# Measurement Error of a Coupled AFM Probe-Elastic Membrane System

Authors: Shueei-Muh Lin; Che-Lun Tsai; Hung-Chang Lee
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

The characteristic of an AFM probe scanning the topography on an elastic substrate is investigated. It is different to the conventional rigid substrate. The analytical solution of the novel system is presented. For an elastic substrate the errors of frequency shift determined by using the force gradient methods and the perturbation method are satisfactory only for larger tip-surface distances. The smaller the interacting distance is, the larger the measurement error due to the amplitude of membrane is.


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## Body

## 1. Introduction

Advances in electromechanical systems are resulting in new applications ranging from mechanical mass or charge detectors to biological imaging. The conventional atomic force microscopy (AFM) is composed of one elastic probe and the sample on a rigid substrate [1–3]. For the future application, the mechanisms of the creative AFM are studied and used to measure the nanowire or particle coated on an elastic microthin plate (membrane). It is very helpful for developing the new generation of micro/nanoactuator and AFM.The conventional methods for determining the frequency shift are the force gradient method, perturbation method, and the analytical method. Relevant literatures are reviewed here. Holscher et al. [4] investigated the frequency shift of AFM scanning the sample on a rigid substrate by the force gradient method. For the small amplitude, the force changed linearly with the tip-sample distanceD, similar to the case of a spring. Hence the influence of the tip-sample forceFts was described by a spring with a spring constant kv, which was equal to the negative force gradient, as kv=-∂Fts/∂D.Giessibl [5] investigated the frequency shift in dynamic force microscopy by the perturbation method. A distributed parameter system was simulated by a lumped mass one. When the tip-surface forceFts was described in terms of the tip-surface distanceD as follows, Fts=-Cn/Dn, the first frequency shift Δf1 could be calculated from the integral Δf1=f1Cn/2πkbA0∫02πcos⁡x/D0+A0cos⁡xndx. Holscher et al. [4] investigated the first frequency shift of noncontact mode by using the lumped mass model and the perturbation method. Lin et al. [6] presented the modified perturbation method for determining the frequency shifts of higher modes.Lin et al. [7] evaluated the force gradient method, perturbation method, and the proposed analytical method for a rigid substrate. The interpretation of frequency shift determined by the force gradient method was unsatisfactory. For a uniform beam the frequency shift determined by the perturbation method and the analytical method was very consistent. But, for a tapered or V-shaped beam at a small tip-surface distance, the frequency shift determined by the perturbation method was overestimated.Although the approximated methods are easily used to determine the dynamic behavior of probe to predict the nanotopography of sample, significant error is unavoidable. The analytical method can provide accurate solution without error. Some analytical methods presented by Lin et al. [6–15] were successfully implemented in the study of the dynamic behavior of probe for scanning the sample on the rigid substrate. However, no analytical solution has been given to the response of a nonuniform beam with a tip mass, subjected to a nonlinear interacting force because of its complexity. So far, there is no literature on the dynamic behavior of AFM scanning the sample on elastic substrate.In this study, the exact frequency shift of an AFM probe scanning a sample on an elastic membrane, subjected to the van der Waal force, is derived. The comparisons of the force gradient method, perturbation method, and the analytical method are presented and assessed. Finally, the effects of several parameters on the sensitivity of measurement are also investigated.


## 2. Coupled Probe-Membrane Model Subjected to Van Der Waal Force

### 2.1. Governing Equation and Boundary Conditions of Beam

The governing equation of a AFM cantilever probe, as shown in Figure1, is [9](1)EbIb∂4Wb∂s4+ρbAb∂2Wb∂t2=0,0<s<Lb.The clamped boundary conditions are at s= 0,(2)Wb=0,(3)and∂Wb∂s=0.The free boundary conditions are at s=Lb,(4)∂2Wb∂s2=0(5)andEbIb∂3Wb∂s3-mt∂2Wb∂t2=Fv.The van der Waals force [13] is(6a)Fv=-AHR6D2,where the tip-surface distance(6b)D=D0-WbL,t-Wmxc,yc,t,and AH is the Hamaker constant,D0 is the distance between the tip of the undeformed beam and the surface of sample, andR is the tip radius.Figure 1
Geometry and coordinate of coupled probe-membrane system.


### 2.2. Governing Equation and Boundary Conditions of Membrane Substrate

#### 2.2.1. Rectangular Membrane

A rectangular membrane as shown in Figure1 is considered. The governing equation of membrane is [16] (7)∂2Wm∂x2+∂2Wm∂y2-m-σh∂2Wm∂t2=-1σhFcx,y,t,where the concentrated interacting force is Fcx,y,t=Fvδx-xcδy-yc.If the membrane is clamped, the boundary condition is(8a)Wm=0.If the membrane is free to deflect transversely at the boundary, the boundary condition is(8b)∂Wm∂n=0,where the derivative is taken normal to the boundary.


## 2.1. Governing Equation and Boundary Conditions of Beam

The governing equation of a AFM cantilever probe, as shown in Figure1, is [9](1)EbIb∂4Wb∂s4+ρbAb∂2Wb∂t2=0,0<s<Lb.The clamped boundary conditions are at s= 0,(2)Wb=0,(3)and∂Wb∂s=0.The free boundary conditions are at s=Lb,(4)∂2Wb∂s2=0(5)andEbIb∂3Wb∂s3-mt∂2Wb∂t2=Fv.The van der Waals force [13] is(6a)Fv=-AHR6D2,where the tip-surface distance(6b)D=D0-WbL,t-Wmxc,yc,t,and AH is the Hamaker constant,D0 is the distance between the tip of the undeformed beam and the surface of sample, andR is the tip radius.Figure 1
Geometry and coordinate of coupled probe-membrane system.


## 2.2. Governing Equation and Boundary Conditions of Membrane Substrate

### 2.2.1. Rectangular Membrane

A rectangular membrane as shown in Figure1 is considered. The governing equation of membrane is [16] (7)∂2Wm∂x2+∂2Wm∂y2-m-σh∂2Wm∂t2=-1σhFcx,y,t,where the concentrated interacting force is Fcx,y,t=Fvδx-xcδy-yc.If the membrane is clamped, the boundary condition is(8a)Wm=0.If the membrane is free to deflect transversely at the boundary, the boundary condition is(8b)∂Wm∂n=0,where the derivative is taken normal to the boundary.


## 2.2.1. Rectangular Membrane

A rectangular membrane as shown in Figure1 is considered. The governing equation of membrane is [16] (7)∂2Wm∂x2+∂2Wm∂y2-m-σh∂2Wm∂t2=-1σhFcx,y,t,where the concentrated interacting force is Fcx,y,t=Fvδx-xcδy-yc.If the membrane is clamped, the boundary condition is(8a)Wm=0.If the membrane is free to deflect transversely at the boundary, the boundary condition is(8b)∂Wm∂n=0,where the derivative is taken normal to the boundary.


## 3. Solution Methods

### 3.1. Coupled Probe-Spring-Membrane Model

For small amplitude of vibration, the interacting forceFv between the tip and sample around the equilibrium position of the tip can be expressed as a Taylor series. This force is approximated as a linear function of the distance between the tip and sample, which is similar to the spring. The influence of the tip-sample force can be described by a spring with a spring constant [4]. Further, the general system can be simulated in the coupled probe-spring-membrane model as shown in Figure 2.Figure 2
Schedule diagram of the beam-spring membrane model.


#### 3.1.1. Governing Equation and Boundary Conditions of Probe

The governing equation of a cantilever beam as shown in Figure1 is [9](9)EbIb∂4Wb∂s4+ρbAb∂2Wb∂t2=0,0<s<Lb.The clamped boundary conditions are at s= 0,(10)Wb=0,(11)and∂Wb∂s=0.The free boundary conditions are at s=Lb,(12)∂2Wb∂s2=0,(13)EbIb∂3Wb∂s3-mt∂2Wb∂t2+kvWmc-Wbt=0,where the effective spring constant kv=-AHR/3d03 andd0 is the initial interacting distance between the probe tip and the membrane.


#### 3.1.2. Governing Equation and Boundary Conditions of Rectangular Membrane

A rectangular membrane as shown in Figure1 is considered. The governing equation of membrane is [16](14)∂2Wm∂x2+∂2Wm∂y2-m-σh∂2Wm∂t2-1σhkvWmc-Wbtδx-xcδy-yc=0.


#### 3.1.3. Displacement of Beam and Membrane

Considering the harmonic motion of the system, the displacement of beam and membrane are(15)Wbs,t=W-bssin⁡Ωt,Wmx,y,t=W-mx,ysin⁡Ωt.Substituting (15) into (9)–(13), one obtains(16)d4W-bds4-λ2W-b=0,0<s<Lbwhere λ2=Ω2ρbAb/EbIb. The clamped boundary conditions are at s= 0,(17)W-b=0,(18)anddW-bds=0.The free boundary conditions are at s=Lb,(19)d2W-bds2=0,(20)EbIbd3W-bds3+Ω2mtW-b+kvW-mc-W-bt=0.The general solution of (16) can be written as(21)W-b=C1V1+C2V2+C3V3+C4V4,where the normalized fundamental solution is(22)V1s=12cosh⁡λs+cos⁡λs,V2s=12λsinh⁡λs+sin⁡λs,V3s=12λcosh⁡λs-cos⁡λs,V4s=12λλsinh⁡λs-sin⁡λs,which satisfies the following normalization condition:(23)V1V2V3V4V1′V2′V3′V4′V1′′V2′′V3′′V4′′V1′′′V2′′′V3′′′V4′′′s=0=1000010000100001.Substituting (21) into (17)–(20), one obtains(24)EbIbV3′′′Lb+Ω2mt-kvV3Lb+EbIbV4′′′Lb+Ω2mt-kvV4Lb-V3′′LbV4′′LbC3=-kvW-mc.The amplitude of vibration of the clamped membrane can be expressed as [10](25)W-mx,y=∑m=1∞∑n=1∞αmnsin⁡βmxsin⁡γnywhere βm=mπ/Lx, γn=nπ/Ly. Substituting (15) into (14), the coefficients are derived as(26)αmn=4kvW-mc-W-btLxLym-Ω2-σhβm2+γn2sin⁡βmxcsin⁡γnyc.Substituting (25) into (24), the frequency equation is obtained:(27)1λλΩ2mt-kv11-1/σhkvχsin⁡λLbcosh⁡λLb-sinh⁡λLbcos⁡λLb-EbIb1+cosh⁡λLbcos⁡λLb=0where χ=4/LxLy∑m=1∞‍∑n=1∞‍(sin2βmxcsin2γnyc/((m-/σh)Ω2−βm2+γn2)), λ=ΩρbAb/EbIb.The natural frequency of the system can be determined via (27).


### 3.2. Force Gradient Method

Holscher et al. [4] investigated the frequency shift of AFM scanning the sample on a rigid substrate by the force gradient method. The effective spring constant is(28)keff=kb+kvwhere the effective spring constant of the cantilever uniform beam kb=3EbIb/Lb3. The natural frequency under the Van der Waal force is ω0′=keff/m=ω01+kv/kb or f0′=f01+kv/kb, wheref0 is the natural frequency of the cantilever. Because kv<<kb, the frequency shift becomes(29)Δf=-f02kbAHR3d3.When the membrane substrate is elastic, the stiffness of the membrane must be considered, but the effect of the inertia of membrane is neglected. The corresponding effective spring constant is keff=kb+k′′, where the composite spring constant k′′=kvkm/kv+km, as shown in Figure 3, in which the effective spring constant of the membrane is (30)km=π2σhLxLy/4∑m=1∞∑n=1∞sin2mπxc/Lxsin2nπyc/Ly/m/Lx2+n/Ly2By neglecting the damping effect, the natural frequency under the van der Waals force is obtained: (31)f=fb01+k′′kb≈fb01+k′′2kb.Further, considering the damping effect between the probe and sample in the viscous model, the damped natural frequency can be approximated by [8](32)fd≅f1-14Q2≅f1-γ2,forγ<<1whereQ is the quality factor of the system and γ is the damping ratio. Obviously, the less the quality factor is, the larger the frequency shift is.Figure 3
Schedule diagram of the mass-spring model.


### 3.3. Generalized Perturbation Method

Giessibl [3] investigated the frequency shift in dynamic force microscopy by the perturbation method. The distributed parameter system is simulated by a concentrated mass one. The frequency shift Δf without the damping effect can be calculated from the integral(33)Δf=-AHR6kbD02-W-bm23/2f0,where W-bm=W-b(L)+W-mxc,yc. The displacement of membrane is derived as follows:(34a)CvkmD0-Wb+Wm2=Wm+CvkmD02.The displacement of membrane can be determined via (34a). If Wb+Wm<<D0, the displacement becomes(34b)Wm=D0-2Wb-2Cv/kmD02-D0-2Wb-2Cv/kmD022-8Cv/kmD01-1-2Wb/D04.It is found from (34b) that if km→∞, Wm=0.Based on (33), the frequency of the system without the damping effect is obtained:(35)f=f0+Δf.Further, substituting (35) into (32), the damped natural frequency is obtained.


### 3.4. Analytical Method

#### 3.4.1. Equation and Conditions of Probe

The dimensionless governing equation and boundary conditions of beam are(36)∂4wb∂ε4+∂2wb∂τ2=0At ε=0,(37)wb=0,(38)∂wb∂ε=0.At ε=1, (39)∂2wb∂ε2=0,(40a)∂3wb∂ε3-μb∂2wb∂τ2=fv,where(40b)fv=-cv6d2,cv=AHRL3E0I0Lc3,d=DLc;Lc=D0→d0=1.


#### 3.4.2. Dimensionless Equation and Conditions of Rectangular Membrane

The dimensionless governing equation and boundary conditions of membrane are(41)∂2∂ξx2+∂2∂ξy2wm-rrigidrmass∂2wm∂τb2=-fcξx,ξy,τb=-rrigidfvδξx-ξxcδξy-ξyc.The clamped boundary condition is(42)wm=0.


#### 3.4.3. Solution Method

In general, it is difficult to derive the analytical solution of the coupled system with a nonlinear and time-dependent boundary condition. If the amplitude is small, the motion of the system is assumed to be harmonic [2, 12].The dynamic solutions of beam and plate can be expressed as(43)wbε,τ=w-bεcos⁡ωτ,wmξx,ξy,τ=w-mξx,ξycos⁡ωτ.Substituting (17) into (6b), the corresponding dynamic interacting distance becomes(44a)dτ=d0-w-bmcos⁡ωτ,where the dimensionless interacting amplitude is(44b)w-bm=w-b1-w-mξxc,ξyc.The minimum interacting distance dmin=d0-w-bm.(a) Displacement Amplitude of Probe. Substituting (17) into the governing equation (9) and the boundary conditions (10)–(13), one obtains(45)d4w-bdε4-ω2w-b=0,0<ε<1;as ε=0,(46)w-b0=0,(47)dw-b0dε=0;as ε=1,(48)d2w-b1dε2=0,(49)d3w-bdε3+μtipω2w-bcos⁡ωτ=-cv6d0-w-bmcos⁡ωτ2.Because condition (49) is time-dependent, the balanced method is considered here [10]. Multiplying (49) by cos⁡ωτ and integrating it from 0 to the period T, 2π/ω, (49) becomes(50a)d3w-bdε3+μtipω2w-b=f-v(50b)f-v=-cvw-bm3d02-w-bm23/2.The general solution of (45) can be written as(51)w-b=C1V1+C2V2+C3V3+C4V4,where V1,V2,V3,V4 are the normalized fundamental solution which satisfies the following normalization condition:(52)V1V2V3V4V1′V2′V3′V4′V1′′V2′′V3′′V4′′V1′′′V2′′′V3′′′V4′′′ε=0=1000010000100001.Four normalized fundamental solutions of (51) are(53)V1ε=12cosh⁡αε+cos⁡αε,V2ε=12αsinh⁡αε+sin⁡αε,V3ε=12α2cosh⁡αε-cos⁡αε,V4ε=12α3sinh⁡αε-sin⁡αε,where α=ω. Substituting (51) into (46)–(48), (50a), and (50b), the beam displacement is(54a)w-bε=f-vηbeamV3ε-V3′′1V4′′1V4ε,where(54b)ηbeam=V3′′′1-V3′′1V4′′1V4′′′1+μtipω2V31-V3′′1V4′′1V41.It should be noted that “ηbeam=0” is the frequency equation of the independent beam.(b) Displacement Amplitude of Membrane. Substituting (43) into (41) and multiplying it by cos⁡ωτ and finally integrating from 0 to the period T, 2π/ω, (41) becomes(55)∂2∂ξx2+∂2∂ξy2w-m+ω2rrigidrmassw-m=-rrigidf-vδξx-ξxcδξy-ξycSubstituting (43) into the clamped boundary condition (42), the associated boundary condition is(56)w-m=0.The amplitude of vibration can be derived [16]:(57)w-mξxc,ξyc=f-v4rrigidrxry∑m=1∞∑n=1∞sin2mπxc/Lxsin2nπyc/Lymπ/rx2+nπ/ry2-rrigidrmassω2.It is found from (57) that the dimensionless natural frequency of an independent clamped membrane is(58)ωmem=mπrx2+nπ/ry2rrigidrmass(c) Frequency Equation. Substituting the probe-membrane amplitudes (54a), (54b), and (57) into the interacting amplitude (44b), the frequency equation of the coupled subsystem is obtained: (59)w-bm-f-vηbeamV31-V3′′1V4′′1V41-f-v4rrigidrxry∑m=1∞∑n=1∞sin2mπxc/Lxsin2nπyc/Lymπ/rx2+nπ/ry2-rrigidrmassω2=0.It is observed from (59) that the frequencies of oscillation are dependent on the amplitude of oscillation at the tip. Given the interacting distance w-bm, the exact corresponding frequencies can be easily determined via (59) by using the numerical method proposed by Lin [10].


## 3.1. Coupled Probe-Spring-Membrane Model

For small amplitude of vibration, the interacting forceFv between the tip and sample around the equilibrium position of the tip can be expressed as a Taylor series. This force is approximated as a linear function of the distance between the tip and sample, which is similar to the spring. The influence of the tip-sample force can be described by a spring with a spring constant [4]. Further, the general system can be simulated in the coupled probe-spring-membrane model as shown in Figure 2.Figure 2
Schedule diagram of the beam-spring membrane model.


### 3.1.1. Governing Equation and Boundary Conditions of Probe

The governing equation of a cantilever beam as shown in Figure1 is [9](9)EbIb∂4Wb∂s4+ρbAb∂2Wb∂t2=0,0<s<Lb.The clamped boundary conditions are at s= 0,(10)Wb=0,(11)and∂Wb∂s=0.The free boundary conditions are at s=Lb,(12)∂2Wb∂s2=0,(13)EbIb∂3Wb∂s3-mt∂2Wb∂t2+kvWmc-Wbt=0,where the effective spring constant kv=-AHR/3d03 andd0 is the initial interacting distance between the probe tip and the membrane.


### 3.1.2. Governing Equation and Boundary Conditions of Rectangular Membrane

A rectangular membrane as shown in Figure1 is considered. The governing equation of membrane is [16](14)∂2Wm∂x2+∂2Wm∂y2-m-σh∂2Wm∂t2-1σhkvWmc-Wbtδx-xcδy-yc=0.


### 3.1.3. Displacement of Beam and Membrane

Considering the harmonic motion of the system, the displacement of beam and membrane are(15)Wbs,t=W-bssin⁡Ωt,Wmx,y,t=W-mx,ysin⁡Ωt.Substituting (15) into (9)–(13), one obtains(16)d4W-bds4-λ2W-b=0,0<s<Lbwhere λ2=Ω2ρbAb/EbIb. The clamped boundary conditions are at s= 0,(17)W-b=0,(18)anddW-bds=0.The free boundary conditions are at s=Lb,(19)d2W-bds2=0,(20)EbIbd3W-bds3+Ω2mtW-b+kvW-mc-W-bt=0.The general solution of (16) can be written as(21)W-b=C1V1+C2V2+C3V3+C4V4,where the normalized fundamental solution is(22)V1s=12cosh⁡λs+cos⁡λs,V2s=12λsinh⁡λs+sin⁡λs,V3s=12λcosh⁡λs-cos⁡λs,V4s=12λλsinh⁡λs-sin⁡λs,which satisfies the following normalization condition:(23)V1V2V3V4V1′V2′V3′V4′V1′′V2′′V3′′V4′′V1′′′V2′′′V3′′′V4′′′s=0=1000010000100001.Substituting (21) into (17)–(20), one obtains(24)EbIbV3′′′Lb+Ω2mt-kvV3Lb+EbIbV4′′′Lb+Ω2mt-kvV4Lb-V3′′LbV4′′LbC3=-kvW-mc.The amplitude of vibration of the clamped membrane can be expressed as [10](25)W-mx,y=∑m=1∞∑n=1∞αmnsin⁡βmxsin⁡γnywhere βm=mπ/Lx, γn=nπ/Ly. Substituting (15) into (14), the coefficients are derived as(26)αmn=4kvW-mc-W-btLxLym-Ω2-σhβm2+γn2sin⁡βmxcsin⁡γnyc.Substituting (25) into (24), the frequency equation is obtained:(27)1λλΩ2mt-kv11-1/σhkvχsin⁡λLbcosh⁡λLb-sinh⁡λLbcos⁡λLb-EbIb1+cosh⁡λLbcos⁡λLb=0where χ=4/LxLy∑m=1∞‍∑n=1∞‍(sin2βmxcsin2γnyc/((m-/σh)Ω2−βm2+γn2)), λ=ΩρbAb/EbIb.The natural frequency of the system can be determined via (27).


## 3.1.1. Governing Equation and Boundary Conditions of Probe

The governing equation of a cantilever beam as shown in Figure1 is [9](9)EbIb∂4Wb∂s4+ρbAb∂2Wb∂t2=0,0<s<Lb.The clamped boundary conditions are at s= 0,(10)Wb=0,(11)and∂Wb∂s=0.The free boundary conditions are at s=Lb,(12)∂2Wb∂s2=0,(13)EbIb∂3Wb∂s3-mt∂2Wb∂t2+kvWmc-Wbt=0,where the effective spring constant kv=-AHR/3d03 andd0 is the initial interacting distance between the probe tip and the membrane.


## 3.1.2. Governing Equation and Boundary Conditions of Rectangular Membrane

A rectangular membrane as shown in Figure1 is considered. The governing equation of membrane is [16](14)∂2Wm∂x2+∂2Wm∂y2-m-σh∂2Wm∂t2-1σhkvWmc-Wbtδx-xcδy-yc=0.


## 3.1.3. Displacement of Beam and Membrane

Considering the harmonic motion of the system, the displacement of beam and membrane are(15)Wbs,t=W-bssin⁡Ωt,Wmx,y,t=W-mx,ysin⁡Ωt.Substituting (15) into (9)–(13), one obtains(16)d4W-bds4-λ2W-b=0,0<s<Lbwhere λ2=Ω2ρbAb/EbIb. The clamped boundary conditions are at s= 0,(17)W-b=0,(18)anddW-bds=0.The free boundary conditions are at s=Lb,(19)d2W-bds2=0,(20)EbIbd3W-bds3+Ω2mtW-b+kvW-mc-W-bt=0.The general solution of (16) can be written as(21)W-b=C1V1+C2V2+C3V3+C4V4,where the normalized fundamental solution is(22)V1s=12cosh⁡λs+cos⁡λs,V2s=12λsinh⁡λs+sin⁡λs,V3s=12λcosh⁡λs-cos⁡λs,V4s=12λλsinh⁡λs-sin⁡λs,which satisfies the following normalization condition:(23)V1V2V3V4V1′V2′V3′V4′V1′′V2′′V3′′V4′′V1′′′V2′′′V3′′′V4′′′s=0=1000010000100001.Substituting (21) into (17)–(20), one obtains(24)EbIbV3′′′Lb+Ω2mt-kvV3Lb+EbIbV4′′′Lb+Ω2mt-kvV4Lb-V3′′LbV4′′LbC3=-kvW-mc.The amplitude of vibration of the clamped membrane can be expressed as [10](25)W-mx,y=∑m=1∞∑n=1∞αmnsin⁡βmxsin⁡γnywhere βm=mπ/Lx, γn=nπ/Ly. Substituting (15) into (14), the coefficients are derived as(26)αmn=4kvW-mc-W-btLxLym-Ω2-σhβm2+γn2sin⁡βmxcsin⁡γnyc.Substituting (25) into (24), the frequency equation is obtained:(27)1λλΩ2mt-kv11-1/σhkvχsin⁡λLbcosh⁡λLb-sinh⁡λLbcos⁡λLb-EbIb1+cosh⁡λLbcos⁡λLb=0where χ=4/LxLy∑m=1∞‍∑n=1∞‍(sin2βmxcsin2γnyc/((m-/σh)Ω2−βm2+γn2)), λ=ΩρbAb/EbIb.The natural frequency of the system can be determined via (27).


## 3.2. Force Gradient Method

Holscher et al. [4] investigated the frequency shift of AFM scanning the sample on a rigid substrate by the force gradient method. The effective spring constant is(28)keff=kb+kvwhere the effective spring constant of the cantilever uniform beam kb=3EbIb/Lb3. The natural frequency under the Van der Waal force is ω0′=keff/m=ω01+kv/kb or f0′=f01+kv/kb, wheref0 is the natural frequency of the cantilever. Because kv<<kb, the frequency shift becomes(29)Δf=-f02kbAHR3d3.When the membrane substrate is elastic, the stiffness of the membrane must be considered, but the effect of the inertia of membrane is neglected. The corresponding effective spring constant is keff=kb+k′′, where the composite spring constant k′′=kvkm/kv+km, as shown in Figure 3, in which the effective spring constant of the membrane is (30)km=π2σhLxLy/4∑m=1∞∑n=1∞sin2mπxc/Lxsin2nπyc/Ly/m/Lx2+n/Ly2By neglecting the damping effect, the natural frequency under the van der Waals force is obtained: (31)f=fb01+k′′kb≈fb01+k′′2kb.Further, considering the damping effect between the probe and sample in the viscous model, the damped natural frequency can be approximated by [8](32)fd≅f1-14Q2≅f1-γ2,forγ<<1whereQ is the quality factor of the system and γ is the damping ratio. Obviously, the less the quality factor is, the larger the frequency shift is.Figure 3
Schedule diagram of the mass-spring model.


## 3.3. Generalized Perturbation Method

Giessibl [3] investigated the frequency shift in dynamic force microscopy by the perturbation method. The distributed parameter system is simulated by a concentrated mass one. The frequency shift Δf without the damping effect can be calculated from the integral(33)Δf=-AHR6kbD02-W-bm23/2f0,where W-bm=W-b(L)+W-mxc,yc. The displacement of membrane is derived as follows:(34a)CvkmD0-Wb+Wm2=Wm+CvkmD02.The displacement of membrane can be determined via (34a). If Wb+Wm<<D0, the displacement becomes(34b)Wm=D0-2Wb-2Cv/kmD02-D0-2Wb-2Cv/kmD022-8Cv/kmD01-1-2Wb/D04.It is found from (34b) that if km→∞, Wm=0.Based on (33), the frequency of the system without the damping effect is obtained:(35)f=f0+Δf.Further, substituting (35) into (32), the damped natural frequency is obtained.


## 3.4. Analytical Method

### 3.4.1. Equation and Conditions of Probe

The dimensionless governing equation and boundary conditions of beam are(36)∂4wb∂ε4+∂2wb∂τ2=0At ε=0,(37)wb=0,(38)∂wb∂ε=0.At ε=1, (39)∂2wb∂ε2=0,(40a)∂3wb∂ε3-μb∂2wb∂τ2=fv,where(40b)fv=-cv6d2,cv=AHRL3E0I0Lc3,d=DLc;Lc=D0→d0=1.


### 3.4.2. Dimensionless Equation and Conditions of Rectangular Membrane

The dimensionless governing equation and boundary conditions of membrane are(41)∂2∂ξx2+∂2∂ξy2wm-rrigidrmass∂2wm∂τb2=-fcξx,ξy,τb=-rrigidfvδξx-ξxcδξy-ξyc.The clamped boundary condition is(42)wm=0.


### 3.4.3. Solution Method

In general, it is difficult to derive the analytical solution of the coupled system with a nonlinear and time-dependent boundary condition. If the amplitude is small, the motion of the system is assumed to be harmonic [2, 12].The dynamic solutions of beam and plate can be expressed as(43)wbε,τ=w-bεcos⁡ωτ,wmξx,ξy,τ=w-mξx,ξycos⁡ωτ.Substituting (17) into (6b), the corresponding dynamic interacting distance becomes(44a)dτ=d0-w-bmcos⁡ωτ,where the dimensionless interacting amplitude is(44b)w-bm=w-b1-w-mξxc,ξyc.The minimum interacting distance dmin=d0-w-bm.(a) Displacement Amplitude of Probe. Substituting (17) into the governing equation (9) and the boundary conditions (10)–(13), one obtains(45)d4w-bdε4-ω2w-b=0,0<ε<1;as ε=0,(46)w-b0=0,(47)dw-b0dε=0;as ε=1,(48)d2w-b1dε2=0,(49)d3w-bdε3+μtipω2w-bcos⁡ωτ=-cv6d0-w-bmcos⁡ωτ2.Because condition (49) is time-dependent, the balanced method is considered here [10]. Multiplying (49) by cos⁡ωτ and integrating it from 0 to the period T, 2π/ω, (49) becomes(50a)d3w-bdε3+μtipω2w-b=f-v(50b)f-v=-cvw-bm3d02-w-bm23/2.The general solution of (45) can be written as(51)w-b=C1V1+C2V2+C3V3+C4V4,where V1,V2,V3,V4 are the normalized fundamental solution which satisfies the following normalization condition:(52)V1V2V3V4V1′V2′V3′V4′V1′′V2′′V3′′V4′′V1′′′V2′′′V3′′′V4′′′ε=0=1000010000100001.Four normalized fundamental solutions of (51) are(53)V1ε=12cosh⁡αε+cos⁡αε,V2ε=12αsinh⁡αε+sin⁡αε,V3ε=12α2cosh⁡αε-cos⁡αε,V4ε=12α3sinh⁡αε-sin⁡αε,where α=ω. Substituting (51) into (46)–(48), (50a), and (50b), the beam displacement is(54a)w-bε=f-vηbeamV3ε-V3′′1V4′′1V4ε,where(54b)ηbeam=V3′′′1-V3′′1V4′′1V4′′′1+μtipω2V31-V3′′1V4′′1V41.It should be noted that “ηbeam=0” is the frequency equation of the independent beam.(b) Displacement Amplitude of Membrane. Substituting (43) into (41) and multiplying it by cos⁡ωτ and finally integrating from 0 to the period T, 2π/ω, (41) becomes(55)∂2∂ξx2+∂2∂ξy2w-m+ω2rrigidrmassw-m=-rrigidf-vδξx-ξxcδξy-ξycSubstituting (43) into the clamped boundary condition (42), the associated boundary condition is(56)w-m=0.The amplitude of vibration can be derived [16]:(57)w-mξxc,ξyc=f-v4rrigidrxry∑m=1∞∑n=1∞sin2mπxc/Lxsin2nπyc/Lymπ/rx2+nπ/ry2-rrigidrmassω2.It is found from (57) that the dimensionless natural frequency of an independent clamped membrane is(58)ωmem=mπrx2+nπ/ry2rrigidrmass(c) Frequency Equation. Substituting the probe-membrane amplitudes (54a), (54b), and (57) into the interacting amplitude (44b), the frequency equation of the coupled subsystem is obtained: (59)w-bm-f-vηbeamV31-V3′′1V4′′1V41-f-v4rrigidrxry∑m=1∞∑n=1∞sin2mπxc/Lxsin2nπyc/Lymπ/rx2+nπ/ry2-rrigidrmassω2=0.It is observed from (59) that the frequencies of oscillation are dependent on the amplitude of oscillation at the tip. Given the interacting distance w-bm, the exact corresponding frequencies can be easily determined via (59) by using the numerical method proposed by Lin [10].


## 3.4.1. Equation and Conditions of Probe

The dimensionless governing equation and boundary conditions of beam are(36)∂4wb∂ε4+∂2wb∂τ2=0At ε=0,(37)wb=0,(38)∂wb∂ε=0.At ε=1, (39)∂2wb∂ε2=0,(40a)∂3wb∂ε3-μb∂2wb∂τ2=fv,where(40b)fv=-cv6d2,cv=AHRL3E0I0Lc3,d=DLc;Lc=D0→d0=1.


## 3.4.2. Dimensionless Equation and Conditions of Rectangular Membrane

The dimensionless governing equation and boundary conditions of membrane are(41)∂2∂ξx2+∂2∂ξy2wm-rrigidrmass∂2wm∂τb2=-fcξx,ξy,τb=-rrigidfvδξx-ξxcδξy-ξyc.The clamped boundary condition is(42)wm=0.


## 3.4.3. Solution Method

In general, it is difficult to derive the analytical solution of the coupled system with a nonlinear and time-dependent boundary condition. If the amplitude is small, the motion of the system is assumed to be harmonic [2, 12].The dynamic solutions of beam and plate can be expressed as(43)wbε,τ=w-bεcos⁡ωτ,wmξx,ξy,τ=w-mξx,ξycos⁡ωτ.Substituting (17) into (6b), the corresponding dynamic interacting distance becomes(44a)dτ=d0-w-bmcos⁡ωτ,where the dimensionless interacting amplitude is(44b)w-bm=w-b1-w-mξxc,ξyc.The minimum interacting distance dmin=d0-w-bm.(a) Displacement Amplitude of Probe. Substituting (17) into the governing equation (9) and the boundary conditions (10)–(13), one obtains(45)d4w-bdε4-ω2w-b=0,0<ε<1;as ε=0,(46)w-b0=0,(47)dw-b0dε=0;as ε=1,(48)d2w-b1dε2=0,(49)d3w-bdε3+μtipω2w-bcos⁡ωτ=-cv6d0-w-bmcos⁡ωτ2.Because condition (49) is time-dependent, the balanced method is considered here [10]. Multiplying (49) by cos⁡ωτ and integrating it from 0 to the period T, 2π/ω, (49) becomes(50a)d3w-bdε3+μtipω2w-b=f-v(50b)f-v=-cvw-bm3d02-w-bm23/2.The general solution of (45) can be written as(51)w-b=C1V1+C2V2+C3V3+C4V4,where V1,V2,V3,V4 are the normalized fundamental solution which satisfies the following normalization condition:(52)V1V2V3V4V1′V2′V3′V4′V1′′V2′′V3′′V4′′V1′′′V2′′′V3′′′V4′′′ε=0=1000010000100001.Four normalized fundamental solutions of (51) are(53)V1ε=12cosh⁡αε+cos⁡αε,V2ε=12αsinh⁡αε+sin⁡αε,V3ε=12α2cosh⁡αε-cos⁡αε,V4ε=12α3sinh⁡αε-sin⁡αε,where α=ω. Substituting (51) into (46)–(48), (50a), and (50b), the beam displacement is(54a)w-bε=f-vηbeamV3ε-V3′′1V4′′1V4ε,where(54b)ηbeam=V3′′′1-V3′′1V4′′1V4′′′1+μtipω2V31-V3′′1V4′′1V41.It should be noted that “ηbeam=0” is the frequency equation of the independent beam.(b) Displacement Amplitude of Membrane. Substituting (43) into (41) and multiplying it by cos⁡ωτ and finally integrating from 0 to the period T, 2π/ω, (41) becomes(55)∂2∂ξx2+∂2∂ξy2w-m+ω2rrigidrmassw-m=-rrigidf-vδξx-ξxcδξy-ξycSubstituting (43) into the clamped boundary condition (42), the associated boundary condition is(56)w-m=0.The amplitude of vibration can be derived [16]:(57)w-mξxc,ξyc=f-v4rrigidrxry∑m=1∞∑n=1∞sin2mπxc/Lxsin2nπyc/Lymπ/rx2+nπ/ry2-rrigidrmassω2.It is found from (57) that the dimensionless natural frequency of an independent clamped membrane is(58)ωmem=mπrx2+nπ/ry2rrigidrmass(c) Frequency Equation. Substituting the probe-membrane amplitudes (54a), (54b), and (57) into the interacting amplitude (44b), the frequency equation of the coupled subsystem is obtained: (59)w-bm-f-vηbeamV31-V3′′1V4′′1V41-f-v4rrigidrxry∑m=1∞∑n=1∞sin2mπxc/Lxsin2nπyc/Lymπ/rx2+nπ/ry2-rrigidrmassω2=0.It is observed from (59) that the frequencies of oscillation are dependent on the amplitude of oscillation at the tip. Given the interacting distance w-bm, the exact corresponding frequencies can be easily determined via (59) by using the numerical method proposed by Lin [10].


## 4. Dynamic Results and Discussion

Considering a rigid substrate, comparison of the frequencies determined by different methods is made and listed in Table 1. It is found that the results determined by the analytical method for rrigid=0 are the same as the exact solutions determined by Lin [9]. The results determined by the perturbation method for rrigid=0 are slightly different from the exact solutions. When the tip-surface distance is small, the results determined by the force gradient method and the beam-spring-membrane model are greatly different from the exact solutions. The errors of frequency shift determined by using the force gradient methods are satisfactory only for larger tip-surface distances. The errors of frequency shift determined by using the perturbation method are smaller than those by the force gradient method for small tip-surface distances.Table 1
Comparison of the first frequency shifts of the AFM probe-membrane system determined by different method [SiO2:E = 70.3×109Pa, ρ = 2.5×103 kg/m3, B =45 μm, H = 3.5 μm, Lb=200 μm, R = 1.5×10-7m, AH = 10-19Joule, mtip=3.18×10-13kg, rrigid=0, rmass=1, rx=ry=1, xc/Lx=yc/Ly=0.5, the first natural frequency of beam fbeam,1 = 74355.803Hz].
- Δ f 1 H z, W-b(Lb)=1nm D0 nm r r i g i d = 0 Exact solution ofbeam only[9] Force gradient method Beam-Spring-Membrane model Perturbation method Analytical method 5 339.983 341.779 361.452 363.420 363.420  6 196.749 197.575 205.241 206.115 206.115  7 123.901 124.352 127.792 128.262 128.262  8 83.004 83.281 84.988 85.274 85.274  9 58.296 58.481 59.393 59.581 59.581  10 42.498 42.628 43.143 43.275 43.275  20 5.312 5.328 5.332 5.349 5.349  30 1.574 1.580 1.577 1.583 1.583  40 0.664 0.668 0.665 0.668 0.668  50 0.340 0.343 0.340 0.342 0.342  60 0.197 0.199 0.197 0.199 0.199  70 0.124 0.126 0.124 0.126 0.126  80 0.083 0.085 0.083 0.085 0.085  90 0.058 0.060 0.058 0.060 0.060  100 0.043 0.044 0.043 0.045 0.045  120 0.025 0.026 0.025 0.026 0.026  150 0.013 0.014 0.013 0.014 0.014  200 0.005 0.007 0.005 0.007 0.007Table2 demonstrates the effect of elastic membrane substrate on the coupled frequencies. The membrane inertia is neglected in both the force gradient method and the perturbation method. It implies that although the coupled frequency depends on the membrane stiffness, but no additional mode except the beam ones occurs. Nevertheless, the membrane inertia is considered in the beam-spring-membrane model and the analytical method. Several additional modes except the beam ones occur for small interacting distanceD0 between the tip and the sample. For D0<8(nm), the frequencies determined by the force gradient method and the perturbation method are significantly different from those by the beam-spring-membrane model and the analytical method. In the conventional measuring methodology, the topography of sample is observed based on the dynamic behavior of beam. Due to the additional modes, it likely results in the error of measuring the topography for small interacting distanceD0. Further, the coupled frequencies determined by the analytical method are shown in Figure 4. Mode A denotes the closest one to the first natural frequency of an independent cantilever; Mode is B next. It is observed from Figure 4(a) that there is multimodes phenomenon for the probe-elastic membrane model. Figure 4(b) demonstrates that, for mode A, the amplitude of beam oscillation dominates. For mode B, the amplitude of membrane oscillation dominates.Table 2
Comparison of the first frequency shifts of the AFM probe-membrane system determined by different method [SiO2:E = 70.3×109Pa, ρ = 2.5×103 kg/m3, B =45 μm, H = 3.5 μm, Lb=200 μm, R = 1.5×10-7m, AH = 10-19Joule, mtip=3.18×10-13kg, rmass=1, rx=ry=1, xc/Lx=yc/Ly=0.5, the first natural frequency of beam fbeam,1 = 74355.803 Hz, ∗: multimodes].
- Δ f 1 H z, W-b(Lb)=1nm, rrigid=15 D0 nm Force gradient method Beam-Spring-Membrane model∗ Mode A Perturbation method Analytical method∗ Mode A 5 437.395 249.616 420.530 254.808  6 225.859 166.653 209.690 170.683  7 134.845 112.323 128.800 114.848  8 87.776 77.975 85.293 79.484  9 60.611 55.894 59.503 56.806  10 43.715 41.162 43.189 41.829  11 32.611 31.256 32.350 31.620  12 24.996 24.209 24.862 24.450  13 19.592 19.115 19.522 19.279  14 15.646 15.348 15.610 15.462  15 12.697 12.504 12.678 12.586  16 10.446 10.319 10.437 10.379  17 8.699 8.613 8.696 8.657  18 7.322 7.262 7.321 7.296  19 6.221 6.179 6.222 6.204  20 5.331 5.300 5.332 5.320  30 1.576 1.570 1.577 1.573  40 0.664 0.432 0.665 0.660  50 0.340 0.335 0.340 0.335  60 0.197 0.191 0.197 0.191  70 0.124 0.118 0.124 0.118Figure 4
Effect of interaction distanceD0 on the frequency shifts of Modes A and B [SiO2:E = 70.3×109Pa, ρ = 2.5×103 kg/m3,B =45 μm,H = 3.5 μm,Lb=200 μm,R = 1.5×10-7m,AH = 10-19Joule, mtip=3.18×10-13kg, rrigid=15, rmass=1, rx=ry=1, xc/Lx=yc/Ly=0.5, the first natural frequency of beam fbeam,1 = 74355.803 Hz, W-bm=1nm].
(a) (b)Figure5 demonstrates the effect of the rigidity ratio rrigid on the frequency shifts of modes A and B. When the interacting distance D0>12nm, the frequency shift is small. When the interacting distance D0=10.5nm, the frequency shift is obvious. There are negligible differences among the frequency shifts for the different rigidity ratio, rrigid. The smaller the interacting distance D0 is, significantly the larger the frequency shift is. For the interacting distance D0<10.5nm, there is great difference among the frequency shifts for the different rigidity ratio, rrigid.Figure 5
Effect of rigidity ratiorrigid on the frequency shift [SiO2:E = 70.3×109Pa, ρ = 2.5×103 kg/m3,B =45 μm,H = 3.5 μm,Lb=200 μm,R = 1.5×10-7m,AH = 10-19Joule, mtip=3.18×10-13kg, rmass=1, rx=ry=1, xc/Lx=yc/Ly=0.5, the first natural frequency of beam fbeam,1= 74355.803 Hz, W-bm=1nm].
(a)
Mode A (b)
Mode BFurther, the characteristic of the associated amplitudes of mode A is investigated and listed in Tables3 and 4. Table 3 demonstrates the effects of the interacting distance and the rigidity ratio, rrigid, on the amplitudes of beam and membrane for mode A. The smaller the interacting distance D0 is, the larger the amplitude of membrane is. The error of measuring the topography on the membrane is increased due to the amplitude of membrane. If the interacting distance D0 is about 10.5 nm, the corresponding amplitude of membrane is about 0.125 nm. Table 4 demonstrates the effects of the interacting distance and the interacting position on the amplitudes of beam and membrane for mode A. It is observed that, for xc/Lx=yc/Ly=0.1, the amplitude of membrane is very small. It means that the beam vibration dominates. The coupled frequency shift is close to that of the system with rigid substrate. If D0=10nm and xc/Lx=yc/Ly=0.5, the corresponding amplitude of membrane is about 0.033 nm.Table 3
Effects of the interacting distance and the rigidity ratio,rrigid, on the amplitudes of beam and membrane for mode A (the other parameters are the same as those of Figure 5).
D 0  (nm) rrigid=5 r r i g i d = 100 r r i g i d = 1000 W - m ( x c , y c ) (nm) W - b ( L b ) (nm) W - m ( x c , y c ) (nm) W - b ( L b ) (nm) W - m ( x c , y c ) (nm) W - b ( L b ) (nm) 5 -0.016 1.016 -2.110 3.110 -0.748 1.748  6 -0.007 1.007 -1.093 2.093 -0.424 1.424  7 -0.004 1.004 -0.619 1.619 -0.263 1.263  8 -0.003 1.003 -0.379 1.379 -0.175 1.275  9 -0.002 1.002 -0.248 1.248 -0.122 1.222  10 -0.001 1.001 -0.171 1.171 -0.089 1.089  11 -0.001 1.001 -0.124 1.124 -0.066 1.066  12 -0.001 1.001 -0.092 1.092 -0.051 1.051  13 -0.001 1.001 -0.071 1.171 -0.040 1.040  14 0.000 1.000 -0.056 1.056 -0.032 1.032  15 0.000 1.000 -0.045 1.045 -0.026 1.026  16 0.000 1.000 -0.037 1.037 -0.021 1.021  17 0.000 1.000 -0.030 1.030 -0.018 1.018Table 4
Effects of the interacting distanceD0 and the interacting position {xc/Lx,yc/Ly} on the amplitudes of beam and membrane for mode A (the other parameters are the same as those of Figure 5).
D 0 (nm) xc/Lx=yc/Ly=0.1 x c / L x = y c / L y = 0.3 x c / L x = y c / L y = 0.5 W - m ( x c , y c ) (nm) W - b ( L b ) (nm) W - m ( x c , y c ) (nm) W - b ( L b ) (nm) W - m ( x c , y c ) (nm) W - b ( L b ) (nm) 5 -0.033 1.033 0.119 0.881 -0.398 1.398  6 0.010 0.990 0.074 0.926 -0.195 1.195  7 0.014 0.986 0.048 0.952 -0.111 1.111  8 0.012 0.988 0.032 0.968 -0.070 1.070  9 0.009 0.991 0.023 0.977 -0.047 1.047  10 0.007 0.993 0.017 0.983 -0.033 1.033  11 0.006 0.994 0.013 0.987 -0.025 1.025  12 0.004 0.996 0.010 0.990 -0.019 1.019  13 0.004 0.996 0.008 0.992 -0.015 1.015  14 0.003 0.997 0.006 0.994 -0.012 1.012  15 0.002 0.998 0.005 0.995 -0.009 1.009  16 0.002 0.998 0.004 0.996 -0.008 1.008  17 0.002 0.998 0.003 0.997 -0.006 1.006Figure6 demonstrates the effects of the interacting distance and the interacting position on the frequency of mode A. When the interacting distance D0>11nm, the frequency shift is small. When the interacting distance D0=10nm, the frequency shift is obvious. There are negligible difference among the frequency shifts for different measuring positions, xc/Lx,yc/Ly. The smaller the interacting distance D0 is, significantly the larger the frequency shift is. For the interacting distance D0<10nm, there are great differences among the frequency shifts for different measuring positions, xc/Lx,yc/Ly.Figure 6
Effect of the interacting position{xc/Lx, yc/Ly} on the frequency shift [SiO2:E = 70.3×109Pa, ρ = 2.5×103 kg/m3,B =45 μm,H = 3.5 μm,Lb=200 μm,R = 1.5×10-7m,AH = 10-19Joule, mtip=3.18×10-13kg, rrigid=15, rmass=1, rx=ry=1, the first natural frequency of beam fbeam,1 = 74355.803 Hz, W-bm=1nm].Figure7 demonstrates the effect of the interacting distanceD0 on the first five natural frequencies of the coupled probe-membrane system. When the interaction distanceD0 is large enough, the coupled natural frequencies approach those of the independent probe or membrane without the coupled effect. It is because the van der Waals force decreases with the interaction distanceD0. If the interaction distanceD0 is decreased less than 6 nm, all the natural frequencies become smaller significantly.Figure 7
Effect of interaction distanceD0 on the frequencies of higher modes [SiO2:E = 70.3×109Pa, ρ = 2.5×103 kg/m3,B =45 μm,H = 3.5 μm,Lb=200 μm,R = 1.5×10-7m,AH = 10-19Joule, mtip=3.18×10-13kg, rrigid=15, rmass=1, rx=0.5,ry=0.4, xc/Lx=yc/Ly=0.4, W-bm=1nm, the first natural frequency of beam fbeam,1 = 74355.80 Hz, the first four natural frequencies of membrane fmem,i = 55361.95, 81567.00, 93121.18, 110723.91Hz].


## 5. Conclusion

In this study, the analytical solution, the force gradient method, the probe-spring-membrane model, and the generalized perturbation method are presented. The comparison of their results is made. Considering a rigid substrate, both the analytical method and the perturbation method are accurate. However, for an elastic substrate the errors of frequency shift determined by using the force gradient method and the perturbation method are satisfactory only for larger tip-surface distances. Several trends about the frequency shift and the induced error of measurement are obtained as follows:(1)
Because the membrane inertia is considered, several additional modes except the beam ones occur for small interacting distanceD0(2)
For larger frequency shift sensitivity and small amplitude of membrane, smaller error of measure will be obtained(3)
The smaller the interacting distanceD0 is, significantly the larger the frequency shift and the amplitude of membrane are(4)
When the interacting distanceD0 is about 10 nm, the frequency shift is obvious. The corresponding amplitude of membrane is less than 0.15 nm. If D0 is decreased, the amplitude of membrane is significantly increased. It will increase the error of measurement



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