Tapping Mode Imaging and Measurements with an Inverted Atomic

This report demonstrates the successful use of the inverted atomic force microscope (i-AFM) for tapping mode AFM imaging of cantilever-supported sampl...
0 downloads 0 Views 272KB Size
Download PDF
Langmuir 2006, 22, 6701-6706

6701

Tapping Mode Imaging and Measurements with an Inverted Atomic Force Microscope Sandra S. F. Chan and John-Bruce D. Green* Department of Chemistry, UniVersity of Alberta, Edmonton, Alberta, Canada, T6G 2G2 ReceiVed January 1, 2006. In Final Form: March 16, 2006 This report demonstrates the successful use of the inverted atomic force microscope (i-AFM) for tapping mode AFM imaging of cantilever-supported samples. i-AFM is a mode of AFM operation in which a sample supported on a tipless cantilever is imaged by one of many tips in a microfabricated tip array. Tapping mode is an intermittent contact mode whereby the cantilever is oscillated at or near its resonance frequency, and the amplitude and/or phase are used to image the sample. In the process of demonstrating that tapping mode images could be obtained in the i-AFM design, it was observed that the amplitude of the cantilever oscillation decreased markedly as the cantilever and tip array were approached. The source of this damping of the cantilever oscillations was identified to be the well-known “squeeze film damping”, and the extent of damping was a direct consequence of the relatively shorter tip heights for the tip arrays, as compared to those of commercially available tapping mode cantilevers with integrated tips. The functional form for the distance dependence of the damping coefficient is in excellent agreement with previously published models for squeeze film damping, and the values for the fitting parameters make physical sense. Although the severe damping reduces the cantilever free amplitude substantially, we found that we were still able to access the lowamplitude regime of oscillation necessary for attractive tapping mode imaging of fragile molecules.

Introduction Development of atomic force microscopy (AFM) has continued unabated since its inception in 1986,1 and numerous excellent reviews have been published.2-6 Many of the initial developments expanded the capabilities of AFM by probing different interfacial properties, such as friction,7 adhesion,8 long-range electromagnetic forces,9 electric double layer forces,10 compliance,11 energy dissipation,12 and heat transfer.13 Still more developments have added capabilities by improving force resolution,14 increasing operational speed,15 and creating parallel platforms16,17 for AFM via microfabrication and instrument design changes. Our lab continues to refine a design known as inverted atomic force * Corresponding author. Phone: (780) 492-7140. E-mail: john.green@ ualberta.ca. (1) Binnig, G.; Quate, C. F.; Gerber, C. Phys. ReV. Lett. 1986, 56, 930-933. (2) Sarid, D. Scanning Force Microscopy with Applications to Electric, Magnetic, and Atomic Forces, revised ed.; Oxford University Press: Oxford, 1994. (3) Procedures in Scanning Probe Microscopies; Colton, R. J., Engel, A., Frommer, J. E., Gaub, H. E., Gewirth, A. A., Guckenberger, R., Rabe, J., Heckl, W. M., Parkinson, B., Eds.; John Wiley & Sons: Chichester, U.K., 1999. (4) Takano, H.; Kenseth, J. R.; Wong, S. S.; O’Brien, J. C.; Porter, M. D. Chem. ReV. 1999, 99, 2845-2890. (5) Applied Scanning Probe Methods; Bhushan, B., Fuchs, H., Hosako, S., Eds.; Springer: New York, 2004. (6) Poggi, M. A.; Gadsby, E. D.; Bottomley, L. A.; King, W. P.; Oroudjev, E.; Hansma, H. Anal. Chem. 2004, 76, 3429-3443. (7) Mate, C. M.; McClelland, G. M.; Erlandsson, R.; Chiang, S. Phys. ReV. Lett. 1987, 59, 1942-1945. (8) Durig, U.; Zuger, O.; Pohl, D. W. J. Microsc. (Oxford) 1988, 152, 259267. (9) Martin, Y.; Wickramasinghe, H. K. Appl. Phys. Lett. 1987, 50, 14551457. (10) Ducker, W. A.; Senden, T. J.; Pashley, R. M. Nature 1991, 353, 239241. (11) Overney, R. M.; Takano, H.; Fujihira, M. Europhys. Lett. 1994, 26, 443447. (12) Cleveland, J. P.; Anczykowski, B.; Schmid, A. E.; Elings, V. B. Appl. Phys. Lett. 1998, 72, 2613-2615. (13) Williams, C. C.; Wickramasinghe, H. K. Appl. Phys. Lett. 1986, 49, 15871589. (14) Rugar, D.; Zuger, O.; Hoen, S.; Yannoni, C. S.; Vieth, H. M.; Kendrick, R. D. Science 1994, 264, 1560-1563. (15) Schaffer, T. E.; Hansma, P. K. J. Appl. Phys. 1998, 84, 4661-4666. (16) Kim, Y. S.; Lee, C. S.; Jin, W. H.; Jang, S.; Nam, H. J.; Bu, J. U. Sens. Mater. 2005, 17, 57-63.

microscopy, i-AFM. In i-AFM, a cantilever-supported sample is characterized with substrate-supported tips located within a microfabricated tip array.18 While the i-AFM design suffers from sample size limitations,19 it possesses several potential advantages with regard to the redundancy and combinatorial applications.20 Unfortunately, combinatorial applications have met with limited success, due in large part to engineering obstacles to the effective patterning of the cantilever with even moderate numbers of chemistries. Recent advances have demonstrated some improved capability in this respect;21 however, much more improvement is required before the promises of this technique can be realized. While all modes of AFM are inherently dynamic, there is a family of techniques known as dynamic AFM.22 These dynamic methods have the capability to simultaneously measure several properties (amplitude, deflection, phase, and frequency), and many are less damaging than their lower-frequency analogues. In some modes, the dynamic signal can provide direct access to both forces and force gradients, and several recent reports have focused on measuring energy dissipation maps.12,22,23 By far, the most popular dynamic mode of AFM is tapping mode (TM).24 As the name implies, the tip repeatedly comes into contact with the sample (tapping). Tapping mode may be operated in either a repulsive or an attractive regime.22,25,26 Repulsive tapping mode can be less damaging than contact mode, due to the lack of (17) Vettiger, P.; Despont, M.; Drechsler, U.; Durig, U.; Haberle, W.; Lutwyche, M. I.; Rothuizen, H. E.; Stutz, R.; Widmer, R.; Binnig, G. K. IBM J. Res. DeV. 2000, 44, 323-340. (18) Green, J. B. D.; Novoradovsky, A.; Park, D.; Lee, G. U. Appl. Phys. Lett. 1999, 74, 1489-1491. (19) Mabry, J. C.; Yau, T.; Yap, H. W.; Green, J. B. D. Ultramicroscopy 2002, 91, 73-82. (20) Green, J. B. D.; Lee, G. U. Langmuir 2000, 16, 4009-4015. (21) Lui, A.; Berkenbosch, R.; Wu, S.-Y.; Green, J.-B. D. Analyst 2006, submitted. (22) Garcia, R.; Perez, R. Surf. Sci. Rep. 2002, 47, 197-301. (23) Schirmeisen, A.; Holscher, H.; Anczykowski, B.; Weiner, D.; Schafer, M. M.; Fuchs, H. Nanotechnology 2005, 16, S13-S17. (24) Zhong, Q.; Inniss, D.; Kjoller, K.; Elings, V. B. Surf. Sci. 1993, 290, L688-L692. (25) Garcia, R.; San Paulo, A. Phys. ReV. B 1999, 60, 4961-4967. (26) Round, A. N.; Miles, M. J. Nanotechnology 2004, 15, S176-S183.

10.1021/la060002i CCC: $33.50 © 2006 American Chemical Society Published on Web 06/14/2006

6702 Langmuir, Vol. 22, No. 15, 2006

lateral forces and a lower average force; however, it is not as gentle as attractive tapping mode.22,27,28 This article is focused on exploring the practical implementation of inverted tapping mode AFM and determining any properties unique to this combination. Imaging the cantilever-supported sample with inverted TMAFM is simple and direct, and it is implemented in a manner essentially identical to conventional TMAFM. The most notable difference between the inverted and conventional tapping mode as observed in our experiments is in the magnitude of cantilever damping that occurs as the tip and sample are approached. The damping is much greater in the inverted mode. This enhanced damping is a direct consequence of the tip geometries available on our tip arrays and not an inherent characteristic of tapping mode in the inverted design. Previous researchers have demonstrated that micromechanical devices such as AFM cantilevers will be susceptible to fluid dynamic damping as the cantilever approaches the sample. This effect, known as squeeze film damping, or SFD,29-36 is not usually a concern when operating in air with conventional AFM tapping mode cantilevers. We demonstrate that SFD is, in fact, the source of the severe damping in i-AFM, and explore its impact on normal operational modes, such as imaging and acquisition of data curves. Experimental Section Chemicals and Materials. Unless otherwise specified, all rinsing steps were performed with 0.2 µm filtered 18.0 MΩ‚cm water (Barnstead-Thermolyne, Nanopure water filtration unit) and all drying steps made use of 0.2 µm filtered N2 gas from liquid N2 boil-off. The following chemicals were used as received: hydrogen peroxide (EMD, 30% H2O2(aq), ACS certified), concentrated sulfuric acid (EMD, 17.8 M H2SO4(aq), ACS certified), hydrofluoric acid (Aldrich, 48 wt % HF in water, >99.99%), 2-propanol, IPA (Fisher, ACS certified), an aqueous solution of 20 nm diameter citrate-capped colloidal gold particles (BBInternational, 7 × 1011 particles/mL), and 3-aminopropyldimethylethoxysilane, APDMES (United Chemical Technologies Inc., >97%). Peroxysulfuric acid, aka piranha, was always freshly prepared by mixing 1 part of 30% H2O2 with 3 parts of concentrated H2SO4. Use extreme caution when handling and disposing of fresh piranha.37 It can reach temperatures in excess of 130 °C within minutes of mixing and will react violently with many organic materials. Furthermore, the solutions will continue to evolve gas for days, potentially leading to the explosion of capped containers! We routinely neutralize our piranha waste with sodium bicarbonate followed by disposal down the drain. Additional caution should also be exercised when working with HF,38 and a tube of calcium gluconate gel should be readily available in case of accidental contact. Cantilevers. The sample cantilevers were tipless silicon levers (MikroMasch, model NSC12), that were 110 µm long, 35 µm wide, and 2.0 µm thick, as verified by SEM imaging. The cantilever resonant frequency, quality factor, and force constant were routinely (27) Anczykowski, B.; Kruger, D.; Babcock, K. L.; Fuchs, H. Ultramicroscopy 1996, 66, 251-259. (28) San Paulo, A.; Garcia, R. Biophys. J. 2000, 78, 1599-1605. (29) Blech, J. J. J. Lubr. Technol. 1983, 105, 615-620. (30) Starr, J. B. Tech. Dig. IEEE Solid-State Sens. Actuator Workshop 1990, 44-47. (31) Andrews, M.; Harris, I.; Turner, G. Sens. Actuators, A 1993, 36, 79-87. (32) Serry, F. M.; Neuzil, P.; Vilasuso, R.; Maclay, G. J. In Electrochemical Society Proceedings: Microstructures and Microfabricated Systems II; Hesketh, P. J., Hughes, H. G., Denton, D. D., Kendall, D. L., Eds.; The Electrochemical Society: Chicago, IL, 1995; Vol. 95, pp 83-89. (33) Hosaka, H.; Itao, K.; Kuroda, S. Sens. Actuators, A 1995, 49, 87-95. (34) Xu, Y.; Smith, S. T. Precis. Eng. 1995, 17, 94-100. (35) Schrag, G.; Wachutka, G. Sens. Actuators, A 2002, 97-98, 193-200. (36) Zhang, C.; Xu, G.; Jiang, Q. J. Micromech. Microeng. 2004, 14, 13021306. (37) Matlow, S. L. Chem. Eng. News 1990, 68, 2. (38) Reinhard, C. F.; Hume, W. G.; Linch, A. L.; Wetherhold, J. M. J. Chem. Educ. 1969, 46, A171-A179.

Chan and Green determined via fitting cantilever resonance data.39 The deflection signal for a cantilever located very far from the tip array driven by ambient and thermal fluctuations was fed into an oscilloscope, and the spectral amplitude was determined as a function of frequency. This spectral amplitude was then squared and fit to the square of the function shown in eq 1 with an additional white noise, normalization constants, and adjustments for the cantilever tilt.40 This particular cantilever has a resonant frequency fo ) 137.3 kHz, quality factor Q∞ ) 165, and a force constant k ) 3.4 N/m. In addition to the tipless sample cantilevers, we also employed conventional cantilevers with integrated tips (MikroMasch, model NSC15) for characterization. According to manufacturer’s specifications the characterization cantilevers have resonant frequencies of ∼325 kHz and force constants of ∼40 N/m, and they were not characterized further. Tip Arrays. Silicon tip arrays were prepared by a microfabrication technique as previously described.19 In this paper tip arrays from different wafers were compared, and while the fabrication procedure has been outlined elsewhere, variations in fabrication procedure produced arrays of tips with different morphologies. Surface Modification. Where possible, sample processing occurred inside of an ULPA-upgraded vertical laminar flow clean hood (Spe´cialite´s Industrielles Canada Inc., model HFLT-542436), with measured particle densities typically less than 35 particles/m3. The tip arrays were treated with concentrated HF for more than 5 min while being agitated. This allowed for the complete removal of a 2 µm thick layer of SiO2, which was in place to protect the tips. The tip arrays were then rinsed thoroughly with water and immersed into fresh piranha for 20 min. Upon removal from the piranha, the arrays were rinsed with copious amounts of the Nanopure water, blown dry with N2 gas, and stored in a cleaned glass vial until required for use. Chemical processing of cantilevers occurred in home-built allglass cantilever holders. Cantilevers were taken as received and immersed into freshly prepared piranha for 20 min. The piranha was poured away, and the cantilevers were rinsed under a strong stream of water for about 1 min. The cantilevers were momentarily stored under water and then blown dry. These cleaned cantilevers were ready for measurement of amplitude-distance curves; however, additional processing was required to create the colloidal gold modified cantilevers. The dried cantilevers were immersed into a freshly prepared 6.4 mM solution of APDMES in IPA, containing ∼5% H2O by volume. After 20 min, the cantilevers were gently rinsed with pure IPA, dried with a stream of N2 gas, and placed into an oven (set to 105 °C) where they remained for 20 min. Following this dehydration step, the cantilevers were immersed (while still hot) into an aqueous solution of the citrate-capped colloidal gold particles, nominal concentration ∼7 × 1011 particles/mL. After 5 min of immersion in the colloidal solution, the cantilevers were rigorously rinsed with the Nanopure water and dried. These colloidal gold modified cantilevers were then stored in a recently cleaned glass Petri dish until ready for measurements. Instrumentation. All of the AFM experiments reported herein were performed with a Digital Instruments Nanoscope IIIa equipped with a Multimode (Veeco Metrology) scanning probe microscope. Amplitude, phase, deflection, and some acoustically driven spectral data were obtained via the manufacturer-supplied software. Higherresolution measurements and oscillations driven by ambient acoustic and thermal forces were acquired on a digital phosphor oscilloscope (Tektronix, model 5034B). Data Curves. The data curves in this report were all obtained by recording a signal as the z-piezoelectric scanner was swept with a simple triangle wave. Because the z-piezo scanner was not a closed loop scanner, there is an inherent distortion of the data curves due to the scanner hysteresis in the z-direction.41 This distortion was removed via software manipulation following data acquisition, by (39) Hutter, J. L.; Bechhoefer, J. ReV. Sci. Instrum. 1993, 64, 1868-1873. (40) Hutter, J. L. Langmuir 2005, 21, 2630-2632. (41) Ge, P.; Jouaneh, M. IEEE Trans. Control Syst. Technol. 1996, 4, 209216.

Tapping Mode InVerted Atomic Force Microscopy

Langmuir, Vol. 22, No. 15, 2006 6703

introducing a single quadratic correction value for all similarly acquired data curves. Three different kinds of data were obtained for this work: (1) the cantilever rms amplitude, A(d); (2) the phase of the cantilever oscillation relative to the drive signal, Φ(d); (3) the low-frequency cantilever deflection, ∆z(d), sometimes converted to average cantilever force, F(d) via multiplication by the cantilever force constant, k. Data curves for the detailed examination of the cantilever oscillation regimes were acquired very near the tip-sample contact. These curves were typically acquired with 8192 data points/curve for a piezo moving over a range of 100 nm at a rate between either 40 or 200 nm/s. The value for the drive amplitude was systematically varied from 50 mV to 8 V in 50 mV increments, capturing the data curves at each step. Software and hardware limitations restricted simultaneous acquisition to two data types at a time; as a result the data were acquired as either (a) amplitude and deflection or (b) amplitude and phase. Data curves for studying the long-range damping were acquired with 8192 data points/curve for a z-piezo excursion of 5.0 µm at a rate of either 16 or 20 µm/s. These data curves had a different hysteresis correction factor. To obtain data curves that extended several tens of micrometers away from contact, the vertical stepper motor was automatically stepped. Following each step the instrument was equilibrated for 5 s, while measuring new data curves roughly twice every second. At the end of 5 s, the latest data curves were captured. This process was automatically repeated until the cantilever response appeared essentially constant. Images. The colloidal gold modified cantilever was loaded, as any conventional cantilever would be, into the AFM cantilever holder, and the tip array was mounted, as any other sample would be, onto a midrange piezoelectric scanner (15 µm in xy and 6 µm in z). The drive amplitude was set to 300 mV, and the resonant peak was found at 132.25 kHz. The imaging conditions were set following inspection of A(d), Φ(d), and ∆z(d) curves just prior to imaging. These curves were inspected for excessive adhesive interactions and attractiverepulsive transitions.25 In the absence of these phenomena, the amplitude set point was adjusted to give a set point ratio Rs ∼ 0.8 of the free amplitude. Images were typically acquired with a fast axis scan rate between 1 and 2 Hz and involved capturing the height and the phase data. SEM measurements were performed with a field emission scanning electron microscope (JEOL, model 6301FXV). The SEM images were used to verify cantilever and tip geometries. Cantilever dimensions (length, width, and thickness) were simply and directly measured perpendicular to the dimension being measured. However, due to the fact that the tips were located in the middle of a 1 cm square chip, the tip heights were measured at an angle of ∼83° relative to the surface normal.

Results and Discussion The following results and discussions focus on three aspects of tapping mode AFM with the inverted design: (1) the demonstration of imaging, (2) the accessibility of attractive and repulsive tapping conditions, and (3) the details of the longrange damping in air. Imaging the Cantilever. In an effort to demonstrate that tapping mode can be used in the i-AFM design, we imaged a tipless silicon cantilever that had been modified with a layer of colloidal gold particles. Figure 1 shows some representative height (Figure 1A) and phase (Figure 1B) images, along with a complementary SEM image (Figure 1C) of the same area. These tapping mode images have an edge length of 1.00 µm, and while the AFM images reveal very little substrate between the particles, the actual particle spacing is better depicted by the SEM image (Figure 1C), which shows the substantial amount of empty space surrounding each particle. This discrepancy is a result of the well-known tip-convolution artifact, which artificially causes the particles to appear wider than their actual diameter, and corresponds to a tip radius of ∼20 nm in this case.

Figure 1. Inverted tapping mode images showing the height (A) and phase (B) images of colloidal gold particles supported on a tipless cantilever. The SEM image (C) shows the same location with a more accurate representation of the colloidal gold particles.

According to manufacturer’s specifications these colloidal gold particles should have a population average diameter of 20 ( 1 nm. From our tapping mode images we obtained an average diameter (from the heights) of 15.1 ( 1.9 nm. It should be mentioned that tapping mode AFM is not always an accurate means of measuring feature heights, especially when the features may have chemical heterogeneity.26,42,43 To address this issue, the set point ratio and the drive frequency were varied. The set point ratio is the ratio of the amplitude during feedback to the “free” amplitude as measured just before surface interactions, (42) Brandsch, R.; Bar, G.; Whangbo, M. H. Langmuir 1997, 13, 6349-6353. (43) Pignataro, B.; Chi, L.; Gao, S.; Anczykowski, B.; Niemeyer, C.; Adler, M.; Fuchs, H. Appl. Phys. A 2002, 74, 447-452.

6704 Langmuir, Vol. 22, No. 15, 2006

Figure 2. Amplitude vs distance curve demonstrating the severe damping of the cantilever oscillation amplitude as the cantilever approached the tip array. The inset SEM images illustrate the range of tip geometries that are available with some of our tip arrays. A commercially available cantilever-supported tip is shown for comparison.

and the ratio was varied from 0.3 to 0.9. The drive frequency was varied from just below to just above the cantilever resonance. The observed heights of the colloidal gold particles were relatively independent of these factors, leading us to conclude that adhesive artifacts did not substantially affect these measurements. In any case, were these adhesive artifacts present they would simply reflect a problem common to conventional tapping mode and would not represent a new inherent characteristic of the inverted design. The SEM data were not of sufficient resolution to confirm or contradict either the i-TMAFM height measurements or the manufacturer’s specifications, and so while there is a discrepancy, we cannot conclude if either value is in error or if this difference is simply a sampling issue. Transmission electron microscopic data would produce more definitive particle geometry; however, the thickness of our cantilevers combined with their crystalline nature rendered that measurement impossible. Future projects that require high-resolution TEM and i-AFM measurements of the same features will explore the use of thinner amorphous cantilevers. Damping of the Cantilever. While imaging the colloidal gold cantilever, we observed substantial damping of the cantilever oscillation prior to feedback. In an effort to better understand the damping, we obtained large amplitude-distance curves. As described in the Experimental Section, we used the stepper motor to generate several 5 µm data curves, which were combined to create composite curves extending out to nearly 100 µm of tipsample separation. A representative composite curve in Figure 2 shows how the cantilever rms amplitude was damped as a tip and cantilever were approached. This was initially a serious concern because even with excessive oscillation of the cantilever base, the resulting free amplitude of the cantilever (at the distal end prior to tapping) was very small. If the damped amplitude was too small, and adhesive forces dominated the tip-sample interaction, then stable tapping mode images would not be attainable. As a result of our fabrication efforts, we have several wafers of tip arrays, and while these wafers were all fabricated with similar process steps, small and sometimes uncontrolled variations in the process produced wafers each with unique tip geometries. The inset in Figure 2 shows SEM images of some representative tips from three of these wafers. Importantly, the tips in our tip arrays are substantially shorter than the tips common to commercially available tapping mode cantilevers. Commercial

Chan and Green

Figure 3. Amplitude (A and D), phase (B and E), and force (C and F) vs distance data from 3.00 and 6.10 V drive excitation, respectively. These data curves exhibit all of the features common to conventional AFM. The inset in (D) shows a more detailed view of the transition between the repulsive (high-amplitude) and attractive (low-amplitude) regimes. For clarity, these curves show a 40 nm section of the 100 nm piezo excursion.

tapping mode tips are typically 15-25 µm tall, while depending upon our tip array the tip height may range from 1 to 8 µm. The most substantial damping was expected and was observed above the shortest tips, and the data shown in Figure 2 is a representative amplitude curve from one of these smaller tips. The specific tip used for this graph had a height of 1.5 µm, based upon SEM images. Measurements were made above numerous tips in each array, and the results were all very similar to those displayed in Figure 2. The damping of the amplitude clearly depends on the separation between the cantilever and the tip array surface. The smallest cantilever-surface separation depends on the tip height and on the relative location of the cantilever above the tip. When the tip is centered below the most distal end of the cantilever, then the point of closest approach of the cantilever to the surface of the tip array will simply equal the tip height. Since the cantilever is tilted (at ∼10°), as the tip location is moved toward the base of the cantilever, the cantilever approaches the tip array surface more closely, and thus the damping will increase. As a result, all of our measurements were made at the same location (∼500 nm from the end) on the same cantilever, so as to limit variability. Attractive or Repulsive. One of the key advantages to tapping mode is the ability to gently image fragile surfaces. 24 By reducing the applied force and eliminating lateral forces, the researcher is able to image samples that would otherwise be too fragile for contact mode AFM. Tapping mode can be operated within two stable regimes, attractive or repulsive.25 The attractive regime affords the least amount of sample damage28,44 and should be used when imaging fragile samples. We examined the frequency, amplitude, phase, and force of the cantilever oscillation as a function of tip-sample distance in order to definitively specify the tapping mode regime. Our goal was to conclusively demonstrate that attractive tapping mode is fully accessible in the inverted design just as it is in the conventional design. Figure 3 shows some relatively high-resolution data curves for two different drive amplitudes, 3.0 and 6.1 V. Careful examination of the amplitude data in Figure 3D (see inset) shows discontinuous jumps in the amplitude. These jumps correspond to the well-documented transitions between the two stable but different regimes of cantilever oscillation: (1) the high-amplitude (44) Thomson, N. H. J. Microsc. (Oxford) 2005, 217, 193-199.

Tapping Mode InVerted Atomic Force Microscopy

Langmuir, Vol. 22, No. 15, 2006 6705

regime that corresponds to the repulsive or intermittent contact tapping mode and (2) the low-amplitude regime that corresponds to attractive tapping mode. In some cases, as in Figure 3A, no clear transition exists, and identification of the regime based on amplitude data is not definitive. The phase curves in Figure 3, parts B and E, show the textbook phase behavior for cantilevers driven near resonance. Although the actual value for the phase can be used to determine the regime as attractive (Φ > 90°) or repulsive (Φ < 90°), ambiguities with some AFM hardware (scale and sign) and software settings (offset) make the actual value of the phase recorded with AFM software questionable. Using the amplitude and phase traces in Figure 3, parts D and E, to definitively identify the low- and high-amplitude regimes, we could then use the phase in Figure 3B to unambiguously determine that the cantilever driven at 3.0 V was operating entirely in the low-amplitude or attractive tapping mode. Force curves are included for completeness. These data simply verify that attractive tapping mode is accessible in the inverted design and further demonstrate behavior similar to conventional tapping mode. Oscillators in Fluid. While we expect that SFD is the dominant factor causing the cantilever damping; to date there are relatively few complete descriptions of SFD of cantilevers. In an effort to elucidate the SFD as it pertains to i-AFM and to dynamic AFM operating with short tips, we modeled the i-AFM cantilever as a one-dimensional driven damped harmonic oscillator. The behavior of this system is described by the well-known differential equation mx¨-bx˘ -kx ) Fosin(ωt), where Fo is the amplitude of a sinusoidal drive force, m is the effective mass of the oscillator, b is the damping term, and ω is the angular frequency of the drive force.45 The solution to this differential equation can be expressed as the resonator amplitude as a function of frequency according to the following expression:

A(ω) )

Fo m

x

(ωo2 - ω2)2 +

) b2ω2 m2

ADCωo2

x

(ωo2 - ω2)2 +

ωo2ω2 Q2 (1)

where ADC, Q, and ωo are the amplitude at zero frequency, the quality factor, and the angular frequency at resonance, respectively. To properly fit the data we will need to know or fit ADC, Q, and ωo. Furthermore, the damping constant is expected to depend on the cantilever-surface separation. An appropriate functional dependence of the damping constant on separation will depend on the fluid dynamics for the oscillator, which will depend on the cantilever and tip array geometry as well as the dynamic properties of the fluid.46 Clearly the viscosity is one of the most important fluid properties for determining the fluid dynamics of the system, and in general the viscosity of a gas will depend on both the pressure of the gas and the geometry of the structure within which the gas is moving. The geometry dependence of the viscosity can be estimated by ηeff ) η/(1 + 9.638Kn1.159), where η is the viscosity and Kn is the Knudsen number. The Knudsen number is the ratio of the mean free path to the separation between the surfaces. If the dimensions are comparable to the mean free path of the gas molecules, then the fluid dynamics will be better described as molecular flow, and the viscosity will be strongly (45) Marion, J. B.; Thornton, S. T. Classical Dynamics of Particles and Systems, 3rd ed.; Harcourt Brace Jovanovich: San Diego, CA, 1988. (46) Landau, L. D.; Lifshitz, E. M. Fluid Mechanics, 2nd ed.; Pergamon Press: Oxford, England; New York, 1987.

affected by changes in the dimensions. The largest possible value for Kn should occur at the point of closest approach, which would corresponds to a separation of ∼1.5 µm for smaller tips. Using the mean free path for air at atmospheric pressure (λ ∼ 71 nm) leads to a value of Kn ∼ 0.047. It should be remembered that this Kn value is an extreme maximum value which corresponds to the point of closest approach and that a more realistic value for the extended cantilever would be the result of the average cantilever-tip array separation when the cantilever and tip are near contact. The model below suggests a doffset ∼ 5 µm, which will correspond to Kn ) 0.014 and an ηeff ) 0.94η. As the cantilever is raised from this point of closest approach the net viscosity is expected to rise from 0.94ηbulk to ηbulk; thus, we can treat η as essentially constant. This suggests that the system should not be in the molecular collision limit but instead will be better described by constant viscosity bulk medium. Fluid flow in and around arbitrary structures can be extremely complex, and the Reynolds number, Re, can be used as a guide to predict the nature of the fluid flow. If Re is greater than about 5000, then turbulent flow is expected to dominate, while for Re less than 2000 laminar flow is expected, and when Re is below 0.1 the flow is in the slow flow or creep flow regime. We can estimate an upper limit of the Reynolds number with the equation (Re ) dVF/η), where d is the separation far from the surface and V, F, and η are the velocity, density, and viscosity of the gas, respectively. The gas velocity can be crudely estimated from the change in volume per unit time divided by the area through which the gas escapes (around the perimeter of the cantilever). At 80 µm of separation, the cantilever has an rms amplitude of 234 nm and a resonant frequency of 137.3 kHz. Thus, the flow velocity perpendicular to the plate motion (parallel to the plate surface) would be less than or equal to ∼7 mm/s, and Re ) 0.04. If the amplitude was maintained as the cantilever approaches the surface, then the linear flow velocity should increase (as 1/d), but since the Reynolds number also depends on the separation as well, the net effect is expected to be independent of separation. If the amplitude decreases as the cantilever approaches the surface, then the velocity will decrease, thereby further reducing Re. As a result, we can expect that the dynamics should not only be in the nonturbulent laminar regime but would be more accurately placed into the slow flow or creep flow regime. The value would need to increase by 5 orders of magnitude before turbulent flow would be expected. Squeeze Film Damping. Several labs have explored the impact of SFD on the dynamic behavior of micromechanical devices33 such as accelerometers30 and even AFM cantilevers.32,34 Squeeze film damping is fundamentally a fluid dynamic phenomena and as such can be fully understood via the compressible gas-film Reynolds equation.47 With several approximations and rough calculations, we can estimate a realistic fluid dynamic regime for these systems, and this can lead us to appropriate squeeze film models.29 As the fluid is squeezed between two surfaces it will impact the dynamics of the surfaces in two ways: as a spring and as a damper. The spring force will arise if the fluid cannot move more rapidly than the cantilever. When the cantilever approaches, the slow moving fluid will be compressed and will exert a repulsive force on the cantilever. The damping forces arise from the fluid that is displaced from between the plates. The appropriate SFD model should correspond to a small Re and small Kn. The spring constant for air trapped between parallel rectangular plates29,31 would be given by k ) 64σ2Paa2/π8d Σ(1/[(mn)2[(m2 + n2)2 + σ2/π4]]), where m and n are odd integers, a is the edge (47) Gross, W. A. Fluid Film Lubrication; Wiley: New York, 1980.

6706 Langmuir, Vol. 22, No. 15, 2006

Chan and Green

Figure 4. Amplitude vs distance curves for tips with different heights of 1.5 µm (red) and 4.7 µm (green), and the least-squares regression fit to the data using eq 3. The fitting parameters correspond to ADCQ∞ ) 234.59 ( 0.07 nm, bo/b∞ ) 2.24 × 1012 ( 0.02 × 1012 nm3, and do ) -5210 ( 30 nm. The inset shows schematic diagrams of the tilted cantilever being oscillated above the tip array as well as a schematic of the model used for the fitting.

length, Pa is the ambient pressure, d is the separation between plates, and σ is the squeeze number. The squeeze number is given by the following equation, σ ) 24πηa2fο/Pad. Using the point of closest approach d ) 1.5 µm, η ) 1.77 × 10-5 Pa s, a ) 35 µm, fο ) 137.3 kHz, and Pa ) 9.4 × 104 Pa, the squeeze number σ ) 1.1. Using σ ) 1.1, the spring constant can be reasonably approximated by k ) 16σ2Paa2/π8d, with an average separation of 5 µm for an area with edge length of 35 µm. This gives a squeeze force constant of 0.04 N/m, ∼100× smaller than the cantilever force constant of 3.4 N/m. If the additional spring constant was substantial it would lead to a detectable shift in the resonant frequency, and none was observed. The damping constant is expected to depend on the separation according to b ) bo/(d - do)3 + b∞, where bο ) 0.419ηa4, b∞ ) mωo/Q∞ ) k/ωoQ∞, and m ) k/ωo2. Rewriting eq 1 in a form where all of the known constants are substituted, and unknowns are either eliminated or remain as fitting parameters, we have

A(ω,d) )

ADCωo2

x

[

]

(2)

ωo 2 2 2 2 2 (ωo - ω ) + + ω Q∞ k(d - do)3 boωo2

Amplitude curves as shown in Figure 2 were obtained when the cantilever was driven at the resonant frequency, and so we can make the further simplification that ω ) ωo on the above equation.

A(ωo,d) )

ADCQ∞ bo/b∞ +1 (d - do)3

(3)

Using this equation we fit data from the same cantilever located above several different tips in the array. Figure 4 shows two sets

of data, obtained over two tips with different heights (1.5 µm and 4.7 µm), and the fit to the 1.5 µm tall tip data set. The fit is in excellent agreement with the data, with the exception occurring at very small separations, where the end of the cantilever approaches the surface very closely. Using the first fitting constant, ADCQ∞ ) 234.59 ( 0.07 nm and the value for Q∞ ) 165, we can determine that the amplitude at low frequencies, when driven at 3.0 V, ADC ) 1.5 nm. The second parameter is the ratio of bo/b∞ ) 2.24 × 1012 nm3, which when combined with b∞ ) k/ωoQ∞ ) 2.30 × 108 kg/s can be used to calculate an experimental value for bo ) 5.14 × 104 kg nm3/s. Using the functional form of bo for a two parallel plate model bo ) 0.42A2η, we obtain an area of the interacting parallel plates that is A ) 2630 µm2. For a 35 µm wide cantilever, this area corresponds to a length of 75 µm, or about half of the cantilever length. The last parameter is an offset to the separation and corresponds to an additional separation equal to 5.2 µm. When considering the fact that the cantilever is tilted and has (in this model) an interacting area that is 75 µm long, the average height of the interacting area will be 75 µm sin(10)/2 ) 6 µm. This is close to the value but perhaps only by chance. The distance dependence is cubed, and as such we might expect that the average height should be a weighted average instead. A superior approach would involve finite-element simulations that include the cantilever tilt; however, this is beyond the scope of this paper. While the quantitative values for the interacting areas and offset distances are by no means conclusive, the quality of the fit of the data with SFD model is compelling and leads us to conclude that the source for this damping of the cantilever oscillation is due to SFD.

Conclusions Under every test, the i-AFM design performed similarly to tapping mode with the conventional design. The only notable difference was the relatively larger amount of damping present in the inverted design. The increased damping was predictable and is a clear result of the relatively shorter tip heights used in our tip arrays as compared to those of commercial tapping mode tips. In some cases, the severe damping of the free amplitude required drive amplitudes to be so excessive that the cantilever amplitude far from the surface was immeasurably large (>20 V). In all cases, the distance dependence of the amplitude damping was fit very well by a coarse model for SFD, wherein the damping coefficient depends on the separation to the inverse cubic power. Furthermore, the area and separation values experimentally determined from the fitting parameters are consistent in magnitude with the dimensions of the cantilever. Acknowledgment. We gratefully acknowledge George Braybrook for his assistance with high-resolution SEM imaging of the tip arrays and of the colloidal gold modified cantilever. J.-B.D.G. thanks Frank Zamborini for helpful discussions with regard to the colloidal gold immobilization. We also recognize that without financial assistance from the Department of Chemistry, the University of Alberta, and the NSERC Discovery Grant this research would not have been performed. LA060002I