Optomechanical terahertz detection with single meta-atom resonator

Most of the common technologies for detecting terahertz photons (>1 THz) at room temperature rely on slow thermal devices. The realization of fast and sensitive detectors in this frequency range is indeed a notoriously difficult task. Here we propose a novel device consisting of a subwavelength terahertz meta-atom resonator, which integrates a nanomechanical element and allows energy exchange between the mechanical motion and the electromagnetic degrees of freedom. An incident terahertz wave thus produces a nanomechanical signal that can be read out optically with high precision. We exploit this concept to demonstrate a terahertz detector that operates at room temperature with high sensitivity and a much higher frequency response compared to standard detectors. Beyond the technological issue of terahertz detection, our architecture opens up new perspectives for fundamental science of light–matter interaction at terahertz frequencies, combining optomechanical approaches with semiconductor quantum heterostructures.


Supplementary
Higher order mechanical modes. a Blue curve: radio frequency spectrum of the second order in-plane (α 2 ) and out-of-plane (β 2 ) flexural cantilever modes. Red curve: fit resulting from the sum of the oscillator's spectral noise with the noise floor of the balanced photo-diode unit (black dashed curve). b Numerical modelling of the amplitude of deformation u max for each mode.

Supplementary Figure 4|Quantum cascade laser emission spectrum and SRR transmission.
Quantum cascade laser emission spectrum (magenta) superimposed to the transmission spectrum of the SRR arrays from Supplementary Fig. 1b. "x/y-polarization" refers to transmission measurements where the incident electric field is either along the SRR gap ("y"), or along the cantilever ("x"). See also Fig. 2b in the main text. In our experiments, the laser electric field was always along the y-direction.

Supplementary Figure 5| Time traces of the cantilever oscillation.
Time traces representative of the incident THz power and the corresponding mechanical response of the cantilever. In red: quantum cascade laser driving current. In blue: signal from the balanced photo detection unit, recorded on an electronic oscilloscope.

Supplementary Figure 6| Data modelling with purely photothermal forces.
Fit of the amplitude a and phase b of the same data as in Fig. 4e and Fig. 4f in the main text, but assuming purely photothermal forces for both α and β modes. The data is reproduced as continuous lines and the model is dotted curves.

Supplementary Note 1: Terahertz optomechanical coupling and effective capacitance
The coupling between the SRR and the cantilever movement can be described in the picture of an equivalent inductance-capacitor circuit [1,2]. The electric field in the gap shown in Fig. 1a is described by an equivalent capacitor C eff (y) which is a function of the cantilever displacement y. Therefore, the dynamical variables of the coupled system are y, the displacement of the cantilever tip, and q, the charge induced on the capacitor plates. Then the system Lagrangian L is written as: The cantilever movement can then be provided, for instance, by the Lagrange equation for the position: Usually, the displacement of the cantilever y is small compared to the gap d gap , therefore the spatial derivative in the above equation can be approximated at zeroth order with its value at y=d gap . Using the expression for the total electrical energy stored in the resonator W eTHz = q²/2C eff (y), and introducing an effective gap through the formula: we arrive at the results stated in the main text (Eq.(1)), with the function The parameter d eff gap defined above allows evaluating the magnitude of the Coulomb force for an arbitrary geometry. However, its exact analytical evaluation is difficult, due to the fringing fields of the capacitance. To determine d eff gap for our system we performed 2D quasi-static numerical simulations of the cross section of the structure in a plane perpendicular to the cantilever, using a finite element method. We evaluated the electric energy of the system W eTHz , considering that the metallic part of the cantilever has a potential of 1 V, while the opposite metallic side is grounded. Performing such simulations for various positions y of the cantilever we obtained the effective gap d eff gap = 800 nm. The electric field maps obtained in that case were very similar to the full electromagnetic simulation of the THz resonance mentioned in the main text. These simulations provided a negligible Coulomb force for cantilever displacements along the z-direction.
The fact that the effective gap (800 nm) is larger than the physical gap (308 nm, see Fig. 1a) is understandable, since the fringing fields which run away from the structure are expected to be less sensitive to the cantilever displacement compared to the field confined in the gap. Note also that owing to propagation effects in the actual resonator the charge density develops along the whole cantilever length. Here we have considered only the charges in the vicinity of the gap, since the effects of the other parts are expected to be small as the distance between the positive and negative charges is much larger than d gap .

Supplementary Note 2: Modelling of the dynamic photothermal force induced by the eddy currents
The THz eddy currents shown in Fig. 1d induce an inhomogeneous heating described by a temperature profile ∆T(x), with the coordinate system O xyz defined in Fig. 2b. Exceptionally, in this part z is a local coordinate and the cantilever displacement is denoted δz. The first step of our analysis is to determine the total elastic energy stored in the cantilever that corresponds to this heating effect. The elastic energy density can be expressed as u(x,z) = -Y i γ i ∆T(x)ε(x,z) [3] with i = 1 or 2 depending on whether z is in the Au or GaAs part, Y i are the corresponding Young's moduli, γ i the thermal expansion coefficients, and ε(x,z) the deformation. Taking the deformation of a bent cantilever as ε(x,z) =(z-z 0 )/R(x), where z 0 =(t 1 +t 2 )/2 is the neutral axis and R(x) is the local curvature radius, and by integrating over the cantilevers thickness we obtain for the total thermoelastic energy: Here Y is the average Young modulus and we replaced the curvature with its approximate value 1/R(x) =d²δz/dx², the function δz(x) describing the shape of the bent cantilever. Actually, the thermal profile ∆T(x) depends not only on x but z as well. In the following, we determine ∆T(x,z) and we define an average over the cantilever thickness : that can be used directly with the expression for the thermoelastic energy in Supplementary  Eq. (4). To determine the full temperature profile we solve the Fourier heat equation in a steady state with a periodic heat source excitation at the top of the gold layer. This is justified as at THz frequencies heat is generated by the eddy currents within the skin depth of the metal. We first consider a harmonic temperature profile ∆T(x,z)exp(iωt) that satisfies the heat equation: With D=(λ 1 t 1 + λ 2 t 2 )/(c 1 ρ 1 t 1 +c 2 ρ 2 t 2 ) the thermal diffusion coefficient, λ i the thermal conductivities and c i the specific heat capacitances. The boundary conditions applied are: zero heat flow on the lateral surfaces of the cantilever and at its free end, except for the top gold surface. Since we consider only the dynamic part of the temperature profile we take ∆T = 0 at the clamping point. The boundary condition on the gold surface is written: Here: are series of dimensionless coefficients describing the projection of the heat source on the spatial harmonics of the temperature profile. First, we include this expression in the formula of the elastic energy above. Second, we define a dynamical profile if the cantilever vibrational mode δz(x) = z max Q β (x), where z max is the amplitude of the displacement of the free end of the cantilever, and Q β (x) is a dimensionless eigen-function which is a solution of the onedimensional equation for elastic waves [3], normalized such that Q β (y) =1 at the free end. The total thermoelastic energy thus becomes: The function Y(ω) is defined in Eq.(3) of the main text with the coefficients A n =p n r n . Here r n are the dimensionless projection integrals: The expression of the effective photothermal force f ph acting on the cantilever end is finally provided by: Since f ph is proportional to the absorbed power, we can define a force f 0 ph per unit absorbed THz power as in Eq.(4). For the models presented in this paper the absorbed power (Joule heating) is modelled with the spatial profile sin²(xπ/L), which fits the current profiles provided by finite difference domain electromagnetic simulations illustrated in Fig. 1d.