Transmission Electron Microscope. Experimental Instruction

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1 Transmission Electron Microscope Experimental Instruction In advanced practical course [F-Praktikum] Date: April 2017

2 Contents 1 Task 3 2 Theoretical Basics Bragg Diffraction Miller indices, reciprocal lattice and Laue condition Structure factor Design of a transmission electron microscope Dark-field and Bright-field Diffraction Pattern High-Resolution TEM (HRTEM) Experimental procedure 14 5 Questions 15 Literature 16 Institute for Material Physics SS

3 1 Task 1 Task On the basis of a transmission electron microscope (TEM) [Zeiss200FE] the diffraction- and imaging behaviour of almost monochromatic electrons is investigated. Differences and similarities considering electron and x-ray diffraction are demonstrated. Parameters from the investigated crystals and the device will be evaluated with the aid of images taken by the CCD-camera using different imaging modes. A focus is set on conventional TEM (ctem) imaging and the role of electron optics in relation to the selected lenses, where darkfield, brightfield and diffraction patterns are interpreted. Furthermore, the influence of the focus on the image quality is discussed using high-resolution TEM (HRTEM). 2 Theoretical Basics Electrons can be described both as particles and waves. The equation after de Broglie gives the wavelength λ e in respect to the acceleration voltage U used by the device: λ e = h 2me Ue (1) With h = Planck s constant, m e = mass of electron and e = elementary charge. Due to the high acceleration voltages, relativistic influences need to be taken into account: λ e = h ( ). (2) 2m e Ue 1 + Ue 2m ec 2 In table 1 the relation between acceleration voltage, velocity and wavelength is displayed. In normal circumstances, the used TEM Zeiss200FE is operated by a voltage of 200 kv with a field emission gun (FEG) as the electron source. Table 1: Electron characteristics as a function of the used acceleration voltage. Accelerationvoltage non-relativistc relativistic λ e Masse e velocity [1 x 10 8 [kv] λ e [nm] [nm] [m e /m 0 ] m/s] Electron-scattering phenomena can be grouped in different ways. When accelerated electrons with a wavelength λ e = 2.51 pm (200 kv) impinge on a sample, several processes might take place (see figure Institute for Material Physics SS

$In analogy to x-rays, electrons are diffracted on lattice planes due to their wave character.$

4 2 Theoretical Basics 1). In ctem only the direct beam, elastic or inelastic scattered electrons are of interest. In analogy to x-rays, electrons are diffracted on lattice planes due to their wave character. However, a major difference occurs in the scattering process: while x-rays are scattered at the outer shell electrons, the electrons interact with the core potential. The stronger reciprocity with the nucleus results in higher diffraction intensities. At the same time, the penetration depth is smaller compared to x-rays. The amount of inelastic scattered electrons should be negligible low. Only thin samples ought to be used to improve the interference contrast (for 200 kv thinner than 100 nm, the thinner the better ). Figure 1: Different kinds of electron scattering from a thin specimen [1]. 2.1 Bragg Diffraction Characteristic for a crystal is its periodicity. For the description of the scattering, the crystal is considered as an infinite expanded, periodical lattice, which structure is characterized by three lattice vectors ( a, b, c basis vectors). Through linear combination of the three basis vectors, it is possible to reach every point in the crystal. The enclosed volume is called unit cell. If the smallest possible volume is enclosed, the unit cell is primitive and the associated crystal lattice is called Bravais lattice. The description of a lattice is thus not always distinct: in the easiest case a crystal can be described by a bravais lattice, whose spatial points consist of one atom. The vectors describing their location, form the basis. These vectors, however, should not be mixed up with the basis vectors! Even for crystals consisting only of one type of atom, one can choose a description more complex than a basis consisting only of one atom at the spatial points. The resulting physical background is schematically described in figure 2. An array of lattice atoms is considered as an atomic plane and the diffraction can be simplified as a reflection of the electron beam on the lattice planes. For Institute for Material Physics SS

5 2 Theoretical Basics constructive interference Bragg s law has to be fulfilled (derive this!): 2d sin ϑ = nλ n N (Bragg Equation), (3) where d is the distance between lattice planes, n the order of the maximum and ϑ the angle of the incident beam. Bragg argued that waves reflected of adjacent scattering planes must have a path difference equal to an integral number of wavelength for constructive interference to occur. This requirement is very concisely, thus leading to sharp interference peaks. The distance between adjacent lattice planes (miller planes, described by (hkl)) depend on the symmetry of the crystal and thus on the lattice constants. In particular, a family of lattice planes is determined by three integers, h, k and l, the miller indices (see next chapter). Figure 2: Constructive interference on atomic planes according to William Lawrence Bragg. 2.2 Miller indices, reciprocal lattice and Laue condition Miller indices are used for the distinct identification of lattice planes. They are a main reference for the interpretation of the obtained diffractograms. A cubic crystal can be described by an orthogonal coordinate system, whose basis is formed by the vectors a, b and c (see figure 4). The Miller plane (hkl) denotes a plane that intercepts the three points a, b and c, or some multiple thereof. The h k l Miller indices are proportional to the inverse of the intercepts of the plane. If one of the indices is zero, the planes do not intersect that axis. Some examples of miller planes are shown in figure 4. Furthermore, any direction can be displayed by linear combination as well. As an example, r 1 is a linear combination consisting of: r 1 = 1 a + 1 b + 0 c. The miller indices are then the prefactor of these linear combination, which need to be integer numbers. Directions are denoted by [hkl]. Negative h, k, l are denoted by h, k, l. The plane (hkl) is orthogonal to the direction [hkl]. Another way to determine the miller indices is the use of the reciprocal space instead of the real space. The Institute for Material Physics SS

2 Theoretical Basics Figure 3: Crystal in a cartesian coordinate system with different atomic planes.

vectors of the reciprocal space are calculated using the real space vectors a, b and c: a = 2π b c a b c b

The reciprocal analogy to the Bragg equation is the Laue equation.

6 2 Theoretical Basics Figure 3: Crystal in a cartesian coordinate system with different atomic planes. Figure 4: Examples of lattice planes in a cubic crystal. vectors of the reciprocal space are calculated using the real space vectors a, b and c: a = 2π b c a b c b c a = 2π a b c c = 2π a b a b c. (6) The denominator is equivalent to the volume of the unit cell. The unit of the numerator is an area. Hence, the reciprocal lattice vector has the unit of an inverse length [m 1 ]. The reciprocal analogy to the Bragg equation is the Laue equation. An incident wave with a wave vector k impinges on the surface. The scattered wave vector is denoted by k. In this case the diffraction condition (Laue condition), where k and k have the same length (in the elastic case) k = k = 2π, is described by: λ (4) (5) k k = G. (7) Institute for Material Physics SS

7 2 Theoretical Basics The difference between incident and scattered wave needs to be a vector of the reciprocal space (scattering vector): G = g 1 a + g 2 b + g 3 c. (8) Where g 1, g 2 and g 3 are equivalent to the miller indices (except for a factor α N + ). The inverse of the distance between planes is the magnitude of G. In order to find k G = d 1 (g 1 g 2 g 3 ) = αd 1 (hkl). (9) the Ewald s sphere is often used. Its aim is to determine which lattice plane, represented by the grid points on the reciprocal lattice, see figure 5, will result in a diffracted signal for a given wavelength λ, of the incident wave. It demonstrates the relationship between the wave vector of the incident and diffracted beams and the diffraction angle for a given reflection. Because of the same length of k and k, the scattering vector must lie on the surface of a sphere of radius 2π λ. This sphere is called Ewald s sphere. Geometrically this means that if the origin of the reciprocal space is placed at the tip of k, then diffraction will occur only for reciprocal lattice points that lie on the surface of the Ewald s sphere. In a TEM, the wavelength of the electrons is much smaller than the spacing between atoms and thus the Ewald sphere radius becomes large compared to the spatial frequency of atomic planes. In this approximation, diffraction patterns in effect illuminate planar slices through the origin of a crystal s reciprocal lattice. In contrary to the theory, the thin TEM samples do not represent an infinite, expanded lattice. This leads to diffraction contrast even at areas where the Laue condition is not fulfilled completely. Furthermore, the diffraction points degenerate to rods orthogonal to the specimens surface. For diffraction contrast it is sufficient, if the Ewald s sphere cuts through such a rod. On the one hand, this results in a lot of reflexes and on the other hand, the orientation of the crystallographic plane might only be deduced up to an accuracy of ±5. Figure 5: Ewald s sphere left: in reciprocal space; centre: for small angles; right: for thin samples. Institute for Material Physics SS

8 3 Design of a transmission electron microscope 2.3 Structure factor The Laue equation determines the occurrence of constructive interference. It does, however, not determine the intensity of the diffraction reflex. The intensity depends on the structure of the unit cell and thus on the scattering of individual atoms (f(θ)) stacked together. This relationship is given by the structure factor F (θ), which is a measure of the amplitude scattered by a unit cell of a crystal structure. F (θ) can be defined by the sum of f(θ)-terms of all the i atoms in the unit cell (with atomic coordinates x i, y i, z i ) multiplied by a phase factor. The phase factor takes account of the difference in phase between waves scattered from atoms on different but parallel atomic planes with the same miller indices (hkl). The scattering angle θ is the angle between the incident and scattered electron wave: F (θ) = f i e 2πi(h x i+k y i +l z i ). (10) i The amplitude of scattering, and hence its square, which equals the intensity of the diffraction reflex, is influenced by the type of atoms f(θ), the position of the atoms in the unit cell (x i, y i, z i ) and the specific atomic planes (hkl) that build the investigated crystal structure. This equation predicts that in certain circumstances the amplitude of scattering is zero. For a bcc lattice consisting of one atom, the structure factor F is determined as F = f{1 + e πi(h+k+l) }. This leads to the following rule: F = 2f (h + k + l) even and (11) F = 0 (h + k + l) odd. (12) For a fcc structure this rule is to be determined for the theoretical examination. 3 Design of a transmission electron microscope One classifies electron microscopes via two criteria: For conventional TEM (ctem) the specimen is radiated by an electron beam, for scanning TEM (STEM) the specimen is scanned line by line. We will restrict the investigation to the use of ctem. The general design of a TEM resembles a light microscope (Figure 6). The illumination source is the emitting cathode, a grid called a Wehnelt cylinder and an anode at earth potential with a hole in its center. The cathode is attached to the high-voltage cable that, in turn, connects to the highvoltage power supply. The electrons coming off the cathode see the negative field and are converged to a point called a crossover between the Wehnelt and the anode. By adjusting this negative field the beam flux can be regulated. Transmission electron microscopes are equipped with at least two condenser lenses to satisfy following requirements: focusing of the electron beam on the specimen in such a way that sufficient image intensity is obtainable even at high magnification; irradiation of a specimen area that corresponds as closely as possible to the viewing screen with a uniform current Institute for Material Physics SS

9 3 Design of a transmission electron microscope density; variation of the illumination aperture and production of a small electron probe (0.2 to 100 nm in diameter). The objective lens creates an intermediate, high magnified image of the specimen. The following lenses (intermediate, projector) are used in imaging mode (Figure 6 A) to magnify the intermediate image further and project it onto the fluorescent screen or the CCD camera. The magnification is set by varying the lens currents and therefore their focal length. By changing the strength of the intermediate lens, the diffraction pattern can be projected on the imaging plane (Figure 6 B). The magnification in diffraction mode is controlled by the camera length, the virtual distance between specimen and screen. Figure 6: Schematic setting of a TEM in a) brightfield/high-resolution mode (imaging mode) and b) diffraction mode [2]. Institute for Material Physics SS

10 3 Design of a transmission electron microscope Because of the small wavelength it is possible to resolve structures in the sub-nanometer scale. This is expressed by the Rayleigh-Abbe criterion: δ = 0.61 λ µ sin β, (13) In this formula δ is the smallest distance resolvable, µ is the reflective index and β the opening half angle of the lens. The expression µ sin β is known as the numerical aperture. The resolution of good light microscopes is in the order of the wavelength, i.e the opening angle reaches almost 180. This is only possible because of the good spherical aberration correction in optics (how?). For an electron microscope it is more difficult to correct spherical aberration and one needs special correctors (Cs). To limit the spherical aberration only very small aperture angles are used. Unfortunately, this method limits also the numerical aperture and therefore the resolution. Nonetheless, modern electron microscopes have a resolution of δ 0.2 nm. Using Bragg s Equation (3) it turns out, that for typical distances d hkl < 0.5 nm and the electron wavelength in the picometre regime the angles are below one degree. So one gets for small angles in a good approximation from fig. 7, that tan θ = R 2L = sin θ applies, (14) where L denotes the distance between specimen and camera (so called camera length) and R denotes the distance between direct beam and diffraction peak. Figure 7: Approximation for small angles with fast electrons. [1] Institute for Material Physics SS

11 3 Design of a transmission electron microscope Inserting this into Bragg s law (3), the basic equation of electron diffraction is derived (for n=1): λ L = R d hkl, (15) at which interferences of higher order correlates linear to integer multiple Miller Indices (difference to X-Ray). 3.1 Dark-field and Bright-field When images are formed in the TEM, one can either form an image using the central spot or one can use some or all of the scattered electrons. This defines either bright-field (central spot) or darkfield (scattered electrons, exclusion of the central spot). In order to obtain dark-field or bright-field images, above the specimen, the 200 µm diaphragm is introduced into the system. Through the objective aperture one can choose which electrons contribute to form the image. Figure 8: The two basic operations of the TEM imaging system involve (A) diffraction mode: projecting the DP onto the viewing screen and (B) imaging mode [1]. Institute for Material Physics SS

12 3 Design of a transmission electron microscope 3.2 Diffraction Pattern The core strength of TEM is that one can obtain both a diffraction pattern and an image from the same part of the specimen. In general there are two different forms of specimens: single crystalline specimen and polycrystalline specimen. In single crystals only one crystal orientation is investigated and the resulting diffraction pattern is a spot pattern. In polycrystalline samples a lot of different grains and hence orientations are present (more or less strongly distinct). These grains have the same lattice constant (for an one-phase material) hence only their orientation differs. This results in a ring pattern. A ring with homogeneous distributed intensities, however, can only be obtained for fine grained materials with a strong variation in orientation. The width of these rings can then be used as an inverse measure of grain size. In general for polycrystalline samples the ring patterns are granular. Nevertheless, they can be used to obtain additional information. For obtaining diffraction pattern, above the specimen, the 200 µm diaphragm is introduced into the system and the condenser lens is removed. The operation is shown on the left in figure 8. In this case, however, the SAD aperture is introduced, which makes it possible to obtain a diffraction pattern only from a smaller part of the specimen. In general, the sizes of this aperture vary between 10 and 80 µm. Most of the intensity is in the direct beam and thus in the center of the pattern. To prevent overexposure of the CCD, a beam stop might be introduced. 3.3 High-Resolution TEM (HRTEM) Highest spatial resolution can be achieved using the highest possible diffraction orders for image formation. In modern devices HRTEM images show atomic interference contrast. To understand this concept one may handle the device analog to an optical microscope, which transfers information from the specimen to the CCD with an instrumental characteristic described by the contrast-transferfunction (CTF). The interference contrast in HRTEM micrographs is basically generated by phaseand not by amplitude-differences (phase-shift of the electron wave in the atomic Coulomb-field, not by amplitude changes arising by scattering and absorption). Phase-contrast HRTEM micrographs can usually not be interpreted directly because abberations are modulating the CTF and thus changing the contrast of structural details in the image. By interference, the phase changes from the specimen are converted into detectable amplitudes. Statistically only a single electron is in the column at one point in time. Consequently the micrograph forms by interference of a single electron with the specimen modulated by the CTF. This image is not necessarily a direct mapping of the crystallographic structure. For example: high (white) intensities might indicate the position of an atomic column. Changing the CTF could change the column contrast to low (black) intensities so that simulations are generally inevitable for interpretation. Institute for Material Physics SS

13 3 Design of a transmission electron microscope Focus The choice of the ideal defocusing is crucial to maximize the capabilities in HRTEM. There is, however, no optimum answer for defocussing. In optics, there are two different approaches: 1. Fresnel-diffraction (near field - huygens principle) and 2. Frauenhofer-diffraction (far field - plane wave). For Gaussian focus the defocusing is zero, see figure 9(C). The contrast information are obtained from the smallest sample area and the contrast is localized (no blurring or overlap of information). For some diffracted rays this results in a flipped contrast in the image, making it harder to interpret the image. In scherzer defocusing a small underfocus is performed. This leads to optimum contrast with optimum spatial resolution. The optimum defocus depends on the spherical aberration Cs (Scherzer, 1949): f Scherzer = 1.2 C s λ, (16) Using scherzer defocusing, the best resolution can be obtained. As shown in figure 9 (D) one can use defocusing to detect astigmatism. A weak underfocused lens results in a parallel electron beam. Institute for Material Physics SS

4 Experimental procedure Figure 9: The image of a hole in an amorphous carbon film illuminated with a parallel beam showing that (A) with the beam underfocused, a bright Fresnel fringe is visible;

14 4 Experimental procedure Figure 9: The image of a hole in an amorphous carbon film illuminated with a parallel beam showing that (A) with the beam underfocused, a bright Fresnel fringe is visible; (B) with the beam overfocused, a dark fringe is visible; (C) at exact focus there is no fringe; and (D) residual astigmatism distorts the fringe. Correcting the astigmatism means changing any image similar to (D) to one similar to (A) or (B). (E) the concept of overfocus in which a strong lens focuses the rays from a point in the object above the normal image plane where a focused image (F) of the object is usually formed. At underfocus (G) the lens is weakend and focuses the rays below the image plane. It is clear from (G) that at a given underfocus the convergent rays are more parallel than the equivalent divergent rays at overfocus (α 2 < α 1 ). [1] 4 Experimental procedure 1. Deduce the diffraction constant λ L from a pattern of the polycrystalline Tantalum sample. The literature lattice constant is a = (0.330 ± 0.005) nm; the crystal structure is base centered cubic (bcc). 2. Calculate the lattice constant a of Aluminum from the obtained diffraction patterns of the polycrystalline sample. Aluminum is face centered cubic (fcc). 3. The polycrystalline Aluminum sample is investigated using bright- and dark-field mode. Use the respective images to determine the grain sizes and their distribution. Institute for Material Physics SS

15 5 Questions 4. Identify the crystal orientation of a Silicon sample. First label the diffraction spots with the corresponding Miller indices. Then use this information to determine the beam direction. For the latter part assume that the sample is not tilted and therefore the surface normal coincide with the electron beam. 5. Take HRTEM-images of the silicon samples using different foci. Discuss the influences. Which focus gives the best resolution? 6. Take an image of the hole to deduce the noise of the detector. Use uniformly illumination of the CCD. 5 Questions The following questions shall be answered in preparation to the experiment as well as in the protocol. The diffraction pattern of a Tantalum sample shows diffraction spots that are missing in the case of an Aluminum sample. What is the explanation for this difference? Why are the observed diffraction rings not smooth? Why is it possible to tilt a single crystalline Silicon sample by several degrees without drastically changing the diffraction pattern? What is the relation between diffraction pattern and the reciprocal lattice? Figure 10 shows a diffraction pattern of Copper (single crystal). The lattice constant of Copper is a = nm [4]. Determine the right indices for the diffraction spots and the incident beam direction. In HRTEM all apertures are removed. Why? What determines the contrast? Why is the focus so important? Institute for Material Physics SS

$5 Questions Figure 10: Diffraction pattern of Copper at 200 kv. References [1] David B. Williams, C. Barry Carter; Transmission Electron Microscopy: A Textbook for Materials Science, Band 2 ; 2009.$

16 5 Questions Figure 10: Diffraction pattern of Copper at 200 kv. References [1] David B. Williams, C. Barry Carter; Transmission Electron Microscopy: A Textbook for Materials Science, Band 2 ; [2] M.v. Heimendahl; Einführung in die Elektronenmikroskopie; Braunschweig: Vieweg; [3] Brümmer; Handbuch Festkörperanalyse mit Elektronen, Ionen und Röntgenstrahlen; Braunschweig: Vieweg; [4] C.B.C.D.B. Williams; Transmission Electron Microscopy; New York: Plenum Press; [5] J.H. Bethge; Elektronenmikroskopie in der Festkörperphysik; Berlin: Springer; [6] Kittel; Festkörperphysik; München: Oldenbourg; [7] M. E. Straumanis und L. S. Yu; Lattice parameters, densities, expansion coefficients and perfection of structure of Cu and of Cu-In α phase; Acta Cryst. A25, pp ; Institute for Material Physics SS

Transmission Electron Microscopy

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