1. Introduction

Efficient electrically-driven solid-state terahertz emitters/oscillators are one of the paramount devices for terahertz applications [1–3]. However, electronic devices become less efficient in generating electromagnetic wave when the frequency approaches 1 THz and above especially at room temperature. As collective charge oscillations in a high-electron-mobility two-dimensional electron gas (2DEG) in the terahertz frequency range, plasmons in semiconductor heterostructures such as AlGaAs/GaAs, AlGaN/GaN, and two-dimensional materials such as graphene, have long been pursued for solid-state terahertz emitters [4–10].

An electrically-driven terahertz plasmon emitter embodies two core processes, namely, the electrical excitation of plasmons and the terahertz emission from plasmons. Unfortunately, both processes are strongly limited by the short lifetime of solid-state plasma wave [11–15]. Inefficient excitation of solid-state plasma wave by electrical current has long been the main hurdle [11–16]. To ease the excitation of plasmon wave by electrical current, plasmon instability mechanisms have been proposed, but was demonstrated in few implementations, such as Dyakonov-Shur instability based on a short ballistic 2DEG channel with specific boundaries [17–19], and streaming plasmon instability driven by direct current (DC) in a long channel with grating couplers [20–23]. However, fast drifting electrons in the channel may result in longitudinal optical (LO) phonons and plasma heating, thus preventing coherent terahertz radiation [24–27]. From the point of view of enhancing the lifetime of the plasmon states, the following two approaches could be tried. First, the boundaries of the plasmon cavity can be properly constructed so that the longitudinal plasma wave sees a highly reflective boundary. Theoretical studies have revealed that plasmon amplification and terahertz oscillations can be achieved in distributed plasmonic resonators [28,29]. Second, strong coupling between plasmon modes and terahertz waves allows for the formation of plasmon polariton states [30–34]. Benefiting from the strong confinement of the plasmon modes in the plasmon cavity, the mixed state (plasmon polariton) may exchange energy with the plasmon mode by absorbing and releasing cavity photons at Rabi frequency Ω_R, which is a key parameter to define the coupling strength between these two systems. In [35], Zhang et al. pointed out the decay rate of the mixed states can be supressed in a strong-coupled plasmon-polariton system during the reversible energy exchange process. When Ω_R is large enough and of the same order of magnitude as the resonance frequency ω₀, a huge number of plasmons are synchronized and coupled to photons collectively, which is known as ultrastrong coupling regime [36, 37]. In this regime, a dense collection of coherent plasmon polariton may decay into terahertz photons and the emission rate grows greatly due to superradiance effect [38, 39]. Thus, such plasmonic device exploiting fast decaying rate of collective plasmon polaritons may become an efficient terahertz emitter.

In this letter, we report on the formation and observation of tunable terahertz plasmons (polaritons) in a grating-gate coupled 2DEG. Plasmon polariton states are formed by coupling the plasmon states in the grating-coupled 2DEG and the terahertz Fabry-Pérot cavity modes. Both plasmon states and plasmon polariton states were probed by reflection spectroscopy with a background illumination from the ambient blackbody radiation. Device temperature was elevated to check the equilibrium plasmon excitation and its relationship with the electron mobility. Simulation results including the reflection spectra and the total reflection power show good agreement with the experimental results. Based on the consistency between theory and experiment, the detected signals are validated to come from the reflected blackbody radiation, rather than terahertz emission induced by electron tunnelling from the gates into the 2DEG as we once interpreted [40]. We find the Rabi splitting is inversely proportional to the resonance frequency and follows a linear relation with the square root of sheet electron density. These findings serve crucial guidance for future device design to achieve ultrastrong coupling between plasmons and cavity photons. The reflection power for two types of plasmonic devices are measured at temperature from 7 K up to 190 K. The Rabi splitting can be observed at a temperature up to 130 K, yet this temperature can be further elevated if a higher-mobility material is exploited to fabricate the grating-coupled plasmonic device.

2. Device information and device physics

Two types of devices, namely the cavity device and the lens device, are constructed based on the same core plasmonic device but with distinctly different terahertz electromagnetic environment. The core plasmonic device is a grating-gate coupled 2DEG based on an AlGaN/GaN heterostructure grown on sapphire substrate, as shown in Fig. 1(a). The ungated 2DEG has a sheet electron density of n₀ ≈ 1.2 × 10¹³ cm⁻² and an electron mobility of μ ≈ 1.5 × 10⁴ cm²/Vs at 7 K. The grating gates biased at a certain negative voltage V_G tune continuously the sheet electron density n_s and the pinch-off voltage at which the electrons under a gate are fully depleted is V_T = −4.2 V. The grating has a pitch distance of L = 4.0 μm (in direction x), a width of 4 mm (in direction y) and a thickness of 200 nm (in direction z). Each grating gate has a length of W = 2.8 μm resulting in ungated 2DEG strips with a length about L − W = 1.2 μm. The grating gate above the 2DEG channel offers two functions: (1), formation of plasmon cavities in 2DEG under each gate; (2), realization of the coupling between terahertz electromagnetic wave and plasmons. The cavity device is made by mounting the device on a gold-plated chip carrier which acts as a gold mirror on the backside of the sapphire substrate, as schematically shown in Figs. 1(a)–1(b). Serving as the main body of a terahertz Fabry-Pérot (F-P) cavity, the sapphire substrate is thinned to a thickness of D = 212 μm to reduce the number of terahertz modes at frequency below 3.0 THz. As a counterpart to the cavity device, the lens device is realized by mounting the same plasmonic device on the flat surface of a hyperspherical silicon lens, as schematically shown in Fig. 1(c).

 

Fig. 1 (a) Schematic of the core terahertz plasmonic device and the simulated terahertz field distribution of the k = 6th cavity mode. Schematics of (b) the cavity device and (c) the lens device. (d) Schematic setup for measuring the modulated reflection signal and the reflection spectra. The incident terahertz power on the device at 7 K comes from the ambient blackbody radiation.

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The grating gates form about 1000 individual plasmon cavities in the 2DEG with the well-known dispersion relation [5]

3. Experimental implementations

The setup shown in Fig. 1(d) was constructed to probe the possible terahertz emission power from the grating-coupled 2DEG by electrical excitation. Here in this work, we find it is suitable for probing the ambient blackbody terahertz radiation reflected by the devices at 7 K. The flat mirror in front of the TPX window of the cryogenic chamber is adjusted to guide the reflected radiation into a silicon bolometer at 4.2 K. When a Fourier-transform spectrometer (FTS) equipped with the silicon bolometer is used to measure the reflection spectra, this mirror is carefully adjusted to guide the reflected radiation into the FTS to maximize the interference. A lock-in amplifier is used to detect the output signal of the silicon bolometer and the devices are modulated by a square-wave gate voltage with its high level fixed at 0 V for half of the period while the low level V_G is tuned for the other half of the period, as schematically shown in Figs. 1(b)–1(d). Driven by such a square-wave gate voltage, the lock-in signal amplitude represents the absolute difference of the reflected terahertz power between the gate voltage set at 0 V and V_G. Hence, the measured spectra represents the absolute difference of the spectral density between the two gate voltages.

4. Results and discussions

The reflection spectra I(f) of the two devices at different gate voltages are measured, as shown in Figs. 2(a)–2(b). The reflection spectra at V_G satisfies such a relation (see Appendix A.1): I(f) ∝ |R_H(f) − R_L(f)| B(f), where B(f) = 2f²k_BT/c² refers to the radiance of the ambient blackbody which follows a Rayleigh-Jeans law, R_H(f) and R_L(f) correspond to the reflectance at gate voltage of high level 0 V and low level V_G, respectively.

 

Fig. 2 The measured reflection spectra of (a) the lens device and (b) the cavity device, are compared with the simulated reflection spectra of (c) the lens device and (d) the cavity device at different gate voltages. The parasitic modes (p2, p3, p4, p5) represent the plasmon modes from reflection spectra at V_G = 0 V. The parabolic dashed curves are plasmon modes (j = 1, 2, · · · , 9) tuned by the gate voltage. The horizontal dashed lines are F-P cavity modes (k = 1, 2, · · · , 11) and the red solid curves are calculated plasmon-polariton modes.

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For the lens device, each discrete plasmon mode with mode number j = 2 – 9 agrees well with the calculated plasmon modes (dashed curves) based on Eq. (1). It’s noted that besides the plasmon modes, some parasitic modes (p2, p3, p4, p5) also appear. These modes are originated from the plasmon modes at V_G = 0 V, which is introduced by our differentiating measurement method. For the cavity device, the reflection spectra are remarkably different: modulated reflection becomes visible when each plasmon mode becomes resonant with one of the terahertz cavity modes as described by Eq. (2). In Fig. 2(b), plasmon modes and F-P cavity modes are marked by the parabolic dashed curves and the horizontal dashed lines, respectively.

Mode splitting resulted from strong coupling between the plasmon modes and the terahertz F-P cavity modes occurs at some of the resonances in Fig. 2(b). Hybridization of the terahertz cavity mode and the plasmon mode can be modelled as two-coupled oscillators and the resulting LPP (${\omega }_{kj}^{-}$) and UPP (${\omega }_{kj}^{+}$) modes can be expressed as [42].

A zoom-in view of the plasmon-polariton spectra formed by the k = 6th cavity mode and the j = 3, 4, 5, 6th plasmon modes are given in Fig. 3(a) with calculated plasmon-polariton modes (${\omega }_{kj}^{-}$,${\omega }_{kj}^{+}$) overlaid. We determined the linewidth of plasmon modes to be γ_P/2π = 117 GHz by fitting a Lorentzian lineshape to reflection spectra of the lens device. The linewidths of the 6th cavity mode γ_C6/2π = 44.1 GHz are also obtained via simulation of a bare cavity device without considering the 2DEG. The quality factors of both plasmons and cavity modes defined by Q_P = ω/γ_P and Q_Ck = ω_Ck/γ_Ck, are displayed in Fig. 3(b). We observed a Rabi splitting of 67.9 GHz between the 6th cavity mode and the 3rd plasmon mode, which is extracted at V_G = −0.85 V, see Fig. 3(c). The linewidth of the two doublet splits (45 GHz), fitted by two Lorentzian peaks, is limited by the cavity mode linewidth. The Rabi splitting Ω_R/2π, extracted from different crossover of cavity modes and plasmon modes, are plotted as bubbles in Fig. 3(d). The splitting values are proportional to the diameters of the bubbles. A normalized coupling ratio $\frac{{\mathrm{\Omega }}_{\text{R}}}{{\omega }_0}\approx 0.13$ is obtained at the crossover between the 2nd plasmon mode and the 4th cavity mode. The Rabi splitting rises as the electron density increases by tuning the gate voltage from −4 V to 0 V, validating the collective behaviour of the light-matter interaction [35,36,43,44]. Moreover, we find the Rabi splittings Ω_R,k=m/2π at m = 4, 5, 6, 7th cavity modes follow a linear relation with the square root of sheet electron density (n_s, in cm⁻²), see Fig. 3(e). The Rabi splitting at the k = 6th cavity mode satisfies such a relation as ${\mathrm{\Omega }}_{\text{R},k=6}(n_s)/2\pi =9.55\times {10}^{-6}\sqrt{n_s}+38.31$ in GHz, by fitting linearly using least square method. The linear function at other cavity mode (k = 4, 5, 7) follows ${\mathrm{\Omega }}_{\text{R},k=m}(n_s)=\frac{{\omega }_{\text{C}6}}{{\omega }_{\text{C}m}}\times {\mathrm{\Omega }}_{\text{R},k=6}(n_s)$ in GHz, where m = 4, 5, 7. We observe the measured Rabi splitting values are in good agreement with the fitted linear lines. It’s indicated that the Rabi splitting is inversely proportional to the resonance frequency. Meanwhile, we notice that the cavity device exhibits a considerable Rabi splitting with a small electron density near the pinch-off voltage, see the 6th cavity mode in Figs. 3(a) and 3(d). It may be attributed to some electrons of the ungated zone near the fringe of grating participate in the gated 2DEG oscillations.

 

Fig. 3 (a) Zoom-in view of the plasmon-polariton modes formed by the k = 6th cavity mode and the j = 3, 4, 5, 6th plasmon modes. The solid curves are the corresponding $\text{LPP}\left({\omega }_{kj}^{-}\right)/\text{UPP}\left({\omega }_{kj}^{+}\right)$ modes. (b) Quality factors of the discrete cavity modes and the continuous plasmon modes at different frequencies. (c) Mode splitting at strong coupling regime with the k = 6th cavity mode and the j = 3rd plasmon mode at V_G = −0.85 V. (d) Extracted Rabi splitting Ω_R/2π plotted as bubbles, with its diameter proportional to the splitting, and the splitting values (in GHz) are labelled below the bubbles. (e) Rabi splitting Ω_R,k=m/2π, m = 4, 5, 6, 7 are plotted as linear functions of the square root of sheet electron densities. The Rabi splitting at the k = 6th cavity mode can be expressed as ${\mathrm{\Omega }}_{\text{R},k=6}(n_s)/2\pi =9.55\times {10}^{-6}\sqrt{n_s}+38.31$ in GHz, fitted by the least square methods. The other linear functions at k = 4, 5, 7th modes are calculated by Ω_R,k=m(n_s) = ω_C6ω_Cm × Ω_R,k=6(n_s), m = 4, 5, 7.

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We measured the temperature dependence of reflection spectra of both devices at fixed gate voltage (for the lens device, V_G = −2.0 V, for the cavity device, V_G = −3.15 V), see Figs. 4(a)–4(b). It’s shown that the linewidth of reflection peaks becomes broader as temperature rises, which corresponds to a faster plasmon decay rate at high temperatures. The plasmon polariton splittings are clearly observed in the cavity device from 6 K up to 130 K. By changing the gate voltage V_G, we measured the the total reflection power at different temperatures, as shown in Figs. 4(c)–4(d). The total reflection power P(V_G), is proportional to $\left|{\int }_0^{\infty }\left[R_{\text{H}}(f)-R_{\text{L}}(f)\right]B(f)\text{d}f\right|$ derived from a lock-in technique. The reflected power for the lens device is rather smooth and varied monotonically shown in Fig. 4(c), which infers a constant number of active plasmon modes coupled to the broadband terahertz radiation. However, the reflection power curve exhibits many oscillations for the cavity device, see Fig. 4(d). For comparison, we plot the simulated total power for both devices in Figs. 4(e)–4(f), which show good accordance with the measured results. A temperature dependence of total reflection power P for both devices is normalized and displayed in Fig. 4(g). The reflected terahertz power reduces monotonically as the device temperature rises, and vanishes at 180 K. By measuring the source-to-drain channel conductance at different temperatures, we obtained the electron mobility (assuming the sheet electron gas density is constant at 1.2 × 10¹³ cm⁻²) which is also displayed in Fig. 4(g). From the similar decreasing curve of the temperature dependence of electron mobility, we indicate the electron scattering rate affects both the plasmon decay rate and the total reflection power. To achieve ultrastrong coupling interaction at an elevated temperature, we may exploit GaAs- or InGaAs-based high-electron-mobility heterostructures to fabricate the grating-coupled plasmonic devices.

 

Fig. 4 The reflection spectrum of (a) lens device at V_G = −2.0 V and (b) cavity device at V_G = −3.15 V at different temperatures. The water absorption peaks are marked with gray triangular symbols. The total reflection power of the (c) lens device and (d) cavity device varied with gate voltages are measured at different temperatures, to compare with the simulated total reflection power of (e) lens device and (f) cavity device. (g) The normalized total reflection power of both devices (symbols) under various gate voltages and the lifetime of channel electrons (solid line) at different temperatures.

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The grating-coupled plasmonic device can be utilized to study the interaction between light and plasmons in the ultrastrong coupling regime. Due to a weak spatial overlap between the cavity modes and the two-dimensional plasmons [45], the coupling strength is small compared with the decay rate of plasmon modes and cavity photons, which leads to a relatively low normalized coupling ratio. To further increase the ratio $\frac{{\mathrm{\Omega }}_{\text{R}}}{{\omega }_0}$, three approaches could be considered. Firstly, design a lower resonance frequency ω₀. Since the Rabi splitting Ω_R is proportional to $\frac{1}{{\omega }_0}$ as discussed above, the ratio satisfies $\frac{{\mathrm{\Omega }}_{\text{R}}}{{\omega }_0}\propto \frac{1}{{\omega }_0^2}$, thus we can achieve a larger Rabi splitting at lower frequency. Secondly, we can make a plasmon mode interacts with a target cavity mode at higher electron density. As we know, the Rabi splitting increases monotonously with the square root of sheet electron density. Thus we can make the crossover falls into a higher electron density region through tailoring the plasmon dispersion relation, such as by adjusting the grating period and the slit width. Finally, a shorter cavity length is beneficial to achieve a giant electric field confinement within the cavity device [37], thus increasing the coupling strength between terahertz photons and plasmons. In addition, the larger frequency spacing between discrete cavity modes owing to a shorter cavity length, will prevent the splitting affected by adjoining modes.

5. Conclusion

In summary, we have observed the plasmon modes and plasmon-polariton modes formed in a lens device and a cavity device by using reflection spectroscopy, respectively. Mode splitting appears only in the cavity device, where plasmon modes coupled with discrete cavity modes, rather than continuous photonic modes in the lens device. We achieve a normalized coupling ratio as high as $\frac{{\mathrm{\Omega }}_{\text{R}}}{{\omega }_0}\approx 0.13$ in the strong coupling regime. The extracted Rabi splitting satisfies a linear relation with the square root of sheet electron density, and is inversely proportional to the resonance frequency. We propose that shifting the splitting crossover to a higher electron density at lower frequency in a shorter-cavity-length device allows for increasing the Rabi splitting. These approaches may provide a platform for exploring the ultrastrong coupled light-matter interaction.

Appendix

A.1. Modulated reflection spectra measured by lock-in technique

Suppose a monochromatic light of wavenumber ν = 1/λ = f/c = ω/2πc is incident into a Michelson interferometer, the intensity of the beam at the detector varied as retardation is

(4)$$I(x)=2\mathit{RT}{I}_0(\nu )\left[1+\text{cos}(2\pi \nu x)\right]$$

where R and T are the reflectance and transmittance of the beamsplitter from the incident beam.

Consider an ideal beamsplitter (R = T = 0.5), the equation becomes

(5)$$I(x)=\frac{1}{2}{I}_0(\nu )\left[1+\text{cos}(2\pi \nu x)\right]$$

note only the modulated part 1/2I₀(ν) cos(2πνx) gives meaningful information in the Fourier transformation.

If such a monochromatic light is modulated by a chopper at a modulation frequency of f_M, then the signal output to the detector at a retardation of x₀ is

(6)$$I(x_0,t)=\frac{1}{2}{I}_0(\nu )\left[1+\text{cos}(2\pi \nu x_0)\right]H(2\pi f_{\text{M}}t)$$

where H(t) is the periodic Heaviside function, which can be expanded as a sum of Fourier serials:

(7)$$H(2\pi f_{\text{M}}t)=\frac{1}{2}+\frac{2}{\pi }\left[\text{sin}(2\pi f_{\text{M}}t)+\frac{1}{3}\text{sin}(2\pi \cdot 3f_{\text{M}}t)+\dots \right]$$

Thus the signal read by the lock-in amplifier at the retardation position of x is

(8)$$I(x)=\frac{1}{\pi }{I}_0(\nu )\left[1+\text{cos}(2\pi \nu x)\right]$$

For a broadband source B(ν), the measured interferogram is the result of the sum of the cosines contributions corresponding to each wavenumber. In this situation the measured interferogram is

(9)$$I(x)={\int }_0^{\infty }\frac{1}{2}B(\nu )\left[1+\text{cos}(2\pi \nu x)\right]\text{d}\nu$$

In our experimental configuration, the silicon bolometer together with a lock-in amplifier are used to detect the reflection terahertz signal. This modulation method is different from the conventional method by using an electromechanical chopper where the terahertz power from the plasmon device is switched on and off at the modulation speed. Thus the terahertz amplitude signal detected by the bolometer at retardation x is in fact the difference between the high and low level of the two gate-voltage states, which can be expressed as follows:

(10)$$\begin{array}{ccc}I(x)& =& H(2\pi f_{\text{M}}t)\cdot {\int }_0^{\infty }\frac{1}{2}{R}_{\text{H}}B(\nu )\left[1+\text{cos}(2\pi \nu x)\right]\text{d}\nu \\ & & +H(2\pi f_{\text{M}}t+\pi )\cdot {\int }_0^{\infty }\frac{1}{2}{R}_{\text{L}}B(\nu )\left[1+\text{cos}(2\pi \nu x)\right]\text{d}\nu \end{array}$$

where B(ν) is the ambient spectral density of the energy flux of blackbody radiation, R_H(ν) and R_L(ν) are the reflection spectra of the plasmon device under high level V_H = 0 V and low level gate voltage V_L, respectively.

The signal read by the lock-in amplifier at retardation x is

(11)$$\begin{array}{ccc}I(x)& =& \frac{1}{\pi }\left|{\int }_0^{\infty }{R}_{\text{H}}(\nu )B(\nu )\left[1+\text{cos}(2\pi \nu x)\right]\text{d}\nu -{\int }_0^{\infty }{R}_{\text{L}}(\nu )B(\nu )\left[1+\text{cos}(2\pi \nu x)\right]\text{d}\nu \right|\\ & =& \frac{1}{\pi }\left|{\int }_0^{\infty }\left[R_{\text{H}}(\nu )-R_{\text{L}}(\nu )\right]B(\nu )\left[1+\text{cos}(2\pi \nu x)\right]\text{d}\nu \right|\end{array}$$

The total reflection power measured in our experiment setup is

(12)$$\begin{array}{ccc}I(x=\infty )& =& \frac{1}{2}I(x=0)\\ & =& \frac{1}{\pi }\left|{\int }_0^{\infty }\left[R_{\text{H}}(\nu )-R_{\text{L}}(\nu )\right]B(\nu )\text{d}\nu \right|\end{array}$$

The terahertz power spectral density incident on the Si bolometer satisfies such a relation:

(13)$$I(\nu )=\left|R_{\text{H}}(\nu )-R_{\text{L}}(\nu )\right|B(\nu )$$

A.2. Simulation method and reflection spectra

To further understand how the two-dimensional plasmons of our plasmonic device interact with the ambient blackbody radiation, we use a commercial finite element method electromagnetic software to simulate the reflection spectra of both devices. Due to the one-dimensional metal grating has a periodicity of L = 4 μm, we choose a unitcell with a square of 4 μm × 4 μm, and a thickness of 212 μm. The incident light radiates along the negative z axis, and reflected back, see Fig 1(a). For the cavity device, the AlGaN/GaN heterostructure is on a sapphire substrate with finite thickness which is plated with gold layer on its backside. As for the lens device, we consider the sapphire substrate as semi-infinite. The boundary surfaces x_min, x_max, y_min, y_max of the model have a periodical boundary condition. The boundary surface z_min, z_max of the lens device are defined as absorption boundaries, while the cavity device are defined as electrical wall and absorption boundary, respectively. The electrons gas, 25 nm under the grating gate, are divided into gated 2DEG and ungated 2DEG. The electron density is tuned by the gate voltage for the gated 2DEG, while remains unchanged for the ungated 2DEG. Both 2DEG follow a Drude model permittivity as $\epsilon (\omega )={\epsilon }_{\infty }-{\omega }_{\text{p}}^2/({\omega }^2+i\omega {\gamma }_0)$, where ε_∞ = 9.2 is the surrounding medium permittivity, ω_p is the bulk plasmon frequency and γ₀ = 0.45 THz is a phenomenological parameter of plasmon decay rate. The bulk plasmon frequency is defined as ${\omega }_{\text{p}}=\sqrt{n_e{e}^2/{\epsilon }_0\epsilon m^{*}}$, where n_e is the bulk electron density. The pinch off voltage of the device is V_T = −4.2 V, and the sheet electron density under gate voltage V_G before fully depleted follows such a relation: n_s = 2.4 × 10¹³/|V_T| × (V_G − V_T) × [1 − 0.32(V_G − V_T)^0.32] cm⁻². We obtain the reflection spectra shown in the graphs below, Figs. 5(a)–5(b) are the measured modulated reflection spectra for the two devices, and Figs. 5(c)–5(d) are direct reflection spectra by simulation.

 

Fig. 5 The measured reflection spectra of (a) the lens device and (b) the cavity device, in comparison with the simulated reflection spectra for (c) the lens device and (d) the cavity device.

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Funding

National Key Research and Development Program of China (2016YFC0801203); National Basic Research Program of China (G2009CB929303); National Natural Science Foundation of China (61775231, 61771466, 61505242, 61611530708, 11403084); Russian Foundation for Basic Research (17-52-53063).

Acknowledgments

The authors acknowledge support from the Nanofabrication Facility at Suzhou Institute of Nano-tech and Nano-bionics (SINANO). H.Q. thanks J. P. Kotthaus, R. A. Lewis and H. Yang for insightful discussions.

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