X-Ray Topography

1. Theory

• Definition: X-ray topography images the diffracted intensity from a single crystal across its area, so dislocations, slip lines, stacking faults and strain fields appear as contrast at micrometre resolution (Kawado et al.; Inaba).
• Contrast origin: The strain field around a defect locally changes the diffraction condition, so it diffracts more or less than the perfect crystal (background knowledge, Authier).
• Burgers vector analysis: A dislocation becomes invisible when the diffraction vector is perpendicular to its Burgers vector, $\mathbf{g}\cdot \mathbf{b}=0$, so comparing reflections identifies b (Inaba; Kawado et al.).
• Geometry: Transmission topographs sample the whole thickness, while reflection topographs sample the near-surface layer set by absorption (Inaba; Ohnishi et al.).

2. Working Principle

• Laboratory topography: A Lang camera or a laboratory topograph with Cu or Mo sources and a digital camera scans whole wafers (Kawado et al.; Inaba).
• Synchrotron topography: White or monochromatic synchrotron beams give short exposures, many reflections and sub-micrometre pixels (Kawado et al.; Ohnishi et al.).
• Reflection selection: Several reflections are recorded so each dislocation’s visibility can be tested (Ohnishi et al.; Inaba).
• Topo-tomography: Rotating the sample about the diffraction vector gives 3-D dislocation shapes (Kawado et al.).

3. Applications

• Silicon wafers: Process-induced slip and dislocations on 300 $\mathrm{mm}$ wafers (Kawado et al.).
• Wide-gap substrates: TSDs, TEDs and BPDs in 4H-SiC and threading dislocations in GaN (Inaba; Ohnishi et al.).
• Oxide and sapphire substrates: Defect structures in sapphire, ScAlMgO4 and MgO (Inaba).
• Bonded and strained structures: Strain fields around features and at bonded interfaces (background knowledge, Bowen and Tanner).

Diffraction Contrast

Name	Formula	Related Formulae	Variables/Constants
Bragg condition for a topograph	$2d_{hkl}\sin {\theta }_B=\lambda$	$\lambda \left[\mathring{\mathrm{A}}\right]=\frac{12.398}{E\left[\mathrm{keV}\right]}$	$d_{hkl}:planespacing$ ${\theta }_B:Braggangle$ $\lambda :wavelength$ $\begin{array}{r}E:photonenergy;XRDpage:Bragglaw;\\ Kawadoetal.:crystaltilted4.36{}^{\circ }to\\ formtheSi004Lauespotwith60\mathrm{keV}\\ white-beamX-rays;Ohnishietal.:1.40\mathring{\mathrm{A}}\\ inreflectionand1.24\mathring{\mathrm{A}}intransmission;\\ Inaba:dualCuandMosource\end{array}$ $\begin{array}{r}(computed:Si004at60\mathrm{keV}gives{\theta }_B=4.36\\ {}^{\circ },matchingKawado;1.40and1.24\mathring{\mathrm{A}}are\\ 8.86and10.0\mathrm{keV})\end{array}$
Invisibility criterion	$\mathbf{g}\cdot \mathbf{b}=0\Rightarrow \text{contrast vanishes}$	$\mathbf{g}\cdot \left(\mathbf{b}\times \mathbf{u}\right)=0(\text{edge residual contrast also vanishes})$	$\mathbf{g}:diffractionvector$ $\mathbf{b}:Burgersvector$ $\begin{array}{r}\mathbf{u}:linedirection;Inaba:contrast\\ disappearswhengandbareperpendicular,\\ g\cdot b=0;Kawadoetal.:nocontrastatg\cdot \\ b=0(citingAuthier);Ohnishietal.:\\ faintvisibilitywheng\cdot (b\times l)\ne 0even\\ atg\cdot b=0;secondcondition(background\\ knowledge,Authier)\end{array}$ $\begin{array}{r}(myreading:twonon-parallelinvisibility\\ reflectionsfixthedirectionofbas\\ {\mathbf{g}}_1\times {\mathbf{g}}_2)\end{array}$
Burgers vectors in Si, SiC and GaN	${\mathbf{b}}_{Si}=\frac{a}{2}⟨110⟩\text{on}\left\{111\right\}$	${\mathbf{b}}_a=\frac{1}{3}⟨11\bar{2}0⟩,{\mathbf{b}}_c=⟨0001⟩,{\mathbf{b}}_{a+c}=\frac{1}{3}⟨11\bar{2}3⟩$	$\begin{array}{r}a,c:latticeconstants;Kawadoetal.:\\ a/2<110>111slipsystemand\\ perfect-dislocationBurgersvectorinSi;\\ Inaba:TSD,TEDandBPDin4H-SiC,BPDsin\\ (0001);Ohnishietal.:edgeE1,E2(bof\\ length1a)andE3(2a),mixedM1andM2,\\ screwpair\pm 1cabout5\mu \mathrm{m}apart,M2\\ written[11-23]/3(seetheQ/A);hexagonal\\ notation(backgroundknowledge,Bowenand\\ Tanner)\end{array}$ $\begin{array}{r}(computed:b_{Si}length3.84\mathring{\mathrm{A}};4H-SiC{b}_a\\ length3.07\mathring{\mathrm{A}}and{b}_{c}length10.05\mathring{\mathrm{A}})\end{array}$

Depth, Resolution and Density

Name	Formula	Related Formulae	Variables/Constants
Penetration and absorption	$T=e^{-\mu t}$	$z_p=\frac{\sin {\alpha }_{in}\sin {\alpha }_{out}}{\mu \left(\sin {\alpha }_{in}+\sin {\alpha }_{out}\right)}(\text{reflection})$	$T:transmission$ $\mu :linearabsorptioncoefficient$ $t:thickness$ $z_p:penetrationdepthinreflection$ $\begin{array}{r}{\alpha }_{in},{\alpha }_{out}:incidenceandexitangles;\\ Ohnishietal.:\mu t=5.4cutting\\ transmissiontoabout1percentofthat\\ at\mu t=1,energychosenneartheGaK\\ edgeat1.20\mathring{\mathrm{A}},grazingincidenceabout5\\ {}^{\circ }inreflection;Inaba:penetrationabout\\ 12\mu \mathrm{m}(Cu4-2-20)and180\mu \mathrm{m}(Mo10-5-50)\\ insapphire;geometryrelation(background\\ knowledge,BowenandTanner)\end{array}$ $\begin{array}{r}(computed:e^{-5.4}/e^{-1}=1.2percent;e^{-5.4}alone\\ is0.45percent)\end{array}$
Geometric resolution	$\delta =\frac{sd}{D}$	$\text{pixel size}=\frac{\text{field}}{N_{pixels}}$	$\delta :imageblur$ $s:sourcesize$ $d:sample-to-detectordistance$ $\begin{array}{r}D:source-to-sampledistance;geometric\\ relation(backgroundknowledge,Bowenand\\ Tanner);Kawadoetal.:220\mathrm{mm}\\ sample-to-filmdistance,cooledCCDof\\ 1024by1024pixelsover10by10\mathrm{mm},Agfa\\ D2film;Ohnishietal.:0.65\mu \mathrm{m}perpixel\\ intransmission,filmresolution\\ insufficientinreflection;Inaba:0.0015\\ {}^{\circ }incident-opticresolution\end{array}$ $\begin{array}{r}(assumed:s=0.1\mathrm{mm},D=80\mathrm{m}\\ (synchrotron)or1\mathrm{m}(laboratory)withd=\\ 10\mathrm{mm};computed:0.28\mu \mathrm{m}atd=220\mathrm{mm}for\\ thesynchrotron,1\mu \mathrm{m}inthelaboratory;\\ Kawad{o}^{\prime }sCCDpixelis9.8\mu \mathrm{m})\end{array}$
Resolvable dislocation density	$\bar{L}\approx \frac{1}{\sqrt{\rho }}$	${\rho }_{max}\sim {\left(\frac{1}{w_{image}}\right)}^2$	$\begin{array}{r}\bar{L}:meanspacingbetweendislocation\\ images\end{array}$ $\rho :dislocationdensity$ $\begin{array}{r}{w}_{image}:dislocationimagewidth;spacing\\ estimate(backgroundknowledge,Bowenand\\ Tanner);Ohnishietal.:GaNthreading\\ densityoforder1\times 10^3\mathrm{c}{\mathrm{m}}^{-2},about7\times 10^3\mathrm{c}{\mathrm{m}}^{-2}\\ intheimagedarea\end{array}$ $\begin{array}{r}(computed:7\times 10^3\mathrm{c}{\mathrm{m}}^{-2}givesabout120\mu \mathrm{m}\\ spacing;imagesabout10\mu \mathrm{m}wideoverlap\\ aboveroughly1\times 10^6\mathrm{c}{\mathrm{m}}^{-2})\end{array}$

Methods and Comparisons

Name	Formula	Related Formulae	Variables/Constants
Lang and white-beam methods	$\text{Lang: slit beam, sample and film translated together}$	$\text{white beam: many Laue spots }(hkl)\text{ at once}$	$\begin{array}{r}Kawadoetal.:Langtransmission\\ topographywithatopographiccamera,\\ camerasaccepting300\mathrm{mm}Siwafersfor\\ process-inducedslip,white-beam\\ topographyacquiringseveralLauespots\\ simultaneously,7by4\mathrm{mm}beam,\\ topo-tomographywith2.4{}^{\circ }CCDand45{}^{\circ }\\ filmsteps;Inaba:laboratoryXRTmicron\\ withshortexposuressuchas10s\end{array}$ $\begin{array}{r}(computed:a45{}^{\circ }stepseriesneeds8\\ filmsperturnagainst150CCDframesat\\ 2.4{}^{\circ })\end{array}$
Reflection against transmission geometry	$\text{reflection: near-surface defects, depth}\sim z_p$	$\text{transmission: whole thickness, all }\mathbf{b}\parallel \text{surface in 400}$	$\begin{array}{r}Inaba:Si(001)220and400in\\ transmission,400recordingall\\ dislocationswithBurgersvectorsparallel\\ tothesurface;sapphireandSAM\\ reflections;Ohnishietal.:six\\ equivalent11-24reflectionsinreflection\\ mode,fiveorsixtransmissionreflections\\ atSPring-8withsix-beamincidenceat27\\ {}^{\circ },combiningbothtodeterminethe\\ Burgers-vectormagnitude\end{array}$ $\begin{array}{r}(myreading:aandccomponentsneed\\ reflectionswithgbothinandoutofthe\\ basalplane,whichonlythecombined\\ geometryprovides)\end{array}$
Comparison with etch-pit counting	$\text{XRT: non-destructive, boules and as-sliced wafers}$	$\text{etching: destructive, toxic etchants}$	$\begin{array}{r}Inaba:comparisontable,nondestructive\\ XRTforboulesorwafers,1.5by1.5by\\ 0.3\mathrm{mm}SiCanalysisvolume,2inch\\ sapphireandSAMwafers;Ohnishietal.:\\ completeBurgers-vectordetermination\\ impossiblebyetchpitsalone;etch-pit\\ practice(backgroundknowledge,Bowenand\\ Tanner)\end{array}$ $\begin{array}{r}(computed:Inab{a}^{\prime }sSiCvolumeis0.675\\ cubicmillimetres)\end{array}$

Important Questions

Ques: How does X-ray topography make a dislocation visible?

Through local changes in diffraction around its strain field. A crystal is set at the Bragg angle for one reflection, and regions whose lattice is tilted or strained by a defect diffract differently, giving dark or bright lines (background knowledge, Authier). Kawado et al. tilted a silicon crystal by 4.36 ${}^{\circ }$ so that 60 $\mathrm{keV}$ white-beam X-rays formed the 004 Laue spot; the Bragg law with $d_{004}$ = 1.358 $\mathring{\mathrm{A}}$ gives exactly 4.36 ${}^{\circ }$ at 0.2066 $\mathring{\mathrm{A}}$ (computed; XRD page). Inaba’s laboratory instrument switches between Cu and Mo sources and records images in about 10 s with a digital camera. Topography sees individual defects at low density, complementing HRXRD, which averages them into peak broadening. Reference: S. Kawado, T. Taishi, S. Iida, Y. Suzuki, Y. Chikaura and K. Kajiwara, ‘Determination of the three-dimensional structure of dislocations in silicon by synchrotron white X-ray topography combined with a topo-tomographic technique’, J. Synchrotron Rad. 11, 304 (2004) (fetch-tool summary); D. K. Bowen and B. K. Tanner, High Resolution X-ray Diffractometry and Topography, Taylor and Francis 1998, and A. Authier, Dynamical Theory of X-Ray Diffraction, Oxford University Press 2001 (background knowledge; not opened); X-Ray Diffractometer (XRD and HRXRD) page of this vault; K. Inaba, ‘Defect structure analysis in single crystal substrates using XRTmicron’, Rigaku Journal 36(2) (2020) (fetch-tool summary).

Ques: How is the Burgers vector of a dislocation determined?

By finding reflections where it vanishes. Inaba and Kawado et al. use the invisibility criterion $\mathbf{g}\cdot \mathbf{b}=0$: contrast disappears when the diffraction vector is perpendicular to the Burgers vector. Two non-parallel invisibility reflections fix the direction of b along ${\mathbf{g}}_1\times {\mathbf{g}}_2$ (my reading). Edge dislocations can keep weak residual contrast unless $\mathbf{g}\cdot (\mathbf{b}\times \mathbf{u})$ also vanishes (background knowledge, Authier), and Ohnishi et al. note exactly this faint visibility in GaN. In silicon the perfect dislocation is $\frac{a}{2}⟨110⟩$ on {111} (Kawado et al.), 3.84 $\mathring{\mathrm{A}}$ long (computed). Inaba’s Si (001) 400 transmission reflection shows all dislocations with Burgers vectors parallel to the surface, since g along [001] is perpendicular to them. Reference: K. Inaba, ‘Defect structure analysis in single crystal substrates using XRTmicron’, Rigaku Journal 36(2) (2020) (fetch-tool summary); S. Kawado, T. Taishi, S. Iida, Y. Suzuki, Y. Chikaura and K. Kajiwara, ‘Determination of the three-dimensional structure of dislocations in silicon by synchrotron white X-ray topography combined with a topo-tomographic technique’, J. Synchrotron Rad. 11, 304 (2004) (fetch-tool summary); D. K. Bowen and B. K. Tanner, High Resolution X-ray Diffractometry and Topography, Taylor and Francis 1998, and A. Authier, Dynamical Theory of X-Ray Diffraction, Oxford University Press 2001 (background knowledge; not opened); K. Ohnishi, K. Iso, H. Ikeda, Y. Tsusaka and Y. Yao, ‘Comprehensive determination of Burgers vectors of threading dislocations in GaN substrates by combining reflection and transmission synchrotron-radiation x-ray topography’, J. Appl. Phys. 139, 195701 (2026), arXiv:2604.04392 (fetch-tool summary).

Ques: Is Ohnishi’s M2 Burgers vector [11-23]/3 a typo?

Probably not. The summary of Ohnishi et al. writes the mixed M2 dislocation as [11-23]/3 and flags it as a possible typo for an a-component of [11-20]/3 plus a c-component of [000-1]. In four-index notation $\frac{1}{3}[11\bar{2}3]=\frac{1}{3}[11\bar{2}0]+[0001]$, the standard a+c Burgers vector (background knowledge, Bowen and Tanner), so the printed form is a valid compact notation; the only possible inconsistency is the sign of the c component, [0001] against [000-1] (my reading). The written form is therefore more likely correct up to that sign. For 4H-SiC the analogous vectors have lengths of 3.07 $\mathring{\mathrm{A}}$ for a and 10.05 $\mathring{\mathrm{A}}$ for c (computed), with TSDs pure c, TEDs and BPDs pure a, as Inaba describes. Reference: K. Ohnishi, K. Iso, H. Ikeda, Y. Tsusaka and Y. Yao, ‘Comprehensive determination of Burgers vectors of threading dislocations in GaN substrates by combining reflection and transmission synchrotron-radiation x-ray topography’, J. Appl. Phys. 139, 195701 (2026), arXiv:2604.04392 (fetch-tool summary); D. K. Bowen and B. K. Tanner, High Resolution X-ray Diffractometry and Topography, Taylor and Francis 1998, and A. Authier, Dynamical Theory of X-Ray Diffraction, Oxford University Press 2001 (background knowledge; not opened); K. Inaba, ‘Defect structure analysis in single crystal substrates using XRTmicron’, Rigaku Journal 36(2) (2020) (fetch-tool summary).

Ques: Why combine reflection and transmission topography?

Because they probe different depths and reflections. Ohnishi et al. used six equivalent {11-24} reflections in reflection mode at 1.40 $\mathring{\mathrm{A}}$ (8.86 $\mathrm{keV}$) with grazing incidence of about 5 ${}^{\circ }$, and transmission reflections at 1.24 $\mathring{\mathrm{A}}$ (10.0 $\mathrm{keV}$) at SPring-8 with 0.65 $\mu \mathrm{m}$ pixels (computed energies). Transmission through thick GaN is limited by absorption: at $\mu t$ = 5.4 the beam is 1.2 per cent of its value at $\mu t$ = 1 (computed), which is why they worked near the Ga K edge and used six-beam diffraction. Reflection sees only a near-surface layer, about 12 $\mu \mathrm{m}$ for Cu and 180 $\mu \mathrm{m}$ for Mo in Inaba’s sapphire examples. Determining both a and c components requires g vectors in and out of the basal plane, which only the combined set provides (my reading). Reference: K. Ohnishi, K. Iso, H. Ikeda, Y. Tsusaka and Y. Yao, ‘Comprehensive determination of Burgers vectors of threading dislocations in GaN substrates by combining reflection and transmission synchrotron-radiation x-ray topography’, J. Appl. Phys. 139, 195701 (2026), arXiv:2604.04392 (fetch-tool summary); K. Inaba, ‘Defect structure analysis in single crystal substrates using XRTmicron’, Rigaku Journal 36(2) (2020) (fetch-tool summary).

Ques: What limits topography resolution and dislocation density?

Source size, geometry, detector pixels and image width. The geometric blur is $\delta =sd/D$ (background knowledge, Bowen and Tanner): an assumed 0.1 $\mathrm{mm}$ source 80 $\mathrm{m}$ away with Kawado’s 220 $\mathrm{mm}$ sample-to-film distance gives 0.28 $\mu \mathrm{m}$, while a laboratory source 1 $\mathrm{m}$ away with 10 $\mathrm{mm}$ gives 1 $\mu \mathrm{m}$ (computed). Detectors often dominate: Kawado’s CCD of 1024 pixels over 10 $\mathrm{mm}$ has 9.8 $\mu \mathrm{m}$ pixels (computed), against Ohnishi’s 0.65 $\mu \mathrm{m}$. Dislocation images are several micrometres wide, so they overlap once spacing $1/\sqrt{\rho }$ approaches the image width: Ohnishi’s $7\times 10^3$ $\mathrm{c}{\mathrm{m}}^{-2}$ gives 120 $\mu \mathrm{m}$ spacing, while roughly $1\times 10^6$ $\mathrm{c}{\mathrm{m}}^{-2}$ is the practical limit for 10 $\mu \mathrm{m}$ images (computed). Highly dislocated epitaxy therefore needs HRXRD or TEM instead. Reference: D. K. Bowen and B. K. Tanner, High Resolution X-ray Diffractometry and Topography, Taylor and Francis 1998, and A. Authier, Dynamical Theory of X-Ray Diffraction, Oxford University Press 2001 (background knowledge; not opened); S. Kawado, T. Taishi, S. Iida, Y. Suzuki, Y. Chikaura and K. Kajiwara, ‘Determination of the three-dimensional structure of dislocations in silicon by synchrotron white X-ray topography combined with a topo-tomographic technique’, J. Synchrotron Rad. 11, 304 (2004) (fetch-tool summary); K. Ohnishi, K. Iso, H. Ikeda, Y. Tsusaka and Y. Yao, ‘Comprehensive determination of Burgers vectors of threading dislocations in GaN substrates by combining reflection and transmission synchrotron-radiation x-ray topography’, J. Appl. Phys. 139, 195701 (2026), arXiv:2604.04392 (fetch-tool summary).

References

1. S. Kawado, T. Taishi, S. Iida, Y. Suzuki, Y. Chikaura and K. Kajiwara, “Determination of the three-dimensional structure of dislocations in silicon by synchrotron white X-ray topography combined with a topo-tomographic technique,” Journal of Synchrotron Radiation, vol. 11, pp. 304-308, 2004. Available: https://journals.iucr.org/s/issues/2004/04/00/ia5002/ia5002.pdf
2. D. K. Bowen and B. K. Tanner, High Resolution X-ray Diffractometry and Topography, Taylor and Francis, 1998; A. Authier, Dynamical Theory of X-Ray Diffraction, Oxford University Press, 2001 (cited from background knowledge).
3. X-Ray Diffractometer (XRD and HRXRD) page of this vault.
4. K. Inaba, “Defect structure analysis in single crystal substrates using XRTmicron,” Rigaku Journal, vol. 36, no. 2, 2020. Available: https://resources.rigaku.com/hubfs/2024%20Rigaku%20Global%20Site/Resource%20Hub/Knowledge%20Library/Rigaku%20Journals/Volume%2036(2)%20-%20Summer%202020/Rigaku%20Journal%2036-2_11-18.pdf
5. K. Ohnishi, K. Iso, H. Ikeda, Y. Tsusaka and Y. Yao, “Comprehensive determination of Burgers vectors of threading dislocations in GaN substrates by combining reflection and transmission synchrotron-radiation x-ray topography,” Journal of Applied Physics, vol. 139, 195701, 2026. Available: https://arxiv.org/pdf/2604.04392

Related pages in this vault: X-Ray Diffractometer (XRD and HRXRD), Transmission Electron Microscope (TEM and STEM), Thermal Stress, Critical-Dimension Small-Angle X-Ray Scattering (CD-SAXS), Cathodoluminescence (SEM-CL), X-Ray Fluorescence Spectrometer (XRF and TXRF), X-Ray Reflectometer (XRR), Rutherford Backscattering Spectrometer (RBS), Electron Backscatter Diffraction (EBSD).