Energy-resolved inelastic electron scattering off a magnetic impurity

We study inelastic scattering of energetic electrons off a Kondo impurity. If the energy $E$ of the incoming electron (measured from the Fermi level) exceeds significantly the Kondo temperature $T_K$, then the differential inelastic cross section $\sigma (E,\omega )$, i.e., the cross section characterizing scattering of an electron with a given energy transfer $\omega$, is well defined. We show that $\sigma (E,\omega )$ factorizes into two parts. The $E$ dependence of $\sigma (E,\omega )$ is logarithmically weak and is due to the Kondo renormalization of the effective coupling. We are able to relate the $\omega$ dependence to the spin-spin correlation function of the magnetic impurity. Using this relation, we demonstrate that in the absence of the magnetic field, the dynamics of the impurity spin causes the electron scattering to be inelastic at any temperature. At temperatures $T$ low compared to the Kondo temperature $T_K$, the cross section is strongly asymmetric in $\omega$ and has a well-pronounced maximum at $\hslash \omega \sim T_K$. At $T⪢T_K$, the dependence $\sigma$ vs $\omega$ has a maximum at $\omega =0$; the width of the maximum exceeds $T_K∕\hslash$ and is determined by the Korringa relaxation time of the magnetic impurity. Quenching of the spin dynamics by an applied magnetic field results in a finite elastic component of the electron scattering cross section. The differential scattering cross section may be extracted from the measurements of relaxation of hot electrons injected in conductors containing localized spins.

Figure 1

(Color online) Differential cross section, $\sigma (E,\omega )$, at large temperatures, $T⪢T_K$, without Zeeman splitting, $B=0$, as given by Eq. (19). The Lorentzian peaks have a width given by the Korringa relaxation rate $1∕{\tau }_K$, Eq. (17).

Figure 2

(Color online) NRG result for $\sigma (E,\omega )$ (solid line) on a logarithmic scale at $T=0$ without Zeeman splitting, $B=0$. A maximum at finite $\omega =T_K$ develops and scattering with small energy transfer $\omega$ is suppressed. Whereas, the high-frequency tail is perturbatively accessible, see Eq. (21) (short-dashed line), the low-frequency tail, Eq. (25) (long-dashed line), is a property of the strong coupling fixed point described by Nozières’ Fermi liquid theory. ($W=0.413\dots$. is Wilson’s number.) The inset shows the temperature correction according to Eq. (25); the contribution for negative $\omega$ is exponentially small for temperatures $0<T⪡T_K$.

Figure 3

(Color online) Weight of the elastic scattering cross section, ${\sigma }_{\mathrm{el}}\left(E\right)=\int d\omega {\sigma }_{\mathrm{el}}(E,\omega )$, see Eq. (32), determined by NRG. The weight increases as $B^2$ for small magnetic fields and saturates logarithmically slowly to the limiting value for large $B$, see Eq. (43).

Figure 4

(Color online) NRG result for $\sigma (E,\omega )$ at $T=0$ for magnetic fields $g{\mu }_{B}B<T_K$. The difference of the curves indicates the scattering weight which for $B>0$ is transfered from the inelastic to the elastic component leading to a delta-function peak at $\omega =0$, as sketched in the inset.

Figure 5

(Color online) Inelastic scattering cross section for several magnetic field values, $B⪢T_K$, at $T=0$ according to Eq. (44), which is applicable unless $\omega ⪢B$. Note that the position of the resonance is shifted away from the Zeeman energy as described by Eq. (30). The inset shows the comparison of the analytical result (44) (dotted lines) with NRG calculation (colored lines). The low-frequency tail matches nicely. However, the NRG overestimates the width of the peak.

Figure 6

(Color online) Experimental setup of Pothier et al.[6] with an additional tunnel barrier which limits the injection of hot electrons into the wire. The voltage drop across the wire is denoted by $U$. The voltage applied at the probing contact is $V$.

Figure 7

Experimental quantum dot setup. A wire leads to a quantum dot in the Kondo regime (indicated by the arrow), which is in addition weakly connected to another reservoir at a voltage $U$. The resonance-level (RL) quantum dot acts as a probe.

Figure 8

(Color online) NRG comparison of the low-frequency asymptote of ${\chi }_{zz}^{″}$ with the prediction of the generalized Shiba relation, Eq. (A6) (dotted lines). The inset shows the numerically evaluated magnetization whose derivative enters Eq. (A6).

Figure 9

(a) First-order Knight-shift diagram; (b) two-loop self-energy correction; (c) one-loop vertex correction. The wiggled line represents the propagator of the Abrikosov fermions and the solid line, the electron propagator. The dot signifies the Kondo interaction.