無阻尼 V 型微探針之分析與討論

合一般均勻樑的現象。

圖3.4 為在長寬比為1的情形下，幾何參數ξ1對系統自然頻率的影響。

ξ₁ 越小自然頻率越高，可以得到結論，V 型樑的共振頻率比均勻樑高，

所以在進行 AFM 量測時，使用 V 型探針可以提高系統的共振頻率，避 免外界低頻的雜訊干擾。

表 3.7 共振頻率值之比較 理論[33]

(kHz)

本論文(kHz) error(%)

33.4 33236.3333 0.55128

表3.8 無因次自然頻率值的比較 模態 本論文 文獻[36]

1 3.51601 3.51562

2 22.03449 22.03363 3 61.69721 61.70102

0 2000 4000 6000 8000 10000 12000

0 0.5 1 1.5 2 2.5 3 3.5 4

B/L

frequency(kHz) 1 mode 1

mode 2 mode 3 mode 4

圖 3.3 ξ₁=0.3時，不同長寬比對自然頻率之影響

0 20 40 60 80 100 120 140 160 180 200

0 0.2 0.4 0.6 0.8 1

frequency shift (kHz)

damped mode 1 damped mode 2 undamped mode 1 undamped mode 2

圖 3.4 ξ₁對系統自然頻率的影響

ξ

3.3 具阻尼 V 型微探針之數值分析與討論

考慮一 V 型均勻厚度微型樑，一端固定在彈性基台，另一端為自由 端，且承受薄膜阻尼力。所使用的微型樑尺寸及其他材料係數等，也是 使用表 3.1的數值，除了w_b =20µm，但是多考慮空氣薄膜厚度10μm， 所對應的阻尼係數a₀ =0.102，圖 3.4 比較在有阻尼與無阻尼況態下共振 頻率變化的情形。可以發現沒有阻尼的樑自然頻率比較大。挖空比率ξ₁ 越小，自然頻率下降越多，這是因為阻尼效應造成的。原因是阻尼係數 跟幾何參數ξ1有關聯，圖 3.5 可以看出ξ1越大，a₀越小，也就是說 V 型 樑面積越大，薄膜阻尼越大。

圖 3.6 顯示樑前四個共振頻率，在薄膜阻尼係數a₀變化時，頻率偏 移之情形。圖中可看出a₀愈大，則頻率偏移也愈大，同時也顯示出，a₀的 變化對較低模態之共振頻率的影響較為明顯。這也表示說，薄膜阻尼對 系統自然頻率的影響，在低模態的時候比較顯著。

在 AFM 探針振動中，如果衰減速率ζ越大，系統會越快達到穩定狀 態。由於阻尼跟衰減速率成正比，阻尼越大振幅衰減越快，但是阻尼係 數跟面積成正比，也就是跟幾何參數ξ₁有關。由圖3.7 中，可以看到在第

間。

另外在圖 3.8 中，發現長寬比越大，衰減速率越大。這是因為在圖 3.9 中，長寬比增加就是面積增加，使得阻尼係數提高。所以會造成長寬 比越大、衰減速率越大的情形。而且，越高的振動模態，衰減越快。

0 0.2 0.4 0.6 0.8 1 1.2 1.4

0 0.2 0.4 0.6 0.8 1

圖 3.5 ξ1對阻尼係數a₀的影響

a

ξ

0 5 10 15 20 25 30

0 2 4 6 8 10

frequency shift (kHz) 1

mode 1 mode 2 mode 3 mode 4

圖 3.6 a₀對頻率偏移之影響 a0

0 2 4 6 8 10 12 14

0 0.2 0.4 0.6 0.8 1

decay rate ζ mode 1

mode 2 mode 3 mode 4

ξ

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0 0.5 1 1.5 2 2.5 3 3.5 4

B/L decay rate ζ 1.

mode 1 mode 2 mode 3

圖 3.8 長寬比對衰減速率ζ的影響

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0 0.5 1 1.5 2 2.5 3 3.5 4

B/L

圖 3.9 B/L 對阻尼係數a₀的影響

a

第四章 結論

本篇論文推導出AFM 均勻厚度之V型探針正解，得到其自然共振頻 率受薄膜阻尼與探針尺寸的影響情形。另外，也求出具有根部彈性拘束 與非線性邊界條件的 AFM非均勻探針的正解，得到探針尖端撓度和探針 尖端與樣品表面距離的關係，還有參數對系統靈敏度的影響。以下是最 後的結論﹕

(1) 在沒有阻尼的時候，ξ1減少會提昇系統的靈敏度和自然頻率，有 助於避免量測時外在多餘的激振。

(2) 在有阻尼的時候，ξ1減少，樑截面積增加，也就是阻尼效應增大，

自然頻率會明顯地下降。

(3) 不論是有阻尼或無阻尼，降低ξ1可以提高自然頻率。

(4) 薄膜阻尼係數a₀愈大，則共振頻率偏移量也愈大。

(5) 在第一模態，ξ1越小衰減速率越大。較高模態下，衰減速率最大 值出現在中間。

(6) 探針自重對靈敏度的效應非常小。

(7) 在ξ1 <0.3時，幾何形狀係數ξ2越大時，系統的靈敏度越大。在ξ1

>0.3時，幾何形狀係數ξ2越小時，系統的靈敏度越大。

參考文獻

[1] Binnig, G., Roher, H., Gerber, H. and Weibel E. “Surface studied by scanning tunneling microscopy,” Physical Review Letters, Vol. 49, pp.

57-61, 1982.

[2] Binnig, G., Quate, C. F., and Gerber, Ch., “Atomic force microscope,”

Physical Review Letters, Vol. 56, pp. 930-933, 1986.

[3] Holscher, H., Schwarz, U. D, Wiesendanger, R., “Calculation of the frequency shift in dynamic force microscopy,” applied surface science, Vol.140, pp. 344-351, 1999.

[4] Sasaki, N., Tsukada, M., Tamura,R., Abe,K.,Sato,N., “Dynamics of the cantilever in noncontact atomic force microscopy,” Applied physics A : Materials, Vol.66, pp. 287-291, 1998

[5] Rabe, U., Janser, K., and Amold,W., “Vibrations of free and

surface-coupled force microscope cantilevers : Theory and experiment,”

American institute of physics, Vol.67, pp. 3281-3293, 1996

[6] Fung, R. F., Huang, S. C., “Dynamic modeling and vibration analysis of the atomic force microscope,” ASME, Journal of Vibration and

Acoustics, Vol.123, pp. 502-509, 2001

[7] Sokolov, I. Y., Henderson, G. S.,Wicks, F. J., “Model dependence of AFM simulations in non-contact mode, ” surface science, Vol.457, pp.

267-272, 2000

[9] Sasaki, N., Tsukada, M., Tamura,R., Abe,K.,Sato,N., “Dynamics of the cantilever in noncontact atomic force microscopy,” Applied physics A Materials, Vol.66, pp. 287-291, 1998.

[10] Rabe, U., Janser, K., and Amold,W., “Vibrations of free and surface-coupled force microscope cantilevers : Theory and experiment,”

American institute of physics, Vol.67, pp. 3281-3293, 1996.

[11] Weigert, S., Dreier, M., and Hegner, M., “Fequency shifts of cantilevers vibrating in various media,” Appl. phys. letter.,Vol.69, pp.2834-2836, 1996.

[12] Fung, R. F., Huang, S. C., “Dynamic modeling and vibration analysis of the atomic force microscope,” transactions of ASME, Vol.123, pp.

502-509, 2001.

[13] Sokolov, I. Y., Henderson, G. S.,Wicks, F. J., “Model dependence of AFM simulations in non-contact mode, ” surface science, Vol.457, pp.

267-272, 2000.

[14] Neumeister J. M., Ducker W. A., “Lateral, normal, and longitudinal spring constants of atomic force microscopy cantilevers,” Rev. Sci.

Instrum., Vol. 65(8), pp.2527-2531, 1994

[15] Cleveland J. P., Manne S., Bocek D., Hansma P. K., “A nondestructive method for determining the spring constant of cantilevers for scanning force microscopy,” Rev. Sci. Instrum., Vol. 64 (2), pp.403-405, 1993 [16] Hazel J. L., Tsukruk V. V., “Spring constants of composite ceramic/gold

cantilevers for scanning probe microscopy,” Thin Solid Films, Vol. 339, pp.249-257, 1999

[17] Gibson C. T., Weeks B. L., Abell C., Rayment T., Myhra S., “Calibration of AFM cantilever spring constant,” Rev. Sci. Instrum., Vol. 97, pp.113-118, 2002

[18] Sader, J. E., “Parallel beam approximation for V-shaped atomic force microscope cantilevers,” Rev. Sci. Instrum., Vol. 66(9), pp.4583-4587, 1995

[19] Sader, J. E., White L., “Theoretical analysis of the static deflection of plates for atomic force microscope applications,” J. Appl. Phys., Vol.

74(1), pp.1-9, 1993

[20] Sader, J. E., “Surface stress induced deflections of cantilever plates with applications to the atomic force microscope: V-shaped plates,” American Institute of Physics, Vol. 91, pp.9354-9361, 2002

[21] Kokubun, K., Murakami, H., Toda, Y. and Ono, M., “A bending and stretching mode crystal oscillator as friction vacuum gauge,” Vacuum, Vol. 34, pp. 731-735, 1984

[22] Blom, F.R., Bouwstra, S., Elwenspoek, M. and Fluitman, J.H.J.,

“Dependence of the quality factor of micro-machined silicon beam resonators on pressure and geometry,” J. vac. Sci. Technol., B10, pp.

19-26, 1992

[23] Hirano, T., Funrehata, T., Gabriel, K. J. and Fujita, “Effect of air damping and suspension compliance on electrostatic comb-drive actuators with sub-micron gap,” Proc. 2^nd Int. Symp. Measurement and Control in

[24] Murakami, Y. and Moronuki, N., “study on the design of micro-machine,” 1^st Report, Proc. Jpn. Soc. Precision Eng., pp. 919-920, 1993.

[25] Matsuda, R. and Fukui, S., “Asymptotic analysis of ultra-thin gas squeeze film lubrication for infinitely large squeeze number,” ASME Paper 94-Trib-2, 1994

[26] Darling, R. B., Hivick, C., and Xu, J., “Compact analytical modeling squeeze film damping with arbibratrary venting conditions using a Green’s function approach,” Sensors and Actuators, Vol. A70, pp. 32-41, 1998

[27] Xu, Y. and Smith, S. T, “Determination of squeeze film damping in capacitance-based cantilever force probes,” Precision Engineering, Vol. 17, pp. 94-100, 1995

[28] Hiroshi Hosaka, Kiyoshi Itao, Susumu Kuroda, “Damping characteristics of beam-shaped micro-oscillators,” Sensors and Actuators A Physical, Vol. 49, pp.87-95, 1995

[29] Chen, G. Y., Warmack, R. J., Thundat, T., Allison, D. P., “Resonance response of scanning force microscopy cantilevers,” Rev. Sci. Instrum., Vol. 65(8), pp.2532-2537, 1994

[30] Fung, R. F., Huang, S. C., “Dynamic modeling and vibration analysis of the atomic force microscope,” ASME, Journal of Vibration and

Acoustics, Vol.123, pp. 502-509, 2001

[31] Lee, S. Y. and Lin, S. M., “Dynamic analysis of non-uniform beams with time-dependent elastic boundary conditions,” ASME Journal of Applied Mechanics, Vol. 63, pp. 474-478, 1996

[32] Lin, S. M., “Dynamic analysis of Rotating nonuniform Timoshenko beams with an Elastically restrained root,” ASME Journal of Applied Mechanics, Vol. 66, pp. 742-749, 1999

[33] Lee, S. Y., Lin, S. M., “Exact solutions for the vibrations of an elastically restrained nonuniform Timoshenko beam with tip mass,” AIAA

Journal, Vol. 30, No. 12, pp. 2930-2934, 1992

[34] Tortonese, M., “Cantilevers and tips for atomic force microscopy,” IEEE Eng. Med. Biol., Vol. 16(2), pp.28-33, 1997

[35] Sader, J. E., Larson I., Mulvaney P., White L. R., “Method of the calibration of atomic force microscope cantilevers,” Rev. Sci. Instrum., Vol. 66(7), pp.3789-3798, 1995

[36] Meirovitch, L., “Analytical methods in vibration,” The Macmillan company, pp.163, 1967

[37] Holscher, H., Schwarz, U. D., Wiesendanger, R., “Calculation of the frequency shift in dynamic force microscopy,” Applied Surface Science, Vol.140, pp. 344-351, 1999

[38] _林昆蔚, “非接觸式原子力顯微鏡探針振動分析” ,國立成功大學機械 工程學系碩士論文, 2003