Heikes formula

He also uses Heikes formula for the thermopower (Heikes and Ure 1961),... 

Extensions of the Heikes formula have been proposed taking into account the other possible origins of entropy which are linked to the spins and orbitals. First, by considering a system containing mixed-valent cations ]V[( )+/M( +i)+ with spin values S and S +i, an extra entropy term coming from spins can be added and ... 

The Heikes formula is valid only at high T, and other calculations including the transport term have been performed at low T (for example, for NaxCo02 ) which show the dominant part of the Heikes formula on the measured thermopower. 

For localised carriers, the Heikes formula states that ... 

Theoretical calculations have been performed for these tzg systems, using the Hubbard model. In this paper, the whole Seebeck coefficient, and not only the high temperature limit, has been calculated. It was shown that, taking into account the spin and orbital terms, the Heikes formula has to be modified by adding an extra contribution ... 

In Figure 4.11, the values of S at 300 K are reported as a function of x. Also shown is the evolution of S x) calculated with the Heikes formula, and with the spin and orbital degeneracy calculated by Marsh et The data are perfectly fitted by the Marsh and Parris formula,showing that an enhancement of thermopower can be obtained by optimising the spin and orbital degeneracies. Combining the resistivity and Seebeck... 

Figure 4.11 S as a function of carrier doping in Pri Ca cCr03, calculated from the Heikes formula (dashed line) or taking into account the spin and orbital degeneracy (solid line)...

One of the major tasks for the development of thermoelectric materials is to find and optimise the thermoelectric properties of both n- and p-type legs. As explained in the Introduction, in the case of oxides, the ZT of n-type materials are always smaller, and it is therefore important to find new ways to generate n-type materials. From the simple Heikes formula, with only the x dependence, and no spin and orbital degeneracy term, it can be shown that n- and p-type materials should be symmetrically obtained by doping an insulator with a small amount of carriers or holes (Figure 4.12). 

Figure 4.12 S calculated from the Heikes formula for electron or hole doping...

One of the most important questions to optimise the figure of merit of these materials is to understand the evolution of S vs T, correlate the value of S at 300 K with doping, and compare this value with the Heikes formula. To investigate the doping effect, several misfit compounds have been synthesised and characterised. 

Optimisation of the Seebeck Coefficient by Doping Following the Heikes formula or the generalised Heikes formula, S should increase when the Co concentration decreases. Following the electroneutrality equation between the two different sublattices, the Co valency in the Cdl2 plane can be written as... 

The introduction of Ti leads to a semiconducting-like behaviour for the substituted compound. More interestingly, in the substituted sample, the Seebeck coefficient has been increased from 120 /iV/K to 165 juV/K. It should be noted that even if the resistivities are strongly modified by the substitution, the Seebeck coefficients present the same T dependence, only shifted in magnitude. This enhancement of S at 300 K shows that the increase in a. by the introduction of Ti in the rock-salt layer has a positive impact as expected from the Heikes formula. 

Before concluding, it should be noted that the Heikes formula should be used also only if a plateau is reached for the S(T) curve. This is not always the case as an increase in S can be observed for some misfits or in Na cCoOi. This increase has not been investigated in detail so far, and it would be very important to understand if this is really an intrinsic increase or if it is related to some oxygen reduction as T increases, depending on the measuring atmosphere. 

In the following, we will show that beyond the Heikes formula, other contributions to thermopower can be added, giving more information on the nature of the carriers in these materials and the origin of correlations. 

This Heikes formula [73] assumes weak coulombic interactions [72] it is used by Nagels [74] in polaronic situations where thermoelectric power is nearly temperature-independent this case involves a weak vibrational energy transfer during hopping and temperature independence for the small polaron density. If N is the number of available sites during hopping and n is the number of carriers, S becomes [72]... 

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