Electron Matter Interaction Fall 2023 Ondrej L. Shanel, Ph.D. with kind help of Andrea R. Konecna, Ph.D. Why Electron Microscopy 2 • Electron benefits • Fundamental • Shorter Wavelength than light at the same energy • Interaction mechanisms with matter (signal types) • Technological • Creation • Manipulation • Detection Proprietary & Confidential Classical Particle description Quantum-mechanical Wave function Non/Relativistic Controlled by fields (electric 𝐄 / magnetic 𝐁) Dr. Konečná – CPO VUT 2022/2023 Electron description Electron properties 4 Proprietary & Confidential | authoremail@thermofisher.com | 21-October-2020 Energy of electron defines its main imaging properties Rayleigh criterion d= 1.22 λ / n.sin d d d Voltage accelerating electron [kV] Speed of electron [v/c] Relative mass of electron [m/m0] Wave length [m] Rayleigh criterion Alpha=14 mrad [nm] Rayleigh criterion Alpha=100mrad [nm] 5 0,14 1,010 1,7E-11 1,51 0,21 10 0,19 1,020 1,2E-11 1,06 0,15 20 0,27 1,039 8,6E-12 0,75 0,11 30 0,33 1,059 7,0E-12 0,61 0,09 60 0,45 1,117 4,9E-12 0,42 0,06 80 0,50 1,156 4,2E-12 0,36 0,05 100 0,55 1,195 3,7E-12 0,32 0,05 120 0,59 1,234 3,4E-12 0,29 0,04 200 0,70 1,391 2,5E-12 0,22 0,03 300 0,78 1,586 2,0E-12 0,17 0,02 α α SEM TEM Objective aperture Objective aperture Sample Sample Electron properties Speed and Mass 5 Proprietary & Confidential | authoremail@thermofisher.com | 21-October-2020 0,1 1 10 0 100 200 300 400 500 600 700 800 900 1000 v/c,me/me0 HT [kV] Speed of electron v Mass of electron me 𝐅 = d𝐩 d𝑡 = d(𝑚𝐯) d𝑡 = −𝑒 𝐄 + 𝐯 × 𝐁 𝑚 = 𝛾𝑚e, 𝛾 = 1 1 − 𝑣2/𝑐2 Lorentz contraction factor Relativistic mass Electron in Classical particle description E – Electrical intensity B – magnetic flux me – electron rest mass c – speed of light 1 2𝑚e −iℏ∇ + 𝑒𝐀 2Ψ − 𝑒Φ∗Ψ = iℏ𝑚 𝑚e 𝜕Ψ 𝜕𝑡 Φ∗ = Φ 1 + 𝑒 2𝑚e 𝑐2 Φ Relativistically corrected scalar potential A – magnetic scalar vector Ψ – wave function Φ- electrical potential me – electron rest mass c – speed of light Wave function description Crystalline Amorphous Described by a potential / scattering probability obtained • From first principles • Quasi-classically • Empirically Sample description Dr. Konečná – CPO VUT 2022/2023 SE… secondary electrons BSE… back-scattered electrons Auger… Auger electrons SE I BSE I BSE III BSE III BSE II 1 nm SE III X- ray hν Light hν AE SE II SE II TE A SE II 5- 50 nm 1- 3 µm Thick sample (SEM)Thin sample (S/TEM ≤ 200nm) Electron – Matter interaction types Dr. Konečná – CPO VUT 2022/2023 Inelastically scattered incident electron Secondary electron Auger electron BSE BSE Dr. Konečná – CPO VUT 2022/2023 Electron – Matter interaction – Energy distribution • Electrons focused to small probe and scanning over the sample • Electron energy: 1-30keV • Resolution ~ 1nm • Thick samples • Signal depends on: • Sample morphology • Sample material • Crystal orientation Scanning electron microscopy - SEM SE I BSE I BSE III BSE III BSE II 1 nm SE III X- ray hν Light hν AE SE II SE II TE A SE II 5- 50 nm 1- 3 µm • Using Focus Beam to Scan over the sample and process signal into Intensity map - Image sample Detector Chamber AC/DC Objective lens Condenser Lens 2 Condenser Lens 1 Gun Deflectors Accelerator Final aperture Deflectors/Stigmator s Scanning electron microscopy - Principle • Electron signals • Secondary electrons – (SE), E<50eV, small escape depth (~10nm) best resolution • Backscattered electrons – (BSE), 50eV50eV, characteristic peaks, surface material composition information • Transmitted electrons (sample must be thin enough) • Absorbed electrons/current • Photons • Cathodoluminescence • X-ray Scanning electron microscopy - Signals http://www4.nau.edu/microanalysis/Microprobe- SEM/Signals.html Vybrané kapitoly z elektronové mikroskopie | 2022 SE I BSE I BSE III BSE III BSE II 1 nm SE III X- ray hν Light hν AE SE II SE II TE A SE II 5- 50 nm 1- 3 µm • Electrons emitted by the sample under electron beam (inner shell ionization effects) • Small escape depth high resolution • Yield depends on local sample tilt Topography contrast • Yield depends on local magnetic or electrostatic fields • Signal is polluted by SE created by BSE in sample – SE2, or on some other surface in specimen chamber (usually final lens) – SE3 noise (information from different part of with different contrast) Secondary electrons Vybrané kapitoly z elektronové mikroskopie | 2022 • Different yield for different materials material contrast • Yield changes with primary beam energy for most materials there is equilibrium point where secondary emission balances primary beam current, i.e. no charging occurs even in case that sample is insulator. Secondary electrons Vybrané kapitoly z elektronové mikroskopie | 2022 • Primary beam electrons reflected by the sample (elastically or inelastically) • Yield depends on atomic number of sample material low loss BSEs reflected close to beam axis – high take off angle • Yield depends on local tilt of sample surface BSEs reflected far from beam axis – low take off angles • Yield depends on crystal orientation channeling contrast & EBSD(P) = Electron Back Scattered Diffraction (Pattern) Backscattered electrons Vybrané kapitoly z elektronové mikroskopie | 2022 • SE image BSE image Examples of SE and BSE images Vybrané kapitoly z elektronové mikroskopie | 2022 Z- contrast Topography Material & topography contrast in BSE signal • Transition of electron in atom filling inner shell vacancy results in release of energy • Energy may be transferred to another electron which is ejected from the atom • Characteristic peaks for elements – analytical method AES- Auger Electron Spectroscopy • Low energies (50eV-3keV)-> small escape depth = surface sensitive method • Extreme surface sensitivity and weakness of signal require usually UHV setup Vybrané kapitoly z elektronové mikroskopie | 2022 Auger electrons • UV to IR light (160nm-2000nm) emitted by the sample under electron irradiation • Effect occurs only in certain materials (semiconductors, minerals, organic molecules) • Direct detection of light emitted by sample, or more complex instruments with monochromator to obtain spectra of emitted light Cathodoluminescence Vybrané kapitoly z elektronové mikroskopie | 2022 • Electron beam induced emission of X-ray has two components • Continuous (“brehmstrahlung”) • Characteristic X-ray – dependent on atomic structure of sample • Peaks of characteristic X-ray corresponds to energy emitted by electron when changing energy levels in atom, thus they enable to determine atomic compound of sample (not chemical structure) Characteristic X-Ray X-ray K O N M L K K K L L Vybrané kapitoly z elektronové mikroskopie | 2022 • EDS or WDS ( also EDX, WDX) • Energy (Wave) Dispersive Spectroscopy (X-ray) • EDS – faster x WDS - more accurate (better energy resolution) • X-ray spectra • X-ray mapping Characteristic X-ray Vybrané kapitoly z elektronové mikroskopie | 2022 • Electrons transmitted through sample without scattering or scattered to space below sample • Only possible for samples with thickness smaller than interaction volume • Electron energy: 30 - 300 keV • Resolution ~ 0.05nm • Signal depends on: • Sample thickness • Sample material • Crystal orientation • Standard imaging - TEM • Scanning transmission electron microscopy – STEM • Electron energy loss spectroscopy - EELS Transmision electron microscopy - TEM • TEM mode – Image of an illuminated sample is magnified onto a camera • STEM Mode - Focused Beam scanning over the sample → processed signal creates an image Camer Data Storage Projection Chamber Diffraction Lens Condenser Lens 2 Condenser Lens 1 Gun Deflectors Accelerator Condensor aperture Image Deflectors/Stigmator s Sample/ Sample holder Sample stage Objective lens Selective Area/Diffraction aperture Intermediate Lens Projective Lens 1+2 Condensor Deflectors/Stigmator s Transmision electron microscopy - TEM TEM - Imaging TEM Diffraction STEM Imaging Back focal plane Diffraction lens Intermediate lens P1 lens P2 lens Sample Upper part of Objective lens Lower part of Objective lens Condensor 2 Gun Filament Sample Condensor 1 Camera/Detector Sample Condensor aperture Transmision electron microscopy – Optical modes Sample Wavefronts representing incoming electron beam Resulting wavefront 𝜓inc 𝐫 𝜓S 𝐫′𝑧 𝐸inc, 𝐪inc 𝐸f, 𝐪f Unscattered electron Elastically scattered electron Inelastically scattered electron Incident electrons Specimen atom Transmitted primary electrons Dr. Konečná – CPO VUT 2022/2023 𝜓 𝐫 = exp(2𝜋i𝑧/𝜆) 𝜓S 𝐫 ≈ exp 2𝜋i𝑧 𝜆S ≈ exp 2𝜋i𝑧 𝜆 1 + eΦS 2𝑚e 𝑐2+2𝑒Φ 2𝑒Φ 2𝑚e 𝑐2+𝑒Φ = exp 2𝜋i𝑧 𝜆 exp ΦS 2𝜋i𝑧 𝜆 e 𝑚e 𝑐2 + 𝑒Φ 𝑒Φ 2𝑚e 𝑐2 + 𝑒Φ = exp 2𝜋i𝑧 𝜆 exp i𝑧𝜎ΦS 𝑣𝑧 𝐑 = ∫ ΦS 𝐫 d𝑧 Suitable for description of thin samples with light atoms. Sample ΦS ≠ 0 Vacuum, ΦS = 0 Vacuum, ΦS = 0 Wave function inside the sample: 𝜎 = 𝑚𝑒𝜆 2𝜋ℏ2Wave function after transmission through the sample: 𝜓S 𝐫 ≈ exp 2𝜋i𝑧 𝜆 exp i𝜎𝑣𝑧 Weak-phase object approximation Dr. Konečná – CPO VUT 2022/2023 𝜓inc 𝐫 , which fulfills 𝐻𝜓inc = 𝐸𝜓inc. 𝜓S 𝐫 = 𝜓inc(𝐫) + 𝑓(𝐫), 𝜓S fulfills 𝐻 + Φ 𝐫 𝜓S 𝐫 = 𝐸𝜓S 𝐫 Let’s assume that prior to the interaction, the beam is described by a wave function: The wave function after scattering on an atom: Electron density 𝜌( 𝐫 ) as a function of distance from nucleus Interaction potential Φ( 𝐫 ) Ref.: Kirkland + 𝜓inc 𝐫 𝜓S 𝐫 =? Dr. Konečná – CPO VUT 2022/2023 Elastic scattering on a signle atom 𝜓S 𝐫 = 𝜓inc 𝐫 + 𝑓𝑒 𝑞 exp i 𝐪 ⋅ 𝐫 𝑟 Final electron wave function after the interaction with an atom: 𝑓𝑒 𝑞 = 2𝜋i 𝜆 න 0 ∞ 𝐽0 𝑞𝑟 1 − exp i𝜎 න Φ 𝐫 d𝑧 𝑟 d𝑟 𝜎 = 𝑚𝑒𝜆 2𝜋ℏ2 For acquiring an image, we propagate 𝜓S 𝐫 through an electron-optical system: 𝐼detector ∝ FT−1 𝜓S 𝐐 TF 𝐐 2 Scattering cross section: -+ 𝜓inc 𝐫 𝜓S 𝐫 Elastic scattering on a signle atom Dr. Konečná – CPO VUT 2022/2023 𝜓S 𝐫 = 𝜓inc 𝐫 + 𝑓𝑒 𝑞 exp i 𝐪 ⋅ 𝐫 𝑟 Final electron wave function after the interaction with an atom: 𝑓𝑒 𝑞 = 2𝜋i 𝜆 න 0 ∞ 𝐽0 𝑞𝑟 1 − exp i𝜎 න Φ 𝐫 d𝑧 𝑟 d𝑟 𝜎 = 𝑚𝑒𝜆 2𝜋ℏ2 C Si Cu Au U Calculation for 𝜓inc ∝ exp(i 2𝜋𝑧/𝜆) 200 keV electrons (Kirkland; Advanced computing in EM) Scattering cross section: Dr. Konečná – CPO VUT 2022/2023 Elastic scattering on a signle atom The simplest approximation: superposition of potentials of independent atoms. ΦS 𝐫 = ෍ 𝑗=1 𝑁 Φ𝑗 𝐫 𝜓S 𝐫 ≈ exp(i𝜎∫ ΦS 𝐫 d𝑧) exp i 2𝜋𝑧/𝜆 Example: Si lattice 10 Å ∫ ΦS 𝐫 d𝑧 Re{exp i𝜎∫ ΦS 𝐫 d𝑧 } Im{exp i𝜎∫ ΦS 𝐫 d𝑧 } Kirkland; Advanced computing in EM Weak-phase object approximation: Si lattice Dr. Konečná – CPO VUT 2022/2023 Kirkland; Advanced computing in EM 𝐼detector = න d2 𝐑det 𝜓prop 𝐑det 2 10 Å Imaging in TEM (simulation): Here approximately pairs of Si atoms Re{exp i𝜎∫ 𝑉S 𝐫 d𝑧 } Weak-phase object approximation TEM: Si lattice Dr. Konečná – CPO VUT 2022/2023 − ℏ2 2𝑚 ∇2 − 𝑒 Φ 𝐫 𝜓tot 𝐫 = 𝐸𝜓tot 𝐫 𝜓tot 𝐫 = 𝜓 𝐫 exp(i 2𝜋𝑧/𝜆) Slowly oscillating term * quickly oscillating term ∇ 𝐑 2 + 4𝜋i 𝜆 𝜕 𝜕𝑧 + 2 𝑚 𝑒 Φ 𝐫 ℏ2 𝜓 𝐫 ≈ 0 Φ 𝐫 = ෍ 𝐆 Φ 𝐆 exp(2𝜋i 𝐆 ⋅ 𝐫)For a periodic crystal: 𝜓 𝐫 = ෍ 𝐆 𝜓 𝐆(𝑧) exp(2𝜋i 𝐆 ⋅ 𝐫) „Paraxial Schrödinger equation“ Thick sample TEM (plane wave on a sample) STEM (focused beam on a sample) 2 layers 20 layers 100 layers 20 layers 100 layers 5 Å Contrast reversal! Dr. Konečná – CPO VUT 2022/2023 Thick sample: GaAs TEM: JEMS https://www.jems-swiss.ch/ Image Simulation SW eels.info 𝐼tot = 𝐼0 + 𝐼inel = 𝐼′ 𝑝0 + 1 − 𝑝0 = 𝐼0 + 𝐼0e 𝑧/Λ 1 − e−𝑧/Λ 𝑝 𝑛 𝑧 = 1 𝑛 𝑧 Λ 𝑛 e−𝑧/Λ • Scattering is quite improbable process; subsequent scattering events can be considered as independent → Poisson statistics • Probability that an electron experiences 𝑛 scattering events after travelling distance 𝑧 inside the sample: • Intensity of the EEL spectrum: 𝐼tot 𝐼0 = e 𝑧/Λ ln 𝐼tot 𝐼0 = 𝑧/Λ Inelastic mean free path Inelastic mean free path Inelastic mean-free path and thickness dependance Dr. Konečná – CPO VUT 2022/2023 Λ = 1/(𝑛 𝜎inel) Number of atoms per unit volume Inelastic scattering cross section 1/Λ = 1 𝜋𝑎0 𝑣2 𝐴 ln 2𝑣2 𝐼 − 7𝐶 2𝑣2 Electron velocity Bohr radius Material-dependent constants: Le and Nguyen-Truong, J. Phys. Chem. C 2021 125 (34), 18946 Inelastic mean-free path Dr. Konečná – CPO VUT 2022/2023 H. Bronw: MeasureIce: accessible on-the-fly measurement of ice thickness in cryo-electron microscopy Inelastic mean-free path in ice Transfer of Image through the optical system 39 Proprietary & Confidential | authoremail@thermofisher.com | 5-August-2020 ψinc Sample A(x,y) – absorbtion φ(x,y) – phase shift Lens W(q) – phase aberration function (lens properties) Back focal plane (bfp) ψs Detector DQE I(R) - Intensity ψdet 𝜓 𝑏𝑓𝑝 (𝑞) = 𝐹𝑇𝜓𝑠 𝑟 Sample Amplitude Influence 𝐴 𝑟 Sample Phase Influence φ 𝑟 = 𝑓𝑒 𝑞 Exit Wave 𝜓𝑠(𝑟) = 𝐴(𝑟)𝜓𝑖𝑛𝑐 𝑟 𝑒 𝑖𝜑(𝑟) Incoming Wave 𝜓𝑖𝑛𝑐(𝑟) when 𝐴 𝑟 ≪ 1 𝑎𝑛𝑑 𝜑 𝑟 ≪ 1, 𝜀 𝑟 = 𝑙𝑛𝐴(𝑟) Exit Wave 𝜓𝑠(𝑟) = 𝜓𝑖𝑛𝑐 𝑟 [1 + 𝜀 𝑟 + 𝑖𝜑(𝑟)] And assumption 𝜓𝑖𝑛𝑐 𝑟 = 1 (𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 𝑖𝑙𝑙𝑢𝑚𝑖𝑛𝑎𝑡𝑖𝑜𝑛) 𝜓 𝑏𝑓𝑝 𝑞 = 𝛿 𝑞 + 𝐸 𝑞 + 𝑖Φ(𝑞) Aberrations addition 𝑊(𝑞) = 𝜋 2 (𝐶𝑠 𝑞4 𝜆3 + Δ𝑓𝑞2 𝜆 ) 𝜓 𝑏𝑓𝑝,𝑎𝑏 𝑞 = 𝛿 𝑞 + 𝐸 𝑞 𝑒−𝑖𝑊(𝑞) + 𝑖Φ(𝑞)𝑒−𝑖𝑊(𝑞) Optical Intensity at Image Plane I 𝑅 = 𝜓 𝑚(𝑅𝑑 𝑒𝑡) 2 = 𝐹𝑇𝜓 𝑏𝑓𝑝,𝑎𝑏 𝐹𝑇𝜓 𝑏𝑓𝑝,𝑎𝑏 ψbfp I 𝑅 = 𝜓 𝑚(𝑅𝑑 𝑒𝑡) 2 = 𝐸𝑡 ∗ {1 − 2𝜑 𝑄 sin 𝑊 𝑄 + 2𝜀 𝑄 cos(𝑊 𝑄 } Michal Brzica bachelor thesis – derived from RICOLLEAU, C., et al. Random vs realistic amorphous carbon models for high resolution microscopy and electron diffraction. Journal of Applied Physics, 2013, 114.21: 213504. ISSN 0021-8979. Available from DOI: 10.1063/1.4831669. 40 Phase shift – Carbon sample Contrast Transfer Function 41 Proprietary & Confidential | authoremail@thermofisher.com | 5-August-2020 ψinc Sample A(x,y) – absorbtion φ(x,y) – phase shift Lens W(q) – phase aberration function (lens properties) Back focal plane (bfp) ψs Detector DQE I(R) - Intensity ψdet I 𝑅 = 𝜓 𝑚(𝑅𝑑 𝑒𝑡) 2 = 𝐹𝑇𝜓 𝑏𝑓𝑝,𝑎𝑏 𝐹𝑇𝜓 𝑏𝑓𝑝,𝑎𝑏 ψbfp I 𝑅 = 𝜓 𝑚(𝑅𝑑 𝑒𝑡) 2 = {1 − 2𝜑 𝑄 sin 𝑊 𝑄 + 2𝜀 𝑄 cos(𝑊 𝑄 } Contrast Transfer Function (CTF) • Describing optical property of TEM CTF Ԧ𝑞′ =𝐸t 𝑞′ 𝐸s Ԧ𝑞′ 𝐸d Ԧ𝑞′ 𝐸u Ԧ𝑞′ ⋅ Intenzita Ԧ𝑞′ ∈ −1; 1 where 𝐸t 𝑞′ - temporal coherency 𝐸s Ԧ𝑞′ - spatial coherency 𝐸d Ԧ𝑞′ - drift impact 𝐸u Ԧ𝑞′ - vibration dumping Michal Brzica bachelor thesis – derived VULOVIĆ, Miloš, et al. Image formation modeling in cryo-electron microscopy. Journal of structural biology, 2013, 183.1: 19-32. ISSN 1047-8477. Available from DOI: 10.1016/j.jsb.2013.05.008. 42 CTF is not seen directly on our PC! Observed Intensity on PC Irn – Read-out noise Idc – dark current CF – Conversion ration e/signal Φe – Primary electron number CTF – Contrast Transfer Function DQE – Detector Quantum Efficiency NTF – Noise Transfer Function Conclusion Electrons are powerful imaging particle Undertading of imaging/interaction principles is the key for understanding of imaged data Next – Design of Transmission Electron Microscopes