Introduction Recently, there are more and more interests in studying and discovering the behavior of low-dimensional system, such as compositional superlattices, doped superlattices, quantum wells, quantum wires and quantum dots. The confinement of electrons and phonons in lowdimensional systems considerably enhances the electron mobility and leads to unusual behaviors under external stimuli. Many attempts have conducted dealing with these behaviors, for examples, electron-phonon interaction

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... ects in two-dimensional electron gases (graphene, surfaces, quantum wells) (Ruker et al., 1992; Richter et al., 2009; Butscher et al., 2006). The dc electrical conductivity (Vasilopoulos et al., 1987; Suzuki, 1992) , the electronic structure (Sager et al., 2007) , the wavefunction distribution (Samuel et al., 2008) and the electron subband (Flores, 2008) in quantum wells have been calculated and analyzed. The problems of the absorption coefficient for a weak electromagnetic wave in quantum wells (Bau&Phong, 1998), in doped superlattices (Bau et al., 2002) and in quantum wires (Bau et al., 2007) have also been investigated and resulted by using Kubo-Mori method. The nonlinear absorption of a strong electromagnetic wave in low-dimensional systems have been studied by using the quantum transport equation for electrons (Bau&Trien, 2011). However, the nonlinear absorption of a strong electromagnetic wave in low-dimensional systems in the presence of an external magnetic field with influences of confined phonons is stills open to study. In this chapter, we consider quantum theories of the nonlinear absorption of a strong electromagnetic wave caused by confined electrons in the presence of an external magnetic field in low dimensional systems which considered the effect of confined phonons. The problem is considered for the case of electron-optical phonon scattering. Analytic expressions of the nonlinear absorption coefficient of a strong electromagnetic wave caused by confined electrons in the presence of an external magnetic field in low-dimensional systems are obtained. The analytic expressions are numerically calculated and discussed to show the differences in comparison with the case of absence of an external magnetic with a specific AlAs/GaAs/AlAs quantum well, a compensated n-p n-GaAs/p-GaAs doped superlattices and a specific GaAs/GaAsAl quantum wire. This book chapter is organized as follows: In section 2, effect of magnetic field on nonlinear absorption of a strong electromagnetic wave in a quantum well. Section 3 presents the effect www.intechopen.com Behaviour of Electromagnetic Waves in Different Media and Structures 276 of magnetic field on nonlinear absorption of a strong electromagnetic wave in a doped superlattice. The effect of magnetic field on nonlinear absorption of a strong electromagnetic wave in a cylindrical quantum wire is presented in section 4. Conclusions are given in the section 5. 2. Effect of magnetic field on nonlinear absorption of a strong electromagnetic wave in a quantum well 2.1 The electron distribution function in a quantum well in the presence of a magnetic field with case of confined phonons It is well known that in quantum wells, the motion of electrons is restricted in one dimension, so that they can flow freely in two dimensions. In this article, we assume that the quantization direction is in z direction and only consider intersubband transitions (n≠n') and intrasubband transitions (n=n'). As well as, we consider a quantum well with a magnetic field B  applied perpendicular to its barriers. The Hamiltonian of the confined electron-confined optical phonon system in a quantum well in the presence of an external magnetic field B  in the second quantization representation can be written as (Mori & Ando, 1989; Bau & Phong, 1998; Bau et al, 2009 ):