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This paper develops the technologies of mechanical characterization of CMOS-MEMS devices, and presents a robust algorithm for extracting mechanical properties, such as Young’s modulus, and mean stress, through the external electrical circuit behavior of the micro test-key. An approximate analytical solution for the pull-in voltage of bridge-type test-key subjected to electrostatic load and initial stress is derived based on Euler’s beam model and the minimum energy method. Then one can use the aforesaid closed form solution of the pull-in voltage to extract the Young’s modulus and mean stress of the test structures. The test cases include the test-key fabricated by a TSMC 0.18 μm standard CMOS process, and the experimental results refer to Osterberg’s work on the pull-in voltage of single crystal silicone microbridges. The extracted material properties calculated by the present algorithm are valid. Besides, this paper also analyzes the robustness of this algorithm regarding the dimension effects of test-keys. This mechanical properties extracting method is expected to be applicable to the wafer-level testing in micro-device manufacture and compatible with the wafer-level testing in IC industry since the test process is non-destructive.

Due to the excellent development of the complementary metal oxide semiconductor (CMOS) technology, many micro-electromechanical systems (MEMS) devices such as comb-fingers [

The property-extraction methods for large-scale implementation in MEMS fabrication require additional measurement and actuation equipment or complicated test structure designs. These methods are not compatible with IC metrology technologies. From the mechanical viewpoint of MEMS devices, the important thin-film material parameters are Young’s modulus [

The electrostatic method employs a bias voltage to deflect the microtest structure downward to the ground plane [

A conceptual diagram of a micro bridge is shown in

The bridge is homogeneous and with uniform cross section.

The bridge is within the Euler-Bernoulli model.

The stress gradient is neglected.

Small deflection and ideal fixed boundary conditions.

The mechanical strain energy of an infinitesimal beam element is:

The total mechanical strain energy of the beam, as shown in _{0}) and the second term is the bending strain energy induced by external loads. The fringing fields are considerable and must be taken into account when modeling the electrostatic loads. For an infinitesimal beam element with length _{e}

It should be mentioned that nonlinearities are simplified with only in the electrostatic part of the model. Indeed, the beam structure is assumed linearly elastic, without any consideration of geometrical nonlinearity in virtue of large deformation. Expanding the electrostatic terms in ^{n} << 1 for

The exact solution for the electrostatic-actuated beam is difficult to obtain since it is a nonlinear system with the nonlinear electrostatic force coupled with the structural deflection. Thus, such problem is often solved by the approximate analytical solution. Using the assumed mode method [_{i}_{i}_{i}

Substituting

The system is in static equilibrium when the first-order derivative of the total potential energy _{j}

The coefficients _{j}^{2}^{2} = 0, then one has:

Substituting

_{PI}

Substituting _{PI}

As shown in _{0}) or Young’s modulus (^{2}) and the residual stress (_{0}) should be replaced by the equivalent residual stress _{0}= _{0}(1 –^{2}). Therefore,

The correlation between the pull-in voltage and the material parameters must be formulated quantitatively to realize the idea of extracting mechanical properties from pull-in voltage of the test beam. An equilibrium equation has been derived based on Euler-Bernoulli beam model and the fringing filed capacitance model. The equilibrium equation,

For a given beam with the pull-in voltage _{PI}_{0} and

By rearranging _{0}) and Young’s modulus (

One can extract Young’s modulus (_{0}) easily by substituting the measured pull-in voltages of the two test beams with different length into

Osterberg [_{0}). Note that the cross-section of the beams of group 1 are in the (100) crystalline plane while that of the group 2 are in the (110) crystalline plane. The Young’s modulus of mono-crystalline silicon in (100) and (110) are 138 GPa and 168 GPa, respectively. The mean stresses of the two beam-samples are 10 MPa [

_{0}) of mono-crystalline silicon in (100) and (110), respectively. It is shown that the extracted values of the present algorithm agree well with the average extracted value of Osterberg’s results [_{SD}_{SD}_{0}) are all within 2%, which are also almost tenth of the deviations of Osterberg’s results [

This paper takes bridge-type structures as test structures. The test structures are fabricated by a TSMC 0.18 μm 1P6M standard CMOS process. The upper electrode is metal 2, and the bottom electrode is a poly layer. The anchors are composed of metal and poly layers where viasconnect columns between each metal layer, and contact is the connecting column between metal 1 and the poly layer. When a driving voltage is applied between the test structure and ground plane, the test structure will deflect downward to the ground and this results in a capacitance variation.

For a deflective microstructure subjected to electrostatic loads, as shown in _{C}_{c}_{C}

The frequency and level of the AC testing signal must be set properly since they will affect the accuracy of the capacitance measurement. The authors chose the root mean square value of the test AC signal level as 25 mV and the frequency 1 MHz. The integration time is set to medium (MED). The instrument parameters setting are listed in

The pull-in voltage is detected by tracking the capacitance sensitivities with respect to the applied bias voltages. Two low noise probes touch the two probing pad of test beam, as shown in

_{PI-ave}_{PI}

We substitute the experimental results in _{0}). _{0}) of the two test beams when the length difference (Δ_{0}) are 132.01 ± 13.48 GPa and 3.4 ± 0.15 MPa, respectively.

This work presents an algorithm for extracting Young’s modulus (_{0}) of structural materials of CMOS-MEMS devices by detecting the pull-in voltages of two micro bridge-type test beams. The overall deviations of the extracted Young’s modulus (_{0}) of the demonstrated materials are within 11% and 5%, respectively, when the two test beams have a length difference (Δ

Since this testing method needs to measure pull-in voltages of two test beams with different length, attention should be paid to the appropriate length design of the two test beams. _{0}). The same procedure is used to deal with the experimental results in _{SD}_{SD_E}_{SD_}_{0}

This paper presents a robust algorithm for extracting Young’s modulus, and mean stress of structural materials of CMOS-MEMS devices. By detecting the pull-in voltages of two bridge-type test beams, and applying the characteristics to the equivalent electromechanical models, and one can know the mechanical properties of thin films. The contributions of this paper may be described in detail as follows:

First, the paper has demonstrated the present method with two common structural materials, such as the material made by the TSMC 0.18 μm standard CMOS process, and mono-crystalline silicon in (100) and (110) orientations. The extracted values by the present method are summarized in _{SD_E}_{SD_} σ_{0}_{SD_E}_{SD_} σ_{0}

The authors are thankful for the IC manufacturing support of our research from Taiwan Semiconductor Manufacturing Company (TSMC).

_{PI}) method and long beam deflection (LBD) method

Schematic of the micro fixed-fixed beam.

Layout of the bridge-type test-key.

Scheme cross-section of the bridge-type testkey, (

SEM picture of the bridge-type test-key after post-processing.

Schematic of the experiment setup for pull-in voltage detection.

Typical sensitivities curves of the capacitances with respect to applied bias voltages of the test beam.

The measured the pull-in voltages of the fixed-fixed beams made of mono-crystalline siliconin (100) and (110) orientations in Osterberg’s work [

The variation of the extracted values by this work.

Summary of actuation and measurement methods for extracting material properties.

Interferometer | [ |
[ |
[ |
[ |
[ |
[ |

Nanoindenter | [ |
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AFM | [ |
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SEM | [ |
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μ strain gauge | [ |
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XRD | [ |
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V-F converter | [ |
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Pull-in | [ |

Geometrical parameters of the mono-crystalline silicon beam samples and the measured pull-in voltages [

Permeability of free space |
8.85 × 10^{−12} | |||||

Initial gap |
1.05 | |||||

Beam width |
50 | |||||

Beam thickness |
2.94 | |||||

| ||||||

Length |
175 | 400 | 225 | 450 | 275 | 500 |

| ||||||

Measured pull-in voltage _{PI} |
77.38 | 16.9 | 47.79 | 13.78 | 32.65 | 11.56 |

| ||||||

Length |
175 | 450 | 225 | 500 | 275 | 550 |

| ||||||

Measured pull-in voltage _{PI} |
85.22 | 14.78 | 52.68 | 12.4 | 36 | 10.61 |

Extracted Young’s modulus and mean stress of the mono-crystalline silicon in (100) crystalline plane and the comparison with Osterberg’s work [

_{1} |
_{2} |
_{0} (MPa) |
_{0} (MPa) | ||

175 | 400 | 135.35 | 9.97 | ||

225 | 450 | 135.21 | 9.68 | 138 ± 4 | 10 ± 2 |

275 | 500 | 134.40 | 9.66 | ||

Average (_{ave} |
134.99 | 9.77 | 138 | 10 | |

Standard Deviation (Δ |
0.42 | 0.14 | 4 | 2 | |

Variety of Standard Deviation _{SD}_{SD}_{ave} |
0.31% | 1.45% | 2.90% | 20.00% |

Extracted Young’s modulus and mean stress of the mono-crystalline silicon in (110) crystalline plane and the comparison with Osterberg’s work [

_{1} |
_{2} |
_{0} (MPa) |
_{0} (MPa) | ||

175 | 450 | 166.88 | 9.50 | 168 ± 6 | 10 ± 1 |

225 | 500 | 168.16 | 9.53 | ||

275 | 550 | 167.40 | 9.69 | ||

Average (_{ave} |
167.48 | 9.57 | 168 | 10 | |

Standard Deviation (Δ |
0.53 | 0.08 | 6 | 1 | |

Variety of Standard Deviation _{SD}_{SD}_{ave} |
0.31% | 0.87% | 3.57% | 10.00% |

Measurement conditions.

Testing Signal Frequency | 1 MHz |

Testing Signal Level | 0.025 V |

Bias Voltage Range | 0–40 V |

Bias Voltage Step | 0.05 V |

Integration Time | Med |

Geometrical parameters of the bridge-type test beams.

Beam width |
5 |

Initial gap |
1.93 |

Beam thickness |
0.53 |

Beam length |
220−300 |

The average and standard deviation of pull-in voltage value of each test beam.

_{PI} |
_{PI-ave} |
_{PI} | |
---|---|---|---|

220 | 12.27,12.32,12.32,12.47,12.82 | 12.44 | 0.20 |

230 | 11.56,11.71,11.76,12.27,12.37 | 11.93 | 0.32 |

240 | 10.91,11.41,11.46,11.51,11.56 | 11.36 | 0.23 |

250 | 9.76,10.46,10.71,10.91,11.11 | 10.59 | 0.47 |

260 | 9.76,10.06,10.16,10.26,10.56 | 10.16 | 0.26 |

270 | 9.01, 9.11,9.52, 9.71, 9.96 | 9.46 | 0.36 |

280 | 8.81, 8.86,8.86 ,9.16, 9.51 | 9.04 | 0.27 |

290 | 8.06, 8.61, 8.86, 8.86,9.51 | 8.78 | 0.47 |

300 | 8.06 ,8.11, 8.31, 8.56,8.61 | 8.33 | 0.22 |

Extracted Young’s modulus and mean stress of structural material fabricated by TSMC 0.18 μm 1P6M standard CMOS process.

Δ |
_{1} |
_{2} |
_{0} (MPa) | |

50 | 220 | 270 | 112.75 | 3.56 |

230 | 280 | 147.87 | 3.15 | |

240 | 290 | 140.74 | 3.41 | |

250 | 300 | 126.68 | 3.48 | |

| ||||

Average (_{ave} |
132.01 | 3.40 | ||

Standard Deviation (Δ |
13.48 | 0.15 | ||

Variety of Standard Deviation _{SD}_{SD}_{ave} |
10.21% | 4.52% |

The extracted results for common structural materials.

metal 2 made by the TSMC 0.18 μm standard CMOS process | _{0}(MPa) | |

132.01 ± 13.48 | 3.4 ± 0.15 | |

mono-crystalline silicon in (100) | 134.9 9± 0.42 | 9.77 ± 0.14 |

mono-crystalline silicon in (110) | 167.48 ± 0.53 | 9.57 ± 0.08 |