Determination of Factors Affecting HRTEM Gate Dielectric
Thickness Measurement Uncertainty
John Henry J. Scott
Surface and Microanalysis Science Division, NIST, 100 Bureau Dr.,Gaithersburg, MD 20899-8371
Abstract. Because high-resolution transmission electron microscopy (HRTEM) relies on a complex contrast
mechanism to produce images of gate dielectric films in cross section, there are many factors affecting the uncertainty of
thickness measurements based on these images. A preliminary survey revealed approximately 50 parameters that affect
the uncertainty in a gate dielectric dimensional metrology experiment using HRTEM, along with approximately 1,200
two-term interactions and almost 20,000 three-term interactions. Using established design-of-experiment (DEX)
methodologies, I performed a screening experiment based on a 2IV(8-4) fractional factorial design to determine which
factors had the greatest impact on the absolute error of the thickness measurements. Absolute error was determined by
simulating HRTEM micrographs using a multislice calculation. The model used for the simulation consisted of a
variable SiO2 film approximately 2 nm thick positioned between two pieces of crystalline Si. This approximation to a
gate stack was built atom-by-atom using commercial molecular modeling software supplemented with custom Tcl
scripts to assemble the gate structures from simpler primitives. By varying the molecular model, sample parameters such
as crystallographic orientation, film thickness, density, and along-beam thickness can be varied precisely. Instrument
parameters and details of the imaging conditions are inputs to the multislice calculation, a simulation technique that has
been vetted by the microscopy community and has been in use for decades. Beam tilt, defocus, and vibration amplitude
were the main factors found to have the largest effects, while beam-tilt↔defocus and defocus↔vibration were the most
important two-term interactions.
multislice HRTEM image simulation techniques.
Measured values of the gate dielectric thickness are
derived from computer-simulated micrographs and
combined with the true thickness of the dielectric
(known exactly from the computer model) to compute
the measurement error [2].
INTRODUCTION
Although HRTEM micrographs appear to provide a
direct, atomic-resolution image of ultrathin films, it is
important to remember that the phase contrast transfer
mechanism employed is interferometric in nature and
can result in complex artifacts that are easily
misinterpreted [1]. An understanding of the errors and
uncertainties introduced by these artifacts is essential
for dimensional metrology of gate dielectrics at the
level now demanded by the semiconductor industry.
EXPERIMENT
Models of gates stacks were built using Cerius2, a
commercial molecular modeling package.† Custom
To determine the error in a thickness measurement
it is necessary to know both the measured thickness
(the experimental outcome) and the true thickness of
the sample:
error = (measured value) – (true value).
†
Certain commercial equipment, instruments, or materials are
identified in this paper in order to specify the experimental
procedure adequately. Such identification is not intended to imply
recommendation or endorsement by the National Institute of
Standards and Technology, nor is it intended to imply that the
materials or the equipment identified are necessarily the best
available for the purpose.
(1)
This work attempts to apply Equation (1) directly by
simulating gate dielectric samples using established
CP683, Characterization and Metrology for ULSI Technology: 2003 International Conference,
edited by D. G. Seiler, A. C. Diebold, T. J. Shaffner, R. McDonald, S. Zollner, R. P. Khosla, and E. M. Secula
2003 American Institute of Physics 0-7354-0152-7/03/$20.00
348
1.3 nm
2.1 nm
2.1 nm cube of SiO2
Figure 1. Schematics showing atom positions in gate stack models with two different gate dielectric thicknesses, 1.3 nm and
2.1 nm (left). Three dimensional projection of the amorphous dielectric phase shown without bordering crystalline Si (right).
gate dielectric thickness which can be compared to the
true value of the thickness defined by the exact atom
positions in the model. Magnification calibration is
performed on the single crystal Si regions of the image
using a two-dimensional least-squares fit of the lattice.
Center-of-mass weighting of the intensity near bright
spots in the image allows sub-pixel determination of
interplanar atomic spacings, measured in pixels. This
information is combined with known Si d-spacings to
obtain a nm/pixel calibration [4].
scripts written in the Tcl scripting language were
developed to extend the Cerius2 feature set for gate
dielectric modeling. Using the Cerius2 application
programming interface (API) and graphical user
interface (GUI) palette, these scripts allow the user to
select gate-specific and HRTEM imaging input
parameters, direct the construction of new gate stack
models, and drive the multislice simulation module.
The gate stack models used here consist of regions
of amorphous SiO2 sandwiched between two pieces of
crystalline silicon, meant to simulate the channel and
polycrystalline silicon (poly-Si) gate electrode. Slabs
of amorphous dielectric and crystalline Si are
juxtaposed, unphysical atom overlaps are eliminated,
and Si-Si and Si-O bonds are permitted to form at the
two interfaces, but the atom positions are fixed and no
energy minimization or relaxation of the structure is
performed (Figure 1).
The thickness of the dielectric layer is extracted
from the micrographs using Lispix, a free image
processing program written in Lisp by David Bright at
NIST [5]. The micrographs are oriented so the two Sidielectric interfaces are vertical. The variance in
intensity is calculated for each vertical row of pixels
and plotted against horizontal position, producing a
vertical-variance plot similar to the plots introduced by
Taylor, et al. [2]. On the left and right extremes of the
image, in the crystalline Si regions, the variance is
large due to high maxima and low minima from the
lattice. In the center of the image the variance is much
smaller since the amorphous dielectric exhibits smaller
swings in intensity and much higher spatial
frequencies. The variance threshold corresponding to
the amorphous-crystalline transition is determined
empirically and used to automatically locate the
horizontal position of the interfaces, and hence the
thickness of the dielectric layer. The definition of this
threshold is subjective and is potentially a source of
interlaboratory bias. A separate research effort is
aimed at understanding the nature of this threshold
Multislice simulations of HRTEM micrographs are
produced from the gate stack models using the
HRTEM module included with Cerius2. This
simulation module is based on earlier code written by
Owen Saxton [3]. Like most multislice programs it
allows the user to specify a large number of
parameters that describe the microscope and imaging
conditions desired for the simulation. All of these
input variables can be controlled through the Tcl
scripts.
The simulated micrographs are processed following
the same steps used for experimentally-obtained
micrographs. This produces a "measured" value for the
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TABLE 1. List of input parameters considered in the twolevel fractional factorial design, and the values chosen for the
two levels for each variable.
Var
X1
X2
X3
X4
X5
X6
X7
X8
Description
beam tilt
tilt azimuth
beam thickness
film thickness
defocus
astigmatism mag
astigmatism azimuth
vibration amplitude
Low (-1)
0 mrad
0º
1.2 nm
1.5 nm
-58.4 nm
0 nm
0º
0 nm
TABLE 2. Composition of the fraction factorial design
showing the combinations of levels chosen for the 16 runs
in the design. The thickness measurement error Y [nm]
represents the outcome of each trial (the response variable).
High (+1)
25 mrad
25º
1.9 nm
2.0 nm
-48.4 nm
0.08 nm
25º
0.04 nm
Y (nm)
X1 X2 X3 X4 X5 X6 X7 X8
----------------------------------------0.470
-1 -1 -1 -1 -1 -1 -1 -1
-0.360
+1 -1 -1 -1 -1 +1 +1 +1
-0.118
-1 +1 -1 -1 +1 -1 +1 +1
0.386
+1 +1 -1 -1 +1 +1 -1 -1
0.650
-1 -1 +1 -1 +1 +1 +1 -1
-0.535
+1 -1 +1 -1 +1 -1 -1 +1
-0.513
-1 +1 +1 -1 -1 +1 -1 +1
-0.470
+1 +1 +1 -1 -1 -1 +1 -1
0.018
-1 -1 -1 +1 +1 +1 -1 +1
0.018
+1 -1 -1 +1 +1 -1 +1 -1
-0.509
-1 +1 -1 +1 -1 +1 +1 -1
-0.487
+1 +1 -1 +1 -1 -1 -1 +1
-0.487
-1 -1 +1 +1 -1 -1 +1 +1
-0.443
+1 -1 +1 +1 -1 +1 -1 -1
0.720
-1 +1 +1 +1 +1 -1 -1 -1
-0.443
+1 +1 +1 +1 +1 +1 +1 +1
assignment and finding an alternative interface
discriminant superior to the vertical-variance plot.
A survey of factors potentially affecting HRTEMbased dimensional metrology produced a list of about
50 scalar parameters. These factors are grouped into
three classes: (1) factors related to the microscope and
imaging conditions; (2) factors describing the sample;
and (3) factors that determine the fidelity of the
multislice simulation used to assess measurement
uncertainties. Class 1 is important for experimental
measurements and includes such factors as lens
aberrations, defocus, defocus spread, beam tilt,
astigmatism, vibration, incident beam energy, and the
location and size of beam apertures. Class 2 is also
important for practical measurements and includes
dielectric film thickness, along-beam "foil" thickness,
interface roughness, and the characteristics of any
damaged surface layers introduced during sample
preparation. Class 3 is only important to the
simulations, comprising factors such as a* and b*
range (number of beams), the number of slices in the
cell, the scattering factors used, the number of pixels
simulated, Debye-Waller temperature factors, and the
nature of approximations used in the code to simplify
the calculation. Because these factors interact with
each other, a proper uncertainty budget should
consider at least the roughly 1200 2-term interactions
as well as the nearly 20,000 3-term interactions.
Although 4-term and higher order interactions have
been documented in complex systems, they are
relatively rare.
this early work is merely to identify important factors,
not determine their quantitative influence, a two-level
screening design was chosen because of its power and
efficiency. Experience was drawn upon to determine
the values of the two levels (high and low, or + and -)
for each of the factors.
The factor and level combinations needed for the
screening were drawn from standard tables [6]. In this
case a 2IV(8-4) fractional factorial design was chosen. In
this notation the "2" indicates a two-level design (two
possible values for each input parameter), the "8"
indicates 8 factors or parameters are considered, and
the "IV" reveals that this design is resolution four.
Resolution IV in this context implies that the main
effects of the 8 variables are not confounded with any
2-term interactions. Some confounding of 2-term
interactions with each other is present, however. This
design requires 2(8-4) = 24 = 16 runs. The outcome or
response variable Y is the thickness error (Equation 1)
in nm. Table 2 shows the 16 runs used and their
outcomes.
RESULTS
The task of determining the importance of such a
large number of factors is daunting, but it can be
simplified by using a priori knowledge of the system,
experimental experience, and statistical design-ofexperiment (DEX) methods. To demonstrate the
process, 8 factors thought to be important were
selected for DEX analysis (Table 1). Since the goal of
While the raw results of each run appear in the Y
column of Table 2, the result of the DEX screening
process (the identification of important factors
affecting HRTEM uncertainty) requires further
processing and interpretation. One useful tool for
quickly visualizing important factors is to plot the
main effects (Figure 2). When displayed in this format,
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FIGURE 2. Main Effects Plot showing the importance of each of the input parameters X1 through X8. The information
conveyed is qualitative: important variables exhibit a large slope. Based on this data, factors X1, X5, and X8 (all circled) have
a significant effect on the thickness measurement error.
from confounding. Recall that a resolution IV design
involves some confounding of 2-term interactions with
each other. In this case the following 2-term effects are
mixed: X1-X3, X2-X7, X4-X6, and X5-X8.
factors that have a large impact on the thickness
measurement error appear as line segments with large
slope. A quick glance at Figure 2 reveals that factors
X1, X5, and X8 (beam tilt, defocus, and vibration) are
significant while the others have a much smaller
effect.
Several other statistical tests and visualization
strategies were independently applied to the 16 runs to
identify important effects, including factor block plots,
stickpin effects plots, half-normal probability plots,
and Youden plots (none shown here). In every case the
same three main effects and two 2-term interactions
were identified as significant.
A similar device (Figure 3) can be used to quickly
spot which of the unique 2-term interactions are
significant. In this plot, columns from left to right
represent factors X1-X8, while rows from top to
bottom represent the same factors X1-X8.
Diagonal
cells represent the same eight 1-term (main) effects
shown in Figure 2. In addition to the three important
effects identified in Figure 2, two more off-diagonal
effects appear here: X1-X5 and X5-X8 (circled).
Several of the other off-diagonal cells have large
slopes but are not circled because they represent the
same effects; this is due to the degeneracy that results
CONCLUSIONS
Following
traditional
design-of-experiment
(DEX) methods, 16 HRTEM multislice simulations
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FIGURE 3. Interaction Effects Matrix showing the importance of the same three main effects as seen in Figure 2 (circled
diagonal elements), as well as two important 2-term interactions (off-diagonal elements). Large slope denotes an important
parameter or 2-term interaction. Uncircled cells with large slope are degenerate with circled cells because of the confounding
structure of the design and carry no new information.
were used to determine conclusively that beam tilt,
defocus, and vibration amplitude have a significant
impact on the measurement error of gate dielectric
thickness. The azimuth of the tilt, the sample
thickness along the beam, the dielectric thickness
itself, and the magnitude and azimuth of the
astigmatism have a much smaller effect. Useful
information about 2-term interactions was also
revealed, but the limited resolution of the screening
design prevents conclusive identification of specific
effects. The same methods are easily extended to
higher resolution (requiring more simulations) to
resolve these issues.
REFERENCES
1. Cowley, J.M., Eyring, L., and Buseck, P.R. (Editors),
High-Resolution Transmission Electron Microscopy and
Associated Techniques, Oxford University Press,
London, 1988.
2. Taylor, S., Mardinly, J., O’Keefe, M., and Gronsky, R.,
“HRTEM Image Simulations for Gate Oxide
Metrology”, Char. and Metrology for ULSI Tech. 2000,
AIP Conference Proceedings 550, 2000, pp. 130-133.
3. Saxton, W.O., O’Keefe, M.A., Cockayne, D.J.H., and
Wilkens, M., Ultramicroscopy 12, 75 (1983).
4. Scott, J.H., Windsor, E.W., “Chemical and Structural
Characterization of Ultrathin Dielectric Films Using
AEM”, Structure and Electronic Properties of Ultrathin
Dielectric Films on Silicon and Related Structures,
edited by D.A. Buchanan, et al. Materials Research
Society Symposium Proceedings, Volume 592, 2000, pp.
153-158.
5. http://www.nist.gov/lispix
6. Box, G.E.P., Hunter, W.G., Hunter, J.S., Statistics for
Experimenters, John Wiley & Sons, New York, 1978.
ACKNOWLEDGMENTS
I gratefully acknowledge James Filliben and
Ivelisse Aviles of the NIST Statistical Engineering
Division for assembling an outstanding internal
training course on DEX for NIST scientists and
engineers. I also acknowledge the support of the NIST
Office of Microelectronics Programs.
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