Modelling and experimental study of roughness in silicon wafer self-rotating grinding

Accepted Manuscript Title: Modelling and experimental study of roughness in silicon wafer self-rotating grinding Authors: Jinglong Sun, Pei Chen, Fei Qin, Tong An, Huiping Yu, Baofeng He PII: DOI: Reference:

S0141-6359(17)30388-4 https://doi.org/10.1016/j.precisioneng.2017.11.003 PRE 6684

To appear in:

Precision Engineering

Received date: Revised date: Accepted date:

3-7-2017 2-10-2017 3-11-2017

Please cite this article as: Sun Jinglong, Chen Pei, Qin Fei, An Tong, Yu Huiping, He Baofeng.Modelling and experimental study of roughness in silicon wafer self-rotating grinding.Precision Engineering https://doi.org/10.1016/j.precisioneng.2017.11.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Modelling and experimental study of roughness in silicon wafer self-rotating grinding Jinglong Suna, Pei Chena,b*, Fei Qina,b, Tong Ana,b, Huiping Yua, Baofeng Hec a

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Research Center for Advanced Electronic Packaging Technology & Reliability, College of Mechanical Engineering & Applied Electronics Technology, Beijing University of Technology, China b Beijing Key Laboratory of Advanced Manufacturing Technology, College of Mechanical Engineering & Applied Electronics Technology, Beijing University of Technology, China c Beijing Precision Metrology Laboratory, Beijing University of Technology, China *Corresponding author: Tel: +8601067392760; Fax: +8601067391617; Email: [email protected]; Highlights:



Surface roughness model in silicon wafer self-rotating grinding is established.



The model reveals the effects of processing parameters, abrasive grain size, material

Roughness model is adopted to predict grinding process in-situ and improve grinding

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

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properties and grinding mark geometry on roughness.



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quality.

The model is potentially employed to monitor the industrial wafer grinding process

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and optimize the grinding parameters for the minimization of surface roughness.

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Abstract

Self-rotating grinding is the most widely used technology to thin silicon wafer. The roughness is an

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important indicator of thinning quality and processing accuracy. To get a better grinding quality, rigid control of roughness is required. Although the models of roughness in metal and ceramic machining were extensively studied, mechanism of roughness formation in silicon wafer self-rotating grinding was not well

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understood. In this article, starting from the mechanism of grinding grooves formation, Rayleigh probability density function was used to characterize the depth of grinding grooves. By establishing a relationship between the roughness and the depth of grooves, a theoretical model of roughness was developed. The overlapping effect of abrasive grains and wheel-workpiece deflection were also considered to improve the accuracy of the model. The model could identify the effects of processing parameters, abrasive grain size,

material properties and grinding mark geometry on roughness quantitatively. Verification experiments under seven grinding conditions were performed to validate the theoretical model. The experimental values agreed with the predictive values with less than 20% deviation. Effects of wheel rotation speed, wafer rotation speed, feed rate and wafer radial distance on roughness were discussed in detail.

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Keywords: Silicon wafer; Thinning process; Roughness; Processing parameters; Wafer radial distance. Nomenclature dV Instantaneous material removal volume

Fg Force exerted on the grain

dA Instantaneous area of ground surface

L

d

Nw

Height of isosceles triangle Maximum grinding depth

Wafer rotation speed

ncut The number of grains covering the segment

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dmax

Circumference of the diamond segment

Ra Arithmetical mean roughness

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E(Ac) Expected value of single cross-sectional

Rq

Mean-square-average roughness

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area of grinding groove

Rt

Maximum roughness height

r

Wafer radial distance

Re

Equivalent radius of grains

Rs

Wheel radius

after wheel deformation

Sa

Cutting area without grains overlapping

Expected value of roughness

Seff

Effective area with grains overlapping

Sr

Area of residual material

t

Material removal thickness

W

Width of diamond segment

γ

Abrasive grains volume ratio

area of grinding grooves

Expected value of grinding grooves depth

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E(h)

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E(Atotal) Expected value of total cross-sectional

Eb

Elastic modulus of wheel binder

Wheel feed rate

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f

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E(Ra)

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Etrue(h) Expected value of grinding grooves depth

f / N s Material removal thickness

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Fb

f(h)

Froce exerted on the wheel binder Probability density function of grinding

2rN w / Ns Material removal width

grooves depth

φ

Effective coefficient of abrasive grains

Ns

Wheel rotation speed

H

Hardness of wafer

N

Number of effective cutting grains

ycl Center line

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1.

Introduction Mono-crystalline silicon wafer is widely used as the substrate material in Integrated Circuit (IC)

manufacturing [1]. Usually, the silicon ingots are sliced into silicon wafers by wire sawing [2], and the initial thickness of silicon wafers are a few hundred micrometers. The silicon wafers need to be thinned

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before IC packaging process. The mechanical backside grinding based on wafer self-rotating is mainly used to thin silicon wafer [3, 4]. During grinding process, the interaction between the abrasive grains and wafer surface can cause surface and subsurface damages [5]. The surface and subsurface damages lead to wafer

deformation and low efficiency in subsequent polishing [6]. With increasing demand of ultra-thin and high quality wafers in recent years [7], the ductile mode machining with crack-free surface is more and more

needed to improve wafer surface quality [8]. Generally, surface roughness is an important indicator of the

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grinding quality and performance [9]. It is a valid way to improve grinding quality by reducing surface

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roughness.

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To improve the surface quality of machining workpiece, numerous researchers have made many

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experimental efforts. Some researchers aimed to improve the grinding quality by conducting parametric studies. Pei et al [10] experimentally found that the roughness is strongly influenced by the processing

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parameters, like the wheel speed, wafer speed and feed rate. Luo et al [11] investigated the effects of the

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wheel speed and wafer speed on roughness. They indicated that the wheel speed presents more influential than wafer speed. Zhang et al [12] investigated the effects of the wheel speed and feed rate on roughness of

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silicon wafers. They found that the feed rate is more influential than wheel speed, hence the feed rate should be controlled primarily. Some researchers attempted to improve the machining performance by introducing

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novel machining method. Ultrasonic vibration assisted machining (USVM) is a novel method to improve machining quality, because it can reduce the machining force to be about 1/4 from that without USVM [13]. Aziz et al [14] indicated that the application of ultrasonic vibration in machining of micro hole could

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produce progressive decrease of burr width and height, which promotes a better machining quality. Guo et al [15] suggested that, under identical grinding conditions, the ground surface assisted by ultrasonic vibration is clearly smoother than that generated without vibration. And the surface roughness was improved to 78 nm from 136 nm. Experimental evaluations give direct readout and an insight into the effects of processing conditions on surface quality. However, few study provides the theoretical relationship 3 / 32

between roughness and processing parameters for silicon wafer processing, which limits the in-situ prediction of roughness and process optimization. Some methods, like nonlinear regression, semi analytical and analytical methods were initially used to develop the theoretical models of roughness in metals and ceramics grinding. By using nonlinear regression,

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Feng et al [16] developed an empirical model of roughness for steel turning, and the model could evaluate the roughness under certain cutting parameters. Sahin et al [17] developed an empirical model of roughness for mild steel turning and investigated the effects of turning parameters on roughness. They found that the feed rate is the main influential factor in roughness. He et al [18] adopted a semi analytical method to

predict roughness in diamond turning and conducted experiments to validate the applicability. For the

regression and semi analytical methods, extensive experiments are still needed to be conducted to perfect

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the model of roughness.

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The analytical models considering the multiple processing parameters were extensively investigated in

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ceramic and metal grinding. The processing parameters and wheel structure were considered as the major

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factors in roughness formation. Sato [19] assumed that the active cutting grains have unified height on wheel surface, and the mean value of the grain protrusion heights was used to predict roughness of ground

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wafer. While the random feature of the grains was neglected, the predicted results were not in the same

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order of magnitude as the experimental results. Younis et al [20] statistically proved that the grain protrusion heights have a random feature and conform to Rayleigh probability density distribution. By

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adopting Younis’ theory, Hecker et al [21] pointed that the depths of grinding grooves produced by cutting grains also conform to Rayleigh probability density distribution. Then, an analytical model of roughness

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was developed by simplifying the grooves as triangles in shape. Based on Hecker’s theory, by simplifying the grooves as spheres in shape, Agarwal et al [22] developed a predictive model of roughness without considering the effect of grains overlapping. Agarwal et al [23] also established a predictive model of

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roughness by considering the effect of grains overlapping. Experimental results showed that the model with considering the grains overlapping can improve the accuracy of rouhness prediction. However, the above models ignored the effect of material removal mechanism on roughness. Shao et al [24] proposed a physics-based model of roughness considering the combined effect of brittle and ductile material removal mechanism. The model furtherly improved the accuracy of roughness prediction in ceramic grinding. In 4 / 32

general, these methods give very comprehensive understanding of the mechanism of roughness formation. However, theoretical prediction of roughness for silicon wafer self-rotating grinding is still under research. The wafer self-rotating grinding has more complex interaction of grain and workpiece than metal or ceramic grinding. During wafer grinding, the grinding wheel and wafer rotate simultaneously, and the

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grinding wheel is fed towards wafer surface, as shown in Fig. 1(a) and (b). The wheel plunges into a wafer from edge to center, and then move away from center to edge, the interaction of grains and wafer leads to

material removal. Different from traditional grinding with pre-set grinding depth, Young et al [25] indicated that the grinding depth of self-rotating grinding changes dynamically during grinding, and the grinding marks are similar to spiral lines whose depth, width and interval keep changing either. Therefore, the

aforementioned roughness models cannot be applied to wafer self-rotating grinding, and further study of

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roughness modelling of silicon wafer self-rotating grinding needs to be performed.

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In this article, a predictive model of arithmetic mean roughness, Ra, is developed. The model considers

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the effects of abrasive grain size, processing parameters and grinding grooves distribution on the roughness.

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Grinding grooves on surface were considered as the main factor in roughness formation. After establishing the theoretical model, verification experiments were performed to validate the model. To characterize the

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quality of wafer comprehensively, the arithmetic mean roughness, Ra, the mean-square-average roughness,

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Rq and the maximum roughness height, Rt were all measured and studied for different grinding conditions. Based on the theoretical model and experimental results, effects of various processing parameters on

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roughness were discussed comprehensively. The model is potentially employed to optimize the processing parameters for the minimization of surface roughness in industry. Experimental procedure

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2.

2.1 Wafer grinding Raw single crystal (100) silicon wafers with thickness of 650 µm and diameter of 150 mm were

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adopted for grinding experiments. The diamond cup-type grinding wheel has a circumference (L) of 450 mm, and grinding segment width (W) of 3 mm for both rough and fine grinding. Wafer grinding experiments were conducted on a wafer grinder (JB-802, CETC), as shown in Fig. 1(a). Wafers were firstly thinned to the thickness of 260 µm by rough grinding, with wheel feed rate of 48 µm/min, wheel speed of 5000 r/min and wafer speed of 200 r/min. Since the main purpose of rough grinding is to improve grinding 5 / 32

efficiency, the surface quality is determined by fine grinding. Seven sets of fine grinding experiments were conducted by varying feed rates, wheel speeds and wafer speeds. The grinding depth of each processing parameter is calculated based on our previously published study [26]. The processing parameters and grinding depths are listed in Table. 1. An on-line thickness measurement device was incorporated into the

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grinding system to monitor wafer thickness, and the target thickness is 200 µm for fine grinding.

2.2 Measurements of roughness

For each ground wafer, rectangular samples with dimension of 6 mm×6 mm were cut by laser at the location of radial distance of 20, 35, 50 and 65 mm, as illustrated by Fig. 2(a). Then all samples were

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cleaned by an ultrasonic bath. First, the samples were observed by a FEI Quanta 650 Scanning Electron

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Microscope (SEM) and the clear wafer surface morphologies were obtained. Then Light Scanning Interferometry (Contour GTK-0) with 0.01 nm resolution under vertical scanning mode was used to

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measure roughness. The scanning area was 0.3×0.2 mm2. Then the surface profile in a length of 100 µm

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was extracted from the scanning image by the post-processing software of Vision, as shown in Fig. 2(b) and (c). All of the surface profiles were perpendicular to grinding marks. For each grinding condition, Ra, Rq

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and Rt values were recorded to specify arithmetical mean deviation, root mean square deviation and total

3.

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repeatability.

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peak to valley height, respectively. Three measurements of the same samples were performed to ensure the

Theoretical model development

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3.1 Material removal mechanism and roughness formation Chen et al [27] indicated that the materials are removed in brittle mode with microcracks and craters,

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when the grinding depth is more than the critical depth of cut. Conversely, the materials are removed in ductile mode with regular plastic grooves. Young et al [25] experimentally investigated the critical depth of cut of single crystal silicon, and it was reported in the range of 20-40 nm. Sun et al [26] developed a new analytical model of grinding depth in silicon wafer self-rotating grinding considering the effect of grinding marks. Based on Sun’s analytical model, the grinding depths are calculated and the maximum grinding depths on the periphery of silicon wafer are listed in Table 1. Clearly, the grinding depths are all less than 6 / 32

20 nm, which reveals that the ductile grinding mode is dominant. Fig. 3(a)-(g) shows the morphologies obtained by the SEM from ground wafers under grinding condition (a) to (g). The surface patterns are mainly composed of the grinding grooves, and brittle craters can be barely observed. There are no cracks in subsurface except for pits induced by grinding marks. The

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SEM images validate the ductile grinding mode furtherly. In industry, roughness Ra is widely adopted in the characterization of surface quality. Therefore, a theoretical model of Ra was developed in the following sections. Fig. 4 shows a flow chart of model

development. Based on four assumptions, the interactions of abrasive grains and wafer can be quantitively described. Then the expected value of the depth of grinding grooves Etrue(h) can be calculated. Correlating the expected value of roughness E(Ra) with Etrue(h), and the analytical expression of expected value of

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roughness E(Ra) can be obtained.

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3.2 Basic assumptions

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To describe the mechanism of roughness formation, the grooves generated by grains need to be

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analyzed, as shown in Fig. 3(h). The formation of grooves is the result of material removal by abrasive grains. The characters of the grooves are determined by the shape, the size, the protrusion height, the

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density and other characters of the abrasive grains. To simplify the analysis, certain assumptions need to be

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made [20, 22]:

(1) The grains could be approximated as spheres in shape, hence the cross-section profile of groove

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generated by a single grain is an arc of a circle. For simplicity, the cross-section profile of groove is approximated as semicircle with radius of h, i.e. the depth of grinding groove.

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(2) The protrusion heights of grains have a random feature and conform to Rayleigh Probability distribution. Hence, the depth of grooves also conform to Rayleigh Probability distribution, as shown in Eq. (1).

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2  h 2 / 2     h /  2  e  f h     0





h0

(1)

h0

where f(h) is the probability density function,  is a parameter that defines the probability density function and it depends on the grinding conditions. (3) Grinding grooves overlap each other on either side only once, with the same types of groove, and 7 / 32

at a distance of two thirds of the depth of grinding groove. (4) The roughness is estimated perpendicular to the grinding grooves. 3.3 Depth of grinding grooves Under the above assumptions, the grinding grooves generated by abrasive grains are shown in Fig. 5. ycl

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is the position of center line and the datum line for calculation of roughnes Ra, the areas above and below ycl are equal. Since the random feature of grain-wafer interaction, two types of grooves are generated depending on whether their depth are less or greater than center line ycl, as shown in Fig. 5. The cross-section area of grinding grooves (Ac) can be calculated from the shape of semicircles, by taking the overlapping effect of

grooves into account, the expected value of cross-section area of grinding grooves, E(Ac), can be expressed by Eq. (2).

 2

E h 2 

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E  Ac  

(2)

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where φ is the effective coefficient of grains to illustrate the overlapping effect.

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By multiplying the total number of effective cutting grains (N) by E(Ac), the expected value of the total area

 2

 

E h2 

(3)

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E  Atotal  NE Ac   N

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of engagement projected onto a plane perpendicular to grinding grooves, E(Atotal), can be obtained as below,

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E(Atotal) is the expected value of the total area removed by grains, as shown in Fig. 6 (a) and (b). At certain location of wafer radial distance r, as shown in Fig. 6(c), the volume of material removal (dV) can be estimated by a volume of a cuboid, and expressed by multiplying E(Ac) by infinitesimal increment,

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dr, as below,

dV  E Atotaldr

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(4)

From the macroscopic point of view, as shown in Fig. 6(c), the relative motion between wheel and wafer creates an instantaneous area on ground surface dA. Therefore, dV can be calculated by the product of dA

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and the removal thickness t, as shown below,

dV  dA  t 

2rN w dr f  Ns Ns

(5)

where dA= 2r  dr  N w / Ns , t=f/Ns, f is the wheel feed rate, Nw is the wafer rotation speed and Ns is the wheel rotation speed. At certain radial distance r, 2rN w / Ns is the material removal width and f / N s is the 8 / 32

material removal thickness. Based on the conservation of volume, Eq. (4) equals Eq. (5), a relationship can be established as below,

E  Atotal  dr 

2r  N w f  dr Ns Ns

(6)

Substituting Eq. (3) into Eq. (6) and rearranging as

4r  N w f  2 N Ns

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 

E h2 

(7)

According to the theoretical investigation by Sharp et al [28], the number of effective cutting grains N can be estimated by Eq. (8).

3L  W  γ d max 4π  Re3

N 

(8)

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where L is the circumference of the diamond segment on the wheel, W is the width of wheel segment, γ is

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the grain volume ratio, which is 0.25 [29], Re is the radius of abrasive grain and dmax is the maximum

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grinding depth.

A theoretical model of the maximum grinding depth considering the effect of grinding marks could be

       

0.4

(9)

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d max

   r  f  Nw  4.14 Re   2  L  W    N 2 1  r s   8Rs2  

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derived from our previously published work [26], as expressed in Eq. (9).

Substituting Eq. (9) into Eq. (8), and then into Eq. (7) gives

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 

E h2

  1 2  r  f  Nw  4.05 Re   2   L  W    N 2 1  r s   8Rs2  

       

0.6

(10)

On the other hand, E(h2) and E(h) can be calculated by the integral of Rayleigh probability [21], as below

 

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E h 2   h 2 f hdh  2 2 

(11)

0

E h    h f h dh  



(12)



2 Substituting Eq. (12) into Eq. (11) gives 0

E h  

E h 2 

2 Correlating Eq. (13) with Eq. (10), the expression of E(h) can be obtained as 9 / 32

(13)

   r  f  Nw 1 E h   1.78 Re     r2 2  L  W    N s 1  2  8 Rs 

       

0.3

(14)

3.4 Evaluation of the effective coefficient of grains φ

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The effective coefficient φ in Eq. (14) should be determined by considering the overlapping effect of

abrasive grains. There are two methods to evaluate the effective coefficient, i.e. analytical and experimental methods. Agarwal et al [23] conducted a series of experiments to calculate the effective coefficient of

grains, and it was found to be approximately 93%. Sharp et al [28] estimated the effective coefficient by the method of coin-tossing analogy, and it was about 66%. The experimental evaluations are time consuming

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and cannot reveal the mechanism of grains overlapping. The method of coin-tossing analogy is a statistical analysis and does not reflect the grinding mechanism.

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In this article, an analytical method is used to calculate φ and the overlapping effect is well considered.

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By considering the overlapping effect, the effective area removed by a single grain is marked as Seff. If no

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overlapping exists, the area removed by a single grain is marked as Sa. Hence, φ is expressed as the ratio of Seff to Sa. Fig. 7 shows a schematic of overlapped grains, W is the width of wheel segment, λ is the fraction,

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height of residual material.

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h is the radius of grains i.e. the depth of grinding grooves, Sr is the area of residual material and d is the

Assuming that there is no overlap or vacancy of grains in the width direction of wheel segment (W), the

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number of cutting grains ncut=W/2h was needed to cover the whole width W. Actually, the grains overlap each other, for the same number of grains, ncut，they can only cover a length of λW, and λ<1. Robbins [30]

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estimated λ by the method of coin-tossing analogy, as follows.



n

2h  cut   W Rearranging and expanding Eq. (15) yields

  1  1 

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(15)

 2h  ln1     ncut ln1    W

(16)

Because 2h<
2h  2h  ncut ln1    ncut  1 W  W

(17) 10 / 32

Substituting Eq. (17) into Eq. (16), and λ=0.632. To obtain φ, the area of residual material Sr in the region of red shadow also needs to be determined, as shown in Fig. 7. The residual region is simplified as isosceles triangle in shape, and its base and height are 2/3h and d, respectively. The height d can be calculated by Eq. (18).

2 2 h  0.057h 3

(18)

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d  h  h 2  1 3h 2  h 

Therefore, for a grain, the area of residual material can be estimated by Eq. (19). Sr 

1 2  h  0.057h  0.019h 2 2 3

(19)

To obtain the effective coefficient φ, we need to find the effective area Seff removed by a grain. As shown in Fig. 7, within the length of λW, subtracting residual material ncutSr from a rectangle λWh, and the

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effective area removed by ncut cutting grains is obtained. Hence, for a single grain, the effective cutting area

Wh  ncut S r ncut

Wh  ncut Sr ncut

 1.264h 2  0.019h 2  1.245h 2

(21)

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Seff 

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Substituting λ, Eq. (19) into Eq. (20) gives

(20)

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Seff 

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Seff can be expressed as below

Sa 

 2

h2

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For a single grain, the cutting area Sa without overlap of grains can be expressed as below (22)

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The effective coefficient φ is the ratio of Seff to Sa, hence φ=0.79. Substituting φ=0.79 into Eq. (14) gives

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   r  f  Nw E h   2.25 Re  2  L  W    N 2 1  r s   8Rs2  

       

0.3

(23)

3.5 Wheel-wafer contact deflection analysis \ Eq. (23) does not consider the contact deflection of wheel-wafer interaction. Agarwal et al [31]

indicated that the elastic contact deflection of wheel-workpiece will influence the grinding depth. And workpiece surface quality is also affected by the contact deflection of wheel-workpiece. Johnson [32] pointed that the elastic deflection of the grain into the binder can be approximated by considering a rigid 11 / 32

grain impressing into an elastic body, as shown in Fig. 8. The binder has a modulus Eb and a Poisson’s ratio ʋ. Thus, the internal force of the binder is that Fb=ηδ, where the stiffness η=2EbRe/(1-ʋ2) and δ is the difference between E(h) and Etrue(h), where E(h) is the depth without wheel deflection, and Etrue(h) is the depth with wheel deflection, as shown below, Fb    2Eb Re /(1 -  2 )Eh  Etrue h

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(24)

Since the internal force of binder, Fb, is equal to the applied normal grinding force, Fg, the latter can be

calculated by Fg=HAcontact, where H is the workpiece hardness, and Acontact is the area of grain-wafer contact. Acontact can be calculated by Re2 sin 2  . In terms of geometry, sinθ is a function of Etrue(h), as expressed below,

Fg  HRe2 sin 2     2 HRe Etrue h   

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2

(25)

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  E ( h)   HRe2 1  1  true  Re    

1

E h   2 HRe



E h 



1   1  2

 EH

b

(26)

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Etrue h  

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Because Eq. (25) equals Eq. (24), hence Etrue(h) can be expressed as a function of E(h), as below,

1

   r  f  Nw 2 H   L  W    N 2 1  r s  2 Eb   8Rs 

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Etrue h   2.25Re

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Substituting Eq. (26) into Eq. (23) gives



1  1

2



       

0.3

(27)

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Compared with Eq. (23), Eq. (27) considered the binder deflection by introducing material properties of the binder, such as H, Eb and ʋ.

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3.6 Roughness Ra calculation The roughness Ra, is defined as the arithmetical average of the absolute values of the deviations of the

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surface profile height from the center line within the sampling length l. Therefore, roughness Ra can be expressed by Eq. (28) [21, 22]. 1 l y  ycl dl l 0 where ycl denotes the center line as mentioned before, and y denotes the random profile height. Ra 

(28)

Based on the assumption (2), the grooves can be divided into two types, either less or greater than center line ycl, as shown in Fig. 5. Because they contribute to the roughness differently, Hecker et al [21] 12 / 32

and Agarwal et al [22] suggested that the expected value of roughness E(Ra) can be expressed in two segments as below.

ERa   p1ERa1  p2 ERa2 

(29)

where p1 and p2 is the probability, E(Ra1) and E(Ra2) are the expected values of roughness induced by

SC RI PT

grinding grooves with depth less and greater than center line ycl, respectively. p1 and p2 are calculated by Rayleigh probability distribution [22], as shown in Eq. (30a) and (30b).

P1   f hdh  1  e ycl ycl

2

2 2

 1  e 4 h  ycl

0

P2   f hdh  e ycl 

2

y cl

(30a)

2 2

 e  4

h  ycl

(30b)

E(Ra1) and E(Ra2) are also given in reference [23], as shown in Eq. (31a) and (31b).

 4

E h1   0.70

U

E Ra1   ycl 

N

  y2  y2  E h2   E  h2 sin 1 1  cl2   yclE  1  cl2   0.40   4 h2  h2      Substituting Eqs. (27), (30a), (30b), (31a) and (31b) into Eq. (29) gives



(31b)

1



1  1

2



   r  f  Nw 2 H   L  W    N 2 1  r s  Eb  8Rs2  

       

0.3

(32)

TE

D

E Ra   0.45Etrue h   1.013Re

M

A

E Ra 2   ycl 

(31a)

Eq. (32) quantitively describes the expect value of roughness with eleven parameters. The processing parameters f, Nw, Ns and geometry parameters of the wheel L, W, γ, the grain size Re, material properties H,

Results and discussion

CC

4.

EP

Eb, υ and the location parameter r are included in the model.

4.1 Experimental results Fig. 9(a)-(g) illustrates one of surface images for each grinding condition from three-dimensional (3D)

A

WLI. Grinding grooves are clearly observed on all the wafer surfaces and paralleled with each other in a local area of 315 μm by 235 μm. One surface profile in the length of 100 μm for each condition is shown in Fig. 9(h). The surface profiles clearly show that the depth of the grooves is repeated periodically with some random features. The experimental values of roughness calculated from 100 μm surface profile under various grinding 13 / 32

parameters are listed in Table 2. By increasing radial distance from 20 to 65 mm (from center to edge), Ra, Rq and Rt are constantly increased. For all processing conditions, Ra and Rq are in the range of 25-30 nm with a standard deviation (SD) of 1.0-2.5 nm, Rt is 100-170 nm with a SD of 2.0-4.0 nm. The effects of the grinding parameters on the roughness Ra, Rq and Rt were shown in Fig. 10. Ra, Rq and

SC RI PT

Rt have the same trend when grinding conditions or radial locations change. When the wheel feed rate (Fig. 10(a)) and wafer rotation speed (Fig. 10(c)) increase, all of Ra, Rq and Rt increase. The reason could be that the increasing of wheel feed rate and wafer rotation speed leads to greater grinding depth. When the wheel rotation speed (Fig. 10(b)) increases, Ra, Rq and Rt decrease. Since Rt is used to specify maximum

roughness height of the ridge between grooves, Rt includes less information than Ra, Rq, and the value of Rt is more scattered. Ra is the average roughness, therefore, the effect of a single peak or valley will have

U

insignificant influence on the value, and Rq has the effect of giving extra weight to the numerically

N

higher values of the surface. Comparing the SDs of Ra with Rq, from 28 sets of data, Ra has less SD than

M

A

Rq in 19 sets of data. Therefore, Ra is a better choice to represent roughness than Rt and Rq for this case.

D

4.2 Comparisons between experimental and predictive results The values of Ra are calculated by Eq. (32) for the grinding conditions (a)-(g) theoretically, as listed in

TE

Table 2. Fig. 11 shows the accuracy of theoretical predictions. If the theoretical values perfectly match the experimental value, the data points will locate in the diagonal line. A 20% deviation band is enclosed by

EP

two dash lines. Most of the datas located within the band, and below the diagonal line, which means the theoretical values are smaller than experiment values with less than 20% deviation. Besides all of

CC

influential parameters considered by Eq. (32), the real grinding process could be more complex than the assumed conditions. Residual chipping on wafer surface and vibration of spindle reported by Chen et al.

A

[33] could be also the possible reasons. More factors might be taken into account if higher accuracy is expected to be achieved. By examining each grinding condition in detail, Fig. 11 (a) and (b) have less deviation than Fig. 11 (c) and (d). The edge (Fig. 11(c) and (d)) may suffer more vibration than center (Fig. 11(a) and (b)) due to high wheel and wafer speed, which are not included in the current model. Therefore, the uncertain factors could 14 / 32

create more deviation. The maximum deviation is 24.6%, the minimum deviation is 11.8%, and the mean deviation is 15.6%. Therefore, with the same trend and relatively reasonable deviation, the theoretical model can effectively predict the roughness. To verify the applicability of the model, the experimental results obtained by Pei et al [10] are applied

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to validate this model. In Pei’s work, the diamond cup wheel of 280mm in diameter with #2000 grain size is adopted. Single crystal silicon wafers of 200 mm in diameter were used for self-rotating grinding. The roughness values near the wafer edge were measured by a surface profiler (KLA-Tencor). The detailed

grinding parameters and roughness values are listed in Table 3. The maximum error is 34.2%, the minimum

U

error is 8.8%. Within the grinding conditions, the roughness model is applicable to Pei’s work.

N

4.3 Parametric study

A

Eq. (32) considers the effects of the grain size, material properties, processing parameters and geometry

M

parameters of the wheel L, W, γ on Ra. Since the geometry parameters of wheel are fixed for specific grinding process, the processing parameters and radial distance are the most interesting parameters.

D

The effects of wafer radial distance (r) on Ra are represented under different feed rates (Fig. 12(a)),

TE

different wheel speeds (Fig. 12(b)) and different wafer speeds (Fig. 12(c)). It can be observed that, the roughness increase when the wafer radial distance increases. Chidambraram et al [34] indicated that, for the

EP

region with same area, the density of grinding marks decreases from wafer center to edge. Since less grains pass the same area, to keep the same removal thickness, each abrasive grain needs to remove more

CC

materials at the edge, which means the grinding depth should increase [25, 26], thus the surface becomes more rough.

The relationship between the roughness and the wheel feed rates is also illustrated in Fig. 12 (a). When

A

Ns and Nw keep constant, Ra increases with the increasing of wheel feed rate. A higher feed rate means each grinding grain needs to remove more materials in the same time, then the plastic deformation zone under grain would be deeper, and the left grinding grooves would also be deeper. Deeper grinding grooves would contribute to a larger Ra. Considering the relative motion between grinding wheel and silicon wafer, the relative speed could affect 15 / 32

Ra in two ways. In one hand, Luo et al [11] indicated that a higher relative speed would create more grinding marks in wafer, therefore the interval distance of two adjacent grinding marks could decrease, which is beneficial for surface quality. In the other hand, more grains participate in grinding process by removing certain amount of materials, which means that each grain will make less damage, thus the surface

SC RI PT

quality could be better. Fig. 12(b) plots that the effects of wheel rotation speed on Ra, when the wheel feed rate and wafer rotation speed are kept constant. It is clearly seen that, at the same radial distance, with the wheel rotation speed increases, which means the relative speed increases, Ra decreases. However, wheel

speed is limited by the capability of the motor and shaft, also high speed always comes with high vibration, which may bring bad surface quality. 4500-5500 r/min is already considered as a high speed grinding

compared with the range of 1500-3400 r/min adopted by Pei et al [35]. Decreasing Nw is another effective

U

way to achieve a lower Ra, as shown in Fig. 12(c). Fig. 12(c) plots that the effects of wafer rotation speed on

N

Ra, when the wheel feed rate and wheel rotation speed are kept constant. With the wafer rotation speed

A

decreases, which means the relative speed increases, the roughness decreases. However, decreasing Nw

M

means less repeating cutting on wafer surface, therefore the form accuracy may be affected negatively, which is harmful to the following IC manufacturing processes.

D

By considering feed rates, wheel speeds and wafer speeds as the variables, and other parameters are

   

0.3

(33)

EP

 f  Nw E Ra   C  2  Ns

TE

constant. Eq. (32) can be simplified as,

where C=461.06 nm for this case. Eq. (33) illustrates that increasing Ns should be the most effective way

CC

to achieve a good surface quality. If Ns cannot be improved any more, lower f and Nw could also be favorable. The interactive effect of feed rate, wheel speed and wafer speed on Ra (at the radial distance of 65 mm) is illustrated by a 4D contour map in Fig. 13. As shown in Fig. 13, the feed rate ranges from 3 to 15 μm/min.

A

At a lower feed rate, whatever the wheel speed and wafer speed change, the roughness is always at a lower level. Since wheel feed rate is dominant in materials removal rate. If a better quality of surface is desired, the grinding efficiency needs to be lower. Since the wheel speed varies in a narrow range from 4500-5500 r/min which is limited by the capacity of grinding machine, the effect of wheel speed is also restrained. Overall, to control Ra within a certain range, the 4D contour map could give a clear illustration on the 16 / 32

selection of grinding parameters. 5.

Conclusions A theoretical predictive model of roughness (Ra) for wafer self-rotating grinding is developed. The

grinding grooves are recognized as the major contributor to Ra. Rayleigh probability density function was

SC RI PT

used to describe the depth distribution of random grinding grooves. Besides processing parameters, the model also considers the effects of abrasive grains overlapping and wheel-workpiece deflection. Validation

experiments of seven sets of variable grinding conditions are conducted. The measurements of Ra, Rt and Rq indicate that Ra is less scattered with sufficient topography information, then Ra is adopted to represent the surface quality of ground wafer. The predictive value is under the same change trend as the experimental value with less than 20% deviation. Most of the predictive values are smaller than experimental results,

U

which indicates that other minor contributors to Ra are not considered in this model.

N

The Ra distributes unevenly in radial direction of wafer. The model indicates Ra is proportional to r0.3,

A

which means the maximum Ra will appear at the edge of wafer. The feed rate, wheel speed and wafer speed

M

are the direct influential factors, which can be dynamic tuning during grinding. By simplifying the model, Ra is found to be proportional to (f Nw/Ns2)0.3. To develop a grinding machine with high wheel speed is the

D

most effective way, and also high-speed grinding is the current trend of machine development. The feed

TE

rate contributes to the depth of the groove, which affects Ra directly. The relative rotating speed affects both the density and the depth of the grinding grooves. Since Ns is limited by machine capacity, and Nw affects

EP

the surface form accuracy, f, Nw and Ns need to be optimized interactively. Further studies should aim at model application in industry, and model refinement by considering more factors, like vibration etc.

CC

Acknowledgments

The authors would like to deeply appreciate the financial support from the National Natural Science

A

Foundation of China (Grant no. 11502005 and 11672009) and National Science and Technology Major Project (Grant no. 2014ZX02504-001-005).

17 / 32

References [1] P. Van-Zant, Microchip fabrication, America: McGraw-Hill Professional 2004. [2] Wu H. Wire sawing technology: A state-of-the-art review. Precis. Eng. 2016; 43: 1-9. [3] Pei ZJ, Fisher GR, Liu J. Grinding of silicon wafers: A review from historical perspectives. Int. J. Mach. Tools Manuf. 2008; 48(12): 1297-1307.

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[4] Zhou LB, Eda H, Shimizu J. State-of-the-art technologies and kinematical analysis for one-step finishing of Φ300 mm Si wafer. J. Mater. Process. Technol. 2002; 129(1): 34-40.

[5] Inoue F, Jourdain A, Peng L, Phommahaxay A, Devos J, Rebibis KJ, Miller A, Sleeckx E, Uedono A. Influence of Si wafer thinning processes on (sub) surface defects. Appl. Surf. Sci. 2017; 404: 82-87. [6] Gao S, Dong Z, Kang R, Zhang B, Guo DM. Warping of silicon wafers subjected to back-grinding process. Precis. Eng. 2015; 40: 87-93.

[7] Kripesh V, Yoon SW, Ganesh VP, Khan N, Rotaru MD, Fang W, Iyer MK. Three dimensional system in package using stacked silicon platform technology. IEEE. Trans. Adv. Packag. 2005; 28(3):

U

377-386.

N

[8] Wu H, Melkote SN. Study of ductile-to-brittle transition in single grit diamond scribing of silicon: application to wire sawing of silicon wafers. J. Eng. Mater. Technol Trans ASME. 2012; 134(4):

A

041011(1-8).

M

[9] Zhang SJ, To S, Wang SJ, Zhu ZW. A review of surface roughness generation in ultra-precision machining. Int. J. Mach. Tools Manuf. 2015; 91: 76-95. [10] Pei ZJ, Strasbaugh A. Fine grinding of silicon wafers: designed experiments. Int. J. Mach. Tools

D

Manuf. 2002; 42(3): 395-404.

[11] Luo SY, Chen KC. An experimental study of flat fixed abrasive grinding of silicon wafers using

TE

resin-bonded diamond pellets. J. Mater. Process. Technol. 2009; 209(2): 686-694. [12] Zhang ZY, Huo FW, Wu YQ, Huang H. Grinding of silicon wafers using an ultra fine diamond wheel of

EP

a hybrid bond material. Int. J. Mach. Tools Manuf. 2011; 51(1): 18-24. [13] Aziz M, Ohnishi O, Onikura H. Novel micro deep drilling using micro long flat drill with ultrasonic vibration. Precis. Eng. 2012; 36: 168-174.

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[14] Aziz M, Ohnishi O, Onikura H. Innovative micro hole machining with minimum burr formation by the use of newly developed micro compound tool. J. Manuf. Processes. 2012; 14: 224-232.

A

[15] Guo B, Zhao QL. Ultrasonic vibration assisted grinding of hard and brittle linear micro-structured surfaces. Precis. Eng. 2017; 48: 98-106.

[16] Feng CX, Wang X. Development of empirical models for roughness prediction in finishing turning. Int. J. Adv. Manuf. Technol. 2002; 20(5): 348-356. [17] Sahin Y, Motorcu AR. Surface roughness model for machining mild steel with coated carbide tool. Mater. Des. 2005; 26(4): 321-326. [18] He CL, Zong WJ, Cao ZM, Sun T. Theoretical and empirical coupled modeling on the surface roughness in diamond turning. Mater. Des. 2015; 82 216-222. 18 / 32

[19] Sato K. On the surface roughness in grinding technology. Reports of Tokohu University. 1995; 20: 59-70. [20] Younis MA, Alawi H. Probabilistic analysis of the surface grinding process. Trans. Can. Soc. Mech. Eng. 1984; 8(4): 208-213. [21] Hecker RL, Liang SY. Predictive modeling of surface roughness in grinding. Int. J. Mach. Tools Manuf. 2003; 43(8): 755-761.

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[22] Agarwal S, Rao PV. A probabilistic approach to predict roughness in ceramic grinding. Int. J. Mach. Tools Manuf. 2005; 45(6): 609-616.

[23] Agarwal S, Rao PV. A new roughness prediction model for ceramic grinding. Proc. Inst. Mech Eng Part B J. Eng Manuf. 2005, 219(1): 811-821.

[24] Shao YM, Li BZ, Steven YL. Predictive modeling of surface roughness in grinding of ceramics. Mach. Sci. Technol. 2015; 19: 325-338.

[25] Young HT, Liao HT, Huang HY. Novel method to investigate the critical depth of cut of ground silicon

U

wafer. J. Mater. Process.Technol. 2007; 182(1): 157-162.

[26] Sun JL, Qin F, Chen P, An T. A predictive model of grinding force in silicon wafer self-rotating

N

grinding. Int. J. Mach. Tools Manuf. 2016; 109: 74-86.

A

[27] Chen JB, Fang QH, Wang CC, Du JK, Liu F. Theoretical study on brittle–ductile transition behavior in elliptical ultrasonic assisted grinding of hard brittle materials. Precis. Eng. 2016; 46: 104-117.

M

[28] Sharp KM, Miller MH, Scattergood RO. Analysis of the grain depth-of-cut in plunge grinding. Precis. Eng. 2000; 24(3): 220-230.

D

[29] Zhou L, Tian YB, Huang H, Sato H, Shimizu J. A study on the diamond grinding of ultra-thin silicon wafers. Proc. Inst. Mech. Eng. Part B J. Eng. Manuf. 2012; 226(1): 66-75.

TE

[30] Robbins HE. On the measure of a random set II. Ann Math Stat. 1996; 16: 342-347. [31] Agarwal S, Rao PV. Predictive modeling of undeformed chip thickness in ceramic grinding. Int. J. Mach. Tools Manuf. 2012; 56: 59-68.

EP

[32] Johnson KL. Contact mechanics, New York: Cambridge University Press. 1985 [33] Chen JB, Fang QH, Li P. Effect of grinding wheel spindle vibration on surface roughness and

CC

subsurface damage in brittle material grinding. Int. J. Mach. Tools Manuf. 2015; 91: 12-23. [34] Chidambraram S, Pei ZJ, Kassir S. Fine grinding of silicon wafers: a mathematical model for grinding marks. Int. J.Mach. Tools Manuf. 2003; 43(15): 1595-1602.

A

[35] Pei ZJ. A study of surface grinding of 300 mm silicon wafers. Int. J. Mach. Tools Manuf. 2002; 42(3): 385-393.

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SC RI PT

(a)

(b)

A

CC

EP

TE

D

M

A

N

U

Fig.1 Schematic of self-rotating grinding mechanism: (a) Experimental set up of wafer grinding; (b) Illustration of the rotating wafer and wheel

20 / 32

15 mm

15 mm 20 mm

(a)

Condition (a) Radial distance 65 mm

0.2

Height (µm)

U

0.1

SC RI PT

6 mm 6 mm

15 mm

0.0

(b)

M

A

N

100 µm

-0.1 -0.2 0

20

40

60

80

100

Sample length (µm) (c)

A

CC

EP

TE

D

Fig.2 Measurements of surface roughness (a) Dicing positions; (b) Surface morphologies; (c) Surface profiles height

21 / 32

SC RI PT U N A M D

Radial distance 65 mm

A

CC

EP

TE

Fig.3 (a) to (g) SEM surface morphologies of wafer under grinding conditions (a) to (g); (h) Schematic of grinding grooves formation.

22 / 32

Basic assumptions (1) The grinding grooves (2) The depth of grooves

(3) Grinding grooves overlap (4) The roughness is estimated each other on either side perpendicular to the grinding conforms to Rayleigh grooves probability density function only once

SC RI PT

are approximated as semicircles in shape

Method of volume conservation Expect value of grooves depth, E(h) Efficient coefficientof φ

Effect of wheel deflection, δ

Expect value of the depth of true grooves considering wheel deflection, Etrue(h) Definition of Ra Expect value of roughness, E(Ra)

U

)

Fig.5 Profile of wafer surface (not to scale)

A

CC

EP

TE

D

M

A

N

Fig. 4 Process flow of roughness model derivation (In the flow chart, E(h) is the expect value of depth of grinding grooves, Etrue(h) is the expect value of depth of grinding grooves after wheel deformation and Ra is the arithmetic mean roughness)

23 / 32

Abrasive grains

Wafer

Chipping

SC RI PT

dr Abrasive grains Grinding marks

Expected area E(Ac)

Enlarged

(b)

(a)

Wafer

dr

dA

U

r

N

t

A

CC

EP

TE

D

M

A

(c) Fig.6 Expression of material removal behavior: (a) Material removal along grinding marks; (b) Enlarged illustration of grains cutting; (c) Material removal along radial direction related to processing parameters.

24 / 32

SC RI PT U

N

Fig.7 Schematic of overlapped grains

A

Fb=ηδ

M

Binder

θ

TE

D

Re

Acontact

Zs

δ Fg=H Acontact Fig.8 Geometry of wheel binder deflection

A

CC

EP

E(h) Etrue(h)

25 / 32

Z1 Z

( 235µm

(b) 315 µm

Grinding marks

(e)

235µm

(c) 315 µm

Grinding marks

Grinding marks

235µm

315 µm

(f) 315 µm

Grinding marks

235µm

235µm

(g) 315 µm

(d)

235µm

315 µm Grinding marks

SC RI PT

(a) 315 µm a)

235µm

Grinding marks

M

A

N

U

Grinding marks

D

(h)

A

CC

EP

TE

Fig.9 Surface morphologies at r=65 mm for grinding conditions (a) to (g) respectively; (h) Surface profiles for grinding conditions (a) to (g).

26 / 32

SC RI PT

TE

D

M

A

N

U

(a)

EP

(b)

(

A

CC

c)

(c) Fig.10 Surface roughness Ra, Rq, Rt of different grinding parameters (a) Various feed rates; (b) Various wheel speeds; (c) Various wafer speeds 27 / 32

SC RI PT

(a) Radial distance 20 mm

(c) Radial distance 50 mm

M

A

N

U

(b) Radial distance 35 mm

(d) Radial distance 65 mm

A

CC

EP

TE

D

Fig.11 Comparision of experimental and theoretical values

28 / 32

Experimental roughness values

35 30 25 20 15 10 5 80

Theoretical roughness values

60 40 20

Wafer radial distance (mm)

0

4

2

8

6

12

10

14

16

Feed rate (µm/min)

(a)

U

f=9 µm/min Nw=150 r/min

Experimental roughness values

N

35 30

A

25 20 15

M

Surface roughness Ra (nm)

SC RI PT

Surface roughness Ra (nm)

Ns=5500 r/min Nw=150 r/min

10 80 60

Theoretical roughness values

D

40

5,500 5250

5,000

20 0

TE

4750

Wafer radial distance (mm)

(b)

4,500

Wheel rotation speed (r/min)

f=9 µm/min Ns=5500 r/min

Experimental roughness values

Surface roughness Ra (nm)

A

CC

EP

35 30 25 20 15

10 80 60

Theoretical roughness values 180

40

200

160 140

20

Wafer radial distance (mm)

120 0

100

Wafer rotation speed (r/min) (c) Fig.12 Experimental and theoretical roughness with different radial distance (a) Various feed rates; (b) Various wheel rotation speeds; (c) Various wafer rotation speeds 29 / 32

Ra (nm)

32

28 5250 26 5,000

SC RI PT

Wheel speed, Ns (r/min)

30 5,500

24

4750

22

4,500 200

20

180 160 140

12

10 8

120 100

4 2

16

Wheel feed rate, f (µm/min)

U

Wafer speed, Nw (r/min)

6

14

A

CC

EP

TE

D

M

A

N

Fig.13 Correlation effect among feed rate, wheel speed and wafer speed

30 / 32

18 16

Table 1 Processing parameters Feed rate

Wheel speed

f (μm/min)

Wafer speed

Ns(r/min)

Grinding depth

Nw(r/min)

(a)

3

5500

150

(b)

9

5500

150

(c)

15

5500

150

(d)

9

4500

150

(e)

9

5000

150

(f)

9

5500

100

(g)

9

5500

200

(nm)

SC RI PT

Conditions

7.47

11.59

14.22 13.61 12.51 9.86

13.00

Wheel parameters: Wheel mesh size, 1200#; Abrasive grain size (Re), ~8 µm;

U

Wheel binder modulus of elasticity, Eb=70 GPa; Poisson’s ratio, ʋ=0.33.

Experimental values(nm)

(mm)

Ra

A

Radial distance Conditions

M

98.12±2.25

11.56

35

16.47±1.54

20.17±2.35

108.52±3.54

13.67

50

18.21±1.11

22.56±1.52

123.57±2.84

15.21

20.12±2.02

25.36±1.92

135.90±3.12

16.46

18.29±1.95

22.19±2.14

115.17±2.85

16.07

22.25±1.42

24.82±1.83

128.87±3.02

18.99

24.49±1.82

28.34±2.05

145.67±2.93

21.16

26.70±1.34

30.23±2.01

158.25±3.12

22.88

20

22.57±2.12

26.07±1.92

135.27±3.21

18.73

35

26.35±1.72

30.22±2.12

149.62±3.42

22.15

50

29.40±2.01

34.12±1.89

165.50±3.54

24.80

65

30.32±1.68

36.26±1.95

175.40±3.68

26.68

20

21.29±1.85

25.34±1.76

130.67±2.88

18.13

35

25.37±1.32

30.02±2.08

142.27±3.42

21.44

50

27.09±2.34

32.54±2.24

158.17±3.55

23.87

65

30.34±2.67

35.45±1.98

172.55±3.78

25.82

20

20.29±1.05

23.86±1.75

120.17±2.16

17.02

35

23.25±1.92

27.44±2.02

134.37±2.98

20.13

50

25.49±2.43

29.12±1.96

147.17±3.24

22.40

65

28.70±1.96

32.05±2.18

162.25±3.75

24.24

EP

CC A

TE

35

65

(e)

Ra

19.25±1.76

50

(d)

Rt

15.35±1.25

20

(c)

Theoretical values(nm)

20

65

(b)

Rq

D

(a)

N

Table 2 Experimental roughness under various grinding parameters (nm)

31 / 32

(g)

17.34±1.14

21.13±1.56

112.52±2.68

14.23

35

20.55±1.56

24.32±1.86

128.52±2.87

16.84

50

22.19±1.78

26.41±2.18

139.57±3.24

18.73

65

24.32±2.04

29.54±2.12

152.90±3.16

20.27

20

21.17±2.44

25.02±1.94

129.67±2.84

17.52

35

25.15±1.68

29.38±1.79

145.27±3.66

20.72

50

27.41±2.28

32.02±1.95

154.17±3.21

23.07

65

29.12±1.86

34.92±2.16

167.55±3.04

Table 3 Grinding conditions and roughness values Wheel speed

Wafer speed

Experimental

f (μm/min)

Ns(r/min)

6

2175

40

12.5

6

4350

40

10.3

6

2175

590

16.4

6

4350

590

16.8

18

2175

40

18

4350

40

18

2175

590

18

4350

590

values, (nm)

Deviation

8.8%

8.5

17.4%

20.9

27.4%

14.5

13.7%

11.1

14.9

34.2%

11.7

9.0

23.1%

18.5

23.5

27.0%

17.7

20.1

13.6%

A

A

CC

EP

TE

D

M

by Eq. (48), (nm) 13.6

N

Nw(r/min)

24.95

Theoretical values

U

Feedrate

SC RI PT

(f)

20

32 / 32