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Nanomaterials (Basel). 2016 May; 6(5): 86.

Published online 2016 May 10. doi: 10.3390/nano6050086

PMCID: PMC5302495

Ho Won Jang, Academic Editor and Soo Young Kim, Academic Editor

Department of Electrical and Computer Engineering, Florida International University, Miami, FL 33174, USA; Email: ude.uif@600arakm (M.K.); Email: ude.uif@100bbavp (P.K.V.); Email: ude.uif@100hnisr (R.S.); Email: ude.uif@alapn (N.P.)

Received 2016 February 16; Accepted 2016 May 3.

Copyright © 2016 by the authors; licensee MDPI, Basel, Switzerland.

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

This work proposes a novel geometry field effect transistor with graphene as a channel—graphene field-effect transistor (GFET), having a hybrid contact that consists of an ohmic source/drain and its extended part towards the gate, which is capacitively coupled to the channel. The ohmic contacts are used for direct current (DC) biasing, whereas their capacitive extension reduces access region length and provides the radio frequency (RF) signal a low impedance path. Minimization of the access region length, along with the paralleling of ohmic contact’s resistance and resistive part of capacitively coupled contact’s impedance, lower the overall source/drain resistance, which results in an increase in current gain cut-off frequency, *f _{T}*. The DC and high-frequency characteristics of the two chosen conventional baseline GFETs, and their modified versions with proposed hybrid contacts, have been extensively studied, compared, and analyzed using numerical and analytical techniques.

Graphene is a promising two-dimensional material exhibiting exceptionally high crystal and electronic qualities, which has fascinated researchers for decades. The richness of its electronic and optical properties include, but are not limited to, its high residual carrier concentration, mobility, Fermi velocity, high thermal conductivity, its perfect 2D body, optical transparency, and high mechanical stability [1,2,3,4,5], which have already revealed a cornucopia of potential engineering and application. Numerous research groups have seen graphene as a descendant of Silicon for analog devices, though the roadblock of graphene—its zero bandgap [6]—made it incapable of switching off field effect transistors (FETs) and, thus, inappropriate for logic devices.

An important metric for a radio frequency transistor’s performance measurement is its current gain cutoff frequency, *f _{T}*. The cutoff frequency inversely depends on source/drain resistance, which is composed of contact resistance and access resistance, and their minimization ensures

The small signal equivalent circuit of a conventional three-terminal GFET, overlaid on the device schematic, is shown in Figure 1.

Small signal equivalent circuit overlaid on top of a conventional graphene field-effect transistors (graphene FET).

The time that it takes the charge carriers to travel from the source to the drain is called the delay time, and can be divided into two parts: transit delay and parasitic delay. The intrinsic and extrinsic gate to source/drain capacitances are responsible for the transit delay and can be expressed as [23,24]:

$${\mathsf{\tau}}_{TR}=\frac{({C}_{GS,EX}+{C}_{GD,EX})}{{g}_{m}}+\frac{({C}_{GS,IN}+{C}_{GD,IN})}{{g}_{m}}$$

(1)

On the other hand, parasitic resistances and capacitances cause a parasitic delay, as their names suggest, and can be expressed as [23]:

$${\mathsf{\tau}}_{PAR}=[1+(1+{C}_{GS,PAR}/{C}_{GD,PAR}){g}_{0}/{g}_{m}]{C}_{GD,PAR}({R}_{S}+{R}_{D})$$

(2)

where *g*_{0} = *1/R _{SD}* is the output conductance,

$${R}_{D}={R}_{S}={R}_{C}+({L}_{A}/\mathsf{\mu}q{n}_{0}W)$$

(3)

where *L _{A}* is the access region length (

$$1/2\mathsf{\pi}{f}_{T}={\mathsf{\tau}}_{TR}+{\mathsf{\tau}}_{PAR}$$

(4)

After substituting Equations (1) and (2) into Equation (4) and rearranging, the current gain cut-off frequency, *f _{T}* of the GFET can be related to the small signal equivalent circuit parameters, as follows:

$${f}_{T}=\frac{{g}_{m}/\left(2\mathsf{\pi}\right)}{[{C}_{GS}+{C}_{GD}]\times \left[1+({R}_{S}+{R}_{D})/{R}_{SD}\right]+{C}_{GD}\times {g}_{m}\times \left({R}_{S}+{R}_{D}\right)}$$

(5)

where, *R _{SD}* is the total channel resistance.

If two capacitively coupled contacts (C3s) are placed on the access regions and connected to the ohmic source/drain, as shown in Figure 2, C3 will make a path for high frequency RF signal parallel to the ohmic contact. The C3 impedance, *Z _{C3}* =

$${Z}_{C}=\frac{{R}_{C}^{2}{R}_{C3}+{R}_{C3}^{2}{R}_{C}+{X}_{C3}^{2}{R}_{C}}{{({R}_{C}+{R}_{C3})}^{2}+{X}_{C3}^{2}}-j\frac{{X}_{C3}{R}_{C}^{2}}{{({R}_{C}+{R}_{C3})}^{2}+{X}_{C3}^{2}}={R}_{C}^{\prime}-j{X}_{C}^{\prime}$$

(6)

Assuming the length of C3 is *L _{C3}*, the access region length of the hybrid contact GFET comes to be ${L}_{A}^{\prime}={L}_{A}-{L}_{C3}$, and the new expression for source/drain impedance and total channel impedance becomes:

$$\begin{array}{l}{Z}_{D}={Z}_{S}={Z}_{C}+{R}_{A}^{\prime}={R}_{C}^{\prime}-j{X}_{C}^{\prime}+({L}_{A}^{\prime}/\mathsf{\mu}q{n}_{0}W)\\ {Z}_{SD}=2{Z}_{D}+{R}_{CH}={R}_{SD}-j{X}_{SD}\end{array}$$

(7)

From the Equation (6), we can see that the total channel resistance has a real part, as well as an imaginary part. A simple matching network can be designed for matching and eliminating the imaginary part of the input impedance. An impedance matching network is an additional circuit that consists of a reactive element of such a value that can effectively nullify the opposite signed reactive element of the device, and thus eliminate the effective reactance of the whole system. It can be achieved with only two reactive elements that transform both the real and imaginary parts. A common two reactive element configuration is referred to as an L-section matching network, as shown in Figure 3a.

Considering network-1, we can quantify the input impedance as:

$${Z}_{in}=j{X}_{L}+\frac{{R}_{SD}+j{X}_{SD}}{1+j{X}_{C}{R}_{SD}-{X}_{C}{X}_{SD}}$$

(8)

To match it with a resistance, *R*, we consider *Z _{in}* =

$$\begin{array}{l}{X}_{L}=\frac{1}{{X}_{CAP}}+\frac{{X}_{SD}R}{{R}_{SD}}-\frac{R}{{X}_{CAP}{R}_{SD}}\\ {X}_{CAP}=\frac{{X}_{SD}\pm \sqrt{\frac{{R}_{SD}}{R}}\sqrt{{R}_{SD}^{2}+{X}_{SD}^{2}-R{R}_{SD}}}{{R}_{SD}^{2}+{X}_{SD}^{2}}\end{array}$$

(9)

By solving these equations, we can determine the capacitor and inductor values required to nullify the imaginary part of the contact impedance. These two relations are derived for network-1 and are valid if *R _{SD}* >

$$\begin{array}{l}{X}_{L}=\pm \sqrt{{R}_{SOURCE-DRAIN}(R-{R}_{SOURCE-DRAIN})}-{X}_{L}\\ {X}_{CAP}=\pm \frac{\sqrt{(R-{R}_{SOURCE-DRAIN})}/{R}_{SOURCE-DRAIN}}{R}\end{array}$$

(10)

For devices working on a wide frequency range, a very common technique in RF/mobile communication, named the “frequency transformation technique”, needs to be used, as reported in Reference [26].

Once the matching network has been used, only the real part of contact resistance *R’ _{c}* remains, and the new expression of source/drain resistance comes out to be:

$${R}_{D}={R}_{S}={R}_{C}^{\prime}+({L}_{A}^{\prime}/\mathsf{\mu}q{n}_{0}W)$$

(11)

One can easily acquire the relation between *f _{T}* and

C3 can be considered as an RC transmission line and its impedance can be analytically calculated [27]. If a C3 is placed on top of the gate dielectric, the contact metal and graphene channel, with the in-between dielectric material, form an RC transmission line. The propagation constant, γ, and characteristics impedance, *Z*_{0}, of this transmission line can be estimated by the following equations:

$$\mathsf{\gamma}=\sqrt{i2\mathsf{\pi}{R}_{sh}C},{Z}_{0}=\frac{1}{W}\sqrt{\frac{{R}_{sh}}{i2\mathsf{\pi}fC}}$$

(12)

where, *R _{sh}* is sheet resistance of the graphene channel,

$${Z}_{in}={Z}_{0}\text{coth}\left(\gamma {L}_{C3}\right)$$

(13)

In simulations, the impedance of C3s can be calculated by using RF transmission line method (TLM) structures, with multiple C3s with various in-between distances. Two C3s and the graphene channel in-between is a two-port network, as shown in Figure 4, and its impedance can be estimated by extracting the two-port S-Parameters and converting them to a B-Parameter [28]. The real and imaginary parts of the B-parameter are actually the real and imaginary parts of total impedance of the two-port network—two C3 impedances, in addition to the in-between graphene channel resistance.

We started our analyses by simulating the DC and RF characteristics of a conventional graphene FET, reported in [29]. This was one of our baseline devices, and we named it GFET-1. The width of this device, as well as that of all other devices simulated in this work, was 100 µm.

The gate length of the baseline GFET-1 was 3 µm, gate dielectric thickness was 24 nm, and the access region length was 1.5 µm, as shown in Figure 5. Chemical vapor deposition (CVD) graphene with a sheet resistance of 210 Ω/□ and a hole (electron) mobility of 530 cm^{2}/V·s (336 cm^{2}/V·s) formed the device channel on 300 nm of SiO_{2}.

We used a commercially available physically-based numerical technology computer aided design (TCAD) device simulation tool (Silvaco Atlas, Santa Clara, CA, USA) [30] and a modified material parameter for graphene to simulate and replicate the reported DC and RF characteristics of the baseline GFET-1. The tool solves electromagnetic and transport differential equations to calculate the electrical performance of a device modeled in DC, AC, or in transient modes of operation [31]. The simulated DC and high frequency characteristics of GFET-1 are shown in Figure 6. The simulated device characteristics are in a very good agreement with the reported ones [29], which also validates our method of simulation.

The I_{d}-V_{d} characteristics and I_{d}-V_{g} characteristics (inset) of the baseline GFET-1. RF characteristics (Current Gain, |h_{21}| and Unilateral Power Gain, UPG) of the baseline GFET-1 plotted in decibel (dB) with respect to frequency.

The sheet resistance of the graphene channel extracted from our simulation was 216 Ω/□. We aim to add two C3s to this device and short them to the ohmic contacts to extensively analyze their effects on the device’s high frequency performance.

As a starting point of capacitive impedance simulation, we first simulated a simple capacitor-like structure. It consisted of 30 nm of SiO_{2} between two metal contacts, and each metal contact had a contact resistance of 0.7 ohm-mm, as shown in Figure 7a.

(**a**) Schematic of the capacitor like structure; (**b**) The real and imaginary parts of impedance, estimated from simulations and analytical calculations.

The real and imaginary parts of this capacitive impedance were estimated using simulations, as well as analytical techniques. In Figure 7b, the real and imaginary parts of the capacitive impedance estimated from simulation and analytical calculations are plotted with respect to frequency. We can see that the results using both methods are in a very good agreement, which validates our simulation technique of estimating capacitive impedance. For further verification, we successfully regenerated the experimental data for III-N RF TLM structures reported in Reference [27]. To estimate the impedance of the capacitance formed between a C3 and graphene channel with a gate dielectric in-between, we simulated an RF TLM structure on graphene having two C3s with various in-between distances.

The C3s were placed on exactly the same structure as in the baseline GFET-1, consisting of 9 nm of SiO_{2} and 15 nm of Al_{2}O_{3} serving as the gate dielectric, deposited on CVD graphene with a carrier mobility the same as that of baseline GFET-1, as shown in Figure 8a. The impedance between contact 1 and 2, 2 and 3, and 3 and 4 were calculated at a specific single frequency, plotted with respect to distance, and extrapolated up to zero distance to extract the real and imaginary parts of a single C3 impedance at that frequency. This procedure was repeated over the frequency range of 5 GHz to 25 GHz, with a step size of 1 GHz. The real and imaginary parts of C3 impedance, plotted with respect to frequency, are shown in Figure 8b.

(**a**) Schematic of RF TLM structure on graphene; (**b**) the real and imaginary part of C3 impedance estimated from both simulation and analytical calculations, plotted with respect to frequency.

Finally, we simulated the proposed GFET, which has two C3s shorted to the ohmic source/drain contacts of the already simulated baseline GFET-1, as shown in Figure 9. The length of capacitively coupled extension was 0.8 µm in this simulation. The current gain, |h21| of the baseline GFET-1 and the proposed modified version with C3s are plotted with respect to frequency in Figure 10a. According to the definition, the frequency at which current gain becomes 0 dB is the current gain cut-off frequency, *f _{T}*. We can see from Figure 10a that, for a C3 length of 0.8 µm, the

(**a**) Current Gain, |h21| of the baseline GFET-1 along with that of the proposed hybrid contact GFET in the electron regime (V_{gs} = +2.0 V and V_{ds} = +5.0 V) plotted with respect to frequency; (**b**) The current gain cut-off frequency (*f*_{T}) of the proposed GFET **...**

The drain to source voltage, as well as the drain side C3 to source voltage, V_{ds}, was 5.0 V during the frequency domain AC simulation. As the drain bias, as well as the drain side C3 bias, were positive, we considered the GFET electron regime operation so that the drain side C3 bias accumulated more major carriers (electrons) underneath. A gate bias of V_{gs} = 2 V was used to operate the GFET in the electron regime.

We later gradually increased the length of the C3s. The approaching C3 towards the gate reduced the access region length, as well as access resistance. Additionally, the increment of capacitive coupling area due to the C3 length increment decreased the capacitive impedance. As a result of access resistance decrease, as well as the decrease in capacitive impedance, the *f _{T}* of the proposed GFET increased further. The effect of increased C3 length over

As the C3 impedance is dependent on frequency and from our results in Figure 6b, it was found that the real part of C3 impedance is reduced at higher frequencies; we intended to quantify the effect of C3 on RF performance for a shorter channel higher mobility GFETs. To do so, as before, a short channel high mobility GFET, reported in Reference [23], was chosen as our short channel high mobility baseline, and was named GFET-2. The device had a CVD-grown graphene channel with a carrier mobility of μ = 2234 cm^{2}/V·s on a sapphire substrate with a gate length of 210 nm, and a source to drain distance of 1.5 μm. We considered the device geometry to be symmetrical and estimated the access region length to be 645 nm on each side of the gate. We simulated the DC and RF characteristics of the baseline GFET-2 as before, and they were in a very good agreement with the reported data [23]. For this simulation, as well as for the following simulations and analytical calculations, the device width was considered to be 100 μm, as before. Later, we simulated our proposed short channel high mobility GFET with a hybrid contact by making a capacitive extension of 245 nm of both the source and the drain towards the gate.

The RF characteristics of the baseline GFET-2 in the electron regime, along with that of the proposed GFET, having a C3 length of 245 nm, are shown in Figure 11a. From Figure 11a, we see that the *f _{T}* of the baseline reported GFET and the proposed GFET with a 245-nm capacitive extension, are 20.05 GHz and 24.4 GHz, respectively. Later, the C3 length was gradually increase up to 550 nm as shown in Figure 8b. Due to the increase of the C3 length, the

(**a**) Current Gain, |h21| of the baseline GFET-2 along with that of the proposed hybrid contact GFET in the electron regime (V_{gs} = +0.6 V and V_{ds} = +1.6 V) plotted with respect to frequency; (**b**) The current gain cut-off frequency (*f*_{T}) of the proposed GFET **...**

In addition to the C3 extension over the access region, we also simulated a GFET with the source to drain distance the same as that of the baseline, but with a longer gate. The new length of the gate was equal to old gate length plus 2*L _{C3}*,

In this work, we have proposed and analyzed a novel geometry GFET with an ohmic source/drain and its capacitive extension towards a gate in order to overcome the set of limitations on its high-frequency performance that arises from contact resistance and access resistance. The extended part of the ohmic contacts over access region, not only reduces access region length and its corresponding access resistance, but also its capacitive coupling to the graphene channel provides a low resistance path for the high frequency signal. From our analyses, we found that our proposed long channel low mobility GFET with hybrid contacts has a current gain cutoff frequency that is 20% higher than the experimental data reported in the literature for the same geometry GFET with conventional ohmic contacts. On the other hand, the improvement for a short channel high mobility GFET with hybrid contacts was even more prominent, and had a current gain cutoff frequency 26.3% higher than that of the reported geometry conventional contact GFETs. The proposed devices would be easier to fabricate with a higher tolerance, and suitable for high frequency analog applications.

This work is supported by National Science Foundation (NSF) CAREER program with the Award Number: 0955013 and Army Research Office (Grant No. W911NF-12-1-0071). Chowdhury Al-Amin gratefully acknowledges the financial support provided by the Florida International University (FIU) Graduate School Dissertation Year Fellowship. Raju Sinha gratefully acknowledges the financial support provided by the Presidential Fellowship from Florida International University Graduate School.

Author Contributions

Nezih Pala conceived the idea. Nezih Pala and Chowdhury Al-Amin discussed the theory and simulation method. Chowdhury Al-Amin designed, systematically investigated the device and carried out the numerical simulations and analytical calculations. In addition to that, Chowdhury Al-Amin wrote the manuscript and prepared the figures. Nezih Pala edited the manuscript. Phani Kiran Vabbina, Mustafa Karabiyik, and Raju Sinha discussed the results and commented on the manuscript. Nezih Pala is the principal investigator of the project.

Conflicts of Interest

The authors declare no conflict of interest.

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