Matthiessen rule

The resistivity of the narrow band materials displays not only an anomalous temperature dependence, but also the resistivity values themselves are extremely high for metallic systems (several hundred gfi cm). The different contributions (e-e, e-p, scattering by impurities) to the resistivity can no longer be taken as additive (the Matthiessen rule is not valid), because the mean free path can be even as short as the interatomic spacing, and this limit is actually approached in some cases. 

Splitting of the transport coefficients into the various contributions is, under certain circumstances, possible using the sum rules , although these rules can only be applied in rather limited cases. In the following we will discuss these sum rules, which are the Matthiessen rule, the Kohler rule and the Nordheim-Gorter rule. 

In the scope of the Boltzmann equation, the Matthiessen rule is valid if the total collision operator can by written as a sum of the collision operators for the different scattering mechanisms. This means that the scattering processes can be treated as independent of each other. 

For the thermopower no such a simple relation as the Matthiessen rule exists. The main reason is that the thermopower (and also the Peltier effect) is a higher -order transport phenomenon. This can be seen from eq. (18), where the matrix element is the dominating quantity. However, it can be shown that within the scope of the variational procedure, if the variational space is two dimensional, the so-called Kohler rule follows, which is ... 

The Kohler rule or the Nordheim-Gorter rule can be used in the same way as the Matthiessen rule to separate the different contributions to the total thermopower S. ... 

It is interesting to present these results schematically. In fig. 1 the temperature variation of po, Pph and Pmag is schematically depicted. Assuming validity of the Matthiessen rule, the total resistivity (p) is given by the full line in that picture. 

The temperature dependence of the electrical resistivity of some AnPt compounds, as measured by Hill and Elliott (1971), is shown in fig. 23. The temperature dependence of the resistivity of UPt has also been studied by Lawson (1970), who found that p(T) can be described at low temperatures by the formula p = po + AT with A = 450nQcm/K. Frings and Franse (1985) obtained an A-value of 190nQcm/K. On the assumption of the Matthiessen rule (eq. (32)), the spin-dependent scattering contribution to the total resistivity can be determined by subtracting p T) of the non-magnetic ThPt. The p ,ag-value of UPt is of the order of about lOO pQcm just... 

Assuming validity of the Matthiessen rule, the electronic thermal resistivity = 1/2.J of a non-magnetic (indicated in formulas by the superscript nm) RAI2 compound can be written as ... 

It should be noted that the carrier mobility in nanowires is lower than that in bulk single-crystalline material due to possible scattering at wire and grain boundaries, uncontrolled impurities, and lattice defects. The overall effect of this additional scattering is taken into account by Matthiessen s rule (Ashcroft and Mermin, 1976b),... 

Figure 26 Normalized longitudinal resistivity p, of a single crystal of (TMTSF)2C104 at 4.2 K versus concentration of irradiation-induced defects (mole %). Initially, linear behavior is observed, corresponding to Matthiessen s rule, followed by an exponential behavior corresponding to Eqs. (11) and (12) (see the text). (From Ref. 112.)...

Figure 27 Transverse resistivity p, versus T2 for (TMTSF)2C104 doped with small quantities of Re04. There is evidence for a T2 law and for Matthiessen s rule. The anomaly associated with ordering of the C104 anions at 24 K is also visible. (From Ref. 104.)...

We shall confine ourselves to a test of formula (9) with particularly pure specimens of copper, gold, and tungsten (Tables II-IV). As usual r denotes the ratio of the resistances at temperatures T and 273 2° jST. The observed values of r are used to calculate the ideal resistance by Matthiessen s rule. In the case of copper... 

The electrical properties of SnO films have been reported by Ehlich and coworkers [177], with high and low mobility samples discussed separately. In high mobility films, the overall mobilitiy can be described by the Matthiessen s rule... 

Matthiessen s rule - The statement that the electrical resistivity p of a metal can be written as p = P +P. where p is due to scattering of conduction electrons by lattice vibrations and p to scattering by impurities and imperfections. If the impurity concentration is small, p. is temperature independent. 

We begin by assuming that the resistivity of non-magnetic RI compounds obeys Matthiessen s rule. This rule states that the temperature dependence of the resistivity is given by... 

Even though deviations from Matthiessen s rule are known to occur in the presence of grain-boundary scattering, this expression can be used as a good approximation to understand the relative importance of the various effects that influence the resistivity in narrow Cu lines. Figure 2.3 shows the variation of the resistivity as a function of Unewidth for the various components and pxotal-... 

In practice, metal lattices are not perfect. Lattice imperfections may include missing atoms, dislocations, and impurities. These imperfections also scatter the charge carriers. It turns out that usually the contributions of thermal vibrations and lattice imperfections can be added up. This is known as Matthiessen s Rule. 

In The presence of additional non-magnetic scattering centers Matthiessen s rule holds only approximately. For details we refer to Fulde and Peschel (1972). 

In the following sections the basic concepts of the transport theories within the scope of the Boltzmann formalism are described. In the succeeding discussion the validity and the applicability of the Matthiessen and the Nordheim-Gorter rule are considered. A description of transport phenomena (electrical and thermal resistivity, thermopower) without an external magnetic field is given in sect. 2.3. There, the influence of the various scattering mechanisms on the temperature dependence of the electrical and the thermal resistivity as well as on the thermopower will be briefly discussed. 

In numerous low- or intermediate-concentration alloys, it has been discovered that Pi and pg are independent. Evidence in support of this property, known as Matthiesseris rule, is the parallelism frequently noted among the p(c) curves for members of an alloy series. Naturally, Matthiessen s rule breaks down when the presence of solute begins to influence pi through its effect on /i( p) and 0p or for other reasons such as ... 

Nximerous Ti-SM and Ti-TM alloys exhibit negative temperatiare coefficients of resistivity. Such gross departures from Matthiessen s rule require detailed knowledge of the electronic structures, and/or the phonon spectra of the alloys concerned, for their explanations. 

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