Normal and Tangential Force Modeling and Control of Single Crystal Silicon Using Wire Saw Velocity Reciprocation

: Single crystal silicon wafer is widely used as the substrate material for integrated circuits, and the wafer is cut by wire saw with fixed diamond abrasive owning to the wire saw ’ s narrow kerf, low cutting force. The cutting force changes during the process operation since the wire change motion direction continuously (i.e., reciprocation) in order to effectively leverage the wire, even if the contact length of wire and part and wire saw tension are fixed, leading to wire saw breakage, wafer collapse, and inferior surface roughness. This paper established the model of cutting force that includes normal force and tangential force, and design the controller for regulating the force via wire velocity control, and the controller obtains the relationship between the normal force and commanded wire velocity. A PI controller experimental studies for Single crystal Silicon wafer wire saw machining are conducted. The results show the controller can well regular the cutting force generated when wire saw machines silicon and on that the wire saw machining process with PI control can decrease wafer surface roughness as compared to the


Introduction
Hard and brittle materials such as single crystal silicon and sapphire are being used more and more in integrated circuitry and photoelectricity.
However, the processing requirements, such as thinner wafer thicknesses, reduced warping and reduced total thickness variation, are becoming more stringent. Fixed diamond abrasive wire saw machining is used for cutting hard and brittle materials due to its flexibility, narrow kerf and low cutting force. The cutting force changes during the process operation due to changes in the contact length of part and wire and wire direction, variations in the process parameters, wire wear, etc. This change in cutting force can lead to wire saw breakage, wafer collapse and increased wafer surface roughness. Therefore, controlling the cutting force is vital to improving operation productive and wafer quality.
Wire saw machining has been developed for slicing hard and brittle materials into thin wafers since the 1990s. Clark et al. [1] observed that the free abrasives in slurry between the wire and part, making hard and brittle materials difficult to machine due to the decreased cutting velocity and uneven wafer thickness resulting from wire wear. Ishikawa et al. [2] observed that when a wafer diameter is close to 6 inch, wire sawing with loose abrasives increases the cutting time as it is difficult to supply abrasives to the wafer center. Watanabe et al. [3] discussed the properties of the wafers sliced using a fixed abrasive wire saw.
Yin Y. et al. [4] , established a mathematical model of diamond wire sawing based on the machining mechanism of brittle material removal and surface generation, and the numerical calculation of the sawing process was carried out. Chung C. et al. [5] established a model to simulate the distribution of diamond grains. The simulation demonstrate that a higher density distribution reduces the rate of material removal because the loading is shared by the abrasives, thereby preventing the grains from penetrating deeply enough into the workpiece to facilitate the removal of material. Lower distribution density was shown to increase the loadings on the abrasives. Clark et al. [1,6] investigated the process monitoring and mechanics of fixed abrasive diamond wire saw machining, show it is the promising machining method for brittle and materials. Li and Tsai [7] presented a machine vision-based scheme to automatically detect saw-mark defects in Si wafer surfaces. Cao [8] described the hydraulic control system of a diamond wire saw. The closed-loop velocity control system was established based on typical process parameters for the diamond wire saw process. Zhu and Kao [9] found the surface roughness of a wafer with wire saw-sliced is usually not uniform, leading to the subsequent lapping and polishing processes less productive.
Liedke and Kuna [10] developed an analytical model for the macroscopic mechanical conditions in the wire saw process, which describes the influence of several process parameters to the cutting pressure.
Li et al. [11] have established a control model to control the normal force by using feed as a control variable. The experiment results showed that the wire saw machining process with adaptive force control can improve the cutting productivity and significantly decrease wafer surface roughness as compared to the cutting process with a constant part feed rate. Li et al. [12] use optic glass as test material, conducted an experiment of normal force control by wire velocity, and experiments showed that the wire velocity controllers are possible.
Based on the previous research [12,13] , this paper investigates the cutting force control of single crystal silicon using wire saw velocity as the control signal. To understand the influence of wire velocity on the cutting force (include normal force and tangential force), this paper develops a model of normal cutting force and tangential force relating wire velocity. Based on the model, an on-line Proportion and Integral (PI) controller is designed to maintain a constant force, and is experimentally implemented. The cutting productivity, wafer topography and surface roughness are investigated when wire saw machining single crystal silicon with and without force control. Figure 1 shows a schematic of the wire saw cutting single crystal silicon, a wire saw wound roller, two work idler pulleys and an moving wheel form the wire feed mechanism. Both ends of the wire saw are fixed at the wire drum roller, and two travel switches in the bottom of the wire drum roller achieve the shifting action for the wire direction. The wire saw tension adjusts by moving wheel with up and down motion, and the single crystal silicon is mounted in the workbench, which feeds towards the wire. The part move in the X and Y directions via stepper motors. The X-direction is the part feed direction and the Y direction is used to regulate the wafer thickness and remains constant during the operation. The X and Y axis ranges are 0-120 mm, the part feed rate range is 0.025-18 mm/min. The wire saw base is made from a high-quality stainless steel and is nickel plated with a JR2 type diamond abrasive having an average abrasive grain size of 30-40 μm. The wire has a length 86 m, an average diameter of 0.24 mm and the velocity range is 0-3 m/s, and the feed rate is 0-3 mm/min. The moving wheel can regulate the wire tension in the range of 0-0.6 MPa. A Gamma SI-32-2.5-type six-component dynamometer from the ATI Company is used to measure the normal force. It has a range of 100 N and a resolution of 1.25×10 -2 N.

Modeling Wafer Cutting Force
Wafer cutting force includes normal force, tangential force and longitude force, the normal force and tangential force, which show in Fig. 2. The force is variable during the whole cutting period, it's difficult to show the force vary by quantitative model. And during the wafer cutting process, the longitude force (in Y direction) is nearly zero. The normal force and tangential force influence the wafer surface roughness. This study considers the normal force and tangential force varies with wire velocity.

The cutting force analysis
(1) Firstly, let's analyzing the relation of the pressure and indentation depth of single abrasive. Fig. 3 shows the SEM picture of wire saw electroplated with diamond abrasives. To simplify the analytic process, the model of single abrasive is considered as wedge, as shows in Fig. 4. Considering the depth calculation of a single abrasive pressed into the part, based on the sliding line field theory [13] , the plastic deformation model formed by the wedge tool pressing, as shows in Fig. 5.
Based on the equation (2) , h is indentation depth of abrasive, it can get Where P n is the penetration force, h is the indentation depth of abrasive.
(2)During the cutting process, the main force is chip forming force and friction force, which are the distribution force of the vertical abrasive to the center of the abrasive at the contact surface with the part and the distribution force along the opposite direction of the contact surface and the speed of the wire saw [14] . Fig. 6 shows the schematic diagram for wire saw cutting process, and l a (μm) is the distance between two abrasives, V s (m/s) is wire velocity and V x (mm/min) is part feed rate. The normal cutting force F n (N) , tangential cutting force F t (N) are sum of all abrasives involved cutting process, respectively.
The abrasive number N a of involved cutting depend on the feed rate and wire saw velocity.
Cutting force F n , F t are sum of varies abrasives penetration force involved into cutting. According equation (5), the force vary with the abrasives size, however, the size of the abrasive varies randomly within a certain range, and the shapes vary in practice. Besides, the properties of single crystal silicon at different positions contacting with the abrasives on the wire saw is different.
As a result, the cutting forces on the abrasives in different positions is different [15] . It is difficult to quantify accurately the cutting force and then to control using the theoretical model. Therefore, the experimental method is used to establish a cutting force model with wire velocity and part feed rate.

The normal cutting force modeling
In order to find relation between rectangle silicon cutting force and wire saw velocity, experiments are designed to build a model. The part is single crystal silicon, and the size is 220 × 20 × 26 mm, the wire saw length is 106 m, and the radius, is 0.12 mm, the contact length between wire saw and part is 26 mm, the part feed rate is 0.75 mm/min. Table 1 lists the process parameters and cutting forces and processing times are in the experiments.  The cutting force profiles show in Fig. 7, when wire saw velocity is 1.0, 1.  The model obtain from the experiments [16] is where F n (t) is the normal cutting force, α and β is the exponential of part feed rate and wire velocity. Least square is used to get the F n (t), α and β which is 2.9641, 0.6941 and -0.7059, respectively. The relation between cutting force and feed rate is increasing feed rate can improve the normal force, however, overlarge feed rate leads the wire bend and more large cutting force which lead to the wafer break and surface roughness inferior; therefore, based on the experience, the feed rate choose 0.75 mm/min. The normal force model is

The tangential cutting force modeling
Similarly，experiments are designed to build a model，to find relation between single crystal silicon cutting force and wire saw velocity. As table 1 lists the process parameters are used for the experiments. The tangential cutting force profiles show in Fig. 9 when the wire velocity is 1, 1.5 and 2.0 m/s and the part feed rate is 0.75 mm/min. From the Fig. 9, the tangential cutting force decreases with the wire saw velocity increases. Fig. 10 is the relation between tangential force and wire velocity of 1.5 m/s, and the bottom of the graph is the zoom in of relation between tangential force and wire velocity of 1.5 m/s during 1000 s to 1200 s.
The wire saw dynamics are where τ = 0.787 s is the time constant, which is determined experimentally, K w = 1 is the gain and the V c is the commanded wire saw velocity. Linearizing Eq. (20) where dt is the sample period (s where V min = 0.5 m/s is the minimum wire saw velocity and V max = 4 m/s is the maximum wire saw velocity. The implemented wire saw velocity is  . The control system block diagram is given in Fig. 11.

Experimental Studies
A schematic of the experimental wire saw machining system is shown in Fig. 12. The wire drum roller motor velocity is regulated by changing the voltage of DC motor driver, which is achieved by NI PXIe-6348 multi-function card. The National Instruments PXIe-6348 multifunction data acquisition system is used to collect process data, send the command, additional, the sample frequency is 10 Hz. Comparing the difference between the measured and reference cutting forces, the computer sends a voltage signal to the potentiometer and adjusts the wire roller velocity accordingly. The program runs by LabView. In each experiment, the part size and process parameters are the same as constant velocities experiments before.

The normal force control experiment
In order to investigate the controller performance during the entire single      Table   2.   Fig. 16-18, the surface roughness with control wire velocity is smaller. Figure 19 shows the control results when reference normal forces is ±0.7 and ±0.45 N, respectively. From the Fig. 19, the controller start to regulate the wire velocity at 300, and 400s when reference tangential forces is 0.45 and 0.7 N, respectively. When each wire saw change direction, the force is up and down which makes the difference between measured force and reference force change constantly which shows the controller work well.    Figure 21 shows the tangential force comparison between control process and no control process. The control experiment time is 30% less than constant experiment.

Analysis control results of the normal and tangential force
Comparing the velocity control effect of the wire saw on the normal force and the tangential force, the corresponding PI controller is designed to conduct the control experiment, and the results show: (1) For normal force control, compared with the constant parameter cutting process, the wire velocity control can reduce the fluctuation of the cutting force that is fluctuate near the reference, and the cutting time is slightly reduced. The surface roughness of the wafer is about 50% higher than that of the constant wire saw velocity.
(2) For tangential force control, the tangential force could be constant during the cutting process, and the wire velocity changes affect the cutting cycle, resulting in non-periodic grooves on the wafer surface, and then gets an improvement of the profile of the wafer surface, and the cutting time is a less than the constant wire velocity.

Summary and Conclusions
This paper investigates the effect of wire velocity on cutting force, which includes normal force and tangential force. In order to make the cutting process more stable, using PI controller to regulate the force via wire velocity control. The experimental studies for crystal silicon wafer wire saw machining are conducted. The following conclusions can be drawn from this work: (1) The results show the PI controller can characterizes the normal force generated when wire saw machining crystal silicon and demonstrate that the wire saw machining process with normal force control significantly decrease wafer surface roughness as compared to the cutting process with a constant wire velocity reciprocation, which is about 50%.
(2) For the wire saw velocity control of tangential force, the PI controller could demonstrate that the wire saw machining process with force control significantly improve the productivity as compared to the cutting process with a constant wire velocity reciprocation, which is from 16.1% to 34.7%.
Meanwhile, the surface roughness becomes better.
version of the manuscript.