Matthiessen’s rule

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 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. 

To a reasonable approximation the impurity resistivity term is independent of temperature, and thus appears merely as an additive constant (Matthiessen s rule). The same is roughly true of the magnetoresistance term pg. The intrinsic resistivity P-, however, varies markedly with temperature, especially at low temperatures. 

The t)q)ical behavior of the resistivity of a metal with temperature is shown in Figure 18.1 and can be expressed by Matthiessen s rule, which can be stated as... 

For temperatures above 20% of the Debye temperature, the resistivity increases linearly with temperature as suggested by Matthiessen s rule. The slope a is pretty much the same for a given metal, regardless of the defects and impurities. The first-order dependence of resistivity on T can be understood from a simple model. The collision cross section between an electron and the core ions in a metal is proportional to the square of the amplitude of its vibration. The square of the amplitude of an oscillator is proportional to its thermal energy kT (assuming T> d, so all of the modes are active). The collision time t is directly proportional to the mean free path A, which is inversely proportional to the collision cross section. Therefore, conductivity is inversely proportional to T. 

Note that the temperature dependence in the Lorenz number has nothing to do with the temperature dependence of resistivity (Matthiessen s rule) that affects the collision time r because this effect is eliminated when taking the ratio of the conductivities. Instead this T originates with the first power dependence of the electronic heat capacity with temperature. 

The resistivity p = l/tr is found to have a linear relationship with temperature in the form p T) = po + aT (Matthiessen s rule), where po is due to collisions with structural and impurity imperfections and the temperature dependence comes about from collisions with the lattice ions whose cross sections increase linearly with temperature. Impurity atoms, such as foimd in solid solution alloys, produce a much larger increase in resistivity than structural defects such as dislocations and grain boundaries or condensed second phases because they are more widely dispersed. Also there is a departure from the linear temperature dependence of the resistivity at low temperatures because all of the phonon modes are active. Griineisen used the Debye theory to develop a universal relationship between reduced resistivity and reduced temperature that holds for all metals. 

Matthiessen s rule gives the total electrical resistivity Pm of (dilute) alloys as ptot = Po + X + Z... 

We see that taking this Xei in (1) gives an expression equivalent to Matthiessen s rule as in (9). It is instructive to express the electron scattering rate not in terms of scattering times x but in the corresponding mean free paths = vx, where v is some properly defined electron velocity at the Fermi energy. (In a model with a spherical Fermi surface, v is the Fermi velocity Vp.) Then the equivalent of Eq. (10) is... 

Matthiessen s rule— for a metal, total electrical resistivity equals the sum of thermal, impurity, and deformation contributions Matthiessen s rule... 

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