I. INTRODUCTION

Section:

ChooseTop of pageABSTRACTI. INTRODUCTION <<II. RESULT AND DISCUSSION...III. CONCLUSIONSUPPLEMENTARY MATERIALREFERENCESTerahertz time-domain spectroscopy facilitates the study of carrier dynamics in semiconductors owing to its time-resolving ability with an ultrafast optical pump to excite carriers.1–31. R. Ulbricht, E. Hendry, J. Shan, T. F. Heinz, and M. Bonn, “Carrier dynamics in semiconductors studied with time-resolved terahertz spectroscopy,” Rev. Mod. Phys. 83, 543–586 (2011). https://doi.org/10.1103/revmodphys.83.5432. H. J. Joyce, C. J. Docherty, Q. Gao, H. H. Tan, C. Jagadish, J. Lloyd-Hughes, L. M. Herz, and M. B. Johnston, “Electronic properties of GaAs, InAs and InP nanowires studied by terahertz spectroscopy,” Nanotechnology 24, 214006 (2013). https://doi.org/10.1088/0957-4484/24/21/2140063. J. Hebling, M. C. Hoffmann, H. Y. Hwang, K.-L. Yeh, and K. A. Nelson, “Observation of nonequilibrium carrier distribution in Ge, Si, and GaAs by terahertz pump–terahertz probe measurements,” Phys. Rev. B 81, 035201 (2010). https://doi.org/10.1103/physrevb.81.035201 During the last two decades, the development of intense terahertz pulse systems has allowed researchers to extend their work to the investigation of many-body interactions, such as electron–electron and electron–phonon scattering, through nonlinear terahertz response mesurements.4–74. P. Gaal, K. Reimann, M. Woerner, T. Elsaesser, R. Hey, and K. H. Ploog, “Nonlinear terahertz response of n-type GaAs,” Phys. Rev. Lett. 96, 187402 (2006). https://doi.org/10.1103/physrevlett.96.1874025. W. Kuehn, P. Gaal, K. Reimann, M. Woerner, T. Elsaesser, and R. Hey, “Terahertz-induced interband tunneling of electrons in GaAs,” Phys. Rev. B 82, 075204 (2010). https://doi.org/10.1103/physrevb.82.0752046. A. T. Tarekegne, K. Iwaszczuk, M. Zalkovskij, A. C. Strikwerda, and P. U. Jepsen, “Impact ionization in high resistivity silicon induced by an intense terahertz field enhanced by an antenna array,” New J. Phys. 17, 043002 (2015). https://doi.org/10.1088/1367-2630/17/4/0430027. A. T. Tarekegne, H. Hirori, K. Tanaka, K. Iwaszczuk, and P. U. Jepsen, “Impact ionization dynamics in silicon by MV/cm THz fields,” New J. Phys. 19, 123018 (2017). https://doi.org/10.1088/1367-2630/aa936b However, in the low frequency regime (0.1–3 THz), the generally achievable field strength is ∼1 MV/cm in a Ti:sapphire amplified system with a 1 kHz repetition rate and in an electron accelerator with a 100 kHz repetition rate.88. B. Zhang, Z. Ma, J. Ma, X. Wu, C. Ouyang, D. Kong, T. Hong, X. Wang, P. Yang, L. Chen, Y. Li, and J. Zhang, “1.4-mJ high energy terahertz radiation from lithium niobates,” Laser Photonics Rev. 15, 2000295 (2021). https://doi.org/10.1002/lpor.202000295 Although the state-of-the-art strength reaches 6 MV/cm with a lower repetition rate,99. B. Green, S. Kovalev, V. Asgekar, G. Geloni, U. Lehnert, T. Golz, M. Kuntzsch, C. Bauer, J. Hauser, J. Voigtlaender, B. Wustmann, I. Koesterke, M. Schwarz, M. Freitag, A. Arnold, J. Teichert, M. Justus, W. Seidel, C. Ilgner, N. Awari, D. Nicoletti, S. Kaiser, Y. Laplace, S. Rajasekaran, L. Zhang, S. Winnerl, H. Schneider, G. Schay, I. Lorincz, A. A. Rauscher, I. Radu, S. Mährlein, T. H. Kim, J. S. Lee, T. Kampfrath, S. Wall, J. Heberle, A. Malnasi-Csizmadia, A. Steiger, A. S. Müller, M. Helm, U. Schramm, T. Cowan, P. Michel, A. Cavalleri, A. S. Fisher, N. Stojanovic, and M. Gensch, “High-field high-repetition-rate sources for the coherent THz control of matter,” Sci. Rep. 6, 22256 (2016). https://doi.org/10.1038/srep22256 it is insufficient for fully studying the nonequilibrium many-body interactions.1010. C. Lange, T. Maag, M. Hohenleutner, S. Baierl, O. Schubert, E. R. J. Edwards, D. Bougeard, G. Woltersdorf, and R. Huber, “Extremely nonperturbative nonlinearities in GaAs driven by atomically strong terahertz fields in gold metamaterials,” Phys. Rev. Lett. 113, 227401 (2014). https://doi.org/10.1103/physrevlett.113.227401 To reach such nonperturbative regimes, where an atomically strong electric field is required to induce interband tunneling,55. W. Kuehn, P. Gaal, K. Reimann, M. Woerner, T. Elsaesser, and R. Hey, “Terahertz-induced interband tunneling of electrons in GaAs,” Phys. Rev. B 82, 075204 (2010). https://doi.org/10.1103/physrevb.82.075204 intervalley scattering,1111. K. Fan, H. Y. Hwang, M. Liu, A. C. Strikwerda, A. Sternbach, J. Zhang, X. Zhao, X. Zhang, K. A. Nelson, and R. D. Averitt, “Nonlinear terahertz metamaterials via field-enhanced carrier dynamics in GaAs,” Phys. Rev. Lett. 110, 217404 (2013). https://doi.org/10.1103/physrevlett.110.217404 or impact ionization,6,126. A. T. Tarekegne, K. Iwaszczuk, M. Zalkovskij, A. C. Strikwerda, and P. U. Jepsen, “Impact ionization in high resistivity silicon induced by an intense terahertz field enhanced by an antenna array,” New J. Phys. 17, 043002 (2015). https://doi.org/10.1088/1367-2630/17/4/04300212. R. D. Schaller and V. I. Klimov, “High efficiency carrier multiplication in PbSe nanocrystals: Implications for solar energy conversion,” Phys. Rev. Lett. 92, 186601 (2004). https://doi.org/10.1103/physrevlett.92.186601 researchers have attempted to combine semiconductors with metamaterials.7,10,11,137. A. T. Tarekegne, H. Hirori, K. Tanaka, K. Iwaszczuk, and P. U. Jepsen, “Impact ionization dynamics in silicon by MV/cm THz fields,” New J. Phys. 19, 123018 (2017). https://doi.org/10.1088/1367-2630/aa936b10. C. Lange, T. Maag, M. Hohenleutner, S. Baierl, O. Schubert, E. R. J. Edwards, D. Bougeard, G. Woltersdorf, and R. Huber, “Extremely nonperturbative nonlinearities in GaAs driven by atomically strong terahertz fields in gold metamaterials,” Phys. Rev. Lett. 113, 227401 (2014). https://doi.org/10.1103/physrevlett.113.22740111. K. Fan, H. Y. Hwang, M. Liu, A. C. Strikwerda, A. Sternbach, J. Zhang, X. Zhao, X. Zhang, K. A. Nelson, and R. D. Averitt, “Nonlinear terahertz metamaterials via field-enhanced carrier dynamics in GaAs,” Phys. Rev. Lett. 110, 217404 (2013). https://doi.org/10.1103/physrevlett.110.21740413. H. R. Seren, J. Zhang, G. R. Keiser, S. J. Maddox, X. Zhao, K. Fan, S. R. Bank, X. Zhang, and R. D. Averitt, “Nonlinear terahertz devices utilizing semiconducting plasmonic metamaterials,” Light: Sci. Appl. 5, e16078 (2016). https://doi.org/10.1038/lsa.2016.78Fabricating metamaterials, which confine and enhance an electric field in their vicinity, on the surface of GaAs enables the evaluation of the nonlinear phenomena. Such metamaterials show field-induced Zener tunneling1414. E. O. Kane, “Zener tunneling in semiconductors,” J. Phys. Chem. Solids 12, 181–188 (1960). https://doi.org/10.1016/0022-3697(60)90035-4 and induce impact ionization near optical hotspots with an enhanced field of up to 30 MV/cm (the maximum incident peak field is 1.5 MV/cm).1010. C. Lange, T. Maag, M. Hohenleutner, S. Baierl, O. Schubert, E. R. J. Edwards, D. Bougeard, G. Woltersdorf, and R. Huber, “Extremely nonperturbative nonlinearities in GaAs driven by atomically strong terahertz fields in gold metamaterials,” Phys. Rev. Lett. 113, 227401 (2014). https://doi.org/10.1103/physrevlett.113.227401 Further, nonlinear terahertz metamaterials that induce carrier generation have been reported for various applications.15–1815. R. Rana, L. Balaghi, I. Fotev, H. Schneider, M. Helm, E. Dimakis, and A. Pashkin, “Nonlinear charge transport in InGaAs nanowires at terahertz frequencies,” Nano Lett. 20, 3225–3231 (2020). https://doi.org/10.1021/acs.nanolett.9b0532816. D. J. Ironside, R. Salas, P.-Y. Chen, K. Q. Le, A. Alú, and S. R. Bank, “Enhancing THz generation in photomixers using a metamaterial approach,” Opt. Express 27, 9481–9494 (2019). https://doi.org/10.1364/oe.27.00948117. A. R. Wright, X. G. Xu, J. C. Cao, and C. Zhang, “Strong nonlinear optical response of graphene in the terahertz regime,” Appl. Phys. Lett. 95, 072101 (2009). https://doi.org/10.1063/1.320511518. I. Al-Naib, G. Sharma, M. M. Dignam, H. Hafez, A. Ibrahim, D. G. Cooke, T. Ozaki, and R. Morandotti, “Effect of local field enhancement on the nonlinear terahertz response of a silicon-based metamaterial,” Phys. Rev. B 88, 195203 (2013). https://doi.org/10.1103/physrevb.88.195203 To achieve such nonlinear phenomena, amplified laser systems have been used to generate intense terahertz pulses,1919. B. J. Kang, D. Rohrbach, F. D. J. Brunner, S. Bagiante, H. Sigg, and T. Feurer, “Ultrafast and low-threshold THz mode switching of two-dimensional nonlinear metamaterials,” Nano Lett. 22, 2016–2022 (2022). https://doi.org/10.1021/acs.nanolett.1c04776 although these systems hamper the realization of extensive nonlinear applications.2020. M. Liu, H. Y. Hwang, H. Tao, A. C. Strikwerda, K. Fan, G. R. Keiser, A. J. Sternbach, K. G. West, S. Kittiwatanakul, J. Lu, S. A. Wolf, F. G. Omenetto, X. Zhang, K. A. Nelson, and R. D. Averitt, “Terahertz-field-induced insulator-to-metal transition in vanadium dioxide metamaterial,” Nature 487, 345–348 (2012). https://doi.org/10.1038/nature11231 Furthermore, such amplified laser systems are not feasible for use with other III–V semiconductors, whose bandgap energies are larger than that of GaAs because of their higher threshold fields (up to tens of MV/cm or higher). Therefore, generating a stronger electric field within an extremely confined volume is indispensable for researchers to study condensed matter in the terahertz frequency regime.

In this study, we induce a terahertz electric field of 46 MV/cm with the help of 5 nm-wide metal–insulator–metal structures on a semi-insulating GaAs substrate. This field strength is far beyond the nonperturbative regime of GaAs and results in an unprecedented nonlinear transmission decrease with an extinction ratio of 60% owing to the field enhancement factor of the nanogaps. This narrow nanogap decreases the minimum incident peak field strength to 0.01 MV/cm for field-induced Zener tunneling in GaAs, thereby suppressing impact ionization. Moreover, we suggest the minimum semiconductor thickness required for the nonlinear response as a function of the gap size via simulations. The corresponding results indicate that the large field enhancement in the nanogap decreases the required semiconductor thickness.

II. RESULT AND DISCUSSIONS

Section:

ChooseTop of pageABSTRACTI. INTRODUCTIONII. RESULT AND DISCUSSION... <<III. CONCLUSIONSUPPLEMENTARY MATERIALREFERENCESFigures 1(a)–1(d) show the fabrication process of our 5 nm nanogap sample.2121. X. Chen, H.-R. Park, M. Pelton, X. Piao, N. C. Lindquist, H. Im, Y. J. Kim, J. S. Ahn, K. J. Ahn, N. Park, D.-S. Kim, and S.-H. Oh, “Atomic layer lithography of wafer-scale nanogap arrays for extreme confinement of electromagnetic waves,” Nat. Commun. 4, 2361 (2013). https://doi.org/10.1038/ncomms3361 First, a 100 nm-thick Au film is deposited by electron beam evaporation over a photoresist patterned on a semi-insulating GaAs (SI-GaAs, resistivity ∼108 Ω·cm) substrate. Rectangular holes [dimensions: 10 µm (lx) × 40 µm (ly)] were exposed after the lift-off process. The pitch sizes of the hole arrays were 20 µm (px) and 50 µm (py) [Fig. 1(a)]. Second, a conformal coating of 5 nm alumina was performed by atomic layer deposition to ensure the formation of a dielectric layer on the sidewall of the metal [Fig. 1(b)]. Third, the same thickness of the metal was deposited on the alumina layer [Fig. 1(c)]. Finally, the excessive metal layer on the second floor was exfoliated using an adhesive tape to level the entire sample surface [Fig. 1(d)], and then, we obtained the vertically oriented insulating layer, resulting in a rectangular loop [Fig. 1(e)]. A cross-sectional scanning electron microscopic image of the 5 nm width (w) metal–insulator–metal structure is shown in Fig. 1(f).Through a nanogap, only a polarized wave in the direction perpendicular to ly is transmitted with a giant field enhancement.2222. M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q. H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics 3, 152–156 (2009). https://doi.org/10.1038/nphoton.2009.22 Further, it was proven that the enhancement rapidly decays along the light propagation axis.2323. J. Jeong, H. S. Yun, D. Kim, K. S. Lee, H.-K. Choi, Z. H. Kim, S. W. Lee, and D.-S. Kim, “High contrast detection of water-filled terahertz nanotrenches,” Adv. Opt. Mater. 6, 1800582 (2018). https://doi.org/10.1002/adom.201800582 As a result, nonlinear phenomena effectively occur in the vicinity of the nanogap.2424. H.-T. Chen, W. J. Padilla, J. M. O. Zide, A. C. Gossard, A. J. Taylor, and R. D. Averitt, “Active terahertz metamaterial devices,” Nature 444, 597–600 (2006). https://doi.org/10.1038/nature05343 In this work, the primarily occurring nonlinear phenomenon is field-induced Zener tunneling, which excites an electron, thereby resulting in a transmission decrease. This transmission decrease occurs when an atomically strong field is induced to tilt the band structure, enabling an electron in the valence band to tunnel to the conduction band of a neighboring unit cell.10,25,2610. C. Lange, T. Maag, M. Hohenleutner, S. Baierl, O. Schubert, E. R. J. Edwards, D. Bougeard, G. Woltersdorf, and R. Huber, “Extremely nonperturbative nonlinearities in GaAs driven by atomically strong terahertz fields in gold metamaterials,” Phys. Rev. Lett. 113, 227401 (2014). https://doi.org/10.1103/physrevlett.113.22740125. H. Hirori, K. Shinokita, M. Shirai, S. Tani, Y. Kadoya, and K. Tanaka, “Extraordinary carrier multiplication gated by a picosecond electric field pulse,” Nat. Commun. 2, 594 (2011). https://doi.org/10.1038/ncomms159826. C. Vicario, M. Shalaby, and C. P. Hauri, “Subcycle extreme nonlinearities in GaP induced by an ultrastrong terahertz field,” Phys. Rev. Lett. 118, 083901 (2017). https://doi.org/10.1103/PhysRevLett.118.083901 Our nanogap sample shown in Fig. 1(f) mainly observes the local permittivity near itself,27,2827. G. Choi, Y.-M. Bahk, T. Kang, Y. Lee, B. H. Son, Y. H. Ahn, M. Seo, and D.-S. Kim, “Terahertz nanoprobing of semiconductor surface dynamics,” Nano Lett. 17, 6397–6401 (2017). https://doi.org/10.1021/acs.nanolett.7b0328928. D. Kim, J. Jeong, G. Choi, Y.-M. Bahk, T. Kang, D. Lee, B. Thusa, and D.-S. Kim, “Giant field enhancements in ultrathin nanoslots above 1 terahertz,” ACS Photonics 5, 1885–1890 (2018). https://doi.org/10.1021/acsphotonics.8b00151 resulting in a decrease in the required semiconductor film thickness of interest down to nanometer size, which facilitates thin-film nonlinear applications.

We performed terahertz time-domain spectroscopy on the nanogaps and observed transmissions that depend on the incident field strength. For our terahertz transmission experiment, a single-cycle terahertz pulse was generated by a prism-cut lithium niobate (LiNbO3) crystal via pulse-front-tilted optical rectification. A 1 kHz Ti:sapphire laser beam (central wavelength of 800 nm, pulse energy of 5.3 mJ, and pulse width of 35 fs; Spitfire, SpectraPhysics) was divided into terahertz generation and detection beams. The generated terahertz beam was guided by four off-axis parabolic mirrors. We controlled the incident terahertz field strengths from 3 to 360 kV/cm by using a pair of wire grid polarizers. Along with the 5 nm-gap sample, we also evaluated 2.5 μm- and 400 nm-gap samples fabricated by electron beam lithography. The corresponding results demonstrate the significant effect of the strong field enhancement in the nanogaps on the nonlinear responses of the material.

Figures 2(a) and 2(b) show the normalized transmitted amplitudes in the time domain for the 5 nm-gap and 2.5 μm-gap samples. To obtain the normalized transmitted amplitude, each transmitted amplitude of the sample in the time domain data is divided by the peak value of the transmitted amplitude of bare GaAs (500 µm thick) in the time domain data (supplementary material). For the frequency-domain analysis, we cut a transmitted pulse before the first echo arrived in the time domain. As the incident field strength increases, the 5 nm-gap sample shows drastically reduced normalized transmitted amplitudes in the time domain. By contrast, the 2.5 μm-gap sample shows a small change in the normalized transmitted amplitude over the entire time domain. Further, Figs. 2(c) and 2(d) present the transmission spectra (obtained by Fourier transformation of the time-domain measurements; see supplementary material for more details) of the 5 nm-gap and 2.5 μm-gap samples for incident field strengths ranging from 3.6 kV/cm to 0.36 MV/cm; the transmission values are divided by the peak value at the minimum incident field strength. In the frequency domain, significant nonlinear effects cause differences in the spectra, such as an extinction ratio of ∼60% at 0.4 THz [Fig. 2(c)]. As the nonlinear effect becomes dominant, distinct redshifts are observed, which can be attributed to an increase in the effective index of the material around the gap.2929. J.-Y. Kim, B. J. Kang, J. Park, Y.-M. Bahk, W. T. Kim, J. Rhie, H. Jeon, F. Rotermund, and D.-S. Kim, “Terahertz quantum plasmonics of nanoslot antennas in nonlinear regime,” Nano Lett. 15, 6683–6688 (2015). https://doi.org/10.1021/acs.nanolett.5b02505 For the 5 nm-gap sample, we restrict the incident field amplitude to 0.29 MV/cm because this nanogap sample can be damaged by a highly localized field and current around the gap when the field strength is 0.32 MV/cm.2929. J.-Y. Kim, B. J. Kang, J. Park, Y.-M. Bahk, W. T. Kim, J. Rhie, H. Jeon, F. Rotermund, and D.-S. Kim, “Terahertz quantum plasmonics of nanoslot antennas in nonlinear regime,” Nano Lett. 15, 6683–6688 (2015). https://doi.org/10.1021/acs.nanolett.5b02505 Excited carriers increase the local index of refraction near the gap, and this high refractive index is responsible for the transmission decrease.2323. J. Jeong, H. S. Yun, D. Kim, K. S. Lee, H.-K. Choi, Z. H. Kim, S. W. Lee, and D.-S. Kim, “High contrast detection of water-filled terahertz nanotrenches,” Adv. Opt. Mater. 6, 1800582 (2018). https://doi.org/10.1002/adom.201800582 Materials that exhibit such responses can be applied to nonlinear switching devices with further parameter optimizations to increase the extinction ratio.Figure 3(a) shows the time-integrated transmission of the 5 nm-, 400 nm-, and 2.5 μm-wide-gap samples; the corresponding values are normalized by the transmission T0 at the minimum incident field (see supplementary material for details on the calculations of the transmission). The time-integrated transmission (T) is calculated using the relation: T=∫Esample2tdt∫EbareGaAs2tdt, where Esample and EbareGaAs are the transmitted time traces of the samples and a bare GaAs substrate, respectively.2929. J.-Y. Kim, B. J. Kang, J. Park, Y.-M. Bahk, W. T. Kim, J. Rhie, H. Jeon, F. Rotermund, and D.-S. Kim, “Terahertz quantum plasmonics of nanoslot antennas in nonlinear regime,” Nano Lett. 15, 6683–6688 (2015). https://doi.org/10.1021/acs.nanolett.5b02505 The 5 nm-gap sample shows a large transmission decrease over the full range of the incident field strength, while the samples with wider gaps show a noticeable transmission decrease above a certain incident field strength, i.e., 150 and 275 kV/cm for the 400 nm-gap and 2.5 μm-gap samples, respectively. This result indicates that the strong field enhancement in the nanogap decreases the incident field strength required for the decrease in transmission due to nonlinear effects, and thus, without the field enhancement of the metamaterial, it is difficult to observe a nonlinear effect via terahertz time-domain spectroscopy.3030. Y.-G. Jeong, M. J. Paul, S.-H. Kim, K.-J. Yee, D.-S. Kim, and Y.-S. Lee, “Large enhancement of nonlinear terahertz absorption in intrinsic GaAs by plasmonic nano antennas,” Appl. Phys. Lett. 103, 171109 (2013). https://doi.org/10.1063/1.4826272 In the 5 nm-gap sample, the transmitted field strength decreases to ∼40% of the reference field strength (i.e., in the absence of nonlinear effects) at the incident field amplitude of 300 kV/cm, indicating the largest transmission reduction observed among all samples.To identify the incident electric field strength that induces nonlinear effects in the GaAs substrate, we systematically decreased the incident field strength [Fig. 3(b)]. For the terahertz electric field strength in the range of 3–10 kV/cm, a 2 mm-thick ZnTe crystal was used for the electro-optic sampling, which increased the signal-to-noise ratio. As evident from Fig. 3(b), the terahertz transmission starts to decline at an incident field of only 6 kV/cm, which implies that the threshold incident field for the 5 nm-gap sample is about 40 times lower than that of the 2.5 μm-gap sample (275 kV/cm). This effective gap-induced drop in the transmission is a major breakthrough in research related to nonlinear phenomena induced by intense electric fields (∼100 kV/cm). The presented metal–insular–metal nanogap fabrication strategy enables the use of intermediate field strengths (∼1 kV/cm) to induce nonlinear responses in semiconductors such as SI-GaAs.11,25,3111. K. Fan, H. Y. Hwang, M. Liu, A. C. Strikwerda, A. Sternbach, J. Zhang, X. Zhao, X. Zhang, K. A. Nelson, and R. D. Averitt, “Nonlinear terahertz metamaterials via field-enhanced carrier dynamics in GaAs,” Phys. Rev. Lett. 110, 217404 (2013). https://doi.org/10.1103/physrevlett.110.21740425. H. Hirori, K. Shinokita, M. Shirai, S. Tani, Y. Kadoya, and K. Tanaka, “Extraordinary carrier multiplication gated by a picosecond electric field pulse,” Nat. Commun. 2, 594 (2011). https://doi.org/10.1038/ncomms159831. J. R. Danielson, Y.-S. Lee, J. P. Prineas, J. T. Steiner, M. Kira, and S. W. Koch, “Interaction of strong single-cycle terahertz pulses with semiconductor quantum wells,” Phys. Rev. Lett. 99, 237401 (2007). https://doi.org/10.1103/physrevlett.99.237401 Although in our experiment, no decrease in the transmission was observed below 3 kV/cm (not shown), other measurement methods could be used to observe any nonlinear change. For instance, a terahertz-pump/optical-probe may reveal nonlinear responses due to the Franz–Keldysh effect.3232. F. Novelli, D. Fausti, F. Giusti, F. Parmigiani, and M. Hoffmann, “Mixed regime of light-matter interaction revealed by phase sensitive measurements of the dynamical Franz–Keldysh effect,” Sci. Rep. 3, 1227 (2013). https://doi.org/10.1038/srep01227For nonlinear response, the voltage applied across the gap for terahertz transmission is important. Figure 4 shows the time-integrated transmission (T/T0) as a function of the voltage applied across the gaps for various antenna gap samples (see supplementary material). The voltage across the gap is estimated from the transient terahertz field strength measured at the output of the gap (more details can be found in the supplementary material). Notably, we found that a voltage of 23 V can result in a 60% transmission decrease, leading to Zener tunneling , which occurs when the applied electric field becomes sufficiently strong to tilt the band structure and allow a valence electron to tunnel through the potential barrier.1414. E. O. Kane, “Zener tunneling in semiconductors,” J. Phys. Chem. Solids 12, 181–188 (1960). https://doi.org/10.1016/0022-3697(60)90035-4 These results demonstrate that several hundreds of voltages are not required to control the transmission if a nanometer-sized gap is used. For an induced gap field that is responsible for the nonlinear effects, decreasing the gap width enables us to lower the voltage across the gap according to the equation: E = V/w, where V is the applied voltage and w is the gap width.More importantly, the underlying mechanism is different from that observed in previous studies on nonlinear metamaterials, in which impact ionization can dominate over Zener tunneling when a strong electric field is applied.10,1110. C. Lange, T. Maag, M. Hohenleutner, S. Baierl, O. Schubert, E. R. J. Edwards, D. Bougeard, G. Woltersdorf, and R. Huber, “Extremely nonperturbative nonlinearities in GaAs driven by atomically strong terahertz fields in gold metamaterials,” Phys. Rev. Lett. 113, 227401 (2014). https://doi.org/10.1103/physrevlett.113.22740111. K. Fan, H. Y. Hwang, M. Liu, A. C. Strikwerda, A. Sternbach, J. Zhang, X. Zhao, X. Zhang, K. A. Nelson, and R. D. Averitt, “Nonlinear terahertz metamaterials via field-enhanced carrier dynamics in GaAs,” Phys. Rev. Lett. 110, 217404 (2013). https://doi.org/10.1103/physrevlett.110.217404 In our case, impact ionization is suppressed owing to the reduction in the active semiconductor area for carrier generation as the gap width decreases. On the contrary, Zener tunneling still occurs because of the enhanced terahertz electric field strength. To analyze the nonlinear effect in our case, the induced gap voltages corresponding to a 10% decrease in the terahertz transmission (from Fig. 4) of the antenna samples are compared. For a 10% transmission decrease, the gap voltages are ∼7.5, 200, and 360 V for the 5 nm-, 400 nm-, and 2.5 μm-gap samples, respectively. When the voltage across the 5 nm-gap is Vgap = 7.5 V, the electric field strength inside the gap is Egap = 1.5 V/nm, which is sufficiently strong to induce Zener tunneling. Because the threshold energy for impact ionization in GaAs is 1.6 eV,1010. C. Lange, T. Maag, M. Hohenleutner, S. Baierl, O. Schubert, E. R. J. Edwards, D. Bougeard, G. Woltersdorf, and R. Huber, “Extremely nonperturbative nonlinearities in GaAs driven by atomically strong terahertz fields in gold metamaterials,” Phys. Rev. Lett. 113, 227401 (2014). https://doi.org/10.1103/physrevlett.113.227401 Vgap = 7.5 V results in less than five ionization steps for a single electron, culminating in a carrier multiplication factor of 25 ≈ 30. In contrast to the 5 nm-gap sample, when the voltage across the 400 nm-gap is near the highest value of Vgap = 200 V, the electric field strength inside the gap becomes Egap = 0.5 V/nm. This low field strength induces an extremely weak Zener tunneling but safely allows at least ten impact ionization steps, which result in a carrier multiplication factor of 1000 according to the previously reported results.10,2510. C. Lange, T. Maag, M. Hohenleutner, S. Baierl, O. Schubert, E. R. J. Edwards, D. Bougeard, G. Woltersdorf, and R. Huber, “Extremely nonperturbative nonlinearities in GaAs driven by atomically strong terahertz fields in gold metamaterials,” Phys. Rev. Lett. 113, 227401 (2014). https://doi.org/10.1103/physrevlett.113.22740125. H. Hirori, K. Shinokita, M. Shirai, S. Tani, Y. Kadoya, and K. Tanaka, “Extraordinary carrier multiplication gated by a picosecond electric field pulse,” Nat. Commun. 2, 594 (2011). https://doi.org/10.1038/ncomms1598 Because of such a large carrier multiplication factor, a large number of carriers are expected to be generated, which can be limited when the number of generated carriers reaches the Pauli blocking regime. Therefore, we can assume that compared with the 2.5 μm-gap and 400 nm-gap samples, the 5 nm-gap sample enables a significant decrease in the transmission with a low gap voltage and a small number of impact ionization steps.33–3533. Y. Okuto and C. R. Crowell, “Energy-conservation considerations in the characterization of impact ionization in semiconductors,” Phys. Rev. B 6, 3076–3081 (1972). https://doi.org/10.1103/physrevb.6.307634. Y. Okuto and C. R. Crowell, “Ionization coefficients in semiconductors: A nonlocalized property,” Phys. Rev. B 10, 4284–4296 (1974). https://doi.org/10.1103/physrevb.10.428435. A. Spinelli, A. Pacelli, and A. L. Lacaita, “Dead space approximation for impact ionization in silicon,” Appl. Phys. Lett. 69, 3707–3709 (1996). https://doi.org/10.1063/1.117196 Although impact ionization is suppressed in the 5 nm gap, the extinction ratio of 60% shown in Fig. 4 originates from the large number of carriers generated by Zener tunneling and impact ionization. We note that the influence of the intervalley scattering or nonparabolicity effect on the transmission drop is insignificant due to the low initial carrier concentration (∼107 cm−3) in SI-GaAs.3636. F. Blanchard, D. Golde, F. H. Su, L. Razzari, G. Sharma, R. Morandotti, T. Ozaki, M. Reid, M. Kira, S. W. Koch, and F. A. Hegmann, “Effective mass anisotropy of hot electrons in nonparabolic conduction bands of n-doped InGaAs films using ultrafast terahertz pump-probe techniques,” Phys. Rev. Lett. 107, 107401 (2011). https://doi.org/10.1103/physrevlett.107.107401 However, an intense terahertz pulse generates free carriers in GaAs as well as enables metal–insulator–metal tunneling,3737. J. G. Simmons, “Generalized formula for the electric tunnel effect between similar electrodes separated by a thin insulating film,” J. Appl. Phys. 34, 1793–1803 (1963). https://doi.org/10.1063/1.1702682 such as Fowler–Nordheim tunneling, in the nanogaps.2929. J.-Y. Kim, B. J. Kang, J. Park, Y.-M. Bahk, W. T. Kim, J. Rhie, H. Jeon, F. Rotermund, and D.-S. Kim, “Terahertz quantum plasmonics of nanoslot antennas in nonlinear regime,” Nano Lett. 15, 6683–6688 (2015). https://doi.org/10.1021/acs.nanolett.5b02505 Nevertheless, the transmission decrease is not governed by the metal–insulator–metal tunneling because it begins above 2.5 V/nm, as demonstrated by the extinction shown by the 5 nm-gap on a quartz substrate (see supplementary material).Finally, we analyze the interacting region, where the nonlinear phenomena occur, via simulations of the electric field amplitude distributions. Without the metamaterial-induced nonlinearity of the semiconductor substrate, the confined field near the gap is proportional to the gap size of the metamaterials.3838. G. Choi, T. Kang, M. Seo, D.-S. Kim, and Y.-M. Bahk, “Enhanced surface carrier response by field overlapping in metal nanopatterned semiconductor,” ACS Photonics 5, 4739–4744 (2018). https://doi.org/10.1021/acsphotonics.8b00724 Figure 5(a) presents the numerically simulated (COMSOL Multiphysics) field distributions of the terahertz waves for the 2.5 μm-gap and 5 nm-gap samples. The corresponding electric field amplitudes, along the z-axis from the gap surface, for various incident field strengths are plotted in Fig. 5(b). The gray dashed line indicates a field strength of 0.1 V/nm with which tunneling rate in the SI-GaAs is above the order of 1010 (cm−3 100 fs−1) for Zener tunneling. This value surpasses the initial carrier density of the SI-GaAs substrate.1010. C. Lange, T. Maag, M. Hohenleutner, S. Baierl, O. Schubert, E. R. J. Edwards, D. Bougeard, G. Woltersdorf, and R. Huber, “Extremely nonperturbative nonlinearities in GaAs driven by atomically strong terahertz fields in gold metamaterials,” Phys. Rev. Lett. 113, 227401 (2014). https://doi.org/10.1103/physrevlett.113.227401 While a field strength below 0.1 V/nm can still induce carrier generation by Zener tunneling, the terahertz transmission shows negligible changes because of the low generation rate. We note that, based on Fig. 3(b), the minimum incident electric field strength required for decreasing the transmission is around 6 kV/cm, and the corresponding gap voltage is 0.08–0.17 V/nm. In the case of the 5 nm-gap, even the incident electric field strength of 6 kV/cm crosses the line indicating the limit. This field strength corresponds to a small change in the transmission spectrum shown in Fig. 3(b). The region where Zenner tunneling occurs is within 2 nm from the gap surface. As the input field strength increases, the interacting volume increases linearly (the electric field strength decays as the distance from the gap surface increases).3939. J. Jeong, D.-S. Kim, and H.-R. Park, “Beyond-hot-spot absorption enhancement on top of terahertz nanotrenches,” Nanophotonics 11, 3159–3167 (2022). https://doi.org/10.1515/nanoph-2022-0214 In the case of the 2.5 μm-gap sample, the field barely crosses the gray line and corresponds to a negligible change in the transmission spectra.