Majorana fermions in Ge/Si hole nanowires

We consider Ge/Si core/shell nanowires with hole states coupled to an $s$-wave superconductor in the presence of electric and magnetic fields. We employ a microscopic model that takes into account material-specific details of the band structure such as strong and electrically tunable Rashba-type spin-orbit interaction and $g$ factor anisotropy for the holes. In addition, the proximity-induced superconductivity Hamiltonian is derived starting from a microscopic model. In the topological phase, the nanowires host Majorana fermions with localization lengths that depend strongly on both the magnetic and electric fields. We identify the optimal regime in terms of the directions and magnitudes of the fields in which the Majorana fermions are the most localized at the nanowire ends. In short nanowires, the Majorana fermions hybridize and form a subgap fermion whose energy is split away from zero and oscillates as a function of the applied fields. The period of these oscillations could be used to measure the dependence of the spin-orbit interaction on the applied electric field and the $g$ factor anisotropy.

Figure 1

Sketch of a Ge/Si core/shell NW (cylinder) placed on top of an $s$-wave superconductor (SC) to induce proximity pairing of the holes in the NW core. The NW axis is chosen to point along the $z$ axis. The applied electric field $\mathit{E}=(E_x,0,0)$ is parallel to the $x$ axis, while the applied magnetic field $\mathit{B}=(B_x,0,B_z)\equiv B(cos\theta ,0,sin\theta )$ is in the $xz$ plane.

Figure 2

The lowest-energy spectra $E_u\left(k_z\right)$ (red) and $E_d\left(k_z\right)$ (blue) as functions of the momentum $k_z$ (a) in the strong SOI regime and (b) in the weak SOI regime. The respective magnitudes of the applied fields are given as insets. The magnetic field $\mathit{B}$ is chosen along the $x$ axis (like the $\mathit{E}$ field). The used NW parameters are $R=7.5\text{nm}$ and $\Delta =23\text{meV}$.

Figure 3

Localization lengths ${\xi }_s^e$ and ${\xi }_s^i$ as functions of the angle $\theta$ defined by $\mathit{B}=B(cos\theta ,0,sin\theta )$, for $E_x=4\mathrm{V}/\mu \mathrm{m}$ and for different magnitudes of $B$. As soon as $\theta$ deviates from 0, ${\xi }_s^i$ increases until its value diverges when the topological gap closes. Furthermore, increasing $B$ above a threshold value given by $|{\Delta }_{-}|=|{\Delta }_{sc}|$, i.e., ${\Delta }_Z^0\ge 2\left|{\Delta }_{sc}\right|$, results in ${\xi }_s^i<{\xi }_s^e$ for a certain range of $\theta$. Since ${\xi }_s^e$ is independent of $\mathit{B}$, its value appears as a constant in the plot. We use the NW parameters $R=7.5\text{nm}$ and $\Delta =23\text{meV}$ and assume a superconductivity pairing potential of ${\Delta }_{sc}=200\mu \text{eV}$.

Figure 4

Localization lengths ${\xi }_w^{\left(e\right)}$ and ${\xi }_w^{\left(i\right)}$ as functions of the angle $\theta$ defined by $\mathit{B}=B(cos\theta ,0,sin\theta )$, for different magnitudes of $E_x$ and $B$. Here we use the combinations $E_x=1\mathrm{V}/\mu \mathrm{m}$ and $B=2\text{T}$ (black), $E_x=2\mathrm{V}/\mu \mathrm{m}$ and $B=1.5\text{T}$ (red). As soon as $\theta$ deviates from 0, ${\xi }_w^{\left(i\right)}$ increases until its value diverges when the topological gap closes. In contrast to ${\xi }_s^e$ (see Fig. 3), ${\xi }_w^{\left(e\right)}$ now shows a dependence on $\theta$. Here, we use the same values for the NW parameters and superconducting pairing parameter as in Fig. 3.

Figure 5

Logarithm (color coded) of the dominating localization length, $\xi =max\{{\xi }_s^e,{\xi }_s^i\}$ and ${\xi }_w^{\left(e\right)}$, in the strong and weak SOI regime, respectively, as functions of the applied fields $E_x$ and $\mathit{B}$. For simplicity we restrict ourselves to $\mathit{B}=(B_x,0,0)$. The diagonal gray area approximately denotes the transitional regime between regions of strong and weak SOI. Here, we use the same values for the NW parameters and superconducting pairing parameter as in Fig. 3.

Figure 6

The fermion energy $E_n^s$ in the strong SOI regime as a function of the electric field $E_x$ for different magnitudes of the applied magnetic field $\mathit{B}=(B_x,0,0)$; see the insets. We consider three different ratios of the localization lengths ${\xi }_s^e/{\xi }_s^i$. If the contribution of the interior branches is dominant, ${\xi }_s^i>{\xi }_s^e,E_n^s$ shows an increasing offset from zero with weak superimposed oscillations on top of it. If the contribution of the exterior branches is dominant, ${\xi }_s^e>{\xi }_s^i,E_n^s$ shows an increasing offset from zero with strong superimposed oscillations on the top of it such that $E_n^s$ periodically returns to zero. Here, we assume a superconducting pairing potential ${\Delta }_{sc}=200\mu \mathrm{eV}$ and use the NW parameters $R=7.5\text{nm},\Delta =23\mathrm{meV}$, and $L=0.7\mu \mathrm{m}$.

Figure 7

The oscillating energy $E_n^w$ as a function of $\mathit{B}=(B_x,0,0)$. We display both the results from solving the implicit Eq. (26) numerically (red) and the simplified solution given in Eq. (27) (blue) and find good agreement. We assume a weak electric field $E_x=0.5\mathrm{V}/\mu \mathrm{m}$ and a proximity induced superconducting pairing potential ${\Delta }_{sc}=200\mu \mathrm{eV}$. We use the NW parameters $R=7.5\text{nm},\Delta =23\text{meV}$, and $L=2\mu \mathrm{m}$.