Finite Thickness Effects on Metallization Vs. Chiral Majorana Fermions

Abstract

The search for chiral Majorana fermions in quantum anomalous Hall insulator/\textit{s}-wave superconductor heterostructures has attracted intense interest, yet remains controversial due to the lack of conclusive evidence. A key issue is that the heterostructure's metallization can produce half-integer conductance signatures resembling those of chiral Majorana fermions, thereby complicating their identification. In this Letter, we investigate how the competition between metallization and chiral Majorana fermions depends on superconductor thickness, revealing its critical role through three distinct regimes: (i) For thin superconductors ($\sim$10 nm), metallization shows periodic oscillations with thickness, matching the Fermi wavelength. (ii) Intermediate thicknesses ($\sim$100 nm) exhibit periodic windows for observing chiral Majorana fermions. (iii) Thick superconductors ($\sim$1000 nm) sustain stable chiral Majorana fermions that are insensitive to thickness variations. These results suggest that superconductor thickness is a key control parameter for advancing efforts to conclusively identify chiral Majorana fermions.

Figures (5)

• Figure 1: Heterostructure composed of a superconductor (SC) layer with thickness $d$ layered atop a Quantum Anomalous Hall (QAH) insulator layer.
• Figure 2: (a,b) A schematic illustration of edge state distribution in the $x-y$ plane. The edge states near $y=0$ propagate to the $+x$ direction, and the edge states near $y=L_y$ propagate to the $-x$ direction. (c,d) Edge state distribution in the $y$ direction, which is plotted by the analytical solution of wave function $\bm{\Psi}$. Only the first component of $\bm{\Psi}$ is plotted. The parameters are $\Delta_{\text{eff}}=1$, $A=B=1$ and $L_y=N_ya=40$ with $a=1$. The system exhibits one edge states ($\mathcal{N}=1$) on each edge when $m=-0.5$ ($|m|<\Delta_{\text{eff}}$), as illustrated in panels (a) and (c). In contrast, it displays two edge state ($\mathcal{N}=2$) on each edge for $m=-1.5$ ($m<-\Delta_{\text{eff}}$), as shown in panels (b) and (d). The edge states localized in left (right) are labeled by solid (dashed) lines.
• Figure 3: Energy spectrum of the holistic Hamiltonian with periodic boundary conditions in the $x$ direction and open boundary conditions in the $y$ direction. The parameters are set as follows: $\mu_m = 0$, $\mu_s = 5 \, \text{eV}$, $m_s = 9.1\times10^{-31} \, \text{kg}$, $\Delta_s = 1.5 \, \text{meV}$, $T=100 \,\text{meV}$, $A = 3 \, \text{eV\AA}$, and $B = 15 \, \text{eV\AA}^2$, with a lattice constant $a = 7.5 \, \mathrm{nm}$ and a length $L_y = N_y a = 1.5 \, \mathrm{\mu m}$ along the $y$ direction. Subplots (a-c) depict different values of the parameter $m$: (a) $m = -3 \, \text{meV}$, (b) $m = 0 \, \text{meV}$, and (c) $m = 3 \, \text{meV}$, corresponding to regions $\mathcal{N}=2$, $\mathcal{N}=1$, and $\mathcal{N}=0$, respectively. (b) illustrates the metallization effect, while (d) is an enlarged view of (b), highlighting the tiny gap at the micro-electronvolt scale. (e) Energy gap as a function of $T$ and $m$, denoted as $E_{\text{gap}}(m, T)$. The color scale represents the $E_{\text{gap}}$ in meV, with regions labeled $\mathcal{N}=0, 1, 2$ indicating different topological phases. (f) Energy gap as a function of $m$ under the condition $T = 100 \, \text{meV}$. The color represent different topological regeion. The parameters used in (e) and (f) are the same as those in (a).
• Figure 4: (a) The supercondutor's chemical potential $\mu_s$ and induced chemical potential $\mu_{\text{ind}}$ versus thickness of superconductor layer $d$. (b) The supercondutor's engergy gap $\Delta_s$ and induced engergy gap $\Delta_{\text{ind}}$ versus $d$. The intrinsic qunatities converge to the bulk value $\mu_s ^{\text{3D}}, \Delta_s ^{\text{3D}}$ soon while the induced qunatities continue to ossilate. (c) The induced chemical potential $\mu_{\text{ind}}$ versus $d$ for wider range (light blue line). The dark blue line is the average of the light blue line. (b) Proximity-induced gap $\Delta_{\text{ind}}$ versus $d$ for wider range (light red line). The dark red line is an average of light red line. The dashed line is the analytical results in 3D limit when the superconductor is thick enough. The parameters are set as: $\mu_m = 0$, $\mu_s ^{\text{3D}} = 5 \, \text{eV}$, $m_s = 9.1\times10^{-31} \, \text{kg}$, $\hbar \omega_D=100K$, $\Delta_s^{\text{3D}} = 1.5 \, \text{meV}$, $T_0^2/k_F = 20000 \, [\text{meV}]^2$, $\kappa =k_F$ .
• Figure 5: Energy gap diagrams for various thicknesses $d$ (a) $d = 5.00$ nm, (b) $d = 5.18$ nm, (c) $d = 200.00$ nm, and (d) $d = 200.12$ nm. Other parameters are set to be: $\mu_m = 0$, $\mu_s = 5 \, \text{eV}$, $m_s = 9.1\times10^{-31} \, \text{kg}$, $\Delta_s = 1.5 \, \text{meV}$, $A = 3 \, \text{eV\AA}$, $B = 15 \, \text{eV\AA}^2$, and $\kappa =k_F$.