Introduction

A transducer based on piezoelectric materials is a widely adopted component that converts an electrical signal to an acoustic phonon. Since the invention of interdigital transducers (IDT) in 19651, surface acoustic wave (SAW) on a piezoelectric substrate has become a promising platform for on-chip elastic wave phonons transducing2. Benefitted from the facilities of the platform, SAW devices are widely used in diverse fields like acousto-optic interaction3,4,5, spintronics6,7,8, sensing9,10, radio frequency (RF) filtering11,12,13, quantum technology14,15,16, and microfluidics17,18 et al. Precise manipulation of phonons may bring many advantages, such as a higher signal-to-noise ratio (SNR), lower energy consumption, and better system stability. In this sense, extensive research has been accomplished on the manipulation of SAW phonon, including wave guide19, acousto-electric amplifier20,21, phase modulator22. Besides manipulation along the propagation of the wave, the transducing process that converts the AC signal to an elastic wave or its reverse, also plays an important role and deserves more research interest.

Generally, IDT will emit elastic wave phonons to both sides along the propagating axis. Considering a lossless communication system with a set of IDTs as transmitter and another set as receiver, bi-directional emission will certainly lead to two consequences intuitively23. First, half of the acoustic energy won’t be received efficiently without a complex wave guide; second, the unreceived acoustic wave will then be reflected or scattered by the media boundaries, and disturb the receiver as noise. Hence, unidirectional emission of SAW phonons will cut off half of the energy consumption and significantly improve the SNR accordingly. The most famous unidirectional transducing structure for SAW is the single-phase unidirectional transducer (SPUDT), invented in 198224. The SPUDT adopts asymmetric electrode structures, introducing a phase shift between the transducing centre and the reflecting centre of the electrode. This phase shift enhances the forward wave and cancels the backward wave by interference25. Following this idea, natural SPUDT (NSPUDT) that unidirectionally emit SAW are realized on substrates, where the phase shifts between the transducing centres and the reflecting centres are inherent, even with normal IDT design26,27. However, the elastic wave phonon emission direction of the above unidirectional transducers is locked into place once they are fabricated, preventing real-time channel selection in space. The development of science and technology raises new demands on the precise and flexible manipulation of phonons. Considering the similarity of the material platform for photonic integrated circuit28,29 and SAW30, phonon transducer with novel functions may also provoke novel acoustic-optic devices.

In this work, a mode-determined unidirectional surface acoustic wave phonon transducer is demonstrated on-chip for the first time. An on-chip frequency splitter is also designed and fabricated based on the frequency-determined wave emission characteristics. Through the ion slicing method, lithium tantalate (LiTaO3) thin film is bonded directly onto 4H silicon carbide (4H-SiC). The handling substrate with high acoustic velocity supports high-velocity modes, such as longitudinal leaky mode SAW (LL SAW). Shear horizontal mode SAW (SH SAW) guided in the piezoelectric layer is distorted by the bonded boundary, exhibiting distinct deformation characteristics compared with the traditional SH SAW. A double-mode natural unidirectional transducer (DMNUDT) of SH SAW and LL SAW is demonstrated accordingly. More interestingly, the designed DMNUDT is capable of emitting SH SAW phonons and LL SAW phonons in opposite directions, which is beyond the capability of traditional SPUDT. In other words, an AC signal of mixed frequency components can be split into two single-frequency acoustic waves, with the radiating directions determined by the vibrating modes. Accordingly, an on-chip frequency splitter is designed and fabricated, realizing a maximum transmission ratio of more than 15 dB in both bands. The proposed transducers offer the researchers an opportunity to manipulate the on-chip elastic wave phonons more flexibly and a new design dimension of acoustic-related devices. The proposed minimal frequency splitter may serve as a compact signal processing device for next-gen mobile communication.

Results

Unidirectional SAW emission

Starting from the normal IDT invented by White in 19651, whose schematic is shown in the left part of Fig. 1a on the top surface of the substrate, the period of the transducer refers to the period of electrical loading. The half-period mechanical loading by metal gratings ensures bi-directional phonon emission, if the displacement and electric field are not detuned from each other (typical bi-directional response and phase matching are shown in Supplementary Fig. 9). In this sense, breaking the matching between mechanical loading and electrical loading OR exciting naturally detuned mode, turns out to be two effective strategies. Following the first idea, the most famous unidirectional transducer, SPUDT was proposed in 198224. The structure is widely adopted in fields of sensing31, acoustofluidics32, radio frequency signal processing33 et al. The second approach leads to the discovery of NSPUDT in 198526, whose structure is the same as normal IDT. This approach requires an intrinsic phase detune between mechanical displacement and electric field27. However, the detune in wave mode is hard to achieve manually, and the applicable circumstances of NSPUDT are greatly limited. However, the progress in heterogeneous integrated substrate provides a novel solution. Through proper design of the substrate, detuned modes can be constructed artificially.

Fig. 1: Overview of platform.
figure 1

a Comparison of SAW phonon emission on bulk piezoelectric crystal and heterogeneous integrated substrate. The heterogeneous substrate not only supports the propagation of high-velocity acoustic modes but also modifies the wave nature of the propagating wave. The modified wave nature comes with novel emitting characteristics of DMNUDT. b–f Specify the emission features of the DMNUDT. c Radar map for max energy emitting ratio of DMNUDT on different orientations. Deviated from x-axis, unidirectional phonon emission behaviours can be observed, and ±20° is selected for the close emitting ratios for each mode. b, f are the simulated emission wave fields of SH SAW and LL SAW when the DMNUDT is set along −20°X. e, d show the simulated fields when the DMNUDT is set along 20°X.

Full size image

Schematics of the mode-determined unidirectional SAW phonon transducer are shown in the right part of Fig. 1a. The proposed DMNUDT has a similar electrode structure to the normal IDT. On a bulk substrate, 42°Y cut LiTaO3, for example, SH SAW propagates with shear strain \({s}_{12}\) dominating the deformation. The normal IDT would only be able to emit SH SAW phonons bi-directionally (discussed in the latter chapter). Besides, high-velocity LL SAW suffers from great attenuation due to the coupling into the bulk wave. When it comes to a heterogeneous integrated piezoelectric substrate (HIP) that is fabricated by directly bonding the 42°Y cut LiTaO3 onto a 4H-SiC, guided SH SAW shows distorted deformation with great \({s}_{23}\) component. The distorted SH SAW grants the subsequent unidirectional emission of SH SAW. Additionally, LL SAW is also supported by the handling layer with a large acoustic impedance mismatch34. Through selecting a proper propagating orientation, SH SAW and LL SAW can be unidirectionally emitted by DMNUDT, with the emission direction determined by modes. Combined with the difference in resonant frequencies of both modes, frequency components are separated in space. Furtherly, a frequency splitter is demonstrated.

On 500 nm 42°Y20°X LiTaO3/4H-SiC, both SH SAW and LL SAW are supported due to the high acoustic phase velocity of the 4H-SiC handling layer. For DMNUDT with a periodicity of 1.4 μm, the emission radar map is shown in Fig. 1c. The energy emitting ratio is calculated by the body integral of mechanical energy in the transmission region (details in Supplementary 1). With the change of in-plane orientation, the emitting radius of each mode varied accordingly. Close emitting ratio of SH SAW and LL SAW can be accessed with in-plane orientation of ±20°. Besides, phonons of SH SAW and LL SAW are radiated to opposite directions, which is guaranteed by the piezoelectric tensor of LiTaO3 (LT) coincidentally. When in-plane orientation is deviated from x-axis lager than 60°, SH SAW can not be excited efficiently. Emitting wave fields at frequency points of maximum unidirectionality are simulated and shown in Fig. 1b–f. Through input frequency selection, different modes are excited with phonons emitted to opposite directions, which is beyond the capability of the previous design of a unidirectional transducer. Details of the simulation are shown in Supplementary Note 1.

Wave nature in the HIP substrate

As declared in the previous chapter, we modify the wave nature of propagating SAW by constructing a HIP substrate. Both SH SAW and LL SAW guided in the HIP substrate are detuned, and natural unidirectional phonon emission is enabled (see Supplementary Note 2 for detailed characteristics of detuned SAW). SH SAW that can be supported by a bulk crystal is chosen for the case study. Eigen frequency study is carried out for the basic features of SH SAW propagating on different substrates. The extracted surface displacement components and voltages are shown in Fig. 2a–c, and the corresponding vibrating profiles are shown in Fig. 2h. Phase detune between surface displacement components and voltages appears when the propagating direction deviates from the x-axis of the piezoelectric crystal. Moreover, the phase detune is then pronounced on the HIP. This can be attributed to that SH SAW guided by HIP is greatly distorted, as shown in Fig. 2g. Different from that on the bulk crystal, the dominant strain component is no longer merely \({s}_{12}\), and the role of \({s}_{23}\) enhances with the stiffness of the handling substrate. Consequently, on HIP, active piezoelectric components for SH SAW extend from the commonly used35 \({e}_{112}\) to \({e}_{112},\,{e}_{323},\) and \({e}_{123},\,{e}_{312}\). With non-zero piezoelectric components of both pairs of \({e}_{112},\,{e}_{323},\) and \({e}_{123},\,{e}_{312}\), respectively, phase detune is guaranteed (details in Supplementary Note 3). The emission characteristics of IDT on different substrates are then simulated and shown in Fig. 2d–f. On 42°YX bulk LT, IDT radiates SH SAW bidirectionally. When the propagating direction is deviated to 42°Y20°X, the emission of IDT becomes slightly asymmetric. Furtherly, through constructing a HIP substrate, NSPUDT of SH SAW is realized on 500 nm 42°Y20°X LT/SiC. In this sense, as shown in Fig. 2h, we may conclude the sufficient and necessary conditions for realizing the unidirectional emission of SH SAW. First, non-zero piezoelectric components of both pairs of \({e}_{112},\,{e}_{323},\) and \({e}_{123},\,{e}_{312}\) need to be satisfied; Second, a high stiffness handling layer that guarantees the mode distortion is also required. SiC handling layer not only enables the distorted SH SAW, but also forms a sufficient waveguide for LL SAW36,37. Similar to that for SH SAW, the phase detune between displacements and voltage of LL SAW can also be mainly attributed to specific elements in the piezoelectric tensor, like pairs of \({e}_{111},\,{e}_{313}\), and \({e}_{113},\,{e}_{311}\). However, due to the complex strain compositions, \({e}_{133}\) and \({e}_{333}\) may also affect the phase matching.

Fig. 2: Wave nature in different substrates and the adopted material platform.
figure 2

a–c are phase-matching characteristics extracted from the SH SAW eigen mode on a 42°YX bulk LT, b 42°Y20°X bulk LT, c 42°Y20°X LT/SiC. The SH SAW eigen modes on different substrates are shown in h. d–f Calculated displacement field of SH SAW on the corresponding substrate. g The distortion of SH SAW on common heterogeneous substrates. The mode distortion factor is defined as the body integral of \({s}_{23}^{2}\) divided by the body integral of \({s}_{12}^{2}\) within the piezoelectric materials. As shown in the inset vibrating diagrams, when the piezoelectric layer is bonded on a stiff substrate, the primary strain of the SH SAW is no longer merely \({s}_{12}\). h Summary of key factors for realizing unidirectional emission of SH SAW.

Full size image

Based on the analysis, the HIP substrate is fabricated by ion slicing technology (details in Methods). The fabricated substrate is then characterized by transmission electron microscopy (TEM). As shown in Fig. 3a, a LiTaO3 film of about 500 nm is split from the bulk crystal and transferred to the SiC substrate. The high crystalline quality is confirmed by the clear reciprocal lattice shown in Fig. 3b. During the surface activation process, the surface of the SiC substrate may be oxidized into SiO2. As shown in Supplementary Fig. 10, this may greatly affect the wave nature of guided SH SAW in the subsequent SAW devices. Hence, the heterogeneous interface is furtherly characterized by scanning TEM (STEM) and shown in Supplementary Fig. 11a. The electron energy loss spectrum (EELS) is adopted for element mapping, characterizing the thickness control of SiO2 that formed during the surface-activated bonding process38. As shown in Supplementary Fig. 11b, the overlapped area of the silicon and oxygen signal refers to the SiO2 layer. As a result, the ultra-thin oxide layer (<5 nm) ensures the performance of the later demonstrated devices.

Fig. 3: Performances of DMNUDT on HIP substrate.
figure 3

a, b TEM images of the fabricated HIP substrates. Inset in (b) is the FFT image for the reciprocal lattice. c Admittance phase responses of resonators on different propagating directions. Typical vibrating profiles of the modes of interest are labelled. The vibrating profiles refer to the antisymmetric mode of SH SAW, symmetric mode of SH SAW, symmetric mode of LL SAW, and antisymmetric mode of LL SAW from i to iv. d Schematic of the SAW resonator, which composes of IDT and reflectors at each side. The resonator is measured by the SG connection as plotted. e Optical image of the fabricated SAW resonator. f, g Extracted effective electromechanical coupling coefficient (\({k}_{{eff}}^{2}\)) of vibrating modes from the admittance responses, by \({k}_{{eff}}^{2}=(\,{f}_{a}^{2}-{f}_{r}^{2})/{f}_{a}^{2}\). \({f}_{r}\) and \({f}_{a}\) are the resonant and anti-resonant frequencies of the mode, respectively. f Variation of SH SAW \({k}_{{eff}}^{2}\) with in-plane orientation. g Variation of LL SAW \({k}_{{eff}}^{2}\) with in-plane orientation. For both SH SAW and LL SAW, the symmetric mode and antisymmetric mode show a seesaw characteristic. Noted that \({k}_{{eff}}^{2}\) of mode ii may not be extracted accurately, because its \({k}_{{eff}}^{2}\) is small, and the frequency spectrum is close to the stop-band resonance of the gratings. Scale bars in a, b, e are 200 nm, 20 nm, and 20 μm, respectively.

Full size image

After characterization of the HIP substrate, SAW resonators (schematically shown in Fig. 3d) with varied in-plane orientations are fabricated on the HIP substrate, validating basic electrical characteristics of the transducer. Wavelength and metallic ratio of the resonators are 1.4 µm and 0.5, respectively. Optical image of the fabricated resonator is shown in Fig. 3e. The benchtop measured electrical performances give a lumped view of the wave feature, and are shown in Fig. 3c. The double-resonance features of both SH SAW and LL SAW validate the unidirectional phonon emission behaviours39,40,41. For each pair of double-resonance, a symmetric mode and an anti-symmetric mode can be obtained (mode i, ii for SH SAW and mode iii, iv for LL SAW). Each resonance is supported by a pair of piezoelectric elements (details in Supplementary Note 3). The corresponding vibrating profile is calculated by FEM and shown in an inset, close to the resonance. When the propagating direction deviates from the X direction, the “double-resonance” feature appears, indicating the first demonstration of natural unidirectional emitting behaviour of SH SAW and LL SAW.

Furtherly, \({k}_{{eff}}^{2}\) of the aforementioned modes is extracted to study the influence of the deviated propagating direction on SAW. As shown in Fig. 3f, g, \({k}_{{eff}}^{2}\) of the aforementioned modes are mirrored by the X direction, and the performances of DMNUDT can be furtherly specified. Increasing the deviate angle from X direction (θ), \({k}_{{eff}}^{2}\) of mode i decreases, while that for mode ii increases (may not be obvious in Fig. 3b for the poor \({k}_{{eff}}^{2}\) extraction accuracy of mode ii, but is verified by simulations and further experiments). In the meantime, \({k}_{{eff}}^{2}\) of mode iv decreases while that of mode iii increases. With the seesaw-like tendency of \({k}_{{eff}}^{2}\) variation for both SH SAW and LL SAW, a close ratio of \({k}_{{eff}}^{2}\) between double resonance is observed at ±20°X, corresponding to the direction that permits close unidirectionality for both modes in the simulation.

Demonstration of the frequency splitter

The frequency-determined phonon emission characteristic may inspire devices with new functions. Power splitter is a component used to split the power of the input signal into multiple channels, and is widely used in radio frequency circuit42,43,44, integrated photonic45,46 et al. This function is easy to access considering the bi-directional emission nature of traditional IDT, which is likely to bi-directionally radiate the power of the input signal equally (transducers in Supplementary Fig. 9, for example). Thanks to the fascinating wave characteristics of the proposed unidirectional transducer, we can split the input signal into a frequency spectrum rather than the power distribution, which can serve for the carrier aggregation47,48 in mobile communication. The function of the frequency splitter is schematically shown in Fig. 4a. Through DMNUDT, the signal with mixed frequency components can be split in space. Then, transducers connected to different channels are placed at different sides of DMNUDT, acting as receivers. However, due to the reciprocity of the transducer49, DMNUDT is not capable of receiving the energy of propagating SAW if the transducers are placed side by side. For instance, if the transducer emits SH SAW to the “+” side unidirectionally, it will only be able to receive SH SAW from the “+” side. In this sense, SPUDT with reversed directivity is adopted for the receivers at port2 and port350. Accordingly, the transducer setting schematic of the proposed frequency splitter is shown in Fig. 4c.

Fig. 4: Frequency splitter on HIP substrate.
figure 4

a, c Function of the proposed frequency splitter and its schematic with electrical connection notation. The unidirectionality of SAW emission will split the frequency components in the mixed AC signal into two channels (colours represent the port name, DUT: device under test). b Simulated responses of the frequency splitter. Within the double-resonance frequency spectrum of the DMNUDT, acoustic modes constrain the emitting direction, furtherly channel selection can be accessed by frequency converge. Port impedances are 50 Ω. d Optical image of the fabricated frequency splitter with port notation. X direction of the transformed coordinate system is marked as “+” direction. Noted that port3 locates at “+” side with port2 at “-” side. The red dotted line refers to position for the cross profile in e. e Cross profile images by FIB-SEM. f, g Measured responses of the fabricated frequency splitter. Ports are matched to 97.5+i*130 Ω for measurement. f Port1 transducers is set for SAW propagate on −20X° direction. g for 20X° direction. Scale bars in d, e are 20 µm and 250 nm, respectively.

Full size image

Performances of the proposed devices are then simulated and shown in Fig. 4b. The substrate is set as 500 nm 42°Y cut LiTaO3/4H-SiC. DMNUDT has 30.5 pairs of transducers, and each receiving port has 10 pairs of SPUDT. Period of transducers is 1.4 µm. The upper block shows the phase of admittance of DMNUDT, and the bottom block shows the transmission characteristics to each receiver. With the double-resonance characteristic, modes confine the emitting directions of the phonons. Furtherly, accompanied by the difference in response frequency of modes, channel selection determined by frequency can be realized. When an AC signal with mixed frequency components (\({f}_{1}+{f}_{2}\)) is loaded to port1, if \({f}_{1}\) and \({f}_{2}\) support the resonances of SH SAW and LL SAW, respectively, DMNUDT will emit SAW phonons of different modes to opposite directions. Received by SPUDT with reversed directivity, \({f}_{1}\) and \({f}_{2}\) are split into two channels.

Subsequently, frequency splitters are fabricated on 500 nm 42°Y cut LiTaO3/4H-SiC by standard electron beam lithography and lift-off process (details in Methods). The transmitting port has 30.5 pairs of DMNUDT, and each receiving port has 20 pairs of SPUDT. Period of transducer is 1.4 µm, and 2 nm Ti/70 nm Al is adopted for the electrode material. An overall optical image of the frequency splitter is shown in Fig. 4d. The function of frequency splitting can be realized within the footprint of a resonator. Focused ion beam (FIB) is used for milling, along the red dotted line in Fig. 4d. The scanning electron microscopy (SEM) images of cross profile are shown in Fig. 4e. Critical sizes of the fabricated frequency splitter are labelled in the figure, and different transducer settings for transmitter and receiver can be seen clearly. Measured performances of the fabricated devices are shown in Fig. 4f, g. About 10 dB TR can be achieved in each channel. It should be noted that the discrepancy between the simulated and measured transmitting curves results from the different port impedances adopted. Low SH wave transmission efficiency in the simulated results can be improved by properly matching impedances between the devices and the ports, as shown by the measured results. Additionally, as demonstrated in Fig. 1c, the directivity can be reversed when the propagating angle is mirrored by x-axis, resulting in a reversed signal distribution in channels. Isolations between the receiving ports are shown in Supplementary Fig. 12. Further improvements in TR can be realized by aligning the passbands of transmitter and receiver, and optimizing the period of transducers for each port. TR in both channels surpassing 15 dB can be achieved, enabling future application in compact on-chip signal processing. Details of the frequency splitter design and optimization are discussed in Supplementary Note 4.

Discussions

SAW phonon transducers based on a piezoelectric substrate attract much research interest for the facility of an electro-mechanical interaction platform. In previous technologies, such as IDT, SPUDT, and NSPUDT, the emission direction of the transducers is locked into place once the structure is defined. This has been regarded as a basic feature of SAW phonon devices, and greatly limits the flexibility in the design of SAW phonon devices.

In this work, we demonstrate a mode-determined unidirectional surface acoustic wave phonon transducer on-chip. Through the ion slicing method, LiTaO3 thin film is bonded directly onto 4H-SiC. The handling substrate with high acoustic velocity supports high-velocity modes, such as LL SAW. SH SAW guided in the piezoelectric layer is distorted by the bonded boundary, exhibiting distinct deformation characteristics compared with the traditional SH SAW. The designed DMNUDT is capable of emitting SH SAW phonons and LL SAW phonons in opposite directions, which is beyond the capability of previous SAW phonon transducers. The proposed transducers offer the researchers an opportunity to manipulate the on-chip elastic wave phonons more flexibly and a new design dimension of acoustic-related devices. We expect that the unique phonon emission characteristics may provoke a breakthrough in acousto-optics, spintronics, RF filtering, and quantum technology.

As a proof-of-concept, an on-chip frequency splitter is designed and fabricated based on the frequency-determined wave emission characteristics. An AC signal of mixed frequency components can be split into two single-frequency acoustic waves, with the radiating directions determined by the vibrating modes. A maximum transmission ratio of more than 15 dB in both bands is realized. We expect that the decent performances and minimal footprint of the frequency splitter may provide a potential solution for a frequency duplexer, which is a key component for carrier aggregation technology and non-standalone operating mode of 5 G devices. More specifically, the operating frequency of LL SAW band is about 1.5 times that of the SH SAW band, enabling the collaboration between 4 G n41 band (2496–2690 MHz) and 5 G n77 band (3300–4200 MHz). Moreover, DMNUDT can separate frequency components spatially through phonon emission, enabling the design of compact signal processing devices.

Methods

Fabrication of HIP substrate

Heterogeneous integration of single-crystalline LiTaO3 thin film and semi-insulating 4H-SiC substrate is achieved by ion-slicing and wafer-bonding process. First, a 4-in 42°YX LiTaO3 wafer is implanted by hydrogen ions (H+) with a 7° tilt to minimize the ion channelling effect. Then, the as-implanted LiTaO3 and SiC wafers are surface activated in ultrahigh vacuum and then aligned, attached, and bonded in ambient atmosphere. After that, exfoliation of LiTaO3 thin film is achieved by post-annealing. Then, LiTaO3 thin film is split from the bulk wafer and transferred onto the SiC substrate. Finally, the chemical–mechanical polishing (CMP) process is applied to smooth the wafer surface and eliminate the damaged layer formed by ion injection.

SAW devices fabrication

The devices are fabricated by electron-beam lithography (EBL), electron-beam evaporation (EBE), and lift-off processes. The samples are first cleaned by acetone, absolute ethyl alcohol, and deionized water, respectively. E-beam resist AR-P 6200 and conductive protective coating AR-PC 5092.02 are successively spin-coated onto the substrate and soft baked. Then, exposure is conducted by EBL. After development in o-xylene, 2 nm Ti and 70 nm Al metal layer is deposited by EBE. The resist is lifted off through immersing in the photoresist stripper AZ-400T at room temperature overnight and ultrasonic exfoliating for 30 s.

Electrical measurement of the fabricated devices

S parameters of the fabricated devices are measured by the vector network analyser Keysight E5071C. GGB 40A-SG(GS)−200-DP-W probes are adopted. Standard short-open-load-through calibration is first conducted at room temperature and ambient atmosphere, with CS-108 calibrating substrate before measurement. SAW resonators can be easily characterized by a 2-port measurement. Moreover, 3-port devices can be characterized by assembling the 2-port results measured by a 2-probe system. The impedance matrix of the devices can be expressed as,

$$\left[\begin{array}{c}{v}_{1}\\ {v}_{2}\\ {v}_{3}\end{array}\right]\,=\left[\begin{array}{ccc}{Z}_{11} & {Z}_{12} & {Z}_{13}\\ {Z}_{21} & {Z}_{22} & {Z}_{23}\\ {Z}_{31} & {Z}_{32} & {Z}_{33}\end{array}\right]\cdot \left[\begin{array}{c}{i}_{1}\\ {i}_{2}\\ {i}_{3}\end{array}\right]$$

For our devices, the third port is terminated as an open circuit. In other words, the current at the third port \({i}_{3}\) is 0. Measured impedance matrix degrades into,

$$\left[\begin{array}{c}{v}_{1}\\ {v}_{2}\end{array}\right]=\left[\begin{array}{cc}{Z}_{11} & {Z}_{12}\\ {Z}_{21} & {Z}_{22}\end{array}\right]\cdot \left[\begin{array}{c}{i}_{1}\\ {i}_{2}\end{array}\right]$$

Vice versa, the 3-port impedance matrix of the whole device can be assembled by the 2-port impedance matrix of each two ports. Similarly, when the third port is terminated short-circuited, an admittance matrix can be adopted for assembling; when the third port is 50 Ω loaded, a scattering matrix can be assembled directly.