass) is considered leading to a relationship between T0 and the Debye temperature. 

On the other hand, we have observed while studying even stoichiometries that JTD do not imply strict localization but, on the contrary, seem to survive rather easily in the metallic state. Trapping in CsC60 is then more probably due to defects than to a large polaron-like effective mass that would be intrinsic to a JT-dis-torted C q. One can then wonder about other types of influence of JTD in metallic compounds. 

The first numerical results from a strictly quantum mechanical calculation were given a few years ago [89]. In particular, P. Kornilovitch formulated a path integral representation of a three-dimensional JT polaron. Applying a QMC algorithm, he calculated the energy of the ground state, the DOS and the effective mass of a single... 

In the polaron problem (or the single-electron system coupled with phonons), both spin degrees of freedom and the electron-electron interaction as described by are irrelevant. The first work on the JT polaron was done by Hock et al. [91] on the E b system [92] which, unfortunately, possesses a too simple internal structure to provide qualitatively different features from those of the H (8> a system. Several works have treated the second simplest E (S> e system and found a quantitative difference in the polaron effective mass from that in the H (gia system [63,93-98]. The r (g) f JT polaron has also been studied and the difference from that in the i (g e system is revealed [99-101]. 

In concluding this section, we emphasize an amazing fact that the internal mathematical structure of the JT center determines the magnitude of the polaron effective mass. This implies that there will be an intrinsic difference in w between the manganese oxides Lai- Sr MnOs with eg electrons and the titanium ones Lai xSr cTi03 with t2g electrons, as may be observed by the difference in the transport mass or the T-linear coefficient in the low-temperature electronic specific heat Cv(T) [101]. The experimental result on Cv(T) obtained by Tokura s group [106] may be relevant to this issue. 

The formation of a bipolaron (or a bound pair of two polarons) is established, if the ground-state energy of the two-electron system is lower than twice the ground-state energy of a polaron. This issue has been studied rather intensively for the Holstein bipolaron [78], but it is not the case for the JT bipolaron. In [96], the electron-electron correlation function and the effective mass of an (gi e bipolaron was studied in one dimension in comparison with the corresponding results for the Holstein bipolaron [107]. In Fig. 5, we plot the phase diagram for the bipolaron formation, from which we find that the JT bipolaron is less stable than the Holstein one. 

In order to obtain the temperature dependence of rim, we consider the situation where small polarons coexist with large polarons of effective mass m [27], The equilibrium condition between large polarons and small polarons may be given by... 

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