Lecture 6. What is contrast? Eyes: < 5% - can t detect <10% - difficult Enhance digital image electronically I I C I

Transcription

1 Phae Contrat

2 Lecture 6 Scattering/Amplitude Contrat BF/DF: thickne, tilt Low-medium magnification Application Crytal defect: dilocation, tacking fault, phae boundarie, precipitate, defect cluter Contrat: g, type of the fault, it depth in the crytal Quantitative determination of b Reolution: trongly excited: 10 nm; weak beam: 1nm Phae Contrat: High magnification Highly coherent beam Large defocuing Reolution: nm Reliable interpretation: Simulation What i contrat? C I I I 1 1 Difference in intenity between two adjacent area I I Eye: < 5% - can t detect <10% - difficult Enhance digital image electronically 1

3 Interference Effect Frenel fringe around a hole in a carbon foil The dicovery of carbon nanotube Single atom and cluter of atom of tungten on the urface of a few atomic layer of graphite

4 Effect of two beam reaching the image Aperture: two beam (the incident and one diffracted) exp Unit amplitude ik r expik r g exp ik r 1g expi k r expik r 1g expig r int R exp i g R g t t Along x: perpendicular to the diffracting plane The total intenity ditribution: I 1 R R co g r 1 R x Rin t d -Lattice reolved (the reolving power of the microcope?) - At leat one order + the incident - More beam: harpen the detail - The viibility: orientation and thickne in inn t n R g t t g n 0 - The poition of the fringe: no imple relation to the poition of atomic plane (Next page) The fringe: not obervable Periodicity: d

5 Lattice Fringe: Thickne and Orientation Thickne variation change the contrat: no imple relation to the atomic plane (111) Lattice fringe Sodium faujaite, d = 1.44 nm Similar effect, when change

6 Summary of Theory A coherent ource of illumination Very thin < 3nm Two or more beam recombined to produce intenity difference from point to point Intrumental factor, phae plate High gun brightne Medium acceleration V, low C Parallel beam illumination: the illumination aperture < 0.1 mrad Highly table - Accelerating Voltage~ 1ppm: low energy preading - Vibration-free mechanical tability Coma-free on-axi alignment Sample height and orientation Simulate through-thickne-through-focu Phae hift ariing : at the pecimen ~ cattering the pherical aberration the defocu of the objective len

7 Phae Shift: Specimen Path difference relative to the wave front in vacuum r E E E E E 0 nr 1dz V rdz 0 Variation in n from point to point acro the pecimen n(r) atomic potential The exit wave amplitude: r rexpi r Ignore the local amplitude modulation: ~1 r 1 i r 1 i r me r h p r V rdz p Strong-phae object Weak-phae object The projected pecimen potential

8 Lack of Contrat at exact focu A phae hift of 90 o between the direct and cattered wave (thin ample) For thin ample, c << I I i ~ I i c i The phae object i inviible If the phae of the cattered wave can be hifted by a further 90 o I i c i I i c 0 Empirical method to enhance the contrat of image Knife edge blocking half of the tranvere patial frequencie Reducing the ize of the objective aperture limited reolution Introducing a focuing error (defocu) Uual method in HRTEM Simple intervention in the BFP (Schlieren contrat) Appeared from time to time Ue of BFP phae plate Optical

9 Phae hift: Intrumental Factor Spherical aberration: Wave front: equal phae Ray: trajectorie orthogonal to the wave front Two ray: R, R+dR Optical path difference due d dr to the angular deviation Total path difference or the phae hift relative to the optical axi Part of the outer zone of a len at a ditance R from the optical axi W R 0 d R 0 dr Angular deviation!!

10 Phae hift: Intrumental Factor Spherical aberration Change of pecimen poition Change of focal length f R C b M C b r f ar a f fr f f a Total angular deviation: The phae hift: z C f R a f f R C dr W R f R underfocu 0, a f z Scherzer formula

11 Phae hift: Intrumental Factor The wave aberration: C and Introducing reduced coordinate * C 1/ 4 ; z * z C 1/ The reduced wave aberration: * *4 * * W 4 z Scattered wave are hifted with a phae of 90 o. Scherzer focu: z * 1 W ha the value 90o over a relatively broad range of cattering angle

12 Image formation Abbe Theory A r B Plane wave: exp ik r 0 u Exit wave: e r ~ 0 a r exp ir expik r r: radiu vector in the pecimen plane q f F q r exp iq r ds S e Fourier tranform q: radiu vector in the diffraction plane Pupil function: aberration/aperture/defocu m 1 M 1 M r Fqexpiq r d q r m S Invere Fourier tranform 1 M r FqH qexpiq r d q S H q exp iw qm q

13 Contrat-Tranfer Function CTF: The imaging propertie of an objective len Independent of any particular pecimen tructure A pecimen with a ingle patial frequency (q = 1/d) The exit wave: 1 The Fourier tranform: F q q iq x 1 coqx i coqx... q The image: x m 1 q iq exp iw qexp iqx 1 q iq exp iw q The intenity: Ix x q q 1 m 1 co 1 D q W q q coiqx inw q q coiqx coiqx B( q) coiqx q q... co iqx D(q) : the CTF of the amplitude tructure of the pecimen B(q): the CTF of the phae tructure * * * * in 4 * B( ) inw z Ideal CTF: B = for all *

14 CTF: Defocu Poitive phae contrat Broad band o patial frequencie with the ame ign of the CTF No main tranfer band for z 0 Negative phae contrat Pa through zero: tranfer gap, for correponding q, no information reache the image Scherzer reolution/point reolution The firt zero: The maximum k for which phae coherency i preerved at Scherzer defocu d p 4 C 3 6

15 Influence of Energy Spread and Illumination Aperture Energy preading: E= 1- ev(thermionic); (Field Emiion) Finite ize of the gun: illumination aperture i. ( i << o, partially patially coherent) E reult in f: electron with different f uperpoed incoherently at the image I x 1 B( q) K q coiqx c q Envelope function: K c q q H exp 4 H C c E 1 E / E0 E 1 E / E 0 Information limit: the damped CTF goe to 0 Chromatic aberration Energy pread Fluctuation in high voltage Beam convergence Spatial incoherence Damp the CTF for increaing q

16 Contrat Tranfer Function (PCTF) Univeral curve Independent of pecimen and microcope Ocillatory nature Depending on: Envelope function: Coherent/partially coherent illumination» Temporal: Focal pread» Spatial: finite divergence» Not affect the poition of the zeroe Focu Scaling factor ued for pecific objective len/beam energy Recap: CTF

17 Reolution Limit Interpretable image reolution: tructural/point re At optimum(scherzer) defocu: PCTF larget band of patial frequencie without any phae reveral the firt zero Intrumental reolution: information limit Envelop function: cut-off 15% for image proceing Fine detail preent, not directly interpretable Focal Serie Recontruction Lattice-fringe reolution: finet pacing' that can be obtained obolete Fine tructure may not contain atomic information

18 Lattice and atomic imaging Comparatively very few lattice image are directly interpretable in term of atomic arrangement Phae reveral: difficult to achieve optimum image defocu Bet focu can be achieved by uing: edge for Frenel/a defect Knowledge of focal tep ize Small-unit-cell material A few diffracted beam contributing Fourier/Self-image: identical image occur periodically in defocu d / Contrat reveral by a half period Thick ample Dynamical complexity In the vicinity of the thickne extinction contour Low direct beam Strong diffraction between diffracted beam: dominating the image contrat Dumbbell different from the projected atomic eparation except for highly pecific thickne and defocu cannot be interpreted a atomic poition

19 Application: an example Accommodation of lattice mifit: Stre relieve reult in: interfacial mifit dilocation Cro ection of Ge/Si(001) heterointerface 1. J. N. Stirman, P. A. Crozier, D. J. Smith, F. Phillipp, G. Brill and S. Sivananthan, Appl. Phy. Lett. 84 (14), (004).

20 Apendix

21 Outline High Reolution TEM Elementary principle of Phae Contrat Weak-phae Object Approximation Multi-lice Method Peudo-weak-phae Object Approximation Reolving Power Application of HRTEM Identify crytal tructure Microtructure and defect Example of HRTEM Recent Progre of HRTEM

22 Brief Hitory of HRTEM 1949 Scherzer: Fundamental HRTEM Contrat Theory 1956 Menter: Experimental obervation TEM: Mechanical, thermal and electrical tabilitie 1930, exceeded the reolution of VLM 1970, the firt tructure image 1980, 0. nm, interpretable reolution limit Mid-1990, FEG information limit 0.1 nm