Elsevier

Ultramicroscopy

Volume 178, July 2017, Pages 62-80
Ultramicroscopy

Measurement of atomic electric fields and charge densities from average momentum transfers using scanning transmission electron microscopy

https://doi.org/10.1016/j.ultramic.2016.05.004Get rights and content

Highlights

  • STEM methods to measure electric fields on nano and subatomic scale are analysed.

  • Limitations of conventional differential phase contrast are worked out.

  • Measuring momentum transfer by from diffraction patterns was related to fields.

  • Electric field can be derived via Ehrenfest theorem or in phase object approximation.

  • Interaction with the specimen is studied via s-state and multislice simulations.

Abstract

This study sheds light on the prerequisites, possibilities, limitations and interpretation of high-resolution differential phase contrast (DPC) imaging in scanning transmission electron microscopy (STEM). We draw particular attention to the well-established DPC technique based on segmented annular detectors and its relation to recent developments based on pixelated detectors. These employ the expectation value of the momentum transfer as a reliable measure of the angular deflection of the STEM beam induced by an electric field in the specimen. The influence of scattering and propagation of electrons within the specimen is initially discussed separately and then treated in terms of a two-state channeling theory.

A detailed simulation study of GaN is presented as a function of specimen thickness and bonding. It is found that bonding effects are rather detectable implicitly, e.g., by characteristics of the momentum flux in areas between the atoms than by directly mapping electric fields and charge densities. For strontium titanate, experimental charge densities are compared with simulations and discussed with respect to experimental artifacts such as scan noise. Finally, we consider practical issues such as figures of merit for spatial and momentum resolution, minimum electron dose, and the mapping of larger-scale, built-in electric fields by virtue of data averaged over a crystal unit cell. We find that the latter is possible for crystals with an inversion center. Concerning the optimal detector design, this study indicates that a sampling of 5mrad per pixel is sufficient in typical applications, corresponding to approximately 10×10 available pixels.

Introduction

The control and measurement of solid state phenomena such as ferroelectricity, magnetism as well as spontaneous or piezoelectric polarisation in crystals is involved in or even paves the way for the engineering of many innovative nanoelectronic devices. Prominent examples are ferroelectric tunnel junctions [1], [2], spin light-emitting diodes [3], [4], [5] or InGaN-based light-emitting diodes [6], [7], [8]. Owing to its high spatial resolution and the strong coupling of electrons to electric and magnetic fields via the Lorentz force, transmission electron microscopy (TEM) is predestined for the characterisation of such fields. The differential phase contrast (DPC) technique [9] has allowed for many impressive measurements in this domain.

As to the investigation of magnetic fields, magnetic domain walls in permalloy have been observed directly [10], [11], the vortex structure and domain wall widths of permalloy particles could be resolved [12] and magnetic vortex cores in permalloy could be shifted and pinned by magnetic fields [13].

Recently, DPC has been employed in studies of electrical properties, such as spontaneous polarisation in GaAs nanowires [14], piezoelectric fields in GaN/InGaN/GaN heterostructures [15], atomic-scale mapping of the electric field in BaTiO3 [16] and dopant-modulated built-in electric fields in pn junctions [17].

As originally proposed for electron microscopy by Rose [9], DPC evaluates the asymmetry of intensity in the central part of diffraction patterns (the Ronchigram) by means of a split four-quadrant detector [10]. Most DPC measurements of both magnetic and electrical properties have relied on the established intuitive interpretation of the DPC signal being caused by a shift of the Ronchigram as a whole, reflecting the classically expected angular deflection of the electron beam in the specimen caused by the Lorentz force.

Some more recent works adopt a quantum mechanical interpretation of diffracted intensities in terms of probability currents [18], [19], ptychographic reconstructions of the object phase [20], [21] or the quantum mechanical expectation value for the electron momentum transfer [22]. The authors of this paper recently related this expectation value to the projection of the electric field using Ehrenfest's theorem [23] in simulation and experiment, which allowed the measurement of atomic electric fields in SrTiO3 [22].

While fields of a large extent of some tens of nanometers could successfully be characterized using segmented detectors and the conventional interpretation assuming a shift of the diffraction pattern as a whole, this approach has been shown to be inadequate for the quantification of atomic electric fields using contemporary aberration-corrected STEM at a resolution below 100pm [22]. In this case the detailed intensity distribution in diffraction patterns has to be considered. On top of that, reliable field strengths are only obtained for very thin specimens. This situation hence demands a comprehensive study investigating the reliability of electric field characterization as a function of the spatial resolution in STEM, the specimen thickness as well as the scale at which electric fields vary, which is the goal of this paper.

In the following, these issues are addressed via detailed simulation studies focused on electric fields in a GaN crystal. In Section 2, we consider the measurement of the angular deflection from electron diffraction patterns in a quantum mechanical approach and point out limitations of DPC when employing segmented detectors. Section 3 treats the relation between angular deflection and electric field distribution in the framework of different models; the equivalence of one model with the phase object approximation is demonstrated. We then investigate the scattering of single atomic columns in Section 4 which has the advantage that electron channeling can be studied apart from Bragg scattering, and interpret results in a simple s-state channeling model. Section 5 deals with the reliability of the electric field determination in GaN crystals as a function of specimen thickness and scattering amplitudes used in the simulation. The mapping of bonding charges is discussed via the comparison of data obtained in isolated atom approximation with density functional theory (DFT) data. Section 6 addresses the influence of inelastic scattering on the diffraction pattern. In Section 7 we suggest a figure of merit for the spatial resolution and discuss requirements towards momentum space sampling and electron dose for given accuracy and precision as a guide for practical setups. Experimental and theoretical charge densities of SrTiO3 are presented in Section 8. We address the mapping of large-scale electric fields from atomic-resolution DPC data in Section 9, followed by a discussion in Section 10.

Section snippets

Diffraction patterns and the expectation value of the electron momentum transfer

In the Schrödinger picture of quantum mechanics, the wave function ψ is the central entity to describe the state of a quantum mechanical system. The wave function has different equivalent representations, for example ψ(r) and ψ(p) describe the same state in real and momentum space, respectively, with position r and momentum p. Both representations are related by Fourier transform, i.e. ψ(p)F[ψ(r)]. Considering image formation in a transmission electron microscope within the framework of

Definitions

If a STEM raster of average momentum transfers is acquired, the maximum information that can be extracted from this two-dimensional field of two-dimensional vectors is another vector field of exactly the same dimensions. The ideal result would be the projected electric fieldEp(r)=1ΔzΔzdzE(r,z),itself, where Δz is the thickness of the specimen along the optical axis. Ep(r) is related to the projected potentialVp(r)=1ΔzΔzdzV(r,z)by means of Maxwell's equations:Ep(r)=gradVp(r).In the

Single atomic columns within the s-state model

To study the average momentum transfer near atomic columns, a single column model system consisting of Ga-atoms stacked in beam direction with the distance characteristic for a Ga-column in a GaN-crystal was investigated. The choice of this model allows for the investigation of the influence of atomic scale electric fields without effects owed to the crystal structure such as Bragg scattering. To gain a deeper understanding of the diffraction pattern dynamics indicated in the previous section,

Reliability of field mapping using the model of an extended, non-propagating probe

In the practice of atomic-scale electric field mapping, Eq. (13) proves beneficial as it directly relates the average momentum transfer p and the projected electric field convolved with the probe intensity, EPC, via a simple proportionality factor. For example, the momentum transfers in Fig. 1(b) simulated for 2.2 nm thick GaN yield the electric field in Fig. 7, where it is superimposed on the mean projected potential VP. While this result is plausible as atomic columns are sources of radial

Thermal diffuse scattering (TDS)

The simulations to this point were conducted with a conventional multislice algorithm using Debye–Waller-dam-ping and therefore broadened atomic potentials to account for thermal movement of the atoms and absorptive potentials for inelastic scattering. While this is accurate for the elastic interaction with the crystal, all inelastically scattered intensity is lost from the simulation and hence has not been considered for the momentum transfer investigations so far. Its influence on the

Investigation of attainable spatial resolution

Until here, all considerations were done for the same probe investigating gallium nitride. To achieve a comprehensive overview of the accuracy of field measurements from average momentum transfer a study of the attainable precision in dependence of the spatial frequency of the electric field was conducted:

To be able to make quantitative statements about achievable spatial resolution, a test-potential as depicted in Fig. 16(a) was used for the accuracy study. This potential oscillates with a

Charge density mapping in strontium titanate from experimental data

As discussed in detail in Ref. [22], the measurement of EPC allows to calculate of the probe-convolved, projected charge density via the flux theorem of Gauß, ρPC(R)=I0(R)1ΔzΔzdzρ(R)=1ε0divEPC,where ε0 represents the vacuum permittivity.

As further demonstrated in Ref. [22], this relation does in fact allow the accurate determination of the projected charge density from simulated average momentum transfers. Prior knowledge of the nuclear charge distribution even gives access to the electron

Mapping large-scale electric fields

Measuring the momentum transfer with the suggested method basically gives access not only to the atomic electric fields but also to influences on a larger scale such as crystal (mis-)orientation and long-ranging electric fields. In this section we investigate whether it is possible to quantify large-scale electric fields as they occur, e.g., within layered structures of polar materials [14] or in case of doping gradients [17].

Quantification of atomic electric fields is limited by thickness due

Discussion and summary

Considering original [9] and recent [16], [59] studies as well as the present one on DPC induced by electric fields, a discussion ought to focus on three main issues. Firstly, there is no doubt that atomic electric fields provide impressive contrast for both conventional segmented ring detectors [10], [60] and a pixel array as simulated here, making DPC especially attractive for the detection of light atoms. However, detecting atoms and understanding the experimental contrast quantitatively are

Conclusion

A comprehensive study on the characterisation of electric fields based on average momentum transfer measurements was presented. The relation of field and momentum transfer was theoretically discussed and analysed in simulations of various model systems as well as realistic crystalline specimens. For thin samples, accurate measurement of atomic electric fields was found to be quantitatively possible. The influence of inelastic scattering processes was simulated and can be neglected for such

Acknowledgements

K.M.-C. acknowledges support from the Deutsche Forschungsgemeinschaft (DFG) under contract MU 3660/1-1. This work was further supported by the DFG under contract RO 2057/4-2 and RO2057/11-1. J.V. and A.B. acknowledge funding from the European Research Council (ERC) under the 7th Framework Program (FP7), and ERC Starting Grant No. 278510-VORTEX. Experimental results are obtained on the Qu-Ant-EM microscope partly funded by the Hercules fund from the Flemish Government. J.V. also acknowledges

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    1

    F. Krause and K. Müller-Caspary contributed equally to this work and thus share first authorship.

    2

    Present address: Department of Materials Science and Engineering, McMaster University, 1280 Main Street West, L8S 4M1 Hamilton, ON, Canada.

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