STEM Calculation¶
STEM image calculation starts from the same convergent-probe representation as CBED. The difference is the observable: CBED displays the disk intensity in the diffraction plane, whereas STEM scans the probe position and integrates the intensity that enters the selected detector at each position.
Observable¶
Let \(\mathbf R_0\) be the probe position, \(\mathbf Q\) the diffraction-plane coordinate, and \(t\) the specimen thickness. If the detector function \(D(\mathbf Q)\) is 1 inside the detector angular range and 0 outside it, the elastic STEM intensity is
BF, ABF, LAADF, and HAADF correspond to different choices of the inner and outer angles in \(D(\mathbf Q)\). Changing the STEM detector angle therefore changes the physical quantity being integrated; it is not only a display setting.
Fourier-Coefficient Acceleration¶
A direct implementation would solve the dynamical problem again for every scanned probe position \(\mathbf R_0\). The convergent-probe expression has a useful structure: the \(\mathbf R_0\) dependence enters as the phase factor
This allows ReciPro to calculate the two-dimensional Fourier coefficients of the image first, rather than calculating \(I_{\mathrm{STEM}}(\mathbf R_0)\) point by point. Conceptually,
so once the coefficients \(F_{\mathbf g,\mathbf h}(t)\) are known, the full scan image can be reconstructed efficiently by an inverse Fourier transform.
This is the main advantage of Bloch-wave STEM for perfect crystals with small unit cells. It can be much faster than repeating a multislice calculation at every probe position.
Reconstructing a Real Image¶
The image is recovered from the coefficients by
Because \(I(\mathbf r)\) is a real intensity, its coefficients must obey Hermitian symmetry exactly,
and the set of \(\mathbf q\) generated by all beam pairs is closed under \(\mathbf q\rightarrow-\mathbf q\). The sum is therefore real by construction, and any imaginary part that survives is numerical error, not physics.
A small imaginary part does survive in practice, because the amplitude at \(\mathbf k+\mathbf q\) is obtained by bilinear interpolation on the finite grid of incident directions (see Angular sampling of the probe). This makes \(I(-\mathbf q)\) and \(I(\mathbf q)^{*}\) differ by an amount of order \(h^{2}\), where \(h\) is the angular step.
Writing a summed pixel as \(a+ib\), the correct way to collapse it to a real image is to take the real part \(a\). That is the orthogonal projection onto the real axis, and it is identical to symmetrising the coefficients first,
and summing afterwards. Taking the magnitude \(\sqrt{a^{2}+b^{2}}\simeq a+b^{2}/2a\) is not equivalent, and it fails in four separate ways:
- the extra term \(b^{2}/2a\) is strictly positive, so it never cancels — it is a bias, not noise;
- it is largest relative to the signal where \(a\) is small, that is in the dark pixels, so it attacks image contrast rather than the overall level;
- it breaks linearity, so the combined image no longer equals elastic + TDS, because \(\lvert z_1+z_2\rvert\neq\lvert z_1\rvert+\lvert z_2\rvert\);
- it hides negative pixels, which are the visible symptom of an insufficient \(\mathbf q\) set and would otherwise warn the user.
ReciPro therefore reconstructs the elastic, TDS and STEM-EDX images from the real part, and clamps to zero only after the source-size blur, so that a genuinely negative pixel remains detectable up to that point.
Note
Versions up to 4.944 summed the magnitude for the elastic and TDS images. On the default angular grid the difference is far below any perceptible level (see the table below); it becomes measurable only on a deliberately coarse grid, and always as a slight brightening of the dark pixels.
Angular Sampling of the Probe¶
The incident cone is sampled on a square grid of directions with step \(\Delta\alpha\) (Angular resolution in the STEM options), covering the convergence semi-angle \(\alpha\) with a small margin. The number of divisions along one axis is
so the number of directions, and hence the eigenvalue problems to be solved, grows as \(N^{2}\). This grid is unrelated to the number of scan points: it discretises the directions inside the probe, not the positions of the probe.
It is also the only source of the Hermitian residual described above, which makes that residual a convenient convergence indicator. The following values were measured for SrTiO₃ [001] at 200 kV with \(\alpha=25\) mrad, 128 beams and 32×32 scan points. "Residual" is \(\max_{\mathbf q}\lvert I(\mathbf q)-I(-\mathbf q)^{*}\rvert\) relative to \(I(\mathbf 0)\), and the last two columns give the brightening that the magnitude sum would have added at the brightest pixel.
| \(N\) | Directions | Elastic residual | TDS residual | Magnitude bias, elastic | Magnitude bias, TDS |
|---|---|---|---|---|---|
| 16 | 256 | 1.2×10⁻³ | 6.1×10⁻³ | 2.4×10⁻⁵ | 1.1×10⁻⁴ |
| 32 | 1024 | 4.1×10⁻⁴ | 2.6×10⁻³ | 1.1×10⁻⁶ | 1.3×10⁻⁵ |
| 64 | 4096 | 5.6×10⁻⁵ | 7.2×10⁻⁴ | 5.8×10⁻⁸ | 4.3×10⁻⁷ |
| 132 | 17424 | 3.8×10⁻⁵ | 1.1×10⁻⁴ | 4.2×10⁻⁸ | 3.6×10⁻⁸ |
The default angular resolution of 0.4 mrad gives \(N=132\) for \(\alpha=25\) mrad, which is already in the converged region. Two points are worth noting:
- The TDS residual is roughly an order of magnitude larger than the elastic one at every grid, because the TDS coefficients carry the additional thickness integral of the detector-selected absorption.
- The residual is a maximum over all \(\mathbf q\), so it scatters somewhat from grid to grid rather than falling perfectly smoothly; the underlying trend is \(O(h^{2})\).
TDS and Detector-Selected Absorption¶
In HAADF-STEM, the inelastic component from thermal diffuse scattering (TDS) is often the main source of image contrast. ReciPro treats TDS as the amount of intensity removed from the elastic channel into a selected angular range, represented by an absorptive potential.
For a detector angular range \(\theta_1\leq\theta\leq\theta_2\), the detector-selected absorptive scattering factor can be written conceptually as
Choosing this range to match a BF, ADF, or HAADF detector evaluates the TDS contribution that enters that detector.
The STEM TDS intensity is the thickness integral of the detector-selected absorption:
where \(\widehat W_{\mathrm{det}}\) represents detector-selected TDS. Once the Bloch-wave eigenvalues and eigenvectors are known, this \(z\) integral can be handled analytically. A numerical slice integration is also possible, and ReciPro uses the appropriate approach for the calculation mode.
Local and Nonlocal Absorption¶
The absorptive potential can be treated in two main ways.
| Form | Meaning | Feature |
|---|---|---|
| Local approximation | Uses an absorptive potential \(U'(\mathbf r)\) that depends only on position. | Usually effective and fast for broad ADF / HAADF detectors. |
| Nonlocal form | Uses \(U'(\mathbf r,\mathbf r')\) or matrix elements \(U'_{\mathbf g,\mathbf h}\) that depend on pairs of incoming and outgoing waves. | More accurate for narrow detectors, heavy elements, or low accelerating voltages, but much more expensive. |
In the local approximation, matrix elements can be evaluated from reciprocal-vector differences such as \(U'_{\mathbf g-\mathbf h}\). In the nonlocal form, each \((\mathbf g,\mathbf h)\) pair requires its own angular integration, so the cost grows rapidly with the number of beams.
Scope of Bloch-Wave STEM¶
Bloch-wave STEM is fast for highly periodic, perfect crystals and is well suited to systematic comparisons of thickness, defocus, and detector angles. For defects, large supercells, or non-periodic structures, methods such as frozen-phonon multislice may be more appropriate because they do not rely on the same small-periodic-cell assumption.
In ReciPro, STEM is easiest to understand as follows: start with the same convergent wave as CBED, then replace the diffraction-disk observable with detector integration over the diffraction plane.
Practical Parameters¶
- Detector angle: BF / ABF / ADF / HAADF are definitions of \(D(\mathbf Q)\) and \(f'_{\kappa}(\mathbf g;\theta_1,\theta_2)\).
- Beam count: High-frequency image components and channeling are sensitive to the number of beams included.
- Thickness step: If numerical slice integration is used, check the change when the slice thickness is halved.
- Angular resolution: Sets the probe direction grid \(N\) (see Angular sampling of the probe). Cost grows as \(N^{2}\), so it is the main lever on calculation time.
- TDS model: For HAADF \(Z\)-contrast, the TDS term is as important as the elastic term.