Channelling effects in atomic resolution STEM

doi:10.1016/S0304-3991(03)00095-0

Ultramicroscopy

Volume 96, Issues 3–4, September 2003, Pages 299-312

https://doi.org/10.1016/S0304-3991(03)00095-0 Get rights and content

Abstract

A Bloch wave theory for incoherent scattering of an incident plane wave has proved successful in predicting the fine detail in 2-D zone axis channelling patterns formed by ADF, BSE and characteristic X-ray detection in beam rocking mode. A previously published example of polarity determination of GaAs by channelling contrast is compared with simulations in order to illustrate the applicability of the theory. Modification of boundary conditions for a focused coherent probe allows lattice-resolution incoherent contrast based on ADF and EELS detection as well as X-ray emissions to be catered for within a similar theoretical framework. Mixed dynamic form factors constitute an integral part of this theory, where quantum-mechanical phase is a core issue. Simulations of lattice-resolution ADF and EELS are discussed with reference to various zone axis projections of GaAs. Issues of single versus double channelling conditions, and local versus nonlocal interactions, are discussed in relation to X-ray, ADF and EELS detection.

Introduction

Incoherent scattering intensities from crystalline samples may be collected in a scanning transmission electron microscopy (STEM) configuration as channelling patterns [1], [2] (derived from a scan in angle of a collimated electron beam) or as real space images [3], [4] (derived from scanning the position of a focused probe).

Firstly, consider channelling contrast: The geometry is identical to that observed in a large angle convergent beam electron diffraction (LACBED) pattern, which may be simultaneously recorded via the zeroth order beam intensity, usually near a zone axis orientation. The locus of an exact Bragg orientation in the LACBED pattern coincides with the edge of an excess or deficit band in the channelling pattern, i.e. the Brillouin zone boundaries are observed in both coherent and incoherent scattering contrast. Bands in the incoherent channelling pattern have much in common with Kikuchi bands, which are generated from spherical waves emitted from local atomic sources of incoherent scattering. Kikuchi bands are related by reciprocity (exchange of source and detector) to contrast in channelling patterns, where the atom may be regarded as a point detector rather than as an emitter of spherical waves, the angular dependence of the incoherent response being derived from beam rocking. Channelling contrast from an individual atomic species may be isolated by variations in the rate of characteristic X-ray emissions. These patterns are related to projected symmetries of the atomic species in question: Both symmetry and contrast in the overall pattern change with displacement of the atomic detector, providing specific evidence for the precise location within a unit cell. Atom location by channelling enhanced microanalysis (ALCHEMI) may be performed by matching of channelling patterns to obtain a distribution of a minority species over available sublattice sites [5], where statistical techniques may be employed for quantitative measures of goodness-of-fit and standard deviation [6]. However the detection of interstitial sites [7], [8], or the determination of specific sublattice site occupancy by a minority species [9], where the majority species are distributed over various sublattice sites, requires the ability to accurately simulate channelling contrast.

Secondly, consider lattice-resolution contrast: This requires a model for the propagation of a coherent focused probe as it disperses within a crystal (see for instance Dwyer and Etheridge [10]), coupled with its interaction with the incoherent scattering potential invoked by the operative detection system. Lattice resolution is obtained by the variation of this interaction with the real space position of the scanned beam when focused within a unit cell. The inclusion of mixed dynamical form factors (MDFFs) for incoherent scattering [11], [12] provides a secure framework within which a full range of local and nonlocal interaction potentials may be described, and where quantum mechanical phase as well as site-dependent phase between non-aligned momentum transfers is correctly addressed. Since MDFFs form an integral part of an existing Bloch wave theory for channelling contrast, it is shown how the theory may conveniently be expanded to a general theory which encompasses both lattice-resolution and channelling contrast. The key to this lies in a change in boundary conditions such that excitation amplitudes of the underlying basis functions (Bloch states) for propagation of the electron wavefunction within the crystal now depend not only on orientation, but also on probe position and phase distortion.

In this paper the theory for the generation of channelling patterns for plane wave incidence, and its extension to lattice-resolution contrast, is sketched to emphasize the overall philosophy. Detailed derivations may be found elsewhere for channelling contrast expressions [13], [14] or lattice-resolution STEM contrast [15], [16]. Previously reported channelling results from X-ray emissions from 〈310〉 GaAs are shown to illustrate the degree of correlation between experiment and theory for plane wave incidence. Revised boundary conditions are then introduced to cater for propagation of a coherent spherical wave focused within a unit cell. This enables annular dark field (ADF) lattice-resolution contrast with reference to different zone axes of GaAs to be calculated, and lattice-resolution contrast derived from electron energy loss spectroscopy (EELS) of Ga L-shell excitations in the 〈110〉 projection is discussed. These have implications regarding cross-talk, i.e. the relative contribution from neighbouring columns when a fine probe is focused on a specific column of atoms. Issues related to a single channelling approximation compared with double channelling theory are discussed with reference to X-ray emission and geometry associated with EELS and ADF detection. The validity of a local approximation (from which the concept of an object function is derived), compared with the full nonlocal interaction theory, is also discussed.

Section snippets

Channelling contrast

In order to generate channelling patterns by beam rocking methods, a collimated probe with small convergence semi-angle β is focused onto the top surface of a crystal and rocked on an N_x by N_y pixel raster near a zone axis orientation, as illustrated in Fig. 1, such that the crystal orientation is effectively varied with respect to the incident wavevector $k$ . Computer control enables simultaneous recording of intensities from ADF or backscattered electron (BSE) detectors, as well as complete

Lattice-resolution contrast

In order to achieve lattice-resolution contrast, the beam-limiting aperture should be expanded such that neighbouring diffracted beams associated with the fundamental unit cell start to overlap when the beam is brought to focus onto the crystal. Essentially, from the Heisenberg uncertainty principle, a large range in lateral momentum hΔq is necessary for a small uncertainty Δx in the lateral position of the electron wavepacket. With an appropriate coherent source and lens characteristics

Excitation amplitudes

The interaction of a $200 keV$ beam with GaAs at room temperature is now described in terms of the variation in excitation amplitude αⁱ as a function of phase distortion and position $R$ of the focused beam within a fundamental unit cell. Similar conditions are assumed for ADF and EELS calculations in the following sections. Spherical aberration coefficients C_s of $1 mm$ and $0.05 mm$ (representing an aberration-corrected STEM) are used, with Scherzer defocus values of $543 A ̊$ or $129 A ̊$ respectively and

ADF contrast

The calculated z-dependence of ADF contrast through scattering symmetrically into an annulus 60 $–160 mrad$ is shown in Fig. 9 for these three zone axes at the symmetrical orientation. The dynamical part alone is considered for a probe located precisely on a column of Ga atoms. For computational convenience, a multislice formulation (which has been shown to be equivalent to our Bloch wave theory of lattice-resolution contrast [15]), based on an 8×8 supercell in each case, was used to obtain these

EELS contrast

The 〈110〉 projection is used for an EELS contrast simulation of Ga L-shell excitations. The intensity is derived for a probe at Scherzer defocus, $C_{s} =0.05 mm$ , scattered within an energy window of $50 eV$ above a $1.3 keV$ threshold into an on-axis aperture. The collection semi-angle is 30 mrad, and an 8×8 supercell is used as the propagating medium. The projected structure is given in Fig. 5, the scan of the fine probe being along 〈001〉 between Ga columns. The calculated instantaneous response as a

Discussion

It should be emphasized that the theory presented in this paper is based on the assumption of single channelling conditions. Essentially, the validity of the single channelling description lies in the range of momentum transfers $h q$ invoked by the detection system. If channelling contrast associated with the final state electron wavefunction is averaged over a number of Kikuchi bands, this approximation is good. For detection of X-rays, the scattered electron $k ′$ is not registered, and the

Conclusion

A Bloch wave theory for incoherent channelling contrast has been extended to include lattice resolution by revision of boundary conditions for a coherent electron probe focused within a fundamental unit cell. This single channelling theory now caters for lattice-resolution for any form of incoherent scattering via mixed dynamic form factors, based on a fully general nonlocal formulation. These MDFFs are integrated over a range of momenta and energy that depend on the operative detector

Acknowledgements

C.J.R. acknowledges a visit to Fujitsu Laboratories in Atsugi, Japan, under the auspices of their Visiting Research Specialist Programme, which in particular stimulated the question of revised boundary conditions. L. J. A. and M. P. O. acknowledge financial support from the Australian Research Council.

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Channelling effects in atomic resolution STEM

Abstract

Introduction

Section snippets

Channelling contrast

Lattice-resolution contrast

Excitation amplitudes

ADF contrast

EELS contrast

Discussion

Conclusion

Acknowledgements

Ultramicroscopy

Mat. Sci. Eng.

Micron

Ultramicroscopy

Philos. Mag. B

Z. Naturf. A

Philos. Mag. A

Optik

EMAG 83

Inst. Phys. Conf. Ser.

Ultramicroscopy

Philos. Mag.

Philos. Mag. Lett.

J. Electron Microsc.

Am. Mineral.

Acta Crystallogr. A

Optik