# The Ultimate Ballistic Drift Velocity in Carbon Nanotubes

Authors: Mohammad Taghi Ahmadi; Razali Ismail; Michael L. P. Tan; Vijay K. Arora
Journal: Journal of Nanomaterials
 (2008)

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## Abstract

The carriers in a carbon nanotube (CNT), like in any quasi-1-dimensional (Q1D) nanostructure, have analog energy spectrum only in the quasifree direction; while the other two Cartesian directions are quantum-confined leading to a digital (quantized) energy spectrum. We report the salient features of the mobility and saturation velocity controlling the charge transport in a semiconducting single-walled CNT (SWCNT) channel. The ultimate drift velocity in SWCNT due to the high-electric-field streaming is  based on the asymmetrical distribution function that converts randomness in zero-field to a stream-lined one in a very high electric field. Specifically, we show that a higher mobility in an SWCNT does not necessarily lead to a higher saturation velocity that is limited by the mean intrinsic velocity depending upon the band parameters. The intrinsic velocity is found to be appropriate thermal velocity in the nondegenerate regime, increasing with the temperature, but independent of carrier concentration. However, this intrinsic velocity is the Fermi velocity that is independent of temperature, but depends strongly on carrier concentration. The velocity that saturates in a high electric field can be lower than the intrinsic velocity due to onset of a quantum emission. In an SWCNT, the mobility may also become ballistic if the length of the channel is comparable or less than the mean free path.


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## Body

## 1. Introduction

Due to the chemical stability  and perfection of the carbon
nanotube (CNT) structure, carrier mobility  is not affected by processing and
roughness scattering as it is in the conventional  semiconducting channel. The
fact that there are no dangling bond states at the surface of CNT allows for a
wide choice of gate insulators in designing a field effect transistor (FET). It
is not surprising that the CNTs are being explored as viable candidates for
high-speed applications. The growing
demand for higher computing power, smaller size, and lower power consumption of
integrated circuits leads to a pressing need to downscale semiconductor
components leading to novel new nanostructures [1]. CNTs, originally discovered by Iijima [2],
have opened a number of applications. Some of these  applications are building
block of nano-VLSI circuit design, including reliable interconnects. The
circuits so designed are shown to be mechanically rigid and able to carry high
current densities. However, almost all synthetic
methods result in bundles of CNTs rather than a well-organized strand requiring
a complicated chemical procedure in separating individual single-walled CNT (SWCNT)
that almost certainly would introduce damages of varying degrees. Peng et al. [3] have directly
obtained soot from a chemical vapor deposition chamber. By fixing one end of
the soot, a CNT (in their case a multiwall CNT) was drawn by using scanning
electron microscope (SEM) nanoprobe. Electronic transport in these CNTs is in
its infancy and needs to be investigated in order to develop novel application,
for example, making a CNT FET. This is the motivation for this work.The most nanoelectronic
applications look for high-speed CNTs that are well known to have very high mobilities.
The low scattering probability in CNTs is responsible for superior mobilities
[4]. However, the work of Ahmadi et al. in
a quantum nanowire (NW) has shown that the ultimate saturation velocity does
not sensitively depend on the low-field mobility. It is, therefore, of interest for us to find
the ultimate velocity that may exist in an SWCNT. This velocity will necessarily depend on
the band structure, the temperature, and the degeneracy level. The carrier
drift is the velocity with which a carrier (electron or hole) can propagate
through the length of the device encountering collisions on the way and
starting its journey fresh on facing a collision. The higher mobility may bring
an electron closer to saturation as a high electric field is encountered, but needs not to elevate the saturation
velocity [5]. The reduction in conducting channel length of the device results
in a reduced transit-time-delay and hence enhanced operational frequency. However, if the length is made smaller than
the mean free path, the mobility may also become ballistic, free from
randomizing scattering events.In the following sections, the
formalism is developed to study the velocity response to the electric field in
an SWCNT as shown in Figure1. The
radius that is a few nanometers in size is comparable to de Broglie wavelength
while length can vary from cm to μm
range. This makes CNT a quasi-1-dimensional
(Q1D) entity.Figure 1
A prototype single-walled carbon
nanotube with lengthL≫λD,
de Broglie wavelength, and d=2R≪λD.


## 2. Q1D-CNT Nanostructure

A single-walled carbon nanotube (SWCNT)
of Figure1 is a sheet of graphite (called graphene) rolled up into a cylinder
with diameter of the order of a nanometer.
For an electron on the surface of the CNT, it is certain to have wave
properties as the diameter is comparable to de Broglie wavelength λD=h/p,
where p is the carrier momentum. The
tube diameter must necessarily contain integer (n) number of de Broglie waves for
an electron to form a standing wave pattern around the rim of the CNT, thereby
giving a resonant condition as the returning wave to the same point reinforces
the electron motion. This constructive interference, leading to the resonant
behavior, gives
(1)2πR=nhp,n=1,2,3…. The quantized energy that depends
on (1) relies on the relationship between energy and momentum that can be
linear or quadratic depending on the chirality of the CNT. In order to understand the
fundamentals of transport parameters without getting into the bandstructure complexities,
we will assume quadratic relationship as in a nanowire [6] for transverse
energy. In this approximation, we consider transverse effective mass m⊥* to be isotropic that depends on the diameter
of the CNT [7] but independent of energy or momentum. The transverse energy of
the CNT is given by
(2)E⊥n=n2ℏ22m*R2,n=1,2,3…. In the longitudinal direction,
the electron waves are propagating that result in the charge transport. With unaltered conduction band at Eco,
the transverse energy in y and z directions is digital in nature as given by
(2), but is of continuous analog-type in the longitudinal direction (taken to
be x axis). The total energy is given by
[8]
(3)E=Eco+E1+ℏ2kx22m*=Ec+ℏ2kx22m*.The chiral vectorC→ specifies the direction of the roll-up:
(4)C→=na→1+ma→2=(n,m). Here, a→1 and a→2 are the basis vectors of the lattice. In the
(n,m) notation for C→,
when (n−m) is a multiple of 3 the
nanotube is metallic, the vectors (n,0) or (0,m) denote zigzag CNTs, whereas
the vectors (n,m) correspond to chiral CNTs. In the semiconducting mode of CNT,
as in (5,3) chirality, the band structure can be manipulated to be parabolic
in the semiconducting mode of operation. Due to the approximation for the
graphene band structure near the Fermi point, the E(k) relation of the CNT is
(5)E(k→)=t3aC–C2|k→CN|=t3aC–C2kcν2+kt2,where kcν is the wave vector component along the circular
direction, which is quantized by the periodic boundary condition, and kt is the wave vector along
the length of the CNT. Also, kcν is minimum for the lowest band of CNT. The minimum value for kcν is zero for the metallic
CNT. Therefore, the density of states (DOS)
is given by
(6)D(E)=D0=83aC–Ctπ,where aC–C=1.42Å is the carbon-carbon (C–C) bond length, t=2.7eV [9, 10] is  the nearest
neighbor C–C tight binding overlap energy, and d is the diameter of the carbon
nanotube, taken to be that of (5,3) chirality.The band structure is indeed
nonparabolic. However, for low-lying states in the vicinity ofk=0, where
most electrons reside, the band structure is approximately parabolic, as shown
in Figure 2. For the semiconducting CNT, the focus of our study, the minimum
magnitude of the circumferential wave vector is kcν=2/3d, where d=2R is the diameter of the CNT. By
substituting this equation into the E(k) approximation for the semiconducting
CNT, we get
(7)E(k)=±t3aC–C2(23d)2+kt2. Near the band minimum, all metallic and semiconducting annotates are equivalent.
According to (5), the conduction and valence bands of a semiconducting CNT are
mirror image of each other and the first band gap is EG=2aC–Ct/d=0.8(ev)/d(nm).Figure 2
The parabolic band
structure of CNT in the vicinity of energy minimum (k=0) with conduction and
valence bands separated by the first bandgap EG.In this description, the energyE(k) and band gap in semiconducting CNT
are function of diameter. There are higher-order subbands that may be populated
as well. However, (5) when kxd≪1 reduces to
(8)E(k)=EG21+(3kxd2)2≈EG2(1+98d2kx2),where EG is the CNT bandgap and d is the diameter. When
expanded to the first order, the E-k relation becomes as in (3) with Eco=EG/2. With this modification, m* in (3) is the longitudinal
effective mass that depends on the diameter of the tube:
(9)m*m0=4ℏ29aC–Cdt=0.08nmd(nm). In one-dimensional carbon
nanotube using (5) for the gradient of k,
definition of density of states DOS, including
effect of the electron spin,
leads to the following
equation which is similar
to that in Q1D nanowire [11]:
(10)DOS=ΔnxΔELx=12π(E−EG2)−1/2(2m*ℏ2)1/2. As in any Q1D, DOS diverges at
the bandedge E=Ec but
drops as square root of the kinetic energy Ek=E−Ec.


## 3. Energy and Velocity Distribution

The distribution function of the
energyEk is given by the Fermi-Dirac distribution function:
(11)f(Ek)=1e(Ek−EF1)/kBT+1,where EF1 is the Fermi
energy that is equivalent to chemical potential and describes the degeneracy
nature of the electron concentration. As
shown in Figure 3, the Fermi energy level runs parallel to the conduction band
edge. The Fermi level is in  the bandgap for the nondegenerate carrier
concentration and within the conduction (or valence) band for degenerate
carrier (electron or hole) concentration. In the absence of electric field, the
bands are flat. The velocity vectors for the randomly moving stochastic
electrons cancel each other giving net drift equal to zero, as shown in the
flat band diagram of Figure 3. In a homogenous CNT, equal numbers from left and
right are entering the free path.Figure 3
The flat-band diagram
for CNT in equilibrium.This random motion does not mean
that the magnitude of a single vector (that we call the intrinsic velocity) is
zero. The average of this intrinsic velocityvi1, as calculated from the average value of
|v| with the distribution function of (11) multiplied by the DOS of (10), is
given by
(12)vi1=vth1ℑo(η1)ℑ−1/2(η1)with
(13)ℑj(η)=1Γ(j+1)∫0∞xje(x−η)+1dx,(14)vth1=1πvth=1π2kBTm*. The normalized Fermi energy η1=(EF1−Ec)/kBT is calculated from the carrier concentration n1 per unit length of the CNT as follows:
(15)n1=Nc1ℑ−1/2(η1)with
(16)Nc1=(2m*kBTπℏ2)1/2. Here, ℑj(η) is the Fermi-Dirac integral of order j and Γ(j+1) is a Gamma function of order j+1.
Its value for an integer j is Γ(j+1)=jΓ(j)=j!.
The value of Γ for j=0 is Γ(1)=1
and for j=−1/2 is Γ(1/2)=π.
The Fermi integral with Maxwellian approximation is always an exponential for
all values of j and is given by [12]
(17)ℑj(η)=eη(nondegenerate). In the strongly degenerate
regime, the Fermi integral transforms to
(18)ℑj(η)=1Γ(j+1)ηj+1j+1=ηj+1Γ(j+2)(deg). The degeneracy of the carriers
sets in at n1=Nc1=2.0×108m−1 for Q1D-CNT with chirality (5,3) (m*=0.189mo)
[13]. The carriers are nondegenerate if the concentration n1 is less
than this value and degenerate if it is larger than this value. The threshold
for the onset of degeneracy will change as chirality changes. The effective
mass is m*=0.099mo [13] for (9,2) chirality,
and degeneracy sets in at n1=1.46×108m−1.Figure4 indicates the ultimate
velocity as a function of temperature. Also shown  is the graph for
nondegenerate approximation. The velocity for low carrier concentration follows T1/2 behavior independent of carrier concentration.
As concentration is increased to embrace degenerate domain, the intrinsic
velocity tends to be independent of temperature, but depends strongly on
carrier concentration. The nondegenerate limit of intrinsic velocity vi1 is vth1 of (14). Figure 5 shows the graph of ultimate
intrinsic velocity as a function of carrier concentration for three
temperatures T=4.2 K (liquid helium),
77 K (liquid nitrogen), and 300 K (room temperature). As expected, at low
temperature, carriers follow the degenerate statistics and hence their velocity
is limited by appropriate average of the Fermi velocity that is a function of
carrier concentration. In the degenerate limit, the intrinsic velocity is only
the function of carrier concentration and is independent of temperature:
(19)vi1Deg=ℏ4m*(n1π). Equation (19) shows that the
intrinsic velocity is a linear function of the linear carrier concentration, as
shown in Figure 5. However, it may be affected by the onset of a quantum
emission. For low carrier concentrations, the velocity is independent of
carrier concentration as the graphs are flat. Most published works tend to use
thermal velocity for modeling as it is independent of carrier concentration.Figure 4
Intrinsic velocity
versus temperature for(5,3) CNT for three concentration values. Also shown is the nondegenerate limit.Figure 5
Velocity versus carrier
concentration forT=4.2 K (liquid helium), T=77 K (liquid nitrogen), and T=300 K (room temperature). The 4.2 K curve is closer to the degenerate limit.


## 4. High-Field Distribution

Arora [14] modified the equilibrium distribution function of (11) by
replacing EF1 (the chemical potential) with the electrochemical
potential EF1+qε→⋅ℓ→.
Here, ε→ is the applied electric field, q is the
electronic charge, and ℓ→ the mean free path during which carriers are
collision free or ballistic. Arora’s distribution function is thus given by
(20)f(Ek)=1e(Ek−EF1+qε→.ℓ→)/kBT+1. This distribution has simpler
interpretation as given in the tilted-band diagram of Figure 6. A channel of CNT can be thought as a
series of ballistic resistors each of length ℓ,
where the ends of each free path can be considered as virtual contacts with
different quasi-Fermi levels separated in energy by qε→⋅ℓ→.
It is clear that this behavior is compatible with the transport regime in a
single ballistic channel where local quasi-Fermi level can be defined on both
ends for the source and drain contacts. This behavior is understandable if we
consider the widely diffused interpretation of inelastic scattering represented
by the Büttiker approach of virtual thermalizing probes [15], which can be used
to describe transport in any regime. Within this approach, carriers are
injected into a “virtual” reservoir where they are thermalized and start their
ballistic journey for the next free path. The
carriers starting from the left at Fermi potential EF1 end the free-path voyage with EF1−qε→ℓ→.
Those starting from right end of the free path end the voyage with the
electrochemical potential EF1+qε→ℓ→.
These are the two quasi-Fermi levels. The current flow is due to the gradient
of the Fermi energy EF1 (x) in the presence of an electric
field. Because of this asymmetry in the distribution of electrons, the
electrons in Figure 6 tend to drift opposite to the electric field ε→ applied in the negative x-direction (right to
left).Figure 6
Partial stream-lining
of electron motion on a tilted band diagram in an electric field.In an extremely large electric field, virtually all the
electrons are traveling in the positivex-direction (opposite to the electric
field), as shown in Figure 7. This is
what is meant by conversion of otherwise completely random motion into a stream-lined
one with ultimate velocity per electron equal to vi. Hence, the
ultimate velocity is ballistic independent of scattering interactions. This interpretation is
consistent with the laws of quantum mechanic where the propagating electron
waves in the direction of the electric field find it hard to surmount the
infinite potential barrier and hence are reflected back elastically with the
same velocity.Figure 7
Conversion of random
velocity vectors to the stream-lined one in an infinite electric field.The ballistic motion in a free
path may be interrupted by the onset of a quantum emission of energyℏωo. This quantum may be an optical phonon or a
photon or any digital energy difference between the quantized energy levels
with or without external stimulation present. The mean-free path with the
emission of a quantum of energy is related to ℓ0 (zero-field mean free path) by an expression [16]
(21)ℓ=ℓ0[1−e−EQ/qεℓ0]=ℓ0[1−e−ℓQ/ℓ0]with
(22)qεℓQ=EQ=(N0+1)ℏω0,N0=1eℏω0/kBT−1.
Here, (No+1)
gives the probability of a quantum
emission. No is the Bose-Einstein distribution function
determining the probability of quantum emission. The degraded mean free path ℓ is now smaller than the low-field mean free
path ℓo.
Also, ℓ≈ℓo in the ohmic low-field regime as expected. In
high electric field, ℓ≈ℓQ.
The inelastic scattering length during which a quantum is emitted is given by
(23)ℓQ=EQqε. Obviously, ℓQ=∞ in zero electric field and will not modify the
traditional scattering described by mean free path ℓ0 as ℓQ≫ℓ0. The low-field mobility and associated drift
motion are, therefore,
scattering-limited. The effect of all possible scattering interactions in the
ohmic limit is buried in the mean free path ℓ0.
The nature of the quantum emitted depends on the experimental set up and the
presence of external stimulations as well as the spacing between the digitized energies. This
quantum may be in the form of a phonon, photon, or the spacing with the
quantized energy of the two lowest levels. For the quantum emission to be
initiated by transition to higher quantum state with subsequent emission to the
lower state, the quantum energy ℏωo=ΔE⊥1−2 is a function of radius of the CNT:
(24)ℏωo=3ℏ22m*R2. This dependence of the quantum on
the radius or diameter of the CNT may give an important clue about the CNT
chirality and could be used for characterization of the CNT.The number of electrons in each direction is proportional
toe±qεℓ/kBT (+ sign is for antiparallel and − sign is for parallel direction). The number of
antiparallel electrons overwhelms due to rising exponential and those in
parallel direction decrease to virtually zero due to the decaying exponential.
The net fraction Fant of the stream-lined electrons is then given by
(25)Fant=eqεℓ/kBT−e−qεℓ/kBTeqεℓ/kBT+e−qεℓ/kBT=tanh⁡(qεℓkBT).When the net velocity response to
the electric filed for the fraction of electrons drifting in the opposite
(antiparallel) direction  of the electric field is considered, the drift
response is obtained as [17]
(26)vd=vsattanh⁡(qEℓkBT)=vsattanh⁡(EEc)with
(27)Ec=kBTqℓ=Vtℓ,where Ec is the critical electric field for the onset
of nonohmic behavior. Also, Vt=kBT/q is the thermal voltage whose value at the room
temperature is 0.0259 V.Figure8 gives the normalized
plot of vd/vsat versus ε/εc.
Also, vsat=vi1 when quantum emitted has energy much higher than the thermal
energy (ℏωo≫kBT).
The saturation velocity vsat=vi1tanh(ℏωo/kBT) when ℏωo is comparable to the thermal energy kBT.
Figure 8 shows the comparison with the empirical equation normally employed in
the simulation programs:
(28)vd=vsat1[1+(E/Ec)γ]1/γ,where γ is a parameter. A wide variety of parameters are quoted in
the published literature. Greenberg and del Alamo [18] give convincing evidence
from measurements on a 5μm InGaAs resistive channel that γ=2.8.
Other values that are commonly quoted are γ=2 for electrons and γ=1 for holes. The discrepancy arises from the
fact that it is impossible to measure directly the saturation velocity that
requires an infinite electric field. No device is able to sustain such a high
electric field. The ultimate saturation velocity then can be obtained only
indirectly. Normally, the highest measured drift velocity is ascribed to be the
saturation velocity which is always lower than the actual saturation velocity. The
plots differ only at the intermediate values of the electric field.Figure 8
Normalized velocity-field characteristics for an
SWCNT as predicted from the theory and compared with empirical models.In the low-field limit,
velocity-field graph is linear from which ohmic mobility can be obtained. Chai
et al. [19] have demonstrated transport of energetic electrons through aligned
tubes with lengths of 0.7–3 mm. These
developments open the possibility of ballistic mobility as suggested by Wang
and Lundstrom [20] and Shur [21]. In fact, Mugnaini and Iannaccone [22, 23]
have done a study
of transport ranging from drift-diffusion to ballistic. Their model is also applicable to nanowires
and CNTs which is similar to the Büttiker approach to dissipative transport
[15]. In principle, our formalism is very similar to that of Mugnaini and
Iannaccone. In all these works, analytical results obtained for nano-MOSFET are
consistent with that for ballistic nanowire transistors [5, 6] and also for the
CNT nanostructures reported here. In this scenario, generic resistive channel of
a CNT can be described as a series of resistive channels each with a finite
scattering length ℓ. Since L can be as low as 0.7μm
that can be lower than the mean free path in a CNT because of extremely high
mobility, it is worthwhile to evaluate how the mobility will change in a
ballistic channel with L<ℓ.
This relation between the mobility, the mean free path ℓ,
and the channel length L is expected
to have deeper consequences on the understanding of transport in nanostructured
devices.The long-channel mobilityμ∞ as obtained from low-field approximation of (26)
is given by
(29)μ∞=qℓm*vm,vm=vth22vsat,where vm is the mobility velocity that is a
combination of thermal hop on thermalization and vsat that arises in
the middle of the free path. In a
short-channel CNT, even the mobility may become ballistic. The length-limited
ballistic mobility μL can be obtained similar to the technique used for the scattering length
for the emission [11] of a quantum and is obtained as
(30)μL=μ∞(1−e−L/ℓ). In the limit that the length L of the CNT is smaller than the mean free path, the mobility will also become
ballistic replacing mean free path ℓ in (30) with L. This gives ballistic
mobility:
(31)μB=qLm*vm. Now the mobility is limited by only the length of the channel and
ballistic intrinsic velocity that is a function of temperature and carrier concentration.Figure9 gives the normalized plot of μL/μ∞ versus L/ℓ.
It is clear from the graph that the relative mobility approaches unity in the
long-channel limit L≫ℓ. However, in the short channel limit (L≪ℓ), the relative mobility is a linear function
of length of the CNT.Figure 9
Normalized length-limited mobility to that in
an infinite sample versus normalized length of the CNT to that of the mean free path in the CNT.


## 5. Conclusion

As the scaling of devices
continues and new nanostructures like CNT are discovered and exploited for
experimentation, more challenges as  well as opportunities appear to extend the
vision encountered in the International Technology Roadmap for Semiconductors [24].
CNT devices have attracted a lot of attention because of their ideal electronic
properties. Both p- and n-type CNT-FET have been fabricated and have exhibited
promising characteristics. However, the device physics and transport mechanisms
of such devices are not yet fully understood. The results presented here will
enhance the efforts that have been put on modeling CNT-FETs. Nonequilibrium Green’s function approach [7]
is too cumbersome to understand the physical processes controlling the
transport in Q1D nanostructure. The developed paradigm provides simple
intuitive description of the Q1D device physics that is easily implementable.Starting from a model for
ballistic one-dimensional FET model and adopting the Büttiker probes interpretation
of inelastic scattering, we have shown that the case of intermediate transport
between fully ballistic transport and drift-diffusion transport can be
described by the series of an equivalent drift-diffusion minichannels with a
ballistic transport, consistently with the earlier results obtained in the ohmic
domain [25–28]. Therefore, this compact macromodel can be considered an
adequate description of transport in nanoscaled CNT. In fact, our model
embraces virtually all nanostructures. In the presence of a magnetic field
[26], the spiraled path of a carrier will have similar transport mechanism as
pointed out here.The asymmetrical distribution function reported here is a very
valuable tool for studying quantum transport in nanostructures. This
distribution function takes into account the asymmetrical distribution of
drifting electrons or holes in an electric field. This distribution function
transforms the random motion of electrons into a stream-lined one that gives
the ultimate saturation velocity that is a function of temperature in
nondegenerate regime and a function of carrier concentration in the degenerate
regime. The ultimate drift velocity is
found to be appropriate thermal velocity for nondegenerately doped CNTs. However, the ultimate drift velocity is the Fermi velocity for degenerately-doped
Q1D CNTs. The inclusion of quantum emission for a given sample may further
highlight the fundamental physical processes that are present. We also show the
insensitivity of the saturation velocity on ohmic mobility that is scattering-limited. Even for the ohmic mobility, we have extended
the vision to embrace ballistic mobility that will be independent of scattering
when the channel length is smaller than the scattering-dependent mean free
path.It is our hope that these results will be liberally utilized for
modeling, simulations, characterization, and extraction of device parameters
without ambiguity.



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*Source: 769250-2008-10-20.xml*