The energy difference (E_{per unit}, eV) between ZB and WZ polymorphs of GaAs, lattice constant, a, and bulk moduli B (GPa) of GaAs (ZB) and Ge (diamond). (^{a} Arabi et al., 2006, ^{b} A. Wronka, 2006, ^{c} BautistaHernandez et al., 2003,^{d} Wang & Ye, 2003 , ^{e}Wang & Ye, 2002, ^{f} Kalvoda et al.,1997,^{g} Yeh et al.,1992, ^{h} Murayama & Nakayama, 1994,^{i}Hellwege & Madelung, 1982, ^{j}Singh, 1993;^{k}CRC, 19971998; ^{l}Levinstein, 1999.)
1. Introduction
In the next thirty years the annual global consumption of energy will rise by more than 50% (Hochbaum & Yang, 2010). Currently most of the energy production comes from the combustion of fossil fuels; nonetheless the deriving CO_{2} emissions represent a real risk for the safety of the environment and for human health. Not secondarily, the prompt availability of fossil fuels is extremely influenced by geopolitical factors. In the very last decades much attention has been devoted to the development of green renewable energetic sources as possible viable alternative. Owing to its almost ubiquitous availability, solar energy seems to be the most promising way to produce alternative energetic sources. The optimal choice of the materials for the device assembling, according to their prompt availability, in conjunction with recycling spent modules and thinning the semiconductor layers (these two latter are the keypoints in reducing the material related sustainability deficits (Fthenakis, 2009)) will help passing from the gigawatt to the terawatt level, that required by the global consumption (Feltrin & Freundlich, 2008). The idea of getting energy from the sun is almost 150 years old. It derives from the discovery of the photovoltaics effect (1839) observed for the first time by the French physicist Edmond Becquerel in an experiment lead with an electrolytic cell made up of two metal electrodes: a weak electrical current was detected by exposing to sunlight a silver coated platinum electrode immersed in electrolyte. From the initial discovery many years have passed until the very first practical application, due to the celebrated Russian physicist Abram Ioffe, consisting in low efficient rectified thallium sulfide cells, has been realized.
The very basic components of a modern solar cell consists in a
The other key performance characteristics are represented by the
Solar cells can operate in a wide range of current (
The initial technology of the modern solar cells based on the wellassessed single and multicrystalline Si cells (the latter cheaper but with reduced quality due to the presence of grain boundaries) has been progressively replaced by the thin film based one, the socalled “Secondgeneration solar cells”, usually constituted by GaAs, CuInSe, CIGS (Copper Indium Gallium Selenide), or CdTe, which represent a cheap and simple alternative technology in photovoltaics (PV). A particular mention is deserved by the emerging technology based on the combination of dyes anchored on metal oxides (DSSC, dye sensitized solar cells, (O’Regan & Graetzel, 1991) consisting in porous layers of TiO_{2} nanoparticles.
The usage of layered materials with different bandgaps is a desirable procedure for increasing the efficiency of the final PV device. Such assembling procedure results in the splitting of the solar spectrum in several parts. In this way, indeed, photons of different energies are absorbed depending on the used material: the stacking of higher bandgap material on the surface able to absorb highenergy photons with lowerenergy photons absorbed by the lower bandgap material beneath is the key step for these MultiJunction (MJ), “Thirdgeneration” solar cells, characterized by the reduction of the transmission and that of the thermalization losses of hot carriers. According to the assembling procedure, MJs aremonolithic or stacked. Owing to the requirement of electronic and lattice match of the constituents, the monolithic ones present additionalintrinsic difficulties; they are epitaxially grown on Ge substrate and constituted by lattice matched (In,Ga)P and GaAs. The reader may understand the structure of similar devices observing Figures 12. In Figure 1, a sketch of (Ga,In)P/(Ga,In)As/Ge is reported, while Figure 2 shows its spectral irradiance of the solar AM1.5 spectrum in conjunction with the parts of the spectrum used. “AM”(AirMass) refers to the spectrum of the incident light and corresponds to the shortest, direct optical path length through the Earth atmosphere, that in conjunction with “1.5” represents the standard for the characterization of terrestrial powergenerating panels. More precisely, AM is 1/cos(z), where zrepresents the zenith angle. The choice of AM1.5 as standard stems from the fact that it corresponds to z~43º, representing the yearly average at midlatitudes. AM0 is the standard for space cells.In the mechanical stacked MJ the separate connections of top and bottom cells do not mandatorily require current matching, making the combination of bandgaps quite flexible. The only tight requirement is thus the transparent contacts. Anyway, the lattice mismatch may cause crystal dislocations introducing levels in the gap and thus mediating the ShockleyReadHall recombination.
Primary role in the rising success of monolithic tandem solar cells must be ascribed to the National Renewable Energy Laboratory (NREL) activity and in particular to the research conducted by J. M. Olson group (Olson et al., 1985). In details, the initial tandem solar cells based on GaAs and GaInP were made via a vapour phase epitaxy growth process and revealed problems related to the purity of source materials. The initial drawbacks were progressively solved during the nineties and subsequently due to their high efficiency and powertomass ratio, systems based on GaInP/GaInAs/Ge have represented and still represent the most reliable choice for communication satellites (Dimroth, 2006).In Figure 3, the calculated conversion efficiency of a triplejunction solar cell as function of the bandgap of the single junctions forming the stacking is reported.
MJ systems are usually constituted by three different possible substrate: Ge, GaAs, and InP. The introduction of chemical components in the different layers is beneficial (Yamaguchi et al., 2005): Al in the top cell has the property of increasing E_{G} to values that cover a larger part of the solar spectrum, while few amounts of In reduce the mismatch between layers.
In the context of the Japanese “New Sunshine Project”, InGaP/InGaAs/Ge monolithic integrated cells have reached an efficiency of 31.7%,while for the stacked InGaP/GaAs//InGaAs ones an efficiency of 33.3% has been reported (Yamaguchi, 2003), representing at the time of their production the efficiency World Guinness for such cells.
Metamorphic triplejunction Ga_{0.44}In_{0.56}P/Ga_{0.92}In_{0.08}As/Ge terrestrial concentrator solar cells (owing to the macroscopic differences between terrestrial spectrum and the conditions in space, the structure of the solar cell must be adjusted. Therefore, the optimum bandgap combination of materials is not the same) have been recently grown and with the usage of a buffer structure, able to minimize the dislocation formation (King et al., 2007), a record of 40.7% in efficiency has been established in 2007 at 240 suns and at AM1.5D. Even more recently (Guter et al., 2009) a new World Guinness has been established for the metamorphic triplejunction solar cell: an efficiency of 41.1% has been achieved under 454 suns and at same standard conditions: this latter cell combines Ga_{0.35}In_{0.65}P (top cell), Ga_{0.83}In_{0.17}As (middle cell) with a Ge bottom cell. An electrically inactive buffer is used in order to make inactive the formation of threading dislocations that in any case have densities below 10^{6} cm^{2}. Such highly efficient cells are integrated by the usage of a Fresnel lens placed ~10 cm over the cells, ensuring a concentration of the sunlight increased by a factor ranging between 400 and 500.
The continuous research of new materials in PV and the optimization of their performances motivated us to study, in recent years, the electronic and the structural properties of the alloys formed by IV group elements and the IIIV compounds. (GaAs)_{1x}(Ge_{2})_{x}, one of the (A^{III}B^{V})_{1x}(C_{2} ^{IV})_{x} nonisovalent alloys, has direct gap ranging between 0.5 and 1.4 eV, ideal for visibleIR light absorption, and lattice constants matching with that of GaAs (5.66 Å (Wang & Ye, 2002)), representing the best candidate in some technologically oriented applications (extra junction of two junction Ga_{0.52}In_{0.48}P/GaAs solar cell (Norman et al., 1999)).
In this chapter we review recent results frommodelling the structural and electronic properties of such nonisovalent alloys widely utilized in MJ solar cells. The relevance of Density Functional Theory (DFT)and GW (G: Green function, W: screened Coulomb interaction) calculations in this field is discussed and the importance of large modellization in accounting the clusterization phenomena is shown. We also review the basic concepts associated with the stabilizing selfcompensation mechanism, both in defective supercells and in alloys.
2. Semiconductor alloys for photovoltaics: The case of (GaAs)_{1x}(Ge_{2})_{x}: A brief overview
The isovalent class of semiconductor alloys represents the most commonone (Wei & Zunger, 1989,Shan et al., 1999, Wei et al., 2000, Janotti et al., 2002, Deng et al., 2010) (i.e., IV/IV, (III−V)/(III−V), and (II−VI)/(II−VI) ). The alloys belonging to this class are characterized by a reduced band offsets between the constituents (< 1 eV), by a small and compositionindependent bowing coefficient, and by a lattice mismatch that is usually below 5% (Zunger, 1999). Differently, the introduction of a lowvalent and highvalent element into a IIIV compound generates a so called “nonisovalent” alloy (Yim, 1969, Bloom, 1970, Funato et al., 1999, Wang & Zunger, 2003, Osorio et al., 1999; Greene, 1983) (i.e., (III−V)/IV, (III−V)/(II−VI)). High carrier mobilities, an enhanced presence of free electrons and holes according to the growth conditions, and the reduced donor−acceptor charge compensation are the attractive characteristics that make this class matter of deep investigation. In the case of ZnSe−GaAs, Wang et al. report only small change in the final alloy bandgap, E_{G}, as result of the introduction of larger bandgap II−VI dopants into the III−V host, while the opposite (GaAs in ZnSe) causes sharp drops in the bandgap of the alloy (Wang & Zunger, 2003).
In recent years, many other papers have been focused on the class of (IIIV)_{1x}(IV_{2})_{x} nonisovalent alloys both at theoretical (Holloway, 2002, Newman et al., 1989, Osorio & Froyen, 1993, Ito & Ohno, 1992, Ito & Ohno, 1993, Newman& Jenkins, 1985, Bowen et al., 1983) and experimental (Green & Elthouky, 1981, Barnett et al., 1982,Alferov et al., 1982, Banerjee et al., 1985, Noreika & Francombe, 1974, Baker et al., 1993,Rodriguez et al., 2000, Rodriguez et al., 2001)level. The employment of nonequilibrium grown techniques that incorporate high dopant concentrations in semiconductors has boosted the attention towards this class of alloys. In particular, despite the mutual insolubility of the constituents, the homogeneous single crystal (GaAs)_{1x}(Ge_{2})_{x} can be synthesized as metastable alloy by sputter deposition technique (Barnett et al., 1982), metalorganic chemical vapor deposition (MOCVD)(Alferov et al., 1982), molecular beam epitaxy (MBE) (Banerjee et al., 1985), and rf magnetron sputtering (Rodriguez et al., 2000, Rodriguez et al., 2001). The direct gap “tailorability” for (GaAs)_{1x}(Ge)_{2x}alloys is observed in a final large, negative, and asymmetric, Vshaped, bowing of the bandgap. Barnett et al. (Barnett et al., 1982) report a minimum value of about 0.5 eV at Ge concentration of about 35%: the optical absorption of homogeneous singlecrystal metastable (GaAs)_{1x}(Ge_{2})_{x} alloys, grown using ultrahighvacuum ionbeam sputter deposition is there investigated.
Herbert Kroemer, Nobel prize in 2001 for his studies in semiconductor heterostructures used in electronics, clearly states “…
Severalother theoretical models have been developed in order to describe such zincblendtodiamond phase transition. They are based on thermodynamic (Newman & Dow, 1983, Newman et al., 1989, Gu et al., 1987, Koiller et al., 1985), percolation(D’yakonov & Raikh, 1982), and stochastic growth approaches (Rodriguez et al., 2000, A Rodriguez at al., 2001, Kim & Stern, 1985, Davis & Holloway, 1987, Holloway & Davis, 1987, Preger et al., 1988, Capaz et al., 1989).
Owing to their intrinsic difficulty in taking into account possible different growth conditions, models based on the percolation method (D’yakonov & Raikh, 1982) predict a critical concentration for the transition at 0.57, quite far from the experimentally reported. In “growth models” the alloy configuration depends on the kinetics of the growth. In such methods no explicit functional minimization is performed (Osorio et al., 1991b) and once that atoms have satisfied the set of growth rules imposed by the model, they are considered as frozen, without any further possibility of including the influence of thermal effects. The prediction of the alloy configuration is based on Monte Carlo (MC) models and the atomic position of single layers depends on the epitaxial growth direction. Thus, Long Range Order (LRO) of the final structure depends on the growth direction. Imposed requirement in such models is the “wrong” bond formation (i.e., III−III and V−V) forbiddance. The absence of SbSb bonds in (GaSb)_{1x}(Ge_{2})_{x} alloy detected via extended Xray absorption fine structure (EXAFS) experiments (Stern et. al., 1985) confirms the appropriateness of such imposed condition.Kim and Stern have proposed a model specifying in the set of growth rules the equivalent probability for the A^{III} and B^{V} species in the site occupancy, obtaining a critical concentration for the phase transition (x_{c}) that is 0.26(+0.03/0.02) on (100) substrate (Kim & Stern, 1985). Such model only considers Short Range Order (SRO), reporting the critical composition below which LRO is present. They also report thex_{c}dependence by the growth morphology: a planar growth along [100], another, still planar along the [111] direction, and finally a spherical growth model are studied. In this last model the critical Ge concentration is calculated to be smaller than 0.18 (even if this last value is affected by computational limitations that lead to possible inaccuracy). Davis and Holloway have developed another model implemented by MC simulation and analyzed via an analytical approximation. In their model the formation of GaGa and AsAs nearest neighbour bonds is forbidden, and additionally every gallium atom forms a bond with As atom present in excess during the growth process (Davis & Holloway, 1987, Holloway & Davis, 1987). They find a value for x_{c} ≈ 0.3 on (100) oriented substrates(Davis & Holloway, 1987). No phase transition is at variance predicted along the <111> direction (Davis & Holloway, 1987), with remnant ZB phase present in all the range of composition. This prediction confirms the experimental findings for (GaAs)_{1x}(Ge_{2})_{x} obtained via High Resolution XRay Diffraction (HRXRD) on several substrate orientations (Rodriguez et al., 2000, Rodriguez et al., 2001).
Rodriguez
where r represents the probability that any Ga (As) occupies its site in the lattice. This corresponds to r=1 (0.5) in the case of perfect LRO (disordered crystal). The analysis of Rodriguez of the optical gap and Raman scattering, shows thatnearneighbour correlations (SRO) extremely influence the optical propertieswhile, at variance, there is no impact of the substrate orientation and the LRO on the optical properties. Figure 4 reports the mean cluster reciprocal length obtained by Rodriguez by the Monte Carlo simulation for different orientations of the alloy. It is evident from the plot the overall tendency of GaAs clusters in the alloys to reduce their size (fragmentation), increasing the Ge concentration, x. That kinetic models better describethe phase transition than thermodynamic ones is well assessed. The reasons stem from the fact that thermodynamic models do not include the details of the critical composition x_{c} as a function of kinetic growth, the phase transition critical concentration must be explicitly added as input, and additionally there are no restrictions on the formation of GaGa and AsAs bonds. Also a determining role in the nature of the alloy is played by the growth temperature (Banerjee et al., 1985): (GaAs)_{1x}Ge_{2x} layers epitaxially grown on GaAs (100) substrates at different temperatures analysed by TEM revealed that at T_{g} = 550 ºC, Ge separated from GaAs into domains of ~100 Å. Differently, singlephase alloys are still detected at T = 430 ºC.
The research described in this chapter has been motivated by the fact that so wide potential applicability of this class of alloys in PV is astonishingly not supported by a deepen knowledge both at Density Functional Theory (DFT) and postDFT level of their electronic, structural, and optical properties. We thus decided to examine the bowing in (GaAs)_{1x}Ge_{2x} alloys, searching the microscopic origin of this intriguing and not yet clarified phenomenon. In particular, at first we have theoretically analyzed the properties of four different intermediate structured compounds that range between “pure” GaAs and “pure” Ge (x_{Ge} = 0.25, 0.50 (two samples), 0.75) (Giorgi et al., 2010). Enlarging our models to ones ranging between 8 and 64 atoms, we have investigated the impact of clustering effects and that of the cluster shape on the bandgap bowing (Kawai et al., 2011).
For the alloy electronic properties calculations two of the methods which are reported to give extremely reliable results have been employed: the Quasiparticle Selfconsistent GW (QSGW) approximation approach (van Schilfgaarde et al., 2006) developed by Mark Van Schilfgaarde and his group at Arizona State University and the frequency dependant GWmethod implemented in the VASP code (VASP) by Shishkin and coworkers. (Shishkin & Kresse, 2006, Shishkin & Kresse, 2007, Fuchs et al., 2007, Shishkin et. al., 2007).
The initial analysis of the 8atoms cell revealed that the reduction of the bandgap for intermediate x values in (GaAs)_{1x}(Ge_{2})_{x} alloys takes place with a lattice constant increase and a symmetry reduction with the formation energies linearly related with the number of bad bonds in each model (Giorgi et al., 2010); the subsequent analysis on the shape and the clusterization effects present in these alloys have confirmed at first the main role that SRO plays on asymmetric bandgap behaviour (Rodriguez et al., 2001), further confirming experimental results, like the tendency of large clusters in the alloy to fragment (McGlinn et al., 1988). Large models have been also employed to refine the shape of the asymmetric bowing.
3. Computational details
Using Blöchl’s allelectron projectoraugmented wave (PAW) method (Blöchl, 1994, Kresse & Joubert, 1999), we performed spinpolarized calculations by using density functional theory (DFT), within both the local density approximation (LDA) (Perdew & Zunger, 1981, Ceperley & Alder, 1980) and the generalized gradient approximation (GGA) of Perdew and Wang (Perdew, 1991, Perdew et al., 1992) and of Perdew, Burke, and Ernzerhof (PBE, Perdew et al., 1996). In details,d electrons in the semicore for both Ga and Ge have been considered. Cutoff energies of 287 and 581 eV were set as the expansion and augmentation charge of the plane wave basis. The force convergence criterion for these models was 0.01 eV/Å.
3.1. The initial case of eight atom unit cells
The initial (GaAs)_{1x}Ge_{2x} models consisting of 8 atoms were optimized with a 10 x 10 x 10centered kpoints sampling scheme. The reliability of our initial structures has been checked by recalculating all the total energies with the generalized LMTO method scheme developed by Methfessel
derived from the productreactant equation:
Owing tothemethodological derived large cancellation of errors, both LDA and GGA are expected to predict reasonable heats of reaction like for that of Eq. (3). At variance with structural properties, optical ones are much less well described, with a well documented underestimation of semiconductor bandgaps. Also dispersion in the conduction band is affected by this DFT shortcoming: for Ge, the LDA gap is negative and _{1c} is lower than L_{1c}in contradiction to experiment. Also the X dispersion is often strongly affected: in GaAs X_{1c} _{1c} is about twice the experimental value; underestimations that are generated by the selfinteraction error (Perdew & Zunger, 1981) and the discontinuities in the derivatives of exchangecorrelation energy(Perdew & Levy, 1983, Sham & Schlüter, 1983).
In the prediction of semiconductor optical properties one of the best method is basedon the GW approximation of Hedin (Hedin, 1965). This approximation is a perturbation theory around some noninteracting Hamiltonian,H_{0}. In particular, the quality of this Hamiltonian highly impacts on the quality of the final GW result. In conjunction with a “safe” choice of the Hamiltonian, it must be stressed that for reliable results the use of an allelectron method is highly recommended (GomezAbal et al., 2008). To satisfy both the requirements, in this initial stage of our calculations, we have adopted here an allelectron method, where not only the eigenfunctions are expanded in an augmented wave scheme, but the screened coulomb interaction W and the selfenergy = iGW are represented in a mixed planewave and localfunction basis (Kotani& van Schilfgaarde,2002, Kotani et al., 2007).In addition, all core states are treated at the HartreeFock level. In the following we briefly describe the methodology key points.
Usually, in literature the initial Hamiltonian (H_{0}) for GW calculations is an LDA derived guess;thus usual GW method may be namedG^{LDA}W^{LDA} approximation. Many limitations characterize thisG^{LDA}W^{LDA} approaches as reported in previous literature (van Schilfgaarde et al., 2006b). The QuasiParticle SelfConsistent GW (QSGW) approximation (van Schilfgaarde et al., 2006a), overcomes most of these limitations. Semiconductor energy band structures are well described with uniform reliability. Discrepancies with experimental semiconductor bandgaps are small and highly systematic and the origin of the error can be explained in terms of ladder diagrams missing in the random phase approximation (RPA) to the polarizability (Shishkin et al., 2007). The RPA results in a systematic tendency for the dielectric constant,_{ }, to be underestimated. The error is very systematic: _{ } is too small by a factor of approximately 0.8, for a wide range of semiconductors and insulators. This fact and also the fact that the static limit of Wmainly controlsthe quasiparticle (QP) excitations, provides a simple and approximate remedy to correct this error:V_{xc} ^{LDA} is scaled by 0.8.
3.2. The extension up to 64 atoms
The Short Range Order effects (i.e., those involvingshape and the clusterization of GaAs/Ge regions) on asymmetric bandgap behaviour was confirmed in the alloys synthesized by ionbeam sputtering techniques(McGlinn et al., 1988) and rf sputtering techniques (Rodriguez et al., 2001). In order to modelalloystructures reproducing the SRO effect and deeply understand their effect on the bandgaps, larger supercells (ranging from 8 to 64 atoms) have to be mandatorily investigated to make reliable the comparison with experiments.
Inthe case of the optimization of larger supercells we made use of similar settings of those of the initial 8atom cells, being the force convergence criterion still 0.01eV/Å. At variance with the initial case, the number of kpoints was still 10×10×10 for models constituted by eight atoms, but in this case we used the MonkhorstPack (MP) scheme (Monkhorst & Pack, 1976), checking in this way the possible impact on the final results:our calculations revealed that MP scheme and Γ−centred sampling schemes gave identical results. For the n_{x}×n_{y}×n_{z} multiplied supercell models derived from the eight atom unit cell we thus used a 8/n_{x}×8/n_{y}×8/n_{z}kpoints sampling scheme. For such larger supercell E_{G} calculation, we employed a GGA+GW_{0} (Fuchs et al., 2007) scheme, using the eigenvalues and wave functions obtained at GGA levelas initial guess for GW_{0} calculations (eigenvalues only updated, screened potential kept fixed). For the GGA calculations, we used the PerdewBurkeErnzerhof (PBE) functional (Perdew et al., 1996). Cutoff energy for response function is 90 eV, and the number of frequency points for dielectric function is 48. The number of unoccupied bands was increased up to 200. A 6×6×6 Γcentred sampling scheme was used for eight atom models. For the n_{x}×n_{y}×n_{z} supercell models of the initial eight atom unit cell we used a Γcentered 4/n_{x}×4/n_{y}×4/n_{z}kpoint sampling scheme.
4. Discussion
4.1. The initial case of eight atom unit cells
A common starting point for both approaches is represented by the calculations at the DFT level of thestructural optimized parameters of the two most stable polymorphs of GaAs, zincblende (ZB, group 216, F43m, Z=4) and wurtzite (WZ, group 186, P63mc, Z=2) and of Ge in its cubic form (group 227, Fd3m, Z=8) Ge and GaAs, reported in Table 1.
The choice of using LDA in all the subsequent calculations stems from the fact that in this context LDA reproduces structural properties closer to experiment than GGA. Both Ga–As and Ge–Ge bond lengths are 2.43 Å in their most stable polymorph.
ZB–GaAs is constituted by interpenetrating fcc sublattices of cations (Ga) and anions (As). The diamond lattice of Ge may be through of as the ZB structure with Ge occupying both cation and anion sites. In this section we consider 8atom (GaAs)_{1x}Ge_{2x} alloy models that vary the Ge composition, including pure GaAs (x=0) to x=0.25 (Ge dimers), x=0.50 (4 Ge atoms), x=0.75 (6 Ge atoms) and finally pure Ge (x=1). Figure 5 reports the structures of the four intermediate alloys. At first we performed an analysis of the Ge dimer in bulk GaAs, at site positions (0.25, 0.25, 0.5) and (0., 0.25, 0.75), i.e. the alloy model I.
GaAs (ZB) 216, F43m, Z=4 
GaAs (WZ) 186, P_{63mc}, Z=2 
Ge (cubic) 227, Fd3m, Z=8 

Our analysis, PAW/LDA ΔE B Lattice constant (Å) 
 66.14 a=5.605 
+0.06 a=3.917, b=3.886, c=6.505 
 71.8 a=5.612 
Our analysis, PAW/PW91 ΔE B Lattice constant (Å) 
 79.01 a=5.739 
+0.03 a=4.040, c=6.668 
 71.0 a=5.747 
Our analysis, PAW/PBE ΔE B Lattice constant (Å) 
 65.94 a=5.744 
+0.023 a=4.045, c=6.670 
 74.8 a=5.741 
Previous study (GGA) B Lattice constant (Å) 
59.96^{a}
a=5.74^{a},5.722^{b} 
a=3.540, c=6.308^{c}  55.9^{d}
a=5.78^{d} 
Previous study (LDA) ΔE B Lattice constant (Å) 
 75.7e , 77.1f a=5.654g,5.53e 5.508f, 5.644h 
+0.0120g a=3.912,c=6.441g a=3.912,c=6.407e 
73.3^{d}, 79.4^{d}
a=5.58^{d}, 5.53^{d} 
Experimentally ΔE B Lattice constant (Å) 
 77.i a=5.649i,5.65j 
+0.0117h 
75.^{l }a=5.678^{k }, 5.66^{l} 
This model can be considered a highly concentrated molecular substitutional Ge_{2} defect in GaAs, for which we predict stability owing to the donor–acceptor selfpassivation mechanism (Giorgi & Yamashita, 2011). For a better understanding of this last aspect concerning selfcompensation mechanism, we invite the reader to take a look at Section 4.3, where the stability of Ge substitutional defects (donor, acceptor, and donoracceptor pairs) in GaAs matrix and its relationship with alloy selfcompensation stabilizing mechanism is deeply discussed.
The overall sequence of Iis a repeated “sandwichlike” structure, ···/As/Ge–Ga/Ge–As/Ga/···along the (001) direction. The bond lengths are 2.38 (Ga–Ge), 2.42 (Ge–Ge), 2.44 (Ga–As), and 2.47 Å (Ge–As) only slightly differing from the calculated values in bulk Ge and GaAs (2.43 Å). The small electronegativity variation (Δχ) is the explanation of this reduced difference in the bondlength, being GaGe and GeAs nearly covalent twocenter bonds (_{Ga}=1.81, _{Ge} =2.01, and _{As}=2.18).
In the alloy Ithe total number of IIIIV and IVV “bad bonds” (Osorio & Froyen, 1993, Kroemer, 2001, Rodriguez et al., 2001) is 12, or 37.5% of the total. According to the Bader analysis (Henkelmann et al., 2006, Tang et al., 2009; Sanville et al., 2007), in the pure host, the difference in electronegativity is responsible for charge transfer from cation to anion.
In the alloy formation process, the introduction of Ge reduces the ionic character of the GaAs bond, while increasing the ionic character of the Ge–Ge bond. When a Ge dimer is inserted in GaAs, 0.32 electrons are transferred away from Ge_{Ga} site, while Ge_{As} gains 0.21 electrons. The charge deficit on Ga, is reduced from 0.6 electrons in bulk GaAs to 0.47e, while the charge excess on As is reduced from 0.6e to 0.5e. The heat of reaction according to Eq. (3) was 0.55 eV, and the optimized lattice parameter was a=5.621 Å. We have also considered Ge donors (Ge_{Ga}) and acceptors (Ge_{As}) in the pure 8atom GaAs host cell, separately. The formation energy has been computed according to the ZhangNorthrup formalism (Zhang & Northrup, 1991). In particular, we calculate ΔE to be 1.03 eV for Ge_{Ga} and 0.84 for Ge_{As}. The sum of the single contributions (1.87 eV) is larger than the heat of formation of the dimer, structure I(0.55 eV). This is ascribable to the fact that in the model alloy at least one correct bond IIIV is formed while in the separate Ge_{Ga} (IVV) and Ge_{As} (IVIII) cases only bad bonds are formed. The isolated Ge_{Ga} is a donor; the isolated Ge_{As} is an acceptor, thus both of them are unstable in their neutral charged state. We have tested it in another work (Giorgi & Yamashita, 2011) where we calculated +1 and 1 as the most stable charged state for Ge_{Ga} and Ge_{As} (both isoelectronic with GaAs), for most of the range of the electronic chemical potential. That the stabilization energy 1.32 eV (i.e., 1.870.55) is only slightly smaller than the host GaAs bandgap establishes that the selfpassivating donoracceptor mechanism is the stabilizing mechanism of this Ge dimer. As previously stated, for a further and deeper analysis of Ge substitutions in GaAs the reader take a look on Section 4.3.
We considered two alternative structures for the x=0.50 case. In the IIa structure Ge atoms are substituted for host atoms at (0.5, 0., 0.5), (0.5, 0.5, 0), (0.75, 0.75, 0.25), and (0.75, 0.25, 0.75). The initial cubic symmetry lowers toward a simple tetragonal one: the optimized lattice parameters were found to be a = 5.590 Å, b = c = 5.643 Å. The 4 intralayer bond lengths were calculated to be Ga–Ge (2.39 Å), Ge–As (2.48 Å), Ge–Ge (2.42 Å), and Ga–As (2.44). Because of the increased amount of Ge, structure IIa was less polarized than I, as confirmed by the slightly more uniform bond lengths. In IIa alloy the number of “bad bonds” is 16 (i.e., 50%) and E rises to 0.72 eV. In the IIb structure Ge atoms are substituted for host atoms at (0.25, 0.25, 0.25), (0.25, 0.75, 0.75), (0.75, 0.75, 0.25), and (0.75, 0.25, 0.75). This structure consists of a stack of pure atomic layers, ···/Ga/Ge/As/Ge···, and thus it contains
We have also made a preliminary calculation of the stability of isolated Ga acceptors (Ga_{Ge}) and As donors (As_{Ge}) vs that of the substitutional molecular GaAs_{Ge2} in Ge pure host supercell consisting of 64 atoms; for such concentrations (0.0312=1Ge/32GaAs unit and 0.0156=1/64GaAs), the molecular substitutional GaAs_{Ge2} is only stabilized by 0.057eV with respect to the separate couple acceptordonor. This small stabilization for GaAs_{Ge2} compared to isolated Ga_{Ge} and As_{Ge} confirms the expected similar probability of finding a mixture of ntype and ptype semiconductors in the “disordered” Gerich phase. We have used the most stable polymorphof the elemental compounds (orthorhombic Ga and rhombohedral As) for the chemical potential, μ, of both elements (Mattila & Nieminen, 1996).
Garich (μ_{Ga}=μ_{Ga} ^{bulk}) and Asrich (μ_{As}=μ_{As} ^{bulk}) conditions have been considered, respectively. In the case of the 8atom cells, the formation energy for Ga_{Ge} and As_{Ge} are 0.26 eV and 0.58 eV, respectively. The model III stabilizes the isolated Ga and As substitutions by 0.30 eV, i.e. the ΔE between the alloy and the isolated substitutionals.
At variance with alloy model I, the large energy difference (~0.4 eV) between the stabilization energy and the Ge host bandgap (0.67 eV at 300 K (Kittel, 2005)) reveals that other factors, and not only a selfcompensating donoracceptor mechanism, impact on the final stability of this III alloy model.
Our calculations reveal an almost exactly linear relationship between the formation energy and the number of bad bonds, as reported in Figure 6. Such relationship is verified at least for systems containingsame number of Ge donors (Ge_{Ga}) and Ge acceptors (Ge_{As}).
This striking result confirms that the electronic structure of these compounds is largely described in terms of independent two center bonds. For stoichiometric compounds, it suggests an elementary model Hamiltonian for the energetics of any alloy with equal numbers of Ge cations and anions. On all the optimized structures QSGW calculationshave been performed and also for the pure GaAs and Ge 8atom cells. From Figure 7, where the QSGW bandgaps as function of [Ge] are reported, one can see the good reproduction of the asymmetric bowing both atand R points. In particular, QSGW calculated bandgaps for pure GaAs and Ge are 1.66 and 1.04 eV on Γ.IIb model (100% of bad bonds), whose bandgap is not reported in Figure 7, has E_{G}< 0 at both the two points, confirming the tight relationship between high concentration of bad bonds and reduced values of the bandgap.
4.2. Extended models: The quantitative description of the asymmetric bowing minimum
The subsequent analysis focused on extended alloy models. In particular, we have taken into account here models ranging between 8 and 64 atoms.
Our choice has the twofold target of confirming the initial results comparing different methodologies for bandgap calculations, and that of investigating the impact that cluster shape and size has in the bandgap itself. It is also supposed that enlarging the size of the models can extremely improve the reproducibility of the asymmetric Vshape of the bandgap bowing. The Special Quasi Random Structures (SQS) methodology (Zunger et al., 1990), developed to incorporate SRO and local latticedistortions in alloy systems, has been widely employed in literature for the description of alloy properties (see Fiorentini & Bernardini, 2001). We stress that the modellization we have here chosen does not lead mandatorily towards a global minimumfor each concentration, stemming this choice from the metastable nature of (GaAs)_{1x}(Ge_{2})_{x}alloys, grown only at nonequilibrium condition (Barnett et al., 1982, Rodriguez et al., 2001, Banerjee et al., 1985, Alferov et al., 1982). The appropriateness of our modellization is confirmed by the high reproducibility of the experimental results we have obtained. McGlinn. (McGlinn et al., 1988)haveexperimentally found that Ge regions start forming networks as the Ge concentration increased in the range between 0. and 0.3, andat [Ge]=0.3suchnetworks are connected with each other. This mechanism is accompanied by the GaAs region fragmentation towards sizereduced clusters. Such experimental finding is our driving force in the choice of the enlarged models: we based our study onmodels reproducing Geclusterized alloys at 0<x<0.3 and GaAsclusterized alloys at 0.3<x<1. As stated, experimental resultsreportonly the formation of size reduced GaAs clustersfor[Ge] >0.3, thuswe decided to compare and discuss local geometry effects and their influence on the bandgap of the two specular models at x = 0.375 (such Ge concentration represents the closest one in our models to the experimentally reported concentration, 0.3, where bandgap minimum is found (Barnett et al., 1982)): a Geclusterized(IIIa) and a GaAsclusterized(IIIb) one. Going back to the discussion regarding our two models at x=0.375, IIIbandIIIa(whose structure is reported in Figure 8), the former alloy has larger bandgap than the latter,revealing thatat low Ge concentrationE_{G} continues decreasing as Ge concentration increases, as long as the alloy geometryis characterized bythe presence of Ge clusters embedded in GaAs host, i.e., in a quantum dotlike fashion. Differently,when GaAs turns to clusterize in Ge network, the bandgap stops decreasing. The relationship between the calculated direct gaps along all the range ofxis reported in Figure 9, in conjunction with the experimental values.
The close resemblance between our theoretical fitting and the experimental one reveals the extreme suitability of our models in order to reproduce the asymmetric bandgap bowing of (GaAs)_{1x}(Ge_{2})_{x} alloys: a sharp E_{G}decreasing at 0<x<0.3 accompanied by the subsequent smooth increasing at 0.3<x<1.
Up to now we were able to explain and demonstrate that the bandgap minimum detection is due to the switching of the embedded cluster in the host from a quantumdotlike fashion (Ge in GaAs) to an antiquantumdot like fashion (GaAs in Ge). Let us go one step further and let us try to understand the chemical origin of this minimum in the bowing.
As we mentioned, in (GaAs)_{1x}(Ge_{2})_{x}alloys, the acceptorsand donors, when nearestneighbors,are subjected to stabilizing selfcompensation mechanism, while between badbond pairssimilar mechanism doesnot occur due to the presence of residual local positive and/or negative charges.However, if such charges can be delocalized, the compensation can be effective even beyond nearestneighbor atomic sites. Thus,since the charge distribution on the bad bonds describes the nearestneighbor atomic chemical environment,it represents a highly valuable analysis to estimate the SRO. In order to investigate the effect of the cluster typeswitchingon bad bonds, we analyzed the charge distribution of IIIa and IIIb models according tothe Bader charge analysis scheme (Henkelmann et al., 2006, Sanville et al., 2007, Tang et al., 2009).
Inoptimized GaAs, charge on Ga (As) atoms is +0.55e (0.55e). In model IIIa, containingGeclusters, the charges range between +0.40 and+0.52 for Ga, 0.53 and 0.48 for As, and between 0.29 and +0.32 for Ge, respectively. The charge reduction for Ga and As with respect to those of pure GaAs are easily explained in terms of electronegativity, χ. Δχare smaller on bad bonds than on correct GaAs bonds.Charges on modelIIIb are +0.30 on Ga, 0.41 on As, and +0.04/+0.08 on Ge, respectively, values extremely reduced if compared with those of IIIa. Such charge lowering is explained in terms of number of bad bonds that single Ga or As atom forms. In IIIa, each Ga and Asatom forms 1 or 2 bad bonds. In IIIb, on the other hand, each Ga and As atom forms 3 bad bonds. The increase of bad bonds on single Ga and As atom causespartial delocalization of charges, as a consequence of the reducedΔχ between their constituting atoms. This result also impacts the VBM charge densitydistribution on Γ; indeed, in the IIIa model it is localized around the As and Ge, whose electronegativities are larger than Ga (strong charge localization on the formed bad bonds). Thus, similarly to the valence band forward shiftascribed to the charge densitylocalization (nonbonding), the sharp bandgap decrease in the range 0< [Ge] < 0.3is ascribed to the enhancement of thenonbonding character of the VBM. Differently, in IIIb the VBM is highly delocalized in the whole crystal.Here, the strong bonding character of VBM stabilizes the system, causing the backward shift of VBM and also the smooth bandgap opening at 0.3<
4.3. On the stability of Ge donor and acceptor defects
In previous sections we have widely taken advantage of the concept of selfpassivation. We here focus on this very basic concept of semiconductor physics, showing that regardless the nature of the cell we are considering, alloys or supercells, the selfpassivation stabilization mechanism between Ge donor and Ge acceptor pairs is the main stabilizing process in these IVdoped/IIIV systems effective also for non nearest neighbor couples. We aim to demonstrate that alloys are superconcentrated defective cells (Giorgi & Yamashita, 2011).The formation energy of the defect is defined as the contribution deriving from the formation energy of the defect in its state of charge, plus the contribution of Ga and As potentials in GaAs, and the potential of the substituting Ge. The thermodynamic stability of the charged substitutional Ge defects is calculated as (Zhang & Northrup, 1991):
whereis the chemical potential difference between bulk GaAs (_{GaAs(bulk)}) and bulk Ga (_{Ga(bulk)}) and As (_{As(bulk)}), respectively, ranges between H_{form}. H_{form} is the heat of formation of bulk GaAs. Beyond the two extreme conditions of = H_{form} (Garich and Asrich conditions, respectively) precipitation takes place. _{e} is the electronic chemical potential and E_{VBM} the energy of the top of the valence band (VBM). n_{Ga(As, Ge)} is the number of atoms of Ga (As, Ge) in the supercell, while q is its total charge. It is straightforward that this equation is the extension (including the charged case and in the stoichiometric case) of Eq. (3) reported in Section 3.1. Similarly for the alloy case,_{Ga(As,)} is calculated from the orthorhombic (trigonal) polymorph for Ga (As) (Mattila & Nieminen, 1996; Giorgi & Yamashita, 2011). A method based on the combination of the Potential Alignment (PA) for the correction of supercells with a net charge and image charge correction (Lany & Zunger, 2008, Makov & Payne, 1995) has been applied.
To improve the description of the bandgap we employed an LDA +U scheme (Dudarev et al., 1998, Giorgi & Yamashita, 2011).VBM and CBM was also corrected and aligned with the LDA +U obtained bandgap. TheGe_{2}quasimoleculedimer defect best reproduces the high Gedoping concentration because of the equally probable substitution of one Ga and one As atom, and represents the starting point for our analysis on the selfcompensation mechanism in such systems. The neutral state is the most stable along the whole bandgap, as a consequence of the stabilization induced by the mentioned donor–acceptor selfpassivating mechanism. The formation energy of the Ge molecular defect is reported in Figure10.
We also considered the case of isolated Ge_{Ga} and Ge_{As} (Ge_{Ga}···Ge_{As}) pairs in the supercell. Comparison between the E of the three structures (Ge_{2GaAs}, d=2.43 Å, Ge_{Ga}···Ge_{As}, d=4.65, 7.30 Å, respectively) reveals the net tendency for Ge to cluster; the first configuration is indeed 0.36 eV more stable than the second and 0.41 eV more stable than the third. This energy difference is due mainly to the formation of one III–IV (Ga–Ge) and one IV–V (Ge–As) bond and the breaking of one IV–IV (Ge–Ge) bond. To evaluate the impact of distance between pairs of substitutionals on the selfcompensating mechanism, we have calculated the Ge–Ge pair correlation J(_{Ge,h,Ge,h}) as:
where E_{tot}(GaAs:Ge_{h}Ge_{h}) is the energy for the double Gesubstituted GaAs and E_{tot}(GaAs:Ge_{h}) is that for the single substituted states (h_{ }and h_{ }). The most correlated pair is the quasimolecular Ge_{2GaAs} defect (–1.55 eV), while for the other two cases we calculate a correlation energy of –1.18 eV for Ge_{Ga}···Ge_{As} at d = 4.65 Å and –1.15 eV for Ge_{Ga}···Ge_{As} at d = 7.30 Å). The SRO included in the difference of the correlation energy between the three distances account for electronic and steric effects. Indeed, the direct formation of Ge–Ge may partly release the stress resulting from substitution of Ga and As in the host. A comparison between the defective supercells and the alloys is here straightforward (see Figure 10, selfcompensation mechanism for the two systems).
As in the case of defective supercells also in alloy models, “bad bond” formation and the selfcompensation mechanism were considered destabilizing/stabilizing driving forces of the final alloy; we may indeed evaluate the SRO effects on both systems and find a unified trend for alloys and defective supercells. The stabilization energy for the quasimolecular defect is (E_{Form}(Ge_{Ga}
^{0}) + E_{Form}(Ge_{As}
^{0})) – E_{Form}(Ge_{2GaAs})= 1.60 eV (= –J_{nn}, the correlation energy). Effects related to the Ge–Ge direct bond formation give a contribution of 1.01 eV to the total stability (1.60 eV – 0.59 eV). The same contribution for one Ge–Ge bond direct formation in an 8atom cell alloy was found to be 0.78 eV. Increasing the number of Ge in the alloy results in an increase in the stabilization energy, i.e., 1.51 (= 1.87 – E_{Form}/2) eV for two Ge_{2}, and 1.69 eV (= 1.87 eV – E_{Form}/3) for three Ge_{2}.In the present case, we evaluate how the distance between pairs influences the stabilization energy, or in other words, how selfpassivation increases the stability of the overall systems. The stabilization energy for the Ge_{Ga}···Ge_{As} at
5. Conclusion
In this chapter we have reviewed the technology behind the MultiJunction technology in solar cell assembling based on IVdoped IIIV alloy showing the importance of Density Functional Theory as a tool for the prediction of the structural and electronic properties of these alloys. After an initial study focusing on eight atom cells, we have extended the analysis to systems constituted by up to 64 atoms. We detected a linear relationship between formation energy and number of bad bonds in the alloys. The relevance of this result stems by the fact that for stoichiometric compounds an elementary model Hamiltonian for the energetics of any alloy with equal numbers of Ge cations and anions as function of the number of bad bonds can be developed. The bandgap bowing for these alloys is confirmed by GW calculations. Increasing the cell size we were able to quantitatively reproduce the asymmetric bandgap bowing of (GaAs)_{1x}(Ge_{2})_{x} alloys.This finding stems from an extremely suitable model choice: moving from previous experimental results, we found that Geclusterized alloys at 0<x<0.3 and GaAsclusterized ones at 0.3<x<1 are the best in reproducing the asymmetric Vshape of the bowing. Turning from Gecluster to GaAscluster embedding alloys at concentrations close to the experimental reported for the bandgap minimum is the keypoint for the interpretation of this controversial phenomenon. The last part of the Chapter has been dedicated to the discussion of the stability of the Ge donoracceptor defects in the GaAs supercells. Regardless the distance between Ge pairs in both defective supercells and alloys, the selfpassivation mechanism results the driving force of the stabilization of IVdoped IIIV systems, being sensitively effective in the former case also for Ge pairs non nearestneighbors.
Acknowledgments
This research was supported by a Grant from KAKENHI (#21245004) and the Global COE Program [Chemical Innovation] from the Ministry of Education, Culture, Sports, Science, and Technology of Japan. GG wants to thanks Dr. G. F. Cerofolini (University of Milano Bicocca) and Dr. A. Korkin (Arizona State University) for the fruitful and stimulating discussions and for a longstanding real friendship.
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