Extension of the Stoney Equation for a Taiko Wafer (Si and SiC)

An extension of the Stoney formula for the case of a back side metallized 8” silicon taiko wafer has been developed, in the elastic regime, within the frame of the theory of elasticity. A good correlation between the calculated warpage, determined by the stress released by a given back side metallization (BSM), and the corresponding experimental warpages of the same thick metal layers deposited on an 8” silicon taiko wafer provides evidences of the correctness of the developed theory. This development suggests the possibility to extend this approach to the case of 8” taiko wafers based on a wide band gap semiconductor such as silicon carbide (SiC).


Introduction
Developments in vertical power MOSFETs devices based on silicon [1] and silicon carbide [2] require, specifically in the automotive sector, a significant cutdown of the switching losses as the technology evolves from one generation to another [3] [4]. A strategy to achieve this goal consists in modifying some key parameters which affect the operations of the device. For example, an improvement of the heat dissipation, along with a lowering of the drain-source on resistance (Rdson) of the device, allows to improve their performances. A large contribution to Rdson and the thermal resistance results from the thickness of the substrate, since the electrical current goes through it. To reduce this quantity a method consists in thinning the semiconductor wafer. Moreover, to improve the thermal coupling as well as the electrical and mechanical properties, an appropriate back side metallization (BSM) is usually required in high power devices [5]. A BSM consists of a multiplicity of metal layers with thicknesses ranging from hundreds of nanometers to microns However, the combination of thinned semiconductor wafer with a metalized multilayer determines that the whole wafer undergoes mechanical stress conditions which results in the warpage of the structure hindering, sometimes severely, the performances and the following manufacturing processes of the devices.
In general, gaining a control on the warpage determined by the BSM residual stress can benefit the whole semiconductor industry and disclose future developments. And indeed, the trend of the increase of the wafer size, which is also occurring in the field of SiC [6] [7], requires a more fundamental control on the warpage effects in the whole wafer as well as an understanding of the resulting warpage in the singled die. Moreover, it is known that with the increase of the size, the handling of a thinned wafer becomes more and more critical. For this reason, in 2008 DISCO proposed the patented taiko process [8] [9] [10], which consists in a back-grinding method that leaves an annular region around the whole wafer (see figure 1) . This solution, which is now a standard, allows an easier handling of the wafer itself and a reduction of the warpage. Indeed, because of the intrinsic structure of the taiko wafer, for a given metal stress and metal film thickness, the resulting warpage of a BSM thinned wafer is mitigated with respect to the case of a canonical (flat) BSM thinned one (see figure 1b).
According to the literature, extended reviews on the use and applications of ultrathin chips (UTC) have been reported , with a focus ranging from UTC for flexible electronics [11] to conventional ones [12] which included also power electronics [13] [14] [15] [16]. Moreover, an attempt to extend the Stoney formula for a non-uniform wafer thickness has been considered in [17] for the case of a wafers having a slight step change in the peripheral region. However, the considered step is treated as a small perturbation with respect to the thickness of the wafer. Therefore, this work does not provide a general solution for the case of a taiko wafer.
Approaches based on the Finite Elements Analysis (FEA) have been also pursued, but for the case of flat canonical substrates by employing standard tools for structural analysis such as ANSYS [18] [19] and ABAQUS [20]. From these works it emerges that it is difficult to predict the final shape of the warped flat substrate. Indeed, the numerical and stable solution usually is not the physical one, unless a small perturbative displacement which will lead to the curl shape [18] is applied. Moreover, simulations have been also reported for the case of a flat and patterned substrate [21] However, and at the best of our knowledge, no extensive essays have been reported for the case of the taiko wafer, in the general case. It is hence unknown how the change of the warpage occurs for these kinds of wafers. In this work a rational and analytical description, developed according to the theory of the elasticity of the resulting warpage, induced by a stressing metal thin film, in the elastic or linear regime, in a taiko wafer, is provided. The calculated warpage for a BSM thinned taiko wafer is benchmarked and assessed according to experimental results gained investigating thick BSM 8" Si taiko wafers. Finally, the utility of a modified Stoney formula is proved and extended for the case of an 8" SiC taiko wafer.

1) Background: the Stoney formula
Internal stresses in a thin BSM film deposited onto the back side of a semiconducting substrate cause the whole set of film and substrate pair to warp until the thermo-mechanical equilibrium is reached. This condition is reached when both the net forces and the resulting bending moments acting on the system reach a null value, at the given operating temperature . A disk-shaped wafer substrate allows a simplification of the theory. Indeed, if the substrate thickness ℎ is constant and negligible with respect to the diameter of the wafer, whenever the thickness of the film, , is uniform and small compared to that of the substrate, the average film stress, can be determined from the curvature of the elastically deformed coated substrate. In 1909 Stoney published a simple formula [22] which provides a straightforward and rational relationship between the measured average curvature and the average film stress : where E, and are the Young's modulus and Poisson's ratio of the substrate. Equation (1.1) constitutes the standard formula which is exploited daily in manufacturing sites, labs and plentifully reported in the literature to infer how film stresses relates with the experimental measurement of the system's curvature (2). After more than a hundred years [23] the Stoney's formula is still the benchmark or reference for the thin film induced stress investigation.
However, eq. 1.1 is not valid for the case of a taiko wafer, because it does not consider the presence of the ring. In this work steps to rationally modify eq. 1.1 for the case of a taiko wafer have been considered. The premises of these developments are the following. Though the abrasive removal of silicon or SiC during the back-grinding process leads to a layer of damaged silicon crystal structure which has high compressive stress [24], we will treat the taiko wafer as an idealized, symmetry driven structure, whose influence on the warpage is determined by the presence of the ring. As in the case of the flat wafer, after thinning we consider that the wafer has approximately a constant curvature. Another important aspect is the influence of the gravitational force. In fact, as the final thickness decreases, the wafer becomes progressively less able to support its weight. However, corrections to the warpage can be applied in order to discriminate the influence of the stress From an instructive point of view the Stoney formula is usually recovered in textbooks in the context of the classical Euler-Bernoulli beam theory [25]. If we consider the cross section of an ultrathin wafer as a beam, its curvature is determined by the ratio between the bending moment applied to the substrate, which is generated by the stressing metal film acting on the wafer, and the flexural rigidity where is the modulus of elasticity of the substrate and is the moment of inertia of the section of the beam with respect to the neutral axis, which results from the intersection between the neutral plane of the wafer and the plane containing the examined cross section. The bending moment is determined by the product of the force applied to the substrate , at the interface between the film and the substrate, times the distance ℎ 2 between the surface of the substrate and the neutral axis, that in this case matches the gravity center axis. Because of the mechanical equilibrium, the resulting forces, acting on the film and on the substrate , as well as the bending moments, on the film and in the substrate , are null: that is + = 0, + = 0. This means that = − ℎ 2 . Finally, the moment of inertia of a section of the beam whose height is ℎ and an infinitesimal width , perpendicular to the height, with respect to the neutral axis, located at ℎ 2 from the interface, is equal to ℎ 3 12 . By combining these quantities, it is easy to recover eq. 1.1.
In the case of a taiko wafer the neutral plane is shifted upward, because of the annular region. This has two outcomes. The first occurs on the bending moments, which changes because of the new position of the neutral axis. The second is that the moment of inertia of the section with respect to the neutral axis will increase. The modified Stoney formula, can be hence written as: In order to determine the quantities that appear in eq. 1.3, a more rigorous point of view must be adopted and the whole dissertation must be developed within the theory of elasticity [26], [27].
2) Taiko wafer According to the theory of elasticity According to the theory of elasticity, (see Landau-Lifshitz, [26]) , if the taiko wafer is not subject to the force of gravity, the vertical displacement with respect to the neutral surface satisfies the biharmonic equation ∆ 2 = 0, which holds both in the ring region as well as in the thinned wafer region.
In the ring region, < ≤ , the simplest solution of the biharmonic equation, that does not consider the dependence on the angle is: where a, b, c and d are four coefficients that must be determined from the boundary conditions. Whereas in the internal and thinned substrate region, 0 ≤ ≤ , the solution of the biharmonic equation is with and two coefficients to be determined, such that to match with eq. 2.1.
In general, the biharmonic equation has the Michell's solution [28], which takes into account also of the dependence on the polar angle . In the case of a taiko wafer, in the experimental practice it results that these wafers warp preferentially in one direction and less in the perpendicular one, for the case of a thick BSM. In the hypothesis that the profile of warpage is parabolic, a simple mathematical solution that takes into account of the dependence on the angle , in the internal region that can be considered is the following: ( , ) = 2 ( where and ⊥ are related with the curvature in the preferential warpage direction and in the perpendicular direction, respectively. On the other hand, in the ring region, a solution which considers the dependence on the angle , compatible with eq. 2.3 is: where and ⊥ are related to the curvatures in the preferential warpage direction and in the perpendicular direction of the ring region, respectively. By considering the average value of equation (2.3) and (2.4) along the angles, the resulting equations are still solutions of the biharmonic equation, but of the simplest form: Hereafter, it is considered that the average warpage of the real taiko wafer does not depend on the angle , and that the behavior of the taiko wafer can be investigated by studying the behavior of the warpage on the radial distance , only, by means of the set of equations 2.5 .
In figure 2 a schematic of half of the vertical cross section of a Taiko wafer (supposed fully cylindrical symmetric) having a thickness of the substrate ℎ, a height of the rim , an internal radius , and external radius , respectively, is reported. A reference frame is fixed at the center of the taiko wafer. The origin is placed at the neutral plane, such that the front of the wafer is at = with respect to the neutral plane, with = −| |. Onto the thinned back side substrate surface a thin metal film having a modulus of elasticity and thickness , indicated in yellow, is deposited. The neutral axis results from the intersection of the neutral plane and the cross-section plane. Once the reference is fixed, ℎ + is the distance between the neutral axis (plane) and the back side substrate surface, ℎ + + is the distance between the neutral axis and the BSM thin film surface, whereas + is the distance of the ring region surface of the taiko wafer with respect to the neutral axis. At the interface between the BSM thin film and the wafer substrate, at a distance from the center, the taiko wafer is subject to a radial and a circumferential force, whose value per unit length is set equal to and , respectively. The wafer is supported at the external ring , whereas and are the reaction forces evaluated per unit length.

Fig.2. Vertical cross section of a Taiko wafer having internal radius
, external radius , thickness of the substrate h and of the rim H, respectively. The taiko wafer is subject to a radial force at the internal step edge of the ring, whose value per unit length is Fr. The wafer is supported at the external ring and Nr is the reaction force evaluated per unit length. The front of the wafer is at = with respect to the neutral plane, with = −| | .
If this is the case, since the taiko wafer is in mechanical equilibrium, the resultant of the forces and the moments must be equal to zero. In particular, the equilibrium of forces and moments holds locally for the thin film deposited in the substrate region: And the reaction moments acting on the ring at are in equilibrium with those of the substrate.
The moments per unit length in the substrate region and ring region are the following: Where and are the flexural rigidity of the substrate and ring regions, respectively. Whereas and are the moments of inertia of a section of the taiko wafer, considered as a beam of width = , with respect to the neutral axis [29]. Being , the forces per unit length acting on the length = , the moment per unit length equals , ( + ℎ).
Moreover, in the limit → ℎ according to the Stoney formula In the event, the set of four equations for the moments can be written as: In general, and are functions of , such that in the limit → ℎ, = = ℎ 3 12 .
The equation is the extension of the Stoney formula for the case of a taiko wafer. It can be solved once that ( ) and are known.

3) Determination of the moment of inertia
By following reference [30] along with the schematic of figure 2, it is possible to gain an expression of the flexural rigidities ( ) and ( ). Indeed, the total bending moment per unit length acting in the substrate region covered with the BSM film can be expressed as: Hence, the flexural rigidity of the substrate region, corrected by the Poisson coefficient of the thin film and of the substrate is (being − = | |) : In the limit → ℎ, since ℎ + = ℎ/2, becomes ℎ 3 12 , which is what we expect.
Analogously, in the ring region it holds and the flexural rigidity in this region becomes

4) Evaluation of
In order to gain the stress from measurement of warpage, the evaluation of , is required. This quantity can be calculated from the equilibrium of the forces.
Substrate region. By following still [30] we can write the total force acting in the substrate region in the radial direction as where ℎ 1 = |ℎ + + | and ℎ 2 = | |.
At a distance we can write, and the total force acting on the substrate region is Ring region. In the ring region we need to evaluate the quantity: Taiko wafer. Hence, the total force acting on the wafer, in the radial direction, equals zero, being in equilibrium: In the limit → ℎ, = , equation (4.5) reduces to the canonical case of a flat disk wafer: where is equal to which is the value of the limit To evaluate z B in the general case, it is necessary to combine, the modified Stoney formula for the taiko wafer (eq 2.7) and calculate the value of the integral (see appendix C) according to the function ( ) evaluated in appendix A. By doing so a linear equation is obtained, whose solution provides the value of the neutral axis z B . has been evaluated and its value is also reported in Appendix C.
Besides the equilibrium of the forces in the radial direction, also the net forces along the circumferential direction must be considered. The net forces in this direction must satisfy eq. 4.11: has been evaluated and reported in appendix C.
In the event, it results that there are two values of , a value determined by eq. 4.8, hereafter indicated as , , which determines a neutral axis along the radial direction, because of the radial net force, and a value determined by eq. 4.11, hereafter indicated with , , which determines an additional neutral axis in the radial direction determined by the equilibrium of the net circumferential forces. Indeed, according to this description, eq. 2.7 splits into a pair of equations:

5) Determination of warpages
For the case of a standard and flat wafer, the warpage can be evaluated from the variation Δ of the value of the measured z-coordinate between the center and the peripheral region. If ( ) = 2 2 , where is the curvature, the Δ between the value at = and = 0 is the warpage, which is equal to 2 2 .
In the case of a taiko wafer, there are two radiuses that must be considered, and . In the substrate region, according to details reported in appendix A, the ( ) function (see eq. A.35) varies with respect to the neutral axis and assumes the value However, the warpage of the taiko wafer, is evaluated with respect to the surface of the ring region, which lies at a distance of + with respect to the neutral axis. Hence, by considering the case of a prevalent warping direction, ⊥ ≈ 0, with respect to the surface of the external ring, the warpage of the taiko wafer is: In

6) Results and discussion
In the standard practice, a taiko wafer typically has a constant step ( − ℎ according to figure 2) of hundreds of microns. In the case of an 8" taiko Si wafer it holds that + ℎ ≤ 725 , being 725 the thickness of an 8" Si wafer. The thickness of the substrate region typically ranges below 100 for the case of ultrathin chips (UTC). In table 2, the values of the main parameters used in this work, have been reported. In order to test our findings we have calculated the resulting curvatures in the substrate region and the warpages for the case of an 8" Si wafer, having a ring width of 3.7 mm and a constant step of 450 , as a function of the thickness ℎ of the substrate region, with ℎ ≤ 295 . The properties of the BSM metallization have been summarized in terms of a stress , which is of the order of hundreds of MPa, and a thickness of the metal film, which ranges between 100 nm and 2000 nm in the case of thick metal layers.  Table 2. Values of the parameters for the external radius , , ring width, internal radius , step height, Poisson's coefficient and Young modulus for an 8" Si taiko wafer. In the last two columns the values of the normalized integrals 1 and 2 is reported.

Experimental
The warpages of six thick BSM 8" Si taiko wafers 90 µm thin, all having a step of 450 µm and a ring width of 3.7 mm, have been measured with an MX-204 equipment (E+H Metrology), at room temperature, and the measurement have been reported in Fig.3a-3d. The BSMs consist of a multilayer of several metals which result in a global metal stress and film thickness that ranges from 230 MPa to 450 MPa and from 1300 nm to 2450 nm, whereas the warpage ranges from 500 µm to about 1300 µm, respectively. The thick BSMs have been deposited by one of our suppliers and the values of the metal stress has been determined independently on thicker chip size substrates, by exploiting the canonical Stoney formula.      It is evident that all the wafers show a preferential direction of warpage which determine a curl-shape. Table  3, summarizes and compares the measured warpages for the cases reported in figure 3 from a to d.
Step ( Table 3. Measured warpages for the case of an 8" Si taiko wafers 90 µm thin, deposited with thick metal layers whose thickness and stress values have been obtained by measurements on equivalent thicker non-taiko wafers. On the last column it is reported the evaluation of the warpage according to the proposed model.

Calculations
The warpages (0) determined by the neutral axis , and , respectively, have been calculated according to eq. 5.2, by exploiting the values of the curvatures , , and , , in the substrate region gained from the modified Stoney's equation 4.15, for the six cases of BSM investigated experimentally. Fig. 4a reports the warpages determined by the neutral axis , as a function of the substrate thickness (in µm), for the six cases of BSMs 8" Si taiko wafer, with thick metallization. The warpages depend on the product ℎ and decreases as the substrate thickness increases. The values of the warpages for a substrate thickness of 90 µm, have been extracted for a comparison with the measured values of the warpages. Analogously, in Fig. 4b the warpages determined by the neutral axis , have been reported as a function of the substrate thickness, for the six cases of BSMs 8" Si taiko wafer, with thick metallization. Also in this case the warpages increase as the product ℎ increase and decreases as the substrate thickness increases. Similarly, also in this case, the values of the warpages for a substrate thickness of 90 µm, have been extracted for a comparison with the measured values of the warpages.
In  Table 4. Extracted values of the warpages determined for the cases of the 90 µm thin substrates by the neutral planes , and , , respectively. The values of the surface of the ring with respect to the neutral axis + , and + , are reported for the case of 90 µm thin BSM 8" Si taiko wafer.
In fig. 5a the linear correlation between the calculated warpages resulting from the neutral axis ,r and the measured warpages referred to the neutral axis, that is the value − ( + , ), for the six measured 8" taiko wafers, 90 µm thin, has been determined. A best fit line corresponding to = 0.53 , very close to the theoretical dependence reported in eq. 5.4, has been determined, with an 2 = 0.99. Very similarly, in Fig. 5b, the linear correlation of the calculated warpage determined by the neutral axis , and the measured warpage with respect to the neutral axis ( − ( + , )) for the six measured 8" taiko wafers, 90 µm thin. The calculated data are correlated with the measured ones, though the best fit line = 0.69 is a slightly further than the theoretical value of y=0.5x.
By considering the good agreement with the hypothesized dependence of the warpage, for the , neutral axis, the linear correlation between the calculated warpages according to eq. 5.4 and the measured warpages has been reported in figure 6. It results that the best fit line has a slope of 1.05, with an interval of confidence of 95% ranging between 0.97 and 1.13. The two quantities show a good linear correlation, being the 2 = 0.99.   . Linear correlation between the calculated warpage determined by the neutral axis , and the measured warpage with respect to the neutral axis, for the six measured 8" taiko wafers, 90 µm thin. The best fit line y=0.53x results very close to the theoretical value of y=0.5x. Figure 5b. Linear correlation between the calculated warpage determined by the neutral axis , and the measured warpage with respect to the neutral axis, for the six measured 8" taiko wafers, 90 µm thin. The calculated data are correlated with the measured ones, though the best fit line y=0.69x is a slightly further the theoretical value of y=0.5x. Figure 6. Linear correlation between the calculated warpage determined by the neutral axis , according to equation 5.4 and the measured warpage with respect to the neutral axis, for the six measured 8" taiko wafers, 90 µm thin. The calculated warpage shows a good linear correlation with the measured warpage. 7) Extension to the case of the 8" SiC taiko wafer.
It is possible to extend the determination of the warpages, for the case of an 8" SiC taiko wafer. In particular we consider a taiko wafer having the same geometrical characteristics of the 8" Si taiko wafer, that is a constant step of 450 µm a ring width of 3.7 mm, a radius of 100 mm. The parameters exploited in this calculation are reported in  Table 5. Values of the parameters for the external radius , , ring width, internal radius , step height, Poisson's coefficient and Young modulus for an 8" SiC taiko wafer. In the last two columns the values of the normalized integrals 1 and 2 is reported.
In figure 7, the calculated warpage according to eq. 5.4 determined by the neutral axis , , for an 8" SiC taiko wafer, as a function of the substrate thickness (in µm), for the six cases of BSMs, examined for the 8" Si taiko wafer, with thick metallization, has been reported. In   In figure 8 the calculated ℎ quantity is compared with the experimental ℎ quantity of the corresponding value for the six BSM examined cases of the 8" Si taiko wafer 90 µm thin. The best fit analysis suggests a linear and positive correlation with a slope of 1.27. the discrepancy with respect to the expected theoretical value of 1 of this quantity can also be determined by the different conditions of measurements exploited. Indeed, the experimental value of the stress is determined by exploiting the canonical Stoney formula in a thicker Si flat wafer. Clearly the two corresponding substrates can differ and determine this discrepancy. Until now, in the normal practice, a further measurement was needed in order to have an estimate of the stress, instead by exploiting eq. 8.1 it is possible to gain a value of the stress determined by the BSM film on the taiko substrate. In fact, from the values reported in table 8 we can determine a set of corresponding values for the stress, directly calculated for the case of the taiko wafers. In table 7 we report the calculated values of the stress determined by the BSM film in MPa. In figure 9, the correlation between the experimental and calculated values of the stresses is reported. In particular, a slope of 1.24 can be determined with an R 2 value of 0.98.

Metal stress (MPa) Experimental
Film thickness (nm) Metal stress (MPa)  Calculated  230  1500  196  450  1300  542  250  2000  367  400  1500  502  310  2450  400  446  2000  582  Table 7. Calculated metal stresses in MPa, calculated according to the extended Stoney equation, for the six 8" BSM Si taiko wafers examined. Finally, the value of the warpage at a die level can be also easily calculated. Indeed, once the ring is removed and the wafer diced into LxL die, the corresponding warpage can be determined according to eq.8.2.

9) Conclusions
In conclusion, an extension of the Stoney formula has been developed for the case of an 8" taiko wafer, according to the theory of elasticity. It has been shown that the taiko wafer mitigates the warpage, because of the presence of the annular region. On one hand the ring region increases the moment applied to the substrate. In fact, the neutral axis is shifted upwardly because of the presence of the ring. On the other hand, the higher moment applied to the substrate is mitigated by the higher moment of inertia of the section of the taiko wafer. Therefore, the resulting warpage, despite a thicker BSM is mitigated and lowered with respect to a canonical and flat substrate.
The developed model has been validated according to experimental values benchmarked for six case of 8" BSM Si taiko wafer, having a substrate thickness of 90 µm and a step of 450 µm. It has been shown that there is a good agreement between the developed theory and the experimental values of the warpage. The modified Stoney equation has been usefully exploited to gain an estimate of the BSM metal stress. The comparison between the measured valued of the stress and the calculated value of the stress shows a linear correlation and states the utility of this formulation. Finally, an extension of the model has been considered for the case of an 8" SiC BSM taiko wafer.

Supporting information Appendix A: Determination of ( )
If the taiko wafer is supported at = by following Landau, the boundary conditions tell us that the there is no vertical displacement at the edge = and that the bending moment is zero: In order to determine the four constants , , , and , two additional equations are required, to solve alongside with the boundary conditions for .
By calculating the first and the second derivative of ( ), we obtain: