Strain and correlation of self-organized ${\text{Ge}}_{1-x}{\text{Mn}}_x$ nanocolumns embedded in Ge (001)

We report on the structural properties of ${\text{Ge}}_{1-x}{\text{Mn}}_x$ layers grown by molecular-beam epitaxy. In these layers, nanocolumns with a high Mn content are embedded in an almost-pure Ge matrix. We have used grazing-incidence x-ray scattering, atomic force and transmission electron microscopy to study the structural properties of the columns. We demonstrate how the elastic deformation of the matrix (as calculated using atomistic simulations) around the columns, as well as the average intercolumn distance can account for the shape of the diffusion around Bragg peaks.

Figure 1

(a) Plane-view TEM micrograph, (c) surface topography measured by AFM, and [(b) and (d)] the central zones of the corresponding FT, each with an inset showing the radial distribution of the FT amplitude. All color scales are linear with the full $z$ scale for the AFM image corresponding to 2.6 nm. In each FT image a more intense ring can be observed with correlation distances equal to $\sim 10\text{nm}$ and $\sim 40\text{nm}$, respectively, in the TEM (intercolumn average spacing) and the AFM (surface roughness) images.

Figure 2

(Color online) Bragg-GISAXS (see text for details) map measured by an angular scan around the in-plane (220) Bragg reflection, using a Vantec position-sensitive detector, at $E=6.5\text{keV}$ and ${\alpha }_i=0.4°$ (intensity plotted on a logarithmic scale, with eight contour levels per order of magnitude). $Q_{ang}$ and $Q_z$ are, respectively, parallel and perpendicular to the surface. Three finite distances effects can be seen, symmetrically about the $Q_{ang}=0$ axis: (i) a correlation distances of about 9–15 nm (marker “D”), corresponding to the average distance between columns, (ii) the oscillations (marker “T”) along $Q_z$ (dotted-dashed line) are due to the finite thickness of the sample (60 nm), and (iii) the two intense streaks (marker “R”) at very small $Q_{ang}$ have been attributed to the surface roughness.

Figure 3

(Color online) (a) Grazing-incidence RSM of the measured intensity in the $hk0$ plane—this map features almost only the (220) and (400) Ge Bragg peaks with a weak diffuse line between them. [(b)–(e)] High-resolution RSM around the (220), (400), (440), and (620) reflections. Around each Bragg peak an asymmetric ring can be seen at $\approx 0.07\text{r}\text{.l}\text{.u}\text{.}$: its positions is given by the average intercolumn distance while the asymmetry of the intensity is due to the compression of the matrix surrounding each column. In (f) and (g) are shown the RSM around reflections (220) and (400), for the same type of sample as for Fig. 2, but after oxidization (due to aging and exposure to air) of the sample: the correlation ring is almost circular due to the decrease in the strain of the Ge matrix around the oxidized columns. The intensity is represented as a logarithmic color scale, the full scale corresponding to 3 orders of magnitude for all plots.

Figure 4

(Color online) (a) Amplitude of the atoms radial displacements (in nm), following an isotropic elastic model around a cylindrical inclusion. (b) Radial pair-distribution function of the nanocolumns, measured experimentally (dots) and fitted using a $\text{gaussian}+\text{error}$ function model. (c) Scattering from a single isolated nanocolumn around the (220) reflection, including the scattering contributions from the displaced atoms as well as the missing atoms inside the column (see text for details). (d) Scattering of an assembly of nanocolumns around the (220) reflection, by taking into account their radial distribution function shown in (b). Intensity is plotted with a logarithmic color scale for (c) and (d).

Figure 5

(Color online) Atomic model obtained using a Stillinger-Weber potential, combining amorphous columns (gray spheres; darker shade indicates larger depth) surrounded by the germanium matrix (Ge atoms as black disks). Ge-Ge bonds of the perfect (undistorted) lattice are shown in gray (green online). Note the atomic displacements of the Ge atoms in the matrix, due to the column expansion.

Figure 6

(Color online) (a) Mapping of the simulated atomic displacements (from the reference perfect Ge lattice, displacements are given in nanometer), for a simulation with $100\times 100\text{unit}$ cells and 42 GeMn nanocolumns (see text for details). Reciprocal space maps have been calculated by summing the intensities from 20 independent atomistic simulations to lower speckle effects. The maps around (b) the (220) and (c) the (400) reflections exhibit the same features as in Figs. 3d, 3e. In (d) and (e) are presented the same maps as for (b) and (c), but where atomic displacements have been reduced by a factor 5. The ring observed around the Bragg reflections is located at the position determined by the average distance between nanocolumns, and the polarization of its intensity (high vs low angle) is determined by the amplitude of the atomic displacements of the atoms in the matrix surrounding the columns. Intensity is plotted with a logarithmic color scale for (b)–(e) with a full scale corresponding to 4 orders of magnitude.

Figure 7

Normalized angular distribution of the intensity on the ring surrounding the (220) reflection. This distribution is characteristic of the amplitude of deformation of the matrix lattice around the nanocolumns with a larger asymmetry in the radial direction (0 vs $180°$) for larger deformations: (a) distribution observed for a sample grown in situ (continuous line) and another (dashed line) with oxidized columns; (b) simulated distribution of intensities, using a single nanocolumn, for various amplitudes of deformations—the relative displacement amplitudes around the columns are given in the inset—for the largest amplitude the maximum displacement is $\approx 0.05\text{nm}$ for atoms near the border of the columns.