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Ballistic Saturation Velocity Modelling Beyond Ohm’s Law
Ismail Saad
1
, Nurmin Bolong
1
, Khairul A. Muhamad.
1
, Abu Bakar A. Rahman
1
and Vijay K. Arora
2
1
Nano Engineering & Material (NEMs) Research Group, School of Engineering & IT, Universiti Malaysia Sabah,
88999, Kota Kinabalu, Sabah
2
Division of Engineering and Physics, Wilkes University, WilkesBarre, PA 18766, U. S. A.
ismail_s@ums.edu.my
/ ismailsaad07@gmail.com
Abstract— The ballistic saturation velocity models were
presented and its impact towards the failure of Ohm’s law was
evaluated on nanocircuits design. The ballistic transport is
predicted in the presence of high electric field and its behavior
is characterized by an onset of the critical voltage, V
c
. 5μ
μ
m
resistor shows deviation from Ohm’s law for relatively low
voltages above V
c
= 1.9 V. When applied to the voltage division
and current division circuits, the lowerlength resistors are
found to have higher resistance as compared to higherlength
resistor even if their ohmic values are the same. Consequently,
the power consumption will not only be lower, but also will
have a linear behavior that affects the figure of merit with
tradeoff between frequency and power in nanocircuits. The
results presented can have profound effect on characterization
and performance evaluation of nanocircuits being considered
for various applications.
KeywordsOhm’s law, Ballistic transport, Velocity saturation,
Nanocircuits
I.
I
NTRODUCTION
As the quest for highspeed devices and circuits for
Ultra Large Scale Integration (ULSI) is continuing [1
4], accurate interpretation of circuit behavior is
essentially needed. The analysis of electronic devices
and circuits relies on the validity of Ohm’s law. This
paradigm is based on the velocity response to an
applied electric field that is linear:
E
0
μν
= (1)
where μ
ο
is the ohmic mobility that describes the ease
with which itinerant carriers are able to roam through
a given device by undergoing the accelerating process
of an electric field, E. Figure 1 shows a typical
experimental setup in which a resistor of length L and
area of crosssection A (A=Wxd) is stimulated by a
voltage V applied across its length. The response is the
current I through the resistor. For a resistor with
electron concentration n and drift velocity v, the
current is I = n q v A, where q is the electronic charge.
The linear currentvoltage (IV) characteristics, with
the application of Eq. (1), are given by:
o
R
V
I =
(2)
where R
o
is the ohmic resistance that depends on the
dimensions of the resistor and is given by:
os
o
1L L L
R
nq A A W
ρρ
μ
===
(3)
where
o
1
nq
ρ
μ
=
is the resistivity (Ω−m) and
s
d
ρ
ρ
=
(Ω/□) the sheet resistivity
.
Fig.1. Typical experiment setup for IV characteristics of Micro
and Nanolength diffused channel
Ohm’s law, given by Eq. (2), has been the basis on
which electronic circuits are designed, characterized,
and their performance evaluated. It worked very well
for macro resistors (L >100 μm). However, Ohm’s law
does not hold for micro/nanoresistors and circuits
when L is few nanometers and electric field, E are
high in scaleddown dimensions. As development of
the devices to nanoscale dimensions continued it
became clear that the saturation velocity plays a
predominant role [57]. The higher mobility brings an
electron closer to saturation as a high electric field is
encountered, but saturation velocity remaining the
same no matter what the mobility. The ballistic
transport [812], was predicted in the presence of a
high electric field. In equilibrium, the band diagram is
flat and randomly oriented velocity vectors cancel
each other. As the applied electric field tilts the band
2012 Fourth International Conference on Computational Intelligence, Modelling and Simulation
21668531/12 $26.00 © 2012 IEEE
DOI 10.1109/CIMSim.2012.72
416
2012 Fourth International Conference on Computational Intelligence, Modelling and Simulation
21668531/12 $26.00 © 2012 IEEE
DOI 10.1109/CIMSim.2012.72
422
diagram, an electron traveling in the direction of
electric field finds it difficult to surmount the barrier.
An electron traveling in the opposite direction
accelerates in a mean free path and collides,
randomizing its velocity and restarting its journey for
another mean free path. The theory of ballistic
saturation velocity models and its application into non
ohmic circuit behavior is described and its impact
towards series, parallel and CMOS design were
evaluated in detailed.
II.
BALLISTIC SATURATION VELOCITY
Velocity response to the electric field results in
velocity saturation in a high electric field. The current
in a resistive (channel) is limited by this saturation
value I
sat3
= n q v
sat
A (3D), I
sat2
= n
s
q v
sat
W (2D), I
sat1
= n
l
q v
sat
(1D) which in turn depends on the doping
concentration n
d
(d=1, 2, 3) and the saturation velocity
v
sat
. It is, therefore, essential to assess the magnitude of
this saturation velocity that results in the current
saturation. The ballistic (B) transport [812], was
predicted in the presence of a high electric field. In
equilibrium, the band diagram is flat and randomly
oriented velocity vectors cancel each other. As the
applied electric field tilts the band diagram (fig 2), an
electron traveling in the direction of electric field finds
it difficult to surmount the barrier. An electron
traveling in the opposite direction accelerates in a
mean free path and collides, randomizing its velocity
and restarting its journey for another mean free path.
The net result is that the random vectors
i
v
r
streamline
in a very high electric field. In the presence of a very
high electric field, all electrons are streamlined
opposite to the direction of the applied electric field.
Fig. 2 Partial streamlining of random motion of the drifting
electrons on a tilted energy band diagram in an electricfield.
The ultimate unidirectional drift velocity is the
saturation velocity that is the average of its absolute
value
*
2 mEv
k
=
, where E
k
is the kinetic energy for a
given dimensionality and m* is the carrier effective
mass. When this averaging is taken by including the
FermiDirac distribution and density of states as a
weight for a given dimensionality, the intrinsic B
velocity for a semiconductor is obtained as
(
)
()
Fdd
Fdd
thdid
vv
η
η
2
2
2
1
−
−
ℑ
ℑ
=
(4)
with
⎟
⎠
⎞
⎜
⎝
⎛
Γ
⎟
⎠
⎞
⎜
⎝
⎛
+
Γ
=
2
2
1
d
d
vv
ththd
(5)
B
th
2k T
v
m*
=
(6)
Fig. 3 Relative intrinsic velocity versus normalized carrier density.
Figure 3 gives a plot of intrinsic B velocity
normalized to the thermal velocity of equation (3). In
the nondegenerate regime, this ratio is
128.1/2 =
π
for a 3D semiconductor, it drops to
886.02/ =
π
for
a 2D nanostructure and further drops to
π
/1
=0.564
for a 1D nanostructure.
10
2
10
1
10
0
10
1
0.5
1
1.5
2
2.5
3
3.5
4
4.5
(n/N)
d
(v
i
/v
th
)
d
3D
2D
1D
417423
III.
NANO

CIRCUITS BEYOND OHM
’
S LAW
The current I in a nanoresistor as shown in fig. 1,
as a function of applied voltage V is given by [11]
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
co
c
c
sat
V
V
R
V
V
V
II tanhtanh
(7)
where
t
c
o
V
VL
=
l
is the critical voltage for the onset of
nonohmic behavior, where V
t
is the thermal voltage
and
o
l is the ohmic mean free path. This critical
voltage is shown to be 2.6 kV for a macroresistor of
length L = 1.0 cm, reducing to a value of 0.26 V for a
nanoresistor of L = 1000 nm.
qTkV
Bt
= = 26 mV
is the thermal voltage at room temperature and a
typical ohmic mean free path is
nm
o
100=l . In fact,
the quantum emission in a high electric field may
lower the value of critical voltage by
)/tanh( Tk
Bo
ω
h where
o
ω
h is the energy of a
quantum. Greenberg and De Alamo [13] measured IV
characteristics of InGaAs heterojunction field effect
transitor (HFET) structure. Their results indicate a
direct experimental verification of Eq. (4) (See Fig. 4).
Their [13] experimental results fit very well the
empirical relation:
c
sat
11
o
cc
V
V
V1
II
R
VV
11
VV
γγ
γγ
==
⎡⎤⎡⎤
⎛⎞ ⎛⎞
⎢⎥⎢⎥
++
⎜⎟ ⎜⎟
⎢⎥⎢⎥
⎝⎠ ⎝⎠
⎣⎦⎣⎦
(8)
The normalized IV characteristics given by Eq. (7)
and (8) are compared in Fig. 4. Fig.4 shows results for
a resistor of ohmic value R
o
= 16.8 Ω. The critical
electric field for the onset of nonohmic behavior is
E
c
= 3.8 kV/cm. The saturation current is I
Wsat
= 565
mA/mm of the width W and
γ
=2.8. V
c
= 0.38 V for a
1μm resistor. An ohmic mean free path
o
l
= 0.7 μm
is extracted from these measurements [11]. The
saturation current
WII
Wsatsat
= depends linearly on
the width of the resistor and V
c
is a linear function of
the length of the resistor.
Fig. 4. Theoretical and experimental IV characteristics of a micro
resistor. Solid curve is from theory of Eq. (7). Dotted curve is from
empirical Eq. (8) with γ = 2.8. Ohmic and saturation current curves
shown intersect at V = V
c
.
When plotted on a scale extending up to 10 V as
shown in figure 5, 5μm resistors clearly shows
deviation from Ohm’s law for relatively low voltages
above V
c
= 1.9 V. For 20μm resistor, V
c
= 7.6 V and
for L = 80 μm resistor V
c
= 30.4 V. On a scale of 10 V,
80μm resistor appears to follow Ohm’s law.
However if voltage is extended beyond 30.4 V,
nonlinear behavior become apparent. V = V
c
marks a
transition from ohmic to nonohmic behavior. All
three curves approach saturation. 5μm resistor
approaches the saturation limit faster than higher
length resistors as the voltage is increased.
Fig. 5. IV characteristics of three resistors with length L = 5, 20 ,
80 μm. W/L = 40 and ohmic resistance R
o
= 16.8 Ω is the same for
all three resistors. I
sat
is 113 mA, 452 mA, and 1808 mA
respectively for the three resistors.
0 1 2 3 4 5
0
0.2
0.4
0.6
0.8
1.0
I/I
sat
V/V
c
Theoretical
Empirical
Saturation
Ohmic
0 2 4 6 8 10
0
100
200
300
400
500
600
CURRENT I (mA)
POTENTIAL V (V)
W/L = 200/5 (Th)
W/L = 200/5 (Emp)
W/L = 800/20 (Th)
W/L = 800/20 (Emp)
W/L = 3200/80 (Th)
W/L = 3200/80 (Emp)
418424
IV. N
ON

OHMIC CIRCUIT BEHAVIOUR
As currentvoltage (IV) characteristics become
nonlinear and resistance no more a constant, it is
natural that familiar voltage divider and current
divider rules may not apply. As the length of a resistor
plays predominant role in transforming IV and
resistive behavior, it is worthwhile to see how the
voltage will be divided between two resistors having
the same ohmic resistance (R
o
= 33.6 Ω, but differing
lengths (say 5 μm and 10 μm). Equation (3) shows
that the ohmic resistance is a function of W/L
provided the diffusion depth d (or sheet resistivity) is
the same. When connected in series, as in Fig. 6, an
applied voltage V will get equally divided across each
resistor according to the voltage division dictated by
Ohm’s law. However, when Eq. (7) is used in place of
Ohm’s law (I=V/R
o
), the voltage V
1
across R
1
is
obtained from
11
c1 c2
c1 c2
VVV
V tanh V tanh
VV
⎛⎞ ⎛ ⎞
−
=
⎜⎟ ⎜ ⎟
⎝⎠ ⎝ ⎠
(8)
with V
c1
= 1.9 V for the 5μm resistor, V
c2
= 3.8 V for
the 10μm resistor. The voltages V
1
and V
2
appearing
across each resistor are shown in Fig. 7. Also, shown
are the results as expected from Ohm’s law.
The 5μm is more resistive as the voltage across the
combination is increased and hence gets more voltage
across it. On the other hand, the voltage drop across
10μm resistor is less than its ohmic value.
Fig. 6. Voltage divider circuit with two microresistors (R
o1
=R
o2
)
and same W/L=20 ratio
Fig. 7. Voltage division in two microresistors with the same
ohmic value (R
o1
=R
o2
) and same W/L=20 ratio
Figure 8 shows a current divider circuit where same
two resistors (R
o1
= R
o2
= 33.6 Ω) are considered. With
the current per unit length of 565 mA/mm, the
saturation current for the resistor 1 is I
sat1
= 56.5 mA
and that for resistor 2 is I
sat2
= 113 mA. As before, the
critical voltages V
c1
= 1.9 V and V
c2
= 3.8 V. Figure 8
shows that the resulting current in each resistor is
substantially below its ohmic value. Only when V<V
c
,
validity of Ohm’s law can be assumed. As V
increases beyond V
c
, the maximum current that can be
drawn from the voltage source is 170 mA as compared
to 595 mA at V = 10 V predicted by the Ohm’s law.
When two parallel channels are conducting, the more
charge will flow in channel through the higher length
resistor even if both resistors have the same ohmic
value as can be observed in figure 9.
Fig. 8. Current divider circuit with two microresistors (R
o1
=R
o2
)
and same W/L=20
0 2 4 6 8 10
0
1
2
3
4
5
6
7
8
9
10
V
1
OR V
2
V
Ohmic
V
1
(Nonohmic)
V
2
(Nonohmic)
419425
Fig. 9. Ohmic and nonohmic currents in a current divider circuit
with two microresistors (R
o1
=R
o2
)
Fig. 10. Relative power consumption in the ohmic and nonohmic
regime
It is now natural to assume that the power
consumption principle P =V I will also be different
when IV characteristics are considered. With Eq. (4)
considered, the power law becomes
c
sat
oc c
VV
VV
P VI tanh VI tanh
RV V
⎛⎞ ⎛⎞
== =
⎜⎟ ⎜⎟
⎝⎠ ⎝⎠
(9)
Power consumption is
oo
RVP
2
= in the ohmic
regime (V<V
c
), but
oc
RVVP = in the regime where
V>>V
c
. In a way, the power consumption will not
only be lower, but also will have a linear behavior, a
welcome reprieve for energy consumption in
integrated circuits as indicated in fig. 10. The power P
consume in a circuit as compared to its ohmic value is
given by
)/(
)/tanh(
c
c
o
VV
VV
P
P
=
(10)
This transformed behavior affects the figure of
merit with tradeoff between frequency and power.
V.
CONCLUSIONS
New paradigm that evaluated the impact of the
failure of Ohm’s law on nanocircuits is presented. Its
analysis was based on ballistic saturation velocity
transport that was predicted in the presence of a high
electric field. When applied to the voltage division and
current division circuits, the lowerlength resistors are
found to have higher resistance as compared to higher
length resistor even if their ohmic values are the same.
Consequently, the power consumption will not only be
lower, but also will have a linear behavior that affects
the figure of merit with tradeoff between frequency
and power in nanocircuits. The results presented can
have profound effect on characterization and
performance evaluation of nanocircuits being
considered for various applications.
A
CKNOWLEDGEMENT
The authors would like to acknowledge the financial
support from FRGS (FRG0248TK2/2010) and
ERGS funds (ERGS0002TK1/2011) of Minister of
Higher Education Malaysia (MOHE). The author is
thankful to the Universiti Malaysia Sabah (UMS) for
providing excellent research environment in which to
complete this work.
R
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