305
8
Quantum Transport in Carbon-Based Devices
High elds tend to reorient carrier moments and velocity vectors under the
inuence of a high electric eld. In this chapter, we explore the modica-
tion in the NEADF because of linear energy–momentum relationship preva-
lent in all carbon-based devices. Ballistic transport comes into play when
the channel length is below the scattering-limited mfp. Quantum effects are
natural ingredients of ballistic nonequilibrium in all carbon-based devices.
8.1 High-Field Graphene Transport
In the presence of an electric eld E, as discussed in Chapter 4, the isotro-
pic equilibrium carrier statistics based on the FD distribution transforms
into anisotropic NEADF given by [1,2]
fE
e
e
EE qkT
xH
FB
(,,)
(( ))/
()
E
E
θ
θ
=
+
=
+
−−⋅
−
1
1
1
1
(8.1)
with
H
EE
kT
o
Fco
B
() ,θηδθ
ηθ
π=−
−
≤≤ cos , =
02
(8.2)
x
EE
kT
kT
q
V
V
kT
q
co
B
o
co
co
Bt
o
t
B
=
−
==
==
∞
,, ,δ
E
E
E
(8.3)
NEADF of Equation 8.1 is highly asymmetric that favors electrons with veloc-
ity vectors antiparallel to the electric eld as compared to those with velocity
vectors parallel to the electric eld. NEADF transforms randomly oriented
velocity vectors or mfps or electric dipoles qℓ into unidirectional ones, yielding
the high-eld saturation velocity equal to v
Fo
in the absence of a quantum emis-
sion [2]. The relative drift velocity v
D
/v
Fo
, as will be shown, is a linear function
of the electric eld below δ
o
= 1 corresponding to the critical electric eld
EE=
co
and is sublinear as the electric eld E > E
co
rises, resulting in saturation for
higher values of the electric eld with
δ
o
1
. The thermal voltage V
t
and ohmic
(o) long-channel LC (∞) mfp ℓ
o∞
in E
co
play prominent roles in this transition [3].
306 Nanoelectronics
NEADF is endowed with a clear voyage from drift diffusion to the ballistic
regime consistent with Buttiker’s paradigm [4,5] with each mfp and resistors
of average length ℓ
o∞
. The ends of the mfp resistor are virtual thermalizing
probes, one end being higher than the other end in the Fermi level (elec-
trochemical potential) by
q
o
E . Within this scenario, carriers are removed
from the device and injected into a virtual reservoir where they are thermal-
ized and reinjected into the ballistic channel of length ℓ
o
. The description is
also consistent with the Natori model [6] of ballistic transport in the ballistic
domain by the Fermi level of the contacts in a MOSFET. In fact, Mugnaini and
Iannaccone [7,8] validate this model by applying it to a nanoscale MOSFET.
Similarly, CNT and several of its variations in the form of graphene sheets
and ribbons are nding a wide variety of applications in monitoring the
environment, sensing chemical elements, developing new indigenous mate-
rials, and providing interconnects for a variety of biochips. This chapter will
open vista for all specialties as NEADF covers a wide range of degeneracy,
phonon emission, and the electric eld, consistent with experimental obser-
vations. The results can be easily converted to obtain current–voltage char-
acteristics as indicated in References 9 and 10.
The drift response v
D
to an applied electric eld is the average of v
FO
cos θ
using the NEADF and DOS, resulting in
v
vu
dH
D
Fo g
=
ℑ
∫
1
2
1
0
2
π
θθ θ
π
co
s(()
)
(8.4)
The reduced Fermi energy η is now a function of the electric eld and is
evaluated from the normalization condition for carrier concentration n
g
udHu
n
N
gg
g
g
=
ℑ=
∫
1
2
1
0
2
π
θθ
π
(()),
(8.5)
The effective density of states N
g
is given by Equation 3.79. The normalized
drift velocity v
D
/v
Fo
as a function of the normalized electric eld δ
o
= E/E
co
is
shown in Figure 8.1 for u
g
= 1, 5, and 10. Three sets of curves almost overlap.
The dashed lines, barely visible in the high-eld domain, are from Equation
8.4. A simplied version of Equation 8.4 is obtained by substitution of cos
θ = ±1/2 in H(θ) as distribution is split into the ±x-direction. θ = −π/2 to + π/2
is for the +x-direction and θ = +π/2 to + 3π/2 is for the –x-direction. 〈cos θ〉 = ±1/d
is for an arbitrary dimensionality d = 3, 2, and 1 [2,11]. The squares in Figure
8.1 are obtained when this approximation is applied, indicating the anisotro-
pic character of carrier distribution. The solid lines are variations of Equation
8.5 as delineated below.
An enhanced perspective of this anisotropy is obtained when Δn
g
/n
g
is also plotted in Figure 8.1, as shown by solid lines. Δn
g
= n
g+
− n
g−
is the
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