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Three-dimensional simulation of single electron transistors

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2004 4th
zyxw
EEE
Conference
on
Nanotechnology
Three-dimensional simulation
of
Single Electron Transistors
G.
Fieri',*,
M.
G.
Pala
and
G.
Iannaccone'?' Dipartimento di Ingegneria dell'hformazione, UniversitA degli Studi di
zyxwvut
isa,
Via
Caruso
1.56122
Pisa, Italy
IEIIT-CNR,
Via
Caruso,
1-56122
Pisa, Italy
email:'g.fiori@iet.unipi.it
zyxw
AbsfnW-
We present
a
three-dimensional
(30)
approach for the simulation of single electron transistor
(SET)
in which subregions with different types
of
confinement
zyxwvu
re
present simultaneously. In particular, we have applied our model, based on the solution
of
the Schrodinger equation with
DFT
to
a
split gate
zyxwvutsrq
nd
a
silicon on insulator
(SOI)
single electron transistor
(SET).
The solution
of
the Schrodinger equation with open boundary conditions
bas
allowed us to compute the three-dimensional conductance in the linear response regime. Index Terms-Single Electron Transistor, PoissodSchrodinger solver, nanoelectronics device simulation.
I.
INTRODUCTION
The
improvement of fabrication techniques in recent years has made possible the realization and characteriza- tion of devices capable of exploiting the discreteness of the electron charge. Single Electron Transistors (SET)
are
one
of
this kind
zyxwvutsr
f
devices, and they represent the final step
of
semiconductor electronics, exploiting the transmission
of
information through
a
single carrier.
In
this
kind
of
devices, it
is
necessary to take into account the effects of quantum confinement, typically three dimensional around
the
dot, two dimensional along the wires connecting the contacts, and one dimensional
in
the two dimensional electron gas (2DEG)
[l].
In
this paper we present a code in which the Pois- sodSchrodinger equation
is
solved self-consistently in
a
zyxwvu
0
domain, by means of
a
multigrid algorithm and with density functional theory, local density approximation
[Z].
The
Poisson equation is solved over the entire
3D
domain, while the solution of the Schrodinger equation is limited to the confined
regions
and solved
in
the momentum space
[31.
In addition, in order to compute the conduc- tance
of
the
SET,
we have developed
a
method based
on
the solution of the
3D
Schrodinger equation with open boundary conditions along the propagation direction by means of the scattering matrix formalism.
11.
PHYSICAL
ODELS
AND NUMERICAL
METHOD
The
PoissodSchrodinger equation has been solved self- consistently, by means of a multigrid algorithm
[4].
While
a
semiclassical approximation is assumed in the whole domain for charge concentrations, in strongly confined regions, the electron concentration is computed by solving the Schrodinger equation with density functional theory, and local density approximation
[Z],
[51,
[61
[71.
In particular, we have considered subregions with
one
dimensional confinement
for
the leads and three dimen- sional confinement for the dot. From
a
numerical point of view, the demand of detailed band
profiles
implies the use of
a
finely discretized grid, which in turn
poses
huge memory requirements
for
the solution of the Schrodinger equation.
Our
approach is able to avoid these drawbacks, solving the Schrodinger equation in the momentum space
[3]
Indeed, once the eigenvalue problem
is
transferred
in
the momentum space by means of the Fast Fourier Transform
(FIT),
we solve
a
reduced eigenvalue problem and then we anti-transform the solution in the direct space. Concerning the 3D quantum region, the number of elec- trons in the confined region
is
fixed,
so
that its chemical potential can be derived
in
a
simple way by means of Slater's formula
[8].
In
order to compute the quasi-equilibrium conductance between two generic regions
of
the structure,
we
have solved the 3D Schrodinger equation with open hound-
ary
conditions along the direction of propagation, using the conduction band previously computed,
in
a post- processing phase.
We
have then obtained the quantum conductance
G
at zero temperature using the Landauer-Biittiker formula
[9],
[IO]
that relates
G
with the transmission matrix
t:
where
zyxwv
is the electron charge and
h
is Planck's constant. In particular, the numerical method
is
based on subdi- viding the propagation direction
in
several slices and in computing the transmission and reflection coefficients of the wave function, imposing the continuity of the wave function and
of
the current density at the interface between adjacent slices.
111.
RESULTS
AND
DISCUSSIONS
zyx
.
Split
Gate
SET
We
have applied
our
model to
a
single electron transistor defined by split gates
on
an
AIGaAdGaAs
modulation 0-7803-8536-5/04/ 20.00 02004
IEEE
337
TABLE
I
zyxwv
OTAL
OT
CAPACITANCE
AND
GATE-DOT
CAPACITANCE
DERIVED
doped heterostructure.
The
layer structure is shown in Fig. la, while the gate layout adopted in the simulation is shown in Fig. Ib. Surface states have been considered
FROM
SIMULATIONS
AS
A
FUNCTION
OF
THE NUMBER
OF
ELECTRONS IN
THE
DOT.
transition
zyxwvu
zyxwv
lectrons)
2
2
zy
3
4
z
9
3.84
zyxw
F
1.68
aF
4.67
aF
CT
105.21
aF
46.93
aF
131.95
zy
F
0.713
aF
0.297
aF
0.805
aF
99.94aF
44 66
aF
125.67
aF
Cj
C
Fig.
1
a)
Layer
smture
of
zyxwvutsrqp
he
simulated device
b
Gate layout
of
th
simulated device.
In
lhe
picture
are
shown
the
regions
n which
1D
and
3D
quantum analysis
have
been
performed.
Gate
1.3.4
and
6
are
the extemal
zyxwvutsrqpon
ates
that
deli
ne
the
ODEG.
while
2
and 3
are
the gates that determine
the
number
of
electrons
in
the
dot.
on the ungated region at the top of the shucture using a model typically applied to metal-semiconductor inter- faces
[Ill
12]. In the two-dimensional electron gas, two kinds
of
con- finement have been considered.
As
shown in Fig. lh, in the center of the ZDEG, the three dimensional Schrodinger equation has been solved, since quantum confinement is strong
along
all the three directions, while, in the rest of the 2DEG, the Schrodinger equation has been solved only in the vertical direction. Fig.
2
shows the chemical potential
Gate Voltage
(V)
Fig.
2.
Chemical
potentials
as
a
function
of
the
inner
gate
voltlge
for
a
number
of elemns
ranging
from
I
to
4,
and equivalent cvpaciwnce circuit
of
the
SET.
as
a
fiinction of the inner gate pair voltage, for a voltage applied
to
the
extemal gates VG=-l.l V. From these curves it is possible to extract the values
of
the
gate capacitance between the dot and the inner gates
C,)
and the total capacitance
zyxw
CT)
efined
as
CT
=
cj
cg
+cm,
(2) where
Cj
is the capacitance between the dot and each reservoir and
Cm
is the capacitance between the dot and the other gates. In Fig. 2 the equivalent circuit of the SET is shown.
In
particular
[13]
we have, where
4Vc
and
4p
are the distance between two curves corresponding to
p N)
and
p N
+
1)
at fixed
VG
and
p
respectively,
as
shown in Fig.
2.
The
values of the capacitance extracted from Fig.
2
are shown in Table
I.
The
capacitance
Cj
has been obtained in
a
similar way
as
4)
where
VD
is the drain voltage, and
the
remaining
C
capacitance can he finally obtained from (2).
Cj
values are quite small, meaning that in this structure, the dot is weakly coupled with the
near
reservoirs.
E.
Silicon-on-insulator
SET
We now focus on the silicon-on-insulator
(SOI)
SET
fabricated and characterized at the University
of
Tiihingen by Augke
et
al.
[141
(Fig.
3).
The
SET consist in two silicon side gates and in a silicon dot in the center of the structure that
is
separated by tunnel barrier from source and drain. In Fig.
3
the equivalent circuit of the SET is also shown, where
Cs
and
CO
are
the
dot-to-reservoir capacitances,
CQ
the side capacitance, and
Cec
s
the
dot to back gate capacitance. In order to estimate the device capacitances, we are primarily interested in the evaluation of the chemical potential of the dot (Fig.
4a).
Since from the
SEM
picture it is not clear whether the central region is prismatic or rounded, we have performed simulations for both geometries. In particular, in Table
11,
we
show
the
capacitance values obtained for the two structures and the experimental measurements given in
338
TABLE
zyxw
I
COMPARISON
BETWEEN SIMULATION
CAPACITANCE
AND EXPERIMENTAI.
DATA
zyxwv
zyx
41.
‘Vback
gale
a)
b)
Fig. 3. SEM micro-gnph
of
the
SO1
SET 1141 and the equivalent Circuit.
0.1
,
, ,
,
,
, , ,
-1
-0.8
-0.6
-0.4
x
10
zyxwvutsrqpo
VG
(v)
yo
01
Fig.
4.
a
The chemical potential
of
the dot
for
different
zyxwvut
umber
of
electrons
as
VG is varied
for
the
prismatic
sl~cture.
The
zyxwvut
eaks
correspond respectively to
zyxwvutsrqpon
=
1
N
=
2,
N
=
3,
N
=
4.
Temperature is
4.2
K.
b)
The coulomb blockade oscillations obtained
as
Vc
is varied for the prismatic smcfure. Temperature
is
from
4.2,
to 7, to
10
and to
15
K. The oscillation Deriod is
0.1
V.
Ref. [I41
:
it is clear that the results are very sensitive to the dot shape.
In
Fig. 4b conductance oscillations at different tem- peratures are plotted for the prismatic structure. Even if the conductance differs by a factor
of
IO
and the oscillation period is approximately twice as compared to the experimental data, we can consider our conductance simulations acceptable, since we do not know the actual geometry of the dot.
IV.
CONCLUSIONS
In this paper we have presented a 3D simulation approach particularly suitable
for
semiconductor single electron transistor that require a huge number of grid points.
We
present a code based on the solution of
the
CG
1.6
zyxw
F
4aF
3.4
aF
CD
laF
6aF
16
aF
Schrodinger equation with
DFT
by means
of
a
multigrid
algorithm,
the solution
of
the single particle Schriidinger equation in a reduced momentum space basis, and the computation of scattering matrices for
3D
structures.
In
the examples shown we have been able to recover
all
sig- nificant microscopic quantities such as charge distribution and conduction band profiles as well as capacitances and transport properties.
ACKNOWLEDGMENT
Financial support from
IST
NanoTCAD project
(EC
contract IST-1999-108’28) is gratefully acknowledged.
Au-
thors acknowledge fruitful discussions with
A.
Schenk and
E
Heinz.
REFERENCES
[I] A. Scholze, A. Schenk. and W. Fichtner, “Single-electron device Simulation’’
IEEE
Trans.
Elect. De”,,
Vol. 47,
No.10,
pp.1811-1818,
2000.
[21
J.
C.
Inkson,
Many body theory
of
solids-
on
nrmduction,
New
York
Plenum,
1984.
[3]
M.
G.
Pala
and
G. lannaccone.
“A three-dimensional
solver
of
the
Schr’odinger equation
in
momentum space for the detailed simulation
of
nano~VUc1ures”
londechnolo~y.
o1.13, pp.369-372, 2002. [4] W. H.
Press,
S.
A. Teukolsky. W. T. Vending and B. P.
Flannery
Nu- merical Recipes in
Fortran
77,
second
edition
New
York
Cambridge University
ss.
1992. [5]
H.
Hohenber and W.
Kohn,
“Inhomogeneous
Electmn
Gas”
Phys. Rev.
.Vol. 136. oo.B8M-B871. 1964. [6] W. Kohn and
‘i.
.
Sham, .“Self-Consistent Equations Including Exchange and Correlation Effects”
Phys. Rev,
Vol.
140,
pp.AI133- A1138, 1965. [7]
U.
von
Bxlh and L. Hedin, “A
Id
xchange-correlation potential
for
the spin polarized
case”.
J
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C
o1.5.pp.1629-1642, 1972. [8]
J.
C.
Slater, “A Simplification of
the
Haruee-Fock
Method“,Phys. Rev.,
Vo1.81, pp.385-390,
1951.
[9]
R.
Landauer,
”Spatial vwivtion
of
currents and
i
elds due to localized
scatten
in metallic conduction”,
IBMJ
Res. Dev.,
Vol.l, p.223, 1957. [IO] M. Buidker. “Symmetry
ofelectrical
conduction”.
IBM
3
Res.
Dev.,
Vo1.32. p.306,
1988.
[Ill
S.
Sze,
Physics
of
Semiconductor Devices.
2nd
Edition.
New
York
[I21
M.
G. Pala, G. lannaccone.
S.
Kaiser,
A.
Schliemann. L. Wonchech
and
A.
Forchel,
”ExuactiOn
of
parameten
of
surface
states from experimental test smctures”
Nanotechnology,
Vol. 13, pp.373-377,
2002.
[I31
M.
Macucci,
K. Hess, and G.
I.
lafnte, “Electronic energy spectrum and the concept
of
capacitance in quantum doe”,
Phys. Rev. B.
Vo1.48. pp.17354-17363, 1993. Cl41
R.
Augke,
W. Eberhardt, C. Single, F. E.
Ens,
D.A.
Wharam.
and
D.
P.
Kem, ‘Doped silicon single
electron
vansiston with single island chancteristics”,
Appl.
Phys.
Len.
Vo1.76, pp.2065-2067,ZWO.
I.
wiley
sans.
1981.
339

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