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- Question : p1 - The stable modification of GaN is the wurtzite (?) structure. Epitaxial layerscan also be grown in the metastable cubic zincblende (?) structure. (a) Determine the lattice mismatch f of ?-GaN on (001)-oriented zincblende GaAs at room temperature. (b) Find a suitable ratio of small integers for a coincidence lattice and determine the respective coincidence-lattice mismatch. (c) On (111)-oriented GaAs the GaN layer tends to grow in the ? phase. What is the wurtzite a lattice parameter of the GaAs(111) plane? Prov that the lattice mismatch of ?-GaN/GaAs(111) is similar to that of ?GaN/GaAs(001), if ?-GaN is assumed to have the same bond length as ?-GaN
- Question : P2 - GaN is grown on various substrates due to a lack of well lattice-matched materials. (a) Calculate the lattice mismatch for growth on the most commonly used basal-plane sapphire Al2O3 in case of an epitaxial relatio [0001]GaN [0001]Al2O3 and [1000]GaN [1000]Al2O3 (i.e., a and c axes of substrate and layer are parallel). Compare this value to that often derived from the alterative definition falternative1 given in the text. The lattice parameters of Al2O3 are aAl2O3 = 4.758
- Question : P3 - (a) How does the Al composition parameter x in the quaternary compound AlxGay In1?x?yAs depend on the Ga composition parameter y for a layer lattice-matched to InP according Vegard
- Question : P4 - Apply in the following problem linear expansion coefficients (which actually underestimate the thermal expansion at higher temperatures) and a linearly weighted quantity for the alloy. (a) Calculate the lattice mismatch f of a ZnS layer on Si substrate at a growth temperature of 360
- Question : P5 - Consider a pseudomorphic ZnSe layer on (001)-oriented GaAs substrate at room temperature. (a) Determine the strain of the ZnSe layer perpendicular to the interface. (b) Calculate the relative change of the unit-cell volume induced by the strain. (c) The strain in the layer changes the distance dhkl between nearest lattice planes. In crystal systems with orthogonal axes and lattice vectors of lengths a, b, and c the distance can be expressed by dhkl = ((h/a)2 +(k/b)2 +(l/c)2)?1/2. Use this relation to calculate the distances between nearest (111) planes in unstrained ZnSe and the pseudomorphically strained ZnSe/GaAs layer. Relate the factor ?3 in the unstrained material to the stacking sequence of the zincblende structure. (d) Express the thickness of a 285
- Question : P6 - The strain in pseudomorphic lattice-mismatched layers is sometimes compensated by inserting additional layers with opposite lattice mismatch, yielding layer stacks which are in total lattice-matched to a substrate. (a) Determine the thickness of an In0.15Ga0.85As layer, which is to be pseudomorphically grown on a 1 monolayer thick (001)-oriented GaP layer to obtain a total lateral lattice constant coinciding with that of GaAs. How many monolayers of In0.15Ga0.85As correspond to this thickness? The stiffness coefficients of GaP are C11 = 141 GPa, C12 = 62 GPa; use a linearly weighted shear modulus for the In0.15Ga0.85As layer determined similar to the unstrained lattice parameter of this layer. (b) The strain in a pseudomorphic superlattice with 10 InxGa1?xAs quantum wells separated by GaAs barriers is to be compensated by the additional insertion of a counteracting GaAs1?yPy layer into the center of each of these barriers. The entire layer stack of the strain-compensated superlattice should adopt the same lateral lattice parameter as the (001)-oriented GaAs substrate. Calculate the composition parameter y for the case that in each of the 20 nm thick barrier layers 12 nm of GaAs is replaced by GaAs1?yPy . The quantum wells have a thickness of 10 nm and a composition of 22 % indium. Apply a linearly weighted shear modulus for the wells, but approximate the value for the barriers by that of GaAs (check if such simplification is justified).
- Question : P7 - Determine the energy per unit length for the following perfect misfit dislocations in GaAs within a radius of 300 nm around the dislocation line. A single dislocation with a dislocation-core radius of 1 3 of the Burgers vector is assumed. (a) Pure screw dislocation. (b) Pure edge dislocation. (c) 60
- Question : P8 - The composition parameter x of a (001)-oriented alloy layer AxB1?x with diamond structure was approximately adjusted for achieving lattice matching to a substrate with lattice parameter aS = 5.5
- Question : P9 - A ZnSe/GaAs(001) layer is characterized by X-ray diffraction using CuK?1 radiation. (a) Calculate the separation between the Bragg angles for the (004) reflections of the substrate and the layer for a completely relaxed layer and a pseudomorphically strained layer. (b) Repeat (a) for the asymmetric (115) reflection. (c) The intensity of the scattered radiation is largely determined by the square of the structure factor. Calculate the approximate intensity ratio I(004)/I(115) of the two reflections, if the ratio of the atomic scattering factors fSe/fZn = 1.17.
- Question : P10 - An InxGa1?xN layer is to be grown lattice-matched on the basal plane of a ZnO substrate. (a) Find the In composition x1 and the Bragg angle of the (00.2) reflection of a lattice-matched InxGa1?xN layer for CuK?1 radiation. (b) Calculate the In composition x2 of a relaxed (not lattice-matched) InxGa1?xN layer producing the (00.2) reflection at a Bragg angle 400 sec below the value of a lattice-matched layer. Compare this value to the composition x3 of a pseudomorphic InxGa1?xN layer, whose (00.2) reflection appears at the same Bragg angle (find x3 by using x2 and interpolation; use for simplicity elastic constants of pure GaN).

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