Intrinsic space-time crystalline order in a hybrid Josephson junction

TL;DR

The paper addresses whether intrinsic space-time crystalline order can arise in a hybrid superconducting–ferromagnetic system without external periodic driving. It develops a coupled sine-Gordon/Landau-Lifshitz-Gilbert model for a long φ0 Josephson junction on a topological insulator, where exchange interaction and Dzyaloshinskii–Moriya interaction internally modulate the critical current via J_c(m_x) and phase shift φ0 = r m_y. The main finding is that intrinsic STC order emerges in the in-plane current pattern, oscillating around 2 Ω_F, and is absent if the critical current is not modulated internally; when external parametric modulation is applied, a classical discrete STC appears at ω_mod/2. The work also proposes NV-center based nanoscale magnetometry as a feasible experimental probe to visualize and validate the STC patterns, highlighting potential routes for observing time-crystalline behavior in SFS heterostructures with broken inversion symmetry.

Abstract

We demonstrate the emergence of an intrinsic space-time crystalline order in a long ferromagnetic $\varphi_{0}$ Josephson junction on a topological insulator without any external periodic modulation. The presence of the exchange and Dzyaloshinskii-Moriya interactions in a ferromagnetic layer with broken inversion symmetry internally modulates the critical current due to the coupling between the magnetic moment and the Josephson phase. This breaks the time translation symmetry, leading to the appearance of the space-time crystalline pattern in the spatiotemporal dependence of the in-plane current, which oscillates with almost twice the ferromagnetic resonance frequency. In the limit where the critical current is not modulated internally, the space-time crystalline order does not occur. In this case, only when an external parametric modulation is applied, the system exhibits a typical classical discrete space-time crystalline order that oscillates at half of the modulation frequency. Considering the still-pending problem of experimental detection, we demonstrate that a recently developed magnetometry device, which visualizes the supercurrent flow at the nanoscale, can be used to detect space-time crystalline patterns in hybrid Josephson junctions experimentally.

Figures (9)

• Figure 1: Proposed long $\varphi _0$ Josephson junction deposited on top of the three-dimensional topological insulator. The arrows show the precession of magnetization in the F-layer.
• Figure 2: (a) Equivalent circuit for the long $\varphi _0$ Josephson junction. Each discrete element in the model is characterized by a Josephson current branch $j_{s}$. The discrete elements are inductively coupled, with currents $j_{1}$ and $j_{2}$ passing through the inductor branches. (b) The same as in (a) but with the quasiparticle current branch characterized by $R$, and the displacement current characterized by $C$. The lossy current is represented by the branch with resistance $R_{s}$.
• Figure 3: (a) Spatio-temporal dependence of the in-plane current $J(y,t)$ in the SFS-TI JJ for $c_{ex}=0.05$, $D_{1}=1.1$, and $D_{2}=0.8$. (b) Magnified view of the part in (a) that demonstrates the STC pattern. (c) The corresponding spatiotemporal averaged current-current correlation function $J^{cor}(\delta y, \delta t)$. (d) The Fast-Fourier transform for the $J(y,t)$ shown in (a). The parameters were fixed to the following values: $\Omega_{F}=1$, $\alpha_{g}=0.05$, $k_{an}=0.5$, $r=0.9$, $G=0.1$, $\beta=0.00024$, $\tilde{d}=0.3$, $\Gamma=0.0762$ and $J_{noise}$$10^{-4}$.
• Figure 4: Spatiotemporal dependence of the in-plane current $J(y,t)$, the current-current correlation function $J^{cor}(\delta y,\delta t)$ and the corresponding FFT analysis for DMI$=0$ and ExI$\neq 0$ (qualitatively the same for ExI$=$DMI$=0$) in (a)-(c), and DMI$\neq0$ and ExI$=0$ in (d-f), respectively. The rest of the parameters are the same as in Fig. \ref{['Fig3']}.
• Figure 5: (a) Magnified view of the spatiotemporal dependence of the in-plane current $J(y,t)$. (b) The corresponding averaged current-current correlation function $J^{cor}(\delta y,\delta t)$. All panels are created for the same parameters as in Fig. \ref{['Fig3']}, but in the extended time domain.