Transport of Charge Carriers

Diffusion and drift are the 2 mechanisms for charge carrier transport in solar cells.

Diffusion refers to the net movement of charge carriers due to a concentration gradient.  Charge carriers perpetually move in random directions, including almost colliding with each other, which changes the direction and velocity of movement.  These near collisions are due to Coulomb-Coulomb interactions, where each charge carrier will bend another charge carrier's movement.  A concentration gradient refers to a non-uniform charge carrier distribution, where charge carriers will move from a region of higher concentration to a region of lower concentration.  The flux or movement of charge carriers in the opposite direction is less.  This will happen until the density of charge carriers is uniformly distributed.

For electrons, the Fick's law of diffusion equates the J_e (electron current density) to the product of q (elementary charge), D_e (diffusion coefficient of electrons), and dn/dx (density gradient in direction x):

J_e = qD_e dn/dx

A similar equation applies for holes:

J_h = qD_h dp/dx

Drift refers to the movement of charged particles under the influence of electric fields.  Holes will experience a force and move in the direction of the electric field, while electrons move in the opposite direction.  The current density induced by the electric field is as follows:

J_e = nqμ_eE
J_h = pqμ_hE

where n and p are the density of electrons and holes respectively (at the start point of the drift), μ is the mobility constant, and E is the electric field.

In an electronic band diagram (see below), the electric field will induce a slope positive (sloping upwards) in the direction of the electric field.  This is applied over both the valence and conduction bands.  Hence on average, excited electrons will move down the slope of the conduction band, while excited holes will climb up the slope of the valence band.

Electrons and holes also suffer from 3 recombination mechanisms, namely radiative recombination, Auger recombination, and the Shockley-Read-Hall (SRH) recombination.  Loss mechanisms such as recombination determine the lifetime of charge carriers.  A high recombination rate will lead to shorter lifetimes τ and correspondingly shorter diffusion lengths L.  Diffusion length is the average distance covered by excited charge carriers, and is shown below:

L_e = √(D_eτ_e)
L_h = √(D_hτ_h)

In a doped semiconductor material, the majority charge carriers will have longer diffusion lengths as compared with minority charge carriers.



Reference:
2.3.3 Transport of Charge Carriers, Delft University of Technology, https://www.youtube.com/watch?v=2EhJh3BQvB0

Excitation of Charge Carriers II

Continuing from Excitation of Charge Carriers I:

An example of p-doping will be the inclusion of Boron, a III-element, to silicon.  A Boron atom forms 3 bonds with silicon atoms, but the 4th bond has a hole because it is filled with only one silicon electron.  Upon excitation to mobile state, electrons will come to fill this hole, thereby moving the hole to other bonds.  The original fixed Boron atoms will thus become negatively charged, and the holes will become the majority charge carriers due to its higher density than electrons.  In an electronic band diagram, Boron atoms are represented by acceptor states, whose energy levels also lie in the forbidden band gap, close to the valence band.  At room temperatures, many silicon electrons from the valence band will be excited to the acceptor states, leaving many holes behind.  The Fermi level will also be closer to the valence band.

The law of mass action states that the product of electron density (electron carrier concentration) n and hole density p is constant at a given temperature, regardless of doping concentration changes.  Hence:

n*p = n_i²

where n_i is the intrinsic density of charge carriers in silicon, and for an intrinsic material:

n = p = n_i = 1.1*10¹⁰ cm^-3

Semiconductor materials can also absorb light energy, where photons from light will excite electrons from the valence band to the conduction band.  The energy from each photon must possess energy that's more than the energy required for each electron to jump the band gap.  If the energy from the photon is less than the band gap energy, the light wave will just pass by without being absorbed by the semiconductor material.  If the photon's energy is much larger than the energy to overcome the band gap, an electron deeper inside the valence band can be excited to the conduction band, or the electron can be excited to a much higher level in the conduction band, after which the electron will quickly relax back to the bottom of the conduction band.  Similar effects are seen for holes - holes will relax to the top of the valence band.  This relaxation releases heat energy and makes the semiconductor material hotter.

For a semiconductor material that has been doped, the majority charge carriers will have a density that is much higher than the concentration of charge carriers excited by light absorption.  However, the density of charge carriers created by light absorption is much higher than the density of minority carriers.  This leads to the important conclusion that solar light absorption will increase minority charge carriers effects.



Referencing:
2.3.2 Excitation of Charge Carriers II, Delft University of Technology, https://www.youtube.com/watch?v=MtzaExAnIrg

Excitation of Charge Carriers I

The Fermi-Dirac distribution function describes the probability of electrons occupying different states or levels of energy, when the material is at thermal equilibrium.  Thermal equilibrium means that no extra energy is included from electrical biasing, light absorption, or heat.

The Fermi level is the energy level where the probability of electrons occupying that level is 50%.  The Fermi level (E_F in the diagram below) is also known as the total chemical potential of an electron.  The Fermi level of a metal can be easily seen due to its single continuous electronic band, but for a semiconductor, the Fermi level lies in the forbidden band gap between the valence and conduction band where no electrons can occupy.  Hence, at a temperature of zero Kelvin, all electrons will occupy the valence band.  At higher temperatures, some electrons will occupy the conduction band.  There will be more electrons in the conduction band as temperature increases, so heat is one way to increase a semiconductor's conductivity.

Besides heat or thermal energy, there are another 2 ways to excite electrons from the valence band to the conduction band: using doping, and using light energy.

Pure semiconductor materials without impurities are called intrinsic, meaning that the density of each mobile carrier, the electrons and holes, are the same.  Doping means the inclusion of impurities in semiconductor materials intentionally, which increases the electrons or holes drastically.  For example, if we dope silicon with Phosphorus, a V-element in the periodic table, 4 of the valence electrons in phosphorus will be bonded to silicon, while the fifth valence electron will become a free electron that is easily excited to the conduction band.

Correspondingly, the bonded fixed phosphorus atom will become positively charged after losing its electron.  This is called n-doping, where electrons are the majority mobile charge carriers due to its higher density.  The minority mobile charge carriers are the holes.  In an electronic band diagram, Phophorus atoms are represented by extra donor states which lie in the forbidden band gap of silicon, close to the conduction band.  Hence, less energy is required to excite electrons from the donor state to the conduction band.  This usually happens at room temperatures.  The Fermi level will also be closer to the conduction band.



Reference:
2.3.1 Excitation of Charge Carriers I, Delft University of Technology, https://www.youtube.com/watch?v=LxRp0YGSWqw

Band Gap II - Electrons in Molecular Bonds

Electrons in the outermost shell are called valence electrons.  A valence bond requires 1 valence electron, each from neighbouring atoms, to share its molecular orbital.  The molecular orbitals of the 4 valence electrons of silicon are shown in the diagram below.  These orbitals merge to form a tetrahedral like structure called the sp3 hybrid.

The orbitals of the an electron each from 2 atoms will merge when they bond, and they must satisfy the Pauli exclusion principle, meaning that one must spin up, while the other spin down.  At this bonding level (bonding state), the energy level is the lowest, more so than the original 2 individual sp3 levels.  This is the valence band level, and it is more stable.  The next energy level is the anti-bonding level, which is higher than the bonding level and the original sp3 level.  This is the conduction band level, and electrons prefer not to be in this state.  The smaller the structural gap between 2 atoms when bonded, the larger the energy required to split the 2 atoms, which means the bonding and anti-bonding levels will be further apart.  This difference in levels is called the band gap.  The valence band and conduction band, formed from the combine effects of many silicon to silicon bonding and anti-bonding levels, are discrete energy levels.

In chemistry, the valence band is also called the Highest Occupied Molecular Orbital (HOMO), while the conduction band is the Lowest Unoccupied Molecular Orbital.  In general, carbon, silicon and germanium are IV materials, but carbon is not a IV-semiconductor material because the bond length is much smaller, and hence the band gap is larger.  The lattice constant is defined as shown in the diagram below.

Materials with semiconductor properties can also be made from III-V materials like gallium arsenide (GaAs) and II-VI materials like cadmium-telluride (CdTe).  These also form the tetrahedral diamond cubic crystalline lattice.

Referencing:
2.2.2 Band Gap II - Electrons in Molecular Bonds, Delft University of Technology, https://www.youtube.com/watch?v=IFZeGsDOCi4

Band Gap I - Electrons in Atoms

The silicon atom has a total of 14 electrons in its 1, 2 and 3 shells.  The electrons are filled going from the lowest energy level 1s, to the highest energy level 3p.  Referring to the diagram, each horizontal line can hold 2 electrons, indicated by the arrows.  The arrows point in opposite directions because of Pauli exclusion principle, which is an important law in quantum mechanics.  It states that electrons have spins, which are rotations of charge, and there are 2 directions or quantum states - spin up and spin down.  These rotations induce magnetic dipoles - up and down.  Each horizontal line must hold a total of just 2 electrons spinning in opposite directions, never spinning in the same direction.  The horizontal line indicates an electron energy level.

The 4 electrons in the outermost shell 3 are the weakest bound to the nucleus, and will hence be the electrons that make bonds with other atoms in the molecules of liquids and gases, or in solids.  When used in bonds, the electrons will be arranged with one electron per electron energy level.

Reference:
2.2.1 Band Gap I - Electrons in Atoms, Delft University of Technology, https://www.youtube.com/watch?v=s-84yDC23HU

How to Transform Light into Electricity

Categorizing based on electrical properties, there are 3 kinds of materials: metals, semiconductors and insulators.  These differences can be explained using the electronic band structures of the materials.  An electronic band is an indicator of the states or levels of potential energy an electron can occupy in the material.

The electrons in metals are weakly bound to the atoms.  Hence, there is an ocean of free electrons in metals, which enable good conduction of electricity.  Metals have a broad electronic band, not fully filled with electrons, and there are no forbidden energy levels.  Electrons in metals are free and mobile.

There are no free electrons in insulators.  The electrons are all bound to the materials' atoms.  Insulators have 2 distinct electronic bands, divided by a large forbidden energy gap between them which is called the band gap.  Almost all electrons are found in the lower electronic band, called the valence band.  This shows that electrons are firmly bound to the materials' atoms in the lattice.  Band gaps in insulators are usually larger than 3eV in energy to cross.

Semiconductors are like insulators, having 2 distinct electronic bands.  However, the forbidden band gap is not as wide as an insulator.  Hence, electrons can acquire energy from heat or light to cross the band gap, going from the valence band to the conduction band.  In the conduction band, electrons are free and mobile, just like electrons in metals.



Reference:
2.1 How to Transform Light into Electricity, Delft University of Technology, https://www.youtube.com/watch?v=d4XO2_u7YYk

Solar light 2

Referencing:

1. ET3034Tux - 1.6.2 – Solar light 2, https://www.youtube.com/watch?v=wizhej5qUNM
2. Solar Energy - the Physics and Engineering of Photovoltaic Conversion, Technologies and Systems, Chapter 5.

Referring to the diagram below, the spectral power density of irradiance from the sun (also known as spectral irradiance) reaching the outer side of Earth's atmosphere is shown in yellow.  This spectrum is the extraterrestrial spectrum and the irradiance is approximately 1361 W/m².  The spectrum of a black body radiator at 5800K temperature is indicated by the grey line.  The spectrum of solar light arriving at the Earth's surface is indicated in red.  Much of the spectral power density is lost in the Earth's atmosphere due to scattering and absorption by molecules and particles, such as ozone, oxygen, water and carbon dioxide.

The depth of the Earth's atmosphere is shortest at the equator, and longer at higher latitudes.  Hence, the path of solar light is shortest at the equator, and correspondingly, the spectral losses are least.  The path of light is indicated by the optical air mass, where an air mass of 1 indicates the shortest path length at the equator.  Air mass AM is described by this equation:

AM = 1 / cosθ

where θ is the latitude in degrees.

An AM of 1.5, or AM1.5, would be the path length of sunlight through the atmosphere at a latitude of 48.2 degrees at noon, ignoring the seasons.  The AM1.5 spectrum is defined as the Standard Test Conditions (STC) for the solar spectrum, where the irradiance is 1000 W/m².  (And under STC, the solar cell temperature is 25°C.)  This spectrum is shown in the diagram below.

When we consider light in terms of photons, the photon flux ϕ is defined as the number of photons per unit time per unit area.  This doesn't come with spectral information, analogous to irradiance.  The spectral photon flux Φ is defined as the number of photons per unit time per unit area per wavelength range.  The spectral photon flux and spectral power density are related by this equation:

Φ(λ) = P(λ) λ/hc

Essentially, the spectral photon flux (also known as spectral photon flux density) is the spectral power density divided by the energy of a photon at a particular wavelength.  If we integrate the spectral photon flux over a wavelength range as follows, we get the photon flux for that wavelength range:

ϕ = ∫₀^λ Φ(λ) dλ

Due to the fact that every photon can create a collected charge carrier in solar cells, the amount of photon flux would in theory determine the maximum current per unit area that can be created in a solar cell.  Hence, the maximum possible conversion efficiencies of solar cells can be also determined.

Another useful tool is the Estimated Sun Hours (ESH), where:

1 ESH = 1 kWh/m²

This means a STC irradiance of 1000 W/m² for one hour on the Earth's surface.  In addition, the Watt-peak (Wp) refers to the maximum power a solar module can deliver under STC.  Hence, for a 100Wp solar module operating at a location on Earth with 5 ESH, the solar energy generated is 500Wh per day. 

Another term is Capacity Factor.  It refers to the time that an electricity generator works at maximum nominal power, and usually averaged over one year.  Hence:

Capacity Factor = ESH * 365 / (24*365)

Solar light 1

Referencing:
- ET3034Tux - 1.6.1 – Solar light 1, https://www.youtube.com/watch?v=lDYFtO9QxsM
- www.google.com

Light is dual natured: it can be described as electromagnetic waves, or as particles which have quantised amounts of energy, called photons.  The photoelectric effect shows that electrons can be ejected from a material upon light absorption by the material.  This ejection requires light with a certain threshold energy, which arises from a material-dependent threshold frequency.  Energies and frequencies below this threshold will not cause electrons to be ejected.  This is shown by the formula for light energy:

E = hv

where h is Planck's constant, and v is the frequency of the incident light.

As an electromagnetic wave, light can be described as propagating in a particular direction.  Perpendicular to this direction, an electric field oscillates in a plane.  Perpendicular to this direction and the plane of the electric field, a magnetic field oscillates.  The distance between the maxima of the electric field's oscillation is called the wavelength λ.  The speed of the light's propagation in vacuum is c.  These are related to the frequency of light by this equation:

c = λv

Hence: E = hv = hc/λ

Irradiance I (also known as solar irradiance, power density or radiant power density) is the power of light per unit area, with units watts per square metre.  The spectral power density P is a quantity with spectral information because it is the incident power of light per unit area per unit wavelength.  I and P are related by this equation:

I = ∫₀^λ P(λ) dλ