Read Back Signals in Magnetic Recording - Research Group Fidler

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Here the matrix = ( M ij ) Numerical Methods M describes a dipole interaction of all boundary nodes of V, with all nodes of V, and is normally fully populated. This matrix has to be calculated only once, because it does not depend on the boundary conditions. For large models this matrix requires a lot of memory. So it is more convenient to calculate only the interaction between boundary and boundary nodes, and evaluate the magnetic potential Ψ S ,2 by solving the Laplace equation in (4.34) with Dirichlet boundary conditions. In addition the interaction matrix is compressed as described in next section. 4.3 Adaptive Cross-Approximation Technique The discretization of boundary integral equations leads to large dense matrices that have no explicit structure in general. However, by suitable renumbering and permuting the boundary nodes, the dense matrix can be written in a form that contains blocks that can be approximated by low-rank matrices [25]. The renumbering of the nodes is done by geometrical criterions. Consecutive boundary nodes should be located close to each other. A simple example of the clustering is given in Figure 4.2. Boundary nodes that are located close to each other are combined in a cluster. A large distance between cluster results in entries in the renumbered boundary matrix that are well separated from each other. For example the cluster with the nodes {6,7,8,9,10} and the cluster {1,2,3,4,5} are admissible cluster pairs. This pair of clusters leads to a block in the boundary matrix which can be approximated by a low-rank matrix. The boundary matrix corresponding to the clustering is also shown in Figure 4.2. The off-diagonal blocks of the matrix represent far-field interactions. This technique leads to compressed interaction matrices, which save a lot of memory. Additionally the algorithmic complexity for setting up the boundary matrix and for matrix-by- vector products is reduced from 2 ON ( ) to ON ( ) . 48
Numerical Methods Figure 4.2: The left figure shows a set of 10 nodes after renumbering and the clustering. The right figure shows the associated boundary matrix structure. The large off-diagonal blocks, corresponding to far field interactions of two clusters, can be approximated by lowrank matrices [25]. 4.4 Conductor Model The sense current through the MR sensor generates a magnetic field, which has to be taken into account. The current is even used for biasing in AMR heads (see Section 6.1.2). Therefore the sense current cannot be neglected. So we have a conductor model in addition to the magnetic model. As there are materials which are magnetic and conductive these two models can overlap. The current model consists of materials with different conductivities and magnetoresistive effects as described in Section 2.8. So the current density must be assumed as function of locus and as function of magnetization. σ=σrM (, ) (4.39) Again we use a tetrahedral mesh and the Finite Element Method to calculate the electric potential and further the current distribution. 4.4.1 Conductivity In case of non-magnetoresistive materials we have constant conductivities. In magnetoresistive materials, however, the conductivity has to be evaluated from the magnetization (see (2.50) or (2.51)). To specify conductivity and magnetoresistance there are 49
Page 1 and 2: DIPLOMARBEIT Read Back Signals in M
Page 3 and 4: Abstract This thesis concentrates o
Page 5 and 6: Contents 3.2.4 Signal of Written Bi
Page 7 and 8: Introduction 1 IntroductionEquation
Page 9 and 10: Introduction In 1955, before the RA
Page 11 and 12: 2 BasicsEquation Section (Next) 2.1
Page 13 and 14: Basics curl H = 0 . (2.15) So H is
Page 15 and 16: Basics This is justified, because t
Page 17 and 18: Basics Here J ij represents the exc
Page 19 and 20: s Basics ∂J α ∂J =−γ J× He
Page 21 and 22: Basics for spin down electrons. An
Page 23 and 24: Basics Figure 2.4: Schematic model
Page 25 and 26: Basics channels in the parallel and
Page 27 and 28: Basics Nowadays there are again eff
Page 29 and 30: Analytical Calculations 3 Analytica
Page 31 and 32: Analytical Calculations developing
Page 33 and 34: Analytical Calculations t C 8 π =
Page 35 and 36: µ 0 Hsig [T] 0.030 0.025 0.020 0.0
Page 37 and 38: Analytical Calculations n n ⎛HH,
Page 39 and 40: z Analytical Calculations Figure 3.
Page 41 and 42: 4 Numerical MethodsEquation Numeric
Page 43 and 44: Numerical Methods ϕ ( x ) =δ . (4
Page 45 and 46: Numerical Methods current flowing o
Page 47: Numerical Methods ∫ ∫ ∫ si =
Page 51 and 52: Numerical Methods j =−σgrad Φ.
Page 53 and 54: Numerical Methods within the volume
Page 55 and 56: Numerical Methods Integration of Am
Page 57 and 58: 4.5 Time integration Numerical Meth
Page 59 and 60: 5 Read Head DesignEquation 5.1 Bias
Page 61 and 62: Read Head Design The exchange bias
Page 63 and 64: Read Head Design free layer (see Fi
Page 65 and 66: FEM Simulations 6 FEM SimulationsEq
Page 67 and 68: FEM Simulations Magnetic Material J
Page 69 and 70: 6.2 Results FEM Simulations In the
Page 71 and 72: Output [V] 0.108 0.107 0.106 0.105
Page 73 and 74: FEM Simulations has a greater influ
Page 75 and 76: Output Voltage [V] 0.1074 0.1072 0.
Page 77 and 78: 7 OutlookEquation Section (Next) Ou
Page 79 and 80: Appendix A In case we have the func
Page 81 and 82: Appendix B Moreover the second inte
Page 83 and 84: List of Figures List of Figures Fig
Page 85 and 86: List of Figures Figure 6.1: The mag
Page 87 and 88: Bibliography [11] H. Katayama, M. H
Page 89 and 90: Bibliography [38] W. F. Brown Jr.,
Page 91: Lebenslauf Otmar Ertl Reinprechtsdo

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