- Email: [email protected]
Offset reduction in silicon Hall sensors
Sensors and Actuators 81 Ž2000. 18–22 www.elsevier.nlrlocatersna
Offset reduction in silicon Hall sensors C. Muller-Schwanneke ¨ a
a,)
, F. Jost b, K. Marx b, S. Lindenkreuz c , K. von Klitzing
a
Max-Planck-Institut fur Heisenbergstr.1, D-70569 Stuttgart, Germany ¨ Festkorperforschung, ¨ b Robert Bosch GmbH, FV r FLT, P.O. Box 106050, D-70049 Stuttgart, Germany c Robert Bosch GmbH, K8 r DIC1, P.O. Box 1342, D-72703 Reutlingen, Germany
Abstract In this paper, we analyse the Anti-Hall ŽAH. method for offset reduction and the spinning-current method. The first method uses Hall plates with an inner and an outer boundary with multiple current injections at the same time. Simultaneously one obtains a magnetic field sensitive voltage at one contact pair and a magnetic-field independent voltage sensitive to an in-plane shear stress at another contact pair. The second method uses consecutive current injections in different orientations of a symmetric Hall plate and the average of the measurement results is the output signal. Both methods exploit the different symmetries of the Hall effect and various effects that contribute to the offset. Nevertheless, both methods have significant differences that can be understood with the reciprocity principle for macroscopic samples concerning the exchange of current and voltage leads on the Hall measurements. Numerical simulations of the potential distribution of the Anti-Hall structures were carried out with a lumped-discrete approach. Experimental data measured with both offset-reduction methods are shown. q 2000 Elsevier Science S.A. All rights reserved. Keywords: Hall plates; Offset; Piezoresistive effect
1. Introduction In many applications magnetic sensors are used, e.g., for the contactless measurement of mechanical quantities like position or angle. Silicon Hall sensors are often prime candidates for such applications due to their cost-effective integration potential. But they are plagued by a stress induced offset voltage which cannot be easily compensated in the production line. Stress drifts in plastic encapsulated sensors will cause drifts of the offset through the piezoresistive effect so that offset reduction methods have to be used. These methods rely on the different symmetries of the Hall effect and various transduction effects which contribute to the offset. The spinning current method w1x averages the results of several consecutive Hall measurements with different orientations in the crystal plane. Very low residual offsets in the microtesla range can be achieved but the complex circuitry required may introduce problems. The method will be explained with the reciprocity principle for macroscopic samples w2x. The so-called Anti-Hall ŽAH. method for Hall plates with an outer and an inner boundary as described by Mani
)
Corresponding author. E-mail: [email protected]
et al. w3x also utilizes different orientations of the current injection. But these currents are injected at the same time at different points on the outer and inner boundaries of the sample. Besides the magnetic-field sensitive signal, a separate stress sensitive signal can be measured simultaneously at another pair of voltage contacts.
2. Sensor devices and measurement set-up The Hall sensors described in this paper were designed and fabricated in a silicon process at the Robert Bosch GmbH in Reutlingen, Germany. The Hall elements were defined by lateral p-diffusions in a 10 mm n-epilayer on a Ž100. p-substrate as in a standard bipolar technology. The epilayer had a typical sheet resistance of 2400 V at room temperature. Small dies of less than 2 mm2 were encapsulated in nonmagnetic plastic packages ŽPLCC28. in order to achieve a high encapsulation stress. 2.1. Anti-Hall All the Anti-Hall structures were derived from a symmetric shape Že.g., octagon. with an inner and outer boundary. A schematic of a sample with an 8-fold symmetry is
0924-4247r00r$ - see front matter q 2000 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 4 - 4 2 4 7 Ž 9 9 . 0 0 1 6 3 - 6
C. Muller-Schwanneke et al.r Sensors and Actuators 81 (2000) 18–22 ¨
19
shown in Fig. 1 and will be called Anti-Hall octagon. The contacts at the outer boundary are labeled with uppercase letters ‘A’, ‘B’, ‘C’, . . . and the contacts at the inner boundary with the corresponding lowercase letters. For the Anti-Hall measurements we used four electrically separated, battery-powered current sources and two voltmeters. The low-potential contact of one current source, the substrate contact and an optional metal shield contact were always connected to ground. The choice of the grounded current source did not influence the measurements significantly. We use the convention that a current IA s I ) 0 entering the sample at ‘A’ and a current Ia s yI leaving the sample at ‘a’ can be denoted by IAa if ‘A’ and ‘a’ are connected to the same current source. This implies that the absolute values of the currents IA , Ia are exactly equal which, in the case of multiple current injection, is only valid for electrically separate current sources.
3. Theoretical results
2.2. Spinning-current
3.2. Spinning-current
The spinning-current Hall plates have an 8-fold symmetry and are shaped like two greek crosses superimposed on each other, with one of them rotated by 458 like the NMOS devices in Ref. w4x. The spinning-current measurements were performed with a current source, a voltmeter and a switching matrix. With this computer-controlled equipment, a full spinning current cycle with eight different orientations could be performed in a few seconds. The residual offsets were computed after the data acquisition. For the residual offset measurements shown in this paper, a magnetic shielding was used. Residual stray fields are estimated to contribute up to 5 mT to the magnetic induction at the sample position. The low-potential contact of the current source, the substrate contact and a metal shield contact were always connected to ground.
Based on the fundamental work of Onsager, Sample et al. w2x proved a reciprocity theorem for macroscopic specimens in the presence of a static magnetic induction B which they refer to as reverse-magnetic-field reciprocity ŽRMFR.. Their only constraint is that the sample be electrically linear. Neither inhomogeneous nor anisotropic Že.g., stressed. samples are excluded from their analysis. Defining the four-terminal resistances as
3.1. Anti-Hall The existence of a diagonal voltage sensitive to the magnetic-field and a magnetic-field independent voltage at a perpendicular contact pair can be derived with line integral arguments. The influence of stress can be included in the model. Due to the different symmetries of the Hall effect and the piezoresistive effect, the corresponding transverse voltages which are generated near the current injection points sum up with opposite signs to give a stress dependent voltage at one contact pair and a Hall voltage at the other diagonal contact pair. These arguments rest on the assumption of homogeneous conditions. Details will be given elsewhere.
≠ ,x ' R AECG
UCG Ž B ≠ , x . IAE
, B ≠ s yB x ,
the RMFR principle states: ≠ sRx R AECG CGAE
Ž 1.
Ž 2.
For the special case of B s 0 we obtain a vanishing orthogonal offset: UCG Ž IAE s I, B s 0 . s UAE Ž ICG s I, B s 0 . Uortho s
1 2
UCG Ž IAE s I, B s 0 .
qUEA Ž ICG s I, B s 0 . s 0
Ž 3.
For B / 0 this orthogonal averaging gives the offset-free averaged Hall signals.
4. Numerical results
Fig. 1. Schematic of an Anti-Hall octagon with respect to the crystallographic axes.
After first performing calculations based on a finite difference approach w3x we favour a lumped-discrete approach w5x to the simulations with the advantage that the problem can be solved with any commercial circuit simulator of the SPICE family. The Hall element is constructed with many instances of a symmetric basic cell which is consistent with the RMFR principle and which allows the
20
C. Muller-Schwanneke et al.r Sensors and Actuators 81 (2000) 18–22 ¨
5. Experimental results All experimental results reported here have been measured on samples encapsulated in plastic packages. Nevertheless, we want to mention that preliminary measurements on samples with a low-stress mounting without encapsulation displayed individual offsets for injection along the ²110: direction reduced by almost one order of magnitude compared to the packaged samples. We conclude that the individual offsets of the packaged samples are dominated by the encapsulation stress.
5.1. Anti-Hall
Fig. 2. A contour plot of the simulated potential distribution of an Anti-Hall square with Bz s 5 T. The current injection IAa s ICc s IeE s IgG s1 mA is indicated by arrows. For the potential contacts at the outer edges we obtain UBF s 0.080 V, UDH s 3.000 V.
simulation of the Hall effect and of anisotropies of the resistivity tensor at B s 0 in the presence of stress. As parameters for the basic cell we used a sensitivity S I s 300 VrAT Ž SI s UHallrŽIB.. which is representative for the n-epi devices. The boundaries were terminated with capacitors whose values did not influence the DC-simulations. 4.1. Anti-Hall In Fig. 2 the potential distribution of an Anti-Hall square with quadruple current injection is shown under the influence of a strong field of 5 T. For the Anti-Hall structures we obtained a Hall voltage UDH s 3 V which is twice the Hall voltage expected for a point-contact Hall plate with the same sensitivity and single current injection. A much smaller voltage UBF s 0.08 V was found at the other diagonal contact pair which depended on the geometry. We also performed simulations with a shear stress via an off-diagonal term in the symmetric resistivity tensor ŽB s 0. and found a stress-dependent voltage UBF whereas the voltage UDH at the Hall contacts vanished exactly.
For most measurements we used the symmetric AntiHall octagons which offer the advantage that the same measurements can be performed with current injection parallel to the ²110: directions or the ²100: directions which makes a large difference concerning the piezoresistive effect. For quadruple injection IAa s ICc s IeE s IgG s 1 mA we could always measure the Hall effect signal UDH whereas the voltage UBF remained magnetic field independent. We also rotated all connections by 458 and obtained different offsets but the same magnetic-field sensitivity at the Žrotated. Hall contacts. In Table 1 we show measurement results on Anti-Hall octagons with quadruple injection at room temperature. The averages and standard deviations were calculated on the basis of 13 samples. First of all, one notices large signals at the stress-sensitive contacts for injection along ²110: which scatter very much from sample to sample. These values are similar to the individual offsets of conventional Hall structures in the same packages. Contrary, the offsets at the Hall contacts scatter around zero with a standard deviation that is about one order of magnitude smaller than for the stress-sensitive contacts. Obviously a large part of the stress-dependent offset is cancelled. For injection along ²100: the signals at the stress-sensitive contacts are much smaller which corresponds to the smaller shear piezoresistive coefficients for that orientation w6x. The offsets at the Hall contacts also scatter less for this orientation. We assume that an inhomogeneous stress con-
4.2. Spinning-current Because the RMFR principle is built into the basic cell, the orthogonal offsets must vanish under linear conditions which could be confirmed by the simulations. More realistic simulations which include the nonlinear sheet resistance dependence on the local voltage caused by the voltage-dependent depletion-layer width will be performed in the future.
Table 1 Mean values and standard deviations of the voltages at the stress-sensitive contacts and the Hall contacts of Anti-Hall octagons at Bs 0 and I s1 mA Quadruple injection
²UST RESS : ŽmV.
²UHALL : ŽmV.
²110: ²100:
y32"40 2.7"1,5
0.6"3.0 y0.4"1.3
1 mV corresponds to 2 mT B-offset.
C. Muller-Schwanneke et al.r Sensors and Actuators 81 (2000) 18–22 ¨
tributes to the offsets at the Hall contacts in order to explain the larger spread for injection along ²110:. Additionally, the influence of current contact displacements limits the performance even in the absence of stress. Measurements on a series of Anti-Hall squares Žwith the same ratio of strip width to hole width as the Anti-Hall octagons. with one current contact deliberately displaced by 1 mm or 2 mm yielded offsets of about 10 mVrmm Žf 20 mTrmm.. Using this number, the Hall offsets for injection along ²100: in Table 1 correspond to contact displacements of several tens of nanometers which cannot be significantly reduced with standard technology. This limit as well as the experimental scattering of the residual offsets translates to B-field equivalent offsets of a few millitesla which cannot be controlled. 5.2. Spinning-current With the spinning current method we could achieve very small residual offsets well below 100 mT for all samples by limiting the current so that the nonlinear contribution to the input resistance remained below 10%. We found it quite remarkable that the method even worked for the Anti-Hall samples Žwith single current injection. as long as we adhered to the RMFR, that is, we exchanged the current and voltage contacts, e.g., in measurements
21
using only the outer boundary contacts as if no hole were present. From a practical point of view, one is interested in a high output voltage of the Hall sensor and therefore wants to use the maximum current compatible with the offset requirements. As the residual offset has contributions from the nonlinearities of the system in addition to other effects which rotate with the current Že.g., Peltier effect w1x., we examined the nonlinear current dependence of the residual offset in more detail. In Fig. 3 we show residual offset data measured on four different samples of the same type. The individual offsets of these samples at 1 mA bias current Žwithout spinning. were Ž31 " 17. mV for injection along ²110: and Žy0.3 " 0.8. mV for injection along ²100:. Despite this large orientation dependence the current dependence of the residual offsets based on measurements with four current injections along ²110: was similar to the case with injections along ²100:, but with the opposite sign. This orientation dependence of the sign of the nonlinear current dependence gives us the hint that the piezoresistive effect is involved. The residual offsets based on eight injections along all possible directions are almost in the range of the stray fields at the sample position of up to 5 mT. We think that a violation of the RMFR by the nonlinear dependence of the sheet resistance on the current via the nonlinear depletion width dependence on the local junction voltage is of importance here. This violation of the RMFR in conjunction with the piezoresistive effect seems to be responsible for the measurement results.
6. Conclusions
Fig. 3. Experimentally determined residual offsets of four spinning-current Hall plates. Residual offsets based on four orientations with current injection along ²100: Žopen symbols., current injection along ²110: Žsolid symbols. and residual offsets based on eight orientations Žsolid lines. are shown. The dashed lines represent an equivalent offset of 100 mT.
For the Anti-Hall method we could verify that separate measurements of a magnetic induction B and of a shear stress are possible at different contact pairs. For injection along ²110: one could observe an offset-reduction of about an order of magnitude. But as the Anti-Hall method does not use any contact commutation, displacements of the current contacts and inhomogeneous stress contribute to a residual offset of a few millitesla which cannot be significantly reduced. In contrast we obtained residual offsets well below 100 mT with the spinning-current method by limiting the current so that the nonlinear contribution to the input resistance remained below 10%. The nonlinear current dependence of the residual offset based on four orientations showed a different sign for injection along the ²110: directions in the Ž100. plane than for injections along ²100:. We propose that a violation of the reciprocity principle for macroscopic samples through the nonlinear dependence of the depletion width on the junction voltage together with the piezoresistive effect may explain such behaviour.
22
C. Muller-Schwanneke et al.r Sensors and Actuators 81 (2000) 18–22 ¨
References w1x P.J.A. Munter, A low-offset spinning-current Hall plate, Sensors and Actuators A 21–23 Ž1990. 743–746. w2x H.H. Sample, W.J. Bruno, S.B. Sample, E.K. Sichel, Reverse-field reciprocity for conducting specimens in magnetic fields, J. Appl. Phys. 61 Ž1987. 1079. w3x R.G. Mani, K. von Klitzing, F. Jost, K. Marx, S. Lindenkreuz, H.P. Trah, Method for simultaneously reducing the misalignment offset
and separating the Hall voltage from the off-diagonal piezoresistive voltage in Hall effect and piezoresistive devices based on silicon, Appl. Phys. Lett. 67 Ž1994. 2223. w4x R. Gottfried-Gottfried, Thermal behaviour of CMOS Hall sensors for different operating modes, Sensors and Actuators A 41–42 Ž1994. 430–434. w5x R.S. Popovic, Numerical analysis of MOS magnetic field sensors, Solid-State Electronics 28 Ž1985. 711–716. w6x Y. Kanda, Piezoresistance effect of silicon, Sensors and Actuators A 28 Ž1991. 83–91.
Offset reduction in silicon Hall sensors C. Muller-Schwanneke ¨ a
a,)
, F. Jost b, K. Marx b, S. Lindenkreuz c , K. von Klitzing
a
Max-Planck-Institut fur Heisenbergstr.1, D-70569 Stuttgart, Germany ¨ Festkorperforschung, ¨ b Robert Bosch GmbH, FV r FLT, P.O. Box 106050, D-70049 Stuttgart, Germany c Robert Bosch GmbH, K8 r DIC1, P.O. Box 1342, D-72703 Reutlingen, Germany
Abstract In this paper, we analyse the Anti-Hall ŽAH. method for offset reduction and the spinning-current method. The first method uses Hall plates with an inner and an outer boundary with multiple current injections at the same time. Simultaneously one obtains a magnetic field sensitive voltage at one contact pair and a magnetic-field independent voltage sensitive to an in-plane shear stress at another contact pair. The second method uses consecutive current injections in different orientations of a symmetric Hall plate and the average of the measurement results is the output signal. Both methods exploit the different symmetries of the Hall effect and various effects that contribute to the offset. Nevertheless, both methods have significant differences that can be understood with the reciprocity principle for macroscopic samples concerning the exchange of current and voltage leads on the Hall measurements. Numerical simulations of the potential distribution of the Anti-Hall structures were carried out with a lumped-discrete approach. Experimental data measured with both offset-reduction methods are shown. q 2000 Elsevier Science S.A. All rights reserved. Keywords: Hall plates; Offset; Piezoresistive effect
1. Introduction In many applications magnetic sensors are used, e.g., for the contactless measurement of mechanical quantities like position or angle. Silicon Hall sensors are often prime candidates for such applications due to their cost-effective integration potential. But they are plagued by a stress induced offset voltage which cannot be easily compensated in the production line. Stress drifts in plastic encapsulated sensors will cause drifts of the offset through the piezoresistive effect so that offset reduction methods have to be used. These methods rely on the different symmetries of the Hall effect and various transduction effects which contribute to the offset. The spinning current method w1x averages the results of several consecutive Hall measurements with different orientations in the crystal plane. Very low residual offsets in the microtesla range can be achieved but the complex circuitry required may introduce problems. The method will be explained with the reciprocity principle for macroscopic samples w2x. The so-called Anti-Hall ŽAH. method for Hall plates with an outer and an inner boundary as described by Mani
)
Corresponding author. E-mail: [email protected]
et al. w3x also utilizes different orientations of the current injection. But these currents are injected at the same time at different points on the outer and inner boundaries of the sample. Besides the magnetic-field sensitive signal, a separate stress sensitive signal can be measured simultaneously at another pair of voltage contacts.
2. Sensor devices and measurement set-up The Hall sensors described in this paper were designed and fabricated in a silicon process at the Robert Bosch GmbH in Reutlingen, Germany. The Hall elements were defined by lateral p-diffusions in a 10 mm n-epilayer on a Ž100. p-substrate as in a standard bipolar technology. The epilayer had a typical sheet resistance of 2400 V at room temperature. Small dies of less than 2 mm2 were encapsulated in nonmagnetic plastic packages ŽPLCC28. in order to achieve a high encapsulation stress. 2.1. Anti-Hall All the Anti-Hall structures were derived from a symmetric shape Že.g., octagon. with an inner and outer boundary. A schematic of a sample with an 8-fold symmetry is
0924-4247r00r$ - see front matter q 2000 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 4 - 4 2 4 7 Ž 9 9 . 0 0 1 6 3 - 6
C. Muller-Schwanneke et al.r Sensors and Actuators 81 (2000) 18–22 ¨
19
shown in Fig. 1 and will be called Anti-Hall octagon. The contacts at the outer boundary are labeled with uppercase letters ‘A’, ‘B’, ‘C’, . . . and the contacts at the inner boundary with the corresponding lowercase letters. For the Anti-Hall measurements we used four electrically separated, battery-powered current sources and two voltmeters. The low-potential contact of one current source, the substrate contact and an optional metal shield contact were always connected to ground. The choice of the grounded current source did not influence the measurements significantly. We use the convention that a current IA s I ) 0 entering the sample at ‘A’ and a current Ia s yI leaving the sample at ‘a’ can be denoted by IAa if ‘A’ and ‘a’ are connected to the same current source. This implies that the absolute values of the currents IA , Ia are exactly equal which, in the case of multiple current injection, is only valid for electrically separate current sources.
3. Theoretical results
2.2. Spinning-current
3.2. Spinning-current
The spinning-current Hall plates have an 8-fold symmetry and are shaped like two greek crosses superimposed on each other, with one of them rotated by 458 like the NMOS devices in Ref. w4x. The spinning-current measurements were performed with a current source, a voltmeter and a switching matrix. With this computer-controlled equipment, a full spinning current cycle with eight different orientations could be performed in a few seconds. The residual offsets were computed after the data acquisition. For the residual offset measurements shown in this paper, a magnetic shielding was used. Residual stray fields are estimated to contribute up to 5 mT to the magnetic induction at the sample position. The low-potential contact of the current source, the substrate contact and a metal shield contact were always connected to ground.
Based on the fundamental work of Onsager, Sample et al. w2x proved a reciprocity theorem for macroscopic specimens in the presence of a static magnetic induction B which they refer to as reverse-magnetic-field reciprocity ŽRMFR.. Their only constraint is that the sample be electrically linear. Neither inhomogeneous nor anisotropic Že.g., stressed. samples are excluded from their analysis. Defining the four-terminal resistances as
3.1. Anti-Hall The existence of a diagonal voltage sensitive to the magnetic-field and a magnetic-field independent voltage at a perpendicular contact pair can be derived with line integral arguments. The influence of stress can be included in the model. Due to the different symmetries of the Hall effect and the piezoresistive effect, the corresponding transverse voltages which are generated near the current injection points sum up with opposite signs to give a stress dependent voltage at one contact pair and a Hall voltage at the other diagonal contact pair. These arguments rest on the assumption of homogeneous conditions. Details will be given elsewhere.
≠ ,x ' R AECG
UCG Ž B ≠ , x . IAE
, B ≠ s yB x ,
the RMFR principle states: ≠ sRx R AECG CGAE
Ž 1.
Ž 2.
For the special case of B s 0 we obtain a vanishing orthogonal offset: UCG Ž IAE s I, B s 0 . s UAE Ž ICG s I, B s 0 . Uortho s
1 2
UCG Ž IAE s I, B s 0 .
qUEA Ž ICG s I, B s 0 . s 0
Ž 3.
For B / 0 this orthogonal averaging gives the offset-free averaged Hall signals.
4. Numerical results
Fig. 1. Schematic of an Anti-Hall octagon with respect to the crystallographic axes.
After first performing calculations based on a finite difference approach w3x we favour a lumped-discrete approach w5x to the simulations with the advantage that the problem can be solved with any commercial circuit simulator of the SPICE family. The Hall element is constructed with many instances of a symmetric basic cell which is consistent with the RMFR principle and which allows the
20
C. Muller-Schwanneke et al.r Sensors and Actuators 81 (2000) 18–22 ¨
5. Experimental results All experimental results reported here have been measured on samples encapsulated in plastic packages. Nevertheless, we want to mention that preliminary measurements on samples with a low-stress mounting without encapsulation displayed individual offsets for injection along the ²110: direction reduced by almost one order of magnitude compared to the packaged samples. We conclude that the individual offsets of the packaged samples are dominated by the encapsulation stress.
5.1. Anti-Hall
Fig. 2. A contour plot of the simulated potential distribution of an Anti-Hall square with Bz s 5 T. The current injection IAa s ICc s IeE s IgG s1 mA is indicated by arrows. For the potential contacts at the outer edges we obtain UBF s 0.080 V, UDH s 3.000 V.
simulation of the Hall effect and of anisotropies of the resistivity tensor at B s 0 in the presence of stress. As parameters for the basic cell we used a sensitivity S I s 300 VrAT Ž SI s UHallrŽIB.. which is representative for the n-epi devices. The boundaries were terminated with capacitors whose values did not influence the DC-simulations. 4.1. Anti-Hall In Fig. 2 the potential distribution of an Anti-Hall square with quadruple current injection is shown under the influence of a strong field of 5 T. For the Anti-Hall structures we obtained a Hall voltage UDH s 3 V which is twice the Hall voltage expected for a point-contact Hall plate with the same sensitivity and single current injection. A much smaller voltage UBF s 0.08 V was found at the other diagonal contact pair which depended on the geometry. We also performed simulations with a shear stress via an off-diagonal term in the symmetric resistivity tensor ŽB s 0. and found a stress-dependent voltage UBF whereas the voltage UDH at the Hall contacts vanished exactly.
For most measurements we used the symmetric AntiHall octagons which offer the advantage that the same measurements can be performed with current injection parallel to the ²110: directions or the ²100: directions which makes a large difference concerning the piezoresistive effect. For quadruple injection IAa s ICc s IeE s IgG s 1 mA we could always measure the Hall effect signal UDH whereas the voltage UBF remained magnetic field independent. We also rotated all connections by 458 and obtained different offsets but the same magnetic-field sensitivity at the Žrotated. Hall contacts. In Table 1 we show measurement results on Anti-Hall octagons with quadruple injection at room temperature. The averages and standard deviations were calculated on the basis of 13 samples. First of all, one notices large signals at the stress-sensitive contacts for injection along ²110: which scatter very much from sample to sample. These values are similar to the individual offsets of conventional Hall structures in the same packages. Contrary, the offsets at the Hall contacts scatter around zero with a standard deviation that is about one order of magnitude smaller than for the stress-sensitive contacts. Obviously a large part of the stress-dependent offset is cancelled. For injection along ²100: the signals at the stress-sensitive contacts are much smaller which corresponds to the smaller shear piezoresistive coefficients for that orientation w6x. The offsets at the Hall contacts also scatter less for this orientation. We assume that an inhomogeneous stress con-
4.2. Spinning-current Because the RMFR principle is built into the basic cell, the orthogonal offsets must vanish under linear conditions which could be confirmed by the simulations. More realistic simulations which include the nonlinear sheet resistance dependence on the local voltage caused by the voltage-dependent depletion-layer width will be performed in the future.
Table 1 Mean values and standard deviations of the voltages at the stress-sensitive contacts and the Hall contacts of Anti-Hall octagons at Bs 0 and I s1 mA Quadruple injection
²UST RESS : ŽmV.
²UHALL : ŽmV.
²110: ²100:
y32"40 2.7"1,5
0.6"3.0 y0.4"1.3
1 mV corresponds to 2 mT B-offset.
C. Muller-Schwanneke et al.r Sensors and Actuators 81 (2000) 18–22 ¨
tributes to the offsets at the Hall contacts in order to explain the larger spread for injection along ²110:. Additionally, the influence of current contact displacements limits the performance even in the absence of stress. Measurements on a series of Anti-Hall squares Žwith the same ratio of strip width to hole width as the Anti-Hall octagons. with one current contact deliberately displaced by 1 mm or 2 mm yielded offsets of about 10 mVrmm Žf 20 mTrmm.. Using this number, the Hall offsets for injection along ²100: in Table 1 correspond to contact displacements of several tens of nanometers which cannot be significantly reduced with standard technology. This limit as well as the experimental scattering of the residual offsets translates to B-field equivalent offsets of a few millitesla which cannot be controlled. 5.2. Spinning-current With the spinning current method we could achieve very small residual offsets well below 100 mT for all samples by limiting the current so that the nonlinear contribution to the input resistance remained below 10%. We found it quite remarkable that the method even worked for the Anti-Hall samples Žwith single current injection. as long as we adhered to the RMFR, that is, we exchanged the current and voltage contacts, e.g., in measurements
21
using only the outer boundary contacts as if no hole were present. From a practical point of view, one is interested in a high output voltage of the Hall sensor and therefore wants to use the maximum current compatible with the offset requirements. As the residual offset has contributions from the nonlinearities of the system in addition to other effects which rotate with the current Že.g., Peltier effect w1x., we examined the nonlinear current dependence of the residual offset in more detail. In Fig. 3 we show residual offset data measured on four different samples of the same type. The individual offsets of these samples at 1 mA bias current Žwithout spinning. were Ž31 " 17. mV for injection along ²110: and Žy0.3 " 0.8. mV for injection along ²100:. Despite this large orientation dependence the current dependence of the residual offsets based on measurements with four current injections along ²110: was similar to the case with injections along ²100:, but with the opposite sign. This orientation dependence of the sign of the nonlinear current dependence gives us the hint that the piezoresistive effect is involved. The residual offsets based on eight injections along all possible directions are almost in the range of the stray fields at the sample position of up to 5 mT. We think that a violation of the RMFR by the nonlinear dependence of the sheet resistance on the current via the nonlinear depletion width dependence on the local junction voltage is of importance here. This violation of the RMFR in conjunction with the piezoresistive effect seems to be responsible for the measurement results.
6. Conclusions
Fig. 3. Experimentally determined residual offsets of four spinning-current Hall plates. Residual offsets based on four orientations with current injection along ²100: Žopen symbols., current injection along ²110: Žsolid symbols. and residual offsets based on eight orientations Žsolid lines. are shown. The dashed lines represent an equivalent offset of 100 mT.
For the Anti-Hall method we could verify that separate measurements of a magnetic induction B and of a shear stress are possible at different contact pairs. For injection along ²110: one could observe an offset-reduction of about an order of magnitude. But as the Anti-Hall method does not use any contact commutation, displacements of the current contacts and inhomogeneous stress contribute to a residual offset of a few millitesla which cannot be significantly reduced. In contrast we obtained residual offsets well below 100 mT with the spinning-current method by limiting the current so that the nonlinear contribution to the input resistance remained below 10%. The nonlinear current dependence of the residual offset based on four orientations showed a different sign for injection along the ²110: directions in the Ž100. plane than for injections along ²100:. We propose that a violation of the reciprocity principle for macroscopic samples through the nonlinear dependence of the depletion width on the junction voltage together with the piezoresistive effect may explain such behaviour.
22
C. Muller-Schwanneke et al.r Sensors and Actuators 81 (2000) 18–22 ¨
References w1x P.J.A. Munter, A low-offset spinning-current Hall plate, Sensors and Actuators A 21–23 Ž1990. 743–746. w2x H.H. Sample, W.J. Bruno, S.B. Sample, E.K. Sichel, Reverse-field reciprocity for conducting specimens in magnetic fields, J. Appl. Phys. 61 Ž1987. 1079. w3x R.G. Mani, K. von Klitzing, F. Jost, K. Marx, S. Lindenkreuz, H.P. Trah, Method for simultaneously reducing the misalignment offset
and separating the Hall voltage from the off-diagonal piezoresistive voltage in Hall effect and piezoresistive devices based on silicon, Appl. Phys. Lett. 67 Ž1994. 2223. w4x R. Gottfried-Gottfried, Thermal behaviour of CMOS Hall sensors for different operating modes, Sensors and Actuators A 41–42 Ž1994. 430–434. w5x R.S. Popovic, Numerical analysis of MOS magnetic field sensors, Solid-State Electronics 28 Ž1985. 711–716. w6x Y. Kanda, Piezoresistance effect of silicon, Sensors and Actuators A 28 Ž1991. 83–91.