electron holography

6 Nanotechnology Issue November 2012 | MicroscopyandAnalysis

Darkfield electron holography for strain mapping at the nanoscale

Martin Hÿtch,1 Christophe Gatel,1 Florent Houdellier,1 Etienne Snoeck 1 and Kazuo Ishizuka 2

1. CEMES, CNRS and Université de Toulouse, Toulouse, France. 2. HREM Research Inc., Higashimatsuyama, Saitama, Japan

IntroductionStrain is an important parameter for a wide range of physical properties at the nanoscale. It can be beneficial, improving performance in the micro-electronic devices or encouraging island growth in quantum dot structures, and detrimental, leading to defect formation and cracking. It is also often unavoidable, notably in the epitaxial growth of layered materials. Many techniques have therefore been developed to measure strain. In the field of electron microscopy these range from convergent-beam electron diffraction (CBED) to high-resolution electron microscopy (HRTEM), and nano-beam electron diffraction (NBED) to darkfield electron holography (DFEH). In this article, we will describe the latter and latest of these techniques [1,2] and the associated software used to analyse the data [3].

Darkfield electron holography was devel-oped to overcome a problem encountered when measuring strain using high-resolution electron microscopy: the question of scale. Geometric phase analysis (GPA) of HRTEM images is very successful in mapping strain in nanometric sized areas [4-6]. It can be extended to larger areas by reducing the magnification and correcting for the distortions of the projector lens which becomes important at lower magnification [7]. These algorithms and other developments have been incorporated into the image analysis software package GPA Phase 3.0 (HREM Research Inc.). Strain in the active region of strained-silicon devices was mapped for the first time in such a way [8]. However, the field of view is still limited with respect to the devices and areas we would like to characterize (see Figure 1a) and cannot be improved much further given the need to image the atomic lattice. This problem led to the devel-opment of darkfield electron holography (DFEH) for strain mapping (see Figure 1b) [1].

Setup for darkfield electron holographyThe setup for darkfield electron holography is shown in comparison to the conventional setup in Figure 2 [9]. A parallel beam is formed by the condenser system to impinge on the specimen, directly in the case of conventional hologra-phy, or at a slight angle in the case of darkfield holography. One part of the beam passes through a reference region, a vacuum in the case of con-ventional holography, or an unstrained region of crystal for DFEH, for example the substrate. The other part travels through the region of interest, for example a strained layer or the active region

Figure 1The problem of the scale of strain measurements: (a) strain map obtained by HRTEM (in colour) superimposed on a brightfield image of the full structure [8]; (b) strain map obtained by DFEH of a similar device [1].

of a transistor. An electrostatic biprism is then used to interfere the two parts of the wave, the transmitted wave in the conventional case, or the diffracted wave for DFEH. Indeed, excepting the biprism, the two setups are identical to a con-ventional brightfield and darkfield experiment, respectively. The interference fringes viewed at the screen encode the phase difference between the two wave paths.

The phase has a number of contributions and can be written as a sum of four principal terms [9]:

(1)

of which the crystalline phase φc (r) encompassing dynamic diffraction phase factors, and phases from magnetic φM (r) and electrostatic fields φE (r) are well known. Geometric phase φG

g (r) which arises from the presence of strain and displace-ment fields, is defined as:

(2)

where g is the diffraction vector and u(r) the displacement field as a function of position. Note that the transmitted beam, for which g=0, does not carry information about displacement fields. Only diffracted beams carry geometric phase, and hence we need to measure the phase of diffracted beams in order to study strain [1].

It can be seen also from Equation (2), that only displacements which are parallel to the diffrac-tion vector are measured. This is fine if you want to measure strains in the diffracted beam direction g. The strain component giving the compression and tension in this direction εgg is

given simply by:

(3)

where ug is the component of the displacement field in the direction of g, and ∇g is the gradient parallel to g. A common example is to use the (004) diffracted beam to measure strains in the growth direction [001] of expitaxial systems.

The full 2D strain tensor requires, however, the measurement of the geometric phase from two diffracted beams. The equation for the 2D displacement field is as follows:

(4)

where a1 and a2 are the real-space basis vectors corresponding to the diffraction vectors g1 and g2. This is the same as for geometric phase analysis of high-resolution images [4]. The strain tensor can then be obtained by numerical differentiation using the standard relations:

(5)

In a similar way the local in-plane rigid body rota-tion ωxy can be determined:

(6)

where, for small rotations, the angle is in radians and anti-clockwise positive. The more developed calculations for large deformations are not neces-

electron holography

7MicroscopyandAnalysis | Nanotechnology Issue November 2012

Figure 4Darkfield hologram analysis. Output from the software for phase and the planar deformation: no corrections applied (raw), camera distortion correction (CCD corrected) and Fresnel fringe filter (Fresnel corrected). The phase is only meaningful in the diffracting regions of the specimen, the rest is noise.

sary here though are generally implemented in the software GPA Phase 3.0 and HoloDark 1.0 [3].

A careful distinction needs to be made between deformation and strain. Strain in a mechanical sense is defined with respect to the stress-free, or undeformed, state. The reference lattice used to measure the deformation (i.e. g1 and g2) might not coincide with this undeformed lattice, in which case the deformation will not be the same as the strain. Secondly, in the presence of compositional variations which change the shape of the unit cell, the undeformed state will not be the same everywhere. Mixtures of strain and composi-tional variations are therefore the most difficult cases to analyse. Strictly speaking, darkfield electron holography, along with all microscopy techniques, only measures deformation.

Experimental requirementsOff-axis holography experiments rely on interfer-ing widely separated parts of the electron wave, so the beam must have sufficient spatial coherence to create interference fringes with sufficient con-trast to exploit. This can only be achieved with a field-emission gun (FEG), either a Schottky field-emission gun, as shown here, or preferably a cold field-emission gun (CFEG). Both these technolo-gies have benefitted from recent improvements from the microscope manufacturers. We have also been working on a carbon-based source to replace the usual tungsten emitter in CFEG, and the initial results promise even higher brightness for holography experiments [10].

The additional requirements for DFEH with respect to conventional holography concern the specimen and its orientation in the microscope. Before switching into holographic mode, the specimen should be oriented into a diffracting condition for the beam of interest. Ideally, this is a two-beam condition, on a systematic row. A slight deviation from the exact Bragg angle is permitted in order to maximize the intensity in the diffracted beam. The angle of rotation about the systematic row should be small, of the order of a degree: enough to reduce multiple scatter-ing but not too large as to incline the viewing geometry adversely. At high angles from the zone axis, interfaces will become smeared in projection, a problem frequently encountered in CBED experiments. The overall aim is to obtain a darkfield image which has strong and uniform contrast over the field of view, and especially in the substrate which will serve as a reference. The difficulty is a question of specimen preparation, as always a vital part of any successful electron microscopy experiment. Other practical aspects and the effect of a series of experimental param-eters such as biprism voltage, sample thickness, exposure time, tilt angle and choice of diffracted beam have been studied in detail on a silicon-germanium layer sample [11].

Once good darkfield conditions have been obtained, the off-axis holography can be set up. Experiments can be carried out in high-resolution mode [12] but for wide fields of view, the Lorentz mode is necessary. Unfortunately, most microscopes are not optimized for darkfield microscopy in Lorentz mode. There are two prin-ciple problems: tilting the beam pre-specimen, and using the objective aperture to select the diffracted beam. In the standard configuration, the relatively high angles of tilt in darkfield mode

Figure 2 Schematic representation of both conventional (centre) and dark-field (right) off-axis electron holography setups. In the conventional setup, the reference is made with the vacuum whereas for dark-field hologra-phy, the reference is a region of unstrained crystal [9].

Figure 3 Darkfield holography of a strained-silicon p-MOSFET device: (a) conventional brightfield image showing the gate, gate oxide and recessed sources and drains of SiGe (darker contrast), measurement (in yellow) and reference (in blue) regions, defined by the biprism position (in red); (b) darkfield hologram using the [11-1] diffracted beam; (c) enlargement showing narrow holographic fringes and larger Fresnel fringes; (d) power spectrum showing side-band and mask radius.

electron holography

8 Nanotechnology Issue November 2012 | MicroscopyandAnalysis

Figure 5Strain analysis of the p-MOSFET: (a) darkfield hologram for (220) diffracted beam; (b) geometric phase de-termined from hologram; (c) strain map determined from phase; (d) simulated strain map by finite element method (FEM). Colour scale identical for experimental and simulated strain. [1].

are compensated by the pre-field of the objective lens. In Lorentz mode, however, the objective is switched off, which limits the tilts that can be used. In the SACTEM-Toulouse (a Tecnai F20 equipped with an imaging aberration corrector from CEOS), we use the corrector as a Lorentz lens [13]. The maximum tilt is about 13 mrad, equiva-lent to the (220) beam in silicon at 200 kV. When acquiring darkfield holograms of the (004) planes, the diffracted beam will not therefore be exactly on the optic axis and some coma is inevitable. The second problem is the use of the objective aperture which is designed to be inserted in the diffraction plane of a microscope in standard mode. In Lor-entz mode, however, the diffraction plane does not usually coincide with this plane. The aperture is therefore blurred and generally the smallest objective apertures, which would be useful for increasing darkfield contrast, cannot be used.

These problems have been addressed in the design of the Hitachi I2TEM-Toulouse (in-situ interferometry TEM), an HF3300 equipped with a CFEG, a double stage for Lorentz microscopy, an imaging aberration corrector (CEOS Aplanator) and multiple biprisms. The second specimen stage is located just above the objective lens which now can serve as a powerful Lorentz lens coupled to the state-of-the-art aberration corrector. Holograms can be obtained with a (008) diffracted beam with a resolution of 0.48 nm, compared with a typical resolution of 2-4 nm in a conventional Lorentz setup [14]. The multiple biprisms allow more flexibility in choosing holographic fringe spaces and field of view, and Fresnel fringes can be eliminated [15].

Strained silicon p-MOSFETThe example used to establish DFEH was a dummy strained-silicon channel p-MOSFET transistor [1]. Uniaxial strain is obtained in the silicon channel by the recessed sources and drains of Si0.80Ge0.20 grown in perfect epitaxy with the silicon (Figure 3). A gate of polycrystalline silicon and gate oxide of silicon oxide complete the dummy device which lacks the subsequent steps of the process such as doping and contacting. The device is capped with a layer of silicon oxide and a thin layer of platinum to protect the surface dur-ing specimen preparation. Samples were prepared using a modified H-bar process in the FIB to a thickness of about 120 nm [8].

The crystal was successively oriented into two-beam diffracting conditions to carry out the measurements. In each case, the biprism was positioned in such a way as to define a part of the substrate as the reference region (in blue on Figure 3) and the active region of the device where the strain is to be measured (in yellow). These regions have been presented on the brightfield image of the device so that the different elements can be seen. After applying a voltage (120 V in this case) to the biprism wire, the diffracted beam from these two regions are interfered to create the hologram. The hologram from the (11-1) diffracted beam is shown in Figure 3b. The Si and SiGe regions are visible because they diffract, whereas the polycrystalline silicon, gate oxide and capping layers are invisible since they do not. Indeed, close inspection of the hologram reveals that the holographic fringes are absent from these regions, with only the Fresnel fringes from the biprism visible (Figure 3c).

Figure 6Alignment of two different holograms: the amplitude image can be usefully used to find the translation vector (magnification change and rotations can also be handled), either manually or by cross-correlation, and applied to the phase images.

The phase of the hologram is then determined by the Fourier method using the HoloDark 1.0 software [3]. First, one of the side-bands is selected on the power spectrum of the hologram with the aid of a pointer, by convention the one pointing towards the reference region (see Figure 3d). The mask size is next chosen to isolate the spot. This is an important parameter as it determines the spa-tial resolution of the phase image and subsequent strain maps (as for GPA). Reducing the mask size reduces the spatial resolution but increases the signal-to-noise of the phase and strain. A compromise needs to be found depending on the structure analysed. In any case it should not

exceed half the carrier frequency. The phase is then calculated by inverse Fourier transform.

The phase needs, in general, to be corrected for the distortions from the CCD camera (see Figure 4). These can be eliminated by measuring the phase of a reference hologram in the vacuum, as for conventional electron holography, and subtracting it from the phase of the darkfield hologram. The drawback of this method is that the reference hologram is necessarily noisy and thus the overall effect is an increase in the noise. The method we use is to measure the distortions of the camera once and for all and then correct the phases automatically. A series of holograms can

electron holography

9MicroscopyandAnalysis | Nanotechnology Issue November 2012

be taken which cover the whole CCD array, with narrow fringe spacings (ideally with a wavelength of about 4 pixels), and wide enough to exclude the Fresnel fringes. The average phase of these holograms can be used to characterize the camera. Alternatively, as was the case here, the data was supplied by the manufacturer (Gatan), which is the best method.

It is also possible at this stage to reduce the artefacts from the Fresnel fringes, manifest in the horizonatal set of fringes seen in the substrate (particularly visible in the deformation maps). One solution is to enlarge the hologram width and analyse only the central zone where their contrast is limited. Unfortunately, the hologram contrast diminishes with the overlap distance which in turn reduces the signal-to-noise of the measured phase. An alternative is to use a suit-able filter in Fourier space, as implemented here (Figure 4).

Figure 5 shows the complete analysis for the (220) diffracted beam. We can see that the phase variations are very strong, up to several multiples of 2p. This is in stark contrast to electrostatic phase measurements which are orders of mag-nitude smaller, and means that the stringent specimen preparation requirements of uniform thickness are relaxed somewhat compared with conventional holography.

The gradient of the phase, according to Equa-tion (3), then gives the local deformation of the lattice in the direction parallel to the diffraction vector [220], chosen as the x-axis. Gradients are determined numerically by subtraction of neighbouring image points in the phase image. The result is shown in Figure 5c. The silicon can be seen to be in compression in the channel region (green and blue on the colour scale), increasing from the substrate to a maximum under the gate. The regions containing SiGe appear as areas of positive deformation (red colours) as the lattice parameter is larger than the Si used as reference.

2D strain componentsIn order to determine the full strain tensor in 2-dimensions, two darkfield holograms need to be acquired and analysed. We chose two of the {111} diffracted beams. The two resulting phase images need then to be aligned, as there is always a shift between the two holograms and corresponding tools are an essential part of the analysis software (Figure 6).

The 2-D strain tensor, determined from the two aligned phase images, is shown in Figure 7 and compared with finite element method (FEM) modeling. The agreement between the experimentally determined strain maps and those coming from the modeling is excellent. Not only the principal strain components along the transistor line (εxx) and the growth direction (εzz) are obtained but also the shear (εxz) and the local rigid-body rotations (ωxz). It would be impossible to distinguish between shear and rotation from only one set of lattice planes.

It can also be seen that at the interfaces between the SiGe and the silicon, particularly for the shear and the rotation, there appears to be strong and localised strain contrast. This is an artifact of the analysis and comes from a difference in mean-inner potential between the alloy SiGe and Si. The electrostatic phase will not therefore be uniform and there will be a (small) step in

experimental values (in blue) are closest to the bulk case since the specimen is relatively thick. For the lateral strain in the channel, εxx, the thin and the thick simulations give identical values in the substrate and gradually diverge as they approach the gate (at the surface) giving values of -1.1% (compression) and -1.3% respectively. As expected, the experimental results lie between these two extremes and are closet to the bulk case.

Strained silicon n-MOSFETTo increase the mobility of electron carriers in an n-type MOSFET, the silicon channel should be in tension. To apply uniaxial stress as for the p-MOSFET, the lattice parameter in the sources and drains should be smaller than silicon. This can be achieved by doping silicon with a small amount of carbon. There is a limit, however, as to the quantity of carbon that can be incorporated substitutionally. With increasing dose, carbon will be incorporated as interstitials, which hardly modifies the lattice parameter, and ultimately will precipitate in the form of SiC, again with limited effect on the lattice parameter. There is therefore uncertainty concerning the injected strain when using the recessed sources and drains with this system and experimental verification is necessary.

Figure 9 shows the analysis of a dummy n-type MOSFET with recessed silicon, nominally doped

the phase at the interface. A large value for the derivative will therefore be registered right at the interface, which explains the localised nature of the contrast [9].

Thin-film effectAt this point, we need to address an important issue inherent to any transmission electron microscopy investigation of strain: the thin-film effect. TEM foils are necessarily thin to allow elec-tron beams to traverse them without too much loss in intensity. The thinning process introduces two free surfaces not present in the bulk sample that will relax some of the stresses and strains present. In general, this relaxation is typically 10% and is never more than 30% of the initial strain but needs of course to be estimated for quantita-tive measurements.

The upper and lower bound for measurements can be established by simulating the strain in an infinitely thin sample (plane stress conditions in elastic theory) and the infinitely thick, or bulk, sample (plane strain conditions). This was carried out for the p-MOSFET. Relaxation depends very much on the strain component and the proximity with the surface, as shown in Figure 8. The dilata-tion reaches a maximum of 0.82% in the bulk simulation (in red) and 0.67% in the thin film (in green), which represents a reduction of 18%. However, the minimum values are identical. The

Figure 8Thin-film relaxation of measured strains: (a) dilatation strain profiles across the transistor array; (b) strain profiles from the substrate to the gate. Experimental curves (blue), plane-strain model (green) and bulk model (red). [1]

Figure 7Strain tensor for p-MOSFET transistor: experimental strain components (left column) and simulated strain components from finite element method modelling (right column). [1]

electron holography

10 Nanotechnology Issue November 2012 | MicroscopyandAnalysis

biographyDr Martin Hytch completed his thesis at the University of Cambridge in 1991, and then moved to France where he has been working ever since for the CNRS. His research centres on developing quantitative electron microscopy techniques, notably high-resolution electron microscopy and electron holography. He now heads the nanomaterials group at the CEMES-CNRS laboratory in Toulouse and was recipient of the European Microscopy Award (FEI-EMA) in 2008.

abstractWe describe the application of the recent method of dark-field electron holography (DFEH) to the measurement of strain in nanostructures and devices. The holographic setup and experimental requirements will be briefly reviewed along with features of the software used to analysis the data. Examples will be given pertaining to strained-silicon transistors.

acknowledgements We are grateful for the support from the European Union Seventh Framework Programme [FP7/2007-2013] under grant agreement n°312483 (ESTEEM2) and the French National Agency (ANR) in the frame of its program in Nanosciences and Nanotechnologies (HD STRAIN Project No. ANR-08-NANO-0 32).

corresponding author details Dr Martin HÿtchCEMES, CNRS and Université de Toulouse, 29 rue Jeanne Marvig, 31055 Toulouse, FranceTel: +33 5 62 25 78 83Email: martin.hytch@cemes.fr

Microscopy and Analysis 26(7):6-10 (AM), 2012

with 1% carbon [16]. Darkfield holography using the (220) diffracted beam shows that indeed the silicon channel region is in tension. The lattice parameter in the region of Si:C can also be seen to be slightly smaller than the silicon. To quantify this, profiles were taken vertically from the substrate to the gate in the channel region and between the gates (Figure 9d). The profiles were averaged over the width of the channel (65 nm) to improve the signal-to-noise ratio. The agreement between the five channels in the field of view is remarkable. The standard deviation between the curves is only 0.02% which is our estimation of the precision of the technique. The diminution of the lattice parameter in the Si:C is small reaches a maximum of about 0.15% at the surface. The value for the strain measured just under the gate is 0.58±0.02%.

These strains were compared with finite element method modelling using the same meth-odology as the p-MOSFET. Assuming that the full 1% of carbon was incorporated substitutionally gave values of strain which were systematically too high. We therefore varied the carbon content until a good fit was found (see Figure 9d). The best fit was obtained for 0.75% carbon on substitu-tional sites. This value is in excellent agreement with the results from the analysis of thin films. There it was found that for 1% of total carbon composition, between 20% and 25% of the carbon was on interstitial sites and only 70-75% on substi-tutional sites.

This example shows the value of carrying out direct experimental verification of the strain in the channel and also the very high precision of the technique. The level of strain in the n-MOS-FET is much lower than the p-MOSFET due to the diminishing returns of injecting carbon.

ConclusionsDarkfield electron holography is a fully quantita-tive technique for the measurement of strain in nanostructures and devices. The analysis is straightforward and gives the strain components directly which can then be compared with mod-

Figure 9Analysis of a n-MOSFET with recessed source and drain of carbon-doped silicon: (a) brightfield image; (b) DFEH map of the horizontal strain component εxx showing the channels to be in tension (red); (c) finite element method modelling with identical colour scale; (d) strain profiles from substrate to gate in the channel (blue), between channels (red, green), and best-fitting model (black).

eling. DFEH is particularly adapted to epitaxial systems as the substrate can act as a reference area. The technique has already been applied success-fully to a number of systems, from the transistors shown here [1,16] and similar strained-silicon devices [9,17,18] to SOI FinFets [19], strained layers [11], misfit dislocations therein [20], and quantum dots [12]. The technique can also be combined with conventional holography to provide a com-plete study of strain and dopants in devices [21]. Future developments include in-situ applications and on-line hologram analysis.

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