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In this work, we investigate the device-to-device variations in the remanent polarization of metal–ferroelectric–insulator–metal stacks based on ferroelectric hafnium–zirconium–oxide (HZO). Our study employs a 3D dynamic multi-grain phase-field model to consider the effects of the polycrystalline nature of HZO in conjunction with the multi-domain polarization switching. We explore the dependence of variations on various design factors, such as the ferroelectric thickness and voltage stimuli (set voltage, pulse amplitude, and width), and correlate the trends to the underlying polarization switching mechanisms. Our analysis reveals a non-monotonic dependence of variations on the set voltage due to the coupled effect of the underlying polycrystalline structure variations and the voltage dependence of polarization switching mechanisms. We further report that collapsing of oppositely polarized domains at higher set voltages can lead to an increase in variations, while ferroelectric thickness scaling lowers the overall device-to-device variations. Considering the dynamics of polarization switching, we highlight the key role of voltage and temporal dependence of domain nucleation in dictating the trends in variations. Finally, we show that using a lower amplitude pulse for longer duration to reach a target mean polarization state results in lower variations compared to using a higher amplitude pulse for shorter duration. 

Measurements are presented of the cross-section for the central exclusive production ofJ/\psi\to\mu^+\mu^- $J/\psi \to {\mu }^{+}{\mu }^{-}$and\psi(2S)\to\mu^+\mu^- $\psi \left(2S\right)\to {\mu }^{+}{\mu }^{-}$processes in proton-proton collisions at\sqrt{s} = 13 \ \mathrm{TeV} $\sqrt{s}=13TeV$with 2016–2018 data. They are performed by requiring both muons to be in the LHCb acceptance (with pseudorapidity2<\eta_{\mu^±} < 4.5 $2<{\eta }_{{\mu }^{\pm }}<4.5$) and mesons in the rapidity range2.0 < y < 4.5 $2.0<y<4.5$. The integrated cross-section results are\sigma_{J/\psi\to\mu^+\mu^-}(2.0 ${\sigma }_{J/\psi \to {\mu }^{+}{\mu }^{-}}(2.0<y_{J/\psi }<4.5,2.0<{\eta }_{{\mu }^{\pm }}<4.5)=400\pm 2\pm 5\pm 12pb,{\sigma }_{\psi \left(2S\right)\to {\mu }^{+}{\mu }^{-}}(2.0<y_{\psi \left(2S\right)}<4.5,2.0<{\eta }_{{\mu }^{\pm }}<4.5)=9.40\pm 0.15\pm 0.13\pm 0.27pb,$where the uncertainties are statistical, systematic and due to the luminosity determination. In addition, a measurement of the ratio of\psi(2S) $\psi \left(2S\right)$andJ/\psi $J/\psi$cross-sections, at an average photon-proton centre-of-mass energy of1\ \mathrm{TeV} $1TeV$, is performed, giving$ = 0.1763 ± 0.0029 ± 0.0008 ± 0.0039,$$ where the first uncertainty is statistical, the second systematic and the third due to the knowledge of the involved branching fractions. For the first time, the dependence of theJ/\psi$ $J/\psi$and\psi(2S) $\psi \left(2S\right)$cross-sections on the total transverse momentum transfer is determined inpp $pp$collisions and is found consistent with the behaviour observed in electron-proton collisions.