1. Introduction

Thin-film lithium niobate (TFLN) is a platform for integrated photonic devices that provides low optical loss, a strong Pockel’s electro-optic (EO) response, and relatively high second- and third-order optical nonlinearity [1]. These valuable properties have motivated significant recent interest in the development of TFLN devices. TFLN EO modulators have demonstrated low drive voltage and modulation bandwidth exceeding 100GHz [2–4]. Optical nonlinearity in TFLN devices has been used to create high-efficiency frequency conversion [5] and on-chip frequency combs [6,7]. The strong optical confinement created with low-index substrates and etched or loaded waveguides in TFLN allows for high integration density. Combined, these properties make TFLN a promising platform to solve a wide range of problems in optical communication [8,9], quantum devices [10–12], microwave photonics [13] and sensing [14].

The strong EO response of TFLN devices – enabled by the excellent material properties of LN and the ability to closely space electrodes [3,15] – is one of their key benefits. However, it has been widely observed that the EO response of TFLN modulators relaxes away on timescales of milliseconds to hours [9,11,16–19]. EO relaxation in TFLN makes it impractical to use the EO effect for many low-frequency or long-timescale applications, including for ubiquitous DC bias tuners in optical modulators. While thermo-optic phase shifters can be used instead [16,20], these come at the cost of lower bandwidth ($<1\, \mathrm{MHz}$), higher power consumption (of order $100\, \mathrm{mW}$ for a $2\pi$ phase shift) [16], and crosstalk between phase shifters [21]. In particular, the high power of thermo-optic phase shifters makes them less suitable for quantum photonics applications at cryogenic temperatures and optical communication applications where transmitter power consumption is a key figure of merit.

In contrast, a capacitive EO bias requires no power dissipation at steady state. For this reason, EO relaxation is an essential challenge to realizing the full potential of low-power TFLN devices. EO relaxation in TFLN devices has yet to be extensively studied. The cause of EO relaxation in TFLN and how it can be improved is poorly understood, although it has recently been shown that changing the cladding process can improve relaxation [22].

In contrast, EO relaxation has been widely studied in bulk LN devices [23–28] and other electro-optic photonics platforms [29,30]. Typically, EO relaxation is caused by the migration of free-charge carriers within and around the EO material. These carriers can completely screen the applied electric field and eliminate the EO response of the modulator on long timescales. In the simplest case where the EO material is separated from the electrodes by an ideal insulator, the EO relaxation occurs at the Maxwell-Wagner dielectric relaxation timescale $\tau =\epsilon _r\epsilon _0/\sigma$ of the EO material, where $\sigma$ is the material’s conductivity, and $\epsilon _r \epsilon _0$ the permittivity. However, the existence of multiple materials, interfaces, and dielectric relaxation phenomena typically make it challenging to gain a complete microscopic understanding of the EO relaxation in any given device. Furthermore, because the carriers are related to material defects, devices with different fabrication process or design often sport widely varying EO relaxation effects. As shown below, the EO relaxation timescale for cladded TFLN devices can be on the order of milliseconds. This is much faster than in bulk devices, where relaxation timescales are typically between hours and years or longer [23–28]. Faster EO relaxation in TFLN than bulk LN is consistent with the observation of faster photorefractive response in TFLN [31] and may be due to higher defect density [32].

This article aims to characterize EO relaxation in TFLN devices quantitatively, investigate the microscopic origin of such relaxation, and demonstrate methods to improve EO stability. By measuring EO relaxation in devices with different electrode geometries, modeling EO relaxation with finite-element methods and simple material models, and comparing our estimated material properties with previous studies in bulk LN, we find evidence that suggests the carrier migration occurs primarily at the top surface of the LN film in our cladded devices, in a thin layer with surface conductivity of approximately $\kappa _{s}{\sim }10^{-11}\, \mathrm{S}$. We perform temperature-dependent measurements to help validate charge carriers as the origin of the reduced EO response and measure the activation energy of the EO relaxation timescale, which may help determine the microscopic cause of the free carriers. Finally, we study several ways to improve low-frequency EO response, including different electrode designs, annealing procedures, and operation at low temperatures. We find that these interventions can improve the EO relaxation timescale in our devices by several orders of magnitude in our devices, from $100\, \mathrm{Hz}$ to less than $10\, \mathrm{mHz}$. Although further work is required to verify the surface-based origin of charge conduction and develop better techniques to enhance low-frequency EO response, here we identify several interventions that can make low-power electro-optic tuning in TFLN devices feasible.

2. Electro-optic relaxation

To probe the EO frequency response in TFLN devices, we use a Mach-Zehnder optical amplitude modulator (MZM) illustrated in Fig. 1(b) as an example testbed EO device. Figure 1(a) shows the fabrication process for our devices. Using a 300nm deep argon-based reactive ion etch, we define $1.5\, {\mathrm{\mu} \mathrm{m}}$ wide rib waveguides into a commercial ion-sliced thin-film lithium niobate wafer from NanoLN with $600\, \mathrm{nm}$ LN film thickness, a $4.7\, {\mathrm{\mu} \mathrm{m}}$ silicon dioxide buried cladding layer formed by thermal growth, and a $525\, {\mathrm{\mu} \mathrm{m}}$ thick high resistivity ($>10\, \mathrm{k\Omega cm}$) silicon substrate. After etching, we use plasma-enhanced chemical vapor deposition to clad the device with $1.2\, {\mathrm{\mu} \mathrm{m}}$ of silicon dioxide and deposit $15\, \mathrm{nm}$ of titanium and $800\, \mathrm{nm}$ of gold for electrodes that are $100\, {\mathrm{\mu} \mathrm{m}}$ wide with $5.5\, {\mathrm{\mu} \mathrm{m}}$ gaps between them.

 

Fig. 1. Setup for measuring EO relaxation in TFLN modulators. (a) Device fabrication steps: (i) the initial TFLN substrate, (ii) a ridge waveguide is patterned and etched into the optical layer, (iii) a silicon dioxide cladding is deposited to cover the waveguide, (iv) metal electrodes are deposited. (b) Left: A schematic top view of a Mach-Zehnder amplitude modulator in TFLN. Right: Cross-section of the electrode. A voltage drive signal $V(t)$ is applied to the electrodes to induce an optical phase change in each arm, which is measured as a change in optical intensity at the output of the Mach-Zehnder modulator. (c) A schematic of the experimental setup, where polarized CW laser light at $1550-1630\, \mathrm{nm}$ is injected into the device-under-test (DUT), a network analyzer applies a probe signal to the device, and the device’s electro-optic response is read out as amplitude fluctuations on a photodetector (PD). (d) The modulator’s optical transmission vs drive voltage transfer function. A single-tone weak electrical drive is applied around the quadrature point of the modulator, where the electro-optic amplitude response is approximately linear, and the optical modulation is measured.

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Figure 1(c) shows the experimental setup. CW laser light (wavelength $1550$ to $1630\, \mathrm{nm}$) with polarization aligned to the TE optical mode is injected onto the chip, passes through the MZM, and is collected at a photodetector. Light is coupled to and from the device via a fiber array and on-chip grating couplers with roughly $10\, \mathrm{dB}$ insertion loss per coupler. A low-frequency network analyzer is used to electrically drive the modulator (drive amplitude $|V|< 0.1V_{\pi }$) and record the modulation response (illustrated in Fig. 1(d)) on the photodetector $V_{\mathrm{PD}}$ to calculate the EO response, which we define as $G(f)=V_{\mathrm{PD}}(f)/V(f)$. We selected this frequency-domain measurement technique over the time-domain step-response techniques used in most prior literature on EO relaxation [23–29] because our method isolates the cause-effect relationship between electrical stimulus and EO response and is not susceptible to extrinsic drift mechanisms like temperature fluctuation or photorefractivity. Additionally, characterizing the frequency-domain EO transfer function allows easy identification of turning points that are related to dielectric relaxation time constants of materials in the device. The sinusoidal EO transfer function of our MZM is shown in Fig. 1(d). To consistently measure the relative EO response across devices, we operate each device at the quadrature point, where the linear EO response is maximized. We use an MZM with unbalanced arm lengths so that we can reach the quadrature point by changing the laser wavelength instead of using a DC bias. We consistently find across all devices that the EO response $G$ is nearly constant at high frequencies (measured at $100\, \mathrm{kHz}$ to $1\, \mathrm{MHz}$) and use this high-frequency value to normalize the data for comparison across devices, except where otherwise noted.

Figure 2(a) shows the frequency dependence of the EO response for one of our fully cladded devices. The EO response is strongly suppressed at frequencies below roughly $100\, \mathrm{Hz}$. In the time domain, the EO effect induced by an applied electrode voltage decays over a few milliseconds. Furthermore, even for frequencies above $100\, \mathrm{Hz}$, we observe a counter-intuitive and unexpected enhancement of the EO response by as much as $2.5\, \mathrm{dB}$ in a broad band around frequency $f_e$. Although all TFLN devices may not have the same behavior, we highlight that such an enhancement feature could confound measurements of the low-frequency halfwave-voltage ($V_\pi$) of TFLN modulators, often performed in this frequency range. To ensure accurate low-frequency $V_\pi$ values, we suggest that the measurement be performed across several decades of frequency and checked for consistency. Additionally, we note that this poor frequency flatness can be difficult to compensate for in some applications. This article seeks to explain these effects – the intermediate enhancement and low-frequency suppression of EO response – and develop methods by which they can be eliminated or improved.

 

Fig. 2. Measurement and modeling of EO relaxation in TFLN devices. (a) Measured EO response (circles) of a device in the low-frequency regime. We observe strong suppression of EO response at frequencies below $100 \, \mathrm{Hz}$ and broad enhancement of the EO response around frequency $f_e$. Solid lines show the results of two-dimensional finite element models of our devices using either bulk conductivity ($\sigma _{LN}{\neq } 0$, $\kappa _s {=} 0$) or surface conductivity ($\sigma _{LN} {=} 0$, $\kappa _s {\neq } 0$) models. (b) Schematic of our finite element model showing material properties. (c) Schematic of a lumped-element model to explain the impact of bulk conduction on the EO response. The dielectric and resistive properties of the silicon oxide cladding and LN waveguide are modeled using lumped-element parameters $C_{\mathrm{ox, LN}}$ and $R_{\mathrm{ox, LN}}$, respectively. The voltage drop across the LN waveguide $\Delta V_{\mathrm{LN}}$ is proportional to the electric field in the waveguide.

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As discussed above, we attribute the anomalous EO frequency response to the movement of charge carriers within our device [23,24]. Here we will compare two possible models for carrier migration in our devices: conduction within the bulk of the LN layer or conduction at the top etched surface layer of the LN at the cladding interface. Figure 2(b) illustrates the material parameters used in these models, where $\sigma _{i}$ and $\epsilon _i$ are the conductivity and relative permittivity of different material layers in our device and $i=\{c, LN, s\}$ is an index referring to the cladding silicon dioxide, LN, or surface layer, respectively. The surface layer conductivity is defined as $\kappa _s=t\sigma _s$, where $t$ is the thickness of the conducting surface layer, which we assume to be small compared to other dimensions in our device so that a boundary condition can be used to model the effects of the surface layer material. If we can determine where and how carrier migration occurs, this information can guide the development of devices with improved low-frequency EO response.

Predictions of the EO response based on lumped-element resistor-capacitor network models have been used in previous works to study the location of free charge carriers within bulk LN devices [28]. Here we follow a similar approach but instead use a two-dimensional finite-element model of the electrostatics in the gate region and fit the simulated EO response (shown in Fig. 2(a)) to our measured data. Before describing the results of our models, we will digress to show that lumped-element models – although they provide a useful intuitive picture – have challenges with quantitative accuracy and are not well suited to studying EO relaxation in TFLN.

Previous works on the low-frequency response of EO devices have typically used lumped-element models [23–25,28] like that shown in Fig. 2(c), which emulates to the bulk conductivity model. In the lumped model, each material in the electrode region is treated as a parallel RC network. The time constant $\tau _i=R_iC_i$ can be compared to the dielectric relaxation time in a distributed model. The voltage drop across the waveguide $\Delta V_{LN}$ is proportional to the electric field in the waveguide and hence also to the EO response. At high frequencies, the admittance of the capacitors dominates that of the resistors, and relatively large values of $\Delta V_{LN}$ can result because the geometry of the electrodes ensures $C_{ox}{\sim }C_{LN}$. However, in the low-frequency regime, resistive admittance dominates so that the waveguide voltage drop is $\Delta V_{LN}{=}\frac {R_{LN}}{R_{LN}+2R_{ox}}V$. If the cladding resistivity is also much higher than that of the LN ($R_{ox}\gg R_{LN}$), then $\Delta V_{LN}$ and the EO response will be suppressed. In a distributed picture, this corresponds to charge carriers inside the LN migrating to the cladding-LN interface and shielding the applied electric field inside the waveguide.

We can directly connect this lumped-element model with the material parameters used in our distributed model by treating the RC network as a capacitance network where the complex relative permittivity $\epsilon ^*$ of each capacitor is given by $\epsilon _i^* = \epsilon _i +\frac {i\sigma _i}{\epsilon _0\omega }$. This permittivity determines the lumped-element parameters through the total complex capacitance $C^* = \epsilon _0 A_i \epsilon _i^*/d_i$, where $A_i$ and $d_i$ are the area and plate separation of a notional lumped-element parallel plate capacitor. Using these definitions, the EO response in the lumped-element model is given by

Having identified the shortcomings of lumped-element modeling for this application, we will now discuss the results of our finite element simulations. The solid lines in Fig. 2(a) show fits of our measured data to the finite-element simulations. The bulk conduction model assumes $\kappa _s{=}0$ and uses $\{\sigma _{LN}, \sigma _c\}$ as free parameters, while the surface conduction model takes $\sigma _{LN}{=0}$ and $\{\kappa _s, \sigma _c\}$ as free parameters.

The fit to the bulk conduction model yields estimates $\sigma _{LN}{\sim }6 \times 10^{-6}\, \mathrm{S/m}$ and $\sigma _c{\sim }2\times 10^{-10}\, \mathrm{S/m}$ and qualitatively reproduces the strong reduction in EO response. However, it is important to note that the bulk conduction model does not explain the EO response enhancement feature near frequency $f_e$. Admittedly, here we use a simple model for the complex permittivity of the materials based on frequency-independent values for the dielectric constant and conductivity. Dielectric relaxation effects can cause the permittivity of real materials to differ significantly from this simple model. However, it is unlikely that a more complicated permittivity model in the LN layer alone (for example, including Debye relaxation observed in bulk LN [33]) could explain the observed enhancement. This is because dielectric relaxation of all kinds tends to increase the permittivity of materials at lower frequencies [34], which would lead to a reduction in EO response for a bulk conduction model.

The fit to the surface conduction model yields estimates $\kappa _{s}{\sim }10^{-11}\, \mathrm{S}$ and $\sigma _c{\sim }4\times 10^{-10}\, \mathrm{S/m}$ and reproduces both the enhancement feature at $f_e$ and reduction in EO response for low frequencies. The EO enhancement feature is due to a geometric effect where the conductive layer can enhance the electric field in the waveguide at intermediate frequencies. This enhancement is analogous to the narrow-band voltage gain produced by some passive RC networks [35]. In our model, the conductive layer fully covers the waveguide and acts as a Faraday shield at lower frequencies, leading to a strongly suppressed EO response. We discuss the evidence for the surface and bulk conduction models below, but the simulation results suggest that the surface conduction model better reproduces key features of the measured data.

3. Temperature dependence

We performed several sets of measurements at cryogenic temperatures to validate charge conduction as the origin of EO relaxation. These measurements also help characterize the improvements in EO relaxation at cryogenic temperatures, which may be valuable for quantum TFLN device applications at such temperatures [10–12,36].

We expect the EO response to improve at low temperatures because most charge carriers should freeze out and become bound to defect sites, causing an increase in resistivity and preventing dielectric relaxation from occurring at the measured timescales. Figure 3(a) confirms that we observe the expected improvement after cooling down our devices to $0.05\, \mathrm{K}$ in a dilution refrigerator with active fiber alignment. Figure 3(b) shows the results of a temperature sweep, illustrating that both the low-frequency response and the frequency flatness are improved, although even at the lowest temperatures, we observe a transition in EO response amplitude near $3 \, \mathrm{Hz}$.

 

Fig. 3. EO relaxation at cryogenic temperatures. (a) Comparison of the low-frequency EO response of our devices at room temperature and $0.05\, \mathrm{K}$. Inset: device cross-section. (b) Impact of temperature on the EO response. Inset: dependence of the enhancement feature frequency $f_e$ on temperature $T$. A fit to an Arrhenius model (black line) yields an estimated activation energy $E_a{=}\, 0.48\, \mathrm{eV}$. (c) Impact of optical power on the EO response at a base temperature of $0.05\, \mathrm{K}$. The legend shows the on-chip input optical power. (d) Photorefractivity induced optical gratings at a base temperature of $0.05\, \mathrm{K}$. The blue curve shows the transmission spectrum shortly after cooldown to base temperature. The orange curve shows the transmission spectrum after 17 hours of measurement with the laser at a fixed wavelength (dashed vertical orange line) and $-11\, \mathrm{dBm}$ on-chip input power.

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The inset of Fig. 3(b) shows the temperature dependence of the enhancement feature frequency $f_e$. A fit to an Arrhenius model yields an activation energy of $E_a{=}\, 0.48\, \mathrm{eV}$ for the enhancement frequency. This energy is comparable to the value of $0.4$ to $0.5\, \mathrm{eV}$ measured for the activation of conductivity in amorphous LN, which is attributed to the transport of lithium ions [37].

Improving low-frequency EO response at low temperatures is beneficial for the operation of cryogenic TFLN devices. However, we show in Fig. 3(c) that the EO response depends on the optical power in the device at low temperatures, and noticeable EO relaxation occurs at moderate optical power levels. As a comparison, we measured the optical-power dependence of the EO response at room temperature and did not see any significant changes up to $0\, \mathrm{dBm}$ on-chip power. This optical-power dependence is likely due to the strong photoconductivity of thin-film lithium niobate, whereby the optical field excites mobile free carriers [31,38]. While this effect could also be due to optical heating, the sample’s base plate remains at a low temperature during these experiments ($T<1\, \mathrm{K}$), and comparison with Fig. 3(b) suggests that local temperatures of $T{\sim }100\, \mathrm{K}$ would be required to reach the $0.1\, \mathrm{Hz}$ cutoff frequency observed for $-1\, \mathrm{dBm}$ on-chip pump power. The TFLN device is well-thermalized to the base plate using silver glue, so it is unlikely to reach such high temperatures while the base plate remains at low temperatures.

In other measurements shown in Fig. 3(d), we see evidence for photorefractive effects, which confirms strong photoconductivity in our devices. Figure 3(d) shows optical transmission spectra of our unbalanced MZM devices at roughly $0.1\, \mathrm{K}$. Immediately after the cooldown, the transmission spectrum (blue) appears as expected, except for a small dip near $\mathrm{1619\, \mathrm{nm}}$ where the laser was used to monitor optical coupling during cooldown. After parking the laser near $1616\, \mathrm{nm}$ (dashed orange line) for $17\, \mathrm{hours}$ with an on-chip optical power of $-11\, \mathrm{dBm}$, the optical transmission spectrum changes dramatically, showing a window of reduced transmission roughly $10\, \mathrm{nm}$ wide imposed onto the usual MZM spectrum. After removing the parked laser, the transmission spectrum does not change over several hours. These changes in the transmission spectrum can be reversed by warming the device up to $T{>}200\, \mathrm{K}$ or by repeatedly scanning the laser at high power ($0\, \mathrm{dBm}$) over a wide wavelength range ($\mathrm{1510\, \mathrm{nm}}$ to $\mathrm{1630\, \mathrm{nm}}$). We attribute these findings to the creation of photorefractive Bragg gratings within the waveguide, which has previously been observed in TFLN ring resonators [39]. Reference [39] provides a full explanation of this effect, but in short: weak optical standing wave patterns (generated, perhaps, by reflection at the grating couplers) can create a photorefractive index grating which in turn enhances the standing wave pattern amplitude. This feedback mechanism can lead to strong Bragg back reflection. These laser-induced gratings create some challenges for using TFLN devices at cryogenic temperatures, but the ability to create in-situ photorefractive index changes in non-resonant devices, as demonstrated here, could also be a useful tool for trimming devices. Furthermore, previously demonstrated techniques for reducing the photorefractive effect in TFLN could be used to avoid this challenge [40].

Note that while performing all measurements at low temperatures, we avoided this photorefractive index grating effect by not leaving the laser at one wavelength long enough to generate changes in the optical transmission spectrum. Optical transmission spectra were checked before and after each measurement to ensure that this effect did not impact the measurement.

4. Improving low-frequency EO response

While operation at cryogenic temperatures can reduce EO relaxation, this is not possible for most device applications. Based on the dielectric relaxation model above, we posit that removing the silicon dioxide layer and making good electrical contact directly to lithium niobate will improve low-frequency EO performance even at room temperature. Such electrical contact would prevent dielectric screening by eliminating Maxwell-Wagner polarization at the LN surface. To test this theory, we use reactive ion etching to create windows in the cladding silicon dioxide of our modulators to expose lithium niobate before depositing metal electrodes, as shown in Fig. 4(a). We carefully calibrated the etch depth to target a 15nm over-etch into the LN layer, ensuring the silicon dioxide cladding was removed without excessive over-etch. To understand the influence of LN surface preparation on the low-frequency EO response, we characterize devices subject to two different types of plasma etching processes – a physical Ar plasma etch and a C$_3$F$_8$ based etch – and also compare them to the fully cladded devices described previously.

 

Fig. 4. Low-frequency response for etched-cladding devices. (a) An illustration of the different cladding-etch procedures for the measured devices. We test three plasma etch conditions: Argon, C$_3$F$_8$, and no etch. (b) The frequency dependence of the EO response. While all devices exhibit similar responses at high frequency ($1\, \mathrm{MHz}$), the EO response amplitude below $10\, \mathrm{kHz}$ differs significantly between the three devices. We normalize the EO response to the $V_\pi$ of each device measured at $200\, \mathrm{kHz}$ to compare devices with different electrode geometry.

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Figure 4(b) shows the EO frequency response of different devices fabricated using the three electrode-interface conditions. Each device has a roughly equivalent response at MHz drive frequencies. The intermediate frequency regime ($1\, \mathrm{Hz}$ to $100\, \mathrm{kHz}$) shows varied behavior between etch conditions. For example, the argon etched device shows a broad enhancement of the EO response, with a peak response at $500\, \mathrm{Hz}$ that is $6\, \mathrm{dB}$ greater than the high-frequency response. The C$_3$F$_8$-etched devices show turning points at similar frequencies. Both etched devices show marked improvement over the low-frequency performance of the fully cladded device, validating the idea that direct electrical contact between the electrodes and the LN layer can improve EO response. The different behavior of the two etched devices, which have nominally identical geometry, suggests that the interface between the metal and the LN layer has different carrier-transport properties that affect the EO relaxation. This might be possible if the two etch methods produce different redeposition byproducts. For example, fluorine-based plasma etching of LN is known to produce a nonvolatile LiF byproduct [41].

Another approach to improve the low-frequency performance would be to modify the conductivity of materials in the electrode region. We introduce a high-temperature annealing step to reduce defect density after the waveguide etch step in the baseline fabrication process. Annealing in oxidizing and reducing environments is commonly used in bulk LN to control types and density of intrinsic defects [27,42–45]. We omit the silicon dioxide cladding for these devices to create devices with minimal EO relaxation and ensure exposure of the lithium niobate to gas during the anneal. We fabricate a set of identical chips, up to the etching of the optical waveguides (Fig. 1(c)(ii)), then introduce chips to an annealing process under either pure oxygen or nitrogen gas flow for one hour at 500$^\circ$C in a tube furnace, and deposit electrodes on each. At higher temperatures above 550$^\circ$C, we experienced delamination of the LN layer from our devices. Prior work in TFLN [40] has shown that annealing at similar temperatures in nitrogen gas can reduce photorefractive effects in TFLN optical ring resonators.

Figure 5(a) compares the frequency response of the annealed and unannealed devices. The unannealed device displays poor frequency flatness with an enhancement feature at a similar frequency to that observed in the fully cladded devices. Both the annealed devices, in contrast, have excellent frequency flatness within the measured frequency range, displaying $<0.2$dB variation of EO response above $1\, \mathrm{Hz}$. At the lowest measured frequencies, we observe a small enhancement in the EO response of the oxygen-annealed device. We also find that oxygen-annealed devices display an aging response not present in nitrogen-annealed devices. Figure 5(b) shows the EO response of the same oxygen-annealed device remeasured after being stored in ambient conditions for 20 days. We observe substantial EO response amplitude enhancement at the low end of the measured frequency range. In contrast, we did not observe any noticeable change in the optical insertion loss or absolute level of the high-frequency EO response.

 

Fig. 5. Low-frequency EO response of uncladded annealed devices. (a) The EO response of an uncladded, unannealed control device (blue) shows a clear mid-frequency response enhancement. The response of a nitrogen-annealed device (red) shows that the frequency response is flattened to less than $0.2$dB variation across the measured frequency range. An oxygen-annealed device also shows an improved response (orange). (b) The EO response of a different oxygen-annealed device is measured after several weeks, revealing a change in the low-frequency behavior over time.

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5. Possible microscopic causes of EO relaxation

Further work is required to confidently identify the microscopic origins of the EO response anomalies we observe. However, based on the above results, we tentatively attribute the EO relaxation in our devices to a thin conductive surface layer on top of the etched LN slab. In this section, we outline the evidence for this conclusion.

Besides the LN bulk and surface conduction models described earlier, the dielectric properties of the cladding or substrate could also explain the measured response. However, the observations of EO response anomalies in the etched-cladding and uncladded devices in Figs. 4 and 5 suggest that the top cladding does not play a critical role. The buried cladding is a high-quality thermal silicon dioxide layer which we expect to behave as a near-ideal dielectric within the measured frequency range. The substrate is separated from the waveguide by the $4.7\, {\mathrm{\mu} \mathrm{m}}$ buried cladding, and simulations suggest that dielectric relaxation in the substrate has a small impact on the measured EO response. Based on these considerations, we identify the electrical properties of the LN bulk or surface as the likely origin of the measured EO anomalies.

As shown above, both the surface and bulk conduction models qualitatively reproduce the reduction in EO response at low frequencies, but only the surface model can explain the observed enhancement of EO response near $f_e$. To assess the validity of these two models, we can also compare the conductivities estimated from the fitting in our finite-element models with previously reported measurements on comparable materials.

The estimated conductivity of the PECVD silicon dioxide cladding ($\sigma _c{\sim }2\times 10^{-10}\, \mathrm{S/m}$ and $\sigma _c{\sim }4\times 10^{-10}\, \mathrm{S/m}$ for the bulk and surface models, respectively) is comparable to that previously measured in low-temperature PECVD deposited silicon dioxide [46].

The estimated LN conductivity of $\sigma _{LN}{\sim }6\times 10^{-6}\, \mathrm{S/m}$ in the bulk model is comparable to that measured in bulk LN wafers which were annealed in vacuum or hydrogen gas to chemically reduce the LN [43,44]. Conduction in reduced lithium niobate is attributed to a system of bound and free polarons [45]. These polarons have wide optical absorption bands that create the black appearance of reduced lithium niobate [42]. However, previous literature suggests that the conductivity estimated in the bulk conduction model is only possible for LN with a high density of polarons that would display substantial optical absorption at the wavelengths used in this study. For example, bulk LN wafers annealed in a 10% hydrogen gas environment at $692\, \mathrm{ ^{\circ }C}$ had a DC conductivity of $\sigma {=}2.1\times 10^{-7}\, \mathrm{S/m}$, a value somewhat lower but comparable to that estimated in our bulk model [43]. However, a similar wafer annealed under 5% hydrogen gas at a lower temperature of $590\, \mathrm{ ^{\circ }C}$ displayed optical absorption of about $100\, \mathrm{dB/cm}$ near the $1.5\, {\mathrm{\mu} \mathrm{m}}$ wavelength used here [47], substantially higher than the roughly $0.3\, \mathrm{dB/cm}$ waveguide attenuation observed in our devices. Hence, the low optical loss in our devices is inconsistent with bulk conduction due to polarons in LN. Other known conduction mechanisms in undoped crystalline LN have much lower conductivity than that predicted for our bulk conductivity model [48].

The estimated surface conductivity of $\kappa _{s}{\sim }10^{-11}\, \mathrm{S}$ in the surface conduction model corresponds to a bulk conductivity of $5\times 10^{-3}\, \mathrm{S/m}$ for a layer with thickness $t{=}2\, \mathrm{nm}$ - an estimate derived from TEM imaging of the waveguide cross-section. This conductivity is comparable to that observed in amorphous lithium niobate [37]. Indeed, as mentioned earlier, the activation energy $E_a{=}0.48\, \mathrm{eV}$ for the enhancement frequency also matches with previous measurements on amorphous LN [37].

A conductive LN surface layer – possibly composed of amorphous LN created during the LN etch process – explains many of the observations described above. While bulk LN conductivity can explain some of the EO response anomalies, the required conductivity has not been observed in bulk LN measurements without optical losses much higher than those observed in our devices. For these reasons, we attribute the observed low-frequency EO response effects to a conductive amorphous LN surface layer.

6. Conclusion

Further work is required to confirm the impact of the LN surface on low-frequency EO response. Here we estimate the electrical material properties in our device by fitting EO response data to finite element simulations of the device. The imperfect correspondence between the measured and modeled EO response and the existence of several materials and interfaces that could harbor multiple types of charge carriers and conduction mechanisms makes it essential to verify the model. Direct transport or dielectric relaxation measurements on individual materials could be used to verify material parameters independently. Further use of direct imaging methods such as TEM or XRF on the surface layer could verify its existence, thickness, and composition. Characterization of EO relaxation of the LN before and during the fabrication process could also help isolate the root cause. We highlight that because EO relaxation is caused by defect-related charge carriers, it is likely that devices using different processing techniques may behave differently than the results shown here. Nonetheless, the characterization techniques, modelling approach and device engineering interventions described here are likely applicable to other TFLN and EO devices.

In addition to further study of the origin of the low-frequency EO response in TFLN devices, other device interventions like new fabrication procedures and annealing steps can be investigated to improve the low-frequency EO response flatness and eliminate EO relaxation. While the interventions described here can reliably produce devices with stable DC tuning over hour-long timescales, we observe drift and relaxation of the EO response in devices on longer timescales (i.e. days). Systematic measurements on such long timescales are beyond the scope of the present investigation, but we expect further improvements in EO response stability will be required to create devices capable of very long-term operation. We highlight that uncladded devices with nitrogen annealing appear particularly promising. Other methods to further enhance EO response flatness and stability could include new annealing procedures, the study of the electrical interface between the metal electrodes and LN slab, and the use of conducting top cladding materials [49]. Using design interventions like those described here to eliminate EO relaxation, TFLN devices can provide compact, low cross-talk, and zero-power EO biasing for high-density and power-critical applications.