Displacive phase transition at the $5∕3$ monolayer of Pb on Ge(001)

At a coverage of $5∕3$ monolayer (ML), Pb adsorbed on Ge(001) forms a ground phase displaying a $\left(\genfrac{}{}{0ex}{}{21}{06}\right)$ symmetry. This phase undergoes two reversible phase transitions $\left(\genfrac{}{}{0ex}{}{21}{06}\right)\leftrightarrow \left(\genfrac{}{}{0ex}{}{21}{03}\right)\leftrightarrow (2\times 1)$ at the critical temperatures $T_{c_1}\sim 178\mathrm{K}$ and $T_{c_2}\sim 375\mathrm{K}$, respectively. We investigated the behavior of the relevant order parameters at the critical temperatures by means of He and in-plane x-ray diffraction (HAS and XRD, respectively). Both phase transitions at the critical temperature put in evidence a clear order-disorder behavior, in agreement with the universality class expected for the corresponding symmetry group transformation. The low-temperature transition yields the critical exponent of the two-dimensional (2-D) Ising universality class, whereas the three-state Potts’ critical exponents are found for the high-temperature transition. By out-of-plane XRD measurements, the low-temperature phase transition is observed to be accompanied by a static surface distortion at room temperature. A complementary HAS study of the temperature evolution of the surface charge corrugation reveals that the complete $\left(\genfrac{}{}{0ex}{}{21}{06}\right)\leftrightarrow \left(\genfrac{}{}{0ex}{}{21}{03}\right)$ transition is of the displacive type. On the contrary, the high-temperature phase transition does not show any change of the surface corrugation up to its irreversible decomposition, thus pointing to a pure order-disorder character.

Figure 1

(Color online) Upper panel: the unit cell of the $\left(\genfrac{}{}{0ex}{}{21}{06}\right)$ phase in the reciprocal space. The open squares represent the surface diffraction peaks of a single domain. Central panel: the $\left(\genfrac{}{}{0ex}{}{21}{06}\right)$ phase has a fourfold degeneracy due to the presence of four 90° rotated domains, shown with four different markers. Lower panel: the $\left(\genfrac{}{}{0ex}{}{21}{03}\right)$ phase is characterized by the disappearance of one peak every two along the [120] surface direction.

Figure 2

(Color online) Upper panel: HAS intensity taken during Pb deposition at $T_s\sim 300\mathrm{K}$ for a few off-specular diffracted peaks. The coverage is expressed in terms of substrate surface atom density. The absolute Pb coverage has been calibrated a posteriori on the maximum of the $(1∕3,1∕3)$ peak. The appearance of the $c(4\times 8)-i$ phase (first layer saturation coverage) is witnessed by the maximum of the $(1∕4,0)$ peak. Lower panel: the specularly reflected x-ray and He intensity are reported on the left and right $y$ axis, respectively. The XRR reflectivity coverage has been calibrated from LT layer-by-layer growth oscillations, according to Ref. 23.

Figure 3

The $(1∕3,1∕3)$ peak intensity is shown across the $\left(\genfrac{}{}{0ex}{}{21}{06}\right)\to (2\times 1)$ transition. Filled and open markers show the phase transition measured with increasing and decreasing temperature, respectively, and evidence the absence of any detectable hysteresis. The sample has been kept only a few minutes at $400\mathrm{K}$, before decreasing the temperature.

Figure 4

(Color online) Temperature behavior of the He-diffracted peaks. The peak intensity is reported on a logarithmic scale to put in evidence the deviation from the Debye-Waller attenuation of the $(5∕12,1∕6)$ and $(1∕3,1∕3)$ peaks, due to the phase transitions. The specularly reflected (0,0) peak has been measured with the [100] surface direction aligned in the scattering plane.

Figure 5

Temperature dependence of the $(1∕2,1∕4)$ He diffraction peak taken along the [210] surface direction. The surface temperature during the acquisition of each point was kept constant within $0.2\mathrm{K}$. The peak intensity (open squares, left $y$ axis) and the Lorentzian FWHM (open circles, right $y$ axis) were obtained by fitting the data to a Voigt function with a fixed Gaussian width of 0.1° due to the instrumental peak broadening. A Lorentzian width of 0.18°, as obtained for the lowest temperature, has been subtracted to better highlight the broadening due to the proliferation of domain walls. The full and shaded line is a fit to the Lorentzian width with the power law, as described in the text.

Figure 6

Temperature dependence of the $(5∕12,1∕6)$ He diffraction peak taken along the [520] surface direction. A constant Gaussian width of 0.13° was used in the Voigt fit. The Lorentzian width at the lowest temperature was 0.12°. The same notation as in Fig. 5.

Figure 7

Temperature dependence of the $(1∕3,1∕3)$ He diffraction peak taken along the [110] surface direction. A Gaussian width of 0.17° was used in the Voigt fit. The Lorentzian width at the lowest temperature was 0.09°. The same notation as in Fig. 5.

Figure 8

Temperature dependence of the $(4∕3,1∕3)$ peak taken by in-plane XRD. Azimuthal scans have been performed with a $6500\mathrm{eV}$ photon energy, while keeping the sample at the critical angle of 0.4°. The peak intensity and the Lorentzian FWHM were obtained by fitting the data to a pure Lorentzian function. The Lorentzian width at the lowest temperature was 0.197°. The same notation as in Fig. 5.

Figure 9

(Color online) Top view of the structural model proposed in Ref. 21 for the $\left(\genfrac{}{}{0ex}{}{21}{06}\right)$ phase. Pb atoms are represented by the large grey circles, the underlying Ge atoms are represented by the small filled circles, and the surface unit cell is also drawn (the lozenge with thick lines). The brighter and darker circles (labeled $A$ and $B$, respectively) are the only Pb atoms in high symmetry sites and, according to Bunk and co-workers, their vertical height differs by $\sim 1.5\mathrm{\AA }$.

Figure 10

(Color online) The XRD $(3∕2,2,L)$ rod scans are shown for three surface temperatures: $173\mathrm{K}$ open squares, $273\mathrm{K}$ filled circles, and $373\mathrm{K}$ open triangles. Upper panel: the integrated intensity along the rod is shown after simple correction for the ALOISA scattering geometry. Lower panel: the same rod scans are shown after correction for the Debye-Waller attenuation (see the text).

Figure 11

(Color online) The temperature dependence of the HAS diffraction peaks is shown after normalization to the specular (0,0) peak, as taken with the same azimuthal orientation of the corresponding diffraction peak. As a guide to the eye, the linear fits to the logarithm of the relative intensity are superimposed to the data to highlight the change of slope.