Biomedical Engineering Reference
In-Depth Information
atoms to diffuse back to the surface to relax the density to a lower, stable sp
2
density. He
also assumed that relaxation occurs by thermally activated diffusion at a rate of:
E
kT
v
=
v
exp
−
0
(2.7)
0
where
E
0
is the activation energy for atomic diffusion. The total number of hops of all the
atoms within one spike is the integral over the spike volume and time:
∞
∞
E
kT r t
∫
∫
2
=
4
n r
v
exp
−
0
d d
t
r
(2.8)
β
π
(
)
0
,
r
t
1
1
where
k
is Boltzmann's constant,
n
0
the atomic number density,
v
0
the phonon frequency,
r
1
the minimum radius of the spike, and
t
1
the minimum time of the spike. It is useful to
put everything in dimensionless atomic units, with an atomic radius
a
. Robertson defines
reduced distances
r
→
=
r
/
a
, times
t
→
=
v
0
t
, and a reduced temperature
τ
=
kT
/
E
0
. The heat
capacity per atom is 3
k
, so
c
= 9
k
/4
π
a
3
,
D
≈
v
0
a
2
, and
n
0
= 3/4
π
a
3
. Integration over
t
→
is carried
out, and the integral over
r
→
is converted to one over
τ
to give:
τ
2
1
∫
−
8 3
/
=
exp
−
d
(2.9)
β
τ
τ
τ
τ
1
In the limits
τ
1
→
0 and
τ
2
→
∞
this gives:
5 3
/
E
E
β
=
0 016
.
p
i
(2.10)
0
where
p
is a dimensionless constant of order 1. All the other quantities cancel out. This is
the equation used in Equation 2.5.
Equation 2.5 gives a good representation of the variation of density or sp
3
fraction with
energy for tetrahedral-amorphous carbon (ta-C) deposited by filtered cathodic vacuum arc
(FCVA) [11,18] with
E
0
= 3.1 eV. The increasing sp
3
fraction at lower ion energy is controlled
by the penetration probability
f
, and the decline in sp
3
fraction at high ion energy is con-
trolled by the relaxation.
Despite this superficial agreement between theory and experiment, this model of depo-
sition is still deficient. There are many faults with the thermal spike concept when applied
to carbon deposition. A thermal spike is only truly valid for much heavier ions and higher
ion energies, where the energy loss rate per distance (stopping power) is much higher, so
the energy density is greater [19,20]. In carbon, energy equipartition does not really hold,
so the spike volume consists of a few excited atoms among much less excited atoms.
A satisfactory model of growth must account for
• The transition temperature to sp
2
bonding, being 400-500 K [21,22], despite the
temperature in a thermal spike's being 10
6
K
• The variation of the transition temperature for sp
3
formation with ion energy [21].
• The variation of transition temperature with instantaneous growth rate [17,23]
