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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

\[I_{\mathrm{STEM}}^{\mathrm{ela}}(\mathbf R_0)= \int D(\mathbf Q)\, \left|\psi(\mathbf Q,t;\mathbf R_0)\right|^2\,d\mathbf Q\]

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

\[\exp(-2\pi i\,\mathbf K\cdot\mathbf R_0)\]

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,

\[I_{\mathrm{STEM}}^{\mathrm{ela}}(\mathbf q)= \sum_{\mathbf g,\mathbf h} F_{\mathbf g,\mathbf h}(t)\, \delta(\mathbf q-\mathbf g+\mathbf h)\]

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.

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

\[f'_{\kappa}(\mathbf g;\theta_1,\theta_2)= \int_{\theta_1}^{\theta_2}\sin\theta\,d\theta \int_0^{2\pi} \left|\Delta f_{e,\kappa}(\mathbf g,\theta,\phi)\right|^2\,d\phi\]

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:

\[I_{\mathrm{STEM}}^{\mathrm{TDS}}(\mathbf R_0)= \int_0^t \langle\psi(z;\mathbf R_0)|\widehat W_{\mathrm{det}}|\psi(z;\mathbf R_0)\rangle\,dz\]

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.
  • TDS model: For HAADF \(Z\)-contrast, the TDS term is as important as the elastic term.

See also