Monte Carlo method in optical atomic force microscopy

Ahemd ElMelegy,Sarwat Zahwi

(Engineering and Surface Metrology Laboratory,National Institute of Standards (NIS),Giza,12211,Egypt)

Abstract：Scanning probe microscopy (SPM)is a branch of microscopy that forms images of surfaces using a physical probe that scans the specimen.Atomic force microscopy is one of the SPM family which is considered as a very versatile tool for surface imaging and measurements.A wide range of various samples can be measured regardless of being conductive,no-conductive,in vacuum,in air or in a fluid as a unique feature.One of the most challenges in atomic force microscopes (AFMs)is to evaluate the associated uncertainty during the surface measurements by AFMs.Here,an optical AFM is calibrated through the calibration of XYZ stage.The approach is to overcome difficulties experienced when trying to evaluate some uncertainty components which cannot be experimentally determined i.e.tip surface interaction forces and tip geometry.The Monte Carlo method is then used to determine the associated uncertainties due to such factors by randomly drawing the parameters according to their associated tolerances and their probability density functions (PDFs).The whole process follows supplement 2 to “the guide to the expression of the uncertainty in measurement”(GUM).The approach validated in the paper shows that the evaluated uncertainty in AFM is about 10 nm.

Key words：measurement;Monte Carlo method;atomic force microscope (AFM);nanometrology

0 Introduction

In 1986,Binnig et al.developed the scanning tunnelling microscope (STM)as a method to measure forces as small as 10-18N[1].The problem is its limitation for measuring conducting surfaces only.The atomic force microscope (AFM)was later invented,A different type of microscope capable of investigating surfaces of insulators on an atomic scale.The AFM is a combination of the principles of the STM and the stylus profilometer.The preliminary results of using AFMs in air demonstrated a lateral resolution of 3 nm and a vertical resolution less than 0.1 nm.Besides the interatomic forces,the AFM can measure electromagnetic forces as well.AFM images are obtained by measurement of the forces acting on a sharp tip at close proximity to the surface of a sample[2-4].

The atomic force microscope (AFM)is an instrument for topographical measurements through scanning the surface by a cantilever probe.It depends on the interaction forces between the probe tip and the surface which will cause deflection as the tip comes to close proximity with the surface.The vertical motion of cantilever is usually detected with external systems to draw the line profile and 3D shape of scanned surfaces,as shown in Fig.1.The measured sample can be moved laterally and vertically using for example a piezoelectric (PZT)tube scanner.A laser beam focused on cantilever upper surface will be reflected into the surface of a position sensitive detector to draw the vertical motion of the cantilever[5].

The calibration of AFMs and evaluation of associated uncertainty is necessary.It is not only required for metrological AFMs,but also for commercial AFMs.The calibration procedure is done by a calibrated transfer standard in both lateral and vertical directions.The uncertainty evaluation in AFM is not simple or straightforward.It is typically estimated using complicated model functions.Evaluation of associated uncertainty by Monte Carlo method is very interesting for making these calculations easily and non-assuming the linearity of the functions.

1 Instrumentation

In this work,a Multitask AutoProbe CP research AFM head is calibrated,as shown in Fig.2.The instrument has technical specifications of 100 μm maximum lateral range inxandydirections and 7.5 μm in vertical scan range.It has 0.25 Å (0.025 nm)and 0.025 Å (0.002 5 nm)maximum resolution in lateral and vertical directions respectively.The Multitask includes a multitask probe head operation in the modes of:contact,non-contact,and Intermittent-contact AFM,magnetic Force microscopy (MFM),lateral force microscopy (LFM),and STM.The calibration is applied here for contact AFM.

Fig.2 Multitask AutoProbe CP research AFM head

(a)AFM calibration grid

In addition,two reference standards are used:a Taylor Hobson©step height standard of 2.28 μm and 0.37 μm is used for vertical calibration,and a Topometrix©AFM calibration grid of different pitches and spaces,as shown in Fig.3,is used for lateral calibration.

2 AFM calibration

The Multitask AutoProbe research CP is calibrated in three motion axes,X,YandZ,as shown in Figs.4-6.The calibration procedure in any axis uses Master Scanner status in Contact AFM mode.The initial setting of scanned sample should consider that the features in eitherXorYaxis is parallel to the scan direction.An image is taken as normally done (scan lines 528×528 at least)whatever the scan axis.The image is edited.The line analysis tool is used then to measure the scanned feature.By comparison to the reference feature value,the error compensation is determined and the correction is entered to the operating software.

Fig.4 Calibration of Multitask AutoProbe CP AFM in X axis

Fig.5 Calibration of Multitask AutoProbe CP AFM in Y axis

(a)

3 AFM and associated uncertainty

The AFM cantilever can be described as a free massmattached to a mechanical spring having a spring constantk,the massmwill move vertically a distance Δz,and the forceFaffecting this mass can be assumed as[7]

f=kΔz.

(1)

The motion behaviour of the AFM cantilever can be described by model Eqs.(1)and (2),

fext=fdriv+fsur+fint=

(2)

wherefextis the sum of external forces,fdrivis the drive forces,fsuris the sum of surrounding and other forces,fintis the sum of interaction forces,meffis effective mass,andγis the damping factor.

The first termfdrivin model equation is neglected in static mode where the cantilever moves mostly under the interaction forces with the scanned surface.The other two terms offsurandfintwill account for other forces affecting cantilever motion based on continuum mechanics.These forces include long-range van der Waals forces,capillary forces,short-range forces,and tip load concept.The van der Waals forces can be the major interacting forcefintaffecting the cantilever while the other forces can be described as the surrounding forcesfsur.

Short-range forces have strong contribution in total tip-surface interactions with a separation distance less than 1 nm or in the special case of contact or static mode of AFMs.Short-range forces may originate from born repulsion,chemical bonding and electrostatic and van der Waals interactions between atoms.The term “tip load concept”refers to the contact shape of the tip apex with the sample area.Tip load is defined as the net normal force acting on the tip apex of AFM cantilever.

In short,the major factors that affect the measurement in AFM are scan rate,gain,measured points,set point in addition to tip shape and scanner resolution.Most of these factors have correlations with each other.

4 Uncertainty evaluation in GUM

The Multitask AutoProbe CP AFM head was calibrated throughx,yandzaxes using two standards,a step height standard as described before.To evaluate associated uncertainty,mathematical model description is used.

Measurand,

H=R+ΔScanner_res+ΔStandard-Cal+ΔTip-shape+

ΔScan_rate+ ΔGain+ Δpoints+ Δset_point,

(3)

whereHis the measurand (pitch or step height);Ris the reading of instrument for the measurand;ΔScanner_res,Δstandard _Cal,ΔTip-shape,ΔScann_rate,ΔGain,Δpoints,and ΔSet_pointare the corrections due to scanner resolution,uncertainty of reference standards,shape of cantilever tip,scan rate,gain,measured points during measurement,and set point value for measurements,respectively.

The evaluation of uncertainties associated with AFM measurements for either step height orXYpitches is based on GUM[7].

Assuming a linear model,the contributory variances are

u2(H)=c12u2(R)+c22u2(ΔScanner_res)+

c32u2(ΔStandard-Cal)+c42u2(ΔTip-shape)+c52u2(ΔScan_rate)+

c62u2(ΔGain)+c72u2(ΔPoints)+c82u2(Δset_point),

(4)

wherec1,c2,c3,c4,c5,c6,c7andc8are sensitivity factors concerning each factor.

For simplicity the following assumptions are considered:each sensitivity factor is considered to be equal 1 and thus

u2(H)=u2(R)+u2(ΔScanner_res)+u2(ΔStandard-Cal)+

u2(ΔTip-shape)+u2(ΔScan_rate)+u2(ΔGain)+

u2(ΔPoints)+u2(Δset_point).

(5)

The uncertainty budget for AFM calibration in vertical and lateral directions are shown in Tables 1 and 2,respectively.

Table 1 Uncertainty budget for AFM calibrationin in vertical direction

Table 2 Uncertainty budget for AFM calibration in lateral direction

5 Uncertainty evaluation by Monte Carlo method

The associated uncertainty for AFM calibration in both vertical and lateral directions is evaluated by using Monte Carlo simulation,as shown in Figs.7 and 8[8-9].The evaluation process for AFM calibration in vertical direction is run with the software specifications:

GUM WorkbenchEdu

Simulator:OMCE V:1.2.3

Mean Value:4 639.6 nm

Standard Uncertainty:2.5 nm

Coverage Interval (p=0.954 5):[4 634.7,4 644.5] nm

(Probabilistically Symmetric)

Expanded Uncertainty Interval (p=0.954 5):(+4.9,

-4.9)nm (Probabilistically Symmetric)

Number of Monte Carlo Trials:2 000 000

Block size:10 000 runs

Fig.7 Probablity distribution fucntion for uncertainty evaluation for AFM in vertical direction,U=±4.9 nm

The evaluation process for AFM calibration in vertical direction is run with the software specifications:

GUM WorkbenchEdu

Simulator:OMCE V:1.2.3

Mean Value:91 823 nm

Standard Uncertainty:5.1 nm

Coverage Interval (p=0.954 5):[91 812.8,91 833.2] nm

(Probabilistically Symmetric)

Expanded Uncertainty Interval (p=0.954 5):(+10,-10)nm

(Probabilistically Symmetric)

Number of Monte Carlo Trials:2 000 000

Block size:10 000 runs

Fig.8 Probablity distribution fucntion for uncertainty evaluation for AFM in lateral direction,U=±10 nm

6 Conclusions

The Multitask AutoProbe CP AFM head is calibrated in three dimensions,X,YandZaxes.The 100 μm scanner is used during the calibration process.A 90 μm scan range forXYgrid is scanned in contact AFM mode forXandYaxes.The 7 μm vertical range is calibrated with reference step height.In All,1 024×1 024 lines as measurement points for better representation of the scanned surface is used.The error in AFM head is compensated in the operating software.The calibrated AutoProbe head is traceable to S.I.units through the reference standards used in calibration process.

The AFM can be considered in its vertical detection as a free massmattached to a mechanical spring having a spring constantk,the massmwill move vertically a distance Δz.The cantilever tip-sample surface separation distance control the free motion of this spring.At nano separation distance,a generated attractive force (van der Waals forces)is driving the cantilever bending across the scanned surface.

There are many factors that dominate or at least strongly drive the surface measurements by AFM.These factors are scan rate,gain,measured points,set point in addition to tip shape and scanner resolution.Most of these factors have correlations with each other.The uncertainty in AFM calibration in both vertical and lateral directions is evaluated.The uncertainty is evaluated using Monte Carlo method in comparative way with GUM.Uncertainties in AFM calibration in vertical and lateral directions are 4.9 nm and 10 nm respectively by Monte Carlo method,while their values in vertical and lateral directions are 4.88 nm and 9.36 nm respectively by GUM.The Monte Carlo method is not only simple and easy in use but it deals easily with the correlations in fcators which is not aviable by GUM.