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MASAUM Journal of Basic and Applied Sciences Vol.1, No. 2 September 2009
233
Abstract
—
The limitations on carrier (holes and electrons)
drift due to high-field streamlining also randomly velocity vector
in equilibrium is reported. Asymmetrical distribution function
that converts randomness in zero-field to streamlined one in a
very high electric field is employed. The ultimate drift velocity is
found to be appropriate thermal velocity for a given
dimensionality for non-degenerately doped nanostructure.
However, the ultimate drift velocity is the Fermi velocity for
degenerately doped nanostructures. Quantum and high-field
effects controlling the transport of carrier in nanostructures are
described. The results obtained are applied to the modeling of a
nanowire transistor.
Index Terms—Silicon Nanowire, Carrier Transport, carrier
Velocity, Nanowire Transistor,
I. I
NTRODUCTION
HE quest for high-speed devices and circuits for Ultra-
Large-Scale-Integration (ULSI) is continuous. The speed
is determined by the ease with which the carrier (electron or
holes) can propagate through the length of the device. In the
earlier designs, the mobility of the carrier was believed to be
of paramount importance. That was the push for Gallium
Arsenide (GaAs) as the mobility of electrons in GaAs is 5-6
times higher than that of electron in silicon. However, as
development of the devices to nanoscale dimensions continued
it became clear that the saturation velocity plays a predominant
role. The reduction in conducting channel length of the device
results in reduced transit-time-delay and hence enhanced
operational frequency. Until today, there is no clear consensus
on the interdependence of saturation velocity on low-field
mobility that is scattering-limited [1], [2]. There are a number
of theories of high-field transport to answer this
interdependence. Among them are Monte Carlo simulations,
energy-balance theories, path integral methods, green function
and many others. No clear consensus has emerged on the true
nature of saturation velocity and its dependence on band
structure parameters, doping profiles or ambient temperature.
Often high mobility is attributed to higher saturation velocity.
This outcome is not supported by experimental observations
1,2,3,4,5
Faculty of Electrical Engineering Universiti Teknologi Malaysia 81310
Skudai, Johor (
*
e-mail: ahmadiph@gmail.com&taghi@ieee.org).
prompting our careful study of the process controlling the
ultimate saturation. It has been confirmed in a number of
works that the low-field mobility is a function of quantum
confinement [3]-[7]. In the following, the fundamental
processes that limit drift velocity are delineated. As devices
are being scaled down in all dimensions, the curiosity towards
ballistic nature of the carriers is elevated. Initially, it was in
the work of Arora [1] that the possibility of ballistic nature of
the transport in a very high electric field for a nondegenerate
semiconductor was indicated. Our floccose is on degenerate
domain in the nanowire where electrons (holes) have analog
type classical spectrum only in one direction while the other
two directions are quantum confined or digital in nature.
When only the lowest digitized quantum state is occupied
(quantum limit), a quantum nanowire shows distinct one-
dimensional character.
II. DISTRIBUTION
FUNCTION
In one dimensional nanowire (see Fig. 1), only one of the
three Cartesian directions is much larger than the De-Broglie
wavelength. The energy spectrum is analog-type only in x-
direction. For N-Type nanowire with rectangular cross-section
the energy spectrum is given by.
*
1
22
2m
k
EE x
c
η
+=
(1)
with
2*
3
22
2*
2
22
22
zy
coc
LmLm
EE ηη
ππ
++=
(2)
Here E
c
is the bandedge in the quantum limit of a nanowire
that is lifted from bulk conduction bandedge E
co
by the zero-
point energy in the y, z-direction where the
length
Dzy
L
λ
<<
,
, the De-Broglie wavelength with a typical
value of 10 nm. k
x
is
the momentum wave vector in the x-
direction. Due to the anisotropic nature of the effective mass
in silicon,
*
3,2,1
m
is derived from longitudinal
(
o
mm 92.0
*
=
λ
) and transverse (
ot
mm 19.0
*
=
) masses in
the ellipsoidal valley model. Only valleys with the higher
longitudinal mass in the y, z directions are important for (100)
Analytical Study of Carriers in Silicon
Nanowires
Mohammad Taghi Ahmadi*
1
, Member, IEEE, Amir Hossein Fallahpour
2
, Javad Allahdadian
3
,
Mojgan Kouhnavard
4
, Razali Ismail
5
, Member, IEEE
T
MASAUM Journal of Basic and Applied Sciences Vol.1, No. 2 September 2009
234
oriented nanowire. The transport or conductivity mass in this
orientation is then
ot
mm 19.0
*
=
[4]. For a cylindrical wire
with circular cross-section, the energy spectrum is the same as
given by (2) except the bandedge is given by [3].
2*
1
22
2
01
2Rm
EE
coc
η
π
α
+=
(3)
where
405.2
01
=
α
is the first zero of the Bessel function of
order zero, i. e.,
0)(
01
=
α
o
J
.
Fig. 1. A prototype nanowire with
,
y z D
L
λ
<<
and
x D
L
λ
>>
for
rectangular cross-section and
D
R
λ
<<
and
D
L X
λ
= >>
for
circular cross-section.
The quantum wave in the x-direction is a traveling
(propagating) wave and that in the y (z)-direction is the
standing wave.
Ω
=z
L
y
L
ezyx
zy
xkj
k
x
ππ
ψ
sinsin
2
),,( ).(
(4)
For the cylindrical wires the standing wave component of the
wave function in (4) is replaced by the Bessel function with
appropriate normalization constant. The distribution function
of the energy E
k
is given by the Fermi-Dirac distribution
function which shows probability of occupied levels
1
1
)( 1
+
=−
Tk
EE
k
B
Fk
e
Ef
(5)
where E
F1
is the one-dimensional Fermi energy at which the
probability of occupation is half and T is the ambient
temperature. In non- degenerately doped semiconductors the
‘1’ in the denominator of (5) is neglected as compared to the
large exponential factor. In this approximation, the distribution
is Maxwellian, which is used in determining the transport
parameters extensively. This simplification is good for a
nondegenerately-doped semiconductor. However, most
nanoelectronic devices these days are degenerately doped.
Hence any design based on the Maxwellian distribution is not
strictly correct and often leads to errors in our interpretation of
the experimental data. In the other extreme, for strongly
degenerate carriers, the probability of occupation is 1 when
E
k
< E
F
and is zero if E
k
> E
F
. Arora [1] modified the
equilibrium distribution function of (5) by replacing E
F1
(the
chemical potential) with the electrochemical
potential
λ
ρ
ρ.
1
ε
qE
F
+
where
ε
r
is the applied electric field,
q the electronic charge and
λ
ρ
the mean free path. Arora’s
distribution function [1] is given by
1
1
)(
.
1
+
=
+−
Tk
qEE
k
B
Fk
e
Ef
λ
ρ
ρ
ε
(6)
This distribution has a simple interpretation [6] as given in the
tilted band diagram of Fig. 2. It can be seen that the Fermi
level on left is
λ
ε
qE
F
−
1
and that on the right
λ
ε
qE
F
+
1
.
These are the two quasi Fermi levels with E
F1
at the point x.
The current flow is due to the gradient of Fermi energy
)(
1
xE
F
when an electric field is applied. Because of this
asymmetry in the distribution of electrons, the electrons tend to
drift opposite to the electric field
ε
ρ
applied in the positive x-
direction (left to right). In an extremely large electric field,
virtually all the electrons are traveling in the negative x-
direction (opposite to the electric field). This is what is meant
by conversion of otherwise completely random motion into a
streamlined one with ultimate velocity per electron equal to the
intrinsic velocity
i
v
. Hence the ultimate velocity is ballistic
independent of scattering interactions.
Fig.2. Partially streamlined electrons on a tilted band diagram in an electric-
field.
T
he ballistic motion in a mean-free path is interrupted by
the onset of a quantum emission of energy
0
ω
η
. This
λ
ε
qE
F
+
1
1F
E q
ε
−
l
X
MASAUM Journal of Basic and Applied Sciences Vol.1, No. 2 September 2009
235
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
0
λ
(zero-
field mean free path) by an expression [7]
]1[]1[
00
00
λ
λ
λ
λλλ
QQ
ee
q
E−−
−=−=
ε
(7)
00
)1(
ωε
ηλ +== NEq
QQ
(8)
1
1
0
0
−
=
Tk
B
e
N
ω
η
(9)
Here
)1(
0
+
N
gives the probability of a quantum emission.
N
o
is the Bose-Einstein distribution function determining the
probability of quantum emission. The degraded mean free
path
λ
is now smaller than the low-field mean free path
o
λ
.
0
λλ ≈
in the Ohmic low-field regime as expected. In the
high electric field,
Q
λλ ≈
. The inelastic scattering length
during which a quantum is emitted is given by
ε
q
E
Q
Q
=λ
(10)
Obviously
∞=
Q
λ
in zero electric field and will not modify
the traditional scattering described by mean free path
0
λ
as
0
λλ >>
Q
. The low-field mobility and associated drift
motion is therefore scattering-limited. The effect of all
possible scattering interactions is now buried in the mean free
path
0
λ
. This by itself may be enough to explain the
degradation of mobility
µ
in a high electric field;
1
*
11
*
1
*
1i
Q
i
c
vm
q
vm
q
m
q
λ
λ≈==
τ
µ
(11)
Here
c
τ
is the mean free time (collision time) during which
the electron motion is ballistic.
1i
v
is the mean intrinsic
velocity for 1D nanowire that is discussed in the next section.
High mobility does not influence the saturation velocity of the
carriers, although it may accelerate the approach towards
saturation as high fields are encountered in nano-length
channels.
III.
N-TYPE
NANOWIRE
The intrinsic velocity is given by
(
)
( )
0 1
1
1 1
2
1
F
i th
F
v v
η
η
π
−
ℑ
=ℑ
(12)
with
*
1
2
B
th
k T
v
m
=
(13)
( )
0
1
( )
( 1) 1
j
jx
x
dx
j e
η
η
∞
−
ℑ = Γ + +
∫
(14)
Here,
( )
j
η
ℑ
is the Fermi-Dirac (FD) integral of order j and
)1(
+
Γ
j
is a Gamma function. Its value for an integer j
is
!)()1( jjjj
=
Γ
=
+
Γ
. For half integer values, it
is
3 1 1 1
2 2 2 2
π
Γ = Γ =
. In the strongly degenerate
regime, the FD integral transforms to
1 1
1
( )
( 1) 1 ( 2)
j j
j
j j j
η η
η
+ +
ℑ ≈ =
Γ + + Γ +
(15)
The carrier concentration per unit length n1 is given by
(
)
1 1 1 1
2
c F
n N
η
−
= ℑ
(16)
For quasi-one-dimensional nanowires the ultimate average
velocity per electron is a function of temperature and doping
concentration.
( )
1
1 0 1
*
1 1
2c
B
i F
N
k T
vm n
η
π
= × ℑ
(17)
with
1
*
2
1
12
2
B
c
m k T
N
π
=
h
(18)
where
1
c
N
is the effective density of states for the conduction
band with
*
1
m
now being the density-of-states effective mass.
n
1
is the carrier concentration per unit length. However, it can
be shown using the distribution function of (6) that the mean
free path for transverse valleys (with ellipsoidal axis
perpendicular to the electric field) is larger due to smaller
effective mass as compared to that in the longitudinal valleys
(with ellipsoidal axis parallel to the electric field). The
electrons transfer to the lower-effective-mass valleys at high
electric field. Therefore, for calculation of the ultimate
MASAUM Journal of Basic and Applied Sciences Vol.1, No. 2 September 2009
236
saturation velocity, appropriate effective mass to use in the
calculations is m
t
= 0.19 m
o
.
Figure 3 indicates the ultimate velocity as a function of
temperature. Also shown is the graph for nondegenerate
approximation. The velocity of electrons for low carrier
concentration follows
2/1
T
behavior independent of carrier
concentration. However for high concentration (degenerate
carriers) the velocity of electrons depends strongly on
concentration and becomes independent of the temperature.
The ultimate saturation velocity is thus the thermal velocity
appropriate for 1D carrier motion :
1 1
*
1
21
B
i ND th th
k T
v v v
m
π
π
= = =
(19)
Fig.3. Velocity versus temperature for nanowire for various
concentrations
.
Figure 4 shows the graph of ultimate intrinsic velocity of
electrons as a function of carrier concentration for three
temperatures (T= 4.2 K, 77 K, and 300 K). As expected, at a
low temperature, carriers follow the degenerate statistics and
hence their velocity is limited by an appropriate average of the
Fermi velocity that is a function of carrier concentration.
Fig.4. Velocity versus doping concentration for T=4.2 K (liquid helium),
T = 77K (liquid nitrogen) and T=300 K (room temperature). The 4.2 K
curve is closer to the degenerate limit.
IV. P-TYPE NANOWIRE
Due to the large band gap between heavy hole and light hole in
the band structure E-K diagram, only high hole plays
effective role in the P-type silicon devices. By conversion of
completely random motion into a streamlined one with
ultimate velocity per holes equal to the intrinsic velocity
i
v
that depends on the carrier concentration and temperature
[9]. Hence the ultimate velocity of holes is ballistic
independent of scattering interactions. The maximum average
velocity per holes is now intrinsic velocity and is given by
i
v
as a function of temperature and doping concentration.
( )
0
*
2
v
B
i F
hh
N
k T
v F
m p
η
π
= ×
(21)
Where
v
N
is the effective density of states for the valance
band and
*
hh
m
is heavy hole effective mass. p is the carrier
concentration per unit length.
1
*
2
2
2
hh B
v
m k T
N
π
=
h
(22)
In high temperature the velocity saturates, but for high
concentration velocity only depends on doping concentration.
In other words similar to the electrons the ultimate drift
velocity is found to be appropriate thermal velocity for a given
dimensionality for non- degenerately doped nanostructure.
(
)
1
*
12
1
2
B
th th
hh
k T
v v
m
π
Γ
= =
Γ
(23)
However the same as the electrons ultimate drift velocity of
holes is the Fermi velocity for degenerately doped
nanostructures.
1 1
*
( )
4
m
hh
v p
m
π
=h
(24)
V. NANOWIRE TRANSISTOR
The velocity saturation effect can conveniently be
implemented in the modeling of a nanowire transistor if the
empirical relation given below is used.
MASAUM Journal of Basic and Applied Sciences Vol.1, No. 2 September 2009
237
0
1
D
c
E
v
E
E
µ
=
+
(25)
With (25) the drain current
D
I
as a function of the gate voltage
GS
V
and drain voltage
D
V
is obtained as
2
2
1
GT D D
D
D
c
V V V
IV
V
β
−
=
+
(26)
with
2
f G
C
L
µ
β
=
l
(27)
GT GS T
V V V
= −
(28)
sat
c
f
v
V L
µ
=
l
(29)
where
G
C
is the gate capacitance per unit length, and L the
effective channel length.
T
V
(=0.32) is the threshold voltage
and
f
µ
l
is the low-field Ohmic mobility that is related to the
mean free path
o
l
. The value of
2
500 / .
f
cm V s
µ
=
l
is
taken.
1
sat i
v v
=
is the intrinsic velocity of (20) in the absence
of quantum emission for
8 1
1
5 10
n m
−
= ×
.
Fig.5. The schematic of a nanowire transistor with gate dielectric.
Because of the unknown nature of the quantum emission, it is
ignored in this calculation. With the simple geometry of the
nanowire transistor of Fig. 5, the gate capacitance is given by
G ins Q
C C C
=
where
ins
C
, the capacitance of the gate
insulator, is given by :
2
2 2
ln 2
ins
ins
ins
C
R R
R
πε
=+
(30)
where R
ins
is the radius of the insulator (taken to be 0.5 nm)
and R (taken to be 0.5 nm) as shown above is the radius of the
wire itself.
ins
ε
is the permittivity of the gate insulator taken to
be 3.9 that of the oxide. The quantum capacitance
Q
C
arises
from the fact that the electron wave function vanishes at the
wire-insulator interface and peaks at the center of the wire to
be calculated according to the permittivity of the wire medium
which is taken to be silicon.
At the onset of current saturation, all carriers leave the
channel at the saturation velocity where electric field is
extremely high. Therefore, the saturation current is given by
( )
Dsat G GT Dsat sat
I C V V v
= −
(31)
When (26) and (31) at the onset of current saturation are
consolidated, the drain voltage at which the current saturates is
given by.
2
1 1
GT
Dsat c
c
V
V V V
= + −
(32)
With this value of
Dsat
V
substituted in (31), the saturation
current is given by
2
Dsat Dsat
I V
β
=
(33)
The design rule for nano-CMOS circuit requires the length
scaled inversely proportional to the low-field mobility and
width scaled proportional to the saturation velocity[10]. This
will bring the PMOS drain characteristic onto the same x-axis
as NMOS as a mirrored image as showed in Figure 6. The
steps in I-V characteristics tend to be equally spaced As
GT
V
becomes much larger than
c
V
. An analysis of the velocity
distribution in the channel indicates that the velocity
throughout the length of the channel is lower than that on the
drain end where carriers are moving with the maximum
permissible velocity. The equally-spaced step behavior of I-V
characteristics is in direct contrast to the behavior expected
from long channel transistors (
GT c
V V
<<
), when steps
separation varies quadratically with the adjusted gate voltage
GT
V
. In fact, if (26) is expanded to second order in
GT
V
, one
can easily notice the complete absence of pinch off effect. The
real strength of nanowire transistor lies in elimination of
optical phonon scattering, as suggested by Sakaki [11].
MASAUM Journal of Basic and Applied Sciences Vol.1, No. 2 September 2009
238
Fig.6. Current -Voltage characteristics of N-type (left) and P-type (right)
silicon nanowire transistor.
VI. CONCLUSIONS
An analytical model that captures the essence of physical
processes in a nanowire transistor is presented. The model
covers seamlessly the whole range of transport from drift-
diffusion to ballistic [2]. In the drift-diffusion regime, the low-
field transport is scattering-limited that is contained in the
mean free path
o
λ
. However, in the high-field domain, the
transport is ballistic with velocity limited to the Fermi velocity
in the degenerate regime and thermal velocity in the
nondegenerate regime. The length of the channel L replaces
o
λ
for a ballistic channel when
o
Lλ<
. It has been clarified
that the intrinsic speed of nanowire and other heterostructure
field-effect transistors (FETs) is governed by the transit time
of electrons (holes). Although the transist time is more
dependent on the saturation velocity than on the weak-field
mobility, the feature of high-electron mobility is beneficial in
the sense that the drift velocity is maintained always closer to
the saturation velocity, at least on the drain end of the
transistor where electric field is necessarily high and controls
the saturation current.. Moreover, it has been indicated that
the saturation velocity in these nanostructures is somewhat
higher than the thermal velocity due to heavy doping level
suitable for FET applications.
In many ways, the distribution function reported [1] is
similar to what is presented by Buttiker [13]. A series of
ballistic channels of length
λ
comprise a macro-channel of
length L. The behavior is well understandable as we consider
ballistic low-field mobility where
λ
is replaced by L.
However, it does not affect the velocity saturation in high
electric field that is always ballistic. The ends of each mean
free path thus can be considered Buttiker’s thermalizing virtual
probes, which can be used to describe transport in any regime.
In high electric field, the electrons (holes) are in a coordinated
relay race, each electron passing its velocity to the next
electron at each virtual probe. The saturation velocity is thus
always ballistic whether or not device length is smaller or
larger than the mean free path. The ballistic saturation
velocity is always independent of scattering-limited low-field
mobility that may be degraded by the gate electric field. The
relation between mobility and the mean free path has deep
consequences on the understanding of the transport in a
nanoscale device.
The importance of developing FET structures with high
electron (hole) concentrations and high current-drive
capabilities is emphasized. The current drive can be enhanced
by an array of nanowire transistors particularly for high speed
digital applications. The switching speed is usually
determined by the charging process of capacitive loads,
including interconnects. The presence of a tradeoff relation
between high mobility and high doping concentration is
required to optimize performance of the nanowire transistor
following the recommendations of Sakaki [12],[ 13]. The
channels can be designed for the benefit of utilizing selectively
doped double-heterojunctions and other FET nanostructures.
The role of the quantum engineering by the gate-field induced
mobility degradation in determining low-field electron
mobility is indicated. Also, the role of quantum waves in
determining the correct capacitance to make quantitative
interpretations of the gate capacitance is required. Although
the technical difficulty encountered in the preparation of a
nanowire or nanodot transistor is far greater than that for 2D
nanostructures, there appear to be several bright prospects if
one develops novel schemes of microfabrications, including
the use of edge of quantum-well nanostructures as the theory
guides the experimenters toward the design of nanowire
transistors and allows them to assess performance accordingly.
It is, therefore, hoped that the present work will encourage
experimenters to correctly interpret the data while assessing
performance metrics of a given nanostructure or an integrated
circuit made of these nanostrutures.
ACKNOWLEDGEMENT
The authors would like to thank Malaysian Ministry of
Science, Technology and Industry (MOSTI) for a research
grant for support of postgraduate students.
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Mohammad Taghi Ahmadi is a lecturer in the
Applied Science Universiti (Takab, West Zarbaijan,
Iran). He received his B.S in 1997 and MSc. in 2006.
He has submitted his Ph.D. thesis on 6
th
July 2009. He
is currently Post Doctoral fellow in the faculty of
Electrical Engineering, Universiti Teknologi Malaysia.
His main research interests are in nanoscale non-
classical device simulation, modeling and
characterization. His research resulted in a number of
publications in high-impact journals for which
he has been recognized by the UTM. His present
research interests are in nanowires , carbon nanotube
and graphite nanoribbon transistors. He is an IEEE
Student Member.
Amir Hosssein Fallahpour was born in Isfahan, Iran,
in 1981. He received the B.Sc degree in electrical
engineering, Isfahan, in 2006. He is currently pursuing
his M.Sc degree in the faculty of Electrical
Engineering, Universiti Teknologi Malaysia. His
research interests are modeling and simulation of
silicon nanowire transistor. He is currently a member of
Computational Nanoelectronics (CoNE) Research
Group in UTM university.
Javad Allahdadian was born in Isfahan, Iran, in 1981.
He received the B.Sc degree in electrical engineering,
Isfahan, in 2007. He is currently pursuing his M.Sc
degree in the faculty of Electrical Engineering,
Universiti Teknologi Malaysia. His research interests are
power system optimization and working on high voltage
Nanoscale insulators.
Mojgan Kouhnavard, was born in Iran in
1984,successfully completed bachelor’s degree program
in solid state physics, at Islamic Azad University, Ghom
Branch ,2007. she is doing her master in physics(mix
mode) in Universiti Teknologi Malaysia. Studying
nanotechnology, physics electronic are Mojgan ‘s
Research interests.
Razali Ismail received the B.Sc. and M.Sc. degrees in
Electrical and Electronic Engineering from the
University of Nottingham, Nottingham, U.K., and the
Ph.D. degree from Cambridge University, Cambridge,
U.K., in 1989. In 1984, he joined the faculty of
Electrical Engineering, Universiti Teknologi Malaysia as a lecturer in
Electrical and Electronic Engineering. He has held various faculty positions
including as head of department and chief editor of the university journal. In
1985, he joined the Department of Electronics and Computer Sciencce,
University of Southampton, Southampton, U.K. where he begins his research
work. In 1987, he continue his research work at the Department of
Engineering, Cambridge University, Cambridge, U.K. where he completed
his Ph.D. degree in microelectronics. His main research interest is in the field
of microelectronics which includes the Modeling and Simulation of IC
Fabrication Process, Modeling of Semiconductor Devices and recently in the
emerging area of Nanoelectronics. He has worked for more than 20 years on
the modeling and simulation of semiconductor devices and has published
various articles on the subject. He is presently with the Universiti Teknologi
Malaysia as an associate professor in Semiconductor Physics and Devices.
