THE EFFECTIVE MASS EQUATION
The concept of effective mass is not only useful for classical concepts like velocity and acceleration, but it can also be used to obtain a more “macroscopic” Schroedinger equation that is approximate but very useful. Let us see this in 1 dimension (ID) first.
We assume that an external electrostatic potential energy V(x) is applied to a 1-dimensional crystal whose eigenstates are the Bloch function Ф„*(х) where n is the band index and к is the wavevector. Then according to the discussion at the beginning of section 2.5, the wavefunctions Ч'(х) of the electrons are no longer individual Bloch functions but are wavepackets described as a sum (or integral) of Bloch functions (see equation 2.37)
Several approximations can be made to equation 2.73. To begin, we may assume that interband transitions are not present and the electrons remain in the same band, then the summation over the band index n can be dropped and each band considered separately. Secondly, we may assume that the periodic part u„ k(x) of the Bloch functions does not vary much with к so that we may replace all unk with the и,|0, the function at the band extremum k= 0. Therefore, we have for the Bloch functions
and for the wavefunction
Substituting 2.76 in the Schroedinger equation 2.74 we get
The first term of 2.77 can be written
where in 2.78 we have used the fact that the Bloch functions ФиА. are eigenfunctions of H(r Using our approximation 2.75, equation 2.78 becomes
Let us now make use of the effective mass approximation for En(k). Then 2.79 becomes (remember we are still within the ID model)
The integral of the RHS of the above equation is an inverse Fourier transform. Remembering
the properties of this transform—that к becomes — in real space—we get
Substituting 2.80 in 2.77 we get
Equation 2.81 is called the effective mass equation. It uses the effective mass approximation for the En(k) to obtain an equation that has the exact form of a Schroedinger equation with energy eigenvalues measured from the band edge Ec. The eigenstates of the corresponding operator in 2.81 are not the actual wavefunctions T^x) but are the envelope functions cp„(x)(see figure 2.18), which constitute all the information we want. We are not really interested in the variation of 'F(x) from site to site, but we are interested in the variation over many sites as this information will reveal either the propagation of an electron through the crystal or the localization of it in a certain portion of the crystal.
The effective mass equation is very useful when we have sandwiches of materials stuck together (which is what devices are made of). Imagine a piece of GaAs of nanometric length L (say L=2nm) sandwiched between two pieces of GaAlAs of much greater length.
FIGURE 2.18 Real and envelope wavefunction in a semiconductor.
Given that GaAlAs has a much higher Eg, it looks as if the electrons in the conduction band of GaAs cannot move into GaAlAs either to the left or to the right, see figure 2.19. We can use the effective mass equation to calculate the energies of this quantum well. What we have called external potential in the lines that lead to the derivation of the effective mass equation need not be a potential energy supplied by a user but it can be any perturbation to the crystal. In this case we can draw a diagram, see figure 2.20, of the band gap variation along the length of the sandwich.
The conduction band difference, or offset, £c(GaAlAs) — £c(GaAs)=A£c can be thought as a quantum well depth that keeps the conduction band electrons of GaAs in GaAs. Likewise for the holes in the valence band. We can therefore write
where A£c is of course a nonzero constant over only the distance L (= 2nm) that GaAs extends. Equation 2.82 is of the form of a quantum well (see chapter 1), albeit of a finite
FIGURE 2.19 Schematic picture of a nanosystem in a one dimension extending infinitely in the other two dimensions.
FIGURE 2.20 Band edges and energy levels of the confined nanosystem of figure 2.19.
barrier height, and therefore it will give a set of discrete levels which electrons will occupy (see figure 2.20), calculated relative to Ec. Note that Ec denotes the conduction band edge of a macroscopic piece of GaAs. If L is of the order of microns, all the £„ will almost coincide with £c, but if L is nanometric, as we have assumed, the (£„- Ec) in equation 2.82 will be positive and substantial. Confinement of carriers creates what we call quantization. In subsequent chapters we will encounter many such cases in our study of devices.
In three dimensions 2.82 is generalized to
and for holes in the valence band
The above equations hold for semiconductors in which Ec and Ev are singly degenerate,
i.e. a simple к point is associated with the band extremum. Such are, for example, both £c and £y in GaAs or only Ev in Si. On the other hand, we have seen that the conduction band-edge £c in Si is six-fold degenerate. In this case we need to ascribe a valley index j to the wavefunction ф,„ i.e. write ф„,; and use the appropriate effective mass m, for each direction (i) in each of the 6 valleys at which the conduction band extremum is located. Furthermore, if the effective mass approximation is to be used at an interface of two different semiconductors further complications exist and a further modification is required. All these are described in section 5.4.