Efficiency limits of photovoltaic energy conversion
One of the most frequently asked questions in photovoltaics is
about efficiencies. Usually people are little impressed by almost
25% for single junction record cells and more than 30% in case of
tandems. Surprisingly, nobody ever askes this question for the
latest sportscar even though the efficiency of typical combustion
engines in cars is probably not much higher. Despite its development
for more than a century and the input of quite significant amounts
of money. Anyway...
The ultimte limit, of course, would be the Carnot limit with the temperature
of the sun as the hot reservoir and the cell at ambient temperature of around
300 K. This will yield an efficiency limit of about 94%. However, we may have some
difficulties with running a process between these reservoirs. Let us
try a more realistic estimation based on the working principle of a
single bandgap semiconductor solar cell.
We will try give an estimation of the maximum power we can draw from
a solar cell and divide it by the incident power. There are some
quite fundamental investigations on this topic [Baruch1995, Schockley1961] but we will
try to stick to illustrative examples.
In the chapter on illumination we discussed the spectral irradiation density in
W/nm.m2. By dividing with the corresponding
photon energy it is easy to calculate the the number of photons per
area and wavelength interval. For the estimation of the photocurrent
we need to know about absorption processes in the semiconductor; photons
with an energy less than the bandgap cannot be absorbed, photons with higher
energies can excite an electron hole pair, their excess energy is lost. The
figure below shows the number of photons per area and time (the flux density)
in dependence of the wavelength for the AM1.5 spectrum.
Let us assume that we have an ideal cell with no optical losses. The
attainable current density for a given bandgap is then obtaind by integrating
the spectral distribution. Starting value is the band gap energy and the
integration runs over all shorter wavelengths. If we plot
the value of the integral for different starting points we get the
maximum current density for a given bandgap. This is the maximum current
that the cell can deliver under short circuit conditions. Lower bandgaps
yield higher currents because they absorb larger parts of the spectrum.
This is shown for Silicon (1.1 eV) and Germanium (0.7eV) in the figure below.
Figure 1: Flux of photons per area and time and wavelength interval (red) and
maximum attainable photocurrent for a given bandgap (blue, upper scale),
assuming an AM1.5 spectrum.
Open circuit voltage
An upper limit for the open circuit voltage of a solar cell will
obviously be the band gap. Lower values are due to recombination
processes in the cell. Even in an otherwise ideal material, black
body radiation will present a minimum amout of unavoidable
radiative recomibation, simply because we operate the solar
cell at some finite temperature. The assumption that all light
with energy higher than the band gap energy is ideally absorbed
defines the solar cell as a black body for this spectral region.
As such it will also emit black body radiation because
it operates at some non-zero temperature and it is in equilibrium
with its environment and the solar irradiation. This concept is
called detailed balance [Schockley1961].
We will consider a less general case which is more illustrative.
Assume that the charge transport is purely by diffusion of minority
carriers. Then, the relationship between current density and voltage
yields the following approximate expression between open circuit voltage
Voc and short circuit current density jsc:
Here, Eg is the bandgap and jsc the short circuit
current. The current prefactor j00 for the diffusion current
of minority carriers in a p-type absorber involves the effective densities
of states NC and NV, the doping concentration
NA, the minority carrier diffusion constant Dn,
and the minority diffusion length Ln.
For the material constants of silicon and germanium and a typical
doping concentration of about 1016 cm-3 we obtain
j00 in the range from 108 to 109
mA/cm2. After putting this into the above equations we find
that the open circuit voltages are about 400 mV lower than the band gap
energy. Indeed, the open circuit voltages for high efficiency
germanium, silicon and gallium arsenide solar cells are up to
245, 706 and 1020 mV, respectively, with their respective
bangaps of 0.67, 1.12 and 1.43 eV.
Note that the diffusive transport is already a very ideal assumption,
even in the best cells other recombination mechanisms are present which
lower the maximum open circuit voltages below the discussed limits.
Now we know the maximum current densitiy and the maximum voltage for a
given band gap energy. However, at both points of the characteristics the
actual output power of the solar cell is zero. We must find the conditions
of maximum output (MPP, maximum power point). In the figure below this
point corresponds to the largest rectangle which we can fit between
the origin and the part of the curve in the fourth quadrant.
Figure 2: Schematic representation of dark (blue) and light (red) current
voltage characteristics of a solar cell. The different quantities like open
circuit voltage Voc and short circuit current density
jsc are shown.
The ratio between the maximum power density and the product of Voc times
jsc is called fill factor (FF). We assume a value of 80% but for high
efficiency GaAs cells it is as high as 87%.
For a calculation of the resulting efficiency for a given bandgap we
use the following equation:
The figure below shows that a broad range of bandgaps between 1.0 and 1.8 eV
are suitable for use under the AM1.5 spectrum and could yield theoretical
efficiencies above 20%.
Figure 3: Maximum theoretical efficiencies under AM1.5 illumination
in dependence of the bandgap energy according to the ideas discussed
on this page. Different assumptions in the literature can yield higher
[Baruch1995] P. Baruch, A. De Vos, P. T. Landsberg, and J. E. Parrot, Sol. En. Mat. 36 (1995) 210-222
[Schockley1961] W. Schockley and H. J. Queisser, J. Appl Phys. 32(3) (1961) 510-519