Kondo effect in graphene with Rashba spin-orbit coupling

We study the Kondo screening of a magnetic impurity adsorbed in graphene in the presence of Rashba spin-orbit interaction. The system is described by an effective single-channel Anderson impurity model, which we analyze using the numerical renormalization group. The nontrivial energy dependence of the host density of states gives rise to interesting behaviors under variation of the chemical potential or the spin-orbit coupling. Varying the Rashba coupling produces strong changes in the Kondo temperature characterizing the many-body screening of the impurity spin, and at half filling allows an approach to a quantum phase transition separating the strong-coupling Kondo phase from a free-moment phase. Tuning the chemical potential close to sharp features of the hybridization function results in striking features in the temperature dependencies of thermodynamic quantities and in the frequency dependence of the impurity spectral function.

Figure 1

(a) Graphene band structure near either Dirac point, plotted schematically as a function of $q_x$ for fixed $q_y=0$, where $\mathbf{q}=\mathbf{k}-{\mathbf{K}}_{\pm }$. Solid (dotted) lines represent the dispersion with (without) Rashba SOI. Each solid line is labeled with its $\alpha ,\beta$ values. (b) The corresponding densities of states $\rho \left(E\right)$. For an impurity adsorbed directly on top of a carbon atom, the hybridization function $\Gamma \left(E\right)$ is simply proportional to $\rho \left(E\right)$.

Figure 2

Kondo temperature $T_K$ vs Rashba parameter $\lambda$ for chemical potential $\mu =0$ and hybridization prefactor ${\Gamma }_0=0.08$, comparing NRG calculations (solid line) with Haldane's formula Eq. (30) (dashed line). The inset makes clear the exponential dependence of $T_K$ on $1/\lambda$, characteristic of a Kosterlitz-Thouless quantum phase transition.

Figure 3

Kondo temperature $T_K$ vs Rashba parameter $\lambda$ for three different combinations of the hybridization prefactor ${\Gamma }_0$ and the chemical potential $\mu$ that, for $\lambda =0$, take the system from its mixed-valence regime [largest value of $\Gamma \left(\mu \right)$, top case] to deep in its Kondo regime [smallest $\Gamma \left(\mu \right)$, bottom case]. Solid lines plot NRG results, while the dashed line shows the prediction of Eq. (30) for the middle value of $\Gamma \left(\mu \right)$.

Figure 4

Kondo temperature $T_K$ vs chemical potential $\mu$ near the jump in $\Gamma \left(\mu \right)$ at $\mu =2\lambda$, calculated for ${\Gamma }_0=0.04$ and two values of the Rashba parameter: $\lambda =0.006$ (circles, lower axis) and $\lambda =0.015$ (squares, upper axis). There is a rapid rise in $T_K$ as $\mu$ crosses the discontinuity in the hybridization function, with a sharper variation for the smaller $\lambda$.

Figure 5

Impurity contribution to the magnetic susceptibility at chemical potential $\mu =2\lambda$, calculated for ${\Gamma }_0=0.04$ and different values of the Rashba parameter $\lambda$. At low temperatures, ${\chi }_{\text{imp}}\left(T\right)$ changes sign, showing a residual negative susceptibility contribution as $T\to 0$.

Figure 6

Impurity spectral function $A_{\text{imp}}\left(\omega \right)$ vs frequency $\omega$ at $T=0$ for $\mu =2\lambda ,{\Gamma }_0=0.04$, and different values of the Rashba parameter $\lambda$. The Kondo peak spanning $\left|\omega \right|\lesssim T_K$ is strongly asymmetric and features a sharp drop around the Fermi level due to the presence of the jump in $\Gamma \left(E\right)$ at $E=\mu$.

Figure 7

Low-temperature variation of temperature times the impurity magnetic susceptibility $T{\chi }_{\text{imp}}$ (upper panels) and impurity entropy $S_{\text{imp}}$ (lower panels) for ${\Gamma }_0=0.04,\lambda =0.006$, and different values of the chemical potential $\mu =2\lambda \pm {10}^{-m}$ close to a jump in $\Gamma \left(\mu \right)$.

Figure 8

Impurity spectral function $A_{\text{imp}}\left(\omega \right)$ vs frequency $\omega$ at $T=0$, calculated for ${\Gamma }_0=0.04,\lambda =0.006$, and different values of the chemical potential $\mu =2\lambda \pm {10}^{-m}$ close to a jump in $\Gamma \left(\mu \right)$. For $T^{*}\simeq |\mu -2\lambda |\lesssim T_K$, the Kondo peak shows a dip near the location $\omega =2\lambda -\mu$ of the discontinuity.

Figure 9

Impurity spectral function $A_{\text{imp}}$ vs frequency $\omega$ at $T=0$ for the noninteracting case $U=0$ with ${\epsilon }_d=0.007,{\Gamma }_0=0.04,\lambda =0.015$, and different values of the chemical potential $\mu =2\lambda \pm {10}^{-m}$ close to a jump in $\Gamma \left(\mu \right)$. A sharp, asymmetric dip structure is centered near the location $\omega =2\lambda -\mu$ of the discontinuity in the hybridization function.