Majorana corner and hinge modes in second-order topological insulator/superconductor heterostructures

Abstract

As platforms of Majorana modes, topological insulator (quantum anomalous Hall insulator)/superconductor (SC) heterostructures have attracted tremendous attention over the past decade. Here we substitute the topological insulator by its higher-order counterparts. Concretely, we consider second-order topological insulators (SOTIs) without time-reversal symmetry and investigate SOTI/SC heterostructures in both two and three dimensions. Remarkably, we find that such novel heterostructures provide natural realizations of second-order topological superconductors (SOTSCs) which host Majorana corner modes in two dimensions and chiral Majorana hinge modes in three dimensions. As here the realization of SOTSCs requires neither special pairings nor magnetic fields, such SOTI/SC heterostructures are outstanding platforms of Majorana modes and may have wide applications in future.Over the past decade, topological superconductors (TSCs) have attracted continuous and tremendous attention[1-9]. Among various TSCs, one-dimensional (1d) and twodimensional (2d) TSCs without time-reversal symmetry (TRS) have attracted particular interest as they harbor Majorana zero modes (MZMs) at their boundaries[10-12] and in the cores of vortices [13][14][15][16], respectively. Owing to their fractional nature, MZMs are ideal candidates to construct nonlocal qubits immune to local decoherence[10]. Moreover, owing to their non-Abelian statistics [17], their braiding operations are found to realize elementary quantum gates. Thus, MZMs are believed to be building blocks of topological quantum computation [18] and have been actively sought in experiments [19][20][21][22][23][24][25][26][27][28][29].As is known, odd-parity superconductors (SCs) provide natural realizations of TSCs, however, they are unfortunately rare in nature. In a seminal paper [14], Fu and Kane pointed out that topological insulator (TI)/SC heterostructures provide an effective realization of odd-parity superconductivity. Accordingly, in the presence of magnetic field, vortices emerging in such heterostructures are found to carry MZMs. In a later influential paper[30], Qi et al pointed out that quantum anomalous Hall insulator (QAHI)/SC heterostructures provide a simple realization of 2d chiral TSCs which harbor not only vortex-core MZMs, but also chiral Majorana edge modes. These two theoretical works have triggered a lot of experimental works on TI(QAHI)/SC heterostructures [28,29,[31][32][33][34][35][36][37][38][39][40][41], and remarkable progress in detecting vortex-core MZMs has been witnessed in recent years [28,29,40,41].Very recently, TIs and TSCs have been generalized to include their higher-order counterparts [42][43][44][45][46][47][48][49][50][51][52][53][54][55][56]. Importantly, higher-order TIs (HOTIs) and TSCs (HOTSCs) have extended the conventional bulk-boundary correspondence. Accordingly, an n-th order TI or TSC in d dimensions host (d − n)dimensional boundary modes. For instance, a second-order TI (SOTI) in 2d and 3d host zero-dimensional (0d) corner modes an...

Higher-order topological insulator in cubic semiconductor quantum wells

“…After the tremendous interest in II–VI and III–V semiconductor QWs induced by prediction and observation of QSHI 39 – 41 , our work shows the importance of cubic semiconductor QWs for the realization of high-order topological states as well. In view of mature growth of IV, II–IV and III–V semiconductor QWs on Si-wafers 64 , 65 as well as device fabrication technology, our results provide an important first step to future realistic electronics operating on the basis of higher-order topological states including higher-order topological superconductors in hybrid devices 66 , 67 .…”

Higher-order topological insulator in cubic semiconductor quantum wells

“…After the tremendous interest in II-VI and III-V semiconductor QWs induced by prediction and observation of QSHI [39][40][41], our work shows the importance of cubic semiconductor QWs for the realization of high-order topological states as well. In view of mature growth of IV, II-IV and III-V semiconductor QWs on Si-wafers [64,65] as well as device fabrication technology, our results provide an important first step to future realistic electronics operating on the basis of higher-order topological states including higher-order topological superconductors in hybrid devices [66,67].…”

Quadrupole spin polarization as signature of second-order topological superconductors

“…SOTSCs with four corner states. Next, we generalize our findings to a different class of SOTSCs and consider a SOTSC hosting a quadruplet of MCSs [39][40][41][42][43][44][45][46][47][48]. The basic ingredient of our construction here is a minimalist version of a 2D TI [3] proximity coupled to an s-wave superconductor with amplitude sc , introduced in Ref.…”