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

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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

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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

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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)

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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.

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0.1

0.2

0.3

0.4

0.5

0.6

0.7

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norm

aliz

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tens

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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!

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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

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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

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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.

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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

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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:

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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:

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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:

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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

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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.

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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:

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|Y(r)

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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

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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