Majorana fermions of a two-dimensional $p_x+i{p}_y$ superconductor

To investigate Majorana fermionic excitations of a $p_x+i{p}_y$ superconductor, the Bogoliubov–de Gennes equation is solved on a sphere for two cases: (i) a vortex-antivortex pair at opposite poles and (ii) an edge near the south pole and an antivortex at the north pole. The vortex cores support a state of two Majorana fermions, the energy of which decreases exponentially with the radius of the sphere, independently of a moderate disorder potential. The tunneling conductance of an electron into the superconductor near the position of a vortex is computed for finite temperature and is compared to the case of an $s$-wave superconductor. The zero-bias conductance peak of the antivortex is half that of the vortex. This effect can be used as a probe of the order-parameter symmetry and as a direct measurement of the Majorana fermion.

Figure 1

(Color online) A vortex-antivortex pair of the $p_x+i{p}_y$ superconductor on the sphere, described by Eq. (6). Thin black lines represent the current flow. Wide green arrows represent the pair relative angular momentum. ${\xi }_p$ is the pairing range. $\xi$ is the coherence length, which determines the vortex core size.

Figure 2

(Color online) BdG spectrum $E_{n,m}$, of the vortex pair on the sphere, depicting the CdGM core states. The inset shows that their double degeneracies are split by weak tunneling between the poles. The positive energy member of the doublet, which saddles zero energy at $m=\frac{1}{2}$, is the Majorana state shared by the vortex and antivortex.

Figure 3

(Color online) Energy of Majorana state $E_{0^{+}}$ as a function of sphere radius $R$. The exponentially decreasing energy indicates tunnel splitting between vortex and antivortex core states.

Figure 4

(Color online) Probability densities of Majorana states ${\left|u_0\right|}^2$ versus latitude on the sphere $\theta$. The smooth parts of the Majorana state wave functions $u_{0^{+}}\left(\theta \right)$ and $u_{0^{-}}\left(\theta \right)$ are approximately symmetric and antisymmetric with respect to reflection about the equator $\theta =\pi /2$. Both ${\left|u_{0^{+}}\right|}^2$ (blue) and ${\left|u_{0^{-}}\right|}^2$ (green) show the exponential localization in the (anti)vortex cores.

Figure 5

(Color online) Probability distributions of Majorana state ${\left|u_{0^{+}}\right|}^2$ as a function of the distance from the (anti)vortex center $r$, at the cores vicinities. The wave functions have exponentially small weights near the equator. The asymptotic (red) and the numeric (blue) wave functions show excellent agreement.

Figure 6

(Color online) The BdG spectrum $E_{n,m}$, of an $s$-wave superconductor, with the same physical parameters of the $p_x+i{p}_y$ in Fig. 2. The distinction from the $p_x+i{p}_y$ case can be noticed in the inset, where the energies of the CdGM states are shifted by half the level spacing, therefore lacking a Majorana state which saddles zero energy.

Figure 7

Three white-noise potentials $W\left(\mathbf{\Omega }\right)$ on the sphere, taken from the same distribution of harmonics components $w_{lm}$ but of increasing high angular momentum cutoff $l_{\Lambda }$ (from top to bottom). Each time $l_{\Lambda }$ is multiplied by 2, while the disorder strength $W_0$ remains constant.

Figure 8

(Color online) Disorder averaged energy of Majorana state $E_{0^{+}}$ versus $R/\xi$ for increasing disorder strength $W_0$ (from top to bottom). The exponential decay survives in the regime $W_0<{ϵ}_F$.

Figure 9

(Color online) BdG spectrum $E_{n,m}$, of an edge without a vortex, depicting edge states. The solid black line is $m{\Delta }_0/l_F$. Here $R/\xi =7.6$, while the fit improves for larger $R/\xi$.

Figure 10

(Color online) Energies of the two lowest edge states without a vortex $E_m^{\text{edge}}$, for $m=\frac{1}{2}$ and $m=\frac{3}{2}$, versus $\xi /R$, show the $1/R$ decay.

Figure 11

(Color online) BdG spectrum $E_{n,m}$, of an antivortex with an edge, showing both the edge states and CdGM states. The Majorana state is present, with an exponentially small energy, as seen in the inset.

Figure 12

(Color online) $\text{log}{\left|u\left(\theta \right)\right|}^2$ of the Majorana state $u_{0^{+}}$ (top), the first excited edge state $u_{3/2}^{\text{edge}}$ (middle), and the first excited CdGM state $u_{-1/2}^c$ (bottom). The Majorana state is split between the edge and the vortex core, while in each branch of excitation the wave function is concentrated either in edge or in the vortex core.

Figure 13

(Color online) Energies of Majorana state $E_{0^{+}}$ (bottom) and first excited edge state $E_{3/2}^{\text{edge}}$ (top) of an antivortex with an edge as a function of the radius of sphere $R$. The energy of the Majorana state decays exponentially, while the energy of the edge state scales as $1/R$.

Figure 14

(Color online) Zero-temperature LDOS $\mathcal{T}\left(E,r\right)$, in the cores of the $p_x+i{p}_y$ antivortex (top) and vortex (middle), and of the $s$-wave vortex (bottom). ${ϵ}_c$ is the spacing between CdGM states, and ${\lambda }_F$ is the Fermi wavelength. The peaks belong to the CdGM states. Notice that the zero energy Majorana state is maximized at the origin in the $p_x+i{p}_y$ antivortex, while it is removed from the origin in the vortex, and is absent in the $s$-wave.

Figure 15

(Color online) Tunneling conductance $\frac{dI}{dV}\left(E,r\right)$, in arbitrary units, of the $p_x+i{p}_y$ antivortex (top) and vortex (middle), and of the $s$-wave vortex (bottom). ${\Delta }_0$ is the energy gap, and $\xi$ is the coherence length, which specifies the radiuses of the cores. The temperature $T=0.15{\Delta }_0$, is about 10 times larger than the CdGM level spacing. The conductance of the $p_x+i{p}_y$ vortex is almost identical to that of the $s$-wave, while the central peak of the antivortex is twice lower.

Figure 16

(Color online) Zero-bias conductance peak at the (anti)vortex core as a function of temperature $T$ in log-log scale. For ${ϵ}_c<T<{\Delta }_0$ the $p_x+i{p}_y$ vortex (green) and $s$-wave (blue) peaks are twice the height of the antivortex peak (red).

Figure 17

(Color online) Effect of spatial resolution. Ratio of vortex to antivortex conductance peak heights (shown in Fig. 16 for perfect resolution) for spatial resolution $\delta r$. ${\lambda }_F$ and $\xi$ are Fermi wavelength and coherence lengths, respectively. Temperature is $7.5{ϵ}_c$.