Heavy hole

Fig. 5. Energy levels of electrons and heavy holes confined to a 6-nm wide quantum well, Iuq 53GaQ 4yAs, with InP valence band, AE and conduction band, AE barriers. In this material system approximately 60% of the band gap discontinuity Hes in the valence band. Teasing occurs between the confined...

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. 

Fig. 24. The computed valence band dispersion E(k) computed from the 6 x 6 Luttinger model for the wave vector parallel and perpendicular to the Mn spin magnetization in (Ga,Mn)As. assuming that the spin splitting of the heavy-hole band at the f point is 0.15 eV.

In equation 3, ran is the effective mass of the electron, h is the Planck constant divided by 2/rr, and Eg is the band gap. Unlike the free electron mass, the effective mass takes into account the interaction of electrons with the periodic potential of the crystal lattice thus, the effective mass reflects the curvature of the conduction band (5). This curvature of the conduction band with momentum is apparent in Figure 7. Values of effective masses for selected semiconductors are listed in Table I. The different values for the longitudinal and transverse effective masses for the electrons reflect the variation in the curvature of the conduction band minimum with crystal direction. Similarly, the light- and heavy-hole mobilities are due to the different curvatures of the valence band maximum (5, 7). 

Here mu, mu, and mch denote effective masses of heavy hole (HH), light hole (LH) and crystal-field split-off hole (CH) bands, respectively. In ZB structure, the hole masses along [kOO], [kkO] and [kkk] directions are given as follows ... 

The valeuce bands, which arise from p atomic orbitals, have sixfold degeneracy and contain at 0 K the six p-orbital Se valence electrons. Due to spiu-orbit coupliug, this degeneracy is split at k = 0 into a fourfold degenerate J = 312 band and a twofold degenerate J = 1/2 split-off valence band where Jis the total unit cell angular momeutum. For ki= = 3/2 band splits into two doubly degenerate components the heavy hole... 

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 stales 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 Lawactz (1971) they are given in Table 6-3 and are compared with the values of m, obtained from... 

Eq. (6-24). We see that they are in reasonably good agreement, though the values given by Lawaetz are preferred. Using them, it is appropriate in many circumstances to treat these two heavy-hole bands as isotropic and identical. 

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... 

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. 

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,... 

Figure 1. Conduction and valence band profiles of the dilute-N W" InAsN/GaSb/InAsN and "M" GaSb/InAsN/GaSb laser structures on InAs substrate. On the upper part, fundamental electron (ei) and heavy hole (hhi) presence probability densities are reported. Tbe ei-hhi optical transitions are expected at 3.3 pm at RT.

Here all the conduction band-edge , and valence-edges , are put into an absolute energy scale. It is clearly shown in the matrix elements of (28) that 3/2, 3/2) heavy-hole (HH) band-edges are shifted by SE =-P -Q and 13/2, 1/2) light-hole (LH) band-edges are shifted by SE =-P +Q, and conduction band-edges are shifted by SE =P from their previous unstrained positions. The corrections to the Kane inteiband matrix element/ on both in-plane and perpendicular directions are trivial. 

It has been shown that the spin-Hall effect may arise from various spin-orbit couphngs, such as a spin-orbit (SO) interaction induced by the electron-impurity scattering potential,a Rashba SO conphng in two-dimensional systems, etc. Murakami et al. also predicted a nonvanishing spin-Hall cnrrent (AHC) in a perfect Luttinger bnlk p -type semiconductors (no impurities or defects)." Experimental observations of the spin-Hall effect have been reported recently in a n -type bnlk semiconductor and in a two-dimensional heavy-hole system. ... 

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. 

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