Intrinsic Character of the $(3\times 3)$ to $(\sqrt{3}\times \sqrt{3})$ Phase Transition in $\mathrm{P}\mathrm{b}/\mathrm{S}\mathrm{i}(111)$

We have investigated the $(3\times 3)$ to $(\sqrt{3}\times \sqrt{3})$ reversible phase transition in $\mathrm{P}\mathrm{b}/\mathrm{S}\mathrm{i}(111)$ by means of variable temperature scanning tunneling microscopy and density functional first-principles calculations. By tracking exactly the same regions of the surface with atomic resolution in a temperature range between 40 and 200 K, we have observed the phase transition in real time. The ability to prepare and track exceptionally large domains without defects has allowed us to detect the intrinsic character of the phase transition at temperatures around 86 K. This intrinsic character is in full agreement with our first-principles calculations. Moreover, our results show that the hypothesis that point defects play a fundamental role as the driving force, reported for similar systems, can be discarded for $\mathrm{P}\mathrm{b}/\mathrm{S}\mathrm{i}(111)$.

Figure 1

(a),(b) STM images showing the general morphology of the sample at 43 K [the lower part of image (a) matches the upper part of image (b)]. (c),(d) The same free-of-defects area highlighted in (a) measured at 43 K and 135 K, respectively. (e),(f) The defective zoom-in region marked in (b) again at 43 K (e) and 135 K (f). Size of the images: $70\times 40{\mathrm{n}\mathrm{m}}^2$ [(a),(b)]; $13\times 13{\mathrm{n}\mathrm{m}}^2$ [(c)–(f)]. Sample bias, $-1.5\mathrm{V}$, and tunneling current, 0.1 nA, for all images. (a) and (b) are displayed in a derivative view [27].

Figure 2

Constant current images extracted from an STM movie [19] spanning in temperature from 40 to 140 K. All the frames of the movie are measured in exactly the same region of the surface while the phase transition undergoes. Size of the images, $20\times 13{\mathrm{n}\mathrm{m}}^2$; sample bias, $-1.5\mathrm{V}$; tunneling current, 0.2 nA. The images are shown in a derivative mode [27].

Figure 3

Plot of the $(3\times 3)$ peak intensity as a function of the temperature. The intensity has been obtained from the Fourier transform (FT) of each of the frames of an STM movie measured in a $10\times 10{\mathrm{n}\mathrm{m}}^2$ free-of-defects area. The insets show 3D views of the typical FT below and above $T_c$ [a circle outlines the position of one $(3\times 3)$ peak]. The full lines correspond to a fit of the intensity obtained using $I\propto |t{|}^{\lambda }$, where, for $T<T_c$, $\lambda =2\beta$ and, for $T>T_c$, $\lambda =-\gamma$. The experimental critical exponents are in very good agreement with the predicted three-state Potts model (see text).

Figure 4

(a) GGA-DFT total energy as a function of ${\mathrm{P}\mathrm{b}}_1$ displacement perpendicular to the surface. The energy of the $\sqrt{3}\times \sqrt{3}$ structure is taken as the reference. (b),(c) Experimental STS average spectra above (b) and below (c) the phase transition.

Figure 5

STM images of the same defective region measured at 40 K (left) and 200 K (right). None of the point defects modifies its position as the temperature is increased. Size of the images, $6\times 13{\mathrm{n}\mathrm{m}}^2$; sample bias, $-1.5\mathrm{V}$; tunneling current, 0.1 nA.