RevSciInstrum_83_103705.pdf

Spring constant calibration of atomic force microscope cantilevers of

arbitrary shape

John E. Sader, Julian A. Sanelli, Brian D. Adamson, Jason P. Monty, Xingzhan Wei et al.

Citation: Rev. Sci. Instrum. 83, 103705 (2012); doi: 10.1063/1.4757398

View online: http://dx.doi.org/10.1063/1.4757398

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Published by the American Institute of Physics.

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Rev. Sci. Instrum. 83, 121101 (2012)

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REVIEW OF SCIENTIFIC INSTRUMENTS

83

, 103705 (2012)

Spring constant calibration of atomic force microscope

cantilevers of arbitrary shape

John E. Sader,

1,2,

Julian A. Sanelli,

Brian D. Adamson,

Jason P. Monty,

Xingzhan Wei,

3,5

Simon A. Crawford,

James R. Friend,

7,8

Ivan Marusic,

Paul Mulvaney,

3,5

and Evan J. Bieske

Department of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia

Kavli Nanoscience Institute and Department of Physics, California Institute of Technology, Pasadena,

California 91125, USA

School of Chemistry, The University of Melbourne, Victoria 3010, Australia

Department of Mechanical Engineering, The University of Melbourne, Victoria 3010, Australia

Bio21 Institute, The University of Melbourne, Victoria 3010, Australia

School of Botany, The University of Melbourne, Victoria 3010, Australia

Melbourne Centre for Nanofabrication, Clayton, Victoria 3800, Australia

MicroNanophysics Research Laboratory, RMIT University, Melbourne, Victoria 3001, Australia

(Received 15 June 2012; accepted 18 September 2012; published online 17 October 2012)

The spring constant of an atomic force microscope cantilever is often needed for quantitative measure-

ments. The calibration method of Sader

et al.

[Rev. Sci. Instrum.

70

, 3967 (1999)] for a rectangular

cantilever requires measurement of the resonant frequency and quality factor in fluid (typically air),

and knowledge of its plan view dimensions. This intrinsically uses the hydrodynamic function for a

cantilever of rectangular plan view geometry. Here, we present hydrodynamic functions for a series

of irregular and non-rectangular atomic force microscope cantilevers that are commonly used in prac-

tice. Cantilever geometries of arrow shape, small aspect ratio rectangular, quasi-rectangular, irregular

rectangular, non-ideal trapezoidal cross sections, and V-shape are all studied. This enables the spring

constants of all these cantilevers to be accurately and routinely determined through measurement of

their resonant frequency and quality factor in fluid (such as air). An approximate formulation of the

hydrodynamic function for microcantilevers of arbitrary geometry is also proposed. Implementation

of the method and its performance in the presence of uncertainties and non-idealities is discussed, to-

gether with conversion factors for the static and dynamic spring constants of these cantilevers. These

results are expected to be of particular value to the design and application of micro- and nanomechan-

ical systems in general.

http://dx.doi.org/10.1063/1.4757398

]

I. INTRODUCTION

Knowledge of the stiffness of microcantilevers used in

the atomic force microscope (AFM) is essential for many ap-

plications of the instrument.

–

Over the past 20 years, many

techniques have been devised for the

in situ

measurement of

these spring constants. These methods allow the user to rou-

tinely and independently calibrate the spring constants of can-

tilevers during operation of the AFM. These calibration meth-

ods include dimensional approaches,

–

methods that probe

the static deflection of the cantilever induced by a calibrated

load,

–

and those that monitor the dynamic vibrational re-

sponse of the cantilever.

–

The performance of these tech-

niques has been widely explored and assessed, and the reader

is referred to Refs.

–

for detailed reviews.

The method of Sader

et al.

for rectangular cantilevers

makes use of the hydrodynamic load experienced by a can-

tilever as it oscillates in a fluid (such as air) – for clarity,

this approach shall henceforth be referred to as the “original

method” in this article. It was originally devised for rectangu-

lar cantilevers, for which the static normal spring constant

Author to whom correspondence should be addressed. Electronic mail:

jsader@unimelb.edu.au.

is determined using the formula,

1906

ρb

(

)

(1)

where

is the density of the fluid surrounding the cantilever,

and

are the cantilever width and length, respectively,

and

are the radial resonant frequency and quality factor in fluid

of the fundamental flexural mode, respectively, and

(

)

is the imaginary part of the (dimensionless) hydrodynamic

function evaluated at the resonant frequency.

To imple-

ment this formula in practice, knowledge of the fluid density

and viscosity, cantilever width and length is required, and the

resonant frequency and quality factor must be measured. The

technique is independent of the thickness and material prop-

erties of the cantilever, which can be difficult to determine

in practice. The technique was extended to calibration of the

torsional spring constant of rectangular cantilevers in Ref.

Subsequently, the original method was generalized in

Ref.

to enable measurement of the spring constant of

any elastic body, including AFM cantilevers of arbitrary

geometry – this shall be referred to as the “general method.”

The general method relies on knowledge of the hydrodynamic

function

for a cantilever of arbitrary shape – a protocol for

its determination was also presented in Ref.

. With the

(dimensionless) hydrodynamic function for a specific type

0034-6748/2012/83(10)/103705/16/$30.00

, 103705-1

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103705-2

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et al.

Rev. Sci. Instrum.

, 103705 (2012)

TR400(L)

BL-RC-150VB(L)

AC240TS

TR400(S)

ASYMFM

BL-RC-150VB(S)

AC160TS

TR800(L)

NCHR

AC240TM

FMR

TR800(S)

FIG. 1. SEM micrographs showing plan view geometries (shapes) of all can-

tilevers used in this study. Details of each cantilever, including dimensions,

are given in Table

of cantilever known (e.g., a particular V-shaped cantilever

model), the spring constant of any one of these cantilevers can

be determined from knowledge of its plan view dimensions,

and measurement of its resonant frequency and quality fac-

tor in fluid (typically air). The general method was tested and

validated for a series of rectangular and V-shaped cantilevers

in Refs.

and

, for which good agreement was found

with independent measurements of the spring constants. The

theoretical framework underpinning the general method also

explained its performance (and that of the original method)

under non-ideal but practical conditions, e.g., presence of an

imaging tip; see Ref.

Continuous advances in atomic force microscopy have

led to the development of a wide array of cantilever de-

signs, as evident from the micrographs presented in Fig.

Many of these cantilevers have designs that deviate strongly

from a rectangular geometry. Ability to calibrate the spring

constants of these cantilevers is thus of critical importance

to quantitative AFM measurements. The primary purpose of

this article is to report the hydrodynamic functions for a se-

ries of cantilevers that are commonly used in the AFM; see

Fig.

. This enables the normal spring constant of these

types of cantilevers to be routinely and accurately determined

through measurement of their resonant frequency and quality

factor in fluid (air). This is regardless of thickness variations,

compositional and size changes, provided the plan view ge-

ometry (shape) remains constant.

As we shall discuss, this

prerequisite is commonly satisfied in practice.

The cantilevers studied here possess significant non-

idealities in the form of arrow shaped ends, irregular rectan-

gular geometries, small aspect ratio rectangular geometries,

non-ideal trapezoidal cross sections of irregular shape and

V-shaped geometries. SEM micrographs of these cantilevers

are given in Fig.

. Importantly, the original method

im-

plicitly assumes that the cantilever plan view is rectangular

and that its aspect ratio (length/width) is large. This is be-

cause the fluid-structure theory

underpinning this method

is derived using this assumption. The method was found to

work well for ideally rectangular cantilevers with aspect ra-

tios (length/width) in the range 3.3

−

13.7, and rectangular

cantilevers of non-ideal geometry (ends slightly cleaved with

imaging tip) for aspect ratios 3.9–10;

the lower limit in as-

pect ratio for which the original method is valid was not de-

termined in Ref.

, and left as an open question. Some of the

cantilevers shown in Fig.

clearly do not satisfy these bounds

– they possess smaller aspect ratios, and have irregular and

non-rectangular shapes. It is important to emphasize that the

general method

is valid for any elastic body, and thus it au-

tomatically includes all non-ideal effects such as those due to

the imaging tip, arbitrary shape, and arbitrary aspect ratio. No

corrections to the reported hydrodynamic functions in this ar-

ticle, to account for any such non-idealities, are required or

warranted.

The AFM is frequently operated in two complementary

modes: (i) “static mode” where the static deflection of the

cantilever due to an applied force is monitored, and (ii) “dy-

namic mode” in which the cantilever is oscillated at or near

resonance.

–

Since the deflection functions of the can-

tilever for these two complementary modes are different, they

probe different spring constants. We remind the reader that the

spring constant of a cantilever at any point along its length is

defined as the second derivative of its potential energy with

respect to amplitude at that point.

The potential energy can

be written in terms of the elastic strain in the cantilever, and

therefore depends explicitly on the mode shape, i.e., the de-

flection function of the cantilever.

Consequently, the static

and dynamic spring constants of an AFM cantilever will dif-

fer, since the deflection functions in these two modes of oper-

ation are not identical. These complementary spring constants

are needed for quantitative analysis of static and dynamic

mode measurements. We therefore present numerical results

allowing for conversion between these two spring constants

for all cantilevers considered in Fig.

. This in turn allows the

general method to be used to determine both the static and dy-

namic normal spring constants. The dynamic spring constant

for only the fundamental flexural mode is considered in this

study.

Experimental protocols for determination of the hydro-

dynamic function

and implementation of the original and

general methods

are summarized. A discussion of the oper-

ation of the general method in the presence of random non-

idealities, such as uncertainty in cantilever dimensions and

cantilever clamping conditions, is also presented.

The article is organized as follows: We begin in Sec.

with a brief exposition of the theory underpinning the gen-

eral method and the experimental protocol for its implemen-

tation. Section

III

focuses on experimental determination of

the hydrodynamic function and implementation of the gen-

eral method. It is divided into several subsections that pro-

vide information on (a) cantilever dimensions, (b) spring con-

stant measurements, (c) measured hydrodynamic functions,

(d) a simplified approximate implementation of the general

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103705-3

Sader

et al.

Rev. Sci. Instrum.

, 103705 (2012)

method valid for any microcantilever, (e) conversion factors

for the static and dynamic spring constants, (f) effect of un-

certainty on the method, and (g) protocols for implementing

the method. Details pertinent to Secs.

and

III

can be found

in the Appendixes.

A. Summary

Readers primarily interested in the hydrodynamic func-

tions for the cantilevers in Fig.

are referred directly to

Table

III

and Eq.

(10)

, which are to be used with Eqs.

(7a)

and

(8)

. This completely specifies the general method for

each cantilever. Conversion factors for the static and dynamic

spring constants of these cantilevers are given in Table

II. THEORETICAL AND EXPERIMENTAL

FRAMEWORK

We first summarize the theoretical framework of the gen-

eral method, which is applicable to any elastic body or de-

vice immersed in a viscous fluid. The device executes reso-

nant oscillations in the fluid. This theoretical framework is

then applied to the present case of interest: a cantilever of ar-

bitrary plan view shape undergoing resonant oscillations in

its fundamental flexural mode. The experimental protocol for

determination of the hydrodynamic function for a cantilever

of arbitrary shape is then given. For a detailed derivation of

this framework and a comprehensive discussion, the reader is

referred to Ref.

A. Arbitrary elastic device in fluid

The principal assumptions of the general method are:

(1) The body behaves as a linearly elastic solid;

(2) Energy dissipation due to vibration of the body occurs

in the fluid;

(3) The oscillation amplitude of the body is small, so that all

nonlinearities due to the body and fluid are negligible;

(4) The fluid flow generated by the oscillating body is in-

compressible.

These assumptions are commonly satisfied in practice,

from which the maximum energy stored in the oscillating

body at resonance directly follows:

stored

(2)

where

is the dynamic spring constant of the oscillation

mode, and

is the oscillation amplitude. The energy dissi-

pated in the fluid due to these resonant oscillations can be

quantified by the (dimensionless) quality factor,

≡

stored

diss

∣

(3)

where

diss

is the energy dissipated per oscillation cycle, at

the resonant frequency

Since the flow is linear, as discussed above, the energy

dissipated per oscillation cycle will depend on the square of

the oscillation amplitude,

. It therefore follows from Eqs.

(2)

and

(3)

that the dynamic spring constant (see definition in the

Introduction) is related to the quality factor by

(

∂

diss

∂A

∣

)

(4)

which is independent of the oscillation amplitude.

In accord with the above-mentioned assumptions, the en-

ergy dissipated per cycle

diss

must depend on (i) the square

of the device oscillation amplitude,

, (ii) the fluid density

and shear viscosity

, (iii) the linear dimension (size) of

the device, denoted

, (iv) the relevant frequency of oscil-

lation, which from Eq.

(4)

is the resonant frequency of the

device immersed in fluid,

, (v) the mode shape of the vi-

brating device, and (vi) the geometry of the device. Note that

the last two quantities are dimensionless. The relationship be-

tween the remaining quantities and

diss

can be rigorously

determined using dimensional analysis

– this gives two di-

mensionless groups. Use of Buckingham’s

theorem

then

yields the required result for the energy dissipated per cycle,

∂

diss

∂A

∣

ρL

(

)

(5)

where

(

) is a dimensionless function that depends on the

dimensionless parameter,

≡

ρL

(6)

which is often termed the inverse Stokes number or Womers-

ley number and is related to the Reynolds number, Re, defined

below. Substituting Eqs.

(5)

and

(6)

into Eq.

(4)

gives the re-

quired expression for the stiffness, as presented in Ref.

Note that

(

) also implicitly depends on the mode

shape and geometry of the body, as is evident from the above

discussion. Provided these dimensionless quantities do not

change,

(

) will remain invariant. This point is examined

further in Sec.

III

B. Cantilevers of arbitrary shape

The hydrodynamic flow induced by the flexural oscilla-

tions of a thin cantilever is dominated by its plan view geom-

etry, with its thickness exerting a negligible effect;

see

Sect.

II C 1

. Since AFM cantilevers typically have both small

and large length scales in their plan view geometry, e.g., the

width

and length

of the cantilever, these can both affect

the flow. We therefore define, without loss of generality,

≡

ρb

(

)

β,

(7a)

(Re)

≡

(

)

(7b)

Note that the linear dimension (size)

is a characteristic

length scale of the flow; see Sec.

II A

. This length scale has

been replaced by a combination of the length,

, and width,

, to accommodate details of the flow generated by a vibrat-

ing microcantilever. Specifically, the dominant hydrodynamic

length scale for the flow is often the smaller length scale, e.g.,

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103705-4

Sader

et al.

Rev. Sci. Instrum.

, 103705 (2012)

the width

of the cantilever.

As such, the flow varies slowly

along the cantilever length,

, and rapidly over its width,

It then follows that the hydrodynamic volume over which

energy dissipation occurs scales as

for viscous bound-

ary layers of comparable size to the cantilever width.

The

rescalinginEq.

(7)

thus ensures that the (dimensionless) hy-

drodynamic function,

(Re),

is an order one quantity in

such situations – this case is often encountered in practice,

and is demonstrated in Sec.

III C

. The Reynolds number Re

contains the width

only, and can thus be formally interpreted

as the squared ratio of the dominant hydrodynamic length

scale to the viscous penetration depth. The redefinitions in

Eq.

(7)

thus facilitate physical interpretation and proper nor-

malization of all dimensionless quantities.

Substituting Eq.

(7)

into Eq.

(5)

and subsequently into

Eq.

(4)

, gives the required formula connecting the dynamic

spring constant to the dissipative properties of the cantilever

at resonance,

ρb

(Re)

(8)

Comparing Eq.

(8)

to Eq.

(1)

reveals that the general method,

which is rigorously applicable to a cantilever of arbitrary

shape, yields an equation of identical form to that for a rect-

angular cantilever. This establishes that the original method,

for rectangular cantilevers of high aspect ratio, can be directly

extended to cantilevers of arbitrary shape. All that is needed is

the hydrodynamic function,

(Re), for the cantilever geom-

etry and mode in question. Equations

(1)

and

(8)

show that

the hydrodynamic functions

(

) and

(Re) are related by a

constant factor for rectangular cantilevers of high aspect ratio.

Since the static and dynamic spring constants differ

by a constant multiplicative factor for a cantilever of fixed

plan view geometry, Eq.

(8)

is equally applicable to the

static spring constant under the appropriate renormalization.

The renormalization factors for all cantilever geometries in

Fig.

are given in Sec.

III E

Throughout we only consider the fundamental flexural

mode of vibration, even though the general method is rigor-

ously applicable to any mode. Note that the material prop-

erties of the cantilever do not enter into the derivation of

Eq.

(8)

, and thus the original and general methods are applica-

ble to cantilevers composed of any elastic material. The appli-

cability of these methods to devices whose thickness and/or

material properties vary along their length is discussed in

Sec.

II C

C. Properties of the general method

In this section, we present a discussion of several features

of the general method that are pertinent to its implementation.

1. Effect of finite cantilever thickness

For thin cantilevers executing flexural oscillations, the

hydrodynamic function,

(Re), depends only on the plan

view geometry of the cantilever and its mode shape, which

are both dimensionless quantities. Cantilever thickness plays

a relatively minor role in the hydrodynamic load (and en-

ergy dissipation) experienced by a cantilever undergoing flex-

ural oscillations, even for quite thick devices.

This is

because the load is dominated by contributions from the hy-

drodynamic pressure rather than the shear stress.

As such,

the cantilever plan view dimension to thickness ratio, e.g.,

the width-to-thickness ratio, does not exert a significant ef-

fect on the hydrodynamic function, and can be ignored.

This property is used in determination of the hydrodynamic

function in Sec.

II D

2. Effect of non-uniform cantilever thickness

and material properties

Spatial variations in thickness and/or material properties

also exert a weak effect on the general method and only enter

via their effect on the cantilever mode shape. This is because

the right hand side of Eq.

(8)

depends on the energy dissi-

pated in the fluid, not the cantilever thickness or material; see

above. The energy dissipated in the fluid is a weighted aver-

age of the mode shape over the cantilever plan view geometry.

Since the fundamental mode shape is a simple monotonically

increasing function of distance from the clamp, spatial varia-

tions in thickness and/or material have a weak effect on this

mode shape and hence the general method. This explains re-

cent theoretical findings demonstrating the robustness of the

original method, with respect to thickness variations along the

cantilever axis, in all but the extreme cases of very strong vari-

ations in thickness

– in these extreme cases, the mode shape

was significantly altered. The same property holds true for the

general method.

For the same reason, the presence of an imaging tip mass

also has a very minor effect on this mode shape, even in the

high tip mass to cantilever mass limit.

Nonetheless, if the

imaging tip is comparable in size to the dominant hydrody-

namic length scale of the flow, its presence will enhance the

true energy dissipation and thus increase the hydrodynamic

function.

While this can lead to an underestimate of the

spring constant obtained using the original method,

the gen-

eral method intrinsically accounts for any such extra energy

dissipation. This is because the hydrodynamic function is de-

termined in the presence of the imaging tip; see Sec.

II D

As such, the general method is rigorous and accurate in such

non-ideal cases.

3. Effect of non-uniform widths and trapezoidal

cross sections

Some of the cantilevers in this study possess strongly

non-ideal geometries, with varying widths along the can-

tilever axis and highly scalloped trapezoidal cross sections;

see SEM micrographs of the NCHR and FMR devices in

Fig.

. Since the pressure load on the plan view area of a

cantilever dominates its hydrodynamic load (see Sec.

II C 1

above), and the original and general methods probe the net

energy dissipated in the fluid, these geometric properties are

inconsequential to the performance of these methods. The

maximum width of the trapezoidal cross section should thus

be employed in both methods; this explains the finding of

Ref.

. The average of the maximum width is used to

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103705-5

Sader

et al.

Rev. Sci. Instrum.

, 103705 (2012)

AC160TS

BL-RC-150VB(L)

NCHR

FMR

TR800(S)

FIG. 2. SEM micrographs showing perspective images for a selection of can-

tilevers used in this study. All cantilevers used are shown in Fig.

. Details of

each cantilever studied, including dimensions, are given in Table

estimate the net energy dissipated by corrugated devices:

NCHR and FMR in Fig.

. The inner width is approximately

one third the outer width, see Fig.

, and is not relevant to the

net energy dissipated and hence implementation of the origi-

nal and general methods.

D. Determination of the hydrodynamic function

We now summarize the experimental protocol

to deter-

mine the hydrodynamic function,

(Re), for a cantilever of

arbitrary shape – this function is needed to close Eq.

(8)

Importantly, the hydrodynamic function is a dimension-

less quantity that remains unchanged as the size and/or com-

position of the cantilever are varied. It is formally the scaled

energy dissipation in the fluid and thus depends only on the

mode shape and plan view geometry. It is also independent

of cantilever thickness for the reasons discussed in Sec.

II C

The hydrodynamic function can therefore be determined for

all cantilevers of the same plan view geometry by studying

a single “test” cantilever immersed in a fluid. Gas is used in

these measurements, since it allows for easy modification of

its transport properties, produces sharp resonance peaks and

thus enables rigorous extraction and measurement of the qual-

ity factor.

While theoretical calculations and simulations can be

used to determine the hydrodynamic function, performing

measurements on a test cantilever (i) automatically accounts

for the true geometry of the cantilever device, and (ii) intrin-

sically includes all complexities such as hydrodynamic cou-

pling between the cantilever and the supporting chip. It also

accounts for all non-ideal structures and other effects due

to the manufacturing process that may be difficult to quan-

tify and thus theoretically model in an accurate fashion, e.g.,

shape of imaging tip.

To proceed, we rearrange Eq.

(8)

to give an expression

for the hydrodynamic function in terms of the properties of

the cantilever and the gas

(Re)

ρb

Lω

(9)

Our goal is to evaluate the hydrodynamic function over a

range of Reynolds numbers, Re, for a single test cantilever.

Varying the gas pressure facilitates systematic variation of

the Reynolds number, Re, because the gas density and de-

vice quality factor are both strongly dependent on gas pres-

sure; the resonant frequency is relatively insensitive to pres-

sure, whereas gas viscosity is invariant. The hydrodynamic

function is measured by placing the test cantilever in a pres-

sure chamber, systematically sweeping the gas pressure and

recording the gas density, resonant frequency, and quality fac-

tor. These results are then substituted into Eqs.

(7a)

and

(9)

This can be performed for a number of different gases to en-

sure consistency between measurements; this is implemented

in Sec.

III C

The dynamic spring constant of each test device is also

needed to complete the determination of the hydrodynamic

function,

(Re); see Eq.

(9)

. Since the general method re-

quires the hydrodynamic function for its implementation, an

alternate calibration method must be employed for this mea-

surement on the test device – the approach used is detailed in

Sec.

III B

and Appendix

We emphasize that once the hydrodynamic function,

(Re), for a particular test cantilever is determined, imple-

mentation of the general method for all cantilevers of that

same type is independent of the above specified gas pressure

and spring constant measurements. This then allows for non-

invasive and accurate spring constant measurements of AFM

cantilevers in practice using Eq.

(8)

III. RESULTS AND DISCUSSION

Hydrodynamic functions for all AFM cantilevers are re-

ported in this section, together with fit functions to facili-

tate their use in practice. The dimensions and properties of

these cantilevers are listed. The apparatus developed for the

gas pressure measurements is detailed in Appendix

. Perfor-

mance of the general method in the presence of non-idealities

is also discussed, together with finite element calculations al-

lowing for conversion between the static and dynamic spring

constants of all cantilevers. We remind the reader that only

the fundamental flexural modes are considered.

A. AFM cantilevers and dimensions

The cantilevers used in this study are from Asylum Re-

search, Nanoworld, and Olympus (Japan); see Fig.

. These

cantilevers are commonly used in practice. They possess sig-

nificant non-idealities, as outlined above. To highlight their

geometric features, perspective SEM images of some of these

devices are given in Fig.

– other devices possess similar

non-idealities to those evident in Fig.

. No cantilever in this

study exhibits an ideal rectangular plan view; such devices

were studied previously.

–

Plan view dimensions of all

cantilevers were measured from SEM images using ImageJ

and a S003 carbon grating replica (2160 lines/mm) as refer-

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103705-6

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et al.

Rev. Sci. Instrum.

, 103705 (2012)

TABLE I. Measured plan view dimensions (in micrometer) of all can-

tilevers, as obtained from SEM micrographs. Definitions of the listed dimen-

sions are illustrated in Fig.

Cantilever

dLL

TIP

AC160TS

51.0

...

151

94.4

151

AC240TM

30.2

...

227

196

227

AC240TS

29.6

...

229

199

229

BL-RC150VB(L)

29.8

10.1

...

93.1

83.8

93.1

BL-RC150VB(S)

29.9

10.1

...

51.7

42.7

51.7

ASYMFM

31.0

...

241

207

241

FMR

30.7

...

242

223

235

NCHR

38.3

...

136

107

128

TR400(S)

15.6

...

110

104

...

100

TR400(L)

29.5

...

164

198

...

194

TR800(S)

15.4

...

109

103

...

99.0

TR800(L)

30.4

...

170

206

...

202

Width tapers from clamp to end-tip in the range 32.1–30.6

Width non-uniform along length, and varies between 28.7 and 32.1

Width non-uniform along length, and varies between 36.6 and 39.5

Average width listed in these cases, and used in analysis.

ence; these dimensions are listed in Table

and Fig.

.The

estimated uncertainty in any given dimension measurement is

less than 1%.

1. Olympus cantilevers

The devices from Olympus, namely, AC160TS,

AC240TM, AC240TS, and the BL-RC150VB, TR400,

TR800 series, all have smooth and uniform edges. The

arrow-shaped cantilevers (AC160TS, AC240TS, AC240TM)

are composed of silicon with reflective aluminum coatings;

AC240TM has an additional platinum coating. The imaging

tips for AC160TS, AC240TS, and AC240TM are positioned

at the very end of the cantilever, i.e., the tip coincides with

the maximum extension of the plan view; see Fig.

.Both

biolevers (BL-RC150VB series) are composed of silicon

nitride with a reflective gold coating; their imaging tips also

coincide with the end of the cantilever and are formed by a

depression in the silicon nitride; see Fig.

. The V-shaped

cantilevers (TR400, TR800 series) are also made of silicon

nitride with a reflective gold coating. However, their imaging

tips are set back from the cantilever end. The TR400 and

TR800 cantilevers possess identical plan view geometries,

but have thicknesses of 400 and 800 nm, respectively, as

specified by the manufacturer. Consequently, they present

ideal candidates for demonstrating the invariance of the

measured hydrodynamic function to thickness variations

between devices.

2. Nanoworld cantilevers

The two Nanoworld devices (FMR, NCHR) exhibit sig-

nificantly different geometric features to the other quasi-

rectangular cantilevers of Asylum Research and Olympus.

These cantilevers are composed of silicon with aluminum re-

flective coatings. However, their widths are non-uniform and

vary significantly along their lengths; see Fig.

. Perspective

images of these devices (in Fig.

) reveal that they possess

TIP

(a)

(b)

FIG. 3. Schematic diagrams of all cantilevers in Fig.

, illustrating dimen-

sions listed in Table

. (a) Quasi-rectangular cantilevers; (b) V-shaped can-

tilevers. Position of imaging tip shown as a square dot.

quasi-trapezoidal cross sections with pronounced scalloping

of their sloped edges along the cantilever length. Their imag-

ing tips are set back from the cantilever end. Given their non-

uniform geometries, these devices allow the robustness of the

original and general methods to be assessed in the presence of

significant non-idealities.

3. Asylum Research cantilever

The cantilever from Asylum Research, ASYMFM, is of

identical geometry to the Olympus AC240TS and AC240TM

devices. It is also composed of silicon but has a CoCr mag-

netic coating. The imaging tip coincides with the end of the

cantilever. It possesses a slightly tapered plan view with its

width slightly narrowing from the clamp to the cantilever end;

see Table

B. Spring constants of test cantilevers

The dynamic spring constant of each test cantilevers was

measured noninvasively by monitoring its Brownian motion

under ambient conditions using a laser Doppler vibrometer

(LDV). This approach eliminates additional uncertainties in-

herent in the standard AFM thermal method

that

arise from calibration of the AFM photodiode displacement

sensitivity. These uncertainties originate from a number of

factors and include required corrections for laser position and

finite spot size, non-ideal contact and friction between the

cantilever tip and sample, compliance of the sample, and con-

version factors relating the cantilever angle-to-displacement

under static and dynamic loads;

these vary with the

cantilever used. Since the spring constant is inversely propor-

tional to the displacement squared in this method, the addi-

tional uncertainty introduced by these effects is doubled in

all AFM thermal method measurements. Elimination of these

additional uncertainties is thus highly desirable in the present

study and is achieved by using a LDV; see Appendix

for

details. Instruments calibrated to the SI standard (e.g., see

Ref.

) provide an alternate approach for measuring the

spring constants of the test cantilevers; these were not imple-

mented in this study.

We emphasize that these LDV measurements are needed

only for the test cantilevers to determine their hydrodynamic

functions. The general method is specified independently of

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103705-7

Sader

et al.

Rev. Sci. Instrum.

, 103705 (2012)

TABLE II. Measured dynamic spring constants

of the test cantilevers,

at their imaging tip positions, using a laser Doppler vibrometer (uncertainty

based on a 95% confidence interval). Resonant frequencies

and quality

factors

in air (1 atm) also shown.

Cantilever

(N/m)

(kHz)

AC160TS

57.3

1.9

370

646

AC240TM

1.65

0.065

65.9

162

AC240TS

2.90

0.13

83.0

213

BL-RC150VB(L)

0.00683

0.00017

12.4

12.5

ASYMFM

2.13

0.055

69.4

187

FMR

2.19

0.085

70.6

163

NCHR

33.9

1.5

299

480

TR400(S)

0.0971

0.0052

34.6

39.7

TR400(L)

0.0293

0.0027

11.8

21.5

TR800(L)

0.194

0.0062

22.9

57.1

the LDV instrument once these measurements have been per-

formed. The determined hydrodynamic functions then allow

for accurate noninvasive calibration of these cantilever types

using the general method, Eq.

(8)

, regardless of their plan

view dimensions, composition, and thickness variations; see

Secs.

and

Several measurement points along the cantilever were in-

terrogated to ensure robust measurements. Details of these

measurements are given in Appendix

. The measured dy-

namic spring constants at the imaging tip position of each

cantilever are listed in Table

; uncertainties due to fitting

the spring constant data at all measurement points are listed

in Table

and discussed in Appendix

. These are typically

between

2% and

5% based on a 95% confidence inter-

val. Since the LDV does not possess additional significant un-

certainties in velocity, due to its inherent calibration relative

to the speed of light, these observed fit uncertainties specify

the total uncertainties in the measured spring constants; see

Appendix

. The measured resonant frequency and quality

factor in air (1 atm) for each cantilever are also given, and

possess smaller uncertainty: between

0.0006% and

0.02%

for the resonant frequency, and

0.3% and

1% for the qual-

ity factor.

The spring constants of two test cantilevers, BL-

RC150VB(S) and TR400(L), could not be measured; also see

Appendix

C. Hydrodynamic functions

The hydrodynamic function for each test cantilever was

determined by measuring its resonant frequency and quality

factor as a function of gas pressure; see Sec.

II D

. These re-

sults were then combined with the measured plan view di-

mensions and spring constant (Secs.

III A

and

III B

), and

substituted into Eqs.

(7a)

and

(9)

to give the required hydro-

dynamic function. The experimental procedure and apparatus

developed for the gas pressure measurements are detailed in

Appendix

To ensure accurate data collection, independent measure-

ments were performed using two different gases: dry nitro-

gen (N

) and carbon dioxide (CO

). One reason for using

is that it possesses a kinematic viscosity (shear viscos-

0.1

0.3

0.2

0.3

0.5

Λ(

)

AC240TS

FIG. 4. Measured hydrodynamic function of the AC240TS device using N

(open circles) and CO

(filled circles). Theoretical result (solid line) calcu-

lated using Eq. (20) of Ref.

ity/density) a factor of two lower than that of air, at 1 atm.

This in turn yields a Reynolds number, Re, a factor of two

higher than the result in air; see Eq.

(7a)

. Any subsequent

measurement on another cantilever in air, with a resonant fre-

quency twice that of the test cantilevers, will thus be within

the characterized range of Re. Importantly, the use of CO

eliminates the need to use gas pressures larger than 1 atm to

increase the upper limit of Re.

1. Quasi-rectangular cantilevers

We first assess the robustness of the characterization pro-

tocol underpinning the general method. This is illustrated for

cantilevers with quasi-rectangular plan views. The hydrody-

namic function of the AC240TS device is given in Fig.

.In-

dependent measurements in N

and CO

overlap precisely,

illustrating the accuracy of the measurements – we remind

the reader that the kinematic viscosities of these gases differ

by a factor of two. Also shown is the theoretical result for

the hydrodynamic function, as calculated using Eq. (20) of

Ref.

. Note the excellent agreement between all three data

sets – there are no adjustable parameters in this comparison.

Since the theoretical result for the hydrodynamic function is

used implicitly in the original method (for rectangular can-

tilevers), the level of agreement in Fig.

demonstrates the

validity of the original method for this type of device. The

original method and the general method coincide in this case

because their hydrodynamic functions overlap.

Figure

compares the measured hydrodynamic func-

tions for the AC240TS, AC240TM, ASYMFM, and FMR de-

vices, which have similar plan view geometries. Note that

(i) these devices have different stiffnesses, resonant frequen-

cies, and quality factors, see Table

, and (ii) measurements

on each device were performed in both N

and CO

.The

measured hydrodynamic functions for all these devices again

overlap with each other and with the theoretical prediction of

Ref.

, see Fig.

. The FMR device possesses a scalloped

trapezoidal cross section, in contrast to the uniform cross sec-

tions of the other devices; see Fig.

. The average of the

maximum width of such cantilevers should be used in the

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103705-8

Sader

et al.

Rev. Sci. Instrum.

, 103705 (2012)

0.1

0.3

0.2

0.3

0.5

Λ(

)

AC240TS, AC240TM

FMR, ASYMFM

FIG. 5. Measured hydrodynamic function of the AC240TS (open circles),

AC240TM (filled circles), ASYMFM (squares) and FMR (diamonds) de-

vices using both N

and CO

. Theoretical result (solid line) calculated using

Eq. (20) of Ref.

original and general method; see Sec.

II C 3

. As discussed in

Sec.

II C

, this non-uniformity in cross section is inconsequen-

tial to both methods – the measured hydrodynamic functions

for all devices overlap over the entire Reynolds number range

studied. This establishes that the original and general methods

coincide for all these devices and demonstrates the robustness

of these methods.

2. Non-rectangular cantilevers

Next, we demonstrate the validity of the general method

for non-rectangular geometries by comparing the measured

hydrodynamic functions for the TR400(L) and TR800(L) de-

vices. These cantilevers possess identical plan view geome-

tries, but differ in thickness by a nominal factor of two. Their

measured resonant frequencies also differ by a factor of ap-

proximately two (11.8 and 22.9 kHz) while their dynamic

stiffnesses are vastly different, in agreement with theoretical

considerations; see Table

. Since the hydrodynamic func-

tion is only sensitive to the plan view geometry, and this is

identical for these two devices, the independently measured

hydrodynamic functions for these two devices are expected

to coincide. This is borne out in Fig.

, where overlap of the

(independent) measurements on these two different devices is

observed. This fundamental property is used in Sec.

III C 3

derive a single fit function for the hydrodynamic function of

devices with identical plan view geometries.

The AC160TS and BL-RC150BV(L) cantilevers also

have strongly non-ideal geometries. The AC160TS device

does not satisfy the fundamental requirement for use of the

original method: a rectangular plan view of large aspect ra-

tio (length/width); it has an arrow shaped geometry. While

the BL-RC150BV(L) device has a rectangular geometry, its

aspect ratio is small (length/width

∼

3), with a significant

imaging tip. The performance of the original and general

methods for these cantilevers, in comparison to higher aspect

ratio quasi-rectangular cantilevers, is discussed in Sec.

Appendix

. Overlap in the measured hydrodynamic func-

tions using N

and CO

, as observed in Figs.

–

, was also

0.01

0.03

0.1

0.3

0.5

TR400(L), TR

00(L)

Λ(

)

FIG. 6. Measured hydrodynamic functions of the TR400(L) (open circles)

and TR800(L) (filled circles) devices using both N

and CO

. These devices

have identical plan view geometries but their thicknesses differ by a nominal

factor of two.

found for the AC160TS and BL-RC150BV(L) devices (data

not shown).

The results in Figs.

–

and measurements for the

AC160TS and BL-RC150BV(L) devices, serve to demon-

strate the robustness of the experimental protocol and valid-

ity of the general method for both quasi-rectangular and non-

rectangular plan view geometries of varying composition.

3. Formulas for hydrodynamic functions

To facilitate their use in practice, analytical formulas for

all hydrodynamic functions were obtained by fitting the mea-

sured data to the following functional form:

(Re)

log

(10)

where

, and

are constant coefficients, specific to each

type of cantilever. This functional form was chosen because

the hydrodynamic function is approximately linear on a log-

log scale and is a monotonically decreasing function of Re;

see Figs.

–

. Fitting the data to a second-order polynomial

on a log-log scale yields the general form in Eq.

(10)

.The

resulting fit coefficients

, and

for all test devices are

presented in Table

III

Three sets of devices have identical plan view geome-

tries: (1) AC240TS, AC240TM, ASYMFM; (2) TR400(L),

TABLE III. Coeffi

cients

,and

in functional form Eq.

(10)

for mea-

sured hydrodynamic functions

(Re) of test cantilevers.

Cantilever

AC160TS

0.7779

−

0.7230

0.0251

AC240TM, AC240TS, ASYMFM

0.8170

−

0.7055

0.0423

BL-RC150BV(L)

1.0025

−

0.7649

0.0361

BL-RC150BV(S)

...

−

0.7613

0.0374

FMR

0.8758

−

0.6834

0.0357

NCHR

0.9369

−

0.7053

0.0438

TR400(S), TR800(S)

1.5346

−

0.6793

0.0265

TR400(L), TR800(L)

1.2017

−

0.6718

0.0383

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103705-9

Sader

et al.

Rev. Sci. Instrum.

, 103705 (2012)

0.1

0.2

0.5

0.2

0.5

Λ(

)

AC240TS, AC240TM

ASYMFM

FIG. 7. Combined data of measured hydrodynamic functions for AC240TS,

AC240TM, and ASYMFM devices (red dots), and resulting fit of this data

to Eq.

(10)

(solid line). Fit coefficients are given in Table

III

. These devices

have identical plan view geometries.

TR800(L); (3) TR400(S), TR800(S). The hydrodynamic

functions within each set must therefore coincide. Single fit

functions for each set were determined in the following man-

ner. Set (1): The fit function was evaluated by combining

independent data from all cantilevers in this set, and fitting

the result to Eq.

(10)

. The combined data and fit function

for Set

(1)

is given in Fig.

; similar fits were obtained for

other sets and devices (not shown). Set (2): Since the mea-

sured spring constant of TR400(L) possesses greater uncer-

tainty than TR800(L) [see Table

and Appendix

], its mea-

sured spring constant was not used but chosen such that its

hydrodynamic function overlapped precisely with that of the

TR800(L) device over the entire Reynolds number regime.

This yielded a spring constant 5% lower than that reported

in Table

, which is within its measured uncertainty. These

combined data were then fit to Eq.

(10)

. Set (3): The spring

constant of the TR800(S) device could not be measured; see

Appendix

. Rather than discarding this data, its spring con-

stant was chosen such that the hydrodynamic function for the

TR800(S) coincided with that of the TR400(S) device over

the entire Reynolds number range studied – again excellent

overlap was observed. The resulting data were fit to Eq.

(10)

While single devices could have been used to determine the

hydrodynamic function for each of these sets, the chosen re-

dundancy of devices facilitates accurate evaluation of the hy-

drodynamic functions.

a. Evaluation of unknown coefficients

Note that the coeffi-

cient

for the BL-RC150VB(S) test device is not spec-

ified, because its spring constant could not be measured

and no other device studied possesses an identical geome-

try. Nonetheless, the results in Table

III

can in the future be

used to generate the hydrodynamic function,

(Re), for this

device model by performing independent measurements on

other test cantilevers, of identical geometry. The dimensions

and materials need not be the same, but the plan view geom-

etry (shape) must remain unchanged. All that is required is

knowledge of the plan view dimensions of the additional de-

vice, measurement of its resonant frequency and quality fac-

tor in gas (at a single known pressure) and its stiffness. From

these measurements, the Reynolds number, Re, of the addi-

tional device follows and the value of the hydrodynamic func-

tion at this Reynolds number is immediately determined using

Eq.

(9)

. This value can then be substituted into the fit formula

for

(Re), in Eq.

(10)

and Table

III

, from which the coeffi-

cient

is uniquely specified. Using this procedure, the hy-

drodynamic function

(Re) for the BL-RC150VB(S) device

can be evaluated.

b. Refinement of hydrodynamic functions

The procedure out-

lined immediately above allows the accuracy of the pre-

sented hydrodynamic functions to be enhanced. This for

example could involve calibrating additional test cantilevers

using methodologies referenced to the SI standard.

These

additional measurements would lead to a refinement in the

coefficient

c. Uncertainty analysis

All fit functions specified in Table

III

possess uncertainties of less than

1% (based on a 95% con-

fidence interval), relative to measured data for the hydrody-

namic function. Note that the data are well represented by a

simple power-law as specified by

(see above discussion

regarding linearity on a log-log scale) – the higher-order cor-

rection due to

exerts a minor influence. Measurements of

the dynamic spring constants introduce additional uncertainty

of approximately

2% to

5% (also based on a 95% confi-

dence interval) into the coefficient

only; the coefficients

and

are unaffected by the spring constant. It is noteworthy

that individual fits to the hydrodynamic functions for devices

with identical plan views differ by only a few percent, e.g., in-

dividual fits to data for AC240TM, AC240TS, and ASYMFM.

This is in line with the measured spring constant uncer-

tainty and the use of an empirical fit function to represent

the measured data; see above. Combining the data from de-

vices with identical plan view geometries yields a more accu-

rate estimate of the hydrodynamic function, and is reported in

Table

III

d. Scaling behavior

The scaling chosen in Eq.

(7)

was mo-

tivated by the expectation that the dominant hydrodynamic

length scale is given by the minimum plan view dimension

of the cantilever. This is strongly supported by the values of

in Table

III

, which are all of order unity, i.e., the hydrody-

namic functions

(Re) are of order unity when the Reynolds

number is one, as required; see discussion in Sec.

II B

Note that the hydrodynamic functions for the V-shaped

cantilevers, TR400 and TR800, are larger in magnitude to

those for other devices. This is expected because V-shaped

cantilevers possess two skewed rectangular arms, enhancing

the net hydrodynamic load and hence energy dissipation.

It is interesting that the hydrodynamic functions of all

devices exhibit a power-law dependence on Reynolds num-

ber of approximately Re

−

, i.e.,

∼−

7; see Table

III

This can be understood by considering the following asymp-

totic limits. In the high Reynolds number limit (Re

1), the

hydrodynamic function is expected to scale as Re

−

due to

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103705-10

Sader

et al.

Rev. Sci. Instrum.

, 103705 (2012)

the presence of thin viscous boundary layers in the vicinity of

the cantilever surface. In the opposite (creeping flow) limit of

low Reynolds number (Re

1), a scaling behavior of Re

−

must be exhibited. Thus, the observation in measurements of

a power-law dependence intermediate to these two limiting

cases is not unexpected, since the Reynolds number is of or-

der unity for all devices studied. This property can be used to

devise an approximate method that circumvents the need for

pressure measurements. This is discussed in Sec.

III D

D. Simplified approximate implementation

of the general method

The general method relies on knowledge of the hydro-

dynamic function,

(Re), which can be measured for a sin-

gle “test” device using the gas pressure protocol discussed in

Sec.

II D

; these measurements are reported in Sec.

III C

.Im-

portantly, the preceding discussion demonstrates that the hy-

drodynamic function of a microscale device (whose Reynolds

number is of order unity) is well approximated by

(Re)

≈

−

(11)

where the coefficient

depends only on the plan view geom-

etry of the device model in question.

Since only one coefficient is unknown in Eq.

(11)

,asin-

gle measurement on a test device is required to determine its

value. For example, by measuring the spring constant, reso-

nant frequency, and quality factor of a test device in air, the

value of

then directly follows from Eqs.

(7a)

(9)

, and

(11)

ρb

Lω

∣

test device

(12)

where all parameters on the right hand side of Eq.

(12)

are

determined from this single test measurement.

Equations

(11)

and

(12)

then uniquely specify the hy-

drodynamic function for arbitrary Reynolds numbers. While

this approach introduces a systematic error into the general

method (since Eq.

(11)

is approximate), this error is expected

to be small since the power-law dependence of the hydro-

dynamic function is bounded between

−

0.5 and

−

1; see

Sec.

III C

. This error is of course minimized if the Reynolds

number of the test device is comparable to (other) devices of

the same geometry to be calibrated.

If the test cantilever has identical plan view dimensions to

any subsequent (uncalibrated) device, Eqs.

(9)

and

(12)

sim-

plify, yielding

test

(

test

)

−

(13)

where

0.7 (see above), the subscript “test” refers to the

(known) test cantilever parameters, and all other parameters

are for the uncalibrated device. Equation

(13)

thus enables

the spring constant

for an uncalibrated device to be eas-

ily determined from measurement of its resonant frequency

and quality factor alone. The actual dimensions of the test

and uncalibrated devices are not required; they simply need

to be identical. Note that small deviations in the plan view

dimensions between the test and uncalibrated devices have a

minimal effect, for reasons discussed in Sec.

III F

TABLE IV. Conversion factors relating the dynamic

and static

spring

constants of all devices in this study. These are evaluated at the imaging tip

positions; see Table

. Finite element (FE) analysis is used for all calcula-

tions based on geometries as measured from SEM micrographs. The FE mesh

is systematically refined to ensure convergence of 99.9%. Poisson’s ratio of

0.25 is used in all calculations.

Cantilever

AC160TS

1.101

AC240TM, AC240TS, ASYMFM

1.043

BL-RC150VB(L)

1.035

BL-RC150VB(S)

1.042

FMR

1.029

NCHR

1.036

TR400(S), TR800(S)

1.054

TR400(L), TR800(L)

1.072

The above approximate implementation of the general

method facilitates the calibration of microscale devices in sit-

uations where equipment for the required pressure measure-

ments is not available.

E. Conversion factors for static and dynamic

spring constants

As discussed in Sec.

, either the static or dynamic spring

constant is needed for quantitative measurements, depending

on the mode of operation. Importantly, the general method

can be applied to measure both the static and dynamic spring

constants. In results for the hydrodynamic function presented

in Sec.

III C

, the dynamic spring constant of the fundamental

flexural mode was used. To calibrate the static spring con-

stant associated with a static force applied at the imaging tip

position, conversion factors between these two spring con-

stants are required. These were calculated using finite element

analysis,

and are given in Table

. Devices with identical

plan view geometries gave identical results to within dimen-

sional uncertainty; the average of these results is reported.

Note that the conversion factors in Table

are dimension-

less and thus independent of the cantilever dimensions.

F. Effect of dimensional uncertainty

on the general method

Next, we study the effect of uncertainty in the plan view

dimensions on the spring constant determined by the general

method. Naive inspection of Eq.

(8)

appears to suggest that

the resulting uncertainty in the spring constant scales with the

cube of the plan view dimensions. However, the width of the

cantilever,

, is embedded in the hydrodynamic function via

the Reynolds number; see Eq.

(7a)

. Since the hydrodynamic

function scales as Re

, to leading order, it follows that the

spring constant scales as

2(1

)

in the general method;

note that

∼−

7 as discussed above. As such, the mea-

sured spring constant exhibits a weak dependence on the can-

tilever width with sub-linear scaling; the dependence on can-

tilever length is linear. This leads to an overall uncertainty in

the measured spring constant that scales with the

∼

3/2 power

of the plan view dimensions. Consequently, knowing the plan

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103705-11

Sader

et al.

Rev. Sci. Instrum.

, 103705 (2012)

view dimensions only approximately, imposes a weak penalty

on the overall uncertainty of the method.

Measurement of the hydrodynamic function

(Re), for

a single test cantilever, also exhibits this robustness to di-

mensional uncertainty. The uncertainty in the measured hy-

drodynamic function again scales with the

∼

3/2 power of the

test device plan view dimensions. This is evident by compar-

ing Eqs.

(7a)

and

(9)

, which are used in the measurement of

(Re), to the power-law dependence, Re

, of the hydrody-

namic function.

The general method has been implemented so that the

dynamic spring constant is determined at the imaging tip po-

sition of all test cantilevers. If the cantilever under consid-

eration has a different tip position, relative to the cantilever

length, the spring constant will need to be adjusted; the imag-

ing tip positions of the test cantilevers are specified in Table

This adjustment can be achieved using the following proper-

ties: (i) the normal spring constant varies approximately with

the cube of the distance along the cantilever length,

–

and

(ii) this spring constant is insensitive to tip position variations

parallel to the clamp.

G. Implementation of the original and general

methods

We now summarize some practical issues relevant to im-

plementation of the original and general methods:

(1) Measurement of the thermal noise spectrum facilitates

determination of the resonant frequency and quality fac-

tor, by eliminating any spurious effects due to the fre-

quency response of the piezoactuator.

Thus, while

active excitation of the cantilever can be used to mea-

sure these fundamental quantities, and may be desirable

for very stiff cantilevers,

interrogation of the thermal

noise poses fewer issues and is simple to implement.

(2) Hydrodynamic functions for the original and general

methods were determined for cantilevers well away from

any surface, i.e., gas surrounding the cantilever was as-

sumed to be unbounded. This implicit assumption must

therefore be satisfied in all implementations of these

methods, since proximity to a surface can reduce the

measured quality factor due to squeeze film damping

– this would lead to an (artificial) underestimate of

the spring constant. Since the dominant hydrodynamic

length scale for many cantilevers is given by their width,

the thermal noise spectra of cantilevers should be mea-

sured at least several widths away from any solid sur-

face. Exploration of these effects and measurement pro-

tocols for implementation of the original and general

methods have been reported in Refs.

, and

(3) The resonance behavior of a cantilever is independent

of the measurement position along its axis, provided a

measurement is not taken at a zero of the slope of its de-

flection function (for which no signal may result). This

is always satisfied for the fundamental flexural mode of a

cantilever, away from the clamp. Consequently, the orig-

inal and general methods are insensitive to position of

the measurement laser on the cantilever plan view and

the laser spot size. These quantities are not required and

can be specified arbitrarily provided the signal-to-noise

is sufficient.

(4) The original and general methods yield the spring con-

stants directly and do not require knowledge of the ab-

solute deflection of the cantilever. Consequently, the

equipartition theorem can be used to calibrate the dis-

placement sensitivity of the optical lever deflection sys-

tem commonly used in the AFM, from the measured

spring constant. This approach was proposed and sys-

tematically studied in Ref.

for a series of rectangu-

lar cantilevers – the same method is applicable to non-

rectangular cantilevers.

(5) Some cantilevers possess a significant “over-hang” at the

clamped end, due to the manufacturing process, e.g., see

TR400(S) and TR800(S) devices in Fig.

. Since the

general method only requires the mode shape of the vi-

brating structure to be identical to the test cantilever,

such non-idealities do not pose an issue. The displace-

ment near the base of the cantilever is always small, and

hence its contribution to the net energy dissipation is

small. The general method is therefore expected to be

robust to such non-ideal variations.

(6) An “under-hang” would shorten the overall length of

the vibrating structure and could affect the net hydro-

dynamic load and energy dissipation. To understand its

effect, consider a long rectangular cantilever of high as-

pect ratio (length/width). The presence of an under-hang

would reduce the overall length of the cantilever, while

not changing its geometry. Since the original method

scales linearly with the cantilever length, this would have

a weak effect for under-hangs of relatively small length

in comparison to the cantilever length, if the original

length were used; use of the shortened length would

eliminate this effect. For non-rectangular cantilevers,

their geometry may also be affected in addition to a

change in cantilever length. Nonetheless, this effect is

still expected to be weak unless the under-hang were a

significant fraction of the cantilever length.

(7) The original and general methods are most easily imple-

mented in air, at 1 atm. The density and viscosity of air

are thus required. These are weakly dependent on atmo-

spheric variations in temperature and pressure, and are

insensitive to humidity. Even so, for more precise mea-

surements the temperature and pressure should be mea-

sured and SI data used to determine the true density and

viscosity.

IV. CONCLUSIONS

Ability to characterize the static and dynamic mechan-

ical properties of AFM cantilevers and in general, micro-

and nanomechanical devices, is critical to many applications.

Manufacturing and measurement specifications often lead to

device geometries that are complex and non-ideal. Dynamic

methods provide a versatile tool for extracting the mechani-

cal properties of such devices. In this study, previous work on

ideal rectangular cantilevers has been extended and applied

to actual devices currently employed. Specifically, we have

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103705-12

Sader

et al.

Rev. Sci. Instrum.

, 103705 (2012)

presented hydrodynamic functions for a series of non-ideal

and non-rectangular cantilevers that are commonly used in

practice. These functions allow the spring constants of these

cantilevers to be easily and routinely determined from mea-

surements of the resonant frequency and quality factor in fluid

(such as air). Performance of the original method

in the

presence of non-idealities was also examined. For all quasi-

rectangular cantilevers of high aspect ratio (length/width), the

original and general methods agreed closely. This demon-

strated the robustness of both methods to the presence of

non-idealities. For highly non-rectangular cantilevers, robust-

ness of the experimental protocol and general method was

also confirmed and deviations between the original and gen-

eral methods were discussed. Simple and accurate formu-

las for the hydrodynamic functions of all cantilevers were

presented to facilitate implementation in practice. A simpli-

fied approximate implementation of the general method was

proposed, facilitating implementation of the general method

when the specified gas pressure measurements are not avail-

able. Finally, conversion factors relating the dynamic and

static spring constants of all cantilevers studied were pre-

sented, together with a discussion of practicalities for imple-

menting the original and general methods.

ACKNOWLEDGMENTS

The authors would like to thank the Melbourne Cen-

tre for Nanofabrication for access to the MSA-400 Micro

System Analyzer, and Toby Ban, Jerome Eichenberger, and

Mario Pineda from Polytec Headquarters, Irvine, CA, for

use of the MSA-500 Micro System Analyzer. This research

was supported by the Australian Research Council Grants

Scheme. An iPhone application implementing the general

method for the cantilevers used in this study is available from:

http://www.ampc.ms.unimelb.edu.au/afm/

APPENDIX A: SPRING CONSTANT MEASUREMENTS

The dynamic spring constants of the fundamental flexural

modes of all test cantilevers were measured using a LDV;

–

MSA-400 and MSA-500 Micro System Analyzers, Polytec

(Waldbronn, Germany). The LDV provides an independently

calibrated measurement of velocity using a laser system that

incorporates a modified Mach-Zehnder interferometer.

The

spot size of the incident LDV laser is nominally 1

m, al-

lowing for precise placement and subsequent measurement of

cantilever velocity at any position on its plan view – all can-

tilever plan view dimensions are an order of magnitude larger

than the optical spot size.

The spring constants were determined by monitoring the

Brownian fluctuations of each cantilever. The equipartition

theorem provides a unique connection between the mean

squared velocity of the device,

, and its dynamic spring

constant,

, at the measurement position

(A1)

where

is Boltzmann’s constant and

is absolute temper-

ature. Note that

and

in Eq.

(A1)

are specified at the

same measurement position on the cantilever.

1. Spring constant at the imaging tip position

LDV measurement sensitivity is intrinsically dependent

upon strong reflection of the incident laser – sensitivity is de-

graded on highly curved surfaces. Curved or slanted surfaces

are thus not easily interrogated, and direct measurement at the

tip position was therefore not always possible. Spring con-

stants at the tip positions were always determined by mea-

suring the dynamic spring constant at a series of defined

positions along the cantilever length – absolute distance cal-

ibration of these points was not required. Interpolation be-

tween these data point values and extrapolation allowed for

the dynamic spring constant at the tip position to be acquired.

This was achieved by fitting the measured spring constants to

the function

tip

−

αx

)

−

(A2)

where

tip

is the required spring constant at the imaging tip

position,

is the spring constant at the LDV measurement

position,

is the (uncalibrated) measurement position along

the cantilever relative to the imaging tip position, and

and

are constants. Choice of this functional form is driven by the

corresponding result for the static spring constant, for which

The dynamic spring constant possesses a

slightly weaker dependence on position, which motivates the

use of an adjustable constant power-law,

To demonstrate the validity of this approach, a compar-

ison is made to calculations from Euler-Bernoulli theory for

a beam with a uniform cross section along its major axis; see

Fig.

. Note that the data point closest to the cantilever end is

a significant distance away (10% of the cantilever length). An

absolute length scale along the cantilever axis is not required

and is automatically determined by fitting the data to Eq.

(A1)

1.2

1.4

1.6

1.8

tip

FIG. 8. Theoretical simulation of fitting procedure, Eq.

(A2)

, to extract the

dynamic spring constant at the imaging tip position. Solid circles are data

calculated using Euler-Bernoulli beam theory. Lines are fits to this data using

Eq.

(A2)

. Each unit on the horizontal axis is 0.02

,where

is cantilever

length; point furthest back from end-tip is at a distance of 0.2

. Imaging tip

coincides with the end-tip here. (Upper fit curve [blue])

2; (Lower fit

curve [red])

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