Elucidation of the strain state of the GaN QDs and Finite Element analysis

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Figure 6.11: (a) Cross-sectional WB-DF CTEM image along the [1123]z.a. obtained from the 10 ML sample under g/3g conditions using g 0002. The inclined TD can be seen clearly coming from the template and originate at the AlN/sapphire interface. The arrows denote such TDs and in between a row of stacked QDs is shown under bright contrast. (b) Cross-sectional HRTEM image along [1123]z.a. obtained from the 5 ML sample showing a region that includes two TDs indicated by dotted lines. This region comprises {1011} facets on which the QDs are nucleated.

Also the white arrows indicate the wetting layer which is clearly resolved.²¹

6.2.3 Elucidation of the strain state of the GaN QDs and Finite Element

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Two masks were employed (g/3 and g/4) that correspond to spatial resolution of strain field determination of 0.71 nm and 0.95 nm respectively. Figure 6.12(b) illustrates the generated strain map of the lattice component along the growth direction using g/3 mask size and having the AlN as reference, meaning that as reference is employed the reduced relative variation of the interplanar spacing (dGaN-dAlN)/dAlN. A depression on the strain at the QD’s upper region is noted. Figure 6.12(c) is the diffractogram where the used spatial frequencies are denoted, and Figure 6.12(d) is the strain profile taken from the central part of the QD with an integration width of 6 nm. The average value of the strain was 3.4% while the best accuracy of strain determination was achieved with the g/4 mask and was found 0.4%.

The determination of the three-dimensional elastic strain state of the QDs is ex- tremely important in order to understand the piezoelectric effects and evaluate the built-in polarization potential and the charge carrier separation.^29,30 Due to the complex shape of the QDs the biaxial stress state cannot be easily approximated, since deviations exist caused by the presence of multiple facets and the small volume size of the dots.^31,32 Moreover, since the thickness of the TEM foil is usually smaller than the thickness of the QDs (an electron transparent TEM sample is needed), the thickness of the TEM foil must be taken into account in order to achieve accurate qHRTEM measurements. Finite ele- ment (FE) analysis then can be employed in order to correctly take into account the strain effects. In this work, continuum-based calculations were performed for the semi- polar QDs and the results were compared to the qHRTEM experimental observations.

The FE analysis was performed by G. Jurczak, T.D. Young and P. Dłużewski from the In- stitute of Fundamental Technological research of the Polish Academy of Sciences in Po- land.

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Figure 6.12: (a) Cross-sectional HRTEM image of a QD along [1123] z.a. (b) Corresponding lat- tice strain map. (c) FFT of the HRTEM image in (a) in which the two spatial frequencies that were used for the generation of the strain map are denoted. (d) Lattice strain profile along the growth direction obtained from the central area of the QD which is denoted in (b). ²¹

Two are the primary fields that affect the optoelectronic properties of the QDs.

The first one is the elastic displacement field ui, that appears due to the lattice mismatch in a heterostructure, which leads to the appearance of stress σij. The second one is the electric displacement field Di, which leads to the appearance of the built-in electric po- tential V. To calculate the elastic-electric problem the following coupled equation set must be solved:³³

( ^{lt ch})

ij Cijkl kl kl eijk kV

σ ==== ε −−−−ε −−−− ϑ (6.1)

( ^{lt ch}) ^spont

i ijk jk jk ij j i

D ====e ε −−−−ε ++++µ ϑV++++P (6.2)

where i, j, k, l=1,2,3; Cijkl is the fourth-order tensor of stiffness moduli, eijk is the third- order piezoelectric tensor, µij is the second-order diagonal dielectric tensor, and P_i^spont is the spontaneous polarization moduli arranged in a vector form. The lattice strain tensor

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ε^lt is composed of an elastic part ε^e (ε^e=ε^lt−ε^ch) and a chemical part ε^ch. The latter corre- sponds to the lattice mismatch and is purely diagonal ⁽ⁱ≡≡^≡≡^j):

( ) /

ch GaN AlN AlN

ii ai ai ai

ε ==== −−−− (6.3)

where a1=ax=a, a2=ay=a, and a3=az=c. a and c are the lattice parameters of the wurtzite structure. Material properties of wurtzite GaN and AlN crystals were taken from Ref.

[34].

The Equations (6.1) and (6.2) were solved in a semi-coupled manner, meaning that the inverse piezoelectric coupling was ignored in Equation (6.1) by setting e=0.

This coupling is minor and does not affect our main results for stress-strain distribution.

The implemented scheme is equivalent to solving at the first step the elastic Equation (6.1). Then, for the resultant strain/stress distribution, the electric Equation (6.2) is solved. Nonlinear elasticity theory based on the use of the logarithmic strain measure was implemented.³⁵ Boundary conditions were determined under the scheme of Mul- tipoint Constraints in order to take into account the influence of the neighboring QDs.

Under this approach, the external faces of the FE model are allowed to move normal to their surface to accommodate elastic relaxation in the QD and its surroundings. The elastic relaxation of the GaN QD causes a trivial expansion of AlN matrix in addition to the local interaction between QD and matrix near interfaces. Tensor transformation law was implemented in order to determine the tensor elements and the induced elastic- electric field for the semipolar orientation at hand, considering their projection in a ro- tated coordinate system.

The motivation for using the FE method is the easiness with which the arbitrary geometries can be constructed with the space grid. The 3D FE grid that describes the geometry of the rectangular-based (1122) -nucleated QD is given in Figure 6.13(a).

Here, the QD with lengths of base edges equal to 20 nm along [1100], and 18 nm along [1123] while the QD height was 3.5 nm. After relaxation, a nonuniform elastic strain dis- tribution was observed in the QD, exhibiting highest strain values at the junctions be- tween the base and the side facets as shown in Figure 6.13(b) and (c).

Depression of the QD strain was found close to the upper (1122) facet at the junctions with the side facets, which is consistent with the experimental observations as shown in Figure 6.12. However, strain concentration was not identified experimentally at the QD base corners, which could be attributed to their being rounded.

Spontaneous polarization and elastic-electric coupling govern electron–hole con- finement in non-polar QDs,²⁹ and the were found equally important in semipolar QDs. A homogeneous potential distribution was calculated along the [1123] projection, due to the QD symmetry [Figure 6.13(d)]. However, a significant localization of positive and negative components of the electrostatic potential was predicted along [1100]as shown

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in Figure 6.13(e). This spatial distribution was affected by the orientations of the side facets. The positive area was well defined, whereas the negative one was spread. For the positive side, the (1126) facet is inclined at 62° to the polar c-axis. On the other hand, the potential peaks negatively on the (1120) facet which has the polar c-axis in-plane, leading to weaker localization.

Figure 6.13:(a) the FE mesh is shown superimposed on the proposed (1122) QD geometry. The capping layer has been removed for illustrative purposes only. (b) and (c) show the elastic strain distribution presented in 2D contour plots f a QD cross-section through the xz- and yz- planes respectively. (d) and (e) are the corresponding distributions of the electrostatic poten- tial (in volts) and are presented in 2D projected contour plots.²¹

Figure 6.14 illustrates line profiles of the elastic and lattice strain, as well as the polarization potential, obtained by cutting through the QD central region along the growth direction. The elastic strain, depicted in Figure 6.14 (a), exhibits small expansion in the growth direction, while it is compressive along x and y. The lattice strain, ex- pressed with respect to relaxed AlN, was much higher along the growth direction than along the other ones (Figure 6.14(b). Furthermore, it was larger than nominal (2.9%) along the growth direction. In particular, 3.7% was calculated at the QD center. We also considered the TEM foil thickness effect, by truncating the FE model to 6 nm thickness along [1123] . This resulted in a decrease of the strain at the QD center by 10% on aver-

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age. The final result is in good agreement with the qHRTEM measurement of Figure 6.12.

As shown in Figure 6.14(c), the potential drop in the growth direction, across the QD center, defined as the difference between the maximum and minimum values of the polarization potential, was 0.2 V, corresponding to an electric field of 0.4 MV/cm. The maximum potential drop was along [1100], reaching 1.3 V (corresponding to electric field of 1.5 MV/cm).

Figure 6.14: Quantification of the results of the FE analysis for a cross section through the cen- ter of the QD on the yz-plane (i.e., [1100] . In (a) and (b), the elastic and lattice normal strain components (εii) are given as a function of position along the growth z-axis. In (c), the variation in the electrostatic potential with position along the z-axis is given.²¹