Theoretical study on photoelectric properties of FAPbI₃ doped with Ge

The photoelectric conversion efficiency of solar cells is related to the band structure and photoelectric property of materials. Figures 2 and 3 show the band structure and DOS of FAPbI₃. It can be seen that there is no obvious difference between the conduction band of the three structures. The conduction band near the Fermi surface is mainly contributed by the electrons of Pb 6p orbital, while the deep conduction band is mainly contributed by the electrons of I 5s and Pb 6s orbitals. The valence band of the three structures is different, mainly contributed by the electrons of I 5p orbital. The valence band DOS near the Fermi surface of the tetragonal structure is significantly higher than that of the other two structures, and the electrons DOS of the 5p orbital of I is also significantly higher. This indicates that the valence band of I near the Fermi surface of the tetragonal structure is more localized than that of the other two structures. The energy states of other elements in FA are mainly distributed in the deep energy level below −5 eV and the DOS distribution is sharp, strong local and not involved in bonding, so their contribution to the DOS near the Fermi surface is almost zero. It shows that the covalent interaction between the organic ions FA⁺ and Pb²⁺, I⁻ is weak and not as strong as that between Pb²⁺ and I⁻, but FA⁺ is crucial to the stability of crystal structure. The stability tolerance factor of ABX₃ is given by the Goldschmidt equation $\left(T=\tfrac{{R}_{A}+{R}_{X}}{\sqrt{2}({R}_{B}+{R}_{X})}\right)$ [36]. The closer T is to 1, the more stable the crystal structure. Since the radius of Pb²⁺ at position B is relatively large, it is difficult to find a single element A with A matching radius. The radius size of organic molecule group FA⁺ is similar to that of Pb²⁺, and the tolerance factor T of FAPbI₃ is 0.99, so it is quite stable.

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It is found that the super strong spin–orbit coupling (SOC) effect will reduce the conduction band when calculating the band structure of materials containing heavy elements. Because FAPbI₃ contains the heavy element Pb, the super strong spin–orbit coupling effect cannot be ignored in the calculation of band structure [37, 38]. As can be seen from figure 4, the energy level drops significantly when the SOC effect is taken into account. The band structure values of three FAPbI₃ structures with and without SOC effect are given in table 1. It can be seen from table 1 that the band gap width of FAPbI₃ of cubic structure is 1.537 eV without considering SOC effect, slightly higher than the experimental value of 1.48 eV, but within the acceptable range. This indicates that it is feasible for us to use PBE in GGA as exchange association functional, and it is not necessary to use HSE06 with higher precision. It can be seen from the band structure diagram that the conduction band is mainly affected by the SOC effect, which destroys the degeneracy of the conduction band and causes the energy level to split and move down significantly, while the valence band is basically unaffected. At the same time, it was found that the symmetry of the geometric structure of the system would be enhanced as the number of delocalized electrons increased after considering the SOC effect. Based on this, we found that the band gap width of three structures were only 1.015 eV, 0.525 eV and 1.179 eV after considering SOC effect. It is worth noting that the band structure of the tetragonal structure is more affected by the SOC effect, and its band gap value is only 0.525 eV. It is obvious that the relativistic effect of the system can not be ignored after the SOC effect is taken into account, so the electronic structure of the system has changed. However, it is far from the experimental value after considering the SOC effect, so we do not consider the SOC effect in the following studies.

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Figure 5 shows the distribution of charge density of three crystal structures of FAPbI₃. Blue represents the outside of the isosurface, gray represents the inside of the isosurface, and yellow points represent the electron clouds (the possible positions of electrons). From the color and shape of the isosurface and the distribution of the electron clouds, we can see that the charge is mainly concentrated on FA⁺, Pb²⁺ and I⁻, in which the electron clouds of Pb²⁺ and I⁻ overlap obviously, while FA⁺ is almost isolated. This indicates that Pb²⁺ and I⁻ have stronger effects, which is consistent with the previous analysis results.

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As can be seen from figure 6, the absorption coefficients of the three structures are not significantly different. The absorption effect of orthogonal structure is the best in the ultraviolet region, while the absorption effect of tetragonal structure is the best in the visible and infrared region. By comparing the solar spectrum, it can be seen that the tetragonal structure is the best for solar energy absorption among the three crystal structures.

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From the calculated data in table 2, it can be seen that with the doping of Ge, the crystal structure changes significantly. The change rules are summarized as follows:

• 1.  
  The change in the lattice constant results in a significant change in the volume of the crystal. With the increase of Ge's doping ratio, the crystal volume first increased and then decreased, and the volume of pure FAGeI₃ was smaller than that of pure FAPbI₃. We know that the radius of Ge²⁺ (0.073 nm) is smaller than the radius of Pb²⁺ (0.119 nm), but both Ge²⁺ and Pb²⁺ have repulsion effects at the same time. When repulsion is dominant, the volume increases and vice versa. From the calculated data in table 2, it can be seen that the repulsion of Ge²⁺ and Pb²⁺ plays a major role as the doping ratio is before 0.5, so the volume increased first and then decreased. It is worth noting that the volume of pure FAGeI₃ is smaller than that of pure FAPbI₃, because we don't care about repulsion, we just care about the radius of the ions.
• 2.  
  The space group changed after Ge's doping, but the pure FAPbI₃ and the pure FAGeI₃ space group were the same, and both were P42/MNM.
• 3.  
  The cell angle doesn't change, and it's all 90°.
• 4.  
  The tolerance factor T in table 2 is solved by the previously mentioned Goldschmidt equation. For organic-inorganic ABX₃ halides, Li et al showed that stable perovskite needs to meet 0.81 < T < 1.11 [39]. We found that they all satisfy with the stability condition.

Figure 8 shows the band structure of FAPb_1−xGe_xI₃ after different doping ratio (x = 0.33, 0.5, 0.67, 1). It can be seen that FAPb_1−xGe_xI₃ is a direct band gap semiconductor and the conduction band and valence band of FAPb_1−xGe_xI₃ have changed after doped with Ge.

• 1.  
  The change of conduction band is more obvious than that of valence band after doping. With the increase of Ge's doping ratio, the conduction band near the Fermi surface become denser first and then sparser. After all Pb are replaced by Ge, the conduction band near the Fermi surface is the same as before without doping. Combined with the partial density of states(pDOS) distribution in figure 9, it can be seen that the 4p orbital electrons of Ge and the 6p orbital electrons of Pb in the conduction band become hybridized after doping with Ge, which leads to the increased contribution of electrons to the conduction band, and the density of electrons becomes dense. With the increase of the Ge's doping ratio, the content of Pb decreases correspondingly, and the hybridization will gradually weaken, which leads to the gradual decrease of electron contribution. When Ge completely replaces Pb, Ge's contribution to the conduction electrons near Fermi surface is similar to that of Pb, so that the conduction band near Fermi surface in the band structure diagram becomes as sparse as before.
• 2.  
  As the increase of doping ratio, the band gap increases first and then decreases gradually. There are two reasons as follows. (1) When x is less than 0.5, Ge's 4p orbital electrons and Pb's 6p orbital electrons are hybridized in the deep energy level of the conduction band, and their contribution to the DOS of the conduction band near Fermi surface is less than that of pure Pb's 6p orbital electrons. (2) When x is less than 0.5, I'5p orbital electrons contribute more to the valence band near the Fermi surface, while when x > 0.5, the corresponding contribution gradually decreases, as shown in table 3. For the valence band near the Fermi surface, the greater the contribution of I, the stronger the nonmetal property, which leads to the decrease of the valence band and the increase of the band gap width. With the increase of the proportion of Ge, the contribution of its 4p orbital electrons to the DOS near the Fermi surface gradually increases, and the hybridization of Ge's 4p orbital electrons with Pb's 6p orbital electrons also becomes stronger and stronger, so that the material shows more and more obvious metallic properties and the conduction band decreases significantly. However, at the same time, the contribution of I's 5p orbital elections to the valence band near Fermi surface also decreases and the valence band does not decrease, so the band gap gradually decreases.

• 3.  
  Doping Ge will affect the smoothness of conduction and valence band. The smoothness is the curvature of the curve. According to the formula:

$$\begin{eqnarray}&&{E}_{{\rm{n}}}(\vec{k})={\alpha }_{1}{k}_{x}^{2}+{\alpha }_{2}{k}_{y}^{2}+{\alpha }_{3}{k}_{z}^{2}\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{\rm{1}}}{{{\rm{m}}}^{\ast }}\right)}_{ij}=\displaystyle \frac{1}{{\hslash }^{2}}\displaystyle \frac{{\partial }^{2}{E}_{n}(\vec{k})}{\partial {k}_{i}{k}_{j}},i,j=x,y,z\end{eqnarray} \tag{ 4 }$$

Where ${\alpha }_{{\rm{1}}},$ ${\alpha }_{{\rm{2}}}$ and ${\alpha }_{{\rm{3}}}$ are the coefficient, $k$ is the wave vector, ${m}^{\ast }$ is the effective mass, $\hslash$ is the reduced Planck Constant. The eigenvalue of the tensor is the inverse of the effective mass and the eigenvector is the direction of the effective mass. The effective mass of holes and electrons can be qualitatively compared from the curvature of the band structure curve. The effective mass of holes and electrons directly affects their separation and propagation in the material layer, and then affects the photoelectric conversion efficiency of the material. As can be seen from figure 8, the conduction band near the Fermi surface does not change significantly after doped with Ge, while the curvature of the valence band significantly increases. This indicates that the effective mass of the hole decreases obviously, while the effective mass of the electron does not, which is beneficial to the separation and transmission of photogenic carriers, thus increasing the short-circuit current J_SC and improving the photoelectric conversion efficiency of solar cells.

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Figure 10 shows the spectral absorption coefficient of FAPb_1−xGe_xI₃, and it is found that its absorption peak ranges from 300 nm to 500 nm. When x = 0, the absorption peak is near 380 nm in near-ultraviolet region, which is not conducive to the absorption of sunlight. As the doping ratio x increases, the absorption coefficient decreases first and then increases. When the doping ratio is large, the absorption peak is significantly red-shifted, which can improve the effective absorption of infrared light with longer wavelength. When x = 1, the absorption peak is around 500 nm, and the absorption of visible light reaches the maximum. For direct band gap semiconductor materials, the absorption coefficient satisfies the following formula:

$$\begin{eqnarray}&&\alpha \left(h\nu \right)=C{(h\nu -{E}_{g})}^{1/2}\end{eqnarray} \tag{ 5 }$$

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The C in above equation is a constant related to the Hall mobility of the electron, hν is the photon energy, E_g is the band gap width. With the increase of Ge's doping ratio, the band gap decreases, the absorption peak increases and the absorption coefficient increases. Figure 11 shows the spectral reflectivity of FAPb_1−xGe_xI₃. It can be seen from the figure that the reflectivity is in the intermediate state without doping. With the doping of Ge, the reflectivity decreases first and then increases. We know that as the material reflects more sunlight, the photoelectron transition decreases, resulting in a decrease in the photoelectric conversion efficiency. In order to improve the photoelectric conversion efficiency, both absorption and reflection must be considered. The absorption of solar photons by the material must satisfy the photoelectric effect equation:

$$\begin{eqnarray}&&{E}_{K}=h\nu -{w}_{o}\end{eqnarray} \tag{ 6 }$$

Where h is the Planck constant, ν is the frequency of the incident light, w_o is the work function (equal to E_g ). According to formula (6), the photon with energy hν less than E_g cannot be absorbed when it enters. When the energy hν enters much greater than E_g , the photon scatters with the lattice after the transition and converts into heat energy, which cannot be converted into carriers. Therefore, the band gap width must be appropriate to maximize the photoelectric conversion efficiency. Experimental studies have shown that the optimal band gap is between 1.3 and 1.4 eV, because this band gap can absorb all visible light and most near-infrared light of the Sun.

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In order to better analyze the relationship between the band gap and the doping concentration, the data points of the band gap in figure 8 were fitted as shown in figure 12. The functional relationship between the band gap and the doping concentration was obtained as follows:

$$\begin{eqnarray}&&{E}_{g}=25.439{{\rm{x}}}^{4}-{\rm{47}}{{\rm{.701x}}}^{3}+25.443{{\rm{x}}}^{2}-3.958{\rm{x}}+1.626\end{eqnarray} \tag{ 7 }$$

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According to this functional relation, the range of doping concentration x is between 0.55 and 0.65 to obtain the optimal band gap. It can be seen that the photoelectric conversion efficiency of FAPbI₃ can be improved after doped with Ge. So far, we haven't seen the experimental report of FAPbI₃ doped with Ge, but there are a lot of theoretical researches. Such as: Diao et al calculated FAPbI₃ doped with Cs, Ge and Sn by first- principles [25]. They found that when Pb was completely replaced by Ge, FAGeI₃ would become an ideal photoelectric material to replace FAPbI₃. Although the photoelectric performance of the former was slightly lower, the former was non-toxic. Tang et al doped Sn to MAPbI3/FAPbI₃ to reduce the band gap to 1.30/1.40 eV and found that the Sn-doped MAPbI₃ and FAPbI₃ had the larger optical absorption coefficient and theoretical maximum efficiency, especially for Sn-doped FAPbI₃ [40].

It is known that the real part of the conductivity corresponds to the actual dissipation of energy. When electrons hit the lattice, they encounter resistance and transfer energy to the lattice, causing the lattice to vibrate and eventually dissipate energy in the form of joule heat. The imaginary part of the conductivity corresponds to the conversion of the energy of the electric field to the kinetic energy of the electron, which makes the electrons move in simple harmonic motion under the action of an external alternating electric field. Figure 13 shows that the incorporation of Ge changes the real and imaginary parts of the conductivity. When there is no doping, the conductivity is in the intermediate state. When doped with Ge, the real part of the conductivity first decreases and then increases, while the imaginary part of the conductivity first increases and then decreases. As the doping ratio increases, the bigger the real part is, the smaller the imaginary part is, which means more energy is lost. Therefore, the energy loss can be effectively regulated and the photoelectric conversion efficiency can be improved as long as the doping ratio of Ge is controlled.

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In order to better understand the relationship between the photoelectron structure and the optical absorption coefficient, the linear optical properties of FAPb_1−xGe_xI₃ compounds will be discussed below according to the Kohn–Sham particle equation and Kramers-Kronig relation theory [1, 41]. Based on DFT, the frequency-dependent complex dielectric function can be expressed as follows:

$$\begin{eqnarray}&&\varepsilon (\omega )={\varepsilon }_{1}(\omega )+i{\varepsilon }_{2}(\omega )\end{eqnarray} \tag{ 8 }$$

In formula (8), ${\varepsilon }_{{\rm{1}}}(\omega )$ and ${\varepsilon }_{{\rm{2}}}(\omega )$ represent the real part and the imaginary part respectively, ω is the photon angular frequency. The real part can be obtained through the Kramers-Kronig relation:

$$\begin{eqnarray}&&{\varepsilon }_{{\rm{1}}}={\rm{1}}+\displaystyle \frac{{\rm{2}}}{\pi }M\displaystyle {\int }_{0}^{\infty }\displaystyle \frac{{\varepsilon }_{2}(\omega ^{\prime} )\omega ^{\prime} }{{\omega ^{\prime} }^{2}-{\omega }^{2}}d\omega \end{eqnarray} \tag{ 9 }$$

In the above formula, M is the principal component value of the integral. The imaginary part is given by the Kohn–Sham particle equation:

$$\begin{eqnarray}&&{\varepsilon }_{{\rm{2}}}(\omega )=\displaystyle \frac{V{e}^{2}}{2\pi \hslash {m}^{2}{\omega }^{2}}{\displaystyle \int {d}^{3}k\displaystyle \displaystyle \sum _{nn^{\prime} }\left|\left\langle kn\right|p\left|kn\right\rangle ^{\prime} \right|}^{2}f(kn)\times (1-f(kn^{\prime} ))\delta ({E}_{kn}-{E}_{kn^{\prime} }-\hslash \omega )\end{eqnarray} \tag{ 10 }$$

Where e represents the charge of the electron, V is the volume of a single cell, p is the momentum transition matrix, $\hslash$ is the reduced Planck Constant. $kn$ and $kn^{\prime}$ are the wave functions of VB and CB. The absorption equation can be expressed as follows:

$$\begin{eqnarray}&&I(\omega )=2\omega {\left(\displaystyle \frac{{\left[{\varepsilon }_{1}^{2}(\omega )+{\varepsilon }_{2}^{2}(\omega )\right]}^{1/2}-{\varepsilon }_{1}(\omega )}{2}\right)}^{1/2}\end{eqnarray} \tag{ 11 }$$

The propagation behavior of electromagnetic fields in a material is described by the real part of the dielectric constant, while the absorption of light in the material is described by the imaginary part. The real and imaginary parts of the dielectric function of FAPb_1−xGe_xI₃ are shown in figure 14. It is clear from the figure that the real and imaginary parts of the undoped dielectric function are in the intermediate state. When doped with Ge, the real and imaginary parts of the dielectric function decrease first and then increase, because the dielectric function is inversely proportional to the width of the band gap. From figure 14, the real and imaginary parts of the dielectric function have two wave peaks, which are located near 200 nm and 500 nm respectively. This is consistent with the previous range of absorption peaks and the real part of the conductivity. It can also be seen from formula (11) that the absorption intensity is related to the dielectric function, so it is reasonable for them to peak at the same wavelength. As can be seen from figure 14, when the wavelength is about 450 nm, the doping ratio of FAPb_1−xGe_xI₃ with x = 1 appears negative. We know that a negative value means that light cannot travel to the perovskite material and that the material exhibits metallic luster. In figure 14, the two peaks of the imaginary part of the dielectric function are determined by electron transitions. Among them, the peak near 500 nm in visible light is determined by the shallow energy level transition of upper valence band to the lower band, while the peak near 200 nm in ultraviolet light is determined by the deep energy level transition of valence band to the conduction band. We find that when Pb is replaced by Ge, the peak value of the imaginary part of the dielectric function is the highest in the visible range among them, which indicates that it has the best dielectric performance and good photoelectric properties.

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