1 ABSTRACT We present an optimization strategy for the offset reduction of integrated magnetic field sensors with simultaneous max- imization of sensitivity. We use the topology optimization method to automatically find a design with maximal sensitiv- ity for given constraints on the offset by changing the con- ductivity of the Hall plate locally. Keywords: Topology optimization, Integrated magnetic field sensor, Hall plate, Offset, Sensitivity 2 INTRODUCTION Magnetic field sensors based on the Hall effect make use of the Lorentz force acting on charge carriers mov- ing in the plane of the device in a direction perpendicular to the carrier velocity vector. Since electrons and holes move to opposite sides of the device they accumulate at the surfaces and build up a potential difference between those two surfac- es, which is a measure for the magnetic field intensity (Fig. 1). Figure 1: Hall device setup The equipotential lines therefore twist by an angle of with denoting the Hall mobility of the electrons. In the production process of an integrated circuit (IC) device, two different lithographic masks are used to define the out- lines of the doped well and the metal contacts on top. Since the Hall displacement of the equipotential lines is rather small for common magnetic fields, a notable variation in the offset appears if the two masks are misaligned [1]. The equivalent magnetic field is for a misalign- ment of and may assume values of several mT. This drawback is compensated applying sophisticated elec- tronic circuits that interchange the operating current and measurement contacts, i.e. spinning current methods [2,3]. With the topology optimization method, so far mainly ap- plied to structural mechanics [4,5,6], it is possible to find a shape which maximizes the sensitivity for a given maximum offset. Thereby the complexity of the driving circuitry is re- duced. 3 OBJECTIVE FUNCTION The aim is to find a dopant distribution which optimizes these objectives. Since it is easier to manufacture a uniform distribution, a design with a 0/1-conductivity distribution (which means no intermediate concentrations) is preferred. Though it is difficult to find an optimal design without the in- termediate concentrations, the use of a penalty function can lead to a clearer design. Therefore, the conductivity is given by a power law , and a constraint for the total dopant dosage is introduced. For the design of mecha- nisms in structural mechanics this approach has proven successfull [7]. The optimization problem is written as: maximize: (signal) subject to: (keep conductivity reasonable) (reduce the offset) (limit the dosage) is the voltage deviation of two possible misaligned measurement contact positions as shown in Fig. 2. The val- ues for both sides of the sensor plate are averaged. For the modeling of the measurement, the Hall plate is con- nected to voltmeters with an input resistance of 10 MΩ at each of the measurement contacts. 4 TOPOLOGY OPTIMIZATION The discretized Hall plate consists of a regular 2D tensor- product finite element (FE) mesh (Fig. 2). A separate mate- rial density , with , is assigned to each ele- ment e. corresponds to the doping concentration . The nonlinearity in the material conductivity is introduced artificially to enforce the binary design. This makes use of the 2D character of the system and corresponds to a variable thickness of the doping perpendic- ular to the Hall plate [6]. These densities form the design variables for the optimization. The simulated sensor is symmetric with respect to the oper- ating current contacts. This ensures that both polarities of the F qv B × = V H B w Contact Offset ∆l V S θ θ B μ B B ( 29 atan – = μ B B ∆l wμ ( 29 ∕ = ∆l σ σ max ρ e p = N c V Hall V Source ∕ = σ min σ x (29 σ max ≤ ≤ o V Offset ( V Hall 29 2 ε 2 ≤ ∕ = N N total ≤ V Offset ρ e ρ min ρ e 1 ≤ ≤ ρ e N e N max ρ e = σ σ max ρ e p = Optimization of Integrated Magnetic Field Sensors Jan Lienemann, Andreas Greiner, Jan G. Korvink and Ole Sigmund * IMTEK -- Institute for Microsystem Technology, Albert Ludwig University Georges-Köhler-Allee 103, 79085 Freiburg, Germany, lieneman@imtek.de, PHONE: ++49 761 203 7381, FAX: ++49 761 203 7382 * Department of Solid Mechanics, Technical University of Denmark DK-2800 Lyngby, Denmark