Spring Constant Calibration of Nanosurface-engineered AFM Cantilevers
O. Ergincana,, G .Palasantzasa and B.J. Kooia
a
Nanostructured Materials and Interfaces — Zernike Institute for Advanced Materials
Nijenborgh 4 9747 AG Groningen The Netherlands
S1 Experimental Setup
S1.1 Noise Measurement
The Q-tuning (or thermal tuning) method, which was developed for the determination of the
cantilever spring constants (k) with accuracy down to ∼ 10 % [1] involves measuring the
cantilever’s mechanical response due to agitations of impinging molecules from the surrounding
fluid (ambient air, gases, liquids) and due to thermal dissipation via internal degrees of freedom.
[1]
Figure S1: The quadruple photodiode captures the thermal motion of the cantilever by the
reflection of the laser beam from the surface of the cantilever. The lock-in amplifier recovers this
signal. The time-varying voltage signal can be converted into PSD vs frequency by Fast Fourier
Transform (FFT).
The AFM hardware measures the cantilever’s fluctuations as a function of time from which,
by Fourier transformation, the frequency dependent power spectral density (PSD) |A(ω)| is
obtained as illustrated in Fig. S1.
|𝐴(𝜔)|2 = 𝐴𝑜 /[(𝜔2 − 𝜔𝑜 2 )2 + (𝜔𝜔𝑜 /𝑄)2 ]
(1)
The averaged Q-factor (both due to intrinsic and fluidic or gas dissipation) is obtained by fitting
the PSD spectrum |A(ω)| with a well-know Lorentzian form (Eq. 1), [1] for multiple PSD
spectrum. The noise measurements were repeated more than eleven times at each pressure to
avoid the influence of instantaneous measurement errors such as jitter effect etc. and to confirm
reproducibility of the measurement (illustrated in Fig. S2). The acquired data was averaged
allowing one to obtain the Q-factor with an accuracy of ∼10 %. Notably, extensive analysis of
cantilever heating effects in both air and vacuum resulted in negligible frequency shifts of the
order of Δf≈10−20Hz (≈10-3% f0).
Figure S2: PSD spectra for various pressures from free molecular to dense regime. The graph on
the left represents the non-modified cantilever.
S1.2 Atomic force microscope (AFM) measurement system
Microfabricated gold coated silicon cantilevers that are used for AFM measurements were
modified using the focused ion beam (FIB) erosion process. The noise and topography analysis
of the cantilevers were performed using the Bruker multimode V ( former Veeco Nanoscope V )
AFM. The topography images of the samples are gathered in contact mode using Veeco-MSNL
super sharp cantilevers which have low spring constant. There are two important properties that
have to be taken into account during topography imaging; the first is to use a probe which has a
natural resonance frequency below the natural frequency of the scanned cantilever surface, and
the second one is to avoid the perturbation on the scanned cantilever with the force acting during
tapping mode (TM).
Figure S3: Q-tuning measurement setup. Left: Schematic is the AFM set-up used for Q-tuning
noise measurements in vacuum. a) illustrates 2 types of pressure gauges; b) solenoid valve in
cooperation with a needle valve; c) the quadruple photodiode d, noise dampers; and b, humidity
and temperature sensors. Right: PSD noise spectra vs. pressure and frequency.
The AFM was modified for high vacuum measurements (Fig. S3). The environment in the
bell jar is monitored using humidity and temperature sensors. The noise created by the turbo
pump for high vacuum measurements has been damped effectively using a pile of sand bags. The
turbo pump was turned off during the measurements for pressures over . The air inlet and the
pressure monitoring are performed with two types of pressure gauges and an electromagnetic
valve. Due to the fact that the accuracy of a full range pressure gauge drops at high pressures, it is
necessary to use a gas independent, diaphragm pressure gauge, for more accurate high pressure
measurements. A difference up to 80 % was experienced between the full range pressure gauge
(Varian FRG-700), and the high pressure gas independent, diaphragm gauge (Vacuubrand DVR2).
All the topography, the spring constant and the thermal noise measurements of both modified,
non-modified cantilever sensors are measured using a Veeco PicoForce Multimode microscope
with Nanoscope V controller measurement system. The topography images of the samples are
gathered in using Veeco-MSNL/SNL super sharp cantilevers which have low spring constants
nominal 0.6−0.07N/m, tip radius nominal 2nm, frequencies ranging from 15−70 kHz and the
back side is reflective gold coated for increased signal to noise ratio. Topography imaging of a
cantilever sensor with another cantilever sensor is a non-trivial operation. There are several
important properties that has to be taken into account during imaging. If the selected mode of
imaging is tapping mode (TM) then it is important to use a probe with natural resonance
frequency well below or above the natural frequency of the scanned cantilever. If contact mode is
used, which is preferable in order to avoid perturbations of the imaging due to the force acting on
the scanned cantilever during TM mode, then it is important to image as close as possible to the
support area of the scanned cantilever to avoid damage of the probe and cantilever bending.
S1.3 Spring constant and resonance frequency shift
Besides the quality factor Q, an accurate measurement of the cantilever spring constant is
essential for a meaningful calibration of the resonator as a sensor. Indeed, is a physical property,
which depends on the elastic modulus (E), and the cantilever dimensions (Table SI). Since all the
samples in this study have been modified on one of the surfaces (opposite to the one on which the
laser beam of the atomic force microscope (AFM) is reflected during noise measurements), an
experimental determination of is crucial. It is possible to acquire by the Q-tuning (Thermal
tuning) method (Supplementary Section S2) and the resonance frequency with an accuracy
down to ∼10 % and ∼1 %, respectively. [1]
First of all, the removed mass calculations, both by simple assumption of the frequency shift
due to mass change ∆m ≈ 2m(∆f/f) (Δm is the mass removed, and m is the total mass), and the
calculations by geometrical means show that for both type-(1) and type-(2) surface modifications
the amount of removed mass is Δm/m ≈ 1.5% (Supplementary Section S2).
S2 Properties of microcantilevers
Table SI: Properties of non-modified and modified microcantilevers
Δm/m
L
w
t
fvacuum
(μm)
(μm)
(μm)
(kHz)
A-1
135.6
30.9
2.2
152.694
0.33
-
A-1(m)
135.6
30.9
2.2
153.707
0.34
1.33
A-2
127.6
31
2.0
134.943
0.30
-
A-2(m)
127.6
31
2.0
135.848
0.33
1.34
A-3
136
31.4
2.0
130.924
0.35
-
A-3(m)
136
31.4
2.0
133.407
0.33
3.79
B-1
95.1
30.7
2.2
326.956
0.35
-
B-1(m)
95.1
30.7
2.2
328.945
0.34
1.21
B-2
87.3
31.4
1.8
270.040
0.33
-
B-2(m)
87.3
31.4
1.8
272.281
0.34
1.66
B-3
95.6
31.0
2.0
272.753
0.35
-
95.6
31.0
2.0
[a] (m) denotes modified cantilever.
279.482
0.34
4.93
Type[a]
B-3(m)
c[b]
(%)
Figure S4: Q-factor vs. pressure P for cantilevers a) Type-(2) surface modification for
dimensions A and B, b) Type-(1) and (3) surface modification for dimension B. The surface
modifications are illustrated in the insets, and the dimensions are given in Supplementary Table
SI and in the manuscript Table I.
Figure S5: L(Re) vs.Re for cantilevers a) Type-(3) surface modification for dimensions A and B,
b) Type-(2) surface modification for dimensions A and B. The dimensions are given in
Supplementary Table SI and in the manuscript Table I
Although the Q-factor of the modified resonators in the intrinsic regime (P>10-4mbar) differ in
some cases 50-100 % with respect to the value of the non-modified resonator (shown in Fig. S6),
the assumption of the hydrodynamic loading being the dominant contribution remains valid for
the viscous regime. In addition our study here supports the Sader model as a non-destructive
dynamic spring constant calculation including the resonators with modified surface structures.
Figure S6: (a-c) Quality factor Q vs. pressure P for cantilever types A and B with dimensions
given in Fig. 1. The corresponding surface modifications (Type 1, 2, 3) are illustrated in the
insets (a-c). The index “m” denotes the modified cantilevers as shown in Fig.1. In the molecular
regime (<102 mbar) (a-c): The red fit lines illustrate the Q~1/P scaling (c) The blue fit lines
illustrate the Q~1/P scaling of Eq.(1) for modified cantilevers A and B. In the continuum regime
(gradient gray background) (≥103 mbar) (a-c):The red fit lines illustrate the Q~1/P-0.5 scaling; and
blue fit lines Q~1/P-0.3 scaling.
S3 Surface modification of cantilevers
S3.1 Focused Ion Beam (FIB)
The focused ion beam (FIB) instrument is similar to a scanning electron microscope (SEM),
except the beam that is rastered over the sample is an ion beam rather than an electron beam (as
illustrated in the Fig. S7). The Ga+ ions are used, and their sputtering action enables precise
machining of samples. In conjunction with the gas-injection capabilities of these systems, which
enable ion-beam-activated deposition and enhanced etching. We have used “The Tescan dual
beam system” which supplement an ion beam column with an additional electron beam column.
This makes the machine versatile for imaging, material removal, and deposition at length scales
of a few nanometers to hundreds of microns.
Figure S7: Schematic illustration of the FIB; Ga ions striking to the surface of a sample
sputtering away matter
S3.2 Modification process
It should be recognized also that modification of the surfaces of the gold coated resonators is a
mass removal process. Fig. S8 above shows the frequency shift Δf after altering the surface
structure of microresonators. Cantilevers used for surface modification process were purchased
from MikroMasch. They are uniformly coated on bottom side with 10nm thick chromium (to
ensure good adhesion of any subsequent overlayer) and a 50nm thick gold layer. As purchased,
the cantilevers were tipless and the RMS surface roughnesses were ∼1nm (in absence of any
surface grooves). Modifications in order to achieve the groove patterns on the detector (back)
side of the resonators were performed using focused ion beam (FIB: Tescan Dual Beam
containing a Field Emission Gun electron source). Using only the detector side for surface
modification prevents losses in the reflection of laser beam since the reflection occurs from the
front side only. The Tescan dual beam microscope has the ability of simultaneous electron beam
and ion microscopy. The ions are accelerated with 30keV extraction voltage and the probe ion
current was 600pA. These parameters give a complete removal of the gold/chromium layer from
the desired location, while reducing possible re-sputtering.
Figure S8: PSD spectra in air of the size A cantilever with various surface modifications (types(1), (2) and (3)). The arrows indicate the frequency shift in air for the different types of surface
modifications (types-(1), (2) and (3)) as shown in the insets (a-c) obtained by AFM.
References
[1]
Benjamin Ohler. Cantilever spring constant calibration using laser doppler
vibrometry. Rev. Sci. Instrum., 78(6):063701–063701–5, June 2007.
[2]
John E. Sader, James W. M. Chon, and Paul Mulvaney. Calibration of rectangular
atomic force microscope cantilevers. Rev. Sci. Instrum., 70(10):3967–3969, October 1999.