Part IV Ge microbridges 77
8.5 Strain mapping @ beamline ID13
with respect to the [110] axis for the large bridges A-1 to L-1. Bridge O-1, however, does not fully follow this trend, suggesting that the real O-1 is somewhat different from the other large bridges. The expected smaller Raman shift for the small bridge A+1 is clearly confirmed.
0 10 20 30 40 50
α (°)
−2.5
−2.0
−1.5
−1.0
−0.5
Raman shift (cm−1)
A+1
A-1 G-1 I-1 L-1
O-1
Raman shifts
large bridges small bridge
Figure 8.4: Raman shifts obtained for the different bridges.
8.5 Strain mapping @ beamline ID13
In order to determine the strain at the centre of the bridges directly, an X-ray nanodiffraction experiment has been conducted at beamline ID13 at the ESRF (detailes on the beamline can be found in Ref. [119]). A similar approach as discussed in Sec. 7.2 has been deployed to map the strain of several bridges and determine the strain variation along the bridges with high precision. The X-ray beam at an energy of 15.198 keV was focused with compound refractive lenses to 600 x 200 nm and the angular range around the nominal Ge Bragg peak was scanned while the individual bridges were moved through the beam. Therefore, the sample was mounted on a piezoelectric-stage on the diffractometer and spatially moved for every angular position. Thus, 2D-maps in real space at different incidence angles were recorded. From the reconstructed RSMs at each real space position the COM was used to determine the Bragg peak position and obtain the lattice constant thereof.
For bridge G-1 the total intensity 2D real space maps recorded around the (008) and (228) Ge Bragg peak are shown exemplary in Fig. 8.5. The step size of the piezo-movements was 1µm, revealing the position of the bridge very nicely. The step-size probing the incidence angle was rather coarse, 0.1◦, for these overview maps.
(a) Total intensity around (008) Ge Bragg peak (b) Total intensity around (228) Ge Bragg peak
Figure 8.5: Exemplary total scattering intensity maps of bridge G-1 obtained around the nominal (008) and (228) Ge Bragg reflection. Total intensities below a certain value are set to white color for clarity.
At every real space position the actual (008) and (228) Bragg peak position is determined by calculating the COM (see Eq.( 7.2)). Possible tilt corrections obtained from the symmetric reflection are applied to the asymmetric reflection. Therefore, it is crucial that the probed material in real space of the symmetric and asymmetric reflection correspond as good as possible to each other. Due to the different incidence angle and therefore slightly differently probed volume of the different reflections, which is discussed in detail in Ref. [95], it was necessary to shift the maps about a few real space measurement points for some bridges in order to match best.
Exemplary projections onto the scattering plane for a (008) and (228) Bragg peak are de-picted in Fig. 8.6. The white circle corresponds to the COM, the white cross represents the statistical errorbar from the COM determination. A typical value of σQ is 0.003 Å−1. The reason for the broad peaks is the divergence of the nanofocused beam, which is in the order of several mrad [119].
From the COM in Q-space the in-plane and out of plane lattice constant (c.f. Eq. (2.5) and (2.4)) and hence the in-plane and out of plane strain (c.f. Eq. (3.3)) is determined.
The determined in-plane and out of plane strain maps are shown exemplary in Fig. 8.7. It can be seen that the out-of-plane strain determined from the symmetric reflection corresponds very well to the one determined from the asymmetric reflection (c.f. (a) and (b)).
8.5 Strain mapping @ beamline ID13 105
(a) Typical (008) reflection. (b) Typical (228) reflection
Figure 8.6: Exemplary symmetric (008) and asymmetric (224) Bragg reflections on the same real space position on the bridge. The white circle corresponds to the COM, whereas the length of the cross in Qxand Qz direction corresponds to the statistical error of the COM determination.
(a)ε⊥from (008) (b)ε⊥from (228)
(c)εk from (228)
Figure 8.7: Exemplary strain maps of bridge G-1.
The empty column on the right hand side is due to a necessary shift of one data point (1µm) along Y of the asymmetric map with respect to the symmetric map to match best the probed material, which results in one empty column in the full dataset.
In order to probe the strain variation along the bridge with a smaller real-space step size and finer angular movements, the same procedure as described above, was performed on a line along the bridge with a piezo step-size of 0.5µm and angular steps of 0.01◦. The detailed strain variation along the bridge for bridge G-1 is shown in Fig. 8.8.
−10 −5Distance from Bridge centre (0 µm) 5 10
−0.5
−0.4
−0.3
−0.2
−0.1
ε⟂ (%)
Bridge G-1: strains along bridge (shift of 1*0.5 um incl.)
out-of-plane from (008) out-of-plane from (228)
0.0 0.2 0.4 0.6 0.8 1.0
ε∥ (%)
in-plane from (228)
(a)ε⊥from (008)
Figure 8.8: Exemplary strain variation on the line along bridge G-1.
Here a shift of one data point (0.5µm) led to to the best match of the result of the symmetric and asymmetric reflection. This means that actually the shift in the map above is too large.
But since the step size in the line along the bridge is half the one of the map, either zero or 1 step shift could be considered in the maps.
In the precision of the lattice constant and thereof on the strain, the statistical error in Q-position, as given above, enters. The uncertainty resulting purely from the statistical Q-uncertainty is plotted for a few datapoints only for clarity.
This statistical error accounts for the major part of the strain uncertainty, but in general also geometrical considerations, uncertainties in angular movements, energy precisions, etc., enters. A thorough discussion on the errorbars can be found in Appendix A.4.2. There, a typical Q-uncertainty σQ = σfit of 0.003 Å−1 was used, which overestimates the ε⊥- and underestimates theεk-uncertainty slightly.
Typical error bars, es discussed in A.4.2, are plotted only for a few datapoints in the following strain distributions along the bridges for all different bridges in Fig. 8.9.
8.5 Strain mapping @ beamline ID13 107
−10 −5 0 5 10
Distance from Bridge centre (µm)
−0.5
−0.4
−0.3
−0.2
−0.1
ε⟂ (%)
A-1: strains along bridge (shift of 0*0.5 um incl.)
out-of-plane from (004) out-of-plane from (117) out-of-plane from (228)
−1.0
−0.5 0.0 0.5 1.0
ε∥ (%)
in-plane from (117) in-plane from (228)
(a) A-1
−10 −5 0 5 10
Distance from Bridge centre (µm)
−0.5
−0.4
−0.3
−0.2
−0.1
ε⟂ (%)
Bridge A+1: strains along bridge (shift of 0*0.5 um incl.)
out-of-plane from (008) out-of-plane from (228)
0.0 0.2 0.4 0.6 0.8 1.0
ε∥ (%)
in-plane from (228)
(b) A+1
−10 −5Distance from Bridge centre (0 µm)5 10
−0.5
−0.4
−0.3
−0.2
−0.1
ε⟂ (%)
Bridge G-1: strains along bridge (shift of 1*0.5 um incl.)
out-of-plane from (008) out-of-plane from (228)
0.0 0.2 0.4 0.6 0.8 1.0
ε∥ (%)
in-plane from (228)
(c) G-1
−10 −5Distance from Bridge centre (0 µm)5 10
−0.5
−0.4
−0.3
−0.2
−0.1
ε⟂ (%)
Bridge I-1: strains along bridge (shift of 7*0.5 um incl.)
out-of-plane from (008) out-of-plane from (228)
0.0 0.2 0.4 0.6 0.8 1.0
ε∥ (%)
in-plane from (228)
(d) I-1
−10 −5Distance from Bridge centre (0 µm)5 10
−0.5
−0.4
−0.3
−0.2
−0.1
ε⟂ (%)
Bridge L-1: strains along bridge (shift of -1*0.5 um incl.)
out-of-plane from (008) out-of-plane from (228)
0.0 0.2 0.4 0.6 0.8 1.0
ε∥ (%)
in-plane from (228)
(e) L-1
−10 −5Distance from Bridge centre (0 µm)5 10
−0.5
−0.4
−0.3
−0.2
−0.1
ε⟂ (%)
Bridge O-1: strains along bridge (shift of 3*0.5 um incl.)
out-of-plane from (008) out-of-plane from (228)
0.0 0.2 0.4 0.6 0.8 1.0
ε∥ (%)
in-plane from (228)
(f) O-1
Figure 8.9: Strain variation on the line along the differently oriented bridges.
As can be seen in Fig. 8.9 the out of plane strain determined from the different probed reflections do coincide very well except for bridge A-1. Due to constrictions of the beamline it was obviously a problem to combine low-index (004) symmetric with high index (117) and (228) asymmetric Bragg reflections. For all the other bridges the (008) and (228) Bragg reflections were probed, leading to consistent strain datasets. For bridge A+1 (Fig. 8.9(b))
unfortunately the Bragg reflection moved off the detector left of the constriction for the (008) reflection and right of the constriction for the (228) reflection. Therefore, a full set of datapoints is only available at the centre of the bridge.
For the strain reference needed for the fitting procedure as described above the spatial centre six datapoints per symmetric and asymmetric reflection are averaged per bridge. I.e. for ε⊥ of A-1, all centre 6 datapoints of the symmetric and both asymmetric reflections are averaged, thus 18 datapoints, for εk only the 6 datapoints of the asymmetric reflection, etc. These numbers with the corresponding errorbars, resulting from the averaging and the individual experimental uncertainties are shown in Fig. 8.10.
0 10 20 30 40 50
α (°)
−0.4
−0.2 0.0 0.2 0.4 0.6 0.8 1.0
ε⟂
(% )
A+1A-1 G-1 I-1 L-1 O-1
Strains
large bridges small bridge
−0.4
−0.2 0.0 0.2 0.4 0.6 0.8 1.0
ε∥
(% )
A+1
A-1
G-1 I-1 L-1
O-1
large bridges small bridge
Figure 8.10: Strains obtained for the different bridges.