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Figure 13.1 The periodic nature of the quantum wire unit cell
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Figure 13.2 The quantum-wire unit cell; note the depth of 1 lattice constant, with sufficient barrier to encompass the wire, and if modelling a single wire, to localise the charge
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can be accommodated in this box would be three lattice constants square with a single lattice constant barrier, thus giving a total of two lattice constants between the wires; the reason for these limitations on the wire geometry being that for even for this small cross-section wire, the number of plane waves required in the expansion would normally be 6625, which would give a Hamiltonian matrix HG,G' occupying 66252 x 8 = 351 Mbytes of computer memory. This is only just in reach of highend desktop computers at the present time, so for the purpose of these illustrative calculations, the expansion set will be reduced by truncating at a maximum reciprocal
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Figure 13.3 The quantum-wire unit cell; note the depth of 1 lattice constant
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Figure 13.4 side nx
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The number of plane waves in the expansion set versus the length of the wire
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lattice vector of 3 x (2'K/Ao), rather than the usual 4. This reduces the expansion set to 2751, which still represents 60 Mbytes. Fig. 13.4 illustrates how the untruncated expansion set increases with the number of lattice constants along the side of the wire unit cell. The accuracy of the fit indicates that the expansion set increases as the square of nx. Given that the number of atoms within the unit cell is also proportional to the area, which in this case of a square unit cell is proportional to nx2, then the expansion set may also be expected to be proportional to the number of atoms in the unit cell. This is confirmed by the linear fit shown in Fig. 13.5. As machine specifications increase, larger wire unit cells will be tractable, thus making this straightforward direct diagonalisation method more useful. For example,
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Figure 13.5 the unit cell
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The number of plane waves in the expansion set versus the number of atoms in
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at the time of writing, desktop machines with 1 Gbyte of RAM are becoming more common, which would allow the full expansion set to be used, or a quantum-wire unit cell of twice the side (four times the area) to be tackled.
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13.3 CONFINED STATES The results of a direct diagonalisation of the Hamiltonian matrix for the quantum-wire unit cell in Fig. 13.3 are shown in Fig. 13.6. The latter illustrates the charge density of the lowest conduction-band state for an area the size of the unit cell and across the z = 0 plane. The origin of the plot has been shifted slightly in order to centralise the wire within the unit cell.
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13.4 V-GROOVED QUANTUM WIRES Due to the limitations on unit-cell size, some 'artistic license' has to be invoked so as to produce a rough approximation to a cross-section of a 'V-groove' quantum wire (illustrated in Fig. 13.7). The same method of calculation as used in the previous section can be employed to generate the corresponding charge density, again for the lowest conduction-band state, as now given in Fig. 13.8. The change in the charge distribution in comparison with a square cross-section wire can be seen by referring back to Fig. 13.6, i.e. a relatively small change in the number of atoms in the wire unit cell, which changes the symmetry of the wire, leads to a quite different charge distribution.
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Figure 13.6 The charge density of the lowest conduction-band state over the cross-section of a Ge quantum wire embedded in a Si host
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Figure 13.7 A 'V-grooved' quantum-wire unit cell
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13.5 ALONG-AXIS DISPERSION The shortest reciprocal lattice vector along any of the mutually perpendicular Cartesian axes in bulk is of the form (0,0,2) when expressed in units of 2H/A0 (see, for example, Table 11.1). Therefore the edge of the Brillouin zone in this direction is (0,0,1), i.e. the point usually referred to as 'X'. However, for quantum wires, the shortest 'along-axis' reciprocal lattice vector is actually (0,0,1), and the edge of the Brillouin zone is therefore (0,0,1/2). This is a little counter-intuitive, since along the axis of the wire, the material might be considered as being just infinitely extended
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