A further positive effect occurs, when the dimensions of the up-converter groove are in the range of the inverse absorption coefficient of the up-converter. This is illustrated in Figure B.4.
It is assumed that light is reaching the up-converter from left and right and is absorbed following the Lambert Beer Law. The intensity within the up-converter groove is then the sum of both illuminations. In the middle of the groove the maximum concentration of 2 is reached, while the magnitude at other distances from the groove surface depends on the absorption coefficient α and the thickness tuc of the groove. In Figure B.4 the concentration (sum of both intensities relative to the illumination only from one side) for different products of α×tuc is shown. The detrimental effect of a high product α×tuc is an enhanced transmission through the up-converter layer. The corresponding transmis-sions are also given in Figure B.4. It should be mentioned that the transmitted light is not necessarily lost, the transmitted photons partly contribute after re-reflection at the adjacent slant. Additionally to this concentration, also light entering directly from the
Figure B.4: Relative intensity within the up-converter groove (width tuc) under illumina-tion from left and right (as drawn schematically on the right). The intensity adds within the up-converter, which leads to a concentration of factor two in the middle of the groove.
The decrease of the concentration aside from the middle depends on the product of the absorption coefficient α and the width tuc. High concentrations have the detrimental effect of high transmission.
front surface of the solar cell (Itop in Figure B.4) contributes as long as the depth of the up-converter groove is in the range of the inverse absorption coefficient. Then also the reflection at the metal covering the rear of the up-converter enhances the concentration.
For BaCl2:Er3+ the exact figure of the absorption coefficient is not reported in literature.
The absorption coefficient of NaYF4:Er3+ amounts to about 3.5 cm−1 at about 1520 nm [20]. A groove of 1 mm width would correspond to a concentration close to the solid line in Figure B.4, but also lead to a transmission of 70%.
Together with the optical confinement when having a slant angle of 20◦ (geometrical con-finement of 2), this leads to a resulting concentration factor of up to 4 in the up-converter.
Assuming a quadratic dependency of the up-conversion emission intensity on the input power, this would lead to 16 times the efficiency compared to a non-focused light (planar up-converter application), without regarding the shading losses in the planar design. This idea can be extended by applying the up-converter in holes instead of grooves. For this concept the structuring of the surrounding of the holes in cone-shape is necessary, which is very difficult to realize. But beside cone-shape also pyramid-shaped pits are expected to enhance the optical concentration.
α absorption coefficient
Anr non-radiative relaxation rate Arad radiative transition rate Atot overall transition rate
APTE addition de photon par transferts d’energie ARC antireflection coating
BSF back surface field
c0 speed of light in vacuum
cps counts per second
CRW combustable renewables and renewable waste
²0 permittivity in vaccuum
η efficiency
E energy
EDX energy dispersion X-ray
EG band gap energy
EQE external quantum efficiency ESA excited state absorption ETU energy transfer up-conversion
f fraction of photons emitted in a 2 photon process fc fraction of radiative recombination
FF fill factor
FGA forming gas anneal
G generation rate
g(ω) line shape function GSA ground state absorption
h Planck’s constant
¯
h Planck’s constant divided by 2π ISC short circuit current
IQE internal quantum efficiency IRSR infrared spectral response
IRUCG infrared up-conversion powder with mainly green emission IUPAC International Union of Pure and Applied Chemistry
J total angular momentum
123
J current density
J0e emitter saturation current density JSC short circuit current density
k Boltzmann constant
˙N temporal change of the population density
n refractive index
∆nav average excess carrier density
Ωt Judd-Ofelt parameters
RCA wafer cleaning procedure developed by Radio Corporation of America
σ absorption cross section
SHG second harmonic generation
τ lifetime
τeff effective lifetime
T transmission
Tcell temperature of the solar cell TS temperature of the sun
TPA two photon absorption
V voltage
VOC open circuit voltage
wCR probability of cross relaxation wCE probability of cooperative excitation wCooR probability of cooperative relaxation wESA probability of excited state absorption wET probability of energy transfer up-conversion
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