Modeling of nanoscale devices with carriers obeying a three-dimensional

density of states

Gino Giusi

1,a)

and Giuseppe Iannaccone

2,b)

1

Dipartimento di Ingegneria Elettronica, Chimica e Ingegneria Industriale, University of Messina,

Contrada di Dio, I-98166 Messina, Italy

2

Dipartimento di Ingegneria dell’Informazione, Universit

a di Pisa, Via G. Caruso 16, 56126 Pisa, Italy

(Received 8 January 2013; accepted 25 March 2013; published online 11 April 2013)

While aggressively nanoscale ﬁeld-effect transistors commonly used in CMOS technology exhibit

strong quantum conﬁnement of charge carriers in one or two dimensions, few devices have been

recently proposed whose operation reminds that of vacuum tube triodes and bipolar transistors,

since charge carriers are ballistically injected into a three-dimensional k-space. In this work we

derive, under the parabolic band approximation, the analytical expressions of the ﬁrst three directed

ballistic moments of the Boltzmann transport equation (current density, carrier density, and average

kinetic energy), suitable to describe ballistic and quasi-ballistic transport in such devices. The

proposed equations are applied, as an example, to describe the ballistic transport in graphene-based

variable-barrier transistors. V

C2013 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4800869]

I. INTRODUCTION

Charge transport modeling is of fundamental importance

to study and predict the behavior of transistors, the basic

building blocks of electronic systems. The ﬁrst amplifying

electron device has been the vacuum tube triode, a three-

terminal device where electrons are thermionically emitted

in the vacuum from a heated electrode, the cathode, while

the number of electrons collected by the anode terminal per

unit time is modulated by the voltage applied to the grid

terminal. Charge transport occurs in the vacuum where

carriers form a three-dimensional (3D) electron gas. The

vacuum tube technology has dominated electronics since the

beginning of the twentieth century until taken over by tran-

sistor technology, i.e., solid-state electronics, in the sixties.

In bipolar transistors, transport from emitter to collector still

occurs in a three-dimensional electron gas. On the other

hand, in nanoscale ﬁeld-effect transistors, carriers are subject

to strong quantum conﬁnement in the channel, typically

along the vertical direction, but more recently along the two

transversal directions, with respect to current ﬂow, in the

so-called nanowire transistors.

1

Recently, nanoscale solid state devices inspired to the

vacuum tube triode have been presented, where charge car-

riers are free to move in all three directions forming a 3D

electron gas.

2,3

As a difference with respect to vacuum tubes

or bipolar transistors, in these recent devices carrier density

can be so high with respect to the density of states of the

materials considered that the free electron gas becomes

degenerate and therefore must be described by Fermi-Dirac

statistics. Let us stress the fact that there is no contradiction

between the nanometer scale, ballistic transport, and the

absence of quantum conﬁnement. Indeed, when one talks

about ballistic transistors, one typically refers to transistors

with nanometer size in the longitudinal direction, so that the

channel length is shorter than the mean free path and the

transport is ballistic or quasi-ballistic. Along the transport

direction the wave function is typically non localized—when

the device is on—and can typically be described by a propa-

gating wave. Quantum conﬁnement in transistors typically

occurs in the transverse direction or directions, typically

imposed by the high electric ﬁeld at the silicon/oxide inter-

face (as in conventional MOSFETs) and/or by the reduced

device dimension in the transverse direction (as in hetero-

structures or in nanowire transistors). In this paper we will

discuss transistors with nanometer scale channel lengths

where lateral quantum conﬁnement is negligible, and there-

fore electrons can be described by a three-dimensional

density of states.

In Sec. II we present the equations describing charge

transport in such devices, which are indeed very similar to

those describing the original thermionic process involved

in vacuum tube triodes. The main differences with respect

to vacuum tube triodes are the fact that carriers move in

a solid state region with a given band-structure and that

the electron gas is degenerate at the emitter and therefore

obeys to Fermi-Dirac statistics. From another point of

view our equations can be used when studying charge

transport with device simulation tools based on a three

dimensional k-space, in particular Monte Carlo methods.

4

In Sec. III we apply the calculated equations to derive a

compact model for ballistic transport in variable-barrier

graphene transistors.

3

Finally, in Sec. IV,wedrawour

conclusions.

II. BALLISTIC DIRECTED MOMENTS FOR 3D

CARRIERS

In this section we develop the equations for the ﬁrst

three directed moments (carrier density, current density, and

average kinetic energy) of the carrier distribution in the case

of ballistic transport for 3D carriers. These equations can

be used to describe the ballistic transport of 3D carriers in

a)

E-mail: ggiusi@unime.it. Tel.: þ390903977560.

b)

E-mail: giuseppe.iannaccone@unipi.it.

0021-8979/2013/113(14)/143711/6/$30.00 V

C2013 AIP Publishing LLC113, 143711-1

JOURNAL OF APPLIED PHYSICS 113, 143711 (2013)

homogenous semiconductor regions, e.g., in the channel of

MOSFETs. In conventional nano-scale MOSFETs carriers

are free to move only in two dimensions (in a plane parallel

to the silicon-oxide interface) while they are conﬁned in the

third direction forming a two-dimensional (2D) electron gas.

Directed moments of the carrier distribution (carrier and cur-

rent density) have been extensively used as basic building

blocks for 2D ballistic

5,6

ad quasi-ballistic

7–14

charge trans-

port models. Two-dimensional carriers occupy discrete

energy levels known as subbands. Making the assumption

that only the lowest sub-band is occupied (energy E

1

in

Fig. 1) and limiting the analysis to one-dimensional transport

(xdirection), the positively directed ballistic current density

(J

þ

) and carrier density (n

þ

) can be calculated, at the top of

the energy barrier in the subband x¼x

max

, by the Natori

equations

5

JþðxmaxÞ¼qN2D

2ﬃﬃﬃﬃﬃﬃﬃﬃ

2kT

pmc

r=1=2

EFS E1ðxmaxÞ

kT

N2D¼kTg2D;

nþðxmaxÞ¼N2D

2=0

EFS E1ðxmaxÞ

kT

;(1)

where qis the electronic charge, kthe Boltzmann constant, T

the absolute temperature, g

2D

the two dimensional density of

states for the considered sub-band, m

C

the effective conduc-

tion mass for the considered sub-band, =jthe Fermi-Dirac

integral of order j, and E

FS

the source quasi Fermi level.

When quantum conﬁnement in the transversal direction

(with respect to transport) is negligible, carriers are better

described by a 3D gas. In order to evaluate the upper per-

formance limit of such devices, we can assume fully ballistic

transport and use a model similar to those proposed by

Natori for 2D carriers (Eq. (1)). In order to calculate the bal-

listic moments for 3D carriers in a homogeneous semicon-

ductor, we make two approximations similar to the one in

the Natori model: one-dimensional analysis (transport along

the direction x) and parabolic bands with effective mass

approximation in the channel.

A. Carrier density

In this subsection we develop the equation for the carrier

density of directed carriers in a homogeneous semiconduc-

tor. Because we are dealing with ballistic transport, the

Fermi-Dirac distribution is still a solution of the Boltzmann

transport equation (BTE) so that the ﬁnal result (Eq. (6))

coincides with the well known expression for the equilibrium

case. However, we proceed in the calculation of the carrier

density because the same procedure is later reused for the

calculation of current density and average kinetic energy.

To start, let us consider a carrier with wave vector k(k

X

,

k

Y

,k

Z

) and total energy E

T

equal to

ET¼ECþE¼ECþh2k2

X

2mXþh2k2

Y

2mYþh2k2

Z

2mZ

;(2)

where E

C

is the bottom energy of the conduction band (indi-

cated above as E

1

in Fig. 1), Eis the kinetic energy, m

j

is the

effective mass in the channel along direction j, and his the

reduced Plank constant. Because scattering is neglected, we

know that at the top of the barrier carriers with k

X

>0 are

injected from the source and carriers with k

X

<0 are injected

from the drain. The concentration of carriers, at x¼x

max

,in

the positive directed ﬂux is

nþ¼X

valley

2

ð2pÞ3ð

þ1

1

dkZð

þ1

1

dkYð

þ1

0

dkXfðET;EFSÞ:(3)

Now, changing the integration variables from (k

X

,k

Y

,k

Z

)to

(q,/,#) in ellipsoidal coordinates

kX

ﬃﬃﬃﬃﬃﬃ

mX

p¼qsin /cos #;kY

ﬃﬃﬃﬃﬃﬃ

mY

p¼qsin /sin #;kZ

ﬃﬃﬃﬃﬃﬃ

mZ

p¼qcos /;

q2½0;1;h2½p=2;þp=2;/2½0;p;(4)

the positive directed carrier density at x

max

can be calculated

as

nþ¼X

valley

2

ð2pÞ3ð

p

0

sin /d/ð

p=2

p=2

d#ð

1

0ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

mXmYmZ

pq2dq

1þexp ECðxmaxÞþh2

2q2EFS

kT

0

@1

A

2

43

5

1

;(5)

from which one obtains

nþ¼N3D

2=1=2ðgSÞ(6)

with

N3D¼X

valley

22pmXYZkT

h2

3=2

;gS¼EFS ECðxmaxÞ

kT ;(7)

where N

3D

is the three dimensional effective density of states

and mXYZ ¼ðmXmYmZÞ1=3. Silicon has six equivalent aniso-

tropic valley minima and the band curvature along a direction

FIG. 1. Energy band diagram along the transport direction xin a device with

two injecting contacts (source and drain). In the case of ballistic transport,

carriers injected by contacts can only be backscattered by the energy barrier

between the contact and the channel. At the top of the barrier (x¼x

max

), pos-

itive directed carriers are all injected by the source, while negative directed

carriers are all injected by the drain.

143711-2 G. Giusi and G. Iannaccone J. Appl. Phys. 113, 143711 (2013)

depends on the minima considered: the two minima along the

xdirection have m

X

¼m

l

and m

Y

¼m

Z

¼m

t

, the two minima

along yhave m

Y

¼m

l

and m

X

¼m

Z

¼m

t

, and the two minima

along zhave m

Z

¼m

l

and m

X

¼m

Y

¼m

t

, where m

t

¼0.19m

0

is the transversal mass, m

l

¼0.91m

0

is the longitudinal mass,

and m

0

is the free electron mass. In any case mXYZ ¼

ðmtmtmlÞ1=3for all minima so that

N3D¼g22pmXYZkT

h2

3=2

¼22pmDOSkT

h2

3=2

;(8)

where gis the number of valleys (6 for silicon) and mDOS

¼g2=3mXYZ is the density-of-states effective mass which is

1.08m

0

for silicon.

In the case of MOSFETs, similar results can be obtained

for carriers injected from the drain, so that the total charge

density in the case of ballistic transport at x

max

is

n¼nþþn¼N3D

2=1=2ðgSÞþN3D

2=1=2ðgDÞ;

gD¼EFD ECðxmaxÞ

kT ;(9)

where E

FD

is the drain quasi Fermi level. Particular cases

are: (i) in saturation (V

DS

kT/q) carriers injected by the

drain are reﬂected back by the channel-drain energy barrier

so that nn

þ

; (ii) in equilibrium (V

DS

¼0) g

S

¼g

D

so that

n¼2nþ¼N3D=1=2ðgSÞ.

B. Current density

The result for the calculation of the positive directed

current density in a homogeneous semiconductor is obtained

following a similar procedure

Jþ¼X

valley

Jþ

v

¼qX

valley

2

ð2pÞ3ð

1

1

dkZð

1

1

dkYð

1

0

dkXfðET;EFSÞvXðEÞ;(10)

where Jþ

vis the current density contribution due to a single

valley and

vx¼1

h

@E

@kX¼hkX

mX

;(11)

is the carrier velocity along the transport direction x.

The value of v

x

depends on the valley considered and

the expression inside the integral of Eq. (10) is not symmetri-

cal with respect to x,y, and z, like in the case of carrier con-

centration (3), so that the sum over the valleys cannot be

replaced by a simple multiplication constant g. By changing

the integration variables like in Eq. (4), the positive-directed

current density due to a single valley is

Jþ

v¼q2pmXYZkT

h2

3=2ﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2kT

pmX

r=1ðgSÞ:(12)

Changing the valley, the band curvature, and the effective

mass along xchange, so that Jþ

vdepends on the particular

valley. The total positive directed current density can be

obtained summing over all valleys

Jþ¼X

valley

Jþ

v¼q2pmXYZkT

h2

3=2ﬃﬃﬃﬃﬃﬃﬃﬃ

2kT

p

r=1ðgSÞX

valley

1

ﬃﬃﬃﬃﬃﬃ

mX

p:

(13)

In the case of silicon, assuming that the 6 band minima are

populated with the same probability, m

X

¼m

t

with probabil-

ity 4/6 and m

X

¼m

l

with probability 2/6, so that the sum over

effective masses can be replaced by

X

valley

1

ﬃﬃﬃﬃﬃﬃ

mX

p¼4

ﬃﬃﬃﬃﬃ

mt

pþ2

ﬃﬃﬃﬃﬃ

ml

p¼6

ﬃﬃﬃﬃﬃﬃ

mC

p;(14)

where m

C

is a three dimensional effective conduction mass,

which results equal to 0.283m

0

. Let us notice that this value

is slightly different from 0.26m

0

which is typically assumed

for mobility-related problems (in this case the effective mass

is calculated as a weighted average of the inverse of the

directional effective masses and not of the inverse of their

square root). Taking into account this effective mass, the

total positive current density and the average carrier velocity

of the positive directed ﬂux are

Jþ¼q2pmDOSkT

h2

3=2ﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2kT

pmC

s=1ðgSÞ;

vþ¼Jþ

qnþ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2kT

pmC

s=1ðgSÞ

=1=2ðgSÞ;(15)

where the term in the square root is the non degenerate

one-dimensional average carrier velocity and the term

=1ðgSÞ==1=2ðgSÞis the velocity degeneration factor.

In the case of MOSFETs similar results can be obtained

for carriers injected from the drain, and the total negative

current density and the average carrier velocity of the nega-

tive directed ﬂux at x

max

are

J¼q2pmDOSkT

h2

3=2ﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2kT

pmC

s=1ðgDÞ;

v¼J

qn¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2kT

pmC

s=1ðgDÞ

=1=2ðgDÞ:(16)

The total current density in the case of ballistic transport is

therefore

J¼JþJ¼q2pmDOSkT

h2

3=2ﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2kT

pmC

s½=1ðgSÞ=

1ðgDÞ:

(17)

It is worth noticing that (i) in saturation (V

DS

kT/q)

carriers injected by the drain are reﬂected back by the

channel-drain energy barrier so that J

0 and JJ

þ

; (ii) in

equilibrium (V

DS

¼0) g

S

¼g

D

so that J¼0.

143711-3 G. Giusi and G. Iannaccone J. Appl. Phys. 113, 143711 (2013)

C. Average carrier kinetic energy

By changing the integration variables like in Eq. (4), the

average kinetic energy of the positive directed carriers in a

homogeneous semiconductor can be calculated as

Eþ¼X

valley

2

ð2pÞ3ð

þ1

1

dkZð

þ1

1

dkYð

þ1

0

dkXEf ðET;EFSÞ

nþ

¼3

2kT =3=2ðgSÞ

=1=2ðgSÞ;(18)

where =3=2is the Fermi-Dirac integral of order 3/2. The term

3/2kT is the well known thermodynamic energy of a 3D elec-

tron gas, whereas the term in the fraction is the energy

degeneration factor.

In the case of MOSFETs similar results can be obtained

for carriers injected from the drain

E¼3

2kT =3=2ðgDÞ

=1=2ðgDÞ:(19)

The reported equations in Subsections II A and II B appear

very similar to the corresponding 2D case developed

by Natori (Eq. (1)). What changes is that the 2D effective

density of states (N

2D

) is substituted with the 3D effective

density of states (N

3D

) and the order of Fermi-Dirac integrals

is increased by 1

=

2. They can be useful for device modeling

and parameter extraction.

15–17

III. APPLICATION TO GRAPHENE VARIABLE-

BARRIER TRANSISTORS

In this section we make an example of application of the

ballistic current equation (15) developed in Sec. II considering

n-type graphene variable-barrier transistors.

3

These are three-

terminal devices based on a hybrid interface formed by a layer

of 2D graphene acting as source electrode and by a layer of n

þ

doped silicon, acting as both a channel and drain. Between

source and channel a Schottky junction is formed. Since gra-

phene is a semi-metal, with low density of states, the junction

barrier height is electrostatically modulated by the voltage

applied to a control metal gate. A simpliﬁed band-diagram

is shown in Fig. 2,wherexdenotes the source to “Silicon

channel” direction. Depending on the position of the drain

contact, the transport can be entirely in the x-direction (back

drain contact) or a combination of x-direction (ﬂux injection)

and transverse (yor z) direction (top drain contact). In the lat-

ter case after injection into/from the silicon channel, transport

continues along the transverse direction till the drain contact.

For simplicity, in the rest of this discussion we will neglect (i)

the scattering in the silicon channel, so as to treat the case of

ballistic transport; (ii) barrier lowering at the Schottky junction

due to image charge effects; (iii) tunneling through the

Schottky junction; (iv) the possible reﬂections at the interface

due to band-structure mismatch. These assumptions allow

us to keep the model simple, and to concentrate on the formal-

ism developed in Sec. II. In support to our assumptions, sev-

eral works on graphene-silicon Schottky junctions show

experimental measurements that agree with the classical equa-

tion of the current in a Schottky junction, supporting the

thermionic assumption for graphene-silicon junctions.

3,21,22

If

some assumptions are not fully veriﬁed, the only change

should be in a pre-factor in the current expression, while the

functional dependence of the current with respect to different

parameters should not change. Moreover, it has been shown

that the interface between chemically inert graphene and a

completely saturated semiconductor surface is free of defects

18

so that we can neglect interfacial states. The Schottky barrier

qUBis modulated by the graphene surface potential V

Gr

3

qUB¼qUB0qVGrðVGÞ;(20)

where qUB0is the equilibrium barrier height when the gra-

phene energy Fermi level is aligned with the silicon energy

Fermi level (qUB00:5eV) and V

G

is the control gate volt-

age. The source is grounded. The barrier height modulation

as function of V

G

is deﬁned by the device electrostatics

which can be expressed in terms of charge balance

QMþQGr þQSi ¼0;

QM¼Cox½VGVGr ¼eox e0

tox ½VGVGr;

QGr ¼2q

p

kT

hvF

2

=1qVGr

kT

=1

qVGr

kT

;

QSi ¼Ud

jUdjﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2eSie0kTND

pexp qUd

kT

þqUd

kT 1

þn2

i

N2

D

exp qUd

kT

qUd

kT 1

1=2

;

(21)

where Q

M

,Q

Gr

, and Q

Si

are the charge (per unit area) in the

metal gate, in the 2D graphene layer (source),

19

and in the

n-doped silicon layer (channel and drain), respectively,

20

C

ox

,e

ox

, and t

ox

are the oxide capacitances per unit area, the

relative dielectric oxide constant, and the oxide thickness,

respectively, e

0

is the absolute dielectric constant, v

f

is the

Fermi velocity of carriers in graphene (10

8

cm/s), e

Si

is the

FIG. 2. Illustration (not in scale) of the energy band diagram for graphene

variable barrier transistors (n-type silicon).

143711-4 G. Giusi and G. Iannaccone J. Appl. Phys. 113, 143711 (2013)

relative dielectric silicon constant, n

i

is the intrinsic electron

concentration, N

D

is the doping concentration of the silicon

layer, and U

d

is the potential drop in the silicon layer cal-

culated as (Fig. 2)

qUd¼qUBþqVDS þEFD ECðxDCÞ;

NDnðxDCÞ¼N3D=1=2

EFD ECðxDCÞ

kT

;(22)

where x

DC

is the abscissa of the drain contact and nis the

electron concentration in the silicon. Let us notice that the

expression for Q

Si

and nare approximations valid when

charge density associated to current ﬂow is negligible. Fig. 3

shows the barrier height modulation effect for different val-

ues of e

ox

and t

ox

(e

ox

¼3.9 stands for SiO

2

oxide, e

ox

¼21

stands for HFO

2

oxide). As expected, the higher is the oxide

capacitance C

ox

¼e

ox

/t

ox

, the stronger is the barrier height

modulation with the gate voltage.

As in Eq. (17) we write the current as the sum of two op-

posite ﬂuxes at the top of the barrier: J

is injected from the

drain and J

þ

is injected from the source. In this case, it is

extremely convenient for us to write the equations at the top

of the barrier in silicon, so that we can use a parabolic 3D

energy dispersion relation. This is the typical approach fol-

lowed in the case of metal-semiconductor junctions. From a

rigorous point of view, we are neglecting in this case possi-

ble reﬂections at the interface due to bandstructure mis-

match; however, since we are dealing with semiclassical

electrons, this is a typical and acceptable approximation. For

convenience, we ﬁrst write the expression for J

JðgÞ¼q2pmDOSkT

h2

3=2ﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2kT

pmC

s=1ðgÞ¼AT2=1ðgÞ;(23)

where Jis current density injected from the silicon layer to

the graphene layer, g¼ðEFD ECÞjx¼xmax =kT ,E

C

is the con-

duction band edge, and E

FD

is the drain Fermi energy. The

normalized electrostatic potential gis related to the drain

and to the gate voltage by

g¼EFD EC

kT x¼xmax ¼qUBðVGÞ

kT qVDS

kT ¼g0qVDS

kT ;

g0¼gðVDS ¼0Þ¼qUB0

kT þqVGrðVGÞ

kT :(24)

The current ﬂux J

þ

injected from the source to the drain is

equal to the ﬂux injected from the drain to the source in the

equilibrium case (V

DS

¼0) when the total current is zero

(J

DS

¼0)

Jþ¼Jðg0Þ¼AT2=1ðg0Þ:(25)

Using this approach we can write the total current without

taking into account for the band-structure of the source (gra-

phene) region

JDS ¼JþJ¼AT2½=1ðg0Þ=

1ðgÞ:(26)

In the case in which carriers in silicon do not form a degener-

ate electron gas, the Fermi-Dirac statistics can be approxi-

mated by the Boltzmann statistics so that =1ðgÞeg

JDS AT2eqUB0

kT e

qVGrðVGÞ

kT 1eqVDS

kT

hi

:(27)

Equation (26) appears very similar but should not be con-

fused with Eq. (17). The latter is used for a homogenous ma-

terial (e.g., silicon) and g

S

,g

D

are related to Fermi levels of

source and drain contacts. Differently, Eq. (26) describes the

transport in a non homogenous structure and g

0

is related to

the Fermi level of the source contact including the modula-

tion effect of the gate potential. At ﬁrst sight, Eq. (26) seems

to lose the information related to the material “graphene.”

However, this info is contained in the relation between the

graphene charge and the potential (the third in Eq. (21)). This

relation is a function of the dispersion relation in the gra-

phene and in particular of the graphene Fermi-velocity v

F

.

We stress again the fact that the validity of Eq. (26) is

subjected to the assumptions discussed above, although ex-

perimental measurements conﬁrm the functional dependence

of the current with respect to parameters. Moreover, the

actual magnitude of the current is function of material de-

pendent parameters such as the Fermi velocity of graphene

and the graphene-silicon conduction band offset which are

normally extrapolated by experiments.

3,21,22

Fig. 4shows the calculated current (with parameters

e

ox

¼21, t

ox

¼1 nm, N

D

¼10

16

cm

3

,V

DS

¼1 V) in the cases

FIG. 3. Barrier height modulation by the control gate voltage for different

oxide thickness and dielectric constant values.

FIG. 4. The calculated current (with parameters e

ox

¼21, t

ox

¼1 nm,

N

D

¼10

16

cm

3

,V

DS

¼1 V) in the cases of (i) Boltzmann distribution and

(ii) Fermi-Dirac distribution. The same ﬁgure shows the current density cal-

culated taking into account the Fermi-Dirac distribution for other values of

the parameters e

ox

,t

ox

,N

D

.

143711-5 G. Giusi and G. Iannaccone J. Appl. Phys. 113, 143711 (2013)

of (i) Boltzmann distribution and (ii) Fermi-Dirac distribu-

tion. It is apparent that, especially in the high current regime,

it is very important to use the Fermi-Dirac distribution. The

same ﬁgure shows the current density calculated taking

into account the Fermi-Dirac distribution for other values

of the parameters e

ox

,t

ox

, and N

D

. Higher channel doping

(N

D

¼10

20

cm

3

) translates in a simple shift of the curve

(a change in threshold voltage), while the use of SiO

2

as

gate oxide (e

ox

¼3.9) results in a strong degradation of the

sub-threshold slope even for very thin oxide thickness

(t

ox

¼1nm).

IV. CONCLUSION

In this paper we developed analytical expressions for

charge transport in ballistic devices with a three-dimensional

degenerate electron gas. Such devices have a size in the

transport direction that is much smaller than the carrier mean

free path, so that transport is ballistic and quasi-ballistic, and

larger size in the transversal directions so that quantum con-

ﬁnement is negligible. Solid-state nanoscale devices inspired

to vacuum tubes or to bipolar transistors can easily belong to

this category. We have shown an application of the proposed

physical description to the development of a semi-analytical

model for ballistic transport in variable-barrier graphene

transistors.

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