Method for extracting the distribution of charge stored in a semiconductor device
View Patent ↗A method is described for extracting the spatial distribution of charge stored in a charge-trapping layer of a semiconductor device. The method comprises the steps of performing a first charge-pumping measurement on a device under test using a variation of the upper level of the pulse and performing a second charge-pumping measurement on this device using a variation of the lower level of the pulse. The data obtained is combined for extracting the spatial distribution. This is done by establishing a relation between a charge pumping current I cp and a calculated channel length L calc of the semiconductor device by reconstructing spatial charge distribution estimates from the charge pumping curves for multiple values of the charge pumping current I cp . From these multiple values of I cp the value is obtained for which the corresponding calculated channel length L calc is substantially equal to the effective channel length L eff of the semiconductor device and the spatial charge distribution is reconstructed from the charge pumping curves using the obtained value of I cp .
1. A method for extracting the spatial distribution of charge (N nt ) stored in a charge-trapping layer of a semiconductor device having a channel underneath the charge trapping layer with an effective length L eff , comprising:
a) determining a varying base-level voltage charge pumping curve,
b) determining a varying top-level voltage charge pumping curve, and
c) combining data from the charge pumping curves to obtain the spatial distribution,
wherein step c) comprises the steps of:
d) establishing a relation between a charge pumping current I cp and a calculated channel length L calc of the semiconductor device by reconstructing spatial charge distribution estimates from the charge pumping curves for multiple values of the charge pumping current I cp ,
e) obtaining from the multiple values of I cp the value for which the corresponding calculated channel length L calc is substantially equal to the effective channel length L eff of the semiconductor device, and
f) reconstructing the spatial charge distribution from the charge pumping curves using the value of I cp obtained in step e).
2. A method according to claim 1 , wherein steps d) to e) comprise:
g) selecting on one of the charge pumping curves one value of the charge pumping current I cp ,
h) determining the calculated channel length L calc corresponding to the selected charge pumping current I cp by reconstructing a spatial charge distribution estimate for the charge pumping current I cp ,
i) comparing the calculated channel length L calc with the effective length L eff and in case of a mismatch, determining from the mismatch a new value for the charge pumping current I cp , and
j) repeating steps h) to i) until the mismatch between the calculated channel length L calc and the effective length L eff is substantially zero.
3. A method according to claim 1 , wherein steps d) to e) comprise:
k) selecting on one of the charge pumping curves at least two charge pumping currents I cp ,
l) determining the calculated channel length L calc corresponding to each of the selected charge pumping currents I cp by reconstructing a spatial charge distribution estimate for each of the selected charge pumping currents I cp , thereby creating a set of at least two datapoints (L calc , I cp ) and
m) determining by interpolation from the set of at least two datapoints the charge pumping current I cp corresponding to a channel length L calc substantially equal to the effective length L eff .
4. A method according to claim 1 , wherein steps d) to e) comprise:
n) selecting on one of the charge pumping curves at least two charge pumping currents I cp ,
o) determining the calculated channel length L calc corresponding to each of the selected charge pumping currents I cp by reconstructing a spatial charge distribution estimate for each of the selected charge pumping currents I cp , thereby creating a set of at least two datapoints (L calc , I cp ),
p) determining an analytical function L calc (I cp )−L eff =0 from the set of at least two datapoints, and
q) determining by solving the analytical function the charge pumping current I cp corresponding to a channel length L calc substantially equal to the effective length L eff .
5. A method according to claim 1 , wherein data from the charge pumping curves is further combined to obtain a spatial distribution of charge in traps (N it ) present at the interface between the channel and a dielectric stack which separates the charge trapping layer from the channel.
6. A method according to claim 1 , further comprising the initial step of bringing the semiconductor device into a reference state by subjecting it to a light programming operation, resulting in a monotonic increase of the charge in the charge-trapping layer which results in a threshold profile V th (x) that monotonically varies along the channel.
7. A method according to claim 6 , wherein the method is performed at least once on the semiconductor device in the reference state and at least once on the semiconductor device in a cycled state after one or more programming/erase cycles, to determine the contribution of charges remaining in the charge-trapping layer after the one or more programming/erase cycles to the threshold voltage along the channel.
8. A method according to claim 1 , wherein step a) comprises applying a pulse train to a gate electrode, which is located above the charge-trapping layer opposite the channel, wherein each pulse in this pulse train has the same top-level voltage and the amplitude of the pulses increases monotonically with time by lowering the base-level voltage of the pulses.
9. A method according to claim 1 , wherein step b) comprises applying a pulse train to a gate electrode, which is located above the charge-trapping layer opposite the channel, wherein each pulse in this pulse train has the same base-level voltage and the amplitude of the pulses increases monotonically with time by increasing the top-level voltage of the pulses.
10. A method according to claim 1 , wherein the difference between the obtained calculated channel length L calc and the effective length L eff is less than 1%.
11. A method according to claim 1 , wherein reconstructing a spatial charge distribution estimate from the charge pumping curves comprises:
r) selecting a starting point on one of the charge-pumping curves,
s) solving the equations
(
V
th_ref
(
x
)
-
V
fb_ref
(
x
)
)
-
(
V
th_cyc
(
x
)
-
V
fb_cyc
(
x
)
)
=
q
Δ
N
it
(
x
)
C
,
and
∫
X
i
-
1
X
i
qfN
it
(
x
)
ⅆ
x
≈
(
(
N
it
(
x
i
)
-
N
it
(
x
i
-
1
)
)
(
x
i
-
x
i
-
1
2
)
≈
I
cp
(
V
th_cyc
(
x
i
)
)
-
I
cp
(
V
th_cyc
(
x
i
-
1
)
)
from the starting point over a selected portion of the charge-pumping curve, and
t) solving at least one of the equations
Δ
V
th
(
x
)
=
V
th_ref
(
x
)
-
V
th_cyc
(
x
)
=
Δ
Q
nt
(
x
)
-
Δ
Q
it
(
x
)
2
C
=
q
Δ
N
nt
(
x
)
-
q
Δ
N
it
(
x
)
2
C
or
Δ
V
fb
(
x
)
=
V
fb_ref
(
x
)
-
V
fb_cyc
(
x
)
=
Δ
Q
nt
(
x
)
+
Δ
Q
it
(
x
)
2
C
=
q
Δ
N
nt
(
x
)
+
q
Δ
N
it
(
x
)
2
C
,
wherein
V th — ref (x) is the threshold voltage curve of a reference device,
V fb — ref (x) is the flatband voltage curve of a reference device,
V th — cyc (x) is the threshold voltage curve of the semiconductor device after at least one program/erase operation,
V fb — cyc (x) is the flatband voltage curve of the semiconductor device after at least one program/erase operation,
q the absolute value of the electron charge,
N it (x) is the number interface traps (number/cm2),
N nt (x) is the number of charge carriers present in the charge-trapping layer (number/cm2),
f is the frequency of the varying-level voltage signal,
Q nt is the charge present in the charge-trapping layer (C/cm2),
Q it is the charge present in the interface traps (C/cm2),
x is a coordinate along the channel of the semiconductor device,
C is the capacitance of the dielectric stack ( 4 ) (F/cm2).
12. A method according to claim 1 , wherein reconstructing a spatial charge distribution estimate from the charge pumping curves comprises:
u) selecting a starting point on one of the charge-pumping curves,
v) solving the equations
N
it
(
x
i
)
=
N
it
,
ref
+
C
q
[
(
V
th_ref
(
x
)
-
V
fb_ref
(
x
)
)
-
(
V
th_cyc
(
x
)
-
V
fb_cyc
(
x
)
)
]
,
and
N
it
(
x
i
)
=
2
[
I
cp
(
V
th_cyc
(
x
i
)
-
I
cp
(
V
th_cyc
(
x
i
-
1
)
)
]
x
i
-
x
i
-
1
-
N
it
(
x
i
-
1
)
from the starting point over a selected portion of the charge-pumping curve, and
w) solving at least one of the equations
Δ
V
th
(
x
)
=
V
th_ref
(
x
)
-
V
th_cyc
(
x
)
=
Δ
Q
n
t
(
x
)
-
Δ
Q
it
(
x
)
2
C
=
q
Δ
N
n
t
(
x
)
-
q
Δ
N
it
(
x
)
2
C
or
Δ
V
fb
(
x
)
=
V
fb_ref
(
x
)
-
V
fb_cyc
(
x
)
=
Δ
Q
n
t
(
x
)
+
Δ
Q
it
(
x
)
2
C
=
q
Δ
N
n
t
(
x
)
+
q
Δ
N
it
(
x
)
2
C
,
wherein
V th — ref (x) is the threshold voltage curve of a reference device,
V fb — ref (x) is the flatband voltage curve of a reference device,
V th — cyc (x) is the threshold voltage curve of the semiconductor device after at least one program/erase operation,
V fb — cyc (x) is the flatband voltage curve of the semiconductor device after at least one program/erase operation,
q the absolute value of the electron charge,
ΔN it (x) is the number interface traps (number/cm2),
ΔN nt (x) is the number of charge carriers present in the charge-trapping layer (number/cm2),
f is the frequency of the varying-level voltage signal,
Q nt is the charge present in the charge-trapping layer (C/cm2),
Q it is the charge present in the interface traps (C/cm2),
x is a coordinate along the channel of the semiconductor device,
C is the capacitance of the dielectric stack (F/cm2).