sensors Article

Optimum Design Rules for CMOS Hall Sensors Marco Crescentini 1,2, *, Michele Biondi 2 , Aldo Romani 1,2 , Marco Tartagni 1,2 and Enrico Sangiorgi 1,2 1

2

*

Department of Electrical, Electronic and Information Engineering “G. Marconi”—DEI, University of Bologna, Cesena Campus, Via Venezia 52, 47521 Cesena, Italy; [email protected] (A.R.); [email protected] (M.T.); [email protected] (E.S.) Advanced Research Center on Electronic Systems (ARCES), University of Bologna, Cesena Campus, Via Venezia 52, 47521 Cesena, Italy; [email protected] Correspondence: [email protected]; Tel.: +39-0547-3-39247; Fax: +39-0547-3-39208

Academic Editor: Vittorio M. N. Passaro Received: 29 January 2017; Accepted: 29 March 2017; Published: 4 April 2017

Abstract: This manuscript analyzes the effects of design parameters, such as aspect ratio, doping concentration and bias, on the performance of a general CMOS Hall sensor, with insight on current-related sensitivity, power consumption, and bandwidth. The article focuses on rectangular-shaped Hall probes since this is the most general geometry leading to shape-independent results. The devices are analyzed by means of 3D-TCAD simulations embedding galvanomagnetic transport model, which takes into account the Lorentz force acting on carriers due to a magnetic field. Simulation results define a set of trade-offs and design rules that can be used by electronic designers to conceive their own Hall probes. Keywords: Hall sensors; current-related sensitivity; power consumption; design rules; 3D physical simulations; Hall effect sensor design

1. Introduction Magnetic field sensors are valuable for many kinds of applications: from industrial to automotive (e.g., position sensors), from home applications (e.g., current sensors) to smartphones (e.g., compass). Among magnetic sensors, the Complementary Metal-Oxide-Semiconductor (CMOS) Hall sensor has a primary role thanks to its compatibility with standard microelectronic processes. A silicon-based Hall probe shows worse performance than a discrete Hall probe implemented with high-mobility compound semiconductors such as GaAs or InSb [1]. However, the possibility of integrating magnetic sensors and signal conditioning circuits into the same substrate allows efficient implementation of several circuit techniques improving Hall performance, such as offset reduction, temperature stabilization and nonlinearity correction [2–4]. Given the pervasive use of CMOS Hall sensors, it will be useful to have a set of simple rules to customize the sensor design to the target application. A macroscopic description of the Hall effect offers a useful theory, though with poor physical insights. On the contrary, quantum theory provides a deep understanding of the processes acting on each single carrier, albeit with an awkward mathematical description unsuitable for sensor design. In this context, numerical simulations of magnetic devices are a promising approach. Although the first research articles about technology-computer-aided-design (TCAD) appeared in the 1990s [5,6], commercial TCAD software coping with galvanomagnetic effects—that is, the full implementation of the Lorentz force—has only recently been made available [7]. The majority of the literature about Hall sensor presents a single and novel Hall device with particular characteristics [8–10] or a novel and better-performing sensing system based on a specific Hall probe [11–16]. There are a few articles investigating the performance of general Hall probes

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focusing on the sole offset voltage [17,18] or comparing a limited set of given geometries [19–21] without providing general rules. Jankovic et al. [18] study the performance of well-defined cross-shaped and vertical Hall sensors so as to define an electrical equivalent SPICE model. The paper is limited to the analysis of the realized sensors without discussing the effects of geometries, bias or other design parameters on the final performance. Paun et al. [20] make use of three-dimensional (3D) TCAD simulations to compare among five different shapes of Hall sensors. The paper mainly analyzes and compares the sensitivities and the offsets so as to define the best sensor shape. However, no discussion on the exact values of length, width, doping or any other design parameter is made; hence, no rules for designers are provided. This manuscript exploits advanced 3D TCAD tools to study the behavior of general Hall probes, irrespective of the chosen geometry. It focuses on the effects of design parameters, such as the aspect ratio or the positions of the contacts, on the final performance of the Hall probe, like sensitivity, power consumption, and bandwidth. The final scope of the article is providing a set of simple rules for designers so they can choose the best geometry, length, and bias for the target application. Offset and temperature are not covered by this manuscript as there is already a significant amount of literature about those effects [3,17,19,22–24]. This manuscript is organized as follows: Section 2 describes the modeled sensors, the basic theory of the Hall effect and the numerical simulator. Section 3 presents all the simulation results and compares some of them with measurements, while Section 4 provides an overall discussion and defines trade-offs and design rules. 2. Materials and Methods 2.1. Devices The simulated Hall probes are composed of lightly doped n-well (yellow region in Figure 1) with constant concentration of donor atoms (ND = 2 × 1016 cm−3 ) and four highly doped contacts placed close to the edges of the n-well. The n-well has a rectangular shape so as to get general results regardless of the actual geometry. Width W and length L of the n-well are swept from 20 µm to 140 µm, which are typical sizes. Two contacts facing each other are used to bias the sensor (named bias contacts) and the other two contacts are used to sense the Hall voltage (named sense contacts). Commonly, bias contacts are distinct from sense contacts. In the following simulations, bias contacts are 1 µm long and Wb = W/2 wide, unless otherwise stated; sense contacts are placed at the middle of the x-axis, that is the bias-current axis, close to the edges along the y-axis, and are small square contacts 1 µm wide.

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depletion region

t

n-well

teff

contacts

p-substrate Figure 1. Three-dimensional (3D) model and cross-section of the Hall sensor. Figure 1. Three-dimensional (3D) model and cross-section of the Hall sensor.

2.2. Theory All the contacts are 0.4 µm thick and they are modeled by three vertical layers (z-axis) with lower Theinto macroscopic description Hall model effect isfor summarized hereafter. When a magn doping concentration going deep the substrate. This isofathe simple a low-energy ion is applied orthogonally to an electric current that is flowing through a material, thus the implantation with projected range close to the surface. A 5 µm thick p-type epitaxial layer surrounds electrons bends from the original path and a small voltage, named Hall voltage V orthogonally to both current and magnetic field. The Hall voltage is given by:

VH  SI  I x  Bz ,

where SI is the current-related sensitivity, Ix is the component of the bias current along the x-

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the n-well by 5 µm along all directions. The voltage of the p-type layer is set to ground by long contacts all around the probe (dark blue in Figure 1). This is a very general structure that can be improved by adopting modern technological solutions, such as the creation of a p-layer above the sensitive n-well for sensitivity enhancement [13,15]. 2.2. Theory The macroscopic description of the Hall effect is summarized hereafter. When a magnetic field is applied orthogonally to an electric current that is flowing through a material, thus the flow of electrons bends from the original path and a small voltage, named Hall voltage VH , arises orthogonally to both current and magnetic field. The Hall voltage is given by: VH = S I · Ix · Bz ,

(1)

where SI is the current-related sensitivity, Ix is the component of the bias current along the x-axis and Bz is the component of the magnetic field along the z-axis. In the following we will generally refer to bias current and orthogonal magnetic field as I and B. The sensitivity SI depends on physical and geometrical parameters. It is usually expressed as: SI = G

rH = Gµ H Rsq qnte f f

(2)

where n is the concentration of quasi-free carriers (which is almost equal to the doping level ND in n-doped silicon), teff is the effective thickness of the n-well, rH is the Hall factor, q is the electron   −1 charge, G is the geometrical correction factor, and Rsq = qµn nte f f is the square resistance of the Hall sensor. The effective thickness teff represents the actual thickness of the sensitive region of the n-well narrowed by the depletion region occurring at the pn-junction (Figure 1). The Hall factor rH expresses the ability of the material to generate the Hall voltage. It mainly relies on scattering effects and material anisotropy and can be combined with the electron mobility so as to get the Hall mobility to µ H = µn · r H . In silicon, the Hall factor is around unity and hence the Hall mobility is equal to the electron mobility (µH ≈ µn ). G is a correction factor introduced to take into account the efficiency of the specific shape of the Hall probe [25,26]. 2.3. Numerical Model Synopsys Sentaurus Device® was used in this work as a physical simulator for the Hall probe. It embeds the common drift-diffusion transport model enhanced with magnetic field-dependent terms taking into account the Lorentz force. The general equation for the electron current density is: → Jn

=

→ J0 + µn

 1 1 + ( µ H B )2

→

µ H B ×

→ J0

µn

→



→

+ µ H B × µ H B ×

→ J0

µn

  ,

(3)

→

where J0 is the current vector due to the applied electric field only, µH is the Hall mobility, µn is the →

electron mobility, B is the magnetic field vector and B is the magnitude of this vector [7]. Equation (3) is a general formulation of carrier transport in the presence of magnetic field, solving the Lorentz equation for each carrier. For this reason, (3) has no validity limitations but numeric convergence is reliable only for B < 10 T [7]. Other than transport equation and Lorentz force, the simulation also takes into account mobility degradation due to doping concentration, temperature and carrier-carrier scattering, high field saturation, and Shockley-Read-Hall recombination. Synopsis Sentaurus Device® has been already proven as a fair and reliable tool for Hall probe analysis [18,20].

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3. Experimental Section In this section, TCAD simulations will be used to: (i) understand the effects of design parameters on sensitivity, power consumption and bandwidth of a general Hall probe and (ii) define a set of design rules. This paper does not deal with the offset since electronic techniques can be used to effectively reduce it [27]. 3.1. Current-Related Sensitivity First of all, we study the effects of geometrical dimensions on current-related sensitivity SI . Simulations were done on the sensors described in Section 2.1, biased with a constant 500-µA current. The current-related sensitivity SI is related to geometrical dimensions by means of the geometrical correction factor G as shown in (2). This factor has been defined as the ratio between the sensitivity of a real Hall probe and that of an infinitely long Hall probe [19,26]: G=

SI S I | L=∞

(4)

Following this definition, G tends to be 1 when L = ∞. The G factor can be more precisely expressed as a complex function of W, L, and Hall angle θ H . The following equation is valid only for rectangular probes with L > 0.85W and 0 ≤ θ H ≤ 0.45, which implies weak magnetic field regime [19] 16 G = 1 − 2 e− π

πL 2W



8 1 − e− 9

πL 2W



θ2 1− H 3

! .

(5)

The effect of the specific geometry on the G factor has been profusely studied in many articles [18–20,22], but the geometries indirectly affect also other parameters. Figure 2 compares (5) with the simulated current-related sensitivity for a Hall probe with constant bias current, W = 40 µm, and sweeping the length from 10 µm to 140 µm. Sensors 2017, 17, 765 5 of 13 700

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Sensitivity rises with the nwell resistance

Sensitivity rather follows G-factor

0.2 0

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Length [µm] Figure 2. 2. Current-related Current-related sensitivity sensitivity SSII at atdifferent differentsensor sensorlengths lengthssimulated simulatedfor foraa40 40µm µmwide widesensor. sensor. Figure Sensitivity follows the G factor only for a small range of lengths. Sensitivity keeps on rising Sensitivity follows the G factor only for a small range of lengths. Sensitivity keeps on rising for L > for 80 L > 80 µm to the lowering of the effective thickness of the n-well. µm due to due the lowering of the effective thickness of the n-well.

Comparing both curves, it is possible to see that simulated SI fits the logarithmic behavior of G factor only for a small range of lengths. Sensitivity falls down very quickly for L < 0.8W = 34 µm. This behavior is not predicted by (5) since it is not valid in this region. On the other hand, sensitivity rises unbound as length increases, while (5) predicts the existence of an asymptotic value for L–>∞.

rather follows G-factor

500 400

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0.4 This discrepancy clearly indicates that geometries alters otherWparameters than the G factor. A possible = 40µm explanation is the increased100 resistivity of the n-well with the length of the sensor (Figure 3a). A longer Sensitivity 0.2 sensor means higher resistance between bias contacts, thus higher bias voltage is needed to keep the G factor 0 0 same current, thereby changing0the effective thickness of the n-well as modulates the depletion region 20 40 60 80 100 120 it140 of the reversed biased pn-junction. As a result, Length a long [µm] n-well is preferable to obtain high current-related sensitivity. This simulation was done for three different sensor widths: 20 µm, 40 µm, and 60 µm, reporting similar results as shown in Figure 3b, where the sensitivity is reported with respect to the Figure 2. Current-related sensitivity SI at different sensor lengths simulated for a 40 µm wide sensor. aspectSensitivity ratio rather thanthe theGsingle dimensions. SensorSensitivity with W =keeps 60 µmonbehaves inLa>slightly follows factorgeometrical only for a small range of lengths. rising for 80 different waytobecause its resistivity showsthickness a steep of increase for L = 100 µm (Figure 3a). µm due the lowering of the effective the n-well.

(a)

(b)

Figure3. 3. (a) (a) Simulated Simulated square square resistance resistance as as aa function function of of the the sensor sensor length length for for different different sensor sensor widths. widths. Figure Higher voltage is needed to keep the same bias current in longer sensors, and this higher voltage Higher voltage is needed to keep the same bias current in longer sensors, and this higher voltage implies implies an effective thickness lowering, resulting in increased square resistance; (b) Simulated an effective thickness lowering, resulting in increased square resistance; (b) Simulated current-related current-related I as a function of geometrical aspect ratio L/W for different sensor widths. sensitivity SI assensitivity a functionSof geometrical aspect ratio L/W for different sensor widths. Lines are Lines are obtained by means of a third-order fitting. obtained by means of a third-order polynomialpolynomial fitting.

3.2. Power Consumption 3.2. Power Consumption The conclusion of the previous section is that the longer the n-well, the higher the sensitivity SI. The conclusion of the previous section is that the longer the n-well, the higher the sensitivity SI . However, for L ≥ 2W, the increase in sensitivity is mainly related to the reduction of the effective However, for L ≥ 2W, the increase in sensitivity is mainly related to the reduction of the effective thickness of the n-well teff (i.e., resistance growing) (Figure 3), leading to higher power consumption thickness of the n-well teff (i.e., resistance growing) (Figure 3), leading to higher power consumption for a fixed bias current. The total resistance of the n-well is given by: for a fixed bias current. The total resistance of the n-well is given by: 1 L L  R  Rsq (1) LW qn1nteff WL R = Rsq = (6) W qµn nte f f W where Rsq depends on the effective thickness of the n-well. Assuming a constant bias current I, the where Rsq R depends on the with effective thickness of thevoltage n-well.V is Assuming constant biasofcurrent I, resistance grows linearly L when the applied small (i.e.,a small values R and L), the resistance grows linearly L when the applied voltagevoltage, V is small (i.e., values of while it growsRmore than linearlywith for higher values of the applied due to thesmall widening of the Rdepletion and L), while it grows n-well more than for higher of the voltage, to the layer between and linearly p-substrate. On thevalues contrary, the applied sensitivity SI hasdue a pseudowidening of the depletion layer between n-well 3.1. and This p-substrate. the contrary,ofthe logarithmic behavior, as discussed in Section suggestsOn the existence ansensitivity optimum Spoint, I has awhich pseudo-logarithmic behavior, as discussed in Section 3.1. This suggests existence of power. an optimum is a value of the aspect ratio that optimizes the trade-off between the sensitivity and point, which is a value of the aspect ratio that optimizes the trade-off between sensitivity and power. For a better understanding it is useful to define an efficiency factor η expressing how much power is required to get a certain Hall voltage VH for a magnetic induction of 1 T: η=

  VH S V = I V·I·B V W·T

(7)

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For a better understanding it is useful to define an efficiency factor η expressing how much power is required to get a certain Hall voltage VH for a magnetic induction of 1 T: Sensors 2017, 17, 765



VH S  I V IB V

 V   W T   

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(6)

where where VV and and II are are the thebias biasvoltage voltageand andcurrent, current,respectively. respectively. The The ηη factor factor defines defines how how much much Hall Hall voltage is generated when consuming 1 W at 1 T of magnetic induction. This parameter can be voltage is generated when consuming 1 W at 1 T of magnetic induction. This parameterreferred can be to as the power-related sensitivitysensitivity of the sensor, since: referred to as the power-related of the sensor, since:

VHH =  η ·VV I· IB· B  = Pη ·BP · B V

(8) (7)

Figure 44 reports Hall sensors with different aspect ratios L/W Figure reports the the simulated simulatedefficiency efficiencyfactor factorη for η for Hall sensors with different aspect ratios and and samesame bias bias current I = 500 high-symmetry square-shaped sensorsensor showsshows the best L/W current I =µA. 500AµA. A high-symmetry square-shaped thepower best efficiency, that is the best trade-off between current-related sensitivity and power consumption. In power efficiency, that is the best trade-off between current-related sensitivity and power consumption. fact, long and narrow sensors provide higher sensitivity due to increased resistance of the n-well, as In fact, long and narrow sensors provide higher sensitivity due to increased resistance of the n-well, described inin Section as described Section3.1, 3.1,but butthey theyalso alsosuffer sufferfrom fromhigher higherpower powerconsumption consumption due due to to the the same same effect. effect. Conversely, short and large sensors dissipate less power but the charge accumulation due tothe theHall Hall Conversely, short and large sensors dissipate less power but the charge accumulation due to effect is negligible. effect is negligible.

Figure4.4. Simulated Simulated efficiency efficiency factor factor ηη for fordifferent differentaspect aspectratios ratiosL/W L/W and and different different width width W W of of the the Figure sensor.The Thebias biascurrent currentisiskept keptconstant constantatat500 500µA. µA.High Highefficiency efficiencyisisreached reachedfor forsquare squaresensors. sensors. sensor.

3.3. Optimum Optimum Square SquareDimension Dimension 3.3. Once high-symmetry high-symmetry shapes shapes are are defined defined as as the the best best choices, choices, (1) (1) and and (2) (2)state stateno noother otherrelations relations Once with the the area areaof ofthe thesensor. sensor. However, However, there there isisanother another trade-off trade-off between between area area and andperformance performance to to with consider. Figure shows the simulated S I for square Hall sensors with increasing areas. Small sensor consider. Figure 5 shows the simulated SI for square Hall sensors with increasing areas. Small sensor suffers from from boundary boundary effects, effects, limiting limiting its its sensitivity sensitivity [25]; [25]; while while large large sensor, sensor, with with an an area area above above aa suffers certain value (let us say 30 µm × 30 µm), does not improve significantly the sensitivity, pointing out certain value (let us say 30 µm × 30 µm), does not improve significantly the sensitivity, pointing out an asymptotic relation as the one shown in (5). Given the square sensor as the optimum geometry, an asymptotic relation as the one shown in (5). Given the square sensor as the optimum geometry,a not depend depend on on the the doping doping acompromise compromisebetween betweenarea areaand andsensitivity sensitivityexists exists and and this this relation relation does does not concentration of the n-well. The blue line in Figure 5 shows the same simulation on a highly doped concentration of the n-well. The blue line in Figure 5 shows the same simulation on a highly doped 3−3 16 n-well ( = 8 10 cm ). Nominal sensitivity has fallen to values between 30 and 50 V/AT, but n-well (ND = 8 × 10 cm ). Nominal sensitivity has fallen to values between 30 and 50 V/AT, the the behavior is still the the same. but behavior is still same.

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Figure Figure 5. 5. Current-related Current-related sensitivity sensitivity S SII simulated simulated for for square square sensors sensors with with different different areas areas and and different different doping concentrations (note two different scales were used on the y-axis. Left scale for the doped lowly doping concentrations (note two different scales were used on the y-axis. Left scale for the lowly doped sensor and right for the doped highlysensor). doped sensor). Small sensors suffer from boundary effects sensor and right scale forscale the highly Small sensors suffer from boundary effects limiting limiting sensitivity. Sensitivity rises as a logarithmic function of the area; hence a 30 µm × 30 µm is a sensitivity. Sensitivity rises as a logarithmic function of the area; hence a 30 µm × 30 µm is a good good trade-off between area and sensitivity. trade-off between area and sensitivity.

3.4. Dimension of Contacts 3.4. Dimension of Contacts Sense contacts must be as small as possible, allowing more punctual and accurate measurements [19]. Sense contacts must be as small as possible, allowing more punctual and accurate They must be placed at the center of the n-well, i.e., x = L/2, where the maximum of the Hall voltage measurements [19]. They must be placed at the center of the n-well, i.e., x = L/2, where the maximum is developed. They must also be placed close to the edges of the n-well, along the y-axis, since the of the Hall voltage is developed. They must also be placed close to the edges of the n-well, along the Hall voltage grows as we go far from the center of the sensor, as shown in Figure 6a, which is the y-axis, since the Hall voltage grows as we go far from the center of the sensor, as shown in Figure 6a, result of a simulation on a square 40 µm × 40 µm sensor. The figure shows the Hall voltage ideally which is the result of a simulation on a square 40 µm × 40 µm sensor. The figure shows the Hall voltage measured at different points on the y-axis. The spike on the y = 19.5 µm curve at x ≈ 20 µm is due to ideally measured at different points on the y-axis. The spike on the y = 19.5 µm curve at x ≈ 20 µm is the presence of the highly doped contact while the oscillations relate to numerical errors. due to the presence of the highly doped contact while the oscillations relate to numerical errors. Dimensions of biasing contacts affect both the power consumption and the sensitivity SI, Dimensions of biasing contacts affect both the power consumption and the sensitivity SI , showing showing another trade-off. Reducing Wb leads to higher contact-to-contact resistance, hence higher another trade-off. Reducing Wb leads to higher contact-to-contact resistance, hence higher applied applied voltage is needed to keep the bias current constant. As said in previous sections, this leads to voltage is needed to keep the bias current constant. As said in previous sections, this leads to a a reduction of the effective thickness of the n-well. Hence, small bias contacts are preferred in reduction of the effective thickness of the n-well. Hence, small bias contacts are preferred in application application requiring high sensitivity, while large bias contacts are preferred in application requiring high sensitivity, while large bias contacts are preferred in application demanding for low demanding for low power consumption. Bias contacts with Wb = W/2 are a good trade-off between power consumption. Bias contacts with Wb = W/2 are a good trade-off between power and sensitivity. power and sensitivity. Figure 6b shows the relation of SI and power consumption with the normalized Figure 6b shows the relation of SI and power consumption with the normalized dimension of Wb /W. dimension of Wb/W. 3.5. Bias Current 3.5. Bias Current Following (2), the current-related sensitivity SI is independent of the bias current. However, is independent of the bias current. However, Following (2),a the current-related sensitivity SI higher simulations show small increase in sensitivity with bias currents (Figure 7a). Increasing the simulations show a small increase in sensitivity with higher bias currents (Figure 7a). Increasing the bias current has not direct effect on SI but lowers the effective thickness of the n-well, since higher bias not direct effect onsame SI buteffect lowers effective thickness of the n-well, 3.1. sinceFigure higher7 bias current voltage has is applied. This is the wethe observed and described in Section bias voltage is applied. This is the same effect we observed and described in Section 3.1. Figure 7 reports both the current-related sensitivity SI and the efficiency factor η for sensors with different reports both the current-related sensitivity S I and the efficiency factor η for sensors with different geometries and sweeping the bias current from 250 µA to 1.25 mA. High bias current leads to high geometries and sweeping the bias from µA to 1.25 mA. current leads the to high power consumption, following P =current RI2 , and high250 sensitivity SI at theHigh samebias time. Therefore, bias 2, and high sensitivity SI at the same time. Therefore, the bias power consumption, following P = RI current defines another sensitivity/power trade-off that must be properly adjusted based on the current defines anotherThese sensitivity/power trade-off that be properly and adjusted based carried on the application constraints. results are in agreement withmust the measurements simulations application constraints. These shapes resultsreported are in agreement with the measurements and simulations on Hall probes with dissimilar in [20]. carried on Hall probes with dissimilar shapes reported in [20].

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x x

y=7.5µm y=7.5µm y=11.5µm y=11.5µm y=15.5µm y=15.5µm y=19.5µm y=19.5µm

10 10 8 8 6 6 4 4 2 2 0 0 0 0

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Current-related Current-relatedSensitivity SensitivitySiSi[V/AT] [V/AT]

y y y y

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0 0

Wb/W [µm] Wb/W [µm]

(a) (a)

(b) (b)

Figure 6. (a) Simulated Hall voltage along x direction for a square 40 µm × 40 µm sensor. Hall voltage Figure6.6.(a)(a) Simulated Hall voltage along x direction a square µmµm × sensor. 40 µm Hall sensor. Hall Figure Simulated Hall voltage along x direction for afor square 40 µm40× 40 voltage is measured at four different distances from the symmetry axis. Hall voltage increases when moving is measured at four different the symmetry Hallincreases voltage increases when isvoltage measured at four different distancesdistances from thefrom symmetry axis. Hallaxis. voltage when moving away from the symmetry axis as the carriers’ path are more bent; (b) Simulations on a square 40 µm moving away the symmetry axis as the carriers’ more (b) Simulations on a 40 square away from the from symmetry axis as the carriers’ path arepath moreare bent; (b)bent; Simulations on a square µm ×4040µm µm sensor with different width W B of bias contacts. × 40 µm sensor with different width W of bias contacts. × 40 µm sensor with different width WB of biasB contacts. L=20 um L=20 um L=40 um L=40 um L=60 um L=60 um

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Efficiency Efficiency[V/WT] [V/WT]

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Figure 7. (a) Sensitivity and (b) efficiency versus bias current simulated for sensors with different Figure 7. (a) Sensitivity and (b) efficiency versus bias current simulated for sensors with different Figureratios 7. (a)but Sensitivity efficiency versus bias current simulated for sensors with different aspect constantand W =(b) 40 µm. aspect ratios but constant W = 40 µm. aspect ratios but constant W = 40 µm.

3.6. Measurement Comparison 3.6. Measurement Comparison 3.6. Measurement Comparison Following the above-defined rules, a square-shaped 30 µm × 30 µm Hall sensor with four Following the above-defined rules, a square-shaped 30 µm × 30 µm Hall sensor with four Following above-defined a square-shaped 30 µm 30 µm Hall identical squarethe contacts placed atrules, the angles of the n-well was×fabricated in sensor CMOS with 0.16 four µm identical square contacts placed at the angles of the n-well was fabricated in CMOS 0.16 µm identical square contacts placed at the thecurrents n-well was measured fabricated and in CMOS 0.16with µm technology. Current-related sensitivity at angles variousofbias compared technology. Current-related sensitivity at various bias currents was measured and compared with technology. Current-related at various proves bias currents wasaccuracy measured andsimulations compared with TCAD simulations in Figure sensitivity 8. The measurement the good of the and TCAD simulations in Figure 8. The measurement proves the good accuracy of the simulations and TCAD simulations in Figureof8.the The measurement proves good accuracy of the of simulations and demonstrates the narrowing effective thickness of thethe n-well since the increase the sensitivity demonstrates the narrowing of the effective thickness of the n-well since the increase of the sensitivity the narrowing the effective thickness of the n-well since the increase of the sensitivity Sdemonstrates I when applying I > 1 mA is of evident. SI when applying I > 1 mA is evident. SI when applying I > 1 mA is evident.

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Figure 8.8.Comparison Comparison between measurements simulations the current-related sensitivity Figure between measurements andand simulations of theofcurrent-related sensitivity SI for S for a square-shaped Hall sensor with four equal contacts placed on the angles of the n-well. I a square-shaped Hall sensor with four equal contacts placed on the angles of the n-well. Both Both measurements and simulations an increase the sensitivity at high current measurements and simulations revealreveal an increase of theofsensitivity at high biasbias current thatthat cancan be be explained by the widening of the depletion layer between the sensitive n-well and the p-substrate. explained by the widening of the depletion layer between the sensitive n-well and the p-substrate. This result result is is in in agreement agreement with with all all the the previous previous simulations. simulations. This

3.7. Bandwidth Bandwidth 3.7. This section section will will analyze analyze bandwidth bandwidthand andstep step response. response. We We define define the the settling settling time time in in response response This to a step change time as the time to reach the steady-state voltage within an error lower than 0.1%. to a step change time as the time to reach the steady-state voltage within an error lower than 0.1%. Figure 99 shows showsthe theHall Hallvoltage voltageVV response a step change magnetic induction from to HH inin response to to a step change of of magnetic induction from 0 to050 Figure 50 mT for two values of capacitive load at the sense contacts, which are floating contacts (blue line) mT for two values of capacitive load at the sense contacts, which are floating contacts (blue line) and andpF-loaded 1.5 pF-loaded contacts of(red 3 pF) (redThe line). The simulations were on a30square 1.5 contacts (for a (for totalaoftotal 3 pF) line). simulations were done on done a square µm × 30 µm µm × 30 µmwith sensor with four identical µmµm × 4.5 µm placed the angles the n-well. 30 sensor four identical contactscontacts of 1 µmof×14.5 placed at the at angles of theof n-well. The The sensor responds faster with reduced capacitive loading, demonstrating the importance of the sensor responds faster with reduced capacitive loading, demonstrating the importance of the interface between between the the sensor sensor and and the the electronic electronic front-end. front-end. In In fact, fact, the the upper upper bandwidth bandwidth limit limit of of Hall Hall interface sensor is set by the capacitive load, as recently proven by the authors in [28,29]. The settling time sensor is set by the capacitive load, as recently proven by the authors in [28,29]. The settling time can canmodeled be modeled asRC an RC constant where is the resistance then-well n-welland andCCisisgiven givenby by the the be as an timetime constant where R isR the resistance ofof the capacitance of of the the n-well n-well and and the the capacitive capacitive loads loads of of the the electronic electronic front-end. front-end. This This RC RC time-constant time-constant capacitance models the rearrangement of carriers inside the sensor in response to a stimulus. It enters into play play models the rearrangement of carriers inside the sensor in response to a stimulus. It enters into regardless ififthe thesource sourceofofthe thestimulus stimulusisisa astep step the magnetic field a step of the current, regardless ofof the magnetic field or or a step of the biasbias current, as as shown in Figure where sensor this casewithout withoutcapacitive capacitiveload) load)responds respondsin inabout about10 10 ns ns shown in Figure 10,10, where thethe sensor (in(in this case to any kind of stimulus. to any kind of stimulus. To further behavior, a step of the was applied to two to similar sensors To further prove provethe theRC-like RC-like behavior, a step of bias the current bias current was applied two similar differentiating each other only by the doping level of the n-well. Doubling the carrier concentration, sensors differentiating each other only by the doping level of the n-well. Doubling the carrier n changes the well resistivity Ωcm to 1.7 Ωcm2.9 keeping the1.7 junction the concentration, n changes thefrom well2.9 resistivity from Ωcm to Ωcm capacitance keeping thealmost junction same. As expected, Figure 11As shows a reduction response time by half. capacitance almost the same. expected, Figureof 11the shows a reduction ofathe response time by a half.

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Hall Voltage [mV] Hall HallVoltage Voltage[mV] [mV]

666 555 444 333 222 111 000 -1 -1 -1 0 0 -2 0 -2 -2

Cload = 0 pF Cload Cload == 00 pF pF Cload==33pF pF Cload Cload = 3 pF

0.05 0.05 0.05

0.1 0.1 0.1

Time [µs] Time Time [µs] [µs]

0.15 0.15 0.15

0.2 0.2 0.2

Figure9.9.Hall Hallvoltage voltage VHHin inresponse responseto toaastep stepchange changeof of 50mT mTin in theapplied appliedmagnetic magneticinduction inductionfor for Figure Figure 9. 9.Hall inresponse responsetoto step change of mT 50 mT in applied the applied magnetic induction Figure Hallvoltage voltage V VH H in aa step change of 50 50 in the the magnetic induction for asensor sensorwith withfloating floatingsense sensecontacts contacts(blue (blueline) line)and andwith with1.5 1.5pF pFcapacitive capacitiveload loadon oneach eachcontact contact(red (red a fora asensor sensor with floating sense contacts (blue pF capacitive load oncontact each contact with floating sense contacts (blue line)line) and and with with 1.5 pF1.5 capacitive load on each (red line). line). (red line). line).

Hall voltage [mV] Hall voltage [mV] Hall voltage [mV]

5 5 5

Hall Voltage [mV] Hall Voltage [mV] Hall Voltage [mV]

4 4 4 3 3 3 2 2 2 1 1 1

≈10 ns ≈10 ns ≈10 ns

0.005 0.005 0.005

2000 2000 2000

1500 1500 1500

1000 1000 1000

500 500 500

0 0 0 -1 -1 -1 00 0

Hall voltage [mV] Hall voltage [mV] Hall voltage [mV]

6 6 6

0.01 0.01 0.01

0.015 0.015 Time [µs] 0.015

Time [µs] Time [µs]

(((aaa)))

0.02 0.02 0.02

0.025 0.025 0.025

0.03 0.03 0.03

0 0 -0.2 0-0.2 -0.2

0.08 0.08 0.08 0.07 0.07 0.07 0.06 0.06 0.06 0.05 0.05 0.05 0.04 0.04 0.04 0.03 0.03 0.03 0.02 0.02 0.02 0.01 0.01 0.010 0 0 -0.01 -0.01 ≈12 ns -0.01 -0.02 ≈12 ns -0.02 ≈12 ns -0.02 -0.03 -0.030.0015 0.0115 -0.03 0.0015 0.0115 0.0015 0.0115

0.0215 0.0215

Time [µs] 0.0215 Time [µs] Time [µs]

0 0 0

0.2 0.2 0.2

0.4 0.4

Time 0.4 [µs] Time [µs] Time [µs]

0.6 0.6 0.6

0.8 0.8 0.8

0.0315 0.0315 0.0315

1 1 1

b) (((b b))

Figure 10.(a) (a) Settlingtime time ofthe the Hallvoltage voltage afteraa50 50 mTstep step changeof of themagnetic magnetic induction; Figure Figure 10. 10. (a) Settling Settling time of of the Hall Hall voltage after after a 50 mT mT step change change of the the magnetic induction; induction; (b) settling time of the Hall voltage after a step change of the bias current. Hall voltage is normalized Figure 10. (a) Settling time of the Hall voltage after a 50 mT step change of the magnetic induction; (b) (b) settling settling time time of of the the Hall Hall voltage voltage after after aa step step change change of of the the bias bias current. current. Hall Hall voltage voltage is is normalized normalized to its steady state value. Transient ends when Hall voltage reaches its steady-state value within 0.1% to (b)to settling time of the Hall voltage after a step change of the bias current. Hall voltage is normalized to its its steady steady state state value. value. Transient Transient ends ends when when Hall Hall voltage voltage reaches reaches its its steady-state steady-state value value within within 0.1% 0.1% error. itserror. steady state value. Transient ends when Hall voltage reaches its steady-state value within 0.1% error. error.

Figure 11.Effect Effect of of the n-well doping concentration (ND =n) n) on theon settling time. Doubling the dopingthe Figure 11. doping concentration (N settling time. the Figure 11. then-well n-well doping concentration = the n) the settling time. Doubling Figure 11.Effect Effect of of the the n-well doping concentration (NDD =(N = n)Don on the settling time. Doubling Doubling the doping doping concentration halves halves the the response response time. time. However, However, higher higher doping doping level level lowers lowers the the sensitivity sensitivity concentration doping concentration the response However, higher doping level lowers sensitivity concentration halveshalves the response time.time. However, higher doping level lowers thethe sensitivity following(3). (3). following following (3). following (3).

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4. Discussion Sensitivity is affected by the geometrical characteristics through the geometrical correction factor G, but it also depends on the effective thickness of the n-well, which is modulated by the bias voltage. For a fixed and constant bias current, a longer Hall probe requires higher bias voltage, thereby reducing the effective thickness of the n-well and increasing the probe resistivity. This relation causes a proportional increase of the sensitivity with the length of the sensor, even after the saturation of the G factor (Figure 2). This resistive-induced increase of the sensitivity is at the detriment of power consumption. The power efficiency factor, which can be referred to as a power-related sensitivity, points out the existence of an optimum aspect ratio which maximizes the sensitivity for a given power consumption (Figure 4). This optimum point is obtained for nearly-square, or high-symmetry, Hall probes. This is a very good result that allows the implementation of the current-spinning technique, which is the state-of-the-art procedure for offset reduction in Hall probes. The technique consists in spatially rotating the bias current by 90◦ , hence the sensor needs to be symmetric so as to correctly implement the current-spinning method [12,26,27]. Ohmic contacts influence sensitivity and power consumption, as well. To maximize the sensitivity, the sense contacts must be small, placed at the middle of the probe along the bias direction, and placed as close as possible to the edges of the probe along the orthogonal axis. Sense contacts have no effect on power consumption since no current flows through them (Figure 6), while bias contacts are prone to sensitivity/power trade-off. The implementation of the current-spinning technique imposes another trade-off on the contacts design: they must have the same shape since they swap continuously. Also, the bias current is prone to a sensitivity/power trade-off, but it is more complicated to analyze since the bias current has a twofold effect: it changes the effective thickness and directly affects the Hall voltage VH , as shown in (1). As a result, the choice of the optimum value for the bias current is driven by the target application. Finally, the dynamic behavior of the sensor can be accurately modeled as an RC circuit where R is the resistance of the sensor and C takes into consideration the well capacitance and all the capacitances connected to the sense contacts. From this model, wide acquisition bandwidth can be achieved by means of (i) minimized capacitive load at sensors contacts; (ii) the use of highly doped n-well as sensing probe. This second solution defines a sensitivity/bandwidth trade-off in Hall sensor design linked to sensor resistance: S I = Gµ H Rsq (9) BW ≈ 2πR1sq C 5. Conclusions This manuscript has exploited modern technology-computer-aided-design (TCAD) simulations to investigate the effects of design parameters on sensor performance. Simulations are based on Synopsis Sentaurus® , which implements galvanomagnetic transport mode, and are verified through experimental measurements on a prototype. The analyses yield the following important trade-offs and design rules for CMOS Hall sensor design:

•

•

• •

Geometric characteristics of the probe affect the current-related sensitivity SI by means of both direct and indirect effects; which are associated with the geometrical correction factor G and the bias-induced change of the well resistivity, respectively. Square sensors, or at least symmetrical geometries, show the optimum sensitivity/power trade-off, i.e., the best energy efficiency. This is a good result since symmetrical sensors easily allow offset cancellation through spinning current technique. Optimum width of bias contacts is half the sensor width. Sense contacts should be as small as possible to have a punctual measure, hence maximize the sensitivity. They must be placed in the middle of the sensor along the bias axis (x-axis) and as far as possible from the center of the sensor along the orthogonal axis (y-axis).

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The intrinsic settling time of Hall sensor can be modeled as an RC time, where R is the resistivity of the n-well and C is the total capacitive effect acting on sense contacts. This is an important result since it reveals that the relatively slow responses of Hall sensor implementations are due to the readout architecture and not to physical constraints of the device. It also outlines an important sensitivity/bandwidth trade-off which is set by the resistivity of the n-well.

Acknowledgments: The software cited in this document is furnished under a license from Synopsis International Limited. Synopsis and Synopsis product names described herein are trademarks of Synopsis, Inc. The authors would like to thank M. Marchesi and D. Cristaudo for their suggestions in sensor modeling. Author Contributions: M.B. and M.C. modeled the sensor and carried out the simulations. M.C. analyzed the results and wrote the manuscript. A.R. and M.T. revised the manuscript and gave keen insights during results analysis. E.S. managed the project. Conflicts of Interest: The authors declare no conflict of interest.

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