Stresses in strained GeSi stripes and quantum structures: Calculation using the finite element method and determination using micro-Raman and other measurements

Stresses in heterostructures containing lattice mismatched GeSi stripes (of half-width l and thickness h) deposited on Si substrates are calculated using the finite element (FE) method. It is shown that the stress distribution is a unique function of l/h, it does not depend on I and h separately or on the substrate dimensions if the substrate dimensions are sufficiently large. Ratio E(E) = E(f)/E(s) of the Young's moduli (E(f) is the Young's modulus of the stripe and E(s) is the Young's modulus of the substrate) changes from 0.78 for Ge/Si to nearly 1 when the Ge fraction in the layer is small. The effect of this change on the stress distribution is calculated and is found to be small but not negligible. Stress distribution in the surface layer of the stripe is a weak function of R(E). Therefore values of stresses given in this paper can also be used for GaAs-based heterostructures. Analytical models are not capable of giving accurate values of the stresses in these structures. These values show that as h increases and I is kept constant, stress at a constant height z in the stripe decreases and at a constant depth z in the substrate increases, first rapidly and then slowly. It saturates and becomes practically constant for l/h < 0.5. Finite element calculations of circular mesas (quantum dots) are also reported. Experimental values of stresses in stripes, quantum wires and quantum dots are found to be in good agreement with the values calculated by the FE method.