Quantum dots (QDs) and quantum wells (QWs) in semiconductors have a typical size in the range of 2-20 nm to be compared to the 0.5-0.6 nm of the atomic unit cells. Attempts of applying *ab initio* calculation, which is deemed the most accurate, usually requires calculation cells including 10000 atoms or more, which is in the limit of what is feasible even with big computers.

The mesoscopic nature of the nanostructures allows applying the integral factorization rule that states that any 3D-integral comprising the product of a periodic function times a function that varies slowly with respect to the period can be separated into two factors: the integral of the periodic function over the unit cell, times the integral of the slowly varying function.

In the k·p methods the one-electron Hamiltonian of the nanostructured system is developed in an orthonormal basis formed by the Bloch functions —which are the solutions of a non-structured semiconductor— calculated at the semiconductor G-point (** k**=0) for a certain band, multiplied by a plane wave of arbitrary wavevector

**[1]. The basis considered in this work comprises the conduction band (cb) and three valence bands —the light holes (lh), the heavy holes (hh) and the spin-orbit (so) band— all spin double-degenerated. This leads to a matrix Hamiltonian of dimension four whose terms are function of**

*k***. In zincblende materials —such as those used in high efficiency cells (GaAs, GaP, InAs, and alloys, among many others) — this matrix is quite easy. However, in this easy form it does not include the spin-orbit interaction or the strain effects due to the inclusion of the nanostructure. We call it the (**

*k**H*0) matrix. We have proposed [2, 3] a Hamiltonian whose eigenvalues are the parabolic dispersion functions characterized by the experimental band edges and effective masses. The use of these experimental dispersion functions, obviously, takes into accounts the spin-orbit interaction, the strain effects, and any other forgotten effect should have been taken into consideration. This is why we call it the empiric k·p Hamiltonian (EKPH) and its matrix (

*HEKP*). The non diagonalized form of this Hamiltonian may be approximately calculated by using the diagonalization matrix for the initial Hamiltonian, without spin-orbit interaction or strain. This matrix is formed with the eigenvectors of (

*H*0). In summary the model based on the EKPH assumes as eigenvalues the empiric ones but as eigenvectors those of (

*H*0).

The introduction of the nanostructures produces offsets in the band edges. We assume them to be square. This is not real but is an acceptable approximation. The shape of the QDs is debatable but we consider it a squat parallelepiped. These two approximations, together with the analytic diagonalization matrix used, lead to a very strong simplification of the calculations (around 106 times faster than the established and more accurate Luttinger-Kahn-Pikus-Bir (LKPB) calculations), and thus to a deep, we think unprecedented, progress in the modeling and interpretation of the solar cells comprising nanostructures. It also leads to a simple labeling of the wavefunctions appearing in the solar cell.

In particular, in the case of QDs, the full spectrum in all the four bands —including bound and extended states— can be calculated. The EKPH model helps interpreting the origin of the peaks found in the solar cell measured quantum efficiency and the calculated spectrum allows to determine the QD size, that coincides with the one observed by TEM.

It also allows determining the eigenfunctions, which are linear combinations of the products of the G-Bloch functions in all the four bands, multiplied by functions that vary in the range of the nanostructure dimensions, called envelope functions. This is necessary to calculate the absorption coefficients in QD intermediate band solar cells and in the QW-tuned bandgap solar cells. They both are in semi-quantitative agreement with the observed quantum efficiency. This method may also be applied to non-structured semiconductors, such as GaAs, leading also to a semi quantitative agreement with the measured absorption coefficient. In all the cases the calculations do not require a full knowledge of the G-Bloch functions but only of the envelope functions and of some symmetry properties of the G-Bloch functions. Thus the absorption coefficients depend mainly on the nanostructure characteristics, although they keep relation with the semiconductor material through its symmetry properties and the effective masses imposed by it to the different bands.

The approximate knowledge of the absorption coefficient for each one of the transitions involved, confirmed also by the quantitative interpretation of the experimental variation of the sub-bandgap quantum efficiency with the temperature [4], allows for a realistic detailed balance treatment and therefore it allows predicting the behavior of nanostructured solar cells in its radiative limit, that, we remind, is almost achieved in the high efficiency multijunction cells of today, and so allows orienting the technological research.

The purpose of the work in this Mega-Grant consists on comparing the calculation with more available experimental data experiments in order to further support the appropriateness of the EPKH, and some additional results have already been achieved within the Mega-Grant: for instance its use has permitted a rather correct calculation of the band to band absorption in GaAs, as compared to classical experimental data (work submitted to publication). It has been used to predict the size of the QDs form sub-bandgap quantum efficiency measurements in InAs/GaS and InAs/GaAlAs QD/host materials (work submitted to publication). It has also been used to suggest the interest of using Type II QDs in the VB in order to prevent a reduction of the voltage (work already published [5]) so giving birth to a new experimental research on Type II QDs under way in this Mega-Grant. Additional work is under way in refining the EKPH method and substantial work is also under way in the comparison of the EKPH method with the more widely used LKPB method in order to compare their respective stresses and wqaeaknesses.

[1] S. Datta, *Quantum Phenomena *(Addison Wesley, Reading (Mass), 1989).

[2] A. Luque* et al.*, Solar Energy Materials & Solar Cells **95**, 2095 (2011).

[3] A. Luque* et al.*, Solar Energy Materials and Solar Cells **103**, 171 (2012).

[4] A. Mellor* et al.*, Advanced Functional Materials **24**, 339 (2014).

[5] A. Luque* et al.*, Applied Physics Letters **103**, 123901 (2013).