Cavity-less on-chip optomechanics using excitonic transitions in semiconductor heterostructures

The hybridization of semiconductor optoelectronic devices and nanomechanical resonators provides a new class of optomechanical systems in which mechanical motion can be coupled to light without any optical cavities. Such cavity-less optomechanical systems interconnect photons, phonons and electrons (holes) in a highly integrable platform, opening up the development of functional integrated nanomechanical devices. Here we report on a semiconductor modulation-doped heterostructure–cantilever hybrid system, which realizes efficient cavity-less optomechanical transduction through excitons. The opto-piezoelectric backaction from the bound electron–hole pairs enables us to probe excitonic transition simply with a sub-nanowatt power of light, realizing high-sensitivity optomechanical spectroscopy. Detuning the photon energy from the exciton resonance results in self-feedback cooling and amplification of the thermomechanical motion. This cavity-less on-chip coupling enables highly tunable and addressable control of nanomechanical resonators, allowing high-speed programmable manipulation of nanomechanical devices and sensor arrays.


Supplementary Note 1: Opto-piezoelectric backaction
We begin by introducing an equation of motion for the cantilever, in which the optopiezoelectric backaction via e-h pairs is taken into account under constant illumination. The derivation is similar to the case of bolometric backaction 1 and described in detail in Supplementary Ref. 2. Assuming that backaction force F p acts in time delay τ with respect to the motion of the cantilever, the equation of motion can be given by where z is the displacement of the cantilever, z 0 is the averaged position, m is the effective mass, Γ (= ω 0 /Q m ) is the bare damping factor, K (= mω 2 0 ) is the bare spring constant, F th is the thermal Langevin force, F p (z(t)) is the opto-piezoelectric backaction force, and h(t) is the response function given by h(t) = 1 − exp(−t/τ). Here, F p (z 0 ) just leads to a static shift of the cantilever's average position. Thus, by selecting the new average position for z, it can be dropped in the following derivation. 1,2 The z-dependent component, F p (z(t)), is caused by the strain effect via the deformation potential and it modifies Γ and K. Under the small displacement approximation, Supplementary Equation 1 can be transformed in the frequency domain as 1,2 where and ∇F p (= ∂F p (z)/∂z| z=z 0 ) is the force gradient around the averaged position. The frequency component of the amplitude is given with the modified frequency ω eff and Γ eff by 1,2 Here, it can be assumed that F p (z) is proportional to the number of electron-hole (e-h) pairs N(z) that contribute to the opto-piezoelectric backaction. Thus where C piezo is the piezoelectric coefficient. N(z) depends on the excitation power P ex and the energy detuning E from the exciton resonance energy E e . Thus, Supplementary Equation 5 is more generally written as For the fixed photon energy, For the fixed laser power, where C deform = ∂Ee ∂z | z=z 0 , which is related to the deformation potential (-8.8 eV in GaAs) 3 and estimated to be C deform ∼ 9 μeV/nm as described in Supplementary Note 5. Assuming N(E) is proportional to the optical absorption, i.e. to the intensity I of the photoluminescence excitation (PLE) spectrum, then ∇F p (E, z) ∝ dI/dE. That is, the backaction is proportional to the slope of the PLE spectrum. Thus, Supplementary Equation 3a has the form where coefficient C has a negative sign for red-detuning and a positive sign for blue-detuning.
For the modulated illumination, on the other hand, opto-piezoelectric force F p (z(t), t) is given as where ε(t) is the modulation amplitude. The equation of motion is then given by 1,2 where F th is negligibly small compared with the driving force F p , and therefore it can be ignored. The frequency component of the amplitude is approximated by 1,2 which shows the resonance amplitude is proportional to F p . The experimental data under the modulated illumination reveals that the resonance amplitude linearly increases with increasing laser power P ex in the excitation regime of P ex < 10 μW (Supplementary Figure 2). This indicates F p ∝ P ex , suggesting the band flattening caused by the screening effect is negligibly small in this power regime. This also suggests ∇F p ∝ P ex , so that Supplementary Equations 3a and 3b have the forms where C Γ and C K are coefficients with a negative sign for red-detuning and a positive sign for blue-detuning. From the equipartition theorem, the damping factor is related to the mode temperature as 1,2 where T eff can be experimentally extracted from the area of Brownian displacement noise power spectrum. Therefore, the theoretical form of the normalized mode temperature is given by T eff /T = Γ/Γ eff = (1 + CdI/dE) −1 . This means the mode temperature reflects the slope of the PLE spectrum. We experimentally confirm this tendency in Fig. 3c in the main text.

Supplementary Note 2: Additional feedback in the strong excitation regime
The laser power dependence of the damping factor (Fig. 4a in the main text) reveals that the additional backaction with the opposite sign, −∇F a , appears in the strong excitation regime of P ex ≥ 15 μW. This suppresses the opto-piezoelectrically induced damping in the blue-detuning condition. The overall experimental data below P ex < 40 μW can be fitted with the form where A Γ is the coefficient for the additional backaction term and r is the exponent. The fitting provides an non-integer value for r (= 1.2), so it does not straightforwardly help us to understand the phenomenon in the high-power regime. The band screening effect is also included in the high-power regime. Such nonlinear optical properties prevent simple understanding of the additional backaction effect. Therefore, the origin of the additional backaction is not clear at present. The bolometric backaction might be the source, 1,4 which comes from the local thermal expansion that would dominantly occur in the GaAs layer by optical absorption. The lattice expansion in the GaAs layer tends to bend the cantilever in the opposite direction with respect to the opto-piezoelectric force. It could therefore be the source of −∇F a .

Supplementary Note 3: The effect of band-gap absorption
With laser irradiation, there exists a noise source that directly increases the mode temperature. This effect can be described by adding the noise force F b (t) to Supplementary Equation 1. Supplementary Equation 14 is modified by T → T + T b as where T b is the noise temperature coming from F b (t). The experimental result (Fig. 4b in the main text) shows T b is negligible for P ex < 3 μW but not for the stronger excitation. T b is not related to the exciton resonance but comes from the band-gap absorption in GaAs, which is confirmed by setting the photon energy above the band-gap energy. The laserpower dependence of the area of the displacement noise power spectrum for P ex = 1.521 eV exhibits the relation The experimental data of T eff /T can be well traced by Supplementary Equation 17 as shown in Fig. 4b in the main text. Note that B T becomes an order of magnitude smaller when the photon energy is set below the band-gap energy (e.g. B T = 8 ×10 −4 μW −2 for P ex = 1.512 eV), indicating T b originates from band-gap absorption. The effect of band-gap absorption also appears through the offset shift in the spring constant. Supplementary Figure 4a shows the experimentally measured change in the spring constant, (K eff − K)/K, as a function of P ex for red-detuning (1.5145 eV) and blue-detuning (1.5160 eV) from the exciton resonance. It shows the negative offset shift (black curve in Supplementary Figure 4a) plus the positive shift for the red-detuning (red plots) or the negative shift for the blue-detuning (blue plots). By subtracting the offset shift, we can extract the change in the spring constant due to the opto-piezoelectric backaction as shown in Supplementary Figure 4b. The optically modified spring constant, K eff , therefore contains two contributions: the offset shift caused by the band-gap absorption, K b /K, and the optopiezoelectric backaction given by Supplementary Equation 3b, i.e., 6 The experimental result shows that the offset shift is P 0.5 ex dependent, while the shift by the opto-piezoelectric backaction is proportional to P ex . Thus, Supplementary Equation 18 has the form where coefficient C K is negative for red-detuning and positive for blue-detuning. The data fitting provides the coefficients as B K = 2.4 × 10 −4 μW −0.5 and C K = ±3 × 10 −5 μW −1 . Note that the offset shift in Supplementary Figure 4a shows good agreement with the laser power dependence of the spring constant measured for P ex = 1.521 eV (above the band-gap energy, see Supplementary Figure 5). This also verifies that K b originates from band-gap absorption.
High-power excitation potentially causes ohmic loss and thermoelastic damping in addition to the shift of the spring constant, but neither of them leads to the reduction of the damping factor as observed in Fig. 4a. Therefore, it is considered that the reduction of the damping factor due to the additional backaction is not linked to the above mentioned frequency shift caused by the band-gap absorption.

Supplementary Note 4: Delay time of the opto-piezoelectric backaction
Delay time τ of the opto-piezoelectric backaction can be extracted from the change in the damping factor and spring constant as 2 Supplementary Figure 1 shows the laser power dependence of τ for red-detuning (1.5145 eV) and blue-detuning (1.5160 eV) from the exciton energy, which is extracted from Supplementary Figure 4b and Fig. 4a in the main text. The result shows τ is in the range of 0.4 -0.8 μs, which is close to ω −1 0 = 2πf −1 0 = 0.41 μs. This means the backaction force is almost π/2-phase shifted with respect to the oscillation phase, which results in the effective self-feedback effect. 2 Note also that the measured τ is on the order of typical non-radiative recombination lifetime in GaAs. 2 This also suggests that the time delay comes from the spatial separation of electrons and holes in the cantilever.

Supplementary Note 5: Optomechanical coupling strength
In our system, optomechanical coupling is based on the strain-induced modulation of the exciton energy, E e . As in Supplementary Refs. 5 and 6, we can extract the opto-mechanical coupling parameter, ∂E e /∂z, which represents the exciton energy shift per displacement and corresponds to the coefficient C deform in Supplementary Equation 8. In our cantilever, the free-edge displacement of 0.1 nm leads to strain of 1 × 10 −7 at the leg. Taking into account this value and the deformation potential of GaAs (∼9 eV), ∂E e /∂z is estimated to be ∼9 μeV/nm. This is similar to the value in a quantum dot-embedded nanowire system in Supplementary Ref. 6 (∂E e /∂z = 10 μeV/nm), getting a benefit from the large deformation potential effect in GaAs. From the motional mass, m = 1.6 × 10 −13 kg, and the mechanical resonance frequency, ω 0 = 2π× 386.68 kHz, the zero-point motion, z zpf , is estimated to be 1.2 × 10 −14 m. Thus, we obtain the opto-mechanical coupling strength as g 0 = 2λ = (1/ )∂E e /∂z · z zpf = 2π× 25 kHz. The ratio of this parameter to the mode frequency, g 0 /ω 0 = 0.07, is smaller than the unity, i.e., smaller than the ultrastrong coupling limit, 5 but still comparable with the value in Supplementary Ref. 6 (g 0 /ω 0 = 0.17).