Reciprocal Space Maps

The complete reciprocal space map of a material is made up of a series of points that represent the various planes of the crystal lattice; these points are defined by the surface normal vectors of the crystal planes. Not all of these points are accessible in the experiment; access to these points is restricted by two physical constraints, namely, the x- ray wavelength and the incident angle, . Scanning in an /2θ range, the scan cannot access crystal planes with a d-spacing less than /2. In addition, the diffractometer cannot access crystal planes that the incident x-ray beam cannot illuminate, i.e. where <0 and

>2θ. Figure 2.4 shows the area of the RSM where these restrictions apply and also shows how the  and /2θ physical scanning directions relate to the RSM directions.

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Figure 2.4 A schematic view of the accessible reciprocal space points from a (001) orientated crystal in HR-XRD reflection mode as shown in figure 2.3. Also shown are the scan axes along which the RSM’s are collected as

well as the positions of the [004] and [224] RSM points.

An out-of-plane and an in-plane RSM needs to be collected to determine the in-plane, ax ,

and the out-of-plane, az , lattice parameters of the layer. Given these lattice parameters, it

is possible to calculate the relaxation and composition of the layer. As alluded to earlier, it is possible to calculate either the relaxation or the composition of the layer from only the (004) rocking curve. However, for this to be achieved, the other parameter has to be known beforehand. For example, if az, calculated from the (004) rocking curve, and the

composition of the layer are known, it is possible to calculate the value of the in-plane lattice parameter, ax , from the change in az ,compared to the bulk material, and how this

change will deform the material in the x and y directions. This is given by the Poission’s ratio of a given material. Knowing both az and now ax it is possible to calculate the

relaxation of the layer. If both the composition and the relaxation of the layer are unknown, both the out-of-plane (004) and in-plane (224) RSM are needed to determine az

and ax directly. The in-plane lattice parameters ax and ay are assumed to be equal as the

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(a)

(b)

Figure 2.5 A schematic view of the (004) and (224) RSM’s showing the positions of the peaks (open circles) of the substrates (a) GaAs and (b) GaSb, and where a generic GaNSb layer peak would be found if it were fully

relaxed (rel) or fully strained (str).

Figure 2.5 shows the positions of the peaks (represented by circles) for the (004) and (224) RSM’s of a layer of GaNSb, which is grown on either GaSb or GaAs. The relative positions of the peaks from each reflection for the substrate and the layer are shown as well as the effect increasing ax and az has on the positions of these peaks. The effect that the relaxation

of the layer has on the position of the layer peak in both RSM’s is also shown. Note that if the layer is fully relaxed then the (224) layer peak will lie on the [224] lattice vector. If the strained peak lies on the left of the [224] vector, then this indicates that the layer is under bi-axial tensile strain (figure 2.5(b)) and if to the right of the vector, that the sample is

38 under compressive strain (figure 2.5(a)). This is the case for samples of GaNSb grown of GaSb and GaAs respectively. Under compressive strain the ax lattice parameter of the layer

is smaller than az and the reverse is true for tensile strain. These two cases are shown

diagrammatically in figure 2.6. The (004) RSM shown in figure 2.5 shows that regardless of the composition or the relaxation of the layer, the peak always lies on the [004] vector. The layer peak will appear closer to the GaAs peak as the nitrogen content increases, i.e. a reduction in the az lattice parameter. Similarly the relaxation of the layer will also move the

layer peak closer to the GaAs peak. From the analysis of the (004) RSM, the vertical component of the shift in the layer peak in the (224) RSM can be determined and, therefore, the change in ax. This is only valid if either the relaxation or the composition of

the layer is known. Where both are unknown, the relative positions of the layer peaks are compared to the positions of the substrate peak to allow for the determination of both relaxation and composition. The high number of dislocations between the layer and the substrate caused by the high degree of lattice mismatch can lead to distortions in the layer peak shape. This spreading of the peak in the omega direction is caused physically by the creation of small mono-crystalline blocks within the layer, each with slightly different orientations, which arise from the interface with a dislocation; this effect is referred to as mosaic spread. There are other effects which cause a spread in the peak in the omega direction such as the surface roughness of the layer at the interface. [9] This explains why the substrate peak is sharp and well defined, indicating few defects and high crystalline quality, compared to that of the layer, which typically has distortions from defects and roughness.