Effects of magnetic anisotropy on the subgap excitations induced by quantum impurities in a superconducting host

We study the magnetic anisotropy effects on the localized subgap excitations induced by quantum impurities coupled to a superconducting host. We establish the ground-state phase diagrams for single-channel and two-channel high-spin Kondo impurities; they unveil surprising complexity that results from the (multistage) Kondo screening in competition with the superconducting correlations and the magnetic anisotropy splitting of the spin multiplets. We discuss the possibility of detecting the Zeeman splitting of the subgap states, which would provide an interesting spectroscopic tool for studying the magnetism on the single-atom level. We also study the problem of two impurities coupled by the Heisenberg exchange interaction, and we follow the evolution of the subgap states for both antiferromagnetic and ferromagnetic coupling. For sufficiently strong antiferromagnetic coupling, the impurities bind into a singlet state that is nonmagnetic, thus the subgap states move to the edge of the gap and can no longer be discerned. For ferromagnetic coupling, some excited states remain present inside the gap.

Figure 1

(a) Magnetic impurity atom adsorbed on the superconductor surface probed by the STM. (b) Idealization of the characteristic differential conductance spectrum as recorded by the STM above a magnetic impurity: in the gap, the LDOS is essentially zero except for the discrete peaks, which correspond to the transitions from the ground state to the subgap excited states.

Figure 2

Many-body ground state as a function of the gap $\Delta$ and the magnetic anisotropy $D$ at fixed exchange coupling $J$ for a range of the impurity spin $S_{\mathrm{imp}}$ in the single-channel case. Magnetic field is zero, $B=0$. The value ${\Delta }_c$ (circle) corresponds to the transition point between the Kondo-screened and -unscreened impurity moment in the isotropic case.

Figure 3

Left panels: Ground-state and subgap many-body excited states as a function of the magnetic anisotropy $D$ for different values of the superconducting gap $\Delta$ for the single-channel spin-1 Kondo models. Zero magnetic field, $B=0$. Top to bottom: “free-spin” regime, transition regime, Kondo screened regime. The energies $E$ are plotted relative to the ground-state energy $E_{\mathrm{GS}}$ and they are rescaled in units of the gap, thus the ground state always lies atop the horizontal axis at $E=E_{\mathrm{GS}}$ and the continuum of excitations starts at $(E-E_{\mathrm{GS}})/\Delta =1$. Right panels: Energies and weights of the subgap spectral peaks in the impurity spectral function. Darker shade corresponds to higher spectral weight of the $\Delta$peak.

Figure 4

Schematic diagram representing the many-body ground state as a function of the exchange coupling constants $J_1$ and $J_2$ for the isotropic ($D=0$) two-channel $S_{\mathrm{imp}}=1/2$ and $1$ Kondo models at fixed superconducting gap $\Delta ={10}^{-5}$. Zero magnetic field, $B=0$.

Figure 5

Energies and weights of the subgap peaks for the two-channel $S_{\mathrm{imp}}=1$ Kondo model as a function of the magnetic anisotropy. Parameters are $\Delta ={10}^{-5}$, $J_1=0.2$, and $J_2=0.1$. Darker shade corresponds to higher spectral weight. The labels $i=1$ and $i=2$ indicate spectral peaks associated with the $i$th scattering channel. Zero magnetic field, $B=0$.

Figure 6

Schematic representation of the relation between the many-body energy levels (shown on the left; $E_{\mathrm{GS}}$ is the ground-state energy, $E_{\mathrm{ES}}$ are the excited-state energies, $\Delta$ is the gap beyond which starts the continuum of the conduction-band states) and the spectral peaks, which can be measured by the STM (shown on the right; $V>0$ indicates tunneling into “empty” states, i.e., the electron addition transitions, while $V<0$ corresponds to the removal of electrons from the system). Only transitions where $S_z$ changes by $1/2$ are spectroscopically visible at very low temperatures. The arrows indicate the spin of the electron being added (positive $V$) or removed (negative $V$) from the system at the given resonance. In the absence of the field, $T_K>\Delta$ and $T_K<\Delta$ cases cannot be easily distinguished spectroscopically since they both exhibit a pair of excitation peaks at $\omega =\pm (E_{\mathrm{ES}}-E_{\mathrm{GS}})$. In the magnetic field, we expect to observe peak splitting for $T_K>\Delta$, while for $T_K<\Delta$ we expect merely a shift of the peak pair concomitant with the weight reduction if the experimental temperature is much lower than the $E_{\mathrm{ES},1}-E_{\mathrm{GS}}$ scale. The example depicted in the plot corresponds to an $S_{\mathrm{imp}}=1/2$ magnetic impurity in the single-channel case.

Figure 7

Energies and weights of the subgap spectral peaks as a function of the external magnetic field for the single-channel spin-$1/2$ and spin-$1$ Kondo models with different magnetic anisotropy terms.

Figure 8

Energies and weights of the subgap spectral peaks as a function of the magnetic field $B$ for the two-channel $S_{\mathrm{imp}}=1$ impurity. Parameters are as in Fig. 5. The spectral peaks are fully spin-polarized, as indicated by the arrows.

Figure 9

Ground-state and subgap many-body excited states in the two-impurity problem with $S_{\mathrm{imp}}=1/2$ impurities coupled by Heisenberg exchange coupling $J$. The inset is a close-up view of the region where the two different singlet ground states cross: the arrow shows the evolution of the local AFM-ordered spin-singlet ground state as $J$ is increased. The gap is $\Delta ={10}^{-5}$.

Figure 10

Ground-state and subgap many-body excited states in the two-impurity problem with $S_{\mathrm{imp}}=1$ impurities coupled by Heisenberg exchange coupling $J$. The gap is $\Delta ={10}^{-5}$.

Figure 11

Close-up view of the small-$J$ region in Fig. 10.

Figure 12

Ground-state and subgap many-body excited states as a function of the superconducting gap $\Delta$ for single-channel ($N=1$) Kondo models with different impurity spin $S_{\mathrm{imp}}$. The NRG calculations have been performed with the NRG discretization parameter $\Lambda =2$. Additional $\Lambda$-scaling calculations suggest that the results for the transition point ${\Delta }_c/T_K$ given in Table  are accurate within a few percent.

Figure 13

Ground-state and subgap many-body excited states as a function of the superconducting gap $\Delta$ for two-channel ($N=2$) Kondo models with different impurity spin $S_{\mathrm{imp}}$. The NRG calculations have been performed with the discretization parameter $\Lambda =4$. The results given in Table  for the two-channel case have been calculated by performing further calculations using a twist parameter $z=0.5$ and averaging the results for $z=1$ and $0.5$. The tabulated results are then accurate within a few percent for both the single-channel and the two-channel case. Note the presence of the additional $S=S_{\mathrm{imp}}-1/2$ excited-state multiplet inside the gap. For nonequal $J_i$, this excited state can become the ground state in some parameter regimes; see Fig. 4.

Figure 14

Ground-state and subgap excited states as a function of the magnetic anisotropy $D$ for different values of the superconducting gap $\Delta$ for the single-channel spin-$1$, spin-$3/2$, and spin-$2$ Kondo models. The results of these (and similar) calculations have been used to establish the schematic phase diagrams shown in Fig. 2.

Figure 15

Ground-state and subgap excitations as a function of the parameter $J_0$ for the two-channel $S_{\mathrm{imp}}=1/2$ Kondo model. We define $J_1=J_0\cos \varphi$ and $J_2=J_0\sin \varphi$; the angles $\varphi$ of the direction in the $(J_1,J_2)$ plane are given (in degrees) as the titles of the subfigures. The gap is $\Delta ={10}^{-5}$. The results of these (and similar) calculations have been used to establish the schematic phase diagram for the $S_{\mathrm{imp}}=1/2$ case in Fig. 4 of the main text. The two-channel calculations with nonequal $J_i$ have to be performed with a higher value of the NRG discretization parameter $\Lambda =8$, otherwise they are intractable. No twist-parameter averaging has been performed here; nevertheless, the results are still qualitatively correct.

Figure 16

Ground-state and subgap many-particle excitations as a function of the parameters $J_1$ and $J_2$ for the two-channel $S_{\mathrm{imp}}=1$ Kondo model. The gap is $\Delta ={10}^{-5}$. The results of these (and similar) calculations have been used to establish the schematic phase diagram for the $S_{\mathrm{imp}}=1$ case in Fig. 4 of the main text.