Characteristics of an armchair graphene nanoribbon tunnel field effect transistor (AGNR-TFET) were modeled quantum mechanically. The transport equation in the AGNR-TFET was solved by using the Dirac-like equation. The potential profile in the AGNR-TFET was determined by solving the Dirac-like equation and the self-consistent Poisson equation. The transfer matrix method (TMM), as a numerical approach, and the Landauer formula were used to calculate the electron transmittance and the tunneling current respectively. The threshold voltage of the device was around 0.01 V. The effect of the AGNR-TFET’s geometry, i.e. width and length of AGNR and oxide thickness, on the tunneling current and the subthreshold swing was also analyzed. It was found that the tunneling current increased with an increase of the width of the AGNR and the oxide thickness while increasing the length of the AGNR made the tunneling current decrease. According to the simulation results, the subthreshold swing of the device can achieve 5 $mV/dec$. Moreover, the AGNR-TFET geometry affects the subthreshold swing of the device.

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- (Program Studi Fisika, Universitas Pendidikan Indonesia, Jalan Dr. Setiabudhi 229, Bandung 40154, Indonesia)
- (Physics of Electronic Material Research Division, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Jalan Ganesa 10, Bandung 40132, Indonesia)
- (Earth Physics and Physics of Complex Systems Research Division, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Jalan Ganesa 10, Bandung 40132, Indonesia)

## I. INTRODUCTION

Miniaturization of silicon complementary metal-oxide-semiconductor (CMOS) devices
has produced remarkable improvements to computer performance characteristics such
as switching speed, density and functionality. However, the continuous reduction of
silicon CMOS device dimensions has resulted in higher power consumption due to high
leakage currents that degrade the switching ratio between on and off currents ($I_{ON}$/$I_{OFF}$)
at low supply voltages. In order to extend computer performance, the search for new
devices beyond CMOS has started, as discussed in recent review papers ^{[1-}^{3]}. One such device, which can compete with or complement CMOS transistors, is the tunnel
field-effect transistor (TFET), which uses a band-to-band tunnel for reducing $I_{OFF}$
in order to enhance $I_{ON}$/$I_{OFF}$ and make the subthreshold swing steeper ^{[4-}^{12]}. The search for new materials with properties that can be better controlled by the
electric field in TFETs continues. Si-based, III-V-based, and carbon nanotube materials
are among them ^{[13-}^{18]}. In addition, two-dimensional materials such as transition metal dichalcogenides
(TMDs) as well as graphene provide other alternatives ^{[19-}^{25]}.

Graphene, which is composed of carbon atoms forming a hexagonal lattice, has interesting
properties, such as one-atom thickness, planar geometry, zero band gap, very light
effective mass, high carrier mobility, long phase coherence length, and its manufacturing
process is compatible with conventional silicon technology ^{[26,}^{27]}. Moreover, its band structure can be changed by limiting its size. When the width
of graphene is much smaller than its length, it is called graphene nanoribbon (GNR)
and it can be viewed as a one-dimensional electronic system ^{[28,}^{29]}. There are two types of GNRs based on their edge shapes: armchair graphene nanoribbons
(AGNRs) and zigzag graphene nanoribbons (ZGNRs). The structure of the energy band
in graphene is determined using the tight binding approach ^{[30-}^{32]}. Analysis of electron behavior around Dirac points or often called a low energy limit
produces Dirac-like Hamiltonian ^{[33,}^{34]} and its application to AGNR produces AGNR bandgap energy ^{[35]}. The AGNRs can be either semiconducting or metallic, depending on their widths, while
the ZGNRs have metallic properties for all widths ^{[35,}^{36]}. By utilizing these properties, GNRs can be used as base material for logic devices
such as field-effect transistors (FETs) and TFETs. Since the first proof of a graphene
FET ^{[26]} further experimental studies have been conducted to obtain better devices ^{[37-}^{42]}.

Apart from the experimental studies, simulation and modelling of these devices has
attracted researchers, either to explain experimental results or to develop future
devices. Simpler atomistic semi-classical transport modelling and simulation have
been performed by employing semi-classical Monte Carlo ^{[43-}^{45]}. More rigorous atomic quantum transport modelling and simulation using the nonequilibrium
Green function (NEGF)-based Schrödinger equation coupled with the Poisson equation
have been implemented ^{[46-}^{50]}. An improved model has been offered by utilizing the NEGF-based Dirac Hamiltonian
coupled with the Poisson equation ^{[51-}^{53]}. On the other hand, analytic quantum-mechanical modelling based on the Schrödinger
equation with the WKB approach has also been applied ^{[54-}^{56]}. However, an analytic model using the Dirac-like equation, which would be simpler
than the NEGF-based Dirac Hamiltonian but more realistic than the Schrödinger equation,
has not yet been applied to the devices. Electronic devices that have very recently
been modelled analytically using the Dirac-like equation are an AGNR-based p-n junction
^{[57]}, double gated FET ^{[58]} and a bipolar transistor ^{[59]}.

In this paper, we report tunneling current characteristics in an AGNR-TFET that was
modeled analytically under the Dirac-like equation. The transmittance was calculated
by applying the transfer matrix method (TMM) and the tunneling current was calculated
by using the Landauer formula. The TMM was utilized to obtain the tunneling current
^{[60]} and it was shown that the tunneling current calculated under the TMM is better than
that under the WKB ^{[61]}. Effects of drain voltage, gate voltage, width, and length of the AGNRs as well as
oxide thickness on the tunneling current and subthreshold swing in AGNR-TFET are presented.

## II. THEORETICAL MODEL

The device structure of the n-channel AGNR-TFET in this paper is shown in Fig. 1(a). The armchair graphene nanoribbon (AGNR) type was utilized and doped it to be p+,
n, and n+, which were utilized as source, channel and drain individually. The p and
n-types of AGNR can be obtained by doping. Boron atoms (B) and Nitrogen (N) are appropriate
dopants for graphene since B and N have similar structures to Carbon (C). The chemical
vapor deposition (CVD) and electrothermal reaction are some experiments that have
been reported to be successful by researchers to create p and n-types of graphene
^{[62-}^{65]}. The potential profile of the AGNR-TFET was determined by solving the Dirac-like
Hamilton equation, self-consistently with the Poisson equation. The finite difference
method was used to discretize the Poisson equation and the Dirac-like Hamiltonian
along the carrier transport direction, defined as the z direction ^{[66]},

##### (1)

$H=v_{F}\left[\begin{array}{cc}{0} & {-i \hbar \partial_{z}-\hbar \partial_{x}} \\ {-i \hbar \partial_{z}+\hbar \partial_{x}} & {0}\end{array}\right]$
where $\hbar$ is the reduced Planck constant and $v_F$ is the Fermi velocity. Poisson’s
equation
solution by applying the surface potential method ^{[54]} used as guess potential. The iteration process
between Dirac-like equation along with Poisson’s equation goes on up to the time that
it reaches
self-consistency. As Fig. 1(b) shows the conduction band of AGNR-TFET studied using the method of
self-consistent at ON-state with 0.1 $V$ of drain voltage, and 0.1 $V$ of gate voltage,
the AGNR
width index N is 43 (around 5 nm), the gate length is 20 nm, and the thickness of
the oxide
is 1 nm ^{[66]}. The AGNR width index N shows the number of atoms creating AGNR wide structure.
These values associate with the AGNR width w according to
$\mathrm{w}=\left(\mathrm{N}- 1\right)\frac{\sqrt{3}}{2}a_{cc}$ wherein $a_{cc}$ is
the C-C bond
length. AGNR is semiconducting for $N=3p +1$ and metallic for $N=3p$, where p is integer
^{[35]}.

Fig. 1. (a) The n-channel AGNR-TFET structure, (b) The Energy band diagram of the
AGNR-TFET at ON-state ^{[66]}.

The followings are the demonstration of the wave function solutions in each region which were acquired by using the Dirac-like equation along the carrier transport direction:

##### (2)

\begin{align} \begin{array}{lll} \psi _{1}=\frac{1}{\sqrt{2}}\left(\begin{array}{l} 1\\ \frac{\left| E- V_{1}\right| }{E- V_{1}}e^{i{\theta _{1}}} \end{array}\right)e^{i{k_{1}}z+i{k_{n}}x}+\frac{B_{1}}{\sqrt{2}}\left(\begin{array}{l} 1\\ - \frac{\left| E- V_{1}\right| }{E- V_{1}}e^{- i{\theta _{1}}} \end{array}\right)e^{- i{k_{1}}z+i{k_{n}}x},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, & & z\leq 0\\ \psi _{S,C,D}=\frac{A_{S,C,D}}{\sqrt{2}}\left(\begin{array}{l} 1\\ \frac{\left| E- V_{S,C,D}\right| }{E- V_{S,C,D}}e^{i{\theta _{SCD}}} \end{array}\right)e^{i{k_{SCD}}z+i{k_{n}}x}+\frac{B_{S,C,D}}{\sqrt{2}}\left(\begin{array}{l} 1\\ - \frac{\left| E- V_{S,C,D}\right| }{E- V_{S,C,D}}e^{- i{\theta _{SCD}}} \end{array}\right)e^{- i{k_{SCD}}z+i{k_{n}}x} & & 0<z\leq 2L\\ \psi _{2}=\frac{A_{2}}{\sqrt{2}}\left(\begin{array}{l} 1\\ \frac{\left| E- V_{2}\right| }{E- V_{2}}e^{i{\theta _{2}}} \end{array}\right)e^{- i{k_{2}}z+i{k_{n}}x}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, & & z>2L, \end{array} \end{align}
The aforementioned formula can be explained where
$B_1$, $A_{S}$, $B_{S}$, $A_{C}$, $B_{C}$, $A_{D}$, $B_{D}$ and $A_{2}$ are constants,
$V$ is the
potential energy, $L$ acts as the channel length (the source length = the drain length
= $L/2$),
$k_{1,S,C,D,2}=$ $\sqrt{\left(E- V_{1,S,C,D,2}\right)^{2}/\left(\mathrm{\hslash }^{2}v_{F}^{2}\right)-
k_{n}^{2}}$, $k_n$ take action as the transverse momentum,
and, $\theta _{1,S,C,D,2}=\tan ^{- 1}\left(k_{n}/k_{1,S,C,D,2}\right)$. Indices $S$,
$C$, and $D$ point out the region in source, channel, and drain respectively, while
indices 1 and 2 shows the region out of AGNR-TFET, in this case is metal. The transmittance
was studied by utilizing the TMM following the method in Refs. ^{[57-}^{61]}. The acquired transmittance was then exerted to compute the tunneling currents by
using the Landauer formula,

##### (3)

\begin{equation} Id=\frac{2g_{v}e}{h}\int _{E_{1}}^{E_{2}}\left[f_{S}\left(E\right)- f_{D}\left(E\right)\right]T\left(E\right)dE \end{equation}where $f_{S}(E)$ and $f_{D}(E)$ are the Fermi-Dirac energy distribution functions for electrons in the source and the drain, respectively, $g_r$ is the degeneration of GNR ($g_r= 1$), $h$ is the Planck constant, $E_{1}$ and $E_{2}$ are limit for energy integration and $T(E)$ is the electron transmittance.

## III. CALCULATED RESULTS AND DISCUSSION

Fig. 2 indicates the study result of implementation of Dirac-like equation on the tunneling
currents (Id) as a function of drain voltage (Vd) for various gate voltages (Vg).
For comparison, we also show the tunneling current calculated by using the Schrödinger
equation through WKB approximation ^{[54]}. AGNR channel length L, AGNR width index N, oxide thickness tox, and the temperature
were 20 nm, 43 (about 5 nm), 1 nm, and 300 K, respectively. The tunneling current
studied by both equations pointed out the same features and the metal-oxide-semiconductor
field-effect transistor (MOSFET) type behaviour was revealed. It was shown that the
tunneling currents initially increase with the drain voltage and then reach saturation.
When gate voltage was added, the conduction band in the channel decreased, so the
valence band in the source was higher than the conduction band in the channel. Electrons
can tunnel from the valence band at the source to the conduction band in the channel.
The tunneling current increased significantly when the gate voltage was increased.
Supposing that the drain voltage is boosted constantly along with the gate voltage
fixed, the amount of electrons in that tunnel remain the same as the valence band
at the source and the conduction band in the channel do not experience alteration
with the escalation of the drain voltage. In this example, the current was somewhat
steady and saturation took place. Moreover, the tunneling current that was examined
by implementing the Dirac-like equation was lower than that acquired by using the
Schrödinger equation.

Fig. 2. The tunneling current as a function of drain voltage for various gate voltages calculated using the Schrödinger equation through WKB approximation and the Dirac-like equation through TMM.

Fig. 3(a) and (b) show the tunneling current as a function of gate voltage calculated using the Schrödinger equation through WKB approximation and the Dirac-like equation through TMM respectively. It can be seen from Fig. 3(a) that the dependence of tunneling current on gate voltage for a fixed drain voltage was linear. We had difficulty determining the threshold voltage of the device. However, the dependence of the tunneling current on the gate voltage for a fixed drain voltage calculated by using the Dirac-like equation through TMM was exponential for low gate voltages. The threshold voltage of the device could easily be determined. The threshold voltage of the device was about 0.01 V.

Fig. 3. The tunneling current as a function of gate voltage calculated using (a) the Schrödinger equation through WKB approximation, (b) the Dirac-like equation through TMM.

Current density as a function of gate voltage is presented in Fig. 4. It also shows the subthreshold swing characteristic of the device. The current density
calculated by using the Schrödinger equation through WKB approximation was higher
than that obtained by the Dirac-like equation through TMM. The subthreshold swing
calculated using the Schrödinger equation through WKB approximation was 0.19 $mV/dec$
^{[54]}, while the subthreshold swing calculated by using the Dirac-like equation was approximately
5 $mV/dec$. We can get the subthreshold swing lower than 5 $mV/dec$ by taking a smaller
threshold voltage as the result of the calculations using the Schrödinger equation.

Fig. 4. The tunneling currents density as a function of gate voltage calculated using the Schrodinger equation through WKB approximation and the Dirac-like equation through TMM.

Fig. 5(a) demonstrates the dependence of the tunneling current on the drain voltage for different
AGNR widths index. AGNR channel length L, oxide thickness tox, the temperature and
the gate voltage are 20 nm, 1 nm, 300 K, and 0.1 $V$ respectively. The AGNR width
index variations were 25, 34, 43 and 52 related to AGNR width about 3 nm, 4 nm, 5
nm and 6 nm. It was shown that the tunneling current increased with increasing AGNR
width. Fig. 5(b) shows the tunneling current as a function of gate voltage for various AGNR widths
on a logarithmic scale. The effect of AGNR width on subthreshold swing can be seen
in Fig. 5(b). The insert explains the differences in subthreshold swing for variations in width.
The subthreshold swing increased with increasing AGNR width. This can be understood
from the fact that the AGNR band-gap is inversely proportional to the AGNR width ^{[35,}^{36]}. As the AGNR width increases, the AGNR band-gap decreases, so more electrons enter
the conduction band and the tunneling currents increases. This condition also affects
the off current ($I_{OFF}$) and the on current ($I_{ON}$) of the device. The $I_{OFF}$
increase is larger than the $I_{ON}$ increase (on current). Consequently, the $I_{ON}$/$I_{OFF}$
ratio decreases and increases the subthreshold swing.

Fig. 5. The tunneling currents as a function of (a) drain voltage, (b) gate voltage for various AGNR widths index.

The dependence of the tunneling current on the drain voltage for different AGNR lengths is given in Fig. 6(a). AGNR width index N, oxide thickness $t_{ox}$, the temperature and the gate voltage are 43, 1 nm, 300 K, and 0.1 $V$ respectively. It can be seen that the tunneling current increased with decreasing AGNR length. Fig. 6(b) represents tunneling current as a function of the gate voltage for various AGNR lengths on a logarithmic scale. The insert in Fig. 6(b) displays the different subthreshold swings for various AGNR lengths. The off current $I_{\mathrm{OFF}}$ decreased when the AGNR length increased so the $I_{\mathrm{ON}}$/$I_{\mathrm{OFF}}$ ratio increased and the subthreshold swing decreased.

Fig. 7(a) explores the tunneling current as a function of drain voltage for different oxide
thicknesses. AGNR channel length L, AGNR width w, the temperature and the gate voltage
are 20 nm, 43, 300 K, and 0.1 $V$ respectively. Fig. 7(b) presents tunneling current as a function of gate voltage for various oxide thicknesses
on a logarithmic scale. It can be observed that the tunneling current of the AGNR-TFET
increased with decreasing oxide thickness. The oxide thickness affects the gate voltage
control on the channel of the device. This also demonstrates that the oxide thickness
is inversely proportional to the conductivity of the AGNR-TFET. The dependence of
the tunneling current on the oxide thickness has a similar characteristic to MOSFET,
nanowire FET and CNT-FET ^{[67]}. The effect of oxide thickness on the subthreshold swing of the device is depicted
in the insert of Fig. 7(b). It can be seen that the subthreshold swing of the device increased with increasing
oxide thickness.

Fig. 6. The tunneling currents as a function of (a) drain voltage, (b) gate voltage for various channel length

Fig. 7. The tunneling currents as a function of (a) drain voltage, (b) gate voltage for various oxide thicknesses

The effects of temperature to the tunneling current AGNR-TEFT depicted on Fig. 8(a) and (b). Fig. 8(a) shows tunneling current in linear scale as a drain voltage function to different
temperature and Fig. 8(b) shows tunneling current in logarithmic scale as a gate function with different temperature.
At this point, AGNR width index, channel length, the thickness of the oxide continuously
is 43, 20 nm and 1 nm. Gate voltage and source voltage is still at 0.1 V. Tunneling
current will increase as the temperature decrease. Mobility factors may hold a dominant
factor to explain the dependence tunneling current to AGNR-TEFT. Charge carrier mobility
in graphene is quiet high, it is around 230.000 cm$^{2 }$V$^{{-}1}$ s$^{{-}1}$ on
5K ^{[68]} and 185.000 cm$^{2 }$V$^{{-}1}$ s$^{{-}1}$ on 100K ^{[69]}. When temperature increases electron mobility decreases due to grating vibration
keeps climbing, therefore the probabilities of electron scattered by grating will
increase. The dependence of tunneling current to temperature for electronic device
based AGNR have been reported by other researchers as well ^{[56,}^{70]}.

## IV. CONCLUSION

In the present paper, the characteristics of an armchair graphene nanoribbon tunnel field-effect transistor (AGNR-TFET) were modeled. The Dirac-like equation was used to explore the characteristics of the device. The electron transmittance and tunneling current were derived by employing the transfer matrix method (TMM) and the Landaur formula. It was found that the tunneling current calculated by using the Dirac-like equation was lower than that obtained using the Schrödinger equation. The threshold voltage and the subthreshold swing of the device calculated by using the Dirac-like equation were about 0.01 $V$ and 5 $mV/dec$ respectively. It was also found that the AGNR-TFET geometry, i.e. width and length of GNR and oxide thickness, affects the tunneling current and the subthreshold of the device. The tunneling current increased as the width of AGNR and the oxide thickness increased, while it decreased with increasing AGNR length and temperature. The subthreshold swing increased with increasing AGNR width while an increase of the AGNR length caused the subthreshold swing to decrease. The subthreshold swing of the device increased with increasing oxide thickness.

### ACKNOWLEDGMENTS

This work was financially supported by “Hibah Penelitian Berbasis Kompetensi” Research Grants, Direktorat Riset dan Pengabdian Masyarakat, Direktorat Jenderal Penguatan Riset dan Pengembangan, Kementrian Riset, Teknologi, dan Pendidikan Tinggi Republik Indonesia in the fiscal year 2018.

### REFERENCES

## Author

Endi Suhendi is an associate professor at Departemen Pendidikan Fisika, Universitas Pendidikan Indonesia, Bandung, Indonesia.

He received his Doctoral degree from Departmen Fisika, Institut Teknologi Bandung, Indonesia, in 2015.

His current research interests are graphene-based nanoelectronic devices and sensors.

Lilik Hasanah received her Doctoral degree from Departmen Fisika, Institut Teknologi Bandung, Indonesia, in 2008.

She is an associate professor at Departemen Pendidikan Fisika, Universitas Pendidikan Indonesia, Bandung, Indonesia.

Her current research interests are semiconductor materials and devices.

Fatimah Arofiati Noor received her B.S., M.S and doctoral degree in Physics from Institut Teknologi Bandung Indonesia, in 2004, 2006 and 2010, respectively.

She is an associate professor at Physics Department, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Indonesia.

Her current research interests are semiconductor materials and devices.

Neny Kurniasih received her B.S. and M.S degree in Physics from Institut Teknologi Bandung Indonesia, in 1985 and 1989.

She received Ph.D degree in global environment engineering from Kyoto University, Japan, in 2001.

She is an associate professor at Departemen Fisika, Institut Teknologi Bandung, Indonesia.

Her current research interests is quantitative non-destructive evaluation of flaws within structures using time-domain boundary integral equation method.

Khairurrijal received his B.S. and M.S degree in Physics from Institut Teknologi Bandung Indonesia, in 1989 and 1993.

He received Doctor of engineering from Graduate Program of Electrical Engineering, Graduate School of Engineering, Hiroshima University, Japan, in 2000.

He is a professor at Departemen Fisika, Institut Teknologi Bandung, Indonesia.

He has authored and co-authored over 30 international journals and conference papers.

His current research interests are electronic materials, devices and instrumentation.