Spin versus charge noise from Kondo traps

Luis G. G. V. Dias da Silva and Rogério de Sousa

Phys. Rev. B 92, 085123 – Published 14 August 2015

Abstract

Magnetic and charge noise have a common microscopic origin in solid-state devices, as described by a universal electron trap model. In spite of this common origin, magnetic (spin) and charge noise spectral densities display remarkably different behaviors when many-particle correlations are taken into account, leading to the emergence of the Kondo effect. We derive exact frequency sum rules for trap noise and perform numerical renormalization-group calculations to show that while spin noise is a universal function of the Kondo temperature, charge noise remains well described by single-particle theory even when the trap is deep in the Kondo regime. We obtain simple analytical expressions for charge and spin noise that account for Kondo screening in all frequency and temperature regimes, enabling the study of the impact of disorder and the emergence of magnetic $1 / f$ noise from Kondo traps. We conclude that the difference between charge and spin noise survives even in the presence of disorder, showing that noise can be more manageable in devices that are sensitive to magnetic (rather than charge) fluctuations and that the signature of the Kondo effect can be observed in spin noise spectroscopy experiments.

Received 30 March 2015
Revised 21 July 2015

DOI:https://doi.org/10.1103/PhysRevB.92.085123

Authors & Affiliations

Luis G. G. V. Dias da Silva

Instituto de Física, Universidade de São Paulo, C. P. 66318, 05315–970 São Paulo, SP, Brazil

Rogério de Sousa

Department of Physics and Astronomy, University of Victoria, Victoria, British Columbia, Canada V8W 2Y2

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Vol. 92, Iss. 8 — 15 August 2015

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Images

Figure 1
Charge noise as a function of frequency for the trap in the symmetric case with $ε_{d} = - U / 2$ . NRG calculations are shown to be well approximated by a mean-field Hartree-Fock decomposition (HF) even when $U / Γ$ is large and the trap is deep in the Kondo regime. This shows that charge noise is well described by single-particle excitations.
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Figure 2
Spin noise as a function of frequency for the trap in the symmetric case with $ε_{d} = - U / 2$ . The NRG results agree with HF only at $U = 0$ . As $U$ increases, NRG shows that the magnetic noise increases, developing a peak at $ω \approx T_{K}$ . In contrast, the single-particle contributions described by HF decrease dramatically as $U$ increases. This shows that magnetic noise is dominated by many-body processes.
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Figure 3
Universal scaling for spin noise in the Kondo regime for $ε_{d} = - U / 2$ . (a) and (b) NRG results for spin noise $S_{s} (ω)$ . Note how all the curves collapse into a single scaling relation when the noise is written as a function of $ω / T_{K}$ . For $ω ≲ T_{K}$ , the magnetic noise scales linearly with $ω$ (Ohmic noise), and for $T_{K} ≲ ω < U$ it decreases with an anomalous power of frequency $\propto 1 / [ω {ln}^{2} (ω / T_{K})]$ . For $ω > U$ , spin noise is cut off $\propto 1 / ω^{2}$ . (c) and (d) NRG results for charge noise $S_{c} (ω)$ do not show universal Kondo scaling and behave just like the single-particle approximation (HF) with noise peaked at $ω \approx M a x {Γ, U}$ with a smooth cutoff $1 / ω^{2}$ at $ω > U$ .
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Figure 4
Comparison of the spin noise fit $S_{s}^{Fit} (ω)$ [Eqs. (14) and (18) with $α = 3$ ; lines] with the NRG results (symbols) at $T = 0$ .
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Figure 5
Spin noise in the presence of trap disorder. The calculated noise for $N$ traps was averaged according to the prescription $Γ = Γ_{0} e^{- λ}$ , with $λ$ being the tunneling distance between the trap and the Fermi gas uniformly distributed in the interval $[0, λ_{\max}]$ . This gives rise to the broad distribution of Kondo temperatures shown in Eq. (22). The resulting noise, shown here for $κ = 10$ and $λ_{\max} = 5$ , displays $1 / f$ behavior over a frequency range that decreases as the temperature increases. This is in contrast to the temperature-independent charge $1 / f$ noise described in the literature [6].
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Figure 6
NRG data for the spin noise $S_{s} (ω)$ at $T = 0$ for $U = 40 Γ$ calculated with DM-NRG and CFS procedures. Inset: A check of the spin sum rule given by Eq. (A1) for the different approaches shows that the CFS fulfils the spin sum rule apart from numerical integration errors.
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