Critical behavior of the dimerized Si(001) surface: Continuous order-disorder phase transition in the two-dimensional Ising universality class

The critical behavior of the order-disorder phase transition in the buckled dimer structure of the Si(001) surface is investigated both theoretically by means of first-principles calculations and experimentally by spot profile analysis low-energy electron diffraction (SPA-LEED). We use density functional theory (DFT) with three different functionals commonly used for Si to determine the coupling constants of an effective lattice Hamiltonian describing the dimer interactions. Experimentally, the phase transition from the low-temperature $c(4\times 2)$- to the high-temperature $p(2\times 1)$-reconstructed surface is followed through the intensity and width of the superstructure spots within the temperature range $78–400\mathrm{K}$. Near the critical temperature $T_{\mathrm{c}}=190.6\mathrm{K}$, we observe universal critical behavior of spot intensities and correlation lengths, which falls into the universality class of the two-dimensional (2D) Ising model. From the ratio of correlation lengths along and across the dimer rows we determine effective nearest-neighbor couplings of an anisotropic 2D Ising model, $J_{\parallel }=(-24.9\pm 0.9_{\mathrm{stat}}\pm 1.3_{\mathrm{sys}})\mathrm{meV}$ and $J_{\perp }=(-0.8\pm 0.1_{\mathrm{stat}})\mathrm{meV}$. We find that the experimentally determined coupling constants of the Ising model can be reconciled with those of the more complex lattice Hamiltonian from DFT when the critical behavior is of primary interest. The anisotropy of the interactions derived from the experimental data via the 2D Ising model is best matched by DFT calculations using the PBEsol functional. The trends in the calculated anisotropy are consistent with the surface stress anisotropy predicted by the DFT functionals, pointing towards the role of surface stress reduction as a driving force for establishing the $c(4\times 2)$-reconstructed ground state.

Figure 1

Dimer reconstruction of Si(001). Atomic structure models of the Si(001) surface for (a) the symmetric $p(2\times 1)$ and (b) the $c(4\times 2)$ reconstruction. (c) Spin model describing the arrangement of alternation of dimer buckling. Effective interaction parameters $J_{\parallel }$ and $J_{\perp }$ along and across the dimer rows are indicated. (d) STM image recorded at $T=300\mathrm{K}$, $U=-2.0\mathrm{V}$, $I=0.7\mathrm{nA}$ (occupied states) exhibiting buckled dimers in the vicinity of a surface defect.

Figure 2

Schematics of dimer tilting patterns of the Si(001) surface. Seven different dimer tilting patterns of the Si(001) surface were studied in DFT modeling to determine up to six interaction parameters between the Si dimers (denoted by the highlighted arrows in the ellipsoids). Here, SDF, TDF, and SDF-2R stand for single dimer flipped, twin dimers flipped (in one row), and single dimer flipped in adjacent rows, respectively. Heads and tails of arrows denote the up- and down-dangling bonds of the tilted surface Si dimers as introduced is Fig. 1.

Figure 3

Energetics of the anisotropic 2D Ising model. (a) Critical temperature $T_{\mathrm{c}}$ as a function of the effective NN coupling constants $J_{\parallel }$ and $J_{\perp }$. (b) Temperature and correlation length ratio dependence of the coupling energy ratio $|J_{\parallel }/J_{\perp }|$ derived from Eq. (8). Equation (6) holds at the black-dotted line with $t=0$ where the relation for $({\xi }_{\parallel }^{+}/a_{\parallel })/({\xi }_{\perp }^{+}/a_{\perp })$ simplifies to Eq. (10). Red-solid lines and the data points indicate results for $T_{\mathrm{c}}=(190.6\pm 0.4_{\mathrm{stat}}\pm 9.6_{\mathrm{sys}})\mathrm{K}$ in (a) and $J_{\parallel }/J_{\perp }=31.2\pm 3.8_{\mathrm{stat}}$ in (b) as derived from the experiment.

Figure 4

SPA-LEED patterns. Patterns were taken at (a) $T=300\mathrm{K}$ and (b) $T=80\mathrm{K}$, respectively. Primitive unit cells of the $p(2\times 1)$ and $c(4\times 2)$ reconstructions are indicated by the yellow rectangles and orange rhombi, respectively. Additional to the spots streaklike intensity is found centered at the $(3/4\overline{1/2})$ spots.

Figure 5

Temperature dependence of diffraction spots. (a) Scaled LEED spot intensities $I$ of the (00) spot, half-order spots, and the $(3/4\overline{1/2})$ spot (shown with its Gaussian and Lorentzian contributions). The spot intensity decreases with increasing temperature due to the Debye-Waller effect according to Eq. (12). Debye temperatures are indicated. The $(3/4\overline{1/2})$ spot clearly deviates from simple Debye-Waller behavior as its intensity is affected by the order-disorder phase transition. (b) The integrated intensity $I_{\mathrm{Int}}$ of the $(3/4\overline{1/2})$ spot drops as function of temperature due to the Debye-Waller effect according to Eq. (12) without being affected by the phase transition. Colored solid lines indicate the expected intensity drop due to the Debye-Waller effect for various Debye temperatures ${\mathrm{\Theta }}_{\mathrm{D}}$ between 325 K and 500 K. The black solid line depicts the best fit with ${\mathrm{\Theta }}_{\mathrm{D}}=(391\pm 7_{\mathrm{stat}})\mathrm{K}$. (c) Intensity of the homogeneous thermal diffuse background $I_{\mathrm{B}}$ and corresponding Debye-Waller fit yielding ${\mathrm{\Theta }}_{\mathrm{D}}=(367\pm {108}_{\mathrm{stat}})\mathrm{K}$. (d), (e) FWHMs (uncorrected for instrumental resolution) of the (00) spot and half-order spots along the $\left[110\right]$ and $\left[1\overline{1}0\right]$ direction, respectively. Insets show line profiles (red dots) and corresponding fits (black lines) through the (00) spot and neighboring half-order spots at $T=78\mathrm{K}$.

Figure 6

Temperature dependence of line profiles through the $(3/4\overline{1/2})$ spot. Profiles through the center of the spot were recorded in the directions across (left panel) and along (right panel) the dimer rows. Data are plotted as bluish to reddish lines while the fit to the data is given by the solid-red lines, see also Eq. (13). For better visibility the experimental data is Fourier-filtered and 80% of the linear background is subtracted. The Gaussian (purple) and Lorentzian (rose) contributions to the spot intensity (corrected for instrumental resolution) are schematically depicted in logarithmic scale closely above $T_{\mathrm{c}}$ at $T=198\mathrm{K}$ in the right panel.

Figure 7

Order-disorder phase transition in Si(001). Critical behavior of the Gaussian (a) and Lorentzian (b) contributions $I_{\mathrm{G},\mathrm{L}}$ to the $(3/4\overline{1/2})$ spot intensity during and above the phase transition, respectively. Data are corrected for the Debye-Waller effect. (c), (d) Temperature-dependent FWHMs of the Lorentzian contribution to the $(3/4\overline{1/2})$ spot profile along (along $\left[110\right]$ direction, intra dimer row coupling) and across (along $\left[1\overline{1}0\right]$ direction, inter dimer row coupling) the dimer rows corrected for the instrumental resolution, respectively. (e) Correlation lengths ${\xi }_{\parallel ,\perp }$ of the Lorentzian contribution vs. reduced temperature $t$ above $T_{\mathrm{c}}$. (f) Correlation length ratio $({\xi }_{\parallel }/a_{\parallel })/({\xi }_{\perp }/a_{\perp })$ of the Lorentzian contribution. Data are fitted for (a) $T\in [95\mathrm{K},T_{\mathrm{c}})$, (b) $T\in [225\mathrm{K},400\mathrm{K}]$, (c) $T\in [220\mathrm{K},300\mathrm{K}],$ and (d) $T\in [210\mathrm{K},300\mathrm{K}]$. Light-pink data points belong to the so-called domain state and are not taken into account for the fits in (b)–(f). Colored lines indicate fits to the critical behavior predicted by Onsager theory, determining the critical temperature $T_{\mathrm{c}}=(190.6\pm 0.4_{\mathrm{stat}}\pm 9.6_{\mathrm{sys}})\mathrm{K}$. First-order corrections are taken into account for the FWHMs and the correlation lengths in (c)–(e), respectively. The asymptotic (linear) behavior is shown by the colored dashed lines in (c)–(e). In (f) the expected behavior for the 2D Ising model (dark-yellow line) is derived by matching the values with the results from the fit to the critical behavior (dark-pink line) at $T_{\mathrm{c}}$. The reduced temperature $t$ is derived from Ising model $T_{\mathrm{c}}=190.6\mathrm{K}$. Shaded areas indicate corresponding systematic errors, accordingly. Black-dotted lines indicate fits to the critical behavior with unfixed exponents $\beta$ and $\gamma$.

Figure 8

Scaling plot of the Lorentzian scaling function $A_{\mathrm{L}}^{+}\left(y_{\delta }^{+}\right)$ from Eq. (13c) above $T_{\mathrm{c}}$, with direction $\delta \in \{\parallel ,\perp \}$ and scaling variable $y_{\delta }^{+}$ from Eq. (13d). The black solid line is the Ornstein-Zernike prediction $A_{\mathrm{OZ}}\left(y\right)={(1+y^2)}^{-1}$, while the red-dashed line is the exact scaling function from Ref. [89 Eq. (B7)].