Performance test of cantilever transfer standard used for AFM tip calibration based on electromagnetic compensation


ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 129 

ACTA IMEKO 
ISSN: 2221-870X 
December 2020, Volume 9, Number 5, 129 - 132 

 
 

PERFORMANCE TEST OF CANTILEVER TRANSFER STANDARD USED 

FOR AFM TIP CALIBRATION BASED ON ELECTROMAGNETIC 

COMPENSATION 
 

J. Jiang1, H. Zhou2, Y. Zhang3, S. Wu4, Z. Zhang5, G. Hu6 

 
1, 3, 4, 5, 6 Laboratory of Force and Torque, Division of Mechanics and Acoustics, National Institute of Metrology, China, 

1 jiangjl@nim.ac.cn, 3 zhangye@nim.ac.cn, 4 wushi@nim.ac.cn, 5 zhiminzhang@nim.ac.cn, 6 hugang@nim.ac.cn  
2 Laboratory of Photoelectric Detection and Image Processing, School of Precision Instrument and Opto-electronic 

Engineering, Tianjin University, China, huiyan9519@tju.edu.cn  

 

Abstract: 

This paper describes stiffness measurement of 

cantilever transfer standards used for Atomic Force 

Microscopy (AFM) tip calibration based on 

electromagnetic compensation. The transfer 

standard of cantilever is designed and manufactured 

based on the bulk fabrication of SOI wafers. The 

measure range of the transfer standard covers from 

0.04 N/m to 16 N/m. The series of transfer standard 

is designed for the calibration test of the cantilever 

used in AFM, along with the test apparatus 

specifically designed. The relative uncertainty of 

the stiffness is smaller than 2.4 % (k = 2). 

Keywords: cantilever; stiffness; small force; 

electromagnetic compensation 

1. INTRODUCTION 

Atomic Force Microscopy (AFM) can be utilised 

to observe the matters under very small dimension 

with high sense of the force below 10-6 N [1-2]. The 

precise measurement depends on the spring 

constant of the cantilever and the displacement of 

the tip measured by positioning detector. In order to 

obtain the spring constant of the AFM tip, high-

accuracy measurements of SI traceable small forces 

below millinewton have been realized by several 

NMIs [3-11]. Besides the small force standard, the 

standard cantilever also needs to be designed and 

manufactured so as to realize the stiffness 

measurement of AFM tip without being measured 

directly on the small force standard. 

In this paper the design and the measurement of 

series of the cantilever transfer standard were 

presented with the small force standard based on 

electromagnetic compensation. The transfer 

standard was designed with a series of lengths for 

covering a wide range of stiffness and manufactured 

based on the bulk fabrication of SOI wafers. The 

uncertainty analysis of the stiffness measurement 

was also conducted. The stiffness of the transfer 

standard can be obtained with relative extended 

uncertainty of 2.4 % (k = 2). Along with the design 

of the test apparatus, the series of transfer standard 

can be used for the AFM tip calibration by pushing 

the cantilever to the transfer standard. 

2. DESCRIPTION OF THE TEST AND THE 
RESULTS 

The measurement of the series of the transfer 

standard was carried out on the measuring system 

based on the electromagnetic compensation balance. 

The cantilever tip can be precisely loaded to the 

weighing part, driven by a piezoelectric moving 

stage with a distance of 100 μm, along with the 

observation of the contact of tip and the loading part 

in different directions. Mounting to the clamp 

specially used in AFM, the tip can be loaded very 

similar to the working condition in the AFM. The 

structure of the measuring system is shown in 

Figure 1. 

 

Figure 1: The structure of the measuring system  

In Figure 2, instead of using the electromagnetic 

compensation balance, the transfer standard 

cantilever is applied to the measuring system. It is 

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mailto:jiangjl@nim.ac.cn
mailto:zhangye@nim.ac.cn
mailto:wushi@nim.ac.cn
mailto:zhiminzhang@nim.ac.cn
mailto:hugang@nim.ac.cn
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ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 130 

calibrated before assembly. This structure is still 

under construction. So with the help of this 

measuring system the transfer standard cantilever 

will be able to be utilised to calibrate the AFM tips 

without the complicated balance system. 

 

Figure 2: Design of the transfer standard test system 

The transfer standard of cantilever is designed 

and manufactured based on the MEMS fabrication 

of SOI wafers. The effective lengths of the transfer 

standard are L0 = 8 mm, 7 mm, 6 mm, 5 mm, 4 mm, 

3 mm, 2 mm, and 1 mm. The width of the transfer 

standard is w = 500 μm. The nominal thickness is 

t = 0.01 mm. This parameter is greatly influenced 

by the SOI wafer selected before the manufacture. 

The nominal stiffness of the transfer standard is 

(0.041 ~ 21.125) N/m. In the test all the cantilevers 

were loaded in a cycle with 11 steps of force, 

repeated at least 10 times. In each step the loading 

was maintained for 30 s to obtain the stable force. 

The temperature range is 21 ℃ to 22 ℃. 

 

 

 

Figure 3: Design of the transfer standard and loading 

zone of transfer standard on the head of the tip 

The loading part is assembled with conical 

diamond probe tip used in nano-indentation with a 

tip diameter of 1 μm and the angle of 60º. This 

structure is for the test of the transfer standard. The 

typical test procedure is shown in Figure 4. 

 

Figure 4: The test process of one loading cycle 

Comparing to the theoretical design of the 

transfer standard cantilever, the actual stiffness is 

lower. According to the comparison shown in 

Figure 5, 10 % - 20 % loss of the stiffness can be 

observed during the tests. Especially at the low 

range of the stiffness, the difference between the 

designed and the actual stiffness is more obvious, 

causing an exponential factor smaller than 1 in 

Figure 5 (bottom). 

 

 

Figure 5: Difference between designed and the measured 

value of the stiffness (Top: Fitted analysis of the stiffness, 

Bottom: Deviation from the designed stiffness) 

3. UNCERTAINTY ANALYSIS 

The uncertainty analysis for the different type of 

the cantilevers was presented in references [12] and 

0.5mm

0.01 mm

0 30 60 90 120 150 180 210 240 270 300 330
20

21

22

23

24

25

26

27

28

29

30

D
e
fo

rm
a
ti

o
n

 (


m
)

Time （s）

 

 

Single loop of loading cycle

0 30 60 90 120 150 180 210 240 270 300 330
0

10

20

30

40

50

60

70

80

90

100

F
o

rc
e
 (


N
)

Deformation

Force

y = 16.77x-2.969

R² = 0.9991

y = 21.125x-3

R² = 1

0.01

0.1

1

10

100

0 1 2 3 4 5 6 7 8 9

S
ti

ff
n
e
ss
（

N
/m

）

Nominal Length L0 （mm）

Test results

Design

Fitting-Test results

Fitting -Design

y = 0.8193x0.9894

R² = 0.9991

0.01

0.1

1

10

100

0.01 0.1 1 10 100

A
c
tu

a
l 

st
if

fn
e
ss
（

N
/m

）

Designed stiffness  (N/m)

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ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 131 

[13]. Here the uncertainty of the stiffness 

measurement for the transfer standard due to the 

repeatability 𝑢1  is estimated by the standard 
deviation of the average value of measurement 

results in 10 cycles. The worst condition for the 

transfer standard is 0.97 × 10-2 throughout the 

whole test. The other aspect regarding to uncertainty 

are originated from force and displacement 

measurements, non-linearity of calibration curves, 

orientation of the cantilevers, location accuracy of 

the measurement point and the stiffness of the 

measurement chain. The uncertainty due to the force 

measurement is estimated in [12]. The uncertainty 

originated from force 𝑢2  is attributed to the 
uncertainty at the lower limit of the force.  

The displacements of the nano-positioning stage 

were measured by a laser interferometer over 

100 μm with 20 steps of 5 μm. The standard 

uncertainty of the measurement results was 68.2 nm. 

The relative uncertainty due to the displacement 

measurement overall the whole range within the 

measurement, 𝑢3, is determined as 0.68 × 10
-4.  

Derived from linear fit of force and 

corresponding displacement data, the residual of the 

linear fit is another source of the uncertainty, 𝑢4 , 
which is no more than 0.5 × 10-2. 

The stiffness of the cantilever is proportional to 

Young’s modulus and the factors of cantilever 

dimensions. Therefore, the location of the contact 

point between the cantilever and the tip of the 

weighing pan should be considered. For the pin 

shown in Figure 3, the radius of the contact area is 

3.3 µm. Assuming that k = 2.58 (Gaussian 

distribution), the relative standard uncertainty due 

to location accuracy 𝑢5 can be given by: 

𝑢5 =
1

2.58

𝛿𝐿

𝐿
 (1) 

where 𝛿𝐿 is the maximum position deviation from 
the ideal location. So 𝑢5  is calculated to be 
1.28 × 10-3, where 𝛿𝐿 = 3.3 μm, 𝐿 = 1000 μm. 

The stiffness of the measurement chain 𝑘b was 
determined to be 1099.1 N/m [12]. Assuming a 

rectangular distribution, the relative standard 

uncertainty due to the stiffness of measurement 

chain 𝑢6 is given by: 

𝑢6 =
1

√3
|

𝑘m
𝑘b − 𝑘m

|
𝛿𝑘b
𝑘b

 (2) 

where 𝛿𝑘b = 120 N/m is the maximum deviation of 
the stiffness of the measurement chain 𝑘b, and 𝑘m 
is the stiffness of the cantilever in each 

measurement. 

In the process of clamping the cantilever, we 

should also ensure that the balance probe and the 

central axis of the cantilever are in exact contact. 

Thus, the torque of the cantilever is 0 and the error 

caused by the torsion of the cantilever is eliminated. 

However, in the actual operation process, it is 

difficult to ensure that the probe is just at the 

position of the central axis. It is known that the 

diameter of the probe used is 20 μm, the Young’s 

modulus is 1140 GPa, and the Poisson’s ratio is 0.07. 

The cantilever Young’s modulus 𝐸  is about 
150 GPa, its Poisson’s ratio 𝜇 is about 0.22, and its 
shear modulus 𝐺 is given by: 

𝐺 =
𝐸

2(1 + 𝜇)
 (3) 

When the probe just aligns with the image of the 

cross centre of the cantilever, the probe is now on 

the cantilever boundary line, from which it can be 

seen that its displacement from the central axis of 

the cantilever is about 240 μm. According to the 

knowledge of material mechanics, its torsion angle 

𝜙 is given by: 

𝜙 =
12𝑇𝑙

𝐺𝑤𝑡3
 (4) 

The torque 𝑇 = 𝐹𝑤/2, where 𝐹 is the force on 
the cantilever under the current position. 

Considering the maximum force 𝐹 of 200 μN in the 
test of the cantilever with lowest stiffness, along 

with the length of the cross in the loading area of the 

cantilever, the width 𝑤 cannot be larger than 15 μm, 
deflection angle 𝜙 = 0.032 rad, so the maximum 
uncertainty caused by torque angle is 0.13 × 10-3. 

The influence can be omitted. 

The combined standard uncertainty 𝑢kc  is 
determined by: 

𝑢kc = √∑ 𝑢𝑖
2

6

𝑖=1

2

 (5) 

The relative standard uncertainty is less than 

1.2 %. 

4. CONCLUSION 

The cantilever transfer standard was fabricated 

based on bulk process and its stiffness has been 

measured based on small force standard based on 

electromagnetic compensation. The relative 

uncertainty of the stiffness is smaller than 2.4 % 

(k = 2). 

5. ACKNOWLEDGEMENT 

This research is sponsored by National Key 

R&D Program of China No.2018YFF0212401. 

6. REFERENCES 

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ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 132 

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