Calibrated microwave reflectance in low-temperature scanning tunneling microscopy

Recent developments of near-field scanning microwave microscopy¹ have enabled the characterization of materials with nanoscale dimensions, surfaces, and nanoelectronics devices² up to GHz frequencies and with sub-micron lateral resolution. In particular, this includes measuring electrostatic force,^3,4 electrical capacitance,^5,6 and electrochemical activity.⁷ Each of these techniques has its specific limitations, such as the frequency of operation.^6,8,9 Recently, microwave reflection spectroscopy with sub-micron lateral resolution has been demonstrated up to 19 GHz.^10,11 Various material properties, including relative dielectric permittivity¹² and dopant concentration,² have been successfully detected by a transmission line implementation realized in an atomic force microscope operated at ambient conditions. Although closely related to the atomic force microscope, the scanning tunneling microscope (STM) has been scarcely demonstrated as a platform for calibrated microwave reflection spectroscopy to date. The present work outlines a straightforward approach, i.e., utilizing the STM tip as a probe for calibrated microwave reflection spectroscopy. In particular, we employ a low-temperature STM instrument, which is well-known to provide ultimate stability and spatial resolution¹³ by operating at ultra-high vacuum conditions and cryogenic temperatures. Our approach offers great potential for achieving microwave impedance spectroscopic fingerprinting even at the scale of single molecules—thus complementing related methods that utilize spin-resonance^14,15 or dielectric relaxation¹⁶ at GHz frequencies. A major advantage of STM over similar AFM based techniques is the ability to select the bias and current setpoint at which the microwave reflection spectroscopy is performed. This opens the door to, for example, studying transient molecular redox reactions.⁷ 

Importantly, the phase and amplitude of S₁₁ strongly depend on the exact frequency chosen. Therefore, the e-parameters have to be determined separately for every measurement frequency.

Quantitative G and C values can only be derived from S_11,m if the values of the e-parameters are precisely known. The e-parameters can be obtained from an in situ retraction curve calibration. Retraction curve calibration requires that both S_11,m(z) and S_11,jun(z) are known [z is the tip–sample separation; see Fig. 1(a)]. In an early approach by Farina et al.,²¹  C(z) was calculated using an assumed analytic function with an absolute tip–sample distance and an ex situ measured tip radius as input parameters. In AFM, a recent retraction calibration procedure relies on simultaneously measuring ∂C(z)/∂z and S_11,m(z) during tip retraction.¹⁰ Using Eqs. (1) and (2) and the assumption that G = 0, S_11,jun(z) is calculated from the independently measured C(z). Inserting S_11,jun(z) and S_11,m(z) into Eq. (3) allows for the determination of the three e-parameters via a least-squares fit routine. In STM, the force detection required to measure ∂C(z)/∂z is not available and an alternate strategy should be used.

In this work, we present a retraction calibration procedure that uses simultaneous G(z) and S₁₁(z) measurements upon the well-controlled variation of the tunnel distance z. We determine the numerical values of the e-parameters of Eq. (3) that define the calibration. Additionally, we present our implementation of microwave reflection spectroscopy in a low-temperature ultra-high vacuum STM. Our STM instrument has been upgraded by a cryogenic microwave transmission line.^20,22 The microwave circuit design described herein incorporates a custom interferometer and lock-in detection, forgoing the use of a vector network analyzer.

Microwave reflectance measurements were performed on our upgraded Createc low-temperature ultra-high vacuum STM system (p < 4 × 10⁻¹¹ mbar, T < 8.5 K),^22,23 shown schematically in Fig. 1(a). All experiments herein were performed over a clean Au(111) surface (Surface Preparation Laboratory), prepared by repeated cycles of Ar ion sputtering (0.61 kV, 10 min) and annealing (703 K, 10 min). A tungsten tip (Bruker DTT10) was used as the STM and microwave probe. The bias voltage V_DC was applied to the tip. The microwave signal was added to V_DC via a bias-tee (SHF BT45R-B). Inside the vacuum chamber, the microwaves are transmitted via microwave compatible cables (Elspec MK5001 and Elspec Stormflex 047Cryo). At the sample side, a second bias-tee (Tektronix PSPL5541A) separates the DC line, which is grounded at the tunnel current amplifier (Femto DLPCA 200), and the microwave line, which is terminated with a 50 Ω load. Note that, as shown in Fig. 1(a), the bias-tees are the only microwave related components inserted in the DC line, which does increase the tunnel current noise by approximately a factor of $<2$⁠. As the rest of the microwave circuit is fully separated from the DC line, this does not affect current noise or the signal-to-noise ratio of G spectra.

We detect amplitude changes and phase shifts of the microwaves that have been reflected at the tunnel junction. In order to increase the signal-to-noise ratio, the background resulting from the near full microwave reflection at the tunnel junction [see Eq. (1)] was subtracted using a Mach–Zehnder-type interferometer.¹¹ The orange shaded section of the schematic in Fig. 1(b) depicts the components and layout of the interferometer. Table I lists the details of the indicated components. The transmission line to the STM tip acts as the measurement signal of the interferometer [see pink arrows in Fig. 1(b)], while the variable voltage attenuator (VVA) acts as the difference signal [see cyan arrows in Fig. 1(b)]. Figure 2(a) shows a schematic representation of the interferometer, including an indication of the microwave amplitude, A, and phase, φ, at different parts of the interferometer. An optimum (i.e., highest) signal-to-noise ratio for S_11,m detection is achieved for destructive interference between the two paths in the interferometer. This happens when the signals from the two parts have the same amplitude (A₃ = A₂) and are antiphase [φ₂ = φ₃ ± (2n + 1)π, where n is an integer]. Amplitude matching is performed by tuning the attenuation of the difference signal A₃ by the VVA input voltage. Antiphase conditions are obtained at fixed frequencies where the total path length of the two arms differs by exactly $n+1/2λ$⁠, where n is an integer and λ is the microwave wavelength. The frequencies for which this condition is satisfied are referred to as notches in the frequency spectrum,¹¹ as shown in Fig. 2(b). Calibration measurements are performed at a frequency f and VVA attenuation that maximizes destructive interference in the interferometer at the tunneling setpoint.

The green shaded section of the schematic in Fig. 1(b) depicts the down-modulation circuit, and Table I lists the details of the indicated components. Before down-modulation, the microwaves coming from the interferometer are amplified using a low-noise amplifier (LNA). Then, the microwave signal is demodulated to an intermediate frequency f_IF since the microwave frequency f is too high to detect with the lock-in amplifier directly. The demodulated signal was detected using a lock-in amplifier at a reference frequency of f_IF obtained by demodulating a fraction of the microwaves going to the interferometer. With this arrangement, continuous phase-stable detection is achieved. In the experiments presented herein, f = 0.908 GHz and f_IF = 500 kHz. The peak-to-peak amplitude of the incident microwaves at the tunnel junction²² was determined to be V_pp,MW = 69 mV. G was measured by using a lock-in amplifier (EG&G 5210) using a sinusoidal modulation of V_DC (V_rms,lock-in = 8 mV, f_lock-in = 775 Hz).

In our concurrent microwave reflection measurement, the amplitude of S_11,m(z) decreases approximately linearly over the entire retraction range, as shown in the upper panel of Fig. 3(b). In contrast, the phase of S_11,m(z), shown in the lower panel of Fig. 3(b), increases sharply in the first nm above the surface, followed by a nearly linear increase over the rest of the retraction curve. Figure 3(c) shows the same S_11,m(z) data in a different representation, i.e., plotted in the complex plane and the z-dependence shown as color gradient. Based on the shape of the S_11,m(z) curve, we distinguish two different regimes, labeled high G (red marker) and G ≈ 0 (cyan marker). The high G regime ranges from z = −1 nm to z ≈ −0.8 nm and is characterized by a conductance of about G > G₀/3. The G ≈ 0 regime starts at z ≈ 4 nm.

In the next step, the values of the parameters e₀₀, e₀₁, and e₁₁ are determined from the experimental S_11,m(z) data with the help of a numerical least-squares fit (⁠$R2≥0.91$ using Eq. (3). We obtain the values e₀₀ = (−1.7 ± 0.4) + (0.0 ± 0.1)i, $e01=0.41±0.08+(−2.3±0.4)i$⁠, and e₁₁ = (0.91 ± 0.02) + (0.79 ± 0.07)i. Here, the numerical error intervals of the e-parameters are obtained from numerical fitting (see also the supplementary material, Fig. 1). The value of e₀₀ depends on the constant C in the calculation of S_11,jun(z) as it compensates for the offset between S_11,m(z) and $S11,junz$⁠. Possible interdependencies between e₀₁ and e₁₁ can be analyzed from the mathematical form of Eq. (3). Since all quantities are complex valued, simple interdependencies are obtained only in certain cases. For instance, in the limit of $S11,jun→±∞$⁠, Eq. (3) reduces to S_11,m = $e00−e01e11$⁠, revealing a trivial interrelation between e₀₁ and e₁₁. For the general case, both e₀₁ and e₁₁ are required for a correct description. Note that the numerical fit procedure concerns the difference between the experimental data [Fig. 3(c)] and the respective values predicted by using Eq. (3) as follows: The scattering parameter at the tunnel junction S_11,jun(z) is calculated using Eqs. (1) and (2). G(z) measured by using the lock-in amplifier is the starting point in the calculation of S_11,jun(z). The exponential fit to G(z) of Eq. (4), as shown in Fig. 3(a), is substituted in for the conductance term in Eq. (2). For the unknown capacitance term, we apply a Taylor series approximation of $Cz≈Cd+C′dz−d+⋯$⁠. Since we perform the numerical fitting in a narrow z-range of 0.54 nm, we consider only the constant and linear terms of the series and neglect higher terms. Note that in the high G regime, the exponential decay of G(z) dominates the linear $C′dz−d$ term in the Taylor expansion of $Cz$⁠. Therefore, we also neglect the linear term and assume $Cz=const$⁠. in Eq. (2) for the rest of our analysis. We remark that after successful calibration, the validity of the assumption is confirmed, i.e., the z-range of z = −1 to −0.46 nm used for calibration is sufficiently narrow (see the section titled “Discussion”).

Figure 4(a) shows our result of the calibration, where the experimental S_11,m(z) data from Fig. 3(c) have been transformed into calibrated admittance Y_jun(z) using the values of the e-parameters obtained from the numerical fit procedure above. The entire measured z-range between z = −1 to 50 nm is calibrated this way. Note that according to Eq. (2), $Yjunz=G+i2πfC$⁠, and thus, the real and imaginary parts of $Yjunz$correspond to G and C, respectively. After calibration, the different conduction regimes can still be clearly distinguished [compare Figs. 3(c) and 4(a)]. As expected, the high G regime (red, z < −0.8 nm) is dominated by a change of G, while C remains almost constant. Conversely, the low-G regime (cyan, z > 4 nm) is dominated by a significant change of C, while the value of G is almost constant at G ≈ 0.

Figures 4(b) and 4(c) show the calibrated data of Fig. 4(a) in a different representation, i.e.,G^MW(z) and ΔC^MW(z), respectively. To avoid confusion with the conventionally obtained G data, the superscript MW indicates that the respective data have been obtained from the microwave reflection procedure. Note that a capacitance shift ΔC^MW(z) is measured since the magnitude of the capacitance between the sample and the tip holder (stray capacitance) is not exactly known.¹⁰ 

The quality of the obtained calibration can be judged by comparing the conductance data, as shown in Fig. 4(b), that has been obtained in two different manners. First, the exponential fit to G(z), directly measured by using the common dI/dV lock-in method [see also Fig. 3(a)], which was used to obtain the e-parameters, is shown as the red curve. Second, G^MW(z) obtained from the microwave reflection according to the procedure described above is shown as the blue curve. As can be seen, both curves are in reasonably good agreement. Note that agreement is particularly good in the calibration interval of z < −0.46 nm, as illustrated in the lower panel of Fig. 4(b). From this good quantitative agreement, we conclude that the calibration was successful. We remark that the minor discrepancy between the fit to G(z) (red) and the G^MW(z) data (blue) in the calibration interval cannot be explained by deviations in C due to the tip apex. Instead, it may be rationalized by a frequency dependence of G^26,27 that is not explicitly included in the calibration procedure and goes beyond the scope of the present work.

In order to verify the validity of our assumption that the change in capacitance is negligible on the calibration interval of z = −1 to −0.46 nm, we compare the relative impact of changes in G and C on $Yjunz$⁠. We obtain a respective change of $ΔGz=ΔReYjunz≈$ 1.6G₀ ≈ 126 µS; see the lower panel of Fig. 4(b). In comparison, a linear fit over the whole z-range from −1 to +50 nm of the ΔC^MW(z) curve, see Fig. 4(c), yields a value for the slope ∂C^MW(z)/∂z of −2.3 fF/nm, resulting in an absolute change of ΔC^MW ≈ 1.2 fF over the respective z-range (Δz = 1–0.46 = 0.54 nm) and, finally, resulting in a value of $ΔCMW2πf=ΔImYjunz≈$ 7 µS. This value is circa 20 times smaller (!) than the above value of 126 µS, demonstrating that Y_jun is, indeed, dominated by G (and not by C), in agreement with the above assumption. Note that within the calibration interval, the slope ∂G(z)/∂z derived from Fig. 4(b) is found to be always larger than the respective slope of 2πf∂C^MW(z)/∂z (see above), which is again in agreement with the above assumption. Repeating the calibration for slightly larger and smaller z-ranges (±0.05 nm) yields values that are in good agreement [see the supplementary material, Fig. 2, for deviations in G^MW(z) and ΔC^MW(z)] to the above values. Either enforcing the G = 0 condition for z > 10 nm or accounting for the linear term in $Cz$ in a self-consistent optimization loop again yields consistent values. This shows that the calibration procedure yields robust values for the capacitance.

The e-parameters obtained in this calibration procedure numerically describe non-idealities, such as absorptions and reflections, of microwave propagation through the transmission line from the calibration plane to the measurement plane. As such, these e-parameters are independent of z and of tip and sample properties.¹⁰ The calculation of S_11,jun(z) involves an exponential fit to G(z), so the tip must be stable to give an exponential G(z) dependence during calibration. The e-parameters can be used to calibrate microwave reflectance measurements to obtain quantitative ΔC^MW values, as shown in Fig. 4(c). It is important to note that a particular set of e-parameters is only valid for the frequency and VVA settings at which the calibration was performed. To obtain a frequency spectrum, a calibration should be performed for each frequency. Currently, this is done with a separate approach curve at each frequency notch, but we envision using a full frequency sweep during a single approach curve in the future.

We present a calibration procedure for microwave reflectance spectroscopy based on the acquisition of conductance retraction curves in an ultra-high vacuum low-temperature STM instrument. While the retraction curve calibration concept based on capacitive force detection has been very successfully employed for microwave reflection measurements in an AFM¹⁰ in the past, in STM, the force detection is not available. Therefore, we acquire the tunneling conductance, which changes exponentially with distance, instead of the capacitance, which has a long-range interaction and changes only slowly with tip–sample distance. We show that in the high G regime, for approximately G > G₀/3 occurring at sub-nm tip–sample distances, the change of the measured scattering parameter S_11,m with respect to the tunnel distance is dominated by changes in the conductance signal. In this narrow z-range of 0.54 nm in the high-conduction regime, the change of the junction capacitance with tunnel distance can be neglected. This allows for a successful calibration of the error parameters describing the distortion of S_11,m relative to the scattering parameter at the tunnel junction S_11,jun. We remark that the employed e-parameter procedure captures all non-idealities of the transmission line and detection circuit and can be readily extended to calibration of an entire range of frequencies.

The calibration of microwave reflectance spectra presented here will pave the way for quantitative capacitance and dielectric spectroscopy measurements with a lateral resolution of a few nm and ultimately single molecule fingerprinting in low-temperature ultra-high vacuum STM. Specifically, we expect that the tip–sample contribution to ΔC^MW can be isolated either in constant height measurements or with an appropriate capacitance model. This may allow us to determine material properties, such as the dielectric permittivity, in future experiments.¹² The imaging capabilities of STM together with frequency dependent microwave reflectance could then be utilized to achieve full spectroscopic imaging.

See the supplementary material for additional details on the dependence of the e-parameters, GMW(z), and ΔCMW(z) on the z-range included in the calibration procedure.

The authors acknowledge the financial support from the European Research Council (ERC) under the European Union’s Horizon 2020 Research and Innovation Program (Grant Agreement No. 771193). This work was additionally supported by funding from EMPIR project 20IND12 Elena.

The authors have no conflicts to disclose.

Bareld Wit: Conceptualization (equal); Formal analysis (lead); Methodology (equal); Validation (equal); Visualization (lead); Writing – original draft (lead); Writing – review & editing (supporting). Georg Gramse: Conceptualization (equal); Funding acquisition (supporting); Methodology (equal); Validation (equal); Writing – original draft (supporting); Writing – review & editing (equal). Stefan Müllegger: Conceptualization (equal); Funding acquisition (lead); Project administration (equal); Supervision (equal); Validation (equal); Writing – original draft (supporting); Writing – review & editing (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.