Chapter 4 Imaging Lecture 18

Transcription

1 Chapter 4 Imain Lecture 18 d (110)

2 Class Announcement Term Project presentation: startin at :03 PM, Monday, Nov. 17, 08 in CHE 10 Presentation time: 10 min./person Submission: submit a PDF file of your presentation. Due day: Dec. 1, Monday. Final submission miht be modified after presentation based on the comments of audience and you can add additional contents includin new data. If you like, please ive me comments and suestions for this class Presentation outline: 1. materials introduction. objective 3. sample preparation 4. experiment: TEM, STEM, EDX, etc data collection 5. data interpretation Conclusion Nov. 19, Wed.: no class

3 Imain Imain in the TEM Diffraction Contrast in TEM Imae HRTEM (Hih Resolution Transmission Electron Microscopy) Imain STEM imain

4 Imain in the TEM What is the contrast? In microscopy, contrast is the difference in intensity between a feature of interest (I s ) and its backround (I 0 ). Contrast is usually described as a fraction such as: I I I s 0 C = = 0 I I 0 You won t see anythin on the screen or on the photoraph, unless the contrast from your specimen exceeds 5-10%.

5 In summary, BF and DF imaes ( for TEM and STEM ) are formed by usin the transmitted and diffracted beam respectively. In order to understand and control the contrast in these imaes, we need to know what features of a specimen cause scatterin and what aspects of TEM operation affect the contrast. Mechanism of Contrast for TEM/STEM imae 1. Mass-thickness contrast: primary contrast source of TEM/STEM imae for noncrystalline materials such as polymers and bioloical materials.. Z-contrast: Hih resolution STEM imae 3. Diffraction contrast: primary contrast source of BF/DF imae especially for crystalline materials such as metals. Amplitude of e-wave contributes to the contrast. 4. Phase contrast: Hih/low resolution TEM imae (atomic lattice imae). 5. Both the amplitude and the phase contrast: are primary contrast source of electron holoraphy imae for manetic materials 6. In some situations, imae contrast may arise from more than one mechanism and one may dominate. 7. Crystalline sample may enerate mass-thickness and diffraction contrast when aperture is removed.

6 Less scatterin electrons in low mass reion More scatterin electrons in hih mass reion Darker contrast Brihter contrast The reverse contrast will be formed in DF imain mode

7 Z contrast Z contrast usually refers to STEM hih resolution imae The imae is formed by HAADF (hih annular anle dark field) detector as shown in fiure Imain beam condition is always away two-beam condition and close to zone-axis orientation STEM Z-contrast is not equivalent to the TEM Z-contrast imae ( mass-thickness contrast). STEM imae always contains some diffraction contrast Hih resolution Z-contrast will be discussed later. Schematic of the HADDF detector set-up for Z-contrast imain in a STEM. The conventional BF detector is also shown alon with the rane of electron scatterin anle athered by the detectors Ө<10 mrad Ө1>50 mrad HADDF BF HADDF

8 Diffraction contrast Diffraction contrast imain uses the coherent elastic scatterin beam to form imae Diffraction contrast imain is controlled by the Bra diffraction throuh variation of crystal structure and orientation of specimen. Diffraction contrast is simply a special form of amplitude contrast because the scatterin occurs at Bra anle. BF and DF imae is diffraction contrast imae by selectin the direct or diffracted beam as seen fiure. Beam condition of diffraction contrast imain: the incident beam must be parallel in order to ive the sharp diffraction spots and thereby stron diffraction contrast. So we need to underfocus C to spread the beam. Parallel beam condition

9 Two-beam conditions for diffraction contrast imain for defects study, especially for metallic materials To et ood stron diffraction contrast in both BF and DF imae, we usually tilt the specimen to two-beam conditions, in which only one diffracted beam is stron. The direct beam is the other stron spot in the pattern. The electrons in the stronly excited {hkl} beam has been diffracted by a specific set of {hkl} plane. So the area that appears briht in the DF imae is the area where {hkl} plane at the Bra condition. The DF imae contains specific orientation information, not just eneral scatterin information as is the case for mass-thickness contrast. Two-beam conditions are not only necessary for ood contrast, they also reatly simplify interpretation of imae.

10 Diffraction contrast can be primarily used for imain the crystalline materials includin defects. The DP with defects is different than the DP without defects. It is analoous to the -D wall structure -dimensional arranements of bricks Brick patterns are represented by commas as motifs showin the difference

11 Two-beam conditions for diffraction contrast imain BF/DF imae pairs alon with two-beam diffraction of a series of dislocation and a stackin fault in Cu- 15at%Al.

12 Settin up of the two-beam conditions While lookin at the DP, tilt around until only one diffracted beam is stron as shown in fiure. The other diffracted beams don t disappear because of the relaxation of the Bra conditions, but they are relatively faint. In this condition, the contrast miht still not be the best (why?). To et the best contrast from defects, the specimen shouldn t be exactly at the Bra condition (s=0). The specimen should be tilted close to the Bra condition, and s is small and positive ( the excess {hkl} kikuchi line is just outside the {hkl} spot). Never form stron-beam imae with s neative; the defects will be difficult to see.

13 (00) (b) (1-11) BF (0-) (31-1) Only two beams are stron and others are faint due to relaxation of Bra condition (s) DF (c) (a) (a) The [001] zone-axis diffraction patter showin many planes diffractin with equal strenth. In the smaller patterns, the specimen is tilted so there are only two stron beams, the direct [000] on-axis beam and a different one of the {hkl} off-axis diffracted beam. (b) and (c) showin the complementary BF and DF imae under two-beam conditions. In (b), the precipitates is diffracted stronly and appears dark. In (c), it appears briht Al-Li Alloy and precipitates

14 Settin up a two-beam CDF (center dark field) imae To et best BF diffraction contrast, tilt to the desired two-beam condition and insert objective aperture on axis to only select direct beam. A two-beam CDF imae is not quite as simple, i.e. just tilt the incident beam so the stron {hkl} diffraction moves onto the optic axis. If doin so, you will find the (hkl) diffraction or 1 becomes weaker as you move it onto the axis and the (3h3k3l) or 3 diffraction becomes stron. This is called amateur mistake. This is also the procedure to set up the weak-beam condition. To set up a stron-beam CDF imain condition, we need to tilt the (-h-k-l) or -1 diffraction which was initially weak, and it becomes stron as it moves on axis. The CDF technique is important to obtain and interpret the diffraction-contrast imae.

15 Procedure to set up a two-beam CDF (center dark field) imae C (A) Standard two-beam conditions involve the (000) spot and (hkl) spot briht because one set of (hkl) planes is exactly at the Bra condition. (B). When the incident beam is tilted throuh ө so that the excited hkl spot moves onto the optic axis, the hkl intensity decreases because the 3h3k3l spot becomes stronly excited. (C) To et a stron (-h-k-l) spot on axis for CDF imae, it is necessary to set up a stron hkl condition first of all, then tilt the initially weak (-h-k-l) spot onto the axis

16 Diffraction contrast in TEM imaes Diffraction contrast and the appearance of features in BF and DF imaes depend sensitively on how the Laue condition is satisfied, which particular diffraction is active, and the specific value of the deviation parameters, s. i.e. K = +s Diffraction contrast is important for imain the defect in crystalline materials. The kinematical diffraction theory, in which electron absorption inside the solid is inored, will be used to semi-quantitatively analyze the diffraction contrast imain. The kinematical theory allows one to quickly interpret the diffraction contrast imain in crystalline materials.

17 A review of the diffracted beam, z direction is like electron un pointin down = = = = = = k k r r s i r r i k at r r s i R k i R at e F s becomes s F e k r f k F e k S k F k S k or e k R f k π π π π ψ ψ ψ ψ,,, The diffracted wave,, where,

18 = = c s c s N b s b s N a s a s N F I I z y x scatt π π π π π π ψ ψ scatt sin sin sin sin sin sin, scattered wave The intensity of The shape factor depends only on s not. The diffracted intensity is not a constant for any position alon the rod. For a rectanular prism -D distribution of I I s

19 For hih enery diffraction we make these simplifications: (1). The deviation vector is very nearly parallel to the z-axis so s is simply equal to sz. () the quantity N z c is the crystal thickness, t. (3) we can inore the widths of the diffractin columns alon x and y, a useful expression for the shape factor intensity as a function of the scalar deviation parameter, s, is S * S () s ( πst) ( ) πsa sin the intensity for the diffraction, I = = ψ, s z = F ( πst) ( πsa ) sin z I, is

20 summary The intensity of diffracted wave, I, depends on the deviation parameter, s, and the thickness, t. The diffraction contrast in a BF or DF imae is then the variation of I for the diffractin columns located in the x,y-planes of the sample. In two-beam conditions, the intensity of the transmitted beam, I 0, and diffracted beam, I, are complementary. With the incident intensity normalized to 1: I 0 =1-I This kinematical theory is valid when I <<I 0, i.e. s is lare and the diffraction remains weak throuh the depth of the sample. When s 0, the kinematical results is incorrect. A dynamic theory is necessary.

21 s Extinction Distance from Howie-Whelan equation From dynamic theory ( π seff t) ( π s ) sin I π = ξ eff This equation is valid even when the diffracted beam is stron and kinematical theory is not valid. The effective deviation parameter, s eff The extinction distance, ξ, defined as : ξ = = s πv λf + ξ, defined as : where V is the volume of the unit cell, eff λ is the electron wavelenth, and F is the structure factor for the diffraction.

22 Extinction Distance ξ = πv λ F The value of ξ decreases as F increases, i.e. the stroner scatterin, the shorter is ξ The value of ξ increases with the indices of the diffractions (hkl) and decreases with increasin atomic number. The value of ξ enerally rane from a few hundred to a few thousand Anstroms. In the exact Bra condition (s=0) for a sinle diffraction, s eff =1/ ξ, I is periodic with depth in the sample, and the period is the distance, ξ. In other words, when s=0, electron intensity transfers one time back and forth between the forward and diffracted beam over the distance ξ as shown in fiure Kinematical theory cannot predict any periodic transfer of electron intensity between the forward beam and diffracted beam when s=0.

23 Extinction Distance ξ = πv λ F = 1 s eff The intensities of transmitted and diffracted beams for a two-beam condition in a thick crystal, showin the intensities oscillates over the distance ξ as well as the complement between the intensity of transmitted and diffracted beams.

24 The Phase-Amplitude Diaram Construction Application 1. A polar representation of a complex number of modulus unity. Usin this polar representation to develop a raphical scheme, known as phase-amplitude diaram to evaluate the diffracted wave ψ, s = i F e r π s r 3. The diffracted wave as seen above equation is a sum of vectors in the complex plane, and each one represents the relative phase factor of a diffracted wavelet. 4. These phase factors of wavelets are summed from unit cells at proressively reater depths, r, in the specimen.

25 Polar representation of a complex number, exp(i ө), on the unit circle i cosө sinө -1 ө 1 -i z = cos + θ sin θ i

26 The diffracted wave ψ, s = i F e r π s r Incident wave, ψ 0 r 0 =(000) r 1 =1a(001) r =a(001) r 3 =3a(001) r 4 =4a(001) r 5 =5a(001) r 6 =6a(001) r 7 =7a(001) When the Laue condition, K=, is satisfied exactly, and s=0, all of the wave scattered by each layer of the specimen add in phase with each other. This situation is richer when there is a deviation from the Laue condition and s is equal to zero Wavelets diffracted from unit cells at increasin depth, r, and s alon the vertical z direction

27 The diffracted wave ψ, s = i F e r π s r (a) i Ө 4 = π4s.r Ө 3 = π3s.r Ө = πs.r Ө 1 = πs.r Ө 0 =0 x Vectors represents the individual terms, F exp(iπs.r ), in equation (a), ψ(s)

28 The diffracted wave ψ, s = i F e r π s r (a) ψ () s i0 iπ1 s r1 iπ s r1 iπ 3 s r1 iπ 4 s r1 = e + e + e + e + e 5 The phase-amplitude diaram is a raphical sum of complex exponentials, and is equivalent to equation (a). The total diffracted wave, ψ(s), is constructed by addin the vectors tail-tohead as shown below fiure. The anle between successive vectors in the sum is πs.r1. If s=0, all the vectors add colinearly and a stron diffracted in constructed. 1 F 3 4 Phase-amplitude diaram showin the sum of five terms

29 The intensity of diffracted wave is I = ψ * ψ = Re + ( ψ ) Im( ψ ) I is real.the intensity does not depend on the orienttation of the wave in complex space, but only its modulus. i ψ Im(ψ) -1 Ψ* -i 1 Re(ψ)

30 Application of phase-amplitude diaram 1. Frines from Sample Thickness Variation 1.1 The same thickness and different s (a) (b) (c) The incident wave is ψ 0. Suppose the extinction distance for a hypothetical material is eiht layers, so the diffracted wave from each layer is about one-eihth the amplitude of ψ 0. When s is small, the amplitude of diffracted wave, ψ, miht approach the amplitude of ψ 0. Kinematical theory is in trouble. As shown in Fi.(b), after eiht layers, the intensity of ψ nearly equal to the intensity of ψ 0. When s>>0, Fi. (c), after eiht layers, the phase-amplitude diaram has slihtly more than one wrap, resultin a weak diffracted wave. Physically this means that there is enouh destructive interference between the top and bottom of the specimen to suppress the intensity of the diffracted wave. So the kinematical theory can be used for thick specimen when s>>0.

31 1. Frines from Sample Thickness Variation 1. Thickness contours in TEM imaes (a) x briht dark z y briht (b) (c) Phase-amplitude diarams are handy for understandin a wide variety of diffraction contrast features in TEM. As shown in Fi. (a), a wede-shaped crystal is oriented with s>>0, so the vectors of the phaseamplitude diaram wrap into a tiht circle. Very close to the tip of the wede, the diffracted wave amplitude, ψ, increases linearly, and the intensity, I, quadratically, with the thickness of the specimen. Not far from the tip of the wede, I beins to decrease as deeper reions contribute diffracted wavelets that out of phase. The DF imae formed by the diffracted wave shows the dark contrast near the tip of wede and briht contrast in the area with hih value of I. The intensity of the BF imae is complementary to that of the DF imae. Since the wede extends alon x-direction in 3-D view, the briht and dark reions extend as bands alon x- direction formin thickness contours or thickness frines Thickness frines are very commonly observed in TEM specimen, because the specimen are often wedeshape near the ede of a hole. Don t confuse diffraction contrast due to thickness chanes with mass thickness contrast. Diffraction contrast chanes with tilt, but the mass-thickness contrast doesn t.

32 1. Frines from Sample Thickness Variation 1. Thickness contours in TEM imaes (a) briht dark briht dark y x z (b) (c) If we tilt the wede-shaped specimen into a smaller value of s eff ( but still s eff >0), the briht and dark bands will move farther apart as seen in Fi. (b) and (c). Compared to the previous s eff, If specimen is tilted so that s eff is reduced by factor of two, for example, the diameters of the phase-amplitude circles will be increase twice, so will the separation of dark and briht bands. The intensity of the contrast variations will be increase four times. In the process of tiltin from s>>0 to s>0 to s~0, the thickness frines starts as dim band close toether. With tilt to smaller s, the frines appear to expand in width and separation, become much more visible, and appear to move away from the hole in the sample.

33 Thickness frines in dynamic condition (s eff ~0) and kinematical condition (s>>0) (a). The mean intensity distribution of the transmitted (BF, I 0 ) and diffracted (DF, I ) beams with depth for dynamic condition (b). Schematic diaram of a wede-shaped defect-free specimen and the position of the thickness frines in the imae. (e). The intensity distribution versus depth for the Io and I in a kinematical condition ( Io>>I )

34 Thickness frines in dynamic condition (s eff ~0) and kinematical condition (s>>0) (c,d) BF and DF dynamical imaes of a wede-shaped Al foil showin dislocations and thickness frines. (f,) BF and DF kinematical imaes of the same area as c,d, showin dislocations but with only faint thickness frines in BF.

35 Thickness frines in dynamic condition (s eff ~0) and kinematical condition (s>>0) (a) (b) Thickness frines and vacancy loops in aluminum alloy showin a set of alternatin dark and liht bands as well as numerous small defects known as vacancy loops. Vacancy loops is hard to see in thin or thick area, and its best visibility is in the intermediate thickness area (b). Thickness frines become less visible in thicker reions, and disappear with increase of thickness. This is because of the effect of absorption or loss coherent electrons in the thick reions.

36 Example of thickness frines for BF/DF imae (a, b), Ti alloy foil. (c,d) are the simulated imae of (a and b)

37 Examples of thickness frines (a). DF imae of a preferentially thinned rain boundary. (b). Micro-twinned GaAs (a) (b) (c) (c). Chemically etched thin film of MO. The white reions are holes in the specimen