The theory of solar cells explains the process by which light energy in photons is converted into electric current when the photons strike a suitable semiconductor device. The theoretical studies are of practical use because they predict the fundamental limits of a solar cell, and give guidance on the phenomena that contribute to losses and solar cell efficiency.
Working explanation Photons in sunlight hit the solar panel and are absorbed by semi-conducting materials. Electrons (negatively charged) are knocked loose from their atoms as they are excited. Due to their special structure and the materials in solar cells, the electrons are only allowed to move in a single direction. The electronic structure of the materials is very important for the process to work, and often silicon incorporating small amounts of boron or phosphorus is used in different layers. An array of solar cells converts solar energy into a usable amount of direct current (DC) electricity.
Photogeneration of charge carriers When a photon hits a piece of semiconductor, one of three things can happen: The photon can pass straight through the semiconductor — this (generally) happens for lower energy photons. The photon can reflect off the surface. The photon can be absorbed by the semiconductor if the photon energy is higher than the band gap value. This generates an electron-hole pair and sometimes heat depending on the band structure.
When a photon is absorbed, its energy is given to an electron in the crystal lattice. Usually this electron is in the valence band. The energy given to the electron by the photon "excites" it into the conduction band where it is free to move around within the semiconductor. The network of covalent bonds that the electron was previously a part of now has one fewer electron. This is known as a hole, and it has positive charge. The presence of a missing covalent bond allows the bonded electrons of neighboring atoms to move into the "hole", leaving another hole behind, thus propagating holes throughout the lattice in the opposite direction to the movement of the negatively electrons. It can be said that photons absorbed in the semiconductor create electron-hole pairs.
A photon only needs to have energy greater than that of the band gap in order to excite an electron from the valence band into the conduction band. However, the solar frequency spectrum approximates a black body spectrum at about 5,800 K,NASA Solar System Exploration - Sun: Facts & Figures retrieved 27 April 2011 "Effective Temperature ... 5777 K" and as such, much of the solar radiation reaching the Earth is composed of photons with energies greater than the band gap of silicon (1.12eV), which is near to the ideal value for a terrestrial solar cell (1.4eV). These higher energy photons will be absorbed by a silicon solar cell, but the difference in energy between these photons and the silicon band gap is converted into heat (via lattice vibrations — called phonons) rather than into usable electrical energy.
The p–n junction
The most commonly known solar cell is configured as a large-area p–n junction made from silicon. As a simplification, one can imagine bringing a layer of n-type silicon into direct contact with a layer of p-type silicon. n-type doping produces mobile electrons (leaving behind positively charged donors) while p-type doping produces mobile holes (and negatively charged acceptors). In practice, p–n junctions of silicon solar cells are not made in this way, but rather by diffusing an n-type dopant into one side of a p-type wafer (or vice versa).
If a piece of p-type silicon is placed in close contact with a piece of n-type silicon, then a diffusion of electrons occurs from the region of high electron concentration (the n-type side of the junction) into the region of low electron concentration (p-type side of the junction). When the electrons diffuse into the p-type side, each one annihilates a hole, making that side net negatively charged (because now the number of mobile positive holes is now less than the number of negative acceptors). Similarly, holes diffusing to the n-type side make it more positively charged. However (in the absence of an external circuit) this diffusion current of carriers does not go on indefinitely because the charge build up on either side of the junction produces an electric field that opposes further diffusion of more charges. Eventually, an equilibrium is reached where the net current is zero, leaving a region either side of the junction where electrons and holes have diffused across the junction and annihilated each other called the depletion region because it contains practically no mobile charge carriers. It is also known as the space charge region, although space charge extends a bit further in both directions than the depletion region.
Once equilibrium is established, electron-hole pairs generated in the depletion region are separated by the electric field, with the electron attracted to the positive n-type side and holes to the negative p-type side, reducing the charge (and the electric field) built up by the diffusion just described. If the device is unconnected (or the external load is very high) then diffusion current would eventually restore the equilibrium charge by bringing the electron and hole back across the junction, but if the load connected is small enough, the electrons prefer to go around the external circuit in their attempt to restore equilibrium, doing useful work on the way.
Charge carrier separation There are two causes of charge carrier motion and separation in a solar cell:
drift of carriers, driven by the electric field, with electrons being pushed one way and holes the other way diffusion of carriers from zones of higher carrier concentration to zones of lower carrier concentration (following a gradient of chemical potential).
These two "forces" may work one against the other at any given point in the cell. For instance, an electron moving through the junction from the p region to the n region (as in the diagram at the beginning of this article) is being pushed by the electric field against the concentration gradient. The same goes for a hole moving in the opposite direction.
It is easiest to understand how a current is generated when considering electron-hole pairs that are created in the depletion zone, which is where there is a strong electric field. The electron is pushed by this field toward the n side and the hole toward the p side. (This is opposite to the direction of current in a forward-biased diode, such as a light-emitting diode in operation.) When the pair is created outside the space charge zone, where the electric field is smaller, diffusion also acts to move the carriers, but the junction still plays a role by sweeping any electrons that reach it from the p side to the n side, and by sweeping any holes that reach it from the n side to the p side, thereby creating a concentration gradient outside the space charge zone.
In thick solar cells there is very little electric field in the active region outside the space charge zone, so the dominant mode of charge carrier separation is diffusion. In these cells the diffusion length of minority carriers (the length that photo-generated carriers can travel before they recombine) must be large compared to the cell thickness. In thin film cells (such as amorphous silicon), the diffusion length of minority carriers is usually very short due to the existence of defects, and the dominant charge separation is therefore drift, driven by the electrostatic field of the junction, which extends to the whole thickness of the cell. .
Once the minority carrier enters the drift region, it is 'swept' across the junction and, at the other side of the junction, becomes a majority carrier. This reverse current is a generation current, fed both thermally and (if present) by the absorption of light. On the other hand, majority carriers are driven into the drift region by diffusion (resulting from the concentration gradient), which leads to the forward current; only the majority carriers with the highest energies (in the so-called Boltzmann tail; cf. Maxwell–Boltzmann statistics) can fully cross the drift region. Therefore, the carrier distribution in the whole device is governed by a dynamic equilibrium between reverse current and forward current.
Connection to an external load Ohmic metal-semiconductor contacts are made to both the n-type and p-type sides of the solar cell, and the electrodes connected to an external load. Electrons that are created on the n-type side, or created on the p-type side, "collected" by the junction and swept onto the n-type side, may travel through the wire, power the load, and continue through the wire until they reach the p-type semiconductor-metal contact. Here, they recombine with a hole that was either created as an electron-hole pair on the p-type side of the solar cell, or a hole that was swept across the junction from the n-type side after being created there.
The voltage measured is equal to the difference in the quasi Fermi levels of the majority carriers (electrons in the n-type portion and holes in the p-type portion) at the two terminals.
Equivalent circuit of a solar cell
An equivalent circuit model of an ideal solar cell's p–n junction uses an ideal current source (whose photogenerated current increases with light intensity) in parallel with a diode (whose current represents recombination losses). To account for resistive losses, a shunt resistance and a series resistance are added as lumped elements. The resulting output current equals the photogenerated current minus the currents through the diode and shunt resistor:
The junction voltage (across both the diode and shunt resistance) is:
where is the voltage across the output terminals. The leakage current through the shunt resistor is proportional to the junction's voltage , according to Ohm's law:
By the Shockley diode equation, the current diverted through the diode is:
exp represents the exponential function
where I0, reverse saturation current n, diode ideality factor (1 for an ideal diode) q, elementary charge k, Boltzmann constant T, absolute temperature the thermal voltage. At 25 °C, volt.
Substituting these into the first equation produces the characteristic equation of a solar cell, which relates solar cell parameters to the output current and voltage:
An alternative derivation produces an equation similar in appearance, but with on the left-hand side. The two alternatives are identities; that is, they yield precisely the same results.
Since the parameters I0, n, RS, and RSH cannot be measured directly, the most common application of the characteristic equation is nonlinear regression to extract the values of these parameters on the basis of their combined effect on solar cell behavior.
When RS is not zero, the above equation does not give directly, but it can then be solved using the Lambert W function:
When an external load is used with the cell, its resistance can simply be added to RS and set to zero in order to find the current.
When is small, we can use the approximation as to produce something much easier to work with
Several further simplifications are now possible, such as when which leads to
When the current generated by the PV is large compared with the current in the shunt, i.e. (because the shunt resistance is large) there is an analytical solution for for any less than :
Otherwise one can solve for using the Lambert W function:
However, when RSH is large it's better to solve the original equation numerically.
The general form of the solution is a curve with decreasing as increases (see graphs lower down). The slope at small or negative (where the W function is near zero) approaches , whereas the slope at high approaches . Therefore for high optimum output power , it is desirable to have large and should be small.