**4. Quantitative measurement ofthe electrical properties of materials on the nanometer‐scale**

For quantitative measurement, the operating frequency of M‐AFM is set at 94 GHz. The high‐ frequency microwaves are easy to propagate in the waveguide and emit from the nano‐slit on the probe tip. Since the width of the nano‐slit is around 100 nm, the field of microwave interacting with the measured materials can be considered to be in 100 nm order. Thus, if the thickness of measured materials is larger than 100 nm, the reflection from the bottom surface of the sample can be neglected. Therefore, only the reflection from the top surface needs to be considered.

**Figure 18.** AFM topography image of the Au/Glass step sample.

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**Figure 19.** Microwave image of the output voltage that was converted from the measured microwave signals.

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Moreover, the diode detector works in a small signal range, where it is considered to be a square‐law detector. Therefore, while keeping the standoff distance between the tip of the M‐ AFM probe and samples constant, the output reflected voltage V, which varies only with the conductivity ofthe sample, has a relationship with the squaredabsolute value ofthe top surface reflection coefficient, |*Γ<sup>s</sup>* | <sup>2</sup> as

$$V = k\_0 \left| \Gamma\_s \right|^2 + b\_0 \tag{1}$$

The two undetermined constants *k*<sup>0</sup> and *b*<sup>0</sup> can be calibrated with two samples whose conduc‐ tivities are known. For good conductors, which are used in this experiment, the surface reflection coefficient |*Γ<sup>s</sup>* | can be written as [34]

$$\left| \Gamma\_s \right| = \left| \frac{1 - \sqrt{\sigma \ / \ jo\varkappa\_0}}{1 + \sqrt{\sigma \ / \ jo\varkappa\_0}} \right| \tag{2}$$

where *ε*<sup>0</sup> and *σ* represent permittivity of free space and the conductivity of the measured material, respectively, and *ω* is the angular frequency of the microwave. For semiconductor

or isolating materials, similar equations can also be constructed. Then, the conductivity can be determined from Eq. (4‐7) as

$$\sigma = \alpha \varepsilon\_0 \left[ 4 \left| \Gamma\_s \right|^2 - \left( \left| \Gamma\_s \right|^2 + 1 \right)^2 \right] / \left[ \left( \left| \Gamma\_s \right|^2 + 1 \right) \sqrt{4 \left| \Gamma\_s \right|^2 - \left( \left| \Gamma\_s \right|^2 - 1 \right)^2} - 4 \left| \Gamma\_s \right|^2 \right] \tag{3}$$

**Figure 20.** Topography and microwave image of measured Cu sample.

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**Figure 21.** Topography and microwave image of measured Pb sample.

**Figure 22.** Topography and microwave image of measured Co sample.

After *k*<sup>0</sup> and *b*<sup>0</sup> in Eq. (1) are calibrated using two reference samples with known conductivities, the conductivities of any samples can be calculated from the measured voltage. Therefore, the M‐AFM allows us to quantitatively evaluate the electrical conductivities of materials on the nanometer scale.

It should be noted that Eq. (2) and (3) are derived under the plane wave condition, while the probe works in near‐field mode. Although near‐field analysis may further improve the precision of evaluation results, it requires more reference samples, which will increase the complexity of the measurement. Since the tested material was very close to the open end of the probe‐tip (the standoff distance of several nanometers was extremely small as compared with the waveguide width (~100 nm) and the wavelength), this problem can be equivalent to the case that the material surface is terminated at the end of the waveguide, which can be represented by the plane wave model. Therefore, the plane wave approximation is used in this study.

There is a limitation of M‐AFM technique we have to face. In the case of that the thickness of measured sample smaller than 100 nm, the reflection of microwave signal from the bottom surface of the sample and the substrate should be considered. Therefore, the M‐AFM can not use the previous mentioned equations to obtain the electrical properties of measured sample quantitatively.

Five different metallic films (Cu, Pb, Al, Co and Zn) with EB fabrication were prepared for the quantitative measurement. The tested electrical conductivities by the four point probe van der Pauw method were obtained as the standard values for calibration and evaluation of M‐AFM results. The tested electrical conductivities of these metal films are in the range of 4.46×106 S/m to 5.68×107 S/m. The measurements were performed in the air, and the AFM worked in non‐contact mode, with a working environmenttemperature of 23.0 °C and a relative humidity of 50%. The resonance frequency of M‐AFM probe was 107 kHz and the Q‐value of it was 675. The scan area was 2×2 μm2 , scanning speed was 1 μm/s. Before scanning, we set the original voltage to be zero while maintaining a constant distance of 2.6 μm between the probe tip and the sample. During the scanning process, the standoff distance between the probe tip and samples was fixed at several nanometers by the atomic force, and the voltage corresponding to the inspected sample was measured.

Figures 20 to 24 show the topographies and microwave images of the five samples. The variations of the measured voltages for the five samples are less than ±0.46 mV, which is much smaller than the dynamic range of the M‐AFM. The signal‐to‐noise ratio of the M‐AFM measurements was evaluated to be 20.14 dB on average.

**Figure 20.** Topography and microwave image of measured Cu sample.

or isolating materials, similar equations can also be constructed. Then, the conductivity can be

Micro-Nano Mechatronics — New Trends in Material, Measurement, Control, Manufacturing and Their Applications in

 

After *k*<sup>0</sup> and *b*<sup>0</sup> in Eq. (1) are calibrated using two reference samples with known conductivities, the conductivities of any samples can be calculated from the measured voltage. Therefore, the M‐AFM allows us to quantitatively evaluate the electrical conductivities of materials on the

It should be noted that Eq. (2) and (3) are derived under the plane wave condition, while the probe works in near‐field mode. Although near‐field analysis may further improve the precision of evaluation results, it requires more reference samples, which will increase the complexity of the measurement. Since the tested material was very close to the open end of the probe‐tip (the standoff distance of several nanometers was extremely small as compared with the waveguide width (~100 nm) and the wavelength), this problem can be equivalent to the case that the material surface is terminated at the end of the waveguide, which can be represented by the plane wave model. Therefore, the plane wave approximation is used in this

There is a limitation of M‐AFM technique we have to face. In the case of that the thickness of measured sample smaller than 100 nm, the reflection of microwave signal from the bottom surface of the sample and the substrate should be considered. Therefore, the M‐AFM can not use the previous mentioned equations to obtain the electrical properties of measured sample

Five different metallic films (Cu, Pb, Al, Co and Zn) with EB fabrication were prepared for the quantitative measurement. The tested electrical conductivities by the four point probe van der Pauw method were obtained as the standard values for calibration and evaluation of M‐AFM results. The tested electrical conductivities of these metal films are in the range of 4.46×106 S/m to 5.68×107 S/m. The measurements were performed in the air, and the AFM worked in non‐contact mode, with a working environmenttemperature of 23.0 °C and a relative humidity of 50%. The resonance frequency of M‐AFM probe was 107 kHz and the Q‐value of it was 675.

voltage to be zero while maintaining a constant distance of 2.6 μm between the probe tip and the sample. During the scanning process, the standoff distance between the probe tip and samples was fixed at several nanometers by the atomic force, and the voltage corresponding

Figures 20 to 24 show the topographies and microwave images of the five samples. The variations of the measured voltages for the five samples are less than ±0.46 mV, which is much smaller than the dynamic range of the M‐AFM. The signal‐to‐noise ratio of the M‐AFM

, scanning speed was 1 μm/s. Before scanning, we set the original

 2 2 22 2 22 2 <sup>0</sup> 4 1 / 14 1 4 *ss s ss s*

(3)

determined from Eq. (4‐7) as

 

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nanometer scale.

study.

quantitatively.

The scan area was 2×2 μm2

to the inspected sample was measured.

measurements was evaluated to be 20.14 dB on average.

**Figure 21.** Topography and microwave image of measured Pb sample.

**Figure 22.** Topography and microwave image of measured Co sample.

**Figure 25.** Variation margins of measured local voltages for samples with the different conductivities.

Thus, the applicability of the evaluation method in my work can be proved.

Figure 26 shows the variation margins of measured local voltages for samples with the square of surface reflection coefficient of them. As previous mentioned, two undetermined constants in Eq. (1) were calculated to be k0 = ‐5.9632 and b0 = 5.9629. That means the method in this work was based on a premise of that the surface reflection coefficient and measured voltage should be kept in a linear relationship. It is noted that the surface reflection coefficient and measured voltage could be provided in a linear relationship (see the fitting straight line in Figure 26).

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**Figure 26.** Variation margins of measured local voltages for samples with the square of surface reflection coefficient

of them.

**Figure 23.** Topography and microwave image of measured Al sample.

**Figure 24.** Topography and microwave image of measured Zn sample.

The Figure 25 shows the variation margins of measured local voltages for samples with different conductivities. Using the measured voltages of two samples obtained from Figures 20 and 21 (4.89 mV for Cu and 18.01 mV for Pb on average) and their tested conductivities (5.68×107 S/m for Cu and 4.46×106 S/m for Pb) for calibration, the two undetermined constants in Eq. (1) were calculated to be k0 = ‐5.9632 and b0= 5.9629. Then, the conductivities of Al, Co and Zn samples were evaluated with Eq. (1) and Eq. (2) by using the measured voltages obtained from Figures 20 to 24.

**Figure 25.** Variation margins of measured local voltages for samples with the different conductivities.

**Figure 23.** Topography and microwave image of measured Al sample.

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**Figure 24.** Topography and microwave image of measured Zn sample.

obtained from Figures 20 to 24.

The Figure 25 shows the variation margins of measured local voltages for samples with different conductivities. Using the measured voltages of two samples obtained from Figures 20 and 21 (4.89 mV for Cu and 18.01 mV for Pb on average) and their tested conductivities (5.68×107 S/m for Cu and 4.46×106 S/m for Pb) for calibration, the two undetermined constants in Eq. (1) were calculated to be k0 = ‐5.9632 and b0= 5.9629. Then, the conductivities of Al, Co and Zn samples were evaluated with Eq. (1) and Eq. (2) by using the measured voltages

Micro-Nano Mechatronics — New Trends in Material, Measurement, Control, Manufacturing and Their Applications in

Figure 26 shows the variation margins of measured local voltages for samples with the square of surface reflection coefficient of them. As previous mentioned, two undetermined constants in Eq. (1) were calculated to be k0 = ‐5.9632 and b0 = 5.9629. That means the method in this work was based on a premise of that the surface reflection coefficient and measured voltage should be kept in a linear relationship. It is noted that the surface reflection coefficient and measured voltage could be provided in a linear relationship (see the fitting straight line in Figure 26). Thus, the applicability of the evaluation method in my work can be proved.

**Figure 26.** Variation margins of measured local voltages for samples with the square of surface reflection coefficient of them.

Figure 27 shows the evaluated results versus the tested values of Al, Co and Zn samples. It is noted from Figures 20 to 24 that no correlation can be observed between the microwave images and their corresponding geometry images. In other words, the variations of the measured local voltages are not caused by the surface morphology. The main causes of error bars of the evaluated conductivities are as follows. Firstly, the film samples prepared by EB evaporation were not homogenous in the microscopic view, and the distribution of conductivity was location‐dependent (local conductivity).

**Figure 28.** Evaluated resistivities of the samples in comparison with tested c resistivities of them.

**a.** We invented out a novel device named of microwave atomic force microscope (M‐AFM), which is a combination of the principles of the scanning probe microscope and the microwave‐measurementtechnique.M‐AFMcan maintain the constant stand‐off distance between the M‐AFM‐probe tip and scanned sample surface, by detecting the deflection of the atomic force between them, and measure the electrical properties of materials with

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**b.** Microwave‐AFM probes were fabricated on the GaAs wafer by using the wet etching process. A waveguide was introducing on the probe by evaporating Au film on the both surfaces of the probes. The open structure (the nano‐slit) of the waveguide at the tip apex of the M‐AFM probe was obtained by using FIB fabrication. SEM was used to observe the fabricated M‐AFM probes. As the results, the average dimensions of the cantilever and the body of the M‐AFM probes are typically 252×31×14 μm and 2742×723×339 μm, respectively. Based on these dimensions, the characteristic impedance of the M‐AFM probes is, on average, 49.3 Ω. In this way, the M‐AFM probe could match well with the co‐axial line, which has an impedance of 50 Ω. The observed tip is located near the front edge of the cantilever and the tip is approximately 7 μm high, and the nano‐slit is

**c.** The AFM topography of the grating sample having 2000 line/mm and 18 nm step height was measured by the fabricated M‐AFM probe. AFM measurements were performed by comparing with the commercial Si AFM probe. The results indicated that GaAs micro‐ wave probe has a capability to catch AFM topography of grating samples and having a high accuracy for lateral and height evaluation, similar as the commercial AFM probe.

**5. Summary**

nanometer scale spatial resolution.

approximately 100 nm in width.

**Figure 27.** Evaluated conductivities of the samples in comparison with the tested conductivities of them.

It is believed that the variation margins of measured local voltages (see Figure 25) by the M‐ AFM caused the error bars of the evaluated conductivities. Secondly, the microwave signal for conductivity measurement was very small, which might be affected by the measurement environment. Therefore, the uncertainty of the microwave measurement may contribute to the error bars. It is also noted from Figure 27 that the deviation of evaluated conductivities from the values tested by the Van der Pauw method is 2.03%, 7.24% and 11.6% for the Zn, Co and Al, respectively. One of the causes of this deviation is that the standoff distance variation between different materials may affect the measured voltage, thereby inducing deviation of evaluated conductivity, especially for high‐conductivity materials such as Al. Another cause of the deviation may be the evaluation equation which was derived under the plane wave approximation rather than the much more complicated near field analysis. The quantitative evaluation was performed three times, and the similar results as shown in Figure 25 were obtained. On the other hand, the evaluated resistivities of the five samples can be presented out as shown in Figure 28.

**Figure 28.** Evaluated resistivities of the samples in comparison with tested c resistivities of them.
