Tight-binding electronic band structure and surface states of Cu-chalcopyrite semiconductors

We report theoretical calculations of the electronic band structure and surface states of Cu-chalcopyrite semiconductors. The systems under consideration are CuGaS2, CuAlSe2, and CuGaS2 doped with Cr. The calculations are carried out using semi-empirical Tight-Binding formalism. By reproducing the band gap of these systems obtained by ab-initio calculations we report the Tight-Binding parameters required for the band structure calculations. We present bulk band structure calculations of the above mentioned semiconductors, and we analyze the (001) and (110) surface states of the corresponding semi-infinite semiconductors.


Introduction
Chalcopyrite crystals derived from the I-III-VI 2 (I = Cu, III = Ga, Al, and VI = S, Se) family have received considerable attention as one of the promising materials to be used for thin film solar cells fabrication [1][2][3]. The energy gap of these materials covers practically all the visible part of the electromagnetic spectrum. The band gap range from 1.7 eV for CuGaSe 2 , i.e. from the deep red, to 2.43 for CuGaS 2 , and to 2.65 eV for CuAlSe 2 , with its main absorption peak located near ultra-violet (UV) region [4][5][6][7][8][9]. Of course, the energy band gap is the most important parameter owing to the fact that it dominates the main optical-absorption peak in semiconductors. The concept of multiband or intermediate band solar cells has recently attracted renewed attention as a viable approach to achieving high solar power conversion efficiencies [10,11]. Several approaches have been employed to demonstrate the concept of intermediate band including quantum dots [12,13] and highly miss-matched alloys [14,15]. Highly miss-matched alloys are a class of materials whose electronic band structures are dramatically modified through the substitution of a relatively small fraction of host atoms with some dopant. Between these I-III-VI 2 materials, and due to its wide gap, CuGaS 2 , is one of most promising candidates to host an intermediate band when it is doped with a transition metal. This is because, as recently suggested, the optimal value of the band gap for a semiconductor that contains an intermediate band, should be around 2.4 eV [14], and CuGaS 2 satisfies this requirement.
The electronic properties of I-III-VI 2 semiconductors have been subject of study for nearly 40 years by different authors using distinct methodologies. As representative examples, we can mention the seminal work of Jaffe and Zunger where they studied selfconsistently within the density-functional theory (DFT), the chemical trends in the electronic structure of six Cu-based ternary chalcopyrite semiconductors including CuGaS 2 [16,17]. They reported 1.65 and 1.25 eV for the energy gap of CuAlSe 2 and CuGaS 2 , far from the accepted experimental values 2.65 and 2.43 eV, respectively. Using full-potential linear muffin-tin orbital method based on the local-density approximation and with the Hedin-Lundqvist parametrization for the exchange and correlation potential, R. Ahuja et al, obtained a similar value for the energy gap of CuGaS 2 (1.2 eV) [18]. Band structure and total energy calculations using the density functional theory within the generalized gradient approximation(GGA) for the exchange correlation functional, were carried out by Chen et al in 2007, to study what they called 'band gap anomalies', due to the unexpected behavior of this parameter with their corresponding lattice constant. Unfortunately, in spite of the methodology used by the authors, they obtained a gap of 0.6932 eV for the CuGaS 2 semiconductor, which is very far from the experimental accepted value (2.43 eV) [8,19]. As it can be seen from all these results, standard Kohn-Sham DFT fails in describing the band structure of chalcopyrite materials, due to the strong underestimation of the band gap. In order to go beyond the standard Kohn-Sham DFT, new results have been published improving the calculated band gaps. Using a self consisted GW scheme based on Hedin's Coulomb hole and screened exchange (scCOHSEX) followed by a perturbative G 0 W 0 step, Aguilera et al, reported an energy gap of 2.65 eV for CuGaS 2 , in very good agreement with the experimental value [20]. Recently, we have done ab-initio DFT theoretical calculations using a modified screened hybrid Heyd-Scuseria-Ernzerhof functional (HSE06) to improve the GGA approximation, and we have obtained a band gap of 2.43 eV, which reproduces excellently the experimental value [21].
On the other hand, due to the growing interest of increase the efficiency of a solar cell by considering CuGaS 2 as the active material hosting an intermediate band, a detailed study of the electronic properties of such system becomes necessary, and in this direction many articles have been published exploring such a possibility [22][23][24][25][26][27]. Of course, a precise knowledge of the band structure of such materials requires ab-initio calculations. Unfortunately, due to the large unit cells required to manage small doping levels, first principles calculations requires very large time consuming computer calculations, which makes this methodology unfeasible. For systems of such complexity, it is very convenient to have simpler methods available as an alternative tool. The Tight-Binding (TB) method is one such possibility, since it gives solutions showing all the correct symmetry properties of the energy bands, and it is rather easy to get solutions for energy bands at an arbitrary point in the Brillouin zone. The time required by this semi-empirical method depends only on the size of the matrix to diagonalize, which in the worst case is very small compared to the time required for a first principles calculation.
In this paper we present a TB study of the electronic band structure of the CuGaS 2 , CuAlSe 2 chalcopyrite semiconductors, together with the corresponding material CuGaS 2 doped with Cr in order to obtain an intermediate band. As a first step, and as a main goal of this paper we adjust all the necessary TB parameters to reproduce the values of the band gap experimentally documented as well as the main features of the band structure obtained from ab-initio first-principles calculations recently reported. Using these parameters, we calculate the electronic band structure of the above mentioned chalcopyrite semiconductors along the principal directions of the corresponding Brillouin zone. The knowledge of all these parameters will allow us to tackle more complicated problems which are not possible to consider from ab-initio methodology. As an example of this kind of difficult problems, we also calculate the electronic surface states of the semi-infinite chalcopyrite semiconductors along the [001] and [011] directions. This last analysis is carried out by using the Surface Green Function Matching (SGFM) method, which is a successful technique employed to calculate the electronic properties of surfaces, interfaces, quantum wells, and superlattices [28,29].

Theoretical model
Chalcopyrites are tetragonal centered crystalline structures with eight atoms in the unit cell basis and spatial groupĪ 4 2d. In figure 1 we present the crystal structure for CuGaS 2 , and the coordinates of the eight atoms in the unit cell appear in table 1. According to the TB formalism, the electronic wave function is written as a linear combination of atomic orbitals as, where ν is the atomic orbital, and the sum over j consider all the atoms in the crystal. The expressions of the TB parameters necessary to diagonalize the matrix appear explicitly in table 20.1 of the book of Harrison [30] and they are not given here. For the chalcopyrites considered in this work, the matrix to diagonalize is of order 42×42, four orbitals (sp 3 ) for each of the two the Ga (Al) atoms and four for each of the four S (Se) atoms, and 9 orbitals (sp 3 d 5 ) for each of the two Cu atoms. The correct form of the Hamiltonian matrix is given in the appendix (A). Similar Hamiltonian has been published before by Rodríguez [31] et al, in their study of the electronic properties of a family of chalcopyrites, however small typos appear in the final form of the matrices reported.

Electronic structure of CuGaS 2 and CuAlSe 2
Recently, we have carried out ab-initio calculations of the structural and electronic properties of these materials, and the results obtained were reported in reference [21]. From the reference of Castellanos et al [21], we observe that the obtained values for the band gap is 2.43 eV for the CuGaS 2 , and 2.65 eV for the CuAlSe 2 . Unfortunately if we introduce the values of the TB parameters obtained from the Harrison's rule, it is not possible to reproduce correctly, neither the experimental values, nor those obtained from first-principle calculations. For example, for the CuGaS 2 chalcopyrite, the band gap obtained using the Harrison's parametrization, is 6.16 eV, which is very different to the value of 2.43 eV obtained from our ab-initio calculations. In order to reproduce the band gap for all the materials considered in this work, we consider a slightly different value for the universal parameters η α β γ , and we carried out as well a fit of the on-site parameters. The adjustment of these last parameters is because their values substantially affect the border of the conduction band, and consequently the values of the corresponding band gaps [16]. The values of the adjusted universal parameters η α β γ and the on-site parameters used all throughout this work appear in tables 2 and 3, respectively. If we introduced these values in our TB Hamiltonian we reproduce better the band gap for all the semiconductors considered in this work. This can be clearly seen in table 4. In the second Table 1. Atomic positions in the CuGaS 2 chalcopyrite. The values of the distortion parameter η=c/2a, and the anion displacement u (in units of a), used in this work were 0.996 and 0.2532 respectively [21].

Atom number
Atom Position 1.36 column of table 4 we show the value of the band gap parameter obtained using the Harrison rule; the third column corresponds to the values obtained from first principles according to the approximations reported in the reference [21]; the fourth column contains the experimental reported values; whereas the last column corresponds the results of the present work.
In figure 2 we show the band structure for the chalcopyrite CuGaS 2 and CuAlSe 2 obtained from our TB calculations using the parameters that better reproduce the experimental values of the band gap appearing in table 4. We can observe that the band structures are very similar. Both materials are semiconductors with a direct band gap; the value of the band gap is 2.43 eV for CuGaS 2 and 2.65 eV for CuAlSe 2 , and match perfectly well with the experimental results. For both semiconductors, the top of corresponding valence band is nondegenerate. The next deeper band is a doublet. The energy difference between them is due to the crystal field splitting, and we found 0.05 eV and 0.122 eV for the CuGaS 2 , and CuAlSe 2 , respectively. This small difference cannot be appreciable due to the scale of the figure; however, these three states are clearly observed by seeing the dispersion relation in the direction Γ-X. The breaking of the degeneration in the valence band is attributed to three main effects: the existence of two different cations; the tetragonal distortion ¹ ( ) c a 2 ; and the displacement of the Table 3. Adjusted 'on-site' parameters in eV, for CuGaS 2 , and CuAlSe 2 , used in our calculations. Reported values are included for comparison (a) [30], (b) [31].  anion which does not lies at the center of the tetrahedron. We have taken into account all these three effects in our TB calculations. The effect of the two different cations was considered by the values of the TB parameters corresponding to the Ga and Cu atoms. On the other hand the effect of the tetragonal distortion and the displacement of the anions were considered by introducing the values of the η and u parameters obtained from our ab-initio calculations [21,33]. Although the splitting due to these three effects is very small, the obtained values are consistent with recent published results [15].

Electronic structure of the chalcopyrite CuGaS 2 doped with Cr
In this section we analyze the electronic properties of the chalcopyrite CuGaS 2 doped with Cr atoms. As has been established before, doping this semiconductor could modify the electronic structure of the semiconductor giving rise to an intermediate band [33]. There have been published many results of chalcopyrite semiconductors doped with metallic elements with the purpose to obtain an intermediate band [12,14,15,[20][21][22][23][24]. Unfortunately, first principles calculations are very expensive from the computational point of view, because the large number of atoms in the unitary cell needed to manage realistic levels of doping. Due to this tough task, ab-initio calculations are carried out considering large fractions of doping only. Even in this case, these studies are mainly concentrated in calculate the band gap, or the density of states at fixed k points; often pending further analysis of the electronic structure. On the other hand, a semi-empirical TB approach allows us to consider very low levels of doping. The only prize to pay for this approach is the cumbersome work in writing the matrix. Fortunately, the time needed to diagonalize the TB matrix, is very small when compared with any first-principles calculation. If we doped the CuGaS 2 chalcopyrite with Cr atoms, the size of the TB matrix increases due to the presence of the metallic atom in the unitary cell. For a 25% of Cr concentration the size of the TB matrix goes from 42×42 to 178×178. For this percentage of Cr atoms, the unitary cell contains 16 atoms: 4 Cu atoms (9 atomic orbitals per atom), 3 and 8 atoms of Ga and S, respectively (4 atomic orbitals per atom), and 1 Cr atom with a sp 3 d 5 orbital basis. In addition to the above, for this doped chalcopyrite, spin-orbit effect must be taken into account to obtain a correct description of the band structure. This is because, for the doped chalcopyrites, previous ab-initio calculations have demonstrated that the spin with a very definite component contributes to the density of states of the intermediate band [23]. We have adjusted the 'on site' parameters for the Cr atom, and the used values appear in table 5. The parameters that take into account the spin-orbit effect λ a have been adjusted to reproduce ab-initio calculations recently reported [21]. The values of these parameters used in the present work, for the d orbitals of the Cu and Cr atoms were 0.012 and 0.2 eV respectively; meanwhile for the p orbitals on the Ga and S atoms were, 0.007 and 0.004 eV respectively. In figure 3 [32].
In figure 4 we show the band structure for the CuGaS 2 :Cr chalcopyrite doped with 25% of Cr atoms, obtained using the TB parameters adjusted to reproduce the LDOS of reference by Castellanos, et al [33].

(001) and (110) Surface states
Finally, with the purpose of showing the power of the TB method in the analysis of problems that are difficult to attack by other methodologies, and to have a deep understanding of the electronic properties of the CuGaS 2 , CuAlSe and CuGaS 2 :Cr chalcopyrites, we study the surface states of these materials. The surfaces under study are those along which chalcopyrites are normally grown when they are considered as the active material in a solar cell: (001) and (110). We suppose that the surfaces are ideal surfaces, i.e., the positions of the atoms in the surfaces created by terminating the crystal, are exactly the same than those in the infinite crystal. The study of surface states is carried out using the Surface Green Function (SGFM) method [28,29], together with the TB Hamiltonian derived here. All the details of the SGFM method appear in the references [28,29], and they will not repeated here. However, it is worth to writting the explicit form of the Green function for the incomplete crystal:  figure 6, the figure denoted with a) corresponding to cation terminated CuGaS 2 surface, we observe that the surface state at 0.84 eV, marked as E1 in figure 5(a), is mostly concentrated in the surface layer. The LDOS of this surface state mainly comes from the Cu atoms with primarily orbital S character. On the other hand, the surface state at 1.85 eV, is mostly concentrated in the Ga atoms but also with primarily S character. These surface state attenuates very quickly, as can be seen into the internal layers. From figure 6(b), corresponding to anion terminated surface, the surface states at 0.80 and 0.93 eV, and denoted as E1 and E2 in figure 5(b), appear mostly concentrated in the S atoms primarily with p orbital character. Finally from figure 6(c), corresponding to the (110) termination surface, we observe that the surface states at 0.32 and 2.05 eV, denoted as E1 and E2 in the top figure 5(c), are mainly localized in the Cu atoms with primarily orbital s character with slightly p orbital character on the S atoms. We observe a quick attenuation of all these states into the internal layers. For the (110) terminating layer, there is a peak at −0.49 eV, i.e. below the top of the valence band. This state is in the continuum and tends to penetrate into the bulk, as is seen in figure 6(c). The attenuation of the surface states for CuAlSe 2 shows similar characteristics as the corresponding cases of CuGaS 2 . Finally, the right column of figures shows the evolution of the surface states of the CuGaS 2 :Cr for the same termination layers. For this semiconductor, although we get a completely different behavior, we observe the rapid decay of the surface states when they penetrate into the crystal. This rapid attenuation is clearly seen in the right-hand column of figures of figure 6.

Conclusions
We have calculated the electronic properties of the chalcopyrites CuGaS 2 , CuAlSe 2 , and CuGaS 2 :Cr, and their corresponding dispersion relation of the surfaces states along the 2D Brillouin zone for the (001) and (110) terminated surfaces. By adjusting the Tight-binding parameters, we were able to reproduce the band gaps obtained by first-principles calculations, and calculate electronic dispersion relation along the principal directions of the Brillouin zone. For the doped CuGaS 2 :Cr chalcopyrite, the electronic structure exhibits an intermediate band located inside the band gap, in accordance with previous ab-initio calculations. Using the TB methodology is possible to calculate the dispersion relation for these kind of semiconductors along the whole Brillouin zone without the expensive computational resources required by first principles. For the incomplete crystal, we have found surface states in the gap for both terminated surfaces; for cation or anion (001) terminated surface, as well as cation and anion (110) terminated surface. The attenuation rate of the surface states is always strong and independent of the cation, anion, or cation and anion terminating layers. These surface states do not penetrate more than four atomic layers for all the cases considered in this work. The present study demonstrates the practical use of the TB method for the study of this kind of systems, where ab-initio methodologies require long time computer calculations.

Appendix. Tight-binding Hamiltonian for CuGaS 2
The general form of the TB Hamiltonian for the chalcopyrite CuGaS 2 is, The diagonal elements of this matrix are in turn 4×4 diagonal matrices corresponding to the S and Ga atoms, and 9×9 diagonal matrices for the Cu atoms. The upper part of the non-diagonal elements of matrix (A1) contains matrices of 4×4 for the interactions between the S and the Ga atoms, and 4×9 matrices for the interaction between the S and the Cu atoms. The non-diagonal Hamiltonians of equation (A.1) are in general written as, 4 .