Electron Backscatter Diffraction (EBSD)

1. Theory

• Definition: Electron backscatter diffraction records patterns of Kikuchi bands formed by backscattered electrons diffracted from lattice planes of a strongly tilted crystalline sample in an SEM, and indexes them to give crystal orientation and phase at each point (Oxford; DTU).
• Geometry: Each band corresponds to a lattice plane satisfying the Bragg condition; band intersections are zone axes and angles between bands are angles between planes (DTU).
• Maps: Scanning the beam gives orientation maps from which grain size, texture, boundary types and phases follow (Oxford).
• Surface sensitivity: The pattern comes from the top 5 to 10 $\mathrm{nm}$, so surface preparation is critical (DTU).

2. Working Principle

• Tilt and detector: The sample is tilted to about 70 degrees, and a phosphor screen and camera capture the pattern (DTU; Oxford).
• Band detection: The Hough transform turns bands into peaks for automatic detection (Oxford).
• Indexing: Combinations of bands are matched to crystal structures, with fit quality expressed as mean angular deviation (Oxford).
• Speed: Modern detectors index up to 3000 patterns per second (DTU).

3. Applications

• Interconnects: Copper grain size and texture in microelectronics (Oxford; Wire Resistance page).
• Thin films and solar cells: Grain structure of polycrystalline films (Oxford).
• Phase identification: Phases and precipitates in alloys and solders (Oxford).
• Deformation: Local misorientation as a measure of plastic strain (Oxford).

Diffraction and Patterns

Name	Formula	Related Formulae	Variables/Constants
Bragg condition	$n\lambda =2d\sin {\theta }_B$	$\text{Kikuchi cone semi-angle}=90^{\circ }-{\theta }_B$	$n:order$ $\lambda :electronwavelength$ $d:planespacing$ $\begin{array}{r}{\theta }_B:Braggangle;DTU:Braggcondition,\\ intersectionsofbandsaszoneaxesand\\ anglesbetweenbandsasanglesbetween\\ planes;Oxford:Braggequation(not\\ legible),conesemi-angle(90-\theta ),\\ example\lambda =0.00859\mathrm{nm}at20\mathrm{kV}andAl\\ (111)spacing0.233\mathrm{nm}giving88.9degrees\end{array}$ $\begin{array}{r}(computed:\sin {\theta }_B=0.0184,{\theta }_B=1.06degrees,\\ conesemi-angle88.94degrees)\end{array}$
Kikuchi band width	$w\approx 2{\theta }_{B}l\approx \frac{n\lambda l}{d}$	$w\propto \frac{1}{d}$	$w:bandwidthonthescreen$ $\begin{array}{r}l:sample-to-screendistance;small-angle\\ relation(computedfromBrag{g}^{\prime }slaw);\\ Oxford:band-widthequationnotlegible,\\ widerd-spacingsgivingthinnerbands\end{array}$ $\begin{array}{r}(assumed:l=20\mathrm{mm};computed:Al(111)\\ gives2{\theta }_B=2.11degreesandw=0.74\mathrm{mm};\\ Cu(111),d=0.209\mathrm{nm},gives2.36\\ degrees)\end{array}$
Hough transform	$\rho =x\cos \theta +y\sin \theta$	$\text{band}\to \text{peak at}\left(\rho ,\theta \right)$	$\begin{array}{r}\rho ,\theta :Hough-spacecoordinatesofa\\ straightbandinpatterncoordinatesx,y;\\ Oxford:thistransform,four-band\\ combinationsinindexing,patternquality\\ fromHoughpeakheights{h}_{i}andstandard\\ deviation{s}_h,meanangulardeviation(MAD)\\ asthefitbetweenpatternandsolution,\\ indexinginlessthanafewmilliseconds\\ perpattern;DTU:upto3000patternsper\\ second\end{array}$ $\begin{array}{r}(computed:a1000by1000pointmapat\\ 3000patternspersecondtakes333\mathrm{s})\end{array}$

Geometry and Resolution

Name	Formula	Related Formulae	Variables/Constants
Sample tilt and resolution	${\theta }_{tilt}\approx 70^{\circ }$	$\frac{d_y}{d_x}\approx \frac{1}{\sin \left(90^{\circ }-{\theta }_{tilt}\right)}=\frac{1}{\sin 20^{\circ }}$	${\theta }_{tilt}:sampletiltfromhorizontal$ $\begin{array}{r}{d}_x,d_y:spatialresolutionacrossanddown\\ thetilt;DTU:tiltofabout70degreesas\\ thebestcompromisebetweenintensityand\\ resolution,y-axisresolution3times\\ worse;Oxford:downhillresolutionreduced\\ bynearlyafactorof3,TKDneedingno\\ morethanabout10degreestilt;geometric\\ relation(computed)\end{array}$ $\begin{array}{r}(computed:1/\sin 20degrees=2.92,matching\\ thefactorof3inbothsources)\end{array}$
Information depth	$z_{EBSD}\approx 5\text{to}10\mathrm{nm}$	$t_{coat}\approx 2\text{to}5\mathrm{nm}(\text{non-conductors})$	$\begin{array}{r}{z}_{EBSD}:depthfromwhichthediffractedsignal\\ originates\end{array}$ $\begin{array}{r}{t}_{coat}:conductivecoatingthickness;DTU:\\ pointsource5to10\mathrm{nm}belowthesurface,\\ surface-sensitivetechnique,coatingsof2\\ to5\mathrm{nm};Oxford:lowerkV,lowerprobe\\ currentandshorterworkingdistancefor\\ betterresolution,nonumericalvalues;\\ NISTIR7814:chemo-mechanicalor\\ electrolyticpolishingtoremovepolishing\\ deformation\end{array}$ $\begin{array}{r}(computed:a5\mathrm{nm}coatingconsumeshalfto\\ allofthe5to10\mathrm{nm}informationdepth,\\ socoatingsmustbeminimal)\end{array}$
Transmission Kikuchi diffraction	$\text{TKD: thin foil, tilt}\lesssim 10^{\circ }$	$d_{TKD}<d_{EBSD}(\text{no tilt elongation})$	$\begin{array}{r}TKD:Kikuchidiffractionfromelectrons\\ transmittedthroughathinspecimen,\\ givingfinerspatialresolutionfor\\ nanocrystallinefilms;Oxford:TKDtiltno\\ morethanabout10degreesdependingon\\ set-up,deformed-steelgrainsabove100\mathrm{nm}\\ resolvedwithintra-grainvariation;\\ resolutioncomparison(myreading)\end{array}$ $\begin{array}{r}(computed:removingthe2.92tilt\\ elongationaloneimprovesthedownhill\\ resolutionalmostthreefold)\end{array}$

Orientation Analysis

Name	Formula	Related Formulae	Variables/Constants
Misorientation angle	${\theta }_{mis}=\arccos \left[\frac{\mathrm{tr}\left({\mathbf{g}}_A{\mathbf{g}}_B^{-1}\right)-1}{2}\right]$	${\theta }_{mis}={\min }_{S\in \text{symmetry}}\theta \left(S{\mathbf{g}}_A{\mathbf{g}}_B^{-1}\right)$	$\begin{array}{r}{\theta }_{mis}:misorientationanglebetweengrainsA\\ andB\end{array}$ $\mathbf{g}:orientationmatrices$ $\begin{array}{r}S:crystalsymmetryoperators;trace\\ formula(myreading);Oxford:24\\ symmetricallyequivalentregionsandupto\\ 24axis-anglepairsforcubic(Lauegroup\\ 11)crystals\end{array}$ $\begin{array}{r}(computed:theminimumover24equivalents\\ givesthereportedangle)\end{array}$
Grain-boundary classes	${\theta }_{mis}<{\theta }_c(\text{low-angle}),{\theta }_{mis}>10\text{ to }15^{\circ }(\text{high-angle})$	$\Sigma 3:60^{\circ }⟨111⟩,\Sigma 5:36.9^{\circ }⟨100⟩$	${\theta }_c:user-definedcriticalangle$ $\begin{array}{r}\Sigma :coincidence-site-latticeindex;Oxford:\\ low-angleboundaries3to10degrees,map\\ thresholdsabove2andabove10degrees,\\ criticalangleuser-defined,\Sigma 3twin60\\ degreesabout[111],\Sigma 536.9degreesabout\\ [100](seetheQ/A);Tongetal.:10\\ degreethreshold,noobjectivelyoptimal\\ value;NISTIR7814:5degreecriterion\end{array}$ $\begin{array}{r}(computed:2\arctan (1/3)=36.87degrees,the\Sigma 5\\ angle)\end{array}$
Grain size and step size	$D_{ECD}=\sqrt{\frac{4A_g}{\pi }}$	$s\le \frac{D}{5\text{to}10}$	$\begin{array}{r}{D}_{ECD}:equivalentcirclediameterofagrain\\ ofarea{A}_g\end{array}$ $\begin{array}{r}s:mapstepsize;Tongetal.:stepabout\\ 10percentofgraindiameter(ISO13067),\\ 20percentadequate;ECDdefinition\\ (computed);Oxford:averageWgrainsize\\ 17\mu \mathrm{m},7.5\mathrm{nm}stepinapattern-quality\\ image,3-DEBSDwithtenstoovera\\ hundredslices;DTU:Fe-3percentSi\\ exampleat50\mathrm{nm}scale,NbafterECAPand\\ 15minutesannealing\end{array}$ $\begin{array}{r}(computed:copperlineswith100\mathrm{nm}grains\\ needstepsof10to20\mathrm{nm})\end{array}$

Important Questions

Ques: How does EBSD determine crystal orientation?

From the pattern of Kikuchi bands formed by diffracted backscattered electrons. Electrons scattered inside the crystal are diffracted by each set of lattice planes that satisfies $n\lambda =2d\sin {\theta }_B$, forming pairs of cones that intersect the phosphor screen as bands (DTU). For aluminium (111), with 0.233 $\mathrm{nm}$ spacing and 0.00859 $\mathrm{nm}$ wavelength at 20 $\mathrm{kV}$, the Bragg angle is only 1.06 degrees, so the cones are nearly flat, with semi-angles of 88.94 degrees (computed), matching Oxford’s 88.9. A Hough transform converts bands into peaks; combinations of four bands are matched to the crystal structure; and the angular fit, the mean angular deviation, measures indexing quality (Oxford). Band intersections give zone axes (DTU). Reference: A. B. da Silva Fanta, ‘Diffraction techniques in the Scanning Electron Microscope (SEM)’, DTU Nanolab NNUM 2019 slides, part 1 (fetch-tool summary); Oxford Instruments, ‘EBSD Explained’ brochure (several equations not legible in extraction) (fetch-tool summary).

Ques: Why is the sample tilted to 70 degrees, and what does it cost?

Because a steep tilt sends more electrons forward toward the detector, giving bright patterns. DTU calls 70 degrees the best compromise between intensity and resolution. The cost is spatial resolution down the tilt direction: the beam footprint is stretched by about $1/\sin (90^{\circ }-70^{\circ })=2.92$ (computed), which matches DTU’s statement that y-axis resolution is three times worse and Oxford’s that downhill resolution falls by nearly a factor of three. Lower voltage, lower probe current and a shorter working distance improve resolution (Oxford). Transmission Kikuchi diffraction through a thin foil needs no more than about 10 degrees tilt and removes this elongation (Oxford), so it is preferred for nanocrystalline films. Reference: A. B. da Silva Fanta, ‘Diffraction techniques in the Scanning Electron Microscope (SEM)’, DTU Nanolab NNUM 2019 slides, part 1 (fetch-tool summary); Oxford Instruments, ‘EBSD Explained’ brochure (several equations not legible in extraction) (fetch-tool summary).

Ques: What does the band width tell about the crystal?

The lattice-plane spacing, since the width is about twice the Bragg angle times the screen distance, $w\approx n\lambda l/d$ (computed from Bragg’s law). Oxford notes that wider spacings give thinner bands. With 20 $\mathrm{kV}$ electrons and an assumed 20 $\mathrm{mm}$ screen distance, aluminium (111) bands span 2.11 degrees, about 0.74 $\mathrm{mm}$ on the screen, and copper (111), with 0.209 $\mathrm{nm}$ spacing, spans 2.36 degrees (computed). Band widths and angles together identify the phase, which is how EBSD distinguishes crystal structures. Because the angle depends on wavelength, the accelerating voltage must be known accurately for phase identification; the angles between bands, which give orientation, do not depend on it. Reference: Oxford Instruments, ‘EBSD Explained’ brochure (several equations not legible in extraction) (fetch-tool summary).

Ques: How surface-sensitive is EBSD, and how should samples be prepared?

Very: DTU places the diffraction source 5 to 10 $\mathrm{nm}$ below the surface. Any damaged, amorphous or oxidized layer of that thickness blurs or destroys the pattern, so NISTIR 7814 calls for chemo-mechanical or electrolytic polishing to remove deformation, and requires the top 50 to 100 $\mathrm{nm}$ to diffract. Non-conducting samples need coatings, which DTU puts at 2 to 5 $\mathrm{nm}$; a 5 $\mathrm{nm}$ coating already consumes half or all of the information depth (computed), so coatings must be as thin as possible, or low-vacuum operation used instead. For films with grains below about 100 $\mathrm{nm}$, TKD on thin lamellae is the better choice (Oxford). Reference: A. B. da Silva Fanta, ‘Diffraction techniques in the Scanning Electron Microscope (SEM)’, DTU Nanolab NNUM 2019 slides, part 1 (fetch-tool summary); Oxford Instruments, ‘EBSD Explained’ brochure (several equations not legible in extraction) (fetch-tool summary); A. Creuziger and M. Vaudin, NISTIR 7814, ‘Report on VAMAS Round Robin of ISO 13067’, 2011: surface 50 to 100 nm must have diffracting lattice planes, chemo-mechanical or electrolytic polishing to remove polishing deformation, 5 degree grain-boundary criterion (fetch-tool summary).

Ques: Is a low-angle boundary below 2, 3 or 10 degrees?

The sources set it differently, because the threshold is a user choice. Oxford describes low-angle boundaries as 3 to 10 degrees, while its example maps draw boundaries above 2 degrees and above 10 degrees, and it states that the critical angle is user-defined. Tong et al. use 10 degrees, compare it with 5, and state there is no objectively optimal threshold, while the NISTIR 7814 round robin used 5 degrees. Oxford’s 3 to 10 degree range is the more likely physical definition of low-angle boundaries, with 2 degrees being a mapping threshold near the noise floor (my reading). Special high-angle boundaries are labelled by coincidence-site index: $\Sigma 3$ twins at 60 degrees about [111] and $\Sigma 5$ at 36.9 degrees about [100] (Oxford; computed 36.87). Reference: Oxford Instruments, ‘EBSD Explained’ brochure (several equations not legible in extraction) (fetch-tool summary); V. Tong, H. Zhang, J. Del Gaudio, K. Mingard and A. Gholinia, ‘Towards standardisation of average grain size measurement of additively manufactured microstructures using EBSD’, arXiv:2605.28852: 10 degree threshold compared with 5, no objectively optimal threshold, step size about 10 per cent of grain diameter per ISO 13067 and 20 per cent adequate, equivalent circle diameter (fetch-tool summary); A. Creuziger and M. Vaudin, NISTIR 7814, ‘Report on VAMAS Round Robin of ISO 13067’, 2011: surface 50 to 100 nm must have diffracting lattice planes, chemo-mechanical or electrolytic polishing to remove polishing deformation, 5 degree grain-boundary criterion (fetch-tool summary).

Ques: How is EBSD used for copper interconnects?

To measure grain size, texture and twin fractions, which the Wire Resistance and Size Effects page links to line resistance (my reading for electromigration). Grain size is reported as an equivalent circle diameter, $\sqrt{4A_g/\pi }$, and Tong et al. give steps of about 10 per cent of the grain diameter (ISO 13067), with 20 per cent adequate, so 100 $\mathrm{nm}$ grains need 10 to 20 $\mathrm{nm}$ steps (computed). Narrow lines are often measured by TKD on plan-view lamellae. Oxford lists microelectronics among EBSD applications and shows a 7.5 $\mathrm{nm}$ step map. At 3000 patterns per second (DTU), a million-point map takes 333 $\mathrm{s}$ (computed). Reference: Oxford Instruments, ‘EBSD Explained’ brochure (several equations not legible in extraction) (fetch-tool summary); A. B. da Silva Fanta, ‘Diffraction techniques in the Scanning Electron Microscope (SEM)’, DTU Nanolab NNUM 2019 slides, part 1 (fetch-tool summary); V. Tong, H. Zhang, J. Del Gaudio, K. Mingard and A. Gholinia, ‘Towards standardisation of average grain size measurement of additively manufactured microstructures using EBSD’, arXiv:2605.28852: 10 degree threshold compared with 5, no objectively optimal threshold, step size about 10 per cent of grain diameter per ISO 13067 and 20 per cent adequate, equivalent circle diameter (fetch-tool summary).

Ques: How is misorientation between grains calculated?

From the rotation that takes one grain’s orientation into the other’s, minimized over crystal symmetry. With orientation matrices ${\mathbf{g}}_A$ and ${\mathbf{g}}_B$, the misorientation angle is $\arccos \{[\mathrm{tr}({\mathbf{g}}_A{\mathbf{g}}_B^{-1})-1]/2\}$, and because a cubic crystal has 24 equivalent orientations, the smallest angle over all symmetry operators is reported (trace formula my reading; Oxford gives the 24 equivalents and up to 24 axis-angle pairs). Small misorientations inside grains map local plastic strain, which Oxford shows in transmission maps of deformed steel with grains above 100 $\mathrm{nm}$. Boundaries are classified by angle and by coincidence index, such as $\Sigma 3$ and $\Sigma 5$. Because the threshold greatly affects measured grain size (Tong et al.), it should be stated with every result (my reading). Reference: Oxford Instruments, ‘EBSD Explained’ brochure (several equations not legible in extraction) (fetch-tool summary); V. Tong, H. Zhang, J. Del Gaudio, K. Mingard and A. Gholinia, ‘Towards standardisation of average grain size measurement of additively manufactured microstructures using EBSD’, arXiv:2605.28852: 10 degree threshold compared with 5, no objectively optimal threshold, step size about 10 per cent of grain diameter per ISO 13067 and 20 per cent adequate, equivalent circle diameter (fetch-tool summary).

References

1. A. B. da Silva Fanta, “Diffraction techniques in the Scanning Electron Microscope (SEM),” DTU Nanolab NNUM 2019, slides part 1. Available: https://www.nanolab.dtu.dk/english/-/media/andre_universitetsenheder/nanolab/events/nnum2019/slides/c3_nnum_diffraction-sem-fanta_part1.pdf
2. Oxford Instruments NanoAnalysis, “EBSD Explained,” brochure. Available: https://nano.oxinst.com/assets/uploads/products/nanoanalysis/documents/Brochures/EBSD_Explained.pdf
3. V. Tong, H. Zhang, J. Del Gaudio, K. Mingard and A. Gholinia, “Towards standardisation of average grain size measurement of additively manufactured microstructures using EBSD,” arXiv:2605.28852. Available: https://arxiv.org/abs/2605.28852
4. A. Creuziger and M. Vaudin, Report on VAMAS Round Robin of ISO 13067: Microbeam Analysis - Electron Backscatter Diffraction - Measurement of Average Grain Size, NISTIR 7814, 2011. Available: https://nvlpubs.nist.gov/nistpubs/Legacy/IR/nistir7814.pdf

Related pages in this vault: Scanning Electron Microscope (SEM), Wire Resistance and Size Effects, Transmission Electron Microscope (TEM and STEM), Focused Ion Beam (FIB), EBIC and EBAC, Cathodoluminescence (SEM-CL), Critical-Dimension Small-Angle X-Ray Scattering (CD-SAXS), X-Ray Topography, Reflection High-Energy Electron Diffraction (RHEED), Electroplating (ECD) Cell.