Quantifying resistivity using scanning impedance imaging

Sensors and Actuators A 137 (2007) 338–344

Quantifying resistivity using scanning impedance imaging Brian G. Buss, Daniel N. Evans, Hongze Liu, Tao Shang, Travis E. Oliphant, Stephen M. Schultz, Aaron R. Hawkins ∗ Electrical and Computer Engineering Department, 459 Clyde Building, Brigham Young University, Provo, UT 84602, United States Received 15 September 2006; received in revised form 1 March 2007; accepted 8 March 2007 Available online 14 March 2007

Abstract Noncontact scanning impedance imaging (SII) has the ability to produce impedance based images of a thin sample. This paper describes how SII can be used to produce quantitative measurements related to the specific electrical impedance (impedivity) of a sample. This is made possible through hardware improvements to previously reported systems and the use of numerical simulations to quantify the degree of current confinement occurring in the coaxial impedance probe. Experiments conducted in homogeneous saline solutions show that measured impedance increases linearly with probe height, confirming simulation results and the concept that current through the sample is being confined into a cylinder of constant area. Microfabricated structures of the photoresist SU-8 are used as a test case to demonstrate that SII probes can accurately determine the quantity ρh (resistivity × sample height). For the experiments described, the dielectric contribution to the measured impedance is negligible; in these cases the sample resistivity is reported instead of the impedivity. This results in a resistivity measurement for SU-8 of ρ = 3 × 106 -cm. Two-dimensional resistivity scans of various samples are also shown. © 2007 Elsevier B.V. All rights reserved. Keywords: Impedance imaging; Resistivity; Conductivity; Noncontact measurement

1. Introduction Specific electrical impedance (impedivity) is one of the most distinguishing characteristics for individual solids and liquids. Resistivity, the real component of impedivity, exhibits differences between good insulators and good conductors of 20 orders of magnitude. The large change in resistivity also exists in biological materials. For example, cancerous tissue exhibits resistivity differences compared to healthy tissue [1,2]. With this in mind, if the impedance of different portions of a heterogeneous media could be measured, natural impedance differences would provide large measurement contrasts. A two- or threedimensional impedance map of a sample would allow for the visualization of material boundaries and compositions, and even the identification of materials based on impedance properties. Impedance is measured by applying a voltage across a sample and then measuring the current flow. The current flow tends to spread out from a measurement electrode resulting in a dramatic reduction in the spatial resolution. To complicate matters, the

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current spreading is dependent on material impedance. Therefore, even though impedance based imaging and measurement provides a very promising technique in terms of image contrast, obtaining images of any significant resolution is a challenge. Despite the issue of current spreading, the technique of electrical impedance tomography (EIT) has been shown to produce impedance images on the macroscopic scale simply from measurements between two or more electrodes that surround a sample [3,4]. Recent research has also extended conventional atomic force microscopy (AFM), combining nanoscale imaging and impedance spectroscopy. A modified conducting AFM tip is used as a conducting electrode to measure frequency-dependent impedance properties [5–8]. This technique is often referred to as scanning impedance or resistance microscopy and is capable of producing high-resolution impedance based images for materials very close to the surface of a sample. The current spreading and contact resistance problems associated with the EIT and resistance microscopy have been addressed by the development of a technique called scanning impedance imaging (SII) [9], which involves immersing a sample in a conducting solution and measuring the current flow through the sample under an applied electric field. The SII system uses a coaxial probe geometry to dramatically reduce

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the current spreading. Immersion of the sample allows noncontact measurement, eliminating the variable contact resistance and facilitating the application of SII to biological and highimpedance materials. Previous work on SII imaging focused on maximizing contrast and spatial resolution [10–12]. However, this previous work only produced relative changes in impedance. The ability to determine the actual resistivity of a sample enables the comparisons between different samples rather than simply a relative measurement within a single sample. Furthermore, a quantitative measurement produces useful information for analyzing biological samples that exhibit a range of resistivity values rather than simply an interface between conductive and insulating regions. This paper describes modifications to the SII system that allow the measurement of quantitative values for sample impedance that relate directly to material impedivity (including its real portion resistivity). Absolute quantitative values can be reported because the role of the probe in current confinement is accurately modeled and measured, confirming that current flow to the probe is confined within a cylinder directly below the probe. The first section of this paper describes the new version of the hardware and measurement processes that enable the system to produce quantitative measurements. The next section describes the coaxial probe, which is the critical component, and describes numerical modeling that reveals how it effectively restricts current spreading. Next is a section that describes how impedivity values can be computed from SII measurements given the current confining properties of the probe. Measurements of known saline solutions are provided to confirm the accuracy of the computed resistivity values. The final section provides quantitative 2-D resistivity scans for three different material systems. 2. System description Fig. 1 illustrates the basic components of the SII system. The scanning setup measures the magnitude of the total impedance from the conducting base plane, through a sample immersed in a conducting water solution, and into the tip of an impedance probe. This is done by applying a known voltage to the base plane and simultaneously measuring the current flux into the

Fig. 1. Diagram of the scanning impedance imaging setup. Current flows from the base plane through the sample to the probe tip. The signal is amplified, digitally filtered, and measured by a spectrum analyzer.

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Fig. 2. Diagram of impedance probe showing (a) the coaxial probe with an inner tip for the signal and an outer shield, and (b) the current flow created by the shield and tip and the placement of the probe over a sample.

probe. This current is amplified with a transimpedance amplifier (Princeton Applied Research 181), and then measured with a fast-Fourier transform spectrum analyzer (Stanford Research 760). The spectrum analyzer effectively isolates the signal components near the input frequency, resulting in a higher current signal-to-noise ratio compared to data measured using an oscilloscope or voltmeter. A Newmark Systems linear XYZ stage (model NLS4-4-16) is used to scan the impedance probe over the sample area. A computer, running a Labview program, controls stage motion and measurement collection at each grid point. The collected data can then be analyzed and used to produce a 2D image. As illustrated in Fig. 1, a sinusoidal voltage is applied across the sample as opposed to a DC voltage. The AC measurement suppresses hydrolysis in the water below the impedance probe. The experimental results presented in this paper were obtained using a 60 kHz signal of amplitude 1 V peak-to-peak. In practice, it has been determined that impedance measurements are independent of signal amplitude for a voltage range between 0.1 and 10 V peak-to-peak. The key element of the SII system is the impedance probe. As described in earlier publications [9,10], it was found that making a coaxial geometry probe significantly increased the 2-D image resolution and the signal-to-noise ratio (the ratio of mean over standard deviation for currents measured through the probe from repeated measurements). Fig. 2 shows that the basic probe design consists of an outer conductor, or shield, separated from an inner conductor, or tip, by an insulator. The intention of surrounding the tip electrode with another conductor is to restrict the current flowing into the tip to a confined volume. With an insulated tip alone, current may flow into the tip from a large region over the sample. The degree of current confinement is discussed in detail in the next section. In practice, impedance probes are made by inserting insulated copper wire into stainless steel syringe needles filled with an insulating epoxy. The tip of the needle is then ground flat and electrical connections are made to the copper wire, which serves as a probe’s tip, and to the syringe, which serves as the shield. The measurements shown in this paper used 38 gauge copper wire and 23 gauge syringe needles to create probes with approximately D = 100 ␮m tip diameter, and Sp = 100 ␮m spacing between the tip and the shield. The test setup shown in Fig. 1 is different from earlier reported SII systems [9,10] due to the use of a transimpedance cur-

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rent amplifier. Using this simple element allows for a dramatic improvement in measurement accuracy because it allows for the tip and the shield to always be biased at the same voltage, preventing current flow between them through the conducting water solution above a sample. Fig. 1 shows that the shield is connected to ground and the tip to the virtual ground created at the input of the current amplifier. Using this arrangement, all of the current flowing into the probe’s tip is insured to arise only from the electric field applied between the tip and the conducting plane beneath the sample. The impedance of the sample material is then simply Zsample =

Vapp , itip

(1)

where Vapp is the voltage applied at the bottom conducting plane and itip is the current that is injected into and then amplified by the transimpedance amplifier. As will be shown, this system arrangement leads to accurate impedivity measurements that match expectations based on numerical models. 3. Current confinement probe As stated above, the primary motive for the coaxial design of the SII impedance probe is the confinement of current within a known volume. While previous measurements demonstrated that the probe was indeed functioning this way by increasing image resolution, no details were provided on the exact shape of the current confining volume. Using a finite difference method (FDM) numerical analysis, those details can now be provided and are critical in calculating sample impedivity. Given the electrode geometry illustrated in Fig. 2, providing a completely accurate model of impedance measurement requires knowledge of the electric field E (or potential φ) and current density J at each point in the region of interest. In linear, isotropic media, J = σE, where σ = σ  + jωε is the complex conductivity, σ  is the (real) conductivity (equal to the inverse of the resistivity) and ε is the permittivity. At a given frequency ω, the conductivity is a scalar function of position. From Maxwell’s equations and Ohm’s law the conductivity σ can be related to the potential by ∇(σ∇φ) = 0.

(2)

Based on Eq. (2), a model was developed using the finite difference method [12]. The model assumed standard lowfrequency approximations, ignoring the electric field outside the conductive solution and limiting current flow at the boundary to the tip, shield, and base plane. From the calculated potential φ(x, y, z) and from Ohm’s law, the electric field E and current density J follow directly. A Fortran code was written to perform numerical solutions for a three-dimensional grid based on these assumptions and boundary conditions. Applying the results of the simulation, current flow into a probe tip can be visualized. Fig. 3 was generated by solving for the electric field in the region below a probe and then tracing current flow from the conducting plane below a sample toward either the tip or shield electrodes. This particular simulation assumed there was a sample 100 ␮m high sitting on the bottom conducting plane, with

Fig. 3. Current flow lines from conducting base plane into the tip and shield of an SII probe. This is a 2-D cross-section taken from a 3-D simulation. The dark lines represent current flowing into the tip while the light lines represent current flowing into the shield. Simulation represents a high conductivity solution over a 100 ␮m thick sample. Conductivity ratio for sample to solution is 1:10.

100 ␮m of a conducting solution between the sample and probe. The sample was assumed to be less conductive than the solution, with a conductivity ratio of 1:10. The full simulation is done in three dimensions with Fig. 3 only showing a 2-D cross-section. The dark lines indicate current flowing from the base plane to the tip, while the lighter lines indicate current flowing into the shield. Readily apparent in the figure is that the current flowing into the tip is confined to a cylindrical volume directly below the tip. The diameter of this cylinder is approximately the midway point between the tip and the shield or Deff = D + Sp ,

(3)

where D and Sp are defined in Fig. 2. Simulations were also run for homogeneous media (corresponding to a situation where only water of a constant conductivity is between the probe and conducting plane) and for other heterogeneous cases (corresponding to water placed over a sample with different conductivity). In each case, the diameter Deff of the current confining cylinder remained approximately equal to the quantity given by Eq. (3). The implications of this result are very significant. First the tip collects current from a confined volume between it and the bottom conductive plane. Second, the volume is cylindrical in shape with an area that is approximately constant versus separation distance h between the probe and the conductive plane. These results will be used in the next section to develop calculations of impedivity for a given sample based on impedance measurements at the probe. Fig. 4 shows FDM simulations comparing a shielded to an unshielded probe. The unshielded case consists of a center electrode surrounded by an insulator (no outer shield electrode). Fig. 4(a) compares the simulated resistance as measured through the probe as a function of probe height in a homogeneous solution of conductivity σ = 0.01 S/m. In this case, only the real part of the impedance is reported because the frequency, ω, was so low that the imaginary part was negligible. The probe dimensions are D = 100 ␮m, and Sp = 100 ␮m. Resistance is determined by dividing the voltage drop across the sample by

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area of the confinement volume no longer remains constant. This still provides a large range of heights h, where the constant area approximation is valid. 4. Extracting resistivity values

Fig. 4. (a) Calculated resistance between the tip and the base plate using a shielded and unshielded probe. (b) The effective area, found by dividing the resistivity times sample height by the resistance from part (a).

the current through the probe tip. Integrating J over the tip surface easily approximates the tip current. Fig. 4(a) shows that the resistance for the shielded probe is nearly linear with height above the sample. This is due to the fact that the shield limits the effective diameter of the cylindrical volume contributing to tip current and the total resistance increases linearly as the cylinder’s volume increases linearly. The unshielded case is nonlinear because the effective probe area increases with height h. The effect of the shielded probe is also demonstrated in Fig. 4(b). In this figure, the total resistance computed in Fig. 4(a) is used together with the standard definition of resistivity R=

ρh , A

(4)

where R is the resistance of a volume of material of constant area A and length h, and ρ is the material’s resistivity. Fig. 4(b) plots ρh/R to determine whether the area A remains constant as h increases so that the simple definition found in Eq. (4) will hold. As shown in the figure, for the unshielded case ρh/R continues to increase with h, indicating that current is being drawn from a volume with a changing cross-sectional area and Eq. (4) will not hold for this case. For the shielded case, however, above 80 ␮m, ρh/R (area) remains approximately constant, indicating that Eq. (4) does accurately describe the resistance relationship for this volume. The simulation results indicate that a shielded probe is able to confine current not only to a confined volume, but also specifically to a cylindrical volume with a nearly constant crosssectional area. This allows total resistance to be related to sample resistivity in the simplest possible way using Eq. (4). Extensions of the model for heights greater than h = 500 ␮m (for a D = 100 ␮m and Sp = 100 ␮m probe), show that gradually current confinement begins to break down and the cross-sectional

It would be ideal to relate measurement data to the physical parameter resistivity in order to classify or compare material compositions. The modeling results discussed earlier establish a theoretical basis for using the simple relationship given in Eq. (4) to relate resistivity to the probe area given by Eq. (3) and the sample resistance Rsample , which is the real part of the impedance given in Eq. (1). For test samples, however, even if the height of the probe above the bottom conducting plane, h, is known, the actual thickness of the sample (Zo − h) may vary along the samples area (h and Zo are defined in Fig. 2). The total resistance Rsample will actually consist of a resistance due to the material under test and the water between this material and the probe. Since, the height of the sample will not always necessarily be known, we will report the quantity (resistivity) × (height) as a quantitative measure, given by ρh = Rsample Aeff ,

(5)

where Aeff = π(Deff /2)2 . If sample height is well known, the ρh quantity can easily be divided by h to yield an absolute material resistivity. This calculation is complicated somewhat by the fact that there is also a conductive water solution between the probe and the sample, but under certain test conditions this can be dealt with as discussed in the next section. To demonstrate how this ρh quantity will be used and to experimentally verify the FDM modeling results, a series of scans were performed for the simplest possible test case – homogeneous saline solutions. Resistivity versus NaCl concentration was measured using a commercial resistivity meter with the results shown in Fig. 5. Rsample was measured as the probe was scanned vertically (increasing h), away from the base plane. The probe size and geometry are described in Section 2. It should be noted that 60 kHz was used as the sinusoidal voltage excitation frequency because it allowed for very high signal-to-noise ratios, as the spectrum analyzer was able to filter out spurious interference

Fig. 5. Resistivity of saline solution versus molar concentration of NaCl.

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Fig. 6. Calculated (dashed) and measured (solid) values for ρh for three different saline solutions versus probe height h.

signals at lower frequencies. Example current levels were 1.8 ␮A with a standard deviation over time of 50 nA, translating to a signal-to-noise ratio of 37. Capacitance terms were measured for different elements in the system for the case described below and for the cases described in Section 5. These terms were found to be so low, that at 60 kHz, impedance was dominated by resistance and the imaginary part of impedance was neglected. Fig. 6 shows a plot of ρh (or Rsample × Aeff ) for three different saline solutions versus height h. These results are similar to those found in Fig. 4(a) in which there is a linear increase in total resistance versus height. Fig. 6 shows direct comparisons of experimental data and computations from the FDM model using different solution conductivities1 . The match between model and experiment is excellent, not only in curve shape but also in absolute value. This match provides confirmation that the degree of current confinement predicted for the shielded probe is occurring in practice.

Fig. 7. Profilometer scan of three SU8 squares indicating their height above a silicon wafer. Squares are approximately 1.5 mm wide.

Fig. 8. SII linescan of the three SU8 squares with results reported as ρh or (Rsample )(Aeff ). The height of the squares as measured using the profilometer is indicated.

Much more interesting from an applications standpoint is the evaluation of material samples below a conductive water solution. The first test case involved a material with constant resistivity but well controlled sample heights. The material chosen was SU8 [13] – a photosensitive polymer that can be applied over a conductive surface in controlled thicknesses with an area defined using photolithography. The total Rsample in this case will be the combination of resistance due to the water and resistance due to the SU8. Given that these resistances would simply add in series, it was decided that the easiest way to deal with this situation was to make the total water resistance very small by using a high conductivity solution when testing the SU8 samples. In the test cases reported here, water of conductivity of σ = 400 ␮S/cm was used, which turned out to be very high compared to the SU8 conductivity. Test samples were constructed by first coating a very flat silicon wafer with a 300 nm thick aluminum layer through evaporation based deposition. The aluminum served as the con-

ductive bottom plane for the sample. SU8 was then applied to the wafer using spin coating and hot plate curing. Using a series of photolithography steps and additional spin coatings, 1.5 mm × 1.5 mm squares of three different thicknesses were formed on the substrate adjacent to each other. Fig. 7 shows a surface profile made of the wafer using a thickness profilometer (Tencor Alphastep 200) showing the three different SU8 thicknesses. Given that the SU8 film has a homogeneous resistivity distribution, and that the water solution used during measurement has a much lower resistivity than the SU8, we would expect the ρh measurement over the different SU8 squares to vary linearly with SU8 height. Fig. 8 shows the results of an SII scan across the middle of the SU8 squares2 . The Rsample values from the scan have been multiplied by the Aeff of the probe to produce ρh values for the sample. Fig. 8 looks very similar to Fig. 7 as expected, as they are both in effect a measure of sample height. Given that we know the actual heights of the SU8 squares from the profilometer measurements, an absolute value of resistivity can be reported for SU8, which is ρ = 3 × 106 -cm. There is

1 The measurement data used in Fig. 6 were smoothed with a linearly-weighted moving average filter to reduce quantization effects in the spectrum analyzer.

2 The data shown in Fig. 8 represent the average of a series of adjacent scans over the region.

5. Example 2-D image scans

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scanned using the SII system. Prominent in the resistivity scan are the veins in the wing and high resistivity regions that could not be obviously predicted from the optical photograph. In both of the images shown in Fig. 10, the gradient scales are for ρh values (units of -cm2 ). 6. Conclusions Fig. 9. 2-D SII scan of SU8 squares. The black and white gradient scale is in terms of ρh, with units of -cm2 , with the shortest square on the left side of the figure.

limited available literature reporting a resistivity value for SU8. In a single case, we did find that involved the addition of silver nanoparticles to SU8 to make it more conducting, our reported resistivity value is within an order of magnitude of that projected for pure SU8 [14]. Fig. 9 shows a two-dimensional area scan of three SU8 squares of variable height. The scan plots the quantity ρh as intensity on a black/white gradient scale with largest ρh (corresponding to the tallest square) the lightest color. The figure illustrates both the large contrast available for an image scan and the high spatial resolution possible, even using a probe tip with a D = 100 ␮m diameter. Fig. 10 shows a further demonstration of the power of SII as an imaging tool by providing images of more complex material samples. This figure shows scans made on a butterfly wing and on a silicon wafer covered with oxide. The conditions used to obtain these scans were similar to those used to create the SU8 images in Figs. 8 and 9. The top left image in Fig. 10 is a photograph of structures etched into the thin (300 nm) oxide layer on the silicon wafer. The bottom left image is an SII scan of the same wafer showing the contrast between the oxide coated regions and those free of oxide. The black box drawn in the upper left picture shows the area of the wafer scanned. The top right image is a photograph of a butterfly wing and below it is the same area

In this paper, we have shown how the original SII test setup has been modified to allow for more accurate impedance measurements including the quantifying of impedivity parameters for an unknown material. One major modification of the system involves using a transimpedance amplifier, which drastically simplifies resistance measurements and eliminates the unknowns associated with the previous use of a feedback resistor for current measurements. The FDM model previously used to study the resolution of the SII technique and probe was used to help quantify the effects of the coaxial probe geometry. It was found that a shielded probe confines the current flowing into the center electrode to be a cylindrical volume with a nearly constant area for samples with both homogeneous and heterogeneous resistivity distributions (neglecting imaginary impedance terms by assuming low-frequency excitation). This allows for the determination of resistivity within this volume using a simple expression (Rsample = ρh/A), where h is the height of the probe above a bottom conducting plane and A is the area of confined current. Unshielded probes yield results that are much less tractable. Simulation predictions were confirmed using experimental results in homogeneous saline solutions. Given that sample height may be an unknown for a given material sample, the quantity ρh is a natural way to report resistivity in SII scans. Scans of SU8 test structures show ρh values that scale with heights as expected. Two other scans were also shown of an oxide structures on silicon and a butterfly wing. The ability to quantify resistivity with this system provides a new tool for classifying material properties and imaging a variety of samples in a new way. References

Fig. 10. 2-D SII scans and optical pictures for oxide coated silicon wafer on left and butterfly wing on the right. The top images are optical photographs. The bottom images are SII scans reported in terms of ρh.

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