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TRANSMISSION MATRICES, AIRY FUNCTIONS

The prerequisite for the application of the formalism of Landauer is the knowledge of the transmission coefficient of the channel in question. A systematic method of obtaining the transmission coefficient is available if only tunnelling with no scattering is present in the channel. The method consists in a) writing down the form of the wavefunctions in the reservoirs and the channel (which may consist of more than one material), b) matching the wavefunctions and their derivatives at the interfaces, and c) deducing the transmission coefficient as the squared modulus of the ratio of the relevant coefficients relating the incident to the transmitted wave times the ratio of the relevant velocities or wavevectors. We have already encountered this method, though not so formally, in the problem of tunnelling through a rectangular barrier in section 1.7. There, we obtained the transmission coefficient |G/A|2, solving a system of 4 equations in 4 unknowns. We assumed without loss of generality that A = 1. Essentially the same method, but using matrix algebra, will be followed here. The method is only useful for a barrier or series of barriers that are either rectangular (i.e. of constant potential) or trapezoidal (i.e. of linear potential). For a general shape barrier one has to revert to the WKB approximation which we initially presented in section 1.8 and a more thorough discussion we will give in the next chapter. However, for the purpose of analyzing the resonant tunneling diode (RTD) in the next section, the approach of the present section is sufficient.

To prepare the ground for the RTD and go one step further than the simple case of the rectangular barrier, we consider the problem of an undoped semiconductor material,

(a) An intrinsic GaAs sample between two metal leads under an applied voltage V

FIGURE 5.9 (a) An intrinsic GaAs sample between two metal leads under an applied voltage V

(b) the band structure of the system in the configuration shown in (a).

say GaAs, with two metal or heavily doped semiconductor contacts at its two ends and a voltage difference Vap applied between the contacts, see figure 5.9a. The length of GaAs is d—the energy band diagram corresponding to this arrangement is shown in figure 5.9b. We assume that the whole of the applied voltage is dropped in the semiconductor region. This is justified considering that the latter is undoped and hence has a very high resistance compared to the metal contacts. Furthermore, if we can neglect the small intrinsic concentration и, in the GaAs semiconductor, then there are no free carriers and the potential variation in the semiconductor is linear according to the Poisson equation. On the other hand, there is no variation of potential energy in the metal contacts if the whole of Vap is dropped in the semiconductor layer.

If the (+) lead of the supply is connected to the right electrode, the Fermi level of layer I is lifted compared to that of layer III or that of layer III is lowered compared to I, see figure 5.9b. We take the origin of energy at the Fermi level of layer III. Then the potential energy in layer I is U(x) = Uap = (~e)Vap, a constant. Note that we use the symbol U for the potential energy so that we can keep on using V for the potential or voltage. The eigenstates of layer I can be written as usual as a linear combination of plane waves

The potential energy in layer II is

where Фв is the Schottky barrier (in eV) between the metal contact and the semiconductor (cf section 3.10) and £x and is the electric field. The Schroedinger equation with a linearly varying potential can be transformed into the Airy differential equation, the solutions of which are the well-known Airy functions Ai(x) and Bi(x) of the first and second kind respectively. The transformation is mathematically elementary but care must be exercised with units. We give this transformation in detail below.

The Schroedinger effective mass equation reads in one dimension in layer II with (for the moment) m'=m

The first term in the brackets in 5.43 has dimensions of [length) " and hence so must have the term next to it. Therefore, the factor

must have dimensions (length) . Hence equation 5.43 transforms into

In 5.44 the left-hand side of the equation is dimensionless, so the right-hand side must also be. Hence the factor

must have the dimensions of (energy)-1. We have

Denoting by and by we get

Since all variables in front of 4х are dimensionless, we can define q = E'-x'. We then get (using simple instead of partial derivatives)

This is Airy’s 2nd order differential equation, and its particular solutions are the two Airy functions Ai(q) and shown in figures 5.10a and 5.10b respectively. The form that У takes in an infinite triangular well is Ai[q) and is shown in figure 5.10c. The general solution in a finite triangular well is

Going back from the mathematical variable q to the physical variables x, фв, Uap and reinstating effective masses we have (noting that Uap = xd)

The definitions of p/; and are obvious. Finally, in layer III the wavefunction can be written as

(a) The Airy functio

FIGURE 5.10 (a) The Airy function Ai(x) of the first kind and (b) the Airy function of the second kind Bi(x), and (c) the solution of the Schroedinger equation for an infinite length linear quantum well. The first 3 eigenfunctions are shown.

where

Requiring that the wavefunction 'R(x) and its derivative 1 ImcW/dx are continuous at the interfaces (according to equations 5.38 and 5.39) we get the following two equations

for the boundary at x = 0 where Ai' and Bi' are the derivatives of Ai and Bi with respect to x, and the following two

for the boundary at x = d.

The above four equations can be written as two matrix equations of 2x 2 size as follows and

Equation 5.55 can be substituted in 5.54 resulting in an equation of the form

The transmission coefficient where

can be obtained from equation 5.56. We normally define a transmission matrix S which is the matrix that relates output to input. If we had allowed an incoming wave coming from the right with coefficient H in layer III, which would have given also a current from right to left, the (2x2) transmission matrix S would obey an equation of the form

As noted previously and as should be apparent now, this technique is only useful for simple barriers. For barriers of a general shape, one has to revert to the WKB approximation discussed in an introductory way in chapter 1 and more thoroughly in the next chapter.

 
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