Buckling of two-dimensional plasma crystals with nonreciprocal interactions

Laboratory realizations of two-dimensional (2D) plasma crystals typically involve monodisperse microparticles confined into horizontal monolayers in radio-frequency (rf) plasma sheaths. This gives rise to the so-called plasma wakes beneath the microparticles. The presence of wakes renders the interactions in such systems nonreciprocal, a fact that can lead to a quite different behavior from the one expected for their reciprocal counterparts. Here we examine the buckling of a hexagonal 2D plasma crystal, occurring as the confinement strength is decreased, taking explicitly into account the nonreciprocity of the system via a well-established point-wake model. We observe that for a finite wake charge, the monolayer hexagonal crystal undergoes a transition first to a bilayer hexagonal structure, unrealizable in harmonically confined reciprocal Yukawa systems, and subsequently to a bilayer square structure. Our theoretical results are confirmed by molecular dynamics simulations for experimentally relevant parameters, indicating the potential of their observation in state-of-the-art experiments with 2D complex plasmas.

Figure 1

(a) Schematic illustration of the interaction forces ${\mathbf{F}}_{\mathrm{int}}\left({\mathbf{r}}_i-{\mathbf{r}}_j\right)$ between the particles $i$ and $j$. (b) Sketch of the elementary hexagonal lattice cell of the monolayer configuration with the frame of reference. Here $\mathrm{\Delta }$ denotes the lattice constant and the angle $\theta$ is the angle between the wave-vector $\mathbf{k}$ and the $x$ axis. (c) Sketch of the reciprocal lattice in $\mathbf{k}$ space with basis vectors ${\mathbf{b}}_{1,2}=2\pi {\mathrm{\Delta }}^{-1}\left(1/\sqrt{3},\pm 1\right)$. The shaded region corresponds to the first Brillouin zone with boundaries $\left|\mathbf{k}\right|=2\pi {\mathrm{\Delta }}^{-1}/\sqrt{3}$ for $\theta =0^{\circ }$ and $\left|\mathbf{k}\right|=4\pi {\mathrm{\Delta }}^{-1}/3$ for $\theta ={30}^{\circ }$. (d) Color plot of the squared eigenfrequency ${\mathrm{\Omega }}_v^2$ of the out-of-plane mode, pointing (see discussion in the text) to the onset of the structural instability for $\tilde{q}=0.2$, ${\tilde{\mathrm{\Omega }}}_{\mathrm{con}}=2.48$, $\tilde{\delta }=0.3$, and $\kappa =1$. The white spots indicate where ${\mathrm{\Omega }}_v^2<0$ and the structural instability sets in (near $\theta ={30}^{\circ }$ and $\left|{\mathbf{k}}^{\left(cr\right)}\right|=4\pi /3$). (e) Color plot of the $z$ value of each particle for the real part of the first unstable eigenmode ${\mathbf{d}}_{\mathrm{soft}}$ as discussed in the text. The particles' positions in the $x\text{−}y$ plane, $s_{x,j}$ and $s_{y,j}$, are depicted, with the color indicating the position of each particle in the $z$ direction.

Figure 2

Sketch of the different candidate equilibrium structures of hexagonal order, expected to emerge at the buckling point and investigated in this work: [(a), (b)] a general hexagonal triple layer structure (111) and [(c), (d)] a hexagonal bilayer structure with a doubly occupied bottom layer (21). The left part shows a top view of each structure, while the right part is a side view. For the triple layer structure, the separation between the middle layer and the top or the bottom layer is denoted by $h_1$ or $h_2$ respectively. For the bilayer structure, the separation between the two layers is denoted simply by $h$. Note that all the particles are identical and the color distinguishes only between particles in the different layers. Also, in all the subfigures, the numbers 0,1,2 stand for the corresponding values of $l$ as discussed in the text [Eqs. (10)] and the shaded green regions mark the unit cells of the corresponding lattices.

Figure 3

Sketch of two further candidate equilibrium structures expected to develop from the structures shown in Fig. 2: [(a), (b)] a bilayer staggered honeycomb structure $\left(2\mathrm{\Delta }\right)$ and [(c), (d)] a bilayer square structure $(2\square )$. The left part shows a top view of each structure, while the right part is a side view. The separation of the two layers in each configuration is denoted by $h$. As in Fig. 2, all the particles here are identical and the color distinguishes only between particles in the different layers. The numbers 0,1,2 stand for the corresponding values of $l$ [(a), (b)] or $w$ [(c), (d)] as discussed in the text [Eqs. (10), (13)] and the shaded green regions mark the unit cells of the corresponding lattices.

Figure 4

[(a)–(d)] Bifurcation diagrams of the hexagonal monolayer configuration in terms of the interlayer distances $h_1^{\left(0\right)}$ and $h_2^{\left(0\right)}$ for two qualitatively different cases [(a), (b)] $\tilde{q}=0$ and [(c), (d)] $\tilde{q}=0.2$. In the first column [(a), (c)], all the possible equilibrium values of $h_1^{\left(0\right)}$ (and consequently also $h_2^{\left(0\right)}$) are depicted as a function of the confinement frequency ${\tilde{\mathrm{\Omega }}}_{\mathrm{con}}$. In the second column [(b), (d)], all the equilibrium pairs ($h_1^{\left(0\right)},h_2^{\left(0\right)}$) obtained when varying ${\tilde{\mathrm{\Omega }}}_{\mathrm{con}}$ are shown, so that we can identify the character of the different equilibria. All the solid light blue (light grey) lines correspond to triple layer configurations [(111) or ($3\mathrm{\Delta }$)], all the dashed brown (dark grey) lines to bilayer (21) configurations, and all the dashed orange (grey) lines to bilayer (12) configurations. Note that the different lines of the same color correspond essentially to the same equilibrium structure and account for all the different permutations of particles 0,1,2 in Figs. 2 and 2. The thicker lines (dashed or not) guide the eye to one of the equivalent structures. [(e), (f)] The values of the equilibrium separations (e) $h_1^{\left(0\right)}$ and (f) $h_2^{\left(0\right)}$ (depicted by color) as a function of ${\tilde{\mathrm{\Omega }}}_{\mathrm{con}}$ and $\tilde{q}$. The dashed-dotted white line marks the critical confinement frequency ${\mathrm{\Omega }}_{\mathrm{con}}^{\left(SI\right)}$ for the structural instability of the monolayer hexagonal structure, while the dashed red line marks the critical confinement frequency for the bifurcation of the (21) to the (111) structure, along the lines of panels (c) and (d). In all the cases, we have used $\kappa =1$ and $\tilde{\delta }=0.3$.

Figure 5

The structural phase diagram of a quasi-2D plasma crystal with nonreciprocal interactions, according to our theoretical findings. The stability regions of the different phases, discussed in the text (Figs. 2 and 3), are depicted here by different colors. In this sense, the overlap of two different colors (e.g., orange region) indicates a regime of bistability. The control parameters are the confinement frequency ${\tilde{\mathrm{\Omega }}}_{\mathrm{con}}$ and the effective wake charge $\tilde{q}$. The other parameters are kept constant at values $\rho =2/\sqrt{3}$, $\kappa =1$, and $\tilde{\delta }=0.3$. The horizontal dashed line marks the value of $\tilde{q}=0.2$ used in our simulations (Fig. 6). In the white region, none of the examined phases is stable. Note, however, that overall the existence of further stable phases is not precluded.

Figure 6

[(a), (b)] MD simulation results for the buckling of the hexagonal monolayer ($1\mathrm{\Delta }$) for experimentally relevant parameters (see Appendix pp3) as the confinement strength decreases slowly in time. In panel (a), the evolution of ${\mathrm{\Psi }}_6$ and ${\mathrm{\Psi }}_8$ [Eq. (21), used as measures for the hexagonal and the square order respectively] is depicted as a function of ${\tilde{\mathrm{\Omega }}}_{\mathrm{con}}$. Panel (b) shows the evolution of the average interlayer separation $h$ in units of $\mathrm{\Delta }$ (light blue markers with error bars) with decreasing ${\tilde{\mathrm{\Omega }}}_{\mathrm{con}}$ (from right to left). For comparison, we also show the theoretically expected values for the interlayer separation of the bilayer hexagonal (21) structure $h^{\left(21\right)}$ (yellow solid line) and the bilayer square $(2\square )$ structure $h^{(2\square )}$ (purple solid line). The square insets are enlarged snapshots of the system at the indicated by the arrows points. The color of the particles encodes their vertical positions $z$ in units of $\mathrm{\Delta }$. [(c), (d)] Same as panels (a) and (b) but for the reverse process in which we start by a square bilayer $(2\square )$ and increase slowly ${\tilde{\mathrm{\Omega }}}_{\mathrm{con}}$ (left to right). The simulation results presented here are for an effective wake charge $\tilde{q}=0.2$, a screening parameter $\kappa =1$, and a coupling parameter $\mathrm{\Gamma }=Q^2/\left(k_{B}T\mathrm{\Delta }\right)=1.187\times {10}^4$. The vertical lines represent the theoretically predicted (Fig. 5) transition points for $\left(1\mathrm{\Delta }\right)\to \left(21\right)$ and $\left(21\right)\to (2\square )$ in panels (a) and (b) and for $(2\square )\to \left(21\right)$ and $\left(21\right)\to \left(1\mathrm{\Delta }\right)$ in panels (c) and (d).

Figure 7

Snapshot of the particles' positions, resulting from MD simulations with slowly decreasing confinement strength, for the case ${\tilde{\mathrm{\Omega }}}_{\mathrm{con}}=1.5$. The color of the particles encodes the values of their vertical positions $z$ in units of $\mathrm{\Delta }$. The parameters used read $\tilde{q}=0.2$, $\kappa =1$ and $\mathrm{\Gamma }=1.187\times {10}^4$ (for more information, see Appendix pp3).