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[T. Hamachi](https://orcid.org/0000-0002-0631-1729), [T. Tohei](https://orcid.org/0000-0002-4113-2566), [Y. Hayashi](https://orcid.org/0000-0001-5672-1497), [S. Usami](https://orcid.org/0000-0002-9710-1718), M. Imanishi, Y. Mori, K. Sumitani, [Y. Imai](https://orcid.org/0000-0003-4686-2629), [S. Kimura](https://orcid.org/0000-0003-1064-7572), [A. Sakai](https://orcid.org/0000-0002-0654-504X)

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[Analysis of local strain fields around individual threading dislocations in GaN substrates by nanobeam x-ray diffraction](https://mdr.nims.go.jp/datasets/ba6eeca7-4c74-462d-a14a-ccaba3d9b820)

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Analysis of local strain fields around individual threading dislocations in GaN substrates by nanobeam x-ray diffractionViewOnlineExportCitationRESEARCH ARTICLE |  JUNE 11 2024Analysis of local strain fields around individual threadingdislocations in GaN substrates by nanobeam x-raydiffraction T. Hamachi  ; T. Tohei   ; Y. Hayashi  ; S. Usami  ; M. Imanishi; Y. Mori; K. Sumitani; Y. Imai  ;S. Kimura  ; A. Sakai  J. Appl. Phys. 135, 225702 (2024)https://doi.org/10.1063/5.0199961 CHORUS 12 June 2024 07:14:10https://pubs.aip.org/aip/jap/article/135/22/225702/3297804/Analysis-of-local-strain-fields-around-individualhttps://pubs.aip.org/aip/jap/article/135/22/225702/3297804/Analysis-of-local-strain-fields-around-individual?pdfCoverIconEvent=citejavascript:;https://orcid.org/0000-0002-0631-1729javascript:;https://orcid.org/0000-0002-4113-2566javascript:;https://orcid.org/0000-0001-5672-1497javascript:;https://orcid.org/0000-0002-9710-1718javascript:;javascript:;javascript:;javascript:;https://orcid.org/0000-0003-4686-2629javascript:;https://orcid.org/0000-0003-1064-7572javascript:;https://orcid.org/0000-0002-0654-504Xhttps://crossmark.crossref.org/dialog/?doi=10.1063/5.0199961&domain=pdf&date_stamp=2024-06-11https://doi.org/10.1063/5.0199961https://pubs.aip.org/aip/jap/article-pdf/doi/10.1063/5.0199961/19990879/225702_1_5.0199961.am.pdfhttps://servedbyadbutler.com/redirect.spark?MID=176720&plid=2408990&setID=592934&channelID=0&CID=883922&banID=521904493&PID=0&textadID=0&tc=1&scheduleID=2327696&adSize=1640x440&data_keys=%7B%22%22%3A%22%22%7D&matches=%5B%22inurl%3A%5C%2Fjap%22%5D&mt=1718176450801081&spr=1&referrer=http%3A%2F%2Fpubs.aip.org%2Faip%2Fjap%2Farticle-pdf%2Fdoi%2F10.1063%2F5.0199961%2F19990878%2F225702_1_5.0199961.pdf&hc=62ea49c4c1766b7c12be45018439f09f3ff4b440&location=Analysis of local strain fields around individualthreading dislocations in GaN substrates bynanobeam x-ray diffractionCite as: J. Appl. Phys. 135, 225702 (2024); doi: 10.1063/5.0199961View Online Export Citation CrossMarkSubmitted: 24 January 2024 · Accepted: 23 May 2024 ·Published Online: 11 June 2024T. Hamachi,1,a) T. Tohei,1,b) Y. Hayashi,1,c) S. Usami,2 M. Imanishi,2 Y. Mori,2 K. Sumitani,3 Y. Imai,3S. Kimura,3 and A. Sakai1,b)AFFILIATIONS1Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama-cho, Toyonaka, Osaka 560-8531, Japan2Graduate School of Engineering, Osaka University, 2-1 Yamadaoka, Suita, Osaka 565-0871, Japan3Diffraction and Scattering Division, Japan Synchrotron Radiation Research Institute (JASRI), 1-1-1 Koto, Sayo, Hyogo 679-5198,Japana)Electronic mail: u679849k@alumni.osaka-u.ac.jpb)Author to whom correspondence should be addressed: tohei@ee.es.osaka-u.ac.jp and sakai@ee.es.osaka-u.ac.jpc)Present address: National Institute for Materials Science, 1-2-1 Sengen, Tsukuba, Ibaraki, 305-0047, JapanABSTRACTPosition-dependent three-dimensional reciprocal space mapping (RSM) by nanobeam x-ray diffraction (nanoXRD) was performed to revealthe strain fields produced around individual threading dislocations (TDs) in GaN substrates. The distribution and Burgers vector of TDs forthe nanoXRD measurements were confirmed by prerequisite analysis of multi-photon excited photoluminescence and etch pit methods.The present results demonstrated that the nanoXRD can identify change in the lattice plane structure for all types of TDs, i.e., edge-, screw-,and mixed TDs with the Burgers vector of b = 1a, 1c and 1m + 1c. Strain tensor components related to edge and/or screw components ofthe TDs analyzed from the three-dimensional RSM data showed a nearly symmetrical strained region centered on the TD positions, whichwere in good agreements with simulation results based on the isotropic elastic theory using a particular Burgers vector. The present methodis beneficial in that it allows non-destructive analysis of screw components of TDs that tend to contribute to leakage characteristics and arenot routinely accessible by conventional structural analysis. These results indicate that nanoXRD could be a powerful way to reveal three-dimensional strain fields associated with arbitrary types of TDs in semiconductor materials, such as GaN and SiC.© 2024 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution-NonCommercial 4.0International (CC BY-NC) license (https://creativecommons.org/licenses/by-nc/4.0/). https://doi.org/10.1063/5.0199961I. INTRODUCTIONWide-bandgap semiconductor materials based power devices,such as silicon carbide (SiC) and gallium nitride (GaN), are under-going widespread implementation in industrial applications, suchas automotive cars.1–4 Commercially available wafers of these typesof materials still contain threading dislocations (TDs), andtheir densities typically range from 103 to 104 cm−2 for SiC and103–106 cm−2 for GaN. Such a high density of the dislocations,especially for GaN, extended defects, and dislocation reactions(coalescence and branch of dislocations) are quite common, whichare closely associated with residual strains and plasticity ofas-grown crystals, leading to wafer-scale problems, such as bowingand clacks. Residual TDs in substrates can be detrimental to deviceproperties: reduction of on-resistance,5 reverse leakage currentpath,6–8 and scattering carriers resulting in a reduction of themobility.9,10 Electrical and electronic behaviors of the dislocationscould vary as a result of segregation and diffusion of impuritiesalong the dislocations,11,12 and elastic strain fields generated by dis-locations themselves would have an essential role of interactionwith the impurities.13,14 Understanding the nature of dislocations,including the strain fields, allows us to better develop novelmethods for producing high-performance power devices.Journal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. Appl. Phys. 135, 225702 (2024); doi: 10.1063/5.0199961 135, 225702-1© Author(s) 2024 12 June 2024 07:14:10https://doi.org/10.1063/5.0199961https://doi.org/10.1063/5.0199961https://pubs.aip.org/action/showCitFormats?type=show&doi=10.1063/5.0199961http://crossmark.crossref.org/dialog/?doi=10.1063/5.0199961&domain=pdf&date_stamp=2024-06-11https://orcid.org/0000-0002-0631-1729https://orcid.org/0000-0002-4113-2566https://orcid.org/0000-0001-5672-1497https://orcid.org/0000-0002-9710-1718https://orcid.org/0000-0003-4686-2629https://orcid.org/0000-0003-1064-7572https://orcid.org/0000-0002-0654-504Xmailto:u679849k@alumni.osaka-u.ac.jpmailto:tohei@ee.es.osaka-u.ac.jpmailto:sakai@ee.es.osaka-u.ac.jphttps://creativecommons.org/licenses/by-nc/4.0/https://creativecommons.org/licenses/by-nc/4.0/https://doi.org/10.1063/5.0199961https://pubs.aip.org/aip/japTypical TDs propagating to the c-direction in SiC and GaNhave Burgers vector of b = 1/3h11�20i = 1a, b = 〈0001〉 = 1c, andb = 1/3h11�23i = 1a + 1c, which are generally classified into three cate-gories: edge, screw, and mixed-types, while some unique mixed TDshaving relatively larger Burgers vectors, such as b = na + 1c (n = 2and 3) and b = h1�101i = 1m + 1c, were also reported.7,15–18 There areseveral techniques to measure the residual strain. Geometric phaseanalysis (GPA) based on a high-resolution transmission electronmicroscope (TEM) enables to analyze strain fields around a core ofdislocations in an atomic scale,19 but the strain around screw disloca-tions not generating in-plane displacement cannot be basically ana-lyzed. Large-angle convergent-beam electron diffraction (LACBED)using TEM has the advantage of determination of an absolute valueof the Burgers vector of dislocations,6–8,15–18 though quantitativeanalysis of strain fields around the dislocation is difficult. In theTEM-based analysis, there are also concerns about damages due tothe TEM sample processing and strain relaxation due to thinningthe sample. Raman spectroscopy and x-ray topography are non-destructive techniques to evaluate the strain and characterize types ofTDs. Raman shift mapping can reveal the in-plane strain of the edgecomponent of pure edge and mixed TDs in GaN.20,21 However, thistechnique does not detect the strain around a screw component ofTDs due to the same reason as the GPA mentioned above. In syn-chrotron radiation x-ray topography, intensity of a diffracted x rayfrom a specific Bragg plane is recorded and all types of TDs can besensitively characterized as a contrast variation in topographicimages depending on the Burgers vector,22,23 but the strain fieldsaround the TDs cannot be quantified.Nanobeam x-ray diffraction (nanoXRD) measurement usingthe incident x ray focused to several-hundred nm in diameter withhigh-brilliance synchrotron sources is an attractive method to evalu-ate microlattice plane structure in crystals due to its penetrativenature for various semiconductor materials.24–27 Position-dependentthree-dimensional reciprocal space (ω−2θ-w) mapping, which candetect local lattice tilt and twist and lattice constant in the crystals,has a potential to reveal microstrain fields in the vicinity of individ-ual TDs. Previously, we investigated three-dimensional variations inthe lattice microstructure along a growth direction in nitride semi-conductor crystals by nanoXRD.24,25 The results clarify modulationsin the lattice plane tilt, twist, and spacing originated from a shift incrystal growth modes, specific void configurations formed at aninterface of the epitaxial layers and substrates, and propagation ofdislocations.24,25 In this study, the nanoXRD technique is used toreveal the microstrain induced by individual TDs in GaN bulk crys-tals with a high spatial resolution. By comparing in-plane ω−2θ−wmapping by nanoXRD with the distribution and the type (i.e.,Burgers vector) of TDs observed by a combination of multi-photonexcited photoluminescence (MPPL) and etch pit methods, modula-tion in the lattice plane structure around all types of TDs can beidentified. The strain fields of the edge and screw components of theTDs analyzed from the ω−2θ−w data are quantitatively discussedbased on simulation results by isotropic elastic theory of dislocations.II. EXPERIMENTALThe Si-doped thick GaN layer over 200 μm was homoepitax-ially grown by hydride vapor epitaxy on a Na-flux-grownsubstrate.28 The surface of the HVPE-GaN was smoothed by chem-ical–mechanical polishing (CMP). Wet chemical etching with aeutectic mixture of sodium hydroxide and potassium hydroxide at450 °C for 20 min was conducted to make TD-related etch pits ona surface of the sample. As reported in our previous study on thesame type HVPE-GaN crystals, sizes of the etch pits werecorrelated with the Burgers vector of the TDs: TDs havingb = 1/3h11�20i, b = 1/3h11�23i, b = 〈0001〉, and b = h1�101i formedextra small (XS), small, medium (M), and large (L) pits, respec-tively.16 Figures 1(a) and 1(b) present scanning electron microscopyimages of two different surface areas referred to as area1 and area2,respectively (taken by an FEI Versa3D dual beam system at anacceleration voltage of 5 kV). The area1 contains one L- and severalXS-sized etch pits and the area2 contains one M- and severalXS-sized etch pits. Thus, we expected that a mixed TD withb = h1�101i and edge TDs with b = 1/3h11�20i existed in the area1while a screw TD with b = 〈0001〉 and edge TDs with b = 1/3h11�20iin the area2. Then, the surface was polished by several micrometersvia CMP again to remove the etch pit. To locate dislocation positionsfor nanoXRD experiments, we performed MPPL observations(Nikon A1MP instrument with an excitation laser wavelength of800 nm and a PL detection range of 352–388 nm). As indicated inFigs. 1(c) and 1(d) showing the MPPL images, it was confirmed thatthe locations of the threading dislocations coincided with those ofthe etch pits. Figures 1(e) and 1(f) show MPPL images at the depthof 9.3 μm in the area1 and area2, and the dislocations’ spots at thesurface are indicated by the dots. From the positional shifts of thedislocation’s spots between the surface and the 9.3 μm depth, we canestimate inclination angles of the dislocations. The inclination anglesof the edge, screw, and mixed dislocations in these regions were esti-mated to be 3.9°, 3.8°, and 2.4° at most, respectively, indicating prop-agation almost parallel to the [0001] axis within a penetration depthof the x-ray beam of 9.1 μm.To analyze the strain fields in the vicinity of each TD, theposition-dependent reciprocal space mapping (RSM) within thearea1 and area2 by a nanoXRD technique was performed at thehard x-ray undulator beamline (BL13XU) in SPring-8. Theprimary x-ray beam having a photon energy of 8 keV monochrom-atized by a Si (111) double-crystal monochromator was focused bya zone plate. The beam size was 480 (vertical) × 770 nm2 (horizon-tal) for the area1 and 370 (vertical) × 820 nm2 (horizontal) for thearea2 measurements. The angular divergence of the incident beamwas estimated to be 0.029°. The diffracted x rays were detected by atwo-dimensional hybrid pixel array detector (HyPix-3000, RigakuCorp.) with a pixel size of 100 μm. The distance between the detec-tor and the GaN sample was 1000.9041 mm. The resolution of 2θand w was calculated to be 5.72 × 10−3°. The detailed optical setupof a nanoXRD measurement has been described in Refs. 27 and 29.The diffraction geometry for the nanoXRD measurement in thisstudy is schematically presented in Fig. 2(a). The x ray was irradi-ated in the direction parallel to [11�20] of the GaN crystal.Symmetric 0004 and asymmetric 11�24 planes were selected asBragg reflection planes. The footprint sizes of the incident x-raybeam in the horizontal direction for the 0004 and 11�24 reflectionwere 1280 and 770 nm for the area1 measurement and 1366 and820 nm for the area2 measurement, respectively. Penetration depthJournal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. Appl. Phys. 135, 225702 (2024); doi: 10.1063/5.0199961 135, 225702-2© Author(s) 2024 12 June 2024 07:14:10https://pubs.aip.org/aip/japFIG. 1. Scanning electron microscopic images of (a) area1 and (b) area2 after the chemical etching. (c)–(f ) MPPL images of (c) and (e) area1, and (d) and (f ) area2after removing the etch pits by CMP. (c) and (d) are observed at the surface, while (e) and (f ) are at the depth of 9.3 μm from the surface. In (a)–(d), the L-pit, M-pit, andXS-pits are indicated by circles with red dashed, yellow broken, and green solid lines. In (e) and (f ), dark spot positions (corresponding to the nonradiative spot due to theTD) at the surface seen in (c) and (d) are marked by dots using the same color as in (a)–(d).Journal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. Appl. Phys. 135, 225702 (2024); doi: 10.1063/5.0199961 135, 225702-3© Author(s) 2024 12 June 2024 07:14:10https://pubs.aip.org/aip/japof the x-ray beam in the GaN crystal is estimated to be approxi-mately 9.1 and 5.0 μm in the case of the 0004 and 11�24 reflection,respectively. The incident beam was scanned on the GaN surface atthe 800 and 500 nm steps along [11�20] and [�1100] directions over20.8 × 19.5 and 12.8 × 11.5 μm2 for areas 1 and 2, respectively, sothat the scanned area1 included one L- (mixed dislocation) andseveral XS-pits (edge dislocations) and the area2 included one M-(screw dislocation) and one XS-pits (edge dislocation), as shown inFigs. 2(b) and 2(c). The lattice plane tilt, spacing, and twist in theGaN crystal are associated with ω, 2θ, and w, respectively, asdescribed in Fig. 2(a). Diffraction intensity spots acquired by ω−2θ−w mapping from respective measurement positions generally havea finite broadening in each angular direction due to imperfect crys-tallinity of GaN. By the same data processing as reported in our pre-vious study,25 three profiles of intensity vs ω, 2θ, and w wereobtained from the raw ω−2θ−w data, and lattice plane tilt (Ω), scat-tering angle (2Θ) corresponding to lattice spacing, lattice plane twist(Φ), and each fluctuation (Δω, Δ2θ, and Δw) were calculated fromthe center and the full width at half maximum of Gaussian fitting ofthe respective profiles. By extracting their values from ω−2θ−w datafor all measurement points in the area1 and area2, in-plane distribu-tions of lattice plane microstructure could be evaluated.III. RESULTS AND DISCUSSIONFigures 3(a)–3(f ) present two-dimensional maps of Ω, 2Θ, Φ,Δω, Δ2θ, and Δw in the area1, respectively. The circles indicatingTD spots at the surface are arranged at the same positions as inFigs. 1(a) and 1(c). In Figs. 3(a)–3(c), it can be found that Ω, 2Θ,and Φ values largely and sharply change around the L-pit regioncompared to those in other regions. The local modulations of thelattice structure around the L-pit were confirmed in the Δω, Δ2θ,and Δw maps in Figs. 3(d)–3(f ). The degree of the modulationaround the XS-pits is comparatively small and apparently unclearin Figs. 3(a)–3(c), but recognized sensitively in the Δω, Δ2θ, andΔw maps in Figs. 3(d)–3(f). Strains of XS-pits show someFIG. 2. (a) A schematic diagram of nanoXRD geometry and position-dependent ω−2θ−w mapping for (b) area1 and (c) area2. In (b) and (c), the irradiated spots of thex-ray beam are superimposed on the corresponding surface MPPL images with the TD spots indicated by the same-colored circles as in Figs. 1(c) and 1(d).Journal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. Appl. Phys. 135, 225702 (2024); doi: 10.1063/5.0199961 135, 225702-4© Author(s) 2024 12 June 2024 07:14:10https://pubs.aip.org/aip/japfluctuations, which might be due to experimental errors since mea-sured strains are close to the detection limit of our experiments,though fluctuations in the crystal possibly affect the strain field dis-tribution. The highest values of Δω, Δ2θ, and Δw for the typicalXS-pits (indicated by an arrow in the maps) are 0.014°, 0.021°, and0.026°, whereas for the L-pit, 0.027°, 0.030°, and 0.042°, suggestingthat the mixed TD produces larger strain in the lattice structure thanedge TDs. Figures 4(a)–4(f) present Ω, 2Θ, Φ, Δω, Δ2θ, and Δw inthe area2, including the M- and XS-pits, respectively. Two strainedregions clearly seen in all the maps coincided with the M- andXS-pit. The highest values of Δω, Δ2θ, and Δw at the M-pit regionare 0.026°, 0.030°, and 0.038°, whereas at the XS-pit region, 0.015°,0.022°, and 0.025°. Again, edge TDs are shown to generate smallerstrain fields than screw TDs. These results demonstrated that thestrain field around not only the edge and mixed TDs but also purescrew TDs having only the c-component Burgers vector could besuccessfully identified by using the present nanoXRD technique.The reciprocal lattice point at a specific measurement point,Qp, can be calculated based on the ω−2θ−w data as follows:Qp ¼QpxQpyQpz0@1A ¼1λ0{�cosΩpc þ cosΦpcos(2Θp �Ωpc )}1λ0sinΦp1λ0{sinΩpc þ cosΦpsin(2Θp � Ωpc )}0BBBBBB@1CCCCCCA: (1)Here, 2Θp and Φp are measured values of 2Θ and Φ at the specificmeasurement point, respectively. Ωpc was calculated as Ωp � αp,where Ωp is a measured value of Ω and αp is an offset angle of the(0001) plane in the [11�20] direction with respect to the samplesurface at the respective measurement point. The αp values werecalculated by 2Θ p,sym/2�Ω p,sym with the 0004 symmetric Braggreflection dataset (Ω p,sym, 2Θ p,sym) measured at the same points asfor the 11�24 asymmetric reflection measurement. λ0 is the wave-length of the incident x rays, which is 0.15498 nm. Respectiveaverage values of Ωpc , 2Θp, and Φp at all measurement points(referred to as Ωpc , 2Θp , and Φp) in each measurement area, i.e.,area1 and area2, were calculated to determine an average 11�24reciprocal lattice point, �Q, for the respective measurement areas bysubstituting the average values of (Ωpc , 2Θp, Φp) for (Ωpc , 2Θp, Φp)in Eq. (1),�Q ¼QxQyQz0@1A ¼1λ0�cosΩpc þ cosΦpcos(2Θp � Ωpc )n o1λ0sinΦp1λ0sinΩpc þ cosΦpsin(2Θp �Ωpc )n o0BBBBBB@1CCCCCCA: (2)Figure 5 schematically shows the relationship among Qp, �Qand strain components of εij. The normal strain components ofε11, ε22, ε33 along the a-, m-, and c-axis are equivalent to theFIG. 3. Two-dimensional maps of Ω, 2Θ, Φ, Δω, Δ2θ, and Δw in the area1 calculated from ω−2θ−w mapping data. The circles indicating TD spots as described inFigs. 1(c) and 2(b) are superimposed.Journal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. Appl. Phys. 135, 225702 (2024); doi: 10.1063/5.0199961 135, 225702-5© Author(s) 2024 12 June 2024 07:14:10https://pubs.aip.org/aip/japdifferences in the qx-, qy-, and qz-axis components of Qp and �Q,respectively, while the shear strain components of ε12 (or ε21), ε23(or ε32), and ε13 (or ε31) are equivalent to the angles between Qpand �Q in the qx−qy, qy−qz, and qz−qx planes, respectively.Therefore, these strain tensor components can be experimentallyobtained by the following equations:ε11 ε12 ε13ε21 ε22 ε23ε31 ε32 ε330@1A ¼QxQpx� 112tan�1 QpyQpx� �� tan�1 QyQx� �� �12tan�1 QpxQpz� �� tan�1 QxQz� �� �12tan�1 QpyQpx� �� tan�1 QyQx� �� �QyQpy� 112tan�1 QpzQpy !� tan�1 QzQy !( )12tan�1 QpxQpz� �� tan�1 QxQz� �� �12tan�1 QpzQpy !� tan�1 QzQy !( )QzQpz� 10BBBBBBBBB@1CCCCCCCCCA: (3)It should be noted that the normal strain component ε22 inthe qy-axis direction, or in the m-axis direction, cannot be calcu-lated from these measurement data alone, because the present mea-surement setup is for 11�24 reflection parallel to the a-axis.Theoretically, the edge component of dislocations generates normalstrains of ε11, ε22 and shear strains of ε12 (or ε21), whereas thescrew component causes shear strains of ε13 (or ε31) and ε23 (orε32). In this study, ε11 and ε12 were calculated for the XS-pit TD,while ε13 and ε23 for the M-pit TD. For the L-pit TD, both strainfields related to the edge and screw dislocation components wereanalyzed.Figures 6(a)–6(d) show the strain tensor component maps ofε11, ε12, ε13, and ε23 for the L-pit in the area1, respectively. A sym-metrically extended strained area around the TD position (indi-cated by circles) was clearly seen in all the maps. According to theisotropic theory described in the supplementary material, the shearstrains of ε13 and ε23 related to the screw dislocation withb = 〈0001〉 can generate two patterns of strain fields depending onFIG. 4. Two-dimensional maps of Ω, 2Θ, Φ, Δω, Δ2θ, and Δw in the area2 calculated from ω−2θ−w mapping data. The circles indicating TD spots as described inFigs. 1(d) and 2(c) are superimposed.Journal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. Appl. Phys. 135, 225702 (2024); doi: 10.1063/5.0199961 135, 225702-6© Author(s) 2024 12 June 2024 07:14:10https://doi.org/10.60893/figshare.jap.c.7250614https://pubs.aip.org/aip/japthe sign of the Burges vector, as presented in Figs. S3(a) and S3(b),in the supplementary material, respectively. Comparison of thesimulation and experimental results revealed that both strain com-ponents were in good agreements with the simulated ones relatedto b = [0001] rather than b = [000�1]. The simulation results of ε13and ε23 using b = [0001] are shown in Figs. 6(g) and 6(h). Weobserve generally good agreements between the experimental andtheoretical strain distribution of the screw components (matchingin the strain field direction may be obtained by rotating the simula-tion results slightly in the clockwise since a screw dislocation hasrotational symmetry about the dislocation line). On the other hand,the TD having the edge dislocation component in the Burgersvector, such as b = 1/3h11�20i = 1a and b = h1�100i = 1m, can theo-retically generate six patterns of the strain fields of ε11 and ε12depending on the direction of the Burgers vector, as presented inFigs. S1 and S2 of the supplementary material. For the L-pit TDestimated to have the 1m component in the Burgers vector, thesimulation results using b = [�1010], presented in Figs. 6(e) and6(f ), are in best agreement with both experimental results of ε11and ε12. Figures 7(a) and 7(b) present strain fields of ε13 and ε23analyzed from the data around the M-pit estimated to haveb = <0001> in the area2. Similarly, the symmetric strain centeredon the TD position was observed. The Burgers vector of this M-pitTD was estimated to be b = [0001] based on the analogy betweenexperimental and simulation results of ε13 and ε23. Figures 8(a)and 8(b) show ε11 and ε12 maps around the XS-pit in area1. It wasfound that the magnitude of the strain of ε11 and ε12 around theXS-pit was comparably smaller than those around the L-pit inFigs. 6(a) and 6(b), indicating a difference in the magnitude of theFIG. 5. The relationship between Qp, �Q and strain components of εij schemati-cally described in reciprocal space.FIG. 6. The experimental strain tensor component maps of (a) ε11, (b) ε12, (c) ε13, and (d) ε23 for the L-pit observed in the area1. In (a)–(d), the TD positions are indi-cated by red circles. The simulation results of (e) ε11 and (f ) ε12 for the edge dislocation component of b = [�1010] and of (g) ε13 and (h) ε23 for the screw dislocation com-ponent of b = [0001].Journal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. Appl. Phys. 135, 225702 (2024); doi: 10.1063/5.0199961 135, 225702-7© Author(s) 2024 12 June 2024 07:14:10https://doi.org/10.60893/figshare.jap.c.7250614https://doi.org/10.60893/figshare.jap.c.7250614https://pubs.aip.org/aip/japFIG. 7. The experimental strain tensor component maps of (a) ε13 and (b) ε23 for the M-pit observed in the area2. The TD positions are indicated by yellow circles.FIG. 8. The experimental strain tensor component maps of (a) ε11 and (b) ε12 for the XS-pit observed in the area1. In (a) and (b), the TD positions are indicated bygreen circles. The simulation results of (c) ε11 and (d) ε12 for the edge dislocation component of b = [11�20].Journal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. Appl. Phys. 135, 225702 (2024); doi: 10.1063/5.0199961 135, 225702-8© Author(s) 2024 12 June 2024 07:14:10https://pubs.aip.org/aip/japin-plane component of the Burgers vector, as discussed later. Asmentioned earlier, the edge component associated with b = 1a cantheoretically generate six patterns of ε11 and ε12 depending on thedirection of the Burgers vector. Comparison of both the experimen-tal results of ε11 and ε12 with the respective simulation ones indi-cated that the Burgers vector of this XS-pit TD was most likelyb = 1/3[11�20].Finally, we attempt to discuss magnitude of the respectivestrain fields of the edge, screw, and mixed TDs. To compare themagnitude of strain fields, we tentatively adopted the strain changeas evaluated from the difference between the highest and lowestvalues of strains measured by nanoXRD. Results of strain changesaround the XS-, M-, and L-pit TDs and the ratio of the values forthe L-pit TD to those for the XS- and M-pit TDs are summarizedin Table S1 of the supplementary material. The results show thatε13 and ε23 for the L-pit TDs were 1.1 times larger than those forthe M-pit TDs, indicating that the strain field of the screw disloca-tion component for the L- and M-pits was almost identical. On theother hand, the degrees of ε11 and ε12 for the L-pit TD were 5.9and 2.0 times larger than those for the XS-pit TDs, respectively.The theoretical strain tensor component based on the elastic theoryis proportional to the Burgers vector. Since the screw componentof the Burges vector of the TD for the L-pit is estimated to be thesame as the one for the M-pit (b = [0001]), the theoretical ratios ofε13 and ε23 values of L-pit TD to those of M-pit TD were both 1.0.Thus, the experimental results of the strain change related to thescrew component showed good agreement with the elastic theory.In the case of the edge component of the L-pit TD with b = 1m,on the other hand, it should produce 1.7 times larger strain of ε11and ε12 than that of the XS-pit TD with b = 1a according to thetheory. The experimental ratios of the strain shift for ε11 and ε12around the L-pit TD to the XS-pit TD were larger than thoseexpected from the elastic theory. These discrepancies may be attrib-uted to the limitation of spatial resolution in the current nanoXRDexperiments. In the simple elastic theory, strain divergentlyincreases at the dislocation core region of the nm scale, so somethreshold radius for the core has to be introduced to define thepeak strain. The present nanoXRD experiments measured strainsaveraged over x-ray beam irradiation volume (≈3 μm3) and theprecise evaluation of peak strain is hindered by a small volumeratio of the core region and dependency on sampling probe posi-tions. This is in contrast to HRTEM based GPA, which observespercent order strain at the nm scale region near the dislocationcore. Nevertheless, the present measurements capture the shapes ofextended strain fields that are similar to those predicted from theelastic theory as seen in Fig. 6. Also, recent progress of the x-raymirror optical system enables focusing of the x-ray beam down toaround 10 nm.30,31 Improvements in spatial resolution due toreduced x-ray beam size in conjunction with 10−4 strain sensitivitywill provide further opportunities for non-destructive analysis ofthree-dimensional strain tensor fields in crystals.IV. CONCLUSIONThis study demonstrated that position-dependent three-dimensional RSM using nanoXRD sensitively detect variation inthe microlattice structure around all types of individual TDs in theGaN bulk crystal, i.e., the edge TDs with b = 1/3h11�20i = 1a,the screw TDs with b = <0001> = 1c, and the mixed TDs withb = h1�101i = 1m + 1c. The strain fields corresponding to the respec-tive strain tensor components around each TD were analyzed usingthe three-dimensional RSM data. The strain fields showing a nearlysymmetrical strained region centered on the respective TD posi-tions were in good agreements with the simulation results based onthe isotropic elastic theory with a particular Burgers vector. Themagnitude of the shear strain tensor components associated withthe screw dislocation component, ε13 and ε23 of the mixed TDwith b = 1m + 1c and the screw TD with b = 1c, was almost thesame, which was reasonably consistent with the isotropic elastictheory. On the other hand, for the edge dislocation related straincomponents, ε11 and ε12, the ratios of the strain shift around themixed TD with b = 1m + 1c to the edge TD with b = 1a, were largerthan the theoretical values expected from the isotropic elasticmodel. Several inconsistencies between the experimental and theo-retical strains in quantitative aspects may be attributed to the limi-tation in the spatial resolution of the current experiments, andfurther improvements are issues in future work. The presentmethod is particularly beneficial in that it allows non-destructiveanalysis of screw components of TDs that tend to contribute toleakage characteristics and are not routinely accessible by otherconventional analysis methods. These results indicate thatnanoXRD could be a powerful way to reveal three-dimensionalstrain fields associated with arbitrary types of TDs in semiconduc-tor materials, such as GaN and SiC.SUPPLEMENTARY MATERIALAdditional simulation results of strain tensor components ofedge and screw dislocations based on the isotropic elastic theoryand results of strain change at TDs are included in thesupplementary material.ACKNOWLEDGMENTSThe nanoXRD measurement was performed at BL13XU atSPring-8 with the approval of JASRI (Proposal Nos. 2019B1797,2019B2101, 2020A1136, 2020A1331, 2020A1402, 2020A1652,2021A1207, 2021A1584, 2021B1345, 2021B1650, 2022B1567,2022B1817, 2023A1695, and 2023A3599). The authors gratefullyacknowledge funding from the Japan Science and TechnologyAgency (Project No. J121052565). A part of this work was also sup-ported by a Kakenhi Grant-in-Aid (No. JP16H06423) and MurataScience and Education Foundation.AUTHOR DECLARATIONSConflict of InterestThe authors have no conflicts to disclose.Author ContributionsT. Hamachi: Conceptualization (equal); Investigation (lead);Methodology (equal); Writing – original draft (lead); Writing –review & editing (equal). T. Tohei: Conceptualization (equal);Investigation (equal); Methodology (equal); Supervision (equal);Journal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. Appl. Phys. 135, 225702 (2024); doi: 10.1063/5.0199961 135, 225702-9© Author(s) 2024 12 June 2024 07:14:10https://doi.org/10.60893/figshare.jap.c.7250614https://doi.org/10.60893/figshare.jap.c.7250614https://pubs.aip.org/aip/japWriting – original draft (equal); Writing – review & editing(equal). Y. Hayashi: Investigation (supporting); Methodology (sup-porting); Writing – original draft (supporting); Writing – review &editing (supporting). S. Usami: Resources (supporting).M. Imanishi: Resources (supporting). Y. Mori: Resources (equal).K. Sumitani: Methodology (supporting); Resources (supporting).Y. Imai: Methodology (supporting); Resources (supporting).S. Kimura: Methodology (supporting); Resources (supporting).A Sakai: Conceptualization (equal); Funding acquisition (lead);Investigation (equal); Methodology (equal); Project administration(lead); Resources (equal); Supervision (lead); Writing – originaldraft (equal); Writing – review & editing (equal).DATA AVAILABILITYThe data that support the findings of this study are availablefrom the corresponding author upon reasonable request.REFERENCES1T. Oka, Jpn. J. Appl. Phys. 58, SB0805 (2019).2T. Kachi, Jpn. J. Appl. Phys. 53, 100210 (2014).3X. She, A. Q. Huang, O Lucia, and B. Ozpineci, IEEE Trans. Ind. Electron. 64,8193 (2017).4J. 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