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In section 1.3 we treated what we call bound states, i.e. states occurring in potential wells which are deep enough. As we saw, all atomic states are like that. We now come to discuss scattering states, i.e. propagating states which encounter a barrier in their way. The problem is linked to what is called tunnelling, i.e. the phenomenon of barrier penetration which is an entirely quantum mechanical phenomenon that frequently occurs in nanoelectronics. Imagine a mobile (a car, anything) of mass m travelling on a horizontal frictionless road with velocity v and encountering a bump

Notation for the eigenvalues E„j in increasing energy

FIGURE 1.10 Notation for the eigenvalues E„j in increasing energy.

of height h(see figure 1.11). The mobile will not go over the bump unless it has enough kinetic energy, in particular it will not go over the barrier unless The concept of tunnelling

FIGURE 1.11 The concept of tunnelling: a classical mobile has to go necessarily over the barrier but a quantum particle can go through the barrier.

Incident and reflected plane waves on a rectangular barrier of height У-

FIGURE 1.12 Incident and reflected plane waves on a rectangular barrier of height У0-

where g is the gravitational constant. Surprisingly it will in quantum mechanics but with a small probability.

To analyze how this is possible we have to solve the Schroedinger equation again, but this time we have to assume an impinging wave to the barrier. So we assume free space everywhere except in the region 0 ^ x ^ L(see figure 1.12) where there is an electrostatic potential barrier of height V0, equivalent to the gravitational barrier of figure 1.11. Onto this barrier impinges a wave e+,kx travelling from left to right. If the energy E of the wave is less than V0, classically we expect that all of the electrons in the wave will be reflected back, forming a wave of the form e~,kx. But as we shall see, a very small portion of the incoming wave will go through.

So the Schroedinger equation in region 1 (see figure 1.12 again) takes the form (assuming V = 0 there)

where k2 = 2mE / h2 and the double upper dash symbol denotes double differentiation. The solutions to this equation are of the form

The critical reader can see that we don’t have to assume an incoming wave as stated. Both incoming and reflected waves result from the Schroedinger equation. Of course, we have to assume that these states are filled, i.e. we have a beam of electrons. In connection with the Pauli principle, one may ask how come there can be at most 2 electrons per energy state and we have a beam which—as the word signifies—implies many electrons, all in the same energy E. The answer is that the energies of electron states in vacuum are very closely packed, they form a continuum, so that we can have many electron states in such a small energy interval ДE, almost infinitesimal, so the Pauli principle is valid on the one hand and the electrons can be considered monoenergnetic on the other.

In region 2 the Schroedinger equation takes the form with solutions of the form

In the remaining region 3, the Schroedinger equation takes exactly the same form as in region 1. So we expect the same solutions as in 1 but we know from physical intuition that only a small number of electrons will penetrate the barrier and there are no electrons coming from the right. (This will change when we encounter the Landauer formalism later). So we write

We now come to the last ingredient of quantum mechanics: the wavefunction Ч1 must be a smooth function of the coordinates. This means that both the function and its derivatives must be continuous. We may recall that the wavefunctions of the quantum well if examined in the entire range of do not have continuous derivatives at the well edges 0,

L but this is only due to the fact that the well-depth is infinite, a highly unrealistic value. For all finite potential problems, the smoothness of the wavefunctions is a required property. Assuming continuity of T* and d'VIdx at the boundaries 0, L we get at x = 0

and at x = L

Equations 1.51a—1.51d constitute a system of 4 equations with 5 unknowns that mathematically is indeterminate. However, the normalization condition provides the 5th equation.

The solution of the Schroedinger equation for the rectangular barrier of figure 1.12; a travelling wave is found in the region to the right of the barrier

FIGURE 1.13 The solution of the Schroedinger equation for the rectangular barrier of figure 1.12; a travelling wave is found in the region to the right of the barrier.

Alternatively, we can set A=l. Then the calculated modulus of the wavefunction is shown in figure 1.13. We can see that this modulus decays exponentially inside the barrier but it does not go to zero. Hence a fraction of the incident electrons will be transmitted through the barrier. We only need the ratio of (D/ A)2 to determine this fraction of electrons. This ratio will also be the ratio of current densities measured in an experiment because the velocities in regions 1 and 3 are the same. The latter stems from the fact that the potentials there are the same. If they were not, we would have to multiply by the ratio of the velocities. Hence if we denote by T the ratio of transmitted to incident current and R the corresponding reflection coefficient

The system of equations 1.51a—1.51d is a problem of straightforward but tedious algebra. An exact solution of В, C, D, and G can be found in terms of A, but we restrict the solution to the case where al»1 when an even simpler solution can be found. Then

The validity of this equation depends on the validity of the approximation aL » 1. Luckily this condition holds for many physical problems. We stress that equation 1.54 is only true for rectangular barriers. In the next section we obtain an approximate formula valid for most barriers of interest in nanoelectronics.

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