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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Five different Hall Effect sensors were modeled and their performance evaluated using a three dimensional simulator. The physical structure of the implemented sensors reproduces a certain technological fabrication process. Hall voltage, absolute, current-related, voltage-related and power-related sensitivities were obtained for each sensor. The effect of artificial offset was also investigated for cross-like structures. The simulation procedure guides the designer in choosing the Hall cell optimum shape, dimensions and device polarization conditions that would allow the highest performance.

One of the most commonly used sensing technologies today consists of CMOS Hall Effect sensors, based on magnetic phenomena. These sensors are primarily employed as current sensors and serve many low-power applications like position sensing and contactless switching within automotive and industrial electronics [

Regarding the Hall Effect device performance investigation, one would need to look, among others, for the highest sensitivity and the lowest offset. The geometry plays an important role in the Hall Effect sensors performance and was studied by the authors in [

The offset and sensitivity are important figures of merit in Hall sensors performance evaluation [

In the microelectronics recent development, the solving of fundamental semiconductor device equations by numerical methods is a productive investigation tool to predict the behavior and assess the performance of various devices.

The present paper analyzes the influence of the shape, dimensions, position of contacts and offset on the Hall Effect sensors performance, including Hall voltage and sensitivity, with the aid of three dimensional physical simulations. In this sense, the study also proposes an analysis of artificially induced offset. In order to ensure Hall Effect sensors optimal design, we use three-dimensional numerical solutions to the system of partial differential equations governing galvanomagnetic carrier transport in magnetic-field-sensitive semiconductors.

Section II presents the motivation behind the simulation approach, the basic physical model of the carrier transport in semiconductors and the methodology used for 3D structures simulation, presenting the design parameters for all analyzed Hall Effect devices. Section III is dedicated to accurate estimation of Hall voltage, different types of sensitivities and influence of geometrical mismatch on the structures offset by performing a comparative study on five different Hall devices. The purpose of this section is to finally reveal which of the simulated magnetic sensors displayed the best performance.

In general, the Hall voltage is defined by the relation:
_{H}_{bias}

The absolute sensitivity _{A}

Relative sensitivities can also be defined. Therefore, the current-related _{I}_{V}_{bias}

Since the Hall voltage and therefore sensitivity are inversely proportional to the n-well doping concentration, a lightly doped n-well is normally used in the fabrication process of the Hall Effect sensors.

Different Hall Effect devices were integrated in a 0.35 μm CMOS technology and evaluated for Hall voltage, sensitivity, offset,

The offset analysis was of particular interest because in reality, Hall Effect sensors have offset. In this sense several samples, each one containing 64 cells (eight different geometries times eight locations), were tested. The experimental data obtained for the offset at room temperature for different biasing currents is presented in [

Among the eight different integrated and tested Hall cells, the minimum offset was obtained for the XL, which is basically a classical Greek-cross but with the dimensions scaled up by a certain factor with respect to a basic shape. This particular geometry will be reproduced by simulation later within the present work, where additional details can be found including design parameters information.

_{bias}_{bias}

In particular, the residual offset mathematical function of the biasing current has a quadratic dependence, for the 2-phases spinning current [_{bias}^{2} in the residual offset, cannot be completely compensated for _{bias}

The aim of the present study is to employ three dimensional simulations for designing and selecting the best Hall device shape to be used in a certain integration process. The performance assessment is conducted by investigating the sensors Hall voltage, sensitivity and offset.

In semiconductor materials, the classical carrier transport model [_{D}_{A}

It is to be mentioned that the magnetic induction effect only manifests in the mathematical relations which define the current density. Equivalently speaking, in the absence of the magnetic field, the continuity equations and the Poisson's

By using the Synopsys Sentaurus TCAD tool [

At each point of the grid, three unknowns will be considered, namely

The magnetic field acting on the semiconductor structure for Hall voltage generation was handled by the galvanic transport model. The analysis of magnetic field effects in semiconductor devices is done by solving the transport equation of electrons and holes inside the device. The usual drift-diffusion model of the carrier densities
_{α}

The physical section of the simulation included a doping dependence mobility model together with Shockley-Read-Hall and Auger recombination processes as was considered in a paper [

For a good tradeoff between accuracy and simulation run time, the three dimensional meshed structures of the Hall effect devices should contain a sufficient number of points. Smaller meshing dimensions and higher number of points increase the accuracy of the simulation results, but would require more CPU time and longer execution. With respect to the previous work the meshing strategy was improved.

The meshes of the simulated geometries contained between 40,000–70,000 points with refinement functions included to ensure maximum convergence and minimum numerical offset. Additional refinement windows have been placed on contacts in order to improve the simulation convergence and to decrease the numerical offset. A mesh step between 0.1 and 1 μm on the three axes was used for the mesh refinement window. In this way, for all structures the numerical offset does not exceed two milivolts for maximum biasing current. To address any further convergence issues, the magnetic field was ramped up to the required value of the magnetic induction.

To assess the Hall Effect sensors performance, FEM lumped circuit models were already developed by the authors [

Therefore, a p-substrate with 10^{+15} cm^{−3} boron concentration and an active n-well region doped with 1.5 × 10^{+17} cm^{−3} arsenic concentration in the form of a Gaussian profile implantation were used. This doping profile allows an average mobility of 0.0630 m^{2}·V^{−1}·s^{−1}. The thickness is 5 μm for the p-substrate and 1 μm for the implantation of the n-doped profile active region, respectively.

Attention should be given to doping profile smoothing in order to ensure a good simulation convergence. Therefore the abrupt edges were avoided by imposing a decay length of a hundred of nanometers at the p-n junction. For testing purposes, each structure was endowed with four electrical contacts, amongst which two are for biasing the device and the other two opposite ones for the measurement of the voltage drop difference.

The geometrical design parameters of all the five simulated Hall Effect devices are given in

The three-dimensional representations of the five simulated geometries are illustrated in

Regarding the polarization of Hall cells, imposing a certain voltage on electrode

For the analysis of the Hall Effect devices behavior, all the structures were simulated using current biasing, without and with magnetic field. In the present study, the biasing current was ramped from 0 to 1 mA.

The effects of the dimensions (input data), respectively the geometrical correction factor, on the Hall Effect sensors technical performance (output data) were analyzed by authors in a recent paper [

The classical cross structures dispose of contacts at the extremities of the four arms. The biasing and sensing will ensure a maximum sensitivity, but the structures can be more affected by any mismatch at the p-n junction. The idea is to increase the dimensions of this classical cross in order to be less prone to border asymmetries. The fourth analyzed shape, the borderless cell, is equipped with very small electrical contacts and they are located closer towards the center of the structure and farther away from the p-n junction. This specific structure could minimize the influence of any border errors but will also affect the sensitivity. The fifth shape, the optimum cell, is a combination of scaled up dimensions and contacts situated half way through with respect to the contacts of XL and borderless cells respectively.

The Hall device I-V characteristic is obtained by simulations for each cell. Its representation for B = 0.5 T is incorporated in

For the Hall Effect sensors, any nonlinearity that might be seen in the I-V characteristic is explained [

The total output voltage of a Hall Effect device is given by the following relation:

Even though the shapes are symmetric we obtain a non-zero offset, which is the numerical offset from the simulator. Therefore, the meshing strategy was adapted to minimize it as much as possible.

We were interested to investigate the offset in order to have accurate information for Hall voltage and sensitivity, as the offset is a parasitic voltage adding to the total output voltage. The offset measurements were performed in the absence of magnetic field while for Hall voltage and sensitivity estimation the magnetic induction was considered B = 0.5 T. This particular value for the magnetic field induction was used to closely reproduce by simulation the magnetic field of B = 0.497 T used for the integrated Hall devices measurements.

When applying a magnetic field of certain intensity, the carriers deviate under the influence of Lorentz force and thus the Hall voltage is forming between the opposite contacts. In

The Hall voltage of all the simulated structures is presented _{Hall}_{Hall}_{Hall}_{Hall}_{Hall}

The same type of graphs is also investigated for the optimum and borderless cell respectively. The peak of the electrostatic potential is 1.8 V for the borderless cell and 2.3 V for the optimum cell. From these graphs we can also deduct the length of the electric contacts, for optimum and borderless cells as the electrostatic potential is constant in that region. On the Ox cuts of L, XL, basic cells, the descent from the peak is not always a straight line due to the non-homogeneity in material mobility, conductivity, sheet resistance,

Using the definitions for the absolute, current-related and voltage-related sensitivities in

The current-related sensitivity is increasing with the biasing current, while the voltage-related sensitivity is decreasing with the biasing current. The explanation of the latter mechanism is the fact that S_{V} can be rewritten as the ratio of the current-related sensitivity to the input resistance _{A}

The selection process of the best Hall Effect device is based on analyzing several parameters behavior such as sensitivity, offset, dissipated power and Silicon surface. By consequence, it is advised to inspect a more complex cost function. There are circuit methods such as the current spinning technique to reduce the offset and the Silicon surface can be traded by the designer for a high sensitivity. Therefore, obtaining the highest sensitivity with a good tradeoff with offset and power dissipated seems to be prevalent.

The dissipated power was calculated within each structure. The ratio of the absolute sensitivity to the power dissipated within the device was also investigated. Even though the relation (6) for power-related sensitivity is not a standalone equation and can be deducted from

The variation of power-related sensitivity _{P}vs._{bias}

We can note that the Greek-like cells and the optimum cell have almost the same power-related sensitivity, with a slightly higher value for the optimum cell. We can observe that for this particular figure of merit, the geometry has less importance, with the best performance belonging to the optimum cell. There is an improvement of almost 10% of the optimum cell with respect to XL structure for constant current of 1 mA. From this perspective, the optimum cell seems like a good candidate.

The above discussion on different types of sensitivities leads to the conclusion that the Hall device polarization is important and dictates which shape should be chosen in order to guarantee the best performance.

Offset voltage can be generated by imperfections in the fabrication process, misalignment of contacts, non-uniformity of material resistivity and thickness, mechanical stress in combination with the piezoresistance effect [_{offset}

Therefore, the offset voltage caused by an asymmetry of the bridge is given by:
_{in}

Previous measurements performed on the Hall Effect devices for offset evaluation released information about how this quantity changes with the shape. Numerical values of the offset for the cross-like integrated cells are displayed in

An analysis of the artificial offset is now intended to see how an induced asymmetry can influence the offset and finally find the shape to guarantee the lowest value of this quantity. At this point, we mainly focus on the offset that might be created by a misalignment in the sensors shape. The Hall devices are equipped with two biasing contacts (

As the offset is a random process, a possible location was presumed on the contact

In the present work, simulated artificial offsets induced by small Ox-axis displacements of 0.5, 0.25 or 0.15 μm on the biasing contact

For the XL cell, the influence of the mismatch on the offset was the least. It seems that by increasing the dimensions of a cell we will be diminishing any errors that might appear on the borders and therefore minimizing the offset. This fact was also analyzed in [

The performances of all the analyzed Hall Effect sensors, including Hall voltage, current-related sensitivity, power-related sensitivity and dissipated power are summarized in the

The influence of the shape, dimensions, offset on the Hall Effect sensors performance was analyzed using 3D physical simulations capable of considering the magnetic field influence on semiconductors.

To this purpose, five different Hall Effect sensors of a certain technological CMOS fabrication process were modeled. The Hall Effect sensor configurations were simulated and evaluated for Hall voltage, absolute, current-related, voltage-related, power-related sensitivities and offset.

The estimations of these important parameters finally allow choosing the best shape depending on the device polarization used in the circuit and the figures of merit priority. In particular, the simulation and experimental results are in good agreement.

The simulation procedure guides the designer in accurately modeling and characterizing specific magnetic sensor shapes of a certain fabrication technology. Estimating their Hall voltage, sensitivity will aid in choosing the Hall cell optimum fabrication process, shape, dimensions and polarization in terms of the performances envisaged to be achieved.

Future investigations on the causes of the offset and a systematic approach that would correlate the causes with quantification in the offset value are envisaged to be performed in future articles.

Measured 4-phases residual offset voltage

3D representation of the simulated basic Hall cell.

3D representation of the L simulated Hall cell.

3D representation of the XL simulated Hall cell.

3D representation of the borderless simulated Hall cell.

3D representation of the optimum simulated Hall cell.

The simulated Hall devices I-V characteristics.

Electrostatic potential (V) for basic structure, B = 0.5 T.

Electrostatic potential (V) for L structure, B = 0.5 T.

Electrostatic potential (V) for XL structure, B = 0.5 T.

Electrostatic potential (V) for borderless structure, B = 0.5 T.

Electrostatic potential (V) for optimum structure, B = 0.5 T.

Hall Effect devices Hall voltage _{bias}

Electrostatic potential (V) in orthogonal cuts on Oy and Ox for the cross-like cells (XL, L and basic respectively).

Electrostatic potential (V) in orthogonal cuts on Oy and Ox for optimum and borderless structures.

Conduction current density for basic cell with emphasis on the biasing contacts

Conduction current density for basic cell (X = 3.25).

Conduction current density for basic cell, with emphasis on the sensing contacts

Hall Effect devices absolute sensitivity _{bias}

Current-related sensitivity of the simulated Hall Effect devices.

Voltage-related sensitivity of the simulated Hall Effect devices.

Power-related sensitivity

The bridge circuit model of a Hall cell.

The measured single phase offset

The induced asymmetry on the biasing contact

Simulated artificial offset of the cross structures induced by an _{1}

Geometrical parameters of the simulated Hall devices.

^{3} | ||||
---|---|---|---|---|

21.6 | 9.5 | 8.8 | 3,645 | |

32.4 | 14.25 | 13.55 | 7,144.2 | |

43.2 | 19 | 18.3 | 11,809.8 | |

50 | 50 | 2.3 | 16,820 | |

54 | 54 | 4.7 | 17,052.8 |

Simulated Hall Effect devices resistance.

2.073 | 2.102 | 2.181 | |

2.212 | 2.235 | 2.298 | |

2.236 | 2.254 | 2.302 | |

1.837 | 1.847 | 1.874 | |

1.393 | 1.400 | 1.422 |

Simulated Hall Effect devices performance summary.

_{HALL} |
_{I} |
_{P} |
_{dissipated} | |
---|---|---|---|---|

42.18 | 84.360 | 38.644 | 2.182 | |

44.75 | 89.500 | 38.907 | 2.300 | |

43.74 | 87.480 | 37.973 | 2.303 | |

38.91 | 77.830 | 41.482 | 1.876 | |

17.70 | 35.390 | 24.840 | 1.424 |