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THE CONCEPT OF EFFECTIVE MASS

An examination of the E(k) curves of GaAs, figure 2.11, reveals that both the conduction band near its bottom E0 and the valence band near its top Ev, occurring both at k= 0, are very nearly parabolic and isotropic. Therefore, their energies can be represented by the modulus к of к only. The energy regions around Ec and Ev are very important because these are the energies that electrons from donors and holes from acceptors will respectively occupy. But these energy ranges are only a small fraction of the whole band so it is acceptable that we should find a simpler representation than the numerical output of a rigorous calculation. This can be done by using Taylor’s theorem. Expanding therefore up to 2nd order around £cfor the conduction band and around Ev for the valence band of GaAs we have

and

Note that in 2.42 and 2.43, as already noted, we have assumed that both bands depend only on the magnitude of k. Also note that there are no first order terms in 2.42 and 2.43 because the expansions have been performed around a minimum for the conduction band and a maximum for the valence band respectively.

The above relations remind us of the free electron case or the particle in a box problem where in both cases the energy is of the form E = Tik2/2m. Now for parabolic isotropic bands the derivatives appearing in 2.42 and 2.43 are constants. It is appealing therefore to rewrite 2.42 and 2.43 in a way that is reminiscent of the above energy-wavevector relation, i.e. put 2.42 and 2.43 in the form

where

and

The parameters m*„, rnp are called the effective mass of electrons and holes respectively and indeed have units of mass so that 2.44 and 2.45 can be reinterpreted as total energies composed of a constant potential energy (Ec in 2.44 and Ev in 2.45) plus a kinetic energy term. Note that the effective mass of the valence band, trip, is negative (all states lie below Ev) so that it is customary, in order to retain the concept of mass, to write

where m*p > 0.

An examination of the Si energy bands in figure 2.11 reveals that the conduction band minimum Ec and the valence band maximum Ev do not occur at the same к point. In particular, Ev occurs at the centre of the BZ, point labelled Г, whereas the conduction minimum occurs on the ГХ axis (see figure 2.8). Furthermore, the conduction band is not isotropic near E0 see figure 2.11, so that a slightly more complicated expression is needed than that of equation 2.44 for Si.

where kt is the (longitudinal) component of к along the ГХ axis and к, the other two transverse to this components, measured from the point of expansion. That is, the constant energy surfaces, in к space, of the conduction band in Si are elongated ellipsoids, as shown in figure 2.16. There are 6 of these ellipsoids corresponding to the positive and negative portions of the 3 к axes.

In equation 2.46 we have treated m* and mj as simple parameters with no formal definition as in equations 2.44a and 2.45a. The effective mass can be generalized to fit any material by making it a tensor as follows.

Then all of the known semiconductor band structures can be expanded near the minimum Ec of the conduction band as follows

The regions in к space where the conduction band minima of Si are located

FIGURE 2.16 The regions in к space where the conduction band minima of Si are located: they are ellipsoidal in shape.

where kx, ky, kz are the components of к measured from the point of expansion and mx, my, mz are the diagonal elements of the effective mass tensor. The valence band maximum on the other hand is usually isotropic.

The concept of effective mass is not only a device which allows simple expansions of the bands near their minimum or maximum. It also allows a description of electrons in a more classical manner as can be seen by the following simple 1-dimensional proof. Imagine an electron in a conduction band with an isotropic effective mass m moving under the action of an electric field £. The electron will move in the opposite direction from which the electric field is applied. So to avoid minus signs we measure the distance travelled ds in time dt in this direction. Then the energy gained under the action of the field is (with v being the velocity)

This is actually a 1-dimensional version of equation 2.41, so it would have been proper to start from there, but we warn the reader that the above mathematical steps do not constitute a proof of equation 2.41. However, from this point onwards there are no loopholes in our derivation. We get

Equation 2.49 looks like Newton’s third law where the force is proportional to the acceleration. Indeed, electrons in the conduction band (or holes in the valence band of a semiconductor) can be thought of as classical particles moving under the action of external forces with the effective mass of the band, however, substituted for the real mass. The 3-dimensional version of equation 2.49a is

We have now reached a stage where we can draw a very important conclusion relating band theory to applications: the lighter the effective mass of the electron in a semiconductor compared to another, the faster these electrons will move and the faster will they respond to signals. As an example, note that the effective mass of electrons in GaAs is 0.067ш whereas in Si it is 0.92m where m is the vacuum mass of electrons. So substantial differences occur between semiconductors which may also explain the interest in GaAs in particular.

 
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