## Introduction

Metal-oxide-semiconductor field-effect transistors (MOSFETs) typically based on silicon have revolutionized modern electronics and informatics, where effective scaling and miniaturization has allowed the production of high-density integrated circuits, such as microprocessors and memory devices1,2. The development of high-performance MOSFETs requires the simultaneous optimization of various performance parameters, including not only semiconductor parameters, such as the mobility and doping concentrations, but also the geometrical parameters of layered devices, such as the channel length and the oxide thickness3,4. These parameters can play a decisive role in determining the electric performance of an individual MOSFET. One of the virtues established in modern MOSFETs manufacture is that the dynamic response in digital and analog circuits can be predictable with the static performance of a single MOSFET, i.e., the device parameters determined under DC operating conditions can be applied to the transient analysis of integrated circuits, which allows an elaborate circuit design of high-density integrated circuits, analog circuits, and functional power devices3,4,5,6. Compact models for semiconductor devices, supported by simulation program with integrated circuit emphasis (SPICE), are a representative example7,8. In the context of their establishment in modern silicon electronics, these sophisticated systems ensure the appropriate circuit design protocols, operational stabilities, and frequency responsibilities of the integrated digital and analog circuits, which should stem from a perfect agreement between the static and dynamic electric properties of a single transistor.

Organic semiconductors (OSCs) are lightweight, flexible, and exhibit a low-temperature processability, which can lead to the production of flexible, functional electronic devices9,10. In particular, their unique solution-processability at relatively low temperatures has opened opportunities in the manufacture of printable electronics11,12. Following various recent developments in printing technologies and in materials science, the mass production of highly integrated OSC devices is expected to lead to challenges in terms of the internet of things (IoT) challenges13,14,15. Recently, various groups have demonstrated the wafer-scale fabrication of ultra-thin single-crystal OSCs via a one-shot solution process16,17,18,19,20,21,22,23,24,25,26,27. The resulting excellent electronic properties, including a field-effect mobility up to 10 cm2 V−1 s−1 originating from coherent band-like transport20,27,28, in conjunction with the miniaturization of organic field-effect transistor (OFET) devices, allows high-speed switching operations at a few tens of a MHz20,29,30. Although the electronic properties of organic devices have improved considerably, in-depth understandings regarding a direct correlation between the static and dynamic responses of OFETs remain limited29,31,32,33,34,35. In general, the dynamic performance of a transistor is evaluated by the current gain cutoff frequency (fT), which is defined as the characteristic frequency at which the current gain is unity, and is given by fT = gm/2πCG, where gm and CG are the transconductance (=∂ID/∂VG, ID: drain current, VG: gate voltage), and the total gate capacitance, respectively1,2,3,4. In reality, CG comprises the gate capacitance and the parasitic capacitance (Cp), whereby the latter is caused by an extrinsic effect for which additional currents flow at the gate-overlapped area, but must be taken into account particularly for a transistor with a short channel length (L) and contact length (LC)1,2,3,4,20,29,30,31,32,33,34,35. Here, 2LC is the characteristic distance by which the gate electrode overlaps the periphery of the source and drain electrodes. In the saturation region, fT can be rewritten as20,29,30,31,32,33,34,35:

$${f}_{{\rm{T}}}\,=\,\frac{{\mu }_{{\rm{eff}}}({V}_{{\rm{G}}}\,-\,{V}_{{\rm{th}}})}{2\pi L(\frac{2}{3}L\,+\,2{L}_{{\rm{C}}})},$$
(1)

where Vth is threshold voltage. According to Eq. (1), fT can be improved by a large effective mobility (μeff), short L, and short LC. The scaling and miniaturization of devices can lead to an effective reduction in both L and LC, but in contrast, this is always in a trade-off relationship with μeff, since the effects of contact resistance (RC) are no longer negligible in the short channel devices, thereby resulting in a reduction in μeff. Although Eq. (1) gives a practical guide to the correlate dynamic response with the static performance, i.e., fT can be predicted from μeff, L, and LC, the validity of the radiofrequency circuit model represented by Eq. (1), which has been established for condensed matter semiconductor devices1,2,3 remains unclear. In particular, for carbon nanotube transistors, the discrepancy between the static and dynamic responses has often been reported since an additional contribution of the parasitic capacitance between the nanotube-gate electrode of an individual nanotube is known to reduce the theoretical fT36,37. It is quantitatively understood that sparse-density nanotubes that are often found in solution-processed thin films behave as the parasitic capacitance, for which even a slight spacing between nanotubes of 100 nm causes fT to half36,37. This motivates the investigation into whether parasitic elements in OFETs related to their unique molecular aggregates may violate the standard radiofrequency circuit model.

In this work, we focus on the manufacture of solution-processed organic single-crystal transistors, and subsequent characterization of the dynamic responses at near very high-frequency band (above 30 MHz). Our original damage-free lithography technique will be expected to allow a fine patterning of electrodes directly on the surfaces of OSCs, with the ultimate aim of producing effective miniaturization of OFETs with a sub-micrometer spatial resolution. Systematic changes in L, and LC in conjunction with full two-port scattering S-parameter measurements will be expected to reveal, whether the standard radiofrequency circuit model established on a solid-state continuous medium can be extended to solution-processed OFETs. In addition, we discuss the importance of the transfer length (LT), which is a characteristic length that determines the contact resistance, and by which the static and dynamic performances can be correlated concomitantly. It is expected that the radiofrequency circuit model presented in this work will allow not only reliable predictions of the operational performances of organic radiofrequency devices, but also the realization of practical, high-speed, organic integrated circuits, radiofrequency rectifiers, and low-noise amplifiers.

## Results

### DC characteristics and evaluation of contact resistance

To assess the static transistor properties, organic single-crystal transistors with a lithographically-defined device geometry were fabricated, and characterized under DC conditions to allow for an accurate determination of the mobility and contact resistance. Our benchmarked OSC 3,11-dinonyldinaphtho[2,3-d:2′,3′-d’]benzo[1,2-b:4,5-b′]dithiophene(C9-DNBDT-NW)20,27,38 (Fig. 1a) was deposited via a continuous edge casting technique17,20,27, a meniscus-guided coating method39, to form a single crystalline bilayer film (for details, see the “Methods” section). Top contact, bottom gate OFETs containing electrodes bearing a submicrometre-scale spatial resolution were fabricated lithographically using a fluorinated photoresist and developer (Fig. 1b, c). A similar device structure was reported in our previous studies20,30, with the exception that the thickness of the gate dielectric layer (AlOx) was reduced to 60 nm to increase the accumulated carrier density at the channel (Fig. 1b, c). We also note that the dual-channel OFET geometry, for which two active transistor channels sharing a common gate and drain electrodes are formed in parallel (Fig. 1d, e), was employed for the purpose of the full two-port S-parameter measurements (typical coaxial ground-signal-ground high-frequency probes were used). A total of 28 OFETs with different L and LC values were fabricated within a monodomain single crystal, and characterization was carried out to correlate the DC and AC electric performances, while maintaining the other geometrical parameters as constants: channel width (W) 1000 μm, OSC thickness 8 nm (bilayer thickness), and dielectric gate thickness 60 nm. It should be noted here that our bilayer single crystalline film covers an area greater than 5 cm × 1 cm, and so all 28 OFETs share an exactly identical single crystalline domain, which allows an unambiguous comparison between the static and dynamic properties. The static device performances were acquired using a semiconductor parameter analyser with controlled applications of the VG and drain voltages (VD).

Figure 1f, g show the static transfer and output characteristics obtained for a typical OFET L = 6 μm and LC = 5 μm under DC conditions, whereby the transfer curve in the saturation region was recorded with the application of a constant drain voltage VD = −7 V. The present OFET with a relatively short L exhibits a textbook-like switching characteristic with negligible hysteresis, a high on-off ratio (>1010), and a clear current saturation behavior, with the exception of Vth. The shift of Vth to a transistor ON state (normally-on state) has often been reported in OFETs with short channels, which is presumably because of an unintentional chemical doping at the channel during lithography processes16. Although this can be problematic in terms of the device stability in an actual integrated circuit, it should be satisfactory for the characterization of dynamic responses as long as the shift of Vth is taken into account during analysis. Next, we discuss the importance of LT, which can project the effect of contact resistance to a length scale. LT is defined as the characteristic length over which 63% of the charge carrier is injected at the contact and semiconductor interface (Fig. 1h).

From the transfer characteristics, the effective filed-effect mobilities μeff were determined to be 8.6 cm2 V−1 s−1 in the saturation regime. When L is reduced, the contribution of the contact resistance relative to the total device resistance increases, and so the value of μeff in the short channel OFETs does not generally agree with the intrinsic mobility μint. Here, we investigated the contact resistance effects by means of the transmission line method (TLM), where OFETs bearing different channel lengths L ranging from 2 to 50 μm were fabricated (Fig. 2a, b). Importantly, an active OSC layer for these OFETs consists of a truly monodomain single crystal with a bilayer thickness; therefore, neither the inhomogeneous distribution of μint, nor spurious effects such as trapping at grain boundaries can contaminate analysis. Figure 2b, c show typical transfer characteristics with various values of L in the linear region (LC = 5 μm, 3 μm, respectively), which confirms that neither hysteresis nor kinks exist, even in the present OFET, which has an L value of only a few μm. Since the monodomain single crystal was used, the values of Vth are essentially independent of L, except for the case where L = 1.5 μm. Hereafter, the shift of Vth was taken into account, i.e., the static and dynamic parameters were compared with standardizing VG − Vth.

Figure 2d–g show the width-normalized total resistance (Rtotal W) as a function of L with various VG − Vth (LC = 5 μm, and 3 μm, respectively). Since the present bilayer single crystalline film covers a length of more than a few centimetres, the contact resistance can be extrapolated with relatively high accuracy, which is evidenced by a high square of the regression coefficient of 0.99 over the VG − Vth. The width-normalized contact resistance (RC W) was determined from the y-intercept of Rtotal W vs. L at a given value of VG − Vth, as shown in Fig. 2f, g. It was found that RC W depends significantly on VG − Vth (Fig. 2h), which is likely to be caused by the trap states in the access region20,30, whereas VG invariant RC W has been previously reported in an ideal Ohmic contact20,30. This discrepancy can be an indicative of the imperfection of dopant insertion. Indeed, RC W was estimated to be 60–70 Ωcm, which is slightly larger than that obtained for the similar top-contact, bottom-gate device of C9-DNBDT-NW20,30. This is presumably because of the poor quality of 2,3,5,6-tetrafluoro-7,7,8,8-tetracyanoquinodimethane (F4TCNQ) deposition found particularly in the present OFET20,30. We also noticed that small molecule dopants such as F4TCNQ require extra care when inserted into the contact. More specifically, the thickness of the dopant layer should be optimized such that the dopant layer itself does not act as a bulk resistance30. Although there will be room for further improvement in the contact resistance obtained herein, this does not cause a fatal error in our analysis as long as RC W is assessed accurately. To address the effects of LC on the contact resistance, an additional key geometrical parameter, namely LT, is introduced29,31,32,33,34,35,40,41. LT is the characteristic length that determines the carrier injection at the contact and semiconductor interface. In principle, carrier injection at a linear regime with a staggered geometry undergoes current crowding29,31,32,33,34,35,40,41 at an effective injection area LT W. LT relates the contact resistivity to the channel sheet resistivity:

$$\frac{{R}_{{\rm{C}}}\cdot W}{{R}_{{\rm{sheet}}}}=2{L}_{{\rm{T}}}\coth \left(\frac{{L}_{{\rm{C}}}}{{L}_{{\rm{T}}}}\right),$$
(2)

where Rsheet is the sheet resistance of the semiconductor layer. Intuitively, a larger part of the contact area can be involved in carrier injection as Rsheet is reduced, i.e., the semiconductor channel becomes more conductive. Note that when LC is assumed to be larger than LT by a factor of 2 (LC > 2LT), $$\coth \left(\frac{{L}_{{\rm{C}}}}{{L}_{{\rm{T}}}}\right)$$ in Equation (2) can be approximated to unity, and it may be controversial whether the current crowding model is applicable when LC is significantly smaller than LT42,43. Analytically, LT can be evaluated from the x-intercept of the TLM plots; the value of L that gives Rtotal W = 0 is equal to $$-2{L}_{{\rm{T}}}\coth \left(\frac{{L}_{{\rm{C}}}}{{L}_{{\rm{T}}}}\right)$$ (Fig. 2f, g). Although previous literature has often highlighted the importance of LT, we hereafter pay more attention to how LT could play a crucial role in the static and dynamic transistor properties.

Figure 2i shows two essential static parameters, namely the intrinsic mobility (μint) and the transfer length (LT) as a function of the gate voltage (VG − Vth), where the VG dependence of LT was determined separately for LC = 3 and 5 μm from the TLM plot shown in Fig. 2d–g. Both μint and LT show similar trends, whereby both exhibit a clear plateau when greater numbers of charge carriers accumulate at the semiconductor channel, i.e., when VG − Vth becomes large. This observation suggests that carrier transport at the semiconductor channel approaches the value of μint, which is free from the effects of contact resistance, thereby resulting in the saturation of LT (Fig. 2i). It should be noted here that VG-invariant static parameters give confidence that μint and LT can be assumed to be constant during analysis of the dynamic response. The values of μint and LT were determined to be 10.7 cm2 V−1 s−1 and 1.8 μm, respectively, and most importantly, they were found to be independent of LC. The effective mobility (μeff) under the influence of the contact resistance can be expressed as follows (for details see the “Methods” section):

$${\mu }_{{\rm{eff}}}={\mu }_{{\rm{int}}}\frac{L}{L\,+\,2{L}_{{\rm{T}}}\coth \left(\frac{{L}_{{\rm{C}}}}{{L}_{{\rm{T}}}}\right)}$$
(3)

With the determined values of μint = 10.7 cm2 V−1 s−1, LT = 1.8 μm, and LC = 3 and 5 μm, the experimentally obtained μeff can be reproduced (see solid curves in Fig. 2g). It should be noted that the current crowding model is likely to be violated when OFETs have a short contact length. The present short contact device with LC = 1 μm is therefore likely to be in this violated regime because the obtained LT = 1.8 μm is notably larger than LC = 1 μm42,43. Therefore, LT was not assessed quantitatively for short contact OFETs, but instead we assumed that the experimentally determined μint = 10.7 cm2 V−1 s−1and LT = 1.8 μm can be extrapolated to short contact OFETs. The validity of this assumption will be discussed later. Overall, the obtained results verify that the static transistor properties, even for OFETs with relatively short L values, can be explained based on the conventional current crowding model. More importantly, only two parameters, namely μint and LT, which had not previously been accurately evaluated in short-channel OFETs, in conjunction with priori-fixed L and LC values, enabled us to describe the static transistor properties, and these parameters could potentially have a close relationship with the dynamic response.

### AC characteristics and evaluation of the cut-off frequency

The dynamic performance of the transistor was then evaluated to obtain the current gain fT. To probe the high-frequency response of the present OFETs, on-chip high-frequency measurements were performed using a vector network analyser in conjunction with a coaxial ground-signal-ground probe in the range of 1–100 MHz. A standard Open-Load-Short-Thru calibration was employed to de-embed the extrinsic signals from the parasitic capacitance and series resistance associated with the pads and cables. Bias-Tee was used to combine the DC and RF signals (Fig. 3b), which ensures that the OFETs are operating in the saturation region. The de-embedded, full scattering S-parameters comprise a complete set of coefficients of the intrinsic input and the output electrical signals of the OFETs. Figure 3c shows a plot of the four S-parameters as a function of frequency (f), by which the short circuit small-signal current gain 20$$\mathrm{log}\,| {h}_{{\rm{21}}}|$$ (in units of dB) was evaluated (see the “Methods” section). As shown in Fig. 3d, the current gain decreases with increasing f, following −20 dB dec−1 (represented as a black line in Fig. 3d), which is consistent with the conventional model expected for FETs; the observation of a decay slope of −20 dB dec−1, equivalent to f−1 dependence, is a consequence that the gate impedance given as jωCG decreases with increasing frequency. Here, ω = 2πf and CG is the total gate capacitance. Thus, this gives us confidence that the present full S-parameter measurements are valid, and can be used to evaluate fT. We note that fT is defined as the frequency at which the current gain is zero (20$$\mathrm{log}\,| {h}_{{\rm{21}}}| =0$$, and so equivalent to h21 = 1). According to Eq. (1), a combination of the large μeff, short L, and short LC gives a large value of fT. Indeed, a large fT of 45 MHz was achieved with the short-channel OFET where L = 1.5 μm and LC = 1 μm, which is the largest value reported to date for OFETs and sufficient enough for use as a wireless power supply for the near-field communication (13.56 MHz) RFID tags. Furthermore, the fT values for 19 transistors with different L and LT values were evaluated systematically (Fig. 4a–c). Similarly, as shown in Fig. 3, the current gains for all 19 OFETs exhibited f−1 dependence, from which fT was evaluated. It should be noted that μint and LT were initially determined within a monodomain of the single crystalline films, and were found to be independent regardless of device geometry, thereby allowing us to link the static parameters μint and LT to the dynamic parameter fT.

Using the experimentally determined values of μint and LC, the expression of fT can be written as follows (see the “Methods” section):

$${f}_{{\rm{T}}}=\frac{{\mu }_{{\rm{int}}}({V}_{{\rm{G}}}-{V}_{{\rm{th}}})}{2\pi (\frac{2}{3}L\,+\,2{L}_{{\rm{C}}})\left\{L\,+\,2{L}_{{\rm{T}}}\coth \left(\frac{{L}_{{\rm{C}}}}{{L}_{{\rm{T}}}}\right)\right\}},$$
(4)

Importantly, all parameters on the right-hand side of Eq. (4) are experimentally-addressable, static parameters. To validate this, fT was plotted as a function of L with various LC values (Fig. 5a), whereby the solid curves represent a theoretical fT derived from Eq. (4) with μint = 10.7 cm2 V−1 s−1 and LT = 1.8 μm, and show an excellent agreement with the experimental fT value. Similarly, a universal trace was observed when fT was displayed as a function of the areal geometrical factor $$(\frac{2}{3}L+2{L}_{{\rm{C}}})\left\{L+2{L}_{{\rm{T}}}\coth \left(\frac{{L}_{{\rm{C}}}}{{L}_{{\rm{T}}}}\right)\right\}$$ (in units of μm2) as shown in Fig. 5b, where all experimentally-determined values of fT can be plotted on the universal line represented by the red line (μint = 10.7 cm2 V−1 s−1). A good agreement between the experimental and theoretical fT values confirms that Eq. (4) is valid, and that the dynamic responses of the OFETs can be predicted using the static device parameters. A slight deviation found particularly in the OFETs with short values of L and LC may be indicative of an underestimation of fT.

Based on the above results, we could summarize our key experimental findings. More specifically, the two static device parameters, namely the intrinsic mobility (μint) and the transfer length (LT), can be evaluated accurately using the TLM. The former of these parameters is identical to the mobility purely unique to the material parameter of C9-DNBDT-NW, while the latter is the characteristic length parameter that visualizes how the effects of contact resistance can be dominant, and can be projected on the length scale. Using the priori-designed channel length (L) and the contact length (LC), the areal factor $$(\frac{2}{3}L+2{L}_{{\rm{C}}})\cdot \left\{L+2{L}_{{\rm{T}}}\coth \left(\frac{{L}_{{\rm{C}}}}{{L}_{{\rm{T}}}}\right)\right\}$$ was introduced for the first time to reproduce the effective mobility (μeff) under the influence of the contact resistance and the cutoff frequency (fT). Here, the smallest areal factor achieved in this work was 26 μm2, where L = 1.5 μm, LC = 1 μm, and LT = 1.8 μm.

Although minimization of the areal factor could be considered the key factor for improving the fT, three length parameters, namely L, LC, and LT, clearly interplay to determine the areal factor, which gives a degree of freedom in the geometrical design of OFETs. We would also like to clarify some practical guidelines that have been already highlighted, in addition to some that have not been recognized. Firstly, LT is a predominant parameter in terms of impacting the whole areal factor, since the term $$\coth \left(\frac{{L}_{{\rm{C}}}}{{L}_{{\rm{T}}}}\right)$$ diverges rapidly as LT increases. Therefore, a reduction in the contact resistance is necessary. Secondly, a reduction of L without scaling down LT has a lesser impact on reduction of the areal factor. For example, when LT is limited to 1.8 μm (the best value obtained herein) only half of the areal factor (14 μm2) is expected to be reached, even when L is 0.01 μm, and this is clearly an ineffective miniaturization despite L being reduced by a factor of 100. In other words, scaling down L and LC is more effective until these two values become comparable to LT. Thirdly, an optimum LC can be found once LT is fixed. As shown in Fig. 5c, the minimum value of the areal factor appears when LC reaches approximately 70% of LT. Our analysis, by which the areal factor is highlighted for the first time, gives practical guidelines to maximize fT, particularly as the latter two guidelines had not been specifically mentioned in previous studies. Finally, we quantified the key areal factor to achieve higher fT values in the OFETs. More specifically, a 100 MHz operation requires an areal factor of approximately 10 μm2, while 1 GHz requires an areal factor of 1 μm2 when μint = 10 cm2 V−1 s−1, in addition to an input voltage of 5 V. Previously, the smallest reported areal factor was evaluated as approximately 20 μm230,44. In the context of materials science, further improvement of the mobility in conjunction with an effective miniaturization of OFETs is necessary and continues to be an ongoing challenge.

## Discussion

We accurately evaluated both the intrinsic mobility and the transfer length of micrometer-scale OFETs by means of the transmission line method, and successfully correlated these two static parameters with the cut-off frequency determined by de-embedded, full scattering S-parameters. The areal factor, which was introduced for the first time in this study, helped to clarify the one-to-one relationship between the static and dynamic parameters in the electrical performances of OFETs, and showed that the radiofrequency model established in a solid-state, continuous medium is extendable to organic single-crystal transistors. Although the validity of this radiofrequency circuit model is limited only to OFETs with single-crystalline OSCs, it allows a reliable prediction of the performances of organic radiofrequency devices. The areal factor can project electrical parameters, such as the contact resistance, on a length parameter, and visualize the effective miniaturization of OFETs, which is expected to provide a useful design guideline not only in terms of further improvements in the cut-off frequency, but also in the integration of organic digital and analog circuits.

## Method

### Device fabrication

All devices were fabricated on a pre-cleaned glass substrate. Thermally evaporated Cr (1.5 nm)/Au (20 nm)/Cr (1.5 nm) layers were patterned by conventional photolithography to form gate electrodes, where a standard positive photoresist (TLOR, Tokyo Ohka Kogyo Co., Ltd.) and developer (NMD-3, Tokyo Ohka Kogyo Co., Ltd.) were used. A 60 nm-thick layer of aluminum oxide was deposited as the gate dielectric layer via atomic layer deposition. The surface of the aluminum oxide was then treated with a 2-(phenylhexyl)phosphonic acid self-assembled monolayer (SAM) by immersing the substrate into a 0.2 mM solution for 18 h. Single crystalline bilayer films of C9-DNBDT-NW were fabricated from a 0.02 wt% 3-chlorothiophene solution with continuous edge casting20,30. The substrate was heated to 81 °C and moved with a shearing rate of 16 μm s−1 while a blade that sustains the meniscus was fixed 95 μm above the substrate. Bilayer single crystalline films were grown selectively by tuning the substrate temperature45. Following annealing of the substrate at 100 °C under vacuum to remove any residual solvent, F4TCNQ and Au were subsequently deposited to form the source/drain electrodes, which were patterned by multiple lithographic processes, whereby a negative photoresist (OSCoR4001, Orthogonal inc.) and Au etchant (AURUM S-50790, Kanto Chemical Co. Inc.) were used. Note that F4TCNQ dopants directly above the channel are removed during the Au etching process, so that the active OSC at the channel remains as an intrinsic semiconductor (undoped semiconductor). All photo-exposure processes were carried out using a maskless aligner (MLA150, Heidelberg Instruments) in a clean room environment. The patterning spatial resolution was evaluated as ±300 nm, which was taken into account during analysis.

### Electrical measurements

All electrical measurements were performed under ambient conditions. The static transistor properties were acquired using a semiconductor parameter analyser (Keithley 4200-SCS) in conjunction with a manual probe station. The on-chip high frequency measurements employed for evaluation of the cut-off frequency (fT) were conducted using a vector network analyser (Agilent E5061B) combined with a manual high frequency probe station, whereby a coaxial ground-signal-ground high frequency probe (GGB Picoprobe 40A-GSG-200 EDP) was used. Before carrying out any measurements, a standard Short-Open-Load-Thru calibration was completed with a standard calibration substrate (GSG CS-5). Bias-Tee (Mini Circuits, ZFBT-4R2GW) was used to combine the DC and RF signals. A source measurement unit (Keithley 2636) was used as an additional DC voltage supply. The short circuit small-signal current gain was defined as $$20\mathrm{log}\,| {h}_{{\rm{21}}}|$$, where $${h}_{{\rm{21}}}=\frac{{i}_{{\rm{D}}}}{{i}_{{\rm{G}}}}$$ was determined from

$$| {h}_{{\rm{21}}}| =| \frac{-2{S}_{{\rm{21}}}}{(1\,-\,{S}_{{\rm{11}}})(1\,+\,{S}_{{\rm{22}}})\,+\,{S}_{{\rm{12}}}{S}_{{\rm{21}}}}| ,$$
(5)

where S11, S12, S21, and S22 are the de-embedded full scattering S-parameters. In addition, fT was evaluated by fitting the f dependence of the current gain (Fig. 4d), where fT was defined as the frequency at 20$$\mathrm{log}\,| {h}_{{\rm{21}}}| =0$$, where h21 = 1.

### Analysis of μeff and fT

We proposed that the intrinsic mobility (μint) and the transfer length (LT) are universal parameters that are independent of the device geometry, thereby enabling a direct link between the static and dynamic electrical properties, as outlined in Eqs. (3) and (4). Equation (3) is derived from:

$${\mu }_{{\rm{eff}}}={\mu }_{{\rm{int}}}\frac{1}{1+\frac{{R}_{{\rm{C}}}\cdot W}{L}{C}_{{\rm{i}}}{\mu }_{{\rm{int}}}\left({V}_{{\rm{G}}}-{V}_{{\rm{th}}}\right)},$$
(6)

Here, the term $${C}_{{\rm{i}}}{\mu }_{{\rm{int}}}\left({V}_{{\rm{G}}}-{V}_{{\rm{th}}}\right)$$ is equivalent to $${({R}_{{\rm{sheet}}})}^{-1}$$. Thus, Eq. (6) can be reduced to

$${\mu }_{{\rm{eff}}}={\mu }_{{\rm{int}}}\frac{L}{L+\frac{{R}_{{\rm{C}}}\cdot W}{{R}_{{\rm{sheet}}}}},$$
(7)

Equation (7) is identical to Eq. (3) considering the given expression of LT in Eq. (2). Combining Eqs. (1) and (3) gives Eq. (4).