25th European Photovoltaic Solar Energy Co OPEN SOURCE ... - ISFH
25th European Photovoltaic Solar Energy Conference, Valencia, Spain, 06-10 September 2010, 20 2DV.1.19
OPEN SOURCE GRAPHICAL GRAPHICA USER INTERFACE IN MATLAB FOR TWO--DIMENSIONAL SIMULATIONS SOLVING THE FULLY COUPLED SEMICONDUCTOR SEMICO EQUATIONS USING ING COMSOL S. Eidelloth1, U. Eitner1, S. Steingrube1, and R. Brendel1,2 Institute for Solar Energy Research Hamelin (ISFH), Am Ohrberg 1, 31860 Emmerthal, Germany Phone: hone: +49(0)5151 999 - 315, Fax: - 400, eMail: [email protected] 2 Institute of Solid State Physics, Leibniz University Hanover, Appelstr. 2, 30167 Hanover, Hanover Germany 1
ABSTRACT: Numerical modeling is useful to understand and improve the device performance of solar cells. The freely available simulation software PC1D is commonly used to model one-dimensional one dimensional semiconductor structures and provides a convenient Graphical User Interface (GUI). Current research, research however, relies on the improvement of twotwo or three-dimensional dimensional cell structures. A free software package for two-dimensional two dimensional cell simulations on an ordinary PC with a convenient GUI is currently not available. The purpose of this work is to close this gap by providing provi such an Open Source GUI. Our GUI can be used to specify solar cell model parameters, to evaluate the simulation data and to save the results in common file formats. The appearance of the GUI can be adapted by modifying the configuration Excel files that at specify the user input panels. Our software solves the fully coupled semiconductor transport equations using COMSOL Multiphysics. The underlying physical model considers diffused layers as conductive boundaries which are characterized by their sheet resistance resistance and by their diode saturation current density. This approach reduces the numerical demands since the space charge regions are not spatially resolved and hence, the number of required mesh points decreases. Keywords: Modeling, Simulation, Software
1
INDRODUCTION
The recently introduced Conductive nductive Boundary (CoBo) model [1] characterizes diffused layers by their sheet resistance and diode saturation current density. This model strongly reduces the numerical demands of two-dimensional dimensional semiconductor simulations. Here, we report on a flexible Graphical User Interface for the CoBo model (CoBoGUI CoBoGUI), which can be downloaded for free from our website www.isfh.de. The purpose of the CoBoGUI is not only to provide a user friendlyy interface to specify the CoBo solar cell model, but also to enable batch simulations, to support the user with predefined post processing functions and display or export the results as a function of the batch parameters. Our flexible software concept enables ena batch simulations for any model that is controlled by a MATLAB function. For example, at ISFH we use the GUI for simulations of thermal boron diffusion profiles where another system of differential equations is solved. In this paper, however, we focus on two-dimensional modeling of the transport of electrons and holes in solar cells. The commercial applications MATLAB and COMSOL Multiphysics are required to run the CoBoGUI.. Please feel free to adapt our open source code package for your own needs under er reference to this paper.
2
SOFTWARE ARCHITECTURE
The software architecture of the CoBoGUI is shown in Figure 1. MATLAB reads configuration files {A} at the program start-up. up. These files contain information about the model variables and the post processing proc functions. After reading the configuration files, MATLAB creates the GUI {B}. The user can specify the simulation parameters and finally start the simulation. The simulation parameters are passed to a MATLAB script
that creates so-called called “FEM structures” stru (FEM: Finite Element Method) ethod) for all batch elements. Each FEM structure contains all information for a single solar cell simulation and is passed to COMSOL Multiphysics {C}. In case of solar cell modeling, COMSOL Multiphysics solves the coupled semiconductor equations with a Finite Element Method approach, adds the solution to the FEM structure and returns it back to MATLAB.
Figure 1: Software architecture of the CoBoGUI. - The interaction of MATLAB with four other components {A}-{D} enables a user friendly simulation environment. MATLAB analyses the JV--curve, performs the CLA (Current density Loss Analysis nalysis, see section 5), and the FELA (Free Energy Loss Analysis nalysis [2]). [2] All information of the batch simulation, including the results of the loss analyses, is stored in a data structure named “dat”. To gain specific information from this data structure, the user can choose from a large amount of predefined post processing functions within the GUI. Alternatively,
25th European Photovoltaic Solar Energy Conference, Valencia, Spain, 06-10 September 2010, 20 2DV.1.19
the advanced user may use any MATLAB function to access the data structure from the MATLAB command line. By default, the results are displayed as MATLAB figures, but they can also be passed to interfaces {D} to obtain other output file formats, such as Excel or SigmaPlot files. All connections between MATLAB and the components {A}-{D} {D} are bidirectional. For example it is possible to save user defined options of cross section plots as a configuration file ({B} => MATLAB => {A}). MATLAB and COMSOL Multiphysics are platformplatform independent dent and so are the JAVA objects used for the GUI. The interface component {D} is the only part of the software architecture that is platform dependent. If you want to use the CoBoGUI on other operation systems than Windows, you might have to adapt the corresponding interface script files.
3
GRAPHICAL USER INTERFACE
In the GUI, the simulation workflow is organized in separate tabs. Thereby, each tab represents a single step in the course of the simulation, as described in the following subsections. 3.1 Geometry The first tab is the “Geometry” tab. It shows a schematic of the predefined unit cell (see Figure 2), and provides input fields for the geometry variables.
Figure 2: Unit cell geometry for the CoBoGUI solar cell simulations. - The first contact point lies at the top left corner (the point of origin), and the second contact point is at the bottom left corner. The boundary parameters starting with “R”” denote sheet resistances and the parameters starting with “J0”” denote saturation saturatio current densities. Symmetry boundary conditions are applied for the left border and the right border. The unit cell that comes with the basic version of the CoBoGUI has two border points that are used as contacts: one at the top left corner and one at the bottom left corner. The boundary regions next to the contact points are denoted by a width in x-direction direction of xfc and xbc, respectively. The second letter “f” (“b”) denotes “front” (“back”) of the cell, and the letter “c” denotes “contacted region”. These se two boundary regions represent metalized surfaces of the solar cell, whereas the rest of the front and the back surface may be passivated. For the left border and the right border of the unit cell symmetry boundary conditions are used. Note that the width wid xw of the unit cell equals the half of the metallization finger pitch.
Geometry variables are specified either as constant values or as functions of other geometry variables. This enables for example batch simulations with constant metallization fraction tion but adapted unit cell width xw. 3.2 Physics The remaining model variables are specified using the input text-fields fields and combo-boxes combo that are available on the “Physics” tab. These variables, e.g. the saturation current density J0fc of the front contact cont region, may be specified either as fixed values or as MATLAB expressions that depend on other model variables. For example, the saturation current density J0fc might be a function of the sheet resistance Rfc, to mention one possible relation. If speciall applications require additional input fields on the “Geometry” or “Physics” tab, the configuration files and the simulation script file can be adapted by advanced users. For example, new useruser defined physical models like mobility-models, mobility etc., can be added ed to the simulation environment. This flexibility is one of the benefits of the CoBoGUI. CoBoGUI 3.3 Generation The “Generation” tab is used to load, view and export several generation profiles. Each generation profile is a function of only the depth coordinate coordinat y and can be applied to a specified x-region, region, e.g. [0, xfc], representing the region underneath neath the front contact.
Figure 3: Screenshot of the Graphical User Interface. The scope of the “Solve” tab includes the voltage range of the simulation, two batch parameter vectors, a simulation description and the path to a simulation data file. 3.4 Solution process Figure 3 shows the he “Solve” tab. tab It contains the voltage range off the simulation, combo boxes and text fields for the batch variables, as well as a text area where a description of the simulation can be defined. The user may also specify the path to a data file where the “dat” structure is saved. Finally the “Solve&Save” “Solve&Sav button runs the simulation. The progress of the solution process can be observed on the MATLAB command line, where the JV-characteristics characteristics and an estimated end time are shown. 3.5 Evaluation & 2D Plots The tab “Evaluation & 2D Plots” provides more than tha
25th European Photovoltaic Solar Energy Conference, Valencia, Spain, 06-10 September 2010, 2DV.1.19
50 predefined post processing functions that display the simulation results as a function of the batch parameters, as a function of the voltage or as a function of the space coordinates x and y. Special features of the CoBoGUI are the CLA and FELA functions (see section 5 and 6). The values of the batch variables span a “batch matrix”, and the post processing functions operate on a sub-matrix of this batch matrix. The user has two options: either, a default sub-matrix for all post processing function can be chosen, or a selection dialog for the selection of a sub-matrix opens up each time a post processing button is pressed. By default, all evaluations are performed at the maximum power point (MPP), however, also any other fixed voltage value can be specified for the evaluation. 3.6 1D Plots If a user likes to gain deep insight in the solution variables, we recommend the cross section plots of the “1D Plots” tab. It is possible to specify user defined plots or to use the supported default plots. These plots are comparable to the plots in PC1D. External data can be plotted as well. 3.7 Data Tools The “Data Tools” tab provides space for further buttons. It is possible to run your own MATLAB scripts from this tab. There are some predefined functions available that show how to control PC1D from the MATLAB environment. 3.8 Script Commands The last tab “Script Commands” has two text areas where MATLAB commands can be entered. With the package a script is provided that shows exemplarily how to access the “dat” structure using the MATLAB language.
4
CONDUCTIVE BOUNDARY MODEL
This section describes the basics of the simplified solar cell model that is controlled by the CoBoGUI. We implement the boundary conditions of the Conductive Boundary (CoBo) model [1] in COMSOL Multiphysics by specifying so called “weak terms” following a special syntax. 4.1 Solution variables The CoBo model includes three solution variables in the simulated semiconductor volume:
virtual front plane is given by a constant value
ϕFn
front
(1)
= Ua
where U a is the applied device voltage.
Figure 4: Schematic diagram of the CoBo model solution variables. - The boundary conditions for the quasi-Fermi potential of the electrons ϕ Fn , the quasiFermi potential of the holes ϕFp and the negative electrostatic potential ϕ i are specified at virtual planes (orange dashed lines). Another fundamental assumption of the CoBo model is that the gradient of ϕFp at the virtual front plane can be indirectly specified by a saturation current density J 0, front :
σp
ϕFn
d ϕFp dy
= jp, ⊥
front
front
−ϕ Fp
front
UT
= J 0, front (e
− 1)
(2)
front
where σ p is the conductivity for the holes, and jp, ⊥ is the normal component of the hole current density, flowing into the virtual front plane. The voltage U T = kB T q is the thermal voltage, calculated from Boltzmann constant k B , temperature T, and elementary charge q. The model parameter J 0, front includes the recombination properties at the virtual front plane. At the virtual back plane in turn the absolute value of ϕFp and the gradient of ϕ Fn are specified:
ϕFp
σn
d ϕFn dy
back
(3)
=0 ϕFn
= jn, ⊥ back
back
= J 0, back (e
back
−ϕFp
UT
back
− 1) (4) ,
• ϕ Fn : quasi-Fermi potential of the electrons, • ϕFp : quasi-Fermi potential of the holes, and • ϕ i : negative electrostatic potential. Figure 4 shows a schematic diagram of these solution variables for a one dimensional solar cell with a p-type base. The orange dashed lines mark virtual planes where the boundary conditions of the CoBo-model apply. 4.2 Boundary conditions The quasi-Fermi potential of the electrons is assumed to be constant throughout the emitter and the space charge region. The boundary condition for ϕ Fn at the
where σ n is the electron conductivity, jn, ⊥ is the normal component of the electron current density and J 0, back is a saturation current density that describes the recombination at the virtual back plane. The nomenclature “Conductive Boundary” is attributed to the features of the virtual planes which allow charge carrier conduction in x-direction, i.e. perpendicular to the plane of projection in Figure 4. The value of ϕ Fn at the virtual front plane is given by a continuity equation in x-direction of our two dimensional model.
25th European Photovoltaic Solar Energy Conference, Valencia, Spain, 06-10 September 2010, 2DV.1.19
We consider a boundary line element at a position x to explain this continuity equation. An electron current density jn, ⊥ ( x) flows perpendicular from the base into the line element. A current density jn, rec ( x) will be lost due to the recombination process at the virtual plane:
jn, ⊥ ( x ) − jn, rec ( x ) = ϕFn
front
jn, ⊥ + J 0, front (e
( x ) − ϕFp UT
(5)
( x) front
− 1)
The difference of these two current densities equals the divergence of the one dimensional current density in x-direction:
jn, ⊥ ( x ) − jn, rec ( x ) =
d 1 d ϕ Fn front ( x ) ( ) d x Rfront dx
(6)
The sheet resistance Rfront characterizes the integrated conductivity of the front diffusion. The vector notation of (5), the corresponding equation for the back surface field and the boundary conditions for ϕ i are found in [1]. In COMSOL Multiphysics, equation (6) can be implemented as boundary condition. For this purpose we transform the differential equation into its so called “weak form”. (The label “weak” might be misleading. In this context it rather means “more general” and the weak form is an alternative mathematical formulation of the differential equation.) This weak form might be of interest for advanced users of the CoBoGUI who wish to expand the physics of the model. The syntax of the weak form is explained in the appendix section 11.1.
5
7
COMPARISON WITH PC1D
A basic assumption of the CoBo model is that the quasi-Fermi potential of the electrons does not change throughout the emitter and the space charge region. Furthermore, the saturation current density that characterizes the recombination at a virtual plane is assumed to be independent of the applied device voltage. Both assumptions are only approximately fulfilled in reality. The CoBo model does not include the voltage dependent behavior of the space charge region, which might dominate the JV-curve at low voltages. However, the deviations of the JV-characteristics are less than 1 % when we compare simulations using the CoBoGUI and PC1D (see Figure 10 in section 11.3) for a quasi one-dimensional geometry.
8
SIMULATION EXAMPLES
The standard screen printing process used at ISFH currently produces solar cells with efficiencies of up to 17.9 %. Typical solar cell parameters of such cells are listed in Table II. Most of the data are directly measured. The value of the saturation current density at the front contact region, however, is an estimate. The base doping density is calculated from the measured resistivity using PC1D. The value of the saturation current density of the back contact region is estimated from a measured surface recombination velocity using the equation J 0back ≈
q ni 2 Seff, back NA
(7)
We calculate the generation profile by fitting a measured reflectance curve using the optical model from Ref. [3].
CURRENT LOSS ANALYSIS
After COMSOL Multiphysics has solved the CoBo model, the simulation script calculates Current density Loss Analysis (CLA) parameters. The calculated current densities characterize how much electron hole pairs are generated and how much electron hole pairs recombine in the base, in the front CoBo segments, and the back CoBo segments. The recombination current density “j_rfc” of the front contact region is calculated for example by integrating the term “1[m]*Jnrec/A” over the front contact region line segment, where “A” is the cell area. The CLA parameters are plotted as a function of the voltage for a single solar cell or as function of the batch parameters at the maximum power points.
6
FREE ENERGY LOSS ANALYSIS
The Free Energy Loss Analysis (FELA) [2] allows us to directly compare transport and recombination losses. It is possible to derive free energy dissipation rate densities in W/m2 for all recombination and transport processes. An alternative to the FELA derivation fom Ref. [2] is given in section 11.2. The CoBoGUI supports FELA plots as a function of the voltage for a single batch element or as function of the batch parameters at the MPPs.
Figure 5: Generation profile for the CoBoGUI simulations. - The cumulated generation of this profile corresponds to a current density of JG = 39.6 mA/cm2. We correct for the reflectance of the screen printed fingers that have an area fraction of 4 % (without busbars) before applying the fit routine. The generation profile is then scaled down according to the metallization fraction to get the total photogeneration right. Here the
25th European Photovoltaic Solar Energy Conference, Valencia, Spain, 06-10 September 2010, 2DV.1.19
generation rate is constant in x-direction. The fact that the CoBo model does not simulate the emitter or the space charge region is ignored for the generation profile: the profile from Figure 5 is applied with its maximum value at y = 0 as defined in Figure 4. The CoBoGUI parameters for the simulation A are listed in Table III and the corresponding results are plotted in Figure 7. The CoBoGUI simulation does not include resistances of the metallization pattern. The FELA pie chart in Figure 6 shows that the main loss mechanism of simulation A is the recombination at the back (“rbc”) with 50 % of the total free energy loss rate. Table I: Typical standard solar cell parameters and characteristics of a single experimental cell. Geometry Thickness yw = 150…160 µm Finger pitch 2 xw = 2.5 mm Finger width 2 xfc = 100…130 µm Electronic parameters Bulk lifetime at an injection level of 1015 1/cm3 before cell process ߬ = 50…150 µs Emitter saturation current density J0e = 175…240 fA/cm2 Emitter sheet resistance Re = 50…60 Ω/ □ Effective back surface recombination velocity: Seff,bsf = 400…500 cm/s Specific base resistance ρ = 2…2.5 Ω cm Characteristics of the best cell Efficiency ߟ = 17.9 % Fill factor FF = 78.4 % Open circuit voltage Voc = 0.628 V Short circuit current density Jsc = -36.4 mA/cm2 Series resistance Rs = 0.7 Ω cm2 Table II: CoBoGUI parameters for simulation A. - The back of the solar cell is fully metalized. Geometry Unit cell height yw = 150 µm Unit cell width xw = 1250 µm Front contact region width xfc = 50 µm Back contact region width (only specified for the sake of completeness; there are constant properties at the back) xbc = 1249 µm Electronic parameters Base lifetime taun = taup = 100 µs Base doping density NA = 6.5 x 1015 cm-3 Fixed electron mobility: 1167 cm2/Vs Fixed hole mobility: 435 cm2/Vs Front saturation current density J0f = 230 fA/cm2 Front sheet resistance Rf = 50 Ω/ □ Front contact saturation current density J0fc = 1000 fA/cm2 Front contact sheet resistance Rfc = 1e-6 Ω/ □ Back saturation current density J0bc = 990 fA/cm2 Back sheet resistance Rbc = 1e-6 Ω/ □ Back contact saturation current density J0bc = 990 fA/cm2 Back contact sheet resistance Rbc = 1e-6 Ω/ □ Table III: CoBoGUI parameters for simulation B. – The solar cell has local back contacts. Geometry Unit cell height yw = 150 µm Unit cell width xw = 1250 µm Front contact region width xfc = 50 µm
Back contact region width xbc = 200 µm Electronic parameters Base lifetime taun = taup = 100 µs Base doping density NA = 6.5 x 1015 cm-3 Fixed electron mobility: 1167 cm2/Vs Fixed hole mobility: 435 cm2/Vs Front saturation current density J0f = 230 fA/cm2 Front sheet resistance Rf = 50 Ω/ □ Front contact saturation current density J0fc = 1000 fA/cm2 Front contact sheet resistance Rfc = 1e-6 Ω/ □ Back saturation current density J0b = 130 fA/cm2 Back sheet resistance Rb = 1e10 Ω/ □ Back contact saturation current density J0bc = 1200 fA/cm2 Back contact sheet resistance Rbc = 1e-6 Ω/ □
Figure 6: FELA pie chart and JV-curve for a standard solar cell simulation A. The Free Energy Loss Analysis chart includes the recombination loss of the back contact region “rbc”, the Shockley Read Hall recombination loss of the base “rsrh”, the transport loss of the passivated front region “tf”, the recombination loss of the passivated front region “rf”, the electron transport loss of the base “te”, the hole transport loss of the base “th” and a combination of other free energy loss rates “others”. The simulation parameters are listed in Table II. The simulation does not include resistances of the metallization pattern. The curve is a spline through the symbols. The second most important loss mechanism is the Shockley Read Hall recombination in the base (“rsrh”) acconting for 20 % of all losses, followed by the transport loss in the emitter (“tf”) with 17 %, the recombination loss in the emitter (“rf”), the transport loss of the electrons in the base (“te”) and the transport loss of the holes in the base (“th”). The simulated fill factor of 82 % neglects resistive losses in the metallization fingers. The simulated efficiency is 19.8 %. Adding a lumped series resistance of 0.6 Ωcm2 would decrease the fill factor to 78.6 %, which is similar to the fill factor of the experimental cell. The JV-characteristics of the experimental cell are listed in Table I. The short circuit current density of the simulation A exceeds that of the experimental cell by 7 %. This may be due to the busbars and the boundary effects of the real solar cell. Another probable
25th European Photovoltaic Solar Energy Conference, Valencia, Spain, 06-10 September 2010, 2DV.1.19
explanation is the Auger recombination loss of the generation current in the emitter which is not included in the CoBo model.
of the bulk that are far away from the rear contact. The transport loss “tf” gets less important. This shows that the rear contact design imposes requirements on the design of the front side.
Figure 7: FELA pie chart and JV-curve for simulation B with local back contacts. - The Free Energy Loss Analysis chart includes the hole transport loss of the base “th”, the Shockley Read Hall recombination loss of the base “rsrh”, the recombination loss of the passivated front region “rf”, the transport loss of the passivated front “tf”, the recombination loss at the back contact region “rbc”, the recombination loss at the passivated back region “rb”, the electron transport loss of the base “te” and a combination of other free energy loss rates “others”. The simulation parameters are listed in Table III. The simulation does not include resistances of the metallization pattern. The curve is a spline through the symbols.
Figure 8: Efficiency and fill factor as functions of the back contact width xbc and the base doping density NA. - The batch simulation does not include resistances of the screen printed metallization pattern. The curve is a spline through the symbols. The unit cell geometry of the simulation is shown in Figure 2.
For the purpose of comparison, simulation B (see Figure 7 and Table III) uses a different back contact design. A large fraction of the back surface is assumed to be passivated and to have a saturation current density of 130 fA/cm2. A smaller fraction is locally contacted. The saturation current density of the contact is assumed to be 1200 fA/cm2. A possible impact of the passivated back region on the generation profile is neglected. For didactical purpose we chose for simulation B the width of the passivated region to have the same cell efficiency as in simulation A. The different rear design results in a different pattern of losses as is shown by comparing the FELA pie chart in Figure 7 with that in Figure 6. The new main loss mechanism now is the transport of the holes in the base (“th”) with 35 %, followed by the Shockley Read Hall recombination in the base (“rsrh”) with 24 %. The recombination loss of the passivated front region (“rf”) is 14 % of the total loss rate of free energy. A solar cell with local back contacts causes longer transport paths for the holes in a p-type base, compared to the paths in a solar cell with a fully contacted back. Additionally, a partially passivated rear side reduces rear side recombination. Therefore, the hole transport losses and at the recombination losses in the volume increase, and the recombination at the back of the cell decreases from simulation A to simulation B. The change from 6 % to 14 % of front side recombination “rf” from simulation A to B is also due to an enhanced quasi-Fermi level splitting in those regions
Figure 9: Free Energy Loss Analysis (FELA) parameters as functions of the back contact width xbc for a base doping density of NA = 6.5 x 1015 cm-3. - Shockley Read Hall recombination loss “rsrh”, Auger recombination loss “raug”, band to band recombination loss “rbb”, hole transport loss “th”, electron transport loss “te”, recombination loss of the front contact region “rfc”, recombination loss of the front region “rf”, transport loss of the front region “tf”, recombination loss of the back contact region “rbc”, and recombination of the back regeion “rb”. Figure 8 shows the efficiency and the fill factor of a batch simulation, where the back contact width xbc and the base doping density NA are varied. The CoBoGUI is
25th European Photovoltaic Solar Energy Conference, Valencia, Spain, 06-10 September 2010, 2DV.1.19
able to vary two batch parameters in a single batch simulation and display the results as an array of curves. A mobility model from PC1D is used to adjust the mobilites of the batch simulation according to the NA values. Possible influences of the base doping on saturation current densities are neglected here but may easily be implemented since the GUI allows for using functions of some parameters as input for other parameters. The variation of NA corresponds to a small variation of the base resistivity of ρ = 2…2.5 Ω cm. The simulations predict that this variation has little influence on the efficiency. The unit cell of the presented basic version of the CoBoGUI has a single back contact. The model yields the largest efficiencies at rather wide contact regions that cover 40 % of the rear side of the cell. Free Energy Loss Analysis plots as a function of the batch parameter xbc are shown in Figure 9 for a doping density of NA = 6.5 x 1015 cm-3. The data of Simulation B correspond to xbc = 200 µm. Figure 8 shows that the efficiency and the fill factor increase if the back contact half width xbc increases from 10 µm to approximately 400 µm. If xbc is further increased, the back contact recombination losses become larger than the hole transport losses (see Figure 9) and the efficiency decreases. A free energy dissipation rate density of f = 10 W/m2 corresponds to one percent of the incident light power of 1000 W/m2. The losses due to Auger recombination and band to band recombination are negligibly small, compared to the other losses in Figure 9.
This equation is multiplied by another scalar function v(u) and integrated over a domain Ω:
∫ v j d L = Ω∫ −v div [c grad(u)] d L
If the domain Ω is a surface, the differential dL would be a small area element and if the domain ∂Ω is a line segment, the differential dL would be a small length element. Additionally, we assume that we can apply the divergence theorem using the domain boundary ∂Ω:
∫ v j d L = Ω∫ grad(v) [c grad(u)] d L +
Ω
(10)
∫ −v [c grad(u)] dP ∂Ω
If the domain Ω is a line element, the integral over ∂Ω equals the summation over the initial point and the final point of the line element. The vector dP is perpendicular to the boundary. With the boundary normal vector n, equation (11) can be written as:
∫ −grad(v) [c grad(u)] + v j d L + Ω
∫ v n [c grad(u)] d P = 0
(11)
∂Ω
We define the boundary condition − n [ c grad( u ) ] = µ
9
(9)
Ω
(12)
SUMMARY for ∂ Ω , and obtain the weak form of equation (8):
The CoBoGUI was designed to offer a flexible Graphical User Interface for the Conductive Boundary model and is realized using MATLAB and COMSOL Multiphysics. The Free Energy Loss Analysis (FELA) helps to identify the main loss mechanisms of solar cells. The CoBoGUI is a tool for device optimization that is numerically not very demanding, since it does not spatially resolve the space charge region. The code package may be downloaded for free from [5]. The model shows limitations, whenever the transport properties in the space charge region dominate the cell behavior.
10 ACKNOLEDGEMENTS We thank B. Beier and T. Dullweber for the experimental cell data. Funding was provided by the State of Lower Saxony.
11 APPENDIXES 11.1 COMSOL Multiphysics weak form To explain the syntax of the weak form, we define an arbitrary non-homogeneous partial differential equation including the scalar functions u(x), j (u), and a constant c: j = − div [ c grad(u ) ]
(8)
0 = ∫ − c grad [ v(u )] grad(u ) + v(u ) j (u ) d L (13)
Ω
−
∫ v(u ) µ d P ∂Ω
More information about weak forms can be found in [4]. The equation (13) is implemented by entering a term called “weak” following a special syntax. The function v(u) is called “test function” and is obtained with the “test” operator: “test(u)”. The term grad[v(u)] is entered by applying the test operator to the two components “ux” and “uy” of the gradient of u for two dimensional simulations: grad[v(u)] => “test(ux) + test(uy)”. Here, we specify a weak term for a solution variable u on a one dimensional domain boundary. The variable u also exists in the two dimensional domain itself. The domain source term jdomain (u ) that flows perpendicular into the boundary is automatically included in the equation system. Therefore, the weak term of the boundary only includes j − jdomain and not j . The so called “Lagrange multiplier” µ is also automatically added to the equation system; hence we do not have to explicitly specify the second integral. Therefore, the “weak” term syntax of equation (10) in COMSOL Multiphysics is: weak = - c * ( test(ux) * ux + test(uy) * uy ) + test(u) * (j- jdomain)
25th European Photovoltaic Solar Energy Conference, Valencia, Spain, 06-10 September 2010, 2DV.1.19
The term (j-jdomain) is zero if the boundary itself has no additional source term. The weak term for the front CoBo equation (5) reads: weak = 1/Rfront * ( test(phiFnTx) * phiFnTx + test(phiFnTy) * phiFnTy ) + test(phiFn) * (-Jnrec) This weak term also works for boundaries that are not parallel to the x-direction since the tangential components (“uTx” instead of “ux”) are used. 11.2 FREE ENERGY LOSS ANALYSIS As an alternative to the derivation in [2] the FELA may be derived as follows: If ∆N particles are transferred from an electrochemical potential µ 1 to an electrochemical potential µ 2, the change of the free energy is
dF = ∆N (µ 2 - µ 1).
(14)
dF ≈ R dA d x q (ϕ Fn − ϕ Fp ) . dt
Thus, it is possible to derive free energy rate densities with the unit W/m2 for all generation / recombination processes and all transport processes in the solar cell, and compare them in a single graph. Please see reference [2] for more details on the Free Energy Loss Analysis. The CoBoGUI supports FELA plots as a function of the voltage for a single batch element or as function of the batch parameters at the MPPs. 11.3 Comparison with PC1D Table IV lists the parameters of a PC1D simulation that we run for a comparison with the CoBo model. We extract a “global saturation current density” J0global from a PC1D simulation by setting the lifetime of the base to a high value of ߬n = ߬p = 1 s and using the equation J sc
J 0global =
For a transportation process during a time dt, with
e
∆N = −
dA jn ( x ) d t q
(15)
(19)
Voc UT
(20)
−1
where J sc is the short circuit current density and Voc is the open circuit voltage of the PC1D simulation.
electrons moving from µ 1 = µ n(x) to µ 2 = µ n(x + dx) through a cross section dA by virtue of a current density jn , the free energy rate can be written as ∂ µ ( x) dF dA ≈− jn ( x) n dx dt q ∂x
(16)
We state that the electrochemical potential of the electrons equals the quasi-Fermi potential ϕ Fn times the elementary charge:
dF ∂ϕ ≈ − dA jn ( x) Fn d x dt ∂x 2
∂ϕ ≈ − dA σ n Fn d x ∂x 1 2 ≈ − dA jn d x
σn
(17a)
(17b)
(17c)
This rate is proportional to the square of the gradient of the quasi-Fermi potential. Equation (17c) is related to the well known equation P = R I2, where P represents the power, R the resistance and I the current of a model circuit. As an example for a FELA parameter we use the free energy loss rate density caused by electron transport within the front contact region, which is given as: tfc = −
1 Acell
xfc
∫0
2
1 ∂ϕ Fn dx . Rfc ∂ x
(18)
For a recombination process the particle rate and the change of the electrochemical potential can be written in terms of a recombination rate R, a volume dV = dA dx, and the quasi-Fermi potentials:
Figure 10: Comparison of the JV-curves of the CoBoGUI and PC1D. - The JV-characteristics agree within 1 %. The CoBo model (red stars) cannot reproduce the PC1D JV-curve (blue circles) at low voltages. The extracted global saturation current density is J0global = 170 fA/cm2. The CoBo simulation then uses this saturation current density value in the front side. The recombination at the rear side is set to zero in both, the PC1D and the CoBo simulation. Figure 10 shows the absolute values of the current density obtained using PC1D and the CoBoGUI, respectively, shifted by the corresponding J sc value and plotted in a semi-log-graph as function of the voltage. At small voltages the J-values of the CoBo model deviate from the values calculated with PC1D by up to 10−2 mA/cm2. The CoBo model does not account for the
25th European Photovoltaic Solar Energy Conference, Valencia, Spain, 06-10 September 2010, 2DV.1.19
voltage dependent behavior of the space charge region. However, the deviations of the characteristic values of the illuminated JV-curve are less than 1 %.
Table IV: PC1D simulation parameters Device Device area: 1 cm2 No surface texturing No surface charge No Exterior Front Reflectance No Exterior Rear Reflectance No internal optical reflectance Emitter contact enabled Base contact enabled No internal shunt elements Region 1 Thickness: 200 µm Material modified from si.mat Fixed electron mobility: 1167 cm2/Vs Fixed hole mobility: 435 cm2/Vs Dielectric constant: 11.9 Band gap: 1.124 eV Intrinsic conc. At 300 K: 1x1010 cm-3 Refractive index: 3.58 Absorption coeff. from internal model No free carrier absorption P-type background doping: 6.5x1015 cm-3 1st front diff. N-type, 1.3x1020 cm-3 peak; erfc; 0.015 µm depth factor, 0.05 µm peak position 2nd front diff. N-type, 3x1019 cm-3 peak; gauss; 0.13 µm depth factor, 0.08 µm peak position No rear diffusion Bulk recombination: ߬n = ߬p = 100 µs No Front-surface recombination No Rear-surface recombination Excitation Excitation mode: Transient, 60 timesteps Temperature: 300 K Base circuit: Sweep from -0.2 to 0.7 V Collector circuit: Zero Photogeneration from cumgen_standard.gen Results Short-circuit Ib: -0.0403 amps Max base power out: 0.0216 watts Open-circuit Vb: 0.6465 volts
REFERENCES [1] R. Brendel, Prog. Photovolt: Res. Appl. (2010), DOI: 10.1002/pip.954 [2] R. Brendel, S. Dreissigacker, N.-P. Harder, and P.P. Altermatt, Appl. Phys. Lett., Vol. 93 (2009) 173503, DOI: 10.1063/1.3006053 [3] R. Brendel, M. Hirsch, R. Plieninger, and J. H. Werner, IEEE Trans. Electron Devices, Vol. 43, No. 7 (1996) 1104, DOI: 10.1109/16.502422 [4] COMSOL Multiphysics Reference Guide, The Weak Form, Version 3.5a (2009) [5] CoBoGUI v1.0 (2010) www.isfh.de/institut_solarforschung/software.php
OPEN SOURCE GRAPHICAL GRAPHICA USER INTERFACE IN MATLAB FOR TWO--DIMENSIONAL SIMULATIONS SOLVING THE FULLY COUPLED SEMICONDUCTOR SEMICO EQUATIONS USING ING COMSOL S. Eidelloth1, U. Eitner1, S. Steingrube1, and R. Brendel1,2 Institute for Solar Energy Research Hamelin (ISFH), Am Ohrberg 1, 31860 Emmerthal, Germany Phone: hone: +49(0)5151 999 - 315, Fax: - 400, eMail: [email protected] 2 Institute of Solid State Physics, Leibniz University Hanover, Appelstr. 2, 30167 Hanover, Hanover Germany 1
ABSTRACT: Numerical modeling is useful to understand and improve the device performance of solar cells. The freely available simulation software PC1D is commonly used to model one-dimensional one dimensional semiconductor structures and provides a convenient Graphical User Interface (GUI). Current research, research however, relies on the improvement of twotwo or three-dimensional dimensional cell structures. A free software package for two-dimensional two dimensional cell simulations on an ordinary PC with a convenient GUI is currently not available. The purpose of this work is to close this gap by providing provi such an Open Source GUI. Our GUI can be used to specify solar cell model parameters, to evaluate the simulation data and to save the results in common file formats. The appearance of the GUI can be adapted by modifying the configuration Excel files that at specify the user input panels. Our software solves the fully coupled semiconductor transport equations using COMSOL Multiphysics. The underlying physical model considers diffused layers as conductive boundaries which are characterized by their sheet resistance resistance and by their diode saturation current density. This approach reduces the numerical demands since the space charge regions are not spatially resolved and hence, the number of required mesh points decreases. Keywords: Modeling, Simulation, Software
1
INDRODUCTION
The recently introduced Conductive nductive Boundary (CoBo) model [1] characterizes diffused layers by their sheet resistance and diode saturation current density. This model strongly reduces the numerical demands of two-dimensional dimensional semiconductor simulations. Here, we report on a flexible Graphical User Interface for the CoBo model (CoBoGUI CoBoGUI), which can be downloaded for free from our website www.isfh.de. The purpose of the CoBoGUI is not only to provide a user friendlyy interface to specify the CoBo solar cell model, but also to enable batch simulations, to support the user with predefined post processing functions and display or export the results as a function of the batch parameters. Our flexible software concept enables ena batch simulations for any model that is controlled by a MATLAB function. For example, at ISFH we use the GUI for simulations of thermal boron diffusion profiles where another system of differential equations is solved. In this paper, however, we focus on two-dimensional modeling of the transport of electrons and holes in solar cells. The commercial applications MATLAB and COMSOL Multiphysics are required to run the CoBoGUI.. Please feel free to adapt our open source code package for your own needs under er reference to this paper.
2
SOFTWARE ARCHITECTURE
The software architecture of the CoBoGUI is shown in Figure 1. MATLAB reads configuration files {A} at the program start-up. up. These files contain information about the model variables and the post processing proc functions. After reading the configuration files, MATLAB creates the GUI {B}. The user can specify the simulation parameters and finally start the simulation. The simulation parameters are passed to a MATLAB script
that creates so-called called “FEM structures” stru (FEM: Finite Element Method) ethod) for all batch elements. Each FEM structure contains all information for a single solar cell simulation and is passed to COMSOL Multiphysics {C}. In case of solar cell modeling, COMSOL Multiphysics solves the coupled semiconductor equations with a Finite Element Method approach, adds the solution to the FEM structure and returns it back to MATLAB.
Figure 1: Software architecture of the CoBoGUI. - The interaction of MATLAB with four other components {A}-{D} enables a user friendly simulation environment. MATLAB analyses the JV--curve, performs the CLA (Current density Loss Analysis nalysis, see section 5), and the FELA (Free Energy Loss Analysis nalysis [2]). [2] All information of the batch simulation, including the results of the loss analyses, is stored in a data structure named “dat”. To gain specific information from this data structure, the user can choose from a large amount of predefined post processing functions within the GUI. Alternatively,
25th European Photovoltaic Solar Energy Conference, Valencia, Spain, 06-10 September 2010, 20 2DV.1.19
the advanced user may use any MATLAB function to access the data structure from the MATLAB command line. By default, the results are displayed as MATLAB figures, but they can also be passed to interfaces {D} to obtain other output file formats, such as Excel or SigmaPlot files. All connections between MATLAB and the components {A}-{D} {D} are bidirectional. For example it is possible to save user defined options of cross section plots as a configuration file ({B} => MATLAB => {A}). MATLAB and COMSOL Multiphysics are platformplatform independent dent and so are the JAVA objects used for the GUI. The interface component {D} is the only part of the software architecture that is platform dependent. If you want to use the CoBoGUI on other operation systems than Windows, you might have to adapt the corresponding interface script files.
3
GRAPHICAL USER INTERFACE
In the GUI, the simulation workflow is organized in separate tabs. Thereby, each tab represents a single step in the course of the simulation, as described in the following subsections. 3.1 Geometry The first tab is the “Geometry” tab. It shows a schematic of the predefined unit cell (see Figure 2), and provides input fields for the geometry variables.
Figure 2: Unit cell geometry for the CoBoGUI solar cell simulations. - The first contact point lies at the top left corner (the point of origin), and the second contact point is at the bottom left corner. The boundary parameters starting with “R”” denote sheet resistances and the parameters starting with “J0”” denote saturation saturatio current densities. Symmetry boundary conditions are applied for the left border and the right border. The unit cell that comes with the basic version of the CoBoGUI has two border points that are used as contacts: one at the top left corner and one at the bottom left corner. The boundary regions next to the contact points are denoted by a width in x-direction direction of xfc and xbc, respectively. The second letter “f” (“b”) denotes “front” (“back”) of the cell, and the letter “c” denotes “contacted region”. These se two boundary regions represent metalized surfaces of the solar cell, whereas the rest of the front and the back surface may be passivated. For the left border and the right border of the unit cell symmetry boundary conditions are used. Note that the width wid xw of the unit cell equals the half of the metallization finger pitch.
Geometry variables are specified either as constant values or as functions of other geometry variables. This enables for example batch simulations with constant metallization fraction tion but adapted unit cell width xw. 3.2 Physics The remaining model variables are specified using the input text-fields fields and combo-boxes combo that are available on the “Physics” tab. These variables, e.g. the saturation current density J0fc of the front contact cont region, may be specified either as fixed values or as MATLAB expressions that depend on other model variables. For example, the saturation current density J0fc might be a function of the sheet resistance Rfc, to mention one possible relation. If speciall applications require additional input fields on the “Geometry” or “Physics” tab, the configuration files and the simulation script file can be adapted by advanced users. For example, new useruser defined physical models like mobility-models, mobility etc., can be added ed to the simulation environment. This flexibility is one of the benefits of the CoBoGUI. CoBoGUI 3.3 Generation The “Generation” tab is used to load, view and export several generation profiles. Each generation profile is a function of only the depth coordinate coordinat y and can be applied to a specified x-region, region, e.g. [0, xfc], representing the region underneath neath the front contact.
Figure 3: Screenshot of the Graphical User Interface. The scope of the “Solve” tab includes the voltage range of the simulation, two batch parameter vectors, a simulation description and the path to a simulation data file. 3.4 Solution process Figure 3 shows the he “Solve” tab. tab It contains the voltage range off the simulation, combo boxes and text fields for the batch variables, as well as a text area where a description of the simulation can be defined. The user may also specify the path to a data file where the “dat” structure is saved. Finally the “Solve&Save” “Solve&Sav button runs the simulation. The progress of the solution process can be observed on the MATLAB command line, where the JV-characteristics characteristics and an estimated end time are shown. 3.5 Evaluation & 2D Plots The tab “Evaluation & 2D Plots” provides more than tha
25th European Photovoltaic Solar Energy Conference, Valencia, Spain, 06-10 September 2010, 2DV.1.19
50 predefined post processing functions that display the simulation results as a function of the batch parameters, as a function of the voltage or as a function of the space coordinates x and y. Special features of the CoBoGUI are the CLA and FELA functions (see section 5 and 6). The values of the batch variables span a “batch matrix”, and the post processing functions operate on a sub-matrix of this batch matrix. The user has two options: either, a default sub-matrix for all post processing function can be chosen, or a selection dialog for the selection of a sub-matrix opens up each time a post processing button is pressed. By default, all evaluations are performed at the maximum power point (MPP), however, also any other fixed voltage value can be specified for the evaluation. 3.6 1D Plots If a user likes to gain deep insight in the solution variables, we recommend the cross section plots of the “1D Plots” tab. It is possible to specify user defined plots or to use the supported default plots. These plots are comparable to the plots in PC1D. External data can be plotted as well. 3.7 Data Tools The “Data Tools” tab provides space for further buttons. It is possible to run your own MATLAB scripts from this tab. There are some predefined functions available that show how to control PC1D from the MATLAB environment. 3.8 Script Commands The last tab “Script Commands” has two text areas where MATLAB commands can be entered. With the package a script is provided that shows exemplarily how to access the “dat” structure using the MATLAB language.
4
CONDUCTIVE BOUNDARY MODEL
This section describes the basics of the simplified solar cell model that is controlled by the CoBoGUI. We implement the boundary conditions of the Conductive Boundary (CoBo) model [1] in COMSOL Multiphysics by specifying so called “weak terms” following a special syntax. 4.1 Solution variables The CoBo model includes three solution variables in the simulated semiconductor volume:
virtual front plane is given by a constant value
ϕFn
front
(1)
= Ua
where U a is the applied device voltage.
Figure 4: Schematic diagram of the CoBo model solution variables. - The boundary conditions for the quasi-Fermi potential of the electrons ϕ Fn , the quasiFermi potential of the holes ϕFp and the negative electrostatic potential ϕ i are specified at virtual planes (orange dashed lines). Another fundamental assumption of the CoBo model is that the gradient of ϕFp at the virtual front plane can be indirectly specified by a saturation current density J 0, front :
σp
ϕFn
d ϕFp dy
= jp, ⊥
front
front
−ϕ Fp
front
UT
= J 0, front (e
− 1)
(2)
front
where σ p is the conductivity for the holes, and jp, ⊥ is the normal component of the hole current density, flowing into the virtual front plane. The voltage U T = kB T q is the thermal voltage, calculated from Boltzmann constant k B , temperature T, and elementary charge q. The model parameter J 0, front includes the recombination properties at the virtual front plane. At the virtual back plane in turn the absolute value of ϕFp and the gradient of ϕ Fn are specified:
ϕFp
σn
d ϕFn dy
back
(3)
=0 ϕFn
= jn, ⊥ back
back
= J 0, back (e
back
−ϕFp
UT
back
− 1) (4) ,
• ϕ Fn : quasi-Fermi potential of the electrons, • ϕFp : quasi-Fermi potential of the holes, and • ϕ i : negative electrostatic potential. Figure 4 shows a schematic diagram of these solution variables for a one dimensional solar cell with a p-type base. The orange dashed lines mark virtual planes where the boundary conditions of the CoBo-model apply. 4.2 Boundary conditions The quasi-Fermi potential of the electrons is assumed to be constant throughout the emitter and the space charge region. The boundary condition for ϕ Fn at the
where σ n is the electron conductivity, jn, ⊥ is the normal component of the electron current density and J 0, back is a saturation current density that describes the recombination at the virtual back plane. The nomenclature “Conductive Boundary” is attributed to the features of the virtual planes which allow charge carrier conduction in x-direction, i.e. perpendicular to the plane of projection in Figure 4. The value of ϕ Fn at the virtual front plane is given by a continuity equation in x-direction of our two dimensional model.
25th European Photovoltaic Solar Energy Conference, Valencia, Spain, 06-10 September 2010, 2DV.1.19
We consider a boundary line element at a position x to explain this continuity equation. An electron current density jn, ⊥ ( x) flows perpendicular from the base into the line element. A current density jn, rec ( x) will be lost due to the recombination process at the virtual plane:
jn, ⊥ ( x ) − jn, rec ( x ) = ϕFn
front
jn, ⊥ + J 0, front (e
( x ) − ϕFp UT
(5)
( x) front
− 1)
The difference of these two current densities equals the divergence of the one dimensional current density in x-direction:
jn, ⊥ ( x ) − jn, rec ( x ) =
d 1 d ϕ Fn front ( x ) ( ) d x Rfront dx
(6)
The sheet resistance Rfront characterizes the integrated conductivity of the front diffusion. The vector notation of (5), the corresponding equation for the back surface field and the boundary conditions for ϕ i are found in [1]. In COMSOL Multiphysics, equation (6) can be implemented as boundary condition. For this purpose we transform the differential equation into its so called “weak form”. (The label “weak” might be misleading. In this context it rather means “more general” and the weak form is an alternative mathematical formulation of the differential equation.) This weak form might be of interest for advanced users of the CoBoGUI who wish to expand the physics of the model. The syntax of the weak form is explained in the appendix section 11.1.
5
7
COMPARISON WITH PC1D
A basic assumption of the CoBo model is that the quasi-Fermi potential of the electrons does not change throughout the emitter and the space charge region. Furthermore, the saturation current density that characterizes the recombination at a virtual plane is assumed to be independent of the applied device voltage. Both assumptions are only approximately fulfilled in reality. The CoBo model does not include the voltage dependent behavior of the space charge region, which might dominate the JV-curve at low voltages. However, the deviations of the JV-characteristics are less than 1 % when we compare simulations using the CoBoGUI and PC1D (see Figure 10 in section 11.3) for a quasi one-dimensional geometry.
8
SIMULATION EXAMPLES
The standard screen printing process used at ISFH currently produces solar cells with efficiencies of up to 17.9 %. Typical solar cell parameters of such cells are listed in Table II. Most of the data are directly measured. The value of the saturation current density at the front contact region, however, is an estimate. The base doping density is calculated from the measured resistivity using PC1D. The value of the saturation current density of the back contact region is estimated from a measured surface recombination velocity using the equation J 0back ≈
q ni 2 Seff, back NA
(7)
We calculate the generation profile by fitting a measured reflectance curve using the optical model from Ref. [3].
CURRENT LOSS ANALYSIS
After COMSOL Multiphysics has solved the CoBo model, the simulation script calculates Current density Loss Analysis (CLA) parameters. The calculated current densities characterize how much electron hole pairs are generated and how much electron hole pairs recombine in the base, in the front CoBo segments, and the back CoBo segments. The recombination current density “j_rfc” of the front contact region is calculated for example by integrating the term “1[m]*Jnrec/A” over the front contact region line segment, where “A” is the cell area. The CLA parameters are plotted as a function of the voltage for a single solar cell or as function of the batch parameters at the maximum power points.
6
FREE ENERGY LOSS ANALYSIS
The Free Energy Loss Analysis (FELA) [2] allows us to directly compare transport and recombination losses. It is possible to derive free energy dissipation rate densities in W/m2 for all recombination and transport processes. An alternative to the FELA derivation fom Ref. [2] is given in section 11.2. The CoBoGUI supports FELA plots as a function of the voltage for a single batch element or as function of the batch parameters at the MPPs.
Figure 5: Generation profile for the CoBoGUI simulations. - The cumulated generation of this profile corresponds to a current density of JG = 39.6 mA/cm2. We correct for the reflectance of the screen printed fingers that have an area fraction of 4 % (without busbars) before applying the fit routine. The generation profile is then scaled down according to the metallization fraction to get the total photogeneration right. Here the
25th European Photovoltaic Solar Energy Conference, Valencia, Spain, 06-10 September 2010, 2DV.1.19
generation rate is constant in x-direction. The fact that the CoBo model does not simulate the emitter or the space charge region is ignored for the generation profile: the profile from Figure 5 is applied with its maximum value at y = 0 as defined in Figure 4. The CoBoGUI parameters for the simulation A are listed in Table III and the corresponding results are plotted in Figure 7. The CoBoGUI simulation does not include resistances of the metallization pattern. The FELA pie chart in Figure 6 shows that the main loss mechanism of simulation A is the recombination at the back (“rbc”) with 50 % of the total free energy loss rate. Table I: Typical standard solar cell parameters and characteristics of a single experimental cell. Geometry Thickness yw = 150…160 µm Finger pitch 2 xw = 2.5 mm Finger width 2 xfc = 100…130 µm Electronic parameters Bulk lifetime at an injection level of 1015 1/cm3 before cell process ߬ = 50…150 µs Emitter saturation current density J0e = 175…240 fA/cm2 Emitter sheet resistance Re = 50…60 Ω/ □ Effective back surface recombination velocity: Seff,bsf = 400…500 cm/s Specific base resistance ρ = 2…2.5 Ω cm Characteristics of the best cell Efficiency ߟ = 17.9 % Fill factor FF = 78.4 % Open circuit voltage Voc = 0.628 V Short circuit current density Jsc = -36.4 mA/cm2 Series resistance Rs = 0.7 Ω cm2 Table II: CoBoGUI parameters for simulation A. - The back of the solar cell is fully metalized. Geometry Unit cell height yw = 150 µm Unit cell width xw = 1250 µm Front contact region width xfc = 50 µm Back contact region width (only specified for the sake of completeness; there are constant properties at the back) xbc = 1249 µm Electronic parameters Base lifetime taun = taup = 100 µs Base doping density NA = 6.5 x 1015 cm-3 Fixed electron mobility: 1167 cm2/Vs Fixed hole mobility: 435 cm2/Vs Front saturation current density J0f = 230 fA/cm2 Front sheet resistance Rf = 50 Ω/ □ Front contact saturation current density J0fc = 1000 fA/cm2 Front contact sheet resistance Rfc = 1e-6 Ω/ □ Back saturation current density J0bc = 990 fA/cm2 Back sheet resistance Rbc = 1e-6 Ω/ □ Back contact saturation current density J0bc = 990 fA/cm2 Back contact sheet resistance Rbc = 1e-6 Ω/ □ Table III: CoBoGUI parameters for simulation B. – The solar cell has local back contacts. Geometry Unit cell height yw = 150 µm Unit cell width xw = 1250 µm Front contact region width xfc = 50 µm
Back contact region width xbc = 200 µm Electronic parameters Base lifetime taun = taup = 100 µs Base doping density NA = 6.5 x 1015 cm-3 Fixed electron mobility: 1167 cm2/Vs Fixed hole mobility: 435 cm2/Vs Front saturation current density J0f = 230 fA/cm2 Front sheet resistance Rf = 50 Ω/ □ Front contact saturation current density J0fc = 1000 fA/cm2 Front contact sheet resistance Rfc = 1e-6 Ω/ □ Back saturation current density J0b = 130 fA/cm2 Back sheet resistance Rb = 1e10 Ω/ □ Back contact saturation current density J0bc = 1200 fA/cm2 Back contact sheet resistance Rbc = 1e-6 Ω/ □
Figure 6: FELA pie chart and JV-curve for a standard solar cell simulation A. The Free Energy Loss Analysis chart includes the recombination loss of the back contact region “rbc”, the Shockley Read Hall recombination loss of the base “rsrh”, the transport loss of the passivated front region “tf”, the recombination loss of the passivated front region “rf”, the electron transport loss of the base “te”, the hole transport loss of the base “th” and a combination of other free energy loss rates “others”. The simulation parameters are listed in Table II. The simulation does not include resistances of the metallization pattern. The curve is a spline through the symbols. The second most important loss mechanism is the Shockley Read Hall recombination in the base (“rsrh”) acconting for 20 % of all losses, followed by the transport loss in the emitter (“tf”) with 17 %, the recombination loss in the emitter (“rf”), the transport loss of the electrons in the base (“te”) and the transport loss of the holes in the base (“th”). The simulated fill factor of 82 % neglects resistive losses in the metallization fingers. The simulated efficiency is 19.8 %. Adding a lumped series resistance of 0.6 Ωcm2 would decrease the fill factor to 78.6 %, which is similar to the fill factor of the experimental cell. The JV-characteristics of the experimental cell are listed in Table I. The short circuit current density of the simulation A exceeds that of the experimental cell by 7 %. This may be due to the busbars and the boundary effects of the real solar cell. Another probable
25th European Photovoltaic Solar Energy Conference, Valencia, Spain, 06-10 September 2010, 2DV.1.19
explanation is the Auger recombination loss of the generation current in the emitter which is not included in the CoBo model.
of the bulk that are far away from the rear contact. The transport loss “tf” gets less important. This shows that the rear contact design imposes requirements on the design of the front side.
Figure 7: FELA pie chart and JV-curve for simulation B with local back contacts. - The Free Energy Loss Analysis chart includes the hole transport loss of the base “th”, the Shockley Read Hall recombination loss of the base “rsrh”, the recombination loss of the passivated front region “rf”, the transport loss of the passivated front “tf”, the recombination loss at the back contact region “rbc”, the recombination loss at the passivated back region “rb”, the electron transport loss of the base “te” and a combination of other free energy loss rates “others”. The simulation parameters are listed in Table III. The simulation does not include resistances of the metallization pattern. The curve is a spline through the symbols.
Figure 8: Efficiency and fill factor as functions of the back contact width xbc and the base doping density NA. - The batch simulation does not include resistances of the screen printed metallization pattern. The curve is a spline through the symbols. The unit cell geometry of the simulation is shown in Figure 2.
For the purpose of comparison, simulation B (see Figure 7 and Table III) uses a different back contact design. A large fraction of the back surface is assumed to be passivated and to have a saturation current density of 130 fA/cm2. A smaller fraction is locally contacted. The saturation current density of the contact is assumed to be 1200 fA/cm2. A possible impact of the passivated back region on the generation profile is neglected. For didactical purpose we chose for simulation B the width of the passivated region to have the same cell efficiency as in simulation A. The different rear design results in a different pattern of losses as is shown by comparing the FELA pie chart in Figure 7 with that in Figure 6. The new main loss mechanism now is the transport of the holes in the base (“th”) with 35 %, followed by the Shockley Read Hall recombination in the base (“rsrh”) with 24 %. The recombination loss of the passivated front region (“rf”) is 14 % of the total loss rate of free energy. A solar cell with local back contacts causes longer transport paths for the holes in a p-type base, compared to the paths in a solar cell with a fully contacted back. Additionally, a partially passivated rear side reduces rear side recombination. Therefore, the hole transport losses and at the recombination losses in the volume increase, and the recombination at the back of the cell decreases from simulation A to simulation B. The change from 6 % to 14 % of front side recombination “rf” from simulation A to B is also due to an enhanced quasi-Fermi level splitting in those regions
Figure 9: Free Energy Loss Analysis (FELA) parameters as functions of the back contact width xbc for a base doping density of NA = 6.5 x 1015 cm-3. - Shockley Read Hall recombination loss “rsrh”, Auger recombination loss “raug”, band to band recombination loss “rbb”, hole transport loss “th”, electron transport loss “te”, recombination loss of the front contact region “rfc”, recombination loss of the front region “rf”, transport loss of the front region “tf”, recombination loss of the back contact region “rbc”, and recombination of the back regeion “rb”. Figure 8 shows the efficiency and the fill factor of a batch simulation, where the back contact width xbc and the base doping density NA are varied. The CoBoGUI is
25th European Photovoltaic Solar Energy Conference, Valencia, Spain, 06-10 September 2010, 2DV.1.19
able to vary two batch parameters in a single batch simulation and display the results as an array of curves. A mobility model from PC1D is used to adjust the mobilites of the batch simulation according to the NA values. Possible influences of the base doping on saturation current densities are neglected here but may easily be implemented since the GUI allows for using functions of some parameters as input for other parameters. The variation of NA corresponds to a small variation of the base resistivity of ρ = 2…2.5 Ω cm. The simulations predict that this variation has little influence on the efficiency. The unit cell of the presented basic version of the CoBoGUI has a single back contact. The model yields the largest efficiencies at rather wide contact regions that cover 40 % of the rear side of the cell. Free Energy Loss Analysis plots as a function of the batch parameter xbc are shown in Figure 9 for a doping density of NA = 6.5 x 1015 cm-3. The data of Simulation B correspond to xbc = 200 µm. Figure 8 shows that the efficiency and the fill factor increase if the back contact half width xbc increases from 10 µm to approximately 400 µm. If xbc is further increased, the back contact recombination losses become larger than the hole transport losses (see Figure 9) and the efficiency decreases. A free energy dissipation rate density of f = 10 W/m2 corresponds to one percent of the incident light power of 1000 W/m2. The losses due to Auger recombination and band to band recombination are negligibly small, compared to the other losses in Figure 9.
This equation is multiplied by another scalar function v(u) and integrated over a domain Ω:
∫ v j d L = Ω∫ −v div [c grad(u)] d L
If the domain Ω is a surface, the differential dL would be a small area element and if the domain ∂Ω is a line segment, the differential dL would be a small length element. Additionally, we assume that we can apply the divergence theorem using the domain boundary ∂Ω:
∫ v j d L = Ω∫ grad(v) [c grad(u)] d L +
Ω
(10)
∫ −v [c grad(u)] dP ∂Ω
If the domain Ω is a line element, the integral over ∂Ω equals the summation over the initial point and the final point of the line element. The vector dP is perpendicular to the boundary. With the boundary normal vector n, equation (11) can be written as:
∫ −grad(v) [c grad(u)] + v j d L + Ω
∫ v n [c grad(u)] d P = 0
(11)
∂Ω
We define the boundary condition − n [ c grad( u ) ] = µ
9
(9)
Ω
(12)
SUMMARY for ∂ Ω , and obtain the weak form of equation (8):
The CoBoGUI was designed to offer a flexible Graphical User Interface for the Conductive Boundary model and is realized using MATLAB and COMSOL Multiphysics. The Free Energy Loss Analysis (FELA) helps to identify the main loss mechanisms of solar cells. The CoBoGUI is a tool for device optimization that is numerically not very demanding, since it does not spatially resolve the space charge region. The code package may be downloaded for free from [5]. The model shows limitations, whenever the transport properties in the space charge region dominate the cell behavior.
10 ACKNOLEDGEMENTS We thank B. Beier and T. Dullweber for the experimental cell data. Funding was provided by the State of Lower Saxony.
11 APPENDIXES 11.1 COMSOL Multiphysics weak form To explain the syntax of the weak form, we define an arbitrary non-homogeneous partial differential equation including the scalar functions u(x), j (u), and a constant c: j = − div [ c grad(u ) ]
(8)
0 = ∫ − c grad [ v(u )] grad(u ) + v(u ) j (u ) d L (13)
Ω
−
∫ v(u ) µ d P ∂Ω
More information about weak forms can be found in [4]. The equation (13) is implemented by entering a term called “weak” following a special syntax. The function v(u) is called “test function” and is obtained with the “test” operator: “test(u)”. The term grad[v(u)] is entered by applying the test operator to the two components “ux” and “uy” of the gradient of u for two dimensional simulations: grad[v(u)] => “test(ux) + test(uy)”. Here, we specify a weak term for a solution variable u on a one dimensional domain boundary. The variable u also exists in the two dimensional domain itself. The domain source term jdomain (u ) that flows perpendicular into the boundary is automatically included in the equation system. Therefore, the weak term of the boundary only includes j − jdomain and not j . The so called “Lagrange multiplier” µ is also automatically added to the equation system; hence we do not have to explicitly specify the second integral. Therefore, the “weak” term syntax of equation (10) in COMSOL Multiphysics is: weak = - c * ( test(ux) * ux + test(uy) * uy ) + test(u) * (j- jdomain)
25th European Photovoltaic Solar Energy Conference, Valencia, Spain, 06-10 September 2010, 2DV.1.19
The term (j-jdomain) is zero if the boundary itself has no additional source term. The weak term for the front CoBo equation (5) reads: weak = 1/Rfront * ( test(phiFnTx) * phiFnTx + test(phiFnTy) * phiFnTy ) + test(phiFn) * (-Jnrec) This weak term also works for boundaries that are not parallel to the x-direction since the tangential components (“uTx” instead of “ux”) are used. 11.2 FREE ENERGY LOSS ANALYSIS As an alternative to the derivation in [2] the FELA may be derived as follows: If ∆N particles are transferred from an electrochemical potential µ 1 to an electrochemical potential µ 2, the change of the free energy is
dF = ∆N (µ 2 - µ 1).
(14)
dF ≈ R dA d x q (ϕ Fn − ϕ Fp ) . dt
Thus, it is possible to derive free energy rate densities with the unit W/m2 for all generation / recombination processes and all transport processes in the solar cell, and compare them in a single graph. Please see reference [2] for more details on the Free Energy Loss Analysis. The CoBoGUI supports FELA plots as a function of the voltage for a single batch element or as function of the batch parameters at the MPPs. 11.3 Comparison with PC1D Table IV lists the parameters of a PC1D simulation that we run for a comparison with the CoBo model. We extract a “global saturation current density” J0global from a PC1D simulation by setting the lifetime of the base to a high value of ߬n = ߬p = 1 s and using the equation J sc
J 0global =
For a transportation process during a time dt, with
e
∆N = −
dA jn ( x ) d t q
(15)
(19)
Voc UT
(20)
−1
where J sc is the short circuit current density and Voc is the open circuit voltage of the PC1D simulation.
electrons moving from µ 1 = µ n(x) to µ 2 = µ n(x + dx) through a cross section dA by virtue of a current density jn , the free energy rate can be written as ∂ µ ( x) dF dA ≈− jn ( x) n dx dt q ∂x
(16)
We state that the electrochemical potential of the electrons equals the quasi-Fermi potential ϕ Fn times the elementary charge:
dF ∂ϕ ≈ − dA jn ( x) Fn d x dt ∂x 2
∂ϕ ≈ − dA σ n Fn d x ∂x 1 2 ≈ − dA jn d x
σn
(17a)
(17b)
(17c)
This rate is proportional to the square of the gradient of the quasi-Fermi potential. Equation (17c) is related to the well known equation P = R I2, where P represents the power, R the resistance and I the current of a model circuit. As an example for a FELA parameter we use the free energy loss rate density caused by electron transport within the front contact region, which is given as: tfc = −
1 Acell
xfc
∫0
2
1 ∂ϕ Fn dx . Rfc ∂ x
(18)
For a recombination process the particle rate and the change of the electrochemical potential can be written in terms of a recombination rate R, a volume dV = dA dx, and the quasi-Fermi potentials:
Figure 10: Comparison of the JV-curves of the CoBoGUI and PC1D. - The JV-characteristics agree within 1 %. The CoBo model (red stars) cannot reproduce the PC1D JV-curve (blue circles) at low voltages. The extracted global saturation current density is J0global = 170 fA/cm2. The CoBo simulation then uses this saturation current density value in the front side. The recombination at the rear side is set to zero in both, the PC1D and the CoBo simulation. Figure 10 shows the absolute values of the current density obtained using PC1D and the CoBoGUI, respectively, shifted by the corresponding J sc value and plotted in a semi-log-graph as function of the voltage. At small voltages the J-values of the CoBo model deviate from the values calculated with PC1D by up to 10−2 mA/cm2. The CoBo model does not account for the
25th European Photovoltaic Solar Energy Conference, Valencia, Spain, 06-10 September 2010, 2DV.1.19
voltage dependent behavior of the space charge region. However, the deviations of the characteristic values of the illuminated JV-curve are less than 1 %.
Table IV: PC1D simulation parameters Device Device area: 1 cm2 No surface texturing No surface charge No Exterior Front Reflectance No Exterior Rear Reflectance No internal optical reflectance Emitter contact enabled Base contact enabled No internal shunt elements Region 1 Thickness: 200 µm Material modified from si.mat Fixed electron mobility: 1167 cm2/Vs Fixed hole mobility: 435 cm2/Vs Dielectric constant: 11.9 Band gap: 1.124 eV Intrinsic conc. At 300 K: 1x1010 cm-3 Refractive index: 3.58 Absorption coeff. from internal model No free carrier absorption P-type background doping: 6.5x1015 cm-3 1st front diff. N-type, 1.3x1020 cm-3 peak; erfc; 0.015 µm depth factor, 0.05 µm peak position 2nd front diff. N-type, 3x1019 cm-3 peak; gauss; 0.13 µm depth factor, 0.08 µm peak position No rear diffusion Bulk recombination: ߬n = ߬p = 100 µs No Front-surface recombination No Rear-surface recombination Excitation Excitation mode: Transient, 60 timesteps Temperature: 300 K Base circuit: Sweep from -0.2 to 0.7 V Collector circuit: Zero Photogeneration from cumgen_standard.gen Results Short-circuit Ib: -0.0403 amps Max base power out: 0.0216 watts Open-circuit Vb: 0.6465 volts
REFERENCES [1] R. Brendel, Prog. Photovolt: Res. Appl. (2010), DOI: 10.1002/pip.954 [2] R. Brendel, S. Dreissigacker, N.-P. Harder, and P.P. Altermatt, Appl. Phys. Lett., Vol. 93 (2009) 173503, DOI: 10.1063/1.3006053 [3] R. Brendel, M. Hirsch, R. Plieninger, and J. H. Werner, IEEE Trans. Electron Devices, Vol. 43, No. 7 (1996) 1104, DOI: 10.1109/16.502422 [4] COMSOL Multiphysics Reference Guide, The Weak Form, Version 3.5a (2009) [5] CoBoGUI v1.0 (2010) www.isfh.de/institut_solarforschung/software.php
