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Revealing ‘invisible’ defects in implanted material with 3D Bragg ptychography
Peng Li, Nicholas Phillips, Steven Leake, Marc Allain, Felix Hofmann, Virginie Chamard
To cite this version:
Peng Li, Nicholas Phillips, Steven Leake, Marc Allain, Felix Hofmann, et al.. Revealing ‘invisible’
defects in implanted material with 3D Bragg ptychography. 2020. �hal-02925871�
Revealing ‘invisible’ defects in implanted material
1
with 3D Bragg ptychography
2
Peng Li1, Nicholas W. Phillips2, Steven Leake3, Marc Allain1, Felix Hofmann2, 3
Virginie Chamard1,* 4
5
1 Aix-Marseille Univ, CNRS, Centrale Marseille, Institut Fresnel, Marseille, France.
6
2 Department of Engineering Science, University of Oxford, Parks Road, Oxford, OX1 3PJ, 7
8 UK
3 ESRF – The European Synchrotron, 71 Avenue des Martyrs, 38000 Grenoble, France.
9
*Correspondence to: V. Chamard, virginie.chamard@fresnel.fr 10
11 12
Abstract 13
Small ion-irradiation-induced defects can dramatically alter material properties and speed up 14
degradation. Unfortunately, most of the defects irradiation creates are below the visibility 15
limit of state-of-the-art microscopy. As such, our understanding of their impact is largely 16
based on simulations with major unknowns. Here we present a novel x-ray crystalline 17
microscopy approach, able to image with high sensitivity, nano-scale 3D resolution and 18
extended field of view, the lattice strains and tilts in crystalline materials. Using this 19
enhanced Bragg ptychography tool, we study the damage helium-ion-irradiation produces in 20
tungsten, revealing a series of crystalline details in the 3D sample. Our results show that few- 21
atom-large ‘invisible’ defects are likely isotropic in orientation and homogeneously 22
distributed. No defect-denuded region is observed close to the grain boundary. These results 23
open up exciting perspectives for the modelling of irradiation damage and the detailed 24
analysis of crystalline properties in complex materials.
25 26 27
Introduction
2829
Atomic defects play a fundamental role in controlling the mechanical and physical properties 30
of crystalline materials, resulting in critical hurdles for advanced applications, as epitomized 31
in e.g., energy generation (nuclear1 or photo-voltaic2), energy storage,3 aerospace,4 32
micromechanics5 and semiconductor miniaturization.6 Native or induced defects localise 33
distortion of the crystal lattice and thereby reduce the overall strain energy.7 Conversely, the 34
behaviour of defects is strongly dependent on the microstructural environment, which 35
provides fantastic potential for tuning material properties.8 36
Understanding and exploiting defects in crystalline materials requires probing the material 37
structure from atomic- to macro-scale. At near atomic resolution, transmission electron 38
microscopy (TEM) is an essential method, which allows the direct visualisation of lattice 39
defects in two dimensions (2D) and even three dimensions (3D).9,10 While it is also sensitive 40
to the associated lattice strains, the investigations of these are restricted to 2D and only 41
possible for a small subset of samples (e.g., straight dislocations of edge character11). This is 42
detrimental for the understanding of the defect-defect interplays, since defects interact via 43
their 3D strain fields and resulting stresses.12 In inherently thin TEM samples, defects may 44
also be lost to nearby surfaces that act as strong defect sinks, thus reducing the apparent 45
defect density.13 46
A further challenge concerns the visibility of small defects: In structures containing more 47
than a few thousand atoms, TEM is insensitive to defects smaller than ~1.5 nm.14 However 48
crystals that contain a population of rather small defects with broad size distribution 49
constitute a large class of materials, such as the ones resulting from intense irradiation 50
exposure (materials for future fusion and fission technologies,1 accelerator targets15 or 51
modified surfaces for biocompatibility16). Defects produced by irradiation range from single 52
atom defects to clusters of tens of nanometres in size, where the number density of defects is 53
typically linked to the defect size by a power law with a negative exponent, i.e., the vast 54
majority of defects are below the visibility limit.17 Although small, these “invisible” defects 55
can dramatically change the mechanical properties of the material18 or its thermal transport,19 56
and may lead to irradiation-induced dimensional change.20,21 Recent simulations have 57
successfully predicted the presence and evolution of large populations of “invisible” defects 58
during irradiation.22 However, these predictions remain unverified by experiments. As such 59
there is an urgent need for techniques able to determine the “invisible” defect number density 60
and spatial distribution.
61
A promising alternative is to infer the concentration of those small defects by measuring the 62
distortions (i.e., strains) they cause in the crystal lattice. This idea has been successfully 63
demonstrated using micro-beam Laue diffraction.20,21 However the spatial resolution of this 64
technique (~0.5 µm in 3D) is insufficient to resolve the nano-scale spatial heterogeneities and 65
defect clustering predicted by simulations.22 66
In this context, we foresee that x-ray lens-less microscopy could play a major role.23 Since its 67
first demonstration in 2001,24 x-ray Bragg coherent diffraction imaging has been proven as a 68
powerful method to investigate crystalline properties of materials25 in various sample 69
environments.26,27 X-ray Bragg ptychography (BP), a recently developed method, combines 70
sensitivity to the atomic displacements of lattice planes and outstanding imaging 71
performances of ptychography. Thereby, it provides the means to image, in 3D, extended 72
crystalline materials, with high sensitivity to weak crystalline displacements and strong 73
robustness to large strain fields, those specifications being mandatory to cope with the 74
distinctive crystalline features of strain induced defects and native strains of e.g., irradiated 75
materials. To date, BP with 3D spatial resolution of 10 - 50 nm, and strain resolution on the 76
order of 10-4 has been demonstrated,28,29 and applied to complex problems in material 77
science, e.g., anti-phase domain boundaries in metallic alloy,30 stacking faults in 78
semiconductor quantum wire,31 and crystalline domains in biomineral32. The advent of 4th 79
generation synchrotrons, where BP will become available at several beamlines, should 80
provide a broader access to this microscopy method.
81
X-ray BP makes use of a series of 3D Bragg diffraction intensity data sets, measured in the 82
vicinity of a Bragg reflection, obtained by scanning the sample across a localised illumination 83
beam. The sample motion is designed to ensure a significant amount of probe overlap, the 84
redundancy in the data set allowing retrieving the otherwise inaccessible phase of the 85
diffracted field from the set of diffracted intensities.33 As long as the probe is known, the 86
phase recovery is performed with iterative algorithms, which further give access to the 87
sample 3D image. An effective complex-valued electron density34, specifically designed to 88
account for the Bragg geometry, is used to describe the sample as a function of r, the 3D 89
spatial coordinate. Its amplitude ⎪ρ(r)⎪, provides information about the morphology (or 90
density) of the scattering crystal domain. The spatially varying phase φhkl(r) associated with a 91
specific hkl Bragg vector Qhkl is linked to the atomic displacement field in the crystal, u(r), 92
by φhkl(r) = Qhkl · u(r).
93
However, current BP approaches suffer from substantial limitations. They require the 94
acquisition of an extended data set, which relies on the stability of the experimental set-up 95
over long measurement times (typically 6-12 hours). This necessitates the samples be 96
resistant to radiation damage, a particular challenge for biologically relevant materials.
97
Moreover, BP requires the 3D probe to be known prior to the sample electron density 98
reconstruction. Although some probe pre-characterisations can be performed via 99
ptychography of a test pattern (or similar) in the transmission geometry, the final BP image 100
quality fully depends on the amount of uncertainties introduced by the lack of detailed 101
knowledge on the probe used during the BP experiment. In forward direction ptychographic 102
imaging, this limitation has long been recognised, and simultaneous retrieval of the probe and 103
object is now the universal measurement standard.35 The additional complexity of the Bragg 104
geometry has thus far prevented this refinement in BP.
105
Here, we used an advanced BP approach to investigate TEM-invisible defects in a tungsten- 106
rhenium alloy sample implanted with helium ions. To enable these measurements, we 107
develop a simultaneous probe refinement strategy that makes it possible to improve the 108
retrieved image sensitivity and map a much larger field of view than previously attainable.
109
Notably, this was achieved without increasing the amount of collected data (and therefore the 110
total acquisition time). This approach allows us to view the extended sample and directly 111
compare implanted and non-implanted regions, highlighting details of the crystalline 112
structure, which include lattice damage from helium irradiation, dislocations and sample 113
preparation damage. The results are discussed in the context of understanding irradiation 114
damage processes within tungsten and their potential impacts for its use in the design of 115
future fusion reactor components.
116 117 118
Results 119
A tungsten, 1% rhenium alloy was manufactured by arc melting, producing a polycrystalline 120
material with grain size of a few hundred microns as determined by scanning electron 121
microscopy (SEM). The addition of rhenium mimics neutron-irradiation-induced 122
transmutation alloying.36 The material was further implanted with helium ions that modified a 123
~3 µm thick surface layer (see Methods Sec. 1) and generate neutron-like collision cascade 124
damage.20 Note that, during fusion reactor operation, helium is generated by transmutation 125
and also diffuses from the plasma. Our use of helium ion implantation effectively mimics 126
these two effects. Finally, using focussed ion beam (FIB) milling, a cross-section sample 127
containing a grain boundary was extracted from the bulk and a region of 15×8 µm2 was 128
thinned to ~ 0.45 µm thickness. The resulting sample is shown in Fig. 1a.
129
BP microscopy was performed at a third generation synchrotron beamline, schematically 130
presented in Fig. 1b and described in detail elsewhere.37 An 8 keV coherent x-ray beam was 131
focused down to about 200 nm at the sample position with a set of Kirkpatrick-Baez (KB) 132
mirrors. The sample was placed on a three-axis translation stage, allowing for nano- 133
positioning and nano-scanning. The stage was mounted on the top of a goniometer cradle, 134
used to orientate the sample to the Bragg conditions and to map the 3D components of the 135
scattered intensity distribution, by angularly scanning the so-called rocking curve.38 The 136
sample was scanned across the beam propagation direction, in steps sufficiently small to 137
collect partially redundant information. For each probe-to-sample position and each angle 138
along the rocking curve, coherent diffraction patterns were recorded on a 2D pixelated 139
detector, placed at an exit angle of twice the Bragg angle, at a sufficiently large distance to 140
ensure far-field regime detection. For our measurements, the (220) specular reflection of the 141
right-hand grain of Fig. 1a was probed. A region of 2×2 µm2 (enclosed in the red rectangle in 142
Fig. 1a) near the grain boundary and the implantation layer was selected for the ptychography 143
scan, so that both implanted and non-implanted areas could be imaged. Full experimental 144
details are provided in Methods Sec. 2.
145
As an example representative of the collected data, a 2D coherent diffraction pattern from the 146
He-implanted region is shown in Fig. 1c. Its asymmetry and extent are strong signatures of 147
the presence of strain and rotation fields within the illuminated sample volume.38 A set of 148
four patterns are further presented, corresponding to the intensity distribution obtained when 149
integrating the 2D patterns along the rocking curve, for each of the extreme positions of the 150
raster scan (Fig. 1d). One observes clearly the motion of the main peak towards lower Bragg 151
angles, resulting from the strain in the implanted region (i.e., on the 20th and 400th positions).
152
This suggests that the implantation results in a crystalline lattice swelling, in line with 153
previous reports.20,21 Moreover, the 3D representation of the intensity distribution allows the 154
investigation of the fringe structure. They arise from the interference from the two surfaces of 155
the sample foil and present a period of about 14 µm-1. This provides an initial estimation of 156
the sample thickness of ~0.45 µm, in excellent agreement with the thickness expected from 157
SEM.
158
To go further in the sample structure analysis, the whole data set is inverted to retrieve the 3D 159
sample electron density. This requires a detailed modelling of the scattering process: Under 160
the kinematic scattering approximation, the Bragg diffraction intensity distribution Ij(q) at the 161
jth given probe position 𝐫! is given by:
162
𝐼! 𝐪 = !𝑃(𝐫−𝐫!)𝜌(𝐫)e!𝐐𝒉𝒌𝒍∙𝐮(𝐫)e!𝐪∙𝐫𝑑𝐫
!
!
,
(1) 163
where q represents the 3D coordinates reciprocal spaces, P is the probe function. For the 164
numerical implementation, the direct and reciprocal spaces are introduced. To preserve the 165
data information, the reciprocal frame follows the measurement scanning space (q1, q2, q3).
166
Using a recently developed formalism39, this sampling depicts a direct space frame (r1, r2, 167
r3). Fig. 2a shows a schematic representation of both space coordinate frames. Those are the 168
spaces into which the inversion is performed before the final object reconstruction is plotted 169
in the orthogonal (x, y, z) sample frame (Fig. 1b).
170
Accessing the phase of the sample scattering function from intensity measurements requires 171
one to overcome the loss of the scattered field phase. In forward ptychography, the 3D 172
problem can be decomposed in a series of 2D problems, each of them being solved in the 173
plane which contains the two scanning ptychographic directions.40 BP inversion is more 174
intricate because it aims at solving a problem that is intrinsically 3D. Indeed, Eq. 1 cannot be 175
simplified into a series of 2D ptychographic problems, by e.g., separating the probe and the 176
object in the integral. However, only two scanning ptychographic directions are available, 177
transverse to the probe propagation direction (note that along the beam direction, the weak 178
focussing power of x-ray lenses produces an extremely elongated depth of focus). Therefore, 179
while forward ptychography allows the simultaneous retrieval of both probe and sample 180
functions,35 3D BP requires strong a priori knowledge of the probe.
181
The structure and size of the probe also have a major impact on the 3D intensity acquisition 182
in BP. To better illustrate this issue, we present the probe used during the experiment (see the 183
cross-section and calculated profile along the beam propagation direction, in Fig. 2b and c).
184
This probe function was characterized prior to our measurement (see Methods Sec. 2). The 185
cross-section presents a rather structured distribution, which includes a central spot of about 186
400×200 nm2 (intensity FWHM) along the horizontal and vertical planes, respectively and 187
some beam tails and secondary maxima that extend to a much bigger area, of about 4×4 µm2. 188
These features have to be considered when defining the rocking curve angular steps. Indeed 189
the numerical analysis of the data set, which is based on the use of the fast Fourier transform, 190
connects reciprocal and direct space through Fourier conjugation relations. Precisely, the 191
sampling angle must be chosen such that the illuminated volume (green line) is fully 192
contained in a cuboid defined in the (r1, r2, r3) reconstruction space (orange line) as shown in 193
Fig. 2a. Depending on the probe structure and on the criterion used to define the probe size, 194
the angular sampling Δθ required along the rocking curve varies substantially, as illustrated 195
in Fig. 2d, for vertical-plane diffraction geometry. This plot presents the sampling condition 196
as a function of the considered (i.e., vertical) probe width W and the sample thickness T, for a 197
Bragg angle θB of 43.9° (see Methods Sec. 3 and Eq. 2 for further details). For our sample, 198
the required angular sampling varies from a few 0.01° when only the probe central lobe is 199
accounted for, to about 0.001° when the full probe extent is considered (interestingly, this 200
behaviour is rather marginally affected by the sample thickness). While 0.001° angular steps 201
are mechanically accessible at Bragg diffraction optimized beamline set-ups, those small 202
steps are still challenging and result in a detrimental linear increase of the total measurement 203
time. To mitigate issues caused by long data collection, a series of compromises are usually 204
made, such as choosing an angular step only accounting for the central lobe sampling 205
condition and/or fewer ptychographic positions and/or smaller angular range, which reduce 206
the field of view and/or degrade the image quality. Furthermore, the structure of the probe 207
secondary maxima that develop far from the central lobe, i.e., corresponding to high 208
frequencies, is more prone to optics instabilities, limiting the use of prior knowledge of the 209
probe for faithful reconstruction. All these underline the need for retrieving the full probe 210
distribution function and the sample scattering function simultaneously, a novel strategy we 211
have implemented and applied in this work, to produce high-fidelity maps of lattice strains 212
and rotations.
213
To succeed with the simultaneous retrieval of the 3D sample scattering function and 214
illumination function, additional information needs to be brought to the inversion problem.
215
Considering the generally small numerical aperture of x-ray optics, which results in a self- 216
similar probe along the beam direction, a straightforward constraint can be derived on the 217
probe invariance. This inversion strategy allows recovering not only the information in the 218
close vicinity of the central probe lobe, but also the information arising from much larger 219
distances where only the tails (the series of secondary maxima) of the probe illuminate the 220
sample. According to the sampling principles described above and illustrated in Fig. 2d, this 221
would imply scanning the rocking curve in extremely small angular steps (in the order of a 222
few 0.001°). We circumvented this issue by further up-sampling the rocking curve and 223
retrieving the information in the missing planes (see Methods Sec. 4). This operation 224
corresponds to virtually inserting angular sampling points in-between the measured ones, and 225
to retrieving their diffraction patterns based on the intensity information in the measured 226
ones. Using this strategy, the field of view was increased and the image quality was 227
improved, revealing unprecedented structural details, as shown hereafter. The whole 228
reconstruction strategy and inversion details are given in Methods Sec. 4.
229
The iso-surfaces of the retrieved 3D electron density and the phase φ220(r) associated with the 230
displacement field (φ220(r) = Q220 · u(r), with Q220 being the Bragg vector for the (220) 231
reflection), are shown in Fig. 3. For sake of clarity, only the relevant part of the retrieved 232
image is shown, using a mask applied on the all presented maps (see Supplementary 233
information S2 for the mask definition). The presence of a strong edge (seen from the top left 234
corner to the bottom middle) in the electron density map corresponds to the expected crystal 235
grain boundary. While the sample density is rather homogenous in most parts of the retrieved 236
field of view, some fluctuations are visible. The pipe-like ones (see black arrows in Fig. 3c 237
and 3d) are well known features, which can be understood when considering the phase map.
238
Here, phase vortices are co-localised with the pipes of missing intensity, a characteristic 239
feature of dislocations in noise-limited BCDI data.27,41–43 Similarly, additional density 240
oscillations appear in regions, where the phase wraps rapidly, e.g., at the top right corner.33 A 241
few density voids are observed in the vicinity of the grain boundary, corresponding to 242
cavities in the crystal. Finally, at the limit of the raster scan, where both the counting statistics 243
and overlap constraint are reduced the reconstruction quality is degraded. The probe function 244
retrieved simultaneously from the BP data set (cross-section profile shown in Fig. 3e) 245
exhibits the expected characteristic features (e.g., the central lobe with size of 450×190 nm2, 246
intensity FWHM), in excellent agreement with the profile obtained separately and previously 247
shown in Fig. 2b. The missing triangular region at the bottom left of the probe reconstruction 248
is due to the limited extent of the crystal set by the grain boundary. Note the extent of the 249
field of views, of about 0.6×6×6 µm3 (z×y×x) for the 3D sample and 5×4 µm2 (p1×p2) for the 250
probe cross section. If the intensity information were not limited by the grain boundary (non- 251
scattering crystal), we estimate the field of view along x could be as large as 7.8 µm. Finally, 252
the spatial resolution of this 3D reconstruction is estimated as 37×40×39 nm3 along z, y and x 253
respectively (see Supplementary information S3).
254
As a matter of comparison, the same data set was used to perform a reconstruction using the 255
formerly employed approach,29,44 i.e., no angular up-sampling and using a reconstructed 256
result of the probe based on transmission ptychography. The 3D reconstruction numerical 257
volume was directly designed according to the conjugation relation applied to the experiment 258
parameters, in particular the angular step. The final results are shown in Figs. 3f-h. The 259
reduction in the field of view is particularly evident along the x direction (maximum field of 260
view of 4.6 µm compared to 7.8 µm for the angular up-sampled approach), which is the 261
direction mostly impacted by the angular sampling along the rocking curve. Although the 262
main focal spot is fully contained in the probe function, the beam tails are clearly truncated.
263
This impacts the quality of the object image, where the grain boundary is still visible, but 264
artefacts, such as fluctuations in both electron density and phase, are more abundant. The 265
previously evident dislocations are barely visible. This direct comparison underlines the merit 266
of the new inversion strategy we have proposed.
267
The high-quality extended reconstruction was used to analyse the structural features present 268
in the implanted crystal. To this aim, we used 3D phase map to extract the lattice strain 269
projected along Q220 and the lattice rotations about the x and y axes, hereafter referred to as 270
εzz, ωx and ωy, respectively (see Methods Sec. 5). The top edge of the reconstructed volume 271
corresponds to the former sample surface during ion-implantation (note that, the strain 272
increase observed at the very top surface is an artefact due to a bit of aliasing resulting from 273
the probe extent). The 3D strain map (Fig. 4 (a) and (b)) clearly shows the He-implantation 274
induced strain of about 3×10-4, evident to a depth of approximately 2.8 µm. In the vicinity of 275
the two FIB-processed surfaces, large positive strains, corresponding to a lattice expansion, 276
can be seen. These strains extend to similar depths (about 150 nm for the top surface and 100 277
nm for the bottom surface) and have similar magnitude (~1.5×10-3 and ~1.1×10-3 for the top 278
and bottom surfaces, respectively). Several dislocations are also observed in both implanted 279
and non-implanted layers. Besides these, εzz presents homogeneous behaviour in the whole 280
implanted layer including in the vicinity of the grain boundary. Finally, we would also 281
encourage the reader to view the Supplementary Videos as these provide an interactive way 282
to visualise the multi-dimensional dataset. SV1 shows the object iso-surface, whilst SV2 and 283
SV3 each show a perpendicular plane propagating through the object with the εzz, ωx and ωy
284
components displayed.
285
To further illustrate the interest of our BP result, characterisation using more routinely 286
applied strain microscopy methods, namely high-resolution electron back scattering 287
diffraction (HR-EBSD) and x-ray micro-beam Laue diffraction measurements, were 288
performed for the same region of the sample. These results are shown in the Supplementary 289
information S4. Whilst these methods are considered a mainstay of analyse for strain and tilt 290
distributions in polycrystalline materials, their limitations are obvious in the present case.
291
HR-EBSD presents a lateral spatial resolution of ~100 nm45 and is only surface sensitive 292
(down to a depth of about 10-20 nm) due to the short mean free path of backscattered 293
electrons.46 On the other hand, micro-beam Laue diffraction is bulk sensitive, but the 294
obtained 2D maps represent the integration of the strain and tilts information along the whole 295
thickness. Its transverse spatial resolution, which integrates the size of the focused beam, the 296
scanning precision and the step size, builds up to about 0.5 µm in the present case. Some 297
indication of positive lattice strain along the upper edge of the sample (where the implanted 298
layer is known to be) was recorded by HR-EBSD and micro-beam Laue diffraction.
299
However, neither approaches were able to definitively capture the presence of the implanted 300
layer with a high level of clarity or any indication of the dislocations and the fine details of 301
the strains as was possible using BP.
302 303 304
Discussion 305
The success of our approach relies on the capability of our new method to retrieve the far- 306
field information in planes, whose intensity distributions are not experimentally measured. Of 307
course, this process has a limit and a dedicated analytical work is on the way to further derive 308
it mathematically. It was numerically determined for this experiment (Supplementary 309
information S5). Qualitatively, the limit can be understood by considering that the numerical 310
aperture of the focusing optic broadens the signal along the corresponding directions in the 311
reciprocal space, in particular along the rocking curve direction. Therefore, the intensity 312
information obtained in a given detector plane carries additional features arising from the 313
broadening of the signal along the direction perpendicular to the detector plane. Interestingly, 314
we observed that the reconstruction quality using 7 angular measurements (up-sampling ratio 315
of 18 and angular step of 0.03°) is comparable to the quality of the image produced by the 316
former reconstruction strategy, using 42 angles. From this comparison, considering the gains 317
with respect to the measurement time (×6) and to the achieved field of view (×1.4 318
considering the limits imposed by the crystalline grain size or ×1.7 considering the whole 319
accessible field of view), a total gain of about 8-10 is estimated. This significant 320
improvement brought by our approach should further benefit from the advent of 4th 321
generation synchrotron sources and their expected gain (of about 100) in coherence flux.
322
Regarding the crystalline properties imaged in the sample, we note that the 3D lattice strain 323
and rotation maps obtained using our new approach are not accessible in any other way and 324
reveal important new features: In the vicinity of the top and bottom sample surfaces, large 325
positive strains, corresponding to a lattice expansion, can be seen. The strains extend to 326
similar depths (~120 nm), have similar values (~1.1-1.5×10-3) and are present over the entire 327
sample surfaces. Their spatial distribution and location are consistent with the effects of 328
residual FIB-induced damage. FIB imaging and machining are known to be able to cause 329
dramatic material changes such as introduction of lattice defects,47 large lattice strains,48 330
amorphization,49 and formation of Ga intermetallic.50 From electron microscopy, FIB damage 331
is often indistinguishable from the intrinsic defects and damage features of interest in the 332
sample.51,52 In the present case, it is worth noting that FIB-induced damage remains after 333
employing preventative and mitigating measures, routinely employed in fabricating strain 334
microscopy samples for electron microscopy including sacrificial capping, glancing ion 335
incidence angle and the removal of surface material via low energy polishing (see Methods 336
Sec. 1 for fabrication details). We observe that even these residual effects lead to large lattice 337
distortions. However, the beauty of our approach is that sample regions affected by FIB 338
damage can be unambiguously identified and then excluded from further analysis. This 339
dramatically simplifies data interpretation and gives certainty that features observed in the 340
sample are not artefacts of the preparation. In the following only sample volumes more than 341
150 nm from sample surfaces, and thereby unaffected by FIB, are considered.
342
The εzz lattice strain map (Fig. 4a and 4b) shows a substantial strain with an average value of 343
3×10-4 in the helium-implanted layer. This is in good agreement with low-resolution micro- 344
beam Laue measurements of helium-implantation-induced lattice strain,21 estimating strains 345
of about 2.4×10-4. Whilst there is a clear change in lattice strain from the He-implanted layer 346
to the un-implanted material beneath, there is no discernible, abrupt change in lattice rotation 347
at the boundary between the implanted and un-implanted material (Fig. 4c-4f). This is a very 348
important evidence, since, in principle, a shear deformation of the lattice during ion 349
implantation could lead to rotations about the x and z axes. Such a shearing would indicate 350
the preferential formation of defects with specific orientation.53 In tungsten, helium 351
implantation is expected to predominantly produce Frenkel pairs that are prevented from 352
recombining as He occupies the vacancies.20,54,55 The lack of sharp change in lattice 353
orientation at the un-implanted/implanted interface shows that defects are randomly oriented, 354
leading to a purely volumetric strain.53,56 This important result should greatly simplify the 355
simulation of irradiation-induced strain in reactors,56 which will be one of the main 356
mechanisms driving in-service degradation of fusion reactor armour.
357
Across the layer, the homogeneous behaviour of εzz indicates a uniform distribution of 358
defects, at least within the resolution and sensitivity limit of the present measurements.
359
Previously, the formation of a zone denuded of large irradiation-induced dislocations loops 360
was reported close to grain boundaries (within 20 to 50 nm) in self-ion irradiated material.57 361
Our results suggest that for small defects and defect clusters, such as those created by helium 362
ion-irradiation,20 this is not the case, limiting the potential effectiveness of grain size 363
reduction as a means of reducing irradiation defect accumulation.
364
Finally, the quality of our crystalline image allows us to consider in more detail the 365
dislocations visible in the reconstructed sample. An iso-surface rendering of the electron 366
density, as well as the associated phase variation in the sample mid-plane are shown in Fig.
367
4h and Supplementary information S6 for dislocation #1 (implanted layer) and #2 (un- 368
implanted region), respectively. Dislocations in tungsten have predominantly b=1/2〈111〉 369
Burgers vector, although dislocations with b=〈100〉 have also been reported.43 For a given 370
Bragg reflection, only dislocations for which Qhkl·b is non-zero are visible.58 For the present 371
(220) reflection, this means that the observed dislocations may have the following Burgers 372
vectors: 1/2[111], 1/2[11-1], [100] or [010]. Dislocation #2 shows a central pipe of missing 373
intensity.26 The phase variation around the dislocation line shows a vortex with a total phase 374
increase of 4π. The phase variation agrees well with the simulated phase for a dislocation in 375
the thickness direction, as shown in Supplementary information S6. In contrast, dislocation 376
#1 shows a more complex and surprising structure: Here two separate pipes of reduced 377
intensity are clearly identified, suggesting that in fact this dislocation corresponds to two 378
parallel dislocations, each with an associated phase vortex of 2π. This is the signature 379
expected from two partial dislocations linked by a stacking fault on the (-110) plane. The 380
presence of dissociated dislocations in tungsten is surprising since bcc metals have 381
comparatively high stacking fault energy.59 However, recent ab-initio calculations show that 382
in fact, the addition of Re can substantially reduce the stacking fault energy in tungsten, a 383
scenario possibly at play here.59 384
385
386 387
Conclusions 388
Thanks to the development of an upgraded BP approach, we obtain quantitative 3D maps of 389
lattice strain and rotation in a He-implanted Tungsten crystal. These maps, of unmatched 390
quality and extent, reveal numerous, otherwise inaccessible, crystalline features. Beyond a 391
FIB-damaged layer, which can now be unambiguously discarded from the analysis, we 392
observed strains and lattice rotations caused by ‘invisible’ helium-implantation-induced 393
defects, identified to be of random orientation. Surprisingly, we do not find a defect-denuded 394
region in the vicinity of the crystal boundary. These results provide new insights, essential for 395
predicting the effects of ion-irradiation on crystals. Moreover, they pave the way for highly 396
detailed investigation of complex next-generation crystalline materials, e.g., refractory high- 397
entropy alloys for extreme environments.60,61 398
399 400
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551 552 553
Methods 554
1 - Sample preparation 555
The sample was produced from a bulk polycrystalline tungsten-1wt% rhenium alloy, 556
manufactured by arc melting from high purity elemental powders. The sample surface was 557
mechanically ground and further polished using diamond paste. A final chemo-mechanical 558
polishing step (with 0.1 µm-colloidal silica suspension) was used to produce a high-quality 559
surface finish. The grain size in the polycrystal was ~500 µm. The sample was implanted 560
with helium ions to produce a 2.8 µm thick implanted layer. To achieve a relatively uniform 561
damage level (0.02 ± 0.003 dpa between 0 µm and 2.8 µm depth) and injected helium 562
concentration (310 ± 30 appm between 0 µm and 2.8 µm depth), implantation was performed 563
using a number of different ion energies up to 2 MeV. The details of the implantation are 564
provided elsewhere.21 For BP measurements, a small specimen (~25×14×0.5 µm3) was 565
extracted from the bulk using a focused ion beam (FIB) lift-out process. This preparation 566
method is adapted from that developed for the preparation of isolated micro-crystals for 567
BCDI.43 In order to minimise the effects of lattice damage induced during the lift-out process 568
a combination of protective capping, glancing angle milling, and low energy polishing was 569
used. Specifically, the surface of the bulk sample was first covered with carbonaceous 570
platinum, to further protect the tungsten from normal incidence angle milling ions, using an 571
electron-beam, which does not introduce any damage in the implanted material. This 572
produces a thin layer of about 0.1 – 0.2 µm, which appears as a dark line in the SEM picture 573
(Fig. 2a). On top of this layer, 2 µm of carbonaceous platinum are further deposited using the 574
gallium-ion-beam, which achieves a balance of sputtering and deposition. This is faster than 575
the electron-beam but more intrusive, justifying the prior deposition of the electron-beam 576
assisted deposition layer. The gallium-deposited platinum appears with a different contrast at 577
the top of the sample. Trenches on either side of the lift-out sample were made using FIB, 578
this was then undercut and attached to a micro-manipulator. The sample was finally attached 579
to a copper TEM-lift-out holder as 90 degrees to its original orientation i.e., the walls of the 580
trench became the top and bottom surfaces of the sample for all measurements. As the sample 581
was still rather thick at this point (1 – 2 µm), FIB was again used to further thin down the 582
sample. This was performed at 30 keV, with a current in the order of 240 pA or slightly 583
lower. Finally, the obtained lamella was polished with a 2 keV gallium ion beam to remove 584
most of the damage from the previous FIB steps. It results in a ~0.35 µm thinned region 585
around the implantation layer region, in the vicinity of the grain boundary. The implanted 586
layer starts from the dark line downwards (Fig. 2a). The (220) lattice planes of the interested 587
grain (further away from the copper support end in Fig. 2a) are parallel to the two lift-out 588
surfaces.
589 590
2 - Experimental details – Bragg ptychography 591
The x-ray beam of the ID01 beamline (ESRF) was filtered with a Si-111 double crystal 592
monochromator to provide a monochromatic beam with energy of 8 keV (or wavelength λ of 593
1.54 Å) with a bandwidth of about 1×10-4. Kirkpatrick-Baez mirrors were used to focus the 594
beam onto the sample, horizontally (H) and vertically (V). A pair of slits placed upstream the 595
mirrors were set to a width of 60×200 µm2 (H ×V), selecting a spatially coherent beam for 596
the experiment. From a dedicated (forward ptychography) beam characterization, the beam 597
profile was extracted and the central spot size (intensity FWHM) was measured as 400×200 598
nm2 (H×V). The resulting numerical aperture (NA) was estimated to be 1.9×10-4 and 3.7×10-4 599
(H×V) using the extent of the calculated over-focused beam and the depth of focus to 600
1950×540 µm2 (H×V). More details are given in the following Methods sections.
601
For the BP measurements, the specular (220) Bragg reflection of the grain away from the 602
copper support was chosen. It corresponds to a Bragg angle θB of 43.9˚. The intensity 603
patterns were measured with an area detector (Maxipix, 516×516 pixels, pitch size of 604
55×55𝜇m2) located at 1.4 m away from the sample. A total of 42 angular positions, with an 605
angular step size of 0.005˚, were taken along the rocking curve. At each of these angles, a set 606
of 20×20 positions, with a step size of 100 nm, was used for the raster scan. It corresponds to 607
an area of 2×2 µm2. The exposure time per frame was set to 0.2 seconds.
608
Immediately after BP measurements, the experimental geometry was changed to the forward 609
geometry to perform a conventional transmission ptychographic scan with a Siemens star 610
pattern, in order to retrieve the illumination profile.35 An Archimedean spiral scan of 513 611
positions, with a step size of 100 nm, was performed and for each position the detector was 612
exposed for 1s to measure the diffraction pattern. A sub-region of 334×334 pixels, centered 613
around the forward beam, was used to retrieve the probe profile at the focal plane using 614
standard approaches based on the extended ptychographical iterative engine algorithm 615
(ePIE).62 The initial guess for the probe was a simulated beam profile based on the 616
experimental setup, i.e. calculated via the Fourier transform of a rectangle that is the 617
numerical aperture (or NA) of the beam. The initial guess for the object was simply a uniform 618
matrix of 1s. The gap between the different sensor modules was masked and left free. The 619
reconstruction was run for 100 iterations, where the difference between the measured 620
diffraction patterns and the calculated ones is very small and further iterations provide very 621
little further convergence.
622 623
3 – Angular sampling along the rocking curve for 3D Fourier transform based Bragg 624
coherent diffraction imaging 625
Due to the use of a 3D discrete Fourier transform to describe the propagation between real 626
and reciprocal spaces, the angular sampling pitch Δθ and the window size R3 of the 627
illumination function along r3 are linked via ∆θ = 𝜆 ∕ 2R3sin(θB)cos(θB), according to the 628
sampling relations,39,63 where, considering the geometry depicted in Fig. 1a, 629
R3 = 𝚃cos(2𝜃B)/sin(𝜃B) + W/tan(𝜃B). Here 𝚃 is the sample thickness and W is the beam width 630
along p1. This leads to the following relation between the beam width W and the sampling 631
pitch Δθ: 632
W = 𝜆 ∕ 2∆θ[cos(θB)]2 - 𝚃cos(2𝜃B) ∕ cos(𝜃B) (2) 633
For the used angular step size of 0.005˚ with a sample thickness of 500 nm, W is about 1.7 634
µm. It is big enough to encompass the focal spot, but not the secondary maxima in the probe 635
tails. Those strong beam tails, which extend beyond the window size, result in an aliasing 636
artefact according to the Nyquist sampling theorem.64 Therefore, with this angular step size, 637
the quality of the reconstruction is limited.
638 639
4 – Reconstruction strategy 640
The overall reconstruction strategy stems from previous works in 3D Bragg 641
ptychography28,33,63 with some pivotal adaptations that are required to obtain the high quality, 642
stable reconstruction shown in Fig. 3. The reconstruction is performed via the following cost- 643
function to minimized over all the probe positions 644
(3)
645
L(⇢, P, b) = XJ
j=1
Lj(⇢, P, b)
where 𝜌, 𝑃 and 𝑏, respectively, the 3D electronic density (i.e., the sample), the 3D probe 646
function and the incoherent intensity background. Individual cost-functions are defined 647
by28,33 648
(4)
649
where 𝐼! 𝐪 and ℎ!(𝐪) = |Ψ!(𝐪)|!+𝑏(𝐪) are the measured and predicted photon counts 650
for the j-th probe position, respectively and Ψ! 𝐪 is the Fourier transform of the exit field.
651
The detector mask ω(q) is designed to discard hot or dead camera pixels but also to provide 652
the expected angular up-sampling factor (described in details below). The second, quadratic 653
term is a thickness-support regularization applied to the retrieved sample preventing the 654
density of the sample to build-up in 𝑆, the set of points outside the sample support. The 655
regularization parameters 𝜇! ≥ 0 adjust how the final solution should comply with this 656
support constraint; in the Bragg geometry, we note that this regularization is pivotal in 657
dealing with the intrinsic lack of diversity of the beam along the propagation direction.63 The 658
reconstruction strategy aims at retrieving simultaneously 𝜌, 𝑃 and 𝑏 from the set of 659
measurements 𝐼!(𝐪)
!!!
! via the minimization of Eq. 3.
660
1 - Presentation of the reconstruction algorithm 661
The reconstruction algorithm was derived from the ePIE62 approach, with specific 662
adaptations that allow the probe function to be accurately retrieved in the Bragg geometry.
663
More specifically, a probe and object updates associated with the current (updated) position 664
𝑗 ∈ 1⋯𝐽 are given by 665
, (5a)
666
667 (5b)
where 𝜕!;! and 𝜕!;! are the gradients of with respect to 𝜌 and 𝑃, respectively 668
(6a)
669
. (6b)
670
The projection and back-projection operators and , respectively, are acting altogether 671
along the direction of the incoming beam 𝐤!; the pair of operators is basically enforcing, in 672
the probe update (Eq. 5b), the invariance of the probe along the direction ki (described in 673
Lj(⇢, P, bj) := w(q)⇥⇣
Ij1/2(q) h1/2j (q)⌘ 2
+ µj
X
r2S
|⇢(r)|2
⇢(r) ⇢(r) + ⇢1(r)⇥@⇢;j(r)
P(r) BkiRki P(r) + P1(r)⇥@P;j(r) L(⇢, P, b) =
XJ
j=1
Lj(⇢, P, b)
<latexit sha1_base64="zrUWBEs8P7B5QdLxBt90Q3QBYaE=">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</latexit>
@⇢;j(r) = P⇤(r rj)⇥ 0
j(r) j(r)⇤
µj[1 S(r)]⇢(r)
@P;j(r) = ⇢⇤(r)⇥ 0
j(r) j(r)⇤
@P;j(r) = B@kPiR;j(r) =ki⇢⇤(r) [ BkiR0(r)ki⇢⇤(r) [ (r)]0(r) (r)]
