中国物理快报 Chinese Physics Letters
Total Page：16
File Type：pdf, Size：1020Kb
ISSN: 0256-307X 中国物理快报 Chinese Physics Letters A Series Journal of the Chinese Physical Society Distributed by IOP Publishing Online: http://www.iop.org/journals/cpl http://cpl.iphy.ac.cn CHINESE PHYSICAL SOCIETY CHIN. PHYS. LETT. Vol. 27, No. 5 (2010) 056201 Frequency Response of the Sample Vibration Mode in Scanning Probe Acoustic Microscope * ZHAO Ya-Jun(ë亚军), CHENG Qian(§0), QIAN Meng-Lu(a梦è)** Institute of Acoustics, Tongji University, Shanghai 200092 (Received 22 December 2009) Based on the interaction mechanism between tip and sample in the contact mode of a scanning probe acoustic microscope (SPAM), an active mass of the sample is introduced in the mass-spring model. The tip motion and frequency response of the sample vibration mode in the SPAM are calculated by the Lagrange equation with dissipation function. For the silicon tip and glass assemblage in the SPAM the frequency response is simulated and it is in agreement with the experimental result. The living myoblast cells on the glass slide are imaged at resonance frequencies of the SPAM system, which are 20 kHz, 30 kHz and 120 kHz. It is shown that good contrast of SPAM images could be obtained when the system is operated at the resonance frequencies of the system in high and low-frequency regions. PACS: 62. 30. +d, 68. 37. Ps, 07. 79. Lh DOI: 10.1088/0256-307X/27/5/056201 The quasi-static detection technology of scanning The sample contacting with the tip can be con- probe microscopy (SPM) has been rapidly develop- sidered as a dynamic load of the tip in the sample ing into a fully dynamic detection technology recently. vibration mode of SPAM. Effects of both the stiffness The sample vibration mode in scanning probe acoustic and the mass of the sample related to the motion of microscopy (SPAM) is known as an imaging and mea- the tip should be taken into account. Therefore an suring technology when the probe tip contacts with active mass of the sample should be introduced in the samples. The experimental setup was presented in mass-spring model of SPAM. Ref. [1]. The vibration applied to the sample is gener- The external force applied to the sample surface ated by a piezoelectric vibration table which is coupled by the cantilever is usually about several nN to µN, to its bottom. The signal is detected by the probe therefore the displacement at the sample surface is tip when it scans the top surface of the sample and very small. The SPAM system should be operated then sent to a lock-in amplifier. Thus an SPAM image at its resonance frequencies in order to obtain good in nano-scale is obtained and the contrast is related signal.[18] Thus it is crucial to develop a more com- to the surface topography, the variation of the tip- plete dynamical model for sample vibration mode in sample interaction forces and the mechanical proper- SPAM and analyze the relevant frequency characteris- ties at the surface and/or the subsurface of the sam- tic in order to improve the performance of the SPAM ple. SPAM imaging has been demonstrated to be a imaging and explore more detailed information of the powerful tool for the surface and subsurface imaging sample. of inorganic and organic materials, such as ferroelec- tric domains of ferroelectric material, properties of biological sample, nanostructure of nano-material as k well as the measurement of the local elastic stiffness R of sample surface.[1−11] The imaging mechanism and m z its spatial resolution of the sample vibration mode in f SPAM have been studied by many researchers using linear mass-spring model, which consists of the mass k of the tip m1, cantilever spring constant k1, its damp- R ing coefficient R1, the tip-surface interaction stiffness [12−17] k and damping coefficient R . High driving m 2 2 z frequency near the resonance frequency of cantilever probe was considered to be helpful in improving image f quality.[1;2] However, some SPAM images with high Fig.1. The new mass-spring model of sample vibration spatial resolution were obtained at low driving fre- mode in SPAM. quency, recently.[4;5] This means that a more complete mass-spring model should be set up for the sample vi- A new mass-spring model of SPAM including an brating mode in SPAM. active mass of sample is shown in Fig. 1. The active *Supported by the National Natural Science Foundation of China under Grant Nos 10774113 and 10834009, and the Doctoral Fund of Ministry of Education of China under Grant No 20070247042. **Email: [email protected] ○c 2010 Chinese Physical Society and IOP Publishing Ltd 056201-1 CHIN. PHYS. LETT. Vol. 27, No. 5 (2010) 056201 h 4 (︁ 2 2 2 !1!2 )︁ 2 2 2i mass of the sample m2 will be numerically analyzed. D = ! − !1 + !2 + !12 + ! + !1!2 Q1Q2 In the mass-spring model illustrated above, m1, R1 2 2 and k represent the tip mass, damp coefficient and [︁(︁! !2 ! !1 )︁ 1 + i! 1 + 2 cantilever spring constant, and m2, R2 as well as k2 Q2 Q1 are the active mass of the sample, damping coefficient (︁ !1 !2 m2!2 )︁ i − + + !2 : (6) and tip-surface interaction stiffness respectively. Here Q1 Q2 m1Q2 f1 is the force applied to the sample surface by the cantilever, and f the force applied on the active mass In Eq. (5) it is clearly shown that the displacement 2 of the tip is related not only to the driving force and m2 by piezoelectric table, z1 and z2 are the displace- ments of the tip and the sample surface respectively. properties of the sample, but also to the frequency Then the frequency characteristics of this system are characteristics of the system. investigated with Lagrange equations. The frequency response of the system is deter- mined by F (!) = 1=jDj. The resonance frequencies of Choosing z1 and z2 as the generalized coordinates, the Lagrange equations with the dissipation function the system can be found from jDj = 0: can be expressed as [︁(︁ ! ! m ! )︁2 !8 + 1 + 2 + 2 2 d @L @L @F d @L @L @F Q1 Q2 m1Q2 − = − ; − = − ; (1) (︁ )︁]︁ dt @z_1 @z1 @z_1 dt @z_2 @z2 @z_2 2 2 2 !1!2 6 − 2 !1 + !2 + !12 + ! where L = T − U, T and Uare the kinetic energy Q1Q2 2 and potential energy of the dynamical system respec- [︁(︁ 2 2 2 !1!2 )︁ 2 2 + !1 + !2 + !12 + + 2!1!2 tively; F is the dissipation function, and they can be Q1Q2 expressed as (︁!2! !2! )︁(︁ ! ! m ! )︁]︁ − 2 1 2 + 2 1 1 + 2 + 2 2 !4 1 2 1 2 Q2 Q1 Q1 Q2 m1Q2 T = m1z_1 + m2z_2 ; 2 2 h 2 2(︁ 2 2 2 !1!2 )︁ − 2!1!2 !1 + !2 + !12 + 1 2 1 2 Q1Q2 U = k1z + k2(z1 − z2) − f1z1 − f2z2; 1 2 2 2 2 2 (︁!1!2 !2!1 )︁ i 2 4 4 1 1 − + ! + !1!2 = 0: (7) F = R z_2 + R (_z − z_ )2; Q2 Q1 2 1 1 2 2 1 2 1 1 1 Obviously, it is more likely to obtain better signal L = m z_2 + m z_2 − k z2 2 1 1 2 2 2 2 1 1 when an SPAM system is operated at its resonance 1 frequencies. − k (z − z )2 + f z + f z : (2) 2 2 1 2 1 1 2 2 In order to simulate the frequency response of the SPAM system operated in sample vibration mode, all Substituting Eq. (2) into Eq. (1), we obtain of the system parameters in Eq. (7) must be deter- m1z¨1 + (R1 + R2)_z1 + (k1 + k2)z1 − R2z_2 − k2z2 =f1; mined. For subsystem of cantilever-tip, its spring − R2z_1 − k2z1 + m2z¨2 + R2z_2 + k2z2 = f2: (3) constant k1 = 0:11 N/m, response frequency f1 = i!t i!t i!t 22:26 kHz and Q1 = 54 are measured with thermal Then let f2 = f20e , z1 = Ae , z2 = Be and tuning technique in the SPAM, respectively. Then substitute them into Eq. (3), the equations about A 2 −12 the mass m1 = k1=!1 = 5:76 × 10 kg and R1 = and B are obtained, 1:47 × 10−8 kg/s can be calculated by Eq. (4a). 2 [19−21] [−m1! + i!(R1 + R2) + (k1 + k2)]A Based on the Hertz contact theory, the tip- surface interaction stiffness k , the radius of contact − (i!R2 + k2)B = 0; 2 volume between the tip and the sample a, and the − (i!R + k )A + (−m !2 + i!R 2 2 2 2 indentation depth on the sample surface 훿 can be es- + k2)B = f20: (4) timated by Using the expressions 1=3 2 * (︁3RFN )︁ a k k k k2 = 2aE ; a = * ; 훿 = ; (8) !2 = 1 ;!2 = 2 ;!2 = 2 ; 4E R 1 m 2 m 12 m 1 2 1 where R is the radius of curvature of the tip, E* is the R ! i = i ; (i = 1; 2) (4a) reduced elastic modulus for sample-tip system, mi Qi 2 2 the solutions to Eq. (4) can be expressed as 1 1 − 휈1 1 − 휈2 * = + ; (9) E E1 E2 A = f20(k2 + i!R2)=∆; which is determined by Young's modulus E and Pois- f20 2 B = [(k1 + k2 − m1! ) + i!(R1 + R2)]; (5) son's ratio 휈 of both the tip and the sample. F = ∆ N GSk is the applied static force, where G is the de- where 1 flection setpoint, S deflection sensitivity and k1 is the ∆ = m1m2D; spring constant of the cantilever.