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Università degli Studi di PisaGiuseppe Iannaccone
G. IannacconeDipartimento di Ingegneria dell’Informazione, Università degli Studi di Pisa
Via Diotisalvi 2, I-56122, Pisa, Italyg.iannaccone@iet.unipi.it
IWCE 2004
Analytical and Numerical Investigation of Noise in nanoscale
Ballistic Field Effect Transistors
Acknowledgments:
Support from EU (SINANO), MIUR (FIRB), Fondazione CRP
Università degli Studi di PisaGiuseppe Iannaccone
Motivation
1. Effects typical of mesoscopic devices can be observed in “mundane” MOSFETs at room temperature as the ballistic component of the drain current increases
2. Suitable noise models are required by circuit designers, especially for analog and mixed signal applications
Here we focus on the limit of ballistic transport.
Università degli Studi di PisaGiuseppe Iannaccone
Noise of the drain current
Transition from Thermal to shot in as the ballistic limit is approached
Shot noise of the gate current
Plus contributions due to defects (not considered here)
Noise in nanotransistors
INTEL - Prototype 20 nm MOSFET
NMOS Gate delay 0.6 ps
Shot noise of the gate tunnel current
Noise of the drain current
Università degli Studi di PisaGiuseppe Iannaccone
Shot Noise Noise is an extremely sensitive probe of electron-electron
interaction. No interactions Poissonian process
Interaction introduce “coordination” in the collective motion of electrons, making the process non Poissonian ( S Sfull).
Interaction: Pauli Exclusion and Coulomb Repulsion
2qISfull
Full shot noise
S: power spectral density of the noise current
Università degli Studi di PisaGiuseppe Iannaccone
Ballistic transport in MOSFETs (I)
Density of states in the first subband in the channel:
Electron density at the subband peak in the channel:zy
tzyD
EEh
mEEN
22 2),(
kTEEDS DFSeEf /)()( )(1
1)(
)()(2 2
0 0
2 MzyDMzySDzyD EEEfEEEfNdEdEn
FSE
FDE
ME
y
0, zy EE
Source Drain
Università degli Studi di PisaGiuseppe Iannaccone
Charge fluctuations in MOSFETs
Fluctuations of n2D as function of fS and fD:
fluctuations electrostatics
in the contacts
Subband maximum EM depends on n2D via electrostatics
Electrostatic effects are included in a single capacitance per unit area CC
M
D
M
SDzyMDSDzyD E
f
E
fNdEdEEffNdEdEn 2
0 0
2
0 0
2 22
C
DM C
nqE 2
2
Università degli Studi di PisaGiuseppe Iannaccone
Equivalent circuit
Add quantum capacitance towards the source and the drain.
Equivalent circuit:
M
SDzyQS E
fNdEdEqC 2
0 0
22
M
DDzyQD E
fNdEdEqC 2
0 0
22
FSEFDE
ME
y
C
Università degli Studi di PisaGiuseppe Iannaccone
Barrier modulation Fluctuation of channel barrier
Current density is modulated by barrier height !
QDQSC
DSDzy
M CCC
ffNdEdEq
E
2
0 0
22
)()(2 2
0 0
MzyDMzySDyzy EEEfEEEfNvdEdEqI
Total
capacitance
yyy mEv /2 Longitudinal velocity
Università degli Studi di PisaGiuseppe Iannaccone
Current fluctuations depend on fluctuations of the occupation factors and of the channel barrier:
fluctuations electrostatics
in the contacts
Current fluctuations expressed as a function of contact fluctuations
Current Fluctuations
M
D
M
SDyzyMDSDyzy E
f
E
fNvdEdEEqffNvdEdEqI 2
0 02
0 0
22
DyQDQSC
QSSQDDS
yQDQSC
QDDQSS
Dyzy
fvCCC
CvCvf
vCCC
CvCv
NvdEdEqI
11
11
2 2
0 0
Università degli Studi di PisaGiuseppe Iannaccone
Shot noise power spectral density Power spectral density:
Far from equilibrium, if fD = 0 , we have CQD = 0, and
SSS fff 1δ 2 DDD fff 1δ 2
DDyQDQSC
QSSQDDSS
yQDQSC
QDDQSS
Dyzy
ffvCCC
CvCvff
vCCC
CvCv
NvdEdEqS
11
111
1
4
22
2
0 0
2
SSyQSC
QSSDyzy ff
vCC
CvNvdEdEqS
11
14
2
2
0 0
2
SDyzy fNvdEdEqI 2
0 0
2
SyQSC
QSS fvCC
Cv
qI
S
1
11
2
2
Università degli Studi di PisaGiuseppe Iannaccone
Noise suppression factor (Fano factor)
Fano factor is always < 1 and
If CC is very large (e.g., large gate capacitance) then Coulomb interaction is completely screened:
For Maxwell-Boltzmann statistics (e.g. below threshold), fS << 1
SyQSC
QSS fvCC
Cv
qI
S
1
11
2
2
Effect of Pauli
Exclusion
Effect of Coulomb Interaction
1CC
Sf 1
SC f 1
22
11
QSC
C
yQSC
QSS
CC
C
vCC
Cv
Università degli Studi di PisaGiuseppe Iannaccone
25 nm “Well tempered” MOSFET
Doping Profile of the 25 nm “well tempered” MOSFET (D. Antoniadis)
Effective channel length 25 nm Super-halo doping in the channel
minimizes charge sharing effects
Lowest subband profile from 2D PS solver (G. Fiori et al., APL 81, 3672 (2002))
With Vg=1V, Vds=0.1 V, 96.5% of current is carried by the 1st subband
Università degli Studi di PisaGiuseppe Iannaccone
Subband Maximum and Source Quantum Capacitance CQS
Università degli Studi di PisaGiuseppe Iannaccone
Shot noise suppression in well tempered MOSFETs
VDS = 0.5 V VDS = 1 V
Università degli Studi di PisaGiuseppe Iannaccone
Noise in the partially ballistic MOSFETs (I)(with G. Mugnaini)
the first N-1 MOSFETs can be aggregated in an equivalent drift-diffusion MOSFET. (G. Mugnaini et al., submitted to IEEE-TED).
The channel of an arbitrary MOSFET is decomposed in a chain of ballistic MOSFETs of length the mean free path.
Università degli Studi di PisaGiuseppe Iannaccone
Noise in the partially ballistic MOSFETs (II)(with G. Mugnaini - preliminary)
Thermal noise source + shot noise source As the ratio between the device length and the mean free path is reduced,
Noise has a transition THERMAL SHOT Presently including the effect of electrostatics on noise
N=L/
Università degli Studi di PisaGiuseppe Iannaccone
Gate currents: Fresh and stressed oxidesExperimental results by F. Crupi –
From G. Iannaccone et al. IEEE-TED 50, 1363 (2003)
Stress voltage 7.8 V(8 V is the breakdown voltage) SILCs should introduce alter also the noise properties
Cu
rre
nt (
A)
2 3 4 510
-13
10-12
10-11
10-10
10-9
10-8
10-7
SILC
Fresh
Voltage (V)20 50 80 110 140 170 20010
-29
10-28
10-27
10-26
10-25
10-24
IDC = 300 nA
IDC = 30 nA
IDC = 3 nA
IDC = 300 pA
Full Shot Noise Experimental Data
Si [
A2 /H
z]
Frequency [Hz]
I-V characteristics6 nm oxide
Noise properties of thecurrent through fresh oxides:full shot noise at large currents
Università degli Studi di PisaGiuseppe Iannaccone
Current through fresh oxides(tunneling + native TAT)
theory and exp.Trap distribution is a gaussian centered at 1.8 eV below the oxide CB, with 0.1 eV standard deviation
3 4 5 610-11
10-10
10-9
10-8
10-7
10-6
10-5
10-4
10-3
tox
= 5.9 nm
Cu
rren
t D
en
sit
y (
A/c
m2 )
Vg (volt)
experimental data fresh native total
5 6 7 8 9 1010-1110-1010-910-810-710-610-510-410-310-2
tox
= 9.9 nm
Vg (volt)
Cu
rren
t D
en
sit
y (
A/c
m2 )
experimental data fresh native total
Exp. By F. Crupi
Università degli Studi di PisaGiuseppe Iannaccone
TAT modelG. Iannaccone et al. IEEE-TED 50, 1363 (2003
• Generation and Recombination rates2g f d
1r f d
• Trap occupation factor
1 2
1 2 1 2
'g g
fg g r r
1 1' 1 ' 'I qg f qr f
1 2
2
1 1 2 2
2' 2 ' 1 2 'SILC
g rS qI qI
g r g r
FANO Factor SILC
' , ' 'TATJ I E x dE dx
' , ' 'TATS S E x dE dx
Università degli Studi di PisaGiuseppe Iannaccone
Extraction of SILC trap distributionComparison with experiments
6 nm oxide Gaussian distribution centered -0.5 eV below Si CB, standard deviation 82 meV
2 3 4 5 610-11
10-10
10-9
10-8
10-7
10-6
10-5
10-4
10-3
2 3 4 5 60.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
Cu
rren
t d
ensi
ty (
A/c
m2 )
VG (volt)
Fan
o F
acto
rV
G (volt)
Università degli Studi di PisaGiuseppe Iannaccone
Extraction of SILC trap distribution (V)
4 5 6 7 8 910-10
10-9
10-8
10-7
10-6
10-5
10-4
experimental not assisted native SILCs total
Cu
rren
t D
en
sit
y (
A/c
m2 )
VG (volt)
For thicker oxides shot noise suppression is due to transitions through native traps
4 5 6 7 8 9 1010
-10
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
tox
= 9.9 nm
fresh after stress
0.5 V
Cu
rre
nt
de
ns
ity
(A
/cm2 )
VG (volt)
5 6 7 8 90.70
0.75
0.80
0.85
0.90
0.95
1.00
VG (volt)
Fa
no
fa
cto
r
Università degli Studi di PisaGiuseppe Iannaccone
Conclusion We have derived an analytical expression of noise in
ballistic MOSFETs with two well defined contributions from Pauli exclusion and Coulomb repulsion.
Noise properties can be computed from a numerical simulation of DC electrical properties.
Numerical results for “well tempered” MOSFETs operating in the ballistic regime have been shown, exhibiting room temperature suppression of shot noise, in typical operating conditions, down to 0.25.
Shot Noise of the gate current: contribution of native traps may be important also for noise properties (experiments here are still missing)
For thicker oxides a distribution of traps can be extracted that reproduces both DC and noise characteristcs
Università degli Studi di PisaGiuseppe Iannaccone
Current Fluctuations
Current fluctuations depend on fluctuations of the occupation factors and of the channel barrier:
fluctuations electrostatics
in the contacts
We can introduce two average velocities vS and vD:
M
D
M
SDyzyMDSDyzy E
f
E
fNvdEdEEqffNvdEdEqI 2
0 02
0 0
22
M
SDyzyQSS E
fNvdEdEqCv 2
0 0
22
M
DDyzyQDD E
fNvdEdEqCv 2
0 0
22
Università degli Studi di PisaGiuseppe Iannaccone
Equilibrium and far from equilibrium If fS = fD, S reduces to 4KTG, as it must be, where
Far from equilibrium, if fD = 0 , we have CQD = 0, and
The noise suppression factor is a weighted average
SSyQSC
QSSDyzy ff
vCC
CvNvdEdEqS
11
14
2
2
0 0
2
SSDyzyS
Dyzy ffNvdEdEqdE
dfNvdEdEqG
122 2
0 0
22
0 0
2
SDyzy fNvdEdEqI 2
0 0
2
SyQSC
QSS fvCC
Cv
qI
S
1
11
2
2
Università degli Studi di PisaGiuseppe Iannaccone
Extraction of native and SILC trap Extraction of native and SILC trap distributiondistribution (I)
• Simulations with a distribution uniform in energy do not provide satisfactory results
' , ' 'TATJ I E x dE dx
,E x E
Trap distribution indipendent of position
• Integral equation with ,E x as the unknown
Hp:
Università degli Studi di PisaGiuseppe Iannaccone
Extraction of SILC trap distribution (II)
Electrons from cathode VB
• Model A: Riccò, Gozzi, Lanzoni, IEEE TED 45, 1998.
“mean” quantities: fluxes and capture cross section
• Model B: Ielmini, Spinelli, Lacaita, IEEE TED 47, 2000.
Transient SILC components
-1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5
0.1
1
B A
E-EV (eV)
-1.0 -0.5 0.0 0.5 1.0 1.5 2.0
0.1
1
E-EV (eV)
Università degli Studi di PisaGiuseppe Iannaccone
Extraction of SILC trap distribution (III)
5.9 nm oxide Comparison with exp. performed
in Pisa
Other thicknesses Comparison with experiments
drawn from the literature (Ricco’ et al.)
Effect of surface traps for very low voltages
1 2 3 4 5 610-11
10-10
10-9
10-8
10-7
10-6
10-5
10-4
10-3
tox
= 5.9 nm
Em = 0.5
= 0.082
MAX
1
= 1e-15 m-1
MAX
2
= 1.7e-15 m-1
Cu
rren
t d
en
sit
y (
A/c
m2 )
VG (volt)
2.5 3.0 3.5 4.0 4.510-11
10-10
10-9
10-8
10-7
10-6
10-5
experimental simulation
tox
= 6.4 nmtox
= 5.3 nm
3.0 3.5 4.0 4.5 5.0 5.510-10
10-9
10-8
10-7
10-6
10-5 experimental simulation
VG (volt)
VG (volt)
J-V Curves
Università degli Studi di PisaGiuseppe Iannaccone
Extraction of SILC trap distribution (IV)
1 2 3 4 5 6
0.70
0.75
0.80
0.85
0.90
0.95
1.00
Fan
o F
acto
r
VG (volt)
experiment E
m = 0.5
Em = 0.4
Em = 0.18
Em = 0.01
Fano Factor
Stronger suppression for Em approaching
the silicon gap center
Università degli Studi di PisaGiuseppe Iannaccone
Understanding the nature of SILCs
Stress-induced leakage currents (SILCs) are the single most important limit to downscaling of non-volatile memories [read disturb, retention degradation]
SILCs are due to tunneling assisted by traps generated by electric field stress.
The energy distribution of traps is not known
We show that detailed modeling, coupled with DC and noise characterization, can provide enough information to extract information about the energy distribution of traps
Università degli Studi di PisaGiuseppe Iannaccone
Tunneling Current: fresh oxides
Electron effective mass in the oxide conduction band
Determination of the oxide thickness
Native traps are required for fitting the current at low fields
3.0 3.5 4.0 4.5 5.0 5.5 6.010-1110-1010-910-810-710-610-510-410-310-2
experiments m
e,ox= 0.6 m
0 t
ox= 6.1 nm
me,ox
= 0.6 m0 t
ox= 6 nm
me,ox
= 0.6 m0 t
ox= 5.9 nm
Tu
nn
elin
g C
urr
en
t (A
/cm
2 )
Vg (volt)
4 5 6 7 8 9 1010-10
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
experiments m
e,ox = 0.6 m
0 t
ox= 10 nm
me,ox
= 0.6 m0 t
ox= 9.9 nm
Tu
nn
elin
g C
urr
en
t (A
/cm
2 )
Vg (volt)
Università degli Studi di PisaGiuseppe Iannaccone
, ,, ' l lJ x T E E
• Transition probability
22,M h E E
2 2
1h E E
E E
• Capture cross section
, k h E E
TAT model (I)• Two-Step tunneling• Anelastic TAT
G. Iannaccone et al.,
IEEE-TED 2003
Università degli Studi di PisaGiuseppe Iannaccone
Tunneling Current: Model The electron density n(x) at
the Si-SiO2 interface is computed by solving the Schrödinger equation for the two-fold and four-fold degenerate conduction band minima.
1D Poisson and Schrödinger equations are solved iteratively.
Once the band profiles and charge densities are obtained, we can compute the tunneling current
i
itFit
tl
i
ilFil
t
kT
EEx
mmkT
kT
EEx
kTmxn
exp1ln)(4
exp1ln)(2
)(
22
22
)()()()(2
22
xExxExm ikilikCik
k
)()()()()( xNxNxnxpq AD
TkEE
TkEEmmTkTq
TkEE
TkEETmkTqJ
BFFGit
BFittlB
i it
it
BFFGil
BFiltB
i il
ilT
/)exp(1
/)exp(1ln4
/)(exp1
/)(exp1ln2
2
2
Università degli Studi di PisaGiuseppe Iannaccone
25 nm “Well tempered” MOSFET
Doping Profile of the 25 nm “well tempered” MOSFET (D. Antoniadis)
Effective channel length 25 nm Super-halo doping in the channel
minimizes charge sharing effects
Quantum confinement in the middle of the channel (z = 45 nm)
With Vg=1V, Vds=0.1 V, 96.5% of current is carried by the 1st subband
Università degli Studi di PisaGiuseppe Iannaccone
Subband profile and characteristics
2D simulation First subband profile in the longitudinal
direction for increasing Vds.
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40
100
200
300
400
500
600
700
800Si super-halo doping
1.21.251.3
1.35
1.4
1.45
1.5VGS
[V] =
I DS [A
/m]
VDS
[V]
Università degli Studi di PisaGiuseppe Iannaccone
Papers on the subject Y. Naveh, A. N. Korotkov, K. K. Likharev, Shot noise suppression in
multimode ballistic Fermi conductors, Phys. Rev. B, 60 (1998), R2169-2172.
O. M. Bulashenko and J. M. Rubì, Shot-noise suppression by Fermi and Coulomb correlations in ballistic conductors, Phys. Rev. B, 65 (2001) 045307.
O. M. Bulashenko and J. M. Rubì, Self-consistent theory of current and voltage noise in multimode ballistic conductors, Phys. Rev. B, 66 (2002), 045310.
G. Gomila, I. R. Cantalapiedra, T. Gonzalez, L. Reggiani, Semiclassical Theory of shot noise in ballistic n+-i-n+ semiconductor structures: Relevance of Pauli and long-range Coulomb correlations, Phys. Rev. B, 66 (2002) 075302.
Università degli Studi di PisaGiuseppe Iannaccone
Far-from equilibrium transport in each ballistic MOSFET, gives a local shot noise [van der Ziel,1986]:
If the chain is long enough, local equilibrium holds in the whole sructure and then local shot noise reduces to conventional thermal noise:
Similarly to the aboveseen current macromodel, a noise macromodel for a device operating in intermediate tranposrt regime, is given by the series of a thermal noise generator with a shot noise generator.We expect that when the ratio between the device length and the mean free path reduces, a more pronounced far-from equilibrium behavior emerges both in the static current and in the noise.
Noise in the partially ballistic MOSFETs (II)(with G. Mugnaini)
Università degli Studi di PisaGiuseppe Iannaccone
Pauli and Coulomb interactions
In most cases interactions make the collective motion more regular
Limits density in real space
Limits density in phase space
Reduced fluctuations
Sub-poissonian process fullSS
Università degli Studi di PisaGiuseppe Iannaccone
Fully ballistic transport regime
Electrons with sufficient energy to overcome the barrier near the source reach the drain conserving energy and transversal momentum
Electron states originating from the source obey the Fermi-Dirac statistics with source Fermi Energy Efs
Electron states originating from the drain obey the Fermi-Dirac statistics with drain Fermi Energy Efd
This ensures continuity of current density per unit energy in each subband
Università degli Studi di PisaGiuseppe Iannaccone
Model
Poisson equation in 2D
The electron density n() in the quantum region is obtained from the solution of the Schrödinger equation with density functional theory
p(), ND+(), NA
-() and of n() out of the quantum region are given by the corresponding semiclassical expressions
)()()()()( DA NNpnqr
Discretization with the box integration method Newton-Raphson method with predictor-corrector iteration scheme
Università degli Studi di PisaGiuseppe Iannaccone
The Schrödinger equation must be solved twice: For the 2 minima along the vertical (x) direction
For the other 4 minima
The quantum electron density becomes
x
yz
lililiClil
yE~Ey,xxmx
1
2
2
tititiCtit
yE~Ey,xxmx
1
2
2
kx
kz
ky
kx
ky
kz
Mass anisotropy and electron density
i i
tili nnn 42
y
EE
FDiyFSiyylli
y
EE
FSiyyllili
dEEEEfEEEf
ED
dEEEEfEDn
ii
ii
~
2
~
0
2
max
max
2
~~
~
Università degli Studi di PisaGiuseppe Iannaccone
Model out of equilibrium
When the Poisson-Schrödinger equation is solved, and charge density and potential profiles are known, we compute the current density in the i-th subband
The total current density is
limaxi E~Ey
FDyliFSylitli dE
kT
EEE~F
kT
EEE~F
h
kTmqJ
21
2122
12
i i
ttii
ltili JJJJ 222
Università degli Studi di PisaGiuseppe Iannaccone
Examples of nanotransistors
INTEL, in production now oxide thickness 2 nm
INTEL test device In production by 2005 (ITRS 2002 update oxide thickness 0.8 nm
Università degli Studi di PisaGiuseppe Iannaccone
Summary
Motivation: VLSI devices are already nanoelectronic devices !!
Mesoscopic Noise in MOSFETs Shot Noise of the drain current in ballistic MOSFETs Shot Noise of the gate current in fresh oxides and in
the case of tunneling assisted by traps Conclusion
Acknowlegments:
F. Crupi, A. Nannipieri, G. Curatola, G. Fiori
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