Yu-Shiba-Rusinov states in real metals

Theoretical descriptions of Yu-Shiba-Rusinov (YSR) states induced by magnetic impurities inside the gap of a superconductor typically rely on a classical spin model or are restricted to spin-$\frac{1}{2}$ quantum spins. These models fail to account for important aspects of YSR states induced by transition-metal impurities, including the effects of higher quantum spins coupled to several conduction-electron channels, crystal or ligand-field effects, and magnetic anisotropy. We introduce and explore a zero-bandwidth model, which incorporates these aspects, is readily solved numerically, and analytically tractable in several limiting cases. The principal simplification of the model is to neglect Kondo renormalizations of the exchange couplings between impurity spin and conduction electrons. Nevertheless, we find excellent correspondence in those cases, in which we can compare our results to existing numerical-renormalization-group calculations. We apply the model to obtain and understand phase diagrams as a function of pairing strength and magnetic anisotropy as well as subgap excitation spectra. The single-channel case is most relevant for transition-metal impurities embedded into metallic coordination complexes on superconducting substrates, while the multichannel case models transition-metal adatoms.

Figure 1

Illustration of crystal-field effects for spin-$\frac{5}{2}$ Mn adatoms on a Pb(001) substrate [11]. (a) Adsorption site of Mn on Pb(001) has $C_{4v}$ point-group symmetry. (b) Crystal-field splitting of the $\ell =2$ manifold. According to Hund's rules for the $d^5$ configuration of Mn, all $d$ orbitals are singly occupied, which results in an $S_i=\frac{5}{2}$ spin. Each ligand-field-split $d$ orbital couples to a separate conduction-electron channel. The point-group symmetry implies that the $d_{xz}$ and $d_{yz}$ orbitals are degenerate and the exchange couplings with their respective conduction-electron channels are identical. (c) Bare anisotropy splitting of an $S_i=\frac{5}{2}$ spin for uniaxial anisotropy $D<0$ (easy-axis anisotropy). The levels are labeled by the projections $M$ of the impurity spin. (For easy-plane anisotropy, $D>0$, the order of the levels would be reversed.) The effect of the anisotropy is strongly modified when the impurity binds quasiparticles on a superconducting substrate.

Figure 2

Phase diagrams for various impurity spins $S_i$ as a function of pairing strength $\mathrm{\Delta }$ and uniaxial anisotropy $D$, based on the single-channel model in Eq. (9). Transverse anisotropy is set to zero and the exchange coupling $J=1$ sets the energy scale. Phases are labeled by $(Q,S_z)$, where $Q$ is the number of bound quasiparticles (fermion parity) and $S_z$ the projection of the total spin. The phases at larger pairing strengths $\mathrm{\Delta }$ have an unscreened impurity spin and no bound quasiparticle ($Q=0$, even fermion parity). The phases at lower pairing strengths $\mathrm{\Delta }$ have a partially screened impurity spin and a bound quasiparticle ($Q=1$, odd fermion parity). See the Appendix for corresponding spectra of the Hamiltonian in Eq. (9).

Figure 3

Subgap tunneling excitation energies of the single-channel model in Eq. (9) as a function of anisotropy $D$ for $S_i=1$ [(a), (b)] and for $S_i=3/2$ [(c), (d)]. Background color indicates the ground state and line color the excited state (same color coding as in the corresponding phase diagram in Fig. 2). Transverse anisotropy is set to zero and the exchange coupling $J=1$ sets the energy scale. Pairing strength $\mathrm{\Delta }$ is chosen as indicated in the panels. The maximal excitation energies shown correspond to the gap edge $\mathrm{\Delta }$ of the superconductor. (a) Tunneling excitations out of the unscreened ground states for $S_i=1$. The corresponding term diagram for $D<0$ uses the same color scheme. Allowed transitions out of the ground state, shown as full arrows, are included in the excitation spectrum. The anisotropy splitting generates additional excitations out of excited states, shown as a dashed arrow. These appear at finite temperatures, with their intensity showing an activated temperature dependence. (b) Tunneling excitations for $S_i=1$ at smaller pairing energy, including excitations out of the screened ground state (green background). The corresponding term diagram for $D<0$ shows that for a screened ground state, the YSR excitations exhibit an anisotropy splitting even at zero temperature. (c) Tunneling excitations out of the unscreened ground states for $S_i=3/2$. (d) Tunneling excitations out of the screened ground states for $S_i=3/2$. Anisotropy splitting is seen at negative anisotropy $D<0$, see left term diagram. For positive anisotropy $D>0$, the anisotropy splitting cannot be observed due to the selection rule $\mathrm{\Delta }{S}_z=\pm \frac{1}{2}$, see right term diagram. In all tunneling spectra, cusps and terminations of excited-state energies are associated with quantum phase transitions.

Figure 4

Subgap excitation spectra of the single-channel model in Eq. (9) for fixed fermion parity as a function of anisotropy $D$ for impurity spins $S_i$ as indicated above the panels. Top row refers to even fermion parity states without quasiparticle ($Q=0$; illustrated by top inset in panel for $S_i=1$) and the bottom row to odd fermion parity states with quasiparticle ($Q=1$; illustrated by bottom inset in panel for $S_i=1$). Background color indicates the ground state and line color the excited state (same color coding as in the corresponding phase diagram in Fig. 2). Gray lines correspond to intermediate total spin projections, which do not become the ground state at any $D$. Dashed lines indicate above-gap excitations (see Sec. 3b2). The term diagrams illustrate the underlying transitions out of the ground states. In experiments, some transitions may be forbidden due to additional selection rules beyond $\mathrm{\Delta }Q=0$, characteristic of the specific probe, and transitions out of excited states may appear at finite temperatures. Transverse anisotropy is set to zero and the exchange coupling $J=1$ sets the energy scale. The excitation spectra are independent of pairing strength $\mathrm{\Delta }$, except that $\mathrm{\Delta }$, here chosen as $\mathrm{\Delta }=1.5$, determines the range, over which the states appear as subgap excitations.

Figure 5

Phase diagrams for an impurity spin with $S_i=1$ coupled to two channels as a function of pairing strength $\mathrm{\Delta }$ and uniaxial anisotropy $D$, for different values of the exchange couplings $J_1$ and $J_2$ in the two channels. Transverse anisotropy is set to zero. Phases are labeled by $(Q,S_z)$. (a) For equal couplings of both channels, there are direct quantum phase transitions between the unscreened phases (0,1) and (0,0) to the fully screened phase (2,0). (b) For similar but different exchange couplings in the two channels, a singly-screened phase $(1,\pm \frac{1}{2})$ appears at small anisotropy, while there is a direct transition between the unscreened and doubly screened phases at larger anisotropy of either sign. (The singly screened phase reappears for intermediate $\mathrm{\Delta }$ at even larger $D<0$, see text.) (c) For significantly different exchange couplings (right panel), the intermediate singly screened phase exists for all negative $D$ but only up to a maximal positive $D$. (d) For very asymmetric exchange couplings, the singly screened phase persists for all values of the anisotropy $D$.

Figure 6

Phase diagrams for $S_i=\frac{3}{2}$ coupled to three channels as a function of pairing strength $\mathrm{\Delta }$ and uniaxial anisotropy $D$. Transverse anisotropy is set to zero. Phases are labeled by $(Q,S_z)$. (a) When the exchange couplings in all three channels are equal, there is a direct transition from the unscreened to the fully screened state for negative uniaxial anisotropy, $D<0$. For positive anisotropy, $D>0$, there is an intermediate phase in which the impurity is screened only in a single channel. (b) If the exchange couplings differ between all channels, there is a cascade of phases at negative anisotropy $D$ in which the impurity spin is successively screened as the pairing strength is reduced. The phases correspond to states in which $Q=0,1,2$, or 3 quasiparticles are bound to the impurity. In contrast, for sufficiently large positive anisotropy $D$, there are only phases with zero, one, or three bound quasiparticles.

Figure 7

Phase diagrams for higher impurity spins with (a), (b) $S_i=2$ and (c,d) $S_i=\frac{5}{2}$ as a function of pairing strength $\mathrm{\Delta }$ and uniaxial anisotropy $D$. The values of the exchange couplings in all $2S_i+1$ channels are given above the panels. Transverse anisotropy is set to zero. Phases are labeled by $(Q,S_z)$. The inset in (c) shows a higher-resolution calculation of the $D<0$ region between the unscreened and the fully screened phases. The color coding of the intermediate phases is as in (d).

Figure 8

Subgap tunneling spectra of the multichannel model for (a)–(d) $S_i=1$ and (e)–(h) $S_i=\frac{3}{2}$ as a function of uniaxial anisotropy $D$, for pairing strengths $\mathrm{\Delta }$ and exchange couplings $J_i$ as indicated in the panels. Transverse anisotropy is set to zero. Background color indicates the ground state and line color the excited state (same color coding as in the corresponding phase diagrams in Figs. 5 and 6). The maximal excitation energy shown corresponds to the gap edge $\mathrm{\Delta }$ of the superconductor.

Figure 9

Subgap excitation spectra of the multichannel model at fixed fermion parity for $S_i=\frac{5}{2}$ as a function of anisotropy $D$. The exchange couplings were chosen as $J_1=1$, $J_2=0.9$, $J_3=0.8$, $J_4=0.7$, and $J_5=0.6$. Transverse anisotropy is set to zero. Background color indicates the ground state and line color the excited state [same color coding as in the corresponding phase diagram in Fig. 7], except for the gray lines for $Q=0,1,5$. The gray lines in the panels for $Q=0$ and 1 correspond to intermediate total spin projections $S_z=\frac{3}{2}$ and $S_z=1$, respectively, which do not become the ground state at any $D$. The gray line in the panel for $Q=5$ corresponds to an $S_z=1$ excitation. There is no panel for $Q=4$, since there are no subgap excitations in that case. The excitation spectra are independent of pairing strength $\mathrm{\Delta }$, except that $\mathrm{\Delta }$ determines the range over which the state still appears as a subgap excitation. The excitations are limited to energies below the maximal $\mathrm{\Delta }$, at which the particular ground state can still be observed at $D=0$.

Figure 10

Representative plots of the eigenenergies $E$ of the single-channel Hamiltonian in Eq. (9) for (a) $S=1$ and (b) $S=\frac{3}{2}$ as a function of anisotropy $D$ for $J=1$ and representative pairing strengths $\mathrm{\Delta }$. (Transverse anisotropy $E=0$.) For even-fermion parity, spectra include only the BCS state of the single-site superconductor. Colors label the number of bound quasiparticles $Q$ and the projection $S_z$ of total spin, see legends. Color coding matches the corresponding phase diagrams in Fig. 2.