Moment-based Analysis of Mechanical Effects on Trench-Hall Devices Stefano Taschini, Jan G. Korvink* and Henry Baltes Physical Electronics Laboratory, ETH Zurich, taschini@ieee.org ETH Hönggerberg HPT-H6, CH-8093 Zürich, Switzerland *Institute for Microsystem Technology, University of Freiburg, Germany ABSTRACT This paper presents a novel method for the assessment of mechanical effects on trench-Hall devices based on a moment expansion combined with two-dimensional finite- element discretization. Analogously to plate theories, a trun- cated moment expansion of the equations in the transverse coordinate is used to approximate the three dimensional problem. The resulting limited set of two-dimensional prob- lems is discretized by means of finite-elements. The param- etrization of the procedure lends itself to the optimization of the device geometry, therefore representing a valuable tool for the designer. Keywords: Hall effect, finite-element method, moment expansion, mechanical stress. 1. INTRODUCTION In integrated Hall sensors, besides stress-induced device off- set due to the piezoresistance effect, a stress dependence of the magnetic sensitivity due to the modulation of the Hall coefficient, the so-called piezo-Hall effect, affects the qual- ity of the device. Trench-Hall devices [1] have been devised to address the cross-sensitivity caused by the poor carrier confinement in conventional vertical Hall devices. At the same time, its compatibility with CMOS fabrication tech- nology allows the integration with read-out and conditioning circuitry, thus enabling cost-effective miniaturization and high-volume production. Unfortunately, during the fabrica- tion, and in particular during the high-temperature steps in the pre-CMOS-processing, the overlying of dielectric and conducting films with different thermomechanical proper- ties introduces sources of mechanical stress. Important steps have been carried out towards the minimization of the offset [2], but there is still much to investigate in the pres- ence of magnetic field. The high aspect ratio of the device introduces problems of accuracy and resource consumption into the numerical sim- ulation. These problems, mainly due to the large number of three-dimensional elements required to provide good inter- polation properties, can be circumvented by using the same approach found in structural mechanics with plate theories. Plate theories postulate a low-degree polynomial depen- dence of the displacement with respect to the transverse coordinate, eventually leading to a moment expansion of the momentum balance equation. 2. PIEZO-HALL-DRIFT TRANSPORT Electronic transport in a n-type semiconductor is dominated by ohmic drift with a correction due to the magnetic induc- tion in the form of , (1) where is the Hall-mobility [3]. Under mechanical stress, the two scalar coefficients and in equation (1) are replaced by two tensors and with a local dependence on the stress field. For not too large stress levels, a linearized dependence is justified. The dependence of the conductivity on the stress tensor is formulated in terms of the fourth- order tensor of the piezoresistance coefficients [4]: . (2) In equation (2), denotes the three-dimensional identity matrix; the double contraction is defined in terms of the cartesian coordinates of the tensors as . (3) Similarly, the piezo-Hall coefficients [5] form the fourth- order tensor relating the Hall mobility to the stress: . (4) By introducing the dimensionless effective magnetic induc- tion , equation (1) becomes . (5) Solving equation (5) for the current density, yields the anisotropic linearity relation with the tensor (6) as the effective conductivity. In equation (6), denotes the tensor that contracted with an arbitrary vector yields the cross product ; the symbol denotes the ten- sor product. Equation (6) is verified immediately by insert- ing Ohm’s law into the left-hand side of equation (5). J Μ * B J  – ΣE = Μ * Μ * Σ Μ * Σ T Π Σ Σ I 3 Π T : – ( 29 = I 3 Π T : Π T : ( 29 ij Π ijkl T kl l 1 = 3  k 1 = 3  = P Μ * Μ * I 3 P Σ : + ( 29 = B * Μ * B  = J B * J  – Σ E  = J Σ * E  = Σ * 1 1 B * ( 29 2 + ----------------------- I 3 B *  B * B *  + + [ ] Σ  = B *  u B * u   J Σ * E  =