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such as Ga1 xA1x As and Cdi-^MnxTe where the effective mass change between the well and barrier is relatively small. Nonetheless, the mechanisms by which the theory presented here can be extended have been mapped out and here lies an opportunity for the interested reader to explore such systems further. 5.9 BAND NON-PARABOLICITY Small well widths and large potential barriers could require the inclusion of the nonparabolicity of the conduction band [106]. Ekenberg [113] described the inclusion of non-parabolicity on the subband structure of quantum wells. This method can account accurately for a variety of physical phenomena, but is analytically complicated. Simpler procedures have been proposed by various authors for the more complex problem of a donor in a quantum well. For example, Chaudhuri and Bajaj [106] have used the following simple replacement:
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where an represents a series of constants. Given the values of these constants, the effects of non-parabolicity can be incorporated into equation (5.17) simply by making the effective mass a function of the energy E. This extension is certainly much more straightforward than the two described in the previous section. In relation to the calculations below, it should be noted that Chaudhuri and Bajaj [106] showed that, even with relatively large potential barriers, band non-parabolicity was only significant for wells narrower than half the Bohr radius of the neutral donor, which, for the case of would be <50 A, and for Cd 1 x Mn x Te, would
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Just as there are many solutions to the Schrodinger equation for an isolated hydrogen atom, there are also many more solutions representing excited energy states of a donor in a heterostructure. Recalling the hydrogen-atom solutions [4]:
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and similarly for 2py and 2pz The corresponding eigenenergy involves only the ground state and the principle quantum number, i.e. so very simply etc. In bulk semiconductors, the neutral donor does also exhibit these states (see [2], p. 314). The situation is more complex in semiconductor heterostructures as the onedimensional potential due to the layer structure breaks the symmetry of the spherical potential, and hence the wave function is more complex than for the hydrogen atom. Much detail has been given as to solving the corresponding ground state under these
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conditions. In an analogy with the hydrogen atom the first excited state of the donor might be written as follows:
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where A2S has been labelled specifically as it cannot be assumed, a priori, that The constant a has been introduced, and is determined by ensuring orthogonality between the ground state and this the first excited state, i.e.
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Studies do exist in the literature of the excited states of donors in heterostructures [114], but to the author's present knowledge the extension utilizing the general form has not yet been attempted. A simple alternative approach which yields reasonable results is presented later in this chapter. 5.11 APPLICATION TO SPIN-FLIP RAMAN SPECTROSCOPY IN DILUTED MAGNETIC SEMICONDUCTORS 5.11.1 Diluted magnetic semiconductors
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Figure 5.20 Zeeman splitting of conduction (CB) and valence (VB) bands of a diluted magnetic semiconductor within an external magnetic field B
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Diluted magnetic semiconductors (DMSs) [115,116] are important because of the strong exchange interaction between the hybridised sp3 d orbitals of the magnetic
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ions and charge carriers. This manifests itself most clearly in the giant Zeeman splittings observed in both the conduction and valence bands when the material is placed in an external magnetic field. At low magnetic ion concentrations, the materials generally exhibit 'frustrated' paramagnetism, with the number of spin singlet states being reduced by nearest-neighbour anti-ferromagnetic spin-pairing [117]. At low magnetic fields (< 8 T) these spins remain locked and cannot contribute to the paramagnetism of the material; however experiments carried out under very high magnetic fields have been able to break these spin-doublets. The magnetic ion itself, usually Mn2+, sits substitutionally on a cation site and can generally be incorporated to high concentrations. The most common DMS is Cd1_xMnxTe, while others include Zn1_xMnxS, and more recently Ga1_xMnxAs. Fig. 5.20 shows the Zeeman effect in a DMS material, with the vertical arrows linking the heavy-hole states ( 3/2) and the electron states, thus illustrating the allowed interband transitions under circularly polarized light, i.e. is the a transition, and is the cr+ transition. The magnitudes of the conduction and valence band splittings are given in terms of the variables A and B as follows:
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where NQU and NoB are constants (220 and 880 meV respectively in Cd1_xMnxTe). The expectation value of the magnetic ion spin along the z-axis (Sz) is given by: where So (x) is the effective spin of the magnetic ions and B j is a Brillouin function describing the response of the spins in a magnetic fields B. The effective spin So accounts for the proportion of magnetic ions which are spin-paired with a nearest neighbour and can not respond to the alignment induced by the magnetic field. Alternatively the effective spin can be considered as the concentration of spin-singlet states, i.e. where the spin of the Mn2+ ions are 5/2. Recent theoretical studies [117] have calculated x in agreement with experiment [118] and shown that for moderate fields (=8 T), i.e. when the splittings have saturated, but before the nearest neighbour spin pairings are broken, the maximum value of x (Sz(x)} occurs at a manganese concentration x = 0.15 and is equal to 0.105. Hence, in Cd1_xMnxTe the maximum splitting in the conduction band is = 23 meV, and for the heavy-holes in the valence band it is =92 meV. The paramagnetic behaviour falls off with increasing temperature. When applying a magnetic field to a semiconductor heterostructure, the direction of the field becomes important. For fields parallel to the growth axis (z-) (the Faraday configuration), the splittings are still well represented by Fig. 5.20. However, application of the magnetic field along the plane of the wells, i.e. the Voigt configuration, leads to mixing of the light- and heavy-hole valence states, thus producing a
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