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Inducing an Incipient Terahertz Finite Plasmonic Crystal in Coupled Two Dimensional Plasmonic Cavities

We measured a change in the current transport of an antenna-coupled, multi-gate, GaAs/AlGaAs field-effect transistor when terahertz electromagnetic waves irradiated the transistor and attribute the change to bolometric heating of the electrons in the two-dimensional electron channel. The observed terahertz absorption spectrum indicates coherence between plasmons excited under adjacent biased device gates. The experimental results agree quantitatively with a theoretical model we developed that is based on a generalized plasmonic transmission line formalism and describes an evolution of the plasmonic spectrum with increasing electron density modulation from homogeneous to the crystal limit. These results demonstrate an electronically induced and dynamically tunable plasmonic band structure.

The terahertz (THz) band of the electromagnetic spectrum (100 GHz -10 THz) presents a unique scientific frontier because of its historically difficult to access position between infrared optics and microwave electronics. To approach the THz band from the microwave domain, the frequency range of electronic circuit elements must be extended. To approach the THz band from photonics, both the low photon energy (4 meV at 1 THz) and comparatively long wavelengths in free space (300 um at 1 THz) pose challenges. Plasmons, fundamental electronic excitations that have light-like characteristics yet drive bound charge oscillations, offer an avenue to bridge this "THz gap." Surface plasmons [1] can scale to THz frequencies provided there is a wavevector defined by geometry [2,3] such as a grating or cavity to compensate for the momentum mismatch between free-space radiation and the sub-wavelength plasmon. Two dimensional (2D) THz plasmons in silicon (Si) [4], III-V materials such as GaAs/AlGaAs [5] and GaN/AlGaN [6] heterostructures, and graphene [7,8] similarly are sub-wavelength compared to free-space electromagnetic waves, but have an additional degree of freedom due to the ability to control electron density, and thus the 2D plasma frequency, through a field effect. It is this latter capability that has generated interest in 2D plasmonic devices [9,10] such as THz detectors [11][12][13][14], mixers [15,16], emitters [17,18], and field enhancement structures [19].
In this letter we present the first experimental observation, to our knowledge, of a coherent coupling between localized plasmonic excitations and formation of a finite onedimensional THz plasmonic crystal in a two dimensional electron gas (2DEG). Whereas in prior work a quasi-infinite plasmonic crystal in a weakly modulated 2DEG was demonstrated [20], here a finite plasmonic crystal is induced in a strongly modulated 2D electron system. The THz plasmonic crystal is observed in an antenna-coupled multi-gate GaAs/AlGaAs high electron mobility transistor (HEMT) where a plasmonic cavity (PC) is formed between device terminals and multiple gate biases provide a 2D electron density modulation in the HEMT channel. Such a solid state system represents the 2D plasmonic analogue of an Esaki-Tsu superlattice [21] or photonic crystal [22]. However, semiconductor superlattices and photonic crystals are defined intrinsically by their material composition and periodicity, whereas the amplitude and period of the 2DEG density modulation and therefore properties of the plasmonic band spectrum can be modified in-situ by manipulation of multiple gate voltages.
The plasmonic system depicted in Figs. 1(a) and 1(b) is based on a multi-gate HEMT with a 14 m long by 10 m wide channel. The HEMT was fabricated from a double quantum well GaAs/AlGaAs heterostructure (Sandia wafer EA1149) with a total electron density of 0 = 4.02 x 10 11 cm -2 and a mobility of 6.14 x 10 5 cm 2 /V s at 12 K. The 20 nm thick quantum wells are 386 nm below the surface and spaced 7 nm from each other. There are three gates, the source gate (G1), center gate (G2), and drain gate (G3), each designed to be 2 m wide and separated by 2 m. The fabricated gate widths and separations differ by approximately 10% from the intended dimensions. The source terminal (S) is referenced to ground potential. The G3 was biased beyond threshold, depleting the electrons underneath it and effectively terminating the two dimensional electron gas (2DEG) at its edges. This forms a 10 m PC between S and G3 where G1 and G2 can tune the electron density beneath them and hence the resonant frequency of the plasma mode.
The gates of the HEMT were biased in a manner similar to previous work [23] with G3 This indicates that the 10 m length from S to G3 and the PC boundary conditions primarily define the resonance condition.
Having established coherence between the plasmonic sub-cavities below G1 and G2, we next examine the bolometric responsivity spectrum at 11.5 K for excitation frequencies from 195 to 450 GHz while identically tuning the gates G1 and G2 such that 1 = 2 ≡ . The spectrum shown in Fig pair of eigenmodes is not strongly excited experimentally while still higher order modes are difficult to resolve. This pairing of modes is indicative of a plasmonic band structure forming in this system. In a strongly modulated regime, low frequency plasmons should be mostly localized in the regions with low electron density, the sub-cavities below G1 and G2 [19]. Thus, the mode pairing may be viewed as the energetic splitting of the degenerate eigenmodes in these subcavities due to coherent coupling between them.
As the basis for modeling the experimentally observed coherent PC phenomena we take a transmission line (TL) description for the device shown in Fig. 1. A PC consisting of alternating gated and ungated regions of 2DEG, each = 2 long and = 10 wide, is formed between the Ohmic source contact and depleted region below G3. The five regions of 2DEG shown in Fig. 1(a) are treated as discrete segments in a two-port TL network. We assume that the voltage at the Ohmic source contact on one side of the cavity goes to zero. This is equivalent to requiring the plasmon charge density fluctuation vanishes at this boundary. In the equivalent circuit shown in Fig. 1(c) the antenna is represented as a voltage generator in series with source impedance determined by its radiation resistance . It is assumed that for the logperiodic antenna on a GaAs substrate in Fig. 1 are the resistance and kinetic inductance of 2DEG per unit channel length respectively, is the electron momentum relaxation time, * is the electron effective mass, and is the width of the 2DEG channel [25,26]. To describe the experiment we assume that a 2DEG is embedded and q is determined by dispersion relation. The coefficient ( ) can be interpreted as electrostatic capacitance per unit length of the equivalent TL. In the fully screened limit ( → 0), ( ) reduces to the standard gate capacitance of a HEMT, = ⁄ . In the unscreened limit ( → ∞) we recover the expression = 2 [27]. In a lossy TL ( ′′ ≠ 0) the imaginary part of the capacitance represents a shunt conductance in the TL equivalent circuit.
With these definitions of the distributed TL elements and the characteristic impedance 0 −1 = , solution of the corresponding telegrapher's equations [28] reproduces the plasmon dispersion law, , as well as spatial distributions of the in-plane electric potential, current and charge density of the plasma wave.
The electric field of the 2D plasmon has a longitudinal component. Therefore, the power carried by the 2D plasma wave cannot be found by direct application of the TL equations [27,29]. The time-averaged complex power carried by the plasma wave can be evaluated using the Poynting vector to yield P ( ) = 1 2 Φ ( , 0)I * ( ) * ( , ) where is the plasmonic current and the form-factor ( , ) is determined as The power transfer can be reduced to the standard TL expression 1 2̃( )̃ * ( ) by defining effective voltage ̃( ) and current ̃( ) [29]. We define them as ̃( ) ≡ Φ ( , 0) and ̃( ) ≡ ( , ) ( ). This definition of effective current is equivalent to redefining the characteristic impedance of the TL as ̃0 = 0 ( , ) ⁄ . In the fully screened limit, the 2D plasma wave becomes purely transverse and ( , ) = 1. In the unscreened limit with ′′ = 0 we obtain ( , ) = 1 2 ⁄ [27]. We assume continuity of the voltage ̃( ) and the power flow P ( ) at the boundaries between segments. This results in the discontinuity of the plasma current accounting for the edge charge accumulation at interfaces between the segments as confirmed by direct numerical solution of Maxwell's equations in the 2DEG with step-like changes of the equilibrium electron density and gate screening [30].
Using a transfer matrix method for a segmented TL [28] with the above plasmonic TL definitions, we calculated power absorption in the inhomogeneous PC at different excitation frequencies as a function of the density modulations (1,2) 0 ⁄ under the gates G1 and G2. In this calculation we assumed that the PC between S and G3 is screened uniformly by the gates and surrounding metallization by taking = 386 , the gate-channel separation, as constant. In evaluating the TL effective resistance responsible for the plasmon resonance linewidth we added radiation damping −1 = 2 2 * ⁄ √ 0 ⁄ to the mobility-related damping [31].  Fig. 4(a). From Fig. 4(a) we conclude that the experimental modes nearest to modes pair into what is the beginning of the band structure, though the plasmon is still wellconfined within the gated region, as illustrated in Fig. 4(b). The fundamental cavity mode near 80 GHz is not paired with another mode. The eigenmode near 460 GHz is associated with the fundamental resonance in the ungated regions of 2DEG and does not pair with a second mode.
In fact the ungated regions represent a second plasmonic sub-lattice, but due to its expected resonant frequency above the measured range and the lack of gate tuning it has not been probed in the present experimental system.
The TL formalism developed in this letter is broadly applicable to any 2D plasmonic system including graphene [33,34] provided the appropriate 2D plasmon dispersion law resulting from linear energy-momentum carrier dispersion is employed [35]. Though a TL approach has been suggested for graphene [27], the effects of screening or inhomogeneity due to metal terminals were not included. To consider graphene HEMTs [36,37] as 2D plasmonic devices, especially designs featuring a top gate or having step-like variation of electron density or screening along the channel, a generalized TL approach is required. The high mobility of suspended graphene at room temperature makes it a particularly promising material system for developing components of an integrated THz plasmonic circuit. In contrast, III-V based plasmonic devices operating below 1 THz require cryogenic temperatures (< 100 K) to operate due to the degradation of the carrier mobility-limited plasmon Q-factor at higher temperatures. In