Electron-Matter Interaction

1. Theory

• Definition: A primary electron entering a solid scatters elastically, changing direction, and inelastically, losing energy, until it stops or escapes; the escaping electrons and photons are the SEM and inspection signals (Thomson et al.; Kanaya and Ono).
• Range: The depth of travel follows the Kanaya-Okayama range, about $E_0^{1.67}$, so lowering beam energy shrinks the interaction volume sharply (HyperSpy source; globalsino).
• Energy loss: The Bethe continuous-slowing-down law sets $dE/dz$ through the mean ionization energy $J$ (Singh and Singh), and it underestimates the loss below about $3J$ (Joy).
• Emitted electrons: Secondary electrons are below 50 $\mathrm{eV}$ and escape from within nanometres to tens of nanometres; backscattered electrons are above 50 $\mathrm{eV}$ and their yield depends strongly on atomic number (Thomson et al.; globalsino).
• Charging: The total yield crosses 1 at $E_1$ and $E_2$; insulators charge negatively outside that window and positively inside it (Thomson et al.).

2. Working Principle

• Signal generation: SEs carry surface and topography information from the escape depth; BSEs carry composition from deeper in the interaction volume (Thomson et al.; Hall and Lloyd).
• Compositional contrast: Compounds backscatter as the weight-fraction average of their elements, and differences of about 0.1 in mean Z can be resolved (Hall and Lloyd).
• Charge balance: The surface potential shifts the landing energy and relaxes with time constant $\rho {\epsilon }_r{\epsilon }_0$ (Thomson et al.).
• X-ray generation: Characteristic lines need an overvoltage of about 2 or more for reliable K-shell data (Joy).

3. Applications

• Low-voltage SEM and CD-SEM: Confining the volume and staying near $E_2$ to limit charging (Thomson et al.; SEM page).
• BSE material contrast: Phase and composition mapping (Hall and Lloyd).
• Simulation: Monte Carlo and range models for metrology tool interpretation (Singh and Singh; HyperSpy source).

Penetration and Energy Loss

Name	Formula	Related Formulae	Variables/Constants
Kanaya-Okayama range	$R=\frac{0.0276A{E}_0^{1.67}}{Z^{0.89}\rho }\mu \mathrm{m}$	$R\propto E_0^n,n\approx 1.35(E_0<5\mathrm{keV}),1.67(\text{above})$	$A:atomicweight$ $E_0:beamenergyin\mathrm{keV}$ $Z:atomicnumber$ $\begin{array}{r}\rho :densitying/c{\mathrm{m}}^3;HyperSpysource:this\\ parameterizationfromKanayaandOkayama\\ (1972);globalsino:thetwoexponentsand\\ tabulateddepths\end{array}$ $\begin{array}{r}(computed:Si(\rho =2.329)32\mathrm{nm}at1\mathrm{keV},\\ 101\mathrm{nm}at2,467\mathrm{nm}at5and2.93\mu \mathrm{m}at15\\ \mathrm{keV};Cu0.90\mu \mathrm{m}andAu0.53\mu \mathrm{m}at15\mathrm{keV},\\ closetoglobalsin{o}^{\prime }s900and540\mathrm{nm})\end{array}$
Bethe stopping power	$\frac{dE}{dz}=-2\pi e^4\frac{\rho N_A}{ME}\ln \left(\frac{1.166E}{ϵ}\right)$	$ϵ=\frac{J}{1+CJ/E},\frac{J}{Z}=9.76+58.8Z^{-1.19}$	$M:molarmass$ $J:meanionizationenergy$ $\begin{array}{r}C:low-energycorrectionbelow1;Singh\\ andSingh,Eqs.5and6;Joy:Bethe\\ adequatedowntoabout3Jandtoolow\\ belowthat\end{array}$ $\begin{array}{r}(computed:J=172\mathrm{eV}forSi,sothe\\ simpleformisreliableaboveabout0.5\\ \mathrm{keV})\end{array}$
Energy along the path	$\frac{E}{E_0}={\left(1-y\right)}^{n/\left(1+n\right)},y=\frac{x}{R}$	$R_{ins}\approx 5x_a$	$y:reduceddepth$ $n:screeningparameter$ $\begin{array}{r}{x}_a:SEescapedepth;KanayaandOno:these\\ relations(OCR-garbled),validity1\mathrm{keV}to\\ 1MeV\end{array}$ $\begin{array}{r}(computed:withn=1.67,halftherange\\ leavesE/E_0=0.65)\end{array}$

Emission and Charging

Name	Formula	Related Formulae	Variables/Constants
Secondary-electron yield curve	$\sigma =\delta +\eta$	$\sigma \left(E_1\right)=\sigma \left(E_2\right)=1$	$\delta ,\eta :SEandBSEyields$ $\sigma :totalyield$ $\begin{array}{r}{E}_1,E_2:crossoverenergies;Thomsonet\\ al.:yieldsasemittedperincident\\ electron,SEbelow50\mathrm{eV},parameters{\sigma }_{max},\\ E_{max},E_1,E_2,gold1.8at600\mathrm{eV}with{E}_{2}10\\ \mathrm{keV},anodizedAl3.0at300\mathrm{eV}with{E}_2\\ 1400\mathrm{eV};vanEdenetal.:glass2.26at\\ 300\mathrm{eV},E_{2}about2100\mathrm{eV};sumofyields\\ (myreadingoftotalyieldincludingboth)\end{array}$ $\begin{array}{r}(computed:gol{d}^{\prime }s{E}_{2}is7timesanodized\\ aluminiu{m}^{\prime }s)\end{array}$
Charging regimes	$E_{land}=E_{beam}-e{\varphi }_s$	$\tau =\rho {\epsilon }_r{\epsilon }_0,{\varphi }_s\left(t\right)={\varphi }_{s0}{e}^{-t/\tau }$	${\varphi }_s:beam-inducedsurfacepotential$ $\begin{array}{r}\tau :chargedecaytime;Thomsonetal.,Eqs.\\ 1to3:negativechargingbelow{E}_{1}and\\ above{E}_2,positivebetween\end{array}$ $\begin{array}{r}(assumed:resist\rho =1\times 10^{12}\Omega m,{\epsilon }_r=3;\\ computed:\tau =27\mathrm{s};myreading:at3\mathrm{keV}\\ onafilmwith{E}_2=1.4\mathrm{keV},negative\\ chargingcontinuesuntilthelanding\\ energyfallstoward{E}_2)\end{array}$
Escape depths	${\lambda }_{SE}<10\mathrm{nm}(\mathrm{Au}),{\lambda }_{SE}\sim \text{tens of nm}(\text{insulators})$	$R\left(\mathrm{Au},500\mathrm{eV}\right)\approx 30\mathring{\mathrm{A}}$	$\begin{array}{r}Thomsonetal.:SEescapelength;Kanaya\\ andOno:Aurangeabout30\mathring{A}at500\mathrm{eV}\\ (seetheQ/A)\end{array}$ $\begin{array}{r}(computed:Kanaya-Okayamaextrapolatedto\\ 0.5\mathrm{keV}gives18\mathring{A}forAu)\end{array}$

Contrast and Data Quality

Name	Formula	Related Formulae	Variables/Constants
Backscatter of compounds	$\bar{\eta }={\sum }_i{C}_i{\eta }_i$	$\text{contrast}=\frac{{\eta }_A-{\eta }_B}{\frac{1}{2}\left({\eta }_A+{\eta }_B\right)}$	$\begin{array}{r}{C}_i:weightfraction;HallandLloyd:these\\ relations,biotite0.1648,Cu-Znphases\\ about0.1apartinZresolved;globalsino:\\ BSEfractionstronglyZdependent\end{array}$ $\begin{array}{r}(assumed:{\eta }_A=0.30,{\eta }_B=0.29;computed:\\ contrast3.4percent)\end{array}$
Data uncertainty	$\frac{\Delta \eta }{\eta }\approx \frac{\Delta \delta }{\delta }\approx 5\%$	$U=\frac{E_0}{E_c}\gtrsim 2(\text{K-shell data reliable})$	$U:overvoltage$ $\begin{array}{r}{E}_c:criticalionizationenergy;Joy:\eta and\\ \delta uncertainbyabout5percentevenfor\\ AlandCu,K-shellcrosssectionsadequate\\ aboveUofabout2,stoppingpowers\\ additiveaboveabout100\mathrm{eV}\end{array}$ $\begin{array}{r}(computed:SiKat1.84\mathrm{keV}needs{E}_{0}ofat\\ least3.7\mathrm{keV}forU=2)\end{array}$
Low-voltage confinement	$\frac{R\left(E_2\right)}{R\left(E_1\right)}={\left(\frac{E_2}{E_1}\right)}^{1.67}$	$\text{information volume}\propto R^3$	$\begin{array}{r}fromtheKanaya-Okayamaexponent\\ (computed;myreadingthatthevolume\\ scalesasthecubeoftherange);SEMpage\\ forimagingconsequences\end{array}$ $\begin{array}{r}(computed:5to1\mathrm{keV}shrinkstherange\\ 14.7timesandthevolumeabout3200\\ times)\end{array}$

Important Questions

Ques: How deep does the electron beam reach in silicon, and how fast does it shrink with voltage?

From tens of nanometres to microns. The HyperSpy code implements the Kanaya-Okayama range $R=0.0276A{E}_0^{1.67}/(Z^{0.89}\rho )$ in micrometres, returning 2.877 $\mu \mathrm{m}$ for copper at 30 $\mathrm{keV}$. For silicon of density 2.329 $g/c{\mathrm{m}}^3$ it gives 32 $\mathrm{nm}$ at 1 $\mathrm{keV}$, 101 $\mathrm{nm}$ at 2, 467 $\mathrm{nm}$ at 5 and 2.93 $\mu \mathrm{m}$ at 15 $\mathrm{keV}$ (computed). The same formula gives 0.90 $\mu \mathrm{m}$ for copper and 0.53 $\mu \mathrm{m}$ for gold at 15 $\mathrm{keV}$, close to globalsino’s 900 and 540 $\mathrm{nm}$ (computed), so the table follows this parameterization. Dropping from 5 to 1 $\mathrm{keV}$ shortens the range 14.7 times and, taking volume as range cubed, shrinks it about 3200 times (computed; my reading). Reference: HyperSpy v1.7.5 source, hyperspy/misc/eds/utils.py, function electron_range (GitHub): Kanaya-Okayama range R (um) = 0.0276 A E^1.67/(Z^0.89 rho) cos(tilt), E in keV, rho in g/cm3, citing Kanaya and Okayama, J. Phys. D 5, 43 (1972), example Cu at 30 keV 2.877 um (read directly); Y. Liao, Practical Electron Microscopy and Database, globalsino.com, page 4795: exponent n about 1.35 below 5 keV and 1.67 above, penetration depths Al 400 nm and 2.4 um at 5 and 15 keV, Fe 35 A, 10, 25, 160 and 990 nm at 0.5, 1, 2, 5 and 15 keV, Cu 150 and 900 nm, Au 88 and 540 nm at 5 and 15 keV, polymer 100 nm and 12 um at 2 and 30 keV; page 4801: BSE fraction highly dependent on Z, SE imaging higher resolution than BSE (fetch-tool summary).

Ques: Is the range exponent 1.67 valid at low beam energy?

The sources suggest not exactly. Globalsino states that the exponent is about 1.35 below 5 $\mathrm{keV}$ and 1.67 above, and Kanaya and Ono give their range relation from about 1 $\mathrm{keV}$ to 1 MeV, with the screening parameter changing as voltage falls. Extrapolating Kanaya-Okayama to 500 $\mathrm{eV}$ gives 18 angstrom for gold (computed), whereas Kanaya and Ono quote about 30 angstrom, though their text is OCR-garbled. Anchoring at silicon’s 467 $\mathrm{nm}$ at 5 $\mathrm{keV}$, an exponent of 1.35 gives 53 $\mathrm{nm}$ at 1 $\mathrm{keV}$ against 32 $\mathrm{nm}$ with 1.67 (computed). Since a smaller exponent predicts longer low-energy ranges, consistent with the 30 angstrom figure, the 1.67 law more likely underestimates the range below a few keV (my reading). Reference: Y. Liao, Practical Electron Microscopy and Database, globalsino.com, page 4795: exponent n about 1.35 below 5 keV and 1.67 above, penetration depths Al 400 nm and 2.4 um at 5 and 15 keV, Fe 35 A, 10, 25, 160 and 990 nm at 0.5, 1, 2, 5 and 15 keV, Cu 150 and 900 nm, Au 88 and 540 nm at 5 and 15 keV, polymer 100 nm and 12 um at 2 and 30 keV; page 4801: BSE fraction highly dependent on Z, SE imaging higher resolution than BSE (fetch-tool summary); K. Kanaya and S. Ono, ‘Interaction of Electron Beam with the Target in Scanning Electron Microscope’, Scanning Electron Microscopy 1982 (1), article 6: range-energy relation valid from about 1 keV to 1 MeV with a screening parameter rising as voltage decreases, E/E0 = (1 - y)^(n/(1+n)), insulator range about 5 times the escape depth, Au range about 30 A at 500 eV (OCR-garbled) (fetch-tool summary); HyperSpy v1.7.5 source, hyperspy/misc/eds/utils.py, function electron_range (GitHub): Kanaya-Okayama range R (um) = 0.0276 A E^1.67/(Z^0.89 rho) cos(tilt), E in keV, rho in g/cm3, citing Kanaya and Okayama, J. Phys. D 5, 43 (1972), example Cu at 30 keV 2.877 um (read directly).

Ques: What sets the energy loss rate, and where does the Bethe law fail?

Ionization of atomic electrons. Singh and Singh write the Bethe continuous-slowing-down law as $dE/dz=-2\pi e^4(\rho N_A/ME)\ln (1.166E/ϵ)$ with $ϵ=J/(1+CJ/E)$ and $J/Z=9.76+58.8Z^{-1.19}$, which gives $J$ = 172 $\mathrm{eV}$ for silicon (computed). Joy states that this simple form is adequate down to about $3J$ but underestimates stopping power below that, that stopping powers add for compounds except below about 100 $\mathrm{eV}$, and that even for aluminium and copper the backscatter and secondary yields are uncertain to about 5 per cent. For silicon, $3J$ is about 0.5 $\mathrm{keV}$ (computed), so modern low-voltage inspection at a few hundred eV sits where the simple model is least reliable (my reading). Reference: M. S. Singh and R. K. B. Singh, ‘Power law nature in electron solid interaction’, arXiv:1601.01993: Bethe continuous slowing down dE/dz = -2 pi e^4 (rho N_A/(M E)) ln(1.166E/epsilon) with epsilon = J/(1 + C J/E), J/Z = 9.76 + 58.8 Z^-1.19 (fetch-tool summary); D. C. Joy, ‘Fundamental Parameters for Microanalysis’, Microscopy and Microanalysis 1996 (Oak Ridge): Bethe stopping power adequate down to about 3J and underestimating below, stopping powers additive except below about 100 eV, K-shell cross sections known above an overvoltage of about 2, eta and delta uncertain by about 5 per cent even for Al and Cu (fetch-tool summary).

Ques: Why do insulators charge, and how is charging controlled?

Because emitted and absorbed electrons do not balance. Thomson et al. parameterize the yield curve by its maximum and the crossover energies $E_1$ and $E_2$: below $E_1$, under 100 V for most insulators, and above $E_2$, over 1 $\mathrm{keV}$, the surface charges negatively, while between them more electrons leave than arrive and it charges positively. The surface potential shifts the landing energy to $E_{beam}-e{\varphi }_s$ and decays with $\tau =\rho {\epsilon }_r{\epsilon }_0$. Measured values span gold, ${\sigma }_{max}$ = 1.8 with $E_2$ = 10 $\mathrm{keV}$, to anodized aluminium, 3.0 with $E_2$ = 1400 $\mathrm{eV}$; van Eden et al. find 2.26 and about 2100 $\mathrm{eV}$ for glass. Imaging insulators near $E_2$ therefore minimizes charging (my reading; SEM page). Reference: C. D. Thomson, V. Zavyalov, J. R. Dennison and J. Corbridge, ‘Electron Emission Properties of Insulator Materials Pertinent to the International Space Station’, 8th Spacecraft Charging Technology Conference (2003), NASA NTRS: SE below 50 eV and BSE above, yield curve with sigma_max, E_max, E1 and E2, negative charging below E1 (under 100 V for most insulators) and above E2 (over 1 keV), positive between, landing energy E_beam - e phi_s, tau = rho eps_r eps_0, phi_s decaying as exp(-t/tau), SE escape length tens of nm in insulators and under 10 nm in gold, gold sigma_max 1.8 at 600 eV with E1 100 eV and E2 10 keV, anodized Al 3.0 at 300 eV with E2 1400 eV (fetch-tool summary); G. van Eden, D. Verheijde and J. Verhoeven, ‘Secondary electron emission yield measurements of dielectrics based on a novel collector-only method’, Nucl. Instrum. Methods B 511, 6 (2022), AMOLF repository: SE below 50 eV, alkali-free glass delta_max 2.26 at 300 eV with second crossover about 2100 eV, SEM imaging affected by surface charging (fetch-tool summary); Scanning Electron Microscope (SEM) page of this vault.

Ques: Where do SE and BSE signals come from, and what contrast do they give?

From different depths. Thomson et al. define secondaries as electrons below 50 $\mathrm{eV}$ generated in the material and backscattered electrons as those above 50 $\mathrm{eV}$ from the beam, with secondary escape lengths under 10 $\mathrm{nm}$ in gold but tens of nanometres in insulators. Kanaya and Ono relate the insulator range to about five escape depths. Globalsino notes that BSE yield depends strongly on atomic number and that SE imaging has higher resolution. Hall and Lloyd give compound backscatter as $\sum C_i{\eta }_i$ and contrast as $({\eta }_A-{\eta }_B)/\frac{1}{2}({\eta }_A+{\eta }_B)$, resolving Cu-Zn phases about 0.1 apart in Z; assumed yields of 0.30 and 0.29 give 3.4 per cent contrast (computed). Reference: C. D. Thomson, V. Zavyalov, J. R. Dennison and J. Corbridge, ‘Electron Emission Properties of Insulator Materials Pertinent to the International Space Station’, 8th Spacecraft Charging Technology Conference (2003), NASA NTRS: SE below 50 eV and BSE above, yield curve with sigma_max, E_max, E1 and E2, negative charging below E1 (under 100 V for most insulators) and above E2 (over 1 keV), positive between, landing energy E_beam - e phi_s, tau = rho eps_r eps_0, phi_s decaying as exp(-t/tau), SE escape length tens of nm in insulators and under 10 nm in gold, gold sigma_max 1.8 at 600 eV with E1 100 eV and E2 10 keV, anodized Al 3.0 at 300 eV with E2 1400 eV (fetch-tool summary); K. Kanaya and S. Ono, ‘Interaction of Electron Beam with the Target in Scanning Electron Microscope’, Scanning Electron Microscopy 1982 (1), article 6: range-energy relation valid from about 1 keV to 1 MeV with a screening parameter rising as voltage decreases, E/E0 = (1 - y)^(n/(1+n)), insulator range about 5 times the escape depth, Au range about 30 A at 500 eV (OCR-garbled) (fetch-tool summary); Y. Liao, Practical Electron Microscopy and Database, globalsino.com, page 4795: exponent n about 1.35 below 5 keV and 1.67 above, penetration depths Al 400 nm and 2.4 um at 5 and 15 keV, Fe 35 A, 10, 25, 160 and 990 nm at 0.5, 1, 2, 5 and 15 keV, Cu 150 and 900 nm, Au 88 and 540 nm at 5 and 15 keV, polymer 100 nm and 12 um at 2 and 30 keV; page 4801: BSE fraction highly dependent on Z, SE imaging higher resolution than BSE (fetch-tool summary); M. G. Hall and G. E. Lloyd, ‘The SEM examination of geological samples with a semiconductor back-scattered electron detector’, Am. Mineral. 66, 362 (1981): eta for compounds as the weight-fraction sum of elemental eta, contrast [eta_A - eta_B]/(0.5(eta_A + eta_B)), biotite eta 0.1648, Cu-Zn phases differing by about 0.1 in Z resolved (fetch-tool summary).

References

1. HyperSpy v1.7.5, hyperspy/misc/eds/utils.py, electron_range function (Kanaya-Okayama parameterization). Available: https://raw.githubusercontent.com/hyperspy/hyperspy/v1.7.5/hyperspy/misc/eds/utils.py
2. Y. Liao, Practical Electron Microscopy and Database, globalsino.com, pages 4795 and 4801. Available: https://www.globalsino.com/EM/page4795.html and https://www.globalsino.com/EM/page4801.html
3. K. Kanaya and S. Ono, “Interaction of electron beam with the target in scanning electron microscope,” Scanning Electron Microscopy, vol. 1982, no. 1, article 6. Available: https://core.ac.uk/download/395016012.pdf
4. M. S. Singh and R. K. B. Singh, “Power law nature in electron solid interaction,” arXiv:1601.01993. Available: https://arxiv.org/pdf/1601.01993
5. D. C. Joy, “Fundamental parameters for microanalysis,” Microscopy Today, reprinted from Microscopy and Microanalysis 1996. Available: https://cambridge.org/core/services/aop-cambridge-core/content/view/ACE68F83F107B6B47C6A618C624829FD/S1551929500063641a.pdf/fundamental_parameters_for_microanalysis.pdf
6. C. D. Thomson, V. Zavyalov, J. R. Dennison and J. Corbridge, “Electron emission properties of insulator materials pertinent to the International Space Station,” 8th Spacecraft Charging Technology Conference, 2003. Available: https://ntrs.nasa.gov/api/citations/20040111072/downloads/20040111072.pdf
7. G. van Eden, D. Verheijde and J. Verhoeven, “Secondary electron emission yield measurements of dielectrics based on a novel collector-only method,” Nuclear Instruments and Methods in Physics Research B, vol. 511, pp. 6-11, 2022. Available: https://ir.amolf.nl/pub/10484/16757OA.pdf
8. M. G. Hall and G. E. Lloyd, “The SEM examination of geological samples with a semiconductor back-scattered electron detector,” American Mineralogist, vol. 66, pp. 362-368, 1981. Available: https://msaweb.org/AmMin/AM66/AM66_362.pdf
9. Scanning Electron Microscope (SEM) page of this vault.

Related pages in this vault: Scanning Electron Microscope (SEM), CD-SEM, E-Beam Inspection and Defect Review, EBIC and EBAC, Cathodoluminescence (SEM-CL), Light-Matter Interaction for Optical Inspection, Electron Optics and the SEM Column, Electron-Beam Lithography Writer, Electron-Beam Evaporator.