Terahertz plasmons in coupled two-dimensional semiconductor resonators

Advances in theory are needed to match recent progress in measurements of coupled semiconductor resonators supporting terahertz plasmons. Here, we present a ﬁeld-based model of plasmonic resonators that comprise gated and ungated two-dimensional electron systems. The model is compared to experimental measurements of a representative system, in which the interaction between the gated and ungated modes leads to a rich spectrum of hybridized resonances. A theoretical framework is thus established for the analysis and design of gated low-dimensional systems used as plasmonic resonators, underlining their potential application in the manipulation of terahertz frequency range signals.


I. INTRODUCTION
The traditional test-beds of terahertz plasmonics are two-dimensional electron systems (2DESs). Early experiments studied plasmon excitation 1,2 , thermal emission 3 , effects of magnetic fields 2 and dc currents 4 . On the other hand, advances in fabrication and measurement constantly enable discovery of new effects 5 . Theoretical work has traditionally concentrated on the dispersion of gated and ungated plasmons 2,6 , magneto-plasmons 2 , and edge-plasmons 7 in unbounded 2DESs. Analytical studies of plasmonic resonators, however, have been limited mainly to the description of single resonators, either with simple boundaries (in which the resonances occur at multiple half-wavelengths 5 ), or with manipulated boundary conditions leading to plasmon instabilities 8 . More complex resonators (realized, for example, by a long periodic grating coupler) have mainly been studied by numerical simulations [9][10][11] .
Several recent experiments, however, have been devoted to systems containing a relatively small number of strongly-coupled plasmonic resonators. Using an antenna to couple an incident electromagnetic wave, Dyer et al. 12,13 studied the collective response of plasmonic crystals comprising up to eleven resonators, with a detector integrated into the crystals.
Wu et al. 14 characterized a system of three coupled resonators by time-domain spectroscopy.
They used on-chip LT-GaAs photo-conductive switches monolithically integrated with the 2DESs. The latter experiments are the first to promise terahertz spectroscopic characterization where signals transit between the ends of a 2DES (as opposed to the traditional excitation of the whole structure by a grating coupler). Theoretical modeling of the measured spectra then requires new approaches that take into account the coupling between resonators.
A general theoretical method to describe plasmonic structures has been developed by Popov and co-workers. An electromagnetic wave is assumed to be incident upon a structure, and the fields are presented in the form of Fourier integrals (for finite structures) [15][16][17][18] or series (for infinitely long periodic plasmonic crystals) 19 . Application of the electromagnetic boundary conditions leads to integral equations for the amplitudes of the harmonics, which can then be found numerically. The structures analyzed include a partially gated infinitely long 2DES 15 , a fully gated 2DES with ohmic contacts 16 , and a partially gated 2DES with ohmic contacts 17 . The resulting terahertz spectra depend on the spectrum of An alternative approach to junctions between gated and ungated 2DESs uses expansions of the electromagnetic fields into eigenmodes 22 , which are rigorous solutions of Maxwells equations across the whole junction. It has been conceived for closed metallic waveguides [23][24][25] and then expanded to include open dielectric waveguides as well [26][27][28] . It has also been applied to waveguides at optical frequencies [29][30][31][32][33] , where metals exhibit plasmonic response, and to plasmonic waveguides in the terahertz range [34][35][36][37][38] . The technique can be used to characterize a single junction between two waveguides and then analyze more complex geometries (comprising multiple junctions) relying, for example, on wave 25 or transfer matrices 39 .
Relying on the strengths of the eigenmode expansions, we present here a complete electromagnetics-based description of systems comprising coupled plasmonic resonators (Sec. II) and compare it directly to the experiments of Wu et al. 14 (Sec. III).

II. THEORETICAL MODEL
The configuration, shown in Fig. 1, contains all the basic components of the experimental plasmonic systems: gated and ungated resonators of different lengths and ohmic contacts terminating the system at both ends. The theoretical model includes three steps. First, we find the eigenmode spectra of the gated and ungated 2DESs. We then use these to calculate the plasmon reflection and transmission coefficients at a junction between two 2DESs. From the coefficients, we can derive the resonator modes and simulate the experimentally measured signals.
To calculate the eigenmode spectrum, we assume TM waves with the magnetic field H y and the electric fields E x and E z . The angular frequency is ω and the longitudinal wavenumber is k z . The 2DESs are embedded in a homogeneous dielectric with the permittivity ε d .
The electromagnetic waves induce a time-varying modulation of the electron density n and the velocity v. We describe the electron dynamics in 2DESs by the equation of motion of the form where τ is the collision time, e is the electron charge, and m is the effective electron mass.
The time-varying current density is J = en 0 v, where n 0 is the dc electron density. Maxwell's boundary conditions then couple the current and electron density to the fields H y and E x .
To find the eigenmode spectra, one has to prescribe the fields is the regions below the 2DES (x < 0), between the 2DES and the gate or between the 2DES and the air (0 < x < d), and in the air for the ungated 2DES (x > d), where d is the thickness of the dielectric above the 2DES. Denoting these regions correspondingly by 3, 2, and 1, we write the magnetic fields of an eigenmode in the form Here q = 1, 2, 3; A q and B q are constants; and the transverse wavenumbers in the dielectric k xd and air k xa can be found from the standard dispersion relations for waves in homogeneous dielectrics.
We omit everywhere the common factor exp[i(k z z − ωt)].
Because the field amplitudes cannot grow at infinity, the permissible values of the transverse wavenumbers k xa,d are either purely imaginary or real. When they are imaginary, the 4 fields decay away from the channel. These wavenumbers correspond to plasmons. In the ungated 2DES they obey the dispersion relation of the form where Γ u = ien 0u k xd /(2mε 0 ε d ω(ω + i/τ )) with n 0u denoting the dc electron density in the ungated 2DES, and ζ = k xd /(k xa ε d ). In the gated 2DES, the plasmons obey the dispersion relation of the form where Γ g = ien 0g k xd /(2mε 0 ε d ω(ω + i/τ )) with n 0g denoting the dc electron density in the gated 2DES. Figure  onality condition The index q denotes the geometrical region (see Fig. 1); the summation goes from 1 to 3 for the ungated and from 1 to 2 for the gated 2DES; the integration limits are determined by the region boundaries; ε q equals either 1 or ε d ; I(k x ,k x ) is a constant; δ(k x ) is the Kronecker delta.
Next, we consider junctions between gated and ungated 2DESs. When a plasmon is incident upon a junction, it will both partially transmit through and partially reflect back from the junction. To find the plasmon transmission and reflection coefficients, we expand the fields at both sides of a junction into the eigenmodes of the gated and ungated 2DESs 22 .
One of the expansions contains the incident and reflected plasmon together with the re- 3. We assume that two perfect magnetic conductors are placed in regions 1 and 3 at the same distance w from the 2DES. Since the plasmonic fields decay away from the channel, sufficiently large values of w do not affect the plasmon dispersions. On the other hand, the presence of the two conductors leads to discrete spectra of the non-plasmonic modes. An advantage of this discretization method is that the effects of radiation can be controlled by the conductor placement. Applying the boundary conditions in each region separately leads for the reflection and transmission coefficients to two algebraic equations of the general form where the summations are over the non-plasmonic modes, and the incidence is from the ungated 2DES. Truncating the summations at a sufficiently large number and using the orthogonality conditions Eq. (4) leads to an algebraic matrix equation for the coefficients R u , T u , t j and r j . Alternatively, Eq. (5) can be solved by the variational method 25 , and we found that both approaches gave the same result.
Once the plasmon transmission and reflection coefficients are known for incidence from the ungated (T u and R u ) and the gated (T g and R g ) 2DESs, the junction between them can be characterized by the wave matrix 25 of the form A section of a 2DES of the length L, on the other hand, is characterised by a 2 × 2 diagonal matrix with the diagonal elements exp(±ik zup,gp L).
Finally, we describe compound plasmonic resonators by multiplying the wave matrices.
The structure shown in Fig. 1 consists of three coupled resonators. To simulate the experiments, we assume that the (perfectly conducting) ohmic contacts, terminating the structure on both ends, leak some signal, in a frequency-independent manner, into and out of the 2DES.
Moderate electron collisions do not affect the mode shapes and, hence, the transmission and reflection coefficients. They do, however, lead to a propagation loss, which can be taken 7 into account by complex-valued plasmon wavenumbers k zup,gp . The resonances of the whole structure can be determined from the frequency variation of the wave matrices.
The modal analysis presented here shares several strengths with the theoretical approach of Popov et al. [15][16][17]19 , based on a Fourier representation of the fields. They both are rigorous solutions of Maxwell's equations, consider the fields everywhere in space, and take effects of retardation into account. However, the modal analysis is able to characterize single junctions, so that complex geometries and be easily analyzed and designed. The price is some loss of generality. We assume, for example, that the ungated resonators of Fig. 1  In addition, commercial numerical solvers have currently a limited ability to self-consistently solve problems involving complex electron dynamics (for example, effects of diffusion and dc currents). Analysis of such problems by purpose-built solvers has so far been limited to rather simple geometries 40 .

III. COMPARISON WITH EXPERIMENTS
Having presented the theoretical model, we will now compare it to our experimental measurements. The experimental setup is described in detail elsewhere 14 . Briefly, a gate (of width L 2 = 4.4 µm) overlaying a 2DES in a high-mobility GaAs/AlGaAs heterostructure formed two ungated resonators either side with lengths L 1 = 19.7 and L 3 = 48.9 µm.
( Fig. 1). The (ungated) electron density, found from magnetotransport experiments, was n 0u = 6.5 × 10 11 cm −2 . A negative dc bias applied to the gate was used to control the electron density in the region of 2DES underneath (the 2DES depth was 75 nm). We take the (empirically determined) relationship between the applied voltage V g and the electron density as n 0g = a 1 (V g + a 2 ) γ , where a 1 = 5.38 × 10 11 cm −2 , a 2 = 2.6, γ = 0.23, and V g is in volts.
In our experiments, two LT-GaAs photoconductive switches were connected to the 2DES by ohmic contacts attached to coplanar transmission lines on either side. THz signals gener-  In the simulations, we calculated the derivative of the transmitted power with the gated electron density. The plasmons were lossy, first due to collisions (τ = 5 ps) and second, due to plasmon leakage into radiation modes at the junctions. Both mechanisms affect the shape of the curves. To simulate the incident THz pulse, we assumed that its power decays exponentially with the frequency 14 .
Even though a common assumption 5,12,13,36,38 , the perfectly reflecting contacts in the simulations is an approximation, since strictly they would not allow coupling between the transmission lines and the 2DESs. The coupling also depends on the fields on both sides of the contact, and it is absent for one of the two modes supported by the transmission line 14 .
However and by putting magnetic conductors close to the 2DESs).
Although the gate-modulated signals are inherently weak for some resonances, they characterize plasmons directly, unaffected, for example, by a direct cross-talk between the transmission lines on either sides of the 2DESs. The gate-modulation technique is, however, sensitive to frequency variations of the input signals and the coupling strength between the transmission lines and the 2DESs. We were able to take the former effect into account, while assuming that the latter effect is weak. The good qualitative agreement between the experiments and simulations of Fig. 3 appears to justify this assumption. Figure 4 shows the same resonances as in Fig. 3(a)  voltage will therefore affect not only the plasmons below the gate but also the plasmons in the ungated 2DESs, an effect previously reported in Ref. [17].
The resonances become more entangled as the frequency increases. For example, the next three resonances (lines 5,6, and 7) stem, at higher gate voltages, from the interaction between three uncoupled resonances (the third-order gated, the third-order longer ungated and the first-order shorter ungated). The interaction creates the broad measured and simulated signals seen in Fig. 3 around 0.3 THz.
Our model could be used to design further configurations that support hybridized resonances generating strong gate-modulated signals. A candidate structure could comprise two gated 2DESs separated by an ungated 2DES, where uncoupled gated plasmons exist in the same frequency range.

IV. CONCLUSIONS
We have presented an electromagnetics-based approach to characterize coupled plasmonic resonators and compared its predictions with experiments. Analysis of the coupled system showed that while some of the resonances may be independent from each other, most of them hybridize as a result of an interaction between gated and ungated 2DESs. We note that our theoretical approach could be extended for the design of more advanced plasmonic devices, in particular resonant oscillators, amplifiers, and detectors carrying dc currents.