Electro-acoustic devices based on surface acoustic waves (SAWs) and Lamb waves are essential for wireless communication and sensing applications, with small form factors due to the slow speed of sound and convenient piezoelectric electromechanical transduction. Recently, acoustic waves have also been shown to couple to superconducting qubits11. Y. Chu, P. Kharel, W. H. Renninger, L. D. Burkhart, L. Frunzio, P. T. Rakich, and R. J. Schoelkopf, “ Quantum acoustics with superconducting qubits,” Science 358, 199–202 (2017). https://doi.org/10.1126/science.aao1511 and spins,22. S. J. Whiteley, G. Wolfowicz, C. P. Anderson, A. Bourassa, H. Ma, M. Ye, G. Koolstra, K. J. Satzinger, M. V. Holt, F. J. Heremans, A. N. Cleland, D. I. Schuster, G. Galli, and D. D. Awschalom, “ Spin–phonon interactions in silicon carbide addressed by Gaussian acoustics,” Nat. Phys. 15, 490–495 (2019). https://doi.org/10.1038/s41567-019-0420-0 control photons33. K. Fang, M. H. Matheny, X. Luan, and O. Painter, “ Optical transduction and routing of microwave phonons in cavity-optomechanical circuits,” Nat. Photonics 10, 489–496 (2016). https://doi.org/10.1038/nphoton.2016.107 and electrons,44. S. Hermelin, S. Takada, M. Yamamoto, S. Tarucha, A. D. Wieck, L. Saminadayar, C. Bäuerle, and T. Meunier, “ Electrons surfing on a sound wave as a platform for quantum optics with flying electrons,” Nature 477, 435–438 (2011). https://doi.org/10.1038/nature10416 and exhibit topological characteristics.55. P. Wang, L. Lu, and K. Bertoldi, “ Topological phononic crystals with one-way elastic edge waves,” Phys. Rev. Lett. 115, 104302 (2015). https://doi.org/10.1103/PhysRevLett.115.104302 As increasingly high frequencies are used in such applications,66. W. Saad, M. Bennis, and M. Chen, “ A vision of 6G wireless systems: Applications, trends, technologies, and open research problems,” IEEE Network 34, 134–142 (2020). https://doi.org/10.1109/MNET.001.1900287 imaging the wave dynamics in these devices with nanoscale spatial resolution, especially in operando, has become a significant challenge.A recent breakthrough is the demonstration of transmission-mode microwave impedance microscopy (T-MIM)7,87. L. Zheng, D. Wu, X. Wu, and K. Lai, “ Visualization of surface-acoustic-wave potential by transmission-mode microwave impedance microscopy,” Phys. Rev. Appl. 9, 061002 (2018). https://doi.org/10.1103/PhysRevApplied.9.0610028. L. Zheng, L. Shao, M. Loncar, and K. Lai, “ Imaging acoustic waves by microwave microscopy: Microwave impedance microscopy for visualizing gigahertz acoustic waves,” IEEE Microwave Mag. 21, 60–71 (2020). https://doi.org/10.1109/MMM.2020.3008240 that can image nanoscale distributions of the electric potentials accompanied by acoustic waves at up to 6 GHz with sub-100 nm spatial resolution, all in a working device. It has been used to visualize wave propagation in phononic circuits,9,109. D. Lee, Q. Liu, L. Zheng, X. Ma, H. Li, M. Li, and K. Lai, “ Direct visualization of gigahertz acoustic wave propagation in suspended phononic circuits,” Phys. Rev. Appl. 16, 034047 (2021). https://doi.org/10.1103/PhysRevApplied.16.03404710. D. Lee, S. Meyer, S. Gong, R. Lu, and K. Lai, “ Visualization of acoustic power flow in suspended thin-film lithium niobate phononic devices,” Appl. Phys. Lett. 119, 214101 (2021). https://doi.org/10.1063/5.0073530 to demonstrate electrical control of SAWs,1111. L. Shao, D. Zhu, M. Colangelo, D. Lee, N. Sinclair, Y. Hu, P. T. Rakich, K. Lai, K. K. Berggren, and M. Lončar, “ Electrical control of surface acoustic waves,” Nat. Electron. 5, 348–355 (2022). https://doi.org/10.1038/s41928-022-00773-3 and to resolve topological edge states in phononic crystals.12,1312. Q. Zhang, D. Lee, L. Zheng, X. Ma, S. I. Meyer, L. He, H. Ye, Z. Gong, B. Zhen, K. Lai, and A. T. C. Johnson, “ Gigahertz topological valley Hall effect in nanoelectromechanical phononic crystals,” Nat. Electron. 5, 157–163 (2022). https://doi.org/10.1038/s41928-022-00732-y13. Y. Nii and Y. Onose, “ Imaging an acoustic topological edge mode on a patterned substrate with microwave impedance microscopy,” Phys. Rev. Appl. 19, 014001 (2023). https://doi.org/10.1103/PhysRevApplied.19.014001 In light of the increasing importance of the technique, a set of systematic circuit-level design principles for T-MIM—as we describe in this Letter—would be a timely addition to the literature and help to guide the development of this nascent technology.MIM brings microwave sensing capabilities to a scanning probe microscope.1414. Y. Shi, J. Kahn, B. Niu, Z. Fei, B. Sun, X. Cai, B. A. Francisco, D. Wu, Z.-X. Shen, X. Xu, D. H. Cobden, and Y.-T. Cui, “ Imaging quantum spin Hall edges in monolayer WTe2,” Sci. Adv. 5, eaat8799 (2019). https://doi.org/10.1126/sciadv.aat8799 Conventional MIM is performed in the reflection mode (herein referred to as R-MIM for clarity), where an ultrasensitive microwave reflectometer measures variations in the tip–sample admittance caused by nanoscale inhomogeneities in the complex permittivity of the sample1515. M. E. Barber, E. Y. Ma, and Z.-X. Shen, “ Microwave impedance microscopy and its application to quantum materials,” Nat. Rev. Phys. 4, 61–74 (2022). https://doi.org/10.1038/s42254-021-00386-3 [Fig. 1(a)]. In R-MIM, an impedance matching network is commonly inserted between the reflectometer and the probe to maximize power delivery and to increase sensitivity. Indeed, we showed in an earlier work1616. J.-Y. Shan, A. Pierce, and E. Y. Ma, “ Universal signal scaling in microwave impedance microscopy,” Appl. Phys. Lett. 121, 123507 (2022). https://doi.org/10.1063/5.0115833 that the R-MIM signal depends quadratically on the voltage enhancement factor of the circuit η=VprobeR/VinR, where VprobeR is the voltage at the probe and VinR is the amplitude of the incident microwave going into the matching network (superscripts indicate the reflection mode). A strong signal can, thus, be achieved near the resonance of a high quality-factor (Q-factor) resonator-style matching network at the cost of bandwidth, but the proportionality holds both on- and off-resonance for any matching network design.The operation of T-MIM is fundamentally different. The microwave source excites the sample, such as a SAW filter, separately from the probe, e.g., by launching a SAW with an inter-digital transducer (IDT). The electric potential propagating with the SAW is then picked up by the probe, which acts as a nano-antenna [Fig. 1(b)]. Therefore, T-MIM measures the potential profile in a working device instead of the intrinsic complex permittivity of the sample. On the circuit level, distinct from R-MIM, the microwave only travels one way from the sample to the homodyne detector through the matching network. One would, thus, expect a different relationship between the T-MIM signal and the properties of the matching-network-probe circuitry.More specifically, a complete T-MIM circuit has two types of parameters: one set is determined by the matching-network-probe design and is shared between transmission and reflection modes; another set is unique to transmission-mode operation. The former includes the voltage enhancement factor η and the complex reflection coefficient S11, the one-port S-parameter that characterizes the quality of the impedance matching looking into the matching network. The latter includes the surface source potential Vs and the effective source output impedance Zs. In the electro-quasistatic limit, applicable here since the tip is much smaller than the relevant microwave wavelength, Vs is simply divided between Zs and the “load impedance” Zin, the input impedance looking into the probe [Fig. 2(h)].We first simulated three distinct T-MIM designs to gain numerical insight (Fig. 2). Figure 2(a) shows a tuning fork sensor with an etched tungsten wire probe and a half-wave transmission line (TL) resonator matching network. Figure 2(b) shows a cantilever-based stripline aluminum probe with a quarter-wave and stub matching network. Figure 2(c) shows a design with the same probe as in Fig. 2(a), but with a direct connection to the detector. For each design, we performed a transmission-mode simulation to obtain the T-MIM signal VMIM and the input impedance Zin and a reflection-mode simulation to obtain η and S11. In the reflection-mode simulations, we included the source output impedance Zs as a regular tip–sample interaction impedance in parallel with the probe [Fig. 2(h)]. See Table I for the detailed simulation parameters.

TABLE I. Simulation parameters.

Vs=0.2 V, G=2×104, Zs corresponds to a 1 fF capacitor77. L. Zheng, D. Wu, X. Wu, and K. Lai, “ Visualization of surface-acoustic-wave potential by transmission-mode microwave impedance microscopy,” Phys. Rev. Appl. 9, 061002 (2018). https://doi.org/10.1103/PhysRevApplied.9.061002CaseProbeR (Ω)L (nH)C (pF)Impedance matching(a)Bare etched W wire with tuning fork1717. Y.-T. Cui, E. Y. Ma, and Z.-X. Shen, “ Quartz tuning fork based microwave impedance microscopy,” Rev. Sci. Instrum. 87, 063711 (2016). https://doi.org/10.1063/1.4954156: l=3 mm and d = 80 μmHalf-wave TL: FM-SR020CU-STR Fairview Microwave0.252.60.04lTL=57 mm, C=0.17 pF(b)Stripline Al cantilever1818. Y. Yang, K. Lai, Q. Tang, W. Kundhikanjana, M. A. Kelly, K. Zhang, Z.-X. Shen, and X. Li, “ Batch-fabricated cantilever probes with electrical shielding for nanoscale dielectric and conductivity imaging,” J. Micromech. Microeng. 22, 115040 (2012). https://doi.org/10.1088/0960-1317/22/11/115040: 5–300 N PrimeNanoQuarter-wave TL: FM-SR118CU-COIL Fairview Microwave50.51lTL=20.5 mm, lstub=24.5 mmThe simulation results show that for resonator-type matching networks [Figs. 2(a) and 2(b)], VMIM and |η| follow similar frequency-dependent profiles with similar maximum values for cases a and b at their respective resonant frequencies [Figs. 2(d) and 2(f)], while the behaviors of Zin and S11 differ markedly from VMIM [Figs. 2(e) and 2(f), inset]. In contrast, the direct connection case [Fig. 2(c)] shows broadband behavior and a lower maximum VMIM, although only by a factor of ∼2. Note that in all these realistic cases, |Zin|≪|Zs|, which results in “weak sampling,” i.e., VMIM/G≪Vs.The similarities in VMIM and η suggest that they may be linearly related. Indeed, the complex dimensionless factor [2VMIM/(GηVs)](Zs/Z0) is unity for all cases at all frequencies within the numerical error [Fig. 2(g)].Following this numerical insight, we set out to analytically derive this linear relationship between VMIM and η and to identify any corrections or necessary approximations. We start with the fact that any linear, reciprocal impedance matching network with an admittance matrix of can be modeled as a two-port π-shaped equivalent circuit [Fig. 2(h)].1919. D. M. Pozar, Microwave Engineering, 4th ed. ( Wiley, Hoboken, NJ, 2011). Thus, we can calculate the input admittance Yin(=1/Zin) as Yin=Yprobe+Y22+Y12−Y12(Y11+Y12+Y0)−Y12+Y11+Y12+Y0,=Yprobe+Y22−Y122Y11+Y0,where Y0=1/Z0 is the characteristic admittance of the system (Z0 is generally 50 Ω) and Yprobe is the probe admittance. Following the voltage division rule, we obtain The T-MIM signal VMIM is, then, VMIM=GV1=GV21/(Y11+Y12+Y0)1/(Y11+Y12+Y0)−1/Y12,=−GV2Y12Y11+Y0,=GVsZ02Zs·[−2Y12Y0(Ys+Y22+Yprobe)(Y0+Y11)−Y122].(1)We note that the last fraction in Eq. (1) is precisely the R-MIM voltage enhancement factor η=VprobeR/VinR, when the tip–sample interaction admittance is Ys.1616. J.-Y. Shan, A. Pierce, and E. Y. Ma, “ Universal signal scaling in microwave impedance microscopy,” Appl. Phys. Lett. 121, 123507 (2022). https://doi.org/10.1063/5.0115833 Therefore, we have VMIM=G·Ys2Y0·η·Vs,(2)showing that the measured T-MIM signal is, indeed, proportional to η. Comparing with the R-MIM result1616. J.-Y. Shan, A. Pierce, and E. Y. Ma, “ Universal signal scaling in microwave impedance microscopy,” Appl. Phys. Lett. 121, 123507 (2022). https://doi.org/10.1063/5.0115833 of ΔVMIMR=−G·ΔY2Y0·η2·Vin,the linear, instead of quadratic, dependence on η, here, can be intuitively understood as the consequence of a single pass through the MIM circuits in T-MIM.One conclusion we can draw from these results is that the ultra-broadband operation is easier for T-MIM than for R-MIM. Two factors contribute here. First, the direct-connection design [Fig. 2(c)] has an |η| of ∼2, as opposed to ∼1 for the Z0 shunt matching for broadband R-MIM.1616. J.-Y. Shan, A. Pierce, and E. Y. Ma, “ Universal signal scaling in microwave impedance microscopy,” Appl. Phys. Lett. 121, 123507 (2022). https://doi.org/10.1063/5.0115833 Second, the signal scales linearly instead of quadratically with η in T-MIM, so the loss in sensitivity compared with typical room-temperature narrow-band |η|max of up to ∼10 is less significant.For narrow-band, sensitivity-limited operation, Eq. (2) is a general result that can guide us beyond the weak sampling limit that has been assumed in the literature so far,77. L. Zheng, D. Wu, X. Wu, and K. Lai, “ Visualization of surface-acoustic-wave potential by transmission-mode microwave impedance microscopy,” Phys. Rev. Appl. 9, 061002 (2018). https://doi.org/10.1103/PhysRevApplied.9.061002 since we did not use the condition |Zin|≪|Zs| in our derivation. We note that given G, Vs, and Zs, boosting VMIM is equivalent to increasing the voltage enhancement factor η, which, as we will show below, benefits from low-loss and critically coupled resonator-type matching networks and is distinct from simply increasing |Zin| or optimizing S11 as suggested previously.77. L. Zheng, D. Wu, X. Wu, and K. Lai, “ Visualization of surface-acoustic-wave potential by transmission-mode microwave impedance microscopy,” Phys. Rev. Appl. 9, 061002 (2018). https://doi.org/10.1103/PhysRevApplied.9.061002To this end, we take the design of a bare metal wire probe with a half-wave resonator matching network [Fig. 2(a)] as a model system and analyze how the TL loss and the coupling capacitance C affect the T-MIM performance. First, we set the TL loss to a lower value of 0.1 Ω/m, which can be achieved using commercially available superconducting co-axial cables (such as COAX SC-086/50-NbTi-NbTi). We then carried out a series of simulations with C varying from 0.17 to 0.001 pF. The results show several features (Fig. 3). First, as C is reduced, the resonant frequency shifts and the resonances become generally narrower. Second, the peak VMIM, |η|, and |S11| values first increase and then decrease [Figs. 3(a) and 3(c)]. Third, the peak |Zin| increases monotonically and surpasses |Zs| for C≲0.05 pF [Fig. 3(b)]. Fourth, Eq. (2) holds exactly in all cases, as expected [Fig. 3(d)].The first two observations can be well understood within the framework of coupled resonators.2020. M. Cai, O. Painter, and K. J. Vahala, “ Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85, 74–77 (2000). https://doi.org/10.1103/PhysRevLett.85.74 Because of the low TL loss, the total round-trip loss of the half-wave resonator is dominated by the coupling to the external circuits through the capacitor when C≳0.05 pF. The system is, thus, over-coupled, leading to broad, shallow resonances and low enhancement factors. As C becomes smaller, the coupling loss decreases and eventually becomes similar to the intrinsic TL loss, at which point the system becomes critically coupled, leading to a narrow and strong resonance with the best possible total coupling (S11) and the highest possible enhancement factor and, thus, T-MIM signal. Finally, as C becomes even smaller, the system becomes under-coupled, leading to a narrow but shallow resonance and reduced voltage enhancement.Indeed, if we plot the peak η and S11 values as a function of C for different TL losses [Figs. 3(e) and 3(f)], the peak positions agree very well with the C values that give rise to equal round-trip TL loss and capacitive coupling loss [short vertical dashed lines in Fig. 3(e)], 1−exp(−2R′lTLZ0)=4Z02/(4Z02+1C2ω2),where R′ is the resistance per unit length of the TL. Following the results for different TL losses in Figs. 3(e) and 3(f), we can conclude that to maximize narrow-band sensitivity, one should choose components with the lowest possible loss first and then tune the coupling strength to achieve critical coupling, using S11 as a convenient guide. On the other hand, following vertical cuts in Figs. 3(e) and 3(f), we can conclude that if the coupling strength C needs to be fixed, one should not aim for critical coupling. Instead, it is preferable to lower the loss of the components as much as possible, even if it makes the system over-coupled and S11 worse (see the supplementary material for more details). These strategies are general because the concept of intrinsic loss and coupling loss can be readily applied to other resonator-type impedance matching networks.2121. H. Kim, H. Choi, S. Jeon, and H. Kim, “ Critical coupling of a planar inverted F-antenna using a ferrite sheet,” Microwave Opt. Technol. Lett. 52, 400–403 (2010). https://doi.org/10.1002/mop.24950Finally, we explain the apparent deviation between the trend of |Zin|max and that of η and VMIM, especially for under-coupled systems [Fig. 3(g)]. From the simple voltage division of Vs between Zin and Zs and the weak sampling limit of |Zin|≪|Zs|, one might expect VMIM to be correlated with Zin, which then needs to be maximized. However, the voltage division only describes the voltage sampled by the probe [V2 in Fig. 2(h)], instead of that coupled into the external detection circuit (V1). Therefore, although a smaller C always leads to a larger |Zin|max and, thus, V2, it does so at the cost of reducing V1 and, thus, VMIM in under-coupled systems. In fact, if |Zin|max is comparable to or larger than |Zs|, VMIM is not maximized at the frequency f0′ that maximizes |Zin|. Instead, it is maximized at f0 which makes Zin the closest to Zs* [Figs. 4(a) and 4(b)]. This is consistent with the maximum power transfer theorem.1919. D. M. Pozar, Microwave Engineering, 4th ed. ( Wiley, Hoboken, NJ, 2011). Practically, because f0′ also minimizes the “unloaded” |S11| (i.e., with the probe away from the sample), there will be an offset between the result of an unloaded |S11| measurement and the actual optimal working frequency for T-MIM [Fig. 4(c)]. This discrepancy persists for critically- and over-coupled systems, albeit less important due to the broader resonances (see the supplementary material).See the supplementary material for the discussion of critical coupling conditions and the resonant frequency differences between unloaded S11 and VMIM.

This work was supported by the Laboratory Directed Research and Development Program of Lawrence Berkeley National Laboratory under U.S. Department of Energy Contract No. DE-AC02-05CH11231.

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Jun-Yi Shan: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Nathaniel Morrison: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Eric Y. Ma: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Project administration (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

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

REFERENCES

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