Strongly coupled Yukawa trilayer liquid: Structure and dynamics

The equilibrium structure and the dispersion relations of collective excitations in trilayer Yukawa systems in the strongly coupled liquid regime are examined. The equilibrium correlations reveal a variety of structures in the liquid phase, reminiscent of the corresponding structures in the solid phase. At small layer separation substitutional disorder becomes the governing feature. Theoretical dispersion relations are obtained by applying the quasilocalized charge approximation (QLCA) formalism, while numerical data are generated by microcanonical molecular dynamics simulations. The dispersions and polarizations of the collective excitations obtained through both of these methods are compared and discussed in detail. We find that the QLCA method is, in general, very satisfactory, but that there are phenomena not covered by the QLCA. In particular, by analyzing the dynamical longitudinal and transverse current fluctuation spectra we discover the existence of a structure not related to the collective mode spectra. This also provides insight into the long-standing problem of the gap frequency discrepancy, observed in strongly coupled layered systems in earlier studies.

Figure 1

The two principal trilayer lattice structures: (a) staggered hexagonal (SH) lattice and (b) overlapping square (OS) lattice. Note that in all figures axis labels represent dimensionless quantities, as defined in the text.

Figure 2

Intralayer (11, 22) and interlayer (12, 23) pair distribution functions for $\mathrm{\Gamma }=160$, at layer separations (a) $d=3.0$, (b) $d=1.5$, (c) $d=0.5$, (d) $d=0.2$. Note that $g_{23}(\rho \to 0)>1$ at larger $d$ values and $g_{23}(\rho \to 0)<1$ at smaller $d$ values. In (a), $g_{11}\left(\rho \right)$ and $g_{22}\left(\rho \right)$ nearly overlap, while in (d) all $g\left(\rho \right)$'s nearly overlap apart from scale; see text for interpretation. Note that $g_{33}\left(\rho \right)$ is not shown as it is identical to $g_{22}\left(\rho \right)$ and $g_{13}\left(\rho \right)$ is not shown because it is identical to $g_{12}\left(\rho \right)$.

Figure 3

The same as Fig. 2 for $\mathrm{\Gamma }=10$.

Figure 4

An overview snapshot of the trilayer system for $\mathrm{\Gamma }=1000$, at $d=0$ from MD simulation. The lattice structure is still nearly hexagonal, but the occupation of the lattice sites is random. Particle symbols indicate the actual layer, which is being occupied by the particle. This image is to be contrasted with Fig. 1, which depicts an ordered trilayer system.

Figure 5

Side view of an arbitrarily chosen vertical row of particles in the SH crystal: (a) ordered and (b) substitutionally disordered.

Figure 6

Illustration of the three longitudinal and three transverse eigenvectors with $\vec{k}$ taken along the $x$ (horizontal) direction. Two sets of modes exist, each either with longitudinal ($\mathcal{L}$) or transverse ($\mathcal{T}$) Cartesian polarizations. $\mathcal{A}$ mode: particles in all the three layers move in the same direction, $\mathcal{M}$ mode: particles in the middle layer (1) move in one direction, while those in layers 2 and 3 move together in the opposite direction, $\mathcal{S}$ mode: particles in the middle layer remain stationary, while those in layers 2 and 3 move in opposite directions with respect to each other. The arrows represent the displacements of the particles within the respective layers.

Figure 7

QLCA dispersion curves of the six collective modes of the trilayer system, for moderate coupling $\mathrm{\Gamma }=10$, and high coupling $\mathrm{\Gamma }=160$ values, at $d=3.0$.

Figure 8

QLCA dispersion curves of the six collective modes of the trilayer system, for moderate coupling $\mathrm{\Gamma }=10$, and high coupling $\mathrm{\Gamma }=160$ values, at $d=1.5$.

Figure 9

QLCA dispersion curves of the six collective modes of the trilayer system, for moderate coupling $\mathrm{\Gamma }=10$, and high coupling $\mathrm{\Gamma }=160$ values, at $d=0.5$.

Figure 10

QLCA dispersion curves of the six collective modes of the trilayer system, for moderate coupling $\mathrm{\Gamma }=10$, and high coupling $\mathrm{\Gamma }=160$ values, at $d=0.2$.

Figure 11

Close-up of the QLCA dispersion of the $\mathcal{A}$ and $\mathcal{M}$ modes for $\mathrm{\Gamma }=160$, at $d=1.5$. Note the avoided crossing and switch of polarizations between the modes.

Figure 12

Close-up of the QLCA dispersion of the $\mathcal{SL}$ and $\mathcal{ST}$ modes for $\mathrm{\Gamma }=160$ at $d=1.5$. The simple crossing indicates that the two modes do not interact.

Figure 13

Longitudinal current fluctuation spectra at $k=0.32$ for $\mathrm{\Gamma }=160$, at $d=3.0$, (a) $L_{11}$, (b) $L_{22}$, (c) $L_{12}$, (d) $L_{23}$. The labeling of the peaks identifies the corresponding mode.

Figure 14

Collective mode dispersion comparison between MD (blue squares) and QLCA (red lines) for the $\mathcal{A}$ mode, for $\mathrm{\Gamma }=10$ at different $d$ values in the range $0.2\le d\le 3.0$. The vertical bars indicate the Gaussian width of the spectral peaks, as explained in the text.

Figure 15

The same as Fig. 14 for $\mathrm{\Gamma }=160$. Compare the avoided crossing points in panels (e) and (g) at $k=2.6$ with their counterparts in panels (a) and (c) in Fig. 17. See also Fig. 12 for the details of the predicted QLCA behavior. Other apparent AC points at lower $d$ values lack simple interpretation, as explained in the text.

Figure 16

The same as Fig. 14 for the $\mathcal{M}$ mode, for $\mathrm{\Gamma }=10$, at $d=1.5$ and $d=3.0$ values. $d<1.5$ layer separations are not shown, for reasons explained in the text.

Figure 17

The same as Fig. 16 for $\mathrm{\Gamma }=160$. Compare the avoided crossing points in panels (a) and (c) at $k=2.6$ with their counterparts in panels (e) and (g) of Fig. 15. See also Fig. 12 for the details of the predicted QLCA behavior.

Figure 18

The same as Fig. 16 for the $\mathcal{S}$ mode, for $\mathrm{\Gamma }=10$.

Figure 19

The same as Fig. 16 for the $\mathcal{S}$ mode, for $\mathrm{\Gamma }=160$.

Figure 20

Comparison of current fluctuation spectra between the liquid for $\mathrm{\Gamma }=160$, (left column) and the lattice for $\mathrm{\Gamma }=1000$, (right column) phases at $d=1.5$. Displayed are (a, b) $L_{11}(k,\omega )$, (c, d) $T_{11}(k,\omega )$, (e, f) $L_{22}(k,\omega )$, and (g, h) $T_{22}(k,\omega )$ spectra. In the liquid spectra black lines show the corresponding QLCA mode dispersions: $\mathcal{ML}$ in (a), $\mathcal{MT}$ in (c), $\mathcal{SL}$ in (e), $\mathcal{ST}$ in (g). The MD simulations are for the OS lattice system and for propagation directions $\varphi =0^{\circ }$ and $\varphi ={45}^{\circ }$, as indicated for the corresponding spectral features. The amplitudes are given in arbitrary units, therefore numerical values at the color bars are omitted. The color scale is logarithmic and covers three orders of magnitude.

Figure 21

Longitudinal current fluctuation spectra $L_{11}$, $L_{22}$, $L_{12}$, and $L_{23}$ at a small layer separation, $d=0.2$, $k=0.32$ for $\mathrm{\Gamma }=160$. Note the different patterns compared to the large $d$ case shown in Fig. 13.

Figure 22

The same as Fig. 21 for the transverse current fluctuation spectra $T_{11}$, $T_{22}$, $T_{12}$, $T_{23}$.

Figure 23

The evolution of $T_{11}-T_{12}$ for $\mathrm{\Gamma }=160$ with decreasing $d$ values. (a) $d=1.5$, (b) $d=1.0$, (c) $d=0.5$, (d) $d=0.2$. Note the transmutation of the $\mathcal{M}$ mode peak into the envelope.

Figure 24

The same as Fig. 23 for $T_{22}-T_{23}$ Note the transmutation of the $\mathcal{S}$ mode peak into the envelope.

Figure 25

Comparison between $T_{11}-T_{12}$ and $Z\left(\omega \right)$ for $\mathrm{\Gamma }=10$, at $d=0$.

Figure 26

The same as Fig. 25 for $\mathrm{\Gamma }=160$.

Figure 27

Gap frequency values for the $\mathcal{S}$ and $\mathcal{M}$ modes at different $d$ values, for $\mathrm{\Gamma }=160$. The S1, M1 points are obtained from the QLCA theory (with the implicit assumption that the gapped modes are excited for all $d$ values.); the S2, M2 points are the results of the MD simulations; the S3, M3 points are obtained from lattice dispersion calculations; at $d=3.0$ and at $d=0.2$ a SH, at $d=1.5$ an OS crystal structure is assumed (cf. Sec. 3). The $\mathcal{S}2$ and $\mathcal{M}2$ points in the shaded area originate from peaks in the current fluctuation spectra generated by the envelope, rather than by a collective mode.