OPTICAL ABSORPTION.

In this section, we will study the optical absorption properties of methylammonium lead iodide perovskite and silicon and discuss why studying this aspect is relevant to our project. Firstly, it can bsaid that obtaining the absorption spectrum of a semi-conductor is interesting because it gives information on whether the semi-conductor is of direct or indirect nature. Indeed, depending on the nature of the transition of a semi-conductor, the absorption of the latter will have a different dependence with photon energy. This will be further explained in the following subsections. Additionally, obtaining the absorption spectrum is useful to determine the bandgap of the material.

The absorbance of a material is related to its transition rate Wi→f . The transition rate is the
probability of a transition per unit time. More precisely, we consider the transition related to exciting an electron from an initial state to a final state when photon absorption occurs. Wi→f is proportional to the square of matrix element and the joint density of states. It is described by Fermi’s golden rule :

The matrix element is proportional to the strength of the coupling between the initial and final
states of our system. However, computing the matrix element is out of the scope of this project.
Hence, we will omit it and just consider the dependence with JDOS. This will be enough to build an approximate idea of the evolution of the absorbance as a function of the energy. Of course, because of this simplification, we will not be able to compute actual values of the absorbance for our perovskite and silicon.

Note that the main developments in this Section are greatly inspired by Fox’s "Optical Properties
of Solids" [36] and Yu et al.’s "Fundamentals of Semiconductors" [52].

OPTICAL ABSORPTION OF MAPI

1 - Joint density of states in momentum and energy space for a direct semi-conductor

In order to derive the absorption spectrum of MAPI, the first step is to compute the joint density of states (JDOS). The JDOS function provides a measure of the number of direct allowed optical transitions between the occupied valence band electronic states and the unoccupied conduction band electronic states separated by photon energy ℏω. [53]. In order to obtain the JDOS, we must first obtain an expression for the density of states. The density of states function g(E) expresses the number of states that electrons are allowed to occupy given a specific energy level or, in other words, the number of electron states per unit volume and energy. For electrons within a band, the density of states per unit range g(E) is obtained from [36] :

with g(k) the density of states in the momentum space. A factor 2 is needed because we must consider that there are 2 electron spin states for each k state. The density of states can then be isolated :

with dE/dk the gradient of the E-k dispersion in the band diagram.

In 3D, the entirety of possible wave vectors k are the vectors starting at the origin and finishing at the cubic lattice nodes. The lattice is constructed in the k-space and is made of elementary cubes of length 2π/L . In order to compute g(k), we consider the incremental volume between 2 shells of radii k and k+dk in the k-space, as seen in Fig. 44.

Figure 44: The allowed wave vector states in k-space form a cubic lattice. A spherical shell gives the number of allowed states at a specific radius |k|.

g(k) is calculated with the ratio between the volume of the shell and the volume associated to one state associated to a specific k wave vector.

The volume of a state associated to a specific k wave vector is given by Vstate = (2π)^3. The volume of the shell is Vshell = surface ∗ thickness = 4πk^2dk. Finally, we obtain that :

NB : Note that if we wanted to compute the density function in the 2D space, for example, we would have used the area between circle of radius k and k+dk

As shown in the pervoskite section of the website, MAPI can be considered a direct gap semiconductor with a bandgap Eg = 1.57 eV in the tetragonal phase.

For a direct bandgap semiconductor, we can approximate the
electronic band structure with an isotropic four-band Kane model as shown in Fig. 45. εF is at the valence band maximum (VBM). The E-k relationships for the bands (conduction, heavy hole, light
hole) are the following, with m∗ the effective masses :

Figure 45: Band structure of a direct gap III-V semiconductor near k=0. E=0 corresponds to the top of the of the valence band and E=Eg corresponds to the bottom of the conduction band. Four bands are shown : the heavy hole (hh) band, the light hole (lh) band, the split-off hole (so) band and the electron (e) band.[36]

Now that we have details on the band structure, we can evaluate the JDOS by evaluating the
density of states g(E) at Ei and Ef . On Fig. 45, we can find that energy conservation during a light
or heavy hole transition necessitates respecting the following relationship :

where m∗h is either m∗lh or m∗hh depending on if we consider a light hole or heavy hole transition. It can be useful to add a new notation to simplify the previous expression: the reduced electron-hole mass μ :

Hence, the energy conservation can be rewritten as :

We now have the relations needed to compute Eq.24. Introducing Eq. 33 and 26 in Eq. 24, we obtain the JDOS :

Note that for photons of energies lower than the bandgap, it is obvious that g = 0, because the photon will not be absorbed. For photons of higher energies, the density of states factors evolves as (ℏω −Eg)^( 1/2)

2 - Absorption spectrum and formula derivation

In the previous point, we have determined an expression of the JDOS function for a direct semiconductor using the isotropic four-band Kane model for the bands.

Considering only its dependence with JDOS and omitting the matrix element, the absorbance
spectrum for a direct semiconductor is expected to follow this trend :

Hence, we expect that for photons of energy ℏω < Eg, no absorption occurs. For photons of energy
ℏω ≥ Eg, the absorbance evolves as (ℏω − Eg) ^(1/2) . In other words, the dependence of α^2 with energy
should be linear when absorption occurs. The reduced mass also affects the absorbance : the higher μ is, the stronger the absorption. The expected dependence of the absorbance squared is plotted in Fig. 46. Note that the absorbance is only plotted for values of energy close to the bandgap. This is due to the fact that the Kane model we have considered for the bands is only valid near k=0. For photon energies too far from the bandgap, the JDOS no longer obeys Eq. 35. At this point, it is better to use the actual band structure of the semiconductor.

Figure 46: Expected dependence of α^2 with energy for the direct semiconductor MAPI. The bandgap values of the orthorhombic, tetragonal and cubic phases are found when the absorbance first reaches 0.

Ziang et al. studied the absorption spectrum of CH3NH3PbI3 in tetragonal phase by using spectroscopic ellipsometry and obtained the results in Fig. 23. Using the free software PlotDigitizer, we were able to extract the data from the figure and use it for our project. Using this data, we obtained Fig. 47. Whilst we understand that some degree of precision is lost when using PlotDigitizer, the obtained results are precise enough for the context of our project.

Figure 47: Experimental absorbance of MAPI as a function of energy. Data obtained by using Plot Digitizer on the figures from Ziang et al.

Finally, using the data obtained from PlotDigitizer, we were able to plot α^2 as a function of the energy near the bandgap. We observe that the experimental data follows the trend expected from theory. Indeed, near the bandgap, the absorbance squared evolves linearly. After extrapolating the experimental data, the absorbance is expected to first reach 0 near 1.56 eV, which is indeed the bandgap of tetragonal MAPI, as expected.

Figure 48: Experimental absorbance of MAPI as a function of energy (near the bandgap). The blue curve represents the data. The red line is a linear approximation of the data.

Hence, we can observe usefulness of obtaining the spectrum of the absorbance of a semi-conductor.
The band-gap can be found as the point at which the absorbance reaches 0. Moreover, if the evolution
of α^2 is linear with the energy, it means the semiconductor is direct.

OPTICAL ABSORPTION OF SILICON.

When considering Si instead of MAPI, our results will differ. Indeed, silicon is an indirect semiconductor. Indirect transitions are a process that involve both photons and phonons. This is hence a second-order process : a photon is destroyed and a phonon is either destroyed or created. In contrast, direct transitions are a first-order process, they only involve the destruction of a photon and phonons do not participate. As a consequence, the transition rate for indirect absorption is smaller than for direct absorption. This will be shown in the following sections.

We examine the contribution to the transition probability R_ind using an extension of the Fermi’s Golden Rule to second order perturbation :

In this equation, |0⟩ represents the initial state of the system and |f ⟩ the final state.

If we convert the summations over kc and kv to an integration over the valence and conduction band energies Ev and Ec with their density of states.

Assuming that the bands are three-dimensional and parabolic, we have that :

By substituting Dv and Dc and integrating over Ev, we obtain :

By using the change of variable we obtain :

Finally, after computing the integral, we obtain the following relation for the imaginary part of the dielectric constant :

The absorption coefficient is connected to the imaginary part of the dielectric function, εi, and the refractive index n(ω) :

Or, as proposed in Mark Fox’s "Optical Properties of Solids" [36], the absorption dependence on energy of an indirect semiconductor can simply be written as :

where ℏω is the energy of the photon and ℏΩ is the energy of the phonon. This expression shows that, unlike direct semiconductors which have a threshold precisely at Eg, the absorbance of an indirect semiconductor will have a slightly different threshold depending on ±ℏΩ (- means the phonon is
absorbed, + means it is emitted).

As seen on Fig. 19, Si’s fundamental bandgap (smallest possible indirect transition) corresponds to Eg = 1.1 eV. Its optical band gap corresponds to the smallest direct transition and has a value of E1 = 3.05 eV. Hence, when phonons of energy higher than E1 interact with Si, we will be able to observe an absorbance spectrum behavior corresponding to a direct semiconductor.

Green et al. have measured the optical properties of intrinsic silicon at 300K and made their data publicly available. The data was obtained using transmission measurements. Using this data, we are able to plot the absorbance of Si as a function of the energy. Since Si has an indirect bandgap at 1.1 eV, in the range of energies near the bandgap, we should observe a linear evolution of α^(1/2).

As per Fig. 49, the experimental data evolves as predicted in theory : α^(1/2) has a linear dependence with energy in the spectral region near the bandgap. This corresponds to the behaviour of an indirect semiconductor. However, we can note that the absorption coefficient does not first reach 0 at Eg = 1.1eV. The data extrapolates approximately to 1.05 eV when α = 0, which means that a phonon of energy≈ 0.05 eV has been absorbed.

Figure 49: Evolution of the absorbance α^(1/2) as a function of the photon energy of intrinsic Si at 300K in the spectral region near the fundamental bandgap Eg = 1.1eV . The data was obtained with transmission measurements from Green et al. [54]

In Fig. 54, we plot the evolution of the absorbance with energy in the spectral region around the optical bandgap, which is the smallest direct transition. Indeed, even though Si is an indirect semiconductor, direct transition might still occur. These transitions happen in Si if the phonon energy
is larger than E1 = 3.05 eV. We would then expect to observe a linear relation between α^2 and energy near the optical bandgap. As per Fig. 50, this is indeed confirmed. Once the threshold of E1 is reached, the direct absorption overthrows the indirect absorption. Indeed, indirect absorption is weaker and negligible when plotted on the same scale with direct absorption. This thus shows the second-order nature of indirect absorption.

Figure 50: Evolution of the absorbance α^2 as a function of the photon energy of intrinsic Si at 300K in the spectral region near E1 = 3.05eV . The data was obtained with transmission measurements from Green et al [54]

COMPARISON OF MAPI AND SILICON

MAPI and Si differ in the nature of their bandgap : MAPI is a direct semiconductor whilst Si is an indirect semiconductor. As mentioned earlier, the indirect transition is a second-order process and the direct transition is a first-order process. This leads to the fact that the direct transition has a higher transition rate. Indeed, on Fig. 23 the absorption coefficients of c-Si and MAPI are compared. It can be noted that the absorption coefficient rises faster with increasing energy (or decreasing wavelength λ) for MAPbI3 than it does for Si. This occurs even though silicon’s bandgap is smaller than MAPI’s. Because of this, it can be generally assumed that MAPI is better at absorbing sunlight than silicon.

It is interesting to note that, as observed on Fig. 23, the absorption coefficient of Si overcomes the absorption coefficient of MAPI at a certain value of energy. This can be explained by the fact that the energy of the photons has become high enough to allow the Si to undergo direct transitions