We theoretically study current dynamics of graphene nanoribbons subject to DC-AC driven fields. We show that graphene exhibits negative differential conductivity (NDC) at high-harmonics. NDC occurs in the neighborhood where a constant electric field is equal to amplitude of ac field. We also observe NDC at both even and odd harmonics and at wave mixing of two commensurate frequencies. The even harmonics are more pronounced than the odd harmonics. A possible use of the present method for generating terahertz frequencies at even harmonics in graphene is suggested.
Graphene has attracted much attention since its discovery in 2004 by Geim and his team [1]. The great interest in studying graphene is driven by its unique band structure [2] and many unusual physical properties like electrical, thermal and mechanical properties [3]. Despite these, there have been difficulties in realizing graphene-based electronic devices, probably, the fact that the graphene sheet lacks band gap, edge defects, disorder, etc. However, various attempts have been made to introduce 2D graphene sheet, for the purpose of overcoming some of these challenges, into a 1D + quantization. This phenomenon results into graphene nanoribbon (GNR) which is assumed as an unrolled single-wall carbon nanotube (SWCNT).
Depending on the nature of the nanoribbon edges, one gets two symmetry groups; armchair graphene nanoribbon (aGNR) and zigzag graphene nanoribbon (zGNR). Armchair and zigzag GNRs show metallic or semicon-ducting electronic properties [4] depending on the number of dimer rows, along transverse direction. is related to the width of the nanoribbon as [5]. Electron dynamics of both ribbons have different properties, mostly due to the berry phase and pseudo spin [6]. Edge states have significant contribution to graphene properties, because in a nano-meter size ribbon, massless Dirac fermions can reach the edges within a femto-second before encountering any lattice effects, and avoiding electron-electron interaction, eletron-phonon interaction, etc.
In this paper, we employ Boltzmann transport equation based on relaxation time approximation to study negative differential conductivity (NDC) in GNRs. In conventional semiconductor devices including semiconductor superlattices, an NDC behavior is known to offer great potential for high frequency applications as Bloch oscillators, frequency multipliers, and fast switching devices. For this reason, the NDC effect has been greatly explored and discussed in several graphene nanostructures, particularly in [7]. NDC can also be observed in other graphene allotropes; carbon nanotubes (CNT) [8]. The unique energy spectrum of holes and electrons in GNRs, especially its narrow gap nature leads to nontrivial features such as NDC in the THz regime [7].
The rest of this paper is organized as follows; In Section 2, we derived the electronic current density of aGNR and zGNR and imposed certain conditions to reduce the equations to forms appropriate for our model. The results obtained in Section 3 are discussed in great details in Section 4. The paper finally concludes in Section 5 where some recommendations for future applications are made.
2. The Theory
Following the approach of refs, [8,9] for SWCNT and semiconductor superlattices, we consider an undoped GNR (both zigzag and armchair) exposed simultaneously to electric field
which is seen as a superposition of n harmonic waves polarized along one direction with angular frequency. The phase difference between the th and th component being is arbitrary, is an integer. are the field amplitudes. We require that . This will correspond to the DC component of the applied field. The dynamics of free -electrons in graphene can be describe by the time-dependent Boltzmann transport equation (BTE) based on relaxation time approximation in zero magnetic field as
We also assume a spatial uniform graphene nanoribbon. Again, the inverse of the relaxation time is momentum independent. In Equation (1), and are the equilibrium and non-equilibrium Fermi electron distribution functions, respectively. is the electronic charge, is the electron wave vector and is the reduced plank constant. In the following section, we will calculate electronic current density for GNRs.
2.1. Armchair and Zigzag Nanoribbon Band Structures
The energy band structure of aGNR and zGNR is characterized by three parameters; band index, phase and wave vector [5,6]. For aGNR,
and for zGNR,
where with (+) for conduction band and (−) for valence band. and, is the lattice spacing which has numerical value of 0.246 nm, is the overlap integral and is the phase perpendicular to the quasi-momentum. The 1 BZ of aGNR is bounded by and the zGNR is. is parallel to the edge and has translational symmetry along this direction. For aGNR, the transverse wave vector (phase) is quantized according to the rule [6] with and . Unlike aGNR, the nature of transverse wave vector quantization is complicated in zGNR, depending on both and as. However, for simplicity we assume is constant, say, so that . Except this little subtlety for zGNRs, all that will be discussed in the following for aGNR are equally applied to the zGNR.
Now, we employed the translational invariance of graphene ribbon in the reciprocal space and expand in Fourier series functions and along the edge having periodicity in. i.e.,
The Fourier coefficient is expressed as
with
and
In (6), counts the number of dimers in the GNRs. The factor in the equation is a central point in this paper and so has to be determined. is an integer and not equal to zero. We consider a classical limit in which energy levels could be excited due to thermal fluctuations, i.e.. This condition is also necessary for large enough field, so that charge carriers can escape low energy scattering [10]. The energy level spacing, is the charging energy, is Boltzmann constant, is lattice temperature and is the graphene nonoribbon width. In what follows, we will compute To do this, (5), (6) and (7) are substituted in (1) following the simplification scheme and. We obtained
where
2.2. Sheet Current Density
The sheet current density of graphene can be determined from the relation
the sheet area., are the spin and valley degeneracies, respectively. The current density can also be written as
the velocity of Dirac fermions in graphene is defined as. In terms of the Fourier coefficients,
Giving
with
Note the dependence of, and the summation over the index in (10). Using the formalism by Litvinov and Manasson [9]. Equation (14) can be put in a Taylor-like expansion in terms of. i.e.,
where
is the differential current density (for), and
is the large-signal dynamic nonlinear conductivity at harmonic with drive frequency.
3. Negative Differential Conductivity3.1. Pure DC Limit
To see immediately that (16) demonstrates NDC, we consider a pure DC limit where. The Bessel functions except the term will vanish. The real part of the differential conductivity become
So that if the differential conductivity is negative and NDC is manifest in GNRs. Electron dynamics may be come more complicated in the presence of high-frequency components in addition to the static electric fields. High negative differential conductivity thus may result in GNRs if an external drive force is applied.
3.2. Monoharmonics
If one component of the AC field in (2) is applied, then and (16) simplifies to
after dropping the subscript on Here, it not clear immediately how NDC can be seen. To observe the phenomenon, we plot with for as shown in Figure 1 (top) for armchair ribbon and in Figure 1 (buttom) for zigzag ribbon,
The effect of NDC is strong in zigzag graphene than in armchair graphene. Especially, in the limit of high harmonic field, NDC is greatly suppressed in aGNR.
3.3. Biharmonics
In practical situations, one can also allow graphene nanoribbon subject to two AC field components. In that case, we let in our general formalism in Section 2. Equation (15) takes the form
and from which (17) follows. We have eliminated the time dependence by averaging over the period of the fields to find the time-independent current. In the left hand side, we replaced, and in the right hand side a delta function emerges which ensures that. If, then one must put and so that. However, we shall generalize this to a case of commensurate frequencies. We exemplified the case by a biharmonic having frequencies which can be periodic or non-periodic, with These two cases were studied in [11,12] for semiconductor superlattices. Simplifying further, we linearize with respect to one of the field amplitudes (say,). For a week field, and, which allows us to take. Finally, the AC current density becomes
. Equation (21) reduces to the monoharmonic case when.
The nature of the NDC is observed for a simultaneously varying harmonic field and phase difference in a three dimensional plot shown in Figure 2.
4. Discussion and Conclusions
In Figure 1 normalized current density is plotted against reduced static electric field for aGNR (left) and zGNR (right) for an applied field. At low fields up to, the quantum derivative of the characteristic yields a positive slope. A negative slope results for. In the whole region, graphene nanoribbon device will operate under Negative Differential Conductivity (NDC). A consequence of NDC in GNR is a formation of electric field domains that impedes a continuous motion of electrons generated by the external electric field and thus blocks high frequency oscillations in the nanoribbon. NDC disappears quite faster in aGNR as compared to zGNR which is more rubust at this limit.
The curves in Figure 3 demonstrate NDC phenomenon. They are obtained at wave mixing of two commensurate frequencies,. Figure 3 (left), and Figure 3 (right),. The onset of NDC in oddharmonics occurs around and in even-harmonics it starts at. In both cases, as in the previous NDC graphs,.
The combined effect of phase shift, between two fields and amplitude on NDC is depicted in Figure 2. There are three peaks at low bias fields at points, and. It is not very clear what these crests and troughs mean, they might be associated with field domains along one direction (for) and others along the opposite direction (for) and vice-versa. This means that electron velocity is higher in the domain with the same polarization along the electron direction and lower in the domain with opposite polarization.
In conclusion, we have demonstrated that graphene nanoribbons exhibit NDC regions in its characteristics at low bias field when. NDC is observed either in the presence of bias field alone or by superimPosing AC fields on the bias field. For one AC field component, NDC occurs around. When two AC fields at commensurate frequencies are applied, highharmonic NDC emerge for both even and odd harmonics at rather very low frequencies. The evenseries gives pronounced high-harmonic NDC than the odd-series. The presence of high-harmonic NDC means that it is possible for high-frequency generation in graphene nanoribbons when electric field domains are suppressed at high enough applied frequencies and. We therefore suggest this approach for the
study of terahertz generation in graphene.
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doi:10.1063/1.125663
along transverse direction. 
is related to the width of the nanoribbon as 
[
electric field
th and 
th component being 
is arbitrary, 
is an integer. 
are the field amplitudes. We require that 
. This will correspond to the DC component of the applied field. The dynamics of free 
-electrons in graphene can be describe by the time-dependent Boltzmann transport equation (BTE) based on relaxation time approximation in zero magnetic field as
is momentum independent. In Equation (1), 
and 
are the equilibrium and non-equilibrium Fermi electron distribution functions, respectively. 
is the electronic charge, 
is the electron wave vector and 
is the reduced plank constant. In the following section, we will calculate electronic current density for GNRs.
, phase 
and wave vector 
[5,6]. For aGNR,
with (+) for conduction band and (−) for valence band. 
and
, 
is the lattice spacing which has numerical value of 0.246 nm, 
is the overlap integral and 
is the phase perpendicular to the quasi-momentum
. The 1 BZ of aGNR is bounded by 
and the zGNR is
. 
is parallel to the edge and has translational symmetry along this direction. For aGNR, the transverse wave vector (phase) is quantized according to the rule [
with 
and 
. Unlike aGNR, the nature of transverse wave vector quantization is complicated in zGNR, depending on both 
and 
as
. However, for simplicity we assume 
is constant, say
, so that 
. Except this little subtlety for zGNRs, all that will be discussed in the following for aGNR are equally applied to the zGNR.
and 
along the edge having periodicity in
. i.e.,
is expressed as
with
counts the number of dimers 
in the GNRs. The factor 
in the equation is a central point in this paper and so has to be determined. 
is an integer and not equal to zero. We consider a classical limit in which energy levels could be excited due to thermal fluctuations, i.e.
. This condition is also necessary for large enough field, so that charge carriers can escape low energy scattering [
, 
is the charging energy, 
is Boltzmann constant, 
is lattice temperature and 
is the graphene nonoribbon width. In what follows, we will compute 
To do this, (5), (6) and (7) are substituted in (1) following the simplification scheme 
and
. We obtained
.
, 
are the spin and valley degeneracies, respectively. The current density can also be written as
. In terms of the Fourier coefficients,
dependence of
, 
and the summation over the index in (10). Using the formalism by Litvinov and Manasson [
. i.e.,
current density (for
), and
harmonic with drive frequency
.
. The Bessel functions except the 
term will vanish. The real part of the differential conductivity 
become
the differential conductivity is negative and NDC is manifest in GNRs. Electron dynamics may be come more complicated in the presence of high-frequency components in addition to the static electric fields. High negative differential conductivity thus may result in GNRs if an external drive force is applied.
and (16) simplifies to
Here, it not clear immediately how NDC can be seen. To observe the phenomenon, we plot 
with 
for 
as shown in 
in our general formalism in Section 2. Equation (15) takes the form
. In the left hand side, we replaced
, and in the right hand side a delta function emerges which ensures that
. If
, then one must put 
and 
so that
. However, we shall generalize this to a case of commensurate frequencies. We exemplified the case by a biharmonic having frequencies which can be periodic 
or non-periodic, 
with 
These two cases were studied in [11,12] for semiconductor superlattices. Simplifying further, we linearize with respect to one of the field amplitudes (say,
). For a week field, 
and
, which allows us to take
. Finally, the AC current density becomes
. Equation (21) reduces to the monoharmonic case when
.
is plotted against reduced static electric field 
for aGNR (left) and zGNR (right) for an applied 
field. At low fields up to
, the quantum derivative of the 
characteristic yields a positive slope. A negative slope results for
. In the whole region, 
graphene nanoribbon device will operate under Negative Differential Conductivity (NDC). A consequence of NDC in GNR is a formation of electric field domains that impedes a continuous motion of electrons generated by the external electric field and thus blocks high frequency oscillations in the nanoribbon. NDC disappears quite faster in aGNR as 
compared to zGNR which is more rubust at this limit.
. 
and 
. The onset of NDC in oddharmonics occurs around 
and in even-harmonics it starts at
. In both cases, as in the previous NDC graphs,
.
, between two 
fields and amplitude on NDC is depicted in 
, 
and
. It is not very clear what these crests and troughs mean, they might be associated with field domains along one direction (for
) and others along the opposite direction (for
) and vice-versa. This means that electron velocity is higher in the domain with the same polarization along the electron direction and lower in the domain with opposite polarization.
characteristics at low bias field when
. NDC is observed either in the presence of bias field alone or by superimPosing AC fields on the bias field. For one AC field component, NDC occurs around
. When two AC fields at commensurate frequencies are applied, highharmonic NDC emerge for both even and odd harmonics at rather very low frequencies
. The evenseries gives pronounced high-harmonic NDC than the odd-series. The presence of high-harmonic NDC means that it is possible for high-frequency generation in graphene nanoribbons when electric field domains are suppressed at high enough applied frequencies 
and
. We therefore suggest this approach for the