Energy relaxation due to magnetic impurities in mesoscopic wires: Logarithmic approach

The transport in mesoscopic wires with large applied bias voltage has recently attracted great interest by measuring the energy distribution of the electrons at a given point of the wire, in Saclay. In the previous theoretical works of different authors extensive numerical calculations were presented, but the relative importance of the different effects was hidden. The main goal of the present paper is to break up the calculation and show how the final result is formed step by step. In the diffusive limit with negligible energy relaxation the energy distribution shows two sharp steps at the Fermi energies of the two contacts. Those steps are, however, broadened due to the energy relaxation which is relatively weak if only a Coulomb electron-electron interaction is assumed. In some of the experiments the broadening is more essential, reflecting an anomalous energy relaxation rate proportional to $E^{-2}$ instead of $E^{-3∕2}$ where $E$ is the energy transfer. Later it has been suggested that such a relaxation rate can be due to the electron-electron interaction mediated by Kondo impurities which is, as has been known for a long time, singular in $E$. In the present paper the magnetic-impurity-mediated interaction is systematically studied in the logarithmic approximation valid above the Kondo temperature. In the case of large applied bias voltage Kondo resonances are formed at the steps of the distribution function and they are narrowed by increasing the bias. An additional Korringa energy broadening occurs for the spins by creating electron-hole pairs in the electron gas out of equilibrium. That smears the Kondo resonances, and the renormalized coupling can be replaced by a smooth but essentially enhanced average coupling and that enhancement can reach the value 8–10. Thus the experimental data can be described by formulas without logarithmic Kondo corrections, but with enhanced coupling. In certain regions of large bias, that averaged coupling depends weakly on the bias. In those cases the distribution function depends only on the ratio of the electron energy and the bias, showing scaling behavior. The impurity concentrations estimated from those experiments and other dephasing experiments can be very different and a possible explanation is also mentioned.

Figure 1

The wire of length $L$. The arrow points to the position $x$ of the attached tunnel junction which is measured in units of $L$.

Figure 2

The diagrams used for calculating (a) the kernel and (b) the Korringa lifetime of the impurity spin. The solid lines denote the conduction electrons, the dotted lines the impurity spin, and the blob is the Kondo coupling.

Figure 3

The renormalized coupling (top) calculated from the sharp double-step distribution function (bottom) at $x=0.5$; $T=0.01\mathrm{K}$, $T_K=0.15\mathrm{K}$ (${\rho }_0{J}_0=0.039$, ${\mathcal{D}}_0=5\times {10}^4\mathrm{K}$): (a) $U=0\mathrm{mV}$, ${\tau }_K=\infty$. (b) $U=0.15\mathrm{mV}$. Solid line: ${\tau }_K=\infty$. Dotted line: $\hslash ∕2{\tau }_K=0.25\mathrm{K}$. (c) $U=0.3\mathrm{mV}$. Solid line: ${\tau }_K=\infty$. Dotted line: $\hslash ∕2{\tau }_K=0.25\mathrm{K}$.

Figure 4

The renormalized coupling (top) calculated from the sharp double-step distribution function (bottom) at $x=0.5$; $T=0.01\mathrm{K}$, $T_K=0.3\mathrm{K}$ (${\rho }_0{J}_0=0.042$, ${\mathcal{D}}_0=5\times {10}^4\mathrm{K}$): (a) $U=0\mathrm{mV}$, ${\tau }_K=\infty$. (b) $U=0.15\mathrm{mV}$. Solid line: ${\tau }_K=\infty$. Dotted line: $\hslash ∕2{\tau }_K=0.4\mathrm{K}$. (c) $U=0.3\mathrm{mV}$. Solid line: ${\tau }_K=\infty$. Dotted line: $\hslash ∕2{\tau }_K=0.4\mathrm{K}$.

Figure 5

The coupling constant (top) and the distribution function (bottom) obtained from the self-consistent calculation using the leading-logarithmic renormalized coupling (solid line) and an appropriate constant coupling ${\rho }_0\overline{J}$ (dotted line). (a) $U=0.15\mathrm{mV}$, ${\rho }_0\overline{J}=0.35$ and (b) $U=0.3\mathrm{mV}$, ${\rho }_0\overline{J}=0.23$. The other parameters are ${\tau }_D=1.8\mathrm{ns}$, $c=4\mathrm{ppm}$, $T_K=0.15\mathrm{K}$, ${\rho }_0=0.21∕\left(\text{site}\mathrm{eV}\right)$, $S=1∕2$, and $T=0.05\mathrm{K}$.

Figure 6

The dependence of the averaged, approximating coupling ${\rho }_0\overline{J}$ on the voltage bias for $T_K=0.15\mathrm{K}$. The other parameters are ${\tau }_D=1.8\mathrm{ns}$, $c=4\mathrm{ppm}$, ${\rho }_0=0.21∕\left(\text{site}\mathrm{eV}\right)$, and $T=0.05\mathrm{K}$.

Figure 7

The dependence of the distribution functions on the impurity concentration is illustrated using the appropriate constant couplings obtained (see Fig. 5). (a) $U=0.15\mathrm{mV}$, ${\rho }_0\overline{J}=0.35$ and (b) $U=0.3\mathrm{mV}$, ${\rho }_0\overline{J}=0.23$. The other parameters are ${\tau }_D=1.8\mathrm{ns}$, $T_K=0.15\mathrm{K}$, ${\rho }_0=0.21∕\left(\text{site}\mathrm{eV}\right)$, and $T=0.05\mathrm{K}$.

Figure 8

Comparison of the solutions of the self-consistent calculation with and without the cross terms in the kernel, Eq. (14). (a) The distribution functions with (solid line) and without the cross terms (dotted line) and (b) The difference between the two solutions. ${\tau }_D=1.8\mathrm{ns}$, $c=4\mathrm{ppm}$, ${\rho }_0=0.21∕\left(\text{site}\mathrm{eV}\right)$, $S=1∕2$, $x=0.5$, $T=0.05\mathrm{K}$, $T_K=0.3\mathrm{K}$, and $U=0.3\mathrm{mV}$.

Figure 9

The distribution functions are illustrated where the Korringa relaxation rates are modified by a multiplying factor; $\gamma =\lambda S(S+1)$. (a) $U=0.1\mathrm{mV}$ and (b) $U=0.3\mathrm{mV}$. The other parameters are ${\tau }_D=2.8\mathrm{ns}$, $c=5\mathrm{ppm}$, ${\rho }_0=0.21∕\left(\text{site}\mathrm{eV}\right)$, $T_K=0.3\mathrm{K}$, and $T=0.05\mathrm{K}$.

Figure 10

The coupling constant (top) and the distribution function (bottom) obtained from the self-consistent calculation at different positions in the wire. The other parameters are $U=0.3\mathrm{mV}$, ${\tau }_D=1.8\mathrm{ns}$, $c=4\mathrm{ppm}$, ${\rho }_0=0.21∕\left(\text{site}\mathrm{eV}\right)$, $T_K=0.15\mathrm{K}$, and $T=0.05\mathrm{K}$.

Figure 11

The coupling constant (top) and the distribution function (bottom) obtained from the self-consistent calculation for different biases. (a) $x=0.25$ and (b) $x=0.5$. The other parameters are ${\tau }_D=1.8\mathrm{ns}$, $c=4\mathrm{ppm}$, $T_K=0.15\mathrm{K}$, ${\rho }_0=0.21∕\left(\text{site}\mathrm{eV}\right)$, and $T=0.05\mathrm{K}$.

Figure 12

The two-electron and one-pseudofermion scattering amplitude $\Gamma$. The solid and dotted lines represent the electrons and the pseudofermion, respectively. One of the electron lines connects continuously the upper corners in the amplitude $\Gamma$ and the other the lower corners.

Figure 13

The two basic diagrams providing singular terms in the energy transfer. The solid and dotted lines represent the electrons and the pseudofermion, respectively. The square box represents the logarithmic electron-pseudofermion vertex correction $J\left(\omega \right)$ and the dotted line the position of the energy denominator.

Figure 14

The two forward-scattering diagrams coming in the golden rule. The solid and dotted lines represent the electrons and the pseudofermion, respectively. The square boxes represents the appropriate vertexes (${\Gamma }_1$ or ${\Gamma }_2$), and the arrows show from where the imaginary parts are taken.