II. THEORY

Near-field microwave microscopy has become a popular technique to characterize the properties of nanoscale materials and devices at high frequency. It has been successfully used to quantitatively image metals, semiconductors, and dielectrics [Reference Usanov12–Reference Imtiaz14]. The basic principle consists of near-field interaction between a source and a sample. The antenna is a sub-wavelength probe. One of the strengths of near-field microscopy is that measurements can be performed without any physical contact between the tip and the sample to characterize.

It is well known that a tip probe in front of a sample can be modeled primarily as a resistance in series with a capacitance depending on the tested sample and the distance separating the sample and the tip apex. In many cases, especially at microwave frequencies, the effect of the resistance is low compared to the impedance of the capacitance close to a few tens of femtoFarad for a tip apex in the order of 100 µm. For apex sizes at the nanoscale, the capacitance may decrease to some attoFarad [Reference Huber9].

For example, a capacitance of 100 fF has a reactance of 1.6 kΩ at a frequency of 1 GHz. Hence, direct determination of the impedance by a traditional measure of its reflection coefficient is difficult because of the high contrast with conventional network analyzer impedance referenced to 50 Ω. To overcome this constraint, indirect methods using a resonator or interferometric techniques are used. These methods have the advantage of being very sensitive to sample variations. However, they are frequency selective and do not allow absolute characterization of the sample.

The original idea developed in this paper is to propose a non-resonant instrumentation with a reference impedance Z ₀ close to the one of the tip probe in order to directly characterize the sample under test through measurement of a reflection coefficient.

In the case of a tip probe modeled only by capacitance C, the reflection coefficient Γ_L is written as

(1)$$\Gamma_L = \displaystyle{1 - j{\rm Z}_0 C\omega \over 1 + j {\rm Z}_0 C\omega}.$$

This expression shows that only the phase φ of Γ_L is usable since its magnitude is constant and equal to unity irrespective of the value of C. By a simple calculation using partial differential we obtain the capacitance measurement error δC depending on the phase error δφ according to the expression

(2)$$\delta C =- \displaystyle{1 + \lpar Z_0 C\omega\rpar ^2 \over 2Z_0 \omega} \delta \phi.$$

This equation, which is a parabolic function of Z ₀, shows that for a pulsation ω and a capacitance C, the error in the measurement of capacitance depends obviously not only on the phase error but also on the reference impedance Z ₀ selected. We show, by calculating the derivative, that there is a particular value Z ₀ for which the measurement error δC is minimum. This optimal reference impedance is given by Z ₀ = 1/Cω.

To show the influence of reference impedance Z ₀ on measurement uncertainty of a near-field microwave microscope, we propose an example of a tip probe designed by using micro-strip lines on Epoxy FR4-1.6 mm substrate (Fig. 1).

Fig. 1. Design of the probe on Epoxy FR4-1.6 mm substrate.

The probe is a very important part of the microscope. It is established that the resolution of the system is determined by the size of the probe and its distance from the sample. In this study, the resolution is not targeted, what is aimed at is the proof of concept of a new instrumentation that does not include resonators. Hence, in this work a simple design using a low cost technology is proposed.

By using a finite element method simulator, Ansoft HFSS, we can estimate the impedance of this probe located at a distance d of a conducting plate. In Fig. 2, the capacitance observed between the probe and the conducting plate for distances going from 1 µm to 5 mm is represented.

Fig. 2. Tip-sample capacitance versus tip-conducting sample distance (HFSS simulations).

The figure shows logarithmic increase of capacitance versus decrease of tip probe-conducting sample distance in the range of 5–500 µm (slope = −18.3 fF/decade). In the particular case of this probe (Fig. 1), we observe that the retrieved capacitance changes between about 40 and 100 fF. Finally, one can see that for a stand-off distance greater than 1 mm, the capacitance remains constant and corresponds to the micro-strip open-ended capacitance estimated to be 40 fF from high frequency structure simulator (HFSS) simulations.

From these simulation results, we present in Fig. 3 the uncertainty on the capacitance versus the reference impedance Z ₀ for a reflection coefficient phase error of 0.1 degree by using equation (2). In this study, we first consider a constant frequency (F = 2 GHz) and the capacitance is varied from 40 to 100 fF (Fig. 3(a)). Then, we estimate the error on the capacitance (C = 100 fF) when the frequency varied from 500 MHz to 2 GHz (Fig. 3(b)).

Fig. 3. Uncertainty of capacitance C versus reference impedance Z ₀.

These results show that for a reference impedance Z ₀ around 1 kΩ, variations as well of the capacitance as of the frequency used do not greatly affect measurement accuracy. Thus, we propose to design a non-resonant measurement system with reference impedance Z ₀ fixed at 1 kΩ for measurement of reflection coefficients.

To do this we strive to reduce the use of reactive elements, such as for example the propagation lines, for benefit of resistive elements [Reference El Fellahi, Glay, Haddadi and Lasri15]. Thus, to form a high impedance reflectometer (HIR), we propose to combine a conventional two-port Vector Network Analyzer (VNA) to a high reference impedance splitter via two impedance matching networks (Fig. 4).

Fig. 4. Bloc diagram of HIR connected to Z _L.

In Fig. 5, we show the schematic and the flow graph of the HIR conceived by considering that the matching networks are made of two resistors (R ₁ = 50 Ω and R ₂ = 1 kΩ). The power divider is based on an Owen splitter [Reference Owen16]. The two resistors are defined to realize a good compromise between coupling factor and isolation (R ₃ = 1.33 kΩ and R ₄ = 2 kΩ).

Fig. 5. Schematic and signal flow graph of HIR connected to Z _L.

By taking into account only all first-order loops, the transmission coefficient S ₂₁ derived from the signal flow graph technique can be written as:

(3)$$S_{21} = \displaystyle{T_1 ^2 [ T_2 \left(1 - \Gamma_L \Gamma _4 \right)+ T_3^2 \Gamma_L\rsqb \over 1 - 2\Gamma_2 \Gamma_3 - T_2^2 \Gamma_2^2 - \Gamma_4 \Gamma_L - 2T_3^2 \Gamma_2 \Gamma_L}\comma \;$$

where the complex reflection coefficient Γ_L is defined by (1).

It has to be noted that the reference impedance is 50 Ω for Γ₁, whereas it is 1 kΩ for Γ₂, Γ₃, Γ₄, and Γ_L. Hence, in ideal case Γ₁ = 0, Γ₂ = Γ₃ = Γ₄ = 0, and T ₂ = T ₃². From these considerations, the transmission coefficient S ₂₁ can be written as:

(4)$$S_{21} = \displaystyle{2T_1^2 T_2 \over 1 + jZ_0 C\omega}\comma \;$$

where T ₁, T ₂, and T ₃ represent respectively the matching network insertion loss, isolation, and coupling factor of the splitter. The different simulations performed lead to the following findings: |T ₁| = −19.1 dB, |T ₂| = −19 dB, and |T ₃| = −9.5 dB.

Given the low capacitance values considered, equation (4) shows that the magnitude of S ₂₁ is almost insensitive to the variation of capacitance C and remains close to −51 dB.

Assuming that the matching networks and the power splitter are made with perfect resistors, T ₁, T ₂, and T ₃ can be considered purely real. Hence, we can approximate the phase of the transmission coefficient by

(5)$$\arg \left(S_{21} \right)=- {\rm atan} \left(Z_0 C\omega \right).$$

From HFSS simulations of the reflection coefficient Γ_L referenced to 1 kΩ, we can give the evolution of S ₂₁ phase-shift versus frequency for several tip-conducting sample distances (Fig. 6). We also report the corresponding value of the capacitance from equation (5).

Fig. 6. S ₂₁ phase-shift versus frequency for different tip-conducting sample distances.

This simulation shows a change in the S ₂₁ phase-shift which is large enough to distinguish a variation of capacitance in the range of 40–100 fF. However, to increase the measurement accuracy, instead of determining the capacitance for a particular frequency, the idea is to take advantage of the non-resonant nature of the measurement system by fitting equation (5) in the frequency band considered.

In the next section we present a microwave measurement system with high reference impedance Z ₀ equal to 1 kΩ.

III. FABRICATION AND EXPERIMENTAL RESULTS

The main problem to develop broadband systems with high characteristic impedance is the difficulty to realize micro-strip lines with high impedance. Thus, to achieve the high reference impedance splitter and the impedance matching networks, we propose to use only resistive lumped elements. Nevertheless, particular attention should be given to reduce connections between these components [Reference Glay, El Fellahi and Lasri17].

As has already been mentioned the objective of this study is not system performance. Features like resolution or high frequency functioning are not the priority of our investigation. The aim is to demonstrate the possibility to achieve near-field microwave microscopy by means of an original high impedance system. Thus, each function of our system is realized using resistive elements in boxes SMD 0402 mounted on Epoxy FR4-1.6 mm substrate.

Owing to the unavoidable propagation phenomena and parasitic capacitance cut-off effect limits of resistors, we restrict the validation tests to a frequency of 500 MHz. To demonstrate these limitation effects, we present in Fig. 7, equivalent impedance (real and imaginary parts) versus frequency of several SMD 0402 resistance values R = 50 Ω (Fig. 7(a)), R = 1 kΩ (Fig. 7(b)) and R = 2 kΩ (Fig. 7(c)). Typical value of the internal shunt capacitance is around 40 fF.

Fig. 7. Equivalent impedance versus frequency of several SMD 0402 resistance values.

The results exhibit that for 50 Ω SMD 0402 resistance, operation up to 1 GHz poses no problem. However, for resistance values of 1 kΩ and 2 kΩ, the results show that in particular the imaginary part is not negligible even for frequencies as low as 100 MHz. We are aware, from these curves that the resistances that are used for the splitter and matching networks are far from perfect especially in microwave. Nevertheless, to prove the concept up to a few hundred MHz, this low-cost technology is sufficient.

In order to measure the characteristics of the resistive matching network, we combine two networks in series so as to be compatible with the impedance 50 Ω of VNA ports. Figure 8 presents the photography of the circuit and the experimental results showing a non-resonant matching impedance (|S ₁₁| < −20 dB) and a relatively constant evolution of |S ₂₁| up to a frequency of 500 MHz.

Fig. 8. Resistive matching networks: photography and experimental results.

As the two networks are in series one can estimate the insertion loss of only one matching network to −19.1 dB, corresponding to the theoretical value given in Section II.

In the same way, concerning the characterization of the high impedance splitter, we have placed in front of each access a resistive matching network in order to be compatible with the VNA in terms of impedance. We present in Fig. 9 the photography of the circuit and the experimental results showing a non-resonant matching impedance (|S ₁₁| < −30 dB) and relatively constant evolutions of |S ₂₁| and |S ₃₂| up to a frequency of 500 MHz.

Fig. 9. Power splitter with resistive matching networks: photography and experimental results.

It has to be mentioned that the resistance values that have been selected to build the splitter are somewhat different to those calculated. These resistances are R ₃ = 1.2 kΩ and R ₄ = 1.8 kΩ instead of R ₃ = 1.33 kΩ and R ₄ = 2 kΩ.

By considering that insertion loss of the resistive matching networks is equal to −19.1 dB, one can estimate the high impedance splitter parameters. We found out a coupling factor equal to −9.8 dB from S ₂₁ and an isolation equal to −17.8 dB from S ₃₂.

Thus, by combining a VNA with two matching networks and a power splitter, as presented in Fig. 4, we obtain an HIR allowing us to measure a reflection coefficient Γ_L with a referenced impedance of 1 kΩ.

To realize the microscope, a probe is associated with this set-up. The microwave circuit part of the near-field microscope is fabricated using hybrid microwave integrated circuit (HMIC) process, where the tip probe (apex = 100 µm) is directly mounted on the circuit (Fig. 10).

Fig. 10. HMIC part of the near-field microwave microscope.

Transitions from SubMiniature version A (SMA)-Connector to micro-strip line on Epoxy FR4 1.6 mm substrate are used to connect the VNA R&S^® ZVL6 via coaxial cables.

The first experiment that is conducted consists of measuring the transmission coefficient when the probe is in air (without sample under test). In Fig. 11, experimental data of the magnitude and the phase-shift of S ₂₁ versus the frequency allow us to emphasize a capacitive behavior.

Fig. 11. S ₂₁ magnitude and phase-shift evolutions versus frequency.

The results show that the magnitude of the transmission coefficient is quite constant up to 200 MHz (around −51.2 dB). Concerning the phase-shift, a response relatively linear with the frequency (such as equation (5), where Z ₀Cω ≪ 1) is observed up to 200 MHz. Above this frequency, there is a resonance phenomenon certainly due to the parasitic elements brought by resistors SMD 0402 boxes.

An estimation of the tip probe to sample capacitance can be easily obtained by inversion of the model given by equation (5).

(6)$$C = \displaystyle{- \tan \left[\arg \left(S_{21} \right)\right]\over Z_0 \omega}.$$

However, weak variations of the capacitance C induced by weak variations of distance d do not offer a sufficiently large phase-shift to guarantee good measurement accuracy by using directly the predictive model (equation 6). Indeed, representation of the S ₂₁ phase-shift versus frequency for different distance d, evolving between 50 and 300 µm exhibit small variations (Fig. 12).

Fig. 12. Measurement of S ₂₁ phase-shift evolution versus frequency for different distance d.

To increase the phase-shift measurement accuracy, we will take advantage of the non-resonant nature of the system. Hence, the transmission coefficient is not measured for only one frequency but in a frequency band. Therefore, to estimate the capacitance C, we have made a linear fitting of the function tan[arg(S ₂₁)] on the frequency range [10–200 MHz] (Fig. 12). These results show that we still can differentiate and quantify variations in capacitance between 50 and 300 µm without any ambiguity.

As has already been shown in Fig. 2, one can note in Fig. 13 the logarithmic increase of capacitance versus decreasing tip-sample distance. We notice a very good agreement between the slopes observed for measurements (slope = −17.8 fF/decade) and HFSS simulations (slope = −18.3 fF/decade). The offset is mainly due to the equivalent capacity of the HIR which is not taken into account in HFSS simulations.

Fig. 13. Measurement of the tip probe to sample capacitance versus tip-sample distance.

Through this example, we demonstrate that our non-resonant method, using a multi-frequency approach (up to 200 MHz in this case), is sensitive enough to distinguish capacitances around 200 fF with an error estimated to 0.5 fF.