Strain measurement in semiconductor FinFET devices using a novel moiré demodulation technique

STEM moiré is a phenomenon caused by interference between a chosen sampling grid and the crystal lattice. The sampling grid is the periodic arrangement of the points sequentially visited by the STEM probe. The illuminated crystal lattice is then sampled at these points. We assume the STEM probe to be infinitely small here, which allows us to express the scanning grid as a 2D Dirac comb.

The continuous image of the sample (independent of specific imaging mode in STEM) can be expressed as an intensity variation of a detector signal (e.g. a high angle annular dark field detector) depending on the probe position $I\left(y,z\right).$

Sampling in STEM is analogous to multiplication of this image function by a 2D Dirac delta comb $S\left(y,z\right)$

$$\begin{eqnarray}&&S\left(y,z\right)=\displaystyle \sum _{m=-\infty }^{m=\infty }\displaystyle \sum _{n=-\infty }^{n=\infty }\delta \left(y-m{p}_{y})\delta (z-n{p}_{z}\right)\end{eqnarray} \tag{ 1 }$$

with ${p}_{y}$ and ${p}_{z}$ the spatial sampling step in y and z directions. We get a resulting discrete image:

$$\begin{eqnarray}&&I\left({y}_{m},{z}_{n}\right)=I\left(m{p}_{y},n{p}_{z}\right).\end{eqnarray} \tag{ 2 }$$

In frequency space, this becomes a convolution of the Fourier transform (FT) of the image $I\left(u,v\right)$ with the FT of the 2D sampling grid which transforms to:

$$\begin{eqnarray}&&S\left(u,v\right)=\displaystyle \sum _{m=-\infty }^{m=\infty }\displaystyle \sum _{n=-\infty }^{n=\infty }\delta \left(u-m{f}_{py})\delta (v-n{f}_{pz}\right),\end{eqnarray} \tag{ 3 }$$

with, ${f}_{py}=\tfrac{1}{{p}_{y}},{f}_{pz}=\tfrac{1}{{p}_{z}}.$

The discrete frequency spectrum of the sampled image is then written as

$$\begin{eqnarray}&&{I}_{c}\left(u,v\right)=I\left(u,v\right)* \displaystyle \sum _{m=-\infty }^{m=\,+\infty }\displaystyle \sum _{n=-\infty }^{n=+\infty }\delta \left(u-m{f}_{py})\delta (v-n{f}_{pz}\right).\end{eqnarray} \tag{ 4 }$$

Note that we have taken the finite size of the sampling grid (the field of view) as a cut-off into the image spectrum rather than in the FT of the grid to keep the delta functions

$$\begin{eqnarray}&&{I}_{c}\left(u,v\right)=\displaystyle \sum _{m=\,-\infty }^{m=\,+\infty }\displaystyle \sum _{n=-\infty }^{n=+\infty }I\left(u-m{f}_{py},v-n{f}_{pz}\right).\end{eqnarray} \tag{ 5 }$$

This results in an infinite series of replicas with the central one occupying the frequency range $\left[\tfrac{-1}{2{p}_{y}},\tfrac{1}{2{p}_{y}}\right]\,\cup \left[\tfrac{-1}{2{p}_{z}},\tfrac{1}{2{p}_{z}}\right].$ If the maximum frequency in the image ${u}_{max}\lt \,1/2{p}_{y}$ and ${v}_{max}\,\lt \,1/2{p}_{z},$ the sampling satisfies the Nyquist criterion and each replica is separated from all other replicas(figure 1(b)). In the case of under-sampling, aliasing happens, leading to the formation of moiré fringes by the replication of frequencies from outside the central portion into this region (figure 1(c)).

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As an example, we can assume an image made of two sinusoids (${g}_{1},$ ${g}_{2}$) as shown in figure 1(a). A sampling frequency ${f}_{py}$ and ${f}_{pz}$ is chosen such that the Nyquist criterion is not satisfied, which results in the moiré effect (figure 1(c)). The yellow window represents the central replica with each red box an identical replication. The index of each window is represented by (m, n) and the frequencies that are outside the yellow window are translated into the yellow window by shifting the window over ($m{f}_{py},$ $n{f}_{pz}$). For example, the frequency g₁ is shifted one window down in v and one window left in u and so on [10]. This leads to a new apparent spatial frequency (the moiré frequency) within the central frequency band:

$$\begin{eqnarray}&&{g}_{y}^{{\prime} }={g}_{y}-m{f}_{py},\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{g}_{z}^{{\prime} }={g}_{z}-n{f}_{pz},\end{eqnarray} \tag{ 7 }$$

where ${g}_{y}$ and ${g}_{z}$ are the frequency components in y and z directions. For a projection image of Si or Ge along the [110] zone axis, we consider the y axis as the $\left[1\bar{1}0\right]$ direction and the z axis along the [001] direction. The lattice distances for Si are different in these two directions ${a}_{1\bar{1}0}=0.384\,{\rm{nm}}$ and ${a}_{001}\,=0.543\,{\rm{nm}}$ (figure 2(a)). This means, for any fundamental frequency in y and z directions, ${g}_{y}$ and ${g}_{z},$ we have ${g}_{y}\,\gt {g}_{z}\in (u,v).$ In conventional STEM, a square scanning grid is used and ${f}_{py}={f}_{pz}.$ In general, this results in the creation of 1D moiré as m and n are not necessarily the same.

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Figure 2(d) shows an experimental image obtained with a square scanning grid, resulting in only one fringe direction to be visible while the other is too high in frequency. These 1D moiré fringes are easily observed for a probe aberration corrected instrument (the probe has to be smaller than the lattice spacing) and readily reveal strain as a slight change of the moiré fringes in different locations of the image.

The resulting distortion of the apparent symmetry by the formation of 1D moiré can be circumvented and a more interpretable situation occurs if the symmetry of the scanning grid matches the symmetry of the lattice by adjusting the scan pattern to obey:

$$\begin{eqnarray}&&\displaystyle \frac{{g}_{y}}{{f}_{py}}=\displaystyle \frac{{g}_{z}}{{f}_{pz}}=U,\end{eqnarray} \tag{ 8 }$$

with $U$ the under-sampling factor >0.5. This can be done by adjusting the strength of the signals driving the scan coils. In this case, all frequencies transform as:

$$\begin{eqnarray}&&{g}_{y}^{{\prime} }={g}_{y}-m{f}_{pz}\displaystyle \frac{{g}_{y}}{{g}_{z}}=\,{g}_{y}\left(1-m\displaystyle \frac{{f}_{pz}}{{g}_{z}}\right)={g}_{y}\left(1-\displaystyle \frac{m}{U}\right),\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&{g}_{z}^{{\prime} }={g}_{z}-n{f}_{py}\displaystyle \frac{{g}_{z}}{{g}_{y}}=\,{g}_{z}\left(1-n\displaystyle \frac{{f}_{py}}{{g}_{y}}\right)={g}_{z}\left(1-\displaystyle \frac{n}{U}\right).\end{eqnarray} \tag{ 10 }$$

If m = n, this transform is an angle preserving scaling factor and the moiré fringes will have the same symmetry as the lattice. An experimental example is shown in figure 3(a).

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When strain occurs in the lattice, frequencies will shift by ${\rm{\Delta }}g$ and a relative shift of the diffracted spots can be obtained in the diffraction pattern (as in NBD). The strain in a particular direction is expressed as:

$$\begin{eqnarray}&&\varepsilon =\displaystyle \frac{\left|{\rm{\Delta }}\mathop{g}\limits^{ \rightharpoonup }\right|}{\left|\mathop{g}\limits^{ \rightharpoonup }\right|},\end{eqnarray} \tag{ 11 }$$

when under-sampling occurs, an apparent strain will appear in the moiré fringe as

$$\begin{eqnarray}&&\varepsilon ^{\prime} =\displaystyle \frac{| {\rm{\Delta }}\mathop{g}\limits^{ \rightharpoonup }| }{\left|\mathop{g^{\prime} }\limits^{ \rightharpoonup }\right|}=\displaystyle \frac{| {\rm{\Delta }}\mathop{g}\limits^{ \rightharpoonup }| }{\sqrt{{\left({g}_{x}-m{f}_{px}\right)}^{2}+{\left({g}_{y}-n{f}_{py}\right)}^{2}}}=\alpha \varepsilon .\end{eqnarray} \tag{ 12 }$$

The strain gets magnified by a boost factor $\alpha$ which makes it much more apparent as can be seen both in figures 3(a) and 4. This boosting is shape preserving (ε' stays parallel to $\varepsilon$) only in the case where the scan grid matches the symmetry of the crystal and a situation is chosen where m = n. The strain boosting factor is plotted as a function of the under-sampling factor U showing how strain boosting is especially strong for low frequency moiré fringes. This also implies a hard trade-off between strain sensitivity and spatial resolution. In order to determine the frequency of the moiré fringe (related to the strain), one needs at least an area of the order of a period of the fringe.

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Another limitation comes from the fact that strain should not shift frequencies outside the central frequency replica. Otherwise m and n may differ for different areas of the sample leading to a much more complicated analysis (in fact a multivalued problem with non-unique solution). This limits the tolerable strain variation in a single image as strain amplification goes up. The change in the aspect ratio of the figures 3(a) and (b) is reflective of the scanning grid matching the symmetry of the lattice by adjusting the scan pattern.

Strain analysis of moiré STEM images is typically performed using GPA [11]. In this method a mask is used in the frequency domain to isolate the side bands around selected frequencies that carry the strain information. This selection mask determines also the spatial resolution that can be obtained and determines the signal to noise ratio in the resulting strain map. The larger the radius of the mask, the higher is the spatial resolution and the more noise enters the estimated strain map [11]. The mask radius is chosen to prevent selecting more than one diffraction spot at a time and is typically chosen smaller than half of the selected reference frequency. This reference lattice vector is used to demodulate the measured signal which then reveals the strain with respect to this reference.

Applying this method to moiré STEM images reveals the apparent strain which needs to be corrected back to the actual strain via equation (12) as shown in figure 5. As the strain magnification is significantly higher for low frequency moiré fringes, it is tempting to choose the undersampling as demonstrated in figure 5(a) where strain is readily apparent from the moiré image due to the boosting effect. The low frequency moiré fringes will however demand a small selection mask and result in poor spatial resolution. On top of this, multiple spatial frequencies, possibly having different m and n values, tend to cluster near the centre and can be very close to each other. This results in unwanted overlaps and the strain maps become unreliable, especially as each of these frequencies should be converted from apparent strain to actual strain with different scaling factors (figure 5).

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In order to overcome this difficulty and increase the practical usefulness of moiré strain mapping, we developed an alternative method that allows significantly suppressing other unwanted replica's.

If we could suppress the unwanted frequencies in the undersampled diffractogram, we could use a much larger mask size before demodulating the signal back to an image of the strain. This would result in higher spatial resolution and the suppression of artefacts. This is achieved by moiré phase stepping interferometry and Quadrature demodulation. The principle of moiré phase stepping interferometry is to phase shift the moiré fringes in different specific directions in repeated experiments.

If we shift the experimental scanning grid over ${\rm{\Delta }}y$ and ${\rm{\Delta }}z$ in y and z directions

$$\begin{eqnarray}&&S\left(y,z\right)\to S\left(y-{\rm{\Delta }}y,z-{\rm{\Delta }}z\right)\end{eqnarray} \tag{ 13 }$$

and apply the Fourier shift theorem we get for the FT of the sampling grid

$$\begin{eqnarray}&&S\left(u,v\right)\to S\left(u,v\right){e}^{i2\pi \left(u{\rm{\Delta }}y+v{\rm{\Delta }}z\right)}.\end{eqnarray} \tag{ 14 }$$

This results in phase shifting of the different components in the spectrum as

$$\begin{eqnarray}&&I\left(u,v\right)=\displaystyle \sum _{m=-\infty }^{m=\infty }\displaystyle \sum _{n=-\infty }^{n=\infty }I\left(u-m{f}_{my},v-n{f}_{mz}\right){e}^{i2\pi \left(u{\rm{\Delta }}y+v{\rm{\Delta }}z\right)}.\end{eqnarray} \tag{ 15 }$$

This equation shows that it is sufficient to shift the sampling grid by a reference lattice fringe spacing ${\rm{\Delta }}y=\tfrac{1}{{g}_{y}}$ and ${\rm{\Delta }}z\,=\tfrac{1}{{g}_{z}}$ to cause a $2\pi$ phase shift of these moiré fringes in the under-sampled image.

Now, that we have the capability to phase shift the frequency components in the recorded moiré images by spatially shifting the sampling grid, we can apply an extended version of quadrature demodulation. Quadrature demodulation is commonly used in telecommunication (e.g. FM radio) [12] and allows to extract a given sideband of a modulated signal. In our case the 'carrier frequency' is a reference spatial frequency and the 'signal' is the strain or deviation with respect to this signal.

In this case we apply a 4 × 4 array of shifts of the scanning pattern, resulting in 16 independent and phase shifted moiré images. The applied shift can be represented as a 4 × 4 matrix S targeted to bring out the strain along 2 non-collinear reference frequencies ${\mathop{g}\limits^{ \rightharpoonup }}_{1}$ and ${\mathop{g}\limits^{ \rightharpoonup }}_{2}$

$$\begin{eqnarray}\begin{array}{l}\mathop{S}\limits^{ \rightharpoonup }=\displaystyle \frac{1}{8}\left[\begin{array}{cccc}\left(0,0\right) & \left(\displaystyle \frac{1}{{g}_{y1}},\displaystyle \frac{1}{{g}_{z1}}\right) & \left(\displaystyle \frac{2}{{g}_{y1}},\displaystyle \frac{2}{{g}_{z1}}\right) & \left(\displaystyle \frac{3}{{g}_{y1}},\displaystyle \frac{3}{{g}_{z1}}\right)\\ \left(\displaystyle \frac{1}{{g}_{y2}},\displaystyle \frac{1}{{g}_{z2}}\right) & \left(\displaystyle \frac{1}{{g}_{y1}}+\displaystyle \frac{1}{{g}_{y2}},\displaystyle \frac{1}{{g}_{z1}}+\displaystyle \frac{1}{{g}_{z2}}\right) & \left(\displaystyle \frac{2}{{g}_{y1}}+\displaystyle \frac{1}{{g}_{y2}},\displaystyle \frac{2}{{g}_{z1}}+\displaystyle \frac{1}{{g}_{z2}}\right) & \left(\displaystyle \frac{3}{{g}_{y1}}+\displaystyle \frac{1}{{g}_{y2}},\displaystyle \frac{3}{{g}_{z1}}+\displaystyle \frac{1}{{g}_{z2}}\right)\\ \left(\displaystyle \frac{2}{{g}_{y2}},\displaystyle \frac{2}{{g}_{z2}}\right) & \left(\displaystyle \frac{1}{{g}_{y1}}+\displaystyle \frac{2}{{g}_{y2}},\displaystyle \frac{1}{{g}_{z1}}+\displaystyle \frac{2}{{g}_{z2}}\right) & \left(\displaystyle \frac{2}{{g}_{y1}}+\displaystyle \frac{2}{{g}_{y2}},\displaystyle \frac{2}{{g}_{z1}}+\displaystyle \frac{2}{{g}_{z2}}\right) & \left(\displaystyle \frac{3}{{g}_{y1}}+\displaystyle \frac{2}{{g}_{y2}},\displaystyle \frac{3}{{g}_{z1}}+\displaystyle \frac{2}{{g}_{z2}}\right)\\ \left(\displaystyle \frac{3}{{g}_{y2}},\displaystyle \frac{3}{{g}_{z2}}\right) & \left(\displaystyle \frac{1}{{g}_{y1}}+\displaystyle \frac{3}{{g}_{y2}},\displaystyle \frac{1}{{g}_{z1}}+\displaystyle \frac{3}{{g}_{z2}}\right) & \left(\displaystyle \frac{2}{{g}_{y1}}+\displaystyle \frac{3}{{g}_{y2}},\displaystyle \frac{2}{{g}_{z1}}+\displaystyle \frac{3}{{g}_{z2}}\right) & \left(\displaystyle \frac{3}{{g}_{y1}}+\displaystyle \frac{3}{{g}_{y2}},\displaystyle \frac{3}{{g}_{z1}}+\displaystyle \frac{3}{{g}_{z2}}\right)\end{array}\right]\,\end{array}\end{eqnarray} \tag{ 16 }$$

each index of the matrix can be written as:

$$\begin{eqnarray}&&{\mathop{S}\limits^{ \rightharpoonup }}_{k,l}=\displaystyle \frac{k}{8{\mathop{g}\limits^{ \rightharpoonup }}_{1}}+\displaystyle \frac{l}{8{\mathop{g}\limits^{ \rightharpoonup }}_{2}}.\,\end{eqnarray} \tag{ 17 }$$

As a spatial shift of 1/g implies a 2π phase shift, the shifts result in a complex factor that is applied to a specific frequency component in the frequency domain:

$$\begin{eqnarray}&&{f}_{kl}(\mathop{g}\limits^{ \rightharpoonup })=\exp (i2\pi \mathop{g}\limits^{ \rightharpoonup }\cdot {\mathop{S}\limits^{ \rightharpoonup }}_{kl}).\end{eqnarray} \tag{ 18 }$$

This means that we can single out any specific spatial frequency of interest by multiplying each sub-diffractogram $I{\left(u,v\right)}_{kl}$ with the complex conjugate of the above matrix and then summing up over all k, l indices. This results in constructive interference of a selected frequency g while destructive interference occurs approximately for all other frequencies

$$\begin{eqnarray}&&{I}_{demod}{(u,v)}_{{g}_{y},{g}_{z}}=\sum _{kl}I{(u,v)}_{kl}{f}_{kl}^{* }({g}_{y},{g}_{z}))\,* \,\delta (u-{g}_{y},v-{g}_{z})\end{eqnarray} \tag{ 19 }$$

with the delta function doing the actual demodulation. Converting this back to real space leads to the demodulated strain component with respect to the selected reference frequency

$$\begin{eqnarray}&&{I}_{demod}{\left(y,z\right)}_{{g}_{y},{g}_{z}}={ {\mathcal F} }^{-1}\left({I}_{demod}{\left(u,v\right)}_{{g}_{y},{g}_{z}}\right).\end{eqnarray} \tag{ 20 }$$

The selectivity improves with a $n\times n$ shifting matrix with increase in $n,$ but this comes at the expense of an increase in the measurement time as ${n}^{2}.$ A minimal choice would be n = 2, but we show in the experimental section that n = 4 provides a more realistic compromise. In fact, the suppression rate for any other frequency $\mathop{g}\limits^{ \rightharpoonup }$ in the discrete Fourier space of the image can be calculated as

$$\begin{eqnarray}&&SR\left(\mathop{g}\limits^{ \rightharpoonup },{\mathop{g}\limits^{ \rightharpoonup }}_{desired}\right)=\displaystyle \frac{1}{16}\displaystyle \displaystyle \sum _{k,l}{f}_{k,l}\left(\mathop{g}\limits^{ \rightharpoonup }\right){{f}^{* }}_{k,l}\left({\mathop{g}\limits^{ \rightharpoonup }}_{desired}\right)\lt 1.\end{eqnarray} \tag{ 21 }$$

Ideally this suppression preserves frequencies close to the desired frequency as these carry the strain information but achieves a high suppression of frequencies that are further out.

In order to appreciate the performance of the present method, we compared it to the diffraction based strain measurement techniques like NBD and Bessel diffraction [3, 4, 6]. The analysis with these methods is applied to a 16 nm FinFet, which is a challenging structure in terms of the spatial resolution requirement but a realistic example of the current semi-conductor technology.

NBD was performed using a quasi-parallel probe with a beam convergence angle of ≈0.3 mrad at 300 kV using an FEI TITAN³ microscope. Bessel diffraction was performed at 300 kV using an aberration corrected FEI TITAN microscope and using an annular aperture in the C2 aperture plane. The convergence angle was chosen to be ≈6 mrad and this gives a diffraction pattern consisting of rings instead of discs (figure 10). Such lower convergence angle is chosen so as to limit the overlapping area between the rings. This improves the precision of the strain measurement but is traded off with the spatial resolution [6].

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The NBD and Bessel diffraction data are collected using a Gatan US1000XP CCD a camera. The NBD data is processed with the Epsilon 1.2.1.40 software from ThermoFischer scientific. The Bessel data is treated with a dedicated python based software, which is based on the autocorrelation of each diffraction pattern and subsequent extraction of the centre of the rings by background fitting and normalisation [6].

The strain maps were also obtained with the discussed phase stepping method and quadrature demodulation, offering an estimated spatial resolution of 2.8 nm (figure 11). Line profiles across the strain maps are averaged over 16 nm width perpendicular to the scan direction. Some scanning artefacts in the slow scan direction are still visible in e.g. the ${{\epsilon }}_{zz}$ strain line profile as stripe artefacts in the horizontal direction. These kinds of artefacts are typical for the STEM based strain methods and can be due to sample drift [15] and electromagnetic noise which results in instabilities of the probe position with respect to time. The precision of the proposed technique in the fast scan direction is estimated as 1.1 × 10⁻³ and the precision in the slow scan direction is 3.5 × 10⁻³. The precision calculated here is for the spatial resolution of 2.8 nm. Improved probe stability in the instrument and faster response of the scan coils (reducing the recording time) could further improve this performance in the future. Rotating the scan direction by 90° provides a direct way to trade precision performance from the y to z axis at the expense of an extra image acquisition.

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The typical trade-off between spatial resolution and precision in GPA is also present here as demonstrated in figure 12 for different low pass filtering choices. The advantage is that this choice can be made during post processing [16, 17]. The typical mask size for low pass filtering option is much higher in the case of the proposed moiré quadrature demodulation as opposed to applying GPA directly on moiré images. The reliability of strain maps has significantly increased alongside the spatial resolution. The precision of strain measurement observed for different mask sizes in figure 12(a) is $1.1\times {10}^{-3},$ $1.7\times {10}^{-3},$ $2.3\times {10}^{-3}$ for mask sizes of $1/5,$ $1/3$ and $1/2$ (which leads to approximately ≈2.8, 1.7 and 1.2 nm spatial resolution) times the sampling frequency. This once again demonstrates the inverse nature of precision versus spatial resolution for GPA [17, 18].

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Figure 12 provides a comparison of the strain maps obtained using moire quadrature demodulation and diffraction techniques like Bessel diffraction and nano-beam diffraction on a 16 nm FinFET. Analysing the line profiles of ${\varepsilon }_{xx}$ reveals that the Ge and Si_0.3Ge_0.7 lattice distances are not the same as is expected for a fully strained Ge layer. This means there is relaxation of the Ge layer in the x direction. However, Ge conforms with the lattice distances of Si_0.3Ge_0.7 in the y direction as seen in the ${\varepsilon }_{yy}$ line profile. Since the relaxed Ge lattice is larger than that of Si_0.3Ge_0.7, it means that the Ge is compressively strained in the y direction. This results in an elongation in the z direction due to the Poisson effect [19] and can be seen in the ${\varepsilon }_{zz}$ line profile. The strain values computed on the Ge channel region with respect to bulk Ge (table 1) for the 16 nm FinFET reveals a uniaxial strain (y-axis) along the channel of the transistor. The strain is almost completely relaxed across (x-axis) the channel and there is a small tensile strain along the growth direction (z-axis). This trend is also confirmed after comparison with the diffraction based strain measurement techniques: NBD and Bessel diffraction [3, 6].

Table 1 shows a comparison of the strain values in the channel, refered to bulk Ge for moiré quadrature demodulation with NBD and Bessel diffraction.

The strain values are averaged over the complete Ge region and the standard deviation represented here is over the full region and does not represent the deviation for each individual pixel in the image. $\mathrm{Standard}\,\mathrm{deviation}\,\mathrm{over}\,\mathrm{an}\,\mathrm{area}\,=\tfrac{{\rm{Std}}.\,{\rm{deviation}}}{\sqrt{area}}.$ This allows to make a fair comparison between methods which have a different spatial resolution and number of measurement points in the strain maps. The epitaxial conditions of growth of Ge on the Si_0.3Ge_0.7 SRB ideally leads to a compressive strain of −1.2% in the ${\varepsilon }_{yy}$ direction and a tensile strain of 0.34% in ${\varepsilon }_{zz}$ direction due to the Poisson effect, in accordance with the elastic theory for materials. Assuming that condition is fulfilled, we can estimate the accuracy for the Bessel diffraction, nanobeam diffraction and moiré quadrature demodulation for these measurements to be better than 0.2%, 0.1% and 0.2%. A better estimate of the accuracy needs further work on samples with well known strain levels.

Table 2 shows the comparison of the moiré quadrature demodulation technique with both diffraction-based techniques. The acquisition time and the analysis time per pixel is significantly lower for the moiré method as opposed to diffraction techniques. This is a direct consequence of the low frame rate that is present in any CCD camera as compared to the very fast readout possible with the HAADF detector used in the moiré method. Even the latest camera technology is still an order of magnitude slower and will probably remain so for the foreseeable future.

Table 2.  Performance comparison of the three techniques evaluated in the strain free reference area: Bessel, NBD and moiré quadrature demodulation.

Method	Spatial resolution	Precision over an area	Precision at nominal resolution	Field of view	Acquisition time (per pixel)	Analysis time (per pixel)	Total dose [${{\rm{e}}}^{-}/{\rm{n}}{{\rm{m}}}^{2}$]
Bessel diffraction	1–3 nm	${\varepsilon }_{xx}$ = 4.2 × 10−5 per 20 × 50 nm2 ${\varepsilon }_{yy}$ = 4.4 × 10−5 per 50 × 50 nm2 ${\varepsilon }_{zz}$ = 9.2 × 10−5 per 50 × 50 nm2	$4\times {10}^{-4}$	>500 × 500 nm2	≈150 ms	≈124 ms	4.6 × 106
Nano-beam diffraction	1–3 nm	${\varepsilon }_{xx}$ = 6.3 × 10−5 per 20 × 50 nm2 ${\varepsilon }_{yy}$ = 7.1 × 10−5 per 50 × 50 nm2 ${\varepsilon }_{zz}$ = 7.2 × 10−5 per 50 × 50 nm2	$7\times {10}^{-4}$	>500 × 500 nm2	≈25 ms	≈156 ms	0.7 × 106
Moiré quadrature demodulation	≥0.7 nm	${\varepsilon }_{xx}$ = 3.1 × 10–4 per 20 × 50 nm2 ${\varepsilon }_{yy}$ = 2.6 × 10−4 per 50 × 50 nm2 ${\varepsilon }_{zz}$ = 2.5 × 10−4 per 50 × 50 nm2	$1\times {10}^{-3}$	<500 × 500 nm2 (instrumental limitations apply due to scan coil response)	≤0.1 ms	<4 μs or real time possible	0.1 × 106

The diffraction techniques involve 4D-STEM mechanism where each individual diffraction pattern is stored per pixel and hence it is also expensive in terms of data size in comparison to the moiré technique. Finally, The precision of the moiré quadrature demodulation technique is $1\times {10}^{-3}$ computed for 3 nm spatial resolution and is in close agreement with the high resolution STEM and standard STEM moiré techniques [8, 20, 21] but with substantial increase in spatial resolution compared to standard moiré and a higher field of view compared to high resolution STEM. The boosting factor that moiré applies on the strain has an added advantage of effectively detecting small amounts of strain.

Due to longer acquisition and readout times, the total electron dose (number of electrons per area $[{{\rm{e}}}^{-}/{\rm{n}}{{\rm{m}}}^{2}]$) impinged on the sample is significantly higher for the diffraction techniques. Shifting to a direct electron detector would significantly improve the signal to noise ratio and would allow for reducing the acquisition time with several orders of magnitude [22–24]. Nevertheless, it is expected that strain precision will scale with the square of the electron dose and we make a brief estimate for the setups presented here by comparing the obtained product of strain precision and the square root of the dose. $({P}_{d}=precision\times \sqrt{Dose}\,[\sqrt{{{\rm{e}}}^{-}}/{\rm{n}}{\rm{m}}]).$ The ${P}_{d}$ for NBD = $1.8\left[\sqrt{{{\rm{e}}}^{-}}/{\rm{n}}{\rm{m}}\right],$ for Bessel = $2.5\left[\sqrt{{{\rm{e}}}^{-}}/{\rm{n}}{\rm{m}}\right]\,$and for moiré quadrature demodulation technique = $0.1\left[\sqrt{{{\rm{e}}}^{-}}/{\rm{n}}{\rm{m}}\right].$ This shows that moiré quadrature demodulation technique is also highly dose efficient in comparison to the presented diffraction based strain measurement techniques although it is expected that upgrading to a more modern detector will significantly improve this performance metric for the diffraction methods.

In summary, the moiré quadrature demodulation technique offers some of the important practical advantages over diffraction techniques such as fast data acquisition, no need for specialised high speed diffraction camera's, very fast data analysis and flexible choice of spatial resolution.