Temperature dependent energy gap for Yu-Shiba-Rusinov states at the quantum phase transition

Abstract

Motivated by recent experiments, which allow for fine tuning of the effective magnetic interaction between the impurity and the superconductor, we investigate the regime around the quantum phase transition where the system's ground state changes from a weakly coupled free spin to a screened spin regime. At this transition we find that the Yu-Shiba-Rusinov (YSR) states remain at finite energies at low temperatures, thereby generating a gap in the spectrum, which is inconsistent with predictions of the original YSR theory. We investigate various gap-generating scenarios and determine that the local suppression of the order parameter, only captured by self-consistent calculations, generates the gap.

Figures (6)

• Figure 1: a) Schematic picture of a superconducting substrate with indicated square lattice and with an absorbed magnetic impurity (red dot), treated within (left) the effective Anderson impurity model, with impurity on-site energy $\varepsilon_{i}$ and spin exchange coupling $J$ for the impurity, and with the impurity coupling to the substrate through the hopping amplitude $t_0$ and (right) classical Kondo model with effective spin exchange coupling $\tilde{J}$ and additional onsite energy of $\tilde{\varepsilon}_{i}$, both induced directly in the substrate. b) Ideal YSR state energies $\epsilon_{YSR}$ with states labeled according to their spin-polarization relative to the impurity spin, where superscript indicates occupied ($-$) and unoccupied ($+$) at zero temperature. The critical coupling $J_C$ marks the QPT.
• Figure 2: Schematic illustration of the two scenarios for opening a gap in the YSR energy spectrum around the QPT as a function of effective coupling $J$. Colors indicate the two spin polarization directions of the YSR states, with spin-down (red) and spin-up (blue). a) Off-diagonal (spin-dependent) interaction with regard to the local impurity Green's function Eq. (\ref{['eqn:greens_fct_classical_ysr']}) causing mixing of the spin polarization. b) Indirect coupling between the two YSR states, caused by a local, discontinuous, change of each spin component of the Green's function, specifically a local suppression of the superconducting order parameter. This does not induce mixing of the YSR spin polarizations, but the energy spectrum can still become discontinuous at $J_C$. With the critical $J_C$ set by the switch in spin-polarization of the occupied state, it coincide with the location of the minimum gap in both scenarios.
• Figure 3: a) YSR state energies $\tilde{\epsilon}_{YSR}$ in a fully self-consistent calculation as a function of coupling $\tilde{J}$ for multiple temperatures. Dotted vertical lines indicate corresponding $\tilde{J}_C$. b) Zoom-in for single temperature $T = 0.0088\, T_C$, with pluses marking the numerical resolution with respect to $\tilde{J}$. Colors represent the magnetization of each state. c) Order parameter at the impurity site for increasing $\tilde{J}$ for the same temperatures as in a), with also same dotted vertical lines as in a). d) Extracted minimum energy of the YSR states (i.e. energy gap) at $\tilde{J}_C$ as a function of temperature.
• Figure 4: a) Total magnetization of the superconducting ground state $\left\langle s_z \right\rangle$ around $\tilde{J}_C$ for the same temperatures as in \ref{['fig:SelfconsistentTemperature']}a),c). Vertical dotted lines denote $\tilde{J}_C$. b) Phase diagram tracking the change in magnetization for different temperatures as a function of $\tilde{J}$, with the transition region (blue shade) defined as the region with magnetization between $5\%$ and $95\%$ of $\left\langle S_Z \right \rangle = -\frac{1}{2}$. Solid black line denotes $\tilde{J}_C$ and dotted orange line denotes $\tilde{J}_\pi$. Dotted horizontal line marks the temperature $T^*$. Zoom-in shows behavior at lowest temperatures.
• Figure 5: Minimum energy of the YSR states at $\tilde{J}_C$ as a function of temperature, normalized to $T_C$ for different parameters: a) Reference case, same as Fig. \ref{['fig:SelfconsistentTemperature']}d). b) Onsite energy added $\tilde{\varepsilon}_{i}=-0.05\,t$. c) Chemical potential changed to $\mu = 3.618\,t$ but same $\Delta_0$ requiring using $V_{sc} \approx 3.6\,t$. d) Bulk order parameter decreased to $\Delta_{0} = 0.1798\,t$ using $V_{sc} \approx 1.66\,t$. e) Bulk order parameter decreased to $\Delta_{0} = 0.1155\,t$ using $V_{sc} \approx 1.48\,t$. The system sizes for calculations other than a) increased to $81 \times 81$.