Characterization of field-effect mobility at optical frequency by microring resonators

1. INTRODUCTION

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.

2. DESIGN AND PRINCIPLE

A. Design of ITiO-gated MOS MRR

To derive the ${\mu }_{\mathrm{op},\mathrm{FE}}$ of ITiO in the accumulation layer, an SOI waveguide MRR is driven by a hybrid $\mathrm{ITiO}\text{-}{\mathrm{SiO}}_2\text{-}\mathrm{Si}$ MOS capacitor operating in the accumulation mode. Figure 1(a) shows the three-dimensional (3D) schematic of the ITiO-gated MOS MRR. The active region consists of a p-type Si (p-Si) waveguide, a silicon dioxide (${\mathrm{SiO}}_2$) insulation layer, and an ITiO gate. The p-Si waveguide, which is based on a $400\mathrm{nm}\times 250\mathrm{nm}$ rib waveguide with a 50 nm partially etched slab, also serves as the bottom substrate of the MOS capacitor with the connection to the ground metal electrodes. The ${\mathrm{SiO}}_2$ covers the p-Si MRR as the oxide layer and the ITiO layer acts as the top gate, which is connected to the gate electrode. The optical properties of TCO materials can be described by the Drude model [32]:

 

Fig. 1. (a) 3D schematic of $\mathrm{ITiO}\text{-}\mathrm{Si}\text{-}{\mathrm{SiO}}_2$ MOS-driven MRR. (b) Cross-sectional schematic in the active region. With the negative $V_g$, it induces the carrier accumulation and refractive index modulation in the ITiO and Si layers. (c) Simulated results with different models: quantum-moment model plots in dashed lines and uniform model plots in solid lines. $Q$ factor (blue line, left $y$ axis) and resonance wavelength shift $\mathrm{\Delta }\lambda$ (red line, right $y$ axis) as a function of $V_g$.

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The plasma frequency (${\omega }_p$) is related to the carrier concentration ($N_c$) by

The plasma collision frequency ($\gamma$) is related to the ${\mu }_{\mathrm{op}}$ by

The value of $a$ is related to the loss $\alpha$ by

The $\mathrm{\Delta }\lambda$ can be calculated by the change of the effective index ($n_{\mathrm{eff}}$):

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

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 $V_g$.

B. Model Setup

In our previous work [38], we compared the free carrier distribution using the quantum-moment model and uniform model. The results showed that a significant difference only occurs at very large $V_g$ that can turn TCOs into the ENZ condition. However, we are not characterizing ${\mu }_{\mathrm{op},\mathrm{FE}}$ of TCO materials close to the ENZ condition in this paper. Approximating the numerical model by a uniform layer ($\mathrm{\Delta }N$) in the accumulation layer can greatly simplify the analysis without sacrificing the accuracy under a moderate $V_g$ [38,39]. To quantify the influence of the uniform concentration, Fig. 1(c) shows the difference between the distribution and uniform concentration in the accumulation layer. Even at the maximum $V_g$ of $-12\mathrm{V}$, it only induces 5% difference, which is comparable to other error sources. Therefore, we believe that the uniform accumulation layer approximation, as illustrated in Fig. 2(a), can provide acceptable accuracy. The FDE module simulates the optical field intensity ($|E{|}^2$) of the bending waveguide with an MOS structure, and the ITiO consists of the bulk layer and accumulation layer, as shown in Fig. 2(b). The $N_c$ and the ${\mu }_{\mathrm{bulk}}$ in the bulk layer are determined by the initial condition ($V_g=0\mathrm{V}$). At the initial condition, there is no field effect on the ITiO, so the carrier concentration and mobility are identical in the bulk and accumulation layer ($\mathrm{\Delta }N=0$; ${\mu }_{\mathrm{op},\mathrm{FE}}={\mu }_{\mathrm{bulk}}$). To achieve the highest $Q$ factor, it assumes the initial condition is at critical coupling, so $r$ equals $a$ at $V_g=0\mathrm{V}$.

 

Fig. 2. (a) Simulation model includes the p-Si layer, ${\mathrm{SiO}}_2$ layer, and the ITiO, consisting of the bulk material and 1 nm accumulation channel. (b) Simulated cross-sectional electric field intensity ($|E{|}^2$) distribution of the ITiO-gated MOS bending waveguide with a 17 nm ${\mathrm{SiO}}_2$ layer and a 17 nm ITiO layer. (c) $Q$ factor maps, with respect to ${\mu }_{\mathrm{op},\mathrm{FE}}$ and $\mathrm{\Delta }\lambda$, in different bulk conditions.

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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 ${\mu }_{\mathrm{op},\mathrm{FE}}$.

3. FABRICATION AND CHARACTERIZATION

A. Fabrication Processes and Testing

The ITiO-gated MRRs are fabricated on an SOI wafer. First, the bus waveguides, microrings, and grating couplers are patterned by two steps of electron beam lithography (EBL) and reactive ion etching (RIE), which has a 250 nm thick rib waveguide and a 50 nm thick slab. The MRRs have a radius of 6 μm, as shown in Figs. 3(a) and 3(b). Further, different gaps of MRRs are fabricated on the same SOI wafer to achieve the critical coupling condition. Figure 3(c) shows the experimental transmission spectrum of the passive MRR. The $Q$ factor is obtained by the Lorentzian fitting to the experimental data, which is a widely adopted method to quantify high-$Q$ resonators [33]. The $Q$ factor is determined by ${\lambda }_{\mathrm{res}}/\mathrm{\Delta }{\lambda }_{-3\mathrm{dB}}$. The passive MRR has a high $Q$ factor of 13,000.

 

Fig. 3. (a) Scanning electron microscope (SEM) image of the fabricated passive Si-MRR with false colors. The microring has a radius of 6 μm. (b) Zoom-in SEM image of microring to show the side-wall roughness. (c) The experimental transmission spectrum of the passive MRR, which is fitted by the Lorentzian function, has a high $Q$ factor of $\sim \mathrm{13,000}$. (d) Optical image of the fabricated ITiO-gated MOS MRR. The ITiO gate, which is highlighted by the white dashed line, covers the active region of the microring except the coupling region to the bus waveguide. The active region covers $\sim 83\%$ of the MRR. The gate electrode lies on ITiO, and the ground electrodes are connected to the p-Si microring through a partially etched Si slab.

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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.

B. Experimental Results

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.

Figure 4(a) shows the experimental spectra of the normalized transmission with different negative $V_g$. Experimental $Q$ factors and $\mathrm{\Delta }\lambda$ as a function of the $V_g$ are plotted in Fig. 4(b), which shows that the $\mathrm{\Delta }\lambda$ is linearly proportional to $V_g$ when the $V_g$ is beyond $-2\mathrm{V}$. It has an average wavelength tunability of $48.5\mathrm{pm}/\mathrm{V}$, and the $Q$ factor is still higher than 4000 when it has a cumulative $\mathrm{\Delta }\lambda$ of 500 pm. Next, we can derive the ${\mu }_{\mathrm{op},\mathrm{FE}}$ by mapping the experimental $\mathrm{\Delta }\lambda$ and $Q$ factor to the simulation results. Figure 4(c) plots the extraction of the ${\mu }_{\mathrm{op},\mathrm{FE}}$.

 

Fig. 4. (a) Lorentzian fitted experimental transmission spectra of ITiO-gated MOS MRR with different $V_g$. (b) Experimental $Q$ factor (blue line, left $y$ axis) and $\mathrm{\Delta }\lambda$ (red line, right $y$ axis). (c) ${\mu }_{\mathrm{op},\mathrm{FE}}$ extraction from experimental $Q$ factor and $\mathrm{\Delta }\lambda$ with errors. (d) Capacitance as a function of $V_g$ for the ITiO-gated MOS MRR.

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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

4. CONCLUSION

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.