You are currently browsing the monthly archive for December 2009.

Materials are often classified into three distinct categories: insulators, semiconductors, and metals.

The three materials are distinguished from one another based on their band gap, or by their resistivities (conductivities).  Roughly speaking, metals are materials with bulk resistivity less than $10^{-3}$ $\Omega \cdot cm$.

Energy band theory can explain these differences in resistivity in terms of differences in band gap.  Every material has a band structure associated with it that results from electrons interacting with one another.  Those band that are filled are called valence bands, and those that are empty are referred to as conduction bands.  The difference between the bottom of the conduction band and the top of the valence band is called the band gap.

If the band gap is above 3 eV, then the material is generally considered an insulator.  If the material has a band gap of less than 3 eV, then it is generally a semiconductor.  Metals do not have a band gap, and instead have partially filled bands.

Semiconductors are materials that have band gap between that of insulators and that of conductors.

We can modify the electrical properties of semiconductors by doping the material.  For example, silicon has four valence electrons.  By doping silicon with boron (which has three electrons), we introduce holes into the valence band of silicon.  By doping silicon with phosphorus (which has five electrons), we introduce electrons in the conduction band of silicon.

Doping can be done with a variety of methods, such as ion implantation or gas phase doping.  Ion implantation is done by bombarding the material with high energy atoms.  The advantage of using ion implantation is that doping concentrations above that of the solid solubility limit can be achieved.  However, the use of ion implantation results in crystal damage, that must be fixed through an annealing step.  Ion implantation is modeled by monte carol methods.

Gas phase doping is performed by flowing the gas phase of the dopant atoms over the material.  Photoresist in photolithography or silicon oxide are often used to define the regions for doping.   This method of doping is limited by the solid solubility limit, and diffusion is used to drive the dopants into the material.

For III-V materials, such as InAs, the prospects of ion implantation do not look so favorable, as the crystal damage is more difficult to fix due to the stoichiometry of the lattice.  Researchers are currently doping these materials in-situ, by gas phase doping, or other exotic methods.

Dimples exist on golf balls for a very good reason.  The efect of dimples were actually discovered accidentally, when it was discovered that rough surface, scarred balls traveled further than smooth ones. Ball makers than began to intentionally put dimples on balls to produce the same effect of rough surface balls.

There are two components to drag.  One is frictional drag, but the other is due to pressure drag.  Frictional drag is due to shear stresses on the object.  Pressure drag is due to pressure differences between the front and rear of the object due to boundary layer separation from the surface of the ball.

For streamlined bodies, the drag is mostly due to frictional drag and it is beneficial to have smooth surfaces to promote laminar flow.  However, for blunt bodies like golf balls, drag is mostly due to pressure differences.

For a smooth surface, the boundary layer has less energy and separates from the ball earlier.  The rear just has a wake region that has decreased pressure. A dimpled ball promotes turbulent boundary layer flow.  A turbulent boundary layer has more energy and thus travels further along the surface before separating from the ball.  A dimpled ball thus has less pressure reduction at the rear, and overall less drag.

You can have a burp or a note, but not both.

The Heisenberg uncertainty principle is an interesting non-classical phenomenon, which states that certain pairs of physical properties cannot be known with arbitrary accuracy.  In it’s most common form, it is expressed as an uncertainty between position and momentum, where the position and momentum of a particle cannot both be known simultaneously with arbitrary accuracy.

The idea sounds very foreign to us, as at the macro level, we are familiar with being able to know both the position and momentum of say, a car at the same time.  What is even more interesting, is that the uncertainty between position and momentum is not a limitation on measurements or observation, but actually a description of physical nature itself.  It is a result of quantum mechanics and the wave-like nature of matter.

There is also an uncertainty principle between energy and time.  This uncertainty is commonly known in Fourier analysis as a frequency-time uncertainty.  One can specify a signal with arbitrary time or arbitrary frequency, but not both.  An example of this uncertainty would be the contrast between a burp and a note. For a burp, this signal has a very narrow standard deviation in time-domain (just a single bump), but the signal is comprised of a many different frequencies and has a large uncertainty (standard deviation) in frequency-domain.  The shorter the pulse, the larger the range of frequencies that are used to make up this pulse.  On the other hand, for a flute playing a note, the frequency of that signal has a narrow standard deviation (a single frequency or note), but in the time-domain, it looks like a sine wave and has a large uncertainty (standard deviation).

Similarly, the wave function associated with a particle such as an electron may have a definite position at some time but the momentum will have some uncertainty, or it may have a definite momentum at a given time, but it’s position will be uncertain.

Mathematically speaking, the position-momentum uncertainty principle is expressed as $\Delta x \Delta p_x >= \frac{\hbar}{2}$ where $\Delta x$ is the standard deviation of the position, and $\Delta p_x$ is the standard deviation of the momentum.  While the position and momentum of a particle can be known with arbitrary accuracy, they cannot both be known at the same time.  $\hbar = 10^{-34} kg m^2/sec$, which is very small.  It is much too small for objects of macroscale size, but is significant for very small objects such as electrons.

Can a butterfly flapping its wings effect a tornado halfway across the world?  Can it greatly alter future events and the evolution of time?

The butterfly effect is the concept that the evolution of a system may be highly sensitive to perturbations in the initial data.  It is one of the main conditions for chaos.  The term was coined by Edward Lorenz, one of the pioneers of chaos theory, in a paper titled Predictability: Does the Flap of a Butterfly’s Wings in Brazil set off a Tornado in Texas? The idea is that a small change in the initial conditions, such as a butterfly’s flap of its wings may cause large scale changes over time leading to a tornado in some other part of the world.

The idea is featured prominently in time travel movies such as appropriately named The Butterfly Effect, Run Lola Run, or the Back to the Future movies, where some small change in the current timeline results in drastically different results at some later time.  It’s interesting to wonder whether in fact, one’s life would be drastically different if some or even one event in one’s life were slightly different.  In Jurassic Park, the chaos theorist Ian Malcom, illustrates the concept by a drop of water rolling down one’s hand when placed near the knuckle.  Even a slight change in where the drop of water is placed results in the drop moving in different directions and larger distances apart than the initial distance.

Systems such as weather exhibit this behavior, and thus make it difficult to accurately predict weather beyond just a few days.

Mathematically speaking, the conditioning of a system is a quantitative measure of it sensitivity–how slight perturbations in the initial conditions propagate.  The condition number is defined as $\frac{| \Delta y /y|}{\Delta x /x}$, and can be understood to be an amplification factor that relates the error or change in a system’s evaluation relative to error or change in a system’s input.  If the relative change in the output of the system is large compared to its input (large condition number), then the system is ill-conditioned or sensitive.  If the relative change in the output of a system is small or commensurate with the input (small condition number), than the system is well-conditioned or insensitive.

The conditioning of a system also depends on the input data.  For example, in the water drop example given above, the system is ill-conditioned near the knuckle, but anywhere else, the water drop will flow in the same direction regardless of small perturbations, and thus, the system is well-conditioned.

Most metals have a crystal structure that is are either body-centered cubic (BCC), face-centered cubic (FCC), or hexagonal close packed (HCP).

The body centered cubic structure consists of one lattice point in the center of the cube in addition to eight corner points.  Many metals such as chromium, iron, tungsten have BCC structure at room temperature.  Visualize it here.

The face centered cubic structure consists of six lattice points at the center of each face of the cube in addition to the eight corner points.  many metals such as aluminum, copper, lead, nickel, and iron crystallize as FCC.  Visualize it here.

The hexagonal close packed structure is similar to the FCC structure, in achieving the highest packing density.  The structures are close packed.    The coordination number or number of closest numbers is maximized (which happens to often minimize the cohesive energy).  Metals such as cadmium, zinc, and titanium have an HCP structure.  Visualize it here.

The simplex hexagonal structure is not close packed.  Visualize the simple hexagonal structure here.  In the simple hexagonal structure, atoms are stacked AAAA.  Atoms are stacked ABAB for hexagonal close packed, and atoms are stacked ABCABC for FCC.

How can one sail against the direction of the wind?  Intuitively, it would seem like one could only move in the direction of the wind.  Terry Tao at UCLA dissects this problem in a very interesting post.  From the analysis, it is the rudder of the boat that allows one to sail against the wind.

Using some assumptions about the effect of tacking the sail (exploiting the crosswind direction) and the use of keels (sails in water), different achievable boat velocities are analyzed.  Through the use of both tacking and keels, one can theoretically achieve arbitrary directions and speeds.

Plasma enhanced chemical vapor deposition is a process used for the deposition of thin films of materials.  Plasma is used to enhance the chemical reaction processes and thus, PECVD can typically be done at lower temperatures than conventional CVD.

PECVD systems include a power supply, match generator, electrodes, and vacuum system to create an RF signal (often at 13.6 MHz).

Silicon oxide is usually deposited using silane, SiH4, and nitrous oxide, N2O.  Nitride is typically deposited using silane and ammonia, NH3.

A nanometer is $10^{-9}$ meters.  To get an idea of how incredibly small a nanometer is, we can take a look at the scale of various things we are familiar with.

At the meter scale, there are humans who are about 1 or 2 meters in height.

One thousand times smaller, at the millimeter scale are fleas or ants or the top of a pin. The diameter of hair is several hundred microns or tenths of a millimeter.

Another thousand times smaller, at the micron scale are cells in our body.

Another thousand times smaller, at the nanometer level, are the diameter of DNA.

An atom diameter is a few Angstroms, or a few tenths of a nanometer.

A good example (borrowed from Richard Feynman‘s Lecture on Physics, Volume 1 Chapter 1 Atoms in Motion) is that if you enlarge an apple to the size of the Earth, then the atoms in that apple are now the size of the original apple.

A nanometer is incredibly small!

According to Richard Feynman in his Lectures on Physics Vol. 1 Chapter 1 Atoms in Motion, which instantly gives the idea some street cred, the most bit of important scientific knowledge that can be expressed in one statement is the “atomic hypothesis (or the atomic fact, or whatever you wish to call it) that all things are made of atoms–little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another.”