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Exploring Semiconductors on the Nanometer Scale:
The Development of X-Ray Reflectivity Analysis Tools
Christopher Bishop PaynePrinceton University
United States Department of EnergyOffice of Science, Science Undergraduate Laboratory Internship Program
SLAC National Accelerator LaboratoryMenlo Park, California
July 20th, 2011
Prepared in partial fulfillment of the requirement of the Office of Science, Department of Energy’s Science Undergraduate Laboratory Internship under the direction of Apurva Mehta and
Matt Bibbe in the SSRL division of SLAC National Accelerator Laboratory.
Participant: __________________________________Signature
Research Advisors: __________________________________Signature
1
Note to draft reader:
Thanks for taking the time to read my paper, in order to make your life a
little easier I would like to point out a few areas of the paper that I know are
lacking/are in development:
1)The matlab figures are hard to read in some cases, something happened
when they were imported to Word so I will fix these.
2) The mathematical connection between oscillations and thickness is not
easily made in the theory section. I am working with my advisor to clarify
this. (the transition from eq 10 to 11)
3)Not all the figures are labeled/labeled neatly, I will remedy this.
4)In the results section, I do not discuss how altering the range affects
either algorithm , this will be inserted in the final draft.
5)The Conclusion/Discussion section is in the process of being overhauled
as it is greatly lacking. I am in the process of preparing a separate
presentation for GE(separate from my talk on thur) telling them what I
found in their data and am going to use parts of this presentation to connect
the development of 2N back to the initial problem GE had.
Thanks,
Chris
2
I. Introduction
The next generation of electronic devices that will power an increasingly technology
dependent world will push the boundaries of our capabilities to manufacture semiconductor chips
that are both smaller and composed of new materials. In terms of being smaller, semiconductor
chips – which are at the heart of all electronics – will need to be manufactured with precision on
the order of a few nanometers in order to keep pace with Moore’s Law. This law, a fundamental
trend in semiconductor manufacturing, corresponds to electronics that are cheaper and have a
higher processing power per area than their predecessors. Additionally, nearly all
semiconductors are now made of silicon, however the use of new semiconductor materials could
yield electronics that are more efficient or can operate under greater extremes.
In order to unlock these favorable attributes of these next generation semiconductors, we
need to be able to characterize and analyze materials with a resolution of a few nanometers.
Many different techniques have been developed to perform this analysis yet the one we will
focus on is called X-Ray Reflectivity(XRR). This technique quantifies parameters of materials
by revealing the number of layers the material is composed of and each layer’s corresponding
thickness, density, and roughness value.i We will discuss these parameters in more detail in the
theory section, for now, it is sufficient to understand that XRR allows the user to understand the
physical structure of a material on the nanometer scale. It should also be noted that unlike other
3
techniques, XRR does not destroy the sample nor does it require specially prepared samples such
Transmission Electron Microscopyii.
Our collaborators on this project, GE, want to realize the potential of these next
generation semiconductors by developing semiconductor’s made from silicon carbide(SiC). GE
believes SiC semiconductor devices would be much more energy efficient than those in use
today and thus make a wide range of applications – from wind turbines to hybrid-electric
vehicles – more efficient.iii Before society can benefit from these favorable SiC semiconductor
attributes, GE needs to understand more about the structure of SiC semiconductors. In order to
do this, GE has been working with Apurva Mehta and Matthew Bibee of SLAC National
Laboratory, who have been taking XRR measurements in order to help answer GE’s questions
about the structure of SiC semiconductors.
I was specifically tasked with developing tools to analyze the XRR data that my
colleagues collected so that we could answer GE with a higher level of confidence, a crucial step
on the path to developing more efficient electronic devices in a world that urgently seeks them.
II. Materials and Methods
i. Reflectivity Theory
In order to fully appreciate and understand the XRR data analysis tools I developed
during this project, we must first establish the theory behind the XRR measurements that were
taken on Beam Line 2-1 at SSRL to provide the XRR data to analyze. XRR is a well established
technique in which a sample is illuminated by an x-ray beam and the reflectivity of the sample is
measured with respect to theta(See Figure 2), with theta typically ranging from zero to eight
degrees.
4
ϴ
Incident Xray Reflected Signal
SampleFigure 2
Detector
d
θ
Detector
Substrate
Thin Layerθ
When the sample is struck in this low theta range, the corresponding reflectivity signal
contains information about the electron density of the surface of the sample.iv Gradients in this
electron density data are correlated with different layers of material existing on the surface of the
sample and thus we are able to detect and characterize surface layers on the order of 1 nm.
To understand the connection between how this layer information is encoded in the reflectivity
versus theta data, we turn to Reflectivity Theory, a model I will now discuss in brief. In Figure 3,
we model a thin layer of thickness d on a substrate:
According to Bragg’s Law, the extra distance traveled by the beam that reflects off the substrate
is:
5
Extra Distance=2 d sin (θ) (1)
This extra distance corresponds to a phase difference at the point that the reflected beams
interfere at the detector, dependent on the wavelength(λ ¿ of the incident beam:
Phase Difference At Detector=2d sin (θ)λ
(2)
As noted in Figure 3, this phase difference causes the relationship between the two waves to be:
(1) Amplitude of Beam Reflected off of ThinTop Layer=R (q )e i (wt +kx ) (3)
(2) Amplitude of Beam Reflected off of Substrate=R(q)ei (wt +kx+2 π 2d sin (θ )
λ) (4)
The coefficient R(q) which modulates the amplitude of both waves is defined to be dependent
on:
q=4 π sin (θ)λ
(5)
sin (ϕ ) ≈ ϕ for small ϕ (6)It is important to note that because of (6) and the fact that λ is constant, q is approximately
linearly dependent on θ . With this in mind, we can now rewrite our equations to be:
(1 ) Amplitude of BeamReflected off of ThinTop Layer=R (q ) ei ( wt+kx )(7)
(2 ) Amplitude of Beam Reflected off of Substrate=R ( q ) ei ( wt+kx +dq)(7)
When these two waves interfere ‘faraway’ at the detector, the detector records the summation of
these waves:
Amplitude Detector Senses=R (q ) ei ( wt+kx )+R (q)ei (wt+kx +dq) (9)
6
Amplitude Detector Senses=ei ( wt+kx )(R (q)+R(q)e i (dq )) (10)
Accounting for all the reflections occurring on the sample, the detector ultimately records an
intensity of the following nature, where σ can be thought of as just a constant:
Intensity (q )=R (q ) e−σ2 Q2
(11)v
The two important points to understand from this result are that the intensity oscillates with
respect to q and that these oscillations exponentially decay with q. Lastly, these oscillations are
so important as their characteristics contain information such as thickness about each layer on the
substrate.
ii. Analysis Tool Development Outline
The challenge that I was tasked with solving was taking the XRR data and extracting the
layer information encoded within the intensity versus theta data. This extraction was made
challenging because the oscillations occur over a range of many decades which both distorts the
oscillations and exacerbates the effects of experimental noise.
In order to meet this task I used MATLAB to create and test algorithms that would
extract the information rich oscillations from the intensity data. A secondary algorithm would
then take the Fourier transform of the extracted oscillations, a procedure that would
quantitatively reveal the different frequency components present in the extracted oscillations.
These frequencies mathematically correspond to the thicknesses of layers present within the
sample. Other data, such as roughness or density of the layers is thought to be contained within
the amplitude of the oscillations in Fourier space, however, this is only a hypothesis at this point
in time.
7
Al
SiC
SiO2
In order to test the accuracy of my MATLAB algorithm, I used a MATLAB program
called Multig, developed by Anneli Munkholm and Sean M. Brennan. This program took as an
input the parameters I was trying to extract from the oscillation data and outputted a simulated
intensity versus theta curve. I would then apply my algorithm to the simulated data and see if I
could determine the same parameters as I had created the curve with. I used this simple process
to develop and tune my algorithm until I could apply the algorithm to a simulated data set and
return the number of layers present in the simulation and their corresponding thickness. As I will
discuss, it is not currently understood how other parameters such as density can be extracted
from the intensity versus theta curves. This entire process is outlined in the four steps below:
1. Data Input: Simulate a simple 2 nm layer of Al between a 20 nm layer of silicon oxide (SiO2)
on a SiC substrate with no roughness.
8
0 1 2 3 4 5 6 7 8 90
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
theta
norm
aliz
ed in
tens
ity
Multig Simulated Intensity Versus Reflectivity Curve
2. Data Analysis: Convert the intensity versus theta data into log(intensity) versus theta and
apply an algorithm that quantifies how much the oscillations in the data are being distorted at
each point. Additionally, the algorithm removes any data below a specified theta, 0.18 in this
case.
3. Distortion Removal: Remove this distortion (plotted in green) from the curve (plotted in red)
and convert the oscillations back to real intensity space (non-log). We now can visually see the
oscillations that were present in the original intensity versus theta curve plotted in blue!
9
0 1 2 3 4 5 6 7 8 90
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
theta
norm
aliz
ed in
tens
ity
0 1 2 3 4 5 6 7 8 9-10
-8
-6
-4
-2
0
theta
log(
inte
nsity
)
0 1 2 3 4 5 6 7 8 90
0.5
1
1.5
2
2.5
3
theta
inte
nsity
- 2N
0 10 20 30 40 50 60 700
100
200
300
400
500
600
nm
amplitu
de o
f signa
l
1
2
3 4
5
0 1 2 3 4 5 6 7 8 9-10
-8
-6
-4
-2
0
theta
log(
inte
nsity
)
0 1 2 3 4 5 6 7 8 90
0.5
1
1.5
2
2.5
3
theta
inte
nsity
- 2N
0 10 20 30 40 50 60 700
100
200
300
400
500
600
nm
ampl
itude
of s
igna
l
1
2
3 4
5
Al
SiC
SiO2
4. Fourier Space Analysis: Lastly, we take the Fourier transform of the extracted oscillation in
order to quantitatively know the number of layers and their corresponding thicknesses. In this
case, this final plot is interpreted to indicate two layers are present on a substrate. Peak four
indicates a layer of 2.01 nm and peak three indicates a layer of 20.1 nm in this particular case.
Peak two is simply the sum of these two layers and is not relevant to our present discussion.
In order to perform the above four step process, I developed two separate algorithms and
tested them on a wide range of simulations in order to ascertain which yielded the most accurate
determination of the parameters used to create the simulated data it operated on. I will first
discuss the dN algorithm followed by the 2N algorithm, focusing on how they implement the
four steps outlined above in detail. It should be noted that the Data Input step is the same for
each method as both operate on the same intensity versus theta data sets.
iii. The dN Algorithm
This algorithm operates on an intensity versus theta data set using four input parameters:
theta_low_clip, theta_high_clip, xray_energy, N_range. During the Data Analysis step, the
algorithm changes the domain of the data to between theta_low_clip and theta_high_clip. This is
10
0 1 2 3 4 5 6 7 8 9-10
-8
-6
-4
-2
0
theta
log(
inte
nsity
)
0 1 2 3 4 5 6 7 8 90
0.5
1
1.5
2
2.5
3
theta
inte
nsity
- 2N
0 10 20 30 40 50 60 700
100
200
300
400
500
600
nm
ampl
itude
of s
igna
l
1
2
3 4
5
necessary to remove the low theta region that lacks pertinent oscillations and sometimes used to
remove high theta data points that usually contain experimental noise as the corresponding
intensities are so low. The second part of the Data Analysis step is to take an averaged
derivative of the intensity versus theta signal as described in Enhanced Fourier Transforms for
X-Ray Scattering Applicationsvi. The discrete version of this, as we are dealing with a finite data
set is described as:
I jdN (θ )= 1
N ∑i=1
N I(θ j+i)−I (θ j−i)θ j+i−θ j−i
(12)
Expressed in words, the algorithm takes the jth intensity & theta data point and replaces it with
the ‘local’ average derivative of the intensity. The ‘local’ area is defined by N and is centered on
the jthdata point.
After applying this transform, the oscillations are immediately extracted and thus no
Distortion Removal step must be taken.
Lastly, Fourier Space Analysis is performed by first converting the theta values into q
values using the function described in (5), note that these new q values have units of m-1 as they
are a function of the wavelength of x-rays used. The standard MATLAB function called the Fast
Fourier Transform is then applied to the intensity versus q data set. The coefficients that result
from this transform are then squared in order to plot amplitude versus thickness (now in m). The
maximums of this plot then indicate the presence of the thickness values they correspond to.
iii. The 2N Algorithm
This algorithm also operates on an intensity versus theta data set using the four input
parameters: theta_low_clip, theta_high_clip, xray_energy, N_range. Additionally, it also
narrows the domain of the data using the clip arguments in the first part of the Data Analysis
11
step. The second part of the Data Analysis step is to simply take the local average of the
intensity versus theta data and record it as the distortion at that given theta value. As alluded to in
Enhanced Fourier Transforms for X-Ray Scattering Applicationsvii:
I j2 Ndistortion (θ )= 1
2 N+1 ∑i= j−N
j+N
I (θi)(13)
The method is called ‘2N’ as the local average includes the jth point plus N closest data points
that correspond to a lower theta and the N closest data points that correspond to a higher theta
than that of the jth point.
After quantifying the distortion value at each point, we know subtract these distortions
from the corresponding original intensity values in the Distortion Removal step:
I j2 N (θ )=I j
Original (θ )−I j2 Ndistortion (θ ) for all j (14)
With the distortion removed, we know can visually see the oscillations that were contained
inside the raw data. One last step in the Distortion Removal step is to take the anti-log of the
oscillation values, which converts the scale of the oscillations from log(intensity) back into
normal intensity. This ‘decompression’ serves to amplify the oscillation signals, provided for a
more robust Fourier transform result.
The final Fourier Space Analysis step is identical to that used in the dN method.
III. Results
I will now discuss the results from testing the above algorithms against Multig
simulations in addition to the sensitivity of parameters such as theta_low_clip, theta_high_clip,
N_range in affecting the accuracy of the algorithms.
12
i. dN Results Under Normal Conditions
Under favorable conditions, such as the simulation of the 2nm of Al and 20 nm of SiO2
discussed earlier, dN successfully extracted the number of peaks present in the model and their
corresponding differences. It should be noted that the oscillations determined by the derivative
are π4 out of phase with the original oscillations as theta values that corresponded to peaks in the
original oscillations locked inside the data now correspond to zeroes of the derivative. This is
simply a natural result of applying the derivative to a function. As seen in Figure Four below, the
oscillations are extracted very cleanly from the original data set in a mathematically sound yet
compact step of taking the derivative:
In terms of the accuracy of the Fourier Space Analysis step associated with the dN method, we
see only two peaks indicating the presence of the two layers. Peak two rightly indicates the
13
0 1 2 3 4 5 6 7 8 9-10
-8
-6
-4
-2
0
theta
log(
inte
nsity
)
0 1 2 3 4 5 6 7 8 9-15
-10
-5
0
5
10
theta
d(in
tens
ity)
0 50 100 1500
500
1000
1500
2000
nm
ampl
itude
of s
igna
l 12
presence of the 20 nm SiO2 layer, while the difference between Peak one and two indirectly
indicates the existence of the 2nm Al layer. Keep in mind, we cannot determine whether the 20
nm layer is really made of Al or SiO2, we only know that it exists and what its thickness is.
Additionally, we cannot certify using this model the order of the layers, such as whether the SiO2
is on top of the Al layer.
Another item to point out, are the second harmonics of the signal present around
approximately 45 nm. The more pronounced these harmonics are, the less sinusoidal are
extracted oscillation is and thus the worse the extraction algorithm is.
The last, and most important result of running the dN method on a simple test simulation
is the low frequency artifact highlighted in the orange box. This should not be there as it
indicates a strong DC (linear) component in the data. Additionally, it drowns out the low
frequency peak that should be present indicating the 2nm layer.
ii. dN Results on simulated roughness
Under actual experimental conditions, roughness is present on the layers of any real
sample being tested since no semiconductor can be perfectly smooth. To simulate this, we ran
the same simulation as above, yet with 1.5 Å of roughness added to the SiO2 layer. Using an
N_range value of 9 to help average out this noise, I received the result in Figure Five:
14
0 1 2 3 4 5 6 7 8 9-20
-10
0
theta
log(
inte
nsity
)
0 1 2 3 4 5 6 7 8 9-20
0
20
theta
d(in
tens
ity)
0 50 100 1500
1000
2000
nm
ampl
itude
of s
igna
l
Note how the roughness greatly attenuates the high frequency oscillation for theta above
approximately four. We do note that the highest peak corresponds to a thickness of 20.0 nm
while peak two corresponds to a thickness of 2.03 nm.
It is interesting to note that the highest peak, peak one, no longer is a summation of the
two layers, rather it is just the largest layer. Also, the low frequency artifact present in the
simulation without roughness seems to have disappeared, leaving the low frequency peak
corresponding to 2.03 nm exposed. Lastly, many smaller peaks are prevalent as a result of the
extracted oscillation not being very sinusoidal – these additional peaks could be mistaken to be
additional layers.
ii. 2N Results Under Normal Conditions
Under favorable conditions, such as the simulation of the 2nm of Al and 20 nm of SiO2
discussed earlier, 2N successfully extracted the number of peaks present in the model and their
corresponding differences. The results of using an N_range of 13 can be seen below in Figure
Six:
15
0 1 2 3 4 5 6 7 8 9-20
-10
0
theta
log(
inte
nsity
)
0 1 2 3 4 5 6 7 8 9-20
0
20
theta
d(in
tens
ity)
0 50 100 1500
1000
2000
nmam
plitu
de o
f sig
nal
12
0 1 2 3 4 5 6 7 8 9-10
-5
0
theta
log(
inte
nsity
)
0 1 2 3 4 5 6 7 8 90
1
2
3
theta
inte
nsity
- 2N
0 10 20 30 40 50 60 700
200
400
600
nm
ampl
itude
of s
igna
l
1
2 3
4 5
Looking at the result of the Fourier Space Analysis, we see three distinct peaks. The first, peak
four, corresponds to a thickness of 2.01 nm while the third corresponds to 20.1 nm. Peak two is
approximately the sum of these two layers. We notice that second harmonics are extremely small
as indicated by peak 5, thus the extracted oscillations are very sinusoidal in nature. One anomaly
is noted for a peak that appears to correspond to 0 nm. The reason for its existence and its
physical meaning is not understood. Additionally, like with dN, the Fourier Space Analysis
cannot currently reveal the density of the layers the peaks correspond to, nor the order of the
layers.
iii. 2N Results on simulated roughness
The following (Figure 7) is the result of running the 2N algorithm on the same roughness
simulation described previously in the dN section:
16
0 1 2 3 4 5 6 7 8 9-10
-5
0
theta
log(
inte
nsity
)
0 1 2 3 4 5 6 7 8 90
1
2
3
theta
inte
nsity
- 2N
0 10 20 30 40 50 60 700
200
400
600
nmam
plitu
de o
f sig
nal
1
2 3
4 5
0 1 2 3 4 5 6 7 8 9-15
-10
-5
0
theta
log(
inte
nsity
)
0 1 2 3 4 5 6 7 8 90
2
4
6
theta
inte
nsity
- 2N
0 5 10 15 20 25 30 350
100
200
300
nm
ampl
itude
of s
igna
l
1
2
3 4 5
Peak five corresponds to a thickness of 1.34 nm while peak two indicates a 20 nm layer, with
peak 3 representing these two layers added together. The difference between peak three and two
is in fact more accurate accounting of the 2nm layer than peak five as it is 2.01 nm. Just as with
the dN method, many ‘false’ peaks emerge in the Fourier Space Analysis, as indicated by peak
four.
iv. Results of altering the theta_low_clip and theta_high_clip bounds on both algorithms
The most sensitive input parameter for both algorithms appears to be the lower theta
bounds the user chooses to input. For example, if the user arbitrarily chooses the lower bound to
be 1̊ and the upper to remain at 9̊, the results of dN and twoN are altered, even when no noise is
introduced, as noted in Table 1.
Algorithm (Lower Bound) dN (1st Osc) dN(1̊) twoN (1st Osc) 2N(1̊)
2 nm Al Result (nm) 1.98 1.85 2.01 1.84
20nm SiO2 Result (nm) 20.1 20.35 20.1 19.95
Note the lower bound called “1st Osc” meaning first oscillation. I developed this lower bound
standard while testing both algorithms when I found that the most accurate extracted oscillations
17
0 1 2 3 4 5 6 7 8 9-15
-10
-5
0
theta
log(
inte
nsity
)
0 1 2 3 4 5 6 7 8 90
2
4
6
theta
inte
nsity
- 2N
0 5 10 15 20 25 30 350
100
200
300
nm
ampl
itude
of s
igna
l
1
2
3 4 5
are taken from data that has all intensity versus theta values that appear before the rise of the first
oscillation removed. This typically occurs around 0.18̊. Additionally, it should be noted that
changing the upper bound of theta only serves to lower the accuracy of both algorithms.
IV. Discussion & Conclusion
Using the simulation test results described in the Results section, it is recommended that
the 2N algorithm be employed to assist in solving the oscillation extraction problem concerning
the XRR data. The key distinction that sets the 2N algorithm apart from the dN algorithm is its
low frequency sensitivity, a characteristic that is critical to our mission of detecting thin layers on
samples. The low frequency artifact that exists in the dN method, as noted by the orange box in
Figure Four is simply unacceptable.
With this said, 2N is certainly an algorithm under development and recommended
avenues to pursue with regards to extracting more information such as density and layer order
are noted in the project summary in Figure 8.
18
While the development of 2N is finished by my colleagues in the future, it is reassuring to note
the progress that 2N has made over the previous algorithm that was used. In figure 9 below, we
see data that was Fourier space analyzed using the previous algorithm before dN. This is real
data that was recorded using GE semiconductor samples, only focus on the blue line for that
matter:
Applying the current version of 2N to the same data set, in Fourier space we get:
19
0 100 200 300 400 5000
0.5
1
1.5
2
2.5
3
3.5
4
4.5
x 10-10 100730 AA53E0-05SY
r (Angstroms)
|Y(r)
|
datamodel
0 1 2 3 4 5 6 7-10
-5
0
theta
log(
inte
nsity
)
0 1 2 3 4 5 6 70
1
2
3
theta
inte
nsity
- 2N
0 20 40 60 80 1000
100
200
300
400
500
nm
ampl
itude
of s
igna
l
1
2 3
4 5
Which reveals the two layer structure of the sample much clearer than the previous model did.
[Will insert GE response to this new analysis, I have a meeting with the company’s semiconductor research division to go over my results this week!]
V. Acknowledgments
This work was supported by the SULI Program, U.S. Department of Energy, Office of Science. I
would like to thank my mentor Apurva Mehta and his graduate student Matthew Bibbe for their
assistance in addition to all the administrators of the SULI program for this wonderful and
enriching experience.
VI. References
20
0 1 2 3 4 5 6 7-10
-5
0
theta
log(
inte
nsity
)
0 1 2 3 4 5 6 70
1
2
3
theta
inte
nsity
- 2N
0 20 40 60 80 1000
100
200
300
400
500
nm
ampl
itude
of s
igna
l
1
2 3
4 5
i Cite th brenannaon paperii http://en.wikipedia.org/wiki/Transmission_electron_microscopy, but apurva was always talking about thisiii 3-27-11 final ge proposaliv Toney&Brennan1989_CarbonThinFilmXRR.pdfv Elements of modern x ray physics p80vi Poust, Goorskyvii Poust, Goorsky
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