Phase diagram of a $d$-wave superconductor with Anderson impurities

We present a self-consistent solution for a model of a $d$-wave superconductor with finite concentration of Anderson impurities at zero temperature using the slave boson method. We show how the phase diagram depends on the strength of interaction between impurity and extended states. For fixed impurity level energy $E_0$ in the Kondo limit there is one superconducting-normal-state transition for all impurity concentrations $n$. When $E_0$ is close the Fermi energy there are three such transitions for impurity concentration exceeding certain minimum value $n\left({\Gamma }_0,E_0\right)$. If hybridization ${\Gamma }_0$ is fixed and the depth of the impurity level is varied, there are always two transitions for concentration above $n\left({\Gamma }_0,E_0\right)$.

Figure 1

The phase diagram of a $d$-wave superconductor with Anderson impurities for four values of impurity concentration. The lines separate the normal state from the SC. The broken line shows the location of the impurity quantum phase transition. Below that line the impurities are decoupled from the superconductor. The slope of this line is approximately $-0.26$ which agrees with the numerical renormalization group (Ref. 9) result and a single-impurity large-$N$ calculation for a $d$-wave superconductor (Ref. 11).

Figure 2

The amplitude of the order parameter as a function of bare hybridization ${\Gamma }_0$ for $E_0=-0.1$ and several impurity concentrations. The approximate location of the critical point where impurities are decoupled from the conduction band is marked with a circle.

Figure 3

Critical concentration for the superconducting-normal transition as a function of ${\Gamma }_0=\pi N_0{V}^2$ for $E_0=-1$ and $E_0=-0.1$. The $E_0=-1$ curve shows the behavior in the Kondo limit.

Figure 4

The phase diagram in the mixed-valence limit, $E_0=-0.01$. The scale of the horizontal axis is changed relative to Fig. 2 to illustrate behavior at large ${\Gamma }_0$.

Figure 5

Superconducting order parameter ${\Delta }_0$ as a function of $E_0$ for several values of ${\Gamma }_0$ at $n=0.01$. The unmarked dashed-dotted curve is the result for ${\Gamma }_0=0.08$. The vertical dotted line marks the impurity quantum phase transition.