Compact SPICE Model for TeraFET Resonant Detectors

Abstract

This paper presents an improved compact model for TeraFETs employing a nonlinear transmission line approach to describe the non-uniform carrier density oscillations and electron inertia effects in the TeraFET channels. By calculating the equivalent components for each segment of the channel—conductance, capacitance, and inductance—based on the voltages at the segment’s nodes, our model accommodates non-uniform variations along the channel. We validate the efficacy of this approach by comparing terahertz (THz) response simulations with experimental data and MOSA1 and EKV TeraFET SPICE models, analytical theories, and Multiphysics simulations.

1. Introduction

Terahertz plasmonic field effect transistors (or TeraFETs) have found versatile applications such as THz detectors [1,2,3,4,5], imagers [6,7,8,9,10,11], and transceivers [12,13,14]. Effective compact models for TeraFETs have been used for designing and simulating THz components, circuits, and systems [15,16,17,18]. Previously developed compact models for TeraFETs have demonstrated qualitative agreement with analytical theories and measured data [19,20,21,22]. In this work, we propose an upgraded compact model for TeraFETs achieving improved quantitative alignment with analytical theories and experimental data. The model uses an improved segmentation approach for the device channel [20,22] to accommodate the non-uniform carrier density oscillations and electron inertia effects. For each segment, the model computes the equivalent components for the channel, including conductance, capacitance, and inductance, based on the voltages at the segment’s nodes. Consequently, these components vary non-uniformly along the channel. We employ our new approach to improve the previous SPICE model frameworks (MOSA1 [23] and EKV [24]) and validate its effectiveness through comparisons with measured data and previous SPICE modeling results, analytical THz response theories, and finite-element simulations using COMSOL Multiphysics software. We observe good quantitative agreement between our new SPICE models, experimental data, analytical theories, and COMSOL simulations, even at resonant detection conditions. The improved TeraFET SPICE model can be effectively used for the design and simulation of THz electronic and optoelectronic devices and circuits.

2. TeraFET SPICE Model

In the presence of THz radiation, field-effect transistors (FETs) with asymmetric boundary conditions demonstrate a notable DC voltage response which results from the rectification of decayed or resonant plasma waves within the FET channel. The hydrodynamic equations governing carrier density in the channel [25,26,27,28,29] analytically describe this THz response. These equations with asymmetric boundary conditions can also be effectively solved using the finite element method. Also, through simulations conducted in COMSOL Multiphysics [30], one can describe the THz response and examine plasma wave profiles, thereby gaining valuable insights into the behavior of TeraFETs within the THz frequency domain [31,32,33].

Previous studies have reported on the compact SPICE models for TeraFET detectors using the model of distributed transmission lines within the channel [20,22]. Figure 1 illustrates the TeraFET structure and the equivalent circuit of the multi-segment compact model for TeraFETs, including parasitic capacitances and series resistances. The THz wave impinging on the TeraFET is represented by a THz AC voltage source applied between the gate and source terminals. The nonlinear transmission line for the channel is implemented by dividing the channel resistances, capacitances, and inductances for the intrinsic FET into multiple segments. Carrier inertia leads to a temporal delay between the terminal voltage variation and the corresponding current response. This effect is captured in the equivalent circuit by adding inductive elements distributed along the channel and forming (along with distributed resistive and capacitive elements) a transmission line. This transmission line approach has proven to be effective for high-frequency MOSFET modeling including terahertz detection applications [34,35]. The channel length of each FET segment is L/N, where L represents the total channel length and N denotes the number of segments, which could be estimated as N ≥ 3L/L_o [14]. L_o is the characteristic length of the plasma wave penetration decaying away from the source terminal. It could be estimated by $L_o=\sqrt{\mu V_{gt}/\omega }$, where $\omega =2\pi f$ is the THz radiation frequency, $V_{gt}=V_{gs}-V_T$ is the gate voltage swing, and $V_T$ is the threshold voltage, if L<<L_o, N can be chosen using empirical values. The Drude inductance characterizing the electron inertia effect in the channel is expressed for each segment as L_drude = τ/g_ch, where g_ch = ∂I_d/∂V_d is the segment channel conductance, I_d and V_d are the drain current and voltage for the segment, respectively, τ = mμ/q is the electron momentum relaxation time, m is the electron effective mass, and μ is the electron mobility [36,37,38]. In prior models, g_ch is calculated based on the voltage at the drain node d and the source node s, subsequently divided by N to enable the same L_drude for each segment [20,22]. In the enhanced model presented in this paper, the channel conductance for segment i (as shown in Figure 1) is determined based on the voltages at the respective drain node d_i and source node s_i. Consequently, L_drude is no longer uniformly distributed along the channel. This varying L_drude accounts for the varying channel resistances across segments. Figure 2 shows a profile for the variation of L_drude defined as $∆L_{drude}=L_{drude}-L_{drude0}$, where L_drude0 represents the Drude inductance calculated without THz radiation and L_drude is the Drude inductance calculated under THz radiation. Here ΔL_drude is plotted as a function of the THz radiation frequency and the drain node for each of the 50 segments, which represents different positions along the 90 nm TeraFET channel. As seen, L_drude is highly nonuniform along the device channel and exhibits resonant features at relatively high frequencies.

To obtain a validated compact SPICE model for TeraFETs, we first adjusted the SPICE parameter values to fit the current-voltage characteristics of an FDSOI MOSFET with a 90 nm channel length simulated in Sentaurus TCAD. Figure 3 shows the good agreement of I-V characteristics between the MOSFET TCAD model and both the single-channel and multi-segment MOSA1 SPICE models.

The validated MOSA1 SPICE model can be used to simulate the THz response of TeraFETs under open-drain boundary conditions. We also add the EKV model as the core for the TeraFET SPICE model to compare with experimental data [39]. Figure 4 shows the comparison of simulated THz response using the single-channel or multi-segment SPICE models and the measured THz response for NMOSFETs with different channel lengths [39]. The good agreement further justifies the use of the SPICE models for TeraFET simulations.

To evaluate the efficacy of the varying L_drude approach, we used both MOSA1 and EKV multi-segment SPICE models to compare simulation results with the analytical THz response $∆U=V_a^{2}f\left(\omega \right)/\left(4V_{gt}\right)$ [25] and the simulated response with COMSOL Multiphysics.

The description of our COMSOL model can be found in [31,32,33]. Figure 5 shows the THz response as a function of the frequency of the THz signal applied between the gate and source for the TeraFET biased above threshold (gate voltage swing V_gt = 0.15 V). The comparison employs different approaches, which include the analytical THz response equations, COMSOL simulations, and SPICE simulations with multi-segment MOSA1 or EKV models using uniform or varying L_drude. Different channel length and mobility values (0.1 m²/Vs for room temperature and 0.4 m²/Vs for low temperature [40]) are chosen for comparison. The comparison reveals that for conditions involving relatively long channels with high mobility or short channels with low mobility, the improved SPICE models using varying L_drude exhibit excellent quantitative agreement with the analytical and COMSOL results. In contrast, the previous SPICE models using uniform L_drude yield significantly reduced response values and only have qualitative agreement with the analytical and COMSOL results. This discrepancy stems from the uniform L_drude model’s inability to account for the nonlinear dependence of L_drude on distance. The response is significantly enhanced by the nonlinearity of L_drude, as the underlying mechanism relies on nonlinearity-induced rectification. For high-mobility TeraFETs with shorter channels, the SPICE models using the varying L_drude demonstrate higher response than the analytical and COMSOL results but still show much better quantitative agreement than the SPICE models using uniform L_drude. The overshot THz response in this parametric range might result from the intensified resonant feature of L_drude, which overestimates the nonuniformity of the device channel. The comparison of THz response in Figure 5 indicates that the TeraFET response is a strong function of the channel length. One must be careful with the selection of models when modeling TeraFETs with different channel lengths.

A high electron mobility of 0.4 m²/Vs achieved at cryogenic temperatures [40] is linked to a high momentum relaxation time and, therefore, to a large quality factor ωτ. This enables a resonant response and leads to a dramatic increase in responsivity.

Since the THz response stems from the rectification of the nonlinear THz current in the channel and the THz current is induced by the modulation of the carrier drift velocity and the carrier density under the THz radiation [25], it is crucial to investigate the distributions of the carrier drift velocity and the carrier density. Figure 6 and Figure 7 show the profiles of drift velocity and electron density in the channel for a relatively long channel and high-mobility TeraFET, simulated by COMSOL and SPICE using multi-segment MOSA1 or EKV models with varying L_drude. The comparison shows similar drift velocity distributions but different electron density profiles between COMSOL and SPICE simulations. This suggests the dominant role of the drift velocity in determining the THz response under the resonant conditions, as the small-signal condition for electron density is retained.

Discrepancies in the electron density profiles may arise from different bias conditions for each segment in COMSOL and SPICE simulations. The analytical THz response expressions and the COMSOL simulations are based on the gradual channel approximation and one-dimensional modeling of boundary conditions, while the SPICE simulations adopt a more complex two-dimensional modeling approach without simplified assumptions.

In the high-mobility or the short-channel low-mobility conditions, TeraFETs exhibit a clear resonant response, where the resonant peaks in the drift velocity profiles may overshadow any difference in the electron density profiles between COMSOL and SPICE simulations when determining the THz response.

Figure 8 shows a comparison of the TeraFET response among the analytical results, COMSOL simulations, and SPICE simulations using multi-segment EKV models with uniform or varying L_drude, for TeraFETs with relatively low mobility (0.1 m²/Vs) and channel lengths of 45 nm and 90 nm. For the 45 nm channel length, the SPICE model with varying L_drude exhibits a smaller response than the analytical results and COMSOL simulations. However, when the channel length increases to 90 nm, the results of the SPICE model with varying L_drude deviate from the analytical results of COMSOL simulations. Figure 9 shows the profiles of drift velocity and electron density in the channel for the 90 nm low-mobility TeraFET simulated by COMSOL and multi-segment EKV model with varying L_drude. The comparison also shows similar drift velocity distributions but different electron density profiles between COMSOL and SPICE simulations.

However, for the condition of 90 nm channel length and 0.1 m²/Vs mobility, drift velocity peaks are considerably smaller compared to high-mobility conditions, where the resonant response is evident. Consequently, differences in electron density profiles between COMSOL and SPICE simulations become significant, potentially playing a larger role in determining the THz response.

The above observations indicate that the proposed varying-L_drude model is valid only in the strong resonant mode. In this mode, the channel length L << L_cr, where L_cr = sτ(1 + (ωτ)⁻¹)^0.5 is a critical length signifying the traveling distance of plasma waves, and s and τ are plasma wave velocity and momentum relaxation time of carriers, respectively [41]. When L approaches L_cr, both the uniform and varying L_drude approaches fail to keep track of the subtle changes in electron inertia, thereby leading to differences observed between the SPICE models and the analytical results or COMSOL simulations in the non-resonant response conditions.

3. Conclusions

The enhanced compact model for TeraFETs significantly improves quantitative alignment with measured data, analytical THz response theories, and FEM numerical simulations. By incorporating non-uniform carrier density oscillations and electron inertia effects in the nonlinear transmission line of the channel, our model accurately captures the behavior of TeraFET detectors, particularly in scenarios involving resonant detection. This advanced TeraFET SPICE model can be used for more accurate predictions and optimizations of TeraFET response for THz technology applications.