One of the major challenges in designing solar cells is to maximize the efficiency. In this example, we consider the optical and electrical factors that reduce the efficiency of a single junction GaAs solar cell below the theoretical ShockleyQueisser limit.
Overview
The ShockleyQueisser limit sets a fundamental theoretical constraint on the efficiency of a single junction solar cell using thermodynamic arguments [Shockley61]. In the derivation of the limit, there are three key assumptions
•all incident light is absorbed below the band gap of the material, and each absorbed photon generates an electronhole pair
•the only source of recombination is direct (bandtoband) radiative recombination (satisfying the principle of detailed balance)
•there are otherwise no limitations on the transport of charge (infinite mobility, perfectly reflecting contacts)
The ShockleyQueisser limit, when combined with the AM1.5G solar spectrum, sets the maximum efficiency for a single junction photovoltaic cell at 33.7% with and ideal band gap of 1.34eV, which is very close to GaAs (with a maximum theoretical efficiency of 32.8%) [Araujo90]. The ShockleyQueisser limit for the AM1.5G solar spectrum, with the location of the GaAs band gap, is plotted in the figure below [Wiki04]
In the consideration of the absolute limit on efficiency, the fundamental sources of loss include blackbody radiation (the solar cell is typically assumed to have temperature T_{earth}), radiative recombination (required by detailed balance), and the nonzero band gap of the material (which reduces the range of wavelengths that can be absorbed). When the AM1.5G spectrum is used as a source (with a total power density specified as 100mW/cm2), the total number of absorbed photons over all wavelengths would be capable of generating a current of
$$\mathrm{J}_{\mathrm{AM} 1.5 \mathrm{G}}=\int_{0}^{\infty} \frac{\lambda}{\mathrm{hc}} \mathrm{S}_{\mathrm{AM} 1.5 \mathrm{G}}(\lambda) \mathrm{d} \lambda=67.3 \mathrm{mA} / \mathrm{cm}^{2}$$
However, the band gap of GaAs is 1.424eV, which limits the range of wavelengths that can be absorbed to those less than 870nm. In the absence of radiative recombination, GaAs would be capable of generating an equivalent current of approximately 32.2mA/cm^{2} (as calculated from the data provided from the solar script command). This is illustrated in the figure below.
To simulate the ideal absorption of the incident light, we solve a simple 1D absorption equation with no reflection. First, we note that the electric field at the surface is related to the spectral irradiance:
$$\mathrm{I}_{0}(\lambda)=\mathrm{S}_{\mathrm{AM} 1.5 \mathrm{G}}(\lambda)=\frac{\mathrm{n}}{2 \mu_{0} \mathrm{c}}\mathbf{E}(0, \lambda)^{2}$$
The absorption coefficient can be found from the refractive index of the material, and the intensity as a function of depth can be found with the simple BeerLambert law,
$$\alpha(\lambda)=\frac{4 \pi \mathrm{k}}{\lambda} \rightarrow \mathrm{I}(\mathrm{z}, \lambda)=\mathrm{I}_{0}(\lambda) \mathrm{e}^{\omega \mathrm{z}}$$
Assuming that each absorbed photon generates and electronhole pair, the generation rate as a function of depth z and wavelength λ
$$G(z)=\frac{\mu_{0} c}{\hbar} \int_{0}^{\infty} \frac{1}{n} S_{A M 1.56} \Im\{\varepsilon\} e^{a z} d \lambda$$
(for more details on the formulation of the generation rate, please see the whitepaper (Optoelectronic Modeling of Photosensitive Devices). By substituting for the electric field and integrating over wavelength, the net generation rate is
$$G(z)=\frac{\mu_{0} c}{\hbar} \int_{0}^{\infty} \frac{1}{n} S_{A M 1.5 G} \Im\{\varepsilon\} e^{\omega z} d \lambda$$
To ensure complete absorption within the spectral range of GaAs (i.e. photons with energies greater than the band gap), an artificial refractive index can be used, which has a large constant value for wavelengths below that of the GaAs band gap, and is zero otherwise (the plot below compares the imaginary index of GaAs to the idealized perfect absorber). Two ideal (no reflection) 1D absorption profiles are shown in the figure below (right), comparing a realistic index for GaAs to the artificial one ensuring complete absorption. These generation rate profiles are used as ideal sources in the following efficiency simulations. In the case of complete absorption, the maximum current density is 32.1mA/cm^{2}.
(a)
(b)
To calculate the 1D generation rate profile and record the result to a data file for import into the CHARGE simulation, the script solar_gaas_ideal_1d_absoprtion.lsf is provided in the associated files. Open the solar_gaas.fsp project and run the script from FDTD to use the material properties for GaAs that correspond to the current example.
Ideal electrical structure and efficiency at the ShockleyQuessier limit
To construct an idealized electrical collection structure, the conditions of the ShockleyQueisser limit must be met. These include
 infinite mobility
 no nonradiative recombination processes
 perfectly reflecting (minority carrier) contacts
This structure can be simulated in CHARGE by carefully controlling the semiconductor material model. Open the solar_gaas.ldev project. In this project file, semiconductor materials for AlGaAs have been added, and a modified version of GaAs ("GaAs (Gallium Aresenide)  Ideal") and AlGaAs has been created. In the initial configuration, the GaAs and AlGaAs layers are defined with the idealized versions of the materials (this can be confirmed by editing the properties of the structures and inspecting the material property). The structure and doping from [Wang13], described in the drawing below, is defined in the project layout.
The material model for the ideal GaAs has been modified such that
 all nonradiative recombination processes are disabled and
 the mobilities for both electrons and holes are set to very large values.
In addition, the hole effective mass has be reduced slightly to give close agreement with the theoretical reverse bias saturation current, which in turn influences the open circuit voltage.
The band offsets in the heterojunctions between the GaAs and the top and bottom AlGaAs layers naturally reflect incident minority carriers, satisfying the perfectly reflecting contacts condition. The simulation layout contains a band structure monitor (monitor), which will record a 1D profile of the band structure in the active region of the simulation. To more accurately interpolate the optical generation rate, two mesh override regions are included at the surface of the GaAs layer. Typically, the automatic mesh refinement will accurately interpolate the optical generation rate, but in the ideal (perfectly absorbing) case, the change near the surface requires additional constraints. An additional mesh constraint is added for the spacecharge layer (SCL), and the minimum and maximum edge lengths in the CHARGE have been reduced in order to provide close agreement with the theoretical result.
The ideal (completely absorbed) optical generation rate from the previous 1D calculation is included as a source of electronhole pairs (ideal_ogr). The object itself is offset by 30nm in z to align it with the top surface of the GaAs layer (recall that the ideal 1D absorbance calculation assumed zero reflection starting from the GaAs surface). The other optical generation rate object (ogr) is disabled in this simulation.
Run the simulation, which sweeps the bias from 0 to 1.13V. Plot the current density and photovoltaic efficiency using the script commands below, which normalize the result to the nominal 1cm^{2} area of the solar cell. Note that in the efficiency calculation, the ratio of the power delivered by the solar cell (P = JV) to the incident power from the AM1.5G illumination is equivalent to the power delivered (as a percentage), because the AM1.5G spectrum is normalized to deliver 100mW/cm^{2}:
$$ \eta=\frac{P_{\text {cell }}}{P_{\text {AM1.5G}}}=\frac{P_{\text {cell, mWlem }}^{2}}{100 \mathrm{mW} / \mathrm{cm}^{2}} \times 100 \%=P_{\text {cell, mW/cm }}^{2} $$
normarea = (getnamed('CHARGE simulation region','x span')*getnamed('CHARGE','norm length'));
base = getresult('CHARGE','base'); ?Jb = 0.1*pinch(base.I)/normarea; # mA/cm2 Ve = base.V_emitter; plot(Ve,Jb,'voltage (V)','current density (mA/cm2)'); plot(Ve,Jb*Ve,'voltage (V)','efficiency (%)');
In the preceding simulation, the short circuit current density is 32.6mA/cm^{2}, the open circuit voltage is about 1.12V, and the maximum photovoltaic efficiency is 32.4%, which is close to the theoretical maximum found from the ShockleyQueisser limit. While care has been taken in the setup of the electrical simulation, some error is introduced in the discretization of the mesh and interpolation of the optical generation rate. For further comparisons in this example, we will use the 32.4% maximum photovoltaic efficiency as the reference.
To view the band structure, select the band structure monitor and visualize the "bandstructure" dataset. The minority carrier blocking layers are clearly visible at the heterojunctions for the top and bottom AlGaAs layers.
Realistic Optical Structure
To account for the realistic optical effects, a 2D FDTD simulation will be run. Aspects of the design that influence the optical response include
 reflections from the front surface (nonideal ARC)
 nonideal back contact reflection (<100%)
 incomplete absorption (realistic refractive index)
 contact shadowing
The first three components can be readily addressed in the optical simulation. Download and open the solar_gaas.fsp. The layered solar cell structure is contained within the "back reflector" structure group. This consists of the AlGaAs/GaAs/AlGaAs stack of the same dimensions as described previously. On the back surface, and aluminium contact layer is included. On the front surface, an antireflective coating (ARC) is added, with a constant refractive index of 1.4. The thickness of this ARC is varied to maximize the optical absorption in the GaAs (a sweep "arc" is included in the project file that can be used to vary the ARC thickness and monitor the net generation of electronhole pairs in the GaAs layer). The optimal thickness is found to be 0.1um.
A plane wave source is used to supply the illumination, and a solar generation rate analsysgroup object (refl_gen) is added to calculate the generation rate and ideal short circuit current density for an AM1.5G solar spectrum source. A mesh override is added at the GaAs surface to improve the resolution for the absorbed power calculation, because light are mostly absorbed in the surface for shorter wavelengths.
Note: It is important to carefully align the solar generation rate analysis object such that no mesh points on or adjacent to the contact are included. Including the aluminum contact in the generation rate analysis will introduce errors into the generation rate calculation, and an artificially large absorption will be reported. 
Run the FDTD simulation. When complete, select the refl_gen generation rate analysis object and run the analysis. The generation rate profile will be calculated from the absorbed power, and the result will be saved to a file OGR_AlGaAs_fine.mat for import into CHARGE. In the properties for the analysis, note that two periods of data will be stored (doubling the x span of the result). In addition, this analysis object has been modified slightly to ensure that the the AM1.5G spectral power is normalized to 100mW/cm^{2} in the wavelength range 0.32.6um. The ideal short circuit current from the generation rate analysis is 31.3mA/cm^{2}, which is a reduction of approximately 5% from the maximum.
Switch to the CHARGE project file. Disable the ideal optical generation rate (ogr_ideal) and enable the imported optical generation rate object (ogr) representing the realistic optical stack. Edit this object and browse to locate the "OGR_AlGaAs_fine.mat" data file generated by FDTD; import this data. Run the CHARGE simulation. This will simulate the response of the system with the realistic optical input, but maintaining the ideal electrical structure.
To include the effect of contact shadowing, we will simply normalize the result to an area that is scaled to include the shadowing loss fraction.
shadow_pct = 0.06; normarea = (getnamed('CHARGE simulation region','x span')*getnamed('CHARGE','norm length')); base = getresult('CHARGE','base'); Jb = pinch(base.I)/normarea; Jbs=Jb/(1 + shadow_pct);# with shadowing Ve = base.V_emitter; plot(Ve,0.1*Jbs,'voltage (V)','current density (mA/cm2)'); plot(Ve,0.1*Jbs*Ve,'voltage (V)','efficiency (%)');
The plots the follow show the effect of the realistic optical structure and shadow loss on the photovoltaic efficiency.
The results are summarized in the table below
Planar optical 
Shadow 


J_{SC} (mA/cm^{2}) 
30.3 
28.6 
V_{OC} (V) 
1.12 
1.12 
η (%) 
29.9 
28.2 
Realistic Electrical Structure
To evaluate the influence of the realistic electrical material, with finite mobility and thermal recombination effects enabled, we will now run a sequence of CHARGE simulations. Using the same project file as before (with the realistic optical generation rate profile enabled), change the material models for the AlGaAs and GaAs structures from the "ideal" versions to their standard counterparts:
Object 
Old material 
New material 

GaAs 
GaAs (Gallium Arsenide)  Ideal 
GaAs (Gallium Arsenide) 
nAlGaAs 
Al(0.3)Ga(0.7)As Ideal 
Al(0.3)Ga(0.7)As 
pAlGaAs 
Al(0.85)Ga(0.15)As Ideal 
Al(0.85)Ga(0.15)As 
The following script commands can also be used to make the change:
switchtolayout; setnamed('GaAs','material','GaAs (Gallium Arsenide)'); setnamed('nAlGaAs','material','Al(0.3)Ga(0.7)As'); setnamed('pAlGaAs','material','Al(0.85)Ga(0.15)As');
Open the material properties for "GaAs (Gallium Arsenide)." Select the mobility tab and note that the nominal values for GaAs mobility are present. Next, select the bulk recombination and generation tab. Note that the models for trapassisted, Auger, and radiative recombination are enabled. The radiative recombination rate coefficient is the same as for the ideal material model.
For the first simulation, disable the trapassisted recombination model for the GaAs and AlGaAs materials. Both the radiative and Auger recombination models should be enabled for all three materials. Click OK to close the material database. The simulation will now be run with realistic material properties for GaAs and AlGaAs, but excluding trapassisted recombination. Run the simulation.
Use the previous script with shadow loss included to plot and analyze the JV and photovoltaic efficiency curves. Record the shortcircuit current density, open circuit voltage, and peak efficiency.
Switch back to layout mode. Expand the material group, and again locate the GaAs and AlGaAs materials. Reenable the trapassisted recombination models for all three materials. In this project file, the base carrier lifetimes and Auger recombination coefficients are equivalent to the values used in [Wang13]. In the GaAs material model, the carrier lifetimes for trapassisted recombination are also corrected for the total dopant density. Click OK to close the material database, and run the simulation. Again, plot and analyze the JV and photovoltaic efficiency, and record the key metrics.
To study the influence of the recombination rate processes, the script solar_gaas_recomb.lsf can be used. The script will read the recombination dataset from the CHARGE results, and using the interptri command, extract a 1D cut line from the recombination profiles for the radiative (Ropt), Auger (Rau), and trapassisted (Rsrh) processes. The plot below is generated. The dominant contribution to the recombination is the radiative process, followed by trapassisted recombination.
Return to the current project, and switch back to layout mode. Another common source of reduced photovoltaic efficiency is series resistive loss. This can be modeled by including a series resistance in the emitter contact model. For this example, following reference [Wang13], we will use the surface resistance R_{surface} = 0.7Ω cm^2, which is specified assuming a nominal device area of 1cm^{2}. To normalize the resistance to the simulation area, we scale it by the inverse of the area normalization,
normarea = getnamed('CHARGE simulation region','x span')* getnamed('CHARGE','norm length')*1e4; # area in centimeter square ?Rse = 0.7/normarea;
which gives a value of 35kΩ . Select the emitter contact and edit its properties. Enable the series resistance (rse) and set the value to 3.5e4. Click OK to close the contact properties editor, and run the simulation. When the simulation is complete, perform the JV and photovoltaic efficiency analysis (again including the shadowing loss) and record the results.
Finally, we would like to include the effects of photon recycling: when electrons and holes recombine radiatively, they must emit a photon with an energy equivalent to the band gap of GaAs, which may then be reabsorbed. The selfconsistent simulation of this process is complicated, but the behaviour can be approximated well by scaling the radiative recombination coefficient by the Asbeck coefficient [Ahrenkiel89]. The Asbeck coefficient is a function of depth and is materialspecific. For a 1.65um thick layer of GaAs, the coefficient has a value of Φ=4.6 such that
$$ \widetilde{C}_{o p t}=\frac{C_{o p t}}{\phi}=1.09 \times 10^{10} \mathrm{cm}^{3} \mathrm{s}^{1} $$
Switch to layout mode and expand the material group. Select the active GaAs material, and change the radiative recombination rate coefficient to match the value above.
Radiative recombination 
Default value 
New value 

ehp capture rate (cm^{3}s^{1}) 
5e10 
1.09e10 
Click OK to close the material database and run the simulation. When the simulation is complete, perform the JV and photovoltaic efficiency analysis (again including the shadowing loss) and record the results. The table below summarizes the results of the analysis for the key solar cell performance metrics. In the last column, the published characteristics for a similar reference design are provided for comparison.
+Auger 
+SRH 
+Rse 
+Photon recycling 
Reference [Green12] 


J_{SC} (mA/cm^{2}) 
28.6 
28.6 
28.6 
29.0 
29.5 
V_{OC} (V) 
1.06 
1.06 
1.06 
1.1 
1.107 
η (%) 
26.9 
26.8 
26.2 
27.5 
28.3 
References
 W. Shockley and H. Queisser, J. Appl. Phys. 32, 510–519 (1961).
 G. Araujo, “Limits to Efficiency of Single and Multiple Bandgap Solar Cells”, in Physical Limitations to Photovoltaic Energy Conversion, A. Luque and G. Araujo, Eds. Adam Hilger, Bristol, 1990, pp. 119–133
 "ShockleyQueisser limit," Wikipedia: The Free Encyclopedia. Wikimedia Foundation, Inc. 22 July 2004. [online]
 X. Wang, et al., "Design of GaAs Solar Cells Operating Close to the ShockleyQueisser Limit," IEEE J. Photovoltaics, 3, 737 (2013)
 R. Ahrenkiel, et al., "Ultralong minority carrier lifetime epitaxial GaAs by photon recycling," Appl. Phys. Lett., 55, 1088 (1989) [doi]
 M. A.Green, et al., “Solar cell efficiency tables (Version 39),” Progr. Photovoltaics, 20, 12 (2012).