X-ray diffraction investigation of a three-dimensional Si/SiGe quantum dot crystal

High-resolution x-ray diffraction was employed to study the structural properties of a three-dimensional periodic arrangement of SiGe quantum dots in a Si matrix. Using extreme ultraviolet lithography at a synchrotron source a two-dimensional array of pits (period $90\times 100{\text{nm}}^2$) was defined and transferred into a (001) Si wafer by reactive ion etching. By molecular-beam epitaxy SiGe islands of about 30 nm diameter and 3 nm height were grown into the pits. Subsequent deposition of Si spacer layers of 10 nm thickness and SiGe island layers results in a three-dimensionally periodic arrangement of quantum dots, mediated by the strain fields of the buried dots. Their so far unmatched structural perfection is assessed by coplanar x-ray diffractometry using synchrotron radiation. Reciprocal-space maps around the (004) and (224) reciprocal-lattice maps were recorded and analyzed to get quantitative information on the disorder of the dot positions and to obtain the mean Ge content of the dots. In addition, information on the strain fields was deduced from the analysis of the diffraction data. Together with atomic force microscopy data on the island shape and size distribution, a complete structural characterization is achieved.

Figure 1

(Color online) AFM micrographs of the first island layer grown in the (a) prepattern pits and in (b) an unpatterned substrate region. The resulting distributions of island height and base diameter are shown in (c) and (d), respectively.

Figure 2

(Color online) Cross-sectional transmission electron micrograph of a (a) three-dimensional dot crystal and of the (b) reference piece without prepatterning. AFM images of the topmost island layer prior to the final capping step are shown in (c) and (d) and the corresponding distributions of island height and diameter are displayed in (e) and (f).

Figure 3

Reciprocal-space maps of diffracted intensity measured in (a) (004) and (b) (224) coplanar diffractions. The oblique streaks are experimental artifacts explained in the text.

Figure 4

Structure model of the dot positions, the $j\text{th}$ and $k\text{th}$ columns of the dots are sketched, $n-1$, $n$, and $n+1$ are the interface indexes.

Figure 5

(Color) (a) The two-dimensional correlation function $G\left(q_x,q_z\right)$ calculated for $\sigma =3\text{nm}$, ${\sigma }_0=3\text{nm}$, and $M=10$, (b) the vertical shape of the first four lateral satellites ($\sigma =3\text{nm}$, normalized to the maximum value), and (c) the correlation function $G\left(q_x,q_z=0\right)$ calculated for various ${\sigma }_0$ (in nanometers).

Figure 6

(a) The dependence of the vertical FWHMs $\Delta q_z$ of the lateral satellites on $q_x$, determined from (004) and (224) reciprocal-space maps (points) and its fit by the theory (line). (b) The ratio of the intensity of the satellites and the intensity of the background between the satellites determined from the measured intensity maps (points) and its fit by the theory (line).

Figure 7

(Color online) The integrated intensities of the satellites calculated for various local Ge concentrations in the dot volumes, using the dot shapes specified in the text (open squares and solid triangles), the measured intensities are denoted by black points.

Figure 8

(Color) Components ${\epsilon }_{xx,zz}$ of the strain tensor calculated in the $xz$ plane and ${\epsilon }_{xx,yy}$ in the $xy$ plane for $z=56\text{nm}$. The contour step is $5\times {10}^{-4}$ and the positive (negative) values are denoted by red (blue) colors. The free surface is at $z=110\text{nm}$; the different shape of the lowermost dot is clearly visible.

Figure 9

The Fourier transformations of AFM pictures (points) and their theoretical simulations (lines). The AFM pictures have been taken from the first and tenth dot arrays (left and right panels, respectively).