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We present a method of coupling free-space terahertz radiation to intersubband transitions in semiconductor quantum wells using an array of meta-atoms. Owing to the resonant nature of the interaction between metamaterial and incident light and the field enhancement in the vicinity of the metal structure, the coupling efficiency of this method is very high and the energy conversion ratio from in-plane to z field reaches values on the order of 50%. To identify the role of different aspects of this coupling, we have used a custom-made finite-difference time-domain code. The simulation results are supplemented by transmission measurements on modulation-doped GaAs/AlGaAs parabolic quantum wells which demonstrate efficient strong light-matter coupling between meta-atoms and intersubband transitions for normal incident electromagnetic waves.
©2011 Optical Society of America
1. Introduction
After the first direct observation of intersubband transitions (ISBs) in semiconductor heterostructures [1], the field has grown rapidly. Based on the transition of electrons between different (bound) states within the conduction band, many practical devices have emerged, such as quantum well infrared photodetectors [2] or quantum cascade lasers [3].
In order to couple light to ISBs, it is first necessary to fulfill the polarization selection rule. Only light polarized along the growth direction of the quantum wells (QW), the z direction, can induce ISBs [4]. Additionally, free-space radiation has to be efficiently coupled into the device. Common ways to achieve both requirements are, for example, incidence under Brewster’s angle [5], coupling into a waveguide geometry [6], or using grating couplers [7].
Metamaterials (MMs), artificial structures consisting of regular arrays of sub-wavelength resonators (meta-atoms) [8], constitute an exciting possibility for coupling free space radiation to QWs [9,10]. Properties such as position and line width of the resonances can be fully controlled by the geometry of the resonant elements. Coupling to the QW is mediated by the near field along the z axis. Thus, it is necessary that the distance between the QW and the meta-atom is smaller than the decay length in the substrate, which is typically on the order of several microns in the THz region.
Strong light-matter coupling in microcavities in the THz range has been achieved recently by Todorov et al. [11]. Thereby, a multi-QW structure has been sandwiched between two gold layers, providing sub-wavelength confinement of the electromagnetic radiation. Geiser et al. [12] have been able to demonstrate strong coupling of a LC resonator to THz ISBs in a parabolic quantum well (PQW) even at room temperature (RT). The strong confinement is in this case also achieved by a vertical double-metal structure which has the drawback of a large capacitance per unit area. In consequence, such structures have to be small and provide only a very limited cross-section for the interaction with free space electromagnetic waves.
In this Letter, we demonstrate efficient coupling of free space THz radiation to ISBs in QWs using an array of meta-atoms. These resonant structures are shown to be convenient mediators for the interaction between light polarized parallel and perpendicular to the growth direction of the QW. The evanescent character of the electric field in the vicinity of the meta-atom structure guarantees its strength amplification and long interaction time in a small volume. The efficiency of this method is thereby sufficient to reach the strong coupling regime. These features make meta-atom based couplers very attractive candidates for large area emitters and beam amplifiers.
2. Model
The basic idea of the coupling mechanism is illustrated in Fig. 1a. Terahertz light is normally incident on the sample surface with the electric field vector parallel to the x axis. The QW is located beneath the meta-atom with its growth direction parallel to the z axis. In this geometry, the incident THz light cannot directly couple to the ISBs due to the selection rules, but it can couple to the meta-atom. Figure 1b shows a cross-section of the meta-atom on resonance along the dashed line in Fig. 1a. The electric field of this resonant mode, illustrated by the red field lines, extends both above and below the meta-atom, thereby possessing a non-zero z component. Hence, the meta-atom couples the incident in-plane field to the z field that is necessary to induce ISBs. Owing to the resonant nature of the interaction and the field enhancement known from such meta-atoms, the conversion efficiency between in-plane and z field easily reaches values on the order of several ten percent.
Fig. 1 Artistic view of the situation. a) The THz pulse is normally incident on the meta-surface. The QW is located underneath. b) Cut through the structure along the x–z plane at the position indicated by the dashed line in a. The arrows indicate the electric field lines at resonance. This provides the z component necessary for efficient coupling to ISBs.
To better understand the role of different aspects of this coupling method, we have developed a finite-difference time-domain (FDTD) code to simulate this interaction of single-cycle THz pulses with meta-atoms on a dielectric substrate coupled to a QW. The 3d FDTD code is based on the standard Yee grid and the leapfrog method for spatial and temporal staggering of the electric and magnetic field components [13, 14]. The temporal step size Δt is related to the spatial grid size Δr by c 0Δt = 𝒞Δr, where c 0 is the vacuum speed of light and 𝒞 the Courant number. To ensure stable computation, the Courant number should be smaller than for 3d simulations [14]. We use 𝒞 = 0.5 throughout the paper. The spatial step size has to be chosen according to the smallest feature size of the meta-atom. The lateral dimension of the simulation volume a × b equals the size of one unit cell of the meta-atom array. We use Δr = 0.5μm and a = b = 40μm. The temporal step size is accordingly Δt = 0.8fs.
The growth direction of the QW is taken as the z axis. The computational space is divided in two parts, air or vacuum on one side and substrate on the other side. The substrate is taken into account as frequency independent and lossless dielectric with permittivity ɛ. The extension to a more realistic, dispersive medium is straightforward, but for the frequency range and substrate material we are interested in, not necessary (the refractive index of semi-insulating GaAs is well approximated by ɛ = 12.96 up to 5 THz). The THz radiation is taken to be normally incident. Therefore, we can use simple periodic boundary conditions at the four side walls, F(i, j,k) = F(i + Nx, j, k) and F(i, j, k) = F(i, j + Ny, k), where F represents any field variable. At the top and bottom plane, we use first-order Engquist-Majda absorbing boundary conditions (ABC) [15, 16], which are exact in 1d. To reduce the load on the ABCs, we use the total-field / scattered-field approach [16, 17]. We have verified that the residual scattering that occurs on the edges and corners is negligible by choosing different sizes of the simulation space and appropriate time windowing. The source is modeled as a modulated Gaussian to simulate a broadband, single-cycle THz pulse:
where E 0 is the peak field amplitude, t 0 a time offset and τ the FWHM of the pulse.The meta-atom is located in the x – y plane at the interface between air and substrate. We simulate the metalized parts of the meta-atom as perfect electric conductor by fixing the tangential components of the electric field to zero. This is a very good approximation in the THz frequency range, where Ohmic losses in the metal can be neglected, provided the thickness of the metal is larger than the skin depth. To minimize the error introduced by approximating the smooth contour of the meta-atom by rectangular grid cells, we use the diagonal split-cell method for approximating round shapes [14].
For the simulations, we use the meta-atom geometry shown in Fig. 2a. This type of meta-atom shows two distinct resonances for the polarization of the incident radiation along the x and y direction, respectively. From Figs. 2b and 2c, it is evident that the resonant mode for polarization along the x direction is a LC mode using the combined capacitance of the splits. The resonance along the y direction is the fundamental dipole mode of the closed ring. The corresponding transmission spectra are shown in Fig. 2d. In the following, we are only interested in the LC resonance along the x direction. This resonance shows a strong enhancement of the electric field at the gap and the two side bars. At a distance of Δr/2 = 250nm to the metal, corresponding to the center of the PQW, the z peak field is about ten times stronger than the incident field. From a least squares fit of a Lorentzian to the simulated frequency domain data for this resonance, shown in Fig. 2d, we determined the resonance frequency and FWHM to be ωc = 2π × 2.30THz and Δωc = 2π × 0.27THz, respectively, giving a Q-factor of 8.5. The energy conversion efficiency
calculated at the z coordinate of the QW, is in this case η = 54%.Fig. 2 a) Layout of the MM. The size of the unit cell is a × b = 40 × 40μm. b,c) Simulated z component of the electric field just beneath the MM for different polarizations of the incident pulse. The polarization is indicated by the arrows. d) Comparison of simulated and measured electric field transmission coefficients of the MM on the undoped structure for both polarizations.
The QW itself is located in the layer just beneath the meta-atom. Due to the confinement of the electrons along the z axis, the response of the QW to the incident electric field is different along the z and the x – y direction, respectively. We approximate the QW by a two-level system with transition energy h̄ω 12. The simplest model is the classical Lorentz model. The z component of the polarization density is related to the z component of the electric field by [14]
where n 0 is the number of electrons per unit cell, e the electron charge, m eff the effective mass, δ the damping coefficient and the oscillator strength. The quantity μ 12 = 〈1|z|2〉 is the dipole matrix element between states |1〉 and |2〉. We chose a finite-difference implementation of Eq. (3) based on the auxiliary differential equation method (ADE) [14]. As can be seen from Eq. (3), the response of the QW is linear in the electric field. To take the field dependence into account, we use the optical Bloch equations [18] where u = (u 1, u 2, u 3) is the Bloch vector, T 1 the excited state lifetime, T 2 the dephasing time, and u 30 the equilibrium population difference. The components of u are the dispersive and the absorptive component of the polarization and the fractional population difference between states |1〉 and |2〉, respectively. The Bloch equations are coupled to the Maxwell equations, as in the previous case, via the z component of the polarization density Pz(t) = −n 0 eμ 12 u 1(t). The finite-difference implementation follows Ref. [18], except that the Bloch vector components are discretized at half time steps n + 1/2. Using this discretization scheme leads to a decoupling of the Maxwell-Bloch equations in the 1d case [19], making the integration scheme fully explicit.The parameters of the QW used for the simulations are chosen according to the experiment (see Sec. 3). In the case of weak electric fields, we expect the Lorentz model to be an adequate description of our system. The transmission coefficient obtained from the simulation is shown in Fig. 3a (light green). The normal mode splitting of the bare resonance of the meta-atoms (black) is clearly visible in the simulation. To get a better understanding, we fit the transmission coefficients with a model coefficient based on a coupled oscillator model. The equations of motion in a one-electron picture can then be written as
where xc(t) and x(t) are the electron displacement from equilibrium, δc the damping coefficient and ωc the resonance frequency of the meta-atom, and Ω the coupling coefficient.Fig. 3 a) Simulated transmission coefficient for the meta-atoms on substrate (S), coupled to the QW using the Lorentz model (L), and using the Maxwell-Bloch model (MB) with 50V/cm incident field using linearly polarized light. b) Transmission coefficient of doped structure at 5K and RT obtained with the Fourier transform spectrometer using unpolarized light. The solid gray lines are fitted curves obtained from the coupled oscillator model. The inset shows the transmission coefficient of the undoped sample.
Only the meta-atom is directly coupled to the incident electric field Ex. The fitted values ω 12 = 2π × 2.3THz and δ = 0.4 × 1012rad/s correspond to the simulation parameters. The coupling coefficient Ω = 5.4 × 1012rad/s is larger than the damping of the QW, δ, which is a clear indication of strong light-matter coupling. The results suggest that the coupling efficiency of the meta-atom to the ISB is already good enough that no further confinement of the electric field, for example in a microcavity [12], is necessary. The transmission coefficient predicted by the Maxwell-Bloch model with an incident field strength of 50V/cm (dark green curve in Fig. 3a) is also in excellent agreement to the simple Lorentz model.
3. Experiments
To support our model calculations, we have performed transmission measurements on PQWs using a Fourier transform spectrometer (Bruker IFS113V) with 4.2K bolometer detector. We have chosen to use the FTIR, as it guarantees the weak electric field condition necessary for the validity of the two-level model we have used in the simulations. Additionally, we avoid any THz induced heating of the PQW that may occur due to the strong field enhancement caused by the meta-atoms.
The samples consist of a single PQW grown by the digital alloy technique using the GaAs/Al0.3Ga0.7As material system. The distance of the well to the surface is 185 nm. We have used both modulation doped and undoped structures for reference [20, 21]. The well width is L = 140nm leading to a theoretical transition frequency of ω 12 = 2π × 2.2THz. The effective sheet density of the doped sample has been determined by Hall measurements to ns = 5×1011cm–2 [22]. The dephasing time has been estimated to be T 2 = 2.5ps [20]. The inter-subband absorption vanishes for temperatures above 240 K, giving us the possibility to use the room-temperature transmission spectra as reference. From self-consistent Poisson-Schrödinger calculations of the wave functions of the lowest and first excited state, we have calculated a dipole matrix element of μ 12 = 8.6nm, leading to an oscillator strength close to one. For coupling the QW to the free-space radiation, we have used an array of meta-atoms consisting of rectangular double-split rings, as shown in Fig. 2a. The rings consist of 150 nm Ag and 10 nm Au and have been fabricated using standard lithography and e-beam evaporation. The spacing between adjacent rings is 40μm × 40μm. The measured frequency response of the bare meta-atoms is in excellent agreement with the simulation (Fig. 2d). These measurements were done with a standard TDS setup on the undoped sample with a frequency resolution of 56GHz. From a Lorentzian fit to the experimental data, we extracted ωc = 2π × 2.28THz and Δωc = 2π × 0.46THz, giving a Q-factor of 4.9, which is lower than expected. We attribute this broadening to geometrical variations between different meta-atoms caused by imperfections in the fabrication process.
For the FTIR measurements, the samples have been wedged under an angle of two degrees to suppress Fabry-Perot fringes in the transmission spectra. Additionally, the QW layer has been removed between the meta-atoms by reactive ion etching using the metal of the meta-atoms as etch mask. Thereby, we could remove any parasitic effects due to the in-plane motion of free electrons. The samples have been held in a liquid He flow cryostat and the spectrometer sample chamber has been evacuated to reduce residual absorption in atmospheric water. The measured transmission coefficient of the doped sample is shown in Fig. 3b for two different temperatures. The measurements have been performed with unpolarized light. Thus, both resonances of the meta-atom are active at the same time and the signal corresponds to the averaged transmission. Additionally, the measurements have been referenced to vacuum. These are the reasons for the lower modulation depth of the measured transmission coefficient compared to the simulation results.
The room temperature result shows the response of the bare meta-atom. Due to the etching, the original resonance position (see Fig. 2d) has been shifted to a slightly higher frequency, as is expected from the lower effective permittivity of the surrounding medium. The single resonance dip splits in two distinct minima when the sample is cooled to 5K. The transmission coefficient of the undoped structure, however, is independent of temperature (see inset of Fig. 3b). This temperature dependence clearly relates the observed splitting to the ISBs in the PQW. At low temperatures, the electrons in the doped structure occupy only the lowest lying subbands, making the ISBs optically active. When temperature increases, the confinement potential for the electrons becomes weaker and, due to increased scattering rates, the electrons finally escape from the PQW. This then destroys the coupling of the incident electric field to the ISBs.
The occurrence of the normal mode splitting is a first indication that we observe strong coupling between the meta-atom and the PQW. A further indication is the ratio between damping and coupling coefficients, δ = 0.5 × 1012rad/s < Ω = 4.7 × 1012rad/s. The values have been extracted from a least squares fit of the coupled oscillator model, Eqs. (7) and (8), with the polarization averaging taken into account (shown as gray lines). The meta-atom transition frequency is determined to ω 12 = 2π × 2.3THz. All these values are in excellent agreement with our simulations.
4. Summary
In summary, we have demonstrated efficient coupling of free space THz radiation to inter-subband transitions in quantum wells by using an array of resonant meta-atoms. Theoretical results obtained from a custom-made 3d FDTD solver have been verified by an experiment on a parabolic quantum well structure. The coupling efficiency has been found sufficient to reach the strong coupling regime even with the weak electric field of the incident THz wave. We think that this will open a new road in the study of light-matter interactions.
Acknowledgments
The authors acknowledge partial financial support by the Austrian Society for Microelectronics (GMe) and the Austrian Science Fund FWF (SFB ADLIS and DK CoQuS, W1210).
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