Spectral properties of Shiba subgap states at finite temperatures

Using the numerical renormalization group (NRG), we analyze the temperature dependence of the spectral function of a magnetic impurity described by the single-impurity Anderson model with a superconducting host. With increasing temperature the spectral weight is gradually transferred from the $\delta$ peak to the continuous subgap background, and both spectral features coexist at finite temperatures: the $\delta$ peak persists to temperatures of order $\mathrm{\Delta }$. The continuous background is due to inelastic exchange scattering of Bogoliubov quasiparticles off the impurity, and it is thermally activated since it requires a finite thermal population of quasiparticles above the gap. In the singlet regime for strong hybridization or away from the particle-hole symmetric point (charge-fluctuation regime) an additional subgap structure is observed just below the gap edges. It has thermally activated behavior with an activation energy equal to the Shiba state excitation energy.

Figure 1

Schematic diagram of the many-particle eigenstates of the Hamiltonian, partitioned into the even- and odd-fermion-parity sectors (i.e., with respect to the parity of the total electron number). This diagram corresponds to the case where the ground state has odd fermion parity (spin doublet). $D_o$ and $D_e$ are the odd-parity (spin-doublet) $|D\rangle$ and the even-parity (spin-singlet) $|S\rangle$ discrete eigenstates. The even-parity continuum $C_e$ starts at energy $\mathrm{\Delta }$ above the odd-parity discrete state $D_o$ since the bottommost states of the continuum are composed of one additional quasiparticle added to $D_o$, thus changing the overall fermion parity. The odd-parity continuum $C_o$ starts at energy $\mathrm{\Delta }$ above the even-parity discrete state $D_e$ for similar reasons. Labels $A$ and $A^{\prime }$ indicate sharp transitions with $\omega =\epsilon$ (contributing to the subgap $\delta$ peak in the impurity spectrum), and the label $B$ indicates diffuse transitions with $\omega \ne \epsilon$ (generating the continuous background in the spectrum). $A$ corresponds to the transition between the many-particle states below the continuum edge, while $A^{\prime }$ form a set of similar transitions in the presence of a thermally excited quasiparticle, but totally elastic as far as the quasiparticles are concerned. In the absence of electron-electron interaction, there are no $B$ transitions. Multiple-quasiparticle states are not shown; they start at $2\mathrm{\Delta }$ above $D_o$.

Figure 2

(a) $\delta$ peak, continuum, and total spectral weight $W$ in the positive-frequency subgap part $\left(0<\omega <\mathrm{\Delta }\right)$ of the impurity spectral function $A(\omega ,T)$. (b) Positions of the $\delta$ peak, ${\omega }_{\delta }$, and of the maximum of the continuous part, ${\omega }_{\mathrm{peak}}$, as well as the mean value of the continuous part ${\overline{\omega }}_c$.

Figure 3

Impurity spectral function $A(\omega ,T)$ at finite temperature $T=\mathrm{\Delta }/4$. The hump at $\omega =10\mathrm{\Delta }=U/2$ is the Hubbard peak. The inset shows a close-up of the subgap region. The position of the $\delta$ peak $\epsilon$ is indicated using the dashed line and almost coincides with the peak of the continuum part.

Figure 4

Width (standard deviation) of the continuum part.

Figure 5

Temperature dependence of the continuum part of the subgap spectrum $A_c(\omega ,T)$. The spectra are normalized by dividing by the total weight of the continuum contribution $W_c\left(T\right)$, shown in Fig. 2.

Figure 6

Subgap-state energy $\epsilon$ as a function of the hybridization strength $\mathrm{\Gamma }$ for fixed $U/\mathrm{\Delta }=20$. The inset shows the $T=0$ spectral weight of the subgap $\delta$ peak.

Figure 7

Temperature dependence of the mean value of the continuum part of the subgap spectrum ${\omega }_c\left(T\right)$ for a range of hybridization strengths $\mathrm{\Gamma }$ in (a) doublet and (b) singlet regimes. The arrow indicates the direction of increasing $\mathrm{\Gamma }$.

Figure 8

(a) Temperature dependence of the continuum-part weight $W_c\left(T\right)$ for a range of hybridization strengths $\mathrm{\Gamma }$. (b) $\mathrm{\Gamma }$ dependence for a range of fixed temperatures.

Figure 9

Temperature dependence of the $\delta$-peak weight $W_{\delta }\left(T\right)$ for a range of hybridization strengths $\mathrm{\Gamma }$. Inset: $\mathrm{\Gamma }$ dependence of $W_{\delta }\left(T\right)$ at finite temperature, which should be compared with the inset in Fig. 6 showing the same quantity at $T=0$.

Figure 10

Temperature dependence of the subgap spectral weight away from the particle-hole symmetric point. We plot separately the negative frequency part (solid lines) and the positive part (dashed lines), further separated into $\delta$-peak and continuum contributions. Here $\delta ={\epsilon }_d+U/2$ is a parameter which measures the departure from the particle-hole symmetric point. We plot two cases close to the singlet-doublet transition point: (a) doublet ground state $\left(\epsilon =0.062\mathrm{\Delta }\right)$ and (b) singlet ground state $\left(\epsilon =0.12\mathrm{\Delta }\right)$. Note the change of the dominant peak from the negative to positive side across the phase transition.

Figure 11

Subgap spectrum for strong hybridization $\mathrm{\Gamma }=0.3U$. The inset shows the temperature dependence of the weight and position of the secondary peak which appears just below the gap edge.

Figure 12

(a) Weight and (b) position of the weakly bound additional resonance just below the gap edge.

Figure 13

Temperature dependence of the quantities characterizing the subgap spectral function in the case of $\mathrm{\Delta }={\mathrm{\Delta }}_{\mathrm{BCS}}\left(T\right)$. The temperature-driven doublet-singlet phase transition is indicated by the arrow. Model parameters are $\mathrm{\Gamma }/U=0.1$ and $U/\mathrm{\Delta }=20$.

Figure 14

Predicted differential conductance $dI/dV$ based on the finite-temperature impurity Green's function from the NRG, calculated in the limit of weak tip-impurity coupling.