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

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.


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.

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.

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 ..
A circuit's differential equation is 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.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: And the relation between Z and Z in Figure 2b is Appl.Sci.2017, 7, x 2 of 14

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 A circuit's differential equation is 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.
( ) ( ) 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: And the relation between Z and Z′ in Figure 2b is  Appl.Sci.2017, 7, 121 3 of 14

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.

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.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; C p is the parasitic capacitance; i is the current; V in is the excitation voltage; p is the electromechanical coupling factor; n, b sa , and k eff 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 (k cl ) of the tip is infinite and the damping (b cl ) 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, v tf = v cl = v tip .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.

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.   Appl.Sci.2017, 7, x 3 of 14

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 So the frequency shift after contact is For ∆f << f 0 , where k tf is the spring constant of the tuning fork's prong, ∆f is the frequency shift, f 0 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.

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.
Appl.Sci.2017, 7, x 4 of 14 And the parameters in Figure 5 are So the frequency shift after contact is 0 probe probe probe probe probe probe probe probe probe For Δf << f0, 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.

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.

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.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; C p is the parasitic capacitance; i is the current; V in is the excitation voltage; p is the electromechanical coupling factor; n, b sa , and k eff 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.

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.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.

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 ( 5), the sample part in Figure 8 can be transformed to the probe as shown in Figure 9.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.

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.
Appl.Sci.2017, 7, x of 14 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: The model can therefore be deemed equivalent to the circuit shown in Figure 11.As the two parts of the circuit are in parallel, the tuning fork prong's electrical parameters are: (10) where b tf , m tf , k tf 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: The sample's electrical parameters are: The model can therefore be deemed equivalent to the circuit shown in Figure 11.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: 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: The final RLC circuit is shown as Figure 5.So when k eff = 0, the resonant frequency of the probe without contact is: The resonant frequency after touching the sample is:

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 (k eff ).Where k eff is The electrical parameters of the equivalent circuit can be calculated as: The final RLC circuit is shown as Figure 5.
So when keff = 0, the resonant frequency of the probe without contact is: The resonant frequency after touching the sample is:

.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  The F ts is the force between the tip and sample, it can be described as follows in DMT model: where F 0 is the adhesive force, z is the distance between the tip and sample, a 0 is the radius of the atom, R is the radius of the tip, and E* is where E s , E t are the Young's modulus of the sample and the tip, v s , v t are the poisson ratio of the sample and the tip.
In the simulation, a constant k eff is used, which means the maximal F ts is 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: The tuning fork prong is always represented by rectangular cantilever, the spring constant, and effective mass can be calculated as: The resonant frequency with contact can be expressed as: The effective mass of the U-shaped cantilever is difficult to calculate.If this mass is 0.001m probe for example, the error in the resonant frequency after ignoring this mass will be: When the value of k eff 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: Using the parameters calculated above, the resonant frequencies for different values of k eff 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 k eff is small.For an average force, between the tip and sample, in tapping mode of typically less than 1 nN, k eff 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.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.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.

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.

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.

Feature of the Second Eigenmode
In Equation (15), the coupling factor n determines the influence of k eff 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 k eff ; 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 k eff increases.
The second eigenfrequency of the Akiyama probe is not sensitive to k eff , 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, ∆A k = 0, ∆A = ∆A b , the amplitude change is caused only by the energy dissipation.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.

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:

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.

Results and Discussion
The sample consists of Au wire on a SiO 2 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 (SiO 2 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.

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.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 SiO 2 substrate.The repeatable results mean this method is credible.

Figure 3 .
Figure 3.The electromechanical coupling model of normal probes.

Figure 4 .
Figure 4.The model of rigid tip.

Figure 5 .
Figure 5.The equivalent RLC circuit of the tuning fork probe.

Figure 3 .
Figure 3.The electromechanical coupling model of normal probes.

Figure 3 .
Figure 3.The electromechanical coupling model of normal probes.

Figure 4 .
Figure 4.The model of rigid tip.

Figure 5 .
Figure 5.The equivalent RLC circuit of the tuning fork probe.

Figure 4 .
Figure 4.The model of rigid tip.

Figure 3 .
Figure 3.The electromechanical coupling model of normal probes.

Figure 4 .
Figure 4.The model of rigid tip.

Figure 5 .
Figure 5.The equivalent RLC circuit of the tuning fork probe.Figure 5.The equivalent RLC circuit of the tuning fork probe.

Figure 5 .
Figure 5.The equivalent RLC circuit of the tuning fork probe.Figure 5.The equivalent RLC circuit of the tuning fork probe.

Figure 6 .
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.

Figure 6 .
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.

Figure 7 .
Figure 7.The electromechanical coupling model of an Akiyama probe.

Figure 8 .
Figure 8.The transformation from cantilever to the tuning fork.

Figure 9 .
Figure 9.The transformation from sample to probe.

Figure 7 .
Figure 7.The electromechanical coupling model of an Akiyama probe.

14 Figure 7 .
Figure 7.The electromechanical coupling model of an Akiyama probe.

Figure 8 .
Figure 8.The transformation from cantilever to the tuning fork.

Figure 9 .
Figure 9.The transformation from sample to probe.

Figure 8 .
Figure 8.The transformation from cantilever to the tuning fork.

Figure 7 .
Figure 7.The electromechanical coupling model of an Akiyama probe.

Figure 8 .
Figure 8.The transformation from cantilever to the tuning fork.

Figure 9 .
Figure 9.The transformation from sample to probe.

Figure 9 .
Figure 9.The transformation from sample to probe.

Figure 10 .
Figure 10.The transformation from mechanical parameters to electrical parameters.
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:

Figure 11 .
Figure 11.The equivalent electrical model of the Akiyama probe.

Figure 10 .
Figure 10.The transformation from mechanical parameters to electrical parameters.

Figure 10 .
Figure 10.The transformation from mechanical parameters to electrical parameters.
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:

Figure 11 .
Figure 11.The equivalent electrical model of the Akiyama probe.Figure 11.The equivalent electrical model of the Akiyama probe.

Figure 11 .
Figure 11.The equivalent electrical model of the Akiyama probe.Figure 11.The equivalent electrical model of the Akiyama probe.

Figure 12 .
Figure 12.The ANSYS model of an Akiyama probe.Figure 12.The ANSYS model of an Akiyama probe.

Figure 12 .
Figure 12.The ANSYS model of an Akiyama probe.Figure 12.The ANSYS model of an Akiyama probe.

Figure 13 .
Figure 13.The measurement of factors using laser Doppler vibrometer.

Figure 13 .
Figure 13.The measurement of factors using laser Doppler vibrometer.

Figure 14 .
Figure 14.The division of the amplitude change in AM mode.

Figure 15 .
Figure 15.The Schematic of bimodal AFM based on an Akiyama probe.

Figure 16 .
Figure 16.The profile of the sample.

Figure 15 .
Figure 15.The Schematic of bimodal AFM based on an Akiyama probe.

Figure 15 .
Figure 15.The Schematic of bimodal AFM based on an Akiyama probe.

Figure 16 .
Figure 16.The profile of the sample.

Figure 16 .
Figure 16.The profile of the sample.

Table 1 .
The corresponding relationship of the electrical and mechanical parameters.

Table 1 .
The corresponding relationship of the electrical and mechanical parameters.

Table 2 .
The parameters of ANSYS simulation.

Table 2 .
The parameters of ANSYS simulation.

Table 3 .
The results of ANSYS simulation.

Table 4 .
The results of ANSYS simulation with different values of k eff .

Table 4 .
The results of ANSYS simulation with different values of keff.

Table 5 .
The second eigenmode results: ANSYS simulation with different values of k eff .

Table 5 .
The second eigenmode results: ANSYS simulation with different values of keff. keff/(N/m)

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.

Table 6 .
The results of tests on the Micro-Electro-Mechanical System (MEMS) sample at intervals of 1 µm.