Transmission Electron Microscope (TEM and STEM)

1. Theory

• Definition: A transmission electron microscope passes a 60 to 300 $\mathrm{kV}$ electron beam through a specimen thinner than about 200 $\mathrm{nm}$ and images the transmitted and diffracted electrons; in STEM mode a focused probe is scanned and transmitted electrons are collected by annular detectors (Voyles; Morgiel).
• Contrast: Diffraction contrast, phase contrast (high resolution), Z-contrast in HAADF STEM and holographic phase all reveal different information (Morgiel; Voyles).
• Spectroscopy: EELS and EDS in STEM give composition, bonding and thickness at near-atomic resolution (Voyles; Iakoubovskii et al.).
• Resolution: Set by lens aberrations and wavelength, about 0.2 $\mathrm{nm}$ for phase contrast in uncorrected 200 $\mathrm{kV}$ instruments (Morgiel).

2. Working Principle

• Specimen: Thin lamellae prepared by FIB or polishing (Morgiel; FIB page).
• Imaging and diffraction: Projector lenses switch between image and diffraction pattern at a chosen camera length, by refocusing the intermediate lens (UCR EE206 notes; Morgiel).
• STEM detectors: Bright-field, annular bright-field, ADF and HAADF detectors at chosen angles (Voyles).
• Spectrometers: EELS spectrometer and EDS detectors record spectra at each probe position (Voyles; Iakoubovskii et al.).

3. Applications

• Device cross-sections: Gate stacks, interfaces and layer thicknesses at atomic scale (Voyles; Morgiel).
• Strain: Lattice spacings and strain contrast at Si/SiO2 and Si/CoSi2 interfaces (Morgiel; Voyles).
• Dopant junctions: Electron holography of p-n junctions (Morgiel).
• Composition: Atomic-resolution EDS and EELS maps (Voyles).

Imaging and Diffraction

Name	Formula	Related Formulae	Variables/Constants
Electron wavelength at TEM voltages	$\lambda \approx \frac{1.226}{\sqrt{V\left(1+0.9785\times 10^{-6}V\right)}}\mathrm{nm}$	$R_E=\frac{1}{\lambda }(\text{Ewald sphere radius})$	$\lambda :relativisticelectronwavelength$ $V:acceleratingvoltageinvolts$ $\begin{array}{r}{R}_E:Ewaldsphereradius;wavelengthform\\ fromtheSEMpage;Morgiel:Ewaldsphere\\ radius1/\lambda ,interactionconstant0.00729\\ pervoltpernanometreat200\mathrm{kV}\end{array}$ $(computed:\lambda =2.51\mathrm{pm}at200\mathrm{kV})$
Point resolution and Scherzer defocus	$d_{Sch}\approx 0.66C_s^{1/4}{\lambda }^{3/4}$	$\Delta f_{Sch}\approx -1.2\sqrt{C_s\lambda }$	$d_{Sch}:Scherzerpointresolution$ $C_s:spherical-aberrationcoefficient$ $\begin{array}{r}\Delta f_{Sch}:Scherzerdefocus;Schrammetal.:\\ d_{Sch}=0.66C_s^{1/4}{\lambda }^{3/4}atScherzerdefocus;\\ GlobalSino:pointresolutionatthefirst\\ CTFzero,extendedScherzern=1.2;the\\ defocusform(myreading);Morgiel:\\ Scherzerdefocusoftheform-(C_s\lambda {)}^{1/2}\\ (symbolsgarbled),contrasttransferT(R)\\ =F{T}^{-1}[FT(\psi )exp(-i\chi )]andI=|T{|}^2,\\ diffraction-contrastresolutionabout2\mathrm{nm}\\ andphase-contrastabout0.2\mathrm{nm},\\ three-foldastigmatismabletocreate\\ spuriouscontrastofseveralnanometres;\\ Voyles:C_s=1\mathrm{mm}at200\mathrm{kV}onaJEOL2010F\end{array}$ $\begin{array}{r}(computed:C_s=1\mathrm{mm}at200\mathrm{kV}gives{d}_{Sch}=\\ 0.23\mathrm{nm}and\Delta f_{Sch}=-60\mathrm{nm})\end{array}$
Camera equation	$Rd=\lambda L$	$\frac{1}{g_{hkl}}=d_{hkl},d_{hkl}=\frac{a}{\sqrt{h^2+k^2+l^2}}(\text{cubic})$	$\begin{array}{r}R:distanceofadiffractionspotfromthe\\ centralbeam\end{array}$ $d:planespacing$ $L:cameralength$ $g_{hkl}:reciprocal-latticevector$ $\begin{array}{r}a:latticeconstant;UCREE206notes:d=\lambda \\ L/r;Morgiel:1/g=dandlattice-fringe\\ spacingequaltotheplanespacing,\\ Si/CoSi2examplewhere{d}_{111}and{d}_{200}differby\\ about1.2percent\end{array}$ $\begin{array}{r}(computed:Si(111),d=0.3135\mathrm{nm},at200\\ \mathrm{kV}andL=1\mathrm{m}givesR=8.0\mathrm{mm})\end{array}$

STEM Contrast and Phase

Name	Formula	Related Formulae	Variables/Constants
HAADF Z-contrast	$I_{HAADF}\propto Z^n,n\approx 1.7$	${\beta }_{inner}\gtrsim 100\mathrm{mrad},{\beta }_{outer}\approx 5{\beta }_{inner}$	$\begin{array}{r}{I}_{HAADF}:high-angleannulardark-field\\ intensityofacolumn\end{array}$ $Z:atomicnumber$ $\begin{array}{r}\beta :detectorangles;Voyles:scattering\\ about{Z}^{1.7}atcommonangles,collection\\ above100\mathrm{mrad}givingZ-contrast,outer\\ anglefivetimestheinnerintheseries\\ shown,24.5\mathrm{mrad}convergence,ADFat25\\ and50\mathrm{mrad}showingstraincontrastat\\ Si/SiO2,straincontrastabout0percent\\ at130angstromsand15percentat340\\ angstroms(100\mathrm{kV},45\mathrm{mrad}),HAADFat100\\ \mathrm{kV}and96\mathrm{mrad},ABFforlightelements,\\ contrastreversalsinthickWplugsabove\\ 45and75\mathrm{mrad}\end{array}$ $\begin{array}{r}(computed:aGecolumnscatters(32/14{)}^{1.7}=\\ 4.1timesmorethanaSicolumn;oneAs\\ atom,4.3timesaSiatom,raisesa\\ 50-atomSicolumnbyonlyabout7per\\ cent)\end{array}$
Exit wave and phase shift	$\psi \left(\mathbf{r}\right)=a\left(\mathbf{r}\right)e^{-i\varphi \left(\mathbf{r}\right)},\varphi =\sigma V_{0}t$	${\varphi }_{pn}=C_E\left[V_{0}t+V_{pn}\left(t-2t_0\right)\right]$	$\psi :exitwave,a$ $\begin{array}{r}\varphi :amplitudeandphase,\sigma (C_E):interaction\\ constant\end{array}$ $V_0:meaninnerpotential$ $t:thickness$ $V_{pn}:junctionpotential$ $\begin{array}{r}{t}_0:electricallydeadsurfacelayer;\\ Voyles:exitwaveanditsFourier\\ transform,intensities|\psi {|}^{2}inimageand\\ diffraction;Morgiel:phaserelationfora\\ p-njunctionunderkinematicconditions,\sigma \\ =0.00729pervoltpernanometreat200\\ \mathrm{kV},V_{0}about12\mathrm{V}forSi,junction\\ potentials0.7to1.2\mathrm{V}(0.5and0.7\mathrm{V}in\\ MOSFETdepletionregions),weak-phase\\ objectsbelowabout10\mathrm{nm},holographyona\\ CM200FEGwith160\mathrm{V}biprism,860\mathrm{nm}field,\\ 3.8\mathrm{nm}fringesanda200\mathrm{nm}FIBlamella\end{array}$ $\begin{array}{r}(computed:100\mathrm{nm}ofSishiftsthephase\\ by8.75\mathrm{rad};a1\mathrm{V}junctionacross200\mathrm{nm}\\ adds1.46\mathrm{rad})\end{array}$
STEM spectrum imaging time	$t_{map}=N_x{N}_y{N}_{frames}{t}_{dwell}$	$\text{dead time}=1-\frac{N_{recorded}}{N_{incident}}$	$t_{map}:acquisitiontime$ $N_x,N_y:pixels$ $N_{frames}:frames$ $\begin{array}{r}{t}_{dwell}:dwellperpixel;relations(computed);\\ Voyles:atomic-resolutionEDSof128by\\ 128pixels,175frames,0.2m/sdwell,\\ about120\mathrm{pA},1119.57countspersecond,\\ realtime9min33s,deadtime3.71per\\ cent,80milliontimesmagnification\end{array}$ $\begin{array}{r}(computed:128x128x175x0.2m/s=\\ 573.4\mathrm{s}=9min33s,exactlythereported\\ time)\end{array}$

EELS and Specimen Thickness

Name	Formula	Related Formulae	Variables/Constants
EELS log-ratio thickness	$\frac{t}{{\lambda }_{in}}=\ln \frac{I_t}{I_0}$	${\lambda }_{in}\approx \frac{106F{E}_0}{E_m\ln \left(2\beta E_0/E_m\right)},F=\frac{1+E_0/1022}{{\left(1+E_0/511\right)}^2},E_m\approx 7.6Z_{ef}^{0.36}$	$t:thickness$ ${\lambda }_{in}:inelasticmeanfreepath(nm)$ $I_t:totalspectrumarea$ $I_0:zero-lossarea$ $E_0:beamenergy(keV)$ $\beta :collectionsemi-angle(mrad)$ $E_m:meanenergyloss(eV)$ $\begin{array}{r}{Z}_{ef}:effectiveatomicnumber;Iakoubovskii\\ etal.:log-ratioandKramers-Kronig\\ models,Egerton-Chenmean-free-path\\ formulawithleadingcoefficientextracted\\ as10^6(seetheQ/A),E_{m}scaling,new\lambda \propto \\ {\rho }^{-0.3}law,Siat200\mathrm{kV}measured142plusor\\ minus5\mathrm{nm}against145\mathrm{nm}(Jin2004),\\ formulasvalidbelowabout20\mathrm{mrad}\\ collection,20percentunderestimateat\\ 160\mathrm{mrad},\alpha =20and\beta =5\mathrm{mrad}used,t/\lambda \\ near1(70to180\mathrm{nm})preferred,EELS\\ usefuloverabout15to500\mathrm{nm}\end{array}$ $\begin{array}{r}(computed:forSiat200\mathrm{keV}and\beta =5\\ \mathrm{mrad},F=0.618,E_m=19.7\mathrm{eV}and{\lambda }_{in}=144\\ \mathrm{nm}withcoefficient106;t/\lambda =\ln 2.7=\\ 0.99meanst=141\mathrm{nm})\end{array}$
Refractive-index sensitivity	$t\propto \frac{1}{1-1/n^2}(\text{Kramers-Kronig sum})$	$\frac{\Delta t}{t}=\frac{2\Delta n/n^3}{1-1/n^2}$	$\begin{array}{r}n:refractiveindexnear1to2\mathrm{eV};\\ Iakoubovskiietal.:Kramers-Kronig\\ thicknessformulawith(1-1/n^2),rounding\\ nto0.1givingabout0.3percenterror\\ forSi(n=3.42)andabout6percentfor\\ SiO2(n=1.5);derivative(computed)\end{array}$ $\begin{array}{r}(computed:\Delta n=0.05gives0.3percent\\ forSiand5.3percentforSiO2,closeto\\ thestatedvalues)\end{array}$
Specimen thickness regimes	$t<10\mathrm{nm}(\text{weak phase object}),t/{\lambda }_{in}\approx 1(\text{EELS thickness})$	$t\approx 15\text{to}500\mathrm{nm}(\text{EELS range})$	$\begin{array}{r}Morgiel:thincrystalsbelowabout10\mathrm{nm}\\ treatedasweakphaseobjects,6\mathrm{nm}\\ Si/CoSi2exampleat-90\mathrm{nm}defocus,200\mathrm{nm}\\ lamellaforholography;Iakoubovskiiet\\ al.:t/\lambda near1toavoidsurfaceplasmons\\ andpluralscattering,range15to500\mathrm{nm};\\ Voyles:straincontrastrisingwith\\ thickness\end{array}$ $\begin{array}{r}(computed:a10\mathrm{nm}weak-phasespecimenof\\ Siisonly0.07inelasticmeanfreepaths)\end{array}$

Important Questions

Ques: How do TEM and STEM differ?

TEM illuminates the whole field with a parallel beam and images it with lenses, while STEM scans a focused probe and records transmitted electrons on detectors at each point. In TEM, contrast comes from diffraction and from phase differences in the exit wave, $\psi =a{e}^{-i\varphi }$, so high-resolution images need careful defocus (Voyles; Morgiel). In STEM, annular detectors select scattering angles: above about 100 $\mathrm{mrad}$, HAADF signals scale as about $Z^{1.7}$ (Voyles), giving direct atomic-number contrast that is easier to interpret. STEM also pairs naturally with EELS and EDS mapping. At 200 $\mathrm{kV}$ the wavelength is 2.51 $\mathrm{pm}$ (computed), so resolution is limited by lens aberrations, not wavelength. Morgiel gives about 0.2 $\mathrm{nm}$ for phase contrast and about 2 $\mathrm{nm}$ for diffraction contrast. Reference: P. M. Voyles, STEM lecture slides, Northwestern University Materials Science (2012) (fetch-tool summary); J. Morgiel, ‘TEM lecture 2: HREM’, IMIM PAN Krakow (2019) (some equations garbled) (fetch-tool summary).

Ques: What sets high-resolution TEM point resolution?

Spherical aberration and wavelength, through the Scherzer condition. The contrast transfer function, $T(R)=F{T}^{-1}\{FT[\psi ]\exp (-i\chi )\}$ with intensity $|T{|}^2$ (Morgiel), passes spatial frequencies with the right phase only up to a limit; at the Scherzer defocus, about $-1.2\sqrt{C_s\lambda }$, the first zero sets the point resolution $0.66C_s^{1/4}{\lambda }^{3/4}$ (Schramm et al.; GlobalSino defines the point resolution at the first zero; the defocus form is my reading). For $C_s=1$ $\mathrm{mm}$ at 200 $\mathrm{kV}$, as on the JEOL 2010F in Voyles’s examples, this gives 0.23 $\mathrm{nm}$ at -60 $\mathrm{nm}$ defocus (computed), consistent with Morgiel’s 0.2 $\mathrm{nm}$ lens-limited phase-contrast resolution. Morgiel warns that three-fold astigmatism can create spurious contrast of several nanometres, so lattice images need image simulation to interpret. Reference: J. Morgiel, ‘TEM lecture 2: HREM’, IMIM PAN Krakow (2019) (some equations garbled) (fetch-tool summary); P. M. Voyles, STEM lecture slides, Northwestern University Materials Science (2012) (fetch-tool summary); S. M. Schramm, S. J. van der Molen and R. M. Tromp, ‘Intrinsic Instability of Aberration-Corrected Electron Microscopes’, arXiv:1209.5340: C3-limited TEM resolution at Scherzer defocus delta = 0.66 C3^(1/4) lambda^(3/4), after Williams and Carter (fetch-tool summary); GlobalSino, ‘Best Defocus Condition for HRTEM Imaging’: point resolution where the phase CTF first crosses zero, extended Scherzer condition with n = 1.2 (fetch-tool summary).

Ques: How is a diffraction pattern indexed?

With the camera equation $Rd=\lambda L$, which converts spot distances to plane spacings (UCR EE206 notes, as $d=\lambda L/r$). For silicon (111), with a spacing of 0.3135 $\mathrm{nm}$, at 200 $\mathrm{kV}$ and a 1 $\mathrm{m}$ camera length, the spot lies 8.0 $\mathrm{mm}$ from the centre (computed). Ratios of spot distances and angles between spots identify the crystal structure and zone axis. Because the Ewald sphere radius $1/\lambda$ is very large, the sphere is nearly flat and many reflections appear at once (Morgiel). Lattice fringes in images have the same spacing as the diffracting planes, $1/g=d$ (Morgiel); in Morgiel’s Si/CoSi2 example, the d111 and d200 spacings differ by only about 1.2 per cent, which careful analysis can resolve. Calibrating L with a known standard is essential for accurate spacings. Reference: J. Morgiel, ‘TEM lecture 2: HREM’, IMIM PAN Krakow (2019) (some equations garbled) (fetch-tool summary); EE206 lecture notes 5, ‘Transmission Electron Microscope (I)’ (UC Riverside course page): d = lambda L/r, intermediate lens refocused on the objective back focal plane for diffraction or image plane for imaging (fetch-tool summary).

Ques: What does HAADF STEM show, and what can distort it?

Column-by-column atomic-number contrast, though strain and thickness also contribute. Voyles explains that scattering scales as about $Z^{1.7}$ at common angles, so with collection above 100 $\mathrm{mrad}$ a germanium column scatters about 4.1 times more than silicon, while a single arsenic atom in a 50-atom silicon column raises its signal by only about 7 per cent (computed). At lower inner angles, such as 25 or 50 $\mathrm{mrad}$, strain contrast appears at Si/SiO2 interfaces, and at a c-Si/a-Si interface it rose from about 0 to 15 per cent as thickness increased from 130 to 340 angstroms. Thick tungsten plugs showed contrast reversals above 45 and 75 $\mathrm{mrad}$. Voyles also notes that annular bright-field imaging reveals light elements such as oxygen that HAADF misses. Reference: P. M. Voyles, STEM lecture slides, Northwestern University Materials Science (2012) (fetch-tool summary).

Ques: How is specimen thickness measured with EELS?

With the log-ratio method, $t/{\lambda }_{in}=\ln (I_t/I_0)$, comparing the total spectrum area with the zero-loss peak (Iakoubovskii et al.). It needs the inelastic mean free path: Iakoubovskii et al. measured 142 plus or minus 5 $\mathrm{nm}$ for silicon at 200 $\mathrm{kV}$, consistent with Jin’s 145 $\mathrm{nm}$. A spectrum with a ratio of 2.7 then means 141 $\mathrm{nm}$ (computed). The method works best for $t/\lambda$ near 1, from about 70 to 180 $\mathrm{nm}$ in their samples, avoiding surface-plasmon effects in thin regions and plural scattering in thick ones, with EELS useful over about 15 to 500 $\mathrm{nm}$. They warn that the standard formulas underestimate thickness by about 20 per cent at a 160 $\mathrm{mrad}$ collection angle. Reference: K. Iakoubovskii, K. Mitsuishi, Y. Nakayama and K. Furuya, ‘Thickness Measurements With Electron Energy Loss Spectroscopy’, Microsc. Res. Tech. 71, 626 (2008) (some equations garbled) (fetch-tool summary).

Ques: Is the mean-free-path coefficient 10^6 or 106?

Iakoubovskii et al.’s extracted text writes the Egerton-Chen approximation with a leading coefficient of $10^6$, but evaluating it for silicon at 200 $\mathrm{keV}$ with a 5 $\mathrm{mrad}$ collection angle, using their F factor and $E_m=7.6Z_{ef}^{0.36}$, gives about $1.36\times 10^6$ $\mathrm{nm}$ with $10^6$ but 144 $\mathrm{nm}$ with 106 (computed). Only 106 agrees with their measured 142 $\mathrm{nm}$, so the superscript is an extraction error and the coefficient is 106, the value in the Malis et al. formula as quoted by Yang et al. (computed). The formula then predicts silicon’s mean free path within 2 per cent. Iakoubovskii et al. also propose a new scaling, $\lambda \propto {\rho }^{-0.3}$, and limit the formula to collection angles below about 20 $\mathrm{mrad}$. Reference: K. Iakoubovskii, K. Mitsuishi, Y. Nakayama and K. Furuya, ‘Thickness Measurements With Electron Energy Loss Spectroscopy’, Microsc. Res. Tech. 71, 626 (2008) (some equations garbled) (fetch-tool summary); Y.-C. Yang, L. Serafini, N. Gauquelin, J. Verbeeck and J. R. Jinschek, ‘Improving the Accuracy of Temperature Measurement on TEM samples using PEET…’, arXiv:2407.20059: Malis et al. lambda = 106 F E0/[Em ln(2 beta E0/Em)], Em = 7.6 Z^0.36, t/lambda = ln(It/I0) (fetch-tool summary).

Ques: How does electron holography image p-n junctions?

By measuring the phase the electron wave picks up in the specimen’s electrostatic potential. The phase is $\varphi =\sigma V_{0}t$, with $\sigma =0.00729$ per volt per nanometre at 200 $\mathrm{kV}$ and a mean inner potential of about 12 $\mathrm{V}$ for silicon, so 100 $\mathrm{nm}$ of silicon shifts the phase by 8.75 $\mathrm{rad}$ (Morgiel; computed). A junction adds its built-in potential, 0.7 to 1.2 $\mathrm{V}$, over the electrically active thickness, $t-2t_0$, allowing for dead surface layers (Morgiel); 1 $\mathrm{V}$ across 200 $\mathrm{nm}$ adds 1.46 $\mathrm{rad}$ (computed). Morgiel’s holography used a 200 $\mathrm{kV}$ CM200FEG with a 160 $\mathrm{V}$ biprism, an 860 $\mathrm{nm}$ field, 3.8 $\mathrm{nm}$ fringes and a 200 $\mathrm{nm}$ FIB lamella, revealing depletion regions of about 0.5 and 0.7 $\mathrm{V}$ in n- and p-MOSFETs. Reference: J. Morgiel, ‘TEM lecture 2: HREM’, IMIM PAN Krakow (2019) (some equations garbled) (fetch-tool summary).

Ques: How long does an atomic-resolution EDS map take?

Minutes, because X-ray signals from single columns are weak. Voyles’s example records 128 by 128 pixels over 175 frames at 0.2 $m/s$ dwell with about 120 $\mathrm{pA}$ probe current, for a real time of 9 min 33 s; multiplying the pixels, frames and dwell gives exactly 573.4 $\mathrm{s}$ (computed). The count rate was 1119.57 counts per second with 3.71 per cent dead time, so the whole map holds only about 640000 counts (computed), a few dozen per pixel, which is why many frames are summed with drift correction. Voyles notes that a 1 $\mathrm{Pa}$ change in air pressure deflects the stage by 1 angstrom, showing why environmental stability matters. EELS maps are faster for light elements, but both demand thin, clean lamellae. Reference: P. M. Voyles, STEM lecture slides, Northwestern University Materials Science (2012) (fetch-tool summary).

References

1. P. M. Voyles, STEM lecture slides, Northwestern University Materials Science, 2012. Available: https://www.numis.northwestern.edu/460/Notes/STEM.pdf
2. J. Morgiel, “TEM lecture 2: HREM,” IMIM PAN, Krakow, 2019. Available: https://www.imim.pl/files/SD/Power/TEM-lecture2_HREM_A.pdf
3. S. M. Schramm, S. J. van der Molen and R. M. Tromp, “Intrinsic instability of aberration-corrected electron microscopes,” arXiv:1209.5340. Available: https://arxiv.org/abs/1209.5340
4. GlobalSino, “Best Defocus Condition for HRTEM Imaging.” Available: https://www.globalsino.com/EM/page4232.html
5. “Transmission Electron Microscope (I),” EE206 lecture notes 5, Winter 2008 (UC Riverside course site). Available: https://intra.ee.ucr.edu/~jianlin/EE206/Winter2008/EE206-lecture-notes-5-TEM-I.pdf
6. Y.-C. Yang, L. Serafini, N. Gauquelin, J. Verbeeck and J. R. Jinschek, “Improving the accuracy of temperature measurement on TEM samples using plasmon energy expansion thermometry (PEET): addressing sample thickness effects,” arXiv:2407.20059. Available: https://arxiv.org/abs/2407.20059
7. K. Iakoubovskii, K. Mitsuishi, Y. Nakayama and K. Furuya, “Thickness Measurements With Electron Energy Loss Spectroscopy,” Microscopy Research and Technique, vol. 71, pp. 626-631, 2008. Available: https://nrds.inl.gov/dataset/0cd47023-c766-44e8-89fd-d798e2898b38/resource/79ba3479-b561-4b0e-84f8-9a88b314b1a4/download/microscopy-res-technique-2008-iakoubovskii-thickness-measurements-with-electron-energy-loss-spec.pdf

Related pages in this vault: Scanning Electron Microscope (SEM), Focused Ion Beam (FIB), Electron Backscatter Diffraction (EBSD), X-Ray Topography, Atom Probe Tomography (APT), Rutherford Backscattering Spectrometer (RBS), X-Ray Diffractometer (XRD and HRXRD), MOS Capacitor, Cathodoluminescence (SEM-CL), Electron Optics and the SEM Column.