Strain-tuning of the optical properties of semiconductor nanomaterials by integration onto piezoelectric actuators

A suitable procedure to introduce strain in optically active nanomaterials—QDs or 2D materials—consists of embedding them in nanomembranes grown/deposited on a carrier substrate. Upon the selective removal of the carrier substrate, these membranes can be then transferred onto the piezoelectric actuator. In this section, we will discuss different procedures in order to obtain nanomembranes consisting of InAs (GaAs) QDs embedded in crystalline matrices, and WSe₂ monolayers embedded in amorphous oxides.

Self-assembled semiconductor QDs in different matrices are obtained by epitaxial growth techniques. In the cases presented in this manuscript, the QDs are grown by MBE in a multi-layer structure (GaAs substrate/Al_0.7Ga_0.3As sacrificial layer (∼100 nm)/nanomembrane (thin active layer containing the QDs)). Two types of QDs will be considered: GaAs and InAs. Specifically, the GaAs QDs are embedded between thin Al_xGa_1−xAs (x < 0.5) layers and are obtained by overgrowing nanoholes fabricated by the droplet etching technique [43]. Their typical emission energy ranges from about 1.55 to 1.75 eV. The InAs QDs are embedded in a GaAs host matrix and present optical emission in the spectral range from 1.33 to 1.39 eV. Interestingly, epitaxial growth techniques allow one to have a very precise control (at atomic scales) on the relative position of the QDs layer with respect to top and bottom surfaces. This, together with the possibility of depositing metallic mirrors on the membrane's surfaces, enables designing thin optical cavities where the QDs can be spatially and spectrally coupled to the cavity modes. The thin active layer containing the QDs has typical thicknesses between 150 and 400 nm depending on the final application.

Nanomembranes containing optically active QDs can be released on the host substrate by selective wet chemical underetching. The procedure consists of four steps: (i) Au rectangular pads are fabricated on top of the epitaxial multi-layer structure by optical lithography and deposition of a bilayer structure (Cr (5 nm)/Au (100 nm)) by thermal evaporation; (ii) non-selective etching with H₂SO₄:H₂O₂:H₂O (1:8:250) of the whole structure down to the GaAs substrate so that the sacrificial layer is exposed; (iii) HF selective etching of the Al_0.7Ga_0.3As sacrificial layer [47] releasing micrometer-sized free-standing nanomembranes coated with Au on the substrate. A sketch of the whole procedure is depicted in figure 2(a).

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2D materials' flakes of different thicknesses are conventionally obtained by the so-called mechanical exfoliation technique developed by Novoselov and Geim to isolate graphene monolayers for the first time in 2004 [48]. This is done by employing a scotch tape to thin down bulk crystals in a repeatable process eventually obtaining a dispersed distribution of crystals with different thicknesses on the tape. Afterwards, the tape is brought into contact with a carrier Si substrate and thin flakes are transferred by Van der Waals forces interaction between the flakes and the substrate. The Si substrate must provide good optical contrast for an easy and fast recognition of monolayer flakes under an optical microscope. This is conventionally accomplished by growing thermally a 280 nm thick SiO₂ layer on top of the Si substrate.

For the processing of WSe₂ monolayers embedded in nanomembranes, a GaAs substrate can be employed on top of which a 270 nm thick SiO₂ layer is previously deposited by plasma-enhanced chemical vapor deposition (PE-CVD). A reference system for flakes localization is used, which consists of gold markers obtained by optical lithography and subsequent Cr/Au deposition. Once the WSe₂ monolayers are exfoliated on the substrate and identified with the help of an optical microscope, a 30 nm thick Al₂O₃ oxide layer is deposited by atomic layer deposition to encapsulate the flakes. This encapsulation provides protection of the flakes for further processing as described below. The substrate employed in this process has to fulfill two basic requirements: the refractive index should be high enough to provide good visibility to identify monolayer flakes and permit a selective removal in order to release the oxide membrane with the embedded 2D flakes. A sketch of the procedure is depicted in figure 2(b). In principle, any encapsulating oxide can be employed for the nanomembrane as long as it is compatible with the rest of the processing in terms of etching selectivity. We have recently performed a systematic study on the effects of the oxide stoichiometry on the optical emission of the encapsulated flakes that should be taken into account to prevent electrical doping of the flakes and obtain similar emission as as-exfoliated uncapped flakes [49].

Once free-standing membranes are obtained on the host substrate, they are transferred and attached onto the piezoelectric actuators by employing flip-chip bonding techniques. For structures with low height/width ratio the stiffness of the interlayer employed to connect the actuator to the membrane plays a minor role on the strain transfer [50], while high stiffness is desirable for high aspect ratio structures. The Au thermo-compression bonding is a suitable technique since it is well-established for semiconductor wafer bonding applications, and Au possesses a high Young modulus value of about 70 GPa which, in principle, is ideal for an efficient strain transfer. In the bonding process, Au-coated parts to be bonded are brought into intimate contact and pressed together while maintaining a temperature typically around 300 °C for 1–2 h. Ideally, due to Au inter-diffusion between both parts a uniform stiff bonding is eventually obtained.

To transfer the membranes discussed in the previous section on PMN–PT substrates coated with a (Cr (10 nm)/Au (100 nm)) thin layer, we follow the procedure sketched in figure 3(a). First, the free standing membranes on the GaAs substrate and PMN–PT substrate are brought into contact while applying a pressure of about 10 MPa at a temperature of 300 °C during 1 h. Upon removal of the GaAs substrate, regularly distributed bonded membranes are ideally transferred on the PMN–PT. It should be noted that wrinkles, curling and/or formation of bubbles often observed on underetched membranes can severely degrade the quality of the bonding, which is especially relevant in the case of large membranes with a size of millimeters. To circumvent this problem, a simple process consisting in Au thermo-compression bonding followed by GaAs substrate back-etching allows for an efficient transfer of flat large membranes. In this case, the underetching step discussed above is avoided and the sample is bonded directly on the PMN–PT substrate, thus ensuring intimate contact between both surfaces and, therefore, a better bonding quality. The substrate back-etching is performed in a three steps process (figure 3(b)): (i) non-selective etching in H₃PO₄(85%):H₂O₂(30%) (7:3) to remove most of the substrate and leave a ∼50 μm thick GaAs layer (∼45 min); (ii) selective etching in citric acid (1:1 weight ratio with deionized water):H₂O₂(30%) (4:1) for the removal of the substrate up to the Al_0.7Ga_0.3As etch-stop layer; (iii) removal of Al_0.7Ga_0.3As layer with HF(49%) during ∼1 min (diluted HF or HCl may be used alternatively for this last step).

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Since the electric field through the piezoelectric actuator is inversely proportional to the substrate thickness for a fixed voltage, it is possible to reduce the working voltages on the actuator by thinning down the piezo to typical values around 300 μm. Unfortunately, mechanical lapping and polishing of PMN–PT material leads to an intrinsic roughness with peak-to-peak values up to ∼10 nm due to a different polishing rate of domains with positive and negative polarities [51]. This eventually prevents intimate contact between PMN–PT and sample during the bonding process and the formation of gaps at the interface [18, 50]. Some alternatives to circumvent this problem like employing polishing liquids presenting pH factors of 2 has been recently demonstrated [51]. Figure 4(c) shows a cross-section scanning electron microscopy (SEM) picture of a GaAs nanomembrane with embedded InAs QDs bonded by gold thermo-compression bonding on a piezoelectric PMN–PT substrate as prepared by focused ion beam cutting. Clear gaps in the joint layer are visible and attributed to a non-uniform contact between both parts. Alternative approaches to attach nanomaterials to the actuators, as well as comparative studies, are discussed in the following.

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Besides a non-uniform bonding, one of the major issues of gold thermo-compression bonding when working with brittle materials like PMN–PT is the relatively high mechanical pressure values required (∼10 MPa). In this regard, polymer-based bonding is a good choice since pressures in the kPa regime are usually employed and provides a voidless bonding layer contrary to gold thermo-compression bonding. Among available polymers, poly(methylmethacrylate) (PMMA), cyanocrylate and SU-8 have been successfully used for bonding applications [52–54]. The emission energy of the excitonic transitions in self-assembled QDs embedded in semiconductor nanomembranes is very sensitive to the strain field at the QD location and can be therefore used as gauge to monitor the transferred strain. A total energy shift of the exciton transition in InAs QDs embedded in GaAs membranes of about 5 meV has been reported when using Au bonding layers and electric fields varying from −25 to 55 kV cm⁻¹ [18, 55]. Unexpectedly, for similar electric fields (voltages) on the actuator, the same energy shift is obtained in the case where the membranes are bonded to the piezoelectric substrate employing cyanoacrylate [56]. The worse strain transfer was reported when using PMMA as bonding layer (with emission energy shifts of about 1.4 meV), attributed to a lower Young modulus of PMMA, in comparison with the others, that combined with the relatively large height/width ratio of the used structures, lead to lateral strain relaxation [57].

SU-8 is a polymer photoresist widely used in MEMS applications that can be patterned by lithography techniques, which also allows a voidless wafer bonding for on-chip applications [54, 58, 59]. Most importantly, the SU-8 hardens upon UV exposure or thermal treatment at temperatures above the glass transition (∼220 °C), leading to Young modulus values up to 5 GPa [58, 60]. The latter, together with high optical transparency for wavelengths above about 400 nm make it very promising for optoelectronic applications, where relatively high bonding strengths are required. We will therefore compare the strain transfer capabilities when SU-8 is employed as bonding layer. The SU-8 processing for the devices presented in this manuscript consists in several steps. First, a 500 nm thick SU-8 layer is spin-coated on the PMN–PT substrate or the sample, depending on the specific application, and soft cured for 5 min at 90 °C to evaporate solvents from the layer. The as-prepared coated PMN–PT (or sample) is afterwards brought into contact with the sample (or PMN–PT) and pressed at about 10 kPa while maintaining a temperature of 220 °C for 15 min (hard cure of the resist). Figure 4(b) shows a SEM cross-section image of a GaAs nanomembrane bonded with SU-8 on a gold-coated PMN–PT substrate. Interestingly, a uniform and thin 150 nm thick bonding interlayer can be observed. A comparative study on the strain transfer from the PMN–PT to the bonded GaAs nanomembrane by Au thermo-compression and SU-8 bonding has been recently reported by performing x-ray diffraction (XRD) studies, where a much better strain transfer efficiency of ∼69% is observed on SU-8 bonded nanomembranes with respect to gold thermo-compression bonding (∼25%) (figure 4(a)) [50]. These findings, which are compatible with those of [61] for epoxy, demonstrate that polymer-based SU-8 bonding is a more suitable technique in comparison with the others, especially for samples presenting relatively high surface roughness like the case of PMN–PT. The remarkable strain transfer on membranes bonded by SU-8 is attributed to the excellent wettability of the polymer when heated up during the bonding process, which fills the gaps between the PMN–PT and the membrane. Moreover, the observed strain transfer efficiency is higher than that reported on epitaxially grown SrTiO₃:Ni²⁺ or NdNiO₃/SrTiO₃ or thin films on PMN–PT substrates [62, 63], and similar to epitaxial La_0.335Pr_0.335Ca_0.33MnO₃ films [64] with a thickness 10 times lower than the GaAs membranes employed in [50]. The differences reported above between gold and cyanocrylate can be attributed to a voidless bonding layer in the latter, similarly to the SU-8 bonding. Figure 5(a) illustrates the approach used to bond Ga(Al)As nanomembranes containing QDs on PMN–PT substrates by SU-8. Contrary to gold thermo-compression bonding, large membranes with lengths up to centimeters can be easily obtained at a lower temperature and mechanical pressure.

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In principle, the best strategy to maximize the strain transfer is to perform the epitaxial growth of films directly on PMN–PT substrates. SrTiO₃ (STO) is known to be a suitable material as a substrate for the epitaxial growth of many complex oxide films. STO has also been used as intermediate layers between PMN–PT (001) and other materials, as it reduces the lattice mismatch between the substrate and the epilayer [63, 65]. For instance, modulation of metal–insulator phase transitions in NdNiO₃ and magnetization control in La_0.7A_0.3MnO₃ thin films subject to strain have been successfully demonstrated upon epitaxial growth on PMN–PT substrates [44, 63]. In addition, strain engineered tuning of upconversion PL in BaTiO₃ films doped with Yb/Er ions and strain-tunable luminescence in STO:Ni thin films have been recently reported [62, 66]. Unexpectedly, the strain transfer magnitudes reported in these works do not exceed 46% efficiencies, which is attributed to the semi-coherent character of the interface between PMN–PT and the grown film that allows for strain relaxation mechanisms in the strained film, i.e. reduced strain transfer [62, 63, 65].

Another example of epitaxial growth of Mn²⁺ doped ZnS films on PMN–PT piezoelectric substrates is reported in [67]. Upon application of dynamical and reversible ac and dc electric fields on the PMN–PT, a strain-induced piezoelectric potential in the ZnS:Mn film is obtained due to the so-called piezo-phototronic effect proposed by Wang [68]. This effect refers to the possibility of tuning the optical properties of materials presenting non-central symmetric structure by a strain-induced piezoelectric potential.

As mentioned above, the visibility of atomically thin 2D materials on the carrying substrate eventually depends on the optical contrast given by the optical reflectivity of the 2D flake with respect to the top PMN–PT substrate's surface. This is especially relevant for the case of mechanically exfoliated flakes with typical sizes between 5 and 20 μm. A suitable procedure consists in growing a transparent oxide layer with the right thickness for maximum visibility in the range of the visible wavelengths around 550 nm. In [69] graphene flakes have been transferred on a 1 μm thick SiO₂ coated PMN–PT substrate, where a thin 60 nm thick PMMA layer is deposited prior to the direct flake transfer by mechanical exfoliation employing Scotch tape. The PMMA layer is afterwards hardened by annealing at 120 °C to ensure efficient strain transfer from the piezoelectric substrate to the attached 2D flake [69].

Large 2D flakes grown by CVD can be directly transferred on top of PMN–PT with no need of any intermediate oxide layer since localization of the flake is not an issue. In [70], trilayer MoS₂ flakes are first grown by CVD on a sapphire substrate and coated by spinning a PMMA layer on top. The coated flakes are then heated at 100 °C for curing the PMMA, and the substrate is selectively etched by using NaOH solution. This process results in a PMMA film with embedded trilayer MoS₂ that can be afterwards transferred on the PMN–PT substrate, previously treated with O₂ plasma to facilitate the adhesion of the film. The PMMA is removed by acetone, isopropyl alcohol and deionized (DI) water treatment. The top electrode for applying electric fields on the PMN–PT is obtained by simply transferring an optically transparent graphene on top of the structure.

An additional strategy is proposed here to integrate 2D flakes onto PMN–PT actuators. It consists in the attachment of an oxide nanomembrane containing encapsulated 2D flakes on the PMN–PT substrate by SU-8 bonding. Figure 5(b) shows a sketch of the nanomembrane with embedded 2D flakes on a GaAs (001) substrate attached to a PMN–PT substrate by SU-8 bonding. To release the membrane on the piezoelectric actuator, the GaAs substrate is removed by employing a wet chemical etching with H₃PO₄(85%):H₂O₂(30%) (7:3), which is selective against the SiO₂ layer.

Strain has a profound effect on the band-structure of semiconductors. In particular, it modifies the energy bandgap of a semiconductor. This effect has been used to control the emission energy of QDs integrated onto piezoelectric actuators. In 2006, Seidl et al reported on the first demonstration of emission energy and FSS tunability in self-assembled QDs by applying uniaxial stress fields parallel to the QD growth plane [83, 84]. In this work, it is successfully demonstrated that uniaxial strain can modify the anisotropies of the exciton wavefunction, and hence the FSS, with negligible decrease of the oscillator strength. In spite this work did not demonstrate suppression of the FSS, it signed the starting point of an intense research activity aiming at investigating the properties of QDs under elastic strains. This approach is based on a PZT piezoelectric stack that allows introducing strain only along a fixed direction so that the strain anisotropy cannot be controlled. Additionally, the relative thick membrane employed in these experiments (0.5 mm) severely limits the magnitude of the exerted stress (∼20 MPa). In this regard, nanometer-thick membranes containing self-assembled QDs integrated onto PMN–PT actuators are an ideal platform to efficiently introduce strain fields.

The first work performed with QDs onto PMN–PT substrates was performed by Zander et al [57], and deals with stretching microring structures with embedded QDs and is discussed in section 3.1.2. The tuning of the excitonic optical transitions in QDs may find interesting applications for quantum networking, in particular when the emission energy is tuned to the D₁ (or D₂) absorption lines of a cloud of rubidium atoms [85]. The main reason is that the atomic cloud can be used as quantum memory for QD photons so as to store and retrieve quantum information, an important ingredient for long distance quantum communication. In the first pioneering work on this hybrid atom-QD interface [83], the authors demonstrate that single photons from GaAs QDs can be slowed down (not stored) when their frequency is finely tuned to the middle of the hyperfine D₁ lines of ⁸⁷Rb. Energy tuning in this experiment was achieved via magnetic field, an approach requiring bulky equipment and hardly suitable for the envisioned applications. In order to overcome this hurdle, it is possible to integrate high quality GaAs QDs onto PMN–PT actuators and tune their emission energy into resonance with a cloud of ⁸⁷Rb atoms by simply using a power supply, as shown in [56]. Figure 6 shows the energy shift of several excitonic lines stemming from a GaAs QD for a varying electric field applied on the actuator. The total energy shift achieved is ∼10.5 meV, a value which is comparable with the inhomogeneous broadening of the QD ensemble. Therefore, it is virtually possible to bring into resonance any arbitrarily selected QD in the sample. It should be noted that the energy shift of the excitonic transitions is mainly due to the strain-induced change of the bandgap of the material, which is proportional to the hydrostatic part of the induced strain field $({\varepsilon }_{{\rm{h}}{\rm{y}}{\rm{d}}}={\varepsilon }_{xx}+{\varepsilon }_{{\rm{y}}{\rm{y}}}+{\varepsilon }_{zz}).$

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These results pave the way towards the implementation of hybrid-atomic interconnects with a scalable technology. A further step in this direction has been accomplished in [86], where the need of an external laser is avoided by injecting carriers into the QD electrically. This is possible by embedding the strain-tunable GaAs QDs into light-emitting diodes (Q-LED) operated in forward bias. The authors demonstrate that it is possible to integrate QD-LEDs onto PMN–PT without degrading the operation of the LED and maintaining the compatibility with the Rb transitions. Recently, the tunability of the excitonic species in QDs integrated on monolithic devices has been successfully exploited in two-photon interference experiments as a first step towards the realization of entanglement swapping between distant QDs [87].

The efficient extraction of light from QDs by their integration in photonic nanostructures is mandatory for many practical applications involving high yield single photon sources and fundamental quantum-electrodynamics experiments [88–90]. In this section, we will review recent advances on strain-tunable optical properties in optical micro-cavities [57, 91, 92] and nanowire waveguides upon integration on monolithic PMN–PT actuators [93, 94].

In [93], the fabrication of a nanowire waveguide is performed by etching nanowire structures by reactive ion etching technique on a nanomembrane containing InAs QDs. The membrane is previously bonded on a gold-coated PMN–PT substrate by gold-thermocompression bonding as shown schematically in figure 7(a). In this case, the membrane is released from the substrate by selective etching of a Al_0.65Ga_0.35As sacrificial layer (see section 2.2 for further details about the processing). The gold layer has a thickness of 200 nm and serves as electrical contact and optical mirror. The QDs are located at a distance D = 110 nm from the mirror, which is found to reduce spectral diffusion due to surface states between the nanowire and the Au mirror. These nanowires structures feature single photon emission with large light extraction efficiency values up to 57% for QDs located at the center of the pillar and optimized structural parameters for the pillar geometry. In particular, in these experiments, a nanowire diameter d = 223 nm is used for an optimized light extraction at the X emission wavelengths of the ground state in the QDs. Finite element simulations on strained nanowires in figure 7(b) show that the strain transfer from the piezoelectric substrate to the pillar decays along the longitudinal direction and the sign is changed from compressive, at the base of the pillar, to tensile due to strain relaxation, which is increased by reducing the pillar diameter. Hence, the magnitude and sign of the strain at the QDs position will eventually depend on their relative position inside the nanowire. Figure 7(c) depicts the X energy shift when the voltage on the PMN–PT substrate is varied up to corresponding electric field values of ∼33 kV cm⁻¹ for several QDs. The slopes obtained for each QD are clearly different and attributed to different radial positions of the investigated QDs inside the nanowire. This is reasonable considering the large non-uniformities in the strain distribution at different positions in the nanowire where the QDs a located. It is reported a maximum X energy shift of about 1.2 meV that can be reversibly applied with no significant hysteresis.

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Another recent work reports on strain-tunable optical emission from InAsP QDs embedded in InP nanowires obtained by vapor–liquid–solid epitaxy technique [94]. In this case, the position of the QD can be controlled deterministically during growth and the geometrical parameters like tapering angle, diameter and length of the nanowire can be controlled precisely by changing the growth conditions in order to enhance the light extraction efficiency up to 42% [95]. Single photon emission in such nanowires is successfully demonstrated. For their integration onto PMN–PT piezoelectric actuators, a direct-transfer approach with a nano-manipulator is employed (figure 7(d)). Interestingly, the authors report a similar X emission energy tuning range up to ∼6.3 meV (F_p = 25 kV cm⁻¹) on as-transferred nanowires on the actuator by van der Waals forces interaction and encapsulated nanowires by depositing a structured Si₃N₄ oxide layer on top (figure 7(f)). The tuning of the X emission from distant nanowires integrated on different chips is shown in figure 7(e). Another interesting aspect is that, upon nanowire transfer, the growth axis of the nanowires is parallel to the piezoelectric substrate's surface which allows in-plane optical emission that might find interesting applications for on-chip integrated devices.

Contrary to the case of nanowire waveguides that allow broadband wavelength operation, in optical micro-cavity structures there are two stringent requirements for an efficient light extraction: spectral and spatial matching of the QD with the cavity mode. The latter can be performed deterministically by in situ lithography or QD location followed by lithography after growth [96–99]. In these approaches, the position of the QD is known before the fabrication of the cavity structure. Other approaches based on the definition of the QD nucleation sites by ex situ lithography/patterning techniques with nanometer resolution [100, 101], followed by regrowth procedures have also been reported [101]. The spectral tuning of the QD emission with the cavity modes is conventionally realized reversibly by applying magnetic fields [102], electric fields [103] or varying the temperature [104]. However, electric fields lead to PL quenching due to an increased electron and hole wavefunctions separation [105] and temperature tuning is detrimental for the emission efficiency and induce dephasing due to the interaction of the excitons with phonons.

Strain-tuning to frequency-match the QD photons with the cavities modes by employing piezoelectric actuators has been demonstrated as a suitable strategy to overcome these problems in an elegant manner. The key ingredient is related to the fact that the QD and the cavity mode shift at a different rate when stress is applied.

Strained QDs inside micro-ring optical cavities have been reported by integration on PMN–PT actuators in [57]. The sample consists of an epitaxial 250 nm thick GaAs layer—containing InAs QDs located at the center—on a Al_0.7Ga_0.3As sacrificial layer grown by MBE on a GaAs(001) substrate. The micro-ring structures are first fabricated by electron-beam lithography leading to patterned features replicating the ring-like shape, which is used as a mask to etch the material around the structures down to the substrate. Figure 8(a) shows a SEM picture of the etched micro-ring where the GaAs layer with embedded QDs and the Al_0.7Ga_0.3As sacrificial layer are well distinguished. The micro-rings are released from the substrate by underetching the sacrificial layer with HF (figure 8(b)) and transferred onto the PMN–PT actuator by PMMA bonding. A 1 μm thick SiO_x layer is deposited on the PMN–PT by thermal evaporation to prevent light absorption losses. Finally, the bonded microrings are coated with a thin PMMA layer to help for an efficient strain transfer. An optical image with an array of microrings bonded on the PMN–PT substrate is shown in figure 8(c).

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These micro-rings cavities present a quality factor exceeding 12 000. Interestingly, the X emission energy (E_x) and optical mode energy (E_M) shift almost linearly with the applied voltage on the PMN–PT with a total shift value of about ±3.8 meV for E_x (F_p ∼ 74 kV cm⁻¹) as shown in figure 8(d). The relatively low strain values with respect to other reported works are attributed to a much softer PMMA bonding layer that does not allow an efficient strain transfer. Most interestingly, E_x and E_M shift in the same direction but with different rates which means that the matching condition E_x = E_M can be reached in order to tune into resonance the QDs emission with the cavity mode. No PL emission linewidth broadening or quenching, usually observed when using electric fields or temperature to tune the optical emission, are observed. As expected, there is some dispersion in the energy shift values for different QDs associated with non-uniformities in the strain field distribution, similarly to the results from the etched nanowire structures shown above. The change of E_x with strain is attributed to the bandgap tuning of the material, being larger (smaller) for compressive (tensile) strain fields, whereas the change of the cavity modes under tension is attributed to both increased size of the resonator and increase of the refractive index because of the photoelastic effect [57]. Similar experiments based on microdisks and reaching also the strong coupling regime were reported in [106].

Photonic crystal optical cavities with embedded QDs have been demonstrated as an efficient strategy for enhancing light extraction. Such a structure essentially consists of a defect in a periodic arrangement of holes fabricated by e-beam lithography followed by reactive ion etching around a single QD in a suspended semiconductor membrane. The design of the holes size and arrangement as well as the thickness of the membrane is selected to couple the optical emission of the QD with the cavity mode. In [91], a L3 photonic crystal cavity is fabricated (quality factor Q ∼ 3310) on a sample grown by MBE on a GaAs(001) substrate that consists of a GaAs nanomembrane with InAs QDs at its center and a Al_1−xGa_xAs sacrificial layer with high Al content (x ∼ 0.7). The GaAs substrate is selectively removed by combining mechanical polishing and wet chemical etching. The resulting slab is afterwards bonded on a PZT piezo stack actuator by using an epoxy resin and the photonic crystal structure is underetched by using HF. The defect line of the photonic crystal is aligned along the main stretching direction of the actuator that allows exerting almost uniaxial strain fields. A sketch of the final device is depicted in figure 8(e). Upon application of a voltage on the piezo actuator from +400 to −300 V (strain variation up to 0.08%) the QD emission lines shifts in energy 5 times faster than the cavity mode. This is relevant, as it allows one to couple the QD emission to the cavity mode as reported in the inset to figure 8(e).

Another approach to introduce strain in photonic crystals is reported in [92]. In this case, the sample containing the photonic crystal (quality factor Q ∼ 12 000) is installed on a device that consists of a L-shaped copper holder coupled to a PMN–PT piezoelectric substrate as shown in figure 8(f). This configuration allows applying uniaxial stress fields. Similar to the previous case, the row defect of the cavity is aligned along the direction of the applied uniaxial stress. The QD exciton emission and cavity mode shift as a function of the electric field applied on the PMN–PT up to 15 kV cm⁻¹, as shown in the inset to figure 8(f). An anticrossing of the cavity mode and QD emission is observed at a field 7.8 kV cm⁻¹, a clear indication of a QD-cavity coupling in the strong coupling regime. Unexpectedly, a redshift instead a blueshift is observed on the QDs emission under compressive strain, a fact probably arising from curling or wrinkling of the suspended structure. It should be noted that also fluctuations in the In content of the QDs may give rise to blue-shifts of the emission under compression, see [107]. Due to a non-uniform distribution of the applied stress field, a different tuning range is observed on various QDs. In line with the works mentioned above, the average shift of the QD emission is higher than the cavity mode with a QD to cavity mode tuning ratio of about 5.8.

The tuning of the FSS with PMN–PT actuators was reported later on in [43], where excitons confined in quantum-well potential-fluctuations (QWPFs) are studied. In this experiment the nanomembrane is bonded on the side of a PMN–PT (001) substrate, which permits delivering highly anisotropic strain fields ${\varepsilon }_{\perp }\approx -0.7\times \varepsilon$ (see figure 1) that are more suitable to modify the anisotropy of the QD confining potential. Figure 9(a) shows the PL polarization-resolved maps for the exciton emission from a QWPF where the splitting of the X bright states (FSS) is well resolved. In addition to the inherent anisotropies in the confining potential in the as-grown QDs, the fabrication process employed to transfer the membrane onto the piezoelectric actuator increases the anisotropy of the strain field in the QD. While continuously varying the applied electric field in the piezoelectric actuator from 33 to −20 kV cm⁻¹, the two split X states undergo a clear anti-crossing accompanied by a rotation of the polarization direction of the exciton emission, which mainly occurs when the FSS reaches its minimum value (see figure 9(b)). Most importantly, the polarization angle saturates for a large strain magnitude, which roughly corresponds to the major strain axis direction of the strain field (strain direction). Based on a continuum model, the observed effects can be attributed to substantial changes of the hole states which eventually condition the polarization of the emission. We refer the reader to [43] for further details. The same trend is observed in the case of GaAs QDs and InAs QDs, as reported by the authors within the same work. A maximum FSS tunability for GaAs QDs and InGaAs QDs of ∼70 μeV and 25 μeV are found, respectively. It should be noted that in this experiment the anisotropy and direction of the strain field are fixed and only the magnitude of the applied stress is varied. In turn, this implies that X bright states generally undergo anti-crossing versus the applied voltage, as the major strain axis is not properly aligned to fully compensate for the QD structural asymmetries.

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Monolithic piezoelectric actuators can be employed to tune the optical emission from InAs QDs through the two absorption lines of D₁ transitions in a Cs vapor of atoms, which has interesting applications as a spectrally selective delay line for single photons [108]. Figure 10(a) shows a sketch of the Cs D₁ transitions that are split into the 6 ²S_1/2 and 6 ²P_1/2 levels. These levels are further split due to hyperfine coupling into levels with total atomic angular momentum F = 3 and F = 4. The optical transmission around the D₁ lines reveals the absorption due to the characteristic four transitions of the hyperfine structure for different vapor temperatures (T_Cs), as shown in figure 10(b). Only two absorption dips separated by 10 GHz are resolved for T_Cs > 100 °C due to Doppler broadening. The splitting of the D₁ lines is significantly larger than the one observed for ⁸⁷Rb atoms [86]. This is advantageous when working with QDs featuring relatively large bandwidth. Moreover, if the FSS value of a given QD is larger than the splitting between the two absorption Cs D₁ lines (41 μeV), it is possible to delay selectively each of the orthogonally polarized split components, horizontal (H) and vertical (V), for the X or XX lines, which would open the possibility of realizing time reordering of both lines. We refer the reader to [108] for further information. Figure 10(c) shows the X split bright states in a QD with a FSS ∼ 58.7 μeV. The corresponding PL amplitude when each component is tuned through the Cs D₁, upon the application an electric field (F_d) on a diode structure, is reported in figure 10(d). When one of the components is tuned to Cs D₁, it can be selectively delayed with respect to the other, as shown in the time-resolved measurements in figures 10(e)–(h). The time delay depends on the Cs atomic vapor temperature, being larger as the temperature is increased. The delay times reachable for a vapor temperature of 143 °C can be as high as 4.9 ns. The observed temporal distribution of the arriving photons is also modified and can be fully explained by considering the spectral broadening of the QD emission and the atomic lines.

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Figure 21(a) shows a sketch of GaAs nanomembrane bonded a 2-legged actuator coated with Au (side A, electrically grounded). The micro-machining of the PMN–PT piezoelectric actuator is performed using a femtosecond laser with 350 fs pulse duration, 25 KHz repetition rate and 6 μJ pulse energy. The sample employed in this work consists of a (GaAs(001) substrate/100 nm thick Al_0.7Ga_0.3As sacrificial layer/330 nm thick GaAs) layered structure grown by MBE. The fabrication procedure is similar to the case of a monolithic device: the sample is bonded on the actuator (gold thermo-compression bonding) by carefully aligning the [110] GaAs crystal direction with respect the [100] direction of the PMN–PT actuator (direction defined by the two legs) followed by a selective wet chemical etching to remove the whole substrate. On the other side (side B), only the legs are coated with Au so as to apply independent voltages on each of the legs. The photoluminescence (PL) measurements are performed in the suspended GaAs region between the legs on an area of around 1 μm (size of the laser spot). The working principle of the device is based on the application of a voltage on the two micro-machined legs, while the top side is set to ground (i.e. an electric field is induced across the piezoelectric material). This produces an out-plane deformation on the piezoelectric legs—in-plane deformation due to the Poisson effect—that in turn introduces an in-plane strain field on the bonded nanomembrane. The tensile/compressive character of the strain field can be controlled by the sign of the applied voltage and by the poling direction of the piezoelectric substrate.

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Semiconductor GaAs is a well-known direct band-gap material with bright PL and known response to deformations. More specifically, the polarization properties of light emitted by the recombination of light- and heavy-hole free excitons in GaAs can be used as a stress gauge. Therefore, standard polarization-resolved micro-PL measurements combined with k.p theory can be used to estimate the strain status of a GaAs nanomembrane subject to arbitrary stress fields, with a spatial resolution of the of order of 1 μm. Since the piezoelectric actuator exerts stress only in the (001) plane of the GaAs crystal, only the in-plane stress is relevant here. The stress status can therefore be fully characterized by the three components of the stress tensor (S_xx, S_yy, S_xy) or, equivalently, by the principal major (S₁)/minor (S₂) stresses and an angle (θ_S1) between S₁ and an arbitrary reference direction (GaAs [110] crystallographic direction in this case). This direction is also taken as a reference for the PL polarization-resolved measurements.

Unstrained GaAs presents degenerate light-hole (LH) and heavy-hole (HH) valence bands at the Γ point of the Brillouin zone and, upon the recombination of light- and heavy-hole free excitons, a single unpolarized PL emission peak should be ideally observed. On the other hand, an in-plane stress (strain) breaks the crystal symmetry and removes the degeneracy between the two, leading to two emission peaks. Polarization-resolved PL maps for different voltages applied on both legs are shown in figure 21(b). The PL spectra show two PL peaks indicating that the nanomembrane at the point of measurement (gap between the legs) is pre-strained, which is likely due to bonding and device processing. Since in the presence of arbitrary in-plane strains the two observed transitions involve an admixture of HH and LH bands, the low and high energy PL peaks are referred as E₁ and E₂, respectively. The splitting in energy between the two free excitons is not the only experimental observable that allows the strain status of the membrane to be estimated. In fact, because of band mixing, the light collected along the perpendicular direction to the membrane is elliptically polarized for both E₁ and E₂ (see figure 23(d)), where the polarization angle of the E₁/E₂ components corresponds to the direction of the minor/major stress axes. Hence, the stress state is fully encoded in the energies E₁, E₂ and the polarization angle (φ). First, the corresponding direct problem is solved by calculating the expected PL spectra for a given stress configuration. This is performed by making use of the 8-band k · p theory where the stress is introduced via the Pikus–Bir Hamiltonian, and the dipole approximation is used for the treatment of the optical properties. The Hamitonian is diagonalized at the point of the Brillouin zone where the minimum/maximum of the conduction/valence bands are located (Γ point), respectively, to obtain the single-particle energies of electrons and holes and the corresponding eigenvalues. The latter is used to extract the transition probabilities as a function of the polarization angle.

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The experimentally measured energies E₁, E₂ and polarization angle φ are then used as input data for a non-linear least-square minimization algorithm capable of retrieving the stress state (S₁, S₂, θ_S1) [29] of the nanomembrane which minimizes the deviations between measured and calculated E₁, E₂, and φ values. The first guess for the fit is obtained from a semi-analytical solution of the Pikus–Bir Hamiltonian without inclusion of the split-off band. The split-off band is then included during the least-square minimization. Self-consistency and uniqueness of the solution were thoroughly checked.

In the case of the 2-legged actuator presented in figure 21, the calculated pre-stress value at zero voltage is characterized by principal stresses S₁ = 150 MPa and S₂ = 76 MPa with S₁ forming an angle θ_S1 = 62° with the [110] crystal direction. An anticrossing between the two free-excitons (with a minimum energy splitting of about 6 meV) is obtained, indicating that the induced stress is not aligned with the pre-stress axes and therefore, the unstrained configuration cannot be reached with these devices (figure 21(c)). The induced major stress angle with respect to the [110] GaAs crystalline direction as a function of the applied voltage is shown in figure 21(d). It should be noted that the pre-strain present in the suspended nanomebrane at the gap between the legs upon processing of the device can induced a slight arbitrary bending of the membrane that should be first compensated. The latter is observed when the major stress angle approaches a zero value (uniaxial character) for voltages larger than 100 V (tensile stress fields), which is the regime where the bending of the nanomembrane is negligible. The angle values dispersion observed for negative voltages (compressive stress fields) is ascribed to an enhanced buckling/bending of the nanomembrane. In this regime, assuming bending of the nanomembrane, the induced stress cannot be properly controlled by the device. The calculated induced hydrostatic stress (S₁ + S₂) and anisotropy (S₁ − S₂) for each of the voltages are shown in figure 21(e). The geometry and gaps between the micro-machined actuating legs eventually condition both tunability of the stress field anisotropy and its magnitude, respectively. Specifically, the latter depends on the ratio between the length of the actuating legs and the gap between them. Hence, by keeping a constant legs length (∼1500 μm), the achievable stress magnitude uniquely depends on the gap between the legs. In the case of the 2-legged actuator, a highly anisotropic stress up to S₁ − S₂ = 263 MPa and hydrostatic stress up to S₁ + S₂ = 419 MPa are calculated for the tensile regime (400 V). The deviation from a purely uniaxial stress is attributed to a possible non-uniform bonding on top of the legs. Remarkably, these values are larger than those obtained for the 6-legged device even for a much lower electric field on the PMN–PT actuator (13 kV cm⁻¹), as shown in section 5.1.2. This is due to a smaller gap between the legs in the case of the 2-legged actuator (∼80 μm) with respect to the 6-legged actuator (∼140 μm).

Figure 22 shows the PL emission shift of the exciton X and multi-exciton (MX) transitions from a GaAs QDs embedded in a AlGaAs nanomebrane integrated onto a 2-legged piezoelectric device. A total PL shift of 41.5 meV is found for this particular device and for a range of applied electric fields up to 22 kV cm⁻¹. Interestingly, due to the large optical emission tunability of such device, the optical emission of as-grown GaAs QDs can be tuned reversibly through both D₁ and D₂ transitions which is very interesting for applications in storage of single photons in thermal Rb atomic clouds [85, 86]. (See also section 3.1.4.) Recently, a QDs based atomic quantum memory at the Rb D₁ line has been successfully demonstrated [127].

In [29], a novel class of piezoelectric actuators that are capable of controlling the three components of the in-plane stress tensor in semiconductor nanomembranes is demonstrated. Most importantly, arbitrary stress configurations including uniaxial and isotropic/anisotropic biaxial stress fields along any direction can be produced. In the following, a variety of micro-machined actuators featuring different designs are presented and their suitability to deliver arbitrary stress fields with relatively high magnitudes is discussed. Generally speaking, these actuators allow exerting strain fields at least one order of magnitude higher than those obtained with the monolithic actuators presented in the previous sections.

The first example we discuss here is a '3-legged device', i.e., a micro-machined piezoelectric actuator featuring three legs oriented radially along directions 60° away from each other, and with a GaAs nanomembrane bonded on its top surface (see figure 23(a)). The legs have a 300 × 300 μm² cross-section and a length of 1500 μm.

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Achieving full control of the in-plane stress field means that the three stress field components can be controlled independently. This can be accomplished by employing three independent 'tuning knobs' capable of generating three independent stress configuration whose axis are not all parallel to each other. In our device, these knobs are provided by the three micron-sized piezoelectric legs. In this case, the working principle of the device is based on the application of three independent voltages (V₁, V₂, V₃) on the three micro-machined legs, while the top side is set to ground.

Figure 23(b) shows an optical image of the bonded GaAs nanomembrane in which the underlying three piezoelectric legs can be distinguished. The PL signal is measured at the central gap between the three legs (white point corresponding to an area with a diameter of ∼10 μm). A cw HeNe laser (632.8 nm) is used for excitation. The PL spectrum at 10 K at the central gap is shown in figure 23(c) for no voltages applied to the piezo-legs. This observation clearly indicates the presence of pre-strain in the nanomembrane, likely due to bonding and device processing (see section 5.1.1). Figure 24(a) shows the corresponding PL spectra shift versus V₃ while keeping the voltages applied to the other legs to zero.

Figure 24(a) shows the corresponding PL spectra shift versus V₃ while keeping the voltages applied to the other legs to zero. From the splitting of the two free exciton at V₃ = 0 it is possible to estimate a pre-stress of S₁ = −8 ± 1 MPa, S₂ = −78 ± 2 MPa, θ_S1 = 110 ± 1°. When V₃ is varied, a clear anticrossing between the two free-excitons is observed (with a minimum splitting value of 2.5 meV), a clear signature that leg 3 is not able to compensate alone for the pre-stress arising during device fabrication. However, by properly tuning V₁ and V₂ it is possible to reach a configuration in which the stress field exerted by the third leg is applied along the same direction of the pre-stress axes. In this case, the symmetry of the crystal can be fully recovered and a crossing of the PL components is observed as V₃ is swept (figure 24(b)). As expected, un-polarized light is measured at the crossing point. Moreover, a 90° rotation of the polarization angle is found on both PL components when the stress is tuned through the crossing point.

Assuming linearity in the elastic strain theory, the total stress state can be expressed as:

$$\begin{eqnarray}&&{\bar{{S}}}_{{\rm{t}}}(\bar{{V}})={\bar{{S}}}_{{\rm{pre}} \mbox{-} {\rm{stress}}}+{\bar{{s}}}_{{\rm{induced}}}(\bar{{V}})={\bar{{S}}}_{{\rm{pre}} \mbox{-} {\rm{stress}}}+\overline{\overline{{T}}}\cdot \bar{{V}},\end{eqnarray} \tag{ 7 }$$

where ${\bar{{S}}}_{{\rm{t}}}=({{S}}_{{xx},}\,{{S}}_{{yy},}\,{{S}}_{{xy}})$ are the stress tensor components in Voigt notation, ${\bar{{S}}}_{{\rm{pre}} \mbox{-} {\rm{stress}}}\,$the stress at zero applied voltage, and $\overline{\overline{{T}}}$ is a 3 × 3 'transfer matrix'. The stress induced by the actuator can be then calculated by subtracting the pre-stress in the membrane to the total stress $({\bar{{s}}}_{{\rm{induced}}}=\bar{{S}}-{\bar{{S}}}_{{\rm{pre}} \mbox{-} {\rm{stress}}}).$

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The capability of the device to compensate the non-controllable pre-stress arising during device processing is a very first indication of the possibility to exert deliberate in-plane stress fields on demand. In order to prove this statement, it is sufficient to use the linearity of the elastic theory, and in particular equation (7). To do that, the device behavior has to be first calibrated to obtain the transfer matrix, which then allows us to calculate the corresponding voltages to be applied on each of the legs so as to achieve the desired strain configuration (encoded via S₁, S₂, θ_S1). Practically, this is done by performing PL measurements while sweeping independently the voltages applied to each leg. The PL data in combination with equation (7) are used to calculate the transfer matrix coefficients and then the induced stress field via ${\bar{{s}}}_{{\rm{i}}{\rm{n}}{\rm{d}}{\rm{u}}{\rm{c}}{\rm{e}}{\rm{d}}}=\overline{\overline{T}}\cdot \bar{V}.$ The capabilities of the device to procedure stress fields on demand is demonstrated by predicting a number of induced stress fields with variable magnitudes and anisotropies on the GaAs nanomembrane. We refer the reader to [29] for detailed information on this procedure. Comparative data between experimental data and theory for different polarization angles is shown in figure 25 by sweeping the voltages on each of the legs in order to induce a uniaxial stress up to 100 MPa (major induced stress s₁) along a fixed direction of ${\psi }_{s1}=15^\circ$ with respect to the [110] direction.

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The 3-legged device is preferable for its simplicity and relatively easy fabrication process. Nevertheless, when stress fields with higher magnitude are required it is more convenient to use piezoelectric actuators featuring 6-legs, as demonstrated in [29, 128]. In this case, the voltages are applied on pairs of aligned legs so as to avoid lateral displacements at the center of the membrane where the PL characterization is carried out. In this 6-legged device the stress field anisotropy can be tuned up to about 200 MPa (for electric fields on the actuator ranging from −2 to 33 kV cm⁻¹) in all the directions (major stress angle) with hydrostatic stress values as large as 350 MPa. This implies that the six-legged devices feature the same capability to control the in-plane stress tensor in nanomembranes as the three-leg device, but the stress magnitude is enhanced by about a factor 2 [29].