Three-dimensional imaging of dislocations by X-ray diffraction laminography Open Access

The effort of semiconductor technology to produce and process larger and thinner wafers leads to revived interest in the study of defects induced in single crystals.¹ Knowledge about the generation, the temporal and spatial evolution, and the propagation of dislocations generated by mechanical handling and various device processing steps during fabrication is one key element for the development of prevention strategies of damage and failure.² 

In this context, X-ray diffraction imaging techniques such as topography have proven their suitability as appropriate methods of defect recognition and characterization: hard X-rays are sensitive to crystal lattice distortions generated by smallest defects and they penetrate matter sufficiently well to provide information about the interior structure of large volumes. Conventional X-ray projection topography^3,4 enables the visualization of single dislocations with a separation down to about 10 μm (dependent upon the specific diffraction contrast mechanisms) in the resulting two-dimensional (2D) projections of the three-dimensional (3D) crystal volume. However, the fact that depth information is lost and that dislocations may easily overlap in the images often complicates or precludes conclusions on the real 3D arrangement.

At first, depth information was retrieved by stereo-pair topography,³ which provides an intuitive 3D impression based on two projections with an appropriate angular separation. More quantitative depth resolution has been achieved by section topography,⁵ where a thin single crystal slice is illuminated by a slender incoming parallel beam narrowed down by slits. Modern step-scanning techniques enable the rendering of scanned volumes by digitally stacking series of such closely spaced section topographs.^6–8 The resulting 3D resolution is limited by the contrast mechanism involved, the minimum feasible beam width, and the intrinsic geometrical projection of the illuminated crystal slices onto the detector.

At synchrotron sources, a technique called topo-tomography⁹ gains 3D access to dislocation networks in crystals by combining X-ray projection topography with tomographic image acquisition and reconstruction principles. It has been applied to bulk crystals of negligible absorption and with lateral sizes (perpendicular to the rotation axis) fitting into the view field of the detector. Concerning laterally more extended specimens, the method has similar drawbacks to those known from ordinary absorption tomography.¹⁰ This hinders its application to 3D imaging of dislocations, e.g., in large wafers.

Intending to overcome the above limitations, in this letter, we introduce an approach based on X-ray laminography.^11,12 We extend the method to topographic diffraction contrast and optimize it for 3D visualization of crystal defects in laterally extended samples by zooming into a selected 3D region of interest (ROI).

In the following, we will describe the principle of diffraction laminography. Then 2D projection acquisition geometries and topographic contrast conditions suited for high spatial resolution will be discussed. We will outline the reconstruction procedure, the necessary experimental conditions, and the main requirements for the instrumental setup. Finally, the method is exemplarily applied to non-destructive high-resolution 3D imaging of dislocations in a semiconductor wafer.

Tomographic approaches enable the reconstruction of an original 3D object function from a set of 2D projections measured along different directions. It has been shown that under certain conditions projection topographs exhibit dislocation contrast which is suited to such 3D imaging methods.⁹ The lattice deformations close to the defects create contrast by inducing a local change of Bragg reflectivity. By setting the crystal bulk volume in Bragg condition for incoming X-rays along $k0$⁠, this local contrast adds up along the mean diffracted beam direction $kh$⁠. Rotating the crystal about a tomographic axis $ρ$ chosen to be parallel to a selected reciprocal lattice vector $h=hhkl$ and tilted by $90°+θB$ with respect to $k0$⁠, we may record projections along different azimuthal sample directions $ϕ$ with a stationary diffracted beam position on the detector plane and comparable contrast conditions, as shown in Fig. 1(a).

For all rotation angles $ϕ$⁠, the detected images $PθB,ϕ$ are equivalent to projections generated by a virtual incoming beam parallel to $kh$⁠, penetrating the sample straight on and always tilted by $90°−θB$ with respect to the rotation axis. Consequently, this acquisition geometry is equivalent to conventional (transmission) laminography, see Fig. 1(b), and we may take advantage of well established dedicated reconstruction algorithms. In our application case, we perform filtered backprojection (FBP) generalized to the tilted rotation axis of laminography.¹² In particular, it has been proven to handle well the rather low signal-to-noise ratio of topographic projections and unavoidable contrast changes during rotation. The projections $PθB,ϕ(u,v)$⁠, with the detector coordinates v aligned parallel to $ρ$ projected on the detector plane, are Fourier filtered, $P̃θB,ϕ=ℱ−1[HθBℱ(PθB,ϕ)]$⁠, the filter being given by $HθB(ku,kv)=|ku|/2×cosθB$⁠. The filtered projections $P̃θB,ϕ(u,v)$ are backprojected through the reconstruction volume along $−kh(ϕ)$ and added up.

These principles of diffraction laminography are applicable to all kinds of crystalline defects which provide usable topographic contrast (enabling the discrimination of defects from the undistorted crystal regions) in a sufficiently large angular range during laminographic rotation. To enhance the applicability to wafers and for high resolution imaging of dislocations, the following three points are worth considering: (A) The lateral sample extension of wafers obstructs projections with small angles to the surface. Missing angular intervals are expected to result in similar artifacts to those are known from limited angle tomography.¹³ (B) 3D reconstruction algorithms assume images describable as line integrals (the so-called projection requirement). Generally, X-ray topographs do not fully satisfy this precondition: the images are combinations of the (for tomography preferable) direct image together with nonlinear and nonlocal contributions of intermediary and dynamical images, as described by dynamical diffraction theory.^4,14 (C) In principle, topographic projections are never integrals of only one single physical property: the contributions of both diffraction and absorption are always inevitably mixed. For each rotation angle $ϕ$⁠, the detected intensity originating from an infinitesimal diffracting sample volume at position r is additionally influenced by the absorption along the path length $lϕ(r)=lϕ0(r)+lϕh(r)$ through the material, with $lϕ0(r)$ along $k0$ to r and $lϕh(r)$ from r along $kh$ to the detector. With μ denoting the absorption coefficient, this effect is only negligible if $μlϕ(r)≪1∀r,ϕ$⁠, otherwise the consistency of 3D reconstruction is affected.

Typical characteristics of wafers, namely, the two coplanar surfaces and the usually small miscut angles with respect to the main crystallographic directions, allow us to geometrically account for (A) and (C). By orienting the lateral sample extension as in conventional laminography perpendicular to the rotation axis $ρ$⁠, i.e., choosing a reflection with a diffraction vector h sufficiently parallel to the crystal surface n, $h∥n$⁠, see Fig. 2(a), comparable imaging conditions are ensured during a full rotation of $ϕ$ about $360°$⁠. Moreover, in this Bragg reflection geometry (BRG), $lϕ0(r)=lϕh(r)=C(d)$ is independent of $ϕ$ for all positions r in the sample and absorption only depends on the depth d(r) below the surface. This directly allows for a consistent 3D reconstruction via FBP. The influence of absorption may be compensated, retroactively, using

Here, $I(x⊥,d)c$ and $I(x⊥,d)$ denote the corrected and uncorrected reconstructed intensities in depth d, with lateral coordinates $x⊥$⁠. $I(d)0$ is a depth dependent background value to be determined by means of statistics. Another possibility to eliminate the parasitic contributions of absorption is to choose $h⊥n$⁠, as shown in Fig. 2(b), resulting in Laue transmission geometry (LTG). Now $lϕ0(r)+lϕh(r)=C(ϕ)$ is independent of r for each $ϕ$ and pure diffraction contrast may be retrieved by adjusting the relative intensities of the individual projections $PθB,ϕ$⁠. The corrected projections $PθB,ϕc$ are given by

with t the wafer thickness and $ϕ=0°$ defined in the laboratory system for $n(ϕ)·(k0×kh)=0$⁠. In LTG, however, two angular intervals of $α$ are blocked for projection acquisition. Nevertheless, both BRG and LTG are advantageous due to their high compatibility with FBP: the absorption corrections are simple and independent of the reconstruction itself, whereas this is not the case for using asymmetric reflections.

At intense sources such as synchrotrons, the so-called weak-beam condition¹⁴ may provide laminographic scans with topographic contrast of dislocations sufficiently satisfying condition (B): excitation of the undisturbed crystal matrix is strongly reduced and only small deformed regions close by dislocations contribute significantly to image formation, diffracting, in good approximation, kinematically. In this way, the tomographic projection requirement is adequately fulfilled and effects of dynamical diffraction (Pendellösung fringes, etc.) are suppressed to a large extent, in fact, being negligible for 3D reconstruction. The lattice displacement field u(r) close by a dislocation¹⁵ gives rise to a shift $δ(r)$ of the angle of incidence, for which the kinematic Bragg condition is locally fulfilled. For a single wavelength $λ$ and a chosen diffraction vector $h=hhkl$⁠, this departure is described by the effective misorientation¹⁴ with respect to $θB$⁠,

where $θB$ denotes the Bragg angle of the undistorted crystal and $∂/∂xh$ the directional derivative along the diffracted beam direction $kh$⁠. For sufficiently monochromatized X-rays, $Δλ/λ≲10−4$⁠, weak-beam contrast is achieved by setting the incidence angle to $θ=θB+Δθ$⁠, where a suitable choice of $Δθ$ is made by moving far enough on one tail of the measured rocking curve so that significant excitation of the bulk crystal is avoided. Consequently, a minimum $Δθmin$ is necessary, which depends on the Darwin width of the reflection and the spatial dispersion along the imaged crystal region. Then, the resulting diffraction contrast dominantly originates from positions r with $δ(r)∈[−Δθ−w/2,−Δθ+w/2]$⁠, where $Δθ$ compensates for the induced deviation according to Eq. (3). Here, we use $w=Δβ+ΔθB$⁠, with $Δβ$ denoting the beam divergence and $ΔθB=Δλ/(2dcosθB)$ describing the influence of finite monochromaticity.

We demonstrate the applicability of diffraction laminography by 3D imaging of dislocation loops in an industrial silicon wafer, which is part of a study of damage induced during manufacturing processes. Here, the determination of critical preconditions and the microscopic laws for dislocation generation and expansion into initially defect-free wafer regions are of special interest. To emulate typical mechanical wafer handling and rapid thermal annealing, the sample had been Berkovich indented with 400 mN and subsequently heated to $T≈1120 K$ for about 70 s with in-situ topographic control.¹⁶ The process had been stopped when the first dislocation loops were emerging. The wafer was 750 μm thick and its surface [001]-oriented, with a miscut of about $0.4°$⁠.

The experiment was performed at ID19, ESRF. Aiming for high spatial resolution, we implemented the scan geometries and weak-beam condition discussed before in this letter. Two independent types of measurements (see Table I) at the same sample position are discussed, one in BRG (004 reflection, Fig. 3) and another in LTG (220 reflection, Fig. 4). A setup equivalent to Fig. 1(a) was realized by mounting a motorized goniometer head on top of a high resolution tomography setup. A double crystal monochromator ensured $ΔE/E≈10−4$⁠. The effective detector pixel size was 0.75 μm. The instrumental setup allowed an alignment of $h∥ρ$ better than $0.005°$⁠. To compensate for a remaining misalignment and wobble during rotation, an automatic feedback loop was implemented, readjusting $Δθ$ every 20 projections by means of intermediate rocking scans. The effects of absorption were corrected according to Eqs. (1) and (2), after and before 3D reconstruction by FBP, respectively.

Slices and 3D renderings of the reconstructed volumes are shown in Figs. 3 and 4. Most of the dislocations are sharply reproduced. The dislocation loops emerge into four ${111}$-planes intersecting at the indentation point, leaving a pyramidal defect-free region below the initial damage.^8,16 The obtained reconstructions offer a conclusive insight into the dislocation arrangement with high resolution. Single dislocations are distinguishable down to a separation of about 10 μm in the 3D volume (while they are overlapping in the 2D projection topographs), enabling an exact determination of their position, orientation, and path within the nascent slip bands.

In individual reconstructions, there might be also poorly visible dislocation segments due to two possible reasons: (i) insufficient detectable contrast in some projections with unfavorable relative orientations of the Burgers vector b, the wave vectors $k0$ and $kh$⁠, and the local dislocation line direction l (in Fig. 3(b), e.g., the in-plane segments are corrupted); (ii) the two intrinsic “dead” angular intervals $α$ (see Fig. 2(b)) of unavailable projections in LTG (that is why the horizontal dislocation segments along the $[1¯10]$-directions are missing in Fig. 4(b)). Dislocations with $b·h=0$ are generally not visible in a reconstruction, but usually for all possible directions of b suitable reflections are selectable, even if (for the reasons discussed above) one restricts the possible scan geometries to the symmetrical BRG and LTG cases.

The reported results demonstrate the feasibility and the potential of diffraction laminography with synchrotron radiation. The method enables 3D imaging of dislocations in flat crystals with, in principle, unlimited lateral extension and zooming into regions of interest. Beyond that, we showed that by paying attention to the peculiarities of the sample geometry, by taking into account effects of absorption, and by employing suitable topographic contrast, high spatial resolution is achievable.

We thank J. Garagorri, M. R. Elizalde, J. Wittge, D. Allen, P. McNally, B. K. Tanner, J. Moosmann, E. Hamann, A. Cecilia, S. Doyle, and W. Ludwig for their various contributions to this work. The allocation of beamtime MI-1079 at the ESRF and financial support by EU-FP7 Project No. 216382 SIDAM are gratefully acknowledged.