## Polar Coordinates

Coordinate systems are tools that let us use algebraic methods to understand geometry. While the rectangular (also called Cartesian) coordinates that we have been using are the most common, some problems are easier to analyze in alternate coordinate systems.

## How to charge your smartphone faster

Complaining about your smartphone's short battery life or the amount of time it takes to charge up are classic examples of the first world problems that can result in impatience and mild irritability. While we've previously looked at ways in which you can improve your smartphone's battery life, we're now taking a look at how you can make the darn thing charge a bit faster.

## What is EMI and how can you prevent it ?

Electromagnetic interference (EMI) is an electromagnetic emission that causes a disturbance in another piece of electrical equipment. EMI can be attributed to a wide span of the electromagnetic spectrum including radio, DC and even microwave frequencies. Because anything that carries rapidly changing electrical currents gives off electromagnetic emissions, it is quite common for one object’s emissions to “interfere” with another’s.

## James Clerk Maxwell - Biography

James Clerk Maxwell is often called one of the world's greatest physicists. He was also a major influence on other important scientists, like **Albert Einstein**. Maxwell's theories were essential in the development of technology we now take for granted, such as: radio, television ,mobile phones.

## Band structure of III-V and group IV semiconductors

**Group IV semiconductors (Si,Ge)**

The band structure of these semiconductors is very similar because:

- They do crystallize in the same crystallographic structure (diamond)
- They have similar electronic outer orbitals

The structure of silicon is purely covalent. The last orbital of atomic silicon has the electronic configuration 3s2p2. There are therefore 4 electrons (2s et 2p) sharing an orbital that could contain 8 (2 for the s orbital, 6 for the p orbital). Silicon has therefore 4 valence bands. The band structure of silicon and germanium, two most important semiconductors formed using the column IV of the periodic table, is shown in Fig. 2.4.

Figure 2.4: Germanium (left), Silicon (center) and Gallium Arsenide (right) band structures.

The valence band maximum is at k = 0 and is degenerate with the heavy and light hole bands. A third important valence band is the “spin-split” called this way because it is splitby the spin-orbit interaction. Finally, most important in the band structure of Silicon and Germanium is the fact that the minimum of the conduction band does not coincide with the maximum of the valence band. The semiconductor is called “indirect”.

The conduction band minimum in silicon is in the direction [010] and, as a result, also in the directions [010], [001], [001], [100], [100] for a total of six minima.

Figure 2.5: Minima of the conduction band of Si, Ge and GaAs

In Germanium, in contrast, the conduction band minimum is in the directions corresponding to the cube’s diagonal, and we have therefore 8 conduction band minima.

**III-V Semiconductors (GaAs, InP, ..)**

The band structure of III-V semiconductors is similar since the tetrahedral bonds have the same structure as the ones in Silicon or Germanium. In fact, the missing electron of the group III with the electron configuration 4s24p (for example Gallium) is provided by the column V element (for example Arsenic) of configuration 4s24p3 and these bonds have a low ionicity.

For a large number of III-V semiconductors, the bandgap is direct.

Figure 2.6: Structure de bande du GaAs

As an example, the computed band structure of GaAs is shown in Fig. 2.6.

## Heinrich Hertz - Biography

The great German physicist, Heinrich Hertz made possible the development of radio, television, and radar by proving that electricity can be transmitted in electromagnetic waves. He explained and expanded the electromagnetic theory of light that had been put forth by Maxwell. He was the first person who successfully demonstrated the presence of electromagnetic waves, by building an apparatus that produced and detected the VHF/UHF radio waves.

## Wavefunctions of the crystal - Bloch Theorem

The Hamiltonian of a semiconductor crystal has the translation symmetry

with being a reciprocal lattice vector.

The Bloch theorem states that the wavefunctions have two “good” quantum numbers, the band index n and a reciprocal vector such that the wavefunctions of the crystal may be written as:

where exhibits the periodicity of the crystal.

It may also be written as:

**Proof of Bloch’s theorem**

Translation operator

If **R** belongs to the Bravais lattice, the operator will commute with the Hamiltionian

and therefore

These two operators form a common set of commuting observable, we can therefore write the wavefunctions using eigenfunctions of both:

Consid´erons maintenant les propri´et´es de c(R)

et donc

To simplify somewhat the notation, a_{i} with i = 1, 2, 3 the basis vectors of the primitive Bravais lattice, and b_{i} the basis vector of the reciprocal lattice. We can always define (because c(a_{i}) is always normalized to unity:

If now **R** is a translation of the Bravais lattice, it can be written as

then,

Using the orthogonality relations between the direct and reciprocal basis vectors, written as

Considering the product between the vector **R** of the real space and the vector k of the

reciprocal space, written in the basis of the b_{i}:

and then

It was in fact the result we had obtained for c(**R**), that was written as:

setting naturally x_{i} = k_{i}. The wavefunctions may well be written as:

that is one of the equivalent formulations of the Bloch theorem.