Measuring and testing – Surface and cutting edge testing – Roughness
Reexamination Certificate
2001-06-13
2003-12-23
Larkin, Daniel S. (Department: 2856)
Measuring and testing
Surface and cutting edge testing
Roughness
Reexamination Certificate
active
06666075
ABSTRACT:
TECHNICAL FIELD OF THE INVENTION
The present invention relates generally to the field of force measurement using scanning probe microscopy (SPM) and, more particularly, to a force measurement system for determining the topography or composition of a local region of interest by means of scanning probe microscopy.
BACKGROUND OF THE INVENTION
Introduction of Terms Used in this Disclosure
In this invention we use cartesian coordinate systems with perpendicular axes as the coordinate system of choice. Nevertheless, one may implement any other well-defined coordinate system including, for example, polar, cylindrical, or spherical coordinate system. The “global” coordinate system X Y Z
40
is fixed with the sample and the “local” coordinate system X
tip
Y
tip
Z
tip
42
is fixed with the apex
45
of the tip
44
of the scanning probe
48
. In general, the scanning probe tip apex
45
may have an arbitrary position and orientation with respect to the sample, therefore, the local coordinate system
42
also may have arbitrary position and orientation with respect to the global coordinate system
40
, as shown in FIG.
1
A. In a special case, the local
42
and global
40
coordinate systems may be aligned with respect to one another, as shown in FIG.
1
B.
The origin of the local coordinate system
42
is at the apex
45
of the tip
44
. The Z
tip
axis
46
is oriented along the length of the tip
44
and is perpendicular to a region of the oscillator
48
surface near the place where the tip
44
is attached. The X
tip
axis
50
is parallel to the long axis of the oscillator
48
. The Y
tip
axis
52
is transverse with respect to the X
tip
axis
50
so as to form a right-handed cartesian coordinate system.
It is known that a dipole-dipole interaction occurs between pairs of atoms located in volumetric regions of the tip
44
and sample
54
when they are in proximity to each other. The associated force is called Van der Walls force. The resulting integrated effect encompasses all dipole-dipole interactions between pairs of atoms in sufficient proximity to generate a measurable interaction between the tip
44
and the sample
54
. This resultant of the integrated dipole-dipole interaction is represented by a three-dimensional “tip-sample interaction force vector”
56
, as shown in
FIG. 2A. A
single point can be used to approximate the volumetric region near the tip apex
45
, and a flat surface can be used to approximate the region of the sample
54
in proximity to the tip
44
, as shown in FIG.
2
B. If the surface of the sample
54
is horizontal (i.e., in the XY plane), the tip-sample interaction force vector
56
will be vertical. However, if the surface of the sample
54
is vertical (e.g., in the XZ plane), the tip-sample interaction force vector
56
will be horizontal. For a general orientation of the surface of the sample
54
, the tip-sample interaction force vector
56
will have three non-zero components, corresponding to the three axes XYZ of the global coordinate system
40
. The tip-sample interaction force vector F
56
can be represented either by its components F
x tip,
F
y tip,
F
z tip
in the local coordinate system
42
or by its components F
X,
F
Y,
F
Z,
in the global coordinate system
40
.
In one possible mathematical representation, the 3×1 vector functions (&PHgr;
i
, for (i=1, 2, 3, . . . ∞), of the spatial coordinates, (e.g., X
tip
Y
tip
Z
tip
) represent mode shapes of the probe structure, and q
i
represent the corresponding generalized coordinates. In one instance of a classical modal analysis, the equations of motion of the probe are
M
j
d
2
q
j
/dt
2
+M
j
&ohgr;
j
2
q
j
−&Sgr;
i=1
to ∞
F
ij
′q
i
=F
0j
Where (j=1, 2, 3, . . . ∞), M
j
is the modal mass, &ohgr;
j
is the resonant frequency, and F
0j
is the static component of the generalized force corresponding to the tip-sample interaction force applied to the probe tip. The term−&Sgr;
i=1
to ∞F
ij
′q
i
can be interpreted as a negative spring force which alters the j
th
resonant frequency of the vibrating probe. The quantity F
ij
′ can be represented in terms of the mode shapes by
F
ij
′=[A]&PHgr;
i
(
tip
)·&PHgr;
j
(
tip
).
Where [A] is a 3×3 coefficient matrix arising from classical modal analysis and the symbol·denotes an inner product of two vectors.
The vector [A] &PHgr;
i
(tip), derived from classical modal analysis, is an example of a more general vector quantity that we call a “resultant surface force interaction.” Our use of the term “resultant surface force interaction” is not limited to any particular physical origin of the tip-sample interaction force and may include, for example, both conservative and non-conservative tip-sample interaction forces.
FIG. 3A
shows typical orientations of three selected mode shape vectors, evaluated at spatial coordinates corresponding to the apex
45
of a probe tip
44
. In this example, &PHgr;
1
(tip)
58
represents the direction in which the tip apex
45
moves when the main bending mode is excited; &PHgr;
2
(tip)
60
represents the direction in which the tip apex
45
moves when the first torsional mode is excited; and, &PHgr;
3
(tip)
62
represents the direction in which the tip apex
45
moves when the second bending mode is excited. For suitably chosen structural design of the probe
48
and tip apex
45
location, and for small-amplitude vibrations, &PHgr;
3
(tip)
62
, &PHgr;
2
(tip)
60
, and &PHgr;
1
(tip)
58
are each substantially aligned with the unit vectors, i
tip
64
, j
tip
66
, and k
tip
68
, respectively, and the modal coordinates q
3
, q
2
, q
1
can be approximated by tip
44
displacements along the in the X
tip
50
Y
tip
52
Z
tip
46
axes, respectively. In this example, the resultants of the surface force interaction can be given a geometric interpretation as vectors aligned along the X
tip
50
Y
tip
52
Z
tip
46
axes.
The resultant surface force interaction vectors F′
x tip
70
, F′
y tip
72
, and F′
z tip
74
can, in some cases, be modeled by the three virtual springs with variable spring constants k
1
76
, k
2
78
, and k
3
80
that are functions of the tip-surface distance, as shown in FIG.
3
B. The vector F′
82
shown in
FIG. 3C
is the sum of the three resultant surface force interaction vectors. As the force axis
84
and the distance axis
86
show in
FIGS. 4A and 4B
, the force-distance curve
88
shows that the resultant surface force is non-linear with respect to the tip-surface distance. Therefore, the modeled spring constants are also non-linear. However, for small amplitudes of vibration of the oscillator tip
44
, the spring constants are linear with respect to the tip-surface distance. To maintain linear response, the oscillator
48
should vibrate with sufficiently small amplitude to keep the oscillator in a linear regime of operation, shown by area of measurement
90
. Contrast that with area of measurement
92
, used in tapping mode. Force axis
84
shows repulsive force
94
and attractive force
96
.
The term “oscillator,” as used in conjunction with the present invention, represents a scanning probe
48
for which multiple resonant modes are intended to be used for force sensing. The term “cantilever” refers to a scanning probe
48
for which only the primary bending (i.e., “cantilever”) mode is intended to be used for force sensing, even though, in general, the probe
48
structure would exhibit multiple resonant modal responses if excited at the appropriate driving frequencies.
The term “force sensor” refers to the resonating oscillator
48
and its sensitivity to surface forces
82
associated with the tip-sample interactions. The purpose of the force sensor is to enable detection of the surface topology or composition by means of coupling the scanning probe tip
44
to the surface of the sample
54
via a tip-sample interaction force
82
. In general, the interaction force
82
between t
Juricic Davor
Mancevski Vladimir
McClure Paul F.
Larkin Daniel S.
Thompson & Knight LLP
Weiss Aaron A.
Xidex Corporation
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