上海交通大学：《探索物质微观世界》课程教学资源（coherent）X-ray diffraction imaging of strain at the nanoscale

nature INSIGHT I REVIEW ARTICLES materials PUBLISHED ONLINE:24 MARCH 2009 DOI:10.1038/NMAT2400 Coherent X-ray diffraction imaging of strain at the nanoscale lan Robinson1*and Ross Harder2 The understanding and management of strain is of fundamental importance in the design and implementation of materials. The strain properties of nanocrystalline materials are different from those of the bulk because of the strong influence of their surfaces and interfaces,which can be used to augment their function and introduce desirable characteristics.Here we explain how new X-ray diffraction techniques,which take advantage of the latest synchrotron radiation sources,can be used to obtain quantitative three-dimensional images of strain.These methods will lead,in the near future,to new knowledge of how nanomaterials behave within active devices and on unprecedented timescales. -ray diffraction methods lie at the foundation of materials cause characteristic patterns of strain,often with associated domain science.Determination of the crystal structures of natural formation on the nanoscale.Classical defect structures,such as free minerals and,later,of crystals grown in the laboratory led dislocations,dislocation loops and stacking-fault tetrahedra all have to our considerable understanding of chemical bonding and cohe- characteristic strain fields that can be used to identify the defect sion that is the basis of the solid-state sciences'.Nanoscale materials modes present,given sufficient resolution.Whereas pure elasticity are emerging that have promising new properties,many of which effects are expected to behave according to classical theory down are associated with previously unknown structures,quantifable to a very small size scale,this is not the case for plasticity.Yield as strains that can be detected by X-ray methods.Here we review stresses are found to increase substantially in crystal grains smaller some of the advances in X-ray diffraction methods and their aug- than a micrometre and continue to deviate on the nanoscale,fol- mentation with the use of the laser-like beams available at the latest lowing a'smaller is stronger'trends.The complete understanding sources of synchrotron radiation,which are now providing spatially of such trends demands suitable methods of strain imaging on resolved structures of nanoscale objects.We will specifically explain the nanoscale. the isostrain method,high-resolution methods and coherent X-ray Looking to the future,imaging of strain dynamicss associated diffraction methods. with transients on the femtosecond timescale will become possible Nanomaterials are defined as those with crystal grain sizes in once the planned hard-X-ray free-electron lasers start operating the range of 100 nm or less.Such materials can possess properties The experimental methods under development today,as described remarkably different from their bulk counterparts,most obviously in this article,will become the methods of choice for producing detected as reduced melting points2,enhanced reactivities'or sig- single snapshots of excited matter at the future facilities?.A shock nificantly modified electronic band structures'.Numerous applica- wave would completely traverse a 100-nm crystal in 100 ps,allow- tions of these new properties are being developed in the new field ing plenty of opportunity to reveal its evolution at the expected of nanotechnology.The simplest explanation of why the properties time resolution of 100 fs.Crystallographic 'difference map'meth- of nanocrystals differ from the bulk is that a considerable fraction ods applied to fully phased starting structures will be very effective of the volume of nanocrystals lies close to external surfaces and for examining localized changes in single-shot applications that can interfaces.The different chemical and physical properties of those destroy the sample. surfaces therefore dominate the bulk behaviour. X-ray diffraction methods are by no means the only method of Strain is the physical concept used to describe the structural measuring strain distributions on the nanoscale.Transmission elec- deviation of the crystal from the ideal bulk state.It is defined as the tron microscopy (TEM)and related off-axis electron holography spatial derivative of the displacement of the material from an ideal methodss can achieve higher resolution than current X-ray meth- lattice,and hence has a tensorial nature.It is connected through ods and may also be quantitative.Transmission electron microscopy elasticity theory with stress,which is generally thought to be the methods are reviewed in a companion article.Both methods meas- origin of strain and can arise from chemical,electrical,magnetic, ure strain as a phase,but one slight distinction is that X-ray diffrac- mechanical and other forces associated with the local environment tion measures the three-dimensional(3D)distribution of a particular of the crystal.Because of the enhanced role of their surfaces,strain component of the strain field whereas electron holography measures in nanomaterials is expected to be significantly enhanced relative the vector strain field in a particular projection.Conventional and to bulk materials,thus opening significant opportunities for nano- aberration-corrected TEM can also map strain at high resolution technology.In this review,we illustrate how strain in nanomateri- by direct imaging.Furthermore,TEM can be used to map both the als can be measured using X-ray diffraction. strain and its relationship to the surrounding device structure that A strong reason for developing the capability to image strain in contains it' nanocrystals in the context of materials science is that the strain can Scanning probe X-ray methods are also not reviewed here. be altered substantially by processing of the material,either as part Spectroscopic methods,reviewed in a related article in this Insight"4, of its designed function or to test fundamental materials science are not directly sensitive to strain.Newly implemented ptycho- principles.Shock waves,ion irradiation tracks,chemical modifica- graphic methods's could be made sensitive to strain,which would tion or external fields applied to a crystalline grain are all expected to appear as a complex density (see below),by introducing Bragg London Centre for Nanotechnology,University College,17-19 Gordon Street,London WC1H OAH,UK and Diamond Light Source,Harwell Campus,Didcot Oxfordshire OX11 ODE,UK.2Advanced Photon Source,Argonne,Illinois 60439,USA."e-mail:i.robinson@ucl.ac.uk NATURE MATERIALS VOL 8|APRIL 2009 www.nature.com/naturematerials 291 2009 Macmillan Publishers Limited.All rights reserved

nature materials | VOL 8 | APRIL 2009 | www.nature.com/naturematerials 291 insight | review articles Published online: 24 march 2009 | doi: 10.1038/nmat2400 X-ray diffraction methods lie at the foundation of materials science. Determination of the crystal structures of natural minerals and, later, of crystals grown in the laboratory led to our considerable understanding of chemical bonding and cohe￾sion that is the basis of the solid-state sciences1 . Nanoscale materials are emerging that have promising new properties, many of which are associated with previously unknown structures, quantifiable as strains that can be detected by X-ray methods. Here we review some of the advances in X-ray diffraction methods and their aug￾mentation with the use of the laser-like beams available at the latest sources of synchrotron radiation, which are now providing spatially resolved structures of nanoscale objects. We will specifically explain the isostrain method, high-resolution methods and coherent X-ray diffraction methods. Nanomaterials are defined as those with crystal grain sizes in the range of 100 nm or less. Such materials can possess properties remarkably different from their bulk counterparts, most obviously detected as reduced melting points2 , enhanced reactivities3 or sig￾nificantly modified electronic band structures4 . Numerous applica￾tions of these new properties are being developed in the new field of nanotechnology. The simplest explanation of why the properties of nanocrystals differ from the bulk is that a considerable fraction of the volume of nanocrystals lies close to external surfaces and interfaces. The different chemical and physical properties of those surfaces therefore dominate the bulk behaviour. Strain is the physical concept used to describe the structural deviation of the crystal from the ideal bulk state. It is defined as the spatial derivative of the displacement of the material from an ideal lattice, and hence has a tensorial nature. It is connected through elasticity theory with stress, which is generally thought to be the origin of strain and can arise from chemical, electrical, magnetic, mechanical and other forces associated with the local environment of the crystal. Because of the enhanced role of their surfaces, strain in nanomaterials is expected to be significantly enhanced relative to bulk materials, thus opening significant opportunities for nano￾technology. In this review, we illustrate how strain in nanomateri￾als can be measured using X-ray diffraction. A strong reason for developing the capability to image strain in nanocrystals in the context of materials science is that the strain can be altered substantially by processing of the material, either as part of its designed function or to test fundamental materials science principles. Shock waves, ion irradiation tracks, chemical modifica￾tion or external fields applied to a crystalline grain are all expected to coherent X-ray diffraction imaging of strain at the nanoscale ian robinson1 * and ross harder2 The understanding and management of strain is of fundamental importance in the design and implementation of materials. The strain properties of nanocrystalline materials are different from those of the bulk because of the strong influence of their surfaces and interfaces, which can be used to augment their function and introduce desirable characteristics. Here we explain how new X-ray diffraction techniques, which take advantage of the latest synchrotron radiation sources, can be used to obtain quantitative three-dimensional images of strain. These methods will lead, in the near future, to new knowledge of how nanomaterials behave within active devices and on unprecedented timescales. cause characteristic patterns of strain, often with associated domain formation on the nanoscale. Classical defect structures, such as free dislocations, dislocation loops and stacking-fault tetrahedra all have characteristic strain fields that can be used to identify the defect modes present, given sufficient resolution. Whereas pure elasticity effects are expected to behave according to classical theory down to a very small size scale, this is not the case for plasticity. Yield stresses are found to increase substantially in crystal grains smaller than a micrometre and continue to deviate on the nanoscale, fol￾lowing a ‘smaller is stronger’ trend5 . The complete understanding of such trends demands suitable methods of strain imaging on the nanoscale. Looking to the future, imaging of strain dynamics6 associated with transients on the femtosecond timescale will become possible once the planned hard-X-ray free-electron lasers start operating. The experimental methods under development today, as described in this article, will become the methods of choice for producing single snapshots of excited matter at the future facilities7 . A shock wave would completely traverse a 100-nm crystal in 100 ps, allow￾ing plenty of opportunity to reveal its evolution at the expected time resolution of 100 fs. Crystallographic ‘difference map’ meth￾ods applied to fully phased starting structures will be very effective for examining localized changes in single-shot applications that can destroy the sample. X-ray diffraction methods are by no means the only method of measuring strain distributions on the nanoscale. Transmission elec￾tron microscopy (TEM) and related off-axis electron holography methods8 can achieve higher resolution than current X-ray meth￾ods and may also be quantitative. Transmission electron microscopy methods are reviewed in a companion article9 . Both methods meas￾ure strain as a phase, but one slight distinction is that X-ray diffrac￾tion measures the three-dimensional (3D) distribution of a particular component of the strain field whereas electron holography measures the vector strain field in a particular projection. Conventional10 and aberration-corrected11,12 TEM can also map strain at high resolution by direct imaging. Furthermore, TEM can be used to map both the strain and its relationship to the surrounding device structure that contains it13. Scanning probe X-ray methods are also not reviewed here. Spectroscopic methods, reviewed in a related article in this Insight14, are not directly sensitive to strain. Newly implemented ptycho￾graphic methods15,16 could be made sensitive to strain, which would appear as a complex density (see below), by introducing Bragg 1 London Centre for Nanotechnology, University College, 17–19 Gordon Street, London WC1H 0AH, UK and Diamond Light Source, Harwell Campus, Didcot, Oxfordshire OX11 0DE, UK. 2 Advanced Photon Source, Argonne, Illinois 60439, USA.*e-mail: i.robinson@ucl.ac.uk nmat_2400_APR09.indd 291 13/3/09 12:04:27 © 2009 Macmillan Publishers Limited. All rights reserved

REVIEW ARTICLES INSIGHT NATURE MATERIALS DOL:10.1038/NMAT2400 ③ 1 Intensity △Reflection 赛Diffraction Radial q, 4 Angular qa Substrate 2=f(4) Decreasing lateral size (000)● a/a. Figure 1|Schematic of the isostrain method used to measure the strain inside quantum dot structures3.The illustration shows how a slice at a given height in the quantum dot is cut out by measuring the transverse diffraction profile at a radial momentum transfer (q)corresponding to the dot's lattice parameter there.a,Cross-section of a quantum dot in which the lattice parameter increases from the bottom to the top.b,Diffraction pattern of the quantum dot,close to one of the substrate reflections.The deviation along the radial q,direction (black slice along the g,direction and colour coding) identifies the contribution from a particular height z,in the structure.c,Intensity as a function of the exit angle a,in units of the critical angle a.The shift of the maximum A,is characteristic of the height z,assuming contributions from all paths 1 to 4.Figure reprinted with permission from ref.21.2005 Elsevier. diffraction,but this has not been reported yet.The most important semiconductor and its substrate,there is a maximum (critical) strain-sensitive scanning instruments are the white-beam Laue thickness of strained material that can form2.The classical example scanning microprobe",beamline 34-ID-E at the Advanced Photon is Ge on Si with a 4%lattice mismatch and a critical thickness around Source (APS),Argonne National Laboratory,Illinois,which can 1 nm,rising to 100 nm for GeoSios alloy29.Critical-thickness effects measure the full strain tensor at every measurement point,and are now a mainstream concept in materials science2.Films greater in the Riso X-ray microscope1s,beamline ID11 at the European thickness than the critical value enter a Stranski-Krastanov growth Synchrotron Radiation Facility(ESRF),Grenoble.These have been mode that eventually leads to isolated islands of relaxed material, reviewed previously.We also do not discuss any of the forward- familiarly known as quantum dots because their size(20-100 nm)is scattering lensless-imaging X-ray work,stemming from the experi- correct for the quantum confinement of electrons. ment of ref.20,which uses the same phasing principles,because We highlight the isostrain method,which has shown that such these techniques are also insensitive to strain. free-standing islands are highly strained with respect to the substrate. A current challenge to the field is the synthesis of well-isolated This method,illustrated in Fig.1,is based on grazing-incidence dif- structures with some indexing system that would allow the same fraction from a large population of free-standing(not buried)quan- crystal to be probed by a number of methods.Most importantly,this tum dots,epitaxially grown on a substrate with a narrow (<10%) would allow the same crystal to be compared before and after some size distribution.As illustrated,it assumes a monotonic variation of treatment is applied.It is to be expected that far more information lattice parameter with height inside the quantum dot,as expected (about shape and strain)would be obtained for a single particle than from its varying composition and strain relaxation.The transverse for an average over a distribution of slightly different ones,even if diffraction profile (Fig.1b)at each radial position,denoted g in the they could all be made to have the same crystallographic orientation. vicinity of an in-plane Bragg peak is the Fourier transform of the This follows the current trend away from interest in thermodynamic shape of the crystal with that particular lattice constant;the exit- average properties towards studying transient and extremely excited angle (a)profile perpendicular to the surface (Fig.Ic)is explained systems far from equilibrium.An important practical consideration by taking into account the four diffraction paths shown in Fig.la is that single-particle methods avoid all the limitations of preparing (ref.30).The isostrain method has been extended by combining it artificially monodispersed samples. with resonant scattering and analysing the intensity profile along the transverse direction to reveal the 3D composition and strain Strain in quantum dots profile in SiGe quantum dots on Si (ref.31). Pattern formation in epitaxial systems of semiconductors has been Recent work has applied the isostrain method to island 'mol- studied using X-ray diffraction,mostly without using the coher- ecules'of InAs quantum dots that form binary pairs aligned with ence properties directly.Extensive work spanning many years has the substrate directions2,pattern formation on GaSb(001)by ion- been published by the groups of Bauer,Stangl and Holy,mostly beam sputtering2 and strong chemical ordering effects in GeSi using beamline ID01 of the ESRE,run by the group of Metzger5. alloys on Si that couple to the strain fields2427.The unique char- Whenever there is lattice mismatch between a thin film of one acteristic of the last work is that the induced chemical ordering is 292 NATURE MATERIALS VOL 8|APRIL 2009 www.nature.com/naturematerials 2009 Macmillan Publishers Limited.All rights reserved

292 nature materials | VOL 8 | APRIL 2009 | www.nature.com/naturematerials review articles | insight NaTure maTerials doi: 10.1038/nmat2400 diffraction, but this has not been reported yet. The most important strain-sensitive scanning instruments are the white-beam Laue scanning microprobe17, beamline 34-ID-E at the Advanced Photon Source (APS), Argonne National Laboratory, Illinois, which can measure the full strain tensor at every measurement point, and the Risø X-ray microscope18, beamline ID11 at the European Synchrotron Radiation Facility (ESRF), Grenoble. These have been reviewed previously19. We also do not discuss any of the forward￾scattering lensless-imaging X-ray work, stemming from the experi￾ment of ref. 20, which uses the same phasing principles, because these techniques are also insensitive to strain. A current challenge to the field is the synthesis of well-isolated structures with some indexing system that would allow the same crystal to be probed by a number of methods. Most importantly, this would allow the same crystal to be compared before and after some treatment is applied. It is to be expected that far more information (about shape and strain) would be obtained for a single particle than for an average over a distribution of slightly different ones, even if they could all be made to have the same crystallographic orientation. This follows the current trend away from interest in thermodynamic average properties towards studying transient and extremely excited systems far from equilibrium. An important practical consideration is that single-particle methods avoid all the limitations of preparing artificially monodispersed samples. strain in quantum dots Pattern formation in epitaxial systems of semiconductors has been studied using X-ray diffraction, mostly without using the coher￾ence properties directly. Extensive work spanning many years has been published by the groups of Bauer, Stangl and Holy, mostly using beamline ID01 of the ESRF, run by the group of Metzger21-27. Whenever there is lattice mismatch between a thin film of one semiconductor and its substrate, there is a maximum (critical) thickness of strained material that can form28. The classical example is Ge on Si with a 4% lattice mismatch and a critical thickness around 1 nm, rising to 100 nm for Ge0.2Si0.8 alloy29. Critical-thickness effects are now a mainstream concept in materials science28. Films greater in thickness than the critical value enter a Stranski–Krastanov growth mode that eventually leads to isolated islands of relaxed material, familiarly known as quantum dots because their size (20–100 nm) is correct for the quantum confinement of electrons. We highlight the isostrain method30, which has shown that such free-standing islands are highly strained with respect to the substrate. This method, illustrated in Fig. 1, is based on grazing-incidence dif￾fraction from a large population of free-standing (not buried) quan￾tum dots, epitaxially grown on a substrate with a narrow (<10%) size distribution. As illustrated, it assumes a monotonic variation of lattice parameter with height inside the quantum dot, as expected from its varying composition and strain relaxation. The transverse diffraction profile (Fig. 1b) at each radial position, denoted qr , in the vicinity of an in-plane Bragg peak is the Fourier transform of the shape of the crystal with that particular lattice constant; the exit￾angle (αf) profile perpendicular to the surface (Fig. 1c) is explained by taking into account the four diffraction paths shown in Fig. 1a (ref. 30). The isostrain method has been extended by combining it with resonant scattering and analysing the intensity profile along the transverse direction to reveal the 3D composition and strain profile in SiGe quantum dots on Si (ref. 31). Recent work has applied the isostrain method to island ‘mol￾ecules’ of InAs quantum dots that form binary pairs aligned with the substrate directions22, pattern formation on GaSb(001) by ion￾beam sputtering23 and strong chemical ordering effects in GeSi alloys on Si that couple to the strain fields24-27. The unique char￾acteristic of the last work is that the induced chemical ordering is z 3 1 2 Substrate Reflection Diraction (000) Decreasing lateral size Intensity 1 Intensity a b c Angular qa Radial qr αf /αc z = f (Δ) Δ αf 4 Figure 1 | schematic of the isostrain method used to measure the strain inside quantum dot structures30. The illustration shows how a slice at a given height in the quantum dot is cut out by measuring the transverse diffraction profile at a radial momentum transfer (qr ) corresponding to the dot’s lattice parameter there. a, Cross-section of a quantum dot in which the lattice parameter increases from the bottom to the top. b, Diffraction pattern of the quantum dot, close to one of the substrate reflections. The deviation along the radial qr direction (black slice along the qa direction and colour coding) identifies the contribution from a particular height z, in the structure. c, Intensity as a function of the exit angle αf in units of the critical angle αc.The shift of the maximum Δ,is characteristic of the height z, assuming contributions from all paths 1 to 4. Figure reprinted with permission from ref. 21. © 2005 Elsevier. nmat_2400_APR09.indd 292 13/3/09 12:04:28 © 2009 Macmillan Publishers Limited. All rights reserved

NATURE MATERIALS DOL:10.1038/NMAT2400 INSIGHT I REVIEW ARTICLES b 46.6 -02 SOI (x10- 46.5 -0.4 BOX 46.4 -0.6 15015050510 463 Si 46.2 -1.0 .1352 46.0 0.6 -12 0.3 -0.4 0.4 -1.4 46.6 -1.0 -05 0.5 1.0 2.7 x (um) 41 0.04 0 1.5 -0.04 -0.2 0.2 0.4 x (um) 46.0 0.6 0.3 0.40.8121.62.02.42.83.23.64.0 (x10) 0.4 0 -0.2 0.2 04 9x (nm) Figure 2 Strain due to patterning of silicon.a,xz cross-section of the measured structure showing the FEA calculation of the g,,strain component shown on the indicated colour scale.A magnified view of the silicon on insulator(SOl)portion is shown below on a finer colour scale.The supporting oxide layer is denoted BOX.b,Measured diffraction pattern surrounding the 004 reflection of the SOl structure as a function of its g.and g,components. c,Corresponding diffraction pattern calculated for the structure shown in a.Figure reprinted with permission from ref.33.2007 AlP. visible at the otherwise forbidden(200)reflection2.An important Again,averaging is used to increase the signal in the experiment, limitation of this X-ray work so far is the need to average the data which was carried out at BM32,a bending magnet beamline at ESRE over a large number of quantum dots,which have to be made with Slight disorder in the relative positions of the wires making up the the same shape and similar size,with identical orientations on the array removes any effect of interference between them;the limited substrate.The situation is helped by the tendency of the islands to coherence of the BM32 beam,in the range of several micrometres, self-assemble at a particular size determined by the strain of the would not be enough to achieve this.However,as the wires are effec- misfit with the substrate.In the most recent work,a microfocused tively floating on amorphous oxide,it is reasonable that any spatial beam was used to isolate the contributions of individual quantum correlations present in the parent SOI layer would be lost. dots".The size distribution can also be controlled by suitable pat- The silicon nitride stress of 1.5 GPa that was found to explain the terning of the starting substrate,which leads to various applications observations is a relatively large value.Without stress,the SOI wires in nanotechnology32 would give diffraction patterns extending less than one-twentieth of the width seen in Fig.2.In effect,many 'phase wraps'are present Strain due to lithographic patterning in the structure(see below).The X-ray methods discussed here are Another perspective of strain measurement using X-rays comes sensitive to stresses orders of magnitude smaller than this,yet use of from the group of Thomas at the Centre National de la Recherche the fabrication method of ref.33 is not uncommon in the construc- Scientifique IM2NP laboratory in Marseille6.They have used tion of semiconductor devices and'strain engineering'is a powerful high-resolution X-ray diffraction to examine the strain distributions method of producing high-performance semiconductor devices". in lithographically prepared micrometre-size structures composed Today's semiconductor industry is producing devices with feature as arrays to enhance the signal3.Finite element analysis (FEA) sizes('design rules')of 45 nm;at this level the resulting strain asso- methods were used to model the strain,followed by a kinematical ciated with the chemically processed interfaces will be much greater diffraction calculation.Like the quantum dot work mentioned than those identified by ref.33.At some point,the FEA bulk model- above,the structures correspond to an array and are averaged over ling will start to break down as the specific structural details of the its periodicity.No attempt was made to model the properties of this crystal interfaces become relevant3,as may already be the case at artificial lattice,but only the shape and strain of the Si bars within 45 nm.Other work by the same group has applied the method to it,which results in the intensity distribution of the diffraction.It Si trench-array structures*and succeeded in using iterative direct- was arguedss that very small deviations from ideal periodicity cause phasing methods (described below)to obtain spatial images of the smearing out of any ultrafine fringes that would arise from the the strain". entire array. In future,strain patterns could be created in model devices The example shown in Fig.2 is from a patterned structure with sizes more relevant to current technology (45 nm)that only engraved in a silicon-on-insulator (SOI)layer lying on its bur- partly penetrate the thickness of the SOI layer,or else in GeSi,as is ied oxide substrate and underlying bulk Si handle.An array of relevant.For example,strain on the scale addressable with X-rays are 1-um-wide wires,spaced 2 um apart,was cut in the SOI(100 nm of expected to result from local heating in the channels ofactive model Si on 200 nm of SiO,)using a silicon nitride lithography mask.After transistors.The embedding environment of a real device will also reactive ion etching,the wire structures are left standing on the bare create strain.As the IM2NP group has found,it is important to use oxide substrate as illustrated in Fig.2.A residual stress of 1.5 GPa SOI methods because the active layer of Si has a different orienta- remains in the silicon nitride layer,causing significant distortions tion from the much thicker handle;the diffraction of interest would in the Si wire,calculated using FEA,as shown.The kinematical dif- be in the shape of the 111 or 220 Bragg peak of the layer,which fraction pattern of the strained wire,shown in Fig.2c,is in good would be completely swamped by the bulk substrate diffraction if agreement with the experimental data in Fig.2b. SOI technologies were not used. NATURE MATERIALS VOL 8|APRIL 2009 www.nature.com/naturematerials 293 2009 Macmillan Publishers Limited.All rights reserved

nature materials | VOL 8 | APRIL 2009 | www.nature.com/naturematerials 293 NaTure maTerials doi: 10.1038/nmat2400 insight | review articles visible at the otherwise forbidden (200) reflection25. An important limitation of this X-ray work so far is the need to average the data over a large number of quantum dots, which have to be made with the same shape and similar size, with identical orientations on the substrate. The situation is helped by the tendency of the islands to self-assemble at a particular size determined by the strain of the misfit with the substrate. In the most recent work, a microfocused beam was used to isolate the contributions of individual quantum dots27. The size distribution can also be controlled by suitable pat￾terning of the starting substrate, which leads to various applications in nanotechnology32. strain due to lithographic patterning Another perspective of strain measurement using X-rays comes from the group of Thomas at the Centre National de la Recherche Scientifique IM2NP laboratory in Marseille33,36,37. They have used high-resolution X-ray diffraction to examine the strain distributions in lithographically prepared micrometre-size structures composed as arrays to enhance the signal33. Finite element analysis (FEA) methods were used to model the strain, followed by a kinematical diffraction calculation. Like the quantum dot work mentioned above, the structures correspond to an array and are averaged over its periodicity. No attempt was made to model the properties of this artificial lattice, but only the shape and strain of the Si bars within it, which results in the intensity distribution of the diffraction. It was argued33 that very small deviations from ideal periodicity cause the smearing out of any ultrafine fringes that would arise from the entire array. The example shown in Fig. 2 is from a patterned structure engraved in a silicon-on-insulator (SOI) layer lying on its bur￾ied oxide substrate and underlying bulk Si handle. An array of 1-μm-wide wires, spaced 2 μm apart, was cut in the SOI (100 nm of Si on 200 nm of SiO2) using a silicon nitride lithography mask. After reactive ion etching, the wire structures are left standing on the bare oxide substrate as illustrated in Fig. 2. A residual stress of 1.5 GPa remains in the silicon nitride layer, causing significant distortions in the Si wire, calculated using FEA, as shown. The kinematical dif￾fraction pattern of the strained wire, shown in Fig. 2c, is in good agreement with the experimental data in Fig. 2b. Again, averaging is used to increase the signal in the experiment, which was carried out at BM32, a bending magnet beamline at ESRF. Slight disorder in the relative positions of the wires making up the array removes any effect of interference between them; the limited coherence of the BM32 beam, in the range of several micrometres, would not be enough to achieve this. However, as the wires are effec￾tively floating on amorphous oxide, it is reasonable that any spatial correlations present in the parent SOI layer would be lost. The silicon nitride stress of 1.5 GPa that was found to explain the observations is a relatively large value. Without stress, the SOI wires would give diffraction patterns extending less than one-twentieth of the width seen in Fig. 2. In effect, many ‘phase wraps’ are present in the structure (see below). The X-ray methods discussed here are sensitive to stresses orders of magnitude smaller than this, yet use of the fabrication method of ref. 33 is not uncommon in the construc￾tion of semiconductor devices and ‘strain engineering’ is a powerful method of producing high-performance semiconductor devices34. Today’s semiconductor industry is producing devices with feature sizes (‘design rules’) of 45 nm; at this level the resulting strain asso￾ciated with the chemically processed interfaces will be much greater than those identified by ref. 33. At some point, the FEA bulk model￾ling will start to break down as the specific structural details of the crystal interfaces become relevant35, as may already be the case at 45 nm. Other work by the same group has applied the method to Si trench-array structures36 and succeeded in using iterative direct￾phasing methods (described below) to obtain spatial images of the strain37. In future, strain patterns could be created in model devices with sizes more relevant to current technology (45 nm) that only partly penetrate the thickness of the SOI layer, or else in GeSi, as is relevant. For example, strain on the scale addressable with X-rays are expected to result from local heating in the channels of active model transistors. The embedding environment of a real device will also create strain. As the IM2NP group has found, it is important to use SOI methods because the active layer of Si has a different orienta￾tion from the much thicker handle; the diffraction of interest would be in the shape of the 111 or 220 Bragg peak of the layer, which would be completely swamped by the bulk substrate diffraction if SOI technologies were not used33. 0 –0.2 –0.4 –0.5 0 0.5 x (μm) z (μm) x (μm) εzz z (μm) –0.6 –0.8 –1.0 –1.2 –1.4 –1.0 –0.04 0 0.04 –0.2 0.2 0.4 0.4 2 0.8 1.2 1.6 .0 2.4 2.8 3.2 3.6 4.0 –0.4 0 1.0 –6.0 –0.4 46.0 46.1 46.2 46.3 46.4 46.5 46.6 46.0 46.1 46.2 46.3 46.4 46.5 46.6 –0.2 0.2 0.3 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 0 0.4 –0.4 –0.2 0 0.2 0.4 0 –4.5 –3.0 –1.5 1.5 3.0 4.5 0 (×10–3) (×10–3) Si BOX SOI Si3N4 qx (nm–1) qz (nm–1) qz (nm–1) a c b Figure 2 | strain due to patterning of silicon. a, xz cross-section of the measured structure showing the FEA calculation of the εzz strain component shown on the indicated colour scale. A magnified view of the silicon on insulator (SOI) portion is shown below on a finer colour scale. The supporting oxide layer is denoted BOX. b, Measured diffraction pattern surrounding the 004 reflection of the SOI structure as a function of its qx and qz components. c, Corresponding diffraction pattern calculated for the structure shown in a. Figure reprinted with permission from ref. 33. © 2007 AIP. nmat_2400_APR09.indd 293 13/3/09 12:04:29 © 2009 Macmillan Publishers Limited. All rights reserved

REVIEW ARTICLES INSIGHT NATURE MATERIALS DOL:10.1038/NMAT2400 Box 1|Solving the phase problem by CXD. The phasing of the data is a critical step that uses a computer methods.A technique known as 'shrink wrap'is sometimes used algorithm that is illustrated schematically.The algorithm takes to allow the support to evolve on successive cycles as the algo- advantage of internal redundancies in the diffraction data when rithm progressesss.The oversampling condition is simply that the the measurement points are spaced close enough together to number of measurement points be at least twice the number of meet the 'oversampling'requirement.Fourier transforms ('F'; unknown density values within this support,as proposed in ref.56 fast Fourier transforms in the computer implementation)con- immediately after the publication of the Shannon theorem7.It was nect real-space arrays of data (left panels)with reciprocal space shown in ref.58 that this is a sufficient condition for a unique set arrays(right panels).Both sides are updated in every cycle of the of phases to be determined,in two or more dimensions.The best algorithm.Inputs to the algorithm are the measured intensity data known method for finding those phases and avoiding 'stagnation' (array amplitudes overwritten in the bottom-right panel)and a problems is the hybrid input-output method,which starts with postulated 3D 'support'volume (translucent box in the top-left a random phase'seed'and propagates a weighted combination of panel)in which all the complex sample density is constrained to the current and previous cycles on the real-space side of the algo- exist.The measurement of diffraction data is shown schematically rithm.This support-constrained phasing of the diffraction patterns on the right:a narrow,coherent X-ray beam illuminates one of of small objects is judged to be reliable because the same structure the Pb nanocrystals grown (in situ)on a substrate(viewed subse- emerges from different random starting phase sets40.One measure quently by SEM)to produce a diffraction pattern on an area detec- of its reliability is that only 50 iterations of hybrid input-output are tor.The dimensions of the support can be estimated directly from often enough to yield a consistent result with good data,whereas an the fringe spacings of the observed pattern or using autocorrelation earlier version of phasing algorithms required several thousand Coherent X-ray diffraction per fringe also ensures an 'oversampling'condition".Obtaining The sample-averaging limitations of the two methods described so enough coherent flux is only practicable with the brightness of far have been overcome by the development of coherent X-ray dif- undulator-based third-generation'synchrotron radiation sources, fraction(CXD)as a 3D structural analysis method for individual such as the ESRF and APS,used for the CXD results reported here. small crystals.As explained in Box 1,the inversion of the diffraction The crystal lattice introduces a powerful new constraint on the pattern yields a 3D image of the density distribution of the sample selection of a grain for imaging.A polycrystalline sample will have and a projection of the strain within it,so it is a genuine form of closely packed grains with numerous different orientations.Its phase-contrast X-ray microscopy.The basic principle of the CXD Bragg diffraction will resemble that of a powder but,with a small experiment is the illumination of the sample by a spatially coherent enough beam and a typical grain size of around a micrometre,the beam of X-rays,meaning that the transverse coherence length (of individual grains can still be separated.Even highly textured sam- up to a few micrometres)should exceed the dimensions of the sam- ples can have orientations distributed widely enough that the grains ple.Under these conditions,scattering from all parts of the sample can be distinguished.Once a Bragg peak is isolated and aligned,its can be expected to interfere in the far-field diffraction pattern. internal intensity distribution can be recorded by means of a CCD The coherence effects we exploit were first documented for hard at the end of a long detector arm.A'rocking'series of CCD images, X-rays in 1991 (ref.38).Coherent X-ray diffraction is usually meas- in which the crystal's angle to the beam is varied by a fraction of a ured in the same way as a normal diffraction experiment,but with a degree around its Bragg peak,yields a complete 3D data set. high-resolution charge-coupled device(CCD)or other X-ray detec- The 3D data are then inverted to form quantitative real-space tor positioned far enough away to resolve the finest fringes,called images using a computational method for solving the phase prob- 'speckles'by analogy with scattering of laser light.As discussed lem,described in Box 1.This computation effectively replaces the further below,the requirement of at least two detector pixel spacings objective lens in a traditional microscope.A typical CXD pattern is 294 NATURE MATERIALS VOL 8|APRIL 2009 www.nature.com/naturematerials 2009 Macmillan Publishers Limited.All rights reserved

294 nature materials | VOL 8 | APRIL 2009 | www.nature.com/naturematerials review articles | insight NaTure maTerials doi: 10.1038/nmat2400 coherent X-ray diffraction The sample-averaging limitations of the two methods described so far have been overcome by the development of coherent X-ray dif￾fraction (CXD) as a 3D structural analysis method for individual small crystals. As explained in Box 1, the inversion of the diffraction pattern yields a 3D image of the density distribution of the sample and a projection of the strain within it, so it is a genuine form of phase-contrast X-ray microscopy. The basic principle of the CXD experiment is the illumination of the sample by a spatially coherent beam of X-rays, meaning that the transverse coherence length (of up to a few micrometres) should exceed the dimensions of the sam￾ple. Under these conditions, scattering from all parts of the sample can be expected to interfere in the far-field diffraction pattern. The coherence effects we exploit were first documented for hard X-rays in 1991 (ref. 38). Coherent X-ray diffraction is usually meas￾ured in the same way as a normal diffraction experiment, but with a high-resolution charge-coupled device (CCD) or other X-ray detec￾tor positioned far enough away to resolve the finest fringes, called ‘speckles’ by analogy with scattering of laser light. As discussed further below, the requirement of at least two detector pixel spacings per fringe also ensures an ‘oversampling’ condition39. Obtaining enough coherent flux is only practicable with the brightness of undulator-based ‘third-generation’ synchrotron radiation sources, such as the ESRF and APS, used for the CXD results reported here. The crystal lattice introduces a powerful new constraint on the selection of a grain for imaging. A polycrystalline sample will have closely packed grains with numerous different orientations. Its Bragg diffraction will resemble that of a powder but, with a small enough beam and a typical grain size of around a micrometre, the individual grains can still be separated. Even highly textured sam￾ples can have orientations distributed widely enough that the grains can be distinguished. Once a Bragg peak is isolated and aligned, its internal intensity distribution can be recorded by means of a CCD at the end of a long detector arm. A ‘rocking’ series of CCD images, in which the crystal’s angle to the beam is varied by a fraction of a degree around its Bragg peak, yields a complete 3D data set. The 3D data are then inverted to form quantitative real-space images using a computational method for solving the phase prob￾lem, described in Box 1. This computation effectively replaces the objective lens in a traditional microscope. A typical CXD pattern is Box 1 | solving the phase problem by CXD. F F–1 500 nm The phasing of the data is a critical step that uses a computer algorithm that is illustrated schematically. The algorithm takes advantage of internal redundancies in the diffraction data when the measurement points are spaced close enough together to meet the ‘oversampling’ requirement39. Fourier transforms (‘F’; fast Fourier transforms in the computer implementation) con￾nect real-space arrays of data (left panels) with reciprocal space arrays (right panels). Both sides are updated in every cycle of the algorithm. Inputs to the algorithm are the measured intensity data (array amplitudes overwritten in the bottom-right panel) and a postulated 3D ‘support’ volume (translucent box in the top-left panel) in which all the complex sample density is constrained to exist. The measurement of diffraction data is shown schematically on the right: a narrow, coherent X-ray beam illuminates one of the Pb nanocrystals grown (in situ) on a substrate (viewed subse￾quently by SEM) to produce a diffraction pattern on an area detec￾tor. The dimensions of the support can be estimated directly from the fringe spacings of the observed pattern or using autocorrelation methods. A technique known as ‘shrink wrap’ is sometimes used to allow the support to evolve on successive cycles as the algo￾rithm progresses55. The oversampling condition is simply that the number of measurement points be at least twice the number of unknown density values within this support, as proposed in ref. 56 immediately after the publication of the Shannon theorem57. It was shown in ref. 58 that this is a sufficient condition for a unique set of phases to be determined, in two or more dimensions. The best known method for finding those phases and avoiding ‘stagnation’ problems is the hybrid input–output method59, which starts with a random phase ‘seed’ and propagates a weighted combination of the current and previous cycles on the real-space side of the algo￾rithm. This support-constrained phasing of the diffraction patterns of small objects is judged to be reliable because the same structure emerges from different random starting phase sets60. One measure of its reliability is that only 50 iterations of hybrid input–output are often enough to yield a consistent result with good data, whereas an earlier version of phasing algorithms required several thousand61. nmat_2400_APR09.indd 294 13/3/09 12:04:30 © 2009 Macmillan Publishers Limited. All rights reserved

NATURE MATERIALS DOL:10.1038/NMAT2400 INSIGHT I REVIEW ARTICLES shown,consisting of characteristic rings resembling the Airy pattern of a compact,solid object and modulated streaks attributed to its Box 2 Sensitivity to lattice strain. prominent facets.The average intensity decays rapidly away from the centre of the Bragg peak,eventually reaching the background Illustration of how the phase in CXD images arises from the level of the detector.This radial cut-off determines the spatial reso- asymmetry of the diffraction pattern.For an ideal crystal,one lution of the resulting real-space image.This is limited,in practice, in which the unit cells lie on a mathematically perfect 3D lattice, by the counting statistics,but more by the stability of the sample and the intensity distribution due to the crystal shape is a periodic instrument,as well as the brightness of the X-ray source.At present, function of reciprocal space.The intensity distribution is identi- the typical resolution of these experiments is around 40 nm,but is cal around every Bragg peak and about the origin of reciprocal expected to improve to 10 nm as the technique develops. space.Overall inversion symmetry of the diffraction(Friedel's Once the diffraction pattern is phased,it can be inverted by law)therefore implies that the diffraction will be locally sym- means of a Fourier transform into a complex density function.As metric about the exact reciprocal lattice points.This results in explained in Box 2,the amplitude and phase of the complex density symmetric intensity patterns about the centre of each Bragg can be interpreted as physical density and lattice deformation,as reflection in the CXD experiment.This is sometimes,but not projected onto the Q vector of the Bragg peak chosen.It is clear that always,observed in practice.When a non-symmetric pattern is the method is sensitive to displacements a small fraction of a unit seen,it can be decomposed into symmetric and antisymmetric cell in magnitude,even though the overall resolution is relatively parts.The symmetric part can be considered to come from the low.It is also obvious that a displacement of a full unit cell (or of physical average electron density,and the antisymmetric part is multiple units cells)will be invisible,so a highly strained crystal will associated with a real-space phase that is equal to the local dis- contain many 2nt wraps of phase,which will have to be unwrapped placement of the atoms from the ideal lattice,projected onto if the full lattice displacement is to be resolved.The full 3D density the Q vector of the Bragg peak concerned 2.This displacement function contains much information and is difficult to view.It is field is imaged as a real-space map of phase values at each posi- useful to examine images of isosurfaces of the density,using a single tion inside the sample.In general,a complex density is required contour level,often between one-quarter and one-half of the maxi- to produce an asymmetric diffraction pattern.In the figure,a mum density.Such images resemble the shape of the crystal and can strained region of the crystal is illustrated as a block of material be compared using scanning electron microscopy(SEM),at least for displaced from the rest by a vector u(r),which depends generally isolated nanocrystals on a substrate.The shapes of individual grains on position.The phase of the X-ray beams scattered by this block of a polycrystalline material can be identified in this way,even is shifted relative to that of the reference crystal by a total amount though they are buried inside a matrix.To view the phase,it is use- =keu-k-u=Q-u.Whenever Q is set to a Bragg condition,all ful to map it onto a colour scale and examine two-dimensional slices unit-cell corners scatter in phase,so this phase shift is manifested cutting through the crystal,as shown in Figs 3 and 4.To examine in the final image as a region of complex density with the same surface strain,it is sometimes interesting to project the colours onto magnitude as the rest of the crystal and a phase o(r). the density isosurface. The presence of strain can be discerned directly from the colour variations in these slices.An unstrained crystal will have constant phase,not necessarily zero,which is uninteresting as it arises from a trivial symmetry of the Fourier transform.A linear variation of phase is usually also disregarded because it can be attributed to false asymmetries introduced in the data by incorrect centring of the 3D data set in the computational array.Because the true centre of the Bragg peak does not usually fall exactly in the middle of one of the CCD pixels,a small centring error is inevitable and must be ignored in the images.Phase curvature is therefore the simplest form of phase structure that can be usefully interpreted as lattice deformation.Regions of positive relative phase in the crystal are due to a component of lattice displacement in the same direction as the Q vector,and imply a compression on the back of the crystal or a lattice expansion on the front,as viewed along Q. We illustrate the imaging capabilities of the CXD method in However,its propagation into the interior must,and does,obey the some examples.The first,shown in Fig.3,is the 3D imaging of Pb laws of elasticity in a defect-free isotropic medium.The maximum nanocrystals,which was the first published demonstration of the strain component seen is a phase shift of the complex density by strain sensitivity.The samples were grown in situ in APS beamline +1.4 rad,corresponding to a total displacement (relative to the 34-ID-Cby evaporation onto a SiO,substrate,melting and recrystal- ideal crystal lattice)of about one-quarter of a Pb{111}spacing,or lizing the molten droplets,then measuring 3D diffraction patterns 0.08 nm (ref.40).This is the cumulative displacement relative to the and constructing the images shown in Fig.3.The physical density bulk at the centre of the crystal;the deviations start about 100 nm (amplitude)of the crystal was almost constant with no defects,but above and increase to this amount at the bottom interface with the there was a prominent phase feature which is attributed to an inter- substrate.The positive sign of the phase deviation at the back of the nal strain field.Figure 3 shows the strain field both as an isosurface crystal (with respect to the direction of Q)indicates a compression and as a variation in colour across a cut-plane.This projection of the of the lattice planes there. strain field onto the (11-1)Q vector could be fitted by a distribu- A further refinement of the result came from the realization that tion of point 'charges'located outside the crystal.This demonstrates the 700-nm size of the Pb crystal was large enough to cause a signifi- that the displacement field decays inside the crystal according to cant phase shift in the object owing to optical refraction of the X-ray the Poisson equation describing continuum elasticity354.The meas- beam.Although much smaller in magnitude(and opposite in sign) ured strain component is apparently caused by some distribution of for X-rays than for visible light,refraction corresponds to a change contact forces due to the substrate upon which the crystal is grown. of phase of the wave field passing through the sample,relative to NATURE MATERIALS VOL 8|APRIL 2009 www.nature.com/naturematerials 295 2009 Macmillan Publishers Limited.All rights reserved

nature materials | VOL 8 | APRIL 2009 | www.nature.com/naturematerials 295 NaTure maTerials doi: 10.1038/nmat2400 insight | review articles However, its propagation into the interior must, and does, obey the laws of elasticity in a defect-free isotropic medium. The maximum strain component seen is a phase shift of the complex density by +1.4 rad, corresponding to a total displacement (relative to the ideal crystal lattice) of about one-quarter of a Pb{111} spacing, or 0.08 nm (ref. 40). This is the cumulative displacement relative to the bulk at the centre of the crystal; the deviations start about 100 nm above and increase to this amount at the bottom interface with the substrate. The positive sign of the phase deviation at the back of the crystal (with respect to the direction of Q) indicates a compression of the lattice planes there. A further refinement of the result came from the realization that the 700-nm size of the Pb crystal was large enough to cause a signifi￾cant phase shift in the object owing to optical refraction of the X-ray beam42. Although much smaller in magnitude (and opposite in sign) for X-rays than for visible light, refraction corresponds to a change of phase of the wave field passing through the sample, relative to shown, consisting of characteristic rings resembling the Airy pattern of a compact, solid object and modulated streaks attributed to its prominent facets. The average intensity decays rapidly away from the centre of the Bragg peak, eventually reaching the background level of the detector. This radial cut-off determines the spatial reso￾lution of the resulting real-space image. This is limited, in practice, by the counting statistics, but more by the stability of the sample and instrument, as well as the brightness of the X-ray source. At present, the typical resolution of these experiments is around 40 nm, but is expected to improve to 10 nm as the technique develops. Once the diffraction pattern is phased, it can be inverted by means of a Fourier transform into a complex density function. As explained in Box 2, the amplitude and phase of the complex density can be interpreted as physical density and lattice deformation, as projected onto the Q vector of the Bragg peak chosen. It is clear that the method is sensitive to displacements a small fraction of a unit cell in magnitude, even though the overall resolution is relatively low. It is also obvious that a displacement of a full unit cell (or of multiple units cells) will be invisible, so a highly strained crystal will contain many 2π wraps of phase, which will have to be unwrapped if the full lattice displacement is to be resolved. The full 3D density function contains much information and is difficult to view. It is useful to examine images of isosurfaces of the density, using a single contour level, often between one-quarter and one-half of the maxi￾mum density. Such images resemble the shape of the crystal and can be compared using scanning electron microscopy (SEM), at least for isolated nanocrystals on a substrate. The shapes of individual grains of a polycrystalline material can be identified in this way, even though they are buried inside a matrix. To view the phase, it is use￾ful to map it onto a colour scale and examine two-dimensional slices cutting through the crystal, as shown in Figs 3 and 4. To examine surface strain, it is sometimes interesting to project the colours onto the density isosurface. The presence of strain can be discerned directly from the colour variations in these slices. An unstrained crystal will have constant phase, not necessarily zero, which is uninteresting as it arises from a trivial symmetry of the Fourier transform. A linear variation of phase is usually also disregarded because it can be attributed to false asymmetries introduced in the data by incorrect centring of the 3D data set in the computational array. Because the true centre of the Bragg peak does not usually fall exactly in the middle of one of the CCD pixels, a small centring error is inevitable and must be ignored in the images. Phase curvature is therefore the simplest form of phase structure that can be usefully interpreted as lattice deformation. Regions of positive relative phase in the crystal are due to a component of lattice displacement in the same direction as the Q vector, and imply a compression on the back of the crystal or a lattice expansion on the front, as viewed along Q. We illustrate the imaging capabilities of the CXD method in some examples. The first, shown in Fig. 3, is the 3D imaging of Pb nanocrystals, which was the first published demonstration of the strain sensitivity40. The samples were grown in situ in APS beamline 34-ID-C by evaporation onto a SiO2 substrate, melting and recrystal￾lizing the molten droplets, then measuring 3D diffraction patterns and constructing the images shown in Fig. 3. The physical density (amplitude) of the crystal was almost constant with no defects, but there was a prominent phase feature which is attributed to an inter￾nal strain field. Figure 3 shows the strain field both as an isosurface and as a variation in colour across a cut-plane. This projection of the strain field onto the (11−1) Q vector could be fitted by a distribu￾tion of point ‘charges’ located outside the crystal. This demonstrates that the displacement field decays inside the crystal according to the Poisson equation describing continuum elasticity35,41. The meas￾ured strain component is apparently caused by some distribution of contact forces due to the substrate upon which the crystal is grown. Illustration of how the phase in CXD images arises from the asymmetry of the diffraction pattern. For an ideal crystal, one in which the unit cells lie on a mathematically perfect 3D lattice, the intensity distribution due to the crystal shape is a periodic function of reciprocal space. The intensity distribution is identi￾cal around every Bragg peak and about the origin of reciprocal space. Overall inversion symmetry of the diffraction (Friedel’s law) therefore implies that the diffraction will be locally sym￾metric about the exact reciprocal lattice points. This results in symmetric intensity patterns about the centre of each Bragg reflection in the CXD experiment. This is sometimes, but not always, observed in practice. When a non-symmetric pattern is seen, it can be decomposed into symmetric and antisymmetric parts. The symmetric part can be considered to come from the physical average electron density, and the antisymmetric part is associated with a real-space phase that is equal to the local dis￾placement of the atoms from the ideal lattice, projected onto the Q vector of the Bragg peak concerned62. This displacement field is imaged as a real-space map of phase values at each posi￾tion inside the sample. In general, a complex density is required to produce an asymmetric diffraction pattern. In the figure, a strained region of the crystal is illustrated as a block of material displaced from the rest by a vector u(r), which depends generally on position. The phase of the X-ray beams scattered by this block is shifted relative to that of the reference crystal by a total amount φ = kf ∙u − ki ∙u = Q∙u. Whenever Q is set to a Bragg condition, all unit-cell corners scatter in phase, so this phase shift is manifested in the final image as a region of complex density with the same magnitude as the rest of the crystal and a phase φ(r). Box 2 | sensitivity to lattice strain. u ki kf nmat_2400_APR09.indd 295 13/3/09 12:04:31 © 2009 Macmillan Publishers Limited. All rights reserved