g′0000 =−g0002 and g′0002 = 0 ). Regarding that according to equations (3.35) and (3.36)
∇kχ|0 = 0 and χ( 0 ) = 0, the wave function in the image plane can be written as ΨI(r) = a0000(r − ξ)eip0000(r−ξ)eiχ(−g0002)e2πi g0002·r +
+a0002(r)eip0002(r)e2πi g0002·R(r) , (3.72) with the abbreviation ξ := 2π1 ∇kχ|−g0002. From this formula, the image intensity can be calculated and is given by
|ΨI(r)|2 = a20000(r − ξ) +a20002(r) + 2a0000(r − ξ)a0002(r)·
·cos
p0000(r − ξ)−p0002(r) +χ(−g0002) + 2π g0002 · r −2π g0002· R(r)
. (3.73) The periodicity dm0002 measured from the image intensity is thus given by the argument of the cosine and amounts to
1 dm0002 =
= 1
2π ∇rp0000(r − ξ)− ∇rp0002(r)− ∇r 2π g0002· R(r)
· ˆe0002+g0002 (3.74)
= 1
2π ∇rp0000(r − ξ)− ∇rp0002(r)− ∇rG0002(r)
· ˆe0002+g0002 (3.75)
=: g0002− 1
2π∇rP0002(r)· ˆe0002 . (3.76)
The measured lattice distance
dm0002 = 1
g0002−2π1 ∇rP0002(r)· ˆe0002 (3.77) is only equivalent to the real 0002 lattice spacing (compare with equation (3.55)) if
∇rP0002(r)· eˆ0002 = ∇rG0002(r)· ˆe0002 , (3.78) i. e. if the gradients of the phases of the 0000 and 0002 beams vanish. As the phases depend on the In concentration and on the thickness of the specimen, this condition is not fullfiled if local variations of the In concentration or of the thickness are present.
For InGaN embedded in GaN, the thickness can be assumed to be constant over com-parable large specimen areas. Therefore specimen orientations are used for imaging where the variation of phases of the used beams with change of In concentration are minimal.
For uncapped InGaN island samples, the influence of the change in thickness from bottom to top of the islands is discussed separately.
Optimised imaging conditions
To ensure that only a minimum of artefacts is present in the measurement of e. g. 0002 lattice fringes, imaging conditions are needed where the gradients of the phases of the transmitted beam and the beam corresponding to the {0002} lattice planes are minimal.
Assuming the sample thickness to be constant in the analysed area, the variation of the phases with respect to the In concentration has to be minimal for a chosen excitation condition. In analogy to [59, 60] these variations of the phases are represented in the following by the standard deviation as calculated for variation of the In concentration. To evaluate the phase changes, Bloch wave calculations were performed for orientations close to theh1120i [59,60] andh1100iZA. The sample was assumed to be tilted by≈6◦ around h0001i, which ensures that the 000lsystematic row is excited. This corresponds to a centre of Laue circle (COLC) (see appendix A.5) of 12 12 0 0 for the h1120i ZA and 7 7 14 0 for the h1100iZA. An additional tilt around an axis perpendicular to h0001i is applied which changes the excitation error of the 000l systematic row of reflections. For specimen tilt between a COLC of h k i3 (corresponding to a strong excitation of the 0006 beam) and h k i3 the logarithm of the standard deviation log(σx) of the phases of the 0000, 0002 and 0002 beams were calculated and the In concentration was varied between 0.00 and 1.00 in steps of 0.01. The results are depicted in figure 3.10 and 3.11 a tod for orientations close to the h1100i and h1120i ZA, respectively.
In figures 3.10 and 3.11 e and f, the standard deviation σt for variation of the (not known) sample thickness from 0 nm to 50 nm in steps of 0.5 nm as calculated from fig-ures 3.10 and 3.11 a to d is shown, i. e. the standard deviation of log(σx). For both ZA orientations, σt of the 0000 beam is small for|L|between 1.5 and 3.0. Nevertheless, σt for the 0002 beam (L < 0) does only show a narrow minimum for L = −2.6 where also σt
of the 0000 beam is low. For L > 0, σt of the 0002 beam has a broad minimum between L= 2.0 and L= 2.6. The broad minimum ensures a minimum of artefacts also for small mistilts around the chosen COLC, which can be present especially at very thin sample areas.
Usually, the In concentration is expected to be well below 1.00. For luminescence in the blue spectral range, an In concentration of about 0.20 is intended. Considering that concentration fluctuations can be present, the concentration range of x up to e. g. 0.40 is of stronger interest. The corresponding evaluations of the Bloch wave calculations for variation of the In concentration between 0.00 and 0.40 are shown in figures 3.12 and 3.13 for orientations close to h1100i andh1120iZA, respectively. It can be seen by comparison with figures 3.10 and 3.11, thatσtof the 0002 beam has an even more pronounced minimum for L between 2 and 2.4, where σt of the 0000 beam is also low. For strain state analysis, images were taken with L≈ 2.2. For this condition, minimal errors of the evaluation are expected.
Furthermore, the described imaging conditions can be obtained fast in comparison
to exact ZA orientations. This is necessary as there are reports in the literature that local lattice distances in InGaN structures change during electron beam irradiation in a TEM already after an exposure of few minutes [61, 62]. These changes are interpreted as redistribution and clustering of In in the InGaN area. The redistribution of In in InGaN QW structures was also observed and analysed by Li et al. [63]. Their experiments were performed using a TEM equipped with a field emission gun (FEG) operated at 200 kV.
They observed only minimum changes in the local In concentration for exposure times up to one minute. Based on their results, they recommend exposure times of one minute or less. In comparison to the CM20 UT microscope operated at 200 kV and equipped with a LaB6 filament, as used in this work for analysis of InGaN structures, a FEG has a higher current density. Therefore, electron illumination times of less than one minute are estimated to sufficiently ensure that the measured lattice spacings are not changed due to electron beam damage. Further analysis concerning electron beam induced damage of InN and GaN can be found in the references [64–68].
Elastic relaxation in the (0001) plane of the InGaN material cannot be accounted for if only the {0002} lattice planes are imaged. If an InGaN island on GaN is to be analysed, which does not contain misfit dislocations, this island can be assumed to be fully strained directly at the InGaN/GaN interface and gradually relax to the top of the island. This relaxation influences the localclattice parameter and thus the calculated In concentration value. In order to gain access to this change of relaxation in growth direction, i. e. to improve the accuracy of the concentration determination, a lattice distance perpendicular to{0001}needs to be measured. The lattice plane distance d1100 = 0.27618 nm of{1100} planes is comparable to d0002 = 0.25925 nm, and can be well resolved in the microscope CM20 UT, which was mainly used for the present work. Thus, the ZA is h1120i. Again, Bloch wave calculations were performed to determine appropriate imaging conditions for 1100 fringe images, analogous to the 0002 fringe imaging conditions. In this case, an excitation of thehh00 systematic row, obtained for a tilt of≈6◦aroundh1100i, corresponds to a COLC of 0 0 0 22 (or equivalently 0 0 0 22). The excitation error of thehh00 systematic row was changed corresponding to a tilt between a COLC of 0 0 0 22 and 3 3 0 22. The logarithm of the standard deviation log(σx) of the phases of the 1100 and the 0000 beams was calculated for variaton of x between 0.00 and 0.40 in steps of 0.01 and is shown in figure 3.14 a and b, respectively. As the {1100} planes are related to each other via a mirror operation, the results are the same for e. g. the 1100 beam. From these two maps, the standard deviationσtfor variation of the sample thickness from 0 nm to 50 nm in steps of 0.5 nm was calculated and is displayed in figure 3.14 c. In this case, either a COLC in the range between 0.5 0.5 0 22 and 0.5 0.5 0 22 (due to the symmetry) or between 0.6 0.6 0 22 and 1.5 1.5 0 22 minimises σt for p1100 and p0000 simultaneously. The latter offers an even more pronounced minimum. Three further calculations of σt for concentration variation between 0.00 and 0.60 (figure 3.14 d), 0.80 (figure 3.14e), and 1.00 (figure 3.14f) exhibit the increase of σt at COLC of about 1 1 0 22 if the concentration variation exceeds 0.80.
Thus, for imaging of 1100 fringes, an excitation corresponding to a COLC of 1.1 1.1 0 22 is chosen if x is surely below 0.80. For larger differences of x, a COLC of 0 0 0 22 has to be used.
It has to be kept in mind, that imaging of 0002 and additional of 1100 fringes requires a second orientation and electron beam exposure of the specimen area of interest. Therefore, special care has to be taken that the electron beam damage is minimised.
a) b)
c) d)
e) f)
Figure 3.10: atodLogarithm of the standard deviation of the phases log(σx) for beam tilt of ≈ 6◦ from the h1100i ZA orientation in dependence of sample thickness and excitation condition. The scalebar is given below the greyscale maps. The maps are obtained from Bloch wave calculations for variation of x between 0.00 and 1.00. a log(σx) of the 0002 and c0000 beam for variation of the excitation corresponding to a COLC between 7 7 14 0 and 7 7 14 3. b log(σx) of the 0002 and d 0000 beam for variation of the COLC between 7 7 14 0 and 7 7 14 3. The standard deviation σt of log(σx) in dependence of the excitation for variation of the specimen thickness from 0 nm to 50 nm, calculated from figures a to d, is shown in e for the 0000 and 0002 beams and in f for the 0000 and 0002 beams.
a) b)
c) d)
e) f)
Figure 3.11: atodLogarithm of the standard deviation log(σx) of the phases for beam tilt of ≈ 6◦ from the h1120i ZA orientation in dependence of sample thickness and excitation condition. The scalebar is given below the greyscale maps. The maps are obtained from Bloch wave calculations for variation of x between 0.00 and 1.00. a log(σx) of the 0002 and c 0000 beam for variation of the COLC between 12 12 0 0 and 12 12 0 3. b log(σx) of the 0002 and d 0000 beam for variation of the COLC between 12 12 0 0 and 12 12 0 3. The standard deviation σt of log(σx) in dependence of the excitation, calculated for variation of the specimen thickness from 0 nm to 50 nm from figures a to d, is shown in e for the 0000 and 0002 beams and in ffor the 0000 and 0002 beams.
a) b)
c) d)
e) f)
Figure 3.12: atodLogarithm of the standard deviation log(σx) of the phases for beam tilt of ≈ 6◦ from the h1100i ZA orientation in dependence of sample thickness and excitation condition. The scalebar is given below the greyscale maps. The maps are obtained from Bloch wave calculations for variation of the In concentration between 0.00 and 0.40. a log(σx) of the 0002 and c 0000 beam for variation of the excitation between COLC of 7 7 14 0 and 7 7 14 3. b log(σx) of the 0002 and d 0000 beam for variation of the excitation between COLC of 7 7 14 0 and 7 7 14 3. The standard deviationσt of log(σx) in dependence of the excitation, calculated for variation of the specimen thickness from 0 nm to 50 nm from figures a to d, is shown in e for the 0000 and 0002 beams and in ffor the 0000 and 0002 beams.
a) b)
c) d)
e) f)
Figure 3.13: atodLogarithm of the standard deviation log(σx) of the phases for beam tilt of ≈ 6◦ from the h1120i ZA orientation in dependence of sample thickness and excitation condition. The scalebar is given below the greyscale maps. The maps are obtained from Bloch wave calculations for variation of the In concentration between 0.00 and 0.40. a log(σx) of the 0002 and c 0000 beam for variation of the excitation between COLC of 12 12 0 0 and 12 12 0 3. b log(σx) of the 0002 and d 0000 beam for variation of the exci-tation between COLC of 12 12 0 0 and 12 12 0 3. The standard deviation σt of log(σx) in dependence of the excitation, calculated for variation of the specimen thickness from 0 nm to 50 nm from figures a to d, is shown in e for the 0000 and 0002 beams and in ffor the 0000 and 0002 beams.
d) a)
e) b)
c) f)
Figure 3.14: Optimised imaging conditions for 1100 fringe images. a and b show the logarithm of the standard deviation log(σx) of the phases for beam tilt of ≈ 6◦ from the h1120i ZA orientation in dependence of sample thickness and excitation condition. The scalebar is given below the greyscale maps. The maps are obtained from Bloch wave calculations for variation of the In concentration x between 0.00 and 0.40. a log(σx) of the 1100 and b 0000 beam for variation of the excitation between COLC of 0 0 0 22 and 3 3 0 22. The standard deviation σt of log(σx) in dependence of the excitation, calculated for variation of the specimen thickness from 0 nm to 50 nm from figuresa andb, is shown in c for the 0000 and 1100 beams. The calculation was repeated for variation of x up to 0.60 (d), 0.80 (e), and 1.00 (f).
Errors due to thickness variation
For e. g. cross section specimens of uncapped InGaN islands, the thickness of the island with respect to the electron beam direction will decrease from bottom to top of the island. The phases of the used beams and thus the measured 0002 lattice plane spacings are influenced by the thickness gradient. Therefore the In concentration derived by strain state analysis will show an additional error. Assume a sample of [0001] growth direction, which shows a thickness change of e. g. the In0.2Ga0.8N islands in electron beam direction corresponding to a wedge of 90◦. The absolute error in the derived In concentration x calculated from equation (3.77) is shown in dependence of the used COLC and specimen thickness t in figure 3.15 a and b as greyscale maps for the imaging conditions close to the h1100i and h1120i ZA, respectively. The phases p0000 and p0002 were calculated by the Bloch wave approach. For COLC hkiL with 2.0≤ L ≤2.4, i. e. for excitation close to the optimised imaging conditions, the absolute error ∆xof the measured In concentration in dependence of the specimen thickness is given in figure 3.15 c and d. For specimens with islands of thickness t ≤16 nm at their basis, ∆x approximately decreases with decreasing specimen thickness for both orientations. This implies an apparent decrease from x+ ∆x(t) at the base to x+ ∆x(0 nm) at the top of the islands. As becomes obvious from figure 3.15 c and d, the error is ∆x≤0.05. The calculation was repeated for constant In concentration x in the islands between 0.0 and 1.0 in steps of 0.1. The maximum position of ∆x shifts from specimen thicknesst≈17 nm tot≈10 nm. For a decrease of the specimen thickness (i. e. t≤(17. . .10) nm), the absolute error of x is still ∆x≤0.05.
a) b)
c) d)
Figure 3.15: Absolute error of the In concentration x caused by a thickness variation cor-responding to a 90◦ wedge of In0.2Ga0.8N for excitation corresponding to a COLC between a 7 7 14 0 and 7 7 14 3 (≈ h1100i ZA) andb 12 12 0 0 and 12 12 0 3 (≈ h1120iZA). aand b show the absolute error of the In concentration. The scalebar is given below the greyscale maps. For different COLC close to the optimum excitation of COLC hkiL with L = 2.2, the absolute error is shown in dependence of specimen thickness inc close to ah1100i ZA and in d close to a h1120i ZA.
Lattice plane bending
A thin TEM specimen can relax elastically along the electron beam direction (figure 3.16).
Due to this relaxation the strain components describing shear are not zero anymore, as
Figure 3.16: A thin TEM specimen can relax elastically along the electron beam direction.
Due to this relaxation, bending of the lattice planes in the InGaN can occur.
was assumed for the derivation of the In concentration in section 3.3.2 (p. 39). Rosenauer et al. [59] used finite element (FE) modelling to simulate a 5 nm thick TEM specimen of an InGaN QW sample with In concentration of 0.20, where lattice plane bending was taken into account. This model was used for HRTEM image simulations with the optimised imaging conditions for 0002 fringe images. The evaluation of the simulated HRTEM images in dependence of the defocus showed only a minor error in the determination of the lattice distances, corresponding to an absolute error in In concentration of less than 0.02. Usually the thickness of the TEM specimen and hence the degree of relaxationr (defined on p. 40) is not known, this would result for the same example in an error of already close to 0.05.
As the contribution to the error due to the unknown thickness, i. e. unknown relaxation of the sample is larger, this value was considered as approximate error for the derivation of the In concentration for this work.
Error due to uncertainties of the elastic constants
One source of error in the derivation of x are the not well known elastic constants Cij. The values given by different authors vary considerably. This holds for GaN [11, 69–75] as well as InN [11, 73–75]. For the assumption of unstrained InGaN, the measured c lattice parameter and the In concentration are related according to Vegard’s rule. As soon as the material is strained, the elastic constants have to be considered for the determination of x. For InN no reliable experimentally obtained elastic constants are reported. In case of GaN, the experimental values reported in references [70–72] are in good agreement with the elastic constants given by Wright [11], which are used throughout this work. The elastic constants for InN are taken from [11] as well. The In concentration x in dependence of the c lattice parameter for the assumption of completely strained InGaN, i. e. completely strained to (0001) GaN, is displayed in figure 3.17 for different sets of elastic constants
as given by [11, 73–75]. In comparison with the curves obtained for Cij of Wright [11], the maximum relative error of x is obtained for the elastic constants given by Azuhata et al. [73] and amounts to 8% in case of r = 0 and 25% in case of r = 1. Typically, the relative errors of the In concentration are less than 5%. The systematic error due to the not well known Cij is not further considered for the evaluated samples in this work.
0.53 0.55 0.57 0.59 0.61
c lattice spacing [nm]
0 0.2 0.4 0.6 0.8 1.0
x
cGaN c
InN
unstrained InxGa1-xN Kim et al., r = 1 Kim et al., r = 0 Chisholm et al., r = 1 Chisholm et al., r = 0 Azuhata et al., r = 1 Azuhata et al., r = 0 Wright, r = 1 Wright, r = 0
Figure 3.17: In concentration x in dependence of the c lattice spacing for thick TEM specimens (r = 1), thin TEM specimens (r = 0) and totally relaxed InxGa1−xN. The results for different elastic constants as given by Kim et al. [74], Chisholm et al. [75], Azuhata et al. [73], and Wright [11] are displayed.