Rutherford Backscattering Spectrometer (RBS)

1. Theory

• Definition: A Rutherford backscattering spectrometer directs a MeV light-ion beam, usually 2 $\mathrm{MeV}$ He, at a sample and measures the energies of ions scattered back near 165 to 170 ${}^{\circ }$, giving composition and areal density against depth without standards (Yu; Goncharova).
• Mass from kinematics: Elastic collision kinematics sets $E_1=K{E}_0$, so heavier target atoms return ions with more energy (Yu).
• Quantity from cross section: The Rutherford cross section is calculable, scaling as $Z_1^2{Z}_2^2/E^2$, so peak areas give absolute areal densities (Yu; Goncharova).
• Depth from energy loss: Ions lose energy on the way in and out, so scattering from deeper atoms appears at lower energy (Yu).

2. Working Principle

• Accelerator: A Van de Graaff or tandem accelerator delivers He+ or H+ at 1 to 3 $\mathrm{MeV}$ (Yu; Goncharova).
• Detection: A silicon surface-barrier detector at a backward angle with about 15 to 20 $\mathrm{keV}$ resolution records the energy spectrum (Yu; Goncharova).
• Dose: Beam charge is integrated in a Faraday cup, and the solid angle is calibrated with standards (Goncharova).
• Channelling: A goniometer aligns crystal axes with the beam to measure crystallinity and damage (Yu).

3. Applications

• Film stoichiometry and thickness: Silicides, high-k oxides and metal films (Yu; background knowledge).
• Implant and damage profiles: Heavy implants and lattice damage by channelling (Yu).
• Interdiffusion: Marker experiments and interface reactions (Yu).
• Light elements: Hydrogen by elastic recoil detection (Goncharova).

Scattering Kinematics and Yield

Name	Formula	Related Formulae	Variables/Constants
Kinematic factor	$K=\frac{E_1}{E_0}={\left[\frac{\sqrt{m_2^2-m_1^2{\sin }^2\theta }+m_1\cos \theta }{m_1+m_2}\right]}^2$	$K\left(180^{\circ }\right)={\left(\frac{m_2-m_1}{m_2+m_1}\right)}^2$	$K:kinematicfactor$ $E_0:incidentenergy$ $\begin{array}{r}{E}_1:energyafterscatteringfromthe\\ surface\end{array}$ $m_1:projectilemass$ $m_2:targetmass$ $\begin{array}{r}\theta :scatteringangle;Yu:relations,2.5\\ \mathrm{MeV}Heat170{}^{\circ }K(Au)=0.923,K(Ag)0.864\\ and0.862,K(Cu)0.783and0.777,K_{Al}=\\ 0.5525;Goncharova:Kasqualitative\\ (mass)information,2\mathrm{MeV}Heat165{}^{\circ }\end{array}$ $\begin{array}{r}(computedat170{}^{\circ }:O0.363,Si0.566,Hf\\ 0.915,Au0.9225andAl0.5525,\\ reproducingYu)\end{array}$
Rutherford cross section	$\frac{d\sigma }{d\Omega }\approx {\left(\frac{Z_1{Z}_2{e}^2}{4E}\right)}^2\frac{1}{{\sin }^4\left(\theta /2\right)}$	$\sigma \propto \frac{Z_1^2{Z}_2^2}{E_0^2}$	$Z_1,Z_2:projectileandtargetatomicnumbers$ $E:energy$ $\begin{array}{r}{e}^2:1.44eVnm;Yu:yieldproportionalto\\ Z_1^2{Z}_2^2/E_0^2,lab-frameform;Goncharova:\\ unscreenedCoulombscattering,sensitivity\\ risingwithZandfallingenergy,cross\\ sectionsin\mathrm{c}{\mathrm{m}}^2/sr;centre-of-mass\\ approximation(backgroundknowledge,Chu,\\ MayerandNicolet)\end{array}$ $\begin{array}{r}(computed:(74/14{)}^2=27.9againstY{u}^{\prime }s\\ tabulatedWtoSicross-sectionratioof\\ 29.6)\end{array}$
Areal density from yield	$A=\sigma \Omega Q\left(Nt\right)$	$\frac{{\left(Nt\right)}_A}{{\left(Nt\right)}_B}=\frac{A_A}{A_B}\frac{{\sigma }_B}{{\sigma }_A}$	$A:peakcounts$ $\sigma :crosssection$ $\Omega :detectorsolidangle$ $Q:numberofincidentions$ $\begin{array}{r}Nt:arealdensity;Yu:relations,Si/W\\ targetat2\mathrm{MeV}and165{}^{\circ }with{\sigma }_{Si}=\\ 2.5\times 10^{-25}and{\sigma }_W=7.4\times 10^{-24}\mathrm{c}{\mathrm{m}}^2/sr,Q=\\ 1.5\times 10^{14},\Omega =1.8\mathrm{msr},A_{Si}=200and{A}_W=\\ 6000,Nt=3\times 10^{15}atoms/c{\mathrm{m}}^2;Goncharova:Q=\\ ItfromaFaradaycup,\Omega calibratedwith\\ standards,arealdensity{N}_0\rho t/M\end{array}$ $\begin{array}{r}(computed:2.96\times 10^{15}and3.00\times 10^{15}atoms/c{\mathrm{m}}^2\\ fromY{u}^{\prime }snumbers)\end{array}$

Energy Loss and Resolution

Name	Formula	Related Formulae	Variables/Constants
Energy loss and depth scale	$\Delta E=\left[S\right]t=\left[KS\left(E_0\right)+\frac{S\left(K{E}_0\right)}{|\cos \theta |}\right]t$	$\left[S\right]=N\left[\epsilon \right]$	$\begin{array}{r}\Delta E:energywidthofalayersignal,[S]:\\ energy-lossfactor\end{array}$ $S:stoppingpower$ $\begin{array}{r}t:thickness,[\epsilon ]:stoppingcross-section\\ factor\end{array}$ $\begin{array}{r}N:atomicdensity;Yu:relationfornormal\\ incidence,Au-markerexamplewith\Delta E_{Au}=\\ 175\mathrm{keV}andK=0.9225giving3945\mathring{\mathrm{A}},Al\\ signal\Delta E=165\mathrm{keV}giving3937\mathring{\mathrm{A}}\end{array}$ $\begin{array}{r}(computed:thetwothicknessestimates\\ agreewithin0.2percent)\end{array}$
Bragg’s rule for compounds	$\epsilon \left(A_m{B}_n\right)=\frac{m}{m+n}{\epsilon }_A+\frac{n}{m+n}{\epsilon }_B$	$\frac{dE}{dx}=N\epsilon$	$\begin{array}{r}\epsilon :stoppingcrosssectionperatom;Yu:\\ \epsilon (Al)=44\times 10^{-15}and\epsilon (O)=35\times 10^{-15}\\ \mathrm{eV}\mathrm{c}{\mathrm{m}}^2/atom,\epsilon (Al2O3)=38.6\times 10^{-15}\\ \mathrm{eV}\mathrm{c}{\mathrm{m}}^2/atom,dE/dx=44.4eV/\mathring{\mathrm{A}},about\\ 2.2\times 10^{9}eV/cmforAlat3\mathrm{MeV};Goncharova:\\ about12eV/\mathring{\mathrm{A}}inSiO2and20eV/\mathring{\mathrm{A}}inSi3N4\\ for95\mathrm{keV}protons\end{array}$ $\begin{array}{r}(computed:44.4eV/\mathring{\mathrm{A}}requiresN=1.15\times 10^{23}\\ atomspercubiccentimetre,thesummar{y}^{\prime }s\\ 1.15\times 10^{22}readasatypesettingslip)\end{array}$
Depth and mass resolution	$\delta x=\frac{\delta E}{\left[S\right]}$	$\delta m=\frac{\delta E}{E_{0}dK/d{m}_2}$	$\delta x:depthresolution$ $\delta E:systemenergyresolution$ $\begin{array}{r}\delta m:massresolution;Yu:\delta E=20\mathrm{keV}at\\ 2\mathrm{MeV},\delta mabout1.48\mathrm{amu}atm=40and\\ 3.84\mathrm{amu}atm=70,Gaisotopes\\ unresolved;Goncharova:RBSdetector15\\ \mathrm{keV}againstMEIS180eV,ERD10to15\mathrm{keV},\\ decreasedmassresolutionforheavy\\ elements(seetheQ/A)\end{array}$ $\begin{array}{r}(assumed:[S]=40.5eV/\mathring{\mathrm{A}}forHeinSiat\\ 165{}^{\circ };computed:37\mathrm{nm}at15\mathrm{keV}and49\mathrm{nm}\\ at20\mathrm{keV};\delta m1.49and3.89\mathrm{amu};Ga\\ isotopes10.3\mathrm{keV}apart)\end{array}$

Thickness, Channelling and Precision

Name	Formula	Related Formulae	Variables/Constants
Areal density and thickness	$t=\frac{\left(Nt\right)}{N}$	$10^{15}atoms/c{\mathrm{m}}^2\approx 2\mathring{\mathrm{A}}\text{of Si}$	$\begin{array}{r}N:atomicdensity;Goncharova:areal\\ densityingpersquarecentimetreor\\ atomspersquarecentimetre,closetothe\\ thicknessinangstromsinabsolutenumbers\\ (seetheQ/A);conversion(background\\ knowledge,Chu,MayerandNicolet)\end{array}$ $\begin{array}{r}(computed:1\mathring{\mathrm{A}}is0.50\times 10^{15}atoms/c{\mathrm{m}}^{2}for\\ Si,0.66\times 10^{15}forSiO2,0.59\times 10^{15}forAuand\\ 0.83\times 10^{15}forHfO2)\end{array}$
Channelling	${\chi }_{min}=\frac{Y_{aligned}}{Y_{random}}$	${\psi }_{1/2}\text{(critical half-angle)}$	${\chi }_{min}:minimumyield$ $Y:backscatteringyield$ $\begin{array}{r}{\psi }_{1/2}:halfwidthoftheangulardip;Yu:{\chi }_{min}\\ about0.02to0.06,{\psi }_{1/2}relatedtothe\\ criticalangle,dechannellingindicating\\ defects\end{array}$ $\begin{array}{r}(computed:{\chi }_{min}=0.03means97percentof\\ thebeamissteeredawayfromlattice\\ atoms)\end{array}$
Precision and companion methods	$\frac{\delta (Nt)}{Nt}\approx 3\text{per cent}$	$\text{ERD for H, MEIS for sub-nm depth}$	$\begin{array}{r}Goncharova:RBSdepth-profilingprecision\\ about3percent,MEIS2percentforhigh\\ Z,ERDat1.6\mathrm{MeV}Hewith75{}^{\circ }incidence\\ and30{}^{\circ }recoil;Yu:normalizedyield\\ independentofQand\Omega ,H^{+}andH{e}^{+}beams\\ at165to170{}^{\circ }\end{array}$ $\begin{array}{r}(computed:countingstatisticsalonegive\\ 1/\sqrt{6000}=1.3percentforY{u}^{\prime }sWpeak)\end{array}$

Important Questions

Ques: How does RBS identify which atoms are present?

By the energy of ions scattered from the surface. Elastic kinematics gives $E_1=K{E}_0$ with K set by the projectile and target masses and the angle (Yu). For He at 170 ${}^{\circ }$, K is 0.363 for O, 0.566 for Si, 0.915 for Hf and 0.9225 for Au (computed), reproducing Yu’s Au and Al values of 0.923 and 0.5525. Heavy atoms cluster near K = 1, so mass resolution worsens with mass: $\delta m=\delta E/(E_{0}dK/dm)$ gives 1.49 $\mathrm{amu}$ at mass 40 and 3.89 $\mathrm{amu}$ at mass 70 for a 20 $\mathrm{keV}$ detector at 2 $\mathrm{MeV}$ (computed), close to Yu’s 1.48 and 3.84. The two Ga isotopes are 10.3 $\mathrm{keV}$ apart (computed), so, as Yu notes, they cannot be resolved. Reference: K. M. Yu, ‘Ion Beam Analysis in Materials Science (Rutherford backscattering and related techniques)’, Lawrence Berkeley National Laboratory slides (2008) (fetch-tool summary; slide text partly garbled).

Ques: Why does RBS not need standards?

Because the Rutherford cross section is calculable. The peak area is $A=\sigma \Omega Q(Nt)$ (Yu), and $\sigma$ follows from Coulomb scattering, scaling as $Z_1^2{Z}_2^2/E^2$ (Yu; Goncharova). Yu’s Si/W example, with Q = $1.5\times 10^{14}$ ions, a 1.8 $\mathrm{msr}$ detector and 200 and 6000 counts, gives $2.96\times 10^{15}$ and $3.00\times 10^{15}$ $atoms/c{\mathrm{m}}^2$ (computed), the stated $3\times 10^{15}$ for each. The cross-section ratio W to Si of 29.6 is close to $(74/14{)}^2$ = 27.9 (computed), the small difference coming from the angular and mass factors. Goncharova notes the charge comes from a Faraday cup and the solid angle is checked with standards; she quotes about 3 per cent precision. Counting statistics alone give 1.3 per cent for 6000 counts (computed). Reference: K. M. Yu, ‘Ion Beam Analysis in Materials Science (Rutherford backscattering and related techniques)’, Lawrence Berkeley National Laboratory slides (2008) (fetch-tool summary; slide text partly garbled); L. V. Goncharova, ‘Ion Beam Analysis and Modification for Current Issues in Surface Science’, Surface Canada 2013 tutorial, Western University (fetch-tool summary; formulas garbled).

Ques: How is depth obtained from the energy spectrum?

From the energy lost on the inward and outward paths. Yu writes $\Delta E=[KS(E_0)+S(K{E}_0)/|\cos \theta |]t$, where the bracket is the energy-loss factor. In his Au-marker example, the Au signal shift of 175 $\mathrm{keV}$ gives 3945 $\mathring{\mathrm{A}}$ and the Al signal width of 165 $\mathrm{keV}$ gives 3937 $\mathring{\mathrm{A}}$, agreeing within 0.2 per cent (computed). Compound stopping uses Bragg’s rule: $\epsilon$(Al2O3) = (2/5)($44\times 10^{-15}$) + (3/5)($35\times 10^{-15}$) = $38.6\times 10^{-15}$ $\mathrm{eV}\mathrm{c}{\mathrm{m}}^2/atom$ (Yu). His stated 44.4 $eV/\mathring{\mathrm{A}}$ requires an atomic density of $1.15\times 10^{23}$ atoms per cubic centimetre (computed), so the summary’s $1.15\times 10^{22}$ is likely a typesetting slip (my reading). Converting energy to depth therefore needs both stopping data and an assumed density, and simulation codes iterate this layer by layer (background knowledge, Chu, Mayer and Nicolet). Reference: K. M. Yu, ‘Ion Beam Analysis in Materials Science (Rutherford backscattering and related techniques)’, Lawrence Berkeley National Laboratory slides (2008) (fetch-tool summary; slide text partly garbled); W.-K. Chu, J. W. Mayer and M.-A. Nicolet, Backscattering Spectrometry, Academic Press 1978, and J. R. Tesmer and M. Nastasi (eds.), Handbook of Modern Ion Beam Materials Analysis, MRS 1995 (background knowledge; not opened).

Ques: Is the RBS detector resolution 15 or 20 keV?

Goncharova cites 15 $\mathrm{keV}$ for an RBS detector and 10 to 15 $\mathrm{keV}$ for ERD, while Yu uses a system resolution of 20 $\mathrm{keV}$ at 2 $\mathrm{MeV}$. Both are realistic: silicon surface-barrier detectors give 12 to 15 $\mathrm{keV}$ intrinsically, and electronic noise and beam energy spread raise the system value toward 20 $\mathrm{keV}$ (background knowledge, Chu, Mayer and Nicolet). For depth, $\delta x=\delta E/[S]$; with an assumed [S] of 40.5 $eV/\mathring{\mathrm{A}}$ for He in Si, 15 $\mathrm{keV}$ gives 37 $\mathrm{nm}$ and 20 $\mathrm{keV}$ gives 49 $\mathrm{nm}$ (computed). The 15 $\mathrm{keV}$ figure is more likely for the detector alone, and 20 $\mathrm{keV}$ a conservative system value (my reading). Goncharova’s MEIS, at 180 eV, reaches sub-nanometre depth resolution, about 0.75 $\mathrm{nm}$ for an assumed 24 $eV/\mathring{\mathrm{A}}$ (computed). Reference: L. V. Goncharova, ‘Ion Beam Analysis and Modification for Current Issues in Surface Science’, Surface Canada 2013 tutorial, Western University (fetch-tool summary; formulas garbled); K. M. Yu, ‘Ion Beam Analysis in Materials Science (Rutherford backscattering and related techniques)’, Lawrence Berkeley National Laboratory slides (2008) (fetch-tool summary; slide text partly garbled); W.-K. Chu, J. W. Mayer and M.-A. Nicolet, Backscattering Spectrometry, Academic Press 1978, and J. R. Tesmer and M. Nastasi (eds.), Handbook of Modern Ion Beam Materials Analysis, MRS 1995 (background knowledge; not opened).

Ques: Is areal density in $1\times 10^{15}$ atoms per square centimetre really close to thickness in angstroms?

Only to within a factor of two. Goncharova states that areal density is close to the thickness in angstroms in absolute numbers. Dividing by atomic density shows the actual conversion: 1 $\mathring{\mathrm{A}}$ corresponds to $0.50\times 10^{15}$ $atoms/c{\mathrm{m}}^2$ for Si, $0.66\times 10^{15}$ for SiO2, $0.59\times 10^{15}$ for Au and $0.83\times 10^{15}$ for HfO2 (computed). So $100\times 10^{15}$ $atoms/c{\mathrm{m}}^2$ of Si is 200 $\mathring{\mathrm{A}}$, not 100, and the rule is a rough mnemonic rather than an equivalence (my reading). Because RBS measures atoms per area directly, a thickness always needs an assumed density, which is why RBS and XRR together give both density and thickness (background knowledge, Chu, Mayer and Nicolet). Reference: L. V. Goncharova, ‘Ion Beam Analysis and Modification for Current Issues in Surface Science’, Surface Canada 2013 tutorial, Western University (fetch-tool summary; formulas garbled); W.-K. Chu, J. W. Mayer and M.-A. Nicolet, Backscattering Spectrometry, Academic Press 1978, and J. R. Tesmer and M. Nastasi (eds.), Handbook of Modern Ion Beam Materials Analysis, MRS 1995 (background knowledge; not opened).

Ques: What does channelling RBS measure?

Crystal quality, damage and lattice location. When the beam is aligned with a major axis, ions are steered between atomic rows and the backscattering yield collapses; the ratio to a random direction is the minimum yield ${\chi }_{min}$, about 0.02 to 0.06 in good crystals (Yu). A value of 0.03 means 97 per cent of the beam avoids close collisions with lattice atoms (computed). Implant damage or disorder raises the aligned yield, and dechannelling behind defects broadens the signal, which Yu notes indicates defects. Substitutional dopants hide in the rows and their yield drops with the host, while interstitials remain visible, so comparing aligned spectra gives lattice location (background knowledge, Tesmer and Nastasi). The angular half width ${\psi }_{1/2}$ characterizes the dip (Yu). Reference: K. M. Yu, ‘Ion Beam Analysis in Materials Science (Rutherford backscattering and related techniques)’, Lawrence Berkeley National Laboratory slides (2008) (fetch-tool summary; slide text partly garbled); W.-K. Chu, J. W. Mayer and M.-A. Nicolet, Backscattering Spectrometry, Academic Press 1978, and J. R. Tesmer and M. Nastasi (eds.), Handbook of Modern Ion Beam Materials Analysis, MRS 1995 (background knowledge; not opened).

References

1. K. M. Yu, “Ion Beam Analysis in Materials Science (Rutherford backscattering and related techniques),” Electronic Materials Program, Lawrence Berkeley National Laboratory, 2008. Available: https://nucleardata.berkeley.edu/research/furnace_pages/downloads/RBS_11_2008_3.pdf
2. L. V. Goncharova, “Ion Beam Analysis and Modification for Current Issues in Surface Science,” Surface Canada 2013 Tutorials, Western University, 2013. Available: https://www.sc2013.uwo.ca/Tutorial/LGoncharova_IonBeams_1.pdf
3. W.-K. Chu, J. W. Mayer and M.-A. Nicolet, Backscattering Spectrometry, Academic Press, 1978; J. R. Tesmer and M. Nastasi (eds.), Handbook of Modern Ion Beam Materials Analysis, Materials Research Society, 1995 (cited from background knowledge).

Related pages in this vault: X-Ray Reflectometer (XRR), Secondary Ion Mass Spectrometer (SIMS and ToF-SIMS), Transmission Electron Microscope (TEM and STEM), X-Ray Fluorescence Spectrometer (XRF and TXRF), Auger Electron Spectrometer (AES), X-Ray Photoelectron Spectrometer (XPS), Scanning Electron Microscope (SEM), Focused Ion Beam (FIB).