Shell formation and two-dimensional nanofriction in three-dimensional ion Coulomb crystals

TL;DR

This work investigates friction between concentric shells in self-organized 3D ion Coulomb crystals in Paul traps, using molecular-dynamics simulations to map shell formation as a function of ion number $N$ and trap aspect ratio $\alpha$, and to quantify a Peierls–Nabarro-type landscape for outer-shell rotation. It finds that small changes in $N$ can cause large changes in the energy barrier for rotation due to shifts in shell commensurability, and it demonstrates diverse dynamical friction regimes, including pinning, stick-slip, and smooth sliding, as well as hysteresis and multidimensional intra-/inter-shell friction. The results connect structural parameters to frictional properties and suggest experimental protocols to observe depinning and spin-like rotational dynamics, with implications for ultra-low-friction nanomechanical devices such as ion-based nanorotors and torque sensors. Overall, the paper highlights the rich, geometry-dependent interplay between shell structure and friction in curved, self-organized crystal interfaces, offering a roadmap for precision control of nanofriction in engineered Coulomb systems.

Abstract

Self-organized three-dimensional (3D) ion Coulomb crystals in linear Paul traps naturally form concentric shells that provide a curved, atomically resolved interface for studying two-dimensional (2D) nanofriction. Building on prior work that used 2D ion crystals to investigate one-dimensional (1D) nanofriction and orientational melting, we leverage this foundation to extend friction studies from linear ion chains and planar rings to 3D shell structures. Using molecular-dynamics simulations, we first map shell formation as a function of ion number $N$ and the trapping aspect ratio, yielding a simple relation that can aid ion-number estimation in experiments. We compute a Peierls--Nabarro-type energy landscape for the rotation of the outer shell against the inner core, showing drastic changes in the effective energy barrier up to a factor of about 60 with only small changes in $N$. Using dynamical simulations, we apply rotational torques to the outer shell of selected systems and show that small changes in $N$ impact the commensurability between shells and can, in some cases, induce a hysteretic response due to torque-induced metastable states. We find that spatially varying coupling to the inner-core corrugation can create coexisting fast and slow moving domains within the rotating outer shell, realizing multidimensional friction where intra-shell shear and inter-shell nanofriction act simultaneously. Our results have implications for stabilizing many-body systems and for the development of ultra-low-friction nanomechanical devices such as ion-based nanorotors and torque sensors.

Figures (15)

• Figure 1: Experimental image of a three-dimensional (3D), Doppler-cooled Coulomb crystal of approximately 200 $^{172}$Yb$^{+}$ ions taken with an EMCCD camera. The aspect ratio of the trapping potential is $\alpha\approx9$. The crystal consists of an inner shell (ions highlighted in red) and a helix shaped outer shell, which wraps around the inner structure.
• Figure 2: Illustrative example of one-dimensional (1D) nanofriction in neighboring ion chains. Ion positions are represented by blue circles and orange triangles. The substrate potential (orange line) created by the triangular-marked ions induces a corrugation, which resists the sliding motion of the blue ion chain. Arrows indicate the motion lateral to the corrugation layer. The corrugation depth $U_0$ determines the transition from a pinned state to a free sliding state. (a) Adjacent ion chains with different periodicities. A similar setup was experimentally realized using a topological defect in Kiethe_Nanofriction_2018 to verify an Aubry-type transition from sliding to pinning when the corrugation depth $U_0$ exceeds a critical value $U_{0,c}$. (b) Ring configuration of two adjacent ion chains. The number of ions directly dictates the periodicity between the two chains and therefore the commensurability. A similar system was used by Duca et al. to investigate orientational melting in 2D Coulomb crystals duca_orientational_2023. We simulate the pinned state by fixing the positions of the corrugation ions and increase the trapping potential, until the corrugation depth exceeds a critical value $U_{0,c}$, causing the system to be pinned.
• Figure 3: Model system of two-dimensional (2D) nanofriction in a three-dimensional (3D) self-organized Coulomb crystal ($N=80$) with two shells. Ions of the outer shell are represented by blue circles. Ions of the inner shell are shown as orange triangles. The ratio of the trapping frequencies is determined by $\omega_r >\omega_z$, causing the crystal to be of a spheroidal shape. The inner shell is approximated by a spheroidal surface (orange shade) for easier distinction of the two shells. The shear rotation around the $z$-axis of the outer shell is illustrated with blue circular arrows.
• Figure 4: (a) Number of shells as different colored regions with respect to the number of ions $N$ and the aspect ratio of the trapping potential $\alpha=\omega_r^2/\omega_z^2$. A brighter color means a larger shell number. Half numbered shell counts describe crystals containing a linear chain of one or more ions on the $z$-axis. Linear fits to the transition regions are plotted as blue dashed lines. The white crosses mark the transition between full shell counts predicted by the linear particle density using $N_s = 0.83 \, \sqrt{\lambda}\,$. (b) The number of shells is plotted with respect to the ratio between $\alpha/N$. A power-law function was fitted to the data (red dashed line). Grey dashed line shows the limit of $N_s=0.5$, which by definition represents the minimum achievable number of shells.
• Figure 5: Rotation of the outer shell (blue wireframe) around the $z$-axis for an $N=80$ ion crystal with trapping aspect ratio $\alpha = 1$. Inner shells are shown in orange. For clarity, only the driving ions (red markers) are plotted explicitly and connected by a dashed red line for easier visualization. The position of each driving ion is described in spheroidal coordinates $(r, \theta, \varphi)$, where $\varphi$ is the azimuthal angle relevant for the rotation. (a) Initial configuration at $\varphi = 0^\circ$. The initial coordinates $r_i$, $\theta_i$, and $\varphi_i$ of a sample ion are shown in black, green, and red, respectively. (b) Configuration after rotation to $\varphi = 90^\circ$. The initial position ($\varphi_i$) is shown with reduced opacity for reference. The rotation angle $\varphi$ is measured relative to $\varphi_i$ and is identical for all driving ions. The outer shell is rotated by incrementally rotating the driving ions (red markers) around the $z$-axis in $\Delta \varphi = 1^\circ$ steps. While radial and polar motion is allowed during relaxation, for the driving ions, the azimuthal position is fixed during each rotation step. The dynamics of all other ions of the outer shell are unrestricted. The ions of the inner shells are kept static throughout the simulation.