Low-temperature dephasing saturation from elastic magnetic spin disorder and interactions

Abstract

We treat the question of the low temperature behavior of the dephasing rate of the electrons in the presence of elastic spin disorder scattering and interactions. In the frame of a self-consistent diagrammatic treatment, we obtain saturation of the dephasing rate in the limit of low temperature for magnetic scattering, in agreement with the non-interacting case. The magnitude of the dephasing rate is set by the strength of the magnetic scattering rate. We discuss the agreement of our results with relevant experiments.An important quantity in disordered electronic systems is the dephasing rate τ −1 φ . It provides a measure of the loss of coherence of the carriers, but in the two-particle channel -c.f. eq. (1) below. Decoherence arises from coulombic interactions, scattering by phonons, magnetic fluctuations etc. The saturation of the dephasing rate at low temperature T seen in numerous experiments 1-13 has attracted a vigorous interest, especially given the longstanding theoretical prediction for a vanishing τ −1 φ as the temperature T → 0. 14-21 . Previous theoretical studies 14-25 have focused on the calculation of τ −1 φ in the absence of spin-scattering disorder. The majority of these studies predict, correctly, a vanishing τ −1 φ (T → 0) . Here we determine and calculate the factors which contribute to dephasing in the presence of spin-scattering disorder. The saturation obtained allows for the consistent elucidation of this puzzle.In the presence of spin-less disorder, the cooperon (particle-particle diffusion correlator -c.f. fig. 1) is given byD is the diffusion coefficient, N F is the density of states at the Fermi level and τ −1 the total impurity scattering rate. We work in the diffusive regime ǫ F τ > 1 (h = 1), ǫ F being the Fermi energy. With spin-disorder present, the cooperon becomes spin-dependent. The relevant terms C i are shown in fig. 1. We start by giving the explicit form of these C o 0,1,2 without a dephasing rate. C o 0 acquires a finite spin-dependent term in the denominator, which is crucial for the determination of the dephasing rate -c.f. below.For the case , τ −1 S > 0, τ −1 so = 0 -with τ −1 S the magnetic impurity scattering rate and τ −1 so the spin-orbit impurity scattering rate -the cooperons are given byBased on eqs. (2), we expect a saturation of the dephasing rate. The simple diffusion pole is "cut-off" by the constant terms proportional to τ −1 S . On symmetry grounds, the spin-conserving Coulomb interaction cannot eliminate these terms.We emphasize that the impurity scattering considered is elastic, bulk-type. Interfacial impurity scattering, though similar to bulk-type, is expected to differ in detail.To calculate the dephasing rate, we write down and solve the appropriate coupled equations for all three renormalized cooperons C i (q, ω), i = 0, 1, 2. We note that usually the terms containing the factors d i and h i below are completely omitted. The equations are shown schematically in fig. 2:

Fermi liquid behaviour in weakly disordered metals close to a quantum critical point

“…The combination (Dq 2 + r) in V (q, ω) is considered here for the first time. The factor Dq 2 indicates disorder induced diffusion of the quasi-particles, with diffusion coefficient D [14,21]. This factor could also originate from antiferromagnetic damping, but this origin would not be consistent with our treatment of ferromagnetic fluctuations -cf.…”

“…below. The factor r may originate from the inadvertent presence of elastic magnetic impurities in the samples [21], and should be equal to the relevant scattering rate τ −1 S . It can also originate from the fermiology of a clean system (without disorder).…”

“…Then, via Cauchy's residue theorem, we obtain W = T ωm P (q, iω m ) = − ∞ −∞ dω π n B (ω) Im P (q, ω). The Bose distribution function n B (ω) = 1/(e (ω/T ) − 1) is commonly approximated [21] as n B (ω) = (T /ω) θ(T − |ω|). We first carry out the ω integration and then the integration over the momentum q in 2 and 3 D.…”

Conductivity of Weakly Disordered Metals Close to a “Ferromagnetic” Quantum Critical Point

“…H, as in the systems of interest mentioned below, like a = h s , h = |H/H c − 1|, s > 0, where H c is the critical field. The factor Dq 2 indicates disorder induced diffusion of the quasiparticles, with diffusion coefficient D 14,17 . For the purpose of our calculations, we will treat ξ and a as independent parameters.…”

Dephasing time in a two-dimensional electron Fermi liquid

“…Among the current opinions on the matter, it has been suggested that the saturated value should depend on the specific sample geometry, 9 the level of disorder in the sample, 10 the microscopic qualities of the defects, 11,12 or e-e scattering mediated by the magnetic exchange interaction. 13 Others argue that the saturation is caused by extrinsic mechanisms, such as magnetic impurities 14 and magnetic spin-spin scattering, 15 hot electron effects, 16 electromagnetic noise sources, 3 or non-equilibrium effects. 17 The possible extrinsic mechanisms urge caution when determining the actual temperature of the twodimensional electron system and ensuring low external radiation is small.…”