Characterizing $p$-wave superconductivity using the spin structure of Shiba states

Cooper pairs in two-dimensional unconventional superconductors with broken inversion symmetry are in a mixture of an even-parity spin-singlet pairing state with an odd-parity spin-triplet pairing state. We study the magnetic properties of the impurity bound states in such superconductors and find striking signatures in their spin polarization which allow one to unambiguously discriminate a nontopological superconducting phase from a topological one. Moreover, we show how these properties, which could be measured using spin-polarized scanning tunneling microscopy (STM), also enable one to determine the direction of the spin-triplet pairing vector of the host material and thus to distinguish between different types of unconventional pairing.

Figure 1

Spin-resolved band structure: We plot the $x$ component of the spin-polarized spectral function as a function of energy and $k_x$. We consider a lattice model with dispersion ${\mathrm{\Xi }}_{\mathit{k}}=-2tcos\left(k_x\right)-2tcos\left(k_y\right)-\mu$, with $t$ the hopping amplitude and $\mu$ the chemical potential. We take $t=1$, $\mu =-3$, ${\mathrm{\Delta }}_s=0.2$, $\varkappa =0.5$, and an inverse quasiparticle lifetime $\delta =0.03$.

Figure 2

Sketch of an out-of-plane magnetic impurity in a $p$-wave TRS superconductor with an in-plane $\mathbf{d}$ vector. In this case, electrons forming Cooper pairs have always some out-of-plane spin components (blue and red arrows) [1]. Furthermore, due to TRS, Cooper pairs have either an angular momentum (green arrow) with $L_z=+1$ and are composed of electrons with spins $s_z=-1/2$ or the opposite [1]. TRS is locally broken by a magnetic impurity. For antiferromagnetic exchange interactions, $J_z>0$, breaking Cooper pairs with electron spins $s_z=-1/2$ (blue arrows) is energetically favored compared to Cooper pairs with spins $s_z=1/2$ (red arrows), hence two nondegenerate in-gap bound states are expected.

Figure 3

Average LDOS (first row) and SP LDOS (second row) (in arbitrary units) as functions of energy and the $p$-wave pairing, $\varkappa$, for an in-plane $\mathbf{d}$ vector. We consider a scalar impurity with $U=6$ in the upper right panel, and magnetic impurities with an impurity strength of $J_z=2$ (left column) and with $J_x=2$ (lower right panel). We set ${\mathrm{\Delta }}_s=0.2$ and $\delta =0.03$. The gap of the system is denoted by the dashed line.

Figure 4

Upper panel: average LDOS (in arbitrary units) as functions of energy for an in-plane $\mathbf{d}$ vector for different values of the $p$-wave pairing, $\varkappa$: blue stands for $\varkappa =0.1$, orange for $\varkappa =0.4$, and red for $\varkappa =0.8$. The dashed lines correspond to the averaged LDOS in the absence of impurities while the plain lines show the LDOS in presence of impurities. We can thus disentangle the spectral features due to the impurity bound states from the gap edges. Lower panel: same as the upper panel but for the SP LDOS. The vertical dashed lines correspond to the two gap edges denoted in the main text by ${\mathrm{\Delta }}_{\mathrm{eff}}^{\pm }$. The subgap Shiba states peaks are marked by filled squares. The other parameters are the same as in Fig. 3.

Figure 5

Average SP LDOS as a function of energy and impurity strength for magnetic impurities with spin along $z$ (left column) and $x$ (right column) for an in-plane $\mathbf{d}$ vector. We set $\delta =0.01$ and we focus on the dominant $p$-wave case ($\varkappa =0.5$, ${\mathrm{\Delta }}_s=0.2$, first row) and the dominant $s$-wave case ($\varkappa =0.1$, ${\mathrm{\Delta }}_s=0.2$, second row). The gap is denoted by the dashed line.

Figure 6

Average SP LDOS as a function of energy and impurity strength for magnetic impurities with spin along $z$ (left column) and along $x$ (right column). We take $\delta =0.01$ and we focus on the pure $p$-wave $\varkappa =0.5$, ${\mathrm{\Delta }}_s=0$ with in-plane ${\mathbf{d}}_{\parallel }$ (first row) and out-of-plane ${\mathbf{d}}_{\perp }$ (second row). The gap is denoted by the dashed line.

Figure 7

The real part of the FT of the $S_z$ SP LDOS component for the hole component of a Shiba BS as a function of momentum ($p_x$, $p_y$) for a magnetic impurity with $J_z=2$ and an in-plane $\mathbf{d}$ vector. We take ${\mathrm{\Delta }}_s=0.2$, $\varkappa =0$ for a pure $s$-wave SC, and ${\mathrm{\Delta }}_s=0$, $\varkappa =0.5$ for a pure $p$-wave SC.

Figure 8

The real part of the FT of the $S_x$ and $S_y$ components of SP LDOS in arbitrary units for the hole component of a Shiba state as a function of momentum ($p_x$, $p_y$), for a magnetic impurity with $J_x=2$. We take $\varkappa =0.5$ for the pure $p$-wave SC with $\mathbf{d}$ in-plane (first row) and $\mathbf{d}$ perpendicular to the plane (second row).

Figure 9

Left: Sketch of the energy levels obtained by perturbation theory in impurity strength $J_z$. The colors stand for the angular momentum: red for $+1/2$ and blue for $-1/2$. Right: Energy levels as a function of impurity strength $J_z$ obtained analytically. We set $\varkappa =0.2$ and used the same color code.