chapter_cnt_growth_placement.tex

%% all about growing, placing, imaging, filtering carbon nanotubes

\chapter{Carbon Nanotube Growth and Placement}
\label{sec:growth}
\chaptermark{Growth and Placement}

In the years since their discovery, many methods of producing electronic devices from carbon nanotubes have been developed. These include random dispersion (Section \ref{sec:random_dispersion}), on-substrate catalyst island growth (Section \ref{sec:catalyst_island}) and stamping \cite{Wu2010, Pei2012}. For this work, the first two methods were successfully used to produce carbon nanotube quantum dots. It was found that the catalyst island growth was much easier to implement and required much less processing time per device at the cost of increased disorder in resulting devices.

\section{Random Dispersion}
\label{sec:random_dispersion}

The first successful method of nanotube device fabrication was what will be referred to here as random dispersion. First, nanotubes are grown in bulk through chemical vapor deposition (or other preferred method). Then, the nanotubes are suspended in a solution. Finally, the nanotubes are cast onto a substrate. Nanotubes can then be located relative to predefined markers on the substrate.

\subsection{Catalyst}
\label{subsec:disperse_catalyst}

All of the devices discussed in this thesis have utilized the same, iron-based catalyst \cite{Kong1998, Kong1998a}. The simplest way to create this catalyst is to combine the ingredients in Table \ref{table:powder_catalyst} in a mortar and pestle and grind until it turns a uniform dark orange color. Adding some additional alumina seems to promote growth of longer tubes, possibly by lowering the density of tubes grown from each alumina/iron/molybdenum cluster.

\begin{table}
	\centering
	\caption{Powder Catalyst}
    \begin{tabular}{ c | c }
    	\hline
        \ce{Fe(NO3)3*9H2O} & \SI{20}{\milli\gram} \\ \hline
        \ce{MoO2(acac)2} & \SI{5}{\milli\gram} \\ \hline
        \ce{Al2O3} & \SI{15}{\milli\gram} \\ \hline
    \end{tabular}
    \label{table:powder_catalyst}
\end{table}

\subsection{Growth}
\label{subsubsec:powder_cvd}

Of the many possible techniques for nanotube growth, we choose chemical vapor deposition for its simple implementation. The process is carried out in a Lindberg Blue tube furnace using a 1 inch diameter quartz tube. The furnace, quartz tube, and exhaust filtering are seen in Figure~\ref{fig:furnace_setup}. This setup has been repeatedly, and successfully, leak checked. Oxygen leaks can be detrimental to the nanotube growth process by forming \ce{CO} and \ce{CO2} with any free carbon, and potentially causing a fiery reaction with free hydrogen from the methane decomposition.

\begin{figure}
    \centering
    \includegraphics[width = 0.8\textwidth]{chapter3/furnace_setup.jpg}
    \caption{The tube furnace fitted with a 1" diameter quartz tube. The tube is sealed at both ends using 1" rubber tubing, cable clamps, and KF25 fittings. Gas flows from left to right in the picture. The gas flows out of the furnace into a mineral oil bubbler to keep hot hydrogen from reaching the air in the room. Gas then flows from the bubbler into the building exhaust.}
    \label{fig:furnace_setup}
\end{figure}

The gases used in the CVD process, argon, hydrogen, and methane, are fed into the furnace using a custom-made gas handling panel. The panel has three gas channels, each with its own analog flowmeter, needle valve, and on/off valve. A digital flowmeter placed at the right side of the panel reads the total flow of combined gas exiting the panel to the furnace. The gas handling panel can be see in Figure~\ref{fig:gas_panel}

\begin{figure}
    \centering
    \includegraphics[width = 0.8\textwidth]{chapter3/gas_panel.jpg}
    \caption{The gas handling panel for our Lindberg tube furnace. Gas flow is from left to right.}
    \label{fig:gas_panel}
\end{figure}

The growth procedure begins with filling a ceramic crucible with the iron catalyst described in \ref{subsec:disperse_catalyst}. The catalyst should be spread in a thin layer across the bottom of the crucible, which is then loaded into the center of a 2-4 foot long quartz tube. The tube is sealed at each end, one side connected to the gas handling panel, and the other connected to the mineral oil filter and building exhaust. Our standard nanotube growth recipe is as follows: 

%can i remove the spacing from between these list items?
\begin{enumerate}
	\item Purge the tube by flowing \SI{2000}{\sccm} of Ar for 20 minutes
	\item Heat tube to \SI{1000}{\degreeCelsius} while flowing \SI{1000}{\sccm} Ar and \SI{200}{\sccm} \ce{H2}
	\item Flow \SI{2000}{\sccm} \ce{CH4} and \SI{200}{\sccm} \ce{H2} for 10 minutes.
	\item Set temperature to \SI{0}{\degreeCelsius} and let the furnace cool while flowing \SI{1000}{\sccm} Ar and \SI{200}{\sccm} \ce{H2}
\end{enumerate}

\noindent The actual nanotube growth occurs during the methane flow step. The \SI{200}{\sccm} \ce{H2} flow can be omitted, but it does seem to help promote nanotube growth. The flow rates do not need to be precise. Most nanotubes grow in the first few seconds of methane flow regardless of the flow rate. The Ar and \ce{H2} are simply to keep \ce{O2} and \ce{H2O} out of the tube. 

\subsection{Nanotube Placement}

After the CVD process, the nanotubes remain attached to the iron\slash alumina\slash molybdenum catalyst particles, which must be removed before depositing onto a silicon substrate. The nanotube\slash catalyst powder is first scraped from the ceramic crucible used in the tube furnace. The powder is then mixed with either dichloroethane or dichlorobenzene in a \SI{1}{\milli\gram} to \SI{10}{\milli\liter} ratio. Dichlorobenzene has been found to leave less residue after deposition, but may promote more damage to nanotubes during sonication. This was noted by Justin Silverman, an undergraduate working with our lab on functionalizing short carbon nanotubes. Some attempts were made to use water along with the surfactant SDS. However, SDS turned out to be difficult to remove and no devices were made in this way.

To remove the catalyst particles from the nanotubes, the solution described above must be placed in an ultrasonic bath for 1-60 minutes. The amount of time needed varied a great deal depending on the equipment and solvent used. The goal of this step is to break up large pieces of catalyst, separate nanotube bundles, and break individual nanotubes away from their catalyst particles. Sonication can be stopped when no large pieces of catalyst\slash nanotube material are visible and the solution has a uniform black color. Leaving the solution in the sonicator for too much time will begin to break long nanotubes. This may be somewhat beneficial in breaking nanotubes at defects that might otherwise affect transport measurements.

When sonication is complete, the solution is transferred to a centrifuge. This step is intended to precipitate the loose catalyst particles from the solution, while leaving the much lighter nanotubes suspended. The centrifuge used in our lab runs at \SI{2200}{\rpm} and nanotube solutions are left inside for 5-10 minutes. Once the centrifuge stops, the precipitate is discarded and the supernatant, containing the suspended nanotubes, is reserved. We also tested using a high-speed, air-powered centrifuge running at \SI{100000}{\rpm} to separate nanotubes and catalyst. This did lead to much better catalyst separation and cleaner results,.

Now the solution is ready for deposition on a silicon substrate. The substrates are typically pre-patterned with a set of reference markers placed by optical (\ref{subsec:optical}) or electron beam lithography (\ref{sec:ebeam_lith}). An example of a patterned substrate can be seen in Figure \ref{fig:markers}

\begin{figure}
    \centering
    \includegraphics[width = 0.8\textwidth]{chapter3/markers.eps}
    \caption{A \ce{Si}/\ce{SiO2} substrate with \SI{1}{\micro\meter} \ce{Au} markers. Left scale bar: \SI{100}{\micro\meter}. Right scale bar: \SI{20}{\micro\meter} }
    \label{fig:markers}
\end{figure}

\subsection{Advantages and Disadvantages}

Despite having some success fabricating devices using randomly dispersed nanotubes and electric force microscopy scans (Section \ref{subsec:EFM}), the process was found to be too time consuming for frequent use. The main failure point was the preparation of nanotube solutions after growth. The concentrations varied significantly, often producing samples with dense nanotube coverage or no nanotubes found in the regions imaged. Additionally, the process of making a suspended nanotube solution is time consuming. Even those solutions with a useful concentration of nanotubes only remain fully suspended for less than 1 hour; meaning the process must be repeated frequently.

\section{Catalyst Island Growth}
\label{sec:catalyst_island}

In 1998 \cite{Kong1998a}, it was discovered that the same type of catalyst used to grow nanotubes in powder form (Section \ref{subsec:disperse_catalyst}), could be suspended in solution, patterned, and used to grow nanotubes directly on silicon substrates. When paired with high melting point metals and optical lithography, nanotubes can be grown directly on patterned substrates in known locations. Devices prepared this way take just a fraction of the time to produce. However, these devices were found to be much more prone to other modes of failure, such as leaks in the gate oxide, amorphous carbon contamination on the substrate from the CVD growth process, and defects along the nanotube length.

\subsection{Catalyst} 

Many different types of catalyst particles can be used in the growth of carbon nanotubes. The ideal catalyst for patterned growth must be compatible with electron beam or optical lithography. Table \ref{table:catalysts} lists most of the catalysts tested in the Markovic lab. To test each catalyst, sputtered molybdenum markers were patterned using the mask aligner and Futurex NR9 resist. Catalyst islands were then patterned using electron beam lithography. Figure \ref{fig:catalyst_islands} shows an example of a substrate with Mo markers/leads and catalyst islands.

\begin{figure}
	\centering
	\includegraphics[width = 0.8\textwidth]{chapter3/catalyst_island.eps}
	\caption{A \ce{Si}/\ce{SiO2} substrate with \SI{3}{\micro\meter} catalyst islands and \ce{Mo} leads. Left scale bar: \SI{100}{\micro\meter}. Right scale bar: \SI{20}{\micro\meter} }
	\label{fig:catalyst_islands}
\end{figure}

\begin{table}
	\centering
	\caption{Patterned Catalysts}
    \begin{tabular}{| r | c || p{70mm} |}
    	\hline
    	\textbf{Catalyst} & \textbf{Suspended In} & \textbf{Results} \\ \hline
        Fe/Mo/alumina \cite{Kong1998} & Methanol & Easy to pattern. Liftoff difficult. Slowly attacks PMMA mask. Appeared to promote gate leaks through the \ce{SiO2} layer. \\ \hline
        Fe/Mo/alumina \cite{Aurich2012} & IPA & Poor adhesion to substrate. \\ \hline
        Fe/Mo/alumina \cite{Ouellette2008} & DI water & Easy to pattern. Excellent adhesion. No gate leak problems. \\ \hline
        \ce{FeCl3} \cite{Hong2005} & DI water & Excellent adhesion to substrate. Not compatible with PMMA mask. Left substrate entirely covered in catalyst. May work well with PDMS stamp. \\ \hline
        thermally evaporated Fe \cite{Biercuk2004, Kang2007} & None & Very easy to pattern. Liftoff is clean. Difficult to control the thickness below \SI{1}{\nano\meter} as required. \\ \hline
    \end{tabular}
    \label{table:catalysts}
\end{table}

All but one of the devices discussed in this thesis were produced using the Fe\slash Mo\slash alumina catalyst suspended in water. The islands were patterned using electron beam lithography. Catalyst is deposited in the following way:

\begin{enumerate}
	\item Add the powder catalyst from Table \ref{table:powder_catalyst} to \SI{15}{\milli\liter} of DI water and stir for 12 hours
	\item Sonicate the solution for 30 minutes
	\item Cover the sample in catalyst solution for 30 minutes
	\item Dry with \ce{N2} gun
	\item Liftoff by sonicating in acetone for 5 minutes, soaking in a clean acetone for 5 minutes, followed by an isopropanol rinse for 1 minute and a DI water rinse for 1 minute
\end{enumerate}

Obtaining reproducible results in the catalyst deposition was the source of much difficulty. The recipe provided here can be repeated before liftoff shows insufficient catalyst coverage on the substrate. Additionally, it is suspected that baking the catalyst on a hot plate to dry the solution leads directly to gate leaks in the \ce{SiO2} layers and later device failure. Thus, there is no baking step in the deposition of our catalyst islands.

\subsection{Growth}
\label{subsubsec:substrate_cvd}

The recipe for on-substrate growth of nanotubes used in this thesis is very similar to the growth recipe for powder catalyst discussed in Section \ref{subsubsec:powder_cvd}. This recipe was developed over the course of several years from many points of reference \cite{Kong1998, Kong1998a, Dirks2010, Huang2003, Huang2004, Zhang2013, Hong2005} and my own notes. The recipe is optimized for the 1 inch Lindberg tube furnace as seen in Figures \ref{fig:furnace_setup} and \ref{fig:gas_panel}. Most samples are placed in a smaller \SI{1}{\centi\meter} diameter, 1 foot long quartz tube, then placed in the larger 1 inch diameter quartz tube. This was done to make the samples easier to load into the 1" tube, as well as to reduce turbulence in the gas flow across the sample \cite{Hong2005}. 

The standard nanotube growth recipe used in this work is below:

\begin{enumerate}
	\item Purge the tube by flowing \SI{2000}{\sccm} of Ar for 20 minutes
	\item Heat the tube to \SI{250}{\degreeCelsius} while flowing \SI{300}{\sccm} Ar and \SI{150}{\sccm} \ce{H2}
	\item Wait for at least 1 hour
	\item Heat the tube to \SI{700}{\degreeCelsius} while flowing \SI{300}{\sccm} Ar and \SI{150}{\sccm} \ce{H2}
	\item Wait for 10 minutes
	\item Heat tube to \SI{950}{\degreeCelsius} while flowing \SI{300}{\sccm} Ar and \SI{150}{\sccm} \ce{H2}
	\item Wait for the temperature to stabilize
	\item Flow \SI{700}{\sccm} \ce{CH4} and \SI{150}{\sccm} \ce{H2} for 10-15 minutes
	\item Set temperature to \SI{0}{\degreeCelsius} and let the furnace cool while flowing \SI{300}{\sccm} Ar and \SI{150}{\sccm} \ce{H2}
\end{enumerate}

In almost every test, this recipe has grown nanotubes successfully. Steps 2 and 3 are included to remove water vapor from the air that might have collected inside the quartz tube on humid days \cite{Dirks2010}. Steps 4 and 5 are meant to remove iron oxide from the iron nanoparticles that make up the catalyst. 

The most common point of failure in growth has been related to the patterned molybdenum leads/markers on the substrate. Molybdenum oxidizes rapidly at high temperatures. Therefore, any oxygen contamination in the tube during the growth process will form a \ce{MoO} layer that is then quickly removed by reacting with the high temperature \ce{H2} flow. This process repeats and can lead to the Mo leads being entirely etched away. Additionally, it has been found that opening the furnace too soon during cooling can lead to the Mo leads peeling off of the substrate. This appears to be caused by some super-heating due to IR radiation reflecting off of the surfaces of the sample and quartz tube. It is a strange phenomenon that is avoided by allowing the furnace to cool to less than \SI{300}{\degreeCelsius} before opening the lid. These problems could also be solved by using a different high temperature metal such as a W/Pt bilayer, common in many other nanotube projects. Molybdenum was chosen for this work because it is easy to sputter and much more affordable.

\subsection{Advantages and Disadvantages}

Growing nanotubes from catalyst islands near predefined leads and markers offers a large improvement in processing time over the method of random dispersion. Nanotubes produced with this method are longer, cleaner, and easier to locate. These improvements made this method the obvious choice for device fabrication. The main disadvantage found in this technique is the introduction of defects along the nanotube length due to growth along the substrate. Because of this, a large number of devices must be fabricated in order to find one with clean transport properties at low temperature. It is also helpful to limit device lengths to less than 200nm to reduce the chances of a defect being included in the resulting device.

\section{Imaging Nanotubes}
\label{subsubsec:imaging_disperse}

Nanotubes on a \ce{SiO2} surface can be located in a number of ways. This section will review a several different methods, focusing on improvements made in the course of this thesis work.

\subsection{Atomic Force Microscopy}

With nanotubes that have been drop cast onto the surface, the standard method is to locate the tubes relative to the predefined markers using a tapping mode atomic force microscope (AFM). An example of an image created this way is seen in Figure \ref{fig:cnt_au_markers}. 

\begin{figure}
	\centering
	\includegraphics[width = 1.0\textwidth]{chapter3/single_cnt_au_markers.pdf}
	\caption{AFM height and amplitude scans of nanotubes dispersed over a substrate with \SI{1.5}{\micro\meter} gold markers. The scale bar is \SI{10}{\micro\meter}.}
	\label{fig:cnt_au_markers}
\end{figure}

This method is very reliable, but extremely time consuming. In order to resolve nanotubes, as well as the predefined markers in the image, AFM scan sizes must be limited to 25$\times$25$\mu$m. Each of these scans takes 30 minutes and many scans are needed to fully image one patterned substrate. Looking closely at Figure \ref{fig:cnt_au_markers}, there are 12 scans covering less than half the substrate. Due to vibrational noise and piezo limits, some images are slightly warped. Stitching the images together is time consuming and inaccurate. One major advantage to AFM imaging is that it is easy to distinguish single tubes from bundles and multiwalled nanotubes, as well as measure nanotube diameter.

\subsection{Electric Force Microscopy}
\label{subsec:EFM}

The Digital Instruments Nanoscope 3 used in our lab is also capable of making electric force microscope (EFM) measurements. An EFM image is made by first measuring the height across the sample in standard tapping mode, then using that height data to run a second `interleave' scan at a fixed height with a bias voltage applied between the tip and sample. By holding the tip at a fixed height, van der Waals interactions between the tip and sample are constant and the only force measured is the electrostatic force from the applied bias voltage. Contrast in the resulting images is related to the different conductivities of the objects on the sample \cite{Bockrath2002}. Thus, conducting (and semiconducting) nanotubes have a high contrast against the insulating \ce{SiO2} substrate. An example of this type of image is shown in Figure \ref{fig:cnt_efm}

\begin{figure}
	\centering
	\includegraphics[width = 0.8\textwidth]{chapter3/cnt_efm.eps}
	\caption{An EFM image of nanotubes dispersed over a substrate with \SI{1.5}{\micro\meter} gold markers. The markers have been automatically located using the height data and highlighted in blue on the EFM image. The scale bar is \SI{10}{\micro\meter}.}
	\label{fig:cnt_efm}
\end{figure}

An entire patterned substrate can be scanned using this method in about 1 hour. The scan size can be increased up to 75$\times$75$\mu$m due to the false contrast provided by the large electrostatic forces between the nanotubes and the tip. Rather than appearing as \SI{1}{\nano\meter} in diameter, the tubes appear in the EFM image to be about 100 times their real diameter. This was a notable improvement over locating nanotubes using AFM height scans alone. Comparing Figure \ref{fig:cnt_au_markers} and Figure \ref{fig:cnt_efm}, it is clear the EFM image is far more useful in locating nanotubes.

\subsection{EFM Through PMMA}

All of the same techniques from Section \ref{subsubsec:imaging_disperse} can be applied to imaging nanotubes grown from patterned catalyst islands. However, because the substrates are not covered in closely spaced markers, it was found that tapping mode AFM height scans were not useful. Scans could only cover a small part of the sample and the resulting images were difficult to orient.

Using electric force microscopy (EFM) made it possible to scan the entire region of interest on the sample in one measurement. An example of this type of scan is shown in Figure \ref{fig:efm_islands}(a). As can be seen in that figure, it was difficult for the AFM tip to avoid crashing into the catalyst islands during the EFM sweep. The catalyst islands are several hundred nanometers in height while the other features on the substrate are less than \SI{10}{\nano\meter}. Such height differences make large area scans difficult in tapping mode. This problem can be avoided by coating the sample in PMMA before scanning with the EFM, as seen in Figure \ref{fig:efm_islands}(b.) The PMMA coating smooths the height differences between the substrate and catalyst islands, without compromising the contrast between the insulating substrate and conducting nanotubes. The idea was adopted from a 2007 paper in which the authors attempted to locate nanotubes suspended in a PMMA layer in three dimensions \cite{Jespersen2007}.

\begin{figure}
	\centering
	\includegraphics[width = 1.0\textwidth]{chapter3/efm_islands.eps}
	\caption{(a) Frequency data collected from an EFM scan of a catalyst island sample after nanotube growth. (b) Frequency data collected from an EFM scan of a similar sample. Prior to the scan this sample was coated with a \SI{250}{\nano\meter} PMMA layer. Both scale bars are \SI{10}{\micro\meter}. }
	\label{fig:efm_islands}
\end{figure}

\subsection{Scanning Electron Microscopy}
\label{subsec:imaging_sem}

In 2002, a paper \cite{Brintlinger2002} was published illustrating that a scanning electron microscope (SEM), operating at a low accelerating potential could provide a similar type of false contrast image as produced by the EFM. The insulating substrate tends to collect charge from the electron beam, while the conducting nanotubes do not. This produces an image in which the nanotubes appear as bright lines about 100 times their actual diameter. An example of this is seen in Figure \ref{fig:sem_islands}. 

\begin{figure}
	\centering
	\includegraphics[width = 0.8\textwidth]{chapter3/sem_islands.eps}
	\caption{A scanning electron micrograph of a catalyst island sample after nanotube growth. The scale bar is \SI{10}{\micro\meter}.}
	\label{fig:sem_islands}
\end{figure}

Typically, an EFM scan of a sample will take 45 minutes. A SEM image of the same sample takes less than 2 minutes. However, the SEM can introduce carbon contamination from the high energy electrons passing through small amounts of oil mist back-streaming into the vacuum chamber from the mechanical roughing pump. Due to the large number of samples that were produced to obtain the data in this thesis, it was decided that contamination from the SEM was an acceptable risk, given the immense time savings. No clear effect from carbon contamination was ever observed in the results. Devices made using EFM imaging showed similar disorder and measurement noise.

\section{Image Filtering}

Once it became clear that the scanning electron microscope was by far the most efficient and reliable way to locate carbon nanotubes on a substrate, it also became important to optimize those images to revel the most information possible. The resolution and contrast in the SEM images produced in our lab are limited by the the use of a thermionic \ce{LaB6} filament. Unlike field emission scanning electron microscopes, which are more common in nanofabrication, the thermionic scanning electron microscope has a large initial crossover size, requiring more electromagntic lens focusing to produce a sufficiently small beam size for imaging. This problem is exasperated when using low accelerating potentials (500-\SI{3000}{\volt}), which are crucial to achieving good contrast of carbon nanotubes on a silicon substrate.

\subsection{Histogram Equalization}

This method was based on two corrections. First, a plane fit to correct for the position of the secondary electron detector. Second, calculating the histogram of pixel brightness then transforming it such that there are equal numbers of pixels at each brightness level. An illustration of this can be seen in Figure \ref{fig:hist_eq}

\begin{figure}
	\centering
	\includegraphics[width = 0.4\textwidth]{chapter3/histogram_equalization_wiki.png}
	\caption{An illustration of the histogram equalization process.}
	\label{fig:hist_eq}
\end{figure}

Figure \ref{fig:hist_eq_data} shows the process of filtering an SEM image using histogram equalization. At the top of figure is the original image after a plane fit (to correct changes in brightness across the substrate). To the right of the image, the histogram of brightness values and the cummulative distribution function are plotted. The center image is after the histogram equalization. Contrast is now dramatically increased in the image, the histogram is nearly flat, and the cumulative distribution function is linear. Finally, the image is Gaussian filtered using a $3 \times 3$ kernel to reduce the high frequency noise, which has been exaggerated by the histogram equalization. The final histogram is not flat quite flat and the cumulative distribution function is not quite linear. However, the contrast in the image has been enhanced significantly

\begin{figure}
	\centering
	\includegraphics[width = 0.8\textwidth]{chapter3/histogram_eq_example.pdf}
	\caption{Top image is the original SEM image after a plane fit. Middle image shows the SEM image after histogram equalization. Bottom image shows the final result after gaussian smoothing. Plots on the right show the histogram of brightness values and cumulative distribution function at each step.}
	\label{fig:hist_eq_data}
\end{figure}

While this filter does succeed in increasing contrast in the image, which was the main problem associated with our SEM, it also increases the background noise to often unacceptable levels. Histogram equalization does not use any specific information about the carbon nanotubes and substrates being imaged. A better filter should be possible by taking advantage of the fact that the features of interest can be well characterized in a sample set of images. 

\subsection{Matched Filter Bank}

Matched filter banks are a well known technique that have been used very successfully to filter retinal images in medicine \cite{Chaudhuri1989}. This technique has previously been adapted to high resolution SEM images of carbon nanotube bundles \cite{Guerrero2014}. The following section describes the implementation of this method to filter images of single walled carbon nanotubes grown on insulating substrates. 
%% nanotube profiles -> kernel -> filtering/thresholding

\subsubsection{Nanotube Profile Model}

To begin building the matched filter bank, the profile shape of a single nanotube from the SEM image must be determined. To find this shape, a random set of 25 SEM images was selected (similar to those in Figure \ref{fig:sem_islands}). From each of these images, images of two nanotubes were cropped. Those nanotubes can be seen in Figure \ref{fig:all_nanotube_sections}a. Each of these images was then rotated such that the longest straight portion (that could be identified by eye) was oriented vertically. The profile of the nanotube was averaged over that long straight section. These results are plotted in Figure \ref{fig:all_nanotube_sections}b.

\begin{figure}
	\centering
	\includegraphics[width = 1.0\textwidth]{chapter3/all_nanotube_sections.eps}
	\caption{a) The original data set of SEM nanotube images used to create and optimize the matched filter bank. b) The same data set rotated and overlaid with extracted nanotube profiles.}
	\label{fig:all_nanotube_sections}
\end{figure}

Based on the shape of these profiles a truncated $sinc$ function, Equation \ref{eq:sinc}, was chosen to fit the nanotube profile.

\begin{equation} 
\label{eq:sinc}
    f(x) = \begin{cases} \frac{A\sin{kx}}{kx} & |x| < \frac{2\pi}{k} \\ 
                         0                    & |x| > \frac{2\pi}{k} 
           \end{cases}
\end{equation}

This function was chosen because it captures the bright nanotube peak and dark regions beside the nanotube while still only requiring one fit parameter. To determine the fit parameters the linear background was subtracted from each profile in Figure \ref{fig:all_nanotube_sections}b and the profiles were  fit using \ref{eq:sinc}. The results of this fitting can be seen in Figures \ref{fig:profile_fits}a and b. 

\begin{figure}
	\centering
	\includegraphics[width = 1.0\textwidth]{chapter3/profile_fits.eps}
	\caption{a) Extracted nanotube profiles with linear background removed (blue). Fits to \ref{eq:sinc} (red). b) Distribution of $k$ values extracted from profile fits. c) Distribution of straight nanotube section lengths.}
	\label{fig:profile_fits}
\end{figure}

\subsubsection{Filter Kernel}

The filter kernel is built starting with the median $k$ value found from the profile fits. To extract each profile, horizontal cuts of the nanotube image were averaged over a straight segment of the nanotube that was identified and labelled by hand. The distribution of these straight segment lengths can be seen in Figure \ref{fig:profile_fits}c. Again, to proceed with building the filter we use the median value of $L$ from that distribution. The kernel is built in an $N \times N$ matrix with $(x,y) = (0,0)$ at the center of the matrix. The full kernel, $K(x,y)$ is defined by Equation \ref{eq:kernel}.

\begin{equation} 
\label{eq:kernel}
K(x,y) = \begin{cases} \frac{A\sin{kx}}{kx} & |x| < \frac{2\pi}{k}, |y| < \frac{L}{2} \\ 
                       0                    & |x| > \frac{2\pi}{k},  |y| > \frac{L}{2}
       \end{cases}
\end{equation}

Once the kernel is defined on the $N \times N$ matrix, the kernel is normalized such that the sum over all of the matrix elements is equal to zero. By convolving this kernel with the SEM image, portions of the image with a shape matching the fit profile and a length $L$ will highlighted in the output. This results in significant background subtraction and accurate nanotube enhancement. 

\subsubsection{Filter Bank}

To build the full filter bank, the kernel, built using Equation \ref{eq:kernel}, is rotated by a set of angles between $0$ and $\pi$. Each of these filters is then convolved with the image separately. By doing this, it is possible to identify nanotube sections lying along any direction on the substrate. A binary result is created by thresholding the convolved image. The filter bank and resulting binary images can be seen in Figures \ref{fig:filter_bank_results}(c) and (d). 

\begin{figure}
	\centering
	\includegraphics[width = 0.8\textwidth]{chapter3/filter_bank_results.eps}
	\caption{(a) Original image. (b) Final filtered image, which is the sum of the binary images in (d). (c) The rotated kernels that form the full matched filter bank. d) Each rotated kernel applied to an image in (a).}
	\label{fig:filter_bank_results}
\end{figure}

After each kernel has been separately convolved with the original image, and the threshold applied, those binary images are added together to produce the final filtered image. Figures \ref{fig:filter_bank_results}a and b show the original and filtered nanotube images. 

In building this filter, many parameters had to be optimized simultaneously and compared by eye. Due to the large parameter space, the matched filters were optimized by iteratively varying a single parameter and choosing the best result. This process is illustrated in Figure \ref{fig:filter_parameter_test}.

\begin{figure}
	\centering
	\includegraphics[width = 0.7\textwidth]{chapter3/filter_parameter_test.pdf}
	\caption{Optimization of the matched filter bank threshold value.}
	\label{fig:filter_parameter_test}
\end{figure}

The results of this optimization are seen in Table \ref{table:filter_parameters}. The optimized values for $k$ and $L$ were very close to those obtained from the original analysis of extracted nanotube profiles. Note that $A$ and $threshold$ are not independent values and $A$ remained fixed for this optimization procedure. 

It is also important to note that the length scale for these parameters is in pixels $(px)$. The randomly selected SEM images that were used for this analysis were not all taken at the same magnification level. However, the average magnification was $850\times$ giving a pixel size of $\sim$\SI{140}{\nano\meter}

\begin{table}
	\centering
	\caption{Matched Filter Bank Parameters}
	%\hfill \\
    \begin{tabular}{| r | p{60mm} | l |}
    	\hline
    	\textbf{Parameter} & \textbf{Description} & \textbf{Optimized Value}  \\ \hline
    	k & inverse length scale for $sinc$ fit &  $1.75 px^{-1}$ \\ \hline
    	A & height of $sinc$ function in $K(x,y)$ &  $10$ \\ \hline
    	L & length of straight nanotube sections to search for &  $16 px$ \\ \hline
    	N & size of the kernel matrix &  $25$ \\ \hline
        R & number of kernels in the filter & $15$ \\ \hline
        threshold & cutoff value for thresholding images convolved with filter kernels & $3.6$ \\ \hline
    \end{tabular}
    \label{table:filter_parameters}
\end{table}

Finally, the filter was tested with the original set of full device SEM images. A selection of the results can be seen in Figure \ref{fig:full_image_filter}. It is clear from this set of images that the filter has the desired result. Nanotubes are much more clearly visible and the background is almost completely removed. Because the filter is very similar in its construction to common edge detection algorithms, it also leaves features at the edges of the optical lithography leads and markers defined on the substrate before nanotube growth. This actually works well, since those features are used to align the SEM images for additional electron beam lithography steps. 

\begin{figure}
	\centering
	\includegraphics[width = 1.0\textwidth]{chapter3/full_image_filter_test.pdf}
	\caption{Two examples applying the matched filter bank to full SEM images of as-grown carbon nanotubes.}
	\label{fig:full_image_filter}
\end{figure}