Impact of impurity scattering on odd-frequency spin-triplet pairing near the edge of the Kitaev chain

We study a Kitaev chain model, which is the simplest model of topological superconductors hosting Majorana fermion, appearing as a zero-energy state at the edge. We analytically calculate the Green's function of the semi-infinite Kitaev chain with a delta-function-type impurity potential within the quasiclassical regime to obtain the spatial dependence of the induced odd-frequency pairing. It is found that if the position of the impurity is not far from the edge, the spatial profile of the local density of states (LDOS) and the odd-frequency spin-triplet $s$-wave pair amplitude is tunable as a function of the strength of the impurity potential. Moreover, the zero-energy LDOS and low-frequency odd-frequency pair amplitude are found to have the same spatial dependence. The spatial profile of the zero-energy LDOS is analyzed based on the wave function of Majorana fermions.

Figure 1

Schematic picture of semi-infinite system with impurity.

Figure 2

Spatial dependence of (a) normalized LDOS at zero energy and (b) normalized odd-frequency component of the pair amplitude for $x=x^{\prime }$ for the semi-infinite Kitaev chain. We choose ${\mathrm{\Delta }}_0/\mu =0.1$ and infinitesimal $\delta ={10}^{-7}\mu$.

Figure 3

Spatial dependence of (a) normalized LDOS at zero energy and (b) odd-frequency component of the pair amplitude for $x=x^{\prime }$ at ${\omega }_n={\mathrm{\Delta }}_0/1000$ of the infinite Kitaev chain with a single impurity at $x=0$. $\tilde{Z}={10}^4$, ${\mathrm{\Delta }}_0=0.1\mu$, and infinitesimal $\delta ={10}^{-7}\mu$ were used in the representative figures.

Figure 4

LDOS at $x=\xi /100$ for infinite continuum Kitaev chain with a single impurity at $x=0$ for $\tilde{Z}=2$ and $\tilde{Z}=4$ with ${\mathrm{\Delta }}_0=0.1\mu$.

Figure 5

Normalized LDOS for $E=0+i\delta$, $L=10\xi$ with several values of $\tilde{Z}$: (a) $\tilde{Z}=0$, (b) 500, and (c) ${10}^4$. ${\mathrm{\Delta }}_0=0.1\mu$. Positive infinitesimal $\delta ={10}^{-7}\mu$. Note that the $y$ axis is given in units of ${10}^6$ (semi-infinite system with impurity).

Figure 6

Normalized LDOS for $E=0+i\delta$, $L=5\xi$ with several values of $\tilde{Z}$: (a) $\tilde{Z}=0$, (b) 500, and (c) ${10}^4$. ${\mathrm{\Delta }}_0=0.1\mu$. Positive infinitesimal $\delta ={10}^{-7}\mu$. Note that the $y$ axis is given in units of ${10}^6$ (semi-infinite system with impurity).

Figure 7

(a) Mean position of wave function as a function of $L$ for (I) $\tilde{Z}=500$ and (II) ${10}^4$. (b) Mean position of wave function as a function of $\tilde{Z}$ for (I) $L=10\xi$ and (II) $5\xi$. Pair potential ${\mathrm{\Delta }}_0=0.1\mu$. (Semi-infinite system with impurity.)

Figure 8

Normalized odd-frequency component of anomalous Green's function for $L=10\xi$ with several values of $\tilde{Z}$: (a) $\tilde{Z}=0$, (b) 500, and (c) ${10}^4$. The other parameters are ${\mathrm{\Delta }}_0=0.1\mu$ and ${\omega }_n={\mathrm{\Delta }}_0/1000$. Note that the $y$ axis is given in units of ${10}^3$ (semi-infinite system with impurity).

Figure 9

Normalized odd-frequency component of anomalous Green's function for $L=5\xi$ with several values of $\tilde{Z}$: (a) $\tilde{Z}=0$, (b) 500, and (c) ${10}^4$. The other parameters are ${\mathrm{\Delta }}_0=0.1\mu$ and ${\omega }_n={\mathrm{\Delta }}_0/1000$. Note that the $y$ axis is given in units of ${10}^3$ (semi-infinite system with impurity).

Figure 10

Schematic picture of dispersion relation of superconductor.

Figure 11

Schematic illustration of the scattering processes where filled (empty) circle indicates electron like quasiparticle (holelike quasiparticle).