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they meet a potential barrier within a crystal. Tunnelling is an everyday phenomena which occurs in a range of semiconductor devices some of which appear in consumer electronics products, see, for example, Sze [44], One way of quantifying the proportion of electrons that tunnel through a barrier is in terms of the transmission coefficient which is defined as the probability that any single electron impinging on a barrier structure will tunnel and contribute to the current flow through the barrier. Ferry has produced a comprehensive analysis of the transmission coefficient for a single-barrier structure (see [45], p. 60). Suffice here to quote Ferry's result in ([45], equation 3.12), for a constant effective mass across the structure, i.e. the transmission coefficient at an energy E for a barrier of width L and height V is given by:
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For values of the carrier energy E greater than the barrier height V, K > ik' (as in Section 2.13), and hence:
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The mathematics show, that for E > V, the transmission coefficient would be expected to oscillate, with resonances when the sine term is zero. These occur when:
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The squared dependence implies that the resonances when T = 1 occur at larger and larger intervals in E, which can be clearly seen in Figs 2.32 and 2.33. Fig. 2.32 displays the transmission coefficient as a function of the energy E and for a range of barrier widths L. For this range of calculations the barrier height V was fixed at 100 meV, and below this energy the thinner the barrier, then the higher the probability of tunnelling. For E > V, the situation is more complex due to the oscillatory nature of T. The trend, however, as highlighted by the curves, is that the
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Figure 2.32 Transmission coefficient as a function of the energy through a single barrier for different barrier widths
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thicker the barrier, the closer the first resonance (T = 1) is to the top of the barrier. This can be understood from equation (2.180):
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Conversely, for a fixed L and a variable V, the first resonance occurs at the same point above the barrier height; this is clearly illustrated in Fig. 2.33.
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2.15 THE DOUBLE BARRIER If two barriers are placed a reasonably small distance apart (in the same crystal, perhaps a few nm) then the system is known as a double barrier (see Fig. 2.31), and has quite different transmission properties to the single barrier. Datta [46], p. 33, has deduced the transmission T(E) dependence for the restricted case of symmetric barriers, while Ferry [45], p. 66, has considered asymmetric barriers. In this formalism, allowance will be made for differing barrier widths as well as the discontinuous change in the effective mass between well and barrier materials. Thus, with the aim of deducing the new T(E), consider the solutions to Schrodinger's equation within
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Figure 2.33 Transmission coefficient as a function of the energy through a single 100 A barrier for different barrier heights
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each region for E < V:
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where k and K have their usual forms as given in equation (2.176) and the positions of the interfaces have been labelled I1, 12, I3, and I4, respectively. Using the standard BenDaniel-Duke boundary conditions at each interface gives the following
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The method of solution is the transfer matrix technique as before, writing the above equations in matrix form:
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Then, as before, the coefficients of the outer regions can be linked by forming the transfer matrix, i.e.
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Clearly, this 2x2 matrix equation still has four unknowns and can't be solved it is at this point that additional boundary conditions have to be imposed from physical intuition. Whereas before, the standard boundary conditions, i.e. w ( z ) > 0 as z > 00, were used to solve for the confined states within quantum wells, in these barrier structures these are not appropriate since the travelling waves in the outer layers can have infinite extent. The standard procedure is to assume, quite correctly, that all of the charge carriers approach the double barrier from the same side, as would occur when as part of a biased device, as illustrated schematically in Fig. 2.34. Furthermore, if it is assumed that there are no further heterojunctions to the right of the structure, then no further reflections can occur and the wave function beyond the structure can only have a travelling wave component moving to the right, i.e. the coefficient L must be zero.