Shockley-Quisser-limit
An Investigation of Shockley-Queisser Limit of Single p-n Junction Solar Cells
2.997 Project Report Bolin Liao Wei-Chun Hsu
Abstract
Shockley-Queisser limit has long been recognized as the theoretical limit of the efficiency of an ideal single p-n junction solar cell, which is based on the most fundamental physical laws.
Despite its profound significance and widely accepted concepts, we found ambiguities in the derivation of the Shockley-Queisser limit in their seminal paper published in 1961. In this project, we look into the basic physical principles of the p-n junction solar cells, move on to clarify the details of the derivation process and finally try to give out a contemporary interpretation of this limit.
1. Introduction
The solid-state solar energy conversion technology utilizing semiconductor solar cells has become an important candidate for the next generation energy exploitation. By the end of 2006,
the total installed PV capacity since 1991 had reached more than 7 GW worldwide [1]. In the meantime, how to further improve the efficiency of photovoltaic solar cells has become an essential issue.
The most prevalent bulk material for solar cells is doped silicon with n-type or p-type impurities.
In 1961, William Shockley and Hans J. Queisser showed that a single p-n junction solar cell has
a theoretical limit of efficiency, which is known as Shockley-Queisser limit or the detailed balance limit [2]. Instead of predicting the efficiency limit based on empirical data and semi-empirical modeling, Shockley and Queisser built up their insight purely on the basis of the detailed balance principle, which is deeply rooted in the second law of thermodynamics. This
limit is one of the most fundamental theories to solar energy production, and is considered to be one of the most important contributions in the field. In addition to the situation with a specific semiconductor model, R. Ross and T. Hsiao generalized this limit to all photochemical solar energy conversion systems in 1977[4]. In 1980, C. Henry [3] further extended the approach of Shockley and Queisser along 4 directions, 1) using an intuitive graphical analyzing method to show the magnitudes of the intrinsic losses, 2) considering both single and multiple energy gap cells, 3) using the actual terrestrial solar spectrum instead of a blackbody spectrum, 4) considering the effect of concentrated sunlight, and finally reached the practical efficiency limit
of an applicable conversion device.
Miscellaneous factors are involved with the efficiency issue of a photochemical solar energy converter. Among these factors, some are not intrinsic, thus can be eliminated theoretically by improved fabrication technology and optimized device design. Examples include the series resistance and reflection loss, etc. Other than these factors, there exists another kind of loss factors, which are either intrinsic to the device characteristics, or determined by fundamental physical laws, which thus cannot be avoided by advanced manufacturing techniques or other methods. It is exactly these intrinsic losses that set the limit of the efficiency for a photochemical solar energy converter. The most important loss due to the device characteristics is the energy gap for photon absorption. Incident photons with energy higher than the energy gap can be absorbed, exciting electrons to higher electronic bands, while those with lower energy are not absorbed, either reflected or transmitted. Moreover, the portion of the absorbed energy greater than the energy gap is dissipated in the process of electrons relaxation by the form of heat, resulting in further loss of the absorbed energy. After absorption, not all absorbed energy can be converted into useful power, which is the electrical power in the case of a solar cell. Only the free energy (the Helmholtz potential) that is not associated with entropy can be extracted from the device, which is determined by the second law of thermodynamics. In the situation of a p-n junction solar cell, this effect manifests itself in the radiative recombination process, the inverse process of photon absorption. In equilibrium, any part of a system must emit exactly the same energy as that absorbed in every direction, at every moment and every frequency interval, which is known as the detailed balance principle, inferred by the second law of thermodynamics. Furthermore, the inevitable irreversibility when actual load is connected will lower the achievable power under the maximum available free energy.
To recapitulate, the main factors considered by Shockley and Queisser to derive the limit efficiency include the energy gap, the radiative recombination and the load matching loss. In this report we will discuss these factors in detail and try to interpret the physical meaning behind the effects. At the last of the report, the discussion is extended to taking other non-ideal factors in practical application into account. Using the same notations as Shockley and Queisser’s original paper, the total efficiency of a single p-n junction solar cell can be expressed as the product of four terms:
( ) ( ) ( )
is called “ultimate efficiency”, only taking into consideration the loss due to the energy gap. denotes the ratio of the operational output voltage to the energy gap.
is the impedance matching factor, and the probability that incident photons with solar cell surrounded by the black body with temperature will produce a hole-electron pair, which is assumed unity when the limit efficiency achieved. We will discuss and derive how these factors come from in the following sections.
2. Energy Gap Loss
Before we start to introduce several factors of the total efficiency of the solar cell, notations and parameters are defined as follows:
: Planck's constant = 6.626068 × 10-34 ;
: Boltzmann constant = 1.3806503 × 10-23
: The electronic charge
: Temperature of sun
: Temperature of solar cell
: Energy gap of the solar cell, and the cut-off frequency
In the original paper, the authors use equivalent voltages to express , , and for convenience:
; ;
Also defined are two ratios among , , and .
;
The ultimate efficiency describes that when photons are incident on a solar cell, only photons with energy larger than the band gap of the solar cell will excite electrons, thus be absorbed. The absorbed electrons only contribute the amount of energy as the band gap, and the absorbed energy can be totally converted into electrical power, by the means of a output voltage , the equivalent voltage of the energy gap. Following this concept, Shockley and Queisser set a situation as Fig.1 to derive this efficiency. We assume a spherical black body with temperature surrounds a spherical pn junction with temperature (this condition is to ensure the there is no radiative recombination inside and the output voltage is exactly . We will discuss this later). Therefore, the ultimate efficiency can be calculated as the ratio of generated photon energy to input power.
( )
where is the number of quanta of frequency greater than incident per unit area per unit time for black body radiation of temperature , and the total incident power.
Figure 1 demonstration of ultimate efficiency situation
and can be calculated from the Plank’s law of blackbody radiation:
( ) ∫ ∫
∫ ∫
Therefore, the ultimate efficiency can be expressed as a function of :
∫ ∫ ∫ ∫ ( ) We plot the efficiency versus the energy gap as follow (consider the sun as a source of blackbody radiation with temperature 5800K):
Figure 2 dependence of the ultimate efficiency on energy gaps The maximum efficiency is 43.96%, corresponding to an energy gap around 1.08 eV.
3. Detailed Balance Principle
Now we consider a more realistic situation, depicted in Fig.3. Two factors will be taken into account, namely, the view factor of the sun seen from the solar cell, and the loss due to radiative recombination.
Assume and are the surface areas of the sun and the solar cell plate respectively. Then the projected area of on the radiation sphere of the sun is , thus the view factor from the
Ultimate Efficiency Vs. Energy Gap
Energy Gap (eV)U l t i m a t e E f f i c i e n c y , (%)
sun to the solar cell is , where L is the distance between the sun and the earth. So the actual incident power is
(*(*()
where R is the radius of the sun, the solid angle subtended by the sun. We define the geometrical factor to lump the all geometric effects:
Similarly, the generation rate of electron-hole pairs due to incident solar radiation can be calculated as:
where is the probability of an incident photon exciting an electron-hole pair.
Figure 3 demonstration of a realistic model
Now we evaluate the crucial point in deriving the Shockley-Queisser limit: the recombination process. Not all excited electron-hole pairs can be extracted by connected load, and a large portion of them will recombine and lose the energy. Generally two kinds of recombination processes will happen inside a solar cell: radiative recombination and non-radiative recombination. The latter one can be eliminated theoretically by ideal fabrication conditions, while the former one is intrinsic due to the detailed balance principle. The detailed balance principle states that: in equilibrium, any part of a system must emit exactly the same energy as it absorbs, in every direction, at every frequency and time interval. In other words, wherever absorption happens, emission (thus radiative recombination) must happen.
To determine the radiative recombination rate, we first imagine the solar cell in equilibrium with a surrounding blackbody radiation field with temperature (without incident solar radiation). Due to the detailed balance principle, the recombination rate must be the same everywhere in the
whole system, including at the surface of the solar cell, where the recombination rate is just equal to the incident radiation exciting rate:
( ) ∫
and is the probability of an incident photon exciting an electron-hole pair, and the coefficient 2 is because both sides of the solar cell can absorb the incident radiation. Then the radiative recombination rate inside the solar cell is just everywhere.
Now we expose the solar cell to solar radiation, and the equilibrium inside the cell is disrupted. Excessive electron-hole pairs are generated due to the incident radiation, and the built-in
potential in the p-n junction will drive the excessive carriers to either side of the cell, resulting in the split of the single Fermi-level in equilibrium to quasi-Fermi levels in non-equilibrium in both sides. The energy band diagrams before and after exposure to solar radiation are shown in Fig.4.
Figure 4 Energy Band Diagram in equilibrium and non-equilibrium
The relations between the quasi-Fermi level and non-equilibrium carrier concentration are given as:
and
To determine the open-circuit voltage that can be obtained, we consider the question that how much the energy of the electron-hole gas system is lowered by removing an electron-hole pair out of the system. If just the internal energy is considered, the answer should be , the energy possessed by an electron-hole pair. But it is not correct. Since there will be an entropy change during this process, the system must exchange some heat with the environment. According to Ross[5], this entropy change is:
Then the total energy change is:
which is not very surprising, since this is exactly the change of the Helmholtz free energy. So instead of , the open circuit voltage that can be obtained is just
.
In non-equilibrium condition, the radiative recombination rate will deviate from the equilibrium value. As a first-order approximation, we assume the recombination rate is proportional to the product of the concentrations of electrons and holes. So the non-equilibrium recombination rate can be written as:
(*
where is the voltage between the two electrodes, as analyzed above. It is obvious to observe that if there are more photons absorbed by the solar cell, the will be larger, and the voltage will become larger as well.
After external load is connected, the output voltage will drop below the open-circuit voltage. To obtain the I-V characteristics of the solar cell, we consider the balance among miscellaneous generation and recombination mechanisms and the external current:
where R(0) and R(V) are the generation rate and recombination rate due to non-radiative processes. To avoid detailed consideration of non-radiative processes, a fraction coefficient is defined as the ratio of the radiative recombination rate to total recombination rate:
We can now obtain the current-voltage relation:
[ (*][ (*]
where is the short circuit current, and is the reverse saturation current. By letting total current be zero, the open circuit voltage can also be derived:
( *( *(*(*
(∫
∫
)
where is defined as the lumped factor of the radiative fraction , geometrical factor , and the exciting probabilities and :
Assume an energy gap E g=1.12eV, and , which is the ideal condition, we plot the open circuit voltage as a function of operational temperature T c:
Figure 5
We can tell from this plot that when T c approaches zero, the open circuit voltage approaches V g =1.12V , which is the largest achievable open circuit voltage. This result can be easily verified by applying L'H?pital's rule to the expression of the open voltage to get the limit behavior.
Accordingly we can define a dimensionless ratio between the open-circuit voltage and the band gap voltage as:
( )
( ∫ ∫ ) ( ∫ ∫ ) If assume T c =300K , , we can plot this factor as below:
Figure 6
Open Circuit Voltage Vs. Operational Temperature
Operational Temperature Tc (K)O p e n C i r c u i t V o l t a g e , V
100
Open Circuit Voltage Ratio Vs. Energy Gap
Energy Gap (eV)O p e n C i r c u i t V o l t a g e R a t i o , (%)
4. Impedance Matching Factor
In a practical device, the maximal values of current and voltage, namely I sh and V op cannot be achieved simultaneously. Instead, an optimized load can gain an optimized power, which is only a portion of the product of I sh and V op, as shown in Fig.7.
Figure 7 the practical achievable power
Before proceeding to consider the optimal load, we first assume the solar cell has not yet been connected to any load, the nominal efficiency is defined as the ratio of open-circuit voltage times the short-circuit current to the incident power.
If the generation rate of electron-hole pairs by the sun is large enough, we can assume:
( ) ( )
Now we define an impedance matching factor as:
( )
Then the practical efficiency can be written as:
( ) ( ) ( )
To find the optimal power, we follow the extremum-finding routine:
(*(*
This is a transcendental equation, and can be solved numerically. Assume T c =300K ,
, the impedance matching factor ( ) can be plotted as follow:
Figure 8
Now that we have all the ingredients to obtain the practical efficiency limit of a single p-n junction solar cell, the Shockley-Quesser limit can be plotted as below, with
T c =300K,400K,500K , and
.
Figure 9
The maximum efficiency is around 30% with an energy gap of 1.28 eV when T c =300K.
100
Impedance matching factor Vs. Energy Gap
I m p e d a n c e m a t c h i n g f a c t o r , %Energy Gap, eV
Energy Gap, eV E f f i c i e n c y , %Schockley-Queisser Limit
References:
[1] Electric Power Reach Institute (EPRI). An EPRI Technology Innovation White Paper. Dec. 2007. https://www.wendangku.net/doc/803373643.html,/docs/SEIG/1016279_Photovoltaic_White_Paper_1207.pdf
[2] Shockley, W., Queisser, H.J., Journal of Applied Physics, 32, 510 (1961).
[3] Henry, C.H., Journal of Applied Physics, 51, 4494 (1980)
[4] Ross, R.T., Hsiao, T.L., Journal of Applied Physics, 48, 4783 (1977)
[5] Ross, R.T., Journal of Applied Physics, 46, 4590 (1967)