Introduction

The electron injection and extraction at the interface of metals and semiconductors constrain the performance of classic charge-based nanodevices1,2,3,4. As the channel length of transistors and diodes scales down to a few tens of nanometers, the tunneling ohmic contacts dominate the active region of the devices, which undermines the speed and energy efficiency of integrated circuits5. One important consequence of this constraint is in high-speed electronics where the limited (trans)conductance defines the maximum cutoff frequency of switches and amplifiers6,7. Considering the case of Gallium Nitride (GaN), which provides the power needed for numerous wireless applications 8,9,10,11, the performance of transistors have not been improved for about a decade12. This impedes the advancement of next-generation telecommunication networks from the perspective of electronic hardware 13,14.

Electronic metadevice is an innovative paradigm in which microscopic manipulation of radiofrequency electric fields and currents results in extraordinary electronic characteristics in a device form factor15,16,17. This is an analogy to optical metamaterials18 in which subwavelength manipulation of electromagnetic fields leads to exotic macroscopic properties. In electronic metadevices, the governing equation of electric fields at the barrier with thickness \(d\) and permittivity \(\varepsilon\), incapsulated between a top metal and a two-dimensional electron gas (2DEG) with sheet resistance \({R}_{{{{\rm{sh}}}}}\), exhibits wave-like eigenfunctions characterized by a wavelength of

$$\lambda =2\pi \sqrt{2}\,{\left|\xi \right|}^{-1},$$
(1)

where \(\xi =\sqrt{j\omega \varepsilon {R}_{{{{\rm{sh}}}}}/d}\), \(\omega\) is the angular frequency, and \(j=\sqrt{-1}\) (see Methods). A periodic metallic structure with feature sizes smaller than λ can manipulate electric fields and currents, altering the macroscopic properties of the device. The wave-like eigenfunctions can collectively interact with subwavelength metallic stripes, leading to non-traditional scaling (see Supplementary Data Fig. 1 and Methods). In this scenario, normalized parameters obtained for 2-stripe or 4-stripe devices do not extend to many-stripe configurations. This contrasts with other classic electronic devices with nanoscale features, such as FinFETs19,20,21,22, where normalized parameters like (trans)conductance remain constant as the number of fins increases.

The concept of electronic metadevices was first demonstrated in the form of switches on GaN with cutoff frequencies exceeding 10 THz 15. Here we present a series of advanced experiments revealing key insights into electronic metadevices, showcasing their potential to impact future telecommunication circuits. Our experimental findings, supported by both theoretical analyses and simulations, demonstrate nearly ideal linear and nonlinear performance in these devices. By utilizing two-terminal electronic metadevices, we have developed an integrated terahertz switch operating in the 220–330 GHz range with sub-20 ps switching times. This switch features an ultra-compact footprint and surpasses the speed of current state-of-the-art terahertz switches. Additionally, we provide the first demonstration of parametric amplification and active mixing in three-terminal electronic metadevices. Finally, we introduce a three-port modulator based on electronic metadevice technology, achieving a bandwidth exceeding 50% and data rates up to 40 Gbps.

Results and Discussion

Figure 1a presents a false-colored scanning electron microscope (SEM) image of a two-terminal electronic metadevice comprising a microstructured metallic stripe array formed on top of an InAlN/GaN heterostructure. Figure 1b shows a schematic cross section of the device, which features a periodic metallic structure separated by a barrier of thickness d from a 2DEG. The metallic stripe array enables radiofrequency field manipulation (Fig. 1c) which confines the current in the 2DEG at the gap between two terminals 17 (Fig. 1d). This enables an excellent coupling between the patterned metal and the 2DEG, where the effective contact resistance (\({R}_{C}={R}_{S}/2\); Fig. 1e) can be smaller than the state-of-the-art tunneling contacts based on metals and semi-metals 15.

Fig. 1: Operation regimes of electronic metadevices.
Fig. 1: Operation regimes of electronic metadevices.The alternative text for this image may have been generated using AI.
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a False-colored scanning electron microscope (SEM) image of a two-terminal electronic metadevice. The state of the device (ON, OFF, or near-threshold) is set by the bias voltage \({V}_{b}\). W and Lch represent the stripe width and the gap between adjacent stripes, respectively, and L represents the stripe length. The scale bar is 2 µm. b Schematic cross section of the periodic stripe array, where d represents the distance between stripe array and the 2DEG. c Vertical electric field \({E}_{z}\) (real part) at the barrier, where the device is excited by a real current. The field manipulation leads to (d), confinement of 2DEG current density at the gap. The scale bars are 2 µm. e Circuit model of the metadevice including \({R}_{{{{\rm{ch}}}}}\) (2DEG channel resistance at the gap), \({R}_{{{{\rm{S}}}}}\) (effective contact resistances), \({C}_{{{{\rm{S}}}}}\) (metal-2DEG capacitances), \({C}_{{{{\rm{P}}}}}\) (metal-metal capacitance), and \({L}_{{{{\rm{S}}}}}\) (access inductances of the pads). f Measured and simulated ON-state resistance (\({R}_{{{{\rm{ON}}}}}\)) of the electronic metadevices (L = 0.9 µm, W = 5.2 µm, and Lch = 0.25 µm) with different number of stripes (N) (see Methods). g Measured \({X}_{{{{\rm{ON}}}}}\) (ON-state reactance; absolute value) of metadevices with 8 and 16 stripes, showing that the imaginary part of the device impedance becomes zero at the operation frequency of the device (blue and red gradients for N = 8 and N = 16, respectively). h ON-state transmission through the electronic metadevices with 8 and 16 stripes in W-band (75–110 GHz) and D-band (110–170 GHz). Measurements (solid lines), simulations (dashed lines). The results indicate the capability of electronic metadevices in providing large bandwidths exceeding 100 GHz (see Methods). i Measured impedance (real part: red, absolute imaginary part: blue) of an 8-stripe electronic metadevice (L = 0.9 µm, W = 5.2 µm, and Lch = 0.25 µm) under a DC bias (\({V}_{b}\)) sweep.

The ON-state of the switch can be modeled by a resistance (\({R}_{{{{\rm{ch}}}}}+2{R}_{C}\); Fig. 1e) in series with an imaginary impedance representing the reactive metal-2DEG coupling, which resonates out with a series inductance L. Therefore, the ON-state impedance at the operation frequency can be solely described by a resistance. Figure 1f, g show the real (\({R}_{{{{\rm{ON}}}}}\)) and imaginary (\({X}_{{{{\rm{ON}}}}}\)) parts of the impedance of electronics metadevices with different number of strips. Ohmic resistance of the metallic stripes are negligible compared to the intrinsic resistance of the device (see Methods).

It is important to note that while XON remains negligible relative to RON within a relatively narrow bandwidth, the device still maintains a low insertion loss across a broad bandwidth. This is because the real part of the impedance considerably influences insertion loss more than the imaginary part (see Methods). Our measurements, simulations, and theoretical analyses demonstrate that electronic metadevices are capable of achieving bandwidths exceeding 100 GHz (Fig. 1h). Additionally, the relative bandwidth of these devices can be further enhanced by increasing the number of stripes (see Methods).

Because of the collective interaction between the wave-like eigenfunctions and the patterned metallic array, the ON-resistance drops rapidly as the number of stripes increases15,17, which enhances the switching cutoff frequency (Supplementary Data Fig. 1d). We note that the device impedance remains stable across a wide range of bias voltages in the ON-state ( − 3 V ≲ Vb ≲ 3 V), facilitating practical matching (Supplementary Data Fig. 2). Electronic metadevices exhibit a capacitive impedance in the OFF state, similar to traditional devices, where COFF linearly scales with the effective width (Supplementary Data Fig. 3).

Figure 1i shows the impedance of an 8-stripe metadevice measured at 105 GHz, representing three operation regimes: ON state, near-threshold, and OFF state. The device resistance is flat at zero bias which leads to a highly linear performance. In the OFF state, thanks to the high breakdown voltage, the device can be biased at high voltages which avoids forced turn ON at high power levels. In addition, the device impedance exhibits an abrupt change close to the threshold, which suggests its potential for a high-performance nonlinear operation like frequency multiplication and mixing.

We characterized the ON and OFF states of two-terminal metadevices at high power levels (Fig. 2a) (see Methods). Figure 2b presents the measured output power of zero-biased metadevices with different number of stripes versus their input power. The devices exhibit ultrahigh-power capability, with 1-dB input compression point exceeding ~1.6 W for the 16-stripe device. In the OFF state, with a bias of Vb = \(-\)20 V, the metadevices can block the same power levels without forced turn-ON or breakdown (Fig. 2c). This represents an important advantage, as the OFF-state linearity in transistor-based switches is typically the primary constraint, limiting the overall performance of the switch23,24,25.

Fig. 2: Near-ideal linear and nonlinear operation of electronic metadevices.
Fig. 2: Near-ideal linear and nonlinear operation of electronic metadevices.The alternative text for this image may have been generated using AI.
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a Schematic of the experimental setup to evaluate high-power linear operation of electronic metadevices (see Methods). b ON-state (DC \({V}_{b}\) = 0 V) output power of metadevices with different number of stripes (N = 2, 4, 8, and 16) versus input power. The 16-stripe metadevice shows an excellent linearity at very high powers above 1 W. c OFF-state (DC \({V}_{b}\) = −20 V) output power of electronic metadevices with different number of stripes (N = 2, 4, 8, and 16) versus input power. The devices demonstrate outstanding high-power blocking capability, avoiding forced turn ON or breakdown. d Measured saturation output power versus effective width (\({W}_{{{{\rm{eff}}}}}\) is shown in the inset) of electronic metadevices with different number of stripes. The dashed line represents an ideal device considering saturation current of 0.6 A mm–1. e Schematic of the experimental setup to evaluate the nonlinear state of electronic metadevices for second harmonic generation. We used a high-frequency oscilloscope with a built-in signal analyzer option to measure the power at the entire spectrum up to 110 GHz (see Methods). f Received waveform at the oscilloscope port. The horizontal and vertical axes are scaled to 25 ps and 1 V per division, respectively. g Digital signal processing (DSP)-filtered second harmonic component. The horizontal and vertical axes are scaled to 25 ps and 0.15 V per division, respectively. h Power spectrum of the signal received at the oscilloscope port. i Output second harmonic power at 102 GHz (Blue). The conversion gain (Red) approaches the theoretical limit of an ideal diode (\(-\)13.4 dB; see Methods), which highlights the excellent dynamic performance of electronic metadevices thanks to their abrupt impedance switching (Supplementary Data Fig. 5).

Figure 2d presents the saturation output power versus effective width of the metadevices. As the number of stripes increases, the performance of metadevices improves, approaching the theoretical limit corresponding to saturation current of 0.6 A mm−1. This improved performance can be attributed to a couple of factors. First, because of the rapid drop in the resistance (Fig. 1f), which boosts the signal transmission and consequently, the output power. Second, because increasing number of stripes reduces the self-resonance frequency, where jX = 0 (Fig. 1g), and shifts it towards the excitation frequency (34 GHz). Our theoretical results confirm the near-ideal linearity of electronic metadevices, where the device impedance can increase until the semiconductor channel reaches its saturation current (see Methods and Supplementary Data Fig. 4). The capability of electronic metadevices in exploiting the full potential of semiconductor materials at high-power levels is an important merit for future telecommunication circuits.

Next, we characterized two-terminal electronic metadevices in their nonlinear regime, with the device biased near its threshold voltage (Fig. 2e–g) (see Methods). Electronic metadevices exhibit abrupt switching close to their threshold voltage which facilities harmonic generation (see Methods and Supplementary Data Fig. 5). We evaluated second harmonic generation above 100 GHz at the bias voltage of Vb = −4 V (Fig. 2h). By increasing the input power up to 20 dBm, the second harmonic output power grew beyond 1 mW (Fig. 2i). The conversion gain of the second harmonic generation approached the theoretical limit of an ideal diode (see Methods), which has the potential to exceed the performance of existing integrated diodes operating at the same frequency (Supplementary Data Fig. 6). The achieved output power of 3 mW at 102 GHz, in this case, was limited by the input power amplifier. The power capabilities of electronic metadevices, combined with their high harmonic generation efficiency, have the potential to enable a broad range of high-frequency signal syntheses.

Integration capability of electronic devices is a critical characteristic which determines the usefulness and impact of the technologies. That is the reason why some potentially high-performance devices like p-i-n diodes26 and phase-change switches 27 are not widely used in today’s microelectronics. Electronic metadevices rely on a planar fabrication which is compatible with standard foundry processes. Figure 3a illustrates false-colored SEM image of a monolithically integrated terahertz switch based on electronic metadevice technology. The voltage at the input signal terminal simultaneously determines the state of the three electronic metadevices, thus governing the state of the entire integrated switch (Fig. 3b, c). Figure 3d shows the vertical field distribution over the integrated switch, indicating that the subwavelength responses are confined within each metadevice and do not interfere. This showcases the integration capability of electronic metadevices as modular components.

Fig. 3: Ultrafast terahertz integrated switches based on electronic metadevices.
Fig. 3: Ultrafast terahertz integrated switches based on electronic metadevices.The alternative text for this image may have been generated using AI.
Full size image

a False-color SEM image of an ultrafast terahertz switch based on monolithically integration of three electronic metadevices (series 8-stripe metadevice S1, and shunt 4-stripe metadevices S2 and S3). b Schematic showing the state of the three metadevices when the integrated switch is in the ON state. In this case, the input and output voltage biases are equal and above the threshold voltage (S1: ON, S2 and S3: OFF). c Schematic showing the state of the three metadevices when the integrated switch is in the OFF state (S1: OFF, S2 and S3: ON). In this case, the output bias remains high (above threshold), while the input terminal is zero biased. d Simulated vertical electric field incapsulated between the top metal and the 2DEG in the ON state, showing field confinement within the layout of each electronic metadevice. e Measured transmission (S21 = S12) of the integrated switch in ON and OFF states. Considering the built-in through feature, which accounts for probe-to-pad losses, the actual insertion loss would be below 2 dB at 310 GHz. f Measured reflection parameters (S11 and S22) showing a good matching and rejection in ON and OFF states, respectively. g Schematic of the experimental setup to characterize the switching speed of the integrated switch. A mm-wave signal at 95 GHz was combined off-chip with a control signal generated by AWG and amplified by a broadband amplifier. The input and output terminals were biased at −3 V and −5 V respectively. h Measured output signal with a switching frequency of 3 GHz. i Measured output signal showing ultrafast switching time below 20 ps, limited by the speed of the control signal.

The philosophy underlying this design is closely linked to the fundamental properties of the devices. The conductance of electronic metadevices exhibits superlinear scaling with the number of stripes. This suggests that the superior performance of electronic metadevices becomes prominent at higher stripe numbers, resulting in ultralow insertion loss values, albeit with potential challenges in isolation. The integrated design illustrated in Fig. 3a, which we refer to as a meta-switch, achieves improved isolation through the incorporation of shunt electronic metadevices. Figure 3e, f present the transmission and reflection of the meta-switch, respectively, both in ON and OFF states. Without any de-embedding, the meta-switch exhibits insertion loss of 2.7 dB and isolation of 15 dB at 310 GHz, and a good matching and rejection, in ON and OFF states, respectively (Fig. 3f).

In addition to the high performance in static regimes (ON and OFF states), the integrated switch exhibits ultrafast switching dynamics. Figure 3g shows the experimental setup to characterize the switching speed of the fabricated meta-switch (see Methods). Figure 3h presents the measured output signal of the switch, where the switching (large signal) and the carrier (small signal) frequencies are 3 GHz and 95 GHz, respectively. A part of the captured waveform with a finer time resolution, shown in Fig. 3i, indicates ultrafast sub-20 ps operation of the integrated switch. The high ON/OFF ratio and the fast dynamics of the metadevice-based switches show great promise for the future telecommunication circuits, and can be a technology enabler to realize high performance metamaterials 28,29. Compared to state-of-the-art switches reported in the literature at the same frequency band (up to 330 GHz), which usually incorporate transmission lines in their design, the presented metadevice-based terahertz switches have advantages of occupying much smaller chip area, as well as providing considerably higher speeds (Supplementary Data Table 1).

Electronic metadevice is a general concept which is not limited to two terminal switches. Microscopic field manipulation can be achieved in three-terminal devices 15 supporting wave-like eigenfunctions, leading to enhanced electrical properties (see Methods and Supplementary Data Fig. 7). Figure 4a shows a false-colored SEM image of a three terminal metadevice fabricated on a high-electron-mobility InAlN/GaN platform (see Methods). We designed two experiments to study active operation of these devices. In the first experiment, terminals 1 and 3 (see Fig. 4a) were excited by two independent radiofrequency signals (pump signal and fundamental signal, respectively), and the output signal at terminal 2 was terminated and measured by a signal analyzer. Using external bias tees, we biased terminal 3 at V \(=\) Vb, while terminals 1 and 2 were zero biased. Figure 4c shows the measured spectrum corresponding to fundamental and pump signals at 1 GHz and 10 GHz, respectively, while Vb = \(-\)3.5 V. The second harmonic of the fundamental signal, and the second and third order mixing of the fundamental and the pump signals are observable in the plot.

Fig. 4: Three-terminal metadevices for parametric amplification and active mixing.
Fig. 4: Three-terminal metadevices for parametric amplification and active mixing.The alternative text for this image may have been generated using AI.
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a False colored SEM image of a three-terminal metadevice. b Schematic of the experimental setup to characterize mixing and parametric amplification of three-terminal metadevices. The optical image depicts the three-terminal metadevice connected to radiofrequency probes (G and S represent ground and signal, respectively). The pump and output terminals were biased at 0 V, and the gate terminal was biased at \({V}_{b}\). The scale bar is 50 µm. c Measurement output spectrum with \({V}_{b}\) = −3.5 V using a spectrum analyzer. d Output power of the fundamental (1 GHz; solid lines) and mixed (9 GHz; dashed lines) signals versus gate bias voltage for different pump power levels \({P}_{{{{\rm{pump}}}}}\) = 5 dBm, 10 dBm, 15 dBm, and 20 dBm. The gradient indicates the gain regime. e Output power of the fundamental (1 GHz; solid lines) and mixed (9 GHz; dashed lines) signals versus pump power for different signal set power levels (\({P}_{{{{\rm{set}}}}}\)). The gradient indicates the gain regime. f, g Scattering parameter measurements on the three terminal metadevice, where terminals 3 and 2 where terminated by the 1st and the 2nd ports of a vector network analyzer. The terminal 1 was excited by a pump signal at 27 GHz. Absolute values of scattering parameters corresponding to (f), Pump OFF and (g) Pump ON (Ppump = 25 dBm). h Square of parametric current gain (\({H}_{21}^{2}\)) of the three-terminal metadevice at different gate bias voltages and frequencies. i Parametric unilateral gain at different frequencies (fsig = 1 GHz, 2 GHz, 5 GHz, and 10 GHz) versus pump power. The inset shows \({H}_{21}^{2}\) versus pump power indicating higher frequency current amplification at higher pump powers. Considering the cutoff frequency of \({f}_{T} > \)10 GHz, we have \({f}_{T}\times {L}^{2} > \) 270 THz nm2.

Figure 4d illustrates the measured power of both fundamental and mixed signals in relation to the bias voltage (Vb) at various pump powers. The fundamental power was fixed at –10 dBm. The findings reveal that as the bias voltage approaches the threshold of the 2DEG depletion, the output fundamental power (depicted by solid lines) experiences an increase, indicating the presence of a gain mechanism when a sufficient pump power is applied. The power measured for the mixed signal (depicted by dashed lines) also shows the same trend. Figure 4e presents the measured output power for fundamental and mixed signals, versus pump power, where the DC bias at terminal 3 was fixed at Vb = \(-\)3.5 V. The results indicate a consistent output fundamental power until the mixed signal power approaches that of the fundamental signal, suggesting a parametric amplification mechanism.

We further studied the parametric gain of three terminal metadevices using a standard scattering parameter measurement using a vector network analyzer. Figure 4f, g present the measured absolute values of scattering parameters, in two scenarios, where the pump signal was ON and OFF, respectively. The DC bias at terminal 3 was fixed at Vb = \(-\)3.5 V in both cases. The results show that the forward and backward transmissions (S21 and S12) become unequal at the presence of an adequate pump power, which indicates a forward parametric gain. Figure 4h shows the extracted parametric current gain square (\({H}_{21}^{2}\)) at different biases and frequencies, presenting amplification at frequencies up to 10 GHz. This is a proof of concept of amplification for a device with a long gate length of 520 nm, resulting in impressive \({f}_{{{{\rm{T}}}}}\times {L}^{2}\) figure-of-merit exceeding 270 THz nm2. Scaling down would improve the frequency performance of three-terminal metadevice parametric amplifiers.

Figure 4i shows the extracted parametric unilateral gain (\({U}_{n}\)) of the device versus input pump power (\({P}_{{{{\rm{pump}}}}}\)) at different fundamental frequencies. The slope \({U}_{n}={P}_{{{{\rm{pump}}}}}^{2}\) suggests a third order parametric amplification mechanism. In this scenario, the fundamental signal, mixed with the pump signal, undergoes an additional mixing with the pump signal, yielding a component with the same frequency as the original fundamental signal. Figure 4i and its inset indicate that both parametric unilateral and current gains at 10 GHz are greater than 1, when adequate pump power is supplied.

We used the fabricated three terminal to realize a three-port modulator for high-speed Amplitude Shift Keying (ASK) and Quadrature Phase Shift Keying (QPSK) modulations. Figure 5a shows the experimental setup to characterize the modulator, where the output signal is directly measured by a 113 GHz oscilloscope (see Methods). Figure 5b presents the waveforms of the ASK modulated signals with data rates of 20 Gbps and 40 Gbps, where the central frequency is 80 GHz, indicating (Fig. 5c). In this case, the baseband data signal was upconverted using an 80 GHz LO signal. This experiment shows the capability of electronic metadevices in ultra-wide-band modulation, as the modulator provides a high bandwidth occupation over 50%. The eye diagrams presented in Fig. 5d, e show excellent modulation capability of the three-terminal electronic metadevices.

Fig. 5: Ultrahigh-speed three-port modulators based on three-terminal electronic metadevices.
Fig. 5: Ultrahigh-speed three-port modulators based on three-terminal electronic metadevices.The alternative text for this image may have been generated using AI.
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a Schematic of experimental setup to realize ultrafast ASK and QPSK data transmitters based on three terminal metadevices. b Measured time domain signal corresponding to ASK modulation at 20 Gbps and 40 Gbps data rates with fLO = 80 GHz. In this case, arbitrary waveform generator (AWG) only streams the raw binary data, which is a base-band signal centered at DC. c Frequency spectrum of the ASK-modulated signals. d, e Eye diagrams of the signals at 20 Gbps and 40 Gbps data rates, respectively, showing an excellent low-jitter modulation. f Frequency spectrum of the QPSK signals. In this case, the AWG generates low frequency QPSK signal centered at 15 GHz. The thee-terminal metadevice up-converts the QPSK signal into the center frequency of 100 GHz, using fLO = 85 GHz. Supplementary Data Fig. 8 shows the complete spectrum of the signal received at the port of the high-frequency oscilloscope. g–j Constellation diagrams of the QPSK signals at data rates of 15 Gbps, 25 Gbps, 35 Gbps, and 40 Gbps, respectively. The data-rate was limited by the AWG.

Broadband linear up conversion capability of three terminal metadevices was used to realize a QPSK transmitter operating at 100 GHz. In this case, the arbitrary waveform generator (AWG) synthesizes a QPSK modulation at low frequency (centered at 15 GHz), and the metadevice linearly upconverts that to 100 GHz. Figure 5f presents the spectrum of the output signal corresponding to the data rates of 15 Gbps, 25 Gbps, and 40 Gbps. The entire spectrum and the power levels of different signals corresponding to the data rate of 40 Gbps are shown in Supplementary Data Fig. 8. Figure 5g–j show the constellation diagrams of the up-converted modulated signals, with data rates of 15 Gbps, 25 Gbps, 35 Gbps, and 40 Gbps, respectively. The data rate was limited by the bandwidth of the oscilloscope.

Conclusion

This paper unveiled the enormous potentials of electronic metadevices for the future high-frequency integrated circuits, particularly in the context of advancing towards next-generation telecommunications. Electronic metadevices exhibit near-ideal linear and nonlinear operations with remarkable power handling and ultrafast response. The transition from individual metadevices to integrated meta-switches highlights their scalability and compatibility with standard processes, promising the realization of high-performance integrated circuits. Implementation of three-terminal metadevices adds another layer of versatility, showcasing parametric amplification and active mixing capabilities, culminating in the development of a three-port modulators with ultra-wide-band modulation capability. As the industry stands on the brink of the 6 G era, electronic metadevices emerge as a crucial and promising paradigm capable of shaping the future landscape of telecommunication hardware.

Methods

Summary of the previous theoretical findings 17

Considering uniformity along \(y\) direction in Fig. 1b, the phasor of electric field \({E}_{z}\) at the angular frequency \(\omega\) at the barrier with thickness \(d\) and permittivity \(\varepsilon\) encapsulated between a top metal and 2DEG with the sheet resistance of \({R}_{{{{\rm{sh}}}}}\) has the following governing equation

$${\left(\partial /\partial x\right)}^{2}{E}_{z}\left(x\right)={\xi }^{2}{E}_{z}\left(x\right),$$
(2)

where \({\xi }^{2}=j\omega \varepsilon {R}_{{{{\rm{sh}}}}}/d\). The solution of (1) can be written as \({E}_{z}\left(x\right)={E}_{+}{e}^{\xi x}+{E}_{-}{e}^{\xi x}\), where \({E}_{+}\) and \({E}_{-}\) are constants. The current density in the 2DEG (\({J}_{x}\)) can be calculated from

$${J}_{x}(x)=d/{R}_{{{{\rm{sh}}}}}\left(\partial /\partial x\right){E}_{z}(x).$$
(3)

At large number of stripes, the total ON-state contact impedance of a metadevice was obtained as

$${Z}_{C}={R}_{{{{\rm{sh}}}}}{\xi }^{-1}{W}_{{{{\rm{eff}}}}}^{-1}\coth \left(\xi L/2\right).$$
(4)

Considering \(|\xi {L|} \, \ll \, 1\), the real part of (4) can be written as

$${R}_{{{{\rm{C}}}}}={R}_{{{{\rm{sh}}}}}({L}_{{{{\rm{ch}}}}}+L/3)/{W}_{{{{\rm{eff}}}}}.$$
(5)

Where \({L}_{{{{\rm{ch}}}}}\) and \(L\) respectively represent the channel length and the stripe length, as depicted in Fig. 1a.

Effect of different parameters on the performance of electronic metadevices

Among the geometrical parameters of electronic metadevices, the stripe width (W), is a scaling parameter as long as the stripe series resistance is negligible compared to the intrinsic resistance of the device. Regarding other parameters (L, Lch, and N) we have:

Subwavelength features (stripe length, L)

As shown in Supplementary Data Fig. 1, decreasing L to values considerably below λ (the subwavelength regime) can reduce the contact resistance of electronic metadevices. The contact resistance is independent of L for \(L\,\gtrsim\, \lambda\).

Channel length (L ch)

The channel length does not affect the electric field pattern under the stripes, and thus, does not modify the contact resistance. One can write the normalized ON-resistance of the device as \({R}_{{{{\rm{ON}}}}}=2{R}_{{{{\rm{C}}}}}+{R}_{{{{\rm{ch}}}}}\), where \({R}_{{{{\rm{C}}}}}={{\mathrm{Re}}}\{{{\mathrm{R}}}_{{{{\rm{sh}}}}}{\xi }^{-1}\coth \left(\xi {{\mathrm{L}}}/2\right)\}\) and \({R}_{{{{\rm{ch}}}}}={L}_{{{{\rm{ch}}}}}{R}_{{{{\rm{sh}}}}}\). The channel length of the device can be designed based on requirements like the power handling.

Number of stripes (N)

In case of L ≳ λ, where there is no subwavelength feature, the devices show traditional scaling: the normalized contact resistance remains constant as the number of stripes increases. In case of L ≪ λ, however, increasing the number of stripes enhances the metal-2DEG coupling. This is where scaling plays an important role in the performance of the devices. Depending on the value of L/λ, a certain number of stripes are needed so that the contact resistance gets close to the lowest possible value.

Device fabrication

The fabrication process of the electronic metadevices involved the growth of an InAlN/GaN epitaxial layer on a 4-inch silicon carbide substrate. The substrate was diced into 1 cm × 1 cm chips. After an initial sample cleaning, a double-layer of polymethyl methacrylate (PMMA) 495 K A8/PMMA 495 K A4 electron-beam resist was coated at speeds of 8000 rpm and 6500 rpm, respectively. Each coating was followed by a 7 min and 30 s bake at 180 °C. Subsequently, an electron-beam lithography step was performed with a dose of 860 µC cm−2. The sample was then developed in a mixture of methyl isobutyl ketone and isopropyl alcohol (IPA) at a 1:3 ratio for 3 min, followed by a 1 min rinse in IPA. Following this, a double-layer metallic stack was deposited, comprising 90 nm of gold and 30 nm of nickel. The nickel layer served dual purposes, acting as an adhesive and forming a Schottky contact. A metal lift-off process was executed using Remover 1165 at 80 °C. After this, a photolithography step was undertaken with nLoF 2020 negative photoresist, followed by a 4-minute inductively coupled plasma etching with chlorine chemistry at 50 W radiofrequency power to define the mesa region.

W-band complex scattering parameter measurements

We used a 67 GHz four-port parametric network analyzer (PNA-X) of Keysight connected two frequency extender modules of Virginia Diodes to measure complex scattering parameters at 70–110 GHz band. The frequency extenders were connected to a couple of GSG terahertz probes. The built-in bias-tee of the probes were used to bias the metadevices in ON and OFF states. We calibrated the system using Short-Open-Load-Through (SOLT) technique. After measuring the scattering parameters, we converted the complex scattering parameter matrix to ABCD matrix, where the B parameter represents the impedance of the metadevice.

Effect of stripe resistances

We can demonstrate that the stripe resistance is negligible compared to the intrinsic resistance of the devices. For the devices shown in Fig. 1, the stripes consist of a 90 nm layer of gold and a 30 nm layer of nickel. Given a length of L \(=\) 0.9 µm and a width of W \(=\) 5.2 µm, the total resistance of a single stripe can be calculated to be \({R}_{{{{\rm{stripe}}}}}\cong 1.3\). This is the effective series resistance adding to the intrinsic resistance of 2-stripe devices. Based on the results shown in Fig. 1, \({R}_{{{{\rm{stripe}}}}}\) is totally negligible compared to the measured resistance of the device. Similarly, for metadevices with numerous stripes, the stripe resistances are configured parallelly, resulting in a negligible resistance.

Operation bandwidth of electronic metadevice switches

The ON-state impedance of two-terminal electronic metadevice switches at zero bias includes both real and imaginary parts. We can write the total impedance as

$$Z=2{Z}_{{{{\rm{C}}}}}+{R}_{{{{\rm{ch}}}}},$$
(6)

Where \({R}_{{{{\rm{ch}}}}}={R}_{{{{\rm{sh}}}}}{L}_{{{{\rm{ch}}}}}\) is the channel resistance. We can re-write (6) as

$$Z={\left(j\omega {C}_{{{{\rm{ON}}}}}\right)}^{-1}+{R}_{{{{\rm{ON}}}}},$$
(7)

where \({C}_{{{{\rm{ON}}}}}\) is the ON-state capacitance of the device

$${C}_{{{{\rm{ON}}}}}={LWN}\varepsilon /(4d).$$
(8)

Considering the operation frequency designed to be \({\omega }_{0}=2\pi {f}_{0}\), one can resonate out the imaginary part of (7) by a series inductor with the inductance of

$${L}_{{{{\rm{S}}}}}={\left({C}_{{{{\rm{ON}}}}}{\omega }_{0}^{2}\right)}^{-1}=4d/({LWN}\varepsilon {\omega }_{0}^{2}).$$
(9)

In this case, the effective impedance of the metadevice in series with the inductance can be written as

$${Z}_{{{{\rm{eff}}}}}=j\omega {L}_{{{{\rm{S}}}}}+{\left(j\omega {C}_{{{{\rm{ON}}}}}\right)}^{-1}+{R}_{{{{\rm{ON}}}}},$$
(10)

Which can be re-written as

$${Z}_{{{{\rm{eff}}}}}=j\sqrt{{L}_{{{{\rm{S}}}}}/{C}_{{{{\rm{ON}}}}}}\,\left(\widetilde{\omega }-{\widetilde{\omega }}^{-1}\right)+{R}_{{{{\rm{ON}}}}},$$
(11)

Where \(\widetilde{\omega }\) is the normalized angular frequency

$$\widetilde{\omega }=\omega /\sqrt{{L}_{{{{\rm{S}}}}}{C}_{{{{\rm{ON}}}}}}.$$
(12)

In a series configuration, the transmission through the device can be written as

$${S}_{21}={\left(1+\frac{{R}_{{{{\rm{ON}}}}}+j\sqrt{{L}_{S}/{C}_{{{{\rm{ON}}}}}}\left(\widetilde{\omega }-{\widetilde{\omega }}^{-1}\right)}{2{Z}_{0}}\right)}^{-1}$$
(13)

Where \({Z}_{0}=\) 50 Ω is the reference impedance. Equation (13) results in

$$|{S}_{21}|={\left(\sqrt{{\left(1+\frac{{R}_{{{{\rm{ON}}}}}}{2{Z}_{0}}\right)}^{2}+\frac{{L}_{{{{\rm{S}}}}}/{C}_{{{{\rm{ON}}}}}}{4{Z}_{0}^{2}}{\left(\widetilde{\omega }-{\widetilde{\omega }}^{-1}\right)}^{2}}\right)}^{-1}$$
(14)

The transmission through the device peaks at \(\widetilde{\omega }=1\). Assuming \({R}_{{{{\rm{ON}}}}}\,\ll\, 2{Z}_{0}\), one can approximately re-write (14) as

$$|{S}_{21}|\cong {\left(\sqrt{1+\frac{{R}_{{{{\rm{ON}}}}}}{{Z}_{0}}+\frac{{L}_{{{{\rm{S}}}}}/{C}_{{{{\rm{ON}}}}}}{4{Z}_{0}^{2}}{\left(\widetilde{\omega }-{\widetilde{\omega }}^{-1}\right)}^{2}}\right)}^{-1}$$
(15)

We define the bandwidth as the frequency range at which the impact of the imaginary part of the impedance is less than the impact of \({R}_{{{{\rm{ON}}}}}\) on the transmission:

$$\frac{{L}_{{{{\rm{S}}}}}/{C}_{{{{\rm{ON}}}}}}{4{Z}_{0}^{2}}{\left(\widetilde{\omega }-{\widetilde{\omega }}^{-1}\right)}^{2} < \frac{{R}_{{{{\rm{ON}}}}}}{{Z}_{0}},$$
(16)

which results in

$$|\widetilde{\omega }-{\widetilde{\omega }}^{-1}| < \frac{2\sqrt{{Z}_{0}{R}_{{ON}}}}{\sqrt{{L}_{{{{\rm{S}}}}}/{C}_{{{{\rm{ON}}}}}}}$$
(17)

We note that this is a stricter definition of bandwidth compared to the commonly used 3-dB bandwidth. To obtain the metadevice switch bandwidth one has to subtract the positive solutions of \(\left|\widetilde{\omega }-{\widetilde{\omega }}^{-1}\right|=\frac{2\sqrt{{Z}_{0}{R}_{{ON}}}}{\sqrt{{L}_{{{{\rm{S}}}}}/{C}_{{{{\rm{ON}}}}}}}\) which results in

$$\Delta \widetilde{\omega }=\frac{2\sqrt{{Z}_{0}{R}_{{ON}}}}{\sqrt{{L}_{{{{\rm{S}}}}}/{C}_{{{{\rm{ON}}}}}}}.$$
(18)

Equation (17) can be approximately written based on the parameters of the metadevice

$$\Delta \widetilde{\omega }=\frac{\pi {f}_{0}{LWN}\varepsilon }{d}\sqrt{{R}_{{{{\rm{sh}}}}}{Z}_{0}\left({L}_{{{{\rm{ch}}}}}+\frac{L}{3}\right)/{NW}}.$$
(19)

The bandwidth can be written as

$$\Delta f=\pi {f}_{0}^{2}L\varepsilon \sqrt{{R}_{{{{\rm{sh}}}}}{Z}_{0}{NW}\left({L}_{{{{\rm{ch}}}}}+\frac{L}{3}\right)}/d.$$
(20)

The number of stripes (N) not only increases super linearly the conductance, but also based on (20), it enhances the operation bandwidth. In addition, devices operating at higher frequency (\({f}_{0}\)) enable a larger relative bandwidth.

To have a sense of this theoretical model, we consider the case of a metadevice with N = 16, W = 5.2 µm, L = 0.9 µm, and Lch = 0.25 µm. The ON-state capacitance can be obtained from (8) to be CON = 220 fF. Considering f0 = 77 GHz we obtain the series inductance to be L = 19.4 pH, inequality (16) results in

$$0.4 \, < \, \widetilde{\omega } \, < \, 2.1$$
(21)

Which is equivalent to

$$30.8{{{\rm{GHz}}}} \, < \, f \, < \, 161.7{{{\rm{GHz}}}}$$
(22)

Which indicates a very wide-band operation corresponding to the relative bandwidth of 136%. Experimental results validate the wide-band operation capability of electronic metadevices (Supplementary Data Fig. 1).

High-power linearity experiment

As shown in Fig. 2a, a high-gain high-power mm-wave amplifier driven by a local oscillator (LO) inject a high-power signal at 34 GHz into the input port of a metadevice (L = 1.2 µm, W = 9 µm, and Lch = 0.32 µm), and the output power was measured through a pre-calibrated power sensor. Two bias-tees were used to control the state of the device-under-test (DUT), and a 10-dB attenuator was used to limit the power at the power sensor port.

Theoretical analysis of high-power operation concerning 1 dB compression point

Here we consider the schematic circuit shown in Supplementary Data Fig. 4, which includes a monotone voltage source \({V}_{0}\left(t\right)=A\cos \left(\omega t\right)\) with series resistance \({R}_{S}\), exciting a two-terminal metadevice switch terminated by a resistive load \({R}_{L}\) (Supplementary Data Fig. 4a). Using first-order approximation, the voltage drop across the metadevice terminals can be written as

$${V}_{{{{\rm{meta}}}}}^{(1)}={R}_{{{{\rm{ON}}}}}{V}_{0}/\left({R}_{{{{\rm{S}}}}}+{R}_{{{{\rm{L}}}}}+{R}_{{{{\rm{ON}}}}}\right),$$
(23)

where \({R}_{{{{\rm{ON}}}}}\) is the small signal resistance of the metadevice at zero bias, and the superscript (1) in \({V}_{{{{\rm{meta}}}}}^{(1)}\) indicates the first approximation. This voltage drop can modify the carrier density and the sheet resistance in the 2DEG, and thus, lead to a nonlinear response in the device.

Electronic metadevices have the advantage of very low \({R}_{{{{\rm{ON}}}}}\) by outbalancing the trade-off between the ON-state resistance and the OFF-state capacitance. Based on (23), a small \({R}_{{{{\rm{ON}}}}}\) not only reduces the insertion loss, but also enhances the linearity by reducing the voltage drop across the terminal. Electronic metadevice can offer very low ON-state resistance values down to a few ohms, which, assuming \({R}_{L}={R}_{S}=50\Omega\), results in only a few percent voltage drop across the device terminals.

To analyze high-power operation of electronic metadevices, we can model their large-signal behavior using a couple of back-to-back diodes (Supplementary Data Figs. 4b and 3c). In this model, \({V}_{{{{\rm{meta}}}}}\) would be the total voltage drop across these two diodes. We write

$${V}_{1}+{V}_{2}={V}_{{{{\rm{meta}}}}},$$
(24)

where \({V}_{1}\) and \({V}_{2}\) represent the voltage drop over diode terminals (Supplementary Data Fig. 4c). Writing Kirchhoff’s Current Law (KCL) considering large signal model of diodes we obtain

$${I}_{S}\left({e}^{{V}_{1}/{V}_{T}}-1\right)=-{I}_{S}\left({e}^{-{V}_{2}/{V}_{T}}-1\right),$$
(25)

where \({I}_{S}\) is a constant and \({V}_{T}\) is the thermal voltage. Equation (25) can be re-written as

$${e}^{{V}_{1}/{V}_{T}}+{e}^{-{V}_{2}/{V}_{T}}=2.$$
(26)

Solving (24) and (26) results in

$${V}_{1}={V}_{T}\,{{\mathrm{ln}}}\left(\frac{2}{1+\exp (-{{\mathrm{V}}}_{{{{\rm{meta}}}}}/{{\mathrm{V}}}_{T})}\right),$$
(27)

and

$${V}_{2}=-{V}_{T}\,{{\mathrm{ln}}}\left(\frac{2}{1+\exp ({{\mathrm{V}}}_{{{{\rm{meta}}}}}/{{\mathrm{V}}}_{T})}\right).$$
(28)

The carrier density at the 2DEG at voltage \(v\) can be linearly approximated as

$${n}_{{{{\rm{S}}}}}(v)={n}_{S}\left(0\right)\left(1-v/{V}_{{{{\rm{th}}}}}\right)$$
(29)

Where \({V}_{{{{\rm{th}}}}}\approx -4\,{{{\rm{V}}}}\) is the threshold voltage at which 2DEG is fully depleted. Considering a constant electron mobility independent from \({n}_{S}\) (ref. 30) and assuming that \({n}_{S}\) has a linear relationship with respect to \(v\) (refs. 31,32,33), one can write the sheet resistance as follows:

$${R}_{{{{\rm{sh}}}}}\left(v\right)={R}_{{{{\rm{sh}}}}}\left(0\right){\left(1-v/{V}_{{{{\rm{th}}}}}\right)}^{-1}$$
(30)

In this case, the total resistance of the metadevice can be written as

$${R}_{{{{\rm{ON}}}}}=\frac{L}{6}{R}_{{{{\rm{sh}}}}}\left({V}_{1}\right)+\frac{L}{6}{R}_{{{{\rm{sh}}}}}\left(-{V}_{2}\right)+{R}_{{{{\rm{sh}}}}}\left(0\right){L}_{{{{\rm{ch}}}}},$$
(31)

which results in

$${R}_{{{{\rm{ON}}}}}=\left(\frac{L/6}{1-{V}_{1}/{V}_{{{{\rm{th}}}}}}+\frac{L/6}{1+{V}_{2}/{V}_{{{{\rm{th}}}}}}+{L}_{{{{\rm{ch}}}}}\right){R}_{{{{\rm{sh}}}}},$$
(32)

Where \({R}_{{{{\rm{sh}}}}}={R}_{{{{\rm{sh}}}}}(0)\) is the zero-bias sheet resistance. We use the first-order approximation of \({V}_{{{{\rm{meta}}}}}\) to identify the consecutive variation in \({R}_{{{{\rm{ON}}}}}\). Supplementary Data Fig. 4d shows the absolute value of voltage drop across back-to-back diodes when the input voltage source is at its maximum. In this case, the first diode is in forward bias and its voltage drop is negligible. The major part of the voltage drops across the metadevice terminals drops over the reverse biased diode, which modifies the sheet resistance under the corresponding terminal, leading to an overall resistance degradation.

Supplementary Data Fig. 4d shows the percentage of resistance degradation for different values of zero-bias \({R}_{{{{\rm{ON}}}}}\). Our results indicate that the increase in \({R}_{{{{\rm{ON}}}}}\) would be relatively minor even at high power excitations up to 1 W. This increased ON-state resistance can only slightly compress ( < 0.1 dB) the output power at high power levels up to 1 W (Supplementary Data Fig. 4e). Our results show that this mechanism does not account for the output power compression shown in Fig. 2b.

Saturation current in the 2DEG channel is possibly responsible for the output power compression of electronic metadevices.

Second harmonic generation experiment

As shown in Fig. 2e, a power amplifier driven by a LO injects mm-wave power at 51 GHz into the first port of an 8-stripe metadevice (L = 0.9 µm, W = 5.2 µm, and Lch = 0.32 µm) and the output port is terminated by the 50 Ω port of a Keysight 113 GHz oscilloscope to characterize the second harmonic at 102 GHz. Two bias tees ensure that the DUT is biased close to the threshold voltage (DC \({V}_{b}\) = −4 V). The loss of the mm-wave cable (8 dB), the bias-tee (1.5 dB) and the ground-signal-ground (GSG) probe (0.7 dB) were de-embedded.

Second harmonic generation in an ideal diode

Assuming an ideal diode (\(I=0\) for \(V\le 0\), and \(V=0\) for \(I\ge 0\)), in series with a load \({R}_{L}\), is excited by a sinusoidal voltage \({v}_{{{{\rm{in}}}}}\left(t\right)={V}_{0}\cos \left(\omega t\right)\), the output voltage at the load can be written as \({v}_{{{{\rm{out}}}}}\left(t\right)={V}_{0}\cos \left(\omega t\right)u\left(\cos \left(\omega t\right)\right)\), where \(u\) represents the step function (\(u\left(x\ge 0\right)=1\) and \(u\,\left(x \, < \, 0\right)=0\)). Using Fourier expansion, we have:

$${v}_{{{{\rm{out}}}}}\left(t\right) = \frac{1}{\pi }{V}_{0}+\frac{1}{2}{V}_{0}\cos \left(\omega t\right)+\frac{2}{\pi }{V}_{0}{\sum }_{m=1}^{\infty }\frac{{\left(-1\right)}^{m}}{1-4{m}^{2}}\cos \left(2m\omega t\right) \\ = \frac{1}{\pi }{V}_{0}+\frac{1}{2}{V}_{0}\cos \left(\omega t\right)+\frac{2}{3\pi }{V}_{0}\cos \left(2\omega t\right)+\ldots$$
(33)

The second harmonic conversion gain equals

$$20{\log }_{10}\frac{2}{3\pi }=-13.46{{{\rm{dB}}}}.$$
(34)

Numerical and theoretical study of switching in electronic metadevices

Using FEM simulations, we aim to elucidate the mechanisms behind the high-performance second harmonic generation observed in electronic metadevices. In a two-terminal device, the sensitivity of impedance to the voltage across device terminals is a critical factor that notably influences the device nonlinear performance, including harmonic generation and mixing.

Supplementary Data Fig. 5a presents a schematic cross session of an electronic metadevice where a negative potential is applied to terminal 1, while terminal 2 is grounded. This arrangement leads to a partial depletion under terminal 1 and a partial accumulation under terminal 2. The back-to-back diode model presented in Supplementary Data Fig. 2c can be used to describe the voltage drop between the terminals and the 2DEG. At high bias voltages, almost the entire voltage drops between terminal 1 and the 2DEG, which represents the reverse biased diode in the model shown in Supplementary Data Fig. 2c.

Considering the 2DEG sheet resistance under terminal 1 to be \({R}_{{{{\rm{sh}}}}}^{{\prime} }\), which is considerably larger than the sheet resistance under terminal 2, and due to the symmetry, we can write the phasor of electric field under terminal 1 using (2) as

$${E}_{z}\left(x\right)={E}_{1}\cosh \left({\xi }^{{\prime} }(x-{x}_{1})\right),\,|x-{x}_{1}| \, < \, \frac{L}{2}$$
(35)

where \({E}_{0}\) is a constant, \({x}_{1}\) is the center position of the stripe, and \({\xi }^{{\prime} }=\sqrt{j\omega \varepsilon {R}_{{{{\rm{sh}}}}}^{{\prime} }/d}\). The 2DEG current density under the stripe can be calculated using (3)

$${J}_{x}(x)=\left(\frac{{E}_{1}{\xi }^{{\prime} }d}{{R}_{{{{\rm{sh}}}}}^{{\prime} }}\right)\sinh \left({\xi }^{{\prime} }(x-{x}_{1})\right).$$
(36)

Considering the 2DEG current density at the channel to be \({J}_{0}\), we have

$$\left(\frac{{E}_{1}{\xi }^{{\prime} }d}{{R}_{{{{\rm{sh}}}}}^{{\prime} }}\right)\sinh \left(\frac{{\xi }^{{\prime} }L}{2}\right)={J}_{0},$$
(37)

which results in

$${E}_{z}\left(x\right)=\frac{{J}_{0}{R}_{{{{\rm{sh}}}}}^{{\prime} }}{{\xi }^{{\prime} }d}\frac{\cosh \left({\xi }^{{\prime} }(x-{x}_{1})\right)}{\sinh \left({\xi }^{{\prime} }L/2\right)},\,|x-{x}_{1}| \, < \, \frac{L}{2}$$
(38)

Depletion of 2DEG increases the sheet resistance, and consecutively the magnitude of \(\xi {\prime}\). However, the variation in electric field remains relatively small when \(\left|{\xi }^{{\prime} }L/2\right|\,\lesssim\, 1\). One can show that for \(\left|{\xi }^{{\prime} }L/2\right|\,\ll\, 1\) we have

$$\left(\partial /\partial {R}_{{{{\rm{sh}}}}}^{{\prime} }\right){E}_{z}\left(x\right)=\left({{{\rm{d}}}}/{{{\rm{d}}}}{R}_{{{{\rm{sh}}}}}^{{\prime} }\right){J}_{x}\left(x\right)=0,$$
(39)

which results in

$$\left(\partial /\partial {R}_{{{{\rm{sh}}}}}^{{\prime} }\right)Z=0,$$
(40)

where \(Z\) is the impedance of the metadevice. This is in agreement with the numerical results showin in Supplementary Data Fig. 3b, where the metadevice impedance is quite flat at high sheet conductance values which corresponds to low values for \(\left|{\xi }^{{\prime} }L/2\right|\). In a sweep of normalized sheet conductance from \(\widetilde{G}=1\) to \(\widetilde{G}=0\), the device undergoes a considerable impedance change from \({Z}_{{{{\rm{ON}}}}}={R}_{{{{\rm{ON}}}}}\) to \({Z}_{{{{\rm{OFF}}}}}=j{X}_{{{{\rm{OFF}}}}}={\left(j\omega {C}_{{{{\rm{OFF}}}}}\right)}^{-1}\). Considering that the impedance variations are minor for high \(\widetilde{G}\) values (\(\widetilde{G}\ge 0.1\) for the results shown in Supplementary Data Fig. 3b), the device is expected to exhibit an abrupt change in the impedance for small \(\widetilde{G}\) values (\(\left|{\xi }^{{\prime} }L\right|\,\gg\, 1\)). This explanation is in agreement with the results shown in Supplementary Data Fig. 3b.

For the considered device in Supplementary Data Fig. 3, the variation in the impedance of the metadevice is negligible compared to the variations in the sheet resistance for depletion up to \(\sim\)90%. After that, the device impedance becomes very sensitive to the sheet resistance. This abrupt switching can be the underlying mechanism explaining the high-performance harmonic generation in electronic metadevices.

The trend of variations in the device impedance is similar to the trend of changing the microscopic fields across the device layout. As shown in Supplementary Data Fig. 3c–h, 60% depletion of the 2DEG under terminal 1 leads to minor changes in the electric fields \({E}_{z}\). Supplementary Data Fig. 3i–k present the simulated electric fields on a cutline at different sheet conductance values. The results show a good agreement with theoretical formula presented in (38), which analytically supports the negligible dependence of microscopic fields and impedance with respect to the depleted sheet resistance for \(\left|{\xi }^{{\prime} }L/2\right|\,\ll\, 1\).

Theory of three-terminal electronic metadevices

Supplementary Fig. 7a shows the cross section of a three-terminal metadevice including top metals, a dielectric barrier with permittivity \(\varepsilon\), and a 2DEG underneath. The length of stripes for terminals 1 and 2 is shown by L, and the length of gate stripe is shown by Lg. The structure is assumed to be uniform along y direction (perpendicular to the plane), and periodic in the x direction. We consider terminals 1 and 2 to be zero biased, while the gate terminal is biased at \(V={V}_{G}\). We consider the scenario in which the gate voltage results in a partial depletion in the 2DEG underneath (\({V}_{G} \, < \, 0\)). We consider the sheet resistance to be \({R}_{{{{\rm{sh}}}}}\) everywhere except under gates where the sheet resistance is \({R}_{{{{\rm{sh}}}}}^{{\prime} }\).

The phasor of vertical electric field encapsulated between the top metal and the 2DEG at the angular frequency \(\omega\) can be written in form of superposition of two eigenfunctions, as presented in (2). The dashed lines in Supplementary Data Fig. 4a are symmetry axes, which constrain the solution (2). In this case, one can write \({E}_{z}\) under terminals 1 and 2 as

$${E}_{z}\left(x\right)={E}_{0}\cosh \left(\xi (x+{x}_{0})\right),\,\left|x+{x}_{0}\right| \, < \, L/2$$
(41)

and

$${E}_{z}\left(x\right)={E}_{0}\cosh \left(\xi (x-{x}_{0})\right),\,\left|x-{x}_{0}\right| \, < \, L/2$$
(42)

respectively, where \({E}_{0}\) is a constant. The 2DEG current density (\({J}_{x}\)) can be directly calculated from \({E}_{z}\) using (3), under terminals 1

$${J}_{x}\left(x\right)=\frac{d\xi {E}_{0}}{{R}_{{{{\rm{sh}}}}}}\sinh \left(\xi (x+{x}_{0})\right),\,\left|x+{x}_{0}\right| \, < \, L/2$$
(43)

and terminal 2

$${J}_{x}\left(x\right)=\frac{d\xi {E}_{0}}{{R}_{{{{\rm{sh}}}}}}\sinh \left(\xi (x-{x}_{0})\right),\,\left|x-{x}_{0}\right| \, < \, L/2$$
(44)

Assuming gate current to be negligible, the 2DEG current density at the right end of terminal 1 should match with the 2DEG current density at the left end of terminal 2:

$${J}_{x}\left(-{x}_{0}+L/2\right)={J}_{x}({x}_{0}-L/2).$$
(45)

which results in \(A=-B\). Now we write \({E}_{z}\) under the gate in its general form

$${E}_{z}\left(x\right)={C}_{1}{e}^{{\xi }_{{{{\rm{g}}}}}x}+{C}_{2}{e}^{-{\xi }_{{{{\rm{g}}}}}x},\,\left|x\right| \, < \, {L}_{{{{\rm{g}}}}}/2$$
(46)

where \({\xi }_{{{{\rm{g}}}}}=\sqrt{j{R}_{{{{\rm{sh}}}}}^{{\prime} }\omega \varepsilon /{d}}\). Equation (44) results in

$${J}_{x}\left(x\right)={\xi }_{{{{\rm{g}}}}}\left(d/{R}_{{{{\rm{sh}}}}}\right)\left({C}_{1}{e}^{{\xi }_{{{{\rm{g}}}}}x}-{C}_{2}{e}^{-{\xi }_{{{{\rm{g}}}}}x}\right),\,\left|x\right| \, < \, {L}_{{{{\rm{g}}}}}/2$$
(47)

Considering the boundary condition of \({J}_{x}\left(-{L}_{{{{\rm{g}}}}}/2\right)={J}_{x}\left({L}_{{{{\rm{g}}}}}/2\right)\), we conclude that \({C}_{1}=-{C}_{2}\), resulting in

$${E}_{z}\left(x\right)=C\sinh \left({\xi }_{{{{\rm{g}}}}}x\right),\,\left|x\right| \, < \, {L}_{{{{\rm{g}}}}}/2$$
(48)

where \(C={C}_{1}/2\). In addition, using 2DEG current continuity at the boundary of terminal 1 and gate terminal we have \({J}_{x}\left(-{x}_{0}+L/2\right)={J}_{x}(-{L}_{{{{\rm{g}}}}}/2)\) which results in

$$A\xi \sinh \left(\xi L/2\right)=-{\xi }_{{{{\rm{g}}}}}C\cosh \left({\xi }_{{{{\rm{g}}}}}{L}_{{{{\rm{g}}}}}/2\right),$$
(49)

Which can be re-written as

$$C=\frac{\xi \sinh \left(\xi L/2\right)}{{\xi }_{{{{\rm{g}}}}}\cosh \left({\xi }_{{{{\rm{g}}}}}{L}_{{{{\rm{g}}}}}/2\right)}{E}_{0}.$$
(50)

Considering \(L\) and \({L}_{g}\) to be small enough to meet \(\left|{\xi }_{{{{\rm{g}}}}}{L}_{{{{\rm{g}}}}}/2\right|\,\ll\, \left|\xi L/2\right|\lesssim 1\), one can approximately re-write (45) as

$${E}_{z}\left(x\right)={E}_{0}\sinh \left(\xi L/2\right)\xi x,\,\left|x\right| \, < \, {L}_{{{{\rm{g}}}}}/2$$
(51)

Supplementary Data Fig. 4b summarizes the theoretical results obtained for a periodic structure with a very large number of stripes. As shown in Supplementary Data Fig. 4c, the field patterns under terminals 1 and 2 in a three-terminal metadevice is similar to that in a two-terminal metadevices, indicating that three-terminal metadevices are capable of providing an excellent coupling between terminals and the 2DEG.

Supplementary Data Fig. 4d, e show FEM simulated electric fields at the barrier and 2DEG current density for a zero-biased three-terminal metadevice at 100 GHz. Increasing the number of sub-wavelength stripes microscopically modifies the electric fields and current densities over the device layout. The device provides a higher current confinement as the number of stripes increases (Supplementary Data Fig. 4f). Supplementary Data Fig. 4g shows the normalized contact resistance of devices with different number of stripes, demonstrating a non-traditional scaling over a wide frequency range.

Our theoretical results show that the difference between two-terminal and three-terminal electronic metadevices is mainly on their switching mechanism. In a three-terminal metadevice, partial depletion of 2DEG under the gate terminal does not change the coupling between terminals 1 and 2 with the 2DEG. Instead, it modifies the channel resistance. In this case, one can write the real part of impedance seen between terminals 1 and 2 as

$${Z}_{12}\left({V}_{{{{\rm{g}}}}}\right)=2{R}_{C}+{R}_{S}+{R}_{{{{\rm{sh}}}}}^{{\prime} }{L}_{{{{\rm{g}}}}}$$
(52)

where \({R}_{C}\) is the contact impedance, \({R}_{{{{\rm{S}}}}}=({L}_{{{{\rm{ch}}}}}-{L}_{{{{\rm{g}}}}})\,{R}_{{{{\rm{sh}}}}}\) is the 2DEG access resistance at the gap between terminals and the gate. The sheet resistance under gate can be written using (30)

$${Z}_{12}\left({V}_{{{{\rm{g}}}}}\right)={R}_{0}+\frac{{R}_{{{{\rm{sh}}}}}{L}_{{{{\rm{g}}}}}}{1-{V}_{{{{\rm{g}}}}}/{V}_{{{{\rm{th}}}}}},$$
(53)

Where \({R}_{0}\) includes contact and access resistances, which are independent from gate bias voltage. Considering the imaginary part of the impedance to be in-resonance with a series inductance, (53) fully represents the device impedance. Assuming \(\frac{{R}_{{{{\rm{sh}}}}}{L}_{{{{\rm{g}}}}}}{1-{V}_{{{{\rm{g}}}}}/{V}_{{{{\rm{th}}}}}}\gg {R}_{0}\) at a partial depletion, and considering terminal 1 to be excited by a voltage source with the phasor of \({V}_{1}\), and terminal 2 to be grounded, the current passing through the device can be written as

$$I=\frac{1}{{R}_{{{{\rm{sh}}}}}{L}_{{{{\rm{g}}}}}}\left(1-\frac{{V}_{{{{\rm{g}}}}}}{{V}_{{{{\rm{th}}}}}}\right){V}_{1}.$$
(54)

Equation (54) shows the capability of the device in mixing the signals applied to terminal 1 and gate.

High-speed ASK and QPSK data transmission experiments

As shown in Fig. 5a, a monotone carrier was injected into terminal 1 of the metadevice (shown in Fig. 4a), and the third terminal was excited by the amplified output of an arbitrary waveform generator (AWG). The second port was directly terminated by a Keysight 113 GHz oscilloscope. Using three bias tees, we zero biased terminals 1 and 2 and biased the terminal 3 at \({V}_{b}\) = −4 V.