Simple model of current flow through a double barrier structure in Java

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Simple model of current flow through a double barrier structure
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Figure 2.37 outlines this model. The electric field dependence is introduced via fFD and p3D. As the electric field is increased, the bottom of the bulk band is increased in energy relative to the centre of the double-barrier structure by an amount AE = eF(L1 +L2/2). Hence, the reference energy (the band minimum) will increase, and by using equation (2.40), the density of states at some energy E, measured from the
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band minimum at the centre of the well, would become
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where the unit step function ensures that the energy of the carriers impinging on the left hand barrier E is greater than AE. At the bottom of the band, E = AE, and p(E) = 0. In addition, the Fermi energy of the bulk carriers increases by the same amount, i.e. from equation (2.49):
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Figure 2.38 Current-voltage curve for the 100 A Gao.sAlo.2As/50 A GaAs/100 A Gao.8Alo.2As of earlier, obtained at different temperatures
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Figure 2.38 illustrates the results obtained by using this model. The current is obtained in arbitrary units and the voltage V has been defined simply as the potential difference across the structure at a particular field, i.e. V = F ( L 1 + L2 + L3). The I-V curves have been plotted for several temperatures for the x = 0.2 structure of Fig. 2.36. The single resonance within this system shows itself as a single peak in the current. Clearly, the current peak broadens as the temperature increases, with this being a direct result of the broadening of the carrier distribution (the number of carriers in a given energy range), given by the product fF D (E)p(E) appearing in the integral for the current, and illustrated schematically in Fig. 2.37. At low temperatures, the carriers occupy a small region of energy space around the band minima. As the field is increased, current only begins to flow when this narrow distribution is brought into
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line with the resonance energy, and hence there is a narrow current peak. As the temperature increases, the carrier distribution broadens, therefore there is a greater range of applied voltages that give some degree of alignment of the carriers with the resonance energy. The peak occurs when the peak of the distribution is aligned with the resonance energy. At voltages above this, the number of carriers available for tunnelling decreases, and hence the current also decreases. The non-zero current at zero field for T = 300 K is a consequence of this simplistic model. It arises because even at zero field, there is a finite number of carriers in the broadened distribution which are aligned with the resonance energy and are therefore able to tunnel.
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Figure 2.39 Current-voltage curves obtained at 77 K for a variety of barrier widths the structure of Fig. 2.38
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Fig. 2.39 displays the results of the same simple model as a function of barrier width for the same structure of Fig. 2.38, but this time at the fixed temperature of 77 K. Again, as noted before, thinner barriers allow the electrons to tunnel more easily and hence give a higher current. An interesting physical point can be inferred from this data. The incomplete nature of the peaks in the 40 and 30 A data is an indication that a significant fraction of the electrons are passing over the top of the barriers. While the current-voltage model advanced in this section can account for such a situation, the transmission coefficient versus energy data is terminated at the top of the barrier, and so only the fraction of electrons below this are included in the integration for the current in equation (2.203). It is left as an exercise for the reader to generalise the transmission coefficient theory to account for the case when E > V, and therefore improve the current-voltage model. This can be achieved by using a similar approach to that of Section 2.13.
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