REVIEW OF SCIENTIFIC INSTRUMENTS
VOLUME 72, NUMBER 5
MAY 2001
A nondestructive technique for determining the spring constant of atomic force microscope cantilevers Christopher T. Gibson,a) Brandon L. Weeks, Jonathan R. I. Lee, Chris Abell, and Trevor Rayment Department of Chemistry, Lensfield Road, Cambridge University, Cambridge CB2 1EW, United Kingdom
共Received 21 August 2000; accepted for publication 11 February 2001兲 We present a simple, accurate, and nondestructive method to determine cantilever spring constants by measuring the resonant frequency before and after the addition of a thin gold layer. The method for resonating the cantilevers uses electrostatic force modulation, which has been described for conductive cantilevers, but we demonstrate it can also be applied to silicon nitride cantilevers. The variations in spring constant for cantilevers of the same type across the same wafer are also explored. © 2001 American Institute of Physics. 关DOI: 10.1063/1.1361080兴
I. INTRODUCTION
nant frequencies for V shaped cantilevers are multiplied by 1.04.14,23 The accuracy for this method is claimed to be in the range 10%–15%, with the principal uncertainties being associated with calibration of the added mass and with its location. The method requires a three-way micropositioner to attach the spheres, and is potentially destructive since glue is sometimes used to fix the spheres to the lever. This makes it difficult to return the lever to its original state after calibration has been completed. In this work we propose a method that relies on the same principle except that instead of gluing spheres to the end of the cantilever the change in mass is accomplished by adding a thin layer of gold. We assume that the layer of gold is thin enough so that it acts as an inelastic body and does not affect the overall mechanical properties of the cantilever. The assumption is supported by recent work performed by Sader et al.14 All cantilevers currently employed have the back surface coated with gold in order to improve their reflectivity since most AFMs use the optical beam technique to detect cantilever deflections. Sader et al.14 observed that if thin gold layers were added to the back of a cantilever the resonant frequency of the lever dropped due to the change in mass. They observed that up to 70 nm could be applied without the gold changing the mechanical characteristics of the cantilever as a whole. We have coated a number of cantilevers, measured their resonant frequency before and after the addition of the gold layer, and determined their spring constant. The results were then compared with those obtained by a technique recently developed by Sader et al.15 To determine the mass of a thin layer of gold applied to a cantilever we use the following equation:
The atomic force microscope 共AFM兲 has emerged as a technique for obtaining qualitative information through a number of different imaging modes such as topographic,1 lateral force,2 or phase contrast3 imaging. The AFM can also provide quantitative information. For these types of measurements a particular cantilever must be calibrated accurately. There are several areas using AFM techniques that either require, or benefit from, levers being calibrated. Recent studies that demonstrate this point include the measurement of surface forces,4 bond strengths,5 elastic properties of surfaces,6 tribology,7 chemical force microscopy,8 and protein unfolding experiments.9,10 AFM cantilevers are also being used as micromechanical sensors to study electrochemistry11 and viscosity in fluids.12 Several techniques have been proposed to calibrate cantilever spring constants. These include resonating the cantilever with or without added masses,13–15 examining the cantilevers thermal noise spectrum,16 loading the cantilever with a known force17–19 and theoretical calculations.20,21 Each technique has advantages and disadvantages that have been discussed elsewhere.22 One of the most commonly used techniques to calibrate AFM cantilevers is that developed by Cleveland et al.13 This technique is based on loading a cantilever with a known mass and measuring the resonant frequency before and after addition of the mass. The masses are usually gold or tungsten spheres, which are available in the appropriate size range 共5–20 m兲. The equation for determining the spring constant is as follows: ⫺2 k N ⫽ 共 2 兲 2 关 M / 共 v ⫺2 1 ⫺ v 2 兲兴 ,
共1兲
where M is the added mass and v 1 and v 2 are the resonant frequencies after and before the addition of the mass at the free end of the lever, respectively. Due to the damping effect air has on the cantilever vibration, resonant frequencies for beam shaped cantilevers are multiplied by 1.02 while reso-
M e ⫽nAt Au Au ,
where M e is the effective mass of the gold layer, A is the surface area of the top or bottom face of the cantilever, t Au is the thickness of gold added, and Au is the density of gold. The variable n is a correction factor required since the gold at the base of the cantilever will contribute less to the effective mass than the gold at the end of the cantilever. The
a兲
Author to whom correspondence should be addressed; electronic mail:
[email protected]
0034-6748/2001/72(5)/2340/4/$18.00
共2兲
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© 2001 American Institute of Physics
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Rev. Sci. Instrum., Vol. 72, No. 5, May 2001
Spring constant of AFM cantilevers
FIG. 1. Resonance spectrum for an AFM cantilever using electrostatic force modulation showing a single and distinct peak. The resonant frequency in air for this cantilever is 17.8 kHz and the Q factor is 29.
factor n depends on the geometry of the cantilever but can be easily obtained from tables supplied by Sader et al.14 共refer in particular to Tables IIa, b, and c兲. Therefore, we can substitute Eq. 共2兲 into Eq. 共1兲. The result is ⫺2 k N ⫽ 共 2 兲 2 关 nAt Au Au / 共 v ⫺2 1 ⫺ v 2 兲兴 .
共3兲
II. EXPERIMENT
The resonant frequency of the cantilevers was measured by a technique similar to that described by Hong et al.24 If an ac voltage is applied between the cantilever and sample then a capacitive electrostatic force between them will produce a mechanical oscillation of the lever. We found the method worked well as long as the cantilever to surface separation was approximately 1 mm or less. Hong et al.24 only used conductive cantilevers such as doped silicon cantilevers, our observations indicate that the method works effectively for silicon nitride cantilevers as well. This is probably due to the electrostatic force acting between the surface and the gold coating on the back of the cantilever rather than the surface and the silicon nitride tip. This is similar to methods used to vibrate cantilevers by attaching magnetic particles to the back of the lever and then applying an oscillating magnetic field.25,26 The frequency response of the cantilever was measured using an Edgerton, Germenshausen, and Grier 共EG and G兲 lock-in amplifier.27 Taking the split photodiode response and feeding it directly into the lock-in amplifier obtained the amplitude. The resonant frequency of the cantilever is then assumed to be the frequency, which produces the peak of highest amplitude. It was found that this method produced resonance curves, which showed a single and distinct peak. This is in contrast to mechanically vibrating the cantilever, which can display a number of vibrational modes.24 Figure 1 shows a typical resonance curve for a V-shaped cantilever. The peak is clearly defined and both the resonant frequency and Q factor can be easily determined. For this particular cantilever, the resonant frequency in air and Q factor was equal to 17.8 kHz and 29, respectively.
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FIG. 2. Schematic showing the two different types of configurations of cantilevers studied, beam and V shaped. L represents the length and w the width of the cantilevers.
The cantilevers used were Thermomicroscope28 microlevers that are composed of silicon nitride with an array of cantilevers at one end of each chip. Figure 2 shows the two different configurations of the cantilevers studied, beam and V shaped. L and w represent the length and width of the cantilevers. These dimensions do not alter significantly between cantilevers from different chips from the same wafer13 and can be determined accurately using optical or electron microscopy. Table I shows the measured length and width for each different type of cantilever as compared to the manufacturers values. The letters B, C, D refer to the different type of cantilevers studied. Thermomicroscopes28 has a standard nomenclature for their cantilevers which we will continue to use for simplicity and ease of reference. Type B are beam shaped while C and D are V shaped. The dimensions were measured using an optical microscope. Once the resonant frequency of the uncoated cantilevers was measured a 25 nm thick gold layer was evaporated onto the tip side. It is assumed that regardless of which side of the cantilever is coated, the mass added will still be the same. The resonant frequency was then measured again, after the gold layer was deposited, and Eq. 共3兲 used to calculate the spring constant of the cantilevers. III. RESULTS
Table II shows the results obtained for cantilevers of type B 共beam shaped兲 using the present method. For comTABLE I. Experimentally measured dimensions of the three types of cantilevers studied compared to the nominal values supplied by the manufacturer. Cantilever B is beam shaped while cantilevers C and D are V shaped. The dimensions were measured using an optical microscopic.
Length 共m兲 experimental width 共m兲 experimental length 共m兲 manufacturer width 共m兲 manufacturer
B
C
D
200
305
218
20
21
21
200
320
220
20
22
22
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Rev. Sci. Instrum., Vol. 72, No. 5, May 2001
TABLE II. Comparison of spring constants for types B cantilevers 共beam shaped兲 from six different chips using the current technique and that developed by Sader et al.15
Chip
Present work 共N/m兲 ⫾10%
Sader et al. 共N/m兲 ⫾15%
1 2 3 4 5 6
0.031 0.025 0.020 0.021 0.022 0.019
0.028 0.025 0.021 0.022 0.024 0.023
parison the spring constants of these cantilevers were also measured using the technique developed by Sader et al.15 This technique works well for beam shaped cantilevers with aspect ratios (L/w) in the range 3–14 and with the Q factorⰇ1. We estimate the uncertainty on our method to be 10% or better while the uncertainty on the measurements using the Sader et al.15 technique were closer to 15%. Both methods are in good agreement and demonstrate that the present method is able to accurately determine the spring constants of AFM cantilevers. The spring constant of V-shaped cantilevers of different geometries was also measured. Table III shows the results for these cantilevers. The numbers 1–6 represent the chip which the cantilevers came from while the letters C and D refer to the type of cantilever. These chips are the same as those used in Table II 共i.e., cantilevers 1-C and 1-D are from the same chip as cantilever 1-B兲. Note, from both Tables II and III, there is a difference between the springs constant for cantilevers of the same type. Recent work by Holbery and Eden29 demonstrated that variations of up to 30% or more could occur for cantilevers of the same type from the same wafer. These variations were attributed to changes in thickness and/or material properties of the cantilevers across the wafer. They were limited to studying cantilevers with spring constants ⬎1 N/m but our results indicate similar trends for the softer cantilevers. The spring constant for two V shaped cantilevers with L⫽140 m and w⫽19 m were also determined using the current technique. These are denoted as type E cantilevers by TABLE III. Spring constants for V-shaped cantilevers of types C and D determined using the current technique. These cantilevers are from the same chips as those studied in Table II. Cantilever 3-D was damaged and could not be measured. Spring constant 共N/m兲 1-D 1-C 2-D 2-C 3-D 3-C 4-D 4-C 5-D 5-C 6-D 6-C
0.048 0.016 0.046 0.014 damaged 0.012 0.032 0.011 0.037 0.012 0.034 0.010
FIG. 3. Plot of resonant frequency vs gold coating thickness for a beam shaped cantilever. The solid line is a fit to the experimental data while the dotted line is a fit to the predicted resonant frequencies using Eq. 共3兲.
Thermomicroscopes.28 The cantilever of this type from chip 5 gave a spring constant of 0.086 N/m while the cantilever from chip 6 was found to be equal to 0.085 N/m. To determine at what point the gold thickness alters the mechanical properties of the cantilevers gold layers were added to a number of levers in 25 nm increments up to a thickness of 100 nm. With knowledge of the spring constant of the cantilevers we substituted that value and the uncoated resonant frequency into Eq. 共3兲 and determined what v 2 should be for any gold coating thickness. Equation 共3兲 assumes the gold acts as an inelastic body. Therefore by comparing values using Eq. 共3兲 to those determined experimentally we found the approximate thickness of gold that affects the overall material properties of the cantilevers. Figure 3 shows a plot of resonant frequency versus gold coating thickness for a beam shaped cantilever with k ⫽0.031 N/m and v 1 ⫽14.2 kHz. The solid line represents a fit to the experimental data while the dotted line is a fit to the predicted resonant frequencies obtained using Eq. 共3兲. From the graph it is obvious that after approximately 58 nm the gold is no longer simply acting as an extra mass. Similar results were found for all other cantilevers.
IV. DISCUSSION
We have demonstrated a simple, accurate, and nondestructive method for determining the spring constant of AFM cantilevers of different geometries. Since many AFM experiments now require the coating of AFM cantilevers8,30 the present technique could be easily incorporated into the preparation of the levers. It does not represent any extra preparative work and only requires the measuring of the resonant frequency, a simple procedure, before and after the gold evaporation. We also show that the resonant frequency and Q factor of silicon nitride cantilevers can be accurately measured using electrostatic force modulation. Variations in spring constant for cantilevers of the same type across the same wafer have also been observed and show that calibration of individual cantilevers is critical in conducting quantifiable experiments.
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Rev. Sci. Instrum., Vol. 72, No. 5, May 2001
ACKNOWLEDGMENTS
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