1. Introduction

Plasma oscillations in two-dimensional electron channels (2DECs) can be used for the detection, frequency multiplication, and generation of terahertz (THz) radiation [1–8] and THz imaging [9–11]. A metal grating of a relatively large area (of the order of a square millimeter) placed in close proximity to the 2DEC plane is an efficient coupler between plasmons in 2DEC and THz radiation [12–18]. It was shown [19] that the coupling between plasmons in 2DEC and THz radiation increases tremendously if the grating coupler has narrow slits between metal fingers. Shaner et al. [20,21] reported on a split-grating-gate THz detector using a periodic grating coupler split by a negatively biased isolated gate finger forming the potential barrier for electrons in 2DEC. Terahertz detection observed in Refs. 20 and 21 was attributed to heating of the electron gas in the 2DEC by the absorbed THz radiation and the pertinent variation of the thermionic current across the barrier under the barrier gate. Two different mechanisms might be responsible for the THz detection in such a detector: the one associated with the electron heating and the other associated with the dynamic variation of the barrier potential by the excited plasma oscillations and the related rectification effect [22]. As found in Ref. 22, the electron heating (bolometric) mechanism dominates at low temperatures (T₀ < 30-40 K), while the dynamic mechanism prevails at higher temperatures. Recently, an improved-performance design of THz plasmonic detector with a split-grating-gate fabricated on a suspended thin membrane-like substrate (of 4-µm thickness) [23] resulted in a 50-fold enhancement in responsivity attributed to a rather slow process of the membrane-substrate lattice heating enhanced due to the detector thermal isolation.

In this paper, we show that the responsivity of the grating-gate plasmonic THz detector can be considerably enhanced due to increased electromagnetic coupling between plasmons and THz radiation in the grating-gate structure on a membrane substrate. The strength of electromagnetic coupling between plasmons in 2DEC and THz radiation depends on both the geometry of the grating-gate coupler and the substrate thickness. We show that the strength of the higher-order plasmon resonances in the grating-gate structure on a membrane substrate is at least four times higher than that for a conventional grating-gate plasmonic detector on a bulk substrate for typical parameters of the grating-gate structure. The coupling strength can be further enhanced by an order of magnitude with a proper design of the grating-gate coupler.

2. Model of the resonant surface layer

To examine the essential physics of the electromagnetic coupling in the grating-gated 2DEC structures schematically shown in the insets in Figs. 1(a) and 1(b), we employ the simplest phenomenological model of the resonant surface layer. Since the both gate-to-channel separation d and the grating gate period L are much smaller than THz radiation wavelength, the response of the grating-gated 2DEC to external in-plane electric field $E=E_0\text{exp}(-i\omega t)$ can be described by the surface admittance [24,25]

 

Fig. 1 Calculated terahertz absorption spectra of the grating-gated 2DEC on (a) a bulk substrate and on (b) the membrane substrate for the gate voltage value – 1 V for L = 4 µm, h = 4 µm, d = 0.4 μm, ε = 12.8, m* = 0.069m ₀, and τ = 8.5×10⁻¹² s. The insets in panels (a) and (b) show schematic views of the grating-gate structures on bulk and membrane substrates, respectively.

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If ${\gamma }_e<<{\omega }_{pn}$ and $\omega \approx {\omega }_{pn},$ the nth term dominates in series Eq. (1). Then, in the vicinity of the nth plasma resonance, one has

Using conventional Fresnel formulas, we can calculate the reflectivity, transmittivity, and absorbance of the entire structure comprising the grating-gated 2DEC with the surface admittance defined by Eq. (1) [or Eq. (2) if ${\gamma }_e<<{\omega }_{pn}$ and $\omega \approx {\omega }_{pn}]$ on the front surface of the substrate slab of thickness h.

If the grating-gated 2DEC lies on the surface of the a semi-infinite bulk substrate [see the inset in Fig. 1(a)], we obtain the absorbance in the neighborhood of the nth plasma resonance as [25,26]

At the resonance, $\omega ={\omega }_{pn},$ Eq. (3) yields

Now, let us consider the structure with a thin membrane-like substrate with thickness $h<<{\lambda }_{\epsilon },$ where ${\lambda }_{\epsilon }$ is the THz wavelength in the substrate material [see the inset in Fig. 1(b)]. To the first-order approximation in respect to a small ratio between the substrate thickness and THz wavelength, the electromagnetic problem is equivalent to that considering the resonant layer with effective admittance

The absorption resonance lineshape in the membrane structure described by the sheet admittance Eq. (8) is formally defined by the same Eq. (3) with R ₀ = 0 and ${\omega }_{pn}$ and ${\gamma }_{rn}$ replaced by ${\omega }_{pn}^{(M)}={\omega }_{pn}[1-{\gamma }_e{k}_{0}h(\epsilon -1)/(2{\omega }_{pn})]$ and ${\gamma }_{rn}^{(M)}\approx e^2{\overline{N}}_s{Z}_0{\beta }_n^2/4m^{\ast },$ where ${\omega }_{pn}^{(M)}$ and ${\gamma }_{rn}^{(M)}$ are the frequency and radiative damping of the nth plasmon mode in the grating-gate structure on a thin membrane substrate, respectively. Since the second term in the square brackets in the right-hand side of the expression for ${\omega }_{pn}^{(M)}$ is the product of two small factors ${\gamma }_e/{\omega }_{pn}$ and k ₀ h, one has ${\omega }_{pn}^{(M)}\approx {\omega }_{pn}$ (the same approximation was used when obtaining the expression for ${\gamma }_{rn}^{(M)}).$ Then Eq. (6) with R ₀ = 0 yields the maximum absorbance $A_{\text{res}}^{\max }=0.5$ for ${\gamma }_e={\gamma }_{rn}^{(M)}$ $\text{[i .e ., }Y({\omega }_{pn})\approx 2/Z_0]$in the structure with a membrane-like substrate. Notice that this value of the maximum absorption is twice as high as that for the structure with a bulk GaAs substrate. The plasmon radiative damping (or, which is the same, the coupling strength between the plasmon modes and THz radiation) also doubles in the membrane structure $({\gamma }_{rn}^{(M)}\approx 2{\gamma }_{rn})$ for $\epsilon \approx 10.$ Therefore, the absorption at higher-order plasmon resonances given by Eq. (7) increases fourfold due to the cumulative effect of reduced reflection $(R_0\to 0)$ and enhanced coupling between plasmons and THz radiation in the membrane structure.

3. Self-consistent electromagnetic modeling

Within the framework of a simple model of the surface resonant layer described in Section 2, the plasmon-resonance frequencies ${\omega }_{pn}$ (or ${\omega }_{pn}^{(M)})$ and radiative damping ${\gamma }_{rn}$ (or ${\gamma }_{rn}^{(M)})$ are phenomenological parameters, which are to be extracted from the measured plasmon absorption spectrum or calculated using a first-principle electromagnetic approach. We calculated the THz absorption spectra using a self-consistent electromagnetic approach described in detail in Ref. 28. Figures 1(a) and 1(b) show the calculated spectra of plasmon resonances in the grating-gated 2DEC on the bulk and membrane substrates, respectively, for different values of the grating-gate aspect ratio w/L, where w is the width of the grating-gate finger. Other parameters are characteristic of AlGaAs/GaAs structures studied in experiments [14,16]: L = 4 µm, h = 4 µm, τ = 8.5×10⁻¹² s, homogeneous electron density in the channel at zero gate voltage is N _s = 4.14×10¹¹ cm^–2, and d = 0.4 μm, which yields the channel pinch-off voltage of –2.3 V (as calculated in the parallel-plate capacitor model). The barrier-layer and substrate dielectric constants are 12.8 each, and m* = 0.069m ₀ with m ₀ being the free electron mass.

In accordance with Eq. (6) obtained in the framework of a simple model of the surface resonance layer, the maximum absorbance at the plasmon resonance reaches $0.5(1-\sqrt{R_0})\approx 0.22$ in the structure with a bulk GaAs substrate [see Fig. 1(a)] and 0.5 in the structure with a membrane substrate [Fig. 1(b)]. As follows from Fig. 1, the absorbance at the second-order plasmon resonance increases roughly by a factor of 5 in the structure on the membrane substrate compared to the structure on a bulk substrate for the same grating aspect ratio 0.5 in accordance with Eq. (7). Absorbance at the higher-order plasmon resonances increases by more than an order of magnitude for narrow slits of the grating gate due to enhanced electromagnetic coupling between the plasmons and THz radiation [19]. The photoresponse of the grating-gate plasmonic detector is linear in respect to the incoming THz power [14]. This means that the responsivity of the slit-grating-gate plasmonic THz detector on a membrane substrate exceeds that of a conventional detector with 0.5-duty-cycle grating gate on a bulk substrate by a factor of 50 at the higher-order plasmon resonances.

4. Conclusions

In conclusion, we have shown that the responsivity of a grating-gate plasmonic detector can be dramatically enhanced by placing the detector onto a membrane substrate and using narrow-slit grating gate due to considerable enhancement of the electromagnetic coupling between the plasmons and THz radiation. Because the electromagnetic coupling is extremely fast process, such enhancement in the responsivity would not deteriorate the detector operation speed as opposed, e.g., to the bolometric enhancement of detector responsivity studied in Ref. 23. These results open up possibilities for improving performance of the grating-gate plasmonic THz detectors.