Download O A RIGINAL RTICLES

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Journal of Applied Sciences Research, 9(3): 1854-1874, 2013
ISSN 1819-544X
This is a refereed journal and all articles are professionally screened and reviewed
ORIGINAL ARTICLES
Graphene Transistor
1
Aziz N. Mina and 2Adel H. Phillips
1
2
Faculty of Science, Beni-Suef University, Beni-Suef, Egypt
Faculty of Engineering, Ain-Shams University, Cairo, Egypt
ABSTRACT
This review on graphene, a one-atom thick, two-dimensional sheet of carbon atoms, starts with a general
description of the graphene electronic structure. The winning quality of graphene is a unique nature of charge
carriers in graphene. Its charge carriers mimic relativistic particles and are easier and more natural to describe
starting with the Dirac equation rather than the Schrodinger equation. This determines a number of unique
phenomena observed in graphene, ranging from the absence of localization to new type of quantum Hall Effect
(QHE). These exceptional properties make graphene a promising candidate for future electronic devices such as
ballistic and single electron transistors, and spin-valve and ultra-sensitive gas sensors. The graphene transistor
could play a role in post CMOS devices that are expected to be able to compute at much faster speeds than
today’s devices. Consequently, we explore the characteristics of graphene quantum dots, graphene field effect
transistor for rf-applications, field emission from graphene nanoribbons transistor. Also, applications of
graphene in the field of optoelectronics are explored in this review. Graphene may be promising for
thermoelectric applications.
Key words: Graphene field effect transistor (GFET), quantum Hall Effect (QHE), graphene nanoribbons (GNR),
graphene quantum dots, Spintronics, Thermoelectric effect, radio-frequency-graphene field effect
transistor (RF-GFET).
Introduction
Nanotechnology can be understood as a technology of design, fabrication, and applications of
nanostructures and nanomaterials. Nanotechnology also includes fundamental understanding of physical
properties and phenomena of nanomaterials and nanostructures. Study on the fundamental relationships between
physical properties and phenomena and material dimensions in the nanometer scale, is also referred to as
nanoscience. The current fever of nanotechnology is driven at least partly by the ever shrinking of devices in the
semiconductor industry and in the nanometer level. The continued decrease in device dimensions has followed
the well-known Moore’s law predicted in 1965 and illustrated in figure 1(Riordan, M., L. Hoddeson, 1998). The
figure shows that the dimension of a device halves approximately every eighteen months and today’s transistors
have fallen well in the nanometer range.
Fig. 1: “Moore’s Law” plot of transistor size versus year. The trend line illustrates the fact that the transistor
size has decreased by a factor of two every 18 months since 1950.
Nanotechnology involves the characterization, fabrication and/ or manipulation of structures, devices or
materials that have at least one dimension (or contain components with at least one dimension) that is
approximately 1–100 nm in length. When particle size is reduced below this threshold, the resulting material
exhibits physical and chemical properties that are significantly different from the properties of macroscale
materials composed of the same substance. In nanometer scale structures in a crystal, the motion of an electron
can be confined in one or more directions in space. When only one dimension is restricted while the other two
remain free, we talk about a quantum well; when two dimensions are restricted, we talk about a quantum wire;
Corresponding Author: Aziz N. Mina, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt
E-mail: [email protected] & [email protected]
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and when the motion in all three dimensions is confined, and we talk about a quantum dot. In solid state
engineering, these are commonly called low dimensional quantum structures. Carbon demonstrates unusually
complicated behavior, forming a number of very different structures. However, only three-dimensional
(diamond, graphite), one-dimensional (carbon nanotubes), two-dimensional (graphene) and zero dimensional
(fullerenes) allotropes of carbon were known. Materials in nanometer scale may exhibit unique physical
properties and have been explored for various applications. Bandgap-engineered quantum devices, such as lasers
and heterojunction bipolar transistors, have been developed with unusual electronic transport and optical effects
(Capasso, F., 1987). The discovery of synthetic materials, such as carbon fullerenes (Krastchmer, W., et al.,
1990) carbon nanotubes (Iijima, S., 1991), graphen (Novoselov, K.S., et al., 2004) and ordered mesoporous
materials (Kresge, C.T., et al., 1992) has further fueled the research in nanotechnology and nanomaterials.
The invention and development of scanning tunneling microscopy (STM) in the early 1980’s (Binnig, G., et
al., 1982) and subsequently other scanning probe microscopy (SPM) such as atomic force microscopy (AFM)
(Binnig, G., et al., 1986) have opened up new possibilities for the characterization, measurement, and
manipulation of nanostructures and nanomaterials. Combining with other well-developed characterization and
measurement techniques such as transmission electron microscopy (TEM), it is possible to study and manipulate
the nanostructures and nanomaterials in great detail and often down to the atomic level.
Graphene is a one-atom-thick sheet of carbon whose strength, flexibility, and electrical conductivity have
opened new horizons for fundamental physics, together with technological innovations in electronic, optical,
and energy sectors (Wu, J., et al., 2007). The production of high quality graphene remains one of the greatest
challenges, in particular when it comes to maintaining the material properties and performance upon up-scaling,
which includes mass production for material/energy oriented applications and wafer-scale integration. Potential
electronic applications of graphene include high-frequency devices and RF communications, touch screens,
flexible and wearable electronics, as well as ultrasensitive sensors, NEMS, super-dense data storage, or photonic
devices. In the energy field, potential applications include batteries and supercapacitors to store and transit
electrical power, and highly efficient solar cells. Some of graphene’s most appealing potential lies in its ability
to transmit light as well as electricity, offering improved performances of light emitting diodes and aid in the
production of next-generation devices like flexible touch screens, photodetectors, and ultrafast lasers. New
horizons have also been opened from the demonstration of high- speed graphene circuits offering highbandwidth suitable for the next generation of low-cost smart phone and television displays. Graphene is
promising as addictive for composite materials, thin films and conducting inks.
In this chapter we shall pay an attention on graphene: its electronic properties and potential of its
application in field of nanodevices.
Hybridization in a carbon atom:
Carbon-based materials, clusters and molecules are unique in many ways (Harris, F., 1999). One distinction
related to the many possible configurations of the electronic states of carbon atom, which is known as the
hybridization of atomic orbitals. Carbon is sixth element of the periodic table and is listed at the top of the
column IV. Each carbon atom has six electrons which occupy 1s2,2s2 ,2p2 atomic orbitals. The 1s2 orbital
contains two strongly bound electrons and they are called core electrons. Four electrons occupy the 2s2 2p2
orbitals and these more weakly bound electrons are called valence electrons. In the crystalline phase the valence
electron give rise to 2s, 2px , 2py and 2pz orbitals which are important in forming covalent bonds in carbon
materials. Since the energy difference between the upper 2p energy levels and the lower 2s level in carbon is
small compared with the binding energy of the chemical bonds; the electronic wave function for these four
electrons can readily mix with each other, thereby changing the occupation of the 2s and three 2p atomic orbitals
so as to enhance the binding energy of the carbon atom with its neighboring atom. This mixing of 2s and 2p
atomic orbitals is called hybridization, whereas the mixing of a single 2s-electron with n=1,2 ,3-2p electrons is
spn hybridization. In carbon, three possible hybridization occur: sp, sp2 and sp3.
Electronic Characteristics of Graphene:
Graphene is composed of sp2-bonded carbon atoms arranged in a two-dimensional honeycomb lattice as
shown in figure 2. The lattice can be seen as consisting of two interpenetrated triangular sub-lattices, for which
the atoms of one sub-lattice are at the center of the triangles defined by the other with a carbon-to-carbon interatomic length, aC–C, of 1.42Å. The unit cell comprises two carbon atoms and is invariant under a rotation of 1200
around any atom. Each atom has one s orbital
and two in-plane p orbitals contributing to the mechanical stability of the carbon sheet. The remaining p
orbital, perpendicularly oriented to the molecular plane, hybridizes to form πthe π* (conduction) and π (valence)
bands, which dominate the planar conduction phenomena (Kelly, B.T., 1981; Mina, A.N., et al., 2012). The
energies of those bands depend on the momentum of the charge carriers within the Brillouin zone. In the low-
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energy regime, i.e. in the vicinity of the K and K’ points, those two bands meet each other producing conicalshaped valleys, as shown in figure 3.
Fig. 2: A hexagonal lattice with the lattice vectors a1 and a2, the grey area corresponds to one choice of unit cell,
rA and rB points to the two atoms in the unit cell. (Mina, A.N., et al., 2012; Geim, A.K and A.H.
MacDonald, 2007).
The lattice constant, a, is given as:
a = a1 = a 2 = 1.42 A 0 x 3 = 2.46 A 0
(1)
The curve below (figure 4) is a slice of figure 3 between the symmetry points. The upper half of the energy
dispersion relation curves is called π* or the antibonding band and π is the bonding band. It is seen that the two
bands are degenerated at the K and Ќ points where they attain the same value. This is also the energy of the
Fermi energy EF . We have two atoms per unit cell and therefore two π electrons per unit cell so the electrons
fully occupy the lower π-band leaving the π* antibonding band empty. A density of states calculation shows that
the density of states at the Fermi surface is zero making graphene a zero-gap semiconductor or semi-metal
(Mina, A.N., et al., 2012).
Fig. 3: π–π* band structure of graphene [Adapted from (Mina, A.N., et al., 2012)]
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Fig. 4: [Adapted from (Mina, A.N., et al., 2012)]
In this low-energy limit, the energy–momentum dispersion relation is linear (relativistic Dirac’s equation)
and carriers are seen as zero-rest mass relativistic particles with an effective speed ,vF ≅ 106 m/s. The peculiar
local electronic structure, thus allows the observations of numerous phenomena which are typical of a twodimensional Dirac fermions gas (Geim, A.K and A.H. MacDonald, 2007). One of the most interesting aspects of
the graphene is its highly unusual nature of charge carriers, which behave as massless relativistic particles
(Dirac fermions). Dirac fermions behavior is very abnormal compared to electrons when subjected to magnetic
fields for example, the anomalous integer quantum Hall effect (QHE) (Zhang, Y.B., et al., 2005; Novoselov,
K.S., et al., 2005).
Quantum Hall effect:
The charge particles behave as massless Dirac fermion in a 2D lattice which exhibit interesting effect on
energy spectrum of the landau levels produced in the presence of magnetic field perpendicular to the graphene
sheets.
The Landau level energies are given by
En =
(n + 0.5  e B)
me
(1)
where n=0,1,2,3,…. and represents the indices for the Landau index, ħ is the reduced Planck’s constant, and
e & me are the electronic charge and its mass respectively. Due to disorder, the Hall conductivity, σxy, of the
 2
2DEG shows the plateaus at
 n 


 2eB 
and is quantized, can be expressed
2e 

 h 
σ xy = ± n
which leads to
integer quantum Hall Effect (IQHE).
In contrast to 2DEG, the energy of Landau levels for graphene is given by the following equation:
E n = ± v F 2eB n
Where
n = 0,1,2,3,...
(2)
is the Landau index and B is the magnetic field applied perpendicular to the
graphene plane. Since, the Landau levels are doubly degenerated for the K & K’ points at n=0, these levels are
shared equally between electrons and holes. This leads to the anomalous IQHE, where the Hall conductivity is
given by:
2
σ xy = ±
4 (n + 0.5) e
h
(3)
The Hall conductivity for the single layer graphene shows a plateau when plotted as a function of carrier
concentration at a fixed magnetic field. Novoselov et al., have first observed anomalous IQHE (shown in Figure
5), measured at B = 14 T and temperature of 4 K (Novoselov, K.S., et al., 2005; 2005).
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Fig. 5: QHE for massless Dirac fermions in single layer graphene (Novoselov, K.S., et al., 2005; 2005)..
Graphene nanoribbons:
Graphene nanoribbons (see Figure 6), in contrast to bulk graphene, exhibit an effective energy gap,
overcoming the gapless band structure of graphene.
High electron mobility and long coherence length make graphene a subject of great interest for nanoscale
electronic applications, even for the realization of room-temperature ballistic (dissipation-free) electronics.
However, a major setback in the development of graphene-based field-effect transistors is the inability to
electrostatically confine electrons in graphene, because a single layer of graphite remains metallic even at the
charge neutrality point. In order to overcome this problem, a way to open a gap in the electronic structure of
graphene has to be found. A straightforward solution is to pattern the graphene sheet into narrow ribbons.
Thanks to recent progresses in preparing single graphite layers on conventional device setups, graphene
nanoribbons (GNRs) with varying widths have been synthesized and characterized experimentally (Han, M.Y.,
et al., 2007). Graphene nanoribbons (GNRs) are elongated strips of graphene with a finite width that can be
obtained by cutting a graphene sheet along a certain direction. Due to the honeycomb structure of graphene, two
prototypical shapes are possible for the edges, namely the armchair edge (Figure 6b) and the zigzag edge
(Figure 6c), with a difference of 300 between the two edge orientations. Depending on the cutting direction, the
edges of the GNRs may thus exhibit either one of these two “ideal” shapes or more complex geometries
composed of a mixture of armchair and zigzag shaped fragments. Graphene nanoribbons, including graphene
nano-constrictions show an overall semiconducting ambipolar behavior, which makes these quasi- 1-D graphene
nanostructures promising candidates for the fabrication of nanoscale graphene transistors (Han, M.Y., et al.,
2007; Liao, L., et al., 2010), tunnel barriers, single electron transistors (Stampfer, C., et al., 2008; Stampfer, C.,
et al., 2009), quantum dots (Ponomarenko, L.A., et al., 2008; Schnez, S., et al., 2009), and double quantum dot
devices (Molitor, F., et al., 2009; Liu, X.L., et al., 2010).
(a)
(b)
(c)
Fig. 6: Honeycomb lattice of graphene (a) with both the zigzag (red) and the armchair (green) directions.
Graphene nanoribbons cut in the ideal sheet along these two highly symmetric directions are called
aGNRs (b) and zGNRs (c), respectively. (Han, M.Y., et al., 2007).
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Authors (Sols, F., et al., 2007) found the geometrical dependence of the energy gap of the graphene
nanoribbons, that is, energy gape, Eg, depends on the constriction width, w, as illustrated by the following
formula:
E g ( w) =
α
w
. exp(− β w)
(4)
where α= 2 eV.nm and β= 0.026 nm-1.
Eq.(4) was found in good agreement with results of Han et al. (2007), that is, the energy gap, Eg decreases
with increasing the constriction width, w,.
Figure 7 shows the mechanism behind the formation of charged islands and quantum dots in graphene
nanoribbons (Molitor, F., et al., 2010). Graphene nanoribbon etched out of a two dimensional graphene flake (as
illustrated on the very top). A disorder potential (arising from both intrinsic and extrinsic effects) leads to the
formation of electron and hole puddles in the near vicinity of the charge neutrality point (thus, at low-carrier
densities). Since graphene has no band gap and carriers mimic relativistic particles Klein tunneling and related
phenomena make the boundary between electron and hole puddles reasonably transparent. The finite width w
introduces a confinement gap (see the black line in the central panel), which isolates individual charge islands or
quantum dots. Due to the confinement gap Klein tunneling gets substituted by real tunneling as highlighted by
the lower panel.
Fig. 7:
5a.Graphene quantum dots:
Quantum dots (Kouwenhoven, L.P., et al., 2001) are small man-made structures in a solid, typically with
sizes ranging from nanometers to a few microns, consisting of 103–109 atoms where the number of free
electrons can be tuned over a wide range. The zero dimension confinement of the electrons in all three spatial
directions results in a quantized energy spectrum and quantum dots are therefore regarded as artificial atoms.
It has been shown recently that cutting out nanostructures from 2-D graphene allows to structurally confine
electrons (Schnez, S., et al., 2002; Guttinger, J., et al., 2008; Stampfer, C., et al., 2008; Todd, K., et al., 2009).
Actually, the functionality of graphene nanodevices can be engineered by designing the structure of the
graphene flake (see Figure 8a). This will be shown with four different examples, discussing (i) a graphene single
electron transistor based on a width-modulated graphene nanoribbon, (ii) graphene single-electron transistor
device with a nearby graphene nanoribbon acting as charge detector, (iii) by discussing a graphene nanodevice
small enough so that quantum confinement effects start to play an important role, and (iv) by a graphene double
quantum dot device.
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Fig. 8: (a) Schematic illustration of an etched graphene quantum dot, highlighting the formation of tunneling
barriers and the charged island (similar to Fig. 9). (b) Illustration of energy levels in Coulomb blockade
regime (off resonance). (c) By aligning an energy level of the island with the transport window the
device is in resonance and transport becomes possible.
(i) Graphene single electron transistor
Single electron transistors (SETs) (Atallah, A.S., et al., 2006; Phillips, A.H., et al., 2007; Phillips, A.H., et
al., 2008), in general, consist of a semiconducting island connected by tunneling barriers to two conducting
leads [see Figure 8(a)] and (at least one) capacitively coupled gate to tune the potential on the island. Electronic
transport through such devices (from source to drain) can be completely blocked by Coulomb interaction for
temperatures and bias voltages lower than the characteristic energy required on adding an electron to the island
(Atallah, A.S., et al., 2006; Phillips, A.H., et al., 2007; 2008; Heinzel, T., 2003; Ihn, T., 2004). This is shown in
Figure 8(b). Since no energy level of the island lies inside the transport window (see dashed horizontal lines),
the device is out of resonance and transport (i.e., current) is suppressed by the so-called Coulomb blockade.
Here, temperature and bias (eVb) are significantly smaller than the charging energy Ec. Additionally, it is
assumed that the tunneling resistance of the two tunneling barriers is significantly larger than e2/h, such that the
island energy level broadening can be neglected (Ihn, T., 2004). By changing the gate potential, we can tune the
single electron transistor into resonance, such that an island energy level is aligned within the transport window
and carriers can resonantly hop from source to drain, as shown in Figure 8(c). Now, concerning graphene
nanoribbons, figure 8(a), shows a tailored (i.e., etched) graphene flake providing the functionality of a graphene
single electron transistor (Guttinger, J., et al., 2008; Stampfer, C., et al., 2008). The width modulated graphene
nanostructure shown in figure 9(a) consists of a central island, two spatially separated constrictions, which act as
effective tunnel barriers and three lateral graphene gates (the plunger gate PG and the barrier gates BG1 and
BG2) are for full electrostatic tunability. It is crucial that narrow graphene constrictions (nanoribbons) have
been found to exhibit an effective transport gap (Han, M.Y., et al., 2007; Sols, F., et al., 2007; Li, X., et al.,
2008) enabling effective tunable tunneling barrier. Although the transport gap is not a clean energy gap and is
likely to be formed by a series of local quantum dots inside the nanoconstriction (Molitor, F., et al., 2009; Liu,
X.L., et al., 2009), the overall energy gap behavior can be modeled by Eq. 4 , as shown above. Thus, the exact
shape of the effective energy gap, Eg(x) (x is the transport direction) is given mainly by the lateral confinement,
that is, by the variation of the width w(x) along the device [see Figure 8(a)]. By assuming that electron–hole
symmetry holds in the confined geometry, we can extract the effective conduction band edge at +Eg(x)/2, and an
effective valence band edge at −Eg(x)/2. Indeed, in this respect we can think of graphene band gap engineering
by tailoring the graphene sheet width (Han, M.Y., et al., 2007). An alternative approach for realizing graphene
single electron transistors is based on etched graphene nanoribbons and top gates for defining the actual charged
islands, as shown and illustrated in figures 9(b) and (c). The flake rests on approximately 300 nm SiO2 and the
highly doped Si substrate is used as a back gate to adjust the overall Fermi level. In figure 9(d) a low bias
source-drain back gate characteristic of this device (see Figure 9(a)) is shown. The transport gap of strongly
suppressed current (–25 V < Vbg < –12 V) separates the region of hole (left side; see also inset) and electron
(right side) transport.
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Fig. 9: scanning force microscope images of an investigated graphene nanodevice and experimental results (Liu,
X.L., et al., 2009).
This transport characteristic is dominated by the two narrow constrictions. Fixing the back gate voltage
(i.e., the Fermi level) to a value inside the transport gap, single electron transport phenomena can be
investigated. Results show current resonances which can be understood as Coulomb blockade resonances of
isolated charge islands inside the corresponding graphene constrictions forming the effective tunneling barrier
(Schnez, S., et al., 2009; Todd, K., et al., 2009). These resonances are therefore attributed to states localized on
the island between the barriers. The authors (Myoung, N., G. Ihm and S.J. Lee, 2010) studied the tunneling of
Dirac fermions through magnetic barriers in armchair graphene nanoribbons (AGNRs). The conductance as a
function of incident energy shows a step-like increase by the height of 4e2/ h; which reflects the number of
transverse modes in AGNRs, superimposed upon characteristic oscillations associated with magnetic barriers.
The results suggest us that the present investigated model with magnetic barrier structures can be used as
building block for single electron tunneling devices or spin-polarized tunneling devices (will be mentioned
below).
(ii) Nanoribbon based charge detection on SETs:
In this section we illustrate an integrated graphene device consisting of a graphene single electron transistor
with a nearby graphene nanoribbon acting as a quantum point-contact like charge detector. Charge detection
techniques (van der Wiel, W.G., et al., 2003) have been shown to extend the experimental possibilities with
single electron devices significantly. They are, powerful for detecting individual charging events and spin–qubit
states (Petta, J.R., et al., 2005; Elzerman, J.M., et al., 2004) and molecular states in coupled quantum dots
(DiCarlo, L., et al., 2004). Furthermore, charge detectors have been successfully used to investigate shot noise
on the single electron level, and full counting statistics (Gustavsson, S., et al., 2006). This makes charge
detection also interesting for advanced investigation of graphene quantum dots and graphene nanodevices in
general. Figure 10(a) shows a SFM image of an integrated graphene device consisting of a graphene single
electron transistor with a graphene nanoribbon nearby. The single electron transistor device consists of two 25–
30 nm wide and 75-nm long graphene constrictions separating source (S) and drain (D) contacts from the
graphene quantum dot (QD). The diameter of the Graphene Island is approximately 100 nm. The constrictions
and the island are electrostatically tuned independently again by two barrier gates (BG1 and BG2) and a central
gate (CG), respectively. The highly doped silicon back gate (BG) allows again adjusting the overall Fermi level.
In addition, it is placed a 30 nm wide graphene nanoribbon approximately 35 nm next to the island, which acts
as a charge detector.
Fig. 10: (a) SFM image of the graphene quantum dot device with charge detector. The dashed lines highlight
the contour of the graphene regions, and results (Stampfer, C., et al., 2009).
In figure 10(b) the main capacitive coupling involved in the measurements. Figure 10 (c) back gate
characteristics of the single electron transistor upper panel and the charge detector lower panel (see figure 10
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(d). The measurements were performed at a source–drain bias voltage of Vb,SET = 2 mV and Vb,cd = 0.5 mV at T ≈
4 K.
(iii) Electron–hole crossover in graphene quantum dots:
For the detection of quantum confinement effects and the identification of individual orbital quantum states
in graphene quantum dots, smaller system sizes and lower temperatures will be considered. Here we discuss a
graphene quantum dot device (see inset of figure 11(a)) consisting of a roughly 50-nm-wide and 80-nm-long
quantum dot which is connected to source (S) and drain (D) via two 25 nm wide and 10-nm-long constrictions,
acting as tunneling barriers. The dot and the leads can be tuned by an in-plane graphene plunger gate (PG) and
the highly doped silicon substrate is used as a back gate (BG) to adjust the overall Fermi level.
Fig. 11: [Adapted from Ref.20].
Figure 11 (a) shows the Coulomb blockade resonances of the graphene quantum as function of plunger gate
voltage VPG at Vbg = −0.9 V (Guttinger, J., et al., 2009) The conductance has been measured at constant bias of
Vb = 100 μV . The studied graphene quantum dot is shown in the inset. The scale bar corresponds to 100 nm.
While figure 11(b) Coulomb diamonds, i.e. finite-bias measurement of the QDs differential conductance GQD =
dI/dVb as function of Vb and VPG at B = 0. Note the rich excited state spectra and in particular the different
vertical distances between the diamond edge and the diagonal structures inside the Coulomb-blockaded region
for positive Vb (Guttinger, J., et al., 2009).
(iv) Graphene double quantum dots:
Beside of graphene single electron transistors and single quantum dot devices more recently also graphene
double quantum dots (Molitor, F., et al., 2009; Molitor, F., et al., 2010; Liu, X.L., et al., 2010) have been
demonstrated successfully. Such coupled double quantum dots have been proven to be particularly interesting
candidates for the successful implementation of solid-state spin qubits (Trauzettel, B., et al., 2007). In figure 12
we show two different realizations of a serial graphene double quantum dot (Molitor, F., et al., 2009; Trauzettel,
B., et al., 2007). Figure 12(a) shows a SFM image of a device where the double quantum dot structure has been
defined by the dry etching based “paper cutting” technique. The left (L) and the right (R) quantum dot, both
having a diameter of ≈ 90 nm, are connected by a 30 nm wide constriction and two additional 20 nm wide
constrictions connect the two dots to the source (S) and drain (D) lead. In total five lateral in-planes graphene
gates and a global back gate can be used to tune the device. In figure 12(b) we show a different graphene double
quantum dot device based on a single graphene nanoribbon (see dark dashed lines) and three top gates (G1, G2
and G3) for tuning the device into different transport regime.
Fig. 12: [Adapted from Ref.48].
6. Field Emission from Graphene Nanoribbons (GNR) Transistor:
The field emission is a unique quantum mechanical effect of electron emission for electrons tunneling from
a condensed matter into the vacuum. This phenomenon occurs in high electric fields (107–108 V/cm) because the
high electric fields narrow the potential barrier and ensure a significant probability of tunneling from the
condensed matter into the vacuum. And a tip with the apex radius of curvature ranging from tens of angstroms
to several microns can ensure a high electric field. Field emission can be applied into vacuum microelectronics,
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fine-resolution electron microscopy, and the modern high-current electronics (Fursey, G.N., 2003). The most
outstanding and widespread application is the production of flat low-voltage displays of high brightness and
high resolution based on field emission arrays (Fursey, G.N., 2003). We know that the average electron energy
in a graphene nanoribbon increases with decreasing width of nanometer, especially when the width is a few
nanometers because electrons in graphene nanoribbons obey the unique properties of band structure (Mao, L.F.,
2009). This means that electrons in graphene nanoribbons with a linear energy-momentum relation may have a
higher average electron energy compared to those with a parabolic energy-momentum relation. Recently, it is
reported finite size effects on the thermionic emission in metal- graphene nanoribbon contacts (Mao, L.F., et al.,
2010). The field-emission current density of a graphene nanoribbon has been investigated by the author (Mao,
L.F., 2011). His results show that field-emission current density at a given external field decreases with
decreasing width of graphene nanoribbon and an increase in the turn-on electric field.
7. Field-effect transistor:
A FET is an electronic device with at least three terminals where the conductivity between two terminals,
the source and drain, is modulated by the electric field created by the third gate terminal(s). This gate control
enables the use of a FET as switch which conducts high current when it is on and very low current when it is
off. Physically this control is achieved by gate modulation of the free carrier density in the channel between the
source and drain contacts beneath the gate, from a high value in the ON state to a low value in OFF state. The
ratio of current in the ON state and that in OFF sate, referred to as a ON/OFF ratio, is a very important
performance metric of digital switches, and the larger the better. The large on current, Ion, allows quick charging
of a capacitive load, which typically includes the gates of one or more subsequent transistors as well at the
metallic, interconnects. Considering only gate capacitance, the switching time becomes approximately CgV/Ion,
where Cg is the gate capacitance and V is the supply voltage. Large on currents can be achieved by improving
carrier velocities via carrier mobility and thermal velocity improvements. There are a range of methods that are
employed for this purpose including channel material choice, straining the channel material, modulation doping,
etc. as an example of typical CMOS logic circuit (see for more details Ref. 54). The semiconductor industry has
been scaling the FETs to improve the performance metrics for logic device such as speed of operation, low
voltage and low power operation. Scaling of the gate length to smaller values results in degradation of gate
control on electrostatic profile in the channel. Although this can be countered by increasing the doping density
of channel, this approach leads to mobility reduction due to increased impurity scattering. Another approach is
to use (ultra)thin body channel FETs like structures. In fact according to the International Technology Roadmap
for Semiconductors (ITRS) such structures will continue the scaling trend for the next 10 years or so
(International Technology Roadmap For Semiconductors (2009 edn)). However, it is becoming increasingly
difficult to scale the devices further due to fundamental physical limitations. Due to very high mobility and
Fermi velocities of electrons in graphene, it offers advantage as a channel material for FETs for very high
frequency circuit applications. Also to counter the loss of electrostatic control and short channel effects as a
result of scaling, thin body or SOI devices are being used, and due to 2D nature of graphene, it is the ultimate
realization of a thin body channel material. However, due to a lack of a band gap, the minimum channel
conductivity in FETs with wide graphene sheet channels is large resulting in poor ON/OFF ratios which are
particularly problematic for CMOS logic applications. Also, poor contacts can still be an issue and, even
relatively more so because of the high channel conductivities possible with graphene. In principle, a band gap
can be opened in graphene by lateral confinement of electrons. In other words, one can use a narrow ribbon of
graphene, which can be semiconductor or metallic, as channel material (Han, M.Y., et al., 2007;Li, X., et al.,
2008). Such ribbons must be only a few nanometers in width to open band gaps of practical interest, which leads
to substantial lithography challenges. And, as we narrow the ribbon and increase the band gap, conductivity is
intrinsically degraded due to an associated decrease in the Fermi velocity of Dirac Fermions charge carriers, and
extrinsically degraded due to factors such as disorder at the edges of these nanoribbons. Another approach is to
use a bilayer of graphene which has an interesting property that its band gap can be tuned by creating a potential
difference between the layers (McCann, E., et al., 2007; Zhang, Y., et al., 2009). However, this requires large
displacement field between the layers up to and order of 2 to 3V per nm, and only moderate gaps on the scale of
200 meV are possible (Zhang, Y., et al., 2009). Yet another approach is to uniaxially strain the graphene along
the transport direction, but this requires strains up to 20% to actually create band gaps (versus simply moving
the band edges in k-space) (Pereira, V.M., et al., 2009).
7.1 Graphene transistor fabrication:
Fabrication of a graphene field-effect transistor (GFET) involves the following three major steps:
• Preparation of a substrate with graphene on it (graphene–substrate),
• Deposition of gate dielectric in case of top-gated FETs (graphene–dielectric),
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• Forming the contacts to graphene (metal–graphene).
The above three steps also represent the three critical interfaces in a GFET. Good quality of these interfaces
is necessary for realizing a high performance FET. The following lists the three major approaches for obtaining
high quality mono- and few layer graphene sheets.
(1) Mechanical exfoliation of highly oriented pyrolytic graphene (HOPG) using an adhesive tape
(Novoselov, K.S., et al., 2004; Geim, A.K. and K.S. Novoselov, 2007).
(2) Epitaxial growth on SiC substrate by desorption of Si (Berger, C., et al., 2004; Berger, C., et al., 2006).
(3) Chemical vapor deposition (CVD) by catalytic decomposition of a gaseous precursor on transition metal
substrates such as nickel (Reina, A., et al., 2008) and copper (Li Xuesong, et al., 2009).
7.2 GFETS for RF applications:
Graphene has extraordinary electronic transport properties like high mobility and high saturation velocity
also make it attractive for radio frequency (RF) electronics applications. The first experimental demonstration of
rf GFET was a 500 nm channel length top-gated FET fabricated using exfoliated graphene (Meric, I., et al.,
2008). The device had current gain up to 14.7 GHz. Recently RF graphene FETs with wide channels showing
cutoff frequencies in GHz range were demonstrated by IBM with an estimated fT of 100 GHz for a 240 nm
channel length device (Lin, Y-M., et al., 2010). The device also showed power gain up to 10 GHz. Graphene
was epitaxially grown on Si terminated SiC wafers in ultra-high vacuum. The graphitized substrate was then
used for the fabrication of 30μm wide top-gated FETs with a gate dielectric stack of 10 nm polymer followed by
a 10 nm hafnium dioxide. Other applications using the ambipolar characteristics of GFET is a single transistorbased frequency-multiplier demonstrated by (Wang et al from MIT in 2009). Back gated FETs fabricated on
exfoliated graphene on SiO2 substrate. When the device is biased to operate near the flat-band, a small signal
applied to the gate results in output signal at the drain which has double the frequency of input signal. The
frequency doubling functionality is usually achieved using complicated frequency doubling circuitry. We can
understand the frequency doubling as follows: For low drain-to-source bias, an ideal graphene FET is expected
to have a linear transfer characteristic which results in a full wave rectification of the input signal with 74%
spectral purity. However, most experimental graphene FETs have a non-linear characteristic near flat-band. In
fact the observed near 90% spectral purity seems to suggest a significant quadratic current–voltage
characteristic, which can explain the frequency doubling with high spectral purity. Using the same principle of
operation, a single transistor based mixer was also demonstrated by (Wang et al in 2010). Scaling down the
channel length plays a critically important role in boosting the RF performance of a field-effect transistor (FET).
Fabrication of graphene transistors with a projected cut-off frequency of 300 GHz for a 140 nm channel (Liao,
L., et al., 2010) and 100 GHz for a 240 nm channel (Lin, Y-M., et al., 2010). Recently, the authors (Jyotsna
Chauhan and Jing Guo, 2011) studied channel length scaling and assessed RF performance limits of graphene
FETs in the sub-100 nm channel length regime by using self-consistent ballistic quantum transport simulations.
The simulated intrinsic cut-off frequency is about 640 GHz at a channel length of 100 nm and increases to about
3.7 THz at channel length equals 20 nm for various gate insulator thickness values.
7.3 Functional Circuits:
Building more complicated circuits beyond just a single transistor is the next step in utilizing graphene in
practical RF applications. AC voltage amplifiers and audio amplification are both simple uses for RF transistors
(Han, SJ., et al., 2011; Guerriero, E., et al., 2012). Frequency multipliers are a bit more complex, but also an
important component in many communications applications. They are used to output harmonic frequencies of an
input frequency, which is useful in generating signals using nonlinear circuit design. Frequency multipliers
based on Schottky diodes and conventional FETs have been popular, with operation as high as 1 THz for the
former. Both have limitations in signal amplification and efficiency respectively. Additionally, these devices
suffer from degraded output signals that need to be further processed and filtered to obtain the desired frequency
(Wang, H., et al., 2009).
The ambipolar nature of graphene makes for unique FETs that can enable novel applications. The “V”
shape of the Id – Vg curve behaves like a full-wave rectifier by itself. With an alternating gate voltage centered
at the charge neutrality point, the output current at the drain of the device will result in full-wave rectification,
alternating between electron and hole conduction. In this way, much simpler frequency doublers can be realized.
This ability is not new, but graphene field effect transistors offer a far less complicated design and potentially
superior performance for rectifier devices (Palacios, T., et al., 2010). Efforts have been made to integrate these
graphene-based devices into higher functioning circuits. (Lin, Y.M., et al., 2011; Habibpour, O., et al., 2012;
Yang, X., et al., 2010). Frequency mixers are useful circuits designed for signal processing, turning two
frequencies into the sum or difference of the inputs. An RF signal, f RF and a local oscillator signal, f LO are
applied to the gate and the drain, modulating the GFET, resulting in an output of the mixed, summed, and
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differenced (intermediate) frequencies. An inductor placed between the input and gate contacts removes the
parasitic capacitances through resonance, while another works as a low-pass filter between the drain and the
output. With the achievement of high extrinsic fT in the self-aligned graphene transistors, GFET-based RF
circuits operating in the gigahertz regime can be readily configured (e.g., a frequency doubler as shown in figure
13a-c). Importantly, RF characterization of the circuit shows a clear doubling signal at 2.1 GHz with the input
signal frequency at 1.05 GHz. Spectrum analysis shows that the frequency-doubler device exhibits a high
spectral purity in the output RF signal, with 90% of the output RF energy at the doubling frequency of 2.1 GHz.
This study clearly demonstrates that a single graphene transistor based frequency doubler can operate in the
gigahertz regime with high-output spectral purity. Similarly, a single graphene transistor-based RF mixer has
also been demonstrated. The two-tone signals with adjacent frequencies are applied to the gate through a power
combiner to manipulate the transistor channel resistance. With an RF input f RF = 6.72 GHz and local oscillator f
LO = 2.98 GHz, the output spectrum shows clear RF mixing function at intermediate frequency (IF) f IF = f RF − f
LO = 3.74 GHz and f IF = f RF + f LO = 9.70 GHz. The above RF circuits were obtained by interconnecting
different elements together. For eventual application, it is highly desired to incorporate all functional
components directly on the chip to achieve a fully integrated RF circuits. To this end, IBM scientists have
fabricated such devices using EG on SiC with inductors integrated on chip (Figure 13 d–g) (Liao, L., et al.,
2012). The frequency spectrum of the mixer output with inputs of 3.8 GHz and 4 GHz shows the 200 MHz and
7.8 GHz outputs. The drain inductor weakens the higher frequency tone, and lowers its amplitude. Power signals
of f IF are proportional to the signal with expected high linearity and conversion loss of about − 27 dBm.
Fig. 13: Circuit designs for a GFET-based frequency doubler (a) and mixer (d). b) Oscilloscope readout of input
(top) and output (bottom). c) Frequency spectrum of the frequency doubler. e) Schematic of a
frequency mixer device and components. f) Optical micrograph of the fabricated device on a SiC
substrate. g) Output frequency spectrum of the mixer. a–c)[ Adapted from (Liao, L., et al., 2012),
(d,f,g) (Lin, Y.M., et al., 2011) and e (Lin, Y.M., et al., 2011).
8. Spintronics:
A novel approach to logic-based devices utilizes particle spin as an extra degree of freedom for signal
processing and propagation, called spintronics. First emerging with the discovery of the giant magnetoresistance
effect, the concept of spintronics is not unique to graphene (Fert, A., Origin, 2008). The long spin-relaxation
time and low spin-orbit coupling in graphene makes it an attractive material for use in spin valves (Hill, E.W., et
al., 2006; Cho, S., et al., 2007) (filters that only let one spin variant through the barrier) and other components
necessary for practical spintronic circuits. Edges can have a significant impact on magnetotransport properties
(Yazyev, O.V. and M.I. Katsnelson, 2008; Wimmer, M., et al., 2008) and nanoribbons have been proposed for
spintronic applications, with unique spin states in zigzag (Son, Y –W., et al., 2006) (Figure 14 a) or armchair
(Saffarzadeh, A., and R. Farghadan, 2011) edges with tunable edge passivation (Hod, O., et al., 2007).
Transverse electric fields can polarize spin in ZGNRs without the need for magnetic contacts for spin filters or
spin FETs (Saffarzadeh, A., and R. Farghadan, 2011; Zhang, Y.T., et al., 2010). Simulations of spin caloritronic
devices show a thermally induced spin current of opposite sign in opposite directions: a spin Seebeck effect in
magnetized ZGNRs (Zeng, M., et al., 2011). Nanoconstriction GFETs are capable of sensing single molecular
magnets on the surface of its graphene channel (Candini, A., et al., 2011). For bulk graphene, figure 14 b–d
shows a four-point device and nonlocal magnetoresistance measurements to separate the voltage and current
probes with ferromagnetic electrodes of varying dimensions; the magnetic polarity of each can be switched
sequentially (Tombros, N., et al., 2007). Antiparallel states will have a higher resistance as precession of spin
polarized electrons needs to be inverted. Spin injection (flow of spin polarized current into graphene, usually
from a magnetized electrode) through both transparent (in direct contact with the electrode) (Han, W., et al.,
2009) and tunneling (through a thin dielectric barrier to improve polarization) (Han, W., et al., 2010) is
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effective, with an asymmetry observed between electron and hole conduction (Han, W., et al., 2009). The robust
spin polarization in graphene is ideal for spin current channels (Shiraishi, M., et al., 2009). Analogous to normal
spintronics, which uses the standard quantum mechanical spin of electrons, pseudospintronics has been
theorized as a way to instill logic operations through unconventional means. Similarly, proposed valleytronics
utilizes the charge carriers distinct position in the K or K ′ valleys (Rycerz, A., et al., 2007). The valley isospin is
a more robust form of information, as it takes a larger (on the order of inverse lattice spacing) momentum to
scatter compared to pseudospin (Mohiuddin, T.M.G., et al., 2009). Creating a pseudospin or isospin valve can
be as simple as distorting graphene in a controllable way. A narrow constriction neck (Rycerz, A., et al., 2007)
or asymmetric bilayer interfaces (Schomerus, H., 2010)] that break symmetry are potential valley filter designs.
Other proposed structures for pseudospin filters and valves utilize line defects (Gunlycke, D and C.T. White,
2011) in monolayers or by alternating perpendicular electric fields in bilayers (San-Jose, P., et al., 2009).
Switching the applied field reverses the pseudospin direction creating a tunable filter (Figure 14 e). A digital
logic bilayer pseudospin FET has been theorized to use ultralow power (Reddy, D., et al., 2010). Strain
engineering and subsequent pseudomagnetic fields have a clear link with this pseudospin polarization, and could
be used as a mechanism for valley filtration (Low, T. and F. Guinea, 2010; Fujita, T., et al., 2010; Zhai, F., et
al., 2010; Chaves, A., et al., 2010; Soodchomshom, B., and P. Chantngarm, 2011; Soodchomshom, B., 2011;
Feng, Z., et al., 2011). These proposed devices have yet to be realized, but potentially add a whole new
dimension to logic devices.
Fig. 14: Spintronics. a) Half-metallicity of ZGNRs. DOS schematics show a polarized spin at the edges (left).
With an in-plane electric fi eld applied, the DOS shifts to remove the bandgap (right). b,c) SEM image
and schematic of four-terminal spin injection measurements. d) Schematic of the diffusion and
precession of spin-polarized currents in the four-terminal device when electrode magnetization is
aligned parallel or antiparallel. e) Schematic of a proposed valley valve using perpendicular electric
fields applied to bilayer graphene to polarize pseudospin.(Adapted from (a) (Son, Y –W., et al., 2004),
( b–d) (Tombros, N., et al., 2007) and (e) (San-Jose, P., et al., 2009).
The authors (Asham, M.D., et al., 2012) investigated Spin dependent transport characteristics through
normal graphene/ ferromagnetic graphene/ normal graphene junction. The spin polarization of the Dirac
fermions tunneled through such junction is pumped by the influence of an AC-field of wide range of
frequencies. The graphene-based such device is modeled as follows: Spin transport in normal graphene/
ferromagnetic graphene/ normal graphene mesoscopic junction is shown in figure 15. Graphene can be
converted into ferromagnetic state by depositing the magnetic insulator EuO on top of it (Asham, M.D., et al.,
2012; Haugen, H., et al., 2008) (see region IV) and a metallic gate. While regions I, II III and V are normal
graphene. On the top of region II, together with the metallic gate, there is a ferromagnetic vector potential
barrier. With such construction of spintronic device, we can inject and manipulate spins of Dirac fermions.
Fig. 15: The proposed device model [Adapted from (Asham, M.D., et al., 2012).
The calculations of spin polarization and giant magneto-resistance show that these parameters could be
modified by the barrier height and the angle of incidence of electrons on the corresponding region of the
investigated device (Asham, M.D., et al., 2012). Quantum pumping by induction of external photons could
enhance spin transport mechanism through such investigated device. Also results show that the cut-off
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frequency for both parallel and anti-parallel spin alignments varies strongly in a wide range of frequencies
(Asham, M.D., et al., 2012).
9. Photonics:
(a) Photodetectors:
Photodetectors convert light signals or optical power into electrical current and are a key component for
optoelectronic devices: absorbed light creates excitons that separate to form a photocurrent. Conventional
photodetectors are mostly based on conventional semiconductor materials; their performance is limited by two
factors arising from intrinsic material properties. The first limitation is cutoff wavelength, which depends on the
bandgap. Photons with energy below the gap will not be absorbed, resulting in a maximal detectable wavelength
(1100 nm for silicon and 2600 nm for InGaAs. Long wavelength transparency limits the far-infrared application
for semiconductor photodetectors. The second limitation is operation speed, which is mainly ruled by charge
carrier mobility and lifetime. In contrast with traditional semiconductors, graphene has no bandgap, leading to
broader wavelength absorption (up to 6 μ m) for broadband detection. Additionally, the exceptionally high
carrier mobility makes graphene ideally suited for ultrafast photodetectors. Absorbing only 2.3% of light per
layer may limit the photocurrent responsivity, but means photodetectors could be built into transparent
electronics.
Recent pump-probe measurements demonstrate that graphene photodetectors can exhibit intrinsically high
response speeds up to the terahertz regime (Prechtel, L., et al., 2012). With the same metal on both electrodes in
the metal-graphene-metal device, the potential profile is symmetric within the channel with opposite
photocurrent at each end, leading to a zero total current if the entire device is illuminated. To break the mirror
band structure, asymmetric metal electrode design to allow observation of an overall photocurrent with titanium
and palladium contacts. This multifinger design can greatly enhance the active area that exists in one focused
laser spot and thus increasing the responsivity. These designs can reach a photocurrent as high as 6.1 mA/W and
were successfully implemented in a 10 Gbit/s data link (Mueller, T., et al., 2010).
(b) Mode-Locked Lasers:
Mode-locking is a technique widely used in lasers to produce a periodic train of ultrashort pulses, usually
on the order of picoseconds to femtoseconds. Saturable absorbers are the most commonly used passive material
for mode-locking techniques, which behave as a nonlinear, intensity-dependent filter. An ideal saturable
absorber should absorb light at low intensity and become transparent when the light is intense enough. The key
features are speed, optical bandwidth and saturation intensity. Conventional semiconductors are widely used in
mode-locked lasers; however, they are limited by small optical bandwidths of only tens of nanometers.
Saturable absorption can be achieved in graphene due to Pauli blocking. At high incident laser intensity, the
concentration of electron-hole pairs increases too much and fills all the states near the conduction or valance
bands, obstructing further absorption. The ultrahigh Fermi velocity and large optical bandwidth make graphene
an ideal material to be integrated into mode-locked laser devices. Bao et al (2009) and Sun et al. (2010)
demonstrated graphene-based lasers, easily integrated into fiber systems.
(c) Optical Modulators:
Optical modulators are used to modulate a beam of light. They are commonly used in on-chip optical
interconnects and have become increasingly important as electrical interconnects face limitations such as high
loss, crosstalk, and limited speed. Key features such as high speed, large optical bandwidth, and small footprint
are desirable for optical modulators and their on-chip integration. However, the field-effect based optical
modulation (e.g., Franz–Keldysh, Kerr and Pockels effects) in silicon-based devices is usually rather weak,
resulting in a millimeter scale footprint. This increases the insertion loss and hinders high-speed performance.
An alternative solution to reduce this footprint is via integration with resonant structures. Unfortunately, devices
with this kind of structure tend to work over very narrow bandwidths (less than 1 nm), thus greatly reducing the
fabrication and environmental tolerance. Compared with contemporary semiconducting materials graphene
shows many advantages in addition to its high carrier mobility and large optical bandwidth, such as its
compatibility with current CMOS processing, low cost, and strong modulation ability, which are all essential
features for high-performance optical modulators. Importantly, a graphene-based optical modulator was recently
demonstrated with an operation frequency up to 1 GHz (Liu, M., et al., 2011). The footprint is only 25 μ m and
the optical bandwidth range demonstrated is from 1.35 μ m to 1.6 μ m, far from exceeding the ability of current
modulators. In this device, silicon waveguides are used to couple in and couple out light, with thin dielectric
layers (7 nm) of Al2 O3 CVD graphene is physically transferred on top of the silicon waveguide as the
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absorption layer. While tuning the gate voltage, the absorption of graphene will change due to the rather strong
electro-absorption effect, resulting in the modulation of light transmission.
(d) Polarizers:
A polarizer converts unpolarized light into well-polarized light, and has broad applications in areas ranging
from optical circuits to car mirrors and sunglasses. In-line fiber-based polarizers that use an evanescent field
substrate processing technology, allow fabrication directly onto the fiber itself. Compared with traditional
absorption or beam-splitting polarizers, this technique shows great advantages in device integration with most
optical systems. However, the current in-line fiber polarizers are made with materials such as metals or crystals,
which limit their optical bandwidth, as well as cause difficulty in changing between transverse electric (TE) and
transverse magnetic (TM) pass modes. Graphene can function as an efficient polarizing material by selective
attenuation of one polarization mode. It can select which mode to pass and which mode to damp by simply
changing its Fermi level, in contrast to metal-based polarizers. Additionally, graphene can be used for polarizers
in an ultra-wide range of wavelengths, from ultraviolet to far-IR. The broadband graphene polarizer
demonstrated by Bao et al. shows the apparent advantages over traditional devices (Bao, Q., et al., 2011). This
simple structure easily integrates graphene within the fiber and works well in a wavelength range from 800 nm
to 1650 nm. A strong s-polarization extinction ratio up to 27 dB is comparable with metal structures. The TE
and TM modes can easily be tuned by changing the gate voltage ( | μ | < ħ ω /2 to support TE mode and | μ | > ħ
ω /2 to support TM mode), resulting in a low-cost, broadband, mode-tunable polarizer.
10. Thermoelectric transport of Dirac particles in graphene:
The Seebeck effect, discovered in 1821 by Thomas Seebeck (Seebeck, T., 1926), is the conversion of a
temperature gradient into an electrical voltage and can be utilized to create thermocouples for temperature
measurements. Thermopower (Seebeck coefficient) has been used as a powerful tool to probe transport
mechanisms in metals and semiconductors. Often the measurement of conductivity is inadequate in
distinguishing among different scattering mechanisms and the thermopower can then be used as a sensitive
probe of transport properties since it provides complementary information to conductivity. Recently, the
thermoelectric properties of graphene sheets and ribbons have attracted experimental as well as theoretical
attention. Theoretical studies on graphene thermopower use linear response theory incorporating the energy
dependence of various transport scattering times (Hwang, E.H., et al., 2009) or atomistic simulations based on
the non-equilibrium Green’s function formalism (Ouyang, Y and J. Guo, 2009). Experimentally (Zuev, Y.M., et
al., 2009; Wei, P., et al., 2009; Checkelsky, J and N. Ong, 2009), the expected change of sign in the
thermopower is found across the charge neutral point as the majority carriers change from electrons to holes.
Away from the charge neutral region the thermopower behaves as
1
n
where
is the carrier density, and
exhibits linear temperature dependence in agreement with the semiclassical Mott’s formula (Cutler, M and N.F.
Mott, 1969).
The Mott’s formula derived mathematically through the Sommerfeld expansion, is only valid at very low
temperatures
T
 1,
TF
where
TF =
EF
kB
is the Fermi temperature, EF is the Fermi energy and kB is
Boltzmann constant.. As the temperature increases deviations from Mott’s formula have been observed (Zuev,
Y.M., et al., 2009; Wei, P., et al., 2009; Checkelsky, J and N. Ong, 2009). For thermoelectric energy
conversion, however, a low thermal conductance is required to increase the efficiency of a thermoelectric
device. The efficiency is characterized by the figure of merit ZT. The performance of thermoelectric materials is
measured through a unitless parameter, ZT, which is called thermoelectric figure of merit and is defined as
GS 2T , where G is the electrical conductance, S is the Seebeck coefficient (or thermopower), K = Ke+Kph is the
Κ
thermal conductance including both electron and phonon contributions, and T is the temperature. The authors
(Yonghong Yan, et al., 2012) investigated the thermoelectric properties of a range of graphene nanoribbons in
the presence of two constrictions (See Figure 16). Thus, the structure can be also viewed as a hexagonal
graphene quantum dot connected to two leads. Results show that with decreasing widths of the constrictions, the
thermal conductance of a nanoribbon can be reduced largely while S2G (power factor) remains still high as
compared with the results for the pristine nanoribbon. As a result, the thermoelectric figure of merit of the
nanoribbon can be enhanced greatly. In the presence of narrowest constrictions, the ZT values of the zigzag
quantum dots can exceed one (or even be up to two) while the ZT values of the armchair quantum dots can be
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close to one, depending on the size of the dot. These results indicate that quantum confined structure of
graphene may be promising for thermoelectric applications.
Fig. 16: Schematic of graphene nanoribbons with two narrowest constrictions.
(a) shows a nanoribbon with zigzag edges while (b) shows a nanoribbon with armchair edges. The left and
the right regions are semi-infinite pristine nanoribbons acting as leads. The number of atom pairs (Nc ) in the
box indicates the width of the constrictions.
The Nernst effect describes the Hall voltage that arises when a temperature gradient and a perpendicular
magnetic field are present in a conductive medium. Normal to the temperature gradient and magnetic field, an
electrical field will then be induced (Behnia, K., 2009). The authors (Yanxia Xing, et al., 2009) investigated the
Nernst and Seebeck effects, in graphene nanoribbon. Results show that due to the electron-hole symmetry, the
Nernst coefficient NC is an even function while the Seebeck coefficient SC is an odd function of the Fermi energy
EF. NC and SC show peaks when EF crosses the Landau levels at high magnetic fields or crosses the transverse
sub-bands at the zero magnetic fields. In the strong magnetic field, due to the fact that high degenerated Landau
levels dominate transport processes the Nernst and Seebeck coefficients are similar for different chirality
ribbons. The peak height of NC and SC , respectively, are [ln 2/|n|] and [ln 2/n] with the peak number n, except
for n = 0. For zeroth peak, it is abnormal. Its peak height is[2 ln 2] for the Nernst effect and it disappears for the
Seebeck effect. While in zero magnetic field, Nernst effect is absent and the Seebeck effect is strongly
dependent on the chirality of the ribbon. For the zigzag edge ribbon, the peaks of SC are equidistance, but they
are irregularly distributed for armchair edge ribbon. Surprisingly, for the insulating armchair edge ribbon, the
Seebeck coefficient SC can be very large near the Dirac point due to the energy gap. When the magnetic field
increases from zero to high values, the irregularly or regularly distributed peaks of SC in different chiral ribbons
gradually tends to be the same. In addition, the nonzero values of the Nernst coefficient NC appear first near the
Dirac point and then gradually in the whole energy region. It is remarkable that for certain crossed ribbons, the
Nernst coefficient NC at weak magnetic fields can be much larger than that in the strong magnetic field due to
small transmission coefficient in the transverse terminals.
Memory Devices:
Graphene-related materials have also been widely explored in memory devices (Zheng, Y., et al., 2009). It
was demonstrated theoretically by Gunlycke et al. that when the graphene nanostrips are under non-equilibrium
state, e.g., in the presence of a ballistic current, both spin polarized and spin-unpolarized nanostrip states exist,
which could be switched through the applied bias in a binary memory device. After that, extensive experimental
investigations on graphene related memory devices were carried out, based on pristine graphene(Zheng, Y., et
al., 2009), graphene oxide (GO), (Wang, S., et al., 2010; He, C.L., et al., 2005) graphene–Au hybrid systems,
(Myung, S., et al., 2010) graphene–polymer composites, (Zhuang, X.D., et al., 2010) and reduced graphene
oxide (rGO)-electrodes. (Liu, J.Q., et al., 2010; Liu, J.Q., et al., 2010).
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