- Email: [email protected]

PII:

S0924-4247(19)30932-X

DOI:

https://doi.org/10.1016/j.sna.2019.111690

Reference:

SNA 111690

To appear in:

Sensors and Actuators: A. Physical

Received Date:

10 June 2019

Revised Date:

17 October 2019

Accepted Date:

21 October 2019

Please cite this article as: Rubin E, Davis S, Bucher I, Multidimensional topography sensing simulating an AFM, Sensors and Actuators: A. Physical (2019), doi: https://doi.org/10.1016/j.sna.2019.111690

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Multidimensional topography sensing simulating an AFM Eyal Rubin, Solomon Davis and Izhak Bucher Dynamics Laboratory, Mechanical Engineering Technion, Israel Institute of Technology

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Haifa, Israel

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Graphical Abstract

HIGHLIGHTS

Exploiting simultaneous multiple vibration mode excitation for 3D sensing. Large scale experimental system FM-AFM simulator

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Multi-mode, Autoresonance control scheme Novel frequency estimation algorithm for faster sensing not affected by transient vibrations Experimental results: 3D topographies - inclined surfaces, steep walls and trenches were measured with 4 (µm) resolution or better.

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ABSTRACT Atomic force microscopy (AFM) is used in the semiconductor industry for inspection and quality control. Frequency modulated AFM (FM-AFM) extracts surface topography by measuring the frequency shift created by the Van der Waals (VdW) interaction forces between the tip and the sample. To improve the measurement speed and address complex geometries emerging in industrial microchip constructions, several enhancements are introduced. While most FM-AFM

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devices operate in a single vibrating mode, this article proposes a method for multidimensional

sensing using frequency modulation of 2 orthogonal vibration modes, simultaneously. The concept

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was tested on a large-scale experimental system, where VdW forces were replaced by magnetic forces, using a magnetic tip and ferromagnetic samples. To emulate the VdW forces accurately,

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the designed ratio between the base frequency and frequency shift was kept to mimic a Nanoscale AFM. By utilizing an Autoresonance (AR) control scheme for faster locking onto resonance

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and a curve fitting frequency estimation algorithm it is possible to sense the minute changes in frequency experienced by several modes, simultaneously. Experimental results employ 3D relevant topographies such as inclined surfaces, steep walls and trenches that were reconstructed

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experimentally with 4 (µm) resolution or better. Downscaling to typical AFM dimensions would theoretically yield sub-nanometer resolution.

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KEYWORDS

3D topology measurement; Fast frequency shift sensing; Autoresonance; Multi-mode sensing

1. INTRODUCTION

This section reviews atomic force microscope (AFM) basic principles of operation and presents the

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motivation and general concept for the multidimensional topographical sensing.

1.1. Atomic force microscope - Some Background

Quality control and defect monitoring are essential technologies in the semiconductor industry.

Inspections are done using several techniques such as scanning electron microscope (SEM), scanning tunneling electron microscope (STEM), and atomic force microscopy (AFM). AFM provides high resolution, but with its relatively slow scanning rate it is not suitable for in-line metrology [1,2]. The AFM sensor in the non-contact configuration (NC-AFM) is based on a -2-

vibrating cantilever that is excited at or near its resonance frequency [3]. A sharp tip is attached to the free end of the cantilever. The tip interacts with the surface, while the distance dependent Van der Waals (VDW) force acts on the cantilever, altering its resonance frequency. One of the most common techniques used today is frequency modulated AFM (FM-AFM) presented by Albrecht et al. [4] in which the frequency shift is measured and correlated to the surface topology. The frequency change takes place instantaneously, hence theoretically, can be noticed within a fraction of an oscillation cycle. The amplitude modulation method (AM-AFM) presented by Martin et Al. [5] is slower and its speed is dependent on the vibrating sensor’s Q factor. In "Tapping

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Mode", the oscillating tip is positioned very close to the surface and repulsive tip-sample

interactions are involved. Tapping mode reduces the lateral forces and the contact forces of the

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interaction significantly relative to static AFM, but involves highly non-linear interaction and is limited in scope and performance [6] [7].

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The majority of AFM systems operate in a single mode of vibration. Some devices use higher modes and even multimode excitation, but all vibrations modes being used vibrate in the same

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direction [8] [9]. When mapping complex geometries such as fins, near-vertical side walls or narrow trenches, a flared tip (T shaped tip) probe is usually used in a single lateral direction [10]

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[11] or by tilting the AFM head while using computational effort for image correlation of several measurements [12]. Another method presented by Dai et al. [13] is based on a vector approach probing, which manipulates the approach angle of the flared tip. In this case, the tip motion can

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always be normal to the surface for lateral probing, using prior knowledge or repetitive measurement, which slows down the measurement time. The work of Heiguci et al [14] shows the potential of using multiple flexure modes of a tuning fork probe by switching between them. Other publications [8] ,[13] show the use of torsional mode and high bending modes. The present paper shows the potential for multi-dimensional sensing by excitation of two (or more) modes

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simultaneously. By designing the frequencies to be close a similar resolution can be achieved for all sensing directions.

1.2. Motivation

Multidimensional scanning is based on simultaneous measurements of 2 or 3 modes of vibration, each producing motion in orthogonal spatial directions. Thus, being able to sense x, y and z (3D), to measure complicated topographies of existing and new generations of Nano-electronics, and can improve scanning speed and methodology. Using the AR method for frequency tracking has a -3-

potential to improve scanning speed dramatically comparing to currently used phase lock loop (PLL) method. In the present work we demonstrate a large-scale realization that proves the concept of a multidimensional sensor based on frequency modulation. The measurement process is performed simultaneously for two modes of vibration – horizontal (x) and vertical (y) as described in Fig 1 . Having obtained scaled parameters (section 3.2), keeps a similar relative frequency shift to base frequency ratio between the large-scale experimental system to a commercial AFM. The nondimensional resolution to image size ratio was similar in the large-scale experimental results and

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b)

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a)

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commercial AFM (section 4.3).

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Fig 1: Multidirectional vibration from FE simulation – a) horizontal motion of the tip (x), b) vertical motion of the tip (y)

2. PROPOSED SYSTEM LAYOUT, ACTUATION AND SENSING This section presents the layout of a large-scale experimental realization of an AFM-like system

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that was used to demonstrate the two-dimensional simultaneous scanning, exploiting a fast AR scheme for excitation and a suitable method for extracting the excitation frequency.

2.1.

Experimental system - configuration

A schematic layout of the experimental system is described in Fig 2.

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XYZ stage Autoresonance

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Frequency Estimation

Fig 2: Experimental System layout showing the vibrating sensor, the digital control system, sensing,

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actuation, and the controlling computer.

The experimental system is based on a vibrating cantilever with cross-section of 12mm X 15mm

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and length of 130 mm. Due to the high stiffness of the cantilever ( 2 0 0 k N

m

), the Van der-Waals

forces that are the main contributors to the frequency shift in Nano-scale AFM are too weak, and

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are therefore replaced with the attraction force that is created between a magnetic tip and the ferromagnetic sample. A cylindrical magnetic tip, with a radius of 1 mm and length of 20 mm is

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attached to the end of the cantilever. The differences and similarities between the magnetic force in this device and the VdW forces in an AFM will be discussed later in section 3.1. The cantilever is excited with two voice-coil actuators placed symmetrically at

45

on both sides of the cantilever.

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Two laser displacement sensors (Keyence LK-H008) are also positioned at

45

on both sides of the

cantilever and measure the displacement of the cantilever close to the tip. Modal filtering [15] is performed digitally in order to project the measured data and excitation forces on the exact modes directions (as explained in section 2.2). All the vibrating parts are mounted on an optical table through a rigid aluminum base in order to decrease the peripheral vibrations of the system.

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An Auto-resonance (AR) control scheme programed on an FPGA (Field-Programmable Gate Array) is used for a fast resonance excitation of the two modes simultaneously. The FPGA enables high sampling rate of 10MHz which is essential for fast and accurate frequency estimation using curve fitting algorithm (section 2.3). During measurements, the sample is attached to a precise 3D piezostage that can move the sample with a 1 micrometer precision for all axes. The mechanical configuration of the experimental system is described in Fig 3.

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2.2.

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Fig 3: a) Experimental system overview b), c) sensing magnetic tip

Modal filtering for vibration mode separation

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The symmetric configuration of the sensors and actuators presented in Fig 3 requires Modal Filtering (MF) [15] in order to separate the response and excitation of the two modes, and use them as inputs and outputs for simultaneous AR algorithms. The MF uses the bi-orthogonality

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between the vibration modes, by taking a linear combination of the two sensors signals so that all modes but one are canceled but one.

An illustration for the modal filtering of the displacement signals, measured by the laser sensors, is presented in Fig 4 (a) where

S1 , S 2

are the measured signals and

Sy,Sx

are the modal filtered

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displacements from the following equation : S x (t ) S1 (t ) S 2 (t ) S y (t ) S1 (t ) S 2 (t )

The measured signal S 1 , S 2 are presented in Fig 4 (b) and the measured signals after MF stage are presented in Fig 4 (c). A similar analysis can be made for the forces actuating on the cantilever by two voice-coil actuators:

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(1) Sy,Sx

F1 ( t ) F2 (t )

1 2 1 2

F

x

(t ) F y (t )

(2)

F

(t ) Fx (t ) y

where F1 , F 2 are the actuation forces that are calculated from the desired input forces for each Fx , Fy

.

b)

a)

s1

c)

s2

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mode

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sx

sz

2.3.

Sy,Sx

Sy,Sx

(b) measured

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signals S 1 , S 2 (c) measured signals after MF -

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Fig 4: (a) Illustration showing the measured signals S 1 , S 2 and the modal filtered signals

Resonance Tracking – Autoresonance and Frequency Estimation

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The change of the natural frequency, attributed to the closeness of the measuring tip to the geometrical features, is tracked by a resonance tracking control scheme. In a typical FM-AFM system, a Phase Locked Loop (PLL) is utilized (see appendix for comparison), whose frequency

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locking speed is greatly reliant on the settling time of the cantilever [16]. The presented system uses two parallel Autoresonance (AR) closed-loop control schemes that operate in parallel on two separate vibration modes. Autoresonance, a self-excitation phenomenon, is a well-known nonlinear feedback method used for automatically exciting a system at its natural frequency [17,18]. It locks onto resonance from the first cycle, and has the potential to increase the imaging

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speed.

The AR feedback loop described in Fig 5 consists of a phase shifting element

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, often realized by a

digitally employed differentiator or integrator, which shifts the phase of the input signal by 90 degrees, and a digital 'relay' or sign function that forces the amplitude of the input signal to constant values [19].

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Fig 5: Schematic AR loop employing an integrator as phase shifting element, a relay and the sensing oscillator realized in G.

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frequency lock while the amplitude increases towards steady-state.

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Simulated response of an oscillator to AR scheme presented in Fig 6 showing the instantaneous

Fig 6: Response of an oscillator to AR scheme from near rest.

Once the cantilever oscillates at resonance, the oscillation frequency is estimated using a new

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algorithm [20] , that uses Linear Least Squares (LLS) [21] to fit an instantaneous phase to the noisy signal for several periods, and then estimates the slope. A schematic diagram of the algorithm is

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presented in Fig 7.

Fig 7: Schematic representation of the frequency estimation algorithm with the two LLS stages

Assuming white noise, the variance of the estimated frequency is [20]:

2 f

tN N SN R 2

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,

(3)

where,

tN

is the total estimation time and

gathered samples and

SN R

is the signal to noise ratio , N the number of

is a system dependendent constant.

Presented in this paper are both resonance tracking and frequency estimation that are performed simultaneously for the two modes of vibration using modal filtering.

2.4.

Reconstructing the geometry by position control

Two main scanning methods can be used [22]: Constant height or constant gap. In the former the

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probe is located at a constant height above the surface, and the change is the resonant frequency is measured and then converted into the change in topography. In the latter, the gap between the

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probe and the sample is kept constant using a standard PI control scheme [23] that keeps a

constant natural frequency for single and dual mode excitation. The XYZ stage’s instantaneous ( 0 .2 m resolution) is used for the surface topology estimation.

AR Loop

Δz Δx

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fnz fnx

Feedback Loop

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Frequency Estimation

XYZ Piezo Stage

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location

PI Controler

fref _z fref _x

Fig 8: Control scheme for position control keeping the gap such that a constant resonance frequency is

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obtained

By keeping the resonance frequency constant, the nonlinear relationship between the sensor to specimen gap and the frequency, need not be known at all. The actual position and hence the geometry is obtained from the precise xyz stage controlling the tip’s position. 0.2 mm gap was maintained for scanning the specimens of the large scale experimental system, as detailed in the next section.

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2.5.

Main Results

Three dimensional samples were scanned using two-mode excitation. The results for the inclined surface, vertical wall and trench are presented in Fig 9 and Fig 10. A comparison of the scanned surfaces to the manufactured dimensions show very good agreement. However, the details of the corners are blurred due to the spatial effect of the magnetic field. This blurring is not expected to occur on the nanoscale where tip-sample interaction, dominated by the local VdW forces, is more localized then the magnetic forces employed here. a)

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d)

Fig 9: Incline specimen: a) CAD model, b) experimental system, c) measurement front view d) measurements

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in 3D

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a)

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c)

Fig 10: measurements: a) steep wall, b) and c) 3D surfaces geometry simulating a trench and a curved wall

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The simultaneous excitation and sensing of two orthogonal modes provide more information for each measuring point. In addition to the surface reconstruction, the additional information can

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also be used for improving the scanning algorithm, e.g. decreasing the scanning step size in advance, while approaching a vertical wall, as illustrated in Fig 11. When the geometrical gradient becomes large, the measuring steps are made smaller to maintain the required accuracy. Clearly,

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near the right-angled corner, small steps were taken, which was made possible by sensing both

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vertical and horizontal gaps.

Fig 11: Scanning step size manipulation using multiple mode information. Shown are the measured locations as set by the sensing algorithm.

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3. ANLYSIS OF THE INTERACTION FORCES AND STABILITY In this section the interaction forces that causes the frequency modulation are discussed and developed for frequency shift assessment, together with other stability issues such as magnetic spatial effects and jump to contact.

3.1.

Tip-surface interaction

In an AFM, the potential energy depends on the gap between the tip and the specimen and it gives

F ts

is the potential and

V ts

z

V ts

.

z

(4)

is the distance between the plane connecting the centers of the

surface atoms and the center of the closest tip atom.

F ts

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Here

z

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rise to a distance dependent interaction force

in an AFM is composed of long range

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forces (up to 100 nm) : (i) Van der Waals, (ii) magnetic and (iii) meniscus [24] in ambient conditions. Short range chemical forces only become relevant at distances of fractions of

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nanometer. The long range VdW interaction is always attractive, and in Vacuum are assumed to contribute the most to the total tip to surface interaction. For better AFM resolution it is common to use a sharp tip to ensure that the VdW interaction is between a single atom at the end of the tip

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to the closest atom of the surface [25]. For a spherical tip with radius R next to a flat surface the

where

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van der Waals potential is given by [26]:

AH

V vdW

AH R 6z

,

(5)

is the Hamaker constant that depends on both tip and sample material types,

spherical tip radius, and

z

R

is the

is the distance between the tip and the closest atom of the surface.

In the large scale experimental system, the VdW potential is too weak with respect to the stiffness

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of the cantilever. Hence, a magneto-static potential has been used instead. The magneto static vector potential of a permanent magnet is given by a surface integral [27]:

where

0

VM

r

is the magnetic permeability,

M

0 4

M nˆ r

da

,

is the volumetric magnetization, nˆ is a unit vector

normal to the surface , r is the distance vector of the measured point from the surface of the magnet, and

da

is the surface differential. -12-

(6)

In addition, the magnetic field is strongly dependent on the geometry of the magnet and the surface, hence the interaction force is also affected by these parameters. Magneto-static simulations were conducted to estimate the anticipated resolution and to find the best working point. The simulations were performed using the Finite Element Method Magnetics (FEMM) software [28] – which solves 2D magneto-static problems employing vector potential formulation in finite elements method. Fig 12 shows an example for the spatial influence of two geometries on the magnetic field, comparing between a flat sample to a wall sample, as calculated using FEMM. a)

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Fig 12: Schematic model and FEMM simulation results: a) Magnetic tip–flat surface model ; b) Magnetic tip–wall model

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Interaction forces in X and Y directions for both models are presented in Fig 13. The spatial influence on the magnetic force can be clearly observed in Fig 13 (b), where forces in the

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horizontal direction (X) change with the change in magnet-surface gap on the vertical direction (Y), although the gap in the horizontal direction remains constant – 0.2 mm from the wall. Non-linear

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decay on the force interaction can be clearly seen in both models’ simulations.

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Fig 13: Magnetic tip to surface interaction forces calculated by FEMM [28] simulations a) Magnetic tip – flat surface model; b) Magnetic tip – wall model

Results show non-linear dependence between the interaction force and stiffness to the gap. The working point of the scanning probe should be as close to the sample as possible for better resolution.

3.2.

Frequency shift and resolution assessment

The dynamics of the vibrating cantilever can be described as a weakly perturbed harmonic

0 Q

where

q

q 0 q F ts F 0 c o s ( d t )

is the displacement of the beam,

2

k0

0

,

(7)

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q

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oscillator with the following equation of motion:

is the natural frequency of the cantilever

m e ff

system , F 0 and

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that depends on the stiffness and the effective mass of the cantilever, Q is the quality factor of the are the amplitude and frequency of the driving force, and

d

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tip to surface interaction force.

Solving the dynamics [26] assuming small amplitudes ( A

gap

F ts V ts q

is the

) yields the frequency shift which

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is linearly dependent on the stiffness ratio of the interaction to cantilever: f f0

k ts V ts q 2

2

(8)

2k0

is the tip to surface interaction stiffness.

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Here,

k ts

Using (8) with the calculated values for the cantilever spatial stiffness and resonance frequencies, together with the calculated interaction force-gap plot from Fig 13 for

k ts

, yields the estimated

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frequency shift plots for the two modes of the experimental system cantilever plotted in Fig 14.

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Fig 14: Frequency shift estimation a) vertical mode b) horizontal mode

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The total resolution of the system is derived from assessment of the measured frequency

the resolution is: RES

f

potential ,

q

f

(9)

V ts f0 3 q 3

are the beam stiffness and natural frequency,

V ts

is the tip-surface interaction

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k0 , f0

2

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q

where

2k0

2 f

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variance, multiplied by the derivative of the frequency shift. Hence the complete expression for

is the displacement (gap) in the relevant direction, and

2 f

is the measured frequency

variance calculated from (3). Using a 10MHz sampling rate, over an estimation time of 10ms with 0 .0 1 2 H z

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2 an assessed SNR of 20dB, yields with an estimation for the frequency variance: f

. Fig

15 shows the expected resolution for the two directional modes of vibration. The difference in resolution between the two modes is mainly due to the difference in the magnetic interaction force between the two modes, together with the difference in stiffness. To get a better resolution, the tip should vibrate as close as possible to the sample, but the working distance is bounded by

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the jump to contact (JTC) - instability state, where the beam stiffness cannot resist the attractive tip to surface force, and causes the tip to jump and stick to the surface of the sample. The instability can be avoided even for soft cantilevers by oscillating the cantilever in a large enough amplitude. Stiff cantilevers can vibrate even in small amplitudes [3].

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Fig 15: Resolution assessment a) vertical mode b) horizontal mode

Larger-Scale System Analysis

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3.3.

The large scale experimental system is compared to a commercial AFM [29] using a non-

f0

f m in

2 f

to the base

. Table 1 shows both ratios are of the same order, hence the up-scaling

of the experimental system is valid.

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frequency of the beam

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dimensional factor of the ratio between minimal measured frequency shift

Table 1: Comparing parameters between a comercial AFM to the larger scale experimental system LARGE SCALE SYSTEM

LARGE SCALE SYSTEM

[29]

MODE 1

MODE 2

k0 (N/m)

35

2·105

3.1·105

kts (N/m)

1

2.5·103

103

330·103

447

549

Δfmin (Hz)

20

0.012

0.012

Δfmin / f0

6.06·10-5

2.68·10-5

2.18·10-5

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f0 (Hz)

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COMERCIAL AFM

4. PERFROMANCE AND SENSITIVITY ANLYSIS This section will present the measured performance and sensitivity of the large-scale system in two directions, which will be compared (using relevant scaling factors) to a commercial AFM.

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4.1.

Resolution

The total measuring resolution of the system is affected by many parameters, such as magnet size, shape and orientation, spatial resolution of XYZ stage, frequency estimation resolution, sensors resolution, external disturbances and noise. Equation (9) shows the effect of the tip-surface interaction forces and the cantilever properties on the theoretical resolution. The resolution of the experimental system was measured by scanning a flat rectangular surface with 10,201 data points, and calculating the standard deviation of the measurements, that can be

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compared after scaling to a commercial FM-AFM [29]. The results in Fig 16 and Fig 17 are similar to the calculated resolution that was presented in Fig 15 for a gap of 0.1 mm. The RMS measured

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resolution of the vertical mode (y) is 1.3µm, and the RMS measured resolution of the horizontal

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mode (x) is 4µm.

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Fig 16: Resolution assessment for the vertical mode: a) scanned data, b) statistical analysis showing the

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histogram, representing the measured gap probability distribution

Fig 17: Resolution assessment for the horizontal mode: a) scanned data, b) statistical analysis showing the histogram, representing the measured gap probability distribution

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4.2.

Repeatability

Frequency vs. gap plots for the modes were measured in multiple experiments to assess the sensitivity of frequency shift to a change in the sample topography features. The results presented

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in Fig 18 show better repeatability for the vertical mode, due to the better resolution of the mode.

Fig 18: Repeatability analysis a) vertical mode b) horizontal mode

4.3.

Performance

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The experimental system is a large-scaled AFM, hence the performance should be compared to a commercial FM-AFM [29] using relevant scaling factor. Since the resolution of the commercial AFM was measured for a specific image size, a suitable non-dimensional comparison is the ratio between the resolution and the image size. The vertical mode is of the same order of magnitude while the horizontal mode is three times larger, due to the lower interaction forces in the

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horizontal direction caused by to the spatial effect of the magnetic field. A different configuration of the tip and beam stiffness could change the measured resolution in the horizontal mode.

Table 2: performance comparing between a comercial AFM [29] to the large scale system

f0 (HZ)

COMERCIAL AFM

LARGE SCALE SYSTEM

LARGE SCALE SYSTEM

[29]

MODE 1

MODE 2

330,000

447

549

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IMAGE SIZE

0.2 (µm)

1 (mm)

1 (mm)

RESOLUTION

0.38 (nm)

1.3 (µm)

4 (µm)

RESOLUTION/IMAGE SIZE

0.0019

0.013

0.004

5. CONCLUSIONS A Large-scale demonstration system simulating an AFM was developed for proving the concept of

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multiple dimension sensing for three-dimensional geometry mapping. An Autoresonance (AR) control scheme for faster excitation and novel frequency estimation algorithm were realized for

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sensing both vibration modes simultaneously without waiting for steady state settling. The

multidirectional data at each measuring point was used for geometry mapping and by adaptive scanning algorithm. A difference in resolution for the two modes was caused by the spatial effects

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of the magnetic forces between the cylindrical tip and the measured ferromagnetic samples, together with the difference in the cantilever stiffness between the modes’ directions. The

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experimental system showed a similar non-dimensional factor of resolution to image size ratio, when compared with a commercial AFM. The multidimensional sensing concept combined with

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fast AR control scheme may be realized in using tuning fork probes (TFP) and may be implemented in the future for Nano-scale implementation, which will improve AFM capabilities for faster

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scanning of complicated geometries.

Conflict of interest declaration

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The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

ACKNOWLEDGEMENTS This work was funded by the Israeli Innovation Authority within the Multi-Dimensional Metrology (MDM) consortium.

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[18]

Eyal Rubin received a MSc. degree in the mechanical engineering faculty at the Technion – Israel institute of technology in 2019. He has completed his Bsc, in mechanical engineering on 2007, and has been working in the industry in the field of dynamic simulations ever since. His fields of interest are vibrations and simulations of dynamic systems.

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Dr. Solomon Davis received a B.S. degree in Mathematics from the University of Oregon in 2010, an M.S. degree in Mechanical Engineering from the University of Washington in 2013, and a PhD in Mechanical Engineering from the Technion – Israel Institute of Technology in 2019. He is currently working on postdoctoral year in the Faculty of Physics at the Technion. In between his Masters and Ph.D. studies, he worked for two years as a mechanical systems and signal processing engineer for a company specializing in automotive testing equipment. These applied skills he carried with him to his doctoral studies where he helped develop high-speed control algorithms implemented on FPGAs. His specific areas of interest are resonance tracking, vibration mode analysis, mechatronics and signal processing.

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Dr Izhak Bucher is a Mechanical Engineering professor at the Technion, Israel Institute of Technology. He is the head of the Dynamics and Mechatronics laboratory since 1996. He was a research associate 1993-96 in the Dynamics Section, Imperial College London UK and completed all his degrees at the Technion, MSc and DSc under Prof. Simon Braun. His research interests are vibration, Dynamics of rotating structures, acoustic and magnetic levitation, system identification and traveling wave control and applications.

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Appendix A Auto-resonance based frequency estimation for a FM-AFM

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Consider an oscillator operating in an Autoresonance loop as shown in Fig A1:

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Fig A1: a) Schematics of the oscillator showing the measured coordinate x and the excitation force u . b) Autoresonance loop showing a relay with amplitude A, and the oscillator’s transfer function with velocity output

y x

.

A resonator connected in an Auto-resonance (AR) loop, automatically produces oscillations at a d

from the first cycle, despite the transient phase where the amplitude continues to

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frequency

grow, the zero-crossing frequency is constant from the first cycle. Figure A1 describes an oscillator

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whose response can be computed for any half cycle where the points indicated i and ii in Fig A2. 1

e

1

2

nt

s in d t

(1)

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A v (t ) -x 0 n n

Fig A2: a) Response of the system in Fig. A1. From initial conditions x ( 0 ) x 0 , x ( 0 ) v 0 , b) response at AR for a longer time

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Consulting (1) we can see that upon changing sign of the velocity, the next time zero crossing will occur is when

s in d t 0

, i.e. when

dt

, which constitutes half a cycle. We can now analyze

the system using the same tools with excitation switching from switching time, therefore a complete period is

T

2 d

A

to

A

with the same

.

Clearly, the amplitude keeps growing until it reaches steady-state, but zero-crossing frequency remains constant and corresponds to the damped natural frequency. -23-

In order to demonstrate the advantage in employing AR over a Phase-locked-loop (PLL), one can examine the measured response of the experimental system employed in this work.

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Fig A3: a) Open-loop step response of the cantilever, decay time is 0.5 sec. b) Estimated frequency upon a step change in the sensed topography using AR on an FPGA (See Fig 2). Time from the actual step to the stable frequency estimate is 11 milliseconds.

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PLL based resonance tracking cannot ignore the sensor’s dynamics and settling time and its

performance depends on the control realization. Judging from Fig A3 AR produces a stable reading

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of the frequency at a rate which is about 50 time faster than the system’s settling time in open loop.

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Comparing AR to PLL based resonance tracking

To better explain the difference between AR and phase-locked loop (PLL) a numerical simulation

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with the parameters of the system presented in Fig 3 was performed. The realization of the PLL based resonance tracking is identical to Fig.1 in [30] with optimized parameters of the PI controller

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to achieve fast convergence to resonance for the specific oscillator being controlled.

Fig A4: Resonance frequency tracking of PLL vs AR under simulated 0.2Hz steps in n .

Due to the low damping of

0 .0 0 5

the performance of the PLL loop controller is limited and

the obtainable convergence settling time to 99% of the change in frequency is related to

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1

n

. Fig

A4 was produced using the instantaneous Hilbert transform phase computation [19]. The results

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were low pass filtered to highlight the faster settling time of AR over PLL.

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