Chiral topological superconductor and half-integer conductance plateau from quantum anomalous Hall plateau transition

We propose to realize a two-dimensional chiral topological superconducting (TSC) state from the quantum anomalous Hall plateau transition in a magnetic topological insulator thin film through the proximity effect to a conventional $s$-wave superconductor. This state has a full pairing gap in the bulk and a single chiral Majorana mode at the edge. The optimal condition for realizing such chiral TSC is to have inequivalent superconducting pairing amplitudes on top and bottom surfaces of the doped magnetic topological insulator. We further propose several transport experiments to detect the chiral TSC. One unique signature is that the conductance will be quantized into a half-integer plateau at the coercive field in this hybrid system. In particular, with the point contact formed by a superconducting junction, the conductance oscillates between $e^2/2h$ and $e^2/h$ with the frequency determined by the voltage across the junction. We close by discussing the feasibility of these experimental proposals.

Figure 1

The hybrid QAH-SC device. In region II, a chiral TSC state is induced through the proximity effect to an $s$-wave SC layer on top of the QAH in magnetic TI. A back-gate voltage $V_{\mathrm{bg}}$ is applied to control the Fermi level in region II. Voltages $V_1$ and $V_2$ are applied on leads 1 and 2, respectively. The SC layer is grounded through a lead in its bulk.

Figure 2

Phase diagram of the QAH-SC hybrid system with typical parameters. (a) ${\mathrm{\Delta }}_1=\mathrm{\Delta }, {\mathrm{\Delta }}_2=0, \mu =0$. (b) ${\mathrm{\Delta }}_1={\mathrm{\Delta }}_2=\mathrm{\Delta }, \mu =0$. (c) ${\mathrm{\Delta }}_1=\mathrm{\Delta }, {\mathrm{\Delta }}_2=0, \mu =0.7$. (d) ${\mathrm{\Delta }}_1=-{\mathrm{\Delta }}_2=\mathrm{\Delta }, \mu =0.7$. Here ${\mathrm{\Delta }}_1, {\mathrm{\Delta }}_2, \mu$ are in the units of $|m_0|$.

Figure 3

(a) Phase diagram of the QAH-SC hybrid system for $\mu =0$ and ${\mathrm{\Delta }}_1=-{\mathrm{\Delta }}_2\equiv \mathrm{\Delta }$. Only $\mathrm{\Delta }\ge 0$ is shown. (b) Without SC proximity effect, the ${\sigma }_{xy}=-1\to 0\to 1$ QAH plateau transition occurs at the coercivity when the magnetization flips. (c) With SC proximity effect to region II in hybrid device Fig. 1, ${\sigma }_{12}$ shows plateau transition $1\to 1/2\to 0\to 1/2\to 1$ in the hysteresis loop. The half-integer plateau in ${\sigma }_{12}$ manifests the $\mathcal{N}=1$ TSC. (d)–(j) The edge transport configuration at A, B, C, D, ${\mathrm{C}}^{\prime }, {\mathrm{B}}^{\prime }$, and ${\mathrm{A}}^{\prime }$ in (c). There is no backscattering for $\mathcal{N}=\pm 2$ TSC in (d), (j), and Majorana backscattering for $\mathcal{N}=\pm 1$ TSC in (e), (i). Red and blue arrows represent $(c\pm c^{†})$ CMEMs, respectively. NSC: normal, topologically trivial SC.

Figure 4

(a) The point contact configuration of two SC islands with SC phases ${\varphi }_1$ and ${\varphi }_2$, across which the reflection and transmission amplitudes of the CMEMs are $r$ and $t$. (b) The conductance ${\sigma }_{12}^{\prime }$ as a function of $\tau$ for different coupling strengths $\xi$. A dc current flows between $a_1$ and $a_2$, an ac voltage between them is measured, with frequency $f=2e{V}_{\text{sc}}/h$.

Figure 5

The phase diagram for ${\mathrm{\Delta }}_2=e^{i{\varphi }_{\alpha }}{\mathrm{\Delta }}_1$ and $\mu =0$. When ${\varphi }_{\alpha }=0$, the $\mathcal{N}=\pm 1$ TSC phases disappear, while when ${\varphi }_{\alpha }=\pi$, the phase spaces of $\mathcal{N}=1$ and $\mathcal{N}=-1$ TSC touch each other, as indicated in Figs. 2 and 3 of the main text, respectively.