1.2 The band gap is shown as function of the lattice parameters. The range of visible spectrum is shown in colors.
1.3 Schema of GaN based HEMT structure. The bi-dimensional electron gas (2DEG) is along the interface between the AlGaN and GaN layers.
1.4 Directions and planes in a hexagonal lattice.
1.5 A 3D picture of the wurtzite structure together with some important projections [12].
1.6 The standard deviation σY as function of the fraction x for the three different systems.
1.7 A 3D picture of the diamond structure together with some important projections [12]. Red lines represent the chemical bonds and the black lines are cell lines.
1.8 The islands A and B separated by a pure tilt boundary represented by the angle ζ and a pure twist boundary represented by the angle χ.
1.9 Changing dislocation type along a single dislocation line with Burgers vector b: screw-type (A), mixed-type (B) and edge-type (C) [80].
2.1 Edge dislocation in a cubic crystal.
2.2 Stress distribution on an infinitesimal volume element.
2.3 Displacement of P to P’ by the displacement vector u.
2.4 (a) Pure shear and (b) simple shear of an area element in the xy plane.
3.1 The dislocation is formed by an offset (defined by the Burgers vector b) of one side S4 of the slip plane δ with respect to the other side S2. The surface S3 encloses the dislocation core region. h denotes the film thickness.
3.2 The core surface S3 in a cylindrical coordinate system.
3.3 The dislocation lies along the y-axis.
3.4 In the hexagonal (and more generally, in non-isotropic crystal) two angles, α and β, are needed for description of the dislocation line direction.
3.5 Pre-logarithmic coefficient K calculated as a function of the angles α and β for c-type dislocations (Burgers vector b = [0001]) in AlN , in GaN , in InN , for a-type dislocations (Burgers vector b = 1∕3 [110]) in AlN , in GaN , in InN , for (a+c) -type dislocations (Burgers vector b = 1∕3 [23]) in AlN , in GaN , in InN .
3.6 The z-axis is perpendicular to the c-plane for the hexagonal symmetry and to the closed packed plane for the cubic one. The dislocation line lies along the y-axis in both cases.
3.7 Dislocation geometry in a hexagonal crystal.
3.8 Some important crystallographic planes of the hexagonal system.
4.1 The straight infinitely long dislocation at the film–substrate interface. The film thickness is h, the slip plane is tilted by an angle ϕ from the normal to the interface.
4.2 The z-axis is perpendicular to the c-plane for the hexagonal symmetry and to the closed packed plane for the cubic one. The dislocation line lies along the y-axis in both cases.
4.3 The z-axis is perpendicular to the c-plane for the hexagonal symmetry and to the closed packed plane for the cubic one. The dislocation line lies along the y-axis in both cases.
4.4 The two most favorable slip systems in the systems AlxGa1-xN/GaN and InxGa1-xN/GaN: a) ⟨113⟩{112} observed by Srinivasan [71] and b) ⟨113⟩{101} determined by Jahnen [33].
4.5 The slip system ⟨110⟩{111} of the 60∘ dislocation shown in the FCC structure.
4.6 d
d∕dy-d
∕dy as function of the Al0.2Ga0.8N
film thickness. dd∕dy is calculated assuming isotropic (Freund model (F))
and anisotropic (Steeds model (S)) elasticity, with or without the evaluation
of the integral along the core surface Ecs. d∕dy is calculated according to
equation (4.9). The critical thickness values of an Al0.2Ga0.8N film grown
on a GaN substrate are indicated by a circle.4.7 Same as in Figure 4.6 but for an In0.2Ga0.8N film grown on a GaN substrate.
4.8 Same as in Figure 4.6 but for a Si0.8Ge0.2N film grown on a Si substrate.
4.9 d
d∕dy is
a function of the film thickness for the material systems AlxGa1-xN/GaN
and InxGa1-xN/GaN. The ⟨113⟩{101} slip system is considered. The two
sets of curves are calculated through the Willis et al. (WJB) and Freund (F)
procedures, respectively.4.10 d
d∕dy is a function of the film thickness for
Si1-xGex/Si. The ⟨110⟩{111} slip system of a 60∘ dislocation is considered.
The two sets of curves are calculated through the Willis et al. (WJB) and
Freund (F) procedures, respectively.4.11 d
d∕dy-d∕dy is a function of
the film thickness. dd∕dy is calculated according to the Freund, Steeds,
Willis et al., and Steeds+Willis et al. approaches. d∕dy is calculated
according to equation (4.9). The critical thickness values an Al0.2Ga0.8N
film grown on a GaN substrate are indicated by a circle.4.12 Same as in Figure 4.11 but for an In0.2Ga0.8N film grown on a GaN substrate.
4.13 Same as in Figure 4.11 but for a Si0.8Ge0.2 film grown on a Si substrate
4.14 The equilibrium critical thickness hc as a function of the AlN mole fraction x calculated through Freund and Steeds+Willis et al. (S+WJB) models including the core energy for two different slip systems. The theoretical curves are compared to experimental data, 1- [41], 2- [77], 3- [6], 4- [20], 5- [15]. Empty and filled circles indicate the absence and the presence of misfit dislocations respectively. Crosses indicate the experimental value of the critical thickness hc.
4.15 The equilibrium critical thickness hc as a function of the InN fraction x calculated through Freund and Steeds+Willis et al. (S+WJB) models including the core energy for two different slip systems. The theoretical curves are compared to experimental data, 1 - [44], 2- [71], 3- [46], 4- [33], 5- [60], 6- [62], 7- [28]. Empty and filled circles indicate the absence and the presence of misfit dislocations respectively. Crosses indicate the experimental value of the critical thickness hc.
4.16 The equilibrium critical thickness hc as a function of the Ge fraction x calculated with the Freund (F – dashed line) and Steeds+Willis et al. (S+WJB – solid line) models including the core energy. The 60∘ misfit dislocation with the ⟨110⟩{111} slip system is considered. The theoretical curves are compared with experimental data from [30]. Empty and filled circles indicate the absence and the presence of misfit dislocations respectively.
5.1 An example of a dislocation reaction (fusion) b1+b2 → b3. Vectors b1, b2 and b3 are the respective dislocation line directions.
5.2 Hexagonal lattice. Burgers vectors of the type 1∕3⟨110⟩ are blue. Burgers vector of the type ⟨0001⟩ is green. Burgers vector of the type 1∕3⟨113⟩ is red.
5.3 Relative motion of inclined TDs as a result of film growth.
5.4 Perspective view schematic of a semipolar GaN film with growth direction [112].
5.5 Geometry of the hexagonal lattice with the (112) plane indicated in blue.
5.6 The total threading dislocation density, called ρ, as a function of the GaN thickness h for different initial conditions.
5.7 The misfit dislocation density ρMD, calculated within the isotropic and anisotropic frameworks, as function of the film thickness h.
5.8 The in-plane strain ε, calculated within the isotropic and anisotropic frameworks, as function of the film thickness h.
5.9 The misfit dislocation density ρMD, calculated within the anisotropic framework, as a function of the film thickness h for different chemical compositions of the film alloy.
5.10 The in-plane strain as a function of the film thickness h for different initial conditions.
5.11 Basic processes of threading dislocation motion in a strained epitaxial film. 5.4 An isolated threading dislocation for h < hc for which no motion is possible. 5.4 Threading dislocation – misfit dislocation system for h > hc: increasing film thickness leads to an increasing configurational force on the threading dislocation which leads to threading dislocation motion and generation of additional misfit dislocation segment length.
5.12 Geometry of an isolated dislocation composed of a threading arm (TD) and a misfit segment (MD).
5.13 Movement and reactions of the threading dislocations
5.14 Geometry of a mixed dislocation in GaN.
5.15 The red solid line shows the threading dislocation density, calculated according to the model described in Section 5.4, as a function of the GaN film thickness h. The GaN film is grown upon an AlN thicker layer. The green and orange lines show the sessile and glissile threading dislocation densities, respectively. The red dotted line shows the threading dislocation density, according to the same model, without threading dislocation blocking. The blue line shows the threading dislocation density, calculated according to Mathis’ model [49], as a function of the GaN film thickness h.
5.16 Threading dislocation density as function of the GaN film thickness h calculated according to the Mathis model [49] and the treatment described in Section 5.4 with different initial amounts of (a+c)-type threading dislocations.
5.17 Threading dislocation density as a function of the GaN film thickness h calculated according to the Mathis model [49] and the treatment described in Section 5.4 using different values for the inclination angle of the (a+c)-type threading dislocations.
5.18 The red line shows the threading dislocation density, calculated using the model of Section 5.4, as a function of the thickness h of the Al1-GafN step-graded layer grown upon an AlN substrate. Dashed lines show the threading dislocation density for structures with a lower number of layers. The black dotted line shows the threading dislocation density according to Mathis [49] for a GaN film grown upon the AlN substrate.
5.19 The red line shows the threading dislocation density, calculated using the model of Section 5.4, as a function of the thickness h of the (AlN/GaN)10 superlattice grown upon an AlN substrate. The dashed lines show the threading dislocation density using a superlattice with a lower number of layers. The black dotted line shows the threading dislocation density according to Mathis’ model for a GaN film grown upon an AlN substrate. The theoretical results are compared to experimental data, 1- [85], 2- [14], 3- [75], 4- [84].