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Mechanical and electrical characterization of quartz tuning fork force sensors
Accepted Manuscript Title: Mechanical and electrical characterization of quartz tuning fork force sensors Author: Bakir Babic Magnus T.L. Hsu Malcolm B. Gray Mingzhen Lu Jan Herrmann PII: DOI: Reference:
S0924-4247(14)00550-0 http://dx.doi.org/doi:10.1016/j.sna.2014.12.028 SNA 9011
To appear in:
Sensors and Actuators A
Received date: Revised date: Accepted date:
3-9-2014 11-11-2014 26-12-2014
Please cite this article as: Bakir Babic, Magnus T.L. Hsu, Malcolm B. Gray, Mingzhen Lu, Jan Herrmann, Mechanical and electrical characterization of quartz tuning fork force sensors, Sensors & Actuators: A. Physical (2015), http://dx.doi.org/10.1016/j.sna.2014.12.028 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Journal
Mechanical and electrical characterization of quartz tuning fork force sensors Bakir Babic,1, a) Magnus T. L. Hsu,1 Malcolm B. Gray,1 Mingzhen Lu,2 and Jan Herrmann1 National Measurement Institute, Bradfield Road, Lindfield NSW 2070,
Australia 2)
National Institute of Metrology, Beijing 100013, P.R. China
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(Dated: 11 November 2014)
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1)
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The results of a systematic experimental characterization of the mechanical and electrical properties of quartz tuning forks (QTFs) are presented. Actuation efficiency and detection sensitivity are introduced as parameters for characterization of the
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QTF with a focus on their use as a force sensor. The spring constants, quality (Q) factors, actuation efficiencies and detection sensitivities of twelve QTFs were mea-
M
sured by combining heterodyne interferometry with an electrical excitation/detection set-up and found to be consistent for all twelve QTFs. Spring constants were determined using geometrical and thermal methods and were found to agree within 3 %.
d
The Q-factor, actuation efficiency and detection sensitivity of the QTFs were mea-
te
sured for the first two vibrational modes. The Q-factor and detection sensitivity were higher in the second vibrational mode while the actuation efficiency was lower. We
Ac ce p
show that the QTF displacement amplitude can be determined from the detection sensitivity and the measured current through the QTF. The performance of a QTF as a force sensor is discussed based on a comparison of the mechanical and electrical parameters measured for the first and second vibrational modes. The results are relevant for QTFs used in sensing applications, especially as force sensors in scanning probe microscopes.
PACS numbers: 68.37, 06.30, 07.07, 07.60 Keywords: AFM, non-contact mode, quartz tuning fork, force sensor, interferometry.
a)
Electronic mail: [email protected]
1
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I.
INTRODUCTION The invention of the atomic force microscope (AFM)1 provided a new tool for the field of
surface science and opened the path for rapid progress in nanoscience and the development
ip t
of nanotechnology. Recently, a metrological scanning probe microscope based on an AFM has been used as a primary standard for length measurement at the nanoscale.2 An AFM
cr
measurement contains a wealth of information about a sample, a tip and their interaction. In addition to mapping the surface topography, an AFM can give quantitative information
us
about local forces between the tip and the sample.1,3 To quantify the interaction forces, it is critical to characterize the force sensor.
an
Quartz tuning forks (QTFs) have been successfully used as force sensors in scanning probe microscopy.3,4 An AFM utilizing a QTF force sensor is operated in dynamic mode. A sharp tip is mounted on one of the tines of a QTF which oscillates at its resonant frequency. The
M
interaction between the tip and the sample surface leads to a change in the effective spring constant and consequently in the resonant frequency and oscillation amplitude of the QTF
d
sensor. These changes can be used as feedback signals to adjust the separation between the
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tip and the sample such that the interaction with the surface is kept constant. There are several advantages for using a QTF as a force sensor compared to other AFM
Ac ce p
sensors. The complexity of the AFM force sensing hardware is significantly reduced compared to the commonly used optical beam-bounce readout since a QTF can be both excited and detected electrically. Quartz oscillators have high frequency stability and low temperature sensitivity at room temperature, and their low heat dissipation makes them attractive as force sensors in low temperature AFMs and room temperature metrological AFM applications.3 In an ambient environment, the quality factor Q of a QTF is typically a few orders of magnitude higher than that of other mechanical resonators, such as the cantilevers commonly used as AFM sensors.5,6 The spring constant of a QTF is much higher than the typical spring constant of a cantilever used in a “beam-bounce” sensing configuration, therefore reducing jump-to-contact problems7 . An illustration of the unprecedented resolution of an AFM based on a QTF sensor operated in the frequency modulation (FM) mode is its ability to image the detailed chemical structure of a single molecule of pentacene.8 The force sensitivity of the QTF force sensor depends on its spring constant. The spring constant k is determined by the geometry of a QTF and by its elastic properties. For small 2
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deflections, the tines of a QTF follow Hooke’s law4 F = −kz, where F is the force acting on a tine and z is the corresponding deflection. To determine the interaction forces in this approximation, one has to determine the spring constant of the QTF while measuring the deflection. Depending on the sensor and on the AFM’s mode of operation, knowledge of the
ip t
spring constant is necessary, but not sufficient, to determine the interaction force between the QTF and the sample. For example, if an AFM is operated in FM-AFM mode, both the
cr
frequency shift and amplitude must be measured to determine the force.9
A QTF can be used as a force sensor in an AFM application thanks to the piezoelec-
us
tric properties of quartz. The QTF is electrically driven by an external voltage to induce deformation and therefore mechanical displacement of the tines. To quantify the coupling
an
between the electrical and mechanical properties of a QTF, one has to measure both of them simultaneously and independently. The relationships between the drive voltage V (t) and the deflection A(t) of a QTF on one hand and between the deflection A(t) and the output
M
current I(t) on the other hand are of particular interest for AFM applications. We define the ratio of the oscillation amplitude A to the amplitude of the electrical drive voltage V as
d
the actuation efficiency. The detection sensitivity is defined as the ratio of the amplitude of
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the current I to the oscillation amplitude A.
When a QTF is used as the force sensor in an AFM, a tip is attached at some location
Ac ce p
to one of the QTF tines. Since the deflection A(x) depends on the coordinate x along the tine, the QTF must be characterized at the position of the tip. As the exact position of the tip may vary, it is necessary to map the mechanical and electrical properties of a QTF force sensor along the tine so that the force for different tip positions can be determined. In this article, we present a systematic quantification of the physical properties of a set of twelve QTF sensors, including their spring constants, quality (Q-)factors, actuation efficiencies and detection sensitivities for the fundamental and second-order vibrational modes using heterodyne interferometry combined with piezoelectric excitation and read-out at ultrasonic frequencies. Experimentally, the anti-phase mode of a QTF is excited and detected at the resonance frequency in a configuration where the QTF is mounted with both tines free to oscillate.10 (This is very different from the widely used qPlus configuration where one of the tines is immobilized by attaching it to a holder and excitation is provided mechanically.3,7 Section II describes the determination of the spring constants, both from interferometric measurements of the displacement of a QTF tine induced by thermal fluctuations and from 3
Page 3 of 20
the tine geometry. In Section III, an electrical model for the QTF is introduced and the results of the electrical characterization of the same QTFs measured in Section II are presented. Optical (interferometric) and electrical methods were used to determine the Q-factors for the first and second vibrational modes. In Section IV, the actuation efficiency and detection
ip t
sensitivity of the QTFs are determined optically and geometrically. The optical method is used to map the actuation efficiency along the QTF tine for the first and second vibrational
cr
modes. The feasibility of determining the QTF displacement amplitude A based on the actuation efficiency and detection sensitivity, respectively, is discussed in Section V. The
us
performance of a QTF force sensor in different vibrational modes is compared. The last
II.
an
section presents conclusions.
THE SPRING CONSTANT
M
There are several methods for the determination of a cantilever spring constant: the geometrical method,4 calibration against a known standard,11 the added mass method,12
the spring constants of QTFs.
d
and the thermal method.13 Here, we use the geometrical and thermal methods to determine
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A QTF can be modeled as two independent cantilevers: each can be described within continuum mechanics as a clamped beam with a free end. The spring constant kg of a
Ac ce p
cantilever is given by its geometry and elastic properties according to kg = E
wt3 4l3
,
(1)
where E is Young’s modulus of the cantilever material, t is the thickness, w is the width and l is the length of the cantilever.14 The geometrical method of determining the spring constant of a QTF is based on measuring the dimensions of the QTF and using a value of E = (75.2 ± 5.7) GPa for Young’s modulus of quartz.4,15 The geometries of twelve QTFs fabricated on the same wafer16 were measured using optical microscopy17 . An optical micrograph of a typical QTF is given in Fig. 1(A) and a transverse view of a QTF tine in Fig. 1(B). The average dimensions of the set of QTFs measured in this experiment were: t = (214.9±0.3) µm, w = (129.5±0.5) µm and l = (2474± 5) µm. Eq. (1) was used to obtain the spring constants from these measured QTF dimensions with results shown in Fig. 2. The average spring constant value is kg = (1596 ± 14) N/m 4
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where the mean is represented in Fig. 2 as the dashed line and the uncertainty value is the experimental standard deviation of the mean. (All uncertainties given in the article are standard deviations unless otherwise stated.)
(C)
x
Vout
FG
t
Vin
le l
(B)
PBS
Laser
y
λ/2
x
MO
λ/4
PBS
λ/2
M
λ/4
PBS
us
PBS
w
QTF
CA
ip t
z
cr
(A)
130.2 µm
Detectors
an
Phasemeter
FIG. 1. (A) Optical microscopy image of a quartz tuning fork used in this experiment. The tine
M
thickness, t, tine length, l and length of the electrode, le , are indicated. (B) Optical micrograph of a QTF tine illustrating measurement of the tine width,w. (C) Schematic of the electrical and
d
optical test bed used for measuring both the deflection of a tuning fork tine using heterodyne
te
laser interferometry and the electrical characteristics. PBS: polarizing beam-splitter, λ/4: quarter wave-plate, λ/2: half wave-plate, M: mirror, CA: charge amplifier, FG: frequency generator, MO:
Ac ce p
microscope objective.
The QTF spring constants were also measured using the thermal method. A vibrating cantilever can be described as a one-degree-of-freedom harmonic oscillator with a resonant frequency ω0 = (k/m)1/2 , where m is the oscillating mass. The QTF with two tines free to oscillate can be described as a system of two weakly coupled harmonic oscillators with two degrees of freedom. In thermal equilibrium (with its surroundings), the QTF tines will vibrate at its resonant frequency due to thermal excitation, with a deflection amplitude, A(t). In this situation, the thermal energy is equivalent to the average mechanical energy of the QTF oscillator: kB T = khA2 i, where kB is the Boltzmann constant, T is the temperature and h...i denotes the average. The thermally excited vibration amplitude was measured using heterodyne interferometry in the Michelson configuration as shown in Fig. 1(C). The all-digital phasemeter used to measure the optical phase difference was implemented on a field-programmable gate array platform and can measure displacements with picometer res5
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olution, large dynamic range and 400 kHz bandwidth.18 The reference and QTF reflection signals were detected using high-speed photodetectors, and the electronic signals were subsequently digitized on high-speed analog-to-digital converters (ADCs). Each ADC signal was then tracked and measured by a phase locked loop. Finally, the signal was filtered and
ip t
the output signal was integrated to provide the phase of the input signal. The difference between the reference and input signal phases yields the phase signal corresponding to the
cr
deflection of the QTF tine. The schematic of the optical measurement system together with the piezoelectric excitation and detection set-up are shown in Fig. 1(C).
us
The measurement of the thermal vibration amplitude, A, of a QTF tine was performed in the following way: The QTF was first fixed on a micromanipulation stage. A HeNe laser
an
(wavelength λ = 633 nm; output power ≈ 100 µW) was focused to a beam diameter of 2 µm with a 40× microscope objective (NA=0.66) and directed onto one of the x-y faces of the QTF tine (Fig. 1(B)) that was visually aligned to be perpendicular to the laser beam. To
M
minimise potential heating of the tine due to laser irradiation, the duration of a measurement during which the tine was illuminated was limited to 1 s. The tine position along its width
te
to the tine width.
d
(y-direction in Fig. 1(B)) was adjusted such that the laser spot was centered with respect
Since the vibration amplitude A(x) depends strongly on the position x along the QTF
Ac ce p
tine (x-direction in Figs. 1(A) and (B)), the QTF thermal amplitude was measured near the end of the tine, where the amplitude for the first two modes is the largest.14 To make results for different QTFs comparable, the end of the tine was determined by shifting the tine along the x-direction until the laser beam seen on a background screen was partially obstructed. The tine was then moved back into the beam path by the fixed amount of 50 µm. This distance was determined experimentally as being closest to the edge of the tine while still giving sufficient signal for reproducible measurements. The resulting beam position was also found to correspond to the typical AFM tip location, following the tip attachment procedure previously reported.19 Two experimental indicators were used to ensure that the alignment (orthogonality) of the tine with respect to the laser beam was optimized and that the measurements were made close to the end of the tine (within 50 µm). The first indicator was that movement of the laser spot along the tine (a few mm in the x-direction) did not lead to a significant change in the intensity of the optical signal. The second was derived from observation of the QTF’s thermal vibration. At the very edge of the tine, the vibration 6
Page 6 of 20
amplitude of the thermal oscillation is close to the limit of detection for the interferometric system and any significant misalignment would lead to a loss of the signal. Using these indicators, we estimate that deviations from perpendicularity between the laser spot and the tine surface are less than 5 degrees, corresponding to a cosine error of < 3 % in the tine
ip t
amplitude. As measurements of the thermal vibration amplitude showed variations of the
cr
order of 25 %, the contribution of the alignment error to this variation is small. 1.6 1.4
kth kg
A (pm)
1.2
us
1.0
0.8
an
0.6
32.2
32.4
32.6
32.8
33.0
f (kHz)
Ac ce p
te
d
M
k
32.0
QTF number
FIG. 2. Experimentally measured spring constants for twelve QTFs determined using geometrical (open) and thermal methods (solid), respectively. The dashed (solid) line indicates the average spring constant kg (kth )determined with the geometrical (thermal) method. Inset: Frequency spectrum of thermally excited vibrations near the first QTF vibrational mode, measured by heterodyne interferometry.
All measurements were performed at room temperature T = (293.8 ± 0.3) K. Based on the spring constant determined from the geometrical model, we expect an average thermal vibration amplitude of A = (1.6 ± 0.2) pm for the QTFs. The average thermal vibration amplitude measured directly with interferometry was found to be A = (1.5±0.4) pm, with an example shown in the inset of Fig. 2. These two values are in agreement and give confidence 7
Page 7 of 20
for the use of this method to undertake further quantitative characterization of QTF force sensors. From the thermal vibration amplitude of a QTF and the ambient temperature we can obtain the QTF spring constant. The corresponding spring constants for the set of QTFs
ip t
together with their uncertainties are shown as solid symbols in Fig. 2. The average spring constant obtained using the thermal method is kth = (1560±210) N/m. This mean value is in
cr
good agreement with the average spring constant kg obtained from the geometry of the QTFs, although the standard deviation for the thermal method is an order of magnitude larger. This
us
is mainly due to the relatively low signal to noise ratio of the interferometric measurement of the tine amplitude in the picometre range.20 An additional contribution comes from the
an
uncertainty of the positioning of the laser spot along the length (x-direction) of the QTF tine since the oscillation amplitude depends on that position (see Fig. 6(B)). Nevertheless, the results indicate that QTFs produced on the same wafer have consistent spring constants
M
and that the geometrical method is suitable to determine the mean spring constant of the QTFs to within 3% of the mean spring constant determined by the thermal method. This is
d
remarkable given that the geometrical method is based on simple assumptions and neglects
ELECTRICAL CHARACTERIZATION AND QUALITY FACTOR
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III.
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several issues, such as the variation in material properties and structural defects.
In non-contact AFM applications employing a QTF force sensors, it is common practice to excite the QTF mechanically by vibrating it using a piezoelectric actuator attached to the QTF and to read out the QTF signal electrically. It is also possible to excite the QTF electrically. This has the advantage that the force sensor assembly is less complex, resulting in fewer parasitic heat and vibrational sources, and provides more comparable sensors since their characteristics are less dependent on the mounting procedures. A QTF can be modeled by an equivalent electrical circuit. The circuit can be described by the LRC resonator model and includes a parallel capacitance which is due to the contacts and leads.5 The schematic of a series LRC circuit is given in the inset of Fig. 3. The frequency-dependent impedance is given by Z(ω) =
iωCp +
1 RTF + iωLTF + 1/iωCTF 8
−1
,
(2) Page 8 of 20
where Cp is the parasitic capacitance, RTF is the resistance, LTF the inductance and CTF the capacitance of the QTF; ω = 2πf where f is the frequency. The electrical response, ITF , i.e. the current flowing through the QTF tine, was measured using a spectrum analyzer.21 A sinusoidal drive voltage with amplitude Vin was applied across
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the QTF. A typical frequency response of a QTF driven at Vin = 10 mV is shown in Fig. 3. The asymmetry in the resonance curve is due to the current flowing through Cp . The
cr
model Z(ω) of equation (2) was adjusted to the measured frequency response yielding the solid line in Fig. 3, with the following parameters determined from the fit : Cp = 4.64 pF,
Vin
M te
Ac ce p 0 32 600
ITF
d
2.10 -8
. -8 110
LTF
Cp
ITF (A)
3.10-8
CTF
an
RTF
4 .10-8
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CTF = 3.99 fF, LTF = 5945 H, RTF = 244 kΩ, ω0 /(2π) = f0 = 32678.9 Hz, Q = 4990.
32 620
32 640
32 660
32 680
32 700
32 720
32 740
f (Hz)
FIG. 3. Amplitude of the current ITF flowing through a QTF driven with a sinusoidal voltage with amplitude Vin = 10 mV as a function of drive frequency f . Inset: Schematic of the equivalent circuit for a piezoelectric resonator.
The Q-factor describes the energy losses in a resonator and is defined as Q = ω0 /δω, where ω0 is the resonance frequency and δω the width of the resonance at half-maximum. A high Q is desirable for the force sensor in an FM-AFM application, since the magnitude of the detectable force is inversely proportional to Q. A higher Q offers higher selectivity and stability of oscillations at the resonant frequency. At room temperature, a typical QTF has a Q-factor that is few orders of magnitude higher than that of other cantilever 9
Page 9 of 20
beams used in AFM5,14 . The QTF Q-factor can be measured both electrically and optically. The electrical measurement of a QTF frequency response gives Qel using the relationship Qel = (ω0 CTF RTF )−1 . In Fig. 4(A), measured Qel values of twelve QTFs for the first and second vibrational modes are shown as circular symbols. For the first mode, we find an
ip t
average value of Q1el = 4700 ± 800, and for the second mode Q2el = 6500 ± 2400. The first
cr
mode has both a lower mean Qel value and a smaller standard deviation.
For the same QTFs, the Q-factor was also measured optically. The interferometric set-up
us
in Fig. 1(C) was used to measure the frequency response of the deflection for the first two vibrational modes of the QTFs. A typical measured frequency spectrum for the first and
an
second vibrational modes is shown in Figures 4(B) and 4(C), respectively. The lines are Lorentzian fits from which the Qop values can be extracted. The average Qop value obtained for the first vibrational mode was Q1op = 3900 ± 500, which is smaller than that for the
M
second mode, Q2op = 4600 ± 700, with a similar standard deviation.
d
The mean Q-factors obtained from both methods are of the order of several thousands
te
and are in agreement with previous reports.5,14 Although the values for both methods agree within their statistical uncertainties, the electrical Q-factors for both vibrational modes are
Ac ce p
higher than the optical Q-factors. In particular, the variability of Q2el is higher than Q2op , the probable reason being that the QTF electrodes are configured highly symmetrically to have a high Q-factor when excited in the first vibrational mode. Due to the electrode design, the operation of the QTF in the second vibrational mode has lower signal to noise ratio and hence the determination of Q2el is less reliable. The experiment also shows that Q1op,el is smaller than Q2op,el in agreement with the literature.22 If one assumes that δωn does not depend on the vibrational mode,23 Qn = ωn /δωn = Q1βn2 /β12, where n is the mode number and βn satisfies cosh(βn )cos(βn )+1=0. In this case, the second vibrational mode is expected to have a higher Q than the fundamental one; Q2 = 6.3 Q1. This is not in agreement with the measured values, hence the assumption that δωn is independent of the vibrational mode is not valid. The modest difference between Q1 and Q2 might be explained by the resonator having a smaller center of mass movement for higher mechanical modes.6 Other damping mechanisms, such as asymmetries, imbalances and acoustic radiation losses, can reduce the Q-factor; however, for a QTF these are higher order contributions. 10
Page 10 of 20
(A)
an
us
cr
ip t
Q1op Q2op Q1el Q2el
M
QTF number
(B) 4.0
(C)
1.6
te
A (nm)
3.0 2.5 2.0
Ac ce p
1.5
1.4
1.0 0.8 0.6
1.0
0.4
0.5
0.2
32.50
32.51
32.52
32.53
Q2op ~5202
1.2
A (nm)
Q1op ~4065
d
3.5
32.54
f (kHz)
196.78
196.82
196.86
196.90
196.94
f (kHz)
FIG. 4. (A) The Q factors of QTFs measured electrically (circles) and optically (squares) for the first (open) and second (solid) modes. Frequency response of a QTF measured optically for the first (B) and second (C) vibrational modes (symbols). The solid lines are Lorentzian fits from which the value of Qop can be extracted.
11
Page 11 of 20
IV.
ACTUATION EFFICIENCY VERSUS DETECTION SENSITIVITY
To control and exploit the characteristics of a QTF for specific applications, it is necessary to determine the relationship between its mechanical and electrical properties. The
ip t
spring constant and displacement amplitude depend on the position, x, along the tine length. When a tip is positioned at a specific location, it is necessary to determine the QTF charac-
cr
teristics at that location so that the force between the sample and the tip can be quantified. Therefore, spatial mapping of the QTF displacement amplitude is necessary.
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Heterodyne interferometry was used to measure the absolute value of the vibration amplitude along the entire length of a QTF. The results shown in Fig. 5 for the first two vibrational modes of one of the twelve QTFs. The vibrational mode shapes are as expected
an
for symmetric coupled beams.14,24 The spatial mapping enables experimental determination of the nodes for the second and higher modes. For the second mode of the QTF shown in
M
Fig. 5, there is a node at x ∼ 0.6 mm where the amplitude drops to zero.
First mode
d
30
Second mode
te
A(nm)
Ac ce p
20
0.5
0.4
15
0.3
10
A(nm)
25
0.6
0.2
5
0.1
0.0
0 0
0.5
1
1.5
2.0
x (mm)
FIG. 5. Mapping of the absolute value of vibration amplitude for the first (open) and second (solid) vibrational modes along the length of the QTF tine. The amplitude of the sinusoidal drive voltage was Vin = 10 mV. The position x = 0 corresponds to the free end of the tine.
12
Page 12 of 20
QTFs can be used both as piezoelectric actuators and as sensors. The actuation efficiency of a QTF, which in previous work has been named sensitivity,4,14 is defined as the ratio between the amplitudes of the tine vibration and of the external drive voltage, at the resonant frequency, γ = A/Vin . The actuation efficiency is related to the piezoelectric coefficient of
ip t
quartz, which is an intrinsic parameter determined by the geometry and crystal orientation of the quartz bar. It can be derived for the first vibrational mode by assuming that a surface
cr
charge density is generated on each electrode when the end of tine is deflected by a distance A. Taking a time derivative and expressing the spring constant via Young’s modulus and
us
the QTF geometry (Eq. (1)), the actuation efficiency near the end of a QTF tine is obtained using the following expression:
an
wtle −1 , γ = 6d21 Eω0 Z 3 l − le /2 l
(3)
where d21 = 2.31 · 10−12 C/N is the relevant piezoelectric coefficient of quartz, Z is the QTF
M
impedance and le is the electrode length (see Fig. 1(A)).25
The actuation efficiency can be determined experimentally by simultaneously measur-
d
ing the vibration amplitude of the QTF tine while driving it electrically with a sinusoidal
te
voltage. The heterodyne interferometry system shown in Fig. 1(C) was used to perform these measurements. The actuation efficiency is determined by a linear fit of the vibration
Ac ce p
amplitude for small drive voltages, shown in Fig. 6(A) as a solid line. We have measured the actuation efficiency near the end of the tine of twelve QTFs, for which the spring constant and the Q-factor were previously measured (Sections II and III) . The results are shown in Fig. 6(B) as open squares. For the first vibrational mode, an average actuation efficiency of γ1op = (3.3 ± 0.5) µm/V was measured for the twelve QTFs. Using Eq. 3, the actuation efficiency can also be determined from the geometry of the QTFs and from their piezoelectric and mechanical properties. These data are plotted as open circles in Fig. 6(A). The average actuation efficiency value obtained from the geometry of the twelve QTFs was γ1geo = (4.0 ± 0.3) µm/V which is in reasonable agreement with the interferometric measurements. Taking into account that the amplitude has been measured interferometrically slightly inward from the end of the tine (∼ 50 µm), this would lower the values found for γ1geo by approximately 4 %, bringing the actuation efficiencies obtained by the two different methods into better agreement. The actuation efficiency of the second vibrational mode has also been measured interfer13
Page 13 of 20
ometrically near the end of the QTF tine, with the results shown in Fig. 6(B) as full squares. The average actuation efficiency was γ2op = (0.10 ± 0.06) µm/V. A lower value is expected for the second mode of a cantilever since the strain is in opposite directions at different points on the tine. Furthermore, the actuation efficiency is inversely proportional to the
ip t
spring constant and the resonant frequency (see Eq. (1) and Eq. (3)), and both these values are roughly an order of magnitude larger for the second mode than for the first mode.26,27
cr
Determination of the vibration amplitude using the actuation efficiency is only meaningful when there is no dissipative interaction of the QTF with its environment, which is not the
us
case in AFM applications. In an AFM, a tip attached to the force sensor interacts with a surface, reducing the oscillation amplitude of the sensor compared with free-air oscillation
an
(for a fixed electrical drive). The supplied drive energy is converted into the mechanical motion of the QTF tines, dissipative losses within the QTF and dissipation due to the interaction with the surface. In this case, deducing the vibration amplitude from A =
M
γVin would be incorrect. However, the detection sensitivity, S, defined as the ratio of the amplitude of the current, through the QTF ITF , and the vibration amplitude A, i.e., S =
d
ITF /A is a more universal quantity. The detection sensitivity and actuation efficiency both
te
depend on the position along the QTF tine due to the position dependence of the vibration amplitude illustrated in Fig. 5. For AFM experiments, their values must be determined
Ac ce p
at the tip position, which in our experiment has been near the end of the QTF tine. The detection sensitivity, Sop , was measured interferometrically at the resonance frequency for the first and second vibrational modes, since the FM-AFM mode of operation requires a QTF to be driven at resonance. Alternatively, it can be calculated using an expression similar to that for the actuation efficiency in Eq. (3), with the difference that the detection sensitivity obtained in this way, Sgeo , is independent of the impedance. The sensitivity for the first vibrational mode is given by:
S1geo =
1 γZ
= 6d21 Eω0
wtle l3
l − le /2 .
(4)
Once the detection sensitivity is determined by either the geometrical or optical method, the vibration amplitude can be determined by simply measuring the amplitude of the current through the QTF. For the set of twelve QTFs, the average detection sensitivity measured by interferometry for the first vibrational mode was S1op = (1.3 ± 0.1) A/m. Using Eq. (4), the 14
Page 14 of 20
(A) 200
ip t
100
cr
A (nm)
150
0 10
20
30
(B)
50
60
70
Vin (mV)
M
5
4
te
d
3
2
γ1op γ1geo γ2op
Ac ce p
γ1, γ2 (µm/V)
40
an
0
us
50
1
0
1
3
5
7
QTF number
9
11
FIG. 6. (A) QTF vibration amplitude versus excitation voltage amplitude at resonance, measured interferometically near the end of the QTF tine (symbols). The line is a linear fit to the measurement data. Its slope gives an actuation efficiency γ1op = (2.87 ± 0.01) µm/V. (B) Actuation efficiency inferred from geometry, γ1geo (open circles) and measured by interferometry, γ1op (open squares) for the first vibrational mode. Also shown is the actuation efficiency for the second vibrational mode measured by interferometry, γ2op (solid squares).
average detection sensitivity determined from the measured geometry was S1geo = (1.1 ± 15
Page 15 of 20
0.1) A/m. The values obtained for the individual QTFs are plotted in Fig. 7 (open symbols, left axis). This value is slight under-estimation due to the fact that the detection sensitivity has not been measured at the very end of the QTF tine. Nevertheless, the results found for
ip t
detection sensitivities with the two measurement methods are in good agreement. The detection sensitivity was also measured interferometrically for the second vibrational mode near the end of the QTF tine. The average value obtained for the second vibrational
cr
mode was S2op = (15.5 ± 4.0) A/m, and the values for the individual QTFs are shown in Fig. 7 (solid symbols, right axis). The detection sensitivities can be used to determine the
us
vibration amplitude of a QTF tine in the corresponding vibrational mode, regardless of the
2.0
M
an
presence or absence of an interaction.
S1op S1geo
30
S2op
25
Ac ce p
20
S2 (A/m)
1.4
d
1.6
te
S1 (A/m)
1.8
15
1.2
1.0
10
0.8
QTF number
FIG. 7. Detection sensitivity for the set of QTFs inferred from the geometry (circles) and measured using interferometry (squares) for the first (open symbols, left axis) and second (solid symbols, right axis) vibrational modes.
16
Page 16 of 20
V.
DISCUSSION
ip t
All reported measurements were performed at room temperature and ambient atmospheric pressure using twelve QTFs fabricated on the same wafer and without a tip attached.
cr
The values obtained for the spring constant, detection sensitivity and Q-factor for some of the QTFs with and without a tip have been compared. The tip attached in our experiment
us
is significantly smaller in size and has a mass that is at least an order of magnitude smaller than that of a tip typically used in qPlus configuration (an etched piece of tungsten wire).
an
The details about the tip and the attachment method applied are described elsewhere.19 After attachment of a tip, the QTF spring constant was found to remain unchanged within the experimental uncertainty; and both the resonant frequency and Q-factor were reduced by
M
less than 1 %. At the resonant frequency, the actuation efficiency and detection sensitivity were found to be unaffected by the attachment of a small mass such as an AFM tip, which
te
d
is consistent with equations (3) and (4).
Ac ce p
Characterizing the mechanical and electrical properties of a QTF provides a quantitative means of comparing the performance when operated in different vibrational modes. In Section III, it was shown that the Q−factor in the second mode can be higher than in the first mode, resulting in lower thermal noise28 and reducing mechanical coupling to parasitic vibrations. Furthermore, operating the QTF sensor in the second mode offers the potential for larger bandwidth operation and ultra-fast scanning.29 However, force sensitivity (defined as the reciprocal of the spring constant, 1/kn = −δz/δFn where n is the mode number), is higher in the first mode, due to its smaller spring constant, than in the second mode.27 The signal-to-noise ratio of the force detector is therefore higher when the QTF is operated in the first vibrational mode than in the second one. This is especially important if the force sensor is operated in the attractive, van der Waals force region, where sub-nanometer vibration amplitudes are necessary for high (atomic) resolution.3,30 Although there are some benefits to operate the QTF in the second vibrational mode, the higher force sensitivity and signal-to-noise ratio favor operation in the first vibrational mode. 17
Page 17 of 20
VI.
CONCLUSION
We have quantitatively characterized the mechanical and electrical properties of a set of QTFs by using heterodyne interferometry combined with piezoelectric excitation and
ip t
detection. The spring constant of twelve QTFs from the same wafer has been systematically measured using thermal and geometrical methods. The thermal method is direct, non-invasive and time-efficient. The results from the thermal and geometrical methods are
cr
consistent, however, the uncertainty for the geometrical method is one order of magnitude
us
lower than that obtained from the thermal method. The Q-factor was measured with electrical and optical methods for the first and second vibrational modes of the QTFs. These measurements show that the Q-factor in the second mode is larger than in the first mode.
an
Defining the detection sensitivity as the ratio of the amplitude of the measured current through a QTF to the vibration amplitude enables an unambiguous determination of the
M
QTF vibration amplitude in the presence of an interaction. The detection sensitivity was measured directly using heterodyne interferometry combined with piezoelectric excitation and detection. Using this set-up, the electrical characteristics such as the actuation effi-
d
ciency and detection sensitivity can be mapped along the QTF tine for the first and second
te
vibrational modes. Finally, we discuss the relative benefit of operating the QTF force sensor in the first rather than the second vibrational mode due to higher force sensitivity and
Ac ce p
signal-to -noise ratio.
ACKNOWLEDGMENTS
The authors acknowledge help from Heather Catchpoole, Andrew Scott, Chris Freund and Malcolm Lawn.
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G. Binnig, C. F. Quate, Ch. Gerber, Phys. Rev. Lett. 56, 930 (1986).
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F. J. Giessibl, Rev. Mod. Phys. 75, 949 (2003).
4
K. Karrai and R. D. Grober, Appl. Phys. Lett. 66, 14 (1995).
5
Y. Qin and R. Reifenberger, Rev. Sci. Instrum. 78, 063704 (2007). 18
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L.B. Sharos, A Raman, S. Crittenden and R. Reifenberger, Appl. Phys. Lett. 84, 4638 (2004). F. J. Giessibl, Phys. Rev. B. 56 (24), 16010 (1997).
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J. E. Sader and S. P. Jarvis, Appl. Phys. Lett. 84, 1801 (2004).
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ip t
7
A. Castellanos-Gomez, N Agrat and G. Rubio-Bollinger, Ultramicroscopy 111(3), 186
cr
(2011).
A. Torii, M. Sasaki, K. Hane and S. Okuma, Meas. Sci. Technol. 7, 179 (1996).
12
J. E. Sader, J. W. M. Chon, and P. Mulvaney, Rev. Sci. Instrum. 70, 3967 (1999).
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M
Phys. Lett. 91, 173124 (2007). Micro Crystal AG, Type E158.
17
Optical microscope Leica model DM6000.
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te
d
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Ac ce p
19
,Journal of Micro/Nanolithography, MEMS, and MOEMS, 11 (1), 011003 (2012). 20
M. T. L. Hsu, B. Babic, D. A. Shaddock, J. Herrmann and M. B. Gray, unpublished.
21
Spectrum analyzer CXA model N9000A, Agilent Technologies.
22
S. Dohn, R Sandberg, W. Svendsen and A. Boisen, Appl. Phys. Lett. 86, 233501 (2005).
23
K. Karrai and R. D. Grober, Ultramicroscopy 61, 197 (1995).
24
R. C. Tung, T. Wutscher, D. Martinez-Martin, R. G. Reifenberger, F. Giessibl, and A. Raman, J. Appl. Phys. 107, 104508 (2010);.
25
F. J. Giessibl, F. Pielmeier, T. Eguchi, T. An and Y. Hasegawa, Phys. Rev. B 84, 125409 (2011).
26
L. D. Landau and E. M. Lifshitz, Theory of elasticity (Pergamon Press, London, 1959).
27
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28
T. R. Albrecht, P. Grutter, D. Horne and D. Rugar, J. Appl. Phys. 69, 668 (1991).
19
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I. Casuso, Jonathan Khao, M. Chami, P. Paul-Gilloteaux, M. Husain, J. Duneau, H. Stahlberg, J. N. Sturgis, and Simon Scheuring, Nature Nanotechnol. 7, 525 (2012).
te
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an
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ip t
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S0924-4247(14)00550-0 http://dx.doi.org/doi:10.1016/j.sna.2014.12.028 SNA 9011
To appear in:
Sensors and Actuators A
Received date: Revised date: Accepted date:
3-9-2014 11-11-2014 26-12-2014
Please cite this article as: Bakir Babic, Magnus T.L. Hsu, Malcolm B. Gray, Mingzhen Lu, Jan Herrmann, Mechanical and electrical characterization of quartz tuning fork force sensors, Sensors & Actuators: A. Physical (2015), http://dx.doi.org/10.1016/j.sna.2014.12.028 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Journal
Mechanical and electrical characterization of quartz tuning fork force sensors Bakir Babic,1, a) Magnus T. L. Hsu,1 Malcolm B. Gray,1 Mingzhen Lu,2 and Jan Herrmann1 National Measurement Institute, Bradfield Road, Lindfield NSW 2070,
Australia 2)
National Institute of Metrology, Beijing 100013, P.R. China
cr
(Dated: 11 November 2014)
ip t
1)
us
The results of a systematic experimental characterization of the mechanical and electrical properties of quartz tuning forks (QTFs) are presented. Actuation efficiency and detection sensitivity are introduced as parameters for characterization of the
an
QTF with a focus on their use as a force sensor. The spring constants, quality (Q) factors, actuation efficiencies and detection sensitivities of twelve QTFs were mea-
M
sured by combining heterodyne interferometry with an electrical excitation/detection set-up and found to be consistent for all twelve QTFs. Spring constants were determined using geometrical and thermal methods and were found to agree within 3 %.
d
The Q-factor, actuation efficiency and detection sensitivity of the QTFs were mea-
te
sured for the first two vibrational modes. The Q-factor and detection sensitivity were higher in the second vibrational mode while the actuation efficiency was lower. We
Ac ce p
show that the QTF displacement amplitude can be determined from the detection sensitivity and the measured current through the QTF. The performance of a QTF as a force sensor is discussed based on a comparison of the mechanical and electrical parameters measured for the first and second vibrational modes. The results are relevant for QTFs used in sensing applications, especially as force sensors in scanning probe microscopes.
PACS numbers: 68.37, 06.30, 07.07, 07.60 Keywords: AFM, non-contact mode, quartz tuning fork, force sensor, interferometry.
a)
Electronic mail: [email protected]
1
Page 1 of 20
I.
INTRODUCTION The invention of the atomic force microscope (AFM)1 provided a new tool for the field of
surface science and opened the path for rapid progress in nanoscience and the development
ip t
of nanotechnology. Recently, a metrological scanning probe microscope based on an AFM has been used as a primary standard for length measurement at the nanoscale.2 An AFM
cr
measurement contains a wealth of information about a sample, a tip and their interaction. In addition to mapping the surface topography, an AFM can give quantitative information
us
about local forces between the tip and the sample.1,3 To quantify the interaction forces, it is critical to characterize the force sensor.
an
Quartz tuning forks (QTFs) have been successfully used as force sensors in scanning probe microscopy.3,4 An AFM utilizing a QTF force sensor is operated in dynamic mode. A sharp tip is mounted on one of the tines of a QTF which oscillates at its resonant frequency. The
M
interaction between the tip and the sample surface leads to a change in the effective spring constant and consequently in the resonant frequency and oscillation amplitude of the QTF
d
sensor. These changes can be used as feedback signals to adjust the separation between the
te
tip and the sample such that the interaction with the surface is kept constant. There are several advantages for using a QTF as a force sensor compared to other AFM
Ac ce p
sensors. The complexity of the AFM force sensing hardware is significantly reduced compared to the commonly used optical beam-bounce readout since a QTF can be both excited and detected electrically. Quartz oscillators have high frequency stability and low temperature sensitivity at room temperature, and their low heat dissipation makes them attractive as force sensors in low temperature AFMs and room temperature metrological AFM applications.3 In an ambient environment, the quality factor Q of a QTF is typically a few orders of magnitude higher than that of other mechanical resonators, such as the cantilevers commonly used as AFM sensors.5,6 The spring constant of a QTF is much higher than the typical spring constant of a cantilever used in a “beam-bounce” sensing configuration, therefore reducing jump-to-contact problems7 . An illustration of the unprecedented resolution of an AFM based on a QTF sensor operated in the frequency modulation (FM) mode is its ability to image the detailed chemical structure of a single molecule of pentacene.8 The force sensitivity of the QTF force sensor depends on its spring constant. The spring constant k is determined by the geometry of a QTF and by its elastic properties. For small 2
Page 2 of 20
deflections, the tines of a QTF follow Hooke’s law4 F = −kz, where F is the force acting on a tine and z is the corresponding deflection. To determine the interaction forces in this approximation, one has to determine the spring constant of the QTF while measuring the deflection. Depending on the sensor and on the AFM’s mode of operation, knowledge of the
ip t
spring constant is necessary, but not sufficient, to determine the interaction force between the QTF and the sample. For example, if an AFM is operated in FM-AFM mode, both the
cr
frequency shift and amplitude must be measured to determine the force.9
A QTF can be used as a force sensor in an AFM application thanks to the piezoelec-
us
tric properties of quartz. The QTF is electrically driven by an external voltage to induce deformation and therefore mechanical displacement of the tines. To quantify the coupling
an
between the electrical and mechanical properties of a QTF, one has to measure both of them simultaneously and independently. The relationships between the drive voltage V (t) and the deflection A(t) of a QTF on one hand and between the deflection A(t) and the output
M
current I(t) on the other hand are of particular interest for AFM applications. We define the ratio of the oscillation amplitude A to the amplitude of the electrical drive voltage V as
d
the actuation efficiency. The detection sensitivity is defined as the ratio of the amplitude of
te
the current I to the oscillation amplitude A.
When a QTF is used as the force sensor in an AFM, a tip is attached at some location
Ac ce p
to one of the QTF tines. Since the deflection A(x) depends on the coordinate x along the tine, the QTF must be characterized at the position of the tip. As the exact position of the tip may vary, it is necessary to map the mechanical and electrical properties of a QTF force sensor along the tine so that the force for different tip positions can be determined. In this article, we present a systematic quantification of the physical properties of a set of twelve QTF sensors, including their spring constants, quality (Q-)factors, actuation efficiencies and detection sensitivities for the fundamental and second-order vibrational modes using heterodyne interferometry combined with piezoelectric excitation and read-out at ultrasonic frequencies. Experimentally, the anti-phase mode of a QTF is excited and detected at the resonance frequency in a configuration where the QTF is mounted with both tines free to oscillate.10 (This is very different from the widely used qPlus configuration where one of the tines is immobilized by attaching it to a holder and excitation is provided mechanically.3,7 Section II describes the determination of the spring constants, both from interferometric measurements of the displacement of a QTF tine induced by thermal fluctuations and from 3
Page 3 of 20
the tine geometry. In Section III, an electrical model for the QTF is introduced and the results of the electrical characterization of the same QTFs measured in Section II are presented. Optical (interferometric) and electrical methods were used to determine the Q-factors for the first and second vibrational modes. In Section IV, the actuation efficiency and detection
ip t
sensitivity of the QTFs are determined optically and geometrically. The optical method is used to map the actuation efficiency along the QTF tine for the first and second vibrational
cr
modes. The feasibility of determining the QTF displacement amplitude A based on the actuation efficiency and detection sensitivity, respectively, is discussed in Section V. The
us
performance of a QTF force sensor in different vibrational modes is compared. The last
II.
an
section presents conclusions.
THE SPRING CONSTANT
M
There are several methods for the determination of a cantilever spring constant: the geometrical method,4 calibration against a known standard,11 the added mass method,12
the spring constants of QTFs.
d
and the thermal method.13 Here, we use the geometrical and thermal methods to determine
te
A QTF can be modeled as two independent cantilevers: each can be described within continuum mechanics as a clamped beam with a free end. The spring constant kg of a
Ac ce p
cantilever is given by its geometry and elastic properties according to kg = E
wt3 4l3
,
(1)
where E is Young’s modulus of the cantilever material, t is the thickness, w is the width and l is the length of the cantilever.14 The geometrical method of determining the spring constant of a QTF is based on measuring the dimensions of the QTF and using a value of E = (75.2 ± 5.7) GPa for Young’s modulus of quartz.4,15 The geometries of twelve QTFs fabricated on the same wafer16 were measured using optical microscopy17 . An optical micrograph of a typical QTF is given in Fig. 1(A) and a transverse view of a QTF tine in Fig. 1(B). The average dimensions of the set of QTFs measured in this experiment were: t = (214.9±0.3) µm, w = (129.5±0.5) µm and l = (2474± 5) µm. Eq. (1) was used to obtain the spring constants from these measured QTF dimensions with results shown in Fig. 2. The average spring constant value is kg = (1596 ± 14) N/m 4
Page 4 of 20
where the mean is represented in Fig. 2 as the dashed line and the uncertainty value is the experimental standard deviation of the mean. (All uncertainties given in the article are standard deviations unless otherwise stated.)
(C)
x
Vout
FG
t
Vin
le l
(B)
PBS
Laser
y
λ/2
x
MO
λ/4
PBS
λ/2
M
λ/4
PBS
us
PBS
w
QTF
CA
ip t
z
cr
(A)
130.2 µm
Detectors
an
Phasemeter
FIG. 1. (A) Optical microscopy image of a quartz tuning fork used in this experiment. The tine
M
thickness, t, tine length, l and length of the electrode, le , are indicated. (B) Optical micrograph of a QTF tine illustrating measurement of the tine width,w. (C) Schematic of the electrical and
d
optical test bed used for measuring both the deflection of a tuning fork tine using heterodyne
te
laser interferometry and the electrical characteristics. PBS: polarizing beam-splitter, λ/4: quarter wave-plate, λ/2: half wave-plate, M: mirror, CA: charge amplifier, FG: frequency generator, MO:
Ac ce p
microscope objective.
The QTF spring constants were also measured using the thermal method. A vibrating cantilever can be described as a one-degree-of-freedom harmonic oscillator with a resonant frequency ω0 = (k/m)1/2 , where m is the oscillating mass. The QTF with two tines free to oscillate can be described as a system of two weakly coupled harmonic oscillators with two degrees of freedom. In thermal equilibrium (with its surroundings), the QTF tines will vibrate at its resonant frequency due to thermal excitation, with a deflection amplitude, A(t). In this situation, the thermal energy is equivalent to the average mechanical energy of the QTF oscillator: kB T = khA2 i, where kB is the Boltzmann constant, T is the temperature and h...i denotes the average. The thermally excited vibration amplitude was measured using heterodyne interferometry in the Michelson configuration as shown in Fig. 1(C). The all-digital phasemeter used to measure the optical phase difference was implemented on a field-programmable gate array platform and can measure displacements with picometer res5
Page 5 of 20
olution, large dynamic range and 400 kHz bandwidth.18 The reference and QTF reflection signals were detected using high-speed photodetectors, and the electronic signals were subsequently digitized on high-speed analog-to-digital converters (ADCs). Each ADC signal was then tracked and measured by a phase locked loop. Finally, the signal was filtered and
ip t
the output signal was integrated to provide the phase of the input signal. The difference between the reference and input signal phases yields the phase signal corresponding to the
cr
deflection of the QTF tine. The schematic of the optical measurement system together with the piezoelectric excitation and detection set-up are shown in Fig. 1(C).
us
The measurement of the thermal vibration amplitude, A, of a QTF tine was performed in the following way: The QTF was first fixed on a micromanipulation stage. A HeNe laser
an
(wavelength λ = 633 nm; output power ≈ 100 µW) was focused to a beam diameter of 2 µm with a 40× microscope objective (NA=0.66) and directed onto one of the x-y faces of the QTF tine (Fig. 1(B)) that was visually aligned to be perpendicular to the laser beam. To
M
minimise potential heating of the tine due to laser irradiation, the duration of a measurement during which the tine was illuminated was limited to 1 s. The tine position along its width
te
to the tine width.
d
(y-direction in Fig. 1(B)) was adjusted such that the laser spot was centered with respect
Since the vibration amplitude A(x) depends strongly on the position x along the QTF
Ac ce p
tine (x-direction in Figs. 1(A) and (B)), the QTF thermal amplitude was measured near the end of the tine, where the amplitude for the first two modes is the largest.14 To make results for different QTFs comparable, the end of the tine was determined by shifting the tine along the x-direction until the laser beam seen on a background screen was partially obstructed. The tine was then moved back into the beam path by the fixed amount of 50 µm. This distance was determined experimentally as being closest to the edge of the tine while still giving sufficient signal for reproducible measurements. The resulting beam position was also found to correspond to the typical AFM tip location, following the tip attachment procedure previously reported.19 Two experimental indicators were used to ensure that the alignment (orthogonality) of the tine with respect to the laser beam was optimized and that the measurements were made close to the end of the tine (within 50 µm). The first indicator was that movement of the laser spot along the tine (a few mm in the x-direction) did not lead to a significant change in the intensity of the optical signal. The second was derived from observation of the QTF’s thermal vibration. At the very edge of the tine, the vibration 6
Page 6 of 20
amplitude of the thermal oscillation is close to the limit of detection for the interferometric system and any significant misalignment would lead to a loss of the signal. Using these indicators, we estimate that deviations from perpendicularity between the laser spot and the tine surface are less than 5 degrees, corresponding to a cosine error of < 3 % in the tine
ip t
amplitude. As measurements of the thermal vibration amplitude showed variations of the
cr
order of 25 %, the contribution of the alignment error to this variation is small. 1.6 1.4
kth kg
A (pm)
1.2
us
1.0
0.8
an
0.6
32.2
32.4
32.6
32.8
33.0
f (kHz)
Ac ce p
te
d
M
k
32.0
QTF number
FIG. 2. Experimentally measured spring constants for twelve QTFs determined using geometrical (open) and thermal methods (solid), respectively. The dashed (solid) line indicates the average spring constant kg (kth )determined with the geometrical (thermal) method. Inset: Frequency spectrum of thermally excited vibrations near the first QTF vibrational mode, measured by heterodyne interferometry.
All measurements were performed at room temperature T = (293.8 ± 0.3) K. Based on the spring constant determined from the geometrical model, we expect an average thermal vibration amplitude of A = (1.6 ± 0.2) pm for the QTFs. The average thermal vibration amplitude measured directly with interferometry was found to be A = (1.5±0.4) pm, with an example shown in the inset of Fig. 2. These two values are in agreement and give confidence 7
Page 7 of 20
for the use of this method to undertake further quantitative characterization of QTF force sensors. From the thermal vibration amplitude of a QTF and the ambient temperature we can obtain the QTF spring constant. The corresponding spring constants for the set of QTFs
ip t
together with their uncertainties are shown as solid symbols in Fig. 2. The average spring constant obtained using the thermal method is kth = (1560±210) N/m. This mean value is in
cr
good agreement with the average spring constant kg obtained from the geometry of the QTFs, although the standard deviation for the thermal method is an order of magnitude larger. This
us
is mainly due to the relatively low signal to noise ratio of the interferometric measurement of the tine amplitude in the picometre range.20 An additional contribution comes from the
an
uncertainty of the positioning of the laser spot along the length (x-direction) of the QTF tine since the oscillation amplitude depends on that position (see Fig. 6(B)). Nevertheless, the results indicate that QTFs produced on the same wafer have consistent spring constants
M
and that the geometrical method is suitable to determine the mean spring constant of the QTFs to within 3% of the mean spring constant determined by the thermal method. This is
d
remarkable given that the geometrical method is based on simple assumptions and neglects
ELECTRICAL CHARACTERIZATION AND QUALITY FACTOR
Ac ce p
III.
te
several issues, such as the variation in material properties and structural defects.
In non-contact AFM applications employing a QTF force sensors, it is common practice to excite the QTF mechanically by vibrating it using a piezoelectric actuator attached to the QTF and to read out the QTF signal electrically. It is also possible to excite the QTF electrically. This has the advantage that the force sensor assembly is less complex, resulting in fewer parasitic heat and vibrational sources, and provides more comparable sensors since their characteristics are less dependent on the mounting procedures. A QTF can be modeled by an equivalent electrical circuit. The circuit can be described by the LRC resonator model and includes a parallel capacitance which is due to the contacts and leads.5 The schematic of a series LRC circuit is given in the inset of Fig. 3. The frequency-dependent impedance is given by Z(ω) =
iωCp +
1 RTF + iωLTF + 1/iωCTF 8
−1
,
(2) Page 8 of 20
where Cp is the parasitic capacitance, RTF is the resistance, LTF the inductance and CTF the capacitance of the QTF; ω = 2πf where f is the frequency. The electrical response, ITF , i.e. the current flowing through the QTF tine, was measured using a spectrum analyzer.21 A sinusoidal drive voltage with amplitude Vin was applied across
ip t
the QTF. A typical frequency response of a QTF driven at Vin = 10 mV is shown in Fig. 3. The asymmetry in the resonance curve is due to the current flowing through Cp . The
cr
model Z(ω) of equation (2) was adjusted to the measured frequency response yielding the solid line in Fig. 3, with the following parameters determined from the fit : Cp = 4.64 pF,
Vin
M te
Ac ce p 0 32 600
ITF
d
2.10 -8
. -8 110
LTF
Cp
ITF (A)
3.10-8
CTF
an
RTF
4 .10-8
us
CTF = 3.99 fF, LTF = 5945 H, RTF = 244 kΩ, ω0 /(2π) = f0 = 32678.9 Hz, Q = 4990.
32 620
32 640
32 660
32 680
32 700
32 720
32 740
f (Hz)
FIG. 3. Amplitude of the current ITF flowing through a QTF driven with a sinusoidal voltage with amplitude Vin = 10 mV as a function of drive frequency f . Inset: Schematic of the equivalent circuit for a piezoelectric resonator.
The Q-factor describes the energy losses in a resonator and is defined as Q = ω0 /δω, where ω0 is the resonance frequency and δω the width of the resonance at half-maximum. A high Q is desirable for the force sensor in an FM-AFM application, since the magnitude of the detectable force is inversely proportional to Q. A higher Q offers higher selectivity and stability of oscillations at the resonant frequency. At room temperature, a typical QTF has a Q-factor that is few orders of magnitude higher than that of other cantilever 9
Page 9 of 20
beams used in AFM5,14 . The QTF Q-factor can be measured both electrically and optically. The electrical measurement of a QTF frequency response gives Qel using the relationship Qel = (ω0 CTF RTF )−1 . In Fig. 4(A), measured Qel values of twelve QTFs for the first and second vibrational modes are shown as circular symbols. For the first mode, we find an
ip t
average value of Q1el = 4700 ± 800, and for the second mode Q2el = 6500 ± 2400. The first
cr
mode has both a lower mean Qel value and a smaller standard deviation.
For the same QTFs, the Q-factor was also measured optically. The interferometric set-up
us
in Fig. 1(C) was used to measure the frequency response of the deflection for the first two vibrational modes of the QTFs. A typical measured frequency spectrum for the first and
an
second vibrational modes is shown in Figures 4(B) and 4(C), respectively. The lines are Lorentzian fits from which the Qop values can be extracted. The average Qop value obtained for the first vibrational mode was Q1op = 3900 ± 500, which is smaller than that for the
M
second mode, Q2op = 4600 ± 700, with a similar standard deviation.
d
The mean Q-factors obtained from both methods are of the order of several thousands
te
and are in agreement with previous reports.5,14 Although the values for both methods agree within their statistical uncertainties, the electrical Q-factors for both vibrational modes are
Ac ce p
higher than the optical Q-factors. In particular, the variability of Q2el is higher than Q2op , the probable reason being that the QTF electrodes are configured highly symmetrically to have a high Q-factor when excited in the first vibrational mode. Due to the electrode design, the operation of the QTF in the second vibrational mode has lower signal to noise ratio and hence the determination of Q2el is less reliable. The experiment also shows that Q1op,el is smaller than Q2op,el in agreement with the literature.22 If one assumes that δωn does not depend on the vibrational mode,23 Qn = ωn /δωn = Q1βn2 /β12, where n is the mode number and βn satisfies cosh(βn )cos(βn )+1=0. In this case, the second vibrational mode is expected to have a higher Q than the fundamental one; Q2 = 6.3 Q1. This is not in agreement with the measured values, hence the assumption that δωn is independent of the vibrational mode is not valid. The modest difference between Q1 and Q2 might be explained by the resonator having a smaller center of mass movement for higher mechanical modes.6 Other damping mechanisms, such as asymmetries, imbalances and acoustic radiation losses, can reduce the Q-factor; however, for a QTF these are higher order contributions. 10
Page 10 of 20
(A)
an
us
cr
ip t
Q1op Q2op Q1el Q2el
M
QTF number
(B) 4.0
(C)
1.6
te
A (nm)
3.0 2.5 2.0
Ac ce p
1.5
1.4
1.0 0.8 0.6
1.0
0.4
0.5
0.2
32.50
32.51
32.52
32.53
Q2op ~5202
1.2
A (nm)
Q1op ~4065
d
3.5
32.54
f (kHz)
196.78
196.82
196.86
196.90
196.94
f (kHz)
FIG. 4. (A) The Q factors of QTFs measured electrically (circles) and optically (squares) for the first (open) and second (solid) modes. Frequency response of a QTF measured optically for the first (B) and second (C) vibrational modes (symbols). The solid lines are Lorentzian fits from which the value of Qop can be extracted.
11
Page 11 of 20
IV.
ACTUATION EFFICIENCY VERSUS DETECTION SENSITIVITY
To control and exploit the characteristics of a QTF for specific applications, it is necessary to determine the relationship between its mechanical and electrical properties. The
ip t
spring constant and displacement amplitude depend on the position, x, along the tine length. When a tip is positioned at a specific location, it is necessary to determine the QTF charac-
cr
teristics at that location so that the force between the sample and the tip can be quantified. Therefore, spatial mapping of the QTF displacement amplitude is necessary.
us
Heterodyne interferometry was used to measure the absolute value of the vibration amplitude along the entire length of a QTF. The results shown in Fig. 5 for the first two vibrational modes of one of the twelve QTFs. The vibrational mode shapes are as expected
an
for symmetric coupled beams.14,24 The spatial mapping enables experimental determination of the nodes for the second and higher modes. For the second mode of the QTF shown in
M
Fig. 5, there is a node at x ∼ 0.6 mm where the amplitude drops to zero.
First mode
d
30
Second mode
te
A(nm)
Ac ce p
20
0.5
0.4
15
0.3
10
A(nm)
25
0.6
0.2
5
0.1
0.0
0 0
0.5
1
1.5
2.0
x (mm)
FIG. 5. Mapping of the absolute value of vibration amplitude for the first (open) and second (solid) vibrational modes along the length of the QTF tine. The amplitude of the sinusoidal drive voltage was Vin = 10 mV. The position x = 0 corresponds to the free end of the tine.
12
Page 12 of 20
QTFs can be used both as piezoelectric actuators and as sensors. The actuation efficiency of a QTF, which in previous work has been named sensitivity,4,14 is defined as the ratio between the amplitudes of the tine vibration and of the external drive voltage, at the resonant frequency, γ = A/Vin . The actuation efficiency is related to the piezoelectric coefficient of
ip t
quartz, which is an intrinsic parameter determined by the geometry and crystal orientation of the quartz bar. It can be derived for the first vibrational mode by assuming that a surface
cr
charge density is generated on each electrode when the end of tine is deflected by a distance A. Taking a time derivative and expressing the spring constant via Young’s modulus and
us
the QTF geometry (Eq. (1)), the actuation efficiency near the end of a QTF tine is obtained using the following expression:
an
wtle −1 , γ = 6d21 Eω0 Z 3 l − le /2 l
(3)
where d21 = 2.31 · 10−12 C/N is the relevant piezoelectric coefficient of quartz, Z is the QTF
M
impedance and le is the electrode length (see Fig. 1(A)).25
The actuation efficiency can be determined experimentally by simultaneously measur-
d
ing the vibration amplitude of the QTF tine while driving it electrically with a sinusoidal
te
voltage. The heterodyne interferometry system shown in Fig. 1(C) was used to perform these measurements. The actuation efficiency is determined by a linear fit of the vibration
Ac ce p
amplitude for small drive voltages, shown in Fig. 6(A) as a solid line. We have measured the actuation efficiency near the end of the tine of twelve QTFs, for which the spring constant and the Q-factor were previously measured (Sections II and III) . The results are shown in Fig. 6(B) as open squares. For the first vibrational mode, an average actuation efficiency of γ1op = (3.3 ± 0.5) µm/V was measured for the twelve QTFs. Using Eq. 3, the actuation efficiency can also be determined from the geometry of the QTFs and from their piezoelectric and mechanical properties. These data are plotted as open circles in Fig. 6(A). The average actuation efficiency value obtained from the geometry of the twelve QTFs was γ1geo = (4.0 ± 0.3) µm/V which is in reasonable agreement with the interferometric measurements. Taking into account that the amplitude has been measured interferometrically slightly inward from the end of the tine (∼ 50 µm), this would lower the values found for γ1geo by approximately 4 %, bringing the actuation efficiencies obtained by the two different methods into better agreement. The actuation efficiency of the second vibrational mode has also been measured interfer13
Page 13 of 20
ometrically near the end of the QTF tine, with the results shown in Fig. 6(B) as full squares. The average actuation efficiency was γ2op = (0.10 ± 0.06) µm/V. A lower value is expected for the second mode of a cantilever since the strain is in opposite directions at different points on the tine. Furthermore, the actuation efficiency is inversely proportional to the
ip t
spring constant and the resonant frequency (see Eq. (1) and Eq. (3)), and both these values are roughly an order of magnitude larger for the second mode than for the first mode.26,27
cr
Determination of the vibration amplitude using the actuation efficiency is only meaningful when there is no dissipative interaction of the QTF with its environment, which is not the
us
case in AFM applications. In an AFM, a tip attached to the force sensor interacts with a surface, reducing the oscillation amplitude of the sensor compared with free-air oscillation
an
(for a fixed electrical drive). The supplied drive energy is converted into the mechanical motion of the QTF tines, dissipative losses within the QTF and dissipation due to the interaction with the surface. In this case, deducing the vibration amplitude from A =
M
γVin would be incorrect. However, the detection sensitivity, S, defined as the ratio of the amplitude of the current, through the QTF ITF , and the vibration amplitude A, i.e., S =
d
ITF /A is a more universal quantity. The detection sensitivity and actuation efficiency both
te
depend on the position along the QTF tine due to the position dependence of the vibration amplitude illustrated in Fig. 5. For AFM experiments, their values must be determined
Ac ce p
at the tip position, which in our experiment has been near the end of the QTF tine. The detection sensitivity, Sop , was measured interferometrically at the resonance frequency for the first and second vibrational modes, since the FM-AFM mode of operation requires a QTF to be driven at resonance. Alternatively, it can be calculated using an expression similar to that for the actuation efficiency in Eq. (3), with the difference that the detection sensitivity obtained in this way, Sgeo , is independent of the impedance. The sensitivity for the first vibrational mode is given by:
S1geo =
1 γZ
= 6d21 Eω0
wtle l3
l − le /2 .
(4)
Once the detection sensitivity is determined by either the geometrical or optical method, the vibration amplitude can be determined by simply measuring the amplitude of the current through the QTF. For the set of twelve QTFs, the average detection sensitivity measured by interferometry for the first vibrational mode was S1op = (1.3 ± 0.1) A/m. Using Eq. (4), the 14
Page 14 of 20
(A) 200
ip t
100
cr
A (nm)
150
0 10
20
30
(B)
50
60
70
Vin (mV)
M
5
4
te
d
3
2
γ1op γ1geo γ2op
Ac ce p
γ1, γ2 (µm/V)
40
an
0
us
50
1
0
1
3
5
7
QTF number
9
11
FIG. 6. (A) QTF vibration amplitude versus excitation voltage amplitude at resonance, measured interferometically near the end of the QTF tine (symbols). The line is a linear fit to the measurement data. Its slope gives an actuation efficiency γ1op = (2.87 ± 0.01) µm/V. (B) Actuation efficiency inferred from geometry, γ1geo (open circles) and measured by interferometry, γ1op (open squares) for the first vibrational mode. Also shown is the actuation efficiency for the second vibrational mode measured by interferometry, γ2op (solid squares).
average detection sensitivity determined from the measured geometry was S1geo = (1.1 ± 15
Page 15 of 20
0.1) A/m. The values obtained for the individual QTFs are plotted in Fig. 7 (open symbols, left axis). This value is slight under-estimation due to the fact that the detection sensitivity has not been measured at the very end of the QTF tine. Nevertheless, the results found for
ip t
detection sensitivities with the two measurement methods are in good agreement. The detection sensitivity was also measured interferometrically for the second vibrational mode near the end of the QTF tine. The average value obtained for the second vibrational
cr
mode was S2op = (15.5 ± 4.0) A/m, and the values for the individual QTFs are shown in Fig. 7 (solid symbols, right axis). The detection sensitivities can be used to determine the
us
vibration amplitude of a QTF tine in the corresponding vibrational mode, regardless of the
2.0
M
an
presence or absence of an interaction.
S1op S1geo
30
S2op
25
Ac ce p
20
S2 (A/m)
1.4
d
1.6
te
S1 (A/m)
1.8
15
1.2
1.0
10
0.8
QTF number
FIG. 7. Detection sensitivity for the set of QTFs inferred from the geometry (circles) and measured using interferometry (squares) for the first (open symbols, left axis) and second (solid symbols, right axis) vibrational modes.
16
Page 16 of 20
V.
DISCUSSION
ip t
All reported measurements were performed at room temperature and ambient atmospheric pressure using twelve QTFs fabricated on the same wafer and without a tip attached.
cr
The values obtained for the spring constant, detection sensitivity and Q-factor for some of the QTFs with and without a tip have been compared. The tip attached in our experiment
us
is significantly smaller in size and has a mass that is at least an order of magnitude smaller than that of a tip typically used in qPlus configuration (an etched piece of tungsten wire).
an
The details about the tip and the attachment method applied are described elsewhere.19 After attachment of a tip, the QTF spring constant was found to remain unchanged within the experimental uncertainty; and both the resonant frequency and Q-factor were reduced by
M
less than 1 %. At the resonant frequency, the actuation efficiency and detection sensitivity were found to be unaffected by the attachment of a small mass such as an AFM tip, which
te
d
is consistent with equations (3) and (4).
Ac ce p
Characterizing the mechanical and electrical properties of a QTF provides a quantitative means of comparing the performance when operated in different vibrational modes. In Section III, it was shown that the Q−factor in the second mode can be higher than in the first mode, resulting in lower thermal noise28 and reducing mechanical coupling to parasitic vibrations. Furthermore, operating the QTF sensor in the second mode offers the potential for larger bandwidth operation and ultra-fast scanning.29 However, force sensitivity (defined as the reciprocal of the spring constant, 1/kn = −δz/δFn where n is the mode number), is higher in the first mode, due to its smaller spring constant, than in the second mode.27 The signal-to-noise ratio of the force detector is therefore higher when the QTF is operated in the first vibrational mode than in the second one. This is especially important if the force sensor is operated in the attractive, van der Waals force region, where sub-nanometer vibration amplitudes are necessary for high (atomic) resolution.3,30 Although there are some benefits to operate the QTF in the second vibrational mode, the higher force sensitivity and signal-to-noise ratio favor operation in the first vibrational mode. 17
Page 17 of 20
VI.
CONCLUSION
We have quantitatively characterized the mechanical and electrical properties of a set of QTFs by using heterodyne interferometry combined with piezoelectric excitation and
ip t
detection. The spring constant of twelve QTFs from the same wafer has been systematically measured using thermal and geometrical methods. The thermal method is direct, non-invasive and time-efficient. The results from the thermal and geometrical methods are
cr
consistent, however, the uncertainty for the geometrical method is one order of magnitude
us
lower than that obtained from the thermal method. The Q-factor was measured with electrical and optical methods for the first and second vibrational modes of the QTFs. These measurements show that the Q-factor in the second mode is larger than in the first mode.
an
Defining the detection sensitivity as the ratio of the amplitude of the measured current through a QTF to the vibration amplitude enables an unambiguous determination of the
M
QTF vibration amplitude in the presence of an interaction. The detection sensitivity was measured directly using heterodyne interferometry combined with piezoelectric excitation and detection. Using this set-up, the electrical characteristics such as the actuation effi-
d
ciency and detection sensitivity can be mapped along the QTF tine for the first and second
te
vibrational modes. Finally, we discuss the relative benefit of operating the QTF force sensor in the first rather than the second vibrational mode due to higher force sensitivity and
Ac ce p
signal-to -noise ratio.
ACKNOWLEDGMENTS
The authors acknowledge help from Heather Catchpoole, Andrew Scott, Chris Freund and Malcolm Lawn.
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