Broadband non-contact characterization of epitaxial graphene by near-field microwave microscopy

Understanding NFMM measurements requires some insights into the interaction between the probe and sample under test. As the sample is placed in the near-field range of the tip and given that the tip radius is much smaller than the wavelength, the probe–sample interaction can be represented by a lumped element network in the quasi-static approximation [20]. We present the schematic of this probe–sample interaction modelling in figure 4.

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In near-field region, the stray capacitance C_str can be neglected because it is much larger than C_c [20]. Thus the lumped element model of such a probe–sample interaction can be simplified as a coupling capacitance C_c in series with the material impedance Z_S. Particularly, in case of a metallized sample, in the first order of approximation, the interaction between the tip and the sample can be simplified and represented by a coupling capacitance C_c.

A calibration procedure is employed to transform the transmission coefficient S₂₁ measured into the impedance of the device/material under test (Z_S). The description of the calibration method and the material impedance extraction from the measured data (magnitude and phase-shift of the transmission coefficient) are illustrated in figure 5.

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As shown in figure 5, the calibration process is derived from a one-port VNA calibration method, requiring three standard impedances. The metalized SiC substrate is selected as the standard material for the calibration process and the probe sample interaction is modeled as said before by a coupling capacitance C_c. In general, there are basically two methods to analytically solve this capacitance: the image-charge method [21] and the surface integration method [22]. In this study, the image-charge method is used to determine the capacitance C_c. As the electromagnetic field is concentrated around the probe tip of semi-sphere shape and according to [23, 24] works, the capacitance C_C between the semi-sphere and the metallic plane is given as follows:

$$\begin{eqnarray}&&{C}_{{\rm{c}}}=4\pi {\varepsilon }_{0}{R}_{0}\,\sin \,h(\alpha )\displaystyle \sum _{n=2}^{\infty }\displaystyle \frac{1}{\sin \,h(n\alpha )},\end{eqnarray} \tag{ 2 }$$

where α = cosh⁻¹(1 + a') with a' = h/R₀. h is the stand-off distance between the tip and the sample. R₀ is the tip radius. The impedance Z_s of the sample can be related simply to the reflection coefficient Γ_s [25, 26] by:

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{s}=\displaystyle \frac{{Z}_{s}-{Z}_{0}}{{Z}_{s}+{Z}_{0}},\end{eqnarray} \tag{ 3 }$$

where Z₀ is the measurement system characteristic impedance (here 50 Ω: VNA impedance). When only the coupling capacitance C_c is considered, Z_s can be written as:

$$\begin{eqnarray}&&{Z}_{s}=\displaystyle \frac{1}{j\omega {C}_{{\rm{c}}}},\end{eqnarray} \tag{ 4 }$$

$\omega$ is the angular frequency. A traditional one port calibration model is used to link the transmission coefficient measured S_21m to the reflection coefficient Γ_s (figure 5(a)).

$$\begin{eqnarray}&&{S}_{21m}={e}_{00}+\displaystyle \frac{({e}_{10}{e}_{01}){{\rm{\Gamma }}}_{s}}{1-{{\rm{\Gamma }}}_{s}{e}_{11}}.\end{eqnarray} \tag{ 5 }$$

The error terms, e₀₀, e₁₁ and e₁₀e₀₁, are respectively the directivity, matching and tracking errors. They can be determined by measuring three reflection coefficients for three known calibration standards Z_S1, Z_S2 and Z_S3 with reflection coefficients Γ_S1, Γ_S2 and Γ_S3. It should be mentioned that the calibration coefficients depend on the operating frequency. For example, if we consider for the calibration procedure, 10, 260, and 410 μm as the stand-off distance, we obtain based on the equation (2) 2.6 fF, 0.22 fF and 0.14 fF respectively for the corresponding coupling capacitance. One can note that the capacitance value decays rapidly with increasing stand-off distance. Then, relying on the equations (3)–(5), the calibration parameters retrieved at 2 GHz are e₀₀ = 0.015/−81.1°, e₁₁ = 3.9 × 10⁻⁶/−178.7° and e₀₁e₁₀ = 0.99/−0.024°. Once these three calibration parameters are determined, the electrical properties are extracted from the transmission coefficient according to the flow-chart given in figure 5(b). This calibration process is repeated for each frequency point considered.

The quality factor of the iNFMM represents one of the most effective parameters to demonstrate the measurement sensitivity [1, 27]. As reported in the literature, the transmission-line-based structures usually exhibit a quality factor in the order of 1000 [3, 28, 29]; while the resonator-based platforms present higher Q values around 5000 [5, 30]. Unfortunately, all these platforms are narrow band and thus enable high measurement sensitivity in a limited frequency band. In contrast, an interferometer-based structure has the potential to achieve a good sensitivity in a broad frequency band. For example, Bakli et al obtained a Q varying from 5300 to 9400 in the frequency band (2–6 GHz) in free space [31] considering a zero level of −60 dB (30 dB above the noise floor of −90 dB of the VNA for an IFBW of 100 Hz). In this work, we extend the working frequency range up to 18 GHz with an excellent sensitivity (Q in the order of 53 000). The zero level is set around −75 dB (15 dB above the VNA noise floor) by carefully tuning the attenuator and the delay line when the probe is placed in air. The intermediate frequency bandwidth (IFBW) and power of the VNA (P₀) are set to 100 Hz and 0 dBm, respectively.

Generally, λ/4 or λ/2 coaxial resonator based NFMMs are used to retrieve the materials/devices under test properties from the measurement of the quality factor and resonant frequency [1]. Nevertheless, since recently, the TLM is exploited more and more in AFM-based NFMM [25, 32]. The proposed iNFMM has the advantage of being operable with both resonator and TLMs. First, the measured quality factor (Q) and the resonance frequency shift (Δf) as a function of the probe–sample distance are shown in figure 6. In fact, Δf represents the difference between the resonance frequency obtained when measuring the sample under test (f_meas) and the reference resonance frequency (f_ref) measured without sample under test. In this study, the reference frequency selected is 2 GHz. The stand-off distance selected goes from 10 μm, to guarantee the tip–sample interaction is in the near field, to 510 μm (h ≫ apex size). In NFMM systems the distance control between the probe and the sample is a key parameter. The resolution and therefore the precision on the quantities measured is directly related to the accuracy with which the distance probe–sample is controlled. Different techniques can be used to ensure a fine control of the distance. For example, for experiments in the nanometer scale generally the probe-tip is mounted onto a quartz tuning-fork oscillator [33]. In our case, for distances in the micrometer range, the tip–sample distance is determined by the following process. First, the probe tip approaches the sample surface gently, and once the tip touches the surface, the contact position is noted as 0. Then, this reference position is used to set the displacement of the tip by using the stage.

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The four samples have different microwave responses in terms of quality factors and resonance frequency shifts qualifying the method for material quality analysis. It is observed in figure 6(a) that the quality factor increases when increasing the tip–sample distance h. This is due to the fact that the quality factor (Q_Ref) is tuned to the max (53 000) for the reference measurement (no sample under test). The total quality factor (Q_t) is given by ${{{Q}}_{{\rm{t}}}}^{-1}={{{Q}}_{{\rm{in}}}}^{-1}+{{{Q}}_{{\rm{c}}}}^{-1},$ where Q_in is the intrinsic quality factor of the material under test and Q_c is the coupling quality factor dominated by the capacitance between tip and materials. Thus, the quality factor variations Q_t measured at the same height indicate the different internal losses including dielectric losses for these four samples. Values of Q_in are proportional to dielectric losses tanδ [34]. By extracting impedance from measured S₂₁ of both SiC and SiCGAlO samples, we obtain values of tanδ(SiCGAlO) = 0.3456 which is much higher than tanδ(SiC) = 9.2 × 10⁻⁵. It corresponds to the results given in figure 6(a) showing that the SiCGAlO quality factor is much lower than that of SiC. Due to the conductivity of graphene and gold, most of the electromagnetic field is reflected, SiC substrates are thus shielded. In other words, dielectric losses from SiC in SiCG and MSiC contribute less efficiently in reducing the total quality factor in this measurement in which Q exhibits similar values in both two samples. It is easy to understand the frequency shift phenomenon when we treat samples as RLC circuit in series. Both SiC and SiCGAlO samples add big capacitance in series in the circuits which makes resonance shift. However, again, the resonance shift difference between SiCG and MSiC are quite close. Compared with the simple metal layer, the graphene layer contributes not only to the inductance but also to the capacitance [35]. The measured imaginary part of graphene impedance (see figure 9) is positive which indicates the contribution of this capacitance effect may be quite small. A further mesoscopic model is required to give precise explanation for the frequency shift difference between SiCG and MSiC. But the clear difference between SiC and SiCG in quality factor and resonance shift measurement demonstrates that this iNFMM technique provides a new method to verify for example the uniformity of graphene grown on SiC substrate instead of using AFM.

If we look now at the second treatment method of the data we can plot the magnitude and phase-shift of the transmission coefficient S₂₁ (figure 7).

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As shown in both the magnitude and phase-shift plots the four samples can be easily distinguished by the proposed iNFMM. One can also note that as the zero level selected (−75 dB) is adjusted when the probe is in air, the ∣S₂₁∣ decays as a function of the stand-off distance. The phase-shift spectra of S₂₁ are also affected by the distance separation between the probe and the materials but in proportions depending on the material under test. Thus, this technique demonstrates its potential for the investigation of the local electromagnetic properties of samples with high sensitivity.

Now, for the demonstration, based on the magnitude and phase-shift of the transmission coefficient, we present the results of the implementation of the calibration process described before to extract the electric properties of the material under test.

When the iNFMM operates in the non-contact mode, the measurement accuracy can be influenced by the separation between the probe and the sample under test. Thus, the extracted electric properties are evaluated as a function of the stand-distance. For the demonstration, the substrate (SiC) is selected as the sample under test. Thanks to the calibration process detailed before, the measured transmission coefficient S₂₁ is translated to the impedance of the sample under test. The probe–SiC interaction is simply modeled as a coupling capacitance (Z_C) in series with the impedance of SiC (Z_SiC), and thus Z_SiC can be extracted by subtracting Z_C from the total impedance Z_S1 (Z_S1 = Z_C + Z_SiC). Then, from the knowledge of Z_SiC the dielectric properties of SiC can be determined. In figure 8, the extracted ε' and tanδ of SiC versus the stand-off distance and also versus the frequency are presented.

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In figure 8(a), the dielectric constant obtained at a small distance, ε' (h = 10 μm) = 9.6, agrees well with the value found in the literature [36]. It is observed that this value diminishes with the stand-off distance between 10 and 30 μm and then is almost constant around 6. Indeed, with the increasing stand-off distance, the electromagnetic coupling between the probe and the sample is weakened, resulting in a reduced wave penetration into the SiC substrate and a larger influence of the air gap between the sample and the tip. Thus, to guarantee a good performance of the platform, the stand-off distance is kept as small as 10 μm for the following. Additionally, the losses level retrieved in figure 8(a) (below 10⁻⁴) also agree well with the literature data. Then, the broadband behavior of the dielectric parameters from 2 to 18 GHz is shown in figure 8(b). The extracted dielectric parameters are also comparable to the theoretical values found in the literature [37].

Thanks to the knowledge of the SiC properties, the impedance of the graphene flake (Z_Gra) can be solved in a similar manner. As modeled in figure 9(a), Z_Gra can be extracted from Z_S2 which consists of three impedances in series: Z_C, Z_Gra and Z_SiC. The real and imaginary parts of impedance retrieved for graphene are presented in figure 9(b).

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As shown in figure 9, the surface resistance of graphene is in the order of 20 kΩ from 2 to 18 GHz. When comparing the resistance obtained with the value measured by other approaches, we note that this resistance extracted by iNFMM is higher than the values retrieved by dc probing and CPW methods, but corresponds to the values acquired by the NFMM method ranging from tens to hundreds of kΩ [6, 14, 37]. Actually, as a 2D material with a thickness in atomic scale, the graphene flake presents a dielectric behavior with low conductivity in the vertical direction [38, 39]. This can explain the high surface resistance measured by iNFMM, in the order of 20 kΩ, as the microwave beam sent out by the probe illuminates vertically the sample (figure 9(a)). On the other hand, the imaginary part of the graphene impedance is also given in figure 9. The reactance of graphene is practically proportional to the frequency varying from 4.9 Ω at 2 GHz to 36.5 Ω at 18 GHz, which corresponds to an average inductance of about 360 pH. It is worth noting that these extracted values represent the local surface impedance of the graphene. Furthermore, the impedance measured represents the value of an area corresponding to the tip projection onto the graphene surface. Considering the tip end as a semi-sphere shape (D = 66 μm), the projected area on the graphene surface is about 3420 μm². This capability of local characterization offers possibility to detect inhomogeneity from the mapping of materials.