A magneto-resistive (MR) stripe element includes a magnetically active body portion and a tri-layer electrical conductor structure arranged proximate the magnetically active body portion. The conductor structure has an alpha-Ta bi-layer film and a chromium capping layer. The alpha-Ta bi-layer film includes a chromium base layer and a tantalum layer. The capping layer caps the alpha-Ta bi-layer film such that the tantalum layer is disposed between the chromium base layer and the chromium capping layer. The tri-layer conductor structure has minimized compressive stress after deposition of the three layers and an even lower compressive stress after annealing. The thickness of the chromium capping layer is dependent at least upon the thickness of the chromium base layer such that the tri-layer conductor has a minimized compressive stress after deposition and annealing. The MR stripe element may be incorporated in a magnetic read head for reading data from a magnetic storage medium.