Localization Length in Anderson Insulator with Kondo Impurities

The localization length, $\xi$, in a two-dimensional Anderson insulator depends on the electron spin scattering rate by magnetic impurities, ${\tau }_s^{-1}$. For antiferromagnetic sign of the exchange, the time ${\tau }_s$ is itself a function of $\xi$, due to the Kondo correlations. We demonstrate that the unitary regime of localization is impossible when the concentration of magnetic impurities, $n_{\mathrm{M}}$, is smaller than a critical value, $n_c$. For $n_{\mathrm{M}}>n_c$, the dependence of $\xi$ on the dimensionless conductance, $g$, is reentrant, crossing over to unitary, and back to orthogonal behavior upon increasing $g$. Sensitivity of Kondo correlations to a weak parallel magnetic field results in a giant parallel magnetoresistance.

Figure 1

(a) Dependence of the dimensionless spin scattering time on the localization length (in the units of $l$) as defined by Eqs. (3, 4) is depicted schematically for different concentrations of magnetic impurities, $n_{\mathrm{M}}$; (b) localization length, $\xi$, is plotted versus dimensionless conductance for $n_{\mathrm{M}}>n_c$. For $g<g_1$ and $g>g_2$ the straight line reflects the “orthogonal” behavior $\ln (\xi /l)=\pi g/2$. In the interval $g_1<g<g_{\mathrm{K}}$ localization length follows the crossover formula Eq. (2). Novel regime Eq. (9) due to the Kondo effect takes place within the interval $g_{\mathrm{K}}<g<g_2$. The inset shows the dimensionless localization length in a quasi-one-dimensional geometry, plotted schematically versus dimensionless conductance for $n_{\mathrm{M}}>{\tilde{n}}_c$.