Origin of the anomalously weak temperature dependence of the magnetic dephasing rate of Kondo impurities in AgFe wires below $T/T_K^{\gamma}\sim 0.1$

Measurements of the magnetic contribution $\gamma_m$ to the electron dephasing rate in Ag wires containing a few ppm of Fe Kondo impurities show a universal scaling with $T/T_K^{\gamma}$ that matches exact numerical-renormalization-group calculations for spin-1/2 single-channel Kondo impurities down to $T/T_K^{\gamma}\approx 0.1$, but below that scale — down to $0.01 T_K^{\gamma}$ — the measured rate decreases only very weakly with temperature instead of showing the predicted Fermi-liquid $T^2$ collapse. The anomaly is proportional to the implanted Fe dose, is reproduced in independent samples, and survives the exclusion of underscreened and overscreened impurity models and of implantation-induced two-level-system defects, yet its microscopic origin has not been identified. The problem is to determine which physical mechanism — a minority population of Fe at defect or grain-boundary sites with strongly reduced local Kondo temperature, or multichannel screening of the nominal spin-2 Fe impurity including crystal-field and spin-orbit effects — produces the anomalously weak low-temperature magnetic dephasing rate, quantitatively and consistently with the established constraints.

Background

In dilute magnetic alloys, a small concentration of magnetic impurities dramatically shortens the phase-coherence time $\tau_\varphi$ of conduction electrons — the timescale over which an electron maintains a well-defined quantum phase. This matters in mesoscopic physics because weak localization (a quantum-interference correction to the conductance of disordered conductors), Aharonov–Bohm oscillations (magnetoconductance oscillations set by the magnetic flux through a ring), and universal conductance fluctuations (sample-specific aperiodic conductance fluctuations of order $e^2/h$) all require phase coherence. In a quasi-one-dimensional wire, $\tau_\varphi(T)$ is extracted by fitting the low-field magnetoresistance to weak-localization theory; in a clean wire it is limited by electron–electron interactions ($\tau_\varphi^{-1}\propto T^{2/3}$) and electron–phonon scattering ($\propto T^3$), so that $\tau_\varphi$ should diverge as $T\to 0$.

A magnetic impurity of spin $S$ in a metal hosts the Kondo effect: below a characteristic Kondo temperature $T_K$ the impurity spin is progressively screened by the conduction electrons, and for a single screening channel and $S=1/2$ the screening is complete, producing a Fermi-liquid ground state in which inelastic scattering dies out as $T\to 0$. Zaránd, Borda, von Delft and Andrei (2004) showed that the inelastic scattering cross section of a Kondo impurity — and hence its contribution $\gamma_m=1/\tau_m$ to the dephasing rate measured in a weak-localization experiment — can be computed exactly from impurity correlation functions, and Micklitz, Altland, Costi and Rosch (2006) evaluated it using the numerical renormalization group (NRG, Wilson's iterative method for quantum impurity models). For spin-1/2 single-channel impurities at low concentration the magnetic dephasing rate per impurity is a universal curve $\gamma_m=(n_S/\nu)\,f(T/T_K)$ with $n_S$ the impurity concentration and $\nu$ the density of states: linear in $T$ for $0.1T_K\lesssim T\lesssim T_K$, crossing over below $0.1T_K$ to Fermi-liquid behavior $\gamma_m\propto T^2$ (or $T^{3/2}$ once weak-localization corrections to the magnetic rate itself are included, as computed by Vavilov, Glazman and Larkin (2003)). Experiments on AuFe wires (Schopfer et al. 2003; Bäuerle et al. 2005) confirmed the linear regime but could not reach far below $0.1T_K$ because $T_K^{\mathrm{Au/Fe}}\approx 0.5$ K.

Mallet et al. (2006) exploited AgFe, where the Kondo temperature in the dephasing-rate convention is $T_K^{\gamma}\approx 4.3$ K: they implanted 2.7–67.5 ppm of Fe into ultra-pure quasi-one-dimensional Ag wires by focused-ion-beam implantation (dosing selected segments of the wire with a scanned ion beam) and measured $\tau_\varphi$ down to 40 mK $\approx 0.01T_K^{\gamma}$. They found that $\gamma_m$ per impurity is a universal function of $T/T_K^{\gamma}$, in remarkable agreement with the $S=1/2$ single-channel NRG theory down to $0.1T_K^{\gamma}$ — and an anomalously weak residual temperature dependence below, where the data lie increasingly above the predicted $T^2$ collapse. Independent measurements by Alzoubi and Birge (2006) on AgFe (2 and 10 ppm) found the same flattening below $T/T_K\approx 0.1$. Several candidate explanations have been excluded. Single-channel underscreened models — a spin $S>1/2$ screened by one channel, which leaves a residual moment with large low-temperature $\gamma_m$ — predict a much broader maximum and are inconsistent with the data for all $1/2\le S\le 5/2$; overscreened models — multichannel models with more conduction-electron screening channels than needed to screen the spin ($n_c>2S$), which likewise do not reach a standard Fermi-liquid ground state — overpredict low-temperature dephasing; and implantation-induced dynamical defects such as two-level systems (TLS, atomic tunneling defects that dephase electrons with little temperature dependence) were excluded by a control experiment implanting nonmagnetic Ag$^+$ ions under identical conditions (Capron et al. 2008). Since $\gamma_m$ scales perfectly with the implanted Fe dose, the anomaly is intrinsic to the Fe impurities — whose nominal configuration, spin $S=2$ coupled through up to five d-orbital channels with unequal couplings split by crystal fields and spin–orbit coupling, is not captured by the single-channel $S=1/2$ model that otherwise works.

Problem Statement

Identify the microscopic origin of the anomalously weak temperature dependence of the magnetic dephasing rate $\gamma_m$ (the magnetic-impurity contribution to the electron dephasing rate, per impurity) observed in AgFe Kondo wires for $T/T_K^{\gamma}\lesssim 0.1$ (down to $0.01T_K^{\gamma}$), where the measured $\gamma_m$ lies increasingly above the spin-1/2 single-channel NRG prediction ($\gamma_m\propto T^2$, or $T^{3/2}$ including weak-localization corrections) even though the same theory is quantitatively accurate above $0.1T_K^{\gamma}$. A complete answer is either (a) a quantitative microscopic theory of dephasing by realistic Fe impurities in Ag — spin $S=2$ coupling through up to five channels with unequal couplings, including crystal-field and spin–orbit effects, and/or a distribution of local Kondo temperatures in which a small fraction of Fe at defect or grain-boundary sites has a much lower local $T_K$ — that reproduces the measured universal $\gamma_m(T/T_K^{\gamma})$ across the full measured range — with agreement taken to mean that the theory curve and the measured $\gamma_m$ lie within the experimental scatter over the entire measured range, including the sub-$0.1T_K^{\gamma}$ regime and not merely above $0.1T_K^{\gamma}$ — and that, rather than merely fitting the curve, determines which of the two mechanisms named in the objective is responsible, by showing that the alternative mechanism fails to account for the same data or, if both contribute, by quantifying their relative contributions; or (b) an experimental identification of the impurity population or mechanism responsible that likewise establishes it as the dominant source of the anomaly. Either form of answer must be quantitatively consistent with the established constraints: the universality of $\gamma_m$ per impurity as a function of $T/T_K^{\gamma}$, its proportionality to the implanted magnetic dose, the exclusion of underscreened single-channel ($S\le 5/2$) and overscreened impurity models, and the exclusion of implantation-induced dynamical defects.

References

1. F. Mallet, J. Ericsson, D. Mailly, S. Ünlübayir, D. Reuter, A. Melnikov, A. D. Wieck, T. Micklitz, A. Rosch, T. A. Costi, L. Saminadayar, C. Bäuerle, 'Scaling of the Low-Temperature Dephasing Rate in Kondo Systems', Phys. Rev. Lett. 97, 226804 (2006), doi:10.1103/PhysRevLett.97.226804; arXiv:cond-mat/0607154, https://doi.org/10.1103/PhysRevLett.97.226804
2. G. M. Alzoubi, N. O. Birge, 'Phase Coherence of Conduction Electrons Below the Kondo Temperature', Phys. Rev. Lett. 97, 226803 (2006), doi:10.1103/PhysRevLett.97.226803; arXiv:cond-mat/0607168, https://doi.org/10.1103/PhysRevLett.97.226803
3. T. Capron, Y. Niimi, F. Mallet, Y. Baines, D. Mailly, F.-Y. Lo, A. Melnikov, A. D. Wieck, L. Saminadayar, C. Bäuerle, 'Low-temperature dephasing in irradiated metallic wires', Phys. Rev. B 77, 033102 (2008), doi:10.1103/PhysRevB.77.033102; arXiv:0711.1810, https://doi.org/10.1103/PhysRevB.77.033102
4. T. Micklitz, A. Altland, T. A. Costi, A. Rosch, 'Universal Dephasing Rate due to Diluted Kondo Impurities', Phys. Rev. Lett. 96, 226601 (2006), doi:10.1103/PhysRevLett.96.226601; arXiv:cond-mat/0509583, https://doi.org/10.1103/PhysRevLett.96.226601
5. G. Zaránd, L. Borda, J. von Delft, N. Andrei, 'Theory of Inelastic Scattering from Magnetic Impurities', Phys. Rev. Lett. 93, 107204 (2004), doi:10.1103/PhysRevLett.93.107204, https://doi.org/10.1103/PhysRevLett.93.107204
6. C. Bäuerle, F. Mallet, F. Schopfer, D. Mailly, G. Eska, L. Saminadayar, 'Experimental Test of the Numerical Renormalization-Group Theory for Inelastic Scattering from Magnetic Impurities', Phys. Rev. Lett. 95, 266805 (2005), doi:10.1103/PhysRevLett.95.266805, https://doi.org/10.1103/PhysRevLett.95.266805
7. F. Schopfer, C. Bäuerle, W. Rabaud, L. Saminadayar, 'Anomalous Temperature Dependence of the Dephasing Time in Mesoscopic Kondo Wires', Phys. Rev. Lett. 90, 056801 (2003), doi:10.1103/PhysRevLett.90.056801; arXiv:cond-mat/0212420, https://doi.org/10.1103/PhysRevLett.90.056801
8. T. Micklitz, T. A. Costi, A. Rosch, 'Magnetic field dependence of dephasing rate due to diluted Kondo impurities', Phys. Rev. B 75, 054406 (2007), doi:10.1103/PhysRevB.75.054406; arXiv:cond-mat/0610304, https://doi.org/10.1103/PhysRevB.75.054406
9. L. Borda, L. Fritz, N. Andrei, G. Zaránd, 'Theory of inelastic scattering from quantum impurities', Phys. Rev. B 75, 235112 (2007), doi:10.1103/PhysRevB.75.235112; arXiv:cond-mat/0703293, https://doi.org/10.1103/PhysRevB.75.235112
10. S. Kettemann, E. R. Mucciolo, 'Disorder-quenched Kondo effect in mesoscopic electronic systems', Phys. Rev. B 75, 184407 (2007), doi:10.1103/PhysRevB.75.184407; arXiv:cond-mat/0609279, https://doi.org/10.1103/PhysRevB.75.184407
11. M. G. Vavilov, L. I. Glazman, A. I. Larkin, 'Electron transport and energy relaxation in dilute magnetic alloys', Phys. Rev. B 68, 075119 (2003), doi:10.1103/PhysRevB.68.075119, https://doi.org/10.1103/PhysRevB.68.075119
12. L. Saminadayar, P. Mohanty, R. A. Webb, P. Degiovanni, C. Bäuerle, 'Electron coherence at low temperatures: The role of magnetic impurities', Physica E 40, 12 (2007), doi:10.1016/j.physe.2007.05.026; arXiv:0709.4663, https://doi.org/10.1016/j.physe.2007.05.026
13. C. Bäuerle, P. Degiovanni, L. Saminadayar, 'Quantum coherence and magnetic scattering', Int. J. Nanotechnol. 7, 403 (2010), doi:10.1504/IJNT.2010.031727, https://doi.org/10.1504/IJNT.2010.031727
14. F. Mallet, F. Schopfer, J. Ericsson, L. Saminadayar, C. Bäuerle, D. Mailly, G. Eska, 'Dephasing in Kondo systems: comparison between theory and experiment', in Controllable Quantum States (World Scientific, 2008), pp. 221–226, doi:10.1142/9789812814623_0035, https://doi.org/10.1142/9789812814623_0035