Nonlinear microwave impedance microscopy

Abstract

A microwave impedance microscopy (MIM) system having multi-harmonic MIM channels has a microwave signal source to produce a microwave signal, a probe tip to transmit the microwave signal to a surface of a sample under test and detect a signal, a scanning setup to scan the sample under test against the probe tip, one or more harmonic separators and demodulators to simultaneously extract multiple harmonic components of the signal, one or more processors configured to execute code that causes the one or more processors to: perform end-to-end phase calibration of the multi-harmonic MIM channels, relate the multiple harmonics components of the signal to corresponding non-linear admittance parameters, and use the non-linear admittance parameters to analyze the sample.

Description

Patent Application U Cal No. BK-2025-102-2-PCT MN No. 407869-0243 NONLINEAR MICROWAVE IMPEDANCE MICROSCOPY TECHNICAL FIELD

[0001] This disclosure relates to microwave impendence microscopy, more particularly to nonlinear microwave impedance microscopy that simultaneously uses multiple harmonics of the incident frequency.BACKGROUND

[0002] Microwave impedance microscopy (MIM) is an emerging scanning probe technique that enables non-contact, nanoscale measurement of local complex permittivity. By integrating an ultra-sensitive, phase-resolved microwave sensor with a near-field probe, MIM has made significant contributions to diverse fundamental and applied fields. These include strongly-correlated and topological materials, two-dimensional and biological systems, as well as semiconductor, acoustic and MEMS devices. Concurrently, notable progress has been made in refining the MIM technique itself and broadening its capabilities.

[0003] Existing literature has focused exclusively on linear MIM based on homodyne architectures, where reflected or trans-mitted microwave is demodulated and detected at the incident frequency. As such, linear MIM lacks the ability to probe local electrical nonlinearity, which is widely present in e.g., dielectrics, semiconductors, and superconductors. Elucidating such nonlinearity with nanoscale spatial resolution can provide critical insights into semiconductor processing and diagnostics as well as fundamental phenomena like local symmetry breaking and phase separation.BRIEF DESCRIPTION OF THE DRAWINGS

[0004] FIGs. 1 A-1D shows circuit schematics and equivalent circuit models for linear and nonlinear microwave impedance microscopy.

[0005] FIGs. 2A-2F show a circuit model and graphs of validation using a parallel nonlinear resistor-capacitor network.

[0006] FIGs. 3 A-3F show a circuit model and graphs of validation using a metal-oxide semiconductor (MOS) point contact.

[0007] FIG. 4 shows an alternative embodiment of a non-linear MIM system.DETAILED DESCRIPTION OF THE EMBODIMENTS

[0008] The embodiments here extend traditional microwave impedance microscopy (MIM) to nonlinear operation to probe local electrical nonlinearity. The embodiments present a quantitative framework relating the multi-harmonic MIM signals to nonlinear tip-sample admittance using a nonlinear circuit responsivity model and provide methods to design, calibrate, and operate the instrument. FIG. 1 A shows a linear MIM system, and FIG. IB shows the equivalent circuit model.

[0009] In conventional MIM, all fields, voltages, and currents are assumed to have antime dependence, so the measurement mechanism can be conveniently described using complex- valued microwave amplitudes and impedance / admittance. In particular, thanks to the deeply sub -wavelength scale, the tip-sample junction can be modeled as a linear two-terminal lumped element with a well-defined admittance Yts that relates the current and voltage of the junction:Z(m0) = rts(co0)7(m0). (1)

[0010] Recent work has shown that the variation in reflected microwave amplitude, AFout, due to the linear tip-sample interaction, compared with an "unloaded" probe, is proportional to the tip-sample junction admittance Yts'.i]2Wout = - ^- YtsVin (2)2y0Where Vtn is the incident microwave amplitude and j'o the system admittance, typically 0.02 S (1 / 50 Q). The voltage enhancement factor t] is the ratio of the total microwave voltage at the tip to that of the incident field, and is determined by the admittance matrix [y] of the impedance matching network and the unloaded probe admittance yprobe.^•YOYAB T] =. (3)YAB ~ (YBB + Yprobe) (To + YAA)

[0011] All quantities above are defined at the incident microwave frequency coo using port definitions in FIG IB. Experimentally, AFout is measured through low-noise amplification and phase-resolved demodulation at coo, achieving thermal-noise-limited detection recently.Table I. Nonlinear admittance for representative 2 -terminal nonlinear lumped elements. Nonlinear Constituent relationInelementtypeCubic I = GiV+ GsV3Gi 0 0 G< 4 nonlinearresistorsQuadratic Q = CiV + C2V2 nicooCi i(D(£2 3ia>cC2 / 2 0 nonlinearcapacitorQuadratic dl / dt ^ / Li+JN / Ld -i / ncooLi ~i / 4a>oL2 -i / 6a>oL2 0 nonlinearinductorMOS ni top C Q-g iM0Coxy 3ia)0Coxy 3ico„C juncti dO IdV —oxoxy^2on Jl - 2y(V - Vm(1 + 2yt>6)V2 2(1 + 2yVfb)N2 4(1 + 2yVfb)N28(1 + 2y(Vfb)N2(n-typej*c*MOS: Metal-oxide-semiconductor; y = —2= C2x where Cox is the oxide capacitance, Nd the doping concentration,q the elemental charge, esthe semiconductor relative permittivity, and A the junction area; Vjb is the flatband voltage; valid for | V| < min(|Vfb|, | Vth|) where Vth is the threshold voltage.

[0012] The framework above begins to break down when the tip-sample interaction becomes nonlinear, as the linearity prerequisites for the concept of impedance / admittance, as well as the assumption of globaltime dependence, are no longer valid. To address these conceptual challenges, the embodiments first introduce a nonlinear admittance formulation to describe weakly-to-moderately nonlinear tip-sample junctions and then solve a system of circuit equations to relate the MIM signals at multi-harmonics of the incident frequency coo to the corresponding nonlinear admittance parameters.

[0013] A nonlinear tip-sample junction can be modeled as a two-terminal lumped element with a nonlinear I -V relationship. In general, the instantaneous current 1(f) is an arbitrary functional of the entire hi-tory of the voltage across the element, including time derivatives and integrals:Z(t) = F[7(T); T < t], (4) which is often intractable to model, especially in the frequency domain.

[0014] However, most scenarios relevant to MIM measurements are expected to be much simpler, involving weak to moderate nonlinearities that allow a perturbative power series expansion, similar to that in nonlinear optics:Z(m) =+ [ T® (mx, m2)V(m1)7(m2)<5(m — — m2)cZm1m2+ j rC3)(m1,m2,m3)7(m1)7(m2)7(m3)x ( — )1— a>2 — m3)cZm1m2m3+..., (5)where Y(n)are herein defined as / / -th order nonlinear admittance parameters that connect different frequencies.

[0015] For nonlinear MIM measurements, the embodiments assume that a single-frequency excitation at coo is used, and thus only harmonics of coo exist in the system at steady state. One can further assume a "feed-forward" response for moderate excitation levels, where the / / -th harmonic in current depends only on voltages and nonlinear admittance terms of the same and lower order. Rewriting I(ncoo) as / «, V(nc o) as Vn, andF^(m1cjoo, m1cjoo,...,mncjoo)as Y^m, results in:Iln =1nvny y / ,1m,m2vmvm2m1+m2=nV V V / t1m1,m2,m3Vm1vm2vm3m1+... (6)where mi = 0,1,2,... This formulation can capture resistive, capacitive, and inductive nonlinearities, with representative examples in Table I.

[0016] In the time domain, the voltages and currents are periodic in the fundamental period 27t / coo, and can be obta (ined via:+ c c\ (7) ( Zrtetrta>°t+ c. c. j (8)n=0 / where c.c. represents the complex conjugate. For zero tip-sample bias (i.e., Vo=O), the first few harmonics are:Ii = YiWVi>I2= Y^V2+ Y^Vf. (9) Z3= r3C1)73+ 2Y^V±V2 +Y^V?.

[0017] Here h represents the linear response from F^) =Yts, as measured in conventional IM. I2 and I3 are 2nd harmonic generation (SHG) and 3rd harmonic generation (THG), arising from various 2nd- (Y2I) and 3rd-order (F^) nonlinear admittance. These nonlinear admittances encode underlying material or device properties and are the key quantities of interest. The discussion below describes how they can be determined through the proposed NL-MIM measurements.

[0018] Experimentally, it is difficult to directly measure the GHz-frequency current and voltage of the tip-sample junction. Instead, one can analyze the reflected signal’s harmonic spectra by demodulating not only at the fundamental frequency coo but also at higher harmonics. Using the latest cancellation-free architecture, this can be accomplished with a single broadband high-linearity low-noise amplifier (LNA), followed by a harmonic separator (one or more of di- / triplexers and filters), and frequency multipliers for generating demodulator references.

[0019] FIG 1C shows an embodiment of a non-linear MIM system having a microwave scanning tip 10 and a sample under test 12. A matching impedance network 14 may be used as part of the scanning setup. The LNA 16 discussed above is followed by harmonic separators and demodulators 18. This allows multiple harmonic components of the signal to be extracted into multi-harmonic MIM channels such as 20.

[0020] Additional instrumentation considerations are discussed after derivation of the quantitative responsivity formulas. The task at hand is therefore to relate the measured complex amplitudes of the reflected signals at various harmonics, Fout(mo) =Fout,-7, Fout(2mo) = Lout, 2, Fout(3mo) =I out;j, etc. to the nonlinear admittance Y(n. To accomplish this, one can solve the circuit that includes the tip-sample junction, the "unloaded" probe, and the impedance matching network, represented by a linear reciprocal admittance matrix [y] where yAB = yBA (Fig. Id). Applying harmonic balance, one can obtain a general system of equations for the / / -th harmonic:r^A;n _ [YAA YABI VA;niJAB YBBIK lVB;n. Yi;n Yin;n + Yout.n< ^A;n ~ Yo(Vin;n ~ Yout;n) (10) IB; TI ~(jYprobe;vVB; Ti 3” In) Tz > f Yin 71 — 1in,n(0 otherwise

[0021] Under the approximations of (i) feed-forward action, which ensures that the nonlinear current In follows Eq. (9), and (ii) the linear tip-sample admittance |F«(7)| being much smallerthan |yProbe;n| and yo, a condition typically satisfied experimentally due to the nanoscale tip area, the following closed-form solutions are derived:^out,-1 = - ^- Y^Vin(11)2 Vout,-2 = - ^ Y^n (12) 1 / — _ y(3) I yo+y AA,-2. y(2)y(2) J z 3 O',yout;3 —z7rl,l,l " T '12rl,2rl.lVin I15! yo ’ ’ yoVAB-2 'where r|« = q(«mo) is the voltage enhancement factor at wcoo (Eq. (3)).

[0022] One or more processors contained within the NL-MIM system may execute code to solve the equations, using the simultaneously obtained multi-harmonic MIM signals, to arrive at the nonlinear admittance parameters to allow the NL-MIM system to analyze a sample being characterized in real time.

[0023] The linear response in Eq. (11) is consistent with prior linear MIM results (Eq. (2)). The nonlinear responses in Eq. (12) and (13) are the main findings of the embodiments. Notably, the 2nd harmonic signal only depends on and scales quadratically with the input voltage, while the 3rd harmonic involves both Y1and Y1y2, and scales cubically with the input voltage. Prefactors in the form of r / nri^ emerge, reflecting the interplay of1 (2) (2) (3) enhancement factors across harmonics. For typical samples where | — Y^2Y[ | « |KX\x|(see examples below), Eq. (13) can be simplified to:3V^1 (3) 3Vout;3 ~ 2y0

[0024] The results were validated through high-precision time-domain circuit simulations combined with phase-calibrated Fourier analysis. The complete circuit including the impedance matching network, probe, and tip-sample junction was simulated under a 2 GHz sinusoidal drive. The simulation used < 1 ps time steps to minimize numerical noise and nonlinear artifacts. To account for baseline linear reflections, experimentally rejected by dynamic mode operation or cancellation circuits, a reference was established by simulating an identical circuit containing only the unloaded (linear) probe. The complex Fourier amplitudes of the difference signal were extracted at 2, 4, and 6 GHz, and compared with predictions from Eq. (11), (12) and (14), using matching network j’-parameters and r] obtained via a separate small-signal frequency-domain analysis. This approach was applied totwo examples: a nonlinear parallel RC network and a realistic metal-oxide-semiconductor (MOS) point contact.

[0025] A first example, shown in FIG. 2A, considers a parallel combination of a cubic nonlinear resistor 22 and a quadratic nonlinear capacitor 24 as the RC network, discussed in Table I. One embodiment uses a half-wavelength resonator 26 as the impedance matching network, which naturally provides low reflection and high q at harmonics of 2 GHz shown in FIG. 2B. The unloaded probe was modeled as a 0.1 pF capacitor in series with a 2 Q resistor. The circuit was simulated in the time domain for a variety of realistic incident powers, with 10 fs time steps, taking data after a settling period of 30 ns.

[0026] FIG. 2C shows the extracted the Fourier components of the time-domain signal at 2, 4, and 6 GHz, representing the 1st, 2nd, and 3rd harmonic NL-MIM signals. FIGs. 2D-F present a quantitative comparison between the time-domain numerical results and the nonlinear admittance theoretical results, with no free fitting parameters. For clarity, both results are normalized by appropriate q terms. Excellent agreements were achieved for all three frequencies for both the real and imaginary channel. The small deviation at the highest power arises from difference frequency generation which is not accurately captured by the feed-forward approximation.

[0027] A possibly more realistic example considers the case of a gold-coated tip 30 in contact with n-type silicon that has a 3 nm native oxide layer on top, shown in FIG. 3 A. In this scenario, the MOS junction can be modeled by a fixed oxide capacitor Cox in series with a voltage-dependent depletion layer capacitor Cdep, shown in Table I and a fixed resistor to account for bulk spreading resistance. The experiment used a broadband 50 Q shunt impedance matching network 32 to demonstrate the generalizability of the framework, shown in FIG. 3 A. The probe admittance is the same as above. The circuit was simulated with 1 ps time steps and a settling period of 10 ns. A larger time step and shorter settling period can be used due to the non-resonant nature of the broadband impedance matching. FIG. 3B shows q and 5n spectra, and FIG. 3C shows the reflected microwave amplitudes for various incident powers.

[0028] The numerical and theoretical NL-MIM results for all three harmonics again show excellent agreements, as illustrated in FIGs. 3D-F, validating the framework. Having established the framework for NL-MIM, the discussion now turns to considerations for its design and operation, as well as implications for new materials and device applications.

[0029] The performance of NL-MIM depends on the matching network design, demodulation phase calibration, and the amount of parasitic nonlinear signals. First, Eq. (11-13) show that the / / -th order NL-MIM signal scales with r / nT]^. Therefore, responsivity is maximized when r| is high for both the first and / / -th harmonic. This can be naturally achieved using e.g., a half-wavelength resonator, such as is shown in FIG. 2). However, large r| values from resonant circuits may result in partial breakdown of the feed-forward approximation and introduce high sensitivity to phase calibrations, complicating data interpretation. Conversely, broadband resistive matching provides simple data interpretation but sacrifices r, illustrated in FIG. 3B, necessitating higher incident power or longer integration times. Matching network design requires a careful balance of trade-offs tailored to specific applications.

[0030] Second, accurate phase measurements of the nonlinear admittance parameters require compensation for phase shifts introduced by the entire MIM circuitry. Direct measurement of component-level phase shift is impractical, so the embodiments propose using appropriate nonlinear reference samples for end-to-end calibration, extending the use of linear reference samples, such as Al-on-SiCh, in conventional MIM. For example, as shown in the MOS junction example, the 2nd and 3rd harmonic signals should appear almost entirely in the imaginary channel for a gold tip on heavily doped n-type silicon at zero bias. Therefore, adjusting the mixer reference phase shifters to maximize the main signal allows robust phase calibration in the mixers such as 21 shown in FIG. 1C.

[0031] Finally, maximizing signal-to-noise ratio involves minimizing parasitic nonlinearities and ensuring effective frequency separation and filtering. The linearity of the LNA is crucial, as nonlinearities can introduce harmonic artifacts from base-line linear reflections, which can be mitigated with tunable matching networks, cancellation circuits, dynamic mode operation, or by placing the triplexer before the LNA, the latter requires a dedicated LNA for each harmonic branch. This is shown in FIG. 4, with dedicated LNAs for each branch, such as 40. Equally importantly, parasitic crosstalk can be minimized using high-extinction-ratio triplexers in the signal path and frequency multipliers in the mixer reference path. Based on the MOS example, total harmonic artifacts should be kept below ~ 30 / 50 dB for 2nd / 3rd harmonic with 5 dBm incident power, which is readily attainable using off-the-shelf components.

[0032] The ability to probe nonlinearities in the GHz regime with nanometer-scale spatial resolution opens up several application avenues. First, nonlinear probes offer inherently higher spatial resolution and enhanced sensitivity to semiconductor response compared tolinear probes. This makes NL-MIM particularly well-suited for nanoscale semiconductor process characterization, where subtle material or structural inhomogeneities manifest as variations in nonlinear responses. Material defects in superconductors may also produce distinct nonlinear signals, further broadening NL-MIM’ s utility as a diagnostic and exploratory tool for classical and quantum microelectronics.

[0033] Beyond applied uses, NL-MIM holds promise for probing fundamental physics. Many emerging phases in condensed matter, such as charge density waves, superconductors, glassy phases, spin liquids, and strange metals, are known or hypothesized to exhibit distinct nonlinear responses that could provide valuable new insights when studied with high spatial resolution. In particular, local nonlinear response may serve as a sensitive indicator of symmetry breaking which is crucial for testing relevant theories in e.g., high-temperature superconductors. The versatility of linear MIM, and its compatibility with wide temperature and magnetic field ranges, gate tuning, DC, or low-frequency biasing, and optical excitation, is fully transferable to NL-MIM, enabling exploration of nonlinear phenomena across diverse materials and external controls.

[0034] All features disclosed in the specification, including the claims, abstract, and drawings, and all the steps in any method or process disclosed, may be combined in any combination, except combinations where at least some of such features and / or steps are mutually exclusive. Each feature disclosed in the specification, including the claims, abstract, and drawings, can be replaced by alternative features serving the same, equivalent, or similar purpose, unless expressly stated otherwise.

[0035] Additionally, this written description makes reference to particular features. It is to be understood that the disclosure in this specification includes all possible combinations of those particular features. For example, where a particular feature is disclosed in the context of a particular aspect, that feature can also be used, to the extent possible, in the context of other aspects. The Appendix submitted with this provisional application is incorporated herein in its entirety.

[0036] Also, when reference is made in this application to a method having two or more defined steps or operations, the defined steps or operations can be carried out in any order or simultaneously, unless the context excludes those possibilities.

[0037] Although specific aspects of this disclosure have been illustrated and described for purposes of illustration, it will be understood that various modifications may be made withoutdeparting from the spirit and scope of the invention. Accordingly, the invention should not be limited except as by the appended claims.

Claims

CLAIMS1. A microwave impedance microscopy (MIM) system having multi-harmonic MIM channels, comprising:a microwave signal source to produce a microwave signal;a probe tip to transmit the microwave signal to a surface of a sample under test and detect a signal;a scanning setup to scan the sample under test against the probe tip;one or more harmonic separators and demodulators to simultaneously extract multiple harmonic components of the signal;one or more processors configured to execute code that causes the one or more processors to:perform end-to-end phase calibration of the multi-harmonic MIM channels; relate the multiple harmonics components of the signal to corresponding non-linear admittance parameters; anduse the non-linear admittance parameters to analyze the sample.

2. The MIM system as claimed in claim 1, wherein the probe tip has an impedance matching network.

3. The MIM system as claimed in claim 2, wherein the impedance matching network comprises a resistor-capacitor (RC) network using a non-linear resistor and a quadratic nonlinear capacitor.

4. The MIM system as claimed in claim 2, wherein the impedance matching network comprises a half-wave resonator.

5. The MIM system as claimed in claim 2, wherein the impedance matching network comprises a 50-ohm shunt impedance network.

6. The MIM system as claimed in claim 1, further comprising one or more harmonic low-noise amplifiers (LNA).

7. The MIM system as claimed in claim 6, wherein the one or more LNAs comprises one LNA in a path of the signal before the harmonic separators and demodulators.

8. The MIM system as claimed in claim 7, wherein the harmonic separators and demodulators are in the path of the signal before the one or more LNAs and separate the signal into harmonic branches, and each harmonic branch has a dedicated LNA.

9. The MIM system as claimed in claim 1, wherein the probe tip detects one of either a transmitted or reflected signal.

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