In this section the dynamics of the semiconductor is described with the help of the Heisenberg equations-of-motions for the different sorts of density functions using an
approximation scheme which was introduced by Fricke [42]. In a first step the density functions are recursively expressed in terms of correlation functions starting with the one-particle expectation values. The infinite hierarchy of equations-of-motions which is formed by these new functions is then truncated by retaining only the correlation functions up to a certain order. The remaining differential equations can either be solved numerically or studied analytically in order to determine the time behavior of the system after an optical excitation.
In the following the interest is focused on the dynamics of the interband polariza-tionhψvl†ψclitand the distribution functions of the valence electrons and the conduction electronshψvl†ψvlitandhψ†clψclitwhose time behavior is reflected in the linear and the nonlinear optical properties of the system. Since the Hamiltonian H(t)conserves the total number of electrons these one-particle densities coincide with their correlated parts. Their dynamics is determined by the equation
id The coefficients of the one-particle energy matrixε˜λ1λ2;l(t)which appear in Equation 2.16are composed of the corresponding coefficients of the bare energy matrixελ1λ2;l(t) and the dynamical Hartree-Fock contributions of the electron-electron interaction:
˜
The two additional contributions on the right-hand side of Equation 2.16 represent corrections to the Hartree-Fock approximation. The first contribution describes the in-fluence of the electron-phonon interaction on the dynamics of the functionhψλ†
1lψλ2lit. where the new functions which appear on the right-hand side of Equation 2.18 are referred to as first order assisted densities. Strictly speaking, these phonon-assisted densities should be expressed with the help of the corresponding correlation functions, for example
hψ†λ
1l+pψλ2lbpit=hψλ†
1l+pψλ2lbpict+hψλ†
1l+pψλ2lithbpit, (2.19)
if the kinetic equations are formulated within the framework of the formalism pre-sented in Reference [42]. However, the expectation valueshbpit andhb†pitvanish for p 6= 0since the HamiltonianH(t)guarantees the conservation of the total quasi mo-mentum modulo a reciprocal lattice vector. For this reason the complete first order phonon-assisted densities are identical with their correlated parts. The second contri-bution in Equation 2.16 is the collision term for the electron-electron scattering pro-cesses. It satisfies the equation The correlation functions with four electronic field operators which appear on the right-hand side of Equation2.20are defined by the following relation
hψ†λ
where the conservation of the total quasi momentum has already been taken into ac-count. If both correction terms are neglected, the dynamics for one-particle densities with different quasi momenta is decoupled and the time behavior is described by the well-known semiconductor Bloch equations. If the electron and the electron-phonon collision terms are taken into account, it is possible to describe the dephasing and relaxation processes which cause the decay of the initially excited state of the sys-tem. While the formation of bound pairs of valence holes and conduction electrons (excitons) in the low density limit can already be described within the framework of the semiconductor Bloch equations the complete description of bound molecule-like complexes of two valence holes and two conduction electrons (biexcitons) requires the consideration of the higher order correlation functions fromδhψλ†1lψλ2liee. These functions are also necessary for the description of the screening of the electron-electron interaction in the high density limit.
In the following the influence of the electron-phonon scattering processes on the relaxation of the excited semiconductor is placed at the center of interest. For this reason the electronic collision term in Equation2.16 is neglected and the attention is now focused on the time behavior of the first order phonon-assisted densities, which is determined by the equation
It is obtained with the help of the same factorization scheme which has already been applied above. As in Equation2.16the different contributions on the right-hand side of Equation2.22can be assigned to two different groups. The terms in the first two lines describe the renormalized one-particle dynamics within the framework of the Hartree-Fock approximation while the two expressions in the last line contain the higher order correction terms. The form of the correction term which is related to the electron-phonon scattering processes is determined by the equation
δhψλ† while the contribution which is due to the electron-electron interaction satisfies the relation The new functions which appear on the right-hand side of Equation2.23 are the cor-related parts of the so-called second order phonon-assisted densities. They are defined by the relations
The new sort of correlation function which has been introduced in Equation 2.24 is Apart from the electronic correlation functions the kinetic equations for the first or-der phonon-assisted density functions contain purely phononic densities, namely the phonon distribution functionhb†pbpitand the two phonon coherencehb−pbpit, which is also called phonon distortion [55]. The dynamics of these densities is determined by the two kinetic equations
The number of one-particle density functions hψλ†
1lψλ2lit, hb†pbpit and hb−pbpit
which have to be taken into account when solving the kinetic equations increases lin-early with the system size N. On the other hand, the number of first order phonon-assisted density functions increases withN2 while the number of the other electronic density functions which appear in Equations2.16 to2.29increases with an exponent which is even larger. In order to keep the influence of the finite-size effects small, it is therefore necessary to truncate the system of kinetic equations at the present level.
Consequently, the correlation functions in the last three lines of Equation2.23and the last two lines of Equation2.24 are not taken into account explicitly in the following calculations. However, the influence of some of theses functions can be considered approximately in the form of correction terms. This will be explained in Section2.3in detail.