# LATTICE IMAGING IN TRANSMISSION ELECTRON MICROSCOPY .LATTICE IMAGING IN TRANSMISSION ELECTRON...

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LATTICE IMAGING IN TRANSMISSION ELECTRON MICROSCOPY

Miroslav Karlk

Department of Materials, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical

University in Prague, Trojanova 13, 120 00 Prague 2, Czech Republic,

E-mail: karlik@kmat.fjfi.cvut.cz

Abstract

With the resolution becoming sufficient to reveal individ-ual atoms, high-resolution electron microscopy (HREM)can now compete with X-ray and neutron methods to deter-mine quantitatively atomic structures of materials, with theadvantage of being applicable to non-periodic objects suchas crystal defects. An introduction to the theory and practi-cal aspects of HREM is given. Principles of other latticeimaging techniques in transmission electron microscopy electron holography and Z-contrast imaging are also de-scribed.

Keywords:

High-resolution electron microscopy, electron-specimeninteractions, electron diffraction, phase contrast, contrasttransfer, electron holography, Z-contrast imaging

1. Introduction

The last decades are characterized by an evolution frommacro- to micro- and more recently to nanotechnology. Ex-amples are numerous, such as nanoparticles, nanotubes,quantum transistors, layered superconducting and mag-netic materials, etc. Since many material properties arestrongly connected to the electronic structure, which in turnis considerably dependent on the atomic positions, it is of-ten essential for the materials science to determine atompositions down to a very high precision. Classical X-rayand neutron techniques fail for this task, because of anon-periodic character of nanostructures. Only fast elec-trons are scattered sufficiently strongly with matter to pro-vide local information at the atomic scale.

One of the most commonly used high-resolution tech-niques in transmission electron microscopy (TEM) is thatof (bright-field) phase-contrast imaging. This relies on theinterference between beams scattered by a specimen and isusually performed under parallel beam illumination condi-tions. The main and significant disadvantage of this latticeimaging method is in the difficulty of image interpretationin terms of the atomic structure of the specimen. The inter-pretation is carried out using computer simulations requir-ing input of a specimen structure model, specimenthickness and microscope parameters.

Unfortunately, even the best electron microscopes arehampered by the fact that only the intensity (i.e. the squareof the amplitude), of the electron wave can be recorded onthe photographic film or CCD-camera and an essential partof the electron wave, the phase information, is lost. Holo-graphic recording overcome this problem. The (aberrated)image wave is superimposed with an unscattered plane ref-erence wave resulting in an interference pattern - electronhologram. After acquisition and transfer to a computer sys-tem, amplitude and phase of the electron wave are recon-

structed using the laws of Fourier optics by sophisticatedimage processing. Electron specimen interaction is thensimulated in the same manner as in the case of conventionalphase-contrast imaging.

On the other hand the Z-contrast technique providesdirectly interpretable images - maps of scattering power ofthe specimen. Allowing incoherent imaging of materials, itrepresents a new approach to high-resolution electron mi-croscopy. The Z-contrast image is obtained by scanning anelectron probe of atomic dimensions across the specimenand collecting electrons scattered to high angles. Simulta-neously, spectroscopic techniques can also be used to sup-plement the image, giving information on atomic-resolution chemical analysis and/or local electronic bandstructure. The resolution of the technique is determined bythe size of the electron probe.

This paper gives an introduction to the theory and prac-tical aspects of high-resolution electron microscopy. Prin-ciples of electron holography and Z-contrast imaging arealso described.

2. Electron crystal interactions

In a transmission electron microscope, a sample in the formof a thin foil is irradiated by electrons having energy of theorder of hundreds of keV. In the interior of the crystal theelectrons are either undeviated, scattered, or reflected(Fig. 1).

Electron scattering may be either elastic or inelastic. Inthe case of elastic scattering the electrons interact with theelectrostatic potential of atomic nuclei. This potential devi-ates the trajectory of incident electrons without any appre-ciable energy loss. In fact a small loss occurs since there is achange in momentum. However, because of the disparity inmass of the scattered electron and the atom the loss is toosmall (E/E ~ 10-9 at aperture angles used in TEM) to affectthe coherency of the beam.

Krystalografick spolenost

Materials Structure, vol. 8, number 1 (2001) 3

Fig. 1. Interaction of an electron beam with a thin foil

Krystalografick spolenost

4 M. KARLK

In the inelastic case energy of the incident electronmay be transferred to internal degrees of freedom in theatom or specimen in several ways. This transfer may causeexcitation or ionization of the bound electrons, excitationsof free electrons, lattice vibrations and possibly heating orradiation damage of the specimen. The most common in-teractions are those with the electrons in the crystal. In thiscase the energy loss E is important, because the interact-ing particles have the same mass m. The fraction of energyE is small as compared to the incident energy E. It is thisprimary process of excitation which is used for the ElectronEnergy Loss Spectrometry (EELS). This kind of spectrom-etry as well as different secondary processes of subsequentdesexcitations of the target X-ray emission, Auger elec-trons emission, cathodoluminiscence etc. permit to linkthe structural aspect of the specimen with the informationabout its chemical nature, if the microscope is equippedwith appropriate detectors.

2. 1. Schrdinger equation and the crystalpotential

It seems obvious that the problem of the diffraction of fastelectrons should require the Dirac equation. However, therole of the spin in the interaction is negligible (about onepercent [1,2]) and it is therefore sufficient to use theSchrdinger equation:

2

2mV r r E r ( ) ( ) ( )

(1)

where E is the total energy of the electron and V r( )

its po-tential energy inside the crystal. For fast electrons (E 50keV), it is necessary to take into account the relativistic ef-fects associated with their motion. The solution of theSchrdinger equation in the crystal, where the movingelectron is subjected to the action of a variable potential, isin general very complicated. In case of transmission elec-tron microscopy it is the very high energy of incident elec-trons (~ 105 eV) with respect to the crystal potential (~ 10eV), which enables to simplify the problem. Due to thevery high energy of the incident beam the isotropy of thespace is broken: the majority of electrons propagate in thedirection of the z axis. The space can be divided into the ax-ial direction (z) and two radial directions (x, y). After thisseparation of variables it is possible to modify the equation(1) by introducing:

i) forward (small angle) scattering approximationii) Glauber (projected potential) approximation

The solution of the Schrdinger equation by the multislicemethod is based on these two approximations. Another ap-proach, often used in the TEM for the solution of theSchrdinger equation is the Bloch wave method, intro-duced by Bethe in 1928 [3]. Unfortunately, this method canonly be used for perfect crystals but not for the crystals con-taining grain boundaries, precipitates or other faults. TheBloch waves theory is described in detail in [4].

i) Forward scattering approximation

By introducing the relation between the wave vectork of

the electron beam and corresponding accelerating potentialE k m 2 2 2/ to the equation (1) we obtain:

k r mV r r2 22

( ) (2)

Owing to the high energy of incident electrons it is possibleto consider the action of the crystal potential as a small per-turbation resulting only in a modulation ( )

r of the primary

electron wave. We are seeking a wave function ( )r in the

form: ( ) exp( . ) ( )

r ik r r (3)

Supposing the expression for the modulation ( )r , it is pos-

sible to write:

2

2

2

2

2

2 22 2 2

2

x y zik

xik

yik

z

mV rx y z

( ) .

. ( )

r 0

In this equation the term

2

22

( ) ( ) r

zik

r

zz may be

neglected since the derivative

( )r

zchanges only slowly

with z. In other words it is possible to neglect backscatteredelectrons. The equation (1) thus acquires the form of theforward scattering approximation:

( )( )

r

z

i

k x yik

xik

y

mV r

z

x y 2

2 222

2

2

2 2

( )

r

Two terms in the square brackets may be considered as in-dependent. The first term:

Ax y

ikx

iky

x y

2

2

2

22 2

represents the propagation of the electron wave, while thesecond one represents the action of the crystal potential.Therefore:

( )( ) ( )

r

z

i

kA

mV r r

z

22

2(4)

The physical meaning of these two terms becomes obviousfrom the following: if in the equation (4) we neglect theterm V r( )

and if we put kx = ky = 0 and k = kz (projection

of k to the directions x a y is very small), we obtain:

( ) ( ) ( ) r

z

i

k

r

x

i

k

r

y

2 2

2

2

2

2(5)

Thi

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