Fluxoid valve effect in full-shell nanowire Josephson junctions

Abstract

We introduce a new type of supercurrent valve based on full-shell nanowires. These hybrid wires consist of a semiconductor core fully wrapped in a thin superconductor shell and subjected to an axial magnetic field. Due to the tubular shape of the shell, the superconductor phase acquires an integer number $n$ of $2π$ twists or \textit{fluxoids} that increases in steps with applied flux. By connecting two such hybrid wires, forming a Josephson junction (JJ), a flux-modulated supercurrent develops. If the two superconducting sections of the JJ have different radii $R_1$ and $R_2$, they can develop equal or different fluxoid numbers $n_1,n_2$ depending on the field. If $n_1\neq n_2$ the supercurrent is blocked, while it remains finite for $n_1=n_2$. This gives rise to a fluxoid valve effect controlled by the applied magnetic field or a gate voltage at the junction. We define a fluxoid-valve quality factor that is perfect for cylindrically symmetric systems and decreases as this symmetry is reduced. We further discuss the role of Majorana zero modes at the junction when the full-shell nanowires are in the topological superconducting regime.

Figures (4)

• Figure 1: Sketch of a S$_1$NS$_2$ short JJ between two full-shell hybrid nanowires in a cylindrical approximation with different mid radii $R_1$ and $R_2$. The semiconductor core is represented in yellow and the thin superconductor shells of thickness $d$ in blue. Radially, most of the core charge density is concentrated close to the superconductor-semiconductor interface in an accumulation layer of thickness $W$. S$_1$ and S$_2$ may have different diffusive coherence lengths $\xi_1$ and $\xi_2$. An axial magnetic field $B$ is applied. $T_{\rm{N}}$ is the normal transmission that characterizes the weak link.
• Figure 2: (a,b) Local density of states (LDOS) at the end of a semi-infinite full-shell hybrid nanowire with mid radius $R_1$ (a) and $R_2$ (b), as a function of energy $\omega$ and applied magnetic field $B$. Several Little-Parks (LP) lobes are displayed characterized by fluxoid numbers $n=1, 2, 3...$ (c) Critical current $I_c$ (normalized to the superconducting quantum unit Beenakker:92$e |\Delta_0^*|/\hbar$, where $|\Delta_0^*|$ is the induced gap at $B=0$) as a function of $B$ for a short junction with normal transmission $T_{\rm{N}}$. Red curve corresponds to a cylindrically symmetric junction. A perfect valve effect is achieved with $I_c=0$ in $B$ intervals $B^{n_1}_{n_2}$ where $n_1\neq n_2$. The valve effect is partially lifted (green and blue curves) for junctions with broken cylindrical symmetry, characterized by mode mixing parameter $\delta\tau$. Inset: $I_c$ vs $\delta\tau$ for two different values of $B$. (d) Fluxoid-valve quality factor $Q^{\rm{FV}}$ versus $B^{n_1}_{n_2}$ for different values of $\delta\tau$. Parameters: $R_1=65$nm, $R_2=60$nm, $W = 20$nm, $d\rightarrow 0$, $|\Delta_0| = 0.23$meV, $\xi_1=\xi_2 = 70$nm, $m^*=0.023 m_e$, $\alpha\rightarrow 0$, $\mu = 0.5$meV, $g = 1$, $\Gamma = |\Delta_0|$, $T_{\rm{N}}=0.7$ and $a_0 = 5$nm.
• Figure 3: Same as Fig. \ref{['fig:triv']} but for a JJ with full-shell hybrid nanowires that can enter the topological superconducting regime. Majorana zero-energy peaks appear in the LDOS of (a,b) for certain flux intervals in odd LP lobes. For high junction transparency (c), $I_c$ is very similar to the trivial JJ case. However, for tunnel JJs (d), $I_c$ is affected by the presence of MZMs (whose contribution is highlighted with red shaded areas). Parameters as in Fig. \ref{['fig:triv']} except for $\alpha = 20$ meVnm and $g = 0$.
• Figure 4: Same as Fig. \ref{['fig:triv']}(a-c) but for superconducting sections with the same radius and with different diffusive coherence lengths $\xi_1\neq\xi_2$. S$_1$ is in the nondestructive LP regime, whereas S$_2$ is in the destructive one. Parameters as in Fig. \ref{['fig:triv']} except for $R_1=R_2=65$nm, $\xi_1=70$nm, $\xi_2=140$nm, and $\mu = 2$ meV.