5-7 Band Structures

A material where the fermi energy lies between two energy bands, but with a small energy gap between the bands that can be overcome by light or thermal excitation.

A material that has semiconducting properties without being doped. Other materials can be made semiconducting by introducing additional holes or electrons into the lattice.

Direct semiconductors have the lowest point of the conducting band and the highest point of the valence band at the same wave vector, allowing a direct excitation of an electron into the conducting band via light.

Indirect semiconductors have these points at different k-vectors, thus, a phonon is needed for the exctitation, adding both momentum and energy. This process is much more unlikely.

Direct semiconductors have a clear energy required to absorb light, which is equivalent to the band gap width.

Indirect semiconductors can start absorbing before the band gap energy due to the additional phonon energy. The absorption curve depends on the temperature since phonon occurrence is linked to this.

Charge carriers are only present if electrons are excited to the conducting band, since the other bands are fully occupied. The excitation can only occur when temperatures are high enough (or by using light).

Besides T, the charge carrier density also depends on the electron/hole effective mass and the valency band and conducting band energies

Excitons are bound electron-hole pairs that are not completely excited into conducting band and valency band. They can be described like hydrogen atoms, where the ionization border is the valency band energy.

Frenkel excitons are strongly bound and occur in organic materials and ionic, molecular or noble gas crystals. They can be described with a hydrogen anology

Wannier-Mott excitons are weakly bound, occuring in materials with a small band gap like semiconductors. The distance between electron and hole is very large due to the weak binding.

Since exciton binding energies impact the bohr radius of them, the size of small semiconducting particles directly influences the exciton energy levels and thus the color of the beads.

For a lattice doped with acceptor atoms, the energy level of the free holes is at E_A above the valency band.

For a lattice doped with donor atoms, the energy level of the free electrons is at E_D below the conducting band

In the intrinsic regime, n ~ exp(-1/T) , the intrinsic semiconducting is most important.

In the extrinsic regime, the thermal energy is not sufficient for instrinsic conduction, but all donor electrons are ionized, so the charge carrier density is constant

In freeze-out, the electrons are successively ionized from their donor atoms, n ~ -1/T

For low T, conductivity increases exponentially with T.

Then, the mobility of the charge carriers begins to decrease (due to scattering), leading to a drop of conductivity.

For very, high T, the intrinsic conductivity sets in, resulting in a sudden jump. (almost not visible in the image)

pn junction that is meant to be in reverse bias near the breakdown voltage. It is used to stabilize voltages.

Recombination of holes and electrons in a pn junction produces photons if the band gap is adjusted accordingly. This process needs direct semiconductors to be likely enough to produce a visible amount of light.

pn junction where light produces additional electron-hole pairs that follow the drift current and create a voltage.

Solar cells work best for a certain current, where they reach their maximum power P = U*I.

a npn junction (or pnp) with different slopes. Electrons diffuse from the emitter to the Base, which can be lowered or increased in energy to modulate the current. From the Base to the Collector, the potential has a steep drop, so any electron arriving at the Base will end up in the Collector. The Voltage at the collector is much higher (hence the potential drop). This setup allows to amplify small voltage modulations at the Base to big voltage modulations at the Collector.

By using a npn junction, electrons can be trapped at the p conducting band, creating a population inversion.

A contact between a semiconductor and a metal.

A type of transistor that can produce small electron channels between the two n contacts. This is used to simulate 1D or 2D electron gases.

When electrons enter a magnetic field, they will follow a curved trajectory, resulting in a circular motion with a certain frequency. This can be used to map surfaces with constant energy in a crystal. The magnetic field is varied with a certain frequency, and when a resonance occurs, a cyclotron frequency was hit.

In metals, the cyclotron resonance peaks are blurry because the magnetic field cannot deeply penetrate the metal. Only for extrema in the fermi surface, when the neighbouring k-vectors have very similar frequencies, a peak will occur. These regions are called extremal orbits.

When applying a B-field to a quasi-free electron, new energy levels form, called Landau levels. They depend quadratically on the k-vector. The resulting allowed k-states form so-called Landau-cylinders parallel to the B-field.

for 2D, the electrons are on discrete Landau levels.

In 3D, the electronic motion parallel to the B-field gives rise to a peak broadening/blurring.

We have a certain number of electrons per Landau level g (degeneracy). If the number of electrons is a integer multiple of g, N=l*g, the l-th landau level is completely occupied and the next is empty, so the fermi energy lies in between the two levels. Since g is proportional to B, a increase would mean that more electrons fit into one level. At the same time, the distance between the levels increases (E ~B). Thus, electrons start moving to the free places in lowel levels until the upper level is completely empty and the fermi energy suddenly jumps to the lower level.

For an increasing B-field at T close to 0 K, the magnetization of a material will oscillate because it depends on the change of internal energy. The internal energy oscillates because of the Landau levels. The warmer it gets, the less sharp are the landau levels, and the effect disappears completely for room temperature.

This effect can be used to find extremal orbits (minimum = neck orbit, maximum=belly orbit)

Due to magnetoresistance, the De-Haas-van-Alphen oscillation cause an oscillation in the conductivity of the material.

If a B-field is applied to electrons travelling though a material, their trajectories are bent, resulting in a Hall voltage perpendicular to the B-fields orientation. In the stationary case, the magnitude of the Hall-E-field is proportional to B, the proportionality constant is called hall constant R_H.

In a 2D electron gas, applying a B field gives rise to discrete Landau levels. For fully occupied Landau levels, no scattering can occur and thus the conductivity (and resistivity!) in the direction of the applied voltage is zero.

The Landau levels in the material might be partially delocalized electrons at the sharp peak and partially localized electrons broadening the peak. The localized electrons can not take part in conducting, but they still need to be “emptied” when the B-field increases. So there is a longer time in which the delocalized electrons are in a fully occupied Landau level, waiting for the Fermi energy to “arrive”.

If the electrons in a material are deviated due to a B-field and arrive at the sample surface, they are elastically reflected. Since the B-field suppresses backscattering, they will be reflected in a less steep angle and end up hitting the wall again due to the Lorentz force. They look as if they skip along the surface.

For very low T and very high B, the landau levels are no longer only at l=0,1,2,… but get fractional numbers. This is because the one particle approximation used for the behaviour of the electrons is no longer valid.

For materials with a very special band structure like graphene, the dispersion relation has a constant derivative for a long time. Thus, the electrons all have the same velocity, which does not depend on mass. Thus, these electrons formally behave like photons.

If it is folded in an armchair structure (30°), it is metallic, otherwise it is a semi-metal or a semiconductor. This is due to a shift in the density of states, creating a well shape in the metallic case:

A material can change its resistance in the presence of a magnetic field (due to magnetization). This is due to spin-orbit coupling and causes changes by a few percent.

Giant magnetoresistance can be achieved by layering ferromagnetic and non.magnetic materials onto each other (only a few atomic layers thick each). The more layers, the stronger the effect.

When no magnetic field is present, the ferromagnetic layers point in different directions, causing a shift in D(E) relative to each other and thus a different conductivity (because it depends on the density of states at the fermi border).

When the magnetic field is present, the ferromagnetic layers point in the same direction, and thus, the conductivity is the same for both.

A device changing its electrical resistance depending on the presence of an electric field. This can be achieved with an antiferromagnet and a ferromagnet or by layering two ferromagnets like for giant magnetoresistance

If the barrier between two ferromagnetic parts is small enough, electrons can tunnel through it. If both are polarized the same way, tunneling is much more likely than the other way around, meaning the resistance is smaller.

This is also used in hard disk read heads. The write head is magnetizing the disc, and the read head has one ferromagnet and a thin layer where electrons can tunnel through when it touches the hard disk. The read and write elements are closely together