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#### Dispersion relation of square lattice waves in a two-dimensional binary complex plasma

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##### Citation

Huang, H., Ivlev, A. V., Nosenko, V., & Lin, Y.-F. (2021). Dispersion relation
of square lattice waves in a two-dimensional binary complex plasma.* Physics of Plasmas,*
*28*(1): 0026106. doi:10.1063/5.0026106.

Cite as: https://hdl.handle.net/21.11116/0000-0008-0D13-5

##### Abstract

Binary complex plasmas consist of microparticles of two different species and can form two-dimensional square lattices under certain conditions. The dispersion relations of the square lattice waves are derived for the longitudinal and transverse in-plane modes, assuming that the out-of-plane mode is suppressed by the strong vertical confinement. The results are compared with the spectra obtained in Langevin dynamics simulations. Furthermore, we investigate the dependence of the dispersion relation on the charge ratio and mass ratio of the two particle species.

Complex plasmas consist of a mixture of weakly ionized gases and microparticles. The latter acquire a charge due to the flow of the surrounding ions and electrons which are negative, owing to the higher thermal velocity of electrons.1 Considering the plasma screening effect, the interaction between the microparticles can be described via the Yukawa potential.2 In the laboratory, the charged particles are usually suspended in the sheath above the lower electrode of a radio frequency (rf) discharge, where the gravity force is balanced by the electric force. In strongly coupled complex plasmas, monodisperse microparticles can be vertically confined to a single layer and form a hexagonal lattice, known as plasma crystal.3,4 Due to the stretched time scales and low damping, two-dimensional (2D) complex plasma crystals provide a unique opportunity to study generic processes in solids and liquids at the kinetic level.5 With external manipulations by electric fields or laser beams, various phenomena such as melting6 and recrystallization,7,8 microstructure under shear,9 Mach cone excitations,10 and entropy production11,12 have been investigated both experimentally and theoretically.

One of the most defining properties of plasma crystals is the dispersion relation of the microparticles' collective oscillations in the form of lattice waves. This has been derived analytically and measured directly using video microscopy in the case of monodisperse complex plasmas.13–15 Remarkably, due to the strong ion flow in the sheath, the interactions between microparticles are altered by the so-called wake effect, resulting in the coupling of the horizontal and vertical modes.16–20 This eventually triggers a mode-coupling instability (MCI) and causes the crystal to melt.21,22

A binary complex plasma consists of microparticles of two different species. With an appropriate selection of their mass and size, these particles can form, in the laboratory, a bilayer23 or a quasi-two-dimensional (q2D) crystalline suspension.24–26 The phonon spectra for these structures have been measured experimentally and studied by a quasi-localized charge approximation approach and molecular dynamics simulations.27,28 A mode coupling between the horizontal modes in the two layers, mediated by the plasma wakes, has been proposed.29

Meanwhile, taking advantage of the plasma etching effect, the two particle species can be suspended at the same height for a certain amount of time.32 Under these conditions, binary complex plasmas have been found to form square lattices with a quadruple symmetry33 as the one presented in Fig. 1. Such a structure has been widely observed and carefully studied in colloids34,35 and recently also discovered in the complex plasmas.28 In general, the realization of certain periodic structures requires the fulfillment of two conditions. First, the force equilibrium for individual particles in the lattice should be satisfied. Second, the stability regime of the equilibrium structure should be determined, for example, using energy-minimization techniques36,37 or the dynamical matrix formalism.38 Apparently, the square lattice structure of Fig. 1 fulfills the first condition in binary mixtures of complex plasmas. Concerning the stability regime, a strong vertical confinement can efficiently suppress the vertical motions and thus the expected MCI.21,22 Alternatively, the MCI can be suppressed by a high damping rate. Besides, the structural stability of the binary square structure depends also on the charge ratio of the two particle species composing the complex plasma mixture, as it will be discussed in the article. For such 2D square lattices of binary complex plasmas, it is interesting to study how the horizontal wave modes are modified in comparison to the well-studied case of hexagonal lattices in monodisperse 2D plasma crystals.

Complex plasmas consist of a mixture of weakly ionized gases and microparticles. The latter acquire a charge due to the flow of the surrounding ions and electrons which are negative, owing to the higher thermal velocity of electrons.1 Considering the plasma screening effect, the interaction between the microparticles can be described via the Yukawa potential.2 In the laboratory, the charged particles are usually suspended in the sheath above the lower electrode of a radio frequency (rf) discharge, where the gravity force is balanced by the electric force. In strongly coupled complex plasmas, monodisperse microparticles can be vertically confined to a single layer and form a hexagonal lattice, known as plasma crystal.3,4 Due to the stretched time scales and low damping, two-dimensional (2D) complex plasma crystals provide a unique opportunity to study generic processes in solids and liquids at the kinetic level.5 With external manipulations by electric fields or laser beams, various phenomena such as melting6 and recrystallization,7,8 microstructure under shear,9 Mach cone excitations,10 and entropy production11,12 have been investigated both experimentally and theoretically.

One of the most defining properties of plasma crystals is the dispersion relation of the microparticles' collective oscillations in the form of lattice waves. This has been derived analytically and measured directly using video microscopy in the case of monodisperse complex plasmas.13–15 Remarkably, due to the strong ion flow in the sheath, the interactions between microparticles are altered by the so-called wake effect, resulting in the coupling of the horizontal and vertical modes.16–20 This eventually triggers a mode-coupling instability (MCI) and causes the crystal to melt.21,22

A binary complex plasma consists of microparticles of two different species. With an appropriate selection of their mass and size, these particles can form, in the laboratory, a bilayer23 or a quasi-two-dimensional (q2D) crystalline suspension.24–26 The phonon spectra for these structures have been measured experimentally and studied by a quasi-localized charge approximation approach and molecular dynamics simulations.27,28 A mode coupling between the horizontal modes in the two layers, mediated by the plasma wakes, has been proposed.29

Meanwhile, taking advantage of the plasma etching effect, the two particle species can be suspended at the same height for a certain amount of time.32 Under these conditions, binary complex plasmas have been found to form square lattices with a quadruple symmetry33 as the one presented in Fig. 1. Such a structure has been widely observed and carefully studied in colloids34,35 and recently also discovered in the complex plasmas.28 In general, the realization of certain periodic structures requires the fulfillment of two conditions. First, the force equilibrium for individual particles in the lattice should be satisfied. Second, the stability regime of the equilibrium structure should be determined, for example, using energy-minimization techniques36,37 or the dynamical matrix formalism.38 Apparently, the square lattice structure of Fig. 1 fulfills the first condition in binary mixtures of complex plasmas. Concerning the stability regime, a strong vertical confinement can efficiently suppress the vertical motions and thus the expected MCI.21,22 Alternatively, the MCI can be suppressed by a high damping rate. Besides, the structural stability of the binary square structure depends also on the charge ratio of the two particle species composing the complex plasma mixture, as it will be discussed in the article. For such 2D square lattices of binary complex plasmas, it is interesting to study how the horizontal wave modes are modified in comparison to the well-studied case of hexagonal lattices in monodisperse 2D plasma crystals.