Calibration of inductance calculations to measurement data for

Calibration of inductance calculations to measurement data for
Calibration of inductance calculations to measurement data
for superconductive integrated circuit process
Coenrad J. Fourie
Department of Electrical and Electronic Engineering, Stellenbosch University, Stellenbosch, South Africa
Introduction
Model and Process
parameters
Importance of inductance calculation often mentioned [1-4].
Calibration:
Segment size
Objective is to match InductEx to measurements
over range of line widths: 0.8 µm - 20 µm.
Manufacturing processes (especially LTS) are characterized
thoroughly, and reliable sheet inductance values are often
available, as for Hypres 4.5 kA/cm2 process [5].
All parameters influence accuracy of results.
For complex layouts, numerical techniques are necessary
(many tool exist [6-10]), but are never completely accurate.
However, model parameters (Fig. 1), which determine
solution speed, also influence accuracy. InductEx [15]
uses segmentation size, ground plane overlap and filament number.
Process parameter (Table 1) effects are obvious.
Modelling technique also affects accuracy [7].
After selection of modelling parameters (segment size, GP
size, etc.) for speed, calculation results should be calibrated
to measurements for reliability over the range of line widths.
Firstly, we fix model parameters (Fig. 1) by finding the largest segment size that gives a flat error
curve when InductEx is used with nominal process parameters (Table 1).
- M1 over ground (M0) error plots in Fig. 2 and 3.
- Select segment size for minimal calculation time
Table I: Hypres layer definitions and nominal process parameters [14].
Description
Ground
Isolation
1st wiring layer
Isolation
2nd wiring layer
Isolation
3rd wiring layer /
We show that FastHenry [11,12] and InductEx [7,13] can
calibrated to withing 2.3 % RMSE of Hypres process measurements for both a mask aligner and a wafer stepper.
We use the Hypres 4.5 kA/cm2 process [14], with measurements:
Mask aligner: 54 test structures repeated over 22 wafers.
Wafer stepper: 48 test structures repeated over 5 wafers.
Mask aligner (3 µm-20 µm): 2.5 µm segments.
Name Thickness (nm) λ (nm) Bias (µm)
M0
100
90
+0.2
I0
150
+0.2
M1
135
90
0
I1B
200
-0.1
M2
300
90
-0.2
I2
500
+0.2
M3
600
90
-0.4
Wafer stepper (0.8 µm-10 µm): 1 µm segments.
Inductive structure
skyplane
Segment
b
Ground
plane
a
a
a
Calibration:
Process parameter adjustment
25
4 um
3 um
20
2.5 um
15
2 um
1.5 um
10
1 um
0.75 um
5
0
0
5
10
15
20
-5
-10
Line width (µm)
Fig. 2: Effect of segment size on InductEx accuracy with
mask aligner structures (nominal process parameters).
6
Select first negative line (-0.05 µm) for width offset (mask-wafer) to adjust curve skew.
Select 6th positive line (103.5 nm) to minimize error.
Recalculate error curves and repeat cycle (also for M2 and M3).
4 um
3 um
2
2.5 um
-2
0
2
4
M0-M1
M0-M2
M0-M3
5.00
M0-M1-M3
M0-M2-M3
0.00
0
5
10
15
20
25
-5.00
-10.00
-15.00
8
10
1 um
0.75 um
0.5 um
-6
0.35 um
-8
-10
Line width (µm)
Mask aligner: calibrated process parameters InductEx difference from measured values
15.00
M0-M1
10.00
Fig. 4: Difference between InductEx (uncalibrated) and
measurements (mask aligner process) vs line width.
Conclusion
Calibration procedure reduces InductEx
calculation result difference from measured results to < 2.3 % RMSE over full
range of line widths from 0.8 µm to 20 µm
for Hypres 4.5 kA/cm2 processes.
M0-M2
M0-M3
5.00
M0-M1-M3
M0-M2-M3
0.00
0
5
10
15
20
Calibrated parameters are artificial, but
very useful.
25
-5.00
-10.00
-15.00
Line width (µm)
Fig. 1: Model parameters (segmentation
described in [16].
1.5 um12
-4
20.00
15.00
10.00
6
Fig. 3: Effect of segment size on InductEx accuracy with
wafer stepper structures (nominal process parameters).
InductEx difference from measured values
Multiple filaments
in height
2 um
0
Difference to average measured inductance (%)
Difference to average measured inductance (%)
Adjustments are made, i.e. M1 (Fig. 5):
5 um
4
20.00 Mask aligner: nominal process parameters -
To adjust process parameters, the effects of variations in
6 parameters are plotted in Fig. 5-7 (for each wiring layer)
from Chang’s analytical equation [17]. (Matlab scripts.)
Difference between InductEx and measurements as function of segment size
(wafer stepper, M1 over ground)
-12
-15
The difference between InductEx calculations and measurements for the mask aligner process, with a segmentation
size of 2.5 µm and nominal process parameters (Table 1), is
plotted in Fig. 4. RMSE = 7.6 %.
8
Difference between InductEx and measurements as function of segment size
(mask aligner, M1 over ground)
5 um
Difference to average measured inductance (%)
Difference to average measured inductance (%)
30
a
Line width (µm)
Fig. 8: Difference between InductEx (calibrated) and
measurements (mask aligner process) vs line width.
Calibration update after each wafer run is
possible, and published on InductEx web
page (www.sun.ac.za/inductex) as layer
definition files (.LDF).
After first cycle, RMSE = 3.06 %.
Final results (Fig. 8): RMSE = 1.87 %.
20.00
InductEx Homepage:
20
10
0
0 µm
10
−5
0
20
−20
0
10
Line width (µm)
20
2
4
6
8
10
-5.00
-10.00
-20%
−5
+10%
0
0%
-10%
0
-20%
0
0%
+20%
−5
−5 -20%
5
Change in λM3
+20%
0%
-20%
+20%
M0-M1-M3
5.00
M0-M2-M3
0.00
0
2
4
0
−5
−10
−5
−10
−15
20
−10
0
-10%
0%
−5
0
20
10
Fig. 5: Effect of parameter variations on M1 line inductance, with required InductEx calibration shift.
20
−10
0
+20%
10
+10%
0
0%
-10%
−20
0
-20%
10
10
−10
0
20
30 Change in M2 width offset
Change in isolation
thickness
−10
10
20
20
-0.3 µm
20
10
0
0 µm
−10
−20
0
10
5 Change in M0 thickness
-20%
0
0%
+20%
−5
+0.3 µm
10
Line width (µm)
20
−20
0
20
−10
0
10
Fig. 6: Effect of parameter variations on M2 line inductance, with required InductEx calibration shift.
20
10
20
-20%
0%
+20%
−5
−15
−20
0
10
20
20 Change in isolation
thickness
20 Change in M3 width offset
10
10
+20%
0
0%
−10
-20%
−20
0
0 µm
20
−20
0
+0.3 µm
10
20
5 Change in M0 thickness
0
−10
10
−20
0
-0.3 µm
0
10
12
Line width (µm)
−10
−15
8
-10.00
0
-20%
6
-5.00
5 Change in M3 thickness
Change in λM0
0%
-20%
0%
+20%
−5
−10
−15
10
20
Line width (µm)
−20
0
References
M0-M2
10.00
Fig. 10: Difference between InductEx (calibrated)
and measurements (wafer stepper) vs line width.
∆LM0-M3 (%)
-10%
5
5
5 Change in M2 thickness
Change in λM0
+20%
∆LM0-M3 (%)
0%
M0-M1
-15.00
Line width (µm)
∆LM0-M3 (%)
0
12
15.00
Required change in L
Positive parameter change
Negative parameter change
∆LM0-M2 (%)
+10%
∆LM0-M2 (%)
∆LM0-M2 (%)
10
5 -20%
0
10
Change in λM2
0%
+20%
+0.3 µm
0
Wafer stepper: Calibrated process parameters InductEx difference from measured values
+20%
10
−10
-20%
−10
0
-20%
30 Change in M1 width offset 15 Change in M0 thickness
-0.3 µm
M0-M2-M3
0.00
∆LM0-M3 (%)
0%
-10%
20
M0-M1-M3
∆LM0-M3 (%)
+10%
10
5.00
∆LM0-M3 (%)
+20%
0
−5
−10
0
M0-M2
Fig. 9: Difference between InductEx (uncalibrated)
and measurements (wafer stepper) vs line width.
∆LM0-M2 (%)
∆LM0-M1 (%)
10
5
20
5
0
-20%
10
Change in isolation
thickness
15
0
0%
−5 -10%
-20%
−10
0
5 +10%
+20%
∆LM0-M2 (%)
-10%
10
∆LM0-M1 (%)
−5
0%
+20%
∆LM0-M2 (%)
5 +10%
10
5
15 Change in M1 thickness
Change in λM0
M0-M1
Required change in L
Positive parameter change
Negative parameter change
∆LM0-M1 (%)
+20%
0
15
∆LM0-M1 (%)
10
Change in λM1
∆LM0-M1 (%)
∆LM0-M1 (%)
15
10.00
-15.00
www.sun.ac.za/inductex
Required change in L
Positive parameter change
Negative parameter change
15.00
Difference to average measured inductance (%)
Difference to average measured inductance (%)
Similar sequence for wafer stepper reduces uncalibrated RMSE (Fig. 9) of 7 % to calibrated RMSE = 2.25 %
(Fig. 10).
20.00
Wafer stepper: nominal process parameters InductEx difference from measured values
10
Fig. 7: Effect of parameter variations on M3 line inductance, with required InductEx calibration shift.
20
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