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UniK/NTNU
Precise 2D Compact Modeling of NanoscaleDG MOSFETs Based on Conformal
Mapping Techniques
T. A. Fjeldly, S. KolbergUniversity Graduate Center (UNIK)
Norwegian University of Science and Technology, Kjeller, Norway
B. IñiguezDepartament d'Enginyeria Electronica, Electrica i Automatica,
Universitat Rovira i Virgili, Tarragona, Spain
UniK/NTNU
Overview
Objective:Establish framework for precise, compact 2D modeling of short-channel nanoscale MOSFETs
Here: Double Gate (DG) MOSFET
• Part 1: Subthreshold and near thresholdconditions where capacitive coupling between contacts dominant electrostatics
• Part 2: Strong inversion where effects of electrons dominant electrostatics. Must betreated self-consistently.
• Determine 2D barrier topography• Calculate electron distribution• Find short-schannel effects including DIBL• Calculate current – drift-diffusion and ballistic
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Modeling procedure• Include gate oxide in extended device
body:
• Map body into upper half plane oftransformed (u,iv)-plane using conformal mapping (Schwartz-Christoffel transform).
• Subthreshold: Solve Laplace’s equation in(u,iv)-plane (low-doped body)
2D potential distribution for capacitivecoupling. Map back to (x,y)-plane
• Near threshold: Determine total potentialdistribution self-consistently by includingelectrostatic effects of electrons.
• Strong inversion: Use long-channel solution. Treat 2D capacitive coupling as perturbation.
• Calculate current (drift-diffusion and ballistic)
• Verify against 2D numerical simulations.
oxSioxox tt εε='
Schwartz-Christoffel transformation( )
)(,
2 kKivukFLiyxz +
=+=
( )( ) )1,()(,'1'1
'),(0 222
kFkKwkw
dwwkFw
=−−
= ∫
UniK/NTNU
Mapping of the body boundary
( ))(
,2 kK
ukFLiyxz =+=
Real part
Imaginary part
G1 G2G2 SD
2xL
y _ t’Si + 2tox
-
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Mapping along symmetry lines
Gate-to-gate symmetry line
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛
+−
−
+=
=
22
2 1,1
1
'2
0
v
vkFkK
tty
x
oxSi
( )( ) ⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+
=
+=
kkF
kkFLx
tty oxSi
12cos,
12
2
22 '
θ
Source-to-drain symmetry line
2xL
Device center
-
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DG MOSFET and regimes of operation
SINANO template device:
• Double gate n-channel SOI MOSFET with aluminum gates
• Specifics: L = 25 nm, tsi = 12 nm, tox = 1.6 nm, εox = 7, body doping Nd =1015 cm-3, contact doping 1019 cm-3,extended body thickness: tsi + 2t’ox
Operating regimes:→ Part 1: In subthreshold, dopant and carrier charges can be neglected
in electrostatics when n or Nd ≤ 1018 cm-3. Potential and electron distributions dominated by the 2D capacitive coupling between the contacts (source, drain and gates)
→ Part 2: Above threshold requires self-consistent analysis- 2a) Near threshold: Use approximate barrier profile- 2b) Strong inversion: Electrons dominate, treat capacitive coupling
as a perturbation
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Part 1: Subthreshold – body potential
Solution of Laplace’s equation in (u,iv)-plane (thin oxide approx):
( ) ( )( )
( )
( )
( ) ⎪⎪⎪⎪⎪
⎭
⎪⎪⎪⎪⎪
⎬
⎫
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
⎨
⎧
⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ +
−⎟⎠⎞
⎜⎝⎛ +
++
⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ −
−⎟⎠⎞
⎜⎝⎛ −
+
⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ +
+⎟⎠⎞
⎜⎝⎛ −
−+
⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ +
−⎟⎠⎞
⎜⎝⎛ −
−−
=
+−=
−−
−−
−−
−−
∞
∞−∫
vu
kvkuVV
vu
kvkuV
vu
vuVV
kvku
kvkuVV
duvuu
uvvu
DSbi
bi
FBGS
FBGS
1tan1tan
1tan1tan
1tan1tan
1tan1tan
1
''
',
11
11
111
112
22
π
π
ϕπ
ϕ
Vds = 0.5 VVGS = -0.45 V
Vds = 0 V
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Comparison with numerical simulations
Subthreshold potential variation along the
symmetry axes
VGS = -0.47 V, VDS = 0 V
Charge density distribution– gate-to-gate symmetry line
• Classic electron distribution:
• QM electron distribution (deep well):
Combine subband states from parabolic well (lowest subbands) and square well potentials (higher subbands).
( ) ( )[ ]⎟⎟⎠
⎞⎜⎜⎝
⎛ −+−⎟
⎠
⎞⎜⎝
⎛=Tk
yqETkmyn
B
SSbgBnc
ϕϕ
π
2exp
22
23
2h
( ) ( ) 2
1ynyn j
l
jsqjq ψ∑
==
VGS = -0.47 V, VDS = 0 V
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Threshold voltage and DIBL
’Classical’ threshold voltage definition: Gate voltage VT where band bending at barrier is 2qφb at VDS = 0 V
For symmetric gate bias: VT = -0.47 V
Drain induced barrier lowering (DIBL):Shift in VT with applied VDS
Asymmetric gate biasing:Apply fixed negative bias at Gate 2 relative to Gate 1. Pro/con effects:
+ reduced short channel effect+ adjustable thereshold voltage- effective channel width halved- added small-signal capacitance
Part 2: Strong inversion – self-consistency
Consider energy barrier at gate-to-gate symmetry axis
φ1(y): 1D potential from electrons
φ2(y): 2D capacitive coupling potential between contacts
Superposition principle: ( ) ( ) ( )yyy 21 ϕϕϕ +=
( ) ( )FBGSbiFBGS VVVvkv
VVv +−⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛−⎟
⎠⎞
⎜⎝⎛+−= −− 1tan1tan2 11
2 πϕ
( ) ( )
( )( )
( )( )
( )( ) ( )⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
⎨
⎧
+<≤⎟⎟⎠
⎞⎜⎜⎝
⎛+
⎟⎟⎠
⎞⎜⎜⎝
⎛
<⎟⎟⎠
⎞⎜⎜⎝
⎛
×−=−=
∫∫
∫
∫
∫
+
+
+
2'2'for''''exp'
''exp'
'for''exp
2'2
''
2'2
'
2'2
'
0 11
oxSioxtt
yth
y
t
tt
tth
ox
oxtt
tth
oy
ttytdyV
ydy
dyV
yt
tydyV
yy
KyEy
oxSi
ox
oxSi
ox
oxSi
ox
ϕ
ϕ
ϕ
ϕ
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UniK/NTNU
2a Near operational threshold
( ) ( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+
−−+−=≈2
'2211
oxSimFBGSp tt
yVVyy ϕϕϕ
Assume that the total gate-to-gate potential φ(y) has a parabolic form, which agrees very well with numerical simulations (Atlas):
Parameter φm is determined by:
• substituting φp(y) into expression for φ1(y)
• performing integrations in expression for φ1(y)
• substituting φ1(y) into total potential φ(y)
• obtaining self-consistent expression for φm by considering device center
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Parameter φm
( ) ( )
( ) ( ) ⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡−⎟⎟
⎠
⎞
⎜⎜
⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛+
−−+⎟⎟
⎠
⎞
⎜⎜
⎝
⎛⎥⎦
⎤⎢⎣
⎡+
−×
⎟⎟⎠
⎞⎜⎜⎝
⎛ −−+−+⎟⎠⎞
⎜⎝⎛−⎥
⎦
⎤⎢⎣
⎡⎟⎠
⎞⎜⎝
⎛ −+−= −
1'2
'21exp'2
'21erf)(
2exp
4'2
211tan4
2
223
21
oxSi
ox
th
m
m
th
oxSi
ox
th
m
m
thm
th
gbFBGSbioxSiBnFBGSbim
ttt
VV
ttt
VVsign
VEVVVttqTkm
kVVV
ϕϕ
ϕϕπϕ
ϕππ
ϕh
VDS = 0 V
2b Strong inversion
Gate-to-gate barrier:
Use long-channel potential φ1(y) for electrons from 1D Poisson’s equation (Taur 2001), adjusted to include effects of body doping:
Treat capacive coupling potential φ2(y) as a perturbation. Find self-consistent total potential by first-order correction using 1D Poisson’s equation
Describes how DIBL-effect fades in strong inversion
( ) ( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−−⎟⎟
⎠
⎞⎜⎜⎝
⎛ −−= 2'
2exp
2cosln21 Siox
th
bc
ths
ithc tty
VVqnVy ϕϕε
ϕϕ
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Current modeling – subthreshold and near threshold
Drift-diffusion transport:Use barrier shape derived self-consistently. Assume that the current is ’small’. Then we have:
nso(x): surface concentration of electrons in all cross-sections from source to drain.
Ballistic transport:
• Current consists of free-flight electrons with sufficient energy to fly over barrier • Electrons ’filtered’ through available quantum states at barrier position• Current limited by ability of source and drain to supply high-energy carriers fast enough
Natori formalism (1994, 2002):
IBT = q(F+-F–) where
( ) ( ) ( )( )∫
−−=−=
L
so
VVthnF
snDD
xndx
eVqWdx
xdVxnqWIthDS
0
1µµ
( )∑ ∑ ⎟
⎟
⎠
⎞
⎜⎜
⎝
⎛ −−=
±±
valleys j B
ijFsi
B
TkqVEE
FmTk
F 2122
232hπ
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Current modeling – strong inversion
Current flows primarily near the Si/SiO2 interfaces.Use classical long-channel expressions for IDD.
Strong inversion threshold voltage (Taur et al. 2002, adjusted for body doping) :
Simulated (Atlas) strong inversion potential contour plot VGS = 0.1 V, VDS = 0 V
( )⎟⎟⎠
⎞⎜⎜⎝
⎛ −+=+=
iox
GSSioxBmsBT qnt
VVtq
TkVVV2
ln2where2 000
εφϕ
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DD and ballistic current – comparision with simulations
Drift-diffusion current Ballistic current
O : simulations: model
O : simulations: model
Observations:
• Good correspondence between modeled and simulated currents• Quite similar results for drift-diffusion and ballistic current• Some more work needed to model ’knee’ region)
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SummaryCompact 2D modeling framework established for short-channel, nanoscale DG MOSFETs by means of conformal mapping techniques
Precise description of:• 2D device barrier topography• Short-channel effects including DIBL• Self-consistency at moderate and high electron densities• Fading of DIBL in strong inversion• Drift-diffusion and ballistic transport
Model verified by numerical simulations