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Mapping of strain fields in GaAs/GaAsP coreshell nanowires with nanometer resolution Eric J. Jones, Sema Ermez, and Silvija Gradecak Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.5b02733 • Publication Date (Web): 30 Oct 2015 Downloaded from http://pubs.acs.org on November 2, 2015
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Nano Letters
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Mapping of strain fields in GaAs/GaAsP core-shell
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nanowires with nanometer resolution
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Eric J. Jones, Sema Ermez, Silvija Gradečak*
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Department of Materials Science and Engineering, Massachusetts Institute of Technology,
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Cambridge, MA 02139
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*Corresponding author. Email:
[email protected] 7
Abstract. We report the nanoscale quantification of strain in GaAs/GaAsP core-shell nanowires.
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By tracking the shifting of higher-order Laue zone (HOLZ) lines in convergent beam electron
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diffraction patterns, we observe unique variations in HOLZ line separation along different facets
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of the core-shell structure demonstrating the non-uniform strain fields created by the
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heterointerface. Furthermore, through the use of continuum mechanical modeling and Bloch
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wave analysis we calculate expected HOLZ line shift behavior, which are directly matched to
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experimental results. This comparison demonstrates both the power of electron microscopy as a
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platform for nanoscale strain characterization and the reliability of continuum models to
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accurately calculate complex strain fields in nanoscale systems.
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Keywords. Nanowire, radial heterostructures, convergent beam electron diffraction, continuum
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modeling, finite element analysis
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The inherently high surface-to-volume ratio of nanowires allows for the efficient
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relaxation of elastic strain enabling the fabrication of heterostructures infeasible in a thin film
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planar geometry. However, the unique geometry of these nanostructures can also significantly
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affect the fabrication and operation of strain-engineered devices. Because the heterointerface is
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no longer planar, it can no longer be assumed that the stress and strain fields will be homogenous
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along any given direction. Therefore, the ability to quantify strain fields of core-shell nanowires
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with both high spatial resolution and high strain sensitivities will be vital to the future
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development and design of strain engineered nanostructures.
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Techniques such as µ-Raman spectroscopy and X-ray diffraction have been used to
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investigate stress and strain in core-shell nanowires, but neither of these techniques have the
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spatial resolution required to probe the strain fields of individual nanostructures.1-3 Transmission
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electron microscopy (TEM), on the other hand, provides a robust platform for the nanoscale
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characterization of strain due to its ability to form nanosized electron probes. TEM-based
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techniques such as convergent beam electron diffraction (CBED), nanobeam electron diffraction
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(NBED), and geometric phase analysis (GPA) have been shown to be powerful tools for strain
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quantification in a number of nanoscale heterostructures.4-8 Convergent beam electron
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diffraction, unlike NBED and GPA, does not require an unstrained reference material to be near
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the region of interest8 and is therefore of particular interest for the study of core-shell
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heterostructures that are unlikely to contain any unstrained material. In CBED, deficient higher-
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order Laue zone (HOLZ) lines are contained within the central disc of a CBED pattern that result
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from the diffraction of electrons from high index crystal planes. As a result, small uniform
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strains can have a significant impact on HOLZ line position whereas non-uniform strain fields
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along the beam direction – due to surface relaxation – have been shown to result in the
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broadening and splitting of HOLZ lines. By tracking the position and width of these lines
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throughout a series of CBED patterns, local variations in a strain field can be directly observed.7,
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9-11
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Due to the inherent complexity of the strain fields created in a core-shell nanowire, it is
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most often necessary to perform some theoretical simulation of the strain field. While examples
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of analytical,12 continuum,1, 2, 13-16 and atomistic16, 17 models can be found throughout literature,
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there exist only a few studies1,
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experimental data and even fewer that demonstrate the nanoscale spatial resolution needed to
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probe the strain fields of individual nanowires. This difficulty in comparing theoretical and
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experimental results has made it challenging to validate the accuracy of any model or decide
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which model is most appropriately used for a given situation.
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that compare the resulting simulated strain field with
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In this work, we demonstrate the use of CBED coupled with a continuum finite element
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analysis (FEA) model to enable strain field characterization with both high spatial and strain
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resolution. We focus this study on GaAs-based nanowires that are a widely studied materials
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platform for the development of advanced opto-electronic devices. Its direct, wide bandgap of
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1.42 eV makes it an ideal material candidate for light emitting diode18,
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applications. Furthermore, the ability to alloy GaAs with other materials such as Al, In, and P
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creates a highly flexible and tunable system allowing for the careful selection of specific
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materials properties, including a broadly tunable band gap. Heterostructures composed of GaAs
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and other higher band-gap materials, such as AlGaAs and GaAsP, are of particular importance
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for the fabrication of GaAs-based opto-electronic devices as the high-bandgap material
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passivates surface states, which are known to reduce radiative recombination of electronic
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carriers. The GaAs/GaAsP system presents unique opportunities for strain engineering of
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and solar cell20-22
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materials properties such as band gap23 and carrier mobilities24 while still providing effective
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passivation of surface states.25, 26 With a maximum lattice mismatch of -3.7% with GaAs for pure
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GaP, GaAsP shells have been used to shift the emission from GaAs nanowires by as much as 260
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meV.2 Additionally, while the critical thickness of GaP thin films grown on GaAs is only 2 nm,27
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defect-free GaAs/GaP core-shell nanowires have been realized with shell thicknesses of 25 nm.2
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By analyzing the positions of HOLZ lines in series of CBED patterns obtained from
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individual GaAs/GaAsP core-shell heterostructures, we directly demonstrate the non-uniform
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nature of the strain fields created by the wrap-around heterointerface. We then show that these
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experimental results are consistent with strain field calculations performed by FEA. Furthermore,
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using the calculated strain fields, we simulate series of CBED patterns using a Bloch wave
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method allowing for a direct comparison to our experimental data. The correlation of these two
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sets of data (1) demonstrates the ability of our techniques to provide strain information with
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nanometer spatial resolution and (2) validates the accuracy of our FEA model. These results not
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only highlight the importance of nanoscale strain characterization but lay the ground work for the
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rational design of advanced strain engineered devices.
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GaAs/GaAsP nanowires were grown using a two-step growth process: particle mediated
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vapor-liquid-solid GaAs nanowire core growth followed by the vapor-solid GaAsP shell
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deposition.2,
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Prior to growth, all substrates were cleaned using a standard triple rinse procedure consisting of
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three, 10 min sonicated rinses in acetone, methanol, and deionized (DI) water. Substrates were
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then blown dry using compressed air. Dried substrates were coated with a 1% poly-l-lysine
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aqueous solution for 10 min, rinsed, and dried. Finally, 90 nm Au nanoparticles were drop-cast
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All nanowires were grown on GaAs substrates with a [111]B surface normal.
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on the substrates: small drops of dilute, aqueous solutions of Au nanoparticles were placed on the
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substrate and allowed to rest for 10 min. The substrates were then rinsed and blown dry.
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Prepared substrates were loaded into a horizontal-flow metalorganic chemical vapor
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deposition reactor. Samples were fist annealed at a temperature of 600 oC for 10 min under
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flowing arsine to remove any native oxide on the substrate surface.29 This step also allows the
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Au seed particles to melt and alloy with the substrate surface causing the seed particles to form
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an epitaxial relationship with the substrate, ensuring vertical nanowire growth. After annealing,
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the reactor temperature was lowered to 420 oC for nanowire core growth. GaAs cores were
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grown using a TMGa flow of 16 µmol/min and an arsine flow of 150 µmol/min resulting in a
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V/III ratio of 9. Wires were grown for a total of 10 min achieving an average length of 12 µm.
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Core growth was then suspended by stopping the flow of TMGa, and the reactor temperature was
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increased to 725 °C for shell deposition. TMGa was flown at a rate of 8 µmol/min while AsH3
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and PH3 were flown at rates of 1267 µmol/min and 1234 µmol/min, respectively, resulting in a
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V/III ratio of 331. After a 5 min shell deposition, the samples were cooled to room temperature
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inside the reactor chamber. The flow of group V precursors was continued until below a
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temperature of 350 oC to prevent degradation of the nanowire sidewall surface.
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To prepare nanowires for cross-sectional strain investigation, nanowires on the substrate
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were first flattened and aligned using a mechanical rolling method similar to other approaches
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described in literature.30 Using a scanning electron microscope (SEM) equipped with a focused
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ion beam (FIB), nanowires were then milled into ~100 nm thick cross-sections suitable for TEM
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analysis. Initial milling was done with a high current and accelerating voltage (30 keV) to
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minimize sample preparation time, whereas the final thinning was performed at lower beam
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currents and accelerating voltages (5 keV) to reduce sample damage from the high-energy Ga
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beam and minimize Ga implantation. Additionally nanowires were coated with protective layers
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of amorphous carbon and platinum prior to FIB milling to ensure the milling process did not
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result in the amorphization of the nanowire shell or core.31
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After cross-sectional TEM sample preparation, GaAs/GaAsP nanowires were imaged in a
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JEOL 2010F equipped with an annular dark field detector for scanning TEM (STEM) image
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acquisition and a CCD camera for bright-field (BF) image and CBED pattern acquisition. CBED
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patterns were formed using an accelerating voltage of 200 keV, a condenser aperture size of 100
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µm, and a convergence angle of approximately 30 mrad. Figure 1(a) shows an annular dark-field
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(ADF) STEM image of a typical GaAs/GaAsP nanowire cross-section while the fast-Fourier
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analysis of a corresponding high resolution TEM (not shown) confirms growth along the [111]
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direction with {110} type sidewall facets. Energy dispersive X-ray spectroscopy (EDS) maps of
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As and P (Figure 1(b) – 1(c)) confirm the formation of a GaAsP shell that is approximately 22
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nm thick and indicate an average concentration of 10 at% P in the shell creating a lattice
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mismatch of -0.3% relative to the GaAs core. EDS maps of Ga (see Supporting Information)
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show a uniform distribution across both the core and shell confirming a lack of significant Ga
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implantation during sample preparation by FIB. These EDS maps also indicate regions of higher
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P concentration (30 at%) located at the corners of the shell, likely the result of in-diffusion that
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occurred after shell growth while the samples cooled in a PH3 environment or preferential
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incorporation in corners, which are the most efficient sites for strain relaxation.13 The excess P at
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the corners leads to higher strains and relaxation in this region of the shell as evidenced by the
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bright contrast in the ADF-STEM image.32 Due to the non-uniform nature of the P concentration
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at the corners, we limited our CBED analysis to those patterns obtained far from the corner
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surfaces.
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CBED studies for the determination of strain require the use of a low symmetry zone axis
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to minimize HOLZ line interaction. In the case of nanowires, care must also be taken to choose a
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zone axis that minimizes any projection effects that result as a consequence of tilting away from
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the nanowire growth axis reducing spatial resolution. Using the JEMS software suite,33 we
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identified the [556] zone axis, which lies approximately 5o off the [111] growth direction
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towards [001], as a suitable zone axis for strain study. Figure 2(a) shows a typical example of a
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CBED pattern obtained along the [556] zone axis in the core of a GaAs/GaAsP core-shell
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nanowire containing a number of prominent HOLZ lines with minimal interaction. In particular,
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the {6¯ 42} and {51¯ 3¯ } HOLZ lines maintain a significant contrast across the entire nanowire
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cross-section and were therefore selected for the analysis. Series of patterns were acquired along
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lines radiating from the center of the nanowire cross-section towards the facet surfaces, as shown
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schematically in Figure 2(b), resulting in over 150 patterns obtained from an individual structure.
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Each series was then analyzed to compare and identify trends in HOLZ line splitting and
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shifting. Facets were numbered as indicated in Figure 2(b) with facet 1 at the top and counting
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clockwise.
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Figure 1. (a) ADF-STEM image of a typical GaAs/GaAsP core-shell nanowire cross-section
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with a core diameter of 54 nm and a shell thickness of approximately 22 nm. (b – c) EDS maps
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of As (green) and P (red) signals indicating a uniform distribution of As throughout the core and
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shell (with a slight reduction of intensity in the shell) and the presence of P only in the shell. The
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P EDS map (c) also indicates regions of high P content near the surface of the shell corners.
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As previously mentioned, by observing the shift and width of HOLZ lines, local
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variations in strain can be directly observed. In our previous studies7, HOLZ line broadening was
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seen to arise in samples with strain variations on the order of 10-4, setting the lower limit of
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detectable strain variations. Among all the series of CBED patterns obtained along the facets in
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this study, almost no HOLZ line broadening was seen indicating limited or no deformation at the
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sample surface due to relaxation (see Supporting Information). We can therefore assume that the
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strain fields are below the detection limit along the beam direction (the growth axis). Despite a 8 ACS Paragon Plus Environment
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lack of broadening, shifts in HOLZ line position were observed. HOLZ line shifting is best
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measured by considering the movement of lines of interest relative to other lines. Therefore, a tie
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line was defined between the intersection points of the (46¯ 2) and (51¯ 3¯ ) lines and the (6¯ 42) and
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(1¯ 53¯ ) lines, as shown in Figure 2(a). The tie line length was then calculated and measured
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throughout each series of CBED patterns using the HANSIS software.34 The normalized tie line
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length was then plotted as a function of the distance from the facet surface (Figures 2(c) – 2(f)),
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from which several key similarities and differences were observed. Firstly, the normalized tie
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line length shows two distinct regimes with a transition around 22 nm from the nanowire surface,
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which roughly corresponds to the GaAsP shell thickness. The only exception is facet 4 (Figure
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2(f)) with a transition at approximately 18 nm. Because the core-shell interfaces lie along the
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{110} faces, any tilting away from the growth axis would result in a loss of spatial resolution
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along certain directions blurring the transition from core to shell. Therefore, this difference in the
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transition point could be due to projection effects when tilted to the [556] zone axis or drift of the
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sample during CBED acquisition. In the case of facets 1 and 2 (Figure 2(c) and 2(d),
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respectively), these regimes can be defined by the distinct increase in tie line length, while along
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facets 3 and 4 (Figures 2(e) and 2(f), respectively) the different regimes are characterized by an
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inflection point in the tie line length variation. We note that measurements of the HOLZ line
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shifting from facets 5 and 6 were attempted; however, due to the thicker shell at facet 5 and the
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degradation of the protective carbon layer around the nanowire, the data collected was
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unreliable.
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Figure 2. (a) Experimental CBED pattern obtained along the [556] zone axis in a GaAs/GaAsP
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core-shell nanowire with two pairs of prominent lines labeled – the {6¯ 42} and {51¯ 3¯ } lines. (b)
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ADF-STEM image of GaAs/GaAsP nanowire cross-section showing approximate locations
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where CBED patterns were obtained (colored markers) and scanning directions (white arrows).
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(c) – (f) Plots of the normalized {46¯ 2}/{51¯ 3¯ } tie line length – yellow line in (a) – versus
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distance from the facet surface for facets 1 – 4. Dotted lines are a guide to the eye and shading
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shows approximate transition between core (green) and shell (red).
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While it is challenging to directly extract the magnitude or direction of a strain field from
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these plots, they do point to the non-symmetric nature of the strain fields around the nanowire
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and unique HOLZ line shifting behavior for each facet. This conclusion, at first, seems
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counterintuitive given the three-fold symmetries of the geometry and the zinc-blende crystal
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structure of GaAs and GaAsP, thus motivating us to formulate an FEA model for a quantitative
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analysis that can be compared with the experimental results.
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The standard geometry used for the FEA models was based on the nanowire cross-section
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shown in Figure 2(a): a GaAs core with a radius of 54 nm and a 22 nm thick GaAsP shell (the
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thicker shell along facet 5 was not taken into account). Corners of the hexagonal core and shell
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were filleted (rounded) with a 10 nm radius. The thickness of each model was 75 nm, but a
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symmetric boundary (no normal displacement) on the back plane of the model creates a virtual
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thickness of 150 nm – the thickness of the wire cross section as measured by CBED.
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To preserve the symmetry of the mechanical properties of the core-shell nanowire, the
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GaAs and GaAsP materials were defined as orthotropic materials with cubic symmetry and
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elastic constants E, G, and ν shown in Table 1. Therefore, in the following discussion the axes
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designated as 1, 2, and 3 correspond to the crystallographic directions of [100], [010], and [001],
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respectively. Similarly to other FEA studies,35 materials were also given a pseudo-thermal
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expansion coefficient to create the lattice misfit between the materials. In all cases, GaAs is used
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as the reference and therefore has a coefficient of 0. The elastic constants for GaAs0.9P0.1 were
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calculated using Vegard’s rule and the elastic constants for GaAs36 and GaP37. In addition to the
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two materials of the core-shell heterostructures, a surrounding amorphous carbon material was
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also included to take into account the protective carbon layer surrounding the nanowire. Due to
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its amorphous nature, this material was described as an isotropic material with elastic constants E
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38
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software package.
and ν and no expansion coefficient. All FEA calculations were performed using the ADINA
E (GPa)
G (GPa)
ν
Lattice mismatch
GaAs
85.92
59.6
0.310
0.00
GaAs0.9P0.1
179.86
123.94
0.308
-0.0036
67
NA
0.300
0.00
Material
Amorphous C
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Table 1. Elastic constants and pseudo-coefficients of thermal expansion for materials used in FEA models.
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Figure 3 shows band plots of the normal and shear strain components at the front surface
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of a GaAs/GaAsP core-shell heterostructures surrounded by a carbon coating. These plots
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directly show the low symmetry of the strain fields; it is therefore expected that the combination
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of strain fields along any facet will be unique leading to unique HOLZ line shifting behavior, as
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observed experimentally. While the strain fields might be expected to reflect the same three-fold
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symmetry of the materials system, it must be remembered that the axes along which these strain
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components have been defined correspond to the crystallographic directions [100], [010], and
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[001]. Indeed, if we instead plot the stress or strain using a cylindrical coordinate system aligned
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to the growth axis, much of the symmetry is recovered (see Supporting Information).
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Figure 3. Band plots of the (a) – (c) normal and (d) – (e) shear strain components calculated for
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a core-shell nanowire heterostructures surrounded by a carbon coating
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To directly correlate the strain fields and the experimentally obtained CBED results,
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CBED patterns were simulated based on the strain values extracted from the FEA calculation.
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Local average strain values were first extracted from the FEA model along lines running through 12 ACS Paragon Plus Environment
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the thickness of the model emulating the path an electron beam travels through the sample. These
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virtual line scans radiate from the surface of each facet towards the center of the nanowire
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similar to the collection of CBED linescans shown in Figure 2(b). The normal strain components
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were then used to calculate the lattice parameters a, b, and c using the following equation: ܽ = ܽ + ߝܽ
(6-1)
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where a is the lattice parameter of the average unit cell, a0 is the reference lattice parameter of
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GaAs, and ε is the appropriate average strain value. Similarly, the unit cell angles α, β, and γ
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were found to play a major role in determining the behavior of HOLZ line shifting within the
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core and shell regions and were determined by the following equation using a small angle
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approximation: α=
π − γ′ 2
(6-2)
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where γ’ is the appropriate average shear strain value. The resulting average unit cell was then
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used as the input for the simulation of a CBED pattern. This process was repeated for each
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position along the line scan. Each simulation assumed a sample thickness of 150 nm – as
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experimentally determined – and included 30 strong reflections in the calculation while weak
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reflections were taken into account using a generalized Bethe approximation.39, 40 All simulations
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were performed using the JEMS software package utilizing a Bloch wave method.33 Figure 4(a)
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shows an example of a typical simulated CBED pattern that accurately reproduces most of the
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important features observed in the experimental pattern (Figure 2(a)). The resulting CBED
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patterns were then analyzed in the same manner as their experimental counterparts in which the
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length of the {6¯ 42}/{15¯ 3} tie line was measured and plotted as a function of distance away
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from the facet surface. Figures 4(c) – 4(f) show a comparison of the simulated HOLZ line shift
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plots (black dots) of a few facets to the experimental data (colored markers) presented earlier. 13 ACS Paragon Plus Environment
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Not only does the simulated data exhibit core and shell regions, but the variation in tie line length
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within each region also show good correlation to experimental measurements. Of particular note
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is the comparison of simulated facet 2 with experimental facet 1 (Figure 4(c)): the core and shell
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regions show a remarkable agreement both in magnitude and shape. Furthermore, the first data
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point of the simulated series also suggests that the sharp decrease in tie line length
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experimentally observed at the facet surface is representative of the HOLZ line behavior, and not
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an experimental artifact. In fact, this first experimental data point would seem to indicate that the
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CBED pattern was indeed obtained from the very top 1 nm of material at the facet surface. The
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fact that this sharp decrease is not observed in other experimental plots could be due to loss of
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spatial resolution and projection effects (not suffered by facet 1) caused by tilting, or that not all
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facets exhibit this kind of behavior. It should also be noted that this type of behavior is also not
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present in all of the simulated patterns, as evidenced by simulated facet 4 (Figure 4(d)).
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However, out of all six facets simulated only two – facets 3 and 4 – did not exhibit this initial
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sharp decrease in tie line length at the facet surface. A fuller description of the simulated results
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can be found in the Supporting Information.
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Figure 4. (a) Simulated CBED pattern along the [556] zone axis with important HOLZ lines and
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tie line labeled. (b) Representative schematic of nanowire cross section used in FEA calculations.
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Protective carbon coating is omitted from the image for clarity. (c) – (f) Comparison of simulated
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HOLZ line shift plots (black dots) to experimental HOLZ line shift data (colored markers) for
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some facets.
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Looking at the other facets, we can see similar levels of agreement between simulated
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and experimental plots. Figure 4(e) shows an excellent agreement between simulated and
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experimental facet 3 in both shape and magnitudes. (The high level of noise in the experimental
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data of facet 3 could be due to projection effects; if we assume that facet 1 suffered the least
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from projection effects, geometry would dictate that facets 2 and 3 would suffer the most.)
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Experimental facet 2 presents (Figures 4(d) and (f)) an interesting case in that its overall
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appearance seems to agree with simulated results, however, matching it to a specific simulated
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facet is difficult. While there is general agreement in shape and magnitude of change between
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core and shell regions, behavior within each region is more difficult to match to any of the
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simulated profiles. The best match may be with simulated facet 5, with discrepancies at the core-
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shell interface explained by loss of spatial resolution due to tilting.
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The fact that the physical nanowire cross-section does vary from the FEA model in a few
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important respects must also be considered. First is the non-uniform shell thickness at facet 5.
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The additional shell material would lead to higher levels of stress on this side of the nanowire
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affecting the surrounding facets. This additional stress could be a reason why no exact match
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could be made for experimental facet 4 (Figure 2(f)), which lies right next to facet 5, yet facet 1
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that lies further away can be matched to a simulated profile nearly perfectly. In addition to the
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non-uniform thickness, Figure 1(a) also demonstrates that the protective carbon layer does not
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completely surround the nanowire. In fact, facet 4 is almost completely free of carbon material,
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which has a significant impact on the strain behavior in the shell. Finally, we could also consider
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that the geometry of the core is not fully correct. ADF-STEM images suggest that there may be
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some amount of facet rounding. While the model includes fillets at the corners, the facets are
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assumed to be perfectly flat and sharp at the interface. Creating a more round interface (for
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example, by in diffusion of P during shell growth) would change the strain fields at that
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interface. Simulations of core-shell heterostructures with a circular core (see Supporting
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Information) demonstrate a flattening of the strain fields in the core. This would naturally lead to
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less variation of HOLZ line splitting in the core region. While strain fields in the shell are mostly
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the same, it is observed that the circular core seems to spread out the strain fields in the shell
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instead of keeping them more confined to a particular facet.
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These results show that small changes in the predicted fields lead to significant changes
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in the predicted HOLZ line shifting. In turn, the predicted strain fields are seen to be highly
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dependent on many factors including chemistry, geometry, and local environment. Therefore we
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believe that the matching of experimental and simulated results (Figure 4(c) – 4(f)) demonstrate
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not only the ability of CBED to accurately measure and map the strain fields in geometrically
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complex heterostructures such as the GaAs/GaAsP core-shell nanowires analyzed in this
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investigation but also the validity of our FEA model.
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In this work we have demonstrated how the combination of CBED and continuum
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simulation techniques enables the mapping of strain fields throughout a core-shell nanowire
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heterstructure with nanoscale spatial resolution. By measuring the shift of HOLZ lines as a
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function of position in the GaAs/GaAsP core-shell nanowires, we reveal the non-uniformity of
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the strain fields created by a wrap-around heterointerface. Using FEA calculated strain fields,
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series of simulated CBED patterns were generated allowing for a more direct comparison with
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the experimental data. This comparison not only provides a quantitative picture of the varying
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strain field components, but also a better understanding of what factors affect the strain fields in
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the heterostructures. Additionally, these results show the accuracy and validity of strain field
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calculations by continuum methods such as FEA providing confidence in their future
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applications for strain prediction in nanoscale systems. Although we focus on a particular
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system, our results provide a foundation for understanding the impacts of geometry,
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composition, and surface relaxation on strain fields in core-shell nanowires in general informing
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both future modeling and experimental studies.
325 326
Conflict of interest. The authors declare no competing financial interest.
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Associated Content
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Supporting
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heterostructure, a more detailed analysis of HOLZ line splitting, and further discussion of the
Information
containing
additional
compositional
maps
of the nanowire
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effects of core geometry on strain field calculations. This material is available free of charge via
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the Internet at http://pubs.acs.org.
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Acknowledgements
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S.G. acknowledges the support of the Engineering Research Center Program of the National
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Science Foundation and the Office of Energy Efficiency and Renewable Energy of the
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Department of Energy under NSF Cooperative Agreement No. EEC
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MIT Energy Initiative Shell Seed Fund Program. E.J.J. acknowledges the support of the National
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Science Foundation Graduate Research Fellowship program. The authors would like to
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acknowledge the use of the shared facilities at the Center for Materials Science and Engineering
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which is supported in part by the MRSEC Program of National Science Foundation under award
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number DMR-08-19762.
1041895 as well as of the
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