Design requirements for group-IV laser based on fully strained Ge1-xSnx embedded in partially relaxed Si1-y-zGeySnz buffer layers

Theoretical calculation using the model solid theory is performed to design the stack of a group-IV laser based on a fully strained Ge1-xSnx active layer grown on a strain relaxed Si1-y-zGeySnz buffer/barrier layer. The degree of strain relaxation is taken into account for the calculation for the first time. The transition between the indirect and the direct band material for the active Ge1-xSnx layer is calculated as function of Sn content and strain. The required Sn content in the buffer layer needed to apply the required strain in the active layer in order to obtain a direct bandgap material is calculated. Besides, the band offset between the (partly) strain relaxed Si1-y-zGeySnz buffer layer and the Ge1-xSnx pseudomorphically grown on it is calculated. We conclude that an 80% relaxed buffer layer needs to contain 13.8% Si and 14% Sn in order to provide sufficiently high band offsets with respect to the active Ge1-xSnx layer which contains at least 6% Sn in order to obtain a direct bandgap

Motivic Decomposition of Projective Pseudo-homogeneous Varieties

Let $G$ be a semi-simple algebraic group over a perfect field $k$. A lot of progress has been made recently in computing the Chow motives of projective $G$-homogenous varieties. When $k$ has positive characteristic, a broader class of $G$-homogeneous varieties appear. These are varieties over which $G$ acts transitively with possibly non-reduced isotropy subgroup. In this paper we study these varieties which we call {\it \mbox{projective pseudo-homogeneous varieties}} for $G$ inner type over $k$ and prove that their motives satisfy Rost nilpotence. We also find their motivic decompositions and relate them to the motives of corresponding homogeneous varieties.Comment: Final version as appeared in "Transformation Groups

Extensions by Antiderivatives, Exponentials of Integrals and by Iterated Logarithms

Let F be a characteristic zero differential field with an algebraically closed field of constants, E be a no-new-constant extension of F by antiderivatives of F and let y1, ..., yn be antiderivatives of E. The antiderivatives y1, ..., yn of E are called J-I-E antiderivatives if the derivatives of yi in E satisfies certain conditions. We will discuss a new proof for the Kolchin-Ostrowski theorem and generalize this theorem for a tower of extensions by J-I-E antiderivatives and use this generalized version of the theorem to classify the finitely differentially generated subfields of this tower. In the process, we will show that the J-I-E antiderivatives are algebraically independent over the ground differential field. An example of a J-I-E tower is extensions by iterated logarithms. We will discuss the normality of extensions by iterated logarithms and produce an algorithm to compute its finitely differentially generated subfields.Comment: 66 pages, 1 figur