Spin polaron

Throughout this book, particularly in the later chapters, we assume that a condensed electron gas can be treated as a Fermi liquid of pseudoparticles, for instance dielectric or spin polarons. We recognize that this is an unproved assumption. 

The conduction band of an antiferromagnetic non-metal spin polarons... 

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... 

Above the Curie or Neel point, a spin polaron will move by a diffusive process. A moment on the periphery of the polaron will reverse its direction in a time t (the relaxation time for a spin wave). Each time it does so, the polaron can be thought to diffuse a distance (a/R)3R, so the diffusion coefficient is... 

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). 

We shall consider in Chapter 9 the attractive interaction between two spin polarons, and the possible relationship to high-temperature superconductivity. 

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. 

There is also evidence for the existence of spin polarons in dilute magnetic semiconductors such as Cdx xMnxTe with x in the range 0-0.8. This is described in the review by Furdyna and Kossut (1988). 

The susceptibility should also increase if the coupling between spin polarons is ferromagnetic—or in other words because of the Zener coupling. The condition for ferromagnetism is that... 

The model of a degenerate gas of spin polarons suggests that if the direct or RKKY interaction between moments is weak and EF too great to allow ferromagnetism then the moments might all resonate between their various orientations. This would mean that it is possible in principle to have a heavily doped magnetic semiconductor or rare-earth metal in which there is no magnetic order, even at absolute zero. This possibility is discussed further in Section 8 in connection with the Kondo effect. 

Even when B U, this is still not large, and if jumps to nearest neighbours were allowed only on account of this term then the mass of a carrier at the extremities of the Hubbard bands would be much enhanced. We think, however, that a more important effect allowing the carrier to move to nearest neighbours is the formation of spin polarons. These will be considered in Section 4. 

As U/I (or U/B) goes to infinity, then so does R/a. Moreover, the energy of the spin polaron at rest tends to —%B. It follows that the bandwidth is B and, in the limit when U/B is large, is unaffected by the Hubbard U. 

Our estimate of the band form should therefore show strong peaks at the extremities due to the (heavy) spin polarons. For these, k is a good quantum number, as long as kR < 1. If this is not so, the polaron concept breaks down and we come into the region investigated by Brinkman and Rice (1970a) where the electron loses energy rapidly to spin excitations and k is a band quantum number. These authors estimate that the bandwidth contracts by about 70%. 

Our conclusion is, then, that near the transition the spin polaron spreads only to the nearest and perhaps next-nearest neighbours. 

Such spin polarons should not have a mass much greater than m in Si P. Moreover, they can pass freely from one atom to another, and are not impeded by the antiferromagnetic order. Thus the bandwidth of each Hubbard band should, we believe, still be of order 2zl, as it is for large values of U/B, and the equation... 

If the spin-polar on model is correct, we must describe the carriers in the antiferromagnetic semimetal formed when the two Hubbard bands overlap as a degenerate gas of spin polarons it should have the following properties. 

Since the strength of the coupling of the moments to the carriers is about r h2jma2, we deduce that, for a temperature above Tthe moments are free, giving entropy NkB In 2. If we think in terms of spin polarons, the polarons have broken up at this temperature. The excitations are mainly magnetic. The carriers will move through an array of disordered spins, and should behave like a non-degenerate gas with entropy... 

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. 

Thermopower measurements due to Kwizera et al (1981) are shown in Fig 6.23. The equation for the thermopower in the metallic state, S=( n2fciT/e)dlnff/d , should not be valid above about 150K it can be seen that d In narrow band in the metallic state, possibly due to Brinkman-Rice enhancement or formation of spin polarons. 

In some of the metal-insulator transitions discussed here the use of classical percolation theory has been used to describe the results. This will be valid if the carrier cannot tunnel through the potential barriers produced by the random internal field. This may be so for very heavy particles, such as dielectric or spin polarons. A review of percolation theory is given by Kirkpatrick (1973). One expects a conductivity behaving like... 

In the case discussed here a Mott transition is unlikely the Hubbard U deduced from the Neel temperature is not relevant if the carriers are in the s-p oxygen band, but if the carriers have their mass enhanced by spin-polaron formation then the condition B U for a Mott transition seems improbable. In those materials no compensation is expected. We suppose, then, that the metallic behaviour does not occur until the impurity band has merged with the valence band. The transition will then be of Anderson type, occurring when the random potential resulting from the dopants is no longer sufficient to produce localization at the Fermi energy. 

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. 

The form of the spin polaron has been calculated, following earlier work for small U, by Sehrieffer et al (1988). It is described as having a cigar shape, lying in the Cu02 planes, as shown in Fig, 9.2. Sehrieffer et al remark that any attractive... 

Fig. 9.1 Binding energy of a pair of spin polarons with (a) opposite and (b) parallel spins.

---