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Article

The Model Analysis of a Complex Tuning Fork Probe and Its Application in Bimodal Atomic Force Microscopy

State Key Laboratory of Precision Measuring Technology and Instruments, Tianjin University, Tianjin 300072, China
*
Author to whom correspondence should be addressed.
Appl. Sci. 2017, 7(2), 121; https://doi.org/10.3390/app7020121
Submission received: 20 November 2016 / Revised: 18 January 2017 / Accepted: 19 January 2017 / Published: 25 January 2017
(This article belongs to the Special Issue Dimensional Micro and Nanometrology)

Abstract

:
A new electromechanical coupling model was built to quantitatively analyze the tuning fork probes, especially the complex ones. A special feature of a novel, soft tuning fork probe, that the second eigenfrequency of the probe was insensitive to the effective force gradient, was found and used in a homemade bimodal atomic force microscopy to measure power dissipation quantitatively. By transforming the mechanical parameters to the electrical parameters, a monotonous and concise method without using phase to calculate the power dissipation was proposed.

Graphical Abstract

1. Introduction

Quartz tuning forks are widely used in clocks, watches, and digital circuit frequency standards. By taking advantage of their extreme stability in frequency, their self-sensing and self-actuating capabilities, their high quality factor, and the ease with which the vibration signal may be obtained with fewer components than the conventional scanning probe microscopy (SPM) probes, etc., they can be used as force sensors in SPM [1,2,3,4,5]. The tuning fork probes are typically realized in two forms (Figure 1). The tip of the probe could be a carbon nanotube, an optical fiber, a conventional Atomic Force Microscopy (AFM) cantilever, or another type of stylus. With the development of some complex tuning fork probes [6,7], the quantitative use of the data (in terms of amplitude, phase, and frequency shifts) becomes challenging for these sophisticated probes. In this article, a kind of electromechanical coupling model with multistage coupling is proposed and used to describe a complex probe.

2. Modelling

The tuning fork probe can be seen as a mechanical oscillator in interaction with the surface by the tip. However, the output signal is current caused by the tuning fork’s piezoelectric effect. Tuning fork probes can be described by electromechanical coupling models [8,9,10].
A system’s differential equation of motion is
m X ¨ ( t ) + b X ˙ ( t ) + k X ( t ) = F ( t )
A circuit’s differential equation is
I ( t ) C + L I ¨ ( t ) + R I ˙ ( t ) = U ˙ ( t )
By comparing the Equations (1) and (2) after Laplace transformation as shown in Equation (3), the corresponding relationship of the electrical and mechanical parameters are shown in Table 1.
m s 2 + b s + k = F ( s ) X ( s ) = F ( s ) s v ( s ) L s 2 + R s + 1 C = U ( s ) s I ( s )
The coupling can be regarded as an ideal transformer. According to the principle of the ideal transformer as shown in Figure 2a, the equivalent impedance Z′ in the left part is:
U 1 2 Z = U 2 2 Z Z = U 1 2 U 2 2 Z = m 2 Z
And the relation between Z and Z′ in Figure 2b is
2 U 1 2 Z = U 2 2 Z U 1 U 2 = m 2 Z = 2 U 1 2 U 2 2 Z = m 2 2 Z

2.1. The Model of Normal Tuning Fork Probes

At first, the multistage coupling model as shown in Figure 3 is utilized to specify normal tuning fork probes. The feature of the new model is two mechanical coupling stages are added to describe the cantilever (or the tip) and the sample respectively. The couplings are regarded as ideal transformers. All the mechanical parameters will be transformed to electrical parameters in an RLC circuit using the principle of ideal transformers.
This electromechanical coupling model consists of three parts: a typical electromechanical coupling part describing the tuning fork, a mechanical coupling part representing the connection between the cantilever (or tip) and the tuning fork, and another mechanical coupling part explaining the contact between the cantilever tip and the sample. In the model, the parameters, b, k, m, F, v, with different subscripts, denote damping, spring constant, mass, force, and the velocity of the tuning fork (subscript tf), and cantilever (subscript cl), respectively; Cp is the parasitic capacitance; i is the current; Vin is the excitation voltage; p is the electromechanical coupling factor; n, bsa, and keff are the mechanical coupling factor, the damping, and effective force gradient between the tip and the sample. If the normal tuning fork’s tip is regarded as a rigid body, the spring constant (kcl) of the tip is infinite and the damping (bcl) is zero. The coupling factor between the tip and the tuning fork, and the factor between the tip and sample (factor n in Figure 3) are one because the tip is adhered to the tuning fork directly, vtf = vcl = vtip. So the model can be transformed to:
With Equation (4), the model in Figure 4 can be transform to a simple RLC circuit as shown in Figure 5.
And the parameters in Figure 5 are
R probe = p 2 ( b tf + b sa ) L probe = p 2 ( m tf + m cl ) C probe = 1 p 2 ( k tf + k eff )
So the frequency shift after contact is
Δ f = f f 0 = 1 2 π L probe C probe 1 2 π L probe C probe = 1 2 π L probe C probe C probe ( C probe C probe ) = f 0 C probe ( C probe C probe )
Δ f f 0 = C probe C probe 1 = k tf + k eff k tf 1 = 1 + k eff k tf 1 ( Δ f f 0 ) 2 + 2 ( Δ f f 0 ) = k eff k t f
For Δf << f0,
k eff 2 Δ f f 0 k tf
where ktf is the spring constant of the tuning fork’s prong, Δf is the frequency shift, f0 is the resonance frequency without contact. This equation is the same with the conclusion that regards the prong as a cantilever [11,12]. That means this model is correct to describe the normal tuning fork probes.

2.2. The Electromechanical Coupling Model of the Akiyama Probe

This kind of model also can be used to describe the tuning fork probes with complex structures. The Akiyama probe symmetrically couples a long, soft, U-shaped cantilever to a quartz tuning fork [7] (Figure 6). In this article, an electromechanical coupling model (Figure 7) with similar symmetrical structure is built to realize the quantitative analysis of the Akiyama probe.
This electromechanical coupling model consists of three parts: a typical electromechanical coupling part describing the tuning fork, a mechanical coupling part representing the connection between the cantilever and the tuning fork, and another mechanical coupling part explaining the contact between the cantilever tip and the sample. In the model, the parameters, b, k, m, F, v, with different subscripts, denote damping, spring constant, mass, force, and the velocity of the tuning fork (subscript tf), and cantilever (subscript cl), respectively; Cp is the parasitic capacitance; i is the current; Vin is the excitation voltage; p is the electromechanical coupling factor; n, bsa, and keff are the mechanical coupling factor, the damping, and effective force gradient between the tip and the sample. The ends of the tuning fork prongs and the cantilever are stuck together, therefore, the coupling factor between the tuning fork and the cantilever is one.

2.2.1. The Transformation of the Model

With Equation (4), the cantilever part in Figure 7 can be transformed to the tuning fork part as shown in Figure 8.
With Equation (5), the sample part in Figure 8 can be transformed to the probe as shown in Figure 9.
With Equation (4), the mechanical part in Figure 9 can be transformed to the electrical part as shown in Figure 10.
As the two parts of the circuit are in parallel, the tuning fork prong’s electrical parameters are:
R tf = p 2 2 b tf L tf = p 2 2 m tf C tf = 2 p 2 k tf
where btf, mtf, ktf are damping, spring constant, mass of one of the tuning fork prongs. Factor p is the electromechanical coupling factor.
The cantilever’s electrical parameters are:
R cl = p 2 2 b cl L cl = p 2 2 m cl C cl = 2 p 2 k cl
The sample’s electrical parameters are:
R sample = n 2 p 2 b sa 4 C sample = 4 n 2 p 2 k eff
The model can therefore be deemed equivalent to the circuit shown in Figure 11.
The electrical parameters of the equivalent circuit can be calculated as:
R probe = R tf + R cl + R sample = p 2 2 b tf + p 2 2 b cl + p 2 n 2 b sa 4 L probe = L tf + L cl = p 2 2 ( m tf + m cl ) C probe = 1 1 C tf + 1 C cl + 1 C sample = 1 p 2 2 k tf + p 2 2 k cl + n 2 p 2 4 k eff
The final RLC circuit is shown as Figure 5.
So when keff = 0, the resonant frequency of the probe without contact is:
f 0 = 1 2 π L probe C probe = 1 2 π p 2 2 ( m tf + m cl ) p 2 2 ( k tf + k cl ) = 1 2 π k tf + k cl m tf + m cl = 1 2 π k probe m probe
The resonant frequency after touching the sample is:
f = 1 2 π k probe + n 2 2 k eff m probe

2.2.2. The Verification of the Model

For proving the validity of this electromechanical coupling model of an Akiyama probe, finite element analysis (FEA) software (ANSYS, 14.0, ANSYS Inc., Canonsburg, PA, USA, 2011), is used to simulate the contact between the tip and the sample [13]. The parameters of the Akiyama probe used to build the model are shown in Table 2.
The contact between the tip and the sample is simulated by a spring as shown in Figure 12 and the spring constant represents the effective force gradient between the tip and the sample (keff). Where keff is
k eff = F ts
The Fts is the force between the tip and sample, it can be described as follows in DMT model:
F ts = { F 0 ( 1 a 0 z a 0 ) 2 ,   z a 0 4 3 E * R ( a 0 z ) 3 2 + F 0 ,   z < a 0
where F0 is the adhesive force, z is the distance between the tip and sample, a0 is the radius of the atom, R is the radius of the tip, and E* is
1 E * = 1 v s 2 E s + 1 v t 2 E t
where Es, Et are the Young’s modulus of the sample and the tip, vs, vt are the poisson ratio of the sample and the tip.
In the simulation, a constant keff is used, which means the maximal Fts is
F max = k eff A
where A is the amplitude of the cantilever.
The results of the non-contact simulation are shown in Table 3.
Coupling factors, p and n, can therefore be calculated as:
p = 2 v tf i = 2 ω D tf ω q = 2 D tf q = 6.76 × 10 5   m / C n = v tip v tf = ω D tip ω D tf = D tip D tf = 26.67
The tuning fork prong is always represented by rectangular cantilever, the spring constant, and effective mass can be calculated as:
k = 1 4 E w ( t l ) 3 = 1571.55   N / m m = 0.24 ρ w t l = 3.01 × 10 8   kg
The resonant frequency with contact can be expressed as:
f = 1 2 π f 0 2 + n 2 k eff 2 m probe
The effective mass of the U-shaped cantilever is difficult to calculate. If this mass is 0.001mprobe for example, the error in the resonant frequency after ignoring this mass will be:
f 1 2 f 2 2 = 1 4 π 2 ( n 2 k eff m probe n 2 k eff 1.001 m probe ) < 0.001 × 1 4 π 2 n 2 k eff m probe
f 1 f 2 = f 1 2 f 2 2 f 1 + f 2 < 0.001 × 1 4 π 2 n 2 k eff m probe 1 2 f 0
When the value of keff is small enough, 0.01 N/m for example, the error is less than 1 Hz, so the effective mass of the U-shaped cantilever can be ignored during the calculation of the resonant frequency and the spring constant of the probe is given by:
k probe = k tf + k cl = 4 π 2 f 0 2 m t f = 3323.84   N / m
Using the parameters calculated above, the resonant frequencies for different values of keff can be obtained from Equation (15). Compared with the results of ANSYS simulation as shown in Table 4, the revised electromechanical coupling model can describe the Akiyama probe correctly when the value of keff is small. For an average force, between the tip and sample, in tapping mode of typically less than 1 nN, keff is less than 0.001 N/m in this simulation: the electromechanical coupling model of the Akiyama probe is accurate as evinced by simulation data.

2.2.3. The Measurement of the Coupling Factors

The calculation of the real probe’s electromechanical coupling factor uses the vibration amplitude of the tuning fork and the output current of the probe. The factor n is calculated by measuring the amplitude of the tuning fork and the cantilever. All the vibration amplitudes are measured by laser Doppler vibrometer (OFV-3001, Polytec, Waldbronn, Germany). The measured points of the tuning fork and the cantilever are shown in Figure 13.

2.3. Feature of the Second Eigenmode

In Equation (15), the coupling factor n determines the influence of keff on the frequency modulation (FM) mode sensitivity. If n > 1, the influence is magnified, otherwise, it is reduced. In the first eigenmode as simulated above, n = 26.67, implying that the resonant frequency is quite sensitive to the value of keff; but in the second eigenmode (Table 5), n < 1, and the sensitivity thereto is very low, and the simulation results show that the resonance frequency will not change as keff increases.
The second eigenfrequency of the Akiyama probe is not sensitive to keff, which means that the eigenfrequency is stable when the tip makes contact with the sample. This is a drawback when studying the relationship between the second eigenfrequency and the contact stiffness. However, the constant resonant frequency means that the amplitude is related to energy dissipation directly.
The amplitude change in amplitude modulation (AM) mode can be divided into two parts as shown in Figure 14. If the resonant frequency is constant, ΔAk = 0, ΔA = ΔAb, the amplitude change is caused only by the energy dissipation.

2.4. The Calculation of the Power Dissipation

With the electromechanical coupling model of an Akiyama probe, the power dissipation caused by damping effects in the sample under test is:
P dis = 1 2 b sa D t i p 2 ω 2 = 1 2 b sa v tip 2 = 1 2 b sa ( 1 2 p n i ) 2 = 1 2 ( 1 4 p 2 n 2 b sa ) i 2 = 1 2 i 2 R sample
Given the definition in Equation (13), with parasitic capacitance (Cp) compensation, the impedance of the circuit in Figure 5 is:
Z ( ω ) = R probe + j ω L probe + 1 j ω C probe = j ω R probe C probe + [ 1 ( ω ω 0 ) 2 ] j ω C probe
When the circuit is resonance oscillating, ω = ω0, the non-contact impedance is:
Z ( ω 0 ) = R probe = p 2 2 ( b tf + b cl )
The contact impedance is:
Z ( ω 0 ) = R probe + R sample = p 2 2 ( b tf + b cl + 1 2 n 2 b sa )
As mentioned above, the resonant frequency is not changed, so ω0 = ω0′, and the change in current is given by:
Δ I = U Z ( ω 0 ) U Z ( ω 0 ) = U R sample R probe + ( R probe + R sample ) = I R sample R probe
where U is the input voltage, I′ is the current after contact. With Equations (26) and (30), the power dissipation is given by:
P diss = 1 2 I 2 R sample = 1 2 I 2 Δ I I R probe = 1 2 I Δ I R probe = 1 2 U I ( I I ) I
where I is the current without contact.
The classical equation using phase to calculate the power dissipation is [14]:
P diss = 1 2 k ω Q ( A 0 A sin φ A 2 ω ω 0 )
where k is the spring constant of the probe, Q is the quality factor, A0 and A are the amplitude with or without contact, ω is the exciting frequency, ω0 is the resonance frequency, ϕ is the phase. When ω = ω0, the equation become:
P diss = 1 2 k ω 0 Q ( A 0 A sin φ A 2 )
For
Q = k m b k = ω 0 2 m
Then,
P diss = 1 2 b ω 0 2 ( A 0 A sin φ A 2 ) = 1 2 b [ ( A 0 ω 0 ) ( A ω 0 ) sin φ ( A ω 0 ) 2 ] = 1 2 b ( v 0 v sin φ v 2 )
where k is the spring constant of the probe, Q is the quality factor, A0 and A are the amplitude with or without contact, ω is the exciting frequency, ω0 is the resonance frequency, ϕ is the phase, b is the damping of the probe, v0 and v are the velocity with or without contact. When the probe is kept at resonance, ϕ = 90°, and with the corresponding relationship in Table 1, the two methods are equivalent.
P diss = 1 2 b ( v 0 v v 2 ) 1 2 R probe ( I I I 2 )
By electromechanical coupling, Equation (31) is equivalent to Equation (32) when the excitation frequency is kept at the resonance frequency during the scanning as the feature of the Akiyama probe is working in the second eigenmode. So Equation (31) is correct.

3. Measuring System

The bimodal AFM measuring system based on the Akiyama probe shown in Figure 15 is built to utilize this feature to measure the power dissipation. Two lock-in amplifications (LIA) excite the probe and demodulate the output signal. The first eigenmode signal amplitude is sent to the feedback system to control the PZT (piezoelectric ceramic transducer) to keep the distance between the tip and the sample constant as normal tapping mode, the position of the PZT describes the morphology of the sample. The second eigenmode signal amplitude is recorded, as mentioned above, this is related to the power dissipation therein.
The setpoint of the first eigenmode signal is 0.6 A01, the amplitude ratio of the two signals is A01/A02 = 0.723 V/0.503 V ≈ 1.437, where A01 and A02 are the amplitude of the first and the second eigenmode signals without contact. Parameters of the probe: the eigenfrequencies are f01 = 46.162 kHz, f02 = 202.646 kHz, cantilever spring constant k ≈ 5 N/m, quality factors: Q1 = 1154, Q2 = 6776. The scanning speed is 250 nm/s. There are 220 measured points along the length of 8760 nm.

4. Results and Discussion

The sample consists of Au wire on a SiO2 substrate, the height of the Au wire is about 150 nm. The sample is coated with a 20 nm thick Cr layer as shown in Figure 14 to obtain the similar surface Young’s modulus (SiO2 substrate: 113.665 ± 4.031 Gpa, Au substrate: 119.382 ± 12.734 Gpa, measured by a nano indenter, Agilent Technologies Inc., Santa Clara, CA, USA). Figure 16 shows the results of a line on the sample and all the lines’ results are averaged and listed in Table 6.
For the uniform surfaces of the two parts, the difference of power dissipation is only caused by the different substrates. The deviation between the two parts are stable in Table 6, the power dissipation of the Au substrate is less than the SiO2 substrate. The repeatable results mean this method is credible.

5. Conclusions

In conclusion, with the help of an electromechanical coupling model, a special feature of a tuning fork probe with complex structure is found and utilized to measure power dissipation. According to the model, the insensitive second eigenfrequency and the relation between the output current and the power dissipation are illustrated. In Equation (31), there are no mechanical parameters of the probe and the electrical parameters are easy to measure. Different to the complex and non-monotonous method using phase to calculate the power dissipation, the Equation (31) is not necessary to calibrate the probe’s spring constant which is still hard to measure accurately. With the monotonous relation between the current and the sample’s damping, the contrast of the image is stable.

Acknowledgments

The authors acknowledge the support of the National Natural Science Fund (Grant No. 51675379), the Tianjin Natural Science Fund (Grant No. 14JCZDJC39400 and 13JCQNJC04700), and the 111 Project fund (Grant No. B07014).

Author Contributions

Zhichao Wu and Tong Guo conceived and designed the experiments; Ran Tao performed the experiments; Jinping Chen and Linyan Xu analyzed the data; Xing Fu and Xiaotang Hu revised the paper.

Conflicts of Interest

The authors declare no conflict of interest.

References

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Figure 1. Normal tuning fork probes.
Figure 1. Normal tuning fork probes.
Applsci 07 00121 g001
Figure 2. The ideal transformers.
Figure 2. The ideal transformers.
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Figure 3. The electromechanical coupling model of normal probes.
Figure 3. The electromechanical coupling model of normal probes.
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Figure 4. The model of rigid tip.
Figure 4. The model of rigid tip.
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Figure 5. The equivalent RLC circuit of the tuning fork probe.
Figure 5. The equivalent RLC circuit of the tuning fork probe.
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Figure 6. The appearance and motion (arrows in the figures) of the Akiyama probe.
Figure 6. The appearance and motion (arrows in the figures) of the Akiyama probe.
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Figure 7. The electromechanical coupling model of an Akiyama probe.
Figure 7. The electromechanical coupling model of an Akiyama probe.
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Figure 8. The transformation from cantilever to the tuning fork.
Figure 8. The transformation from cantilever to the tuning fork.
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Figure 9. The transformation from sample to probe.
Figure 9. The transformation from sample to probe.
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Figure 10. The transformation from mechanical parameters to electrical parameters.
Figure 10. The transformation from mechanical parameters to electrical parameters.
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Figure 11. The equivalent electrical model of the Akiyama probe.
Figure 11. The equivalent electrical model of the Akiyama probe.
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Figure 12. The ANSYS model of an Akiyama probe.
Figure 12. The ANSYS model of an Akiyama probe.
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Figure 13. The measurement of factors using laser Doppler vibrometer.
Figure 13. The measurement of factors using laser Doppler vibrometer.
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Figure 14. The division of the amplitude change in AM mode.
Figure 14. The division of the amplitude change in AM mode.
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Figure 15. The Schematic of bimodal AFM based on an Akiyama probe.
Figure 15. The Schematic of bimodal AFM based on an Akiyama probe.
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Figure 16. The profile of the sample.
Figure 16. The profile of the sample.
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Table 1. The corresponding relationship of the electrical and mechanical parameters.
Table 1. The corresponding relationship of the electrical and mechanical parameters.
ElectricalMechanical
Inductance (L)Mass (m)
Capacitance (C)Reciprocal spring constant (1/k)
Resistance (R)Damping (b)
Voltage (U)Force (F)
Current (I)Velocity (v)
Table 2. The parameters of ANSYS simulation.
Table 2. The parameters of ANSYS simulation.
ParametersTuning ForkCantilever
Length/μm2400175
Thickness/μm22031.9
Width/μm103.73.7
Density/kg·m−322902300
Young modulus/GPa78.7180
Table 3. The results of ANSYS simulation.
Table 3. The results of ANSYS simulation.
Resonance Frequency/HzTuning Fork Amplitude Dtf/mCharge q/CCantilever Amplitude Dcl/m
52,8881.26 × 10−73.73 × 10−133.36 × 10−6
Table 4. The results of ANSYS simulation with different values of keff.
Table 4. The results of ANSYS simulation with different values of keff.
keff (N/m)ANSYS/HznModel/HzDeviation/Hz
052,88813.3852,8880
0.00152,89012.9252,8911
0.00552,89611.1852,8982
0.0152,9019.4952,9021
0.0252,9097.1552,904−5
Table 5. The second eigenmode results: ANSYS simulation with different values of keff.
Table 5. The second eigenmode results: ANSYS simulation with different values of keff.
keff (N/m)Resonance Frequency/HzCharge/CTuning Fork Amplitude/mCantilever Amplitude/m
0206,1532.07 × 10−132.93 × 10−81.03 × 10−9
0.0001206,1532.07 × 10−132.93 × 10−81.03 × 10−9
0.001206,1532.07 × 10−132.93 × 10−81.03 × 10−9
0.01206,1532.07 × 10−132.93 × 10−81.03 × 10−9
0.1206,1532.07 × 10−132.93 × 10−81.03 × 10−9
Table 6. The results of tests on the Micro-Electro-Mechanical System (MEMS) sample at intervals of 1 μm.
Table 6. The results of tests on the Micro-Electro-Mechanical System (MEMS) sample at intervals of 1 μm.
SubstrateAu/nWSiO2Deviation
Line 10.5990.60910
Line 20.5940.5995
Line 30.5910.5965
Line 40.5890.5989
Line 50.5850.59712
Average0.5910.6008
Standard Deviation5103

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MDPI and ACS Style

Wu, Z.; Guo, T.; Tao, R.; Xu, L.; Chen, J.; Fu, X.; Hu, X. The Model Analysis of a Complex Tuning Fork Probe and Its Application in Bimodal Atomic Force Microscopy. Appl. Sci. 2017, 7, 121. https://doi.org/10.3390/app7020121

AMA Style

Wu Z, Guo T, Tao R, Xu L, Chen J, Fu X, Hu X. The Model Analysis of a Complex Tuning Fork Probe and Its Application in Bimodal Atomic Force Microscopy. Applied Sciences. 2017; 7(2):121. https://doi.org/10.3390/app7020121

Chicago/Turabian Style

Wu, Zhichao, Tong Guo, Ran Tao, Linyan Xu, Jinping Chen, Xing Fu, and Xiaotang Hu. 2017. "The Model Analysis of a Complex Tuning Fork Probe and Its Application in Bimodal Atomic Force Microscopy" Applied Sciences 7, no. 2: 121. https://doi.org/10.3390/app7020121

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