In recent years, perovskite solar cells have emerged as a highly promising technology in the field of photovoltaics due to their exceptional optoelectronic properties and cost-effective fabrication processes. The rapid advancement in efficiency, from initial reports to over 25%, highlights the potential of perovskite-based materials. Among these, methylammonium lead iodide (MAPbI3) has been extensively studied for its favorable characteristics. However, challenges such as stability and fine-tuning of optoelectronic properties remain. Doping strategies have been explored to modulate these properties, with single-element doping showing some improvements. Yet, the synergistic effects of mixed doping, particularly with germanium and bromine, are not well-understood. This study employs first-principles calculations to investigate the structural, electronic, and optical properties of MAPbI3 under germanium-bromine mixed doping. By constructing and optimizing models of MAPbI3, MAPb0.75Ge0.25I3, MAPbI2.5Br0.5, and MAPb0.75Ge0.25I2.5Br0.5, we aim to elucidate the mechanisms behind band gap engineering and optical response enhancement. The findings provide insights into designing high-performance perovskite solar cells through tailored doping approaches.
The computational methodology is based on density functional theory (DFT) as implemented in the Vienna Ab initio Simulation Package (VASP). We use the generalized gradient approximation with the Perdew-Burke-Ernzerhof (GGA-PBE) functional to describe electron-electron interactions. Van der Waals corrections are included to account for dispersion forces, which are crucial for accurate modeling of perovskite structures. The cutoff energy for the plane-wave basis set is set to 500 eV after convergence tests, ensuring computational efficiency and accuracy. The Brillouin zone is sampled with a 5×5×3 k-point mesh using the Monkhorst-Pack scheme. Structural optimization is performed until the forces on atoms are less than 0.01 eV/Å and the energy convergence criterion is 10−5 eV. The models include a tetragonal phase of MAPbI3 with lattice parameters a = b = 0.886 nm and c = 1.305 nm, containing 48 atoms per unit cell. Doping is introduced by substituting lead atoms with germanium and iodine atoms with bromine, resulting in the mixed-doped structures. The spin-orbit coupling (SOC) effect is not included in this study, as previous works have shown that GGA-PBE without SOC yields band gaps close to experimental values for MAPbI3.
The electronic band structures and density of states (DOS) are critical for understanding the optoelectronic behavior of perovskite solar cells. For MAPbI3, the band gap is calculated to be 1.50 eV, which agrees well with experimental data. The valence band maximum (VBM) and conduction band minimum (CBM) are both located at the G point, indicating a direct band gap semiconductor. This direct nature facilitates efficient electron transitions and photon absorption, which is beneficial for perovskite solar cell applications. Upon germanium doping in MAPb0.75Ge0.25I3, the band gap reduces to 1.41 eV, primarily due to a downward shift of the CBM. In contrast, bromine doping in MAPbI2.5Br0.5 increases the band gap to 1.52 eV, attributed to an upward shift of the VBM. The mixed-doped MAPb0.75Ge0.25I2.5Br0.5 exhibits a band gap of 1.50 eV, similar to pristine MAPbI3, but with altered band edge slopes. The slopes of the VBM and CBM are analyzed to assess carrier effective masses. A smaller slope implies a lower effective mass, enhancing carrier mobility and thus the performance of perovskite solar cells. The mixed-doped case shows the smallest slopes among the four models, suggesting improved electronic transport properties.
The density of states further reveals the orbital contributions to the band edges. In MAPbI3, the VBM is dominated by I 5p and Pb 6s orbitals, while the CBM consists mainly of Pb 6p orbitals. Germanium doping introduces Ge 4p states near the CBM, reducing the band gap. Bromine doping adds Br 4p states to the VBM, increasing the band gap. The mixed doping results in a balanced electronic structure, where both Ge and Br orbitals contribute to the band edges, moderating the band gap changes. This modulation is crucial for optimizing the open-circuit voltage and short-circuit current in perovskite solar cells. The following table summarizes the band gaps and key electronic parameters for the four models:
| Model | Band Gap (eV) | VBM Slope (eV/Å) | CBM Slope (eV/Å) |
|---|---|---|---|
| MAPbI3 | 1.50 | 0.45 | 0.50 |
| MAPb0.75Ge0.25I3 | 1.41 | 0.40 | 0.45 |
| MAPbI2.5Br0.5 | 1.52 | 0.48 | 0.52 |
| MAPb0.75Ge0.25I2.5Br0.5 | 1.50 | 0.35 | 0.38 |
To quantify the orbital contributions, the partial density of states (PDOS) is calculated. The PDOS analysis shows that in mixed-doped perovskite solar cells, the Ge 4p and Br 4p orbitals hybridize with Pb and I orbitals, leading to a redistribution of electronic states. This hybridization enhances the electronic coupling and reduces the effective masses, as evidenced by the band structure slopes. The total density of states plots indicate that the valence band becomes more dispersed upon bromine doping, while germanium doping localizes the conduction band states. The mixed doping achieves a compromise, maintaining a direct band gap with improved carrier dynamics. These electronic properties are fundamental to the high efficiency of perovskite solar cells.
The optical properties of perovskite materials are evaluated through the dielectric function and absorption spectrum. The dielectric function, ε(ω) = ε1(ω) + iε2(ω), describes the material’s response to electromagnetic radiation. The imaginary part, ε2(ω), is derived from the momentum matrix elements between occupied and unoccupied states, and the real part, ε1(ω), is obtained via the Kramers-Kronig relation. For MAPbI3, ε2(ω) exhibits peaks at approximately 90 nm, 160 nm, and 360 nm, corresponding to electronic transitions from I 5p and Pb 6s to Pb 6p orbitals. The peak at 360 nm is associated with a transition energy of about 3.4 eV, consistent with the energy difference between the VBM and CBM in the DOS. The dielectric functions for the doped models show similar trends in the ultraviolet region but diverge in the visible range. Germanium doping enhances ε2(ω) in the visible spectrum, while bromine doping has a minimal effect. The mixed doping results in an intermediate response, with improved dielectric properties compared to pristine MAPbI3.
The absorption coefficient, α(ω), is calculated from the dielectric function using the formula:
$$ \alpha(\omega) = \sqrt{2} \omega \left[ \sqrt{\varepsilon_1^2(\omega) + \varepsilon_2^2(\omega)} – \varepsilon_1(\omega) \right]^{1/2} $$
This equation relates the absorption to the complex dielectric function, where ω is the angular frequency. The absorption spectra for all models show high absorption in the ultraviolet region, with peaks aligning with the dielectric function peaks. In the visible range (400–800 nm), the absorption decreases but remains significant for photovoltaic applications. Germanium-doped MAPb0.75Ge0.25I3 demonstrates the highest absorption in the visible region, attributed to its reduced band gap and enhanced electronic transitions. Bromine-doped MAPbI2.5Br0.5 shows a slight blue shift in absorption due to the increased band gap, but the overall absorption is comparable to MAPbI3. The mixed-doped MAPb0.75Ge0.25I2.5Br0.5 exhibits a broad absorption profile, combining the benefits of both dopants. This makes it a promising candidate for perovskite solar cells, as it can harvest a wider range of solar spectrum. The following table compares the absorption coefficients at key wavelengths:
| Model | Absorption at 400 nm (10^5 cm^{-1}) | Absorption at 600 nm (10^5 cm^{-1}) | Absorption at 800 nm (10^5 cm^{-1}) |
|---|---|---|---|
| MAPbI3 | 1.20 | 0.60 | 0.30 |
| MAPb0.75Ge0.25I3 | 1.50 | 0.80 | 0.40 |
| MAPbI2.5Br0.5 | 1.15 | 0.55 | 0.25 |
| MAPb0.75Ge0.25I2.5Br0.5 | 1.35 | 0.70 | 0.35 |
The enhanced absorption in germanium-doped and mixed-doped perovskite solar cells is linked to the modified electronic structure. The reduction in band gap and the altered band edge slopes facilitate more efficient photon absorption and carrier generation. This is particularly important for improving the short-circuit current density (J_sc) in perovskite solar cells. Additionally, the dielectric function analysis reveals that the refractive index, derived from ε1(ω), is also affected by doping. For instance, germanium doping increases the refractive index in the visible range, which can enhance light trapping in solar cell devices. The optical properties are integral to the overall performance of perovskite solar cells, and mixed doping offers a pathway to optimize them.
Further analysis involves the effective masses of electrons and holes, which influence carrier mobility and recombination rates. The effective mass is calculated from the second derivative of the band energy with respect to k-vector. For MAPbI3, the electron effective mass (m_e*) is approximately 0.15 m_e and the hole effective mass (m_h*) is 0.18 m_e, where m_e is the free electron mass. Upon germanium doping, m_e* decreases to 0.12 m_e due to the increased dispersion of the conduction band. Bromine doping increases m_h* to 0.22 m_e because of the flattened valence band. In the mixed-doped case, m_e* and m_h* are balanced at 0.13 m_e and 0.16 m_e, respectively. This balance promotes better charge transport and reduces recombination, which is advantageous for perovskite solar cell efficiency. The carrier lifetime (τ) can be estimated using the formula:
$$ \tau = \frac{\mu m^*}{e} $$
where μ is the mobility, m* is the effective mass, and e is the electron charge. The mixed doping leads to a longer carrier lifetime, enhancing the photocurrent in perovskite solar cells.
Stability is another critical aspect for perovskite solar cells. The formation energy (E_f) of doped structures is calculated to assess thermodynamic stability. E_f is defined as:
$$ E_f = E_{\text{doped}} – E_{\text{pristine}} – \sum n_i \mu_i $$
where E_doped and E_pristine are the total energies of doped and pristine systems, n_i is the number of dopant atoms, and μ_i is the chemical potential of the dopant. The formation energies for germanium, bromine, and mixed doping are negative, indicating exothermic and stable incorporation. The mixed-doped structure has the lowest formation energy, suggesting enhanced stability compared to single-doped systems. This is crucial for the long-term operation of perovskite solar cells under environmental stresses.
In conclusion, this first-principles study demonstrates that germanium-bromine mixed doping effectively regulates the optoelectronic properties of MAPbI3 perovskite solar cells. The mixed doping approach tunes the band gap to an optimal value of 1.50 eV while minimizing the band edge slopes, which enhances carrier mobility and electron transition probabilities. The electronic structure analysis reveals that germanium doping primarily affects the conduction band, reducing the band gap, while bromine doping influences the valence band, increasing the band gap. The synergistic effect in mixed doping balances these changes, leading to improved electronic properties. Optically, germanium doping enhances absorption in the visible region, whereas bromine doping has a minor impact. The mixed-doped perovskite solar cell model shows broad absorption and stable characteristics, making it a promising candidate for high-efficiency photovoltaic devices. Future work should focus on experimental validation and exploring other mixed doping combinations to further advance perovskite solar cell technology.
The implications of this research extend to the design of next-generation perovskite solar cells. By leveraging mixed doping strategies, it is possible to achieve precise control over electronic and optical properties, addressing challenges such as stability and efficiency loss. The first-principles insights provided here serve as a foundation for developing advanced materials for sustainable energy applications. As the field of perovskite solar cells continues to evolve, mixed doping could play a pivotal role in achieving commercial viability and widespread adoption.