Biomedical Engineering Reference
In-Depth Information
14
0.13 eV
10
0.67 eV
100 Hz
1000 Hz
10,000 Hz
6
2
1
1.5
2
2.5
3
3.5
1000/
T
(K)
FIGURE 4.19
Temperature and frequency dependence of the bulk grain interior resistance obtained by fitting the impedance
spectra. The activation energy is estimated from the slope as 0.13 eV below 250°C and 0.67 eV above 250°C,
respectively. (From Ye, H.,
J. Appl. Phys.
, 94, 7878, 2003. With permission.)
(right) becomes more complete and begins to dominate the spectra with increasing tem-
perature. Comparing the diameters of the semicircles in each Cole-Cole plot, it is apparent
that the ratio of the low-frequency to the high-frequency semicircle diameter increases
with an increase in temperature. This indicates that impedance from grain boundaries
becomes more significant at higher temperatures; that is, electrical conduction through
grain boundaries dominates at temperatures above 250°C.
Theoretically, a double
R
-
C
parallel circuit model in series [2] could be used to simulate
the electrical conduction of diamond films contributed from both grain interiors and grain
boundaries. Each parallel
R
-
C
equivalent circuit model accurately fits each Cole-Cole
semicircle. The fitting procedure used here is the same as the one described by Kleitz and
Kennedy [89] and allows the determination of resistance and relaxation frequencies with a
good precision. Here the resistor
R
represents ionic or electronic conduction mechanisms,
whereas the capacitor
C
represents the polarizability of the diamond. The symbols
R
gi
,
R
gb
,
C
gi
, and
C
gb
are defined as before. The complex impedance Z* measured by the RCL meter
can be expressed as the following function of the
R
gi
,
R
gb
,
C
gi
, and
C
gb
of the specimen:
Z* =
Z
→
- j Z
→
(4.58)
R
R C
R
R C
gi
gb
=
+
(4.59)
Z
2
2
2
2
2
2
1
+
1
+
ω
ω
gi gi
gb
gb
2
2
ω
ω
R C
R C
ω
ω
R C
R C
gi
gi
gb
gb
=
+
(4.60)
Z
2
2
2
2
2
2
1
+
1
+
gi
gi
gb
gb
where
Z
→
and Z
→
represent the real and imaginary portions of the impedance, and
ω
is the
angular frequency. When plotted in a complex plane, Z
→
versus
Z
→
takes the form of two
semicircles. In this representation, the grain interior and grain boundary contributions are
easily identified, and the electrical conduction paths of the bulk material can be studied
separately from grain boundary interference; this task has been performed above.
