1. Theory

Definition: A scanning electron microscope scans a focused electron beam of about 0.5 to 30 $keV$ across a sample and builds an image from the electrons and X-rays emitted at each point (Trieste; MIT).
Signals: Secondary electrons, from the top few nanometres, show topography; backscattered electrons, from deeper, show atomic-number contrast; characteristic X-rays give composition (Trieste; MIT).
Interaction volume: Beam electrons scatter through a pear-shaped volume whose depth follows the Kanaya-Okayama range (MIT).
Resolution: Set by the probe size and the volume from which the signal escapes, not by the electron wavelength alone (Cornell CNF; MIT).

2. Working Principle

Electron source and column: A field-emission or thermionic gun, condenser and objective lenses demagnify the source to a small probe (Cornell CNF; Mahapatro).
Scanning: Deflection coils raster the probe; the recorded signal intensity at each point builds the image (Cornell CNF).
Detectors: Everhart-Thornley and in-lens detectors for secondaries, solid-state or scintillator detectors for backscattered electrons, silicon drift detectors for X-rays (Cornell CNF; Bruker; Trieste).
Vacuum: High vacuum for conductors; variable pressure or low voltage for insulators (Cornell CNF).

3. Applications

Cross-sections and morphology: Device structures, films and defects (Trieste).
Composition: EDS mapping of contaminants and phases (MIT).
Failure analysis: Imaging after FIB cross-sectioning, with EBIC and voltage contrast (Focused Ion Beam and EBIC and EBAC pages).
Process monitoring: Linewidths in the CD-SEM variant (CD-SEM page).

Beam and Probe

Name	Formula	Related Formulae	Variables/Constants
Electron wavelength	$λ = \frac{h}{2 m _{0} e V ( 1 + \frac{e V}{2 m _{0} c ^{2}} )} \approx \frac{1.226}{V ( 1 + 0.9785 \times 1 0 ^{- 6} V )} nm$	$d_{d} = \frac{1.22 λ}{α}$	$λ : e l ec t r o n w a v e l e n g t h$ $h : P l an c k co n s t an t$ $m_{0} : e l ec t r o n r es t ma ss$ $e : e l e m e n t a r y c ha r g e$ $V : a cce l er a t in g v o l t a g e in v o l t s$ $c : s p ee d o f l i g h t$ $d_{d} : d i f f r a c t i o n co n t r ib u t i o n t o t h e p r o b e s i z e$ $α : p r o b e co n v er g e n ce se mi - an g l e; L M U M u ni c h : r e l a t i v i s t i c w a v e l e n g t h h c / (e V)^{2} + 2 e V m_{0} c^{2}, r e a r r an g e d h er e (co m p u t e d); d i f f r a c t i o n d i sc f or m (m y r e a d in g); C or n e l l C N F : d i f f r a c t i o n a s a f u n d am e n t a l l imi t$ $(co m p u t e d : 38.8 pm a t 1 kV, 17.3 pm a t 5 kV an d 9.9 pm a t 15 kV; w i t h α = 5 mrad a t 1 kV, d_{d} = 9.5 nm)$
Probe size	$d_{p}^{2} = d_{g}^{2} + d_{s}^{2} + d_{c}^{2} + d_{d}^{2}$	$d_{s} = \frac{1}{2} C_{s} α^{3}, d_{c} = C_{c} \frac{Δ E}{E _{0}} α$	$d_{p} : p r o b e d iam e t er$ $d_{g} : d e ma g ni f i e d so u r ce s i z e$ $d_{s} : s p h er i c a l - ab er r a t i o n d i sc$ $d_{c} : c h r o ma t i c - ab er r a t i o n d i sc$ $C_{s}, C_{c} : ab er r a t i o n coe f f i c i e n t s$ $Δ E : e n er g y s p r e a d$ $E_{0} : b e am e n er g y; C or n e l l C N F : ab er r a t i o n s a dd in g in q u a d r a t u r e, s p h er i c a l f r o m f oc u s in g, c h r o ma t i c f r o m e n er g y s p r e a d; e x pl i c i t d i sc f or m s (m y r e a d in g)$ $(a ss u m e d : C_{c} = 2 mm, Δ E = 0.7 eV, α = 5 mrad; co m p u t e d : d_{c} = 7 nm a t 1 kV b u t 0.47 nm a t 15 kV)$
Depth of field	$D o F \approx \frac{2 r}{α}$	$r \approx pixel size at the sample$	$D o F : d e pt h o f f i e l d$ $r : a cce pt ab l e b l u r r a d i u s$ $α : co n v er g e n ce se mi - an g l e; C or n e l l C N F : s ma l l co n v er g e n ce an g l e g i v in g g oo d d e pt h o f f oc u s; g eo m e t r i c r e l a t i o n (co m p u t e d)$ $(a ss u m e d : r = 0.1 μ m, α = 5 mrad; co m p u t e d : 40 μ m, a g ain s t 0.68 μ m f or a 100 x /0.95 o pt i c a l o bj ec t i v e o n t h e O pt i c a l M i cr osco p e p a g e)$

Interaction Volume and Backscattering

Name	Formula	Related Formulae	Variables/Constants
Kanaya-Okayama range	$R_{K O} = \frac{0.0276 A E _{0}^{1.67}}{Z ^{0.889} ρ} μ m$	$R_{K O} \propto E_{0}^{1.67}$	$R_{K O} : e l ec t r o n r an g e in mi cr o m e t r es$ $A : a t o mi c w e i g h t (g / m o l)$ $E_{0} : b e am e n er g y (k e V)$ $Z : a t o mi c n u mb er$ $ρ : d e n s i t y (g / c m^{3}); M I T s l i d es : K ana y a - O k a y ama r an g e (f or m u l a g a r b l e d, s t an d a r d f or m u se d), in cr e a s in g w i t h e n er g y an d d ecr e a s in g w i t h Z, t y p i c a l r an g es a t 15 kV o f 1.8 μ m (C), 1.1 μ m (F e) an d 0.8 μ m (U) (see t h e Q / A); T r i es t e : X - r a y s g e n er a t e d f r o m a r e g i o n o f t h e or d er o f mi cr o m e t r es$ $(co m p u t e d f or S i : 32 nm a t 1 keV, 0.47 μ m a t 5 keV, 2.93 μ m a t 15 keV an d 9.3 μ m a t 30 keV)$
Elastic scattering cross section	$Q_{e} = 1.62 \times 1 0^{- 20} \frac{Z ^{2}}{E ^{2}} cot^{2} \frac{θ _{e}}{2}$	$Q_{e} \propto \frac{Z ^{2}}{E ^{2}}$	$Q_{e} : cr oss sec t i o n f or sc a tt er in g b ey o n d an g l e θ_{e} (c m^{2} p er a t o m p er e l ec t r o n)$ $E : e n er g y (k e V); M I T s l i d es : t hi s f or m, in cr e a s in g w i t h Z an d d ecr e a s in g w i t h E$ $(co m p u t e d : d o u b l in g E_{0} c u t s Q_{e} f o u r f o l d; g o l d sc a tt er s ab o u t 32 t im es m or e s t r o n g l y t han s i l i co n p er a t o m)$
Backscatter coefficient	$η = η (Z), \frac{d η}{d Z} > 0$	$η (θ) = η^{'} cos θ$	$η : f r a c t i o n o f b e am e l ec t r o n s ba c k sc a tt er e d, η^{'} : coe f f i c i e n t a l o n g t h e s u r f a ce n or ma l$ $θ : e mi ss i o n an g l e; C or n e l l C N F : B S E y i e l d r i s in g w i t h Z; T r i es t e : cos in e an g u l a r d i s t r ib u t i o n, η r i s in g w i t h Z an d o n l y w e ak l y w i t h v o l t a g e, B S E f r o m ab o u t 10 t o 20 p er ce n t o f t h e K ana y a - O k a y ama r an g e; M I T : B S E e n er g i es t y p i c a l l y 10 t o 20 keV f or a 15 keV b e am, B S E co n t r a s t b e tt er am o n g l o w - Z e l e m e n t s$ $(co m p u t e d : B S E co m e f r o m ab o u t 0.44 t o 0.59 μ m in S i a t 15 keV)$

Secondary Electrons, Charging and X-Rays

Name	Formula	Related Formulae	Variables/Constants
Secondary-electron yield	$δ = \frac{N _{S E}}{N _{B}}, δ (θ) \approx \frac{δ _{0}}{c o s θ}$	$E_{S E} \approx 1 to 15 eV, λ_{S E} \approx 2 to 5 nm (metals)$	$δ : seco n d a r y y i e l d$ $N_{S E}, N_{B} : n u mb er s o f seco n d a r i es an d b e am e l ec t r o n s$ $θ : l oc a l t i l t$ $E_{S E} : seco n d a r y e n er g y$ $λ_{S E} : esc a p e d e pt h; T r i es t e : t h ese r e l a t i o n s, e n er g i es 1 t o 15 eV (b e l o w 50 eV), esc a p e d e pt h 2 t o 5 nm f or m os t m e t a l s an d g e n er a l l y b e l o w 10 nm, y i e l d p e ak in g a t ab o u t 300 t o 800 eV, S E 1 f r o m t h e b e am e n t r y, S E 2 f r o m e x i t in g B S E, S E 3 f r o m B S E s t r ik in g t h e c hamb er; M I T : S E t y p i c a l l y b e l o w 10 eV$ $(co m p u t e d : a 60 d e g r ee f a ce t g i v es tw i ce t h e y i e l d o f a f l a t a r e a, t h e or i g in o f e d g e an d t o p o g r a p hi c co n t r a s t)$
Charging and crossover	$σ_{t o t} = η + δ$	$σ_{t o t} = 1 at E_{I} and E_{I I}$	$σ_{t o t} : t o t a l e l ec t r o n y i e l d$ $E_{I}, E_{I I} : f i r s t an d seco n d cr osso v er e n er g i es (n o n e t c ha r g in g); T r i es t e : cr osso v er e n er g i es E_{I} an d E_{I I} an d ma x im u m - y i e l d e n er g y E_{m} nam e d w i t h o u t v a l u es; M aha p a t r o : E 2 f r o m 500 eV t o 3.5 keV f or m os t in s u l a t or s, ima g in g a t E 2 t o p r e v e n t c ha r g in g; C or n e l l C N F : l o w v o l t a g e n e a r E 2 an d v a r iab l e p r ess u r e$ $(a ss u m e d : δ ab o u t 0.1 an d η = 0.16 f or s i l i co n a t 15 keV; co m p u t e d : σ_{t o t} i s 0.26, so a g r o u n d e d co n d u c t or m u s t ab sor b 74 p er ce n t o f t h e b e am)$
Characteristic X-rays and EDS	$E_{0} > E_{c}, U = \frac{E _{0}}{E _{c}} ≳ 2$	$ν \propto (Z - 1) (Moseley, K lines), F W H M_{E D S} \approx 125 to 130 eV (Mn K α)$	$E_{c} : cr i t i c a l e x c i t a t i o n e n er g y$ $U : o v er v o l t a g e r a t i o$ $ν : X - r a y f r e q u e n cy$ $F W H M_{E D S} : e n er g y r eso l u t i o n o f s i l i co n d r i f t d e t ec t or s; T r i es t e : c ha r a c t er i s t i c l in es f r o m ab o u t 1 t o 100 keV, co n t in uu m u p t o E_{0} (D u an e - H u n t l imi t), p r ima r y e n er g y e x cee d in g t h e t r an s i t i o n e n er g y, X - r a y s f r o m mi cr o m e t r e - sc a l e v o l u m es; M I T : E D S s p ec t r a w i t h maj or - e l e m e n t p e ak s; Gl o ba l S in o : o v er v o l t a g e a t l e a s t 2 f or t h e hi g h es t - e n er g y l in e; B r u k er : S D D r eso l u t i o n 121 t o 129 eV a t M n K - a l p ha; M ose l e y^{'} s l a w (m y r e a d in g)$ $(a ss u m e d : S i K e d g e 1.84 keV; co m p u t e d : U = 8.2 a t 15 kV b u t o n l y 2.7 a t 5 kV)$

Important Questions

Ques: How does an SEM form its images?

By scanning a fine electron probe and recording the electrons emitted at each point. Secondary electrons, with energies of about 1 to 15 $eV$ , escape only from the top 2 to 5 $nm$ of metals and generally below 10 $nm$ , so they map topography (Trieste). Their yield rises roughly as $1/ cos θ$ with surface tilt, so a 60 degree facet appears twice as bright as a flat area (computed), giving edge contrast. Trieste distinguishes SE1 from the beam entry point, which carry the high-resolution information, SE2 produced by exiting backscattered electrons, and SE3 from the chamber walls. Backscattered electrons, typically 10 to 20 $keV$ for a 15 $keV$ beam (MIT), give compositional contrast. Reference: University of Trieste, ‘SEM part 2’ lecture slides on electron-specimen interactions (author not stated) (fetch-tool summary); N. Chatterjee, ‘12.141 Electron Microprobe Analysis’, MIT OpenCourseWare, January IAP 2012, day 1 slides (Kanaya-Okayama formula garbled in extraction) (fetch-tool summary).

Ques: How deep does the beam penetrate?

About as far as the Kanaya-Okayama range, $0.0276 A E_{0}^{1.67} / (Z^{0.889} ρ)$ micrometres, which rises steeply with energy and falls with atomic number (MIT; formula in standard form). In silicon it is only 32 $nm$ at 1 $keV$ but 0.47 $μ m$ at 5 $keV$ , 2.93 $μ m$ at 15 $keV$ and 9.3 $μ m$ at 30 $keV$ (computed). X-rays are therefore generated from micrometre-scale volumes (Trieste), limiting EDS spatial resolution to roughly the range, while backscattered electrons emerge from 10 to 20 per cent of it, about 0.44 to 0.59 $μ m$ in silicon at 15 $keV$ (computed). Low voltages confine the signal near the surface, sharpening images of thin films and reducing charging, at the cost of more chromatic blur. Reference: N. Chatterjee, ‘12.141 Electron Microprobe Analysis’, MIT OpenCourseWare, January IAP 2012, day 1 slides (Kanaya-Okayama formula garbled in extraction) (fetch-tool summary); University of Trieste, ‘SEM part 2’ lecture slides on electron-specimen interactions (author not stated) (fetch-tool summary); Cornell NanoScale Facility, ‘Basics of Scanning Electron Microscopy (SEM)’ slides: lens aberrations (spherical from focusing, chromatic from energy spread, diffraction as a fundamental limit) adding in quadrature to the probe size, small convergence angle giving good depth of focus, BSE yield rising with Z, Everhart-Thornley (SE2) and in-lens (SE1) detectors, cold and Schottky field-emission sources, rastering with recorded signal intensity, low voltage near E2 and variable pressure for insulators (fetch-tool summary).

Ques: Do the Kanaya-Okayama ranges match the MIT slide’s typical values?

Only roughly. The MIT slide lists typical ranges at 15 $kV$ of 1.8 $μ m$ for carbon, 1.1 $μ m$ for iron and 0.8 $μ m$ for uranium, while the Kanaya-Okayama formula on the same slide gives 2.75, 1.00 and 0.57 $μ m$ with standard atomic weights and densities (computed, graphite density 2.26 $g/c m^{3}$ ). Iron agrees within 10 per cent, but carbon differs by 34 per cent and uranium by 29 per cent. The slide’s values are probably read from Monte Carlo plots of the region of most energy deposition, which is smaller than the full range for light elements and broader for heavy ones (my reading). For estimating maximum depth, the Kanaya-Okayama formula is the more likely reference; for where most signal arises, the smaller Monte Carlo figure is better (my reading). Reference: N. Chatterjee, ‘12.141 Electron Microprobe Analysis’, MIT OpenCourseWare, January IAP 2012, day 1 slides (Kanaya-Okayama formula garbled in extraction) (fetch-tool summary).

Ques: Why do heavy elements look bright in backscattered images?

Because the backscatter coefficient rises strongly with atomic number (Cornell CNF; Trieste). The elastic cross section scales as $Z^{2} / E^{2}$ (MIT), so gold scatters about 32 times more strongly than silicon per atom (computed). Trieste notes that $η$ depends only weakly on voltage and that backscattered images distinguish nickel from aluminium oxide where secondary images do not. MIT notes that contrast is better among low-Z elements, because $η$ changes faster per unit Z there. Emission follows a cosine distribution about the surface normal (Trieste). Backscattered electrons also leave from deeper than secondaries, about 0.44 to 0.59 $μ m$ in silicon at 15 $keV$ (computed from the Kanaya-Okayama range), so backscattered images carry some subsurface composition contrast (my reading). Reference: University of Trieste, ‘SEM part 2’ lecture slides on electron-specimen interactions (author not stated) (fetch-tool summary); N. Chatterjee, ‘12.141 Electron Microprobe Analysis’, MIT OpenCourseWare, January IAP 2012, day 1 slides (Kanaya-Okayama formula garbled in extraction) (fetch-tool summary); Cornell NanoScale Facility, ‘Basics of Scanning Electron Microscopy (SEM)’ slides: lens aberrations (spherical from focusing, chromatic from energy spread, diffraction as a fundamental limit) adding in quadrature to the probe size, small convergence angle giving good depth of focus, BSE yield rising with Z, Everhart-Thornley (SE2) and in-lens (SE1) detectors, cold and Schottky field-emission sources, rastering with recorded signal intensity, low voltage near E2 and variable pressure for insulators (fetch-tool summary).

Ques: What limits SEM resolution?

The probe size, set by source size and lens aberrations, and the signal escape volume. The probe diameter combines source image, spherical aberration $C_{s} α^{3} /2$ , chromatic aberration $C_{c} (Δ E / E_{0}) α$ and diffraction $1.22 λ / α$ in quadrature (Cornell CNF; the explicit forms are my reading). The wavelength is tiny, 9.9 $pm$ at 15 $kV$ and 38.8 $pm$ at 1 $kV$ (computed from LMU’s relativistic formula), but at the small apertures used, diffraction and chromatic terms matter at low voltage: with an assumed 2 $mm$ $C_{c}$ and 0.7 $eV$ spread at 5 $mrad$ , the chromatic disc is 7 $nm$ at 1 $kV$ but 0.47 $nm$ at 15 $kV$ (computed). Field-emission sources and immersion lenses keep the probe near a nanometre at low voltage, where shallow SE1 escape gives the sharpest surface detail. Reference: Cornell NanoScale Facility, ‘Basics of Scanning Electron Microscopy (SEM)’ slides: lens aberrations (spherical from focusing, chromatic from energy spread, diffraction as a fundamental limit) adding in quadrature to the probe size, small convergence angle giving good depth of focus, BSE yield rising with Z, Everhart-Thornley (SE2) and in-lens (SE1) detectors, cold and Schottky field-emission sources, rastering with recorded signal intensity, low voltage near E2 and variable pressure for insulators (fetch-tool summary); LMU Munich physics didactics, ‘relativistic de Broglie wavelength’ page: lambda = h c/sqrt((e V_a)^2 + 2 e V_a m_e c^2) (fetch-tool summary); University of Trieste, ‘SEM part 2’ lecture slides on electron-specimen interactions (author not stated) (fetch-tool summary).

Ques: How does EDS identify elements, and how small an area does it analyse?

By measuring the energies of characteristic X-rays, which are fixed by each element’s electron shells. A beam must exceed the critical excitation energy of a shell to ionize it; characteristic lines span about 1 to 100 $keV$ , on a continuum extending up to the beam energy, the Duane-Hunt limit (Trieste). For efficient excitation the overvoltage $E_{0} / E_{c}$ should be at least 2 for the highest-energy line (GlobalSino): for silicon K at 1.84 $keV$ , 8.2 at 15 $kV$ but 2.7 at 5 $kV$ (computed). Silicon drift detectors resolve 121 to 129 $eV$ at Mn K-alpha in Bruker’s XFlash 6-10 versions. Since X-rays come from micrometre-scale volumes (Trieste), 2.9 $μ m$ deep in silicon at 15 $kV$ (computed), EDS cannot analyse a 50 $nm$ particle on its own; thin lamellae in the TEM are needed. Reference: University of Trieste, ‘SEM part 2’ lecture slides on electron-specimen interactions (author not stated) (fetch-tool summary); GlobalSino, ‘Overvoltage Ratio in EDS Measurements’: overvoltage E0/Ei at least 2 for the highest-energy line in SEM-EDS, greater than 1.3 for effective generation (fetch-tool summary); Bruker, ‘XFlash 6-10’ detector page: energy resolution 121 eV at Mn K-alpha (limited edition), other versions 123, 126 and 129 eV (fetch-tool summary).

Ques: Why do insulators charge, and how is it avoided?

Because they cannot conduct away the difference between incoming and emitted electrons. The total yield is backscattered plus secondary, and only at two crossover energies, $E_{I}$ and $E_{I I}$ , does it equal one (Trieste names these without values). On silicon at 15 $keV$ , with an assumed backscatter coefficient of 0.16 and secondary yield of 0.1, only 26 per cent leaves, so a grounded conductor absorbs 74 per cent of the beam (computed); an insulator would charge negatively. Secondary yield peaks at about 300 to 800 $eV$ (Trieste), so imaging near $E_{I I}$ , which Mahapatro puts at 500 $eV$ to 3.5 $keV$ for most insulators, balances the charge. Conductive coatings, variable-pressure gas that neutralizes charge, and fast scanning are alternatives. Reference: University of Trieste, ‘SEM part 2’ lecture slides on electron-specimen interactions (author not stated) (fetch-tool summary); A. K. Mahapatro, ‘Low Voltage Surface Electron Microscopy’, chapter 10 of Characterization Techniques for Materials II (INFLIBNET e-PG Pathshala, CC BY-NC 4.0): E2 from 500 eV to 3.5 keV for most insulators, accelerating voltage equal to E2 to prevent charging, LaB6 and field-emission sources (fetch-tool summary).

Ques: Why does an SEM have so much more depth of field than an optical microscope?

Because its probe converges at very small angles. Depth of field is about $2 r / α$ , with r the acceptable blur and $α$ the probe semi-angle (computed from the cone geometry; Cornell CNF ties good depth of focus to a small angle). With a blur of 0.1 $μ m$ and 5 $mrad$ it is 40 $μ m$ (computed), whereas a 100x/0.95 optical objective gives only 0.68 $μ m$ (Optical Microscope page). The small angle is possible because the electron wavelength is thousands of times shorter than light, so diffraction stays small even at milliradian apertures. This is why SEM images of rough fracture surfaces, wire bonds and 3-D structures appear sharp from top to bottom. Smaller apertures or longer working distances increase depth further, at the cost of probe current and resolution. Reference: Cornell NanoScale Facility, ‘Basics of Scanning Electron Microscopy (SEM)’ slides: lens aberrations (spherical from focusing, chromatic from energy spread, diffraction as a fundamental limit) adding in quadrature to the probe size, small convergence angle giving good depth of focus, BSE yield rising with Z, Everhart-Thornley (SE2) and in-lens (SE1) detectors, cold and Schottky field-emission sources, rastering with recorded signal intensity, low voltage near E2 and variable pressure for insulators (fetch-tool summary); Optical Microscope page of this vault.

References

University of Trieste, “SEM part 2,” lecture slides on electron-specimen interactions (author not stated). Available: https://moodle2.units.it/pluginfile.php/762256/mod_resource/content/1/SEM%20part%202.pdf
N. Chatterjee, “12.141 Electron Microprobe Analysis,” MIT OpenCourseWare, January IAP 2012, day 1 slides. Available: https://ocw.mit.edu/courses/12-141-electron-microprobe-analysis-january-iap-2012/05c8da30b00f7a0612fbdffa2167c1f5_MIT12_141IAP12_slides_day1.pdf
Cornell NanoScale Science and Technology Facility, “Basics of Scanning Electron Microscopy (SEM),” slides. Available: https://www.cnfusers.cornell.edu/sites/default/files/Equipment-Resources/CNF_SEM_basicsv2.pdf
LMU Munich, Physics Didactics, “Relativistic de Broglie wavelength,” electron diffraction module. Available: https://www.didaktik.physik.uni-muenchen.de/elektronenbahnen/en/elektronenbeugung/wellenlaenge/de-broglie-relativistisch.php
A. K. Mahapatro, “Low Voltage Surface Electron Microscopy,” chapter 10 in Characterization Techniques for Materials II, INFLIBNET e-PG Pathshala. Available: https://ebooks.inflibnet.ac.in/msp11/chapter/low-voltage-surface-electron-microscopy
GlobalSino, “Overvoltage Ratio in EDS Measurements.” Available: https://www.globalsino.com/EM/page4612.html
Bruker, “XFlash 6-10” EDS detector page. Available: https://www.bruker.com/en/products-and-solutions/elemental-analyzers/eds-wds-ebsd-SEM-Micro-XRF/detectors/xflash-6-10-detector.html
Optical Microscope, page in this vault.

Quartz 5

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Scanning Electron Microscope (SEM)