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Báo cáo toàn văn Kỷ yếu hội nghị khoa học lần IX Trường Đại học Khoa học Tự nhiên, ĐHQG-HCM

Dinh Sy Hien
University of Science, VNU-HCM
Graphene has been one of the most vigorously studied research materials. Graphene nanoribbon
material has been briefly reviewed. Top-gate graphene nanoribbons field effect transistor used for
digital IC applications is modeled. Self-consistent atomistic simulations based on the non-equilibrium
Green’s function method are employed. The current-voltage characteristics of the graphene
nanoribbon field-effect transistor are studied. The effects of the geometrical parameters of channel
material on the current-voltage characteristics of the graphene nanoribbon FET are explored.
Especially, the room temperature on-off current ratio by top-gate voltage of GNR-FET has been
calculated and reached 104.
Key words: Graphene, Graphene nanoribbon FET, non-equilibrium Green’s function, currentvoltage characteristics.
Graphene [1-8] has been one of the most vigorously studied research materials since its inception in 2004.
Graphene has attracted considerable attention from scientific community due to its excellent electronic
properties, such as high electron and hole mobilities even at room temperature and at high doping concentration
[9], high thermal conductivity [10], and its interesting optical properties [11]. 2D graphene is a gapless material,

which makes it unsuitable for digital IC applications. However, an energy bandgap can be induced by tailoring a
graphene sheet into graphene nanoribbons (GNR) called 1D graphene (GNR) [12]. Depending on the orientation
of the ribbon edges, GNR can have edges with zigzag shape, armchair or a combination of these two [13]. In
order to obtain a suitable bandgap for transistor applications, the width of GNR must be scaled to extremely
small values. Bandgap energy of narrow GNR is inversely proportional to the width of the GNR. In narrow
GNR, line-edge roughness plays an important role in the device characteristics [14-20]. The effect of line-edge
roughness on the device performance of GNR field-effect transistor (GNR-FET) has been numerically studied in
[14-15, 21].
In this paper, using top-gate GNR-FET model, device performances are investigated. The electronic
transport in the GNR-FET used narrow GNR as channel of sub-10 nm is studied. The device characteristics are
explored by using the non-equilibrium Green’s function method. Basing on the obtained results, on-off current
ratio of the GNR-FET for digital IC applications has been calculated. This work is organized as follows: section
2 describes channel materials used for GNR-FET, simulation method, and results of simulations. Concluding
remarks are drawn in section 3.
Graphene channel materials
Bandgap engineering. In modern electronics, bandgap formation is the key concept for switching current,
and thus, for processing electric signals.
Although graphene has great advantages for use in electronics applications, including atomically thin
channels, high mobility, and large electric field effects, its semi-metallic electronic band structure makes the
creation of a graphene transistor quite challenging.
So far, several methods have been proposed for introduction of bandgap in graphene. Among them the
most promising are graphene nanoribbons. In this section, we briefly review theoretical predictions,
experimental results, and the major challenges of the formation of bandgap in graphene.
Graphene nanoribbons. In quantum mechanical systems, the confinement of carriers leads to discrete
energy levels. This also the case in graphene; however, some diffences are seen because of its peculair lattice
Thin graphene wires are called graphene nanoribbons. Two common structures, armchair and zigzag
nanoribbons (Figure 1), have been intensely studied theoretically.
Theoretical predictions. In the following theoretical treatment of graphene nanoribbons, the graphene
edges are assumed to be passivated by hydrogen, as illustrated in Figure 1.

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Báo cáo toàn văn Kỷ yếu hội nghị khoa học lần IX Trường Đại học Khoa học Tự nhiên, ĐHQG-HCM
In the tight binding (TB) approximation for π-electrons in graphene, armchair graphene nanoribbons are
metallic when the number of carbon atoms in the ribbon width, N a satisfies the relation, Na = 3p+2 (where p is a

positive integer), and are semiconducting otherwise. The energy gap Δ Na is inversely proportional to the width in
each group, Na = 3p or Na = 3p+1.
Zigzag nanoribbons in the TB approximation are metallic and have flat bands at  = 0.
In the first-principles calculation using the local spin density approximation (LSDA), the result is
significantly different from that discussed above. Specially, all of the armchair and zigzag nanoribbons are
semiconducting with gaps depending on the ribbon width.
The energy gap of zigzag nanoribbons in the LSDA calculation, Δ, is well fitted by
for the ribbons width w > 1 nm.

Figure 1 Two kinds of graphene nanoribbons: a) armchair and b) zigzag. N a and Nz denote the
number of carbons in ribbon width in armchair and zigzag nanoribbons, respectively. White circles
indicate hydrogen atoms passivating the graphene edges.
The magnitude of the gaps is presented in Figure 2.

Figure 2. Energy gaps in graphene nanoribbons.
Experiments. Graphene nanoribbons have been made by various methods, including electron beam
lithography followed by oxygen plasma etching [22-25], and chemical derivation [26-29]. The main challenge in
gap formation in graphene nanoribbons is suppression of structural disorder. Structural disorder causes weak
localization and the Coulomb blockade effect, and suppresses the mobility.
Lithographically defined graphene nanoribbons were first reported by Han et al in 2007 [22]. After
contacting a graphene flake with Cr/Au (3/50 nm) electrodes, they produced a graphene nanoribbon from the
flake by oxygen plasma etching. They estimated the magnitude of the energy gap, and found that the energy gap
g is well fitted by
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Báo cáo toàn văn Kỷ yếu hội nghị khoa học lần IX Trường Đại học Khoa học Tự nhiên, ĐHQG-HCM
where w is the ribbon width, a = 0.2 eVnm, and w* = 16 nm. Han et al. attributed inactive width w* to
contribution from localized edge state near the ribbon edge caused the structural disorder from etching process.
Graphene nanoribbons have also been made by chemical exforliation. Li et al. [26] obtained graphene
nanoribbons with edges that appeared smoother than those obtained lithographically.
Graphene nanoribbons with various widths ranging from 50 nm down to sub-10 nm scale were obtained by
this method. The room temperature on-off current ratio Ion/Ioff induced by the back-gate voltage increased
exponentially with decreasing ribbon width; it reached 10 7 in sub-10 nm ribbons. Here, the on (off) current I on
(Ioff) is defined as the maximum (minimum) value of the source-drain current I for a fixed bias (source-drain)
voltage V within a measured gate voltage range. The energy gap g estimated from relationship

was converted into an empirical form
and falls between the limits of theoretical results (Figure 2).
Wang et al. [28] reported that even in smooth, chemically graphene nanoribbons with widths of sub-10 nm,
the mobility was limited to 200 cm2/Vs and the mean free path was limited. These values are significantly
smaller than those for wider graphene devices. These values were attributed to scattering at the edges caused by
edge roughness.
Top-gate graphene nanoribbons FET
In this sub-section, the effect of the geometrical parameters on the transfer characteristics and performance
of GNR-FET is investigated. A top-gate GNR-FET with gate oxide of Al2O3 with relative dielectric constant, r
= 9.8 is assummed [30]. Graphene monolayer flake is exfoliated from bulk natural graphite crystals by the
micromechanical cleavage. The substrate consists of a highly-doped, n-type Si(100) wafer with an arsenic
doping concentration of ND > 1020 cm-3, on which a 300 nm-thick SiO2 layer is grown by thermal oxidation.
Metal contacts on the sample is defined by using electron beam lithography (EBL) followed by a 50 nm-thick
metal (Ni) layer evaporation and a lift-off process. A graphene FET with source-drain separation and top-gate
length is shown in Figure 3 [30].

Figure 3. Structure of top-gate graphene field-effect transistor [30] is used in our simulations.
For all simulation, the widths of source and drain contacts of 1 nm, the length of channel of 10 nm, room
temperature are assummed. The top-gate GNR-FET having channel of a highly-doped, n-type with NH3 doping
concentration is also assummed for suppressing Schottky effect in the source-semiconducting-drain contacts of
the device.
The flow of current is due to the difference in potentials between the source and the drain, each of which is
in a state of local equilibrium, but maintained at different electro-chemical potentials

1, 2 and hence with two

distinct Fermi functions [31]:

f1 E   f 0 E  1  

ISBN: 978-604-82-1375-6



expE  1  / kBT   1


Báo cáo toàn văn Kỷ yếu hội nghị khoa học lần IX Trường Đại học Khoa học Tự nhiên, ĐHQG-HCM

f 2 E   f 0  E   2  



expE  2  / kBT   1

by the applied bias V: 2  1  qV . Here, E- energy, kB - Boltzmann constant, T- temperature.
The density matrix is given by




dE n
 2 G E    2 A1E  f1E   A2 E  f2 E 


The current ID flows in the external circuit is given by Landauer formula:

I D  q / h  dETE  f1 E   f 2 E 



The quantity T(E) appearing in the current equation (4) is called the transmission function, which tells us
the rate at which electrons transmit from the source to the drain contacts by propagating through the device.
Knowing the device Hamiltonian [H] and its coupling to the contacts described by the self-energy matrices

1, 2

, we can calculate the current from (8). For coherent transport, one can calculate the transmission from the
Green’s function method, using the relation

T E   Trace 1G2G  Trace 2G1G


The appropriate NEGF equations are obtained:

 A  f E   A  f E , A  iG  G   A   A 

G  EI  H  1   2  , 1, 2  i 1, 2  1, 2 , A1 E   G1G  , A2 E   G2G  ,







where H is effective mass Hamiltonian, I is an identity matrix of the same size,

1, 2 are the broadening


functions, A1,2 are partial spectral functions, A(E) are spectral function, G is correlation function. We use a
discrete lattice with N points spaced by lattice spacing ‘a’ to calculate the eigenenergies for electrons in the
Results and discussion
The main goal of the project was to make a user-friendly simulation program that provides as much control
as possible over every aspect of the simulation. Flexibility and ease of use are difficult to achieve
simultaneously, but given the complexity of quantum device simulations became clear that both criteria were
vital to program success. Consequently, graphic user interface development was major part of the program.
We start by simulating ID-VD characteristics of top-gate GNR-FET. Figure 3 shows the schematic of the
device used in our simulations. Top-gate GNR-FET with one-dimensional graphene as the channel is simulated.
The device is simulated with Al2O3 as the dielectric which has been predicted to be one of the promising
dielectrics for GNR-FETs in recent experiment [30]. All the simulations have been done for channel length of
GNR-FET, L = 10 nm.
Figure 4 shows the ID-VD characteristics of the GNR-FET having the length of 10 nm versus different gate
voltages. It can be noted that when the gate voltage is increased the saturated drain current exponentially
increased. This behavior is in agreement with experimental results [31].

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Báo cáo toàn văn Kỷ yếu hội nghị khoa học lần IX Trường Đại học Khoa học Tự nhiên, ĐHQG-HCM

Figure 4. The ID-VD characteristics of the top-gate GNR-FET at different gate votage, VG = 0.1 V,
0.4 V, 0.6 V, 0.8 V (bottom to up).
Figure 5 shows the ID-VD characteristics of the top-gate GNR-FET having the length of 10 nm under
ballistic transport and that with phonon scattering. It is shown that scattering can have an appreciable affect on
the on-current. At VGS = 0.8 V, the on-current is reduced by 9% due to the phonon scattering.

Figure 5. The ID-VD characteristics of the gate top GNR-FET at VG = 0.8 V for ballistic, scattering,
where the length of the gate is LG=10 nm.
Figure 6 shows ID-VD characteristics of GNR-FET versus the gate voltage, VG. When the gate voltage is
small, the drain current is gradually increased. When the gate voltage is greater than VG = 0.3 V, the drain
current is exponentially increased. The modeling results agree well with experimental data [31].

Figure 6. The 3D plot of ID-VD characteristics of the top gate GNR-FET versus VG, where the length
of the gate is LG=10 nm.
Figure 7 shows the 3D plot of ID-VD characteristics of the GNR-FET versus the temperature, T. It can be
noted that as the temperature increases the saturated drain current gradually increases. We also observe that the
off-current is about 1×10-9 nA at very low temperature and the low gate voltage, V g = 0.1 V. From Figure 4 and 7
we can calculate on/of-current ratio, Ion/Ioff = 1×10-5 nA/1×10-9 nA = 104.

ISBN: 978-604-82-1375-6


Báo cáo toàn văn Kỷ yếu hội nghị khoa học lần IX Trường Đại học Khoa học Tự nhiên, ĐHQG-HCM

Figure 7. The 3D plot of the ID-VD characteristics of the top-gate GNR-FET versus temperature. The
GNR-FET parameters are: material, Al2O3, the gate length is LG = 10 nm, the gate thickness is tox = 2
nm, at the gate voltage, VG = 0.1 V.
The effect of the channel length scaling on the device characteristics is investigated. ID-VD characteristics
of GNR-FET versus the length of the gate layer at room temperature are shown in Figure 8. Apparently, as the
length of the GNR-FET decreases, the saturated drain current gradually increases.

Figure 8. The 3D plot of the ID-VD characteristics versus the gate length of the top-gate GNR-FET at
room temperature, T = 300 K. The parameters of the GNR-FET: material, Al2O3, the gate thickness,
tox= 2 nm.
Figure 9 shows ID-VD characteristics of the top-gate GNR-FET versus the gate thickness at room
temperature. Apparently, as the gate thickness, tox of the GNR-FET is increased, the saturated drain current is
gradually decreased.

Figure 9. The 3D plot of ID-VD characteristics of the top-gate GNR-FET versus the gate thickness, tox
at room temperature, T = 300 K. The parameters of the GNR-FET: material, Al2O3, the gate length is
LG = 10 nm.
ISBN: 978-604-82-1375-6


Báo cáo toàn văn Kỷ yếu hội nghị khoa học lần IX Trường Đại học Khoa học Tự nhiên, ĐHQG-HCM
A model for the top-gate GNR-FET using NEGF written in GUI of Matlab has been reported. The top-gate
GNR-FET has been simulated. Typical simulations is then successfully performed for various parameters of the
GNR-FET or the electronic transport of GNR-FET has been investigated. The model is not only able to
accurately describe ID-VG, ID-VD characteristics of the GNR-FET, but also effects of channel materials, gate
materials, size of GNR-FET, temperature on the characteristics. The obtained results indicate that the
performance of GNR-FET in terms of on/off-current ratio is improved in narrow ribbons, while the conductance
is degraded in longer channel. We also observe that the on/off-current ratio of the GNR-FET is 104 as the GNRwidth of 1 nm and the GNR-length of 10 nm.

Đinh Sỹ Hiền
Đại học Khoa học Tự nhiên, ĐHQG-HCM
Graphene là một trong các vật liệu được nghiên cứu sôi động nhất. Trong bài báo này, vật liệu
dải nano graphene được tổng quan một cách ngắn gọn. Transistor hiệu ứng trường cổng trên sử dụng
cho các ứng dụng vi mạch số được mô hình. Những mô phỏng mức nguyên tử tự-tương thích dựa
trên phương pháp hàm Green không cân bằng được sử dụng. Đặc trưng dòng thế của transistor hiệu
ứng trường dải nano graphene được nghiên cứu. Những ảnh hưởng của các thông số hình học của
vật liệu kênh lên đặc trưng dòng thế của transistor hiệu ứng trường dải nano graphene được nghiên
cứu kỹ. Đặc biệt là tỷ số dòng on-off tại nhiệt độ phòng theo thế cổng trên của GNR-FET đã được tính
toán và đạt tới 104.
Từ khóa: Graphene, GNR-FET, hàm Green không cân bằng, đặc trưng dòng-thế.
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