图形化薄膜等效弹性模量的声表面波表征研究

陈龙; 肖夏; 张立; 戚海洋

doi:10.21656/1000-0887.430050

图形化薄膜等效弹性模量的声表面波表征研究

doi: 10.21656/1000-0887.430050

陈龙^1,,
肖夏^1, ,,
张立¹,
戚海洋²

1.
天津大学微电子学院，天津 300072
2.
浙江省计量科学研究院，杭州 310063

基金项目:

国家自然科学基金项目 61571319

详细信息

作者简介:
陈龙(1997—)，男，硕士生(E-mail: chenlong_cl@yeah.net)

通讯作者:
肖夏(1971—)，女，教授，博士，博士生导师(通讯作者. E-mail: xiaxiao@tju.edu.cn)

中图分类号: O341;TN47
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- 被引次数: 0
出版历程
- 收稿日期: 2022-02-21
- 修回日期: 2022-09-04
- 刊出日期: 2023-04-01

Surface Acoustic Wave Characterization of Equivalent Young's Moduli for Patterned Films

CHEN Long^1
,,
XIAO Xia^{1
, ,},
ZHANG Li¹,
QI Haiyang²

1.
School of Microelectronics, Tianjin University, Tianjin 300072, P.R.China
2.
Zhejiang Institute of Metrology, Hangzhou 310063, P.R.China

摘要

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.
- 声表面波 /
- 大马士革结构 /
- 图形化薄膜 /
- 等效弹性模量 /
- 数值模拟
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.
- surface acoustic wave /
- damascene structure /
- patterned film /
- equivalent Young's modulus /
- numerical simulation

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图 1 分层材料及其代表性体积单元^[23]

Figure 1. The schematic diagram of the layered material and the representative volume unit^[23]

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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

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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

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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

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图 5 膜厚为300 nm下表面波在表 1的不同图形化薄膜中沿Si[100]晶向传播的色散曲线

Figure 5. Dispersion curves of surface acoustic waves propagating along the Si[100] direction in different patterned films in table 1 with a film thickness of 300 nm

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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

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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

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图 8 膜厚为300 nm下表面波在表 1编号⑥对应双大马士革薄膜中沿Si[100]晶向传播的理论和仿真色散曲线

Figure 8. Theoretical and simulated dispersion curves of surface acoustic waves propagating along the Si[100] direction in the dual damascene film corresponding to case ⑥ in table 1 with a film thickness of 300 nm

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图 9 表面波沿Si[100]晶向传播时单大马士革薄膜(表 4编号⑤)厚度对色散关系的影响

Figure 9. Influences of the single damascene film thickness (case ⑤ in table 4) on the dispersion relation for the surface wave propagating along the Si[100] direction

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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(h_f=300 nm)等效弹性模量匹配结果

Table 3. Matching results of the dual damascene structure with thickness h_f=300 nm

case No.	volume(Cu: low-k)	SAW method	strain energy method	(ΔE_f/E_f^energy)/%
case No.	volume(Cu: low-k)	E_f/GPa	E_f^energy/GPa	(ΔE_f/E_f^energy)/%
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(h_f=100 nm)等效弹性模量匹配结果

Table 4. Matching results of the single damascene structure with thickness h_f=100 nm

case No.	volume(Cu: low-k)	SAW method	strain energy method	(ΔE_f/E_f^energy)/%
case No.	volume(Cu: low-k)	E_f/GPa	E_f^energy/GPa	(ΔE_f/E_f^energy)/%
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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