The Solar Cell
4.4 Solar Cell Design and Analysis
The design of a practical silicon solar cell can now be considered. In order for light to reach the junction area of the p-n junction, the junction should be close to the surface of the semiconductor. The junction area must be large enough to capture the desired radiation.
This dictates a thin n- or p-region on the illuminated side of the solar cell. A significant challenge is to enable the thin region to be sufficiently uniform in potential to allow the junction to function over its entire area. If a contact material is applied to the surface of the cell, sunlight will be partly absorbed in the contact material. The common solution to this is to make the thin region of the silicon as conductive as possible by doping it heavily. In this way, the highly doped thin region simultaneously serves as a front electrode with high lateral conductivity (conductivity in a plane parallel to the plane of the junction) and as one side of the p-n junction. Since n-type silicon has higher electron mobility and therefore
106
105
104
103
102
10
3 2 1.5 1 0.7
0.2 0.6
Amorphous Si GaP
InP Si
GaAs
Absorption coefficient (cm–1)
Ga0.3In0.7As0.64P0.36
InGaAs Ge
Energy (eV)
Wavelength (μm)
1 1.4 1.8
Figure 4.5 Absorption coefficients covering the solar spectral range for a range of semicon- ductors. Note the absorption tails in silicon and germanium arising from two-step absorption processes. Amorphous silicon is a non-crystalline thin film that has different electron states and hence different absorption coefficients compared with single-crystal silicon. Reprinted from Shur, M., Physics of semiconductor devices. Copyright (1990) with permission from Prentice Hall, USA
higher conductivity than is achieved by the lower mobility of holes in p-type material, the thin top layer in silicon solar cells is preferably n-type in practice.
A crystalline silicon solar cell is shown in Figure 4.7. It consists of a thin n+front layer.
A metal grid is deposited on this layer and forms an ohmic contact to the n+material. The areas on the n+ front layer that are exposed to sunlight are coated with an antireflection
Visible
UV Infrared
O2 H2O
H2O H2O
H2O
Absorption bands CO2 H2O
Sunlight at top of the atmosphere
5250°C blackbody spectrum
Radiation at sea level 2.5
2
1.5
Spectral irradiance (Wm–2nm–1) 1
0.5
0
250 500 750 1000 1250
Wavelength (nm)
1500 1750 2000 2250 2500 O3
Figure 4.6 Solar radiation spectrum for a 5250◦C blackbody, which approximates the space spectrum of the sun, as well as a spectrum at the earth’s surface that survives the absorp- tion of molecules such as H2O and CO2 in the earth’s atmosphere. Note also the substan- tial ozone (O3) absorption in the UV part of the spectrum. Reproduced from www.global warmingart.com/wiki/File:Solar_Spectrum_png. Copyright (2011) globalwarmingart.com
Antireflection coating
Sunlight
xn xs 0 0 xb xp
W
Metal grid n+-layer p-region Back metal contact
+ + + + + + + + +
- - - - - - - - - -
Figure 4.7 Cross-section of a silicon solar cell showing the front contact metal grid that forms an ohmic contact to the n+-layer. The depletion region at the junction has width W
layer. The simplest such layer is a quarter wavelength in thickness such that incident light waves reflecting off the front and back surfaces of this layer can substantially cancel each other (see Problem 4.3). The metal grid does block some sunlight; however, in practice the grid lines are narrow enough to prevent excessive light loss. A thick p-type region absorbs virtually all the remaining sunlight, and is contacted by a rear metal ohmic contact.
Because most of the photons are absorbed in the thick p-type layer, most of the minority carriers that need to be collected are electrons. The goal is to have these electrons reach the front contact. There will, however, also be some minority holes generated in the n+region that ideally reach the p-region.
Sunlight entering the solar cell will be absorbed according to the relationship introduced in Section 1.12
I (x)=I0e−αx
In order to simplify the treatment of the solar cell we will assume that the optical generation rate G is uniform throughout the p-n junction. This implies that the absorption constantα is small. Real solar cells are approximately consistent with this assumption for photons of longer wavelengths of sunlight very close in energy to the bandgap. Shorter wavelengths, however, should really be modelled as a rapidly decaying generation rate with depth.
We will also start by assuming that the relevant diffusion lengths of minority carriers in both the n-type and p-type regions are much shorter than the thicknesses of these regions.
This means that the p-n junction may be regarded as possessing semi-infinite thickness as far as excess minority carrier distributions are concerned, and in Figure 4.7 the front surface and back surface at xn=xsand xp=xbrespectively are far away from regions containing excess carriers.
For the n-side, the diffusion equation (Equation 1.64a) for holes may be rewritten as Dp
d2δp(xn)
dx2 = δp(xn) τp
−G (4.1)
The term G must be subtracted from the hole recombination rate δp(xn) τp
because it is the additional hole generation rate. The solution to this equation where
Lp= Dpτp
is
δp(xn)=Gτp+C exp x
Lp
+D exp −xn
Lp
(4.2) Note that the solution is the same as Equation 1.65a except for the added term Gτp. See Problem 4.4.
The boundary conditions we shall satisfy are:
δp(0)= pn
expq V
kT −1 and
δp(xn→ ∞)=Gτp
Concentration (log scale)
W
n or p
pn
p(xn) Gτp
Gτn
np
xn xp
n(xp) pp
n+ p
nn
Figure 4.8 Concentrations are plotted on a log scale to allow details of the minority carrier concentrations as well as the majority carrier concentrations to be shown on the same plot.
Note that the n+-side has higher majority carrier concentration and lower equilibrium minority carrier concentration than the more lightly doped p-side corresponding to Figure 4.7. Diffusion lengths are assumed to be much smaller than device dimensions. The p-n junction is shown in a short circuit condition withV=0
The first boundary condition is as discussed in Section 2.5. The dynamic equilibrium in the depletion region still determines the carrier concentrations at the depletion region boundaries. For the second boundary condition Gτp is the excess carrier concentration optically generated far away from the depletion region (see Equation 1.51).
Substituting these two boundary conditions into Equation 4.2 we obtain (see problem 4.4) δp(xn)=Gτp+
pn
exp
q V kT
−1
−Gτp
exp
−xn
Lp
(4.3a) The analogous equation for the p-side is
δn(xp)=Gτn+
np
exp
q V kT
−1
−Gτn
exp
−xp
Ln
(4.3b) Note that if V =0, Equations 4.3a and 4.3b yieldδp(xn)=Gτp andδn(xp)=Gτn for large values of xn and xp respectively. In addition at V = 0 these equations give zero for bothδp(xn=0) andδn(xp =0). We can show this more clearly in Figure 4.8 for an illuminated p-n junction under short circuit conditions (V =0). We use Equation 4.3 to plot p(xn)=δp(xn)+pn and n(xp)=δn(xp)+np, where pn and np are the equilibrium minority carrier concentrations.
Having determined the minority carrier concentrations we can now determine diffusion currents In(x) and Ip(x) using the equations for diffusion currents (Equation 1.54). By substitution of Equations 4.3a and 4.3b into Equation 1.54 we obtain
Ip(xn)= q A Dp
Lp
pn
exp q V
kT
−1
exp
−xn
Lp
−q AG Lpexp xn
Lp
(4.4a) for holes diffusing in the n-side and
In(xp)=q A Dn
Ln np
exp
q V kT
−1
exp
−xp
Ln
−q AG Lnexp xp
Ln
(4.4b) for electrons diffusing in the p-side (see Problem 4.5). Note that the first terms in these equations are identical with Equations 2.21d and 2.21c for an unilluminated diode.
Since there is uniform illumination, we need also to consider generation in the depletion region. We shall neglect recombination of electron-hole pairs since W is much smaller than the carrier diffusion lengths. This means that every electron and every hole created in the depletion region contributes to diode current. The generation rate G must be multiplied by depletion region volume WA to obtain the total number of carriers generated per unit time in the depletion region. Carrier current optically generated from inside the depletion region therefore becomes the total charge generated per unit time or
I(depletion)=qGWA (4.5)
Although both an electron and a hole are generated by each absorbed photon, each charge pair is only counted once: one generated electron drifts to the n-side metal contacts, flows through the external circuit, and returns to the p-side. In the meantime, one hole drifts to the p-side metal contact and is available there to recombine with the returning electron.
It therefore follows using Equation 2.22 that the total diode current becomes I =I0
exp
q V kT
−1
−IL (4.6)
where IL, the current optically generated by sunlight, has three components, from the n- side, from the depletion region, and from the p-side respectively. Using Equation 4.5 as well as the second terms from Equations 4.4a and 4.4b at xn=xp=0, we obtain
IL=q AG(Ln+W+Lp) (4.7)
which confirms that Figure 4.2 is valid and the I−V characteristic is shifted vertically (by amount IL) upon illumination. Of the three terms in Equation 4.7, the second term is generally smallest due to small values of W compared to carrier diffusion lengths, and since electron mobility and diffusivity values are higher than for holes the first term will be larger than the third term. It is reasonable that diffusion lengths Lnand Lpappear in Equation 4.7:
carriers must cross over the depletion region to contribute to solar cell output current. They have an opportunity to diffuse to the depletion region before they drift across it, and the diffusion lengths are the appropriate length scales over which this is likely to occur.
Operating point for maximum output power ISC
IMP I
VMP VOC V
Figure 4.9 Operating point of a solar cell. The fourth quadrant in Figure 4.2 is redrawn as a first quadrant for convenience. Open circuit voltage VOC and short circuit current IOCas well as current IMPand voltage VMPfor maximum power are shown. Maximum power is obtained when the area of the shaded rectangle is maximized
The solar cell short circuit current ISCcan now be seen to be the same as ILby setting V=0 in Equation 4.6. Hence
ISC=q AG(Ln+W+Lp) (4.8)
In addition, the solar cell open circuit voltage VOCcan be found by setting I=0 in Equation 4.6 and solving for V to obtain
VOC=kT q ln
IL
I0 +1
(4.9) These quantities are plotted in Figure 4.9 together with the solar cell operating point.
The fill factor FF is defined as
FF= IMPVMP
ISCVOC
(4.10) In crystalline silicon solar cells FF may be in the range of 0.7 to 0.85.
Example 4.1
An abrupt silicon p-n junction solar cell at room temperature is exposed to sunlight.
Assume that the sunlight is uniformly intense throughout the silicon yielding an optical generation rate of 5×1019 cm−3s−1. The solar cell has a junction area of 100 cm2, a depletion region width of 3μm and reverse saturation current density of J0=1×10−11A cm−2. The silicon has a carrier lifetime of 2×10−6s.
(a) Find the optically generated current that is generated inside the depletion region.
(b) Find the total optically generated current.
(c) Find the short-circuit current.
(d) Find the open-circuit voltage.
(e) If the solar cell fill factor is 0.75, find the maximum power available and discuss this in terms of the expected total available sunlight.
Solution (a)
I(depletion)=qGWA=1.6×10−19C×5×1019 cm−3s−1×3×10−4cm
×100 cm2=0.24 A (b) From Example 2.3,
Ln=
Dnτn =
3.51×101cm2s−1×2×10−6 s=8.38×10−3cm and
Lp=
Dpτp =
1.25×101 cm2s−1×2×10−6s=5.00×10−3cm and therefore
IL=q AG(Ln+W +Lp)
=1.6×10−19C×100 cm2×5×1019 cm−3s−1(8.38×10−3+3×10−4 +5.00×10−3cm)=10.9 A
(c) The short circuit current ISCis the same as IL. Therefore ISC=10.9 A.
(d) Open-circuit voltage:
I0=J0A=1×10−11A cm−2×100 cm2=1×10−9A and now
VOC= kT q ln
IL I0
+1
=0.026 V ln
10.9 A 1×10−9A+1
=0.601 V (e) The maximum output power is obtained at the operating point VMPand IMP.
Therefore
IMPVMP=FF×ISCVOC=0.75×10.9 A×0.601 V=4.91 W
The available sunlight per square metre for full sun conditions on the earth’s surface is approximately 1000 W, which yields 10 W over an area of 100 cm2. The best silicon solar cell, however, is not more than 25% efficient and therefore the most electrical power that we could expect to be available from the solar cell should be closer to 2.5 W. The model we have used assumes a uniform optical generation rate inside the silicon. This is not realistic since sunlight will be absorbed and a decreasing optical generation rate with depth will exist in reality, which will decrease the available power.
In practice achieving an electrical output power of 4.91 W from a silicon solar cell of 100 cm2would require a concentration of sunlight by a factor of approximately two using reflective or refractive optical concentrators. See Section 4.14.