Chapter 4 Imaging Lecture 4 d (110)
Final Exam Notice Time and Date: :30 4:30 PM, Wednesday, Dec. 10, 08. Place: Classroom CHEM-10 Coverage: All contents after midterm Open note Term project is due today
Imaging Imaging in the TEM Diffraction Contrast in TEM Image HRTEM (High Resolution Transmission Electron Microscopy) Imaging or phase contrast imaging STEM imaging
Basic Imaging Theory 1.The imaging process As a first approximation, we consider coherent image, i.e. a plane parallel, monochromatic electron wave falls in the object, and is scattered, the scattered waves interfere and recombined by an electron lens (objective lens) to form an image.. The process consists of two parts: (a). The interaction of the incident wave with the object, defining the exit wave from the object, transmission function q (x); (b) the action of the objective lens in forming the image, transfer function t (x). ψ(x)= q(x)*t(x), and I (x)= q(x)*t(x) ^ or ψ(u)= Q(u)*T(u). 3. The essential question: how is the images intensity distribution related to the object structure? 4. For detailed discussion of imaging and electron scattering, see Diffraction Physics by John M. Cowley, North-Holland, Amsterdam, 1981, nd edition.
Many beams were selected by an aperture (ring) to form image showing more information about the structure of specimen. the question is where are the atoms with respect to the bright and dark contrast dots? Do the dark spots or bright spot correspond what atoms (Ga or As), etc.? Beautiful image sometimes is difficult to interpret. a. Simulation b. SAD pattern c. FFT pattern
5. Imaging in WPOA Imaging function ψ I I I ( x) = ( 1 iσv ( x) ) t( x) ( ) ( ) [ ( )] [ ( ) ] iχ u t x = F T u = F A u e = F[ A( u) ( cos χ( u) + i sin χ( u) )] c( x) = F[ A( u) cos χ( u) ] s( x) = F[ A( u) sin χ( u) ] ψ ( x) = ( 1 iσv ( x) ) ( c( x) + is( x) ) ( x) = ψ ( x) ψ ( x) * = ψ ( x) where Let and so ignore high order term, σ ( x) = 1+ σv ( x) s( x)...( a) ( u) = δ ( u) + σq( u) S( u)...( b) The effect of equation (a) is that mulyiplying diffraction pattern in BFP by the phase contrast transfer function S ( u) = F[ s( x) ] = A( u) sin χ( u) or in Fourier space
S ( u) = A( u) sin χ( u) where ( ) A u ( ) 1 3 χ u = π fλu + πcsλ u u = θ / λ, reciprocal vector χ ( u) may be called phase distortion function Phase contrast transfer function is aperture function the phase contrast transfer function (PCTF) is oscillatory, showing maximum transfer of contrast when χ(u)=±π/, and zero contrast when χ(u)=±π When S(u) is negative, positive phase contrast results, meaning that atoms would appear dark against a bright background. When S(u) is positive, negative phase contrast results, meaning that atoms would appear bright against a dark background. when S(u) =0, there is no detail in the image for this value of U 4
Cs=1. mm, and E=300 kv 1.5 delta f, -58nm delt f, -150 nm delta f, -10nm sinx(u) 1 0.5 0-0.5-1 0 4 6 8 10-1.5 - u (1/nm) u If we fix Cs and E, u increases as f decreases achieving a higher spatial resolution. Further decreasing f will cause u suddenly jump to cross the u-axis and decrease u
Scherzer defocus the CTF could be optimized by balancing the effect of spherical aberration against a particular negative of f, called Scherzer defocus. At this defocus, all beams will have nearly constant phase out to the first crossover of the zero axis. This crossover point is defined as the instrumental resolution limit. This is the best performance that can be expected from a microscope unless we use imaging processing method to extract more information. The implications of resolution require as many beams as possible being transferred through the optical system with identical phase.
( ) ( ) 4 3 4 1 1 4 3 3 3 3 3 4 3 0.66 Scherzer resolution 3 4 Scherzer defocus and (b) combining (a)...(b) 1 3 - so be near -1 will sin, 3 - when as shown in figure, ) 0...( 0, let 1 λ λ λ π λ π π χ π χ λ λ π λ π λ π λ π χ λ π λ π χ s Sch Sch s s s s s C r Cs f u C u f a u C f u C u f u C u f du u d u C u f u = = + = = = + = + + = + = d c s g p d d d d d + + + = 4 3 4 1 min 4 1 0.8 0.9 λ λ α s s optimum C d C = = Based on the electron probe size Resolution
(a) (b) Envelop Damping Function The plots of sinχ(u) as a function of u could extend out as far as you want to plot them. In practice, they don t because of the envelop damping function, i.e. the microscope is unable to image the finest detail due to reasons other than the simple transfer characteristics of a linear system. Envelop damping function results in the information retrieval limit or instrumental resolution limit. If we set up the Scherzer defocus or damping function, i.e. whichever equals zero first, the phase-contrast images are directly interpretable. Beyond the limit, we have to use computer simulation for interpretation.
The minimum contrast defocus condition all contrast is minimized and cannot see anything in minimum contrast condition : sin χ and ( u) = 0.3 1 ( λ) f MC = 0.44 C s Imaging using first passbands using minimum contrast defocus as a reference point, we can adjust the defocus to Scherzer defocus. As shown in figure, the first passband condition settings for CTF allow higher spatial frequencies to contribute to the image, and X is constant or dx/du is small over a range of u which includes the reflection of interest.
Cs=1. mm, and E=300 kv delt f, -150 nm sinx(u) 1.5 1 0.5 0-0.5-1 -1.5-0 4 6 8 10 u (1/nm) As shown in above figure, we may obtain a pretty picture using high order passbands settings, but it may not give a true structure image of specimen. at the high-order passbands setting, imaging the specimen is beyond the instrumental resolution limit so we cannot use the intuitive approach for image interpretation. we should exactly know where the zeros are in the CTF, and use computer simulation to assist the image interpretation.
Simulation of high resolution TEM image Thick specimen Slices Multi-slice method is used to simulate HRTEM image a specimen of thickness, t, is modeled as n slice, and each of thickness is z, or t=n z The crystal potential of each slice is replaced by its two-dimensional projected potential. The effect of the first slice on the phase of incident wave-front is calculated, and the resulting wave function is then propagated through free space to the next slice. This process is repeated until the desire specimen thickness is achieved. Assuming z is sufficiently small, this method is highly accurate.
Schematic procedure of simulation of HRTEM
Thickness, nm defocus, nm The output of simulation is always in this type of table showing variation of image with thickness and defocus Simulated HRTEM images of a Ti/TiH interface over a range of crystal thickness and defocus values. The zone axis is [0001] of Ti
HRSTEM imaging Spectroscopy Other technologies in TEM Basic SEM Review
Schematic view of the various signals and their detectors used on a STEM STEM imaging or Z-contrast imaging The Scanning Transmission Electron Microscope ( STEM) provides the facility of scanning a small probe across a specimen in the TEM, comparable to a Scanning Electron Microscope (SEM). The interaction between the electron beam and the specimen generates different signals like transmitted electrons, backscattered electrons, secondary electrons and X-rays, containing information about the actually illuminated part of the specimen. In the STEM, the beam is scanned across the specimen in a rectangular pattern. At the same time, the intensity of one or more signals is measured and translated into brightness values. An image containing specific information about the specimen is collected while scanning the beam. Z-contrast images are formed by collecting high angle ( 75-150 mrad, an order magnitude larger than Bragg angle) elastically scattered electron with an annular dark field detector ( HAADF).
Using the STEM detector to form BF and DF image STEM image formation: A Bright Field (BF) detector is placed in a conjugate plane to the back focal plane to intercept the direct beam while a concentric Annular Dark Field (ADF) detector intercepts the diffracted electrons. The signals from either detector are amplified and modulate the STEM CRT. The specimen (Au islands on a C film) gives complementary ADF and BF images as can be seen by clicking the button opposite.
CBED mode Ray diagram and beam condition of STEM In the coherent nano-probe mode (sub-nanometer diameter) with the smallest possible probe size ( for filed emission gun d = 0.-0.4 nm) To form coherent CBED pattern and the coherent CBED disks are allowed to overlap STEM image is formed by scanning these overlapped CBED pattern ( nano-probe) Nano-probe to form CBED pattern STEM mode is good for spectroscopic analysis
Small convergent angle, α Ray diagram showing how to form CBED pattern
large convergent angle, α The pattern is overlapped due to high convergent angle
CBED pattern without overlapping
CBED whole pattern (small camera length) without overlapping
CBED pattern with overlapping
Lattice imaging in STEM using overlapping CBED orders If CBED orders overlap, and if the angular range over which the illumination is coherent exceeds the Bragg angle, it becomes possible to form a phase contrast STEM lattice image or called high resolution Z-contrast image. The image is formed by detecting part of the CBED pattern ( the overlap region) and by displaying this intensity as a function of probe position, as the probe is scanned over the sample. As shown in figure, two illumination angles allow the overlap of CBED, so a detector at D record a twobeam lattice image as the probe is scanned. Interchanging S and D point shows that this is equivalent to inclined illumination TEM lattice imaging. This is called reciprocity theorem, which states if we swap a point source of illumination between the TEM and the STEM configurations, the intensity at the detector will be the same. In STEM, the focusing of incident electrons into a small spot requires converging ray paths, in analogy to the requirement in TEM that rays are included that have undergone scattering to large diffracted vector and contain information on short spatial periodicities.
Axial three-beam lattice imaging in STEM. By opening up the illumination angle to twice the Bragg angle, three orders overlap at the axial detector D. The appearance of a two-dimensional coherent CBED pattern used for axial five-beam lattice imaging is shown at the right
Reciprocity theorem applied to TEM and STEM: If the direction of electrons in STEM is reversed and the source and detector interchanged, then the BF-STEM and the TEM are identical. the image in the BF-STEM is equivalent to the BF image in the CTEM. ADF-STEM image would be equivalent to the BF-TEM image using incoherent hollow cone illumination.
A B Idealized relationship between two imaging modes: the wave amplitude at point B in a system due to a point source at point A is the same as the amplitude that would be produced at A by a point source at B. All contrast mechanism possible in CTEM are also possible in STEM, I,e, phase contrast, diffraction contrast, Fresnel fringes etc.
Rutherford Scattering The relative angular distribution of the transmitted electrons With a HAADF detector, the scattered electron to extreme large angle are collected and form a Z-contrast image.
Mechanism of STEM imaging The specimen in (a) consists of an array of atomic columns ([110] Si for example), for which the potential for high-angle scattering can be represented by an object function consisting of weighted spikes, as in (b). The STEM image can ed interpreted as a convolution of the nano-probe and the objection function, as in (c). As probe scans, it maps out the spikes, producing a direct image of high-angle scattering power. It acts as the single probe lightens up the bulbs (atom columns) while scanning. The bright spots are location of atoms. Comparing to HRTEM, the resolution in STEM imaging is less affected by phase errors induced by objective lens. The incoherence of the STEM images (or Z-contrast image) is a consequence of their large K (scattering vector). For the incoherent imaging, we sum the intensities, I, from the atoms, rather than the wave-function amplitudes, ψ, from the atoms as in coherent HRTEM imaging. Each atom can be considered as an independent scatter because there is no constructive and destructive interference between the wave amplitudes scattered by the different atoms. At high angles, the scattering scales with atomic number approximately as Z^, as expected for Rutherford scattering from an unscreened nucleus.
Mechanism of STEM imaging (cont ) When the probe is smaller than the spacing of aligned atomic columns in a crystal, the atom columns are illuminated sequentially as the probe is scanned over the specimen. This Z-dependence scattering carries composition information about specimen independent of the geometry of the unit cell and the presence of forbidden diffractions or defects. The interpretation of the Z-contrast image is almost intuitive.
(a) (a). Simulated defocus series for Si<110> with corresponding probe intensity profiles STEM image and simulated Si dumbbells structure Mechanism of STEM imaging (cont ) Changing the focus of the objective lens alters the intensity distribution of the probe on the specimen surface as shown in above figure for Si<110>. Near Scherzer defocus, -70 nm, the probe most closely resembles a diffraction-limited Airy disk. At smaller defocus the central peak broadens, and at larger defocus, the central peak sharpens, but more intensity appears in the subsidiary maxima. While these conditions result in significantly different images, the contrast away from Scherzer defocus is reduced substantially in both cases. So in practice, the Scherzer defocus can be found readily. The good contrast image is only from one defocus, allowing intuitive interpretation. As shown in figures, although the expected Si dumbbells are not resolved in this <110> images, by knowing that atoms act as sharp objects, one can infer from the oval shape of the bright spots that at least two atomic columns must be present closer together.
STEM imaging theory (1-D) ψ t (u, x p ) the aberration wave function : e max iχ ( u ) ( x, x p ) = Ap e ( x, x ) dx = 1 [ iχ ( u )] The aberrated electron probe wave function in the plane of the specimen when deflected to position x is : ψ A p ψ 0 [ ] [ πiu( x x p )] e du is a normalization constant chosen to yield p ψ t ( x, xp ) is ( x, x ) = t( x) ψ ( x, x ) function form CBEDpattern u the electron probepasses through the specimen and the transmission function is t(x),so the transmitted wave ψ t this wavefunction is then diffractedonto thedetector planeand Ψ p πiux ( u, xp ) = F[ ψ t ( x, xp )] = e ψ t ( x, xp ) dx this wavefunction Ψ( u, x ) theintensity of p scatteringangleφ = λu is the CBEDpattern p p p as a function of
The CED pattern is incoherently integrated over the detector geometry and the result is the final STEM image signal g(x p ) for one probe position x p. ( ) ( ) x Ψ u, x D( u) g = p p du where D(u) is the detector function ( ) D u = 1 umin < u < u 0 otherwise max Where (λu min ) and (λu max ) are the inner and outer angles of the ADF detector. This process is repeated for each position x p This equation is difficult to intuitively relate to any specific structure in the specimen. An approximated linear image model for thin specimens assuming an incoherent image process is
g ( x) = f ( x) * h ( x) where the specimen function f(x) is approximatedly the probability for scattering to the large angles of f The outer dimension of ADF detector is large enough (infinite comparing to the inner dimension ADF, and then image produced by ADF is an incoherent image ( x) D( u) σ u ( x) s position x of ADF σ u umax ( x) σ ( x) s du u is the partial cross section for scattering to angle u the specimen. s = min u ADF detector With the incoherent image assumption an ADF-STEM image of s du a very thion specimen is essentially a mass thickness map of the specimen s s at
The point spread function is just the intensity distribution in the focused probe h where J max iχ ( u ) ( x) = ψ ( x) = A e max iχ ( u ) () r = A e ( x) [ ] ignore astigmatism and for 1- D, the probe intensity is h ADF ADF 0 p p u 0 p u [ ] J ( πur) 0 0 udu is the zeroth order Bessel function, and r is the radial coordinate. This integral can only be done numerically du
Circular aperture In the -D, the transmission function of a circular aperture is f ( ) ( 1 if x + y ) < ( a / ) x, y = Then 0 ( πau) elsewhere πa J1 F( u) = πau u is a radial coordinate and J the first order Bessel function J 1 ( z) = 0 k k (-1) z k k! ( k ) z = + 1! k 1 (x) is the first order Bessel function.
Defocus is in Scherzer condition, D=1. and umax=1.56 D=0.8 and umax=1. D=1.5 and umax=1.5 D=.5 and umax=.5 Calculated STEM probe intensity ( approximate point spread function) when astigmatism is negligible) versus normalized radius, R, defocus D and obj. aperture umax and umin.
The of H transfer function is the spread function : ADF 0 ( u) A h ( r) J ( πur) = rdr p ADF just the inverse Fourier transform 0
Calculated approximate STEM transfer function corresponding to the defocus and apertures used in the previous probe intensity figure. K is normalized reciprocal vector
R FWHM= 0.5 Defocus is in Scherzer condition, D=1. and umax=1.56 Rres.=1/u=1/3 Resolution loss u D=0.8 and umax=1. u h ADF Probe intensity function u p 0 max iχ ( u ) () r = A e [ ] J ( πur) 0 udu Resolution loss Transfer function H ADF 0 ( u) A h ( r) J ( πur) = rdr p ADF 0
Cs=1. mm, and E=300 kv Transfer function sinx(u) 1.5 1 0.5 0-0.5-1 delta f, -58nm delt f, -150 nm delta f, -10nm S ( u ) = A ( u ) sin χ ( u ) 0 4 6 8 10 TEM: where χ STEM: 1 3 4 ( u) = π fλu + πc u sλ -1.5 - u (1/nm) H ADF 0 ( u) A h ( r) J ( πur) = rdr p ADF 0 u u
Obtaining STEM image and alignment of STEM forming CBED pattern in a very large aperture for HRSETM, orientate crystal on zone axis accurately focusing the probe so that a over-lapped CBED pattern (for crystalline specimen ) or ronchigrams (for amorphous specimen) are formed. Ronchigrams is named by John Cowley, because the geometry used to obtain them is identical to that used to test optical lenses and mirrors. Pioneer work for calculation of Ronchigrams, J. Electron Micro Tech, 3:5-44, (1986): J.M. Cowley.
Calculated Ronchigrams, 00
Experimental Ronchigrams in amorphous area CBED pattern with large converge angle in crystalline area
(a) (b) Overlapped CBED pattern and the probe ready for STEM imaging (a). Si <110>; (b). Steel
(a) (b) Using Ronchigrams to align STEM optics (a): Misaligned probe (b): Astigmatic probe, x/y are not symmetric (c): Optimized probe (c)
Examples of STEM image Z-contrast image
Z-contrast image
HRSTEM image HRTEM image
Probe condition and HRSTEM image showing atom location
Mapped HRSTEM image of Si showing the dumbbell structure
<110> simulated and experimental images of interfacefacial ordering I a (Si 4 Ge 8 ) 4 superlattice grown on Ge <001> at 350C
HRSTEM image of Si and FFT
50 nm Simulated HRSTEM images of <110> GaAs at 00 Kev for a thickness of (a) nm, and (b). 0 nm at Scherzer condition
1 nm Simulated HRSTEM defocus series of silicon nitride (β-si 3 N 4 ), 5 nm thick at 00 Kev. The defocus values are: (a): 70 nm; (b): 90 nm; (C): 110 nm; (d): 130 nm Silicon atom positions appears white in (a) and do not reverse contrast unlike the HRTEM BF phase contrast image. However, there are still significant artifacts in the image that appear as the defocus is increased from Scherzer defocus (70 nm). The artifacts are consistent with the tails of the probe intensity. STEM is less sensitive to changes in defocus. Both HRTEM and HRSTEM images are not straightforward to interpret. Image simulation is one approach to verify the image is interpreted correctly.
The artifacts due to increasing the defocus are consistent with the tails of the probe intensity.
N Si N Si HRTEM image of silicon nitride (β-si 3 Ni 4 ) with different defocus (a) 70 nm and (b) 90nm showing the contrast reverse HR-STEM image of silicon nitride (β-si 3 Ni 4 ) with different defocus (a) 70 nm and (b) 90nm showing the contrast do not reverse but artifacts are introduced N 1 nm N Si Si artifacts
Next lecture: Electron Energy Spectroscopy in TEM EDX: energy-dispersive X-ray spectroscopy EELS: Electron energy loss spectroscopy