High Resolution Analysis of Embedded Quantum Dots
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Mat. Res. Soc. Symp. Proc. Vol. 583 © 2000 Materials Research Society
THE MODEL
The dot shape that has been modelled for this work is a truncated pyramid. Such a dot along with a WL is shown in Figure 1 in three different projections. The dimensions used for
Figure 1 Three projections of the quantum dot modelled here the pyramid have been estimated from HAADF images, an example of which is illustrated in Figure 2. The base, top and height of the dot have been taken as 28nm, 1Onm and 6nm respectively.
Figure 2 Bright-field and HAADF images of a dot The WL is taken to be nmm thick and for these preliminary experiments the composition is assumed to be constant throughout. The position of the dot within the foil is chosen and then the geometry of the specimen of the specimen containing the WL and the dot is defined using the following elements (shown in Figure 3). The wetting layer region consists of three laminar elements with separately defined thickness and composition. We can therefore model WLs of fixed or variable composition.
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Figure 3 Construction of the WL and dot in the model (not to scale) The dot is constructed from a number of square laminar elements the size and composition of which are separately defined. Each element has 5 segments which can have user-defined thickness and composition. Complex patterns of composition variation within a single dot can therefore be modelled, although in the initial experiments reported here, the dot composition is uniform. A typical laminar thickness for the results presented here is 0.1 nm, so that a dot of thickness 6nm consists of 60 laminae. The classic t 312 beam broadening equation of Goldstein et al [5] defines the geometry of the electron beam that interrogates the specimen. Material parameters appropriate to GaAs have been used. The number of X-rays produced within the volume of the electron beam, at a given position s along the line scan, is then described by the equation P(s) = k JJ(x,y, z)on((x - s), y, z)dxdydz where k is the fraction of X-rays detected, J is the electron flux produced by the electron beam, a-is the scattering cross-section for the production of X-rays and n describes the composition and shape of the dot. X-ray emission has been assumed to occur linearly with composition and electron dose. The effect of absorption (of X-rays in the specimen) has been calculated to be less than 5% in all circumstances and has been ignored. The calculation of a single X-ray line
scan takes tens of minutes in MathCAD on a PC. All computations have been for a cross-sectional specimen with the WL parallel to the electron beam and the line scan perpendicular to the WL, passing through the centre of the dot (dotted line in Figure 3). The model was first used to establish the profiles of the WL alone and a dot alone. These are shown in Figure 4. The WL profile is symmetric, as would be expected,
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Figure 4 Computed X-ray p
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