Influence of Rashba spin-orbit coupling on the Kondo effect

An Anderson model for a magnetic impurity in a two-dimensional electron gas with bulk Rashba spin-orbit interaction is solved using the numerical renormalization group under two different experimental scenarios. For a fixed Fermi energy, the Kondo temperature $T_K$ varies weakly with Rashba coupling ${\lambda }_R$, as reported previously. If instead the band filling is low and held constant, increasing ${\lambda }_R$ can drive the system into a helical regime with exponential enhancement of $T_K$. Under either scenario, thermodynamic properties at low temperatures $T$ exhibit the same dependencies on $T/T_K$ as are found for ${\lambda }_R=0$. Unlike the conventional Kondo effect, however, the impurity exhibits static spin correlations with conduction electrons of nonzero orbital angular momentum about the impurity site. We also consider a magnetic field that Zeeman splits the conduction band but not the impurity level, an effective picture that arises under a proposed route to access the helical regime in a driven system. The impurity contribution to the system's ground-state angular momentum is found to be a universal function of the ratio of the Zeeman energy to a temperature scale that is not $T_K$ (as would be the case in a magnetic field that couples directly to the impurity spin), but rather is proportional to $T_K$ divided by the impurity hybridization width. This universal scaling is explained via a perturbative treatment of field-induced changes in the electronic density of states.

Figure 1

Schematic plots of (a) the dispersion relations ${\epsilon }_h\left(k\right)$ and (b) the densities of states per helicity channel ${\rho }_h\left(\epsilon \right)$, in the presence of Rashba SO interaction. The middle curve in (a) represents the dispersion $\epsilon \left(k\right)$ in the absence of Rashba interaction. In (b), the combined density of states $\rho \left(\epsilon \right)$ (dashed line) is constant and equal to its no-Rashba value ${\rho }_0\left(\epsilon \right)$ throughout the energy range ${\epsilon }_0<\epsilon <D_{-}$.

Figure 2

Helicity-resolved conduction-band filling fractions ${\eta }_{\pm }$ and overall filling fraction $\eta =({\eta }_{+}+{\eta }_{-})/2$, plotted as functions of Rashba energy ${\epsilon }_R$ for ${\epsilon }_0=-0.08D$ and (a) fixed Fermi energy ${\epsilon }_F=0$, (b) constant filling fraction $\eta =-{\epsilon }_0/(D-{\epsilon }_0)\approx 0.074$.

Figure 3

Schematic plots of the effective angular-momentum-resolved densities of states ${\rho }_{\tau }\left(\epsilon \right)$ obtained from Eq. (43) for (a) weak Zeeman splitting ${\epsilon }_Z<2{\epsilon }_R$, (b) strong Zeeman splitting ${\epsilon }_Z>2{\epsilon }_R$. In both panels, ${\rho }_{\tau }\left(\epsilon \right)$ is constant and equal to its no-Rashba value ${\rho }_0\left(\epsilon \right)$ throughout the energy range ${\epsilon }_0+{\epsilon }_Z<\epsilon <D_{-}$.

Figure 4

Impurity contribution to (a) temperature times magnetic susceptibility $T{\chi }_{\mathrm{imp}}$, and (b) entropy $S_{\mathrm{imp}}$, both plotted vs scaled temperature $T/T_K$ for five different cases: a no-Rashba reference (black solid line), and Rashba energies ${\epsilon }_R/D=0.04$ (red and orange curves) and 0.08 (green and blue curves) under the scenarios of fixed Fermi energy (dotted lines) and constant band filling (dashed lines). The two dotted lines lie directly on top of one another on the scale of this plot. The Kondo temperature $T_K$ is as defined in Eq. (44). Data are for ${\epsilon }_0=-0.08D$, $U=-2{\epsilon }_d=0.1D$, and $\mathrm{\Gamma }=0.005D$.

Figure 5

(a) Scaled Kondo temperature $T_K/T_K^0$ vs Rashba energy ${\epsilon }_R$ for fixed Fermi energy ${\epsilon }_F=0$ and for different values of ${\epsilon }_d$ expressed in the legend in units of $D$. Symbols represent NRG data and lines are the result of the perturbative treatment described in the text. (b) Perturbative shifts ${\tilde{\epsilon }}_d-{\epsilon }_d$ and $\tilde{U}-U$ in the effective impurity parameters vs ${\epsilon }_R$ for the same cases shown in (a). Data are for ${\epsilon }_0=-0.08D$, $U=0.1D$, and $\mathrm{\Gamma }=0.005D$.

Figure 6

Kondo temperature for a constant band-filling fraction $\eta \simeq 0.074$: (a) $T_K/T_K^0$ vs ${\epsilon }_R$ for different values of ${\epsilon }_d$ and $U$ expressed in the legend in units of $D$; (b) results for ${\epsilon }_R\ge 0.04D$ plotted as $-1/ln(T_K/D)$ vs $\rho \left[{\epsilon }_F\left({\epsilon }_R\right)\right]/{\rho }_0\left(0\right)$. Data are for ${\epsilon }_0=-0.08D$ and $\mathrm{\Gamma }=0.005D$.

Figure 7

Static correlations between the impurity spin and the total angular momentum in different conduction-band channels, calculated for (a), (c) fixed Fermi energy ${\epsilon }_R=0$, or (b), (d) constant band filling $\eta \approx 0.074$. The upper panels show $\langle {\mathbf{S}}_d·{\mathbf{J}}_h\rangle$ for helicity $h=\pm$, while the lower panels plot the correlation of ${\mathbf{S}}_d$ with ${\mathbf{J}}^{m=0}$ (the total angular momentum of all electrons with orbital angular momentum $m=0$) and with ${\mathbf{J}}^{m\ne 0}$ defined in Eq. (49). Data are for ${\epsilon }_0=-0.08D$, $U=-2{\epsilon }_d=0.1D$, and $\mathrm{\Gamma }=0.005D$.

Figure 8

(a) Zero-temperature impurity polarization $M_{\mathrm{imp}}$ vs Zeeman energy ${\epsilon }_Z$ for a system with fixed Fermi energy ${\epsilon }_F=0$. Data are for ${\epsilon }_0=-0.08D$, $U=-2{\epsilon }_d=0.15D$, and the values of $\mathrm{\Gamma }$ and ${\epsilon }_R$ labeled on the plot in units of $D$. (b) Same data as in (a), replotted vs $2f\mathrm{\Gamma }{\epsilon }_Z/T_{K}D$, where $f=2.4$ and 1.8 for ${\epsilon }_R/D=0$ and 0.1, respectively.

Figure 9

Quantity $f$ entering the scaling collapse of the impurity polarization for a system with fixed Fermi energy ${\epsilon }_F=0$: (a) $f$ vs ${\epsilon }_R$ for different values of $U$ and ${\epsilon }_d$ expressed in the legend in units of $D$. Each symbol was obtained via an NRG calculation of the compensating local field for $\mathrm{\Gamma }{\epsilon }_Z/T_{K}D=0.1$, while the lines represent algebraic results based on Eq. (50). (b) NRG values of $f$ vs $\mathrm{\Gamma }{\epsilon }_Z/T_{K}D$ for $U=0.15D$ and different values of ${\epsilon }_R$ and ${\epsilon }_d$ expressed in the legend in units of $D$. Lines are guides to the eye showing that $f$ is independent of ${\epsilon }_Z$ for $\mathrm{\Gamma }{\epsilon }_Z/T_{K}D\lesssim 10$. Data in both panels are for ${\epsilon }_0=-0.08D$ and $\mathrm{\Gamma }=0.005D$.

Figure 10

(a) Zero-temperature impurity polarization $M_{\mathrm{imp}}$ vs Zeeman energy ${\epsilon }_Z$ for constant band filling $\eta \simeq 0.074$. Data are for ${\epsilon }_0=-0.08D$, $U=-2{\epsilon }_d=0.15D$, and the values of $\mathrm{\Gamma }$ and ${\epsilon }_R$ labeled on the plot in units of $D$. (b) Same data as in (a), replotted vs $2f\mathrm{\Gamma }{\epsilon }_Z/T_{K}D$ with values of $f$ shown in the legend. In this case, $f$ depends on $\mathrm{\Gamma }$ as well as ${\epsilon }_R$.

Figure 11

Quantity $f$ entering the scaling collapse of the impurity polarization for a system with constant band filling $\eta \simeq 0.074$: (a) $f$ vs ${\epsilon }_R$ for different values of $U$ and ${\epsilon }_d$ expressed in the legend in units of $D$. Each symbol was obtained via an NRG calculation of the compensating local field for $\mathrm{\Gamma }{\epsilon }_Z/T_{K}D=0.1$, while the lines are guides to the eye. (b) NRG values of $f$ vs $\mathrm{\Gamma }{\epsilon }_Z/T_{K}D$ for $U=0.15D$ and different values of ${\epsilon }_R$ and ${\epsilon }_d$ expressed in the legend in units of $D$. Lines are guides to the eye showing that $f$ is independent of ${\epsilon }_Z$ for $\mathrm{\Gamma }{\epsilon }_Z/T_{K}D\lesssim 10$. Data in both panels are for ${\epsilon }_0=-0.08D$ and $\mathrm{\Gamma }=0.005D$.