X‑ray Bragg Ptychography on a Single InGaN/ GaN Core−Shell Nanowire Dmitry Dzhigaev,*,† Tomaš Stankevič,‡,⊥ Zhaoxia Bi,¶ Sergey Lazarev,†,§ Max Rose,† Anatoly Shabalin,†,# Juliane Reinhardt,† Anders Mikkelsen,¶ Lars Samuelson,¶ Gerald Falkenberg,† Robert Feidenhans’l,‡ and Ivan A. Vartanyants*,†,∥ †
Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, D-22607 Hamburg, Germany Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 2100 Copenhagen, Denmark ¶ NanoLund, Department of Physics, Lund University, P.O. Box 118, SE-221 00 Lund, Sweden § National Research Tomsk Polytechnic University (TPU), pr. Lenina 30, 634050 Tomsk, Russia ∥ National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Kashirskoe shosse 31, 115409 Moscow, Russia ‡
S Supporting Information *
ABSTRACT: The future of solid-state lighting can be potentially driven by applications of InGaN/GaN core− shell nanowires. These heterostructures provide the possibility for fine-tuning of functional properties by controlling a strain state between mismatched layers. We present a nondestructive study of a single 400 nm-thick InGaN/GaN core−shell nanowire using two-dimensional (2D) X-ray Bragg ptychography (XBP) with a nanofocused X-ray beam. The XBP reconstruction enabled the determination of a detailed three-dimensional (3D) distribution of the strain in the particular nanowire using a model based on finite element method. We observed the strain induced by the lattice mismatch between the GaN core and InGaN shell to be in the range from −0.1% to 0.15% for an In concentration of 30%. The maximum value of the strain component normal to the facets was concentrated at the transition region between the main part of the nanowire and the GaN tip. In addition, a variation in misfit strain relaxation between the axial growth and in-plane directions was revealed. KEYWORDS: Bragg ptychography, nanowire, GaN, InGaN, Finite Element Method he field of solid-state lighting is one of the rapid growing directions in applications of III−V semiconductor materials and their compounds.1 Nowadays, the best performance in terms of external quantum efficiency (EQE) in the blue range of the visible spectrum is achieved by III-nitride semiconductors. As an example, InGaN/GaN-based light-emitting diodes (LEDs) have a high luminance of 1 × 106 cd/m2, EQE of about 80%, and a lifetime of 1 × 105 h,2,3 exceeding those of other common light sources. A significant improvement in performance of LEDs across the whole visible range is expected with the use of III-nitride based nanowires (NWs). They possess outstanding mechanical and optoelectronic properties because of a high surface-tovolume aspect ratio.4 One of the main advantages of the NWs is their ability to stand mechanical strain without plastic relaxation.5 While the mechanical properties are tightly connected with optoelectronic characteristics, there is a high demand for tools providing information about the inner structure of NWs without destructive sample preparation.
T
© 2017 American Chemical Society
Techniques based on electron diffraction provide a powerful tool for characterization of nanostructures with atomic resolution.6 Scanning electron microscopy (SEM) provides a detailed study of the surface of a NW. Transmission electron microscopy (TEM) resolves the atomic structure of a crystal, however, the low penetration depth of electrons requires a NW to be sliced and prepared out of its as grown environment. X-ray diffraction based methods overcome the limitations of electron microscopy, due to a higher penetration depth of photons. Reciprocal space mapping allows to obtain averaged strain profiles of ensembles of nanostructures.7 Scanning X-ray diffraction microscopy (SXDM)8 provides information about the strain both for ensembles of NWs and for single nanoobjects, however the resolution is limited by the beam size.9−12 In the latter approach, strain distribution is obtained locally by Received: December 5, 2016 Accepted: March 6, 2017 Published: March 6, 2017 6605
DOI: 10.1021/acsnano.6b08122 ACS Nano 2017, 11, 6605−6611
Article
www.acsnano.org
Article
ACS Nano direct evaluation of Bragg peak shifts at each position of the beam on the sample. Bragg coherent X-ray diffraction imaging (CXDI) (for reviews see refs 13 and 14) is based on recording far-field intensity patterns of the coherently scattered radiation and solving a phase problem iteratively.15,16 This method provides local information about the shape and strain of a NW in two and three dimensions (2D and 3D) with the resolution reaching tens of nanometers.17,18 The main advantage of Bragg CXDI is that samples can be measured with a resolution not limited by the beam size, but only by the extent of the scattered radiation. The technique, however, can be only applied on a localized area of a sample, with a coherent beam size limit fulfilling the sampling conditions.14 A coherent scanning microscopy technique called X-ray ptychography19,20 was developed in order to solve the problem with the extended size of samples. In this technique, an object is scanned in the beam with the overlapping adjacent illumination spots, and diffraction patterns are recorded in the far-field. In the case of a crystalline sample, a natural extension of this technique is scanning of the sample in Bragg scattering conditions.21−25 To determine full 3D information about the strain field in the sample, it is necessary to perform a set of ptychography scans at different incidence angles close to Bragg conditions. A number of results of X-ray Bragg ptychography (XBP) applications for the study of strain fields in the nanostructured samples were reported recently.26,27 Here we present results of the 2D XBP performed on a single InGaN/GaN core−shell NW at selected angular positions near the Bragg angle. Based on theoretical analysis, we extract information about 3D distribution of the strain field in the NW by performing the finite element method (FEM) analysis and comparing it with the results of our XBP measurements.
Figure 1. (a) Experimental set up of the XBP measurements. A single NW consisting of the GaN core (red) and the InGaN shell (green) was mounted under the Bragg condition of the 101̅0 GaN peak. The xy plane is parallel to the detector plane, while the scattering plane is yz. The inset shows the normalized dependence of integrated intensity of the GaN peak on the angular position of the sample. (b) Diffraction pattern averaged over all positions of the ptychographical scan at the Bragg angle θ = 8.47°. Regions of the scattered intensity corresponding to different parts of the NW are indicated as A (core), B (shell), and C (rough surface of the shell).
RESULTS AND DISCUSSION Experiment. The scheme of the XBP experiment on a selected NW is shown in Figure 1a (for experimental details see Methods section). Our sample was a wurtzite (WZ) core−shell InGaN/GaN NW (see Figure 2) (for sample description see Methods section). The sample substrate containing an isolated NW at the corner was mounted on a scanning stage horizontally, providing vertical orientation of the NW. Next, a single NW was located and aligned with respect to the beam and rotation axis x using the X-ray fluorescence signal from Ga. The XBP scan was performed with the detector positioned 2.3 m downstream from the sample in Bragg condition corresponding to the 101̅0 GaN Bragg peak with twice the Bragg angle 2θB = 16.94°. A series of 2D ptychographical scans were performed for a set of angular positions θ of the sample shown in the inset of Figure 1a. The angular range of the measurement was ±0.3° with 0.1° increment. At each rotational step an area of 500 × 500 nm2 in the upper part of the NW was scanned in the uv plane (see Figure 1a), resulting in a rectangular raster grid of 11 × 11 points with 50 nm step size. In total, 121 diffraction patterns were recorded with 75 s exposure time per pattern with an incident flux of 6 × 107 photons/s. An example of the averaged diffraction pattern recorded over all scan positions at a single angular position of the sample θ = 8.47° is shown in Figure 1b. It contains signal from different parts of the InGaN/GaN NW. The strongest peak in region A in Figure 1b with diagonal streaks corresponds to the 101̅0 Bragg reflection from the GaN core. The streaks originate from the facets of the tip of the NW. The elongated horizontal streak
in region B in Figure 1b originates from the InGaN/GaN interface and the InGaN shell. Diffuse signal around the InGaN peak (region C in Figure 1b) comes from the rough surface of the InGaN shell. The integrated intensity of the region A in 2D scans as a function of the sample angular position is shown in the inset of Figure 1a. A clear maximum at θ = 8.47° was observed, which indicates that this angular position is close to Bragg angle. Ptychographical reconstruction was performed in 2D with the ePIE algorithm28 for every angular position. Positions of the sample on a scanning grid were corrected for the Bragg angle Δy = Δu cos(2θB), since the reconstruction is evaluated in the detector plane. Random amplitude and phase distributions were used as a starting guess for the object function. The phase offset between 2D data sets for different detuning angles was neglected by bringing the center of mass of the averaged data set to the center of the calculation frame. The beam profile reconstructed from a Siemens star measurement (see Supporting Information) has improved the convergence of the algorithm and provided a stable solution for the object function. It was sufficient to perform 100 iterations, after which the reciprocal error function did not change significantly. 6606
DOI: 10.1021/acsnano.6b08122 ACS Nano 2017, 11, 6605−6611
Article
ACS Nano
X-ray Bragg Ptychography Modeling. Results of 2D XBP (see Figure 3) were compared with the outcome of simulations based on FEM model described in the Methods and Experimental Section. Results of these simulations for the GaN core of the NW are presented in Figure 4. To perform this
Figure 4. Results of the 2D XBP modeling. Amplitudes (a) and phases (b) of the 2D XBP model based on FEM simulations with adjusted detuning angle Δθ = −0.03°, and plastic relaxation parameters ρad = 0.95 and ρcd = 0.5. Scale bars are equal to 100 nm.
Figure 2. Scanning electron microscopy measurements of a single core−shell InGaN/GaN NW. (a and b) Typical surface morphology of the NW under investigation from the top (a) and from the side (b). Cross-sectional SEM images taken at the top (c) and bottom (d) parts of a NW similar to the investigated one.
comparison, the displacement field u(x, y, z) obtained from the FEM model together with the shape function s(x, y, z) of the NW were substituted in eq 10, and integration was performed numerically for different values of Qz(Δθ) as a function of the detuning angle Δθ. Two parameters of the FEM model significantly influence results of our simulations. The first one is the detuning angle Δθ, and the second one is the plastic relaxation parameter ρd. We determined that the amplitude of the object function O(x, y, z) defined in eq 10 (see Methods and Experimental Section) is especially sensitive to the detuning angle Δθ and its phase to the plastic relaxation parameter ρd. Varying the detuning angle and comparing the amplitude of the object function determined from the experiment in Figure 3a and from the model, we observed that the best matching between results for the central slice at θ = 8.47° is obtained at a detuning angle of Δθ = −0.03 ± 0.005°. By this, we concluded that our measurements at the central angular position were performed at a small offset angle from the exact Bragg position. Other angular positions were determined according to this angular shift (see Figure 4). As soon as the phase of the object function φ(x, y, Δθ) defined in eq 10 (see Methods and Experimental Section) is strongly influenced by the parameter ρd, the amount of plastic relaxation in the model was found by adjusting this parameter to fit the phase distribution φ(x, y, Δθ) obtained from XBP, as presented in Figure 3b, while the value of the detuning angle Δθ was fixed. The reconstructed phase map from the central angular position (see Figure 3b) was used for comparison with the FEM model. We determined that the best match between the experiment and model was obtained for the plastic relaxation parameters ρad = 0.95 in a radial growth direction and ρcd = 0.5 in an axial growth direction (see Figure 4b). This is an expected result as soon as an energy of the dislocation formation is proportional to the square modulus of the Burgers vector.29,30 As soon as lattice parameters in a- and c-directions
The amplitude and phase of five 2D XBP reconstructions, corresponding to different angular positions near the Bragg angle of the sample, are shown in Figure 3. As it can be seen from these results, both the amplitude and phase have a stripelike structure with stronger variations at larger offsets from the Bragg angle. In the following, we provide a model of a strained GaN NW that produces similar features.
Figure 3. Results of 2D XBP reconstructions. Amplitudes (a) and phases (b) of the NW reconstruction for five angular positions of the rocking curve scan (see inset in Figure 1a). Scale bars are equal to 100 nm. 6607
DOI: 10.1021/acsnano.6b08122 ACS Nano 2017, 11, 6605−6611
Article
ACS Nano
about 0.15%. This is similar to the results obtained by the SXDM method.11 These results confirm the validity of the parameters used in the FEM analysis as well as the final 3D strain distribution obtained in our model (see Figure 5a).
are different, it leads to different magnitudes of the Burgers vectors of the corresponding misfit dislocations. Due to the geometry of our experiment, we are not sensitive to the strain field component along the c-direction. However, the values of the plastic relaxation in both directions may be retrieved by fitting the parameters of the 3D FEM simulation to the results of the 2D XBP experiment. In this way both components ρad and ρcd are determined from the model, since they influence the 3D strain field distribution in the simulation. Distribution of the phase of the object function at other values of the detuning angle Δθ was obtained by direct calculation of the integral in eq 10 with the obtained values of the plastic relaxation parameters and corresponding angles Δθ (see Figure 4b). A comparison of Figures 3 and 4 shows that our model well represents the main features obtained from the experiment. A positive phase gradient in the tip region of the NW is reproduced (see central angular position in Figures 3b and 4b). The central part of the NW with the alternating phase profile is also fitted well in the simulation. The final 3D model of the strain field in the GaN core of the NW determined from our FEM simulations is presented in Figure 5a. The Cartesian coordinate system x*, y*, z* related to the NW is introduced as the following
CONCLUSIONS In conclusion, we demonstrated the application of a 2D XBP technique to strain imaging in a single heterostructured NW. By fitting the FEM model to 2D XBP results, we obtained the 3D image of the strain distribution within the NW. Adjusted parameters of the model indicated the amount of plastic relaxation ρad = 0.95 for the in-plane and ρcd = 0.5 for the out-ofplane directions. The results of ptychographical reconstructions revealed a complicated profile of the strain distribution that varies in the range of ±1 × 10−3 and was induced by the mismatch between the GaN core and the InGaN shell with In concentration of 30%. Asymmetry in the strain relaxation obtained by 2D XBP and FEM simulations has an influence on the electrical properties of the NWs. Our results for the strain component εy*y* in the GaN core are in good agreement with SXDM measurements performed on the same NW. Together with the complementary studies performed by SXDM, the 2D XBP can provide detailed information about a single NW. The nonuniformity of the strain distribution changes the band gap and, consequently, alters the current flow in a semiconductor device. The results of our work provide important information for the understanding of the relation between structural and optoelectronic properties of semiconductor core−shell NWs with high spatial resolution. We believe that the proposed approach will open possibilities for improved performance of core−shell NW-based devices.
x* = x y* = y cos θB z* = z cos θB
(1)
METHODS AND EXPERIMENTAL SECTION Sample Description. The structure under investigation was a wurtzite (WZ) core−shell InGaN/GaN NW (see Figure 2) grown at Lund University using selective area metal−organic chemical vapor deposition (MOCVD).31 The GaN NWs were grown to be 340 nm thick and 2 μm long on a 1−2 μm-thick GaN template on top of a Si (111) substrate. An array of openings in a Si3N4 mask was used to control the position and size of the NWs. After the axial growth of the GaN NWs, the temperature was lowered to 610 °C in order to grow the InGaN shell. The In concentration in the gas phase was about 30%. The total diameter of the NW from facet to facet estimated by SEM measurements is equal to 390 nm (see Figure 2a). From SEM image shown in Figure 2b, one can see that the surface of the InGaN shell is rough. This surface morphology can be explained by the uneven process of the radial growth of the shell. Misfit dislocations are likely to appear at the interface between core and shell in order to relax the misfit strain. Cross-sectional SEM measurements were performed on the sample grown under similar conditions in order to measure variation of the shell thickness along the NW. Two SEM images show the top (see Figure 2c) and the bottom (see Figure 2d) parts of the NW. The following layers inside the NW can be observed: an axially grown GaN core, a radially grown GaN, the InGaN shell, and a ZnO + AlO layer to support cleaving. The ZnO + AlO layer was not present during the Xray measurements. The contrast between the axially and radially grown GaN may be caused by unintentional doping and hence difference in conductivity. The thickness of the InGaN shell was measured to range from 15 ± 3 nm at the bottom to 29 ± 5 nm at the top of the NW. The sample was prepared for the coherent X-ray diffraction measurements by cleaving a sharp corner of the substrate so that a single NW at the tip would be not obscured by the others. No additional sample preparation was performed. Experimental Setup. The measurement was performed at the nanoprobe end-station of the P06 beamline32 at the PETRA III
Figure 5. Result of FEM simulation. (a) The 3D representation of the FEM simulation result for εy*y* strain component. Here the coordinate axes x*, y*, z* are introduced with respect to the model symmetry axes. Two cuts in y*z* plane at different x* positions are shown to the left. (b) Projected strain ⟨εy*y*⟩z within the modeled NW. This result can be obtained from experimental 2D XBP with Δθ = 0°. Scale bar is equal to 100 nm.
Distribution of the strain field in the cross sections (see Figure 5a) is symmetrical around the axis perpendicular to the reciprocal lattice vector H101̅0 as soon as we are presenting the strain field component calculated along the H101̅0 vector. The maximum strain values εy*y* of about 0.2% are observed in the region between the main NW part and the tip. Below that region, the NW is more relaxed, and the strain values do not exceed ±0.05%. Projection of the strain field obtained in the 3D FEM model on the direction along the scattered beam ⟨εy*y*⟩z is presented in Figure 5b. As we can see from these results, the projected strain reveals two peaks in the region of the NW close to the transition to the tip with the maximum value of the strain of 6608
DOI: 10.1021/acsnano.6b08122 ACS Nano 2017, 11, 6605−6611
Article
ACS Nano synchrotron radiation facility at DESY in Hamburg, Germany. The photon energy of the X-ray beam was 15.25 keV. A sample stage with translation and rotation motors was used to perform scanning measurements. A Vortex EM X-ray fluorescence detector was located close to the sample and perpendicular to the beam. A Pilatus 300k 2D hybrid pixel detector with a pixel size of 172 × 172 μm2 was used to record far-field diffraction patterns. The incident beam was focused using nanofocusing lenses (NFLs) (see Supporting Information for beam characterization). Theory. We consider in the following Bragg scattering on a crystalline sample close to the Bragg reciprocal lattice vector H. We assume also that the kinematical approximation for scattering in the far-field is valid. The scattering amplitude An(Q) from such a sample at a selected position rn of the incidence beam Ein(r) can be described by14,33 A n(Q) =
∫ S(r)Ein(r − rn)e−iQ×rdr
Q z(Δθ) = |H|cos θB × Δθ =
A n(Q x , Q y , Δθ) ≈
∬ ⎡⎣⎢∫ S(x , y , z)e−iQ (Δθ)zdz⎤⎦⎥Ein(x − xn , y − yn ) z
× e−i(Q xx + Q yy)dx dy
(7)
In conventional 2D ptychography, performed in transmission geometry, the sample, described by the complex object function O(x, y), is scanned through the beam, described by the complex probe function P(x, y), with overlapping adjacent positions. For a single position n in the ptychographical measurement, we can write for a scattered amplitude:
Here, Q = q − H, where q = kf − ki is a momentum transfer vector, and ki and kf are the incident and scattered wave vectors. In eq 2, a complex valued function S(r) is introduced as (3)
A n(Q x , Q y , Δθ ) ≈
It describes a sample with the shape function s(r):
∬ O(x , y , Δθ)P(x − xn , y − yn ) × e−i(Q xx + Q yy)dx dy
⎧1 if r ∈ V s(r) = ⎨ ⎩ 0 if r ∉ V
(6)
where λ is the wavelength of the radiation. In the case when the incoming wavefield Ein(x, y, z) is weakly dependent on the z-coordinate, we can use the approximation Ein(x, y, z) ≃ Ein(x, y). It is well satisfied in experiments described in this work. The X-ray beam is generated by the NFLs with a depth of focus on the order of a 100 μm, which is much larger than the size of the nanosamples used in the measurements. With this approximation we obtain from eq 5
(2)
S(r) = s(r)e−iH × u(r)
2π sin 2θB × Δθ λ
(8)
Far-field diffraction intensities for each probe position comprise the data set for ptychography measurements:
(4)
where V is the sample volume, and the displacement of the lattice points from the nominal positions is u(r). A single diffraction pattern measured by the detector corresponds to a slice of the 3D intensity distribution in reciprocal space made by the Ewald sphere (see Figure 6). We adopt in the following the Cartesian
In(Q x , Q y , Δθ ) = |A n(Q x , Q y , Δθ)|2
(9)
The set of N diffraction patterns is then inverted by the ptychographical phase retrieval algorithm, providing information on the projected structure of the sample through the reconstruction of the object function O(x, y). A comparison of eqs 7 and 8 shows that in the case of Bragg ptychography, the probe function is represented, similar to conventional ptychography, by the incoming wavefield P(x, y) = Ein(x, y) and the object function at each detuning angle Δθ by the following complex function:
O(x , y , Δθ ) = |O(x , y , Δθ )|eiφ(x , y , Δθ) =
∫ s(x , y , z)e−iH×u(x ,y ,z)e−iQ (Δθ)zdz z
(10)
This expression shows, first of all, that the object function in most cases is a complex function. When the displacement field in a crystalline sample is equal to zero u(x, y, z) = 0 and the measurements are performed in exact Bragg conditions Qz(Δθ) = 0, it will be an amplitude function representing a projected shape of the crystalline object. Second, this expression shows that both amplitude and phase of the object function have a nontrivial dependence on the shape, strain field, and detuning angle. In the case of exact Bragg position Qz(Δθ) = 0, expression eq 10 is simplified to
Figure 6. Scheme of the diffraction geometry. (a) The incoming beam with the wave vector ki is scattered from hkl planes with the wave vector kf. A rocking curve scan denoted by the angle θ is performed around x axis. The shape function of the NW s(r) is represented by hexagon. (b) The Ewald sphere is constructed on the basis of ki and kf vectors. In the experiment, a coherent scattering intensity is measured around the hkl Bragg point.
O(x , y , Δθ ) = |O(x , y , Δθ )|eiφ(x , y , Δθ) coordinate system with its center aligned with the center of the detector, axes x and y defined in the plane of the detector and axis z aligned along the scattered wavevector kf (see Figure 6a). Reciprocal space coordinates Q = {Qx, Qy, Qz} are introduced accordingly (see Figure 6b). In this coordinate system, eq 2 can be represented in the following form: A n(Q x , Q y , Q z) =
=
(11)
Even at these scattering conditions, the displacement field u(x, y, z) can not be directly determined from the phase of the object function φ(x, y, Δθ) (see ref 34 for a detailed theoretical discussion of these specific scattering conditions). FEM Model. In order to reveal the inner structure of the investigated sample, we performed simulations based on the FEM approach. A 3D shape of the NW was constructed using results of the SEM measurement. The model consisted of a hexagonal shaped GaN core with a diameter of 390 nm from facet to facet and the InGaN shell with a thickness of 25 nm. The pyramidal GaN tip in the model had an inclination of 30°, as determined from the SEM studies. The
∬ ⎡⎣⎢∫ S(x , y , z)Ein(x − xn , y − yn , z)e−iQ zdz⎤⎦⎥ z
× e−i(Q xx + Q yy)dx dy
∫ s(x , y , z)e−iH×u(x ,y ,z)dz
(5)
We note here that Qz is equal to zero if the X-ray incidence angle is tuned to the exact Bragg condition θB. Detuning of the incidence angle θ by Δθ = θ − θB introduces the Qz component which is given by 6609
DOI: 10.1021/acsnano.6b08122 ACS Nano 2017, 11, 6605−6611
Article
ACS Nano shell layer with a thickness of several nanometers was added to the tip, as it is also visible in SEM cross-section in Figure 2c. The simulation was performed in the frame of linear theory of elasticity. The lattice constants obtained from ref 35 were aGaN = 3.189 Å, cGaN = 5.178 Å for wurtzite GaN and aInN = 3.533 Å, cInN = 5.693 Å for InN. Since the GaN has a transversely isotropical structure of the unit cell, the full stiffness matrix is given by five independent components. Elasticity constants Cij of the GaN and InN used in the model are presented in Table 1.
Characterization of the focused X-ray beam by transmission ptychography (PDF)
AUTHOR INFORMATION Corresponding Authors
*E-mail:
[email protected]. *E-mail:
[email protected]. ORCID
Dmitry Dzhigaev: 0000-0001-8398-9480
Table 1. Elastic Constants for the GaN and the InN Used in FEM Simulationsa material
C11, [GPa]
C12, [GPa]
C13, [GPa]
C33, [GPa]
C44, [GPa]
GaN InN
374 237
138 106
101 85
395 236
98 53
a
Present Addresses ⊥
MAX IV Laboratory, Lund University, Box 118, SE-221 00 Lund, Sweden # Department of Physics, University of California San Diego, 9500 Gilman Dr., La Jolla, California 92093, United States
Ref 36.
Notes
The authors declare no competing financial interest. The shell of the NW is an alloy of the InN and GaN. The parameters of a crystal scale linearly according to the Vegard’s law with the indium concentration x in the shell:36−38
pInxGa1 − xN = xpInN + (1 − x)pGaN
ACKNOWLEDGMENTS This work was supported by European project 280773 “Nanowires for solid state lighting” (NWs4LIGHT), Danish National Research Council DANSCATT, and the Virtual Institute VH-VI-403 of the Helmholtz Association. The authors are thankful to E. Weckert for fruitful discussions and support and V. Kaganer and D. Novikov for a careful reading of the manuscript and important comments.
(12)
where parameter p is either lattice constants a and c or elastic constants Cij. For our sample, the In concentration value was set during the sample preparation stage to be 30% (x = 0.3). This value was also supported by the location of the satellite peak corresponding to InGaN shell (see Figure 1b). The mechanism of elastic and plastic strain relaxations was appears at the simulated in the FEM model. A misfit strain εmisfit ij interface between the core and shell as the result of lattices mismatch. It can be expressed as εijmisfit =
akInxGa1 − xN − akGaN akGaN
⎛ a InN ⎞ = x·⎜ kGaN − 1⎟ ⎝ ak ⎠
REFERENCES (1) Crawford, M. H. Leds for Solid-State Lighting: Performance Challenges and Recent Advances. IEEE J. Sel. Top. Quantum Electron. 2009, 15, 1028−1040. (2) Jiang, H. X.; Lin, J. Y. Nitride Micro-LEDs and Beyond - a Decade Progress Review. Opt. Express 2013, 21, A475−A484. (3) Narukawa, Y.; Ichikawa, M.; Sanga, D.; Sano, M.; Mukai, T. White Light Emitting Diodes with Super-High Luminous Efficacy. J. Phys. D: Appl. Phys. 2010, 43, 354002. (4) Yan, R.; Gargas, D.; Yang, P. Nanowire Photonics. Nat. Photonics 2009, 3, 569−576. (5) Glas, F. Critical Dimensions for the Plastic Relaxation of Strained Axial Heterostructures in Free-Standing Nanowires. Phys. Rev. B: Condens. Matter Mater. Phys. 2006, 74, 121302. (6) Dasgupta, N. P.; Sun, J.; Liu, C.; Brittman, S.; Andrews, S. C.; Lim, J.; Gao, H.; Yan, R.; Yang, P. 25th Anniversary Article: Semiconductor Nanowires-Synthesis, Characterization, and Applications. Adv. Mater. 2014, 26, 2137−2184. (7) Stankevič, T.; Mickevičius, S.; Schou Nielsen, M.; Kryliouk, O.; Ciechonski, R.; Vescovi, G.; Bi, Z.; Mikkelsen, A.; Samuelson, L.; Gundlach, C.; Feidenhans’l, R. Measurement of Strain in InGaN/GaN Nanowires and Nanopyramids. J. Appl. Crystallogr. 2015, 48, 344−349. (8) Mocuta, C.; Stangl, J.; Mundboth, K.; Metzger, T.; Bauer, G.; Vartanyants, I.; Schmidbauer, M.; Boeck, T. Beyond the Ensemble Average: X-ray Microdiffraction Analysis of Single Sige Islands. Phys. Rev. B: Condens. Matter Mater. Phys. 2008, 77, 245425. (9) Kriegner, D.; Persson, J.; Etzelstorfer, T.; Jacobsson, D.; Wallentin, J.; Wagner, J.; Deppert, K.; Borgström, M.; Stangl, J. Structural Investigation of GaInP Nanowires Using X-ray Diffraction. Thin Solid Films 2013, 543, 100−105. (10) Stankevič, T.; Hilner, E.; Seiboth, F.; Ciechonski, R.; Vescovi, G.; Kryliouk, O.; Johansson, U.; Samuelson, L.; Wellenreuther, G.; Falkenberg, G.; Feidenhans’l, R.; Mikkelsen, A. Fast Strain Mapping of Nanowire Light-Emitting Diodes Using Nanofo- cused X-ray Beams. ACS Nano 2015, 9, 6978−6984. (11) Stankevič, T.; Dzhigaev, D.; Bi, Z.; Rose, M.; Shabalin, A.; Reinhardt, J.; Mikkelsen, A.; Samuelson, L.; Falkenberg, G.; Vartanyants, I. A.; Feidenhans’l, R. Strain Mapping in an InGaN/
(13)
Due to significant thickness of the epitaxial shell, misfit dislocations play a major role in strain relaxation process. Following theoretical predictions,39−41 the critical thickness of the shell in core−shell NWs does not exceed 20 nm for the core with a diameter of 340 nm. Therefore, the relaxation of misfit strain via dislocations has to be considered. The plastic relaxation can be taken into account as an effective reduction of εmisfit ij εijplastic = ρd εijmisfit
(14)
Here, the control parameter is ρd∈[0; 1]. It is related to the relaxation of the strain in the shell through formation of misfit dislocations in the way that ρd = 0 is a fully coherent elastic strain between the core and shell, and ρd = 1 is the fully relaxed strain in the shell without straining the core. As soon as the presence of stacking faults, as well as crystal twinning, is not expected in GaN NWs, the parameter ρd depends only on crystallographic directions. A value of ρd close to unity was expected due to the large amount of defects in the shell. Finally, the general expression for the strain at the interface was defined as follows εij = εijmisfit − εijplastic = (1 − ρd )εijmisfit
(15)
This relation was used as a boundary condition for determining a strain in the GaN core in the FEM model.
ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsnano.6b08122. Strain εy*y* slices along X direction (MPG) Strain εy*y* slices along Y direction (MPG) Strain εy*y* slices along Z direction (MPG) 6610
DOI: 10.1021/acsnano.6b08122 ACS Nano 2017, 11, 6605−6611
Article
ACS Nano GaN Nanowire Using a Nano-Focused X-ray Beam. Appl. Phys. Lett. 2015, 107, 103101. (12) Stankevič, T.; Dzhigaev, D.; Bi, Z.; Rose, M.; Shabalin, A.; Reinhardt, J.; Mikkelsen, A.; Samuelson, L.; Falkenberg, G.; Vartanyants, I. A.; Feidenhans’l, R. Nanofocused X-ray Beams Applied for Mapping Strain in Core-Shell Nanowires. Proc. SPIE 2015, 9592, 95920D−95920D-7. (13) Nugent, K. A. Coherent Methods in the X-ray Sciences. Adv. Phys. 2010, 59, 1−99. (14) Vartanyants, I.; Yefanov, O. In X-ray Diffraction. Modern Experimental Techniques; Seeck, O. H., Murphy, B. M., Eds.; Pan Stanford Publishing: Singapore, 2015; Chapter 12, pp 341−384. (15) Fienup, J. R. Phase Retrieval Algorithms: A Comparison. Appl. Opt. 1982, 21, 2758−2769. (16) Marchesini, S. A Unified Evaluation of Iterative Projection Algorithms for Phase Retrieval. Rev. Sci. Instrum. 2007, 78, 049901. (17) Favre-Nicolin, V.; Eymery, J.; Koester, R.; Gentile, P. CoherentDiffraction Imaging of Single Nanowires of Diameter 95 Nanometers. Phys. Rev. B: Condens. Matter Mater. Phys. 2009, 79, 195401. (18) Dzhigaev, D.; Shabalin, A.; Stankevič, T.; Lorenz, U.; Kurta, R. P.; Seiboth, F.; Wallentin, J.; Singer, A.; Lazarev, S.; Yefanov, O. M.; Borgström, M.; Strikhanov, M. N.; Samuelson, L.; Falkenberg, G.; Schroer, C. G.; Mikkelsen, A.; Feidenhans’l, R.; Vartanyants, I. A. Bragg Coherent X-ray Diffractive Imaging of a Single Indium Phosphide Nanowire. J. Opt. 2016, 18, 064007. (19) Rodenburg, J.; Hurst, A.; Cullis, A.; Dobson, B.; Pfeiffer, F.; Bunk, O.; David, C.; Jefimovs, K.; Johnson, I. Hard-X-ray Lensless Imaging of Extended Objects. Phys. Rev. Lett. 2007, 98, 034801. (20) Thibault, P.; Dierolf, M.; Menzel, A.; Bunk, O.; David, C.; Pfeiffer, F. High-Resolution Scanning X-ray Diffraction Microscopy. Science 2008, 321, 379−382. (21) Godard, P.; Carbone, G.; Allain, M.; Mastropietro, F.; Chen, G.; Capello, L.; Diaz, A.; Metzger, T.; Stangl, J.; Chamard, V. ThreeDimensional High-Resolution Quantitative Microscopy of Extended Crystals. Nat. Commun. 2011, 2, 568. (22) Hruszkewycz, S.; Holt, M.; Murray, C.; Bruley, J.; Holt, J.; Tripathi, A.; Shpyrko, O.; McNulty, I.; Highland, M.; Fuoss, P. Quantitative Nanoscale Imaging of Lattice Distortions in Epitaxial Semiconductor Heterostructures Using Nanofocused X-ray Bragg Projection Ptychography. Nano Lett. 2012, 12, 5148−5154. (23) Hruszkewycz, S.; Holt, M.; Murray, C.; Bruley, J.; Holt, J.; Tripathi, A.; Shpyrko, O.; McNulty, I.; Highland, M.; Fuoss, P. Quantitative Nanoscale Imaging of Lattice Distortions in Epitaxial Semiconductor Heterostructures Using Nanofocused X-ray Bragg Projection Ptychography. Nano Lett. 2012, 12, 5148−5154. (24) Takahashi, Y.; Suzuki, A.; Furutaku, S.; Yamauchi, K.; Kohmura, Y.; Ishikawa, T. Bragg X-ray Ptychography of a Silicon Crystal: Visualization of the Dislocation Strain Field and the Production of a Vortex Beam. Phys. Rev. B: Condens. Matter Mater. Phys. 2013, 87, 121201. (25) Holt, M. V.; Hruszkewycz, S. O.; Murray, C. E.; Holt, J. R.; Paskiewicz, D. M.; Fuoss, P. H. Strain Imaging of Nanoscale Semiconductor Heterostructures with X-ray Bragg Projection Ptychography. Phys. Rev. Lett. 2014, 112, 165502. (26) Chamard, V.; Allain, M.; Godard, P.; Talneau, A.; Patriarche, G.; Burghammer, M. Strain in a Silicon-on-Insulator Nanostructure Revealed by 3D X-ray Bragg Ptychogra- phy. Sci. Rep. 2015, 5, 9827. (27) Hruszkewycz, S.; Allain, M.; Holt, M.; Murray, C.; Holt, J.; Fuoss, P.; Chamard, V. High-resolution Three-Dimensional Structural Microscopy by Single-Angle Bragg Pty- chography. Nat. Mater. 2016, 16, 244−251. (28) Maiden, A. M.; Rodenburg, J. M. An Improved Ptychographical Phase Retrieval Algorithm for Diffractive Imaging. Ultramicroscopy 2009, 109, 1256−1262. (29) Hull, R.; Bean, J. C. Misfit Dislocations in Lattice-Mismatched Epitaxial Films. Crit. Rev. Solid State Mater. Sci. 1992, 17, 507−546. (30) Jain, S.; Harker, A.; Cowley, R. Misfit Strain and Misfit Dislocations in Lattice Mismatched Epitaxial Layers and Other Systems. Philos. Mag. A 1997, 75, 1461−1515.
(31) Hersee, S. D.; Sun, X.; Wang, X. The Controlled Growth of GaN Nanowires. Nano Lett. 2006, 6, 1808−1811. (32) Schroer, C. G.; Boye, P.; Feldkamp, J. M.; Patommel, J.; Samberg, D.; Schropp, A.; Schwab, A.; Stephan, S.; Falkenberg, G.; Wellenreuther, G.; Reimers, N. Hard X-ray Nanoprobe at Beamline P06 at PETRA III. Nucl. Instrum. Methods Phys. Res., Sect. A 2010, 616, 93−97. (33) Vartanyants, I.; Robinson, I. Partial Coherence Effects on the Imaging of Small Crystals Using Coherent X-ray Diffraction. J. Phys.: Condens. Matter 2001, 13, 10593. (34) Dzhigaev, D.; Stankevič, T.; Besedin, I.; Lazarev, S.; Shabalin, A.; Strikhanov, M. N.; Feidenhans’l, R.; Vartanyants, I. A. Theoretical Analysis of the Strain Mapping in a Single Core-Shell Nanowire by Xray Bragg Ptychography. Proc. SPIE 2015, 9592, 95920S. (35) Levinshtein, M. E.; Rumyantsev, S. L.; Shur, M. S. Properties of Advanced Semiconductor Materials: GaN, AIN, InN, BN, SiC, SiGe; John Wiley & Sons: New York, 2001. (36) Morales, F.; González, D.; Lozano, J.; García, R.; HauguthFrank, S.; Lebedev, V.; Cimalla, V.; Ambacher, O. Determination of the Composition of InxGa1−xN from Strain Measurements. Acta Mater. 2009, 57, 5681−5692. (37) Caetano, C.; Teles, L. K.; Marques, M.; Dal Pino, A.; Ferreira, L. G. Phase Stability, Chemical Bonds, and Gap Bowing of InxGa1−xN Alloys: Comparison between Cubic and Wurtzite Structures. Phys. Rev. B: Condens. Matter Mater. Phys. 2006, 74, 045215. (38) Kehagias, T.; Dimitrakopulos, G. P.; Becker, P.; Kioseoglou, J.; Furtmayr, F.; Koukoula, T.; Häusler, I.; Chernikov, A.; Chatterjee, S.; Karakostas, T.; Solowan, H.-M.; Schwarz, U. T.; Eickhoff, M.; Komninou, P. Nanostructure and Strain in InGaN/GaN Superlattices Grown in GaN Nanowires. Nanotechnology 2013, 24, 435702. (39) Raychaudhuri, S.; Yu, E. Critical Dimensions in Coherently Strained Coaxial Nanowire Heterostructures. J. Appl. Phys. 2006, 99, 114308. (40) Chu, H.; Zhou, C.; Wang, J.; Beyerlein, I. An Analytical Model for the Critical Shell Thickness in Core/Shell Nanowires Based on Crystallographic Slip. J. Mech. Phys. Solids 2013, 61, 2147−2160. (41) Yan, X.; Fan, S.; Zhang, X.; Ren, X. Analysis of Critical Dimensions for Nanowire Core-Multishell Heterostructures. Nanoscale Res. Lett. 2015, 10, 389.
6611
DOI: 10.1021/acsnano.6b08122 ACS Nano 2017, 11, 6605−6611
