a) b)
Figure A.2: Schematic to explain the COLC. a shows an Ewald sphere intersecting a two dimensional reciprocal lattice with reciprocal basis vectors a∗1 and a∗2. The two dimensional crystal is oriented in a low indexed ZA. The reciprocal lattice does not consist of points but of reciprocal lattice rods (relrods) due to the finite size of the TEM specimen.
The incident electron beam with wave vector k0 and one scattered wave vector k is shown.
The Ewald sphere intersects relrods of the zero order Laue zone (ZOLZ) (lower row), and one relrod of the first order Laue zone (FOLZ) (upper row), giving rise to reflections gh0
and g71 in the diffraction pattern with intensities as schematically indicated by the size of the black circles at the bottom of the image (not all reflections are labeled). In b the crystal is tilted. Now the Ewald sphere intersects the relrods corresponding to g00 and g40 in their centres, i. e. the excitation error is zero in both cases. In three dimensions, the intersection of the Ewald sphere and the ZOLZ is described by a circle, whose centre is called centre of Laue circle (COLC). The COLC corresponding to this two dimensional example is g20 and is indicated by the cross in b.
Fourier transform, Dirac’s delta function, and convolution
The Fourier transform of a function f is defined as Fr→kf(r) = f˜(k) =
Z
r
e2πi k·rf(r)d r . (B.1)
The inverse Fourier transform is given by F−k1→rf˜(k) =
Z
k
e−2πi k·rf(˜ k)d k . (B.2)
Hence it is obtained
F−k1→rFr→kf(r) = f(r) . (B.3)
The convolution of a function f(r) with a function g(r) is given by f(r)⊗g(r) =
Z
r′
f(r′)g(r − r′)d r′ . (B.4)
The following relations are found:
multiplication theorem Fr→k[f(r)g(r)] = [Fr→kf(r)]⊗[Fr→kg(r)] (B.5)
and (B.6)
convolution theorem Fr→k[f(r)⊗g(r)] = [Fr→kf(r)][Fr→kg(r)] . (B.7) 141
The Dirac δ function is defined by its properties
δ(r − r0) = 0 for r 6= r0 and (B.8)
r2
Z
r1
δ(r′− r0)f(r′)d r′ = f(r0) for r0 ∈[r1, r2] . (B.9)
Important relations concerning the δ-function are
Fr→kδ(r) = 1 , (B.10)
Fr→kδ(r − r0) = e2πi k·r0 , (B.11)
F−k1→rδ(k − k0) = e−2πi k0·r , (B.12)
δ(a r) = 1
|a|δ(r) , and (B.13)
f(r)δ(r − r0) = f(r0)δ(r − r0) . (B.14)
Fit functions
The used fit functions with fit parameters ai and bi for the averaged projected displace-ments and normalised local distances were chosen as they result in a good reproduction of the experimental values. There is no physical model for them. For fitting the projected displacementsu in [0002] (fringe numbern), the following fit function was used (InGaN in area n > a0):
u =
−(a1+ 0.5a2(a3 ∗(a0−n)+
+a3(a0−n)erf(−a3(a0−n))−e−a23(a0−n)2π1 for n ≤ a0
−(a1+ 0.5a2(a3 ∗(a0−n)+
+a3(a0−n)erf(−a3(a0−n))−e−a23(a0−n)2π1 + +a4(a0−n)2+a5(a0−n)3+a6e−a7(a0−n)+a8
for n > a0
. (C.1)
For the fit of the normalised local (1100) distances, the fit function was d1100(x)
d1100(0) =
b1erf(b2(n−b0)) for n ≤ b0
b1erf(b2(n−b0)) +b3+a4eb5(n−b0) for n > b0 . (C.2)
143
List of island samples
To study the InxGa1−xN islands, various samples were analysed for this work. The im-portant growth parameters are given in chapter 5, as soon as the samples are introduced.
Nevertheless, all island samples are listed here to allow a better overview to the reader.
For the “InxGa1−xN sample series with QD like PL”, schematic sketches are displayed.
Detailed information about the individual samples can be found in chapter 5. In the fol-lowing, Tg denotes the growth temperature of the InxGa1−xN or of the GaN cap layer and tg is the duration of InGaN growth. In addition, the nominal GaN cap layer thicknesses are given for the MBE samples.
Analysed island samples grown by MBE
sample Tg(InxGa1−xN ) tg(InxGa1−xN ) In+InGa Tg(GaN, cap) nom. cap layer
no. [◦C] [s] flux ratio [◦C] thickness [nm]
n0921 450 90 0.5 − − − − − −
n0944 450 110 0.5 450 40
n1018 510 70 0.7 − − − − − −
n1023 510 70 0.7 510 2
n1021 510 70 0.7 510 8
The samples n0921 and n0944, where the InxGa1−xN was grown at 450◦C, are refered to as
“low temperature series”. MBE samples grown at 510◦C belong to the “high temperature series”.
145
Samples with geometric InGaN islands grown by MOVPE sample Tg(InxGa1−xN ) tg(InxGa1−xN ) In+InGa Tg(GaN, cap) sample
no. [◦C] [s] ratio [◦C] series
g0509-6 600 100 0.848 − − −
g0538 780 75 0.848 − − − C1
g0540 780 250 0.848 − − −
g0622 600 22 0.736 − − −
g0623 600 33 0.736 − − − C2
g0652 600 17 0.582 − − −
g0653 600 35 0.582 − − −
g0654 600 52 0.582 − − − C3
g0690 600 52 0.582 820
MOVPE grown InxGa1−xN sample series with QD like PL
g0770 g0768
a) b)
g0769 g0764
c) d)
For the InxGa1−xN sample series with QD like PL, the hatched areas in the above figure correspond to material grown at Tg = 700◦C, whereas the GaN cap layers marked in grey are grown at 820◦C. For each sample the sample number is given above the sketch.
The nominal In concentrations and layer thicknesses are displayed. The two different InxGa1−xN layers were grown using an In+InGa ratio of 0.736 and 0.218, resulting in nominal In concentration of 0.4 and 0.1, respectively.
2.1 Wurtzite and sphalerite unit cells . . . 6 2.2 Derivation of binodal and spinodal curves from Gibbs free energy G . . . . 9 2.3 Binodal and spinodal curves for InxGa1−xN . . . 10 2.4 Bandgap energies of InxGa1−xN . . . 12 2.5 Band alignment and influence of piezoelectric effect . . . 12 2.6 Density of states for bulk material, QW and QD . . . 13 2.7 MOVPE reactor . . . 15 2.8 MBE chamber . . . 16 2.9 Growth modes of epitaxial films . . . 17 3.1 Electron scattering at a single atom . . . 21 3.2 Atomic electron scattering amplitudes . . . 23 3.3 Electron scattering at a periodic lattice . . . 24 3.4 Normalised beam intensities for GaN . . . 28 3.5 Image formation . . . 30 3.6 Aberration functionχ(k) . . . 32 3.7 Effect of spatial and temporal incoherence envelope functions . . . 35 3.8 Projected displacements . . . 36 3.9 Indium concentration in dependence of c lattice spacing for different states
of strain relaxation . . . 41 3.10 Optimum imaging condition for 0002 fringe images (h1100i ZA orientation,
0.00≤x≤1.00) . . . 47 3.11 Optimum imaging condition for 0002 fringe images (h1120i ZA orientation,
0.00≤x≤1.00) . . . 48 147
3.12 Optimum imaging condition for 0002 fringe images (h1100i ZA orientation, 0.00≤x≤0.40) . . . 49 3.13 Optimum imaging condition for 0002 fringe images (h1120i ZA orientation,
0.00≤x≤0.40) . . . 50 3.14 Optimum imaging condition for 1100 fringe images . . . 51 3.15 Absolute error of x due to specimen thickness variations . . . 53 3.16 Elastic relaxation of a thin TEM specimen . . . 54 3.17 Indium concentration in dependence of clattice spacing for different sets of
elastic constants . . . 55 3.18 Intensity difference between 0002 and 0002 beams of wurtzite GaN . . . . 58 4.1 Schematic of a pyramidal defect . . . 60 4.2 Antibixbyite structure . . . 61 4.3 STEM images of PDs . . . 63 4.4 EDS of a PD . . . 64 4.5 HRTEM image of a single PD . . . 65 4.6 Diffractogram of amorphous material . . . 66 4.7 Simulated HRTEM images in comparison to the basal plane IDB of a PD . 68 5.1 AFM and SEM images of InGaN surfaces before and after etching . . . 83 5.2 HRTEM image of a droplet . . . 84 5.3 EDS linescan of a droplet . . . 84 5.4 Topography of sample n0921 . . . 87 5.5 TEM overview of n0921 . . . 87 5.6 Islands types of n0921 . . . 88 5.7 Averaged projected displacement of a wurtzite island of n0921 . . . 89 5.8 0002 WBDF image of sample n0944 . . . 90 5.9 Analysis of high resultion images of n0944 . . . 91 5.10 Experimental segregation profile of n0944 . . . 92 5.11 Strained epitaxial layer . . . 94 5.12 Topography of sample n1018 . . . 97 5.13 TEM overview of n1018 . . . 97
5.14 Z-contrast image of n1018 . . . 98 5.15 TEM overview image of sample n1023 . . . 98 5.16 Averaged projected displacements of n1023 in between islands . . . 99 5.17 HRTEM image and map of strain of sample n1021 . . . 100 5.18 Z-contrast image of n1021 . . . 101 5.19 SEM image of sample g0509-6 . . . 104 5.20 0002 WBDF image of g0509-6 . . . 105 5.21 SAD pattern of g0509-6 . . . 105 5.22 0002 fringe image of sample g0622 . . . 107 5.23 Averaged projected displacements along [0001] in the centre of one island of
g0622 . . . 107 5.24 HRTEM image in h1120i ZA orientation of sample g0623 . . . 109 5.25 In concentration evaluation from 0002 fringe image for g0623 . . . 109 5.26 SEM image of sample g0652 . . . 112 5.27 HRTEM overview image of g0652 . . . 112 5.28 SEM image of sample g0653 . . . 113 5.29 HRTEM overview image of g0653 . . . 113 5.30 SEM image of sample g0654 . . . 114 5.31 HRTEM overview image of g0654 . . . 114 5.32 Bright field TEM image of sample g0690 . . . 115 5.33 Comparison of the C3 sample series . . . 116 5.34 Schematic structure of sample g0770 . . . 118 5.35 Cross sectional TEM overview image of sample g0770 . . . 119 5.36 In concentration profile and Z-contrast image of g0770 . . . 120 5.37 Schematic structure of samples g0768, g0769, and g0764 . . . 120 5.38 Cross sectional TEM overview image of sample g0768 . . . 121 5.39 TEM overview images of g0769 and g0764 . . . 121 5.40 Averaged In concentration profiles of sample g0769 and g0764 . . . 122 5.41 In concentration of one island of g0622 . . . 126 5.42 Focus variation image reconstruction of g0654 . . . 128
5.43 In concentration map for an island of g0654 . . . 129 5.44 In concentration profile along [0001] within one island of g0654 . . . 130 A.1 Planes in a hexagonal unit cell . . . 139 A.2 Centre of Laue circle . . . 140
2.1 Material parameters . . . 7 2.2 Precursors for MOVPE . . . 14 3.1 Defocus values for delocalisation free two beam imaging . . . 33 5.1 Island samples grown with MBE . . . 86 5.2 Samples with geometric InGaN islands grown by MOVPE . . . 104 5.3 In concentration measured along [0001] within the centre of islands of g0622 108 5.4 WL thickness of uncapped samples of the C3 series . . . 116 5.5 Muraki fit parameters for g0769 and g0764 . . . 122 A.1 Physical constants . . . 137 A.2 Vacuum classification . . . 137 A.3 Notation for directions, planes, and reflections . . . 138
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