Band Structure In Quantum Wells

In order to interpret correctly the optical absorption experiments in quantum wells, we need to know the band structure. Figure 8.2 of Chapter 8 shows the absorption spectra of a 40 period multiple quantum well (MQW) GaAs-AlGaAs, in which the barriers have a width of 7.6nm [3]. Observe that the spectrum follows in general the steps of the DOS curve in 2D semiconductors (Section 4.2). At the edge of each step there is a sharp maximum that, as will be shown in the next section, is attributed to excitonic effects. It can also be observed in Figure 8.2 that at the edge of the first transition for electrons between the conduction and valence bands for n = 1, there is a peak at 1.59 eV which corresponds to the heavy hole (HH) valence band and one at 1.61 eV for the light holes (LH), which is below the heavy ones (Section 4.8).

The reason for observing the above splitting corresponding to holes is that the one-dimensional potential due to the quantum well breaks the cubic symmetry of the crystal, and consequently lifts the degeneracy of the hole band in GaAs, in a similar manner as did strain in the previous section. Detailed calculations, too long to be included in this text, show that the presence of the well potential causes the LH states to move downwards in energy more than the HH (Figure 4.13). It is interesting to know that if the calculations do not take into account very small terms in the expansion of the perturbed Hamiltonian, the hole bands cross each other because, then, the heavy hole band moves faster downwards. The resulting crossing of the two bands, produced as a consequence of their different curvature, would cause the phenomenon known as mass reversal. If this does not happen, it is because, when very detailed calculations are performed, it can be shown that the crossing effect is removed, appearing instead as an effect known as anticrossing.

We can therefore appreciate that the band structure in quantum wells, especially those corresponding to holes, are fairly complicated and most results can only be numerical. An additional complication comes from the fact that the square wells have barriers of finite height (Section 4.3). Figure 4.14 shows the calculated band structure for MQW of the type AlGaAs-GaAs [4]. Note that some of the bands have a shape far from the ideal

Heavy hole valence band

Light hole valence band

Figure 4.13. (a) Valence bands of GaAs bulk crystal; (b) position of the valence bands in a GaAs in a quantum well.

Heavy hole valence band

Light hole valence band

Figure 4.13. (a) Valence bands of GaAs bulk crystal; (b) position of the valence bands in a GaAs in a quantum well.

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HH1

^---- .

HH2

^____ LHlX

\ ■

HH3

HH4 HLHH42

0.01 0.00 0.01 Wavevector

0.04

0.01 0.00 0.01 Wavevector

0.04

Figure 4.14. Structure of the valence bands in a GaAs-AlGaAs MQW. Observe that some bands (HH2) show negative hole masses close to k = 0 and that some of the bands (HH3) have a shape far from parabolic. After [4].

parabolic one and that the sign of the curvature changes, implying the existence of hole states with a negative effective mass.

As we have seen in this section, quantum confinement can cause a change of the energy levels corresponding to the HH and LH bands. In addition, we can change the energy of these bands by applying tensions or compressions (Section 4.8). In some cases, we can move the LH band above the HH, or even cause degeneracy. As a consequence of the degeneracy, a very high concentration of hole states is produced, a phenomenon which is exploited in optoelectronic modulators.

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