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The dynamic behaviors of an Atomic Force Microscope are of interest, and variously unpredictable phenomena are experimentally measured. In practical measurements, researchers have proposed many methods for avoiding these uncertainties. However, causes of these phenomena are still hard to demonstrate in simulation. To demonstrate these phenomena, this paper claims the tip-jump motion is a predictable process, and the jumping kinetic energy results in different nonlinear phenomena. It emphasizes the variation in the eigenvalues of an AFM with tip-sample distance. This requirement ensures the phase transformations from one associated with the oscillation mode to one associated with the tip-jump/sample-contact mode. Also, multi-modal analysis was utilized to ensure the modal transformation in varying tip-sample distances. In the presented model, oscillations with various tip-sample distances and with various excitation frequencies and amplitudes were compared. The results reveal that the tip-jump motion separates the oscillation orbit into two regions, and the jumping kinetic energy, comparing with the superficial potential energy, leads the oscillation to be bistable or intermittent. The sample-contact condition associates to bifurcation and chaos. Additionally, the jumping is a strong motion that occurrs before the tip-sample contacts, and this motion signal can replace the sample-contact-signal to avoid destroying the sample.

Dynamic atomic force microscopy (AFM) is widely used in high resolution imaging on a nanometer scale. The most commonly used operating mode of dynamic AFM involves a feedback system of amplitude modulation and exploits the fact that the tip of the microcantilever oscillates with amplitudes of a few tens of nanometers. A hard interaction between tip and sample introduces a strong nonlinearity in the motion of the tip; such nonlinearity includes tip-jump, bistability [

This paper claims that the tip-jump is a required condition that results in different nonlinear phenomena, and those can be predicted. The tip-jump is caused by the asymmetric two-wall potential that is determined using Liapunov stability theory [

In addition to the nonlinear phenomena, the superficial force that governs the microcantilever of an AFM yields two significant characteristics. The natural frequency of the microcantilever changes directly with the tip-sample distance [

This investigation involves a multi-modal analysis of AFM microcantilever, in which the natural frequencies vary with the tip-sample distance, to ensure the accuracy of oscillation of AFM microcantilever suffering from superficial forces. The tip-jump mechanism was based on force disequilibrium, and a force-displacement diagram helped explain the tip-beginning and tip-ending positions on the superficial potential force curves. Then the discretization method [

This investigation makes three main contributions. It provides an easy understanding of the tip-jump mechanism by demonstrating the force equilibrium. The method used also elucidates the cause of the zero-eigenvalue and points out the tip begin/end positions of the jumping. The second contribution is that this paper claims the tip-jump motion is a predictable process, and the jumping energy comparing with the superficial potential force energy, results in bistability for large kinetic energy and snapping for low kinetic energy. Furthermore, if the tip contacts the sample, the bistable motion may become hysteresis and the snapping motion may become intermittency or chaos. The third contribution is that this paper proposes the tip-jump motion is a strong signal occurred before the tip contacts the sample, and the detected jumping-signals can replace the traditional sample-contact signal to avoid destroying the sample.

The typical microcantilever in an AFM is constructed from a piezoelectric oscillator at one end with amplitudes that the tip at the free end can tap samples. As shown in

The elastic Bernoulli-Euler equation of the microcantilever motion is:
_{i}_{i}_{1}_{1}_{2}_{2}

The deflection ξ(_{n}(_{n}_{i}_{i}^{IV}_{i}_{i,n}_{n}_{n}(_{n}

The solutions to _{n}_{n}, D_{n}, E_{n}, F_{n}_{n}_{0}), _{n}_{0})] can be obtained by applying the initial conditions in every piecewise linear segment. Note that the integration intervals in

The clamped end of the microcantilever (_{f}t_{f}_{1} and _{2} are the Hamaker constants. ^{i}^{i}

Let Δ_{1} = _{f}t

The polynomials of _{1}_{2}

Applying _{n}_{n}_{n}_{n}_{n} can be numerically determined by solving:

This equation describes a clamped-free condition when
_{n}, depending on

When the tip of the microcantilever oscillates from one segment to another one, the displacement and velocity conditions in the former segment are regarded as the initial conditions in the new segment. The orthogonality conditions derived in _{0}, the initial conditions _{0}(_{0}) and _{0}(_{0}) those were obtained from previous segment are expanded in new modal basis:

Applying _{0} of the interchange:

_{f}t_{m}_{f}t_{m}_{f}_{n}_{i}^{W}_{1} = _{m}_{f}t_{2} = 0. Rather,

Its solution can be obtained:

Applying the orthogonality conditions to _{i}

Coefficients _{1,n}_{2,n}

Substituting

In

In _{1}(

To estimate the displacements and the velocities of the microcantilever after jump occurred, a model was developed by using the conservation of energy. The tip-jump from Point

After jump, the velocity of the microcantilever is:

Consider the variation of the kinetic energy of the microcantilever:

The total potential energy between Points

Finally, the total potential energy equals the kinetic energy Δ_{k}^{E}^{E}

The eigenvalues of the microcantilever with respect to the tip-sample distance were obtained numerically by using

When the AFM base was located at a particular elevation, the microcantilever bent and the tip moved to the equilibrium point (Point A or C in

In the following numerical discussion, 30 piecewise linear segments were considered within a certain region (11 nm) from the sample to AFM base. Note that the interval in each piecewise segment is setup to be around 1/3 times small than the jumping interval to ensure the simulation accuracy. In this case, the base was arbitrarily set up at Δ_{1} = 8.084 nm, and the equilibrium at Points A and C were obtained at 7.2 and 1.935 nm. _{1} or IC_{2} represents an initial disturbance of displacement and velocity of the tip. In these two cases, motion begins in Region II, and the tip travels into Region I or Region III, oscillates around the equilibrium points.

A demarcation line (D-line) in Region II is introduced to distinguish the oscillation vibration around Point A from that around Point C. When the tip is initially located above the D-line, the tip finally oscillates around Point C; otherwise, the tip finally oscillates around Point A. The D-line can be obtained by finding the points whose potential energy equal to the potential energy at Point B, where Point B is the cross-points as discussed in

Unlike damped driven spring systems that have been described elsewhere, this system exhibited a jump region that separated AFM dynamics into two oscillation systems. One oscillation system was in Region I and exhibited noncontact oscillation, and its behavior was simple. The other oscillation system was in Region III and involved noncontact and contact states; the oscillating behavior in contact state was complicated and was the most likely cause of bifurcation and chaos.

When the AFM base was set at a certain height elevation, two forced vibrations were driven at two equilibrium points with various amplitudes. In the cases considered herein, area A_{ab} was set to exceed area A_{bc}, as shown in _{m}

_{m}_{m}_{m}

The excitation frequency was set to
_{m}

With a constant driving amplitude of _{m}

In

Comparing

Numerous nonlinear phenomena occur in AFM experiments, but numerical models have until now failed to be useful in simulating them. Researchers have proposed many methods for avoiding uncertainties in the practical measurements. The tip-jump is one of such nonlinear phenomena, and this investigation proposed a mechanism of jumping. Its initial kinetic energy and the superficial potential energy obtained from the tip-sample distance affect its behavior, which may involve snapping, bistability, hysteresis, intermittency, bifurcation, or chaos.

This investigation noted that characteristics of the microcantilever that varies with the tip-sample distance is a significant requirement in the numerical simulations. This requirement ensures that the eigenvalue of the microcantilever transitions continuously from one associated with the oscillation mode to one associated with the jump mode. Three regions are defined to separate the tip motion range. Region I represents the tip motion before jump, Region II is the region where jump occurred and the first eigenvalue is vanished, and Region III is the interval between jump and contact. This achievement sets the model herein apart from other damped-spring models. This paper reveals that the elevation of the base of AFM can markedly influences the static characteristics of the microcantilever, including its natural frequency and shape functions. The results of forced oscillation show that a jump occurred if and only if the acquired kinetic energy exceeds the superficial potential energy. The results show large excitation amplitudes lead to the acquisition of much kinetic energy. A change in the excitation frequency can increase/reduce its kinetic energy, and to an extent that depends on whether the tip is where the resonant frequency of microcantilever is close to the excitation frequency.

The simulation results reveal that snapping occurs following a jump when the kinetic energy obtained in the region after jump region is too low to enable to jump back. Bistability is occurred by periodic jumps when the kinetic energy obtained in Region I and Region III suffices to maintain periodic jumps. Bistability with contacts may result in hysteresis, and snapping with contacts may lead intermittency or chaos. Moreover, the results indicate that the excitation frequency obtained in Region III can increase the sensitivity of measurement relative to that obtained in Region I. That also can eliminate the uncertainties in AFM detection and can avoid the tip to contacting or destroying the sample.

The author would like to thank the National Science Council of the Republic of China, Taiwan, for financially supporting this research under Contract No. NSC 100-2212-E-390-007. Ted Knoy is appreciated for his language editorial assistance.

Schematics of the deflection of AFM microcantilever. Δ_{1}(t) denotes elevation of AFM base, Δ_{2}(

(

Schematics of the three equilibrium points for soft stiffness microcantilever.

Schematics of tip-jump mechanism. When the microcantilever restoring force curve meets the tangent line at Point

Relationship between first three eigenvalues and tip-sample distance.

First mode shape functions for various tip-sample distances in

Phase portrait of tip under free vibration with two tip initial conditions and various damping ratios.

Tip phase portraits with various driving amplitudes at
_{m}_{m}_{m}_{m}

Phase portraits with various driving amplitudes at
_{m} = 0.05 nm. (_{m}_{m}_{m}

Time histories associated with phase portraits in _{m}_{m}_{m}_{m}

Phase portraits for various tip-sample distances. (

Time histories for various tip-sample distances. (

Material properties of microcantilever.

Description | Symbol | Si-Si (111) case |
---|---|---|

Length | 449 μm | |

Width | 46 μm | |

Thickness | 1.7 μm | |

Tip radius | 150 nm | |

Material density | 2,330 kg m^{−3} | |

Static stiffness | 0.11 N m^{−1} | |

Elastic modulus | 176 GPa | |

Q factor (air) | 20 | |

Hamaker (rep.) | _{1} |
1.3596 × 10^{−70} J m^{6} |

Hamaker (att.) | _{2} |
1.865 × 10^{−19} J |