ICSE2010 Proc. 2010, Melaka, Malaysia

Ballistic Transport in Nanowires and Carbon

Nanotubes

Vijay K. Arora, Senior Member, IEEE Division of Engineering and Physics, Wilkes University, Wilkes-Barre, PA 18766, U. S. A.

Ibnu Sina Institute of Fundamental Sciences, Universiti Teknologi Malaysia, UTM Skudai 81310

E-Mail: vijay.arora@wilkes.edu

Abstract-The charge carriers in nanowires (NWs), carbon

nanotubes (CNTs), and those confined to a very high magnetic

field have one-dimensional (1D) character as quasi-free

propagation of electron waves with analog energy spectrum exists

only in one direction. The energy spectrum is quantum (or digital)

in other two cartesian directions where electron waves are

standing waves. In the quantum limit, an electron (hole) occupies

the lowest (highest) digitized/quantized state giving it a distinct 1D

character. The energy | |FE v k= in carbon-based devices is

linearly dependent on the wave vector k, where 610 /Fv m s≈ .

This is in direct contrast to parabolic character 2 2

/ 2 *E k m= h in

solids with effective mass m*, for example in silicon NWs. The

probability of changing wavevector from +ve to –ve direction

through scattering or vice versa is greatly reduced and hence high

mobility is expected, especially at low temperatures. The crucial

outcome of this paper is the answer to the question: Does a higher

mobility leads to a higher ultimate saturation velocity? The

distribution function in a high electric field E is then naturally

asymmetrical affected by the energy q± lE absorbed or emitted by

a carrier of charge q during its ballistic flight in a mean free

path l . The ultimate drift in response to a high electric field

results in unidirectional streaming of the otherwise randomly-

oriented velocity vectors in equilibrium. The high-field drift

limited by the intrinsic velocity is ballistic, unaffected by

scattering-limited processes. The ultimate velocity is further

limited to an emission of a quantum either in the form of an

optical phonon or a photon by an electron excited to a higher state

by the applied electric field. The velocity does not depend on

scattering parameters. Ballistic processes as a result of reduction

in length of a CNT or NW below the scattering-limited mean free

path l in the quasi-free direction are also discussed.

I. INTRODUCTION

Nanowires (NWs) and carbon nanotubes (CNTs) are quasi-

one-dimensional (Q1D) nanostructures where electrons are

confined in say y-z plane while x-direction is quasi-free.

Electron waves are standing waves in the y-z plane and

propagating waves in the x-direction. In equilibrium the

velocity vectors are randomly oriented giving zero net drift of

the carriers. However, in response to an applied electric field,

the electron velocity vectors tend to realign opposite to the

electric field. This unidirectional streaming of the carriers

gives drift velocity Dv proportional ( D ov µ= E ) to a low-

magnitude electric field E ( /c sat ov µ< =E E ), where satv is

the saturation velocity in an extremely large electric field when

carries are in extreme nonequilibrium. The saturation of the

drift velocity arises because of the intrinsic velocity vectors

being realigned parallel to an applied electric field for holes

and antiparallel for electrons [1]. The magnitude of the intrinsic

velocity is the ultimate drift velocity causing saturation. The

emission of a quantum in the form of a phonon or a photon

may further lower the saturation velocity depending on the

energy of the quantum involved.

The energy spectrum 2 2 / 2 *kE k m= h as a function of

momentum vector k is parabolic in the effective mass

approximation that is valid for most semiconducting materials.

However, for carbon-based graphene devices the energy

spectrum | |k FE v k= is linear. Density of states is different

in two configurations and so are the ultimate drift velocities

that depend on the temperature and carrier concentration. In

the following, using the nonequilibrium function developed by

Arora [2], drift response to a high electric field is delineated

including the effect on mobility due to limited length of the

channel.

II. ELECTRONS IN A MAGNETIC FIELD

Carriers, for example electrons, in a magnetic field [3-4]

offer many similarities to nanowires and other Q1D

configurations explored. An electron travelling at an angle to

an applied magnetic field makes a spiral motion with the axis

of the spiral in the direction of the magnetic field, as shown in

Fig. 1. Only the velocity component perpendicular to the

magnetic field is affected by the magnetic field Br

. The

component parallel to the magnetic field is unaffected. The

velocity */x x ev k m= h is therefore analog type. However, the

perpendicular component follows the periodic circular motion

that is quantum in character when the radius of the orbit is

comparable to electron’s de Broglie wavelength.

The magnetic force qv B×rr

provides the centripetal force

in the circular orbit. The magnetic force exists due to the

perpendicular-to-the-magnetic-field component v⊥ of the

velocity vector vr

and is given by

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ICSE2010 Proc. 2010, Melaka, Malaysia

* 2

*

e

e

m v qRBqv B v

R m

⊥⊥ ⊥= ⇒ = (1)

Fig. 1. (a) The velocity vector in a magnetic field applied in the x-direction

being decomposed into components parallel and perpendicular to the magnetic

field. (b) Perpendicular motion is affected by the magnetic field resulting in a

circular orbit, the circle being carried away by parallel component resulting in

a spiral.

Each orbit contains only the integer number of de Broglie

waves for sustained and stable orbits. This resonance condition

for constructive superposition of quantum waves in an orbit

yields

* *

2 n

e e n

hR n v n

m v m Rπ ⊥

⊥

= ⇒ =h

(2)

where n is an integer (n=1, 2, 3,…). Elimination of v⊥ from (1)

and (2) yields for the quantized orbit the radius

1/ 21/2

1n

nR n R

qB

= =

h,

1/ 2

1RqB

=

h (3)

The quantized kinetic energy nE⊥ is given by

* 2

*

1,

2n e n c c

e

qBE m v n

mω ω⊥ ⊥= = = (4)

Here cω is known as the cyclotron frequency of the orbit. In

addition to the kinetic energy, the harmonic oscillations at

either y or z axis give potential energy (known as zero point

energy / 2cωh ) in addition to unperturbed kinetic energy

parallel to the magnetic field). The complete energy spectrum

in the quantum limit (n=1) with respect to the conduction band

edge is given by

2 2

*2

x

xk c

e

kE E

m= +

h (5)

with

1

2C co cE E ω= + h (6)

III. QUANTUM NANOWIRES

A Q1D quantum well or nanowire [5-7] emerges as two (y-

and z-direction) are squeezed so that lengths andy z DL L λ< ,

the de Broglie wavelength while the remaining one (x) remains

classical (analog) with x DL λ>> . Two prototype NWs, one

with rectangular cross-section and other with circular cross-

section, are shown in Fig. 2. In this Q1D quantum well, analog

type levels appear only in one dimension while the other two

go quantum [7]:

2 2

2 *

xk c

e

kE E

m= +

h (7)

with 2 2 2 2

* 2 * 22 32 2

c co

y z

E Em L m L

π π= + +

h h(rectangular NW) (8)

2 2201 * 22

c coE Em R

πα= +

h (circular NW) (9)

Here Ec is the elevated conduction band edge in the quantum

limit of a nanowire.

Fig. 2. A prototype nanowire with y,z DL λ<< and x DL λ>> for rectangular

cross-section and DR λ<< and L>> Dλ for circular cross-section.

Ec >Eco is higher than Eco of the bulk conduction band edge

by the zero-point energy in the y, z-plane. Eqs. (8) and (9) are

valid in the quantum limit where only the lowest quantized

state is occupied. kx is the analog-type continuous momentum

wave vector in the x-direction. Its value can be any value

between −∞ to +∞ . Due to the anisotropic nature of the

effective mass in silicon, *1,2,3m is derived from the longitudinal

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ICSE2010 Proc. 2010, Melaka, Malaysia

( * 0.98 om m=l ) and transverse ( * 0.19t om m= ) masses in the

ellipsoidal valley model of silicon, for example. 01 2.405α =

is the first zero of the Bessel function of order zero, i.

e., 01( ) 0oJ α = . The quantum limit is assured by the

condition DR λ<< .

IV. CARBON NANOTUBES

Carbon nanotubes (CNTs), originally discovered by Iijima

[8], have opened a new vista for design of compact

nanostructures and their applications. It is now well known that

a CNT is rolled-up cylinder of graphene, a one-atom-thick

allotrope of carbon, whose properties depend on chirality of the

roll-up [9-10]. The unzipping of a CNT in to grapheme has

been recently demonstrated by a number of labs. The

fabrication of single- and multiple-wall nanotubes is now

possible with a wide variety of procedures.

Fig. 3. A prototype single wall carbon nanotube with length DL λ>> , the de

Broglie wavelength and Dd 2R λ= << .

The carrier mobility of the CNT nanostructure is not

affected by processing and roughness scattering due to the

chemical stability and perfection 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).

Outer wall in a double-wall CNT can also be designed as a gate

that will control the charge on the inner CNT and hence the

flexibility of making it n- or p-type. It is no surprise that the

CNTs are being explored as viable candidates for high-speed

applications.

Nonparabolic energy E dependence on wavevector k and

the diameter d of the tube is given by [11-13]

23

( ) 12 2

gE dk

E k

= +

(10)

with Eg the bandgap between the conduction band and valence

band. The diameter of the tube is related to the bandgap

through

2cc

g

a tE

d= (11)

where acc=0.142 nm is the length of the carbon-to-carbon bond

and t= 2.7 eV is carbon-to-carbon binding energy leading to

0.8 / ( )g

E eV d nm= . The diameter is related to chirality (n, m)

by the relation [13]

2 23 ccad n m nm

π= + + (12)

With chirality such that n-m is a multiple of 3 times an integer,

the CNT is metallic. Otherwise, the CNT is semiconducting.

In a metallic CNT (with circumferential wave vector zero), the

energy dependence on the energy is a linear function of k with

FE v k= h , where 610 /

Fv m s≈ . The CNT then behaves as a

relativistic particle of rest mass zero, with the velocity 610 /

Fv m s≈ akin to the velocity of a photon [11, 13]. In the

other extreme (kd<<1), (10) can be transformed to a parabolic

form 2 2 / 2 *E k m= h with m*/mo=0.08 nm/d(nm). In this

extreme, the effective mass depends on the chirality.

V. DRIFT RESPONSE TO ELECTRIC FIELD IN A NW

Since electrons are confined to move only in x-direction,

transport behavior of electrons in NWs is perhaps easy to

understand. An electron, in traversing a mean free path l ,

finds the electrochemical potential (the Fermi energy) changed

by an amount q lE . This change in Fermi energy results in a

distribution function which has forward-backward asymmetry

in the direction of an electric field and is given by

1( )

1

k F

B

kE E q

k T

f E

e

− +=

+

r rlE.

(13)

This distribution has simpler interpretation as given in the tilted

band diagram of Fig. 4. A channel of CNT can be thought as a

series of ballistic resistors each of length , where the ends of

each free path can be considered as virtual contacts with

different quasi Fermi levels separated in energy by qE . 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 at the source and drain contacts. This behavior is

understandable if we consider the widely diffused

interpretation of inelastic scattering represented by the

Büttiker’s virtual thermalizing probes[14] , which can be used

to describe transport in any regime. Within this approach,

carriers are injected into a “virtual” reservoir where these are

thermalized and start their ballistic journey for the next free

path. The carriers starting from the left at Fermi potential EF

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ICSE2010 Proc. 2010, Melaka, Malaysia

complete the free-path voyage with FE q− lE . Those starting

from right complete the voyage with the electrochemical

potential FE q+ lE . These are the two quasi Fermi levels.

The current flow is due to the gradient of the Fermi energy

( )F

E x in the presence of an electric field. Because of this

asymmetry in the distribution of electrons, the electrons in Fig.

4 tend to drift opposite to the electric field rE applied in the

negative x-direction (right to left).

Fig. 4. Partial streaming of electron motion on a tilted band diagram in an

electric-field.

Fig. 5. Conversion of random velocity vectors to the streamlined one in an

infinite electric field.

In an extremely large electric field, virtually all the

electrons are traveling in the positive x-direction (opposite to

the electric field), as shown in Fig. 5. This is what is meant by

conversion of otherwise completely random motion into a

streamlined one with ultimate velocity per electron equal to iv .

Hence the ultimate velocity is ballistic independent of

scattering interactions. This interpretation is consistent with

the laws of quantum mechanics 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.

The ballistic motion in a free path may be interrupted by

the onset of a quantum emission of energy 0ω . 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 ∞l (zero-field

mean free path in a long sample with L>> ∞l ) by an expression

[15]

[1 ] [1 ]

Q QE

qe e∞ ∞

− −

∞ ∞= − = −

l

l ll l l

E (14)

with

0 0( 1)Q Qq E N ω= = +l hE (15)

00

1

1Bk T

N

e

ω=

−

h (16)

Here 0( 1)N + gives the probability of a quantum emission. No

is the Bose-Einstein distribution function. The degraded mean

free path is now smaller than the low-field mean free path

∞l . In the low-field ohmic limit ∞≈l l as expected. In the

high electric field limit, Q≈l l . The inelastic scattering length

during which a quantum is emitted is given by

Q

Q

E

q=lE

(17)

Obviously Q = ∞l in zero electric field and will not modify

the traditional scattering described by mean free path ∞l as

Q ∞>>l l . The low-field mobility and associated drift motion

is therefore scattering-limited. The effect of all possible

scattering interactions in the ohmic limit is buried in the mean

free path ∞l . The nature of the quantum emitted depends on

the experimental set-up and the presence of external

stimulations and 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 a transition to a higher

quantum state with subsequent emission to the lower state, the

quantum energy 1 2o Eω ⊥ −= ∆h is a function of confinement

dimensions.

The distribution function of (13) can be simplified for an

electric field cE E> with /c tV= lE , where /t BV k T q= is the

thermal voltage. In this regime, the number of itinerant

electrons in each direction is proportional to e± cEE /

(+ sign is

for antiparallel and – sign is for parallel direction). The

number of antiparallel electrons overwhelms owing to the

rising exponential and the number of those in parallel direction

decline to virtually nonexistent owing to the decaying

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ICSE2010 Proc. 2010, Melaka, Malaysia

exponential. The net fraction antF of the streamlined electrons

parallel to the electric field is given by /

/tanh( / )

c

cant c

e eF

e e

−= =

+

-

-

c

c

E/E E E

E/E E EE E (18)

For linear transport, the distribution function obtained from the

Boltzmann transport equation has been extremely successful in

predicting mobility behavior. Thornber [16] did a study of

scattering rate and momentum scaling transformations of the

distribution function. It was determined that the saturation

velocity is invariant under scaling of the magnitude of

scattering rates, which alters mobility, while mobility is

invariant under scaling of the magnitude of momentum, which

alters saturation velocity. The mobility and saturation velocity

are thus independent parameters in velocity-field profiles of

heterostructures. Consistent with the observations of Thornber,

the electric field tends to organize in its direction the otherwise

completely random motion of the electrons that is invariant to

scattering parameters. The electric dipole moments ql due to

the quasi-free electron motion in a mean free path l between

two collisions tend to organize in the direction antiparallel to

the applied field. High field effects thus become important

when the dipole energy is comparable to the thermal energy

(nondegenerate samples) or to the Fermi energy (degenerate

samples). At each collision, the randomness is re-established.

If the electric field is strong, this randomness is destroyed, and

an electron exhibits a quasi-ballistic behavior appropriate to the

thermal motion of the electron.

The above arguments give for drift response to an electric

field an expression

tanh( / ) tanh( / )D sat B sat cv v q k T v= =lE E E (19)

with

tanh( / )sat i Q Bv v E k T≈ (20)

/c tV= lE , /t BV k T q= (21)

The intrinsic velocity iv is the magnitude of the velocity vector

given by

11

1

( )ci th o

Nv v

nη= ℑ (22)

with

1

1=th thv v

π

2=

*

Bth

k Tv

m

(23)

( )0

1( )

( 1) exp 1

i

i

xdx

i xη

η

∞

ℑ =Γ + − +∫ (24)

= F c

B

E E

k Tη

− (25)

( )1/ 2

*1 2 / 2c BN m k T π= h (26)

n1 is the linear carrier concentration per unit length.

( ) ln[1 exp( )]o η ηℑ = + can be obtained in a closed form. The

reduced Fermi energy is obtainable from

1 1 1/2 ( )cn N η−= ℑ (27)

The linear carrier concentration in a magnetic field is 2

1 3 1n n Rπ= × , where 3n is the bulk carrier concentration per

unit volume. The imposition of a magnetic field in a bulk

semiconductor is an excellent way to study 1D transport as

cross-sectional area 2

1Rπ can be changed by changing the

magnitude of the applied magnetic field.

Fig. 6 gives the normalized plot of d satv v versus cE E .

Also, shown is the plot from the empirical equation that has

been tested in a number of experiments. This empirical

equation is

( )1

1

1 /

d sat

c

v v

γ γ

=

+ E E

(28)

where γ is a parameter. A wide variety of parameters are

quoted in the published literature. Greenberg and Del Alamo

[17] give convincing evidence from measurements on a 5-µm

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 that is impossible to be present in any functional

device. Normally the highest measured drift velocity is

ascribed to be the saturation velocity which is always lower

than the actual saturation velocity. γ =1 is found convenient in

modeling a transistor. As seen from Fig. 6, the value of γ

does not affect the extreme behavior in low- or high-field. The

plots differ only at the intermediate values of the electric field.

As electrons are drifting with velocity closer to saturation

velocity in a nanowire transistor, the error in using γ =1 is

negligible.

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

v/v

sa

t

V/Vc

Theory

Emp (γγγγ = 2.8)

Emp (γγγγ = 2.0)

Emp (γγγγ = 1.0)

Linear

Saturation

Fig. 6 Normalized velocity-field characteristics for a nanowire as predicted

from the theory and compared with empirical models. The γ=2.8 curve is closer

to theoretical limit.

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ICSE2010 Proc. 2010, Melaka, Malaysia

VI. DRIFT RESPONSE TO ELECTRIC FIELD FOR A CNT

The density of state function for a CNT is given by [13]

( )2

2

( )2

/ 2

g

o

g

EED E D E

E E

= >

−

(29)

8

3o

cc

Da tπ

= (30)

The intrinsic velocity vi is the weighted average of

(1/ )( / )v dE dk= h with product of the density of states and the

Fermi-Dirac distribution function as the statistical weight. The

resulting velocity following the steps delineated in Refs. [1]

and [18] is found to be

0. ( ) / ( / )

i CNT C TF Nv n Nv η= ℑ (31)

with 3 / 4F g

v d E= h . CNT

n is the carrier concentration per unit

length. The effective density of states isCNT o B

N D k T= . The

Fermi-Dirac integral of order zero ( ) ln(1 )o

eηηℑ = + is

reducible in a closed form. The reduced Fermi energy

( ) /F c B

E E k Tη = − with / 2c g

E E= is obtained from

x η20

x ( / 2 ) 1dx

e 1x x( / )

g B

CNT CNT

g B

E k Tn N

E k T

∞

−

+ =

+ +∫ (32)

In strong degeneracy limit, (32) simplifies to 1/ 2

2

1/22 2

( / )

/

CNT CNT

CNT

g B

F c B

E k T

E

n

E

N

N k T

η η +

= −

=

(33)

The reduced Fermi energy η as a function of carrier

concentration is plotted in Fig. 7 for three chiralities (n, m)=(9,

2), (6,1), and (3,1). In the nondegenerate limit ( 2η < − ) for

low carrier concentration, η is independent of chirality.

However, in the degenerate limit ( 2η > ), η depends on

chirality.

105

106

107

108

109

1010

-20

0

20

40

60

80

100

120

140

160

CARRIER CONCENTRATION (m-1)

ηη ηη

(n,m)=(9,2)

(n,m)=(6,1)

(n,m)=(3,1)

Fig. 7. The reduced Fermi energy ( ) /F c B

E E k Tη = − as a function of

carrier concentration nCNT for (n, m)=(9,2), (6, 1), and (3,1).

Fig. 8 is a plot of intrinsic velocityi

v as a function of carrier

concentration nCNT per unit length of the CNT at room

temperature (T = 300 K) for (n, m)=(9, 2), (6,1), and (3,1).

105

106

107

108

109

1010

0

1

2

3

4

5

6

7

8

CARRIER CONCENTRATION (m-1)

INT

RIN

SIC

VE

LO

CIT

Y (

10

5 m

/s)

(n,m)=(9,2)

(n,m)=(6,1)

(n,m)=(3,1)

Fig. 8. The intrinsic velocity vi as a function of carrier concentration nCNT for (n,

m)=(9,2), (6, 1), and (3,1).

At low carrier concentration, the intrinsic velocity is

constant, independent of the carrier concentration and rises

sharply when the carrier concentration enters the degenerate

regime, reaching the ultimate limit 60.96 10 /F

v m s= × . The

ballistic nature of the mobility and saturation velocity is

explored by Saad et. al [19] for a parabolic band structure.

Durkop, Getty, Cobas, and Fuhrer[20] explore the transport

characteristics of CNT and report extraordinary high mobility 27.9 / .m V sµ∞ = with the intrinsic mobility estimate exceeding

210 / .m V sµ∞ = at room temperature [21]. These values

exceed those reported for all known semiconductors, an

encouraging sign for applications of CNT in high-speed

transistors, single- and few-electron memories, and chemical

and biochemical sensors. Mobility determines the change in

conductivity per charge, and hence the sensitivity of such

devices. However, to date the relationship of the mobility to the

saturation velocity in quantum-well nanostructure is poorly

understood. The interpretation of device data has been

complicated by short devices with non-ohmic contacts, where it

is difficult to identify injection velocity under ballistic

conditions. The mobilityL

µ in a sample of length L as

compared to the mobility µ∞ in a long sample ( L = ∞ ) is given

by [1]

( )L B[1 exp L / ]µ µ∞= − − l (34)

where bracket is the probability of electron scattering as

detailed in [1]. The ballistic mean free path

B inj chv / v∞ ∞= >l l l as the transit time injt L / v= in a

ballistic channel is restricted by the injection velocity vinj

appropriate for a degenerately-doped bulk contacts for the

conducting channel while the collision time c ch/ vτ ∞= l is

restricted by the channel intrinsic velocity. The probability of

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ICSE2010 Proc. 2010, Melaka, Malaysia

electron scattering is unity (scattering certainly to happen) for a

long conductor. The probability of electron going ballistic is

proportional to /B

L l as the channel length is reduced

indicating that the mean free path is limited by the device

length. Fig. 9 shows the mobility degradation due to limited

size of the CNT as compared to the ballistic mean free path.

Also, shown are the results obtained from the ballistic mobility

model of Shur [22] The drop in the size-limited mobility given

by (34) is much steeper than that of Shur’s model.

Fig. 9. Ballistic Mobility

Lµ as a function of normalized CNT length /L l .

As shown in Fig. 10, the drift response to the applied electric

field is linear for low values of the field. However, the drift

velocity saturates to an appropriate intrinsic velocity of Fig. 8.

The saturation velocity does not sensitively depend on the low-

field mobility indicating the fact that the intrinsic velocity

controls the speed behavior of the device in a high electric field

that is necessarily present in scaled down CNT devices. The

saturation velocity indicated in Fig. 10 is that appropriate for

nondegenerate statistics that is independent of its chirality.

10-3

10-2

10-1

100

10-1

100

101

ELECTRIC FIELD (V/µµµµm)

DR

IFT

VE

LO

CIT

Y (

10

5 m

/s)

µµµµL=10 m

2/v-s

µµµµL=5 m

2/v-s

µµµµL=1 m

2/v-s

Fig. 10. Drift velocity response to the electric field for three values of the low-

field mobility leading to the same saturation velocity.

ACKNOWLEDGMENTS

This work is supported by Distinguished Visiting Professor

Grant of the Universiti Teknologi Malaysia and Brain Gain

fellowship awarded by the Academy of Sciences Malaysia with

support from the Ministry of Science, Technology, and

Innovation (MOSTI).

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