In order to provide the information on their Hall voltage, sensitivity, and drift with temperature, a new simpler lumped circuit model for the evaluation of various Hall cells has been developed. In this sense, the finite element model proposed by the authors in this paper contains both geometrical parameters (dimensions of the cells) and physical parameters such as the mobility, conductivity, Hall factor, carrier concentration, and the temperature influence on them. Therefore, a scalable finite element model in Cadence, for behavior simulation in circuit environment of CMOS Hall effect devices, with different shapes and technologies assessing their performance, has been elaborated.
Hall effect sensors are largely used in the actual industrial context as magnetic sensors aimed primarily at sensing the current in a large variety of applications, and also for proximity switching, positioning, or speed detection [
There is a strong connection between the geometry of the Hall device and the performance [
To achieve high sensitivity, small offset, and drift, various Hall effect sensors configurations were integrated in a CMOS technology. Extensive measurements on the new proposed cells [
The motivation of the current work is to provide a circuit model able to predict the temperature effects on the Hall effect sensors and their influence on the performance. Under the assumed research objective, a different elementary cell, with a slightly modified design, including the temperature effects, is proposed and implemented in Cadence.
Section
Section
In Section
The Hall voltage is defined as follows [
The current-related sensitivity
Different Hall effect sensors have been integrated in a 0.35
The following experimental results presented in Table
Specific parameters evaluation for some of the integrated Hall cells.
Integrated geometry | XL | Optimum | Borderless |
---|---|---|---|
Shape |
|
|
|
Measured |
2.2 | 1.8 | 1.3 |
Measured |
80.6 | 62.4 | 31.1 |
Measured offset drift ( |
0.039 | 0.328 | 0.526 |
|
|
|
|
|
|
|
|
Contacts length ( |
18.3 | 4.7 | 2.3 |
Measurements for the resistance dependence with temperature of two integrated Hall effect sensors (XL and borderless cells) are depicted in Figures
The measured input resistance versus temperature for XL cell.
The measured input resistance versus temperature for borderless cell.
In the subsequent graphs, Figures
In order to be able to correctly investigate the Hall cells behavior and predict their performance, an FEM lumped circuit model was developed.
As it is known, a lumped circuit model, also named lumped element model, simplifies the behaviour description of spatially distributed physical systems. In fact, this model is used to recreate the topology of a specific physical system with the aid of discrete entities that would approximate its conduct. The advantages of the finite element model (FEM) consist in the possibility to test many different shapes with the aid of a single elementary cell and the fact that the desired accuracy can be tuned by the choice of the number of elementary cells. In addition, different integration processes can be simulated. The parameters used by the model are grouped in geometrical parameters (chosen by the designer) and technological parameters (specific to the fabrication process). The latter can be either obtained by parameters extraction or by theoretical prediction where possible.
To model the Hall effect sensors, an FEM lumped circuit model containing a new elementary cell was developed. With respect to the previous papers introducing FEM lumped circuit models for Hall effect sensors [
The elementary cell in Figure
The elementary cell (
The elementary cell representation as a port with eight input/output pins.
To build the Hall cell, interconnections of several elementary cells will be used in order to recreate the layout of the structure. Therefore, the representation in Figure
The XL cell representation by FEM model.
The resistances on each branch are given by the following equations, where
By consequence, from the equation above, we can observe that each current flowing through a branch can be defined by the current through the opposite (orthogonal) branch multiplied by certain gains,
The elementary cell in Figure
The particular circuit model for the XL Hall cell is presented in Figure
In the case of the Hall cell, the Hall voltage
High impedances
In order to polarize the Hall cell, we use current bias on the electric path, from left to right. Tensions will be created on the two independent magnetic paths, in the form of a
This polarization scheme, as shown in Figure
To assess the performance of Hall effect sensors, three-dimensional physical simulations were also performed by the authors in recent papers [
The selection of either physical or circuit model is dictated by the objectives of the user. For example, when one needs to integrate in circuit environment the Hall cell, the lumped circuit model should be used. For detailed analysis of the physical behaviour, the three-dimensional simulations should be performed. The advantage of the latter is that they offer more faithful reproduction of the internal physical processes but require more time, whilst the circuit model’s accuracy is given by the choice of the corresponding parameters and their physical association. Between the two models, the circuit model is definitely faster.
In Figure
Three-dimensional representation of XL Hall cell with the electrostatic potential distribution.
The Hall cells temperature behavior is of importance in their performance assessment. The temperature drift of the current-related sensitivity is of particular interest.
To be able to model the temperature dependence of the current-related sensitivity, one would need to take into account the parameters that vary with temperature from (
Graphical representation of carrier concentration
The temperature dependence of the carrier concentration
The temperature dependence of
In Figure
The Hall scattering factor
Therefore, it is expected for
In (
However, in our case, for the absolute room temperature
The temperature dependence of the Hall factor
The Hall factor temperature dependence.
The finite element model developed in CADENCE containing the new proposed elementary cell was used to simulate different integrated Hall sensors. All additional blocks requiring modeling were coded in VERILOG-A. We can mention at this point that, even for a large number of elementary cells (for example a FEM model of the XL cell consisting of 64 elementary cells), the simulation time is reasonable, less than 1 s, and the use of CPU resources is reasonable.
In this section, we validate the developed model by showing that there is good agreement between the simulated and measured data. Simulation results are given at this point for the XL, borderless, and optimum Hall cells, with the emphasis on the temperature behavior of the current-related sensitivity.
The Hall voltage, absolute, and current-related sensitivity are some of the figures of merit predicted by simulation for the Hall devices. The temperature influence on the figures of merit governing the sensors performance was also extensively investigated.
Two integration CMOS processes were analyzed. The values of the parameters used to simulate the Hall cells within CMOS Process 1 are summed up in Table
Process 1 parameters.
Parameter | Symbol | Numerical value |
---|---|---|
Length |
|
|
Width |
|
|
Thickness |
|
|
Donor concentration |
|
|
Acceptor concentration |
|
1021 m−3 |
Conductivity |
|
933 Sm−1 |
Mobility |
|
0.0715 cm−2V−1s−1 |
Magnetic field |
|
0.5 T |
Process 2 parameters.
Parameter | Symbol | Numerical value |
---|---|---|
Length |
|
|
Width |
|
|
Thickness |
|
|
Donor concentration |
|
|
Acceptor concentration |
|
1021 m−3 |
Conductivity |
|
382.8 Sm−1 |
Mobility |
|
0.1 cm−2V−1s−1 |
Magnetic field |
|
0.5 T |
Both voltage bias and current bias were used in the considered Hall cells simulation, but we are focusing at this moment on current polarization. Figure
The simulated absolute sensitivity
Figure
The simulated absolute sensitivity
As it was presented by authors in a recent paper, for the same integration process and current polarization, a maximization of the geometrical correction factor
There is an increase of approximately 20% of the XL cell’s absolute sensitivity with respect to the optimum cell. This is explained by the decrease of the optimum cell’s absolute sensitivity due to the specific square structure with contacts located further away from the p-n junction.
Figure
The simulated Hall voltage
In order to investigate the temperature drift of the current-related sensitivity, (
Simulations were performed in CADENCE to investigate the current-related sensitivity temperature dependence. The curve obtained in Figure
The simulated current-related sensitivity
The relative variation
In Figure
The simulated relative variation of the current-related sensitivity versus the temperature (a) and the residuals of the fitting curve (b).
The obtained graph in Figure
Mathematically, the residual for a specific predictor value is the difference between the response value
There is an excellent accordance of the simulations results obtained with the theoretical prediction and also with experimental results. The same parabolic allure, but for the temperature characteristic measurements of the “intrinsic” sensitivity (in fact the measured relative variation of the current-related sensitivity of the Hall plate), is announced by Manic [
We can mention that our simulations and both the measured and simulated relative variations of the “current-related sensitivity related to the value at room temperature” as a function of temperature,
The FEM model developed can be applied to a variety of Hall effect sensors shapes and different integration processes, by changing the specific parameters. However, the present model is destined to analyze horizontal Hall cells and it is not valid at the moment for vertical sensors but can be changed, however, to serve this purpose, by changing the internal structure of the elementary cells, primary for the cells in the borders.
Different Hall effect sensors were integrated in a CMOS technology. The XL cell displayed the best results and proved to have the highest sensitivity and the minimum offset drift. To predict the performance of Hall effect sensors, a finite element lumped circuit model containing a new elementary cell with a slightly modified design was developed.
The proposed model implemented and tested contains both geometrical and physical parameters and is able to predict the Hall voltage, sensitivity, and their temperature drift. The temperature dependence of the Hall factor and the carrier concentration including freeze-out effect were carefully addressed by a detailed analytical analysis. In this way, the quadratic behavior of the current-related sensitivity with the temperature was also proven by simulation.
Simulations were performed for structures which reproduce the previously integrated Hall cells and the results obtained are in agreement with both the theory and the experimental results.
Due to the general character of applicability and versatility, even at this stage, our model can be used for other CMOS Hall effect devices integration processes, by adjusting the specific parameters such as doping concentration, conductivity, and thickness of the implantation profile.
In the future, after a specific calibration, the actual model will also be used for Hall effect sensors offset prediction and numerical evaluation.