Cryogenic Memory Element Based on an Anomalous Josephson Junction

We propose a nonvolatile memory element based on a lateral ferromagnetic Josephson junction with spin-orbit coupling and out-of-plane magnetization. The interplay between the latter and the intrinsic exchange field of the ferromagnet leads to a magnetoelectric effect that couples the charge current through the junction and its magnetization, such that by applying a current pulse the direction of the magnetic moment in $F$ can be switched. The two memory states are encoded in the direction of the out-of-plane magnetization. With the aim to determine the optimal working temperature for the memory element, we explore the noise-induced effects on the averaged stationary magnetization by taking into account thermal fluctuations affecting both the Josephson phase and the magnetic moment dynamics. We investigate the switching process as a function of intrinsic parameters of the ferromagnet, such as the Gilbert damping and strength of the spin-orbit coupling, and propose a nondestructive readout scheme based on a dc superconducting quantum interference device. Additionally, we analyze a way to protect the memory state from external perturbations by voltage gating in systems with a both linear-in-momentum Rashba and Dresselhaus spin-orbit coupling.

Figure 1

$S$-$F$-$S$ Josephson junction driven by a rectangular bias current pulses, $I_{\mathrm{bias}}$, with amplitude $I_{\max }$. The $z$ component of the magnetization, $M_z$, is the observable used to define the logic memory states 0 and 1.

Figure 2

Stationary magnetization, $m_z^{\mathrm{st}}$, as a function of $r$ and $\gamma$, at different values of the amplitude $I_{\max }$ of the current pulse, in the absence of noise fluctuations. The other parameters are $\epsilon =10$, $\omega =1$, $\sigma =5$, and $m_z(t=0)=+1$.

Figure 3

Stationary magnetization, $m_z^{\mathrm{st}}$, as a function of the current pulse width, $\sigma$, at different values of the amplitude $I_{\max }\in [0.8–1.5]$, in the absence of noise fluctuations. The other parameters are $r=0.1$, $\gamma =0.25$, $\epsilon =10$, $\omega =1$, and $m_z(t=0)=+1$.

Figure 4

(a) Stationary magnetization, $m_z^{\mathrm{st}}$, as a function of $r$ and $\gamma$. (b) Average stationary magnetization, $\overline{m_z^{\mathrm{st}}}$, as a function of $r$ and $\gamma$, at $D_I=0.01$, calculated by averaging over $N_{\exp }=100$ independent numerical repetitions. (c) Average stationary magnetization, $\overline{m_z^{\mathrm{st}}}$, as a function of the thermal-current intensity, $D_I$, at $\gamma =0.25$ and $r=0.1$ [namely, the $(\gamma ,r)$ values highlighted with a red circle in (b)], calculated by averaging over $N_{\exp }=1000$ independent numerical repetitions. The inset shows the normalized temperatures corresponding to the noise intensities $D_I$. For all panels $I_{\max }=0.9$ and $\sigma =5$, whereas the values of other parameters are the same as those used to obtain Fig. 2.

Figure 5

Current pulse (a) and following time evolution of phase and Josephson current, see (b), and magnetization components, see (c), in the absence of noise and including a thermal-current contribution with amplitude $D_I=0.05$, see (e),(d). The values of other parameters are $I_{\max }=0.9$, $\sigma =5$, $r=0.1$, and $\gamma =0.25$. The legends in (b),(c) refer also to (d),(e), respectively.

Figure 6

Time evolution of the magnetization $m_z$, in response to a sequence of three current pulses shown in the top panel, in the presence of a thermal-current noise with amplitude $D_I=0.05$. The values of the other parameters are $I_{\max }=0.9$, $\sigma =5$, $r=0.1$, and $\gamma =0.25$.

Figure 7

Average stationary magnetization, $\overline{m_z^{\mathrm{st}}}$, as a function of the noise intensity, $D_I$, calculated by taking into account both the thermal-current and the thermal-field noise contribution, and by averaging over $N_{\exp }=1000$ independent numerical repetitions. The values of other parameters are $I_{\max }=0.9$, $\sigma =5$, $r=0.1$, and $\gamma =0.25$.

Figure 8

Time evolution of phase and Josephson current (a) and magnetization components (b) as the system is excited by the current pulse in Fig. 5. Here we are taking into account both a thermal-current and a thermal-field contribution, with noise intensity $D_I=0.005$. The values of other parameters are $I_{\max }=0.9$, $\sigma =5$, $r=0.1$, and $\gamma =0.25$.

Figure 9

Stationary magnetization, $m_z^{\mathrm{st}}$, as a function of $r$ and $\gamma$, at different values of the relative Dresselhaus coefficient $\tilde{\beta }=\beta /\alpha$, in the absence of noise fluctuations, by imposing $I_{\max }=0.9$ and $\sigma =5$.

Figure 10

SQUID-based memory readout and cartoon showing the critical current interference pattern of the SQUID, in the cases of both positive and negative orientation along the $z$ axis of the magnetic moment, see (a),(b), respectively.