The formation of solid solutions between the non-isotype compounds ZnC and CuBIIIC2 (B-Al, Ga, In; C-S, Se) enables the band-gap energy to be changed from the large value of the binary wide-gap semiconductor ZnC to the band gap of the chalcopyrite-type end member CuInC2 [37]. The possible application of 2(ZnS)- CuInS2 mixed crystals as absorbers in thin-film solar cells was introduced by Bente et al. [38]. However, these solid-solution series are characterized by a relatively — large miscibility gap, where a tetragonal and a cubic phase coexists [37, 39-41], thus the applicable range is limited.

For tetragonal Zn2x(CuB)1-xC2 mixed crystals, the question of the distribution of the three cations Zn[2]+, Cu+ and B3+ on the two cation positions of the chalcopyrite — type structure arises. Like in CIGSe, the cation distribution influences the opto­electronic properties of the solid-solution compounds. The Cu-Zn differentiation problem existing in X-ray diffraction due to a nearly-equal atomic form-factor f for Cu+ and Zn2+, can be solved by neutron diffraction because of the different neutron scattering cross-sections for Cu and Zn, which are based on the different neutron scattering lengths (bZn = 5.68 fm, bCu = 7.718 fm [18]).

The most detailed analysis was performed for chalcopyrite-type Zn2x(CuIn)1-xS2 mixed crystals [42]. It should be noted that the study was performed using powder samples with stoichiometric composition (the chemical composition was determined by wavelength-dispersive X-ray spectroscopy using an electron microprobe system). For the interpretation of the average neutron-scattering lengths of the cation sites 4a and 4b (b^f and Zgb), derived from the cation SOFs determined by Rietveld analysis of neutron powder diffraction data, the principle of the conservation of tetrahedral bonds (CTB) for ternary ABC2 chalcopyrites [43] was applied. In the chalcopyrite-type structure a displacement of the anion from its ideal position (У, i. e. the middle of the cation tetrahedron) by u — У (u is the anion x-coordinate) can be observed. Hence different bond lengths RAC ф RBC result, which in turn cause different-sized anion tetrahedra AC4 and BC4, resulting in a tetragonal deformation П = c/2a parallel to the crystallographic c-axis. The bond lengths are [43]

The parameters n and u are considered as the degrees of freedom of the chal­copyrite-type structure [Jaffe, Zunger 84]. The Abrahams-Bernstein relation [44]

1 c2 1

u = 2 — 32? -16 (5J)

correlates the tetragonal distortion u with the lattice parameters a and c. However, there is a limitation: only one of the anion tetrahedra is assumed to be deformed, whereas the other is taken as regular.

According to the CTB, the degrees of freedom (n and u) would attain values that simultaneously minimize the difference between the bond lengths RAC and RBC and the sums of the elemental radii as

RAC(a, g, u) — га — rc= 0 and RBC(a, g, u) — Гв — rC = 0 (5.8)

By applying Eqs. (5.6) and (5.7) the solutions for Eq. (5.8) can be written as о 12a2

a

2p + a — (2p + a)2-18a2

g2 = 8(P — a)

g = 3a2

Here a is the bond-mismatch parameter and P the mean-square-bond

a = rax _ RBx = (rA + rX)2-(rB + rX)2 ^5 11^

P = RAX + RBx = (rA + rX)2-(rB + rX)2

The CTB model can be extended to quaternary chalcopyrite-type compounds, assuming the covalent radii in the equations above as the average radius of the cations on the two cation positions, according to

rA = ZnArZn2+ + CuArCu+ + InArIn3+ and (5 12)

rB = ZnBrZn2+ + CuBrCu+ + InBrIn3+

Here ZnA, CuA, and InA are the mole fractions of the cations on the Wyckoff position 4a (A) and 4b (B) according to the cation-distribution model. These fractions corre­spond to the total amount ofZn, Cu, and In in (2ZnS)x(CuInS2)1-x (i. e. ZnA + ZnB = 2x). Thus the average cation-radii are influenced by the cation distribution.

For the calculation of the average neutron-scattering lengths of the cation sites 4a and 4b (b^ and b4a1c), a certain cation distribution has to be assumed. A first comparison with the experimentally-determined average neutron-scattering lengths show, that Zn is not statistically distributed on the sites 4a and 4b. Thus three different aspects have to be taken into account for modelling the cation distribution:

(i) Zn is non-statistically distributed

(ii) If Zn prefers the 4a position a CuIn anti-site is enforced and if Zn prefers the 4b position an InCu anti site is enforced (enforced anti-sites)

(iii) Independent of the Zn distribution, CuIn and InCu anti-sites may exist (spontaneous anti-sites)

The evaluation criteria for the cation distribution in tetragonal (2ZnS)x(CuInS2)1-x mixed crystal were formulated as:

(1) u(exp) = u(calc) (u(exp) is determined by Rietveld analysis of the powder diffraction data, u(calc) applying the CTB rule)

(2) bZ = b4f and b4bP = fo4f

Applying both criteria, and taking into account the aspects (i)-(iii), the cation distribution was evaluated in two steps. First, the possible cation distributions fulfilling criterion (1) were derived. As can be seen from Fig. 5.11, a variety of different cation distributions are possible.

In the second step criterion (2) is also taken into account. The graphical solution is shown in Fig. 5.12. It becomes clear, that Zn occupies the 4b site preferentially, enforcing *1.8-4.5 % InCu. Moreover, there is a small fraction of spontaneous Cu-In anti-sites (i. e. CuIn and InCu).

Taking into account the experimental error of the average neutron-scattering lengths Ь^ and bb it can be deduced, that 27.5 % of the Zn occupies the 4a site,

Fig. 5.11 Possible cation distributions fulfilling the criterion u(exp) = u(calc) for the sample Zn018Cu092In090S2. Closed symbols refer to the 4a position, open symbols to the 4b position

Fig. 5.12 Average neutron-scattering lengths calculated for possible cation distributions taking into account a non-statistical Zn distribution as well as enforced and/or spontaneous Cu-In anti­sites. Both limits (all Zn would be ZnCu or ZnIn) narrow the possible cation distributions. The dotted lines give the calculated average neutron-scattering length for the sites 4a and 4b for different cation distributions (with the experimental error), taking into account both enforced and spontaneous Cu-In anti-sites. The blue lines mark the distributions which fulfil criterion (1). Both criteria only meet within the region marked by the green vertical lines

whereas the rest of the Zn occupies the 4b site. This leads to 3.6 % enforced InCu anti-site. These results are in a good agreement with the cation distribution eval­uated from X-ray powder diffraction data ((Zn + Cu)Cu = 0.91(3); InCu = 0.09(3); (Zn + Cu)In = 0.14 and InIn = 0.86(3)), but here Cu and Zn could not be

Table 5.3 Cation distribution in Zn2x(CuIn)1-xS2 mixed crystals (the values are given as mole — fractions) ZnCu CuCu InCu ZnIn CuIn InIn Zn0.11Cu0.95In0.94S2 0.0173 0.9500 0.031 0.0927 0.0 0.9090 Zn0.18Cu0.92In0.9S2 0.0495 0.9035 0.047 0.1305 0.0165 0.853

distinguished. The cation distribution in Zn2x(CuIn)1-xS2 mixed crystal is sum­marized in Table 5.3.

The CTB rule was only applied for the sulfide mixed-crystals due to the rela — tively-well known cation radii in sulfides [23]. For the selenide and telluride Zn2x(CuIn)1-xC2 mixed crystals the differences between bff and bc4fc as well as beJf and b>4afc, assuming a statistical Zn distribution in the calculation of the average neutron-scattering lengths, were considered [42, 45]. Here, a non-statistical distri­bution of Zn on the both cation sites was also found. With increasing ZnX-content in Cu0.5In0.5X there is a propensity for a more statistical distribution of the cations, indicating a tendency for disorder in the cation substructure.

It can be assumed that the non-statistical distribution of Zn and the associated Cu-In anti-sites are related to the limited solubility of ZnX in Cu0.5B0.5X. This fact can be discussed within the framework of formation energies of intrinsic point — defects in copper chalcopyrites. The Cu-In anti-site occupancy, resulting in CuIn and InCu, are the defects with the lowest formation energies (in CuInSe2: 1.3 eV for CuIn and 1.4 eV for InCu [46]). Thus, these defects can be formed relatively easily. Nevertheless, the formation energies of ZnCu or ZnIn are not known. If one of the occupancies were energetically unfavourable, the solubility would be affected.

Using the approach of the average neutron-scattering length and the CTB rule it was possible to determine the cation distribution for various stoichiometric chal- copyrite-type Zn2x(CuB)1-xC2 compounds. It was clearly shown that Zn tends to occupy the 4b site preferentially, resulting in the formation of InCu and CuIn defects, resulting in a partially disordered chalcopyrite-type crystal structure.