## Permittivity Measurement with HF2IS Impedance Spectroscope

## Introduction

Dielectric spectroscopy is a form of impedance spectroscopy where the dielectric properties i.e. permittivity of a medium is characterized as a function of frequency. This technique allows us to evaluate the energy storage and dissipation properties of a capacitor filled with a certain material. For example, dielectric spectroscopy is a common experimental method of measuring an electrochemical system for battery storages. In this blog, an example of how to measure the dielectric constant of a polymer with the HF2IS Impedance Spectroscope will be detailed.

**Theoretical Permittivity Calculation**

To derive the dielectric constant (i.e. permittivity) of a polymer, one way is to use the parallel plate measurement method ^{[1]}. A graphical representation and its equivalent circuit model is illustrated below.

Basically, a thin film dielectric material is sandwiched between two conducting metal plates. The plate has a surface area of A with a spacing d which corresponds also to the thickness d of the dielectric material. This forms in essence a capacitor. In general, a capacitor can be modeled in a very complex manner, especially in cases low and high frequency behaviors need to be considered. However in this case, we will just model the capacitor as a parallel G//C_{p} circuit, where C_{p }and G represents the parallel capacitance value and the dielectric loss of the material, respectively.

The relative permittivity ε*_{r} of the dielectric film is a complex parameter consists of real and imaginary parts ^{[1]}:

ε*_{r} = ε’_{r} – jε”_{r}

where ε*_{r} is a ratio between the permittivity of the film ε^{* }and the permittivity of free space ε_{o} (1/36π x 10^{-9} F/m):

ε*_{r} = ε^{*}/ε_{o}

To calculate this relative permittivity, one must first formulate the equivalent impedance Z of the capacitor:

Y = 1/Z = G + jωC_{p}

which can be re-written as:

Y = jω[C_{p}/C_{o} – jG/(ωC_{o})]C_{o}

where C_{o} is the free air capacitance i.e. capacitance measured with no polymer between the metal plates. Note that a parallel plate capacitance is calculated from ε*_{r}(ε_{o}A/d) = ε*_{r}C_{o}.Therefore the expression [C_{p}/C_{o} – jG/(ωC_{o})] is in fact the complex relative permittivity ε*_{r}. Now we can define,

ε’_{r }= C_{p}/C_{o}

and

ε”_{r }= G/(ωC_{o})

ε’_{r} indicates the capacity of a material to store energy in the presence of an electric field, while ε”_{r} indicates how lossy/conductive the material is. ε”_{r }is normally much smaller than ε’_{r }in a capacitor, luckily. Here, we are interested here is the energy storage capacity parameter ε’_{r}. To measure ε’_{r} of the polymer, we must first measure the impedance of the parallel plates structure.

**Experimental Setup**

The measurement was carried out using a translation stage test fixture with parallel plate electrodes containing a polystyrene film. This forms a capacitor with polystyrene as the dielectric. The polystyrene sample was made using a hot press, where pellets of a commercial polystyrene were pressed between plates into a central geometry plate. The central plate is 100 μm thick with a 25 mm disc cavity for the polymer to flow into. The plates were then heated to 160 °C and a load of 6 T was applied for about 5 minutes to make the sample. Once the disc was made, the surface of the sample was gold coated on both sides in a sputter coater to ensure good electrical contact with the electrode plates. Care was taken to ensure the edges of the sample were not gold coated (as this would create a conducting path across the sample).

Each of the parallel plates of the test fixture has two wires brought out for the connection to the HF2IS Impedance Spectroscope and the HF2TA Transimpedance Amplifier. The 4-terminal setup is basically applied to measure the impedance of the capacitor. The overall setup is illustrated below.

With the HF2IS, both the HF2TA output and the differential voltage across the capacitor can be recorded simultaneously. The measured voltage divided by the measured current will then give the measured impedance.

## Measurement Result and Analysis

The measured impedance and phase are plotted in the graph below. The blue line is the magnitude and the red line is the phase. The measurement is performed over 3 frequency regions: 1 to 10 Hz, 10 Hz to 1 kHz, and 1 kHz to 1 MHz. The reason for the measurement segmentation is to accommodate a very large dynamic range of the measurement signal due to the impedance variation over 8 decades. By making measurement with smaller frequency ranges, one can optimize the measurement signal-to-noise ratio better. The small discontinuities of data points which occurred across segment transitions due to the different gain and bandwidth settings of the instrument were fitted for display purposes.

As one can see, the impedance magnitude stays fairly linear (in log-log scale) as one would expect from a capacitor measurement. However, we do see a deviation of phase from 90° at frequencies close to 1 MHz. This could be due to two reasons:

- The parasitic elements in the setup (e.g. cables, connectors, input ports)
- The un-calibrated phase shift in the setup either due to either the cables or to the amplifiers inside the instrument

Nevertheless, this will have no significant impact to the final result as it will be seen later. To calculate the relative real permittivity ε’_{r}, we simply need to extract C_{p} from the imaginary part of the Z measurement and then divide by the free air capacitance C_{o} = ε_{o}A/d. The imaginary part of Z can be represented by the following expression:

jωR^{2}C_{p}/(1 + ω^{2}R^{2}C_{p}^{2}), where R = 1/G

In fact, G is generally very small for a capacitor such that the above expression can be approximated by j/ωC_{p}. We can then simply calculate C_{p} from the imaginary part of the measured Z. The calculated C_{p} and ε’_{r} are polynomial fitted (to remove small discontinuities) and plotted below.

The relative real permittivity ε’_{r} is calculated to be about 2.82 which is close to the typically reported values of 2.4 to 2.7 for the polystyrene. One important factor that may have affected slightly the measured value would be the actual thickness of the polystyrene film sample i.e. not exactly 100 μm thick. In addition, the higher frequency distortion behavior we see corresponds to the phase shift away from 90º observed in the impedance plot. This is most likely due to the influence of the setup parasitics as mentioned before. The assumption we made to calculate C_{p} from the imaginary of the Z measurement may start to lose validity at this point.

## Conclusion

In this blog, we described the procedure of measuring permittivity using the parallel plate measurement method. The obtained permittivity value of polystyrene was very close to the expected theoretical value, despite some high frequency measurement distortions. One advice for minimizing the high frequency distortion would be to consider de-embedding the fixture and the instrument’s influence on the setup. For more information on how to best de-embed or pre-calibration the HF2IS or HF2TA, please contact Zurich Instruments for support. I hope that you have been able to apply the technique here to measure permittivity using the HF2IS Impedance Spectroscope.

## Acknowledgement

We would like to thank Dr. Stephen Boothroyd from the Department of Chemistry of Durham University for preparing the sample, providing the measurement data and describing the setup, as well as his colleague Dr. Bryan Denton for fabricating the permittivity measurement fixture.

**References**

[1] Agilent Application Note 1369-1.

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