Analysis of Magnetic Field of the Conducting Line with cladding layer

Abstract In the structure of Magnetic Random Access Memory (MRAM), there exists orthogonal conducting wires clad with high permeability magnetic materials, and the Tunneling Magnetoresistance (TMR) cells is located between them. By passing the current into wires, they will produce two orthogonal magnetic …elds on the cells, so the magnetic moment of the free layer in TMR will rotate to parallel or antiparallel the magnetic moment of the pinned layer in order to write the cell into 0 or 1. The high permeability magnetic materials is used to concentrate the magnetic flux, once the wires are clad with them, we can pass less current to produce the same magnetic field. It has advantages of saving the power and preventing the current over the load of the wires. The main topic of the thesis is to calculate the magnetic field produced by the conducting wires clad with high permeability magnetic materials. The geometry of the conducting wire and the high permeability magnetic materials is larger in one dimension (along the current direction, say z-direction) than that in the other two dimensions, so we approximately take the two dimensional real space into account. It implies that the magnetic field is independent of z-axis. According to this fact, we make use of one way which transforms the real space into the imaginary space to evaluate the magnetic field on TMR cells. That is "Schwarz-Christo¤el Transformation". Schwarz-Christos¤fel Transrmation provides a method that maps the interior of the polygon to the upper-half plane, so we can deal with the problem in the upper-half plane. Finally, we integrate directly the current and the image current source in the real space to obtain the magnetic field. In addition we study the features of magnetic …eld under several kinds of geometry of cladding layer.Contents 1 Introduction . . . . . . .. . . . . . . . . . . . .. . .11 1.1 Introduction to MRAM . . . . . . . . . . . . . . . . . . . . . . . . . . .11 1.2 Introduction to coherent rotation . . . . . . . . . . . . . . . . . . . . . . .. . 13 2 The Theory of Applied Mathematics . . . . . . . . . . . 17 2.1 Conformal Mapping . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 The Schwarz-Christo¤el Transformation . . . . . . . . . . . . . . . . . . . . . 21 3 Calculation of the Magnetic Field . . . . . . . . . . . 27 3.1 Transformthe boundary of cladding layer by S-C Transformation. . . . . . .. . . . . . .. . . . . . .. . 27 3.2 The image current distribution . . . . . . . . . . . . . . . .. . . . . . . 37 3.3 Calculation of the magnetic …field . . . . . . . . . . . . . . . . . . . . . . . .. . 39 3.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . 42 4 Summary. . . . . . . . . . . . . . . . . . . . . . . . .5