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Figure 12.24 The necessary periodic nature of any electric field accounted for with pseudopotential theory
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, then the potential energy due to the electric field would be:
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where z1 and zNa are the z-coordinates of the first (1) and last (Na) atoms, respectively. With this definition, the field does look like that in Fig. 12.24, with small zero-field regions of width A0/2 in between the regions of linear sloping potential.
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Figure 12.25 The possible unit cells for a pseudopotential study of a superlattice
The pseudopotential calculations so far have all centred around true superlattices, i.e. systems of quantum wells with significant overlap between the wave functions of adjacent wells. Single isolated quantum wells (SQWs) can be considered by using pseudopotential theory just by making the barriers within each period thick in size, thus producing a large distance between the wells. Hitherto it was not relevant where the atoms were exchanged within the unit cell to produce a superlattice, for example, as shown in Fig. 12.25 the well could be formed at the beginning of the unit cell, the middle or the end. However, when incorporating an electric field as well, it is important that the field extends either side of the 'region of interest' this is achieved simply by ensuring the quantum wells are centred in the unit cell.
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The effect of an electric field can be calculated for a single quantum well, or a system of several quantum wells, by the appropriate choice of unit cell (see for example Fig. 12.26).
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Figure 12.26 The unit cells required to study the effect of an electric field on a single quantum well (SQW) or a multiple quantum well (MQW)
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Therefore, an electric field can be considered as an additional perturbation with the same periodicity as the superlattice unit cell, whether that cell contains one or more quantum wells. Thus, the original Schrodinger equation for the superlattice (equation 12.21) i.e. would have an addition term representing the perturbation due to the electric field, i.e. the consequence of which is that the potential term in equation (12.38) becomes:
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The manipulation of the first term (Vs1) clearly proceeds as before, thus giving the original perturbing potential V due to the superlattice potential, as defined in equation (12.49), so therefore:
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Consider just the integral component, and again writing g =
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Given the form of VF in equation (12.55) and writing the origin of the electric field potential as then obtain:
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The in-plane (x-y) integrals only have value when the x- and y-components of g are zero, and are then equal to the length of the crystal in that dimension, i.e.
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where the the vector coefficients gx and gy are defined by g = gxi + gyj + gzk, and thus the Dirac 6-functions ensure that the integral is non-zero for g vectors along the axis of the field only. If there are Ns1z unit cells along the z-direction, then:
However, the total volume of the crystal Q = L x L y N s1z n z Ao, where, of course, nzAo is the superlattice period. Hence:
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Integrating by parts, then obtain:
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which upon evaluation, gives:
The total potential term for both the superlattice perturbation and the electric field perturbation is obtained by substituting equation (12.68) into equation (12.59), thus giving:
Using the definition for the superlattice perturbing potential V' in equation (12.49), then the final form for the Hamiltonian matrix elements including an electric field is:
which is an extension of the earlier form in equation (12.50). Inspection of equation (12.70) does raise a small problem, namely divergence of the electric field perturbation when gz = 0. For this particular instance, it is necessary to revisit equation (12.64) and put gz 0, i.e.
and then:
Recalling that
then this becomes:
Therefore, it has been shown that an electric field can be included in the pseudopotential formalism, with the result being an additional potential term in the Hamiltonian matrix. Consider the application of an electric field to a single 10 ml (28.25 A) thick GaAs quantum well surrounded by 20 ml (56.5 A) thick Gao.8Alo.2 As barriers. Thus, in the periodic formalism characteristic of pseudopotentials, this would imply a separation of 2 x 20 ml (113 A), which should be enough for them to act as independent quantum wells. Fig. 12.27 displays the results of calculations of the change in the energy (AE) of the lowest conduction-band energy level as a function of the electric field, as deduced by the empirical pseudopotential (EPP) method. For comparison the figure shows also data obtained by the envelope function approximation (EFA) for the same system. The parabolic nature of this, i.e. the quantum-confined-Stark effect,is clearly