# Excursion to the Col Dei Rossi

Last week, between 15 – 19 March I went to Italy, attending international conference in Canazei, about 180 km north of Venice. The name of the conference is The 18th Molecular Beam Epitaxy (http://web.nano.cnr.it/eurombe2015/)

Canazei is located in the upper part of the Val di Fassa, and this region is part of the Dolomites, a mountain range located in the northeastern of Italy. This is my first time of my life seeing Alps as it is just standing before my eyes, so huge! According to the wikipedia, The Dolomites form a part of Southern Limestone Alps and extend from the River Adige in the west to the Plave Valley in the east. A huge range of lining mountain in the north of Italy.

Those mountain suppose d to be Ciampac (2100 m)

The Dolomites is acknowledged as The Natural World Heritage Sites by UNESCO. Basically this place is heaven for those who can ski, both down hill and cross country. It has almost 260 km range to do ski activity: Ski until you drop :)

It has hundred of squared kilometres of walls, towers, pinnacles and valleys. A well known phenomenon, called Enrosadira, takes place at dawn and dusk when the sun light reflects on the walls of the Dolomites. Unfortunately, I was not able to see this phenomenon. The break I had was unfortunately insufficient to find out. It is supposed look like this:

In winter, the Enrosadira will appear like this

This is the map for the Val di Fassa:

And here I was, standing on the Col Dei Rossi, after taking some breathtaking cablecar from Canazei. The transportation was almost made me crazy, the wind blew and shaked our cable car! Who is not afraid of those disturbance?? Anyway, I relieved I could made to the one of the top peak in here.

On top of Col Dei Rossi, Canazei (2382 m)

The first two days were cloudy, but the last two days were awesome. Unfortunately, I did not have good schedule on the last two days. I wish I could climbed the mountain on that days :(

Well, eventhough I was not able to see Enrosadira, I was able to see the sun rays were splitted by the mountain behind hotel I stayed and I guess it was Col Dei Rossi.

Sunrays were splitted!

That was wonderful 4-day stay in Canazei. The last photo was taken before I left Canazei to Venice, where my flight back to Trondheim departed. I enjoyed and surprised with the great hospitality I received there. I met some of the people who were hardly speaking English, but they tried their best, and the Italian (or Ladin?) language was used eventhough I did not understand what they wanted to say. I wonder why they continue talking in Italian to me? Maybe, they wanted to make me welcomed in Canazei.

Grazie!

# Plasma Optics (I)

When we talk about optics, we always relate it with the interaction between light and matters. The interaction will give varying result as it depends on what kind of material are being interacted with. One important properties of material is called dielectric function $\epsilon(\omega,K)$, a function whose frequency and wavevector has impact on the physical interaction between light and matters.

We have two fascinating interaction probabilities in here: the light can be reflected or propagate from/through matter. Before we start with everything, it is better to have understanding of what plasma is. Basically, plasma, one of the fundamental state of matters (others are solid, liquid and gas), is medium with equal concentration of positive and negative charges, of which at least one charge type is mobile. Plasma takes form in gas which composed of free electrons and ions. Plasma has high energy.

Plasma has frequency, called plasma frequency $\omega_{p}$ which becomes a guideline for deciding whether the light will be reflected or propagate. I will try to explain this later. All of us know the relation between energy E and wavelength $\lambda$: $E=h\frac{c}{\lambda}$. It is also obvious that metal, in the visible light, is reflecting incoming light. But how does it can be explained by plasma optics?

The answer lies on the value of incoming wavelength light $\lambda$. When $\lambda>\lambda_{p}$, incident light will be reflected. In the other hand, as $\lambda<\lambda_{p}$, incident light will propagate through matter. Illustration in figure \ref{fig:vis} will give a brief explanation of where should the light get reflected or propagate through material. The example is group of alkali metal, with wavelength ranging from 155 – 362 nm.

Visible wavelength, with group of alkali metal wavelength (155 – 362 nm)

We will come back later to this concept in the next section.

# 3. Fiber optic: application of total internal reflection

The first and second part should be enough to provide illustration of how the fiber optic works: first, it depends heavily on the Snell’s law in governing how the light behaves after passing the interface of two materials with different refractive index number. Second, total internal reflection phenomenon is established owing to the fact that light coming at certain angle, defined as critical angle, is able to have reflection instead of refraction, as the light comes from the denser medium to less denser medium ($n_{1}>n_{2}$). Mathematical expression for the first and second part are described in here and here, respectively.

Now, we can go back to the figure 1. First, we need to realize that there are three different refractive index, as depicted below:

Figure 5. Fiber optic with three different refractive indexes

Refractive index for air (where, later, the source of light-laser- comes from), core and cladding of fiber optic are designed as it is so that light with certain range of angle, can be guided (controlled) from one point to another. To get an idea of the value of critical angle need to be fulfilled, we can start first inside the fiber optic, in the interface between core and cladding. We can start describe it as it in the figure 4 in the right picture. To have total internal reflection means that refractive index of core must be larger than refractive index of cladding ($n_{2}>n_{3}$).

Figure 6. Total internal reflection in the interface between core and cladding

Then we follow the same procedure as in the second part,

Next, we can determine the correlation between air and core of the fiber optic (figure 7). Since total internal reflection in this boundary is not needed (or precisely, avoided), we can proceed with the refractive index of air smaller than core of the fiber optic ($n_{1}).

Figure 7. Incident light and its refraction in the interface between air and core of fiber optic

Applying Snell’s law, we can calculate the incident angle required to achieve total internal reflection,

$n_1 \sin \theta_1 = n_2 \sin \theta_4$

We do not know $\sin \theta_4$ but we do know the value of $\sin \theta_2$ or $\sin \theta_{c(i)}$. Based on the figure 7, we can figure the value of $\theta_4$ from ordinary trigonometry case,

Figure 8. Trigonometry case to find out the value of Θ4

Mathematically, $\sin \theta_{4}$ is equal to $\sin \left( \frac{\pi}{2}-\theta_2 \right)$. To re-define $\sin \theta_{4}$, trigonometry identity of $\sin (\frac{\pi}{2}-\theta)=\cos \theta$ can be utilized. Then the continuation of the last equation is,

There it is, we find already the incident angle required to conduct total internal reflection in the fiber optic. {\textit{Numerical aperture (NA)} is defined as the qualitative limitation of receiving angle of the fiber optic with respect to the incident angle of light. Therefore, the numerical aperture for fiber optic can be defined as follows:

# 2. Total Internal Reflection

As it has been discussed in the first part that, the incident light coming from one medium to another as it strikes the boundary with certain angle, will be refracted in definite manner according to the $n_{1} \sin \theta_{1}=n_{2} \sin \theta_{2}$. Let’s just start this section by increasing incident angle of light in figure 2, first part. This is what will happen if the incident angle is increased:

Figure 3. Refraction phenomenon, with n1 is smaller than n2

As we know from the behavior described in the first part, we observed the light being refracted in increasing manner when the angle of incident light is increased as well. It is less likely that we can notice something intriguing in here.

How about if medium one has larger value than medium two ($n_{1}>n_{2}$)? We should have light will have lower velocity in medium 1 than in medium 2 ($v_{1} < v_{2}$) and thus angle of light with respect to the normal in medium 1 is smaller than in medium 2 ($\theta_{1}<\theta_{2}$). As we increase the angle of incident in medium 1, we will this result:

Figure 4. Refraction phenomenon evolves to total internal reflection, with n1 larger than n2

Total internal reflection is depicted in the right illustration of figure 2. In this phenomenon, the medium boundary acts as a reflector, instead of refractor under two conditions:

• Light propagates from medium with higher refractive index to the medium with lower refractive index ($n_{1}>n_{2}$).
• The angle of incident light has to satisfy minimum of certain angle, i.e. critical angle $\theta_{c}$.

Total internal reflection can be considered when the refracted light is in parallel direction with the interface (figure 2, middle illustration), i.e. $\theta_{2}=90^{\circ}$. Using Snell’s law equation described earlier in the first part and two conditions stated above, we can quantify the required incident angle so that it can trigger total internal reflection:

$n_{1} \sin \theta_{1}=n_{2} \sin \theta_{2}$

$n_{1} \sin \theta_{c}=n_{2} \sin 90^{\circ}$

$n_{1} \sin \theta_{c}=n_{2}$

$\sin \theta_c=\frac{n_2}{n_1}$

$\theta_c=\sin^{-1} \frac{n_2}{n_1}$

Total internal reflection is occured when the incident angle fulfills the critical angle whose value is determined by the medium 1 and 2, as it is expressed in the last formula of the above equations, called the critical angle equation:

$\theta_c=\sin^{-1} \frac{n_2}{n_1}$

This equation establishes the requirement for total internal reflection to take place: medium 2 must have refractive index smaller than medium 1. If it happened in opposite way, there will be no solution for this equation, meaning that total internal reflection does not take place.

# Let the density and its respective radius being known, then a critical beam angle shall be found!

It was happened when I read a paper from V. Consonni et al published in Physical Review B (or you can find it in here) about the shadowing effect part, in the page 4. V. Consonni et al gave a mathematical description (modelling) for the growth rate of self-induced GaN nanowires, and extracted several important parameters (for instance effective diffusion length on the sidewalls and substrate surface, desorption rate, driving forces for the diffusion of gallium adatoms to the nanowire top) based on the finding of equations from the fitting with the experimental results, which are as a function of growth time, gallium rate and growth temperature.

So, I found this “magic” at page 4, on the sub-part of “Theoretical modelling of the NW axial growth rate for the self-induced approach”, namely “shadowing effects”. Beforehand, I agreed with them regarding the geometrical considerations in the molecular beam epitaxy chamber determining the final form of nanowires which are heavily influenced by the incident angle of the gallium effusion cell and nitrogen plasma source with respect to the substrate (read here). It is important also to realize that the incident angle of these sources will be having an impact in the shadowing of the grown nanowires after they reach certain height.

To my surprise, V. Consonni et al provided three examples, just straight answer, of how shadowing effects took a role in affecting structural morphology of the nanowire (density, radius, spacing, height) and the respective angle of gallium beam. In all cases, they assumed nanowire to have regular square area. The first case is nanowire having density, radius, spacing of 100 ${\mu}m^{-2}$, 30 nm and 51 nm, respectively. They found out that with an angle of 21 degree, the gallium impinged on a nanowire with height of 133 nm. The second example is nanowire with density and radius of 100 ${\mu}m^{-2}$ and 30 nm, a critical gallium beam angle of 49 degree was found. What… The last example, nanowire with radius and gallium beam angle of 30 nm and 21 degree, the density was deduced to be 200 ${\mu}m^{-2}$. I just don’t understand that straight answer.. It is just BOOM! Just let the radius and density to be known, then gallium beam angle will be discovered by some calculations which I do not understand.

To elucidate this matter, I make simple self-explanation. I used plain figure, based on the first case, which I think the most reasonable example I want to approach due to the more data in it. The objective is simple, before shadowing plays role, I want to find the maximum gallium beam angle and density, while radius, spacing and height are known. The figure has same scale measurement as with nm:

Nanowires configuration based on the first example

Let’s consider that gallium beam angle comes to the nanowires in a single line (of course in reality, the nanowires are received bunch of gallium flux on them). The consideration of “just before” shadowing effect means that this effect does not occur, i.e. maximum height of nanowires are found to be around 133 nm. The figure below gives an illustrative idea of how the maximum gallium beam angle “just before” the shadowing effect.

Maximum gallium beam angle with respect to the nanowire before shadowing effect

If I made a larger angle, than shadowing effect would take place. This is the maximum angle before shadowing. We can calculate that angle by doing a little hack. We can consider those figure with this:

The alternative approach for angle calculation

I doubt the angle will be 21 degree. With $\tan \theta = \frac{111 nm}{133 nm}$, the maximum gallium beam angle is 39.84 degree.

The density of the nanowire itself is lower with my calculation. Since 1 ${\mu}m^{-2}$ is equal with $10^{-6}$ $nm^{-2}$. We have 1 nanowire every 111 nm, meaning that in 2000 x 500 $nm^{-2}$, we have only 81 nanowire for 1 ${\mu}m^{-2}$.

See! The protractor also shows around 39 degree

It might be that V. Consonni et al have lower maximum limit to ensure that shadowing effect really will not be occurred, while my illustration really push the maximum angle of gallium beam angle. Using my approximation, with gallium beam angle of 21 degree, the maximum height of nanowire can be around 192 nm.

Maximum height of nanowire with gallium incident angle of 21 degree

Ok, I will leave the shadowing part here.

# Numerical Aperture on The Fiber Optic (1)

Yesterday, I re-arranged my bookshelf containing my notes I had made during Master course in NTNU. I felt nostalgic with these notes and they brought me back to the time when I tried to write them as neat as possible so that they can be useful (readable) for me in the future. I thought that it must be better to document them digitally to prevent these notes to be permanently damaged when they are just in the form of paper.

In the following note, I will write the study on the numerical aperture of fiber optic. Fiber optic is a cable made of glass (or plastic) threads (fibers), used for transmitting data in guided mode (for example: internet connection) utilizing light waves. We can see how it looks like in figure 1. I find the concept of fiber optic is very interesting because it uses simple optical phenomenon in our daily life. Surprisingly, the basic knowledge for explaining the guided-light has been attained in junior high school. Unfortunately, what we learnt during that time was only simple phenomenon without realizing there is an advanced technology depending on it.

Figure 1. Fiber optic

I will divide this explanation into three sections: first we deal with the Snell’s law. This is quite basic of refraction phenomenon, but I think it is very important to grasp the fundamental part. Next, we will move to the total internal reflection, where we will make use of the trait of Snell’s law to control the light. Finally, the last part will deal with fiber optics, by utilizing total internal reflection property.

# 1. Snell’s law

The basic law governing guided transmitted light in fiber optic is Snell’s law. This law deals with light coming from point P one media (with refraction index 1, $n_{1}$) with certain angle relative to the normal axis ${\theta_1}$, to point Q another media (with refraction index 2, $n_{2}$) where the light will have change its angle relative to the normal axis ${\theta_2}$ (figure 2). Alteration of light direction (refraction) is begun at the interface between two different media. We can meet with this phenomenon daily by observing how the straw in the glass of water looks bent or the strange form of stair in the swimming pool.

Figure 2. Refraction of light. Taken from wikipedia

How does direction of light change? Roughly speaking because of the atomic structure composing the media. The mass of the atom will create certain density in that media leading to the specified interaction between light and the atom and it will determine speed of light penetrating in that media. We can say that refraction characteristics of material are varying from one to another. Therefore, in figure 2, propagation of light with speed of ${v_1}$ in media ${n_1}$ changes its speed to ${v_2}$ in media ${n_2}$. Since there is an alteration of speed meaning there is a slightly change in the direction of light, relative to the normal axis, and in this case is media with ${\theta_1}$ to ${\theta_2}$.

To get rough qualitative measurement of the angle in each media, we can use this rule of thumb (based on figure 2: the incidence angle in medium 1 is larger than the refraction angle in medium 2 (${\theta_1} > {\theta_2}$) shows that light propagate faster in medium 1 than in medium 2 (${v_1} > {v_2}$). Since light propagate faster in medium 1 compared to medium 2, we can conclude that refraction index in medium 1 is smaller than medium 2 (${n_1} < {n_2}$). The denser the material is, the higher is the value of refractive index and the light will propagate slower depicted with the smaller angle of refraction (it will move closer to the normal axis).

For the case of refraction index in medium 1 is larger than medium 2 (${n_1} > {n_2}$), the result will show exactly in the reverse way: light will propagate slower in medium 1 than in medium 2 (${v_1} < {v_2}$) due to the fact that medium 1 is denser than medium 2 and this will lead to the incidence angle in medium 1 is smaller than the refraction angle in medium 2 (${\theta_1} < {\theta_2}$).

Based on the relationship we have established so far, this part will be closed with the formula between angle of incidence ${\theta_1}$ and refraction ${\theta_2}$, speed v and refraction index n for medium 1 and 2,

$\frac{\sin \theta_1}{\sin \theta_2}=\frac{v_1}{v_2}=\frac{n_2}{n_1}$

# Van der Waals (VdW) Bonding, GaAs on Si and Graphene as a Buffer Layer

Figure 1: a) Atomic geometry of GaAs/multi-layer graphene/Si interface showing only top-most graphene layer is strained by heteroepitaxial growth, b) schematic for structure with GaAs grown on top of single-layer graphene buffer layer/Si substrate. (Image source: http://www.semiconductor-today.com/news_items/2014/SEP/UCLA_100914.shtml and http://onlinelibrary.wiley.com/doi/10.1002/adfm.201400960/abstract)

Ok, In this post I will try to summarize this paper, one sentence at every question.

# Introduction

## Why is this important?

Photonics devices based on the integration of III-V materials (in this case is GaAs) on Si.

## What needs to be done?

Heteroepitaxial growth using MBE utilizing 2D material, owning to its VdW bonding which yields to accomodation to lattice mismatch between 3D-2D material.

## What has been done already?

MBE-grown GaAs NWs on graphene/Si substrate (A. M. Munshi et al, Nano Lett.2012124570), MOCVD-grown InAs/In(x)Ga(1-x)As NWs on graphene (Nano Lett. 2013131153) and GaAs/Si using layered GaSe ( J. Cryst. Growth 1995150685).

# Goal of the Research Project

Using graphene as a buffer layer in realizing VdW epitaxial growth of GaAs on Si

# Method

First, Y. Alaskar et al calculated surface energy of the graphene buffer layer with Ga and As prelayer; adsorption and migration energy of Al, Ga, In and As on graphene; The experimental is divided in fabrication (mechanical exfoliation of graphene and growing using MBE) and characterization (FESEM, AFM, Raman spectroscopy and XRD)

# Result

Theoretical result will not be presented in here, and instead, we focus on experimental result: (i) due to the low surface energy of graphene triggering high tension toward GaAs and (ii) low adsorption and mitigation energy of Ga and As on graphene at high temperature, island growth (3D) is formed instead of single crystal (2D) of GaAs

# Challenges and Bottlenecks

Developing understanding of VdW interaction between GaAs/graphene/Si

# Opportunities and Workarounds

Two-step growth is employed to investigate graphene using Ga- or As- prelayer.

Important: Growth morphology-surface energy, nucleation step-thin film properties, low surface energy of graphene, low adsorption energy of Ga and As, nucleation site, surface energy-adsorption energy, surface energy-growth temperature-wettability-nucleation process,