Benefits of Weak Disorder in One-Dimensional Topological Superconductors

Majorana bound states are zero-energy modes localized at the ends of a one-dimensional (1D) topological superconductor. Introducing disorder usually increases the Majorana localization length, until eventually inducing a topological phase transition to a trivial phase. In this Letter, we show that in some cases weak disorder causes the Majorana localization length to decrease, making the topological phase more robust. Increasing the disorder further eventually leads to a change of trend and to a phase transition to a trivial phase. Interestingly, the transition occurs at ${\xi }_0\gg l$, where $l$ is the disorder mean free path, and ${\xi }_0$ is the localization length in the clean limit. Our results are particularly relevant to 1D topological superconductors formed in planar Josephson junctions.

Figure 1

(a) Phase diagram of the planar Josephson junction Eq. (1) in the clean limit. In the topological phase ($\mathcal{Q}=-1$), the system supports zero-energy MBSs at each end of the junction. (b) The Majorana localization length $\xi$ versus the disorder-induced inverse mean free time ${\tau }^{-1}$ for different points inside the topological phase [see markers in (a)]. At weak disorder, $\xi$ decreases with disorder. For stronger disorder, $\xi$ increases until eventually diverging at the phase transition to the trivial phase. Here, $\xi$ is averaged over 100 disorder realizations. The system parameters are $E_{\mathrm{SO}}=m_{\mathbf{\text{e}}}{\alpha }^2/2=1$, ${\mathrm{\Delta }}_0=1$, ${\mu }_{\mathbf{\text{J}}}={\mu }_{\mathrm{SC}}=2.5$, $E_{\mathbf{\text{Z}},\mathrm{SC}}=0$, $l_{\mathrm{SO}}=1/m_{\mathbf{\text{e}}}\alpha =0.2W$, $W_{\mathrm{SC}}=W$. In (b), $\tau$ is normalized by the overall gap in the clean system $E_{\mathrm{gap}}$, which is 0.032, 0.032, 0.022, 0.008, and 0.004 for the red, brown, blue, green, and black plots, respectively. Similarly, $\xi$ is normalized by that of the clean system ${\xi }_0$, which is $24W$, $34W$, $36W$, $93W$, and $194W$, in the same order.

Figure 2

(a) Andreev reflection (AR) in the $m$th channel. (b) With disorder, the electron can first scatter to the $n$th channel, perform AR, and then scatter back to the $m$th channel.

Figure 3

(a) Example of the low-energy spectrum of the normal strip ($|y|<W/2$) before it is coupled to the SCs. Red arrows show the spin expectation value averaged over $y$, $\int dy⟨\mathit{\sigma }(y)⟩$. The direction of the arrow indicates the direction of the spin in the ($xy$) plane. (b) Upon introducing the superconductors, a gap is induced. For the parameters considered here, the junction supports five gapped transverse channels. (c),(d) Localization length versus disorder in a $p$-wave SC ${\xi }_{\mathbf{\text{p}}}$ (black dotted line) and in an $s$-wave SC ${\xi }_{\mathbf{\text{s}}}$ (black dashed line). Disorder increases ${\xi }_{\mathbf{\text{p}}}$ and decreases ${\xi }_{\mathbf{\text{s}}}$. A multichannel TSC can sometimes be viewed as a combination of a $p$-wave SC and an $s$-wave SC. The overall localization length is determined by the larger between ${\xi }_{\mathbf{\text{p}}}$ and ${\xi }_{\mathbf{\text{s}}}$. (c) When the $p$-wave gap $|{\mathrm{\Delta }}_{\mathbf{\text{p}}}|$ exceeds the $s$-wave gap $|{\mathrm{\Delta }}_{\mathbf{\text{s}}}|$, the overall $\xi$ shows nonmonotonic behavior as a function of disorder strength (blue solid line). This is the situation in the planar JJ studied here. (d) When $|{\mathrm{\Delta }}_{\mathbf{\text{p}}}|\le |{\mathrm{\Delta }}_{\mathbf{\text{s}}}|$, disorder causes $\xi$ to increase monotonically, reaching a phase transition at $1/2{\tau }_{\mathbf{\text{c}}}=|{\mathrm{\Delta }}_{\mathbf{\text{p}}}|$.

Figure 4

(a) Majorana localization length versus disorder in a 2DEG coupled to an $s$-wave SC in two different geometries. In the first geometry (blue), the smaller superconducting gap is at the large-momentum channels, while in the second geometry (red), the gaps of the different channels are of the same order. The system’s parameters are as in Fig. 1, with $E_{\mathbf{\text{Z}},\mathbf{\text{J}}}=1$ and with ${\mathrm{\Delta }}_0=1$ (${\mathrm{\Delta }}_0=0.25$) for the first (second) geometry (the reduced ${\mathrm{\Delta }}_0$ in the second geometry is necessary for the system to be topological). (b) Majorana localization length for the planar JJ [see Eq. (1) and Fig. 1] for different ratios of magnetic-disorder strength to potential-disorder strength. Magnetic disorder couples opposite-spin modes. The rest of the system’s parameters are as in Fig. 1, with $\varphi =\pi$, $E_{\mathbf{\text{Z}},\mathbf{\text{J}}}=1$.