Atomic Resolution Transmission Electron Microscopy

  • Angus I. KirklandEmail author
  • Shery L.-Y. Chang
  • John L. Hutchison
Part of the Springer Handbooks book series (SHB)


This chapter provides an overview of the essential theory and instrumentation relevant to high-resolution imaging in the transmission electron microscope together with selected application examples. It begins with a brief historical overview of the field. Subsequently, the theory of image formation and resolution limits are discussed. We then discuss the effects of the objective lens through the wave aberration function and coherence of the electron source. In the third section, the key instrument components important for HRTEM imaging are discussed; namely, the objective lens, electron sources and monochromators, energy filters and detectors. The theory and experimental implementation of exit wavefunction reconstruction from HRTEM images is detailed in the fourth section, including examples taken from studies of complex oxides. The final section treats the simulation of HRTEM images with particular reference to the widely adopted multislice method.

1.1 Introduction and Historical Context

High-resolution transmission electron microscopy ( ) uses a self-supporting thin sample (typically tens of nanometers in thickness) illuminated by a highly collimated electron beam at energies of between \({\mathrm{30}}\,{\mathrm{keV}}\) and \({\mathrm{1}}\,{\mathrm{MeV}}\). A series of magnetic electron lenses image the electron wavefield at the exit face of the sample onto a detector at high magnification. HRTEM has evolved from initial instrumentation constructed by Knoll and Ruska [1.1, 1.2, 1.3] to its current state where individual atom columns in a wide range of materials and orientations can be routinely imaged using sophisticated computer-controlled microscopes (Fig. 1.1). For this reason HRTEM is integral to characterization of materials and modern instrumentation now occupies a central place in many laboratories worldwide and has made a substantial contribution to key areas of materials science, physics and chemistry, and instrument development for HRTEM also supports a substantial commercial industry of manufacturers [1.4, 1.5, 1.6, 1.7]. HRTEM has also made substantial contributions to structural biology for which the 2017 Nobel prize in Chemistry was awarded (see [1.10, 1.11, 1.12, 1.13, 1.8, 1.9] for reviews of selected representative examples from this field; see also Chap.  4 in this volume). However, because of limitations of space we will not consider this aspect further herein.

Fig. 1.1

A modern \(30{-}300\,{\mathrm{kV}}\) HRTEM fitted with a cold field-emission gun, probe- and image-forming aberration correctors, and a range of digital detectors for HRTEM and (scanning transmission electron microscopy) and configured for full remote operation

Numerous HRTEM studies of bulk semiconductors [1.14, 1.15, 1.16, 1.17, 1.18], defects (Fig. 1.2) [1.19, 1.20] and interface structures (Fig. 1.3) [1.21, 1.22] in these materials, of metals and alloys [1.23, 1.24, 1.25, 1.26], and of ceramics, particularly oxides [1.27, 1.28, 1.29, 1.30], have been reported in a vast literature spanning several decades (for additional general reviews see [1.31, 1.32, 1.33, 1.34, 1.35, 1.36, 1.37, 1.38]). An excellent collection of representative HRTEM images can be found in [1.31, 1.39, 1.40]. More recently HRTEM has become an essential tool in the characterization and discovery of nanoscale (Fig. 1.4) [1.41, 1.42] and most recently low dimensional materials, particularly graphene [1.43, 1.44, 1.45] (Fig. 1.5a-f) largely facilitated by the availability of low-voltage high-resolution instruments.

Fig. 1.2

HRTEM image of a [110] oriented CVD-deposited diamond film showing twins, stacking faults, and nanograins. Note the local disorder at the intersection of the stacking faults and twins

Fig. 1.3

(a) HRTEM image of a lattice-matched heterojunction between InP and (Ga,In)As. At the defocus used and for this particular foil thickness differences in scattering between these two isostructural materials allows them to be distinguished. (b) HRTEM image of a heterojunction between InAs and InAsSb that have a significant lattice mismatch. In this case the lattice misfit is accommodated as a regular array of Lomer dislocations marked

Fig. 1.4

HRTEM image of a nanocrystalline gold particle supported on a {111} cerium oxide surface. The gold particle shows an almost perfect cubeoctahedral morphology and both particle and substrate are in an epitaxial [110] orientation despite the large lattice mismatch

Fig. 1.5a-f

Real-time dislocation dynamics. False-color HRTEM images showing changes in the position of an edge glide dislocation in graphene with time under continuous electron-beam irradiation. (a) Time \(={\mathrm{0}}\,{\mathrm{s}}\). (b) Time \(={\mathrm{141}}\,{\mathrm{s}}\). (c) Time \(={\mathrm{321}}\,{\mathrm{s}}\). (df) Atomic models illustrating the structures inferred from (ac), respectively. The white T indicates the position of the dislocation. Adapted from [1.44]. Reprinted with permission from AAAS

Finally, we note that HRTEM and in particular the development of in situ capabilities (see later) has made a substantial contribution to the study of heterogeneous catalysts [1.46, 1.47].

Of crucial importance to its success in all of these areas is the ability of HRTEM to provide real-space images of the atomic configuration at localized structural irregularities and defects in materials, that are inaccessible to broad-beam bulk diffraction methods and which largely control their properties.

Advances in instrumentation for HRTEM over the same timescale have enabled this information to be recorded with increasing resolution and precision (see Chap.  12 in this volume) leading to improvements in the quantification of the data obtained.

This chapter concentrates on HRTEM at atomic resolution. Following a brief historical overview of the development of HRTEM (for more detailed articles outlining some of the key events in the broad history of electron microscopy, see [1.48, 1.49, 1.5, 1.7]) we begin by outlining some of the theory pertinent to image formation at high resolution and the effects on recorded images of the aberrations introduced by imperfect objective optics. We also provide various definitions of resolution.

The second section surveys the key instrumental components affecting HRTEM and provides an outline of currently available solutions. The final section describes computational approaches to both the recovery of the specimen exit-plane wavefunction (coherent detection) from a series of images and methods available for HRTEM image simulation.

1.1.1 A Brief Historical Summary

Ernst Ruska and Max Knoll constructed the first prototype transmission electron microscope, capable of ca. \({\mathrm{400}}\times\) magnification and demonstrating the principles of electron microscopy  [1.1]. Subsequently, the resolving power of the electron microscope rapidly matched and then exceeded that of the optical microscope in 1934 [1.50]. The first commercial electron microscope was produced in 1939 by Siemens based on Ruska's design [1.51]. However, further improvements in resolution were somewhat slower due to the need to identify and overcome various instrumental limitations.

Alongside these early instrumental developments, a number of groups applied HRTEM to studies of a variety of materials problems. Marton showed early images of biological samples [1.52] and the first subnanometer lattice-fringe images from phthalocyanine crystals were obtained in the 1950s [1.53]. This was later extended to lattice images of metal foils showing crossed fringe patterns [1.54].

Concurrently, the first published HRTEM images of complex transition-metal oxides provided preliminary evidence that many of these (specifically those of Mo, W, Ti, and V) were not perfect structures ([1.55]; see also [1.30, 1.31] for reviews). These observations of planar faults in oxides possibly represent the first occasion in which useful information about defects at the atomic level was provided by HRTEM. This work created much interest among solid-state chemists, who for the first time saw a new scientific tool that would enable them to overcome the barriers to structural determinations of these materials imposed by their large unit cells and often extensive disorder. It also immediately provided an explanation for nonstoichiometry in these materials and entirely changed the way in which thermodynamic properties of oxides were modeled.

The work briefly summarized above was possible with the typical instrumental resolutions available in most laboratories at that time. However, it was not until this improved that it became possible to resolve individual cation columns in oxides and certain other classes of material. In the 1970s, the first images showing the component \(\mathrm{MO_{6}}\) octahedra were published with a resolution of \({\mathrm{0.3}}\,{\mathrm{nm}}\) for a series of mixed Ti-Nb structures that demonstrated a direct correspondence between the lattice image and the projected crystal structure [1.56].

The typical spatial resolution (slightly better than ca. \({\mathrm{0.5}}\,{\mathrm{nm}}\)) provided by most commercial instruments in the 1960s and 1970s was largely limited by mechanical and electrical instabilities. Subsequent improvements in instrument design and construction led to a generation of microscopes becoming available in the mid 1970s with point resolutions of less than \({\mathrm{0.3}}\,{\mathrm{nm}}\) operating at intermediate voltages around \({\mathrm{200}}\,{\mathrm{kV}}\) [1.57]. Toward the end of this period the dedicated \({\mathrm{600}}\,{\mathrm{kV}}\) Cambridge HREM [1.58] and several other high-voltage instruments also became operational [1.59, 1.60], providing resolutions better than \({\mathrm{0.2}}\,{\mathrm{nm}}\).

This generation of instruments was capable of resolving the separation of individual atomic columns in fcc metals in several projections which notably led to extensive studies of the structures of nanoscale metal particles (see [1.61] for a review). They were also used to directly identify the cation arrangements in several families of complex mixed-metal oxides [1.62].

The following two decades saw further significant improvements in microscope design with commercially available high-resolution instruments being produced by several manufacturers [1.63, 1.64]. One outcome of these developments was the installation of high-voltage HRTEMs (operating at ca. \({\mathrm{1}}\,{\mathrm{MV}}\)) in several laboratories worldwide [1.65, 1.66]. These machines were capable of point resolutions of ca. \({\mathrm{0.12}}\,{\mathrm{nm}}\), significantly higher than that available in intermediate-voltage instruments. Concurrently, commercial instrument development also started to concentrate on improved intermediate-voltage instrumentation (at up to \({\mathrm{400}}\,{\mathrm{kV}}\)) [1.67] with interpretable resolutions between \(\mathrm{0.2}\) and \({\mathrm{0.15}}\,{\mathrm{nm}}\). This improvement in resolution enabled detailed studies of defects and surfaces in semiconductors [1.17, 1.20] for which a resolution \(<{\mathrm{0.14}}\,{\mathrm{nm}}\) was required to resolve individual atomic columns in \(\langle 110\rangle\) projections.

In the 1990s, further progress was made in improving resolution through a combination of instrumental and theoretical developments. For the former, the successful design and construction of improved high-voltage instrumentation [1.68, 1.69] demonstrated interpretable resolutions close to the long sought after goal of \({\mathrm{0.1}}\,{\mathrm{nm}}\) and successfully demonstrated atomically resolved images of grain boundaries in metals [1.37]. Perhaps more significantly, field-emission sources became widely available on intermediate-voltage microscopes [1.70, 1.71] improving the absolute information limits of these machines to values close to the point resolutions achievable at high voltage.

This new generation of instruments with more coherent sources led to renewed theoretical and computational efforts aimed at reconstructing the complex specimen exit wavefunction using either electron holograms [1.72, 1.73, 1.74, 1.75] or extended focal or tilt series of HRTEM images (see a subsequent section). These latter indirect approaches extended the interpretable resolution beyond conventional axial image limits and provided both the phase and modulus of the specimen exit wavefunction, free from effects due to the objective lens rather than the aberrated intensity available in a conventional HRTEM image.

More recently, in one of the most significant instrumental developments in the history of electron microscopy the successful completion and testing of the necessary electron optical components for direct correction of the spherical aberration present in all electromagnetic round lenses (see [1.49] for a review) has been accomplished. From initial corrector designs for the TEM, based on two coupled hexapole elements a number of more complex designs have emerged which are able to correct both axial and off-axial aberrations and most recently first-order chromatic aberrations. (Although not discussed here a parallel development enabled aberration correction in STEM based on corrector designs using series of quadrupole and octapole elements. A more detailed review of the history of aberration correction for both geometries is provided by Hawkes and Krivanek in Chap.  13 in this volume.) These corrected instruments are now widely installed and are capable of directly interpretable resolutions of \({\mathrm{50}}\,{\mathrm{pm}}\) at \({\mathrm{300}}\,{\mathrm{kV}}\) [1.76] and close to \({\mathrm{100}}\,{\mathrm{pm}}\) at \({\mathrm{30}}\,{\mathrm{kV}}\) [1.77, 1.78], the latter being of particular use in studies of radiation-sensitive organic and inorganic materials including graphene.

1.2 Essential Theory

In this section, we outline some of the essential theory required for understanding HRTEM image contrast. Many of the topics described here are treated in more detail elsewhere [1.79, 1.80, 1.81] and only selected frameworks directly relevant to HRTEM imaging are discussed further as examples.

We begin with a general review of the essentials of the HRTEM image formation process and subsequently treat the key areas of resolution, and the effects of the objective lens and source in more detail.

1.2.1 Image Formation

As shown schematically in Fig. 1.6 (from a simplified ray optical perspective and from a wave optical perspective as described subsequently) the formation of an HRTEM image involves three steps:
  1. 1.

    Electron scattering in the specimen

  2. 2.

    Formation of a diffraction pattern in the back focal plane of the objective lens

  3. 3.

    Formation of an image in the image plane. (It can easily be shown that the specimen, back focal, and image planes are mathematically related by Fourier transform operations.)

Hence, to understand the relationship between contrast recorded in an HRTEM image and the atomic arrangement within the specimen it is essential to develop theoretical frameworks describing each of these steps.
Fig. 1.6

(a) Schematic optical ray diagram showing the principles of the imaging process in HRTEM and indicating the reciprocal relationships between specimen, diffraction and image planes. (b) Schematic diagram illustrating the wave optical relationship between the recorded image intensity in HRTEM and the specimen exit-wave through the objective lens aberration function

To describe the general case (of arbitrary specimens) each of the above steps requires a complex mathematical and computational treatment describing dynamical theory that is outside the scope of the section (comprehensive accounts can be found elsewhere [1.31, 1.38, 1.79, 1.80, 1.81, 1.82]). We therefore restrict ourselves to treatment of the simplest cases for illustrative purposes and subsequently follow the nomenclature and sign conventions given in [1.79].

In the simplest case of a thin specimen, neglecting absorption, the effect of the specimen on an incident electron wave is to alter only its phase leaving the amplitude unchanged. Under this phase object approximation ( ), which ignores Fresnel diffraction within the specimen but includes the effects of multiple scattering, the specimen exit-wave complex amplitude can be written as
where \({\upsigma}\) is an interaction constant given by
$$\sigma=2\uppi me\frac{\lambda_{\mathrm{r}}}{h^{2}}\;,$$
in which both \(m\) and \({\lambda}_{\mathrm{r}}\) are relativistically corrected values of the electron mass and wavelength and
is the two-dimensional () projection of the specimen potential along the beam direction. The interaction constant decreases with accelerating voltage (with values of \({\mathrm{0.00729}}\,{\mathrm{V^{-1}{\,}nm^{-1}}}\) at \({\mathrm{200}}\,{\mathrm{kV}}\) decreasing to \({\mathrm{0.00539}}\,{\mathrm{V^{-1}{\,}nm^{-1}}}\) at \({\mathrm{1000}}\,{\mathrm{kV}}\)), whereas the specimen inner potential generally increases with atomic number, although this also depends on the density [1.39] (Table 1.1).
Table 1.1

Mean inner potential of representative materials in volts


\(Z\) (atomic number)

Mean inner potential




















Equation (1.1) shows that within this model the effect of the specimen is to advance the phase of the electron wave by \(\sigma\phi_{\mathrm{p}}(x,y)\) over the wave in vacuum. For extremely thin specimens composed of low atomic number elements the values of the mean inner potential are such that this phase advance is small and hence the exponential term in (1.1) can be expanded and approximated as
$$\psi_{\mathrm{e}}(x,y)\approx 1-\mathrm{i}\sigma\phi_{\mathrm{p}}(x,y)$$
in the weak phase object approximation ( ), which assumes kinematic scattering within the specimen requiring that the intensity of the central unscatttered beam is significantly stronger than that of the diffracted beams. Strictly speaking it is the variation in the phase change produced by different parts of the specimen that is important, which supports this approximation.

It is important to note that both of the above formulations are projection approximations such that atoms within the specimen can be moved along the incident beam direction without affecting the exit wavefunction.

The complex amplitude of the scattered wave in the back focal plane of the objective lens is given by the Fourier transform of (1.3).

With \(\phi_{\mathrm{p}}(x,y)\) real this gives
Equation (1.4) is subsequently modified by the presence of a limiting objective aperture and by phase shifts introduced by the objective lens.
The former can be included through the simple function
$$\begin{aligned}\displaystyle P(\boldsymbol{k}_{x},\boldsymbol{k}_{y})=1\quad|\boldsymbol{k}|\leq r\;,\\ \displaystyle P(\boldsymbol{k}_{x},\boldsymbol{k}_{y})=0\quad|\boldsymbol{k}|\geq r\;.\end{aligned}$$
The phase shifts introduced by the objective lens are parameterized by the coefficients of a wave aberration function, \(W(u,v)\), which is described in detail in a subsequent section.
Thus, including aperture and lens effects the complex amplitude, under the WPOA, is given by
$$\begin{aligned}\displaystyle\psi^{\prime}_{\mathrm{d}}(\boldsymbol{k}_{x},\boldsymbol{k}_{y})&\displaystyle=\delta(\boldsymbol{k}_{x},\boldsymbol{k}_{y})-\mathrm{i}\,\sigma\mathfrak{F}\{\phi_{\mathrm{p}}(x,y)\}\\ \displaystyle&\displaystyle\quad\,\times P(\boldsymbol{k}_{x},\boldsymbol{k}_{y})\exp[\mathrm{i}\,W(\boldsymbol{k}_{x},\boldsymbol{k}_{y})]\;.\end{aligned}$$
A further Fourier transform of (1.6) finally gives the image amplitude (in the image plane) as
$$\begin{aligned}\displaystyle\psi_{\mathrm{i}}(x,y)&\displaystyle=1-\mathrm{i}\,\sigma[\phi_{\mathrm{p}}(-x,y)]\\ \displaystyle&\displaystyle\quad\,\times\mathfrak{F}^{-1}\{P(\boldsymbol{k}_{x},\boldsymbol{k}_{y})\exp[\mathrm{i}\,W(\boldsymbol{k}_{x},\boldsymbol{k}_{y})]\}\;.\end{aligned}$$
Since the cosine terms in the expansion of (1.7) cancel, the recorded image intensity, to first order, is
$$\begin{aligned}\displaystyle I(x,y)&\displaystyle=\psi_{\mathrm{i}}(x,y)\psi_{\mathrm{i}}^{\ast}(x,y)\\ \displaystyle&\displaystyle\approx 1+2\sigma\phi_{\mathrm{p}}(-x,-y)^{\ast}\\ \displaystyle&\displaystyle\quad\,\times\mathfrak{F}^{-1}\{\sin[W(\boldsymbol{k}_{x},\boldsymbol{k}_{y})]P(\boldsymbol{k}_{x},\boldsymbol{k}_{y})\}\;.\end{aligned}$$
The above expression shows that for this simplest theory the image contrast is proportional to the projection of the specimen potential convolved with an impulse response function arising from the instrument.

Detailed treatment of the latter also requires the inclusion of the effects due to the partial coherence of the electron source, which acts to damp higher spatial frequencies, and of the detector, which also modifies the recorded contrast through its modulation transfer function. The effects of both of these are described in subsequent sections.

A useful modification to the above treatment makes the potential \(\phi_{\mathrm{p}}(x,y)\) complex. This complex projected specimen potential [1.82, 1.83] provides a description of the attenuation of the image wavefield through either scattering outside a limiting aperture or more usefully for unfiltered HRTEM images by the depletion of the elastic wavefield by inelastic processes [1.84].

A number of further extensions to this basic treatment have previously been proposed to overcome the limitations of a projection approximation in the thick phase grating approximation [1.85]. We do not give a detailed derivation here but note that this approximation successfully accounts for multiple scattering and a degree of curvature of the Ewald sphere (Fresnel diffraction within the specimen) and is thus more generally applicable to HRTEM imaging under less restrictive conditions than the WPOA.

The projected charge density ( ) approximation is an alternative extension that provides a tractable analytic expression for the image intensity including the effects of multiple scattering (unlike the weak phase object) but retaining the restriction of a projection approximation.

Starting from the expression for the specimen exit-wave complex amplitude given by the POA in the absence of an objective aperture and with no wave aberration function we can write
The amplitude in the back focal plane of the objective lens is given by
$$\begin{aligned}\displaystyle\psi_{\mathrm{d}}(\boldsymbol{k}_{x},\boldsymbol{k}_{y})&\displaystyle=\mathfrak{F}\{\exp[-\mathrm{i}\,\sigma\phi_{\mathrm{p}}(x,y)]\}\\ \displaystyle&\displaystyle\quad\,\times\exp\left[\mathrm{i}\,\uppi\Updelta C_{1}\lambda\left(u^{2}+v^{2}\right)\right]\\ \displaystyle&\displaystyle\approx\phi_{\mathrm{p}}(\boldsymbol{k}_{x},\boldsymbol{k}_{y})\left[1+\mathrm{i}\,\uppi\Updelta C_{1}\lambda|\boldsymbol{k}|^{2}\right],\end{aligned}$$
if only a small defocus, \(\Updelta C_{1}\) is allowed and where \(\phi_{\mathrm{p}}(\boldsymbol{k}_{x},\boldsymbol{k}_{y})\) represents the Fourier transform of \(\exp[\mathrm{i}\,\sigma\phi_{\mathrm{p}}(x,y)]\). More generally the PCD approximation can also be used if all aberration coefficients are small which is particularly relevant to HRTEM images formed using corrected instruments.
Thus, the image amplitude (in the image plane) is given by
$$\begin{aligned}\displaystyle\psi_{\mathrm{i}}(x,y)&\displaystyle=\exp[-\mathrm{i}\,\sigma\phi_{\mathrm{p}}(x,y)]\\ \displaystyle&\displaystyle\quad\,+\mathrm{i}\,\uppi\Updelta C_{1}\lambda\mathfrak{F}^{-1}\left\{\left(\boldsymbol{k}_{x}^{2}+\boldsymbol{k}_{y}^{2}\right)\phi(\boldsymbol{k}_{x},\boldsymbol{k}_{y})\right\}.\end{aligned}$$
A standard theorem from Fourier analysis is now used [1.86], which states that if \(f(x,y)\) and \(\phi(u,v)\) are a Fourier transform pair then
Applying this the image amplitude is given by
$$\begin{aligned}\displaystyle\boldsymbol{\psi}_{\mathrm{i}}(x,y)&\displaystyle=\exp\left[-\mathrm{i}\,\sigma\phi_{\mathrm{p}}(x,y)\right]\\ \displaystyle&\displaystyle\quad\,-\frac{\mathrm{i}\Updelta\mathrm{C}_{1}\lambda}{4\uppi\nabla^{2}}\left\{\exp\left[-\mathrm{i}\,\sigma\phi_{\mathrm{p}}(x,y)\right]\right\}\\ \displaystyle&\displaystyle=\exp\left[-\mathrm{i}\,\sigma\phi_{\mathrm{p}}(x,y)\right]\\ \displaystyle&\displaystyle\quad\,+\left(\frac{\mathrm{i}\Updelta\mathrm{C}_{1}\lambda\sigma}{4\uppi}\right)\exp\left[-\mathrm{i}\,\sigma\phi_{\mathrm{p}}(x,y)\right]\\ \displaystyle&\displaystyle\qquad\,\times\left[\sigma\nabla\phi_{\mathrm{p}}(x,y)+\mathrm{i}\nabla^{2}\phi_{\mathrm{p}}(x,y)\right],\end{aligned}$$
which yields an image intensity to first order as
$$I(x,y)\approx 1-\left(\frac{\Updelta\mathrm{C}_{1}\lambda\sigma}{2\uppi}\right)\nabla^{2}\phi_{\mathrm{p}}(x,y)\;.$$
From Poisson's equation
we finally obtain
$$I(x,y)=1+\left(\frac{\Updelta C_{1}\lambda\sigma}{2\uppi\epsilon_{0}\epsilon}\right)\rho_{\mathrm{p}}(x,y)\;,$$
in which \({\rho}_{\mathrm{p}}(x,y)\) is the projected total charge density including the nuclear contribution.

Historically, the restriction of limited defocus and spherical aberration (or other uncorrected aberrations) meant that the application of the PCD approximation was restricted to relatively low-resolution imaging. However, this approximation would now seem to be ideal for the interpretation of aberration-corrected images in which these restrictions can be experimentally achieved at high resolution [1.87]. In this regard, we note that this theory has also been modified [1.88, 1.89] to include effects due to higher order lens aberrations and the presence of a limiting objective aperture.

Further extensions to the models outlined above require solution of the dynamic electron diffraction problem using one of several possible computational algorithms, a description of which is outside the scope of this section [1.100, 1.101, 1.102, 1.38, 1.79, 1.82, 1.90, 1.91, 1.92, 1.93, 1.94, 1.95, 1.96, 1.97, 1.98, 1.99]. However, for generalized HRTEM image simulation the multislice algorithm  [1.103, 1.104, 1.105, 1.85, 1.90] has been most commonly employed for the simulation of HRTEM images, to a large extent due to its computational efficiency compared to alternative methods [1.106] such as Bloch wave calculations , and a summary of this particular algorithm is therefore provided in a subsequent section.

1.2.2 Resolution Limits

Unlike their optical equivalents there is no simple measure of resolution for the electron microscope, as the resolution depends on both the instrument and also on the scattering properties of the sample used. (For a more detailed treatment of resolution see Chap.  12 in this volume.)

The ultimate resolution of any optical system is the diffraction limit imposed by the wavelength of the radiation, \(\lambda\), and the aperture angle of the objective lens, \(\alpha\), and the refractive index, \(n\), which can be formalized through Abbe's equation as
where the value of the constant \(k\) lies between \(\mathrm{0.6}\) and \(\mathrm{0.8}\) depending on the coherence of the illumination. However, because of imperfections in the objective lens and limited coherence (as discussed subsequently) experimental resolution limits are far lower than that set by (1.16) and hence a single useful figure definition of resolution for HRTEM is not possible.

Two independent definitions of attainable resolution are commonly used, defined, respectively, by the key optical properties of the objective lens and those of the source.

The first of these is the directly interpretable or Scherzer limit  [1.107] or point resolution, which defines the maximum width of a pass band transferring all spatial frequencies from zero, without phase reversal, and is determined by the coefficients of the wave aberration function of the objective lens.

Ignoring the phase shifts due to higher order aberrations the phase contrast transfer function ( ) [1.79] due to defocus and spherical aberration is given by
$$\sin W(\boldsymbol{k})=\sin\left\{\uppi C_{1}\lambda|\boldsymbol{k}|^{2}+\frac{\uppi}{2}C_{3}\lambda^{3}|\boldsymbol{k}|^{4}\right\}.$$
For HRTEM this defines a focus setting [1.107] that offsets the phase shift due to spherical aberration , \(\mathrm{C}_{3}\) through a suitable choice of defocus
which leads to a broad band of phase contrast transfer without zero crossings (Fig. 1.7a,b) up to a frequency of \(\boldsymbol{k}_{\text{max}}={\mathrm{1.6}}(C_{3}\lambda^{3})^{-1/4}\). In this definition of the Scherzer focus, the passband in the contrast transfer function (CTF ) contains a local minimum \(={\mathrm{0.7}}\), whereas the original definition, \(C_{1,\text{Scherzer}}=(C_{3}\lambda)^{1/2}\), avoids a local minimum in the passband at the cost of a slightly poorer point resolution \(d_{1}={\mathrm{0.707}}(C_{3}\lambda^{3})^{1/4}\).
Fig. 1.7a,b

Phase contrast transfer functions (PCTFs) plotted in two dimensions calculated for a modern \({\mathrm{300}}\,{\mathrm{kV}}\) FEGTEM illustrating the interpretable resolution (\(d_{1}\)) and information limits (\(d_{2}\)). (a) Calculated at the Scherzer defocus (\({\mathrm{-34.4}}\,{\mathrm{nm}}\)). (b) Calculated at the higher Lichte underfocus (\({\mathrm{-174.4}}\,{\mathrm{nm}}\)). In all cases CTFs are calculated for \({\mathrm{300}}\,{\mathrm{kV}}\), \(C_{3}={\mathrm{0.6}}\,{\mathrm{mm}}\), \(\Updelta E={\mathrm{0.8}}\,{\mathrm{eV}}\), beam divergence \(={\mathrm{0.1}}\,{\mathrm{mrad}}\)

The reciprocal of (1.18) gives the point resolution as
Thus, HRTEM images of thin specimens recorded at the Scherzer defocus (or its extended variant) will have components that are directly proportional to the (negative of) the projected potential of the specimen extending to spatial frequencies equal to the interpretable resolution limit [1.108, 1.56, 1.79]. For higher spatial frequencies, the contrast is partially reversed as the PCTF starts to oscillate (Fig. 1.7a,b). The above definitions of point resolution assume a fixed positive \(C_{3}\). Variable \(C_{3}\) in corrected microscopes modifies these results as detailed in a subsequent section.

Given the form of (1.19) it is evident that a decrease in electron wavelength has a greater effect than an equivalent decrease in the spherical aberration, and for this reason high-voltage instrumentation (see earlier references) was historically the preferred route to achieving higher interpretable resolutions.

A higher resolution limit, the information limit , defines the highest spatial frequency transferred from the specimen exit wavefunction to the image intensity. This is determined by the effects of spatial and temporal coherence in the illumination and by mechanical acoustic and electrical instabilities that also act to damp the transfer of higher spatial frequencies.

As will be discussed in detail in a subsequent section the effects of temporal and spatial coherence (see also [1.79]) can be treated through respective envelope functions of form
$$\begin{aligned}\displaystyle&\displaystyle E_{\mathrm{f}}(\boldsymbol{k})=\exp\left\{-0.5\uppi^{2}\Delta^{2}(\lambda\boldsymbol{k}^{2})^{2}\right\},\\ \displaystyle&\displaystyle E_{\mathrm{s}}(\boldsymbol{k})=\exp\left\{-\uppi^{2}q_{0}^{2}\left(C_{1}\lambda\boldsymbol{k}+C_{3}\lambda^{3}\boldsymbol{k}^{3}\right)^{2}\right\},\end{aligned}$$
in which the expression for the spatial envelope includes only defocus and spherical aberration, with \(q_{0}\), the standard deviation of a Gaussian modeling the convergent cone of illumination at the specimen surface, and where
In (1.21) \(C_{\mathrm{C}}\) is the chromatic aberration , \(E_{0}\) is the spread in electron energies arising from the source, \(V_{0}\) is the accelerating voltage, and \(I_{0}\) is the objective lens current, which affects the objective lens magnetic field [1.109, 1.110]. The last term is written here in terms of the objective lens current \(I_{0}\) following convention. However, this is not strictly correct, particularly if the lens is operated close to saturation where the magnetic field is not proportional to the current. It may seem surprising that the objective lens current influences the variation of focus only in the presence of chromatic aberration. However, this is because of the general scaling rule that electron trajectories are identical when energy and magnetic field are changed according to \(E^{\prime}=k^{2}E\) and \(B^{\prime}=kB\). Hence, in a microscope corrected for chromatic aberration, the focus cannot be changed by changing the currents in all lens elements by the same factor.

The form of these distributions is far less important than their width and in general, as above, Gaussian distribution functions are used, as this makes any further derivation analytically tractable. For the defocus spread, the terms due to voltage and lens current instabilities are well described by Gaussian function, whereas the intrinsic source energy spread would be more accurately described by a Maxwellian distribution for thermionic emitters or the Fowler–Nordheim equation [1.111] for field emitters. The above definition assumes that the fluctuations are independent of each other.

The form of these two envelope functions leads to definitions of information limits in which the information transfer drops to a level of \(e^{-2}\) or \({\mathrm{13.5}}\%\) [1.109].
$$\begin{aligned}\displaystyle&\displaystyle d_{2}=\sqrt{\frac{\uppi\lambda d}{2}}\\ \displaystyle&\displaystyle d_{3}=S_{+}^{\frac{1}{3}}+S_{-}^{\frac{1}{3}}\\ \displaystyle&\displaystyle S_{\pm}=\frac{\frac{3.3}{4\uppi q_{0}}\pm\left[\frac{C_{1}^{3}}{27C_{3}\lambda^{2}}+\left(\frac{3.3}{4\uppi q_{0}}\right)^{2}\right]^{\frac{1}{2}}}{C_{3}\lambda^{2}}\end{aligned}$$
However, it should be noted that the limit defined by \(d_{3}\) increases with defocus and does not therefore define an absolute information limit per se. In particular, for the case of field-emission sources where the source size is small, the spatial coherence is not limiting and thus the information limit due to temporal coherence determines the information limit for all defoci.

Traditionally, both of these limits have been experimentally measured using power spectra calculated from amorphous specimens or from a Young's fringe pattern  [1.112] calculated from the power spectrum of two images with a small real-space displacement. However, Bartel and Thust [1.113] have reported that for the typical resolution limits now attainable it is necessary to separate the effects of linear and nonlinear components of the contrast which cannot be distinguished in a Young's fringe pattern and have proposed an alternative method based on tilting the illumination.

1.2.3 The Wave Aberration Function

The key optical component affecting HRTEM image formation is the objective lens and in this section, we review its influence in terms of the wave aberration function (see also [1.114]).

For an ideal lens, a point object at a position \((x,y)\) in the object plane leads to a spherical wavefront in the diffraction plane, contracting to a conjugate point in the image plane. It should be noted that for a pure phase object this condition would, however, lead to zero contrast at the Gaussian focus. However, all electromagnetic lenses suffer from aberrations causing deviations from this ideal spherical wavefront thus reducing the sharpness of an image point much more severely than the diffraction limit.

For HRTEM, a wave aberration function \(W(u,v)\) is therefore defined that describes the distance between the ideal and actual wavefronts in the diffraction plane as a function of the position of the point object in the diffraction plane, \((u,v)\) [1.115] (Fig. 1.8).

Fig. 1.8

Schematic diagram showing the origin of the wave aberration function, \(W(u,v)\), which describes the complex deviation from an ideal spherically diffracted wave

The wave aberration function, \(W(\omega)\), written in terms of a complex position variable \(\omega=u+\mathrm{i}\,v\) can be Taylor expanded to third order in terms of only the axial aberrations as
$$\begin{aligned}\displaystyle W(\omega)&\displaystyle={\Re}\left\{\frac{2\uppi}{\lambda}\left(A_{0}\lambda\omega^{\ast}+\frac{1}{2}A_{1}\lambda^{2}\omega^{\ast 2}\right.\right.\\ \displaystyle&\displaystyle\quad\,+\frac{1}{2}C_{1}\lambda^{2}\omega^{\ast}\omega+\frac{1}{3}A_{2}\lambda^{3}\omega^{\ast 3}\\ \displaystyle&\displaystyle\quad\,+\frac{1}{3}B_{2}\lambda^{3}\omega^{\ast 2}\omega+\frac{1}{4}A_{3}\lambda^{4}\omega^{\ast 4}\\ \displaystyle&\displaystyle\quad\,+\frac{1}{4}S_{3}\lambda^{4}\omega^{\ast 3}\omega+\frac{1}{4}C_{3}\lambda^{4}\omega^{\ast 2}\omega^{2}+\ldots\left.\left.\!\vphantom{\frac{1}{4}}\right)\right\}.\end{aligned}$$
Converting to polar notation, with \(\omega=\boldsymbol{k}\mathrm{e}^{\mathrm{i}\phi},A_{n}=|A_{n}|\mathrm{e}^{\mathrm{i}{\alpha_{n}}}\) and \(B_{n}=|B_{n}|\mathrm{e}^{\mathrm{i}{\beta_{n}}}\), (1.23) can be more conveniently rewritten as
$$\begin{aligned}\displaystyle W(\boldsymbol{k},\phi)&\displaystyle=\frac{2\uppi}{\lambda}\left(|A_{0}|\lambda k\cos(\phi-\alpha_{0})\phantom{\frac{1}{2}}\right.\\ \displaystyle&\displaystyle+\frac{1}{2}|A_{1}|\lambda^{2}k^{2}\cos 2(\phi-\alpha_{1})+\frac{1}{2}C_{1}\lambda^{2}k^{2}\\ \displaystyle&\displaystyle+\frac{1}{3}|A_{2}|\lambda^{3}k^{3}\cos 3(\phi-\alpha_{2})\\ \displaystyle&\displaystyle+\frac{1}{3}|B_{2}|\lambda^{3}k^{3}\cos(\phi-\beta_{2})\\ \displaystyle&\displaystyle+\frac{1}{4}|A_{3}|\lambda^{4}k^{4}\cos 4(\phi-\alpha_{3})\\ \displaystyle&\displaystyle+\frac{1}{4}|S_{3}|\lambda^{4}k^{4}\cos 2(\phi-\sigma_{3})\\ \displaystyle&\displaystyle+\frac{1}{4}C_{3}\lambda^{4}k^{4}+\ldots\left.\vphantom{\frac{1}{4}}\right),\end{aligned}$$
which makes the azimuthal and radial dependence of the various aberration terms more apparent.

More rigorously \(W\) is a function of both position of the point object in the diffraction and image planes and of the energy i. e., \(W(x,y,u,v,E)\). Until recently HRTEM experiments were carried out at high magnification with a correspondingly small field of view and the dependence of \(W\) on \((x,y)\) was neglected in the isoplanatic approximation. However, the increasing use of large pixel array detectors requires the measurement and correction of low-order off-axial aberration coefficients [1.116, 1.117]. Similarly, the dependence of \(W\) on \((E)\) has generally been neglected in the isochromatic approximation although the development of \(C_{\mathrm{C}}\) correctors [1.118, 1.119] has necessitated measurement of the first-order chromatic focus and astigmatism.

Table 1.2 lists the axial aberration coefficients important in HRTEM to third order using the notation due to Typke and Dierksen [1.120]. We note that an alternative nomenclature due to Krivanek et al is also in common use [1.121]. The seemingly counterintuitive notation (e. g., \(C_{1}\) in Table 1.2 for a second-order term in the wave aberration function stems from the ray-optical theory of Seidel aberrations [1.114, 1.122], which are described in terms of displacements of ray path intersections with the image plane as a function of \((u,v)\). These displacements are proportional to the gradient of the wave aberration function and hence an \(n\)-th-order Seidel aberration corresponds to a term of order \(n+1\) in the wave aberration function.

All of the aberration coefficients listed in Table 1.2 except \(C_{1}\) and \(C_{3}\) are due to lens imperfections (either mechanical or electrical) and would not appear in a perfect round lens [1.107]. However, they can be corrected using combinations of multipole fields with appropriate symmetry and in recent years this has enabled the correction of \(C_{3}\) using long hexapole fields (see Chap.  13 in this volume) [1.123, 1.124, 1.125, 1.126], for reviews see [1.127, 1.49].

Table 1.2

Axial aberration coefficients important for HRTEM to third order


Order in \(\boldsymbol{\upomega}\)

Azimuthal symmetry

Name and description




Image shift




Two-fold astigmatism




Three-fold astigmatism




Four-fold astigmatism




Axial coma




Axial star








Spherical aberration

Table 1.3

Accuracy of aberration coefficients\({}^{\mathrm{a}}\)



\(K_{\text{max}}\) (\(\mathrm{nm^{-1}}\))

\(d_{\text{min}}\) (nm)

\(A_{1}\), \(C_{1}\) (nm)

\(A_{2}\), \(B_{2}\) (nm)

\(A_{3}\), \(S_{3}\), \(C_{3}\) (\(\mathrm{\upmu{}m}\))

Tilt (\(\mathrm{\upmu{}rad}\))































\({}^{\mathrm{a}}\) Accuracy to which the aberration coefficients need to be determined such that each of them causes a maximum root mean square () error in the wave aberration function of less than \(\lambda/16\), i. e., a phase change of less than \(\uppi/8\) for a given target resolution. The values are calculated for an accelerating voltage of \({\mathrm{300}}\,{\mathrm{kV}}\) (\(\lambda\approx{\mathrm{2}}\,{\mathrm{pm}}\)). The necessary accuracy for the beam tilt \(\tau\) is calculated from that for \(B_{2}\) using \(\Updelta\tau=-\Updelta B_{2}/(3C_{3})\) with \(C_{3}={\mathrm{0.57}}\,{\mathrm{mm}}\)

For HRTEM imaging the effects of the coefficients of the wave aberration function are to introduce phase shifts given by multiplying each term in \(W\) by \(2\uppi/\lambda\). These phase shifts can also be used to estimate the maximum tolerable value in any coefficient for a particular target resolution (Table 1.3) [1.128] and an overall stability budget for aberration correctors [1.129].

Few of the aberration coefficients defined above are directly observable under axial illumination and their determination therefore relies on measurements taken as a function of known injected beam tilts.

If the illumination is tilted by an angle \(\tau\) the observed aberration coefficients up to \(C_{3}\) (marked with a prime) are given by
$$\begin{aligned}\displaystyle A^{\prime}_{0}&\displaystyle=A_{0}+A_{1}\tau^{\ast}+A_{2}\tau^{\ast 2}+C_{1}\tau+\frac{1}{3}B_{2}^{\ast}\tau^{2}\\ \displaystyle&\displaystyle\quad\,+\frac{2}{3}B_{2}\tau^{\ast}\tau+C_{3}\tau^{\ast}\tau^{2}\;,\\ \displaystyle A^{\prime}_{1}&\displaystyle=A_{1}+2A_{2}\tau^{\ast 2}\tau+\frac{2}{3}B_{2}\tau+C_{3}\tau^{2}\;,\\ \displaystyle C^{\prime}_{1}&\displaystyle=C_{1}R\left(\frac{4}{3}B_{2}\tau^{\ast}\right)+2C_{3}\tau^{\ast}\tau\;,\\ \displaystyle A^{\prime}_{2}&\displaystyle=A_{2}\;,\\ \displaystyle B^{\prime}_{2}&\displaystyle=B_{2}+3C_{3}\tau\;,\\ \displaystyle C^{\prime}_{3}&\displaystyle=C_{3}\;.\end{aligned}$$
From (1.25) it is immediately apparent that the shift \(A^{\prime}_{0}-A_{0}\) between two images taken at beam tilts differing by \(\tau\) depends on all the other aberration coefficients. Hence, measuring the image shifts induced by a suitable set of beam tilts provides a method for measuring all coefficients.

Alternatively, the tilt-induced changes, in defocus (\(C^{\prime}_{1}-C_{1}\)) or two-fold astigmatism (\(A^{\prime}_{1}-A_{1}\)) measured from diffractograms (Fig. 1.9a-d), depend on \(A_{1}\), \(C_{1}\), \(A_{2}\), \(B_{2}\), and \(C_{3}\), thus providing an alternative measure of these coefficients. If the spherical aberration is independently determined then measurement of the orientations only of diffractograms also provides another simple measure of \(A_{1}\), \(A_{2}\), and \(B_{2}\) [1.130].

Fig. 1.9a-d

Individual axial diffractograms calculated from uncorrected HRTEM images of a thin amorphous germanium foil showing the effects of defocus (\(C_{1}\)) and two-fold astigmatism (\(A_{1}\)). (a) From an axial image recorded at large underfocus with the two-fold astigmatism corrected. (b) From an axial image in the presence of a small amount of two-fold astigmatism giving rise to an elliptical diffractogram. (c) As for (b) but with a larger two-fold astigmatism giving rise to a Maltese Cross-shaped diffractogram showing overfocus and underfocus along orthogonal axes. (d) Representative tableau of diffractograms recorded for a set of differently titled illumination conditions illustrating the variation in \(A_{1}\) and \(C_{1}\) with illumination tilt. Each experimental diffractogram is shown merged with a simulated diffractogram calculated from the fitted values of \(A_{1}\) and \(C_{1}\). The illumination tilt axes are indicated

Measurements of diffractograms [1.115, 1.120, 1.131, 1.132, 1.133, 1.134] or image shifts [1.115, 1.120, 1.135, 1.136, 1.137, 1.138], however, have their own particular experimental advantages and disadvantages.

A practical difficulty with the application of tilt-induced displacements arises from the measurement of the image shifts using the peak position in the cross-correlation function ( ) defined in terms of the image contrast \(c(\boldsymbol{k})\) as
The tilt-induced change in \(A_{0}\) introduces a linear phase variation in the cross spectrum \(c_{1}^{\ast}c_{2}\) which leads to a displacement of the XCF peak to a position given by the shift vector between the images. As already shown, the other imaging parameters also change as the beam is tilted and this causes the phase variation to become nonlinear at higher spatial frequencies leading to distorted cross-correlation peaks. However, when the imaging conditions for both images are approximately known, these nonlinear phase shifts can be compensated and a sharp XCF peak can be recovered [1.139]. Shift measurements also fail for clean perfectly periodic specimens in which image positions differing by any integer multiple of a lattice vector cannot be distinguished. A more severe problem associated with the tilt-induced shift method is that any displacement due to specimen drift is indistinguishable from the tilt-induced displacement sought and hence this approach is most frequently used at low resolution or for initial coarse alignment at high resolution.

Conversely, diffractogram measurements require the presence of an area of thin disordered material and are thus less applicable to many specimens. However, they are insensitive to specimen drift and their measurement is relatively straightforward even under tilted illumination conditions. Hence diffractogram measurements are best suited to fine adjustment of the aberrations at high resolution [1.134].

Historically, the use of diffractograms for the determination of aberrations was first suggested by Thon [1.140] for the measurement of defocus from ring positions in an optically generated diffractogram. This method was later extended through the use of diffractogram tableaus acquired for a known set of beam tilt azimuths [1.141] (Fig. 1.9a-d) to the measurement of the spherical aberration, \(C_{3}\), axial coma, \(B_{2}\), and later threefold astigmatism, \(A_{2}\) [1.131, 1.133].

However, the diffractogram tableau method was computationally too demanding for routine use at this time and was therefore restricted to a demonstration that the alignment using current reversal or voltage centering was inadequate for HRTEM [1.142]. The current reversal center alignment involves reversing the current of the objective lens and is no longer practical with the strongly excited lenses used in modern instruments. It should not be confused with the current center alignment, where the objective lens current is oscillated by a small amount. Similarly, in the voltage center alignment, the high tension is oscillated. Generally, the axes found by the three methods are distinct and the voltage center provides the closest approximation to the coma-free axis required for HRTEM.

Fitting \(C_{1}\) and \(A_{1}\) to match simulated diffractogram patterns with experimental ones can be done manually to relatively high accuracy (Fig. 1.9a-dd; [1.143]). This, however, is a lengthy and tedious process and automation of this task is necessary for practical use. This problem has been addressed by Baba et al [1.144] and Fan and Krivanek [1.145] in an algorithm where the diffractogram is divided into \(\mathrm{32}\) sectors and the defocus along the directions in each sector is determined by cross-correlating the rotational sector average with an array of theoretical diffractograms. The sinusoidal focus variation as a function of the azimuth angle obtained is subsequently fitted to the measured focus values to determine \(C_{1}\) and \(A_{1}\) and automatic alignment can be achieved using a tableau with as few as four tilt azimuths.

More generally, the major challenges for implementing a robust automated diffractogram fitting procedure are as follows:
  1. 1.

    The most abundant amorphous material in the microscope, carbon, is a weak scatterer. Hence, at high spatial frequencies, the signal is weak compared to the noise background affecting overall diffractogram quality.

  2. 2.

    The strength of the observed signal depends on the linear and nonlinear components of the contrast which, as already noted [1.113] cannot be distinguished in the phase contrast transfer function.

  3. 3.

    Diffractograms taken close to Scherzer or Gaussian defocus show few rings and are prone to errors in the fitting process.

  4. 4.

    In the presence of large two-fold astigmatism and at close to Gaussian defocus the diffractograms are cross- rather than ring-shaped, which leads to difficulties for algorithms based on the evaluation of rotationally averaged sectors of the diffractogram.

  5. 5.

    It is often difficult to distinguish between overfocus and underfocus.

  6. 6.

    Most automated algorithms fail when a significant amount of crystalline material is present, because of strong reflections at positions unrelated to the ring pattern.

More recently, the advent of \(C_{3}\)-corrected microscopes has made automated aberration measurement critical because the nonround lens elements introduce a multitude of high-order aberrations (up to six-fold astigmatism) that require correction in an elaborate alignment procedure. Uhlemann and Haider [1.146] have implemented an algorithm that can evaluate the apparent defocus and astigmatism from a diffractogram in less than \({\mathrm{400}}\,{\mathrm{ms}}\) based on a comparison of the experimental diffractogram with an extensive library of precalculated diffractograms and which is robust to the effects arising from the presence of crystalline material.

However, given the limitations of both of these traditional methods an alternative method has also been proposed, which is both applicable to a wide range of specimens and has the high accuracy for use in atomic-resolution imaging [1.147, 1.148, 1.149].

In this approach a phase-correlation function ( ) (Fig. 1.9a-d) [1.150] is initially calculated (defined as the conventional XCF with the modulus set to unity)
where \(c_{\mathrm{i}}(\boldsymbol{k})\) are the image Fourier transforms and \(F(\boldsymbol{k})\) is a rotationally symmetric weighting factor used to suppress high-frequency noise. The modulus normalization is essential as it suppresses the crystal reflections that make the conventional XCF periodic, thus allowing aberration measurement in the absence of amorphous material.
The PCF calculated between two images recorded at different defocus levels consists of a centrosymmetric ring pattern, the exact form of which depends on the relative defocus between them (assuming that the other aberration coefficients remain constant between the two images) (Fig. 1.10a,b). It is possible to compensate for the phase shifts giving rise to this ring pattern by the application of a phase factor dependent on the defocus difference, thus defining a phase-compensated PCF
$$\begin{aligned}\displaystyle&\displaystyle\text{PCF}_{\mathrm{w}}(\boldsymbol{x})\\ \displaystyle&\displaystyle\quad=\mathfrak{F}^{-1}\left\{F(\boldsymbol{k})\frac{\cos[W_{\mathrm{D}}(\boldsymbol{k})]c_{1}^{\ast}(\boldsymbol{k})c_{2}(\boldsymbol{k})}{|\cos[W_{\mathrm{D}}(\boldsymbol{k})]c_{1}^{\ast}(\boldsymbol{k})c_{2}(\boldsymbol{k})+h|}\right\},\end{aligned}$$
in which the very small positive number, \(h\), prevents a zero denominator and where \(W_{\mathrm{D}}(\boldsymbol{k})=\uppi\Updelta C_{1}\lambda|\boldsymbol{k}|^{2}\) describes the propagation from the first to the second image in the presence of only a defocus difference \(\Updelta C_{1}\). When the value of \(\Updelta C_{1}\) matches the actual focus difference the phase compensated PCF collapses to a sharp localized correlation peak. Hence the relative defocus difference between two images can be determined by simply maximizing peak height in the phase-compensated PCF as a function of the compensated defocus difference (Fig. 1.11), which with subsequent refinement can yield relative defoci to an accuracy better than \({\mathrm{1}}\,{\mathrm{nm}}\) [1.148].
Fig. 1.10a,b

Comparison of (a) cross- and (b) phase-correlation functions (XFC and PCF) between two images of a crystalline material with a defocus difference of \({\mathrm{69}}\,{\mathrm{nm}}\). Because of the periodicity of the crystalline specimen, the XCF peak repeats periodically but the PCF dose not show this repetitive pattern and consists of a single peak broadened into a concentric ring pattern due to the defocus difference. Reprinted from [1.148], Copyright 2002, with permission from Elsevier

Fig. 1.11

(a) PCF as a function of compensated focus difference indicated. (b) Peak height of the PCF (solid line) and XCF (dashed line) (shown against the left-hand side and right-hand side scales) between two images as a function of compensated focus difference showing a sharp maximum at the correct compensating focus difference in the PCF. Reprinted from [1.148]. Copyright 2002, with permission from Elsevier

In the second step an initial restored image wavefunction (see later for details of the restoration process), \({\psi}_{\mathrm{i}}(\boldsymbol{k})\), in the plane of a reference image is restored and the absolute values of \(A_{1}\) and \(C_{1}\) are determined using a phase contrast index function ( ) \(f_{\text{PCI}}\) given by
$$\begin{aligned}\displaystyle f_{\text{PCI}}(\boldsymbol{k},C_{1},A_{1})&\displaystyle=-\cos\{\arg[\psi_{\text{si}}(\boldsymbol{k})]\}\\ \displaystyle&\displaystyle\quad+\arg[(\psi_{\text{si}}(-\boldsymbol{k})]+2W(\boldsymbol{k},C_{1},A_{1})\;.\end{aligned}$$
$$\begin{aligned}\displaystyle W(\boldsymbol{k},C_{1},A_{1})&\displaystyle=\frac{1}{2}|A_{1}|\lambda\boldsymbol{k}^{2}\cos 2(\phi-\alpha_{1})+\frac{1}{2}C_{1}\lambda\boldsymbol{k}^{2}\\ \displaystyle&\displaystyle\quad+\frac{1}{2}C_{3}\lambda^{3}\boldsymbol{k}^{4}\;,\end{aligned}$$
is the symmetric part of the wave aberration function defined earlier with trial parameters \(C_{1}\), \(A_{1}\) and fixed spherical aberration \(C_{3}\). This function measures the conjugate symmetry in the corrected image wavefunction and has a value of \(+1\) when this is conjugate antisymmetric and \(-1\) when conjugate symmetric. Therefore, within the weak phase object approximation, when the trial values of \(C_{1}\) and \(A_{1}\) are correct \(f_{\text{PCI}}\) tends to 1 for all spatial frequencies, whereas for mismatched parameters, \(f_{\text{PCI}}\) shows dark bands (Fig. 1.12a-da–c). In practical applications, the correct values of \(C_{1}\) and \(A_{1}\) are readily determined from a plot of \(f_{\text{PCI}}\) averaged over \(\boldsymbol{k}\) (Fig. 1.12a-dd).
Fig. 1.12a-d

Phase contrast index function \(f_{\text{PCI}}\). For mismatched values of \(C_{1}\) (a\({\mathrm{-50}}\,{\mathrm{nm}}\), (b\({\mathrm{-10}}\,{\mathrm{nm}}\), \(f_{\text{PCI}}\) shows dark rings, whereas at the correct values of \(C_{1}\), \(f_{\text{PCI}}\) is close to one (white) at all spatial frequencies (c). (d) The \(f_{\text{PCI}}\) averaged over \(\boldsymbol{k}\) and plotted as a function of \(C_{1}\) showing a sharp maximum at the correct focus value of \({\mathrm{-128}}\,{\mathrm{nm}}\). The values \(\Updelta C_{1}\) given in (ac) are relative to this focus. Reprinted from [1.148]. Copyright 2002, with permission from Elsevier

This general method is best implemented experimentally using a combined tilt-defocus dataset geometry as illustrated in Fig. 1.13, which comprises a set of short focal series recorded at a number of tilt azimuths. This dataset is over redundant and can be used to calculate values of \(C_{1}\) and \(A_{1}\) for each of the recorded tilt azimuths using the PCF/PCI approach.

Fig. 1.13

Schematic representation of the combined tilt/defocus dataset used for both aberration determination using the PCF/PCI approach and tilt azimuth reconstruction described in a subsequent section. A three-member focal series is recorded for each of six different tilt azimuth angles and also for axial illumination. Image numbers refer to the experimental recording sequence designed to minimize hysteresis in the objective lens and beam tilt coils and to enable measurement of focus drift during acquisition

For the measurement methods described, data recorded for a suitable number of defined illumination directions and magnitudes provide the full set of aberration coefficients as in the case of traditional diffractogram tableaus and these can be conveniently and reliably determined in practice by least-squares fitting of the parameters to be determined to the observations available. Where the injected tilt is known accurately the dependence of the observables on the aberration coefficients is linear yielding an easily found, unique solution [1.115, 1.149]. Alternatively, if the calibrations of the beam tilt coil magnitudes and orientations are not known, these unknowns can be fitted in an additional nonlinear iterative minimization of the residual misfit from the linear fitting of the aberration coefficients [1.115]. More recently, Barthel and Thust [1.151] have applied pattern recognition to diffractogram analysis, developing an optimized high-precision approach to aberration measurement which has subsequently been used to evaluate the stability of individual aberrations coefficients affecting the corrected state [1.152].

1.2.4 Coherence

As already described, contrast in an HRTEM image formed arises from the interference of electron waves that have been scattered by the specimen. The degree to which these waves interfere is often referred to as the degree of coherence [1.153]. Hence, coherence is vital in determining high-resolution image contrast (for a review see [1.154]).

For all electron sources used in HRTEM imaging [1.81], the source is only partially coherent [1.114, 1.154]. There are two aspects to this partial coherence: partial spatial coherence , due to the finite dimensions of the electron source [1.153], and partial temporal coherence , which is associated with the finite energy distribution of the source and fluctuations in both the accelerating voltage and the objective lens current [1.155]. The overall effect of this partial coherence of the source is to restrict the amount of information that can be extracted from high-resolution images [1.154, 1.156, 1.157]. Thus, the source represents the second major optical component that directly affects HRTEM image contrast.

Frequently, the illumination system of the electron microscope is treated as an incoherently filled effective source [1.158, 1.159] with an intensity distribution \(S(\boldsymbol{q})\). Within this model the effects of partial spatial coherence on the HRTEM image intensity can be calculated by incoherent summation of intensities over all the incident angles from the effective source. For cases in which this basic assumption does not hold the mathematical description of coherence is best treated in terms of the mutual dynamic object spectrum and associated mutual coherence function. This treatment is outside the scope of this chapter and excellent overviews can be found in [1.114, 1.37, 1.81].

For this source model the temporal partial coherence effect can also be treated incoherently [1.108] and it is generally assumed that the spatial and temporal distributions of the source are not correlated (for a detailed treatment of the case in which this is not valid see [1.114]). Under these conditions the image intensity can be written as
$$\begin{aligned}\displaystyle I(\boldsymbol{x})&\displaystyle=\sum_{\boldsymbol{q}}\sum_{\mathrm{f}}S(\boldsymbol{q})F(f)\\ \displaystyle&\displaystyle\quad\,\times\left|\mathfrak{F}^{-1}\left\{\psi_{\mathrm{d}}(\boldsymbol{k};\boldsymbol{q})\exp\left[-\mathrm{i}W(\boldsymbol{k},C_{1}+f,\boldsymbol{q})\right]\right\}\right|^{2},\end{aligned}$$
in which the incoherent summations of the coherent image intensities at incident angles \(\boldsymbol{q}\) and focus spread levels are weighted by the spatial and temporal intensity distributions \(S(\boldsymbol{q})\), \(F(f)\), respectively.

The individual coherent image intensities are thus given as the modulus square of the convolution of the Fourier transform of the specimen exit wavefunction \(\psi_{\mathrm{e}}(\boldsymbol{k},\boldsymbol{q})\) at an incident angle \(\boldsymbol{q}\) and the objective lens aberration function \(W(\boldsymbol{k},C_{1}+f,\boldsymbol{q})\) with corresponding beam tilt and defocus. General computation of the image intensity therefore involves a full dynamical calculation of coherent image intensities over an equally spaced mesh of beam tilts, \(\boldsymbol{q}\) and focal spread values, \(f\). This can be optimally tackled through the use of numerical Monte Carlo integration [1.161].

A number of approximations may also be made to make this calculation more tractable and which provide analytical solutions. For small beam tilts of the incident wave and for modest specimen thickness, the specimen exit wavefunction can be approximated as a single exit wavefunction at the mean incident illumination angle. A further approximation can also be made by expanding the aberration function to only first order which is valid if the beam tilt and focal spread are small as is the case for many modern instruments. Using these approximations and assuming that the focal spread and beam divergence distributions are Gaussian with \(\mathrm{e}^{-1}\) values of \(q_{0}\) and \(C_{1}\) respectively the image intensity can be written as [1.162]
$$\begin{aligned}\displaystyle T(\boldsymbol{k}_{1},&\displaystyle\boldsymbol{k}_{2})\\ \displaystyle&\displaystyle=\exp\left\{-\mathrm{i}\left[W(\boldsymbol{k}_{1};C_{1})-W(\boldsymbol{k}_{2};C_{1})\right]\right\}\\ \displaystyle&\displaystyle\quad\times\exp\left[-\uppi^{2}q_{0}^{2}\left(W^{\prime}_{\boldsymbol{k}_{1}}-W^{\prime}_{\boldsymbol{k}_{2}}\right)^{2}\right]\\ \displaystyle&\displaystyle\quad\times\exp\left[-\uppi^{2}\Updelta^{2}\left(\frac{\partial W}{\partial C_{1}}|_{\boldsymbol{k}={\boldsymbol{k}_{1}}}-\frac{\partial W}{\partial C_{1}}|_{\boldsymbol{k}={\boldsymbol{k}_{2}}}\right)^{2}\right].\\ \displaystyle\end{aligned} $$
The Fourier transform of (1.33) is given by the integral over all pairs of diffracted beams with \(T\) representing a transmission cross coefficient ( ) (Fig. 1.14a,b). Within this expression for the TCC the first exponential term describes the spatial coherence for a Gaussian spatial intensity distribution with an even aberration function. This term is a function of the square difference of the gradient of the aberration function at two spatial frequencies and allows maximum transfer when the slopes of the aberration function are identical at \(\boldsymbol{k}_{1}\) and \(\boldsymbol{k}_{2}\). The second exponential term describes the temporal coherence, and is a function of the square difference of the derivative of the aberration function with respect to focus at two spatial frequencies. This function has maximum transfer when \(|\boldsymbol{k}_{1}|=|\boldsymbol{k}_{2}|\).
Fig. 1.14a,b

Moduli of the transmission cross-coefficient (TCC) functions defined in the text for combined spatial and temporal coherence and calculated under (a) the weak object approximation and (b) for a strong object. In both case the functions are shown for defoci ranging from \(\mathrm{-1}\) to \(\mathrm{-9}\) Scherzer defocus (Sch). All calculations were carried out at \({\mathrm{300}}\,{\mathrm{kV}}\) with \(C_{3}={\mathrm{0.6}}\,{\mathrm{nm}}\) and with \(\mathrm{e}^{-1}\) values of Gaussian beam divergence and focal spread distributions of \({\mathrm{0.28}}\,{\mathrm{mrad}}\) and \({\mathrm{4.7}}\,{\mathrm{nm}}\), respectively. (Adapted from [1.160])

If a further assumption of a weak scattering object is made, such that only interference between the transmitted beam and diffracted beams is considered, the TCC can be further simplified as [1.157]
$$\begin{aligned}\displaystyle T(k)&\displaystyle=\exp[-\mathrm{i}\,W(k)]\\ \displaystyle&\displaystyle\quad\,\times\exp\left[-\uppi^{2}q_{0}^{2}\left(C_{1}\lambda k+C_{3}\lambda^{3}k^{3}\right)^{2}\right]\\ \displaystyle&\displaystyle\quad\,\times\exp\left[-\uppi^{2}\Updelta^{2}\left(\frac{\lambda k^{2}}{2}\right)^{2}\right].\end{aligned}$$
In this limit both spatial and temporal coherence functions for weak objects decrease with increasing spatial frequencies (Fig. 1.14a,b), and can be recognized as the simple damping envelopes described in an earlier section defining the information limit.

1.3 Instrumentation

This section briefly describes certain aspects of instrumental design that are important for HRTEM in that they directly contribute to the resolution attainable. Inevitably, because of limitations of space, not all aspects are covered comprehensively and the reader is referred to one of several dedicated texts on HRTEM and electron optics for more detailed treatments of individual areas [1.114, 1.122, 1.163, 1.164, 1.165, 1.79, 1.80, 1.81] (see also Chap.  13 in this volume and references therein).

1.3.1 Electron Sources

For HRTEM electron sources have to fulfil a number of key requirements summarized as follows [1.114, 1.122, 1.166, 1.79]:
  1. 1.

    High brightness and coherence

  2. 2.

    High current efficiency

  3. 3.

    Long life under available vacuum conditions

  4. 4.

    Stable emission characteristics

  5. 5.

    Low energy spread.

An ideal source would therefore provide independent control of the illumination intensity and coherence for any given illuminated area, a situation that is currently best approximated by the properties of field-emission sources.
For any source, the current, \(I\), passing through an area, \(A\), is proportional to both \(A\) and to the solid angle subtended by the illumination aperture at the source [1.80]. A beam brightness can be defined as the constant of proportionality, \(\beta\), in the above relationship and as the area and angle tend to zero this is given by
$$\beta=\frac{I}{\uppi A\theta^{2}}\;,$$
for an aperture semiangle, \(\theta\).

The effect of an ideal lens [1.122, 1.164, 1.166, 1.80] below the source forming the illumination system is to reduce the current density by \(M^{2}\) but to increase the angular aperture by \(1/M^{2}\). Thus, in the absence of any lens aberrations, the brightness remains constant at all conjugate planes in the microscope.

To compare different source types a theoretical brightness for a particular source can be expressed in terms of the emission current density at the cathode (filament), \(\rho\), the cathode temperature, \(T\), and the relativistic high voltage, \(V\), as
$$\beta_{\mathrm{m}}=\frac{\rho eV}{\uppi k_{\mathrm{B}}T}\;.$$
This requires optimal operating conditions and in practice is often reduced to lower values for thermionic sources of between \(\mathrm{0.1}\) and \(\mathrm{0.5}\)\(\beta_{\mathrm{m}}\). Table 1.4 lists values of \(\beta_{\mathrm{m}}\) together with other physical and operational characteristics for the four sources that have been used in HRTEM instruments.
Table 1.4

Physical properties of modern electron sources important for HRTEM [1.79]


Virtual source diameter

Measured brightness at \({\mathrm{100}}\,{\mathrm{kV}}\)


Energy a


Melting point


Vacuum required


Emission current


Heated field-emission ZrO\(/\)W







Room-temperature field emission


\(\mathrm{2\times 10^{9}}\)





Hair-pin filament


\(\mathrm{5\times 10^{5}}\)







\(\mathrm{7\times 10^{6}}\) (at \({\mathrm{75}}\,{\mathrm{kV}}\))





a Full-width-at-half-maximum (FWHM)

Tungsten hairpin and \(\mathrm{LaB_{6}}\) cathodes rely on thermionic emission of electrons, where at suitably high temperatures electrons in the tail of the Fermi distribution acquire sufficient kinetic energy to overcome the work function of the cathode material. Tungsten hairpin sources are the oldest electron sources in commercial use but are only marginally useful for HRTEM and have largely been replaced by \(\mathrm{LaB_{6}}\) cathodes  [1.167] and more recently by Schottky field-emission sources ([1.68, 1.69, 1.70, 1.71]; for a comprehensive review of Schottky field emitters see [1.168, 1.70]) which are now employed on almost all commercial instruments. The choice of single-crystal \(\mathrm{LaB_{6}}\) cathodes as thermionic emitters for HRTEM is largely due to their higher brightness while still being operable under relatively poor vacuum conditions. They also have the advantage of operating at a lower temperature than W filaments (because of a lower workfunction), thus providing lower energy spreads. For these reasons until the relatively recent development of commercial field-emission sources \(\mathrm{LaB_{6}}\) cathodes were most widely deployed for HRTEM.

The detailed physics of field emission and the design of field-emission guns for HRTEM are beyond the scope of this section and the article by Crewe [1.169] and the chapter in Reimer [1.81] provide detailed reviews. These sources are now universal in HRTEM instruments and consist of either a heated or an unheated single crystal of tungsten in a \(\langle 111\rangle\) or \(\langle 310\rangle\) orientation. In operation, the tip of the source is held in a region of high electrostatic field enabling electrons to tunnel through the lowered potential energy barrier at the surface. In many HRTEM instruments the alternative Schottky emitter comprising a ZrO-coated W tip [1.168, 1.170, 1.171], in which the work function is lowered to \({\mathrm{2.8}}\,{\mathrm{eV}}\) (from \({\mathrm{4.6}}\,{\mathrm{eV}}\)), is frequently used allowing electrons to overcome the potential barrier at a lower temperature than for conventional thermionic sources leading to a reduced energy spread. True field-emission sources operating at room temperature have been less widely employed due to the need for stringent ultrahigh vacuum ( ) conditions at the tip region to avoid contamination [1.172] but with advances in vacuum technology are now commercially available [1.173, 1.174]. These sources provide the highest brightness and coherence and lowest energy spread.

Although a detailed discussion is outside the scope of this chapter it should be noted that there have recently been significant advances made in the development of pulsed electron sources using laser-stimulated photocathodes that can operate in both stroboscopic and single-shot modes ([1.175, 1.176, 1.177]; see also Chap.  8).

The brightness and energy spread of the source affect HRTEM images through image recording times and the resolution limit imposed by temporal coherence.

The detail in an HRTEM image is strongly affected by the spatial coherence of the electrons emitted from the source and certain physical parameters contribute to this. A theoretical treatment of coherence and its influence on HRTEM images has been given in a previous section and we therefore confine discussion here to the relevant physical characteristics of the source itself.

As already discussed an effective source can be defined as an imaginary electron emitter filling the illuminating aperture [1.158, 1.159]. In this case each point within the aperture represents a point source of electrons giving an emergent spherical wave, becoming approximately planar at the specimen. Thus, at the specimen plane each electron can be specified by its direction in an incident plane wave. Hence increasing the size of the illuminating aperture increases the size of the central diffraction spot and decreases the spatial coherence. This can be described as a coherence width, which defines the transverse distance at the object plane over which the illuminating radiation may be treated coherently. Thus, waves scattered from atoms separated by less than this distance will interfere coherently and their complex amplitudes must be added, whereas atoms separated by more than this distance scatter incoherently and the intensities of the scattered radiation are added. For sources to fulfill the assumption that the illuminating source can be replaced by an incoherently filled aperture (see earlier) the coherence width in the plane of this aperture must therefore be small compared to its size, thus making the degree of coherence dependent only on the size of the illuminating aperture. The effect of source size on coherence hence becomes important only for sources smaller than ca. \({\mathrm{1}}\,{\mathrm{\upmu{}m}}\), e. g., the smallest \(\mathrm{LaB_{6}}\) or pointed tungsten sources available. Field-emission sources do not conform to this model and for these the degree of coherence is critically dependent on the excitation of the condenser and gun lens system.

1.3.2 Optics

The optical column of a modern HRTEM instrument typically consists of between two and four condenser lenses, an objective lens, and up to six imaging lenses below the specimen. The science of magnetic lens design dates back to the late 1920s [1.178] when it was realized that rotationally symmetric magnetic fields could be used to focus electrons and that to produce a lens of high refractive power the magnetic field along the axis of rotational symmetry needed to be confined to a small region in which the field strength was sufficiently high. This and other elegant early work using algebraic expressions for the lens field has been extensively reviewed elsewhere [1.119, 1.122, 1.163, 1.164, 1.166, 1.179, 1.180, 1.4] and due to limitations of space will not be considered further here. Modern lens design is guided by computed solutions of the Laplace equation (see [1.181] for a review) and subsequent numerical solution of the ray equation [1.182]. Electron optical design has also been comprehensively reviewed elsewhere [1.122, 1.164, 1.166, 1.180, 1.183, 1.184] (see also [1.119, 1.185]) including not only the calculation of the lens polepiece shape but also that of the overall magnetic circuit. Because of this extensive literature, we provide here only a brief summary of selected aspects of electron optics relevant to HRTEM and further restrict our discussion to the properties of the objective lens only. (Lenses which use the image formed by a preceding lens as an object are classed as projector lenses and include intermediate lenses in modern instruments. These are thus distinguished from the objective lens which uses the physical specimen as its object.)

For all lenses, the aberrations increase with angle and it is the optical properties of the objective lens that dominate the final quality of the image formed. These can be treated analytically through either the eikonal [1.186] or trajectory methods (for collected references to these see [1.164]). Alternatively, a numerical approach has been described by [1.179, 1.187] based on finite element calculation (FEM ) of the magnetic flux density distribution for a trial model lens geometry (see also [1.188, 1.189]). Subsequently, this can be used to calculate the axial magnetic field distributions and the paraxial rays, which can be finally combined to calculate the aberration integrals [1.179, 1.184].

Until the advent of aberration correction, the most important lens aberrations for HRTEM were third-order spherical aberration and first-order chromatic aberration, which as already noted are finite positive for all round electromagnetic lenses (Fig. 1.15a-c). Crucially, they cannot be corrected by simple refinement of the mechanical design of the objective lens polepiece. This leads to the situation where in corrected instruments the limiting aberration is now often the fifth-order spherical aberration which is also always positive finite for round electromagnetic lenses.

Fig. 1.15a-c

Illustration of certain lens aberrations. (a) A perfect lens focuses a point source to a single image point. (b) Spherical aberration causes rays at higher angles to be overfocused. (c) Chromatic aberration causes rays at different energies (indicated by color) to be focused differently

Third-order spherical aberration (\(C_{3}\)) arises when rays leaving an object at large angle, \(\theta_{0}\), are refracted too strongly in the outer regions of the lens and are brought to a focus before the Gaussian image plane. If all rays from such an object are considered then the radius of the circle of least confusion due to the spherical aberration is \(r=MC_{3}\theta_{0}^{3}\) in the Gaussian image plane, with \(M\) the magnification. The value of \(C_{3}\) depends on the lens geometry and excitation and on the object position within the lens field and typically takes values between \(\mathrm{0.5}\) and \({\mathrm{2}}\,{\mathrm{mm}}\) for modern HRTEM instruments (at \(200{-}300\,{\mathrm{kV}}\)) and sets a limit to the interpretable resolution in the absence of corrector elements.

For a known lens geometry with a defined magnetic field distribution, values for \(C_{3}\) can be obtained from either computed solutions to the full (nonparaxial) ray equation [1.189] or through the use of the expression below due to Glaser [1.190]
$$\begin{aligned}\displaystyle C_{3}&\displaystyle=\frac{e}{16m_{0}V_{\mathrm{r}}}\int_{z1}^{z2}\left[\left(\frac{\mathrm{d}B_{z}(z)}{\mathrm{d}z}\right)^{2}+\frac{3e}{8m_{0}V_{\mathrm{r}}}B_{z}^{4}(z)\right.\\ \displaystyle&\displaystyle\qquad\qquad\quad\left.-B_{z}^{2}(z)\left(\frac{h^{\prime}(z)}{h(z)}\right)^{2}\right]h^{4}(z)\mathrm{d}z\;,\end{aligned}$$
where \(h(z)\) is the paraxial trajectory of a ray that leaves an axial object with unit slope and \(B_{z}(z)\) is the magnetic field. The spherical aberration can be measured using either diffractograms of image shifts as detailed earlier. The presence of derivatives in (1.37) explains why the value of \(C_{3}\) is sensitive to the shape of the field and for this reason the detailed geometry of the polepiece [1.187] and the location of the specimen within the polepiece gap are key design parameters of the objective lens [1.184].

First-order chromatic aberration arises from variations in the lens focal length with wavelength and hence electron energy. As all electron sources are polychromatic to a greater or lesser extent (see earlier) a series of in-focus images are formed on a set of planes normal to the optic axis, one for each wavelength present in the illumination. A geometrical optics treatment, as a first approximation, can be used to determine the effects of this focal length variation, which leads to an extended disk in the Gaussian image plane (of a point object) with radius, \(M\theta_{0}C_{\mathrm{C}}[(\Updelta V_{0}/V_{0})-(2\Updelta I/I)]\). Typical values for \(C_{\mathrm{C}}\) in modern instruments are similar to those for \(C_{3}\) and contribute to setting the absolute information limit for HRTEM.

In a similar fashion to the spherical aberration the value of the chromatic aberration can also be evaluated if the field distribution is known [1.191] as
For applications to HRTEM both of these aberrations can be reduced by introducing the specimen into the field of an immersion objective lens for which the magnetic field on the illumination side of the electron optical column (the prefield) does not directly affect image formation. This saturated (condenser objective) [1.192] geometry also has the significant practical advantage of enabling switching from HRTEM using a relatively broad parallel beam to probe-forming modes e. g., for analysis or STEM imaging from the same area of the sample and has largely replaced earlier asymmetric top-entry lens designs [1.193].

Both spherical and chromatic aberration are sensitive to the overall geometry of the polepiece, which is often parameterized in terms of the dimensions defining the upper and lower polepiece faces and bores and the gap between the two polepieces. However, the behavior of both \(C_{3}\) and \(C_{\mathrm{C}}\) (and of focal length) can be readily calculated for a given geometry as a function of the lens excitation [1.122, 1.164, 1.166, 1.184, 1.192, 1.194] and specimen position enabling these coefficients to be optimized, subject to constraints imposed by machining tolerances and requirements for specimen movement and tilt.

1.3.3 Specimen Stages

The most important purely mechanical components of modern HRTEM instruments are the specimen holder and the goniometer into which it is fitted. A number of complex designs for these, retaining maximum flexibility in specimen movement and access to peripheral devices under constraints of extremely high mechanical and thermal stability, have been proposed and are reviewed elsewhere [1.195, 1.196, 1.197, 1.198].

The original designs for instruments used for high resolution were based on top-entry goniometers into which a conical cartridge was inserted fitting into the upper tapered bore of the objective lens polepiece with the specimen supported in a cup at the base of the cartridge. However, this arrangement leads to restrictions in objective lens design as the polepiece bore must be grossly asymmetric with an upper bore of sufficient diameter to allow the specimen to pass through [1.184, 1.193]. Translate and tilt movements also require complex, high-precision mechanical mechanisms as the specimen is located at the base of the cartridge surrounded by the polepiece. The geometry of top-entry stages also precludes the use of energy-dispersive X-ray ( ) detectors and the provision of specimen heating or cooling is extremely difficult [1.197]. Despite these limitations this design has inherent advantages in that the specimen cartridge is mechanically and acoustically isolated from the external environment and its cylindrical symmetry leads to high thermal stability.

In the majority of current commercial instruments, a side-entry design is far more common in which the specimen is attached to a rod that is inserted into the goniometer. This allows the use of EDX and secondary electron detectors providing additional analytical and imaging capability but introduces potential mechanical and thermal instability. However, this is outweighed for many applications by the flexibility of this type of design (for example, in providing specimen biasing, heating and cooling, and gas or even liquid environments) and modern commercial implementations are sufficiently stable for atomic-resolution imaging. In the majority of cases commercial side-entry goniometers provide high-precision computer control of all five axes of specimen movement with stepper motors or donator drives [1.199]. This design aspect has recently been further improved to subnanometer precision in movement using additional piezo controls for translation [1.200]. Recently, the capability of this design has been significantly enhanced through advances in (microelectromechanical systems) technology fabrication by which dedicated cells within which the specimen temperature and environment can be controlled and stimuli can be applied ([1.201]; for a review see [1.202]; see also Chap.  3 in this volume).

It is also worth noting that several designs for fully bakeable UHV HRTEM stages have been proposed. In one example, a side-entry mechanism is employed with a detachable tip that locates within the goniometer allowing the transfer rod to be withdrawn [1.203] and in another a fully bakeable top-entry design for a dedicated corrected UHV STEM has been described [1.204].

1.3.4 Energy Filters

In many applications of HRTEM, energy filtering is of great importance, particularly for improving quantification and interpretability of images by removal of inelastically scattered electrons ([1.205, 1.206, 1.207, 1.208]; see [1.165], for a review). At this point we note that the energy filters outlined here also find widespread application as spectrometers for electron energy-loss spectroscopy ( ) and in energy-filtered imaging at medium resolution [1.165, 1.184, 1.209]. However, this section is restricted to briefly reviewing their use for imaging.

Imaging filters are best differentiated by the nature and geometry of their constituent optical elements and can be classified as purely electric, purely magnetic, or combined electric/magnetic [1.184, 1.210]. To separate electrons according to their energy loss dipole fields are required and hence all filters have a curved optic axis with the notable exception of the Wien filter  [1.211]. For HRTEM purely electric [1.212] and combined electric/magnetic filters have the disadvantage of being limited to operation at relatively low accelerating voltages due to difficulties associated with the large electric field strengths required at higher voltage. However, as discussed later, these have found recent alternative application as monochromators in which the electrons have an energy of only a few electronvolts.

Magnetic filters have been constructed as both straight vision (in column) types [1.213, 1.214] and more commonly in a postcolumn geometry, which can be retrofitted to conventional instruments [1.215].

In-column filters can be classified further as either \(\Upomega\)-geometries in which the deflection angles of the sector magnets cancel to zero and \(\alpha\)-filters in which the sum of the deflection angles equals \(2\uppi\) [1.216]. For routine operation and ease of alignment of in-column filters relatively simple filter geometries are required while any residual aberrations should not appreciably affect the quality of the final HRTEM image. This can be achieved through careful choice of suitable symmetries of the magnetic deflection field and of the paraxial rays about the midplane of the system by which second-order axial aberrations and distortions can be canceled making these designs suitable for energy-filtered HRTEM. On the basis of this principle several commercial filters have been successfully produced [1.213, 1.214]. A fully corrected \(\Upomega\)-geometry filter has also been proposed and constructed, initially using curved polepiece faces [1.217] but subsequently revised to use straight polepiece faces and additional hexapole elements [1.218]. Finally, we note that a corrected in-column filter, with an extremely high dispersion achieved through the use of inhomogeneous magnetic fields created by conical magnets has also been described [1.219].

The alternative postcolumn filter geometry based on a single \(90^{\circ}\) sector magnet with curved polepiece faces has also been produced commercially and has found widespread application [1.215]. For energy-filtered HRTEM this design has the disadvantage of nonzero second-order distortions and chromatic aberration of magnification. These filters therefore operate at a large intermediate magnification so that these do not appreciably affect the image quality. However, because of the large magnifications required the third-order distortions and higher order chromatic aberrations limit the number of clearly resolved object elements. For these reasons, a sequence of quadrupole and hexapole elements is incorporated after the sector magnet and alignment to third order is carried out by automated computer control. A significant practical advantage, however, of this geometry is that it can be retrofitted to any column.

Finally, we also note that investigations into the contribution of phonon scattering to the Stobbs factor [1.207, 1.208] in off-axis electron holograms [1.75] provide compelling evidence for intrinsic elastic energy filtering (within experimental limitations) in this mode. This issue has been rigorously treated [1.220] in a quantum mechanical framework using a global Hamiltonian describing the electron, source, and object to prove that although inelastic interference is possible it is too small to be observed. Hence, electron holography provides de facto perfect energy filtering, which therefore suggests that electron-optical filters may be redundant in this mode.

1.3.5 Detectors

Image recording should not impair the overall resolution of recorded HRTEM data. However, unless the detector used is carefully optimized with respect to microscope operating conditions this component can have a negative effect on overall performance. Historically, photographic plates using one of several possible specialist emulsions were universally used for image recording [1.221, 1.222] but these have now been largely superseded by digital detectors, initially slow-scan charge-coupled device ( ) cameras [1.223, 1.224, 1.225, 1.226] and more recently (complementary metal-oxide semiconductor) sensors ([1.227, 1.228, 1.229, 1.230, 1.231, 1.232]; for a review see [1.233]).

Particular benefits associated with these detectors are that the images are instantly available, in digital form, and the camera response is practically linear over a large dynamic range. Their sensitivity is also higher than that of most photographic emulsions, making single electron detection possible.

Although limited early experiments using CCD chips as direct transmission electron microscopy ( ) electron detectors have been performed [1.234], these were not viable because of the sensitivity of the gate insulator to radiation damage. For low energies (\(<{\mathrm{10}}\,{\mathrm{keV}}\)), this damage can be avoided using back-thinned CCDs [1.235], but higher energy electrons penetrate through to the sensitive gate oxide on the front side. Moreover, a primary electron of energy \(E\) generates \(E/{\mathrm{3.64}}\,{\mathrm{eV}}\) electron–hole pairs in silicon [1.236] leading to saturation of the sensor well capacity after the detection of only a few electrons at the typical primary energies used in HRTEM.

For this reason, indirect detection was initially used for both CCD and CMOS detectors for HRTEM by which the primary electrons impinge on a YAG single crystal (YAG: yttrium-aluminum-garnet \(\mathrm{Y_{3}Al_{5}O_{12}}\), a transparent crystal that is made scintillating by doping with impurity atoms, usually europium (YAG:Eu) or cerium (YAG:Ce)) or phosphor powder scintillator for which a wide range of materials are available. Choices for electron detection include P22 (\(\mathrm{Y_{2}Q_{2}S}\):Eu) and P43 (\(\mathrm{Gd_{2}Q_{2}S}\):Tb). The generated light is relayed to the sensor via a lens- or fiber-optical coupling (Fig. 1.16). However, within this complex coupling, scattering of both the primary incident electrons and the emitted photons in the scintillator material occurs. Together, these processes blur the image, attenuating high spatial frequencies leading to relatively poor modulation transfer functions ( s) and limited detector quantum efficiency ( ) [1.237, 1.238, 1.239].

Fig. 1.16

Schematic diagram of an indirect digital detector used for HRTEM. The incoming high-energy electrons are scattered along their trajectory within the scintillator where they give rise to photon emission. Electrons that penetrate through the scintillator may be backscattered into the scintillator from the fiber plate or a mechanical support layer. A lens- or fiber-optical coupling system conveys the generated light to the sensor where the photons generate electron–hole pairs. The electrons of these pairs are collected in pixelated potential wells. MTF and (noise transfer function) signify modulation and noise transfer functions, respectively and the subsript S refers to these in absence of a pixelated detector. For details see [1.237]

Electron-sensitive imaging plates have also found application as an alternative digital recording media [1.224, 1.240]. These consist of a thin embedded layer (ca. \({\mathrm{40}}\,{\mathrm{\upmu{}m}}\) thick) of a photostimulable phosphor. Luminescence is activated postexposure by a scanning laser in a separate processing system that includes a photomultiplier to convert the output light into a digitized electronic signal. The exposed plate can be subsequently erased by exposure to a suitable light source. These systems provide excellent recording linearity over a wide dynamic range with a sensitivity about three times that of photographic emulsions, but which is voltage dependent. However, their performance also depends strongly on electron dose and drops rapidly at low dose levels. Hence, the most promising application of this technology reported to date is in recording quantitative diffraction information [1.224].

Recently, significant progress has been made in direct electron detection following early experiments by several groups [1.228, 1.229, 1.230, 1.231, 1.232, 1.241]. This had led to a revolution in the detectors used for HRTEM in which direct injection of beam electrons into solid-state CMOS devices can now be achieved leading to substantial increases in both resolution and sensitivity [1.227, 1.233]. This has had particular impact in cryoelectron microscopy and diffraction from radiation-sensitive biological material [1.227, 1.228, 1.229, 1.230, 1.241]. Importantly direct detectors have demonstrated direct electron imaging with performance characteristics exceeding indirect CCD and CMOS sensors. At the time of writing, two distinct sensor geometries have been employed for HRTEM imaging. Hybrid sensors [1.242, 1.243] (Fig. 1.17) consist of small arrays of large (typically several tens of \(\mathrm{\upmu{}m}\)) pixels fabricated in thick Si with the sensitive readout electronics bump bonded to the back of the sensitive array. These have the advantage of being intrinsically radiation hard as the primary electrons do not penetrate the pixel array through to the sensitive readout. The large pixel sizes lead to an MTF that is almost invariant with primary voltage [1.243] and which is close to the theoretical limit at low voltage. The alternative monolithic active pixel sensors ( ) are based on larger arrays of smaller pixels (typically less than \({\mathrm{10}}\,{\mathrm{\upmu{}m}}\)) with the readout electronics and amplification [1.227, 1.232, 1.244, 1.245] integrated at each pixel. For this class of sensor, the Si containing the pixel array must be mechanically backthinned [1.246] to avoid the charge generated by the incident electrons spreading across many pixels, degrading the MTF. However, their performance is often superior to hybrid sensors at high voltage. Importantly, both sensor types can be operated at high frame rates (of several \({\mathrm{1000}}\,{\mathrm{frame{\,}s^{-1}}}\)) which has led to their recent use in the acquisition of four-dimensional ( ) STEM datasets ([1.247, 1.248]; see also Chap.  2 in this volume). Finally, we note that direct electron detectors generally operate in a counting rather than an integrating mode, which significantly reduces readout noise leading to significant improvements in the DQE, a feature which is particularly important for lose dose imaging [1.227, 1.233].

Fig. 1.17

(a) Schematic diagram and (b) photograph of modern high-speed direct, counting electron detector based on the hybrid Medipix sensor. In this form of detector primary electrons are directly detected in a thick sensitive Si substrate which is bump bonded to radiation-sensitive readout electronics

Detector Characterization

All digital detectors can be characterized in terms of three key parameters that, in combination, describe their overall performance. These involve measurement of the gain, resolution, and detector quantum efficiency. The linearity and uniformity of the response is also important, but in currently available CCD and CMOS devices are extremely well controlled and hence not practically limiting.

Detector resolution is described by the MTF, which describes the spatial-frequency-dependent signal transfer, that is affected by both electron and photon scattering within the detection chain. This parameter can be evaluated by integration (over the scintillator area) of the ratio of the output to the input signal in reciprocal space \((u,v)\), normalized to unity at zero spatial frequency [1.237].

More formally
$$\text{MTF}(u,v)=\frac{1}{G}\int g_{u}(u,v)\mathrm{d}\mu\;,$$
where the Fourier transform of the light intensity (number of photons per unit area) collected at a particular position on the detector in its focal plane is given by \(\hat{g}_{\mu}(u,v)\), \(\mathrm{d}{\mu}\) defines the probability of a particular electron hitting the scintillator at this point, and \(G\) is the total gain, i. e., the average number of detectable CCD well electrons per primary electron given by
The inverse Fourier transform of this MTF is the point spread function ( )
The DQE measures the statistical performance of radiation detectors [1.249] generally defined as the quotient of the squared signal-to-noise ratio ( ) at the output and input of the detector (in general as a function of spatial frequency [1.237]).
This can also be expressed in terms of experimentally accessible quantities as
where \(\hat{p}(0,0)\) is the total number of electrons recorded in each exposure, \(V(u,v)\) is the variance, (in units of number of CCD well electrons squared) and \(G\) is the gain defined previously. It should be noted that unless the dose is very low, in which case dose-independent noise sources such as readout noise become important, the observed variance \(V(u,v)\) is proportional to the electron dose and therefore the DQE is dose independent.
The above expression can be conveniently rewritten through a simple rescaling of units into the digital numbers (DN ) that are read out directly from the camera to give the expression
where \(G_{\mathrm{DN}}\) is the gain in digital numbers per incident electron, \(I_{\text{DN}}=\hat{p}(0,0)G_{\mathrm{DN}}\) is the total intensity in digital numbers, and \(V_{\mathrm{DN}}(u,v)\) is the variance in \((\mathrm{DN})^{2}\).

Experimentally these performance characteristics can be readily measured from a series of controlled exposures of a deterministic input signal (such as a knife edge) [1.239] for a range of experimental conditions as shown in Figs. 1.18 and 1.19 for respectively indirect and direct sensors, illustrating the almost ideal performance of the direct detector at low voltage.

Fig. 1.18

(a) MTF measured for \({\mathrm{1024}}\times{\mathrm{1024}}\) (solid line), \({\mathrm{2048}}\times{\mathrm{2048}}\) (dashed line) CCD cameras and \({\mathrm{2048}}\times{\mathrm{2048}}\) with \({\mathrm{2}}\times{\mathrm{2}}\) pixel binning (dotted line). (b) DQE corresponding to (a). In all cases plots extend from \(\mathrm{0}\) spatial frequency to the spatial frequency at the corner of Fourier space. For details see [1.239]

Fig. 1.19

(a) MTF and (b) DQE as a function of spatial frequency, recorded at \({\mathrm{80}}\,{\mathrm{kV}}\) using the Medipix hybrid direct detector at various digital to analog conversion ( ) threshold values (red \(={\mathrm{5.6}}\,{\mathrm{kV}}\), green \(={\mathrm{50.4}}\,{\mathrm{kV}}\), black \(={\mathrm{60.9}}\,{\mathrm{kV}}\)). The theoretical responses are shown as open circles for comparison. (Adapted from [1.243])

1.3.6 Aberration Correctors

One of the most important long-standing goals in electron optics has been the correction of the positive spherical aberration that is present in all round lenses [1.107]. The essential optical elements used in all correctors that have been developed are nonround (multipole) lenses as originally proposed by Scherzer [1.250]. (For a more detailed treatment of aberration correctors see Chap.  13 in this volume.) A range of designs for various correctors have been proposed (for comprehensive reviews see [1.119, 1.122, 1.124, 1.166, 1.180, 1.49]). However, it was only in the late 1990s that these components were developed to a level at which improvements to the highest resolution attainable in uncorrected instruments was demonstrated. For the most part, this has been due to the extreme mechanical precision and electrical stability required in the corrector elements and the need for sophisticated computer control of a complex corrector alignment. However, in the 1990s significant improvements in resolution for both HRTEM [1.251, 1.252] and STEM [1.121] were demonstrated, although for the purpose of this section we will not consider the latter further.

The majority of correctors used in HRTEM are based on a pair of strong electromagnetic hexapole elements together with two pairs of round lenses [1.253] that act to couple the hexapoles and to couple the corrector to the objective. Correction is achieved because of the fact that the primary, nonrotationally symmetric second-order aberrations of the first hexapole (a three-fold astigmatism) are exactly compensated by the second hexapole element. Because of their nonlinear diffraction power, the two hexapoles additionally induce a residual secondary, third-order spherical aberration that is rotationally symmetric [1.254] and proportional to the square of the hexapole strength. This aberration has a negative sign, thus canceling the positive spherical aberration of the objective lens. For HRTEM applications it is essential that the corrector is aplanatic to provide a sufficiently large field of view, which is achieved by matching the coma-free plane of the objective lens to that of the corrector using a round transfer lens doublet. To reduce the azimuthal (anisotropic) component of the off-axial coma the current direction of the first transfer lens doublet is opposite to that in the objective lens. In addition to these primary optical elements commercial correctors [1.251, 1.255] contain a number of additional multipole elements for alignment and correction of any residual parasitic aberrations.

More recently more complex designs based on three hexapole elements have been demonstrated that also enable the correction of higher order axial aberrations and certain low-order off-axial aberrations [1.116, 1.117]. Finally, correctors for both spherical aberration and chromatic defocus and first-order two-fold astigmatism has been successfully constructed at both high voltage [1.118, 1.119] based on designs due to Rose [1.256], that use a complex combination of electromagnetic and electrostatic multipole elements.

All of these are described in considerable detail in several texts (see Chap.  13 in this volume and [1.119, 1.49]) to which the reader is referred for further detail.

As already noted, the main difficulties associated with the practical operation of these complex electron optical systems are the requirements for sophisticated computer control of the various elements and a systematic alignment procedure providing rapid, accurate measurement of the required coefficients of the wave aberration function. In practice, this is achieved using measurements of the two-fold astigmatism, \(A_{1}\), and defocus, \(C_{1}\), from tableaus of diffractograms recorded at known beam tilts (see earlier). These measured values are subsequently used to calculate the aberrations present and the appropriate currents to the optical elements are then applied using reference relationships linking the aberration coefficients to the field strengths in the optical elements under direct computer control.

Aberration-Corrected Imaging Conditions for HRTEM

The optimization of imaging conditions for corrected instruments and the interpretation of the images obtained have also required renewed analysis. It is possible to identify three conditions for HRTEM imaging when spherical aberration is a variable parameter [1.128].

If the spherical aberration of the objective lens is exactly corrected and the defocus is also set to zero, then the phase contrast transfer function equals 0 for all spatial frequencies up to the information limit set by partial coherence, while the amplitude contrast transfer function equals 1. However, the above treatment ignores the effects of higher order aberrations such as \(C_{5}\), which also contribute to high-resolution phase contrast. Under these conditions HRTEM can be carried out under pure amplitude contrast conditions in a mode that is not available in a standard uncorrected TEM [1.128].

Alternatively, phase contrast HRTEM images can be obtained using several conditions in which \(C_{3}\) and \(C_{1}\) are balanced against either \(C_{5}\) or the chromatic aberration coefficient, \(C_{\mathrm{C}}\) [1.257, 1.258].

For the first of these conditions, in the presence of positive \(C_{5}\), restricting the phase shifts due to the wave aberration function to lie between 0 and \(\uppi/2\) gives optimum values for \(C_{1}\) and \(C_{3}\) as
$$\begin{aligned}\displaystyle&\displaystyle C_{1}=1.56\left(C_{5}\lambda^{2}\right)^{\frac{1}{3}}\;,\\ \displaystyle&\displaystyle C_{3}=-2.88\left(C_{5}^{2}\lambda\right)^{\frac{1}{3}}\;,\end{aligned}$$
with a corresponding point resolution, limited by \(C_{5}\), given by
Under circumstances in which \(C_{5}=0\) the choice of \(C_{1}\) reduces to the conventional Scherzer defocus \((C_{1}=-(C_{3}\lambda)^{1/2})\) (for finite positive \(C_{3}\)) with a corresponding point resolution (as defined previously) of
Alternatively, matching the first zero of the phase contrast transfer function to the information limit of the instrument, determined by \(C_{\mathrm{C}}\), and again constraining the phase shifts due to the aberration function to lie between 0 and \(\uppi/2\), yields alternative optimum values of
$$\begin{aligned}\displaystyle&\displaystyle C_{1}=1.7\uppi\Delta\;,\\ \displaystyle&\displaystyle C_{3}=-3.4\frac{(\uppi\Delta)^{2}}{\lambda}\;,\\ \displaystyle&\displaystyle C_{5}=1.3\frac{(\uppi\Delta)^{3}}{\lambda^{2}}\;,\end{aligned}$$
where \(\Delta\) is the focal spread defined in a previous section.
By analogy with the \(C_{5}\) limited condition when \(C_{5}\) is zero the above reduce to [1.258]
$$\begin{aligned}\displaystyle&\displaystyle C_{1}=-\uppi\Delta\;,\\ \displaystyle&\displaystyle C_{3}=\frac{(\uppi\Delta)^{2}}{\lambda}\;.\end{aligned}$$
For either of these conditions when \(C_{5}\) is zero or has a negligible effect at the experimentally achievable resolution, the signs of both \(C_{3}\) and \(C_{1}\) can be inverted, and this has been shown to differentially affect the transfer of the linear and nonlinear components of the image intensity in images of complex oxides, providing higher contrast at the weakly scattering anion sites [1.259].

1.3.7 Monochromators

As already discussed, one of the limiting factors in determining the information limit for HRTEM is the energy spread of the source. In addition to the benefits of a reduced energy spread for HRTEM there are also substantial benefits in EELS and related techniques where a lower energy spread improves the spectral energy resolution.

One possible route to reducing the energy spread is in the use of high brightness sources with intrinsically low energy spreads such as nanometer-sized cold-field emitters [1.260, 1.261] or more ambitiously, ballistic emission sources [1.262], negative electron affinity photocathodes [1.263, 1.264], and metal ion sources [1.265]. However, none of these has currently been manufactured with sufficient brightness and/or stability for routine use in HRTEM.

For this reason, monochromation of the electron beam produced from conventional Schottky or cold field-emission sources using dedicated electron optical components provides an attractive alternative that has now been realized experimentally using several different designs operating between \(\mathrm{100}\) and \({\mathrm{300}}\,{\mathrm{kV}}\) ([1.266, 1.267, 1.268, 1.269, 1.270]; see also Chap.  13 in this volume).

The number of possible monochromator geometries that have been proposed and tested can broadly be divided into those with straight optical axes (typically Wien filters) [1.184, 1.266, 1.269, 1.271] in either single- or double-focusing configurations and those with curved axes (typified by \(\Upomega\)-geometries) [1.266, 1.267, 1.270]. The former requires both electromagnetic and electrostatic elements to satisfy the Wien condition and to achieve the required energy dispersion, whereas the latter can be constructed from purely electromagnetic or electrostatic elements. Further classification of monochromators can be made by considering whether the monochromator and/or the energy-selecting slit are operated at high potential or at ground and whether the monochromator is a single-dispersion or double-dispersion design. Finally, monochromators can be operated in either short-field (deflector) mode, in which the monochromator field length is short compared to the distance to the beam focus, or in long-field mode, in which case the monochromator also acts as a focusing element. Within these various classes there are advantages and disadvantages associated with each configuration that have been reviewed by Mook and Kruit [1.266].

For all of these design possibilities the monochromator must provide sufficient current density for HRTEM at a given energy resolution determined by the energy-selecting slit width. Monochromation inevitably reduces brightness as it filters electrons with unwanted energies and thereby reduces the current. However, the optical design of the monochromator itself also affects the brightness through the aberrations introduced by the filter and effects due to Coulomb interactions within the monochromator [1.184, 1.266, 1.267].

1.4 Exit-Wave Reconstruction

Although HRTEM now routinely achieves atomic resolution, the relationship of the image contrast to the underlying object structure for a single image is complex due to the nature of the scattering and imaging processes, as already outlined. A number of alternative approaches to inversion of the imaging process to recover the specimen exit-plane wavefunction, including holography, first proposed by Gabor [1.272] and more recently demonstrated at atomic resolution [1.273, 1.274, 1.275, 1.72, 1.74, 1.75]. The approach described in this section combines the complementary information available from images taken under different conditions in indirect exit-wave reconstruction, which is directly related to the more general problem of coherent detection [1.79].

As already discussed, the wave aberration function of the objective lens defines a passband of uniformly transmitted spatial frequencies up to the instrument point resolution (at the Scherzer defocus) and beyond this limit the transfer function oscillates. At higher values of defocus this passband extends to higher frequencies due to the form of the spatial coherence envelope function, which suggests, in principle, that the addition of several images recorded at different defoci should enable the deconvolution of the effects of the objective lens transfer function and provide a wider passband extending to the axial information limit. This forms the basis of focal series restoration (reconstruction). A simpler alternative using only a single image has also been proposed [1.276] but suffers from difficulties associated with the positions of the zero crossings in the transfer function that can prevent deconvolution and from amplification of noise.

1.4.1 Theory

The recorded image plane intensity can be written as
where \(\boldsymbol{k}\) and \(\boldsymbol{x}\) are Fourier space and real-space variables rather than \((u,v)\) and \((x,y)\) as in Sect. 1.2. For conditions in which the quadratic term is small then the Fourier transform of the image contrast (the fractional intensity deviation) is given as
The Fourier transforms of the object wave (in the back focal plane of the objective lens) and image waves are related by the previously described phase shifting wave aberration function and thus
Hence in terms of the object wave (in the back focal plane of the objective lens) the Fourier transform of the image contrast is described by
in which the term \(n(\boldsymbol{k})\) represents the observational noise in the image.

The essence of all reconstruction schemes is to find an estimate \(\psi^{\prime}_{\mathrm{d}}(\boldsymbol{k})\) of \(\psi_{\mathrm{d}}(\boldsymbol{k})\) given a set of observed image contrast Fourier transforms \(c(\boldsymbol{k})\) and measurements of the individual \(w(\boldsymbol{k})\).

Given data from several differently aberrated images (obtainable from focal or tilt azimuth dataset geometries as described later), an optimum solution for \(\psi^{\prime}_{\mathrm{d}}(\boldsymbol{k})\) can be defined in various ways [1.277]. In particular, a Wiener filter applied to a series of images [1.278] in the presence of noise gives an optimal analytical estimate of the reconstructed wavefunction, expressed in the form of a weighted superposition of the individual transforms as
in which the restoring filters , \(r_{i}(\boldsymbol{k})\), depend on the \(w_{i}(\boldsymbol{k})\) for the set of images as
$$\begin{aligned}\displaystyle r_{i}(\boldsymbol{k})&\displaystyle=\frac{\Omega(-\boldsymbol{k})w_{i}^{\ast}(\boldsymbol{k})-C^{\ast}(\boldsymbol{k})w_{i}(-\boldsymbol{k})}{\Omega(-\boldsymbol{k})\Omega(\boldsymbol{k})-|C(\boldsymbol{k})|^{2}+v(\boldsymbol{k})}\;,\\ \displaystyle\Omega(\boldsymbol{k})&\displaystyle=\sum_{i}|w_{i}(\boldsymbol{k})|^{2}\;,\\ \displaystyle C(\boldsymbol{k})&\displaystyle=\sum_{i}w_{i}(\boldsymbol{k})w_{i}(-\boldsymbol{k})\;.\end{aligned}$$
From (1.54) above it can be seen that the effect of the restoring filters on a Fourier component transmitted in only a single image is simply to retain it after division by the corresponding transfer function and for Fourier components present in multiple images to average the estimates. For a Fourier component not transferred in any image the value of the filter tends to zero due to the inclusion of the noise-to-object power ratio, \(v(\boldsymbol{k})\). This is the major advantage of the Wiener filter in preventing noise amplification where \(w_{i}(\boldsymbol{k})\) is close to zero. In the final step of the overall reconstruction process the exit-plane wavefunction itself is obtained simply by inverse transformation.

For completeness, we also provide an alternative formulation of through-focal series restoration due to Spence [1.79], which gives useful insight into the behavior of the various terms and clearly illustrates the general power of coherent detection methods.

The image amplitude under the weak phase object approximation for a single image at a defocus, \(\mathrm{C}_{1n}\) is given as
$$ \begin{aligned}\displaystyle\psi_{\mathrm{i}}(\boldsymbol{x},C_{1n})&\displaystyle=1-\mathrm{i}\sigma\phi_{\mathrm{p}}(-\boldsymbol{x})\,\mathfrak{F}\{P(\boldsymbol{k})\exp[\mathrm{i}W(\boldsymbol{k},C_{1n})]\}\\ \displaystyle&\displaystyle=\phi_{\mathrm{p}}(\boldsymbol{x})\otimes\mathfrak{F}\{P(\boldsymbol{k})\exp[\mathrm{i}W(\boldsymbol{k},C_{1n})]\}\;,\end{aligned}$$
with an image intensity
$$\begin{aligned}\displaystyle I(\boldsymbol{x},C_{1n})&\displaystyle=|1+\psi_{\mathrm{i}}(\boldsymbol{x},C_{1n})|^{2}\\ \displaystyle&\displaystyle=|1+\psi_{\mathrm{i}}(\boldsymbol{x},C_{1n})|+\psi_{\mathrm{i}}^{\ast}(\boldsymbol{x},C_{1n})+h\;,\end{aligned}$$
where \(h\) are the higher order, nonlinear terms.

We are now able to extract the wanted second term in (1.56) and discriminate against the conjugate and the higher order terms.

To effect this, we first multiply the transform of the intensity by the conjugate of the transfer function to deconvolve it and then sum over \(N\) images recorded at different defoci.

This gives
$$\begin{aligned}\displaystyle S(\boldsymbol{k})&\displaystyle=\sum_{n}I(\boldsymbol{k},C_{1n})\exp[-\mathrm{i}W(\boldsymbol{k},C_{1n})]\\ \displaystyle&\displaystyle=\boldsymbol{\delta}(\boldsymbol{k})+\phi_{\mathrm{p}}(\boldsymbol{k})\sum 1+\phi_{\mathrm{p}}^{\ast}(-\boldsymbol{k})\\ \displaystyle&\displaystyle\quad\,\times\sum\exp[-2\mathrm{i}\,W(\boldsymbol{k},C_{1n})]\\ \displaystyle&\displaystyle\quad\,+\sum_{n}H_{n}(C_{1n})\exp\left[-\mathrm{i}W(\boldsymbol{k},C_{1n})\right]\;,\end{aligned}$$
where \(\phi(\boldsymbol{k})=1-\mathrm{i}\,\sigma\mathfrak{F}\{\phi_{\mathrm{p}}(\boldsymbol{x})\}\) is the image we wish to restore.

Equation (1.57) now clearly shows the power of coherent detection. The second term sums as \(N\) times the wanted image, whereas the unwanted third and fourth terms sum as \(N^{1/2}\) times their initial values and the final term behaves as a two-dimensional random walk. Overall, the effects of the coefficients of the wave aberration function appear to have been eliminated and the only remaining resolution-limiting effect is the function \(P(\boldsymbol{k})\). We note that the above describes a general approach of integration against a kernel to provide a stationary phase condition for a wanted signal.

A further independent formulation of a linear restoration scheme similar to that described above, known as the paraboloid method, has also been reported [1.279, 1.280, 1.281]. This is based on a three-dimensional () Fourier transform of a through-focal series of images with the third variable conjugate to the defocus. The exit-wave sought is localized on a parabolic shell in this three-dimensional space and can readily be separated from the conjugate wave and the nonlinear terms. This approach has the attraction of providing intuitive insight into how the information from a series of images is gathered to yield the complex exit-wave, although unlike the Wiener filter it is nonoptimal in its suppression of the conjugate wave and experimental noise [1.147].

The approaches to exit-plane wavefunction restoration described above assume linear imaging. An alternative method has also been developed for a more general case, including the nonlinear contributions to the image intensity. In the original implementation of this latter approach [1.282, 1.283, 1.284] the nonlinear image reconstruction is accomplished by matching the intensities calculated from the restored wave to the measured intensities of images in a focal series through minimization of a least-squares functional (the multiple input maximum a posteriori (MIMAP )). A more recent improved maximum likelihood ( ) description [1.280, 1.285, 1.286] provides a computationally efficient and numerically optimized recursive solution and explicitly includes the coupling between the exit wave and its complex conjugate. More recently, focal series reconstruction has been extended to use aberration-corrected image intensities [1.287].

Finally, it should be noted that several alternative reconstruction schemes using various combinations of axial images and holograms have more recently been reported offering advantages in certain areas over those described above [1.288, 1.289, 1.290].

An alternative imaging geometry for use in exit wave reconstruction uses a dataset recorded for several different illumination tilts (of the same magnitude but with different azimuths) [1.291, 1.292] in a super-resolution scheme also referred to as aperture synthesis. The basis of this approach relies on the fact that for an incident beam tilt of a Bragg angle \(\theta_{\mathrm{b}}\) the optic axis bisects the angle \(\theta=2\theta_{\mathrm{b}}\) between the incident beam and a first-order Bragg beam. In this geometry, the two beams define the diameter of an achromatic circle (for positive \(C_{3}\)) on which all even-order terms on the wave aberration function cancel. However, only one sector of the diffraction pattern contributes to each image and it is therefore necessary to record images at several (minimally four) different tilt azimuths. The method is also restricted to relatively thin samples due to the effects of parallax [1.293, 1.294, 1.295] which becomes more severe at higher tilt angles. Unwanted phase shifts across the synthetic aperture arising from variations in the axial aberration coefficients with tilt and the effects of uncorrected chromatic aberration also potentially limit the resolution extension that can be achieved [1.293, 1.294]. The restoring filters required are of a form similar to those used for focal series reconstruction but with the transfer functions modified so as to be appropriate to tilted illumination [1.277]. The advantage of this geometry is in improved flat information transfer beyond the axial limit, albeit in only one direction in a single image, with the disadvantage of poorer low-frequency transfer in the restored wave. In principal spatial frequencies up to twice the conventional Scherzer resolution limit can be recovered from a dataset composed of four tilted illumination images recorded with orthogonal tilt directions [1.291]. Tilt series restoration has also been applied to aberration-corrected data [1.294, 1.295] demonstrating resolution improvements over corrected axial imaging.

It is also worth noting that although the reconstructed wavefunction is free of artifacts due to the lens aberrations, there still exists a significant discrepancy between absolute values of the experimental and simulated exit wavefunctions [1.296]. The cause of this discrepancy (frequently known as the Stobbs factor  [1.207, 1.208]), which is also observed in comparisons between conventional experimental HRTEM images and simulations remains an active research area and holographic data [1.297] suggest that phonon scattering may make a substantial contribution. Work from the Jülich group has shed further light on this problem with an analysis of HRTEM imaging on an absolute contrast scale [1.298].

1.4.2 Experimental Geometries

The most widely applied procedure for exit-wave reconstruction is, as already described, to record several (or many) images at different focus levels for which the frequencies of weak transfer are different. This provides almost continuous transfer in the recovered exit-wave extending to the ultimate limits set by the effects of partial coherence (Fig. 1.20a-e). In practice, the dataset for a focal series reconstruction is relatively easily obtained and typically some \(\mathrm{20}\) or more images separated by close focal increments can conveniently be recorded on a digital detector utilizing external computer control of the microscope. The subsequent processing involves image registration across the series (to account for specimen drift) and the determination of the individual imaging conditions (using one of the methods described earlier). It should be noted that image registration is challenging at the resolutions now available and in general modified correlation functions and nonrigid registration schemes are required [1.299].

Fig. 1.20a-e

A comparison of transfer functions . (a) Conventional phase contrast transfer function (PCTF) for a single Scherzer defocus image. (b) Effective wave transfer function ( ) for a \(\mathrm{21}\)-member focal series with a defocus step between images of \({\mathrm{5}}\,{\mathrm{nm}}\). (c) Effective wave transfer function for a \(\mathrm{6}\)-member tilt azimuth series with a tilt of \({\mathrm{7.7}}\,{\mathrm{mrad}}\). (d) Two-dimensional representation of the effective wave transfer function in (b). (e) Two-dimensional representation of the effective wave transfer function in (c). In all cases electron optical parameters appropriate to an uncorrected \({\mathrm{300}}\,{\mathrm{kV}}\) FEGTEM were used (\({\mathrm{300}}\,{\mathrm{kV}}\), \(C_{\mathrm{s}}={\mathrm{0.57}}\,{\mathrm{nm}}\), focal spread \(={\mathrm{4}}\,{\mathrm{nm}}\), beam divergence \(={\mathrm{0.1}}\,{\mathrm{mrad}}\)). (a) Scaled \(-1{-}1\); (b,c) scaled \(0{-}1\); (d,e) scaled \(0{-}1\) (black to white)

The basic methodology for recording data for use in tilt series reconstruction is the same as that for focal series but particular attention must be paid to the image registration (because of the distorted form of cross-correlation functions involving tilted illumination images) and to the initial defocus condition. In general, for this geometry optimal transfer is achieved in the tilted illumination mode (with positive \(C_{3}\)) when \(C_{1}=\theta^{2}\) with \(C_{1}\) and \(\theta\) being the defocus and illumination tilt measured in reduced units of \((\sqrt{(C_{3}\lambda)})\) and \((\lambda/C_{3})^{1/4}\), respectively [1.300] (Fig. 1.20a-e). With the development of aberration-corrected instruments, this method is simplified by the removal of these restrictions in the absence of \(C_{3}\). In practice, a combined defocus/tilt geometry provides optimal transfer and has the additional benefit of providing the required data for the determination of the aberration coefficients (see earlier) and for image registration.

1.4.3 Example Exit-Wave Restorations of Complex Oxides

To illustrate the benefits of exit-wave restoration over conventional HRTEM imaging we have chosen two examples of complex oxides. Figure 1.21a shows a structural model of \(\mathrm{Nb_{16}W_{18}O_{94}}\) projected along [001] together with an axial image recorded at the Scherzer defocus (Fig. 1.21b) and the restored exit-plane wavefunction (Fig. 1.21c,d) recovered to a resolution of ca. \({\mathrm{0.11}}\,{\mathrm{nm}}\). It is apparent that although the basic cation lattice can be determined from the axial image the restored modulus shows the positions of the cation columns in projection at substantially higher resolution. Moreover, the modulus remains directly interpretable to a greater specimen thickness than the axial image. The restored phase shows a more complex contrast changing rapidly with specimen thickness. In addition to the strong positive contrast (white, corresponding to a phase advance) located at the cation sites and corresponding directly to the positions of strong negative (black) contrast in the modulus there is additional weak contrast at positions between the cations at the anion sublattice sites.

Fig. 1.21

(a) Structural model of the complex oxide \(\mathrm{Nb_{16}W_{18}O_{94}}\) projected along [001]. (b) Conventional axial HRTEM image recorded at the Scherzer defocus of a thin crystal. (c) Reconstructed modulus of the exit-plane wavefunction of \(\mathrm{Nb_{16}W_{18}O_{94}}\) with the marked area enlarged (inset), which directly shows the cation positions (black) with improved resolution compared to the axial image. The line indicates a stacking fault with a shift of a third of a unit cell along [010]. (d) Reconstructed phase of the exit-plane wavefunction with the marked area enlarged (inset). The cation sites in the phase are recovered with positive (white) contrast and additional weak between the cation atomic columns which indicate the positions of the oxygen anions are also resolved. The reconstructed phase and modulus are shown at the same scale

Our second example illustrates an experimental tilt azimuth series restoration of an inorganic perovskite with \({\mathrm{0.1}}\,{\mathrm{nm}}\) information transfer.

The basic perovskite structure is cubic with general formula \(\mathrm{ABO_{3}}\) and can be considered as an array of corner-sharing \(\mathrm{BO_{6}}\) octahedra where B is typically a small transition-metal cation (Fig. 1.22a-c). The cubeoctahedral interstices generated by this basic lattice are occupied by larger A cations, typically alkali earth or rare earth metals. Few perovskites are cubic and most distort to a lower symmetry to stabilize the structure.

Fig. 1.22a-c

Simplified structural models of \(\mathrm{A_{\mathit{n}}B_{\mathit{n}}O_{3\mathit{n}+2}}\) compounds represented by corner-sharing \(\mathrm{BO_{6}}\) octahedra and isolated A cations. Octahedra and cations drawn with light gray shading are half a lattice plane below those with darker shading. (a) The basic perovskite \(\mathrm{ABO_{3}}\). (b,c) The \(n={\mathrm{5}}\) structure \(\mathrm{A_{5}B_{5}O_{17}}\) with layers five octahedra wide, projected in the [100] and [010] directions

The layered structure described here (of general formula \(\mathrm{A_{\mathit{n}}B_{\mathit{n}}O_{3\mathit{n}+2}}\)) is composed of slabs of perovskite sliced along the [110] plane with the two excess oxygen anions accommodated at the interface region between two slabs (Fig. 1.22a-c). These structures show similar distortions to bulk perovskites but with additional degrees of freedom at the interface [1.301, 1.302].

The restored exit-plane wavefunction for \(\mathrm{Nd_{5}Ti_{5}O_{17}}\) in the [010] projection is shown in Fig. 1.23. The restoration again shows improved resolution in both modulus and phase compared to the conventional axial image and an enhanced sensitivity to the weakly scattering oxygen sublattice in the restored phase. Comparison with the structural model shown in Fig. 1.22a-c shows that the Ti are bridged by O anions and correspondingly in the restored phase, the Nd rows show distinct cations whereas the Ti rows also show weak contrast between the cation sites (Fig. 1.23) corresponding to the anion positions. The reconstruction also reveals local distortions present in this material by which the Nd cations at the outside of the perovskite slabs are displaced by small amounts in alternate directions (Fig. 1.23) [1.302].

Fig. 1.23

(a) Enlarged region taken from the reconstructed modulus calculated from a tilt azimuth dataset of an \(\mathrm{Nd_{4}SrTi_{5}O_{17}}\) crystal edge in the [010] projection showing a small difference in positions of the Nd(4) and Nd(5) cations at the interface between adjacent perovskite slabs. (b) Reconstructed phase of the same region as (a) showing details of the oxygen anion sublattice between the Ti sites. The experimental data were recorded using the tilt series geometry described in the text with a JEOL JEM-3000F FEGTEM, \({\mathrm{300}}\,{\mathrm{kV}}\), \(C_{3}={\mathrm{0.57}}\,{\mathrm{mm}}\) with an injected tilt of \({\mathrm{1.9}}\,{\mathrm{mrad}}\)

1.5 HRTEM Image Simulation

A key step in the process of obtaining quantitative structural information from HRTEM images is the calculation of image simulations based on defined structural models and imaging conditions for comparison with experimental images. As already discussed, simple models for electron scattering and imaging at high resolution suffer from limitations and in general, computationally tractable N-beam dynamic calculations are essential to this process.

For this purpose, the multislice method is most commonly used to compute the electron wave at the exit-plane of a specimen with known atomic structure. This approach was first suggested by Cowley and Moodie [1.303] and has found extensive use in HRTEM simulation [1.90]. In particular, the availability of efficient fast Fourier transform ( ) algorithms and the general increase in readily available computing power have made earlier constraints on its accuracy due to limitations in the number of slices or the number of diffracted beams immaterial.

An alternative to the multislice method is the Bloch wave approach, first introduced by Bethe [1.304] and described in detail, for instance, in [1.31]. By analogy with the Bloch theorem in solid-state physics, the solution of the Schrödinger equation in a periodic crystal potential is written as a product of a plane wave and a function that has the same periodicity as the crystal. The latter function is then expanded into its Fourier components from which the Schrödinger equation reduces to a matrix equation in these Fourier coefficients. For simple crystals, relatively accurate solutions can be obtained using only a few of these Bloch waves, with the minimal case requiring only two (the two-beam approximation). This approach has the advantage of providing valuable insight into the working of dynamic electron diffraction and explains phenomena such as thickness fringes. However, for complex crystals with large unit cells, the method becomes impractical, as a large number of beams has to be used in the calculation and the computation time for the matrix solution scales with \(N^{3}\), where \(N\) is the number of beams included. For this reason, it has only rarely been applied to calculations of HRTEM images.

Finally, we note that for the majority of cases matching of experimental and simulated images is carried out purely visually with limited attempts to define quantitative figures of merit ( ) describing the differences between experimental and simulated images (or restored exit wavefunctions). In part, this is because of the large number of parameters involved in HRTEM simulations (including atomic coordinates and those describing imaging conditions) giving rise to a large multiparameter optimization problem with many local minima. However, a number of possible FOMs have been reported [1.305, 1.306, 1.307, 1.308] based on comparisons in both image and Fourier space.

1.5.1 The Multislice Formalism

The formal basis of the multislice method (for an extensive treatment see [1.102]) is the division of the specimen into a number of thin slices perpendicular to the direction of the incident beam. The effects of the specimen potential (transmission, in real space) and of Fresnel diffraction (propagation, in Fourier space) are then treated separately for each slice (Fig. 1.24). The specimen potential is itself calculated from electron structure factors using one of several possible approximations [1.102, 1.309]. It is a requirement that the individual slices used in the simulation must be thin enough to be weak phase objects and that they obey periodic boundary conditions perpendicular to the incident beam direction.

Fig. 1.24

(a) Schematic diagram showing the real-space transmission and Fourier space propagation steps. (b) Flow chart illustrating the steps involved in a multislice simulation

The equations central to the multislice algorithm can be formally derived starting from the Schrödinger equation for the wavefunction \(\psi_{\mathrm{f}}\), of an electron in the electrostatic potential \(V(x,y,z)\) of the specimen
where \(m\) is the relativistically corrected electron mass. For high-energy electrons, the motion is predominantly in the \(z\)-direction. Hence, it is convenient to separate the full wavefunction, \(\psi_{\mathrm{f}}\), into a product of the solution of the free Schrodinger equation (with \((V\equiv 0)\), which is a plane wave propagating in the \(z\)-direction, and a wavefunction \(\psi\) that represents the effects due to the specimen and varies much more slowly with \(z\)
$$\psi_{\mathrm{f}}(x,y,z)=\mathrm{e}^{2\uppi kz}\,\psi(x,y,z)\;,$$
where \(k=1/\lambda\) is the inverse wavelength of the free electron. Substituting this into (1.58) and using \(E=h^{2}k^{2}/(2m)\) yields. (It should be noted that \(k\) is defined here as \(1/\lambda\) rather than \(2\uppi/\lambda\) and therefore \(\hbar\) rather than \(h\) is used in the expression for \(E\).)
$$\begin{aligned}\displaystyle&\displaystyle-\frac{\hbar^{2}}{2m}\left[\nabla_{xy}^{2}+\frac{\partial^{2}}{\partial z^{2}}+4\uppi k\frac{\partial^{2}}{\partial{z}}+\frac{2meV(x,y,z)}{\hbar^{2}}\right]\\ \displaystyle&\displaystyle\quad\times\psi(x,y,z)=0\;.\end{aligned}$$
In an elastic scattering process, away from the \(z\)-direction, \(k_{{x}}\) and \(k_{{y}}\) are proportional to the scattering angle, while the change, \(\Updelta k_{{z}}\), is proportional to its square. For small scattering angles, therefore the term
$$\left|\frac{\partial^{2}\psi}{\partial z^{2}}\right|\ll\left|{\nabla^{2}}_{xy}\psi\right|$$
can be neglected in the paraxial approximation.
What remains is a first-order differential equation in \(z\)
$$\frac{\partial\psi(x,y,z)}{\partial z}=[A+B]\psi(x,y,z)\;,$$
with the operators
$$A =\frac{\mathrm{i}\lambda}{4\uppi}\nabla_{xy}^{2}$$
$$B =\mathrm{i}\sigma V(x,y,z)\;,$$
and the interaction parameter \(\sigma\) defined here as
$$\sigma=\frac{2\uppi me\lambda}{h^{2}}\;.$$
This differential equation has the formal solution
$$\begin{aligned}\displaystyle&\displaystyle\psi(x,y,z+\Updelta z)\\ \displaystyle&\displaystyle\quad=\exp\left\{\int_{z}^{z+\Updelta z}[A(z^{\prime})+B(z^{\prime})]\mathrm{d}z^{\prime}\right\}\psi(x,y,z)\;,\end{aligned}$$
and when \(\Updelta z\) is small, this reduces to
$$\begin{aligned}\displaystyle&\displaystyle\psi(x,y,z+\Updelta z)\\ \displaystyle&\displaystyle\quad=\exp\left[\frac{\mathrm{i}\lambda}{4\uppi}\Delta z\nabla_{xy}^{2}+\mathrm{i}\,\sigma V_{\Updelta z}(x,y,z)\right]\psi(x,y,z)\;,\end{aligned}$$
where \(V_{\Updelta z}\) is the projected specimen potential between \(z\) and \(z+\Updelta z\)
$$V_{\Updelta z}(x,y,z)=\int_{z}^{z+\Updelta z}V(x,y,z^{\prime})\mathrm{d}z^{\prime}\;.$$
The two operators in the exponent in (1.66) do not commute and therefore this function cannot be rewritten as a product of two exponential functions. However, as both operators are small (to the order of \(\Updelta z\)), the approximation
$$\begin{aligned}\displaystyle\mathrm{e}^{\varepsilon A+\varepsilon B}&\displaystyle=\mathrm{e}^{\varepsilon A}\mathrm{e}^{\varepsilon B}+\frac{\varepsilon^{3}}{2}[B,A]+O\left(\varepsilon^{3}\right)\\ \displaystyle&\displaystyle=\mathrm{e}^{\varepsilon A}\mathrm{e}^{\varepsilon B}+O\left(\varepsilon^{3}\right)\;,\end{aligned}$$
can be used.
This leaves the expression
$$\begin{aligned}\displaystyle\psi(x,y,z&\displaystyle+\Updelta z)\\ \displaystyle&\displaystyle=\exp\left(\frac{\mathrm{i}\lambda\Updelta z}{4\uppi}\nabla_{xy}^{2}\right)t(x,y,z)\psi(x,y,z)\;,\end{aligned}$$
where \(t(x,y,z)\) is the transmission function for the specimen slice between \(z\) and \(z+\Updelta z\)
$$t(x,y,z)+\exp[\mathrm{i}\sigma V_{\Updelta z}(x,y,z)]\;.$$
The first operator can be further separated into components for \(x\) and \(y\) coordinates as
$$\exp\left(\frac{\mathrm{i}\lambda\Updelta z}{4\uppi}\nabla_{xy}^{2}\right)=\exp\left(\mathrm{i}\alpha\frac{\partial^{2}}{\partial x^{2}}\right)\exp\left(\mathrm{i}\alpha\frac{\partial^{2}}{\partial y^{2}}\right),$$
where \(\alpha=\lambda\Updelta z/(4\uppi)\).
Considering the Fourier transform of the operator for the \(x\)-direction applied to a function \(f(x)\)
$$ \begin{aligned}\displaystyle&\displaystyle\mathfrak{F}\left\{\exp\left(\mathrm{i}\alpha\frac{\partial^{2}}{\partial x^{2}}\right)f(x)\right\}\\ \displaystyle&\displaystyle=\int\mathrm{e}^{-2\uppi kx}\exp\left(\mathrm{i}\alpha\frac{\partial^{2}}{\partial x^{2}}\right)f(x)\mathrm{d}x\end{aligned}$$
$$ =\sum\frac{(\mathrm{i}\alpha)^{n}}{n!}\int\mathrm{e}^{-2\uppi kx}f^{{(2n)}}(x)\mathrm{d}x$$
$$ =\sum\frac{(\mathrm{i}\alpha)^{n}}{n!}(-2\uppi k)^{{2n}}\int\mathrm{e}^{-2\uppi kx}f(x)\mathrm{d}x$$
$$ =\mathrm{e}^{-4\mathrm{i}\,\uppi^{2}k^{2}\alpha}\mathfrak{F}\{f(x)\}\;.$$
The above can be repeated for the \(y\)-direction, finally yielding
$$\begin{aligned}\displaystyle&\displaystyle\mathfrak{F}\left\{\exp\left(\frac{\mathrm{i}\lambda\Delta z}{4\uppi}\nabla_{xy}^{2}\right)f(x,y)\right\}\\ \displaystyle&\displaystyle\quad=\exp\left[-\mathrm{i}\,\uppi\lambda\Delta z\left(k_{x}^{2}+k_{y}^{2}\right)\right]\mathfrak{F}\{f(x,y)\}\;.\end{aligned}$$
Using this result a single multislice step can be written as
$$\begin{aligned}\displaystyle\psi(x,y,z&\displaystyle+\Updelta z)\\ \displaystyle&\displaystyle=\mathfrak{F}^{-1}\{P(k_{x},k_{y})\}\mathfrak{F}\{[t(x,y,z)\psi(x,y,z)]\}\;,\end{aligned}$$
where the real-space transmission function, \(t(x,y,z)\), and the Fourier space propagation function, \(P(k_{x},k_{y})\), are given by
$$t(x,y,z) =\exp[\mathrm{i}\sigma V_{\Updelta z}(X,Y,Z)]\;,$$
$$p(k_{x}k_{y}) =\exp\left[-\mathrm{i}\uppi\lambda\Updelta z\left(k_{x}^{2}+k_{y}^{2}\right)\right].$$
It should be noted that the propagator has the same form as a wave aberration with a defocus of \(\Updelta z\).

The algorithm can be implemented very efficiently using FFTs and is particularly suitable for simulations of crystals, as only one unit cell has to be calculated and the periodic boundary conditions necessary for the Fourier transforms are automatically fulfilled. The density of sampling points in the unit cell determines the maximum spatial frequency in the Fourier transform and therefore the maximum number of diffracted beams included in the simulation. To avoid aliasing artifacts, it is also necessary to exclude all beams above a limit less than the Nyquist frequency in the propagation step [1.102].

Although the multislice method provides an accurate and computationally tractable method for calculating HRTEM images the approximations made in the above derivation can, under certain conditions, limit the accuracy achieved.

The number of beams included in the simulation is ultimately limited by the discrete sampling of the unit cell. Unlike the Bloch wave approach [1.304], which gives self-consistent results even with a relatively small number of beams, the multislice method is reliable only when scattering into beams that are not included in the simulation is negligible. For any given simulation, this latter effect can be tested by verifying that the total intensity of the wavefunction does not decrease significantly as the wave propagates through the specimen. A decrease of \({\mathrm{5}}\%\) over the complete specimen is considered acceptable [1.102], but with modern computers it is possible to choose a sufficiently large number of beams to give a loss of intensity smaller than \({\mathrm{0.1}}\%\).

The paraxial approximation implies the neglect of the term
$$\frac{\mathrm{i}\lambda}{4\uppi}\frac{\partial^{2}\psi}{\partial z^{2}}\;.$$
A wave scattered by an angle, \(\theta\), toward the \(x\)-direction can be written as
$$\psi=\mathrm{e}^{2\uppi\mathrm{k}[\sin\theta x+(\cos\theta-1)z]}\;,$$
with \(k=1/\lambda\), hence
$$\left|\frac{\partial^{2}\psi}{\partial z^{2}}\right|<\uppi^{2}k^{2}\theta^{4}_{\text{max}}\;.$$
Therefore, the error due to the paraxial approximation accumulated over a specimen thickness, \(t\), can be estimated as
$$\Updelta\psi<\frac{\lambda\mathrm{t}}{4\uppi}\left|\frac{\partial^{2}\psi}{\partial z^{2}}\right|_{\text{max}}<\frac{\uppi t}{4\lambda}\theta^{4}_{\text{max}}\;.$$
This term is vanishingly small for typical situations in HRTEM (e. g., \(E={\mathrm{300}}\,{\mathrm{keV}}\), \(\lambda={\mathrm{2}}\,{\mathrm{pm}}\), \(t={\mathrm{20}}\,{\mathrm{nm}}\), \(\theta_{\text{max}}={\mathrm{20}}\,{\mathrm{mrad}}\) yields \(\Updelta\psi={\mathrm{0.16}}\%\)) and therefore any error introduced by the paraxial approximation can generally be neglected.

In principle, the error due to the paraxial approximation can be made arbitrarily small by making the slices sufficiently thin. However, as the potential of each atom is usually projected into the slice that contains the atom center, the accuracy is still limited when propagation over the range of the atomic potential has a noticeable effect. In this case, the atomic potential has to be divided across several slices.

A useful feature of the exit-plane wavefunction calculated using the multislice method is that it depends qualitatively on the local potential of the crystal within a small cylinder whose axis is aligned to the incident beam direction with a diameter of typically a few nanometers for HRTEM imaging conditions. This suggests that HRTEM images of defects within a crystalline supercell can be simulated without perfectly smooth periodic continuation of the crystal potential at the supercell boundaries, as any abrupt changes in the potential will influence the exit-plane wavefunction only laterally, within the radius of the above cylinder. Thus, simulated images of large scale defects can be separately computed in a montage of patches and subsequently joined provided the small allowance described is made at the borders for discontinuities in the potential. Significantly this is not the case for simulations of dynamic electron diffraction patterns from equivalent structures and these cannot be calculated using this patching approach.

Finally, we observe that the derivation of the multislice algorithm described here is based on the Schrödinger equation, whereas a fully relativistic treatment would require the use of the Dirac equation. However, it has been shown [1.310, 1.311] that provided relativistically correct expressions are used for both electron mass and wavelength, the expressions derived from the Schrodinger equation are an extremely good approximation of those derived from the Dirac equation.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Dept. of MaterialsUniversity of OxfordOxfordUK
  2. 2.Eyring Materials CenterArizona State UniversityTempe, AZUSA

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