Mass enhancement

These formulae can lead to a large mass enhancement of order 10, even if WH is of order so that hopping is unlikely to be observed even at high temperatures... 

We shall in this book use the concept of a degenerate gas of small—or at any rate heavy—polarons. Clearly we should not expect these to be formed unless the number of carriers is considerably less than the number of sites. We also remark, as mentioned earlier, that in all metals, at temperatures less than B phonons lead to a certain mass enhancement, of order less than 2. A treatment is given by Ashcroft and Mermin (1976, p. 520). This affects the thermopower some results for an amorphous alloy (Ca AlJ from Naugle (1984) are shown in Fig. 2.2. A theoretical treatment of the range between this situation and the polaron gas has not yet been given. 

The polaron radius, if greater than (2), will be very sensitive to m —or, more exactly, to the bandwidth of the undistorted lattice. A particularly striking effect is that in materials like NiO doped with lithium, where the carriers are Ni3 + ions and in which the hole moves from one Ni2+ ion to another. The mass enhancement for a free carrier is rather small (about 5), while a bound carrier hopping round the Li+ ion on the sites available to it behaves like a small polaron with an activation energy for motion (see Bosman and van Daal (1970) and Chapter 6 below). 

The Brinkman-Rice mass enhancement (2 ) 1 thus occurs, but E is the Fermi energy for the current carriers, which is proportional (in a crystalline lattice) to tj2f3 and may be written j2/3 f. So an enhancement by (2jj5/3) 1 is expected for the susceptibility, which could be of order 50, nearer to the large values observed. 

However, we do not think that this model is applicable to V203 and its alloys, chiefly because we do not think a conduction band necessarily lies near the lower Hubbard band, and it does not describe many important features, particularly the large mass enhancement (approximately 50) in the metallic phase. In our view, to understand the behaviour of such materials as the temperature is raised, and in... 

We now suppose that the carriers can form dielectric polarons, with mass enhancement up to mp, but no hopping. If so, our temperature 7i becomes T, where... 

Interaction of electrons with phonons, and the fact that the presence of a trapped electron can deform the surrounding material, allows the radius of an empty localized state to change when the state is occupied. Also, in a condensed electron gas phonons lead to a mass enhancement near the Fermi energy, or in some circumstances to polaron formation. For the development of the theory, and comparison with experiment, it is therefore desirable to begin by choosing a system where these effects are unimportant. The study of doped semiconductors provides such a system. This is because the radius aH of a donor is given, apart from central cell corrections, by the hydrogen-like formula... 

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. 

Figure 6.8 shows the reflectivity of crystals of V203 obtained by Fan (1972). From the low-frequency behaviour, Fan deduced m 9me since for a bandwidth /v 1 eV, mm me, the factor 9 represents the mass enhancement. The inflexion above 1 eV may be due to the transition to the unenhanced mass discussed in Chapter 4, Section 6. Results for V02 are also shown in Fig. 6.9 they give m 3me. 

We think that all these observations are to be explained by the assumption that metallic V203 is a highly correlated electron gas, as first suggested by Brinkman and Rice (1970b) and described in Chapter 4. The very low degeneracy temperature suggests that there may also be some mass enhancement of the carriers by polaron formation. Two electrons per atom would just half fill an ej band, so that the number of electron-like and hole-like carriers would be... 

Various authors (Fuchs 1965, Lightsey 1973, Webman et al 1976) have discussed the conductivity in terms of classical percolation theory, but we think this is unlikely to be correct for particles for which there is no evidence for mass enhancement. 

The assumption that the carriers are small or intermediate polarons in no way militates against discussions of ths band structure of the ground state (see e.g. Camphausen et al. 1972, Cullen and Callen 1971,1973). The absence of Jahn-Teller distortion (Goodenough 1971) also, in our view, indicates not the absence of a polaron mass-enhancement but rather a value.of V0jB not too far from the critical value. These conclusions seem to be in agreement with the considerations of Sokoloff (1972), who used a description in terms of a degenerate band of small polarons. Samara (1968) showed that pressure lowers the temperature of the Verwey transition. If this depended only on e2/ ca then the opposite should be the case. But pressure will increase B, and push the substance nearer to the critical value for the metal-insulator transition. 

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. 

Where strong-correlation fluctuations are present in an itinerant-electron matrix, the magnetic susceptibility may be interpreted as a coexistence of Curie-Weiss and mass-enhanced Pauli paramagnetism. 

G, whereas cell-free extracts acting on penicillin N had been stimulated by ATP [19], Increasing cell mass enhanced the concentration of DAOG formed from penicillin G, the optimum concentration being 19 mg/ml. Higher cell concentrations inhibited the reaction, probably because oxygen supply was limiting under such conditions. In the studies of Cho et al. [14], the buffer used for bioconversion was 50 mM Tris-HCl [Tris = tris(hydroxymethyl)aminomethane] at pH 7.4. It was later found that 50 mM MOPS buffer or 50 mM HEPES [N-(2-hydroxyethyl) piperazine-N -(2-ethanesulfonic acid] buffer at pH 6.5 improved activity [41]. 

Recent measurements and LDA-FPLO calculations performed by the groups of Wosnitza and Rosner, respectively, (Bergk et al., 2007) showed such a large mass-enhancement anisotropy up to a factor of five for YNi2B2C. 

As already discussed for / H-(ET)2l3, the question remains why with the ESR p-value of 2.007 [330] a bare electron mass /ib larger than pc would be extracted. If no mass-enhancement due to electron-phonon interaction is considered, i. e., A = 0, a lower limit of the g value of 2.25 would be obtained from the dHvA data for / -(ET)2lBr2. This might hint at antiferromagnetic fluctuations or an appreciable electron-electron interaction which... 

The authors in [351] neglected a possible mass-enhancement due to electron-phonon interaction and attributed the spin-splitting zeros to a p value of 1.55. 

Fig. 2 Inverse of the polaron mass enhancement factor, m/m, as a function of for the T (8) a (HP Holstein polaron) and the (8) e JT polaron. In the latter, the result in the infinite chain d = 1) is compared with that in the two-site system as weU as the analytic result in (26). The anti-adiabatic condition of (Uo// = 5 is assumed...

Fig. 3 Inverse of the mass enhancement factor, mim, as a function of g

Physically the polaron mass enhancement is brought about by the virtual excitation of phonons. In the H (g a Holstein model no restriction is imposed on exciting multiple phonons, implying that all the terms in Fig. lb for the vertex function contribute, while in the g e JT model, there is a severe restriction due to the existence of the conservation law intimately related to the 50(2) rotational symmetry in the pseudospin space. Actually, among the first- and second-order terms for the vertex function, only the term T2/ contributes, leading to the smaller polaron mass enhancement factor m jm than that in the Holstein model in which the correction r 1 is known to enhances m /m very much. In this way, the applicable range of the Migdal s approximation [48] becomes much wider in the g e JT system [63]. 

If the scattering mechanism of the electrons is T-independent, , is constant and S T) becomes proportional to T [5.103]. Many-body effects, on the other hand, may cause non-linearities at low temperatures due to electron-phonon mass-enhancement effects, giving S(T) = [1 + A(T)] Sb(T) [5.104]. Sb is now the bare thermopower of (5.19) without mass-enhancement. Higher correction terms have been proposed by Kaiser [5.105]. The resistivity should not be influenced by mass-enhancement effects [5.106]. 

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