Modeling the tip-sample interaction in an atomic force microscope ...

Modeling the tip-sample interaction in an atomic force microscope: evaluating theLennard-Jones potential by direct integrationMary C. ScottDepartment of Physics, North Carolina State University, Raleigh, NC 27695-8202(Dated: April 25, 2008)Atomic force microscopy (AFM) is one of the most powerful tools in use today for surface characterization.AFM can be used to measure a variety of different types of samples, compared toscanning tunneling measurements, which require a conducting sample. Recent advances in sensitivityhave even demonstrated the ability to discern different elements in heterogenous samples. Here,basic modeling of the AF microscope tip-sample interaction is presented in the context of the vander Waals interaction. Amplitude and frequency response are derived for dynamic measurementmodes. Theoretical results are qualitatively discussed, as is the application of this method to otherforces.I. INTRODUCTIONII.BASIC OPERATING PROCEDUREScanning probe microscopes are powerful tools forimaging surface topography in real space. The fieldsof nanoscale science, tribology and many other surfacesciences have made great advances through the use ofthese devices. Scanning probe microscopes have a hugerange of magnification capability, varying on the order of10 −3 x to true atomic resolution [1]. These microscopesoperate in general by bring a sharp tip close to a sampleand monitoring the sample-tip interaction. This idea wasfirst implemented by Gerd Binnig and Heini Rohrer, thecreators of the scanning tunneling microscope (STM) in1982 [2]. This device monitored the current of electronstunneling from sample to tip (or vice versa) [2]. In 1986,Gerg Binnig, Calvin Quate and Christopher Gerber expandedon this idea and created a microscope sensitiveto atomic forces, the atomic force microscope (AFM) [2].Atomic force microscopes, unlike scanning tunnelingmicroscopes, which require a conducting sample, canmeasure surface properties of almost any material [1].In fact, AFM can resolve individual atoms on a varietyof sample types (conducting, insulating, hard minerals,soft biological samples) in a variety of environments(air, liquid, vacuum)[1]. While it is true that the interactionforces between the sample atoms and tip vary fromelement to element, a classical drawback of the atomicforce measurement is the lack of sensitivity to differentelements. However, recent dynamic AFM measurementsin ultra-high vacuum environment have been performedwith a sensitivity that is element specific [4]. While differentnoise sources and device instabilities limit resolution,it is imperative to use an appropriate model to describethe interaction between the sample and the measuringtip. This letter will outline the basic operation of theatomic force microscope, focusing on the physical modelsused to glean topographical, electrical, magnetic andeven elemental information from the sample of interest.As with all scanning probe microscopes, the basicprinciple behind atomic force microscopy is to bring anatomically sharp tip close to a sample, causing the tipto experience attractive and repulsive forces from thesample[1]. Sample-tip distances are typically on the orderof nanometers [2][3]. In the case of AFM, the measuringprobe is a sharp tip attached to a cantilever. This cantileverhas a known spring stiffness, quality factor andresonant frequency. The vertical position of the lever-tipsystem is controlled via a piezoelectric device. A commonway to measure the position of the measuring tipis to reflect a laser beam off of a polished surface on theback of the lever to which the tip is attached. Beamdeflection or frequency and phase of beam position canthen be measured[1], and through the use of a physicalforce model, related to the height of the tip above thesample. Relative topographical data is then obtained bytranslating the sample via piezoelectronic control in thex, y, and z directions. A schematic of a typical AFMsetup is shown in Figure 1.Project/setup.JPGFIG. 1: Schematic of a typical AFM setup, including piezoelectriccontroller, sample on translating stage, lever-tip system,and measurement laser.The AF microscope can be used in many different operationalmodes. Perhaps the simplest is the constantforce or “contact” mode, in which feedback from the

2force measurement is used to keep the force, and thereforethe distance above the sample, constant. The sampleheight can then be measured directly[2]. This method requiresindependent measurement of the force exerted onthe sample tip and the tip position. There are two maindynamic modes of operation, frequency modulation andamplitude modulation, also known as non-contact andtapping modes, respectively[6]. During operation in a dynamicmode measurement, the piezoelectronic controlleroscillates the lever-tip system. Frequency and amplitudeinformation are measured through phase-sensitiveoptical measurement (i.e., lock-in detection)[2][1]. Amplitudemodulation (tapping mode), which typically involvesmoving the tip from the non-contact (attractiveforce) region to the contact (repulsive force) region. Thismeasurement is most often performed in gas or liquidmedia, whereas the frequency modulation mode (noncontact)is more conducive to measurement in vacuumconditions [6]As always, sources of noise should be considered. Thermaldrift of the sample and especially of any piezoelectricpositioning devices are a concern. Measures can betaken to thermally stabilize the system, but speeding upthe measurement is the primary way to deal with thisproblem. Other sources include laser beam stability andmeasurement noise[2].III.MODELING THE TIP-SAMPLEINTERACTIONBeyond limiting any sources of noise in the device, resolutionin AF microscopes depends on the model usedto relate the response of the cantilever-tip system to theheight of the tip above the sample. For this section, dynamicmode operation only is considered. While the contactmode of operation directly measures height, dynamicmodels are considered more sophisticated and more readilygive atomic resolution[6]. This is in part due to reducedphysical interaction between the tip and sampleand due to the increase in measureable parameters in adynamic mode. Frequency, amplitude, phase and overalloffset of the tip can all be measured. Any one of theseparameters alone could characterize the surface, but togetherthey result in higher resolution[6].Many models exist to describe the response of an oscillatinglever with an external force field. These includeperturbation theory, Fourier analysis, and variationalapproach[6]. All of these methods strive to makethe connection between the tip response and the heightof the tip above the sample, which will ultimately leadto the “picture” of the sample’s surface. Common toall of these methods is the development of a governingequation of motion. In the method below, the force willbe incorporated into the equation of motion via an effectivespring constant. Then, the harmonic oscillatorproblem for the cantilever-tip system can be solved bydirect integration[2]. This direct integration method requiresan interaction potential that obeys a power law,i.e.,w(r) = −Cr n (1)Where r is the distance between interacting objects andC is a material- and geometry-specific constant. Thisgeneral force can then be integrated over various volumesand defined in terms of the tip-sample distance, z. Forexample, integrating a point particle’s interaction with a2D sheet of molecules with number density ρ 1 results in[2]:2πρ 1 C 1W (r) = −. (2)(n − 2)(n − 3) zn−3 This would be an ideal model if the tip were truly atomicallysharp. However, actual tips have a finite radiusof curvature. A better model is to consider the tip asphere of radius R, where R is the radius of curvature ofthe actual tip. Performing a pairwise integration over asphere of radius R and number density ρ 2 and a sheetwith number density ρ 1 gives, for z

3Project/geom.JPG= -C 0 [ σ − 1 z 2 300A =σ 7z08 ] (8)− 2 σ 715 z09 ] (9)Project/response.JPG∂t + k(z − u)FIG. 2: Geometry to solve the driven, forced harmonic oscillatorproblem.where k is the spring constant of the lever and z and u areare the positions of the tip and the driving piezoelectronicdevice, respectively. The height of the sample is set tozero. At this point the problem is solved in the sense thatthe tip position (which is also the relative tip height) isnow related to the position of the driving piezoelectronicdevice, which is known. However to extract frequencyand amplitude data from this model, we Taylor expandthe force about z 0 and take the first two terms:F 0 = −C 0 [ σ z02 − 130F 1 = 2C 0 [ σ z 3 0with C 0 = − 2 3 π2 ɛρ 1 ρ 2 σ 5 . The effective spring constantwill be defined ask ′ = k − F 1 (10)for a geometry with z positive towards the sample. Theequation of motion to solve ism ∂2 z+ γ ∂z + k(z − u) = F (z) (11)2∂twith the positions of the base of the lever driven by thepiezoelectric controller, and the tip given byandu = u 0 + ae iwt (12)z = z 0 + Ae iwt−iθ (13)respectively. γ is mω 0 /Q, the damping coeffecient withω 0 = sqrt(k/m) the natural resonant frequency. Q isthe quality factor of the spring. With the Taylor seriesapproximation of the force, Eq. 11 ism ∂2 z∂t 2 + γ ∂zσ 7] + 2Cz08 0 [ σ − 2 z03 15σ 7](14)z09Applying the condition that at equilibrium F 0 = k(z 0 −u 0 ), we see that the amplitude is defined byA(k ′ − ω 2 m + iωγ) = ake iθ . (15)The resonant frequency for this system is ω ′ 0 =sqrt(k ′ /m). Solving for the amplitude givesaω02[(ω 0 ′ 2 − ω 2 ) 2 + ω 2 ω0 2 . (16)/Q2 ]1/2For operation near resonance, the resonant frequency isω 0 ′ ∼ = √ (k/m 1 − F )12k(17)and the amplitude reduces to A = a(k/k ′ )Q for operationnear ω ′ 0 [2].FIG. 3: Frequency and amplitude response as a function ofoverall height, z 0, for van der Waals interaction. All constantshave been set to 1.It should be noted that for this approximation, the elementspecific attributes of the system (damping, etc.)have been absorbed into quantities like γ and k. Forelemental resolution, these values are still constant withrespect to the z direction but are no longer constants withrespect to the x and y directions. Substituting into theseresults the force derivative for the van der Waals interactiongives the response as a function of the force derivative,F 1 , and therefore as a function of z 0 , the overallheight above the sample. These are plotted in Figure 3.It can also be seen from Figure 3 that, as one would intuitivelyexpect, an attractive force increases the amplitudeand decreases the resonant frequency, while a repulsiveforce decreases the amplitude and increases the resonantfrequency. This is evident in that as z/σ becomes smalland the force becomes repulsive, the amplitude decreasesand the frequency increases.

4In general, this procedure could also be applied toa surface where electric or magnetic interactions werepresent. For a thin magentic film, the magnetic forcewill be nearly constant when compared with electric andatomic forces. The electric force, which falls off moreslowly than the van der Waals for the case of a sphereinteracting with a plane[2], will be dominant at longerdistances. However, if only magnetic and atomic forcesare considered, there may be a distance at which the magneticforce will begin to outweigh the van der Waals force[2]. These different force behaviors become particularilyimportant when element-specific topography is desired.Figure 4 shows the qualitative behavior of the three mainforces under consideration. The electrostatic force hasbeen derived from the electrostatic interaction betweena sphere of radius R and a two dimensional plane. Themagnetic force was derived from considering two interactingmagnetized spheres [2].Project/forces.JPGFIG. 4: Comparison of various interaction forces. Interactionpotentials have been integrated over a sphere and a plane.IV.CONCLUDING REMARKSThe resolution limit of a scanning force microscope dependsnot only on experimental noise, but also on the validityof the model used to describe the system. Here, wehave modeled the cantilever-tip system as a mass springsystem. The microscopic van der Waals potential wasintegrated over the macroscopic sample and tip, and incorporatedinto the governing equation of motion as aneffective spring constant. Resonant frequencies and amplitudeswere determined for operation with a drivingfrequency near the natural (unforced) resonant frequencyof the cantilever-tip system. This simple model outlinesthe basic procedure that allows the connection betweentip response and height above the sample to be made.This model can be expanded to include other forces,such as electric or magnetic, and to account for a samplecomprised of more than one element. A basic orderof-magnitudecomparison was made between the electric,magnetic and atomic forces. The implementation of theseexpanded models, along with a stable experimental device,allow for true atomic resolution and elemental sensitivity.These features, combined with the ability of AFmicroscopes to measure a wide variety of samples, makeAFMs some of the most important characterization toolsfor surface scientists today.[1] C.B. Prater, H.J. Butt, P.K. Hansma, Nature. 345, 6278(1990).[2] D. Sarid, Scanning force microscopy with applications toelectric, magnetic and atomic forces. New York: OxfordUniversity press (1994).[3] J.N. Israelachvili and D. Tabor, Proc. R. Soc. Lond. 331(1972).[4] Y. Sugimoto, P. Pou,M. Abe, al., Nature. 446, 7131(2007).[5] A. S. Foster and W. A. Hofer, Scanning Probe Microscopy.New York: Springer (2006).[6] R. Garcia and R. Perez, Surface Science Reports. 47(2002).