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THE EFFECTIVE MASS EQUATION FOR HETEROSTRUCTURES

In analyzing low-dimensional systems by the effective mass equation, we were careful to apply this equation to the calculation of the wavefunctions of the same material, be it a planar layer in a sandwich structure or a wire surrounded by a dielectric. We did not examine the variation of a wavefunction across two materials with a band edge offset Д£c or AEy between them. To do so we will need boundary conditions for the wavefunction and its derivative, as we did in chapter 1 for the calculation of the tunneling current of an electron beam hitting a barrier in vacuum. However the boundary conditions for the interface of two materials, see figure 5.8, with different effective masses m and m2 are not the usual ones when it comes to the envelop functions.

Let X(x) denote the envelop function along the direction perpendicular to an interface of two materials at x = 0. X(x) is the x-component of the total wavefunction as in equation 5.2. Equality of the envelop wavefunction is still required

But the relation between the derivatives must be modified. Instead of the usual equality we require

Why is this? Basically to conserve current. In chapter 1 we saw that the derivative in a certain direction corresponds to the momentum operator in that direction, so the above relation 5.39 constitutes an equality of velocities and hence current is conserved. In fact,

Matching of a 1-dimensional wavefunction X at an interface of two materials with different effective masses

FIGURE 5.8 Matching of a 1-dimensional wavefunction X at an interface of two materials with different effective masses.

the modified boundary condition 5.39 guarantees that the actual wavefunction has a continuous slope when the interface is crossed.

Another complication which arises with heterostructures is the following: in deriving the effective mass approximation (EMA) we have omitted the к dependence of the cellular part unk of the wavefunction, cf. equation 2.69, keeping only u„0, the cellular part at the band extremum, assumed to be at к = 0 as in GaAs. However, when an electron crosses an interface, it may encounter a semiconductor where the band extremum is not at the same к point. Such is the case, for example, for the interface GaAsIGaP and the interface GaAs/GaAs-xPx for x > 0.4. In this case, the effective mass approximation does not hold, so that great care must be exercised before it is applied.

On the assumption that we have an interface with the band extrema at the same к point we cannot continue to use the EMA of chapter 2 before a further modification in the equation itself is made. We have to write the EMA in the following form (in 1 dimension)

In equation 5.40 the meaning of m’(x) is that it is piecewise constant—it changes from a constant m,* to another constant m2*. The reason for this mathematical trick is that the hamiltonian of this form preserves its hermiticity. It does not do so if a space varying mass is used in its usual form.

 
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