ISO 11775:2015

INTERNATIONAL ISO
STANDARD 11775
First edition
2015-10-01
Surface chemical analysis — Scanning-
probe microscopy — Determination of
cantilever normal spring constants
Analyse chimique des surfaces — Microscopie à sonde à balayage —
Détermination de constantes normales en porte-à-faux de ressort
Reference number
©
ISO 2015
© ISO 2015, Published in Switzerland
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ii © ISO 2015 – All rights reserved

Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 Symbols and abbreviated terms . 2
5 General information . 4
5.1 Background information . 4
5.2 Methods for the determination of AFM normal spring constant . 5
6 Dimensional methods to determine k . 5
z
6.1 General . 5
6.2 k using formulae requiring 3D geometric information . 5
z
6.2.1 Method . 5
6.2.2 Measuring the required dimensions and material properties of the cantilever . 7
6.2.3 Determining k for the rectangular cantilever . 8
z
6.2.4 Determining k for the V-shaped cantilever . 8
z
6.2.5 k for the trapezoidal cross-sections . 9
z
6.2.6 k to account for coatings . 9
z
6.3 k using plan view dimensions and resonant frequency for rectangular
z
tipless cantilevers .10
6.3.1 Determining k .10
z
6.3.2 Uncertainty .11
7 Static experimental methods to determine k .11
z
7.1 General .11
7.2 Static experimental method with a reference cantilever .11
7.2.1 Set-up .11
7.2.2 Determining k .
z 12
7.2.3 Uncertainty .14
7.3 Static experimental method using a nanoindenter .15
7.3.1 General.15
7.3.2 Determining k for a tipped or tipless cantilever .15
z
7.3.3 Uncertainty .16
7.4 Measurement methods .18
7.4.1 Static deflection calibration.18
8 Dynamic experimental methods to determine k .18
z
8.1 General .18
8.2 Dynamic experimental method using thermal vibrations using AFM .18
8.2.1 Determining k .18
z
8.2.2 Uncertainty .20
Annex A (informative) Inter-laboratory and intra-laboratory comparison of AFM cantilevers .21
Bibliography .24
Foreword
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bodies (ISO member bodies). The work of preparing International Standards is normally carried out
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different types of ISO documents should be noted. This document was drafted in accordance with the
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The committee responsible for this document is ISO/TC 201, Surface chemical analysis, Subcommittee
SC 9, Scanning probe microscopy.
iv © ISO 2015 – All rights reserved

Introduction
Atomic force microscopy (AFM) is a mode of scanning probe microscopy (SPM) used to image surfaces
by mechanically scanning a probe over the surface in which the deflection of a sharp tip sensing the
surface forces mounted on a compliant cantilever is monitored. It can provide amongst other data,
topographic, mechanical, chemical, and electro-magnetic information about a surface depending
on the mode of operation and the property of the tip. Accurate force measurements are needed for
a wide variety of applications, from measuring the unbinding force of protein and other molecules
to determining the elastic modulus of materials, such as organics and polymers at surfaces. For such
force measurements, the value of the AFM cantilever normal spring constant, k , is required. The
z
manufacturers’ nominal values of k have been found to be up to a factor of three in error, therefore
z
practical methods to calibrate k are required.
z
This International Standard describes five of the simplest methods in three categories for the
determination of normal spring constants for atomic force microscope cantilevers. The methods are
in one of the three categories of dimensional, static experimental, and dynamic experimental methods.
The method chosen depends on the purpose and convenience to the analyst. Many other methods may
also be found in the literature.
INTERNATIONAL STANDARD ISO 11775:2015(E)
Surface chemical analysis — Scanning-probe microscopy —
Determination of cantilever normal spring constants
1 Scope
This International Standard describes five of the methods for the determination of normal spring
constants for atomic force microscope cantilevers to an accuracy of 5 % to 10 %. Each method is in one
of the three categories of dimensional, static experimental, and dynamic experimental methods. The
method chosen depends on the purpose, convenience, and instrumentation available to the analyst. For
accuracies better than 5 % to 10 %, more sophisticated methods not described here are required.
2 Normative references
The following documents, in whole or in part, are normatively referenced in this document and are
indispensable for its application. For dated references, only the edition cited applies. For undated
references, the latest edition of the referenced document (including any amendments) applies.
ISO 18115-2, Surface chemical analysis — Vocabulary — Part 2: Terms used in scanning-probe microscopy
3 Terms and definitions
For the purposes of this document, the terms and definitions given in ISO 18115-2 and the following apply.
3.1
normal spring constant
spring constant
force constant
DEPRECATED: cantilever stiffness
k
z
quotient of the applied normal force at the probe tip (3.2) by the deflection of the cantilever in
that direction at the probe tip position
Note 1 to entry: See lateral spring constant, torsional spring constant.
Note 2 to entry: The normal spring constant is usually referred to as the spring constant. The full term is used
when it is necessary to distinguish it from the lateral spring constant.
Note 3 to entry: The force is applied normal to the plane of the cantilever to compute or measure the normal
force constant, k . In application, the cantilever in an AFM may be tilted at an angle, θ, to the plane of the sample
z
surface and the plane normal to the direction of approach of the tip to the sample. This angle is important in
applying the normal spring constant in AFM studies.
3.2
probe tip
tip
probe apex
structure at the extremity of a probe, the apex of which senses the surface
Note 1 to entry: See cantilever apex (3.3).
3.3
cantilever apex
end of the cantilever furthest from the cantilever support structure
Note 1 to entry: See probe apex (3.2), tip apex (3.2).
4 Symbols and abbreviated terms
In the list of abbreviated terms below, note that the final “M”, given as “Microscopy”, may be taken
equally as “Microscope” depending on the context. The abbreviated terms are:
AFM Atomic force microscopy
FEA Finite element analysis
PSD Power spectral density
SEM Scanning electron microscopy
SPM Scanning probe microscopy
The symbols for use in the formulae and as abbreviated terms in the text are:
A amplitude of cantilever at a certain frequency
A amplitude of a cantilever at its fundamental resonant frequency
A mean amplitude of a cantilever associated with white noise
white
1/3
B gradient determined from a straight line fit to values of L versus Φ
Φ x x
−13/
L
x
B gradient determined from a straight line fit to values of L versus
k x k
( z )
C correction factor for the thermal vibration method described in 8.2
C correction factor for the thermal vibration method described in 8.2
d distance between the probe tip and the cantilever apex
D height of the probe tip
e width of the V-shaped cantilever at a distance L from the apex
E Young’s modulus of the material of a cantilever
E Young’s modulus of the base material of a cantilever
B
E Young’s modulus of the coating material on a cantilever
C
f frequency
f fundamental resonant frequency of a cantilever
F force of a nanoindenter
h displacement of a nanoindenter
i index of P , where i = 1 to 5
i
k Boltzmann constant
B
k normal spring constant
z
L
x
normal spring constant at the position L along a cantilever
k x
z
2 © ISO 2015 – All rights reserved

R
k normal spring constant of a reference cantilever
z
W
k normal spring constant of a working cantilever
z
k normal spring constant of a cantilever with a coating thickness of 0
z(tc=0)
L length of a rectangular cantilever or the effective length of a V-shaped cantilever
L distance between the base of a cantilever and the effective position of a V-shaped cantilever
x
L length of a V-shaped cantilever between the apex and the start of the arms
L length of a V-shaped cantilever between the base and the start of the arms
P label of one of the five positions on the reference cantilever axis
i
Q quality factor of a cantilever
r term defined by Formula (7)
t thickness of a cantilever
t thickness of the bulk material of a cantilever
B
t thickness of a coating on a cantilever
C
T absolute temperature of the cantilever measured in Kelvins
u standard uncertainty in A
A0 0
u standard uncertainty in B
B
u standard uncertainty in C
C1 1
u standard uncertainty in C
C2 2
u standard uncertainty in the distance between the probe tip and the cantilever apex
d
u standard uncertainty in the Young’s modulus of a cantilever
E
u standard uncertainty due to the calibration of force in the nanoindenter
F
u standard uncertainty in the resonant frequency
f0
u standard uncertainty due to the calibration of displacement in the nanoindenter
h
u standard uncertainty in the normal spring constant
kz
u standard uncertainty in the normal spring constant of the reference cantilever
kzR
u standard uncertainty in the length of a cantilever
L
u standard uncertainty in the quality factor of a cantilever
Q
u standard uncertainty in the thickness of a cantilever
t
u standard uncertainty in the absolute temperature
T
u standard uncertainty in the width of a cantilever
w
u standard uncertainty in x
x1 1
u standard uncertainty in α
α1 1
u standard uncertainty in the density of a cantilever
ρ
w width of a cantilever
w width of one side of a trapezium
w width of one side of a trapezium
w wcosθ
t
x offset to account for the small uncertainty in the true position of the base of the cantilever com-
pared to an arbitrary reference point
x offset to account for the uncertainty in the true position of the probe tip compared to an arbi-
trary reference point
Z term defined by Formula (4)
Z term defined by Formula (5)
α angle of the working cantilever with respect to the reference cantilever or surface
α numeric constant used in Formula (11)
δ average inverse gradient of the force-distance curve obtained with the working cantilever
R
pressing on the reference cantilever or device
δ average inverse gradient of the force-distance curve obtained with the working cantilever
W
pressing on a stiff surface
θ half angle between the arms of a V-shaped cantilever
Θ term defined by Formula (6)
ν Poisson’s ratio of the cantilever material
ρ density of a cantilever
φ
term defined by Formula (16)
x
5 General information
5.1 Background information
The spring constant, k , of an AFM cantilever is needed for quantitative force measurement. It is used to
z
convert the deflection of the cantilever into a force. Applications that need this include the measurement
of material properties at the nanoscale, such as elastic modulus, adhesive forces, and for studying the
breaking of covalent bonds and protein unfolding. Depending on the application, k will be chosen
z
−1 −1
to be in the range between 0,005 Nm and 200 Nm . There are two main shapes of cantilever: the
rectangular “diving board” shape and the V-shaped. Both types vary slightly in basic shape and design
between manufacturers and can be rectangular or trapezoidal in cross-section. Some cantilevers are
also coated with a thin metallic layer. These factors all influence the value of k .
z
Many manufacturers provide data sheets for their cantilevers giving nominal values of k . Unfortunately,
z
these values can be routinely in error by up to a factor of 3. One reason why similar cantilevers have very
different values of k is that the spring constant is proportional to the thickness cubed and the thickness
z
4 © ISO 2015 – All rights reserved

of AFM cantilevers is difficult to control accurately during manufacture. Since the cantilevers wear out,
break, and need regular replacement, quick and accurate methods to determine k are required.
z
5.2 Methods for the determination of AFM normal spring constant
There are many methods to determine the normal spring constant and these are classified as the following.
a) The dimensional methods where k is determined from the cantilever material and the geometrical
z
properties. In this method, any structural defects are not included.
b) The static experimental methods where k is determined by measurement of the static deflection
z
of the cantilever under an applied force.
c) The dynamic experimental methods where k is determined by measurement of the dynamic
z
properties of the cantilever.
In this International Standard, we describe procedures for a total of five methods with one or
two methods in each category. Use one or more of the methods to determine k and its associated
z
uncertainty, u . Which method or methods are used depends on the time, equipment, and the accuracy
kz
that the user requires the spring constant to be measured to. Some advantages and disadvantages of
the methods are given in Table 1.
Table 1 — Summary of the advantages and disadvantages of the methods in ISO 11775
Clause Method Advantages Disadvantages
Dimensional measurement Simple. Allows one to see why k Does not include defects.
z
varies from cantilever to cantilever. Slow and time consuming.
Static experimental measure- Can be made traceable to SI. May potentially damage the
ment using a reference cantile- cantilever. Can be time con-
ver or a nanoindenter suming and in some cases,
requires a nanoindenter.
Dynamic experimental meas- Fast if AFM instrument contains Uncertainty can be higher.
urement – thermal vibrational relevant software and hardware.
8 method Gives very good cantilever-to-canti-
lever comparability for cantilevers
of a given design.
NOTE This International Standard does not include all the methods for calibrating k that are described in
z
the literature.
6 Dimensional methods to determine k
z
6.1 General
The dimensional methods involve accurate measurements of a cantilever’s geometry and knowledge
of the material properties to determine k . The procedures described here use analytical formulae and
z
are only applicable if the geometry is suitable. For other geometries, finite element analysis (FEA) is
required and is not described here. Defects in the material, such as cracks or non-ideal geometry are
not generally included.
6.2 k using formulae requiring 3D geometric information
z
6.2.1 Method
In order to determine k for a rectangular beam with a rectangular cross-section, as shown in Figure 1,
z
measure the thickness t, the width w, and the distance (L - d), which is the length of the cantilever, L,
minus the distance from the free end of the cantilever to the probe tip, d. The measurement methods for
these are given in 6.2.2. Also, obtain or measure, using an appropriate method, the value for the Young’s
modulus E of the cantilever.
Make at least seven independent measurements of those parameters that you are measuring by
removing and replacing the cantilever. Evaluate the average values for these parameters and use them
to calculate k using Formula (1) as detailed in 6.2.3.1, incorporating the averages of these independent
z
measurements.
L-d
L
Figure 1 — Schematic of a rectangular shape cantilever with a probe tip a distance d from its
free end
Similarly, if you are using a V-shaped cantilever, as shown in Figure 2, measure L , the length of a V-shape
cantilever between the (virtual) apex and the start of the arms; L , the length of a V-shape cantilever
between base and the start of the arms; d, the distance between the probe tip and the cantilever apex;
e, the width of the V-shaped cantilever at the distance L from the apex; and θ, the half angle between
the arms. Also, obtain or measure, using an appropriate method, Young’s modulus E and Poisson’s ratio
of the cantilever ν. Make at least seven independent measurements of those parameters that you are
measuring by removing and replacing the cantilever. Evaluate the average values for these parameters
and use them to calculate k using Formulae (3) to (7).
z
d
L₀
e L
L₁
w
t
2θ
Key
1 apex
2 base
Figure 2 — Schematic of a V-shaped cantilever
6 © ISO 2015 – All rights reserved

If the cross-section of the cantilever is trapezoidal and not rectangular, apply Formula (9) and follow
the method given in 6.2.5.
If the cantilever has a significant coating, then account for this in the k calculation by following the
z
method given in 6.2.6.
6.2.2 Measuring the required dimensions and material properties of the cantilever
6.2.2.1 Measuring the plan view dimensions of the cantilever
The plan view dimensions of the cantilever, including width and (L - d) for rectangular cantilevers or
length and the offset of the probe tip from the cantilever apex for V-shaped cantilever, shall be measured
using an appropriate method, for example, optical microscopy or SEM. The measurement instrument
chosen shall be in calibration and operated in accordance with the manufacturer’s documented
instructions. Measure the width of the cantilever in at least three places along the length and determine
an average width. More measurements will be required if the width of the cantilever is uneven in order
to obtain a more accurate average width. A similar procedure applies in measuring L , d, and other
dimensions for the V-shaped cantilever. In measuring d, the distance measured shall be from the apex
or virtual apex of the V-shaped to the probe tip.
NOTE Using optical microscopy on typical commercial cantilevers, uncertainties in length and width are
approximately 1 %. SEM can prove more accurate but is likely to be more time consuming and expensive.
6.2.2.2 Measuring the thickness of the cantilever
The thickness of the cantilever shall be measured using an appropriate method, for example, using SEM
on the edge or side of the cantilever. The measurement instrument chosen shall be in calibration and
operated in accordance with the manufacturer’s documented instructions. Measure the thickness in a
number of different locations along the cantilever’s edge or side, and determine an average thickness.
NOTE 1 With careful, calibrated measurement, the uncertainty in thickness can be approximately 3 %.
NOTE 2 The number of measurements depends on the unevenness of the thickness. In addition, Formula (1)
given in 6.2.3.1 assumes a rectangular cross-section with no taper along the length. Analytically, it can be seen
that if there is an even taper in the thickness of 1 % change from end to end, then k is uncertain to approximately
z
1 %. Similarly, if it tapers in the middle and then returns to the original thickness, a 1 % change in thickness
results in a change in k of approximately 1 %, as discussed in Reference [4].
z
6.2.2.3 Measuring the material properties of the cantilever
The Young’s modulus, Poisson’s ratio, and the density of the cantilever material, including coatings, if
required, shall be determined from reference values if the cantilever is composed of known materials in
a known crystal orientation. Otherwise, these values need to be measured by another suitable method.
If no accurate values exist for these parameters and they cannot be measured, alternative methods to
calibrate k shall be used as detailed in Clauses 7 and 8.
z
NOTE Cantilevers are typically made from silicon or silicon nitride. Silicon is highly anisotropic, so
knowledge of the crystal orientation is critical. For the [110] direction, the Young’s modulus is 168,9 GPa, with
[4]
an uncertainty of approximately 1 %. The modulus of silicon nitride cantilevers is less certain and can depend
on the manufacturing technique. For example, the values of 146 GPa to 290 GPa have been reported for low
[5]
pressure CVD growth and around 400 GPa for single crystal material. The Poisson’s ratio and density of silicon
−3 −3
at room temperature are ~0,28 g cm and 2,329 g cm , respectively, but the Poisson’s ratio depends on the
−3
crystal orientation. The Poisson’s ratio and density of silicon nitride at room temperature are ~0,27 g cm and
−3
~3,3 g cm , respectively, but both depend on the form and growth of the material.
6.2.3 Determining k for the rectangular cantilever
z
6.2.3.1 Determining k
z
For a rectangular beam with a rectangular cross-section, as shown in Figure 1, composed of a single
material, once values for the Young’s modulus, E, thickness, t, width, w, and (L - d), which is the cantilever
length, L, minus the tip distance, d, from the free end, have been determined; calculate the cantilever
spring constant, k , using Formula (1).
z
Ewt
k = (1)
z
4 Ld−
()
Formula (1) involves the assumption that the bowing of the cantilever across the width, w, is negligible
and is therefore applicable to practical cantilevers where w << L. The cantilever must be attached to a
base and end effects here are usually small and ignored.
NOTE Formula (1) is reviewed in Reference [1].
6.2.3.2 Uncertainty
Determine the standard uncertainty in the spring constant, u , by using
kz
12/
 
2 2 2 2 2
         
 
u u 3u 3u 3u
    
    
E wt L d
    
     
uk= + + + + (2)
    
kzz     
      
    
 E   w   t  Ld−  Ld− 
   
     
 
 
where the uncertainties of the model are ignored. The main uncertainty arises from the measurement
of t and to a lesser extent, d, L, and E.
NOTE Typical values often given are approximately u /E = 0,03, u /w = 0,01, u /t = 0,04, u /(L-d) = 0,01,
E w t L
and u /(L-d) = 0,01 so that u /k = 0,12 and u is seen to be very important. u /E will be higher for silicon
d kz z t E
nitride cantilevers.
6.2.4 Determining k for the V-shaped cantilever
z
6.2.4.1 Determining k
z
For a V-shaped cantilever, shown in Figure 2, determine the dimensional and material properties of the
cantilever and then calculate k using
z
−1
 
 
w


 t 


kZ=+Z +−Θ d (3)

  
z1 22

sinθ 

 
 
 
where
 
 
 
    
Ld−
12L () L
 d 
 
0  
0    0 
 
 
Z = +−dL d ln −1 ++Ldln (4)
 ()  
   
1 0 0
 
   
2 L  d 
 
 
Eet     
 
 
 
 
L 2L
1 1
 
Z =+3 wdcotcθθ−−os rsinθ (5)
()
2  t 
cosθ
Ew t cosθ  
 
t
 
31L +ν
w
()
1 

t


Θ = −+drcotθ (6)

2 


sinθ 
 
Ew t cosθ
t
8 © ISO 2015 – All rights reserved

Lwtansθθ+−d in 1−νθcos
()()
1 t
r = (7)
21−−νθcos
()
Here, L is the length of a V-shaped cantilever between the (virtual) apex and the start of the arms, L
0 1
is the length of a V-shaped cantilever between base and the start of the arms, d is the distance between
the probe tip and the cantilever apex, e is the width of the V-shaped cantilever at the distance L from
the apex, w is wcosθ, where θ is the half angle between the arms, and ν is the Poisson’s ratio of the
t
cantilever material.
NOTE Formula (3) is reviewed in Reference [1], but note that there is a small change in the symbols that are
defined by Formulae (4), (5), and (6) in order to simplify Formula (3).
6.2.4.2 Uncertainty
The uncertainty calculation for k for a V-shaped cantilever based on Formula (3) is complex. However,
z
to a first approximation for the uncertainty calculation, this cantilever may be considered to be an
unskewed rectangular beam of width 2w, length L, and tip position d.
This simplified model can be used to calculate to a good approximation the uncertainty in k of a
z
V-shaped cantilever. Hence, determine the uncertainty in k using
z
12/
 
2 2 2 2 2
         
 
u u 3u 3u 3u
    
    
E 2wt L d
    
 
    
uk= + + + + (8)
    
    
kzz
    
 
 E   2w   t  Ld−  Ld− 
    
         
 
 
where the main contribution to the uncertainty is typically from t and L. Formula (8) differs from
Formula (2) in the second term in the square brackets related to the uncertainty in w, which is generally
small. The contribution arising from the uncertainty in θ is also generally small and is ignored here.
NOTE 1 Typical values often given are approximately u /E = 0,03, u /2w = 0,01, u /t = 0,04, u /(L-d) = 0,01,
E 2w t L
and u /(L-d) = 0,01 so that u /k = 0,12 and u is seen to be very important. u /E will be higher for silicon
d kz z t E
nitride cantilevers.
NOTE 2 These formulae are described in Reference [1].
NOTE 3 More complex methods to calculate the uncertainty in k may be appropriate for those requiring
z
higher accuracy than those given by this International Standard.
6.2.5 k for the trapezoidal cross-sections
z
Some cantilevers, rather than having a rectangular cross-section, have a trapezoidal cross-section of the
upper and lower widths, w and w . Measure these widths using the methods given in 6.2.2.1. Determine,
1 2
to a first approximation, the width of the cantilever for use in 6.2.3 and 6.2.4 using Formula (9).
w + w
w= (9)
Determine k for a diving board cantilever with a trapezoidal cross-section, using Formula (9) to
z
calculate w and use Formula (1) to determine k . Follow a similar procedure for a V-shaped cantilever
z
with a trapezoidal cross-section, but calculate k using the formulae detailed in 6.2.4.
z
NOTE Formula (9) leads to a small overestimation in k , which is less than 2 % if w is in the range
z 1
[2]
0,6 < w < 1,6 and less than 1 % if w is in the smaller range 0,7 < w < 1,4 .
2 1 2
6.2.6 k to account for coatings
z
The changes in k arising from a coating on the cantilever, which is often added to improve the
z
reflectivity from the laser beam used for monitoring the cantilever deflections, may be described by a
simple equation. Coating thicknesses, which are small compared to the thickness of the cantilever, k ,
z
can be calculated as a linear combination using the relationship
 
t E  t  E 
C C C C


kk=+13 +3 (10)
      
zz()t =0
C
 t E t E 
 B  B   B   B 
 
where t and t are the thicknesses, and E and E are the Young’s moduli of the coating and the bulk
C B C B
material of the cantilever, respectively. Here, k is simply the calculated spring constant ignoring
z(tc=0)
the coating. Therefore, to determine k , measure or obtain, using appropriate methods, t , t , E , and E
z C B C B
then determine k using Formula (10). The uncertainty arising from the coating is small compared with
z
the uncertainty of the uncoated cantilever and its contribution can therefore often be ignored.
NOTE Formula (10) is derived in Reference [1]. This shows that, for example, a V-shaped cantilever adding a
40 nm gold coating increases the spring k by approximately 5 %.
z
6.3 k using plan view dimensions and resonant frequency for rectangular tipless
z
cantilevers
6.3.1 Determining k
z
The determination of k by the dimensional methods described in 6.2 and 6.3 requires accurate
z
information about the cantilever dimensions and, in particular, the thickness. The spring constant
is proportional to the cube of the thickness. It is also the most difficult dimension to control during
manufacturing and so it is subject to the greatest variability in its value leading to large uncertainties
in k . If the measurement of the cantilever thickness is difficult or an SEM is unavailable, the method
z
detailed in this section can be used to determine k .
z
Measure the plan dimensions (w and L) of a rectangular cantilever using the methods described in
6.2.2.1. Determine the resonant frequency f of the cantilever by mechanically vibrating the cantilever
as a function of frequency. Obtain or measure the Young’s modulus E and density ρ of the cantilever
material following the method described in 6.2.2.3.
The resonant frequency f of a rectangular tipless cantilever in vacuum of Young’s modulus E, density ρ,
thickness t, width w, and length L, is calculated to be
 
α
t E



f =  (11)



43π  L  ρ
 
where α = 1,875 1. Solving for t and substituting into Formula (1) yields
3 3 32/
48π 3 wL ρ
k = f (12)
z 0
6 3
E
α
1−dL/
1 ()
where d is the distance of the tip from the free end of the cantilever.
Determine the cantilever spring constant using Formula (12). This formula is based on the resonant
frequency in vacuum. If it is not possible to measure this, then measure the resonant frequency in air
and apply a small correction factor. This correction factor increases the measured frequency value by
around 1 %, depending on the values of E, L, w, ρ, and t.
The value of α is for a rectangular cross-section cantilever beam with no tip that is clamped at
one end. For different shapes and types of cantilever, the value of α should be determined by an
appropriate method.
NOTE 1 Formulae (11) and (12) assume a cantilever comprised of a single material with a constant cross-
section and does not account for defects. The method is reviewed in Reference [1].
10 © ISO 2015 – All rights reserved

NOTE 2 The increase in resonant frequency between air and vacuum has been measured to be approximately
[3]
0,4 % to 0,6 %, although others found it to be 2 % to 4 %, as reviewed in Reference [1]. Unless the resonant
frequency is measured in vacuum, a modest uncertainty will be introduced.
NOTE 3 The resonant frequency is determined here via a driven method rather than the thermal method as
detailed in 8.2 in order to obtain a better signal quality. An interlaboratory study showed that the scatter in
measured f using this method was 0,7 %.
o
NOTE 4 For cantilevers with a tip, further considerations are required. The tip adds mass to the cantilever,
−2
which decreases the measured resonant frequency to an extent proportional to f . The volume of the tip can be
estimated using microscopy, and the mass is then the product of this volume, and the tip density and a suitable
correction factor may be determined. Further details and a method to determine different values for α from the
ratio of first and second harmonic frequencies can be found in Reference [2].
6.3.2 Uncertainty
Determine the standard uncertainty in the spring constant of a tipless cantilever using
12/
2 2
2 2 2 2
 
3uu 3 
6u  u 3u
u    3u   
ρ f 0
α1 w d
E L
 
uk= + + + + + + (13)
   
 
kzz        
   
 αρ2E 2 f w L Ld 
−
   
   
 1     0 
 
in the usual case where d/L is small.
NOTE Typical values often given are approximately 6u /α = 0,03, u /2E = 0,015, 3u /2ρ = 0,005 to
α1 1 E ρ
0,02 (depending on the cantilever material with the latter number for non-silicon cantilevers), 3u /f = 0,03
f0 0
(including the uncertainty due to f being measured in air), u /w = 0,01, 3u /L = 0,03 and 3u /(L-d) = 0,03 so
0 2w L d
that u /k = 0,06 and u is seen to be important.
kz z α
7 Static experimental methods to determine k
z
7.1 General
Static experimental methods involve the application of a constant force or a set of constant forces applied
to the cantilever and the subsequent measurement of the deflection. These methods generally, but not
exclusively, use a pre-calibrated reference beam or device to push on the working cantilever or vice versa.
There are a number of different static experimental methods that are described using
a) one or more calibrated reference cantilevers, and
b) a calibrated nanoindenter.
7.2 Static experimental method with a reference cantilever
7.2.1 Set-up
R
Obtain or calibrate a reference cantilever of known spring constant, k . A tipless rectangular cantilever
z
with constant cross-section along its length is preferred. The spring constant of the reference cantilever
W
should approximately match that of the working cantilever, k . The piezoelectric z-scanner of the AFM
z
shall be in dimensional calibration and the AFM shall be operated in accordance with the manufacturer’s
documented instructions. This should be operated in closed loop mode or an alternative method used
to deal with the effects of the piezoelectric scanner nonlinearity, hysteresis, creep, and drift. An AFM
equipped with top view optics is recommended for the highest positional accuracy of the reference
cantilever on the working cantilever.
NOTE 1 One method to calibrate a reference cantilever is to use the nanoindenter on cantilever measurement
method as detailed in 7.3. Another method is to acquire or calibrate a cantilever that has been traceably calibrated
for instance via mass artefacts or quantum-based standards using electromechanical force balances.
NOTE 2 If a tipless cantilever with marked probing positions along its length is available, it may aid the ease or
accuracy of undertaking this method.
7.2.2 Determining k
z
The deflection constant for the working cantilever shall be measured using the method outlined in
7.4.1. This shall be done at the start and end of the experiment.
The reference cantilever shall be mounted securely in the sample position and aligned to be
perpendicular to the long axis of the working cantilever, which shall be mounted in the cantilever holder,
as shown in Figure 3. The perpendicular alignment makes it easy to move the working cantilever along
the reference cantilever and aids the location of the tip position. Land the working cantilever tip as near
the axis of the reference cantilever as possible and near the free end. Conduct five force-distance curves,
recorded as cantilever deflection in volts versus piezoelectric z-scanner extension in nanometres. The
cantilever deflection should be kept within the elastic limit of the cantilever. The cantilever should bend
less than 5 % of its total length and be kept within the linear region of the photodiode detector. For
most cantilevers, a movement of less than 200 nm is suggested. Calculate the average gradient of each
resultant plot of deflection voltage signal versus z-scanner displacement. This should be measured over
the maximum range over which the curve linearity is less than a desired target uncertainty. Measure
the distance between a suitable reference point on the test cantilever and the reference cantilever
base. Ideally, this should be the tip position of the test cantilever, but if this cannot be seen, then use
a reference point on the symmetry axis of the working cantilever, as shown by point A in Figure 3 b).
Repeat this procedure for at least three to five evenly spaced positions, L , as far apart as possible along
x
the outermost 70 % of the length of the reference cantilever, as shown by the P ’s in Figure 3 a).
i
x
L α
x
A
D
L
P P P P P
5 3 2 1 4
x
a) Plan view b) Higher magnification of a) c) Side-view
Key
1 ref. cantilever axis 4 tip under working cantilever
2 ref. cantilever 5 long axis of working cantilever
3 working cantilever
NOTE In this particular example for the working cantilever position L denotes the distance between P and
x 2
the base of the cantilever
Figure 3 — Schematic of the static experimental method using a reference cantilever
12 © ISO 2015 – All rights reserved

At each position, calculate the spring constant of the working cantilever using
   
 
Lx+ δ
  3D
 
W R 1 R 2 

 
  
kk= −1 cos ααα1− tan (14)
 
  
z z
  
 
Lx++x  δ  2L
  
   
x 12 W
where L is the length of the calibrated cantilever, D is the height of the probe tip, L is the distance of
x
the working cantilever along the reference cantilev
...