Exciton Spin Dynamics in Semiconductor Quantum Wells
The spin properties of excitons in nanostructures are determined by their fine structure. Before analyzing the exciton spin dynamics, we give first a brief description of the exciton spin states in quantum wells. We will mainly focus in this chapter on Ga
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3.1 Two-dimensional Exciton Fine Structure The spin properties of excitons in nanostructures are determined by their fine structure. Before analyzing the exciton spin dynamics, we give first a brief description of the exciton spin states in quantum wells. We will mainly focus in this chapter on GaAs or InGaAs quantum wells which are model systems. For more details, the reader is referred to the reviews in [1, 2]. As in bulk material, exciton states in II–VI and III–V quantum wells correspond to bound states between valence band holes and conduction band electrons. As will be seen later, exciton states are shallow twoparticle states rather close to the nanostructure gap, i.e., their spatial extension is relatively large with respect to the crystal lattice, so that the envelope function approximation can be used to describe these states. A description of the exciton fine structure in bulk semiconductors can be found in [3]. In quantum well structures, as in bulk material, a conduction electron and a valence hole can bind into an exciton, due to the Coulomb attraction. However, the exciton states are strongly modified due to confinement of the carriers in one direction. As we have seen, this confinement leads to the quantization the single electron and hole states into sub-bands (cf. Chap. 2), and to the splitting of the heavy- and lighthole band states. The description of the excitons is obtained, through the envelope function approach, and the fine exciton structure is then deduced by a perturbation calculation performed on the bound electron–hole states without electron–hole exchange. However, this approach becomes then more complex in the context of two dimensional structures [4]. The full electron–hole wave function is usually approximated by eiK ⊥ .R⊥ φj nl (r ⊥ )us (r e )umh (r h ), Ψα (r e , r h ) = χc,νe (ze )χj,νh (zh ) √ A
(3.1)
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T. Amand and X. Marie
where, α represents the full set of quantum indexes characterizing the exciton quantum state, e.g., explicitly: |α = |s, mh ; νe , νh , K ⊥ , j, n, l . Here χc,νe (z) and χj,νh (z) are the single particle envelope functions describing the electron, heavy-hole (j = h) or light-hole (j = l) motion along the z-growth axis, R ⊥ and K ⊥ are the exciton center of mass position and wave vector, respectively, A is the quantum well quantization area, and φj nl (r ⊥ ) characterizes the electron–hole relative motion in the quantum well plane. This is in fact the function basis we shall take to formulate the electron–hole exchange in a quantum well exciton. The electron–hole exchange is determined through the evaluation of the direct and exchange integrals:
e2 Ψα (r e , r h ) dr e dr h , b |r e − r h | structure e2 Ψα (r h , r e ) dr e dr h . =− Ψβ∗ (r e , r h ) b |r e − r h | structure
Dβ,α = −Eβ,α
Ψβ∗ (r e , r h )
(3.2a) (3.2b)
In the calculations of integrals (3.2), two contributions appear: a short-range one, which corresponds to the case where the electron and the hole are in the same Wigner cell in the structure, and a long-range one, which corresponds to th
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