semi-classical transport theory
DESCRIPTION
Semi-Classical Transport Theory. Outline:. What is Computational Electronics? Semi-Classical Transport Theory Drift-Diffusion Simulations Hydrodynamic Simulations Particle-Based Device Simulations Inclusion of Tunneling and Size-Quantization Effects in Semi-Classical Simulators - PowerPoint PPT PresentationTRANSCRIPT
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Semi-Classical Transport Theory
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Outline: What is Computational Electronics?
Semi-Classical Transport Theory Drift-Diffusion Simulations Hydrodynamic Simulations Particle-Based Device Simulations
Inclusion of Tunneling and Size-Quantization Effects in Semi-Classical Simulators Tunneling Effect: WKB Approximation and Transfer Matrix Approach Quantum-Mechanical Size Quantization Effect
Drift-Diffusion and Hydrodynamics: Quantum Correction and Quantum Moment Methods
Particle-Based Device Simulations: Effective Potential Approach
Quantum Transport Direct Solution of the Schrodinger Equation (Usuki Method) and Theoretical Basis
of the Green’s Functions Approach (NEGF) NEGF: Recursive Green’s Function Technique and CBR Approach Atomistic Simulations – The Future
Prologue
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Direct Solution of the Boltzmann Transport Equation
Particle-Based Approaches Spherical Harmonics Numerical Solution of the Boltzmann-Poisson
Problem
In here we will focus on Particle-Based (Monte Carlo) approaches to solving the Boltzmann Transport Equation
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Ways of Solving the BTE Using MCT
Single particle Monte Carlo Technique Follow single particle for long enough time to
collect sufficient statistics Practical for characterization of bulk materials
or inversion layersEnsemble Monte Carlo Technique
MUST BE USED when modeling SEMICONDUCTOR DEVICES to have the complete self-consistency built in
Carlo Jacoboni and Lino Reggiani, The Monte Carlo method for the solution of charge transport in semiconductors with applications to covalent materials, Rev. Mod. Phys. 55, 645 - 705 (1983).
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Path-Integral Solution to the BTE
The path integral solution of the Boltzmann Transport Equation (BTE), where L=Nt and tn=nt, is of the form:
1( )
0
( ) ( ') ( ', ( ) )N
N m tN m eff
m
f t t f p S p p eE N m t e
( ( ) )mg p eE N m t
K. K. Thornber and Richard P. Feynman, Phys. Rev. B 1, 4099 (1970).
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The two-step procedure is then found by using N=1, which means that t=t, i.e.:
1 0'
( ) ( ') ( ', ) teff
p
f t t f p S p p eE t e
0 ( )g p eE t
Intermediate function that describesthe occupancy of the state (p+eEt)at time t=0, which can be changeddue to scattering events (SCATTER)
Integration over a trajectory, i.e.probability that no scattering occurred within time integral t (FREE FLIGHT)
+
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Monte Carlo Approach to Solving the Boltzmann Transport Equation Using path integral formulation to the
BTE one can show that one can decompose the solution procedure into two components:
1. Carrier free-flights that are interrupted by scattering events
2. Memory-less scattering events that change the momentum and the energy of the particle instantaneously
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Particle Trajectories in Phase Space
Particle trajectories in k-space and real space
yk
xk
-e
x
x
x x
x x
x x
x
x
xEy
x
y
x
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Carrier Free-Flights The probability of an electron scattering in a small time interval dt is
(k)dt, where (k) is the total transition rate per unit time. Time dependence originates from the change in k(t) during acceleration by external forces
where v is the velocity of the particle. The probability that an electron has not scattered after scattering at t =
0 is:
It is this (unnormalized) probability that we utilize as a non-uniform distribution of free flight times over a semi-infinite interval 0 to . We want to sample random flight times from this non-uniform distribution using uniformly distributed random numbers over the interval 0 to 1.
t
ttd
n etP 0)(k
/0 tet BvEkk
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Generation of Random Flight TimesHence, we choose a random number
t
ttd
i er 01,
k
Ith particle first random number
We have a problem with this integral!
We solve this by introducing a new, fictitious scattering process which does not change energy or momentum:
)()()()( kkxx Ek Sss
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Generation of Random Flight Times
t
ttd
i er 01,
k
i
i kk )()( The sum runs over all the real scattering processes. To this we add the fictitious self-scattering which is chosen to have a nice property:0new
scatterersreal
iss kk )()( 0
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• The use of the full integral form of the free-flight probability density function is tedious (unless k is invariant during the free flight).
• The introduction of self-scattering (Rees, J. Phys. Chem. Solids 30, 643, 1969) simplifies the procedure considerably.
• The properties of the self-scattering mechanism are that it does not change either the energy or the momentum of the particle.
• The self-scattering rate adjusts itself in time so that the total scattering rate is constant. Under these circumstances, one has that:
dtedtedttPtt ttd
self
t
0kk
Self-Scattering
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Self-Scattering• Random flight times tr may be generated from P(t) above using
the direct method to get:
where r is a uniform random between 0 and 1 (and therefore r and 1-r are the same).
• must be chosen (a priori) such that > (k(t)) during the entire flight.
• Choosing a new tr after every collision generates a random walk in k-space over which statistics concerning the occupancy of the various states k are collected.
rrter rtr lnln
111
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Free-Flight Scatter Sequence for Ensemble Monte Carlo
= collisions
1n23456
N
1,it
1,itHowever, we need a second time scale, which provides the times at which the ensemble is “stopped” and averages are computed.
Particle time scale
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Free-FlightScatterSequence
dte=dtau
dte ≥ t?
no yes
dt2 = dte dt2 = t
Call drift(dt2)
dte ≥ t?yes
dte2 = dte
Call scatter_carrier()
Generate free-flight dt3
dtp=t-dte2
dt3 ≤ dtp?
no yes
dt2 = dtp dt2 = dt3
Call drift(dt2)
dte2=dte2+dt3dte=dte2
no
yesdte < t ?
dte=dte-t
dtau=dte
R. W. Hockney and J. W. Eastwood, Computer Simulation Using Particles, 1983.
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Choice of Scattering Event Terminating Free Flighto At the end of the free flight ti, the type of scattering which ends the
flight (either real or self-scattering) must be chosen according to the relative probabilities for each mechanism.
o Assume that the total scattering rate for each scattering mechanism is a function only of the energy, E, of the particle at the end of the free flight
where the rates due to the real scattering mechanisms are typically stored in a lookup table.
o A histogram is formed of the scattering rates, and a random number r is used as a pointer to select the right mechanism. This is schematically shown on the next slide.
EEEE popaciself
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Choice of Scattering Event Terminating Free-Flight
We can make a table of the scattering processes at the energy of the particle at the scattering time:
itE121
321
4321
Self
54321
0
1
32
4
5
Selection process for scattering
0r
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21 30EE2E3
E4
Look-up table of scattering rates:
Store the total scattering rates in a table for a grid in energy
………
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Choice of the Final State After Scattering
Using a random number and probability distribution function
Using analytical expressions (slides that follow)
0 0.5 1 1.5 2 2.5 3 3.50
1
2
3
4
5
6x 10
4
Polar angle
Arb
itrar
y U
nits
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1. Isotropic scattering processescos 1 2 , 2r r
2. Anisotropic scattering processes (Coulomb, POP)
kx
ky
kz
k0
0
Step 1:Determine 0 and 0
kx’
ky’
kz’
kStep 2:Assume rotatedcoordinatesystem
Step 3:perform scattering
0
2
0
21 1 2cos ,
rk k
k k
E E
E E
POP
2 2
2cos 11 4 (1 )D
rk L r
Coulomb
=2r for both
kx’
ky’
kz’
k
k’
Step 4:kxp = k’sin cos, kyp = k’sin*sin, kzp = k’cos
Return back to the original coordinate system:kx = kxpcos0cos0-kypsin0+kzpcos0sin0ky = kxpsin0cos0+kypcos0+kzpsin0sin0kz = -kxpsin0+kzpcos0
k’≠k forinelastic
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1. Isotropic scattering processescos 1 2 , 2r r
2. Anisotropic scattering processes (Coulomb, POP)
kx
ky
kz
k0
0
Step 1:Determine 0 and 0
kx’
ky’
kz’
kStep 2:Assume rotatedcoordinatesystem
Step 3:perform scattering
0
2
0
21 1 2cos ,
rk k
k k
E E
E E
POP
2 2
2cos 11 4 (1 )D
rk L r
Coulomb
=2r for both
kx’
ky’
kz’
k
k’
Step 4:kxp = k’sin cos, kyp = k’sin*sin, kzp = k’cos
Return back to the original coordinate system:kx = kxpcos0cos0-kypsin0+kzpcos0sin0ky = kxpsin0cos0+kypcos0+kzpsin0sin0kz = -kxpsin0+kzpcos0
k’≠k forinelastic
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Representative Simulation Results From Bulk Simulations - GaAs
k-vector
-valley
X-valley [100]L-valley [111]
Conduction bands
Valence bands
-valley table-Mechanism1-Mechanism2- …-MechanismN
L-valley table-Mechanism1-Mechanism2- …-MechanismNL
X-valley table-Mechanism1-Mechanism2- …-MechanismNx
Define scattering mechanisms for each valley
Call specified scattering mechanisms subroutines
Renormalize scattering tablesSimulation Results Obtained byD. Vasileska’s Monte Carlo Code.
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parameters initializationreadin()
scattering table constructionsc_table()
histograms calculationhistograms()
Free-Flight-Scatterfree_flight_scatter()
histograms calculationhistograms()
write datawrite()
?
carriers initializationinit()
t t t Time t exceeds maximum simulation time tmax
yes
no
Optional
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Energy [eV]
Cum
ulat
ive
rate
[1/s
]
-6 -4 -2 0 2 4 6
x 108
0
5
10
15
20
25
30
35
40
Wavevector ky [1/m]
Arb
itrar
y U
nits
Initial Distribution of thewavevector along the y-axisthat is created with thesubroutine init()
0 0.05 0.1 0.15 0.2 0.25 0.3 0.350
10
20
30
40
50
60
70
80
90
Energy [eV]
Arb
itrar
y U
nits
Initial Energy Distribution createdwith the subroutine init()
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110
10
1011
1012
1013
1014
1015
Energy [eV]
Sca
tterin
g R
ate
[1/s
]
acoustic
polar optical phonons
intervalley gamma to L
intervalley gamma to X
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parameters initializationreadin()
scattering table constructionsc_table()
histograms calculationhistograms()
Free-Flight-Scatterfree_flight_scatter()
histograms calculationhistograms()
write datawrite()
?
carriers initializationinit()
t t t Time t exceeds maximum simulation time tmax
yes
no
Optional
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110
10
1011
1012
1013
1014
1015
Energy [eV]
Sca
tterin
g R
ate
[1/s
]
acoustic
polar optical phonons
intervalley gamma to L
intervalley gamma to X
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Energy [eV]
Cum
ulat
ive
rate
[1/s
]
-6 -4 -2 0 2 4 6
x 108
0
5
10
15
20
25
30
35
40
Wavevector ky [1/m]
Arb
itrar
y U
nits
Initial Distribution of thewavevector along the y-axisthat is created with thesubroutine init()
0 0.05 0.1 0.15 0.2 0.25 0.3 0.350
10
20
30
40
50
60
70
80
90
Energy [eV]
Arb
itrar
y U
nits
Initial Energy Distribution createdwith the subroutine init()
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Transient Data
0 1 2
x 10-11
0
0.5
1
1.5
2
2.5
3
3.5
4x 10
5
time [s]
velo
city
[m/s
]
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Steady-State Results
0 1 2 3 4 5 6 7
x 105
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2x 10
5
Electric Field [V/m]
Drif
t Vel
ocity
[m/s
]
0 1 2 3 4 5 6 7
x 105
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
Electric Field [V/m]
Con
duct
ion
Ban
d V
alle
y O
ccup
ancy gamma valley occupancy
L valley occupancy
X valley occupancy
k-vector
-valley
X-valley [100]L-valley [111]
Conduction bands
Valence bandsGunn Effect
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Particle-Based Device Simulations
In a particle-based device simulation approach the Poisson equation is decoupled from the BTE over a short time period dt smaller than the dielectric relaxation time Poisson and BTE are solved in a time-marching
manner During each time step dt the electric field is
assumed to be constant (kept frozen)
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Particle-Mesh CouplingThe particle-mesh coupling scheme consists of the following steps:
- Assign charge to the Poisson solver mesh- Solve Poisson’s equation for V(r) - Calculate the force and interpolate it to the particle locations - Solve the equations of motion:
r1 Ek krk
qdtdtE
dtd ;
Laux, S.E., On particle-mesh coupling in Monte Carlo semiconductor device simulation, Computer-Aided Design of Integrated Circuits and Systems, IEEE Transactions on, Volume 15, Issue 10, Oct 1996 Page(s):1266 - 1277
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Assign Charge to the Poisson Mesh
1. Nearest grid point scheme
2. Nearest element cell scheme
3. Cloud in cell scheme
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Force interpolation
The SAME METHOD that is used for the charge assignment has to be used for the FORCE INTERPOLATION:
,r r r r F Ei i p pp
q W
xp-1 xp xp+1
epxw
px
wp
ppep
pp
xVV
xVV 11
21
pE
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Treatment of the Contacts
From the aspect of device physics, one can distinguish between the following types of contacts:
(1) Contacts, which allow a current flow in and out of the device
- Ohmic contacts: purely voltage or purely current controlled
- Schottky contacts
(2) Contacts where only voltages can be applied
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Calculation of the Current
The current in steady-state conditions is calculated in two ways: By counting the total number of particles that
enter/exit particular contact By using the Ramo-Shockley theorem
according to which, in the channel, the current is calculated using
1
( ),N
xi
eI v idL
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Current Calculated by Counting the Net Number of Particles Exiting/Entering a Contact
Electrons that naturally came out in time interval dt (N1)Electrons that were deleted (N2)Electrons that were injected (N3)
dq = q(N1+N2-N3), q(t+dt)=q(t) + dq, current equals the slope of q(t) vs. t
Source DrainGate
Mesh nodeElectronDopant
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Device Simulation Results for MOSFETs: Current Conservation
0
1000
2000
3000
4000
5000
6000
7000
1 1.5 2 2.5 3
source contactdrain contact
net #
of e
lect
rons
exiti
ng/e
nter
ing
cont
act
time [ps]
WG = 0.5 mm
(a)
0
0.1
0.2
0.3
0.4
0.5
0.6
0 50 100 150
Cur
rent
ID [
mA
/mm
]Distance [nm]
(b)
VG=1.4 V, VD=1 V
Cumulative net number of particlesEntering/exiting a contact for a 50 nmChannel length device
Drain contactSource contact
Current calculated using Ramo-Schockleyformula
1
( ),N
xi
eI v idL
X. He, MS, ASU, 2000.
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Simulation Results for MOSFETs: Velocity and Enery Along the Channel
0
5x106
1x107
1.5x107
2x107
2.5x107
0 50 100 150
Drif
t vel
ocity
[cm
/s]
Distance [nm]
(a)
Mean Drift Velocity Along the Channel
0
0.1
0.2
0.3
0.4
0.5
0.6
0 50 100 150
Ave
rage
ene
rgy
[eV
]Distance [nm]
(b)
Average Kinetic Energy Along the Channel
VD = 1 V, VG = 1.2 V
Velocity overshoot effect observed throughout the whole channel length of the device – non-stationary transport.
For the bias conditions used average electron energy is smaller that 0.6 eV which justifies the use of non-parabolic band model.
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Simulation Results for MOSFETs: IV Characteristics
0
0.1
0.2
0.3
0.4
0.5
0 0.2 0.4 0.6 0.8 1
Dra
in c
urre
nt [
mA
/mm
]
Drain voltage VD [V]
1.2 V
0.8 V
1.0 V
Silvaco simulations
VG = 1.0 V
VG = 1.4 V
2D MCPS
VG = 1.4 V
The differences between the Monte Carlo and the Silvaco simulations are due to the following reasons:
• Different transport models used (non-stationary transport is taking place in this device structure).
• Surface-roughness and Coulomb scattering are not included in the theoretical model used in the 2D-MCPS.
X. He, MS, ASU, 2000.
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Simulation Results For SOI MESFET Devices – Where are the Carriers?
Nd = 1019 Na =3-10x 1015 Nd = 1019
Si Substrate
Lg =60, 100nm Lg =60, 100nm
Oxide Layer
Nd = 1019 Nd = 3-10x1015 Nd =1019
Si Substrate
Oxide Layer
SOIMOSFET
SOIMESFET
Applications:
Low-power RF electronics.T.J. Thornton, IEEE Electron Dev. Lett., 8171 (1985).
Micropowercircuits basedon weakly inverted
Implantablecochlea and retina
Digital Watch
Pacemaker
MOSFETs
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Proper Modeling of SOI MESFET Device
Gate current calculation: WKB Approximation Transfer Matrix Approach
for piece-wise linear potentials
Interface-Roughness: K-space treatment Real-space treatment
Goodnick et al., Phys. Rev. B 32, 8171 (1985)
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107
108
109
1010
1011
1012
0.1 1 10 100 1000
Lg =25nm simulated resultsLg =50nm simulated resultsLg = 90nm simulated resultsLg = 0.6um Experimental resultsLg = 50nm Projected Experimental results
Drain Current Id [ µA/µm]
Cut
off F
requ
ency
f T [Hz]
Output Characteristics and Cut-off Frequency of a Si MESFET Device
-1000
-800
-600
-400
-200
0
200
400
-0.2 0 0.2 0.4 0.6 0.8 1
Vgs
= 0.3VV
gs = 0.4V
Vgs
=0.5VV
gs = 0.6V
I d [µA
/µm]
Vds
[Volts]
Tarik Khan, PhD, ASU, 2004.
![Page 40: Semi-Classical Transport Theory](https://reader036.vdocuments.mx/reader036/viewer/2022081511/56815b62550346895dc94ba0/html5/thumbnails/40.jpg)
Output Characteristics and Cut-off Frequency of a Si MESFET Device
-1000
-800
-600
-400
-200
0
200
400
-0.2 0 0.2 0.4 0.6 0.8 1
Vgs
= 0.3VV
gs = 0.4V
Vgs
=0.5VV
gs = 0.6V
I d [µA
/µm]
Vds
[Volts]107
108
109
1010
1011
1012
0.1 1 10 100 1000
Lg =25nm simulated resultsLg =50nm simulated resultsLg = 90nm simulated resultsLg = 0.6um Experimental resultsLg = 50nm Projected Experimental results
Drain Current Id [ µA/µm]
Cut
off F
requ
ency
f T [Hz]
Tarik Khan, PhD, ASU, 2004.
![Page 41: Semi-Classical Transport Theory](https://reader036.vdocuments.mx/reader036/viewer/2022081511/56815b62550346895dc94ba0/html5/thumbnails/41.jpg)
Modeling of SOI Devices
When modeling SOI devices lattice heating effects has to be accounted for
In what follows we discuss the following: Comparison of the Monte Carlo, Hydrodynamic
and Drift-Diffusion results of Fully-Depleted SOI Device Structures*
Impact of self-heating effects on the operation of the same generations of Fully-Depleted SOI Devices
*D. Vasileska. K. Raleva and S. M. Goodnick, IEEE Trans. Electron Dev., in press.
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FD-SOI Devices:Monte Carlo vs. Hydrodynamic vs. Drift-Diffusion
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
2.5
3
Drain Voltage [V]
Dra
in C
urre
nt [m
A/u
m]
DD SRHD SRMonte Carlo
0 0.2 0.4 0.6 0.8 1 1.20
0.5
1
1.5
2
2.5
Drain Voltage [V]
Dra
in C
urre
nt [m
A/u
m]
DD SRHD SRMonte Carlo
0 0.2 0.4 0.6 0.8 1 1.2 1.40
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Drain Voltage [V]
Dra
in C
urre
nt [m
A/u
m]
DD SRHD SRMonte Carlo
Source Drain
Gate oxide
BOX
tox
tsi
tBOX
LS Lgate LD
feature 14 nm 25 nm 90 nm
Tox 1 nm 1.2 nm 1.5 nm
VDD 1V 1.2 V 1.4 V
Overshoot EB/HD
233% / 224% 139% / 126% 31% /21%
Overshoot EB/DD with series resistance
153%/96% 108%/67% 39%/26%
Source/drain doping = 1020 cm-3 and 1019 cm-3 (series resistance (SR) case) Channel doping = 1E18 cm-3
Overshoot= (IDHD-IDDD)/IDDD (%) at on-state
Silvaco ATLAS simulations performed by Prof. Vasileska.
90 nm
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FD-SOI Devices:Monte Carlo vs. Hydrodynamic vs. Drift-Diffusion
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
2.5
3
Drain Voltage [V]
Dra
in C
urre
nt [m
A/u
m]
DD SRHD SRMonte Carlo
0 0.2 0.4 0.6 0.8 1 1.2 1.40
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Drain Voltage [V]
Dra
in C
urre
nt [m
A/u
m]
DD SRHD SRMonte Carlo
Source Drain
Gate oxide
BOX
tox
tsi
tBOX
LS Lgate LD
feature 14 nm 25 nm 90 nm
Tox 1 nm 1.2 nm 1.5 nm
VDD 1V 1.2 V 1.4 V
Overshoot EB/HD
233% / 224% 139% / 126% 31% /21%
Overshoot EB/DD with series resistance
153%/96% 108%/67% 39%/26%
Source/drain doping = 1020 cm-3 and 1019 cm-3 (series resistance (SR) case) Channel doping = 1E18 cm-3
Overshoot= (IDHD-IDDD)/IDDD (%) at on-state
Silvaco ATLAS simulations performed by Prof. Vasileska.
0 0.2 0.4 0.6 0.8 1 1.20
0.5
1
1.5
2
2.5
Drain Voltage [V]
Dra
in C
urre
nt [m
A/u
m]
DD SRHD SRMonte Carlo
25 nm
![Page 44: Semi-Classical Transport Theory](https://reader036.vdocuments.mx/reader036/viewer/2022081511/56815b62550346895dc94ba0/html5/thumbnails/44.jpg)
FD-SOI Devices:Monte Carlo vs. Hydrodynamic vs. Drift-Diffusion
0 0.2 0.4 0.6 0.8 1 1.20
0.5
1
1.5
2
2.5
Drain Voltage [V]
Dra
in C
urre
nt [m
A/u
m]
DD SRHD SRMonte Carlo
0 0.2 0.4 0.6 0.8 1 1.2 1.40
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Drain Voltage [V]
Dra
in C
urre
nt [m
A/u
m]
DD SRHD SRMonte Carlo
Source Drain
Gate oxide
BOX
tox
tsi
tBOX
LS Lgate LD
feature 14 nm 25 nm 90 nm
Tox 1 nm 1.2 nm 1.5 nm
VDD 1V 1.2 V 1.4 V
Overshoot EB/HD
233% / 224% 139% / 126% 31% /21%
Overshoot EB/DD with series resistance
153%/96% 108%/67% 39%/26%
Source/drain doping = 1020 cm-3 and 1019 cm-3 (series resistance (SR) case) Channel doping = 1E18 cm-3
Overshoot= (IDHD-IDDD)/IDDD (%) at on-state
Silvaco ATLAS simulations performed by Prof. Vasileska.
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
2.5
3
Drain Voltage [V]
Dra
in C
urre
nt [m
A/u
m]
DD SRHD SRMonte Carlo
14 nm
![Page 45: Semi-Classical Transport Theory](https://reader036.vdocuments.mx/reader036/viewer/2022081511/56815b62550346895dc94ba0/html5/thumbnails/45.jpg)
FD-SOI Devices:Why Self-Heating Effect is Important?
1. Alternative materials (SiGe)2. Alternative device designs (FD SOI, DG,TG, MG, Fin-FET transistors
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FD-SOI Devices:Why Self-Heating Effect is Important?
dS
~ 300nm
A. Majumdar, “Microscale Heat Conduction in Dielectric Thin Films,” Journal of Heat Transfer, Vol. 115, pp. 7-16, 1993.
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Conductivity of Thin Silicon Films – Vasileska Empirical Formula
300 400 500 600
20
40
60
80
Temperature (K)
Ther
mal c
ondu
ctivi
ty (
W/m
/K)
experimental datafull lines: BTE predictionsdashed lines: empirical modelthin lines: Sondheimer
20nm
30nm50nm
100nm
/ 23
00
2( ) ( ) sin 1 exp cosh2 ( )cos 2 ( ) cos
a a zz T dT T
0( ) (300 / )T T
0 2
135( ) W/m/KTa bT cT
0 2 4 6 8 10105
7.5
10
12.5
15
17.5
2020
Distance from Si/gate oxide interface (nm)
Ther
mal c
ondu
ctivi
ty (
W/m
-K)
300K400K600K
SiBOX
10nm
0 20 40 60 80 1000102030405060708080
Distance from Si/gate oxide interface (nm)
Ther
mal c
ondu
ctivi
ty (
W/m
-K)
20nm30nm50nm100nm
![Page 48: Semi-Classical Transport Theory](https://reader036.vdocuments.mx/reader036/viewer/2022081511/56815b62550346895dc94ba0/html5/thumbnails/48.jpg)
Particle-Based Device Simulator That Includes Heating
Average and smooth: electron density, drift velocity and electron
energy at each mesh point
end of MCPS phase?
Acoustic and Optical Phonon Energy Balance Equations Solver
end of simulation?
end
no
yes
Define device structure
Generate phonon temperature dependent scattering tables
Initial potential, fields, positions and velocities of carriers
t = 0
t = t + t
Transport Kernel (MC phase)
Field Kernel (Poisson Solver)
t = n t?
yes
Average and smooth: electron density, drift velocity and electron
energy at each mesh point
end of MCPS phase?
end of MCPS phase?
Acoustic and Optical Phonon Energy Balance Equations Solver
end of simulation?
end of simulation?
end
no
yes
Define device structure
Generate phonon temperature dependent scattering tables
Initial potential, fields, positions and velocities of carriers
t = 0
t = t + t
Transport Kernel (MC phase)
Field Kernel (Poisson Solver)
t = n t?
Define device structure
Generate phonon temperature dependent scattering tables
Initial potential, fields, positions and velocities of carriers
t = 0
t = t + t
Transport Kernel (MC phase)
Field Kernel (Poisson Solver)
t = n t?t = n t?
yes
![Page 49: Semi-Classical Transport Theory](https://reader036.vdocuments.mx/reader036/viewer/2022081511/56815b62550346895dc94ba0/html5/thumbnails/49.jpg)
Heating vs. Different Technology Generation
25 nm FD SOI nMOSFET (Vgs=Vds=1.2V)
10 20 30 40 50 60 70
3
8
131345 nm FD SOI nMOSFET (Vgs=Vds=1.2V)
20 40 60 80 100 120
3
12
212160 nm FD SOI nMOSFET (Vgs=Vds=1.2V)
20 40 60 80 100 120 140 160 180
3
15
2727
300
400
500
300
400
500
300
400
500
source contact drain contact
source region drain region
80 nm FD SOI nMOSFET (Vgs=Vds=1.5V)
50 100 150 200
3
20
353590 nm FD SOI nMOSFET (Vgs=Vds=1.5V)
50 100 150 200 250
3
21
3939100 nm FD SOI nMOSFET (Vgs=Vds=1.5V)
50 100 150 200 250 300
3
23
4343
300
400
500
600
300400500
300
400
500
120 nm FD SOI nMOSFET (Vgs=Vds=1.8V)
0 120 240 360360
4
28
5252140 nm FD SOI nMOSFET (Vgs=Vds=1.8V)
50 100 150 200 250 300 350 400
4
33
6060
x (nm)
180 nm FD SOI nMOSFET (Vgs=Vds=1.8V)
100 200 300 400 500
4
41
7676400
600
400
600
300400500600
25 nm FD SOI nMOSFET (Vgs=Vds=1.2V)
10 20 30 40 50 60 70
3
8
131345 nm FD SOI nMOSFET (Vgs=Vds=1.2V)
20 40 60 80 100 120
3
12
212160 nm FD SOI nMOSFET (Vgs=Vds=1.2V)
20 40 60 80 100 120 140 160 180
3
15
2727 300
400
500
300
400500
300
400
500
600source contact drain contact
80 nm FD SOI nMOSFET (Vgs=Vds=1.5V)
50 100 150 200
3
20
353590 nm FD SOI nMOSFET (Vgs=Vds=1.5V)
50 100 150 200 250
3
21
3939100 nm FD SOI nMOSFET (Vgs=Vds=1.5V)
50 100 150 200 250 300
3
23
4343 300
400
500
600
400
600
300
400500600
120 nm FD SOI nMOSFET (Vgs=Vds=1.8V)
50 100 150 200 250 300 350
4
28
5252140 nm FD SOI nMOSFET (Vgs=Vds=1.8V)
50 100 150 200 250 300 350 400
4
33
6060
x (nm)
180 nm FD SOI nMOSFET (Vgs=Vds=1.8V)
100 200 300 400 500
4
41
7676400
600
400
600
300400500600700
T=300K on gate T=400K on gateAcoustic Phonon Temperature Profiles
![Page 50: Semi-Classical Transport Theory](https://reader036.vdocuments.mx/reader036/viewer/2022081511/56815b62550346895dc94ba0/html5/thumbnails/50.jpg)
Higher Order Effects Inclusion in Particle-Based Simulators Degeneracy – Pauli Exclusion Principle
Short-Range Coulomb Interactions
Fast Multipole Method (FMM)V. Rokhlin and L. Greengard, J. Comp. Phys., 73, pp. 325-348 (1987).
Corrected Coulomb ApproachW. J. Gross, D. Vasileska and D. K. Ferry, IEEE Electron Device Lett. 20, No. 9, pp. 463-465 (1999).
P3M MethodHockney and Eastwood, Computer Simulation Using Particles.
![Page 51: Semi-Classical Transport Theory](https://reader036.vdocuments.mx/reader036/viewer/2022081511/56815b62550346895dc94ba0/html5/thumbnails/51.jpg)
Potential, Courtesy of Dragica Vasileska, 3D-DD Simulation, 1994.
MOTIVATION
![Page 52: Semi-Classical Transport Theory](https://reader036.vdocuments.mx/reader036/viewer/2022081511/56815b62550346895dc94ba0/html5/thumbnails/52.jpg)
Length [nm]
100
110
120
130
140
150
Wid
th [n
m]
60 80 100 120 140
Current Stream Lines, Courtesy of Dragica Vasileska, 3D-DD Simulation, 1994.
MOTIVATION
![Page 53: Semi-Classical Transport Theory](https://reader036.vdocuments.mx/reader036/viewer/2022081511/56815b62550346895dc94ba0/html5/thumbnails/53.jpg)
Discrete Impurities(Short-Range Interaction)
Efficient 3D PoissonEquation Solvers
3D Monte CarloTransport Kernel
Effective Potential (Space Quantization) 3DMCDS3DMCDS
Discrete Impurities(Short-Range Interaction)
Efficient 3D PoissonEquation Solvers
3D Monte CarloTransport Kernel
Effective Potential (Space Quantization) 3DMCDS3DMCDS
The ASU Particle-Based Device Simulator
(1) Corrected Coulomb Approach
(2) P3M Algorithm (3) Fast Multipole
Method (FMM)
(1) Ferry’s Effective Potential Method
(2) Quantum Field Approach
Statistical Enhancement: Event Biasing Scheme
Short-Range Interactions and Discrete/Unintentional
Dopants
Quantum Mechanical Size-quantization Effects
Boltzmann Transport Equations(Particle-Based Monte Carlo Transport Kernel)
Long-range Interactions(3D Poisson
Equation Solver)
![Page 54: Semi-Classical Transport Theory](https://reader036.vdocuments.mx/reader036/viewer/2022081511/56815b62550346895dc94ba0/html5/thumbnails/54.jpg)
Significant Data ObtainedBetween 1998 and 2002
![Page 55: Semi-Classical Transport Theory](https://reader036.vdocuments.mx/reader036/viewer/2022081511/56815b62550346895dc94ba0/html5/thumbnails/55.jpg)
MOSFETs - Standard Characteristics
0
50
100
150
200
250
300
350
40 60 80 100 120 140
VD=1.0 [V], V
G=1.0 [V]
VD=0.5 [V], V
G=1.0 [V]
Ele
ctro
n en
ergy
[m
eV]
Distance [nm]
LG= 80 nm
0
5x106
1x107
1.5x107
2x107
40 60 80 100 120 140
VD=0.5 [V], V
G=1.0 [V]
VD=1.0 [V], V
G=1.0 [V]
Drif
t vel
ocity
[cm
/s]
Distance [nm]
LG=80 nm
(a)
The average energy of the carriers increases when going from the source to the drain end of the channel. Most of the phonon scattering events occur at the first half of the channel.
Velocity overshoot occurs near the drain end of the channel. The sharp velocity drop is due to e-e and e-i interactions coming from the drain.
W. J. Gross, D. Vasileska and D. K. Ferry, "3D Simulations of Ultra-Small MOSFETs with Real-Space Treatment of the Electron-Electron and Electron-Ion Interactions," VLSI Design, Vol. 10, pp. 437-452 (2000).
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MOSFETs - Role of the E-E and E-I
0
100
200
300
400
100 110 120 130 140 150 160 170 180
with e-e and e-imesh force only
Ele
ctro
n en
ergy
[m
eV]
Length [nm]
VD=1 V, V
G=1 V
channel drain
0
100
200
300
400
500
600
700
800
0.12 0.13 0.14 0.15 0.16 0.17 0.18
Ene
rgy
[meV
]
Length [nm]
0
100
200
300
400
500
600
700
800
0.12 0.13 0.14 0.15 0.16 0.17 0.18
Length [nm]
Ene
rgy
[meV
]mesh forceonly
with e-e and e-i
Individual electron trajectories over
time
-1x107
-5x106
0
5x106
1x107
1.5x107
2x107
2.5x107
0 40 80 120 160
with e-e and e-imesh force only
Drif
t vel
ocity
[cm
/s]
Length [nm]
VD=V
G=1.0 V
source drainchannel
![Page 57: Semi-Classical Transport Theory](https://reader036.vdocuments.mx/reader036/viewer/2022081511/56815b62550346895dc94ba0/html5/thumbnails/57.jpg)
MOSFETs - Role of the E-E and E-I Mesh force onlyMesh force only With With ee--ee and and ee--ii
Short-range e-e and e-i interactions push someof the electrons towards higher energies
10-3
10-2
0 50 100 150 200 250 300 350 400
sourcechanneldrain
Ele
ctro
n di
strib
utio
n(a
rb. u
nits
)Energy [meV]
VG=0.5 V, V
D=0.8 V
10-3
10-2
0 50 100 150 200 250 300 350 400
SourceChannelDrain
Ele
ctro
n di
strib
utio
n(a
rb. u
nits
)
Energy [meV]
VG=0.5 V, V
D=0.8 V
D. Vasileska, W. J. Gross, and D. K. Ferry, "Monte-Carlo particle-based simulations of deep-submicron n-MOSFETs with real-space treatment of electron-electron and electron-impurity interactions," Superlattices and Microstructures, Vol. 27, No. 2/3, pp. 147-157 (2000).
![Page 58: Semi-Classical Transport Theory](https://reader036.vdocuments.mx/reader036/viewer/2022081511/56815b62550346895dc94ba0/html5/thumbnails/58.jpg)
Degradation of Output Characteristics
0
10
20
30
40
50
60
70
80
0.0 0.2 0.4 0.6 0.8 1.0 1.2
with corrected Coulombmesh force only
Dra
in c
urre
nt I D
[mA
]
Drain voltage VD [V]
increasing VG
LG = 35 nm, WG = 35 nm, NA = 5x1018 cm-3, Tox = 2 nm, VG = 11.6 V (0.2 V)
The short range e -e and e -i interactions have significant influence on the device output characteristics.
There is almost a factor of two decrease in current when these two inte-raction terms are considered.
LG = 50 nm, WG = 35 nm, NA = 5x1018 cm-3
Tox = 2 nm, VG = 11.6 V (0.2 V)
0
10
20
30
40
50
60
70
80
0 0.2 0.4 0.6 0.8 1 1.2
with corrected Coulombmesh force only
Dra
in c
urre
nt I D [
mA]
Drain voltage VD [V]
increasing VG
W. J. Gross, D. Vasileska and D. K. Ferry, "Ultra-small MOSFETs: The importance of the full Coulomb interaction on device characteristics," IEEE Trans. Electron Devices, Vol. 47, No. 10, pp. 1831-1837 (2000).
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Mizuno result:(60% of the fluctuations)
Stolk et al. result:(100% of the fluctuations)
Fluctuations in thesurface potential
Fluctuations in theelectric field
Depth-Distributionof the charges
![Page 60: Semi-Classical Transport Theory](https://reader036.vdocuments.mx/reader036/viewer/2022081511/56815b62550346895dc94ba0/html5/thumbnails/60.jpg)
MOSFETs - Discrete Impurity Effects
0
20
40
60
80
100
0 1 2 3 4 5
sVth
=> approach 1s
Vth => approach 2
sVth
=> our simulation results
s Vth [
mV
]
Oxide thickness Tox
[nm]
05
1015
2025
3035
40
1x1018 3x1018 5x1018 7x1018
sVth
=> approach 1s
Vth => approach 2
sVth
=> our simulation results
s Vth [
mV
]
Doping density NA [cm-3]
10
20
30
40
50
60
20 40 60 80 100 120 140
sVth
=> approach 1s
Vth => approach 2
sVth
=> our simulation results
s Vth
[mV
]Device width [nm]
effeff
A
ox
ox
ABSi
BBSiVth WL
NTNqq/Tkq 44 3
434
sApproach 2 [2]:
si
ABB
effeff
A
ox
oxBSiVth n
Nln
qTk
;WLNTq
44 3
2Approach 1 [1]:
[1] T. Mizuno, J. Okamura, and A. Toriumi, IEEE Trans. Electron Dev. 41, 2216 (1994).
[2] P. A. Stolk, F. P. Widdershoven, and D. B. Klaassen, IEEE Trans. Electron Dev. 45, 1960 (1998).
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020406080
100120140160180200
160
170
180
190
200
210
220
230
240
250
260
270
Number of Atoms in Channel
Num
ber o
f Dev
ices
5 samples at maximum
5 samples at minimum
5 samples of average
Depth Correlation of sVT To understand the role that the position
of the impurity atoms plays on the threshold voltage fluctuations, statistical ensembles of 5 devices from the low-end, center and the high-end of the distribution were considered.
Significant correlation was observed between the threshold voltage and the number of atoms that fall within the first 15 nm depth of the channel.
0.9
1
1.1
1.2
1.3
1.4
160 180 200 220 240 260 280 300
Thre
shol
d vo
ltage
[V]
Number of channel dopant atoms
low-end
center
high-end
(a)LG=50 nm, W
G=35 nm
NA=5x1018 cm-3, T
ox=3 nm
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20 25
V T cor
rela
tion
Depth [nm]
depth
Moving slab
rangeND ND
(b) LG=50 nm, W
G=35 nm, T
ox=3 nm
NA=5x1018 cm-3
Number of atoms in the channel
Num
ber o
f dev
ices
Number of atoms in the channel
Depth [nm]
VT
corr
elat
ion
Thre
shol
d vo
ltage
[V
]
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Fluctuations in High-Field Characteristics
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20 25
average velocitycorrelationdrain currentcorrelation
Cor
rela
tion
Depth [nm]
(c)
LG=50 nm
WG=35 nm
Tox
=3 nm
NA=5x1018 cm-3
Impurity distribution in the channel also affects the carrier mobility and saturation current of the device.
Significant correlation was observed between the drift velocity (saturation current) and the number of atoms that fall within the first 10 nm depth of the channel.
0
5 106
1 107
1.5 107
160 180 200 220 240 260 280
Drif
t vel
ocity
[cm
/s]
Number of channel dopant atoms
(a)low-end
center
high-endV
G=1.5 V, V
D=1 V
LG=50 nm, W
G=35 nm
NA=5x1018 cm-3
0
5
10
15
20
160 180 200 220 240 260 280
Dra
in c
urre
nt [
mA]
Number of channel dopant atoms
low-end
center
high-end
(b)
Number of atoms in the channel
Drif
t vel
ocity
[cm
/s] Number of atoms in the channel
Depth [nm]
Cor
rela
tion
Dra
in c
urre
nt [
mA] VG = 1.5 V, VD = 1 V
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Current Issues in NovelDevices – Unintentional Dopants
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THE EXPERIMENT …
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Results for SOI DeviceSize Quantization Effect (Effective Potential):
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
2 4 6 8 10 12 14 16Channel Width [nm]
Thre
shol
d Vo
ltage
[V]
Experimental
Simulation
S. S. Ahmed and D. Vasileska, “Threshold voltage shifts in narrow-width SOI devices due to quantum mechanical size-quantization effect”, Physica E, Vol. 19, pp. 48-52 (2003).
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Results for SOI Device
Due to the unintentional dopant both the electrostatics and the transport are affected.
-10000
10000
30000
50000
70000
90000
110000
130000
0 20 40 60 80Distance Along the Channel [nm]
Aver
age
Velo
city
[m/s
]0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Aver
age
Kin
etic
Ene
rgy
[eV]
VelocityEnergy
Dip due to the presence of the impurity. This affects the transport of the carriers.
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Results for SOI Device
Unintentional Dopant:
D. Vasileska and S. S. Ahmed, “Narrow-Width SOI Devices: The Role of Quantum Mechanical Size Quantization Effect and the Unintentional Doping on the Device Operation”, IEEE Transactions on Electron Devices, Volume 52, Issue 2, Feb. 2005 Page(s):227 – 236.
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Results for SOI Device
Channel Width = 10 nmVG = 1.0 VVD = 0.1 V
32.54%
47.62%
33.01%
26.98%
42.85%
27.18%
11.90%
26.19%
11.11%
01
23
45
67
89
10
0 10 20 30 40 50Distance Along the Channel [nm]
Dis
tanc
e A
long
the
Wid
th [n
m]
.
Sour
ce
Drai
n34.13%
47.62%
42.06%
16.66%
42.85%
26.19%
9.93%
26.19%
19.46%
0
1
2
3
4
5
6
70 10 20 30 40 50
Distance Along the Channel [nm]
Dis
tanc
e A
long
the
Dep
th [n
m]
.
Sour
ce
Dra
in
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Results for SOI Device
86.30%
96.76%
86.52%
87.39%
96.09%
86.96%
69.57%
88.26%
67.39%
0
1
2
3
4
5
0 10 20 30 40 50Distance Along the Channel [nm]
Dis
tanc
e A
long
the
Wid
th [n
m]
Sou
rce
Dra
in81.09%
96.76%
88.48%
79.78%
96.09%
88.26%
59.78%
88.26%
76.09%
0
1
2
3
4
5
6
70 10 20 30 40 50
Distance Along the Channel [nm]
Dis
tanc
e A
long
the
Dep
th [n
m]
Sou
rce
Dra
in
Channel Width = 5 nmVG = 1.0 VVD = 0.1 V
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Results for SOI Device
Impurity located at the very source-end, due to the availability of Increasing number of electrons screening the impurity ion, has reduced impact on the overall drain current.
0%
10%
20%
30%
40%
50%
60%
0 10 20 30 40 50Distance Along the Channel [nm]
Cur
rent
Red
uctio
n
Impurity position varying along the center of the channel
V G = 1.0 VV D = 0.2 V
Source end Drain end
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Results for SOI Device
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
2 4 6 8 10 12 14 16Channel Width [nm]
Thre
shol
d Vo
ltage
[V]
ExperimentalSimulation (QM)Discrete single dopants
D. Vasileska and S. S. Ahmed, IEEE Transactions on Electron Devices, Volume 52, Issue 2, Feb. 2005 Page(s):227 – 236.
S. Ahmed, C. Ringhofer and D. Vasileska, Nanotechnology, IEEE Transactions on, Volume 4, Issue 4, July 2005 Page(s):465 – 471.
D. Vasileska, H. R. Khan and S. S. Ahmed, International Journal of Nanoscience, Invited Review Paper, 2005.
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Results for SOI DeviceElectron-Electron Interactions:
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
0 0.2 0.4 0.6 0.8 1
Electron Kinetic Energy [eV]
Dis
tribu
tion
Func
tion
[a.u
.]
PMFMM
V G = 1.0 VV D = 0.3 V
0.0E+00
5.0E+04
1.0E+05
1.5E+05
2.0E+05
2.5E+05
3.0E+05
0 20 40 60 80 100
Distance Along the Channel [nm]
Ele
ctro
n V
eloc
ity [m
/s]
PMFMM
V G = 1.0 VV D = 0.3 V
Source Drain
D. Vasileska and S. S. Ahmed, IEEE Transactions on Electron Devices, Volume 52, Issue 2, Feb. 2005 Page(s):227 – 236.
S. Ahmed, C. Ringhofer and D. Vasileska, Nanotechnology, IEEE Transactions on, Volume 4, Issue 4, July 2005 Page(s):465 – 471.
D. Vasileska, H. R. Khan and S. S. Ahmed, International Journal of Nanoscience, Invited Review Paper, 2005.
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Summary
Particle-based device simulations are the most desired tool when modeling transport in devices in which velocity overshoot (non-stationary transport) exists
Particle-based device simulators are rather suitable for modeling ballistic transport in nano-devices
It is rather easy to include short-range electron-electron and electron-ion interactions in particle-based device simulators via a real-space molecular dynamics routine
Quantum-mechanical effects (size quantization and density of states modifications) can be incorporated in the model quite easily with the assumption of adiabatic approximation and solution of the 1D or 2D Schrodinger equation in slices along the channel section of the device