Anal. Chem. 2005, 77, 1192-1195
Correspondence
A Method for Calculating the Spring Constant of Atomic Force Microscopy Cantilevers with a Nonrectangular Cross Section Mark A. Poggi,† Andrew W. McFarland,‡ Jonathan S. Colton,‡ and Lawrence A. Bottomley*,†
School of Chemistry and Biochemistry, and George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332
An important figure of merit in atomic force microscopy is the spring constant of the cantilever probe. To create high aspect ratio tips, cantilever probe manufacturers commonly use dynamic etch techniques that modify the shape and dimensions of both the tip and the beam. Most of the methods used to determine beam stiffness assume a rectangular cross section even though the commonly used etches produce beams with trapezoidal cross sections. We present herein an improved method for determining beam stiffness that takes into account the actual geometry of the cantilever. Atomic force microscopy (AFM) has become an invaluable metrological tool for science and technology enabling researchers to characterize surface topology on the nanometer scale and visualize the orientation and spatial distribution of molecules adsorbed to surfaces.1 AFM also has been used to measure the mechanical properties of single molecules, molecular ensembles, and surface structures as well as adhesive interactions between chemically modified tips and a substrate. Knowledge of cantilever beam stiffness (spring constant) is required for proper interpretation of images acquired under constant force and for extracting the mechanical properties of samples from force curve data. Several methods for determining cantilever beam stiffness have been presented, each with advantages and limitations.2 Cleveland’s method involves measurement of the resonance of the beam before and after loading it with a known mass.3 Although highly accurate, this method is tedious and time-consuming and can lead to damage of the cantilever tip. Gibson et al. as well as Tortonese and Kirk have developed a “beam on beam” approach, where the cantilever is brought into contact with a calibrated standard * Corresponding author. E-mail:
[email protected]. Phone: (404)-894-4014. Fax: (404)-385-6447. † School of Chemistry and Biochemistry. ‡ George W. Woodruff School of Mechanical Engineering. (1) Poggi, M. A.; Gadsby, E. D.; Bottomley, L. A.; King, W. P.; Oroudjev, E.; Hansma, H. Anal. Chem. 2004, 76, 3432-3446. (2) Burnham, N. A.; Chen, X.; Hodges, C. S.; Matei, G. A.; Thoreson, E. J.; Roberts, C. J.; Davies, M. C.; Tendler, S. J. B. Nanotechnology 2003, 14, 1-6. (3) Cleveland, J. P.; Manne, S.; Bocek, D.; Hansma, P. K. Rev. Sci. Instrum. 1993, 64, 403-405.
1192 Analytical Chemistry, Vol. 77, No. 4, February 15, 2005
cantilever beam.4,5 The accuracy of this straightforward method is limited by the certainty to which the reference beam stiffness is known, the positioning of the tip onto the end of the reference beam, and the calibration of the photodiode sensitivity and scanner vertical movement. A third method involves measurement of the dimensions of the beam and calculation of its stiffness by assuming a known density and Young’s modulus using the following equation for a rectangular cantilever with a rectangular cross section
k ) Ewt3/4l3
(1)
where k is the spring constant, E is the Young’s modulus of the beam, and w, l, and t are the beam width, length, and thickness, respectively. The simplicity of this method serves as the basis for its widespread popularity. A fourth method involves measurement of the thermal resonance spectrum of the cantilever and fitting a simple harmonic oscillator model to it.6,7 A fifth method models the dynamic deflections of a cantilever resonating freely in space.8 The latter two methods provide estimates of the spring constant of the beam, k, and quality factor, Q, of the beam resonance under the experimental conditions (in a vacuum, air, or fluid). The theoretical foundation of three of these methods assumes a cantilever beam with a rectangular cross section. The reader is referred to the review by Burnham et al. for further details.2 High-resolution imaging applications require tips with a small radius of curvature. AFM probe manufacturers now use dynamic micromachining techniques after the beam and the tip have been lithographically defined and etched to produce high aspect ratio tips. The proprietary dynamic etch process modifies the shape of both the tip and the cantilever. Dynamic etching of a rectangular (4) Gibson, C. T.; Watson, G. S.; Myhra, S. Nanotechnology 1996, 7, 259-262. (5) Tortonese, M.; Kirk, M. Proc. SPIE-.Int. Soc. Opt. Eng. 1997, 3009, 5360. (6) Sader, J. E.; Chon, J. W. M.; Mulvaney, P. Rev. Sci. Instrum. 1999, 70, 3967-3969. (7) Sader, J. E. J. Appl. Phys. 1998, 84, 64-76. (8) Hutter, J. L.; Bechhoefer, J. Rev. Sci. Instrum. 1993, 64, 1868-1873. 10.1021/ac048828h CCC: $30.25
© 2005 American Chemical Society Published on Web 01/12/2005
Figure 1. SEM images of several FESP probes fabricated by Veeco Probes. (a) Top view of the entire cantilever beam; (b) an increased magnification of the beam in (a) showing the two apparent widths of the beam (marked in white); (c) and (d) angled views of a cantilever probe showing the trapezoidal cross-sectional shape.
beam produces a cantilever with a trapezoidal cross section;9,10 triangular-shaped cantilevers are similarly effected.11 Since several of the methods delineated above assume uniform cross sections, significant error results when using these to determine the spring constant of the cantilever with a sharpened tip (e.g., FESP probes from Veeco Metrology, Golden probes from NT-MDT, and most of the rectangular probes from MikroMasch). In this correspondence, we present a new and straightforward method to determine the spring constant of the beam based on knowledge of the true geometry of the cantilever and its measured resonance. Cantilevers fabricated from single crystal 〈100〉 silicon wafers were obtained from NanoDevices and Veeco Probes. The geometry of cantilevers with sharpened tips was determined by scanning electron microscopy. Cantilever chips were mounted onto standard aluminum specimen mounts (Ted Pella) pretreated with a small amount of Aquadag colloidal graphite (Ted Pella). Images were acquired with a Hitachi model S-800 scanning electron microscope (SEM). The microscope was calibrated using a 0.463 µm/line grid (SPI Supplies, West Chester, PA). Distances measured in the SEM were accurate to within (10%. Figure 1 presents a set of images acquired on a FESP cantilever probe and is typical of those obtained on all probes examined. Clearly visible in these images is the trapezoidal cross section of the beam. Ten different beams from two different manufacturers (9) Chen, X.; Zhang, S.; Dikin, D. A.; Ding, W.; Ruoff, R. S.; Pan, L.; Nakayama, Y. Nano Lett. 2003, 3, 1299-1304. (10) Rabe, U. J. K.; Arnold, W. Rev. Sci. Instrum. 1996, 67, 3281-3293. (11) Grow, R. J.; Minne, S. C.; Manalis, S. R.; Quate, C. F. J. Microelectromech. Syst. 2002, 11, 317-321.
and five different fabrication batches were examined. Careful inspection of the images and cantilever dimensions reveals a high degree of uniformity among cantilevers from different manufacturers and fabrication batches. The average ratio of the bottom face to the top face (tip side) of the beam was 1.69 ( 0.15. The incline angle from the bottom face to the top face was on average 22.3 ( 1°. From Euler-Bernoulli beam theory,12 the stiffness, k, of rectangular beams with trapezoidal cross sections is proportional to I, the second moment of its cross sectional area
k ) 3EI/l3
(2)
Geometrically, the second moment of a beam with a trapezoidal cross section, Itrapezoid,13 is
Itrapezoid )
t3(a2 + 4ab + b2) 36(a + b)
(3)
where t is the beam thickness, a is the width of the top face (tip side) of the beam, and b is the width of the bottom face (see Figure 2). Stiffnesses for 10 beams were determined based on the EulerBernoulli model (eqs 2 and 3) and an elastic modulus14 of 130 GPa and are presented in Table 1. (12) Timoshenko, S. Theory of Elasticity, 3rd ed.; McGraw-Hill: New York, 1970. (13) Oberg, E.; Jones, F. D.; Horton, H. L.; Ryffel, H. H.; Grenn, R. E. Machinery’s Handbook; Industrial Press: New York, 1996. (14) Franca, D. R.; Blouin, A. Mea. Sci. Technol. 2004, 15, 859-868.
Analytical Chemistry, Vol. 77, No. 4, February 15, 2005
1193
Table 1. Summary of the Dimension of Ten Different Cantilever Beams That Were Characterized from Five Different Fabrication Batches beam second spring moment of consta obsd calcd calcd beam dimensions (µm) cantilever cross-sectional ktrapezoidal fair fvac fair, (Hz) no. l wtop wbottom t area (N/m) (Hz) (Hz) 1 2 3 4 5 6 7 8 9 10 a
249.6 247.1 210.2 205.9 206.4 206.4 211.2 208.2 210.2 206.4
23.8 19.4 28.7 19.5 21.7 20.2 20.8 28.8 21.8 19.8
41.3 35.9 40.5 34.8 36.2 33.5 37.4 42.8 38.8 36.7
1.13 × 10-22 9.34 × 10-23 4.79 × 10-23 7.56 × 10-23 4.55 × 10-23 4.03 × 10-23 1.17 × 10-22 5.30 × 10-23 9.92 × 10-23 8.06 × 10-23
3.49 3.47 2.56 3.25 2.68 2.64 3.67 2.62 3.43 3.28
2.82 2.42 2.01 3.38 2.02 1.79 4.83 2.29 4.17 3.58
67 800 67 184 69 569 92 700 78 252 75 632 98 456 72 456 96 200 92 200
66 775 67 549 69 572 91 268 75 111 74 007 97 922 72 465 92 434 91 507
66 399 67 262 68 961 90 854 74 628 73 562 97 463 71 867 92 011 91 069
A relative uncertainty of 15% is found from a propagation of error calculation.
Figure 2. Graphical depiction of the location of the center of mass of a trapezoid and the corresponding measurable lengths. a is the width of the top portion of the beam (tip side), b is the width of the bottom portion, t is the beam thickness, and the point (cx, cy) represents the center of mass of the trapezoid.
The fundamental resonance mode of the beam in a vacuum, fvac, is related to its stiffness by
fvac )
( )x Ri L
2
1 EI ) FA 2π
k x0.2427m
(4)
where m is the mass of a free cantilever beam oscillating in space.3 The effective mass of the beam is a function of the density of the silicon as well as the volume of the beam (function of beam geometry). Calculated values of fvac (using a density15 for silicon of 2.33 g/cm3) are listed in Table 1. The impact of fluid dampening on cantilever resonance in air can be calculated using the model reported by Chon and co-workers:16
fair ) fvac[1 + (πFft/4Fcl)]-1/2
(5)
where fair is the primary resonant mode of the cantilever in air, fvac is the primary resonant mode of the cantilever in a vacuum, Ff is the density of the fluid environment that the cantilever resides (15) Sader, J. E.; Larson, I.; Mulvaney, P.; White, L. R. Rev. Sci. Instrum. 1995, 66, 3789-3798. (16) Chon, J. W. M.; Mulvaney, P.; Sader, J. E. J. Appl. Phys. 2000, 87, 39783988.
1194
Analytical Chemistry, Vol. 77, No. 4, February 15, 2005
(Fair ) 1.18 kg/m3), t is the beam thickness, Fc is the density of the cantilever (Fc ) 2330 kg/m3), and l is the length of the cantilever. This model of fluid dampening was developed for beams with a rectangular cross section. Our use of this model implicitly assumes that the dampening phenomenon exerted on a beam of trapezoidal cross section is approximately that of a rectangular beam. Calculated values of fair are also listed in Table 1. It is interesting to note that the impact of fluid dampening by air for the cantilevers investigated herein is minimal. (A reviewer correctly pointed out that even though damping is not significant for this resonance mode, the magnitude of damping scales with resonance frequency. Thus, if one were to operate the cantilever at a higher frequency, the damping will increase significantly.) A relative difference of only 0.5% was found between the predicted resonance frequencies in air and in a vacuum. The calculated values compare favorably with measured values.17 An average difference of only 2% was observed between our measured and predicted resonance values (in air). The small difference between the computed and observed resonance frequencies is likely due to the unaccounted mass of the tip. To estimate the impact of calculating beam stiffnesses for cantilevers with sharpened tips using the rectangular cross section model (method 3 and eq 1), the base width of the trapezoid was used as the width of the beam. Calculated values of the spring constant based on this model, krectangular, are listed in Table 2 along with the values computed using the trapezoidal model, ktrapezoidal. Differences in the values computed using a rectangular moment ranged from 15 to 25% larger than the stiffness calculated for a trapezoidal moment (eqs 2 and 3). Thus, the widespread use of method 3 for cantilevers of this type results in a substantial and systematic overestimate of the spring constant, the magnitude of which scales with cantilever thickness. (17) Thermal spectra were acquired using a Nanoscope IIIa (Veeco Metrology, Santa Barbara, CA) scanning probe microscope with extender electronics. The raw cantilever motion/deflection and the piezoscanner drive voltages were sampled from a Signal Access Module (Veeco Metrology) placed between the extender box and the base of the microscope. Thermal spectra were acquired by feeding the raw cantilever deflection signal to a SRS 785 Dynamic Signal Analyzer using a Blackman-Harris window (Stanford Research Systems). The piezoscanner was calibrated in x, y, and z using NIST certifiedcalibration gratings (MikroMasch).
Table 2. Comparison of Spring Constants Calculated Using Three Methods for Ten Cantilever Beams cantilever no.
ktrapezoidal
krectangular
kSader
1 2 3 4 5 6 7 8 9 10
2.82 2.42 2.01 3.38 2.02 1.79 4.83 2.29 4.17 3.58
3.66 3.24 2.38 4.45 2.58 2.28 6.37 2.77 5.48 4.79
2.34 3.39 3.39 5.18 2.79 2.28 6.38 4.03 6.05 5.31
Similarly, Sader’s method for calculating the spring constant, educing from analysis of a rectangular cantilever’s thermal spectrum and knowledge of its dimensions, employs the following equation:
kSader ) 0.1906Ffw2lQfΓi(ff)ff 2
(6)
where Γi(ff) is the imaginary component of the hydrodynamic function and ff is the primary resonant mode of the cantilever in fluid. Computed values of the spring constant using Sader’s method based on the bottom width of the trapezoid are also listed in Table 2. Spring constant values computed using Sader’s method were 21-43% larger than the stiffness calculated for a trapezoidal moment (eqs 2 and 3). Again, the use of a rectangular model for trapezoidal cantilevers results in an overestimate of the spring constant. In light of these findings, interpretations of force-based
measurements acquired with cantilevers having nonrectangular cross sections whose spring constants were determined using method 3, Sader’s method, or other methods that assume a uniform width, should be reevaluated. In summary, a modified method for calculating the stiffness of an AFM cantilever is presented that takes into account the nonrectangular cross section of the cantilever beam produced by dynamic etch processes currently used to sharpen the probe tip. The current practice of computing beam stiffness assuming a rectangular cross section results in a gross overestimation of the actual value. This geometrical method for determining cantilever stiffness requires only that the dimensions of the cantilever be accurately determined if the density and elastic modulus of the beam material is known. ACKNOWLEDGMENT We thank Nick Schacher of Veeco Instruments Probe Division for helpful discussions. M.A.P. gratefully acknowledges support from the ACS Analytical Division Fellowship sponsored by the Society for Analytical Chemists of Pittsburgh. A.W.M. acknowledges a NaST Fellowship from the Georgia Tech Center for Nanotechnology. The NASA Graduate Student Researchers Program (NGT-1-02002) and the National Institutes of Health (EB000767) provided financial support for this research.
Received for review August 9, 2004. Accepted October 11, 2004. AC048828H
Analytical Chemistry, Vol. 77, No. 4, February 15, 2005
1195
