Quantum Wells, Wires and Dots by Harrison Paul; Valavanis Alex; & Alex Valavanis

Author:Harrison, Paul; Valavanis, Alex; & Alex Valavanis
Language: eng
Format: epub
Publisher: John Wiley & Sons, Incorporated
Published: 2016-04-27T00:00:00+00:00

Figure 9.3 Run time as a function of the cube root of the order of the matrix to be diagonalised (i.e. the number of wave functions along each direction). The measured run times have been rescaled to a 0–1 range to preserve generality of results between different computational hardware. The line is a merely a quartic polynomial fitted to guide the eye

Also, considering matrices of the same order, the larger the size of the single matrix element (in terms of memory), the longer the diagonalisation time. The matrix elements in the plane wave calculation are complex quantities, and therefore occupy twice as much memory as the (real) elements of the sine wave method.

Figure 9.4 compares the ground state energies calculated with the two methods, for a pyramid with and (the barrier dimension is ): in this case, the plane wave calculation gives an energy at least 7 meV larger than that obtained with the sine wave calculation, for all the considered. This means that, in order to obtain the same accuracy, the plane wave calculation requires more wave functions (i.e. bigger matrices) and takes much longer to run than the sine wave method. This proves that sine waves are, from all points of view, a more efficient basis set than plane waves for the expansion of the envelope function when performing such calculations.

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