ECOLE POLYTECHNIQUE DE LOUVAIN
QUALITY INDEX.
1 - Generalities
An interesting aspect to study when optimizing a solar cell is studying the effect of varying thick-
nesses of the active layer. Two competing phenomenon will play a role in the choice of the optimal
thickness of the cell : absorption and recombination mechanisms.
One the one hand, in order to maximize the absorbance, the active layer should be as thick as
possible in order to absorb as much photons as wanted. Should this be the only deciding factor in
designing the thickness of the active layer, the latter would have to respect the following equality :
where α is the absorption coefficient of the largest absorbed wavelength.
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However, on the other hand, another phenomenon that affects the cell’s efficiency is the scattering
of the charge carriers. The thicker the cell, the bigger the chance of scattering of the charges. If scattering occurs, it will be assumed that it leads to the decay of the particle, thus inhibiting these
charges from contributing to the current of the photovoltaïc module. Another way to look at this is by studying the mean free path of the charges. The thicker the cell, the more probable the thickness will be bigger than the mean free path. Hence, the higher the probability of collision between an electron and a hole. If this phenomenon were the only one impacting the cell efficiency, the thickness would have to respect this inequality :
where λ is the mean free path of the minority carriers.
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The true optimal thickness is a value that lies between both of these inequalities. During this
section, our objective will be to determine the optimal thickness for both a MAPI-based cell and a
traditional silicon cell. It will also allow us to observe how scattering affects the device performance.
2 - Total probability of survival of the electron/hole pair
Our objective is to calculate the probability that holes and electrons make it to the contacts at a given frequency : P(eh|ν). Expressing this as a function of d the device thickness will allow us to optimize our thickness by choosing the one that leads to the highest probability of survival. To achieve this, we will use the following variables :
Before anything else, the first aspect that must be considered is whether the solar cell we will use in our calculations is in a n-i-p or a p-i-n configuration. As illustrated in Fig. 55, in the n-i-p configuration, the cell is illuminated on the side of the electron transport layer (ETL) and in the other configuration, it is illuminated on the side of the hole transport layer (HTL).
With a n-i-p configuration, the holes have a greater distance to reach the contacts whilst in the p-i-n, the distance from the holes to the contact is reduced. Depending on the relative mobilities of
the charges, we can consider one configuration over the other. For example, if the holes were to be less mobile than the electrons, it would be preferable to work with a n side illumination. Indeed, that way, the holes will need to cover a smaller distance to the contacts than it would have to in a p side illumination. Depending on the choice of configuration, the choice of variables in our calculations will change, as will be explained later.
Figure 55: Left : Scheme of a solar cell in a n-i-p configuration. Right : Scheme of a solar cell in a p-i-n configuration [55]
To calculate the total probability, we must both consider the total probability of survival of the electrons P(e|ν) and the total probability of survival of the holes P(h|ν). They are given by the following expressions :
In Eq 49 and 50, P(e|z) and P(h|z) are the probabilities, given a position z at which the carriers are created, that they survive to the contacts located at a position d. Depending on the configuration of the cell, they have different expressions. Indeed, the holes and the electrons will have to follow
different paths depending on the chosen configuration, as illustrated on Fig. 56 and Fig. 57
Figure 56: Solar cell with n side illumination (nip configuration). The electron path is z and the hole path is d-z where d is the thickness of the active layer of the solar cell.
Figure 57: Solar cell with p side illumination (pin configuration). The electron path is d-z and the hole path is z where d is the thickness of the active layer of the solar cell
Finally, after taking these considerations into account, we have the probabilities for each type of
configuration :
n-i-p configuration
p-i-n configuration
The mean free paths, le and lh can be calculated using the Drude model. The Drude model is the
kinetic theory of gases applied to the electrons of a metal. Several hypothesis are made in this theory :
• Independent electron approximation : the electrons in the metal behave like particles in an ideal
gas, which means there are no interaction or collisions between the electrons.
• Cations are immobile and electrons can collide with them, thus changing their velocity and
direction.
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This model leads to several results related to the transport properties of electrons in materials. The
one we are interested in are the following :
with μ the mobility, e the charge of an electron, m∗ the effective mass, l the mean free path, vd the
drift velocity of the particle and τ the relaxation time (=average time between collisions).
To compute the mean free path of the holes and the electrons, we need to compute both their drift velocity and their relaxation time. In order to compute the drift velocity, we must first calculate the thermal velocity. Normally, the thermal velocity of the carriers is a spectrum but we will make the supposition that it is constant. The thermal velocity considered will be the most probable velocity. This occurs at an energy 3/2kBT above the CBM for electrons and an energy 3/2kBT below the VBM for holes. The proof that electrons of energy 3/2kBT are the most probable has been shown by Poncé et al. in [56]. Indeed, when considering an idealized system with a parabolic band of mass m∗, the average scattering rate can be written as :
The expression of τ (x) can be found in [56]. By plotting the average scattering rate, Poncé et al.
obtained Fig. 58. It can be deduced that the average scattering rate reaches a maximum at x=3/2.
Hence, it means that when computing the free mean path of the carriers, the most relevant energy is indeed 3/2kBT .
Figure 58: Most representative electron energy. x = εkBT represents the electron energy in units of the thermal energy. The weight factor x^(3/2)exp(−x) appearing in the integral peaks at x = 3/2, therefore the most representative electrons are those found at an energy (3/2)kBT from the band edge. [56]
We then have that :
By isolating the thermal velocity, we end up with :
Here, in order to calculate the effective mass, we will use a harmonic mean :
The drift velocity is obtained with the following assumption :
Whilst not rigorous, this 10^(−2) factor was found to be close enough for the example of Si [57]. The same assumption will be made for CH3NH3PbI3.
In order to compute the mean free paths le and lh, we must also take into consideration the charge carrier lifetime τ . In the section of our website calculating the mobility, we have determined the lifetime between collisions τ m. Using this value would not be correct as it is the lifetime between scattering events and not actually the charge carrier lifetime. Since the charge carrier lifetime is difficult to model and also depends on the doping of non-intrinsic configurations, we will make the following assumption:
This is not generic but it will allow us to obtain the correct order of magnitude in our calculations.
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Finally, the second term in Eq 49 and 50, p(pair|z,ν), is the normalised probability density that the pair is excited at a distance z from the illuminated side with a photon of frequency ν. It is given
by :
Injecting Eq. 51, 52, 62 in Eq. 49 and 50, we obtain these final expressions for respectively the total
probability of survival of the electrons and the holes at a given frequency for an n-side illumination.
Similarly, we can also obtain equations for a p-side illumination.
n-i-p configuration
p-i-n configuration
From P (e|ν) and P (h|ν), we can now compute the total probability of both charges making it to
the contacts P(eh|ν). It is simply found by doing :
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The total probability is just the minimum between the probability of survival of the electrons and of the holes. Indeed, if the least probable carrier survives, the other carrier also survives by default.
QUALITY INDEX OF SILICON.
For our calculations, the absorption spectrum of silicon is needed. It was measured by Green et al. [54] and is represented in Fig. 59 as a function of the frequency of the incident photons.
Figure 59: Evolution of the absorbance α of silicon at 300K as a function of incident light frequency. The data was obtained with transmission measurements by Green et al. as a function of the wavelength and was then modified to be expressed as a function of frequency. [54]
We have the following data for the calculation of the survival probabilities :
Table 11: Data used for calculating the probability of survival of charge carriers in the silicon active layer
The mean effective masses of silicon were calculated with the harmonic mean as previously men-
tioned. Due to unsatisfying results in our section concerning the mobility, we have chosen not to use the lifetime between collisions τm calculated there. Instead, we have found appropriate values of τm by using the Drude formula : μ = eτm/m∗ and testing several values of τm until reaching values of the mobility that corresponded to experimental values found in the scientific literature.
As seen in Tab. 11, for silicon, we have found that le > lh, this means that electrons have a better
mobility than the holes. Hence, it is better to chose a configuration in which the electrons have the
biggest distance to cross to reach the contacts. For silicon, we thus consider a pin configuration for our calculations.
We can first compute the total probability as a function of the frequency of the photon. This
depends on the value of the active layer thickness d. To begin, we first plotted the probabilities for
different values of the thickness, ranging from 10μm to 300μm, as seen on Fig. 60, 61 and 62.
Figure 60: Total probability of survival of the electrons in silicon as a function of the incident photon frequency for a pin configuration. The probabilities are plotted for different values of the layer thickness ranging from 10 μm to 300 μm.
​​Figure 61: Total probability of survival of the holes in silicon as a function of the incident photon frequency for a pin configuration. The probabilities are plotted for different values of the layer thickness ranging from 10 μm to 300 μm
Figure 62: Total probability of survival of both charges at the same time in silicon as a function of the incident photon frequency for a pin configuration. The probabilities are plotted for different values of the layer thickness ranging from 10 μm to 300 μm
The optimal thickness can be found with the total probability of survival of the pair. As seen on
Fig. 62, depending on the frequency of the incident photon, the optimal thickness will vary. In order to find an appropriate optimal thickness for the whole spectrum, we need to chose a thickness such as it is the best for the a majority of the frequency range. In order to do so, we can calculate the area under the curve for each thickness and we will determine the optimal thickness by choosing the curve associated to the thickness with the biggest area. Indeed, since all curves follow the same trend, if the area is the biggest for one thickness compared to the others, it means that the probability of survival is the largest for a majority of the time for this particular thickness of the active layer.
After doing so, we have calculated the following optimal thickness for a silicon solar cell :
This value of the optimal thickness should be taken with a grain of salt. Whilst it gives a coherent
enough order of magnitude, several assumptions had to be made in order to obtain it so it is not a
rigorous value. Finally, we can plot the total probability of survival of the pair in silicon at the optimal thickness on Fig 63.
Figure 63: Total probability of survival of both charges at the same time in silicon as a function of the incident photon frequency for a pin configuration at the optimal thickness d_opt = 18.18 [μm]
QUALITY INDEX OF MAPI.
In the mobility section of this website, we have only focused on orthorhombic MAPI. This is mainly due to the fact that it is the easiest phase to work with. In the orthorhombic form, the methylammonium ion has a fixed known position, which makes calculations much easier than if we had to consider all possible configurations for the other phase.
Hence, in this section, we will also only consider
the orthorhombic phase. The absorption spectrum for orthorhombic MAPI was calculated using first
principle calculations by Wang et al.
Figure 64: Evolution of the absorbance α as a function of the photon frequency for orthorhombic MAPI. This graph was obtained by Wang et al. by using first principle calculations. Optical absorption properties are calculated using the independent particle approximation with 20×20×20 k-meshes.[58]
Table 12: Data used for calculating the probability of survival of charge carriers in the orthorhombic MAPI active layer
The data used for calculating the survival probabilities for charge carriers in orthorhombic MAPI
are present in Tab. 12. As we did not get satisfying results for calculating τm in our mobility calculation section, the lifetime between collisions τm for orthorhombic MAPI comes from experimental data, cfr. Fig. 36 of the mobility section. The effective masses are the same as in the mobility section of this website. From this, the mean free paths lh and le were computed. The mobility of the holes is slightly higher, making them more mobile than the electrons. Hence, in theory, the nip configuration would be the best in this case scenario.
First, we have plotted the survival probabilities for thicknesses ranging from 0 μm to 300 μm and
obtained the following results.
Figure 65: Total probability of survival of the electrons in orthorhombic MAPI as a function of the incident photon frequency for a nip configuration. The probabilities are plotted for different values of the layer thickness ranging from 0 μm to 300 μm
Figure 66: Total probability of survival of the holes in orthorhombic MAPI as a function of the incident photon frequency for a nip configuration. The probabilities are plotted for different values of the layer thickness ranging from 0 μm to 300 μm
Figure 67: Total probability of survival of both charges at the same time in orthorhombic MAPI as a function of the incident photon frequency for a nip configuration. The probabilities are plotted for different values of the layer thickness ranging from 0 μm to 300 μm
In order to find the optimal thickness, we used the same logic as used for silicon. Finally, we
obtained the following optimal thickness for orthorhombic MAPI:
Similarly to the optimal thickness obtained for silicon, this result is to be taken with precaution.
It is not a precise answer as many assumptions were made. However, it can give an idea of the order of magnitude. This result can also be used to be compared to the results obtained for silicon. This will be done further on in the discussion.
Finally, the total probability of survival of the pair can be plotted for the optimal thickness
d_(opt,MAPI) as seen on Fig. 68
Figure 68: Total probability of survival of both charges at the same time in orthorhombic MAPI as a function of the incident photon frequency for a nip configuration at the optimal thickness d_opt = 3.03 [μm]
DISCUSSION
From our results, we can observe a sizeable difference between the optimal thickness for silicon and for MAPI. Silicon requires a thickness of 18 [μm] and MAPI requires a thickness of 3 [μm]. The difference between both thicknesses is of approximately an order of magnitude of 10.
However, these are just theoretical results. In reality, the silicon layer must be thicker than that.
Indeed, silicon is an indirect semi-conductor. This means that in order to absorb a photon, there must be some amount of photon-phonon interaction. In order for this to be possible, it is preferable to have a thicker active layer, as this will make the probability of photon-photon interaction higher. MAPI, on the other hand, is a direct semi-conductor so a thicker layer is not necessary as no photon-phonon interaction is necessary for photon absorption to occur. For non-wafer based silicon, the optimal values of the active layer thickness lie in the range 20 - 100 [μm] [59]. A thin film is defined as a film of thickness ranging from fractions of nanometer up to several micrometers of the thickness. As such, silicon cannot be considered a thin film. Moreover, most silicon based solar cells use silicon wafers. For those, thicknesses between 200 and 500 [μm] are generally used, partially due to practical reasons (manufacturing and handling the wafers).[60]
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Using the total probability of survival that we have calculated, we can go back to our results from
the Shockley-Queisser section and try to observe what impact the probability of survival will have on the efficiency. The probability is integrated in our calculations by simply multiplying Qs (the number of photons absorbed per unit time and area) by the probability of survival of the pair. For this discussion, we will consider 3 thicknesses : the optimal thickness d_{opt,Si} = 18.18 [μm],
d = 100 [μm] and d = 200 [μm].
By changing the thickness, we change the total probability of survival which in turn, of course affects the Shockley-Queisser limit. We obtain the results in Fig. 69, 70, 71. Firstly, we can notice that the efficiency curve as a whole decreases. This is expected. Now that we consider the total probability of survival of the charges, we no longer consider that all charges will reach the contact.
This means that less carriers will be able to contribute to the current, thus decreasing the efficiency of the cell.
By increasing the thickness, we considerably decrease the efficiency of the solar cell. This is ex-
pected. The optimal thickness calculated was of approximately 18 [μm]. By changing the thickness
to 100 [μm] and 200 [μm], we are going further from the optimal thickness. This is why the efficiency decreases.
We can also compute a new efficiency limit for orthorhombic MAPI as seen on Fig. 72. For MAPI,
we obtain lower efficiencies than we did with the regular Shockley-Queisser in Fig. 35 : we now have a maximum efficiency of 25.92% for orthorhombic MAPI. As just explained, obtaining lower efficiencies was to be expected. According to our calculations, the maximum efficiency is now obtained at a bandgap of 1.33 eV, which is higher than in Fig. 35. Or, in other words, the maximum efficiency is obtained at higher frequencies than before :
Finally, all results are assembled in Tab. 13.
Table 13 : Optimal parameters obtained after taking into account the probability of survival of the charges. MAPI is considered at d_{opt} = 18 [μm]. Silicon is considered at d_{opt} = 3.13 [μm], 100 [μm] and 200 [μm]. We also consider the efficiency at the bandgap of each material : 1.12eV for silicon and 1.71 for orthorhombic MAPI.
Figure 69: Shockley-Queisser efficiency limit for silicon taking into account the probability of survival of the charge carriers for a p-side illumination at thickness d_opt = 3.13[μm]
Figure 70: Shockley-Queisser efficiency limit for silicon taking into account the probability of survival of the charge carriers for a p-side illumination at thickness d = 100[μm]
Figure 71: Shockley-Queisser efficiency limit for silicon taking into account the probability of survival of the charge carriers for a p-side illumination at thickness d = 200[μm]
Figure 72: Shockley-Queisser efficiency limit for orthorhombic MAPI taking into account the probability of survival of the charge carriers for a n-side illumination at optimal thickness d_opt = 18[μm]
In conclusion, by using the probability of survival of the charges, we obtain lower efficiencies and higher frequencies in comparison to the results we obtained in the Shockley-Queisser section of the website.