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^{1}, Xiaoshuang Chen

^{1,a)}, Weida Hu

^{1,a)}, Anqi Yu

^{1}and Wei Lu

^{1}

### Abstract

The ability to manipulate plasma waves in the two-dimensional-(2D)-electron-gas based plasmonic crystals is investigated in this work. It is demonstrated that the plasmon resonance of 2D plasmonic crystal can be tuned easily at terahertz frequency due to the wavevector quantization induced by the size effect. After calculating self-consistently by taking into account several potential mechanisms for the resonant damping of plasma waves, it can be concluded that the plasmon-plasmon scattering plays the dominant role. Based on the calculations, we can predict the scattering or inter-excitation among the oblique plasmons in the 2D crystal. The results can be extended to study 2D-electron-gas plasmonic waveguides, terahertz modulators, and detectors with electrostatic gating.

The authors acknowledge the support provided by the State Key Program for Basic Research of China (2013CB632705 and 2011CB922004), the National Natural Science Foundation of China (10990104, 61006090, 61290301, and 11274331), the Fund of Shanghai Science and Technology Foundation (10JC1416100), and Shanghai Rising-Star Program.

### Key Topics

- Plasmons
- 85.0
- Plasma waves
- 14.0
- Dielectrics
- 6.0
- III-V semiconductors
- 6.0
- Linewidths
- 6.0

## Figures

Schematic of plasmonic crystals, (a) 1D crystal with AlGaN barrier d and periodic grating gate serving as electrodes, (b) 2D plasmonic crystal with etched multichannel (type I), (c) 2D plasmonic crystal with etched dielectric layer d 2 (type II), (d) the turning of plasma wave vector at the boundary between gated and ungated channels in the unit cell of 2D crystal indicates the oblique plasmon, the polarization of incident wave is perpendicular to the gate finger, and the dimensions of the unit cell are also indicated. The length L of the electrode strip is 1 μm, and the thickness of AlGaN barrier is around 30 nm. s is the slit width between gate fingers.

Schematic of plasmonic crystals, (a) 1D crystal with AlGaN barrier d and periodic grating gate serving as electrodes, (b) 2D plasmonic crystal with etched multichannel (type I), (c) 2D plasmonic crystal with etched dielectric layer d 2 (type II), (d) the turning of plasma wave vector at the boundary between gated and ungated channels in the unit cell of 2D crystal indicates the oblique plasmon, the polarization of incident wave is perpendicular to the gate finger, and the dimensions of the unit cell are also indicated. The length L of the electrode strip is 1 μm, and the thickness of AlGaN barrier is around 30 nm. s is the slit width between gate fingers.

The plasmonic resonant spectra of 2D (type I in Fig. 1(b) ) and 1D crystal devices and plasmon-induced field distribution. (a) The sheet electron density n s and the slit width s are 6.1 × 1012 cm−2 (V g = 0 V) and 1 μm for both crystal devices. The width W and period D of the unit cell in the 2D devices are 0.3 μm and 0.9 μm (blue line), 0.6 μm and 1.2 μm (red line), respectively. The purple dotted line is for the 1D crystal. Meanwhile, Drudebackground absorption is also shown in dashed line. (b) The plasmon resonance after eliminating the Drude absorption from the spectra in (a) (Aq − Ab). (c) The half width at half maximum (HWHM) of resonance a for the 1D and 2D crystal devices with different slit widths s and channel widths W: open circle is the 1D crystal with the change of slit width s, solid circle and squares are the 2D crystal with the change of slit width s and channel width W, respectively. The limitation of collision term (electron relaxation) is also indicated. (d) Field distributions of resonances a and b under the continuous wave (CW) excitation in the unit cell of 2D crystal: upper panels are the field distributions cutting at y axis, and lower panels are the field distributions cutting at z axis. The dipole distributions along the channels are shown by thered symbols for positive charge and pink symbols for negative charge, in the meantime, the electric field lines are shown by the dotted lines.

The plasmonic resonant spectra of 2D (type I in Fig. 1(b) ) and 1D crystal devices and plasmon-induced field distribution. (a) The sheet electron density n s and the slit width s are 6.1 × 1012 cm−2 (V g = 0 V) and 1 μm for both crystal devices. The width W and period D of the unit cell in the 2D devices are 0.3 μm and 0.9 μm (blue line), 0.6 μm and 1.2 μm (red line), respectively. The purple dotted line is for the 1D crystal. Meanwhile, Drudebackground absorption is also shown in dashed line. (b) The plasmon resonance after eliminating the Drude absorption from the spectra in (a) (Aq − Ab). (c) The half width at half maximum (HWHM) of resonance a for the 1D and 2D crystal devices with different slit widths s and channel widths W: open circle is the 1D crystal with the change of slit width s, solid circle and squares are the 2D crystal with the change of slit width s and channel width W, respectively. The limitation of collision term (electron relaxation) is also indicated. (d) Field distributions of resonances a and b under the continuous wave (CW) excitation in the unit cell of 2D crystal: upper panels are the field distributions cutting at y axis, and lower panels are the field distributions cutting at z axis. The dipole distributions along the channels are shown by thered symbols for positive charge and pink symbols for negative charge, in the meantime, the electric field lines are shown by the dotted lines.

The inter-excitation/scattering between oblique plasmons and voltage modulation effect. (a) The upper panel is the plasmon resonances after eliminating Drude-absorption: the blue and green lines are the plasmon resonances in the type II 2D crystal with dielectric layer thickness d 2 = 250 nm, the red line is for the type I 2D crystal with period D = 0.9 μm and channel width W = 0.3 μm in the unit cell. The slit width s is 1 μm each in (a). The field distribution corresponding to the resonance a of blue line is shown in lower panel with the arrows indicating the turning of plasma wavevector. (b) The transmission (right axis)/absorption (left axis) of type I 2D crystal under different gate voltages, the transmission T th when the channel is pinched off is also shown (green line), the slit width s is 0.5 μm. (c) The modulation depth of resonances a and b under different gate voltages in 1D crystal and type I 2D crystal. (d) The absorption enhancement of resonance a in type I 2D crystal with slit width s = 0.5 μm (squares) and 1 μm (circles) when normalized to the one with channel width W = 0.3 μm, the gate voltage V g is 0 V.

The inter-excitation/scattering between oblique plasmons and voltage modulation effect. (a) The upper panel is the plasmon resonances after eliminating Drude-absorption: the blue and green lines are the plasmon resonances in the type II 2D crystal with dielectric layer thickness d 2 = 250 nm, the red line is for the type I 2D crystal with period D = 0.9 μm and channel width W = 0.3 μm in the unit cell. The slit width s is 1 μm each in (a). The field distribution corresponding to the resonance a of blue line is shown in lower panel with the arrows indicating the turning of plasma wavevector. (b) The transmission (right axis)/absorption (left axis) of type I 2D crystal under different gate voltages, the transmission T th when the channel is pinched off is also shown (green line), the slit width s is 0.5 μm. (c) The modulation depth of resonances a and b under different gate voltages in 1D crystal and type I 2D crystal. (d) The absorption enhancement of resonance a in type I 2D crystal with slit width s = 0.5 μm (squares) and 1 μm (circles) when normalized to the one with channel width W = 0.3 μm, the gate voltage V g is 0 V.

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