Describing Broadband Terahertz Response of Graphene FET Detectors by a Classical Model

Direct power detectors based on field-effect transistors are becoming widely used for terahertz applications. However, accurate characterization at terahertz frequencies of such detectors is a challenging task. The high-frequency response is dominated by parasitic coupling and loss associated with contacts and overall layout of the component. Moreover, the performance of such detectors is complicated to predict since many different physical models are used to explain the high sensitivity at terahertz frequencies. This makes it hard to draw important conclusions about the underlying device physics for these detectors. For the first time, we demonstrate accurate and comprehensive characterization of graphene field-effect transistors from 1 GHz to 1.1 THz, simultaneously accessing the bias dependence, the scattering parameters, and the detector voltage responsivity. Within a frequency range of more than 1 THz, and over a wide bias range, we have shown that the voltage responsivity can be accurately described using a combination of a small-signal equivalent circuit model, and the second-order series expansion terms of the nonlinear dc I–V characteristics. Without bias, the measured low-frequency responsivity was 0.3 kV/W with the input signal applied to the gate, and 2 kV/W with the input signal applied to the drain. The corresponding cutoff frequencies for the two cases were 140 and 50 GHz, respectively. With a 300-GHz signal applied to the gate, a voltage responsivity of 1.8 kV/W was achieved at a drain-source current of 0.2 mA. The minimum noise equivalent power was below 30 pW/$\sqrt{\rm Hz}$ in cold mode. Our results show that detection of terahertz signals in graphene field-effect transistors can be described over a wide frequency range by the nonlinear carrier transport characteristics obtained at static electrical fields. This finding is important for explaining the mechanism of detection and for further development of terahertz detectors.

and space sciences [3]. The direct power detector is an indispensable component for such applications [4], [5]. However, the required high sensitivity and stability of the detectors, over a wide frequency range and, preferably, at room temperature, are not readily available at the THz frequencies [6]. Superconducting detectors, such as hot electron bolometers [7], can provide high sensitivity, but operate at cryogenic temperatures using relatively complex cooling systems, which can be justified only for specific applications. Nonlinear semiconductor devices, such as Schottky-barrier diodes [8], [9], backward tunnel diodes [10], heterojunction bipolar transistors [11], and field-effect transistors (FETs) [12], have been proved to be good candidates for high-sensitivity room-temperature detectors. In particular, development of broadband THz detectors based on FETs is being a target of intensive theoretical and experimental studies [13]- [16] since the mid 1990s. As a result, the complementary metal oxide semiconductor (CMOS) detectors with the noise equivalent power (NEP) as low as 10 pW/ √ Hz at 300 GHz have been demonstrated [17]. The NEP of the best published detectors based on the GaN high electron mobility transistors (HEMTs) is, currently, 26-31 pW/ √ Hz in the frequency range of 450-650 GHz [16]. Since the first demonstration of the graphene-based [18] THz detector by Vicarelli et al. [19], detectors based on the two-dimensional (2-D) materials have drawn increasing attention due to their unique properties, which open up possibilities for applications in a number of emerging areas requiring flexibility [20], and transparency while being compatible with state-of-the-art CMOS technology [21]. Apparently, a better understanding of device principles and the main limiting factors is important in order to exploit the full potential of graphene detectors. To explain the detection mechanisms and corresponding modeling at THz frequencies [22], many researchers have examined plasma wave mixing [19], thermoelectric [23], bolometric effects [24], and rectification or resistive self-mixing, due to the nonlinearity of the dc I-V characteristics [25].
However, these studies are not able to explain the underlying physics for detection at THz frequencies. Most studies utilize antenna-coupled detectors [16], [26] and it is therefore difficult to perform accurate calibration of the available and reflected RF power. In addition, it is necessary to vary frequency, bias, temperature, and other conditions to distinguish between different types of detection mechanisms. For example, the bias dependence of the detector responsivity was analysed by Khan et al. [27] at 500 GHz and by Sun et al. [28] at 900 GHz for This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see http://creativecommons.org/licenses/by/4.0/ antenna-coupled CMOS detectors and GaN HEMT detectors, respectively. On-wafer characterization [29] of FET detectors was recently demonstrated [30], but within a limited frequency band. In the research by Andersson and Stake [25], on-wafer characterization was systematically applied to characterise GFET detectors using a Volterra series expansion of the dc drain current, but only for a frequency range below 67 GHz and in cold mode. Nevertheless, no previous work has shown accurate broadband measurements that reveal the overall detection characteristics and only few reports on classical quasi-static modeling of THz detectors.
In this work, we use recent advancements in THz on-wafer probe [31] characterization, which allows us to simultaneously measure the dc I-V dependence, high frequency scattering parameters (S-parameters), and the GFET detector response from 1 GHz to 1.1 THz. The method provides accurate calibration of power in close vicinity of the device under test, and hence accurate characterization of the responsivity over an ultrabroad frequency range. The responsivity of the GFET versus frequency and bias with the input signal applied either to the gate or the drain terminals is investigated. Finally, it is shown that a standard FET equivalent circuit together with a series expansion of the nonlinear static I-V characteristics [25], can reproduce the experimental data up to 1.1 THz and over a wide bias range. The main advantage of an empirical model is that no prerequisite knowledge of physical parameters or the level of model details is required in order to make conclusions about the main operating principles.

II. ELECTRICAL NONLINEAR MODEL
In this section, the rectified output current of the GFET THz detector will be modeled using the nonlinear properties of the GFET as expressed by a second-order Taylor expansion of the dc I-V characteristics of the GFET [27], i. e., where v g and v d are the intrinsic voltages of the ac signal applied to the gate and the drain terminals respectively, and where θ is the phase difference between them. The v g , v d , and θ can be extracted using the equivalent circuit in Fig. 1. The g 2 parameters are the second-order Taylor series coefficients obtained from the intrinsic voltage derivatives of the dc drain-source current (I DS ) with respect to the intrinsic drain-source and gate-source voltages where the intrinsic drain-source and gate-source voltages are given by V GSi = V GS − R S I DS and V DSi = V DS − (R S + R D )I DS , respectively, as can be derived from the equivalent circuit in Fig. 1. This model shows the intrinsic GFET (inside For the voltage output, the detector can be modeled as a current source with an intrinsic source-drain resistance (r ds ). Thus, the rectified voltage measured between the drain and the source terminals can be expressed as [32] v where R M is the impedance of the voltmeter. The expression is simplified due to the relatively large value of R M . Finally, the voltage responsivity of the detector is where P ava is the available input signal power to the detector, and S ii is the complex reflection coefficient S 11 or S 22 depending on whether the input signal is applied to the gate or the drain. In the experimental section, the g 2 parameters will be extracted from the measured dc I-V characteristics and used for modeling the THz output voltage and responsivity. Finally, approximative analytical expressions for the g 2 parameters will be given in the appendix. Fig. 2 shows an optical micrograph of a GFET detector. The gate length (L) is 1.2 μm, and the gate width (W ) is 5 μm (two 2.5-μm wide gate fingers in parallel). The devices were fabricated using single-layer chemical vapor deposition graphene (provided by Graphenea). Graphene was transferred onto Si/SiO 2 substrates (high-resistivity silicon with 1 μm SiO 2 ), and covered with a 6-nm Al 2 O 3 layer to prevent any contaminations during following processing steps. The Al 2 O 3 layer was deposited by repeating four times a sequence of e-beam evaporation and natural oxidation in air of a 1-nm Al layer. The area for the graphene mesa was defined by e-beam lithography, whereafter the Al 2 O 3 and graphene outside the mesa were removed by buffered oxide etch and oxygen plasma, respectively. Source and drain electrodes were patterned by a standard sequence of the e-beam lithography, Al 2 O 3 etch, Ti/Pd/Au (1/15/120 nm) deposition, and lift-off. After patterning of the gate area, a uniform Al 2 O 3 gate dielectric was deposited by repeating six times a sequence of depositing a 1-nm Al  layer and natural oxidizing in air. Then Al/Au (100/80 nm) gate electrodes were formed by using standard deposition and lift-off technology. Finally, Ti/Au (20/280 nm) pad electrodes were formed by using deposition and lift-off.

III. DESIGN, FABRICATION, AND EXPERIMENTAL METHODS
The devices were characterized on-wafer at room temperature. Fig. 3 shows the schematic block diagram of the onwafer measurement setup, together with a photograph of T-wave probes in WR-3.4 waveguide band. The signal obtained from a Keysight N5247 A PNA-X microwave network analyzer with frequency extender modules from Virginia Diodes Inc. (VDI) was applied to the gate/drain using ground-signal-ground (GSG) probes. The rectified voltage between the drain and the source was measured using an SR830 lock-in amplifier with a modulation frequency of 1 kHz. The dc voltage and current were provided and measured by a Keithley 2604B source meter.
For the frequency range of 1 to 67 GHz, the output signal power (P ) was measured using a PNA power sensor 1-67 GHz at the end of coax cables. For other higher frequency ranges, the P was measured using a VDI Erickson PM5 power meter with taper waveguide section at the waveguide flange of the extenders. The available signal power (P ava ) was calculated from the measured P after subtraction of measured insertion loss and return loss of the probes. The S-parameters were measured by the network analyzer with different on-wafer calibration procedures for different frequency ranges. For the frequency range of 1 to 67 GHz, a short-open-load-thru procedure was used by a CS-5 calibration kit from Picoprobe. For other higher frequency ranges, the on-wafer calibration was performed with thru-reflect-line (TRL) calibration kits which were designed and fabricated on the same substrate as GFET detectors. The TRL calibration kits consist of three kinds of standards, i.e., through, short, and λ/4 line. To achieve a 50 Ω line impedance and fit the device dimension at the same time, the width and the gap of the TRL calibration kits were designed to be 9 and 6 μm, respectively. Different extender modules, probes, power calibration tools, S-parameters calibration procedures, and lengths of the λ/4 line for corresponding frequency ranges are listed in Table I.

A. Second-Order Taylor Series Coefficients
The second-order Taylor series coefficients, i.e., g 2,gs , g 2,gs,ds , and g 2,ds , were calculated from (2)-(4) based on the measured dc I-V characteristics. To reduce the effects of noise on extracted data, a five-point moving average filter was applied while calculating the first-order derivatives. Fig. 4(a)-(c) show the g 2,gs , g 2,gs,ds , and g 2,ds as a function of V GS for a set of different I DS in the range from −0.2 to 0.2 mA, respectively. Under cold condition (no bias), the g 2,gs is zero, and the g 2,gs,ds and g ds2 are nonzero, which are displayed with black dashed curves in Fig. 4. The observed increase of g 2,gs with I DS , in the vicinity of the Dirac point, can be explained by analyzing the corresponding expression derived from the analytical quasi-static GFET model (see Appendix). It can be seen that for V DSi much less than V GSi , the g 2,gs is linearly proportional to V DSi . Analysis of the (8) in Appendix indicates that away from the Dirac point, i.e., at larger V GSi , the g 2,gs goes to zero, which is in agreement with dependence on Fig. 4(a). It can be shown that at larger V DSi one can expect saturation of the g 2,gs , which can be associated with the charge carrier velocity saturation. With I DS increasing, the curves of g 2,gs,ds and g 2,ds have positive shifts due to channel doping by the drain bias [see (9) and (10) in Appendix]. Note that the polarities of the g 2,gs,ds and g 2,ds are opposite, which means that generated rectified output currents with opposite polarities can counteract each other. , we observed relatively large standing waves due to less accurate calibration and parasitcs associated with the device. The parasitic capacitances (i.e., C PG and C PD ) and inductances (i.e., L G , L D , and L S ) were extracted from measured S-parameters using open and short de-embedding structures, respectively [33]. The remaining circuit elements were then extracted from the measured GFET two-port S-parameters [34], as shown in Table II. Based on these elements, the intrinsic ac signals were modeled using   the equivalent circuit (see Fig. 1). Fig. 6(a) and (b) shows the modeled v g , v d , and cos θ with 1 μW signal applied to the gate and the drain, respectively. With the signal applied to the gate, v g is much larger than v d ; while with the signal applied to the drain, v g is much lower than v d . Fig. 7(a) and (b) shows the R V versus frequency from 1 GHz to 1.1 THz with the signal applied to the gate and the drain, respectively. The R V shows 1/f 2 dependence at higher frequencies in agreement with the simulation results. Note that the R V increases at lower frequencies with the signal applied to the gate because high-impedance mismatch at lower frequencies may reduce the accuracy of the characterization. The dash-dot lines indicate that the cut-off frequencies are around 140 and 50 GHz with the signal applied to the gate and the drain, respectively. The cut-off frequency is mainly determined by the parasitic elements of the equivalent circuit in Fig. 1 [25]. Fig. 7 also shows the modeling results assuming only half of the parasitic resistance or/and gate capacitance, which demonstrates that further optimization can be achieved by increasing v g and v d via reducing the parasitic resistance and the gate capacitance.

D. Responsivity at 300 GHz Versus Bias
Due to effects of the high impedance and capacitive elements of the device at high frequencies, we assume that S 11 , S 22 , v g , and v d are bias independent at 300 GHz. The top row of Fig. 8 shows the experimental and simulation voltage responsivity R V versus bias of the device with 300 GHz signal applied to the gate. The R V , at V GS close to the Dirac point (V Dir ), is proportional to I DS , which follows the same trend as the g 2,gs in Fig. 4(a) except that under cold condition, as show in Fig. 8(c). This is  because, with the signal applied to the gate, v g is much larger than v d , as shown in Fig. 6(a), so the g 2,gs term in (1) becomes the dominant part with the increase of the drain bias. When I DS is zero, the R V is not zero as g 2,gs due to the contribution of the g 2,gs,ds term. For negative biasing, the maximum value of the R V increases up to 1.8 kV/W with I DS = −0.2 mA. Compared with previously reported record value of 70 V/W [35], this is a breakthrough in GFET THz detection.
The bottom row of Fig. 8 shows the experimental and simulation R V versus bias of the device with 300-GHz signal applied to the drain. The sign of the R V changes from positive to negative with increasing V GS . The maximum of R V is approximately 300 V/W. As I DS increases, there is a positive shift of the R V curves with no obvious changes of the maximum of R V , which follows exactly the trend of the g 2,ds . This is because v g is much larger than v d in the case of the signal applied to the drain (see Fig. 6). So the g 2,ds term in (1) is the dominant part. As can be seen from Fig. 8, the nonlinear empirical quasi-static model shows excellent agreement with the experimental data.
According to above analysis, in the case of the signal applied to the gate, the response depends on the g 2,gs and g 2,gs,ds terms in the drain-biased and unbiased mode, respectively; while with the signal applied to the drain, the response depends on the g ds2 term only. Therefore, further performance improvements can be achieved using following approaches: 1) increasing the g 2,gs , g 2,gs,ds , and g 2,ds by reducing the residual carrier density [see (8)- (10) in Appendix] and experimentally demonstrated by Generalov et al. [35]; 2) increasing the v g and v d by reducing the contact resistance and the gate capacitance; 3) applying the signal at both the gate and the drain ports with 180 o relative phase difference; 4) increasing g 2,gs by increasing the drain biasing with the signal applied to the gate. Particularly, we can reduce the contact resistance and the residual carrier density via modifying the fabrication process [36], and a lower gate capacitance can be achieved by reducing the gate length.

E. NEP at 300 GHz Versus Bias
The sensitivity of a detector is evaluated by its NEP, which is given by NEP = √ S V /R V where S V is the noise spectral density. Fig. 9(a) shows the S V at V GS − V Dir = −0.2 V and V DS = −0.1 V, which was measured using a Keysight E4727 A advanced low-frequency noise analyzer. The S V reveals a 1/f spectral dependence with modulation frequency in the range of 1 Hz to 1 MHz. Therefore, the S V noise with other bias points can be estimated by the Hooge's model [37]. Fig. 9(b) and (c) shows the estimated minimum NEP at different V GS (NEP min ) as a function of I DS with 300 GHz signal applied to the gate and the drain, respectively. The solid and dashed lines indicate the NEP min at modulation frequencies of 1 kHz and 1 MHz, respectively. The noise is lower at higher modulation frequency since the effect of 1/f noise is negligible and only dominated by the thermal noise, as shown by the dotted lines in Fig. 9(b) and (c). At low modulation frequencies, the NEP increases with the drain biasing, since 1/f noise is the main noise of the detector. While at high modulation frequencies NEP  decreases with the current biasing when the THz signal is applied to the gate, which is due to the increase of responsivity with the drain bias. At 1 kHz modulation frequency, the minimum NEP of 26 and 24 pW/ √ Hz with 300 GHz signal applied to the gate and the drain, respectively, appears at zero drain basing. To achieve a lower NEP, both higher R V and lower noise are required. We have discussed four methods to increase the R V above. However, the last method, i.e., increase of the drain biasing, can introduce extra noise source.

V. CONCLUSION
In conclusion, we have demonstrated accurate and comprehensive on-wafer characterization of GFET detectors, which allows for corrections of the impedance mismatch and thereby opens up possibility for detailed measurements up to THz frequencies. It is shown that the GFET response versus frequency and bias can be well predicted by a nonlinear empirical model derived from the dc I-V characteristics and the S-parameters. Previous research on large signal quasi-static device models [38] of GFETs has successfully been applied to design THz heterodyne mixers [39] and detectors. This indicates that drift-diffusive type charge-carrier transport or quasi-static models are sufficient to describe the operation of GFETs from dc to THz frequencies.
Finally, we reported a voltage responsivity of 1.8 kV/W at I DS of 0.2 mA and a minimum NEP less than 30 pW/ √ Hz under cold condition at 300 GHz. Table III compares the THz voltage responsivity and NEP achieved in this work with THz detectors fabricated by different technologies. The presented GFET detector reveals significant improvement in performance at THz frequencies, in comparison with previously published graphene-based detectors, and approaching the semiconductorbased counterparts. This work provides a basis for understanding of the broadband THz detection in the GFET-based detectors, which will advance the development of FET-based THz electronics.

APPENDIX ANALYTICAL QUASI-STATIC GFET MODEL
According to the drain resistance model [44] of GFET, the dc drain current can be expressed as where q is the elementary electron charge, μ is the carrier mobility, n 0 is the carrier density at the Dirac point, and C ox is the gate capacitance per unit area. Then the Taylor coefficients in (2)-(4) can be expressed as Thus, the rectified voltage at the signal applied to the gate and under drain current basing can be approximated as The rectified voltage at the signal applied to the gate and without drain current basing can be approximated as The rectified voltage response at the signal applied to the drain can be approximated as v THz ≈ − 1 2