THE DOUBLE BARRIER in Java

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THE DOUBLE BARRIER
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Figure 2.34 The wave function after imposition of the boundary conditions Therefore, if the 2x2 matrix of equation (2.199) is written as M, then:
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and the ratio of transmitted to incident current, i.e. the transmission coefficient, is simply:
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Figure 2.35 Transmission coefficient as a function of the energy through a double barrier of width 100 A and height 100 meV, separated by a distance L2
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Fig. 2.35 gives an example of the form of T(E) for barriers of height 100 meV and width 100 A as a function of the distance L2 between them. The effective masses
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were both taken to be equal to the bulk T valley electron mass of 0.067 mo- It can be seen from the figure that the curves contain Dirac O-functions at certain energies E below the potential barrier height V. This is quite unlike the single barrier case. At these resonance energies, the double-barrier system appears transparent and has a transmission coefficient of 1. The wave functions of these states are localised between the barriers and are often referred to as quasi-bound states since they resemble the bound states of quantum well structures. However, they are not stationary states in that electrons or holes in such states will eventually scatter into the lower energy states outside of the barriers. The effect of an increasing barrier height V is shown in Fig. 2.36. It can be seen that, away from a resonance, an increasing barrier height leads, as would be expected, to a decrease in the transmission coefficient T. The classical explanation for this would be, 'it is harder for the electrons to tunnel through higher barriers'. The resonance energies increase with increasing barrier height due to confinement effects, and the appearance of the second resonance at higher energies is a reflection on the existence of a second quasi bound state.
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Figure 2.36 Transmission coefficient as a function of the energy through a 100 A Gai_ x Al x As /50 A GaAs/100 A Gai_ x Al x As double barrier, for x=0.1, 0.2, 0.3, and 0.4
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While the transmission coefficient represents a very important parameterisation of the properties of a double-barrier structure, it itself is not a measurable quantity. In fact, the properties of such two-terminal electronic devices are generally inferred (or summarised) from their current-voltage characteristics (I-V curve). When an electric field is placed across such a double barrier structure, any charge carriers present in the semiconductor, intrinsic or extrinsic, constitute a current which approaches the left-hand barrier. These charge carriers have a distribution of energy
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THE DOUBLE BARRIER
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and momenta, often a Fermi-Dirac distribution. Those carriers that are of the same energy as the resonance are able to pass right through the double barrier without hindrance a phenomenum which has become known as resonant tunnelling. As the applied electric field (applied voltage) is increased, the number of carriers with the resonance energy increases and peaks as the Fermi level of the semiconductor to the left of the first barrier is brought into alignment. Therefore the current gradually increases. At higher fields, the current falls away and a period of negative differential resistance ensues [47,48]. Such a current-voltage characteristic has been exploited in high frequency circuits, thus bringing the resonant tunnelling diode to prominence as a very useful electronic device [49]. There are various models of the current-voltage properties of different levels of complexity, the simplest of which would probably be to return to the idea that the current at any particular field would be equal to the number of carriers that tunnelled through the structure. This in turn would be the probability of a particular carrier tunnelling, multiplied by the number of carriers at that energy, i.e.
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As the carriers approaching the barrier structure are in a bulk band, then the integral is over their energies, and the Fermi-Dirac distribution function and density of states have the bulk (3D) forms.
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