Reflection high-energy positron diffraction study on $\mathrm{Si}\left(111\right)\text{−}\sqrt{3}\times \sqrt{3}\text{−}\mathrm{Ag}$ surface

We carried out a reflection high-energy positron diffraction study on a $\mathrm{Si}\left(111\right)\text{−}\sqrt{3}\times \sqrt{3}\text{−}\mathrm{Ag}$ surface at temperatures ranging from $50\text{to}800\mathrm{K}$. We found that the rocking curves obtained both below and above the critical temperature of the phase transition $\left(T_c\right)$ can be explained with the inequivalent triangle (IET) model by introducing an order parameter between the IET(+) and IET(−) phases. We determined the length and rotation angle of the Ag triangle to be $2.76\pm 0.02\mathrm{\AA }$ and $5.3\pm 0.5°$, respectively. We also determined the surface parallel and normal Debye temperatures of the topmost Ag atoms to be 110 and $140\mathrm{K}$, respectively. We found that the temperature dependence of the total reflection intensities below $T_c$ can be completely reproduced by considering the order-disorder phase transition for a critical exponent of $\beta$ given by $0.28\pm 0.05$.

Figure 1

(Color online) Schematic drawings of (a) honeycomb-chained-triangle (HCT) and two topologically different inequivalent triangle (IET) models, (b) IET(+) and (c) IET(−), for the $\mathrm{Si}\left(111\right)\text{−}\sqrt{3}\times \sqrt{3}\text{−}\mathrm{Ag}$ surface (top view). The rotation angles $\left(\phi \right)$ of the Ag atoms for the IET(+) and IET(−) models are $+6°$ and $-6°$ on the basis of the HCT model, respectively. The side view of the structure model is displayed in (d). $r$, $\phi$, and $d$ denote the cylindrical coordinates of the Ag atoms.

Figure 2

(Color online) RHEPD patterns measured from the $\mathrm{Si}\left(111\right)\text{−}\sqrt{3}\times \sqrt{3}\text{−}\mathrm{Ag}$ surface at $293\mathrm{K}$ (upper panel) and at $103\mathrm{K}$ (bottom panel). The glancing angle and azimuth of the incident positrons are 3.0° and the $\left[11\overline{2}\right]$ direction, respectively. The incident positron energy is $10\mathrm{keV}$. The patterns are displayed by the superposition of the left and right parts.

Figure 3

(Color online) RHEPD rocking curves for the specular spot on the $\mathrm{Si}\left(111\right)\text{−}\sqrt{3}\times \sqrt{3}\text{−}\mathrm{Ag}$ surface under the one-beam condition. The incident positron energy is $10\mathrm{keV}$. The blue and red circles denote the experimental curves at 110 and $293\mathrm{K}$, respectively. The solid lines show the rocking curves that were calculated based on the dynamical diffraction theory by using various values of heights $\left(d\right)$ for the Ag triangle from the underlying Si trimer. “TR” stands for the total reflection region.

Figure 4

(Color online) RHEPD rocking curves for the (a) (0 0) spot, (b) $(1∕31∕3)$ spot, and (c) $(2∕32∕3)$ spot on the $\mathrm{Si}\left(111\right)\text{−}\sqrt{3}\times \sqrt{3}\text{−}\mathrm{Ag}$ surface at $110\mathrm{K}$. The incident positron energy is $10\mathrm{keV}$. The circles indicate the measured intensities. The solid lines show the rocking curves determined using various values of $p$ for the Ag atoms at the IET(+) and IET(−) sites. $p$ indicates the order parameter between the IET(+) and IET(−) phases.

Figure 5

(Color online) RHEPD rocking curves for the (a) (0 0) spot, (b) $(1∕31∕3)$ spot, and (c) $(2∕32∕3)$ spot on the $\mathrm{Si}\left(111\right)\text{−}\sqrt{3}\times \sqrt{3}\text{−}\mathrm{Ag}$ surface at $293\mathrm{K}$. The incident positron energy is $10\mathrm{keV}$. The circles denote the measured intensities. The solid lines show the rocking curves determined by using various rotation angles $\left(\phi \right)$ for the Ag triangle with regard to the HCT structure.

Figure 6

(Color online) Calculated RHEPD intensities for the (0 0), $(1∕31∕3)$, and $(2∕32∕3)$ spots on the $\mathrm{Si}\left(111\right)\text{−}\sqrt{3}\times \sqrt{3}\text{−}\mathrm{Ag}$ surface as a function of $p$ for the Ag atoms at the IET(+) and IET(−) sites. The intensities are calculated based on (a) dynamical and (b) kinematical diffraction theories. The incident condition is the same as that in Fig. 4.

Figure 7

(Color online) Temperature dependences of the RHEPD intensities for the (0 0) spot at $\theta =1.6°$ and $\theta =3.5°$ under the one-beam condition and for the $(2∕32∕3)$ spot at $\theta =1.0°$ along 1.5° with regard to the $\left[11\overline{2}\right]$ direction. The solid lines show the intensities calculated by using the Debye temperatures of $110\mathrm{K}$ (surface parallel) and $140\mathrm{K}$ (surface normal) for the topmost Ag atoms and the bulk Debye temperature of $610\mathrm{K}$ for the bulk Si atoms. The incident positron energy is $20\mathrm{keV}$.

Figure 8

Temperature dependence of the totally reflected RHEPD intensities for the (0 0), $(1∕31∕3)$, and $(2∕32∕3)$ spots on the $\mathrm{Si}\left(111\right)\text{−}\sqrt{3}\times \sqrt{3}\text{−}\mathrm{Ag}$ surface. The glancing angle is 2.0°, which satisfies the total reflection condition. The azimuthal angle corresponds to 1.5° with regard to the $\left[11\overline{2}\right]$ direction. The incident positron energy is $10\mathrm{keV}$. The circles denote the experimental intensities. The solid line shows the intensity calculated on the basis of the dynamical diffraction theory using the critical exponent of 0.28.