Monte Carlo study of the honeycomb structure of anthraquinone molecules on Cu(111)

Using Monte Carlo calculations of the two-dimensional (2D) triangular lattice gas model, we demonstrate a mechanism for the spontaneous formation of honeycomb structure of anthraquinone (AQ) molecules on a Cu(111) plane. In our model long-range attractions play an important role, in addition to the long-range repulsions and short-range attractions proposed by Pawin, Wong, Kwon, and Bartels [Science 313, 961 (2006)]. We provide a global account of the possible combinations of long-range attractive coupling constants which lead to a honeycomb superstructure. We also provide the critical temperature of disruption of the honeycomb structure and compare the critical local coverage rate of AQ’s where the honeycomb structure starts to form with the experimental observations.

Figure 1

(a) Long-range attractions $E_6$, $E_{12}$, and $E_{19}$ and (b) mappings of the attractive coupling constants ($E_1$, $E_{\mathrm{LT}}$, $E_{\mathrm{IT}}$) and the trio repulsion constant ($E_{\mathrm{T}}$). Without the trio repulsion, very dense, compact clusters appear, contrary to experiment.

Figure 2

Pinwheels experimentally (a) not observed and (b) observed. Note that they are on Cu(111).

Figure 3

Percentage of $E_6$ hexagons as a function of the magnitude of $E_{19}$ at $\rho =0.4$. The ground state site configuration is also shown for $E_{19}=0.06\epsilon$. The other ground state site configuration shown is for $E_{19}=0.08\epsilon$, $\rho =0.3$, where $E_{19}$ hexagons form.

Figure 4

Percentage of $E_{12}$ hexagons as a function of the magnitude of $E_{12}$ at $\rho =0.33$ with $E_{\mathrm{T}}=0.06\epsilon$ and with $E_{\mathrm{T}}=0.03\epsilon$, and at $\rho =0.4$ with $E_{\mathrm{T}}=0.06\epsilon$. The ground state site configurations are also shown at $E_{12}=0.11\epsilon$, $\rho =0.33$ with $E_{\mathrm{T}}=0.06\epsilon$ and with $E_{\mathrm{T}}=0.03\epsilon$.

Figure 5

Percentage of $E_{12}$ hexagons as a function of the magnitudes of $E_{12}$ and $E_{19}$ at $\rho =0.33$ with $E_{\mathrm{T}}=0.06\epsilon$ and with $E_{\mathrm{T}}=0.03\epsilon$, and at $\rho =0.40$ with $E_{\mathrm{T}}=0.06\epsilon$. The ground state site configuration is shown at $E_{12}=0.06\epsilon$, $E_{19}=0.03\epsilon$, $\rho =0.33$ with $E_{\mathrm{T}}=0.06\epsilon$, of which data point (73 $E_{12}$ hexagons, 97.3%) is not shown in the percentage plot since it is too big compared to the others. The other ground state site configuration is at $E_{12}=0.05\epsilon$, $E_{19}=0.08\epsilon$, $\rho =0.40$ with $E_{\mathrm{T}}=0.06\epsilon$, where $E_6$ hexagons form instead of $E_{12}$.

Figure 6

Percentage of $E_6$ hexagons as a function of the magnitude of $E_6$ at $\rho =0.4$ ($T=0.001\epsilon /k_B$). The ground state site configuration is also shown at $E_6=0.10\epsilon$, which makes 143 (81.3%) $E_6$ hexagons. The other ground state site configuration shown is at $E_6=0.06\epsilon$, $\rho =0.4$, $T=0.1\epsilon /k_B$, where also many $E_6$ hexagons (141, 80.1%) form.

Figure 7

The ground state site configurations for $E_6$ repulsions of (a) $E_6=-0.06\epsilon$ and (b) $E_6=-0.03\epsilon$. Only straight branches form.

Figure 8

Percentage of $E_6$ hexagons as a function of the magnitude of $E_{19}$ with a fixed $E_6=0.06\epsilon$ at $\rho =0.3$ (green asterisks) and at $\rho =0.4$ (red solid circles). $E_{\mathrm{T}}=0.06\epsilon$ for both cases. The solid circles are scaled to the right axis while the asterisks to the left. The ground state site configurations are shown at $E_6=0.06\epsilon$, $E_{19}=0.02\epsilon$, $\rho =0.3$ and at $E_6=0.06\epsilon$, $E_{19}=0.07\epsilon$, $\rho =0.4$, where there are 28 (15.9%) and 156 (88.6%) $E_6$ hexagons, respectively. There are many more $E_6$ hexagons for $\rho =0.4$ than for $\rho =0.3$.

Figure 9

Percentage of $E_6$ hexagons as a function of the magnitude of $E_{12}$ with a fixed $E_6=0.06\epsilon$ at $\rho =0.33$ (green asterisks) and at $\rho =0.4$ (red solid circles). $E_{\mathrm{T}}=0.06\epsilon$ for both cases. The ground state site configurations are shown at $E_6=0.06\epsilon$, $E_{12}=0.03\epsilon$, $\rho =0.33$ and at $E_6=0.06\epsilon$, $E_{19}=0.02\epsilon$, $\rho =0.4$, where there are 108 (61.4%) and 101 (57.4%) $E_6$ hexagons, respectively. Generally there are more $E_6$ hexagons for $\rho =0.4$ than for $\rho =0.33$.

Figure 10

Percentage of $E_6$ hexagons as a function of the magnitudes of $E_6$ and $E_{12}$ at $\rho =0.33$ (green asterisks) and at $\rho =0.4$ (red solid circles). $E_{\mathrm{T}}=0.06\epsilon$ for both cases. The ground state site configurations are shown at $E_6=0.04\epsilon$, $E_{12}=0.01\epsilon$, $\rho =0.33$ and at $E_6=0.03\epsilon$, $E_{12}=0.03\epsilon$, $\rho =0.33$. For the former there are $E_6$ hexagons and long straight branches (marked with red ovals) while for the latter both $E_6$ and $E_{12}$ hexagons are present. Here also there are generally more $E_6$ hexagons for $\rho =0.4$ than for $\rho =0.33$.

Figure 11

The ground state site configurations for (a) $E_6=E_{12}=E_{19}=0.06\epsilon$ and (b) $E_6=E_{12}=E_{19}=0.03\epsilon$. $\rho =0.4$ for both cases. For the former there are 160 (90.9%) $E_6$ hexagons, the most in our investigation.

Figure 12

Percentage of $E_6$ hexagons as a function of the magnitudes of $E_{\mathrm{LT}}$ and $E_{\mathrm{IT}}$ at $\rho =0.3$ (green asterisks) and at $\rho =0.4$ (red solid circles). $E_{\mathrm{T}}=0.06\epsilon$ for both cases. The ground state site configurations are shown at $E_{\mathrm{LT}}=0.04\epsilon$, $E_{\mathrm{IT}}=0.04\epsilon$, $\rho =0.4$ and at $E_{\mathrm{LT}}=0.06\epsilon$, $E_{\mathrm{IT}}=0.03\epsilon$, $\rho =0.3$. For the former and $E_{\mathrm{LT}}=0.05\epsilon$, $E_{\mathrm{IT}}=0.05\epsilon$, $\rho =0.4$, there are the same 128 (72.7%) $E_6$ hexagons, while for the latter there are 67 (38.1%) $E_6$ hexagons. Obviously, there are more $E_6$ hexagons for $\rho =0.4$ than for $\rho =0.3$.

Figure 13

Percentage of $E_6$ hexagons as a function of the magnitude of $A$ with a fixed $E_6=0.06\epsilon$ at $\rho =0.33$ (green asterisks) and at $\rho =0.4$ (red solid circles). $E_{\mathrm{T}}=0.06\epsilon$ for both cases. The solid circles refer to the right axis while the asterisks to the left. The ground state site configurations are shown at $A=0.02\epsilon$, $\rho =0.33$ and at $A=0.08\epsilon$, $\rho =0.4$, for which there are 35 (19.9%) and 143 (81.3%) $E_6$ hexagons, respectively. There are many more $E_6$ hexagons for $\rho =0.4$ than for $\rho =0.33$.

Figure 14

Venn diagram of coupling constants summarizing all of our results. It shows which components are necessary to make which hexagons, also which hexagons form when two or three long-range attractions are present at the same time. For clarity the repulsions are denoted with a negative sign.

Figure 15

Percentage of $E_6$ hexagons as a function of temperature at $\rho =0.4$. The ground state site configurations are also shown at the temperatures $T=0.1\epsilon /k_B$, $T=0.7\epsilon /k_B$, and $T=0.495\epsilon /k_B$. $T_c=0.495\epsilon /k_B$ is the transition temperature. Each point is an average over 100 different runs.

Figure 16

Percentage of $E_6$ hexagons as a function of particle density $\rho$ at temperature $T=0.001\epsilon /k_B$. Here also each point is an average over 100 different runs. The ground state site configurations are also shown at $\rho =0.30$, $\rho =0.40$, and $\rho =0.55$. Hexagons start to appear at $\rho =0.30$, peak at $\rho =0.40$, and decrease slightly after that.