Line-shape model for the modulated reflectance of multiple quantum wells

In this paper we describe a model for the reflectance and modulated reflectance line shapes of quantum-well (QW) systems. In particular, we concentrate on electroreflectance (ER) of multiple quantum wells (MQW’s). The work is an extension of our previous line-shape model for single quantum wells (SQW’s), which is summarized briefly. In both cases we show that optical interference of light reflected from the well(s) and from the front surface of the sample has a dramatic effect upon the line shape. A simple analytical formula is derived for both the SQW and MQW cases that relates the ER to the QW dielectric function. Line shapes are calculated for a single exciton with different numbers of wells in the stack, which are then compared with an exact analysis using a transfer-matrix method. For SQW’s the line shape is a mixture of the modulated real and imaginary parts of the QW dielectric function, which depends on the depth of the well in the sample. Therefore, in a MQW, each of the wells has a different contribution to the ER and the line shape is not just a scaled-up version of that for a SQW, as it is for absorption spectroscopies. As the stack thickness increases, the line shape becomes increasingly complicated, essentially due to the variation of the light penetration depth across the exciton profile. Our explicit analysis allows a clear understanding of the influence of each sample parameter on the line shape. It is shown that the center of the line shape is determined by the front of the MQW, while extra features appear in the wings as the number of wells increases, which originate from light reflected by the back of the stack. Finally, we discuss how our predictions compare with our recent measurements on GaAs/${\mathrm{Al}}_{0.3}$ ${\mathrm{Ga}}_{0.7}$As MQW samples. This comparison demonstrates the importance of including interference effects, while also highlighting the limitations of the model.