Modeling of nanoscale devices with carriers obeying a three-dimensional density of states

Article (PDF Available)inJournal of Applied Physics 113(14) · April 2013with 174 Reads
DOI: 10.1063/1.4800869
Abstract
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.
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
714
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 /;
q0;1;hp=2;þp=2;/0;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
0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
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=2ffiffiffiffiffiffiffiffi
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=2ffiffiffiffiffiffiffi
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=2ffiffiffiffiffiffiffiffi
2kT
pmC
s=1ðgSÞ;
vþ¼Jþ
qnþ¼ffiffiffiffiffiffiffiffi
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=2ffiffiffiffiffiffiffiffi
2kT
pmC
s=1ðgDÞ;
v¼J
qn¼ffiffiffiffiffiffiffiffi
2kT
pmC
s=1ðgDÞ
=1=2ðgDÞ:(16)
The total current density in the case of ballistic transport is
therefore
J¼JþJ¼q2pmDOSkT
h2

3=2ffiffiffiffiffiffiffiffi
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.
1517
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
jUdjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
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=2ffiffiffiffiffiffiffiffi
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 ¼qUBðVGÞ
kT qVDS
kT ¼g0qVDS
kT ;
g0¼gðVDS ¼0Þ¼qUB0
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 =1ðgÞeg
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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