Surface Acoustic Wave Characterization of Equivalent Young's Moduli for Patterned Films
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摘要: 基于分层材料等效弹性性质理论，建立了单大马士革和双大马士革结构的细观力学等效模型. 通过引入图形化结构等效弹性常数，建立了声表面波在图形化薄膜/衬底分层结构中的传播模型，利用Green函数和矩阵法计算得到了表面波理论色散曲线. 采用有限元法计算了24种不同体积比的大马士革结构数值算例，并与能量法计算结果进行了对比验证. 结果表明，厚度为300 nm的双大马士革薄膜和厚度为100 nm的单大马士革薄膜等效弹性模量平均相对误差分别为2.06%和2.27%. 验证了等效图形化结构模型的正确性以及声表面波方法表征图形化薄膜机械特性的可行性，研究结果可为开发适用于图形化薄膜、低压力的化学机械抛光工艺提供参考.Abstract: Based on the equivalent elasticity theory for layered materials, the micro-mechanics equivalent models for single and dual damascene structures were established. The equivalent elastic constant of the patterned structure was introduced, to establish the propagation model for the surface acoustic waves propagating in the layered structure of the patterned film/substrate, and the theoretical dispersion curves of the surface acoustic waves were calculated with Green's function and the matrix method. The finite element method was used to calculate 24 numerical examples of damascene structures with different volume ratios, and the results were compared with those of the strain energy method. The results show that, the average relative errors of the equivalent Young's moduli of the 300 nm-thick dual damascene film and the 100 nm-thick single damascene film are 2.06% and 2.27%, respectively. The research verifies the correctness of the equivalent patterned structure model and the feasibility of the surface acoustic wave method to characterize the mechanical properties of patterned films, and provides a reference for the development of suitable chemico-mechanical polishing technologies for patterned films under low pressure.
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图 2 单大马士革结构代表体积单元
Figure 2. The representative volume unit of the single damascene structure
图 3 双大马士革结构代表体积单元
Figure 3. The representative volume unit of the dual damascene structure
图 4 表面波在双大马士革薄膜/硅衬底结构中的传播
Figure 4. The schematic diagram of SAWs propagating in the dual damascene film/substrate
图 6 图形化薄膜/硅衬底有限元网格化模型
Figure 6. The finite element meshing model for the patterned film/silicon substrate
图 7 模型的表面振型及特征频率
注 为了解释图中的颜色，读者可以参考本文的电子网页版本.
Figure 7. The surface modes and eigenfrequencies of the model
表 1 双大马士革薄膜参数(参数标记见图 3)和对应编号
Table 1. Dual damascene film parameters (see fig. 3 for parameter labels) and corresponding numbers
case No. w1/nm h1/nm w2/nm h2/nm ① homogeneous low-k material ② 50 100 50 200 ③ 50 100 60 200 ④ 50 100 90 200 ⑤ 50 100 120 200 ⑥ 50 100 150 200 ⑦ 50 100 180 200 ⑧ 50 100 210 200 ⑨ 50 100 240 200 ⑩ 50 100 270 200 ⑪ homogeneous copper material 
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表 2 仿真模型的波长-频率-速度值
Table 2. Wavelength-frequency-velocity values for the simulation model
wavelength λ/μm average eigenfrequency f/MHz velocity v/(m/s) 990 4.964 4 914.36 480 10.23 4 910.40 210 23.32 4 897.20 129 37.87 4 885.23 99 49.24 4 874.76 81 60.06 4 864.86 60 80.76 4 845.60 51 94.75 4 832.25 45 107.12 4 820.40 42 114.61 4 813.62 39 123.21 4 805.19 36 133.20 4 795.20 33 144.96 4 783.68 30 158.99 4 769.70 27 176.02 4 752.54 24 197.12 4 730.88
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表 3 双大马士革薄膜(hf=300 nm)等效弹性模量匹配结果
Table 3. Matching results of the dual damascene structure with thickness hf=300 nm
case No. volume(Cu: low-k) SAW method strain energy method (ΔEf/Efenergy)/% Ef/GPa Efenergy/GPa ① 0∶1 (full of low-k) 9.90 10.00 1.00 ② 1∶5.0 12.87 13.23 2.72 ③ 1∶4.3 13.37 13.64 1.98 ④ 1∶2.9 14.63 15.01 2.53 ⑤ 1∶2.1 16.94 16.72 1.32 ⑥ 1∶1.6 18.74 18.96 1.16 ⑦ 1∶1.2 22.01 22.08 0.32 ⑧ 1.1∶1 26.20 26.71 1.91 ⑨ 1.4∶1 33.00 34.34 3.90 ⑩ 1.9∶1 47.74 49.39 3.34 ⑪ 1∶0 (full of Cu) 129.90 130.00 0.08
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表 4 单大马士革薄膜(hf=100 nm)等效弹性模量匹配结果
Table 4. Matching results of the single damascene structure with thickness hf=100 nm
case No. volume(Cu: low-k) SAW method strain energy method (ΔEf/Efenergy)/% Ef/GPa Efenergy/GPa ① 1∶9 11.78 12.06 2.32 ② 1∶7 12.07 12.48 3.29 ③ 1∶5 12.87 13.23 2.72 ④ 1∶4 13.56 13.83 1.95 ⑤ 1∶3 14.73 14.79 0.41 ⑥ 1∶2 16.76 16.59 1.02 ⑦ 1∶1 21.79 21.58 0.97 ⑧ 2∶1 30.52 30.27 0.83 ⑨ 3∶1 37.24 37.68 1.17 ⑩ 4∶1 42.87 44.02 2.61 ⑪ 5∶1 47.32 49.42 4.25 ⑫ 7∶1 56.28 58.69 4.11 ⑬ 9∶1 63.50 66.06 3.88
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