Impurity scattering

The total electrical resistance at room temperature includes tire contribution from scattering of conduction electrons by the vacancies as well as by ion-core and impurity scattering. If the experiment is repeated at a number of high temperarnre anneals, then the effects of temperarnre on tire vacancy conuibu-tion can be isolated, since the other two terms will be constant providing that... 

The lattice may be distorted because of several reasons as vacancies, interstitials, dislocations and impurities. These lattice defects cause the so-called impurity scattering which produces the term i ei. At low temperatures, i ei is the constant electronic thermal resistance typical of metals. 

For lattice acoustic-mode deformation potential scattering, s =, giving r = /8 = 1.18. For ionized-impurity scattering, s = —f, giving rn0 = 315 /512 = 1.93. For a mixture of independent scattering processes we must... 

Consider a mixture of acoustic-mode (rL) and ionized-impurity (r,) scattering. For tL t, we would expect r 0 = 1.18 and for r, tl, rn0 = 1.93. But for intermediate mixtures, r 0 goes through a minimum value, dropping to about 1.05 at 15% ionized-impurity scattering (Nam, 1980). For this special case (sL = i, s, = — f), the integrals can be evaluated in terms of tabulated functions (Bube, 1974). For optical-mode scattering the relaxation-time approach is not valid, at least below the Debye temperature, but rn may still be obtained by such theoretical methods as a variational calculation (Ehrenreich, 1960 Nag, 1980) or an iterative solution of the Boltzmann equation (Rode, 1970), and typically varies between 1.0 and 1.4 as a function of temperature (Stillman et al., 1970 Debney and Jay, 1980). 

Fig. 20. (a) Temperature dependence of the upper critical field calculated within a two-band model for several impurity scattering rates yjmp (cm-1). (b) calculated Hc2(0)-vi.-ylmp curve illustrating the transition from the clean to the dirty limit. Dotted line Hc2(0)-y,mp dependence in the dirty limit. (Drechsler et al. 2000 Fuchs... 

The proportionality factor is the electron mobility, xn, which has units of square centimeters per volt per second. The mobility is determined by electron-scattering mechanisms in the crystal. The two predominant mechanisms are lattice scattering and impurity scattering. Because the amplitude of lattice vibrations increases with temperature, lattice scattering becomes the dominant mechanism at high temperatures, and therefore, the mobility decreases with increasing temperature. 

Theory predicts that the mobility decreases as T 3/2 because of lattice scattering (8). But because electrons have higher velocities at high temperatures, they are less effectively scattered by impurities at high temperatures. Consequently, impurity scattering becomes less important with increasing temperature. Theoretical models predict that the mobility increases as T3/2/nj, in which nx is the total impurity concentration (8). The mobility is related to the electron diffusivity, Dn, through the Einstein relation... 

Summary. We study how the single-electron transport in clean Andreev wires is affected by a weak disorder introduced by impurity scattering. The transport has two contributions, one is the Andreev diffusion inversely proportional to the mean free path i and the other is the drift along the transverse modes that increases with increasing . This behavior leads to a peculiar re-entrant localization as a function of the mean free path. 

Here we report how the single electron transport in Andreev wires at low temperatures T weak disorder introduced by impurity scattering assuming that inelastic processes are negligible. The Andreev wire is clean in the sense that the mean free path is much longer than the wire diameter, 3> a. 

The multiband superconductivity shows up only in the clean limit , where the single electron mean free path for the interband impurity scattering satisfies the condition / > hvF/Aav where vF is the Fermi velocity and Aav is the average superconducting gap [24,28,30,35],... 

The criterium that the mean free path should be larger than the superconducting coherence length must be met. This is a very strict condition that implies also that the impurity interband scattering rate yab should be very small yah (1/2 )(KB/ft)Tc. Therefore most of the metals are in the dirty limit where the interband impurity scattering mixes the electron wave functions of electrons on different spots on bare Fermi surfaces and it reduces the system to an effective single Fermi surface. 

The critical temperature Tc obtained by aluminium [143], carbon [183,184] and scandium [177] substitutions are reported in Fig. 12. The universal scaling of the critical temperature Tc, versus z show the negligible effects of impurity scattering in different substitutions because of the suppression of interband impurity scattering in the superlattice. 

These results show that the two gaps behaviour is present over all the range of the reduced Lifshitz parameter -0.8< z <+0.8. These results support the predictions that the doped materials remain in the clean limit for interband pairing although the large number density of impurity centres. This results falsifies the predictions of the multigap suppression because of impurity scattering due to substitutions. 

Impurity scattering leads to the formation of the gapless state at some sectors zero energy, N(0), where N(0) is the 2D density of states per spin at the Fermi level, and leads to a universal quasiparticle interlayer conductivity crg(0,0) ... 

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