IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 46, NO. 1, JANUARY 1999 89

An Analytical Model of MAGFET SensitivityIncluding Secondary Effects Using a Continuous

Description of the Geometric Correction FactorJohannes W. A. von Kluge and Werner A. Langheinrich

Abstract—An analytical model of the sensitivity of magneticfield-effect transistors (MAGFET’s) is presented. The model in-cludes secondary and parasitic geometric effects as well as operat-ing point dependencies. In order to get a continuous mathematicaldescription for the sensitivity, we introduce a continuous functionfor the geometric correction factorG: This description ofG is notlimited to MAGFET’s and can be used for any magnetic device.

Index Terms—Geometric correction factor, MAGFET, model-ing.

I. INTRODUCTION

M AGNETIC field-effect transistors show great poten-tial in applications where magnetic fields have to be

measured, but where a standard CMOS process has to beutilized. N-channel dual-drain MAGFET’s made of silicon,which are the examined devices in this paper, have ratherhigh sensitivities compared to their layout area, but suffer fromflicker noise, which limits their resolution [1].

Until now, the MAGFET was mainly examined numerically[3], [4]. Lau et al. [5] developed an analytical model whichdescibes the dependency of the MAGFET sensitivity only onprimary geometric parameters and operating point but whichshows the common discontinuities due to the usage of thediscontinuous modeled geometric correction factor. With ourmodel we want to overcome these problems by introducing acontinuous function for as well as modeling secondary andparasitic geometric effects.

Fig. 1 shows an n-channel dual-drain MAGFET togetherwith its geometric parmeters. These parameters are channellength , channel width , and drain distance as primaryparameters, source narrowing as secondary parameter, andthe overlap of the poly gate as a parasitic parameter. Thedevice works electrically like a standard MOS transistor. Itsmagnetic field measuring capability originates in the splitdrain. If a magnetic field is applied perpendicular tothe device (shown in Fig. 1), the Lorentz force will deflectthe charge carriers to one side. Thus the current in thecorresponding drain will increase while the current in theother drain will decrease, leading to a current difference

Manuscript received March 26, 1998. This work was supported by theDeutsche Forschungsgemeinschaft (DFG). The review of this paper wasarranged by Editor W. Weber.

The authors are with the Solid State Electronics Labs, Darmstadt Uni-versity of Technology, D-64289 Darmstadt, Germany (e-mail: kluge@iht.tu-darmstadt.de).

Publisher Item Identifier S 0018-9383(99)00254-3.

Fig. 1. Layout of dual-drain MAGFET with geometric parameters.

[2]. The magnetic field sensitivityof the MAGFET is defined as

(1)

where is the magnetic field and the total current

The dependency of this sensitivity on the primary geometricparameters of the device is reported by many authors [2]–[7].In this article we want to develop an analytical expressionfor which includes the dependencies on and alsoon the secondary parameter (source narrowing) and theparasitic parameter (gate overlap). Furtheremore, we areinterested in the operating point dependency of whichmeans the dependency on the gate–source–voltageandthe drain–source–voltage As result, we obtain as afunction of seven parameters

(2)

II. THE MODEL

Modeling is based on n-channel MAGFET’s, however holdssimilarly for p-channel devices. Substrate currents, minorities,diffusion currents, and generation/recombination are neglected.We assume small magnetic fields which areperpendicular to the surface

0018–9383/99$10.00 1999 IEEE

90 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 46, NO. 1, JANUARY 1999

In order to get an equation for the sensitivity, we need anequation for the current difference The general equationfor the current density in an n-type semiconductor includingthe magnetic effects can be taken from [2]

(3)

In this equation is the current density, themagnetic field dependent conductivity, isthe electic field, the Hall mobility and the magnetic fielddensity. When using the axes of coordinates shown in Fig. 1,the -component of is responsible for the current difference

After neglecting the term due to small magneticfields, this component can be written as

(4)

originates in the Lorentz force and can thus be written aswhere is the geometric correction factor

and the velocity of the charge carriers. is introducedsince we examined nonideal magnetic devices which havetheir fieldlines disturbed by contacts [2]. Having in mind that

with being the scattering factor andthe chargecarrier mobility, (4) can be developed into

(5)

Integrating this equation over the geometry of the MAGFET-channel results in the current difference This integrationcan be performed as long as the following two conditions aremet.

• Geometry of the MAGFET channel should be easy tointegrate.

• Geometric correction factor should have a continuous,integrable form.

A. Geometry of the MAGFET Channel

In order to meet the first condition, the MAGFETchannelis transformed into a rectangle (Fig. 2). In this equivalentMAGFET, the drain distance is chosen to be zero in orderto get a rectangular shape. The resulting channel lengthand channel width are

(6)

with and being fit parameters, which render the in-fluence of the source narrowing on the rectangular MAGFET-form.

The transformed channel length can be calculated fromthe potential distribution in the normal MAGFET: A draindistance pulls down the potential near the drain, leadingto an inhomogeneous electric field [5]. Based on simulations,

Fig. 2. Transformed channel region of the MAGFET with geometric dimen-sions (The transformed drain distance is zero!).

we propose the following model for the potential in theMAGFET:

(7)

with being the coordinate from Fig. 1. After calculating theelectric field by derivation at the drain itequals

(8)

This is equivalent to a homogeneous electric fieldof a rectangular MAGFET with channel

length

(9)

Obviously, an increase in drain distanceof the originalMAGFET leads to a decrease in channel lengthof theequivalent MAGFET with no drain distance.

B. Geometric Correction Factor

The geometric correction factor is used to take into accountthe difference in the field distribution of a real Hall sample(e.g., a Hall plate), compared to an ideal Hall sample. Anideal Hall sample is an infinite long slab with point contacts tosense the Hall voltage. In a real device, the field distributionis changed by nonideal current- and sense contacts due toshort-circuiting.

therefore is defined as [2], [6]

(10)

with being the Hall voltage of the real and thatof the ideal device. The second part of (10) is only validfor devices with constant dimensions in-direction. It is

VON KLUGE AND LANGHEINRICH: ANALYTICAL MODEL OF MAGFET SENSITIVITY 91

important to note that is a ratio of Hall voltages and thusnot dependent on the thickness of the device it is used upon.This is an important fact for the MAGFET model since the“thickness” of the transformed MAGFET-slab (its channeldepth) is unknown.

Under certain conditions, can be calculated analytically.This is done by conformal mapping methods used on rectan-gular Hall plates. The Hall plate has a length between thecurrent contacts and a width between the sense contacts.The sense contacts are positioned at Thus thefollowing equations are only valid for the middle part of theHall plate. can be calculated for different and asfollows [6]

1) For is given by

(11)

2) For is given by

(12)

These equations can be used for the MAGFET. But sincethe model should cover the total geometric range, a transitionfunction for the missing partis needed. We developed the following weighted averagingfunction, which yields a good result:

(13)

This function connects any two functions and aslong as

(14)

(15)

lies in the middle of the definition gap of the functionsand and defines the position, where is the

mean value of and if their domain of definitionwould extend into the gap. The parametersand in (13)define the interpolation accuracy.

When using (13) with the two functions for the geomet-ric correction factor, has to be calculated as the two-dimensional mean value of the boundaries.

(16)

Fig. 3. Two-dimensional plot of the geometric correction factorGtot in themiddle (x = (L�=2)) of a Hall plate.

The other values are taken as

(17)

As result we obtain the function

(18)

which is shown in Fig. 3 and which gives a continuousdescription of the magnitude of the geometric correction factorin the middle of a Hall device depending on any width orlength.

As indicated above, it is not only the magnitude ofwhich changes due to different geometries of the MAGFET.Another point are the conducting contacts which short-circuitthe Hall field near the ends of the probe. This leads to aninhomogenous behavior of over the length of the device:

becomes a function of the-coordinate. Simulations showthe expected results of the behavior of [6]. The Hallvoltage, and therefore as well, is zero at the current contacts

due to the short-circuiting. G(x) equalsat and the function does certainly not growabove 1 which leads to a flattening of the usually paraboliccurves if increases above the point at which is1. Based on these premises, we developed a function, whichagrees with them and which is also integrable. This function is

(19)

92 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 46, NO. 1, JANUARY 1999

Fig. 4. Calculated functions ofG(x) with different (L�=W �) ratios.Dashed lines are simulated values of [2], [6].

Fig. 4 shows the calculated functions of with differentratios. Comparing it with the simulated results from

[2], [6], which are drawn in dashed lines to show theirtendencies, it can be estimated that the overall error ofis under 5%. This is also true for which liesinside the interpolation region. For ratios greater than1 however, function overestimates the simulated resultsin the region near This error can be estimated to beunder 25%, getting smaller while further increases.For ratios smaller than 0.5 both functions are nearlyidentical.

The above derivation of the geometric correction factoris a general approach and is therefore not restricted to

MAGFET’s. The function of can be used for anyrectangular Hall device.

C. Calculation of the Current Difference

The current difference can be calculated by integrating(5) over the geometry of the transformed channel of theMAGFET (Fig. 2). Doing so, one has to take into account,that the part to of the originalMAGFET channel does not produce a current difference whichinfluences the sensitivity. Therefore we integrate in-directionfrom to

In this integration process, stands for the channel depthand compensates the errors of Since we now havean integrable form of it is possible to solve the aboveproblem. This leads to

(20)

In (20), stands for the solved integral. The generalsolution for is

D. Operating Point Dependency

When calculating the sensitivity from (20) by dividingit by the only remaining parameter depending on theMAGFET voltages is the mobility This is due to the fact thatthe MAGFET behaves electrically like a MOSFET. Thereforeits mobility is modeled like the MOSFET moblility. The MOS-FET mobility decreases with increasing gate-source-voltagedue to surface scattering and it decreases with increasing drain-source-voltage due to velocity saturation [8]. A good modelfor which includes both effects but neglects the bulk effectis the Level 6 model from HSPICE [9]

(21)

Here, is the mobility without voltages, is thegate–source–voltage, is the threshold voltage andis the oxid thickness. is the drain–source–voltage in thelinear region and the saturation voltage in the saturation region.

and are fit parameters which can be taken from thetechnology in which the MAGFET has been processed.

Introducing to (20) and using (1), the sensitivity results in

(22)

The parameter is used to compensate for the neglectionsmade during modeling.

III. V ERIFICATION OF MODEL

The model was verified by measurements of 21 n-channelMAGFET’s of an industrial CMOS process, which were alsoused in [10]. The measuring setup consists of Helmholtzcoils which produce a magnetic field density of 5 mT. AllMAGFET’s were biased to operate with a total currentof50 A. Figs. 5–11 show the results. The dots are the measured

VON KLUGE AND LANGHEINRICH: ANALYTICAL MODEL OF MAGFET SENSITIVITY 93

Fig. 5. Model and measured points of sensitivityS depending on channellength L:

Fig. 6. Model and measured points of sensitivityS depending on channelwidth W:

values, while the lines show the calculation with the model.If not mentioned otherwise, the dimensions of the parametersare m, m, m, m,

m. The mean error of the model is in the rangeof 5% which lies within the accuracy of reading. The valuesof the parameters could be extracted from themeasurements, while originates from a comparison of themodel with the curves in [2], [6]. Since compensates theintegration error, it has the value of 1 for and

• .• .• .• .

IV. DISCUSSION

When using the model to optimize the sensitivity of aMAGFET, the following results are obtained: It is obvious,that long and thin channels are advantageous. Looking atFig. 12 however, where a channel width of m isused m, m, m), it can be observedthat the increase in sensitivity by increasingis limited forthis ideal device to a point where

(23)

Fig. 7. Model and measured points of sensitivityS depending on draindistanced:

Fig. 8. Model and measured points of sensitivityS depending on sourcenarrowingD with constantW + 2D:

Fig. 9. Model and measured points of sensitivityS depending on gateoverlap u:

Beyond that point the increase in the geometric correctionfactor becomes dominant, which leads to a decrease insensitivity for a further increase of Certainly, this ratiofor maximum sensitivity also depends on the other geometricparameters. They decrease the ratio, leading tofor a realistic device (e.g., m, m, m).

Fig. 7 shows that the drain distanceof the MAGFETshould be made as small as possible.

When looking at the secondary parameter, the source nar-rowing it becomes apparent that the sensitivity can beincreased by using a large when the total widthis kept constant. This originates in the fact, that a source

94 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 46, NO. 1, JANUARY 1999

Fig. 10. Model and measured points of sensitivityS depending on thegate–source–voltageVGS :

Fig. 11. Model and measured points of sensitivityS depending on thedrain–source–voltageVDS :

Fig. 12. Sensitivity calculated for large channel lengthsL with a constantchannel widthW = 50 �m (d = 0 �m, D = 25 �m, u = 0 �m).

narrowing increases the current density at the source, leadingto an increase in sensitivity.

An increase of the parasitic gate overlapon the other handdecreases sensitivity. The reason for the decrease lies in thefact that an increasing enlarges the part of the channel whichdoes not contribute to the current difference However, itincreases the geometric correction factorsince the overallchannel length increases. As a result, the sensitivity becomeslessened.

Another interesting point is how is influenced by anincrease in channelarea. This dependency is plotted in Fig. 13.If channel length equals the channel width and thus bothare increased simultaniously, the sensitivity increases fast until

Fig. 13. Sensitivity calculated for changing channel area(d = 0 �m,D = 0 �m, u = 0 �m).

and reach approximatly 200 m. A further increase ofchannel area is quite useless, sincenearly remains at thatvalue even if the area is increased enormously.

When looking at the operating point, it becomes clear thatthe MAGFET should be biased at small voltages, since thisdecreases the degradation ofkeeping the sensitivity high.

V. CONCLUSIONS

This paper describes an analytical model for the sensitivityof MAGFET’s. The model shows the influence of sevenparameters including channel length, channel width, draindistance, source narrowing and gate overlap as well as theMAGFET voltages for the operating point. It uses a continuousdescription of the geometric correction factor in order toprovide one model for all kinds of parameter values. Thiscontinuous description of may also be used for any othertype of magnetic sensor, which is modeled with the use of(e.g., Hall plates). The model was verified with measurementsof 21 MAGFET’s of an industrial CMOS technology andshowed a good agreement.

ACKNOWLEDGMENT

The authors would like to thank Dialog SemiconductorGmbH for the fabrication of the MAGFET testchips.

REFERENCES

[1] H. P. Baltes and R. S. Popovic, “Integrated semiconductor magneticfield sensors,”Proc. IEEE, vol. 74, pp. 1107–1132, Aug. 1986.

[2] C. S. Roumenin,Handbook of Sensors and Actuators—Solid StateMagnetic Sensors. Amsterdam, The Netherlands: Elsevier, 1994.

[3] A. Nathan, L. Andor, H. P. Baltes, and H. G. Schmidt-Weinmar,“Modeling of a dual-drain NMOS magnetic-field sensor,”IEEE J. Soild-State Circuits, vol. SSC-20, pp. 819–821, Mar. 1985.

[4] A. Nathan, A. J. Huiser, and H. P. Baltes, “Two-dimensional numericalmodeling of magnetic-field sensors in CMOS technology,”IEEE Trans.Electron Devices, vol. ED-32, pp. 1212–1219, July 1985.

[5] J. Lau, P. K. Ko, and P. C. H. Chan, “Modeling of split-drain magneticfield-effect transistor (MAGFET),”Sens. Actuators A, vol. 49, pp.155–162, 1995.

[6] R. S. Popovic,Hall Effect Devices,. London, U.K.: IOP, 1991.[7] D. Killat, J. v. Kluge, F. Umbach, W. Langheinrich, and R. Schmitz,

“Measurement and modeling of sensitivity and noise of MOS magneticfield effect transistors,”Sens. Actuators A, vol. 61, pp. 346–351, 1997.

[8] Y. P. Tsividis, Operating and Modeling of the MOS Transistor. NewYork: McGraw-Hill, 1988.

[9] Meta-Software,HSPICE User’s Manual H9001, Meta-Software, Inc.,1990.

VON KLUGE AND LANGHEINRICH: ANALYTICAL MODEL OF MAGFET SENSITIVITY 95

[10] J. v. Kluge and W. Langheinrich, “Optimierung des signal-rauschabstandes von magnetfeldempfindlichen MOS-transistoren(MAGFET’s),” Proc. Fachtagung “Informations- undAutomatisierungstechnik,” Otto-von-Guericke-Univ., Magdeburg,Germany, 1998.

Johannes W. A. von Kluge was born in Diez,Germany, in 1970. He received the Dipl.-Ing. de-gree from the Darmstadt University of Technology,Darmstadt, Germany, in 1996. He is currently pur-suing the Ph.D. degree at the Solid State ElectronicsLaboratories, Darmstadt University of Technology.

His main interests lie in the fields of optimizationof solid-state sensors, as well as the design of signalprocessing circuits for sensors.

Werner A. Langheinrich was born in Ulm, Ger-many, in 1934. He received the Dipl. and Dr.rer.nat.degrees in physical chemistry from Technical Uni-versity of Stuttgart, Germany, in 1960 and 1962,respectively.

He then joined AEG-Telefunken Research Labs.In 1977, he became a Full Professor at Darm-stadt University of Technology, Darmstadt, Ger-many, where he is managing director of Solid StateElectronics Labs. His main field of intertest is sili-con–semiconductor and microsystems technology.