Instrumentation & Measurement Magazine 25-4 - 18
Fig. 6. Calculated resistance and capacitance for HfO2 films with thicknesses ranging from 1nm up to 1 μm (on the x-axis) using circular top electrodes with
different diameters (shown in the legend). Shaded areas are not possible to measure accurately due to equipment limitations.
parallel RL in series [10]. This combined set of graphs indicate
that there are two important parameters: one is the value of R
in each case, and the second is the value of the corresponding
time constant. As Fig. 5 indicates, when the values of R1
=R2
and R2
,
the complex impedance graph shows perfect circles made up
of the R1
. Even though one cannot see it on these simulations,
the order of appearance from low to high frequency is
different depending on whether the time constant is τLR
or τLR > τRC. This is more clearly seen on the parts where R1
or vice-versa.
It should be added that because many materials and devices
have components with resistivity that can vary over 20
orders of magnitude (when including the most insulating
to the most conducting components), interpretation can be
very complicated. In fact, for the same material or device, the
complex graphs and frequency explicit spectra can be quite
different and lead to confusion unless the user has been trained
in the proper analysis methods. Additional characteristics associated
with the dielectric, optical or magnetic response of a
given material or device can further complicate the analysis
and must be taken into consideration.
Obtaining Trustworthy Responses
Electrical measurements are extremely sensitive to many different
factors in addition to the specific R, C and L responses
they may have. As already mentioned, because the resistivity
of materials can range from 10−6
to 1014
ohm-cm, it is sometimes
difficult to detect the correct properties for either end
of the spectrum. This means that when one makes impedance
measurements, one must already have an idea of what
range of values one expects to obtain based on the base properties
of the materials/components involved by taking the
geometric corrections needed. For resistivity, ρ, one can estimate
R by taking the expected length l and electrode contact
area A using:
R
18
l
A
(5)
< τRC
Instrumentation & Measurement Magazine 25-4
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