### Operating principle of a piezoelectric nanobender

Consider a slab of piezoelectric material sandwiched between two electrodes that are separated by a length *L* and have a potential difference of *U* (Fig. 1a). The piezoelectric property of a material is represented by its charge piezoelectricity tensor ** d**, a third-rank tensor that relates strain to electric field inside a material (

**=**

*S***⋅**

*d***). The terms**

*E**d*

_{11(22,33)}(Voigt notation) couple

*S*

_{kk}to

*E*

_{k}, for

*k*=

*x*,

*y*,

*z*, causing compressional/tensile strain to build up in the direction of the electric field. This leads to a displacement

*Δ*=

*S*

*L*=

*d*

_{11}

*U*. Considering that

*d*≈ 10 pm V

^{−1}in standard piezoelectric materials such as aluminum nitride (AlN) and lithium niobate (LiNbO

_{3})

^{26,27}, such a transducer would only generate displacements at the atomic scale for voltages

*U*~ 1 V which are easily produced by CMOS circuits.

The above expression *Δ* = *d*_{11}*U* also shows that the generated displacement does not depend on the size of the transducer. This applies to any piezoelectric actuator. We write the constitutive relation between the strain and the electric field ** S** =

**⋅**

*d***, as ((**

*E*

*∇***)**

*u*^{T}+

*∇***)/2 =**

*u***⋅**

*d*

*∇**U*where

**(**

*u***) is the displacement field and**

*r**U*(

**) is the electric potential distribution for a given actuator geometry, applied voltage and boundary conditions. If the geometry of both the actuator and the boundary conditions are scaled by a factor**

*r**ϵ*while keeping the same applied voltage, then ({boldsymbol{u}}^{prime} ({boldsymbol{r}})={boldsymbol{u}}({boldsymbol{r}}/epsilon )) and (U^{prime} ({boldsymbol{r}})=U({boldsymbol{r}}/epsilon )) are solutions to the new equations. In other words, the magnitude of the displacement stays constant as the actuator is shrunk leading to an increase in relative displacement that favors smaller actuators.

As illustrated above, the diagonal elements of ** d**,

*d*

_{11(22,33)}give rise to tens of picometers of displacement at a potential of around one volt. A much larger displacement can be generated with transverse

**(**

*d**d*

_{12(13,23)}) components (Fig. 1a). In this situation, the potential

*U*gives rise to an electric field

*U*/

*w*across the width

*w*, which generates strain along the length

*L*of the beam. This leads to a displacement (Delta approx dcdot UL/w sim (0.01 {rm{nm}} {{rm{V}}}^{-1})cdot {mathcal{R}}U), where we have defined the aspect ratio of the actuator ({mathcal{R}}equiv L/w). Compared to the previous case, the displacement is enhanced by ({mathcal{R}}). However, reaching

*Δ*~ 100 nm with one volt still requires ({mathcal{R}}=1{0}^{4}), roughly on the same order as that of a long strand of human hair, or sheet of paper, making it impractical.

Is there a configuration that results in a displacement which scales faster than linear with ({mathcal{R}})? Bending of a beam generates a displacement proportional to *L*^{2}, where contraction occurs in one half of the beam and expansion in the other half. Looking back to the corresponding electric field *E*, we recognize that bending can be actuated by flipping the direction of *E* across the width of the beam. Assuming that the derivative of the *E* field is constant across the width of the beam ∂_{z}*E* ~ *U*/*w*^{2} (*z* is transverse to the beam), the end-point displacement can be approximated by (Supplementary Note 1)

$$Delta approx frac{1}{2}dcdot {partial }_{z}Ecdot {L}^{2} sim (0.01 {rm{nm}} {{rm{V}}}^{-1})cdot {{mathcal{R}}}^{2}U.$$

(1)

We see that the displacement in this case is enhanced by the square of the aspect ratio ({{mathcal{R}}}^{2}). The required ({mathcal{R}}) for *Δ* ~ 100 nm is drastically decreased to a practical value ({mathcal{R}}=100). As an example, such a non-uniform *E* field on a *w* ~ 400 nm wide beam with length *L* ~ 40 *μ*m would enable actuation of 100 nm displacement per volt—a displacement on the same order as the width of the beam. We emphasize that the nanoscale aspect of the nanobender is important for achieving such a large relative displacement. By the scale-invariance arguments above, a larger structure would generate the same displacement, leading to a less appreciable relative motion.

A strongly inhomogeneous *E* field is naturally generated by the fringing fields of a submicron-scale electrode configuration. We consider a simple device, which we call the nanobender, where a pair of parallel electrodes lies on the top surface of a beam made of a thin piezoelectric LiNbO_{3} film. For a beam oriented along the crystal axis *X*, the inhomogeneous *E*_{Y} field induces a varying strain *S*_{XX} and results in bending of the beam (Fig. 1b) via the piezoelectric tensor element *d*_{21}. For *Y*-cut LiNbO_{3} where the crystal *Y* axis is perpendicular to the chip, this bending gives rise to an in-plane displacement that scales quadratically with *L*. Finite-element simulations^{28} of the nanobender are shown in Fig. 1c. The simulated *E*_{Y} (arrowheads) changes sign along *Z*, causing expansion in one half of the beam and contraction in the other. For all the simulations and experiments, we use a nanobender width *w* = 450 nm, LiNbO_{3} thickness *t* = 300 nm, electrode width *w*_{m} = 150 nm, electrode–electrode gap *w*_{g} = 150 nm and an aluminum electrode thickness *t*_{m} = 50 nm. A more detailed study of how these parameters affect nanobender performance is presented in Supplementary Note 2. Once the nanobender’s cross-section is fixed, the length *L* ultimately determines the maximum displacement *Δ* generated at the end of the beam. Through simulations (Fig. 1d) we are able to confirm the quadratic length–displacement relationship in equation (1). The simulated displacement along the other two directions are more than one order of magnitude smaller.

Actuation that induces bending is commonly adopted by macroscopic piezoelectric actuators, realizing displacement per volt values similar to the nanobenders (~ 0.1 *μ*m V^{−1})^{29}. These structures can induce bending by utilizing multiple electrodes to switch the electric field direction, or multilayered piezoelectrics. Such approaches are possible in an integrated platform^{30} but relatively difficult to realize, especially for in-plane actuation. Remarkably, electrostatic forces can also be used to generate bending with large travel^{31}, though scaling actuators down to a few microns is challenging, and current demonstrations require much larger footprints for similar displacements (~50 × 2000 *μ*m^{2} for ~ 0.1 nm V^{2}). Note that the flexoelectric effect, distinct from the piezoelectric effect, is also capable of generating bending from a homogeneous electric field^{32,33}. However, the piezoelectric effect in the nanobender exceeds the flexoelectric effect by orders of magnitude, which is thus neglected (see Supplementary Note 3 for estimates).

### Integration with a zipper cavity

By integrating the nanobender with a nanophotonic “zipper” cavity^{24,25} on the thin-film LiNbO_{3} material platform, we demonstrate its potential for realizing photonic devices with wide low-voltage tunability. A zipper cavity is a sliced photonic crystal consisting of two nano-patterned beams separated by a gap ({tilde{g}}_{0} sim 200 {rm{nm}}) that confines an optical resonance. The *E*_{Z} component of the fundamental optical cavity mode is plotted in Fig. 1e. Due to the sub-wavelength confinement of the mode, the resonance wavelength of the cavity is strongly dependent on the gap between the two beams (Supplementary Note 4). A voltage applied to the nanobenders moves the two halves of the zipper cavity (Fig. 1f), tuning the optical resonance wavelength. In Fig. 1g we present the simulated voltage–wavelength tuning curve. The tuning curve is nonlinear due to the large changes in ({tilde{g}}_{0})—a smaller ({tilde{g}}_{0}) increases the optical mode confinement and optomechanical coupling, increasing the slope of the tuning curve.

To couple light into the device, we use an edge coupling scheme where a lensed fiber is aligned to a tapered waveguide (Fig. 2a, b). Light is guided to a reflector and evanescently couples to both an active and a zipper cavity that acts as an experimental control. This control cavity is attached to nanobenders without metallization. The reflection spectrum of the device is recorded for all subsequent measurements (see methods) and the background is removed through normalization (see Supplementary Note 5). The bender-zipper cavity is positioned such that the nanobenders are parallel to the crystal *X* axis, necessary for the nanobenders to operate as designed. We also fabricate and measure a device with nanobenders perpendicular to the crystal *X* axis and measure two orders of magnitude lower tuning (see Supplementary Note 6). We attach the nanobenders to the zipper cavity with and without narrow tethers and measure larger tuning in the untethered devices (highlighted in blue and red in Fig. 2b). The motivation and effect of adding the tether is discussed in more detail in Supplementary Note 4. To apply a voltage to the nanobenders, we use electrical probes to make contact with on-chip aluminum pads.

We apply voltages to the nanobenders in steps of 50 mV and obtain the reflection spectrum for each voltage (Fig. 2c). We observe wavelength tuning for three different optical modes O_{i} of the active cavity. No tuning for the control cavity is observed. Additionally the linear wavelength–voltage relationship around 0 V indicates that tuning originates from the piezoelectric effect, in contrast to electrostatic, thermo-optical, and thermo-mechanical tuning. Reflection spectra near the fundamental optical mode O_{1} around 0 V and at 2 V are shown in Fig. 2d. The linewidth of O_{1} is around 90 pm corresponding to a quality factor *Q* of 1.7 × 10^{4}. The linewidth is limited by thermal mechanical broadening and decreases by almost an order of magnitude at 4 K (Supplementary Note 7). The shallower dip at 2 V is due to a decrease of the external coupling rate *κ*_{e} as the separation between the zipper cavity and the coupling waveguide is increased by actuation of the nanobender. It may be possible to compensate for this effect by using a secondary nanobender on the coupling waveguide or actuate the two halves of the zipper cavity independently. In Fig. 2e we show the extracted resonance wavelength shift versus DC voltage for O_{1} and O_{2} of the active zipper cavity. We can tune over tens of nanometers with CMOS-level voltages, corresponding to hundreds of optical linewidths. We perform a linear fit on this tuning curve for small voltages (∣*U*∣ < 0.5 V) and obtain a tuning coefficient quantifying the change in wavelength per volt of 5 nm V^{−1}. All tuning coefficients are reported at 0 V.

We also investigate how tuning coefficient scales with nanobender length (Fig. 2f). For this purpose we fix all other parameters within fabrication imperfections which mostly affect ({tilde{g}}_{0}). More than 40 devices with different *L* are measured. As expected, the zipper cavities with longer nanobenders tune more. The tuning coefficients are higher on devices without the tethers. This is partly supported by simulations. Hence optimizing the way nanobenders are attached is important for composite mechanical systems.

### Modulation speed of the bender-zipper cavity

In addition to slowly tuning the bender-zipper cavity using a DC voltage, we also apply a small AC voltage. This allows us to learn about the AC modulation strength as well as the mechanical resonance frequencies of the bender-zipper device. As shown in Fig. 3a, the total voltage applied on the nanobenders is ({V}_{text{dc}}+{V}_{text{ac}}sin (Omega t)) where *Ω* is the modulation frequency. These voltages lead to wavelength shifts of the cavity given by (Delta {lambda }_{text{dc}}+Delta {lambda }_{text{ac}}(Omega )sin (Omega t+phi )) where *ϕ* is a phase offset. In the DC measurements, we sweep the laser wavelength across the resonance of the cavity. For AC measurements, we instead fix the wavelength of the laser and sweep the cavity using the bias voltage (see Fig. 3b), while using the AC voltage to modulate the cavity resonance. The measurement result is the convolution of the cavity’s Lorentzian lineshape with the probability distribution that samples the sinusoidally modulated cavity center frequency.

Sweeping the modulation frequency (Fig. 3c), we observe that the AC tuning coefficient *α*_{ac} ≡ *Δ**λ*_{ac}/*V*_{ac} is enhanced at certain frequencies close to 1 MHz. These correspond to the mechanical resonances of the bender-zipper device (Supplementary Note 8). The data were taken with *V*_{ac} = 50 mV. Cut-lines of the dataset are shown in Fig. 3d, both off-resonance (*Ω*_{1}) and close to resonance (*Ω*_{2}). We also show a measurement without AC modulation where we recover a simple Lorentzian. We fit the reflection spectra to extract the AC tuning coefficient and plot it as a function of the modulation frequency (Fig. 3e). Consequently, we are not only able to observe the mechanical resonance frequencies of the system but also directly extract the strength of the modulation. On mechanical resonance, the tuning coefficient is enhanced by a factor ~ 10, amounting to *α*_{ac} ~ 50 nm V^{−1}. This corresponds to *V*_{π} = *κ*/(2*α*_{ac}) ~ 1 mV. As expected, the frequency dependence of the AC tuning coefficient closely matches with the thermal-mechanical spectrum (Fig. 3f). We obtain the thermal-mechanical spectrum by detuning the laser from the cavity by around half a linewidth where the cavity frequency fluctuations are transduced to intensity fluctuations that we detect with a high-speed detector and record on a real-time spectrum analyzer. The mechanical quality factor *Q*_{m} ≈ 20 is relatively low due to air damping. This is verified by measurements in low pressure conditions which show several orders of magnitude enhancement in *Q*_{m} (Supplementary Note 7). Thus, modulation experiments at low pressures could enable even larger resonant AC tuning coefficients (over a smaller bandwidth), reducing *V*_{π} to ~ 20 *μ*V.

### Mechanical contact and hysteresis

We have shown that tens of millivolts are sufficient to tune the optical cavity by more than its linewidth. The small gap and large displacement per volt, taken together, means that the two halves of the zipper touch for voltages on the order of 5 V.

We demonstrate continuous wavelength tuning of a bender-zipper cavity with *L* = 22 *μ*m by reducing the gap between the two halves of the zipper down to the point when they come into contact (Fig. 4a). Focusing on the fundamental mode *O*_{1}, we measure a tuning range of 63 nm with 4.5 V. To the best of our knowledge, this is the largest tuning range demonstrated for an on-chip optical cavity using CMOS-level voltages. From the initial gap size, we infer a displacement actuation of ~ 25 nm V^{−1} from each pair of nanobenders. After the contact, the tuning stops regardless of increasing voltage.

As we begin decreasing the voltage, the resonance wavelength shifts ten times less than before the contact because the zipper halves are stuck. We find that the voltage needs to be reduced lower than the contact voltage for the tuning to be restored. This hysteresis is likely due to the van der Waals force that keeps the zippers attached. When we further decrease the voltage, the nanobenders exert a force opposite in direction which eventually manage to detach the zippers. The whole process is reversible as the mode recovers its original wavelength after detaching.

The hysteresis behavior could be applied as an optical memory^{34} which necessitates hysteresis for functioning. We test the reliability of the hysteresis loop by repeating the contact–detach process. In Fig. 4b we show nine successive contact–detach cycles, which were preceded by ~ 40 cycles. The hysteresis loop is apparent and there is relatively good overlap between the cycles. However, the voltage at which the zippers detach is not consistent across cycles and drifts to lower voltages. After several cycles, the nanobenders are not able to get the zippers to detach (not shown here) although we have found that applying a short AC pulse on mechanical resonance is able to detach them reliably, acting as a reset operation. After the reset, for several cycles the zippers are again able to detach with a DC voltage. The reason for this behavior will be subject to future investigations.

In Fig. 4c we show measurements of the thermal power spectral density of the bender-zipper cavity before contact, during contact and after detaching. We see a clear difference in the spectra between the detached zipper and the attached one. In the latter case, the lateral relative motion between the two halves of the zipper cavity is effectively suppressed. The higher noise floor measured during the contact is likely from laser phase noise, which is more efficiently transduced due to a narrower optical linewidth. We are thus able to reversibly modify both the optical and mechanical properties of the zipper cavity using the nanobenders.