## Abstract

Surface roughness is a critical indicator to evaluate the quality of 4H-SiC grinding surfaces. Determining surface roughness experimentally is a time-consuming and laborious process, and developing a reliable model for predicting surface roughness is a key challenge in 4H-SiC grinding. However, the existing models for surface roughness in wafer rotational grinding fail to yield reasonable results because they do not adequately consider the processing parameters and material characteristics. In this study, we proposed a new analytical model for predicting surface roughness in 4H-SiC wafer rotational grinding, which comprehensively incorporates the grinding conditions and material characteristics of brittle substrate. This model derives and calculates the material's elastic recovery coefficient based on contact mechanics and elastic contact theory. Subsequently, we modified the grain depth-of-cut model by incorporating elastic recovery coefficient. Additionally, we analyze the distribution of the failure mode (ductile or brittle) on the surface of a material when the depth at which the material is cut instead follows a random distribution known as the Rayleigh distribution. To validate the accuracy of the established model, a series of grinding experiments are conducted using various grain depth-of-cut to produce 4H-SiC wafers with different surface roughness values. These results are then compared with those predicted by both this model and the traditional model. The findings demonstrate that the calculated data obtained from the proposed model exhibit better agreement with the measured data. This research addresses the need for an improved surface roughness model in 4H-SiC wafer rotational grinding.

## 1 Introduction

As a new type of semiconductor material, single-crystal silicon carbide (4H-SiC) possesses several outstanding advantages that make it suitable for high-power electronic and high-frequency components [1]. In order to efficiently back-thin and flatten 4H-SiC wafers while maintaining high surface integrity, wafer rotational grinding is considered an ideal method [2,3]. However, due to the 4H-SiC's high hardness and brittleness, damage to the substrate wafer surface is inevitable in the grinding process [4,5]. Surface roughness *R*_{a} is a key indicator for assessing ground surface quality, and the ability to accurately predict *R*_{a} can help control damage, reduce costs, and improve productivity [6].

Researchers have conducted extensive studies to develop reliable surface roughness models, which can be broadly categorized as empirical methods and analytical methods [7]. Empirical methods， which rely on empirical dependency solutions, are based on specific initial surfaces, grinding wheel specifications, and operating conditions. Agrawal et al. [8] proposed a model for predicting the *R*_{a} of AISI 3040 steel turning using multiple regression methods. Zhu et al. [9] determined an exponential model for *R*_{a} of SiCp/Al composite materials by fitting and analyzing orthogonal experimental results. Guo et al. [10] established a method for predicting *R*_{a} on the basis of a hybrid feature selection method and a sequence deep learning framework of long short-term memory networks. While empirical models are relatively easy to develop, their applicability is limited to a specific range of parameters, and they cannot accurately predict surface roughness under different conditions.

Analytical methods, on the other hand, provide a better comprehension of the formation mechanism of surface roughness and have broader applicability as they are derived from basic principles. Hecker et al. [11,12] considered the randomness of grains and established a *R*_{a} model based on the probability distribution of grain depth-of-cut (*d*_{g}) in the grinding process. Agarwal et al. [13] further incorporated factors such as overlapping grooves, microstructure of the grinding wheel, grinding motion, dynamic conditions, and material properties to develop a *d*_{g} model and a *R*_{a} model in ceramic grinding. Wu et al. [14] distinguish the grinding stage between ductile and brittle regimes on the basis of the brittle-plastic transition critical cutting depth and proposed a *R*_{a} model for hard and brittle materials that accounts for both brittle and ductile removal. They validated their model through silicon carbide ceramic cylindrical grinding experiments. These studies highlight the importance of considering the *d*_{g} as a comprehensive parameter for predicting surface roughness. Zhang et al. [15] conducted the precision grinding experiment of silicon wafers and measured the nano-grinding chips. They developed a prediction model for *d*_{g} in nano-grinding based on measured data sets. However, this method is inefficient and requires precise measuring equipment and skilled operators [16]. To address these limitations, Zhang et al. [17] proposed a *d*_{g} model for grinding that incorporates grinding parameters, wheel grinding section structure, effective grain number, grain shape, and the elastic deflection of grain and substrate during the grinding process. Their model was validated through grinding experiments and provides valuable insights for the design and fabrication of brittle materials during grinding. It is worth noting that even in the brittle removal state, hard and brittle substrates undergo elastic recovery during the material removal process [18,19]. Chai et al. [20] found that considering elastic recovery in the modeling and prediction of scratch normal force for 4H-SiC resulted in more accurate results compared to traditional models. However, the current prediction model for *R*_{a} based on *d*_{g} does not consider the impact of material elastic recovery.

In this paper, a comprehensive *R*_{a} model has been proposed for wafer rotational grinding of 4H-SiC. The model considers both grinding conditions and substrate material properties. The model incorporates a grain depth-of-cut model that incorporates grinding motion conditions, grinding wheel geometric parameters, and the probability distribution of *d*_{g}. The elastic recovery coefficient of the material is deduced and calculated based on contact mechanics and elastic contact theory, and the correctness of the elastic recovery parameters is confirmed through a single-point diamond scratch test. Based on this, the *d*_{g} model that confirms the Rayleigh distribution is modified to account for both brittle and ductile removal states of the ground substrate surface in different *d*_{g}. Finally, the proposed surface roughness model is validated through grinding experiments conducted on 4H-SiC wafers with different grain depth-of-cut. This research will provide valuable guidance for the design and fabrication of hard-brittle materials during grinding.

## 2 Models

### 2.1 Surface Roughness Model.

Figure 1 is the process of wafer rotational grinding, where both the grinding spindle and the chuck table rotate around their respective centroid axles. The wheel grinding section contacts the substrate in a quarter region. During grinding, the diamond grain penetrates the 4H-SiC wafer surface at a cutting depth *d*_{g}, resulting in the removal of substrate material through mechanical force. Due to the randomness of the interaction of the grinding wheel grains embedded into the substrate surface during the grinding process, it is difficult to comprehensively and accurately describe the macroscopic and microscopic morphology characteristics of the ground substrate surface. For establishing a surface roughness prediction model for grinding substrates, it is necessary to simplify the formation and morphology characteristics of the ground substrate surface in this model. The specific assumptions and simplified conditions are as follows:

The grains with the same blade radius are randomly distributed in the grinding section, the depth of the grooves generated by the grains on the substrate surface is the same as the grains' depth-of-cut that undergo elastic recovery, and the grains' depth-of-cut conforms to the Rayleigh distribution.

When grinding substrate in the plastic domain, the cross-sectional shape of the grinding grooves generated by grains on the substrate surface is semicircular, while the cross-sectional shape of the grinding grooves on the substrate surface in the brittle domain is semi-elliptical [11,21].

When the grooves on the substrate surface caused by grains overlap, they only overlap once on both sides of the grooves.

*d*

_{g}on the substrate accords with the Rayleigh probability density function [22]:

*σ*is a variable that completely defines the Rayleigh probability function.

*d*

_{g}during grinding also leads to elastic recovery phenomena related to the material properties. Therefore, in Eq. (1), the expectation of the Rayleigh distribution is equal to the average

*d*

_{g}of the substrate surface after elastic recovery as follows:

*d*

_{r}is the

*d*

_{g}after elastic recovery, and

*φ*

_{r}is the elastic recovery coefficient of the substrate.

*d*

_{g}[23,24]. Figure 2 illustrates a schematic diagram of the microgeometric morphology of the ground wafer surface. Areas I, II, and III are three surface morphologies that have different effects on the surface roughness of the wafer, with

*p*

_{1},

*p*

_{2}, and

*p*

_{3}being their corresponding distribution probabilities and can be calculated as follows:

*p*

_{1}is lower than the position

*y*

_{cl}of the surface contour centerline,

*p*

_{2}is the probability that the depths of surface scratch morphology are between

*y*

_{cl}and the critical cutting depth

*h*

_{cr}, and

*p*

_{3}is the probability that the depths of surface scratch morphology exceeds

*h*

_{cr}. The

*h*

_{cr}can be expressed as follows [25]:

*t*

_{1},

*t*

_{2}, and

*t*

_{3}denote the depths of surface scratch morphology below

*y*

_{cl}, between

*y*

_{cl}and

*h*

_{cr}, and above

*h*

_{cr}, respectively.

*A*

_{1}represents the shaded area above

*y*

_{cl}in Area I, while

*A*

_{2u}and

*A*

_{2d}represent the shaded areas above and below

*y*

_{cl}in Area II, respectively.

*A*

_{3u}and

*A*

_{3d}represent the shaded areas above and below

*y*

_{cl}in Area III. Areas I and II exhibit a plastic removal state, whereas Area III undergoes brittle removal for material removal.

*C*

_{t}represents the transverse crack lengths of microcracks formed by grinding wheel particles on the substrate surface in region III. The

*R*

_{a}values of Areas I, II, and III are denoted as

*R*

_{a1},

*R*

_{a2}, and

*R*

_{a3}, respectively. The following equations can be derived from geometric relationships:

*R*

_{a}of the ground substrate can be expressed as follows:

*y*

_{cl}as 1.257

*σ*by deriving the model and summarized the expectations of

*R*

_{a1},

*R*

_{a2}, and

*R*

_{a3}in wafer rotational grinding, and the details of the derivation process are shown in Appendix.

*ϕ*of the wafer rotational grinding is 0.822 [27]. The process of obtaining parameter

*ϕ*can be found in Appendix.

This model highlights the significance of *d*_{g} and the elastic recovery coefficient *φ*_{r} as crucial parameters in predicting surface roughness.

### 2.2 Grain Depth-of-Cut *d*_{g}.

*d*

_{g}is a comprehensive parameter that reflects the grinding conditions and directly affects the

*R*

_{a}. In previous work, from the perspective of the motion relationship of rotary grinding, we analyzed the material removal cross-sectional area (CSA)

*A*

_{w-p}at a specific radial position

*R*

_{1}on the wafer per revolution of the grinding wheel through the contact length

*L*and removal height

*H*

_{p}, as shown in Fig. 3. However, when considering the material removal by a single grain, the CSA of material removal (

*A*

_{x−z}) represents the total cutting area of the effective grains in the grinding process, as shown in Fig. 1(b). As the material removal CSAs obtained by two different methods are equal, the prediction model for

*d*

_{g}can be established as follows [17]:

*r*

_{g}denotes the grain radius,

*f*denotes the wheel feed rate,

*R*

_{1}signifies the radial position on the wafer surface,

*n*

_{w}represents the wafer rotation speed,

*θ*signifies the grain cutting angle,

*n*

_{s}is the grinding wheel rotation speed,

*η*represents the grain volume fraction,

*κ*signifies the effective grain fraction,

*W*denotes the width of the grinding section of the cup-shaped grinding wheel,

*D*represents the grinding wheel diameter, and

*E*

_{t}* represents the reduced elastic modulus of wheel bond.

### 2.3 Elastic Recovery Coefficient *φ*_{r}.

*P*and the projection of the contact area

*A*

_{T}can be expressed as [28,29]:

*P*

_{m}in the contact area can be denoted as follows [30]:

*E*represent the hydrostatic pressure in the contact area and the elastic modulus of the substrate, respectively.

*σ*

_{u}represents the yield strength of the substrate, and

*γ*is the angle between the conical indenter surface and the contact surface. In certain calculations, it is permissible to treat the indenter as a simplified conical indenter with 70.3 deg half angle. Thus,

*γ*can be calculated as 90–70.3 deg = 19.7 deg [31,32].

Figure 4 illustrates the progress of indenter pressing into the substrate. The definition of *A*_{T} is shown in Fig. 4(b). The gray area represents the projection of the contact area between the pressed part of the indenter and the substrate, while the red area represents the projection of the substrate elastic recovery part and the contact part of the indenter.

*h** is the difference between the actual spherical tip of the indenter and the assumed triangular pyramid tip,

*r*denotes the indenter spherical tip radius, the angle

*β*between the indenter centerline and side edge is 77 deg, and

*A*

_{T}can be expressed as follows:

*h*and residual depth

*h*

_{r}respectively, denote the depth of the indenter pressed into the surface of the wafer and the groove residual depth, as shown in Fig. 4(a). Through simultaneous Eqs. (11)–(13) and Eqs. (15)–(17),

*P*can be expressed as a function of

*h*:

*δ*is associated with the indenter geometric shape, specifically for standard Berkovich indenters where

*δ*equals 1.034. The reduced modulus

*E*

_{r}can be calculated as follows:

*E*and

*v*denote the elastic module and Poisson's ratio of the substrate material, and the

*E*

_{i}and

*v*

_{i}of the diamond indenter are denoted as 1140 GPa and 0.07, respectively.

*A*can be obtained by contact depth as follows:

*P*over

*h*, we can obtain the first-order derivative function expression of

*P*as follows:

## 3 Experimental Details

### 3.1 Nano-Indentation Experiments.

Commercially available 4H-SiC (off <11–20>4 ± 0.15 deg, N-type Nitrogen) substrates with dimensions of 10 × 10 × 0.35 mm^{3}, obtained from Beijing Innotronics Technology Ltd., was utilized for this research. The experiments were conducted on the C-face of the substrate, and after chemical mechanical polishing (CMP), the substrate *R*_{a} < 1 nm. The mechanical properties of the 4H-SiC substrate were determined by the continuous stiffness method with a nano-indentation instrument (100BA-1C, MTS) equipped with a Berkovich indenter. The indentation depth was set to 1100 nm, and the load was held for 10 s. Each subsequent indentation position was spaced 100 *μ*m apart and was repeated nine times. Throughout the experimental process, the system continuously measured the stiffness value of the substrate, thereby obtaining the hardness and modulus at each displacement point. Consequently, a single indentation process could be employed to measure the changes in hardness and elastic module corresponding to the indentation depth.

According to Eq. (25), a scratch test is conducted to verify the accuracy of the elastic recovery coefficient by measuring the *h* and *h*_{r} values. The KLA G200 nano-indentation instrument, as depicted in Fig. 6(a), was employed for the nano-scratch experiment using a diamond indenter with a spherical tip radius *r* of 20 nm. The scratch process consisted of three stages. In the first stage, known as the pre-scanning stage, the indenter scans the substrate surface at a constant speed under a small load of 0.06 mN. Once the scanning is complete, the indenter returns to the zero position. The second stage is the uniformly variable load scratch stage, as depicted in Fig. 6(b). Scratch parameters can be found in Table 1. Throughout this stage, the indenter lateral displacement *x* and *h* are continuously recorded. The third stage is the residual depth measurement stage, where the indenter returns to the zero position and traverses the scratches under a constant micro load of 0.06 mN to measure the residual depth *h*_{r} of the scratches. The micro-surface morphology of the substrate was observed using scanning electron microscopy after the experiment.

### 3.2 Grinding Experiment.

Ultraprecision grinding test was conducted on an grinding machine (VG 401MK2, Okamoto Machinery Co., Ltd., Japan), which operates on the basis of the theory of wafer rotation grinding, as shown in Fig. 7. The machine employed a continuous feed method with a minimum feed speed of 1 *μ*m/min. 4H-SiC substrates were bonded with paraffin at a radius of *R*_{1} = 50 mm on a 200 mm diameter single-crystal silicon wafer to serve as grinding experimental specimens.

Equation (10) comprehensively describes the effects of grinding parameters, radial position, grain radius, and grinding wheel geometric factors on *d*_{g}. Grinding substrates were performed using resin-bonded diamond grinding wheels of #325 (average grain radius of 24.2 *μ*m), #600 (average grain radius of 12.3 *μ*m), and a vitrified-bond diamond grinding wheel of #1500 (average grain radius of 6.5 *μ*m) to investigate the impact of different *d*_{g} on *R*_{a}. The grinding wheel's diameter *D* is 350 mm, and the grinding section's width *W* is 3 mm. The grain cutting angle *θ* is 70 deg [35], the effective grain number *κ* is 0.15 [36], and the grain volume fraction *η* is 0.25. The reduced elastic modulus *E*_{t}* of resin and vitrified bond are 42.3 [23] and 81.6 GPa [37], respectively. In addition, the values of the grinding parameters are shown in Table 2. When the grinding experiment was completed, the micro-ground surface morphology of the substrates was observed through SEM (JSM-7610F Plus, JEOL Ltd., Japan), and *R*_{a} was measured using a three-dimensional profilometer (NewView9000, ZYGO Corporation, America).

Condition | Grain size | Wheel speed n_{s} (r/min) | Wafer speed n_{w} (r/min) | Feed rate f (μm/min) | Grain depth-of-cut (d_{g}) by Eq. (10) (nm) |
---|---|---|---|---|---|

(a) | #325 | 2399 | 240 | 10 | 204.38 |

(b) | #325 | 2399 | 240 | 4 | 150.59 |

(c) | #325 | 2399 | 240 | 2 | 119.52 |

(d) | #600 | 2399 | 240 | 10 | 103.88 |

(e) | #600 | 2399 | 240 | 4 | 76.54 |

(f) | #600 | 2399 | 240 | 2 | 60.75 |

(g) | #1500 | 2399 | 240 | 10 | 44.69 |

(h) | #1500 | 2399 | 240 | 4 | 32.93 |

(i) | #1500 | 2399 | 240 | 2 | 26.14 |

Condition | Grain size | Wheel speed n_{s} (r/min) | Wafer speed n_{w} (r/min) | Feed rate f (μm/min) | Grain depth-of-cut (d_{g}) by Eq. (10) (nm) |
---|---|---|---|---|---|

(a) | #325 | 2399 | 240 | 10 | 204.38 |

(b) | #325 | 2399 | 240 | 4 | 150.59 |

(c) | #325 | 2399 | 240 | 2 | 119.52 |

(d) | #600 | 2399 | 240 | 10 | 103.88 |

(e) | #600 | 2399 | 240 | 4 | 76.54 |

(f) | #600 | 2399 | 240 | 2 | 60.75 |

(g) | #1500 | 2399 | 240 | 10 | 44.69 |

(h) | #1500 | 2399 | 240 | 4 | 32.93 |

(i) | #1500 | 2399 | 240 | 2 | 26.14 |

## 4 Results and Discussion

### 4.1 Mechanical Parameters.

Figure 8 shows the relationship between the average hardness *H* (as depicted in Fig. 8(a)) and elastic modulus *E* (as depicted in Fig. 8(b)) obtained from the nano-indentation system as a function of *h*. It is evident that both *H* and *E* exhibit similar trends as the indentation depth increases. Within the range of 0∼300 nm, both *H* and *E* rapidly increase with increasing indentation depth, followed by a gradual decrease and stabilization after 300 nm, ultimately converging to a stable value beyond 800 nm. *H* and *E* initially increase as the *h* increases before reaching a stable point, a phenomenon referred to as the reverse size effect [38,39]. The reverse size effect is frequently observed in experiments employing the continuous stiffness method and is influenced by factors such as the shape of the indenter [40], the magnitude of the load, the loading rate [41], the properties of the material, and the release of stress from indentation cracks [42]. The experiment applied a relatively high load (622.5 mN) and a significant indentation depth (1100 nm), along with a rapid loading rate. On the other hand, the material's critical cutting depth is small (11.98 nm), leading to early crack formation and stress release. These factors lead to the appearance of reverse size effects in the experiment. Select consistent values for hardness and elastic modulus, denoted as *H* and *E* respectively, that remain constant regardless of the indentation depth. The hardness *H* of 4H-SiC used in this experiment was measured to be 40.26 GPa, while the elastic modulus *E* was determined to be 492.47 GPa.

Figure 9 showcases the surface morphology of the substrate following the nano-indentation test. *C*_{1}, *C*_{2}, and *C*_{3} are three radial cracks that propagate along the edges of the indenter. The fracture toughness *K*_{c} of the substrate can be determined as [43]:

*C*represents the radial crack length from the indented center to the crack bottom, and

*ψ*denotes the half-opening angle of the indenter, which is 65.3 deg for the Berkovich indenter. The maximum load

*P*applied by the indenter during the experiment is 622.5 mN. The average fracture toughness, calculated based on nine indentation radial cracks, is 2.74 ± 0.38 MPa·m

^{1/2}.

### 4.2 Confirmation of Elastic Recovery Coefficient *φ*_{r}.

Figure 10 presents the experimental results of nano-scratch. Figure 10(a) displays the surface microstructure of the scratch groove, while Fig. 10(b) presents the measured *h* and *P* values during the scratch process. The loading method of uniformly variable load determines that the value of lateral displacement is equal to the value of the normal load. As explained by Zhang [44], there are four different states before a brittle fracture occurs during the scratch process: no wear, adhesion, plowing, and cutting. These states can be observed when the lateral displacement of the indenter is less than 10 *μ*m. We focus on the brittle-plastic deformation and consider the Berkovich indenter as a single diamond grain. In this context, the indentation depth *h* of the scratch corresponds to the *d*_{g}. When the lateral displacement of the indenter reaches about 15 mN, corresponding to *P* = 15 mN and *h* = 130 nm, microcracks appear in the groove, and bulk chips on both sides of the groove indicate brittle removal. This is attributed to the residual stress on the substrate caused by the indenter, leading to the extension of cracks toward the groove surface and the formation of microcracks. Under these conditions, material removal primarily occurs through brittle fracture. The experiment measures the residual depth of the groove by repeated scanning of the indenter. It is noteworthy that at high indentation loads, more cracks and brittle debris may be present in the groove, resulting in significant fluctuations in the measured residual depth. To ensure measurement accuracy, this article primarily focuses on stable experimental data for indentation depth *h* less than 130 nm. In the mathematical model described by Eqs. (18)–(25), the Berkovich indenter tip radius *r* is 20 nm. For 4H-SiC, the Poisson's ratio *v* is 0.231 [45]. The yield strength of the substrate *σ*_{u} is typically taken as *H*/2.8∼*H*/2.2 [46,47], and in this model, the median value of H/2.5 is used. According to Eq. (20), the reduced modulus *E*_{r} of the substrate is calculated to be 361.75 GPa. Under these conditions, the elastic recovery coefficient *φ*_{r} of the substrate was determined to be 30.42% using Eqs. (21) and (25), which aligns with the findings of Chai et al. [20].

The comparison between the calculated values obtained from Eq. (24) and the calculating data of residual depth is illustrated in Fig. 11. It is evident that the measured values align well with the calculation values, validating the accuracy of the calculated elastic recovery. Notably, for an indentation depth *h* of 100 nm, the measured data are slightly lower than the calculated data. Using a high-power scanning electron microscope, we observed microcracks and chip accumulation in the grooves near the anomaly point at the specific location where the discrepancy occurred, as depicted in Fig. 12. The size of these chips is approximately 10 nm. As the indenter glides along the groove bottom with nearly zero contact load, the presence of these chips leads to a discrepancy where the measured residual depth appears smaller than the true residual depth, causing a deviation at *h* = 100 nm in Fig. 11.

### 4.3 Verification of Grinding Results.

*R*

_{a}of each set of ground substrates was measured by a surface profilometer, and the results are presented in Fig. 13. It is evident that, across all cases, the ground surface roughness decreases with a reduction in grinding wheel grain size and feed rate. By examining the calculated values of

*d*

_{g}in Table 2, it can be observed that

*R*

_{a}the ground substrate diminishes as the

*d*

_{g}decreases, indicating a significant correlation. This correlation has led many researchers to simplify

*R*

_{a}as a function of

*d*

_{g}, as demonstrated in the classical surface roughness prediction model proposed by Agarwal et al. [13] as follows:

*χ*= 9.6% [13]. represents the overlap coefficient of substrate surface grooves created by grains.

A comparison was made between the classical surface roughness prediction model (as depicted in Eq. (28)), the surface roughness model that considers elastic recovery coefficient *φ*_{r} (as depicted in Eq. (9)), and the model without considering *φ*_{r}, by examining their consistency with the experimental measurement results. The comparison between the measurement values obtained from the grinding test and the predicted values from each model is presented in Fig. 14. The prediction curve of the model that accounts for elastic recovery demonstrates improved agreement with the experimental data. Particularly when the *d*_{g} is below 100 nm, there is an average error of approximately 7% between the predicted and experimental values after the inclusion of elastic recovery. This finding emphasizes the significance of material elastic recovery as an influential factor in the grinding process.

When the *d*_{g} exceeds 100 nm, there is a significant discrepancy between the predicted data and the experiment data. This discrepancy can be attributed to the brittle removal of the surface material when the cutting depth exceeds 100 nm (as presented in Figs. 15(a)–15(c)). The ground surface of the substrate displays numerous irregularly shaped fragments of varying sizes, which cover and damage the original grinding texture. The morphology of these randomly formed crushing pits is challenging to describe using analytical models and can only be approximated. Consequently, there exists an inherent error between the observed surface morphology on the substrate and the ideal scenario illustrated in Fig. 5, resulting in poor prediction outcomes. However, predictions that take into account elastic recovery demonstrate relatively higher accuracy.

According to experimental results in Fig. 14, the prediction accuracy of this model surpasses that of the traditional Eq. (28) model. The traditional model underestimates experimental results because it only considers the grinding surface profile under the pure plastic removal state during the modeling process, neglecting the influence of the brittle removal state on *R*_{a}. The *d*_{g} decreases along with an increase in the proportion of plastic removal. The prediction curve shows that the results of the traditional model and this model will converge. Hence, the model established in this manuscript is more appropriate for predicting the surface roughness in grinding when both brittle and plastic removal states coexist. Figure 15 depicts the surface morphology of the substrate ground under nine different parameter sets. Observation of grinding with #325 and #600 grinding wheels (as depicted in Figs. 15(a)–15(c) and 15(d)–15(f)) reveal notable surface damage, characterized by irregular crushing pits and fragments (highlighted by arrows), alongside limited plastic grooves. At this stage, brittle fracture is the primary method of material removal, with the crushing pits nearly entirely covering the material's surface. With a decrease in the feed rate, the size and distribution area of individual crushing pits diminish gradually, while plastic grooves indicative of the plastic removal state emerge (highlighted by lines) [48]. The microstructure of the substrate surface post-grinding with #1500 grinding wheel is depicted in Figs. 15(g)–15(i). It is evident that with the decrease of the particle radius, the size and quantity of fracture pits and fragments symbolizing brittle removal on the substrate surface are consistently diminishing, suggesting a reduction in the extent of brittle removal. Meanwhile, the count and length of plastic grooves denoted by lines are steadily rising, signaling a growing prevalence of plastic removal states. The discrepancy between the measured roughness values and those predicted by the traditional model decreases gradually, suggesting that the traditional model is better suited for predicting *R*_{a} in plastic grinding conditions.

The proposed model suggests that the variables influencing ground surface roughness can be categorized into two groups. The first group comprises grinding wheel type and grinding parameters, like grinding wheel geometric parameters, effective number of grains, wheel feed rate, grinding wheel and wafer speed. These effects can be comprehensively reflected by the grain depth-of-cut model. Another group includes substrate characteristics, such as the brittle-plastic grinding state of the substrate, substrate hardness, and elastic recovery coefficient, some of which were not previously considered in models. The average prediction error of this model is approximately 10%, with significantly higher accuracy than the classic prediction model and the prediction model that disregards elastic recovery. The model presented in this article predicts grinding surface roughness by considering the *d*_{g} and the mechanical properties of the substrate. It is hoped that this model can also be applied to grinding hard and brittle substrate materials, where the mechanical properties of other materials are known, and the cutting depth of abrasive particles can be kept constant and controlled during the grinding process. Nevertheless, discrepancies persist between the model predicted *R*_{a} and the measured experimental findings. These discrepancies can be traced back to the assumptions of the simplified model, particularly the reduction of the intricate surface morphology during brittle removal to an ideal contour. This limitation renders the model less effective for grinding surfaces experiencing severe brittle removal states with abrasive particle cutting depths exceeding 120 nm. Additionally, the model approximates the grinding wheel grain radius as the average grain radius, potentially introducing inaccuracies in determining *d*_{g} and *R*_{a} [49].

## 5 Conclusions

This article introduces a model for predicting surface roughness in wafer rotational grinding. The model considers various parameters of the grinding process, including machining parameters, grains shape, grooves overlap, wheel structure, and other grinding conditions. Moreover, it takes into account material properties of the substrate, such as material elastic recovery, as well as the distribution of surface brittleness and ductility at different grain depth-of-cut. The detailed conclusions are as follows:

Depending on the derivation and calculation of contact mechanics and elastic contact theory, the

*φ*_{r}of the 4H-SiC substrate is 30.42%, and the accuracy of the elastic recovery coefficient was confirmed through nano-scratch experiments.Precision grinding experiments were conducted using 4H-SiC substrates, and grinding surfaces with different roughness values were obtained under different parameters. The predictive accuracy of the proposed model is compared to both a traditional model and a model without considering elastic recovery. The results demonstrate that the proposed model exhibits superior agreement with the experimental findings, indicating that the elastic recovery of 4H-SiC substrate during grinding cannot be ignored.

## Authorship Contribution Statement

**Hongyi Xiang**: Methodology, writing—original draft, formal analysis, visualization, and validation. **Haoxiang Wang**: Writing—review & editing, validation. **Renke Kang**: resources, supervision, and funding acquisition. **Shang Gao**: methodology, writing—review & editing, investigation, project administration, supervision, and funding acquisition.

## Conflict of Interest

The authors declare that they have no conflict of interest.

## Acknowledgment

This research is financially supported by the National Key Research and Development Program of China (2022YFB3404304), the National Natural Science Foundation of China (52375411 and 51991372), and the Major Science and Technology Project of Henan Province of China (221100230100).

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

## Nomenclature

*f*=grinding wheel feed rate,

*μ*m/min*h*=indentation depth, nm

*k*=half the length of the chord of the groove in area II

*r*=spherical tip radius of indenter, nm

*A*=contact area

*D*=grinding wheel diameter, mm

*H*=hardness of the substrate, GPa

*L*=contact arc length

*P*=indenter normal force, mN

*S*=contact stiffness

*W*=width of grinding section of the cup-shaped grinding wheel, mm

*b*_{s}=width of the projected rectangular segment

*d*_{g}=grain depth-of-cut, nm

*d*_{r}=grain depth-of-cut after elastic recovery, nm

*h*_{c}=contact depth, nm

*h*_{cr}=critical cutting depth, nm

*h*=_{r}residual depth, nm

*l*_{cut}=length of the projected rectangular segment

*l*_{g}=single abrasive cutting contour projection length

*n*=_{s}grinding wheel speed, r/min

*n*_{w}=workpiece speed, r/min

*r*_{g}=grain radius of grinding wheel

*y*_{cl}=centerline distance, nm

*A*_{all}=total CSA of

*N*_{min}effective abrasive cutting*A*_{eff}=actual cutting CSA

*A*_{g}=theoretical cutting CSA

*A*_{r}=residual CSA of an effective grain cutting substrate surface material in Fig. 16(b)

*A*_{r}=residual CSA of an effective grain cutting substrate surface material in Fig. 16(c)

*A*_{w−p}=material removal area obtained from macroscopic methods

*A*_{x−z}=material removal area obtained from microscopic methods

*A*=_{T}projection area in the contact area

*C*=_{t}transverse crack length

*H*_{p}=material removal height

*K*_{c}=fracture toughness, MPa·m

^{1/2}*N*_{min}=minimum number of abrasives cutting contour projections reaching the length of

*b*_{s}*P*=_{m}average pressure in contact area

*R*_{1}=radial distance on the wafer surface, mm

*R*_{a}=surface roughness, nm

*h*=^{*}difference between the actual spherical tip of the indenter and the assumed triangular pyramid tip, nm

- $P\xaf$ =
hydrostatic pressure

- $dr\u2032$ =
residual height of single effective abrasive cutting substrate surface material in Fig. 16(c)

*p*_{1},*p*_{2},*p*_{3}=probability of distribution corresponding to cutting depths in different areas

*t*_{1},*t*_{2},*t*_{3}=depths of surface scratch morphology lower than

*y*_{cl,}between*y*_{cl}and*h*_{cr}, and higher than*h*_{cr}*v*,*v*=_{i}the substrate and the diamond Poisson's ratio

*C*_{1},*C*_{2},*C*=_{3}three radial cracks that propagate along the edges of the indenter,

*μ*m- CSA =
cross-sectional area

*E*,*E*_{i}=elastic modulus of the substrate and diamond indenter, GPa

*E*_{r},*E*_{t}^{*}=reduced elastic modulus of substrate material and grinding wheel binder

*β*=the angle between the indenter centerline and the side edge

*γ*=the angle between the indenter surface and the contact surface

*δ*=geometric shape coefficient of the indenter

*ε*=shape coefficient of the Berkovich indenter

*η*=grain volume fraction of grinding wheel

*θ*=grain cutting angle

*κ*=effective grain fraction of grinding wheel

*ξ*=overlap ratio coefficient

*ϕ*=overlap coefficient of the wafer rotational grinding

*σ*=parameter of Rayleigh function

*σ*_{u}=yield strength of the substrate, Gpa

*φ*_{r}=elastic recovery coefficient

*χ*=overlap coefficient of substrate surface grooves in ultraprecision grinding

*ψ*=half-opening angle of the indenter

## Appendix

#### Calculation of Centre Line Position *y*_{cl}, and *E*(*R*_{a1}), *E*(*R*_{a2}), *E*(*R*_{a3}) (Eq. (8))

*A*

_{1},

*A*

_{2}, and

*A*

_{3}in areas I, II, and III as depicted in Fig. 2. Specifically, the expected value of shadow area

*A*

_{1}below the centerline in Area I as follows:

where the formula *ϕ* represents the overlap coefficient of the CSA of the grooves generated by grains on the surface of the substrate.

*A*

_{2u}and

*A*

_{2d}can be summarized as:

*k*represents half the length of the chord of the groove formed by the grains on the surface of the substrate in area II, intersecting with the contour's centerline.

*A*

_{3u}and

*A*

_{3d}can be summarized as:

*y*

_{cl}as follows:

*h*

_{cr}is much larger than

*σ*[26], $e\u2212hcr22\sigma 2\u22480,Erfc(hcr2\sigma )\u22480,Erf(hcr2\sigma )\u22481$, in Eq. (A7). Then, substitute the approximate values above into Eq. (A7) and further calculate to obtain the equation as:

*R*

_{a}can also be estimated by dividing the enclosed area between the surface contour line and

*y*

_{cl}by the sampling length, the expected values of

*R*

_{a1},

*R*

_{a2}, and

*R*

_{a3}for areas I, II, and III are as follows:

#### Mathematical Calculation of *ϕ*

*A*

_{eff}of a single effective grain on the grinding wheel abrasive layer end face taking into account the overlapping effect can be obtained as:

*A*

_{g}is the theoretical CSA of a single effective grain cutting substrate surface material,

*ϕ*is the overlap coefficient of the wafer rotational grinding.

*l*

_{cut}and

*b*

_{s}, as illustrated in Fig. 16(a). Here,

*l*

_{g}represents the projection length of the substrate material effectively cut by the grinding wheel's abrasive layer end face on its cross-section (as depicted in Fig. 16(a)). The length

*l*

_{cut}of each segment cut by the grinding wheel's abrasive layer along its circumferential direction must ensure that it contains the minimum abrasive grain

*N*

_{min}required to cut substrate materials with a cross-sectional width of

*b*

_{s}in a rectangular area of length and width

*l*

_{cut}and

*b*

_{s}, respectively. Due to the probability distribution of effective grains on the end face of the wheel's abrasive layer, the length

*l*

_{cut}of each segment cut along its circumferential direction varies. However, the length

*l*

_{cut}of any cut segment of the abrasive layer is much smaller than the circumferential length of the wheel's abrasive layer end face. Further analysis demonstrates that when the CSA of the effective abrasive grains cutting the substrate before and after the grinding layer end face of the grinding wheel does not completely overlap, the required effective grains are minimal, as shown in Fig. 16(b). Therefore, the minimum number of grains

*N*

_{min}contained in each rectangular area of length

*l*

_{cut}and width

*b*

_{s}cut by the grinding wheel's abrasive layer along the circumference can be expressed as:

*l*

_{g}between

*N*

_{min}abrasive grains and the cross section of the substrate surface in a rectangular area of length

*l*

_{cut}and width

*b*

_{s}onto the front end of the cross section, it can be covered by the width of

*b*

_{s}, as depicted in Fig. 16(c). Here,

*δ*is a proportional coefficient and

*δ*≤ 1. Due to the fact that

*l*

_{g}/

*b*

_{s}is much smaller than 1, according to probabilistic statistical methods [50],

*δ*can be represented as:

*d*

_{g}of grains is much smaller than the radius

*r*

_{g}. The calculation formulas for the CSA

*A*

_{g}of a single effective grain cutting the surface material of a substrate, the total CSA

*A*

_{all}of

*N*

_{min}effective grains cutting the surface material of a substrate in a rectangular area of length

*l*

_{cut}and width

*b*

_{s}, and the residual CSA

*A*

_{r}of a single effective grain cutting the surface material of a substrate can be respectively expressed as follows

*A*

_{eff}of a single effective abrasive grain cutting substrate surface material can be expressed as: