We have spatially investigated lattice spacing, twist, and bending in individual laterally (110)-oriented Ge nanowires (NWs) on pre-patterned Si(001) substrates. A combination of synchrotron-based scanning x-ray diffraction microscopy with an x-ray focus size of 50 nm and numerical finite element calculations on the elastic strain reveals a three-dimensional relaxation scenario, which becomes particularly complex next to NW nucleation points. Despite a lattice mismatch of 4.2%, lattice compliance is preserved, since strain can effectively be released close to the seeding window. Areas in the NWs other than that appear fully relaxed. The resulting NW twist, i.e., lattice rotations around the growth axis, amounts to less than 0.1°.

High-quality, dislocation-free low-dimensional Ge structures integrated on Si are experiencing a renaissance due to their realistic potential in future quantum technologies.1,2 Especially, the development of compatible high-κ dielectrics replacing the naturally unstable Ge oxide gives a boost to Ge-based CMOS strategies without the need of expensive substrates. In that context, low-dimensional objects such as nanowires (NWs) bear great potential for many applications, such as, e.g., in thermoelectrics,3 enabling all-electrical spin manipulation through an exceptionally strong spin–orbit interaction4 or in low-power devices such as tunneling field-effect transistors (TFETs).5 

Major obstacles toward dislocation-free Ge nanostructures on Si are the different thermal expansion coefficients and a lattice mismatch of 4.2%. Epitaxial growth performed close to thermodynamic equilibrium on planar substrates follows the Stranski–Krastanow growth mode in which the accumulated strain energy will be relieved through formation of pseudomorphic three-dimensional islands on top.6 Elastic strain as mediated through the substrate may further result in self-assembled ensembles of highly uniform objects.7 Beyond a critical thickness, plastic relaxation sets in by formation of misfit dislocations. Another approach uses patterned substrates as an ensemble of well-defined nucleation spots.8 Here, the effective interface area remains limited to the seeding windows, and consequently, the stored strain energy becomes effectively decreased compared to the planar (i.e., non-structured) case. This route has been widely used to ensure a high degree of ordering within the NW ensemble, e.g., for in-plane Ge NWs on nanostructured Si(001)/SiO2 substrates9,10 and self-assembled in-plane Ge NWs on rib-patterned Si substrates.11–13 

Toward a more comprehensive understanding of growth-related phenomena, but also from an application point of view, local information on morphology, strain, and chemical composition become highly important. Recently synchrotron-based scanning x-ray diffraction and fluorescence with foci size on the order of 100 nm or below attracts a growing community to study nondestructively low-dimensional objects with highest spatial resolution, e.g., core/shell14–16 and axial NWs.17 This method has also been applied to probe tilt and bending in single, suspended pure Ge NWs under geometrical constraints18 and to detect doping levels in NWs in the single ppm range.19 

In this Letter, we report on in-plane NWs on pre-patterned Si(001)/SiO2 substrates consisting of ultra-sharp conical Si pillars, which are formed by reactive ion etching (RIE) on masked Si(001) wafers.20 Subsequently, an amorphous SiO2 layer was deposited by chemical vapor deposition to cover the pillar ensemble, thus providing regular seeding points for the NW growth with a pitch separation of 500 nm. In a following step, these seeding points serve as nucleation sites for Au droplets acting as a catalyst during the final vapor–liquid–solid NW growth. We found that a substrate temperature of 550 °C effectively suppresses Au precipitation on SiO2, while it ensures droplet formation exclusively on the circular tips of the Si pillars. Details of the NW growth and their preliminary characterization have been described previously.10 

We have used the hard x-ray nanoprobe setup at beamline I14,21 Diamond Light Source Ltd. (Didcot, UK) to investigate lattice strain, tilt, and bending in Ge NWs via scanning x-ray diffraction microscopy (SXDM). These lattice properties are available through a precise determination of Bragg peak positions in reciprocal space. The radial component depends on the inter-planar spacing d while angular components are sensitive to both the NW bending (either along the growth direction and/or twisting around it) and anisotropic strain. Combining measurements of several Bragg reflections, SXDM gives access to the full strain tensor and elastic constants, which depend on a potential alloy composition.

A rotation stage combined with a high-precision scanning stage, see Fig. 1, situated at the heart of the experimental setup at I14, allows an unrestricted, full ω-rotation of the sample around the vertical axis. A range of scattering angles (2Θ) is available by area detectors (Excalibur and Merlin) on two orthogonal translation stages transversal to the incoming beam. Since no sample rotation other than ω is available, the sample needs to be manually turned to reach other zones of Bragg reflections that involve a large rotation around the incoming beam direction. In order to exactly recover the same location after such intervention, we have used fluorescence markers on the sample that could be locally resolved with the focused x-ray beam using an energy dispersive fluorescence detector sitting upstream, i.e., in front of the sample. The markers are trenches with an approximate size of 2 μm and have been prepared by focused ion beam using a gallium ion source. Hence, we were able to track the Ga-fluorescence as well as the diffraction signal caused by the damaged lattice around the marker.

FIG. 1.

Experimental setup at beamline I14 of the Diamond Light Source. A well-focused x-ray beam of 50 nm diameter hits the sample from its backside and thereby excites the slightly asymmetric Si(331) reflection. The inclination of the (331) lattice planes with respect to the [001]-direction and the respective Bragg angle results in a nearly normal incidence [cf. Fig. 2(d)]. This is rather important in order to minimize absorption losses. The primary beam passes the whole substrate, while the x-ray fluorescence yield has to travel through the substrate again due to the upstream location of the fluorescence detector. The two-dimensional detector probes a section of reciprocal space covering reflections from the Si substrate, the area close the (Si)Ge seeding point (not shown here) and the Ge NWs themselves. The photograph below depicts the central part of the experiment: the ω-circle carrying the sample and the fluorescence detector upstream.

FIG. 1.

Experimental setup at beamline I14 of the Diamond Light Source. A well-focused x-ray beam of 50 nm diameter hits the sample from its backside and thereby excites the slightly asymmetric Si(331) reflection. The inclination of the (331) lattice planes with respect to the [001]-direction and the respective Bragg angle results in a nearly normal incidence [cf. Fig. 2(d)]. This is rather important in order to minimize absorption losses. The primary beam passes the whole substrate, while the x-ray fluorescence yield has to travel through the substrate again due to the upstream location of the fluorescence detector. The two-dimensional detector probes a section of reciprocal space covering reflections from the Si substrate, the area close the (Si)Ge seeding point (not shown here) and the Ge NWs themselves. The photograph below depicts the central part of the experiment: the ω-circle carrying the sample and the fluorescence detector upstream.

Close modal

The given geometry of the experiment and the detector positions imply that either the diffraction or the fluorescence signal has to pass through the whole substrate once. Therefore, we used a silicon substrate with a reduced thickness of 0.2mm. Moreover, it was highly desirable to minimize the geometrical path length by realizing a nearly normal incidence of the beam onto the sample. At an x-ray energy of 18 keV, the Bragg angle ΘB of 15.4° for (331)-type reflections and an inclination of 76.7° between those netplanes and the [001] surface normal add up to a value very close to normal incidence. The optimized transmission geometry is shown in Fig. 1(a), in which the sample side carrying the NWs is facing downstream. Thus, the primary x-ray beam needs to travel through the silicon substrate before it can get diffracted by the NWs. The simultaneously caused Ge-Kα and Kβ fluorescence yield from the NWs has to pass the substrate again to be recorded by the upstream (backscatter) fluorescence detector. Even though the NW diameter is only 50nm and, thus, the fluorescing volume is very small, the Ge-fluorescene was sufficient to identify the positions of individual NWs during SXDM scans.

The SXDM measurements consist of a raster scan of the sample through the beam while continuously taking images with the area detectors with a rate of 10s1. The achievable resolution is equal to the beam size of 50 nm. Each detector image represents a curved 2D cut through 3D reciprocal space. By subsequently rotating the sample about the vertical axis ω, a 3D volume is spanned. This way, a 5D data set is obtained, i.e., intensity I is recorded as a function of sample positions x and y, the angle ω as well as pixel positions (rows and columns) of the detector. It turned out to be crucial to measure the Ge-fluorescence yield from the NWs in order to correct the positional drift that occurs during each raster scan and the positional offset during rotation as the NW cannot be centered on the axis of rotation. The spatial distribution of Ge-fluorescence is automatically correlated with the previous scan, and the observed displacement is corrected for by adapting the scan ranges for the following scan. The data evaluation involves several steps of correction and reduction. The intermediate goal is to obtain spatially resolved information about the center position of the measured Bragg reflections in 3D reciprocal space, which then results in three 2D data sets. One should note that the measurements are operated at the limit for detecting any diffraction signal by NWs. Therefore, a careful analysis of the rich 5D data became essential to extract useful information. In the course of analysis, projections on real and reciprocal space have been screened, and empty regions have been masked out in order to increase the signal to background ratio. In particular, regions of the sample without Ge-fluorescence, and hence presumably no Ge NWs, have been discarded. The intensity of those regions containing NWs has then been projected onto a 3D reciprocal space map (RSM). For each, thus, obtained RSM, the smallest 3D regions containing significant diffraction signal were marked. Three of such regions have been identified (see below). Now, within these regions, we compute the center of mass of intensity for each point on the NW after subtracting the background. Eventually, this yields peak position in reciprocal space as a set of triplets (ω, detector row, detector column) at any probed sample location (x, y). Based on detector calibration, these coordinates are then translated into spherical coordinates in reciprocal space (ω, χ, Q) vs (x, y) with Q=2π/d being the absolute value of the scattering vector Q. For detector calibration, we recorded diffraction rings of Ceria reference powder that was placed at the sample position. In order to probe both lattice parameters across and along the individual NW growth direction, the measurement was repeated (not shown) after flipping the orientation of the two NW subsets by a 90° manual sample rotation χ around its surface normal.

As schematically shown in Fig. 1(a), several isolated diffraction peaks show up on the detector during ω-rotation of the sample. These can be attributed (i) to Ge NWs, (ii) a transition region near the seeding point, and (iii) the Si substrate. The exact origin of each signal is revealed by looking at the real-space distribution of intensity over the sample surface.

In order to visualize a majority of the Ge NWs and nano-clusters (NCs) with SXDM, a sufficiently large rocking scan Δω is required to cover the range of the anticipated mosaicity. For this purpose, a sample area of 5 × 6 μm2 [see Fig. 2(a)] has been probed within a Δω range of 2.1° at a step size of 0.06°. The Ge-fluorescence yield and the integrated diffraction intensity after drift correction are displayed in Figs. 2(b) and 2(c), respectively.

FIG. 2.

(a) Scanning electron micrograph (SEM) of the investigated area containing two subsets of NWs with either horizontal (H) or vertical (V) orientations, lying within the diffraction plane or perpendicular. Occasionally, there are initial Ge NCs where no extended NW growth took place. This distribution can be directly compared to a spatially resolved Ge-Kα x-ray fluorescence map (b) at the very same location. The white bar corresponds to 1 μm. (c) The integrated rocking curve intensities close to the Ge(331) reflection as a composite image. In contrast to an ideal case, real NWs are bent (along the NW axis) and/or twisted (around this axis). In the case of horizontally oriented NWs, the first effect exceeds the angular acceptance window of the rocking curve, and respective NWs are barely noticeable. Complementary to that, the twist component (within the vertically oriented objects) is obviously smaller, and consequently, most of those NWs become visible. (d) A cross section of a typical NW as derived by atomic force microscopy (yellow dots) and a stereographic projection with [110]-pole. In average, the vertical height of the NWs amounts for about 40 nm.

FIG. 2.

(a) Scanning electron micrograph (SEM) of the investigated area containing two subsets of NWs with either horizontal (H) or vertical (V) orientations, lying within the diffraction plane or perpendicular. Occasionally, there are initial Ge NCs where no extended NW growth took place. This distribution can be directly compared to a spatially resolved Ge-Kα x-ray fluorescence map (b) at the very same location. The white bar corresponds to 1 μm. (c) The integrated rocking curve intensities close to the Ge(331) reflection as a composite image. In contrast to an ideal case, real NWs are bent (along the NW axis) and/or twisted (around this axis). In the case of horizontally oriented NWs, the first effect exceeds the angular acceptance window of the rocking curve, and respective NWs are barely noticeable. Complementary to that, the twist component (within the vertically oriented objects) is obviously smaller, and consequently, most of those NWs become visible. (d) A cross section of a typical NW as derived by atomic force microscopy (yellow dots) and a stereographic projection with [110]-pole. In average, the vertical height of the NWs amounts for about 40 nm.

Close modal

By accumulating all the diffraction images within the covered angular range, individual Ge structures on the surface are more clearly identified in both the x-ray fluorescence (XRF) and SXDM maps in Figs. 2(b) and 2(c). Especially, the Ge-Kα fluorescence yield, Fig. 2(b), displays very well the two subsets of NWs, horizontal and vertical ones. However, when more carefully examining XRF and SDXM mappings, a noticeable broadening of the (Si)Ge-related features perpendicular to the diffraction plane (i.e., in vertical direction) becomes visible. An averaging artifact can be excluded since the individual SXDM mappings at particular ω-positions contain such a broadening as well, indicating a more systematic effect. Based on the magnitude of the effect, an instrument-related broadening can be concluded. A convolution with an instrumental resolution function in the vertical direction affects, in particular, the visibility of horizontally extended NWs having small dimension in the vertical direction.

One way to interpret the 5D data set (i.e., diffracted intensity as a function of ω, χ, Q, x, and y) is to extract spatially resolved data about the crystal lattice orientation and strain. To do so, we reduce the 3D RSM, which we have for every point (x, y) to maps of the corresponding Bragg peak positions and their intensity: Qcen(x,y),I(x,y). The radial component of the peak position gives information about the interplanar spacing d=2π/|Q|, and the two angular components describe the tilt of the lattice plane normal. The tilt component perpendicular to the scattering plane Δχ is purely due to variations in orientation whereas the component in the scattering plane Δω contains an additional contribution due to anisotropic strain. Figure 3 shows color-coded maps of ξ(x,y) with ξ=[Δω,Δχ,d], where the local intensity I(x, y) is represented via the saturation of the color.

FIG. 3.

(a) SEM image of the investigated area. The scheme on top shows the potential deformation of the NWs: twisting around the growth axis and a bending perpendicular to it. Based on the Ge x-ray scattering signal [rf. Fig. 4(a)], (b) and (c) depict spatially resolved deviations from an ideal diffraction condition with respect to ω and χ. The horizontal bar in (b) corresponds to 1 μm. (d) The distribution of lattice spacings dGe(331).

FIG. 3.

(a) SEM image of the investigated area. The scheme on top shows the potential deformation of the NWs: twisting around the growth axis and a bending perpendicular to it. Based on the Ge x-ray scattering signal [rf. Fig. 4(a)], (b) and (c) depict spatially resolved deviations from an ideal diffraction condition with respect to ω and χ. The horizontal bar in (b) corresponds to 1 μm. (d) The distribution of lattice spacings dGe(331).

Close modal

For the two principal NW growth directions [see Fig. 3(a)], the two axes, ω and χ have different meaning. Variations about the vertical axis Δω signify a twist for vertically grown NWs, yet an out-of-plane bending for horizontally grown ones. The axis χ is oriented almost perpendicular to the sample surface. Therefore, it corresponds to an in-plane bending of both NW orientations (see sketch in Fig. 3). The fact that Δω variations [Fig. 3(b)] are much more severe within the horizontal NWs indicates that out-of-plane bending plays a bigger role than twist. It also causes difficulties for the SXDM measurement of horizontal NWs: since the Bragg condition is changing along the NW, only segments are diffracting at a time and become subsequently visible at different ω positions. Even after combining all information for the series of scans at different ω, the horizontal NWs are not imaged with the same quality as vertical ones, as seen in Fig. 2(c). The presented values correspond to absolute deviations from the ideal position of the Ge(331) reflection with respect to that of the Si-substrate. Here, Δω=0 corresponds to the exact Bragg condition, whereas Δχ=0 describes a situation in which the diffraction plane contains exactly the Si[001] and [110] directions, i.e., the horizontally oriented NWs are perfectly aligned to the diffraction plane.

Ge NCs, which accidentally did not serve as nucleation spots for NW growth, show the largest deviation in Δω of up to ±0.8°, Fig. 3(b), whereas the ensemble of vertically oriented NWs follows a comparatively narrow Δω distribution with a full width at half maximum (FWHM) of only 0.1° indicating an upper limit for twist. Rarely two NWs starting from adjacent nucleation spots are getting interconnected. Those objects, like the ones enclosed in white rectangular areas in Figs. 3(b) and 3(c), provide an excellent model case to probe the lattice coherence between different NWs. Both objects do not feature any discontinuity across the Δω- and Δχ-distributions, demonstrating the capabilities of a monolithic growth approach in which the template-based lattice coherence among the different seeding windows will be inherited to the individual Ge NWs. The total twist Δω along these NWs is below 0.1°, whereas the in-plane bending Δω of the left NW reaches values of 0.3°, which would correspond to a radius of 140μm.

Eventually, Fig. 3(d) shows the distribution of interplanar d-spacing around an average value of 1.296 Å, which is slightly off the expected value of 1.298 Å in the case of Ge(331) net planes. As shown in the previous work,10 the NWs should, on average, consist of fully relaxed Germanium. The small offset may well be within the precision of the detector calibration, but a slight lattice compression due to Silicon alloying cannot be excluded. Moreover, we probe strain in different directions of the NWs: for horizontal ones, we are mostly sensitive to the growth direction whereas for vertical ones, the lattice parameters perpendicular to the growth direction is probed. It can be seen that vertical NWs tend to have a slightly larger d-spacing indicating lattice relaxation perpendicular to the wire. It is also worth mentioning that the experimentally derived lattice spacing d along the NWs does not fluctuate within the given accuracy of about ±0.05%.

Up to here, we have mainly focused on scattering by the NWs themselves. Looking at another region in reciprocal space, we can inspect the nucleation region as well. Figure 4(a) shows a superposition of detector images recorded during raster scan over the surface, which shows the (331) reflections of relaxed Si and Ge. Interestingly, a third contribution flares up in-between but rather close to the scattering cloud of the substrate. If taking exactly this region of interest of reciprocal space and projecting it onto real space, one can recognize that this additional feature originates from the seeding points. It is important to note that this was not visible in integrating lab RSM measurements but only becomes accessible through the real-space resolution.

FIG. 4.

(a) Integrated detector images (respectively, 2D sections in reciprocal space) taken during a raster scan covering a region close to the Si(331) and Ge(331) bulk positions. Additionally, there is a third feature, labeled seed, which stems from a strained area next to the NW nucleation site. (b)–(d) Locally resolved the variations in the Δω,Δχ and d-spacing probed at the very same location as shown in Fig. 3(a) but reconstructed from a region in reciprocal space related to the seeding window.

FIG. 4.

(a) Integrated detector images (respectively, 2D sections in reciprocal space) taken during a raster scan covering a region close to the Si(331) and Ge(331) bulk positions. Additionally, there is a third feature, labeled seed, which stems from a strained area next to the NW nucleation site. (b)–(d) Locally resolved the variations in the Δω,Δχ and d-spacing probed at the very same location as shown in Fig. 3(a) but reconstructed from a region in reciprocal space related to the seeding window.

Close modal

An analogous analysis as for the relaxed Ge signal yields the respective spatial maps of Δω,Δχ, and the d-spacing, Figs. 4(b)–4(d). The spread of ω and χ is much lower for this seed region. Yet again, the ω-distribution is broader due to the additional component of anisotropic strain (e.g., a changing ratio of εxx/εzz). The mean value d of 1.255 Å compared to the Si-bulk value of 1.246 Å indicates a local compositional intermixing toward relaxed Si0.83Ge0.17. Since the gold droplets are initially saturated with Si, they might provide conditions for a certain compositional transition region during initial growth stages. If elastic lattice strain was the main origin of this deviation, we would expect some anisotropy in the spatial distribution of strain around the seed center since we mostly probe the horizontal component of strain. Nevertheless, it can be seen that both effects contribute by looking at maps of Δχ. Here, a continuous transition of the sign of bending is visible when going vertically through the seed region (top → positive, bottom → negative). This lattice bending with opposite sense on opposing sides of the seed is a signature of elastic strain connected to the gradient in composition. We may notice that the subsets of seeds along various columns in Figs. 4(b)–4(d) depict different extensions, e.g., the third-left column contains more localized spots compared with the adjacent ones. This indicates local fluctuations, which are stable within a column and might be related to the fabrication process.

In the major parts of the extended NWs, we do not observe a gradient in the Ge concentration along the growth axis. Instead, the SXDM data show a rather abrupt change of total strain (composition). Therefore, we conclude that the size of the transition region is at the resolution limit of 50 nm, which fits well to what is expected by equilibrium calculations. In this case, the signal of the transition region is spread out between the Bragg peaks of Ge and Si and, therefore, too weak to be distinguished.

As a numerical approach, finite elements (FE) become the method of choice especially for complex geometries to calculate elastic properties in low-dimensional objects.22,23 Very close to the real structure, we have built a model that considers an 80 × 80 nm2 Si seeding pillar with (001)-orientation embedded in a matrix material and a Ge NW growing on in the 110-direction, Fig. 5(a). The Ge NW forms a hetero-interface exclusively with the top of the Si pillar, while the FEM nodes forming the NW hull are only subject to a geometrical (and not a heteroepitaxial) constraint. Due to a vertical mirror plane at y =0, it would be sufficient to calculate only one half when applying corresponding mirror forces as a boundary condition. However, we have considered the whole structure and decided to plot afterwards the strain distributions exactly at this mirror plane, because such an intersection gives an insight into the situation at the nucleation point within the seed and the initially grown part of the NW. Please note that we neglect effects due to different thermal expansion coefficients (CTEs) for Ge and Si since they differ by approximately a factor of two resulting in an order of magnitude smaller amount of CTE-induced strain compared by strain due the lattice mismatch between both constituents. Figures 5(a)–5(c) depict two normal and one shear component of the total strain tensor εtotal. Please be aware of the difference between elastic and total strain: the first quantity refers to the particular material itself, whereas total strain provides the relative difference to a general reference (in most cases, the substrate). Therefore, strain-related peak shifts in reciprocal space with respect to a reference (a substrate reflection) are directly connected to a non-zero total strain.

FIG. 5.

Normal and shear components of the total strain tensor, εxxtotal (a), εzztotal (b), and εzxtotal (c) considering the bulk Si lattice as a reference as derived by the numerical finite element method. An individual nano-patterned pillar has been modeled by a squared Si seeding window 80 × 80 nm2 (a), which forms a coherent interface to the lateral Ge NW growing along 110. Since there is a mirror plane at y =0, only the back half of the whole structure is shown here and, therefore, depicts the strain state within the seeding window and the NW. The asterisks indicate regions in (a), where the effect of the shear component results in a dilated Ge lattice within the NW next to the seeding point. For a better visibility, the displacements have been enlarged by a factor of 5.

FIG. 5.

Normal and shear components of the total strain tensor, εxxtotal (a), εzztotal (b), and εzxtotal (c) considering the bulk Si lattice as a reference as derived by the numerical finite element method. An individual nano-patterned pillar has been modeled by a squared Si seeding window 80 × 80 nm2 (a), which forms a coherent interface to the lateral Ge NW growing along 110. Since there is a mirror plane at y =0, only the back half of the whole structure is shown here and, therefore, depicts the strain state within the seeding window and the NW. The asterisks indicate regions in (a), where the effect of the shear component results in a dilated Ge lattice within the NW next to the seeding point. For a better visibility, the displacements have been enlarged by a factor of 5.

Close modal

The two normal components εxxtotal and εzztotal in Figs. 5(a) and 5(b) prove a complex strain situation near the seeding point, which results from an energy minimization under the given boundary conditions (geometry, elastic anisotropy, and lattice constants). As a direct consequence of the heteroepitaxial misfit fGe/Si between Ge and Si of about 4.2%, the topmost region within the pillar (up to about 50 nm in depth) undergoes a tensile dilatation, εxxtotal=0.7%,,1.2%; however, directly linked areas within the NW are laterally compressed (εxxtotalfGe/Si). This numerical result fits very well with the observed lattice distortion in Fig. 4(d) of about 0.7%, which comes as an integral information within the x-ray spot size of about 50 nm. An experimental local resolution of sub-features as derived from FEM might require even smaller x-ray foci. Both εxxtotal and εzztotal only establish their equilibrium value of 4.2% far away from the seeding window, see the left-hand tail of the NW in Figs. 5(a) and 5(b). Since the elastic strain field remains locally restricted close to the seeding area and the NW itself, we conclude that there are no strain-mediated competing sub-processes during growth between adjacent seeds, and thus, the underlying growth scenario does not depend in our case on the spatial distribution function, i.e., on the seed pitch.

The particular geometry yields a remarkable relaxation feature within the NW next to the seeding window and directly above the pillar as denoted by asterisks in Fig. 5(a). Here, the Ge lattice becomes tensile strained, i.e., along the growth direction, while along z the Ge lattice replies by compression as shown in (b), thus εzztotal<fGe/Si<εxxtotal. From a naive point of view, this effect might be surprising since the lattice spacing of the material with larger bulk lattice parameter (Ge) becomes locally dilated beyond its bulk value within the plane of the heterointerface. The non-diagonal components of the strain tensor, which describe shear strains, are responsible for this behavior. Figure 5(c) depicts the coupling component εzxtotal with non-zero values only close to the seeding window. This behavior comes as a direct consequence of the particular geometry placing a vertical constraint at the bottom facet of the NW and a local constraint within the seeding window.

In conclusion, we have probed individual laterally oriented Ge nanowires on pre-patterned Si(001) substrates by means of scanning x-ray diffraction microscopy. A spatial resolution of about 50 nm was achieved by a sufficiently small x-ray focus in conjunction with a sophisticated, on-the-fly drift correction using the Ge-fluorescence signal. This approach enabled one to scan strain, tilt, and bending along single objects and to clearly distinguish it from the strain state near the seeding point. Finite element calculations on the total strain provide a detailed complementary view on the complex relaxation behavior.

The authors thank S. Cecchi (Paul-Drude-Institut für Festkörperelektronik, Leibniz-Institut im Forschungsverbund Berlin e.V.) and M. Schmidbauer (Leibniz-Institut für Kristallzüchtung, Berlin) for internal reviews of the manuscript. We are indebted to Oliver Skibitzki (IHP Leibniz-Institut für innovative Mikroelektronik, Frankfurt/Oder) for providing nano-patterned templates. Beamtime access through Project No. MG-22484 at the Hard X-ray Nanoprobe beamline I14, Diamond Light Source Ltd. (UK) is highly appreciated.

The authors have no conflicts to disclose.

M.H. and C.R. contributed equally to this work.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

1.
F. N. M.
Froning
,
L. C.
Camenzind
,
O. A. H.
van der Molen
,
A.
Li
,
E. P. A. M.
Bakkers
,
D. M.
Zumbuehl
, and
F. R.
Braakman
,
Nat. Nanotechnol.
16
,
308
(
2021
).
2.
G.
Scappucci
,
C.
Kloeffel
,
F. A.
Zwanenburg
,
D.
Loss
,
M.
Myronov
,
J.-J.
Zhang
,
S.
De Franceschi
,
G.
Katsaros
, and
M.
Veldhorst
,
Nat. Rev. Mater.
6
,
926
(
2021
).
3.
G.
Gadea
,
M.
Pacios
,
Á.
Morata
, and
A.
Tarancón
,
J. Phys. D: Appl. Phys.
51
,
423001
(
2018
).
4.
C.
Kloeffel
,
M. J.
Rančić
, and
D.
Loss
,
Phys. Rev. B
97
,
235422
(
2018
).
5.
A. M.
Ionescu
and
H.
Riel
,
Nature
479
,
329
(
2011
).
6.
M.
Hanke
,
M.
Schmidbauer
,
D.
Grigoriev
,
H.
Raidt
,
P.
Schäfer
,
R.
Köhler
,
A.-K.
Gerlitzke
, and
H.
Wawra
,
Phys. Rev. B
69
,
075317
(
2004
).
7.
M.
Hanke
,
H.
Raidt
,
R.
Köhler
, and
H.
Wawra
,
Appl. Phys. Lett.
83
,
4927
(
2003
).
8.
M.
Hanke
,
R.
Köhler
,
P.
Schäfer
,
D.
Lübbert
,
I.
Häusler
,
W.
Neumann
,
P.
Modregger
,
T.
Boeck
, and
T.
Baumbach
,
Phys. Rev. B
83
,
125312
(
2011
).
9.
R.
Bansen
,
J.
Schmidtbauer
,
R.
Gurke
,
T.
Teubner
,
R.
Heimburger
, and
T.
Boeck
,
CrystEngComm
15
,
3478
(
2013
).
10.
F.
Lange
,
O.
Ernst
,
T.
Teubner
,
C.
Richter
,
M.
Schmidbauer
,
O.
Skibitzki
,
T.
Schroeder
,
P.
Schmidt
, and
T.
Boeck
,
Nano Future
4
,
035006
(
2020
).
11.
G.
Chen
,
G.
Springholz
,
W.
Jantsch
, and
F.
Schäffler
,
Appl. Phys. Lett.
99
,
043103
(
2011
).
12.
L.
Du
,
D.
Scopece
,
G.
Springholz
,
F.
Schaeffler
, and
G.
Chen
,
Phys. Rev. B
90
,
075308
(
2014
).
13.
J.-H.
Wang
,
T.
Wang
, and
J.-J.
Zhang
,
Nanomaterials
11
,
2988
(
2021
).
14.
T.
Krause
,
M.
Hanke
,
L.
Nicolai
,
Z.
Cheng
,
M.
Niehle
,
A.
Trampert
,
M.
Kahnt
,
G.
Falkenberg
,
C. G.
Schroer
,
J.
Hartmann
,
H.
Zhou
,
H.-H.
Wehmann
, and
A.
Waag
,
Phys. Rev. Appl.
7
,
024033
(
2017
).
15.
A.
Al Hassan
,
R. B.
Lewis
,
H.
Küpers
,
W.-H.
Lin
,
D.
Bahrami
,
T.
Krause
,
D.
Salomon
,
A.
Tahraoui
,
M.
Hanke
,
L.
Geelhaar
, and
U.
Pietsch
,
Phys. Rev. Mater.
2
,
014604
(
2018
).
16.
T.
Krause
,
M.
Hanke
,
Z.
Cheng
,
M.
Niehle
,
A.
Trampert
,
M.
Rosenthal
,
M.
Burghammer
,
J.
Ledig
,
J.
Hartmann
,
H.
Zhou
,
H.-H.
Wehmann
, and
A.
Waag
,
Nanotechnology
27
,
325707
(
2016
).
17.
M.
Keplinger
,
B.
Mandl
,
D.
Kriegner
,
V.
Holý
,
L.
Samuelsson
,
G.
Bauer
,
K.
Deppert
, and
J.
Stangl
,
J. Synchrotron Radiat.
22
,
59
(
2015
).
18.
M.
Keplinger
,
R.
Grifone
,
J.
Greil
,
D.
Kriegner
,
J.
Persson
,
A.
Lugstein
,
T.
Schülli
, and
J.
Stangl
,
Nanotechnology
27
,
055705
(
2016
).
19.
A.
Troian
,
G.
Otnes
,
X.
Zeng
,
L.
Chayanun
,
V.
Dagytė
,
S.
Hammarberg
,
D.
Salomon
,
R.
Timm
,
A.
Mikkelsen
,
M. T.
Borgström
, and
J.
Wallentin
,
Nano Lett.
18
,
6461
(
2018
).
20.
G.
Niu
,
G.
Capellini
,
M. A.
Schubert
,
T.
Niermann
,
P.
Zaumseil
,
J.
Katzer
,
H.-M.
Krause
,
O.
Skibitzki
,
M.
Lehmann
,
Y.-H.
Xie
,
H.
von Känel
, and
T.
Schroeder
,
Sci. Rep.
6
,
22709
(
2016
).
21.
P. D.
Quinn
,
L.
Alianelli
,
M.
Gomez-Gonzalez
,
D.
Mahoney
,
F.
Cacho-Nerin
,
A.
Peach
, and
J. E.
Parker
,
J. Synchrotron Radiat.
28
,
1006
(
2021
).
22.
T.
Krause
,
M.
Hanke
,
O.
Brandt
, and
A.
Trampert
,
Appl. Phys. Lett.
108
,
032103
(
2016
).
23.
O.
Marquardt
,
T.
Krause
,
V.
Kaganer
,
J.
Martín-Sánchez
,
M.
Hanke
, and
O.
Brandt
,
Nanotechnology
28
,
215204
(
2017
).