Yu-Shiba-Rusinov states and ordering of magnetic impurities near the boundary of a superconducting nanowire

We theoretically study the spectrum induced by one and two magnetic impurities near the boundary of a one-dimensional nanowire in proximity to a conventional $s$-wave superconductor and extract the ground-state magnetic configuration. We show that the energies of the subgap states, supported by the magnetic impurities, are strongly affected by the boundary for distances less than the superconducting coherence length. In particular, when the impurity is moved towards the boundary, multiple quantum phase transitions periodically occur in which the parity of the superconducting condensate oscillates between even and odd. We find that the magnetic ground-state configuration of two magnetic impurities depends not only on the distance between them, but also explicitly on their distance away from the boundary of the nanowire. As a consequence, the magnetic ground state can switch from ferromagnetic to antiferromagnetic while keeping the interimpurity distance unaltered by simultaneously moving both impurities away from the boundary. The ground-state magnetic configuration of two impurities is found analytically in the weak coupling regime and exactly for an arbitrary impurity coupling strength using numerical tight-binding simulations.

Figure 1

One-dimensional nanowire with superconducting gap aligned along the $x$ axis with two magnetic impurities with classical spins ${\mathbf{S}}_1$ and ${\mathbf{S}}_2$ separated by the interimpurity distance $x_{12}=x_2-x_1$. The impurity spins are exchange coupled to the quasiparticles in the nanowire. This gives rise to YSR subgap bound states (not shown) and effective interactions between the magnetic impurities which are both affected by the boundaries of the nanowire.

Figure 2

The energy of the YSR state $E_s/\mathrm{\Delta }$ [see Eq. (7)] as a function of the distance $k_F{x}_1$ of the impurity away from the boundary for (a) $\alpha =0.55$ and (b) $\alpha =0.96$. The energy $E_s$ oscillates around its bulk value. (b) If the YSR energy is close to zero energy for a magnetic impurity placed far away from the boundary, then moving it closer to the boundary can induce a quantum phase transition and change the occupancy of the YSR bound state. The coherence length is fixed to $k_F{\xi }_{sc}=40$.

Figure 3

Color plot of the RKKY coefficient $F_{\text{sum}}$ as a function of $x_2/{\lambda }_F$ on the $x$ axis and $x_1/{\lambda }_F$ on the $y$ axis, where ${\lambda }_F=2\pi /k_F$ is the Fermi wavelength. The red (yellow) color denotes the region in which $F_{\text{sum}}$ is negative (positive) and hence the ground state is FM (AFM). The green dotted line and the blue dashed line parallel to the diagonal are two lines of constant $x_{12}$, while $X$ remains constant along the black dashed-dotted line perpendicular to them. The figure exhibits oscillations between the FM and AFM ground states along the constant $x_{12}$ lines, unlike the conventional RKKY interaction obtained far away from boundaries.

Figure 4

The figure shows YSR energy $E_s/\mathrm{\Delta }$ as a function of the distance $x_1/a$ of the impurity away from the boundary for (a) $\tilde{J}=1.0t$ and (b) $\tilde{J}=0.644t$. We used the following values of the parameters: $N=1000, {\mu }_l=-1.9t, \mathrm{\Delta }=0.005t$. As the impurity approaches the boundary, the YSR energy oscillates with a periodicity of ${\lambda }_F/2$ around its value obtained deep inside the bulk. The amplitude of oscillations decays exponentially due to the presence of the superconducting gap. In (b), close to the boundary, the YSR energy crosses the chemical potential indicating the quantum phase transition.

Figure 5

The energies $E_s$ for two YSR states (blue and red) for (a) the FM configuration and (b) the AFM configuration of the magnetic impurities as a function of the distance $x_1/a$ between the first impurity ${\mathbf{s}}_1$ and the boundary for a fixed interimpurity distance $x_{12}=28a$. Note that in (a) the two YSR states are strongly hybridized, whereas this is not the case in (b). In (a) the hybridized energy levels of the YSR states oscillate around their bulk value with an increasing amplitude of oscillation close to the boundary. The YSR energies for the AFM configuration in (b) do not hybridize with each other but exhibit oscillations induced by the boundary. Each of the YSR energies oscillates with a different amplitude as they originate from impurities sitting at different distances from the boundary. The various parameters are chosen as $\tilde{J}=1.0t, N=1000, {\mu }_l=-1.9t$, and $\mathrm{\Delta }=0.005t$.

Figure 6

The energy differences $\delta E_{\text{YSR}}$ (left panel), $\delta E_{\text{qp}}$ (middle panel), and $\delta E_{\text{g}}$ (right panel) between FM and AFM configuration as a function of $x_1/a$ along the $x$ axis and $\tilde{J}/t$ along the $y$ axis. In the first and second rows of the figure, the interimpurity distance is $x_{12}=28a$ and $40a$, respectively, such that $2k_F{x}_{12}$ is approximately an odd integer multiple of $\pi /2$ in the first row and an integer multiple of $\pi$ in the second row. The red (yellow) patches indicate a FM (AFM) configuration of impurities. The magnetic configuration alternates between the FM and AFM phases with a periodicity of ${\lambda }_F/2$. The amplitude of oscillations decays exponentially as a function of $x_1$ due to the presence of the superconducting gap $\mathrm{\Delta }$. The blue dashed line in the figures denotes the exponentially decaying envelope function. The phase diagram for the total magnetic ground state is a sum of the contributions coming from the YSR and the quasiparticle states. Evidently, the oscillations in $\delta E_{\text{g}}$ are observed for a larger range of $\tilde{J}$ in (c) than in (f) in the second row, indicating the stronger effect of boundary when $2k_F{x}_{12}$ is an odd-integer multiple of $\pi /2$. The various parameters are chosen as $\mu =-1.9t, \mathrm{\Delta }=0.005t$, and $N=500$.

Figure 7

The energy differences $\delta E_{\text{YSR}}$ (left panel), $\delta E_{\text{qp}}$ (middle panel), and $\delta E_{\text{g}}$ (right panel) as a function of $x_1/a$ along the $x$ axis and $x_2/a$ along the $y$ axis. In the first, second, and third rows, $\tilde{J}=0.05t, 0.5t$, and $1.5t$, respectively. The red regions in the phase diagrams indicate a FM configuration while the yellow regions indicate an AFM one. The energy difference $\delta E_{\text{YSR}}$ exhibits oscillations around zero along the lines of constant $x_{12}$, originating from the boundary-induced hybridization between the YSR states. The plots for $\delta E_{\text{qp}}$ also show similar oscillations except close to the region $x_{12}\approx 0$. Both the YSR states and the quasiparticle states contribute significantly in the phase diagram of $\delta E_{\text{g}}$ for the total magnetic ground state. When $|x_{12}|<{\lambda }_F$, the phase diagram of $\delta E_{\text{g}}$ is governed by the quasiparticle contribution, while beyond this regime, the YSR contribution dominates. The other parameters are fixed as $\mu =-1.9t, \mathrm{\Delta }=0.005t$, and $N=500$.