**3. DFT methods in polymer solar cells**

The density-functional theory (DFT) has proven massively popular among the massive panel of current theoretical techniques. This success is mainly because no changeable inputs parameters are required, efficient numerical codes exist, and a high level of adaptability, particularly in representing semiconductor and metal ground state features. The ability of DFT approaches to handle larger systems has grown as computing power has increased. However, the present limit does not yet reach the 10000–10 million atom window involved in the active device region of PV cells. Modern semiconductor optoelectronic devices, such as quantum wells [39] and quantum dots [40], have features with a feature size of a few nanometers. Such systems are composed of various materials and alloys' complex two-dimensional (2D) and three-dimensional (3D) geometries. Nonetheless, DFT approaches can be used to get insight into the physical phenomena of individual device components, such as particular materials or tiny heterostructures. Quantitative PV design, for example, necessitates a valid prediction of electronic bandgap's band-lineups and effective masses.

DFT based on the local density approximation (LDA) [41] or the generalized gradient approximation (GGA) [42] is well recognized for failing to replicate the excited states of molecules adequately. Hybrid approaches that contain a fraction of Hartree-Fock exchange, on the other hand, may be able to avoid the band-gap problem, but their results are highly dependent on the material of interest. Even while Heyd et al. [43]. HSE06 hybrid functional is a good option for computing band gaps, band offsets, or alloy characteristics [44]. It fails to simulate the direct– indirect crossover in GaAsP alloys [45]. Many-body perturbation theory (MBPT) can also provide factual findings, mainly when using the GW technique (GW, where G stands for Green's function and W for the screened potential), which can be utilized in a perturbative scheme [46] or self-consistently [47]. The ionization potential and electron affinity of the donor and acceptor materials, respectively, are essential parameters for charge separation because they determine the relative alignment of electron and hole levels. DFT inside a super-lattice (SL) approximation can be used to estimate the drop of the interface's potential in heterostructures, resulting in a reasonable estimate of the band-lineup. The DFT potential drop at the interface can be quickly compensated by the GW eigenvalues derived for the bulk valence band states [48], but a GW treatment for a complicated stack is out of reach.

In DFT simulation, a reasonable estimation of alloy electrical characteristics is likewise a difficult task. Indeed, even typical semiconductors experience significant band-gap bowing; that is, the band-gap energy follows:

*Applications of Current Density Functional Theory (DFT) Methods in Polymer Solar Cells DOI: http://dx.doi.org/10.5772/intechopen.100136*

$$E\_{\mathbf{g}}(\mathbf{x}) = \overline{E}\_{\mathbf{g}}(\mathbf{x}) - b \,\boldsymbol{\upchi}. (\mathbf{1} - \boldsymbol{\upchi}) \tag{1}$$

Where b is a parameter for bowing, super-cell techniques are better than virtual crystal approximations for statistically random alloys. However, DFT with specific quasi-random structures (SQS) (small super-cells) that recreate mixing enthalpies and atomic correlations of extremely large super-cells can provide identical results for specific alloy compositions. Chemical mixing, strain, and atomic relaxation effects are all included in SQS models.

Furthermore, semiempirical approaches for studying mechanical or electronic properties, such as the valence force field (VFF) and the tight-binding approximation [49] or elasticity and the K.P method [50], require precise electronic parameters as input. They can be calculated using DFT or discovered through experiments. For example, the density functional perturbation theory (DFPT) [51] is used to get quantitative estimates of electromechanical tensors of bulk materials. When just byproducts of a first-order perturbation computation are required, an efficient application of the "2n + 1" theorem yields second-and third-order derivatives of the total energy, provided that atomic-displacement variables are removed. Various physical responses of insulating crystals, such as elastic constants, linear piezoelectric tensors, and linear dielectric susceptibility, as well as tensor properties related to internal atomic displacements like Born charges and phonons, can be obtained using second-order derivatives [51]. Interference techniques and symmetry analysis must be used in conjunction with the DFPT method for third-order components linked to physical qualities such as nonlinear electrical susceptibilities, nonlinear elasticity, or photoelastic and electrostrictive effects [52].
