Supporting Information for:
Inversion Domain Boundaries in GaN Nanowires Revealed by Coherent Bragg Imaging.
Stéphane Labat1,*, Marie-Ingrid Richard1,2, Maxime Dupraz3,4, Marc Gailhanou1, Guillaume Beutier3,4, Marc Verdier3,4, Francesca Mastropietro1, Thomas W. Cornelius1, Tobias U. Schülli2, Joel Eymery5,6,*, Olivier Thomas1 *Corresponding authors’ email: stephane.labat@univ-‐amu.fr,
[email protected] 1
Aix Marseille Univ., CNRS, Univ. Toulon, IM2NP UMR 7334, F-13397 Marseille, France 2
3
Univ. Grenoble Alpes, SIMAP, F-38000 Grenoble, France 4
5
6
ID01 ESRF, F-38043 Grenoble, France
CNRS, SIMAP, F-38000 Grenoble, France
Univ. Grenoble Alpes, F-38000 Grenoble, France
CEA, INAC-SP2M, "Nanophysique et semiconducteurs" group, F-38000 Grenoble, France
Table of Content 1: Reconstruction process 2: Additional measurements on other nanowires 3: Molecular statics simulations
Supporting Information 1: Reconstruction process The data treatment of the (004) Bragg peak will be detailed as an example of the procedure for sample reconstruction. The error metric that quantifies the matching between the retrieved intensity 𝐴 𝑞 ²!"#!$"%"& , and the measured intensity 𝐼 𝑞
𝐸𝑟 =
𝐴 𝑞
!"#$%&"' is chosen as:
!"#!$"%"&
𝐼 𝑞
− 𝐼 𝑞 !"#$%&"'
!"#$%&"'
²
The measured intensity consists in a regular array of 512x256 values or pixels. A large square support of 90x90 pixels is used as a primary support in the Shrink-‐Wrap procedure. After 5 cycles of 50 ER followed by 100 HIO, a shrinking procedure is used to decrease the size of the support: we remove pixels from the edge of the support with a threshold of 25 % of the average modulus. A new iteration is done with the updated support (5 cycles of 50 ER followed by 100 HIO) ended with the same shrinking procedure. This shrinking procedure slows down around the 140th iteration as shown by the drawing of the support size as a function of the iteration number (Fig. S1). This corresponds to the iteration for which the support starts including non-‐physical solution for our samples in terms of voids. The shrinking procedure is stopped. Then, ten supports with a slight variation of 10-‐pixels size from the one retrieved after 140 iterations are used with 20 cycles of 50 ER + 100 HIO. For each support, one hundred initial random phase sets are associated to the measured intensity and give one hundred different reconstructions. The noise of the measured intensity precludes discriminating the best candidate from the error metric criterion alone (Fig. S2). Indeed, for intensities ranging between 0 and 105, a Poisson distribution noise involves 10-‐3 error bar on the metric values. Thus the 82 reconstructed intensities of Fig. S2 giving an error metric between 4.5 10-‐3 and 5.5 10-‐3 match the measured intensity in the same way. To discriminate these solutions, the standard deviation of the modulus maps is used. Samples with the best homogeneous modulus map and similar metric errors are retained.
Figure S1 ⏐ Support size determination. The Shrinking procedure used a threshold of 25 % of the average value of the modulus. The boxed reconstructed object is chosen as a support for the following procedure using ER and HIO only.
0.016
Error metric
0.014 0.012 0.010 0.008 0.006 0.004
20
25
30
35
40
H L
45
Relative modulus standard deviation %
50
Figure S2 ⏐ Error metric and modulus map homogeneity for hundred samples reconstructed from 004 Bragg peak. More than 80 % of the results give the same metric error, but have significant differences in reconstructed modulus homogeneity. From the phase maps, one evidences the phase shift between domain 1 and 2 (see Fig. S3). The 12 best reconstructions in terms of modulus homogeneity are used to calculate the average phase shift and the error bar of this value is estimated from the difference between the maximum and minimum value divided by 2.
Average phase difference radian
H L
- 2.6 Φ1 -‐ Φ2-‐3= -‐2.80 ± 0.05 radians
- 2.7 - 2.8 - 2.9 - 3.0
ΔΦexpected for IDB*= -‐3.07 radians
- 3.1 - 3.2
0.10
0.15
0.20
0.25
H L
0.30
Standard deviation on the phase values of the largest domain radian Figure S3⏐Phase shift between domain 1 and 2-‐3 (Φ 1 -‐ Φ 2-‐3 ) and phase map homogeneity for 82 reconstructed samples from 004 Bragg peak. The 12 best reconstructions in terms of modulus homogeneity are reported in purple.
Supporting Information 2 : additional measurements on other nanowires The 004 reflection has been measured on several nanowires. The reconstructed real-‐space images show a variety of domain configurations, but always the same phase shift between +c and -‐c domains, equal to -‐2.8 radians, with a flat phase inside each domain. This observation suggests that the displacement of (c/2+8) pm across the IDB is an intrinsic property of the material and independent of the domain configuration.
Figure S4⏐Reconstruction of two nanowires from their 004 Bragg reflections. Reconstructed modulus a-‐b and phase c-‐d from 004 Bragg peaks measured for two different nanowires. The nanowires present different domain structures but the same phase shift between the +c and –c domains.
Supporting Information 3 :Molecular statics simulations Introduction Atomistic simulations were carried out in order to investigate the mutual interactions between several IDBs*. Because of the very large number of atoms involved in a realistic configuration, ab initio calculations cannot be performed and we decided to carry out molecular statics calculations using a Tersoff-‐Brenner empirical potential. As shown by W.H. Moon et al.,2 the Tersoff-‐Brenner potential is able to reproduce quite well the IDB* formation energy determined from first principles by Northrup et al.1 The Tersoff-‐ Brenner potential parameters of Nord et al. were used 3 and it was verified that the lattice parameters (a= 3.1809 Å, c = 5.1944 Å) and cohesion energy ( -‐ 4.528 eV/atom ) were within the precision of those shown in table 3 of their publication. The c/a ratio is found very close to the ideal wurtzite axial ratio because only first neighbour interactions are considered. However experimental values of c/a are also close to this value.4
Single infinite planar IDB First, a single ideal planar IDB* configuration was studied. For that purpose a GaN box with periodic boundary conditions (PBC) in the directions [001] and [210] was used. In the direction [010] perpendicular to the IDB* surface no PBC was used because it would have constrained the system. As this introduces two surfaces perpendicular to this direction, the length of the box was taken large enough to be able to separate surface relaxation effects from the IDB* relaxation.
A formation energy of 39 meV/Å2 for IDB* was obtained, in the same range as the value of 25 meV/Å2 obtained from first principles calculations by Northrup et al.1 After relaxation the displacements induced by the defect in the direction perpendicular to the IDB plane was found equal to 9.3 pm, close to the value of 10 pm deduced from the bond lengths indicated by Northrup et al.1 Along the c-‐axis, the Ga terminated crystal on one side of the defect is, after relaxation, translated by (c/2 -‐ 1.3 pm) with respect to the N terminated crystal on the other side. The Simulations show also that the strain is localized in the 2-‐3 atomic planes on both sides of the boundaries.
Multiple IDBs in a nanowire This simulation was carried out with a nanowire size and geometry, and an IDB configuration as close as possible to the experimental ones. Periodic boundary conditions were applied along the nanowire axis corresponding to the +c or -‐c axis for the GaN crystal. The simulation box contained around 1.4 107atoms. In the case of a complex configuration of IDBs such as the one encountered in the GaN wire, a complicated displacement field is expected, as each of the two domains with a Ga-‐ terminated surface (labelled 2 and 3, see Fig. 3b) is separated from the main N-‐terminated domain (labelled 1) by two IDB planes with a different orientation. Molecular statics was used to relax the structure and determine this displacement field. The uz component along the c-‐axis predicted by our atomistic simulations (about -‐1 pm) is much smaller than the experimental value of +8 pm obtained experimentally from the 004 reflection, which is independent from the in plane (ux, uy) displacement field. For five reflections, the displacement induced phase shift was added to the structure factor and the resulting phase maps are shown in Fig. S5. Because of the displacement field, there is a significant difference
-‐ as high as 0.3 radian -‐ between the domains 2 and 3, in particular for the hkl reflections with h≠0, which is also not observed experimentally. 1. Northrup, J.E., Neugebauer, J. &Romano, L.T. Inversion Domain and Stacking Mismatch Boundaries in GaN.Phys Rev. Lett.77, 103-‐106 (1996). 2. Moon, W.H. &Choi, C.H. Molecular-‐dynamics study of inversion domain boundary in w-‐ GaN.Physics Letters A 352, 538–542, (2006). 3. Nord, J., Albe, K., Erhart, P. &Nordlund, K. Modelling of compound semiconductors : analytical bond-‐order potential for gallium, nitrogren and gallium nitride. J. Phys. :Condens. Mater. 15, 5649–5662 (2003). 4. Paszkowicz, W., Podsiadło, S., Minikayev, R. Rietveld-‐refinement study of aluminium and gallium nitrides, Journal of Alloys and Compounds382,100–106 (2004).
FIGURE S5⏐ Calculated phase maps of the GaN cross-‐section for different reflections. The maps include the effect of the structure factor, of a c/2 displacement along z of the N terminated regions and of the additional displacement field.
