Size optimization for observeing Majorana fermions

Abstract

Majorana fermions (zero modes) are predicted to emerge in nanowire-superconductor heterostructures. This theoretical prediction typically relies on an oversimplified model, where both the nanowire and the superconductor are idealized as one-dimensional systems. In reality, heterostructures have finite sizes that deviate from this idealization-and as a result, smoking-gun evidence confirming the existence of these zero modes remains elusive. Here, we investigate the finite-size effects of both the nanowire and the superconductor, and optimize their sizes to ensure that only one Majorana fermion exists at each end of the heterostructure. It is discovered that the optimal transverse sizes of the nanowire are less than 100nm in width and approximately 1nm in thickness. For the superconductor layer, its optimal thickness (a key aspect of its size) must exceed its coherence length. We also present the optimal sizes of the two types of materials used in the experiment in a quantitative manner. Notably, the identified optimal thickness of the superconductor (Al films, $\sim$1000nm)--a critical size parameter--is two orders of magnitude larger than the thickness of Al films currently utilized in experimental devices (e.g., InSb-Al and InAs-Al heterostructures). Our findings could explain why Majorana fermions have not been observed in current experiments, and offer guidance for the size selection of heterostructures to implement Majorana fermions in future studies.

Figures (2)

• Figure 1: A finite-size nanowire is in contact with a finite-size $s$-wave superconductor. The sizes in the $y$ and $z$ directions are finite and significantly smaller than the size in the $x$ direction. The sizes in the $x$ and $y$ direction are identical for both the nanowire and the superconductor, i.e., $L_{x}^{w}=L_{x}^{s}$ and $L_{y}^{w}=L_{y}^{s}$.
• Figure 2: (a) For an InSb nanowire with $L_{w}^{y}=L_{w}^{z}=70\mathrm{nm}$, coated with a 10 nm thick Al film, the number of Majorana fermions in the parameter of $\mu_w$ and $B$. (b) For an InSb nanowire with $L_{w}^{y}=70$ nm and $L_{w}^{z}=1$ nm, the number of Majorana fermions varies with changes in superconducting thickness and chemical potential in the nanowire at a constant magnetic field $B=0.8$T. (c) For a given chemical potential of the nanowire $\mu_w=0.5\Delta_s$, the number of Majorana fermions is determined by the superconducting thickness and the applied magnetic field. For a given subband of nanowire characterized by $\boldsymbol{k}_{w}=(2\pi/(N_{y}^{w}+1),k_{z}^{w})$, the induced chemical potential (d) and the induced gap (e) vs. thickness of superconductor. (f) and (g) display the constraint relation for the chemical potential window and the induced energy gap for InAs-Al and InSb-Al. The maximum of chemical potential window reaches 1.73(4.63) meV for InAs-Al (InSb-Al), as indicated by the red circular dots. The maximum of the induced energy gap is 0.48$\Delta_{s}$ (0.67$\Delta_{s}$) for InAs-Al (InSb-Al), indicated by green triangles. The parameters are set as $\Delta_{s}=0.34\,\mathrm{meV}$, $\mu_{s}=1\,\mathrm{eV}$, $a=0.85\mathrm{\AA}$, $t_s \simeq 0.015t_w=10\,\mathrm{eV}$ and $T_{k_{z}^{w}}\simeq 27.5\,\mathrm{meV}$.