ISO 11775:2015
(Main)Surface chemical analysis — Scanning-probe microscopy — Determination of cantilever normal spring constants
Surface chemical analysis — Scanning-probe microscopy — Determination of cantilever normal spring constants
ISO 11775:2015 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.
Analyse chimique des surfaces — Microscopie à sonde à balayage — Détermination de constantes normales en porte-à-faux de ressort
General Information
Standards Content (Sample)
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 11775:2015(E)
©
ISO 2015
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ISO 11775:2015(E)
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ISO 11775:2015(E)
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
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ISO 11775:2015(E)
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards
bodies (ISO member bodies). The work of preparing International Standards is normally carried out
through ISO technical committees. Each member body interested in a subject for which a technical
committee has been established has the right to be represented on that committee. International
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ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters of
electrotechnical standardization.
The procedures used to develop this document and those intended for its further maintenance are
described in the ISO/IEC Directives, Part 1. In particular the different approval criteria needed for the
different types of ISO documents should be noted. This document was drafted in accordance with the
editorial rules of the ISO/IEC Directives, Part 2 (see www.iso.org/directives).
Attention is drawn to the possibility that some of the elements of this document may be the subject of
patent rights. ISO shall not be held responsible for identifying any or all such patent rights. Details of
any patent rights identified during the development of the document will be in the Introduction and/or
on the ISO list of patent declarations received (see www.iso.org/patents).
Any trade name used in this document is information given for the convenience of users and does not
constitute an endorsement.
For an explanation on the meaning of ISO specific terms and expressions related to conformity
assessment, as well as information about ISO’s adherence to the WTO principles in the Technical
Barriers to Trade (TBT) see the following URL: Foreword - Supplementary information.
The committee responsible for this document is ISO/TC 201, Surface chemical analysis, Subcommittee
SC 9, Scanning probe microscopy.
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ISO 11775:2015(E)
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.
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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).
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ISO 11775:2015(E)
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
0
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
1
C correction factor for the thermal vibration method described in 8.2
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
0
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
0
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
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ISO 11775:2015(E)
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
0
L length of a V-shaped cantilever between the base and the start of the arms
1
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
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ISO 11775:2015(E)
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
1
w width of one side of a trapezium
2
w wcosθ
t
x offset to account for the small uncertainty in the true position of the base of the cantilever com-
1
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-
2
trary reference point
Z term defined by Formula (4)
1
Z term defined by Formula (5)
2
α angle of the working cantilever with respect to the reference cantilever or surface
α numeric constant used in Formula (11)
1
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)
2
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
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ISO 11775:2015(E)
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
6
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-
7
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
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ISO 11775:2015(E)
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
0
cantilever between the (virtual) apex and the start of the arms; L , the length of a V-shape cantilever
1
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
0
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
1
d
L₀
e L
L₁
w
t
2θ
Key
1 apex
2 base
Figure 2 — Schematic of a V-shaped cantilever
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ISO 11775:2015(E)
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
0
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.
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ISO 11775:2015(E)
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
3
Ewt
k = (1)
z
3
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
...
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