Measuring and testing – Surface and cutting edge testing – Roughness
Reexamination Certificate
2001-08-10
2003-06-17
Larkin, Daniel S. (Department: 2856)
Measuring and testing
Surface and cutting edge testing
Roughness
Reexamination Certificate
active
06578410
ABSTRACT:
FIELD OF THE INVENTION
The present invention relates generally to surface forces measurement instrumentation, and more particularly is a cantilever spring assembly used in probe microscopy.
BACKGROUND OF THE INVENTION
Scanning Probe Microscopes (SPM's) are research instruments that have been in use in universities and industrial research laboratories since the early 1980's. These instruments allow for various imaging of surfaces as well as measurement of the intermolecular forces between two surfaces (or a small tip and a flat surface) in vapors or liquids. The distance resolution is 1 Å, which means that images and forces can be obtained at the atomic level. Over the years, the technique has been improved and its scope extended so that it is now capable of measuring many different surface properties and phenomena.
SPM's such as AFM's (Atomic Force Microscope) and STM's (Scanning Tunneling Microscope) generally consist of a sample surface and a fine round “tip” that is supported at the end of a force-measuring cantilever spring. They operate by first bringing (positioning) the tip near the surface and then moving the tip or surface vertically (contact or tapping mode) or laterally (scanning mode) while measuring the force produced on the tip by the surface. The force is calculated by measuring the deflection of the cantilever spring supporting the tip. These displacements are measured by one of three methods: (1) The most common method is the optical or beam deflection method (bouncing a laser light beam from the end of the cantilever spring and measuring its deflection from where it falls on a quadrant photo detector). (2) A less common method is the resistive method (using resistance, semi-conductor or piezo-resistive strain gauges in a half-bridge configuration. This method is illustrated in U.S. Pat. No. 5,444,244, by Michael D. Kirk et al., issued Aug. 22, 1995). (3) The least common method is the capacitance method (using a standard capacitance bridge). Both normal and lateral (friction) forces acting on the tip can in principle be measured by any of these methods. In cases where friction forces are measured, the AFM is often referred to as a Friction Force Microscope (FFM).
When making “force measurements” at different locations of a surface (i.e., on scanning) with an AFM or FFM tip, one is also recording topographical images of the surface, i.e., using the tip as a microscope (hence the origin of the name Atomic Force Microscope and Friction Force Microscope).
LIMITATIONS OF THE PRIOR ART
The limitations of the prior art may be best understood by first considering the equations that describe the response of a simple cantilever spring to different kinds of forces, both normal and lateral, that give rise to different kinds of spring deflections (such as bending, twisting, and buckling). Considering the equations that pertain to optical and resistive detectors, it can be seen that these two detection systems have inherently different sensitivities to the spring deformation. This analysis also explains why similarly shaped cantilever springs may exhibit high sensitivity to lateral forces when measured with the optical technique, but low sensitivity when measured with a resistive bridge. The criteria needed for a cantilever spring or spring system to have high resolution for both normal and lateral forces can then be established. The present invention illustrates these principles with a new cantilever design that constitutes the preferred embodiment of the invention.
FIG. 1A
shows a “simple cantilever” spring of length L, width b, and thickness t. The spring is clamped at one end, with a rigid tip of length h at the free end. This is the basic design of a typical AFM or FFM cantilever, although other versions, for example, triangular or double (side-by-side) cantilevers (FIG.
1
B and FIG.
1
C), are more commonly used. These design modifications, however, do not change the basic analysis presented here regarding the optimization of cantilever design to measure normal and lateral forces independently and at high sensitivity, which is the object of this invention.
When a normal force F
z
or a lateral force F
x
or F
y
acts normally or horizontally on the tip at point P in
FIG. 1A
, the angular deflections &Dgr;&thgr; at Q may be determined according to the following equations (where &thgr;
ij
refers to deflections of the cantilever about the j-axis due to a force applied along the i-direction, and where E is the Elastic Modulus of the cantilever material):
&Dgr;&thgr;
zx
=L
2
F
z
/2
Ebh
3
in bending mode due to a vertical force
F
z
(1)
&Dgr;&thgr;
xy
=LhF
x
/Ebh
3
in twisting mode due to a lateral force
F
x
(2)
&Dgr;&thgr;
yx
=3
LhF
y
/Ebh
3
in buckling mode due to a lateral force
F
y
(3)
Since in OPTICAL mode the light beam bounces off the surface at Q, the above angles give the angle by which the light is deflected (the total change of angle on reflection is actually 2&Dgr;&thgr;), which in turn is proportional to the sensitivity by which the forces can be measured using a quadrant photodetector. Since in many applications we have F
x
≈F
y
≈F
z
(i.e., for friction coefficients close to 1), we see that high sensitivity to lateral forces (in twisting and buckling modes) compared to normal forces (in bending mode) can be achieved only for high values of h/L (h/L≈1).
In the case of cantilevers in RESISTIVE mode, the angular deflections at the tip P or surface Q given by Eqs. 1-3 are not what is measured by the resistive elements on the cantilever. (If resistive elements were to be placed at Q, they would hardly measure anything when the spring bends because the surface at Q is essentially rigid to local bending and remains flat when the other parts of the cantilever bend). Instead, the resistive elements must be placed along the compliant length of the cantilever, as in the Kirk et al. reference, U.S. Pat. No. 5,444,244. The cantilever bends into arcs of circles whose relevant angles of curvature are given by the following equations (cf. FIG.
1
A):
&Dgr;&thgr;
zx
∝L
2
F
z
/2
Ebh
3
in bending mode due to a vertical force
F
z
(4)
&Dgr;&thgr;
xx
∝2
hF
x
Eh
3
in twist-bending mode due to a lateral force
F
x
(5)
&Dgr;&thgr;
yx
∝3
LhF
y
/Ebh
3
in buckling mode due to a lateral force
F
y
(6)
These angles are proportional to the sensitivity by which the forces can be measured using a resistance detector such as a Wheatstone bridge, and it should be noted that whereas Equations (4) and (6) are proportional to Equations (1) and (3), the expression for the twist-bending deflection, Equation (5), is different from Equation (2), the equation for pure twist. Thus, when measured with a resistive cantilever, high sensitivity to lateral forces in twist-bending mode compared to normal forces (in bending mode) can be achieved only for high values of 2hb/L
2
. This compares with the factor of h/L for the optical detection method.
Thus, if the tip length h is much smaller than L, which is the norm in all current cantilevers used in AFM's and FFM's, the sensitivity to measuring friction forces in the twist mode using the OPTICAL method will be less than the sensitivity to measuring normal forces (by a factor that is proportional to h/L). But using the RESISTIVE method, the loss in sensitivity is usually even worse, being now proportional to 2hb/L
2
(unless b>L/2). The inventor believes that this is the reason for the very low sensitivity of current resistive cantilevers when measuring lateral forces, so much so that resistive cantilevers are not in common use for such purposes.
It may appear that this problem may be solved by increasing the length of the tip h and cantilever width b, i.e., making h≈b≈L. However, increasing h introduces a new problem. As shown in
FIG. 2A
, a longer tip at the end of a cantilever spring will move significantly out of the vertical when a normal (vertical) force F
z
or displacement
Larkin Daniel S.
The Kline Law Firm
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