Design and Analysis of Self-Repairable MEMS ... - Semantic Scholar
Design and Analysis of Self-Repairable MEMS Accelerometer Xingguo Xiong, Yu-Liang Wu*, and Wen-Ben Jone Department of ECECS, University of Cincinnati, Cincinnati, OH 45221, USA Department of Computer Science and Engineering*, The Chinese University of Hong Kong, Shattin, Hong Kong Email: xiongx(wjone)@ececs.uc.edu, [email protected]*
Abstract In this paper, a self-repairable MEMS (SRMEMS) accelerometer design is proposed. The accelerometer consists of identical modules: of them serve as the main device, while the remaining modules act as the redundancy. If any of the working module in the main device is found faulty, the control circuit will replace it with a good redundant module. In this way, the faulty device can be self-repaired through redundancy. The sensitivity loss due to device modularization can be well compensated by different design alternatives. The yield model for MEMS redundancy repair is developed. The simulation results show that the BISR (built-in self-repair) design leads to effective yield increase compared to non-BISR design, especially for a moderate non-BISR yield. By implementing the fault tolerance feature into MEMS devices, the yield as well as the reliability of a MEMS device implemented in a SoC can be improved. Keywords: Built-in self-repair (BISR), MEMS, micro-accelerometer, yield analysis, redundancy repair.
1. Introduction As Micro Electro Mechanical System (MEMS) is utilized in more and more applications, fault-tolerant MEMS design becomes extremely crucial with the following two reasons. First, with increasing applications of MEMS to safety-critical fields, such as aerospace, automobile and medical apparatus, MEMS reliability becomes an urgent need. Especially, many MEMS devices have movable parts and their repeated movements (vibrations, etc.) may lead to different kinds of structural material fatigues. Thus, even if a MEMS device is tested as fault-free, it still may fail after serving for a certain lifetime. Second, there is increased tendency that MEMS is going to be integrated into system-on-chip (SoC) designs using a standard CMOS process [1]. It will be uneconomical to get rid of the entire SoC chip, if there exist minor MEMS defects. Thus, it is emergent to find a solution to have a defective MEMS device fix itself, whenever the test process (in-field or manufacturing) finds defects existent. A few efforts on the structural trimming (somewhat hard repairing) of MEMS devices with certain physical or chemical processes have been reported [2][3], e.g., laser repair of stiction-failed microcantilevers has been proposed [4]. The hard repairing of an individual MEMS device leads to high cost. Further, it is not suitable for in-field self-repair, because specific apparatus is required. The necessity and possibility for fault tolerance and redundancy repair for array-based MEMS devices (such as MEMS microvalve array) have been addressed in [5]. However, to the best of our knowledge, the redundancy repair approach for non-array-based MEMS devices has never been reported. In this paper, built-in self-repair through modular redundancy (soft repairing) of a non-array-based MEMS comb accelerometer is explored. We revise a MEMS design by partitioning it into identical modules such that healthy modules can work jointly to guarantee the normal function of the entire device. The sensing mass in each BISR module falls to about that of the non-BISR device. If we maintain the same beam width for both non-BISR/BISR devices, the sensitivity of a BISR device falls to about that of the non-BISR device. Thus, in this paper, several techniques are proposed to compensate the sensitivity loss due to the proposed modularized design. In the BISR mode, the device first performs the built-in self-test (BIST) process to check the function of each individual module. The BISR control circuit will then restructure some good modules as a main device for normal operation. The remaining good modules will serve as redundant devices for reliability enhancement. A BIST technique based on dual-mode (sensitivity and symmetry) testing [6] will
be applied to guarantee the success of identifying defective modules. The results obtained based on computer simulation demonstrate that, with a tolerable device area overhead, the proposed BISR technique can improve the manufacturing yield of a comb accelerometer from 0.698 to 0.947 without sacrificing the sensitivity. This technique is especially powerful for a manufacturing process that has moderate yield where a sufficient number of modules are healthy. Further, this technique can be efficiently applied to safety-critical applications, since the reliability of a BISR MEMS device for in-field applications can be enhanced. Simulation results also demonstrate the feasibility of the BISR technique. The major application of the proposed BISR technique is to enhance both the manufacturing yield and in-field reliability of MEMS accelerometers implemented in a SoC design.
2. Non-BISR MEMS Comb Accelerometer fixed finger (left)
fixed finger (right)
anchor
beam
central movable mass
movable finger
Figure 1. The general design of MEMS comb accelerometer
A typical surface-micromachined comb accelerometer [7] is shown in Figure 1. The comb accelerometer is made of a thin layer of poly-silicon on the top of a silicon substrate. In the static state, each movable finger stays in the middle position between the left and right fixed fingers, and the capacitance gap between each pair of movable and fixed fingers equals to . If there is an acceleration along the sensitive direction, the central mass and the movable fingers will move for a certain displacement due to the inertial force. Squarewave modulation voltages and with complimentary phases are applied to the corresponding left and right fixed fingers separately. Thus, we have
where represents the modulation voltage amplitude, denotes the frequency of the modulation voltage, and gives the time for operation. Given , the voltage level sensed at the central mass will be:
By measuring the voltage level of the movable fingers and the central mass, we can find the displacement of , which in turn is directly proportional to the acceleration. Thus, we can derive the value of the experienced acceleration.
3. BISR MEMS Comb Accelerometer Design Due to the tiny size (in the range of microns) of MEMS accelerometers, its overall capacitance is generally below 1pF. In order for the tiny capacitance to be detected by the signal sensing circuit, it is desirable to have a large number of comb finger groups. For example, an ADXL50 accelerometer contains 42 differential comb finger groups (i.e., 84 capacitance gaps) [7]. On the other hand, such a highly dense comb structure with many long and narrow capacitance gaps is extremely vulnerable to various defects such as particle contamination and stiction [8]. If a conductive particle falls into any of the 84 capacitance gaps and lead to a short-circuit, it will result in a failure of the entire device. Thus, a large number of finger groups unavoidably leads to the decrease in yield as well as reliability.
anchor
beam
mass
Mb
Ma
Mc
Md
Me
Mf
Figure 2. Modularized comb accelerometer structure
The modularized comb accelerometer design with BISR feature is shown in Figure 2. Here, the device consists of six identical modules, and each module has its own beams, mass and finger structures (fixed and movable). By assumption, four modules are connected together as the main device, while the remaining two modules serve as redundancy. The movable parts of each module are physically connected to those of adjacent modules through the common anchors, and signals sensed by all movable fingers in the device are connected to the sensing circuit directly. However, the fixed fingers of each module are connected to the modulation signal circuit through switches made of analog MUXes. By turning on or off these switches, we can determine whether a module works as part of the main device or the redundant device. For example, if the modulation signals of module are turned off, then the movable fingers of cannot sense any signal. Thus, is electronically disconnected from the MEMS device (though it is still connected to the entire MEMS structure physically), and is not involved in the MEMS function. During the BISR mode, the device will first perform the BIST process for each individual module. The dual-mode BIST technology presented in [6] will be used here to ensure a thorough test. The BIST result of each module is compared with the pre-stored good module response to judge whether the module is good or faulty. This information is fed to the BISR control circuit as input for self repairing. If a module is tested as faulty, the control circuit will permanently exclude the module from the main device and replace it with a good redundant module (if there is any). Thus, after repairing, the main device can still be ensured to work properly. In the following discussion, we call the MEMS comb accelerometer with (without) the BISR feature as a BISR (non-BISR) accelerometer. In order for a fair comparison, we assume that the total number of finger groups of the non-BISR accelerometer should equal to that of the main device in the BISR accelerometer. In this way, the area overhead of the modularized BISR accelerometer includes the area for two redundant modules, plus the area for extra beams and gaps between modules in the BISR accelerometer, as well as the area for control circuits.
4. Performance Analysis 4.1. Sensitivity Analysis A MEMS comb accelerometer actually can be simplified by a spring-mass model. The width and length of each tether (seismic mass) are represented by ( ) and ( ) separately, while the width and length of each movable finger are denoted by and respectively. There are totally number of movable fingers, and the thickness of the device is . Assume the density and the Young’s modulus of poly-Si as and respectively. The sensing mass of the accelerometer, which includes the seismic mass and all the movable fingers attached to it, can be expressed as follows:
The total spring constant of all four beams is:
The displacement sensitivity of the device, which is defined as the displacement of movable fingers per unit gravity acceleration (g) along the sensitive direction, can be expressed as:
Assume all other parameters are fixed, the relationships between the device displacement sensitivity and the beam width , central mass width are shown in Figure 3.
Figure 3. The relationship between displacement sensitivity of accelerometer and beam/mass width
We denote the displacement sensitivity of the BISR (non-BISR) accelerometer by ( ). In the BISR device, we partition all comb finger groups into modules in the main device, and add number of modules as redundancy. That is, each BISR module contains comb finger groups. Correspondingly, the sensing mass of each BISR module also falls to of the non-BISR version. If we keep the same geometry parameters (i.e., width , length and thickness ) for the beams of both BISR and non-BISR designs, we have:
where and are the sense mass and spring constant of the non-BISR accelerometer. For the BISR accelerometer design, we can recover its sensitivity by reducing the beam width of each module, by using folded beams instead of straight beams to increase the effective length ( ) of each beam, or by enlarging the width of the central mass, etc. However, tuning down the beam width is the most efficient method to recover the sensitivity value back to normal. Take the beam width of the non-BISR accelerometer as . If we reduce the beam width of the BISR accelerometer to 0.63 , the sensitivity of the BISR accelerometer will become approximately equal to that of the non-BISR design (in the case of n=4). The designer can compensate the sensitivity loss of a high-sensitivity BISR accelerometer by the combination of the above three alternatives, according to the individual device requirements. Even if all the device parameters (e.g., beam width and mass width) are fixed, the sensitivity of an accelerometer in the open-loop mode can still be conveniently enhanced by electrostatic force with an appropriate DC biasing voltage [9][10]. The electrostatic force acts as a spring with a negative spring constant. This will help reduce the effective spring constant of the accelerometer and increase its sensitivity. Since this electrostatic force actually deflects the mass, it is equivalent to the effect of an inertial force acting on the mass. Thus, the electrostatic force is a powerful tool to enhance the sensitivity. A small DC biasing voltage (several volts) can increase the sensitivity to infinity, and this offers a great flexibility in the sensitivity recovery. Thus, if electrostatic force is used for sensitivity compensation, the beam width and mass width can be kept the same as those of the non-BISR device.
4.2. Frequency Analysis Using the simplified spring-mass model discussed above for the comb accelerometer, we have the resonant frequency of the non-BISR design given by the following equation [7]:
where and are defined in the last section. Assume the dimension of each beam in the BISR design remains the same as that of the non-BISR design, but the comb finger groups are divided into identical modules. The resonant frequency of the BISR accelerometer can thus be given by
Further, the relationship between resonant frequency and displacement sensitivity can be given by:
Consequently, if the displacement sensitivity of the BISR accelerometer is compensated to the same as that of the non-BISR design (e.g., by adjusting the value), the resonant frequency will also remain the same as that of the non-BISR device.
5. Yield analysis of BISR comb accelerometer 5.1. Yield Model for MEMS Redundancy Repair Assume a set of defects can occur to number of locations in a MEMS device, i.e., there are possible defects in the MEMS device. Further, assume every defect occurs independently of each other, and the probability for each defect to occur is equal and defined as . Thus, based on the defect distribution discussed in [11], the probability that number of indistinguishable randomly distributed defects occurring to the MEMS device can be expressed as a Poisson distribution:
The simple Poisson distribution is too pessimistic for yield estimation because the defect clustering effect is not considered. Hence, the compound Poisson distribution is more popular by considering the normalized distribution of a chip defect density clustering factor. The next problem is how the average defect distributes. In this work, we assume that the defect distribution function for is a gamma function given by the following equation [11]:
where is the device area, is a defect density coefficient, is the average defect, and is the clustering parameter. Further, the average defect density is given by:
The probability that defects occur in a MEMS device with area can be given by the following equation [11]:
½
½
For a non-BISR MEMS accelerometer, the yield ! is the probability that no defect occurs (i.e.,
),
which can be expressed as:
!
where is the area of the non-BISR accelerometer. From this equation, we see that the yield drops as the device area increase. This is reasonable because the larger the device area is, the more likely it may suffer from some defects. Assume there are totally " possible defects in the BISR MEMS accelerometer, number of modules in the main device, and number of redundant modules as shown in Figure 4. The yield of the BISR MEMS accelerometer after redundancy repairing, denoted as ! , equals the probability that none of the " defects occurs, plus the probability that some among the " defects do occur but they all fall into no more than number
i faults
m redundant modules
n main modules
Figure 4. Fault distribution among the modules of BISR accelerometer.
of modules. The former is the case where all the number of modules are healthy, while the latter is the case in which some modules are faulty, but the device can still be self-repaired into a good one through redundancy. Each of both cases will be investigated in the following discussions. Assume the main device of the BISR accelerometer has the same number of finger groups as the corresponding non-BISR design. Let denotes the area of the non-BISR accelerometer, denotes the area of beams in each module of the BISR accelerometer (and also the non-BISR accelerometer) plus the minimum gap between two modules/devices set by the design rules. In our simulation, we select # . The area of the BISR accelerometer is given by:
The probability that none of the " number of defects occurs can be expressed as:
For the second case in which some defects do occur but the device can still be self-repaired through redundancy, it can be further divided into the following two sub-cases:
The total number (") of defects is smaller than or equal to the number () of redundant modules, i. e., " . For this sub-case, regardless of whatever distribution for the defects, the BISR comb accelerometer can always be repaired into a good device. The total number (") of defects is larger than the number of the redundant modules ; however, all the defects fall into number of modules or less. For this sub-case, the BISR accelerometer can still be repaired into a good device.
Both the above two sub-cases must be counted into the yield of the BISR accelerometer. The first sub-case can be easily solved, while the second sub-case requires an extensive analysis. For the first sub-case, the probability that " number (" ) of defects occur in the BISR MEMS accelerometer can be expressed as:
For the second sub-case (" $ ), the faulty device can be repaired into a good device only if all the " number of defects fall into number of modules or less. First, we examine the probability that " defects are distributed into % (% ) MEMS modules and each of the % modules contains at least one defect. So, we distribute one defect to each of the % modules to ensure that each of the % modules contains at least one defect. Thus, there are " % defects remaining to be distributed to the % modules with any number of defects (maybe distinct 0) for this distribution. As we know, there are ¼ ways to distribute identical balls into cells with any number of balls per cell. Here, we have " % and % and the total number of ways of distribution is . Finally, the probability that the BISR accelerometer (with modules in the main device, and modules in the redundancy device) can be repaired when a certain " number (" $ , sub-case 2) of defects occur is:
&
j " j
Thus, the probability that more than number of defects occur to a MEMS device, but it still can be self-repaired into a good device can be expressed as follows.
½
½
&
j
j
Hence, the yield for the BISR accelerometer after redundancy repair is:
½
j j
The yield increase '! of the BISR accelerometer, when compared with the corresponding non-BISR accelerometer, can be given by:
½
j j
We can see that under this theory, if we set and , then ! comes back to which is exactly the yield of the non-BISR device. Thus, the general case of ! includes the yield of the nonBISR device as a special case. Based upon the MEMS yield model for redundancy repair, we can derive the relationship between the yield increase and the non-BISR device yield for different and numbers. Figure 5 shows the simulation result for and separately. From this figure, we can observe that the BISR device always gives a positive yield increase regardless of the non-BISR yield. This demonstrates the effectiveness of the redundancy repair technique for MEMS devices. If the non-BISR yield is too low (approaches 0) or too high (approaches 1), the yield increase by redundancy repair is not significant. This is a reasonable result. For a very low non-BISR yield (approaches 0), the defect density is extremely high and there are too many faulty modules in the main device. Compared with so many defective modules in the main device, the redundancy is relatively deficient to repair all of them. Thus, the yield increase by redundancy BISR is not significant. For a very high non-BISR yield (approaches 1), the main device itself is highly likely to be fault-free, and hence repair is not necessary.
Figure 5. The yield increase vs. non-BISR yield for different m numbers
In order to verify the effectiveness of redundancy repair for moderate initial yield, we randomly select # , # , and , the yield of the non-BISR accelerometer is 0.698. For the BISR accelerometer with and , the yield becomes 0.947 which is an increase of 35.7% (when compared with the non-BISR yield 0.698). This demonstrates an effective improvement on the yield for a comb accelerometer through redundancy repair.
6. Design and Simulation of BISR Accelerometer The geometry parameters of the BISR comb accelerometer with and are listed in Table 1, using a set of design rules comparable to ADXL accelerometers [7]. For comparison, a none-BISR accelerometer with the same number of capacitance groups (as that at the main device of the BISR accelerometer) is also designed. The geometry parameters of the non-BISR accelerometer are also listed in the same table. The simulation results for the performance of both BISR and none-BISR accelerometers are shown in Table 2. From Table 2, we can see that, by narrowing the beam width, the sensitivity loss of the BISR accelerometer due to device modularization can be fully compensated. The BISR accelerometer shows approximately the same displacement sensitivity as that of the none-BISR accelerometer. Table 1. Design of BISR/non-BISR accelerometers Design Parameters device area(( ) thickness t (() no. of capacitance groups capacitance gap ( beam width ( beam length ( mass width ( mass length ( finger width ( finger length (
BISR device 1500900 6 206 2 2 300 200 2206 4 200
non-BISR device 980900 6 80 2 3.2 300 200 880 4 200
Table 2. Simulation results of BISR/non-BISR accelerometers
Performance
BISR non-BISR main device device sensing mass ( 0.844=3.36 3.36 capacitance ) * 0.1034=0.41 0.41 sensitivity 6.8 6.64 sensitivity 0.74=2.8 2.74 spring constant 1.214=4.84 4.95 frequency +, 6.05 6.12 In order to demonstrate the effectiveness of the sensitivity compensation by adjusting the beam width, ANSYS [12] simulation has been performed for both non-BISR and BISR accelerometers. Due to the limitation in node number imposed by the available ANSYS version, simplified models for both non-BISR and BISR accelerometers are used for simulation. ANSYS simulation results for the displacement sensitivity in response to acceleration from 0 to 50 are shown in Figure 6. For the non-BISR accelerometer, the beam width is 3.2(m. For the BISR MEMS accelerometer, the beam width is shrunk to 2(m. According to Figure 6, the sensitivity of the non-BISR accelerometer is 6.68 , while the sensitivity of the BISR accelerometer after compensation is 6.83 . The sensitivity of the BISR device is about the same as that of the non-BISR device. Further, the electrostatic force can act as a powerful tool in compensating the sensitivity loss of the BISR design as illustrated in [9]. The sensitivity of the accelerometer can be increased till infinity by applying an appropriate DC biasing voltage given the accelerometer works in the open-loop mode. Thus, as suggested by the simulation results, the sensitivity loss is not a concern for the BISR design.
7. Conclusion and Future Research In this paper, a redundancy-based BISR technique suitable for MEMS devices is proposed. Such a technique can improve the fabrication yield as well as the in-field reliability of MEMS devices. The basic idea of the proposed method is to partition a MEMS device into modules such that defective modules can be virtually eliminated from the working machine. The sensitivity loss (due to device modularization) and its compensation are discussed in detail. A MEMS yield model for redundancy repair has also been developed. Simulation results demonstrate effective yield improvements for MEMS fabrication with a moderate initial yield. The optimized design of a BISR MEMS device can be achieved by adjusting the area of the main and redundant devices. Based on the proposed method, both BISR and non-BISR MEMS accelerometers have been designed and simulated using ANSYS for performance evaluation.
Figure 6. ANSYS simulation results for sensitivity of BISR/non-BISR accelerometers
Compared to the hard repair methods of MEMS, the BISR method through modular redundancy eliminates the need of expensive physical/chemical processing. The entire BISR process is performed automatically by the BISR control circuit. Thus, it is effective for both the manufacturing and the in-field MEMS device repairing. Furthermore, the redundancy BISR scheme is virtually applicable to any local defect. If any of the working modules in the main device is found faulty, regardless whatever local defect it is, the entire faulty module will be replaced by a good redundant module. Thus, the faulty MEMS device can be self-repaired without digging into the detail of the defects in the faulty module. In this way, the redundancy repair method can be an effective and promising solution to fault-tolerant MEMS design. In the BISR design, we trade the hardware overhead for yield and reliability, while the performance remains unchanged. Although the sensitivity compensation techniques (reducing beam width, increasing mass width, electrostatic force compensation, etc.) can also be applied to the non-BISR device to enhance its sensitivity, the yield and reliability of the non-BISR device cannot be improved. Due to fabrication process variations or parametric defects, the performance of one BISR module cannot be perfectly the same as others. Thus, after a redundant module replaces a faulty module, the device must be recalibrated. Our future research is to develop a built-in self-calibration (BISC) technique to ensure a smooth switch between defective and healthy modules, without introducing a significant performance change after redundancy repairing.
References [1] URL: http://public.itrs.net. [2] J. Ramirez-Angulo, and R. L. Geiger, ”New Laser-trimmed Film Resistor Structures for Very High Stability Requirements”, IEEE Transactions on Electron Devices, Vol. 35, Issue 4, pp. 516-518, Apr. 1988. [3] K. Tanaka, Y. Mochida, M. Sugimoto, K. Moriya, T. Hasegawa, K. Atsuchi, and K. Ohwada, ”A Micromachined Vibrating Gyroscope”, Sensors and Actuators A: Physical, Vol. 50, pp. 111-115, 1995. [4] S. B. Koppaka and L. M. Phinney, ”Release Processing Effects on Laser Repair Stiction-Failed Microcantilevers”, Journal of Microelectromechanical Systems, Vol. 14, No. 2, Apr. 2005, pp. 410-418. [5] S. K. Tewksbury, ”Challenges Facing Practical DFT for MEMS”, Proceedings of the 2001 IEEE International Symposium on Defect and Fault Tolerance in VLSI Systems (DFT’01), Oct. 24-26, 2001, pp. 11-17. [6] X. Xiong, Y. Wu, and W. Jone, ”A Dual-mode Built-in Self-test Technique for Capacitive MEMS Devices”, Proceedings of 22nd IEEE VLSI Test Symposium’04, pp. 148-153, 25-29 May, 2004. [7] W. Kuehnel, and S. Sherman, ”A Surface Micromachined Silicon Accelerometer with On-chip Detection Circuitry,” Sensors and Actuators A: Physical, Vol. 45, pp. 7-16, 1994. [8] N. Deb, and R. D. Blanton, ”Analysis of Failure Sources in Surface-micromachined MEMS”, Proc. of International Test Conference, pp. 739-749, Oct. 2000. [9] B. Li, D. Lu, and W. Wang, ”Open-loop Operating Mode of Micromachined Capacitive Accelerometer”, Sensors and Actuators A: Physical, Vol. 79, pp. 219-223, 2000. [10] M. Bao, H. Yang, H. Yin, and S. Shen, ”Effects of Electrostatic Forces Generated by the Driving Signal on Capacitive Sensing Devices”, Sensors and Actuators A: Physical, Vol. 84, pp. 213-219, 2000. [11] J. Hirase, ”Yield Increase of VLSI after Redundancy-repairing,” Proc. of 10th Asian Test Symposium, pp.353-358, Nov. 2001. [12] URL: http://www.ansys.com.
Abstract In this paper, a self-repairable MEMS (SRMEMS) accelerometer design is proposed. The accelerometer consists of identical modules: of them serve as the main device, while the remaining modules act as the redundancy. If any of the working module in the main device is found faulty, the control circuit will replace it with a good redundant module. In this way, the faulty device can be self-repaired through redundancy. The sensitivity loss due to device modularization can be well compensated by different design alternatives. The yield model for MEMS redundancy repair is developed. The simulation results show that the BISR (built-in self-repair) design leads to effective yield increase compared to non-BISR design, especially for a moderate non-BISR yield. By implementing the fault tolerance feature into MEMS devices, the yield as well as the reliability of a MEMS device implemented in a SoC can be improved. Keywords: Built-in self-repair (BISR), MEMS, micro-accelerometer, yield analysis, redundancy repair.
1. Introduction As Micro Electro Mechanical System (MEMS) is utilized in more and more applications, fault-tolerant MEMS design becomes extremely crucial with the following two reasons. First, with increasing applications of MEMS to safety-critical fields, such as aerospace, automobile and medical apparatus, MEMS reliability becomes an urgent need. Especially, many MEMS devices have movable parts and their repeated movements (vibrations, etc.) may lead to different kinds of structural material fatigues. Thus, even if a MEMS device is tested as fault-free, it still may fail after serving for a certain lifetime. Second, there is increased tendency that MEMS is going to be integrated into system-on-chip (SoC) designs using a standard CMOS process [1]. It will be uneconomical to get rid of the entire SoC chip, if there exist minor MEMS defects. Thus, it is emergent to find a solution to have a defective MEMS device fix itself, whenever the test process (in-field or manufacturing) finds defects existent. A few efforts on the structural trimming (somewhat hard repairing) of MEMS devices with certain physical or chemical processes have been reported [2][3], e.g., laser repair of stiction-failed microcantilevers has been proposed [4]. The hard repairing of an individual MEMS device leads to high cost. Further, it is not suitable for in-field self-repair, because specific apparatus is required. The necessity and possibility for fault tolerance and redundancy repair for array-based MEMS devices (such as MEMS microvalve array) have been addressed in [5]. However, to the best of our knowledge, the redundancy repair approach for non-array-based MEMS devices has never been reported. In this paper, built-in self-repair through modular redundancy (soft repairing) of a non-array-based MEMS comb accelerometer is explored. We revise a MEMS design by partitioning it into identical modules such that healthy modules can work jointly to guarantee the normal function of the entire device. The sensing mass in each BISR module falls to about that of the non-BISR device. If we maintain the same beam width for both non-BISR/BISR devices, the sensitivity of a BISR device falls to about that of the non-BISR device. Thus, in this paper, several techniques are proposed to compensate the sensitivity loss due to the proposed modularized design. In the BISR mode, the device first performs the built-in self-test (BIST) process to check the function of each individual module. The BISR control circuit will then restructure some good modules as a main device for normal operation. The remaining good modules will serve as redundant devices for reliability enhancement. A BIST technique based on dual-mode (sensitivity and symmetry) testing [6] will
be applied to guarantee the success of identifying defective modules. The results obtained based on computer simulation demonstrate that, with a tolerable device area overhead, the proposed BISR technique can improve the manufacturing yield of a comb accelerometer from 0.698 to 0.947 without sacrificing the sensitivity. This technique is especially powerful for a manufacturing process that has moderate yield where a sufficient number of modules are healthy. Further, this technique can be efficiently applied to safety-critical applications, since the reliability of a BISR MEMS device for in-field applications can be enhanced. Simulation results also demonstrate the feasibility of the BISR technique. The major application of the proposed BISR technique is to enhance both the manufacturing yield and in-field reliability of MEMS accelerometers implemented in a SoC design.
2. Non-BISR MEMS Comb Accelerometer fixed finger (left)
fixed finger (right)
anchor
beam
central movable mass
movable finger
Figure 1. The general design of MEMS comb accelerometer
A typical surface-micromachined comb accelerometer [7] is shown in Figure 1. The comb accelerometer is made of a thin layer of poly-silicon on the top of a silicon substrate. In the static state, each movable finger stays in the middle position between the left and right fixed fingers, and the capacitance gap between each pair of movable and fixed fingers equals to . If there is an acceleration along the sensitive direction, the central mass and the movable fingers will move for a certain displacement due to the inertial force. Squarewave modulation voltages and with complimentary phases are applied to the corresponding left and right fixed fingers separately. Thus, we have
where represents the modulation voltage amplitude, denotes the frequency of the modulation voltage, and gives the time for operation. Given , the voltage level sensed at the central mass will be:
By measuring the voltage level of the movable fingers and the central mass, we can find the displacement of , which in turn is directly proportional to the acceleration. Thus, we can derive the value of the experienced acceleration.
3. BISR MEMS Comb Accelerometer Design Due to the tiny size (in the range of microns) of MEMS accelerometers, its overall capacitance is generally below 1pF. In order for the tiny capacitance to be detected by the signal sensing circuit, it is desirable to have a large number of comb finger groups. For example, an ADXL50 accelerometer contains 42 differential comb finger groups (i.e., 84 capacitance gaps) [7]. On the other hand, such a highly dense comb structure with many long and narrow capacitance gaps is extremely vulnerable to various defects such as particle contamination and stiction [8]. If a conductive particle falls into any of the 84 capacitance gaps and lead to a short-circuit, it will result in a failure of the entire device. Thus, a large number of finger groups unavoidably leads to the decrease in yield as well as reliability.
anchor
beam
mass
Mb
Ma
Mc
Md
Me
Mf
Figure 2. Modularized comb accelerometer structure
The modularized comb accelerometer design with BISR feature is shown in Figure 2. Here, the device consists of six identical modules, and each module has its own beams, mass and finger structures (fixed and movable). By assumption, four modules are connected together as the main device, while the remaining two modules serve as redundancy. The movable parts of each module are physically connected to those of adjacent modules through the common anchors, and signals sensed by all movable fingers in the device are connected to the sensing circuit directly. However, the fixed fingers of each module are connected to the modulation signal circuit through switches made of analog MUXes. By turning on or off these switches, we can determine whether a module works as part of the main device or the redundant device. For example, if the modulation signals of module are turned off, then the movable fingers of cannot sense any signal. Thus, is electronically disconnected from the MEMS device (though it is still connected to the entire MEMS structure physically), and is not involved in the MEMS function. During the BISR mode, the device will first perform the BIST process for each individual module. The dual-mode BIST technology presented in [6] will be used here to ensure a thorough test. The BIST result of each module is compared with the pre-stored good module response to judge whether the module is good or faulty. This information is fed to the BISR control circuit as input for self repairing. If a module is tested as faulty, the control circuit will permanently exclude the module from the main device and replace it with a good redundant module (if there is any). Thus, after repairing, the main device can still be ensured to work properly. In the following discussion, we call the MEMS comb accelerometer with (without) the BISR feature as a BISR (non-BISR) accelerometer. In order for a fair comparison, we assume that the total number of finger groups of the non-BISR accelerometer should equal to that of the main device in the BISR accelerometer. In this way, the area overhead of the modularized BISR accelerometer includes the area for two redundant modules, plus the area for extra beams and gaps between modules in the BISR accelerometer, as well as the area for control circuits.
4. Performance Analysis 4.1. Sensitivity Analysis A MEMS comb accelerometer actually can be simplified by a spring-mass model. The width and length of each tether (seismic mass) are represented by ( ) and ( ) separately, while the width and length of each movable finger are denoted by and respectively. There are totally number of movable fingers, and the thickness of the device is . Assume the density and the Young’s modulus of poly-Si as and respectively. The sensing mass of the accelerometer, which includes the seismic mass and all the movable fingers attached to it, can be expressed as follows:
The total spring constant of all four beams is:
The displacement sensitivity of the device, which is defined as the displacement of movable fingers per unit gravity acceleration (g) along the sensitive direction, can be expressed as:
Assume all other parameters are fixed, the relationships between the device displacement sensitivity and the beam width , central mass width are shown in Figure 3.
Figure 3. The relationship between displacement sensitivity of accelerometer and beam/mass width
We denote the displacement sensitivity of the BISR (non-BISR) accelerometer by ( ). In the BISR device, we partition all comb finger groups into modules in the main device, and add number of modules as redundancy. That is, each BISR module contains comb finger groups. Correspondingly, the sensing mass of each BISR module also falls to of the non-BISR version. If we keep the same geometry parameters (i.e., width , length and thickness ) for the beams of both BISR and non-BISR designs, we have:
where and are the sense mass and spring constant of the non-BISR accelerometer. For the BISR accelerometer design, we can recover its sensitivity by reducing the beam width of each module, by using folded beams instead of straight beams to increase the effective length ( ) of each beam, or by enlarging the width of the central mass, etc. However, tuning down the beam width is the most efficient method to recover the sensitivity value back to normal. Take the beam width of the non-BISR accelerometer as . If we reduce the beam width of the BISR accelerometer to 0.63 , the sensitivity of the BISR accelerometer will become approximately equal to that of the non-BISR design (in the case of n=4). The designer can compensate the sensitivity loss of a high-sensitivity BISR accelerometer by the combination of the above three alternatives, according to the individual device requirements. Even if all the device parameters (e.g., beam width and mass width) are fixed, the sensitivity of an accelerometer in the open-loop mode can still be conveniently enhanced by electrostatic force with an appropriate DC biasing voltage [9][10]. The electrostatic force acts as a spring with a negative spring constant. This will help reduce the effective spring constant of the accelerometer and increase its sensitivity. Since this electrostatic force actually deflects the mass, it is equivalent to the effect of an inertial force acting on the mass. Thus, the electrostatic force is a powerful tool to enhance the sensitivity. A small DC biasing voltage (several volts) can increase the sensitivity to infinity, and this offers a great flexibility in the sensitivity recovery. Thus, if electrostatic force is used for sensitivity compensation, the beam width and mass width can be kept the same as those of the non-BISR device.
4.2. Frequency Analysis Using the simplified spring-mass model discussed above for the comb accelerometer, we have the resonant frequency of the non-BISR design given by the following equation [7]:
where and are defined in the last section. Assume the dimension of each beam in the BISR design remains the same as that of the non-BISR design, but the comb finger groups are divided into identical modules. The resonant frequency of the BISR accelerometer can thus be given by
Further, the relationship between resonant frequency and displacement sensitivity can be given by:
Consequently, if the displacement sensitivity of the BISR accelerometer is compensated to the same as that of the non-BISR design (e.g., by adjusting the value), the resonant frequency will also remain the same as that of the non-BISR device.
5. Yield analysis of BISR comb accelerometer 5.1. Yield Model for MEMS Redundancy Repair Assume a set of defects can occur to number of locations in a MEMS device, i.e., there are possible defects in the MEMS device. Further, assume every defect occurs independently of each other, and the probability for each defect to occur is equal and defined as . Thus, based on the defect distribution discussed in [11], the probability that number of indistinguishable randomly distributed defects occurring to the MEMS device can be expressed as a Poisson distribution:
The simple Poisson distribution is too pessimistic for yield estimation because the defect clustering effect is not considered. Hence, the compound Poisson distribution is more popular by considering the normalized distribution of a chip defect density clustering factor. The next problem is how the average defect distributes. In this work, we assume that the defect distribution function for is a gamma function given by the following equation [11]:
where is the device area, is a defect density coefficient, is the average defect, and is the clustering parameter. Further, the average defect density is given by:
The probability that defects occur in a MEMS device with area can be given by the following equation [11]:
½
½
For a non-BISR MEMS accelerometer, the yield ! is the probability that no defect occurs (i.e.,
),
which can be expressed as:
!
where is the area of the non-BISR accelerometer. From this equation, we see that the yield drops as the device area increase. This is reasonable because the larger the device area is, the more likely it may suffer from some defects. Assume there are totally " possible defects in the BISR MEMS accelerometer, number of modules in the main device, and number of redundant modules as shown in Figure 4. The yield of the BISR MEMS accelerometer after redundancy repairing, denoted as ! , equals the probability that none of the " defects occurs, plus the probability that some among the " defects do occur but they all fall into no more than number
i faults
m redundant modules
n main modules
Figure 4. Fault distribution among the modules of BISR accelerometer.
of modules. The former is the case where all the number of modules are healthy, while the latter is the case in which some modules are faulty, but the device can still be self-repaired into a good one through redundancy. Each of both cases will be investigated in the following discussions. Assume the main device of the BISR accelerometer has the same number of finger groups as the corresponding non-BISR design. Let denotes the area of the non-BISR accelerometer, denotes the area of beams in each module of the BISR accelerometer (and also the non-BISR accelerometer) plus the minimum gap between two modules/devices set by the design rules. In our simulation, we select # . The area of the BISR accelerometer is given by:
The probability that none of the " number of defects occurs can be expressed as:
For the second case in which some defects do occur but the device can still be self-repaired through redundancy, it can be further divided into the following two sub-cases:
The total number (") of defects is smaller than or equal to the number () of redundant modules, i. e., " . For this sub-case, regardless of whatever distribution for the defects, the BISR comb accelerometer can always be repaired into a good device. The total number (") of defects is larger than the number of the redundant modules ; however, all the defects fall into number of modules or less. For this sub-case, the BISR accelerometer can still be repaired into a good device.
Both the above two sub-cases must be counted into the yield of the BISR accelerometer. The first sub-case can be easily solved, while the second sub-case requires an extensive analysis. For the first sub-case, the probability that " number (" ) of defects occur in the BISR MEMS accelerometer can be expressed as:
For the second sub-case (" $ ), the faulty device can be repaired into a good device only if all the " number of defects fall into number of modules or less. First, we examine the probability that " defects are distributed into % (% ) MEMS modules and each of the % modules contains at least one defect. So, we distribute one defect to each of the % modules to ensure that each of the % modules contains at least one defect. Thus, there are " % defects remaining to be distributed to the % modules with any number of defects (maybe distinct 0) for this distribution. As we know, there are ¼ ways to distribute identical balls into cells with any number of balls per cell. Here, we have " % and % and the total number of ways of distribution is . Finally, the probability that the BISR accelerometer (with modules in the main device, and modules in the redundancy device) can be repaired when a certain " number (" $ , sub-case 2) of defects occur is:
&
j " j
Thus, the probability that more than number of defects occur to a MEMS device, but it still can be self-repaired into a good device can be expressed as follows.
½
½
&
j
j
Hence, the yield for the BISR accelerometer after redundancy repair is:
½
j j
The yield increase '! of the BISR accelerometer, when compared with the corresponding non-BISR accelerometer, can be given by:
½
j j
We can see that under this theory, if we set and , then ! comes back to which is exactly the yield of the non-BISR device. Thus, the general case of ! includes the yield of the nonBISR device as a special case. Based upon the MEMS yield model for redundancy repair, we can derive the relationship between the yield increase and the non-BISR device yield for different and numbers. Figure 5 shows the simulation result for and separately. From this figure, we can observe that the BISR device always gives a positive yield increase regardless of the non-BISR yield. This demonstrates the effectiveness of the redundancy repair technique for MEMS devices. If the non-BISR yield is too low (approaches 0) or too high (approaches 1), the yield increase by redundancy repair is not significant. This is a reasonable result. For a very low non-BISR yield (approaches 0), the defect density is extremely high and there are too many faulty modules in the main device. Compared with so many defective modules in the main device, the redundancy is relatively deficient to repair all of them. Thus, the yield increase by redundancy BISR is not significant. For a very high non-BISR yield (approaches 1), the main device itself is highly likely to be fault-free, and hence repair is not necessary.
Figure 5. The yield increase vs. non-BISR yield for different m numbers
In order to verify the effectiveness of redundancy repair for moderate initial yield, we randomly select # , # , and , the yield of the non-BISR accelerometer is 0.698. For the BISR accelerometer with and , the yield becomes 0.947 which is an increase of 35.7% (when compared with the non-BISR yield 0.698). This demonstrates an effective improvement on the yield for a comb accelerometer through redundancy repair.
6. Design and Simulation of BISR Accelerometer The geometry parameters of the BISR comb accelerometer with and are listed in Table 1, using a set of design rules comparable to ADXL accelerometers [7]. For comparison, a none-BISR accelerometer with the same number of capacitance groups (as that at the main device of the BISR accelerometer) is also designed. The geometry parameters of the non-BISR accelerometer are also listed in the same table. The simulation results for the performance of both BISR and none-BISR accelerometers are shown in Table 2. From Table 2, we can see that, by narrowing the beam width, the sensitivity loss of the BISR accelerometer due to device modularization can be fully compensated. The BISR accelerometer shows approximately the same displacement sensitivity as that of the none-BISR accelerometer. Table 1. Design of BISR/non-BISR accelerometers Design Parameters device area(( ) thickness t (() no. of capacitance groups capacitance gap ( beam width ( beam length ( mass width ( mass length ( finger width ( finger length (
BISR device 1500900 6 206 2 2 300 200 2206 4 200
non-BISR device 980900 6 80 2 3.2 300 200 880 4 200
Table 2. Simulation results of BISR/non-BISR accelerometers
Performance
BISR non-BISR main device device sensing mass ( 0.844=3.36 3.36 capacitance ) * 0.1034=0.41 0.41 sensitivity 6.8 6.64 sensitivity 0.74=2.8 2.74 spring constant 1.214=4.84 4.95 frequency +, 6.05 6.12 In order to demonstrate the effectiveness of the sensitivity compensation by adjusting the beam width, ANSYS [12] simulation has been performed for both non-BISR and BISR accelerometers. Due to the limitation in node number imposed by the available ANSYS version, simplified models for both non-BISR and BISR accelerometers are used for simulation. ANSYS simulation results for the displacement sensitivity in response to acceleration from 0 to 50 are shown in Figure 6. For the non-BISR accelerometer, the beam width is 3.2(m. For the BISR MEMS accelerometer, the beam width is shrunk to 2(m. According to Figure 6, the sensitivity of the non-BISR accelerometer is 6.68 , while the sensitivity of the BISR accelerometer after compensation is 6.83 . The sensitivity of the BISR device is about the same as that of the non-BISR device. Further, the electrostatic force can act as a powerful tool in compensating the sensitivity loss of the BISR design as illustrated in [9]. The sensitivity of the accelerometer can be increased till infinity by applying an appropriate DC biasing voltage given the accelerometer works in the open-loop mode. Thus, as suggested by the simulation results, the sensitivity loss is not a concern for the BISR design.
7. Conclusion and Future Research In this paper, a redundancy-based BISR technique suitable for MEMS devices is proposed. Such a technique can improve the fabrication yield as well as the in-field reliability of MEMS devices. The basic idea of the proposed method is to partition a MEMS device into modules such that defective modules can be virtually eliminated from the working machine. The sensitivity loss (due to device modularization) and its compensation are discussed in detail. A MEMS yield model for redundancy repair has also been developed. Simulation results demonstrate effective yield improvements for MEMS fabrication with a moderate initial yield. The optimized design of a BISR MEMS device can be achieved by adjusting the area of the main and redundant devices. Based on the proposed method, both BISR and non-BISR MEMS accelerometers have been designed and simulated using ANSYS for performance evaluation.
Figure 6. ANSYS simulation results for sensitivity of BISR/non-BISR accelerometers
Compared to the hard repair methods of MEMS, the BISR method through modular redundancy eliminates the need of expensive physical/chemical processing. The entire BISR process is performed automatically by the BISR control circuit. Thus, it is effective for both the manufacturing and the in-field MEMS device repairing. Furthermore, the redundancy BISR scheme is virtually applicable to any local defect. If any of the working modules in the main device is found faulty, regardless whatever local defect it is, the entire faulty module will be replaced by a good redundant module. Thus, the faulty MEMS device can be self-repaired without digging into the detail of the defects in the faulty module. In this way, the redundancy repair method can be an effective and promising solution to fault-tolerant MEMS design. In the BISR design, we trade the hardware overhead for yield and reliability, while the performance remains unchanged. Although the sensitivity compensation techniques (reducing beam width, increasing mass width, electrostatic force compensation, etc.) can also be applied to the non-BISR device to enhance its sensitivity, the yield and reliability of the non-BISR device cannot be improved. Due to fabrication process variations or parametric defects, the performance of one BISR module cannot be perfectly the same as others. Thus, after a redundant module replaces a faulty module, the device must be recalibrated. Our future research is to develop a built-in self-calibration (BISC) technique to ensure a smooth switch between defective and healthy modules, without introducing a significant performance change after redundancy repairing.
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