Analyzing method of semiconductor device, designing method thereof, and design supporting apparatus
A design supporting apparatus of a semiconductor device, includes sections to perform: setting an impurity concentration with respect to a channel direction and a depth direction to node points arranged discretely in a channel region of a model transistor based on a predetermined concentration distribution rule; calculating an electric characteristic of the model transistor by using the impurity concentration; and storing the impurity concentration as a model parameter of the model transistor in a storage unit, when the calculated electric characteristic and an electric characteristic prepared previously are coincident with each other within a predetermined range. The device characteristic calculating section calculates a surface potential to each of the node points by reducing a dimension of the impurity concentration in the depth direction, corrects the surface potential based on interaction between the node points adjacent to each other, and calculates the electric characteristic by using the corrected surface potential.
1. A method of analyzing a semiconductor device, comprising:
setting an impurity concentration with respect to a channel direction and a depth direction to node points arranged discretely in a channel region of a model transistor based on a predetermined concentration distribution rule;
calculating a surface potential to each of said node points by reducing a dimension of said impurity concentration in the depth direction;
correcting the surface potential based on interaction between said node points adjacent to each other to generate the corrected surface potential;
calculating an electric characteristic of said model transistor by using the corrected surface potential; and
storing said impurity concentration as a model parameter of said model transistor in a storage unit, when the calculated electric characteristic and an electric characteristic prepared previously are coincident with each other within a predetermined range,
wherein said calculating a surface potential comprises:
calculating the surface potential to each of said node points by solving a Poisson equation in the depth direction on said node point,
wherein the Poisson equation is expressed by the following equation:
C
ox
(
V
G
-
V
F
B
-
ϕ
so
)
=
C
N
sub
N
sub
0
(
β
(
ϕ
s
0
-
V
)
-
1
+
exp
(
-
β
(
ϕ
s
0
-
V
B
)
)
)
+
exp
(
β
(
ϕ
s
0
-
ϕ
f
-
2
Φ
B
)
)
where φ s0 is the surface potential, V G is a gate voltage, V FB is a flat band voltage, V B is a substrate voltage, N sub0 is an effective impurity concentration on a surface in the channel region, Φ B is an energy difference between a quasi-Fermi level and an intrinsic Fermi level, φ f is the quasi-Fermi level, ∈ ox is a dielectric constant of a gate oxide film, T ox is a film thickness of the gate oxide film, q is an elementary charge amount, ∈ si is a dielectric constant of silicon, β is a thermal voltage, k B is the Boltzmann constant, T is an absolute temperature on an operation, n i is an intrinsic carrier density, and N(x, z) expresses said impurity concentration set to each node point,
wherein the following equations are met,
C
ox
=
ɛ
ox
T
ox
C
=
2
ɛ
Si
q
N
sub
0
β
Φ
B
=
1
β
log
(
N
sub
0
n
i
)
β
=
q
k
B
T
ϕ
s
0
=
q
ɛ
Si
∫
0
W
dep
N
(
x
,
z
)
z
ⅆ
z
+
1
β
+
V
B
N
sub
=
(
∫
0
W
dep
N
(
x
,
z
)
ⅆ
z
)
2
2
∫
0
W
dep
N
(
x
,
z
)
z
ⅆ
z
N
sub
0
=
N
(
x
,
0
)
.
2. The method according to claim 1 , wherein said correcting the surface potential comprises:
calculating the corrected surface potential based on the interaction, by solving simultaneous equations of a quasi 2-dimensional Poisson equation based on a charge density in an inversion layer surface and a current continuation equation indicating that a current between the node points is constant, through an iterative calculation method by using the surface potential as an initial value,
wherein when W is a gate width, μ is a mobility of carrier, and Q i is a channel charge density, the current I D is expressed by the following equation:
I
D
=
W
μ
Q
i
ⅆ
ϕ
f
ⅆ
x
wherein when φ s is the corrected surface potential, Q g a gate charge density, and Q b0 is a charge density in the inversion layer surface, the quasi 2-dimensional Poisson equation is expressed by the following equation:
Q
g
-
Q
i
+
E
Q
b
0
=
C
2
N
sub
N
sub
0
(
β
(
ϕ
s
-
V
B
)
-
1
+
exp
(
-
β
(
ϕ
s
-
V
B
)
)
)
+
E
Q
b
0
2
here, the following equations are met:
Q
g
=
C
ox
(
V
G
-
V
F
B
-
ϕ
s
)
Q
i
=
Q
g
-
Q
g
2
-
C
2
exp
(
β
(
ϕ
s
-
ϕ
f
-
2
Φ
B
)
)
E
=
ɛ
Si
W
dep
Q
b
0
ⅆ
2
ϕ
s
ⅆ
x
2
Q
b
0
=
C
N
sub
N
sub
0
(
β
(
ϕ
s
0
-
V
B
)
-
1
+
exp
(
-
β
(
ϕ
s
0
-
V
B
)
)
)
.
3. The method according to claim 1 , wherein said correcting the surface potential comprises:
calculating the corrected surface potential based on the interaction, by solving simultaneous equations of a quasi 2-dimensional Poisson equation based on a charge density in an inversion layer surface and a current continuation equation indicating that a current between the node points is constant, by an iterative calculation method by using the surface potential as an initial value,
wherein when W is a gate width, μ is a mobility of carrier, and Q i is a channel charge density, the current I D is expressed by the following equation:
I
D
=
W
μ
Q
i
ⅆ
ϕ
f
ⅆ
x
wherein when φ s is the corrected surface potential, Q g a gate charge density, Q b0 is a charge density in the inversion layer surface, and ΔE is a correction term for a short channel effect, the quasi 2-dimensional Poisson equation is expressed by the following equation:
Q
g
-
Q
i
+
E
Q
b
0
=
C
2
N
sub
N
sub
0
(
β
(
ϕ
s
-
V
B
)
-
1
+
exp
(
-
β
(
ϕ
s
-
V
B
)
)
)
+
E
Q
b
0
2
here, the following equations are met:
Q
g
=
C
ox
(
V
G
-
V
F
B
-
ϕ
s
)
Q
i
=
Q
g
-
Q
g
2
-
C
2
exp
(
β
(
ϕ
s
-
ϕ
f
-
2
Φ
B
)
)
E
=
ɛ
Si
W
dep
Q
b
0
ⅆ
2
ϕ
s
ⅆ
x
2
+
Δ
E
Q
b
0
=
C
N
sub
N
sub
0
(
β
(
ϕ
s
0
-
V
B
)
-
1
+
exp
(
-
β
(
ϕ
s
0
-
V
B
)
)
)
.
4. The method according to claim 3 , wherein when a distance from an end of a drain region in said channel region is X d and a distance from an end of a source region in said channel region is X s , said correction term is expressed by the following equation:
Δ
E
=
max
(
1
-
x
X
s
,
0
)
+
max
(
1
-
L
gate
-
x
X
d
,
0
)
.
5. A design supporting apparatus of a semiconductor device, comprising:
a model parameter setting section configured to set an impurity concentration with respect to a channel direction and a depth direction to node points arranged discretely in a channel region of a model transistor based on a predetermined concentration distribution rule;
a device characteristic calculating section configured to calculate an electric characteristic of said model transistor by using said impurity concentration; and
a determining section configured to store said impurity concentration as a model parameter of said model transistor in a storage unit, when the calculated electric characteristic and an electric characteristic prepared previously are coincident with each other within a predetermined range,
wherein said device characteristic calculating section calculates a surface potential to each of said node points by reducing a dimension of said impurity concentration in the depth direction, corrects the surface potential based on interaction between said node points adjacent to each other to generate the corrected surface potential, and calculates the electric characteristic of said model transistor by using the corrected surface potential,
wherein said device characteristic calculating section
calculating the surface potential to each of said node points by solving a Poisson equation in the depth direction on said node point,
wherein a surface potential the Poisson equation is expressed by the following equation:
C
ox
(
V
G
-
V
F
B
-
ϕ
so
)
=
C
N
sub
N
sub
0
(
β
(
ϕ
s
0
-
V
)
-
1
+
exp
(
-
β
(
ϕ
s
0
-
V
B
)
)
)
+
exp
(
β
(
ϕ
s
0
-
ϕ
f
-
2
Φ
B
)
)
where φ s0 is the surface potential, V G is a gate voltage, V FB is a flat band voltage, V B is a substrate voltage, N sub0 is an effective impurity concentration on a surface in the channel region, Φ B is an energy difference between a quasi-Fermi level and an intrinsic Fermi level, φ f is the quasi-Fermi level, ∈ ox is a dielectric constant of a gate oxide film, T ox is a film thickness of the gate oxide film, q is an elementary charge amount, ∈ Si is a dielectric constant of silicon, β is a thermal voltage, k B is the Boltzmann constant, T is an absolute temperature on an operation, n i is an intrinsic carrier density, and N(x, z) expresses said impurity concentration set to each node point,
wherein the following equations are met,
C
ox
=
ɛ
ox
T
ox
C
=
2
ɛ
Si
q
N
sub
0
β
Φ
B
=
1
β
log
(
N
sub
0
n
i
)
β
=
q
k
B
T
ϕ
s
0
=
q
ɛ
Si
∫
0
W
dep
N
(
x
,
z
)
z
ⅆ
z
+
1
β
+
V
B
N
sub
=
(
∫
0
W
dep
N
(
x
,
z
)
ⅆ
z
)
2
2
∫
0
W
dep
N
(
x
,
z
)
z
ⅆ
z
N
sub
0
=
N
(
x
,
0
)
.
6. The design supporting apparatus according to claim 5 , wherein said device characteristic calculating section calculates the corrected surface potential based on the interaction, by solving simultaneous equations of a quasi 2-dimensional Poisson equation based on a charge density in an inversion layer surface and a current continuation equation indicating that a current between the node points is constant, through an iterative calculation method by using the surface potential as an initial value,
wherein when W is a gate width, μ is a mobility of carrier, and Q i is a channel charge density, the current I D is expressed by the following equation:
I
D
=
W
μ
Q
i
ⅆ
ϕ
f
ⅆ
x
wherein when φ s is the corrected surface potential, Q g a gate charge density, and Q b0 is a charge density in the inversion layer surface, the quasi 2-dimensional Poisson equation is expressed by the following equation:
Q
g
-
Q
i
+
E
Q
b
0
=
C
2
N
sub
N
sub
0
(
β
(
ϕ
s
-
V
B
)
-
1
+
exp
(
-
β
(
ϕ
s
-
V
B
)
)
)
+
E
Q
b
0
2
here, the following equations are met:
Q
g
=
C
ox
(
V
G
-
V
F
B
-
ϕ
s
)
Q
i
=
Q
g
-
Q
g
2
-
C
2
exp
(
β
(
ϕ
s
-
ϕ
f
-
2
Φ
B
)
)
E
=
ɛ
Si
W
dep
Q
b
0
ⅆ
2
ϕ
s
ⅆ
x
2
Q
b
0
=
C
N
sub
N
sub
0
(
β
(
ϕ
s
0
-
V
B
)
-
1
+
exp
(
-
β
(
ϕ
s
0
-
V
B
)
)
)
.
7. The design supporting apparatus according to claim 5 , wherein said device characteristic calculating section calculates the corrected surface potential based on the interaction, by solving simultaneous equations of a quasi 2-dimensional Poisson equation based on a charge density in an inversion layer surface and a current continuation equation indicating that a current between the node points is constant, through an iterative calculation method by using the surface potential as an initial value,
wherein when W is a gate width, μ is a mobility of carrier, and Q i is a channel charge density, the current I D is expressed by the following equation:
I
D
=
W
μ
Q
i
ⅆ
ϕ
f
ⅆ
x
wherein when φ s is the corrected surface potential, Q g a gate charge density, Q b0 is a charge density in the inversion layer surface, and ΔE is a correction term for a short channel effect, the quasi 2-dimensional Poisson equation is expressed by the following equation:
Q
g
-
Q
i
+
E
Q
b
0
=
C
2
N
sub
N
sub
0
(
β
(
ϕ
s
-
V
B
)
-
1
+
exp
(
-
β
(
ϕ
s
-
V
B
)
)
)
+
E
Q
b
0
2
here, the following equations are met:
Q
g
=
C
ox
(
V
G
-
V
FB
-
ϕ
s
)
Q
i
=
Q
g
-
Q
g
2
-
C
2
exp
(
β
(
ϕ
s
-
ϕ
f
-
2
Φ
B
)
)
E
=
ɛ
Si
W
dep
Q
b
0
ⅆ
2
ϕ
s
ⅆ
x
2
+
Δ
E
Q
b
0
=
C
N
sub
N
sub
0
(
β
(
ϕ
s
0
-
V
B
)
-
1
+
exp
(
-
β
(
ϕ
s
0
-
V
B
)
)
)
.
8. The design supporting apparatus according to claim 7 , wherein when a distance from an end of a drain region in said channel region is X d and a distance from an end of a source region in said channel region is X S , said correction term is expressed by the following equation:
Δ
E
=
max
(
1
-
x
X
s
,
0
)
+
max
(
1
-
L
gate
-
x
X
d
,
0
)
.
9. A non-transient computer-readable recording medium in which a computer-executable program code is stored to attain an analyzing method of a semiconductor device, which said analyzing method comprises:
setting an impurity concentration with respect to a channel direction and a depth direction to node points arranged discretely in a channel region of a model transistor based on a predetermined concentration distribution rule;
calculating a surface potential to each of said node points by reducing a dimension of said impurity concentration in the depth direction;
correcting the surface potential based on interaction between said node points adjacent to each other to generate the corrected surface potential;
calculating an electric characteristic of said model transistor by using the corrected surface potential; and
storing said impurity concentration as a model parameter of said model transistor in a storage unit, when the calculated electric characteristic and an electric characteristic prepared previously are coincident with each other within a predetermined range,
wherein said calculating a surface potential comprises:
calculating the surface potential to each of said node points by solving a Poisson equation in the depth direction on said node point,
wherein the Poisson equation is expressed by the following equation:
C
ox
(
V
G
-
V
FB
-
ϕ
so
)
=
C
N
sub
N
sub
0
(
β
(
ϕ
s
0
-
V
)
-
1
+
exp
(
-
β
(
ϕ
s
0
-
V
B
)
)
)
+
exp
(
β
(
ϕ
s
0
-
ϕ
f
-
2
Φ
B
)
)
where φ s0 is the surface potential, V G is a gate voltage, V FB is a flat band voltage, V B is a substrate voltage, N sub0 is an effective impurity concentration on a surface in the channel region, Φ B is an energy difference between a quasi-Fermi level and an intrinsic Fermi level, φ f is the quasi-Fermi level, ∈ ox is a dielectric constant of a gate oxide film, T ox is a film thickness of the gate oxide film, q is an elementary charge amount, ∈ Si is a dielectric constant of silicon, β is a thermal voltage, k B is the Boltzmann constant, T is an absolute temperature on an operation, n i is an intrinsic carrier density, and N(x, z) expresses said impurity concentration set to each node point,
wherein the following equations are met,
C
ox
=
ɛ
ox
T
ox
C
=
2
ɛ
Si
qN
sub
0
β
Φ
B
=
1
β
log
(
N
sub
0
n
i
)
β
=
q
k
B
T
ϕ
s
0
=
q
ɛ
Si
∫
0
W
dep
N
(
x
,
z
)
z
ⅆ
z
+
1
β
+
V
B
N
sub
=
(
∫
0
W
dep
N
(
x
,
z
)
ⅆ
z
)
2
2
∫
0
W
dep
N
(
x
,
z
)
z
ⅆ
z
N
sub
0
=
N
(
x
,
0
)
.
10. The non-transient computer-readable recording medium according to claim 9 , wherein said correcting the surface potential comprises:
calculating the corrected surface potential based on the interaction, by solving simultaneous equations of a quasi 2-dimensional Poisson equation based on a charge density in an inversion layer surface and a current continuation equation indicating that a current between the node points is constant, through an iterative calculation method by using the surface potential as an initial value,
wherein when W is a gate width, μ is a mobility of carrier, and Q i is a channel charge density, the current I D is expressed by the following equation:
I
D
=
W
μ
Q
i
ⅆ
ϕ
f
ⅆ
x
wherein when φ S is the corrected surface potential, Q g a gate charge density, and Q b0 is a charge density in the inversion layer surface, the quasi 2-dimensional Poisson equation is expressed by the following equation:
Q
g
-
Q
i
+
EQ
b
0
=
C
2
N
sub
N
sub
0
(
β
(
ϕ
s
-
V
B
)
-
1
+
exp
(
-
β
(
ϕ
s
-
V
B
)
)
)
+
EQ
b
0
2
here, the following equations are met:
Q
g
=
C
ox
(
V
G
-
V
FB
-
ϕ
s
)
Q
i
=
Q
g
-
Q
g
2
-
C
2
exp
(
β
(
ϕ
s
-
ϕ
f
-
2
Φ
B
)
)
E
=
ɛ
Si
W
dep
Q
b
0
ⅆ
2
ϕ
s
ⅆ
x
2
Q
b
0
=
C
N
sub
N
sub
0
(
β
(
ϕ
s
0
-
V
B
)
-
1
+
exp
(
-
β
(
ϕ
s
0
-
V
B
)
)
)
.
11. The non-transient computer-readable recording medium according to claim 9 , wherein said correcting the surface potential comprises:
calculating the corrected surface potential based on the interaction, by solving simultaneous equations of a quasi 2-dimensional Poisson equation based on a charge density in an inversion layer surface and a current continuation equation indicating that a current between the node points is constant, by an iterative calculation method by using the surface potential as an initial value,
wherein when W is a gate width, μ is a mobility of carrier, and Q i is a channel charge density, the current I D is expressed by the following equation:
I
D
=
W
μ
Q
i
ⅆ
ϕ
f
ⅆ
x
wherein when φ S is the corrected surface potential, Q g a gate charge density, Q b0 is a charge density in the inversion layer surface, and ΔE is a correction term for a short channel effect, the quasi 2-dimensional Poisson equation is expressed by the following equation:
Q
g
-
Q
i
+
EQ
b
0
=
C
2
N
sub
N
sub
0
(
β
(
ϕ
s
-
V
B
)
-
1
+
exp
(
-
β
(
ϕ
s
-
V
B
)
)
)
+
EQ
b
0
2
here, the following equations are met:
Q
g
=
C
ox
(
V
G
-
V
FB
-
ϕ
s
)
Q
i
=
Q
g
-
Q
g
2
-
C
2
exp
(
β
(
ϕ
s
-
ϕ
f
-
2
Φ
B
)
)
E
=
ɛ
Si
W
dep
Q
b
0
ⅆ
2
ϕ
s
ⅆ
x
2
+
Δ
E
Q
b
0
=
C
N
sub
N
sub
0
(
β
(
ϕ
s
0
-
V
B
)
-
1
+
exp
(
-
β
(
ϕ
s
0
-
V
B
)
)
)
.