2.3 The generalized Planck law of luminescence
2.3.4 Solving electroluminescence boundary conditions
The solution for the electroluminescence spectrum of a particular sample is obtained using the generalized Planck law by integrating the emission spectrum over the thickness
of the p type region of the solar cell to account for reabsorption (see Equation 2.3.3).
Since the silicon material emits primarily via the indirect bandgap, photon recycling does not saturate the emission which would make the spectrum independent of the reabsorption, since reabsorption of a photon from the primary optical emission does not frequently lead to another optical recombination event. Thus, the depth at which the photon originates in the cell is retained.
Here a formalism proposed by W¨urfel [204] is used where the injection of carriers is treated by assuming the pn junction supplies carriers, but is very close to the front surface of the solar cell. In this case, the thickness of the emitter is negligible. The excess minority carrier concentration is solved using the boundary conditions with the front surface of the solar cell (i.e. the boundary of the n and p region at the pn junction facing into the base of the solar cell) having a concentration determined by an applied voltageVa, and the rear surface of the solar cell having a concentration dependent upon the rear surface recombinationSr. A p-type silicon wafer is assumed.
The probability of a photon emission at a location in the solar cell can be calculated from the generation rate in Equation 2.3.6. As noted in Section 2.3.3, the n type region provides a negligible amount of radiative recombination compared to the p type region, and is very thin. Thus, this region is ignored and the spatial integral may be performed by integrating over the thickness of the solar cell. Electroluminescence results in a concentration of electrons injected on the p-side of the solar cell (wherez≈0)
ne(0) = n2i
NAeeVakT (2.3.8)
which obeys the differential equation [65, 204]
d2ne
dx2 − ne
L2e = 0. (2.3.9)
The boundary condition at the rear assumes injected carriers obeySrne(d) =−Dedndxe(d) where Sr is the rear surface recombination velocity, d is the cell thickness, and Le =
√Deτe relates the diffusion length, lifetime, and diffusivity by the Einstein relation.
Assuming the paraxial approximation and an untextured surface so that photons are detected by the camera along a line normal to the samples surfaces, the angular emission of photons is ignored, and integration is applied through the cell depth alongz.
Using Equation 2.3.3, and accounting for photons either transmitted through the upper surface of the sample without reflection, or photons originating in the sample, reflecting off the rear then being transmitted through the upper surface, the detected photon count rate may be determined. The detected rate will be dependent on the emitted rate of Equation 2.3.3 for djγ,emittedd¯hω and the instrument function of Equation 2.3.4 as
djγ,detected(¯hω)
d¯hω =FI(¯hω)djγ,emitted(¯hω)
d¯hω . (2.3.10)
Substitution of Equation 2.3.6 and integration are then performed. The excess carrier concentration profile is solved at a depthz. The solution is
ne(z) =Aez/Le+Be−z/Le (2.3.11)
whereA= ne(0)r−
r−+r+e2d/Ld andB = ne(0)r+
r++r−e−2d/Ld. The parameters are defined asr±= 1±r, andr =SrLe/De [350]. Substituting Equation 2.3.8 into the solutionsA and B gives
A= r−(n2i/NA)eeVakT
r−+r+e2d/Ld , (2.3.12)
B= r+(n2i/NA)eeVakT
r++r−e−2d/Ld. (2.3.13) Upon substitution of the solutions into the integral, and subsequent expansion of the terms, a number of integrals result. The following equations summarize the results of
these integrals [351, 352]
X1= Z d
0
ez/Le−zα(λ)
dz=−Le ed(1/Le−α(λ))−1
Leα(λ)−1 (2.3.14) X2=
Z d 0
e−z/Le−zα(λ)
dz= Le 1−e−d(1/Le+α(λ))
Leα(λ) + 1 (2.3.15) X3 =
Z d 0
ez/Le−(2d−z)α(λ)
dz= Lee−2dα(λ) ed(1/Le+α(λ))−1
Leα(λ)−1 (2.3.16)
X4= Z d
0
e−z/Le−(2d−z)α(λ)
dz = Lee−2dα(λ) ed(α(λ)−1/Le)−1
Leα(λ) + 1 . (2.3.17) which can be substituted to obtain the analytical solution of the electroluminescence spectrum. The analytical solution written in units of wavelength is
jγ= Z λ1
λ1
FI(λ)(1−R(λ))α(λ)2πc
λ4 e(−2π¯hc/λ−eVa)/kT× X1r−
r−+r+e2d/Le − X2r+
r++r−e−2d/Le − Rr(λ)X3r−
r−+r+e−2d/Le − Rr(λ)X4r+ r++r−e2d/Le
dλ.
(2.3.18)
To consider the solid angle Ω of a cone with apex angle 2θ, the area as a spherical cap on a unit sphere is
Ω = 2π(1−cosθ) (2.3.19)
which assumes a Lambertian emission of light. By setting θ= 90◦ the solution Ω = 2π results, representing emission into a half sphere above the solar cell, as presented in Equation 2.3.18.
This solution gives an emission flux depending on the applied voltage to the solar cell, where the density of minority carriers into the base of the solar cell decreases exponentially with the diffusion length. The diffusion length for a high performance solar cell is larger than the thickness of the solar cell (≈170−230àm). Thus, taking the generation rate gγ out of the integral is appropriate when a low defect density is assumed, or when the diffusion length is large enough to allow generation of photons uniformly across the thickness of the solar cell. For large defect densities the excess minority carrier distribution is retained in the integral (via the generation rate as a function of depth in the solar cell).
To summarize, the solution of the electroluminescence flux shown in Equation 2.3.18 has assumed the low injection condition, that the emitter is negligible in thickness com- pared to the base of the solar cell, and that the light emission is observed under the paraxial approximation [305] and with negligible refraction at the upper surface of the solar cell. The distribution of the photons is solved from the distribution of the minority carriers at depth z in the silicon wafer solar cell, while the density of photons in the depth of the solar cell result from the recombination of excess minority carrier electrons, and majority carrier acceptors. The photons generated are integrated over their path in the silicon, ignoring the thin n-type region, to account for reabsorption.