Page 47 - Principles and Applications of NanoMEMS Physics

1. NANOELECTROMECHANICAL SYSTEMS 33

§ 1 1 ·
γ ¨ ¨ r ij σ − ′ a + r ik σ − ′ a § 1· 2
¸
¸
=
e
h ( ,rr ,θ ) λ © ¹ ¨ cosθ + ¸ for r < aσ′ , else . 0 (3)
ij ik jik jik ij ,ik
© 3¹
The optimal parameters, in terms of experimental agreement for a silicon tip
on a silicon sample, was found by Stillinger and Weber to be as follows: A =
7.049556277, p = 4, Ȗ = 1.20, B = 0.6022245584, q = 0, Ȝ = 21.0, E bo nd =
u
3.4723 aJ, a = 1.8, ıƍ = 2.0951 Å, and σ = 2 1 6 σ ′.
Similarly, at distances under 100nm, long-range forces, namely, van der
Waals, electrostatic, and magnetic forces are operative. The van der Waals
forces, are characterized by a potential given by Eq. (4)

α d 2
V = − 1 . (4)
vdW 6
z

For the tip-sample situation found in AFM, namely, a spherical tip with
radius R separated a distance z from a flat surface (where z is the effective
distance between the plane connecting the centers of the surface atoms and
the center of the closest tip atom) the van der Waals potential is given by
[42] Eq. (5)

V = − HR , (5)
vdW
z 6
where H is the Hamaker constant embodying the atomic polarizability and
density of the tip and sample material pair and, for the majority of solids and
interactions across vacuum, has a value of H = 1 eV . For tip-sample
materials characterized by this value of Hamaker constant, and with a spherical
tip of radius R~100nm separated from flat sample by ~0.5nm, the respective
van der Waals potential and force are approximately -30eV and -10nN,
respectively.
When both the tip and the sample are conductive and at separations of
~100nm, they may also experience electrostatic forces, characterized by the
potential, Eq. (6) [42-45]:

πε RV 2
F () z = − 0 , (6)
electrosta tic
z
where V is the electrostatic potential difference. Accordingly, a potential
difference V~1Volt, between a spherical tip of radius R~100nm a distance