Effect of the anomalous dispersion on the optical characteristics of a quantum well

Abstract

Frequency dependencies of optical characteristics (reflection, transmission and absorption of light) of a quantum well are investigated in a vicinity of interband resonant transitions in a case of two closely located excited energy levels. A wide quantum well in a quantizing magnetic field directed normally to the quantum-well plane, and monochromatic stimulating light are considered. Distinctions between refraction coefficients of barriers and quantum well, and a spatial dispersion of the light wave are taken into account. It is shown that at large radiative lifetimes of excited states in comparison with nonradiative lifetimes, the frequency dependence of the light reflection coefficient in the vicinity of resonant interband transitions is defined basically by a curve, similar to the curve of the anomalous dispersion of the refraction coefficient. The contribution of this curve weakens at alignment of radiative and nonradiative times, it is practically imperceptible at opposite ratio of lifetimes . It is shown also that the frequency dependencies similar to the anomalous dispersion do not arise in transmission and absorption coefficients. Optical methods are widely used at research of electronic properties of low-dimensional semiconductor systems during last decades 1,2,3,4 . It is connected mainly that after interactions with such system, the light wave contains an information about electronic processes, in particular, about an electronic spectrum, lifetimes of excited states and scattering mechanisms. Interesting results turn out, when energy levels of the electronic system are discrete that takes place in quantum dots and quantum wells. Discreteness of energy levels in a quantum well is provided by excitonic states (if incident light is perpendicular to the quantum-well plane), or by quantizing magnetic field directed perpendicularly to the quantum-well plane. In high-quality quantum wells, a radiative broadening of absorption lines at low temperatures and weak doping can be comparable with the contribution of nonradiative relaxation mechanisms or to exceed them. In this case, it is impossible to be limited by linear approximation on interaction of electrons with electromagnetic field, and it is necessary to consider all the orders of this interaction 5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24 . Reflection, absorption and transmission of an electromagnetic wave, which interacts with discrete energy levels of electronic system in a quantum well in the region of frequencies corresponding to interband transitions, were considered in 13,14,15,16,17,18,19 . In these works, a stimulating light could be monochromatic 19 or pulse irradiation 13,14,15 . One 16 , two 17,19 and large number of excited energy levels 18 were considered. Results of these works are applicable for narrow quantum wells, when the inequalityis carried out, where d is the quantum well width, κ is the module of the light wave vector. Actually above mentioned works are true in a zero approximation on the parameter κd.On the other hand...

10.1007/s11451-008-2020-1

“…In our recent paper [15], we analyzed the situation where the interaction of the electromagnetic wave with the electronic system cannot be considered a small perturbation, the nonequalities kd ≠ 0 and ν ≠ ν 1 are satisfied, and the quantum well contains two closely spaced excited energy levels. It was demonstrated that the above functions can be written in the form…”

“…The function N involves the complex quantity ε = ε' + iε'', which has the following form [3,13,15]: (13) This quantity accounts for the influence of the spatial dispersion on the radiative broadening ε'γ rj and the shift ε''γ rj of the energy levels of the doublet.…”

“…Relationship (13) was derived under the assumption that the Lorentz force determined by the external magnetic field is large as compared to the Coulomb and exchange forces in the electron-hole pair and that the dependence of the wave function of the pair on the z coordinate can be represented in the form of the factor Φ(z) [15].…”

“…The electric fields of the electromagnetic pulse involve the reciprocals of the lifetimes γ ri of the energy levels of the doublet in the form ε'γ ri and ε''γ ri . In the case of monochromatic irradiation and pulsed irradiation in the homogeneous medium, the products ε'γ ri and ε''γ ri determine the broadening and the shift of the energy levels of the doublet, respectively [4,13,15]. In the aforementioned general case of pulsed irradiation and the inhomogeneous medium, the relationships for the poles Ω 1 and Ω 2 contain the quantities f 1 γ ri and f 2 γ ri , which depend on the parameters kd and ζ and on the quantities ε'γ ri and ε''γ ri .…”

“…For kd = 0, the shift of the energy levels is absent and their broadening is governed by the quantities ζγ ri . Since γ ri ~ ν -1 at kd ≠ 0, the quantity ζγ ri ~ is the reciprocal of the lifetime at a zero width of the quantum well [15]. Figure 1a depicts the dependences of the function f 1 (determined by relationship (23) and related to the shift of the energy levels of the doublet) on the parameter kd for different ratios ζ.…”