Metal–oxide–semiconductor (MOS) capacitors are one of the most prevailing electronic device structures, and have laid the foundation of modern transistors that have transformed the entire industry of microelectronics [1]. In recent years, MOS devices have also gained increasing utility in photonic applications, which could pave the way for a new generation of hybrid electronic–photonic systems [2,3]. MOS-driven silicon photonic devices in particular have rapidly become one of the most promising building blocks for future optical interconnect systems due to their enhanced performance in electro-optic (E-O) modulation and scalability of fabrication [4–6]. Photonic devices based on the MOS structure usually operate in the accumulation mode. When a negative bias voltage ($V_g$) is applied, it induces the field effect and modifies the refractive indices of the semiconductor materials through the plasma dispersion effect so that an optical phase shift is induced to the guided light. In addition to their intrinsic advantages, MOS structures provide feasibility of heterogeneous integration with other materials such as graphene, III-V, and transparent conductive oxides (TCOs) on silicon photonics [7–9]. Of these heterogeneously integrated photonic devices, an MOS device with a TCO gate can achieve unity-order refractive index changes in the accumulation layer [10]. Several ultra-efficient Si-TCO photonic devices have been reported using a Mach–Zehnder interferometer, an electro-absorption modulator, a photonic crystal nanocavity, and a microring resonator (MRR) [9,11–13].

Carrier mobility is one of the most pivotal properties of semiconductors, as it can determine the performance of solid-state devices. Carrier mobility represents the velocity of electrons or holes under certain electric fields, and therefore it determines the conductivity and frequency response of electronic devices such as transistors. Thus, the high mobility of semiconductors is critical to achieving high bandwidth and low power dissipation [14]. For photonic devices, the impact of carrier mobility reaches even further. As described by the Drude model, the collision frequency [Eq. (1c)], which is the collision process between free carriers and ionized impurities in TCOs, is inversely proportional to the carrier mobility at the optical frequency [15,16]. Furthermore, the optical loss due to free carrier absorption is determined by the imaginary part of the complex permittivity [Eq. (1a)], which is influenced by the collision frequency as well. Hence, high-mobility semiconductors are critical to low optical loss waveguides. For instance, previous research has shown that high-mobility TCOs can significantly enhance the performance of photonic modulators by increasing the extinction ratio, improving the energy efficiency and quality factor ($Q$ factor) [13,17,18].

The carrier mobility of semiconductors at electrical frequency (DC or RF) is usually measured by the Hall effect. It actually measures the bulk mobility (${\mu }_{\mathrm{bulk}}$), which is the average mobility of the entire thin film layer [19,20]. For many electronic devices, field-effect mobility (${\mu }_{\mathrm{FE}}$) is even more critical to determining the device performance. When a bias $V_g$ is applied to the gate, the field effect induces accumulation or inversion layer at the surface of the semiconductor with the insulator, forming a channel of free charges that are drastically different than those in the bulk materials [10,21]. The carrier mobility in the accumulation or inversion layer, which is also called the field-effect mobility ${\mu }_{\mathrm{FE}}$, is generally higher than ${\mu }_{\mathrm{bulk}}$ because the high concentration of free carriers in the channel layer brings an electrostatic screening effect that reduces impurity coulomb scattering [22]. This phenomenon has been verified by thin-film transistors (TFTs) [23], and TFTs have been used to measure the electric frequency ${\mu }_{\mathrm{FE}}$. The measurements of the gate voltage, drain voltage, and drain current are used to extract the electric frequency ${\mu }_{\mathrm{FE}}$. For example, experimental results show that the electric frequency ${\mu }_{\mathrm{FE}}$ of TCOs increases as the $V_g$ increases [24–28].

In contrast to electric frequency mobility, which is limited by ionized-impurity scattering and grain-boundary scattering, the optical frequency mobility (${\mu }_{\mathrm{op}}$) is insensitive to grain-boundary scattering. It is only determined by ionized-impurity scattering because the average electron path length, which is in the range of a few nanometers and under the application of a rapidly oscillating electric field, is much smaller than the grain size [29]. By comparing the difference between electrical and optical frequency carrier mobility, we can observe the contribution from the grain-boundary scattering and ionized-impurity scattering separately [30]. The optical frequency bulk mobility (${\mu }_{\mathrm{op},\mathrm{bulk}}$) of a semiconductor film on a thick substrate is usually characterized by a spectroscopic ellipsometry [31]. However, ellipsometry cannot effectively measure the optical frequency field-effect mobility (${\mu }_{\mathrm{op},\mathrm{FE}}$) due to the ultrathin accumulation layer ($\sim 1\mathrm{nm}$). The accumulation layer is only around 0.1% of the probing wavelength used in the ellipsometry, which cannot induce meaningful light–matter interaction to calculate the film’s refractive index and thickness. Therefore, a fundamentally different method is needed for the measurement of ${\mu }_{\mathrm{op},\mathrm{FE}}$ in the ultra-thin accumulation layer.

In this paper, we propose a novel characterization method to extract the ${\mu }_{\mathrm{op},\mathrm{FE}}$ of TCO materials using an MRR on a silicon-on-insulator (SOI) wafer. This method works for all TCOs and can even be applied to other types of semiconductor materials. In this paper, titanium-doped indium oxide (ITiO) is used in the experiment for ${\mu }_{\mathrm{op},\mathrm{FE}}$ characterization due to its potential for high mobility. An $\mathrm{ITiO}\text{-}{\mathrm{SiO}}_2\text{-}\mathrm{Si}$ MOS-driven MRR is fabricated through heterogeneous integration, which can provide orders of magnitude stronger light–matter interaction compared with ellipsometry measurement. By operating the MRR in the accumulation mode with negative $V_g$, the $Q$ factors and resonance wavelength shift ($\mathrm{\Delta }\lambda$) values are measured and subsequently used to derive the ${\mu }_{\mathrm{op},\mathrm{FE}}$ in the ultra-thin accumulation channel. Experimental results in this work demonstrate that the ${\mu }_{\mathrm{op},\mathrm{FE}}$ of ITiO increases from 25.3 to $38.4{\mathrm{cm}}^2\text{⋅}{\mathrm{V}}^{-1}\text{⋅}{\mathrm{s}}^{-1}$ with increasing negative $V_g$. This proposed ${\mu }_{\mathrm{op},\mathrm{FE}}$ measurement technique will provide an effective characterization method for field-effect electro-optic devices, especially for heterogeneously integrated silicon photonic devices.

The plasma frequency (${\omega }_p$) is related to the carrier concentration ($N_c$) by $${\omega }_p=\frac{N_c{e}^2}{{\epsilon }_0{m}^{*}},$$(1b)where $e$ is the electron charge, ${\epsilon }_0$ is the vacuum permittivity, and $m^{*}$ is the effective mass of charge carriers.

The plasma collision frequency ($\gamma$) is related to the ${\mu }_{\mathrm{op}}$ by $$\gamma =\frac{e}{m^{*}{\mu }_{\mathrm{op}}}.$$(1c)The cross-sectional schematic in the active region of the device is shown in Fig. 1(b). Applying a negative $V_g$ on the ITiO gate induces electron accumulation at the $\mathrm{ITiO}/{\mathrm{SiO}}_2$ interface and hole accumulation at the p-$\mathrm{Si}/{\mathrm{SiO}}_2$ interface. This field effect changes the optical permittivities of ITiO and Si, which influences the resonance wavelengths (${\lambda }_{\mathrm{res}}$) and $Q$ factors of the silicon MRR. The $Q$ factor can be written as [33] $$Q=\frac{\pi n_{g}L\sqrt{ra}}{{\lambda }_{\mathrm{res}}(1-ra)},$$(2a)where $r$ is the self-coupling coefficient, $a$ is the single-pass amplitude transmission, $L$ is the circumference of a ring, and $n_g$ is the group index of the ring waveguide.

The value of $a$ is related to the loss $\alpha$ by $$a^2=e^{-\alpha L}.$$(2b)The values of $r$ and $a$ are crucial to the $Q$ factor. $r$ is determined by the coupling between the bus waveguide and microring and can be adjusted by changing the waveguide gap or coupling length [34,35]. $a$ is affected by the loss from the accumulation layers of ITiO and p-Si when the negative $V_g$ is applied. Hence, applying a moderate $V_g$ changes $a$ while not affecting $r$. At the critical coupling ($r=a$) condition, the transmission at ${\lambda }_{\mathrm{res}}$ decreases to zero [33,36]. When the loss of the MRR is fixed, the $Q$ factor can be improved by working at the critical coupling condition [37].

The $\mathrm{\Delta }\lambda$ can be calculated by the change of the effective index ($n_{\mathrm{eff}}$): $$\mathrm{\Delta }\lambda =-\frac{\mathrm{\Delta }{n}_{\mathrm{eff}}}{n_{\mathrm{eff}}}{\lambda }_{\mathrm{res}}.$$(3a)

As shown in Fig. 1(a), the ITiO does not cover the whole ring. Therefore, the $n_{\mathrm{eff}}$ depends on the length of the microring covered by the ITiO electrode, which can be written as $$n_{\mathrm{eff}}=P\times n_{\mathrm{eff},\mathrm{active}}+(1-P)\times n_{\mathrm{eff},\mathrm{coupling}},$$(3b)where $n_{\mathrm{eff},\mathrm{active}}$ is the effective index in the active region covered by ITiO and the $n_{\mathrm{eff},\mathrm{coupling}}$ is the effective index in the coupling region without the coverage of ITiO. $P$ is the ITiO coverage percentage on the MRR.

To understand how the $Q$ factor and $\mathrm{\Delta }\lambda$ are affected by $V_g$, we simulated an ITiO-gated MOS MRR with a radius of 6 μm by the finite-difference-eigenmode (FDE) solver in Lumerical MODE software. The carrier concentration distribution is simulated by Silvaco and imported into Lumerical MODE. The simulation results are plotted with dashed lines in Fig. 1(c). When a negative $V_g$ is applied, it increases the $N_c$, and changes the relative permittivity [Eqs. (1a) and (1b)] of ITiO, which will further modulate the effective index $n_{\mathrm{eff}}$ of the guided mode in the microring waveguide calculated by Lumerical. The reduction of the real part of $n_{\mathrm{eff}}$ blueshifts the resonance wavelength as given in Eq. (3a), while the increase of the imaginary part of the $n_{\mathrm{eff}}$ increases the optical loss and reduces the $Q$ factor as explained in Eqs. (2a) and (2b). Figure 1(c) shows the downward trend of the $Q$ factor and blueshift of $\mathrm{\Delta }\lambda$ by applying the $V_g$.

When a negative $V_g$ is applied, it induces the field effect and changes carrier concentration in the accumulation layer. We can sweep different $\mathrm{\Delta }N$ to simulate different external $V_g$. We have already known that the electric frequency ${\mu }_{\mathrm{FE}}$ increases under the field effect because an electrostatic screening effect reduces the ionized-impurities scattering when the concentration of accumulated free carriers increases [22]. As the ${\mu }_{\mathrm{op},\mathrm{FE}}$ is also affected by ionized-impurities scattering, we expect that the ${\mu }_{\mathrm{op},\mathrm{FE}}$ also changes under the field effect. Hence, we can sweep $\mathrm{\Delta }N$ and ${\mu }_{\mathrm{op},\mathrm{FE}}$ in the simulation, which will induce different $\alpha$, $n_g$, and $n_{\mathrm{eff}}$ while running the FDE solver. $\alpha$ and $n_g$ are used to calculate the $Q$ factor with Eqs. (2a) and (2b), and the $\mathrm{\Delta }\lambda$ can be obtained from Eq. (3). After $Q$ factors and $\mathrm{\Delta }\lambda$ are obtained from the simulation, we can plot the $Q$ factor map with respect to ${\mu }_{\mathrm{op},\mathrm{FE}}$ and $\mathrm{\Delta }\lambda$, as shown in Fig. 2(c). However, we can see that the $Q$ factor maps are influenced by the initial conditions, i.e., $N_c$ and ${\mu }_{\mathrm{bulk}}$. Therefore, the final $Q$ factor map will be known when the initial condition is measured from the experiment. Finally, we can measure the experimental $Q$ factor and $\mathrm{\Delta }\lambda$ from the tunable MRR with negative $V_g$ to derive the ${\mu }_{\mathrm{op},\mathrm{FE}}$ by mapping the $Q$ factor with the simulation results. Also, we can observe how the field effect changes the ${\mu }_{\mathrm{op},\mathrm{FE}}$.

Next, a 17 nm thick ${\mathrm{SiO}}_2$ layer is formed by dry oxidation at 1000°C, and a 17 nm ITiO gate is deposited by radio frequency (RF) sputtering at room temperature, followed by a lift-off photolithography process. The ITiO is characterized by Hall effect measurement, which has the $N_c$ of $2.63\times {10}^{19}{\mathrm{cm}}^{-3}$ and ${\mu }_{\mathrm{bulk}}$ of $26.5{\mathrm{cm}}^2\text{⋅}{\mathrm{V}}^{-1}\text{⋅}{\mathrm{s}}^{-1}$. The ${\mathrm{SiO}}_2$ layer on the Si contact region is etched by hydrofluoric (HF) acid. Finally, the Ni/Au electrodes are thermally evaporated and patterned by regular photolithography. For characterization of the ITiO-gated MOS MRRs, the input and output fibers have a tilt angle of 8°, and the polarization controller is used to make the input light in the TE mode. The light is coupled into and out from the silicon bus waveguide through the waveguide grating couplers. The gate voltage is applied through the GSG electrodes from the GSG probe. Finally, the transmission spectra with different $V_g$ are detected by an optical spectrum analyzer.

In this work, the initial condition of ITiO is measured, which has the $N_c$ of $(2.624\pm 0.014)\times {10}^{19}{\mathrm{cm}}^{-3}$ and ${\mu }_{\mathrm{bulk}}$ of $26.5\pm 0.15{\mathrm{cm}}^2\text{⋅}{\mathrm{V}}^{-1}\text{⋅}{\mathrm{s}}^{-1}$. Hence, we can build the experimental $Q$ factor map with these parameters ($N_c$ and ${\mu }_{\mathrm{bulk}}$), and this $Q$ factor map can be used to derive the ${\mu }_{\mathrm{op},\mathrm{FE}}$ with the experimental results.

Since this method is an indirect method to estimate the ${\mu }_{\mathrm{op},\mathrm{FE}}$, we need to discuss its accuracy. The major error sources come from the experimental results in Fig. 4(b) with the simulation in Fig. 1(c). For the wavelength tunability, the experiment (48.5 pm/V) matches the simulation (51.9 pm/V) with a standard deviation of 7%. For the $Q$ factor, we can first compare it at the initial condition ($V_g=0\mathrm{V}$) because it does not have any change of $\mathrm{\Delta }N$ and ${\mu }_{\mathrm{op},\mathrm{FE}}$ in the accumulation layer. Therefore, we can directly see the difference between experiment and simulation when we use the same parameters. The experiment matches very well with the simulation at $V_g=0\mathrm{V}$, which only has an error of $<1\%$. Even though the $Q$ factor error increases when a larger gate bias is applied, it is still less than 5%. The error from the mismatch causes the error of ${\mu }_{\mathrm{op},\mathrm{FE}}$ ($\mathrm{\Delta }{\mu }_{\mathrm{op},\mathrm{FE}}$), which is $2.5{\mathrm{cm}}^2\text{⋅}{\mathrm{V}}^{-1}\text{⋅}{\mathrm{s}}^{-1}$. The other source of errors comes from the experiment measurement. The experimentally measured $Q$ factors are $Q\pm 50$. This standard deviation can cause a $\mathrm{\Delta }{\mu }_{\mathrm{op},\mathrm{FE}}$ of $\sim 3{\mathrm{cm}}^2\text{⋅}{\mathrm{V}}^{-1}\text{⋅}{\mathrm{s}}^{-1}$. However, in the small $V_g$ region, it can even be as large as $\mathrm{\Delta }{\mu }_{\mathrm{op},\mathrm{FE}}$ of $5-10{\mathrm{cm}}^2\text{⋅}{\mathrm{V}}^{-1}\text{⋅}{\mathrm{s}}^{-1}$ due to the small relative change. In addition, the ${\lambda }_{\mathrm{res}}$ may have $\pm 2\mathrm{pm}$ difference during the measurement, which induces $\mathrm{\Delta }{\mu }_{\mathrm{FE}}$ of $1{\mathrm{cm}}^2\text{⋅}{\mathrm{V}}^{-1}\text{⋅}{\mathrm{s}}^{-1}$. The measurement errors from $N_c$ and ${\mu }_{\mathrm{bulk}}$ are minor and cause $\mathrm{\Delta }{\mu }_{\mathrm{op},\mathrm{FE}}$ of 0.7 and $1{\mathrm{cm}}^2\text{⋅}{\mathrm{V}}^{-1}\text{⋅}{\mathrm{s}}^{-1}$, respectively. The overall error of $\mathrm{\Delta }{\mu }_{\mathrm{op},\mathrm{FE}}$ is combined with $$\mathrm{\Delta }{\mu }_{\mathrm{op},\mathrm{FE}(\mathrm{all})}=\sqrt{\mathrm{\Delta }{\mu }_{\mathrm{op},\mathrm{FE}(\mathrm{mismatch})}^2+\mathrm{\Delta }{\mu }_{\mathrm{op},\mathrm{FE}(Q)}^2+\mathrm{\Delta }{\mu }_{\mathrm{op},\mathrm{FE}(\mathrm{\Delta }\lambda )}^2+\mathrm{\Delta }{\mu }_{\mathrm{op},\mathrm{FE}(\mathrm{Nc})}^2+\mathrm{\Delta }{\mu }_{\mathrm{op},\mathrm{FE}(\mu \mathrm{bulk})}^2}.$$Finally, the error bars are plotted in Fig. 4(c) together with the mobility results. Figure 4(c) results show that the ${\mu }_{\mathrm{op},\mathrm{FE}}$ has a large fluctuation in the small $V_g$ region (0 to $-2\mathrm{V}$). We can determine the accumulation mode region ($V_g<-2\mathrm{V}$) from Fig. 4(b) since the $\mathrm{\Delta }\lambda$ is linearly proportional to $V_g$ in the accumulation mode [39]. The flat band voltage ($V_{\mathrm{FB}}$) is also found to be around $-2\mathrm{V}$ from the capacitance–voltage curve of the device, as shown in Fig. 4(d). This method can only achieve meaningful results when the field-effect is obvious. When the $V_g$ is small (0 to $-2\mathrm{V}$), the change of the carrier concentration is relatively minor compared to the bulk concentration. Therefore, the change of the $Q$ factor and $\mathrm{\Delta }\lambda$ are difficult to measure accurately, which induces a large fluctuation when $V_g$ is low. In the obvious accumulation mode region ($V_g<-2\mathrm{V}$), when the $\mathrm{\Delta }N$ becomes larger, the measurement error does not have a significant influence. Hence, we can see that the ${\mu }_{\mathrm{op},\mathrm{FE}}$ increases steadily in the moderate to strong accumulation mode. It shows a trend of increasing ${\mu }_{\mathrm{op},\mathrm{FE}}$ from 25.3 to $38.4{\mathrm{cm}}^2\text{⋅}{\mathrm{V}}^{-1}\text{⋅}{\mathrm{s}}^{-1}$ as the negative $V_g$ increases. Interestingly, a similar phenomenon is also mentioned in the TFT measurement when measuring the electric frequency ${\mu }_{\mathrm{FE}}$, and it shows a stable growth of electric frequency ${\mu }_{\mathrm{FE}}$ in accumulation mode but not in the depletion mode [40,41]. When the larger negative $V_g$ is applied, it has a higher ${\mu }_{\mathrm{op},\mathrm{FE}}$ in the accumulation layer, reducing the optical absorption loss. Therefore, it can help the ITiO-gated MOS MRR maintain a good $Q$ factor even though a larger negative $V_g$ is applied.

In conclusion, we invented a new characterization method for quantifying the ${\mu }_{\mathrm{op},\mathrm{FE}}$ in the accumulation channel by a tunable $\mathrm{ITiO}\text{-}{\mathrm{SiO}}_2\text{-}\mathrm{Si}$ MOS-driven MRR. The proposed integrated photonic platform provides dramatically stronger light–matter interaction compared with the traditional ellipsometry measurement. By constructing a comprehensive numerical model, we generated the contour map of the $Q$ factor of the MRR with respect to ${\mu }_{\mathrm{op},\mathrm{FE}}$ and $\mathrm{\Delta }\lambda$ by sweeping $\mathrm{\Delta }N$ and ${\mu }_{\mathrm{op},\mathrm{FE}}$ in the simulation. Experimental results of the $Q$ factor and $\mathrm{\Delta }\lambda$ were measured under the negative $V_g$ and subsequently used to derive the ${\mu }_{\mathrm{op},\mathrm{FE}}$ by mapping the data into the simulation results. Our experimental results demonstrated that the ${\mu }_{\mathrm{op},\mathrm{FE}}$ of ITiO increases from 25.3 to $38.4{\mathrm{cm}}^2\text{⋅}{\mathrm{V}}^{-1}\text{⋅}{\mathrm{s}}^{-1}$ with increasing $V_g$, which shows a similar trend in the electric frequency ${\mu }_{\mathrm{FE}}$. This method provides a novel pathway to precisely obtain the in-device ${\mu }_{\mathrm{op},\mathrm{FE}}$ from an integrated photonics platform that has never been explored. Our approach fills the gap of existing carrier mobility characterization methods for field-effect electro-optic devices, especially for heterogeneously integrated silicon photonic devices.

Acknowledgment. The authors would like to acknowledge the Oregon State University Materials Synthesis and Characterization Facility (MASC) and Electronic Microscopy Facility for their support in device fabrication, and Prof. Janet Tate at the Department of Physics for the Hall measurement.