The Solar Cell
4.5 Thin Solar Cells
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.
and at xp=xb,δnpwill depend on the value of Sb. If we examine the case in which Sbis very large then from Equation 4.12 it follows that
δnp(xp=xb) ∼=0
These two boundary conditions may be substituted into the general solution of the diffusion equation for electrons. In Equation 1.65a we have written this for holes. For electrons it becomes
δnp(xp)=Aexp −xp
Ln
+Bexp xp
Ln
(4.13) where Ln=√
Dnτn. We now need to consider both terms since the length of the p-type material is finite and we are not justified in assuming that B =0. The two boundary conditions give us two equations
np
exp q V
kT
−1
= A+B (4.14)
and
0=Aexp −xb
Ln
+Bexp
xb
Ln
(4.15) Multiplying Equation 4.14 by exp
xb Ln
and subtracting it from Equation 4.15 we can solve for A and obtain
A=
exp xb
Ln
exp xb
Ln
−exp
−xb Ln
np
exp q V
kT
−1
By multiplying Equation 4.14 by exp
−xb
Ln
we can similarly solve for B and obtain
B=
exp
−xb
Ln
exp
−xb
Ln
−exp xb
Ln
np
exp
q V kT
−1
Substituting A and B into Equation 4.13 we obtain
δnp(xp)= np
exp
q V kT
−1
exp xb
Ln
−exp
−xb Ln
exp xb
Ln
exp
−xp
Ln
−exp
−xb
Ln
exp
xp
Ln
(4.16) The identical method may be used to find the minority hole concentration in the n+ material. We will assume that Sf is very large, which implies that
δpn(xn=xs)∼=0
+ + + + +
- - - - -
xb xs
xn
δpn δnp
0
0 xp
Figure 4.10 Excess minority carrier concentrations for a solar cell having dimensionsxsand xbthat are small compared to the carrier diffusion lengths. Very high values of surface recom- bination velocity are present. Note that a forward bias is assumed, and there is no illumination
and we obtain
δpn(xn)= pn
exp
q V kT
−1
exp xs
Lp
−exp
−xs
Lp
exp xs
Lp
exp
−xn Lp
−exp
−xs Lp
exp
xn Lp
(4.17) Equations 4.16 and 4.17 become meaningful if they are plotted. If we substitute xp= xb into Equation 4.16 we obtain δnp(xb)=0 as expected, and if we substitute xn=xs
into Equation 4.17 we obtain δpn(xn)=0 as expected. The resulting plots are shown in Figure 4.10 for the p-n junction in a forward bias condition without illumination. If xband xsare much smaller than Lnand Lprespectively, thenδnp(xp) andδpn(xn) become almost straight lines as shown. This can be understood from Equation 4.13 since exponential functions are approximately straight lines for small arguments. (The function exp(x) may be approximated as 1+x for small values of x.)
It is clear that the carrier concentrations of both minority electrons and minority holes decrease rapidly as we move away from either side of the depletion region. This means that minority diffusion currents are enabled that flow away from the junction. The thinner the solar cell becomes, the steeper these decreases become. Minority currents flow in the opposite directions to the directions that we require: in a solar cell the minority currents should flow towards the junction.
If illumination were incident uniformly throughout the solar cell in Figure 4.10, excess carriers would be generated. If the solar cell was operated with an electrical load, the
solar cell voltage would decrease and the minority carrier concentrations near the depletion region edges would decrease. The result is that upon illumination of the solar cell of Figure 4.10, surface recombination would compete strongly with the desired carrier drift across the depletion region and solar cell performance would be poor.
For this reason, thin solar cells having high values of SBand SFare not ideal. We will now examine a second limiting case in which SBand SFare assumed to be zero. If we once again look at the p-side of the junction, minority current flow to the back surface at xB=0 is zero, and from Equation 4.12
−Dn
dδn(xb) dx =0
This implies that the excess minority carrier concentrationδn is independent of xb.δn(xb) is therefore a constant and is equal to the value ofδn at the edge of the depletion region.
From Equation 2.17b we have in the p-side δnp(xp)=np
exp
q V kT
−1
and in the n-side
δpn(xn)=pn
exp
q V kT
−1
These straight lines are shown in Figure 4.11 for a solar cell without illumination in a forward bias condition. Note that if the solar cell were connected to an electrical load
+ + + + +
- - - - -
xb
xs
xn 0 0 xp
δpn δnp
Figure 4.11 Excess minority carrier concentrations for a solar cell having dimensionsxsand xbthat are small compared to the carrier diffusion lengths. Zero surface recombination velocity is assumed, which means that there is no drop-off of carrier concentrations towards the front and back surfaces
and illuminated, the minority carrier concentrations would slope downwards towards the depletion region to approach (but not reach) the condition at V =0 of Figure 4.8, and the generated minority carriers would have an opportunity to diffuse towards the junction without competition from surface recombination. Although some bulk recombination is always present, the reduced thickness of the solar cell would reduce recombination in general. The thickness of the solar cell could now be set to allow adequate optical absorption of the sunlight without concern for surface recombination. In practice effective surface recombination velocities must be minimized. This will be further discussed in Section 4.6.