DESIGNS

Design of Advanced Microscopic Cantilevers: Improving AFM Cantilever Spring Constants

Robby – Mech. Eng. & Mgt. – 2120870 Supervisor: Dr Phillip Dobson

University of Glasgow, charity number SC004401

INTRODUCTION

Background Atomic force microscopy (AFM) works by scanning a probe over a sample in horizontal direction while controlling (and mostly maintaining) the deflection. However, as frequently found, other reaction forces and moments may also simultaneously occur and cause the AFM cantilever to be deflected in unwanted directions. Therefore, it is crucial to design an advanced AFM cantilever that has not only relatively low natural frequency and normal spring constant but also high spring constants in other directions.

!

!

!

!

!

!

!

!!

!

!

!

!

Longitudinal

Normal

Lateral Torsional or Twisting

Longitudinal Bending

Lateral Bending

Figure 1: Coordinate system used for cantilever. It comprises 3 linear directions and 3 rotational directions.

Aims of the Project •  To design an AFM cantilever that has relatively low natural frequency, low

normal spring constant, and high spring constants in other directions; •  To design an AFM cantilever that has high spring constant ratios.

This project aims to geometrically improve the AFM cantilever by analysing several structural variations.

Parameters First parameter is the natural frequency, f, that indicates the dominant oscillation of the tip in normal direction. The deflections of one particular node of the tip are inspected in order to calculate the spring stiffness. Each of the spring constants is equal to the load divided by its associated deflection. Other parameters are the ratios of each of the other spring constants to the normal spring constant, e.g. klat/knorm.

Table 1: Spring constants calculated by dividing the load with the associated deflection.

!"#$%&'()*#)%

*%!+,-./012+"3%+,-.

'1)%4

&'(5%0!().,16,2$#!217 !80%" #9

444

444

:;;4<=<9>;?><:

01--.=!",=%"3+,%'2+,@,=2)+/3($$(/,*1-!1'

<;;

A;;

<

:;

!"#$%&'()*#)%

*%!+,-./012+"3%+,-.

'1)%4

&'(5%0!().,16,2$#!217 !8)0( #9

444

444

:;;4<=<9>;?><:

01--.=!",=%"3+,%'2+,@,=2)+/3($$(/,*1-!1'

<;;

A;;

A:

<

:;

!"#$%&'()*#)%

*%!+,-./012+"3%+,-.

'1)%4

&'(5%0!().,16,2$#!217 !8*#2)0( #9

444

444

:;;4<=<9>;?><:

01--.=!",=%"3+,%'2+,@,=2)+/3($$(/,*1-!1'

<;;

A;;

<;

:;

<

:;

!"#$%&'()*#)%

*%!+,-./012+"3%+,-.

'1)%4

&'(5%0!().,16,2$#!217 !8*#2"(0 #9

444

444

:;;4<=<9>;?><:

01--.=!",=%"3+,%'2+,@,=2)+/3($$(/,*1-!1'

<;;

<A:

0:;

:;

<

!"#$%&'()*#)%

*%!+,-./012+"3%+,-.

'1)%4

&'(5%0!().,16,2$#!217 !8/#7' #9

444

444

:;;4<=<9>;?><:

01--.=!",=%"3+,%'2+,@,=2)+/3($$(/,*1-!1'

<;;

<A:

:;

0:;

0:;

:;

0B<

<

!"#$%&'()*#)%

*%!+,-./012+"3%+,-.

'1)%4

&'(5%0!().,16,2$#!217 !89%0(! #:

444

444

;<<4=>=:?<@?=;

01--.>!",>%"3+,%'2+,A,>2)+/3($$(/,*1-!1'

==<<

B<<

0B<

0;<

0;<

0;<

0B<

0;<

;<;<

B;B;

a b c d e f

a b c d e f g h i

!"#$%&'()*#)%

*%!+,-./012+"3%+,-.

'1)%4

&'(5%0!().,16,2$#!217 '89 #:

444

444

;<<49=9:><?>9;

01--.=!",=%"3+,%'2+,@,=2)+/3($$(/,*1-!1'

99<

9<;

0A

!"#$%&'()*#)%

*%!+,-./012+"3%+,-.

'1)%4

&'(5%0!().,16,2$#!217 '89 #:

444

444

;<<4=>=:?<@?=;

01--.>!",>%"3+,%'2+,A,>2)+/3($$(/,*1-!1'

=< =

=<<

=<<

0B

!"#$%&'()*#)%

*%!+,-./012+"3%+,-.

'1)%4

&'(5%0!().,16,2$#!217 '89 #9

444

444

:;;4<=<9>;?><:

01--.=!",=%"3+,%'2+,@,=2)+/3($$(/,*1-!1'

<<;

<;A:

A:

A: A:

!"#$%&'()*#)%

*%!+,-./012+"3%+,-.

'1)%4

&'(5%0!().,16,2$#!217 '89!/( #9

444

444

:;;4<=<9>;?><:

01--.=!",=%"3+,%'2+,@,=2)+/3($$(/,*1-!1'

<<;

<;

A: A:

A:A:

<;9? 9:

a b c d

a b c

d e f

!"# $

!"#$

%!"#&$

!""# $

'""#$

!#$

!#$

!#$

'#$

!#$

("# $

!""#$

!#$

!"# $ a

b

RESULTS

!"#

!""

!$%&

#"

'()('*

!##

'""

+,

'+%-

'()('$

#

#

"%&

"%*

$!

,,'!

'%$, &" '%$

!-#'&"

,"'%+

!""

$"'%+

,""

!"#

!""

!$%&

#"

'()('*

!##

'""

+,

'+%-

'()('$

#

#

"%&

"%*

$!

,,'!

'%$, &" '%$

!-#'&"

,"'%+

!""

$"'%+

,""

!"#

!""

!$%&

#"

'()('*

!##

'""

+,

'+%-

'()('$

#

#

"%&

"%*

$!

,,'!

'%$, &" '%$

!-#'&"

,"'%+

!""

$"'%+

,""

a b c

Discussions Non-uniform thickness cantilever (T-NonUni) has smaller spring constant ratios than uniform thickness cantilever. Rectangular-shaped cantilever (S-Rec) has marginally superior parametric values among all shapes. The cantilever with shifted-round perforation (H-CirS) shows slightly better values as well. Assembling multiple beams into one AFM cantilever enhance non normal spring constant values, yet this also increases the normal spring constant, resulting in lower spring constant ratios. Adding a preventive structure on top of the cantilever proves effective in raising the torsional spring constant.

Conclusions The chosen final design notably increases the other spring constant values while only slightly raises the natural frequency and the normal spring constant of the AFM cantilever compared with common ones. The design also improves all spring constant ratios.

Cantilever Type

f (kHz)

knorm (N/m)

klat (N/m)

klong (N/m)

ktor (10-9) (Nm/rad)

klatbend (10-6) (Nm/rad)

klongbend (10-9) (Nm/rad)

Rectangular 34.02 0.58 111.43 69.63 12.60 123.15 14.58 Dagger 36.23 0.58 107.89 67.91 12.19 116.91 14.53

V-Shaped 31.64 0.16 26.65 16.59 3.01 76.71 3.32 F-H-CirS-N-2 38.09 2.75 2,170.43 349.32 426.59 6,237.45 11,376.03 Table 2: Comparison of natural frequencies and spring constants between the chosen

final design and common AFM cantilevers.

Figure 2: Comparison of spring constant ratios between the chosen final design and common AFM cantilevers.

!"#

!""

!$%&

#"

'()('*

!##

'""

+,

'+%-

'()('$

#

#

"%&

"%*

$!

,,'!

'%$, &" '%$

!-#'&"

,"'%+

!""

$"'%+

,""

!"#

!""

!$%&

#"

'()('*

!##

'""

+,

'+%-

'()('$

#

#

"%&

"%*

$!,,

'!

'%$, &" '%$

!-#'&"

,"'%+

!""

$"'%+

,""

!"#

!""

!$%&

#"

'()('*

!##

'""

+,

'+%-

'()('$

#

#

"%&

"%*

$!

,,'!

'%$, &" '%$

!-#'&"

,"'%+

!""

$"'%+

,""

a b c Thickness Variations: uniform thickness (T-Uni) (a) and non-uniform thickness (T-NonUni) (b).

Shape Variations: S-Rec (a), S-Tri (b), S-DagTri(c), S-DagCir (d), S-Pawn (e), and S-Keris (f) shaped. Hollowed Structure Variations: H-Solid (a), H-All (b), H-UniJack

(c), H-Rec (d), H-RecS (e), H-Tri (f), H-Cir (g), H-CirS (h), and H-HonCom (i).

Arrangements of Single and Multiple Beams: N-1 (a), N-2 (b), N-4(c), and N-Spi (d).

Preventive Structure Variations Against Rotational Deflections: P-Chip (a), P-Mid (b), P-

Mid-t-10 (c), P-Mid+ (d), P-Mid2 (e), and P-Mid3 (f).

Common Designs: Rectangular (a), Dagger (b), and V- (c) shaped.

Final Designs: F-Solid (a), F-H-CirS (b), and F-H-CirS-N-2 (c).

1.00

10.00

100.00

1,000.00

10,000.00

(log 1

0)

Spring Constants

Rectangular Dagger V-Shaped F-H-CirS-N-2

klat/knorm klong/knorm ktor/knorm (!10-9) (m2/rad)

klatbend/knorm (!10-6) (m2/rad)

klongbend/knorm (!10-9) (m2/rad)

Normal

Spring

Constant

Lateral

Spring

Constant

Longitudinal

Spring

Constant

Torsional

Spring

Constant

Lateral

Bending

Spring

Constant

Longitudinal

Bending

Spring

Constant

!!"#$

! !!"#$!!"#$

!!"#

! !!"#!!"#

!!"#$

! !!"#$!!"#$

!!"#

! !!"#!!"#

!!"#$%&'

! !!"#$%&'!!"#$%&'

!!"#$%&#'

! !!"#$%&#'!!"#$%&#'

Chosen Design