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VSRD

IJEECE, Vol. 2 (9), 2012
,
1

5
____________________________
1
,2
Research Scholar
,
3
Assistant
Professor,
1,2,
3
Department of Electronics & Communication Engineering
,
Subharti Institute of
Technology &
Engineering
, Meerut, Uttar Pradesh,
INDIA.
*Correspondence :
mohinisingh2008@gmail.com
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H
H
A
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I
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C
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E
Quantum Confinement Effects on Gate Capacitance
in GAA

SNWT
1
Mohini Preetam Singh
*
,
2
Prashant
Dixit
and
3
Vivek Gupta
ABSTRACT
In this paper we introduce some
corrections in gate capacitance of Silicon Nanowire Transistor using
cylindrical
coordinate system with NEGF approach.
The
behaviour of the gate capacitance
(C
g
)
is expected to be strongly
influenced by quantum effects in ultrascaled devices. A
ccording to
the structure of silicon nanowire transistor,
inversion charge centroid shifts with gate voltage. The consideration of this shift results in the degradation of
depletion capacitance
(C
d
)
. Further gate capacitance
(C
s
)
will also change according to the corr
ections in the
charge density at the mid of
channel. Thus gate capacitances
,
which
is a combination of
C
d
,
C
ox
and
C
s
will
lower substantially
.
Keywords :
Non

Equilibrium Green’s Function, Self

Energy, Quantum Transport, Uncoupled Mode Space
Approach
.
1.
INTRODUCTION
Fig. 1
:
GAA SNWT
Semiconductor nanostructures are unique in offering the possibility of studying quantum transport in an
artificial potential landscape. This is
regime of ballistic transport
in which scattering with impurities can be
neglec
ted. The transport properties can then be tailored by varying the geometry of the conductor.
The nanowire
FETs are a particular case of multiple gate FETs
or gate all

around FETs
(see Fig.1)
, in which quantum
confinement occurs in the transverse cross sect
ion. Nanowire FETs are basically quasi

one dimensional
M
ohini Preetam
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et al
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International Journal of Electrical, Electronics & Comm. Engg. Vol. 2 (9), 2012
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transistors where tran
sport occurs in a set of loosly
\
coupled propagating modes.
For decades,
progress in device
scaling has followed an exponential curve: device density on a
microprocessor doubles every three years.
This has come to be known as Moore’s law. We
focus on quantum effects and non

equilibrium, near

ballistic
transport in extremely
scaled transistors (in contrast to quasi

equilibrium, scattering

dominant transport i
n long
channel devices), where a non

equilibrium Green’s functions formalism (NEGF) has
been used to deal with the
quantum transport problem. Work on gate capacitances has already been simulated for Cartesian coordinates with
MATLAB code
[1]
. Here we are fi
nding out the same with cylindrical coordinates using a more sophisticated
simulator based on NEGF formalism.
Section II describes the NEGF approach used in simulator to express drain
currents in terms of electron density. Section III
indicates the paramet
ers involved for calculation of gate
capacitance. Section IV represents the simulated results in case of cylindrical coordinates. Section V finally shows
the comparison.
2.
NEGF APPROACH
As MOSFETs scale to the nanometer regime, canonical carrier transport th
eories are no longer capable of
describing carrier transport accurately. The canonical theories are basically derived from Boltzmann transport
equation (BTE), with more or fewer approximations being made
[2]
. These models focus on scattering

dominant trans
port, which typically occurs in long channel devices. Nanoscale transistors, however, operate in a
quasi ballistic

transport regime. Simulations using conventional models may either under

predict or over

predict
the device performance
[3,4,5]
. As we mention
ed earlier, the BTE assumes a classical approach in describing
carrier dynamics, so quantum features prevailing in nanoscale devices can never be captured in the solutions.
NEGF
formalism express the ballistic quantum transport in terms of G (retarded gree
n’s function) of the
Schrodinger equation used to compute local density of states and solving the single particle Schrodinger
equation with open boundary conditions.
Fig.
2
:
One
Dimensional Discretization With
A Uniform Mesh., a Is The Grid
Spacing
In this paper we have used a NEGF based sophisticated tool. The calculations involves a self
–
consistent
solution of a 3D Poisson equation and a 3D Schrodinger equation by mode space approach NEGF in an attempt
to reduce simulation time due calculations in
volved in differential channel charge versus
which requires a
precision.
3D Schrodinger equation is based on an expansion of active device Hamiltonian in the subband eigenfuction
space. The same can be obtained by solving a 2D Schrodinger equation by F
DM at each slice of SNWT to
obtain subband eigen energy levels and eigenfunctions (modes).
In 3D full stationary equation is given by:
M
ohini Preetam
Singh
et al
/ VSRD
International Journal of Electrical, Electronics & Comm. Engg. Vol. 2 (9), 2012
Page
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7
(
)
(
)
…
(1)
where
is the 3D device Hamiltonianin real space and is given by:
(
)
(
(
)
)
(
(
)
)
(
)
…
(2)
In this expression
is the electron effective mass in 3 different directions.
(
)
is the electron potential
energy in the device coming from self

consistent solution of Poisson’s equation. Solution of the Green’s
function equation related to the real
space Hamiltonian of (1) can be simplified by solving a 2D Schrodinger
equation in the transversal direction and a 1D Green’s
function equation in the
longitudinal (x) direction
(see
Fig.2)[6]
. We first expand the 3D electron wave function in the subband eigenfunction space:
(
)
∑
(
)
(
)
…
(3)
where
(
)
is the n

th eigenfunction of the transversal 2D Schrodinger eq
uation at the slice
of
the nanowire transistor:
(
)
(
)
(
)
…
(4)
a
nd
(
(
)
)
(
(
)
)
(
)
…
(5)
where
(
)
is the n

th subband energy level at
r =
Inserting (2) and (3) into (1), using the relationship
describedby (4) and (5), and integrating along the cross section, we obtain the 1D longitudinal uncoupled
Schrodinger equation:
(
∑
(
)
)
(
)
∑
(
)
(
)
∑
(
)
(
)
(
)
(
)
(
)
…
(6)
W
here
(
)
=
∮
(
)
(
)
(
)
…
(7a)
(
)
=
∮
(
)
(
)
(
)
…
(7b)
A
nd
(
)
=
∮
(
)
(
)
(
)
…
(7c)
In practical, due to strong quantum confinement in SNWTs, usually only a few of lowest subbands (i.e.
m,n=1,…,M, M < N
YZ
) are occupied and need to be included in calculation therefore if we increase mo
de
number,
M , the device characteristics such as electron density
profile and terminal current wil
l not change any
more
[7]
.
Thus with the first M subbands considered (m,
n=1,2,..M)
.
Equation 7 represents an equation that
contains M equations , each repres
enting a selected mode. We can write down these M equations in a matrix
format:
M
ohini Preetam
Singh
et al
/ VSRD
International Journal of Electrical, Electronics & Comm. Engg. Vol. 2 (9), 2012
Page
4
of
7

(
)
(
)




(
)
(
)


(
)
(
)

…
(8)
Where
[
(
)
(
)
]
(
)
(
)
(
)
…
(9)
Also since we are using uncoupled mode space approach for our calculations
,
we have assumed that the shape
of silicon body is uniform along the r
direction. As a result , the confinement potential profile (in
)
varies very slowly along
the channel direction.
Also, for an instance the conduction band edge V(r,
) takes same shape but different values at different r.
Therefore the eigenfunction
(
)
are approximately same along the channel although eigenvalue
(
)
is different.
(
)
(
)
…
(10)
Or
(
)
(
)
…
(11)
which infers
(
)
∮
(
)

(
)

…
(12)
(
)
and
(
)
,(m,n=1,2….,M)
…
(13)
Inserting equation (12)
and (13) in equation (9) we obtain:
(
)
and (m,n =1,2,…,M)
which means
that coupling between modes is negligible (all modes are uncoupled)
[8]
. Thus the device Hamiltonian H
becomes a block

diagonal matrix.


After the device Hamiltonian H is obtained,
electron concentration and cuurent can be calculated by iterative
procedure
followed in applying the NEGF formalism consid
ering the interaction(see Fig. 3
).
Fig.
3
:
Flowchart
Representation o
f Evaluation
of NE
GF
Approach
M
ohini Preetam
Singh
et al
/ VSRD
International Journal of Electrical, Electronics & Comm. Engg. Vol. 2 (9), 2012
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For the subband i , with a plane
wave eigen
energy
E, we write
rel
evant to the 1D transport as [9
]
(
)
[
[
(
)
]
∑
]
[
[
(
)
]
∑
]
…
(1
4)
Where, we define the longitudinal (x) energy
.The third term in the
bracket is called the self

energy
matrix, which is given as
:
∑

∑
∑

…
(1
5)
The two corner entries in
∑
(
)
represent the effects on the finite device Hamiltonian due to the interactions of
the device with the contacts [10]. The self

energy c
oncept allows us to eliminate the huge reservoir and work
solely within the device subspace whose dimensions are much smaller.
∑
(
)
can be expressed in terms of
known quant
ities[10
]. At the source contact,
∑
(
)
where E =
(
)
(
)
…
(1
6)
(
)
is the subband energy at the contact boundary.
∑
(
)
can be obtained in a similar fashion. It is very
important to note that the self

energies are functions of longitudinal energy
as shown before. This
allows us to focus on the longitudinal energy in all our calculations. Under ballistic conditions, the transverse
mode contributions (planewaves in the y direction) can be treated independently
of the longitudinal contribution.
Once
th
e Green’s function is obtained
internal electron density and terminal current of the device under study
can be computed [30,32].We define a new quantity in terms of self

energies
Γ
(
∑

∑
)
…
(1
7)
Physically this function determines the electron
exchange rates between the source/drain
reservoirs and the
active device region [10]. But in general it can be viewed as the
measure of interaction strength due to any
perturbation source. Although the device itself
may
be
in
a
non

equilibrium
state,
elec
trons
are
injected
from
the
equilibrium
source/drain reservoirs. The spectral density functions due to the source/drain contacts
can be
obtained as
and
…
(1
8)
where
(
∑
∑
)
and
(
∑
∑
)
(For clarity, here we use
to denote matrices the same
size as G, with nonzero diagonal entries
(
∑
∑
)
or
(
∑
∑
)
). Note that the spectral functions are
matrices and the diagonal entries represent the local density

of

states at each node. The source related
spectral function is filled up according to the Fermi energy in the source contact, while the drain related spectral
function is filled up according to the Fermi energy in the drain contact. The 2D electron density m
atrix is
obtained as
M
ohini Preetam
Singh
et al
/ VSRD
International Journal of Electrical, Electronics & Comm. Engg. Vol. 2 (9), 2012
Page
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of
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(
)
∫
√
[
(
)
]
[
(
)
]
…
(1
9)
Where f is the
Fermi

Dirac function, and
√
representsthe transverse mode state density (including the
spin degeneracy)
[11]
. Since the spectral functions depend on the longitudinal energy only, they can be moved
out of the integration sign. Therefore above equation
reduces to,
(
)
√
[
⁄
(
)
]
[
⁄
(
)
]
…
(2
0)
Where the Fermi

Dirac integral of
⁄
accounts for all trans
verse mode contributions (see [
1
0

11
]) for
analytical approximation for
⁄
and also note that all quantities appearing as arguments of Fermi

Dirac
integrals are normalized to
). To o
btain the total 2D electron density, we need to integrate above equation
over
. We also need to sum contributions from every conduction band valley and subband. Finally, we can get
a 3D electron density by multiplying the corresponding distribution fun
ction

(
)

to the 2D density matrix
at each longitudinal lattice node. The 3D electron density is fed back to the Poisson equation solver for the self

consistent solution.
Once self

consistency is achieved, the terminal current can be expressed as a
function of the transmission
coefficient [10]. The
transmission coefficient from the source contact to the drain contact is defined in terms of
Green’s function as
:
[
]
…
(2
1)
It is straightforward to write the
transmitted current as
:
(
)
∫
√
[
(
)
(
)
]
(
)
…
(2
2)
where the 2 in the numerator is for spin degeneracy. Note that
i
s the independent of transverse energy
and can therefore be moved out of the integration sign. The above equation is then reduces to,
(
)
√
[
⁄
(
)
⁄
(
)
]
(
)
…
(2
3)
T
he total current is obtained by integrating over
and summing over all
valleys and subbands.
3.
MODELING OF GATE CAPACITANCE
The behaviour of the gate capacitance
is expected to be strongly influenced by quantum effects in ultrascaled
devices. Indeed,
results from the series combination of three capacitances :
,
,
. Capacitance
behaviour for Cartesian coordinates has already simulated [1].
In this paper we try to configure the same in
cylin
drical coordinates (see Fig. 4
). To obtain an analytical model of intermediate transport, we assume that
M
ohini Preetam
Singh
et al
/ VSRD
International Journal of Electrical, Electronics & Comm. Engg. Vol. 2 (9), 2012
Page
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of
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screening
can be considered constant which is sound enough for small cross

section channels and low electron
densities. In inversion layer centroid approach, we consider the charge accumulated in the centroid layer and its
geometrical screening is included in the ef
fective gate capacitance as a series contribution
.
Fig.
4
:
4*4 SNWR with tox=1nm , tsi=4m representing the ser
ies combination of
Cox and Cd
For cylindrical coordinates
canbe given by
:
(
)
…
(24)
where capacitances
are given by
:
(
)
…
(25)
(
)
…
(26)
…
(27)
Here
is calculated using the concept of centroid position (Zi) variation with
appli
ed (see Fig.7
).
According to our cylindrical approach it is observed that centroid position shifts from surface as comparative to
Cartesian coordinates i.e. around 0.04nm.
depends on the geometry. In the subthreshold region, the charge
in the channel isvery small, and the channel potential follows the variations of gate voltage.
(
)
At higher Vg however, electrons start to populate the first subband in t
he center of the channel, thereby
screening Vo, the potential in the center of the channel, from Vg. As a result, the potential in the center of the
wire varies slower with
, while it increases faster with
close to the surface where the electron
c
oncentration is lower. For
, it is necessary to calculate the derivate of
versus
(a small gate voltage
step) which can increase computation time. Therefore we switched to mode space NEGF rather than real space.
4.
RESULTS AND DISCUSSION
Below
threshold, the first sub band is barely occupied, and thus, the electron concentration, and therefore
,
is
M
ohini Preetam
Singh
et al
/ VSRD
International Journal of Electrical, Electronics & Comm. Engg. Vol. 2 (9), 2012
Page
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of
7
very low. When
is increased, however, the electron concentration and
first increase monotonically.
increases monotonically,
with
Vg
until it reaches a first maximum as the peak of the first sub band of the DOS
coincides with the Fermi level. As long as only the first subband is occupied, the centroid and therefore the
depletion capacitance remains constant as the shape
of the channe
l remains constant
. At this the number of
electrons increase with Vg but it does not have any
impact on depletion capacitance
.
A comparison is shown
between
with Cartesian coordinates and cyl
indrical coordinates (see Fig. 8
).The simulated current
–
vol
tage
curve is obtained a
t different values of
for
equals to 0.4V
(see Fig. 6)
.The simulated electron concentration
versus channel length curve shows the variation of electron concentration with channel length for
=0.4V.(see
Fig. 5
).
The
and
curves show th
e variation due to
(see Fig. 9
).The lowering of semiconductor
capacitance is due to fact that we have considered a 3 D sophisticated NEGF based tool which has taken
different electrons concentration in the channel. Around 5 %
decrease
in gate capacitance is due to
this is
because of the fact that due to quantum confinement effects, the carrier centroid in channel is shifted at a
distance
from silicon/oxide interface and hence called
as dark space. The impact of
is also observed as
it depends on the geometry of the Silicon Nanowire Transistor. So
three times lowering of gate capacitance is
observed after simulation.
Fig. 5 :
It plots 1D electon density profile along the channel of simulated cylindrical SNWT
Fig.
6
:
Simulated
Id
–
curve at
= 0.4V and at T=300K for [100]
channel orientated for 4*4 GAA SNWR with tox=1nm, tsi=4nm
M
ohini Preetam
Singh
et al
/ VSRD
International Journal of Electrical, Electronics & Comm. Engg. Vol. 2 (9), 2012
Page
9
of
7
Fig.
7
:
Position of centroid (nm) versus
(V)
Fig. 8 :
(F) versus
(V) for
Rectangular And Cylindrical
Coordinates
(a)
0.655
0.66
0.665
0.67
0.675
0.68
0
0.2
0.4
0.6
Zi(nm)
Vg(V)
0.00E+00
5.00E01
1.00E+00
1.50E+00
2.00E+00
0
0.2
0.4
0.6
Cd(F) e

17
Vg(V)
Cdrec
Cdcyl
0
5
10
15
20
0
0.2
0.4
0.6
Cs(F)
Vg(V)
0
5
10
15
20
0
0.2
0.4
0.6
Cg(V)
Vg(V)
M
ohini Preetam
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/ VSRD
International Journal of Electrical, Electronics & Comm. Engg. Vol. 2 (9), 2012
Page
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(b
)
Fig.
9
:
(a)

graph
(b)

graph
5.
CONCLUSION
Here we have reached a conclusion that Vg has an impact on Cg both depending upon Cs due to our advanced
NEGF based sophisticated tool and also
and
due change in geometrical parameters.
6.
REFRENCES
[1]
Aryan Afzalian
, Member, IEEE
, Chi

Woo Lee, Nima Dehdashti Akhavan, Ran Yan, Isabelle Ferain, and
Jean

Pierre Colinge, “Quantum Confinement Effects in Capacitance Behavior of Multigate Silicon
Nanowire MOSFET
s,” IEEE TRANSACTIONS ON NANOTECHNOL OGY, VOL. 10, NO. 2, MARCH
2011.
[2]
A Martinez1, A. R. Brown and A. Asenov, Dept of Electronics & Electrical Engineering, University of
Glasgow Rankine Building, Oakfield Ave, Glasgow G12 8LT, UK, “Full

Band NEGF simulatio
ns of
surface roughness in SiNanowires”, 2nd Workshop on Theory, Modelling and Computational Methods for
Semiconductors IOP Publishing Journal of Physics: Conference Series 242 (2010) 012016
doi:10.1088/1742

6596/242/1/012016.
[3]
Haiyan Jiang a, Sihong Shao a
, Wei Cai b,*, Pingwen Zhang a, “Boundary treatments in non

equilibrium
Green’s
function (NEGF) methods for quantum transport in nano

MOSFETs,” Journal of Computational
Physics 227 (2008) 6553
–
6573.
[4]
Paolo Michetti, Giorgio Mugnaini, Giuseppe Iannaccone
Mem
ber, IEEE,
“
Analytical model of nanowire
FETs in a partially
ballistic or dissipative transport regime
[5]
C. W. J. Beenakker and H. van Houten Philips Research Laboratories, Eindhoven, The Netherlands,
“Quantum Transport in Semiconductor Nanostructures”, Publ
ished in Solid State Physics, 44, 1

228 (1991).
[6]
Suprio Dutta, “Electronics Transfer in Mesoscopic Systems”, Cambridge Studies in Semiconductor Physics
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[7]
Mark Lundstrom Jing Guo, “Nanosc
ale Transistors Devices Physics
, Modell
ing and Simulation”, Springer
International Edition.
[8]
M.S. Lundstrom, Fundamentals of Carrier Transport, 2nd ed., Cambridge
University Press, Cambridge,
UK, 2000.
[9]
F. Assad, Z. Ren, D. Vasileska, S. Datta, and M.S. Lundstrom, “On the
performance limits for
Si
MOSFET’s:
A theoretical study,” IEEE Trans.
Electron Dev., 47, pp. 232

240, 2000.
[10]
Z.
Ren
and
M.S.
Lundstrom,
“Simulation
of
nanoscale
MOSFETs:
A
scattering theory interpretation,”
Superlattices and Microstructures, 27, pp.
177

189, 2000.
[11]
K. Banoo,
J.

H. Rhew, M.S. Lundstrom, C.
W. Shu, and J.W. Jerome,
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Ballistic
Transport in Si Nanotransistors,” presented at the 7
th
International Workshop on Computational Electronics
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25, 2000.
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