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# Modelling Photovoltaic Systems Using PSpice - 45271 02

(Parte **1** de 3)

Spectral Response and Short-Circuit Current

Summary

This chapter describes the basic operation of a solar cell and uses a simplified analytical model which can be implemented in PSpice. PSpice models for the short circuit current, quantum efficiency and the spectral response are shown and used in several examples. An analytical model for the dark current of a solar cell is also described and used to compute an internal PSpice diode model parameter: the reverse saturation current which along with the model for the short-circuit current is used to generate an ideal Z(V) curve. PSpice DC sweep analysis is described and used for this purpose.

2.1 Introduction

This chapter explains how a solar cell works, and how a simple PSpice model can be written to compute the output current of a solar cell from the spectral irradiance values of a given sun spectrum. We do not intend to provide detailed material on solar cell physics and technology; many other books are already available and some of them are listed in the references [2.1], [2.2], [2.3], [2.4] and [2.5]. It is, however, important for the reader interested in photovoltaic systems to understand how a solar cell works and the models describing the photovoltaic process, from photons impinging the solar cell surface to the electrical current produced in the external circuit.

Solar cells are made out of a semiconductor material where the following main phenomena occur, when exposed to light: photon reflection, photon absorption, generation of free camer charge in the semiconductor bulk, migration of the charge and finally charge separation by means of an electric field. The main semiconductor properties condition how effectively this process is conducted in a given solar cell design. Among the most important are:

(a) Absorption coefficient, which depends on the value of the bandgap of the semiconductor and the nature, direct or indirect of the bandgap.

20 SPECTRAL RESPONSE AND SHORT-CIRCUIT CURRENT

(b) Reflectance of the semiconductor surface, which depends on the surface finishing: shape and antireflection coating.

(c) Drift-diffusion parameters controlling the migration of charge towards the collecting

(d) Surface recombination velocities at the surfaces of the solar cell where minority carriers junction, these are carrier lifetimes, and mobilities for electron and holes.

recombine .

2.7.7 Absorption coefficient a@)

The absorption coefficient is dependent on the semiconductor material used and its values are widely available. As an example, Figure 2.1 shows a plot of the values of the absorption coefficient used by PClD for silicon and GaAs [2.6 1. Values for amorphous silicon are also plotted.

As can be seen the absorption coefficient can take values over several orders of magnitude, from one wavelength to another. Moreover, the silicon coefficient takes values greater than zero in a wider range of wavelengths than GaAs or amorphous silicon. The different shapes are related to the nature and value of the bandgap of the semiconductor. This fact has an enormous importance in solar cell design because as photons are absorbed according to

Lambed’s law:

4(x) = 4(0) ePX

1 ,E+07 1 ,E+06

3 1,E+05

A r E

* c a .-

0 lSE+O4 0

1,E+03

1 ,E+02 1 ,E+01

0 .- n

1,E+0 0 500 1000 1500

Wavelength (nm)

_--_ a-Si

Figure 2.1 wavelength. Data values taken from PClD [2.6] Absorption coefficient for silicon, GaAs and amorphous silicon as a function of the

INTRODUCTION 21 if the value of n is high, the photons are absorbed within a short distance from the surface, whereas if the value of u. is small, the photons can travel longer distances inside the material. In the extreme case where the value of a is zero, the photons can completely traverse the material, which is then said to be transparent to that particular wavelength. From Figure 2.1 it can be seen that, for example, silicon is transparent for wavelengths in the infrared beyond

1.1 micron approximately. Taking into account the different shapes and values of the absorption coefficient, the optical path length required inside a particular material to absorb the majority of the photons comprised in the spectrum of the sun can be calculated, concluding that a few microns are necessary for GaAs material and, in general, for direct gap materials, whereas a few hundreds microns are necessary for silicon. It has to be said that modern silicon solar cell designs include optical confinement inside the solar cell so as to provide long photon path lengths in silicon wafers thinned down to a hundred micron typically.

2.7.2 Reflectance R(A)

The reflectance of a solar cell surface depends on the surface texture and on the adaptation of the refraction coefficients of the silicon to the air by means of antireflection coatings. It is well known that the optimum value of the refraction index needed to minimize the reflectance at a given wavelength has to be the geometric average of the refraction coefficients of the two adjacent layers. In the case of a solar cell encapsulated and covered by glass, an index of refraction of 2.3 minimizes the value of the reflectance at 0.6pm of wavelength. Figure 2.2 shows the result of the reflectance of bare silicon and that of a silicon solar cell surface described in the file Pvcell.pnn in the PClD simulator (surface textured 3 pm deep and single AR coating of 2.3 index of refraction and covered by 1 m

glass).

As can be seen great improvements are achieved and more photons are absorbed by the solar cell bulk and thus contribute to the generation of electricity if a proper antireflection design is used.

Wavelength (nm)

Figure 2.2 Reflectance of bare silicon surface (thick line) and silicon covered by an antireflection coating (thin line), data values taken from PClD [2.6]

2 SPECTRAL RESPONSE AND SHORT-CIRCUIT CURRENT 2.2 Analytical Solar Cell Model

The calculation of the photo-response of a solar cell to a given light spectrum requires the solution of a set of five differential equations, including continuity and current equations for both minority and majority carriers and Poisson’s equation. The most popular software tool used to solve these equations is PClD [2.6] supported by the University of New South Wales, and the response of various semiconductor solar cells, with user-defined geometries and parameters can be easily simulated. The solution is numerical and provides detailed information on all device magnitudes such as carrier concentrations, electric field, current densities, etc. The use of this software is highly recommended not only for solar cell designers but also for engineers working in the photovoltaic field. For the purpose of this book and to illustrate basic concepts of the solar cell behaviour, we will be using an analytical model for the currents generated by a illuminated solar cell, because a simple PSpice circuit can be written for this case, and by doing so, the main definitions of three important solar cell magnitudes and their relationships can be illustrated. These important magnitudes are:

(a) spectral short circuit current density; (b) quantum efficiency; (c) spectral response.

A solar cell can be schematically described by the geometry shown in Figure 2.3 where two solar cell regions are identified as emitter and base; generally the light impinges the solar cell by the emitter surface which is only partially covered by a metal electrical grid contact. This allows the collection of the photo-generated current as most of the surface has a low reflection coefficient in the areas not covered by the metal grid.

0 a a

_, Emitter Base _,

Figure 2.3 Schematic view of an externally short circuited solar cell

ANALYTICAL SOLAR CELL MODEL 23

As can be seen, in Figure 2.3, when the solar cell is illuminated, a non-zero photocurrent is generated in the external electric short circuit with the sign indicated, provided that the emitter is an n-type semiconductor region and the base is a p-type layer. The sign is the opposite if the solar cell regions n-type and p-type are reversed.

The simplified model which we will be using, assumes a solar cell of uniform doping concentrations in both the emitter and the base regions.

2.2. I

Our model gives the value of the photocurrent collected by a 1 cm2 surface solar cell, and circulating by an external short circuit, when exposed to a monochromatic light. Both the emitter and base regions contribute to the current and the analytical expression for both are given as follows (see Annex 2 for a summary of the solar cell basic analytical model).

Short-circuit spectral current density

Ernifter short circuit spectral current density -nW

Base short circuit spectral current density where the main parameters involved are defined in Table 2.1.

Table 2.1 Main parameters involved in the analytical model Units sb R

Absorption coefficient Photon spectral flux at the emitter surface

Photon spectral flux at the base-emitter interface Electron diffusion length in the base layer

Hole diffusion length in the emitter layer Electron diffusion constant in the base layer Hole diffusion constant in the emitter layer Emitter surface recombination velocity Base surface recombination velocity Reflection coefficient cm-’ Photodcm’pm s Photodcm’pm s cm cm cm2/s cm2/s cds cds -

24 SPECTRAL RESPONSE AND SHORT-CfRCUfT CURRENT

The sign of the two components is the same and they are positive currents going out of the device by the base layer as shown in Figure 2.3. As can be seen the three magnitudes involved in equations (2.2) and (2.3) are a function of the wavelength: absorption coefficient a, see Figure 2.1, reflectance R(X), see Figure 2.2 and the spectral irradiance Ix, see Chapter 1, Figure 1.9. The spectral irradiance is not explicitly involved in equations (2.2) and (2.3) but it is implicitely through the magnitude of the spectral photon flux, described in Section 2.2.2, below.

The units of the spectral short-circuit current density are A/cm2pm, because it is a current density by unit area and unit of wavelength.

2.2.2 Spectral photon flux

The spectral photon flux qhO received at the front surface of the emitter of a solar cell is easily related to the spectral irradiance and to the wavelength by taking into account that the spectral irradiance is the power per unit area and unit of wavelength. Substituting the energy of one photon by hclX, and arranging for units, it becomes:

16g [ photon ] qhcl = 10 19.8 cm2pm.s with Z, written in W/m2pm and X in pm. Equation (2.4) is very useful because it relates directly the photon spectral flux per unit

area and unit of time with the spectral irradiance in the most conventional units found in textbooks for the spectral irradiance and wavelength. Inserting equation (2.4) into equation (2.2) the spectral short circuit current density originating from the emitter region of the solar cell is easily calculated.

The base component of the spectral short circuit current density depends on q5'0 instead of qhO because the value of the photon flux at the emitter-base junction or interface has to take into account the absorption that has already taken place in the emitter layer. 4'0 relates to 40 as follows.

where the units are the same as in equation (2.4) with the wavelength in microns.

2.2.3 Total short-circuit spectral current density and units

Once the base and emitter components of the spectral short-circuit current density have been calculated, the total value of the spectral short-circuit current density at a given wavelength is calculated by adding the two components to give:

PSPICE MODEL FOR THE SHORT-CIRCUIT SPECTRAL CURRENT DENSlTT 25

Subcircuit SILICON-ABS.LIB

(1) (10) with the units of Ncm’pm. The photocurrent collected at the space charge region of the solar cell has been neglected in equation (2.6).

It is important to remember that the spectral short-circuit current density is a different magnitude than the total short circuit current density generated by a solar cell when illuminated by an spectral light source and not a monochromatic light. The relation between these two magnitudes is a wavelength integral as described in Section 2.3 below.

Qr;l I b

2.3 PSpice Model for the Short-circuit Spectral Current Density

The simplest PSpice model for the short-circuit spectral current density can be easily written using PWL sources to include the files of the three magnitudes depending on the wavelength: spectral irradiance, absorption coefficient and reflectance. In the examples shown below we have assumed a constant value of the reflectance equal to 10% at all wavelengths.

2.3.1 Absorption coefficient subcircuit

The absorption coefficient for silicon is described by a subcircuit file, ‘silicon-abs.lib’ in Annex 2, having the same structure as the spectral irradiance file ‘aml5g.lib’ and ma access nodes from the outside: the value of the absorption coefficient at the internal node (1 I) and the reference node (10). The block diagram is shown in Figure 2.4.

As can be seen a PWL source is assigned between internal nodes (1 1) and (10) having all the list of the couples of values wavelength-absorption coefficient in cm-’.

(1 1)

PWL voltage source Vabs-si c

Figure 2.4 Block diagram of the subcircuit for the absorption coefficient of silicon and the internal schematic representation

26 SPECTRAL RESPONSE AND SHORT-CIRCUIT CURRENT

h b

@) ’ 4,

2.3.2 Short-circuit current subcircuit model

(201) (205) - b JscdV

Jscb(h) Subcircuit (2M) b Jsc (202) JSC.LIB (207) .

(208) QW)

(209) b SRO (203)

(203) 200

The PSpice short-circuit model is written in the file ‘jsc.lib’, shown in Annex 2, where the implementation of equations (2.2) and (2.3) using equations (2.4) and (2.5) is made using voltage controlled voltage sources (e-devices). This is shown below.

egeomO230200value={1.6e-19*v~202~*v~203~*~1000/962.5~*v~201~*(le16/19.8)* +lp*( 1-v(204) )/(v(202) *lp+l) } egeoml231200value={cosh(we/lp)+se*(lp/dp)*sinh(we/lp)} egeom2232200value={se*(lp/dp)*cosh(we/lp)+sinh(we/lp)} egeom3 233 200value={~se*(lp/dp)+v(202)*lp-exp~-v(202)*we)*v~232))} ejsce205 200value={v(230)/(v(202)*lp-1)*(-v(202)*lp*

+exp(-v(202)*we)+v(233)/v(23l~~};shortcircuit

Note that equation (2.2) has been split in four parts and it is very simple to recognize the parts by comparison. The e-source named ‘ejsce’ returns the value of the emitter short circuit spectral current density, this means the value of JscE~ for every wavelength for which values of the spectral irradiance and of the absorption coefficient are provided in the corresponding PWL files. For convergence reasons the term ( a2Lp2-1) has been split into (aLp+l) (cuLp-1) being the first term included in ‘egeom3’ and the second in ‘ejsce’ sources.

A similar approach has been adopted for the base. It is worth noting that the value of the photon flux has been scaled up to a loo0 W/m2 AM1.5 G spectrum as the file describing the spectrum has a total integral, that means a total irradiance, of 962.5 W/m2. This is the reason why the factor (1000/962.5) is included in the e-source egeomO for the emitter and egeom33 for the base. The complete netlist can be found in Annex 2 under the heading of ‘jsc.lib’ and the details of the access nodes of the subcircuit are shown in Figure 2.5. The meaning of the nodes QE and SR is described below.

(Parte **1** de 3)