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Advances in Condensed Matter Physics
Volume 2018, Article ID 7812743, 24 pages
https://doi.org/10.1155/2018/7812743
Research Article

An Analytical Theory of Piezoresistive Effects in Hall Plates with Large Contacts

Infineon Technologies, Villach, Austria

Correspondence should be addressed to Udo Ausserlechner; moc.noenifni@renhcelressua.odu

Received 13 February 2018; Accepted 21 March 2018; Published 4 June 2018

Academic Editor: Yuri Galperin

Copyright © 2018 Udo Ausserlechner. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Four-terminal transducers can be used to measure the magnetic field via the Hall effect or the mechanical stress via the piezoresistance effect. Both effects are described by an anisotropic conductivity tensor with small offdiagonal elements. This has led other authors to the conclusion that there is some kind of analogy. In both cases the output voltage depends on the geometry of the device and the size of the contacts. For Hall plates this influence is accounted for by the Hall-geometry factor. The alleged analogy proposes that the Hall-geometry factor also applies to four-terminal stress transducers. This paper shows that the analogy holds only for a limited class of devices. Moreover, it is shown that devices of different geometries may have identical magnetic field sensitivity but different mechanical stress sensitivities. Thus, shape optimization makes sense for mechanical stress sensors. In extreme cases the output voltages of vertical Hall-effect devices may have notable magnetic field sensitivity but zero mechanical stress sensitivity. As byproduct, exact analytical formulae for the equivalent resistor circuit of rectangular and circular devices with two perpendicular mirror symmetries are given. They allow for an accurate description of how mechanical stress and deformation affect the output offset voltage and the magnetic sensitivity of Hall-effect devices.

1. Introduction

Resistive devices with four terminals have been proposed as mechanical stress sensors for a long time[1, 2]. They are called four-terminal piezotranducers [3] or van der Pauw stress sensors [4]. The very same phenomenon is observed in Hall plates at zero magnetic field, where the zero-point error (= initial offset error prior to spinning current scheme [5]) is very sensitive to mechanical stress. This was occasionally called Kanda effect [6] or pseudo-Hall effect [7]. An alternative view on the same topic is how to determine anisotropic resistivity from measurements on van der Pauw samples [8]. Over the past five decades a large number of papers have been published on various aspects of this bundle of problems. However, it seems that no analytical closed form expression was found, which relates the anisotropic conductivity to the output voltage of such a device with four contacts of arbitrary size. Conversely, the analogous problems for isotropic van der Pauw measurement at zero magnetic field and for Hall plates at small magnetic field have been solved recently [912]. In the following we apply similar methods to the piezoresistive effect in these devices. Thereby we use the affine transformation of van der Pauw [8] in order to replace the original device having anisotropic resistivity by an equivalent device having isotropic conductivity. For these devices with isotropic conductivity we compute the equivalent resistor circuit (ERC), which describes the behavior of the devices under the action of mechanical stress and magnetic field. For the ERC of rectangular devices we use our own results of former works [912], whereas we derive new formulae for the ERC of circular devices.

The outline of this work is as follows. Section 2 sums up basic phenomenological facts of the Hall effect and of piezoresistivity as they are known in the literature. The goal is to contrast similarities against differences of both effects. Section 3 briefly repeats the affine transformation of van der Pauw and specifies it for (100)-planes of cubic crystals, like silicon. With these formulae it becomes apparent that vertical Hall-effect devices are very slightly affected by mechanical stress. Section 4 elaborates on how rectangular and circular devices respond to biaxial inplane stress. There we make ample use of the ERC as it was developed in prior works [9, 10]. Section 4 closes with a comparison of several device shapes versus Wheatstone bridge circuits of resistor stripes in orthogonal directions: which one has the largest sensitivity to mechanical stress? Section 5 discusses the effect of biaxial inplane stress on the Hall-geometry factor of Hall plates. This leads to a drift of magnetic sensitivity versus mechanical stress, which is caused by piezoresistance and not by piezo-Hall effect. Finally, Section 6 compares the proposed analytical theory with results of other authors.

2. Similarities of Piezoresistance and Hall Effect

In a thin plate-like material with isotropic ohmic resistivity the Hall effect is described by the relation between electric field and current density with the odd symmetric resistivity tensor in ()-coordinates is the resistivity at zero magnetic field, is the Hall mobility, and is the magnetic flux density component perpendicular to the Hall plate, which lies in the ()-plane [13]. The inverse relation uses the conductivity tensor is the conductivity at zero magnetic field. Equations (1a) and (1b) assume negative charge carriers. Alternatively to the literature also uses the Hall coefficient , where is the sign of the charge carriers. The static electric field is longitudinal, i.e., its curl vanishes. Therefore it can be expressed as the negative gradient of a scalar electric potential . In the quasistatic case the divergence of the current density vanishes . Combining these equations with (1b) gives whereby we assumed that , , and are constant versus spatial coordinates. The mixed-order terms cancel out and therefore (2) is the Laplace equation. Thus, the Hall effect does not enter (2); with and without magnetic field it is the same partial differential equation for the potential. The magnetic field affects only the boundary conditions: the potential at the contacts or the current flow via the contacts is forced by the external circuitry. At the isolating boundary the current density must flow parallel to the boundary and this means or in other words (where and denote normal and tangential direction and the plus/minus sign depends on the direction of the magnetic field, the sign of the charge carriers, and if the boundary is left or right to the current flow). In other words, at the isolating boundary there is the Hall angle between the electric field and the boundary. To sum up, the potential in a Hall plate and the output voltage of a Hall plate can be computed by solving the Laplace equation and satisfying these boundary conditions. For Hall plates with two perpendicular mirror symmetries this was done in the seminal work [14] and the result iswhere is the current related magnetic sensitivity, is the sheet resistance (equal to one over the product of conductivity times thickness of the Hall plate), is the current flowing through the Hall plate, and is the Hall geometry factor. At small magnetic field we write and then the Hall geometry factor depends only on the lateral geometry (= the layout) of the Hall plate, not on its thickness and not on any material parameters. We can compute for any geometry of Hall plates from scratch (in rare cases analytically and in general numerically), but it is simpler to measure the input and output numbers of squares as shown in [9, 10] and to use the following formula (which is derived in [12]):Thereby is the incomplete elliptic integral of the first kind, is the complete elliptic integral of the first kind, and is the modular lambda function defined by for and . The input number of squares is and the output number of squares is , whereby the input resistance is measured between the two current supplying contacts of a Hall plate (while the sense contacts are floating) and the output resistance is measured between the two sense contacts (while the other two contacts are floating). The role of input and output contacts is arbitrary – does not change if we swap it [12]. Equation (4) expresses the Hall geometry factor as a function of input and output numbers of squares without need to know any geometrical details of the device. This means that Hall plates with different geometry have identical if they have identical and . The reason why different geometries may have the same number of squares and the same Hall output voltage at identical supply current is that we can transform one into the other via conformal mapping [14]. Since the conformal mapping is done by analytic functions which satisfy Cauchy-Riemann differential equations, the mapping functions are also solutions of the Laplace equations. Therefore, if we have a solution of the potential in one Hall plate, we can apply a conformal transformation to this geometry and to the solution. Thereby the same current will flow through the transformed device and the same potentials will appear at its contacts. Consequently, we can study a specific device geometry (e.g., a circular disk) and we can be sure that there is no other ingeniously shaped device boundary (e.g., Greek cross or octagon…) that leads to better device performance. This was already noted by Wick in [14]. Of course this holds only within the limited scope of linear electrostatic theory; beyond that scope devices will be different: different shapes lead to more or less homogeneous electric fields and therefore to more or less pronounced velocity saturation effects. Although their total power may be identical, the spatial distribution of power density differs and this leads to different temperature distributions. The temperature coefficient of the resistivity acts back on the current distribution. Finally the contacts may have different temperatures, which leads to different thermoelectric voltages and different residual offsets after spinning current schemes are applied [15].

In the case there is piezoresistance at zero magnetic field, the conductivity tensor , and its inverse the resistivity tensor becomes anisotropic due to mechanical stressThe significant difference between (1a), (1b), and (5) is that the conductivity tensor of the Hall effect is odd symmetric, whereas it is even symmetric for the piezoresistance. In the language of nonequilibrium thermodynamics both Hall effect and piezoresistance are irreversible dissipative processes, but the relation between current density and electric field is reciprocal for piezoresistance whereas it is antireciprocal for the Hall effect [16]. Therefore, a rotation of the coordinate system around the -axis does not change the Hall-resistivity tensor (1a), yet it changes the piezoresistance resistivity tensor (5) in such a way that for certain orientations (the principal axes) the elements in the minor diagonal even vanish. In network theory it is well known that any reciprocal (i.e., symmetric) impedance matrix of a linear, lumped, and passive N-port can be realized by a network of resistors, inductances, capacitors, and ideal transformers [17], however a nonreciprocal impedance matrix needs at least one additional linear, lumped, passive element: the gyrator [18]. Instead of a gyrator one may also use the series connection of two current controlled voltage sources (CCVS) or the parallel connection of two voltage controlled current sources (VCCS). Therefore the equivalent lumped circuit of a Hall effect device comprises resistors and at least one gyrator [19, 20], whereas the equivalent lumped circuit of a piezoresistive device consists only of resistors. Hence, the gyrator makes an essential difference between piezoresistance devices and Hall effect devices, and this already denies any perfect analogy between piezoresistance and Hall effect.

In a conventional plastic encapsulated package the surface of a semiconductor chip is exposed to inplane mechanical stress [21, 22]. For inplane normal stress , and inplane shear stress components on a (100) silicon chip it holds [23]with , , and being the three piezoresistive coefficients of the cubic m3m single crystal. Thereby the - and -axes of the chip are aligned along -crystal directions as shown in Figure 1 (see also [24]).

Figure 1: Reference frame of the single crystal and chip reference frame for a (100)-silicon wafer.

Computing the conductivities from (6a) to (6c) gives is the determinant of the resistivity matrix, and its name will become apparent later in (12a)–(12c). For we getwith and . In the general case we can apply the Mohr circle construction [25] to the second-rank conductivity tensor in (5): we rotate the -coordinate system by an angle in mathematically positive direction (counterclockwise) to obtain the rotated coordinate system [26]. In the conductivity tensor has equal values on the main diagonal and maximum value on the minor diagonalIn (9b) the plus sign is valid for and the minus sign is valid for and denotes the sign of . Inserting (7a)–(7d) into (9b) givesSo in any case we get a symmetric conductivity tensor with identical values in the major diagonal and with small values in the minor diagonal. With (10a) and analogous to the Hall effect we get a second-order partial differential equation for the electric potential(11) is not Laplace’s equation, because the mixed derivative does not vanish and this is a consequence of the piezoresistive conductivity tensor being even symmetric. As for the boundary conditions they are similar to the Hall effect: the potential has constant values at the contacts, whereas it holds at isolation boundaries parallel to -axis or -axis. So seems to take over the role of – at least at these isolation boundaries. However, inside the sample there is a striking difference between Hall effect and piezoresistance [27, 28]: inside the Hall-effect region the angle between electric field vector and current density vector is homogeneous versus spatial position and equal to the Hall angle , whereas in the piezoresistive region in general the angle between and is inhomogeneous versus spatial position .

The similarities between Hall effect and piezoresistance led some authors to presume an analogy [3, 2931], since it is tempting to drop the mixed derivatives in (11). Then the piezoresistive potential would become a solution of Laplace’s equation just like the Hall potential and since both fulfill the same boundary conditions, we would only have to replace by to fulfill the skew-derivative boundary conditions. Then the output voltage of a van der Pauw stress sensor would be analogous to (3), whereby the sign denotes small stress and there we may neglect the stress dependence of . If both normal and shear stress are present in (10b) they would mix quadratically. If only inplane shear stress is present on (100)-silicon we would get . These relations may hold for certain device geometries, like the one in [2] with small sense contacts and homogeneous current density (which is shown in Figure 2(a)), or for devices with (as in [32]), but they do not hold in the general case. Figure 2(b) shows a device geometry with 90° symmetry and large contacts, where these simple formulae lead to an error of 36% in the output voltage when compared to an accurate finite element simulation.

Figure 2: (a) Rectangular device with large supply contacts and point-sized sense contacts with potential distribution and current streamlines at . At zero stress it has two squares input resistance and according to (4) this gives . (b) Cross-shaped device with large supply contacts and large sense contacts with potential distribution and current streamlines at . At zero stress it has 1.0359 squares input resistance and according to (4) this gives . The plots were obtained by a finite element simulation (FEM) with commercial software code COMSOL MULTIPHYSICS. This FEM simulation gives a stress output signal of device (a) which differs only 0.4 ppm from from [2]. An FEM calculation of device (b) gives an output signal that is 57% larger than .

Where does this error come from? In (11) we had to neglect the mixed derivative in order to obtain the analogy with the Hall effect, and this was not allowed. Of course we may neglect the -term in (11) if we want to compute the potential up to zero-th order in . However, if we compute the output voltage of the pseudo-Hall device, we subtract the potential at both output contacts in order to get the small imbalance of potential due to . Hence, this output voltage is proportional to ; it is of first-order in and so we must not neglect terms in (11) which are linear in . Therefore in general we cannot use the Hall geometry factor for van der Pauw stress sensors. However, in the special case of Figure 2(a) at small stress the potential is constant versus x. Therefore the mixed derivative in (11) vanishes and (11) becomes the Laplace equation. For this particular symmetry (11) and (2) are identical and the piezoresistive output is analogous to the Hall output when we replace by .

Figures 3(a) and 3(b) show the angle between electric field and current density for the device of Figure 2(b) operated in the same way as in Figure 2(b). However in Figure 3(a) there is small shear stress and no magnetic field whereas in Figure 3(b) there is large magnetic field and no mechanical stress. Obviously, in Figure 3(a) the angle is small but inhomogeneous: it varies from −0.446° at the supply contacts to +0.446° at the sense contacts. Conversely, in Figure 3(b) the angle is large and perfectly homogeneous (51.078°). Figures 4(a) and 4(b) show the same angle for the device of Figure 2(a). In Figure 4(a) there is again small shear stress at zero magnetic field and in Figure 4(b) there is a large Hall angle at zero stress. Figure 4(b) shows again a perfectly homogeneous large Hall angle. In Figure 4(a) the angle is within a very narrow range of −0.445766° and −0.445752°; nevertheless the color coding reveals a systematic pattern of this angle. This is a striking contrast to Figure 3(a). The specific geometry of the device of Figure 4(a) gives a homogeneous current density and thus a homogeneous angle at small mechanical stress just like for the Hall effect. This seems to be the deeper reason why one may use the Hall geometry factor in Figure 4(a) to compute the output voltage at shear stress, although this is not allowed for other device geometries like in Figure 3(a). At larger stress on the device in Figure 4(a) the spatial dependence of gets more pronounced, yet this inhomogeneity is still smaller than for Figure 3(a) and the piezoresistive angle will never change sign as in Figure 3(a).

Figure 3: (a, b) Angle between electric field and current density in a device of Figure 2(b) operated like in Figure 2(b). The plots were obtained by FEM simulations with COMSOL MULTIPHYSICS. In (a) we have zero magnetic field and small mechanical stress . In (b) we have zero stress and even large magnetic field with 51.078° Hall angle. As predicted by [28] the Hall angle is perfectly homogeneous throughout the device (b), whereas in the piezoresistive case (a) the angle varies between −0.446° at the supply contacts and in the major portion of the device and +0.446° at the sense contacts. Therefore the analogy [2] between Hall effect and piezoresistive effect does not apply in this case.
Figure 4: (a, b) Angle between electric field and current density in a device of Figure 2(a) operated like in Figure 2(a). The plots were obtained by FEM simulations with COMSOL MULTIPHYSICS. In (a) we have zero magnetic field and small mechanical stress . In (b) we have zero stress and even large magnetic field with 51.078° Hall angle. As predicted by [28] the Hall angle is perfectly homogeneous throughout the device (b), but surprisingly in the piezoresistive case (a) the angle is also highly homogeneous (it varies only between −0.445766° and −0.445752°). This seems to be the reason why the analogy [2] between Hall effect and piezoresistive effect works for this device.

3. The Affine Transformation

Since (11) is not Laplace’s equation, we cannot directly apply a conformal transformation to solve it. However, (11) can be transformed into Laplace’s equation by an affine transformation . This procedure has been worked out in [8, 24, 33] (the affine transformation is also used in [34]; however, it seems that conductivities in - and -directions are mixed up. [7] uses again a different transformation which does not seem to comply with (12a)–(12c).). We summarize it here briefly in a notation that will we used in the rest of this work. In the transformed geometry the equivalent device has an isotropic conductivityThe potentials in corresponding points before and after the transformation are identical: . The matrix is chosen such that the currents through corresponding contacts remain constant under the transformation, because only then the transformation preserves the resistances. Also the total power dissipation in the device and the power density are not affected by the transformation. This implies that the thickness and the area of the device remain constant under the transformation [24].

Inserting (7a)–(7c) into (12a) and developing into a MacLaurin series for small stress terms gives the approximationThe affine transformation (12a), (12b), and (12c) applies to all crystal classes; however, (13a), (13b), and (13c) use (7a), (7b), (7c), and (7d) and therefore they apply only to (100)-planes of cubic crystal classes (e.g., silicon).

Note that an affine transformation is definitely not a conformal transformation: conformal transformations preserve the angles between two curves, whereas affine transformations do not, because they skew the geometry. Equation (13a) shows that the geometry is skewed for shear stress. The difference of normal inplane stresses scales the - and -axes differently. This also skews most devices except those ones with edges parallel to -axes. The sum of normal inplane stresses scales the equivalent isotropic conductivity. Isotropic stress does not change the shape of the equivalent device; it only changes .

So far we neglected the pure deformation of the geometry (= strain) caused by the stress on the device. This is commonly justified because for many materials the piezoresistance effect dominates. However, the affine transformation (12a) already describes a deformation of the device, and so we can add the strain terms without unduly increasing the complexity of the calculation. A homogeneous two-dimensional strain with zero displacement of the origin is described by the transformation [35]whereby , , and are the tensor strains (as opposed to the engineering strains [36]). The strain is related to the stress via the fourth rank compliance tensor, which in silicon has only three nonvanishing terms , , and [37]. Transforming the tensors into the chip coordinate system of Figure 1 givesThe last equation (15d) means that the thickness of the conducting region changes, too. We take account of this by keeping the unstrained thickness but changing the resistivity accordingly . The combined effect of piezoresistance and strain is obtained by multiplying both matrices and , whereby the order of multiplication is irrelevant for small stress.

In the literature of the Hall geometry factor it is a common technique to map finite device geometries like disks, squares, rectangles, crosses, or octagons via conformal mapping onto the infinite upper half space of the ()-plane [11, 14, 38]. If the original devices have contacts only at their perimeter, the conformal mapping transfers these contacts onto the -axis boundary. Let us for a moment assume an original device with an arbitrary number of contacts on the -axis, its active region being the entire upper half plane with anisotropic resistivity (6a), (6b), and (6c). We can transform it via affine transformation (12a), (12b), and (12c) into an equivalent problem with isotropic resistivity . However, since all contacts are on the -axis and the geometry is infinite, matrix just scales the -axis by the factor . The different scaling factor along the -axis is irrelevant, because there are no features (neither contacts nor boundaries) except on the -axis. So the affine transformation (12a), (12b), and (12c) effectively just scales the entire geometry by a single factor, and this does not change ratios of voltages on (or currents through) the contacts or ratios of resistances between the contacts. In the end the absolute values of resistances of the transformed device indeed change due to in (7d). This shows a clear difference between Hall effect and piezoresistance effect: for the Hall effect we can transform devices onto the upper half plane with conformal transformation and preserve the Hall output voltage, whereas for the piezoresistance effect any direct conformal transformation onto the upper half plane would make the piezoresistive output voltage vanish. The correct procedure for a piezoresistive potential problem is to apply the affine transformation (12a), (12b), and (12c) prior to conformal transformation(s): the affine transformation establishes the validity of Laplace’s equation and then it is allowed to go on with conformal transformations. Figure 5 shows numerical simulations of such a device in the upper half plane with piezoresistive coefficients of low n-doped silicon. The input voltage is forced by an external voltage supply between the outer contacts. Then the change of output voltage at the center contact caused by mechanical stress is close to zero (except for numerical inaccuracies) while the input resistance changes with stress according to in (7d). This geometry has also some practical importance, because it is similar to vertical Hall devices [39, 40], where all contacts are on the top surface of a rectangular Hall region. In the early days this Hall region was the entire wafer and therefore it was close to infinite. The offset voltage of such a device can drift over lifetime due to drift of mechanical stress on the chip. With the above discussion we can say that for large Hall tubs this drift will get very small and for small Hall tubs it is caused by the isolating boundary at the sidewalls and bottom of the tub but not by the top surface. This was already stated in [41] based on the same theory using affine and conformal transformations. It was also experimentally observed that the offset of vertical Hall devices is more stable than the offset of conventional horizontal Hall plates [42].

Figure 5: Finite element model of a resistive device with three contacts at the lower boundary (= -axis). The plots were obtained by FEM simulations with COMSOL MULTIPHYSICS. They show the potential distribution and the current streamlines. The entire upper half plane is conductive with piezoresistance according to (5) and (6a)–(6c) with and the piezoresistive coefficients for low n-doped silicon (, , ). Pure strain is neglected. The outer contacts are 0.15 m long, the center contact is 0.1 m long, and the spacing between contacts is 0.25 m. The thickness into the drawing plane is 1 m. The left contact is forced to 1 V, the right contact is forced to 0 V, and the center contact is floating (i.e., no current flows in or out). The infinite boundary is modeled with infinite elements. The mesh has 1.4 million elements. At zero mechanical stress the center contact is at 0.5 V and the input resistance is . At the output voltage increases 57 nV and the input resistance decreases −2.443%. At the output voltage decreases −56 nV and the input resistance decreases −2.441%. At the output voltage decreases −2.5 µV and the input resistance decreases −1.218%. All these resistance changes deviate less than 10 ppm from theory (12a)–(12c). The changes in output voltage should be zero according to theory and the small deviations are probably due to insufficiently fine mesh.

4. Devices with Mirror Symmetries to - and -Axes and Contacts Not Lying on the - and -Axes, at Zero Shear Stress

In standard microelectronic packages the main part of the chip (except its perimeter) is under pure inplane biaxial stress in the order of 100 MPa [22]. Hence, this case is most relevant in practice. Then it holds for a (100)-silicon chip with (7a), (7d)For very large stress difference in the GPa-range the terms under the square-roots in (17a), (17b) become negative, and either or reverse sign (see (6a), (6b)), which is impossible for passive devices (); there the first-order piezoresistive theory breaks down.

4.1. Rectangular Devices with Four Large Contacts

With (6a)–(6c), (12a), and (12b) and with the results of [10, 11] we can immediately give the equivalent resistor circuit (ERC) and its dependence on stress for rectangular devices shown in Figure 6. We summarize these findings in (18a)–(18m):The parameters , , and are length, width, and spacing between contacts, respectively, of the device at zero stress (see Figure 6). They undergo the affine transformation (12a) and to account for the mechanical stress. The ratio is independent of stress, because both lengths are along -direction and the common scaling factor of the affine transformation cancels out. is the constant thickness of the device. is the sheet resistance of the equivalent device with isotropic resistivity after the affine transformation. , , , and are dimensionless parameters which appeared in the sequence of conformal transformations in [10, 11]. In (18i) the Jacobi sine-amplitude function is used. It is defined by . Equations (18a)–(18f) are given in (14a–c) and (A11–13) in [10]; (18g)–(18j) are equivalent to (8, 9, 14) in [11]. The parameters , , depend on mechanical stress; if we refer to their values at zero stress, we add a zero in the indices: , , and . The ERC in Figure 6 and (18a)–(18m) is not an approximation; they are valid in a strict sense even for large mechanical stress and arbitrary size of contacts as long as we stay in the linear electrostatic and piezoresistive theory. With the ERC one can compute the input resistances, the ratios of output over input voltages, and the transimpedances of Hall- and van der Pauw-operating modes (all at zero magnetic field). For the Hall-operating mode it holds thatFor the van der Pauw-operating mode it holds thatwhereby is the current which enters the device through contact and leaves through contact and is the voltage across both contacts .

Figure 6: Rectangular stress sensor in (100)-silicon with two perpendicular mirror-symmetry axes , . No magnetic field is applied. An input voltage is forced between contacts and an output voltage is tapped between . All contacts have the same size . (a) shows the top view of the potential distribution and the current streamlines in the plane device. (b) shows the equivalent resistor circuit (ERC).

Only for carefully chosen contact spacing does the output voltage vanish at zero stress in the Hall-operating mode and : we get these offset-free devices with at zero stress (see (19b)) and this means , from which follows the contact spacing (see [12])Offset-free devices are only possible if the width is smaller or equal to the length . Figure 7 shows the sensitivity of the output voltage in Hall-operating mode with respect to small changes in stress at fixed supply voltage for devices of Figure 6 and operated according to Figure 6. There the stress sensitivity function has a flat maximum of 46.839%/GPa at , . At zero stress the number of squares between contacts or between contacts is 1.3386 (with , ). The output voltage at zero stress is 14.81% of the supply voltage; this device is not offset-free (). All these values were checked with a finite element simulation. If we restrict the search to offset-free devices the maximum stress sensitivity is only 2.5% smaller, namely, 45.674%/GPa at and (with , see Figure 8); then the resistance between even or odd contacts has 1.2712 squares. The value of both maxima is computed for of low p-doped silicon; in case of other materials of a cubic crystal class the stress sensitivity scales linearly with [43].

Figure 7: Sensitivity of output voltage versus mechanical stress (normalized to supply voltage ) for a device of Figure 6 operated in the same way as in Figure 6. All other components of the stress tensor vanish, as well as . (a) is a 3D-plot and (b) shows the contour lines thereof. In (a) the steep slopes of the surface near and and are only sketched due to numerical problems. The plots are graphical representations of (18a)–(18m) in the limit of small stress and with the piezoresistive coefficient for low p-doped silicon. Largest stress sensitivities are obtained for aspect ratios . The dependence on contact size is small, but tends to be best. The black curve denotes offset-free devices with , where at zero stress. The maximum of the surface does not lie on this curve. The contours in (b) are for 46.75, 46.5, 46, 45, 44, 42.5, 40, …, 22.5, 20%/GPa.
Figure 8: Sensitivity of output voltage versus mechanical stress (normalized to supply voltage ) for an offset-free device of Figure 6 operated in the same way as in Figure 6. All other components of the stress tensor vanish, as well as . The plot is a graphical representation of (18a)–(18m) in the limit of small stress and with the piezoresistive coefficient for low p-doped silicon. Largest stress sensitivity of 45.674%/GPa is obtained for aspect ratios . There satisfies (20).

The theory was verified by finite element simulations at small and large stress (Figure 9). The device had the geometric parameters , , and . For the conductance a linear material law was used according to (7a)–(7d) with and , , and . The two-dimensional simulation had a fine mesh of 1.76 million elements. It used Lagrange multiplier for improved accuracy of currents into the contacts. Table 1 compares the results of this numerical simulation with results of the analytical theory comprising (17a), (17b), (18a)–(18m), (19a), and (19b). Even up to very large stress levels FEM and the theory agrees better than 0.1%.

Table 1: Mechanical stress sensitivity of rectangular device from Figure 9: comparison of numerical simulation (FEM) versus analytical theory. , , . The small residual output voltage of the FEM in the first line is probably due to insufficiently fine mesh. In the last line the stress difference is so large that gets nearly zero. Then gets large (~48.9), gets small (~0.13), gets small (~0.0017), and the output voltage approaches the supply voltage. For even larger stress difference the first-order piezoresistive theories (6a) and (6b) break down and also the numerical simulation gives meaningless results.
Figure 9: Rectangular device from Figure 6 with , for comparison of the analytical theory with a finite element simulation with COMSOL MULTIPHYSICS. The plots show the potential and current streamlines. (a) The left plot is for small mechanical stress difference (third line of Table 1); (b) the right plot is for large mechanical stress difference (last line of Table 1).
4.2. Rectangular Devices with Four Point-Sized Contacts

If the contacts of the device in Figure 6 get very small we have , which we plug into (18i) and develop that into a series in small . Inserting this into (18g), (18h), and (18d)–(18f) givesInserting this into (19a), (19b) gives(22a)–(22c) hold for small contacts and arbitrary stress. changes with stress according to (18j). For small stress we write , where is the value at zero stress and is the small stress variation of . For point-sized contacts and then the only offset-free geometry with at zero stress is , which gives (cf. (20)). From (18k), (18l), and (17b) we get for small stressPlugging this into (18j) and developing for small stress givesIn (23b) we used the differentiation rule for the modular lambda functionwhich can be proven by with the substitution and . The denominator at the right hand side of (24b) is equal to according to Legendre’s relation [44]. In (24b) is the complete elliptic integral of the second kind, defined by and . Inserting (23b) into (22c) and developing for small stress giveswhereby is the gamma function and we used [45]. (25a) is identical with (16) in [46]. It was also checked by an FEM simulation. If the devices are not square but general rectangles with they are not offset-free and the transconductance is A plot of (25b) reveals that square devices have smaller stress sensitivity at given input current than devices with large aspect ratio.

4.3. Circular Disk Shaped Devices with Four Large Contacts

Figure 10(a) shows a circular device with two pairs of contacts of different size: each contact covers an aperture angle of and , respectively. Thus, its layout has two DoFs just like the device of Figure 6. The device has two perpendicular mirror symmetries, namely, the - and -axes. We assume that it is implemented in (100)-silicon with an alignment between the ()-coordinate system and the crystal system like in Figure 1. Under a biaxial mechanical load along the symmetry axes the resistivity tensor becomes anisotropic () but the offdiagonal elements vanish (). As explained above we may apply the affine transformation (12a), which makes an ellipse in the -plane out of the circle in the original -plane (see Figure 10(b)). This device has the same potentials and currents at all contacts as the device in Figure 10(a), even though in Figure 10(b) the resistivity is isotropic and the potential is a solution of the Laplace equation, whereas in Figure 10(a) the resistivity is anisotropic and the potential is no solution of the Laplace equation. Thereby the stress acts along a line, where the contacts with angles are arranged. The stress acts along a line, where the contacts with angles are arranged. In a second step we use a conformal transformation with being the imaginary unit, which rotates the device by 90° in clockwise direction and scales it isotropically with the factor such that the foci of the ellipse are at . A second-conformal transformation maps the interior of the ellipse onto the upper half of the -plane (Figure 10(d)), as explained in [48]. At the boundary of the elliptical region the mapping is and are complex numbers and . (26a) and (26b) are only valid for , which means . If we move on the -axis from over to this means a walk in the -plane in counterclockwise direction starting at and moving through the lower -plane over back to through the upper -plane. Thus, positive real corresponds to the upper half of the edge of the ellipse. Then the end points of the contacts in the -plane arefor and , , and being sine and cosine of the Jacobi amplitude. Next we map the interior of the rectangular device of Figure 10(e) from its -plane onto the upper half of the -plane in Figure 10(d) via the Schwartz-Christoffel transformation with . Thereby the center of the lower partial contact is mapped onto the origin in the -plane. The device in Figure 10(d) is identical to the even symmetry device in [10]. Therefore the flush contacts at positions have the resistance with between them, and the resistance between the partial contacts is (see [10]). Applying the Schwartz-Christoffel transformation to gives . However, this is identical to . Therefore, it holds , and with (27b) and (27c) we getWith the half-period addition theorem it follows that [49]Finally, we apply the Schwartz-Christoffel transformation to : . With (27a) and (27b) this givesWith (28b) and (28c) and with we immediately know from (18i), and with (18d)–(18h) we can compute , , and . With these parameters we know all resistors in the ERC of the even symmetry device in Figure 10(a) (see [10]) and also of the odd symmetry device in Figure 12(a). With (19a)–(19i) we get all impedance, voltages, and transresistances in Hall- and van der Pauw-operating modes.

Figure 10: (a–d) Mapping a circular device with anisotropic conductivity by one affine and three conformal transformations onto an equivalent rectangular device with isotropic conductivity. (a) shows a circular disk shaped device with two pairs of contacts and two perpendicular mirror-symmetry axes and . Its resistivity is anisotropic with the principal axes x and y. This describes the action of biaxial load along - and -axes on chips in (100)-silicon aligned like in Figure 1. The elliptical device in (b) with isotropic resistivity is equivalent to the device in (a). The shape in (b) is obtained by the affine transformation (12a). The conformal transformation rotates the device clockwise by 90° and scales it isotropically: and with . A second-conformal transformation   (26a) and (26b) maps the interior of the ellipse in (c) onto the upper half of the t-plane in (d). A third-conformal transformation maps the upper half of the t-plane into the interior of the rectangle in (e). The contacts along the perimeter of the shapes relate to the contacts on the real t-axis. Corresponding points have equal indices. The sequence of points with indices 0, 1, 2, 3, and 4 is always counterclockwise while the conductive region is at the left hand side of this path. The asterisk means the conjugate complex.

We checked (28b), (28c), and (18d)–(18h) with finite element calculations of the circular device in Figure 11 with and with isotropic conductivity. For zero stress we set and obtained and as input/output numbers of squares. Then we stretched the device in - and -directions according to Figure 10(b) with and obtained , , , , and . Both times the results of FEM and analytical theory matched up to 0.04% for and . Thereby, the FEM model had a mesh with 0.6 million elements per device.

Figure 11: FEM-computation of input and output resistances of circular and elliptical devices with isotropic conductivity. Due to symmetry only a quarter of the real devices had to be modeled. The elliptical device is obtained from the circular one by stretching it times 1.5 in vertical direction and times 2/3 in horizontal direction. Contacts are thick black lines on the perimeter. The color coding denotes the electric potential (red = 1 V, blue = 0 V) and the gray lines are current streamlines.
Figure 12

For we must change the transformation between Figures 10(b) and 10(c) to . This means to swap with in (28a)–(28c) and to replace in (26b). Since the new transformation does not rotate the device, -contacts are now mapped onto the partial contacts of the rectangular device in Figure 10(e), so that we also have to swap with .

We checked the correctness of (26b), (28b), (28c), and (18d)–(18i) with a series of finite element simulations given in Table 2.

Table 2: Mechanical stress sensitivity of a circular device from Figure 10(a): comparison of numerical finite element simulation (FEM) versus analytical theory. is the resistance between the -contacts in Figure 10(a) and is the resistance between the -contacts in Figure 10(a). The mesh of the FEM-simulation had 2.66 million elements and the model used Lagrange multiplier and the piezoresistivity matrix from (6a) to (6c) with , , and . The theory used (18d), (18e), (18g), (18h), (18i), (28b), (28c), and (26b) for . For we swapped with in (28b) and (28c) and with , and we replaced in (26b).

With these foundations we can study the mechanical stress sensitivity of symmetric circular devices with arbitrary ratio of input/output resistances. Output voltages of devices from Figure 10 do not respond to , because their contacts are on the axes of mirror-symmetry and even after the affine transformation (12a) they will stay on these symmetry axes so that their output signals remain zero. One can prove this by inserting , , and from (18d) to (18f) into (2a–c) of [10] to compute the equivalent resistor circuit (ERC) that applies to a device from Figure 10. This ERC is shown in Figure 2b of [10] – it is different from the ERC of Figure 6 in this current work. For the measurement of we have to consider the complementary devices where contacts and isolating boundaries are swapped (see Figure 12(a)). In Figure 12(b) we checked the analytical theory against numerical simulations for various contact geometries in the wide range of mechanical stress up to ±640 MPa. In an experiment the silicon wafer would very likely break at smaller stress levels, and the linear piezoresistive theory would describe the measured results insufficiently. Nevertheless, we chose the large stress to show the excellent agreement between analytical theory and numerical simulation.

The mechanical stress sensitivity at small stress is plotted in Figure 13, where we note a symmetry when and are swapped. Offset-free devices have . They correspond to the black line and this line goes right through the global maximum of the function at the point . This is one distinct difference to rectangular devices from Figure 5. This maximum sensitivity is 48.3501%/GPa, which is more than 3% larger than the maximum sensitivity with systematic offset in Figure 5 and nearly 6% larger than for offset-free rectangular devices in Figure 5. The mechanical stress sensitivity of offset-free circular devices is shown in Figure 14. The examples of rectangular versus circular devices show that different shapes of devices lead to different maximum achievable stress sensitivities. This is another distinct difference to Hall plates, where all shapes of devices, which can be derived from each other by conformal mapping, lead to the same magnetic sensitivity. However, there is another similarity between Hall plates and circular pseudo-Hall devices: at constant supply voltage their sensitivity to magnetic field and mechanical stress, respectively, is maximized, if the input and output resistances are identical to squares: (compare Figure 14 with [12]). Yet this similarity does not hold for rectangular devices or inverted Greek crosses (see Figure 18(a)).

Figure 13: Sensitivity of output voltage versus mechanical stress (normalized to supply voltage ) for a circular device of Figure 12. All other components of the stress tensor vanish, as well as . The plots are graphical representations of (26b), (28b), (28c), (18a)–(18i), and (19b) in the limit of small stress and with the piezoresistive coefficient for low p-doped silicon. Largest stress sensitivities are obtained for contact size with a flat maximum. The black curve denotes offset-free devices with at zero stress; they are the only ones, where at zero stress. The black curve goes through the maximum of the surface. The contours in (b) are for 48.25, 48, 47.5, 47, 46.5, 46, 45, 42.5, 40, …, 22.5, 20%/GPa. In (a) the steep slopes of the surface near and are only sketched due to numerical problems.