Heavy-hole state

In both equations the second term can be large, giving a small mass. In contrast, there are no such matrix elements between wave functions for the conduction band and the heavy-hole states, so their masses remain large. 

What we should like to have is a suitable average over the two bands and all directions, for this will give the correct density of heavy-hole states per unit energy. Since the heavy holes dominate the total density of states, the corresponding mass will be appropriate for a description of properties that depend directly on the density of states. Such values have been estimated by Lawaetz (1971) they are given in Table 6-3 and are compared with the values of nth obtained from... 

The ability to create and observe coherent dynamics in heterostructures offers the intriguing possibility to control the dynamics of the charge carriers. Recent experiments have shown that control in such systems is indeed possible. For example, phase-locked laser pulses can be used to coherently amplify or suppress THz radiation in a coupled quantum well [5]. The direction of a photocurrent can be controlled by exciting a structure with a laser field and its second harmonic, and then varying the phase difference between the two fields [8,9]. Phase-locked pulses tuned to excitonic resonances allow population control and coherent destruction of heavy hole wave packets [10]. Complex filters can be designed to enhance specific characteristics of the THz emission [11,12]. These experiments are impressive demonstrations of the ability to control the microscopic and macroscopic dynamics of solid-state systems. 

The simplified theory is adequate to obtain qualitative agreement with experiment [1,16]. Comparisons between the simplified and more advanced versions of the theory show excellent agreement for the dominant (electronic) contribution to the time-dependent dipole moment, except during the initial excitation, where the k states are coupled by the laser field [17]. The contributions to the dipole from the heavy holes and light holes are not included in the simplified approach. This causes no difficulty in the ADQW because the holes are trapped and do not make a major contribution to the dynamics [1]. This assumption may not be valid in the more general case of superlattices, as discussed below. 

We do this by using the k p method, (called k-dol-p), which is based upon the perturbation theory of Eq. (1-14). In this method, energy is calculated near a band maximum or minimum by considering the wave number (measured from the extremum) as a perturbation. (The method is described in many solid state texts, such as Kittel, 1963, p. 186, or Harrison, 1970, p. 140.) The method was used for a study of effective masses by Cardona (1963, 1965). It was also usetl in the more extensive study by Lawaetz (1971) referred to in the discussion of heavy-hole bands. We shall discuss here only the conduction band and the light-hole band where the effects of interaction are great. 

Here the matrix element is that of the momentum operator p, in the x-direction, since we have chosen k to lie in the x-direction. The matrix element is taken between the state Fi of the conduction-band minimum and any other state T at r the denominator is the energy difference between the two states. We drop all terms in this sum except those with the valence-band maximum, for which the energy denominator is the smallest and the contribution the largest. It can be shown by symmetry that the matrix clement vanishes for the two heavy-hole bands (they correspond to p orbitals with an orientation perpendicular to the x-axis), so only the matrix element between wave functions for the conduction band and the light-hole band remains. The denominator is the band gap Eq, so wc may extract a conduction-band mass from Eq. (6-26). This mass is given by... 

The hole states are more complicated as there are both heavy hole and light hole bands to consider. Nevertheless, conceptually it is very similar to the classic particle-in-the-box problem. The allowed optical transitions occur between energy levels in which An = 0, i.e. transitions between electron and hole states with the same quantum number. Since the energy levels of the electron and hole states scale the same with length, the shift in the exciton transitions show a very good correlation with the 1/L dependence expected from the simple particle-in-the-box picture. 

The states u> and c> are obtained from solution of the Hamiltonian matrix, Eq. (6-10), for k = 0, in which case all off-diagonal matrix elements take the same value. It can be easily verified that the two eigenvectors (other than the doubly degenerate ones corresponding to heavy holes) are (1, 1, 1, l)/2 and (1, 1, — 1, — l)/2. The Bond Orbital Approximation turns out to be exact for these states at r. The second of these eigenvectors corresponds to the light-hole band (the first corresponds to the bottom of the valence band). Thus the valence-band state at T that enters the calculation can be written as a sum of bond orbitals, as in Eq. (3-20), with k = 0 ... 

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