Diffraction from disordered vicinal surfaces with alternating terraces

Vicinal surfaces with terraces of alternating stress develop inhomogeneous distributions of terrace sizes which sometimes leads even to the formation of double steps. Both vicinal Si(001) and vicinal Ge(001) are typical examples for this behavior. However, vicinal surfaces of some alloys show this effect, too. It is well established that average terrace sizes can be evaluated from the splitting of peaks in surface sensitive diffraction experiments. More parameters, however, are necessary to obtain an improved characterization of the morphology of the vicinal surface. Therefore, we present a detailed analysis of diffraction patterns from alternating vicinal surfaces to extract more statistical data, e.g., standard deviations of the terrace size distributions, step rms widths, step correlation lengths, and kink densities. This analysis considers both profiles of (split) diffraction peaks and the profile of the diffuse scattering. In addition, the diffraction analysis is applied to vicinal Ge(001) to characterize the morphology in full detail.

Figure 1

Schematic drawing of a cross section of a vicinal surface, which ascends in the positive $x$ direction to clarify the nomenclature. The $m\text{th}$ row is presented. The $n\text{th}$ terrace (emphasized by dark dots) at height $z=nd$ is confined by the ascending step at position $R_{n,m}$ and the descending step at position $R_{n-1,m}$.

Figure 2

Phase dependence of the diffraction pattern. Open circles: diffraction intensity calculated from Eq. (14) for ${\beta }_A={\beta }_B=0$ (no step roughness). Solid lines: best fitting of the sum of Lorentzians to the diffraction intensities. Gamma distributions are assumed for the terrace size distribution. The average size of the terraces of type $A$ is three times larger than the average size of the terraces of type $B$ $({\Gamma }_B∕{\Gamma }_A=1∕3)$. The standard deviations of both distributions of $A$-type terraces and $B$-type terraces are ${\sigma }_A∕{\Gamma }_A={\sigma }_B∕{\Gamma }_B=1∕5$ of the average terrace size. The first line scan is for $S=0.05$ and the last for $S=0.45$. The first line scan shows diffraction peaks of orders $\nu =-1$, $\nu =0$, and $\nu =1$ (from left to right). For higher scattering conditions $S$, the second order peak $(\nu =2)$ appears on the right side.

Figure 3

Phase dependence of the diffraction peaks. Open symbols: parameters obtained from fitting peak profiles to Lorentzians (see Fig. 2). Top: integrated intensities of the peaks $(G_0^{\mathit{int}}=2\kappa G_0)$. The solid lines show the behavior obtained from Eq. (24). Bottom: full width at half maximum $(\mathrm{FWHM}=2\kappa )$. The solid lines show the quadratic behavior as received from Eq. (21).

Figure 4

Roughness dependence of the diffraction pattern for diffraction condition $S=1∕4$. Open circles: diffraction intensity calculated from Eq. (14) for ${\beta }_A={\beta }_B=0$ (no step roughness). A logarithmic scale is used because of the large dynamic range of the diffraction intensity. In addition, the line scans are rescaled for reasons of clarity. Solid lines: best fitting of the sum of Lorentzians to the diffraction pattern. Gamma distributions are assumed for the terrace size distribution. The average size of the terraces of type $A$ is three times larger than the average size of the terraces of type $B$. The standard deviation of the distribution of $A$-type terraces varies from ${\sigma }_A∕{\Gamma }_A=1∕15$ (bottom) to ${\sigma }_A∕{\Gamma }_A=10∕15$ (top). The standard deviation of $B$-type terraces is kept constant at ${\sigma }_B∕{\Gamma }_B=2∕5$.

Figure 5

Roughness dependence of the diffraction peaks. Open symbols: analysis of the Lorentzians of Fig. 4. Top: integrated intensity $(G_0^{\mathit{int}}=2\kappa G_0)$ of the peaks, which is almost constant. Center: full width at half maximum $(\mathrm{FWHM}=2\kappa )$, which follows the quadratic behavior of Eq. (21). The FWHM of the peaks is almost identical due to the symmetry of the diffraction condition. Bottom: peak positions which are almost constant except for large values of the standard deviations ${\sigma }_A∕{\Gamma }_A$ due to higher order effect of the cumulant approximation.

Figure 6

Influence of the step fluctuations on the diffraction pattern. Open symbols: diffraction intensity calculated from Eq. (14) for scattering phase $S=0.25$ with ${\Gamma }_B∕{\Gamma }_A=1∕3$ and ${\sigma }_A∕{\Gamma }_A={\sigma }_B∕{\Gamma }_B=1∕5$. Gamma distributions are assumed for the terrace size distribution. Open dots: no step fluctuations $(w_A=w_B=0)$. Open squares: with step fluctuations $w_A∕{\Gamma }_A=w_B∕{\Gamma }_B=1∕2$. Solid lines: best fitting of the peaks with Lorentzians. The peak profiles are not affected by step fluctuations but by the intensity of the peaks. Please note that a linear scale is used for the intensity in contrast to the logarithmic scale of previous figures of diffracted intensities.

Figure 7

Influence of the step fluctuations on the diffraction peaks at $S=0.25$, ${\Gamma }_A∕{\Gamma }_B=3$, and ${\sigma }_A∕{\Gamma }_A={\sigma }_B∕{\Gamma }_B=1∕5$. Open symbols: analysis of peak profiles to which Lorentzian profiles are fitted. Top: integrated intensity $(G_0^{\mathit{int}}=2\kappa G_0)$ of the peaks, which decreases with increasing amplitude $w_{A,B}$ of the step fluctuations. The solid line shows the Gaussian dependency of the peak intensity on the step fluctuations. Bottom: full width at half maximum $(\mathrm{FWHM}=2\kappa )$, which is almost constant.

Figure 8

Model of the vicinal Ge(100) surface with both $1\times 2$ reconstructed $A$ terraces and $2\times 1$ reconstructed $B$ terraces. The terraces are separated by both straight $S_A$ steps and rough $S_B$ steps with different structures due to the different reconstructions of the upper terrace.

Figure 9

(Color online) Mapping of the reciprocal space in the plane opened by the vertical [001] vector and the lateral [110] vector. The inclined surface diffraction rods are caused by the $5.4°$ inclination toward the $\left[\overline{1}\overline{1}0\right]$ direction of the surface with respect to the Ge(001) plane. The vertically running rods at $K_x=\pm 50\%Bz$ are due to $1\times 2$ reconstructed terraces.

Figure 10

Diffraction profiles obtained from Ge(001) with 5.4° miscut toward the $\left[\overline{1}\overline{1}0\right]$ direction. The profiles are recorded in the direction of miscut. One observes peaks of the orders $\nu =3$ (left peaks) and $\nu =4$ (right peak) for scattering phases below $S=1.75$. The positions of the peaks shift with increasing scattering condition, as expected for vicinal surfaces. Above $S=2$, the peak of order $\nu =3$ vanishes while the peak of order $\nu =5$ appears at the right side. The peak of order $\nu =6$ appears close to $S=2.5$ at the right side.

Figure 11

FWHM of diffraction peaks of orders $\nu =4$, $\nu =5$, and $\nu =6$ obtained from Ge(001) with 5.4° miscut toward the $\left[\overline{1}\overline{1}0\right]$ direction. The FWHM of the peaks follows the square dependence predicted by Eq. (21).

Figure 12

Attenuation of the intensity of the diffraction peaks of fourth and sixth orders with respect to the step roughness $w_B$. From diffuse diffraction data, $w_B=0.15({\Gamma }_A+{\Gamma }_B)$ is estimated. Therefore, the attenuation is negligible within the error of the experimental data.

Figure 13

Two-dimensional diffraction pattern recorded from a vicinal Ge(001) surface with 2.7° miscut toward the $\left[\overline{1}\overline{1}0\right]$ direction for scattering phase $S=3.7$. The diffraction peaks are of the orders $\nu =7$ (left) to $\nu =9$ (right). The dashed lines show line scans of the diffuse diffraction presented and analyzed in Fig. 14.

Figure 14

(a) Line scans extracted from Fig. 13 recorded in the $\left[\overline{1}10\right]$ direction ($y$ direction) for different values of $K_x$. (b) $K_x$ dependence of the FWHM of the line scans. The line shows the dependence described by Eq. (29).