21922
J. Phys. Chem. B 2005, 109, 21922-21927
Miscibility of Zinc Sulfide and Zinc Phosphide Karl Jug,* Igor P. Gloriozov,† and Bettina Heidberg Theoretische Chemie, UniVersita¨t HannoVer, Am Kleinen Felde 30, 30167 HannoVer, Germany ReceiVed: July 28, 2005; In Final Form: September 2, 2005
Solid solutions in the system zinc sulfide/zinc phosphide (Zn2+xS2-2xP2x) were investigated using the cyclic cluster model within the semiempirical MSINDO method. Results of cyclic cluster calculations for binding energies of the perfect ZnS and Zn3P2 are presented and compared with the experimental data. The miscibility of ZnS and Zn3P2 over the whole composition range of 0 < x < 1 was investigated by calculating the Gibbs free energy of mixing ∆MG for different values of x. A miscibility gap was found at both ends of the composition range and compared with experimental data.
1. Introduction Zinc chalcogenides have band gaps in a range of technological importance.1 Modification of their properties by impurity incorporation is currently debated in the context of possible applications in ultraviolet optoelectronics and spin electronics.2,3 Cation substitution has been widely studied, but anion substitution has received less attention. Recently we have started to study the effects of anion substitution by quantum chemical methods. We have investigated the miscibility of zinc chalcogenides ZnS/ZnSe and ZnS/ZnO4 by cyclic cluster model (CCM) calculations.5 Whereas the first system shows miscibility over the whole composition range, the second system has a large miscibility gap. Both facts were related to the relative sizes of the anions. In a subsequent work we studied the interaction of doping anions in these systems and compared these trends with the interaction of anion vacancies.6 Whereas the doping anions show predominantly a repulsion, the vacancies favor attraction. A more complicated case arises if the doping is done with an anion of a different valency compared to the chalcogen anion. The doping of zinc sulfide with phosphorus constitutes such a case. In several studies by Locmelis and Binnewies7-9 based on chemical transport of solid solutions, mixed phases ZnS:P and Zn3P2:S were generated and their properties analyzed. Heating of the two solid phases ZnS and Zn3P2 resulted in up to 0.7 atom % P in ZnS:P and 1.2 atom % S in Zn3P2:S at 900 °C.7 In another set of experiments8 the mixed phase ZnS:P was prepared under phoshorus pressure. Here up to 8 atom % phosphorus was found under 3 bar pressure at 900 °C. The substitution of phosphorus atoms in ZnS is accompanied by an increase of the zinc content and leads to an increase of the unit cell. In the present work we study the change of properties over the whole composition range.
Coulombic effects. To achieve this goal the cyclic cluster was embedded in an infinite field of point charges13 using the Ewald summation technique. The cluster shape and size is measured by the relative average coordination number k.14 This number is 1 for the infinite three-dimensional periodic crystal and less than 1 for free clusters. The coordination number of each atom is determined by the number of nearest neighbors. Then the average coordination number of all atoms in the free cluster is calculated and divided by the average coordination number in the bulk. This number is kept also for the corresponding cyclic cluster for consistency reasons. In the (2 × 2 × 2) Zn32S32 cluster the sum of coordination numbers of all 64 atoms is 172. The average coordination number (per atom) of Zn32S32 is therefore 2.69. Since this number is 4 in the ZnS bulk, the relative average coordination number in Zn32S32 is 0.672. In the present work we focus on the miscibility of zinc sulfide and zinc phosphide. Therefore cyclic clusters of the type Zn2mS2m and Zn3mP2m with various m values were studied. 2. Mechanism of Mixing As already discussed earlier4,6 zinc sulfide prevails in cubic zinc blende structure, but the most stable form of zinc phosphide is the tetragonal structure.15 A transition from ZnS to Zn3P2 can be achieved under conservation of electroneutrality. Since there is a formal substitution of S2- ions by P3- ions, lattice defects are the consequence. Two kinds of lattice defects could occur: (a) two P3- ions replace three S2- ions and one sulfur vacancy is generated; (b) two S2- ions are replaced by two P3ions and an additional Zn2+ ion occupies an interstitial site. The first case would relate to a mixing of the type
(1 - x)ZnS + xZnP2/3 f ZnS1-xP2x/3
(1)
whereas the second leads to
2. Methods and Models All quantum chemical calculations presented in this work were performed with MSINDO,10,11 a semiempirical MO method with a documented accuracy for structure and energy, which was extended to third-row transition metal elements.12 The CCM5 was incorporated in MSINDO in order to achieve the effect of periodicity of a perfect crystalline solid in a finite cluster. For ionic systems it is important to include long-range † Visiting scientist. Permanent adress: Department of Chemistry, Moscow State University, Moscow 119899, Russia.
(1 - x)ZnS + xZn3/2P f Zn1+x/2S1-xPx
(2)
The composition x varies from 0 to 1. From experimental analysis9 on the size of the volume of the unit cell, the insertion of Zn2+ ions into vacant lattice sites is observed. In particular, the occupation of tetrahedron holes is deduced. For the purpose of cyclic cluster studies, we rewrite eq 2 in the following form
(1 - x)Zn2S2 + xZn3P2 f Zn2+xS2-2xP2x
10.1021/jp0541720 CCC: $30.25 © 2005 American Chemical Society Published on Web 10/21/2005
(3)
Miscibility of Zinc Sulfide
J. Phys. Chem. B, Vol. 109, No. 46, 2005 21923
As already outlined earlier,4 the stability of solid solutions depends on the Gibbs free energy of mixing ∆MG
∆MG ) ∆MH - T∆MS
(4)
∆MH and ∆MS denote the enthalpy and entropy of mixing, respectively. Negative ∆MG values indicate thermodynamic stability of the solution, whereas positive ∆MG values indicate a miscibility gap. The enthalpy of mixing ∆MH and the entropy of mixing ∆MS are calculated in the following way
∆MH ) ∆ME0 + P∆MV ) E0(Zn2+xS2-2xP2x) - (1 - x)E0(Zn2S2) xE0(Zn3P2) + P[V(Zn2+xS2-2xP2x) (1 - x)V(Zn2S2) - xV(Zn3P2)] (5) ∆MS ) -R(1 - x) ln(1 - x) - Rx ln(x)
(6)
The influence of the additional term P∆MV in the enthalpy of mixing, can be estimated. For pressures of a few atmospheres, its energy contribution to ∆MH is negligible. The entropy of mixing in eq 6 refers to the statistical exchange17 between sulfur and phosphorus atoms. Since each of the additional zinc atoms is usually linked to one of the substituted phosphorus atoms in the most stable arrangement, no additional degrees of freedom are introduced for the zinc atoms in the statistical part of the entropy. Equation 6 is consistent with eq 3 but does not contain the vibrational contribution to the entropy. The latter could be obtained by a molecular dynamics calculation,17 according to an algorithm described by Goddard.18 For large systems with over 100 atoms, this would require an enormous amount of computer time and is therefore planned in a separate investigation. 4. Cyclic Cluster Studies 4.1. Energetics. To have a good starting point for the miscibility investigation, it was necessary to adjust some parameters to fit the structure and heat of formation of zinc sulfide and zinc phosphide. Compared to the parametrization previously used for ZnO, ZnS, and ZnSe,4 the screening parameter KSC of the core integrals for 3d orbitals was introduced also for phosphorus and the value for sulfur was modified for improved band gaps. Besides the orbital exponents ζ3dU of the one-center two-electron integrals for phosphorus and the exponents κ3 and κ4 of the antipenetration factors of twocenter core integrals for phosphorus and sulfur were optimized. The final results are in Table 1 together with the experimental binding energy.19 Here EB′ ) EB/m is defined for Zn2mS2m and Zn3mP2m clusters. From the table it is apparent that the ZnS cyclic cluster energy is already converged within 1 kJ/mol for the smallest considered cluster Zn32S32. For zinc phosphide clusters the energy convergence is slower and is reached for the Zn96P64 cluster. Since we wanted to study the transition from Zn32S32 to Zn48P32, we have adjusted the parameters to these clusters. Subsequently we have studied the stepwise substitution of sulfur atoms by phosphorus atoms together with the addition of zinc atoms. The corresponding equation to eq 5 without the pressure term is
∆ M E0 )
TABLE 1: Relative Average Coordination Number k and Binding Energy EB′ (kJ/mol) for a Series of Cyclic Zinc Sulfide and Zinc Phosphide Clusters
1 S P ) - (1 - x)E0(Zn32S32) [E (Zn 16 0 32+16x 32-32x 32x xE0(Zn48P32)] (7)
It is now more convenient to work with the binding energies
cluster
shape
k
EB′
Zn32S32 Zn64S64 Zn108S108 Zn128S128 Zn256S256 2ZnS bulk (exp)a
2×2×2 2 x2 × 2 x2 × 2 3×3×3 2 x2 × 2 x2 × 4 4×4×4
0.672 0.766 0.771 0.820 0.824 1.000
-1230.0 -1231.0 -1231.6 -1231.2 -1231.6 -1230.0
Zn24P16 Zn48P32 Zn96P64 Zn192P128 Zn3P2 bulk (exp)a
1×1×1 x2 × x2 × 1 2×2×1 2×2×2
0.646 0.672 0.760 0.810 1.000
-1218.9 -1234.4 -1245.4 -1245.0 -1234.3
a
Reference 19.
TABLE 2: Mixing Parameter x, Binding Energy EB′ (kJ/mol), Enthalpy of Mixing ∆ME0, Entropy of Mixing Contribution T∆MS (kJ/mol), Free Energy of Mixing ∆MG (kJ/mol) at 900 °C, and Band Gap Eg (eV) for Mixed Cyclic Zinc Sulfide/Zinc Phosphide Clusters (a)
(b)
cluster
x
E B′
∆ME0
T∆MS
∆MG
Eg
Zn32S32 Zn33S30P2 Zn34S28P4 Zn35S26P6 Zn36S24P8 Zn40S16P16 Zn44S8P24 Zn45S6P26 Zn46S4P28 Zn47S2P30 Zn48P32
0.000 0.063 0.125 0.188 0.260 0.500 0.750 0.813 0.875 0.938 1.000
-1229.99 -1226.41 -1225.10 -1223.40 -1221.54 -1217.96 -1223.95 -1225.94 -1229.34 -1231.27 -1234.36
0 3.862 5.449 7.428 9.564 14.265 9.395 7.686 4.566 2.911 0
0 2.280 3.675 4.717 5.484 6.760 5.484 4.717 3.675 2.280 0
0 1.582 1.774 2.711 4.080 7.505 3.910 2.969 0.891 0.631 0
3.77 2.98 2.86 2.71 2.59 2.08 1.58 1.53 1.46 1.46 1.45
EB instead of the total energies E0 of the systems. EB is defined here as
EB(Z32+16xS32-32xP32x) ) E0(Zn32+16xS32-32xP32x) (32 + 16x)E0(Zn) - (32 - 32x)E0(S) - 32xE0(P) (8) By definition the binding energy of neutral atoms is zero. Substitution of eq 8 into eq 7 yields
∆ M E0 )
1 S P )[E (Zn 16 B 32+16x 32-32x 32x (1 - x)EB(Zn32S32) - xEB(Zn48P32)]
) EB′(Zn32+16xS32-32xP32x) - (1 - x)EB′(Zn32S32) xEB′(Zn48P32) (9) where we have denoted
EB′ ) EB/16
(10)
We have optimized all significant cyclic cluster structures between Zn32S32 and Zn48P32 to obtain the structures with the lowest energies. This required the calculation of a variety of choices for the mixed zinc sulfide/zinc phosphide clusters. The relative energies of cluster isomers differed only by a few kJ/ mol. The normalized values according to eq 10 differ usually by no more than 1 kJ/mol. This justifies the statistical entropy term of eq 6. The results for binding energies, enthalpies of mixing, entropies of mixing, and Gibbs free energies of mixing at 900 °C of the lowest-energy clusters are collected in Table 2. During the mixing process the binding energy EB′ increases
21924 J. Phys. Chem. B, Vol. 109, No. 46, 2005
Jug et al.
Figure 2. Volume increase ∆V′ (Å3) from Zn32S32 to Zn48P32 in dependence of mixing parameter x.
about 1 atom % of phosphorus. ∆ME0 will now be calculated as
∆ME0 )
1 53 E (Zn S P ) - EB(Zn108S108) 54 B 109 106 2 54 1 1 E (Zn P ) 54 16 B 48 32
[
) EB′(Zn109S106P2) -
Figure 1. Labeling of atoms in (a) Zn32S32 and (b) Zn48P32 clusters.
and reaches a maximum at x ) 0.5, which is about 12 kJ/mol larger than two times the binding energy EB′ of pure ZnS. This increase suffices to cause a miscibility gap at 900 °C. No thermodynamically stable mixture is found at both ends indicated by a positive Gibbs free energy of mixing. Locmelis and Binnewies7 reported the substitution of 0.7 atom % phosphorus in zinc sulfide and of 1.2 atom % sulfur in zinc phosphide in a transport reaction. But our mixed system Zn33S30P2 contains about 3.1 atom % phosphorus and Zn47S2P30 contains 2.5 atom % sulfur. It was therefore necessary to study larger clusters, at least Zn108S108 and Zn109S106P2, and to take Zn48P32 as the reference for zinc phosphide. Zn109S106P2 contains
]
53 E ′(Zn108S108) 54 B 1 E ′(Zn48P32) (11) 54 B
The energy value obtained from eq 11 is 1.095 kJ/mol. The entropy term for x ) 1/54 at 900 °C is 0.899 kJ/mol. This results in a ∆MG value of 0.20 kJ/mol. This value is so small that it can easily be turned into a negative value by the temperaturedependent vibrational contribution to the entropy of mixing. In the case of Zn32S31O, the value of the vibrational contribution is 1 kJ/mol.17 We therefore conclude that 1 atom % mixing of phosphorus in zinc sulfide can be thermodynamically stable in line with the experimental data. 4.2. Structure and Substitution Pattern. Structures of the initial cubic Zn32S32 cluster and the final tetragonal Zn48P32 cluster are given in Figure 1. The labeling 1, 2, ..., 64 is kept throughout the course of substitution of S atoms (dark gray circles) by P (black circles) atoms. The additional Zn atoms are labeled 65, 66, ..., 80. The structural parameters and the volume of the unit cell of the cyclic clusters Zn32S32 to Zn48P32 are given in Table 3 together with the experimental values for ZnS, Zn2S2, and Zn3P2.20-22 The lengths a, b, and c of the lattice vectors are the standard structural parameters. a ) b ) c holds for the cubic lattice of zinc sulfide and a ) b * c for the tetragonal lattice of zinc phosphide. Since we want to compare the increase in volume when going from Zn32S32 to Zn48P32, we have chosen a unit cell for zinc sulfide with a lattice parameter a′ ) 2a. The standard lattice parameter a corresponds to a unit cell Zn4S4. In the case of zinc phosphide the lattice parameter for Zn48P32 is a′ ) ax2. The lattice parameter a refers to a unit cell Zn24P16. In our procedure the increase in bond length from the Zn-S bond to the Zn-P bond can be followed step by step. The volume V′ refers to a unit cell a′ × a′ × a′ for ZnS and to a unit cell a′ × a′ × c for Zn3P2. Whereas zinc sulfide can be characterized by a single bond length r with an experimental
Miscibility of Zinc Sulfide
J. Phys. Chem. B, Vol. 109, No. 46, 2005 21925
TABLE 3: Lattice Parameters a, a′, and c (Å), Bond Lengths r (Å), and Corresponding Volumes V and V′ (Å3) for Zn32S32 and Zn48P32 (Experimental Bulk Values in Parentheses) cluster a ) 5.443 (5.410),a V ) 161.25 (158.34)a a′ ) 10.886, V′ ) 1290.04 (1266.72) r(Zn-S) ) 2.357 (2.343) a ) 8.197 (8.0889),b c ) 11.510 (11.4069),b V ) 773.36 (746.36)b a′ ) 11.592 (11.4394), c ) 11.510 (11.4069),b V′ ) 1546.65 (1492.71) r1(Zn-P) ) 2.368 (2.352),c r2(Zn-P) ) 2.497 (2.468),c r3(Zn-P) ) 2.935 (2.764)c
Zn32S32 Zn48P32
a
Reference 20. b Reference 21. c Reference 22.
TABLE 4: Bond Valencies VAB and Atomic Valencies VA of Zn-P Bonds in the Cyclic Cluster Zn48P32 with Atomic Labels (see Figure 1b) in Parentheses type
label
VAB
Zn
(1) (20) (55) (78) (9) (25) (48) (63)
0.13 (9) 0.85 (10) 0.14 (45) 0.73 (25) 0.13 (2) 0.74 (17) 0.73 (33) 0.82 (1)
P
VA 0.74 (16) 0.34 (11) 0.74 (48) 0.15 (28) 0.14 (19) 0.34 (20) 0.73 (40) 0.88 (7)
0.74( 58) 0.34 (25) 0.74 (62) 0.90 (42) 0.85 (60) 0.74 (34) 0.74 (50) 0.88 (49)
value of 2.343 Å,21 there are three bond length ranges r1, r2, r3 in zinc phosphide determined by experiment:22 2.332-2.376 Å, 2.406-2.592 Å, and 2.761-2.768 Å with average values of 2.352, 2.468, and 2.764 Å, respectively. One bond length is substantially larger than the single Zn-S bond length. So the increase in volume from V′(Zn32S32) to V′(Zn48P32) can be explained. For the mixed clusters the volume increase
∆V′ ) V'(Zn32+16xS32-32xP32x) - V′(Zn32S32)
(12)
is almost linear with increasing mixing parameter x (Figure 2). We found that
∆MV ) )
1 ∆ V′ 16 M 1 [V′(Zn32+16xS32-32xP32x) - (1 - x)V′(Zn32S32) 16 xV′(Zn48P32)] (13)
is not exceeding 1.5 Å3. This supports our neglect of the pressure term in eq 7. To understand the coordination and bonding in the cyclic clusters, we analyze the situation in Zn48P32. From experimental data22 it was concluded that each zinc atom is coordinated to four phosphorus atoms and each phosphorus atom is coordinated to six zinc atoms. Since these bonds refer to different bond lengths, an analysis by means of a bond valence concept23,24 is desirable. This concept assigns exclusive bond contributions to each pair of atoms. We list those bond valencies VAB of representative pairs of Zn-P bonds, which are significantly different from zero, in Table 4. It is apparent that there are three ranges of valencies: 0.13-0.14, 0.34, 0.73-0.90. The first range (0.13-0.14) refers to almost negligible covalent bonds. They refer to the very long bond distances. We have omitted a bonding dash in such cases in Figure 1. A typical example is the bond Zn(1)-P(9). The second range (0.34) shows weak covalent bonding and the third range (0.73-0.90) strong covalent bonding. In the latter two ranges bonding dashes are introduced in the figures. The sum of the valencies VAB from a
0.82 (63) 0.88 (28) 0.82 (63) 0.73 (43) 0.90 (68) 0.34 (35) 0.74 (55) 0.82 (55)
0.90 (73) 0.73 (73) 0.34 (79) 0.15 (68)
0.86 (75) 0.73 (78) 0.34 (34) 0.16 (72)
2.43 2.41 2.44 2.51 3.78 3.62 3.62 3.71
reference atom A to coordinated atoms B is the atomic valency VA coord
VA )
∑B VAB
(14)
VZn is slightly larger than 2.4, and VP is slightly larger than 3.60. The ratio 2.4/3.6 reflects the coordination ratio 4/6. From Table 4 it is apparent that each Zn atom has usually two, sometimes three, strong bonds and that phosphorus has four strong bonds. The substitution pattern involves the replacement of two S atoms by two P atoms which are frequently bound by the additional Zn atom. They form P-Zn-P subunits in the cluster. These two new P-Zn bonds are strong bonds. The arragement of the atoms can been seen in Figures 3 and 4 from another perspective. The figures make clear that Zn, S, and P atoms are located in parallel planes. 4.3. Band Gap. The experimental band gap of solid ZnS is 3.85 eV25 and that of solid Zn3P2 is 1.6 eV.26 These band gaps were included in our parametrization and could be well reproduced with values of 3.77 and 1.45 eV, respectively. The calculated band gap for the clusters listed in Table 2 is presented in Figure 5 over the whole range 0 e x e 1. Altogether there is a decrease of the band gap with increasing x. Close to the value x ) 0 there is a sharp band gap drop of 0.79 eV from x ) 0 to x ) 0.063. This drop is due to the increase of the highest occupied molecular orbital (HOMO) level, whereas the energy of the lowest unoccupied molecular orbital (LUMO) level is almost unchanged. In the defect-free Zn32S32 cluster the HOMO is delocalized over all S atoms. The increase of the HOMO for x ) 0.063 is caused by the substitution of two sulfur atoms by two phosphorus atoms. The HOMO is here mainly localized at the two phosphorus atoms which have higher lying atomic orbitals than the replaced sulfur atoms. Further substitution with P atoms causes a much smaller effect, because the interaction between the P atoms is small. We have found the same localization of the HOMO on the P orbitals for x ) 0.018 in the larger system Zn109S106P2 and a corresponding drop of 0.74 eV for the band gap compared to Zn108S108. This means that the extrapolation of the band gap below x ) 0.063 is suggestive.
21926 J. Phys. Chem. B, Vol. 109, No. 46, 2005
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Figure 3. Plane diagrams of Zn32S32 cluster. Zinc atoms are marked by squares and sulfur atoms by circles. The planes are (a) z ) 0 (Zn) and z ) 1/8 (S), (b) z ) 1/4 (Zn) and z ) 3/8 (S), (c) z ) 1/2 (Zn) and z ) 5/8 (S), and (d) z ) 3/4 (Zn) and z ) 7/8 (S) in units of lattice parameter c.
Figure 4. Plane diagrams of Zn48P32 cluster. Zinc atoms are marked by squares and phosphorus atoms by circles. The planes are (a) z ) 0 (Zn) and z ) 1/8 (P), (b) z ) 1/4 (Zn) and z ) 3/8 (P), (c) z ) 1/2 (Zn) and z ) 5/8 (P), and (d) z ) 3/4 (Zn) and z ) 7/8 (P) in units of lattice parameter c.
But we cannot make any definitive prediction for the shape of the band gap curve very close to x ) 0. It seems therefore reasonable to omit the point for x ) 0 from the interpolation
curve. A bowing by interpolation with a parabolic curve starting from x ) 0 is not visible. At the other end for x ) 1, there is a smooth transition of the band gap from Zn47S2P30 to Zn48P32,
Miscibility of Zinc Sulfide
J. Phys. Chem. B, Vol. 109, No. 46, 2005 21927 to x ) 1, and the sharp drop close to x ) 0 is explained by the increase of the HOMO due to the substituted P atoms. Acknowledgment. This work was partially supported by the Deutsche Forschungsgemeinschaft. We thank Professor M. Binnewies and his group and Professor J. Heidberg for frequent discussions. Part of the figures were drawn with SCHAKAL. References and Notes
Figure 5. Band gap change for mixed zinc sulfide/zinc phosphide systems.
because the higher lying P orbitals determine the band gap. Altogether, the band gap change is dominated by the large difference between the band gaps of ZnS and Zn3P2. 5. Conclusion The cyclic cluster model (CCM) as implemented in MSINDO has proved to be a suitable tool for the calculation of bulk properties of zinc sulfide and zinc phosphide. The calculated structure and energetics are in good agreement with the experimental data. The mechanism of the mixing process can be described by an increase of zinc atoms accompanied by the substitution of sulfur atoms by an equal number of phosphorus atoms. The additional zinc atoms occupy interstitial sites which are tetrahedral holes in the zinc sulfide lattice. The new zinc atoms form frequently P-Zn-P subunits with the substituted phosphorus atoms. A miscibility gap is found at room temperature. At both ends of the composition range about 1 atom % of phosphorus or sulfur can be mixed into zinc sulfide or zinc phosphide, respectively. The calculated band gap shows a continuous decrease along the composition range from x ) 0
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