Analytic Solution for the Drain Current
Analytic Solution for the Drain Current
of Undoped Symmetric DG MOSFET
A. Ortiz-Conde, F.J. García-Sánchez,
S. Malobabic, and J. Muci
Caracas, Venezuela
E-mail addresses: {ortizc, fgarcia, smalobabic}@ieee.org; jmuci@usb.ve
Workshop on Compact Modeling 2005: Anaheim, California, USA, May 10-12, 2005.
Motivation:
• Undoped DG MOSFETs are very attractive for
scaling devices because they permit the
lessening of SCE by means of an ultra thin body.
• Surface potential-based description seems to be
the way to attain model robustness for reliable
circuit simulation.
• Surface potential descriptions require that the
channel potential be expressed in terms of the
terminal voltages, ideally by exact explicit
analytic expressions.
• Highly accurate and physics based compact
models which are computationally efficient are
required.
WCM'2005. Analytic Solution for the Drain Current of Undoped Symmetric Dual-Gate MOSFET. Ortiz-Conde et al.
2
Previous work on DG MOSFET:
• In 2000, Taur [1] obtained a system of two
equations to describe the potential.
• In 2004, He [2] developed a charge-based drain
current model.
• In 2004, Taur [3] derived a drain current model.
• In 2005, Sallese [4] obtained an approximate
expressions for the current.
[1] Y. Taur,. IEEE Electron Device Lett., 21, 245–247, 2000.
[2] J . He et al, WCM, NSTI-Nanotech, Boston, 2, 124-127, 2004.
[3] Y. Taur et al, IEEE Electron Device Lett, 25, 107-109, 2004.
[4] J.M. Sallese et al, Solid-St. Electron. 49, 485–489, 2005.
WCM'2005. Analytic Solution for the Drain Current of Undoped Symmetric Dual-Gate MOSFET. Ortiz-Conde et al.
3
Contents:
• Part I: A potential-based, analytic,
continuous, and fully consistent physical
description of the drain current of the
undoped body symmetric double gate
MOSFET.
• Part II: Considerations about the threshold
voltage of undoped DG MOSFETs.
WCM'2005. Analytic Solution for the Drain Current of Undoped Symmetric Dual-Gate MOSFET. Ortiz-Conde et al.
4
Symmetric DG n-MOSFET:
Both gates have same: work function, oxide thickness and applied bias
WCM'2005. Analytic Solution for the Drain Current of Undoped Symmetric Dual-Gate MOSFET. Ortiz-Conde et al.
5
ψ ο ≈ ψS
Below threshold
qVG
qψ S
≈
VG
EC
qψ ο
EF
qVG
Ei
Ev
Ei
Ec , Ev
EF
qψ S
Above threshold
qψ ο
EF
EC
Ei
qVG
qVG
ψο < ψS < VG
Ev
WCM'2005. Analytic Solution for the Drain Current of Undoped Symmetric Dual-Gate MOSFET. Ortiz-Conde et al.
6
where:
V GF ≡ V GS - V FB
ψ o = U − U 2 − V GFψ o max
Our analytic solution [6]:
⎞
⎛C
r = ( A tox + B ) ⎜⎜ + D ⎟⎟ e − E V
⎠
⎝ tSi
1
U = [V GF + (1 + r )ψ o max ]
2
Electric potential ( V )
β (ψ o −V )
⎧ ⎡
⎤⎫
2
t
q
⎪
ni
Si ⎥ ⎪
2
e
ψ S = ψ o − ln ⎨cos ⎢⎢
⎬
2kT ε s
2 ⎥⎪
β ⎪
⎥⎦ ⎭
⎩ ⎢⎣
2
)
0.5
ψO
0.4
0.3
tSi = 20 nm
tox = 2 nm
V=0
Rigorous solution
Analytic solution
0.2
0.1
0.0
8
4
0
−e
βψ o
Error inψ ( mV )
(e
βψ S
( mV )
e
−β V
S
2kT ni ε s
=
+
ψ
V GF
S
Co
ψS
0.6
Error in ψ
Rigorous solution of ψs and ψo by Taur [5]:
0
-4
1.0
0.5
0.0
-0.5
0.0
0.5
1.0
1.5
VGF (V)
[5] Y. Taur, "An analytical solution to a double-gate MOSFET with undoped body“. IEEE Electron
Device Lett., 21, 245–247, 2000.
[6] S. Malobabic et al, "Modeling the Undoped-Body Symmetric Dual-Gate MOSFET", IEEE Int.
Caracas Conf. on Cir. Dev. and Sys., (República Dominicana), pp. 19-25, Nov. 2004.
WCM'2005. Analytic Solution for the Drain Current of Undoped Symmetric Dual-Gate MOSFET. Ortiz-Conde et al.
7
Assumptions:
• Quasi-Fermi level is constant across thickness
(x direction)
• Current flows only along channel length (y
direction)
• The energy levels are referenced to the electron
quasi-Fermi level of the n+ source.
• Our formulation does not account for shortchannel effects, carrier confinement energy
quantization, interface roughness, ballistic-type
transport, and mobility degradation.
WCM'2005. Analytic Solution for the Drain Current of Undoped Symmetric Dual-Gate MOSFET. Ortiz-Conde et al.
8
Pao-Sah's type formulation:
Under the assumption that
the mobility is independent
of position in the channel:
where:
Q I ≡ - 2q
∫
tSi
∫
0
2
( n - ni ) dx = - 2q
0
substituting:
W
ID = 2 µ
L
VDS ψ S
W
ID= 2 µ εs
L
V DS ψ S
∫ ∫
0
ψo
qn
∫ F dψ dV
ψo
ψS n-n
i
∫
ψo
F
dψ
⎛ 1 dα ∂F ⎞
−
⎜
⎟ dψ dV
⎝ 2 F dV ∂V ⎠
WCM'2005. Analytic Solution for the Drain Current of Undoped Symmetric Dual-Gate MOSFET. Ortiz-Conde et al.
9
Pierret and Shields's type model:
In 1983, Pierret and Shields [7] transform the Pao-Sah's
double integral formulation, for bulk MOSFET, into a
single integral without introducing any additional
approximation.
In 1992, we obtained [8] a single integral current
equation for a SOI MOSFET by using the previous ideas.
The charge coupling
factor, α , was used:
α ≡ Fs − G (ψ s , V )
2
2
where G(ψ,V) is the Kingston function.
Now, we will use the same procedure for a DG MOSFET.
[7] R.F. Pierret, J.A. Shields, Solid-St. Electron. 26, 143-147, 1983.
[8] A. Ortiz-Conde et al, Solid-St. Electron. 35, 1291-1298, 1992.
WCM'2005. Analytic Solution for the Drain Current of Undoped Symmetric Dual-Gate MOSFET. Ortiz-Conde et al.
10
Pierret and Shields's type model:
Electric field:
F = −
2 k T ni
εs
where: α ≡ Fs 2 − G 2 (ψ s , V ) = −
is the charge coupling factor.
Taking the partial derivative:
recalling that:
e
β (ψ −V )
+α
2 k T ni β (ψ −V )
e
εs
o
∂F
1 dα 1 q ni β (ψ −V )
e
=
−
2 F dV F ε s
∂V
n = n i e β (ψ −V )
qn
⎛ 1 dα ∂F ⎞
=εs⎜
−
we obtain:
⎟
F
F
dV
V
2
∂
⎝
⎠
Substituting into the double integral:
ψ
W V DS S ⎛ 1 dα ∂F ⎞
−
⎟ dψ dV
ID= 2 µ εs ∫ ∫ ⎜
L
⎝ 2 F dV ∂V ⎠
0 ψ
o
WCM'2005. Analytic Solution for the Drain Current of Undoped Symmetric Dual-Gate MOSFET. Ortiz-Conde et al.
11
Pierret and Shields's type model:
0.70
ΨsL
0.55
Ψs0
0.50
0.45
tio
gr
a
In
te
V=0
ΨoL
n
Ψs
0.60
gi
on
Breaking the integral into
different parts and then
solving all of them, we
obtain an analytic solution
without introducing any
additional approximation.
Re
Ψs and Ψo ( V )
0.65
of
The region of integration
is shown in the figure.
V = VDS
Ψo
VGF = 0.7 V
tox = 2 nm
tSi = 20 nm
Ψo0
0.0
0.1
0.2
0.3
0.4
V (V)
WCM'2005. Analytic Solution for the Drain Current of Undoped Symmetric Dual-Gate MOSFET. Ortiz-Conde et al.
12
{
Drain current equation: [9]
W
ID= µ
L
(
)
1 2
⎡
⎤
2
(
)
V
ψ
ψ
ψ
ψ
2C o ⎢ GF SL S 0
SL
S 0 ⎥ first term
2
⎣
⎦
Third term
+ 4
kT
Co (ψ SL - ψ S 0 )
q
second term
βψ ⎤
β (ψ oL −VDS )
⎡
+ t Si k T ni ⎢e
− e o0 ⎥
⎣
⎦
}
where VGF is the difference between the gate-to-source voltage and the flat-band voltage.
This equation is equivalent to that of Taur [10] , although the derivation was different.
−α
This equivalence is established, after some algebraic
βT ≡
2
manipulations, by noting that the beta, βΤ , introduced in [10]
⎛ kT ⎞
⎜⎜ 4
⎟⎟
is related to our charge coupling factor α by:
q
t
Si ⎠
⎝
[9] A. Ortiz-Conde et al, "Rigorous analytic solution for the drain current of undoped symmetric dualgate MOSFETs", Solid-State Electronics, 49, 640-647, 2005.
[10] Y. Taur et al, "A Continuous, Analytic Drain-Current Model for DG MOSFETs " , IEEE Electron
Device Lett, 25, 107-109, 2004.
WCM'2005. Analytic Solution for the Drain Current of Undoped Symmetric Dual-Gate MOSFET. Ortiz-Conde et al.
13
tSi = 20 nm
-2
ID L / ( W µ ) ( mC V m )
Comparison:
4
Normalized drain current
as a function of drain
voltage
3
VGF = 1.0 V
tox = 2 nm
Numeric
Analytic
0.9 V
2
0.8 V
0.7 V
1
0.6 V
0
0.5 V
Errors
-16
10
0
-16
-10
0.0
0.2
0.4
0.6
0.8
1.0
VDS (V)
WCM'2005. Analytic Solution for the Drain Current of Undoped Symmetric Dual-Gate MOSFET. Ortiz-Conde et al.
14
Semiconductor film thickness:
150
tSi = 20 nm
5 nm
ID L/(Wµ) ( µC V m-2 )
10-4
ID L/(Wµ) ( C V m-2 )
Effect of semiconductor
film thickness on drain
current at low drain
voltage.
10-5
100
10-6
10-7
50
10-8
tox = 2 nm
10-9
VDS =10 mV
10-10
0.2
0.4
0.6
VGF (V)
0.8
0
1.0
WCM'2005. Analytic Solution for the Drain Current of Undoped Symmetric Dual-Gate MOSFET. Ortiz-Conde et al.
15
Gate oxide film thickness:
150
10-4
VDS =10 mV
10-5
ID L/(Wµ) ( µC V m-2 )
tSi = 20 nm
ID L/(Wµ) ( C V m-2 )
Effect of gate oxide
thickness on drain
current at low drain
voltage.
100
10-6
10-7
50
10-8
10-9
tox = 2 nm
1 nm
10-10
0.2
0
0.4
0.6
0.8
1.0
VGF (V)
WCM'2005. Analytic Solution for the Drain Current of Undoped Symmetric Dual-Gate MOSFET. Ortiz-Conde et al.
16
Current components:
In weak conduction: the first term is negligible, the second term is
positive, the third term is negative.
The combination of the second and third term gives a positive value
approximated by:
β (ψ oL −VDS ) ⎤
W
⎡ βψ o 0
−e
I Dw ≈ µ t Si k T ni ⎢e
⎥⎦
L
⎣
The strong conduction term can be obtained by subtracting the
weak component from the total current:
I Ds = I D − I Dw
WCM'2005. Analytic Solution for the Drain Current of Undoped Symmetric Dual-Gate MOSFET. Ortiz-Conde et al.
17
-2
Current equation
and its two terms
IDw and IDs, vs gate
voltage.
Normalized current (C V m )
Current components:
10-4
VDS = 10 mV
10-5
tox = 2 nm
tSi = 20 nm
10-6
IDs L/(Wµ)
10-7
IDw L/(Wµ)
ID L/(Wµ)
10-8
10-9
0.2
0.3
0.4
0.5
0.6
0.7
0.8
VGF ( V )
WCM'2005. Analytic Solution for the Drain Current of Undoped Symmetric Dual-Gate MOSFET. Ortiz-Conde et al.
18
Threshold:
The intersection of the two current components (IDw and IDs ) may
be understood as the transition threshold from weak to strong
conduction.
Below threshold:
Above threshold:
ψ SL ≈ ψ S 0 ≈ ψ oL ≈ ψ o 0 ≈ VGF
W
I Ds = µ Qbulk VDS
L
Qbulk
I Dw = µ
βV
W
t q ni e GF V
DS
L Si
⎛q 2
=
W⎜⎜
β
⎝ 2 Co
4 Co
ε s ni e
βVGF
2
kT
⎞
⎟
⎟
⎠
where W is the principal branch of the Lambert Function and Qbulk
is the solution [11] for bulk devices. Equating IDw and IDs gives:
kT ⎡ ⎛ 8 kT ε s
VTx = ⎢ln⎜⎜ 2 2
q ⎣ ⎝ q ni tSi
⎛ 4 ε s ⎞ ⎤ kT ⎛ 8 kT ε s
⎞
⎟⎟ ⎥ ≈ ln⎜⎜ 2 2
⎟⎟ − W⎜⎜
⎝ Co tSi ⎠ ⎦ q ⎝ q ni tSi
⎠
⎞
⎟⎟
⎠
[11] Ortiz-Conde et al, "Exact analytical solution of channel surface potential as an explicit function of
gate voltage in undoped-body MOSFETs using the Lambert W function and a threshold voltage
definition therefrom“, Solid-State Electronics, 47, 2067-2074, 2003.
WCM'2005. Analytic Solution for the Drain Current of Undoped Symmetric Dual-Gate MOSFET. Ortiz-Conde et al.
19
VTx =
Threshold:
kT
q
⎡ ⎛ 8 kT ε s
⎢ln⎜⎜ 2 2
⎣ ⎝ q ni t Si
⎞
⎛ 4 ⎞ ⎤ kT ⎛ 8 kT ε s
⎟⎟ − W⎜⎜ ε s ⎟⎟ ⎥ ≈
ln⎜⎜ 2 2
C
t
q
⎠
⎝ o Si ⎠ ⎦
⎝ q ni t Si
⎞
⎟⎟
⎠
Intersection of the two
current components:
lines from plots and
symbols from equation.
The difference
between the two
results is
approximately VDS .
VTx ( V )
0.5
0.4
Symbols: equation
tSi = 20 nm
10 nm
5 nm
VDS = 10 mV
0.3
2
4
6
8
10
tox ( nm )
WCM'2005. Analytic Solution for the Drain Current of Undoped Symmetric Dual-Gate MOSFET. Ortiz-Conde et al.
20
For strong conduction, all cases
approaches bulk. Therefore, the
threshold between strong and
weak conduction approaches
that of bulk devices.
For large tSi , there are two subthreshold regions ( 60 and 120
mV/dec) and VTx is the threshold
between these two regions.
6
10-2
tSi = 10 nm
10-3
100 nm
1µm
bulk
0
12
4
c
de
/
V
m
ec
10-6
V = 0V
Q (mC/m2)
10-5
3
2
mV
/d
10-7
5
tox=2nm
100 nm
1µm
bulk
Linear Extrapolation VTLE
is weakly dependent on tSi .
For tSi > 10 nm, VTLE
approaches that of bulk.
60
Q (C/m2)
10-4
tSi = 10 nm
10-8
10-9
0.0
VTx decreases as tSi increases
0.2
0.4
VGF (V)
0.6
1
0
0.0
V = 0V
tox=2nm
0.2
0.4
VGF (V)
0.6
WCM'2005. Analytic Solution for the Drain Current of Undoped Symmetric Dual-Gate MOSFET. Ortiz-Conde et al.
21
Conclusions:
• Exact potential-based analytic description
of the drain current in the undoped-body
symmetric DG MOSFET.
• Threshold voltage increases with
decreasing tox (and tSi in DG).
• For large tSi , there are two sub-threshold
regions ( 60 and 120 mV/dec).
WCM'2005. Analytic Solution for the Drain Current of Undoped Symmetric Dual-Gate MOSFET. Ortiz-Conde et al.
22
END of presentation
Thank you for your attention
WCM'2005. Analytic Solution for the Drain Current of Undoped Symmetric Dual-Gate MOSFET. Ortiz-Conde et al.
23
of Undoped Symmetric DG MOSFET
A. Ortiz-Conde, F.J. García-Sánchez,
S. Malobabic, and J. Muci
Caracas, Venezuela
E-mail addresses: {ortizc, fgarcia, smalobabic}@ieee.org; jmuci@usb.ve
Workshop on Compact Modeling 2005: Anaheim, California, USA, May 10-12, 2005.
Motivation:
• Undoped DG MOSFETs are very attractive for
scaling devices because they permit the
lessening of SCE by means of an ultra thin body.
• Surface potential-based description seems to be
the way to attain model robustness for reliable
circuit simulation.
• Surface potential descriptions require that the
channel potential be expressed in terms of the
terminal voltages, ideally by exact explicit
analytic expressions.
• Highly accurate and physics based compact
models which are computationally efficient are
required.
WCM'2005. Analytic Solution for the Drain Current of Undoped Symmetric Dual-Gate MOSFET. Ortiz-Conde et al.
2
Previous work on DG MOSFET:
• In 2000, Taur [1] obtained a system of two
equations to describe the potential.
• In 2004, He [2] developed a charge-based drain
current model.
• In 2004, Taur [3] derived a drain current model.
• In 2005, Sallese [4] obtained an approximate
expressions for the current.
[1] Y. Taur,. IEEE Electron Device Lett., 21, 245–247, 2000.
[2] J . He et al, WCM, NSTI-Nanotech, Boston, 2, 124-127, 2004.
[3] Y. Taur et al, IEEE Electron Device Lett, 25, 107-109, 2004.
[4] J.M. Sallese et al, Solid-St. Electron. 49, 485–489, 2005.
WCM'2005. Analytic Solution for the Drain Current of Undoped Symmetric Dual-Gate MOSFET. Ortiz-Conde et al.
3
Contents:
• Part I: A potential-based, analytic,
continuous, and fully consistent physical
description of the drain current of the
undoped body symmetric double gate
MOSFET.
• Part II: Considerations about the threshold
voltage of undoped DG MOSFETs.
WCM'2005. Analytic Solution for the Drain Current of Undoped Symmetric Dual-Gate MOSFET. Ortiz-Conde et al.
4
Symmetric DG n-MOSFET:
Both gates have same: work function, oxide thickness and applied bias
WCM'2005. Analytic Solution for the Drain Current of Undoped Symmetric Dual-Gate MOSFET. Ortiz-Conde et al.
5
ψ ο ≈ ψS
Below threshold
qVG
qψ S
≈
VG
EC
qψ ο
EF
qVG
Ei
Ev
Ei
Ec , Ev
EF
qψ S
Above threshold
qψ ο
EF
EC
Ei
qVG
qVG
ψο < ψS < VG
Ev
WCM'2005. Analytic Solution for the Drain Current of Undoped Symmetric Dual-Gate MOSFET. Ortiz-Conde et al.
6
where:
V GF ≡ V GS - V FB
ψ o = U − U 2 − V GFψ o max
Our analytic solution [6]:
⎞
⎛C
r = ( A tox + B ) ⎜⎜ + D ⎟⎟ e − E V
⎠
⎝ tSi
1
U = [V GF + (1 + r )ψ o max ]
2
Electric potential ( V )
β (ψ o −V )
⎧ ⎡
⎤⎫
2
t
q
⎪
ni
Si ⎥ ⎪
2
e
ψ S = ψ o − ln ⎨cos ⎢⎢
⎬
2kT ε s
2 ⎥⎪
β ⎪
⎥⎦ ⎭
⎩ ⎢⎣
2
)
0.5
ψO
0.4
0.3
tSi = 20 nm
tox = 2 nm
V=0
Rigorous solution
Analytic solution
0.2
0.1
0.0
8
4
0
−e
βψ o
Error inψ ( mV )
(e
βψ S
( mV )
e
−β V
S
2kT ni ε s
=
+
ψ
V GF
S
Co
ψS
0.6
Error in ψ
Rigorous solution of ψs and ψo by Taur [5]:
0
-4
1.0
0.5
0.0
-0.5
0.0
0.5
1.0
1.5
VGF (V)
[5] Y. Taur, "An analytical solution to a double-gate MOSFET with undoped body“. IEEE Electron
Device Lett., 21, 245–247, 2000.
[6] S. Malobabic et al, "Modeling the Undoped-Body Symmetric Dual-Gate MOSFET", IEEE Int.
Caracas Conf. on Cir. Dev. and Sys., (República Dominicana), pp. 19-25, Nov. 2004.
WCM'2005. Analytic Solution for the Drain Current of Undoped Symmetric Dual-Gate MOSFET. Ortiz-Conde et al.
7
Assumptions:
• Quasi-Fermi level is constant across thickness
(x direction)
• Current flows only along channel length (y
direction)
• The energy levels are referenced to the electron
quasi-Fermi level of the n+ source.
• Our formulation does not account for shortchannel effects, carrier confinement energy
quantization, interface roughness, ballistic-type
transport, and mobility degradation.
WCM'2005. Analytic Solution for the Drain Current of Undoped Symmetric Dual-Gate MOSFET. Ortiz-Conde et al.
8
Pao-Sah's type formulation:
Under the assumption that
the mobility is independent
of position in the channel:
where:
Q I ≡ - 2q
∫
tSi
∫
0
2
( n - ni ) dx = - 2q
0
substituting:
W
ID = 2 µ
L
VDS ψ S
W
ID= 2 µ εs
L
V DS ψ S
∫ ∫
0
ψo
qn
∫ F dψ dV
ψo
ψS n-n
i
∫
ψo
F
dψ
⎛ 1 dα ∂F ⎞
−
⎜
⎟ dψ dV
⎝ 2 F dV ∂V ⎠
WCM'2005. Analytic Solution for the Drain Current of Undoped Symmetric Dual-Gate MOSFET. Ortiz-Conde et al.
9
Pierret and Shields's type model:
In 1983, Pierret and Shields [7] transform the Pao-Sah's
double integral formulation, for bulk MOSFET, into a
single integral without introducing any additional
approximation.
In 1992, we obtained [8] a single integral current
equation for a SOI MOSFET by using the previous ideas.
The charge coupling
factor, α , was used:
α ≡ Fs − G (ψ s , V )
2
2
where G(ψ,V) is the Kingston function.
Now, we will use the same procedure for a DG MOSFET.
[7] R.F. Pierret, J.A. Shields, Solid-St. Electron. 26, 143-147, 1983.
[8] A. Ortiz-Conde et al, Solid-St. Electron. 35, 1291-1298, 1992.
WCM'2005. Analytic Solution for the Drain Current of Undoped Symmetric Dual-Gate MOSFET. Ortiz-Conde et al.
10
Pierret and Shields's type model:
Electric field:
F = −
2 k T ni
εs
where: α ≡ Fs 2 − G 2 (ψ s , V ) = −
is the charge coupling factor.
Taking the partial derivative:
recalling that:
e
β (ψ −V )
+α
2 k T ni β (ψ −V )
e
εs
o
∂F
1 dα 1 q ni β (ψ −V )
e
=
−
2 F dV F ε s
∂V
n = n i e β (ψ −V )
qn
⎛ 1 dα ∂F ⎞
=εs⎜
−
we obtain:
⎟
F
F
dV
V
2
∂
⎝
⎠
Substituting into the double integral:
ψ
W V DS S ⎛ 1 dα ∂F ⎞
−
⎟ dψ dV
ID= 2 µ εs ∫ ∫ ⎜
L
⎝ 2 F dV ∂V ⎠
0 ψ
o
WCM'2005. Analytic Solution for the Drain Current of Undoped Symmetric Dual-Gate MOSFET. Ortiz-Conde et al.
11
Pierret and Shields's type model:
0.70
ΨsL
0.55
Ψs0
0.50
0.45
tio
gr
a
In
te
V=0
ΨoL
n
Ψs
0.60
gi
on
Breaking the integral into
different parts and then
solving all of them, we
obtain an analytic solution
without introducing any
additional approximation.
Re
Ψs and Ψo ( V )
0.65
of
The region of integration
is shown in the figure.
V = VDS
Ψo
VGF = 0.7 V
tox = 2 nm
tSi = 20 nm
Ψo0
0.0
0.1
0.2
0.3
0.4
V (V)
WCM'2005. Analytic Solution for the Drain Current of Undoped Symmetric Dual-Gate MOSFET. Ortiz-Conde et al.
12
{
Drain current equation: [9]
W
ID= µ
L
(
)
1 2
⎡
⎤
2
(
)
V
ψ
ψ
ψ
ψ
2C o ⎢ GF SL S 0
SL
S 0 ⎥ first term
2
⎣
⎦
Third term
+ 4
kT
Co (ψ SL - ψ S 0 )
q
second term
βψ ⎤
β (ψ oL −VDS )
⎡
+ t Si k T ni ⎢e
− e o0 ⎥
⎣
⎦
}
where VGF is the difference between the gate-to-source voltage and the flat-band voltage.
This equation is equivalent to that of Taur [10] , although the derivation was different.
−α
This equivalence is established, after some algebraic
βT ≡
2
manipulations, by noting that the beta, βΤ , introduced in [10]
⎛ kT ⎞
⎜⎜ 4
⎟⎟
is related to our charge coupling factor α by:
q
t
Si ⎠
⎝
[9] A. Ortiz-Conde et al, "Rigorous analytic solution for the drain current of undoped symmetric dualgate MOSFETs", Solid-State Electronics, 49, 640-647, 2005.
[10] Y. Taur et al, "A Continuous, Analytic Drain-Current Model for DG MOSFETs " , IEEE Electron
Device Lett, 25, 107-109, 2004.
WCM'2005. Analytic Solution for the Drain Current of Undoped Symmetric Dual-Gate MOSFET. Ortiz-Conde et al.
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tSi = 20 nm
-2
ID L / ( W µ ) ( mC V m )
Comparison:
4
Normalized drain current
as a function of drain
voltage
3
VGF = 1.0 V
tox = 2 nm
Numeric
Analytic
0.9 V
2
0.8 V
0.7 V
1
0.6 V
0
0.5 V
Errors
-16
10
0
-16
-10
0.0
0.2
0.4
0.6
0.8
1.0
VDS (V)
WCM'2005. Analytic Solution for the Drain Current of Undoped Symmetric Dual-Gate MOSFET. Ortiz-Conde et al.
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Semiconductor film thickness:
150
tSi = 20 nm
5 nm
ID L/(Wµ) ( µC V m-2 )
10-4
ID L/(Wµ) ( C V m-2 )
Effect of semiconductor
film thickness on drain
current at low drain
voltage.
10-5
100
10-6
10-7
50
10-8
tox = 2 nm
10-9
VDS =10 mV
10-10
0.2
0.4
0.6
VGF (V)
0.8
0
1.0
WCM'2005. Analytic Solution for the Drain Current of Undoped Symmetric Dual-Gate MOSFET. Ortiz-Conde et al.
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Gate oxide film thickness:
150
10-4
VDS =10 mV
10-5
ID L/(Wµ) ( µC V m-2 )
tSi = 20 nm
ID L/(Wµ) ( C V m-2 )
Effect of gate oxide
thickness on drain
current at low drain
voltage.
100
10-6
10-7
50
10-8
10-9
tox = 2 nm
1 nm
10-10
0.2
0
0.4
0.6
0.8
1.0
VGF (V)
WCM'2005. Analytic Solution for the Drain Current of Undoped Symmetric Dual-Gate MOSFET. Ortiz-Conde et al.
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Current components:
In weak conduction: the first term is negligible, the second term is
positive, the third term is negative.
The combination of the second and third term gives a positive value
approximated by:
β (ψ oL −VDS ) ⎤
W
⎡ βψ o 0
−e
I Dw ≈ µ t Si k T ni ⎢e
⎥⎦
L
⎣
The strong conduction term can be obtained by subtracting the
weak component from the total current:
I Ds = I D − I Dw
WCM'2005. Analytic Solution for the Drain Current of Undoped Symmetric Dual-Gate MOSFET. Ortiz-Conde et al.
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-2
Current equation
and its two terms
IDw and IDs, vs gate
voltage.
Normalized current (C V m )
Current components:
10-4
VDS = 10 mV
10-5
tox = 2 nm
tSi = 20 nm
10-6
IDs L/(Wµ)
10-7
IDw L/(Wµ)
ID L/(Wµ)
10-8
10-9
0.2
0.3
0.4
0.5
0.6
0.7
0.8
VGF ( V )
WCM'2005. Analytic Solution for the Drain Current of Undoped Symmetric Dual-Gate MOSFET. Ortiz-Conde et al.
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Threshold:
The intersection of the two current components (IDw and IDs ) may
be understood as the transition threshold from weak to strong
conduction.
Below threshold:
Above threshold:
ψ SL ≈ ψ S 0 ≈ ψ oL ≈ ψ o 0 ≈ VGF
W
I Ds = µ Qbulk VDS
L
Qbulk
I Dw = µ
βV
W
t q ni e GF V
DS
L Si
⎛q 2
=
W⎜⎜
β
⎝ 2 Co
4 Co
ε s ni e
βVGF
2
kT
⎞
⎟
⎟
⎠
where W is the principal branch of the Lambert Function and Qbulk
is the solution [11] for bulk devices. Equating IDw and IDs gives:
kT ⎡ ⎛ 8 kT ε s
VTx = ⎢ln⎜⎜ 2 2
q ⎣ ⎝ q ni tSi
⎛ 4 ε s ⎞ ⎤ kT ⎛ 8 kT ε s
⎞
⎟⎟ ⎥ ≈ ln⎜⎜ 2 2
⎟⎟ − W⎜⎜
⎝ Co tSi ⎠ ⎦ q ⎝ q ni tSi
⎠
⎞
⎟⎟
⎠
[11] Ortiz-Conde et al, "Exact analytical solution of channel surface potential as an explicit function of
gate voltage in undoped-body MOSFETs using the Lambert W function and a threshold voltage
definition therefrom“, Solid-State Electronics, 47, 2067-2074, 2003.
WCM'2005. Analytic Solution for the Drain Current of Undoped Symmetric Dual-Gate MOSFET. Ortiz-Conde et al.
19
VTx =
Threshold:
kT
q
⎡ ⎛ 8 kT ε s
⎢ln⎜⎜ 2 2
⎣ ⎝ q ni t Si
⎞
⎛ 4 ⎞ ⎤ kT ⎛ 8 kT ε s
⎟⎟ − W⎜⎜ ε s ⎟⎟ ⎥ ≈
ln⎜⎜ 2 2
C
t
q
⎠
⎝ o Si ⎠ ⎦
⎝ q ni t Si
⎞
⎟⎟
⎠
Intersection of the two
current components:
lines from plots and
symbols from equation.
The difference
between the two
results is
approximately VDS .
VTx ( V )
0.5
0.4
Symbols: equation
tSi = 20 nm
10 nm
5 nm
VDS = 10 mV
0.3
2
4
6
8
10
tox ( nm )
WCM'2005. Analytic Solution for the Drain Current of Undoped Symmetric Dual-Gate MOSFET. Ortiz-Conde et al.
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For strong conduction, all cases
approaches bulk. Therefore, the
threshold between strong and
weak conduction approaches
that of bulk devices.
For large tSi , there are two subthreshold regions ( 60 and 120
mV/dec) and VTx is the threshold
between these two regions.
6
10-2
tSi = 10 nm
10-3
100 nm
1µm
bulk
0
12
4
c
de
/
V
m
ec
10-6
V = 0V
Q (mC/m2)
10-5
3
2
mV
/d
10-7
5
tox=2nm
100 nm
1µm
bulk
Linear Extrapolation VTLE
is weakly dependent on tSi .
For tSi > 10 nm, VTLE
approaches that of bulk.
60
Q (C/m2)
10-4
tSi = 10 nm
10-8
10-9
0.0
VTx decreases as tSi increases
0.2
0.4
VGF (V)
0.6
1
0
0.0
V = 0V
tox=2nm
0.2
0.4
VGF (V)
0.6
WCM'2005. Analytic Solution for the Drain Current of Undoped Symmetric Dual-Gate MOSFET. Ortiz-Conde et al.
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Conclusions:
• Exact potential-based analytic description
of the drain current in the undoped-body
symmetric DG MOSFET.
• Threshold voltage increases with
decreasing tox (and tSi in DG).
• For large tSi , there are two sub-threshold
regions ( 60 and 120 mV/dec).
WCM'2005. Analytic Solution for the Drain Current of Undoped Symmetric Dual-Gate MOSFET. Ortiz-Conde et al.
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END of presentation
Thank you for your attention
WCM'2005. Analytic Solution for the Drain Current of Undoped Symmetric Dual-Gate MOSFET. Ortiz-Conde et al.
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