PRL 112, 165502 (2014)
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PHYSICAL REVIEW LETTERS
Strain Imaging of Nanoscale Semiconductor Heterostructures with X-Ray Bragg Projection Ptychography
1
Martin V. Holt,1,* Stephan O. Hruszkewycz,2 Conal E. Murray,3 Judson R. Holt,4 Deborah M. Paskiewicz,2 and Paul H. Fuoss2
Center for Nanoscale Materials, Argonne National Laboratory, Argonne, Illinois 60439, USA 2 Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA 3 IBM T. J. Watson Research Center, Yorktown Heights, New York 10598, USA 4 IBM Semiconductor Research and Development Center, Hopewell Junction, New York 12533, USA (Received 15 January 2014; revised manuscript received 4 March 2014; published 23 April 2014) We report the imaging of nanoscale distributions of lattice strain and rotation in complementary components of lithographically engineered epitaxial thin film semiconductor heterostructures using synchrotron x-ray Bragg projection ptychography (BPP). We introduce a new analysis method that enables lattice rotation and out-of-plane strain to be determined independently from a single BPP phase reconstruction, and we apply it to two laterally adjacent, multiaxially stressed materials in a prototype channel device. These results quantitatively agree with mechanical modeling and demonstrate the ability of BPP to map out-of-plane lattice dilatation, a parameter critical to the performance of electronic materials. DOI: 10.1103/PhysRevLett.112.165502
PACS numbers: 61.05.cp, 42.30.Rx, 68.35.bg, 68.37.Yz
The realization of advanced functionality in many nanostructures depends on the manipulation and control of material properties near heterogeneous interfaces. For example, in nanoscale semiconductor systems, strain enhancement of silicon carrier mobility via lithographically deposited epitaxial heteromaterial stressors has yielded dramatic performance improvements in device electronics over the last ten years [1,2] and has enabled continued scaling of device dimensions to follow Moore’s law. While transport behavior in these systems has been studied empirically, the precise details of how strain-driven piezoelectric phenomena— including band splitting, band warping, or intraband scattering [3]—dictate the behavior of real-world nanoscale semiconductor channel devices have been difficult to uncover. This hurdle is, in part, due to the difficulty in imaging nanoscale distributions of strain with sufficient sensitivity to enable structural properties to be correlated with electronic behavior. For example, a local lattice expansion or compression of 0.1% can correspond to a change in electron mobility of as much as a 20% in silicon [3,4]. To date, high-resolution transmission electron microscopy methods [5,6] have resolved strains of order 0.1%, and emerging coherent x-ray diffraction imaging (CXDI) methods have been successfully used to image the diffracted phase of single crystal nanostructures with a sensitivity to lattice displacement of better than 0.01% [7–11]. However, in order to maximize the utility of CXDI techniques for mapping specific lattice responses, methods of separating the different lattice distortions that contribute to a phase image reconstructed by CXDI must be developed. Here, we use Bragg projection ptychography (BPP), a recently developed coherent x-ray diffraction imaging technique, to image lattice distortions in the constituent components of a lithographically engineered silicon-on-insulator 0031-9007=14=112(16)=165502(6)
embedded silicon germanium (SOI=eSiGe) nanoscale semiconductor channel system. We present a new method by which components of lattice strain and rotation can be separately extracted from the reconstructed BPP phase images without the use of a priori knowledge of either component, and we show that these strain and rotation maps are consistent with nanoscale linear elastic mechanical models of both the eSiGe stressor and strained SOI channel regions to strain levels of ∼10−4 . The analysis discussed provides a means to independently estimate two distinct types of the lattice distortion from a single reconstructed BPP diffraction phase map—a component of the rigid body rotation and the out-ofplane strain that is critical to device performance. Bragg projection ptychography is an x-ray imaging technique capable of mapping lattice perturbations in single crystal thin films with nanoscale spatial resolution [12,13]. The technique entails measuring a series of coherent Bragg x-ray diffraction patterns with a nanofocused beam from overlapping positions of the film. The resulting coherent Bragg diffraction patterns are extremely sensitive to perturbations of the crystalline lattice in a given illuminated area [14,15], and, therefore, encode information about local crystal structure, morphology, internal strain fields, and lattice rotation. Using iterative optimization algorithms [16–18], BPP generates a two-dimensional image of the sample's projected diffracted structure factor that is consistent with the overlapping data at a resolution finer than the size of the focused x-ray beam. Furthermore, if Bragg peaks from different material phases in the region of interest are sufficiently separated in angle, BPP can be used to reconstruct complementary images of crystal lattice distortions in each constituent phase. In this Letter, we investigated two materials in a nanoscale semiconductor device prototype composed of single crystal
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© 2014 American Physical Society
PRL 112, 165502 (2014)
PHYSICAL REVIEW LETTERS
silicon on insulator that was lithographically etched into parallel 100-μm long, 60-nm wide strips separated by 460 nm. Epitaxial Si0.8 Ge0.2 stressor structures were deposited into the etched trenches, inducing in-plane compressive stress along h110i in the SOI channel regions. The SOI channels and eSiGe stressors were 75 nm and 60 nm thick, respectively, and the SOI channels were capped with a polycrystalline Si gate [19]. This processing resulted in a device prototype with a cross section shown schematically in Fig. 1 (poly-Si gates not depicted). The out-of-plane lattice parameters of the eSiGe stressors and the strained SOI channels were sufficiently different (∼0.7%) that independent 004 symmetric Bragg nanodiffraction measurements suitable for BPP imaging could be made of both the SOI and eSiGe components of the device prototype [20]. By comparing BPP reconstructions of those measurements, localized lattice distortions were identified in both materials arising from their shared interfaces. The BPP measurements were made at the Hard X-ray Nanoprobe beamline operated by the Center for Nanoscale Materials at the Advanced Photon Source [21,22]. The sample was oriented such that the scattering plane was parallel with the long (∼100 μm) dimension of the channel structures, as shown in Fig. 1. A hard x-ray Fresnel zone plate focusing optic [23] was used to create a ∼35 nm FWHM focused beam at the sample surface with a photon energy of 9 keV. Far-field coherent nanodiffraction patterns were recorded from the device prototype at the eSiGe 004 diffraction condition and at the SOI 004 diffraction condition (θBragg ¼ 30.24° and 30.49° respectively) by an x-ray sensitive charge-coupled device area detector [24]. For BPP imaging, the beam was scanned at each Bragg condition in a spiral pattern with a virtual step size of ∼13 nm, smaller than the beam diameter. In order to mitigate the sample damage known to occur under continuous exposure SOI 004
SiGe 004
x-ray detector z [001] y [110]
x [110]
SiGe BOX 100 µm
125 nm
SOI channels
Fig. 1 (color online). Schematic of the experimental scattering geometry—materials as labeled are the embedded silicon germanium (eSiGe) stressor region, the silicon-on-insulator (SOI) channel region, and the buried oxide (BOX) region. The diffraction plane is in the y − z plane, oriented normal to the channel direction. Coherent x-ray diffraction patterns were collected at two separate scattering conditions (SiGe 004, SOI 004) allowing for ptychographic lattice visualization of both stressor and strained materials across a nanoscale heteromaterial interface.
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of a buried SiO2 layer to hard x-rays [25,26], a fixed offset of 60 nm per point along the self-similar device direction was added to the motion of the spiral. For image reconstruction, these offsets were removed to enforce a “virtual” overlap of the beam positions, which leverages the observed selfsimilarity of the morphology, composition, and strain state of both materials along the channels (in the y direction— see Supplemental Material [27]). A similar approach that exploited self-similar sample structure and virtual beam overlap was successfully employed to enable ptychographic imaging at free electron laser sources [28,29]. It is important to note that while this methodology allowed us to measure ∼10 times higher diffracted intensity at each point, the reconstructed images are insensitive to possible spatial variations of strain or morphology along the offset direction and, therefore, represent an average in that direction. Sixteen virtual spiral scans, each having 101 points and a ∼160 nm diameter, were acquired over an area containing three 460-nm-wide SiGe structures. Since the spirals themselves were offset from each other by 110 nm, the data from all 16 spirals were used collectively to simultaneously reconstruct a single, extended BPP field of view containing 3.5 stressor periods. Similarly, two spiral scans of 101 points were measured at the SOI 004 diffraction condition to map the complementary lattice structure in one interstitial SOI channel. Typical coherent nanodiffraction patterns measured from several regions of the prototype electronic device are shown in Fig. 2. Near the center of the eSiGe stressors, the diffraction patterns are centered and symmetric about the qx , qy origin of the CCD detector [Fig. 2(b)] and represent the focused beam propagated to the far field, modulated by the crystal truncation rod of a perfect film. Near the center of the SOI channel regions [Fig. 2(f)], additional features in the diffraction pattern arise from the fact that the beam illuminates the entire width of the channel to some degree, including the edges where the strain fields intensify. In both materials, as the beam position changes, the diffraction patterns change shape and position in the CCD detector as the lattice distortion fields induced by the multiaxial stress state near the eSiGe=SOI interfaces [30] are sampled by the beam [Figs. 2(c), 2(d)/Figs. 2(e), 2(g)]. The polycrystalline silicon (polySi) gate above the SOI channel and the epitaxy of the eSiGe=SOI interfaces serve as boundary conditions that govern the nanoscale distribution of lattice distortions that can be measured with BPP and modeled using the boundary element method (BEM). Results of the BPP image reconstructions are summarized in Fig. 3. The BPP data sets for both materials were independently reconstructed to produce spatially resolved images of the complex amplitude [Figs. 3(a), 3(b)] and phase [Figs. 3(c), 3(d)] of the 004 diffraction condition. Together, these quantities map the complex volumetric structure factor (F004 ) in the film, projected onto a plane parallel to the CCD detector [12]. The reconstructed amplitude away from the interfaces is mostly a representation of the object density; however, because it represents a
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PRL 112, 165502 (2014)
(a) BPP SiGe reconstruction amplitude
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Fig. 2 (color online). (a) Cross sectional, scanning electron microscopy (SEM) image of the semiconductor channel region. Representative nanofocused coherent x-ray diffraction patterns are shown taken from both the eSiGe stressor region (b)–(d) and strained SOI region (e)–(f). Differences are evident in the intensity distribution of diffraction images taken from the edges of the regions (c), (d), (e), (g) compared to the center of the regions (b), (f) due to lattice perturbations near the lithographically defined interface.
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projection of the structure factor of the film through its thickness, it will deviate from the projected density envelope of the material if local thickness-depended strain fields are present. In the eSiGe reconstruction, the amplitude shows well-resolved gaps [Fig. 3(a)], corresponding to the known spacing and periodicity of the SOI channels, and also shows distinctly “rounded” edges arising from the complex projection effect. A BPP reconstruction was also done of the SOI channel between the second and third imaged eSiGe stressors, as indicated in Figs. 3(a)–3(b), and its amplitude FWHM is 70 nm, in good agreement with the expected 60 nm channel width. We also note that the field of view of the virtual spirals end near the top and bottom edges of the reconstructions shown. Thus, the most numerically constrained region of the reconstruction is near the center of the images in the y direction [used for the lineouts in Figs. 3(e)–3(h)]. The reconstructed BPP phase images [Figs. 3(c), 3(d)] spatially resolve lattice distortions to which the 004 Bragg diffraction condition is sensitive. Using an analysis of the spatial derivatives of the reconstructed BPP phase map described in the Supplemental Material [27], the phase maps of eSiGe and SOI were converted to images of both the out-of-plane lattice strain [Figs. 3(e), 3(f)] and lattice tilt [Figs. 3(g), 3(h)]. The independent extraction of the out-ofplane strain from the phase of the BPP reconstruction is particularly relevant in characterizing the piezoresistive properties of current- and next-generation electronic materials
Amplitude 400
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Fig. 3 (color online). Results of Bragg projection ptychography (BPP) analysis showing an iteratively reconstructed image of both the complex amplitude (a), (b) and phase (c), (d) of the diffracting sample. The x=y gradients in the phase image were interpreted using methods described in the text to estimate the lattice strain (e), (f) and rotation (g), (h) as a function of position transverse to the channel direction. These results are found to be quantitatively consistent with lattice strain and rotation predicted by linear elastic mechanical modeling [solid lines (e)–(h)].
and devices [1]. It also represents an advance of the BPP technique, which had previously been demonstrated to image only lattice rotation in these systems, and depended on the use of mechanical modeling to estimate the effects of strain [12]. The extraction of tilt and strain maps [Figs. 3(e), 3(g)] from the BPP phase images is key to our analysis and enables a comparison to linear elastic models. To separate these phenomena, we take advantage of the fact that local tilt and strain result in different signatures in the diffraction patterns, as depicted by the two limiting cases in Fig. 4. In this example, a model system consisting of a 65-nm-thick single crystal film was considered. Two separate perturbations to the crystal lattice structure were introduced as a function of x position corresponding to an out-of-plane lattice strain and a lattice tilt. Without changing the diffraction condition, far-field 004 Bragg coherent nanodiffraction patterns were simulated from the undistorted and perturbed regions [Figs. 4(b), 4(c), 4(d)]. The resulting
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(a)
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PRL 112, 165502 (2014)
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Fig. 4 (color online). (a) Model system of a 65 nm thick continuous silicon film with two simple lattice perturbations introduced as a function of position along x—the first is a constant lattice strain along the out-of-plane (z) lattice vector, and the second is an out-of-plane lattice tilt. The strain perturbation leads to a qy displacement in the coherent diffraction pattern (c) and the tilt perturbation leads to a qx displacement (d) when comparing to the ideal lattice coherent diffraction pattern (b). The magnitude of this displacement allows us to calculate the linear relationship between BPP phase gradients in the x direction and lattice tilt, and phase gradients in the y direction and lattice strain.
diffraction patterns contain the coherent diffraction fingerprints characteristic of lattice tilt and out-of-plane strain [31,32]. Specifically, a z-axis lattice dilatation or contraction shifts the centroid of the diffraction pattern along qy , and also changes the shape of the Bragg peak [Fig. 4(c)]. By contrast, a small angular deflection of the lattice in a direction perpendicular to the diffraction plane shifts the Bragg peak along qx [Fig. 4(d)] [33]. We note that the sensitivity to Bragg peak shifts along qy due to out-of-plane strain is made possible by the high convergence angle of the focused beam, which enables the Bragg condition to be satisfied in the material over a range of lattice parameters. In BPP, a shift of the experimental Bragg peak away from the qx , qy origin at a given scan point is taken into account by a local phase gradient in the reconstruction that displaces the zero-frequency component of the Fourier transform. Therefore, ramps in a BPP phase image are the result of specific perturbations in the crystal lattice that alter the local Bragg condition relative to a region in the sample for which the Bragg condition is exactly satisfied. For the system studied here, regions of the film that are slightly tilted or strained relative to the center of the eSiGe and SOI structures are expected to contain phase gradients along the x and y directions in the BPP reconstructed image, which, in turn, can be independently interpreted as strain and tilt, respectively. The eSiGe=SOI channel structures imaged here are in a state of plane strain in which lattice deformation along the y direction is forbidden by the 100 μm extent of the structures, thereby limiting the permitted lattice distortions and simplifying the interpretation of local phase gradients in
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the BPP phase reconstruction. In this case, a phase gradient observed along the y direction is a result of z axis lattice strain evolving in the x − z plane (not a result of structural distortion in the y direction), whereas a phase gradient along x comes about from local lattice tilting [9]. Given these constraints and the geometry of the experiment, lattice rotation in the x − z plane and strain along z can be expressed, to first order, as functions of Δqx and Δqy , respectively, which in turn are related to dφBPP =dx and dφBPP =dy (Supplemental Material [27]). Without the plane-strain constraint of this system, higher-order analysis of the reconstructed phase image would be necessary to differentiate lattice distortions that could contribute to reconstructed structure factor phase (i.e., strain vs z − y plane rotation) [34]. Having separated these components in both the eSiGe and SOI, lineouts in the x direction taken near the center of each BPP strain and tilt map were compared with linear elastic modeling predictions. For both the eSiGe and SOI structures, BEM simulations were performed using the nominal cross section of the system to generate continuum mechanical predictions of lattice rotation and strain in both materials due to their respective boundary conditions [35]. The BEM results contained no adjustable parameters, allowing for a direct comparison to experimental results. The cross sectional BEM results were projected along the z-thickness direction for comparison with the BPP measurements, and the lattice strain [solid lines Figs. 3(e) and 3(f)] and tilt [solid lines Figs. 3(g) and 3(h)] profiles from the linear elastic models were found to be consistent with most of the measured tilt and strain behavior of both materials. The overall evolution of strain in the eSiGe stressors relative to the center of the stressor is small (0.02%) and concentrated in a 50 nm region near the lateral interfaces, and the parabola-shaped strain distribution measured within the SOI channel is qualitatively consistent with the BEM model. The lattice tilt of the eSiGe stressors also largely matches the BEM model up to the near-edge regions; however, the agreement of the observed tilt in the SOI channel with the model is poor. These discrepancies are partially due to the fact that the BEM models assume perfect vertical interfaces and sharp corners, whereas it is evident from the cross sectional SEM image [Fig. 2(a)] that the corners of the stressor structures are rounded. However, the partial agreement of the BEM model to the BPP results suggests that continuum mechanical modeling can successfully predict nanoscale deformation in such systems, and that incorporating more realistic boundary conditions is necessary for advanced modeling of the lattice responses in highly confined and stressed nanoscale materials. In conclusion, we have generated maps of out-of-plane lattice strain and lattice tilt in adjacent constituent materials of a device prototype, and we demonstrated a means of untangling the structural information contained in the phase of a BPP reconstruction. Measurements with the sensitivity to strain demonstrated here (
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PHYSICAL REVIEW LETTERS
Strain Imaging of Nanoscale Semiconductor Heterostructures with X-Ray Bragg Projection Ptychography
1
Martin V. Holt,1,* Stephan O. Hruszkewycz,2 Conal E. Murray,3 Judson R. Holt,4 Deborah M. Paskiewicz,2 and Paul H. Fuoss2
Center for Nanoscale Materials, Argonne National Laboratory, Argonne, Illinois 60439, USA 2 Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA 3 IBM T. J. Watson Research Center, Yorktown Heights, New York 10598, USA 4 IBM Semiconductor Research and Development Center, Hopewell Junction, New York 12533, USA (Received 15 January 2014; revised manuscript received 4 March 2014; published 23 April 2014) We report the imaging of nanoscale distributions of lattice strain and rotation in complementary components of lithographically engineered epitaxial thin film semiconductor heterostructures using synchrotron x-ray Bragg projection ptychography (BPP). We introduce a new analysis method that enables lattice rotation and out-of-plane strain to be determined independently from a single BPP phase reconstruction, and we apply it to two laterally adjacent, multiaxially stressed materials in a prototype channel device. These results quantitatively agree with mechanical modeling and demonstrate the ability of BPP to map out-of-plane lattice dilatation, a parameter critical to the performance of electronic materials. DOI: 10.1103/PhysRevLett.112.165502
PACS numbers: 61.05.cp, 42.30.Rx, 68.35.bg, 68.37.Yz
The realization of advanced functionality in many nanostructures depends on the manipulation and control of material properties near heterogeneous interfaces. For example, in nanoscale semiconductor systems, strain enhancement of silicon carrier mobility via lithographically deposited epitaxial heteromaterial stressors has yielded dramatic performance improvements in device electronics over the last ten years [1,2] and has enabled continued scaling of device dimensions to follow Moore’s law. While transport behavior in these systems has been studied empirically, the precise details of how strain-driven piezoelectric phenomena— including band splitting, band warping, or intraband scattering [3]—dictate the behavior of real-world nanoscale semiconductor channel devices have been difficult to uncover. This hurdle is, in part, due to the difficulty in imaging nanoscale distributions of strain with sufficient sensitivity to enable structural properties to be correlated with electronic behavior. For example, a local lattice expansion or compression of 0.1% can correspond to a change in electron mobility of as much as a 20% in silicon [3,4]. To date, high-resolution transmission electron microscopy methods [5,6] have resolved strains of order 0.1%, and emerging coherent x-ray diffraction imaging (CXDI) methods have been successfully used to image the diffracted phase of single crystal nanostructures with a sensitivity to lattice displacement of better than 0.01% [7–11]. However, in order to maximize the utility of CXDI techniques for mapping specific lattice responses, methods of separating the different lattice distortions that contribute to a phase image reconstructed by CXDI must be developed. Here, we use Bragg projection ptychography (BPP), a recently developed coherent x-ray diffraction imaging technique, to image lattice distortions in the constituent components of a lithographically engineered silicon-on-insulator 0031-9007=14=112(16)=165502(6)
embedded silicon germanium (SOI=eSiGe) nanoscale semiconductor channel system. We present a new method by which components of lattice strain and rotation can be separately extracted from the reconstructed BPP phase images without the use of a priori knowledge of either component, and we show that these strain and rotation maps are consistent with nanoscale linear elastic mechanical models of both the eSiGe stressor and strained SOI channel regions to strain levels of ∼10−4 . The analysis discussed provides a means to independently estimate two distinct types of the lattice distortion from a single reconstructed BPP diffraction phase map—a component of the rigid body rotation and the out-ofplane strain that is critical to device performance. Bragg projection ptychography is an x-ray imaging technique capable of mapping lattice perturbations in single crystal thin films with nanoscale spatial resolution [12,13]. The technique entails measuring a series of coherent Bragg x-ray diffraction patterns with a nanofocused beam from overlapping positions of the film. The resulting coherent Bragg diffraction patterns are extremely sensitive to perturbations of the crystalline lattice in a given illuminated area [14,15], and, therefore, encode information about local crystal structure, morphology, internal strain fields, and lattice rotation. Using iterative optimization algorithms [16–18], BPP generates a two-dimensional image of the sample's projected diffracted structure factor that is consistent with the overlapping data at a resolution finer than the size of the focused x-ray beam. Furthermore, if Bragg peaks from different material phases in the region of interest are sufficiently separated in angle, BPP can be used to reconstruct complementary images of crystal lattice distortions in each constituent phase. In this Letter, we investigated two materials in a nanoscale semiconductor device prototype composed of single crystal
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© 2014 American Physical Society
PRL 112, 165502 (2014)
PHYSICAL REVIEW LETTERS
silicon on insulator that was lithographically etched into parallel 100-μm long, 60-nm wide strips separated by 460 nm. Epitaxial Si0.8 Ge0.2 stressor structures were deposited into the etched trenches, inducing in-plane compressive stress along h110i in the SOI channel regions. The SOI channels and eSiGe stressors were 75 nm and 60 nm thick, respectively, and the SOI channels were capped with a polycrystalline Si gate [19]. This processing resulted in a device prototype with a cross section shown schematically in Fig. 1 (poly-Si gates not depicted). The out-of-plane lattice parameters of the eSiGe stressors and the strained SOI channels were sufficiently different (∼0.7%) that independent 004 symmetric Bragg nanodiffraction measurements suitable for BPP imaging could be made of both the SOI and eSiGe components of the device prototype [20]. By comparing BPP reconstructions of those measurements, localized lattice distortions were identified in both materials arising from their shared interfaces. The BPP measurements were made at the Hard X-ray Nanoprobe beamline operated by the Center for Nanoscale Materials at the Advanced Photon Source [21,22]. The sample was oriented such that the scattering plane was parallel with the long (∼100 μm) dimension of the channel structures, as shown in Fig. 1. A hard x-ray Fresnel zone plate focusing optic [23] was used to create a ∼35 nm FWHM focused beam at the sample surface with a photon energy of 9 keV. Far-field coherent nanodiffraction patterns were recorded from the device prototype at the eSiGe 004 diffraction condition and at the SOI 004 diffraction condition (θBragg ¼ 30.24° and 30.49° respectively) by an x-ray sensitive charge-coupled device area detector [24]. For BPP imaging, the beam was scanned at each Bragg condition in a spiral pattern with a virtual step size of ∼13 nm, smaller than the beam diameter. In order to mitigate the sample damage known to occur under continuous exposure SOI 004
SiGe 004
x-ray detector z [001] y [110]
x [110]
SiGe BOX 100 µm
125 nm
SOI channels
Fig. 1 (color online). Schematic of the experimental scattering geometry—materials as labeled are the embedded silicon germanium (eSiGe) stressor region, the silicon-on-insulator (SOI) channel region, and the buried oxide (BOX) region. The diffraction plane is in the y − z plane, oriented normal to the channel direction. Coherent x-ray diffraction patterns were collected at two separate scattering conditions (SiGe 004, SOI 004) allowing for ptychographic lattice visualization of both stressor and strained materials across a nanoscale heteromaterial interface.
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of a buried SiO2 layer to hard x-rays [25,26], a fixed offset of 60 nm per point along the self-similar device direction was added to the motion of the spiral. For image reconstruction, these offsets were removed to enforce a “virtual” overlap of the beam positions, which leverages the observed selfsimilarity of the morphology, composition, and strain state of both materials along the channels (in the y direction— see Supplemental Material [27]). A similar approach that exploited self-similar sample structure and virtual beam overlap was successfully employed to enable ptychographic imaging at free electron laser sources [28,29]. It is important to note that while this methodology allowed us to measure ∼10 times higher diffracted intensity at each point, the reconstructed images are insensitive to possible spatial variations of strain or morphology along the offset direction and, therefore, represent an average in that direction. Sixteen virtual spiral scans, each having 101 points and a ∼160 nm diameter, were acquired over an area containing three 460-nm-wide SiGe structures. Since the spirals themselves were offset from each other by 110 nm, the data from all 16 spirals were used collectively to simultaneously reconstruct a single, extended BPP field of view containing 3.5 stressor periods. Similarly, two spiral scans of 101 points were measured at the SOI 004 diffraction condition to map the complementary lattice structure in one interstitial SOI channel. Typical coherent nanodiffraction patterns measured from several regions of the prototype electronic device are shown in Fig. 2. Near the center of the eSiGe stressors, the diffraction patterns are centered and symmetric about the qx , qy origin of the CCD detector [Fig. 2(b)] and represent the focused beam propagated to the far field, modulated by the crystal truncation rod of a perfect film. Near the center of the SOI channel regions [Fig. 2(f)], additional features in the diffraction pattern arise from the fact that the beam illuminates the entire width of the channel to some degree, including the edges where the strain fields intensify. In both materials, as the beam position changes, the diffraction patterns change shape and position in the CCD detector as the lattice distortion fields induced by the multiaxial stress state near the eSiGe=SOI interfaces [30] are sampled by the beam [Figs. 2(c), 2(d)/Figs. 2(e), 2(g)]. The polycrystalline silicon (polySi) gate above the SOI channel and the epitaxy of the eSiGe=SOI interfaces serve as boundary conditions that govern the nanoscale distribution of lattice distortions that can be measured with BPP and modeled using the boundary element method (BEM). Results of the BPP image reconstructions are summarized in Fig. 3. The BPP data sets for both materials were independently reconstructed to produce spatially resolved images of the complex amplitude [Figs. 3(a), 3(b)] and phase [Figs. 3(c), 3(d)] of the 004 diffraction condition. Together, these quantities map the complex volumetric structure factor (F004 ) in the film, projected onto a plane parallel to the CCD detector [12]. The reconstructed amplitude away from the interfaces is mostly a representation of the object density; however, because it represents a
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PRL 112, 165502 (2014)
(a) BPP SiGe reconstruction amplitude
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Fig. 2 (color online). (a) Cross sectional, scanning electron microscopy (SEM) image of the semiconductor channel region. Representative nanofocused coherent x-ray diffraction patterns are shown taken from both the eSiGe stressor region (b)–(d) and strained SOI region (e)–(f). Differences are evident in the intensity distribution of diffraction images taken from the edges of the regions (c), (d), (e), (g) compared to the center of the regions (b), (f) due to lattice perturbations near the lithographically defined interface.
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projection of the structure factor of the film through its thickness, it will deviate from the projected density envelope of the material if local thickness-depended strain fields are present. In the eSiGe reconstruction, the amplitude shows well-resolved gaps [Fig. 3(a)], corresponding to the known spacing and periodicity of the SOI channels, and also shows distinctly “rounded” edges arising from the complex projection effect. A BPP reconstruction was also done of the SOI channel between the second and third imaged eSiGe stressors, as indicated in Figs. 3(a)–3(b), and its amplitude FWHM is 70 nm, in good agreement with the expected 60 nm channel width. We also note that the field of view of the virtual spirals end near the top and bottom edges of the reconstructions shown. Thus, the most numerically constrained region of the reconstruction is near the center of the images in the y direction [used for the lineouts in Figs. 3(e)–3(h)]. The reconstructed BPP phase images [Figs. 3(c), 3(d)] spatially resolve lattice distortions to which the 004 Bragg diffraction condition is sensitive. Using an analysis of the spatial derivatives of the reconstructed BPP phase map described in the Supplemental Material [27], the phase maps of eSiGe and SOI were converted to images of both the out-of-plane lattice strain [Figs. 3(e), 3(f)] and lattice tilt [Figs. 3(g), 3(h)]. The independent extraction of the out-ofplane strain from the phase of the BPP reconstruction is particularly relevant in characterizing the piezoresistive properties of current- and next-generation electronic materials
Amplitude 400
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Fig. 3 (color online). Results of Bragg projection ptychography (BPP) analysis showing an iteratively reconstructed image of both the complex amplitude (a), (b) and phase (c), (d) of the diffracting sample. The x=y gradients in the phase image were interpreted using methods described in the text to estimate the lattice strain (e), (f) and rotation (g), (h) as a function of position transverse to the channel direction. These results are found to be quantitatively consistent with lattice strain and rotation predicted by linear elastic mechanical modeling [solid lines (e)–(h)].
and devices [1]. It also represents an advance of the BPP technique, which had previously been demonstrated to image only lattice rotation in these systems, and depended on the use of mechanical modeling to estimate the effects of strain [12]. The extraction of tilt and strain maps [Figs. 3(e), 3(g)] from the BPP phase images is key to our analysis and enables a comparison to linear elastic models. To separate these phenomena, we take advantage of the fact that local tilt and strain result in different signatures in the diffraction patterns, as depicted by the two limiting cases in Fig. 4. In this example, a model system consisting of a 65-nm-thick single crystal film was considered. Two separate perturbations to the crystal lattice structure were introduced as a function of x position corresponding to an out-of-plane lattice strain and a lattice tilt. Without changing the diffraction condition, far-field 004 Bragg coherent nanodiffraction patterns were simulated from the undistorted and perturbed regions [Figs. 4(b), 4(c), 4(d)]. The resulting
165502-3
PHYSICAL REVIEW LETTERS
(a)
Undistorted lattice planes
Strained lattice, ∆d/d = 1×10-3
Tilted lattice planes
Out-of-plane z (nm)
PRL 112, 165502 (2014)
Beam position 1
Beam position 2
Beam position 3
50
0
tilt = 1 mrad
Beam position 1 qx=0
(b)
qy qx
Beam position 2 (c)
∆qy
Beam position 3 (d)
∆qx
sqrt( intensity )
Coherent nanodiffraction patterns
In-plane distance x (µm)
1
0
Fig. 4 (color online). (a) Model system of a 65 nm thick continuous silicon film with two simple lattice perturbations introduced as a function of position along x—the first is a constant lattice strain along the out-of-plane (z) lattice vector, and the second is an out-of-plane lattice tilt. The strain perturbation leads to a qy displacement in the coherent diffraction pattern (c) and the tilt perturbation leads to a qx displacement (d) when comparing to the ideal lattice coherent diffraction pattern (b). The magnitude of this displacement allows us to calculate the linear relationship between BPP phase gradients in the x direction and lattice tilt, and phase gradients in the y direction and lattice strain.
diffraction patterns contain the coherent diffraction fingerprints characteristic of lattice tilt and out-of-plane strain [31,32]. Specifically, a z-axis lattice dilatation or contraction shifts the centroid of the diffraction pattern along qy , and also changes the shape of the Bragg peak [Fig. 4(c)]. By contrast, a small angular deflection of the lattice in a direction perpendicular to the diffraction plane shifts the Bragg peak along qx [Fig. 4(d)] [33]. We note that the sensitivity to Bragg peak shifts along qy due to out-of-plane strain is made possible by the high convergence angle of the focused beam, which enables the Bragg condition to be satisfied in the material over a range of lattice parameters. In BPP, a shift of the experimental Bragg peak away from the qx , qy origin at a given scan point is taken into account by a local phase gradient in the reconstruction that displaces the zero-frequency component of the Fourier transform. Therefore, ramps in a BPP phase image are the result of specific perturbations in the crystal lattice that alter the local Bragg condition relative to a region in the sample for which the Bragg condition is exactly satisfied. For the system studied here, regions of the film that are slightly tilted or strained relative to the center of the eSiGe and SOI structures are expected to contain phase gradients along the x and y directions in the BPP reconstructed image, which, in turn, can be independently interpreted as strain and tilt, respectively. The eSiGe=SOI channel structures imaged here are in a state of plane strain in which lattice deformation along the y direction is forbidden by the 100 μm extent of the structures, thereby limiting the permitted lattice distortions and simplifying the interpretation of local phase gradients in
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the BPP phase reconstruction. In this case, a phase gradient observed along the y direction is a result of z axis lattice strain evolving in the x − z plane (not a result of structural distortion in the y direction), whereas a phase gradient along x comes about from local lattice tilting [9]. Given these constraints and the geometry of the experiment, lattice rotation in the x − z plane and strain along z can be expressed, to first order, as functions of Δqx and Δqy , respectively, which in turn are related to dφBPP =dx and dφBPP =dy (Supplemental Material [27]). Without the plane-strain constraint of this system, higher-order analysis of the reconstructed phase image would be necessary to differentiate lattice distortions that could contribute to reconstructed structure factor phase (i.e., strain vs z − y plane rotation) [34]. Having separated these components in both the eSiGe and SOI, lineouts in the x direction taken near the center of each BPP strain and tilt map were compared with linear elastic modeling predictions. For both the eSiGe and SOI structures, BEM simulations were performed using the nominal cross section of the system to generate continuum mechanical predictions of lattice rotation and strain in both materials due to their respective boundary conditions [35]. The BEM results contained no adjustable parameters, allowing for a direct comparison to experimental results. The cross sectional BEM results were projected along the z-thickness direction for comparison with the BPP measurements, and the lattice strain [solid lines Figs. 3(e) and 3(f)] and tilt [solid lines Figs. 3(g) and 3(h)] profiles from the linear elastic models were found to be consistent with most of the measured tilt and strain behavior of both materials. The overall evolution of strain in the eSiGe stressors relative to the center of the stressor is small (0.02%) and concentrated in a 50 nm region near the lateral interfaces, and the parabola-shaped strain distribution measured within the SOI channel is qualitatively consistent with the BEM model. The lattice tilt of the eSiGe stressors also largely matches the BEM model up to the near-edge regions; however, the agreement of the observed tilt in the SOI channel with the model is poor. These discrepancies are partially due to the fact that the BEM models assume perfect vertical interfaces and sharp corners, whereas it is evident from the cross sectional SEM image [Fig. 2(a)] that the corners of the stressor structures are rounded. However, the partial agreement of the BEM model to the BPP results suggests that continuum mechanical modeling can successfully predict nanoscale deformation in such systems, and that incorporating more realistic boundary conditions is necessary for advanced modeling of the lattice responses in highly confined and stressed nanoscale materials. In conclusion, we have generated maps of out-of-plane lattice strain and lattice tilt in adjacent constituent materials of a device prototype, and we demonstrated a means of untangling the structural information contained in the phase of a BPP reconstruction. Measurements with the sensitivity to strain demonstrated here (
