Transcript Titre de la Communication

Quantum Transport Simulation in
DG MOSFETs using a Tight
Binding Green’s function
Formalism
M. Bescond, J-L. Autran, M. Lannoo
4th European Workshop on Ultimate Integration of Silicon, March 20 and 21, 2003
Laboratoire Matériaux et Microélectronique de Provence
UMR CNRS 6137 - Marseille/Toulon (France) - www.l2mp.fr
Outline
 Overview of the problem
Device considered
Theory: Tight Binding Green’s function formalism
Results and discussion
Conclusion
Laboratoire Matériaux et Microélectronique de Provence
UMR CNRS 6137 - Marseille/Toulon (France) - www.l2mp.fr
Overview of the problem
 Device dimensions scale into the nanometer regime.
 The Green’s function formalism represents a basic method to
describe the quantum behavior of the transistors : capacity to
describe interactions and semi-infinite contact (source, drain).
 However, most of the studies consider this formalism in the EMA,
whose validity in the nanometer scale is debatable :
E(k)
Parabolic approximation
of a finished system of
atoms
0
Laboratoire Matériaux et Microélectronique de Provence
UMR CNRS 6137 - Marseille/Toulon (France) - www.l2mp.fr
Parabolic approximation
of an homogeneous
medium
k
Device considered
 Single Atomic conduction channel DG MOSFET.
VG
VS
SiO2, TOX = 1.5 nm
 
²
2m*sca²
VD
TC=0.4 nm
SOURCE
SiO2, TOX = 1.5 nm
DRAIN
x
Semiconductor
VG
y
L = 10 nm
 Mixed-mode approach :
 The axis source-channel-drain is represented by an atomic linear chain treated
in tight binding (1).
 The other parts of the system are classically treated from a dielectric point of
view.
(1)
M. Bescond, M. Lannoo, D. Goguenheim, J-L. Autran, Journal of Non-Cristalline Solids (2003) in press.
Laboratoire Matériaux et Microélectronique de Provence
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Device considered
 Band profile versus position :
Source
Channel
Drain
E (eV)
E (eV)
EFS
VDS
E (eV)
0.11 eV
X
 Hypothesis :
 Source and drain are considered as metallic reservoirs.
 We consider a negative Schottky barrier of –0.11 eV.
Laboratoire Matériaux et Microélectronique de Provence
UMR CNRS 6137 - Marseille/Toulon (France) - www.l2mp.fr
EFD
Tight binding Green’s function formalism
 Retarded Green’s function : (2) S. Datta, Superlatt. Microstruct., 28, 253
(2000).
G(E) 
1
lim E  H  D  S  i
0
Energy
One defines
Hamiltonian
S  i(S  S)
and
Self energies(2)
D  i(D  D)
Electron density can be computed as :
f : Fermi-Dirac distribution

n 


dE [f (E   )A  f (E   )A ]
S
S
D
D
2
AS = GSG+
AD = GDG+
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 Spectral functions
Tight binding Green’s function formalism
 The current :
 The device is virtually cleaved into two regions :
VG V G
1
VS V S
2
VD VD
SOURCE
DRAIN
VG VG
 The transmitted current I through the plane separating the two parts is :
I
dQ
dt
, where Q is the charge density of the system.
Laboratoire Matériaux et Microélectronique de Provence
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Tight binding Green’s function formalism
 In the tight binding set, hamiltonian operator has the following form :
 H11 H12 

H
 H21 H22 
Include the self energies of the semi-infinite
source and drain.
 The associated retarded Green’s function of the uncoupled system is :
0
 E  i  H11


g  lim 
0 
0
E  i  H 22 

1
 The final expression of the current is :



I   4e dE Tr1 n VnV  fS(E)  fD(E)
 

-2in = g-g*

=(I-gVgV))-1
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 0 H12 

V
 H21 0 
 Coupling matrix
Tr1 trace restricted to part 1
Results and Discussion
Simulation code :
Electron density profiles :
Electrostatic Potential +
Electron density
27
10
1V
25
Selfconsistent
coupling
-3
n [m ]
10
Poisson
2D
23
10
21
Increase VG
10
Green
New Electrostatic
Potential
VDS = 0.4 V
T = 300 K
0V
19
10
-10
0
x [nm]
New Electron density
CURRENT
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10
Results and discussion
 IDS versus VG at two different temperatures :
-3
10
-5
10
-7
IDS [A]
10
T=300 K
-9
10
T=100 K
-11
10
-13
10
VDS = 0.4 V
-15
10
0.0
0.2
0.4
0.6
VG [V]
 Tunneling current affects :
- the magnitude of the current in the subthreshold region,
- the quantitative shape of the curve.
Laboratoire Matériaux et Microélectronique de Provence
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0.8
Results and discussion
I
[A]
SubthresholdDS
slope [mV/decade]
 IDS (VG) for several values of the channel length :
VDS= 0.4 V
74-6
10
72
Ideal slope
-8
10
70
increase L
-10
68
10
66
10
-12
64-14
nm
IdealL=10
slope
10
62
L=12 nm
L=20 nm
-16
10
60
-18
10 10
0.0
12
0.2
14
0.4
16
0.6
Channel length [nm]
18
0.8
20
VG [V]
 For a 20 nm device, the curve has a nearly perfect slope of 60 mV/decade.
 In smaller devices, the increase of the subthreshold current is due to
electron tunneling through the ‘bump’ of the electric potential profile.
Laboratoire Matériaux et Microélectronique de Provence
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Results and discussion
 IDS vs VDS. Dashed line represents the current obtained with a
quantum of conductance G0 = 2e²/h (3).
12
(2e²/h)VDS
10
VG= 0.9 V
0.8 V
8
IDS [A]
Reflections due to the
drop voltage
0.7 V
6
4
0.6 V
2
0.5 V
0
0.0
0.1
0.2
0.3
VDS [V]
0.4
(3)
R. Landauer, J. Phys. :Condens.
Matter, 1, 8099 (1989).
 In thin channels, the conductance is quantified in units of G0.
 Saturation shows up only when the electron potential energy maximum in the
channel is suppressed by positive gate voltage, and is due to the exhaustion of
source electrons.
Laboratoire Matériaux et Microélectronique de Provence
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Results and discussion
Transmission coefficient (T)
 Transmission coefficient for VG = 0.7 V :
1.0
0.8
increase VDS
0.6
VDS=0.1 V
VDS=0.2 V
VDS=0.3 V
VDS=0.4 V
0.4
0.2
0.0
-0.02
0.00
0.02
0.04
0.06
Energy (eV)
 Even if injected ballistic particles transmit freely from source to drain
without channel potential barrier, reflections due to the drop voltage VDS
still exist.
 Transparency attenuation is all the more pronounced as the applied
voltage VDS increases .
Laboratoire Matériaux et Microélectronique de Provence
UMR CNRS 6137 - Marseille/Toulon (France) - www.l2mp.fr
Conclusion
 Single conduction channel MOSFET device using tight
binding Green’s function formalism has been simulated.
 « Tunneling transistor » : tunneling effect changes the
overall shape of the current characteristics : the
subthreshold curve is no longer exponential.
 Even in the strong-tunneling regime the transistor is still
responsive to gate voltage.
 Because of the decrease of the transverse number, the
resonant level energies of the channel have to be
determined with a high precision.
Next step : include the 3D silicon atomic structure.
Laboratoire Matériaux et Microélectronique de Provence
UMR CNRS 6137 - Marseille/Toulon (France) - www.l2mp.fr