Introduction
Strain is a key parameter for understanding many physical phenomena at the nanoscale. The mechanical and electronic properties of a material are directly related to the strain in the material, and the response of a material to an applied strain is fundamental to the engineering of mechanical or electronic properties. Strain fields in materials have always been one of the main contrast mechanisms in a transmission electron microscope (TEM), generating contrast by altering the diffracting conditions around a defect, for example. In recent years, TEM methods have been used and developed specifically to study strain fields, either to measure strain directly or to apply strains and observe the evolution of materials in situ. This article will review the current state of the art in this field. The first part will concentrate on the measurement of strain from imaging and diffraction, and the second on the quantitative measurement of strain during in situ experiments.
The concept of elastic strain engineering for electronic properties came originally from the field of microelectronics, and naturally many of the examples will be taken from this area. By straining silicon, the mobility of carriers can be increased significantly. Engineering strain in the active region of transistors has been, and continues to be, essential in helping the industry upgrade performance year after year. Different methods have been employed to inject strain into the channel region of devices, including recessed sources and drains of alloys of silicon and stress liners.Reference Thomson, Armstrong, Auth, Alavi and Buehler1 The complex geometry of devices also leads to non-uniform strain distributions. It is therefore important to be able to measure the strain at a very local level and ideally to be able to map the strain over the whole device.
It is also important to understand that the physical phenomena of interest, such as stress, piezoresistivity, and piezoelectricity, are all related by tensor relations, which means that there is a complex interplay between the components in the different directions of the crystal.Reference Nye2 Stress, for example, does not depend uniquely on the strain in one particular direction:
$$\sigma _{ij} = c_{ijkl} \varepsilon _{kl}$$,where σij are the stress components, c ijkl are the constants of elasticity, and εkl are the different strain components. To determine stress from strain, it is therefore necessary to measure all the strain components. Naturally, the most significant contribution for the σxx stress component is εxx, but not uniquely so. The same goes for mobility and the piezoelectric effect; even mobility is affected by shear components.
A distinction needs to be made between strain, which has a mechanical sense in Equation 1, and deformation. Mechanical strain is defined with respect to the relaxed, unstrained state of the material. TEM techniques, on the other hand, measure the change in local lattice parameter and orientation of the lattice planes with respect to some reference material, usually the substrate or an undeformed part of the material. A deformed sample can have both regions of elastic (recoverable) strain as well as inelastic strain. TEM contrast also arises from a local change in the chemical composition. Without knowing the local composition, it is therefore not possible to determine the local mechanical strain. This also motivates in situ experiments, where the sample can be loaded within the microscope, thus providing knowledge of both the undeformed and deformed states.
Strain measurements by transmission electron microscopy
Four main TEM techniques have been developed to measure strain and will be described in this article: convergent-beam electron diffraction (CBED),Reference Jones, Rackham and Steeds3,Reference Armigliato, Balboni, Carnevale, Pavia, Piccolo, Frabboni, Benedetti and Cullis4 nano-beam electron diffraction (NBED),Reference Usuda, Numata, Irisawa, Hirashita and Takagi5,Reference Uesugi, Hokazono and Takeno6 high-resolution TEM (HRTEM),Reference Bierwolf, Hohenstein, Phillipp, Brandt, Crook and Ploog7,Reference Hÿtch, Snoeck and Kilaas8 and dark-field electron holography (DFEH).Reference Hÿtch, Snoeck, Houdellier and Hüe9,Reference Hÿtch, Houdellier, Hüe and Snoeck10 In addition, in situ mechanical testing techniques have been developed to measure both applied and local strains. They can be diffraction or image based. A change in the lattice parameter will change the diffraction angle. At their most basic level, diffraction techniques, such as NBED, are based on measuring the displacement of the diffraction spots. Imaging techniques rely on quantifying the displacement of atomic columns from micrographs obtained via HRTEM or high-angle annular dark-field scanning TEM (HAADF-STEM), or by directly mapping the distortion of a sample with digital image correlation (DIC). These are then related to strain by taking the spatial derivatives. Diffraction is therefore a direct measurement of strain and imaging techniques of displacements.
There also exist hybrid techniques that are a mix of diffraction and imaging based on imaging a particular diffracted beam in dark-field mode, either to map its intensity, as in conventional TEM, or its phase through DFEH. Unlike the intensity of a diffracted beam, the phase can be related directly to the strain, or more precisely to the displacement field, through the following relation:
$${\rm{\varphi }}_g^G = - 2{\rm{\pi }}{\bf{g}}.{\bf{u}}$$, where 
${\rm{\varphi }}_g^G$ is the so-called geometric phase of the diffracted beam, g is the corresponding reciprocal lattice vector, and u is the local displacement vector.Reference Hÿtch, Houdellier, Hüe and Snoeck11
It should be noted that displacements can only be measured in the direction of the reciprocal lattice vector of the diffracted beam in question. This is general to all TEM techniques, diffraction and imaging, and excludes measurement of strain in the direction of propagation of the beam except for CBED, which makes use of diffraction to high-order Laue zones (HOLZ).
Specimen preparation and thin-film relaxation
A common feature of TEM techniques is that the specimen needs to be thinned to become sufficiently transparent to electrons, typically to the order of 100 nm for silicon. This means that TEM specimen preparation techniques are necessarily destructive, unless the original object was of these dimensions (such as a nanocrystal). Second, the sample studied will not be in the same strain-state as the original “bulk” material. The two new surfaces of the thin lamella allow for relaxation of stress and are commonly referred to as the “thin-film effect.”Reference Gibson and Treacy12 In addition, the method of specimen thinning itself (ion-beam or mechanical polishing) may also modify strain in the sample. Every effort is generally made to reduce these phenomena, but the thin-film effect is inherent and cannot be avoided. In general, imaging techniques, which can use thicker specimens, are preferred, and measurements need to be compared with modeling to evaluate the corrections to be applied to the measured data.
For studying devices, focused ion beam (FIB) is the required technique for specimen preparation because of its site specificity. It also has the advantage of providing samples of uniform (and specified) thickness. However, obtaining thicknesses under 100 nm through FIB preparation is demanding, and care needs to be taken to protect surfaces and limit the amorphous surface layers caused by ion milling. Specimen geometry considerations for in situ tests will be described later.
The projection problem, dynamic scattering, and imaging aberrations
The position of the diffracted spot in a diffraction pattern can be related directly by the Bragg law to the local spacing of the atomic lattice, which is simplistic. Similarly, directly relating the lattice fringes in a HRTEM image to the position of the atomic lattice is problematic.
First, there is the projection problem: in TEM, the fast electron travels through a three-dimensional sample to produce two-dimensional information. For example, specimens are never uniformly strained along the viewing direction. This is a direct result of the thin-film effect mentioned previously. Therefore, the strain measured at a particular point on the specimen, viewed in projection, is some average over the thickness of the sample.Reference Lubk, Javon, Cherkashin, Reboh, Gatel and Hÿtch13 If electron scattering was kinematical, the result would be the linear average, but electron scattering is highly dynamical, and multiple scattering events occur. Effects are particularly pronounced at interfaces between materials where strain gradients are high.
Second, electron diffraction does not occur only at the Bragg angle due to the thinness of the specimens or through dynamical scattering.Reference Hirsch, Howie, Nicholson, Pashley and Whelan14 A certain spread of scattering angles, therefore, is always present. Again, the average scattering angle does not always coincide exactly with the reciprocal lattice vector of the atomic lattice that we wish to measure.
Finally, the strain in the samples is never uniform across the field of view. A measurement is always a local average spatially, and not always a simple one. The most striking case is for HRTEM imaging.Reference Spence15 Lenses are not perfect, even in the age of aberration correction. There are therefore slight shifts between the image of the atomic lattice and its actual position.Reference Hÿtch and Plamann16 Indeed, the position of neighboring atomic columns can even influence the apparent position of the atomic column of interest.
Precision, accuracy, and spatial resolution
The precision and accuracy of strain measurements depend on their spatial resolution (Table I). In general, the higher the spatial resolution, the less accurate the technique. Measuring the strain, atom column by atom column at an interface, is not the same as measuring strain broadly over a few nanometers in a more uniform region of the specimen. High strain gradients mean lower precision and accuracy. Similarly, the thickness of the sample should be as uniform as possible. It all comes down to the required level of accuracy.
Comparison of transmission electron microscopy strain measurement techniques.
CBED, convergent-beam electron diffraction; NBED, nano-beam electron diffraction; HR(S)TEM, high-resolution (scanning) TEM; DFEH, dark-field electron holography.
Diffraction-based TEM methods
Convergent-beam electron diffraction
Historically, CBED was the first TEM technique to be used to measure strain in nanostructures. As the name suggests, the beam is focused into a spot, as small as one or two nanometers, onto the specimen. The diffraction pattern is then formed of large disks representing the intensity at different angles of incidence represented by the probe. Within the transmitted disk, dark HOLZ lines can be seen that represent the scattering to HOLZ. Strain, or more generally speaking, lattice parameters are measured from the shift of the positions of these lines.Reference Jones, Rackham and Steeds3
The larger the scattering angle, the higher the sensitivity to strain. CBED, which uses HOLZ lines, is therefore extremely sensitive to strain, typically on the order of 10–4. By scanning the probe over the sample, a map can be obtained (see Figure 1a–b).Reference Armigliato, Balboni, Carnevale, Pavia, Piccolo, Frabboni, Benedetti and Cullis4 Nevertheless, to obtain sufficient HOLZ scattering, the crystal needs to be oriented into a relatively high-order zone axis. Structures are usually grown at low-order zone axes, so interfaces can appear blurred, limiting the spatial resolution and interpretability of the results (see Figure 1c–d).Reference Armigliato, Balboni and Frabboni17
Strain mapping with convergent-beam electron diffraction (CBED). (a) Conventional TEM image of a shallow trench isolation (STI) structure, probe positions marked; (b) interpolated map of εxx horizontal strain component in silicon; strain units are ×10–4. Reprinted with permission from Reference 4. © 2003 American Institute of Physics. (c) On-axis high-angle annular dark-field scanning TEM image of a STI structure; (d) tilted zone-axis used for CBED analysis makes the interfaces blurred. Reprinted with permission from Reference 17. © 2005 American Institute of Physics.
It is also true that the high convergence angle and scattering angles mean that quite a wide region of the specimen is actually sampled, despite the small size of the focused probe, further limiting the actual spatial resolution of the technique.Reference Chuvilin and Kaiser18 But the real problem with CBED is that it is too sensitive to bending of the atomic columns. The relaxation of the thin film leads to splitting of the fine HOLZ lines.Reference Clément, Pantel, Kwakman and Rouvière19 This phenomenon is now well-understood, but the simulations are cumbersome and require the exact knowledge of the geometry and the strain state of the sample, which is of course what we are trying to measure.Reference Houdellier, Roucau, Clément, Rouvière and Casanove20 Nevertheless, the technique can work in symmetric positions on devices, like the center of a channel, where bending is minimized.Reference Huang, Kima, Chidambaram, Irwin, Jones, Weijtmans, Koontz, Wang, Tang and Wise21–Reference Müller, Rosenauer, Schowalter, Zweck, Fritz and Volz24 In addition, large-angle CBED is a useful technique to visualize strain in a sample and can even be used quantitatively in certain cases, as demonstrated in Figure 2.Reference Liu, Duan, Xu and Liu25
Large-angle convergent-beam electron diffraction pattern of a strained-silicon transistor. (a) Bending of the black 4
$$\overline 4 $$0 diffraction line in the channel is indicative of strain; (b) corresponding shear component, units of ×10–3. Reprinted with permission from Reference 25. © 2008 Elsevier.
Nano-beam electron diffraction
CBED was progressively abandoned in favor of the more widely applicable technique of NBED.Reference Hÿtch, Snoeck, Houdellier and Hüe9 The idea is to form a spot pattern from a small area on the sample (see Figure 3). First, a pattern is taken from a reference area of the sample, typically the substrate, and the spot positions are determined. Patterns are then taken successively from different areas, analyzed, and the spot positions are compared with the reference pattern.Reference Hÿtch, Houdellier, Hüe and Snoeck10 The shifts in the spot positions are then used to determine the deformation tensor at each of the probe positions.
Nano-beam electron diffraction (NBED) experiment: (a) acquisition area; (b) high-angle annular dark-field scanning TEM (HAADF-STEM) and electron diffraction patterns acquired simultaneously during the scan; (c) HAADF-STEM intensity map of the specimen to identify where NBED patterns were acquired; (d) NBED patterns; and (e) resulting strain tensor point by point across the acquisition area. Reprinted with permission from Reference 6. © 2011 Elsevier.
The challenge was to be able to form narrow beams of relatively parallel illumination (convergence angle typically less than 0.5 mrad). This can be achieved with small objective apertures (1–10 microns), as originally shown in Reference 5, or illumination systems with three condenser lenses.Reference Béché, Rouvière, Clément and Hartmann26 The accuracy and precision of the technique has been evaluated to be 10–3 strain resolution, and the advantages and disadvantages with respect to CBED are analyzed in Reference 27.
With NBED, a low-order zone axis can be used. The technique is conceptually simple and is less affected by bending of the atomic columns in regions of higher strain gradients than CBED. Dynamic scattering still limits the accuracy, and the precision is limited by an additional factor that is the size of the detector used. For example, a strain of 10–3 would represent only a shift of 1 pixel if the spots are spaced by 1000 pixels. Due to the current size of charge coupled devices, sub-pixel precision is required, which can prove problematic.
An alternative to measuring the position of the maximum intensity of the spot is to detect the position of the disks in the NBED patterns, since the convergence is quite high, on the order of 0.5 mrad,Reference Uesugi, Hokazono and Takeno6 or indeed, to use highly convergent illumination in CBED configuration.Reference Müller, Rosenauer, Schowalter, Zweck, Fritz and Volz24 Another problem inherent to probe methods is knowing exactly where the NBED pattern (or CBED pattern) was taken with respect to the structure analyzed. This can be countered by simultaneously recording the HAADF-STEM image (Figure 3c).Reference Uesugi, Hokazono and Takeno6,Reference Armigliato, Balboni and Frabboni17 NBED has been successfully used to study a number of strained-silicon devices.Reference Liu, Li, Pandey, Benistant, See, Zhou, Hsia, Schampers and Klenov28–Reference Sato, Matsumoto, Nakano, Konno, Fukui, Nagaoki and Taniguchi30 NBED is usually carried out to create strain profiles, as two-dimensional mapping can be too time consuming.
Imaging techniques
Any technique that can create images of the atomic lattice can be used to measure strain, by HRTEM, or HAADF-STEM. The two main techniques for measuring the positions of the lattice from the image are peak-findingReference Bierwolf, Hohenstein, Phillipp, Brandt, Crook and Ploog7 and geometric phase analysis (GPA).Reference Hÿtch, Snoeck and Kilaas8 Peak-finding, as its name suggests, relies on identifying the positions of intensity peaks related to the atomic column positions. Similar algorithms are used in identifying the spot positions in NBED. A region of substrate is identified as the reference region, and the displacement of the peaks with respect to this extended lattice defines the displacement field. GPA relies on determining the phase of periodicities in the image through Fourier filtering. The phase is related to the displacement field through Equation 2. The spatial resolution is determined by the size of the mask used in the filtering step of the process, with a maximum of two times the lattice spacing. Spatial resolution is typically 1–2 nm for a reasonable strain resolution of 10–3.Reference Hÿtch, Putaux and Pénisson31
High-resolution transmission electron microscopy
The first method used to study strained-silicon devices was analyzing the local fast Fourier transform in HRTEM images, analogous to NBED.Reference Ang, Chui, Bliznetsov, Tung, Du, Balasubramanian, Samudra, Li and Yeo32 Geometric phase analysis was subsequently used to map the different strain components over the active region of a strained-silicon transistor test structure (Figure 4).Reference Hüe, Hÿtch, Bender, Houdellier and Claverie33 The challenge was to obtain sufficient contrast from a FIB prepared specimen of about 100 nm in thickness, whereas HRTEM normally requires a foil thickness of less than 50 nm. In this case, aberration correction was used but is not strictly necessary.Reference Chung, Lian and Rabenberg34 According to the finite-element modeling method, the difference between the strain, εxx, in the relaxed TEM sample and the bulk structure is rather small (Figure 4c). This is generally the case for components parallel to the surface, which are constrained by the geometry of the source and drain region. The largest relaxation is observed for the components in the growth direction, z, and can reach 20–30% of the value of the strain in extreme cases.
Strain mapping by geometric phase analysis of high-resolution transmission electron microscope (HRTEM) images: (a) HRTEM image of strained-silicon test device, enlarged to show the lattice contrast; (b) strain maps compared with finite element method (FEM) modeling of the thin-film sample; (c) horizontal strain, εxx, as a function of distance from the gate, z, and modeled thin-film and bulk strain. Reprinted with permission from Reference 33. © 2008 American Physical Society. (d) Strain profiles of εxx from a similar type of structure obtained by geometric phase analysis (GPA) of HAADF-STEM images (blue) and extrapolated profile from convergent-beam electron diffraction (CBED) (orange). Reprinted with permission from Reference 36. © 2010 Wiley.
As an alternative to HRTEM, HAADF-STEM can be used.Reference Chung, Lian and Rabenberg35,Reference Diercks, Lian, Chung and Kaufman36 Thicker specimens can be used, but STEM images suffer from additional distortions from the scanning process itself that are difficult to correct. Both HRTEM and HAADF-STEM imaging suffer from relatively small fields of view partly from the limited size of detectors and the difficulties of obtaining uniform contrast over large areas. This means that the reference region is not always in a truly unstrained region of the device, as a comparison with CBED measurements reveals (Figure 4d).Reference Diercks, Lian, Chung and Kaufman36
Diffraction imaging in dark-field mode
Since the very beginning of TEM, the dark-field technique of imaging with a diffracted beam has allowed for the study of strain, in particular by revealing the presence of dislocations or precipitate strain fields. A common sign of strain in a sample is bend contours, which occur due to flexing of the sample. This phenomenon is described in any text book on conventional TEM (e.g., see Reference 37). For quantitative mapping of strain fields, the intensity of the diffracted beam can be fitted to dynamical scattering simulations based on finite-element modeling of the strains.Reference Demarest, Hull, Schonenberg and Janssens38 The technique, called quantitative electron diffraction contrast, suffers from heavy reliance on modeling, specimen bending, and the nonlinear relationship between strain and diffracted intensity.
Dark-field electron holography
DFEH is the most recent of the TEM techniques presented here and was developed to map strain by directly measuring the phase of diffracted beams.Reference Hÿtch, Snoeck, Houdellier and Hüe9,Reference Hÿtch, Houdellier, Hüe and Snoeck10 The phase of the diffracted beam is related to the strain, through Equation 2, and depends on other factors, such as the specimen thickness, the mean-inner potential of the material, and dynamic scattering.Reference Hÿtch, Houdellier, Hüe and Snoeck11 It is therefore important to have specimens prepared with uniform specimen thickness and uniform diffraction conditions (i.e., without significant bending).
The setup for DFEH is similar to conventional off-axis electron holography, except that the experiment is carried out in dark-field mode (i.e., the diffracted beam is oriented along the optical axis by tilting the incident beam) (Figure 5). A part of the diffracted beam emerging from an unstrained part of the specimen, typically the substrate, is interfered with the beam emerging from the region of interest. With the precautions mentioned previously, the resulting phase difference is due only to strain. With DFEH, a much larger area can be mapped compared to HRTEM (Figure 6a–b), even up to several microns. By capturing the phase of two non-collinear diffracted beams, the two-dimensional strain tensor can be determined (Figure 6c).
Dark-field electron holography. (a) A diffracted beam from an unstrained region of crystal (in blue) is interfered with the same diffracted beam emanating from the region of interest (in yellow) with the aid of a bi-prism wire (red). Provided the crystal is of uniform thickness, the non-geometric phase terms cancel, or only produce a uniform phase change to a good first approximation, thus yielding the geometric phase component. (b) Typical transistor geometry with reference zone in substrate and active region at the surface. Reprinted with permission from Reference 11. © 2011 Elsevier.
The problem of the scale of strain measurements. (a) Strain map obtained by high-resolution transmission electron microscopy (in color) superimposed on a bright-field image of the full structure (G, gate electrode); (b) strain map obtained by dark-field electron holography (DFEH) of a similar device. Reprinted with permission from Reference 45. © 2013 Wiley. (c) Strain tensor maps obtained by DFEH: experimental strain components (left column) and simulated strain components from finite element method (right column). Reprinted with permission from Reference 10. © 2008 Nature Publishing Group.
While the technique was aimed at fairly low spatial resolutions, the technique can be pushed to a spatial resolution of 1 nm.Reference Cooper, Rouvière, Béché, Kadkhodazadeh, Semenova, Yvind and Dunin-Borkowski39,Reference Wang, Bruley, van Meer, Li, Domenicucci, Murray and Rouvière40 The real limitations to spatial resolution are the approximations and assumptions underlying the method rather than instrumental. The precision can reach 2 × 10–4 in strain profiles if lateral averaging is applied.Reference Hüe, Hÿtch, Houdellier, Bender and Claverie41 DFEH has found a number of applications, thanks to its mapping capability and precision.Reference Cooper, Béché, Hartmann, Carron and Rouvière42–Reference Wang, Yao, He, Yang, Gu, Wang and Duan44 A brief review of different applications can be found in Reference 13. In addition to using off-axis holographic techniques, dark-field inline holography (DFIH) is possible using variations in intensity of dark-field images with defocus. The results of DFIH are similar to off-axis techniques but with more relaxed requirements on the spatial coherence of the electron beam.Reference Hÿtch, Houdellier, Cherkashin, Reboh, Javon, Benzo, Gatel, Snoeck, Claverie and Claverie45
As mentioned in the introduction, the simple relation between the phase of the diffracted beam and the displacement field described by Equation 2 is not exact, notably due to the projection problem and dynamic scattering. However, analysis of the simple diffraction conditions relevant to DFEH has allowed an analytical formula for the projection rule and the averaging of strain over different depths in the thin foil to be determined.Reference Koch, Ozdol and van Aken46
In situ techniques for measuring strain
The basic mechanisms that determine materials deformation behavior occur at nanometer length scales, and the defects that control mechanical properties are best characterized through electron microscopy. Therefore, it is not surprising that mechanical testing methods were among the first in situ methods to be developed for the TEM.Reference Hirsch, Howie, Nicholson, Pashley and Whelan14,Reference Butler47 Starting in the late 1950s, in situ straining stages were developed for dynamic observations of dislocation motion in metals.Reference Wilsdorf48 Throughout the last 50 years, there have been significant developments in the field of in situ TEM mechanical testing,Reference Robertson, Ferreira, Dehm, Hull and Stach49 including the evolution of mechanical probing techniques such as in situ nanoindentation.Reference Minor, Stach and Morris50 However, there is a large difference between deforming a sample in situ in a TEM and measuring the imposed strain in the material. In order to measure strain, it is necessary first and foremost to have an accurate measurement of the gauge section of the deforming volume, something that requires precise sample manufacturing. Thus it was not until recent developments in the field of microfabrication and FIB preparation that quantitative measurements of the actual strain imposed on a material during an in situ TEM mechanical test were routinely achieved.Reference Minor, Syed Asif, Shan, Stach, Cyrankowski, Wyrobek and Warren51,Reference Shan, Mishra, Syed Asif, Warren and Minor52 In situ TEM mechanical testing with quantitative force versus displacement measurements have been demonstrated using multiple testing configurations and types of samples, including nanoindentation, compression, tension, and bending, as described in a recent review article.Reference Legros, Gianola and Motz53
In general, there are two methods to measure strain during an in situ TEM mechanical test, either through measurement of the displacement imposed on the sample by the actuation device, or through direct visualization techniques using markers and DIC. In both cases, it is critical that the geometry of the deforming volume is well known. One method to know the sample geometry precisely is to use microfabrication, such as when a thin film of known thickness is deposited on a microfabricated structure.Reference Haque and Saif54 By monitoring the displacement in the device, one can obtain precise measurements of the strain applied to the suspended thin film. More recently, FIB microfabrication of TEM test samples has been shown to be a more generally applicable method to prepare precise sample geometries.
Using FIB, it is possible to prepare nanopillars where the cross-section of the sample is easily measured, although achieving a uniform cross-section throughout a pillar is not easy at small scales. As a further extension of FIB preparation of mechanical test samples, it has been recently shown that in situ TEM tensile testing can be achieved by milling a dog-bone-shaped sample out of a bulk material with a FIB and also milling an inverted diamond gripper to pull the sample.Reference Kiener and Minor55 Through this method, uniaxial quantitative tensile testing is possible. Tensile tests have numerous advantages over compression testing, including decreased specimen taper, increased flexibility in sample geometry, and a homogeneously deforming gauge section.
Figure 7 shows a comparison of a nanocompression and nanotensile test of a FIB-prepared Ti sample. As shown in Figure 7a–b, the nanocompression test resulted in a relatively non-uniform deformation at the top of the pillar. In this case, it is difficult to describe the strain in the sample since the initial length of the sample is not an accurate estimation of the length of the deforming volume. Figure 7c–d shows the results of a nanotensile test on the same sample, where more homogeneous deformation across the gauge length of the sample makes the initial sample length more precise for strain measurement. Figure 7e shows the resulting quantitative data for both tests, where the nanocompression test is most accurately represented by engineering stress versus displacement of the compression device, while the nanotensile test can accurately plot engineering stress versus engineering strain. In the case of the nanocompression test, the engineering stress is calculated using the contact area, whereas the engineering stress in the nanotensile test can be calculated using the uniform gauge area of the sample and assuming homogeneous deformation, which until necking occurs is a reasonable approximation.
Comparison of in situ nanocompression and nanotensile tests of pure Ti. (a) Image of a Ti nanopillar before testing. The pillar is tapered, hence the deformation is inhomogeneous and localized at the top of the pillar, as shown in (b). (c) Gauge section of a nanotensile test before testing. (d) After the nanotensile test. (e) Quantitative data for the two tests, where the nanocompression test is plotted as engineering stress versus displacement and the nanotensile test is plotted as engineering stress versus engineering strain. Courtesy of Q. Yu.
The second general method of measuring strain during in situ testing is to rely on analysis of images using DIC. The DIC technique uses post-experiment image processing to measure the displacement between features such as FIB-deposited markers.Reference Guo, Chen, Oh, Wang, Dejoie, Syed Asif, Warren, Shan, Wu and Minor56,Reference Tian, Cheng, Shan, Li, Wang, Han, Sun and Ma57 In the example shown in Figure 8, an 8-µm-long Mo-alloy nanofiber of known geometry is pulled in tension using a microfabricated “push-to-pull” device and an in situ TEM nanoindentation system.Reference Chisholm, Bei, Lowry, Oh, Syed Asif, Warren, Shan, George and Minor58 Since the sample geometry is known and the instrumented nanoindenter can provide calibrated displacement during the test, the strain of the nanowire can be calculated directly, similar to the FIB-prepared nanotensile test in Figure 7. However, during the test, it was observed that the actual deformation in the sample was highly localized, and therefore the strain calculation for the entire 8-µm-long nanofiber does not give an accurate representation of the strain in the actively deforming region of the sample, which is only a few hundred nanometers in length. Therefore, DIC was applied to the still frames extracted from the recorded videos to measure the local instantaneous projected area and the local elongation of the sample. Figure 8b shows the local true stress-strain curve for the necking region, as calculated using DIC with images of the fiber at notable points overlaid on the curve. By calculating the strain directly from the TEM images, it is possible to get a much more accurate measurement and actually report that the sample shows an increase in strength as the dislocation source is exhausted (the source stops emitting dislocations to carry the strain).Reference Chisholm, Bei, Lowry, Oh, Syed Asif, Warren, Shan, George and Minor58 Without the DIC calibration of the local strain, the local hardening of the sample would go unnoticed in a stress-strain curve of the entire nanofiber.
In situ tensile test of a Mo-alloy nanofiber demonstrating the application of digital image correlation to calculate local stress and strain in the deforming volume. (a) TEM micrograph of an 8-µ-long Mo-alloy nanofiber before deformation. (b) Local true stress-strain curve generated using digital image correlation, with a time-series of the local area being analyzed shown along the curve. From left to right: Initial dislocation structure; at 1.56 GPa, a spiral arm source begins to activate and emit dislocations; dislocation structure after load drop; final necked region at 3 GPa after the spiral arm source deactivates. Reprinted with permission from Reference 58. © 2012 Elsevier.
Summary
Within the context of semiconductor devices, nano-beam electron diffraction (NBED) and dark-field electron holography (DFEH) are the most commonly used techniques today for measuring strain. In a wider context, high-resolution transmission electron microscope (HRTEM) is the most used technique because of its availability. Convergent-beam electron diffraction (CBED) is used for more specialized case studies or as a benchmark for other techniques. A brief summary of the different techniques can be found later in the text (and see Table I).
CBED is still probably the most accurate technique for measuring strain in samples with small strain gradients. Otherwise, line splitting causes the analysis to be too complicated for practical use. Thick samples can be used for CBED, and it is the only technique providing information on the three-dimensional strain components, notably those in the incident beam direction. NBED is a practical and versatile technique that is conceptually simple but with sensitivity limited to about 10–3. Some specialized instrumentation is required, including small objective apertures or three condenser lenses. HRTEM, whether in TEM or STEM mode, offers the possibility of strain mapping with precision similar to NBED but requires demanding specimen preparation and is limited in field of view. In principle, any high-resolution image can be analyzed for strain. DFEH is capable of mapping strain over micron fields of view with a precision similar to CBED of 10–4. However, dedicated equipment is required, notably an electrostatic bi-prism, and operators need to be trained in electron holography, which has slowed wider usage of the technique.
In situ TEM mechanical testing can report accurate values of strain, but the key experimental parameters rely on precise knowledge of the dimensions of the deforming gauge section of the sample. There are two general methods to report strain from an in situ test: (1) conversion of displacement of the actuating mechanism to strain through knowledge of the sample geometry or (2) direct imaging techniques such as digital image correlation.
Acknowledgments
M.J.H. acknowledges financial support from the European Union under the Seventh Framework Programme under a contract for an Integrated Infrastructure Initiative Reference 312483-ESTEEM2 and the European Metrology Research Programme (EMRP) Project IND54 Nanostrain. The EMRP is jointly funded by the EMRP participating countries within EURAMET and the European Union. A.M.M. acknowledges support from the National Center for Electron Microscopy, Lawrence Berkeley National Laboratory, which is supported by the US Department of Energy under Contract #DE-AC02–05CH11231.
